4140
Ind. Eng. Chem. Res. 2004, 43, 4140-4148
Experimental Analysis of Heat Transfer in Packed Beds with Air Flow Joa˜ o. C. Thome´ o,*,† Cla´ udia O. Rouiller,‡ and Jose´ T. Freire‡ Departamento de Engenharia e Tecnologia de Alimentos, Universidade Estadual Paulista, Rua Cristo´ va˜ o Colombo 2265, Sa˜ o Jose´ do Rio Preto SP, Brasil 15054-000, and Departamento de Engenharia Quı´mica, Universidade Federal de Sa˜ o Carlos, Rod Washington Luiz km 235, Sa˜ o Carlos SP, Brasil 13565-905
Heat-transfer studies were carried out in a packed bed of glass beads, cooled by the wall, through which air percolated. Tube-to-particle diameter ratios (D/dp) ranged from 1.8 to 55, while the air mass flux ranged from 0.204 to 2.422 kg/m2‚s. The outlet bed temperature (TL) was measured by a brass ring-shaped sensor and by aligned thermocouples. The resulting radial temperature profiles differed statistically. Angular temperature fluctuations were observed through measurements made at 72 angular positions. These fluctuations do not follow a normal distribution around the mean for low ratios D/dp. The presence of a restraining screen, as well as the increasing distance between the temperature measuring device and the bed surface, distorts TL. The radial temperature profile at the bed entrance (T0) was measured by a ring-shaped sensor, and T0 showed to be a function of the radial position, the particle diameter, and the fluid flow rate. Introduction Packed beds are often used as chemical reactors where strongly exothermic irreversible reactions take place. Under such conditions, knowledge of the temperature profile within the bed is crucial to avoid low selectivity, catalyst deactivation, or runaway. An engineer needs reliable heat-transfer parameters to predict both the heat removal rate from the bed through the tube wall and the local temperatures within the porous media. Unfortunately, the equations available in the literature give a wide range of thermal parameter values for a given experimental condition. Many possible sources of this mismatch have been discussed in the literature, from parameters lacking physical meaning to misuse of statistical methods to fit the models to experimental data and improper conception of these models.1-8 However, there are reasons to believe that part of the disagreement arises in the experimental data from which the correlations have been generated. This work discusses the most commonly used techniques to measure the temperature of the thermal boundaries of a packed bed of glass beads percolated by air. Different techniques have been used to measure both the inlet and outlet bed temperatures, and possible factors that would affect the measured temperatures are analyzed in detail. Two thermocouple configurations have been used to measure the temperature at the exit of the bed. Results from ring-shaped thermocouples are compared with those from aligned probe thermocouples. Review of Packed-Bed Temperature Measurement Techniques A typical packed-bed measuring cell is presented in Figure 1. It is composed of an entrance section, a heating * To whom correspondence should be addressed. Fax: 5517-2212262. E-mail:
[email protected]. † Universidade Estadual Paulista. ‡ Universidade Federal de Sa˜o Carlos.
Figure 1. Typical cell for packed-bed heat-transfer measurements.
section, and an assembly to measure the bed temperature. The entrance section is a tube without a jacket having the same diameter as the inner tube of the heating section. Its function is to develop the fluid velocity profile before the fluid enters the heating section. It is packed with the same particles as the heating section and can be made of metal or plastic. The heating section is generally surrounded by a jacket, which supplies or removes heat from the fluid flowing inside the tube. To estimate representative parameters of the whole bed, the entrance temperature and temperature at several positions inside the porous media must be known. Essentially, four techniques have been reported in the literature for measuring bed temperature profiles. These are shown schematically in Figure 2: a. Axial thermowell. An axial thermowell is a tube inserted longitudinally along the axis of the bed through which several thermocouples pass. Because the tips of these sensors are placed at different radial and axial positions, it allows the radial and axial temperature profiles to be recorded simultaneously in a single experiment. According to Hansen and Jørgensen,9 this is the only method suitable for small-diameter tubes.
10.1021/ie030759u CCC: $27.50 © 2004 American Chemical Society Published on Web 06/11/2004
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Figure 2. Techniques to measure the temperature in packed beds: a, axial thermowell; b, radial thermowell; c, axial insertion; d, ladder frame.
