research article combustion rate of solid carbon...

Hindawi Publishing CorporationJournal of CombustionVolume 2013 Article ID 790672 22 pageshttpdxdoiorg1011552013790672

Research ArticleCombustion Rate of Solid Carbon inthe Axisymmetric Stagnation Flowfield Established overa Sphere andor a Flat Plate

Atsushi Makino1 Masahiro Hojo1 and Masahito Shintomi2

1 Aerospace Research and Development Directorate Japan Aerospace Exploration Agency 7-44-1 Jindaiji-HigashiChofu Tokyo 182-8522 Japan

2Department of Mechanical Engineering Numadu National College of Technology 3600 O-oka Numadu 410-8501 Japan

Correspondence should be addressed to Atsushi Makino amakinochofujaxajp

Received 29 November 2012 Revised 4 February 2013 Accepted 4 February 2013

Academic Editor Yiguang Ju

Copyright copy 2013 Atsushi Makino et alThis is an open access article distributed under the Creative CommonsAttribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Carbon combustion in the forward stagnation flowfield has been examined through experimental comparisons by conductingaerothermochemical analyses with the surface C-O

2and C-CO

2reactions and the gas-phase CO-O

2reaction taken into account

By virtue of the generalized species-enthalpy coupling functions close coupling of those reactions has been elucidated Explicitcombustion-rate expressions by use of the transfer number in terms of the natural logarithmic term just like that for dropletcombustion have further been obtained for the combustion response in the limiting situations It has been confirmed that beforethe establishment of CO flame the combustion rate can fairly be represented by the expression in the frozen mode that afterits establishment by the expression in the flame-attached or flame-detached modes and that the critical condition derived bythe asymptotics can fairly predict the surface temperature for its establishment The formulation has further been extendedto include the surface C-H

2O and gas-phase H

2-O2reactions additionally so as to evaluate the combustion rate in humid

airflow Since those expressions are explicit and have fair accuracy they are anticipated to make various contributions not onlyfor qualitativequantitative studies but also for various aerospace applications including propulsion with high-energy-densityfuels

1 Introduction

Carbon combustion has been a research subject indispen-sable for practical utilization of coalchar combustion aero-space applications with carbon-carbon composites (CC-composites) ablative carbon heat shields andor propul-sion with using high-energy-density fuels Because of thispractical importance extensive research has been conductednot only experimentally but also theoreticallynumericallyand several comprehensive reviews [1ndash12] summarize accom-plishments in this field Nevertheless because of complexitiesinvolved there still remain several problems indispensablefor understanding the basic nature of the combustion Someof them also command fundamental interest because ofsimultaneous existence of surface and gas-phase reactionsinteracting with each other

Generally speaking the carbon combustion consists ofthe following processes

(1) diffusion of oxidizing species to the solid surface(2) adsorption of molecules onto active sites on the

surface(3) formation of products from adsorbed molecules on

the surface(4) desorption of solid oxides into the gas phase(5) migration of gaseous products through the boundary

layer into the freestream

The slowest of them determines the combustion rate becausethese steps occur in series Note that steps (2) and (4) areextremely fast in general

2 Journal of Combustion

When the surface temperature is low step (3) is known tobe much slower than steps (1) or (5) so that the combustionrate defined as mass being transferred in unit area and timeis determined solely by chemical kinetics Since the processin this regime is kinetically controlled the combustion rateonly depends on the surface temperature exponentially Tothe contrary the process of diffusion that proceeds in theboundary layer is irrelevant so that the combustion rateis independent of its thickness thereby concentrations ofoxidizing species at the reacting surface are not too differentfrom those in the freestream Furthermore since solid carbonis more or less porous in general combustion proceedsthroughout the sample specimen

When the surface temperature is high step (3) is knownto be much faster than steps (1) and (5) so that the combus-tion rate is controlled by the diffusion rate of oxidizing species(say oxygen) to the solid surface where their concentrationsare negligibly small In this diffusionally controlled regimetherefore the combustion rate exhibiting surface regressionstrongly depends on the boundary layer thickness while itsdependence on the surface temperature is weak (prop119879

05sim10)Since oxygen transfer to the carbon surface can occur via

O2 CO2 and H

2O the major surface reactions can be

C +O2997888rarr CO

2 (R1)

2C +O2997888rarr 2CO (R2)

C + CO2997888rarr 2CO (R3)

C +H2O 997888rarr CO +H

2 (R4)

At higher temperatures say higher than 1000K it is generallyrecognized that CO formation is the preferred route and thatthe relative contribution of (R1) can be negligible [13] Thusin the following reaction (R2) will be referred to as the C-O

2

reactionComparing (R2) and (R3) as alternate routes of CO

production the C-O2reaction is the preferred route for CO

production at low temperatures in simultaneous presence ofO2and CO

2 It can be initiated around 600K and saturated

around 1600K proceeding infinitely fast eventually relativeto diffusion The C-CO

2reaction of (R3) is the high temper-

ature route initiating around 1600K and becoming saturatedaround 2500K It is of particular significance because CO

2

in (R3) can even be the product of the gas-phase water-catalyzed CO oxidation

2CO +O2997888rarr 2CO

2 (R5)

referred to as the CO-O2reactionThus the C-CO

2and CO-

O2reactions can form a loopSimilarly the C-H

2O reaction (R4) generating CO and

H2 is also important when the combustion environment

consists of an appreciable amount of water This reaction isalso of significance because H

2O is the product of the H

2-

oxidation

2H2+O2997888rarr 2H

2O (R6)

referred to as the H2-O2reaction which then constitutes a

loop with the C-H2O reaction

The present study is intended to shed more light on thecarbon combustion with putting a focus on its combustionrate under an interaction of the surface and gas-phasereactions It is therefore not intended as a collection ofengineering data or an exhaustive review of all the pertinentwork published Rather it has an intention to represent thecarbon combustion by use of some of the basic characteristicsof the chemically reacting boundary layers under recognitionthat flow configurations are indispensable for proper evalu-ation of the combustion rate especially under the situationin which the gas-phase reaction can intimately affect overallcombustion response through its coupling to the surfacereactions

Among various flow configurations it has been reportedthat the stagnation-flow configuration has various advan-tages because it provides a well-defined one-dimensionalflow characterized by a single parameter [14] called as thestagnation velocity gradient It has even been said thatmathe-matical analyses experimental data acquisition and physicalinterpretations have been facilitated by its introduction Sincecarbon combustion in the two-dimensional stagnation flowestablished over a cylinder has already been summarizedsomewhere in a set of review papers [15 16] in whichextensive comparisons have been conducted between exper-imental and theoreticalnumerical results here we confineourselves to studying carbon combustion in the axisymmetricstagnation flow over a sphere or a flat plate by comparingexperimental data in the literaturewith theoreticalnumericalresults newly obtained From the practical point of view wecan even say that it simulates the situations of ablative carbonheat-shields andor the strongly convective carbon burningin the forward stagnation region of a particle

In the following formulation of the governing equationsis first presented in Section 2 based on theories on thechemically reacting boundary layer in the forward stagnationfield Chemical reactions considered are the surface C-O

2

and C-CO2reactions and the gas-phase CO-O

2reaction

Generalized species-enthalpy coupling functions are thenderived without assuming any limit or near-limit behaviorswhich not only enable us tominimize the extent of numericalefforts needed for generalized treatment but also provideuseful insight into the conserved scalars in the carbon com-bustion In Section 3 it is also shown that straightforwardderivation of the combustion response can be allowed in thelimiting situations so thatwe have those for the frozen flame-detached and flame-attached modes

In Section 4 a further analytical study is made aboutthe ignition phenomenon in the gas phase related to finite-rate kinetics by use of the asymptotic expansion method toobtain a critical condition Appropriateness of this criterionis further examined by comparing surface temperatures atwhich the CO flame can appear After having constructedthe theory evaluation of kinetic parameters of the globalgas-phase reaction is conducted for further experimentalcomparisons

In Section 5 it is endeavored to obtain explicit com-bustion-rate expressions even though they might be approx-imate because they are anticipated to contribute muchto the foundation of theoretical understanding of carbon

Journal of Combustion 3

combustion offering mathematical simplifications just likethat in droplet combustion and to the practical applicationsin the fields of aerospace andor others Further experimentalcomparisons are conducted by use of results reported in theliterature

After having examined appropriateness of the explicitexpressions carbon combustion in humid airflow is thenexamined in Section 6 relevant to basic research for erosiveattacks of water vapor to CC-composite in rocket nozzlesfor example Endeavor has been made for extending formu-lations for the system with three surface reactions and twoglobal gas-phase reactions in order to conduct experimentalcomparisons at high concentrations of the water vapor

Concluding remarks are made in Section 7 with nomen-clature tables and references cited

2 Formulation

Among previous studies [17ndash21] it may be noted thatAdomeitrsquos group has made a great contribution by clarifyingwater-catalyzedCO-O

2reaction [18] conducting experimen-

tal comparisons [19] and investigating ignitionextinctionbehavior of the CO flame [20] Here an extension ofthe worthwhile contributions is made along the followingdirections First simultaneous presence of the surface C-O

2


2reaction is

included so as to allow studies of surface reactions over anextended range of its temperatures as well as to examinetheir coupling with the gas-phase reaction Second a set ofgeneralized coupling functions [22] are conformed to thepresent flow configuration in order to facilitatemathematicaldevelopment andor physical interpretation of the resultsThird an attempt ismade to identify effects of thermophysicalproperties as well as other kinetic and system parametersinvolved Note here however that explanations cannot helpbut be similar to those in the previous work [15] because theformulation of the carbon combustion in the stagnation flow-field has been constructed in a generalizedmanner regardlessof the flow configuration whether it is axisymmetric or two-dimensional

21 Model Definition The present model simulates the iso-baric carbon combustion of constant surface temperature119879s in the stagnation flow of temperature 119879

infin oxygen mass-

fraction 119884Oinfin and carbon dioxide mass-fraction 119884Pinfin ina general manner [23] The major reactions considered arethe surface C-O

2and C-CO

2reactions and the gas-phase

CO-O2reaction The surface C + O

2997888rarr CO

2reaction

is excluded [13] because our concern is the combustion attemperatures above 1000K Crucial assumptions introducedare conventional constant property assumptions with unityLewis number constant average molecular weight constantvalue of the product of density 120588 and viscosity 120583 one-step overall irreversible gas-phase reaction and first-ordersurface reactions Surface characteristics such as porosityand internal surface area are grouped into the frequencyfactors for the surface reactions

22 Governing Equations The steady-state axisymmetricandor two-dimensional boundary-layer flows with chemicalreactions are governed as follows [24 25]Continuity

120597 (120588119906119877119895)

120597119909

+

120597 (120588V119877119895)

120597119910

= 0 (1)

Momentum

120588119906

120597119906

120597119909

+ 120588V120597119906

120597119910

minus

120597

120597119910

(120583

120597119906

120597119910

) = 120588infin119906infin(

120597119906

120597119909

)

infin

(2)

Species

120588119906

120597119884119894

120597119909

+ 120588V120597119884119894

120597119910

minus

120597

120597119910

(120588119863

120597119884119894

120597119910

) = minus119908119894

(119894 = FO)

120588119906

120597119884P120597119909

+ 120588V120597119884P120597119910

minus

120597

120597119910

(120588119863

120597119884P120597119910

) = 119908P

120588119906

120597119884N120597119909

+ 120588V120597119884N120597119910

minus

120597

120597119910

(120588119863

120597119884N120597119910

) = 0

(3)

Energy

120588119906

120597 (119888119901119879)

120597119909

+ 120588V120597 (119888119901119879)

120597119910

minus

120597

120597119910

(120582

120597119879

120597119910

) = 119902119908F (4)

where 119879 is the temperature 119888119901the specific heat 119902 the heat

of combustion per unit mass of CO 119884 the mass fraction 119906the velocity in the tangential direction 119909 V the velocity in thenormal direction 119910 and the subscripts C F O P N g s andinfin respectively designate carbon carbonmonoxide oxygencarbon dioxide nitrogen the gas phase the surface and thefreestream

In these derivations use has been made of assumptionsthat the pressure and viscous heating are negligible in (4)that a single binary diffusion coefficient119863 exists for all speciespairs that 119888

119901is constant and that the CO-O

2reaction can be

represented by a one-step overall irreversible reaction witha reaction rate

119908F = (]119894119882119894) 119861g(

120588119884F119882F

)

]F

(

120588119884O119882O

)

]O

exp(minus119864g

119877119900119879

) (5)

where 119861 is the frequency factor 119864 the activation energy 119877119900the universal gas constant ] the stoichiometric coefficientand 119882 the molecular weight We should also note that 119877119895in (1) describes the curvature of the surface such that 119895 = 1

and 0 designate axisymmetric and two-dimensional flowsrespectively and the velocity components 119906

infinand Vinfin

of thefrictionless flow outside the boundary layer are given by useof the velocity gradient 119886 as

119906infin= 119886119909 V

infin= minus (119895 + 1) 119886119910 (6)

23 Boundary Conditions The boundary conditions for thecontinuity and the momentum equations are the well-knownones expressed as

at 119910 = 0 119906 = 0 V = Vs

as 119910 997888rarr infin V = Vinfin

(7)


For the species conservation equations we have in the frees-tream

(119884F)infin = 0 (119884119894)infin= 119884119894infin

(119894 = OPN) (8)

At the carbon surface components transported from gas tosolid by diffusion transported away from the interface byconvection and producedconsumed by surface reactions areto be considered Then we have

(120588V119884F)s minus (120588119863120597119884F120597119910

)

s= 2119882F(

120588119884O119882O

)

s119861sO exp(minus

119879119886sO

119879s)

+ 2119882F(120588119884P119882P

)

s119861sP exp(minus

119879119886sP

119879s)

(120588V119884O)s minus (120588119863120597119884O120597119910

)

s= minus119882O(

120588119884O119882O

)

s119861sO exp(minus

119879119886sO

119879s)

(120588V119884P)s minus (120588119863120597119884P120597119910

)

s= minus119882P(

120588119884P119882P

)

s119861sP exp(minus

119879119886sP

119879s)

(120588V119884N)s minus (120588119863120597119884N120597119910

)

s= 0

(9)

24 Nondimensional Conservation Equations In boundarylayer variables the conservation equations for momentumspecies 119894 and energy are respectively

1198893119891

1198891205783+ 119891

1198892119891

1198891205782+

1

2119895

120588infin

120588

minus (

119889119891

119889120578

)

2

= 0 (10)

119871 (F + P) = 119871 (O + P) = 119871 (P minus119879) = 119871 (N) = 0

(11)

119871 (119879) = minus119863119886g120596g (12)

where the convective-diffusive operator is defined as

119871 =

1198892

1198891205782+ 119891

119889

119889120578

(13)

The present Damkohler number for the gas-phase CO-O2

reaction is given by

119863119886g = (

119861g

2119895119886

)(

120588infin

]P119882P)

]F+]Ominus1

(]F)]F(]O)

]O (14)

with the nondimensional reaction rate

120596g = (

119879infin

119879

)

]F+]Ominus1

(F)]F(O)

]O exp(minus119879119886g

119879

) (15)

In the above the conventional boundary-layer variables 119904 and120578 related to the physical coordinates 119909 and 119910 are

119904 = int

119909

0

120588infin(119909) 120583infin(119909) 119906infin(119909) 1198772119895119889119909

120578 =

119906infin(119909) 119877119895

radic2119904

int

119910

0

120588 (119909 119910) 119889119910

(16)

The nondimensional streamfunction 119891(119904 120578) is related tothe streamfunction 120595(119909 119910) through

119891 (119904 120578) =

120595 (119909 119910)

radic2119904

(17)

where 120595(119909 119910) is defined by

120588119906119877119895=

120597120595

120597119910

120588V119877119895= minus

120597120595

120597119909

(18)

such that the continuity equation is automatically satisfiedVariables and parameters used are as follows

119879 =

119879

119902 (119888119901120572F)

119879119886 =

119864119877119900

119902 (119888119901120572F)

120572F =]P119882P]F119882F

F =]P119882P]F119882F

119884F

O =

]P119882P]O119882O

119884O N = 119884N 120575 =

119882P119882C

(19)

Here use has been made of an additional assumptionthat the Prandtl and Schmidt numbers are unity Since weadopt the ideal-gas equation of state under an assumptionof constant average molecular weight across the boundarylayer the term (120588

infin120588) in (10) can be replaced by (119879119879

infin) As

for the constant 120588120583 assumption while enabling considerablesimplification it introduces 50ndash70 errors in the transportproperties of the gas in the present temperature rangeHowever these errors are acceptable for far greater errors inthe chemical reaction rates Furthermore they are anticipatedto be reduced due to the change of composition by thechemical reactions

The boundary conditions for (10) are

119891 (0) = 119891s (

119889119891

119889120578

)

s= 0 (

119889119891

119889120578

)

infin

= 1 (20)

whereas those for (11) and (12) are

at 120578 = 0 119879 =

119879s

119894= (119894)s (119894 = FOPN)

as 120578 997888rarr infin

119879 =

119879infin F = 0

119894= (119894)infin

(119894 = OPN)

(21)


which are to be supplemented by the following conservationrelations at the surface

minus(

119889F119889120578

)

s+ (minus119891s) Fs = 120575 (minus119891sO) + 2120575 (minus119891sP)

= 120575 (minus119891s) + 120575 (minus119891sP)

(22)

minus(

119889O119889120578

)

s+ (minus119891s) Os = minus120575 (minus119891sO)

= minus120575 (minus119891s) + 120575 (minus119891sP)

(23)

minus (

119889P119889120578

)

s+ (minus119891s) Ps = minus120575 (minus119891sP) (24)

minus (

119889N119889120578

)

s+ (minus119891s) Ns = 0 (25)

where

120575 (minus119891s) = 120575 (minus119891sO) + 120575 (minus119891sP) = 119860 sOOs + 119860 sPPs

(26)

119860 sO equiv 119863119886sO (119879infin

119879s

) exp(minus119879119886sO119879s

)

119863119886sO equiv

119861sO

radic2119895119886 (120583infin120588infin)

119860 sP equiv 119863119886sP (119879infin

119879s

) exp(minus119879119886sP119879s

)

119863119886sP equiv119861sP


(27)

and 119863119886sO and 119863119886sP are the present surface Damkohlernumbers based only on the frequency factors for the C-O

2

andC-CO2reactions respectively Here these heterogeneous

reactions are assumed to be first order for simplicity andanalytical convenience As for the kinetic expressions fornonpermeable solid carbon it is considered that they incor-porate effects of porosity andor internal surface area whilesurface reactions are generally controlled by combinations ofchemical kinetics and pore diffusions

For self-similar flows the normal velocity Vs at the surfaceis expressible in terms of (minus119891s) by

(120588V)s = (minus119891s)radic2119895119886120588infin120583infin (28)

If we remember the fact that the mass burning rate ofsolid carbon is given by = (120588V)s which is equivalentto the definition of the combustion rate [kg(m2sdots)] wesee that the streamfunction (minus119891s) can be identified as thenondimensional combustion rate Note also that the surfacereactions are less sensitive to velocity gradient variations thanthe gas-phase reaction because119863119886ssim119886

minus12 while119863119886g sim 119886minus1

25 Coupling Functions With the boundary conditions forspecies cast in the specific forms of (22) to (24) the couplingfunctions for the present system are given by

F + P =(Pinfin + 120575120573) + (Pinfin minus 120575) 120573120585

1 + 120573

(29)

O + P =(Oinfin+Pinfinminus120575120573)+(Oinfin+Pinfin+120575) 120573120585

1 + 120573

(30)

O +119879 = Os +

119879s + (Oinfin minus Os +

119879infinminus119879s) 120585

Os =Oinfin +

119879infinminus119879s minus 120574

1 + 120573 + 119860 sO [120573 (minus119891s)]

(31)

P minus119879 = Ps minus

119879s + (Pinfin minus Ps minus

119879infin+119879s) 120585

Ps =Pinfin minus

119879infin+119879s + 120574

1 + 120573 + 119860 sP [120573 (minus119891s)]

(32)

N = Ninfin1 + 120573120585

1 + 120573

(33)

where

120585 =

int

120578

0

exp(minusint120578

0

119891119889120578)119889120578

int

infin

0

exp(minusint120578

0

119891119889120578)119889120578

120574 =

(1198791015840)s

(1205851015840)s 120573 =

(minus119891s)

(1205851015840)s

(1205851015840)s=

1

int

infin

0

exp(minusint120578

0

119891119889120578)119889120578

(34)

and a prime indicates 119889119889120578 Using the new independentvariable 120585 the energy conservation (12) becomes

(

1198892 119879

1198891205852) = minus

119863119886g120596g

(119889120585119889120578)2 (35)

Therefore the equations to be solved are (10) and (35) subjectto the boundary conditions in (20) and

(119879)120585=0

=119879s (

119879)120585=1

=119879infin (36)

by use of (minus119891s) given by (26) and the coupling functions in(29) to (32) Key parameters in solving those are 119863119886g 119863119886sO119863119886sP and (minus119891s)

It may be informative to note that the parameter 120573introduced as (minus119891s)(120585

1015840)s in the formulation and indispens-

able in obtaining combustion rate (minus119891s) coincides with theconventional transfer number presented by Spalding [26]This has already been demonstrated by considering elementalcarbon (119882C119882F)119884F + (119882C119882P)119884P taken as the transferred


substance and by evaluating driving force and resistancedetermined respectively by the transfer rate in the gas phaseand the ejection rate at the surface [27 28] that is

(119882C119884F119882F+119882C119884P119882P)sminus(119882C119884F119882F+119882C119884P119882P)infin1minus(119882C119884F119882F+119882C119884P119882P)s

=

(F + P)sminus (F + P)

infin

120575 minus (F + P)s

equiv 120573

(37)

Note that use has been made of the coupling function F + Pin (29) in deriving the final relation

3 Combustion Behavior in the Limiting Cases

Here we discuss analytical solutions for some limiting casesof the gas-phase reaction since several limiting solutionsregarding the intensity of the gas-phase CO-O

2reaction

can readily be identified from the coupling functions Inaddition important characteristics indispensable for funda-mental understanding are obtainable

31 Frozen Mode When the gas-phase CO-O2reaction is

completely frozen the solution of the energy conservation(12) readily yields

119879 =

119879s + 120574120585 120574 =

119879infinminus119879s (38)

Evaluating (31) and (32) at 120585 = 0 to obtain the surfaceconcentrations of O

2and CO

2 and substituting them into

(26) we obtain an implicit expression for the combustion rate(minus119891s) as

120575 (minus119891s) = 119860 sOOinfin

1 + 120573 + 119860 sO [120573 (minus119891s)]

+ 119860 sPPinfin

1 + 120573 + 119860 sP [120573 (minus119891s)]

(39)

which is to be solved numerically from (10) because ofthe density coupling The combustion rate in the diffusioncontrolled regime becomes the highest with satisfying thefollowing condition

120573max =Oinfin + Pinfin

120575

(40)

32 Flame-Detached Mode When the gas-phase CO-O2

reaction occurs infinitely fast two types of flame-sheetburning modes are possible One involves a detached flame-sheet situated away from the surface and the other anattached flame sheet situated on the surface The flame-detached mode is defined by

O (0 le 120585 le 120585f) = F (120585f le 120585 le infin) = 0 (41)

By use of the coupling functions in (29) to (32) it can beshown that

120575 (minus119891s) = 119860 sPOinfin + Pinfin minus 120575120573

1 + 120573

(42)

119879f =

119879s + (Oinfin +

119879infinminus119879s) 120585f

120585f =2120575120573 minus Oinfin

(2120575 + Oinfin) 120573

(43)

Once (minus119891s) is determined from (42) and (10) 120585f canreadily be evaluated yielding the temperature distribution as

0 le 120585 le 120585f 119879 =

119879s + (Oinfin +

119879infinminus119879s) 120585

120585f le 120585 le infin

119879 =

119879infinminus

119879infinminus119879s + (

Oinfin minus 2120575120573

1 + 120573

) (1 minus 120585)

(44)

In addition the infinitely large 119863119886g yields the followingimportant characteristics as reported by Tsuji and Matsui[17]

(1) The quantities119884F and119884O in the reaction rate120596g in (15)become zero suggesting that fuel and oxygen do notcoexist throughout the boundary layer and that thediffusion flame becomes a flame sheet

(2) In the limit of an infinitesimally thin reaction zoneby conducting an integration of the coupling functionfor CO and O

2across the zone bounded between

120578fminus lt 120578 lt 120578f+ where 120578f is the location of flame sheetwe have

minus (

119889F119889120578

)

fminus= (

119889O119889120578

)

f+

or

minus (

119889119884F119889120578

)

fminus=

]F119882F]O119882O

(

119889119884O119889120578

)

f+

(45)

suggesting that fuel and oxidizer must flow into theflame surface in stoichiometric proportions Herethe subscripts f+ and fminus respectively designate theoxygen and fuel sides of the flame Note that inderiving (45) use has been made of an assumptionthat values of the individual quantities such as thestreamfunction119891 and speciesmass-fraction119884

119894 can be

continuous across the flame(3) Similarly by evaluating the coupling function for CO

and enthalpy at the flame sheet we have

(

119889119879

119889120578

)

fminusminus (

119889119879

119889120578

)

f+= minus(

119889F119889120578

)

fminus

or

120582(

119889119879

119889120578

)

fminusminus 120582(

119889119879

119889120578

)

f+= minus119902120588119863(

119889119884F119889120578

)

fminus

(46)


suggesting that the amount of heat generated is equalto the heat conducted away to both sides of thereaction zone

33 Flame-Attached Mode When the surface reactivity isdecreased by decreasing the surface temperature then thedetached flame sheet moves towards the surface until it iscontiguous to it (120585f = 0) This critical state is given by thecondition

120573a =Oinfin

2120575

(47)

obtained from (43) and defines the transition from thedetached to the attached mode of the flame Subsequentcombustion with the flame-attached mode is characterizedby 119884Fs = 0 and 119884Os ge 0 [20 22 29] with the gas-phasetemperature profile

119879 =

119879s + (

119879infinminus119879s) 120585 (48)

given by the same relation as that for the frozen modebecause all gas-phase reaction is now confined at the surfaceBy using the coupling functions from (29) to (32) with 119884Fs =0 it can be shown that

120575 (minus119891s) = 119860 sOOinfin minus 2120575120573

1 + 120573

+ 119860 sPPinfin + 120575120573

1 + 120573

(49)

which is also to be solved numerically from (10) The maxi-mum combustion rate of this mode occurs at the transitionstate in (47) which also corresponds to the minimumcombustion rate of the flame-detached mode

34 Diffusion-Limited Combustion Rate The maximumdiffusion-limited transfer number of the system can beachieved through one of the two limiting situations The firstappears when both of the surface reactions occur infinitelyfast such that 119884Os and 119884Ps both vanish in the limit of thefrozen mode yielding (40) The second appears when thesurface C-CO

2reaction occurs infinitely fast in the limit

of the flame-detached mode which again yields (40) It isof interest to note that in the first situation the reactivityof the gas-phase CO-O

2reaction is irrelevant whereas in

the second the reactivity of the surface C-O2reaction is

irrelevantWhile the transfer numbers 120573 are the same in bothcases the combustion rates thereby the oxygen supply ratesare slightly different from each other because of the differentdensity coupling related to the flame structures Note that thelimiting solutions identified herein provide the counterpartsof those previously derived [22 29 30] for the carbon particleand generalize the solution of Matsui and Tsuji [21] withincluding the surface C-CO

2reaction

4 Combustion Rate and the CO Flame

A momentary reduction in the combustion rate reportedin theoretical works [19 21ndash23 31 32] can actually beexaggerated by the appearance of CO flame in the gas phase

bringing about a change of the dominant surface reactionsfrom the faster C-O

2reaction to the slower C-CO

2reaction

due to an intimate coupling between the surface and gas-phase reactions In spite of this theoretical accomplishmentthere are very few experimental data that can support it

In the literature in general emphasis has been put onexamination of the surface reactivities with gaseous oxidizerssuch as O

2 CO2 and H

2O (cf [10]) although surface

reactivities on the same solid carbon are limited [33ndash35]As for the gas-phase CO-O

2reactivity which is sensitive to

the H2O concentration main concern has been put on that

of the CO flame [36] called as the ldquostrongrdquo CO oxidationwhich is however far from the situation over the burningcarbon especially for that prior to the appearance of COflame because some of the elementary reactions are tooslow to sustain the ldquostrongrdquo CO oxidation Furthermore ithas been quite rare to conduct experimental studies fromthe viewpoint that there exist interactions between chemicalreactions and flow so that studies have mainly been confinedto obtaining combustion rate [33 34 37ndash39]

41 Combustion Rate and Ignition Surface Temperature Hereexperimental results of Visser and Adomeit [34] for thecarbon combustion are first presented which are closelyrelated to the coupled nature of the surface and gas-phasereactions In their experiment the oxidizer flow at roomtemperature after being set for a flow rate and variousspecies concentrations such as O

2and H

2O issued into

the atmosphere with a uniform velocity (up to 95ms) andimpinged on a graphite hemisphere to establish an axisym-metric stagnation flow This flowfield is well established andis specified uniquely by the velocity gradient 119886 (= 3119881

infin119889)

where 119881infin

is the freestream velocity and 119889 the diameterof graphite sphere or hemisphere [40] The spherical testspecimen was heated by an electromagnetic high frequencygenerator and the surface temperature was measured by aradiation thermometer During each experimental run thetest specimen was set to burn in oxidizer flow at constantsurface temperature Since the surface temperature is keptconstant with external heating quasi-steady combustion canbe accomplished The experiment involved taking picturesof the test specimen in the forward stagnation region andanalyzing them to obtain surface regression rate which wasused to determine the combustion rate Note that use hasbeen made of the value evaluated by use of the well-knownrelation 119886 = 3119881

infin119889 as far as the velocity gradient is con-

cerned even though it is specified in another way in theliterature

Figure 1(a) shows the combustion rate in oxidizer flow of170 sminus1 with the O

2mass fraction of 07 [34 41] as a function

of the surface temperature when the H2O mass fraction

is 00002 The combustion rate increases with increasingsurface temperature up to a certain surface temperatureThecombustion in this temperature range is anticipated to bethat with negligible CO oxidation and hence the combustionrate in the frozen mode can fairly predict the experimentalresults A further increase in the surface temperature causesthe momentary reduction in the combustion rate because


0

001

002

003

004

1000 1500 2000 2500

Data from Visser and Adomeit 1984

Visser 1985

Flame-detached

Flame-attachedFrozen

119886 = 170 sminus1

Com

busti

on ra

te119898

(kg

m2s) 119884Oinfin = 07

119884Ainfin = 00002

119879si

g=

1480

K

Surface temperature 119879s (K)

(a)

0

001

002

003

004

1000 1500 2000 2500


Frozen

Flame-detached

Flame-attached

119886 = 170 sminus1

Com

busti

on ra

te119898

(kg

m2s) 119884Oinfin = 07

119884Ainfin = 00038


119879si

g=1249

K

(b)

Figure 1 Combustion rate of the graphite hemisphere in oxidizer flow with the velocity gradient of 170 sminus1 as a function of the surfacetemperature (a) for the O

2mass-fraction 119884Oinfin = 07 and the H

2Omass-fraction 119884Ainfin = 00002 (b) for 119884Oinfin = 07 and 119884Ainfin = 00038 Data

points are experimental [34 41] and curves are calculated from the theory in [23] The ignition surface-temperature 119879sig is calculated basedon the ignition analysis [32]

appearance of the CO flame alters the dominant surfacereaction from the C-O

2reaction to the C-CO

2reaction

The surface temperature when the CO flame first appearsis called as the ignition surface temperature [32] Abovethe ignition surface temperature the combustion is in theflame-detached mode with the ldquostrongrdquo CO oxidation Thesolid curve is the predicted combustion rate with the surfacekinetic parameters as follows for the C-CO

2reaction 119861sP =

15 times 108ms and 119864sP119877 = 31 times 10

4 K reported whilefor the C-O

2reaction 119861sO = 41 times 10

6ms and 119864sO119877 =

21 times 104 K used in another work [42] because of a lack

of those reported In numerical calculations use has beenmade of the formulation presented in Section 2 as well asthe explicit combustion-rate expressions to be explained inSection 5 for convenience Values of thermophysical proper-ties are those for oxygen at 119879

infin= 32K which yields 120588

infin120583infin=

263 times 10minus5 kg2(m4 sdot s) and 120583

infin120588infin

= 183 times 10minus5m2s just

for simplicityIt may be informative to note that a decrease in the com-

bustion rate observed at temperatures between 1500K and2000K has generally been called as the ldquonegative temperaturecoefficientrdquo of the combustion rate which has also been aresearch subject in the field of carbon combustion Nagle andStrickland-Constable [43] used the ldquositerdquo theory to explainthe peak rate while Yang and Steinberg [44] attributedthe peak rate to the change of reaction depth at constantactivation energy Other entries relevant to the ldquonegativetemperature coefficientrdquo can be found in some survey papers[10 11] However another explanation can bemade as will beexamined in the next section that this phenomenon can beinduced by the appearance of CO flame established over theburning carbon thereby the dominant surface reaction hasbeen altered from the C-O


2reaction

[28 42 45 46]The same trend is also observed in humid oxidizer-flow

of 170 sminus1 as shown in Figure 1(b) The ignition surface tem-perature can be reduced because of the increased humidity

0

001

002

003

004

1000 1500 2000 2500


119886 = 390 sminus1CO2 flow

119884Pinfin = 1

Com

busti

on ra

te119898

(kg

m2s)


