162
Ind. Eng. Chem. Fundam., Vol. 18, No. 2, 1979
with reasonable removal of oxygen. Phenolic and furan oxygens are likely to remain unaffected during the liquefaction process unless a catalyst is present. Of course, some mineral matter present in coal can serve as a catalyst as has been demonstrated by Tarrer et al. (19761,Granoff et al. (1978), and Hamrin (1978). The work also suggests that when coal slurries are pumped through a preheater, the level of solvation and molecular weight reduction which occurs at low temperatures may be due, in part, to the fracturing of C-0 bonds. However, with the presence of free radicals a t these low temperatures, adduction reactions can occur that tie up a portion of both the cracked coal products and donor solvent. Acknowledgment Funding of this research was provided by the U S . Department of Energy under Project E(49-18)-2305. We also acknowledge H. Podall and L. Kindley of DOE and
A. B. King, R. G. Goldthwait, H. G. McIlvried, and K. A. Kueser for helpful suggestions.
Literature Cited Brucker, R., Koiiing, G., Brennst. Chem., 46, 41 (1965). Cronauer, D. C., Jewell, D. M., Shah, Y. T., Kueser, K. A,, Ind. Eng. Chem. Fundam., 17, 291 (1978). Eisenbraun, E. J., et al., J . Org. Chem., 35, 1260 (1970); 36, 686 (1971). Emert, J., Goldenberg, M.. Chiu. G. L., Valeri, A,, J. Org. Chem., 42, 2012 (1977). Granoff, B., Baca, P. M., Thomas, M. G., Noles, G. T.. Final Report entitled, "Chemical Studies of the Synthoil Process: Mineral Matter Effects", Sandia Laboratories, SAND-78-1113, June 1978. Hamrin. C. E., "Catalytic Activity of Coal Mineral Matter", presented at the Fifth Annual DC€/Fossil Energy Conference on University Coal Research, Lexington, Ky., Aug 22-24, 1978. Reifarth, K., Ph.D. Thesis, University of Berlin, 1977. Ruberto, R. G., Cronauer, D. C., "Oxygen and Oxygen Functionalities in Coal and Coal Liquids", presented at the 174th National Meeting of the American Chemical Society, Chicago, Ill., Aug 28-Sept 2, 1977. Tarrer, A. R., Guin, J. A., Pitts, W. S., Henley, J. P., Prather, J. W., Styles, G. A., Am. Chem. SOC.Div. Fuel Chem. Prepr., 21, No. 5. 59 (1976).
Received for review July 10, 1978 Accepted January 29, 1979
Diffusion and Reaction in a Stagnant Boundary Layer about a Carbon Particle. 3. Stability Eduardo Mon and Neal R. Amundson" University of Minnesota, Minneapolis, Minnesota 55455,and University of Houston, Houston, Texas 77004
A simplified transient model of diffusion and reaction is considered in the stagnant boundary layer that surrounds a carbon particle during combustion. The model is used to develop local asymptotic stability criteria for analysis of the multiple steady states. A special case arises when carbon monoxide burns outside the boundary layer as the linearized conservation equations are then solvable by a finite Fourier transform. In the more general case, results are obtained by Galerkin's method. Regions of instability created by variation of the particle size and ambient oxygen concentration are presented.
Introduction This paper is a continuation of two previous papers. The general problem under consideration is the burning of a char or carbon particle in an oxygen-containing atmosphere. It is assumed that the particle is impervious to gaseous species. Oxygen which must diffuse through the stagnant boundary layer about the particle reacts with the carbon at the surface to form carbon monoxide. The latter then must diffuse outwards from the surface and on the way it meets the incoming oxygen, reacting homogeneously to produce carbon dioxide. The carbon dioxide diffuses to the interstitial fluid but also to the surface where it reacts with the carbon to produce more carbon monoxide. Thus there are three diffusing species and three reactions, two being heterogeneous at the surface with one exothermic and one endothermic and there is an exothermic homogeneous reaction in the boundary layer itself. Thus the mathematical model is of diffusion-reaction type for the three reacting species coupled with the heat conduction in the boundary layer. The boundary conditions at the particle surface are nonlinear since the reaction rates there are of Arrhenius type. It is assumed that whatever *Address correspondence to this author at the Department of Chemical Engineering, University of Houston, Houston, Texas 77004. 0019-7874/79/1018-0162$01,00/0
