- Email: [email protected]
Non-newtonian (purely viscous) fluid flow through packed beds: Effect of particle shape
Powder Technology,
67 (1991) 15-19
1.5
Non-Newtonian (purely effect of particle shape R. P. Chhabra* Department
(Received
viscous)
fluid flow through
packed
beds:
and B. K. Srinivas
of Chemical
Engineering
Indian Institute of Technoloa,
May 4, 1990; in revised form October
Kanpur
208016
(India)
22, 1990)
Abstract The effect of particle shape on the flow of purely viscous type non-Newtonian fluids through beds of non-spherical particles has been investigated experimentally. Extensive pressure drop measurements are reported for scores of Newtonian and non-Newtonian fluids through beds of three different types of packing materials (two sizes of Raschig rings and one size of gravel chips). It is established that the widely used Ergun equation provides a satisfactory representation of data for non-Newtonian fluids also if a volume equivalent diameter and a sphericity factor are used to account for non-spherical shape of particles. The experimental results reported herein encompass the following ranges of conditions: 0.0016
Introduction Owing to their wide-ranging applications, considerable effort has been expended in exploring and understanding the hydrodynamics of non-Newtonian fluid flow in packed beds. There is no question that most of the research efforts have been directed at developing what might be called an ‘Ergun equation’ for the flow of non-Newtonian fluids, i.e., at devising a method for the prediction of pressure loss for a given liquid and bed combination. Notwithstanding the fact that the calculation of the pressure loss associated with the flow of Newtonian fluids in packed columns has defied predictions from first principles, it is not at all surprising that most developments in this area involving the flow of even purely viscoustype non-Newtonian fluids have been largely of dimensional and/or empirical nature. Consequently, there is no paucity of predictive expressions available in the literature. Exhaustive and informative accounts of the developments in this area are available in the literature [l-3]. More recently, Srinivas and Chhabra [4] have evaluated the relative performance of some of the widely used expressions which have been proposed for the estimation of pressure loss for nonNewtonian ilow in packed tubes. Most of the aforementioned investigations have, however, dealt exclusively with non-Newtonian fluid *Author
to whom correspondence
0032-5910/91/$3.50
should be addressed.
flow in beds of uniformly sized spherical particles, and very little is known about the effect of particle shape on pressure drop for non-Newtonian fluids. Indeed, there have been only three studies reported in the open literature pertaining to the flow of nonNewtonian fluids through beds of non-spherical particles. Yu et al. [5] used plastic cubes to study the effect of particle shape on pressure drop in packed beds. Kumar and Upadhyay [6] investigated the flow of one weakly non-Newtonian solution through a column packed with glass cylinders. Machac and Dolejs [7], on the other hand, examined the influence of a range of particle shapes (such as cylinders, polyhedrons, etc.). Both Yu et al. [S] and Kumar and Upadhyay [6] concluded that the effect of nonspherical shape is adequately accounted for by using a sphericity factor whereas Machac and Dolejs [7] introduced a bed factor; the latter, representing the ratio of the total drag to the friction drag, was reported to be a weak function of non-Newtonian parameters. Thus, it is safe to conclude that not much is known about the effect of particle shape on pressure drop in packed columns. The present work aims to fill this gap in our knowledge. Extensive pressure drop measurements for the flow of non-Newtonian solutions through beds of non-spherical particles embracing wide ranges of kinematic and rheological properties are reported. It is demonstrated that the conventional Reynolds number-friction factor approach provides a satis-
0 1991 -
Elscvier Sequoia,
Lausanne
16
factory representation of the non-Newtonian fluid flow through packed columns of non-spherical particles, provided that an equivalent diameter together with a sphericity factor is used.
