Ind. Eng. Chem. Fundam., Vol. 18, No. 1, 1979 Fisher, M. E., J . Math. Phys., 5 (7), 944 (1964); "Conference on Phenomena in the Neighborhood of Critical Points", p 21, 1965. Gammon, B. E., Douslin, D. R., J . Chem. Phys., 64(1), 203 (1976). Goodwin, R. D., J . Res. Narl. Bur. Stand., 74A (2), 221 (1970). Gosman, A. L., McCarty, R. D., Hust, J. G., NSRDS-NBS 27 (1969). Guggenheim, E. A., J . Chern. Phys., 13(7), 253 (1945). Hall, K. R., Eubank, P. T.. Ind. Eng. Chem. Fundam., 15, 323 (1976). Keenan, J. H., Keyes, F. G., Hill, P. 0.. Moore, J. G., "Steam Tables", Wiley, New York, N.Y., 1969. Kubicek, A. J.. Eubank, P. T., J . Chem. Eng. Data, 17(2), 232 (1972). Levelt-Sengers, J. M. H., hi. Eng. Chem. Fundam., 9, 470 (1970). Levelt-Sengers, J. M. H., Chen, W. T., J . Chem. Phys., 56, 595 (1972). Lin, D. C.-K., Silberberg, I.H I McKetta, J. J., J . Chem. Eng. Data, 15(4), 483 (1970). Lydersen, A. L., Greenkorn, R. A,, Hougen, D. A., University of Wisconsin, College of Engineering, Eng. Exp. Sta. Rept. 4, 1955. Martin, J. J., Campbell, J. A , Seidei. E. M., J . Chem. Eng. Data, 8 (4). 560 (1963). Michels, A., Blaisse, B., Michels, C., Proc. R. SOC.London, Ser. A , 160, 358 (1937). Michels, A., Levelt, J. M., DB Graaff, W., Physica, 24, 659 (1958). Osborne. N. S.. Stimson. H. I-..Ginninas. D. C.. J . Res. Natl. Bur. Stand.. 18. 369 (1937). Perel'shtein, I.I., in "Thermophysical Properties of Matter and Substances", Val . - .. 4. , V . . .A.. .Rahinnvich .. .- ..-. ., Ed --. , 1975 .- . - . Pitzer, K. S., Lippmann, D. Z.,Curl, R. F., Jr., Huggins, C. M., Petersen, D. E., J . Am. Chem. SOC..77. 3433 (19.55). Reid, R. C., Prausnitz, J.' M., Sherwood, T. K., "The Properties of Gases and Liquids", 3rd ed, McGraw-Hill, New York, N.Y., 1977. Riedel, L., Chem.-Ing.-Tech., 26, 83 (1954).
7
Roach, P. R., Phys. Rev., 170 (I), 213 (1968). Roder, H. M., Weber, L. A,, NASA SP-3071 (1972). Rowlinson, J. S., in "Handbuch der Physik", Vol. 12, Springer, Berlin, 1958. Rowlinson, J. S.,"Conference on Phenomena in the Neighborhood of Critical Points", p 9, 1965. Rowiinson, J. S.,Ber. Bunsenges. Phys. Chem.. 76, 281 (1972). Shank, R. L., J . Chem. Eng. Data, 12(4), 474 (1967). Stewart, R. B.,Jacobsen. R. T., Myers, A. F.,NASA CR-128527 (1972a); NASA CR-128528 (1972b). Street, W. B., Sagan, L. S., Staverley, L. A. K., J. Chem. Thermodyn., 5, 633 (1 973). Street, W. B., Staverley, L. A. K., J . Chem. Phys., 55(5), 2495 (1971). Thompson, P. A., Thompson, J. R., Bauer, J., Thompson, C., Katsch, W., Meier, G., Max-Pianck-Institut fur Stromungsforschung, Bericht 125/1975. Tsonopoulos, C., AlChE J . , 20, 263 (1974). Vargaftik. N. B., "Tables of the ThermophysicalProperties of Liquids and Gases", Wiley, New York, N.Y., 1975. Vohra, S. P., Kang, T. L., Kobe. K. A., McKetta, J. J., J . Chem. Eng. Data, 7(1), 150 (1962). Wagner, W., Cryogenics, 12, 214 (1972). Wallace, B., Jr., Meyer, H., Phys. Rev. A , 2(4), 1563 (1970). Washburn, E. W., Ed., "International Critical Tables of Numerical Data, Physics, Chemistry and Technology", McGraw-Hill, New York. N.Y.. 1928. Weber, L. A., J . Res. Natl. Bur. Stand., 74A (1). 93 (1970). Young, S., J. Phys., 8, 5 (1909). Young, S., 2. Phys. Chem., 70, 620 (1910).
