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Gas pressure drop and heat dispersion in a layer of fibrous material
Chemical Engineering and Processing,
32 (1993) 191- 198
191
Gas pressure drop and heat dispersion in a layer of fibrous material P. Vychodil Institute
and V. Sta&k*
of Chemical Process Fundamentals,
Czechoslovak
Academy
of Sciences,
165 02 Prague 6 (Czech Republic)
G. A. Fateev Heat and Mass
Transfer Institute,
Byelorussian
Academy
of Sciences, Minsk 220728 (Belarus)
N. Vegyte Research Institute for Thermal Insulating and Acoustic (Received
December
Construction
Materials
and Products,
Vilnius 232657 (Lithuania)
19, 1992)
Abstract The pressure drop and effective radial heat conductivity have been measured in fibrous layers of insulation materials exhibiting unisotropic stratified structure. Both quantities have been found to vary with the two principal orientations of strata in the fibrous material with respect to the direction of the flow. Pressure losses have been successfully interpreted in terms of the Ergun equation, valid for beds of particulate solids, applying the concept of the equivalent diameter to the fibrous material. The equivalent diameters were correlated with the void fraction of the material, allowing direct use of the Ergun equation for calculation of the energy losses. The heat dispersion has been found detectably unisotropic but, for technical purposes, the problem may be solved on the basis of an isotropic model with an error of about 20%. A correlation of the effective thermal conductivity of the material for this case also has been presented. The effective heat conductivities were correlated as Peclet numbers, using the effective ‘particle’ diameter as the scale length both in the Peclet and the Reynolds number. The effective ‘particle’ diameters for the fibrous materials were obtained by fitting the Ergun equation to pressure drop IX. gas velocity data. A joint plot of the Peclet US. Reynolds number, interpreting the effective thermal conductivities of layers of fibrous material as well as particulate beds, suggests that the adopted approach is also physically meaningful. The results presented represent a contribution to the process of thermally self-sustained curing of adhesive in the manufacture of fibrous insulating materials.
Introduction
The problem of heat dispersion in layers of solid particles passed by flowing gas, or their effective conductivity;has been the subject of continuing research interest, owing to its potential in the elucidation of more complex phenomena involving chemical reactions as well as direct relevance for design purposes. The problem of the thermal conductivity of packed beds has been reviewed, for example, by Tsotsas and Martin [l], and dealt with in a monograph by Schliinder and Tsotsas [2]. Correct concepts of heat transfer and chemical reactions permit re-evaluation of certain technological processes traditionally carried out under isothermal conditions ruling out transport of heat. Packed beds, as systems exhibiting typically low effective conductivities and large gas-solid interfacial areas, can be operated under the regime with local heat zones and reaction
*Author
to whom correspondence
0255-2701/93/$6.00
should be addressed.
autowaves, with the aim to carry out reactions only weakly exothermic, solely at the expense of the reaction heat. Catalytic oxidation of lean mixtures of gaseous organic contaminants under such conditions has been described, for example, by Boreskov et al.
[3].
An example of a process of this type is the manufacture of insulating materials from mineral gauze. Analysis of this process, on the basis of mathematical models of the chemical reaction and heat transfer involved, has been presented by Fateev et al. [4] and Vegyte and coworkers [5, 61. Non-isotropic
fibrous material
(gauze)
Fibrous materials are frequently used for thermal insulation. Mineral gauzes are usually formed by fibres 2- 15 pm in diameter, the mean being usually 8 pm. The layer of gauze is formed by the sedimentation of individual fibres, with a small amount of resin adhesive
0
1993 -
Elsevier Sequoia.
All rights reserved
192
added to fix the layer. A sufficiently deep layer of fibres is then cured, while its density is crucial for the thermal insulation properties as well as mechanical strength of the product. The technology of manufacture gives rise to a typical stratified structure of the gauze, while the orientation of the fibres in individual strata is largely random. In traditional technologies, a relatively shallow layer of gauze (approximately 0.1 m) is exposed to the flow of hot gas (air with combustion gases), which passes through the layer at a velocity of about 1 m s-i and at a temperature between 180 and 250 “C. Under these conditions, the cementing resin cures (usually polycondensation of phenolformaldehyde resin), releasing a small amount of reaction heat. The cured carpet of the insulation material is then cooled by air. Using this technology, the material heats up fairly quickly and the curing takes place practically at the temperature of the hot gas. The new technology of thermal processing presumes that the fibrous layer moves through a curing chamber, while the gas flows countercurrently to its motion. A high temperature curing zone forms in the centre of the chamber and both streams of material-the layer and the gas-exit essentially cool. In the design and calculation of such an energy-saving process, one has to respect the following factors. The process takes place under non-isothermal conditions in a layer of fibrous material. Depending on the orientation of the strata, the gas may flow either parallel or perpendicular to the stratified structure of the fibrous layer. Heat transfer then may also take place in two principal directions with respect to the strata, thus affecting the temperature regime in the equipment.
Method and aim Figure 1 shows two principal possible orientations of the strata in the fibrous layer with respect to the flow of gas. For brevity, the first orientation, where the direction of the flow is perpendicular to the individual strata, and, hence, to the individual fibres, shall be termed
“11111
“111!1
radial
ax%al
Fig. 1. Two principal layer.
orientations
of the strata in the fibrous
‘radial’ orientation. The second orientation shall be termed ‘axial’ and the velocity vector is now parallel to the individual strata of the fibrous layer. The principal aim of this paper is to determine experimentally the pressure losses and effective heat conductivities of layers of unisotropic fibrous materials in the flow configurations shown in Fig. 1. An attempt shall be made to interpret the heat conductivities on the basis of characteristics evaluated from pressure drop measurements, employing concepts valid for particulate fixed beds. This concept originates from the idea that the stratified structure of the gauze and its anisotropy are related to the pressure drop and heat transfer through similar mechanisms. In addition, both quantities investigated are indispensable for the design of the above outlined technology of manufacture of fibrous insulation carpets.
