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by means of a commercial finite element code, ANSYS/FLOTRAN. The geometry model, representing a fixed bed, consisted of an arrangement of eight sphere...
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Ind. Eng. Chem. Res. 1998, 37, 739-747

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Computational Fluid Dynamics Studies of the Effects of Temperature-Dependent Physical Properties on Fixed-Bed Heat Transfer Simon A. Logtenberg and Anthony G. Dixon* Department of Chemical Engineering, Worcester Polytechnic Institute, 100 Institute Road, Worcester, Massachusetts 01609

The influence of the temperature profile on the fluid flow and heat transfer in a fixed bed of tube to particle ratio of 2.86 was studied by solving the 3D Navier-Stokes and energy equations by means of a commercial finite element code, ANSYS/FLOTRAN. The geometry model, representing a fixed bed, consisted of an arrangement of eight spheres in a tube. The difference in heat-transfer parameters between a wall-cooled and wall-heated reactor was studied, using air as a fluid. The dimensionless wall heat-transfer coefficient, Nuw, and the radial effective conductivity ratio, kr/kf, were evaluated from the calculated temperatures at different locations in the bed by comparing these with the analytical solution of a two-dimensional pseudohomogeneous model, using a nonlinear least-squares analysis. Results were obtained for Reynolds numbers in the range 9-1450. Though for high Re there was no real difference between a wallcooled or a wall-heated tube, for low Re a significant difference was found. The effect of the viscosity, conductivity, and density variations, as a consequence of temperature variations, was studied using a hydrocarbon mixture as the fluid. Results indicated that the temperature profile had an influence on the fluid and heat flow and thus on the effective parameters, although at high Re numbers the influence became less. Introduction Fixed-Bed Heat-Transfer Behavior. Fixed-bed reactors are still one of the most used gas-solid reactors in industry, in which heat-transfer plays an important role. The relatively simple construction and behavior of this reactor make it still favored over other types of more advanced and complicated reactors. Up to now, much has been written in the literature about the heattransfer behavior in fixed-bed reactors. Many studies have been done over the last 40 years in order to predict the heat-transfer behavior in these types of reactors, using simple or more complicated models. Today, reactor engineers still prefer to describe the heattransfer behavior by means of a two-dimensional pseudohomogeneous model (Vortmeyer and Haidegger (1991)), normally referred to as the standard model, which neglects temperature differences between the fluid and solid phases. The overall heat-transfer resistance in the bed is described in this model by an effective radial thermal conductivity, lumping together all heattransfer mechanisms, and a wall heat-transfer coefficient, first introduced by Coberly and Marshall (1951), explaining the well-known “temperature jump” near the wall:

qr ) hw(T|r)R - Tw)

(1)

The overall heat-transfer coefficient can be determined by the following equation: * Author to whom correspondence should be addressed. Phone: (508) 831-5350. Fax: (508) 831-5853. E-mail: [email protected].

dt/2 1 1 + ) U hw βkr

(2)

Various choices for the value of β were recently discussed by Dixon (1996). Although this model is simple, the comparison of experimental data with model predictions has been quite inconsistent over the last 4 decades. Also the enormous number of correlations available in the literature to predict the effective heat-transfer parameters creates confusion for the chemical reactor design engineer; i.e., which correlation should they use and how accurate is this? Recent reviews (Tsotsas and Schlu¨nder (1990), Freiwald and Paterson (1992)) showed that there is still no answer to the question of how to predict the heat-transfer behavior in packed-bed reactors. The consequence is that expensive pilot plants still have to be built in order to determine the optimum and safe operating conditions. Although results for the effective radial conductivity (normally expressed as kr/kf) showed rather good agreement between different researchers, no consistent results have been obtained for the wall heat-transfer coefficient (normally expressed as Nuw), especially at low to moderate Reynolds numbers (Vortmeyer and Haidegger (1991), Li and Finlayson (1977), Tsotsas and Schlu¨nder (1990)). This is the reason why lately there seems to be a tendency among researchers to include a nonconstant velocity profile that depends on the local radial void fraction rather than an assumed constant velocity over a cross section in a packed bed (Tsotsas and Schlu¨nder (1990), Vortmeyer and Haidegger (1991), Borkink and Westerterp (1994)). Additionally, a reactor that has a low tube-to-particle diameter ratio (N) is an

