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Elucidation of pressure drop in packed-bed systems
~
Applied Thermal Engineering Vol. 16, No. 3, pp. 195-202, 1996 Copyright © 1995 Elsevier Science Ltd 1359-4311(95)00002-X Printed in Great Britain. All rights reserved 1359-4311/96 $15.00 + 0.00
Pergamon
ELUCIDATION OF PRESSURE DROP IN PACKED-BED SYSTEMS E. A. Foumeny, A. Kulkarni, S. Roshani and A. Vatani Department of Chemical Engineering, University of Leeds, Leeds LS2 9JT, UK (Received for publication 29 June 1995)
Abstract--Studies on pressure-drop characteristics of beds of equilateral and non-equilateral solid cylinders of aspect ratios ranging from 0.5 to 2.0 have been carried out. These studies have exposed the shortcomings of the commonly used Ergun type correlations for predicting the pressure drop in packed systems. An empirical correlation to predict pressure drop in beds of cylindrical particles has been developed. In addition, derivatives of cylinders, such as hollow cylinders, have been examined, and comparisons have been made with particles, such as spheres. Keywords-~Packed bed, voidage, Reynolds number, friction factor, pressure drop. NOMENCLATURE A B dp dp~ dt f' l L AP Re' Sp u Vp
intercept, equation (2) slope, equation (2) particle diameter (m) equivalent diameter, 6Vo/S p (m) tube diameter (m) modifiedfriction factor length of solid cylinder (m) length of packed bed (m) pressuredrop (kPa/m) modifiedReynolds number surface area of particle (m2) superficialfluid velocity (m/s) volume of particle (m3)
Greek letters
E p #
mean voidage density of fluid (kg/m3) absolute viscosity of fluid (N/m.s)
1. I N T R O D U C T I O N
Energy loss, as characterised by a pressure drop of the process fluid, is an important consideration
in the design and operation of fixed-bed systems and has consequently been a subject of great interest for few decades. A vast amount of information in the form of empirical and semi-empirical correlations which relate the pressure d r o p to the h y d r o d y n a m i c conditions o f the packed beds is available. The E r g u n correlation [1], as given by equation (1), has been widely used for design or treatment o f raw experimental data extracted from certain geometrical configurations. E r g u n ' s correlation accounts for viscous and inertial energy losses and relates them to the dynamic variable, velocity o f the fluid, as well as the structure o f the bed, as characterised by the bed mean voidage, E. AP T
150/~u(1 - e)2 ~---
d2£ 2
1.75pu2(1 - E) ~
dp(5 3
(l)
In spite o f its general acceptability, m a n y workers have questioned the applicability o f the constants 150 and 1.75 for all types o f packing shapes. F o r example, Handley and Heggs [2] p r o p o s e d a correlaiton having constants 368 and 1.24, while M c D o n a l d et al. [3] rearranged the E r g u n equation in terms o f a linear relationship (2) between the modified friction factor, f ' , and the modified Reynolds number, Re', with constants A and B representing the intercept and the 195
E.A. Foumeny et al.
196
slope, respectively. They proposed a fixed value of 180 for A, and suggested that slope B varied between 1.8, for smooth particles, and 4.0, for rough particles. APd~c3
L#u(1--¢)2 +
B
pudp
+ A.
(2)
#(l-E)
The left-hand side of equation (2) is referred to as the modified friction factor, f ' , and the term pudv/l~(1 - E ) is the modified Reynolds number, Re'. The constants A and B could, hence, be related to certain bed characteristics, such as shape and size of the packed particles. Based on this approach, Foumeny et al. [4] carried out a series of experiments on beds of spheres. Their study covered a wide range of tube to particle diameter ratios, 3 ~< d,/dp <~24, and modified Reynolds number, 5 ~< Re' ~< 8500. Through an optimisation exercise they established a constant value for A of 130, while the slope B varied functionally in relation to the diameter ratio, dt/do, as represented by equation (3): f ' = BRe' + 130,
(3)
where e =
4/a 0 . 3 3 5 4 / ¢ + 2.28
.
It appears that for certain beds, dt/dp < 10, the total pressure drop across the bed can be greatly influenced by the diameter ratio, dt/dp. This finding clearly demonstrates the limitation of the Ergun equation. Although there are a number of correlations available for packings used for gas-liquid contacting operations [5], there is very little information available on the shapes of catalyst pellets employed in gas or vapour phase reactions. It is the aim of this work to carry-out pressure-drop experiments on commonly used particle shapes and to assess how shape and size influence the overall pressure losses of fixed beds. Since nearly two-thirds of the particles used in the industry are of a cylindrical nature [6], cylindrical pellets and their derivatives are considered in this work. 2. E X P E R I M E N T A L
2.1. Measurement of pressure drop The experimental set-up used for the study of the pressure drop across packed beds is as shown in Fig. 1. The packed bed is made of a perspex tube of an internal diameter of 50 mm. The desired packing is supported by a wire gauze fixed at the bottom of the tube. Two pressure tappings have been provided, one just below the wire gauze and the other 30 cm above it. The tube is gently and consistently filled up to a height of 50 cm with the desired packings, so that a stable and uniform
PR Pressure regulator PG Pressureguage FM Flowmeter
CV Controlvalve CZ Calmingzone
_ ~
9 PG
~C
PR
To atn~sphere
(a)Rig
(b) Externallygrooved
(c) Doublegrooved
Fig. 1. Experimentalset-up and cross-section of typical cylindricalparticles.
