402
Ind. Eng. Chem. Fundam. 1988, 25, 402-405
A Transient Technique To Determine Heat Transfer in Small Packed Beds Yao-Jyh R. Huang, James A. SChwarz; and John C. Heydwelller Department of Chemical EngInWng and Materials Sclence, Syracuse Unlverstfy, Syracuse, New York 13244
A transient technique is proposed for measuring the mean thermal conductivity of a small bed of solid particles. Such beds are used in different catalytic rate studies or for a tempwature-programmed desorption apparatus. The lower bound of conductivity for a thin bed of 47% by weight nickel/sillca catalyst was 2.6 W/(m K), In orderof-magnitude agreement with published data. If methane were produced catalytically from a CO/H, mixture in such a catalytic bed at typical flow condltlons, the calculated difference between center line and wall temperatures should not exceed 4 OC.
Introduction Investigations of radial heat-transfer rates in packed beds employed as nonadiabatic catalytic reactors have followed two different methods of approach. In the first, only boundary temperatures have been measured and the results were correlated as heat-transfer coefficients or overall effective thermal conductivities reported by Argo and Smith (1953). The second approach is based upon the direct measurement of radial temperature transverses within the bed. Here results are reported in terms of effective thermal conductivities of the gas-solid bed by Coberly and Marshall (1951). Using this method, Leva (1950) found that resistance to heat transfer increased greatly near the wall. Because of the wall resistance, it is not possible to make an exact comparison between overall effect thermal conductivities and effective thermal conductivities applicable within the bed. A resolution to this dilemma was proposed by Froment and Bischoff (1979). They suggested that a mean effective thermal conductivity be defined which would embody all thermal resistances between the bed wall and center. The use of a mean effective thermal conductivity removes the spatial dependence of this parameter that arises due to variations in the packing density and flow velocity near the wall. Small packed beds of supported metal catalyst with high carrier gas flow rates are typically employed in laboratory-scale reactors such as those used in temperature-programmed desorption (TPD) apparatus reviewed by Falconer and Schwarz (1983). Since these experimental reactors are used to study chemical reaction rates, the temperature distribution within the catalyst bed must be known under reaction conditions. Calculation of the temperature profile requires a knowledge of the so-called effective thermal conductivity. The application of correlations to estimate this quantity is often subject to error because of the necessity to extrapolate experimental data outside the range in which measurements were made to obtain the correlations. Furthermore, the measurement of this quantity by typical multiple thermocouple probes is not practical for the thin beds that are used as "differential reactors". A transient technique is proposed to determine the mean effective thermal conductivity of a small packed bed of supported metal catalyst such as that used in the TPD apparatus. A sinusoidal temperature wave that is symmetrical about an average temperature is employed as the
* To whom correspondence should be addressed. 0198-4313/86/1025-0402$01.50/0
perturbation. Use of an unreactive gas eliminated any reaction, even though a catalyst was present. Under transient operation, the temperatures in the bed changed with time periodically. A comparison between experimental results obtained from thermocouples embedded in the particulate bed and the appropriate energy balance relationships, parametric in the mean effective thermal conductivity, was carried out over the entire cycle. To demonstrate the method for employing these transient procedures, experiments were conducted at two modulation frequencies at a fixed set of reactor conditions of flow rate, gas composition, and catalyst particle size. The amplitude of the temperature perturbation (k7.7 K) about the average temperature was small to ensure that the physical properties of the gas phase remained constant over the transient period. The metal loading of the catalyst was high enough so that the combined effects of the low thermal conductivity of the insulating support and the high thermal conductivity of the metal would result in a mean effective thermal conductivity reflecting both contributions. Note that lower weight loading catalysts should have a smaller mean effective thermal conductivity, and thus heat-transfer effects under reaction conditions may be important in evaluating point temperatures necessary for determining reliable kinetic data. The results of this study are compared to the effective thermal conductivities of the gas-solid bed derived from earlier correlations reported by Coberly and Marshall (1960), Deissler and Eian (19521, Kunii and Smith (1960), and Zehner and Schlunder (1970,1972). It was found that the best available correlations provide a scatter of values that range over an order of magnitude. The value determined directly by the transient procedures falls within this scatter.