Figure 2 Combustion rate of the graphite hemisphere in CO2-flow

with the velocity gradient of 390 sminus1 as a function of the surfacetemperature Data points are experimental [34] and the curve iscalculated from the theory in [23]

that can facilitate the establishment ofCOflameNonethelessbecause of a small difference between combustion ratesbefore and after the ignition of CO flame the abrupt changein the combustion rate does not appear clearly for this case

Figure 2 shows the combustion rate in CO2flow of

390 sminus1 [34] as a function of the surface temperature Thecombustion rate increasesmonotonicallywith increasing sur-face temperature suggesting appropriateness of the surfacekinetic parameters for the C-CO

2reaction reported

42 Establishment of the CO Flame While studies relevantto the ignitionextinction of CO flame over the burningcarbon are of obvious practical utility in evaluating protectionproperties from oxidation in reentry vehicles as well as thecombustion of coalchar they also command fundamentalinterests because of the simultaneous existence of the surfaceand gas-phase reactions with intimate coupling [21 22 34]


As mentioned in the previous section at the same surfacetemperature the combustion rate is expected to be momen-tarily reduced upon ignition because establishment of theCO flame in the gas phase can change the dominant surfacereactions from the faster C-O


2

reaction By the same token the combustion rate is expectedto momentarily increase upon extinction These conceptsare not intuitively obvious without considering the couplednature of the gas-phase and surface reactions

Fundamentally the ignitionextinction of CO flame incarbon combustion must necessarily be described by theseminal analysis done by Linan [47] for the ignitionextinction and structure of diffusion flames as indicatedby Matalon [48ndash50] Specifically as the flame temperatureincreases from the surface temperature to the adiabatic flametemperature there appears a nearly frozen regime a partial-burning regime a premixed-flame regime and finally a near-equilibrium regime Ignition can be described in the nearly-frozen regime while extinction in the other three regimes

For carbon combustion Matalon [49] analytically ob-tained an explicit ignition criterion when the O

2mass

fraction at the surface is O(l) When this concentration isO(120576) the appropriate reduced governing equation and theboundary conditions were also identified [50] Here puttingemphasis on the ignition of CO flame over the burningcarbon an attempt has first beenmade to extend the previoustheoretical studies so as to include analytical derivationsof various criteria governing the ignition with arbitrary O

2

concentration at the surface Note that these derivations aresuccessfully conducted by virtue of the generalized species-enthalpy coupling functions [22 23] identified in the previ-ous Section Furthermore it may be noted that the ignitionanalysis is especially relevant to situations where the surfaceO2concentration is O(120576) because in order for gas-phase reac-

tion to be initiated sufficient amount of carbon monoxideshould be generated This requires a reasonably fast surfacereaction and thereby low O

2concentration The second ob-

jective is to conduct experimental comparisons relevant tothe ignition of CO-flame over a carbon hemisphere in anoxidizing stagnation flow at various surface temperaturesof the test specimen freestream velocity gradients and O

2

concentrations

43 Asymptotic Analysis for the Ignition Here we intend toobtain an explicit ignition criterion without restricting theorder of magnitude of the O

2concentration 119884Os First we

note that in the limit of 119879119886g rarr infin the completely frozensolutions for (11) and (12) are

(119879)0=119879s + (

119879infinminus119879s) 120585

(119894)0= 119894s + (119894infin minus

119894s) 120585 (119894 = FOP) (50)

For finite but large values of 119879119886g weak chemical reactionoccurs in a thin region next to the carbon surface whenthe surface temperature is moderately high and exceeds theambient temperature Since the usual carbon combustion

proceeds under this situation corresponding to the condition[47] of

119879s + Fs gt

119879infin (51)

we define the inner temperature distribution as

(119879)

in= (

119879)0+ 120576

119879s120582120579 (120585) +O (120576

2)

=119879s [1 minus 120576 (120594 minus 120582120579)] +O (120576

2)

(52)

where

120576 =

119879s119879119886g

120582 =

Oinfin119879s minus

119879infin

120585 = 120576(

119879s

119879s minus

119879infin

)120594 (53)

In the above 120576 is the appropriate small parameter forexpansion and 120594 and 120579 are the inner variables

With (52) and the coupling functions from (29) to (32)the inner species distributions are given by

(O)in= Os + 120576

119879s120582 (120594 minus 120579) (54)

(F)in= (

2120575120573 minus Oinfin

1 + 120573

) + Os

+ 120576(

119879s

119879s minus

119879infin

)(


1 + 120573

120594 minus Oinfin120579)

(55)

Thus through evaluation of the parameter 120574 expressed as

120574 equiv (

119889119879

119889120585

)

s=[

[

119889120594

119889120585

119889(119879)

in119889120594

]

]s

= minus (119879s minus

119879infin) + Oinfin(

119889120579

119889120594

)

s+O (120576)

(56)

the O2mass fraction at the surface is obtained as

Os =Oinfin

1 + 120573 + 119860 sO [120573 (minus119891s)]1 minus (

119889120579

119889120594

)

s (57)

Substituting 120594 (52) (54) and (55) into the governing(12) expanding and neglecting the higher-order convectionterms we obtain

1198892120579

1198891205942= minusΔ(120594 minus 120579 + 120578O)

12 exp (120582120579 minus 120594) (58)

where

Δ = 119863119886g exp(minus119879119886g

119879s

)

120573

(minus119891s)

119879s

119879s minus

119879infin

2

times (

119879s119879119886g

)

32

(

119879infin

119879s

)

12

Fs

(120582119879s)12

(59)

120578O =

Os

120576119879s120582

(60)


It may be noted that the situation of 119884Fs = O(120576) is notconsidered here because it corresponds to very weak carboncombustion such as in lowO

2concentration or at low surface

temperatureEvaluating the inner temperature at the surface of con-

stant 119879s one boundary condition for (58) is120579 (0) = 0 (61)

This boundary condition is a reasonable one from theviewpoint of gas-phase quasi-steadiness in that its surfacetemperature changes at rates much slower than that of the gasphase since solid phase has great thermal inertia

For the outer nonreactive region if we write

(119879)

out=119879s + (

119879infinminus119879s) 120585 + 120576

119879sΘ (120585) +O (120576

2) (62)

we see from (12) that Θ is governed by 119871(Θ) = 0 with theboundary condition that Θ(infin) = 0 Then the solution isΘ(120585) = minus119862I(1 minus 120585) where 119862I is a constant to be determinedthrough matching

By matching the inner and outer temperatures presentedin (52) and (62) respectively we have

120582120579 (infin) = minus119862I (

119889120579

119889120594

)

infin

= 0 (63)

the latter of which provides the additional boundary condi-tion to solve (58) while the former allows the determinationof 119862I

Thus the problem is reduced to solving the single gov-erning equation (58) subject to the boundary conditions (61)and (63) The key parameters are Δ 120582 and 120578O Before solving(58) numerically it should be noted that there exists a generalexpression for the ignition criterion as

2Δ I120582 = (radic120578O + radic120587120582

2

119890120582120578O

times erfc (120582120578O)+(120582minus1) intinfin

120594I

119890(120582minus1)120594erfc (119911) 119889120594)

minus1

(64)

erfc (119911) = 2

radic120587

expintinfin

119911

(minus1199052) 119889119905 (65)

corresponding to the critical condition for the vanishment ofsolutions at

(

119889120579

119889120594

)

s=

1

120582

or (

119889(119879)

in119889120585

)

s

= 0 (66)

which implies that the heat transferred from the surface tothe gas phase ceases at the ignition point Note also that (64)further yields analytical solutions for some special cases suchas

at 120582 = 1 2Δ I =1

radic120578O + (radic1205872) 119890120578O erfc (120582120578O)

as 120578O 997888rarr infin 2Δ I120582 =1

radic120578O

(67)

the latter of which agrees with the result of Matalon [49]

In numerically solving (58) by plotting 120579(infin) versusΔ fora given set of 120582 and 120578O the lower ignition branch of the S-curve can first be obtained The values of Δ correspondingto the vertical tangents to these curves are then obtainedas the reduced ignition Damkohler number Δ I After that auniversal curve of (2Δ I120582) versus (1120582) is obtained with 120578Otaken as a parameter Recognizing that (l120582) is usually lessthan about 05 for practical systems and using (64) and (67)we can fairly represent (2Δ I120582) as [32]

2Δ I120582 = (radic120578O +

radic120587

2

[119890120578O erfc (radic120578O)

+

1

119865 (120582)

minus 1 exp(minusradic120578O

2

)])

minus1

(68)

where

119865 (120582) = 056 +

021

120582

minus

012

1205822+

035

1205823 (69)

Note that for large values of (l120582) (68) is still moderatelyaccurate Thus for a given set of 120582 and 120578O an ignitionDamkohler number can be determined by substituting thevalues of Δ I obtained from (68) into (59)

It may be informative to note that for some weaklyburning situations in which O

2concentrations in the reac-

tion zone and at the carbon surface are O(1) a monotonictransition from the nearly frozen to the partial-burningbehaviors is reported [31] instead of an abrupt turning-pointbehavior with increasing gas-phase Damkohler numberHowever this could be a highly limiting behavior that is inorder for the gas-phase reaction to be sufficiently efficientand the ignition to be a reasonably plausible event enoughCO would have to be generated at the surface which furtherrequires a sufficiently fast surface C-O

2reaction and hence

the diminishment of the surface O2concentration fromO(l)

For these situations the turning-point behavior can be amoreappropriate indication for the ignition

44 Dominant Parameters for the Ignition of CO FlameFigure 3(a) shows the ignition surface-temperature 119879sig [34]as a function of the velocity gradient The velocity gradienthas been chosen for the abscissa as originally proposed byTsuji and Yamaoka [51] for the diffusion flame and its appro-priateness has been examined by Makino and Law [32] forthe two-dimensional flow configuration by varying both thefreestream velocity and graphite rod diameter that can exertinfluences in determining velocity gradient It is seen that theignition surface-temperature 119879sig increases with increasingvelocity gradient and thereby decreasing residence time Thehigh surface temperature as well as the high temperature inthe reaction zone causes the high ejection rate of CO throughthe surface C-O

2reaction These enhancements facilitate the

CO flame by reducing the characteristic chemical reactiontime and hence compensating a decrease in the characteristicresidence time

A solid curve in Figure 3(a) is the ignition surface-temperature for the prescribed conditions predicted by


1000

1200

1400

1600

1800

2000

10 100 1000

Data from Visser and Adomeit

1985

119884O = 07

119884A = 00003

Velocity gradient 119886 (sminus1)

Igni

tion

surfa

ce te

mpe

ratu

re119879

sig

(K)

(a)

1000

1200

1400

1600

1800

2000

00001 0001 001


1985

Water-vapor mass-fraction 119884A

119884O = 07

119886 = 170 sminus1

Igni

tion

surfa

ce te

mpe

ratu

re119879

sig

(K)

(b)

1000

1200

1400

1600

1800

2000

0 02 04 06 08 1Oxygen mass-fraction 119884O

119886 = 170 sminus1119884A = 00003

Igni

tion

surfa

ce te

mpe

ratu

re119879

sig

(K)

(c)

Figure 3 Surface temperature at the establishment of CO flame as a function of (a) the stagnation velocity gradient (b) the H2O mass-

fraction in the freestream (c) the O2mass-fraction Data points are experimental in [34] curves are calculated from the theory in [32] with

gas-phase kinetic parameters [42]

the ignition criterion [32] presented in the previous sectionwith using kinetic parameters [42] to be mentioned in thenext section The density 120588

infinof the oxidizing gas in the

freestream is estimated at 119879infin

= 323K It is seen thatfair agreement is demonstrated suggesting that the presentignition criterion has captured the essential feature of theignition of CO-flame over the burning carbon

Figure 3(b) shows the ignition surface-temperature 119879sigas a function of the H

2O mass fraction in the freestream It

is seen that the ignition surface-temperature decreases withincreasing H

2O mass fraction because the CO-O

2reaction

is facilitated with increasing concentration of H2O as is well

known and so-called as the ldquocatalytic effectrdquo of water-vaporon the CO-oxidation rate Figure 3(c) shows the ignitionsurface temperature as a function of the O

2mass fraction

in the freestream It is also seen that the ignition surface-temperature decreases with increasing O

2mass fraction In

this case the CO-O2reaction is facilitated with increasing

concentrations of O2 as well as CO because more CO is

now produced through the surface C-O2reaction Although

agreement is not so good as far as the approximatemagnitudeis concerned both the experimental and predicted resultsexhibit a decreasing trend with increasing O

2mass fraction

45 Kinetic Parameters for the Global Gas-Phase ReactionEstimation of gas-phase kinetic parameters has already beenconducted [32 42] with experimental data for the ignitionsurface temperature in the two-dimensional stagnation flowand the ignition criterion for the CO flame over the burningcarbon Here reaction orders are a priori assumed to be 119899F =1 and 119899O = 05 which are the same as those of the global rateexpression presented byHoward et al [36] It is also assumed

that the frequency factor 119861g is proportional to the half orderof H2O concentration that is

119861g = 119861lowast

g (120588119884A119882A)12

[(molm3)12

sdot s]minus1

(70)

where the subscript A designates water vapor The H2Omass

fraction at the surface is estimated with 119884As = 119884Ainfin(l + 120573)with water vapor taken as an inert because it acts as a kindof catalyst for the gas-phase CO-O

2reaction and hence its

profile is not anticipated to be influenced Thus for a givenset of 120582 and 120578O an ignition Damkohler number can bedetermined by substituting Δ I in (68) into (59)

Figure 4 shows the Arrhenius plot of the global gas-phase reactivity [42] obtained as the results of the ignitionsurface temperature In data processing data in a seriesof experiments [32 42] have been used by use of kineticparameters for the surface C-O

2reaction With iteration in

terms of the activation temperature required for determiningΔ I with respect to 120578O 119864g = 113 kJmol is obtained with

119861lowast

g = 91 times 106[(molm3) sdot s]

minus1

(71)

This activation energy is within the range of the global CO-O2

reaction (cf Table II in [36])It is noted that 119861lowastg obtained is one order of magnitude

lower than that of Howard et al [36] which is reported tobe

119861lowast


minus1

(72)

because the present value is that prior to the appearance ofCO flame and is to be low compared to that of the ldquostrongrdquo


Pressure (MPa)0028 01 079

Air

O2

Graphite densityOpen 182 times 103 kgm3

Solid 125 times 103 kgm3

100

102

104

106

3 4 5

Oxidizer

lowast(119879

s119879119886

g)32

exp(minus119879119886

g119879

s)

(m3m

oL)middot

sminus1

1s

119884Oinfin = 0533

119861g

Figure 4 Arrhenius plot of the global gas-phase reaction [42]obtained from the experimental results of the ignition surface-temperature in two-dimensional stagnation flows with variouspressures O

2and H

2O concentrations and velocity gradients

CO oxidation in the literature As for the ldquoweakrdquo CO oxida-tion Sobolev [52] reported

119861lowast


minus1

(73)

by examining data of Chukhanov [53 54] who studiedthe initiation of CO oxidation accompanied by the carboncombustion We see that the value reported [52] exhibits alower bound of the experimental results shown in Figure 4

It is also confirmed in Figure 4 that there exists noremarkable effects of O

2andor H

2O concentrations in the

oxidizer thereby the assumption for the reaction orders isshown to be appropriate within the present experimentalconditions The choice of reaction orders however requiresa further comment because another reaction order for O

2

concentration 025 in place of 05 is recommended inthe literature Relevant to this an attempt [42] has furtherbeen conducted to compare the experimental data withanother ignition criterion obtained through a similar igni-tion analysis with this reaction order However its result wasunfavorable presenting a much poorer correlation betweenthem

5 Experimental Comparisons

51 Other Sources for the Combustion Rate Experimentalcomparisons have already been conducted in Figure 1 and

a fair degree of agreement has been demonstrated as far asthe trend and approximate magnitude are concerned Herefurther experimental comparisons are to be made for resultsof Golovina and Khaustovich [55] with keeping both thegas-phase and surface kinetic parameters fixed Figure 5(a)compares predicted results with experimental data in airflowof 120 sminus1 at an atmospheric pressure the freestream velocity119881infin

= 06ms and the sphere diameter 119889 = 15mmmade of electrode carbon The ignition surface temperatureis estimated to be 119879sig asymp 1479K obtained by assuming theH2Omass fraction to be 0003 the dewpoint ofwhich is 271K

(minus2∘C) Values of thermophysical properties are those for airat 119879infin= 320K which yields 120588

infin120583infin= 212 times 10

minus5 kg2(m4 sdots)and 120583

infin120588infin

= 178 times 10minus5m2s used in the previous work

[28 42] We see from Figure 5(a) that up to the ignitionsurface temperature 119879sig the combustion proceeds under theldquoweakrdquo or negligible CO oxidation and that the ldquostrongrdquo COoxidation prevails at temperatures higher than the ignitionsurface temperature Unfortunately because of the lack of theexperimental data the abrupt change in the combustion ratedoes not appear clearly although the general behavior seemsto be similar to that in Figures 1(a) and 1(b)

Figure 5(b) compares predicted results with experimentaldata in CO

2flow of 120 sminus1 with the CO

2mass fraction of 05

A fair degree of agreement is demonstrated in the trend andapproximate magnitude except for the temperature rangefrom 2000K to 2500K The discrepancy in this temperaturerange has been attributed to the dust separation from thesurface the density change of carbon and so forth [33 55]which have not been taken into account in the presentformulation described in Section 2

Figure 5(c) compares predicted results with experimentaldata in H

2O flow of 120 sminus1 with the H

2O mass fraction

of 05 The solid curve is predicted with the surface kineticparameters for the C-H

2O reaction as 119861sA = 20 times 10

7msand 119864sA119877 = 33 times 10

4 K (119864sA = 271 kJmol) evaluatedand used in the previous work [56] Again fair agreement isdemonstrated

52 Approximate Explicit Expressions for the CombustionRate In general in order to calculate the combustion ratetemperature profiles in the gas phase must be obtained bynumerically solving the energy conservation equation forfinite gas-phase reaction kinetics However if we note thatcarbon combustion proceeds with nearly frozen gas-phasechemistry until the establishment of the COflame [42 45 46]and that the combustion is expected to proceed under nearlyinfinite gas-phase kinetics once the CO-flame is establishedanalytically-obtained combustion rates [23 27] presented inSection 3 are still useful and valuable for practical utility

However it should also be noted that the combustion-rate expressions thus obtained are implicit so that furthernumerical calculations are required by taking account ofthe relation 120573 equiv (minus119891s)(120585

1015840)s which is a function of the

streamfunction 119891 Since this procedure is slightly compli-cated and cannot be used easily in practical situations explicitexpressions are anxiously required in order to make theseresults more useful


0

0002

0004

0006

0008

001

1000 1500 2000 2500 3000

Air

Frozen

Data from Golovina and Khaustovich 1961

Com

busti

on ra

te119898

(kg

m2s)

119879si

g=

1479

K

Flame-detached

Flame-attached

119886 = 120 sminus1


(a)

0

0002

0004

0006

0008

001

1000 1500 2000 2500 3000

Data from Khitrin and Golovina 1964

Com

busti

on ra

te119898

(kg

m2s) 119886 = 120 sminus1

C-CO2


(b)

0

0005

001

0015

002

1000 1500 2000 2500 3000


C-H2O119886 = 120 sminus1

Com

busti

on ra

te119898

(kg

m2s)


(c)

Figure 5 Experimental comparisons for the combustion rate under an atmospheric pressure (a) in airflow (b) in CO2-flow with CO

2mass

fraction of 05 (c) inH2O-flowwithH

2Omass fraction of 05 Data points are experimental [33 55] and curves are calculated from the theory

In order to elucidate the relation between (minus119891s) and120573 dependence of (1205851015840)s on streamfunction 119891 is first to beexamined by introducing a simplified profile of 119891 as

119891 =

119891s (0 le 120578 le 120578lowast)

119887120578 + 119888 (120578lowastle 120578 le 120578

lowastlowast)

120578 + 119889 (120578 ge 120578lowastlowast)

(74)

before conducting an integration Here 119887 119888 and 119889 areconstants 119891(120578

lowast) = 119891s and 119891(120578lowastlowast) = 119891

119900

Recalling the definitions of 120573 and (1205851015840)s andmaking use ofa relation (minus119891s) ≪ 1 as is the case formost solid combustionwe have the following approximate relation [28]

1 + 120573 asymp exp [119870 (minus119891s)] or (minus119891s) asympln (1 + 120573)

119870

(75)

where

119870 = 120578lowast+ radic

120587

2

[exp((119887 minus 1) 119891

2

119900

2119887

) minus 1 erfc(119891119900

radic2

)

+

1

radic119887

erf (119891119900

radic2119887

) minus erf (119891119900

radic2

)] + radic

120587

2

(76)

Equation (75) shows that the combustion rate (minus119891s) canbe expressed by the transfer number 120573 in terms of thelogarithmic term ln(1 + 120573) Note that the first and secondterms in (76) are one order of magnitude smaller than thethird term (1205872)

12In order to obtain the specific formof the transfer number

120573 a two-term expansion of the exponential function isexpected to be sufficient because (minus119891s) ≪ 1 so that use hasbeen made of the following relation [27 28]

120573

1 + 120573

= 1 minus exp [minus119870 (minus119891s)] asymp 119870 (minus119891s) (77)

By virtue of this relation (39) (42) and (49) yield thefollowing approximate expressions for the transfer number[28] respectivelyFrozen mode

120573 asymp (

119870119860 sO

1 + 119870119860 sO)(

2119882C119882O

119884Oinfin)

+ (

119870119860 sP

1 + 119870119860 sP)(

119882C119882P

119884Pinfin)

(78)


Flame-detached mode

120573 asymp (

119870119860 sP

1 + 119870119860 sP)(

2119882C119882O

119884Oinfin +

119882C119882P

119884Pinfin) (79)

Flame-attached mode

120573 asymp (

119870119860 sO

1 + 2119870119860 sO minus 119870119860 sP)(

2119882C119882O

119884Oinfin)

+ (

119870119860 sP

1 + 2119870119860 sO minus 119870119860 sP)(

119882C119882P

119884Pinfin)

(80)

Although these are approximate the transfer number canbe expressed explicitly in terms of the reduced surfaceDamkohler numbers 119860 sO and 119860 sP and O

2and CO

2con-

centrations in the freestreamIn addition we have

119870119860 s119894 = 119896s119894119870


(81)

where 119896s119894 is the specific reaction rate constant for the surfacereaction expressed as

119896s119894 equiv 119861s119894 (119879infin

119879s) exp(minus

119879119886s119894

119879s) (119894 = OP) (82)

Note that the factor 119870[2119895119886(120583infin120588infin)]

12 in (81) alsoappears in the combustion rate defined in (28) by use of therelation in (75) as

asymp 120588infin


119870

ln (1 + 120573) (83)

53 Correction Factor 119870 and Mass-Transfer Coefficient Inorder to elucidate the physical meaning of the factor

119870

[2119895119886 (120583infin120588infin)]12

(84)

let us consider the situation that 120573 ≪ 1 with the frozenmode combustion taken as an example Then (83) leads tothe following result

asymp

1

1119896sO + 119870radic2119895119886 (120583infin120588infin)

120588infin(

2119882C119882O

119884Oinfin)

+

1

1119896sP + 119870radic2119895119886 (120583infin120588infin)

120588infin(

119882C119882P

119884Pinfin)

(85)

We see that this expression is similar to the well-knownexpression for the combustion rate of solid

=

1

1119896s + 1ℎ119863(120588119884O)infin (86)

for the first-order kinetics [57ndash60] Here ℎ119863is the overall

convective mass-transfer coefficient It is seen that the factor

[2119895119886(120583infin120588infin)]

12K corresponds to the mass-transfer coeffi-cient ℎ

119863 suggesting that the specific form of ℎ

119863is of use in

determining a form of the correction factor119870Furthermore by evaluatingmass fluxes at the surface and

in the gas phase with the elemental carbon (119882C119882F)119884F +(119882C119882P)119884P taken as the transferred substance and by use ofthe coupling function in (29) we have

ℎ119863= (

119879119877

119879infin

) (1205851015840)sradic2119895119886 (120583infin120588infin) (87)

suggesting that the correction factor119870 depends on both (1205851015840)sand the representative temperature 119879

119877in the boundary layer

54 Expression of the Correction Factor 119870 In obtaining anapproximate expression for the factor119870 it seems that we canuse the accomplishment in the field of heat andmass transferThe mass-transfer coefficient is given as [61 62]

ℎ119863= (

119879119877

119879infin

)

119862mtSc06


fromNu119909

radicRe119909

= 119862mtSc04

(88)

where 119862mt = 0763 and 0570 for the axisymmetric andtwo-dimensional stagnation flows respectively based on theanalogy between heat and mass transfers where

Nu119909=

ℎ119863119909

119863119877

Re119909=

120588119877119880119909

120583119877

119880 = 119886119909 Sc =120583120588

119863

(89)

However this kind of expression is far from satisfactionbecause (88) is originally obtained for heat-transfer problemswithout mass transfer In addition this relation is obtainedunder an assumption that there is no density change eventhough there exist temperature andor concentration distri-butions in the gas phase

Because of the simultaneous existence of temperature andconcentration distributions in the carbon combustion we arerequired to obtain an approximate expression for the factor119870 in another way In this attempt [28] (1205872)12 in the factor119870 in (76) is kept as it is because it is 1(1205851015840)s for the inviscidstagnation flow without mass ejection from the surface Theremaining part of the factor 119870 is then determined by use ofnumerical results [23 42 45] In this determination use hasbeen made of a curve-fitting method with (119879

infin119879s) taken as

a variable to have a simple functionIt has turned out that we can fairly represent the combus-

tion rate for the frozen andor flame-attached modes in theaxisymmetric stagnation flow with

119870 = radic

2

3

(

119879infin

119879s)(1 minus

119879infin

2119879s) + radic

120587

2

(90)

within 3 error for 119884Oinfin = 0233 [28] within 6 error for119884Oinfin = 07 (cf Figure 6) Examinations have been made


0

001

002

003

004

Frozen

1000 1500 2000 2500

Flame-detached

Flame-attached

Com

busti

on ra

te119898

(kg

m2s) 119884O = 07

119884A = 00003

119886 = 170 sminus1


Figure 6 Combustion rates for the three limiting modes in thestagnation oxidizer flow as a function of the surface temperatureThe solid curves are numerical results and dashed curves are thoseof the explicit expressions

in the range of the surface Damkohler numbers 119863119886sO and119863119886sP from 106 to 1010 that of the surface temperature 119879sfrom 1077K to 2424K and that of the freestream temperature119879infin

from 323K to 1239K The frozen and flame-attachedmodes can fairly be correlated by the single equation (90)because the gas-phase temperature profiles are the sameNote that the combustion rate in high O

2concentrations

violates the assumption that (minus119891s) ≪ 1 Nonetheless theexpressions appear to provide a fair representation becausethese expressions vary as the natural logarithm of the transfernumber

In the flame-detached mode not only the surface andfreestream temperatures but also the oxidizer concentrationthat must be taken into account It has turned out that

119870 = radic

2

3

(

119879infin

119879s)(1 minus

119879infin

2119879s)

minus 005 (1 +

4119882C119882O

119884Oinfin) + radic

120587

2

(91)

can fairly represent the combustion rate in axisymmetricstagnation flow within 4 error when the O

2mass-fraction

119884Oinfin is 0233 and 0533 although the error becomes 6 nearthe transition state for the flame attaches In an oxygen flowthe error is within 6 except for the transition state while itincreases up to 15 around the state

As for the two-dimensional stagnation flow it has alreadybeen shown that the combustion rate in the frozen andorflame-attached modes can fairly be represented with

119870 = (

119879infin

119879s)(1 minus

119879infin

2119879s) + radic

120587

2

(92)

and that in the flame-detached mode with

119870 = (

119879infin

119879s)(1 minus

119879infin

2119879s)

minus 005 (1 +

4119882C119882O


120587

2

(93)

0

0005

001

0015

1000 1500 2000 2500

Data fromVisser and Adomeit 1985Parker and Hottel 1936Tu Davis and Hottel 1934

Air119884Ainfin = 0003

Com

busti

on ra

te119898

(kg

m2s)

119886 = 300 sminus1

119886 = 180 sminus1

119886 = 60 sminus1


(a)

0

002

004

006

008

1000 1500 2000 2500


Com

busti

on ra

te119898

(kg

m2s) 119886 = 1000 sminus1

119886 = 320 sminus1

119886 = 88 sminus1


119884Oinfin = 07

119884Ainfin = 00002

(b)

Figure 7 Combustion rate as a function of the surface temperaturewith the velocity gradient taken as a parameter (a) in airflow withthe H

2O mass-fraction 119884Ainfin = 0003 (b) in oxidizing-flow with

119884Oinfin = 07 and 119884Ainfin = 00002 Data points are experimental results[34 59 63] in airflow and those [34] in oxidizing flow curves arecalculated by use of the explicit combustion-rate expressions

The errors in these modes are nearly the same as those for theaxisymmetric case respectivelyThe difference in expressionsin (90) and (92) and that in (91) and (93) can be attributed tothe difference in the flow configurations

55 Combustion Rates at Various Velocity Gradients Inorder to verify the validity of the explicit combustion-rateexpressions further experimental comparisons have beenconducted Kinetic parameters and values of thermophysicalproperties are those evaluated and used in the previoussections Experimental data used are those in the literature[34 59 63]

Figure 7(a) shows the combustion rate in airflow as afunction of the surface temperature with the velocity gradi-ent taken as a parameter The H

2O mass fraction in airflow

is set to be 0003 in the same manner as that in Figure 5(a)


When the velocity gradient is 60 sminus1 it is predicted thatwith increasing surface temperature the combustion rate firstincreases then decreases abruptly and again increases asshown in Figure 1(b) As the velocity gradient is increasedeven up to 300 sminus1 the trend is still the same although thecombustion rate becomes high due to an enhanced oxidizersupply

Note that use has also been made of the combustion-rate expressions Up to the ignition surface temperature areasonable prediction can be made by (75) with the transfernumber 120573 for the frozen mode in (78) and the correctionfactor 119870 in (90) for axisymmetric case When the surfacetemperature is higher than the ignition surface temperature(75) with 120573 for the flame-detached mode in (79) and 119870

in (91) can fairly represent the experimental results exceptfor the temperatures near the ignition surface temperatureat low velocity gradients say 60 sminus1 in Figure 7(a) In thistemperature range we can use (75) with 120573 for the flame-attached mode in (80) and 119870 in (90) although accuracy ofthis prediction is not so high compared to the others Thisis attributed to the fact that we cannot assume infinitely fastgas-phase reaction rates because the combustion situation isthat just after the establishment of CO flame

Data points are experimental Tu et al [59] made anexperiment under a condition of 119886 = 60 sminus1 using a sphericalspecimen (brush carbon 119889 = 25mm) in airflow (05ms)Parker and Hottel [63] did an experiment under conditionsof 119886 = 60 sminus1 (998771) and 180 sminus1 (∙) using a hemisphericalspecimen (brush carbon 119889 = 25mm) in airflow (up to15ms) Visser and Adomeit [34] did an experiment underconditions of 119886 = 170 sminus1 using a hemispherical specimen(artificial graphite 119889 = 15mm) in oxidizer flow (085ms)with O

2mass-fraction 119884Oinfin = 018 Experimental data of

Visser and Adomeit [34] have been used because no otherresults in airflow can be found for the axisymmetric caseNonetheless it can even be justified if we take account ofa possibility that the ambient air has been entrained by theoxidizer flow thereby the oxygen concentration is enrichedThe scatter of the combustion rate in the temperature rangefrom 1500K to 1800K can be attributed to a fluctuation ofwater-vapor concentration in the oxidizing flow which canexert influences on the appearance of CO flame

Figure 7(b) is the similar plot of the combustion rate inoxidizer flow with the O

2mass fraction YOinfin = 07 The

H2O mass fraction in oxidizer flow is set to be 00002 in

accordance with the experimental condition [34] The sametrend as that observed in Figure 7(a) can be shown even inthe oxygen-enriched condition However a further increasein the velocity gradient under which no experiments haveever been reported in the axisymmetric stagnation flow canchange the trend When the velocity gradient is 1000 sminus1which is even higher than that ever used in the previousexperiment [37] the combustion rate first increases thenreaches a plateau and again increases after an abrupt increaseas surface temperature increases Since the ignition surfacetemperature is as high as 1912 K at which the combustionrate without CO flame is smaller than that with CO flamenot a decrease but an increase is predicted to occur abruptly

0

001

002

1000 1500 2000 2500

Axisymmetric stagnation flow

Data from Matsui Tsuji and Makino 1983

Matsui 1983

Com

busti

on ra

te119898

(kg

m2s)

119886 = 300 sminus1

119884Ainfin = 0001


119884Ainfin = 0004

Figure 8 Combustion rate as a function of the surface temperaturewith the H

2O mass-fraction 119884Ainfin taken as a parameter Data

points are combustion ratemeasured in the axisymmetric stagnationairflow over a flat plate [38 64] curves are calculated by use of theexplicit combustion-rate expressions

in the combustion rate just after the appearance of the COflame Itmay be informative to note that this kind of an abruptincrease in the combustion rate has already been observed inthe two-dimensional stagnation flow [28]

6 Combustion in Humid Airflow

Here carbon combustion in humid airflow has been exam-ined focusing on promoting and suppressing effects of thewater vapor in the airflow From the practical point of viewthe carbon combustion in airflow at highH

