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TECHNICAL NOTE NASA TN D-4747

A SIMPLIFIED EQUILIBRIUM

HYDROCARBON-AIR COMBUSTION

GAS MODEL FOR USE IN AIR-BREATHING

ENGINE CYCLE COMPUTER PROGRAMS

by Vincent R. 3Iascitti

Langley Research Center

Langle 5 Station, Hampton, Va.

NATIONALAERONAUTICSAND SPACEADMINISTRATION• WASHINGTON,D. C. • SEPTEMBER1968

https://ntrs.nasa.gov/search.jsp?R=19680025643 2020-05-12T07:23:30+00:00Z

NASA TN D-4747

A SIMPLIFIED EQUILIBRIUM HYDROCARBON-AIR

COMBUSTION GAS MODEL FOR USE IN AIR-BREATHING


By Vincent R. Mascitti


Langley Station, Hampton, Va.

NATIONAL AERONAUTICS AND SPACE ADMINISTRATION

For sale by the Clearinghouse for Federal Scientific and Technical Information

Springfield, Virginia 22151 - CFSTI price $3.00

A SIMPLIFIED EQUILIBRIUM HYDROCARBON-AIR

COMBUSTION GAS MODEL FOR USE IN AIR-BREATHING


By Vincent R. Mascitti


SUMMARY

This paper presents a simplified hydrocarbon-air combustion gas model, including

the effects of dissociation, for convenient use in engine cycle computer programs. The

generalized model reduces to the hydrogen-air system as well as to the dissociating-air

system. The exclusion of chemical species containing atomic nitrogen allows a consider-

able simplification of the composition equations. The thermodynamic properties of stoi-

chiometric combustion of the kerosene-air and hydrogen-air systems are computed with

the simplified model and compared with those of more comprehensive gas models. In

addition, the effect of the neglected chemical species on the performance of an idealized

subsonic combustion ramjet is presented. The simplified gas model has been used to

define the limiting conditions for solid carbon and ammonia formation for fuel-rich gas

mixtures. A computer program listing of the calculation procedure for the simplified

gas model is presented.

INTRODUCTION

The current interest in supersonic and hypersonic vehicles has emphasized the need

for an accurate yet convenient method of calculation of the thermodynamic properties of

combustion gas mixtures. At high temperatures considerable iteration difficulties arise

in calculating the chemical composition and thermodynamic properties of combustion gas

mixtures, difficulties which stem from the phenomenon of internal energy excitation of

the species coupled with chemical dissociation.

In the past, methods such as the method presented in reference 1 have been used to

calculate the thermodynamic properties of combustion gases for air-breathing engine

cycle computations. The method of reference 1 is based on separate calculations of ther-

modynamic properties of air and the products of combustion of a stoichiometric fuel-air

mixture. It is assumed that the properties of the combustion products for any fuel-air

ratio that is less than stoichiometric may be obtained by linear interpolation between the

two extreme cases. Results of this methodare exact for nodissociation. Application ofthis methodto conditions in which dissociation occurs will only give approximate results.The deviation from anexact calculation is small for turbojet or turbofan calculationssince temperatures are relatively low anddissociation is not extensive. However, thehigh operating temperatures of hypersonic enginecycles may result in extreme dissocia-tion and causelarge errors in calculated engineperformance.

The methodof reference 1does not allow engine cycle calculations for equivalentratios aboveunity. However,at high Machnumber flight, the fuel required to cool aero-dynamic surfaces and enginecomponentsmayforce engineoperation into this region.

In short, the methodof reference 1does not provide a sufficiently general basisfor engine cycle calculations under all conditions of current andfuture interest.

Chemicalequilibrium theory which governsdissociation phenomenonhas beenfor-mulatedfor manyyears (for example, ref. 2). However, its application to complex com-bustion gasmixtures has proved so tedious and complicated as to preclude direct use inenginecycle computer programs. The combustiongas modelsof references 3 and4,which serve as a basis of comparison for the simplified gas modelpresentedherein, areelaborate treatments of the hydrogen-air andkerosene-air systems.

The generalized hydrogen-air modelof reference 3 assumes 12chemical speciesrequiring eight independentchemical reactions leadingto eight equilibrium expressions.Sincethe initial proportions of hydrogenandair define four mass balanceequations,themathematical solution (12equationsand 12unknowns)is demonstrated. The reduction ofthe system of equationsto one equationand oneunknown,as recommendedin reference 5,is extremely tedious. With the exclusion of the techniqueof reference 5, solution of thissystem by a single-level iteration is eliminated. The adoptionof a two-level iteration oroneof the methodssummarized in reference 6 to solve for gas composition hasa greateffect on the utility of the computationprocedure as a subroutine for anengine cycle com-puter program. Consequently,the most efficient meansof representing the thermody-namic properties of thesegas models in an enginecycle program is by an elaborate sys-tem of fitted curves which sacrifices both computer storage andaccuracy.

