the combustion of liquid fuels

COMBUSTION OF LIQUID FUELS 847

Symbol

a

B

b

C

D

d

E

g

H

h

h

k

?n

,h r'

~h pr~

ne ~ n H ,

etc.

P Q

~ttt

R

Re

106

THE COMBUSTION OF LIQUID FUELS

B y D. B. SPALDING

NOMENCLATURE

Meaning Unils

Generalised diffusing substance or property

Transfer Number Generalised diffus-

ing property ap- pearing in the essential equations

Specific heat at constant pressure cal/gm~

Molecular diffusion co-efficient cm2/sec

Diameter cm Heat of activation

of a chemical reaction cal/mol

Acceleration due to gravity cm/sec 2

Calorific value of fuel cal/gm

Heat of formation cal/gm Heat transfer co-

efficient cal/cm 2 sec~

Thermal conduc- tivity cal/cm~

s e c

Weight concentration gm/gm

Mass flow per unit area times a linear dimension gm/cmsec

Mass flow per unit area gm/cm2sec

Mass flow into unit volume gm/cm~sec

Fractional weight of element in substance gm/gm

Pressure gm/cm 2 Heat reaching fuel

surface from gas per gm fuel vaporised cal/gm

Heat produced in unit volume and time cal/cmasec

Universal gas constant cal/mol~

Reynolds Number

Symbol Meaning

r Weight of oxygen required for combustion of unit weight of fuel

T Temperature U Velocity of gas

stream u Velocity in x-direc-

tion V Normal laminar

flame speed in a homogeneous fuel/ air mixture

v Velocity in y-direction

w Velocity in z-direction

x Direction along fuel surface and direction of gas flow

y Direction at right angles to fuel surface and gas flow

z Direction along fuel surface and at right angles to gas flow

~ Thickness of stagnant gas film

~' Thickness of material or thermal boundary layer

Thermal diffusivity Dynamic viscosity

v Kinematic viscosity o Density # Prandtl Number =

Z Rate constant of a chemical reaction

Suffix Reference

o, 0~ Oxygen CO, H~O, Carbon monoxide,

etc. water, etc. Fuel

g Gas stream, far re- moved from surface

Any substance Reaction zone (in-

finitely thin)

Units

~ or ~

cm/sec

cm/sec

cln/sec

cm/sec

cm/sec

c m

C I11

C Ill

c m

CHI2/sec

grn/cmsec c m 2 / s e c

gm/cm ~

cm3/gmsec

848 BURNING OF FUEL DROPLETS

Suffix References s Surface engaging in

mass transfer a Any element

This work was commenced before the international adoption of the joule as the fundamental unit of heat. I t is sufficiently accurate for the purpose of the paper to treat a calorie as 4.2 joule.

SECTION 1

1. In troduct ion

In many combustion processes, the fuel is present in the combustion space in solid or liquid form, and the rate of burning is largely determined by the rates of mass and heat transfer between the phases. Although mass and heat transfer have

reaction zone from opposite sides. Parallel, but less advanced work on wake flames of the type shown in figure lc and d, has recently been reported by the present author (Spalding, 1951).

The designer of combustion equipment frequently has to burn a specified amount of fuel in the smallest possible space and with the minimum pressure drop across the system. To do this he needs information about the rate of burning of his fuels as a function of the design variables within his control. Similar problems arise in the design of heat transfer apparatus, and the present investigation is largely an application to combustion of the theoretical and experimental methods which have proved of value in heat transfer work. I t has been found that these methods are adequate for many purposes, and that detailed knowledge of the chemical reactions is often not required.

To know the rate of combustion if a given type of flame occurs in a specified system is not sufficient for the designer. He must also know whether that type of flame can indeed exist under those conditions. Here mass transfer theory is unable to help and an excursion must be made into chemical reaction theory.

One example of this type of problem has been examined, namely the conditions under which an envelope flame can exist, particularly around a sphere. The method put forward may be applica- ble to other examples.

FIG. 1. Flames supported by a 1-inch sphere of kerosene. Forced convection, a. Top. Re = 695 envelope flame, b. Second from top. Re = 1860 envelope flame, c. Third from top. Re = 1000 wake flame, d. Bottom. Re = 3220 wake flame.

separately undergone considerable investigation, processes involving chemical reaction as well have been relatively neglected. There is therefore no accepted method by which the combustion rate, particularly of liquid fuels, can be predicted, nor �9 is there general agreement as to what are the deci- sive factors.

Nevertheless, it is desirable to have a theory of combustion rate so that new combustion systems can be designed in a scientific manner, and so that the limits of improvement by varying fuel characteristics or operating conditions in existing systems can be determined. The present paper is offered as a contribution to this end. Discussion is restricted in the main to envelope flames of the type shown in figure l a and b; these are diffusion flames in that the fuel and oxygen approach the

SECTION 2

2.1 Balance Equat ions

g . l l M a t e r i a l B a l a n c e s . The diffusion of a component through a gaseous medium may be described, if the flow is laminar and steady and the co-ordinate system is rectangular, by an equation of the form

doom 02m O2m l Lax' + ~ + az ~ J

( i) Ora a m Om

- - U - - - - Y ~ - - W - - ---- 0 OX Oy OZ

where m is the weight of the component in unit weight of the local mixture and D is its diffusion co-efficient.

This equation expresses the fact that the amount of the component diffusing in through all the walls of a small cube in unit time is exactly equal to the net rate at which the component is transported out again; that is, that the local concentration does not change with time, and that no


sinks or sources of the component are contained in the cube. In the form presented, the equation is not quite correct, firstly because D may not be constant through the space so that Dcg~m/~x 2 should be replaced by a/Ox(DOm/Ox), and secondly because, as Ackermann has shown, the concentrations should enter as partial pressures and not fractional weight concentrations when large tem- perat'ure gradients are present. Thermo-diffusion is also not accounted for.

However equations of the form (1) will be used in the following analysis because of their sim- plicity, because the errors are not large if the components do not differ greatly in molecular weight from, and are of low concentration in, the bulk of the fluid, and because lack of knowledge of the true values of the diffusivities and the mathematical complexity of considering their variation make it desirable later to treat these properties, and also the gas density, as constant throughout the fluid.

When a component is reacting chemically in the medium, equation (l) is no longer true, because sinks and sources of the component are distributed throughout the field. I t is however possible to write down equations expressing the fact that the net weight of any particular element, contained in whatever component, diffusing in through the walls of unit volume is equal to the net rate of convection outwards. The equation so describing the conservation of the element a which forms a frac- tion n, i by weight of the component j , has the form

n~i i = 0 (2)

where m'" is the L.H.S. of equation (l) multi- plied by the gas density and means the rate of chemical formation of the component in unit time and volume. In general m'" # 0.

The number of equations of type (2) which can be obtained is equal to the number of participating elements. In the most common cases these will be carbon, hydrogen, oxygen and nitrogen. More- over, further equations can be obtained by combining the elemental balances to eliminate particular components.

2.12 Heat balance. When heat flows by conduction in a moving medium, the equation describing steady laminar flow in a rectangular co-ordinate system without sources or sinks is

- -q" ' FO2T 02T 02T 1

c-~- = K LOx2 + ~ + -~/z~ 3 (3)

OT OT OT - u Ox v Oy w ~ - 0

where 0 " is the rate of production of heat in unit time and volume, K is the thermal diffusivity, c is the specific heat and p is the density.