Received f o r review July 11, 1977 Accepted October 5 , 1978
Effect of Nonuniform Activity Distribution on Catalyzed Gas-Solid Reactions Guilleirmo L. Guzman and Eduardo E. Wolf" Department of Chemical Engineering, University of Notre Dame, Notre Dame, Indiana 46556
The effect of nonuniform activity distribution in a solid-catalyzed gas-solid reaction under mass transfer limitations is analyzed. The theoretical analysis includes two models of the pore structure, diffusivity profiles and uniform,
linear, and parabolic activity profiles. The results reveal that in gas-solid reactions, nonuniform catalyst distribution is detrimental to solid conversion and, consequently, efforts should be made to attain a uniform catalyst distribution.
Introduction Gas-solid reactions are important in many processing and chemical industries. Noncatalytic gas-solid reactions have been extensively studied and recently Szekely et al. (1976), have published a book on the subject. Among the many examples of gas-solid reactions, the gasification of coal has acquired new relevance as an alternative for coal conversion to high BTU gas. Several technical problems exist due to the high temperatures and pressures required. To overcome these difficulties, researchers have studied the possibility of using solid catalysts to reduce the severity of the operating conditions. Several studies about the catalytic gasification of coal have been published recently (Gardner et al. 1973; Otto and Shelef, 1976), including an excellent review by Johnson (1976). The results obtained indicate that the steam gasification of coal is catalyzed by many metals and metal salts such as KHC03, K2C03,ZnCl,, etc. The economic feasibility of catalytic coal gasification depends on a balance between additional conversion gained by the catalyst vs. the catalyst cost and its recovery. Studies carried out so far are mostly screening experiments designed to determine the most active catalyst without detailed analysis of the phenomena involved. The objective of this paper is to model the interaction between 0019-7874/79/1018-0007$01 .OO/O
the physical and chemical processes involved in a general solid-catalyzed gas-solid reaction. In such a reaction, a reactive solid impregnated with a solid catalyst reacts with a gas to form a gas product and a solid ash residue. Thus the situation is different than in the case of a heterogeneous catalytic reaction where the catalyst is deposited in an inert support which does not undergo chemical reqction. In a solid-catalyzed gas-solid reaction, the support area decreases with time as the solid is consumed. Several models of pore structure have been used to describe the relationship between surface area and solid porosity (Petersen, 1957; Szekely and Evans, 1970). Recently, Guzman and Wolf (1978) showed that, using a single-pellet diffusion reactor, it is possible to determine the pore model that describes the solid best. Another important parameter to account for is the distribution of the catalyst within the reactive solid. In many instances, the intimacy of the contact between reactive solid and solid catalyst determines the catalytic effects to a large extent. Thus a major parameter considered in this work is the catalyst distribution within the solid. The effect of catalyst distribution on activity has been studied in the case of heterogeneous catalysts but no such studies have appeared for a solid catalyzed gas-solid reaction. Nonuniform activity distribution is important in @ 1979 American Chemical Society
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Ind. Eng. Chem. Fundam., Vol. 18, No. 1, 1979
diffusion-controlled heterogeneous catalytic reactions. Shadman-Yazdi and Petersen (1972) have analyzed the case of a catalyst pellet with different activity profiles for the reaction A B C. They found that when B is the desired product the selectivity toward B can be improved substantially. Luss and Corbett (1974) have shown that for a series self-poisoning reactions, concentrating the activity in the exterior shell often yields maximum selectivity, but it also yields minimal resistance to deactivation. Wolf (1977) has shown that longer catalyst life can be obtained by using a catalyst composed of an inert protective layer and an active layer. The following theoretical analysis combines the concepts used in gas-solid reactions with those of nonuniform activity distribution used in heterogeneous catalysis. Theory Let us consider the following gas-solid reaction: A + bB, cC, + dD,; solid B is envisioned as a pseudo-tomogeneous material with a slab geometry. The following assumptions are made. (1) Diffusion of reactants inside the catalyst is fast compared to solid consumption, so that a pseudo-steady state condition exists. (2) The resistance due to diffusion in the gas film is neglected; however, diffusion