2Oconcentrations

is related to evaluating protection properties of rocket noz-zles made of carbonaceous materials from erosive attacks ofwater vapor contained in the working fluid for propulsion aswell as the coalchar combustion in such environments withan appreciable amount of water vapor

61 Combustion in Relatively Dry Airflow In order to have aclear image for the effect ofwater vapor on the combustion letus first examine combustion behavior in relatively dry airflowor oxidizer flow As shown in Figures 1(a) 1(b) 5(a) 7(a) and7(b) the combustion rate increases with increasing surfacetemperature up to the ignition surface temperature whichis quite sensitive to the water-vapor concentration in the gasphase At the ignition surface temperature there appears anabrupt change in the combustion rate because appearanceof the CO-flame alters the dominant surface reaction fromthe C-O


2reaction A further increase

in the surface temperature raises the combustion rate andit reaches the diffusion-limited value It should howeverbe noted that this is the case that the H

2O mass-fraction

119884Ainfin is strictly controlled When its control is poor theappearance of CO flame can necessarily cause a scatter inthe combustion rate as shown in Figure 8 beingmeasured inthe axisymmetric stagnation airflow over a flat plate [38 64]due to the ldquounsteadyrdquo combustion that proceeds without COflame at one time while with CO flame at the other time


0

0005

001

0015

1000 1500 2000 2500


Data from Matsui Tsuji and Makino 1986C

ombu

stion

rate119898

(kg

m2s)

119886 = 100 sminus1

119884Oinfin = 02

119884Ainfin = 014


119879si

g=1094

K

Figure 9 Combustion rate in humid airflow with the H2O mass-

fraction 119884Ainfin = 014 as a function of the surface temperature 119879sData points are experimental [39] and curves are calculated by useof the explicit combustion-rate expressions

Predicted results by use of the explicit combustion-rateexpressions are also shown in Figure 8 as a function of thesurface temperature with the H

2Omass-fraction 119884Ainfin taken

as a parameter In this prediction use has been made of thesurface kinetic parameters described in the previous works[15 42 65] for the C-O


6ms and119864sO119877=22 times 10

4 K (119864sO =180 kJmol) for the C-CO2reac-

tion 119861sP = 60 times 107ms and 119864sP119877 = 32 times 10

4 K (119864sP =

269 kJmol) for the C-H2O reaction 119861sA=20 times 10

8ms and119864sA119877 = 33 times 10

4 K (119864sA = 271 kJmol) This is due to thefact that the type of artificial graphite of their test specimensis exactly the same as that in the previous works It shouldbe informative to note that 119884Ainfin = 0001 in air correspondsto the dew point of 260K (minus13∘C) and 119884Ainfin = 0004 to thedew point of 275K (+2∘C) both of which are usual humiditythat we encounter As far as the trend and approximatemagnitude are concerned fair agreement is demonstratedIn this context the scatter of the combustion rate [64] notincluded in the previous report [38] for 119884Ainfin = 0001 can beattributed to the poor control of water-vapor concentrationin the airflow

62 Combustion Rate in Humid Airflow A further increasein the H

2O mass fraction can considerably change the

combustion behavior [39] The H2O mass-fraction 119884Ainfin is

now increased to be 014 the dew point of which is ashigh as 334K (61∘C) The airflow temperature is raised to119879infin

= 370K for preventing condensation of water vaporaccordingly Figure 9 shows the combustion rate in humidairflowof 119886 = 100 sminus1 as a function of the surface temperature119879s The O

2mass fraction is reduced because of the increased

H2O concentration It is seen that the combustion rate

increases first gradually and then rapidly with increasingsurface temperature This trend is quite different from thatin Figure 1(a) Therefore we can say that the high H

2O

mass fraction is unfavorable for the enhancement of com-bustion rate especially in the medium temperature rangebecause establishment of the CO flame is facilitated and

hence suppresses the combustion rate To the contrary athigh surface temperatures the high H

2O mass fraction is

favorable because the water vapor can participate in thesurface reaction as an additional oxidizer

In order to elucidate causes for this trend theoreticalresults are obtained by use of the formulation [56] to bedescribed in the next section with the surface C-H

2O and

global gas-phase H2-O2reactions taken into account addi-

tionally Not only results in the frozen and flame-detachedmodes but also that in the flame-attached mode are shown Itis seen that experimental results at temperatures lower thanabout 1550K are in accordance with the theoretical result ofthe flame-attached mode while those at temperatures higherthan about 1600K are in accordance with the result of theflame-detachedmode Since the ignition surface temperaturepredicted is as low as 1094K no abrupt change in thecombustion rate can be observed eventually even though theCO flame is established From these comparisons we candeduce that because of the high H

2O mass fraction in the

airflow the CO flame established at 1094K adheres to thecarbon surface so that the combustion in the flame-attachedmode prevails until CO ejection becomes strong enoughto separate the CO flame from the surface The differencebetween the combustion rate experimentally obtained andthat in the flame-attachedmode can be attributed to the finiterate of gas-phase reaction because the airflow is neither fastin the velocity nor high in the temperature As the surfacetemperature is further increased the CO flame detachesso that the combustion proceeds in the flame-detachedmode The rapid increase in the combustion rate at hightemperatures can be attributed to the participation of the C-H2O reaction

63 Extended Formulation Theoretical study [56] hasalready been conducted for the system with three surfacereactions and two global gas-phase reactions by extendingthe previous formulation Although some of the assumptionsintroduced in Section 2 are not appropriate for systems withhydrogen species use has been made of those as they are fortractability in order to capture fundamental aspects of thecarbon combustion under prescribed situations

By extending (26) to have contribution of the C-H2O

reaction the combustion rate (minus119891s) can be expressed as

120575 (minus119891s) = 119860 sOOs + 119860 sPPs + 119860 sAAs (94)

Again use has been made of an assumption that all thesurface reactions are the first order The reduced surfaceDamkohler number 119860 s119894 the surface Damkohler number119863119886s119894 and the stoichiometrically weighted mass fractionrelevant to the oxidizing species 119894 (=O P A) are also definedin the same manner as those in Section 2

Although 119884119894s must be determined through numerical

calculations when the gas-phase kinetics is finite they canbe determined analytically for some limiting cases as men-tioned One of them is the frozen mode in which we have[28]

119894s =

119894infin

1 + 120573 + 119860 s119894 [120573 (minus119891s)](119894 = OPA) (95)


Another is the flame-attached mode in which CO and H2

produced at the surface reactions are immediately consumedso that it looks that the CO flame adheres to the surface Inthe same manner [28] we have

Os =Oinfin minus 2120575120573

1 + 120573

Ps =Pinfin + 120575120573

1 + 120573

As =Ainfin

1 + 120573

(96)

The third is the flame-detached mode in which the gas-phase reaction is infinitely fast and the CO flame locates inthe gas phase Although a coupling function

Os + Ps + As =Oinfin + Pinfin + Ainfin minus 120575120573

1 + 120573

(97)

can easily be obtained and we can also put 119884Os = 0 forthis combustion situation a separation of 119884As from 119884Ps isnot straightforward For this aim an introduction of anotherspecies-enthalpy coupling function say

119879 + O + (1 minus 119876) A (98)

is necessary which yields

As =1

1 minus 119876

119879infinminus119879s + Oinfin + (1 minus 119876) Ainfin minus 120574

1 + 120573 + 119860 sA [120573 (minus119891s)] (99)

Here 119876 is the ratio of the heats of combustion of the H2-O2

and CO-O2reactions in the gas phase For evaluating 120574 use

has been made of the temperature profile

119879 = 119879s + (119879f minus 119879s) (120585

120585f) (100)

inside the flame from which we have

120574 =119879infinminus119879s + Oinfin + (1 minus 119876) Ainfin

+ (1 minus 119876)(As1 minus 120585f120585f

minus

Af

120585f)

(101)

where use has been made of the coupling function in(98) evaluated at the flame By further using 120585f and 119884Afdetermined by use of other coupling functions O minus F minus Hand H + A respectively we have from (99) as [56]

As =Ainfin

1 + 120573 + 119860 sA [120573 (minus119891s)] (1 minus Oinfin2120575120573) (102)

64 Approximate Explicit Expressions for the CombustionRate By use of the approximate relation in (77) analyticalexpressions for the transfer number 120573 can be obtained asfollows

Frozen mode

120573 asymp (

119870119860 sO

1 + 119870119860 sO)(

2119882C119882O

119884Oinfin) + (

119870119860 sP

1 + 119870119860 sP)

times (

119882C119882P

119884Pinfin) + (

119870119860 sA

1 + 119870119860 sA)(

119882C119882A

119884Ainfin)

(103)

Flame-detached mode

120573 asymp

1

2

119870119860 sP

1 + 119870119860 sP(

2119882C119882O

119884Oinfin +

119882C119882P

119884Pinfin)

+

119870119860 sA

1 + 119870119860 sA(

119882C119882O

119884Oinfin +

119882C119882A

119884Ainfin)

+

1

2

[

119870119860 sP

1 + 119870119860 sP(

2119882C119882O

119884Oinfin +

119882C119882P

119884Pinfin)

minus

119870119860 sA

1 + 119870119860 sA(

119882C119882O

119884Oinfin +

119882C119882A

119884Ainfin)

2

+ 4

119870119860 sP

1 + 119870119860 sP(

119882C119882O

119884Oinfin +

119882C119882P

119884Pinfin)

times

119870119860 sA

1 + 119870119860 sA(

119882C119882A

119884Ainfin)]

12

(104)

Flame-attached mode

120573 asymp (

1

1 + 2119870119860 sO minus 119870119860 sP)

times (119870119860 sO2119882C119882O

119884Oinfin + 119870119860 sP119882C119882P

119884Pinfin

+119870119860 sA119882C119882A

119884Ainfin)

(105)

As the correction factor 119870 for the axisymmetric flow wehave (90) for the frozen and flame-attached modes and (91)for the flame-detached mode As shown in Figure 9 experi-mental comparisons have already been conducted and a fairdegree of agreement is demonstrated in general suggestingappropriateness of the present analysis including the choiceof thermophysical properties used

7 Concluding Remarks

In the present study combustion of solid carbon has beenexamined through experimental comparisons In order tohave a clear understanding the carbon combustion consid-ered here is only that in the forward stagnation flowfieldbeing established by inserting a spherical specimen into auniform flow or by impinging a uniform flow on a flat platethe situation ofwhich is closely related to those of ablative car-bon heat shield andor strongly convective carbon burningIn the formulation an aerothermochemical analysis has been


conducted based on the chemically reacting boundary layerwith considering the surface C-O

2and C-CO

2reactions and

the gas-phase CO-O2reaction By virtue of the generalized

species-enthalpy coupling functions derived successfullyit has been demonstrated that there exists close couplingbetween the surface and gas-phase reactions that exertsinfluences on the combustion rate Combustion response inthe limiting situations has further been identified by using thegeneralized coupling functions

After confirming the experimental fact that the com-bustion rate momentarily changes upon ignition becauseestablishment of theCOflame in the gas phase can change thedominant surface reaction from the faster C-O

2reaction to

the slower C-CO2reaction focus has been put on the ignition

of CO flame over the burning carbon in the prescribedflowfield and theoretical studies have been conducted byusing the generalized coupling functions The asymptoticexpansion method has been used to derive the explicit igni-tion criterion from which in accordance with experimentalresults it has been shown that ignition is facilitated withincreasing surface temperature and oxidizer concentrationwhile suppressed with increasing velocity gradient Then anattempt has been made to estimate kinetic parameters for thegas-phase reactions prior to the appearance of the CO flameFor this aim use has been made of the ignition criteriontheoretically obtained by evaluating it at the ignition surface-temperature experimentally determined

Experimental comparisons have further been conductedand a fair degree of agreement has been demonstrated Inaddition an attempt has been made to obtain explicit com-bustion-rate expressions presented by the transfer number interms of the natural logarithmic term just like that for dropletcombustion For the three limiting cases explicit expressionshave further been obtained by making an assumption ofsmall combustion rate It has even been found that before theestablishment of CO flame the combustion rate can fairly berepresented by the expression in the frozen mode and thatafter the establishment of CO flame the combustion rate canbe represented by the expression in the flame-attached andorflame-detached modes Since the present expressions areexplicit and have fair accuracy they are anticipated to makevarious contributions not only for qualitative and quantitativestudies in facilitating understanding but also for practicalutility such as designs of furnaces combustors ablativecarbon heat-shields and high-temperature structures withCC-composites in various aerospace applications as well aspropulsion with using high-energy-density fuels

Finally carbon combustion in humid airflow has beenstudied which can even be a basic research for erosive attacksofwater vapor toCC-composite in rocket nozzles It has beenfound that the high H

2Omass fraction is unfavorable for the

enhancement of combustion rate especially in the mediumtemperature range because establishment of the CO flame isfacilitated and hence suppresses the combustion rate To thecontrary at high surface temperatures the high H

2O mass-

fraction is favorable because the water vapor can participatein the surface reaction as an additional oxidizer Theoreticalresults obtained by additionally introducing the surface

C-H2Oreaction and the global gas-phaseH

2-O2reaction into

the previous formulation have also suggested the usefulnessof the explicit expressions for the combustion rate which caneven be used in experimental comparisons for the H

2Omass

fraction of 014Although essential feature of the carbon combustion has

been captured to some extents further progresses are stronglyrequired for its firm understanding because wide attentionhas been given to carbonaceous materials in various fields

Nomenclature

119860 Reduced surface Damkohler number119886 Velocity gradient in the stagnation

flowfield119861 Frequency factor119887 Constant119862 Constant119888119901 Specific heat capacity of gas

119863 Diffusion coefficient119863119886 Damkohler number119889 Diameter or constant119864 Activation energy119865 Function defined in the ignition

criterion119891 Nondimensional streamfunctionℎ119863 Mass-transfer coefficient

119895 119895 = 1 and 0 designate axisymmetric andtwo-dimensional flows respectively

119870 Factor119896 Surface reactivity119871 Convective-diffusive operator Dimensional mass burning (or

combustion) rate119902 Heat of combustion per unit mass of CO119899 Exponent for the order of reaction119877119900 Universal gas constant

119877 Curvature of surface or radius119904 Boundary-layer variable along the

surface119879 Temperature119879119886 Activation temperature119905 Time119906 Velocity component along 119909119881 Freestream velocityV Velocity component along 119910119882 Molecular weight119908 Reaction rate119909 Tangential distance along the surface119884 Mass fraction119910 Normal distance from the surface

Greek Symbols

120572 Stoichiometric CO2-to-reactant mass

ratio120573 Conventional transfer number120574 Temperature gradient at the surface


120575 Product CO2-to-carbon mass ratio

120576 Measure of the thermal energy in thereaction zone relative to the activationenergy

120578 Boundary-layer variable normal to thesurface or perturbed concentration

Θ Perturbed temperature in the outer region120579 Perturbed temperature in the inner region120582 Parameter defined in the ignition analysis120583 Viscosity] Stoichiometric coefficient120585 Profile function120588 Density120594 Inner variable120595 Streamfunction120596 Reaction rate

Subscripts

A Water vapor or C-H2O surface reaction

a Critical value at flame attachmentC CarbonF Carbon monoxidef Flame sheetg Gas phaseig Ignitionin Inner regionmax Maximum valuemt Mass transferN NitrogenO Oxygen or C-O

2surface reaction

out Outer regionP Carbon dioxide or C-CO

2surface reaction

R Representative values Surfaceinfin Freestream or ambience

Superscripts


119899 Reaction ordersim Nondimensional or stoichiometrically

weighted1015840 Differentiation with respect to 120578lowast Without water-vapor effect

References

[1] H R Batchelder R M Busche andW P Armstrong ldquoKineticsof coal gasificationrdquo Industrial and Engineering Chemistry vol45 no 9 pp 1856ndash1878 1953

[2] M Gerstein and K P Coffin ldquoCombustion of solid fuelsrdquo inCombustion Processes B Lewis R N Pease and H S TaylorEds pp 444ndash469 Princeton University Press Princeton NJUSA 1956

[3] P L Walker Jr F Rusinko Jr and L G Austin ldquoGas reactionof carbonrdquo in Advances in Catalysis and Related Subjects D D

Eley P W Selwood and P B Weisz Eds vol 11 pp 133ndash221Academic Press New York NY USA 1959

[4] T J Clark R E Woodley and D R De Halas ldquoGas-graphitesystemsrdquo in Nuclear Graphite R E Nightingale Ed pp 387ndash444 Academic Press New York NY USA 1962

[5] L N Khitrin The Physics of Combustion and Explosion IsraelProgram for Scientific Translations Jerusalem Israel 1962

[6] M F Mulcahy and I W Smith ldquoKinetics of combustion ofpulverized fuel a review of theory and experimentrdquo Pure andApplied Chemistry vol 19 no 1 pp 81ndash108 1969

[7] H G Maahs ldquoOxidation of carbon at high temperaturesreaction-rate control or transport controlrdquo NASA TN D-63101971

[8] D E Rosner ldquoHigh-temperature gas-solid reactionsrdquo AnnualReview of Materials Science vol 2 pp 573ndash606 1972

[9] R H Essenhigh ldquoCombustion and flame propagation in coalsystems a reviewrdquo Symposium (International) on Combustionvol 16 no 1 pp 353ndash374 1977

[10] R H Essenhigh ldquoFundamentals of coal combustionrdquo in Chem-istry of Coal Utilization M A Elliott Ed pp 1153ndash1312 Wiley-Interscience New York NY USA 1981

[11] K Annamalai and W Ryan ldquoInteractive processes in gasifica-tion and combustion-II Isolated carbon coal and porous charparticlesrdquo Progress in Energy and Combustion Science vol 19no 5 pp 383ndash446 1993

[12] K Annamalai W Ryan and S Dhanapalan ldquoInteractiveprocesses in gasification and combustionmdashpart III coalcharparticle arrays streams and cloudsrdquo Progress in Energy andCombustion Science vol 20 no 6 pp 487ndash618 1994

[13] J R Arthur ldquoReactions between carbon and oxygenrdquo Transac-tions of the Faraday Society vol 47 pp 164ndash178 1951

[14] H Schlichting and K Gestin Plane Stagnation-Point FlowBoundary LayerTheory Springer Berlin Germany 8th edition2000

[15] AMakino ldquoMass transfer related to heterogeneous combustionof solid carbon in the forward stagnation regionmdashpart 1combustion rate and flame structurerdquo in Mass Transfer inChemical Engineering Processes J Markos Ed pp 251ndash282InTech Rijeka Croatia 2011 httpwwwintechopencomarti-clesshowtitlemass-transfer-related-to-heterogeneous-com-bustion-of-solid-carbon-in-the-forward-stagnation-region-1

[16] AMakino ldquoMass transfer related to heterogeneous combustionof solid carbon in the forward stagnation regionmdashpart 2combustion rate in special environmentsrdquo in Mass Transfer inChemical Engineering Processes J Markos Ed pp 281ndash306InTech Rijeka Croatia 2011 httpwwwintechopencomarti-clesshowtitlemass-transfer-related-to-heterogeneous-com-bustion-of-solid-carbon-in-the-forward-stagnation-region-2

[17] H Tsuji and K Matsui ldquoAn aerothermochemical analysis ofcombustion of carbon in the stagnation flowrdquo Combustion andFlame vol 26 pp 283ndash297 1976

[18] G Adomeit G Mohiuddin and N Peters ldquoBoundary layercombustion of carbonrdquo Symposium (International) on Combus-tion vol 16 no 1 pp 731ndash743 1977

[19] G Adomeit W Hocks and K Henriksen ldquoCombustion of acarbon surface in a stagnation point flow fieldrdquoCombustion andFlame vol 59 no 3 pp 273ndash288 1985

[20] K Henriksen W Hocks and G Adomeit ldquoCombustion of acarbon surface in a stagnation point flow fieldmdashpart II ignitionand quench phenomenardquo Combustion and Flame vol 71 no 2pp 169ndash177 1988


[21] K Matsui and H Tsuji ldquoAn aerothermochemical analysis ofsolid carbon combustion in the stagnation flow accompaniedby homogeneous CO oxidationrdquo Combustion and Flame vol70 no 1 pp 79ndash99 1987

[22] A Makino and C K Law ldquoQuasi-steady and transient com-bustion of a carbon particle theory and experimental compar-isonsrdquo Symposium (International) on Combustion vol 21 no 1pp 183ndash191 1988

[23] A Makino ldquoA theoretical and experimental study of carboncombustion in stagnation flowrdquo Combustion and Flame vol 81no 2 pp 166ndash187 1990

[24] P Chung ldquoChemically reacting nonequilibrium boundary lay-ersrdquo in Advances in Heat Transfer J P Hartnett and T F IrvineJr Eds vol 2 pp 109ndash270 Academic Press New York NYUSA 1965

[25] C K Law ldquoOn the stagnation-point ignition of a premixedcombustiblerdquo International Journal of Heat and Mass Transfervol 21 no 11 pp 1363ndash1368 1978

[26] D B Spalding ldquoCombustion of fuel particlesrdquo Fuel vol 30 no1 pp 121ndash130 1951

[27] A Makino ldquoAn approximate explicit expression for the com-bustion rate of a small carbon particlerdquo Combustion and Flamevol 90 no 2 pp 143ndash154 1992

[28] A Makino T Namikiri and N Araki ldquoCombustion rate ofgraphite in a high stagnation flowfield and its expression as afunction of the transfer numberrdquo Symposium (International) onCombustion vol 2 pp 2949ndash2956 1998

[29] P A Libby andT R Blake ldquoTheoretical study of burning carbonparticlesrdquo Combustion and Flame vol 36 pp 139ndash169 1979

[30] S K Ubhayakar and F A Williams ldquoBurning and extinction ofa laser-ignited carbon particle in quiescent mixtures of oxygenand nitrogenrdquo Journal of the Electrochemical Society vol 123 no5 pp 747ndash756 1976

[31] KHenriksen ldquoWeakhomogeneous burning in front of a carbonsurfacerdquo Symposium (International) on Combustion vol 22 no1 pp 47ndash57 1989

[32] A Makino and C K Law ldquoIgnition and extinction of CO flameover a carbon rodrdquo Combustion Science and Technology vol 73no 4ndash6 pp 589ndash615 1990

[33] L N Khitrin and E S Golovina ldquoInteraction between graphiteand various chemically active gases at high temperaturesrdquoin High Temperature Technology pp 485ndash496 ButterworthsLondon UK 1964

[34] W Visser and G Adomeit ldquoExperimental investigation of theignition and combustion of a graphite probe in cross flowrdquoSymposium (International) on Combustion vol 20 no 1 pp1845ndash1851 1985

[35] D J Harris and I W Smith ldquoIntrinsic reactivity of petroleumcoke and brown coal char to carbon dioxide steam and oxygenrdquoSymposium (International) on Combustion vol 23 no 1 pp1185ndash1190 1991

[36] J B Howard G C Williams and D H Fine ldquoKinetics ofcarbon monoxide oxidation in postflame gasesrdquo Symposium(International) on Combustion vol 14 no 1 pp 975ndash986 1973

[37] KMatsui AKoyama andKUehara ldquoFluid-mechanical effectson the combustion rate of solid carbonrdquoCombustion and Flamevol 25 pp 57ndash66 1975

[38] K Matsui H Tsuji and A Makino ldquoThe effects of water vaporconcentration on the rate of combustion of an artificial graphitein humid air flowrdquo Combustion and Flame vol 50 pp 107ndash1181983

[39] K Matsui H Tsuji and A Makino ldquoA further study of theeffects of water vapor concentration on the rate of combustionof an artificial graphite in humid air flowrdquo Combustion andFlame vol 63 no 3 pp 415ndash427 1986

[40] H Schlichting and K Gestin Sphere Boundary Layer TheorySpringer Berlin Germany 8th edition 2000

[41] W Visser Verbrenung Einer Umstromten Graphitoberflache[PhD thesis] Institute fur Allgemaine Mechanik RWTHAachen Germany 1984

[42] A Makino N Araki and Y Mihara ldquoCombustion of artificialgraphite in stagnation flow estimation of global kinetic param-eters from experimental resultsrdquo Combustion and Flame vol96 no 3 pp 261ndash274 1994

[43] J Nagle and R F Strickland-Constable ldquoOxidation of Carbonbetween 1000ndash2000 ∘Crdquo in Proceedings of the 5th Conference onCarbon pp 154ndash164 Pergamon New York NY USA 1962

[44] R T Yang andM Steinberg ldquoA diffusion cell method for study-ing heterogeneous kinetics in the chemical reactiondiffusioncontrolled region Kinetics of C + CO

2rarr 2CO at 1200ndash

1600 ∘Crdquo Industrial and Engineering Chemistry Fundamentalsvol 16 no 2 pp 235ndash242 1977

[45] A Makino I Kato M Senba H Fujizaki and N ArakildquoFlame structure and combustion rate of burning graphite inthe stagnation flowrdquo Symposium (International) on Combustionvol 26 no 2 pp 3067ndash3069 1996

[46] A Makino M Senba M Shintomi H Fujizaki and N ArakildquoExperimental determination of the spatial resolution of CARSin the combustion fieldmdashCARS thermometry applied to thecombustion field of solid carbon in a stagnation flowrdquo Nenshono Kagaku to Gijutsu (Combustion Science and Technology) vol5 no 2 pp 89ndash101 1997 (Japanese)

[47] A Linan ldquoThe asymptotic structure of counterflow diffusionflames for large activation energiesrdquo Acta Astronautica vol 1no 7-8 pp 1007ndash1039 1974

[48] M Matalon ldquoComplete burning and extinction of a carbonparticle in an oxidizing atmosphererdquo Combustion Science andTechnology vol 24 no 3-4 pp 115ndash127 1980

[49] M Matalon ldquoWeak burning and gas-phase ignition abouta carbon particle in an oxidizing atmosphererdquo CombustionScience and Technology vol 25 no 1-2 pp 43ndash48 1981

[50] M Matalon ldquoThe steady burning of a solid particlerdquo SIAMJournal onAppliedMathematics vol 42 no 4 pp 787ndash803 1982

[51] H Tsuji and I Yamaoka ldquoThe counterflow diffusion flame inthe forward stagnation region of a porous cylinderrdquo Symposium(International) on Combustion vol 11 no 1 pp 979ndash984 1967

[52] G K Sobolev ldquoHigh-temperature oxidation and burning ofcarbon monoxiderdquo Symposium (International) on Combustionvol 7 no 1 pp 386ndash391 1958

[53] Z Chukhanov ldquoThe burning of carbon 1 The sequence ofprocesses in the combustion of air suspensions of solid fuelsrdquoTechnical Physics of the USSR vol 5 pp 41ndash58 1938

[54] Z Chukhanov ldquoThe Burning of Carbonmdashpart II oxidationrdquoTechnical Physics of the USSR vol 5 pp 511ndash524 1938

[55] E S Golovina andG P Khaustovich ldquoThe interaction of carbonwith carbon dioxide and oxygen at temperatures up to 3000 ∘KrdquoSymposium (International) on Combustion vol 8 no 1 pp 784ndash792 1991

[56] AMakino andNUmehara ldquoCombustion rates of graphite rodsin the forward stagnation field of the high-temperature humidairflowrdquo Proceedings of the Combustion Institute vol 31 no 2pp 1873ndash1880 2007


[57] K Fischbeck ldquoUber das Reaktionsvermogen der Festen StofferdquoZeitschrift fur Elektrochemie und Angewandte PhysikalischeChemie vol 39 no 5 pp 316ndash330 1933

[58] K Fischbeck L Neundeubel and F Salzer ldquoUber das Reak-tionsvermogen von Kristallartenrdquo Zeitschrift fur Elektrochemieund Angewandte Physikalische Chemie vol 40 pp 517ndash5221934

[59] C M Tu H Davis and H C Hottel ldquoCombustion rate ofcarbon combustion of spheres in flowing gas streamsrdquo Indus-trial and Engineering Chemistry vol 26 no 7 pp 749ndash757 1934

[60] D A Frank-KamenetskiiDiffusion and Heat Transfer in Chem-ical Kinetics edited by J P Appleton Plenum New York NYUSA 2nd edition 1969

[61] Y Katto An Outline of Heat Transfer 1982[62] FMWhiteHeat andMass Transfer Addison-Wesley Reading

Mass USA 1988[63] A S Parker and H C Hottel ldquoCombustion rate of carbon

study of gas-film structure by microsamplingrdquo Industrial andEngineering Chemistry vol 28 no 11 pp 1334ndash1341 1936

[64] K Matsui Private communications 1983[65] A Makino H Fujizaki and N Araki ldquoCombustion rate of bur-

ning graphite in a stagnation flow of water vaporrdquo Combustionand Flame vol 113 no 1-2 pp 258ndash263 1998

International Journal of

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Active and Passive Electronic Components

Control Scienceand Engineering

Journal of



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Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design



Shock and Vibration


Civil EngineeringAdvances in

Acoustics and VibrationAdvances in



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Journal of

Advances inOptoElectronics


Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of


Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Chemical EngineeringInternational Journal of Antennas and

Propagation




Navigation and Observation



DistributedSensor Networks



When the surface temperature is low step (3) is known tobe much slower than steps (1) or (5) so that the combustionrate defined as mass being transferred in unit area and timeis determined solely by chemical kinetics Since the processin this regime is kinetically controlled the combustion rateonly depends on the surface temperature exponentially Tothe contrary the process of diffusion that proceeds in theboundary layer is irrelevant so that the combustion rateis independent of its thickness thereby concentrations ofoxidizing species at the reacting surface are not too differentfrom those in the freestream Furthermore since solid carbonis more or less porous in general combustion proceedsthroughout the sample specimen

When the surface temperature is high step (3) is knownto be much faster than steps (1) and (5) so that the combus-tion rate is controlled by the diffusion rate of oxidizing species(say oxygen) to the solid surface where their concentrationsare negligibly small In this diffusionally controlled regimetherefore the combustion rate exhibiting surface regressionstrongly depends on the boundary layer thickness while itsdependence on the surface temperature is weak (prop119879

05sim10)Since oxygen transfer to the carbon surface can occur via

O2 CO2 and H

2O the major surface reactions can be

C +O2997888rarr CO

2 (R1)

2C +O2997888rarr 2CO (R2)

C + CO2997888rarr 2CO (R3)

C +H2O 997888rarr CO +H

2 (R4)

At higher temperatures say higher than 1000K it is generallyrecognized that CO formation is the preferred route and thatthe relative contribution of (R1) can be negligible [13] Thusin the following reaction (R2) will be referred to as the C-O

2

reactionComparing (R2) and (R3) as alternate routes of CO

production the C-O2reaction is the preferred route for CO

production at low temperatures in simultaneous presence ofO2and CO

2 It can be initiated around 600K and saturated

around 1600K proceeding infinitely fast eventually relativeto diffusion The C-CO

2reaction of (R3) is the high temper-

ature route initiating around 1600K and becoming saturatedaround 2500K It is of particular significance because CO

2

in (R3) can even be the product of the gas-phase water-catalyzed CO oxidation

2CO +O2997888rarr 2CO

2 (R5)

referred to as the CO-O2reactionThus the C-CO

2and CO-

O2reactions can form a loopSimilarly the C-H

2O reaction (R4) generating CO and

H2 is also important when the combustion environment

consists of an appreciable amount of water This reaction isalso of significance because H

2O is the product of the H

2-

oxidation

2H2+O2997888rarr 2H

2O (R6)

referred to as the H2-O2reaction which then constitutes a

loop with the C-H2O reaction

The present study is intended to shed more light on thecarbon combustion with putting a focus on its combustionrate under an interaction of the surface and gas-phasereactions It is therefore not intended as a collection ofengineering data or an exhaustive review of all the pertinentwork published Rather it has an intention to represent thecarbon combustion by use of some of the basic characteristicsof the chemically reacting boundary layers under recognitionthat flow configurations are indispensable for proper evalu-ation of the combustion rate especially under the situationin which the gas-phase reaction can intimately affect overallcombustion response through its coupling to the surfacereactions

Among various flow configurations it has been reportedthat the stagnation-flow configuration has various advan-tages because it provides a well-defined one-dimensionalflow characterized by a single parameter [14] called as thestagnation velocity gradient It has even been said thatmathe-matical analyses experimental data acquisition and physicalinterpretations have been facilitated by its introduction Sincecarbon combustion in the two-dimensional stagnation flowestablished over a cylinder has already been summarizedsomewhere in a set of review papers [15 16] in whichextensive comparisons have been conducted between exper-imental and theoreticalnumerical results here we confineourselves to studying carbon combustion in the axisymmetricstagnation flow over a sphere or a flat plate by comparingexperimental data in the literaturewith theoreticalnumericalresults newly obtained From the practical point of view wecan even say that it simulates the situations of ablative carbonheat-shields andor the strongly convective carbon burningin the forward stagnation region of a particle

In the following formulation of the governing equationsis first presented in Section 2 based on theories on thechemically reacting boundary layer in the forward stagnationfield Chemical reactions considered are the surface C-O

2


2reaction

Generalized species-enthalpy coupling functions are thenderived without assuming any limit or near-limit behaviorswhich not only enable us tominimize the extent of numericalefforts needed for generalized treatment but also provideuseful insight into the conserved scalars in the carbon com-bustion In Section 3 it is also shown that straightforwardderivation of the combustion response can be allowed in thelimiting situations so thatwe have those for the frozen flame-detached and flame-attached modes

