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Ind. Eng. Chem. Process Des. Dev. 1980, 19, 680-688
Analysis and Design of Catalyst-Coated Findspines Selim M. Senkan Department of Chemical Engineering, Massachusetts Institute of Technoiogy, CambrMge, Massachusetts 02 139
The feasibility of using catalyst-coated fins and/or spines to handle highly exothermic and fast chemical reactions is demonstrated theoretically. Finlspine aspect ratios are correlated to the other parameters characterizing the system in order to identify regions that would permit effective reactor operation and control. Effects of several finkpine shapes and various operating conditions are examined for the case of a first-order irreversible reaction.
Introduction and Background Reactors in which the catalyst is coated on and cooled through the walls of the reactor vessel are very suitable for carrying out highly exothermic and fast chemical reactions. The tube-wall-reactor concept, in which the walls of heat-exchanger tubes are coated with the catalyst, was utilized by the Pittsburgh Energy Technology Center to conduct the CO methanation reaction in their coal-tomethane Synthane pilot plant (Haynes et al., 1970). The wall-coating is attractive because it is well known that the capacity of conventional fixed-bed catalytic reactors is limited by heat transfer. Therefore, by bonding a catalyst layer onto a cooling surface, this limitation can be significantly improved by reducing the large gas-solid thermal resistance (Senkan et al. 1979). Even though the wall-supported catalytic reactors potentially offer low thermal sensitivity, which leads to reduced sintering rates and thus a longer catalyst life, their commercial use was limited due to problems related to application of the catalyst coating and to its activation, removal, and regeneration. Also, the relatively low catalyst loadings per unit gross volume of the reactor deterred their widespread use. However, with the aid of flame- and plasma-spraying techniques, in conjunction with alloy catalysts (e.g., Raney nickel), catalytic heat-transfer surfaces can be routinely coated, activated, and regenerated (Forney and Demeter, 1966; Peters, 1967; Yasumura and Yoshino, 1972). Moreover, the low catalyst loading in these reactors can be circumvented by use of catalyst-coated fins or spines (Strakey et al., 1975; Schultz and Hemsath, 1976). Therefore, the use of reactor systems involving catalysts coated on and cooled through the supporting surfaces appears to be an attractive alternative to packed beds. Although the general concept of fins/spines with heat generation on their surfaces is not new (Jacob, 1949), our present understanding of these systems is insufficient for design in chemical reaction engineering (Callinan and Burford, 1977; Kern and Kraus, 1972; Luss and Ervin, 1972). In particular, it is important to know a priori the allowable fin/spin configurations and dimensions that will permit adequate control of the temperature along the structure while at the same time providing the maximum catalytic surface area. In this paper, various heat-transfer characteristics of catalytic rectangular/cylindrical fin/spine structures are investigated in an attempt to identify the important design variables and to examine their effects on temperature control. Formulation of the Problem Consider an exothermic chemical reaction of the form A B taking place on an arbitrarily shaped catalyst-
coated fin or spine surface (Figure 1). In developing the mathematical model, the following assumptions are made: (1)The fin/spine is coated with an infinitesimally thin catalyst; that is, only the external geometrical surface area of the structure is catalytically active. (2) Only one-dimensional heat conduction is considered through the fin/spine (in the x direction, see Figure 1). That is, uniform temperature is assumed across the structure at any location. This assumption would be realistic if the thickness of the fin/spine were small when compared to its length and if conduction across the structure were rapid when compared to convection and heat generation rates at the catalytic surfaces. For the case of a fin, this also implies that conduction in the third dimension (direction perpendicular to the sheet of paper) is insignificant. Clearly this condition would be realistic when the width of the fin is smaller than its length, and when axial temperature variations are not large. (3) Thermophysical properties of the system are constant and uniform. (4) Bulk gas temperature and composition changes are insignificant. (5) Convective heat and mass transport coefficients are constant and uniform. (6) The reaction is irreversible and exothermic, and the rate can be expressed by power law kinetics with an Arrhenius temperature dependency. (7) Heat transfer at the base of the fin is expressed by Newton’s law of cooling. (8) The tip of the fin is insulated with respect to heat and mass transfer. Therefore, the governing equations for heat and mass transfer, respectively, are
k,$(X)(cb - c,) = k,e-E/RTCsn$(x)
(2)
where A&) and A,(x) are the cross-sectional and surface areas of the fin/spine structure, respectively, and A,(x) = dA,(x)/dx. For a rectangular fin of thickness 2 b and length L (unity in the third dimension),A, = 2b and A&) = 2x. For a cylindrical spine of radius r, A, = n? and A,(x) = 2nrx. Similarly, for a triangular fin of base dimension 2 b and length L , A,(x) = 2 b ( l - ( x / L ) and A,(x) = (1 + ( b / L ) 2 ) 1 / 2 xThe . associated boundary conditions for eq 1 are
-+
0196-4305/80/1119-0680$01 .00/0 0 1980 American Chemical Society
Ind. Eng. Chem. Process Des. Dev., Vol. 19, No. 4, 1980 681 FinlSpirNe with a r b i t r a r y shape
B u l k Gas Stream C b a n d T b
It is important to note that the forms of eq 6, 7, or 9 are similar to the ones encountered in the analysis of catalytic wires (Pismen and Kharkats, 1968; Luss and Ervin, 1972; Jackson, 1972). However, for nonrectangular shapes, no such resemblance exists; for example, for the case of a triangular fin, the equation corresponding to eq 9 will be
C a t a l y s t Coated Surface A S ( x )
(10)
+
Rectangular Fin
QCy1 i n d r i c a l S p i n e
Figure 1. Configurations of various catalyst-coated fins and spines.