Pressure drop in packed-bed systems
197
Table 1. Properties o f beds o f equilateral solid cylinders Particle shape Solid Solid Solid Solid Solid
cylinder cylinder cylinder cylinder cylinder
dr~ (m)
(l/dp = (I/dp = (l/dp = (l/dp = (l/d o =
1) 1) 1) 1) 1)
3.00 4.00 6.00 10.0 14.0
x x x x x
10 -3 10 a 10 3 10 -3 10 -3
dt/d~
M e a n voidage
Range of Re'
16.67 12.50 8.33 5.00 3.57
0.0360 0.362 0.382 0.420 0.482
70~50 90--650 140-1000 250-1950 400-2800
Table 2. Properties o f beds o f non-equilateral solid cylinders Particle shape Solid Solid Solid Solid Solid Solid Solid Solid
cylinder cylinder cylinder cylinder cylinder cylinder cylinder cylinder
dp~ (m)
(l/dp = (l/dp = (l/dp = (l/dr, = (I/dp = (I/dp = (l/dp = (l/dp =
0.5) 0.5) 2.0) 2/3) 0.5) 1.5) 3/4) 2.0)
7.500 11.25 12.00 12.86 15.00 16.88 18.00 18.12
x x x x x x x x
10 -3 I0 - 3 10 3 10 3 10 -3 10 -3 10 -3 10 -3
dt/dw.
M e a n voidage
Range of Re'
6.67 4.44 4.17 3.89 3.33 2.96 2.78 2.76
0.438 0.448 0.450 0.468 0.490 0.491 0.494 0.495
200-1400 300-2120 320-2270 360-2520 440-3060 490-3450 530-3700 540-3740
bed is ensured on each occasion. The 20 cm of packing above the upper pressure tapping is designed to act as a calming zone for the air entering the column from the top at a regulated pressure of 70 psig and at room temperature. The flow of air is measured by means of two flowmeters connected in parallel and varied in order to cover a range of modified Reynolds number, Re', between 70 and 3740. The pressure drop is measured by means of a water manometer. Five beds of different particle sizes of equilateral solid cylinders made of glass, three further beds comprised of hollow cylinders, externally-grooved cylinders and double-grooved cylinders of alumina, as well as nine beds of different sizes of non-equilateral solid cylinders made of PVC have been studied. An externally grooved cylinder is a hollow cylinder with grooves on its outer side, as shown in Fig. 1. A hollow cylinder which has grooves on its inner, as well as outer, surface is called a double-grooved cylinder. It is mentionable that the equivalent diameter of particles, d~, is calculated using a~pe= 6;~/Sp, where Sp and Vp represent, respectively, the surface area and the volume of the particles. For greater accuracy and reproducibility, each experimental run has been repeated three times, where the deviations were found to be less than 5%. The results presented here are the average of three measurements. Details of the beds of solid cylinders and their properties are shown in Tables 1 and 2. 2.2. Measurement of mean voidage It is evident from the Ergun equation that the pressure drop in fixed beds is strongly related to the value of the mean voidage. Small differences in the values of mean voidage can give rise to big differences in the constants A and B of the modified Ergun equation. It is therefore important that the mean voidage of the bed is accurately k n o w n so that reliable pressure drop correlations can be formulated. In order to obtain such information, the mean voidage values of the beds considered here have been determined experimentally using the water-displacement procedure. Each bed is repacked at least three times and the mean values recorded. It has been found that the values of the mean voidage of repeated attempts varied by less than 5%, which is regarded as within acceptable limits. The data presented in Table 1 are the average values of the mean voidage. Particles made of porous alumina have been soaked overnight in water in order to be saturated, and only then are packed in the tube, in order to obtain the mean voidage of the desired beds. 3. RESULTS AND DISCUSSION 3.1. Effect of particle size Figures 2 and 3 show the experimentally obtained results of the pressure drop experiments conducted on equilateral solid cylinders. Figure 2 shows how particle size, d~ and particle Reynolds number, pud~/#, influence the pressure drop in packed beds. As expected, it is evident that, for a given bed, increasing the Reynolds number increases the pressure drop across the bed. It can also be seen that, under similar hydrodynamic conditions, decreasing the particle size or increasing
198
E.A. Foumeny et al. 5O 40
J~
Equilateral solid cylinder o dt/dpe- 16.67
9
I I ?
~30
,~ f
--
o dt/dpe- 12.50 A dt/dpe~ 8.33
?
