Ind. Eng. Chem. Fundam. 1983, 22,62-71
62
Diffusion and Reaction in a Stagnant Boundary Layer about a Carbon Particle. 8. Comparison of Lumped and Distributed Chemical Kinetics Enlo Kumplnsky and Neal
R. Amundson'
University of Houston, Houston, Texas 77004
A simplified model of heat and mass transfer was developed, describing the diffusion and reaction phenomena in a stagnant boundary layer surrounding a burning carbon particle. The model comprises thermal radiation and heterogeneous reactions with intraparticle effects lumped at the surface. Langmuir-type or lumped first-order kinetics was employed for the heterogeneous chemlcal transformations. The degree that carbon reactivity and competiion between O2 and C02 for carbon active sites affect the pseudo-steady-state structure of the model was determined. Results for a large range of parameters showed that carbon reactivity is of prime importance in the solution morphology: the competition for sites, however, is of l i b consequence. An important aspect of the present study is the comparison between the resutts obtained by means of Langmuir and lumped first-order heterogeneous kinetic expressions. For a given set of parameters, it was verified that computations with both reaction rate categories essentially display the same qualitative features. However, significant quantitative discrepancies may occur, especially for combustion rates at low surface temperatures.
Introduction Due to its utilization as an alternative energy source to oil and natural gas, coal has drawn the attention of many investigators. In particular, the search for a more efficient performance during the combustion of carbonaceous materials has been the reason for several experimental and theoretical works, summarized in bibliographic reviews by Caram and Amundson (1977) and Laurendeau (1978). Carbon combustion at high temperatures is a process that involves the diffusion of oxygen through a stagnant film, followed by its reaction at the carbon surface, generating mainly CO. The CO diffuses away from the surface and may be in part oxidized to C02. This in turn may either disseminate toward the ambient or react with carbon to produce CO. Carbon combustion is certainly a transient phenomenon, but a pseudo-steady-state analysis preserves the qualitative features of the dynamic process (Sundaresan and Amundson, 1980a). In the past 60 years, many pseudosteady-state mathematical models were examined. The majority of these utilized either a single-film or a double-film formulation, in which the location where the CO oxidation takes place must be a priori specified. With very few exceptions, to discern the validity of each of these models for specific circumstances is not a trivial matter. To circumvent this inconvenience,Caram and Amundson (1977) developed simplified heat and mass conservation balances for the boundary layer surrounding a carbon slab or sphere, restricted to a fixed ambient, lumping the intraparticle effects at the surface or assuming an impervious sphere. It was disclosed from the solution locus in the plane having ambient temperature as the abscissa and surface temperature as the ordinate that one can determine the presence of pathological behavior, Le., the existence of one, three or five pseudo-steady states. Natural extensions have followed the work by Caram and Amundson (1977): Mon and Amundson (1978,1979, 1980), Sundaresan and Amundson (1980a,b, 1981) and Srinivas and Amundson (1980, 1981a,b) which are prerequisites to this one. In all of these studies, lumped first-order kinetic expressions were used for the C-O2 and C-C02 heterogeneous reactions. Moreover, the transient model proposed by Sundaresan and Amundson (1981) agreed only in part
with existing experimental data. These authors reasoned that a better agreement between the results of their computations and the experimental information might be achieved possibly by utilizing a more sophisticated description of the heterogeneous reaction phenomena. Specifically, they as well as others (e.g., Williams, 1979) suggested that the study of the competition among the reactive species for active sites at the carbon surface would probably be needed. As an extension to the previous work, we propose to examine a simplified pseudo-steady-state model using Langmuir-type kinetic expressions for the heterogeneous reactions. By utilizing appropriate elementary mechanisms, we can include or neglect the competition for active sites. Qualitative and quantitative comparisons with a model that makes use of lumped first-order kinetic expressions for C-O2 and C-C02 reactions can be accomplished. The effect of carbon reactivity will now be investigated in more detail than before (Caram and Amundson, 1977; Mon and Amundson, 1978). Important aspects of the model such as boundary layer thickness, ambient oxygen concentration, and role of radiation will also be covered. T h e Models We will consider a stagnant boundary layer surrounding a spherical carbon particle with internal diffusional and reactive effects lumped at the surface, although the analysis is equally applicable to hard impervious coke particles. The molecular species present in the boundary layer are COP,CO, and 02,respectively labeled by subscripts 1,2, and 3, as well as N2, regarded as an inert. We assume that only O2 and N2 are present in the ambient. A tiny amount of water vapor appears in the gaseous phase and its only role is to catalyze the reaction of CO oxidation. The reactions of the model are R1: C + C02 = 2CO (heterogeneous, endothermic)
R2: CO
RS: C
+ 1/202 = C02
+ 1/202 = CO
(homogeneous, exothermic)
(heterogeneous, exothermic)
The transport and the reaction models in the boundary layer will now be established.
