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Ind. Eng. Chem. Fundam. 1983, 22, 305-308
of the flow of non-Newtonian fluids in porous media. It is obvious that more studies are needed to further evaluate this point of view and to more completely exploit the basic concepts presented here. Although capillary models may be quite adequate for many cases, this work suggests that large deviations from these conventionally accepted theories will occur in the analysis of the flow of polymer solutions in enhanced oil recovery processes. The importance of this energy related application suggests that further studies are highly desirable. Acknowledgment This research was conducted as a part of the Penn State Cooperative Research Project in Enhanced Oil Recovery which is financially supported by the following petroleum and chemical companies: Amoco, Arco, DOW,Exxon, Marathon, Mobil, Sohio, Texaco, and Witco. Also, the authors are indebted to Professor J. S. Vrentas for his valuable suggestions. Nomenclature hp = pressure drop D, = characteristic particle diameter of the porous media as defined by eq 2 f = friction factor defined by eq 3 k = permeability K = parameter in power law model L = length of porous media n = parameter in power-law model NE,= Ellis number NRe= Reynolds number defined in Table I R1 = radius in the pore model shown in Figure 6 VI = average velocity in small radius section of the pore model shown in Figure 6 Vo = superficial velocity Greek Symbols
p =
viscosity of Newtonian fluid
= porosity p = density t
a = parameter in Ellis model
P = geometric parameter defined in Figure 6 y = geometric parameter defined in Figure 6 6 = geometric parameter defined in Figure 6 T
~ =/parameter ~
+ = shear rate
in Ellis model
viscosity of a nowNewtonian fluid vo = parameter in Ellis model r = shear stress Registry No. Sodium carboxymethyl cellulose, 9004-32-4; xanthan gum, 11138-66-2. Literature Cited 7 =
Bird, R. B.; Stewart, W. E.; Lightfoot, E. N. "Transport Phenomena"; Wiley: New York, 1960. Booer, D. V.; Gupta, R.; Tanner, R. I. J. Non-Newtonian F/uMMech. 1978. 4, 239. Christopher, R. H.; Middleman, S. Ind. Eng. Chem. Fondem. 1965, 4 , 422. Deiber, J. A.; Schowatter. W. E. AIChE J. 1981, 2 7 , 912. Duda, J. L.; Vrentas, J. S. Can. J . Chem. Eng. 1872, 50, 671. Duda. J. L.; Vrentas, J. S. Trans. SOC.Rheol. 1873, 17, 89. Duda. J. L.; Klaus, E. E.; Fan, S. K. SOC.Pet. Eng. J. 1981, 2 7 , 613. Dullien, F. A. "Porous Media-Fluid Transport and Pore structure"; Academic Press: New York, 1979. Hong, S. A. Ph.D. Thesis, The Pennsylvania State University, University Park, PA, 1982. Kemblowski, Z.; Michniewicz, M. Rhml. Acta 1979, 78, 730. Kumar, S.; Upadhyay, S. N. Ind. Eng. Chem. Fundam. 1981, 2 0 , 186. Marshall, R. J.; Metzner, A. B. Ind. Eng. Chem. Fundam. 1967, 6 , 393. Payatakes, A. C.; Tien, C.; Turian, R. M. AIChE J . 1873. 79, 58. SheffieM, R. E.; Metzner, A. B. A I C E J. 1878, 2 2 , 736. Unsai, E.; Duda, J. L.; Klaus, E. E. "Chemistry of Oil Recovery"; ACS Sympw sium Series 91, 1978; Chapter 8. Vrentas, J. S.; Duda, J. L. Appl. Sc/. Res. 1973, 2 8 , 241. Wang, F. H. L.; Duda, J. L.; Klaus, E. E. Society of Petroleum Engineering Paper 8418, 1979.
Received for review February 28, 1982 Revised manuscript received January 27, 1983 Accepted February 28, 1983
Simplified Solution Technique for Moving Boundary Problems in Gas-Solid Noncatalytic Reactionst P. A. Ramachandran' Chemical Engineering Division, National Chemical Laboratory, Poona 4 1 7 008 Indla
I n many gas-solid noncatalytic reactlons the solid reactant is depleted near the surface. The prediction of the conversion-time behavior of these systems then involves the solution of a moving boundary problem, for which a simplified procedure has been proposed in this paper.
Introduction Gas-solid noncatalytic reactions are of importance in many metallurgical and chemical processes. The model equations describing these systems generally require a numerical solution due to the interaction of diffusion with reaction and also due to the inherent transient nature of these systems* The local rate Of at any given psition in the pellet is Often represented by the kinetic model NCL Communication Number: 3011.