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740 Ind. Eng. Chem. Res., Vol. 37, No. 3, 1998

interesting and complex case. This reactor is important due to the large number of industrially important reactions that are highly exothermic and must be carried out in narrow reactor tubes immersed in a cooling medium (Dixon (1988), Freiwald and Paterson (1992)). The problem with this low ratio is the extra difficulty in modeling this specific type of reactor, due to the strong influence of the wall region. Explanations for the discrepancies between the different results for Nuw have been given already many times. Most researchers try to explain the discrepancies by considering the true meaning of the heat-transfer coefficient at the wall and what it actually represents. In addition, a recent discussion cast some doubts on the applicability of the use of a wall heat-transfer coefficient in the region of smaller Reynolds numbers (Vortmeyer and Haidegger (1991)). However, others (Li and Finlayson (1977), Zio´lkowski and Legawiec (1987)) tried to explain the discrepancy by the fact that different experimental techniques were used to determine the effective heat-transfer parameters. A theoretical analysis of the heat-transfer behavior at the wall was given by Yagi and Kunii (1961), who stated that the extra heat-transfer resistance at the wall was influenced by three different mechanisms: (1) A decrease in the thermal conductivity of the granulated solid phase due to bed porosity increase at the wall. (2) A reduction in the turbulent thermal conductivity arising from changes in the bed geometry in the vicinity of the wall. (3) A heat-transfer resistance for the fluid adjacent to the apparatus wall. Later these mechanisms were evaluated by Tobı´s and Zio´lkowski (1988). In their analysis they recommended using a modified Reynolds number for the third mechanism, defined by the near-wall hydrodynamics, rather than an averaged Reynolds number of the whole bed. Tsotsas and Schlu¨nder (1990) stated that the use of a heat-transfer coefficient at the wall is only justified when the spatial extent of the region wherein the heattransfer is located is small with respect to the tube diameter. In addition, they stated that at low Re (or molecular Peclet number) no resistance at the wall exists and a Nuw-Gz plot should be used rather than a Nuw-Re plot. All in all one can conclude that so far no unique answer exists for predicting the heat-transfer behavior in all cases. Besides the fact that researchers try to use more advanced models by including a radial velocity profile that depends on the local void fraction, still not all phenomena can be explained. The question therefore arises as to whether there are some other aspects that cannot be lumped and should therefore be included in the model that predicts the heat-transfer behavior of fixed-bed reactors. One of these aspects is the influence of the temperature profile on the velocity profile. Although it is a known fact that the fluid properties are dependent on the temperature, hardly any attention has been paid to the problem of how the interaction between radial temperature distribution and flow in a fixed-bed reactor affects the results for predicting the heattransfer behavior of these reaction systems. It could very well be that the discrepancies between models and applicability of the wall heat-transfer coefficient at the wall can be partially explained by the fact that the fluid properties in the fixed-bed reactor are changing with