0196-4313/83/1022-0062$01'.50/0 @ 1983 American Chemical Society
Ind. Eng. Chem. Fundam., Vol. 22, No. 1, 1983
The Transport Models In a previous work, Sundaresan and Amundson (1980a) concluded that the qualitative features of the pseudosteady-state structure were weakly sensitive to the complexity of the transport model in the boundary layer. Hence, for simplicity, we adopt Fick's law for molecular diffusion and Fourier's law for heat conduction, together with the assumptions of temperature-independent transport properties and overall gaseous concentration (calculated at To = 1000 K). T h e Reaction Models (1) Carbon and Carbon Dioxide. The oxygen-exchange mechanism described by Ergun (1956,1961) will be chosen. The reduction of COP to CO at the reactive carbon sites is reversible and is expressed by
c02 + Cf
++co + rl hl
C(0)
The transfer of carbon from the solid phase to the gas phase, namely, the gasification process, may be given by
(2) Oxygen and Carbon Monoxide. This reaction occurs in the presence of water vapor as a catalyst. Howard et al. (1973) discuss that carbon monoxide burnout rate can be approximately described by the use of the radical mechanism CO + O H - C02 + H
with the following equilibration equations H + 0 2 F' OH + 0
0 + H2 2 OH + H OH + H, 2 H 2 0 + H The equilibration conditions are used to eliminate the concentrations of hydroxil and hydrogen radicals. (3) Carbon and Oxygen. We will make use of the mechanism proposed by von Fredersdorff and Elliott (1963),but considering constant the number of active sites. Initially, oxygen is dissociated into C*(O),the chemisorbed atomic oxygen, and the free radical 0 C*f + 02 2% C*(O)
+0
Table I. Physical Data c = 1.22 X kgmol/m3 cD = 1.559 x kg-mol/(m s) E,/R= 11424 K EJR = 15098 K E,/R= 19000 K E,/R= 29190 K E , / R = 19000 K KO,= 4.15 X IO3 k,, = 8.1 X lo's-' (m3/kgmol)1'2 k,, = 3.28 X 1014m3/(s kg-mol) k, = lot4s-'
63
k,, = 8 x 10" s-' n, = 1.661 X lo-'' kg-m ol /site 2.5 X lo2' atom/m2 To= 1000 K (-AH,) = -1.725 X 10' J / k g m o l of C (-AH,)= 2.830 X 10' J/kg-mol of CO (-AH,)= 1.106 X 10' J/kg-mol of C E = 0.93 h = 5.44 x w/ (m K ) a = 5.676 X lo-' W / ( m Kz)* sg =
The DAS model deals with the above presented Langmuir-type mechanisms, assuming disjoint sets of actiue sites. Some sites are exclusively attacked by O2 whereas the remaining sites are attacked solely by COP The CFS model is characterized by partial competition for sites. Some sites are attacked only by O2whereas the remaining sites are shared by COPand 02.The reason for not considering sites exclusively attacked by C02 in the CFS model is that, as it has been widely publicized in the literature, the carbon reactivity to O2is much larger than the carbon reactivity toward COP. Hence, it is plausible to assume that most sites attacked by C 0 2 can also be attacked by 02,but not the other way around. The LFO model makes use of lumped first-order kinetic expressions for the heterogeneous reaction rates. The lumping process is performed only after solving the problem with the DAS and CFS models. DAS Model. For the C-C02 reaction, the molar fluxes of C02 and CO at the surface are respectively Nco, = -rl + rl' and Nco = r1 - rl' r4. If a pseudosteady state is assumed, then rc(o) = rl - rl' - r4 = 0. Hence, Nco, = -r4 and Nco = 2r4. Developing the rates we obtain
+
Knowing that 6 I +:6 = 1 and assuming pseudoequilibrium, i.e.
In the next step the atomic oxygen is chemisorbed
(3) then we can write
Since the latter step is extremely fast (Laurendeau, 1978) we assume that the atomic oxygen does not interfere in the equilibration mechanism of the CO-O2 reaction. Finally, the carbon atom is gasified
Kinetic Expressions (1) Homogeneous Reaction. Based on Howard et al. (1973),we will employ the following rate expression, valid between 840 and 2360 K R2 = k 2 [ C O ] [ 0 2 ] 1 / 2
60' =
1
(4)
The assumption of pseudoequilibrium rather than equilibrium for eq 3 avoids its contradiction with eq 2. For the C-O2 reaction, the molar fluxes at the surface are No, = -r3, No = r3 - r i , Nc*(o)= r3 + r3' - r5, and Nco = r5. With the assumption of a pseudosteady state, No = Nc.(o) = 0, which implies that No, = -r5/2. The pseudosteady-state balances are
(1)
where k 2 = k,, exp(-E2/R7') and kO2= k'02[H20]1/2.The brackets represent molar concentrations of the respective species. The value of kO2given in Table I corresponds to a water vapor mole fraction of 0.0032 at 1000 K. (2) Heterogeneous Reactions. We are going to develop expressions for the DAS, CFS, and LFO models.