Experimental
set-up,
materials
under water as the manometer fluids. It included a 100-litre stainless steel jacketed tank to maintain the temperature of test liquid constant within +O.S “C during an experimental run. The bed voidage for each packing material was estimated by measuring the volume of water needed to fill the void space for a known weight of packing in a tube of same diameter as the packed bed. Each value of voidage represents an average of five to six repeat measurements which were within f3% of the mean value; the resulting values of voidage for each of the three packings are: 0.472, 0.654 and 0.704. These measurements are in general agreement with the values reported in the literature [8]. Water and aqueous solutions of glucose syrup and those of a commercial grade carboxymethyl cellulose (manufactured by Cellulose India, Ltd., Ahmedabad) were used as test liquids. The rheological properties, namely, steady state shear stress-shear rate data of glucose syrup and polymer solutions were measured using a RV-12 Haake viscometer at the same temperature as that encountered in the packed bed experiments. The solutions of glucose syrup displayed constant viscosities and hence were classed as Newtonian fluids, whereas the solutions of carboxymethyl cellulose exhibited varying levels of shear-thinning behaviour. The density of each test liquid was measured using a constant-volume density bottle. Altogether, fourteen test liquids comprising six Newtonian and eight non-Newtonian systems were used in this study.
and procedure
A glass column of 100 mm diameter and 1800 mm long packed with Raschig rings or gravel chips was used to measure pressure loss for flow through packed beds of non-spherical particles. The three different types of packing used were characterised by assigning a volume equivalent diameter (Dpe) which was estimated by measuring the volume displaced by a known number of particles. Each value of D,, represents an average of not less than fifteen independent determinations, and the variation in the resulting values of D,, did not exceed 4%, thereby indicating the uniform size of particles. A schematic diagram of the apparatus used is shown in Fig. 1. Test liquids were circulated through fixed beds (101.6 mm dia, and 891 mm deep) using a 0.5hp positive displacement pump. The liquid flow rate was measured by weighing a liquid sample collected at the discharge end in a known time. There was also provision to use a magnetic flowmeter for high liquid rates such as those encountered for the flow of water. Pressure drop across 891 mm of bed height (located in the middle of the column) was measured using simple U-tube manometers with carbon tetrachloride
Magnetic flowmeter Tank
1
*
Flexible _ joint
I Magnet,c flowmeter
Is ’
Flexible joint ng water Tank
To drain
L-
-s+
- -
Fig.
1. Schematic
diagram
of experimental
set-up.
Auxiliary
arrangement
1,
17
Results
and discussion
Rheological
characteristics of test liquids
As mentioned earlier, the aqueous solutions of glucose syrup behaved like Newtonian fluids by exhibiting a constant value of viscosity in each case, and the resulting values of viscosity and density are given in the Table, together with the other physical and operating conditions; a thirty two-fold variation in viscosity is evident. On the other hand, the aqueous solutions of carboxymethyl cellulose exhibited shearthinning characteristics. Figure 2 shows the representative shear stress-shear rate data for three of the test liquids. The usual two-parameter power law fluid model was found to adequately represent the pseudoplastic behaviour in the shear rate range of interest here as ascertained by using the expression proposed by Gogarty [9]. Thus, one can write T=K ($J)
(I)
The best values of n and K have been using a non-linear regression approach, resulting values of these along with the other physical properties are summarised in the
obtained and the relevant Table.
Pressure drop-flow rate behaviour
It is customary to use dimensionless numbers to present pressure drop results as a function of flow rate and bed characteristics (voidage, particle size and shape). For the flow of Newtonian fluids through a bed of spherical particles, the following definitions of Reynolds number and a friction factor, proposed by Ergun [lo], have gained wide acceptance:
pvoD,
Re=
I.4 f=%
-c> _P
PVo2 i
A__ L ) l--E
(3)
Based on extensive experimental results, Ergun [lo] presented the following equation: f = s
+ 1.75
Equation (4) has been found to be applicable up to about Re- 1 000. It is, however, appropriate to point out here that considerable confusion exists in the literature regarding the numerical constants appearing in eqn. (4), especially the value of 150. Indeed, the values ranging from 115 to 180 have been suggested in the literature [ll], for beds of uniform size spherical particles. Even wider variations in the values of constants in eqn. (4) have been observed for beds of non-spherical particles [12]. Furthermore, there is some evidence, however ten-
18
analyse the results for non-Newtonian polymer solutions. Most workers have used the same definition (eqn. (2)) of friction factor for power law liquids as well. However, on the other hand, a range of definitions of Reynolds number have been used for power law fluids by different workers, but most of these are inter-related through functions of voidage and power law index [4]. In order to maintain the continuity and consistency with our previous work [4], the following definition introduced by Kemblowski and Michniewicz [13] has been used in this work: 1
I
1
I
I
I
I
10 SHEAR
I
100 RATE
1
500
(5)
(S’)
Fig. 2.