temperature and therefore have an influence on the prediction of the effective heat-transfer parameters. Temperature gradients introduce density, viscosity, and fluid conductivity gradients, which could influence the fluid and heat flow. Vortmeyer et al. (1992) studied the influence of the density temperature dependence on the temperature location of the hot spot at Re ) 32. They concluded that, under typical industrial conditions (high Re numbers), the effect of the varying fluid density on the velocity field can be neglected, because they found only a small difference when they included a temperature dependency of the density at this low Re number. However, in their study constant viscosity and fluid conductivity were assumed. Earlier studies of Schertz and Bischoff (1969) showed that under isothermal conditions a different flow pattern was observed than under nonisothermal conditions. They attributed this to the increase in the viscosity of gases with temperature. Also Radestock and Jeschar (1971), Stanek and Szekely (1973), Lederbrink and Starnick (1978), and Stanek and Vychodil (1987) have treated some aspects of the temperature profile influence on the flow in fixedbed reactors. To the authors knowledge, no one has been able to really test this temperature profile influence on the prediction of the heat-transfer parameters. Reasons for this are most likely based on the limited practical means that are available. With this paper we hope to give some more insight and understanding of this issue by means of a computational fluid dynamics study. Computational Fluid Dynamics (CFD). Now that computers are becoming more and more powerful, computational modeling is more developed and therefore getting more interesting. Computational fluid dynamics is a fast-growing field in the study of fluid flow and heat transfer (Logtenberg and Dixon (1998)). A recent review about the capabilities of CFD for the chemical reaction engineering field has been given by Harris et al. (1996). Previous studies on heat transfer in fixedbed reactors, using CFD, were done by Lloyd and Boehm (1994) and in a series of publications by Dalman et al. (1984a,b, 1986a,b). However, these studies were limited to a 2D CFD study of a very simplified geometry. A first attempt to use a 3D CFD approach in a simple geometry to determine a Nuw for Re in the range of 25101 was done by Derkx and Dixon (1996). In this study, they studied fluid and heat flow around three spheres in a tube with N ) 2.14. The wall heat-transfer coefficient was obtained from the calculated heat flux and temperature profiles, giving the temperature jump from eq 1. The Nuw showed agreement with values predicted by a model-matching theory using correlations based on experimental measurements. A more advanced method to obtain Nuw and kr/kf values in a more complex geometry was presented in a recent study (Logtenberg and Dixon, 1998). Here Nuw and kr/kf values were evaluated from the calculated temperatures at different locations in the bed by fitting these with the analytical solution of the usual two-dimensional pseudohomogeneous model, using a nonlinear leastsquares analysis. The Nuw and kr/kf values showed reasonable qualitative agreement with both experimental values and values predicted by a model-matching theory based on experimental measurements (Dixon and Cresswell, 1979). The effects of pressure and magnitude of the wall temperature on the effective heat-transfer parameters were shown to have no influence, which was

Ind. Eng. Chem. Res., Vol. 37, No. 3, 1998 741

Figure 1. Geometry model of eight spheres, representing a packed bed.

Figure 2. Vertical section of the eight spheres in a tube, showing the element mesh.

in qualitative agreement with results from an experimental study (Borman et al., 1992). Also, no actual depth dependence of the Nuw and kr/kf was found at Re ) 131. In this paper we discuss a computational fluid dynamics (CFD) analysis approach, using a finite element method (FEM), to study the influence of the temperature profile on the fluid properties and velocity profile and how these changes affect the effective heat-transfer parameters. The analysis will be done by comparing a cooling versus a heating experiment and comparing results for constant and temperature-dependent physical properties of the fluid. With this analysis we try to shed light upon the issue of how these changes in these fluid parameters actually influence the fluid and heat flow in a fixed-bed reactor. Model Generation CFD Geometry. A fixed-bed reactor was represented by an arrangement of eight spheres (see Figure 1), consisting of two layers of four spheres on top of each other in a tube (indicated as volumes 1 and 2 in Figure 2). By solving the 3D Navier-Stokes equations, temperature and velocity values were calculated at numer-