The simpler form KO2T/Ox ~ has been preferred to the more accurate a/ax(KaT/ax) for reasons similar to those of section 2.11.

When exo- or endo-thermic chemical reactions occur, equation (3) does not hold, for in general 0 'p' # 0. Instead we have an equation relating ~ " to the rate of production of the various gaseous components and their heats of formation, namely

r ~E h~,~'", = 0 (4) P

where hj is the hcat liberated when unit weight of component j is formed from its elements. Heat production by viscous dissipation or loss by gaseous radiation is neglected.

Equations of this sort have been used, for example by Lewis and von Elbe, and Zeldovich and Frank-Kamenetsky, as the foundation of theories of flame speed in homogeneous combustible mixtures.

2.2 Heterogeneous Combustion: A Bom~dary Condition

This paper is concerned with combustion from a solid or a liquid fuel surface. In the general case a gas stream containing oxygen flows past a fuel body in a manner either mainly determined arbi- trarily (forced convection), or induced by the density changes associated with combustion (natural convection). Chemical reaction occurs at, or in the immediate neighbourhood of, the fuel surface, and fuel is transferred in some form or other into the gas stream. Apart from density changes, this matter transfer modifies the pattern of gas flow near the fuel surface. The rate of matter transfer from unit area of the fuel surface is of major interest.

2.21 Conditions at the fuel surface. It has been found experimentally that there is no appreciable relative tangential velocity at the interface between a fuid and a rigid body. If, as is usual in these problems, y and v are respectively the direction and velocity at right angles to the surface, then the terms of the differential equations con- raining u and w may be ignored in the region close to the fuel surface.

Discussion will be restricted to the common case in which the temperature and composition of the gas at the fuel surface do not differ significantly


from place to place. In this region therefore diffusion in the x and z directions may be ignored.

With these simplifications, equations (2) and (4) may be integrated in the y-direction, yielding expressions for the rate of transfer of material and heat across the fuel-gas interface.

From equation (2) we obtain

] o . ( 5 )

= c o n s t a n t

close to the fuel surfacc, where #t," is the net rate of transfer of element a normal to the surface.

If the net rate of mass transfer across the interface is m,'P, then we also have the two relations

�9 I)'

�9 y lllo, s ,~ . = p ~ , - (6)

not f

where v~ is the normal gas velocity at the surface, and n,s is the proportion of element a in the fuel. These relations hold because fuel leaves the surface in the form of vapour or reaction products, bu t no material is absorbed into the bulk of the fuel.

(5) and (6) can be combined to give an important boundary condition linking the velocity field with the material field. I t is

n o~ Dj \ -~-y/~ (7)

Y* Z nc, j mjs - - r~a!

The heat balance equation (4) may be integrated in a similar way. The resultant is

Kc o--y - - cv T - ~ hj D i O~jj v m j (8)

= const , close to t h e fuel surface.

We shall suppose that no reaction takes place at the interface, so that for all other components than fuel

( 0 m ) n,i~ = -- D i - - - vmi = 0 (9)

Oy

This does not exclude the case of solid fuel combustion, but treats it as a special ease of liquid fuel combustion. Solid fuels will for the moment be considered to react in the gas phase just outside the solid-gas interface. This is a mathematical con- venience without physical significance.

The assumption of no reaction at the interface

further implies that the sensible heat conducted through the interface provides the latent heat of vaporisation of the fuel and any heat lost to the interior of the fuel. The sum of these latter quan- tities will be denoted by the symbol Q, in cal/gm. of fuel vaporised.

We can then deduce from (6), (8) and (9) the boundary condition

Y'. hj Dj \ 0 y / . -- \ 0 y I . (10)

v~ = ~ h i m i , - - Q - hj

where the suffrx s now denotes "close to" rather than "at" the interface.

2.22 Diffusivities: the essential equations. The solution of the differential equations can be greatly simplified if the various thermal and molecular diffusivities may be assumed equal. Experi- ment and kinetic theory alike show that they must be approximately equal, and the paucity of experimental data, particularly at elevated temperatures, makes the assumption of r = D i = D often the most reasonable that can be made. For instance the D of oxygen in air at 0~ is 0.178 cm2/sec, and the K of air is 0.187 cm2/sec. Similar good agreement is found with most diatomic gases.

When this assumption is made, equations (2) and (4) may be transformed into equations of the form

5'" = 0 (11)

where a = ~ n,mj for matter transfer (12)

or a = ~ hjmj - cT for heat transfer (13)

The boundary conditions (7) and (10) become

D (O~)~ for m a t t e r t r a n s f e r (14) V s - -

a 8 - - n l

or v.. = (15) a.~ - - (Q 4- h i - cT.,)

for h e a t t r a n s f e r

These may be given the common form

(0;) , ( 1 6 )

V s - - a , - - a /

where af is the value of the expression a within the bulk of the fuel close to the interface.


We can equally well replace (11) and (16) by equations involving a new variable b, namely

b'"= o (17)

v~ = K (18)

where b is the dimensionless expression a/(a~ - as). Equations (17) and (18) are the foundation of

the subsequent theory of heterogeneous combustion.

Combustion in a homogeneous combustible mixture, or in a gaseous diffusion flame, can also be described by equation (17), but (18) is not valid; indeed surfaces are not involved. The character- istic of the first of these problems is that b is constant throughout the field, whereas b varies in heterogeneous combustion and in diffusion flames.

2.3 Implications of the Equations

2.31 Form of the solution. Writing (17) out in full, we have

"L~l~ + ~ . + Oz'2.J (19)

Ob Ob Ob u - ~ - - r o y 0

which will be recognised as the equation normally encountered in heat or mass transfer problems, wherein temperature or concentration takes the place here occupied by b.

The same methods of solution as are used in heat transfer problems may therefore also be used here. Indeed, where the boundary conditions are similar, the same solutions may be used.

The usual problem is that a stream of gas, of uniform composition and temperature, sweeps a surface of fuel. The property b has a constant value a t a large distance from the fuel surface, and a different constant value at all points of the surface.

If the velocity field is determined independently, the value of b at any point in the gas is determinate when the values of b at the fuel surface and at infinity are known. This follows because (19) is a linear partial differential equation of the second order, so that two arbitrary func- tions are involved in its solution. These two func- tions are determined by the boundary conditions, which are b = b0 at infinity in the gas stream and b = b, at the fuel surface.

In general, however, the velocity field is not determined independently, if for no other reason

than that the normal velocity at the interface is determined by the local gradient of b, as equation (18) shows. In this the general case differs from that of pure heat transfer wherein v, = 0. The velocity field is determined by the Navier-Stokes differential equations and by the boundary conditions of which (18) is one. In this case also, however, a unique solution may be obtained if the values of b at the fuel surface and at infinity are known.

In general, we shall be interested in the mass transfer rate at the fuel surface. For this it is not necessary to know the absolute values of b at infinity and at the fuel surface, but only their difference which determines the slope of b at the surface.