The purposeof the simplified gasmodel presentedherein is to incorporate the sub-stantial effects of dissociation consistentwith convenientuse in computerizedengine cyclecalculations. The proposedgasmodel is simplified by neglecting the formation of speciescontaining atomic nitrogen. This assumptionenablesthe solution for chemical composi-tion to be obtainedwith a single-level iteration. Althoughthe original intent of this studywasdirected toward the hydrogen-air system, it was foundthat the model could be gen-eralized to anyhydrocarbon-air system, as well as to a dissociating-air system, withlittle additional complexity. The computer program, presented in the appendix,canbereadily incorporated as a subroutine in an enginecycle program or usedalone to generate

2

Mollier diagrams. Inputs to the program are the ratio of carbon atoms to hydrogenatoms in the fuel molecule, equivalenceratio, temperature, andpressure. Outputsfromthe program are the mole fractions of the chemical species assumed,enthalpy, entropy,and molecular weight.

SYMBOLS

A,B

D, D1,D2__D3,D4 _)

Fnet

coefficients of iteration functions

initial atomic proportions

net thrust, lbf (N)

fuel-air ratio

fs

G

H

stoichiometric fuel-air ratio

acceleration due to gravitational field of earth, 32.174 ft/sec 2

Gibbs free energy, Btu/lbm (J/kg)

enthalpy, Btu/lbm (J/kg)

(9.807 m/sec2)

Isp

J

specific impulse, sec

mechanical equivalent of heat, 778.20 ft-lbf/Btu (9.991 N-m/J)

K equilibrium constant in terms of partial pressures; numerical subscripts

indicate individual reaction

M mean molecular weight

n ratio of hydrogen atoms to carbon atoms in fuel molecule

P

q

pressure, atm

dynamic pressure, lbf/ft 2 (N/m2)

R

rch

req

S

universal gas constant, 1.98588Btu/mole-°R (8.314J/mole-°K)

ratio of carbonatoms to hydrogenatoms in fuel molecule, 1/n

equivalenceratio, f/fs

entropy, Btu/lbm-°R (J/kg-°K)

T temperature, OR (OK)

V velocity, ft/sec

wA air flow rate, lbm/sec

X mole fraction

Subscripts:

oo free-stream conditions

J Jth species

j jet

lim

(m/sec)

(kg/sec)

limiting conditionfor solid carbon or ammonia formation

s static conditions

stagnation conditions

Superscript:

O at reference pressure of 1 atmosphere (1.01325 x 105 N/m 2)

DESCRIPTION OF GAS MODEL

The temperatures of interestfor air-breathing engine applicationare assumed to be

below 7000 ° R (3889 ° K) and the pressures are assumed to be between 0.001 and 100 atm.

For these conditions, references 3 and 4 indicate that the formation of nitrogen species,

4

suchas N, NH, NH3,and NO,occurs in negligible amountsand,therefore, has a verysmall effect on the thermodynamic properties of combustiongas mixtures. The assump-tion to neglect these speciesgreatly reduces the complexity of the calculation proceduresince molecular nitrogen canbe consideredinert. The results of this assumptionon thethermodynamic properties of gas mixtures andon the performance of an idealized ram-jet are discussed in a subsequentsection.

Figure 1is a diagram illustrating the chemical species and required chemicalreactions considered in this gasmodel. The figure is divided into regions of tempera-ture and equivalenceratio. Above Tcut_off, the gas is consideredto bedissociating.(Tcut_off is an arbitrarily definedlimit belowwhich dissociation canbe considerednegligible.) The chemical species assumedfor pure air (req = 0)are identical to those

of reference 3 and include atomic nitrogen species. For the combustion gas model

(req > 0), the dissociated nitrogen species (N, NH, NH3, NO)are not included. It is appar-

ent that by excluding the species and reaction containing carbon, the model reduces to

the hydrogen-air system.

Below Tcut_off , there is no dissociation and combustion is complete. Therefore,

the initial proportions of elements are sufficient to define the gas composition. However,

for req > 1, the gas model of reference 4 indicates that CO may form even below

Tcut_of f. As a result, the initial proportions of elements are not sufficient to define the

gas composition and one chemical reaction is required.

In this region of fuel-rich operation, solid carbon and ammonia may form. The

limiting pressure for the formation of solid carbon depends upon the relative proportions

of carbon dioxide CO 2 and carbon monoxide CO, equivalence ratio, and temperature.

However, there is a limiting equivalence ratio, for a given hydrocarbon-air system, above

which solid carbon can form under any condition.