r-
Keeping in mind that Bf*'
+ O0*I
= 1 we get
64
Ind. Eng. Chem. Fundam., Vol. 22, No. 1, 1983
(7)
The surface rates of carbon gasification due to reactions C-C02 and C-O2 in kg-mol of C/(m2 s) are respectively R1 = r4 = r~$~k,[C~I]8gI and R, = r5 = n$gk5[CT*']f?O*'. CFS Model. Expressions for sites attacked only by O2 are identical with the ones described above, with superscript I in [CT*'] replaced by 11. In order to develop the formulas for sites shared by COz and 02,we assume that the mechanisms for the individual heterogeneous reactions remain valid. Knowing that el1 + 02' + Bo*" = 1 and substituting superscripts I for I1 in eq 2, 3, 5, and 6, we obtain a system of equations whose solution for 8," and do*" is eoII
=
1
{
(8)
[COI 1 + 1+--2k~l."21)Kl[C02] (&*I1
1
=
(
Kl[COZI
1 + l+--
Lc01
(9)
k, )2k3[021
The gasification rates are R1 = r4 = n$ kI[CT'I]etI and R3 = r5 = n$gk,{[CT*11]80*1 [cT1']80*'f. There are two comparison possibilities between DAS and CFS models. If we set [CT*I'] = [CT*'], then the concentration of sites available to O2 in CFS model, compared to DAS, is increased by [CTII] and the global concentration of sites in both models is kept the same. On the other hand, if we make = [CT*I] - [Gn],the number of sites available to O2 in CFS model remains the same as in DAS, but the global concentration of active sites is reduced by_ [CT'']. LFO Mode!. For this case we have R1 = r4 = k4[C02] and R3 = r5 = k5[02).Solution of the DAS and CFS models showed that the Arrhenius configuration was the best one for k4 and k5. All the remaining reaction rate and equilibrium constants utilized in this work have Arrhenius form. The numerical data for K1 was obtained from Ergun (1956, 1961) and covers the range 1073 < T < 1673 K. An estimate of It4 was based on the work of Menster and Ergun (1973). The rate constants k3 and k5 can be obtained from the study of Lewis and Simons (1979),who used a different mechanism from that presently employed. With the adopted assumptions, however, both mechanisms lead to equivalent expressions for Bo* when surface temperatures do not exceed 2000 K. The evaluation of ko4 and k05 involved the initial selection of [CT] and [(&*], respectively. Yet, as observed by Menster and Ergun (1973),the choice , of the total concentration of active sites is immaterial for comparison of relative reaction rates. If we assume that one out of lo5 carbon atoms is active to C 0 2 as our standard case, we will be provided with the flexibility of varying [CT] over several orders of magnitude. Laurendeau (1978) argues that the speed of the C-O2 reaction cannot be explained unless [cT*] is substantially higher than [CT]. Hence, we arbitrarily set [cT*] = 1oo[cT] without loss of generality. Dutta et al. (1977) and Dutta and Wen (1977) report different reactivities of carbon toward COPand 02, respectively, noting that samples presenting the higher reactivity to C02 were also more reactive toward 02. Mathematically, it is possible to vary [C,] keeping [CT*] unchanged or vice versa. However, to be physically consistent, reactivity to C 0 2 and O2 must be increased or lowered concomitantly during the solution of the problem.
+
Table I provides the numerical data required for this study. For all the calculations it was assumed the pressure was constant at 101 kPa. Boundary Layer Problem The mass and energy balances in the boundary layer of radius b that surrounds a spherical carbon particle of radius a can be written as
Appropriate boundary conditions at the particle surface (r = a) are
(12)
dT -A= (-AHl)Ri + (-AH3)R3 -qr (13) dr At the external edge of the boundary layer (r = b ) the values are fixed x i. = x i.b i* T = Tb (14) The term qr denotes the heat flux which accounts for the radiant interaction between the surface of the particle and the ambient. It takes the following form (Mon and Amundson, 1978): qr = m(T,4 - Tm4).Three cases have been considered: radiation equilibrium, T , = T, (Caram and Amundson, 1977), together with single particle radiation, T, = Tb, and interacting particle radiation, both defined by Mon and Amundson (1978). In the latter case T, is regarded as a parameter. We set in general T, = Tf + pTb, where Tf and p are constants. When Tf = 0 and p = 1 we yield the case of single particle radiation. When Tf and p are positive but otherwise arbitrary constants we produce interacting particle radiation. In this work computations were carried out with T, = T, and T, = Tb. Solution Performing appropriate combinations between eq 10 and 11 and subsequent integrations we can reduce the original system to one nonlinear second-order differential equation and three algebraic equations. We choose oxygen as the key component and treat the problem as an initial value one, with unknowns guessed a t the particle surface. Casting the equations in dimensionless form with dimensionless groups shown in Table I1 we obtain
Ind. Eng. Chem. Fundam., Vol. 22, No. 1, 1983 65 Table 11. Dimensionless Groups
A
h
h
[(aa)’R21/ P j = Poj e x ~ ( - @ j / ~ ~=) ;4, . i5 [ c D ( 1- a t ) ‘ ] w = (-AH,)cD/T,h y = [ W O E To‘1 / [ cD(-AH,)] a = 1 - (a/&) E = [1 - ( u / r ) I / a 7 = T/T, Pol = Mol Po2 = [ ( u ~ ) ~ ~ , , c ” ~ ] / c D= - E , / R T , 0, = kos/2ko,c @ j =Ej/RT,; j = 2, 4 , 5 P w = ( a a n $ g k w [ C ~ l ) / C D f 3 = (hEs-E,)/RT, Pos= (aan,s,k,,[C~])/cD @ j = Ej/RT,;j = 4 , 5 dl =
physical feasibility region by imposing the obvious condition that the mole fractions of C02, CO, and O2 be simultaneously positive. It is the domain in the f b vs. T, plane wherein the locus must lie. The three limiting curves of the feasibility region are defined as follows. Limit A (xls= 0). This corresponds to a single film model with very slow homogeneous reaction and thus negligible generation of C02. DAS and CFS:
fbA
=
7,
-w
(
h*-
where B = p5* for DAS, B = p5*
- QRA
l:!5 x3s
)
(26)
+ P5 for CFS, and
Rl and R 3 are presented below, according to the heterogeneous kinetic model DAS:
2 1
.,
84
=
%3=-
x2s
1 + 81-
w1=
2)-
04
1
+ (1 +
(21)
83
\
- + 7
2 P5 Limit B (x%= 0). This double f i i model refers to the case of infinitely fast homogeneous reaction so that CO and O2cannot coexist in the boundary layer. For this limiting situation
1+x3s
Xls
CFS:
85*
; 933 =
81x2s
Xls
85*
83
1+x3s
+
85
1+
( L:2a):a 1+-
(22)
-
LFO: R1 b4xls; %3 = bBX3, (23) Equations 15 were solved by means of a semiimplicit Runge-Kutta method, described by Villadsen and Michelsen (1978). The procedure was successfully applied by Sundaresan and Amundson (1980a) and can be adapted to the present system of equations. The scheme requires that x % < x3b be fixed and r Sbe guessed using sensitivity functions. When a desired accuracy is achieved, successively smaller values of x% are adopted, so that the (fb,f,) and (fb,%,) loci are established. Knowing f s for each iteration step and the fixed value x3, we are able to obtain x l S and x2, in order to compute the reaction rates. With this intention we combined eq 10 for C02 and O2 in dimensionless form, followed by integration from t = 0 to 1, yielding -xls 2(x3b- x33 = R1 R, (24) If we now introduce eq 21, 22, or 23 into eq 24, followed by the elimination of x2 at the surface (DAS and CFS), Le., xa = 2 ( - x3, ~ - ~xle),~we get an algebraic equation in xls as a function of known x3, and guessed T,. The resulting equation is linear for LFO, quadratic for DAS, and cubic for CFS. For DAS and CFS the roots are real, but only one is blessed with physical significance. If the radiant flux depends on f b , we combine eq 10 for O2 and eq 11 in dimensionless form, integrating from 5 = 0 to 1
+
y(ff
+
Tb + pfb)4 + + 2(xgb- x3,) + hR,- y f t - ’W.= 0 W
(25) A Newton-Raphson procedure is used to solve eq 25 for f b . In view of eq 24, we can also write 3,= xls + xzs. Hence, when diffusion totally controls the process, R,,, = 2x3b. The Region of Feasible Solutions Caram and Amundson (1977) conceived the idea of a
where the dimensionless position of the flame front is given by
34
LFO:
Limit C ( x P s= 0). This is in conformity with a single film model for infinitely fast oxidation of CO, generated by the reactions of carbon with C02and O2so that the net reaction is C + O2 = C 0 2 DAS and CFS: fbC
=
T~
-w
[
(h* +l)( p4 +
3)] - QR‘
1+_ & .