Pressure
losses in fibrous layers
The measurements of the pressure losses were carried out simultaneously with the measurements of the effective thermal conductivities of the samples of fibrous materials in a cylindrical column 0.05 m in internal diameter. The depth of the samples of fibrous materials placed in the column was 0.112 m. In all cases, the gas was air at a superficial velocity ranging between 0.2 and 2 m s-‘. More details about the experimental set-up employed may be found in earlier papers relating to beds of particles [7, 81. A series of samples was investigated experimentally for both orientations of the strata designated earlier as ‘radial’ and ‘axial’. The samples differed considerably in mechanical strength from very soft to very rigid, mechanically strong materials. We took the void fraction as the principal quantity characterizing the samples. The void fraction seems to characterize best the stability of the shape of the sample, independently of the density of the material forming the fibre. This density was found to be 2610 kg m-j for the greenish materials, while several greyish samples exhibited a density of 2800 kg mP3. The void fraction of the samples investigated ranged between 0.93 and 0.98. Samples of void fraction around 0.94 maintained a stable shape and were mechanically strong. In contrast, samples of void fraction about 0.97 were soft and of unstable shape. Regardless of the void fraction, all the samples displayed a typical stratified structure of the fibres. Samples with a void fraction in excess of 0.98 were dropped from the investigation, owing to their changing shape (deformation) during measurement. Figures 2 and 3 show plots of the pressure drops per unit height for the ‘radial’ and ‘axial’ configuration,
193 TABLE 1. Optimum equivalent diameters of the Ergun equation for samples of fibrous material of different void fraction and for two configurations of the layer ‘Radial’
0
1 u
Cm/s1
configuration
‘Axial’ configuration
Void fraction
Equivalent diameter (km)
Void fraction
Equivalent diameter (pm)
0.944 0.950 0.96 1 0.969 0.973 0.973 0.976
15.4 15.2 22.0 24.1 21.2 23.4 21.8
0.935 0.938 0.974 0.974 0.980
15.2 14.6 35.3 37.7 34.1
2
Fig. 2. Pressure drop per unit length as a function of superficial velocity for ‘radial’ configuration of samples of different void fraction. Curves computed from the Ergun equation with optimized (d,) parameters.
u [m/s1 Fig. 3. Pressure drop per unit length as a function of superficial velocity for ‘axial’ configuration of samples of different void fraction. Curves computed from the Ergun equation with optimized (d,) parameters.
respectively, and for samples of different void fraction. Both figures indicate clearly that the pressure losses depend very strongly on the void fraction of the sample, as well as on the orientation of the strata of the sample with respect to the direction of the flow; the differences being as much as an order of magnitude. In an attempt to generalize the description of the pressure losses, the experimental results were fitted by the Ergun equation in the following form:
with the unknown equivalent ‘particle’ diameter $e characterizing the fibrous structure being an adjustable parameter. The optimum values of the diameter d, are shown in Table 1 for various samples, as characterized by the void fraction and for the two orientations of strata in the sample with respect to the flow. Optimum curves of the dependence of the pressure losses on the superficial velocity of gas are shown in Figs. 2 and 3 by solid lines, indicating a very good fit with the experimental points. In fact, the Ergun equation with optimum values of the effective diameter d, is capable of describing the pressure losses with an error between lo/oand 3”/0,only in exceptions (for extremely low velocities) does the error reach 10%. Therefore, the concept of looking at the stratified fibrous structure as a ‘particulate’ bed seems to work well. A significant obstacle in direct use of the Ergun equation for the calculation of pressure losses in fibrous materials is in the fact that we have no a priori knowledge of the equivalent diameter dpe for a given sample of the fibrous material. This can be resolved if we note that the optimum equivalent diameter, as may be seen from Table 1, is a strong function of the void fraction of the material. Unfortunately, this function is different for the two orientations of strata of the fibres with respect to the direction of the flow. From Table 1, it is seen that the optimum equivalent diameter of the fibrous samples is higher for ‘axial’ orientations of the strata with respect to the flow direction (see Fig. 1). With decreasing void fraction and, hence, increasing compactness of the fibrous samples, however, the differences between the ‘axial’ and ‘radial’ configurations lessen, while, for the void fraction around 0.93, the pressure losses are virtually independent of the configuration. An attempt to correlate the effective diameters of the fibrous samples with the void fraction gave the following results:
194
for the ‘axial’ configuration dPe = -4.64
x 1O-4 + 5.132 x 1O-4a
(2)
and for the ‘radial’ configuration d, = -2.234
x 1O-4 + 2.535 x 10-4a
(3)
Both correlations hold for dpe in metres and 0.93
01
0.93
I
as&
I
I
0.95
I
I
0.96
0.97
of
2 u
of fibrous layers
In an earlier paper, Stanek and Vychodil [7] presented a method of determining the effective radial thermal conductivity of packed beds passed by flowing gas. This parameter is evaluated from comparison of the experimental and theoretical gas temperature profiles measured at the bed outlet, corresponding to a known inlet gas temperature profile. This method is similar to experiments involving heating the bed across the walls but suppresses the effect of the region near the wall, where irregularities of the bed structure may be dominant. The experimental results are compared with the theoretical solution of the pseudohomogeneous model of heat dispersion in cylindrical coordinates in the following form: (4)
E
1
Effective thermal conductivity
098
Fig. 4. Optimum equivalent ‘particle’ diameter as a function void fraction for ‘radial’ ( 0) and ‘axial’ ( 0) configurations.
0
a comparison of the experimental dependence of the pressure losses on the superficial velocity of air with the optimum curve computed on the basis of optimum dper on the other hand, and with the curve computed on the basis of dpe from correlation for a sample of void fraction 0.974. on the other hand. As expected, the agreement impairs with increasing air velocity but, even in the worst case, the error does not exceed 13%.
Cm/s1
Fig. 5. Test of direct applicability of the Ergun equation for fibrous layers. Pressure drop per length as a function of superficial velocity for ‘axial’ layer with E = 0.974: 1, computed from Ergun equation with optimized dp; 2, computed from Ergun equation with d,, from correlation (worst case).
The closed form solution of this model with appropriate boundary and initial conditions may be found in the same paper by Stanek and Vychodil [7]. The formulation of the mathematical model in eqn. (4) is such that, rigorously, it applies only to an isotropic system with radial symmetry. With the ‘radial’ orientation of the layer shown in Fig. 1, the sample essentially satisfies these conditions. Even then, however, its effective radial thermal conductivity may vary periodically along the axial coordinate, owing to the stratified structure of the fibrous material. Since the thickness of individual strata is substantially smaller than the depth of the sample, the application of eqn. (4) to ‘radial’ samples should not bring about excessive errors. With the ‘axial’ orientation of the sample, when the principal direction of the flow of gas is perpendicular to individual strata of the fibres, however, the situation is more complicated. As may be seen from Fig. 1, the sample of fibrous material is now radially unisotropic and one can envision at least two principal directions of different effective radial thermal conductivity (k,,, , k,,,). Therefore, the problem becomes three-dimensional and the pseudohomogeneous model of heat dispersion should be in the tensoral form.
