226
I n d . E n g . C h e m . Res. 1988,27, 226-233
Heat Transfer in Packed-Tube Reactors Patrick E. Peters,+Rinaldo S. Schiffino, and Peter Harriott* School of Chemical Engineering, Cornell University, Ithaca, New York 14853
Radial heat transfer was investigated for three tubes packed with alumina spheres, cylinders, or rings. Overall heat-transfer coefficients, determined from the average gas temperature for different packed heights, were used with expressions for the effective thermal conductivity to calculate wall-film heat-transfer coefficients. The Nusselt number for the wall film increased with D,/D, as well as with the particle Reynolds number, but the effective thermal conductivity decreased with Dp/D, for a given Reynolds number. These trends led to a fairly broad maximum in the curve of overall coefficient versus D,/D,, with the maximum shifting to lower values of D,/D, a t higher flow rates. For the same size packing, the overall coefficients for the different shapes were nearly the same over a 10-fold range of flow rates. In the design of packed-tube catalytic reactors, the problem of radial heat transfer has generally been treated by using pseudohomogeneous models. These models do not consider explicitly the presence of the catalyst and treat the system as if only one phase was present with characteristic effective transport properties. The radial temperature profiles depend on an effective thermal conductivity, k,, which varies with the fluid flow, the physical properties of the solid and fluid phases, and the geometry of the system. Although k , decreases near the tube wall, because of the increase in void fraction, it is usually considered independent of radial position, and the extra resistance is accounted for using a wall film coefficient, h,. The parameters k , and h, are used in two-dimensional models to predict radial and axial temperature profiles in the reactor tube. Most workers agree on the general form of the equations for k, and h,, but there are significant differences in the exponents and empirical coefficients of published correlations. The differences become more pronounced when these correlations are extrapolated to high Reynolds numbers and to large ratios of particle diameter to tube diameter. In commercial reactors, very high Reynolds numbers are often used to get good heat transfer, and the particle size may be 0.1-0.3 times the tube diameter to avoid high pressure drop. Heat transfer for large D,/D, ratios (from 0.1 to 0.6) and for high Reynolds numbers (up to 8000) is the main focus of this study. Some packed-tube reactors use ring-shaped catalyst particles to get a greater effectiveness factor and lower pressure drop than solid particles of the same size, but there are only a few studies of heat transfer in beds of rings. In two studies of commercial hydrocarbon reformers (Hyman, 1968; Singh and Saraf, 1979) using ring-shaped catalyst in 0.1-m tubes at Reynolds numbers of 5000 or higher, Beek’s (1962) correlation for h, had to be modified by a 0.4 factor to make the calculated conversion and exit temperature match the measured values. However, in a similar study (DeDeken et al., 1982), a good fit to the reformer data was found by using an equation that gives relatively high values for h,. One goal of this work was to compare heat-transfer coefficients for rings with those for other packings at the higher Reynolds numbers typical of industrial practice.