This technique has the disadvantage of heat conduction along the well and disturbance of the bed structure caused by the well. b. Radial thermowells. Radial wells may be placed in several axial and radial positions,10-13 each carrying one or more thermocouples. This method allows data sampling in a single test, and it has the same detrimental features as the axial thermowell. Although Kuczinski et al.10 estimated the heat fin effect for thermocouple readings, they did not include the heat transferred from the wall to the bed by both the wells and thermocouples in the energy balance. Mongkhonsi et al.12 showed that these wells distort the radial temperature profile, which becomes radially asymmetric, and built a sophisticated model to take into account both the heat fin effect and flow disturbance promoted by the wells. c. Thermocouples above the bed of particles. This is a widely used technique.2,7,14-22 A set of thermocouples is placed above the top layer of particles, with the sensors located at different radial and angular positions. According to some authors,6,23-26 the flow pattern changes when the fluid leaves the porous media, and measurements made above the top layer are not representative of the actual phenomena within the porous matrix. Measuring the axial temperature profile via this technique requires different bed heights, so having a complete temperature map requires opening of the vessel many times and frequent repacking of the bed. This repacking may alter the bed structure and compromise the reproducibility of the measurements. Moreover, there is no experimental confirmation that a long bed is a summation of shallow ones, although Thome´o and Freire7 had made measurements at the entrance of the thermal section, z ) 0 in Figure 1, using several bed lengths, and they did not observe significant variation of the measured temperature. d. Ladder frame. This is the preferred technique of the Westerterp group.23,27-29 A ladder frame holds several thermocouples, with the ladder placed inside the bed. The sensor tips are positioned at many axial, radial, and angular positions, allowing complete temperature profiles to be maeasured in a single experiment. Disturbance of the bed structure is the most evident problem of this method.
In this work, technique c was used because of both its operational flexibility and its noninvasive character because it does not disturb the porous media like the other methods and does not introduce any additional heat-transfer mechanism to the system. Even this method has some variations. Many authors have used a cross to hold several sensors disposed at various radial and angular positions. Some have used a metal cross,14,15,30 while others have employed a cross made up of a thermal insulator.2,16,18,19,22,31 Dixon2 and Freiwald and Paterson19 discussed heat conduction effects, from the tube wall to the thermocouple tips, with metal crosses, and showed that the measurements tend to be highly affected by these effects. Thome´o and Freire7 aligned seven sheathed thermocouples at different radial positions and rotated the set to measure the temperature at different angular positions. To filter the angular temperature variation, Giudici and Nascimento20 proposed a concentric-ring sensor, a set of concentric copper rings with a thermocouple soldered to each ring. The angular temperature variation is a consequence of angular velocity fluctuations promoted by the nonhomogeneity of the porous matrix.4,26,32-39 According to Cresswell,40 one might expect angular temperature differences up to 40 °C at different angular positions for the same radial position, depending on the operating conditions, particle shape, and tube-to-particle diameter ratio (D/dp). Such angular variations are generally considered to be random, and only the model of Mongkhonsi et al.12 takes them into account. All other models employ an average temperature taken over the angular values and are fitted to a single radial temperature profile. Dixon18 stated that for D/dp below 4 the temperature from at least eight different angular positions, 45° apart each other, must be averaged to give a representative mean of the whole cross section of the tube. The distance of the thermocouple tips from the bed surface has also varied from one group to another. As the flow pattern changes when the fluid leaves the porous media, the temperature profile also changes. High clearances could lead to flat temperature profiles, while at short distances, the tips may touch the particles, changing the structure of the bed upper layer. Dixon16-18 left a gap of 3-6 mm, while De Wasch and Froment14 adopted a spacing