195
Details of the tensoral treatment of unisotropic media have been available for some time in the literature (see, for example, refs. 9 and IO). Nevertheless, as a first approximation, we shall use the isotropic model, i.e. eqn. (4), even for this case. In this particular case, this approximation need not be as bad as it may appear at first, thanks to the low thermal conductivity of the material of the fibres (glass, 0.8 W m-’ K-‘) and the small equivalent diameter of the fibres. The outlet temperature profiles were monitored by means of 20 thermistors located in various radial positions at the outlet of the bed. These thermistors were located in a strip about 8 mm wide across the column diameter. For the ‘radial’ samples, repeated experiments with the strip of thermistors in various angular positions proved the correctness of the assumption of the radial symmetry of the problem. For the ‘axial’ samples, each experimental run was repeated twice; once with the strip of the thermistors oriented parallel to the strata of fibres and once perpendicular to the strata. The results of these repeated measurements always differed detectably. Therefore, the dispersion of heat in this configuration is not quite isotropic.
Results
and discussion
Figure 6 shows plots of the effective radial thermal conductivity of three samples of the fibrous mineral gauze of different void fraction with dependence on the superficial velocity of air. The dependence is seen to be rather weak and the values of kRR are almost the same for the two less compact samples. 0.6
I
I
I
i
2
p”’
I
0.4
0.3
0
0.4
= = 0.944
0
C =
0.961
dpe
=
22.0
un,
0
c
0.976
dpe
=
21.8
,,,I,
=
dpe
=
IS.4
I
I
I
05
1
25
,,n
2
u [m/s1 Fig. 6. Effective radial thermal conductivity as a function of superficial gas velocity for fibrous layer of ‘radial’ configuration and different void fraction.
2
1
Re Fig. 7. Eflective radial thermal conductivity in the form of Peclet number as a function of the Reynolds number for ‘radial’ configuration of fibrous layers. Optimum equivalent ‘particle’ diameter (d,) was used to evaluate Pe and Re. Data correlated by the of correlation: Pea, = Re Pr/ Gunn and Khalid type (8.2 + 0.87 Re Pr).
For all three samples, the measurements were carried out in the ‘radial’ configuration and the results ranged between 0.4 and 0.5 W m-’ K-‘. The observed dependence of the effective conductivity on the velocity of air is rather weak, apparently owing to the small effective diameter of the material and, therefore, low convective contribution to the effective conductivity. Figure 7 shows the same dependence in the form of a plot of the Peclet number us. the Reynolds number. The characteristic dimension needed for evaluation of both dimensionless criteria was the effective diameter L& resulting from the pressure losses and the Ergun equation. With this characteristic dimension, the Reynolds number falls in a very low region between 0.15 and 1.5, this region being virtually inaccesible to measurement in classic particulate packed beds. All the experimental points in Fig. 7 fall essentially on a single line and were correlated using the well-known Gunn and Khalid [ 1 l] type of correlation with the following results: PeRR =
0
0.6 08
Re Pr 8.2 + 0.87 Re Pr
(5)
This optimum curve is also shown in Fig. 7 by a solid line. Figure 8 shows analogous results of the effective radial thermal conductivity of the fibrous mineral gauze as a function of the superficial gas velocity for ‘axial’ orientation of the strata in the sample of fibrous material. The open and full circles distinguish between measurements when the strip of the thermistors below the
196
0.2
I
2 x
ti ii
O O O
00
0
0
00
a5
0.1
0
lm-
0
0
.
.
0
0 n
m
l
-.
0
1
1.5 u
2
[m/e1
Fig. 8. Effective radial thermal conductivity as a function of superficial gas velocity for fibrous layer of ‘axial’ configuration. Full and open symbols distinguish samples of different void fraction. Circles and squares distinguish between measurement of the conductivity in the plane perpendicular and parallel to the strata of the fibrous material respectively.
sample was located parallel or perpendicular to the strata in the fibrous material. The difference in the results indicates a detectable anisotropy of the material given by different contributions of the principal components of radial effective thermal conductivity k,,, and k RU (see Fig. 1). From Fig. 8, it is seen that the component of the radial thermal conductivity perpendicular to the strata of the sample is lower than that in the direction parallel to the strata. (Of course, the results in Fig. 8 are not directly the components k,,, and kRLZ.) This observation confirms the expectation that, in the direction parallel to the strata in the fibrous layer, the conductive contribution (conductance along the fibres) is higher, while the ‘thinner’ regions between individual strata enhance the convective contribution to heat dispersion. Figure 9 shows plots of the radial effective heat conductivity for the ‘axial’ configuration of the fibrous sample in the form of the Prandtl US. the Reynolds number. The length scale again is the effective ‘particle’ diameter from the pressure drop measurement. Different effective conductivities found for the two different orientations of the strip of the thermistors remain visible in these plots. A joint correlation, neglecting these differences, yielded the following result: Pe,,
=
0.04
l
a5
I1
O.OB 0.06
.
*..a* l
=I Om
Re Pr 8.9 + 0.39 Re Pr
This correlation, of course, is less accurate; the error in some regions reaches as much as 20%. A comparison of the correlations in eqns. (5) and (6) indicates less curvature of the results with eqn. (6).
ff
Ml 01
I a2
I I Illll 0.6 (18 1 0.4
Re
I 2
3
Fig. 9. Effective radial thermal conductivity in the form of Peclet number as a function of the Reynolds number for ‘axial’ configuration of fibrous layers. Optimum equivalent ‘particle’ diameter (d,) was used to evaluate Pe and Re. Full and open symbols distinguish samples of different void fractions. Circles and squares distinguish between measurements of the conductivity in the plane perpendicular and parallel to the strata of the fibrous material respectively. Data correlated by the Gunn and Khalid type of correlation: Pea, = Re Prl(8.9 + 0.39 Re Pr).
However, the difference between the results from both correlations is small in the given range of the Reynolds number and, therefore, one can use eqn. (5) as a general correlation.