Effective Thermal Conductivity Several studies have shown that the effective thermal conductivity can be calculated as a sum of stagnant (or zero-flow) and dynamic contributions (Bunnell et al., 1949;
Coberly and Marshall, 1951; Yagi and Kunii, 1957; Agnew and Potter, 1970). The stagnant contribution can be estimated by combining in series and parallel the resistances corresponding to conduction through the fluids in the voids, through the contact surface between particles, through the stagnant film near the contact surface, and through the particles. The stagnant contribution decreases with increasing void fraction and with decreasing ratio of solid to fluid conductivity. The dynamic contribution comes mainly from the random radial motion of the fluid elements as they pass through the bed and depends on the particle Reynolds number, Re,, the Prandtl number, Pr, and the Peclet number for radial heat transfer, Pek In dimensionless variables, the effective thermal conductivity is expressed as k, k,O Re$r
_ -- kf
kf
+- Peh
In a packed bed, it is usually assumed that the radial effective diffusivities for mass and heat transfer are the same. This assumption implies an analogy in the mechanism for radial heat and mass transfer which is due to the random radial motion of fluid elements. At high mass velocities, an increase in the particle-fluid heat transfer and in the conduction-in-series contribution might introduce some error in this assumption. Calculations, however, indicate these effects to contribute only 5% to k , for a Reynolds number of about 1000. This assumption allows for the substitution of Peh by Pe, in eq 1. The Peclet number for mass transfer depends on D /Dt and on the particle shape. Random walk theory preJicts a value of about 10 for small values of D,/D,. Studies of radial diffusion in air by Bernard and Wilhelm (1950) gave Pe, = 10-12 for spheres with D,/D, = 0.047. Fahien and Smith (1955) measured Pe, for several particle sizes and proposed the following relationship for the rkgion Re, > 100, where Pe, is independent of Re,: Pe, = 8.0[1 + 19.4(D,/D,)2] (2) The increase in Pe, with D,/D, was attributed to the increase in void fraction, but there is no theoretical basis for the exponent of 2 , so the values of Pe, at high D,/Dt are uncertain. The published data can be fitted within 5% to a linear equation: Pe, = 6.3 + 34.7(DP/D,) (3) An alternate correlation for the effect of particle size was proposed by Bauer and Schlunder (1978), who measured heat-transfer rates for several types and sizes of particles:
Presently with Lawrence Livermore Laboratories, Livermore, CA 94550. 0888-5885/88/2627-0226$01.50/0
(1)
0 1988 American Chemical Society
Ind. Eng. Chem. Res., Vol. 27, No. 2, 1988 227 TIT, aT/ar = 0
at z = O at r = 0
-ke aT/ar = h,(TR - T,)
b Yagi +Wahoo
at r = R
(6)
The last boundary condition defines the wall heattransfer coefficient, h,, in terms of the wall temperature, Tw,and the bed temperature at the wall, TR. The temperature at the wall is obtained by evaluating the radial temperature profile at r = R. Similarly, a bed heat-transfer coefficient, hbed,can be defined in terms of TR and an average bed temperature, Tav,as
5 i Rings England+Gunn 0 Yagi+Kunii
0
0
01
D,
'
02
03
-he (aT/ar),=R = hbed(Tav - TR)
(7)
If eq 7 is written for hbed, and with r* = r/R,
Dt
Figure 1. Peclet number for radial dispersion.
This equation was fitted to data for spheres, cylinders, and rings, with D, defined as the diameter of a sphere with the same volume as the particle. The factor X F / D , was reported to be 1.15 for spheres, which gives a limiting value of Peh = 7.0 at low De / D t , in comparison with 8.0 from eq 2. Equations 3 and 4 agree approximately for intermediate values of D,/D,, but eq 4 predicts lower values for D,/D, > 0.1. The equations given by Yagi and Wakao (1959) to fit their data for spheres and cement clinkers correspond to Pe, = 9.0 for D,/D, = 0.021-0.072 and Pe, = 11.0 for D p / D , = 0.12-0.17. These results are in reasonable agreement with those of Fahien and Smith (1955)) as shown in Figure 1. For cylinders, Agnew and Potter (1970) used heattransfer measurements to determine ke and found Peh to be about 20% lower for cylinders than for spheres, but the values decreased as D IDt increased. Analysis of the same data by Olbrich and $otter (1972) gave different values of Peh but still 20% lower values for cylinders. Other heat-transfer studies with beds of cylinders by Bunnell et al. (1949) and by DeWasch and Froment (1972) gave values of Peh as large or larger than typical values for spheres. However, Bauer and Schlunder (1978) found Peh to be about 40% lower for cylinders than for spheres. Because of the large differences, none of the values for cylinders are included in Figure 1. For rings, England and Gunn (1970) measured radial dispersion of argon in small fixed beds and found Pe, for rings to be about 4.8 compared to 6.5 for solid cylinders. In the summary of heat-transfer studies by Yagi and Kunii (1957), values of Peh for Rasching rings ranged from 4.2 to 8.3, increasing with D,/D, as shown in Figure 1. Bauer and Schlunder (1978) studied the effects of L I D , and Di/D, on Peh for beds of rings and cylinders, but all the data are for DplD, < 0.1. For a typical ring catalyst with L I D , = 1 and Di/D, = 0.6, their correlation gives Peh = 3.3 at D,/Dt = 0.1, which is even lower than the other values in Figure 1, and it is only one-third of the corresponding value for spheres. T h e Heat-Transfer Model In the pseudohomogeneous model, the gas and bed temperatures are the same, and the two-dimensional heat balance with plug flow of gas, in cylindrical geometry, is written as