Representative shear stress-shear rate data shown for three test liquids. Key to the symbols is same as in Fig. 3.
uous, of these constants being weakly dependent on particle size, shape, roughness and porosity, etc. [12]. However, based on extensive comparisons, Mac Donald et al. [12] concluded that despite these uncertainties, the Ergun equation predicts the values of pressure drop with an uncertainty of 550% under most conditions of practical interest. Ergun [lo] asserted that eqn. (4) was also applicable to flow through beds of non-spherical particles, provided that an equivalent particle diameter (D,,) and a sphericity factor (A) are used to correct for the non-spherical shape. There are several ways of calculating an equivalent or effective diameter (D& for a non-spherical particle, the simplest of all being the one based on a sphere of the same volume as that of the particle. This definition has been used here. Similarly, there does not appear to be a unique way of estimating the value of a sphericity factor. In this work, it has been calculated back from the Ergun equation (i.e., eqn. (4) with D, replaced by (Dpc&)) using the experimental values of (APL) and V, for three Newtonian solutions. The experimental results (corresponding to run Nos. 1-3) obtained with glucose syrup solutions were used for this purpose, and the resulting values of the spheric@ factor (&) of the three types of particles used herein are given in the Table; the values for Raschig rings compare favourably with the literature values [S]. The values of 4, given in the Table represent an average of at least fourteen individual pressure drop-velocity measurements, and the maximum deviation from the mean value is only 5.0%. Furthermore, the values of & so obtained predicted the results for water with an average error of ll%, which is regarded to be quite satisfactory and acceptable. These values of D,, and & will now be used to
where for non-spherical particles D, = Dpe& Attention is drawn to the fact that for a Newtonian fluid, i.e., n=l, eqn. (5) reduces correctly to eqn. (2). The values of friction factor and Reynolds number calculated using these definitions are plotted in Fig. 3; the predictions of the Ergun equation are also included in this figure. An examination of this figure suggests that the Ergun equation successfully correlates f-Re’ data for power law liquids flowing through beds of non-spherical particles over three orders of magnitude of Re’ (0.0016-2.5) and for a range of values of II as seen in the Table. There are no discernable trends present in Fig. 3 either with respect to particle shape or power law index. Equation (4) predicts the results for 146 individual pressure drop measurements with an average error of 13% with only 11 data points showing error larger than 20%, the maximum error being 26%. The fact that the data for two sizes of Raschig rings and gravel chips are indistinguishable from each other in Fig. 3 suggests that D,, and & provide an adequate description of non-spherical shape at least in the low Reynolds number regime.
Conclusions The flow of power law liquids through beds of non-spherical particles has been investigated experimentally. The resulting values of friction factor are well correlated by the widely used Ergun equation using a suitably modified Reynolds number for power law liquids. Furthermore, the present results indicate that the additional effects arising from the nonspherical shape of particles can be adequately accounted for by introducing a volume equivalent diameter and a sphericity factor. However, this conclusion is presently limited to the low Reynolds number regime only. The deviations between the experimental and predicted values of friction factor
19
16
162 REYNOLDS
Fie. 3. Friction
factor-Reynolds
100 NUMBER
number
data
102
lo1
Re
for non-Newtonian
fluid
flow through
beds
of non-spherical
particles.