In Section 4 a further analytical study is made aboutthe ignition phenomenon in the gas phase related to finite-rate kinetics by use of the asymptotic expansion method toobtain a critical condition Appropriateness of this criterionis further examined by comparing surface temperatures atwhich the CO flame can appear After having constructedthe theory evaluation of kinetic parameters of the globalgas-phase reaction is conducted for further experimentalcomparisons

In Section 5 it is endeavored to obtain explicit com-bustion-rate expressions even though they might be approx-imate because they are anticipated to contribute muchto the foundation of theoretical understanding of carbon





2 Formulation




2


2reaction is



infin oxygen mass-


2and C-CO



2997888rarr CO

2reaction



120597 (120588119906119877119895)

120597119909

+

120597 (120588V119877119895)

120597119910

= 0 (1)

Momentum

120588119906

120597119906

120597119909

+ 120588V120597119906

120597119910

minus

120597

120597119910

(120583

120597119906

120597119910

) = 120588infin119906infin(

120597119906

120597119909

)

infin

(2)

Species

120588119906

120597119884119894

120597119909

+ 120588V120597119884119894

120597119910

minus

120597

120597119910

(120588119863

120597119884119894

120597119910

) = minus119908119894

(119894 = FO)

120588119906

120597119884P120597119909

+ 120588V120597119884P120597119910

minus

120597

120597119910

(120588119863

120597119884P120597119910

) = 119908P

120588119906

120597119884N120597119909

+ 120588V120597119884N120597119910

minus

120597

120597119910

(120588119863

120597119884N120597119910

) = 0

(3)

Energy

120588119906

120597 (119888119901119879)

120597119909

+ 120588V120597 (119888119901119879)

120597119910

minus

120597

120597119910

(120582

120597119879

120597119910

) = 119902119908F (4)





2reaction can be


119908F = (]119894119882119894) 119861g(

120588119884F119882F

)

]F

(

120588119884O119882O

)

]O

exp(minus119864g

119877119900119879

) (5)



infinand Vinfin


119906infin= 119886119909 V

infin= minus (119895 + 1) 119886119910 (6)


at 119910 = 0 119906 = 0 V = Vs


(7)




(119894 = OPN) (8)


(120588V119884F)s minus (120588119863120597119884F120597119910

)

s= 2119882F(

120588119884O119882O

)

s119861sO exp(minus

119879119886sO

119879s)

+ 2119882F(120588119884P119882P

)

s119861sP exp(minus

119879119886sP

119879s)

(120588V119884O)s minus (120588119863120597119884O120597119910

)

s= minus119882O(

120588119884O119882O

)

s119861sO exp(minus

119879119886sO

119879s)

(120588V119884P)s minus (120588119863120597119884P120597119910

)

s= minus119882P(

120588119884P119882P

)

s119861sP exp(minus

119879119886sP

119879s)

(120588V119884N)s minus (120588119863120597119884N120597119910

)

s= 0

(9)


1198893119891

1198891205783+ 119891

1198892119891

1198891205782+

1

2119895

120588infin

120588

minus (

119889119891

119889120578

)

2

= 0 (10)

119871 (F + P) = 119871 (O + P) = 119871 (P minus119879) = 119871 (N) = 0

(11)

119871 (119879) = minus119863119886g120596g (12)


119871 =

1198892

1198891205782+ 119891

119889

119889120578

(13)



119863119886g = (

119861g

2119895119886

)(

120588infin

]P119882P)

]F+]Ominus1

(]F)]F(]O)

]O (14)


120596g = (

119879infin

119879

)

]F+]Ominus1

(F)]F(O)


119879

) (15)


119904 = int

119909

0


120578 =

119906infin(119909) 119877119895

radic2119904

int

119910

0

120588 (119909 119910) 119889119910

(16)


119891 (119904 120578) =

120595 (119909 119910)

radic2119904

(17)


120588119906119877119895=

120597120595

120597119910

120588V119877119895= minus

120597120595

120597119909

(18)


119879 =

119879

119902 (119888119901120572F)

119879119886 =

119864119877119900

119902 (119888119901120572F)

120572F =]P119882P]F119882F

F =]P119882P]F119882F

119884F

O =

]P119882P]O119882O

119884O N = 119884N 120575 =

119882P119882C

(19)



infin) As



119891 (0) = 119891s (

119889119891

119889120578

)

s= 0 (

119889119891

119889120578

)

infin

= 1 (20)


at 120578 = 0 119879 =

119879s

119894= (119894)s (119894 = FOPN)


119879 =

119879infin F = 0

119894= (119894)infin

(119894 = OPN)

(21)



minus(

119889F119889120578

)


= 120575 (minus119891s) + 120575 (minus119891sP)

(22)

minus(

119889O119889120578

)



(23)

minus (

119889P119889120578

)


minus (

119889N119889120578

)

s+ (minus119891s) Ns = 0 (25)

where


(26)

119860 sO equiv 119863119886sO (119879infin

119879s

) exp(minus119879119886sO119879s

)

119863119886sO equiv

119861sO


119860 sP equiv 119863119886sP (119879infin

119879s

) exp(minus119879119886sP119879s

)

119863119886sP equiv119861sP


(27)


2









1 + 120573

(29)


1 + 120573

(30)

O +119879 = Os +


119879infinminus119879s) 120585

Os =Oinfin +


1 + 120573 + 119860 sO [120573 (minus119891s)]

(31)



119879infin+119879s) 120585

Ps =Pinfin minus

119879infin+119879s + 120574

1 + 120573 + 119860 sP [120573 (minus119891s)]

(32)

N = Ninfin1 + 120573120585

1 + 120573

(33)

where

120585 =

int

120578

0

exp(minusint120578

0

119891119889120578)119889120578

int

infin

0

exp(minusint120578

0

119891119889120578)119889120578

120574 =

(1198791015840)s

(1205851015840)s 120573 =

(minus119891s)

(1205851015840)s

(1205851015840)s=

1

int

infin

0

exp(minusint120578

0

119891119889120578)119889120578

(34)


(

1198892 119879

1198891205852) = minus

119863119886g120596g

(119889120585119889120578)2 (35)


(119879)120585=0

=119879s (

119879)120585=1

=119879infin (36)








=


infin

120575 minus (F + P)s

equiv 120573

(37)




2reaction




119879 =

119879s + 120574120585 120574 =

119879infinminus119879s (38)


2and CO



120575 (minus119891s) = 119860 sOOinfin

1 + 120573 + 119860 sO [120573 (minus119891s)]

+ 119860 sPPinfin

1 + 120573 + 119860 sP [120573 (minus119891s)]

(39)



120575

(40)






1 + 120573

(42)

119879f =

119879s + (Oinfin +

119879infinminus119879s) 120585f

120585f =2120575120573 minus Oinfin

(2120575 + Oinfin) 120573

(43)


0 le 120585 le 120585f 119879 =

119879s + (Oinfin +

119879infinminus119879s) 120585

120585f le 120585 le infin

119879 =

119879infinminus



1 + 120573

) (1 minus 120585)

(44)






minus (

119889F119889120578

)

fminus= (

119889O119889120578

)

f+

or

minus (

119889119884F119889120578

)

fminus=

]F119882F]O119882O

(

119889119884O119889120578

)

f+

(45)


119894 can be



(

119889119879

119889120578

)

fminusminus (

119889119879

119889120578

)

f+= minus(

119889F119889120578

)

fminus

or

120582(

119889119879

119889120578

)

fminusminus 120582(

119889119879

119889120578

)

f+= minus119902120588119863(

119889119884F119889120578

)

fminus

(46)




120573a =Oinfin

2120575

(47)


119879 =

119879s + (

119879infinminus119879s) 120585 (48)



1 + 120573

+ 119860 sPPinfin + 120575120573

1 + 120573

(49)








2reaction





2reaction



2 CO2 and H







2and H

2O issued into


infin119889)

where 119881infin









0

001

002

003

004

1000 1500 2000 2500


Visser 1985

Flame-detached


119886 = 170 sminus1

Com

busti

on ra

te119898

(kg

m2s) 119884Oinfin = 07

119884Ainfin = 00002

119879si

g=

1480

K


(a)

0

001

002

003

004

1000 1500 2000 2500


Frozen

Flame-detached

Flame-attached

119886 = 170 sminus1

Com

busti

on ra

te119898

(kg

m2s) 119884Oinfin = 07

119884Ainfin = 00038


119879si

g=1249

K

(b)







2reaction






6ms and 119864sO119877 =




infin120583infin=


infin120588infin





2reaction



0

001

002

003

004

1000 1500 2000 2500



119884Pinfin = 1

Com

busti

on ra

te119898

(kg

m2s)







2reaction reported





2




2mass


2





2

concentrations




(119879)0=119879s + (

119879infinminus119879s) 120585

(119894)0= 119894s + (119894infin minus

119894s) 120585 (119894 = FOP) (50)



119879s + Fs gt

119879infin (51)


(119879)

in= (

119879)0+ 120576

119879s120582120579 (120585) +O (120576

2)

=119879s [1 minus 120576 (120594 minus 120582120579)] +O (120576

2)

(52)

where

120576 =

119879s119879119886g

120582 =

Oinfin119879s minus

119879infin

120585 = 120576(

119879s

119879s minus

119879infin

)120594 (53)



(O)in= Os + 120576

119879s120582 (120594 minus 120579) (54)

(F)in= (


1 + 120573

) + Os

+ 120576(

119879s

119879s minus

119879infin

)(


1 + 120573


(55)


120574 equiv (

119889119879

119889120585

)

s=[

[

119889120594

119889120585

119889(119879)

in119889120594

]

]s



119889120579

119889120594

)

s+O (120576)

(56)


Os =Oinfin

1 + 120573 + 119860 sO [120573 (minus119891s)]1 minus (

119889120579

119889120594

)

s (57)


1198892120579

1198891205942= minusΔ(120594 minus 120579 + 120578O)

12 exp (120582120579 minus 120594) (58)

where

Δ = 119863119886g exp(minus119879119886g

119879s

)

120573

(minus119891s)

119879s

119879s minus

119879infin

2

times (

119879s119879119886g

)

32

(

119879infin

119879s

)

12

Fs

(120582119879s)12

(59)

120578O =

Os

120576119879s120582

(60)








(119879)

out=119879s + (

119879infinminus119879s) 120585 + 120576

119879sΘ (120585) +O (120576

2) (62)



120582120579 (infin) = minus119862I (

119889120579

119889120594

)

infin

= 0 (63)



2Δ I120582 = (radic120578O + radic120587120582

2

119890120582120578O


120594I

119890(120582minus1)120594erfc (119911) 119889120594)

minus1

(64)

erfc (119911) = 2

radic120587

expintinfin

119911

(minus1199052) 119889119905 (65)


(

119889120579

119889120594

)

s=

1

120582

or (

119889(119879)

in119889120585

)

s

= 0 (66)


at 120582 = 1 2Δ I =1


as 120578O 997888rarr infin 2Δ I120582 =1

radic120578O

(67)



2Δ I120582 = (radic120578O +

radic120587

2

[119890120578O erfc (radic120578O)

+

1

119865 (120582)


2

)])

minus1

(68)

where

119865 (120582) = 056 +

021

120582

minus

012

1205822+

035

1205823 (69)





2reaction and hence








1000

1200

1400

1600

1800

2000

10 100 1000


1985

119884O = 07

119884A = 00003


Igni

tion

surfa

ce te

mpe

ratu

re119879

sig

(K)

(a)

1000

1200

1400

1600

1800

2000

00001 0001 001


1985


119884O = 07

119886 = 170 sminus1

Igni

tion

surfa

ce te

mpe

ratu

re119879

sig

(K)

(b)

1000

1200

1400

1600

1800

2000


119886 = 170 sminus1119884A = 00003

Igni

tion

surfa

ce te

mpe

ratu

re119879

sig

(K)

(c)












2reaction



2mass fraction


2mass fraction In





2mass fraction



119861g = 119861lowast

g (120588119884A119882A)12

[(molm3)12

sdot s]minus1

(70)








119861lowast


minus1

(71)




119861lowast


minus1

(72)




Air

O2



100

102

104

106

3 4 5

Oxidizer

lowast(119879

s119879119886

g)32

exp(minus119879119886

g119879

s)

(m3m

oL)middot

sminus1

1s

119884Oinfin = 0533

119861g


2and H



119861lowast


minus1

(73)



2andor H



2









infin120588infin









2O mass fraction



7msand 119864sA119877 = 33 times 10







0

0002

0004

0006

0008

001

1000 1500 2000 2500 3000

Air

Frozen


Com

busti

on ra

te119898

(kg

m2s)

119879si

g=

1479

K

Flame-detached

Flame-attached

119886 = 120 sminus1


(a)

0

0002

0004

0006

0008

001

1000 1500 2000 2500 3000


Com

busti

on ra

te119898

(kg

m2s) 119886 = 120 sminus1

C-CO2


(b)

0

0005

001

0015

002

1000 1500 2000 2500 3000


C-H2O119886 = 120 sminus1

Com

busti

on ra

te119898

(kg

m2s)


(c)


2mass




119891 =

119891s (0 le 120578 le 120578lowast)

119887120578 + 119888 (120578lowastle 120578 le 120578

lowastlowast)

120578 + 119889 (120578 ge 120578lowastlowast)

(74)



119900



119870

(75)

where

119870 = 120578lowast+ radic

120587

2

[exp((119887 minus 1) 119891

2

119900

2119887

) minus 1 erfc(119891119900

radic2

)

+

1

radic119887

erf (119891119900

radic2119887

) minus erf (119891119900

radic2

)] + radic

120587

2

(76)




120573

1 + 120573



120573 asymp (

119870119860 sO

1 + 119870119860 sO)(

2119882C119882O

119884Oinfin)

+ (

119870119860 sP

1 + 119870119860 sP)(

119882C119882P

119884Pinfin)

(78)


Flame-detached mode

120573 asymp (

119870119860 sP

1 + 119870119860 sP)(

2119882C119882O

119884Oinfin +

119882C119882P

119884Pinfin) (79)

Flame-attached mode

120573 asymp (

119870119860 sO

1 + 2119870119860 sO minus 119870119860 sP)(

2119882C119882O

119884Oinfin)

+ (

119870119860 sP

1 + 2119870119860 sO minus 119870119860 sP)(

119882C119882P

119884Pinfin)

(80)


2and CO

2con-


119870119860 s119894 = 119896s119894119870


(81)


119896s119894 equiv 119861s119894 (119879infin

119879s) exp(minus

119879119886s119894

119879s) (119894 = OP) (82)



asymp 120588infin


119870

ln (1 + 120573) (83)


119870

[2119895119886 (120583infin120588infin)]12

(84)


asymp

1


120588infin(

2119882C119882O

119884Oinfin)

+

1


120588infin(

119882C119882P

119884Pinfin)

(85)


=

1

1119896s + 1ℎ119863(120588119884O)infin (86)



[2119895119886(120583infin120588infin)]



119863is of use in



ℎ119863= (

119879119877

119879infin





ℎ119863= (

119879119877

119879infin

)

119862mtSc06


fromNu119909

radicRe119909

= 119862mtSc04

(88)


Nu119909=

ℎ119863119909

119863119877

Re119909=

120588119877119880119909

120583119877

119880 = 119886119909 Sc =120583120588

119863

(89)






119870 = radic

2

3

(

119879infin

119879s)(1 minus

119879infin

2119879s) + radic

120587

2

(90)



0

001

002

003

004

Frozen

1000 1500 2000 2500

Flame-detached

Flame-attached

Com

busti

on ra

te119898

(kg

m2s) 119884O = 07

119884A = 00003

119886 = 170 sminus1





2concentrations



119870 = radic

2

3

(

119879infin

119879s)(1 minus

119879infin

2119879s)

minus 005 (1 +

4119882C119882O


120587

2

(91)


2mass-fraction



119870 = (

119879infin

119879s)(1 minus

119879infin

2119879s) + radic

120587

2

(92)


119870 = (

119879infin

119879s)(1 minus

119879infin

2119879s)

minus 005 (1 +

4119882C119882O


120587

2

(93)

0

0005

001

0015

1000 1500 2000 2500



Com

busti

on ra

te119898

(kg

m2s)

119886 = 300 sminus1

119886 = 180 sminus1

119886 = 60 sminus1


(a)

0

002

004

006

008

1000 1500 2000 2500


Com

busti

on ra

te119898

(kg

m2s) 119886 = 1000 sminus1

119886 = 320 sminus1

119886 = 88 sminus1


119884Oinfin = 07

119884Ainfin = 00002

(b)




















0

001

002

1000 1500 2000 2500



Matsui 1983

Com

busti

on ra

te119898

(kg

m2s)

119886 = 300 sminus1

119884Ainfin = 0001


119884Ainfin = 0004







2Oconcentrations






2O mass-fraction



0

0005

001

0015

1000 1500 2000 2500



ombu

stion

rate119898

(kg

m2s)

119886 = 100 sminus1

119884Oinfin = 02

119884Ainfin = 014


119879si

g=1094

K







6ms and119864sO119877=22 times 10



4 K (119864sP =


8ms and119864sA119877 = 33 times 10










2O



2O mass fraction is



2O and












119894s =

119894infin

1 + 120573 + 119860 s119894 [120573 (minus119891s)](119894 = OPA) (95)





1 + 120573

Ps =Pinfin + 120575120573

1 + 120573

As =Ainfin

1 + 120573

(96)



1 + 120573

(97)


119879 + O + (1 minus 119876) A (98)


As =1

1 minus 119876


1 + 120573 + 119860 sA [120573 (minus119891s)] (99)




119879 = 119879s + (119879f minus 119879s) (120585

120585f) (100)



+ (1 minus 119876)(As1 minus 120585f120585f

minus

Af

120585f)

(101)


As =Ainfin



Frozen mode

120573 asymp (

119870119860 sO

1 + 119870119860 sO)(

2119882C119882O

119884Oinfin) + (

119870119860 sP

1 + 119870119860 sP)

times (

119882C119882P

119884Pinfin) + (

119870119860 sA

1 + 119870119860 sA)(

119882C119882A

119884Ainfin)

(103)

Flame-detached mode

120573 asymp

1

2

119870119860 sP

1 + 119870119860 sP(

2119882C119882O

119884Oinfin +

119882C119882P

119884Pinfin)

+

119870119860 sA

1 + 119870119860 sA(

119882C119882O

119884Oinfin +

119882C119882A

119884Ainfin)

+

1

2

[

119870119860 sP

1 + 119870119860 sP(

2119882C119882O

119884Oinfin +

119882C119882P

119884Pinfin)

minus

119870119860 sA

1 + 119870119860 sA(

119882C119882O

119884Oinfin +

119882C119882A

119884Ainfin)

2

+ 4

119870119860 sP

1 + 119870119860 sP(

119882C119882O

119884Oinfin +

119882C119882P

119884Pinfin)

times

119870119860 sA

1 + 119870119860 sA(

119882C119882A

119884Ainfin)]

12

(104)

Flame-attached mode

120573 asymp (

1

1 + 2119870119860 sO minus 119870119860 sP)

times (119870119860 sO2119882C119882O

119884Oinfin + 119870119860 sP119882C119882P

119884Pinfin

+119870119860 sA119882C119882A

119884Ainfin)

(105)






2and C-CO

2reactions and




2reaction to







2O mass-



2-O2reaction into


2Omass



Nomenclature










Greek Symbols








Subscripts



2surface reaction


2surface reaction


Superscripts




References









































































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation













2 Formulation




2


2reaction is



infin oxygen mass-


2and C-CO



2997888rarr CO

2reaction



120597 (120588119906119877119895)

120597119909

+

120597 (120588V119877119895)

120597119910

= 0 (1)

Momentum

120588119906

120597119906

120597119909

+ 120588V120597119906

120597119910

minus

120597

120597119910

(120583

120597119906

120597119910

) = 120588infin119906infin(

120597119906

120597119909

)

infin

(2)

Species

120588119906

120597119884119894

120597119909

+ 120588V120597119884119894

120597119910

minus

120597

120597119910

(120588119863

120597119884119894

120597119910

) = minus119908119894

(119894 = FO)

120588119906

120597119884P120597119909

+ 120588V120597119884P120597119910

minus

120597

120597119910

(120588119863

120597119884P120597119910

) = 119908P

120588119906

120597119884N120597119909

+ 120588V120597119884N120597119910

minus

120597

120597119910

(120588119863

120597119884N120597119910

) = 0

(3)

Energy

120588119906

120597 (119888119901119879)

120597119909

+ 120588V120597 (119888119901119879)

120597119910

minus

120597

120597119910

(120582

120597119879

120597119910

) = 119902119908F (4)





2reaction can be


119908F = (]119894119882119894) 119861g(

120588119884F119882F

)

]F

(

120588119884O119882O

)

]O

exp(minus119864g

119877119900119879

) (5)



infinand Vinfin


119906infin= 119886119909 V

infin= minus (119895 + 1) 119886119910 (6)


at 119910 = 0 119906 = 0 V = Vs


(7)




(119894 = OPN) (8)


(120588V119884F)s minus (120588119863120597119884F120597119910

)

s= 2119882F(

120588119884O119882O

)

s119861sO exp(minus

119879119886sO

119879s)

+ 2119882F(120588119884P119882P

)

s119861sP exp(minus

119879119886sP

119879s)

(120588V119884O)s minus (120588119863120597119884O120597119910

)

s= minus119882O(

120588119884O119882O

)

s119861sO exp(minus

119879119886sO

119879s)

(120588V119884P)s minus (120588119863120597119884P120597119910

)

s= minus119882P(

120588119884P119882P

)

s119861sP exp(minus

119879119886sP

119879s)

(120588V119884N)s minus (120588119863120597119884N120597119910

)

s= 0

(9)


1198893119891

1198891205783+ 119891

1198892119891

1198891205782+

1

2119895

120588infin

120588

minus (

119889119891

119889120578

)

2

= 0 (10)

119871 (F + P) = 119871 (O + P) = 119871 (P minus119879) = 119871 (N) = 0

(11)

119871 (119879) = minus119863119886g120596g (12)


119871 =

1198892

1198891205782+ 119891

119889

119889120578

(13)



119863119886g = (

119861g

2119895119886

)(

120588infin

]P119882P)

]F+]Ominus1

(]F)]F(]O)

]O (14)


120596g = (

119879infin

119879

)

]F+]Ominus1

(F)]F(O)


119879

) (15)


119904 = int

119909

0


120578 =

119906infin(119909) 119877119895

radic2119904

int

119910

0

120588 (119909 119910) 119889119910

(16)


119891 (119904 120578) =

120595 (119909 119910)

radic2119904

(17)


120588119906119877119895=

120597120595

120597119910

120588V119877119895= minus

120597120595

120597119909

(18)


119879 =

119879

119902 (119888119901120572F)

119879119886 =

119864119877119900

119902 (119888119901120572F)

120572F =]P119882P]F119882F

F =]P119882P]F119882F

119884F

O =

]P119882P]O119882O

119884O N = 119884N 120575 =

119882P119882C

(19)



infin) As



119891 (0) = 119891s (

119889119891

119889120578

)

s= 0 (

119889119891

119889120578

)

infin

= 1 (20)


at 120578 = 0 119879 =

119879s

119894= (119894)s (119894 = FOPN)


119879 =

119879infin F = 0

119894= (119894)infin

(119894 = OPN)

(21)



minus(

119889F119889120578

)


= 120575 (minus119891s) + 120575 (minus119891sP)

(22)

minus(

119889O119889120578

)



(23)

minus (

119889P119889120578

)


minus (

119889N119889120578

)

s+ (minus119891s) Ns = 0 (25)

where


(26)

119860 sO equiv 119863119886sO (119879infin

119879s

) exp(minus119879119886sO119879s

)

119863119886sO equiv

119861sO


119860 sP equiv 119863119886sP (119879infin

119879s

) exp(minus119879119886sP119879s

)

119863119886sP equiv119861sP


(27)


2









1 + 120573

(29)


1 + 120573

(30)

O +119879 = Os +


119879infinminus119879s) 120585

Os =Oinfin +


1 + 120573 + 119860 sO [120573 (minus119891s)]

(31)



119879infin+119879s) 120585

Ps =Pinfin minus

119879infin+119879s + 120574

1 + 120573 + 119860 sP [120573 (minus119891s)]

(32)

N = Ninfin1 + 120573120585

1 + 120573

(33)

where

120585 =

int

120578

0

exp(minusint120578

0

119891119889120578)119889120578

int

infin

0

exp(minusint120578

0

119891119889120578)119889120578

120574 =

(1198791015840)s

(1205851015840)s 120573 =

(minus119891s)

(1205851015840)s

(1205851015840)s=

1

int

infin

0

exp(minusint120578

0

119891119889120578)119889120578

(34)


(

1198892 119879

1198891205852) = minus

119863119886g120596g

(119889120585119889120578)2 (35)


(119879)120585=0

=119879s (

119879)120585=1

=119879infin (36)








=


infin

120575 minus (F + P)s

equiv 120573

(37)




2reaction




119879 =

119879s + 120574120585 120574 =

119879infinminus119879s (38)


2and CO



120575 (minus119891s) = 119860 sOOinfin

1 + 120573 + 119860 sO [120573 (minus119891s)]

+ 119860 sPPinfin

1 + 120573 + 119860 sP [120573 (minus119891s)]

(39)



120575

(40)






1 + 120573

(42)

119879f =

119879s + (Oinfin +

119879infinminus119879s) 120585f

120585f =2120575120573 minus Oinfin

(2120575 + Oinfin) 120573

(43)


0 le 120585 le 120585f 119879 =

119879s + (Oinfin +

119879infinminus119879s) 120585

120585f le 120585 le infin

119879 =

119879infinminus



1 + 120573

) (1 minus 120585)

(44)






minus (

119889F119889120578

)

fminus= (

119889O119889120578

)

f+

or

minus (

119889119884F119889120578

)

fminus=

]F119882F]O119882O

(

119889119884O119889120578

)

f+

(45)


119894 can be



(

119889119879

119889120578

)

fminusminus (

119889119879

119889120578

)

f+= minus(

119889F119889120578

)

fminus

or

120582(

119889119879

119889120578

)

fminusminus 120582(

119889119879

119889120578

)

f+= minus119902120588119863(

119889119884F119889120578

)

fminus

(46)




120573a =Oinfin

2120575

(47)


119879 =

119879s + (

119879infinminus119879s) 120585 (48)



1 + 120573

+ 119860 sPPinfin + 120575120573

1 + 120573

(49)








2reaction





2reaction



2 CO2 and H







2and H

2O issued into


infin119889)

where 119881infin









0

001

002

003

004

1000 1500 2000 2500


Visser 1985

Flame-detached


119886 = 170 sminus1

Com

busti

on ra

te119898

(kg

m2s) 119884Oinfin = 07

119884Ainfin = 00002

119879si

g=

1480

K


(a)

0

001

002

003

004

1000 1500 2000 2500


Frozen

Flame-detached

Flame-attached

119886 = 170 sminus1

Com

busti

on ra

te119898

(kg

m2s) 119884Oinfin = 07

119884Ainfin = 00038


119879si

g=1249

K

(b)







2reaction






6ms and 119864sO119877 =




infin120583infin=


infin120588infin





2reaction



0

001

002

003

004

1000 1500 2000 2500



119884Pinfin = 1

Com

busti

on ra

te119898

(kg

m2s)







2reaction reported





2




2mass


2





2

concentrations




(119879)0=119879s + (

119879infinminus119879s) 120585

(119894)0= 119894s + (119894infin minus

119894s) 120585 (119894 = FOP) (50)



119879s + Fs gt

119879infin (51)


(119879)

in= (

119879)0+ 120576

119879s120582120579 (120585) +O (120576

2)

=119879s [1 minus 120576 (120594 minus 120582120579)] +O (120576

2)

(52)

where

120576 =

119879s119879119886g

120582 =

Oinfin119879s minus

119879infin

120585 = 120576(

119879s

119879s minus

119879infin

)120594 (53)



(O)in= Os + 120576

119879s120582 (120594 minus 120579) (54)

(F)in= (


1 + 120573

) + Os

+ 120576(

119879s

119879s minus

119879infin

)(


1 + 120573


(55)


120574 equiv (

119889119879

119889120585

)

s=[

[

119889120594

119889120585

119889(119879)

in119889120594

]

]s



119889120579

119889120594

)

s+O (120576)

(56)


Os =Oinfin

1 + 120573 + 119860 sO [120573 (minus119891s)]1 minus (

119889120579

119889120594

)

s (57)


1198892120579

1198891205942= minusΔ(120594 minus 120579 + 120578O)

12 exp (120582120579 minus 120594) (58)

where

Δ = 119863119886g exp(minus119879119886g

119879s

)

120573

(minus119891s)

119879s

119879s minus

119879infin

2

times (

119879s119879119886g

)

32

(

119879infin

119879s

)

12

Fs

(120582119879s)12

(59)

120578O =

Os

120576119879s120582

(60)








(119879)

out=119879s + (

119879infinminus119879s) 120585 + 120576

119879sΘ (120585) +O (120576

2) (62)



120582120579 (infin) = minus119862I (

119889120579

119889120594

)

infin

= 0 (63)



2Δ I120582 = (radic120578O + radic120587120582

2

119890120582120578O


120594I

119890(120582minus1)120594erfc (119911) 119889120594)

minus1

(64)

erfc (119911) = 2

radic120587

expintinfin

119911

(minus1199052) 119889119905 (65)


(

119889120579

119889120594

)

s=

1

120582

or (

119889(119879)

in119889120585

)

s

= 0 (66)


at 120582 = 1 2Δ I =1


as 120578O 997888rarr infin 2Δ I120582 =1

radic120578O

(67)



2Δ I120582 = (radic120578O +

radic120587

2

[119890120578O erfc (radic120578O)

+

1

119865 (120582)


2

)])

minus1

(68)

where

119865 (120582) = 056 +

021

120582

minus

012

1205822+

035

1205823 (69)





2reaction and hence








1000

1200

1400

1600

1800

2000

10 100 1000


1985

119884O = 07

119884A = 00003


Igni

tion

surfa

ce te

mpe

ratu

re119879

sig

(K)

(a)

1000

1200

1400

1600

1800

2000

00001 0001 001


1985


119884O = 07

119886 = 170 sminus1

Igni

tion

surfa

ce te

mpe

ratu

re119879

sig

(K)

(b)

1000

1200

1400

1600

1800

2000


119886 = 170 sminus1119884A = 00003

Igni

tion

surfa

ce te

mpe

ratu

re119879

sig

(K)

(c)












2reaction



2mass fraction


2mass fraction In





2mass fraction



119861g = 119861lowast

g (120588119884A119882A)12

[(molm3)12

sdot s]minus1

(70)








119861lowast


minus1

(71)




119861lowast


minus1

(72)




Air

O2



100

102

104

106

3 4 5

Oxidizer

lowast(119879

s119879119886

g)32

exp(minus119879119886

g119879

s)

(m3m

oL)middot

sminus1

1s

119884Oinfin = 0533

119861g


2and H



119861lowast


minus1

(73)



2andor H



2









infin120588infin









2O mass fraction



7msand 119864sA119877 = 33 times 10







0

0002

0004

0006

0008

001

1000 1500 2000 2500 3000

Air

Frozen


Com

busti

on ra

te119898

(kg

m2s)

119879si

g=

1479

K

Flame-detached

Flame-attached

119886 = 120 sminus1


(a)

0

0002

0004

0006

0008

001

1000 1500 2000 2500 3000


Com

busti

on ra

te119898

(kg

m2s) 119886 = 120 sminus1

C-CO2


(b)

0

0005

001

0015

002

1000 1500 2000 2500 3000


C-H2O119886 = 120 sminus1

Com

busti

on ra

te119898

(kg

m2s)


(c)


2mass




119891 =

119891s (0 le 120578 le 120578lowast)

119887120578 + 119888 (120578lowastle 120578 le 120578

lowastlowast)

120578 + 119889 (120578 ge 120578lowastlowast)

(74)



119900



119870

(75)

where

119870 = 120578lowast+ radic

120587

2

[exp((119887 minus 1) 119891

2

119900

2119887

) minus 1 erfc(119891119900

radic2

)

+

1

radic119887

erf (119891119900

radic2119887

) minus erf (119891119900

radic2

)] + radic

120587

2

(76)




120573

1 + 120573



120573 asymp (

119870119860 sO

1 + 119870119860 sO)(

2119882C119882O

119884Oinfin)

+ (

119870119860 sP

1 + 119870119860 sP)(

119882C119882P

119884Pinfin)

(78)


Flame-detached mode

120573 asymp (

119870119860 sP

1 + 119870119860 sP)(

2119882C119882O

119884Oinfin +

119882C119882P

119884Pinfin) (79)

Flame-attached mode

120573 asymp (

119870119860 sO

1 + 2119870119860 sO minus 119870119860 sP)(

2119882C119882O

119884Oinfin)

+ (

119870119860 sP

1 + 2119870119860 sO minus 119870119860 sP)(

119882C119882P

119884Pinfin)

(80)


2and CO

2con-


119870119860 s119894 = 119896s119894119870


(81)


119896s119894 equiv 119861s119894 (119879infin

119879s) exp(minus

119879119886s119894

119879s) (119894 = OP) (82)



asymp 120588infin


119870

ln (1 + 120573) (83)


119870

[2119895119886 (120583infin120588infin)]12

(84)


asymp

1


120588infin(

2119882C119882O

119884Oinfin)

+

1


120588infin(

119882C119882P

119884Pinfin)

(85)


=

1

1119896s + 1ℎ119863(120588119884O)infin (86)



[2119895119886(120583infin120588infin)]



119863is of use in



ℎ119863= (

119879119877

119879infin





ℎ119863= (

119879119877

119879infin

)