ous positions within the bed. The solid spheres were ceramic materials with a thermal conductivity of 1.2 W/(m‚K). A specified inlet flow was given at the bottom, which exited at the top of the tube. The spheres did not touch each other nor the wall, due to the fact that this would give problems near the contact points for the mesh generation and the stability of the numerical solution. The finite element mesh is visible in Figure 2, which shows a finer mesh in the vicinity of the spheres and the wall. The tube was extended after the last layer of four spheres (indicated as volumes 4 and 5 in Figure 2) in order to allow the flow to approximately develop before it exited. In order to make sure that the velocity gradients did not become too large, which could lead to instability of the solution, a small extension was used at the inlet (indicated as volume 3 in Figure 2). The length of the tube was a total of 0.16 m, and the diameter was 0.12 m. The diameter of the spheres was 0.042 m, giving an N of 2.86. The center locations of the spheres were set at the following positions: (x, y, z) ) ((0.024, (0.024, 0.035), (0, (0.033 941 1, 0.085), and ((0.033 941 1, 0, 0.085), with all coordinates in SI units. The total number of elements that was used in this study was about 49 000 elements. The elements were three-dimensional four-node tetrahedra. CFD Equations. The fluid was taken to be incompressible, Newtonian, and in laminar or in turbulent flow depending on the value of Re. All fluid properties were made temperature dependent with the wall-cooled and wall-heated CFD calculations. The ideal gas law was used for the density temperature dependence, and Sutherland’s law was used for thermal fluid conductivity and viscosity temperature dependencies, when air was used as the fluid. For the hydrocarbon mixture a polynomial equation was used for conductivity and viscosity temperature dependence. The boundary conditions that were applied to the model were (1) a constant temperature and velocity at the inlet, (2) a constant temperature at the wall, and (3) a no-slip condition around the spheres and at the wall. The turbulent model, which was used in some CFD calculations, was the k- model, which is based on the two-equation eddy viscosity model of Jones and Launder (1972). However, the k- model is not valid immediately adjacent to the walls. Therefore, a wall turbulence model is necessary for the elements at the wall. Given the current value of the velocity parallel to the wall at a certain distance from the wall, an approximate solution can be obtained for the wall shear stress. In order to approximate the turbulent boundary layer velocity profile, the log-law-of-the-wall model was used (Launder and Spalding (1974), White (1991)). Due to the fact that it is unknown when to activate the turbulence model, most cases were solved, using both the laminar and turbulent model, so an evaluation could be done afterward. Studies at what Reynolds number the turbulent eddies start to become visible in a fixed-bed reactor were done by Jolls and Hanratty (1966). They found that at Reynolds numbers between 60 and 130 turbulence became visible. Tobı´s and Zio´lkowski (1988) stated that a transition from laminar to turbulent flow occurs at Re ) 100. Although this was an indication of when the turbulence model should be activated, it could be expected that in the present case the transition to turbulence would probably occur at higher Re numbers, due to the fact that the void fraction was higher in the model, because of nontouching

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spheres. An indication of the extent of turbulence in the fluid, which could be used as a guideline, was the value of the eddy viscosity, calculated from the k- model, compared to the value of the fluid viscosity. The steady-state temperatures at different locations in the bed that were calculated can be described by a two-dimensional pseudohomogeneous axially dispersed plug-flow (ADPF) model, giving the following equation in dimensionless form (Dixon, 1988):

[

]

1 dp ∂2θ ∂θ 1 dp ∂2θ 1 ∂θ + + ) ∂ξ Per R ∂ζ2 ζ ∂ζ Pea R ∂ξ2

(3)

with the following boundary conditions, applicable to the situation simulated by CFD:

θ)1 at ξ ) 0 ∂θ )0 at ζ ) 0 ∂ζ ∂θ + Biθ ) Bi at ζ ) 1 ∂ζ

}

(4)

The analytical solution for this set of equations was given by Wakao and Kaguei (1982):

Tw - T Tw - T0

n

J0(anζ) exp(-an2ψ)

n)1

an[1 + (an/Bi)2]J1(an)

) θ(ζ,ξ) ) 2

∑

(5)

where,

ψ)

2y an 2 1+ 1+4 kk GcpR r a

[ ( ) ] y)

1/2

krξ GcpR

with an being the roots of the characteristic equation

anJ1(an) - BiJ0(an) ) 0

(6)

This ADPF analytical solution was compared with the temperatures obtained from the CFD calculations, using a nonlinear least-squares approach with a modified Levenberg-Marquardt algorithm and a finite difference Jacobian, so that hw and kr could be evaluated. Pea was kept constant and calculated from an empirical correlation, assuming Peaf(∞) ) 2.0 and using the formula of Zehner and Schlu¨nder (1973) for the solid conductivity. For the void fraction a value of 0.73 was used, which was obtained from the total volume of the particles and volumes 1 and 2 (see Figure 2). Numerical Solution. A numerical solution was obtained when the convergence monitors for all the variables were no longer monotonically decreasing and the average, maximum, and minimum values of the solution variables no longer had a monotonic trend. The convergence monitor was defined as N

convergence monitor )