of reactants inside the solid is significant. (3) Transport of reactant inside the solid occurs by diffusion only. (4)The solid catalyst is continuously distributed inside the solid so that an activity profile exists. ( 5 ) The activity profile and the catalyst activity remains the same during the reaction. Assumption 1 is generally acceptable for most gas-solid reactions (Bischoff, 1963; Szekely and Evans, 1970). Operating conditions can be adjusted to minimize external mass transfer effects in accord with assumption 2. The third assumption is valid for most cases of diffusion of gases in porous solid. Assumption 4 is valid in the same sense that is used in heterogeneous catalysis, namely that by appropriate catalyst preparation, one can deposit the reactive metal in such a way as to create an activity profile. Assumption 5 is a simplifying assumption due to the lack of information about the effect of the solid consumption on the catalyst dispersion. This point will be discussed further later. Under the assumptions listed above, the continuity equations for reactant A and solid B are
--
-
where CA is the molar concentration of A within the solid, Deffis the effective diffusivity of A in the porous solid, r A is the reaction rate per unit volume of the pellet, CB is the solid apparent density, and b is a stoichiometric coefficient. The initial and boundary conditions are
(5) In the analysis of gas-solid reactions, the reaction rate should be expressed per unit surface area, since the reaction takes place on the surface. Thus as the solid is consumed, the surface area decreases and the rate falls to zero. In solid catalyzed reactions, the specific rate is expressed in terms of reaction rate per unit area of metal catalyst. As the reaction proceeds, the solid is consumed,
the support-catalyst contact decreases, and the reaction rate decreases with time. If no reaction takes place in the absence of catalyst, and the catalyst morphology remains constant, we can assume that the solid-catalyst contact area is proportional to the solid surface area. The proportionality constant depends on the catalyst dispersion. Consequently, for a reaction which is first order with respect to gas concentration, we can write rA
= akCA
(6)
To account for the activity distribution, we assumed that the local reaction rate decreases toward the center of the pellet (Shadman-Yazdi Petersen (1972). Two situations were distinguished: (a) the catalyst uniform activity is optimal, and (b) the catalyst uniform activity is not optimal. In the first case, addition of more catalytic agent does not increase the reaction rate; thus one can assume that the rate constant varies with distance according to
k = ko( 1 -
t)’
where ko is the maximum rate constant under conditions of uniform distribution. Different distributions can be obtained by assigning 0 the value of 0 (uniform distribution), 1 (linear distribution), or 2 (parabolic distribution). However, the total activity (Ktotd= .foLhdx) is different for each catalyst distribution, the catalyst with uniform distribution having the highest total activity. Case b, corresponding to nonoptimal activity, can be described by
which allows calculations based on the same total activity. However, the combined rate constant hb(1 + p) increases as increases, which is only possible if the catalyst loading has not been optimized. Obviously one should always try to optimize the catalyst loading first. Consequently, all calculations were carried out using eq 7a except in one case to compare results a t constant activity. As the solid is consumed and the surface area decreases, the porosity and the effective diffusivity increases. For a porous solid, the effective diffusivity is assumed to change according to Deff = DAB^/^ (8) The tortuosity factor R is expected to decrease as the porosity increases. A relation between R and 0 often invoked in heterogeneous catalysis and in Walker’s (1959) experimental work in coal combustion is R = 1/0. In general, one can propose the following relation between Deffand 0
where De,, = DABd0/R0and Bo, Qo are the initial porosity and tortuosity. Thus, depending on the value of M , eq 9 allows one to account for different functionalities between Deffand 0: (a) M = 0, constant effective diffusivity; (b) M = 1,constant tortuosity factor; and (c) M = 2, R = 1/6. The solid reactant concentration can be related directly to OB by
In order to account for the formation of a solid ash product, the following relation between 0B and 0 has been developed. The total void is considered to be equal to the initial void
Ind. Eng. Chem. Fundam., Vol. 18, No. 1, 1979
9
I .oo
0.90 EACTION FRONTS
0.80 0.70
SOLID REACTANT
0.60 X
SOLID PRODUCT
0.50
0.40
0.30
0.20 0.10
0
T Figure 2. Conversion vs. dimensionless time, random-pore model: hi = 2.5; CY = 0.4;00 = 0.3;G = 4.105.
