34
Ind. Eng. Chem. Fundam. 1984, 23,34-37
Diffusion and Reaction in a Stagnant Boundary Layer about a Carbon Particle. 8a. Effect of the Carbon Dioxide Reduction Reaction Enlo Kumplnrky and Neal R. Amundson’ University of Houston, Houston, Texas 77004
A pseudo-steady-state model for carbon combustion has been used to discern whether the C-COP reaction plays a major role In the solution of the problem. I t was shown that under many circumstances, typical of practical situations, such a chemical transformation coukl indeed be disregarded; however, as the numerical results indicate, this Is far from being a general principle. Before discarding the C-CO? reaction in the determlnatlon of the transient paths, it is recommended that the pseudo-steady-state loci be generated in order to ascertain the relevance of this heterogeneous reaction.
Introduction In a recently published paper, the authors verified how the reactivity of carbon toward Oz and COz affects the pseudo-steady-state structure in the combustion of carbonaceous spheres. On that occasion, in order to be consistent with the literature, the speeds of the C-O2 and C-COz reactions were concomitantly increased or lowered. Later, the role of the C-C02 reaction was assessed for conditions typical of pulverized fuel furnaces and atmospheric fluidized bed combustors (Kumpinsky, 1983). It was evident that under such circumstances the only significant heterogeneous reaction was that between C and O2 We then inquired whether we could extend this finding to other parameters. A pseudo-steady-state analysis enables us to answer this question. In the sequel we make use of a model that has been previously developed (Kumpinsky and Amundson, 1983). The concentration of active sites available to Oz is fixed so that the reactivity level between C and O2 is held constant. The effect of the C-COz reaction is measured by varying the concentration of sites accessible to carbon dioxide. The Model The theoretical development of the problem has been carried out in our earlier publication and will not be repeated at this time. However, a summary of the model is presented for the reader’s convenience. We consider a stagnant boundary layer surrounding a spherical carbon particle with the internal effects lumped at the surface; alternatively,this might be a hard impervious coke particle. COz,CO, Oz, and Nz are present in the boundary layer, the latter being regarded as inert. It is assumed that only O2 and Nz are in the ambient, In addition, a minute amount of water vapor appears in the gaseous phase so as to catalyze the reaction between CO and Oz. The chemical transformations of the model are the following. A t the particle surface R1: C + C02 = 2CO (heterogeneous, endothermic) R3: C +l/202 = CO (heterogeneous, exothermic) In the boundary layer Rz: CO + ‘ / 2 0 2 = C02 (homogeneous, exothermic) We will employ the DAS model of our former work, which postulates that the active sites are disjoint with respect to O2and C02;i.e., there is no competition for sites. The form of the homogeneous reaction is preserved as well.
A s before, we adopt Fick’s law for molecular diffusion and Fourier’s law for heat conduction and also assume that the transport properties and overall gaseous concentration are temperature independent (evaluated at To= 1000 K and at an absolute pressure of 101 kPa). We perform mass and energy balances in the boundary layer of radius b suurrounding a spherical carbon particle of radius a. We then obtain a set of four second-order nonlinear ordinary differential equations in xl, x2, x3, and T. Combinations among them reduce the system to one differential equation and three algebraic equations. We select the oxygen as the key component and treat the problem as an initial value one, guessing the unknowns at the particle surface. When an equation is enumerated twice, the first number refers to the present work while the second one corresponds to our earlier publication. In dimensionless form we can write
with initial conditions (5 = 0)
f
=
3 3 2; x3 = x3,;
(2,161
{=0
The algebraic equations are the following
where
In addition -xls
+ 2(x3b- x3,) = Rl + R 3
(7,24)
(8) x 2 s = Z(x3b - x3s - xl3 If the radiant flux is dependent, we determine the dimensionless ambient temperature by means of y
(
+~~
7b 7~ b ) ~
+ +2
( - 3c3J ~ + ~ hR, ~ - y7,4 -
0 ~96-43~3/84/~023-0034~0~ .50/0 0 1984 American Chemical Society
W
=0
(9,251
Ind. Eng. Chem. Fundam., Vol. 23, No. 1, 1984 35
The values of the constants, the dimensionless groups,
and the mathematical symbols are given respectively in Table I, Table 11, and Nomenclature of our former publication. The numerical procedure has been previously discussed and will be omitted here. T h e Region of Feasible Solutions The region of feasible solutions is the domain in the 7b vs. 7, plane wherein the locus of solutions must lie. It is bounded by three curves, succinctly described as follows. Limit A (xis = 0). This is a single film model for very slow homogeneous reaction and as a consequence negligible generation of C02.