3s
(32) where
66
Ind. Eng. Chem. Fundam., Vol. 22,
No. 1, 1983
0 751
E
053-
3 25-
000!
c 70
r,, larbilrary units1
b
(e)
050
075
iG0 Tb
I25
I50
'670
095
i20
145
I70
b'
Figure 1. Evolution of the feasibility region, molar ratio C02/(C02+ CO) leaving the boundary layer and combustion rate with increasing B, E, F: T , = Tb;A and B: dashed arrows indicate reactivity increase. carbon reactivity: a = 5 X lo4 m; b = 2a; XQb = 0.10; A, C, D: T, = T,;
It can be shown that, as the surface temperature increases, equations for limiting cases A and B asymptotically tend to TbAB
=
T,
- w(2X3&* - QRAB)
(35)
Curves generated by equations of the limits B and C will cross at the intersection of their restrictions, Le., iaBC= $ ~ ~ / (&4/r3b) ln for DAS as well as CFS and 7sBC = 44/ln Po4 for LFO. At these particular T,,6 of limit B and xQSof limit C become exactly zero and it can be demonstrated that TbBC
=
TsBC
-W[X3b(h
+ 2) - QR~']
(36)
As the surfase temperature decreases, the groups p4, p5, p5*, p4,and p5 go to zero and thus equations for limits A and C reduce to TbAC
=
T,
iwQR*'
(37)
Due to the exponential nonlinearities, in order to generate the feasibility region it is simpler to fix T , and then calculate Tb. When Q R is Tb-dependent,a Newton-Raphson scheme is employed. Results and Discussion We initially compare DAS and CFS models in order to analyze how the competition for carbon active sites affects the pseudo-steady-state structure of the problem. We set [cT*] = 1oo[cT] according to a previous discussion and keep all the conditions fixed. We now vary the heterogeneous kinetic model from DAS to the two possibilities of CFS model, namely, to hold the overall concentration of active sites constant and retain unchanged the total amount of sites available to 02.Comparing the results, we verified that the discord between surface temperatures at the same bulk temperature was at most of the order 1 K in the ( T b , T , ) locus for the large range of parameters
examined. There were no noticeable qualitative and quantitative dissimilarities in the (7b,n,) locus, either. The lack of meaningful differences between the models was further corroborated by imposing severe conditions. Indeed, by keeping [CT*] constant and promoting a large increase in [C,] ,the number of shared sites is substantially augmented. Under these conditions, no morphological changes were noticed due to the choice of different model; i.e., the number of pseudo steady states was not altered. Quantitative differences were minor also. We now will perform carbon reactivity and parametric studies by means of the DAS model. The following results are equally valid for the CFS model, since the figure sizes do not allow the detection of such minuscule disparities. Figures 1A and 1B show how the reactivity of carbon affects the feasibility region while keeping all the other conditions fixed. Each limiting case is treated separately in those figures, in order to avoid any overlap among the curves. In actuality, lines I, 11, and IV match lines 111, V, and VI, respectively. The boundary curves, described by eq 35, 36, and 37, correspond to lines 11-V, 1-111, and IV-VI, in this sequence. Clearly, the boundary curves do not depend on the carbon reactivity. The solid lines are obtained from eq 26, 29, and 32 corresponding respectively to the limits A, B, and C. As the carbon reactivity increases, the curves of the three limiting cases move downward and to the left, enhancing the sigmoid features of limits A and C. This raises the possibility of multiple pseudosteady states for the solution locus. Lines I through VI in Figure 1A are all parallel to the straight line 7, = T~ since Q R = 0 (radiation equilibrium). This is not true for Figure lB, where it is assumed that single particle radiation obtains. Figures 2 and 3 show the feasibility regions (dashed lines) and the reaction loci (solid lines) for certain values of the parameters. Some figures contain results obtained
Ind. Eng. Chem. Fundam., Vol. 22, No. 1, 1983 87
0.75
0.75
F
I
1 I
0.00 0.25 0.50 0,75
1.00 1,25
1
1
I
l
000 0 2 5 0 5 0 0 7 5
1.50
l
1
1.00 1,25
1.50
Tb
Tb
1
' o
i
(E)
"
,
,
I
I
5x10
200 I 75 r
I .75
I
I50
-
1.50
e" I25
1.25
1.00 r
d ,i/
i
075
0 5 0 075
I00
125
150
0.75
-
1 G . LP
L---LL
050
175
_
L
075
_
_
-
Lanpmuir L # n e ' # c s LdmDed k # n e t # C i
L
1 1
-
I 0 0 1.25
L
1.50
1.75
Tb
T.