of one particle diameter (5.7-9.5 mm). Freiwald and Paterson19 rested the tips on the bed surface and reported that the results were similar to those where the tips were a few millimeters above the bed surface. The entrance bed temperature (T0) is other important experimental information needed for the thermal balance in order to calculate the thermal parameters, and its measurement is more difficult than that of the outlet bed temperature (TL). The entrance temperature profile should be evaluated at z ) 0 in Figure 1. Usually, either a flat profile for T05,41-45 or T0 as a function of the radial coordinate is assumed.1,2,15-18,22,23,27,46 Some authors have claimed that the thermal entrance effects originate as a result of either an erroneous measurement or a misinterpretation of T0.2,7 These entrance effects include the anomalous axial variation of the global thermal parameters, which are supposed to be constant for the entire bed. Heat flows from the heating section to the calming section through the tube wall. Any temperature sensor
4142 Ind. Eng. Chem. Res., Vol. 43, No. 15, 2004
at z ) 0 will be influenced by this heat flow, and a temperature measurement error will occur. Dixon et al.30 estimated parameters using the information provided by a thermocouple inserted radially at z ) 0, but the parameters were quite scattered. As an alternative for measuring T0 at z ) 0, these authors measured the temperature far upstream of the entrance of the heating section (z f -∞ in Figure 1) and proposed a model that includes the entrance section in the heat-transfer balance. Dixon2 improved this approach by measuring the wall temperature of the entrance section and added this information to the model. Westerterp and co-workers23,29 and Chalbi et al.46 measured the radial temperature profile at a position above z ) 0 (z ) δ in Figure 1) and considered this position as the beginning of the heat section in order to calculate the parameters. Both of these techniques have the same disadvantage of losing thermal information in an important region for parameter estimation. The heat-transfer measuring cell of Stanek and Vychodil47 did not have an entrance section. Instead, they placed an array of electrical resistances in the plane z ) 0 to generate a known T0 profile. This approach is not realistic and has the drawback of superposing entrance heat and hydrodynamic effects. Thome´o and Freire7 placed L-shaped thermocouples at the entrance section with the tips of those thermocouples at z ) 0. Because the entrance section of Thome´o and Freire’s equipment was made of brass, heat conduction still occurred along the sheath, and the presence of the thermocouples might modify the bed structure and disturb the flow. Material and Methods The equipment used in this work is described in detail by Rouiller.48 Air supplied by a compressor percolated through a bed of glass beads. The pressure in the line was regulated by a pressure regulator equipped with a filter, which intercepted oil and dirt. The air flow rate was measured at a double-orifice plate flowmeter, connected to differential manometers. Before entering in the bed, the air temperature was adjusted by an electrical heat exchanger. After percolating through the bed, the air was vented to the atmosphere. The thermal measuring cell is similar to that illustrated in Figure 3. The entrance section was made of stainless steel of 50 mm diameter and 150 mm length. The heating section was composed of two concentric stainless steel tubes of 50 and 100 mm diameter, both 100 mm in length. The annular space between them was used as a jacket, through which water, kept at a constant temperature by a thermostatic bath, was circulated. Measurements of TL were made in a device composed of two nylon tubes held by a nylon tray, coupled to the top of the heating section (see Figure 3a). Each tube was 50 mm in diameter and 100 mm in length and carried a temperature-measuring assembly. One was formed by seven sheathed iron-constantan thermocouples, with each stainless steel sheath being 1.5 mm in outer diameter and 100 mm in length (Figure 3b). These thermocouples, located in seven radial positions (r/R ) 0, 0.16, 0.35, 0.50, 0.65, 0.80, and 0.95), were held in place by a nylon plate. The other assembly was a ringshaped sensor containing six brass rings (Figure 3c), with a copper-constantan thermocouple soldered onto each ring (Figure 3d). Each ring had a circular cross section of 1.5 mm diameter. The rings were kept in place
Figure 3. Outlet bed temperature measuring devices: a, nylon tray; b, aligned thermocouples; c, ring-shaped sensors; d, detail of the circular ring.