The
soid line represents the prediction of eqn. (4). fluid density shear stress sphericity factor
for power law liquids are well within the uncertainty associated with the Ergun equation. Acknowledgements References Financial support received from the Department of Science and Technology (Government of India) is gratefully acknowledged (Grade No. III-4(2)/86ET). List of symbols
DPC fp K L n AP Re or Re’ VO
volume equivalent diameter spherical particle diameter friction factor power law consistency coefficient length of bed power law flow behaviour index pressure drop Reynold number, eqn. (2) or eqn. (5) superficial liquid velocity
Greek symbols 5,
E P
shear rate voidaga fluid viscosity
1 J. G. Savins, Ind. Eng. Chem., 61 (1969) 18. S. N. Upadhyay and P. Mishra, 2 S. Kumar, K. Kishore, J. Sci. Ind. Res., 40 (1981) 238. M. Dziubinski and J. Sck, Adv. Trans. 3 Z. Kemblowski, Proc., 5 (1987) 117. 4 B. K. Srinivas and R. P. Chhabra, Inr. J. Eng. Fluid Me&, (1991) in press. 5 Y. H. Yu, C. Y. Wen and R. C. Bailie, Can. 1. Chem. Eng., 46 (1968) 149. Eng. Chem. Fundum., 6 S. Kumar and S. N. Upadhyay,Ind. 20 (1981) 186. and V. Dolejs, Chem. Eng. Sci., 36 (1981) 7 I. Machac 1679. 8 Penys Chemical Engineers Handbook, McGraw-Hill, New York, 5th edn., 1973, pp. 5-53. 9 W. B. Gogarty, Sot. Pet. Eng. J., 7 (1967) 161. 10 S. Ergun, C&m. Eng. Prog., 48 (1952) 89. and K. L. Pinder, Can. J. Gem. Eng., 11 T. F. Al-Fariss 65 (1987) 391. 12 I. F. MacDonald, M. S. El-Sayed, K. Mow and F. A. L. Dullien, Ind. Eng. Chem. Fundam., 18 (1979) 199. and M. Michniewicz, Rheo. Acta, I8 13 Z. Kemblowski (1979) 730.
67 (1991) 15-19
1.5
Non-Newtonian (purely effect of particle shape R. P. Chhabra* Department
(Received
viscous)
fluid flow through
packed
beds:
and B. K. Srinivas
of Chemical
Engineering
Indian Institute of Technoloa,
May 4, 1990; in revised form October
Kanpur
208016
(India)
22, 1990)
Abstract The effect of particle shape on the flow of purely viscous type non-Newtonian fluids through beds of non-spherical particles has been investigated experimentally. Extensive pressure drop measurements are reported for scores of Newtonian and non-Newtonian fluids through beds of three different types of packing materials (two sizes of Raschig rings and one size of gravel chips). It is established that the widely used Ergun equation provides a satisfactory representation of data for non-Newtonian fluids also if a volume equivalent diameter and a sphericity factor are used to account for non-spherical shape of particles. The experimental results reported herein encompass the following ranges of conditions: 0.0016
Introduction Owing to their wide-ranging applications, considerable effort has been expended in exploring and understanding the hydrodynamics of non-Newtonian fluid flow in packed beds. There is no question that most of the research efforts have been directed at developing what might be called an ‘Ergun equation’ for the flow of non-Newtonian fluids, i.e., at devising a method for the prediction of pressure loss for a given liquid and bed combination. Notwithstanding the fact that the calculation of the pressure loss associated with the flow of Newtonian fluids in packed columns has defied predictions from first principles, it is not at all surprising that most developments in this area involving the flow of even purely viscoustype non-Newtonian fluids have been largely of dimensional and/or empirical nature. Consequently, there is no paucity of predictive expressions available in the literature. Exhaustive and informative accounts of the developments in this area are available in the literature [l-3]. More recently, Srinivas and Chhabra [4] have evaluated the relative performance of some of the widely used expressions which have been proposed for the estimation of pressure loss for nonNewtonian ilow in packed tubes. Most of the aforementioned investigations have, however, dealt exclusively with non-Newtonian fluid *Author
to whom correspondence
0032-5910/91/$3.50
should be addressed.