~
[] dt/dpe - 5.00 • dt/dpe~3.57
/
m--m"r'm
0
200
400 600 800 1000 1200 1400 Particle Reynolds number (Rep)
Fig. 2. Dependency of pressure drop on diameter ratio and Reynolds number.
6° L
././.
5OOO 4OOO
j
m
./
3oo0 :
2o0o
.,~ / / /
../
Equilateral solid cylinder - o at/dpe-16.67 0 dt/dpe- 12.50
,.~//
.#,y/ 1000
dt/d-s.33
~ : -
n dt/dl~ ~ 5.00 • dt/dpe - 3.57 I
500
I
I
I
I
1000 1500 2000 2500 Modified Reynolds number (Re')
3000
Fig. 3. Plot of modified friction factor, f ' , vs modified Reynolds number, Re' (experimental).
the tube to particle diameter ratio results in a very large increase in the pressure drop. This is to be expected, since the mean voidage of the bed goes down significantly as the particle size decreases, thereby increasing the resistance to flow of the fluid. Furthermore, recent studies [7, 8] on beds of solid cylinders have clearly shown that, for a given tube diameter, the wall effect increases with increasing particle size. Thus, the fluid experiences more channelling in a bed of large-size particles than small ones and, therefore, provides a lower pressure drop. The displayed experimental results of Fig. 3 show the linear functionality between modified friction factor and modified Reynolds number. Inspection of the data reveals that the slope, B, of the lines directly increases with the diameter ratio, while the intercept, .4, varies randomly over a rather narrow range of values, as compared with the full range in the y-axis. It is further observed that the increase in the slope of the lines is not a linear function of the tube to particle diameter ratio but rather tends to a limiting value as dt/dp~ increases. The small variation in the intercept is believed to be due to the experimental and fitting errors, which tend to be more pronounced at extreme operating conditions. Such a behaviour was also observed by Foumeny et al. [4] in their study on spherical particles. Moreover, such errors can be treated statistically and optimisation routines may be used to overcome this. The optimum value of the intercept, based on the Marquardt non-linear regression algorithm [9], which uses the least-squares fit method, has been found to be equal to 211. Figure 4 shows the
Pressure drop in packed-bed systems
199
4~
"~ 2 -
1-
0
0
I
I
2
I
4
6
I
I
I
I
I
8 10 12 14 Diameter ratio (dt/dpe)
16
I
18
20
Fig. 4. Variation of slope with tube to particle diameter ratio for equilateral solid cylinders.
variation of slope in relation to d t/d~. From Fig. 4, it is evident that the experimental values of the slopes are not very much different from their corresponding fitted values. The non-linear nature of the relationship between slope B and dt/d w is also clearly seen and for this the Marquardt optimisation routine [9] has been used to obtain a functional description between B and dt/dr~. The type of function found to represent such a dependency best is mathematically described by the following equation: 5.265 7.047 B = 3.81 dt/d~ (dt/d~) 2" (4) Substituting for B and A in equation (2) results in a generalised pressure drop correlation applicable to packed beds of solid cylinders: APd2pE3 =(3.81 L#u(1 --E) 2 _
5 . 2 6 5 7.047 "~ pud v dt/dp~ ( d t ~ ) 5 ] / t ( i SE) 1-211.0.
(5)
The relationship between the modified friction factor and the modified Reynolds number, according to equation (5), is shown in Fig. 5. The trend again is very similar to that observed by Foumeny et al. [4], bearing in mind that this represents solid cylinders and not spheres. 6000
5OO0
,
4ooo 3ooo
/ p/ ./" f / , o /
2000 ~
'
/
/°./,./ ./ oO/*
/
Equilateral solid cylinder ................ dt/dpe ~ 16.67 . . . . . . . .
1000 ~**//-
500
I 1000
I 1500
dt/dpedt/dpe ~12.50~ 8.33 dt/dpe - 5.00 dt/dpe ~ 3.57
.... 0
/,'/"
I 2000
I 2500
3000
Modif'~d Reynolds number (Re') Fig. 5. Plot of modified friction factor, f ' , vs modified Reynolds number, Re' (optimised).
200
E . A . Foumeny et al. 4.0
3.5 3.0 .5
~
2.0 I
/
1.5 --
f
O ¢
O
1.0
0.50
I 2
0
I 4
I 6
1/d ~ 2.00 1/d ~ 3/4
A
lid
a
l i d ~ 0.50 l i d - 1.0 from equation 4
~ 213
I I I I 8 10 12 14 Diameter ratio (tit/ripe)
I 16
I 18
20
Fig. 6. Comparison between the optimised slopes for non-equilateral solid cylinders and the predictive correlation for equilateral solid cylinders.
In order to check the applicability of equation (5) to non-equilateral cylinders, the data of all non-equilateral cylinders have been subjected to the same optimisation exercise as the equilateral ones. For the range of particle aspect ratio, 0.5 ~
3.0[................ Sphere 2.5}-~ Solidcylinder [ /
0 ~-
Hollow cylinder Externally grooved '" Double grooved /
?