∑ i)1

|φki - φk-1 | i |φki |

(7)

where φi is vx, vy, vz, P, or T. It must be clear that there is no single exact answer to the 3D Navier-Stokes equations, especially in com-

Figure 3. Nuw values at different Reynolds numbers.

plex geometries, since nature does not guarantee that a single exact answer will exist. Problems which are oscillatory in nature (e.g., vortex shedding behind a sphere) may not yield stationary results from a steady state or a transient solution algorithm. It is important that the mass and energy balances are solved within acceptable limits, in this study set at 5%. It appeared that at higher Re numbers the solution became more unstable, which could indicate unstable eddies or vortex shedding behind or in between the spheres. At very low Re numbers (Re < 50) the energy balance was not solved within the limits of 5% and the error went as high as 15%. Results and Discussion Cooling versus Heating Experiments. Values for Nuw and kr/kf were obtained from the temperatures of the nodes selected at z/dp ) 1.19 and 2.38, just above the two layers of spheres (see Figure 2), which were fitted by eq 5 with a least-squares fitting program. For the calculations of Nuw and kr/kf, kf was kept as a constant reference value (kf ) 0.03), and for the calculation of Re, the mass flux G ) Fv and the average values for the viscosity in the bed were used. In Figure 3 values for Nuw and in Figure 4 values for kr/kf are shown at different Re. In these figures the effective heattransfer parameters are shown for both cooled and heated wall experiments. The predictions of a theoretical model for the effective heat-transfer parameters (Dixon and Cresswell (1979)) are included for reference. The 95% confidence intervals are not shown in these figures, since they were smaller than the sizes of the symbols. This is due to fitting to so many “data” points. Values for both the laminar and turbulent solutions are shown for the APDF model (eq 3) for Re between 20 and 720 to show the influence of the model choice on the values of Nuw and kr/kf. Both Figures 3 and 4 show that the choice of turbulent or laminar model did influence the calculation of the effective parameters. In the case of cooling the laminar model predicted a lower value for Nuw compared to the turbulent model, and in the case of heating it predicted a higher value for Nuw. In Figure 4 a straight line is drawn through the kr/kf values

Ind. Eng. Chem. Res., Vol. 37, No. 3, 1998 743

Figure 4. kr/kf values at different Reynolds numbers.

at high Re number, where the heat transfer will be dominated by the mixing of the fluid in the voids of the packing, and a horizontal line is drawn through the kr/ kf values at the low Re, where the heat transfer will take place primarily by molecular conduction. This change of heat-transfer behavior was also mentioned by Tsotsas and Schlu¨nder (1990). The “intersection point” of these two lines could now be taken as the point where the turbulent model should be activated, because radial mixing of the fluid is becoming important, which can be best described by a turbulence model. For the cooled and the heated wall the “intersection point” is around Re ) 80. So, above Re ) 80 the turbulent solutions and below Re ) 80 the laminar solutions are believed to be the best estimates for the effective parameters. Figure 4 shows that the comparison of kr/kf values with the theoretical model is in quite good agreement, although the values are systematically below the model, which is most likely the consequence of the higher void fraction in this geometry. In Figure 3 a discrepancy at low Re numbers for Nuw, i.e., an increase in Nuw values, compared to the theoretical model, was found. This increase in Nuw could be a consequence of the following facts: (1) The ∆T over the entire bed is much higher at low Re numbers than at higher Re numbers. (2) At the lower Re numbers the error in the energy balance for the CFD calculations became quite large (i.e., >5%). (3) The geometry model is a simplified version of the actual fixed-bed reactor at this N. The void fraction is overestimated in the model, because there is a “gap” between the two layers of spheres. However, it could also be that the upturn of Nuw at very low Re is actually occurring in fixed-bed reactors. Such an increase could be explained by the fact that, at lower Re, the molecular heat conduction is becoming more important. The fluid boundary layer will disappear at these low Re and therefore also the extra resistance at the wall, due to the boundary layer. However, there will still be an extra resistance at the wall due to the increased local void fraction, causing a lower effective stagnant conductivity. Future CFD studies where spheres will be touching could give more insight into this phenomenon.