where R(x,T) is a general chemical reaction expression. The boundary conditions are
The term in the brackets defines a dimensionless parameter, p, which contains all the temperature dependence of hbed:
The Biot number can be expressed in terms of p and a ratio of coefficients: (10) If the reaction rate is approximately constant in the radial direction, and the temperature is not a strong function of the axial position, solving eq 5 results in a parabolic temperature profile and a value of 4 for fi. When there is an exothermic reaction with a moderately large overall temperature difference, the higher reaction rate near the center can be allowed for by using a lower hM, and a value of equal to 3 is sometimes recommended for a conservation design. For heat transfer in packed beds without reaction, the bed coefficient is high in the inlet region where the temperature profile is not fully developed. The coefficient decreases with length and approaches an asymptotic value, which is more appropriate to use for packed-bed reactor design than the value for short lengths (Li and Finlayson, 1977). However, the asymptotic value of hbedin the heat-transfer tests depends on the Biot number, on boundary conditions, and on the velocity profile. Crider and Foss (1965) showed that, for plug flow and constant wall temperature, the value of is 2.9 at high Biot number and 4.0 at low Biot number. For low Biot number, the temperature at the outer boundary of the bed increases appreciably with distance from the inlet, and the value of p is nearly the same as for heat transfer to a fluid in laminar plug flow with constant heat flux (Nu= 8.0 or p = 4.0). The velocity profile in a packed tube deviates appreciably from plug flow. The velocity profile usually has a pronounced maximum about one particle diameter from the wall, a minimum in the center, and goes to zero at the wall. The detailed profile may be quite eratic, with more than one peak and a strong dependence on angular position and on the method of packing the tube (Lerou and Froment, 1977). The deviation from plug flow increases with the ratio of particle size to tube sue, and for D,/D, = 0.125, the maximum velocity is about twice the central minimum (Schwartz and Smith, 1953). Increasing the proportion of fluid flowing near the wall should increase the bed coef-
228 Ind. Eng. Chem. Res., Vol. 27, No. 2, 1988
ficient for heat-transfer tests or the value of 0,but there has been no quantitative evaluation of 0 for different DJD, ratios. Evidence that this effect may be large comes from comparing the parabolic flow and plug flow solutions for heat transfer to fluids in pipe flow. For the constant flux case, the limiting Nusselt numbers me 4.4 for parabolic flow and 8.0 for plug flow, which correspond to 6 = 2.2 and 4.0 ( p is based on R and Nu on DJ. When these results are extended to a packed tube, the surplus flow near the wall and the low velocity in the center could make /3 larger than 4.0 for some cases. Velocity profiles similar to those for packed beds have been observed for laminar upflow of liquid in an electrically heated vertical pipe (Scheele and Hanratty, 1962$1963). The velocity profiles become distorted because of natural convection, and above a certain Grashof number, there is a minimum velocity at the center and a maximum about 0.2-0.3R from the wall. Under these conditions the Nusselt numbers were about 6-9 (0 = 3-4.5) and increased with the ratio of maximum to minimum velocity. Although these results do not apply directly to the packed tube, they show that values of greater than 4.0 are possible with a distorted velocity profile. The resistances to the radial heat transfer in the bed and at the wall can be added to define an overall heat-transfer coefficient, ho: 1 -1= - 1 (11)
+-
hO
hbed
hw
The overall heat-transfer coefficient can be used in a one-dimensional model for reactor design to predict axial profiles of the average temperature in the packed tube. Equation 11 can also be used to calculate wall-film coefficients from measured overall coefficients, which is the approach taken here. In this study, the temperature and velocity profiles were not determined, and the heat-transfer coefficients were obtained from the measured average gas temperatures for different bed lengths beyond the entrance region. The amount of heat transferred was determined without making any assumption about the velocity profile. Values of Pe, were determined in separate experiments by measuring the dispersion of a tracer from a point source in the center of the packed tube. The results for Pe, were used in calculating k,. A value of ,f3 equal to 4 was assumed for the calculation of hM. The correct value would be less than 4 for runs at low Reynolds number (high Bi) and probably greater than 4 for runs at the highest Reynolds number (low Bi). The wall film coefficient was calculated by subtracting the bed resistance from the measured overall resistance. The major resistance was in the wall film,and using different values for p would not have much effect on the calculated values of h,.