happens within the boundary layer occurs very rapidly so that the boundary layer is in a quasi-steady state. In the first article (Caram and Amundson, 1977), a simplified model without radiation and with independent diffusion was investigated and a computational scheme developed. It was shown that there were multiple steady states. The computations are not trivial and the necessary concept of feasible regions of solutions initiated. In a second paper (Mon and Amundson, 1978), the problem was generalized to include multicomponent diffusion and radiation and some parametric studies were made. In the present paper the stability of the steady states is studied and the stable feasible regions are defined. Since the burning of a particle is always a transient problem, stability here means the stability of the quasi-steady state boundary layer concentration and temperature profiles. An order of magnitude analysis reveals that perturbations within the boundary layer will dissipate substantially faster than those within the solid. This paper depends heavily on the two previous ones and we advise the reader to consult those before proceeding with this one. The Transient Model Aside from the fact that our model considers the unsteady-state combustion, our assumptions, notation, parameter values, and reaction kinetics will be identical with 0 1979 American
Chemical Society
Ind. Eng. Chem. Fundam., Vol. 18, No. 2, 1979
those of the previous paper unless otherwise indicated. They will not be restated here. We assume, however, that the particle is in radiation equilibrium (qR = O), independent diffusion (y = l),and make use of the approximation
163
thickness of one particle radius. The boundary may then be immobilized by the coordinate transformation
Further simplification can be attained by introduction of the new variables In this form, the qualitative behavior of the pseudosteady-state solution can resemble the results obtained without the above additional assumptions by appropriate selection of the particle size and ambient gas composition. We assume that any stability conclusions are also applicable to the more rigorous model. The mass and energy balances for a ( t ) < r < b ( t ) take the form
_ at
,-2
where P = k/DQ3. The boundary conditions a t r = a(t) are ac, - Rz (4) ar a -2 = -(R2 ar a
+ R,)
a”2
ar
i = 1, 2, 3 ui = ci(2 - p ) u = T(2 - p ) In the fixed domain 0
(14) (15)
< p < 1, eq 2 and 3 then become
au _ -
a0
(3)
ar
ac3
ds
a2u
- + -(p,O)
P
ap2
where
The transformed boundary conditions, eq 4-6 and 11,a t p = 1 are
- R1 cy
au3 -
-aR1
a P
CY
u3s
(21)
(7)
In addition, the particle radius is given by the shrinkage equation
where CYPCCS
p=3M,K
At r = b ( t ) ,the ambient gas composition and temperature are taken to be constant c . = (;. rb i = 1, 2, 3 (9)
T = Tb (10) Carrying out the differentiation of the left-hand side in ( 7 ) and using (8) leads to
where Q1 and Q2 are the heats of reaction evaluated a t the solid temperature Q1 = (-AH,) = H3s + 2Hc, - 2Hzs Q2
= (-flz)
= His + Hcs- 2Hzs
The above equations represent a moving boundary diffusion problem of the type discussed by Bankoff (1964). In the absence of the homogeneous reaction term in eq 2 and 3, it is analogous to the classic problem of diffusion in a catalyst pellet for which exact solution is not possible (Luss, 1968) and only approximations are available (Bischoff, 1965; Theofanous and Lim, 1971). In our case the problem is simplified by taking a fixed boundary layer
R 1 = k, ex.( R 2 = k 2 exp(
-:) -:)
u3s uls
A t the external edge of the boundary layer, p = 0, we have ui = Uib i = 1, 2, 3 (25) u = ub (26) and the new form of (8) after integration is
The stability analysis will be simplified if we note that a temperature disturbance will dissipate significantly slower with respect to the solid than the gas. An order of magnitude estimate of the relative time constants may be obtained as follows. Linearization of (22) about the steady state yields dus _- - 9& + A d0 P
164
Ind. Eng. Chem. Fundam., Vol. 18,
No. 2,
1979
where A is a remainder term and
The absolute value of the solid time constant is thus T, = Ip/q1. The time constants for diffusion and for reaction may be similarly estimated by writing the energy balance equations in linearized and homogeneous form independently of the solid
The solution of eq 36-38 is straightforward and need not be presented here. We simply note that for any finite initial condition, the solution is bounded and asymptotically zero. Equations 39-41 then represent a system of three equations in the four unknowns ul, u2, us, and u. Specifying one, we can solve for the remaining three. The balances are thus effectively uncoupled. Stability Analysis. We proceed by writing eq 36-41 in terms of deviation variables
Ul = u, - u,,,
w, = w, w,,, v = u - us,
(42)
-
av
p =
- = qu, ap
1, u = 0 , p = 0
(31)
where R is the average value of [(l/P)(a 9 / a ~ ) ] , in ~ the boundary layer. The solution of (30)-(31) can be written as m
u =
C a, exp(R -An2)yn
(32)
,=l
where a, is a constant and m j nare the eigenfunctions of the problem
- -n - -An2y,; A, cot A, a P2
=q
(33)
The time constant for diffusion may then be defined in terms of the smallest eigenvalue as 7 d = IX1-21 and the time constant for reaction as 7 R = IR-lI, Two limiting cases may be examined. At low temperatures (1000 K) 7s 10'; 7 D 1; TR io3