Experimental Section The tubular reactor used in this study was made of stainless steel and consisted of an outer shell with a small microreactor insert. Approximately 100 mg of silica-supported nickel catalyst with a 0.225" particle diameter was placed in the microreactor to form a loosely packed catalyst bed. The dimensions of the reactor and the physicochemical properties of the catalyst used are reported by Lee and Schwarz (1982). The bed depth was 5 mm, and the porosity of the bed was 0.54. The reactor was contained within an IR oven that was controlled by a Micricon 823 microprocessor. The temperature control point was at the surface of the outer shell of the reactor so that the temperature at the outer surface of the reactor was the preprogrammed temperature. A cooling coil 0 1986 American Chemical Society
Ind. Eng. Chem. Fundam., Vol. 25, No. 3, 1986 403 Table I. Literature Values of the Effective Thermal Conductivity for 47% by Weight Ni/Si02 Catalysts
'
*
O
m
ken,
W/(m K) source 75.49 2.77 Kunii and Smith (1960) 2.60
this work
2.25
1.21
Zehner and Schlunder (1970, 1972) Coberly and Marshall (1951) Deissler and Eian (1952)
0.21
Argo and Smith (1953)
1.61
0.03
comment bulk nickel effective thermal conductivity mean effective thermal conductivity effective thermal conductivity effective thermal conductivity effective thermal conductivity effective thermal conductivity pure silica
around the outer shell was used to increase the heat removal rate of the shell. Thus, the desired sinusoidal temperature wave applied on the outer shell of the reactor was achieved by varying the flow rate of the cooling water. The temperature of the oven could be controlled to an accuracy within 0.1 OC. The temperature of the bed was measured at three points: the center of the bed at half-depth, the bottom of the inner wall of the microreactor, and the center of the bottom of the bed. The O.OO95-in.-diameter (0.24 mm) Omega chromel/alumel thermocouples used had a time constant estimated to be 50 ms. The thermocouples were calibrated in an ice bath before use and connected to a Hewlett-Packard 3467A logging multimeter which had four input channels and was capable of recording the voltage readings from thermocouples simultaneously and continuously. The channels of the logging multimeter connected with the thermocouples were zeroed and grounded before use to provide the same reference to all the readings from the detectors. The system with the temperature detectors was open and was operated under nonreaction conditions to obtain temperature profiles. Helium was used in the gas stream at a flow rate of 100 cm3/min. The particle Reynolds number was 0.118. The system was brought to a steady state at 510 K while He perfused the bed and cooling water flowed through the cooling coil outside the outer shell. The temperature of the outer surface of the reactor was then modulated by a sinusoidal wave controlled by the microprocessor. The temperature was programmed to be 510 + 40 sin (0.01745s) K, which resulted in a cycle time of 360 s and a sinusoidal perturbation to the bed of amplitude *7.7 K. Results identical with those reported were found for a cycle time of 240 s. The readings of the three thermocouples were recorded every 4 s by the logging multimeter. After 10 cycles, the system reached a cyclic steady state and the temperatures at the detecting points were recorded and analyzed to yield the temperature waveforms at the points. Results Effective thermal conductivity data for packed beds have been estimated by theoretical calculations and determined by experimental measurements. Table I shows a range of expected values of the conductivity for the dimensions of the silica-supported nickel catalyst and the reactor used in this study. Values ranged from 0.03 to 75.49 W/(m K). The upper and lower bounds of this range are the thermal conductivities of bulk nickel and pure silica, respectively. The value of 1.61 W/(m K) was determined from the correlation given by Coberly and Marshall (1951), which also considers the heat-transfer resistance near the wall. It was expected that this re-
K
-4
-I
4ou
t 0;
; E F F E;C N ;
MEAN
1 I;
;1 I1 M R M L CoNxETIVITY(W/MK)
Figure 1. Values of error, E , vs. the value of the mean effective thermal conductivity of the bed. See Table I and text for details of determining the best lower bound for the mean effective thermal conductivity.