by a wood frame. Their radial positions were the same as those of the aligned thermocouples, except for the extra thermocouple at r/R ) 0. The thermocouple sheath and the ring diameters were chosen based on the smallest particle diameter used in this work, 0.9 mm. According to Bear,49 a representative elementary volume (REV) of a porous media comprehends the particle in its vicinity. In terms of temperature measuring, a sensor larger than the REV would lose some important thermal information and a smaller one would be sensitive to topologic bed characteristics that vary from one position to another. Hence, for 0.9 mm particles, a sensor of 1.5 mm averages well the thermal information above a particle, the void spaces around it, and the contact points among particles. Both measuring devices had a brass shaft to introduce them into the nylon tube and keep the sensor at the desired height above the bed. Because Freiwald and Paterson19 reported problems of heat conduction along the thermocouple assembly when it is inside the bed for long periods, the assemblies in this work were placed over the bed only 5 min before the measurements were taken. The distance of the sensor above the bed was varied from 0 to 35 mm. When the aligned thermocouples were used, they were rotated 360° at intervals of 5°. Measurements of T0 were made by inserting a circular ring-shaped sensor at z ) 0. This assembly is similar to that used to measure TL and was crushed in a nylon flange, with the radial positions being the same as those for the sensors used to measure TL. The thermocouple wires were connected to the outside through the bottom of the entrance section, at the same radial positions as those of the corresponding thermocouples at z ) 0. The particles were soda-lime glass beads of diameters 0.9, 2.6, 4.4, 20.7, and 28.5 mm, so the ratio D/dp varied from 1.8 to 55.6. The smaller particles were sieved, and the two larger sizes were measured using a vernier caliper. The particles were visually spherical.
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Bed porosities () were measured by the weighting method, and they are nearly constant for dp ) 0.9, 2.6, and 4.4 mm ( ) 0.38) and increased for the larger particles (for dp ) 20.7 mm, ) 0.52, and for dp ) 28.5 mm, ) 0.68). The air mass flux (G) ranged from 0.204 to 2.42 kg/ m2‚s. Gas entered the bottom of the entrance section at 65 °C, and the wall temperature was fixed at 22 °C. This small temperature difference means that radial thermal equilibrium is reached at a low bed depth. Thus, a shallow bed was used in this work to observe high radial and angular temperature variations. Nevertheless, the thermal behavior observed here is expected in longer beds with larger temperature differences. Usually, small temperature differences are found in biochemical reactors. The method described by Thome´o and Freire7 was used to densely pack the bed. The entrance section and the thermal section were packed with the same technique and particles. Two stainless steel screens were used, at the top of the thermal section and at the bottom of the entrance section, to avoid particle pneumatic transport and to support the pack, respectively. The screens were tightly fixed on the tube flanges, and the particles touched the top screen. A grid with a mesh opening slightly smaller than the particle diameter was employed for each particle diameter. Results and Discussion Influence of the Pack Holding Screen. The screen used to restrain the packing influences the measured TL, as can be seen in Figure 4. The presence of the screen decreases the temperature of the radial position nearest the tube wall for distances from the bed surface to the thermocouple tips up to 5 mm. The decrease is more substantial for larger particle diameters and appears to be independent of the fluid flow rate. Because the screen was made of stainless steel, heat was conducted from the bed to the tube wall through the screen, decreasing the upper bed temperature. One way of avoiding this problem would be to make the measurements higher above the bed surface, but as shown in the next section, the temperature profile is distorted when the measurements are taken far above the bed surface. Another option would be to employ a lower thermal conductivity screen (e.g., plastic), but this would limit the outlet bed temperatures. Some authors6,50 pointed out that the wall region is of high parameter sensitivity, and the decrease in the temperature creates uncertainty in the estimated parameters. Influence of the Clearance between the Bed Surface and Temperature Sensor. The influence of the distance ZPM of the temperature sensor above the bed surface on the measured TL was assessed using both the circular ring-shaped sensor and the restraining screen at the bed exit. For high D/dp, the clearance ZPM did not change TL significantly, as shown in Figure 5a. As D/dp was reduced, the influence of ZPM increased, as shown in Figure 5b,c. However, for D/dp ) 1.8, TL appears to be insensitive to ZPM up to 25 mm, as shown in Figure 5d. This behavior can be explained by the mixing of the fluid inside and above the bed and is intrinsically related to both the bed structure and the fluid velocity profile. For high D/dp, the radial velocity profile is flat over almost the entire cross section of the bed26,36,37 and the interstitial spaces are small and randomly distrib-
Figure 4. Influence of the screen on measured TL for aligned thermocouples: (a) dp ) 4.4 mm, D/dp ) 11.4, G ) 1.30 kg/m2‚s; (b) dp ) 20.7 mm, D/dp ) 2.4, G ) 1.20 kg/m2‚s.