flow in beds of uniformly sized spherical particles, and very little is known about the effect of particle shape on pressure drop for non-Newtonian fluids. Indeed, there have been only three studies reported in the open literature pertaining to the flow of nonNewtonian fluids through beds of non-spherical particles. Yu et al. [5] used plastic cubes to study the effect of particle shape on pressure drop in packed beds. Kumar and Upadhyay [6] investigated the flow of one weakly non-Newtonian solution through a column packed with glass cylinders. Machac and Dolejs [7], on the other hand, examined the influence of a range of particle shapes (such as cylinders, polyhedrons, etc.). Both Yu et al. [S] and Kumar and Upadhyay [6] concluded that the effect of nonspherical shape is adequately accounted for by using a sphericity factor whereas Machac and Dolejs [7] introduced a bed factor; the latter, representing the ratio of the total drag to the friction drag, was reported to be a weak function of non-Newtonian parameters. Thus, it is safe to conclude that not much is known about the effect of particle shape on pressure drop in packed columns. The present work aims to fill this gap in our knowledge. Extensive pressure drop measurements for the flow of non-Newtonian solutions through beds of non-spherical particles embracing wide ranges of kinematic and rheological properties are reported. It is demonstrated that the conventional Reynolds number-friction factor approach provides a satis-
0 1991 -
Elscvier Sequoia,
Lausanne
16
factory representation of the non-Newtonian fluid flow through packed columns of non-spherical particles, provided that an equivalent diameter together with a sphericity factor is used.
Experimental
set-up,
materials
under water as the manometer fluids. It included a 100-litre stainless steel jacketed tank to maintain the temperature of test liquid constant within +O.S “C during an experimental run. The bed voidage for each packing material was estimated by measuring the volume of water needed to fill the void space for a known weight of packing in a tube of same diameter as the packed bed. Each value of voidage represents an average of five to six repeat measurements which were within f3% of the mean value; the resulting values of voidage for each of the three packings are: 0.472, 0.654 and 0.704. These measurements are in general agreement with the values reported in the literature [8]. Water and aqueous solutions of glucose syrup and those of a commercial grade carboxymethyl cellulose (manufactured by Cellulose India, Ltd., Ahmedabad) were used as test liquids. The rheological properties, namely, steady state shear stress-shear rate data of glucose syrup and polymer solutions were measured using a RV-12 Haake viscometer at the same temperature as that encountered in the packed bed experiments. The solutions of glucose syrup displayed constant viscosities and hence were classed as Newtonian fluids, whereas the solutions of carboxymethyl cellulose exhibited varying levels of shear-thinning behaviour. The density of each test liquid was measured using a constant-volume density bottle. Altogether, fourteen test liquids comprising six Newtonian and eight non-Newtonian systems were used in this study.
and procedure
A glass column of 100 mm diameter and 1800 mm long packed with Raschig rings or gravel chips was used to measure pressure loss for flow through packed beds of non-spherical particles. The three different types of packing used were characterised by assigning a volume equivalent diameter (Dpe) which was estimated by measuring the volume displaced by a known number of particles. Each value of D,, represents an average of not less than fifteen independent determinations, and the variation in the resulting values of D,, did not exceed 4%, thereby indicating the uniform size of particles. A schematic diagram of the apparatus used is shown in Fig. 1. Test liquids were circulated through fixed beds (101.6 mm dia, and 891 mm deep) using a 0.5hp positive displacement pump. The liquid flow rate was measured by weighing a liquid sample collected at the discharge end in a known time. There was also provision to use a magnetic flowmeter for high liquid rates such as those encountered for the flow of water. Pressure drop across 891 mm of bed height (located in the middle of the column) was measured using simple U-tube manometers with carbon tetrachloride
Magnetic flowmeter Tank
1
*
Flexible _ joint
I Magnet,c flowmeter
Is ’
Flexible joint ng water Tank
To drain
L-
-s+
- -
Fig.