We shall call this difference bo - b~ the Transfer Number, and denote it by the letter B. In Section 4 it will appear that mass transfer rates, which here usually mean combustion or vaporisation rates, can be calculated in terms of B, which may be thought of as a driving force for mat ter transfer, expressed in non-dimensional form. Apart from their presence in B, such characteristics of the system of chemical reaction as the composition of the gas streams, the carbon-hydrogen ratio of the fuel, its heat of combustion, or the gas temperature do not appear in the solution.

In terms of the quant i ty a, which was introduced in 2.22, the Transfer Number appears as

B = b o - - b8 = a g - - a~ ( 2 0 ) as - a l

2.32 Conclusions. The foregoing discussion jus- tifies the following statements:

(a) The rate of mat ter transfer from a fuel or liquid surface of given shape in a given stream of gas depends only on the Transfer Number B.

(b) There are as many different forms of Trans- fer Number possible as differential equations can be formed from the fundamental balance equations by elimination and substitution. The number of fundamental equations (Equations (2) and (4)) is equal to the number of participating chemical elements, plus one for heat.

(c) Since there can only be one value of v , , each of the many forms of Transfer Number valid for a given system must have the same numerical value.

(d) In calculations of the combustion rate, that form of Transfer Number will be used which contains concentrations and temperatures which are among the data. Uusually the properties of the gas stream will be known, but the gas composition


and temperature at the fuel surface may not be known fully.

(e) Since the number of possible gas components is very great, some must be eliminated by common sense reasoning. For a start, those components which for some reason (e.g. the low temperature) cannot be present at the fuel surface or at infinity, may be omitted from consideration al- together. Even though these components, e.g. H, OH, are present in the flame zone, they exert no influence on the mass transfer rate, unless of course their diffusion coefficients differ greatly from that of the rest of the gas, which is contrary to the assumption of section 2.22. In the case of H radicles this assumption is invalid, but there is no simple way of doing anything about it.

(f) When solid fuels are in question, diffusion processes alone govern the combustion rate, and no equation involving heat need be used in finding this rate. (See 3.1 below.)

(g) When liquid fuels are in question, vaporisation is an important component process of the whole, and the Transfer Number must be formed from a heat balance equation. (See 3.3 below.)

(h) A large number of gas components may be present near a liquid fuel surface, formed by thermal decomposition of the fuel. Each component will in general have its own heat of formation, and the simplest possible form of Transfer Number must still be very unwieldy and contain many terms that cannot be known. In this case, since the heats of formation of the decomposition products will all be small compared with those of the products of combustion, it is best to ignore their presence. The resultant error cannot be large.

SECTION 3

In this section the Transfer Numbers appropriate to common forms of mass transfer with o r without combustion will be presented, though not rigidly derived.

3.1 Combustion of Carbon

If we consider carbon burning in an atmosphere containing oxygen, carbon dioxide and steam, we can set up balance equations, as in Section 2, for the elements O, C, and H. By algebraic manipula- tion it is possible to eliminate two gas components from these equations, and it is convenient to let these be CO and H2, the gases whose surface concentrations are not normally known. We then

arrive at an equation and boundary condition of the form of 2.22 (17) and 2.22 (18) in which

b ~ 12/I'6m~ + 12~4m~-~ + 12/~8m"2

1 "Jr- 12/~6mo~ , + 12/~4mco2,

-J- 12/i8m~20,

where the suffix s denotes the surface concentration.

The Transfer Number is equal to the difference of the values of this expression in the gas stream and at the surface. Therefore surface concentrations appear, and must be known if the mass transfer rate is to be calculated. If sufficient is known of the kinetics of the chemical reaction at the carbon surface, the surface concentration can be calculated in any particular case. Here we shall deal merely with the simple but important case in which the surface temperature is so high that the surface concentrations of 02, CO2 and H20 are effectively zero.

For this case the Transfer Numbe= i

. . . . . . , . . . . I

~B 12/~6mo2 ~ -~- 12~4mco,o + t2/ismH~ool

There is nothing of fundamental novelty in this result. I t is included to show that all transfer processes, familiar and unfamiliar, can be put in a common form. The method of derivation shows that this B is valid whether reaction takes place in the gas phase or not; that is to say that the burning rate should be the same, apart from sec- ondary influences, whether the carbon monoxide produced at the fuel surface is oxidised in the boundary layer or escapes unburned. Only surface and gas stream conditions matter.

Most commonly no CO2 or H20 are present in the incoming gases. Then the Transfer Number is simply

B = 12~6mo%

This is based on the assumption that all C02 reaching the surface is instantly assumed while CO is the carrier of carbon.

A lower temperature condition may be that no monoxide is formed at all, but the carbon is carried away in the form of C02. For this case the Transfer Number is

B = 1}~2mo~o

that is, just one half the previous value.


3.g Combustion of Metals

Metals differ from other solid fuels in that their oxides do not normally vaporise at the temperatures reached by the metal surface. Consequently the mass transfer is predominantly inwards to the surface. A particular case is that of steel, i.e., iron and carbon. Its Transfer Number may be derived from the balance equations for oxygen and carbon in the form

B = mo2• - - mo2.

_ f l + 4~2n= ~ too2, ) , 1 - 7~6n~ f

where nc is the weight of carbon in unit weight of steel.

When the chemical reaction rate is very high the surface concentration of oxygen tends to zero, giving

- - ~ ?mo2o B = --~ 4~/12nc)

With very pure oxygen and iron, the Transfer Number tends to - 1 , which corresponds to an infinite inward transfer rate.

313 Combustion of Liquid Fuels

In the full version of this paper a method of de- riving the Transfer Number of a liquid fuel from the fundamental balance equations has been presented. Here we shall simply indicate a less formal method giving the same result.

We start from the differential equation

q,,, + H . , , , - - 7 ~ 1 o ~ ~ 0 r

which implies that heat is only evolved within the gas where oxygen disappears, and that the amount corresponds to the full heat of combustion of the fuel H. r is the weight of oxygen combining with unit weight of fuel.

If we then assume that Do2 = ~, which is very nearly true as has been seen, we may introduce a new variable

Hmo~ ~_ cT r

b = Q

which satisfies the equation b " = 0 and the boundary condition v. = ~(Ob/Oy),, the latter

holds because K(Ob/Oy), is simply k/O.(OT/Oy),, since too,, and (amoJOy)8 are both zero for

thermodynamic equilibrium and no reaction at the surface.

The Transfer Number thus becomes

_ H mo~o -b c

Q r

Normally all the magnitudes occurring in this expression are contained in the data of the problem so that the Transfer Number can be evaluated. I t is true that T, is not known precisely but it is always sufficiently accurate to insert thc boiling temperature of the fuel.

In this derivation the heats of the reactions of hydrocarbon decomposition have been ignored, and the existence of reactions involving inter- mediate reaction products has also been neglected. The latter assumption is fulfilled everywhere but in the thin high temperature reaction zone, which forms such a small proportion of the whole field that non-fulfilment cannot cause important errors. For the reaction zone treated as a whole it is cer- tainly true that the disappearance of oxygen involves evolution of the whole heat of reaction.