uted over the bed cross section, providing good fluid mixing inside the packed bed. Consequently, radial mixing above the bed surface is weak, and TL does not change strongly with ZPM. As the ratio D/dp is reduced to less than 20, wall effects increase, reinforcing fluid channeling. When the gas emerges from the porous media, it migrates toward the tube center and mixing of fluid elements having quite different temperatures takes place, resulting in a strong variation of TL with ZPM. However, for D/dp ) 1.8, each bed layer is composed of only one particle and the velocity profile is dictated by the relative position of the particles.36 At this D/dp ratio, fluid mixing occurs in the large interstitial space beside the last particle and TL is not very sensitive to ZPM up to 25 mm. Dixon18 discussed the radial thermal dispersion in a similar manner. This author related the radial thermal conductivity (kr), a global effective parameter, to the static (kr0) and to the dynamic (krf) contributions, both individual parameters. For moderate and high flow rates, Dixon showed that the modified radial Pe´clet number for the fluid at high flow rates (Perf∞ ) Gcpdp/ krf, where cp is the fluid specific heat) was nearly constant at a minimum value from large ratios of D/dp to 3 and that Perf∞ has an oscillatory dependence on D/dp between 3 and 1.14. Perf∞ presents minimum values at D/dp ) 3 and 2, values corresponding to structured
4144 Ind. Eng. Chem. Res., Vol. 43, No. 15, 2004
Figure 5. Effect of the distance above the bed surface on measured TL.
nonrandom packings, and maximum values at D/dp ) 2.5 and 1.5. A high Perf∞ is associated with poor radial dispersion, mainly due to bypassing, while a low Perf∞ denotes high radial heat transfer, due to good fluid mixing. Considering the influences of both the restraining screen and the distance ZPM, the distance between the temperature sensor and the bed surface should be 5 mm. Effect of the Sensor Device on the Measured Outlet Bed Temperature. The shape and average value of the radial temperature profile at the bed exit are influenced by the measuring device. Figure 6 shows the average radial profiles for a test replicated nine times with the same packing (Figure 6a,c) and replicated nine times after repacking (Figure 6b,d). The mean integrated temperature over the cross section differs statistically from the results obtained with both sensors. This mean was numerically calculated through the Simpson rule from TL measured in tests replicated nine times using the same packing, and the Tukey test was used to compare the results. As a post hoc comparison test, the Tukey test does not imply a normal distribution among replications. On the other hand, all of the replications were around the mean within a 95% confidence interval, and they were not rejected by the Student’s t test at any radial position for the nine consecutive tests carried out with the same packing or for different packings, as shown in Table 1. The Student’s t statistics is the ratio of the standard deviation divided by the mean value of the temperature. Note that the Student’s t increases from the center to the wall,
Table 1. Calculated Student’s t for Nine Replications for Either the Same Packing or a Different Packing (D/dp ) 11.4 and G ) 1.51 kg/m2‚s) same packing
aligned rings
r/R ) 0.95
different packing
r/R ) 0.16
r/R ) 0.5
r/R ) 0.16
r/R ) 0.5
r/R ) 0.95
0.0050 0.0094
0.0081 0.0106 0.0045 0.0188 0.0351 0.0036 t(8,5%) ) 1.8595a
0.0107 0.0107
0.0020 0.0159
a Critical t value with 8 degrees of freedom and a 95% confidence interval.