(H) 200 1.75 -
e" L P - Lumped k 8 n e l ~ C I
0.751
,
000 025
,
/
0 5 0 0.75
100-
,
,
1.00
1.25
, 1.50
075
-
000 0 2 5
050 0 7 5
100
1.25
000 0 2 5
150
0 5 0 075
rb
1.00 1.25
1.50
Tt
Figure 2. Influence of carbon reactivity and kinetics on the feasibility region and reaction locus for various parameter sets.
by way of lumped kinetics, but for the moment we are just concerned about Langmuir kinetics. We warn that, when temperatures are outside the validity range of the kinetic parameters, our results do not have physical meaning; they just express the mathematical behavior of the system. Figures 2A-C show that, for radiation equilibrium, the area bounded by limits A, B, and C slightly diminishes as the reactivity of carbon increases. This is the case when curve c is totally contained in the q, X 7,plane. If curve c is discontinued, however, the feasibility region will substantially narrow with ascending [C,], such as in Figures 3A-D. Conversely, Figures 2D-F show that, by assuming single particle radiation, the area inside the feasibility region expands with increasing [CT]. This occurs as long as limit c remains totally contained inside the q,X T~ plane. Independently of reactivity, we noticed that by diminishing the oxygen ambient mole fraction and/or increasing the radiative effect the feasibility region will narrow. The effects of the particle radius and boundary layer thickness on the feasibility domain can be simultaneously analyzed. Equations 26 to 34 are affected by pj. When j = 4 or 5 three dimensionless groups linearly depend on
a, a,[CT],and [cT*]. If we increase or decrease a and/or a while keeping the remaining parameters fixed, the effect
on the feasibility region is similar to increasing or decreasing the carbon reactivity by the same amount. For the adopted transport models that effect will be exactly the same when we assume radiation equilibrium (QRdepends on aa but not on [CT] or [cT*]). For the same reactivity level, two particles will generate an identical feasibility region if the product aa is kept unchanged. In this way, by labeling two different particles as 1and 2 we can write alal= a2aZ, whence a2
b2 = 1
+ 4(: a2
; a2 Iala*
(38)
- 1)
Equation 38 shows a nonlinear relation and a restriction between particle and boundary layer radii. As an example, m and bl = m. we take a particle with al = 5 X Hence, for a particle with a2 = 5 X m to have the same feasibility region as the former, its boundary layer radius m, according to eq 38. Thus, should be b2 = 5.26 X the regions of feasible solutions in Figures 2A-C ( a = 5 X
68
Ind. Eng.
Chem. Fundam., Vol.
22, No. 1, 1983
2 00 I75 I50 c” I
IO0
.’i I00
000 0 2 5 050 0 7 5
25
P ,
,umped
kqetcs
075
125
I50
000 0 2 5
050
075
100
125
150
rb
rc
(C1
(D)
I I 7 --7----
i
-
125-
--_ II O O -
5
- _-. O
L
J 075-_ - L-.000 0 2 5 0 5 0 0 7 5
. . . l
100
I25
z
150
L
1
030 025
.-lL100 I 2 5 I
050 0 7 5
1
~
L
150
70
Figure 3. Effect of carbon reactivity and kinetics on the feasibility region and reaction locus for a = 5 = T,.
m, b = 2a) correspond to the feasibility regions of a particle with a = 5 X m and b = 1 . 0 5 2 ~ .As to the (7b,Ts) locus, we cannot establish a relation similar to eq 38 since the dimensionless homogeneous reaction rate B2 presents a nonlinear relation between a, cy, and E. For particles under radiation equilibrium, eq 35 shows that the asymptote for high temperatures is independent of the particle radius and boundary layer thickness. Similarly, eq 37 indicates that particles under radiation equilibrium or single particle radiation will display the same feasibility region at low temperatures ( T , = T b ) , independently of a and b. The effect of carbon reactivity on the solution locus has a characteristic trait: the higher the carbon reactivity, the closer the ( q , , ~locus ~ ) gets to limit A. This is due to the fact that the COPproduced is more rapidly consumed when [C,] increases. It is interesting to study the effect of reactivity on a particle immersed in an atmosphere with high oxygen concentration, Figures 3A-D. For carbons with low reactivity, Figure 3A, the C-C02 reaction is slow and the (Tb,Ts) locus tends to remain far from limit A between asymptotes 35 and 37. Carbons with higher reactivities than the one previously discussed, shown in Figures 3B-C, present a (Tb,Ts) locus with up to five pseudo steady states. When T* is low, CO is produced by the C-O2 reaction and the homogeneous reaction is comparatively slow, which means that only small amounts of COz are produced. Furthermore, the consumption of COz a t the surface is enhanced when [C,] becomes larger, so the locus stays close to limit A. As the surface temperature rises, more and more CO is generated. This CO diffuses toward the
X
m, b = 2a, xQb= 0.21, and T ,
boundary layer, promoting a large increase in the homogeneous reaction. Subsequently, the C 0 2 production is intensified, moving the (Tb,Ts) locus away from limit A. As an immediate consequence, the locus approaches limit B due to the elevated consumption of O2 in the boundary layer, folding itself and bringing about the five pseudosteady states. For carbons with even higher reactivity, Figure 3D, the COz production in the boundary layer cannot compensate for the high effect of the C - C 0 2 reaction. Thereby, the C 0 2 surface mole fraction is kept low and the ( T ~ , T locus ~) remains always close to limit A. Thus, there exist at most three pseudosteady states for this set of parameters. The approach of the locus toward limit A when the carbon reactivity is magnified does not necessarily imply that less C02is generated. It may indicate that more COz is consumed at the carbon surface. This point is corroborated by means of Figures 1C and E, which exhibit the behavior of the ( T b , m c ) locus when the carbon reactivity is varied. Other factors, acting singularly or in combination, that make the (7b,TS) locus approach limit A are the following: decrease of the boundary layer thickness, particle size as well as ambient oxygen concentration, and increase of the thermal radiation. Figures 1D and F show how carbon reactivity affects the rate of gasification for the prescribed conditions. Mie clearly notice that the curves tend to ?.?,, = 2X3b. We also perceive that, as carbon reactivity increases, the loci progress toward the left and the stable, upper branches move upward. Comparing our (Tb,%,) loci with e d e r ones, we discern a feature that only appeared in Srinivas and
Ind. Eng. Chem. Fundam., Vol. 22, No. 1, 1983 69
LG
-
Lonqmusr kinetics
L P - Lumped hInc1ics
os
10
12
I4
16
07
II
13
15
06
OS
IO
09
Tb
LG - Lonpmuir kinCtiCI LP- Lumped h e t i c s
I15
0II'
I3
I
i
17
19
02
04
Tb
Tb
Figure 4. Influence of kinetics on the combustion rate for different data sets.