3.4 Vaporisation Without Combustion

For this case the heat conservation and fuel conservation equations apply independently and lead to two Transfer Numbers

and

8 - c(ro Q r.) 1

B - m r~ _--_ rnj,

rrt]. -- 1

which must be numerically equal in any given case. The equality may be used, in conjunction with the vapor pressure relation of the diffusing substance, to determine the values of T, and mj, .

3.5 Typical Values o.f the Transfer Number

In order to fix ideas, numerical values of the Transfer Numbers valid for some typical combustion and other mass transfer processes are presented in Table 1.

The latent heats'of vaporisation of the hydro- carbons have been taken from calculations by Haggenmacher, and other data used in calculating the table from the Handbook of Chemistry and Physics.

8 5 4 B U R N I N G OF F U E L D R O P L E T S

3.6 Solid and Liquid Fuels as One Family

Table 1 shows that there is a general tendency for the less volatile fuels to have lower Transfer Numbers, partly because of the increased quantity of heat required to vaporise them, and partly because of the decrease in the term (To - T,). We can imagine a continuous series of fuels of in- creasing boiling points (T,) but with other properties constant. As T, rises, B falls, until when (T, - To) has the value moo/r. (H - Q)/c the Transfer Number becomes simply moo~r, all heat terms having disappeared.

TABLE 1

Fuel Formula

n-Pentane C6H~2

n-Hexane C6H~,

n Heptane CTH~n

n-Octane C,H~,

n-Decane

Condition i. B

8,23 !

Combustion in ! 9.00 atmospheric air, 7" 0 - T 8 9.15 =0, Q =la- [ tent heat of 9.70 fuel i

CO formation 10.02 Ct0H22

Benzene C~H~

Toluene CH,C6HI

Cyclohexane ICH,)6

Methyl alcohol CH,OH

Ethyl alcohol C,H~OH

Solid carbon C

Water H~O

wDecane CnHm

neglected

moo = 0 . 2 3 2

mc02~ = 0

Vaporising into gas stream at 2000~C

Dit to

7.74

8.35

8.25

2.67

3.50

0.17

0.77

2.98

T �9 Condition B

As before except

Q = latent heat and sens ib le heat to raise liquid from 15~ to B.P.

mo o = 0.232

raCO s = 0

This is the expression already shown to be valid for a solid fuel, and we may consider therefore that solid fuels belong to the same family as liquid fuels, but have the extreme property that the boiling temperature is equal to or above the reaction temperature.

SECTION 4

The essential equation of mass transfer 2.22 (17) has been solved in conjunction with the boundary condition 2.22 (18) for the following systems:

(1) A stagnant film of gas interposed between a plane surface of fuel and a plane reservoir of gas.

(2) A sphere of fuel in an infinite stagnant atmosphere of gas.

(3) A flat plate of fuel in a gas atmosphere: forced convection.

(4) A vertical flat plate of fuel in a gas atmosphere: natural convection.

(5) The forward stagnation point of a sphere in a laminar gas stream: forced convection.

The solutions will be reported elsewhere. Sys- tems (3), (4) and (5) were solved by a method based on those of Eckert and Lieblein and of Squire. The results only will be presented here.

4.1

The rate of mass transfer through a stagnant ] _ _ film of thickness ~ is given by

i 7 .78 . t t m, 8,c/k = ln(1 + B) 6.39 I in obtaining which and later results it has been 5.45 necessary to assume a constant value of k/c, or 5.02 alternatively of course of DO, throughout the gas.

4.11 4.2

.o9 The rate of mass transfer from a sphere ill an infinite stagnant atmosphere is given by

.85 .f! m~ dc/k = 21n(1 + B)

.22

where d is the sphere diameter. Assuming the proc- .37 ess to be quasi-stationary, the burning time tb .95 of a fuel particle can be calculated in the form

Ib" k 1 ~.0872 2 - -

do COy 8 In (1 + B)

where do is the initial diameter and p: the fuel density. The "evaporation constant" used by workers at the British National Gas Turbine Establish- ment therefore becomes

k X = 8~00/ln (1 + B)

Since B contains properties of the atmosphere, we see that it is not a constant of the fuel alone.

4.3

The rate of mass transfer from a fiat plate in a laminar longitudinal gas stream is given by

k B - f t ( B , a)

where ~h # is the average rate of mass transfer per unit area, x is the length of the plate and cr is the Prandtl Number.


The left hand side of the equation may be considered as a Nusselt Number. Except when B is zero its value (lifters from the value for simple heat transfer in the same circumstances. This "ordi- na ry" Nusselt Number we will denote by Nu*. Figure 2 shows the ratio of Nu in mass transfer to Nu* plotted against B for positive values only. (r is taken as 0.710. We see that Nu falls as B increases, so that the mass transfer rate increases with B but at a slower rate.

" x : ~ i

~t, '~ PRATE rO~cCt~ CONWCnO~, ~,

B

FIG. 2. Reduction of Nusselt number by mass transfer, comparison of "stagnant film" with boundary layer theory.

Only outward mass transfer (B ~ O) has been considered since this is of major interest in combustion, For metals B ~ - 1 , but this case could not be dealt with by the present methods for mathematical reasons.

4.4

The rate of mass transfer from a vertical flat plate by natural convection is given by

m xc gx - k - B - = f2(B, o-)

where g is thc acceleration due to gravity, while other symbols have their previous meanings. The expression under thc fourth root sign is a modified form of Grashof No. Thc usual term containing the temperature difference is omi t ted , because it depends on the assumption that no large temperature differences exist. This leads to considerable ovcrestimates of the buoyancy forces when combustion occurs, and it is more accurate to assume that the buoyancy forces everywhere have their theoretical maximum value, namely that corresponding to zero density.

N u / N u * is plotted for this case also in figure 2.

4.5

Mass transfer at the forward stagnation pCnt of a sphere occurs at a rate determined by

kB - f ~ ( B , ~)

Once again the relation between the transfer rate and the Transfer Number is represented by a plot of N u / N u * in figure 2.

4.6 A General Method of Estimating Mass Transfer Rates

4.61 The "stagnant film" theory. A popular and useful, though sometimes misleading conception of convective heat transfer rests on the "s tagnant film hypothesis". The sole resistance to heat transfer is supposed to reside in a film of stagnant fluid surrounding the body and of constant thickness. Since the whole temperature difference is supposed to occur across this film, and heat transfer is by conduction alone, the thickness 6.~ of the film may be easily related to the heat transfer coefficient by

h = k / & (1)

from which it follows that the Nusselt No. is the ratio of a typical dimension to the thickness of the effective stagnant film.

hx x N u - -- (2)

/c 6~

If this fictional film is thought of as real and moreover, not to vary in thickness when matter transfer takes place, we may use the results of 4.1 to predict the mass transfer rate from a body of which the Nusselt No. is known. We obtain

~h tt x c .'C - N u * (3)

kin(1 + B) &

where the asterisk denotes the "ordinary" Nusselt No. of the body for pure heat transfer.

Equation (3) could be used to calculate the rate of matter transfer from any body for which heat transfer data were available. This has frequently been done with satisfactory results for systems where the rate of matter transfer is low, so that the velocity field does not appreciably differ from that of pure heat transfer. I t is necessary to test the accuracy of the method when the matter transfer rate is high, as for instance in the combustion of liquid fuels.