The formation of ammonia was considered only for the hydrogen-air system since

the formation of sotid carbon precluded the formation of ammonia for the combustion

products of all the hydrocarbon-air systems studied. The limiting pressure for the for-

mation of a given amount of ammonia depends upon the relative proportion of nitrogen N 2

and hydrogen H2, equivalence ratio, and temperature.

DERIVATION OF GAS MODEL

Overall Stoichiometry and Gas Model Constants

In order to apply the simplified gas model to the combustion of any hydrocarbon

fuel, a generalized statement concerning the atomic composition of the fuel is required.

5

Consider the following stoichiometric reactions:

CH+_O2= CO2+21H20

CH2 + 6 02

CH3 + 7 02

In general terms,

where in a fuel molecule

Thus, in the general system,

= CO 2 + _ H20

= CO 2 + 3 H2 O

n + 4 n H20CH n+_O2= CO 2 +

No. atoms Hn=

No. atoms C

No. moles CHn_ 4recl_

No. moles 02 n + 4

where req is the equivalence ratio.

The initial composition of air incorporated in this gas model is the same as that of

reference 3; that is, combustion is assumed to have occurred with dry air of the following

composition by volume:

O 2 20.9495 percent

N 2 78.0881 percent

Ar 0.9624 percent

The following constants and parameters define, in general terms, the initial atomic pro-

portions of the elements in the gas mixture:

D1 = No. atoms Ar = 0.00616227No. atoms N

D2 = No. atoms N = 3.727445No. atoms O

D3 = No. atoms C _ 2reqNo. atoms O n + 4

D4 = No. atoms H_ 2nreqNo. atoms O n + 4

and a convenient parameter

D=I+D2(I+2D1)

1. thenDefine rch = _,

D 3 - 2reqrch1 + 4rch

2req

D4 - 1 + 4rch

For the special system of hydrogen-air combustion,

2H 2 + O 2 = 2H20

No. moles H2

No. moles O 2 2req

rch = 0 D 3 = 0 D4 = 2req

The stoichiometric fuel-air ratio is given by

fl 008 +

fs=0"028931 l" 1 +4rchl2"01rchl/

It is of value to represent the equilibrium constants for the assumed reactions in a

convenient manner. The treatment of chemical equilibria in reference 2 leads to the fol-

lowing equilibrium expression for a general reaction:

aA +bB = cC +dD

The equilibrium expression is

where

{ico o oGC GD GA - bK = exp _-_ + d _ - a _ RTJ/

Figure 2 shows the variation of lOgl0K with 1/T for the chemical reactions consid-o

Gj for the species were takenered in this gas model. The values of Gibbs free energy

from the data tabulations in references 3 and 4. Figure 2 indicates that the equilibrium

constants can be represented as linear functions.

With the initial proportions of the gas mixture elements defined and a convenient

representation of the equilibrium constants determined, the governing relations for chem-

ical composition can be formulated. Referring to figure 1 will aid the reader in the fol-

lowing derivations.

Hydrocarbon-Air Gas Model With Dissociation

The following gas model applies to the region T > Tcut_off,

Chemical species assumed

H20 , CO2, 02, H2, N2, Ar, O, H, CO, OH

Chemical reactions

2H + O = H20

2H = H 2

20 = O 2

O + CO = CO 2

H+O=OH

Mass conservation equations

10

(1) _ Xj=l

J=l

(2) XA = DI(2XN2 )

(3) 2XN2 = D2(2Xco 2 + XCO + XH20 + 2Xo2 + X O + XOH)

XCO 2 + XCO = D3(2Xco 2 + XCO + XH20 + 2Xo2 + X O + XOH )

X H + 2XH2 + 2XH2 O + XOH = D4(2Xco 2 + XCO + XH2 O + 2Xo2

(4)

req > 0:

(5) ÷ Xo ÷XoH)

Equilibrium expressions

(6) XH2 O = p2KlXoX2 H

(7) XH2 = PK2 x2

(8) Xo2 = PK3 x2

(9) XCO 2 = PK4XoXco

(10) XOH = PK5XoXH

Solving for two equations and two unknowns in X O and

o_= (A22X2 +A21Xo + A20)X2 + (A12X_) + A11Xo +A10)XH

/3= (B224 + B21Xo + B20)X2 +

where

A22=p3K1K4(1- 2D3 +D)

A_._: _(_ ÷D)+2K_._(_- _._31

A12 = p2K4K5(I - 2D 3 + D)

All=P_K5(I+D)+2K4(1-2D31

A01 =

A00 =

Setting

system.