again as a consequence of the bed nonhomogeneity, and that it is quite small, less than 5% in all situations analyzed, indicating that the measurements are repeatable. Table 2 shows radial temperature profiles measured by the aligned thermocouples, averaged over 72 angular positions, and those measured by the circular ringshaped sensor. The differences between the values registered by the aligned thermocouples and by the rings (β) are not affected strongly by the flow rate but are quite dependent on the particle diameter. A likely reason for β requires an analysis of the area taken up by the rings above the bed. The area occupied by the rings corresponds to nearly 20% of the total tube cross section, while for the aligned thermocouples, it is less than 1%. Therefore, there is a sudden change in the air flow pattern when the fluid leaves the porous media, mixing portions of fluid coming from regions at quite different temperatures and modifying the actual fluid
Ind. Eng. Chem. Res., Vol. 43, No. 15, 2004 4145
Figure 6. Outlet bed radial temperature profile for D/dp ) 11.4 and G ) 29.6 kg/m2‚s: (a) ring-shaped sensor, same packing; (b) ringshaped sensor different packings; (c) aligned thermocouples, same packing; (d) aligned thermocouples, different packings. Table 2. Differences in TL Due to the Type of Sensor D/dp ) 11.4 G ) 12.7 kg/m2‚s, Re ) 157 sensor
G ) 22.0 kg/m2‚s, Re ) 272
difference
sensor
difference
r/R
aligned
rings
β
aligned
rings
β
0 0.16 0.35 0.65 0.95
37.5 37.5 36.5 33.3 29.6
39.6 39.5 37.7 34.6 28.1
-2.1 -2.0 -1.2 -1.3 1.5
43.4 43.1 41.2 37.4 32.2
45.2 45.0 42.7 39.2 31.4
-1.8 -1.9 -1.5 -1.8 0.8
D/dp ) 1.8 G ) 14.6 kg/m2‚s, Re ) 1177 G ) 27.4 kg/m2‚s, Re ) 2213 sensor
difference
sensor
difference
r/R
aligned
rings
β
aligned
rings
β
0 0.16 0.35 0.65 0.95
40.6 40.4 38.9 39.9 34.1
39.2 39.8 39.9 39.3 33.2
1.4 0.6 -1.0 0.6 2.5
44.6 44.4 43.0 43.8 41.1
43.5 44.0 44.1 43.8 38.4
1.1 0.4 -1.1 0.0 2.7
temperature. Probably, this temperature difference will be smaller, or even absent, if fewer rings are used for the measurements. β is positive for the nearest radial position in both data sets presented. Considering that for r/R ) 0.95 the outer edge of the ring is only 1.75 mm from the tube wall, the heat transfer between the wall and that ring is enhanced and the measured temperature is lower
than the actual temperature at r/R ) 0.95. For D/dp ) 11.4, β becomes more negative toward the center of the bed, while for D/dp ) 1.8, no specific trend is observed. Understanding these behaviors requires a deep analysis on the fluid distribution inside the porous matrix and the changes promoted by the devices in the flow pattern, and this is beyond the scope of this paper. Moreover, the cross used by some authors to hold the thermocouples for measurement of TL may also occupy a large fraction of the bed cross section. For instance, some crosses used by Dixon16,18 had eight arms. This author did not give many details of the crosses, but considering that each arm was at least 1 mm wide and that their tips were 1 mm from the 2 in. tube wall, the arms occupied about 19% of the total tube area, similar to the area occupied by the rings. Thus, the aligned thermocouple device is the best solution to assess the angular temperature fluctuation at the exit of the bed. Angular Temperature Fluctuation. It is often assumed in the literature that the angular fluctuations of temperature around the mean are distributed normally,7,18,23 but so far only Dixon17 confirmed this behavior through a Kolmogorov-Smirnov test. A goodness-of-fit test is suitable to indicate whether the sample is normally distributed or not, but it should be coupled with a graphical data visualization to provide more confidence to this hypothesis. Shapiro-Wilk (SW) and skewness (SK) tests were applied to the temperature data measured at 72 angular positions for each radial position, and normal probability plots were built up. SW
4146 Ind. Eng. Chem. Res., Vol. 43, No. 15, 2004 Table 3. Calculated Statistics for the Normality Test for Angular Temperature Distribution at the Bed Exit D/dp 55 1.8
55 1.8 a
G (kg/m2‚s) 0.74 1.56 0.74 1.40 0.74 1.56 0.74 1.40
r/R 0 -0.22 0.10 1.14 0.25 0.941/NS 0.971/NS 0.812/S 0.768/S
0.16
0.35
Skewness -0.16 0.35 -0.55 -0.55 Shapiro-Wilka,b 0.956/NS 0.924/NS 0.816/S 0.852/S
-0.36 -0.20 -0.28 0.01 0.981/NS 0.941/NS 0.908/S 0.912/S
0.65 0.12 0.16 -0.38 -0.37 0.959/NS 0.940/NS 0.853/S 0.871/S
0.95 0.20 0.01 0.39 0.60 0.963/NS 0.932/NS 0.918/S 0.914/S
First entry: W statistics. b Second entry: NS, nonsignificant test; S, significant test.