2.01.........0
.~1.0
~ 0.5 0
+°"*+'°'**°'°'+°*°°*" I 500 I000 1500 Particle Reynolds number (Rep) Fig. 7. Effect of catalyst shape on pressure drop.
2000
Pressure d r o p in p a c k e d - b e d systems
201
2.0 x 104
1.5x104 -
................ ----o---------0 ~
Sphere Solid cylinder Hollow cylinder Externally grooved Double grooved
/
o •~ l.Ox 104
~ 5 . 0 x 103
0
1000
2000 3000 4000 5000 Modified Reynolds number (Re')
6000
Fig. 8. R e l a t i o n s h i p between modified friction f a c t o r , f ' , and modified R e y n o l d s number, Re', for different particle shapes.
The sizes of hollow, externally grooved, and double-grooved cylinders and spheres are chosen, so as to approximate each other as well as at least one of the different sizes of solid cylinders considered. This investigation focuses its attention on particles which give a particle diameter of around 16 mm (based on equivalent volume) and their properties are listed in Table 3. It is apparent from Fig. 7 that, for the same value of Reynolds number, the pressure drop in a bed of spheres is much lower than that of other particle shapes, while that in the bed of solid cylinders is the highest. The pressure drops in beds made of other cylindrical shapes are very close to each other. From Table 3 it can be seen that the values of mean voidage of the beds of hollow, externally grooved and double-grooved cylinders are very close to each other and, hence, their pressure drop curves are also very close to each other. Solid cylinders give the highest pressure drop because the mean voidage of their bed has the lowest value amongst all cylindrical particles. This, however, does not explain the pressure drop characteristics of a bed of spheres. If the mean voidage of the bed were to be the only reason for the fluid to lose its pressure, then the energy losses, for the same value of Reynolds number, should be higher in beds of spheres than in other beds. The results actually go against this explanation. One of the possible reasons for the anomalous behaviour of beds of spherical particles could be the structure of the bed in terms of its tortuosity. It is possible that beds of cylindrical particles have a more tortuous structure than that of spherical particles. It is well known that flowing fluids lose their mechanical energy in the form of pressure when they follow a curved path. Cylinders can orientate themselves at various angles with respect to the axis of the bed, as well as to the horizontal base. Spheres, by virtue of their unique shape, are incapable of influencing the structure of the bed by their orientation. Moreover, frictional losses in bed of grooved cylinders could be higher than the rival shapes because of the surface irregularity of the particles themselves. It is important to note here that the pressure drop in beds of spheres was found to be much higher than those of hollow and grooved cylinders, when compared on the basis of flow-rate of the fluid. Hence, if spheres were to be replaced with either hollow or grooved cylinders and the reactor operated for the same throughput, then the significant savings in energy consumption could be made. The raw data obtained for beds of grooved and hollow cylinders, along with those of spheres, Table 3. Properties of beds of common catalyst shapes Shape Sphere Solid cylinder Hollow cylinder Externally-grooved cylinder Double-grooved cylinder
d~ (m) 16.00 x 14.00 × 10.54 × 6.71 x 6.36 x
10 -3 l0 -3 10 3 10 -3 10-3
dt/d ~
Mean voidage
Range of Re'
Surface/volume mm:/mm 3
3.13 3.57 4.74 7.45 7.86
0.461 0.482 0.613 0.666 0.650
500-3800 400-2800 560-4970 580--6000 430-5620
0.375 0.429 0.489 0.894 0.943
202
E.A. Foumeny et al.
are converted to a modified friction factor and a modified Reynolds number and compared with each other in Fig. 8. As per equation (2), each curve exhibits a linear relationship. Extensive investigation into each of the particle shapes, together with a wide range of tube to particle diameter ratios and Reynolds numbers, could then lead to an individual correlation similar to the one obtained for beds of spheres and solid cylinders. Such generalised predictive correlations would enable design engineers make comparison between the performance of various catalyst pellets at a very early stage in the development of fixed-bed reactors and make a choice depending upon their transport characteristics, namely heat and mass transfer coefficients [10]. 4.
CONCLUSIONS
Detailed study on beds of equilateral and non-equilateral solid cylinders has revealed the similarity between their qualitative behaviour and that of beds of spherical particles. The findings of this work have highlighted the inadequacy of the commonly used Ergun correlations. As a general rule, decreasing the particle size reduces the mean voidage of the bed and, thereby, increases the pressure drop across it. A predictive correlation for pressure drop in packed beds incorporating the size of solid cylindrical particles and their proportion to the tube diameter has been developed. It has been shown that the predictive correlation developed for equilateral solid cylinders can also be useful to determine the pressure drop across beds of non-equilateral cylinders for the range of tube to particle diameter ratios considered. In addition to this, the effect of particle shape on the energy losses has also been investigated. It has been found that mean voidage is not sufficient to characterise the structure of the bed and additional parameters, such as tortuosity, are required to explain the pressure drop characteristics of beds packed with different shapes. Acknowledgements--The authors are very grateful to the financial support provided by the University of Leeds (AK) and the Ministry of Higher Education of Iran (AV).
REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9. 10.
S. Ergun, Chem. Engng Prog. 48, 89 (1952). D. Handley and P. J. Heggs, Trans. Inst. Chem. Engrs 46, 251 (1968). F. MacDonald, M. S. El-Sayed, K. Movand and F. A. L. Dullien, Ind. Engng Chem. Fundam. 18, 199 (1979). E. A. Foumeny, F. Benyahia, J. A. Castro, H. A. Moallemi and S. Roshani, Int. J. Heat Mass Transfer 36, 536 (1993). Design information for packed bed towers. The Norton Company, U.S.A. (1977). E. A. Foumeny and S. Roshani, Chem. Engng Sci. 46, 2363 (1991). Y. Cohen and A. B. Metzner, A. L Chem. J. 27, 705 (1981). S. Roshani, Ph.D. Thesis, University of Leeds (1990). D. W. Marquardt, J. Soc. Ind. Appl. Math. 11, 431 (1963). E. A. Foumeny, P. J. Heggs and J. Ma, Heat Exchange Engineering: Design Aspects o f Fixed Beds, Vol. 6 (in press).
Applied Thermal Engineering Vol. 16, No. 3, pp. 195-202, 1996 Copyright © 1995 Elsevier Science Ltd 1359-4311(95)00002-X Printed in Great Britain. All rights reserved 1359-4311/96 $15.00 + 0.00
Pergamon
ELUCIDATION OF PRESSURE DROP IN PACKED-BED SYSTEMS E. A. Foumeny, A. Kulkarni, S. Roshani and A. Vatani Department of Chemical Engineering, University of Leeds, Leeds LS2 9JT, UK (Received for publication 29 June 1995)
Abstract--Studies on pressure-drop characteristics of beds of equilateral and non-equilateral solid cylinders of aspect ratios ranging from 0.5 to 2.0 have been carried out. These studies have exposed the shortcomings of the commonly used Ergun type correlations for predicting the pressure drop in packed systems. An empirical correlation to predict pressure drop in beds of cylindrical particles has been developed. In addition, derivatives of cylinders, such as hollow cylinders, have been examined, and comparisons have been made with particles, such as spheres. Keywords-~Packed bed, voidage, Reynolds number, friction factor, pressure drop. NOMENCLATURE A B dp dp~ dt f' l L AP Re' Sp u Vp
intercept, equation (2) slope, equation (2) particle diameter (m) equivalent diameter, 6Vo/S p (m) tube diameter (m) modifiedfriction factor length of solid cylinder (m) length of packed bed (m) pressuredrop (kPa/m) modifiedReynolds number surface area of particle (m2) superficialfluid velocity (m/s) volume of particle (m3)
Greek letters
E p #
mean voidage density of fluid (kg/m3) absolute viscosity of fluid (N/m.s)
1. I N T R O D U C T I O N
Energy loss, as characterised by a pressure drop of the process fluid, is an important consideration
in the design and operation of fixed-bed systems and has consequently been a subject of great interest for few decades. A vast amount of information in the form of empirical and semi-empirical correlations which relate the pressure d r o p to the h y d r o d y n a m i c conditions o f the packed beds is available. The E r g u n correlation [1], as given by equation (1), has been widely used for design or treatment o f raw experimental data extracted from certain geometrical configurations. E r g u n ' s correlation accounts for viscous and inertial energy losses and relates them to the dynamic variable, velocity o f the fluid, as well as the structure o f the bed, as characterised by the bed mean voidage, E. AP T
150/~u(1 - e)2 ~---
d2£ 2
1.75pu2(1 - E) ~
dp(5 3
(l)
In spite o f its general acceptability, m a n y workers have questioned the applicability o f the constants 150 and 1.75 for all types o f packing shapes. F o r example, Handley and Heggs [2] p r o p o s e d a correlaiton having constants 368 and 1.24, while M c D o n a l d et al. [3] rearranged the E r g u n equation in terms o f a linear relationship (2) between the modified friction factor, f ' , and the modified Reynolds number, Re', with constants A and B representing the intercept and the 195
E.A. Foumeny et al.
196
slope, respectively. They proposed a fixed value of 180 for A, and suggested that slope B varied between 1.8, for smooth particles, and 4.0, for rough particles. APd~c3
L#u(1--¢)2 +
B
pudp
+ A.
(2)
#(l-E)
The left-hand side of equation (2) is referred to as the modified friction factor, f ' , and the term pudv/l~(1 - E ) is the modified Reynolds number, Re'. The constants A and B could, hence, be related to certain bed characteristics, such as shape and size of the packed particles. Based on this approach, Foumeny et al. [4] carried out a series of experiments on beds of spheres. Their study covered a wide range of tube to particle diameter ratios, 3 ~< d,/dp <~24, and modified Reynolds number, 5 ~< Re' ~< 8500. Through an optimisation exercise they established a constant value for A of 130, while the slope B varied functionally in relation to the diameter ratio, dt/do, as represented by equation (3): f ' = BRe' + 130,
(3)
where e =
4/a 0 . 3 3 5 4 / ¢ + 2.28
.