119862mtSc06


fromNu119909

radicRe119909

= 119862mtSc04

(88)


Nu119909=

ℎ119863119909

119863119877

Re119909=

120588119877119880119909

120583119877

119880 = 119886119909 Sc =120583120588

119863

(89)






119870 = radic

2

3

(

119879infin

119879s)(1 minus

119879infin

2119879s) + radic

120587

2

(90)



0

001

002

003

004

Frozen

1000 1500 2000 2500

Flame-detached

Flame-attached

Com

busti

on ra

te119898

(kg

m2s) 119884O = 07

119884A = 00003

119886 = 170 sminus1





2concentrations



119870 = radic

2

3

(

119879infin

119879s)(1 minus

119879infin

2119879s)

minus 005 (1 +

4119882C119882O


120587

2

(91)


2mass-fraction



119870 = (

119879infin

119879s)(1 minus

119879infin

2119879s) + radic

120587

2

(92)


119870 = (

119879infin

119879s)(1 minus

119879infin

2119879s)

minus 005 (1 +

4119882C119882O


120587

2

(93)

0

0005

001

0015

1000 1500 2000 2500



Com

busti

on ra

te119898

(kg

m2s)

119886 = 300 sminus1

119886 = 180 sminus1

119886 = 60 sminus1


(a)

0

002

004

006

008

1000 1500 2000 2500


Com

busti

on ra

te119898

(kg

m2s) 119886 = 1000 sminus1

119886 = 320 sminus1

119886 = 88 sminus1


119884Oinfin = 07

119884Ainfin = 00002

(b)




















0

001

002

1000 1500 2000 2500



Matsui 1983

Com

busti

on ra

te119898

(kg

m2s)

119886 = 300 sminus1

119884Ainfin = 0001


119884Ainfin = 0004







2Oconcentrations






2O mass-fraction



0

0005

001

0015

1000 1500 2000 2500



ombu

stion

rate119898

(kg

m2s)

119886 = 100 sminus1

119884Oinfin = 02

119884Ainfin = 014


119879si

g=1094

K







6ms and119864sO119877=22 times 10



4 K (119864sP =


8ms and119864sA119877 = 33 times 10










2O



2O mass fraction is



2O and












119894s =

119894infin

1 + 120573 + 119860 s119894 [120573 (minus119891s)](119894 = OPA) (95)





1 + 120573

Ps =Pinfin + 120575120573

1 + 120573

As =Ainfin

1 + 120573

(96)



1 + 120573

(97)


119879 + O + (1 minus 119876) A (98)


As =1

1 minus 119876


1 + 120573 + 119860 sA [120573 (minus119891s)] (99)




119879 = 119879s + (119879f minus 119879s) (120585

120585f) (100)



+ (1 minus 119876)(As1 minus 120585f120585f

minus

Af

120585f)

(101)


As =Ainfin



Frozen mode

120573 asymp (

119870119860 sO

1 + 119870119860 sO)(

2119882C119882O

119884Oinfin) + (

119870119860 sP

1 + 119870119860 sP)

times (

119882C119882P

119884Pinfin) + (

119870119860 sA

1 + 119870119860 sA)(

119882C119882A

119884Ainfin)

(103)

Flame-detached mode

120573 asymp

1

2

119870119860 sP

1 + 119870119860 sP(

2119882C119882O

119884Oinfin +

119882C119882P

119884Pinfin)

+

119870119860 sA

1 + 119870119860 sA(

119882C119882O

119884Oinfin +

119882C119882A

119884Ainfin)

+

1

2

[

119870119860 sP

1 + 119870119860 sP(

2119882C119882O

119884Oinfin +

119882C119882P

119884Pinfin)

minus

119870119860 sA

1 + 119870119860 sA(

119882C119882O

119884Oinfin +

119882C119882A

119884Ainfin)

2

+ 4

119870119860 sP

1 + 119870119860 sP(

119882C119882O

119884Oinfin +

119882C119882P

119884Pinfin)

times

119870119860 sA

1 + 119870119860 sA(

119882C119882A

119884Ainfin)]

12

(104)

Flame-attached mode

120573 asymp (

1

1 + 2119870119860 sO minus 119870119860 sP)

times (119870119860 sO2119882C119882O

119884Oinfin + 119870119860 sP119882C119882P

119884Pinfin

+119870119860 sA119882C119882A

119884Ainfin)

(105)






2and C-CO

2reactions and




2reaction to







2O mass-



2-O2reaction into


2Omass



Nomenclature










Greek Symbols








Subscripts



2surface reaction


2surface reaction


Superscripts




References









































































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation












(119894 = OPN) (8)


(120588V119884F)s minus (120588119863120597119884F120597119910

)

s= 2119882F(

120588119884O119882O

)

s119861sO exp(minus

119879119886sO

119879s)

+ 2119882F(120588119884P119882P

)

s119861sP exp(minus

119879119886sP

119879s)

(120588V119884O)s minus (120588119863120597119884O120597119910

)

s= minus119882O(

120588119884O119882O

)

s119861sO exp(minus

119879119886sO

119879s)

(120588V119884P)s minus (120588119863120597119884P120597119910

)

s= minus119882P(

120588119884P119882P

)

s119861sP exp(minus

119879119886sP

119879s)

(120588V119884N)s minus (120588119863120597119884N120597119910

)

s= 0

(9)


1198893119891

1198891205783+ 119891

1198892119891

1198891205782+

1

2119895

120588infin

120588

minus (

119889119891

119889120578

)

2

= 0 (10)

119871 (F + P) = 119871 (O + P) = 119871 (P minus119879) = 119871 (N) = 0

(11)

119871 (119879) = minus119863119886g120596g (12)


119871 =

1198892

1198891205782+ 119891

119889

119889120578

(13)



119863119886g = (

119861g

2119895119886

)(

120588infin

]P119882P)

]F+]Ominus1

(]F)]F(]O)

]O (14)


120596g = (

119879infin

119879

)

]F+]Ominus1

(F)]F(O)


119879

) (15)


119904 = int

119909

0


120578 =

119906infin(119909) 119877119895

radic2119904

int

119910

0

120588 (119909 119910) 119889119910

(16)


119891 (119904 120578) =

120595 (119909 119910)

radic2119904

(17)


120588119906119877119895=

120597120595

120597119910

120588V119877119895= minus

120597120595

120597119909

(18)


119879 =

119879

119902 (119888119901120572F)

119879119886 =

119864119877119900

119902 (119888119901120572F)

120572F =]P119882P]F119882F

F =]P119882P]F119882F

119884F

O =

]P119882P]O119882O

119884O N = 119884N 120575 =

119882P119882C

(19)



infin) As



119891 (0) = 119891s (

119889119891

119889120578

)

s= 0 (

119889119891

119889120578

)

infin

= 1 (20)


at 120578 = 0 119879 =

119879s

119894= (119894)s (119894 = FOPN)


119879 =

119879infin F = 0

119894= (119894)infin

(119894 = OPN)

(21)



minus(

119889F119889120578

)


= 120575 (minus119891s) + 120575 (minus119891sP)

(22)

minus(

119889O119889120578

)



(23)

minus (

119889P119889120578

)


minus (

119889N119889120578

)

s+ (minus119891s) Ns = 0 (25)

where


(26)

119860 sO equiv 119863119886sO (119879infin

119879s

) exp(minus119879119886sO119879s

)

119863119886sO equiv

119861sO


119860 sP equiv 119863119886sP (119879infin

119879s

) exp(minus119879119886sP119879s

)

119863119886sP equiv119861sP


(27)


2









1 + 120573

(29)


1 + 120573

(30)

O +119879 = Os +


119879infinminus119879s) 120585

Os =Oinfin +


1 + 120573 + 119860 sO [120573 (minus119891s)]

(31)



119879infin+119879s) 120585

Ps =Pinfin minus

119879infin+119879s + 120574

1 + 120573 + 119860 sP [120573 (minus119891s)]

(32)

N = Ninfin1 + 120573120585

1 + 120573

(33)

where

120585 =

int

120578

0

exp(minusint120578

0

119891119889120578)119889120578

int

infin

0

exp(minusint120578

0

119891119889120578)119889120578

120574 =

(1198791015840)s

(1205851015840)s 120573 =

(minus119891s)

(1205851015840)s

(1205851015840)s=

1

int

infin

0

exp(minusint120578

0

119891119889120578)119889120578

(34)


(

1198892 119879

1198891205852) = minus

119863119886g120596g

(119889120585119889120578)2 (35)


(119879)120585=0

=119879s (

119879)120585=1

=119879infin (36)








=


infin

120575 minus (F + P)s

equiv 120573

(37)




2reaction




119879 =

119879s + 120574120585 120574 =

119879infinminus119879s (38)


2and CO



120575 (minus119891s) = 119860 sOOinfin

1 + 120573 + 119860 sO [120573 (minus119891s)]

+ 119860 sPPinfin

1 + 120573 + 119860 sP [120573 (minus119891s)]

(39)



120575

(40)






1 + 120573

(42)

119879f =

119879s + (Oinfin +

119879infinminus119879s) 120585f

120585f =2120575120573 minus Oinfin

(2120575 + Oinfin) 120573

(43)


0 le 120585 le 120585f 119879 =

119879s + (Oinfin +

119879infinminus119879s) 120585

120585f le 120585 le infin

119879 =

119879infinminus



1 + 120573

) (1 minus 120585)

(44)






minus (

119889F119889120578

)

fminus= (

119889O119889120578

)

f+

or

minus (

119889119884F119889120578

)

fminus=

]F119882F]O119882O

(

119889119884O119889120578

)

f+

(45)


119894 can be



(

119889119879

119889120578

)

fminusminus (

119889119879

119889120578

)

f+= minus(

119889F119889120578

)

fminus

or

120582(

119889119879

119889120578

)

fminusminus 120582(

119889119879

119889120578

)

f+= minus119902120588119863(

119889119884F119889120578

)

fminus

(46)




120573a =Oinfin

2120575

(47)


119879 =

119879s + (

119879infinminus119879s) 120585 (48)



1 + 120573

+ 119860 sPPinfin + 120575120573

1 + 120573

(49)








2reaction





2reaction



2 CO2 and H







2and H

2O issued into


infin119889)

where 119881infin









0

001

002

003

004

1000 1500 2000 2500


Visser 1985

Flame-detached


119886 = 170 sminus1

Com

busti

on ra

te119898

(kg

m2s) 119884Oinfin = 07

119884Ainfin = 00002

119879si

g=

1480

K


(a)

0

001

002

003

004

1000 1500 2000 2500


Frozen

Flame-detached

Flame-attached

119886 = 170 sminus1

Com

busti

on ra

te119898

(kg

m2s) 119884Oinfin = 07

119884Ainfin = 00038


119879si

g=1249

K

(b)







2reaction






6ms and 119864sO119877 =




infin120583infin=


infin120588infin





2reaction



0

001

002

003

004

1000 1500 2000 2500



119884Pinfin = 1

Com

busti

on ra

te119898

(kg

m2s)







2reaction reported





2




2mass


2





2

concentrations




(119879)0=119879s + (

119879infinminus119879s) 120585

(119894)0= 119894s + (119894infin minus

119894s) 120585 (119894 = FOP) (50)



119879s + Fs gt

119879infin (51)


(119879)

in= (

119879)0+ 120576

119879s120582120579 (120585) +O (120576

2)

=119879s [1 minus 120576 (120594 minus 120582120579)] +O (120576

2)

(52)

where

120576 =

119879s119879119886g

120582 =

Oinfin119879s minus

119879infin

120585 = 120576(

119879s

119879s minus

119879infin

)120594 (53)



(O)in= Os + 120576

119879s120582 (120594 minus 120579) (54)

(F)in= (


1 + 120573

) + Os

+ 120576(

119879s

119879s minus

119879infin

)(


1 + 120573


(55)


120574 equiv (

119889119879

119889120585

)

s=[

[

119889120594

119889120585

119889(119879)

in119889120594

]

]s



119889120579

119889120594

)

s+O (120576)

(56)


Os =Oinfin

1 + 120573 + 119860 sO [120573 (minus119891s)]1 minus (

119889120579

119889120594

)

s (57)


1198892120579

1198891205942= minusΔ(120594 minus 120579 + 120578O)

12 exp (120582120579 minus 120594) (58)

where

Δ = 119863119886g exp(minus119879119886g

119879s

)

120573

(minus119891s)

119879s

119879s minus

119879infin

2

times (

119879s119879119886g

)

32

(

119879infin

119879s

)

12

Fs

(120582119879s)12

(59)

120578O =

Os

120576119879s120582

(60)








(119879)

out=119879s + (

119879infinminus119879s) 120585 + 120576

119879sΘ (120585) +O (120576

2) (62)



120582120579 (infin) = minus119862I (

119889120579

119889120594

)

infin

= 0 (63)



2Δ I120582 = (radic120578O + radic120587120582

2

119890120582120578O


120594I

119890(120582minus1)120594erfc (119911) 119889120594)

minus1

(64)

erfc (119911) = 2

radic120587

expintinfin

119911

(minus1199052) 119889119905 (65)


(

119889120579

119889120594

)

s=

1

120582

or (

119889(119879)

in119889120585

)

s

= 0 (66)


at 120582 = 1 2Δ I =1


as 120578O 997888rarr infin 2Δ I120582 =1

radic120578O

(67)



2Δ I120582 = (radic120578O +

radic120587

2

[119890120578O erfc (radic120578O)

+

1

119865 (120582)


2

)])

minus1

(68)

where

119865 (120582) = 056 +

021

120582

minus

012

1205822+

035

1205823 (69)





2reaction and hence








1000

1200

1400

1600

1800

2000

10 100 1000


1985

119884O = 07

119884A = 00003


Igni

tion

surfa

ce te

mpe

ratu

re119879

sig

(K)

(a)

1000

1200

1400

1600

1800

2000

00001 0001 001


1985


119884O = 07

119886 = 170 sminus1

Igni

tion

surfa

ce te

mpe

ratu

re119879

sig

(K)

(b)

1000

1200

1400

1600

1800

2000


119886 = 170 sminus1119884A = 00003

Igni

tion

surfa

ce te

mpe

ratu

re119879

sig

(K)

(c)












2reaction



2mass fraction


2mass fraction In





2mass fraction



119861g = 119861lowast

g (120588119884A119882A)12

[(molm3)12

sdot s]minus1

(70)








119861lowast


minus1

(71)




119861lowast


minus1

(72)




Air

O2



100

102

104

106

3 4 5

Oxidizer

lowast(119879

s119879119886

g)32

exp(minus119879119886

g119879

s)

(m3m

oL)middot

sminus1

1s

119884Oinfin = 0533

119861g


2and H



119861lowast


minus1

(73)



2andor H



2









infin120588infin









2O mass fraction



7msand 119864sA119877 = 33 times 10







0

0002

0004

0006

0008

001

1000 1500 2000 2500 3000

Air

Frozen


Com

busti

on ra

te119898

(kg

m2s)

119879si

g=

1479

K

Flame-detached

Flame-attached

119886 = 120 sminus1


(a)

0

0002

0004

0006

0008

001

1000 1500 2000 2500 3000


Com

busti

on ra

te119898

(kg

m2s) 119886 = 120 sminus1

C-CO2


(b)

0

0005

001

0015

002

1000 1500 2000 2500 3000


C-H2O119886 = 120 sminus1

Com

busti

on ra

te119898

(kg

m2s)


(c)


2mass




119891 =

119891s (0 le 120578 le 120578lowast)

119887120578 + 119888 (120578lowastle 120578 le 120578

lowastlowast)

120578 + 119889 (120578 ge 120578lowastlowast)

(74)



119900



119870

(75)

where

119870 = 120578lowast+ radic

120587

2

[exp((119887 minus 1) 119891

2

119900

2119887

) minus 1 erfc(119891119900

radic2

)

+

1

radic119887

erf (119891119900

radic2119887

) minus erf (119891119900

radic2

)] + radic

120587

2

(76)




120573

1 + 120573



120573 asymp (

119870119860 sO

1 + 119870119860 sO)(

2119882C119882O

119884Oinfin)

+ (

119870119860 sP

1 + 119870119860 sP)(

119882C119882P

119884Pinfin)

(78)


Flame-detached mode

120573 asymp (

119870119860 sP

1 + 119870119860 sP)(

2119882C119882O

119884Oinfin +

119882C119882P

119884Pinfin) (79)

Flame-attached mode

120573 asymp (

119870119860 sO

1 + 2119870119860 sO minus 119870119860 sP)(

2119882C119882O

119884Oinfin)

+ (

119870119860 sP

1 + 2119870119860 sO minus 119870119860 sP)(

119882C119882P

119884Pinfin)

(80)


2and CO

2con-


119870119860 s119894 = 119896s119894119870


(81)


119896s119894 equiv 119861s119894 (119879infin

119879s) exp(minus

119879119886s119894

119879s) (119894 = OP) (82)



asymp 120588infin


119870

ln (1 + 120573) (83)


119870

[2119895119886 (120583infin120588infin)]12

(84)


asymp

1


120588infin(

2119882C119882O

119884Oinfin)

+

1


120588infin(

119882C119882P

119884Pinfin)

(85)


=

1

1119896s + 1ℎ119863(120588119884O)infin (86)



[2119895119886(120583infin120588infin)]



119863is of use in



ℎ119863= (

119879119877

119879infin





ℎ119863= (

119879119877

119879infin

)

119862mtSc06


fromNu119909

radicRe119909

= 119862mtSc04

(88)


Nu119909=

ℎ119863119909

119863119877

Re119909=

120588119877119880119909

120583119877

119880 = 119886119909 Sc =120583120588

119863

(89)






119870 = radic

2

3

(

119879infin

119879s)(1 minus

119879infin

2119879s) + radic

120587

2

(90)



0

001

002

003

004

Frozen

1000 1500 2000 2500

Flame-detached

Flame-attached

Com

busti

on ra

te119898

(kg

m2s) 119884O = 07

119884A = 00003

119886 = 170 sminus1





2concentrations



119870 = radic

2

3

(

119879infin

119879s)(1 minus

119879infin

2119879s)

minus 005 (1 +

4119882C119882O


120587

2

(91)


2mass-fraction



119870 = (

119879infin

119879s)(1 minus

119879infin

2119879s) + radic

120587

2

(92)


119870 = (

119879infin

119879s)(1 minus

119879infin

2119879s)

minus 005 (1 +

4119882C119882O


120587

2

(93)

0

0005

001

0015

1000 1500 2000 2500



Com

busti

on ra

te119898

(kg

m2s)

119886 = 300 sminus1

119886 = 180 sminus1

119886 = 60 sminus1


(a)

0

002

004

006

008

1000 1500 2000 2500


Com

busti

on ra

te119898

(kg

m2s) 119886 = 1000 sminus1

119886 = 320 sminus1

119886 = 88 sminus1


119884Oinfin = 07

119884Ainfin = 00002

(b)




















0

001

002

1000 1500 2000 2500



Matsui 1983

Com

busti

on ra

te119898

(kg

m2s)

119886 = 300 sminus1

119884Ainfin = 0001


119884Ainfin = 0004







2Oconcentrations






2O mass-fraction



0

0005

001

0015

1000 1500 2000 2500



ombu

stion

rate119898

(kg

m2s)

119886 = 100 sminus1

119884Oinfin = 02

119884Ainfin = 014


119879si

g=1094

K







6ms and119864sO119877=22 times 10



4 K (119864sP =


8ms and119864sA119877 = 33 times 10










2O



2O mass fraction is



2O and












119894s =

119894infin

1 + 120573 + 119860 s119894 [120573 (minus119891s)](119894 = OPA) (95)





1 + 120573

Ps =Pinfin + 120575120573

1 + 120573

As =Ainfin

1 + 120573

(96)



1 + 120573

(97)


119879 + O + (1 minus 119876) A (98)


As =1

1 minus 119876


1 + 120573 + 119860 sA [120573 (minus119891s)] (99)




119879 = 119879s + (119879f minus 119879s) (120585

120585f) (100)



+ (1 minus 119876)(As1 minus 120585f120585f

minus

Af

120585f)

(101)


As =Ainfin



Frozen mode

120573 asymp (

119870119860 sO

1 + 119870119860 sO)(

2119882C119882O

119884Oinfin) + (

119870119860 sP

1 + 119870119860 sP)

times (

119882C119882P

119884Pinfin) + (

119870119860 sA

1 + 119870119860 sA)(

119882C119882A

119884Ainfin)

(103)

Flame-detached mode

120573 asymp

1

2

119870119860 sP

1 + 119870119860 sP(

2119882C119882O

119884Oinfin +

119882C119882P

119884Pinfin)

+

119870119860 sA

1 + 119870119860 sA(

119882C119882O

119884Oinfin +

119882C119882A

119884Ainfin)

+

1

2

[

119870119860 sP

1 + 119870119860 sP(

2119882C119882O

119884Oinfin +

119882C119882P

119884Pinfin)

minus

119870119860 sA

1 + 119870119860 sA(

119882C119882O

119884Oinfin +

119882C119882A

119884Ainfin)

2

+ 4

119870119860 sP

1 + 119870119860 sP(

119882C119882O

119884Oinfin +

119882C119882P

119884Pinfin)

times

119870119860 sA

1 + 119870119860 sA(

119882C119882A

119884Ainfin)]

12

(104)

Flame-attached mode

120573 asymp (

1

1 + 2119870119860 sO minus 119870119860 sP)

times (119870119860 sO2119882C119882O

119884Oinfin + 119870119860 sP119882C119882P

119884Pinfin

+119870119860 sA119882C119882A

119884Ainfin)

(105)






2and C-CO

2reactions and




2reaction to







2O mass-



2-O2reaction into


2Omass



Nomenclature










Greek Symbols








Subscripts



2surface reaction


2surface reaction


Superscripts




References









































































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











minus(

119889F119889120578

)


= 120575 (minus119891s) + 120575 (minus119891sP)

(22)

minus(

119889O119889120578

)



(23)

minus (

119889P119889120578

)


minus (

119889N119889120578

)

s+ (minus119891s) Ns = 0 (25)

where


(26)

119860 sO equiv 119863119886sO (119879infin

119879s

) exp(minus119879119886sO119879s

)

119863119886sO equiv

119861sO


119860 sP equiv 119863119886sP (119879infin

119879s

) exp(minus119879119886sP119879s

)

119863119886sP equiv119861sP


(27)


2









1 + 120573

(29)


1 + 120573

(30)

O +119879 = Os +


119879infinminus119879s) 120585

Os =Oinfin +


1 + 120573 + 119860 sO [120573 (minus119891s)]

(31)



119879infin+119879s) 120585

Ps =Pinfin minus

119879infin+119879s + 120574

1 + 120573 + 119860 sP [120573 (minus119891s)]

(32)

N = Ninfin1 + 120573120585

1 + 120573

(33)

where

120585 =

int

120578

0

exp(minusint120578

0

119891119889120578)119889120578

int

infin

0

exp(minusint120578

0

119891119889120578)119889120578

120574 =

(1198791015840)s

(1205851015840)s 120573 =

(minus119891s)

(1205851015840)s

(1205851015840)s=

1

int

infin

0

exp(minusint120578

0

119891119889120578)119889120578

(34)


(

1198892 119879

1198891205852) = minus

119863119886g120596g

(119889120585119889120578)2 (35)


(119879)120585=0

=119879s (

119879)120585=1

=119879infin (36)








=


infin

120575 minus (F + P)s

equiv 120573

(37)




2reaction




119879 =

119879s + 120574120585 120574 =

119879infinminus119879s (38)


2and CO



120575 (minus119891s) = 119860 sOOinfin

1 + 120573 + 119860 sO [120573 (minus119891s)]

+ 119860 sPPinfin

1 + 120573 + 119860 sP [120573 (minus119891s)]

(39)



120575

(40)






1 + 120573

(42)

119879f =

119879s + (Oinfin +

119879infinminus119879s) 120585f

120585f =2120575120573 minus Oinfin

(2120575 + Oinfin) 120573

(43)


0 le 120585 le 120585f 119879 =

119879s + (Oinfin +

119879infinminus119879s) 120585

120585f le 120585 le infin

119879 =

119879infinminus



1 + 120573

) (1 minus 120585)

(44)






minus (

119889F119889120578

)

fminus= (

119889O119889120578

)

f+

or

minus (

119889119884F119889120578

)

fminus=

]F119882F]O119882O

(

119889119884O119889120578

)

f+

(45)


119894 can be



(

119889119879

119889120578

)

fminusminus (

119889119879

119889120578

)

f+= minus(

119889F119889120578

)

fminus

or

120582(

119889119879

119889120578

)

fminusminus 120582(

119889119879

119889120578

)

f+= minus119902120588119863(

119889119884F119889120578

)

fminus

(46)




120573a =Oinfin

2120575

(47)


119879 =

119879s + (

119879infinminus119879s) 120585 (48)



1 + 120573

+ 119860 sPPinfin + 120575120573

1 + 120573

(49)








2reaction





2reaction



2 CO2 and H







2and H

2O issued into


infin119889)

where 119881infin









0

001

002

003

004

1000 1500 2000 2500


Visser 1985

Flame-detached


119886 = 170 sminus1

Com

busti

on ra

te119898

(kg

m2s) 119884Oinfin = 07

119884Ainfin = 00002

119879si

g=

1480

K


(a)

0

001

002

003

004

1000 1500 2000 2500


Frozen

Flame-detached

Flame-attached

119886 = 170 sminus1

Com

busti

on ra

te119898

(kg

m2s) 119884Oinfin = 07

119884Ainfin = 00038


119879si

g=1249

K

(b)







2reaction






6ms and 119864sO119877 =




infin120583infin=


infin120588infin





2reaction



0

001

002

003

004

1000 1500 2000 2500



119884Pinfin = 1

Com

busti

on ra

te119898

(kg

m2s)







2reaction reported





2




2mass


2





2

concentrations




(119879)0=119879s + (

119879infinminus119879s) 120585

(119894)0= 119894s + (119894infin minus

119894s) 120585 (119894 = FOP) (50)



119879s + Fs gt

119879infin (51)


(119879)

in= (

119879)0+ 120576

119879s120582120579 (120585) +O (120576

2)

=119879s [1 minus 120576 (120594 minus 120582120579)] +O (120576

2)

(52)

where

120576 =

119879s119879119886g

120582 =

Oinfin119879s minus

119879infin

120585 = 120576(

119879s

119879s minus

119879infin

)120594 (53)



(O)in= Os + 120576

119879s120582 (120594 minus 120579) (54)

(F)in= (


1 + 120573

) + Os

+ 120576(

119879s

119879s minus

119879infin

)(


1 + 120573


(55)


120574 equiv (

119889119879

119889120585

)

s=[

[

119889120594

119889120585

119889(119879)

in119889120594

]

]s



119889120579

119889120594

)

s+O (120576)

(56)


Os =Oinfin

1 + 120573 + 119860 sO [120573 (minus119891s)]1 minus (

119889120579

119889120594

)

s (57)


1198892120579

1198891205942= minusΔ(120594 minus 120579 + 120578O)

12 exp (120582120579 minus 120594) (58)

where

Δ = 119863119886g exp(minus119879119886g

119879s

)

120573

(minus119891s)

119879s

119879s minus

119879infin

2

times (

119879s119879119886g

)

32

(

119879infin

119879s

)

12

Fs

(120582119879s)12

(59)

120578O =

Os

120576119879s120582

(60)








(119879)

out=119879s + (

119879infinminus119879s) 120585 + 120576

119879sΘ (120585) +O (120576

2) (62)



120582120579 (infin) = minus119862I (

119889120579

119889120594

)

infin

= 0 (63)



2Δ I120582 = (radic120578O + radic120587120582

2

119890120582120578O


120594I

119890(120582minus1)120594erfc (119911) 119889120594)

minus1

(64)

erfc (119911) = 2

radic120587

expintinfin

119911

(minus1199052) 119889119905 (65)


(

119889120579

119889120594

)

s=

1

120582

or (

119889(119879)

in119889120585

)

s

= 0 (66)


at 120582 = 1 2Δ I =1


as 120578O 997888rarr infin 2Δ I120582 =1

radic120578O

(67)



2Δ I120582 = (radic120578O +

radic120587

2

[119890120578O erfc (radic120578O)

+

1

119865 (120582)


2

)])

minus1

(68)

where

119865 (120582) = 056 +

021

120582

minus

012

1205822+

035

1205823 (69)





2reaction and hence








1000

1200

1400

1600

1800

2000

10 100 1000


1985

119884O = 07

119884A = 00003


Igni

tion

surfa

ce te

mpe

ratu

re119879

sig

(K)

(a)

1000

1200

1400

1600

1800

2000

00001 0001 001


1985


119884O = 07

119886 = 170 sminus1

Igni

tion

surfa

ce te

mpe

ratu

re119879

sig

(K)

(b)

1000

1200

1400

1600

1800

2000


119886 = 170 sminus1119884A = 00003

Igni

tion

surfa

ce te

mpe

ratu

re119879

sig

(K)

(c)












2reaction



2mass fraction


2mass fraction In





2mass fraction



119861g = 119861lowast

g (120588119884A119882A)12

[(molm3)12

sdot s]minus1

(70)








119861lowast


minus1

(71)




119861lowast


minus1

(72)




Air

O2



100

102

104

106

3 4 5

Oxidizer

lowast(119879

s119879119886

g)32

exp(minus119879119886

g119879

s)

(m3m

oL)middot

sminus1

1s

119884Oinfin = 0533

119861g


2and H



119861lowast


minus1

(73)



2andor H



2









infin120588infin









2O mass fraction



7msand 119864sA119877 = 33 times 10







0

0002

0004

0006

0008

001

1000 1500 2000 2500 3000

Air

Frozen


Com

busti

on ra

te119898

(kg

m2s)

119879si

g=

1479

K

Flame-detached

Flame-attached

119886 = 120 sminus1


(a)

0

0002

0004

0006

0008

001

1000 1500 2000 2500 3000


Com

busti

on ra

te119898

(kg

m2s) 119886 = 120 sminus1

C-CO2


(b)

0

0005

001

0015

002

1000 1500 2000 2500 3000


C-H2O119886 = 120 sminus1

Com

busti

on ra

te119898

(kg

m2s)


(c)


2mass




119891 =

119891s (0 le 120578 le 120578lowast)

119887120578 + 119888 (120578lowastle 120578 le 120578

lowastlowast)

120578 + 119889 (120578 ge 120578lowastlowast)

(74)



119900



119870

(75)

where

119870 = 120578lowast+ radic

120587

2

[exp((119887 minus 1) 119891

2

119900

2119887

) minus 1 erfc(119891119900

radic2

)

+

1

radic119887

erf (119891119900

radic2119887

) minus erf (119891119900

radic2

)] + radic

120587

2

(76)




120573

1 + 120573



120573 asymp (

119870119860 sO

1 + 119870119860 sO)(

2119882C119882O

119884Oinfin)

+ (

119870119860 sP

1 + 119870119860 sP)(

119882C119882P

119884Pinfin)

(78)


Flame-detached mode

120573 asymp (

119870119860 sP

1 + 119870119860 sP)(

2119882C119882O

119884Oinfin +

119882C119882P

119884Pinfin) (79)

Flame-attached mode

120573 asymp (

119870119860 sO

1 + 2119870119860 sO minus 119870119860 sP)(

2119882C119882O

119884Oinfin)

+ (

119870119860 sP

1 + 2119870119860 sO minus 119870119860 sP)(

119882C119882P

119884Pinfin)

(80)


2and CO

2con-


119870119860 s119894 = 119896s119894119870


(81)


119896s119894 equiv 119861s119894 (119879infin

119879s) exp(minus

119879119886s119894

119879s) (119894 = OP) (82)



asymp 120588infin


119870

ln (1 + 120573) (83)


119870

[2119895119886 (120583infin120588infin)]12

(84)


asymp

1


120588infin(

2119882C119882O

119884Oinfin)

+

1


120588infin(

119882C119882P

119884Pinfin)

(85)


=

1

1119896s + 1ℎ119863(120588119884O)infin (86)



[2119895119886(120583infin120588infin)]



119863is of use in



ℎ119863= (

119879119877

119879infin





ℎ119863= (

119879119877

119879infin

)

119862mtSc06


fromNu119909

radicRe119909

= 119862mtSc04

(88)


Nu119909=

ℎ119863119909

119863119877

Re119909=

120588119877119880119909

120583119877

119880 = 119886119909 Sc =120583120588

119863

(89)






119870 = radic

2

3

(

119879infin

119879s)(1 minus

119879infin

2119879s) + radic

120587

2

(90)



0

001

002

003

004

Frozen

1000 1500 2000 2500

Flame-detached

Flame-attached

Com

busti

on ra

te119898

(kg

m2s) 119884O = 07

119884A = 00003

119886 = 170 sminus1





2concentrations



119870 = radic

2

3

(

119879infin

119879s)(1 minus

119879infin

2119879s)

minus 005 (1 +

4119882C119882O


120587

2

(91)


2mass-fraction



119870 = (

119879infin

119879s)(1 minus

119879infin

2119879s) + radic

120587

2

(92)


119870 = (

119879infin

119879s)(1 minus

119879infin

2119879s)

minus 005 (1 +

4119882C119882O


120587

2

(93)

0

0005

001

0015

1000 1500 2000 2500



Com

busti

on ra

te119898

(kg

m2s)

119886 = 300 sminus1

119886 = 180 sminus1

119886 = 60 sminus1


(a)