Apparatus and Procedures Heat Transfer. Heat was transferred from low pressure steam (4psig) to dry air in double-pipe heat exchangers. In the inner pipe, air entered at the bottom, passed through the packed bed and a mixing cup, and was vented to the atmosphere. In the outer pipe, steam entered at the top, and water drained from a stream trap. The air flow rate was measured with a rotameter at low flow rates or with an orifice meter at high flow rates. Air temperatures were measured before the flow meters and inside the mixing cups with copper-constantan thermocouples. Temperature readings were taken from a digital multichannel temperature indicator. The heat exchangers were made of standard steel pipe sections that were welded to steel plates to create dou-
11
*--- Suspending Rod
Temperature +Reading Section 7
Thermoc w ple
:l
C- Teflon Ring
n n
I
,\
Insulating Material
dt Figure 2. Temperature measurement system.
ble-pipe steam jackets. Three heat exchangers were built with inner pipe sizes of 26-, 52-, and 102-mm diameter and 61-, 91-, and 213-cm length. The inner pipes extended beyond the steam jackets and were flanged at both ends. Alumina catalyst supports supplied by Norton and United Catalyst were used as packing materials. Five sizes of spheres (4.8, 6.4, 9.5, 12.7, and 19.0 mm), two sizes of cylinders (4.8 and 6.4 mm), and one size of ring (16-mm o.d., 7.9-mm i.d., and 9.5-mm length) were tested in the three heat exchangers. The D,/D, range covered in this study was from 0.091 to 0.362 for spheres, from 0.063 to 0.24 for cylinders, and from 0.156 to 0.60 for rings. The packing was loaded into the heat exchangers from the top and was supported at the bottom by a wire screen. The mixing cup was placed at the top of the packing inside the tube and was fastened to the upper flange through a threaded rod and a wire screen. The mixing cup consisted of two sections of copper pipe (schedule 40) and a Teflon O-ring, as shown in Figure 2. The dimensions of the mixing cups varied proportionally to the tube size. For an inside tube diameter, D,, the mixing cup diameter was D,,the mixing section length was Dt, and the temperature reading section length was 20,. The mixing section was closed at the bottom by a copper orifice plate with a D,/4 center hole. The temperature reading section was closed at the top by a perforated copper plate fastened to the threaded rod. The Teflon O-ring was placed in between the two sections when they were brought together. The inside diameter of the mixing cup was D,/4, and layers of cork, cardboard, and Fiberglas were used as insulating material. The thermocouple was placed in the temperature reading section at 0.50, from the bottom of the section. Preliminary tests showed that plots of log (T, - Tav)vs L were steeper near the entrance but became linear for L 2 2D,, as the heat-transfer coefficient approached the asymptotic value. Steady-state temperature measurements were then made at two different packing lengths for each run to obtain data free from entrance effects. The coefficients were calculated from the difference in mixing cup temperatures for the two lengths, the air flow rate, and the steam temperature. These asymptotic coefficients were corrected for the condensate resistance and the wall re-
Ind. Eng. Chem. Res., Vol. 27, No. 2, 1988 229
500r----I 5001
200
N
E
200
X
I
\
i%100 0
c
, " iL
e
48 0 9 5 0 127 @ 19
50t
E,n
I
2
I
5
6.4
IO
G kg/m2.s
Figure 3. Overall heat-transfer coefficients for spheres in the 52.5-mm pipe.
d
5001
!