It appears that for certain beds, dt/dp < 10, the total pressure drop across the bed can be greatly influenced by the diameter ratio, dt/dp. This finding clearly demonstrates the limitation of the Ergun equation. Although there are a number of correlations available for packings used for gas-liquid contacting operations [5], there is very little information available on the shapes of catalyst pellets employed in gas or vapour phase reactions. It is the aim of this work to carry-out pressure-drop experiments on commonly used particle shapes and to assess how shape and size influence the overall pressure losses of fixed beds. Since nearly two-thirds of the particles used in the industry are of a cylindrical nature [6], cylindrical pellets and their derivatives are considered in this work. 2. E X P E R I M E N T A L
2.1. Measurement of pressure drop The experimental set-up used for the study of the pressure drop across packed beds is as shown in Fig. 1. The packed bed is made of a perspex tube of an internal diameter of 50 mm. The desired packing is supported by a wire gauze fixed at the bottom of the tube. Two pressure tappings have been provided, one just below the wire gauze and the other 30 cm above it. The tube is gently and consistently filled up to a height of 50 cm with the desired packings, so that a stable and uniform
PR Pressure regulator PG Pressureguage FM Flowmeter
CV Controlvalve CZ Calmingzone
_ ~
9 PG
~C
PR
To atn~sphere
(a)Rig
(b) Externallygrooved
(c) Doublegrooved
Fig. 1. Experimentalset-up and cross-section of typical cylindricalparticles.
Pressure drop in packed-bed systems
197
Table 1. Properties o f beds o f equilateral solid cylinders Particle shape Solid Solid Solid Solid Solid
cylinder cylinder cylinder cylinder cylinder
dr~ (m)
(l/dp = (I/dp = (l/dp = (l/dp = (l/d o =
1) 1) 1) 1) 1)
3.00 4.00 6.00 10.0 14.0
x x x x x
10 -3 10 a 10 3 10 -3 10 -3
dt/d~
M e a n voidage
Range of Re'
16.67 12.50 8.33 5.00 3.57
0.0360 0.362 0.382 0.420 0.482
70~50 90--650 140-1000 250-1950 400-2800
Table 2. Properties o f beds o f non-equilateral solid cylinders Particle shape Solid Solid Solid Solid Solid Solid Solid Solid
cylinder cylinder cylinder cylinder cylinder cylinder cylinder cylinder
dp~ (m)
(l/dp = (l/dp = (l/dp = (l/dr, = (I/dp = (I/dp = (l/dp = (l/dp =
0.5) 0.5) 2.0) 2/3) 0.5) 1.5) 3/4) 2.0)
7.500 11.25 12.00 12.86 15.00 16.88 18.00 18.12
x x x x x x x x
10 -3 I0 - 3 10 3 10 3 10 -3 10 -3 10 -3 10 -3
dt/dw.
M e a n voidage
Range of Re'
6.67 4.44 4.17 3.89 3.33 2.96 2.78 2.76
0.438 0.448 0.450 0.468 0.490 0.491 0.494 0.495
200-1400 300-2120 320-2270 360-2520 440-3060 490-3450 530-3700 540-3740
bed is ensured on each occasion. The 20 cm of packing above the upper pressure tapping is designed to act as a calming zone for the air entering the column from the top at a regulated pressure of 70 psig and at room temperature. The flow of air is measured by means of two flowmeters connected in parallel and varied in order to cover a range of modified Reynolds number, Re', between 70 and 3740. The pressure drop is measured by means of a water manometer. Five beds of different particle sizes of equilateral solid cylinders made of glass, three further beds comprised of hollow cylinders, externally-grooved cylinders and double-grooved cylinders of alumina, as well as nine beds of different sizes of non-equilateral solid cylinders made of PVC have been studied. An externally grooved cylinder is a hollow cylinder with grooves on its outer side, as shown in Fig. 1. A hollow cylinder which has grooves on its inner, as well as outer, surface is called a double-grooved cylinder. It is mentionable that the equivalent diameter of particles, d~, is calculated using a~pe= 6;~/Sp, where Sp and Vp represent, respectively, the surface area and the volume of the particles. For greater accuracy and reproducibility, each experimental run has been repeated three times, where the deviations were found to be less than 5%. The results presented here are the average of three measurements. Details of the beds of solid cylinders and their properties are shown in Tables 1 and 2. 2.2. Measurement of mean voidage It is evident from the Ergun equation that the pressure drop in fixed beds is strongly related to the value of the mean voidage. Small differences in the values of mean voidage can give rise to big differences in the constants A and B of the modified Ergun equation. It is therefore important that the mean voidage of the bed is accurately k n o w n so that reliable pressure drop correlations can be formulated. In order to obtain such information, the mean voidage values of the beds considered here have been determined experimentally using the water-displacement procedure. Each bed is repacked at least three times and the mean values recorded. It has been found that the values of the mean voidage of repeated attempts varied by less than 5%, which is regarded as within acceptable limits. The data presented in Table 1 are the average values of the mean voidage. Particles made of porous alumina have been soaked overnight in water in order to be saturated, and only then are packed in the tube, in order to obtain the mean voidage of the desired beds. 3. RESULTS AND DISCUSSION 3.1. Effect of particle size Figures 2 and 3 show the experimentally obtained results of the pressure drop experiments conducted on equilateral solid cylinders. Figure 2 shows how particle size, d~ and particle Reynolds number, pud~/#, influence the pressure drop in packed beds. As expected, it is evident that, for a given bed, increasing the Reynolds number increases the pressure drop across the bed. It can also be seen that, under similar hydrodynamic conditions, decreasing the particle size or increasing
198
E.A. Foumeny et al. 5O 40
J~
Equilateral solid cylinder o dt/dpe- 16.67
9
I I ?