4.62 Comparison with boundary layer calculations. I t is simplest to compare the results of the


"stagnant film" theory with those of sections 4.3, 4.4, and 4.5 in terms of the ratio Nu/Nu*, i.e. the actual over the "ordinary" Nusselt Number.

For the "stagnant film" theory, this ratio is seen from (3) to be

Nu/Nu* = 1/B ln(1 4 -B) (4)

For the flat plate in forced and natural convection, and for the forward stagnation point of a sphere, the ratio for any value of B is found by dividing the calculated expression (rM~xc/kB)/x/Re by its value when B is zero.

The ratios have been calculated and are plotted against B in figure 2 for ~ = 0.710. I t will be seen that the four curves do not differ by much more than 30 per cent over the whole range, and are particularly close when B is small. This indicates that the effective "s tagnant film" is not greatly changed in thickness, whatever the Transfer Number.

4.63 The general method. This fortunate agreement suggests a method of estimating the rate of matter transfer from a body to a gas flowing in a manner for which only heat transfer data are available. Although its universal val idi ty cannot be proved, it is likely to give results of reasonable accuracy, not only in laminar but also in turbulent flow.

Divided into steps, the method is this: (a) From the data find the Reynolds Number,

if forced convection is in question, or the Grashof Number if natural convection occurs. In the la t ter case the ratio (T~ -- T~)/To must be taken as unity if combustion takes place.

(b) From known heat transfer data, find the corresponding Nusselt Number, Nu*.

(c) From the data calculate the Transfer Num- ber B appropriate to the type of mass transfer.

(d) Calculate the mean rate of mass transfer from equation (3) using suitable mean values of the fluid properties

Such an extrapolation of the theoretical results of this chapter, which may be modified by some empirical multiplier if found necessary, greatly limits the experimental work which must be done before the whole field of combustion is covered, and may be found helpful wherever rough predictions of matter transfer rate are required.

SECTION 5

So far we have made no assumption about the manner or rate of chemical reaction other than that it exists and that it is sufficiently complete to

prevent oxygen from passing unconsumed right through the reaction zone. Although this has enabled us to calculate combustion rates, a closer examination of the reaction mechanism is needed if we are to explain extinction phenomena. Even now however we need go no further than to assume that the chemical reaction rate per unit volume depends on the oxygen and fuel concentrations and on the temperature according to a law of the Arrhenius type

~h/tl = Zmomsp 2 exp ( - E/RT)

Z and E are constants of the reaction. I t is clear that the reaction rate increases as the re- actant concentrations rise, and very steeply as the temperature rises, since E / R is of the order 20,000~

F'~s $VRFAC~

Tr . 2220~

~ *~5B9 ~ GAS STREAM

m ~ . .H2

T=- 20'~

FIG. 3. The boundary layer of a burning liquid fuel temperature and concentration curves. Thin reaction zone: B = 2.0.

If we now consider a section through a diffusion flame, as idealised in figure 3, it becomes obvious that the oxygen and fuel concentratioff curves cannot reach zero at the same point, for this would imply a reaction zone of negligible thickness in which, moreover, the reaction rate would be negligible. The reactants diffusing there could not be consumed, and the fuel and oxygen would tend to interpenetrate. This is in fact what happens, and the reaction zone becomes sufficiently thick for the oxygen and fuel concentrations to rise and to allow the above expression, integrated over the reaction zone, to equal the rate of diffusion of reactants. The actual situation is illustrated in figure 4, in which the reaction product m~m/ is plotted.

However we observe that, as the reaction zone


becomes thicker, the peak temperature in it falls. Since the reaclion rate is strongly dependent on temperature, this effect soon counters the increase of reaction rate consequent upon the increased concentrations. There is therefore a maximum intensity of reaction which is permitted to a flame, expressible in gms/cm2sec. If the rate of mass transfer exceeds this, the flame will be extinguished.

I t is possible to calculate the value of this maximum intensity, which we shall call the flame strength, in terms of the constants of the reactions. This has been done in the full version of the paper. Here we shall merely state that it is a constant for a given fuel and atmosphere, that it is independent of the particular pattern of flow but

f

I IL TM KrK ~1 ESS OF A~"TIIAI. Ii

; I .E*~r ,o . z o . s [ /

FUI tL YGEH S I D E

l

Fro. 4. Illustration of reaction zone of small but finite thickness.

dependent in a way not yet investigated upon the turbulence, that it increases rapidly with temperature, and that it is directly related to the flame speed of a homogeneous stoichiometric mixture of the fuel and the atmosphere by the law: flame strength = Cm:pV, where C is of order unity. A similar result has been reached by Zeldovich in a paper recently made available as N.A.C.A. Tech. Memo. 1296.

This theory may be used to explain the extinction of the flame at the forward stagnation point of a sphere as the air velocity is increased. We have seen that for this case

m"d V' a other things being equal, so that

. H

m ~, ~ / ~ / d

Since there is a maxinmm possible value of ~i*" fLxed by considerations of flame strength, there is likewise a maximum possible value of U/d, which is to say that the flame will be extinguished at an air velocity which is directly proportional to the diameter. I t is noteworthy that there is nothing particularly fundamental about the form U/d: for example if the transfer rate had happened to be proportional to the 0.8 power of the Reynolds No., as in pipe flow, the criterion for breakdown would be U/d~

In order to calculate the critical value of U/d from the flame strength, it is necessary to dis- tinguish between the rate of transfer of fuel from the surface, which we shall denote by #z",, and the rate of transfer of fuel into the reaction zone ~h"~, which alone is relevant to the extinction of

. . . . . . . i

B

FIG. 5. Thin reaction zone data for moJr = .0667, ,r = 0.710.

the flame. This latter quanti ty might be termed the flame stress, for when it exceeds the strength the flame is broken, i.e. extinguished.

The calculation of the ratio rh"r/~h", involves a rather long chain of argument. I t is plotted against B in figure 5, which also contains tr, i.e. (br - c,)/B, y,/6' which is the ratio of the distance between the reaction zone and the fuel surface and the boundary layer thickness, and also rh't, in dimensionless form for the stagnation point of a sphere, all being valid for atmospheric air and a hydrocarbon fuel.

In this way the theoretical curve of U/d has been plotted in figure 6. We see that the larger B is, the higher is the velocity at which an envelope flame can persist on a sphere of given size.

Another interesting consequence of the theory of flame strength is that very small isolated droplets cannot have flames at all. Tha t is to say that a droplet will burn down to a critical size and then


be extinguished. Solid fuel particles will behave in a similar way. This critical diameter is quite small, about 5V for ordinary fuels in atmospheric air, but is approximately inversely proportional to pressure. If the droplets are in motion relative to the gas, the flames will of course blow out while still considerably larger. It seems probable that in gas turbine combustion chambers droplets are continually being extinguished and relighted as a result of the violent turbulence; and indeed that the reason for the presence of a flame tube is not, as is so often stated, that fuels will only burn if the air/fuel ratio is within certain narrow limits (for what is the air/fuel ratio in figure 1?), but rather that it is necessary to isolate a region of generally

J

-, - -4 �9 ~ 4 B

FIG. 6. Extinction condition at forward stagnation point of a sphere. Kerosine, atmospheric air.

high temperature so that an extinguished droplet may be relighted.