For a rectangular configuration, eq 1 and 2 can be nondimensionalized with respect to bulk gas temperature T b and composition Cb, and the length of the fin L to give
+
where Bil, = Bih(l (b/L)2)1/2and d'2 = d2(1 (b/L)2)1/2. Notice now that unlike eq 9, eq 10 is a nonautonomous differential equation. However, to illustrate the computational methods involved, a rectangular geometry with a first-order surface reaction will first be examined in detail, followed by a brief comparative analysis of the triangular shape. The design parameters for the rectangular geometry include Bih and d2, which are both related to the aspect ratio (L2/b). For a given reaction (k0,E, and n) and a set of operating conditions (h,k,, Tb,c b , and k), the design problem requires a suitable choice of the aspect ratio (L2/b), for which fin tip can be cooled effectively by conduction a t the base of the structure. As with catalytic pellets, an effectiveness factor can also be defined for catalytic fins as the ratio of the actual reaction rate along the structure to the rate at the corresponding bulk conditions, that is
or using eq 7 1
and 2k,Cb(l - x) = 2koe-E/RTboCbnXn
(5)
where 5 = x/L, 0 = ?'/Tb, and x = c,/cb. By rearranging eq 4 and 5, the following can be shown
and 1 - ;y = Dae-Y(l/o-')Xn
(7) where Bih = hL2/bk is the Biot number for heat transfer, 4' = koerCb"(-AH,)L2/bkTb is the ratio of the heat generation rate on the surface to the heat conduction rate in the fin, y = E/RTb is the dimensionless activation energy, and Da = kocYCbn-l/k, is the Damkohler number for mass transfer. The boundary conditions (3a) and (3b) will become
where Bi, = h&/k if; the corresponding Biot number a t the base of the fin. For a first-order reaction, X can be solved in eq 7 and substituted into eq 6 to yield
Physical Interpretation of the Governing Equations For an exothermic reaction, the maximum temperature would be located a t the fin tip because of the imposed boundary conditions (3a, 3b). Any other location for this maximum would imply an unrealistic design situation. To show this, one can multiply eq 9 by dO/d[, integrate once, and using the boundary condition (8b) and O(1) = Om, the following relationship between O(t) and dO/dt(, can be obtained
Note that the right-hand side of eq 13 represents the heat duty for the conduction heat transfer (see Figure 2). The integrand is the difference between the heat generation rate (Figure 2, curve G) and the convection cooling rate (Figure 2, curve C), the integral of which must be compensated by conduction cooling along the fin under steady-state conditions. Clearly this amount must be positive for a realistic design, since a negative integral would imply complex dO/dt values, which would have no physical significance. For relatively small values for Bih/d2(Figure 2, line Cl), the heat generation rate is always greater than the con-
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Ind. Eng. Chem. Process Des. Dev., Vol. 19, No. 4, 1980 Table I. Ranges of Parameters of Physical Interest ~
~~
dimensionless group
typical numerical range
1. activation energy group 2. apparent reaction order
5 < 7 4 30 first-order is considered here 0 < Bi,
Heat
3. Biot number for heat transfer ( a t the fin/ spine base) 4. Biot number for heat transfer (gas/fin/ spine) 5. Damkohler number for mass transfer 6. ratio of coolant t o bulk gas temperature 7. ratio of heat generation t o conduction rate (exothermic)
Genera t i o n or Convection Cooling
Figure 2. Heat generation and removal from a catalytic fin/spine.
vection cooling rate for 0 < el, which is a situation commonly encountered in gas-solid catalytic systems. Therefore, for 8-, Iel, the integral in eq 13 is always positive for all e < Om=. In the limiting case when Bih = 0, the governing equation (9) and the associated boundary conditions (8a, 8b) would be equivalent to the problem of heat transfer with chemical reaction in a porous rectangular slab, a subject which has been studied extensively. At moderate cooling conditions (Figure 2, line CJ, negative values of the integral would be possible for a range of t9values, for which A - > A l + A2+. Finally, for relatively large values of Bih/42 (Figure 2 , line C3), integral (13) would be positive only for 0 < e,, I1. Computational Method The split-boundary-value problem as given by eq 8a, 8b, and 9 can be coverted into a tractable initial-value problem by the tranformation p = (1- [)@. Therefore the resulting heat transport equation will be
+
Note that the quantity B i h / 4 2 = hTb/koeYCb"(-A",) represents the ratio of the convective cooling rate to the heat generation by reaction and is independent of the aspect ratio L 2 /b. The computational algorithm which requires no trial and error involves the the following steps: (1) Set the parameters e,, y,Da,and B i h / 4 2 . (2) Select a value for e(0) = Omax, for which dO/dp(O) = 0 is always true. (3) Integrate eq 14 from 0 = 8-, and dO/dp = 0 as an initial value problem until 6 = 8, - (p/Bi,)(dO/dp) at which point p = 4. This procedure can then be repeated as many times as needed. The algorithm requires continuous monitoring
0 < Da