~30
,~ f
--
o dt/dpe- 12.50 A dt/dpe~ 8.33
?
~
[] dt/dpe - 5.00 • dt/dpe~3.57
/
m--m"r'm
0
200
400 600 800 1000 1200 1400 Particle Reynolds number (Rep)
Fig. 2. Dependency of pressure drop on diameter ratio and Reynolds number.
6° L
././.
5OOO 4OOO
j
m
./
3oo0 :
2o0o
.,~ / / /
../
Equilateral solid cylinder - o at/dpe-16.67 0 dt/dpe- 12.50
,.~//
.#,y/ 1000
dt/d-s.33
~ : -
n dt/dl~ ~ 5.00 • dt/dpe - 3.57 I
500
I
I
I
I
1000 1500 2000 2500 Modified Reynolds number (Re')
3000
Fig. 3. Plot of modified friction factor, f ' , vs modified Reynolds number, Re' (experimental).
the tube to particle diameter ratio results in a very large increase in the pressure drop. This is to be expected, since the mean voidage of the bed goes down significantly as the particle size decreases, thereby increasing the resistance to flow of the fluid. Furthermore, recent studies [7, 8] on beds of solid cylinders have clearly shown that, for a given tube diameter, the wall effect increases with increasing particle size. Thus, the fluid experiences more channelling in a bed of large-size particles than small ones and, therefore, provides a lower pressure drop. The displayed experimental results of Fig. 3 show the linear functionality between modified friction factor and modified Reynolds number. Inspection of the data reveals that the slope, B, of the lines directly increases with the diameter ratio, while the intercept, .4, varies randomly over a rather narrow range of values, as compared with the full range in the y-axis. It is further observed that the increase in the slope of the lines is not a linear function of the tube to particle diameter ratio but rather tends to a limiting value as dt/dp~ increases. The small variation in the intercept is believed to be due to the experimental and fitting errors, which tend to be more pronounced at extreme operating conditions. Such a behaviour was also observed by Foumeny et al. [4] in their study on spherical particles. Moreover, such errors can be treated statistically and optimisation routines may be used to overcome this. The optimum value of the intercept, based on the Marquardt non-linear regression algorithm [9], which uses the least-squares fit method, has been found to be equal to 211. Figure 4 shows the
Pressure drop in packed-bed systems
199
4~
"~ 2 -
1-
0
0
I
I
2
I
4
6
I
I
I
I
I
8 10 12 14 Diameter ratio (dt/dpe)
16
I
18
20
Fig. 4. Variation of slope with tube to particle diameter ratio for equilateral solid cylinders.
variation of slope in relation to d t/d~. From Fig. 4, it is evident that the experimental values of the slopes are not very much different from their corresponding fitted values. The non-linear nature of the relationship between slope B and dt/d w is also clearly seen and for this the Marquardt optimisation routine [9] has been used to obtain a functional description between B and dt/dr~. The type of function found to represent such a dependency best is mathematically described by the following equation: 5.265 7.047 B = 3.81 dt/d~ (dt/d~) 2" (4) Substituting for B and A in equation (2) results in a generalised pressure drop correlation applicable to packed beds of solid cylinders: APd2pE3 =(3.81 L#u(1 --E) 2 _
5 . 2 6 5 7.047 "~ pud v dt/dp~ ( d t ~ ) 5 ] / t ( i SE) 1-211.0.
(5)
The relationship between the modified friction factor and the modified Reynolds number, according to equation (5), is shown in Fig. 5. The trend again is very similar to that observed by Foumeny et al. [4], bearing in mind that this represents solid cylinders and not spheres. 6000
5OO0
,
4ooo 3ooo
/ p/ ./" f / , o /
2000 ~
'
/
/°./,./ ./ oO/*
/
Equilateral solid cylinder ................ dt/dpe ~ 16.67 . . . . . . . .
1000 ~**//-
500
I 1000
I 1500
dt/dpedt/dpe ~12.50~ 8.33 dt/dpe - 5.00 dt/dpe ~ 3.57
.... 0
/,'/"
I 2000
I 2500
3000
Modif'~d Reynolds number (Re') Fig. 5. Plot of modified friction factor, f ' , vs modified Reynolds number, Re' (optimised).