Figure 7. Normal probability plot for angular temperature fluctuation at the bed outlet: (a and b) D/dp ) 55, G ) 0.74 kg/m2‚s; (c and d) D/dp ) 1.8, G ) 1.40 kg/m2‚s.
statistics51 is a hypothesis test for normality that computes the W statistics at a chosen significant level. The null hypothesis for normality should be rejected for low values of W and p values < 0.05. SK statistics close to zero suggests a nearly symmetrical distribution. Positive SK indicates a distribution with an asymmetric tail extending toward positive values, while negative SK refers to asymmetry extending toward negative values. Table 3 presents W and SK values for some experimental conditions of this work, and Figure 7 shows some normal probability plot. In Figure 7, Tθ is the measured angular temperature. SW and SK tests indicated reliable normal distributions for D/dp ) 55 for all radial positions, while these tests did not authorize this assumption for D/dp ) 1.8 for any radial position. Normal probability plots confirmed these trends because the experimental data were closer to a straight line for large D/dp. The few points observed in Figure 7c mean that from all 72 angular temperature measurements only five different temperature values were registered.
One may note from Table 3 and Figure 7d that, even for SK close to zero and high values of W, the probability plots do not attest the normality for low D/dp at r/R ) 0.95. This suggests that the average value of TL depends strongly on the sample location for D/dp ) 1.8. However, as pointed out by Dixon,18 the average TL calculated from the measurements made 45° apart from each other gave the same result independent of the set chosen for this calculation. The difference between the average calculated with all 72 data points and that based on only 8 points is no bigger than 0.5 °C for any data set. Any other alternative subset of eight data 45° apart gave worse results. Entrance Radial Temperature Profile. The entrance radial temperature profile (T0) depends strongly on the flow rate and particle diameter, as shown in Figures 8 and 9. The temperature profiles in these figures are the average for nine repetitions of each test with the same packing. For low flow rates, T0 drops continuously toward the wall, except for D/dp ) 1.8,
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employing the actual measured temperature in numerical solutions of the partial differential equation of the thermal balance. Conclusion
Figure 8. Entrance radial temperature profile for high D/dp: open symbols, D/dp ) 55.6; closed symbols, D/dp ) 11.4.
Figure 9. Entrance radial temperature profile for low D/dp: open symbols, D/dp ) 2.4; closed symbols, D/dp ) 1.8.
where the central core is flat. For high flow rates, the profile tends to be flatter but still drops steeply close to the wall. As pointed out previously, some authors represented T0 as a function of the radial coordinate only. Borman et al.23 suggested that T0 might be fitted by a seconddegree polynomial function, as follows:
Tr,z)0 - Tw r 2 )1-A T0,z)0 - Tw R
()
(1)
Fitting eq 1 to the present data did not lead to good results for all of the experimental conditions. The correlation coefficient was low for the majority of the conditions, and the uncertainty over the parameter A was sometimes larger than the parameter itself. Thus, a correlation to represent T0 must contain the flow rate and the particle diameter as independent variables. However, considering that the entrance temperature profile, at the axial position z ) 0 in Figure 1, varies with the material of the entrance section, with the percolating fluid, and with the particle shape, a simple correlation is unable to represent all of the possible variables even if G and dp were to be included. Hence, adopting an analytical function to fit T0 is not the best way to represent this boundary. A single value of T0 measured at the entrance of the calming section can be used instead, as in some works of Dixon’s group,18,30,31 despite doubts about the physical meaning of the effective axial thermal conductivity. Another option is