Conclusions
Experimental measurement of pressure losses incurred under the flow of gas through a layer of unisotropic fibrous material indicated substantial differences, depending on the orientation of individual strata of the fibres in the sample material with respect to the direction of the flow of air. Both the ‘axial’ and ‘radial’ configurations are of practical importance from the standpoint of the technology of manufacture of the fibrous insulation carpets. For the ‘axial’ configuration of strata, the corresponding pressure loss is higher by as much as an order of magnitude. This difference is attributed to the lower resistance of the material between individual strata, owing to the locally decreased void content of the material. The Ergun equation originally formulated for beds of particulate solids has been used successfully to correlate the pressure losses against the superficial velocity of the flowing air, with the effective ‘particle’ diameter being an adjustable parameter. As expected, the effective di-
197
ameter exceeds the mean diameter of the fibres (in this study, this is by a factor of 3-4). For direct use of the Ergun equation, the ‘optimum’ values of characteristic dimension of the fibrous materials were successfully correlated with the void fraction of the material. The pressure losses computed on the basis of the eqns. (2) and (3) and the Ergun equation are sufficiently accurate for technical purposes. The effective thermal conductivity of the fibrous materials was experimentally investigated and interpreted on the basis of the pseudohomogeneous heat dispersion model for an isotropic medium. It is readily conceded that, for fibrous materials, the idea of an isotropic medium is not a very accurate one. The experimental results for the ‘axial’ configuration of the strata with respect to the flowing air confirmed the tensoral character of the dispersion. ‘Radial’ configurations offer results acceptable from the standpoint of an isotropic model. Nevertheless, a common correlation for the effective thermal conductivity of the fibrous material (eqn. (5)) may be used for a general configuration in connection with the isotropic model, if an error of about 20% can be tolerated. It follows that the idea of applying the concepts and models originally developed for beds of particles to unisotropic fibrous materials proved fairly effective. Physically, these concepts also seem meaningful, as may be seen from Fig. 10. This figure plots the dependence
of the Peclet number on the Reynolds number for the ‘axial’ and ‘radial’ samples of fibrous insulation gauze layers obtained in this study, together with the results of Yagi and Wakao [ 121, and Stanek and Vychodil [8] for spherical particles, our earlier results, and the result of Plautz and Johnstone [ 131. Although the two sets of data (particulate-fibrous) do not overlap, the results with layers of fibrous materials, reaching substantially lower ranges of the Reynolds number, seem to be reasonable extrapolations of the results with particulate beds. Nomenclature
c Pg
dPC
k,, k, kRR- k,,, k,,, 7
(Jm --Is-’ K-‘)
AP PeRR
7 PeRL
Pr
-I
1oc
,.-5
Re
/‘1 .’
5_,,M
,’
1’ /%
l-
,’
4’
u
T
3.
CL1
1
Z
2
E
-A, 01
I 1
10
pressure drop (Pa) Peclet number for effective radial thermal conductivity of the fibrous material in ‘radial’ and ‘axial’ configurations Prandtl number based on d, as scale length radial coordinate (m) Reynolds number based on dp, as scale length superficial velocity of gas (m s-r) temperature (K) axial coordinate (m)
Greek letters
Pg PLa ml
specific heat capacity of gas (J kg-’ K-‘) equivalent diameter of the fibrous material (m) axial and radial effective heat conductivity (J m-’ s-’ K-‘) effective radial heat conductivity of the fibrous material for ‘radial’ configuration and two principal components for the ‘axial’ configuration
void fraction (-) density of gas (kg m-3) viscosity of gas (kg m-r s-r)
-
100
1000
Re Fig. 10. Effective radial thermal conductivity in the form of Peclet number as a function of the Reynolds number for fibrous and particulate materials: 1, fibrous layer with ‘radial’ configuration, i.e. Pen, = Re Pr/(8.2 f0.87 Re Pr); 2, fibrous layer with ‘axial’ configuration, i.e. Pea, = Re Pr/(8.9 + 0.39 Re Pr); 3, glass balls, i.e. Pea = Re Pr/( 13.9 + 0.13 Re Pr) [8]; 4, glass particles, i.e. Pen = Re Pr/(27.5 + 0.08 Re Pr) [ 131; 5, glass particles. i.e. Pen = Re Pr/(6.0 + 0.11 Re Pr) [ 121.
References I E. Tsotsas and H. Martin, Thermal conductivity of packed beds. A review, Chem. Eng. Process., 22 (1987) 19. 2 E. U. Schliinder and E. Tsotsas, Wiirmeiibertragung in Festbetten, durchmischten Schiittgiitem und Wirbelschichten, Georg Thieme, Stuttgart, 1988. 3 G. K. Boreskov, Yu. Sh. Matros and 0. V. Kiselev, Heterogeneous catalytic process under unsteady-state regime, Dokl. Akad. Nauk SSSR, 237 ( 1977) 91 (in Russian).
198
4 G. A. Fateev, N. Vegyte and L. P. Petrova, Calculation of the temperature and concentration fields under thermofiltration in a reacting system, Sbornik Trudov Instituta Teplo-i Massoobmena Akademii Nauk BSSR, Minsk, 1983, Belorussian Academy of Sciences, Minsk, 1983, p. 129 (in Russian). 5 N. Vegyte, A. Ju Skrinska and G. A. Fateev, Polycondensation of resin in the layer of thermal insulation material under the self-sustaining regime of reaction zones, Proc. First ANUnion Symp. on Macroscopic Kinetics and Chemical Gasodynamics, Chernogolovka, Institute of Chemical Physics AN SSSR, Vol. I, Part I, 1984, Academy of Sciences of USSR, Moscow, 1984, p. 114 (in Russian). 6 N. Vegyte and G. A. Fateev, Analysis of the conditions of thermal curing of adhesive under the conductive and convective regime in a thermal insulation material, Sbornik Trudov VNZI Teploisoliatsia (UDK 666.198:532.546) (1986) (in Russian).
7 V. Stantk and P. Vychodil, On the observability of heat dispersion parameters of the pseudohomogeneous model of heat transfer in packed beds, CON. Czech. Chem. Commun., 48 (1983) 2484. 8 V. Stanitk and P. Vychodil, On the length effect of the heat dispersion parameters, Chem. Eng. Commun., 27 (1984) 69. 9 J. Bear, Dynamics of FIuids in Porous Media, Elsevier, New York, 1972, ch. 5. 10 F. A. L. Dullien, Single phase flow through porous media and pore structure, Chem. Eng. .I., 10 (1975) 1. 11 D. J. Gunn and M. Khalid, Thermal dispersion and wall heat transfer in packed beds, Chem. Eng. Sci., 30 (1975) 261. 12 S. Yagi and N. Wakao, Heat and mass transfer from wall to fluid in packed beds, AIChE J., 5 (1959) 79. 13 D. A. Plautz and H. F. Johnstone, Heat and mass transfer in packed beds, AIChE J., I (1955) 193.