sistance would be small for these small beds in which the carrier gas flow rate was high. The mathematical model used to describe the thermal distribution in the bed employs the concept of a mean effective thermal conductivity as proposed by Froment and Bischoff (1979). The appropriate energy balance equations are described in the Appendix. Substitution of values for this mean effective thermal conductivity into the energy balance equations yields different theoretical temperature profiles obtained under the transient conditions employed. The theoretical and experimental temperature data are compared by using the following error calculation
where the subscripts, bed and shell, represent the measurement points at the center of the bed bottom and the bottom of the inner wall of the microreactor insert, respectively. Figure 1 shows the plot of the value of the error, E, as a function of the mean effective thermal conductivity chosen for calculating the temperatures. Figure 1shows that a value of 2.6 W/(m K) is close to the point of minimum error. The experimental value is close to the results calculated from the equations proposed by Kunii and Smith (1960) and Zehner and Schlunder (1970, 1972). Because of the wall resistance in their case, it is not possible to make an exact comparison between these calculated values of the thermal conductivities and the mean effective thermal conductivity (applicable within the bed itself) determined in this work. The calculated values can be viewed as average conductivities weighted in favor of the low conductivities existing near the wall. They may be used to predict the bulk mean temperature of the gas leaving the bed but not the point temperature necessary to evaluate temperature uniformity under reaction conditions. Discussion When reactions are studied in a TPD apparatus to obtain kinetic data for modeling reaction mechanisms or to provide data for reactor design, it is important to ensure temperature uniformity. A reliable measure for the lower bound of the mean effective thermal conductivity is required for temperature uniformity under reaction conditions, To assess the effect on radial temperature gradients by underestimating the mean effective thermal conduc-
404
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Ind. Eng. Chem. Fundam., Vol. 25, No. 3, 1986
m
Ken
(w/MK)
Figure 2. Calculated temperature gradient in the radial direction as a function of the mean effective thermal conductivity under methanation conditions (H2/C0 = l O / l , flow rate 100 cm3/min). The indicated temperatures are those chosen experimentally to be the wall temperature.
tivity for catalysts with different metal loadings, particle sizes, and weights, the following procedures are proposed. The transient technique described earlier is used to determine the mean effective thermal conductivity of the catalyst bed to be used in the reaction studies. The carrier gas used should be inert and have a heat capacity comparable to that of the reactive gas mixture. A reliable rate expression should be used in the pseudohomogeneous energy balance with the experimentally determined mean effective thermal conductivity. The numerical simulation should be confined to reaction conditions producing less than 10% conversion to ensure the absence of masstransfer effects. To demonstrate the application of these procedures, we have considered the methanation reaction occurring over Ni/Si02 and have used the chemical rate expression reported by Vannice (1975) and Polizzotti and Schwarz (1982). This reaction has been studied extensively in laboratory studies employing small beds of group VI11 (groups 8-10 in 1985 notation) metal catalysts on various supports. For convenience, the wall temperature was chosen as the desired temperature for obtaining kinetic data. Numerical simulation as a function of the mean effective thermal conductivity at various wall temperatures was performed, and the temperature of the bottom center of the bed was used to estimate the radial temperature gradient. Conversion was less than 10% at T < 513 K. Figure 2 shows the calculated temperature difference between the center of the bed and the wall of the reactor as a function of the mean effective thermal conductivity for various wall temperatures. The temperature gradient in the axial direction at the bed center was comparable to that in the radial direction. As the mean effective thermal conductivity decreases, which would occur for lower weight loading catalysts,the temperature gradient increases. The telling feature of Figure 2 is the dramatic increase in the temperature gradient as the wall temperature is increased. The evaluation of rate data based on temperature uniformity is impossible. For example, a rough estimate at a wall temperature of 473 K shows that 70% error occurs when the rate is assigned to the wall temperature and the center line temperature excess is 10 K.