Amundson (198la). In previous boundary layer work with intraparticle effects lumped at the surface, in the presence of three pseudo steady states, the upper branch displayed a steady increase in R,with rising 7,,. In the present work we notice that the upper branches may have Rc stabilized or even decreased as 7 b rises. When this happens, Rc eventually starts to increase as 7 b is elevated, up to 2 ~ This can be explained by the following argument. Based on experimentalwork such as the one by Dutta et al. (1977) and Dutta and Wen (1977),we adopted an Rl/R3relation several times smaller than Mon and Amundson (1978) and Sundaresan and Amundson (1980a) did. Now, if the production of COPin the boundary layer is intense, highly reactive O2is replaced by slowly reactive C02as the main gasifying agent. Hence, as 7 b increases, R1becomes more relevant while R3loses its importance. If the increase in Rl cannot properly compensate the decrease in R3 the gasification rate will drop. At high temperatures, R1 eventually becomes intense and 3, starts to increase again as 7 b rises. We can understand the process by means of Figure 4A. We observe that, between the points labeled 1and 2, the gasification rate is primarily due to the C-Oz reaction. The dashed lines represent the contributions of R1 and R3 for the carbon combustion rate after point 2 is reached. We notice that R1 increases as 7 b rises while R3lessens. When we attain point 3, Bc practically becomes equal to R1. A more dramatic decrease in Rc for the upper branch as 7 b is increased can be visualized in Figure 4B. For larger b, the production of CO, by means of the homogeneous reaction is intensified. Hence, more
O2 is replaced by COz as gasifying agent and the decay in 32, is abrupt for 7 b ranging from 0.76 to 0.86 (curve labeled LG). For 7 b larger than 0.86, 2, steadily increases up to the limit 2x3bdue to the significant increase in Rl. Srinivas and Amundson (1981a) refer to work in which this interesting phenomenon was experimentally evidenced. ~ ~The. lumping effect can be studied by comparing the solutions of the DAS and LFO models. The lumping of DAS must be performed for a specific set of parameters that will be considered standard. For the standard conditions we obtain the lumped equations that generate a feasibility region as well as (7b,78) and ( 7 b , R c ) loci that are as close as possible to their respective Langmuir-type counterparts. Once we succeed in establishing a good match between DAS and LFO models for the standard conditions, identical kinetic data must be tested for other circumstances. Amundson and co-workers employed the same lumped kinetic information, independently of the parameters. We want to verify the qualitative and quantitative consistency of that assumption. We adopted the data of Figure 2B as our standard matching conditions. We initially tried to match the rates at an assigned 7 s , fixing Ej at the respective values utilized by Caram and Amundson (19771, in order to determine the preexponential factors. The results were very poor. However, solving the equations for the standard case, we verified that In ( R J x ~ and J In ( R 3 / x g S )lie almost 0: a straight line when plotted against 7 ~ We ~ computed . k , and Ej, j = 4,5 by means of linear regression using the least-square method and exponential regression utilizing Newton-Gauss, with
70
Ind. Eng. Chem. Fundam., Vol. 22,
No. 1, 1983
advantage for the latter. Hence, we adopted the follo-wing values generated by the Newton-Gauss erocedure: kO4= 5.614-X lo8 m/s, k05 = 1.629 X lo7 m/s, E J R = 36634 K, and E,/R = 21843 K. Figure 2B shows that the (Tb,T,) locus as well as the limits A and C obtained by way of the LFO model are in very good agreement with their respective DAS counterparts. The limit B tends to differ for lower values of Tb, but the discrepancies diminish as Tb rises. Limits B for the DAS and LFO models fail to agree at lower Tb for two main reasons. First, eq 29 with 31 produce lines that deviate less from straightness than curves generated by means of eq 29 with 30. Second, the restrictions accompanying eq 30 and 33 are structurally different from restrictions for eq 31 and 34, since only the former ones depend on the oxygen bulk mole fraction. Hence, it is not possible to equalize the point that limit B encounters limit C for both DAS and LFO models without seriously affecting the good agreement for limit C. Concordance for limit B as well as (Tb,T8)and (sb,%c) loci are also considerably worsened. Figure 4A shows that the (Tb,%,) locus was hardly affected by the lumping effect when computations were performed with our standard data. In order to test the LFO model against DAS, we initially perturbed one variable at a time. Later, all parameters were simultaneously changed. Comparison can be visualized in Figure 2C, altered carbon reactivity; 2G and 4B, increased boundary layer thickness; 3B and 4E, x3b equal to 0.21 instead of 0.10; 2H and 4C, particle radius of 5 X m; 2E and 4D, assumption of m replaced by 5 X single particle radiation; 21 and 4F, where we combine all the previous perturbations. We clearly notice that there is no qualitative disagreement. If quantitative discrepancies occur in the (7b,T8)locus, they are small in most cases. However, relatively high quantitative disconformities may take