It can be seen in Figures 3 and 4 that the type of experiment, cooling or heating, shows a difference in magnitude for Nuw and a slight difference for kr/kf at the lower to moderate Re. The difference at the low to moderate Re between the heated and the cooled wall can be explained by the fact that different mechanisms are taking place. Due to cooling at the wall, the density at the wall is higher, and the viscosity is lower, than that within the bed, therefore increasing the local Reynolds number and thus promoting local turbulence near the wall. The higher viscosity within the bed creates a retardation of the axial flow in the bed center. This retardation of the flow within the bed together with the lower viscosity at the wall, causing an increase in the local turbulence, increases the radial heat flux near the wall. On the other hand, with a heated wall the flow gets retarded near the wall and increased within the bed. This enhancement of the flow in the bed center and decrease of the local turbulence at the heated wall decreases the radial heat flux near the wall and therefore gives lower values of Nuw compared to a cooled wall. However, the observation that with a heated wall the Nuw is larger at very low Re numbers can then be explained by the fact that the fluid conductivity at the wall is in this case much higher than that with a cooled wall and at these low Re molecular conduction will start to play a significant role. The idea of increased flow near the wall for a cooled wall is confirmed if we look at Figure 5a, where the relative flow rate (G/G0) profile is shown for a low flow rate (Re ≈ 20), and at Figure 5b, where the profile is shown for a high flow rate (Re ≈ 1300). In this figure, the relative radial flow rate profile was calculated by dividing the magnitude of the velocity times the density at the different radial positions at z/dp ) 1.19 and 2.38 (see Figure 2) by the magnitude of the inlet velocity times the inlet density. No account was taken of the direction of the local velocities. The large scatter of the flow rate profile is the consequence of not averaging out the velocities over the angular coordinate of the tube. In addition, the profiles were taken from two different axial positions, where the radial flow rate distribution is not necessarily the same. In Figure 5a,b we see an increase of the flow in the bed center (r/R ) 0) and near the wall (r/R ) 1), which can be attributed to the radial void fraction profile of the geometry. In Figure 5a we can see that the G/G0 ratio is higher for a heated wall than for a cooled wall in the bed center and lower near the wall, which confirms our previous discussion above. By comparing Figure 5a with Figure 5b, it is seen that the retardation of the fluid within the bed with a cooled wall is much more pronounced at low Re than at high Re. Influence of Temperature-Dependent G, µ, and k on Effective Parameters. In order to study the effect of the temperature-dependent properties on the effective parameters hw and kr, an equimolar hydrocarbon mixture of methane, ethane, propane, and butane was used as the fluid. The reason for this was that this mixture has a larger dependence on the temperature than air and also is more realistic for actual fixed-bed reactor applications. Nuw and kr/kf values were determined for three different inlet velocities (i.e., Reynolds numbers), for which the following five different CFD calculations were done: (1) all fluid properties are constant (calculated with an average temperature); (2) density (F) is a function of temperature, with other

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Figure 5. Relative radial flow profile for heated and cooled fixed-bed reactors at (a) Re ) 20 and (b) Re ) 1300.

properties constant; (3) viscosity (µ) is a function of temperature, with other properties constant; (4) fluid conductivity (k) is a function of temperature, with other properties constant; (5) all properties are a function of temperature. The average temperature at which the constant properties were evaluated was chosen to be the average temperature of the CFD calculation with the same inlet velocity, where air was used as a fluid. However, in order to see what the impact of the chosen value of this average temperature was, two different values were chosen for the case of an inlet velocity of 0.02 m/s. The equations for the temperature-dependent properties were obtained by fitting a second-order polynomial function through experimental values, which were taken from standard reference work (Perry and Green, 1986). These polynomial property functions were then used as input to the CFD program. For the heat capacity a constant value was used in all cases. In all cases the inlet temperature was set at 300 K and the wall temperature at 600 K, similar to the previous study with the heated wall, where air was used as the fluid. In