Conclusion The mean effective thermal conductivity for small packed beds such as those used in a TPD apparatus can be determined experimentally for conditions of reactor
operation. Correlations to estimate the effective thermal conductivity obtained for larger beds provide, at best, an order-of-magnitude estimate of this value. Since conventional methods for direct measurement on these small beds are impractical, a transient technique employing a periodic low-frequency perturbation to the applied temperature is proposed. The results provide a lower bound for the mean effective thermal conductivity. For a bed having a small mean effective thermal conductivity,a low modulation frequency is required. In other words, the temperature modulation frequency for a catalyst containing a small amount of metal has to be lower than that for a catalyst containing a high concentration of metal. Procedures are described to evaluate the effect of the mean effective thermal conductivity on temperature gradients under reaction conditions. Severe temperature gradients occur when the mean effective thermal conductivity is low and the reaction temperature is high. Computation of kinetic data based on assumed temperature uniformity is invalid under these conditions. The procedures described herein provide a method to extend this transient technique to other catalyst systems and to define operating conditions which ensure temperature uniformity under reaction conditions.
Acknowledgment Portions of this work are supported by the Department of Chemical Sciences, Basic Energy Science, under DOE Contract DE-AC02-84ER13158 and the Syracuse University Institute for Energy Research. We acknowledge the additional support and interest of the Texaco Foundation. We also acknowledge discussions with Professor P. Y. Chen and Dr. P. I. Lee.
Appendix The reactor system modeled is a cylindrical differential catalyst bed. The energy balance equation for the shell is
and the pseudohomogeneousenergy balance equation used by Hill (1977) for the packed bed is
where T and 0 are the temperatures of the bed and shell, respectively, tb is the porosity of the bed, pg and pa are the densities of gas and catalyst particles, respectively, (Cp)g and (C,,), are the specific heat of gas and catalyst particles, respectively, u, is the superficialvelocity per unit bed area, and hdf is the mean effective thermal conductivity of the bed. Under transient conditions, a sinusoidal temperature wave with a common temperature TI, an amplitude HI, and frequency w is applied to the outer shell as an input perturbation. The boundary conditions and the initial conditions for this system are r = 0 ; aT - 0 (A3) ar
--
r = b; 0 = T I
+ HI sin ( w t )
r=a; T=8
(A4)
(A5)
Ind. Eng. Chem. Fundam. 1986, 25, 405-409
z = 0;
T = 8 = TI + HI sin
(wt)
405
the largest deviation in the axial direction would occur near the bed entrance due to cooling effects from the incoming gas. This was found to be the case. (A7)
where b and a are the radius of the outer stainless steel shell and the catalyst bed, respectively, and L is the bed depth. Recall, our analysis is based on the concept of a mean effective thermal conductivity. Boundary conditions A5 and A6 reflect this approach. These two-dimensional time-dependent differential equations with associated boundary and initial conditions allow us to calculate the temperature profile within the bed under the transient conditions. The orthogonal collocation technique described by Finlayson (1980) with third-order Legendre polynomial used by Kreyszig (1979) and the biased-difference methods proposed by Heydweiller and Patel (1982) are used to eliminate the radial and axial direction dependences of eq A1 and A2. The software package developed by Byrne and Hindmarsh (1975,1976) is used to solve the resulting ordinary differential equations numerically. The temperature distribution in the axial direction was calculated layer by layer. Thus, the number of layers used for calculation in the axial direction may affect the final numerical results. To ensure that the temperature gradients are not affected by the choice of the number of