1. Schematic
diagram
of experimental
set-up.
Auxiliary
arrangement
1,
17
Results
and discussion
Rheological
characteristics of test liquids
As mentioned earlier, the aqueous solutions of glucose syrup behaved like Newtonian fluids by exhibiting a constant value of viscosity in each case, and the resulting values of viscosity and density are given in the Table, together with the other physical and operating conditions; a thirty two-fold variation in viscosity is evident. On the other hand, the aqueous solutions of carboxymethyl cellulose exhibited shearthinning characteristics. Figure 2 shows the representative shear stress-shear rate data for three of the test liquids. The usual two-parameter power law fluid model was found to adequately represent the pseudoplastic behaviour in the shear rate range of interest here as ascertained by using the expression proposed by Gogarty [9]. Thus, one can write T=K ($J)
(I)
The best values of n and K have been using a non-linear regression approach, resulting values of these along with the other physical properties are summarised in the
obtained and the relevant Table.
Pressure drop-flow rate behaviour
It is customary to use dimensionless numbers to present pressure drop results as a function of flow rate and bed characteristics (voidage, particle size and shape). For the flow of Newtonian fluids through a bed of spherical particles, the following definitions of Reynolds number and a friction factor, proposed by Ergun [lo], have gained wide acceptance:
pvoD,
Re=
I.4 f=%
-c> _P
PVo2 i
A__ L ) l--E
(3)
Based on extensive experimental results, Ergun [lo] presented the following equation: f = s
+ 1.75
Equation (4) has been found to be applicable up to about Re- 1 000. It is, however, appropriate to point out here that considerable confusion exists in the literature regarding the numerical constants appearing in eqn. (4), especially the value of 150. Indeed, the values ranging from 115 to 180 have been suggested in the literature [ll], for beds of uniform size spherical particles. Even wider variations in the values of constants in eqn. (4) have been observed for beds of non-spherical particles [12]. Furthermore, there is some evidence, however ten-
18
analyse the results for non-Newtonian polymer solutions. Most workers have used the same definition (eqn. (2)) of friction factor for power law liquids as well. However, on the other hand, a range of definitions of Reynolds number have been used for power law fluids by different workers, but most of these are inter-related through functions of voidage and power law index [4]. In order to maintain the continuity and consistency with our previous work [4], the following definition introduced by Kemblowski and Michniewicz [13] has been used in this work: 1
I
1
I
I
I
I
10 SHEAR
I
100 RATE
1
500
(5)
(S’)
Fig. 2.
Representative shear stress-shear rate data shown for three test liquids. Key to the symbols is same as in Fig. 3.
uous, of these constants being weakly dependent on particle size, shape, roughness and porosity, etc. [12]. However, based on extensive comparisons, Mac Donald et al. [12] concluded that despite these uncertainties, the Ergun equation predicts the values of pressure drop with an uncertainty of 550% under most conditions of practical interest. Ergun [lo] asserted that eqn. (4) was also applicable to flow through beds of non-spherical particles, provided that an equivalent particle diameter (D,,) and a sphericity factor (A) are used to correct for the non-spherical shape. There are several ways of calculating an equivalent or effective diameter (D& for a non-spherical particle, the simplest of all being the one based on a sphere of the same volume as that of the particle. This definition has been used here. Similarly, there does not appear to be a unique way of estimating the value of a sphericity factor. In this work, it has been calculated back from the Ergun equation (i.e., eqn. (4) with D, replaced by (Dpc&)) using the experimental values of (APL) and V, for three Newtonian solutions. The experimental results (corresponding to run Nos. 1-3) obtained with glucose syrup solutions were used for this purpose, and the resulting values of the spheric@ factor (&) of the three types of particles used herein are given in the Table; the values for Raschig rings compare favourably with the literature values [S]. The values of 4, given in the Table represent an average of at least fourteen individual pressure drop-velocity measurements, and the maximum deviation from the mean value is only 5.0%. Furthermore, the values of & so obtained predicted the results for water with an average error of ll%, which is regarded to be quite satisfactory and acceptable. These values of D,, and & will now be used to
where for non-spherical particles D, = Dpe& Attention is drawn to the fact that for a Newtonian fluid, i.e., n=l, eqn. (5) reduces correctly to eqn. (2). The values of friction factor and Reynolds number calculated using these definitions are plotted in Fig. 3; the predictions of the Ergun equation are also included in this figure. An examination of this figure suggests that the Ergun equation successfully correlates f-Re’ data for power law liquids flowing through beds of non-spherical particles over three orders of magnitude of Re’ (0.0016-2.5) and for a range of values of II as seen in the Table. There are no discernable trends present in Fig. 3 either with respect to particle shape or power law index. Equation (4) predicts the results for 146 individual pressure drop measurements with an average error of 13% with only 11 data points showing error larger than 20%, the maximum error being 26%. The fact that the data for two sizes of Raschig rings and gravel chips are indistinguishable from each other in Fig. 3 suggests that D,, and & provide an adequate description of non-spherical shape at least in the low Reynolds number regime.