32 (1993) 191- 198
191
Gas pressure drop and heat dispersion in a layer of fibrous material P. Vychodil Institute
and V. Sta&k*
of Chemical Process Fundamentals,
Czechoslovak
Academy
of Sciences,
165 02 Prague 6 (Czech Republic)
G. A. Fateev Heat and Mass
Transfer Institute,
Byelorussian
Academy
of Sciences, Minsk 220728 (Belarus)
N. Vegyte Research Institute for Thermal Insulating and Acoustic (Received
December
Construction
Materials
and Products,
Vilnius 232657 (Lithuania)
19, 1992)
Abstract The pressure drop and effective radial heat conductivity have been measured in fibrous layers of insulation materials exhibiting unisotropic stratified structure. Both quantities have been found to vary with the two principal orientations of strata in the fibrous material with respect to the direction of the flow. Pressure losses have been successfully interpreted in terms of the Ergun equation, valid for beds of particulate solids, applying the concept of the equivalent diameter to the fibrous material. The equivalent diameters were correlated with the void fraction of the material, allowing direct use of the Ergun equation for calculation of the energy losses. The heat dispersion has been found detectably unisotropic but, for technical purposes, the problem may be solved on the basis of an isotropic model with an error of about 20%. A correlation of the effective thermal conductivity of the material for this case also has been presented. The effective heat conductivities were correlated as Peclet numbers, using the effective ‘particle’ diameter as the scale length both in the Peclet and the Reynolds number. The effective ‘particle’ diameters for the fibrous materials were obtained by fitting the Ergun equation to pressure drop IX. gas velocity data. A joint plot of the Peclet US. Reynolds number, interpreting the effective thermal conductivities of layers of fibrous material as well as particulate beds, suggests that the adopted approach is also physically meaningful. The results presented represent a contribution to the process of thermally self-sustained curing of adhesive in the manufacture of fibrous insulating materials.
Introduction
The problem of heat dispersion in layers of solid particles passed by flowing gas, or their effective conductivity;has been the subject of continuing research interest, owing to its potential in the elucidation of more complex phenomena involving chemical reactions as well as direct relevance for design purposes. The problem of the thermal conductivity of packed beds has been reviewed, for example, by Tsotsas and Martin [l], and dealt with in a monograph by Schliinder and Tsotsas [2]. Correct concepts of heat transfer and chemical reactions permit re-evaluation of certain technological processes traditionally carried out under isothermal conditions ruling out transport of heat. Packed beds, as systems exhibiting typically low effective conductivities and large gas-solid interfacial areas, can be operated under the regime with local heat zones and reaction
*Author
to whom correspondence
0255-2701/93/$6.00
should be addressed.
autowaves, with the aim to carry out reactions only weakly exothermic, solely at the expense of the reaction heat. Catalytic oxidation of lean mixtures of gaseous organic contaminants under such conditions has been described, for example, by Boreskov et al.
[3].
An example of a process of this type is the manufacture of insulating materials from mineral gauze. Analysis of this process, on the basis of mathematical models of the chemical reaction and heat transfer involved, has been presented by Fateev et al. [4] and Vegyte and coworkers [5, 61. Non-isotropic
fibrous material
(gauze)
Fibrous materials are frequently used for thermal insulation. Mineral gauzes are usually formed by fibres 2- 15 pm in diameter, the mean being usually 8 pm. The layer of gauze is formed by the sedimentation of individual fibres, with a small amount of resin adhesive
0
1993 -
Elsevier Sequoia.
All rights reserved
192
added to fix the layer. A sufficiently deep layer of fibres is then cured, while its density is crucial for the thermal insulation properties as well as mechanical strength of the product. The technology of manufacture gives rise to a typical stratified structure of the gauze, while the orientation of the fibres in individual strata is largely random. In traditional technologies, a relatively shallow layer of gauze (approximately 0.1 m) is exposed to the flow of hot gas (air with combustion gases), which passes through the layer at a velocity of about 1 m s-i and at a temperature between 180 and 250 “C. Under these conditions, the cementing resin cures (usually polycondensation of phenolformaldehyde resin), releasing a small amount of reaction heat. The cured carpet of the insulation material is then cooled by air. Using this technology, the material heats up fairly quickly and the curing takes place practically at the temperature of the hot gas. The new technology of thermal processing presumes that the fibrous layer moves through a curing chamber, while the gas flows countercurrently to its motion. A high temperature curing zone forms in the centre of the chamber and both streams of material-the layer and the gas-exit essentially cool. In the design and calculation of such an energy-saving process, one has to respect the following factors. The process takes place under non-isothermal conditions in a layer of fibrous material. Depending on the orientation of the strata, the gas may flow either parallel or perpendicular to the stratified structure of the fibrous layer. Heat transfer then may also take place in two principal directions with respect to the strata, thus affecting the temperature regime in the equipment.
Method and aim Figure 1 shows two principal possible orientations of the strata in the fibrous layer with respect to the flow of gas. For brevity, the first orientation, where the direction of the flow is perpendicular to the individual strata, and, hence, to the individual fibres, shall be termed
“11111
“111!1
radial
ax%al
Fig. 1. Two principal layer.
orientations
of the strata in the fibrous
‘radial’ orientation. The second orientation shall be termed ‘axial’ and the velocity vector is now parallel to the individual strata of the fibrous layer. The principal aim of this paper is to determine experimentally the pressure losses and effective heat conductivities of layers of unisotropic fibrous materials in the flow configurations shown in Fig. 1. An attempt shall be made to interpret the heat conductivities on the basis of characteristics evaluated from pressure drop measurements, employing concepts valid for particulate fixed beds. This concept originates from the idea that the stratified structure of the gauze and its anisotropy are related to the pressure drop and heat transfer through similar mechanisms. In addition, both quantities investigated are indispensable for the design of the above outlined technology of manufacture of fibrous insulation carpets.