I
20 0 2
I
I
I
I 2 G kg/m' s
I
I
2 G kg/m2.s
I
102
I
I
5
10
Figure 5. Overall heat-transfer coefficients for cylinders.
500---
D, =26.6mrnp
0 5
20 0.5
I
I
5
IO
Figure 4. Overall heat-transfer coefficients for spheres in the 26.6-mm and 102-mm pipes.
sistance to get the overall heat-transfer coefficients for the bed, ho. Mass Transfer. The mass-transfer experiments consisted of measuring the radial dispersion of a helium tracer in air flowing upward through a packed column. The helium was injected at the center of the column, and samples were withdrawn from different radial positions for different packing lengths and analyzed by thermal conductivity. The air and helium flow rates were measured with rotameters and an orifice meter. The two columns used were made of 6.3-mm-thick Plexiglas and were 41.2and 165-mm inside diameter and 80- and 150-cm length. Two sizes of spheres (6.4 and 12.7 mm), one size of cylinder (6.4 mm), and one size of ring (16" o.d., 7.9-mm i.d., and 9.5-mm length) were tested in two columns. The packing was loaded from the top up to about one column diameter, and the injection pitot tube was inserted through a tap in the column wall and placed in the middle of the packing. More packing was added to the desired length, and samples were withdrawn from nine radial positions in the large column and from five positions for the small column by using 0.8-mm copper pitot tubes. The columns were closed a t the top with a flange and a back-pressure regulator. The steady-state measurements were made 2 or 3 times to test reproducibility of the profiles. Results Overall Coefficients. The overall heat-transfer coefficients for spheres are shown in Figures 3 and 4. The coefficients increased with the 0.55 and 0.69 power of the superficial mass velocity, with lower exponents for the larger particles, where the wall film resistance is more important, as will be shown later. The film coefficient
201
'
0.5
I
I
I
2
I
5
1
10
G kg/m2.s Figure 6. Overall heat-transfer coefficients for rings.
increases with Re,'I2, whereas at the high Reynolds numbers of this study, the bed coefficient is nearly proportional to Re,. More extensive data would probably show a gradual decrease in slope at very high velocity for the same reason. The average slope and values of h, are in fair agreement with the data of Leva et al. (1948) for cooling of gases in packed tubes (slope = 0.7) but not with the more widely quoted results for heating of gases, which gave a slope of 0.9 (Leva, 1947, 1950; Holt, 1984). The results for the 52.5-mm (2-in.) pipe, given in Figure 3, show a maximum coefficient for D, = 9.5 mm or D,/D, = 0.18. The coefficients for 6.4" spheres were almost spheres and were not plotted. The the same as for 4,s" decrease in coefficient for sizes past the maximum is not quite as pronounced as expected from previous correlations (Leva, 1950). With the 26.6-mm (1-in.) pipe, the highest coefficient was found for the smallest particles, with D,/D, = 0.18. For the 102-mm (4-in.) pipe, only the two largest sizes of spheres were used, and the coefficients were higher for the 12.7-mm spheres, with D,/D, = 0.12 than for the 19-mm spheres. The overall coefficients for 4.8- and 6.4-mm cylinders in three pipe sizes are shown in Figure 5. In the 52.5" pipe, the overall coefficients for the two cylinder sizes were almost the same and were correlated by the same line. The slopes of the lines were 0.75 and 0.76 for the largest and 0.58 for the smallest pipes. The decrease in coefficients on increasing the pipe size was slightly greater for cylinders than for spheres (Figures 4 and 5). The coefficients for cylinders were about the same as for spheres of the same size at low mass velocities and 10% higher at high mass velocities.
