Diffusion and Reaction in a Stagnant Boundary Layer about a Carbon Particle Hugo S. Caram and Neal R. Amundson' University of Minnesota, Minneapolis, Minnesota 55455
A simplified model for diffusion and reaction in the boundary layer surrounding a burning carbon particle is considered. The model accounts for the homogeneous combustion of carbon monoxide and the heterogeneous reaction of carbon with oxygen and with carbon dioxide, the latter two reactions appearing in the model as nonlinear boundary conditions. It provides an insight into the double and single film models, proposed by others, as well as of the distribution of the products in the combustion of carbon.
Introduction Due to its practical importance the combustion of carbon has been the center of a large number of studies and several good reviews are available, i.e., Field (1967), Mulcahy and Smith (1969), Gray et al. (1973), and Walker et al. (1959). We will be concerned here with the efforts to model the combustion of a particle in a fixed environment. Although the char or carbon is a porous substance and intraparticle diffusion probably plays a role in the combustion process, we will be considering the process to be lumped in the external surface of the particle (Khitrin, 1957; Smith, 1972). Two basic models will be considered. The single film model was first proposed by Nusselt in 1924 and assumes (Figure 1)that the oxidation of carbon is controlled by diffusion of oxygen through a stationary film to the suirface of the carbon particle where it reacts to form CO and/or CO2. This last point was investigated by Arthur in 1951 who found, after eliminating the masking effect caused by the homogeneous combustion of CO, that above 1273 K the main product of the reaction was CO. T o complete the picture and explain the concentration profiles observed in fixed bed combustion we can include the COz reduction reaction a t the carbon surface and consider that at high temperatures the process is diffusion controlled (Hougen and Watson, 1947). The double film model was proposed by Burke and Schuman (Figure 2) in 1931. In it carbon reacts at the surface with COz and the CO produced is burned in a thin flame front in the boundary layer. No oxygen will reach the carbon surface and no CO will reach the external edge of the boundary layer. Once we recognize the fact that the main product of the reaction between carbon and oxygen at high temperatures is CO, the difference between the models resides in the place where the oxidation of 120 occurs. Research to determine the validity of the models has been carried out and the models developed are summarized in Table I. It must be mentioned that single and double-film models will predict the same surface temperature and rate of combustion, so that we cannot assess the validity of the models by measuring these quantities and we must measure temperature and concentration distributions through the film. Some experimental evidence was introduced by Wicke and Wurzbacher in 1962 when measurements of the concentration profiles of CO, COZ, and 0 2 in the thin film surrounding a burning carbon rod were made. The results show a maximum in the concentration of COz. Temperature measurements were made by De Graaf (1965) and by Kish (1967) who found temperature maxima several hundred degrees above the surface temperature. It is usually assumed that large particles (>2 mm) will burn according to the double film theory while small particles (1373 K). Three zones are r = kCcoCo2 in a tration of CO, (1961) ( 2 models) the surface assumed: Zone I close to the thin isotherAal is a model ( 2 ) 0, reaches the surface no 0,;zone 11. thin isoflame surface but parameter thermal flame; zone 111. no CO ( 2 ) CO/CO, equili- ( 2 ) Reaction zone CO,, 0 out of the flame. Stefan flow includes all b.1. brium in surface assumed except in flame. ( 2 ) Low temperature model ( < l o 7 3 K ) most of the product of the reaction is CO, and there is little CO t o be burned. Not considered First order w.r.t. CO Isothermal film. Used t o justify Hugo and Wicke Not considered CO concentration profiles and zero order with (1962) reference to 0, described by Wicke and Wurzbacher. Stefan flow. Upper bound used to Non isothermal model Field First order Not considered check critical size (1967) after which it becomes important Unsteady state model considering Second order Zero order and First order Kurylko et al. intraparticle diffusion. Finite (1972) intraparticle r = kCcoCo, element model. Extremely slow diffusion to run. Very fast, flame thickness neglected