where XAs
=
(11,27)
Limit B (x3, = 0). This is a double film model for infinitely fast homogeneous reaction, so that CO and O2 cannot coexist in the boundary layer.
where
Limit C (x2, = 0). This is a single film model for infinitely fast oxidation of CO, generated by the C-O2 and C-C02 reactions, so that the net result is C + O2= CO,.
where
Results and Discussions The effect of the C-C02 reaction on the pseudosteady-state analysis will be investigated by means of four types of graphics: (7b,Ts), (7b,%c), (Tb,mc) and ( a , ~ , ) . Figures lA, D, and G show the (7b,7s) loci for the prescribed conditions. All the parameters are held constant, with exception of [CT]. We notice that the lower the v d u e of [c,],the higher the limit B. This means that as [CT] diminishes, only at greater T , will the role of the C-C02
reaction become significant. In particular, for [CT] = 0 the limit B is totally absent since the only reactant consuming carbon is oxygen. In any case, we perceive that at lower surface temperatures the role of the C-C02 reaction is meaningless. Minuscule amounts of carbon dioxide are generated by the homogeneous reaction, so that the availability of C02 a t the surface is inconsequential. Furthermore, even with high xis, B1would be imperceptible at low 7,. As T~ escalates, there occur clear differentiations among the several loci in each figure. Such discrepancies are more conspicuous for the cases in which radiation equilibrium prevails. We observe that, for the same 7b and at higher T,, the lower [CT], the greater T,, since the endothermic consumption of carbon by C02 is curtailed. In Figures 1A and D the pathology of the problem is not altered by the decrease in [C,]. This means that the number of pseudo steady states, namely, 1-3-1, is not changed. However, impressive modifications take place if there exist five pseudo steady states in a certain range of Tb, such as in Figure 1G. When [CT] = lo4 sites/atom, as 7,advances upward, the combined consumption of C by C02 and O2 provides the boundary layer with large quantities of CO. Eventually, ignition occurs and the locus folds on itself, approaching the limit B. The C-C02 reaction is sufficiently high to generate the required CO to sustain ignition and most of the oxygen is consumed in the boundary layer. The structure is clearly 11-3-5-3-1. If [CT] = sites/atom, the morphology is 1-3-5-3-1 as well, but the upper fold is much less noticeable. Now, the C-C02 reaction is not as intense as in the previous case and a higher 7,is necessary for ."R1to become progressively more relevant. When [CT] = lo4 sites/atom, the structure imis 1-3-1. An even larger 7, is required to make portant. The gain in significance, however, is very gradual so that the locus is unable to fold on itself in order to yield up to five pseudo steady states. For [C,] = 0 the only heterogeneous reaction is R3and the absence of the limit B will impede the formation of the upper fold. Thereby, the only morphology possibility in this case is 1-3-1. Figures l B , E, and H present the (7b,Bc) loci, corresponding to Figures lA, D, and G. We observe that, until ignition occurs, there is absolutely no difference in the curves, regardless of the value of [CT]. After ignition, holding 7b fixed, the combustion rate is elevated as [CT] is magnified. It is interesting to consider the case [CT] = 0 in the (?b,."Rc) locus. Following ignition, 2, clearly tends to the value x3,,. This is an indication that the limit C is approached asymptotically,causing an increase in the heat generation close to the surface. The outcome would be a considerable change in the combustion time if the transient problem were solved. As we know from our earlier publication, as 7b escalates and the limit A or B is neared, %, tends toward 2 ~ This ~ is ~ a .clear modification in the pseudo-steady-state structure and it shows that the C-C02 reaction cannot always be neglected. Even if [CT] is small, such as lo4 sites/atom, still B, tends to 2~~~after passing through a local minimum, as is illustrated in Figures 1B and H. The molar ratio COz/(COz+ CO) leaving the boundary layer is plotted aggainst 7 b and is displayed in Figures lC, F, and I, corresponding to Figures l A , D, and G. Great dissimilarities between the curves for the various [CT] occur after ignition obtains. Hoever, at very high bulk temperatures they all tend to the limit m, = 1, since ignition takes place in every case examined here. Special attention sould be devoted to Figures 1D-F, where single particle radiation is considered. That is
36 Ind. Eng. Chem. Fundam., Vol. 23, No. 1, 1984
1 000
/ 025
050
075
IO0
I25
IX
rh
40/
30t
L
'OM
IW
120
1.40
t60
'b
loo WO
025
050
075
100
125
150
02
04
rb
06
08
O W ' 0 '2
06
04
OB
'b
Figure 1. Effect of carbon reactivity towards COz on the locus of solutions, combustion rate, and molar ratio COp/(COz + CO) leaving the boundary layer.