* Currently on leave at Washington University, St. Louis, MO 63130. 0198-4313/83/1022-0305$01.50/0
-db/de = ~ f ( b ) (1) Equation 1 is based on the assumption of a first-order reaction with respect to the gas which is observed in many practical systems.The termf ( b) representsthe dependency of the rate on the solid concentration (b). This may be an empirical power law of the type: f ( b ) = b" (volume reaction models) or the form could be complex derived from suitable structural models such as the grain model (Calvelo and Smith, 1970; Szekely and Evans, 1970) or the random pore model (Bhatia and Perlmutter, 1980). For many of the commonly encountered rate forms the solid reactant is completely depleted at some positions in the pellet after a finite time (for example for n < 1in case 0 1983 American Chemical Society
306
Ind. Eng. Chem. Fundam., Vol. 22, No. 3, 1983
modulus equal to @A, which has experienced a cumulative concentration OC. As this quantity can be readily calculated (from, say, a chart of conversion at time 8, vs. Thiele modulus or from suitable approximate analytical representation of the same) the conversion X corresponding to position X can be calculated from eq 3. To complete the solution of the problem it is necessary to know at what time the ash layer has reached the position A, Le., the rate of movement of the interface. The rate of movement of the interface at time 8, (i.e., at the time when the ash layer has just formed) is given by the equation (Dudukovie and Lamba, 1978)
S O L I D PELL
,
INNER CORE
ASH LAYER
1.0
where d+/dy represents the flux of the cumulative concentration at the surface. This is related to the conversion which has occurred in the pellet (which now consists of only the inner core) by the equation
-GAS CONCENTRATION PROFILE
UNREACTED
B
o
x
(
I
Figure 1. Concentration profile at time 0 > Be illustrating the two zones formed in the pellet.
of volume reaction models). The problem then becomes of the “moving boundary type” because the boundary between the partly reacted and completely reacted regions of the solid moves inward in the pellet with progress of time. A rigorous technique for numerical solution of such problems has been obtained by Dudukovic and Lamba (1978). The present paper proposes a simplified technique for solution of these types of problems. Theory The method will be illustrated by considering a reaction which is fractional order with respect to b although it is equally applicable to the grain model. Also a spherical geometry is assumed, for illustration and external mass transfer gradients are assumed to be negligible. The effect of this resistance need not be explicitly included in the analysis for reactions which are first order with respect to the gas because the law of addition of reaction and external diffusion times can be used to account for this resistance (Sohn, 1978). The complete conversion of b at the pellet surface occurs at time BC given by BC = 1/(1 - n) (2) The cumulative concentration (Soeade) at the surface is now equal to 8,. The conversion of the pellet at this time is a unique function of the Thiele modulus (4) of the system and can be calculated numerically (DudukoviE and Lamba, 1978) or by approximate analytical equations derived in a recent paper (Ramachandran, 1982). Consider a time 8 > 8,. The pellet will now consist of two zones: an outer zone of completely reacted solid (ash layer) extending from X to 1 and inner core of partially reacted solid from 0 to X (see Figure 1). Let the partial conversion which has taken place in the inner core be denoted as Xi.Then the total conversion X at time 8 for a spherical pellet is equal to
x = X3Xi + 1 -
A3
(3)
The value of the cumulative gas concentration at position X is now equal to 8, as the solid B has been completely depleted at A. Hence the conversion in the inner core Xi is the same as that for a hypothetical pellet, with a Thiele
= 42Xi/3
(5)
Substituting in (4)we obtain
A similar relation can be used for dh/d8 at time 8 noting the difference that the inner core has an effective radius A, effective Thiele modulus of 4X, and a dimensionless gas concentration equal to ax. Hence by analogy to eq 6 we have dX _ ---3Xax (7) d9 &PXi A rigorous mathematical derivation of eq 7 is also possible although it is not presented in this note. The unknown concentration ax can be found by equating the instantaneous rate of diffusion through the ash layer and the rate of reaction in the inner core (characterized by the rate of movement of the inner core).
Substituting for dh/d8 from eq 7 and rearranging, we obtain X2Xi Uh
=
X2Xi + (1
r L
r -.l x *.
1
(9)
Hence the rate of movement of the interface can be obtained by substituting this value of a, in eq 7. Integrating the resultant equation gives
%=e..+
(10) Thus the time 0 at which the interface has reached position X can be calculated from eq 10 and the corresponding conversion can be calculated by use of eq 3. We shall now illustrate the method by applying it to a number of cases.