A10=2(1-D3)

A03 = 2Dp2K3K4

1 + D- 2pK4(1 - 2D3)

-2(1-D31

D 3 and

X H gives

+ (A03X30 +A02_ + A01Xo +A00) = 0

(B12X2 + B1 lXo + BI0)XH - (B034 + B02X2 + BoIXO) = 0

B22 = p3K1K4_C1- 2D31-D41

B21 = p2_1_(1- D3)-D4_ + 2K2K4(1- 2D3)_

B2o--2pK2(1-D_

B12 = p2K4K5(1 - 2D 3 - D4)

BI, = PIK5(1-D3- D4)+ K4(I-2D3)_

B10 = 1 - D 3

B03 = 2D4p2K3K 4

B02 = D4P(2K3 + K4)

B01 = D4

K4 equal to zero reduces the system to identically the hydrogen-air

9

The expressionsforiteration scheme. Newton'siteration schemefor determining XOwritten as

and fl are satisfied simultaneously by use of the Newton

and X H can be

Xo(L + 1) = Xo(L) -

where Xo(L) is the Lth approximation of X 0 and the derivative of a(Xo, XH) with

respect to X 0 is

With Xo(L + 1) determined,

theorem to the expression for

do_ _ oo_ 8ot /_\81oAO I

dX 0 OX0 aXH_ 0/3 /

\0XH/

XH(L + 1) can be calculated by applying the binomial

/3. Although there may be a number of pairs of values

Xo, XH) which satisfy c_ = 0, there is only one pair which satisfies the condition

0 < Xj =<1. The initial value of Xo(L ) is taken to be 0 and the iteration scheme is

used until X O is determined to five significant figures.

For req > 1 the iteration function a is insensitive to changes in X O. As a

result, X H is used as the iteration variable in this region of computation. Unfortu-

nately, if X H is assumed, the expression for fl is cubic in X O and an additional

iteration is required. For fuel-rich hydrogen-air mixtures, B03 = 0, /3 is quadratic in

XO, and the single-level iteration is maintained. However, for fuel-rich mixtures con-

taining carbon, a two-level iteration is necessary for solution.

Air Gas Model With Dissociation

The following gas model applies to the region T > Tcut_off,


O2, N2, Ar, O, NO, N

Chemical reactions

20 = O 2

2N = N 2

req = O:

N +O= NO

10


6

(1) _ Xj = 1J=l

(2) X A = DI(2XN2 + XNO + XN)

(3/ 2x_2+X_o+xN--_2(2x02+Xo+XNo)

Equilibrium expressions

(4) XO2 = PK3X2

(5) XN2 = PK6 x2

(6) XNO = pKTXoXN

Solving for two equations and two unknowns in

a= A20X2 + (AllX O + A10)X N

_ _ox_+(_o +_o)x_-(_o_x_where

A20 = (1 + 2D1)PK 6

All = (1 + D1)PK 7

A10 = 1 + D 1

A02 = pK 3

A01 = 1

A00 = -1

X O and X N yields

+ (A02X _ + A01X O + A00 ) =0

+ B 0 lXo) = 0

The simultaneous solution of a and

for the hydrocarbon-air gas model.

B20 = 2pK 6

Bll = (1- D2)PK 7

B10 = 1

B02 = 2D2PK 3

B01 = D2

/3 is subject to the same iterationprocedure used

II

Fuel-Rich Gas Model With Solid Carbon Formation

The following gas model applies to the region T < Tcut_off, req > 1:


H20, CO2, H2, N2, At, CO

Chemical reaction

H 2 +CO 2= CO+H20


6

(1) _, Xj = 1

J=l

(2) X A = DI(2XN2 )

(3) 2XN2 = D2(2Xco 2 + XCO + XH20)

(4) XCO 2 + XCO = D3(2Xco 2 + XCO + XH2OI

(5) 2XH2 + 2XH2 O = D4(2Xco 2 + XCO + XH20)

Equilibrium expression

(6) XH2oXco = K8XH2Xco2

Solving for XCO yields

AAX20 + BBXco + CC = 0

where

BB=(D+D4+2D3- I)E2(I-2D3) +Ks(D4+6D3-21

CC= 2D3K8(2 - D 4 - 4D3)

With the chemical composition in this region defined, the criteria for solid carbon

formation can be formulated. Consider the reaction

CO 2 + C(Solid) = 2CO

and equilibrium expression

XCO 2

Plim = K9 X 2CO

12

If p > Plim' solid carbonwill form. There is, however, an equivalenceratio for thecombustionproducts of a given hydrocarbon(rch), where solid carbon can form even ina near vacuum. Consider the following equilibrium expression:

Setting XCO2gives

2PlimXCO = K9Xco 2

= 0 in the system for the fuel-rich gas model with solid carbon formation

1 + 4rch

req, lim - 2rch

Nondissociating Gas Model

The following gas model applies to the region T ---Tcut_off:


H20 , CO2, 02, H2, N2, Ar


6

(1) _-" Xj= 1L.._,'

J=l

(2) XAr = DI(2XN2 )

(3) 2xN2=D2(2Xco2÷X,20÷2Xo21

(4) XCO 2 = D3(2Xco 2 + XH20 + 2X02 )

(5) 2XH2 + 2XH20 = D4(2Xco 2 + XH20 + 2Xo2 )

In each of the regions req= 0, 0< req< i, and req= l, one or more of the species

can be eliminated; thus, for req = 0,

XH2 = 0

XCO 2 = 0

XH2 O = 0

Xj = Constants

13

for 0<req< 1,

XH2 = 0

Xj= Xj_rch, req/

andfor req= 1,

XH2 = 0

XO2 = 0

Xj= Xj(reh )

For the hydrogen-air system (rch = 0) with req > 1,

XO2 = 0

XCO 2 = 0

Xj= Xj(req )

With the chemical composition in this region, the condition for the formation of a given

amount of ammonia can be determined. Consider the reaction

N 2 + 3H 2 = 2NH 3

and equilibrium expression

X 2NH 3

= >

P2im K10X3H2XN2 (XNH 3 0.01, p> Plim)

Thermodynamic Properties of Gas Mixtures

The previous derivations have outlined the solution for gas mixture composition.

With the chemical composition defined and a tabulation of the thermodynamic properties

of the pure constituents provided, the thermodynamic properties of a gas mixture can be

calculated by the following equations (ref. 3):

-fi-J

S = R_X_R- in p - In Xj)J

= _ XjMjM

J

14

where _ and _- are the thermodynamicproperties of the pure constituents at refer-encepressure (1 atm). Thesethermodynamic expressions inherently assumethat thepure constituents obey the perfect gas law.

The standardreference state of the elements At, N, O, I-I,Ctaken from reference 3are as follows:

Ar as Ar

N as N2

Oas 02

H as H2

C as C(Solid)

By definition, the energy content of these elements in their standard reference states

(T = 0o R) is zero. Equations for the computation of fuel enthalpies, consistent with this

thermodynamic basis, are presented in reference 4.

A listing of the computer program to calculate the composition and thermodynamic

properties of this gas model is presented and discussed in the appendix. Computational

time for obtaining the chemical composition and thermodynamic properties with this pro-

gram has been estimated at 6000 cases/min on the IBM 7094 data processing system at

the Langley Research Center.

RESULTS AND DISCUSSION

Comparison of Gas Models

The computer program for the simplified combustion gas model has been used to

calculate the thermodynamic properties of the stoichiometric kerosene-air and hydrogen-

air systems. The purpose of this calculation is to compare the results of the simplified

gas model with the more extensive treatments of references 3 and 4. Since the thermo-

dynamic properties of the pure constituents used in the simplified model were taken from

these references, the differences between the results of the simplified model and the ref-

erence models are due to the formation of the neglected chemical species N, NH, NH3,and NO.

Figure 3 is a Mollier diagram for the stoichiometric products of combustion of

kerosene with air (rch = 0.5). The solid-line curves are values plotted from the tabulated

data of reference 4. The dash-line curves are values obtained from the computer pro-

gram for the simplified model. Unfortunately, the data presented in reference 4 are lim-

ited to temperatures below 5000 ° R (2778 ° K). Since the formation of species including

15

atomic nitrogen (N, NH, andNH3)is more important at temperatures above5000° R, thegoodagreementbetweengasmodels is not surprising.

Figure 4 is a Mollier diagram for the stoichiometric products of combustionof

hydrogenwith air .(rch= 0). The solid-line curves are values plotted from the tabulateddata of reference 3. The dash-line curves are values obtained from the computer pro-

gram for the simplified model. The thermodynamic properties tabulated in reference 3

cover temperatures as high as 10 000 ° R (5560 ° K). The substantial disagreement above

7000 ° R (3889 ° K) at low pressures is due to the formation of species containing atomic

nitrogen. An interesting result is that the agreement between lines of constant pressure

is much better than the agreement between lines of constant temperature. This result is

fortunate since, for thermodynamic processes such as isentropic nozzle expansion, tem-

perature is not used directly to calculate performance.

The simultaneous conditions of high temperatures and low pressures, the area of

substantial disagreement between the simplified gas model and the reference gas models,

are beyond the realm of operation of typical hypersonic engine cycles. For example, for

subsonic combustion ramjets, combustion pressures and temperatures are high since the

airstream is brought nearly to stagnation conditions before combustion. For supersonic

combustion ramjets, combustion pressures and temperatures are low since a large por-

tion of the total air enthalpy remains in the form of kinetic energy. Hence, the conditions

of high temperatures and low pressures do not occur simultaneously for engine cycles

currently considered feasible for hypersonic flight.