Table 1. Effective Heat-Transfer Parameters at High Re Number with Tav ) 510 properties

vin (m/s)

Re

kr/kf

Nuw

constant F ) f(T) µ ) f(T) κ ) f(T) all ) f(T)

0.64 0.64 0.64 0.64 0.64

1409 1198 1401 1409 1192

61.9 54.7 59.6 69.1 60

15.2 17.0 13.1 15.0 16.3

Table 2. Effective Heat-Transfer Parameters for Moderate Re Number with Tav ) 470 properties

vin (m/s)

Re

kr/kf

Nuw

constant F ) f(T) µ ) f(T) κ ) f(T) all ) f(T)

0.08 0.08 0.08 0.08 0.08

206 161 194 206 153

10.7 11.7 10.2 11.4 11.9

6.4 7.7 6.0 7.9 9.2

Table 1 data are shown for an inlet velocity of 0.64 m/s (i.e., Re ) 1190-1400), in Table 2 data are shown for an inlet velocity of 0.08 m/s (i.e., Re ) 150-210), and in Tables 3 and 4 data are shown for an inlet velocity of 0.02 m/s (i.e., Re ) 40-60), where in Table 3 the

Ind. Eng. Chem. Res., Vol. 37, No. 3, 1998 745

Figure 6. Relative radial flow profile for constant and varying viscosity and conductivity at (a) Re ) 45 and (b) Re ) 1300. Table 3. Effective Heat-Transfer Parameters for Low Re Number with Tav ) 400

Table 4. Effective Heat-Transfer Parameters for Low Re Number with Tav ) 460

properties

vin (m/s)

Re

kr/kf

Nuw

properties

vin (m/s)

Re

kr/kf

Nuw

constant F ) f(T) µ ) f(T) κ ) f(T) all ) f(T)

0.02 0.02 0.02 0.02 0.02

58 39 62 58 41

2.6 2.7 2.7 3.1 3.2

12.8 11.5 12.4 10.4 9.4

constant F ) f(T) µ ) f(T) κ ) f(T) all ) f(T)

0.02 0.02 0.02 0.02 0.02

54 58 54 54 57

3.2 3.2 3.1 3.3 3.2

18.4 15.6 17.4 12.1 9.4

average temperature for evaluation of fluid properties was set at 400 K and in Table 4 at 460 K. The reason for the variation in Re, with the same inlet velocity, is the changing of the viscosity and inlet density of the fluid, thereby changing the value of Re. From Tables 1-4 it can be concluded that there is an influence of the temperature profile on the effective parameters. At high Re, Table 1 shows that the relative variation in kr/kf is not very large, in contrast to the values for Nuw. The reason for this lies in the fact that the temperature at which the constant properties were evaluated was closer to the bed center temperature than the wall temperature. So, at the wall the constant properties were evaluated at 510 K, although the wall

temperature was 300 K, causing a wrong prediction of the fluid properties at the wall. The bed center temperature was around 510 K, so the fluid properties were estimated well using a constant average temperature of 510 K. Table 2 shows the same behavior as in Table 1, although the difference in Nuw was not as large, because of the lower Tav. Tables 3 and 4 again show this behavior, where the error in kr/kf became higher if Tav was set at a lower temperature (see Table 3). The cause for the bigger discrepancy in kr/kf at the lower Tav lies in the fact that the calculated average temperature, from the CFD analysis for the bed with all properties varying, is closer to the value of 460 K than to the value of 400 K. However, with a lower temper-