Pressure
losses in fibrous layers
The measurements of the pressure losses were carried out simultaneously with the measurements of the effective thermal conductivities of the samples of fibrous materials in a cylindrical column 0.05 m in internal diameter. The depth of the samples of fibrous materials placed in the column was 0.112 m. In all cases, the gas was air at a superficial velocity ranging between 0.2 and 2 m s-‘. More details about the experimental set-up employed may be found in earlier papers relating to beds of particles [7, 81. A series of samples was investigated experimentally for both orientations of the strata designated earlier as ‘radial’ and ‘axial’. The samples differed considerably in mechanical strength from very soft to very rigid, mechanically strong materials. We took the void fraction as the principal quantity characterizing the samples. The void fraction seems to characterize best the stability of the shape of the sample, independently of the density of the material forming the fibre. This density was found to be 2610 kg m-j for the greenish materials, while several greyish samples exhibited a density of 2800 kg mP3. The void fraction of the samples investigated ranged between 0.93 and 0.98. Samples of void fraction around 0.94 maintained a stable shape and were mechanically strong. In contrast, samples of void fraction about 0.97 were soft and of unstable shape. Regardless of the void fraction, all the samples displayed a typical stratified structure of the fibres. Samples with a void fraction in excess of 0.98 were dropped from the investigation, owing to their changing shape (deformation) during measurement. Figures 2 and 3 show plots of the pressure drops per unit height for the ‘radial’ and ‘axial’ configuration,
193 TABLE 1. Optimum equivalent diameters of the Ergun equation for samples of fibrous material of different void fraction and for two configurations of the layer ‘Radial’
0
1 u
Cm/s1
configuration
‘Axial’ configuration
Void fraction
Equivalent diameter (km)
Void fraction
Equivalent diameter (pm)
0.944 0.950 0.96 1 0.969 0.973 0.973 0.976
15.4 15.2 22.0 24.1 21.2 23.4 21.8
0.935 0.938 0.974 0.974 0.980
15.2 14.6 35.3 37.7 34.1
2
Fig. 2. Pressure drop per unit length as a function of superficial velocity for ‘radial’ configuration of samples of different void fraction. Curves computed from the Ergun equation with optimized (d,) parameters.
u [m/s1 Fig. 3. Pressure drop per unit length as a function of superficial velocity for ‘axial’ configuration of samples of different void fraction. Curves computed from the Ergun equation with optimized (d,) parameters.
respectively, and for samples of different void fraction. Both figures indicate clearly that the pressure losses depend very strongly on the void fraction of the sample, as well as on the orientation of the strata of the sample with respect to the direction of the flow; the differences being as much as an order of magnitude. In an attempt to generalize the description of the pressure losses, the experimental results were fitted by the Ergun equation in the following form:
with the unknown equivalent ‘particle’ diameter $e characterizing the fibrous structure being an adjustable parameter. The optimum values of the diameter d, are shown in Table 1 for various samples, as characterized by the void fraction and for the two orientations of strata in the sample with respect to the flow. Optimum curves of the dependence of the pressure losses on the superficial velocity of gas are shown in Figs. 2 and 3 by solid lines, indicating a very good fit with the experimental points. In fact, the Ergun equation with optimum values of the effective diameter d, is capable of describing the pressure losses with an error between lo/oand 3”/0,only in exceptions (for extremely low velocities) does the error reach 10%. Therefore, the concept of looking at the stratified fibrous structure as a ‘particulate’ bed seems to work well. A significant obstacle in direct use of the Ergun equation for the calculation of pressure losses in fibrous materials is in the fact that we have no a priori knowledge of the equivalent diameter dpe for a given sample of the fibrous material. This can be resolved if we note that the optimum equivalent diameter, as may be seen from Table 1, is a strong function of the void fraction of the material. Unfortunately, this function is different for the two orientations of strata of the fibres with respect to the direction of the flow. From Table 1, it is seen that the optimum equivalent diameter of the fibrous samples is higher for ‘axial’ orientations of the strata with respect to the flow direction (see Fig. 1). With decreasing void fraction and, hence, increasing compactness of the fibrous samples, however, the differences between the ‘axial’ and ‘radial’ configurations lessen, while, for the void fraction around 0.93, the pressure losses are virtually independent of the configuration. An attempt to correlate the effective diameters of the fibrous samples with the void fraction gave the following results:
194
for the ‘axial’ configuration dPe = -4.64
x 1O-4 + 5.132 x 1O-4a
(2)
and for the ‘radial’ configuration d, = -2.234
x 1O-4 + 2.535 x 10-4a
(3)
Both correlations hold for dpe in metres and 0.93
01
0.93
I
as&
I
I
0.95
I
I
0.96
0.97
of
2 u
of fibrous layers
In an earlier paper, Stanek and Vychodil [7] presented a method of determining the effective radial thermal conductivity of packed beds passed by flowing gas. This parameter is evaluated from comparison of the experimental and theoretical gas temperature profiles measured at the bed outlet, corresponding to a known inlet gas temperature profile. This method is similar to experiments involving heating the bed across the walls but suppresses the effect of the region near the wall, where irregularities of the bed structure may be dominant. The experimental results are compared with the theoretical solution of the pseudohomogeneous model of heat dispersion in cylindrical coordinates in the following form: (4)
E
1
Effective thermal conductivity
098
Fig. 4. Optimum equivalent ‘particle’ diameter as a function void fraction for ‘radial’ ( 0) and ‘axial’ ( 0) configurations.
0
a comparison of the experimental dependence of the pressure losses on the superficial velocity of air with the optimum curve computed on the basis of optimum dper on the other hand, and with the curve computed on the basis of dpe from correlation for a sample of void fraction 0.974. on the other hand. As expected, the agreement impairs with increasing air velocity but, even in the worst case, the error does not exceed 13%.
Cm/s1
Fig. 5. Test of direct applicability of the Ergun equation for fibrous layers. Pressure drop per length as a function of superficial velocity for ‘axial’ layer with E = 0.974: 1, computed from Ergun equation with optimized dp; 2, computed from Ergun equation with d,, from correlation (worst case).
The closed form solution of this model with appropriate boundary and initial conditions may be found in the same paper by Stanek and Vychodil [7]. The formulation of the mathematical model in eqn. (4) is such that, rigorously, it applies only to an isotropic system with radial symmetry. With the ‘radial’ orientation of the layer shown in Fig. 1, the sample essentially satisfies these conditions. Even then, however, its effective radial thermal conductivity may vary periodically along the axial coordinate, owing to the stratified structure of the fibrous material. Since the thickness of individual strata is substantially smaller than the depth of the sample, the application of eqn. (4) to ‘radial’ samples should not bring about excessive errors. With the ‘axial’ orientation of the sample, when the principal direction of the flow of gas is perpendicular to individual strata of the fibres, however, the situation is more complicated. As may be seen from Fig. 1, the sample of fibrous material is now radially unisotropic and one can envision at least two principal directions of different effective radial thermal conductivity (k,,, , k,,,). Therefore, the problem becomes three-dimensional and the pseudohomogeneous model of heat dispersion should be in the tensoral form.