-
-
-
whereas a t higher temperatures (1500 K) 7,
-
IO5;
TD
-
1;
TR
-
10
where it must be stressed that these are order of magnitude quantities only. In both cases the value of the solid time constant is sufficiently high relative to the others to justify the writing of (22) for our purposes are
Uncoupling t h e T r a n s i e n t Balances. If we notice that the heats of reaction are related as follows (35)
Equations 36 and 37 retain the same form when written in terms of W,, but the last condition becomes w,=o p = o (43) Our analysis will be based on the adiabatic perturbation. That is, we consider a perturbation that originates from the steady state such that w,=o d = O (44) It is possible to show that this perturbation is the only one that need be considered for this problem. The proof is given in the Appendix following an argument similar to that used by Amundson (1965) for the tubular reactor. The solution of eq 36 and 37 in terms of W,, with the conditions (43) and (441, then reduces to the trivial one W,(d,p) = 0 0 5 0 I a; 0 I p I 1 (45) The system of eq 39-41 when written as deviation variables, together with eq 35 and 45, can be solved for the concentration variables in terms of the temperature to give U , = v,/3V i = 1, 2, 3 (46) To determine the stability of the system, it is then only necessary to examine the temperature deviation equation aV a2V 4 _ -- +O < p < l (47) ad ap2 P
v=o p=o (49) where rl = R1 - Rlss;r2 = Rz - RSss;and 4 = 9 - 9,. When the above reaction terms are linearized about the steady-state values, it follows with the aid of eq 46 that
it is possible to take linear combinations of the balances, eq 16 and 17, together with the corresponding boundary conditions in order to eliminate the nonlinear reaction terms. Three independent equations result
a wi _ - -wis
p =
1
aP
w3
=
Q2 - u1
81
+ u3 +
k -u DQi
(37)
aV -
aP
= (17 - l)Vs
p = l
v=o p = o (52) where the additional terms normally expected on the right-hand side of eq 50 are zero because of eq 45, and where
A Special Case. In our previous paper we noted that carbon particles in the pulverized full range ( a I175 Fm) did not develop a carbon monoxide flame when exposed
Ind. Eng. Chem. Fundam., Vol. 18, No. 2, 1979 E l
20 r
165
N
c/ 4
!k
‘“.s,
16
4e%
r 1.0
0.8
0.2
0.4
0.6
0
DIMENSIONLESS DISTANCE, p
Figure 2. The three different steady-state temperature profiles that occur a t the points labeled 1, 2, and 3 in Figure 1.
5
AMBIENT TEMPERATURE ( K )x l O z
Figure 1. Region of instability in the TB-Tbplane created by variation of the particle size a t a fixed ambient oxygen content of 10%.
to low oxygen concentrations such as found in fluid bed combustors. Under these circumstances, eq 50 reduces to Fick’s second law. Its solution, with boundary conditions (51) and (52), is obtainable by a finite Fourier transform and is also given by Carslaw and Jaeger (1959). It has the form m
V = 2C exp(-A,,%) x n=l
(55) where 6 ( p ) is the temperature disturbance a t 0 = 0. The eigenvalues A, are the roots of the equation A, cot A, + 1 = 7 n = 1, 2, ... (56) The roots of this equation are all real when 7 I2. For 7 > 2 there is a pair of pure imaginary roots, in which case any perturbation would grow, rather than damp out, as revealed by visual inspection of (55). A sufficient requirement for instability is therefore 7 > 2 (57) The steady-state behavior of the model has previously been presented as a locus of solutions in the T,-Tb plane. A typical set of loci that result in the absence of CO oxidation in the boundary layer are shown in Figure 1. We have selected an ambient gas composition of 10% 02,no CO or C02, and varied the particle radius in the range 15 pm Ia I 175 pm. Three different steady states are possible, and application of the stability criteria given by eq 57 indicates that the intermediate steady state is unstable. In general, the unstable portion of the locus corresponds to the segment with negative slope, a result which reassures us that increased ambient temperatures will not lead to lower carbon surface temperatures. The range of instability can be seen to decrease as the particle size decreases, the entire locus being stable for particles with radius slightly smaller than 15 pm. In addition, the unstable range moves upward and to the right with decreasing particle size, as the loci shift toward higher ambient and surface temperatures. Figure 2 shows the different temperature profiles that are possible for a 50-pm particle exposed to an ambient temperature of 1300 K. They correspond to the points labeled 1 , 2 , and 3 in Figure 1. The temperature distributions are seen to be concave with their maxima reached a t the carbon surface, the site of the exothermic C-O2 reaction. In this case, a difference of 570 K in carbon temperatures can result with a given ambient temperature.