0

002

004

006

008

1000 1500 2000 2500


Com

busti

on ra

te119898

(kg

m2s) 119886 = 1000 sminus1

119886 = 320 sminus1

119886 = 88 sminus1


119884Oinfin = 07

119884Ainfin = 00002

(b)




















0

001

002

1000 1500 2000 2500



Matsui 1983

Com

busti

on ra

te119898

(kg

m2s)

119886 = 300 sminus1

119884Ainfin = 0001


119884Ainfin = 0004







2Oconcentrations






2O mass-fraction



0

0005

001

0015

1000 1500 2000 2500



ombu

stion

rate119898

(kg

m2s)

119886 = 100 sminus1

119884Oinfin = 02

119884Ainfin = 014


119879si

g=1094

K







6ms and119864sO119877=22 times 10



4 K (119864sP =


8ms and119864sA119877 = 33 times 10










2O



2O mass fraction is



2O and












119894s =

119894infin

1 + 120573 + 119860 s119894 [120573 (minus119891s)](119894 = OPA) (95)





1 + 120573

Ps =Pinfin + 120575120573

1 + 120573

As =Ainfin

1 + 120573

(96)



1 + 120573

(97)


119879 + O + (1 minus 119876) A (98)


As =1

1 minus 119876


1 + 120573 + 119860 sA [120573 (minus119891s)] (99)




119879 = 119879s + (119879f minus 119879s) (120585

120585f) (100)



+ (1 minus 119876)(As1 minus 120585f120585f

minus

Af

120585f)

(101)


As =Ainfin



Frozen mode

120573 asymp (

119870119860 sO

1 + 119870119860 sO)(

2119882C119882O

119884Oinfin) + (

119870119860 sP

1 + 119870119860 sP)

times (

119882C119882P

119884Pinfin) + (

119870119860 sA

1 + 119870119860 sA)(

119882C119882A

119884Ainfin)

(103)

Flame-detached mode

120573 asymp

1

2

119870119860 sP

1 + 119870119860 sP(

2119882C119882O

119884Oinfin +

119882C119882P

119884Pinfin)

+

119870119860 sA

1 + 119870119860 sA(

119882C119882O

119884Oinfin +

119882C119882A

119884Ainfin)

+

1

2

[

119870119860 sP

1 + 119870119860 sP(

2119882C119882O

119884Oinfin +

119882C119882P

119884Pinfin)

minus

119870119860 sA

1 + 119870119860 sA(

119882C119882O

119884Oinfin +

119882C119882A

119884Ainfin)

2

+ 4

119870119860 sP

1 + 119870119860 sP(

119882C119882O

119884Oinfin +

119882C119882P

119884Pinfin)

times

119870119860 sA

1 + 119870119860 sA(

119882C119882A

119884Ainfin)]

12

(104)

Flame-attached mode

120573 asymp (

1

1 + 2119870119860 sO minus 119870119860 sP)

times (119870119860 sO2119882C119882O

119884Oinfin + 119870119860 sP119882C119882P

119884Pinfin

+119870119860 sA119882C119882A

119884Ainfin)

(105)






2and C-CO

2reactions and




2reaction to







2O mass-



2-O2reaction into


2Omass



Nomenclature










Greek Symbols








Subscripts



2surface reaction


2surface reaction


Superscripts




References









































































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation












=


infin

120575 minus (F + P)s

equiv 120573

(37)




2reaction




119879 =

119879s + 120574120585 120574 =

119879infinminus119879s (38)


2and CO



120575 (minus119891s) = 119860 sOOinfin

1 + 120573 + 119860 sO [120573 (minus119891s)]

+ 119860 sPPinfin

1 + 120573 + 119860 sP [120573 (minus119891s)]

(39)



120575

(40)






1 + 120573

(42)

119879f =

119879s + (Oinfin +

119879infinminus119879s) 120585f

120585f =2120575120573 minus Oinfin

(2120575 + Oinfin) 120573

(43)


0 le 120585 le 120585f 119879 =

119879s + (Oinfin +

119879infinminus119879s) 120585

120585f le 120585 le infin

119879 =

119879infinminus



1 + 120573

) (1 minus 120585)

(44)






minus (

119889F119889120578

)

fminus= (

119889O119889120578

)

f+

or

minus (

119889119884F119889120578

)

fminus=

]F119882F]O119882O

(

119889119884O119889120578

)

f+

(45)


119894 can be



(

119889119879

119889120578

)

fminusminus (

119889119879

119889120578

)

f+= minus(

119889F119889120578

)

fminus

or

120582(

119889119879

119889120578

)

fminusminus 120582(

119889119879

119889120578

)

f+= minus119902120588119863(

119889119884F119889120578

)

fminus

(46)




120573a =Oinfin

2120575

(47)


119879 =

119879s + (

119879infinminus119879s) 120585 (48)



1 + 120573

+ 119860 sPPinfin + 120575120573

1 + 120573

(49)








2reaction





2reaction



2 CO2 and H







2and H

2O issued into


infin119889)

where 119881infin









0

001

002

003

004

1000 1500 2000 2500


Visser 1985

Flame-detached


119886 = 170 sminus1

Com

busti

on ra

te119898

(kg

m2s) 119884Oinfin = 07

119884Ainfin = 00002

119879si

g=

1480

K


(a)

0

001

002

003

004

1000 1500 2000 2500


Frozen

Flame-detached

Flame-attached

119886 = 170 sminus1

Com

busti

on ra

te119898

(kg

m2s) 119884Oinfin = 07

119884Ainfin = 00038


119879si

g=1249

K

(b)







2reaction






6ms and 119864sO119877 =




infin120583infin=


infin120588infin





2reaction



0

001

002

003

004

1000 1500 2000 2500



119884Pinfin = 1

Com

busti

on ra

te119898

(kg

m2s)







2reaction reported





2




2mass


2





2

concentrations




(119879)0=119879s + (

119879infinminus119879s) 120585

(119894)0= 119894s + (119894infin minus

119894s) 120585 (119894 = FOP) (50)



119879s + Fs gt

119879infin (51)


(119879)

in= (

119879)0+ 120576

119879s120582120579 (120585) +O (120576

2)

=119879s [1 minus 120576 (120594 minus 120582120579)] +O (120576

2)

(52)

where

120576 =

119879s119879119886g

120582 =

Oinfin119879s minus

119879infin

120585 = 120576(

119879s

119879s minus

119879infin

)120594 (53)



(O)in= Os + 120576

119879s120582 (120594 minus 120579) (54)

(F)in= (


1 + 120573

) + Os

+ 120576(

119879s

119879s minus

119879infin

)(


1 + 120573


(55)


120574 equiv (

119889119879

119889120585

)

s=[

[

119889120594

119889120585

119889(119879)

in119889120594

]

]s



119889120579

119889120594

)

s+O (120576)

(56)


Os =Oinfin

1 + 120573 + 119860 sO [120573 (minus119891s)]1 minus (

119889120579

119889120594

)

s (57)


1198892120579

1198891205942= minusΔ(120594 minus 120579 + 120578O)

12 exp (120582120579 minus 120594) (58)

where

Δ = 119863119886g exp(minus119879119886g

119879s

)

120573

(minus119891s)

119879s

119879s minus

119879infin

2

times (

119879s119879119886g

)

32

(

119879infin

119879s

)

12

Fs

(120582119879s)12

(59)

120578O =

Os

120576119879s120582

(60)








(119879)

out=119879s + (

119879infinminus119879s) 120585 + 120576

119879sΘ (120585) +O (120576

2) (62)



120582120579 (infin) = minus119862I (

119889120579

119889120594

)

infin

= 0 (63)



2Δ I120582 = (radic120578O + radic120587120582

2

119890120582120578O


120594I

119890(120582minus1)120594erfc (119911) 119889120594)

minus1

(64)

erfc (119911) = 2

radic120587

expintinfin

119911

(minus1199052) 119889119905 (65)


(

119889120579

119889120594

)

s=

1

120582

or (

119889(119879)

in119889120585

)

s

= 0 (66)


at 120582 = 1 2Δ I =1


as 120578O 997888rarr infin 2Δ I120582 =1

radic120578O

(67)



2Δ I120582 = (radic120578O +

radic120587

2

[119890120578O erfc (radic120578O)

+

1

119865 (120582)


2

)])

minus1

(68)

where

119865 (120582) = 056 +

021

120582

minus

012

1205822+

035

1205823 (69)





2reaction and hence








1000

1200

1400

1600

1800

2000

10 100 1000


1985

119884O = 07

119884A = 00003


Igni

tion

surfa

ce te

mpe

ratu

re119879

sig

(K)

(a)

1000

1200

1400

1600

1800

2000

00001 0001 001


1985


119884O = 07

119886 = 170 sminus1

Igni

tion

surfa

ce te

mpe

ratu

re119879

sig

(K)

(b)

1000

1200

1400

1600

1800

2000


119886 = 170 sminus1119884A = 00003

Igni

tion

surfa

ce te

mpe

ratu

re119879

sig

(K)

(c)












2reaction



2mass fraction


2mass fraction In





2mass fraction



119861g = 119861lowast

g (120588119884A119882A)12

[(molm3)12

sdot s]minus1

(70)








119861lowast


minus1

(71)




119861lowast


minus1

(72)




Air

O2



100

102

104

106

3 4 5

Oxidizer

lowast(119879

s119879119886

g)32

exp(minus119879119886

g119879

s)

(m3m

oL)middot

sminus1

1s

119884Oinfin = 0533

119861g


2and H



119861lowast


minus1

(73)



2andor H



2









infin120588infin









2O mass fraction



7msand 119864sA119877 = 33 times 10







0

0002

0004

0006

0008

001

1000 1500 2000 2500 3000

Air

Frozen


Com

busti

on ra

te119898

(kg

m2s)

119879si

g=

1479

K

Flame-detached

Flame-attached

119886 = 120 sminus1


(a)

0

0002

0004

0006

0008

001

1000 1500 2000 2500 3000


Com

busti

on ra

te119898

(kg

m2s) 119886 = 120 sminus1

C-CO2


(b)

0

0005

001

0015

002

1000 1500 2000 2500 3000


C-H2O119886 = 120 sminus1

Com

busti

on ra

te119898

(kg

m2s)


(c)


2mass




119891 =

119891s (0 le 120578 le 120578lowast)

119887120578 + 119888 (120578lowastle 120578 le 120578

lowastlowast)

120578 + 119889 (120578 ge 120578lowastlowast)

(74)



119900



119870

(75)

where

119870 = 120578lowast+ radic

120587

2

[exp((119887 minus 1) 119891

2

119900

2119887

) minus 1 erfc(119891119900

radic2

)

+

1

radic119887

erf (119891119900

radic2119887

) minus erf (119891119900

radic2

)] + radic

120587

2

(76)




120573

1 + 120573



120573 asymp (

119870119860 sO

1 + 119870119860 sO)(

2119882C119882O

119884Oinfin)

+ (

119870119860 sP

1 + 119870119860 sP)(

119882C119882P

119884Pinfin)

(78)


Flame-detached mode

120573 asymp (

119870119860 sP

1 + 119870119860 sP)(

2119882C119882O

119884Oinfin +

119882C119882P

119884Pinfin) (79)

Flame-attached mode

120573 asymp (

119870119860 sO

1 + 2119870119860 sO minus 119870119860 sP)(

2119882C119882O

119884Oinfin)

+ (

119870119860 sP

1 + 2119870119860 sO minus 119870119860 sP)(

119882C119882P

119884Pinfin)

(80)


2and CO

2con-


119870119860 s119894 = 119896s119894119870


(81)


119896s119894 equiv 119861s119894 (119879infin

119879s) exp(minus

119879119886s119894

119879s) (119894 = OP) (82)



asymp 120588infin


119870

ln (1 + 120573) (83)


119870

[2119895119886 (120583infin120588infin)]12

(84)


asymp

1


120588infin(

2119882C119882O

119884Oinfin)

+

1


120588infin(

119882C119882P

119884Pinfin)

(85)


=

1

1119896s + 1ℎ119863(120588119884O)infin (86)



[2119895119886(120583infin120588infin)]



119863is of use in



ℎ119863= (

119879119877

119879infin





ℎ119863= (

119879119877

119879infin

)

119862mtSc06


fromNu119909

radicRe119909

= 119862mtSc04

(88)


Nu119909=

ℎ119863119909

119863119877

Re119909=

120588119877119880119909

120583119877

119880 = 119886119909 Sc =120583120588

119863

(89)






119870 = radic

2

3

(

119879infin

119879s)(1 minus

119879infin

2119879s) + radic

120587

2

(90)



0

001

002

003

004

Frozen

1000 1500 2000 2500

Flame-detached

Flame-attached

Com

busti

on ra

te119898

(kg

m2s) 119884O = 07

119884A = 00003

119886 = 170 sminus1





2concentrations



119870 = radic

2

3

(

119879infin

119879s)(1 minus

119879infin

2119879s)

minus 005 (1 +

4119882C119882O


120587

2

(91)


2mass-fraction



119870 = (

119879infin

119879s)(1 minus

119879infin

2119879s) + radic

120587

2

(92)


119870 = (

119879infin

119879s)(1 minus

119879infin

2119879s)

minus 005 (1 +

4119882C119882O


120587

2

(93)

0

0005

001

0015

1000 1500 2000 2500



Com

busti

on ra

te119898

(kg

m2s)

119886 = 300 sminus1

119886 = 180 sminus1

119886 = 60 sminus1


(a)

0

002

004

006

008

1000 1500 2000 2500


Com

busti

on ra

te119898

(kg

m2s) 119886 = 1000 sminus1

119886 = 320 sminus1

119886 = 88 sminus1


119884Oinfin = 07

119884Ainfin = 00002

(b)




















0

001

002

1000 1500 2000 2500



Matsui 1983

Com

busti

on ra

te119898

(kg

m2s)

119886 = 300 sminus1

119884Ainfin = 0001


119884Ainfin = 0004







2Oconcentrations






2O mass-fraction



0

0005

001

0015

1000 1500 2000 2500



ombu

stion

rate119898

(kg

m2s)

119886 = 100 sminus1

119884Oinfin = 02

119884Ainfin = 014


119879si

g=1094

K







6ms and119864sO119877=22 times 10



4 K (119864sP =


8ms and119864sA119877 = 33 times 10










2O



2O mass fraction is



2O and












119894s =

119894infin

1 + 120573 + 119860 s119894 [120573 (minus119891s)](119894 = OPA) (95)





1 + 120573

Ps =Pinfin + 120575120573

1 + 120573

As =Ainfin

1 + 120573

(96)



1 + 120573

(97)


119879 + O + (1 minus 119876) A (98)


As =1

1 minus 119876


1 + 120573 + 119860 sA [120573 (minus119891s)] (99)




119879 = 119879s + (119879f minus 119879s) (120585

120585f) (100)



+ (1 minus 119876)(As1 minus 120585f120585f

minus

Af

120585f)

(101)


As =Ainfin



Frozen mode

120573 asymp (

119870119860 sO

1 + 119870119860 sO)(

2119882C119882O

119884Oinfin) + (

119870119860 sP

1 + 119870119860 sP)

times (

119882C119882P

119884Pinfin) + (

119870119860 sA

1 + 119870119860 sA)(

119882C119882A

119884Ainfin)

(103)

Flame-detached mode

120573 asymp

1

2

119870119860 sP

1 + 119870119860 sP(

2119882C119882O

119884Oinfin +

119882C119882P

119884Pinfin)

+

119870119860 sA

1 + 119870119860 sA(

119882C119882O

119884Oinfin +

119882C119882A

119884Ainfin)

+

1

2

[

119870119860 sP

1 + 119870119860 sP(

2119882C119882O

119884Oinfin +

119882C119882P

119884Pinfin)

minus

119870119860 sA

1 + 119870119860 sA(

119882C119882O

119884Oinfin +

119882C119882A

119884Ainfin)

2

+ 4

119870119860 sP

1 + 119870119860 sP(

119882C119882O

119884Oinfin +

119882C119882P

119884Pinfin)

times

119870119860 sA

1 + 119870119860 sA(

119882C119882A

119884Ainfin)]

12

(104)

Flame-attached mode

120573 asymp (

1

1 + 2119870119860 sO minus 119870119860 sP)

times (119870119860 sO2119882C119882O

119884Oinfin + 119870119860 sP119882C119882P

119884Pinfin

+119870119860 sA119882C119882A

119884Ainfin)

(105)






2and C-CO

2reactions and




2reaction to







2O mass-



2-O2reaction into


2Omass



Nomenclature










Greek Symbols








Subscripts



2surface reaction


2surface reaction


Superscripts




References









































































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation












120573a =Oinfin

2120575

(47)


119879 =

119879s + (

119879infinminus119879s) 120585 (48)



1 + 120573

+ 119860 sPPinfin + 120575120573

1 + 120573

(49)








2reaction





2reaction



2 CO2 and H







2and H

2O issued into


infin119889)

where 119881infin









0

001

002

003

004

1000 1500 2000 2500


Visser 1985

Flame-detached


119886 = 170 sminus1

Com

busti

on ra

te119898

(kg

m2s) 119884Oinfin = 07

119884Ainfin = 00002

119879si

g=

1480

K


(a)

0

001

002

003

004

1000 1500 2000 2500


Frozen

Flame-detached

Flame-attached

119886 = 170 sminus1

Com

busti

on ra

te119898

(kg

m2s) 119884Oinfin = 07

119884Ainfin = 00038


119879si

g=1249

K

(b)







2reaction






6ms and 119864sO119877 =




infin120583infin=


infin120588infin





2reaction



0

001

002

003

004

1000 1500 2000 2500



119884Pinfin = 1

Com

busti

on ra

te119898

(kg

m2s)







2reaction reported





2




2mass


2





2

concentrations




(119879)0=119879s + (

119879infinminus119879s) 120585

(119894)0= 119894s + (119894infin minus

119894s) 120585 (119894 = FOP) (50)



119879s + Fs gt

119879infin (51)


(119879)

in= (

119879)0+ 120576

119879s120582120579 (120585) +O (120576

2)

=119879s [1 minus 120576 (120594 minus 120582120579)] +O (120576

2)

(52)

where

120576 =

119879s119879119886g

120582 =

Oinfin119879s minus

119879infin

120585 = 120576(

119879s

119879s minus

119879infin

)120594 (53)



(O)in= Os + 120576

119879s120582 (120594 minus 120579) (54)

(F)in= (


1 + 120573

) + Os

+ 120576(

119879s

119879s minus

119879infin

)(


1 + 120573


(55)


120574 equiv (

119889119879

119889120585

)

s=[

[

119889120594

119889120585

119889(119879)

in119889120594

]

]s



119889120579

119889120594

)

s+O (120576)

(56)


Os =Oinfin

1 + 120573 + 119860 sO [120573 (minus119891s)]1 minus (

119889120579

119889120594

)

s (57)


1198892120579

1198891205942= minusΔ(120594 minus 120579 + 120578O)

12 exp (120582120579 minus 120594) (58)

where

Δ = 119863119886g exp(minus119879119886g

119879s

)

120573

(minus119891s)

119879s

119879s minus

119879infin

2

times (

119879s119879119886g

)

32

(

119879infin

119879s

)

12

Fs

(120582119879s)12

(59)

120578O =

Os

120576119879s120582

(60)








(119879)

out=119879s + (

119879infinminus119879s) 120585 + 120576

119879sΘ (120585) +O (120576

2) (62)



120582120579 (infin) = minus119862I (

119889120579

119889120594

)

infin

= 0 (63)



2Δ I120582 = (radic120578O + radic120587120582

2

119890120582120578O


120594I

119890(120582minus1)120594erfc (119911) 119889120594)

minus1

(64)

erfc (119911) = 2

radic120587

expintinfin

119911

(minus1199052) 119889119905 (65)


(

119889120579

119889120594

)

s=

1

120582

or (

119889(119879)

in119889120585

)

s

= 0 (66)


at 120582 = 1 2Δ I =1


as 120578O 997888rarr infin 2Δ I120582 =1

radic120578O

(67)



2Δ I120582 = (radic120578O +

radic120587

2

[119890120578O erfc (radic120578O)

+

1

119865 (120582)


2

)])

minus1

(68)

where

119865 (120582) = 056 +

021

120582

minus

012

1205822+

035

1205823 (69)





2reaction and hence








1000

1200

1400

1600

1800

2000

10 100 1000


1985

119884O = 07

119884A = 00003


Igni

tion

surfa

ce te

mpe

ratu

re119879

sig

(K)

(a)

1000

1200

1400

1600

1800

2000

00001 0001 001


1985


119884O = 07

119886 = 170 sminus1

Igni

tion

surfa

ce te

mpe

ratu

re119879

sig

(K)

(b)

1000

1200

1400

1600

1800

2000


119886 = 170 sminus1119884A = 00003

Igni

tion

surfa

ce te

mpe

ratu

re119879

sig

(K)

(c)












2reaction



2mass fraction


2mass fraction In





2mass fraction



119861g = 119861lowast

g (120588119884A119882A)12

[(molm3)12

sdot s]minus1

(70)








119861lowast


minus1

(71)




119861lowast


minus1

(72)




Air

O2



100

102

104

106

3 4 5

Oxidizer

lowast(119879

s119879119886

g)32

exp(minus119879119886

g119879

s)

(m3m

oL)middot

sminus1

1s

119884Oinfin = 0533

119861g


2and H



119861lowast


minus1

(73)



2andor H



2









infin120588infin









2O mass fraction



7msand 119864sA119877 = 33 times 10







0

0002

0004

0006

0008

001

1000 1500 2000 2500 3000

Air

Frozen


Com

busti

on ra

te119898

(kg

m2s)

119879si

g=

1479

K

Flame-detached

Flame-attached

119886 = 120 sminus1


(a)

0

0002

0004

0006

0008

001

1000 1500 2000 2500 3000


Com

busti

on ra

te119898

(kg

m2s) 119886 = 120 sminus1

C-CO2


(b)

0

0005

001

0015

002

1000 1500 2000 2500 3000


C-H2O119886 = 120 sminus1

Com

busti

on ra

te119898

(kg

m2s)


(c)


2mass




119891 =

119891s (0 le 120578 le 120578lowast)

119887120578 + 119888 (120578lowastle 120578 le 120578

lowastlowast)

120578 + 119889 (120578 ge 120578lowastlowast)

(74)



119900



119870

(75)

where

119870 = 120578lowast+ radic

120587

2

[exp((119887 minus 1) 119891

2

119900

2119887

) minus 1 erfc(119891119900

radic2

)

+

1

radic119887

erf (119891119900

radic2119887

) minus erf (119891119900

radic2

)] + radic

120587

2

(76)




120573

1 + 120573



120573 asymp (

119870119860 sO

1 + 119870119860 sO)(

2119882C119882O

119884Oinfin)

+ (

119870119860 sP

1 + 119870119860 sP)(

119882C119882P

119884Pinfin)

(78)


Flame-detached mode

120573 asymp (

119870119860 sP

1 + 119870119860 sP)(

2119882C119882O

119884Oinfin +

119882C119882P

119884Pinfin) (79)

Flame-attached mode

120573 asymp (

119870119860 sO

1 + 2119870119860 sO minus 119870119860 sP)(

2119882C119882O

119884Oinfin)

+ (

119870119860 sP

1 + 2119870119860 sO minus 119870119860 sP)(

119882C119882P

119884Pinfin)

(80)


2and CO

2con-


119870119860 s119894 = 119896s119894119870


(81)


119896s119894 equiv 119861s119894 (119879infin

119879s) exp(minus

119879119886s119894

119879s) (119894 = OP) (82)



asymp 120588infin


119870

ln (1 + 120573) (83)


119870

[2119895119886 (120583infin120588infin)]12

(84)


asymp

1


120588infin(

2119882C119882O

119884Oinfin)

+

1


120588infin(

119882C119882P

119884Pinfin)

(85)


=

1

1119896s + 1ℎ119863(120588119884O)infin (86)



[2119895119886(120583infin120588infin)]



119863is of use in



ℎ119863= (

119879119877

119879infin





ℎ119863= (

119879119877

119879infin

)

119862mtSc06


fromNu119909

radicRe119909

= 119862mtSc04

(88)


Nu119909=

ℎ119863119909

119863119877

Re119909=

120588119877119880119909

120583119877

119880 = 119886119909 Sc =120583120588

119863

(89)






119870 = radic

2

3

(

119879infin

119879s)(1 minus

119879infin

2119879s) + radic

120587

2

(90)



0

001

002

003

004

Frozen

1000 1500 2000 2500

Flame-detached

Flame-attached

Com

busti

on ra

te119898

(kg

m2s) 119884O = 07

119884A = 00003

119886 = 170 sminus1





2concentrations



119870 = radic

2

3

(

119879infin

119879s)(1 minus

119879infin

2119879s)

minus 005 (1 +

4119882C119882O


120587

2

(91)


2mass-fraction



119870 = (

119879infin

119879s)(1 minus

119879infin

2119879s) + radic

120587

2

(92)


119870 = (

119879infin

119879s)(1 minus

119879infin

2119879s)

minus 005 (1 +

4119882C119882O


120587

2

(93)

0

0005

001

0015

1000 1500 2000 2500



Com

busti

on ra

te119898

(kg

m2s)

119886 = 300 sminus1

119886 = 180 sminus1

119886 = 60 sminus1


(a)

0

002

004

006

008

1000 1500 2000 2500


Com

busti

on ra

te119898

(kg

m2s) 119886 = 1000 sminus1

119886 = 320 sminus1

119886 = 88 sminus1


119884Oinfin = 07

119884Ainfin = 00002

(b)




















0

001

002

1000 1500 2000 2500



Matsui 1983

Com

busti

on ra

te119898

(kg

m2s)

119886 = 300 sminus1

119884Ainfin = 0001


119884Ainfin = 0004







2Oconcentrations






2O mass-fraction



0

0005

001

0015

1000 1500 2000 2500



ombu

stion

rate119898

(kg

m2s)

119886 = 100 sminus1

119884Oinfin = 02

119884Ainfin = 014


119879si

g=1094

K







6ms and119864sO119877=22 times 10



4 K (119864sP =


8ms and119864sA119877 = 33 times 10










2O



2O mass fraction is



2O and












119894s =

119894infin

1 + 120573 + 119860 s119894 [120573 (minus119891s)](119894 = OPA) (95)





1 + 120573

Ps =Pinfin + 120575120573

1 + 120573

As =Ainfin

1 + 120573

(96)



1 + 120573

(97)


119879 + O + (1 minus 119876) A (98)


As =1

1 minus 119876


1 + 120573 + 119860 sA [120573 (minus119891s)] (99)




119879 = 119879s + (119879f minus 119879s) (120585

120585f) (100)



+ (1 minus 119876)(As1 minus 120585f120585f

minus

Af

120585f)

(101)


As =Ainfin



Frozen mode

120573 asymp (

119870119860 sO

1 + 119870119860 sO)(

2119882C119882O

119884Oinfin) + (

119870119860 sP

1 + 119870119860 sP)

times (

119882C119882P

119884Pinfin) + (

119870119860 sA

1 + 119870119860 sA)(

119882C119882A

119884Ainfin)

(103)

Flame-detached mode

120573 asymp

1

2

119870119860 sP

1 + 119870119860 sP(

2119882C119882O

119884Oinfin +

119882C119882P

119884Pinfin)

+

119870119860 sA

1 + 119870119860 sA(

119882C119882O

119884Oinfin +

119882C119882A

119884Ainfin)

+

1

2

[

119870119860 sP

1 + 119870119860 sP(

2119882C119882O

119884Oinfin +

119882C119882P

119884Pinfin)

minus

119870119860 sA

1 + 119870119860 sA(

119882C119882O

119884Oinfin +

119882C119882A

119884Ainfin)

2

+ 4

119870119860 sP

1 + 119870119860 sP(

119882C119882O

119884Oinfin +

119882C119882P

119884Pinfin)

times

119870119860 sA

1 + 119870119860 sA(

119882C119882A

119884Ainfin)]

12

(104)

Flame-attached mode

120573 asymp (

1

1 + 2119870119860 sO minus 119870119860 sP)

times (119870119860 sO2119882C119882O

119884Oinfin + 119870119860 sP119882C119882P

119884Pinfin

+119870119860 sA119882C119882A

119884Ainfin)

(105)






2and C-CO

2reactions and




2reaction to







2O mass-



2-O2reaction into


2Omass



Nomenclature










Greek Symbols








Subscripts



2surface reaction


2surface reaction


Superscripts




References









































































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










0

001

002

003

004

1000 1500 2000 2500


Visser 1985

Flame-detached


119886 = 170 sminus1

Com

busti

on ra

te119898

(kg

m2s) 119884Oinfin = 07

119884Ainfin = 00002

119879si

g=

1480

K


(a)

0

001

002

003

004

1000 1500 2000 2500


Frozen

Flame-detached

Flame-attached

119886 = 170 sminus1

Com

busti

on ra

te119898

(kg

m2s) 119884Oinfin = 07

119884Ainfin = 00038


119879si

g=1249

K

(b)







2reaction






6ms and 119864sO119877 =




infin120583infin=


infin120588infin





2reaction



0

001

002

003

004

1000 1500 2000 2500



119884Pinfin = 1

Com

busti

on ra

te119898

(kg

m2s)







2reaction reported





2




2mass


2





2

concentrations




(119879)0=119879s + (

119879infinminus119879s) 120585

(119894)0= 119894s + (119894infin minus

119894s) 120585 (119894 = FOP) (50)



119879s + Fs gt

119879infin (51)


(119879)

in= (

119879)0+ 120576

119879s120582120579 (120585) +O (120576

2)

=119879s [1 minus 120576 (120594 minus 120582120579)] +O (120576

2)

(52)

where

120576 =

119879s119879119886g

120582 =

Oinfin119879s minus

119879infin

120585 = 120576(

119879s

119879s minus

119879infin

)120594 (53)



(O)in= Os + 120576

119879s120582 (120594 minus 120579) (54)

(F)in= (


1 + 120573

) + Os

+ 120576(

119879s

119879s minus

119879infin

)(


1 + 120573


(55)


120574 equiv (

119889119879

119889120585

)

s=[

[

119889120594

119889120585

119889(119879)

in119889120594

]

]s



119889120579

119889120594

)

s+O (120576)

(56)


Os =Oinfin

1 + 120573 + 119860 sO [120573 (minus119891s)]1 minus (

119889120579

119889120594

)

s (57)


1198892120579

1198891205942= minusΔ(120594 minus 120579 + 120578O)

12 exp (120582120579 minus 120594) (58)

where

Δ = 119863119886g exp(minus119879119886g

119879s

)

120573

(minus119891s)

119879s

119879s minus

119879infin

2

times (

119879s119879119886g

)

32

(

119879infin

119879s

)

12

Fs

(120582119879s)12

(59)

120578O =

Os

120576119879s120582

(60)








(119879)

out=119879s + (

119879infinminus119879s) 120585 + 120576

119879sΘ (120585) +O (120576

2) (62)



120582120579 (infin) = minus119862I (

119889120579

119889120594

)

infin

= 0 (63)



2Δ I120582 = (radic120578O + radic120587120582

2

119890120582120578O


120594I

119890(120582minus1)120594erfc (119911) 119889120594)

minus1

(64)

erfc (119911) = 2

radic120587

expintinfin

119911

(minus1199052) 119889119905 (65)


(

119889120579

119889120594

)

s=

1

120582

or (

119889(119879)

in119889120585

)

s

= 0 (66)


at 120582 = 1 2Δ I =1


as 120578O 997888rarr infin 2Δ I120582 =1

radic120578O

(67)



2Δ I120582 = (radic120578O +

radic120587

2

[119890120578O erfc (radic120578O)

+

1

119865 (120582)


2

)])

minus1

(68)

where

119865 (120582) = 056 +

021

120582

minus

012

1205822+

035

1205823 (69)





2reaction and hence








1000

1200

1400

1600

1800

2000

10 100 1000


1985

119884O = 07

119884A = 00003


Igni

tion

surfa

ce te

mpe

ratu

re119879

sig

(K)

(a)

1000

1200

1400

1600

1800

2000

00001 0001 001


1985


119884O = 07

119886 = 170 sminus1

Igni

tion

surfa

ce te

mpe

ratu

re119879

sig

(K)

(b)

1000

1200

1400

1600

1800

2000


119886 = 170 sminus1119884A = 00003

Igni

tion

surfa

ce te

mpe

ratu

re119879

sig

(K)

(c)












2reaction



2mass fraction


2mass fraction In





2mass fraction



119861g = 119861lowast

g (120588119884A119882A)12

[(molm3)12

sdot s]minus1

(70)








119861lowast


minus1

(71)




119861lowast


minus1

(72)




Air

O2



100

102

104

106

3 4 5

Oxidizer

lowast(119879

s119879119886

g)32

exp(minus119879119886

g119879

s)

(m3m

oL)middot

sminus1

1s

119884Oinfin = 0533

119861g


2and H



119861lowast


minus1

(73)



2andor H



2









infin120588infin









2O mass fraction



7msand 119864sA119877 = 33 times 10







0

0002

0004

0006

0008

001

1000 1500 2000 2500 3000

Air

Frozen


Com

busti

on ra

te119898

(kg

m2s)

119879si

g=

1479

K

Flame-detached

Flame-attached

119886 = 120 sminus1


(a)

0

0002

0004

0006

0008

001

1000 1500 2000 2500 3000


Com

busti

on ra

te119898

(kg

m2s) 119886 = 120 sminus1

C-CO2


(b)

0

0005

001

0015

002

1000 1500 2000 2500 3000


C-H2O119886 = 120 sminus1

Com

busti

on ra

te119898

(kg

m2s)