195
Details of the tensoral treatment of unisotropic media have been available for some time in the literature (see, for example, refs. 9 and IO). Nevertheless, as a first approximation, we shall use the isotropic model, i.e. eqn. (4), even for this case. In this particular case, this approximation need not be as bad as it may appear at first, thanks to the low thermal conductivity of the material of the fibres (glass, 0.8 W m-’ K-‘) and the small equivalent diameter of the fibres. The outlet temperature profiles were monitored by means of 20 thermistors located in various radial positions at the outlet of the bed. These thermistors were located in a strip about 8 mm wide across the column diameter. For the ‘radial’ samples, repeated experiments with the strip of thermistors in various angular positions proved the correctness of the assumption of the radial symmetry of the problem. For the ‘axial’ samples, each experimental run was repeated twice; once with the strip of the thermistors oriented parallel to the strata of fibres and once perpendicular to the strata. The results of these repeated measurements always differed detectably. Therefore, the dispersion of heat in this configuration is not quite isotropic.
Results
and discussion
Figure 6 shows plots of the effective radial thermal conductivity of three samples of the fibrous mineral gauze of different void fraction with dependence on the superficial velocity of air. The dependence is seen to be rather weak and the values of kRR are almost the same for the two less compact samples. 0.6
I
I
I
i
2
p”’
I
0.4
0.3
0
0.4
= = 0.944
0
C =
0.961
dpe
=
22.0
un,
0
c
0.976
dpe
=
21.8
,,,I,
=
dpe
=
IS.4
I
I
I
05
1
25
,,n
2
u [m/s1 Fig. 6. Effective radial thermal conductivity as a function of superficial gas velocity for fibrous layer of ‘radial’ configuration and different void fraction.
2
1
Re Fig. 7. Eflective radial thermal conductivity in the form of Peclet number as a function of the Reynolds number for ‘radial’ configuration of fibrous layers. Optimum equivalent ‘particle’ diameter (d,) was used to evaluate Pe and Re. Data correlated by the of correlation: Pea, = Re Pr/ Gunn and Khalid type (8.2 + 0.87 Re Pr).
For all three samples, the measurements were carried out in the ‘radial’ configuration and the results ranged between 0.4 and 0.5 W m-’ K-‘. The observed dependence of the effective conductivity on the velocity of air is rather weak, apparently owing to the small effective diameter of the material and, therefore, low convective contribution to the effective conductivity. Figure 7 shows the same dependence in the form of a plot of the Peclet number us. the Reynolds number. The characteristic dimension needed for evaluation of both dimensionless criteria was the effective diameter L& resulting from the pressure losses and the Ergun equation. With this characteristic dimension, the Reynolds number falls in a very low region between 0.15 and 1.5, this region being virtually inaccesible to measurement in classic particulate packed beds. All the experimental points in Fig. 7 fall essentially on a single line and were correlated using the well-known Gunn and Khalid [ 1 l] type of correlation with the following results: PeRR =
0
0.6 08
Re Pr 8.2 + 0.87 Re Pr
(5)
This optimum curve is also shown in Fig. 7 by a solid line. Figure 8 shows analogous results of the effective radial thermal conductivity of the fibrous mineral gauze as a function of the superficial gas velocity for ‘axial’ orientation of the strata in the sample of fibrous material. The open and full circles distinguish between measurements when the strip of the thermistors below the
196
0.2
I
2 x
ti ii
O O O
00
0
0
00
a5
0.1
0
lm-
0
0
.
.
0
0 n
m
l
-.
0
1
1.5 u
2
[m/e1
Fig. 8. Effective radial thermal conductivity as a function of superficial gas velocity for fibrous layer of ‘axial’ configuration. Full and open symbols distinguish samples of different void fraction. Circles and squares distinguish between measurement of the conductivity in the plane perpendicular and parallel to the strata of the fibrous material respectively.
sample was located parallel or perpendicular to the strata in the fibrous material. The difference in the results indicates a detectable anisotropy of the material given by different contributions of the principal components of radial effective thermal conductivity k,,, and k RU (see Fig. 1). From Fig. 8, it is seen that the component of the radial thermal conductivity perpendicular to the strata of the sample is lower than that in the direction parallel to the strata. (Of course, the results in Fig. 8 are not directly the components k,,, and kRLZ.) This observation confirms the expectation that, in the direction parallel to the strata in the fibrous layer, the conductive contribution (conductance along the fibres) is higher, while the ‘thinner’ regions between individual strata enhance the convective contribution to heat dispersion. Figure 9 shows plots of the radial effective heat conductivity for the ‘axial’ configuration of the fibrous sample in the form of the Prandtl US. the Reynolds number. The length scale again is the effective ‘particle’ diameter from the pressure drop measurement. Different effective conductivities found for the two different orientations of the strip of the thermistors remain visible in these plots. A joint correlation, neglecting these differences, yielded the following result: Pe,,
=
0.04
l
a5
I1
O.OB 0.06
.
*..a* l
=I Om
Re Pr 8.9 + 0.39 Re Pr
This correlation, of course, is less accurate; the error in some regions reaches as much as 20%. A comparison of the correlations in eqns. (5) and (6) indicates less curvature of the results with eqn. (6).
ff
Ml 01
I a2
I I Illll 0.6 (18 1 0.4
Re
I 2
3
Fig. 9. Effective radial thermal conductivity in the form of Peclet number as a function of the Reynolds number for ‘axial’ configuration of fibrous layers. Optimum equivalent ‘particle’ diameter (d,) was used to evaluate Pe and Re. Full and open symbols distinguish samples of different void fractions. Circles and squares distinguish between measurements of the conductivity in the plane perpendicular and parallel to the strata of the fibrous material respectively. Data correlated by the Gunn and Khalid type of correlation: Pea, = Re Prl(8.9 + 0.39 Re Pr).
However, the difference between the results from both correlations is small in the given range of the Reynolds number and, therefore, one can use eqn. (5) as a general correlation.