J
I
1
1
8 IO 12 14 16 AMBIENT TEMPERATURE ( K ) x I O - ’
Figure 3. Region of instability in the Ts-Tb plane created by variation of the particle size a t a fixed ambient oxygen content of 15%.
lo
J 4 6 8 IO 12 AMBIENT TEMPERATURE ( K )
X
10.‘
Figure 4. Region of instability in the T,-Tb plane created by variation of the ambient oxygen concentration a t a fixed particle radius of 150 wm.
The effect of increasing the ambient oxygen content to 15% is shown in Figure 3. The region of instability becomes significantly wider as does the locus segment of negative slope. Whereas a 5-pm particle did not possess an unstable steady state with 10% 02,it now displays a significant unstable range. Similar regions of instability may be obtained by variation of the ambient oxygen concentration with a fixed particle size. Such effect is shown in Figure 4 for a 300-pm particle. The result of decreasing ambient oxygen is to shift the upper boundary of the unstable region downward and to the right, while the lower boundary moves upward and to the right in what
166
Ind. Eng. Chem. Fundam., Vol. 18, No. 2, 1979 a=50pm
22 r
10
6
8
IO
12
AMBIENT TEMPERATURE ( K ) x
AMBIENT TEMPERATURE ( K ) x l d z
14
16'
Figure 5. Region of instability in the T,-Tb plane created by variation of the ambient oxygen concentration at a fixed particle radius of 50
w. is essentially a straight line. The range of unstable carbon temperatures as expected decreases with decreasing oxygen levels. For a smaller particle size, 100 pm, (Figure 5) the unstable region retains similar dimensions but is shifted toward higher ambient and surface temperatures. The General Case. When carbon monoxide oxidation occurs in the boundary layer, eq 50 cannot be simplified as in the previous section. To determine local stability we follow Galerkin's method (Perlmutter, 1972). It is reasonable to suppose a solution of (50)-(52) in terms of the normalized eigenfunctions of the simpler problem
where An are the roots of (56). Equation 58 satisfies the boundary conditions (511, (52) but leaves the gn(8) undetermined. Substitution of (58) into (50) yields
When (60) is multiplied by X,, and integrated with respect to p from the surface to the external edge of the boundary layer, we obtain
where we have also used the orthogonality property of the eigenfunctions and defined In,as
If the above procedure is repeated for m = 1, 2, ..., 1, eq 61 is equivalent to a set of linear ordinary differential equations with constant coefficients (63) where
(64)
The above matrix is symmetric and its eigenvalues are
Figure 6. Feasibility region and locus of steady-state solutions showing the stable and unstable portions of the locus. Reactivity of the particle is ten times greater to O2 and C 0 2 than previous cases. Table I. Largest Eigenvalues of t h e Matrix (64)a t Two Selected Points in the Locus of Figure 6 ~~
~
no. of terms in trial soln ~~~
point 5
point 6
33.68 33.81 34.04 34.06 unstable
-41.71 -47.63 -41.62 -47.61 stable
~
4 6 8 10 conclusion
therefore always real. The system stability is determined by the sign of the largest eigenvalue of (64) and requires a series of tests with increasing values of 1. An example of the steady-state multiplicity behavior that results when carbon monoxide burns in the boundary layer is shown in Figure 6. The choice of parameter values (i.e., increased reactivity, ambient gas composition) was made to best illustrate the behavior obtained when thermal radiation and multicomponent diffusion were considered for a similar sized particle. As the slope of the locus changes sign, it may be divided into four segments. Along the lower segment of negative slope, CO oxidation does not occur to any significant extent and since 7 > 2, this segment is unstable. The remaining portions of the locus were analyzed by the general method and the conclusion was that the stable and unstable segments alternate as shown in the figure. Table I shows the value of the largest eigenvalue of the matrix (64) obtained for increasing number of terms, I , in the trial solution (58) a t the points marked 5 and 6 in the locus. Ten terms appeared to provide sufficient accuracy, although the same stability conclusion could have been obtained with 1 as low as 4. In addition to the four shown steady states, the present model also predicts an unignited (stable) steady state below 1000 K. However, the more complete model considered in our last paper showed that it is possible to obtain five different solutions, all representing an ignited carbon particle above 1000 K. Extending the results obtained here to that case, we may conclude that three different ignited, stable solutions are possible. Fig.ure 7 shows the five temperature profiles predicted by an ambient temperature of 500 K. The lower three remind us of the previously discussed case (Figure 2), except now there is an additional pair of solutions that display a sharp temperature maximum due to CO oxidation in the boundary layer. Conclusions Application of the derived local stability criteria to small carbon particles (