layers used in the calculation, a reasonable number of layers has to be employed. We assume a hypothetical configuration of the catalyst particles in the bed to be in the form of individual particle layers. A bed with 5-mm depth will consist of 23 layers of 0.225-mm-diameter catalyst particles in the axial direction. To explore how the number of layers in the calculation affects the calculated derivative of temperature with respect to axial distance, the calculation was made for 10 layers and 26 layers. The results showed that differences between the 10-layer calculation and 26-layer calculation were negligible. In other words, the results are independent of the number of layers used in the axial direction when the number of layers is larger than 10. I t is interesting to note that because the gas residence time above the bed is short (-2 s), it was expected that
Nomenclature a = radius of catalyst bed, m b = radius of outer stainless steel shell, m (C,),, (C,), = specific heat of gas and catalyst particles, respectively, kJ/(kg K) HI = amplitude of temperature modulation, K k,ff = mean effective thermal conductivity of catalyst bed, W/(m K) kshell = thermal conductivity of reactor shell, W/(m K) r = radius position in catalyst bed, m T = temperature of catalyst bed, K t = time, s TI = common temperature, K u, = superficial velocity per unit area, m/s z = axial position in catalyst bed, m Greek Letters tb
= porosity of catalyst bed
8 = temperature of outer shell, K pg, ps = densities of gas and catalyst particles, kg/m3 w = modulation frequency, s-l 7 = cycle time, s
Literature Cited Argo, W. B.; Smith, J. M. Chem. Eng. Prog. 1953. 4 9 , 443. Byrne. G. D.; Hindmarsh, A. C. Report UCRL-75868, 1975; Lawrence Livermore Laboratory, Livermore, CA; Report UCID-30312. 1976; Lawrence Livermore Laboratory, Livermore, CA. Coberly, C. A.; Marshall, W. R., Jr. Chem. Eng. hog. 1951. 4 7 , 141. Deissler, R. G.; Eian. C. S. National Advisory Committee for Aeronautics: RM-E52c05, Washington, DC, 1952. Falconer, J. L.; Schwarz, J. A. Catal. Rev.-Sci. Eng. 1983, 2 5 , 141. Finlayson, B. A. Nonllnear Analysis in Chemical Engineering; McGraw-Hill: New York, 1980; p 117. Froment. G. F.; Bischoff, K. B. Chemical Reactor AnafLsls and Des@n; Wiiey: New York, 1979; p 532. Heydweiller, J. C.; Patel, H. S. Comput. Chem. Eng. 1982. 6 , 101. Hill, C. G., Jr. An Intrductlon to Chemical Engineering Kinetics & Reactor Design; Wiiey: New York, 1977; p 496. Kreyszig, E. Advanced Engineering Mathematics; Wiiey: New York, 1979; p 195. Kunii, D.; Smith, J. M. AIChE J . 1980, 6 , 71. Lee, P. 1.; Schwarz. J. A. J . Catal. 1982, 73, 272. Leva, M. Ind. Eng. Chem. 1950, 4 2 , 2498. Polizzotti, R. S.; Schwarz, J. A. J . Catal. 1982, 77, 1. Vannlce, M. A. J . Catal. 1975, 3 7 , 449. Zehner, P.; Schiunder, E. U. Chem.-Ing.-Tech. 1970, 42, 333. Zehner, P.; Schlunder, E. U. Chem.-Ing.-Tech. 1972, 44, 1303.
Receiued for review September 25, 1984 Revised manuscript receiued September 11, 1985 Accepted October 10, 1985
Packed Column Efficiency Fundamentals Gordon A. Hughmark Ethyl Corporation, Baton Rouge, Louisiana 7082 1
Turbulent liquid-film hydrodynamics are used with experimental liquid-film data for dumped and structured packings to develop relationships for liquidfilm thickness and gas-phase pressure drop analogous to vertical gas-liquid flow in a circular pipe. The turbulence contribution to liquid-phase mass transfer is found to be represented by the shear velocity of the liquid film. The analysis can be extended to the flooding point as experimental flooding data show a consistent relationship of the dimensionless film thickness and the hydraulic radius for flow in the packing.
A reliable design method for packed column masstransfer efficiencies should logically depend upon the two-resistance model with a vapor-liquid interfacial area and calculated vapor- and liquid-phase mass transfer
coefficients based upon fundamentals. Such a design method would be more likely to extrapolate correctly to systems outside the range of experimental efficiency data than methods based upon empirical models. Packed 0 1986 American Chemical Society