-
2h ( g l - g I s )- -( T - T,) = Arrrx (23) M1 w 1 T h e constants A‘, A“, and A”’ can be found using the boundary conditions (110) t o ( 1 3 ) as
A” = -(2R1+ Rz) A” = 2R1
(24)
+ Rz
(25)
Art’ = w1
=- ( - M I )
+
( 2 R 1 Rz)
(26)
w 1
and from the proposed kinetic expressions one obtains
+
W A = -AT = A” = -LA”’ = 2 k l ~ o , B k3cCo2, = (--AH11
+ M3
Pgl = :!k1 Pg3 8 k3 2 Mi
where k l = k l ( T , ) and k3 = k 3 ( T s ) .I f we substitute back into eq 21-23 and use the dimensionless quantities defined in Table I1 it follows that (gi - gis) + a i ( g z - 6‘2s) = -S[2@1&!3s+ P2gisI
(27)
+ 2W&3 - g 3 ~ =) s [ 2 P i g 3 ~+ @&is] (gl - gls) W ( 7 - 7s) = sY[2Plg3s + &&’IS]
(28)
(gl - &?is)
(29) Using the boundary values a t s = 1 we obtain three equations (27,28,29) with four unknowns (glS,gzS,g3s, T ~ so ) that knowing one of them we can solve for the other three. The simplest choice is T~ because the set of equations is then linear. Explicitly gls =
gib
+ 2a2g3b - [ g i b + a3(1 - 7 s ) ] ( a Z + Pi)/’YPi 1 + Pz - ( 1 + rPz)(P1 + (YZ)/YPl (30)
g3s = [gib - gls + a3(1 - 7 s ) - Y18?€!1s]/%”Y1 Ind. Eng. Chem., Fundam., Vol. 16, No. 2, 1977
(31) 173
Table I1 ~~
~~
~
L= 5x
-
Dimensionless parameters for the plane case
Y=-
mm
K , :3007
x IO5 EXP[-17966/TS)
-AH1 w1
In the spherical case 1 is substituted by Ro and we define the additional parameters
Dimensionless variables s = r/l or s = r/Ro 7 = T/Th Reference quantities 1 = boundary layer thickness; Ro = particle radius; T b = ambient temperature 0 I
gZs = k l b
- gls + alg2b + 2Plg3s + PZ&!lsI/al (32)
Thus if we know the surface temperature we can determine the surface mass fractions and by using eq 27,28, and 29 we can find any three of the four dimensionless dependent variables as a function of the fourth. If we choose carbon dioxide as a key component, we can rewrite eq 6 using the kinetic relation described by ( 5 ) in dimensionless form and, with the additional hypothesis of constant physical properties, we obtain
0
The Region of Feasible Solutions If we fix the ambient concentration and the boundary layer thickness we can consider the surface temperature to be a function of the ambient temperature. Our interest is to determine not only a solution of the problem described by (33) and its associated equations but also the locus of solutions in the plane (Tb,T,) when conditions at s = 1 remain constant. Once we fix the ambient temperature T b , any assumed value of the surface temperature will generate from eq 30, 31, and 32 three surface mass fractions gl,, g2s,and g3,. A necessary condition that Ts be a feasible temperature is that g1,L 0, g2, 174
Ind. Eng. Chem., Fundam., Vol. 16, No. 2, 1977
/
4
I
5
I
I
I
7 8 Ambient Temperature , Tb [ K ) x 1 6 ~
6
9
Figure 3. Typical feasibility region (shaded area).
I 0, and g3, L 0. Thus it is clear that the locus of the solutions (Le., the (T,, Tb) locus) must lie in the region where these conditions are satisfied. The boundaries of the region will be determined by the equations gls(Tb,Ts) = 0
(344
gZs(Th, Ts) = 0
(34b)
Ts) = 0
(344
g3s(Tb,
They can be easily solved for Tb as a function of T , to obtain from (34a,b,c) Tb
Equation 33 is to be solved using the boundary conditions (10) and (14) dgl/ds = Pgl, when s = 0 and gl = g l b when s = 1. The obvious procedure to obtain the solution once the ambient conditions have been fixed would be to guess the surface temperature 7s = T/Tb and find the surface concentrations using eq 30-32. We can then integrate eq 33 using the initial condition eq 10. If the proper guess of the temperature was made then gl(1) = gib, necessarily. If not, a new value of Ts must be tried by some systematic iteration procedure. The proposed method, although simple in principle presents some severe computational difficulties and a more sophisticated analysis is required. This provides a t the same time considerable insight into the structure of the problem.