place in the ( T b , n , ) locus, especially at carbon gasification rates that correspond to lower surface temperatures. In this situation, small deviations in T* may cause large variations in the exponentials of the rate constants. We can utilize lumped instead of Langmuir kinetics when we are working under the conditions that the lumping was performed. Nevertheless, a careful investigation must be carried out if the same lumped rates are going to be used with different parameters, in order to avoid relatively large quantitative errors. These discrepancies, however, cannot explain why the numerical computations of Sundaresan and Amundson (1981) displayed a considerable quantitative disagreement in the burnoff time of large carbon particles, when compared to the available experimental data. When x3b was held constant on a wet basis, the model of Sundaresan and Amundson (1981) predicted that the burning time was virtually independent of the water vapor level. When x3b was maintained constant on a dry basis, the model showed that the particle would burn at a slower rate in moist air. However, the authors conceded that the differences in the combustion rates in dry and moist air conditions predicted by their model were much smaller than the experimentally observed values of Smith and Gudmundsen (1931). This disagreement may be explained a t least in part by the following argument. For small particles, C02 formation is negligible and the C-O2 reaction is responsible for the carbon combustion. For larger particles, however, ignition does occur in the boundary layer, if water vapor is present. If the CO-O2 reaction is intense, COz replaces O2 as the main gasifying agent. It has been observed that the relation was excessively high in previous studies of Amundson and co-workers. For a smaller ratio between
the C-C02 and C-O2 reactions, we have shown that the combustion rate may disclose a decay as soon as the CO oxidation in the boundary layer becomes intense. Hence, under this circumstance the differences in the combustion rates for large particles under dry and moist air conditions predicted by Sundaresan and Amundson (1981) would be magnified. Thus, comparisons with the experimental results would show a better agreement. In the present study with DAS, CFS, and LFO models we notice that the curve corresponding to limit C intercepts the curve of limiting case B with positive derivative dT8/dTb. In previous work we see interceptions with negative slopes, for example, Figure 4 of Caram and Amundson (1977) and Figure 6B of Mon and Amundson (1978). Differentiating eq 32 with respect to T , and evaluating at x3, = 0 we obtain for the DAS and CFS models, after inversion
(5)= dTbC
x3a=0
Repeating the procedure with eq 34 we get for the LFO model 1 + 4: 1 - (h*
+ 1)w-7- X3b
( TqBC)
4'4
+ 4yW(T,BC)3
P5
The case of radiation equilibrium in eq 39 and 40 is yielded by setting y = 0. For any choice of parameters, the term in the denominator with negative sign was less than approximately 0.1 in the present study. Again, this is due to the fact that we adopted an ratio considerably smaller than others who found negative jnterceptions. If we now impose a large rise in [C,] and kO4while keeping [C,*] and ko5 unchanged, the negative term may increase significantly. If it overcomes in absolute value the quantities that compose the positive part of the denominator of eq 39 and 40, negative slopes dT8/dTbca t 3c3, = 0 will be generated. Additional information regarding this work can be found elsewhere (Kumpinsky, 1983). Summary and Conclusions Heat and mass conservation equations for the stagnant boundary layer surrounding a burning carbon particle were derived. Fixed ambient conditions, equal diffusivities,and constant physical properties were adopted. The intraparticle effects were lumped at the surface with thermal radiation and Langmuir-type or lumped first-order kinetic expressions appearing as nonlinearities in the boundary conditions. Initially, the importance of the competition between O2 and COPfor active sites a t the carbon surface was examined. With that purpose, two models for the heterogeneous reactions employing Langmuir kinetics were derived. The DAS model assumed sites active either to O2 or to COz, while the CFS model took into account the competition for sites. Numerical computations revealed that the discrepancies between the results obtained by means of both models were not significant. By varying the concentrations of active sites for the DAS model, it was verified that the carbon reactivity exerts a fundamental influence on the feasibility region and reaction locus. The sensitivity of the DAS model to the ra-
Ind. Eng. Chem. Fundam., Vol. 22,