Ind. Eng. Chem. Fundam., Vol. 22, No. 3, 1983 307
Table I. Equations for Analytical Prediction of Xi Based on Asymptotic Approximation reaction eq for Xi as a function of @A kinetics 1
i
I
Table 111. Comparison of the Analytical Representation for Xi with Exact Numerical Solution asymp single point Thiele approx soln exact mod, @ A (Table I) (Table 11) num soln n = 0.5 0.9124 0.6641 0.4099 0.1832
2 5
10 25 Thiele
mod
(Table I)
0.9753 0.6902 0.4322 0.3231
0.9759 0.7153 0.4243 0.1856
(Table 11)
numerical
n = 0.67 2 5 10 25
Table 11. Equation for Analytical Prediction of Xi Based on Single Point Collocation Approximation reaction
kinetics
eq for X i as a
[$$y
function of
+(F)
110.25
I”
-]+
110.25
0.9975 0.8389 0.4972 0.3347
0.9975 0.8523 0.5332 0.2382
10
@A
n = 0.5
1.398
n = 0.67
0.6991 $ - $ 12/3+ $ ,’/27] + 0.3, where $ is the real root of the cubic equation: $ - 9$ 1’ + $ ,[27 + 283.5/(@2A’)] 850.5/(@’h2) =0
-
0.9453 0.7558 0.5015 0.2330
0.3
Applications (1) n = 0.0. Here Xi can be analytically predicted by the following equation (Ishida and Wen, 1968).
08 X
Z
0
06
: K
0 4
0 2
3
4
5
7
10
20
DIMENSIONLESS TIME 7
Using this relation and the corresponding value of dXi/dA in eq 10, it can be shown that the resulting equation for 8 is the same as that derived by Ishida and Wen (1968). Thus the correctness of eq 7 in predicting the rate of movement of the interface is established by this comparison. (2) n = 0.5,0.67. An analytical representation to Xi vs. 4A is extremely useful as it simplifies the evaluation of the integral term in eq 10 considerably. Ramachandran (1982) has proposed a technique for analytical prediction of conversion-time behavior of gas-solid reactions which is sufficiently accurate for intermediate and high values of Thiele modulus. The use of this method leads to approximate analytical solutions shown in Table I for Xi as a function of $A. These solutions are sufficiently accurate for all practical purposes for +A > 10. As this solution is based on an asymptotic property of the differential equation for the cumulative concentration it may be called an asymptotic approximation. For lower value of Thiele modulus a single interior collocation point solution of the problem appears to be more satisfactory (Ramachandran and Kulkarni, 1980). This method applied for n = 0.5 and 0.67 yields the equations presented in Table I1 for Xi as a function of 4A. The accuracy of equations in Tables I and I1 in predicting Xi is demonstrated in Table I11 by comparing it with exact numerical solution. The conversion-time behavior (Xvs. 8) for 8 > 8, can then be calculated by using appropriate equations either from Table I or I1 for Xi vs. X relationship in eq 10 and 3 for calculating 8 and X, respectively. The calculated results are shown in Figure 2. Equations from Table I were used for 4 = 25 and equations from Table I1 for I$ = 5 and 10 in these calculations. Also a comparison with a complete numerical
Figure 2. Computed results for n = 0.5 and 0.67: (-) present method, n = 0.5; (X) exact numerical solution of DudukoviE and Lamba (1978) for n = 0.5; (- - -) n = 0.67.
solution of the problem is shown in Figure 2. The agreement is very close and hence the method represents a sound procedure for calculation of conversion-time pattern for moving boundary problems of this type. (3) Grain Model. An approximate analytical solution for X vs. 8 behavior for 8 > BC has been derived by Ramachandran (1982) based on the asymptotic behavior of the system equations expressed in terms of cumulative concentrations. (See Table I). By use of these relationships for Xi vs. 4 A, the conversion-time behavior for the moving boundary case can be predicted in a manner similar to that for a fractional order reaction. Conclusions A eimple method has been proposed for prediction of the conversion-time behavior of gas-solid noncatalytic reactions occuring with a moving boundary separating the ash layer (reacted solid) and the partly reacted inner core. The method is shown to be sufficiently accurate for use in most of the practical situations. Nomenclature a = dimensionless concentration of gas A ax = dimensionless concentration of A at the interface b = dimensionless concentration of solid B Bi = Biot number for intragrain diffusion f ( b ) = kinetic rate form with respect to solid B h = Thiele modulus for the grain model h+ = parameter defined in Table I n = order of reaction with respect to solid X = conversion in the solid at time 0 Xi = conversion in the inner core at time 8 y = dimensionless radial position in the pellet 4 = Thiele modulus for the gas-solid reaction 0 = time elapsed since start of reaction
308
Ind. Eng. Chem. Fundam. 1983,22, 308-3 1 1
Oc = time at which the ash layer begins to form X = position at time 0, of the reaction interface separating the
fully and partly reacted portions of the solid
4 = cumulative concentration defined as S l a d0
= cumulative concentration at the interior cohcation point Literature Cited
Ishide, M.; Wen, C. Y. A I C M J . 1968, 14, 311. Ramchandran, P. A. Eng. SCl., to be published. Ramchandran, P. A.; Kulkarnl, B. D. I d . Eng. Chem. Process Des. Dev. 1980, 19, 717. Sohn. H. Y. Met. Trans. 1978, QB, 89. Szekev, J.; Evans, E. W. Chem. Eng. Sci. 1971, 25, 1091.