0
I
3
From eq 27,28, and 29,r, gz, and g3 can be written as functions of g1 so
+ a1g2s - Smlg3s + P2glslJ/a1
0
/
0-
0
(33)
gz = b l s - gl
0
/
0
=
[glb(l
+ Tsl/a3'{ l = (a2 + Pd/rP1
- 700 K), 3 solutions for (242 K < T < 268 K, 348 K < T < 700 K) and 5 solutions for (268 K < T < 348 K). It is of some interest to analyze some of the results of the computations. The locus of the system steady states in the (Tb,T,) plane is shown in Figure 7 . As before, the unignited steady state has been omitted from the diagram. At low temperatures, as before, the carbon-oxygen reaction is the dominating process. At low surface temperatures (-930 K) the model predicts almost no C02 production due to the combustion of CO. (If the situation obtained by Arthur (1951) is satisfied the main product of the reaction at this low temperature would be (202.) The concentration of 0 2 a t the surface is about 20% of the bulk concentration and any increase in the surface temperature will imply an increase in the rate of reaction so that to keep the thermal balance of the system the ambient temperature must be reduced thus explaining the negative slope of the curve. At higher surface temperatures
(-1100 K) the oxygen concentration has become so small that any increase in that temperature must be accompanied by an increase in the ambient temperature (points 2, 3 in Figure
7). The changes can be followed in Figures 8,9, and 10 where we have presented the rate of combustion 41,the mass fraction of C02 at the surface gl,, the amount of heat generated a t the surface HS and released by the whole process HT. In the combustion rate diagram a t low surface temperature an increase in the surface temperature will imply an increase in the rate of combustion (point 1in Figure 7,8,9, and 10). We have seen that a low surface temperature will be on the negative slope branch of the (T,,Tb) locus and this implies a negative slope in the (ql,Tb) diagram. As soon as dT,/dTb becomes positive dq,/dTb will be positive too (points 2 , 3 in Figures), but as the surface mass fraction of oxygen diminishes the process begins to be diffusion controlled so the slope dql/dTb becomes smaller and smaller. If we look now a t the C02 mass fraction a t the surface (Figure 9) we see that it will increase with the surface temperature because of the increase in the amount of CO burnt in the boundary layer. s o the (C02,Tb)line will in principle change slopes as the (q1,Tb) line does (points 1 and 2). The difference arises when the surface temperature goes beyond 1080 K when the surface becomes more reactive toward the C02 and its surface mass fraction begins to decrease with an increasing surface temperature (point 3 in Figure 9). Looking at the curves of heat generated at the surface ( H S ) and total heat generated (HT) against the ambient temperature, we can expect that a t low temperatures as the process is confined to the carbon surface both curves will almost coincide and have qualitatively the same shape as the combustion'rate curve (points 1and 2 in Figure 10).As the surface temperature increases, however, the total amount of heat will become greater than the heat generated at the surface although both lines will have similar slopes. But as we have seen the Boudouard reaction begins to gain in importance at T, = 1080 K (point 2) and with a decaying surface concentration of 0 2 the heat generated at the surface begins to decrease. Still the combustion rate is increasing and with it the total amount of heat generated (point 3 in Figures 8 and 10). When we reach a surface temperature of about 1195 K (point 4 in Figures 7, 8, 9, and 10) there is no heat generated at the surface, the Ind. Eng. Chem., Fundam., Vol. 16, No. 2, 1977
177
30I
e: g .e
\
\
\
6 .7
.6 L
\
\\\
2
3
4
5
Ambient Temperature, Tb(K)x16*
e=
Ambient Temperature,
Figure 8. Rate of combustion vs. ambient temperature for a highly reactive carbon.
-20
L
Figure 10. Rate of heat generation a t the carbon surface and total heat generated vs. ambient temperature for a highly reactive carbon.