Ramjet Performance

In applying the simplified gas model to hypersonic engine calculations, it is of inter-

est to determine the effect of the use of the simplified gas model on the calculated per-

formance of an idealized subsonic combustion ramjet.

Since the purpose of this calculation is to show the effect of small differences in

thermodynamic properties on ramjet performance, the following simplifying assumptions

were made:

(1) Free-stream Mach number chosen along a constant dynamic-pressure path

(q = 1500 lb/ft2 (718 200 N/m2))

(2) Airstream decelerated to stagnation conditions with total pressure recovery

degraded to 10 atm (due to internal duct pressure limitations at high Mach numbers)

(3) Completely mixed stoichiometric hydrogen-air combustion

(4) No pressure losses during combustion ,(Pt, nozzle= 10 atm)

16

(5) Enthalpyof injected hydrogen equals zero

Molecular hydrogen is a reference element; thus, I_H2 = 0 at T = 0 ° R.

(6) Combustion gas isentropically expanded to free-stream static conditions by

using so-called "shifting equilibrium"

The 1962 Standard Atmosphere (ref. 7) yields free-stream static conditions and with

assumption (1) gives the variation of altitude with Mach number shown in figure 5. For

steady adiabatic flow, the total energy in the airstream is given by

v5Ht, air= Hoo, air +_-_

For an adiabatic combustion process,

Ht, products= Ht_ air + fHt_hydrogenl+f

With two properties (Ht, products and Pt, nozzle) of the combustion gas defined, the

Mollier diagram can be entered. Figure 6 is a schematic of a Mollier diagram showing

lines of constant pressure for the simplified gas model and model of reference 3. Since

the simplified gas model neglects the dissociated nitrogen species, the entropy level is

slightly less than that obtained from reference 3. In expanding the combustion gas to

free-stream static pressure, the gas enters the region of almost exact agreement between

gas models. Consequently, the expansion using the simplified gas model gives a slightly

larger value of _H = Ht - H s than that from reference 3. However, the effect of the

difference in AH for the two calculations is reduced by the fact that

vj CHt-Hs

The performance parameters are given by the following expressions:

Fnet= (1 + f)Vj - Voow A

Fnet

Isp = wag f

The variation of specific impulse with Mach number for this idealized ramjet is presented

in figure 7. The solid-line curve was calculated by using the thermodynamic properties

of reference 3. The dash-line curve was calculated by using the computer program for

the simplified gas model. The maximum deviation is 1 percent at a Mach number of 12.

17

Solid CarbonandAmmonia Formation

The formation of solid carbon extracts useful energy from the combustiongas. Ifthis phenomenonoccurs during a nozzle expansionprocess, the useful energy absorbedinforming solid carbon is not recoverable and a loss in performance results.

The derivation of the nondissociating(T < Tcut_off) fuel-rich gasmodel requiredone chemical reaction in order to include the formation of carbon monoxideCO. Theequilibrium expression resulting from the required reaction is independentof pressure.Consequently,the composition of the gas in this region is independentof pressure. Thisfact simplifies the treatment of solid carbonformation, since the limiting pressure, abovewhich solid carbon can form, is related to temperature andcomposition only.

The computer program for the simplified gas model indicates the formation of solidcarbon by anerror statement. Whenthe limiting pressure is exceeded,the error state-ment is printed, but the program computesthe thermodynamic properties as thoughsolidcarbon had not formed. Whenthe limiting equivalenceratio is exceeded,the error state-ment is printed andno calculations are made. The limiting pressure for solid carbonformation is readily extracted from the thermodynamic calculations. Figures 8 to 10present the variation of limiting pressure with equivalenceratio andtemperature for

methane-air (rch = 0.25), kerosene-air (rch = 0.5), and benzene-air (rch = 1) combustion

products, respectively. For all systems, as

req - req, lim, Plim -* 0 since XCO 2 - 0

and as

req- 1, Plim- _ since XCO-0

The formation of ammonia is considered only for the hydrogen-air, nondissociating,

fuel-rich system. The composition of the gas, under these conditions, depend_ upon

equivalence ratio only. Therefore, the limiting pressure, above which the mole fraction

of ammonia is greater than some arbitrary amount, is related to temperature and

composition.