The complete set of equations can be written in terms of dimensionless variables as
(b)
Figure 1. Schematic diagram of pore model geometrical structure: a, grain model; b, rando:m-pore model.
plus the void of the solid product, the latter being proportional to the porosity of the solid product and the difference between (dB - do). Thus the total porosity is given by 0
00 = 1 + -
.( $
=1
- 1)
The relation between dB and the solid surface area is provided by the mo'del of pore structure. The packed sphere or grain model and random-pore models are used in this work. (See Figure 1.) Grain Model. See Calvelo and Cunningham (1970) and Szekely and Evans (11970). According to the grain model (GM), if the reaction proceeds in a well defined concentric front in the grain, tho reactive area per unit volume of the solid is given by
Random-Pore Model. See Petersen (1957). In the random-pore model (RPM), the surface area per total pellet volume is related to dB by the following equations
00
where [ = r/ro is the pore radius relative to the initial pore radius. G is a constant related to the initial solid porosity by 27 - 27 = 0 G3 - -G (15) 480 480
To obtain an explicit relation between a and CB, it is more convenient to use 5 as a parameter rather than OB. Then CBcan be related to $, instead of dB by the following relation
.( $
- 1)
where
the function f(dBl60) depends on the pore model used (eq 1 2 for the grain model or eq 13-15 for the random-pore model). X is the conversion and dav is the average porosity. The initial and boundary conditions in dimensionless form are
t > 0;
and
+
aJ/
= 0; J/(O,r)= 1;7 = 1; -(l,r)
=0 (23) ao The system of eq 17 to 20 is nonlinear and an analytical solution is not feasible. A numerical solution was obtained using the quasilinearization technique described by Lee (1968), along with a four-point finite-difference substitution of the derivatives. Results and Discussion The numerical solutions of eq 17-20 are displayed in Figures 2-5. do and CY have the same effect in all results; i.e., the time required to attain complete solid conversion decreases as these parameters increase. Consequently, only one set of values (do = 0.3, CY = 0.4) is shown hereafter. In all cases, except in Figure 3, the activity profile corresponding to optimal catalyst loading (eq 7a) was used. Figure 2 shows the conversion vs. dimensionless time curves obtained using the RPM and hl = 2.5 which cor-
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Ind. Eng. Chem. Fundam., Vol. 18, No. 1, 1979
0
I
a
3
4
7
Figure 3. Conversion vs. dimensionless time, RPM and nonoptimal catalyst loading: hl = 2.5; CY = 0.4; Bo = 0.3; G = 4.105; M = 2. 1.0
0.9
0.8 0.7
0.6 X
0.5 0.4
0.3 0.2 0.1 n
h, = 2.5;
1816-
--- OM -RPM M.2
Figure 5. Relative rate vs. conversion: h, = 2.5; CY = 0.4; Bo = 0.3.
responds to an intermediate value between complete reaction control (h, < 1)and diffusion control (h, > 5). The results indicate that solids with uniform catalyst distribution (0 = 0) exhibit almost complete conversion, whereas solids with nonuniform distribution, e.g., /3 = 1 or /3 = 2, the conversion approaches an asymptotic value less than unity. The asymptotic conversion decreases as /3 increases. The incomplete conversion obtained with nonuniform distributions arises because according to eq 7a, as the reaction progresses the solid inner core has less catalyst to activate the gaseous reactant. Thus, under the conditions of optimal catalyst loading, nonuniform distribution is detrimental to the solid conversion.