Figure 9. Surface mass fraction of carbon dioxide as a function of the ambient temperature for a highly reactive carbon.
surface concentration of COz has passed through a local minimum (Figure 9) and the rate of combustion and surface temperature reaches a local maximum in what seemingly is a cusp (Figures 8 and 7). If we now follow the curve in the (T,,Tb) plane we enter into a new branch. The carbon surface is absorbing heat due to the carbon-carbon dioxide reaction (point 5 in Figures 7,8,9, and 10) and a temperature maximum will appear inside the boundary layer. The homogeneous combustion of CO will be increased as can be seen from the steep increase in the surface mass fraction of COZ. But now the reaction zone will be absorbing the oxygen that formerly reached the surface of the carbon. The result will be that the rate of combustion (Figure 8) will decrease along this branch. The controlling factor here will be the surface reduction of COz and this is shown in the steep rise in the surface mass fraction of Con. If we keep going down this curve until we reach a surface temperature of 1180 K we find what is seemingly another cusp (point 6 in Figures 7, 8, 9, and 10). The new branch beginning there will be characterized by the fact that no oxygen will reach the surface. Now as the surface temperature increases the condition will approach that of the two-film theory. The mass fraction of COZ will decrease to a value close to zero and the rate of combustion will approach the diffusion limit a t the same time that the heat absorbed at the surface and generated by the system reach their maxima. In Figure 11we present the five steady-state temperature profiles within the film obtained for an ambient temperature of 300 K. The 2nd, 3rd, 4th and 5th steady-state profiles correspond to the points 1, 3, 5, 7 in Figures 7,lO. It should be noted that there is little difference between the surface tem178
Ind. Eng. Chem.. Fundam., Vol. 16, No. 2, 1977
peratures for the 4th and 5th steady states. We can compare the temperature profiles with those predicted by the two-film theory. These can be easily obtained in this case by assuming: (a) all of the oxygen is consumed in a thin diffusion flame, (b) the process is controlled by diffusion of oxygen to the diffusion flame and COz to the carbon surface, and (c) the only reaction taking place at the surface is carbon dioxide reduction. Using this hypothesis in our simplified model we find that the temperature profile can be represented by two straight lines connecting the maximum temperature found in the flame with the surface and bulk temperatures, respectively. The surface and flame temperature will be given by
The temperature predicted by (40) is the same one that the carbon surface burning to CO would have if there were no reaction in the boundary layer. We can now compare these results with the temperature profiles obtained by solving eq 33 for the case presented in Figure 4 with 10%0 2 and different ambient temperatures. The results are presented in Figure 12.
The Spherical Particle We will solve eq 1and 2 for particles of carbon of radius Ro surrounded by a quiescent boundary layer of finite thickness. The standard approach is to find the solution with conditions given at infinity but if the boundary conditions are not in chemical equilibrium the system defined by (1) and (2) has no solution. The boundary conditions will be written as (42)
(44)
when R = Ro. A t the external edge of the boundary layer (R = Rb)
t
“i
21
We can now integrate the equations to obtain
NQ 17
k l
Mi - g1,P’) + MZ -(g2 - gz,P’)
5 L l l L A 3
2
J
3
4
5
7
6
8
9
IO
Dimensionless Distance, s = x / l
Figure 11. Multiple solutions for the temperature distribution in the boundary layer for a highly reactive carbon ( T h = 300 K , 1 = 5 mm).
or, written in dimensionless form, Two Film Model
1
Converqing to
.6
.I
.7 .0 Dimensionless Distance, s . x / l .2
.3
.9
Figure 12. Comparison between the temperature distributions obtained from a simplified ‘doublefilm model and those obtained in this paper (2 = 5 mm, 10%02).
and we can easily show that g i = gib, gz = gZb, g3 = g3b, T = Tb
(46)
It must be noted that in the boundary condition ( 4 5 ) we are implicitly assuming that all the solid is a t the surface temperature T,. We can combine eq 6 , 7 , 8 and 9 and after our integration obtain equations analogous to ( 1 8 ) ,(19), and ( 2 0 )
where these are applicable to a unit of solid angle instead of a unit of area. These expressions cannot be integrated again unless we make use of the Schwab-Zeldovich approximation (Williams, 1965)
x =D D 1 := D z = D 3 = PCP
The integrating facto’r will be e x p [ q / D p r ]and the constants A’, A”, and A’” are analogous to the ones found in ( 2 4 ) ,( 2 5 ) , and ( 2 6 ) but applied to a unit solid angle.