diation form and to parameters such as particle radius, boundary layer thickness, and ambient oxygen mole fraction was also inspected. In order to substantiate qualitative and quantitative aspects of the lumping effect, the LFO model was developed, considering lumped first-order kinetic expressions for the heterogeneous reactions. The LFO rate constants were calculated by means of the solution of the DAS model for a fixed set of parameters. For the assigned conditions, the LFO model and ita prototype DAS disclosed excellent qualitative and quantitative agreements. When parametric variations were performed, computations with the LFO model were executed by means of the previously determined rate constants. The solutions with the DAS and LFO models were characterized by a very good qualitative agreement. However, some relatively high quantitative disparities for combustion rates did occur, especially when the surface temperatures were low. The quantitative discrepancies between the computations of Sundaresan and Amundson (1981) and the experimental results of Smith and Gudmundsen (1931) were explained by means of a smaller ratio between C-C02 and C-02 reactions, rather than by way of competition for sites. It is apparent that the reaction rate intensities govern the pathological nature of the pseudo-steady-state structure rather than the pattern choice of the heterogeneous rate expressions. Acknowledgment We are indebted to Dr. L. E. Arri for his cooperation on the establishment of the kinetic models. Nomenclature a = particle radius, m b = value of r at the edge of the boundary layer, m c = concentration of the gas mixture, kg-mol/m3 Cf = free carbon site C(0) = occupied carbon site [C,] = concentration of active sites, sites/carbon atom D = average diffusivity of COz, CO, and O2 in N2, m2/s Ej = activation energy of the jth rate constant, J/kg-mol f = defined by eq 15 and 16 h,h* = see Table I1 koj,k, = frequency factor and rate constant of the jth reaction or step, respectively; m3/2/(kg-mo11/2 s),j = 2; DAS and CFS models: m3/(kg-mols), j = 1, 3; s-l, j = 4, 5 io,, kj = respectively frequency factor and rate constant of the heterogeneous reaction rates for the LFO model, m/s, j=4,5
KOl, K, = frequency factor and equilibrium constant for the
oxygen-exchangestep of the C-COz reaction, respectively m, = molar ratio COZ/(CO2+ CO) leaving the boundary layer n, = number of kg-mol of carbon per site, kg-mol/site p = parameter for the radiant energy flux qr = radiant energy flux, W/m2 QR = dimensionless radiant energy flux (Table 11) r = radial coordinate, m r . = jth elementary reaction step, (kg-mol/m2s) d = universal gas constant, J/(kg-mol K) R, = dimensionless rate of carbon combustion, eq 20 Ri= overall ith reaction rate, kg-mol of C/(m2s) for i = 1and 3, kg-mol of CO/(m3 s) for i = 2
No. 1, 1983 71
W i = dimensionless rates, i = 1, 2, 3 (Table 11) = number of carbon atoms per unit surface area, atoms/m2 = temperature, K w = see Table I1 xi = mole fraction of ith species: i = 1, CO,; i = 2, CO; i = 3, 0 2
?
Greek Letters a,Po,,Pi, y, 4, =
see Table I1
6 = value at the flame front of the double-film model, eq
30 and 31 = heat of the overall ith reaction, J/kg-mol t = emissivity of carbon particle = defined by eq 15 and 16 Of = fraction of free carbon sites Bo = fractional surface coverage of active sites X = thermal conductivity of the gas mixture, W/(m K) ( = dimensionless radial coordinate (Table 11) u = Stefan-Boltzmann constant, W/(m K2), T = dimensionless temperature (Table 11) (-Mi)
Subscripts b = value of the respective variable in the bulk f = a fixed value of the temperature, defined for the radiant energy flux m = a mean value of the temperature, defined for the radiant energy flux s = value of the respectie variable at the carbon surface Superscripts A, B, C = refer to the limiting cases A, B, C, respectively I, I1 = refer to the kinetic models DAS and CFS, respectively * = oxygen-related variables or parameters ^= refers to variables or parameters related to the LFO kinetic
model Literature Cited
Caram, H. S.; Amundson, N. R. Ind. Eng. Chem. Fundam. 1977, 16, 171. Dutta, S.; Wen, C. Y.; Beit, R. J. Ind. Eng. Chem. Process Des. Dev. 1977, 16, 20. Dutta, S.; Wen, C. Y. Ind. Eng. Chem. Process Des. Dev. 1977, 16, 31. Ergun. S. J . fhys. Chem., 1956, EO, 480. Ergun, S. U . S . Bur. Mines Bull. 1981, 598. Howard, J. E.; Wllliams, G. C.; Fine, D. H. "Fourteenth Symposium (International) on Combustion", Pennsylvania State University, University Park, PA: The Combustlon Institute. Ptttsburgh. PA, 1973; p 975. Kumpinsky, E., Ph.D. Dissertation, University of Houston, Houston, TX, 1983. Laurendeau, N. M. Pmg. EnergyComb. Sci. 1978, 4 , 221. Lewis, P. F.; Simons, G. A. Combust. Sci. Techno/. 1979, 2 0 , 117. Menster, M.; Ergun, S. U S . Bur. Mines Bull. 1973, 664. Mon, E.; Amundson, N. R. Ind. Eng. Chem. Fundam. 1978, 17, 313. Mon, E.: Amundson, N. R. Ind. Eng. Chem. Fundem. 1979, 18, 162. Mon, E.; Amundson, N. R. Ind. Eng. Chem. Fundam. 1980, 19, 243. Smith, D. F.; Gudmundsen, A. Ind. Eng. Chem. 1931, 23, 277. Srlnivas, E.; Amundson, N. R. Can. J . Chem. f n g . 1980, 5 8 , 476. Srinivas, E.; Amundson, N. R. Can. J . Chem. Eng. W a l e , 5 9 , 60. Srlnivas, E.; Amundson, N. R. Can. J . Chem. Eng. 198lb. 5 9 , 728. Sundaresan, S.; Amundson, N. R. Ind. Eng. Chem. Fundam. 19806, 19, 344. Sundaresan, S.; Amundson, N. R. Ind. Eng. Chem. Fundam. 1980b, 19, 351. Sundaresan. S.; Amundson, N. R. AIChE J . 1981, 2 7 , 679. Villadsen, J.; Mlcheisen, M. L. "Solution of Differential Equation Models by Polynomial Approxlmatlon"; Prentice-Hall: Englewood Cliffs, NJ, 1978; Chapter 8. von Fredersdorff, C. G.; Elliott, M. A. I n "Chemistry of Coal Utilization: Coal Gasiflcatlon"; Lowry, H. H., Ed.; Wiby: New York, 1963; Supplementary Volume. D 892. Williams, G. 5. private communicatlon, Massachusetts Institute of Technoiogy, 1979.
Received for review December 14, 1981 Accepted August 11, 1982