possibly the most realistic form for the radiant flux (Sundaresan and Amundson, 1980). A t conditions typical of fluidized beds, say, 1000-1200 K, we compare the reasonable value [c,] = 10" sites/atom with [c,] = 0. For that range of temperature, in both situations we note minor differences, indicating that in the solution of the corresponding transient problems the C-C02 reaction could be
neglected. This fact is explored in more detail in Figues 2A-H, where most of the conditions in Figures 1D-F are preserved. T~ is plotted as a function of a for a given Tb. The (Tb,T,) loci are generated for distinct particle radii, keeping the remaining parameters fixed, Le., b = 2a, X3b = 0.21, T, = T b and [c,*]= sites/atom. Then, for each particular q,,the values of T~ are obtained from the
Ind. Eng. Chem. Fundam., Vol. 23, No. 1, 1984
I50
[c+
id5r i t e r / a t a SO rb= 1.00
1.00
1.40
I .40 c-
c-
0.50
(C)
I .90
1 2.5
5.0
7.5
10.0 0
axlo4(m)
.2.5
95.0
07.5
k ' I(
-
, 0.900
axlo4(m)
1.90 -
2.5
9
.
5.0 7.5 a x io4h)
10.0
0
37
(0) I
I
I
2.5
5.0
7.5
I(
3
ox1o4(m)
I .5
/
"-y[cT1
ID
5.0
;5
10.0O. -O. I
u 2.5 5.0 7.5 10.0 0x10~
_ _
215 510 715 Id0 ax lo4(m 0 x 1 0(m) ~ (m) 0x104 ( m ) Figure 2. Dimensionless surface temperature as a function of the particle size. Pseudo-steadystate trajectories for different values of the bulk temperature: b = 2a; .%3b = 0.21, T, = Tb; [C,*] = sites/atom. i.5
loci ( T b , T s ) generated for every radius, resulting in the graphics displayed in Figures 2A-H. A qualitative discussion about this structure is provided by Sundaresan and Amundson (1980) and will not be repeated. Here, we are particularly interested in determining whether the C-C02 reaction can be neglected during the calculations. Clearly, [CT] has no influence on the unignited branch and on the intermediate unstable pseudo steady state. Small discrepancies can be visualized in the ignited pseudo steady state, and they become more discernible as the particle radius is reduced and/or Tb is elevated. In general, however, the differences in T~ are always small when we compare the results obtained with the moderate value [CT] = sites/atom to those acquired by means of [CT] = 0. This is a clear indication that, for the prescribed conditions, the single particle quantities required for reactor design, namely, the heat and mass fluxes at the external edge of the boundary layer, are hardly affected by the presence of the C-C02 reaction. Obviously, this affirmation is generally untrue when radiation equilibrium obtains, according to the previous figures, unless we are dealing with the unignited branch. Such possibility, nevertheless, is unacceptable from an engineering point of view. Summary and Conclusions A previously developed pseudo-steady-state model for the combustion of carbon has been applied to determine whether the C-C02 reaction may be totally neglected. It was substantiated that under certain conditions the role
of this reaction is minor. However, in many situations, especially if high bulk temperatures are considered, large discrepancies may occur in the (Tb,&) locus between [c,] # 0 and [CT] = 0. Under such circumstances, the transient paths could be pronouncedly different from one another. Before solving the unsteady-state problem without the C-C02 reaction, it is wise to establish the pseudo-steadystate structure including and disregarding R1in order to discern the validity of this assumption. Some morphological changes can be detected in the pseudo-steady state structure when the only heterogeneous reaction taken into account is the one between C and 02. First, the ( T b , T 8 ) locus approaches limit c when ignition occurs, rather than l i i i t B. As a consequence, at very high surface temperatures the net reaction is C + O2 = COz. Secondly, a maximum of three pseudo steady states are found instead of five. Acknowledgment The early part of this work was supported by the Department of Energy. The later parts were supported by no one except the University of Houston. Registry No. Carbon, 7440-44-0; carbon dioxide, 124-38-9.
Literature Cited Kumplnsky, E. Ph.D. Dissettatlon, University of Houston, Houston, TX, 1983. Kumpinsky, E.; Amundson, N. R. Ind. Eng. Chem. Fundam. 1983, 22, 62. Sundaresan, S.; Amundson, N. R. Ind. Eng. Chem. fundam. 1980, 19, 344.
Received for review August 23, 1982 Accepted December 10, 1983