Received for review April 15, 1982 Revised manuscript received January 31, 1983 Accepted February 28, 1983
Bhetk, S. K.; Perlmutter. D. D. AIChE J . 1980, 26. 379. Calvelo, A.; Smith, J. M. "Proceedings of Chemeca 70". 1970; No. 3. DudukoviE, M. P.; Lamba, H. S. Chem. Eng. Sci. 1978, 33. 303.
Novel Kinetic Analysis of Coal Liquefaction Takeshl Okutanl' and Neil R. Foster
CSIRO Dlvision of Fossil Fuels, PO Box
136, North Ryde. NSW, Australia 21 13
The application of a nonisothermal method of kinetic analysis to the study of coal hydrogenation is described. Kinetic parameters obtained by this technique compare favorably with those obtained with a conventional approach. The nonisothermal method is convenient and provides a means of obtaining reaction rate data for the complex initial stages of the coal liquefaction process.
Introduction Kinetic parameters for the coal hydrogenation process, although of a purely empirical nature due to the complex chemical structure of coal and ita hydrogenation products, are essential in order to design reactors and to facilitate process optimization. Kinetic analysis of coal hydrogenation is usually carried out in batch autoclaves, which enables the measurement of the conversion of coal with time at a constant temperature, and is referred to in this paper as the conventionalapproach. However, this method is laborious, and because of the excessive heat-up times required in standard autoclaves it is impossible to determine reaction rates at the initial stages of hydrogenation. In the area of solid-state chemistry, Kubo et al. (1966) proposed a method of nonisothermal analysis whereby kinetic data are estimated from the relationship between a conversion, measured a t a constant heating rate, and temperature. This method is as accurate and much simpler than the conventional approach. If the method is applicable to studies of coal liquefaction it will provide a valuable tool for kinetic analysis of this complex reaction. When combined with rapid quenching of the products the method will also provide a potential means of obtaining accurate reaction rate data for the initial stages of the liquefaction process. The results obtained by the application of Kubo's nonisothermal method to coal liquefaction using Wandoan coal (Australia) and an cu-Fe203-S catalyst (5% Fe203 + 2% S,based on coal) are reported in this paper. The kinetic parameters obtained are compared with those determined by conventional analysis. Experimental Section Materials. Proximate, ultimate, and maceral analyses of Wandoan coal (-100 mesh) are presented in Table I. The coal samples were dried for 12 h at 107 f 5 OC'under
* The Government Industrial Development Laboratory,Hok-
kaido, 2-17 Tsukisamu-Higashi,Toyohira-ku,Sapporo, 061-01,
Japan
Table I. Proximate, Ultimate, and Maceral Analyses of Wandoan Coal Proximate Analysis (wt %, Air Dry Basis) moisture 7.7 ash 22.1 volatile matter 37.1 fixed carbon 33.1 Ultimate Analysis (wt %, daf) carbon 75.5 hydrogen
6.2
nitrogen
1.0
sulfur
0.3
oxygen (diff.) 17.0 Maceral Analysis (wt %) vitrinite 84 exinite 9 inertinite 7 vacuum. Analytical grade a-FezO3 (Merck) was used as catalyst without purification; the sulfur has been described previously (Okutani et al., 1979). Both the a-Fe203and sulfur were in powder form (-300 mesh). Tetralin (purity greater than 99.590, Ajax Chemicals, Australia) was used as a vehicle oil without further purification. Commercial hydrogen gas was of analytical grade. Apparatus and Sample Preparation. The batch-type 115-mL unstirred autoclave (Hastelloy C)has previously been described (Okutani et al., 1979). In order to maintain contact between coal particles and hydrogen, a method was used (Yokayama et al., 1979) in which a-A1203 particles were mixed with coal particles; 10 g of coal powder, 0.5 g of a-F%03, 0.2 g of S,and 10 g of a-Alz03(80mesh spheres) were mixed. Yokoyama et al. (1979) have shown that mass transfer resistances, as reflected by lack of hydrogen availability, are negligible when this method of sample preparation is adopted. a-A1203was prepared by calcination of -y-Alz03 (analytical grade, Merck) for 1h at 1150 OC. For experiments using Tetralin as vehicle oil, the sample was wetted by adding 6 g of Tetralin to a mixed solid sample of 4 g of coal, 0.2 g of a-Fe203, 0.08 g of S, and 10 g of cu-A1203.
0196-4313f83f1022-0308$0l.50fO 0 1983 American Chemical Society