Figure 2 also shows that as the activity becomes less uniform: as P increases, the diffusivity profiles are less significant since the solid overall activity is lower and therefore the diffusion resistance is less important. For a given value of /3 and 7 , a solid with variable tortuosity ( M = 2) attains higher conversions than a solid with constant tortuosity ( M = 1)or constant diffusivity ( M = 0)* The calculations were repeated for the nonoptimal catalyst loading distributions (eq 7b) to compare results on the basis of equal total activity. The results shown in Figure 3 indicate that nonuniform distribution gives higher initial conversion than uniform distribution due to higher activity at the pellet external face. However, as the reaction progresses the reactant encounters less catalyst resulting in lower conversions. It follows that for total constant activity, except for initial conversions, the nonuniform activity distribution is also detrimental to conversion. These results differ from those obtained in diffusion-controlled heterogeneous catalysis wherein, if the activity distribution and concentration profiles are matched, nonuniform distribution leads to catalyst savings and equal overall activity (Shadman-Yazdi and Petersen, 1972). The detrimental effect found in this work occurs because after the solid has reacted, the catalyst-solid contact area is lost and the catalyst is left unused. Thus, in processes where solid-catalyzed gas-solid reactions occur, all efforts should be made to obtain a uniform catalyst distribution. Since the remedy for nonoptimal distribution is an obvious one, the calculations presented hereafter (Figures 4 and 5) are carried out using eq 7a. Conversion vs. time curves obtained using the grain model, shown in Figure 4,are similar and exhibit the same trends as those obtained using the RPM (Figure 2). In all cases, for the same set of reaction and material parameters (P, M , h,) and T , the grain model predicts higher conversions than the RPM. The relative reaction rates vs. conversion curves obtained with both model and M = 2 are shown in Figure 5. The reaction rate R was calculated as the rate of diffusion at 7 = 0 and divided by the initial rate R, to give the relative rate R/Ro. Figure 5 shows that, according to both models, initially the increase in the gas diffusion rate due to pore opening leads to higher rates ( R / R o> 1). As the reaction proceeds, the catalyst-solid contact area decreases and the rate decreases after reaching a maximum. The maximum decreases as M decreases and P increases. Figure 5 reveals more differences between the pore models than the conversion vs. time plots; consequently this type of plot can be used, along with experimental data, to select the pore model that describe the solid best. Figure 5 also shows that the RPM predicts higher relative rates than the grain model; yet according to Figure 2 it also predicts lower conversion for the same parameters and r . This apparent inconsistency occurs because the GM predicts higher porosities and lower a l a o values than the RPM. As a consequence the GM gives higher conversions (eq 21) and lower rates (eq 6) in accord with Figures 2 and 5. The effect of the Thiele parameter on conversion was also investigated. The results obtained were similar to those shown in Figure 2 except that the dimensionless time required to reach a given conversion increases as hl increases. Conclusions The theoretical analysis of a solid-catalyzed gas-solid reaction under diffusion influence reveals that a nonuniform catalyst distribution is detrimental to conversion
Ind. Eng. Chem. Fundam., Vol. 18, No. 1, 1979
and reduces the overall reaction rate. The results were obtained using two different models of the pore structure, diffusivity profiles and three different activity profiles under optimal and nlonoptimal conditions; therefore, they are applicable to a variety of reactions and solids. These results are relevant to some new advanced catalyst coal gasification processes under development. Literature Cited Bischoff, K., Chem. Eng. Sci., 18. 711 (1963). Calvelo, A,, Cunningham, R. E., J . Catal., 17, 1 (1970). Gardner, N., Samuels, E., Wilks, K., Am. Chem. SOC.,Div. FuelChem., Prepr., 18, 217 (1973). Guzman, G., Wolf, E. E., Chem. Eng. Sci., submitted for publication, 1978.
11
Johnson, J. L., Catal. Rev. Sci. Eng., 14 (l),131 (1976). Lee, E. S.,"Quasilinearization and Invariant Imbedding," Academic Press, New York, N.Y., 1968. Luss, D., Corbett, E. W., Chem. Eng. Sci., 29, 1473 (1974). Otto, K., Shelef, M., 6th International Congress of Catalysis, Prepr. €347, London,
1976. Petersen, E., AIChE J . , 3, 443 (1957). Shadrnan-Yazdi. F., Petersen, E. E., Chem. Eng. Sci., 27, 227 (1972). Szekely, J., Evans, J. W., Sohn, H., Gas-Solid Reactions," Academic Press, New York, N.Y., 1976. Szekeiy, J., Evans, J. W., Chem. Eng. Sci., 25, 1091-1107 (1970). Walker, P. L., Rusinko, F., Austin, L. G., Adv. Caral.. 11, 178 (1959). Wolf, E. E., J . Catal., 47, 85 (1977).