The computer program for the simplified gas model indicates the formation of

ammonia. (XNH 3 > 0.01) by an error statement. When the limiting pressure is exceeded,

the error statement is printed, but the program computes the thermodynamic properties

as though ammonia had not formed. Figure 11 presents the variation of limiting pressure

with equivalence ratio and temperature for hydrogen-air combustion products.

temperatures, as

req "* % Plim "* _ since

and as

For all

XN2 - 0

req-. 1, Plim- _ since XH2- 0

18

CONCLUDINGREMARKS

A simplified equilibrium hydrocarbon-air combustiongas model, for use in enginecycle computer programs, has beenpresented. The generalized gasmodel includes theeffects of dissociation andreduces to the special systems of hydrogen-air combustionproducts, as well as dissociating air. The associatedcomputer program canbe readilyincorporated as a subroutine in a general engine cycle computer program or used aloneto generateMollier diagrams.

With the exceptionof pure air, this gas modelneglects the formation of chemicalspecies containingatomic nitrogen; this assumptionallows a considerable simplificationof the solution for chemical composition. The effect of this assumptionon the thermo-dynamic properties of stoichiometric kerosene-air andhydrogen-air combustionproductsis presented. The importance of this assumptionis shownin terms of the performanceof an idealized subsoniccombustionramjet. Goodagreementbetweenthe simplifiedmodel and more comprehensivetreatments is obtainedin the range of temperaturesapplicable to hypersonic engine cycles.

The computer program hasbeenusedto calculate the limiting pressure for solidcarbon and ammoniaformation in fuel-rich gasmixtures. The results of this calculationare presentedas a function of temperature andequivalenceratio for the combustionprod-ucts of various hydrocarbonfuels.

Langley ResearchCenter,National Aeronautics and SpaceAdministration,

Langley Station, Hampton,Va., May 7, 1968,722-03-00-02-23.

19

APPENDIX

COMPUTER PROGRAM FOR CALCULATION OF GAS COMPOSITION

AND THERMODYNAMIC PROPERTIES OF

SIMPLIFIED GAS MODEL

The calculation procedure for determining gas composition and thermodynamic

properties for the simplified gas model has been programed for and used with the

IBM 7094 data processing system at the Langley Research Center. A printout of the

program is presented in this appendix. The program is written in FORTRAN IV lan-

guage (ref. 8). The symbols used for the program are as follows:

RCH rch X(1) XH20

REQ req X(2) XCO 2

TEMP W X(3) X02

P p X(4) XH2

M M X(5) XN2

H H X(6) X A

s s x(7) x o

MT(J) Mj X(8) X H

TT(I) T I X(9) Xco

HT(I,J) I_j X(10) XOH

ST(I,J) _j X(ll) XNO

X(12) X N

Input

The input is read into the IBM 7094 data processing system by the FORTRAN

statement

READ (5,100) RCH, REQ, TEMP, P

100 FORMAT (4E 16.8)

For example, an input car would be

Col.- 1 17

+0.00000000E+00 +0.10000000E+01

23 49

+0.50000000E+04 +0.10000000E+00

2O

APPENDIX

Output

The outputs for this program are the mole fractions of each constituent and the

thermodynamic properties of the gas mixture. For example, output for the input example

would be as follows:

RCH= 0.00000000E-38 REQ=- 0.10000000E+01 TEMP= 0.50000000E 04R P= 0. 100000000E 00ATM

XH20 XCO2 XO2 XH2

0.18315569E 00 0.00000000E-38 0.27084661E-01 0.71114589E-01

XN2 XA XO XH

0.57613321E 00 0.71005768E-02 0.25695777E-01 0.63610586E-01

XCO XOH XNO XN

0.00000000E-38 0.46109576E-01 0.00000000E-38 0.00000000E-38

M=0.21993865E 02 H= 0.14316146E 04BTU/LBM S=0.30918686E 01BTU/LBM-R

Built-In Data

The thermodynamic properties of the pure constituents (R_--J, -_) are taken from ref-

erences 3 and 4 and built in the program as two-dimensional arrays named HT(I,J) and

ST(I,J), where I is the index on temperature and J is the index on constituents. The

thermodynamic properties are built in for the discrete temperatures given in TT(I).

The enthalpy array HT(I,J) is interpolated linearly, whereas the entropy array ST(I,J)

is interpolated logarithmically.

Complete Program

The complete program including comments is reproduced in the following pages.

21

APPENDIX

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APPENDIX

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APPENDIX

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34

RE FERENCES

1, Hall, Eldon W.; and Weber, Richard J.: Tables and Charts for Thermodynamic Cal-

culations Involving Air and Fuels Containing Boron, Carbon, Hydrogen, and Oxygen.

NACA RM E56B27, 1956.

2o Clarke, J. F.; and McChesney, M.: The Dynamics of Real Gases. Butterworths, 1964.

3. Browne, W. G.; and Warlick, D. L.: Properties of Combustion Gases - System:

H2-Air. R62FPD-366, Advan. Engine Technol. Dep., Gem Elec. Co., Nov. 1962.