(c)


2mass




119891 =

119891s (0 le 120578 le 120578lowast)

119887120578 + 119888 (120578lowastle 120578 le 120578

lowastlowast)

120578 + 119889 (120578 ge 120578lowastlowast)

(74)



119900



119870

(75)

where

119870 = 120578lowast+ radic

120587

2

[exp((119887 minus 1) 119891

2

119900

2119887

) minus 1 erfc(119891119900

radic2

)

+

1

radic119887

erf (119891119900

radic2119887

) minus erf (119891119900

radic2

)] + radic

120587

2

(76)




120573

1 + 120573



120573 asymp (

119870119860 sO

1 + 119870119860 sO)(

2119882C119882O

119884Oinfin)

+ (

119870119860 sP

1 + 119870119860 sP)(

119882C119882P

119884Pinfin)

(78)


Flame-detached mode

120573 asymp (

119870119860 sP

1 + 119870119860 sP)(

2119882C119882O

119884Oinfin +

119882C119882P

119884Pinfin) (79)

Flame-attached mode

120573 asymp (

119870119860 sO

1 + 2119870119860 sO minus 119870119860 sP)(

2119882C119882O

119884Oinfin)

+ (

119870119860 sP

1 + 2119870119860 sO minus 119870119860 sP)(

119882C119882P

119884Pinfin)

(80)


2and CO

2con-


119870119860 s119894 = 119896s119894119870


(81)


119896s119894 equiv 119861s119894 (119879infin

119879s) exp(minus

119879119886s119894

119879s) (119894 = OP) (82)



asymp 120588infin


119870

ln (1 + 120573) (83)


119870

[2119895119886 (120583infin120588infin)]12

(84)


asymp

1


120588infin(

2119882C119882O

119884Oinfin)

+

1


120588infin(

119882C119882P

119884Pinfin)

(85)


=

1

1119896s + 1ℎ119863(120588119884O)infin (86)



[2119895119886(120583infin120588infin)]



119863is of use in



ℎ119863= (

119879119877

119879infin





ℎ119863= (

119879119877

119879infin

)

119862mtSc06


fromNu119909

radicRe119909

= 119862mtSc04

(88)


Nu119909=

ℎ119863119909

119863119877

Re119909=

120588119877119880119909

120583119877

119880 = 119886119909 Sc =120583120588

119863

(89)






119870 = radic

2

3

(

119879infin

119879s)(1 minus

119879infin

2119879s) + radic

120587

2

(90)



0

001

002

003

004

Frozen

1000 1500 2000 2500

Flame-detached

Flame-attached

Com

busti

on ra

te119898

(kg

m2s) 119884O = 07

119884A = 00003

119886 = 170 sminus1





2concentrations



119870 = radic

2

3

(

119879infin

119879s)(1 minus

119879infin

2119879s)

minus 005 (1 +

4119882C119882O


120587

2

(91)


2mass-fraction



119870 = (

119879infin

119879s)(1 minus

119879infin

2119879s) + radic

120587

2

(92)


119870 = (

119879infin

119879s)(1 minus

119879infin

2119879s)

minus 005 (1 +

4119882C119882O


120587

2

(93)

0

0005

001

0015

1000 1500 2000 2500



Com

busti

on ra

te119898

(kg

m2s)

119886 = 300 sminus1

119886 = 180 sminus1

119886 = 60 sminus1


(a)

0

002

004

006

008

1000 1500 2000 2500


Com

busti

on ra

te119898

(kg

m2s) 119886 = 1000 sminus1

119886 = 320 sminus1

119886 = 88 sminus1


119884Oinfin = 07

119884Ainfin = 00002

(b)




















0

001

002

1000 1500 2000 2500



Matsui 1983

Com

busti

on ra

te119898

(kg

m2s)

119886 = 300 sminus1

119884Ainfin = 0001


119884Ainfin = 0004







2Oconcentrations






2O mass-fraction



0

0005

001

0015

1000 1500 2000 2500



ombu

stion

rate119898

(kg

m2s)

119886 = 100 sminus1

119884Oinfin = 02

119884Ainfin = 014


119879si

g=1094

K







6ms and119864sO119877=22 times 10



4 K (119864sP =


8ms and119864sA119877 = 33 times 10










2O



2O mass fraction is



2O and












119894s =

119894infin

1 + 120573 + 119860 s119894 [120573 (minus119891s)](119894 = OPA) (95)





1 + 120573

Ps =Pinfin + 120575120573

1 + 120573

As =Ainfin

1 + 120573

(96)



1 + 120573

(97)


119879 + O + (1 minus 119876) A (98)


As =1

1 minus 119876


1 + 120573 + 119860 sA [120573 (minus119891s)] (99)




119879 = 119879s + (119879f minus 119879s) (120585

120585f) (100)



+ (1 minus 119876)(As1 minus 120585f120585f

minus

Af

120585f)

(101)


As =Ainfin



Frozen mode

120573 asymp (

119870119860 sO

1 + 119870119860 sO)(

2119882C119882O

119884Oinfin) + (

119870119860 sP

1 + 119870119860 sP)

times (

119882C119882P

119884Pinfin) + (

119870119860 sA

1 + 119870119860 sA)(

119882C119882A

119884Ainfin)

(103)

Flame-detached mode

120573 asymp

1

2

119870119860 sP

1 + 119870119860 sP(

2119882C119882O

119884Oinfin +

119882C119882P

119884Pinfin)

+

119870119860 sA

1 + 119870119860 sA(

119882C119882O

119884Oinfin +

119882C119882A

119884Ainfin)

+

1

2

[

119870119860 sP

1 + 119870119860 sP(

2119882C119882O

119884Oinfin +

119882C119882P

119884Pinfin)

minus

119870119860 sA

1 + 119870119860 sA(

119882C119882O

119884Oinfin +

119882C119882A

119884Ainfin)

2

+ 4

119870119860 sP

1 + 119870119860 sP(

119882C119882O

119884Oinfin +

119882C119882P

119884Pinfin)

times

119870119860 sA

1 + 119870119860 sA(

119882C119882A

119884Ainfin)]

12

(104)

Flame-attached mode

120573 asymp (

1

1 + 2119870119860 sO minus 119870119860 sP)

times (119870119860 sO2119882C119882O

119884Oinfin + 119870119860 sP119882C119882P

119884Pinfin

+119870119860 sA119882C119882A

119884Ainfin)

(105)






2and C-CO

2reactions and




2reaction to







2O mass-



2-O2reaction into


2Omass



Nomenclature










Greek Symbols








Subscripts



2surface reaction


2surface reaction


Superscripts




References









































































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation












2




2mass


2





2

concentrations




(119879)0=119879s + (

119879infinminus119879s) 120585

(119894)0= 119894s + (119894infin minus

119894s) 120585 (119894 = FOP) (50)



119879s + Fs gt

119879infin (51)


(119879)

in= (

119879)0+ 120576

119879s120582120579 (120585) +O (120576

2)

=119879s [1 minus 120576 (120594 minus 120582120579)] +O (120576

2)

(52)

where

120576 =

119879s119879119886g

120582 =

Oinfin119879s minus

119879infin

120585 = 120576(

119879s

119879s minus

119879infin

)120594 (53)



(O)in= Os + 120576

119879s120582 (120594 minus 120579) (54)

(F)in= (


1 + 120573

) + Os

+ 120576(

119879s

119879s minus

119879infin

)(


1 + 120573


(55)


120574 equiv (

119889119879

119889120585

)

s=[

[

119889120594

119889120585

119889(119879)

in119889120594

]

]s



119889120579

119889120594

)

s+O (120576)

(56)


Os =Oinfin

1 + 120573 + 119860 sO [120573 (minus119891s)]1 minus (

119889120579

119889120594

)

s (57)


1198892120579

1198891205942= minusΔ(120594 minus 120579 + 120578O)

12 exp (120582120579 minus 120594) (58)

where

Δ = 119863119886g exp(minus119879119886g

119879s

)

120573

(minus119891s)

119879s

119879s minus

119879infin

2

times (

119879s119879119886g

)

32

(

119879infin

119879s

)

12

Fs

(120582119879s)12

(59)

120578O =

Os

120576119879s120582

(60)








(119879)

out=119879s + (

119879infinminus119879s) 120585 + 120576

119879sΘ (120585) +O (120576

2) (62)



120582120579 (infin) = minus119862I (

119889120579

119889120594

)

infin

= 0 (63)



2Δ I120582 = (radic120578O + radic120587120582

2

119890120582120578O


120594I

119890(120582minus1)120594erfc (119911) 119889120594)

minus1

(64)

erfc (119911) = 2

radic120587

expintinfin

119911

(minus1199052) 119889119905 (65)


(

119889120579

119889120594

)

s=

1

120582

or (

119889(119879)

in119889120585

)

s

= 0 (66)


at 120582 = 1 2Δ I =1


as 120578O 997888rarr infin 2Δ I120582 =1

radic120578O

(67)



2Δ I120582 = (radic120578O +

radic120587

2

[119890120578O erfc (radic120578O)

+

1

119865 (120582)


2

)])

minus1

(68)

where

119865 (120582) = 056 +

021

120582

minus

012

1205822+

035

1205823 (69)





2reaction and hence








1000

1200

1400

1600

1800

2000

10 100 1000


1985

119884O = 07

119884A = 00003


Igni

tion

surfa

ce te

mpe

ratu

re119879

sig

(K)

(a)

1000

1200

1400

1600

1800

2000

00001 0001 001


1985


119884O = 07

119886 = 170 sminus1

Igni

tion

surfa

ce te

mpe

ratu

re119879

sig

(K)

(b)

1000

1200

1400

1600

1800

2000


119886 = 170 sminus1119884A = 00003

Igni

tion

surfa

ce te

mpe

ratu

re119879

sig

(K)

(c)












2reaction



2mass fraction


2mass fraction In





2mass fraction



119861g = 119861lowast

g (120588119884A119882A)12

[(molm3)12

sdot s]minus1

(70)








119861lowast


minus1

(71)




119861lowast


minus1

(72)




Air

O2



100

102

104

106

3 4 5

Oxidizer

lowast(119879

s119879119886

g)32

exp(minus119879119886

g119879

s)

(m3m

oL)middot

sminus1

1s

119884Oinfin = 0533

119861g


2and H



119861lowast


minus1

(73)



2andor H



2









infin120588infin









2O mass fraction



7msand 119864sA119877 = 33 times 10







0

0002

0004

0006

0008

001

1000 1500 2000 2500 3000

Air

Frozen


Com

busti

on ra

te119898

(kg

m2s)

119879si

g=

1479

K

Flame-detached

Flame-attached

119886 = 120 sminus1


(a)

0

0002

0004

0006

0008

001

1000 1500 2000 2500 3000


Com

busti

on ra

te119898

(kg

m2s) 119886 = 120 sminus1

C-CO2


(b)

0

0005

001

0015

002

1000 1500 2000 2500 3000


C-H2O119886 = 120 sminus1

Com

busti

on ra

te119898

(kg

m2s)


(c)


2mass




119891 =

119891s (0 le 120578 le 120578lowast)

119887120578 + 119888 (120578lowastle 120578 le 120578

lowastlowast)

120578 + 119889 (120578 ge 120578lowastlowast)

(74)



119900



119870

(75)

where

119870 = 120578lowast+ radic

120587

2

[exp((119887 minus 1) 119891

2

119900

2119887

) minus 1 erfc(119891119900

radic2

)

+

1

radic119887

erf (119891119900

radic2119887

) minus erf (119891119900

radic2

)] + radic

120587

2

(76)




120573

1 + 120573



120573 asymp (

119870119860 sO

1 + 119870119860 sO)(

2119882C119882O

119884Oinfin)

+ (

119870119860 sP

1 + 119870119860 sP)(

119882C119882P

119884Pinfin)

(78)


Flame-detached mode

120573 asymp (

119870119860 sP

1 + 119870119860 sP)(

2119882C119882O

119884Oinfin +

119882C119882P

119884Pinfin) (79)

Flame-attached mode

120573 asymp (

119870119860 sO

1 + 2119870119860 sO minus 119870119860 sP)(

2119882C119882O

119884Oinfin)

+ (

119870119860 sP

1 + 2119870119860 sO minus 119870119860 sP)(

119882C119882P

119884Pinfin)

(80)


2and CO

2con-


119870119860 s119894 = 119896s119894119870


(81)


119896s119894 equiv 119861s119894 (119879infin

119879s) exp(minus

119879119886s119894

119879s) (119894 = OP) (82)



asymp 120588infin


119870

ln (1 + 120573) (83)


119870

[2119895119886 (120583infin120588infin)]12

(84)


asymp

1


120588infin(

2119882C119882O

119884Oinfin)

+

1


120588infin(

119882C119882P

119884Pinfin)

(85)


=

1

1119896s + 1ℎ119863(120588119884O)infin (86)



[2119895119886(120583infin120588infin)]



119863is of use in



ℎ119863= (

119879119877

119879infin





ℎ119863= (

119879119877

119879infin

)

119862mtSc06


fromNu119909

radicRe119909

= 119862mtSc04

(88)


Nu119909=

ℎ119863119909

119863119877

Re119909=

120588119877119880119909

120583119877

119880 = 119886119909 Sc =120583120588

119863

(89)






119870 = radic

2

3

(

119879infin

119879s)(1 minus

119879infin

2119879s) + radic

120587

2

(90)



0

001

002

003

004

Frozen

1000 1500 2000 2500

Flame-detached

Flame-attached

Com

busti

on ra

te119898

(kg

m2s) 119884O = 07

119884A = 00003

119886 = 170 sminus1





2concentrations



119870 = radic

2

3

(

119879infin

119879s)(1 minus

119879infin

2119879s)

minus 005 (1 +

4119882C119882O


120587

2

(91)


2mass-fraction



119870 = (

119879infin

119879s)(1 minus

119879infin

2119879s) + radic

120587

2

(92)


119870 = (

119879infin

119879s)(1 minus

119879infin

2119879s)

minus 005 (1 +

4119882C119882O


120587

2

(93)

0

0005

001

0015

1000 1500 2000 2500



Com

busti

on ra

te119898

(kg

m2s)

119886 = 300 sminus1

119886 = 180 sminus1

119886 = 60 sminus1


(a)

0

002

004

006

008

1000 1500 2000 2500


Com

busti

on ra

te119898

(kg

m2s) 119886 = 1000 sminus1

119886 = 320 sminus1

119886 = 88 sminus1


119884Oinfin = 07

119884Ainfin = 00002

(b)




















0

001

002

1000 1500 2000 2500



Matsui 1983

Com

busti

on ra

te119898

(kg

m2s)

119886 = 300 sminus1

119884Ainfin = 0001


119884Ainfin = 0004







2Oconcentrations






2O mass-fraction



0

0005

001

0015

1000 1500 2000 2500



ombu

stion

rate119898

(kg

m2s)

119886 = 100 sminus1

119884Oinfin = 02

119884Ainfin = 014


119879si

g=1094

K







6ms and119864sO119877=22 times 10



4 K (119864sP =


8ms and119864sA119877 = 33 times 10










2O



2O mass fraction is



2O and












119894s =

119894infin

1 + 120573 + 119860 s119894 [120573 (minus119891s)](119894 = OPA) (95)





1 + 120573

Ps =Pinfin + 120575120573

1 + 120573

As =Ainfin

1 + 120573

(96)



1 + 120573

(97)


119879 + O + (1 minus 119876) A (98)


As =1

1 minus 119876


1 + 120573 + 119860 sA [120573 (minus119891s)] (99)




119879 = 119879s + (119879f minus 119879s) (120585

120585f) (100)



+ (1 minus 119876)(As1 minus 120585f120585f

minus

Af

120585f)

(101)


As =Ainfin



Frozen mode

120573 asymp (

119870119860 sO

1 + 119870119860 sO)(

2119882C119882O

119884Oinfin) + (

119870119860 sP

1 + 119870119860 sP)

times (

119882C119882P

119884Pinfin) + (

119870119860 sA

1 + 119870119860 sA)(

119882C119882A

119884Ainfin)

(103)

Flame-detached mode

120573 asymp

1

2

119870119860 sP

1 + 119870119860 sP(

2119882C119882O

119884Oinfin +

119882C119882P

119884Pinfin)

+

119870119860 sA

1 + 119870119860 sA(

119882C119882O

119884Oinfin +

119882C119882A

119884Ainfin)

+

1

2

[

119870119860 sP

1 + 119870119860 sP(

2119882C119882O

119884Oinfin +

119882C119882P

119884Pinfin)

minus

119870119860 sA

1 + 119870119860 sA(

119882C119882O

119884Oinfin +

119882C119882A

119884Ainfin)

2

+ 4

119870119860 sP

1 + 119870119860 sP(

119882C119882O

119884Oinfin +

119882C119882P

119884Pinfin)

times

119870119860 sA

1 + 119870119860 sA(

119882C119882A

119884Ainfin)]

12

(104)

Flame-attached mode

120573 asymp (

1

1 + 2119870119860 sO minus 119870119860 sP)

times (119870119860 sO2119882C119882O

119884Oinfin + 119870119860 sP119882C119882P

119884Pinfin

+119870119860 sA119882C119882A

119884Ainfin)

(105)






2and C-CO

2reactions and




2reaction to







2O mass-



2-O2reaction into


2Omass



Nomenclature










Greek Symbols








Subscripts



2surface reaction


2surface reaction


Superscripts




References









































































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation
















(119879)

out=119879s + (

119879infinminus119879s) 120585 + 120576

119879sΘ (120585) +O (120576

2) (62)



120582120579 (infin) = minus119862I (

119889120579

119889120594

)

infin

= 0 (63)



2Δ I120582 = (radic120578O + radic120587120582

2

119890120582120578O


120594I

119890(120582minus1)120594erfc (119911) 119889120594)

minus1

(64)

erfc (119911) = 2

radic120587

expintinfin

119911

(minus1199052) 119889119905 (65)


(

119889120579

119889120594

)

s=

1

120582

or (

119889(119879)

in119889120585

)

s

= 0 (66)


at 120582 = 1 2Δ I =1


as 120578O 997888rarr infin 2Δ I120582 =1

radic120578O

(67)



2Δ I120582 = (radic120578O +

radic120587

2

[119890120578O erfc (radic120578O)

+

1

119865 (120582)


2

)])

minus1

(68)

where

119865 (120582) = 056 +

021

120582

minus

012

1205822+

035

1205823 (69)





2reaction and hence








1000

1200

1400

1600

1800

2000

10 100 1000


1985

119884O = 07

119884A = 00003


Igni

tion

surfa

ce te

mpe

ratu

re119879

sig

(K)

(a)

1000

1200

1400

1600

1800

2000

00001 0001 001


1985


119884O = 07

119886 = 170 sminus1

Igni

tion

surfa

ce te

mpe

ratu

re119879

sig

(K)

(b)

1000

1200

1400

1600

1800

2000


119886 = 170 sminus1119884A = 00003

Igni

tion

surfa

ce te

mpe

ratu

re119879

sig

(K)

(c)












2reaction



2mass fraction


2mass fraction In





2mass fraction



119861g = 119861lowast

g (120588119884A119882A)12

[(molm3)12

sdot s]minus1

(70)








119861lowast


minus1

(71)




119861lowast


minus1

(72)




Air

O2



100

102

104

106

3 4 5

Oxidizer

lowast(119879

s119879119886

g)32

exp(minus119879119886

g119879

s)

(m3m

oL)middot

sminus1

1s

119884Oinfin = 0533

119861g


2and H



119861lowast


minus1

(73)



2andor H



2









infin120588infin









2O mass fraction



7msand 119864sA119877 = 33 times 10







0

0002

0004

0006

0008

001

1000 1500 2000 2500 3000

Air

Frozen


Com

busti

on ra

te119898

(kg

m2s)

119879si

g=

1479

K

Flame-detached

Flame-attached

119886 = 120 sminus1


(a)

0

0002

0004

0006

0008

001

1000 1500 2000 2500 3000


Com

busti

on ra

te119898

(kg

m2s) 119886 = 120 sminus1

C-CO2


(b)

0

0005

001

0015

002

1000 1500 2000 2500 3000


C-H2O119886 = 120 sminus1

Com

busti

on ra

te119898

(kg

m2s)


(c)


2mass




119891 =

119891s (0 le 120578 le 120578lowast)

119887120578 + 119888 (120578lowastle 120578 le 120578

lowastlowast)

120578 + 119889 (120578 ge 120578lowastlowast)

(74)



119900



119870

(75)

where

119870 = 120578lowast+ radic

120587

2

[exp((119887 minus 1) 119891

2

119900

2119887

) minus 1 erfc(119891119900

radic2

)

+

1

radic119887

erf (119891119900

radic2119887

) minus erf (119891119900

radic2

)] + radic

120587

2

(76)




120573

1 + 120573



120573 asymp (

119870119860 sO

1 + 119870119860 sO)(

2119882C119882O

119884Oinfin)

+ (

119870119860 sP

1 + 119870119860 sP)(

119882C119882P

119884Pinfin)

(78)


Flame-detached mode

120573 asymp (

119870119860 sP

1 + 119870119860 sP)(

2119882C119882O

119884Oinfin +

119882C119882P

119884Pinfin) (79)

Flame-attached mode

120573 asymp (

119870119860 sO

1 + 2119870119860 sO minus 119870119860 sP)(

2119882C119882O

119884Oinfin)

+ (

119870119860 sP

1 + 2119870119860 sO minus 119870119860 sP)(

119882C119882P

119884Pinfin)

(80)


2and CO

2con-


119870119860 s119894 = 119896s119894119870


(81)


119896s119894 equiv 119861s119894 (119879infin

119879s) exp(minus

119879119886s119894

119879s) (119894 = OP) (82)



asymp 120588infin


119870

ln (1 + 120573) (83)


119870

[2119895119886 (120583infin120588infin)]12

(84)


asymp

1


120588infin(

2119882C119882O

119884Oinfin)

+

1


120588infin(

119882C119882P

119884Pinfin)

(85)


=

1

1119896s + 1ℎ119863(120588119884O)infin (86)



[2119895119886(120583infin120588infin)]



119863is of use in



ℎ119863= (

119879119877

119879infin





ℎ119863= (

119879119877

119879infin

)

119862mtSc06


fromNu119909

radicRe119909

= 119862mtSc04

(88)


Nu119909=

ℎ119863119909

119863119877

Re119909=

120588119877119880119909

120583119877

119880 = 119886119909 Sc =120583120588

119863

(89)






119870 = radic

2

3

(

119879infin

119879s)(1 minus

119879infin

2119879s) + radic

120587

2

(90)



0

001

002

003

004

Frozen

1000 1500 2000 2500

Flame-detached

Flame-attached

Com

busti

on ra

te119898

(kg

m2s) 119884O = 07

119884A = 00003

119886 = 170 sminus1





2concentrations



119870 = radic

2

3

(

119879infin

119879s)(1 minus

119879infin

2119879s)

minus 005 (1 +

4119882C119882O


120587

2

(91)


2mass-fraction



119870 = (

119879infin

119879s)(1 minus

119879infin

2119879s) + radic

120587

2

(92)


119870 = (

119879infin

119879s)(1 minus

119879infin

2119879s)

minus 005 (1 +

4119882C119882O


120587

2

(93)

0

0005

001

0015

1000 1500 2000 2500



Com

busti

on ra

te119898

(kg

m2s)

119886 = 300 sminus1

119886 = 180 sminus1

119886 = 60 sminus1


(a)

0

002

004

006

008

1000 1500 2000 2500


Com

busti

on ra

te119898

(kg

m2s) 119886 = 1000 sminus1

119886 = 320 sminus1

119886 = 88 sminus1


119884Oinfin = 07

119884Ainfin = 00002

(b)




















0

001

002

1000 1500 2000 2500



Matsui 1983

Com

busti

on ra

te119898

(kg

m2s)

119886 = 300 sminus1

119884Ainfin = 0001


119884Ainfin = 0004







2Oconcentrations






2O mass-fraction



0

0005

001

0015

1000 1500 2000 2500



ombu

stion

rate119898

(kg

m2s)

119886 = 100 sminus1

119884Oinfin = 02

119884Ainfin = 014


119879si

g=1094

K







6ms and119864sO119877=22 times 10



4 K (119864sP =


8ms and119864sA119877 = 33 times 10










2O



2O mass fraction is



2O and












119894s =

119894infin

1 + 120573 + 119860 s119894 [120573 (minus119891s)](119894 = OPA) (95)





1 + 120573

Ps =Pinfin + 120575120573

1 + 120573

As =Ainfin

1 + 120573

(96)



1 + 120573

(97)


119879 + O + (1 minus 119876) A (98)


As =1

1 minus 119876


1 + 120573 + 119860 sA [120573 (minus119891s)] (99)




119879 = 119879s + (119879f minus 119879s) (120585

120585f) (100)



+ (1 minus 119876)(As1 minus 120585f120585f

minus

Af

120585f)

(101)


As =Ainfin



Frozen mode

120573 asymp (

119870119860 sO

1 + 119870119860 sO)(

2119882C119882O

119884Oinfin) + (

119870119860 sP

1 + 119870119860 sP)

times (

119882C119882P

119884Pinfin) + (

119870119860 sA

1 + 119870119860 sA)(

119882C119882A

119884Ainfin)

(103)

Flame-detached mode

120573 asymp

1

2

119870119860 sP

1 + 119870119860 sP(

2119882C119882O

119884Oinfin +

119882C119882P

119884Pinfin)

+

119870119860 sA

1 + 119870119860 sA(

119882C119882O

119884Oinfin +

119882C119882A

119884Ainfin)

+

1

2

[

119870119860 sP

1 + 119870119860 sP(

2119882C119882O

119884Oinfin +

119882C119882P

119884Pinfin)

minus

119870119860 sA

1 + 119870119860 sA(

119882C119882O

119884Oinfin +

119882C119882A

119884Ainfin)

2

+ 4

119870119860 sP

1 + 119870119860 sP(

119882C119882O

119884Oinfin +

119882C119882P

119884Pinfin)

times

119870119860 sA

1 + 119870119860 sA(

119882C119882A

119884Ainfin)]

12

(104)

Flame-attached mode

120573 asymp (

1

1 + 2119870119860 sO minus 119870119860 sP)

times (119870119860 sO2119882C119882O

119884Oinfin + 119870119860 sP119882C119882P

119884Pinfin

+119870119860 sA119882C119882A

119884Ainfin)

(105)






2and C-CO

2reactions and




2reaction to







2O mass-



2-O2reaction into


2Omass



Nomenclature










Greek Symbols








Subscripts



2surface reaction


2surface reaction


Superscripts




References









































































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










1000

1200

1400

1600

1800

2000

10 100 1000


1985

119884O = 07

119884A = 00003


Igni

tion

surfa

ce te

mpe

ratu

re119879

sig

(K)

(a)

1000

1200

1400

1600

1800

2000

00001 0001 001


1985


119884O = 07

119886 = 170 sminus1

Igni

tion

surfa

ce te

mpe

ratu

re119879

sig

(K)

(b)

1000

1200

1400

1600

1800

2000


119886 = 170 sminus1119884A = 00003

Igni

tion

surfa

ce te

mpe

ratu

re119879

sig

(K)

(c)












2reaction



2mass fraction


2mass fraction In





2mass fraction



119861g = 119861lowast

g (120588119884A119882A)12

[(molm3)12

sdot s]minus1

(70)








119861lowast


minus1

(71)




119861lowast


minus1

(72)




Air

O2



100

102

104

106

3 4 5

Oxidizer

lowast(119879

s119879119886

g)32

exp(minus119879119886

g119879

s)

(m3m

oL)middot

sminus1

1s

119884Oinfin = 0533

119861g


2and H



119861lowast


minus1

(73)



2andor H



2









infin120588infin









2O mass fraction



7msand 119864sA119877 = 33 times 10







0

0002

0004

0006

0008

001

1000 1500 2000 2500 3000

Air

Frozen


Com

busti

on ra

te119898

(kg

m2s)

119879si

g=

1479

K

Flame-detached

Flame-attached

119886 = 120 sminus1


(a)

0

0002

0004

0006

0008

001

1000 1500 2000 2500 3000


Com

busti

on ra

te119898

(kg

m2s) 119886 = 120 sminus1

C-CO2


(b)

0

0005

001

0015

002

1000 1500 2000 2500 3000


C-H2O119886 = 120 sminus1

Com

busti

on ra

te119898

(kg

m2s)


(c)


2mass




119891 =

119891s (0 le 120578 le 120578lowast)

119887120578 + 119888 (120578lowastle 120578 le 120578

lowastlowast)

120578 + 119889 (120578 ge 120578lowastlowast)

(74)



119900



119870

(75)

where

119870 = 120578lowast+ radic

120587

2

[exp((119887 minus 1) 119891

2

119900

2119887

) minus 1 erfc(119891119900

radic2

)

+

1

radic119887

erf (119891119900

radic2119887

) minus erf (119891119900

radic2

)] + radic

120587

2

(76)




120573

1 + 120573



120573 asymp (

119870119860 sO

1 + 119870119860 sO)(

2119882C119882O

119884Oinfin)

+ (

119870119860 sP

1 + 119870119860 sP)(

119882C119882P

119884Pinfin)

(78)


Flame-detached mode

120573 asymp (

119870119860 sP

1 + 119870119860 sP)(

2119882C119882O

119884Oinfin +

119882C119882P

119884Pinfin) (79)

Flame-attached mode

120573 asymp (

119870119860 sO

1 + 2119870119860 sO minus 119870119860 sP)(

2119882C119882O

119884Oinfin)

+ (

119870119860 sP

1 + 2119870119860 sO minus 119870119860 sP)(

119882C119882P

119884Pinfin)

(80)


2and CO

2con-


119870119860 s119894 = 119896s119894119870


(81)


119896s119894 equiv 119861s119894 (119879infin

119879s) exp(minus

119879119886s119894

119879s) (119894 = OP) (82)



asymp 120588infin


119870

ln (1 + 120573) (83)


119870

[2119895119886 (120583infin120588infin)]12

(84)


asymp

1


120588infin(

2119882C119882O

119884Oinfin)

+

1


120588infin(

119882C119882P

119884Pinfin)

(85)


=

1

1119896s + 1ℎ119863(120588119884O)infin (86)



[2119895119886(120583infin120588infin)]



119863is of use in



ℎ119863= (

119879119877

119879infin





ℎ119863= (

119879119877

119879infin

)

119862mtSc06


fromNu119909

radicRe119909

= 119862mtSc04

(88)


Nu119909=

ℎ119863119909

119863119877

Re119909=

120588119877119880119909

120583119877

119880 = 119886119909 Sc =120583120588

119863

(89)






119870 = radic

2

3

(

119879infin

119879s)(1 minus

119879infin

2119879s) + radic

120587

2

(90)



0

001

002

003

004

Frozen

1000 1500 2000 2500

Flame-detached

Flame-attached

Com

busti

on ra

te119898

(kg

m2s) 119884O = 07

119884A = 00003

119886 = 170 sminus1





2concentrations



119870 = radic

2

3

(

119879infin

119879s)(1 minus

119879infin

2119879s)

minus 005 (1 +

4119882C119882O


120587

2

(91)


2mass-fraction



119870 = (

119879infin

119879s)(1 minus

119879infin

2119879s) + radic

120587

2

(92)


119870 = (

119879infin

119879s)(1 minus

119879infin

2119879s)

minus 005 (1 +

4119882C119882O


120587

2

(93)

0

0005

001

0015

1000 1500 2000 2500



Com

busti

on ra

te119898

(kg

m2s)

119886 = 300 sminus1

119886 = 180 sminus1

119886 = 60 sminus1


(a)

0

002

004

006

008

1000 1500 2000 2500


Com

busti

on ra

te119898

(kg

m2s) 119886 = 1000 sminus1

119886 = 320 sminus1

119886 = 88 sminus1


119884Oinfin = 07

119884Ainfin = 00002

(b)




















0

001

002

1000 1500 2000 2500



Matsui 1983

Com

busti

on ra

te119898

(kg

m2s)

119886 = 300 sminus1

119884Ainfin = 0001


119884Ainfin = 0004







2Oconcentrations






2O mass-fraction



0

0005

001

0015

1000 1500 2000 2500



ombu

stion

rate119898

(kg

m2s)

119886 = 100 sminus1

119884Oinfin = 02

119884Ainfin = 014


119879si

g=1094

K







6ms and119864sO119877=22 times 10



4 K (119864sP =


8ms and119864sA119877 = 33 times 10










2O



2O mass fraction is



2O and












119894s =

119894infin

1 + 120573 + 119860 s119894 [120573 (minus119891s)](119894 = OPA) (95)





1 + 120573

Ps =Pinfin + 120575120573

1 + 120573

As =Ainfin

1 + 120573

(96)



1 + 120573

(97)


119879 + O + (1 minus 119876) A (98)


As =1

1 minus 119876


1 + 120573 + 119860 sA [120573 (minus119891s)] (99)




119879 = 119879s + (119879f minus 119879s) (120585

120585f) (100)



+ (1 minus 119876)(As1 minus 120585f120585f

minus

Af

120585f)

(101)


As =Ainfin



Frozen mode

120573 asymp (

119870119860 sO

1 + 119870119860 sO)(

2119882C119882O

119884Oinfin) + (

119870119860 sP

1 + 119870119860 sP)

times (

119882C119882P

119884Pinfin) + (

119870119860 sA

1 + 119870119860 sA)(

119882C119882A

119884Ainfin)

(103)