Received for review November 29, 1977 Accepted September 6, 1978
The Effect of Pulsations on Heat Transfer 0. Erdal Karamercan and John
L. Gainer*
Department of Chemical Engineering, University of Virginia, Charlottesville, Virginia 2290 1
The effect of pulsating the water stream in a double-pipe heat exchanger, with steam on the shell side, was investigated. Pulsation frequencies ranged up to 300 cycles per minute, and five different displacement amplitudes were used at each flow rate investigated. The heat transfer coefficient was found to increase with pulsations, with the highest enhancement observed in the transition flow regime.
Background The effect of pulsating one of the flows in a heat exchanger has been studied previously, with conflicting conclusions as to the effects seen. This probably indicates a lack of proper understanding of the pertinent variables involved. There is evidence in the literature (Baird et al., 1966; Darling, 1959; Havemann and Rao, 1954; Keil and Baird, 1971; Lemlich, 1961; Lemlich and Hwu, 1961; Ludlow, 1975; Martinelli et al., 1943; West and Taylor, 1952) that pulsating the flow in a heat exchanger enhances the heat transfer, although there is disagreement on this point. Furthermore, most of the previous investigators considered only a small number of operating variables in their studies and usually confined their studies to relatively narrow ranges of these variables. The aim of this study, therefore, was to investigate the effects of a combination of independent variables on the performance of pulsed-flow heat exchangers. These variables included thle frequency and the amplitude of the pulsations, the Reynolds number, the location of the pulsator with respect to the heat exchanger, and the length of the heat exchanger area. This study also investigated a broader range of these operating variables than covered in most previous works. The effect of pulsed flow on heat and mass transfer has been investigated by various researchers. In the area of heat transfer, theoretical and experimental studies of Mueller (1957) showed that, for pulsating flows which encompassed a frequency range of 2.3 to 14.9 cycles/min and a Reynolds number range of 53000 to 76000, the average Nusselt number was found to be less than the corresponding steady flow Nusselt number. McMichael and Hellums (1975) have presented a theoretical development which concludes that for laminar flow, pulsations cause a decrease in the rate of heat transfer at flow am0019-7874/79/1018-0011$01 .OO/O
plitudes which do not allow flow reversal to take place. Experimental studies under laminar flow conditions were conducted by Martinelli et al. (1943), using semisinusoidal velocity disturbances on the tube side fluid in a concentric tube heat exchanger. They found that the overall heat transfer coefficient was increased over the steady flow coefficient by 10% at the most. Lemlich and Hwu (1961) observed increases of up to 51% in the Nusselt number for air flowing at Reynolds numbers of 560 to 5900 in a horizontal, double-pipe, stream-to-air heat exchanger when acoustic vibrations were imposed on the air. Experiments performed by Lemlich (1961) and later by Lemlich and Armour (1965) on a double-pipe, steam-towater heat exchanger showed increases up to 80% in the overall heat transfer coefficient when the pulsator was installed upstream. However, when installed downstream, a decrease in the overall heat transfer coefficient was observed. The Reynolds numbers investigated were between 500 and 5000 and the pulsation frequencies ranged from 30 to 200 cycles/min. Darling (1959) reported similar observations. He obtained a 90% increase in the heat transfer coefficient, at a Reynolds number of 6000 and a pulsation rate of 160 cycles/min, when pulses were introduced upstream of the heaters. No improvement in the heat transfer coefficient was observed with the interrupter value downstream of the heater. In the turbulent flow regime, Baird et al. (1966) concluded that pulsations will improve heat transfer, particularly if the flow can be made to reverse in direction for part of the cycle. The maximum improvement they found was 225% for the water-side heat transfer coefficient in a double pipe, steam-to-water vertical heat exchanger. Keil and Baird (1971) observed as much as a 100% increase in the overall heat transfer coefficient using pulsating frequencies of 24 to 66 cycles/min in a commercial shell0 1979 American Chemical Society