230 Ind. Eng. Chem. Res., Vol. 27, No. 2, 1988
E
'em
05-
I
I
6 0 Spheres A Cvlinders 0 Rings R Rings, cex,
03t 0 2
01
0
02
03
04
05
06
Figure 7. Void fractions for beds of spheres, cylinders, and rings. I
I
I
I
Figure 9. Peclet number for different packings.
I 200
60
I
IO0
i
I /
00 10
08
06
04
02
1
A Cylinder 0 Ring
Dp' Dt
I
0 Sphere
aL
00
r/R
Figure 8. Radial concentration profile for rings.
The overall coefficients for rings increased with about the 0.6 power of the mass velocity as shown in Figure 6. The highest coefficients were for the 52-mm pipe and the lowest for the 26" pipe. When the rings were compared with 12.7-mm spheres, the rings gave 15% higher coefficients in the 102-mm pipe but about the same values as spheres in the 52.5-mm pipe. These results can be related to the void fractions in the beds, which are plotted in Figure 7 as a function of D /De For this plot, the nominal diameter of the 5/s-in. X Bl8-in. rings was taken to be in. or 12.7 mm. Cylinders pack more closely than spheres in large tubes, but the difference disappears when D,/D, is 0.2 or greater. The rings have a larger total void fraction, but the external void fraction is about the same as for the cylinders at the same D,/De Thus,the fact that rings gave higher coefficients than spheres in the largest pipe might be attributed to the lower external void fraction of 0.35 compared to 0.41 for spheres. In the 52.5" pipe, these packings had about the same external void fraction and the same coefficients. Mass Transfer. Radial concentration profiles were measured at distances of about one or two column diameters from the injection point for the 6.4-mm spheres in the small column (D / D t = 0.153), the 12.7-mm spheres in the large column (bp/Dt = 0.077), the 6.4-mm cylinders in both columns (D,/D, = 0.038 and 0.153), and the rings in the large column (D,/D, = 0.077). The concentration profiles for rings are shown in Figure 8. The lines were obtained from the solution of the plug flow mass balance equation for constant radial Pe, (as in Fahien and Smith (1955)). The fit by least squares between the theoretical and experimental concentration profiles was generally within l o % , being more accurate for the large column, where the plug flow assumption holds best. The Reynolds numbers ranged from 70 to 880, and for Re, > 100, Pe, was independent of Re,.
2o
D, = 2 6 6mm
t
i
1
10 200
500
1000
2000
5000
10.000
ReP
Figure 10. Nusselt numbers for spheres, constant Pe, approach.
Figure 9 shows Pe, for different packings as a function of D,/D,. The data for 12.7-mm spheres in the large column gave Pe, = 10.4, in agreement with published results. For 6.4-mm spheres in the small column, Pe, = 16.9 is somewhat above the data of Fahien and Smith (1955). The data for the small column had appreciable scatter, and the results were not considered accurate enough to warrant adjusting the correlation given by eq 3. For cylinders, Pe, were less than those for spheres and within the wide range of published data. The linear correlation that best fitted that data was Pe, = 3.2 + 49.4(DP/D,) (12) The one point for rings, Pe, = 7.5, is an average of six values taken in the large column. The value is about the same as for cylinders, and eq 12 is assumed to hold for both of these packings. Film Coefficients: Constant Peclet Approach. The value of k:/k, in the equation for the effective thermal conductivity, eq 1, depends on the void fraction of the bed and the ratio of the thermal conductivity of the solid and fluid phases. For typical porous catalysts at moderate temperature, k 2 / k f is between 3 and 7 . For the alumina supports used in this study, the average k:/kf was estimated to be 5, and this value was used in calculating k , and hbed. For the high Reynolds numbers used in the experiments, k , was not sensitive to the value of k 2 / k f . A simple approach to getting wall coefficients,h,, from the overall coefficients is to use eq 1 with k:/kf = 5 and a constant Pe, of 10.0 to get k,. Then hbedis calculated from eq 9 with /3 = 4, and h, is obtained by using eq 11. The wall coefficients for beds of spheres are shown as
Ind. Eng. Chem. Res., Vol. 27, No. 2, 1988 231
200
i l
I
e e