(54) If we now make s = sb and use the boundary condition a t the external edge of the boundary layer, we obtain a system of four equations, ( 4 9 ) ,(50),( 5 1 ) ,and (53),and five unknowns e , gis, g2,, g3s, and T~ so that fixing one of them fixes the other four. The simplest procedure is to fix the surface temperature and solve for t, eq 53, eliminating g3, and g l s by the use of eq 50 and 51. We obtain
This algebraic equation has a unique solution with physical meaning (e 2 0, P’ 2 1 ) since d 1 >0 dP’ while the derivative of the right hand term is
-(P’ In P’) = In P’
+
Ind. Eng. Chem., Fundam., Vol. 16, No. 2, 1977
179
a
b 13
h
-X
Y
I-*
f
c
12
Ea E
l-
al 0 r L
?.f
I1
IO Ambient Temperature ,Tb(K 1 X IO-‘
Figure 13. (a) Comparison between the feasibility regions and reaction locus for a spherical particle with Stephan flow and the flat case with no Stephan flow considered ( 1 = 5 mm, 10% 0 2 ) . (b) Same as (a), but for a highly reactive carbon.
because 0 < y < 1for the given data. Once we have solved eq 55, we can go back to eq 49,50, and 51 to obtain gls = [glb t a3(1 - 7 s )
- yC(P’ - 1)]/P’ g2s = [(I + y)C(P’ - 1) - W(1 - 7 s ) + alg2b]/P’al g3s = [(y - 1)c(P’- 1)- a3(1 - T s ) + k&3b]/2LY2P’ From here on we can follow the procedure described before for the flat plate case with no Stefan flow. The conservation equation can be written in dimensionless form as
to be solved with the boundary conditions
&tgl = ds
whens = 1
when S = S b g l = gib The values of g2, g3, and 7 are obtained as a function of g l and s by solving eq 49, 50, and 51. We can now determine the feasibility regions in which all the surface concentrations are positive as before. The calculations were carried out for external conditions similar to those used in Figure 4 but where the flat slab with a 5 mm boundary layer was replaced by a spherical particle with Ro = 5 mm and a stationary spherical boundary layer of radius R b = 15mm. It was found in principle that further increases in the thickness of the boundary layer had little effect in the structure or behavior of the system. For comparison these have been plotted against the feasibility regions obtained for the flat slab. As can be seen in Figure 13a and b only small changes in the feasibility regions obtain. For the integration of the differential equations the same numerical method used earlier can be applied here. Shooting methods are used for the low-temperature branches but for the cases where there is little or no oxygen arriving a t the surface and there is a definite maximum of the temperature 180
Ind. Eng. Chem., Fondam., Vol. 16, No. 2. 1977
inside the film they proved inadequate and the problem had to be solved by the method of finite differences and quasilinearization. The solution in all cases required more time than in the previous problem. With the finite difference method a much finer mesh was required. (The number of grid points at which the solution became insensitive to the number of grid points taken went from 50 in the previous case to 200 in the spherical case.) In this case the guess of the initial profile was found to be important especially at low ambient temperatures where the combustion was easily quenched and the trivial solution a t almost ambient temperature obtained. This solution method tended to become unstable requiring the use of underrelaxation. The results of the computations are plotted in Figure 13a. It was of some interest to determine if the pathology of the model would be affected by the change in geometry and the addition of the convective term. Computations were carried out for the highly reactive model and are presented in Figure 13b superimposed on the results for the flat plate case. In all cases the results are qualitatively similar and there are only small quantitative differences between them. Note that there is an increase in the range of ambient temperatures at which there is multiplicity. This can be explained easily since the possibility of more than three steady-state solutions arises from the fact that at a given point an increase in the surface temperature implies an increase in the amount of CO produced which would choke off the oxygen flow to the surface. This effect is enhanced by the presence of the Stefan flow opposing the diffusion of oxygen towards the surface.