746 Ind. Eng. Chem. Res., Vol. 37, No. 3, 1998

ature a better estimate was obtained for Nuw, as can be seen when comparing Tables 3 and 4. It is difficult to explain the exact influence of the temperature dependence of a single fluid property on the values for the effective heat-transfer parameters, because all these properties will influence each other. It appears that the choice of Tav has a big influence on the final calculation of the effective heat-transfer parameters. So, the choice of the average temperature at which the properties are evaluated can influence the calculation of the effective heat-transfer parameters. It can be concluded that there is an influence of the temperature profile on the heat flow, although this becomes less at higher Re numbers. To see what influence the temperature profile has on the radial distribution of the fluid flow, the relative radial flow profile was compared for constant and varying viscosity with a Tav ) 460 K for a low (Re ≈ 45) and high (Re ≈ 1300) flow rate (see Figure 6), similar to the experiment done by Schertz and Bischoff (1969). At low Re (Figure 6a) it appears that there is an influence, although it is not large. At higher Re (Figure 6b) the influence of the temperature profile becomes practically negligible for the fluid flow. In the experiment of Schertz and Bischoff the case with constant properties was done at a much lower temperature than the average temperature for the case with varying properties, most likely causing the bigger differences that were found in that specific study.

Nomenclature

Conclusion

Subscripts

This study has shown the influence of the temperature profile on the fluid flow and heat-transfer in a low-N fixed-bed reactor tube. It has been found that the influence of the temperature profile is not negligible at low Re numbers. In this study it was found that a gas-solid wall-cooled fixed-bed reactor gas behaves differently than a gas-solid wall-heated fixed-bed reactor. Wall-cooled fixed-bed reactors give a slightly higher value for kr/kf than wall-heated reactors, which is attributed to the viscosity and density effects. Also, at high Re the cooling of a gas gives a higher value for Nuw than heating of a gas, although at very low Re the opposite trend was found. It can be concluded that the flow profile in the bed and the heat-transfer rate are affected by the shape of the temperature profile (wall-cooled or wall-heated) in the bed. At low Re the temperature profile can have a strong influence on the heat-transfer in the fixed-bed reactor, depending on how sensitive the fluid parameters are to temperature. At high Re, the temperature profile does not affect the velocity profile and heattransfer results very much. If constant fluid properties are used in order to evaluate the effective heat-transfer parameters, it was found that the choice of the temperature at which the fluid properties are evaluated is important. It is expected that the geometry of the fixed-bed reactor will also have a strong influence on the flow and the prediction of the heat-transfer behavior. Future CFD studies, with contacting spheres and at higher N, should give more insight into the interaction between temperature and fluid flow in fixed-beds.

a ) axial f ) fluid p ) particle r ) radial s ) solid t ) turbulent w ) wall

Acknowledgment The authors thank ANSYS, Inc., for their support in this research study.

cp ) fluid heat capacity [J/(kg‚K)] d ) diameter [m] G ) mass flow rate [kg/(m2‚s)] hw ) wall heat-transfer coefficient [W/(m2‚K)] k ) conductivity [W/(m‚K)] kr ) effective radial conductivity, [W/(m‚K)] kf ) fluid molecular conductivity [W/(m‚K)] l ) distance above the bed [m] N ) tube to particle ratio (dt/dp) q ) wall heat flux [W/m2] R ) tube radius [m] r ) radial coordinate [m] T ) temperature [K] T0 ) inlet temperature [K] U ) overall heat-transfer coefficient [W/(m2‚K)] v ) superficial gas velocity [m/s] x ) Cartesian x-coordinate [m] y ) Cartesian y-coordinate [m] z ) Cartesian z-coordinate [m] Greek Letters β ) lump constant in eq 2  ) void fraction µ ) fluid viscosity [(N‚s)/m2] F ) density of fluid phase [kg/m3] θ ) dimensionless temperature [(Tw - T)/(Tw - T0)] ζ ) dimensionless radial coordinate (r/R) ξ ) dimensionless axial coordinate (z/R)

Dimensionless Groups Biot number ) Bi ) hwR/kr wall Nusselt number ) Nuw ) hdp/kf Peclet number ) Pe ) Gcpdp/kr Prandtl number ) Pr ) cpµ/kf Reynolds number ) Rep ) Fvdp/µ Graetz number ) Gz ) [Per/(L/D)](D/d)

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Received for review May 30, 1997 Revised manuscript received August 19, 1997 Accepted August 20, 1997 IE970382Q