(Reprinted Sept. 1964.)

4. Anon.: Properties of Combustion Gases - System: CnH2n-Air.

Volume I - Thermodynamic Properties. AGT Develop. Dep., Gen. Elec. Co.,

c. 1955.

Volume II - Chemical Composition of Equilibrium Mixtures. McGraw-Hill

Book Co., Inc., c.1955.

5. Erickson, Wayne D.; Kemper, Jane T.; and Allison, Dennis O.: A Method for Com-

puting Chemical-Equilibrium Compositions of Reacting-Gas Mixtures by Reduction

to a Single Iteration Equation. NASA TN D-3488, 1966.

6. Zeleznik, Frank J.; and Gordon, Sanford: An Analytical Investigation of Three Gen-

eral Methods of Calculating Chemical-Equilibrium Compositions. NASA TN D-473,

1960.

7. Anon.: U.S. Standard Atmosphere, 1962. NASA, U.S. Air Force, and U.S. Weather

Bur., Dec. 1962.

8. McCracken, Daniel D.: A Guide to FORTRAN l>rograming. John Wiley & Sons, Inc.,

c. 1961.

35

0

0

d

z

_q0_q

0

o_

.o

_ 0

0 tt

0 +

0

oII II

0 0+ +

0rj

o_Z

d

Z_q0

0

o 0•,'_ II

_, II II 0

0r_

0rj

• d•r..I t2_

2:

0

+

_ 0 _._ r..)._ A

II _._

_ o ur..)+

0

0

0

00

II _,1

_ 0o r..)

!

_ _xl V

r_ vgl

Z o

0

0

0°_,,.i

._...

E

g_

_UT:FeToosgT(I _-_ _ui_TaOSSTpuo N

a.m:l_,_adtuej.

36

I00 -

om

o.J

8O

60

4O

2O

0

-2O

=Nz

2H+O=H20

O+N-- NO

i+CO = C0220 "-- 02

H H2= OH

r---H2+ CO 2 = CO + H20

3H2 +N2 = 2NH3

C02+ C(S) = 2C0

I I I I I.0002 .0004 .0006 .0008 .001

I / T, I/°R

I 1 I I

7' 000:3000 2000 I000

Temperoture, °R

I I I I3000 2000 I000 600

Temperoture, °K

Figure 2.- Variationof equilibrium constantswith temperaturefor gasmodelreactions.

3q

6:1062500

5

2000

4

1500

0 0

- 500

-I000

- 15001.4 1.6 1.8 2.0 2.2 2.4 2.6 2.8

Entropy, S, Btu/Ibm-°RL i I _ i 16 7 8 9 I0 II

Entropy,S, J/kg-OK

3.0

t12

5.2

13XlO 3

Figure 3.- Mollier diagramfor kerosene-aircombustionproducts(rch = 0.51. req = 1.0.

38

o0o0

"oX0

]ii]

!ii:

]]ii

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ihl

iil

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......... I I

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Lii iiiii!il iiii_iii

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i_ ...ikL! .... iil

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39

%

oo

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0 IJ,' apnl!llV

0 o 0ID o ID

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40

HI

0¢.-

CILl

Hs

Ref. 3

m--Simplified

Vj = _/2Jg (Ht-Hs)

mode I

PI=IO otm

P "P0o

Entropy

Figure 6.- Isentropic nozzle expansion process.

41

m

0

E

"I0G>

® E_-_

//

/

L ) ....... _ ...... J _

o ooo0as ' dSl

_ ___L ......... ]

o ooo

O,J

m

o

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CP

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t-

O

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E

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,,.n.,

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42

E0

d

c

EJ

100.0

I0.0

.0

.Ol

Equivalence ratio, req

Figure 8.- Limiting pressure for solid carbon formation for methane-air combustion products (rch : 0.25). req,lim : 4.0.

43

Figure g.- Limiting pressure for solid carbon formation for kerosene-air combustion products (rch = 0.5). req, lim = 3.0.

44

E

,m

(3.

i/)¢/)

C

.B

EJ

I00.0

I0.0

1.0

.I

.01

TemperatureoR 2400• K 1333

I 2 3

Equivalence ratio, req

Figure 10.- Limiting pressurefor solid carbonformationfor benzene-aircombustionproducts(rch = ]..01).req,lirn = 2.5.

45

I00.0

I0.0

.01

Equivalence rotio, req

Figure ].].- Limiting pressure for ammonia formation (XNH 3 = 0.01)for hydrogen-air combustion products (rch = 0t.

a simplified equilibrium hydrocarbon-air combustion …€¦ · a simplified equilibrium...

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