Summary and Conclusions The mass and energy conservation equations were solved for the boundary layer surroundihg flat and spherical burning carbon surfaces. The intraparticle phenomena were lumped at the surface as nonlinear boundary conditions and fixed conditions were assumed outside the boundary layer. There was only a small quantitative difference between the solutions
+
k3 = rate constant for the C CO2 reaction, m/s 1 = boundary layer thickness, m koz = 1.3 X 10" [ C ~ ~ 0 ] ~ / ~ m ~ M, /~= / molecular weight of the i t h component, kg/kg-mol P = function defined by eq 52 kg-mol% P' = function defined by eq 56 -AH1 = 1.12 X los J/kg-mol q = rate of combustion per unit solid angle, kg/s - A H 2 = -1.70 X los J/kg-mol ql = rate of combustion, kg/m2 s - A H 3 = 2.82 X los J/kg-mol
Table 111. Physical Data Used for the Calculation D1 = D Z= D3 = 2 X m2/s E1IR = 17966 K Ez/R = 29790 K Ez/R = 15098 K kol = 3.007 X lo5 m/s yp = 6.7 X J/m s K ko:3 = 4.1 X lo9 m/s p = 0.35 kg/mz Average MW = 29.0 kg/kgmol
for the sphere of radius R surrounded by a quiescent film of radius 3R with a Stefan flow and the solution for a flat carbon surface surrounded by a boundary layer of thickness R when Stefan flow was neglected. By appropriate manipulation of the equations the problem can be reduced to the solution of a single second-order ordinary differential equation with split boundary conditions which requires a guess of the surface temperature. Based on the essential condition that the mass fractions of the compounds must be positive a t the surface, it is possible to determine the regions of the (T,,Tb) plane where the solution of the problem can be found along with upper and lower bounds for extinction and ignition. The solution of the problem yields the loci of solutions in the (T,!Tb) plane and the concentration profiles inside the film. According to the choice of parameters the system may present one or three or five possible steady state solutions. Large particles (5 mm) will present an upper steady state in which the particle will be surrounded by a CO flame. For very small particles (50 wm) a flame will not develop while for intermediate sizes there will be a smooth transition between the two possibilities. For highly reactive carbon as many as five possible steady states were calculated. They can be explained by the possibility of having two superimposed ignition phenomena, one corresponding to the carbon surface ignition and the other corresponding to the ( 2 0 flame ignition. Due to the choking off of the oxygen supply to the surface this could lead to combustion oscillatioiis as the ones observed by Kurylko et al. (1972), although it is difficult to extrapolate the results of the steady-state model to the unsteady-state behavior of the model. No conclusions were drawn about the character of the steady states, that is to their stability or parametric sensitivity. This will be presented later in a separate paper. Acknowledgment This work was not supported by any alphabetical private or public agency except the U of M (whom we applaud). Nomenclature A', A", A"' = constants defined in eq 27, 28, 29 in the flat cases, and in eq 47 for the spherical case C, = concentration ad the ith component, kg-mol/m3 cp = specific heat of the gas, J/kg-mol K C = constant defined in (54) D, = diffusivity of the i t h component, m2/s E, = activation energy for the ith reaction, kcal/kg-mol g, = mass fraction of the ith component HS = heat generated a t the surface, W/m2 H T = total heat generated, W/m2 kl = rate constant for the C 0 2 reaction, m/s kol = frequency factlor for the C 0 2 reaction, m/s k2 = rate constant f a r the CO combustion, m3/s kg-mol koz = frequency factor for the CO combustion, m3/s kgmol k03 = frequency factor for the C C 0 2 reaction, m/s