SECTION 6

Experimental

6.1 Apparatus. In order to investigate the steady combustion of liquid fuels, it was necessary to provide and maintain fuel surfaces of specified shape. The method adopted was to cause liquid fuel to flow continuously over the surface of a solid body of suitable form, any excess of fuel supplied over fuel burned being withdrawn continuously from the lowest point of the body. In general it was necessary to provide excess fuel because of the difficulty of supplying exactly sufficient fuel for combustion to each part of the solid surface without leaving any part dry.

The burners used in most of the experiments on envelope flames are illustrated in figure 7a, b and c. Their dimensions are contained in table 2.

In each case fuel entered from below, passed upwards inside the burner, and on reaching the

top spilled over the outside surface, burning as it went. Any excess fuel was collected in a trough at the base. The fuel was therefore pre-heated during its upward path, so that the exposed liquid tended to remain at an even temperature.

The flat plate burner and the 11/~ inch sphere have been described, together with the system of measuring the fuel consumption, in a previous publication (Spalding, 1950, I), which contained an early and limited form of the theory of combustion presented here.

The 1 inch sphere of the forced convection experiments was not hollow because the non-uni-

I

t 1

Fro. 7

L

T A B L E

C h a r a c t e r i s t i c W e t t e d B u r n e r D i m e n s i o n s d i m e n s i o n area

F l a t p l a t e , n a t u r a l

c o n v e c t i o n

1~$" sphere, n a t u r a l

c o n v e c t i o n 1" sphere , forced 1.025 ~ d iam.

c o n v e c t i o n ~l * sphere , forced 0.775 * d iam.

c o n v e c t i o n [$n sphere , forced 0.529 ~ d iam.

c o n v e c t i o n ]

2 . 5 0 " X 2 . 0 8 #

X 0 .16 # 1.560 ~ d iam.

h t . = 6 .36 cm.

d = 3 .97 cm.

d = 2 .60

d = 1.97

d = 1.34

74.3 c m 3

49.2

21.3

12.2

5.65

formity introduced by the horizontal air stream made good heat conduction within the sphere desirable. The burner was accordingly turned from solid copper, and the fuel passed upwards through a central hole. Since the air stream tended to blow fuel away from the leading surface, the whole sphere was covered with silk stocking material, stretched tight and stuck with Durofix. In addi- tion, grooves were filed in the leading face to increase the flow of fuel to it, because of the high local rate of vaporisation. Surface tension in the capillaries of the material then ensured that the whole surface remained wet. Fuel was supplied


to the burner through a tube which was flattened to reduce the disturbance to the air stream. For the same reason the trough in which excess fuel was collected was of elongated section and fixed to the supply tube, down the outside of which the excess fuel flowed, with its top edge }J~ inch below the bottom of the sphere.

It has been seen in the theoretical sections that the rate of mass transfer in liquid fuel vaporisation or combustion is greatly influenced by the amount of heat Q which must be supplied from the boundary layer to the fuel surface to vaporise unit weight. Now the excess fuel rises in temperature in passing over burners of the type used, and in doing so draws heat from the flame. This fact has been used to vary O through a range much wider than would be possible by using fuels of different latent heats of vaporisation, namely by varying the rate of excess fuel flow and measuring its temperature rise. Q appears in the denominator of B, the Transfer Number, so that the greater the rate of fuel flow is, the smaller is the Transfer Number, and with it the mass transfer rate.

In some experiments on combustion in forced convection, spherical burners were used, consisting of steel ball bearings silver-soldered to vertical steel wires. Fuel was supplied down the top wire, and any excess ran down the bottom. The spheres were covered with brass mesh pressed into hemi- spheres and soldered along the great circle at right angles to the air stream. This simple arrangement was not as satisfactory as that of (c) eventually proved to be, but enabled small excess fuel rates to be used.

The wind tunnel used has been described in a publication dealing with wake flames (Spalding, 1951). The manometer used for measuring the comparatively low air velocities encountered has also been reported elsewhere (Spalding II, 1950).

Temperatures were measured by copper-con- stantan thermo-couples.

It was desired to estimate the radiation emitted by the flames of the 1-inch sphere in forced convection. The simple device used for this purpose consisted of a square copper plate of 1/~-inch side with its lower face blackened by smoke. This was held with its plane horizontal at a point 2 ~ inches vertically above the mid-point of the sphere. To its trailing edge was soldered a thermocouple. Another plate interposed below the .1,~-inch plate shielded the latter from radiation from the tail of the flame. The temperature attained by the plate was a measure of the radiation received by

it, and therefore of the radiation emitted by the sphere if this is assumed to be uniformly distributed. This assumption is obviously over simple since the flame is usually blue and non-radiant over the front half, but yellow and highly radiant over the rear half.

I

Fie. 8. Kerosine flames on ll/~-inch sphere in natural convection, a. Upper left. b. Upper right, c. Lower left. d. Lower right. Explanation in text.

6.2 Experiments

6.21 Combustion rate in natural convection. The appearance of the flame of kerosene on the l l/~ - inch sphere is illustrated in figure 8a, b, c and d, for four different rates of total fuel flow. In figure 8a the fuel flow was just sufficient to cover the whole sphere; the flame was tall, yellow, turbulent and smoky and exhibited vortices, rendered visible by particles of carbon between the flame and the liquid surface, near the upper hemisphere. As the


total fuel flow over the burner was increased the flame changed in appearance through b and c to d in figure 8. In d the flame was completely blue and would be extinguished if the fuel flow rate increased further; the heat carried away by the excess fuel under this condition amounted to about one quarter of the heat evolved in combustion. Other fuels behaved similarly.

The experimental results previously reported, together with some new data are plotted in figure 9a, b and c in a form appropriate for comparison

ha ~J

t4 o, ~a

~ l f I ~

e,

a l

a~

. / / .

./" t

J J

f J / -

/ I

~TUR.~ C04"Vs162 FROM

f / a , 9 (~

l i l " /

,, ~rl _A tIATURAL r F ~ a ~ --

t~ ",.$PHE,~.E: V,4RI#r162 Fffd. $ ]l l[ I .,, olll

FIO. 9

with the theory of section 4.43 The Transfer Num- ber for each case has been evaluated from the known fuel and atmospheric data, and from the measured amount of heat carried away by the excess fuel. In the group ( m ' c / k ) ~ / ~ , the typical dimension x has been taken as the plate height or the sphere diameter for the two burners. ra' is the average mass transfer rate per unit surface area times x. The values of k and K are those appropriate to ro(~m temperature; no a t tempt to calculate a mean value for the flame has been made.

The theoretical curve of section 4.4 derived for

1All experimental results are reported in full in a thesis submitted for the degree of Ph.D. at the Uni- versity of Cambridge, 1951.

a vertical flat plate has been plotted in figure 9, even though the majori ty of the experiments were carried out on the spherical burner. The justifica- tion for making the comparison is that heat transfer data by natural convection from plates, spheres or cylinders are known to be correlated well by a single relation, and we may expect the same to be true of combustion.