From this experimental study of a packed bed filled with glass beads, percolated by air and cooled by the wall, it is verified that the experimental data are strongly affected by the system characteristics, by the measuring device, and by the sampling method. The presence of a metallic restraining screen at the bed exit decreases the temperature near the tube wall. Average temperatures measured at the top of the porous media with ring-shaped sensors and with aligned thermocouple probes differ statistically. The rings perturb the flow, distorting the radial temperature profile. If the measuring sensor is too far above the bed surface, the outlet bed temperature profile, TL, is distorted, with the degree of distortion dependent on the particle diameter and the fluid flow rate. A clearance of about 5 mm appears to be optimal. Measurements made at 72 angular positions showed that TL was normally distributed around the mean for high D/dp but not for low D/dp. Eight measurements made 45° apart resulted in averages close to those based on 72 angular measurements. The shape and average value of the entrance radial temperature profile depend on the particle size and fluid flow rate. For the solution of the thermal balance equation, this thermal boundary should take into account both dp and G effects. Simple polynomial functions to represent T0, such as the one proposed by Borman et al.,23 which includes only the radial position influence, do not lead to good results for all experimental conditions. Literature Cited (1) Borkink, J. G. H.; van der Watering, C. G.; Westerterp, K. R. The statistical character of bed-scale effective heat transport coefficients for packed bed. Chem. Eng. Res. Des. 1992, 70, 610619. (2) Dixon, A. G. The length effect on packed bed effective heat transfer parameters. Chem. Eng. J. 1985, 31, 163-173. (3) Gunn, D. J.; Misbah, M. M. A. Estimation of heat transport parameters from the dynamic response of fixed beds. Chem. Eng. Res. Des. 1992, 70, 620-626. (4) Lerou, J. J.; Froment, G. F. Estimation of heat transfer parameters in packed beds from radial temperature profiles. Chem. Eng. J. 1978, 15, 233-237. (5) Li, C. H.; Finlayson, B. A. Heat transfer in packed bedssa revaluation. Chem. Eng. Sci. 1977, 32, 1055-1066. (6) Sklivianiotis, M.; Castro, J. A. A.; McGreavy, C. Characteristic features of parametric sensitivity in a fixed bed heat exchanger. Chem. Eng. Sci. 1988, 43, 1517-1522. (7) Thome´o, J. C.; Freire, J. T. Heat transfer in fixed bed, a model nonlinearity approach. Chem. Eng. Sci. 2000, 55, 23292338. (8) Tsotsas, E.; Schlu¨nder, E. U. Heat transfer in packed beds with fluid flow, remarks on the meaning and the calculation of a heat transfer coefficient at the wall. Chem. Eng. Sci. 1990, 45, 819-837. (9) Hansen, K. W.; Jørgensen, S. B. Dynamic modelation of gasphase catalytic fixed-bed reactor I, experimental apparatus and determination of reaction kinetics. Chem. Eng. Sci. 1976, 31, 579586. (10) Kuczynski, M.; Oyevaar, M. H.; Pieters, R. T.; Westerterp, K. R. Methanol synthesis in a countercourrent gas-solid-solid trickle flow reactor, an experimental study. Chem. Eng. Sci. 1987, 42, 1887-1898. (11) Lamine, A. S.; Colli Serrano, M. T.; Wild, G. Hydrodynamics and heat transfer in packed beds with liquid upflow. Chem. Eng. Sci. 1992, 47, 3493-3500.
4148 Ind. Eng. Chem. Res., Vol. 43, No. 15, 2004 (12) Mongkhonsi, T.; Lo´pez-Isunza, H. F.; Kershenbaum, L. S. The distortion of measured temperature profile in fixed bed reactors. Chem. Eng. Res. Des. 1992, 70, 255-264. (13) Willhite, G. D.; Kunii, D.; Smith, J. M. Heat transfer in beds of fine particles, heat transfer perpendicular to flow. AIChE J. 1962, 8, 340-345. (14) De Wasch, A. P.; Froment, G. F. Heat transfer in packed beds. Chem. Eng. Sci. 1972, 27, 567-576. (15) Dixon, A. G. Thermal resistance models of packed-bed effective heat transfer parameters. AIChE J. 1985, 31, 826-834. (16) Dixon, A. G. Angular temperature variations in fixed beds of spheres. 29th National Heat Transfer Conference, Atlanta, GA, 1993; ASME HTD-236; ASME: Fairfield, NJ, 1993; pp 55-64. (17) Dixon, A. G. Heat transfer in packed beds of spheres with dt/dp < 4. Proceedings of the International HT Conference, Brighton, U.K., 1994; Institution of Chemical Engineers Symposium Series 135; Institution of Chemical Engineers: Brighton, U.K., 1994; pp 225-230. (18) Dixon, A. G. Heat transfer in fixed beds at very low (