Conclusions The flow of power law liquids through beds of non-spherical particles has been investigated experimentally. The resulting values of friction factor are well correlated by the widely used Ergun equation using a suitably modified Reynolds number for power law liquids. Furthermore, the present results indicate that the additional effects arising from the nonspherical shape of particles can be adequately accounted for by introducing a volume equivalent diameter and a sphericity factor. However, this conclusion is presently limited to the low Reynolds number regime only. The deviations between the experimental and predicted values of friction factor
19
16
162 REYNOLDS
Fie. 3. Friction
factor-Reynolds
100 NUMBER
number
data
102
lo1
Re
for non-Newtonian
fluid
flow through
beds
of non-spherical
particles.
The
soid line represents the prediction of eqn. (4). fluid density shear stress sphericity factor
for power law liquids are well within the uncertainty associated with the Ergun equation. Acknowledgements References Financial support received from the Department of Science and Technology (Government of India) is gratefully acknowledged (Grade No. III-4(2)/86ET). List of symbols
DPC fp K L n AP Re or Re’ VO
volume equivalent diameter spherical particle diameter friction factor power law consistency coefficient length of bed power law flow behaviour index pressure drop Reynold number, eqn. (2) or eqn. (5) superficial liquid velocity
Greek symbols 5,
E P
shear rate voidaga fluid viscosity
1 J. G. Savins, Ind. Eng. Chem., 61 (1969) 18. S. N. Upadhyay and P. Mishra, 2 S. Kumar, K. Kishore, J. Sci. Ind. Res., 40 (1981) 238. M. Dziubinski and J. Sck, Adv. Trans. 3 Z. Kemblowski, Proc., 5 (1987) 117. 4 B. K. Srinivas and R. P. Chhabra, Inr. J. Eng. Fluid Me&, (1991) in press. 5 Y. H. Yu, C. Y. Wen and R. C. Bailie, Can. 1. Chem. Eng., 46 (1968) 149. Eng. Chem. Fundum., 6 S. Kumar and S. N. Upadhyay,Ind. 20 (1981) 186. and V. Dolejs, Chem. Eng. Sci., 36 (1981) 7 I. Machac 1679. 8 Penys Chemical Engineers Handbook, McGraw-Hill, New York, 5th edn., 1973, pp. 5-53. 9 W. B. Gogarty, Sot. Pet. Eng. J., 7 (1967) 161. 10 S. Ergun, C&m. Eng. Prog., 48 (1952) 89. and K. L. Pinder, Can. J. Gem. Eng., 11 T. F. Al-Fariss 65 (1987) 391. 12 I. F. MacDonald, M. S. El-Sayed, K. Mow and F. A. L. Dullien, Ind. Eng. Chem. Fundam., 18 (1979) 199. and M. Michniewicz, Rheo. Acta, I8 13 Z. Kemblowski (1979) 730.