In fact, figure 9 shows that the experimental results for both burners are in good agreement with theory. We may therefore conclude:

(a) The relation between combustion rate and Transfer Number which is predicted by the boundary layer theory for a vertical fiat plate is in good agreement with experiment, both in form and, when the gas properties are inserted with their atmospheric values, in magnitude.

(b) The same relation predicts the combustion rate from a sphere by natural convection with equal accuracy.

(c) The difference observed between the results with fuels of widely differing volatil i ty are of the same order as the experimental error.

(d) Over the range of Transfer Number used, from 0.25 to 3, the rate of mass transfer by natural convection may be expressed by the empirical law

rh 'c -o '45Ba/4V~-k

6.3 Burning by Forced Convection Preliminary experiments on combustion rate,

flame thickness and air velocity for extinction were carried out on the burners consisting of spheres supported on vertical wires. Since, however, thesc permitted no variation of Q, the 1-inch sphere was made and used for the majori ty of experiments.

5.31 Qualitative. The appearance of flames of the type investigated is shown in figure la and b, which are photographs of kerosene burning from the 1-inch diameter copper sphere in an air stream of two different velocities. Around the leading face of the sphere the flame is usually blue and close to the liquid surface. The long tail of flame stretches many diameters downstream and is always yellow, even with ethyl alcohol as fuel; for the vapours leaving the fuel surface must travel far in a region of high temperature without oxygen before they can reach the reaction zone, and consequently conditions are favourable to cracking and the formation of carbon particles, which may lose so much heat by radiation as to be unable to burn when their turn comes.

As is to be expected from the theory of section 5,


the leading half of such flames is suddenly extinguished as the air velocity is increased to a critical value. Itowever, the trailing half is stabi- lized downstream of the line of boundary layer separation, and may take the form of figure lc or d, according to the velocity and size of the sphere. The flame of c is converted gradually into the flame of d by a gradual increase of velocity which causes the upstream edge of the diffusion flame to pass downstream while its tail is drawn into the sphere. Flame d is therefore of the pre- mixed type.

6.32 Combustion rate. The measured combustion rates are plotted in the dimensionless form Ot//~)/x/R-e against B in figure 10. Once again

i .o

o o 07 06

oo

I 1 I S I t l I I I I t Itl

o . i O.L 2 3 a ~ t , ' r 6 9 , o ~ ~ ~t 6 6 7 6 , ) t o

15

FIO. 10. Combustion rate from the 1-inch sphere

the known fuel properties, measured heat extrac- tion ratio, and air properties at room temperature have been used to evaluate these expressions, and ~il' is taken to be the mean rate of mass transfer per unit area times the diameter.

Also plotted is the curve of section 4.3 which is valid for a flat plate. This is used as a comparison because of the present impossibility of calculating transfer rates all round a bluff body and because it is known from experiment that heat transfer data far spheres and plates follow approximately the same law within the range of Reynolds Num- bers in question.

Figure 11 shows the dependence of combustion rate on Reynolds Number, the ordinate being ~#c/kln(1 + B), which according to the stagnant film theory should be equal to the Nusselt Number for heat transfer without mass transfer. Also plotted are the curves of Nu against Re presented by Sokolsky and Kramers for spheres and the theoretical straight line valid for flat plates. The experiments differ from the latter curve by

amounts of the same order as the difference between the observers.

The scatter of the results is greater than in the natural convection experiments. This may be mainly attributed to an experimental difficulty connected with the collection of the excess fuel quanti ty: in order to avoid aerodynamic disturbance of the flame, the collection trough could not be immediately at the base of the sphere, but the excess fuel had first to trickle down the flattened outside surface of the fuel inlet tube. Thus exposed to the air flow, part of the excess fuel must have vaporised and the remainder have cooled. Even so it was not always possible to prevent the flame from licking down close to the collection trough and so heating its contents. As a result the meas-

~e

FIG. 11. Combustion from sphere by forced convection.

urement of Q cannot be considered as very accurate.

From figure l0 it appears that the experimental points lie 10 to 15 per cent higher than the theoretical curve derived for a flat plate. This discrepancy is of little significance since the choice of 1~, k, etc. at their 20~ values was quite arbitrary. Moreover when compared with the predic- tion of the stagnant film theory in figure l l , the results appear to be an equal amount too low.

We may conclude: (a) The relation between combustion rate and

Transfer Number derived for the flat plate in longitudinal laminar flow is in good agreement with experiments performed on spheres in forced convection, over a range of B from 0.6 to 5 and of Re from 800 to 4,000.

(b) Over the same range the experimental data may also be adequately correlated by the empirical relation

~h'/# = 0.53 B "Vs.Re 1/2

(c) The results of both natural and forced convection experiments may equally well be taken as confirming the general method of estimating


mass transfer rates outlined in section 4.6 and based on the stagnant film theory, since the dis- crepancies between the curves of figure 2 are of the same order as those between theory and experiment.

6.4 Experiments on Flame Extinction

6.41 Forced convection. I t was predicted in section 5 that there must be a maximum air velocity at which a flame can persist at the forward stagnation point of any given fuel sphere, depending on the sphere diameter, the Transfer Number and the air temperature.

The effect of diameter was tested by a series of experiments in which kerosine was burned from the spheres of various sizes mounted on vertical wires. The fuel was supplied at a rate just sufficient to keep the sphere wet. The corresponding value of B is 3.25. The velocity of the air stream was

because of the unsatisfactory manner in which it was necessary to calculate the rate of fuel consumption within the reaction zone. Moreover the dis- agreement is exaggerated by comparing velocities rather than flame strengths, which vary as U 1/2.

The distance of the reaction zone from the liquid surface at the stagnation point was measured by means of a travelling telescope, the outer boundary of the visible zone being taken as defining the reaction surface. The results are also recorded in table 3, and it may be seen that the distances are fairly constant, as is to be expected if U/d = constant and (d/6) ~c x / U d / v . Once again, however, the absolute magnitude is less satisfactory, because according to section 4.4 for Re = 4,000, d = 1.025 inches, B = 3.25, the theoretical value of /5 is 0.077 inches, which would imply a flame thickness of about 0.08 inches, i.e. about twice the experimental value.

TABLE 3

Flame Extinction for Spheres

Air Velocity at Flame Distance at Sphere Diameter Extinction ( 2 0 ~ Extinction

+ Diameter

inches

1.025 0.775 O. 505 0.400 0.275

sec- 1

9O 91 97 98

101

inxhes

0.037 0.036 0.037 0.036 0.034

increased until the leading half of the envelope flame was extinguished, and this extinction velocity was recorded, care being taken that flame breakdown was not prematurely caused by such acci- dental agents as sudden draughts, vibration, carbon deposition, etc. The air temperature was also measured.

The results of these experiments are contained in table 3, in which the air velocities at extinction have been corrected to the common air temperature of 20~ in accordance with the findings presented below. According to the theory of section 5, U/d should be independent of the diameter; in fact there is only a 10 per cent variation over the range considered, but this may be accepted as a reasonable confirmation of the theory.

The agreement as to the absolute value of U/d is less satisfactory, because according to figure 6 the theoretical U/d for B = 3.25 is about 27 sec -1 compared with the average experimental value of 95 sec -1. However, this is not surprising

d~ ~

~ 20' 2

K

/ 7 , p, / I

. / i I o ~ /I I J o/ ,2.7 a} / B = 4 . 5 (b}

./. 10 l l 12 13 14 15 16

MA~IOMETE~ RE~ADING, CM.