+
+
+
q o = rate of combustion of the carbon, kg/s r = radial coordinate, m 2 = reaction rate for the reaction CO 02, kg-mol/m3 s R = universal gas constant R1 = rate of the C 0 2 , kg-mol/m2 s R2 = rate of reaction C COZ, kg-mol/m2 s Ro = particle radius, m s = dimensionless distance T = temperature, K T,,, = maximum temperature inside the boundary layer,
+
+
+
K
W1 = heat of combustion of two moles of C O , J/kg-mol x = distance measured from the surface, m
Greek Letters a3'
= definedineq36
+ 0 2 (l), J/ heat of reaction for the reaction C + C 0 2 (2) J/
(-AHl) = heat of reaction for the reaction C kg-mol of (-M2) =
0 2
kg-mol of C 0 2
X = thermal conductivity, J/m s K vi = stoichiometric coefficient in the reaction CO 0
+ .LhOz =
2
defined in eq 38 density of the gas, kg/m3 = definedineq37 { = defined in eq 35
cp = p =
+
L i t e r a t u r e Cited Arthur, J. R., Trans. faraday SOC., 47, 164 (195 1). Avedesian, M. M., Davidson, J. M., Trans. Inst. Chem. Eng., 51, 121 (1973). Burke, S.P., Schuman. T. E. W., Proc. 3rdInt. Conf. Bituminous Coal, 2, 485 (1931). Coffin, K. P., Brokaw, R. S.,N.A.C.A. Tech. Notes, No. 3929 (1957). DeGraaf, J. G. A., 6rennst.-Warme-Kraff, 17, 227 (1965). Dutta. S.,Wen. C. Y., Belt, J., Prepr., Div. FeIChem., Am. Chem. SOC., 20(3), 103 (1975). Ergun, S.,Mentser, M., Chem. Phys. Carbon, 1, 204 (1965). Field, M. A., Gill, D. W., Morgan, B. B., Hawksley, P. G. W., "Combustion of Pulverized Coal," BCURA Leatherhead, Cherey and Sons, Ltd., Banbury, England, 1967. Frank-Kamenetskii, D. A., "Diffusion and Heat Exchange in Chemical Kinetics," Plenum Press, New York, N.Y., 1969. Gray, D., Cogoli, J. G.,Essenhigh, R. H., Adv. Chem. Ser., No. 131, 72 (1973). Hedden, K.. Lowe, A., Carbon, 5, 339 (1967). Heid, E. F. M. van der, Chem. Eng. Sci., 14, 300 (1961). Hougen, 0.A., Watson, K. M. "Chemical Process Principles Part Ill-Kinetics and Catalysis," Wiley, New York. N.Y., 1947. Howard, J. B., Williams, G. C., Fine, D. H., "FourteenLb Symposium (International) on Combustion," p 975, The Combustion Institute, Pittsburgh, Pa., 1973. Hugo, P., Wicke, E., Wurzbacher, G., Int. J. Heat Mass Transfer, 5, 929 (1962). Khitrin, L. N.. "physics of Combustion and Detonation." Moscow University Press, 1957. Kish, D., Ber. Bunsenges. Phys. Chem., 71, 60 (1967). Kurylko, L., Essenhigh, R. M., Symp. (Int.) Combust., 14th, Pittsburgh, 1375 (1972). Lee, E. S.,"Quasilinearization and Invariant Imbedding," Academic Press, New York, N.Y., 1968. Mulcahy, M. F. R., Smith, I. W., Rev. Pure Appl. Chem., 19, 81 (1969). Nusselt, W. Z., Ver. Deut. lng., 68, 124 (1924). Roberts, S.M., Shipman, J. S.,"Two Point Boundary Value Problems: Shooting Methods," American Elsevier, New York, N.Y., 1972 Smith, I. W., Combust. Flame, 17, 421 (1971). Smith, I. W.. Tyler, R. J., fuel, 51, 312 (1972). Spalding, D. B., fuel, 30, 121 (1951). Spalding. D. B., "Some Fundamentals of Combustion," Butterworths, 1955. Varma, A., Amundson, N. R., Can. J. Chem. Eng., 51, 206 (1973). Walker, P. L., Rusinko, F., Austin, L. G., Adv. Catal., 11, 134 (1959). Wicke. E., Wurzbacher. G..Int. J. Heat Mass Transfer, 5, 277 (1962). Williams, F. A.. "Combustion Theory," p 9, Addison Wesley, Reading, Mass., 19665.
Receiued for review May 17,1976 Accepted January 28,1977
Ind. Eng. Chem., Fundam., Vol. 16, No. 2, 1977
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