Flame-detached mode

120573 asymp

1

2

119870119860 sP

1 + 119870119860 sP(

2119882C119882O

119884Oinfin +

119882C119882P

119884Pinfin)

+

119870119860 sA

1 + 119870119860 sA(

119882C119882O

119884Oinfin +

119882C119882A

119884Ainfin)

+

1

2

[

119870119860 sP

1 + 119870119860 sP(

2119882C119882O

119884Oinfin +

119882C119882P

119884Pinfin)

minus

119870119860 sA

1 + 119870119860 sA(

119882C119882O

119884Oinfin +

119882C119882A

119884Ainfin)

2

+ 4

119870119860 sP

1 + 119870119860 sP(

119882C119882O

119884Oinfin +

119882C119882P

119884Pinfin)

times

119870119860 sA

1 + 119870119860 sA(

119882C119882A

119884Ainfin)]

12

(104)

Flame-attached mode

120573 asymp (

1

1 + 2119870119860 sO minus 119870119860 sP)

times (119870119860 sO2119882C119882O

119884Oinfin + 119870119860 sP119882C119882P

119884Pinfin

+119870119860 sA119882C119882A

119884Ainfin)

(105)






2and C-CO

2reactions and




2reaction to







2O mass-



2-O2reaction into


2Omass



Nomenclature










Greek Symbols








Subscripts



2surface reaction


2surface reaction


Superscripts




References









































































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











Air

O2



100

102

104

106

3 4 5

Oxidizer

lowast(119879

s119879119886

g)32

exp(minus119879119886

g119879

s)

(m3m

oL)middot

sminus1

1s

119884Oinfin = 0533

119861g


2and H



119861lowast


minus1

(73)



2andor H



2









infin120588infin









2O mass fraction



7msand 119864sA119877 = 33 times 10







0

0002

0004

0006

0008

001

1000 1500 2000 2500 3000

Air

Frozen


Com

busti

on ra

te119898

(kg

m2s)

119879si

g=

1479

K

Flame-detached

Flame-attached

119886 = 120 sminus1


(a)

0

0002

0004

0006

0008

001

1000 1500 2000 2500 3000


Com

busti

on ra

te119898

(kg

m2s) 119886 = 120 sminus1

C-CO2


(b)

0

0005

001

0015

002

1000 1500 2000 2500 3000


C-H2O119886 = 120 sminus1

Com

busti

on ra

te119898

(kg

m2s)


(c)


2mass




119891 =

119891s (0 le 120578 le 120578lowast)

119887120578 + 119888 (120578lowastle 120578 le 120578

lowastlowast)

120578 + 119889 (120578 ge 120578lowastlowast)

(74)



119900



119870

(75)

where

119870 = 120578lowast+ radic

120587

2

[exp((119887 minus 1) 119891

2

119900

2119887

) minus 1 erfc(119891119900

radic2

)

+

1

radic119887

erf (119891119900

radic2119887

) minus erf (119891119900

radic2

)] + radic

120587

2

(76)




120573

1 + 120573



120573 asymp (

119870119860 sO

1 + 119870119860 sO)(

2119882C119882O

119884Oinfin)

+ (

119870119860 sP

1 + 119870119860 sP)(

119882C119882P

119884Pinfin)

(78)


Flame-detached mode

120573 asymp (

119870119860 sP

1 + 119870119860 sP)(

2119882C119882O

119884Oinfin +

119882C119882P

119884Pinfin) (79)

Flame-attached mode

120573 asymp (

119870119860 sO

1 + 2119870119860 sO minus 119870119860 sP)(

2119882C119882O

119884Oinfin)

+ (

119870119860 sP

1 + 2119870119860 sO minus 119870119860 sP)(

119882C119882P

119884Pinfin)

(80)


2and CO

2con-


119870119860 s119894 = 119896s119894119870


(81)


119896s119894 equiv 119861s119894 (119879infin

119879s) exp(minus

119879119886s119894

119879s) (119894 = OP) (82)



asymp 120588infin


119870

ln (1 + 120573) (83)


119870

[2119895119886 (120583infin120588infin)]12

(84)


asymp

1


120588infin(

2119882C119882O

119884Oinfin)

+

1


120588infin(

119882C119882P

119884Pinfin)

(85)


=

1

1119896s + 1ℎ119863(120588119884O)infin (86)



[2119895119886(120583infin120588infin)]



119863is of use in



ℎ119863= (

119879119877

119879infin





ℎ119863= (

119879119877

119879infin

)

119862mtSc06


fromNu119909

radicRe119909

= 119862mtSc04

(88)


Nu119909=

ℎ119863119909

119863119877

Re119909=

120588119877119880119909

120583119877

119880 = 119886119909 Sc =120583120588

119863

(89)






119870 = radic

2

3

(

119879infin

119879s)(1 minus

119879infin

2119879s) + radic

120587

2

(90)



0

001

002

003

004

Frozen

1000 1500 2000 2500

Flame-detached

Flame-attached

Com

busti

on ra

te119898

(kg

m2s) 119884O = 07

119884A = 00003

119886 = 170 sminus1





2concentrations



119870 = radic

2

3

(

119879infin

119879s)(1 minus

119879infin

2119879s)

minus 005 (1 +

4119882C119882O


120587

2

(91)


2mass-fraction



119870 = (

119879infin

119879s)(1 minus

119879infin

2119879s) + radic

120587

2

(92)


119870 = (

119879infin

119879s)(1 minus

119879infin

2119879s)

minus 005 (1 +

4119882C119882O


120587

2

(93)

0

0005

001

0015

1000 1500 2000 2500



Com

busti

on ra

te119898

(kg

m2s)

119886 = 300 sminus1

119886 = 180 sminus1

119886 = 60 sminus1


(a)

0

002

004

006

008

1000 1500 2000 2500


Com

busti

on ra

te119898

(kg

m2s) 119886 = 1000 sminus1

119886 = 320 sminus1

119886 = 88 sminus1


119884Oinfin = 07

119884Ainfin = 00002

(b)




















0

001

002

1000 1500 2000 2500



Matsui 1983

Com

busti

on ra

te119898

(kg

m2s)

119886 = 300 sminus1

119884Ainfin = 0001


119884Ainfin = 0004







2Oconcentrations






2O mass-fraction



0

0005

001

0015

1000 1500 2000 2500



ombu

stion

rate119898

(kg

m2s)

119886 = 100 sminus1

119884Oinfin = 02

119884Ainfin = 014


119879si

g=1094

K







6ms and119864sO119877=22 times 10



4 K (119864sP =


8ms and119864sA119877 = 33 times 10










2O



2O mass fraction is



2O and












119894s =

119894infin

1 + 120573 + 119860 s119894 [120573 (minus119891s)](119894 = OPA) (95)





1 + 120573

Ps =Pinfin + 120575120573

1 + 120573

As =Ainfin

1 + 120573

(96)



1 + 120573

(97)


119879 + O + (1 minus 119876) A (98)


As =1

1 minus 119876


1 + 120573 + 119860 sA [120573 (minus119891s)] (99)




119879 = 119879s + (119879f minus 119879s) (120585

120585f) (100)



+ (1 minus 119876)(As1 minus 120585f120585f

minus

Af

120585f)

(101)


As =Ainfin



Frozen mode

120573 asymp (

119870119860 sO

1 + 119870119860 sO)(

2119882C119882O

119884Oinfin) + (

119870119860 sP

1 + 119870119860 sP)

times (

119882C119882P

119884Pinfin) + (

119870119860 sA

1 + 119870119860 sA)(

119882C119882A

119884Ainfin)

(103)

Flame-detached mode

120573 asymp

1

2

119870119860 sP

1 + 119870119860 sP(

2119882C119882O

119884Oinfin +

119882C119882P

119884Pinfin)

+

119870119860 sA

1 + 119870119860 sA(

119882C119882O

119884Oinfin +

119882C119882A

119884Ainfin)

+

1

2

[

119870119860 sP

1 + 119870119860 sP(

2119882C119882O

119884Oinfin +

119882C119882P

119884Pinfin)

minus

119870119860 sA

1 + 119870119860 sA(

119882C119882O

119884Oinfin +

119882C119882A

119884Ainfin)

2

+ 4

119870119860 sP

1 + 119870119860 sP(

119882C119882O

119884Oinfin +

119882C119882P

119884Pinfin)

times

119870119860 sA

1 + 119870119860 sA(

119882C119882A

119884Ainfin)]

12

(104)

Flame-attached mode

120573 asymp (

1

1 + 2119870119860 sO minus 119870119860 sP)

times (119870119860 sO2119882C119882O

119884Oinfin + 119870119860 sP119882C119882P

119884Pinfin

+119870119860 sA119882C119882A

119884Ainfin)

(105)






2and C-CO

2reactions and




2reaction to







2O mass-



2-O2reaction into


2Omass



Nomenclature










Greek Symbols








Subscripts



2surface reaction


2surface reaction


Superscripts




References









































































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










0

0002

0004

0006

0008

001

1000 1500 2000 2500 3000

Air

Frozen


Com

busti

on ra

te119898

(kg

m2s)

119879si

g=

1479

K

Flame-detached

Flame-attached

119886 = 120 sminus1


(a)

0

0002

0004

0006

0008

001

1000 1500 2000 2500 3000


Com

busti

on ra

te119898

(kg

m2s) 119886 = 120 sminus1

C-CO2


(b)

0

0005

001

0015

002

1000 1500 2000 2500 3000


C-H2O119886 = 120 sminus1

Com

busti

on ra

te119898

(kg

m2s)


(c)


2mass




119891 =

119891s (0 le 120578 le 120578lowast)

119887120578 + 119888 (120578lowastle 120578 le 120578

lowastlowast)

120578 + 119889 (120578 ge 120578lowastlowast)

(74)



119900



119870

(75)

where

119870 = 120578lowast+ radic

120587

2

[exp((119887 minus 1) 119891

2

119900

2119887

) minus 1 erfc(119891119900

radic2

)

+

1

radic119887

erf (119891119900

radic2119887

) minus erf (119891119900

radic2

)] + radic

120587

2

(76)




120573

1 + 120573



120573 asymp (

119870119860 sO

1 + 119870119860 sO)(

2119882C119882O

119884Oinfin)

+ (

119870119860 sP

1 + 119870119860 sP)(

119882C119882P

119884Pinfin)

(78)


Flame-detached mode

120573 asymp (

119870119860 sP

1 + 119870119860 sP)(

2119882C119882O

119884Oinfin +

119882C119882P

119884Pinfin) (79)

Flame-attached mode

120573 asymp (

119870119860 sO

1 + 2119870119860 sO minus 119870119860 sP)(

2119882C119882O

119884Oinfin)

+ (

119870119860 sP

1 + 2119870119860 sO minus 119870119860 sP)(

119882C119882P

119884Pinfin)

(80)


2and CO

2con-


119870119860 s119894 = 119896s119894119870


(81)


119896s119894 equiv 119861s119894 (119879infin

119879s) exp(minus

119879119886s119894

119879s) (119894 = OP) (82)



asymp 120588infin


119870

ln (1 + 120573) (83)


119870

[2119895119886 (120583infin120588infin)]12

(84)


asymp

1


120588infin(

2119882C119882O

119884Oinfin)

+

1


120588infin(

119882C119882P

119884Pinfin)

(85)


=

1

1119896s + 1ℎ119863(120588119884O)infin (86)



[2119895119886(120583infin120588infin)]



119863is of use in



ℎ119863= (

119879119877

119879infin





ℎ119863= (

119879119877

119879infin

)

119862mtSc06


fromNu119909

radicRe119909

= 119862mtSc04

(88)


Nu119909=

ℎ119863119909

119863119877

Re119909=

120588119877119880119909

120583119877

119880 = 119886119909 Sc =120583120588

119863

(89)






119870 = radic

2

3

(

119879infin

119879s)(1 minus

119879infin

2119879s) + radic

120587

2

(90)



0

001

002

003

004

Frozen

1000 1500 2000 2500

Flame-detached

Flame-attached

Com

busti

on ra

te119898

(kg

m2s) 119884O = 07

119884A = 00003

119886 = 170 sminus1





2concentrations



119870 = radic

2

3

(

119879infin

119879s)(1 minus

119879infin

2119879s)

minus 005 (1 +

4119882C119882O


120587

2

(91)


2mass-fraction



119870 = (

119879infin

119879s)(1 minus

119879infin

2119879s) + radic

120587

2

(92)


119870 = (

119879infin

119879s)(1 minus

119879infin

2119879s)

minus 005 (1 +

4119882C119882O


120587

2

(93)

0

0005

001

0015

1000 1500 2000 2500



Com

busti

on ra

te119898

(kg

m2s)

119886 = 300 sminus1

119886 = 180 sminus1

119886 = 60 sminus1


(a)

0

002

004

006

008

1000 1500 2000 2500


Com

busti

on ra

te119898

(kg

m2s) 119886 = 1000 sminus1

119886 = 320 sminus1

119886 = 88 sminus1


119884Oinfin = 07

119884Ainfin = 00002

(b)




















0

001

002

1000 1500 2000 2500



Matsui 1983

Com

busti

on ra

te119898

(kg

m2s)

119886 = 300 sminus1

119884Ainfin = 0001


119884Ainfin = 0004







2Oconcentrations






2O mass-fraction



0

0005

001

0015

1000 1500 2000 2500



ombu

stion

rate119898

(kg

m2s)

119886 = 100 sminus1

119884Oinfin = 02

119884Ainfin = 014


119879si

g=1094

K







6ms and119864sO119877=22 times 10



4 K (119864sP =


8ms and119864sA119877 = 33 times 10










2O



2O mass fraction is



2O and












119894s =

119894infin

1 + 120573 + 119860 s119894 [120573 (minus119891s)](119894 = OPA) (95)





1 + 120573

Ps =Pinfin + 120575120573

1 + 120573

As =Ainfin

1 + 120573

(96)



1 + 120573

(97)


119879 + O + (1 minus 119876) A (98)


As =1

1 minus 119876


1 + 120573 + 119860 sA [120573 (minus119891s)] (99)




119879 = 119879s + (119879f minus 119879s) (120585

120585f) (100)



+ (1 minus 119876)(As1 minus 120585f120585f

minus

Af

120585f)

(101)


As =Ainfin



Frozen mode

120573 asymp (

119870119860 sO

1 + 119870119860 sO)(

2119882C119882O

119884Oinfin) + (

119870119860 sP

1 + 119870119860 sP)

times (

119882C119882P

119884Pinfin) + (

119870119860 sA

1 + 119870119860 sA)(

119882C119882A

119884Ainfin)

(103)

Flame-detached mode

120573 asymp

1

2

119870119860 sP

1 + 119870119860 sP(

2119882C119882O

119884Oinfin +

119882C119882P

119884Pinfin)

+

119870119860 sA

1 + 119870119860 sA(

119882C119882O

119884Oinfin +

119882C119882A

119884Ainfin)

+

1

2

[

119870119860 sP

1 + 119870119860 sP(

2119882C119882O

119884Oinfin +

119882C119882P

119884Pinfin)

minus

119870119860 sA

1 + 119870119860 sA(

119882C119882O

119884Oinfin +

119882C119882A

119884Ainfin)

2

+ 4

119870119860 sP

1 + 119870119860 sP(

119882C119882O

119884Oinfin +

119882C119882P

119884Pinfin)

times

119870119860 sA

1 + 119870119860 sA(

119882C119882A

119884Ainfin)]

12

(104)

Flame-attached mode

120573 asymp (

1

1 + 2119870119860 sO minus 119870119860 sP)

times (119870119860 sO2119882C119882O

119884Oinfin + 119870119860 sP119882C119882P

119884Pinfin

+119870119860 sA119882C119882A

119884Ainfin)

(105)






2and C-CO

2reactions and




2reaction to







2O mass-



2-O2reaction into


2Omass



Nomenclature










Greek Symbols








Subscripts



2surface reaction


2surface reaction


Superscripts




References









































































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










Flame-detached mode

120573 asymp (

119870119860 sP

1 + 119870119860 sP)(

2119882C119882O

119884Oinfin +

119882C119882P

119884Pinfin) (79)

Flame-attached mode

120573 asymp (

119870119860 sO

1 + 2119870119860 sO minus 119870119860 sP)(

2119882C119882O

119884Oinfin)

+ (

119870119860 sP

1 + 2119870119860 sO minus 119870119860 sP)(

119882C119882P

119884Pinfin)

(80)


2and CO

2con-


119870119860 s119894 = 119896s119894119870


(81)


119896s119894 equiv 119861s119894 (119879infin

119879s) exp(minus

119879119886s119894

119879s) (119894 = OP) (82)



asymp 120588infin


119870

ln (1 + 120573) (83)


119870

[2119895119886 (120583infin120588infin)]12

(84)


asymp

1


120588infin(

2119882C119882O

119884Oinfin)

+

1


120588infin(

119882C119882P

119884Pinfin)

(85)


=

1

1119896s + 1ℎ119863(120588119884O)infin (86)



[2119895119886(120583infin120588infin)]



119863is of use in



ℎ119863= (

119879119877

119879infin





ℎ119863= (

119879119877

119879infin

)

119862mtSc06


fromNu119909

radicRe119909

= 119862mtSc04

(88)


Nu119909=

ℎ119863119909

119863119877

Re119909=

120588119877119880119909

120583119877

119880 = 119886119909 Sc =120583120588

119863

(89)






119870 = radic

2

3

(

119879infin

119879s)(1 minus

119879infin

2119879s) + radic

120587

2

(90)



0

001

002

003

004

Frozen

1000 1500 2000 2500

Flame-detached

Flame-attached

Com

busti

on ra

te119898

(kg

m2s) 119884O = 07

119884A = 00003

119886 = 170 sminus1





2concentrations



119870 = radic

2

3

(

119879infin

119879s)(1 minus

119879infin

2119879s)

minus 005 (1 +

4119882C119882O


120587

2

(91)


2mass-fraction



119870 = (

119879infin

119879s)(1 minus

119879infin

2119879s) + radic

120587

2

(92)


119870 = (

119879infin

119879s)(1 minus

119879infin

2119879s)

minus 005 (1 +

4119882C119882O


120587

2

(93)

0

0005

001

0015

1000 1500 2000 2500



Com

busti

on ra

te119898

(kg

m2s)

119886 = 300 sminus1

119886 = 180 sminus1

119886 = 60 sminus1


(a)

0

002

004

006

008

1000 1500 2000 2500


Com

busti

on ra

te119898

(kg

m2s) 119886 = 1000 sminus1

119886 = 320 sminus1

119886 = 88 sminus1


119884Oinfin = 07

119884Ainfin = 00002

(b)




















0

001

002

1000 1500 2000 2500



Matsui 1983

Com

busti

on ra

te119898

(kg

m2s)

119886 = 300 sminus1

119884Ainfin = 0001


119884Ainfin = 0004







2Oconcentrations






2O mass-fraction



0

0005

001

0015

1000 1500 2000 2500



ombu

stion

rate119898

(kg

m2s)

119886 = 100 sminus1

119884Oinfin = 02

119884Ainfin = 014


119879si

g=1094

K







6ms and119864sO119877=22 times 10



4 K (119864sP =


8ms and119864sA119877 = 33 times 10










2O



2O mass fraction is



2O and












119894s =

119894infin

1 + 120573 + 119860 s119894 [120573 (minus119891s)](119894 = OPA) (95)





1 + 120573

Ps =Pinfin + 120575120573

1 + 120573

As =Ainfin

1 + 120573

(96)



1 + 120573

(97)


119879 + O + (1 minus 119876) A (98)


As =1

1 minus 119876


1 + 120573 + 119860 sA [120573 (minus119891s)] (99)




119879 = 119879s + (119879f minus 119879s) (120585

120585f) (100)



+ (1 minus 119876)(As1 minus 120585f120585f

minus

Af

120585f)

(101)


As =Ainfin



Frozen mode

120573 asymp (

119870119860 sO

1 + 119870119860 sO)(

2119882C119882O

119884Oinfin) + (

119870119860 sP

1 + 119870119860 sP)

times (

119882C119882P

119884Pinfin) + (

119870119860 sA

1 + 119870119860 sA)(

119882C119882A

119884Ainfin)

(103)

Flame-detached mode

120573 asymp

1

2

119870119860 sP

1 + 119870119860 sP(

2119882C119882O

119884Oinfin +

119882C119882P

119884Pinfin)

+

119870119860 sA

1 + 119870119860 sA(

119882C119882O

119884Oinfin +

119882C119882A

119884Ainfin)

+

1

2

[

119870119860 sP

1 + 119870119860 sP(

2119882C119882O

119884Oinfin +

119882C119882P

119884Pinfin)

minus

119870119860 sA

1 + 119870119860 sA(

119882C119882O

119884Oinfin +

119882C119882A

119884Ainfin)

2

+ 4

119870119860 sP

1 + 119870119860 sP(

119882C119882O

119884Oinfin +

119882C119882P

119884Pinfin)

times

119870119860 sA

1 + 119870119860 sA(

119882C119882A

119884Ainfin)]

12

(104)

Flame-attached mode

120573 asymp (

1

1 + 2119870119860 sO minus 119870119860 sP)

times (119870119860 sO2119882C119882O

119884Oinfin + 119870119860 sP119882C119882P

119884Pinfin

+119870119860 sA119882C119882A

119884Ainfin)

(105)






2and C-CO

2reactions and




2reaction to







2O mass-



2-O2reaction into


2Omass



Nomenclature










Greek Symbols








Subscripts



2surface reaction


2surface reaction


Superscripts




References









































































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










0

001

002

003

004

Frozen

1000 1500 2000 2500

Flame-detached

Flame-attached

Com

busti

on ra

te119898

(kg

m2s) 119884O = 07

119884A = 00003

119886 = 170 sminus1





2concentrations



119870 = radic

2

3

(

119879infin

119879s)(1 minus

119879infin

2119879s)

minus 005 (1 +

4119882C119882O


120587

2

(91)


2mass-fraction



119870 = (

119879infin

119879s)(1 minus

119879infin

2119879s) + radic

120587

2

(92)


119870 = (

119879infin

119879s)(1 minus

119879infin

2119879s)

minus 005 (1 +

4119882C119882O


120587

2

(93)

0

0005

001

0015

1000 1500 2000 2500



Com

busti

on ra

te119898

(kg

m2s)

119886 = 300 sminus1

119886 = 180 sminus1

119886 = 60 sminus1


(a)

0

002

004

006

008

1000 1500 2000 2500


Com

busti

on ra

te119898

(kg

m2s) 119886 = 1000 sminus1

119886 = 320 sminus1

119886 = 88 sminus1


119884Oinfin = 07

119884Ainfin = 00002

(b)




















0

001

002

1000 1500 2000 2500



Matsui 1983

Com

busti

on ra

te119898

(kg

m2s)

119886 = 300 sminus1

119884Ainfin = 0001


119884Ainfin = 0004







2Oconcentrations






2O mass-fraction



0

0005

001

0015

1000 1500 2000 2500



ombu

stion

rate119898

(kg

m2s)

119886 = 100 sminus1

119884Oinfin = 02

119884Ainfin = 014


119879si

g=1094

K







6ms and119864sO119877=22 times 10



4 K (119864sP =


8ms and119864sA119877 = 33 times 10










2O



2O mass fraction is



2O and












119894s =

119894infin

1 + 120573 + 119860 s119894 [120573 (minus119891s)](119894 = OPA) (95)





1 + 120573

Ps =Pinfin + 120575120573

1 + 120573

As =Ainfin

1 + 120573

(96)



1 + 120573

(97)


119879 + O + (1 minus 119876) A (98)


As =1

1 minus 119876


1 + 120573 + 119860 sA [120573 (minus119891s)] (99)




119879 = 119879s + (119879f minus 119879s) (120585

120585f) (100)



+ (1 minus 119876)(As1 minus 120585f120585f

minus

Af

120585f)

(101)


As =Ainfin



Frozen mode

120573 asymp (

119870119860 sO

1 + 119870119860 sO)(

2119882C119882O

119884Oinfin) + (

119870119860 sP

1 + 119870119860 sP)

times (

119882C119882P

119884Pinfin) + (

119870119860 sA

1 + 119870119860 sA)(

119882C119882A

119884Ainfin)

(103)

Flame-detached mode

120573 asymp

1

2

119870119860 sP

1 + 119870119860 sP(

2119882C119882O

119884Oinfin +

119882C119882P

119884Pinfin)

+

119870119860 sA

1 + 119870119860 sA(

119882C119882O

119884Oinfin +

119882C119882A

119884Ainfin)

+

1

2

[

119870119860 sP

1 + 119870119860 sP(

2119882C119882O

119884Oinfin +

119882C119882P

119884Pinfin)

minus

119870119860 sA

1 + 119870119860 sA(

119882C119882O

119884Oinfin +

119882C119882A

119884Ainfin)

2

+ 4

119870119860 sP

1 + 119870119860 sP(

119882C119882O

119884Oinfin +

119882C119882P

119884Pinfin)

times

119870119860 sA

1 + 119870119860 sA(

119882C119882A

119884Ainfin)]

12

(104)

Flame-attached mode

120573 asymp (

1

1 + 2119870119860 sO minus 119870119860 sP)

times (119870119860 sO2119882C119882O

119884Oinfin + 119870119860 sP119882C119882P

119884Pinfin

+119870119860 sA119882C119882A

119884Ainfin)

(105)






2and C-CO

2reactions and




2reaction to







2O mass-



2-O2reaction into


2Omass



Nomenclature










Greek Symbols








Subscripts



2surface reaction


2surface reaction


Superscripts




References









































































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation




















0

001

002

1000 1500 2000 2500



Matsui 1983

Com

busti

on ra

te119898

(kg

m2s)

119886 = 300 sminus1

119884Ainfin = 0001


119884Ainfin = 0004







2Oconcentrations






2O mass-fraction



0

0005

001

0015

1000 1500 2000 2500



ombu

stion

rate119898

(kg

m2s)

119886 = 100 sminus1

119884Oinfin = 02

119884Ainfin = 014


119879si

g=1094

K







6ms and119864sO119877=22 times 10



4 K (119864sP =


8ms and119864sA119877 = 33 times 10










2O



2O mass fraction is



2O and












119894s =

119894infin

1 + 120573 + 119860 s119894 [120573 (minus119891s)](119894 = OPA) (95)





1 + 120573

Ps =Pinfin + 120575120573

1 + 120573

As =Ainfin

1 + 120573

(96)



1 + 120573

(97)


119879 + O + (1 minus 119876) A (98)


As =1

1 minus 119876


1 + 120573 + 119860 sA [120573 (minus119891s)] (99)




119879 = 119879s + (119879f minus 119879s) (120585

120585f) (100)



+ (1 minus 119876)(As1 minus 120585f120585f

minus

Af

120585f)

(101)


As =Ainfin



Frozen mode

120573 asymp (

119870119860 sO

1 + 119870119860 sO)(

2119882C119882O

119884Oinfin) + (

119870119860 sP

1 + 119870119860 sP)

times (

119882C119882P

119884Pinfin) + (

119870119860 sA

1 + 119870119860 sA)(

119882C119882A

119884Ainfin)

(103)

Flame-detached mode

120573 asymp

1

2

119870119860 sP

1 + 119870119860 sP(

2119882C119882O

119884Oinfin +

119882C119882P

119884Pinfin)

+

119870119860 sA

1 + 119870119860 sA(

119882C119882O

119884Oinfin +

119882C119882A

119884Ainfin)

+

1

2

[

119870119860 sP

1 + 119870119860 sP(

2119882C119882O

119884Oinfin +

119882C119882P

119884Pinfin)

minus

119870119860 sA

1 + 119870119860 sA(

119882C119882O

119884Oinfin +

119882C119882A

119884Ainfin)

2

+ 4

119870119860 sP

1 + 119870119860 sP(

119882C119882O

119884Oinfin +

119882C119882P

119884Pinfin)

times

119870119860 sA

1 + 119870119860 sA(

119882C119882A

119884Ainfin)]

12

(104)

Flame-attached mode

120573 asymp (

1

1 + 2119870119860 sO minus 119870119860 sP)

times (119870119860 sO2119882C119882O

119884Oinfin + 119870119860 sP119882C119882P

119884Pinfin

+119870119860 sA119882C119882A

119884Ainfin)

(105)






2and C-CO

2reactions and




2reaction to







2O mass-



2-O2reaction into


2Omass



Nomenclature










Greek Symbols








Subscripts



2surface reaction


2surface reaction


Superscripts




References









































































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










0

0005

001

0015

1000 1500 2000 2500



ombu

stion

rate119898

(kg

m2s)

119886 = 100 sminus1

119884Oinfin = 02

119884Ainfin = 014


119879si

g=1094

K







6ms and119864sO119877=22 times 10



4 K (119864sP =


8ms and119864sA119877 = 33 times 10










2O



2O mass fraction is



2O and












119894s =

119894infin

1 + 120573 + 119860 s119894 [120573 (minus119891s)](119894 = OPA) (95)





1 + 120573

Ps =Pinfin + 120575120573

1 + 120573

As =Ainfin

1 + 120573

(96)



1 + 120573

(97)


119879 + O + (1 minus 119876) A (98)


As =1

1 minus 119876


1 + 120573 + 119860 sA [120573 (minus119891s)] (99)




119879 = 119879s + (119879f minus 119879s) (120585

120585f) (100)



+ (1 minus 119876)(As1 minus 120585f120585f

minus

Af

120585f)

(101)


As =Ainfin



Frozen mode

120573 asymp (

119870119860 sO

1 + 119870119860 sO)(

2119882C119882O

119884Oinfin) + (

119870119860 sP

1 + 119870119860 sP)

times (

119882C119882P

119884Pinfin) + (

119870119860 sA

1 + 119870119860 sA)(

119882C119882A

119884Ainfin)

(103)

Flame-detached mode

120573 asymp

1

2

119870119860 sP

1 + 119870119860 sP(

2119882C119882O

119884Oinfin +

119882C119882P

119884Pinfin)

+

119870119860 sA

1 + 119870119860 sA(

119882C119882O

119884Oinfin +

119882C119882A

119884Ainfin)

+

1

2

[

119870119860 sP

1 + 119870119860 sP(

2119882C119882O

119884Oinfin +

119882C119882P

119884Pinfin)

minus

119870119860 sA

1 + 119870119860 sA(

119882C119882O

119884Oinfin +

119882C119882A

119884Ainfin)

2

+ 4

119870119860 sP

1 + 119870119860 sP(

119882C119882O

119884Oinfin +

119882C119882P

119884Pinfin)

times

119870119860 sA

1 + 119870119860 sA(

119882C119882A

119884Ainfin)]

12

(104)

Flame-attached mode

120573 asymp (

1

1 + 2119870119860 sO minus 119870119860 sP)

times (119870119860 sO2119882C119882O

119884Oinfin + 119870119860 sP119882C119882P

119884Pinfin

+119870119860 sA119882C119882A

119884Ainfin)

(105)






2and C-CO

2reactions and




2reaction to







2O mass-



2-O2reaction into


2Omass



Nomenclature










Greek Symbols








Subscripts



2surface reaction


2surface reaction


Superscripts




References









































































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation













1 + 120573

Ps =Pinfin + 120575120573

1 + 120573

As =Ainfin

1 + 120573

(96)



1 + 120573

(97)


119879 + O + (1 minus 119876) A (98)


As =1

1 minus 119876


1 + 120573 + 119860 sA [120573 (minus119891s)] (99)




119879 = 119879s + (119879f minus 119879s) (120585

120585f) (100)



+ (1 minus 119876)(As1 minus 120585f120585f

minus

Af

120585f)

(101)


As =Ainfin



Frozen mode

120573 asymp (

119870119860 sO

1 + 119870119860 sO)(

2119882C119882O

119884Oinfin) + (

119870119860 sP

1 + 119870119860 sP)

times (

119882C119882P

119884Pinfin) + (

119870119860 sA

1 + 119870119860 sA)(

119882C119882A

119884Ainfin)

(103)

Flame-detached mode

120573 asymp

1

2

119870119860 sP

1 + 119870119860 sP(

2119882C119882O

119884Oinfin +

119882C119882P

119884Pinfin)

+

119870119860 sA

1 + 119870119860 sA(

119882C119882O

119884Oinfin +

119882C119882A

119884Ainfin)

+

1

2

[

119870119860 sP

1 + 119870119860 sP(

2119882C119882O

119884Oinfin +

119882C119882P

119884Pinfin)

minus

119870119860 sA

1 + 119870119860 sA(

119882C119882O

119884Oinfin +

119882C119882A

119884Ainfin)

2

+ 4

119870119860 sP

1 + 119870119860 sP(

119882C119882O

119884Oinfin +

119882C119882P

119884Pinfin)

times

119870119860 sA

1 + 119870119860 sA(

119882C119882A

119884Ainfin)]

12

(104)

Flame-attached mode

120573 asymp (

1

1 + 2119870119860 sO minus 119870119860 sP)

times (119870119860 sO2119882C119882O

119884Oinfin + 119870119860 sP119882C119882P

119884Pinfin

+119870119860 sA119882C119882A

119884Ainfin)

(105)






2and C-CO

2reactions and




2reaction to







2O mass-



2-O2reaction into


2Omass



Nomenclature










Greek Symbols








Subscripts



2surface reaction


2surface reaction


Superscripts




References









































































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











2and C-CO

2reactions and




2reaction to







2O mass-



2-O2reaction into


2Omass



Nomenclature










Greek Symbols








Subscripts



2surface reaction


2surface reaction


Superscripts




References









































































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation














Subscripts



2surface reaction


2surface reaction


Superscripts




References









































































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation




























































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation









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