FIG. 12. Dependence of extinction velocity on air temperature.

The effect of the air stream temperature on the flame strength was investigated by observing the change with temperature of the air velocity caus- ing extinction. The burners used were the 1-inch copper sphere burning petrol with a Transfer Num- ber of 2.7, and the 1-inch steel sphere burning petrol with a Transfer Number of 4.5.

Figure 12 shows the results for the two burners. Lines of slope approximately 0.08 cm manometer

scale/~ change in air temperature can be drawn

through the points. Since the average manometer

reading is 12.5 cm, and the mass transfer rate

varies as the fourth root of the velocity head, we

may deduce

. l ! 1 amm

- - = ~/~ X ~ ~ ' l ! m,, a T o

= 0.167 p e r c e n t p e r ~


I t can be shown by the theory outlined in section 5 that

. i t

1 am,~ Cv/Cr 1 �9 t ! ~,,,, OT o 1 + moo/r 0-~

where G , c~ are respectively the specific heats of the gases at the temperatures of the gas stream and the reaction zone, while

2R(T~ - - 0,,) ~ E

and is the amount by which the peak temperature in the reaction zone falls below the "thin reaction zone temperature" Tr.

Inserting mog/r = 0.0667, c.j = 0.24, c~ = 0.304, we find

0,, = 444~

The decrease in maximum temperature from its theorelical "thin reaction zone" value is therefore large. If the latter temperature is taken as 2560~ from figure 13, since the Transfer Number of petrol burning without fuel excess is 4.5. the above equation for 0,,, leads to a value for the heat of activation

E = 40,500 cal /mol

This is of the order which is valid for a great many chemical reactions, and is close to the value of 38,000 cal /mol which Dugger found to agree well with his experimental flame speeds in propane-air mixtures.

No thorough investigation of the variation of U/d with B was carried out, partly because the local B at the stagnation point was probably not the same as the measured average over the whole sphere; however the following inferences may be drawn from the records of the experiments carried out on combustion rate:

(a) U/d decreases with B until, when B = 0.6, U/d = 50 sec-lapprox. This tendency is in accordance with the calculated relation shown in figure 6.

(b) There appears to be no appreciable difference between the U/d's of petrol and kerosine at the same Transfer Number.

(c) Benzene has a slightly lower U/d ratio than petrol for the same Transfer Number, despite having a slightly higher homogeneous flame speed, according to Gerstein, Levine and Wong. Perhaps this may be explained by the greater heat loss by radiation from the yellow benzene flame.

(d) Ethyl alcohol has a higher U/d ratio than petrol for a given Transfer Number. No calculation has been made of U/d for alcohol, because no values of the homogeneous flame speed have been found in the literature for comparison.

6.4Z Natural convection. We will here analyse in detail only the extinction of the flame of the 1 inch sphere burner in atmospheric air by increase of the kerosine flow rate. Extinction occurred at ~i Transfer Number of 0.246 and a mean mass transfer rate of */l~ 'p of 0.000325 gm/cm 2 sec. According to figure 5, #z/'/ th/t for B = 0.246 is 0.77.

~ K

2~00

20~

B-5 3

IE;O0

I

�9 . 7

"4

�9 / ~ .~_~~

O ~*L/~.~ FIG. 13. Combustion temperature of a hydrocarbon.

H = 10,300 cal/gm. Tu = 20~

We will assume that the local mass Transfer rate at the base of the sphere, where the boundary layer is thinnest and where extinction occurs, is 50 per cent greater than the mean. We find then that the maximum reaction rate before extinction is n%, t' = 0.000375 gms/cm ~ sec.

By using the relation between flame strength and flame speed derived in section 5, and making allow- ance for the difference in reaction temperature between the homogeneous and heterogeneous flame temperatures, we find that the theoretical flame strength based on a flame speed of 35 cm/sec is 0.00047 gms/cm 2 sec.

This is in fairly good agreement with the experiment, although the discrepancy is in the opposite direction from that of 6.41, and it may be taken as a further indication that the theory presented in section 5 contains some grains of truth. I t is noteworthy that the experimental scatter of the natural convection combustion rate experiments is lower than for forced convection, and also that


the margin of error in calculating the actual value of *hr" at the extinction point is smaller. For in the natural convection experiment concerned the total area of flame was not much greater than that of the liquid surface, and the flame was of almost constant thickness around the lower half; whereas the area of the forced convection flame was always many times that of the liquid, so that estimation of the rate of reaction at the forward stagnation point can only be made by way of the rather doubtful calculation of section 5. We have also seen that although neglecting the temperature dependence of the gas transport properties does not seriously impair the accuracy of mass transfer rate predictions, the predicted thickness and struc- ture of the boundary layer may be more seriously in error.

Acknowledgments. The author is grateful to the Managers of the Imperial Chemical Industries Fellowship Fund for his tenure of a Fellowship during the first two years of the work described here, and to the Depar tment of Scientific and In- dustrial Research for a contract for the investigation placed with the University of Cambridge on behalf of the Mechanical Engineering Research Organisation of the United Kingdom. Approval for publication has been given by the Director of Mechanical Engineering Research. He also wishes to thank numerous members of the staff of the University Engineering Depar tment for assistance

of all sorts, and particularly Mr. E. C. Deverson for carrying out a large number of experiments.

REFERENCES

1. ACKERMANN, G.: V.D.I., Forschungsheft, 382 (1937).

2. BARTOLOME, E.: Naturwissenschaften, 36, 171, 206 (1949).

3. DIJGGER, G. L.: J. Am. Chem. Soc., 72, 5271 (1950).

4. ECKERT, E., AND LIEBLEIN, V.: Forschung, 16, 33 (1949).

5. GERSTEIN, M., LEVINE, O., AND WONG, E.: J. Chem. Phys., 73, 418 (1951).

6. HAC~ENMACUER, J. E.: Ind. Eng. Chem., 437 (1948).

7. Handbook of Chemistry and Physics. Chemical Rubber Publishing Co. (1949).

8. LEwis, B., and YON ELBE, G.: J. Chem. Phys., 2, 537 (1934).

9. Scnlm, H.: Z. angew, Math. Mech., 27, 54 (1947). 10. SEMENOV, N. N.: Progress of Physical Science

(U. S. S. R.), 24, 433 (1940). Translated as N. A. C. A. Tech. Memo. 1026.

11. SOKOLSKY, A. P.: Cited by L. A. Vulis. J. Tech. Phys. (U. S. S. R.), 16, 89 (1946).

12. SPALDING, D. B.: I. Fuel, 29, 2, 25 (1950). 13. SPALDING, D. B.: II. j . Sci. Inst., 27, 310 (1950). 14, SPALDING, D. B.: I. Mech. E. General Discussion

on Heat Transfer, Section IV (1951). 15. SQUIRE, H. B.: Quoted by S. Goldstein, p. 641. 16. ZELDOVlCH, FRANK-KAMENETSKY: J. Phys. Chem.

(U. S. S. R.), 12, 100 (1938).

the combustion of liquid fuels

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