Conclusions
Experimental measurement of pressure losses incurred under the flow of gas through a layer of unisotropic fibrous material indicated substantial differences, depending on the orientation of individual strata of the fibres in the sample material with respect to the direction of the flow of air. Both the ‘axial’ and ‘radial’ configurations are of practical importance from the standpoint of the technology of manufacture of the fibrous insulation carpets. For the ‘axial’ configuration of strata, the corresponding pressure loss is higher by as much as an order of magnitude. This difference is attributed to the lower resistance of the material between individual strata, owing to the locally decreased void content of the material. The Ergun equation originally formulated for beds of particulate solids has been used successfully to correlate the pressure losses against the superficial velocity of the flowing air, with the effective ‘particle’ diameter being an adjustable parameter. As expected, the effective di-
197
ameter exceeds the mean diameter of the fibres (in this study, this is by a factor of 3-4). For direct use of the Ergun equation, the ‘optimum’ values of characteristic dimension of the fibrous materials were successfully correlated with the void fraction of the material. The pressure losses computed on the basis of the eqns. (2) and (3) and the Ergun equation are sufficiently accurate for technical purposes. The effective thermal conductivity of the fibrous materials was experimentally investigated and interpreted on the basis of the pseudohomogeneous heat dispersion model for an isotropic medium. It is readily conceded that, for fibrous materials, the idea of an isotropic medium is not a very accurate one. The experimental results for the ‘axial’ configuration of the strata with respect to the flowing air confirmed the tensoral character of the dispersion. ‘Radial’ configurations offer results acceptable from the standpoint of an isotropic model. Nevertheless, a common correlation for the effective thermal conductivity of the fibrous material (eqn. (5)) may be used for a general configuration in connection with the isotropic model, if an error of about 20% can be tolerated. It follows that the idea of applying the concepts and models originally developed for beds of particles to unisotropic fibrous materials proved fairly effective. Physically, these concepts also seem meaningful, as may be seen from Fig. 10. This figure plots the dependence
of the Peclet number on the Reynolds number for the ‘axial’ and ‘radial’ samples of fibrous insulation gauze layers obtained in this study, together with the results of Yagi and Wakao [ 121, and Stanek and Vychodil [8] for spherical particles, our earlier results, and the result of Plautz and Johnstone [ 131. Although the two sets of data (particulate-fibrous) do not overlap, the results with layers of fibrous materials, reaching substantially lower ranges of the Reynolds number, seem to be reasonable extrapolations of the results with particulate beds. Nomenclature
c Pg
dPC
k,, k, kRR- k,,, k,,, 7
(Jm --Is-’ K-‘)
AP PeRR
7 PeRL
Pr
-I
1oc
,.-5
Re
/‘1 .’
5_,,M
,’
1’ /%
l-
,’
4’
u
T
3.
CL1
1
Z
2
E
-A, 01
I 1
10
pressure drop (Pa) Peclet number for effective radial thermal conductivity of the fibrous material in ‘radial’ and ‘axial’ configurations Prandtl number based on d, as scale length radial coordinate (m) Reynolds number based on dp, as scale length superficial velocity of gas (m s-r) temperature (K) axial coordinate (m)
Greek letters
Pg PLa ml
specific heat capacity of gas (J kg-’ K-‘) equivalent diameter of the fibrous material (m) axial and radial effective heat conductivity (J m-’ s-’ K-‘) effective radial heat conductivity of the fibrous material for ‘radial’ configuration and two principal components for the ‘axial’ configuration
void fraction (-) density of gas (kg m-3) viscosity of gas (kg m-r s-r)
-
100
1000
Re Fig. 10. Effective radial thermal conductivity in the form of Peclet number as a function of the Reynolds number for fibrous and particulate materials: 1, fibrous layer with ‘radial’ configuration, i.e. Pen, = Re Pr/(8.2 f0.87 Re Pr); 2, fibrous layer with ‘axial’ configuration, i.e. Pea, = Re Pr/(8.9 + 0.39 Re Pr); 3, glass balls, i.e. Pea = Re Pr/( 13.9 + 0.13 Re Pr) [8]; 4, glass particles, i.e. Pen = Re Pr/(27.5 + 0.08 Re Pr) [ 131; 5, glass particles. i.e. Pen = Re Pr/(6.0 + 0.11 Re Pr) [ 121.
References I E. Tsotsas and H. Martin, Thermal conductivity of packed beds. A review, Chem. Eng. Process., 22 (1987) 19. 2 E. U. Schliinder and E. Tsotsas, Wiirmeiibertragung in Festbetten, durchmischten Schiittgiitem und Wirbelschichten, Georg Thieme, Stuttgart, 1988. 3 G. K. Boreskov, Yu. Sh. Matros and 0. V. Kiselev, Heterogeneous catalytic process under unsteady-state regime, Dokl. Akad. Nauk SSSR, 237 ( 1977) 91 (in Russian).
198
4 G. A. Fateev, N. Vegyte and L. P. Petrova, Calculation of the temperature and concentration fields under thermofiltration in a reacting system, Sbornik Trudov Instituta Teplo-i Massoobmena Akademii Nauk BSSR, Minsk, 1983, Belorussian Academy of Sciences, Minsk, 1983, p. 129 (in Russian). 5 N. Vegyte, A. Ju Skrinska and G. A. Fateev, Polycondensation of resin in the layer of thermal insulation material under the self-sustaining regime of reaction zones, Proc. First ANUnion Symp. on Macroscopic Kinetics and Chemical Gasodynamics, Chernogolovka, Institute of Chemical Physics AN SSSR, Vol. I, Part I, 1984, Academy of Sciences of USSR, Moscow, 1984, p. 114 (in Russian). 6 N. Vegyte and G. A. Fateev, Analysis of the conditions of thermal curing of adhesive under the conductive and convective regime in a thermal insulation material, Sbornik Trudov VNZI Teploisoliatsia (UDK 666.198:532.546) (1986) (in Russian).
7 V. Stantk and P. Vychodil, On the observability of heat dispersion parameters of the pseudohomogeneous model of heat transfer in packed beds, CON. Czech. Chem. Commun., 48 (1983) 2484. 8 V. Stanitk and P. Vychodil, On the length effect of the heat dispersion parameters, Chem. Eng. Commun., 27 (1984) 69. 9 J. Bear, Dynamics of FIuids in Porous Media, Elsevier, New York, 1972, ch. 5. 10 F. A. L. Dullien, Single phase flow through porous media and pore structure, Chem. Eng. .I., 10 (1975) 1. 11 D. J. Gunn and M. Khalid, Thermal dispersion and wall heat transfer in packed beds, Chem. Eng. Sci., 30 (1975) 261. 12 S. Yagi and N. Wakao, Heat and mass transfer from wall to fluid in packed beds, AIChE J., 5 (1959) 79. 13 D. A. Plautz and H. F. Johnstone, Heat and mass transfer in packed beds, AIChE J., I (1955) 193.