200
E . A . Foumeny et al. 4.0
3.5 3.0 .5
~
2.0 I
/
1.5 --
f
O ¢
O
1.0
0.50
I 2
0
I 4
I 6
1/d ~ 2.00 1/d ~ 3/4
A
lid
a
l i d ~ 0.50 l i d - 1.0 from equation 4
~ 213
I I I I 8 10 12 14 Diameter ratio (tit/ripe)
I 16
I 18
20
Fig. 6. Comparison between the optimised slopes for non-equilateral solid cylinders and the predictive correlation for equilateral solid cylinders.
In order to check the applicability of equation (5) to non-equilateral cylinders, the data of all non-equilateral cylinders have been subjected to the same optimisation exercise as the equilateral ones. For the range of particle aspect ratio, 0.5 ~
3.0[................ Sphere 2.5}-~ Solidcylinder [ /
0 ~-
Hollow cylinder Externally grooved '" Double grooved /
?
2.01.........0
.~1.0
~ 0.5 0
+°"*+'°'**°'°'+°*°°*" I 500 I000 1500 Particle Reynolds number (Rep) Fig. 7. Effect of catalyst shape on pressure drop.
2000
Pressure d r o p in p a c k e d - b e d systems
201
2.0 x 104
1.5x104 -
................ ----o---------0 ~
Sphere Solid cylinder Hollow cylinder Externally grooved Double grooved
/
o •~ l.Ox 104
~ 5 . 0 x 103
0
1000
2000 3000 4000 5000 Modified Reynolds number (Re')
6000
Fig. 8. R e l a t i o n s h i p between modified friction f a c t o r , f ' , and modified R e y n o l d s number, Re', for different particle shapes.
The sizes of hollow, externally grooved, and double-grooved cylinders and spheres are chosen, so as to approximate each other as well as at least one of the different sizes of solid cylinders considered. This investigation focuses its attention on particles which give a particle diameter of around 16 mm (based on equivalent volume) and their properties are listed in Table 3. It is apparent from Fig. 7 that, for the same value of Reynolds number, the pressure drop in a bed of spheres is much lower than that of other particle shapes, while that in the bed of solid cylinders is the highest. The pressure drops in beds made of other cylindrical shapes are very close to each other. From Table 3 it can be seen that the values of mean voidage of the beds of hollow, externally grooved and double-grooved cylinders are very close to each other and, hence, their pressure drop curves are also very close to each other. Solid cylinders give the highest pressure drop because the mean voidage of their bed has the lowest value amongst all cylindrical particles. This, however, does not explain the pressure drop characteristics of a bed of spheres. If the mean voidage of the bed were to be the only reason for the fluid to lose its pressure, then the energy losses, for the same value of Reynolds number, should be higher in beds of spheres than in other beds. The results actually go against this explanation. One of the possible reasons for the anomalous behaviour of beds of spherical particles could be the structure of the bed in terms of its tortuosity. It is possible that beds of cylindrical particles have a more tortuous structure than that of spherical particles. It is well known that flowing fluids lose their mechanical energy in the form of pressure when they follow a curved path. Cylinders can orientate themselves at various angles with respect to the axis of the bed, as well as to the horizontal base. Spheres, by virtue of their unique shape, are incapable of influencing the structure of the bed by their orientation. Moreover, frictional losses in bed of grooved cylinders could be higher than the rival shapes because of the surface irregularity of the particles themselves. It is important to note here that the pressure drop in beds of spheres was found to be much higher than those of hollow and grooved cylinders, when compared on the basis of flow-rate of the fluid. Hence, if spheres were to be replaced with either hollow or grooved cylinders and the reactor operated for the same throughput, then the significant savings in energy consumption could be made. The raw data obtained for beds of grooved and hollow cylinders, along with those of spheres, Table 3. Properties of beds of common catalyst shapes Shape Sphere Solid cylinder Hollow cylinder Externally-grooved cylinder Double-grooved cylinder
d~ (m) 16.00 x 14.00 × 10.54 × 6.71 x 6.36 x
10 -3 l0 -3 10 3 10 -3 10-3
dt/d ~
Mean voidage
Range of Re'
Surface/volume mm:/mm 3
3.13 3.57 4.74 7.45 7.86
0.461 0.482 0.613 0.666 0.650
500-3800 400-2800 560-4970 580--6000 430-5620
0.375 0.429 0.489 0.894 0.943
202
E.A. Foumeny et al.
are converted to a modified friction factor and a modified Reynolds number and compared with each other in Fig. 8. As per equation (2), each curve exhibits a linear relationship. Extensive investigation into each of the particle shapes, together with a wide range of tube to particle diameter ratios and Reynolds numbers, could then lead to an individual correlation similar to the one obtained for beds of spheres and solid cylinders. Such generalised predictive correlations would enable design engineers make comparison between the performance of various catalyst pellets at a very early stage in the development of fixed-bed reactors and make a choice depending upon their transport characteristics, namely heat and mass transfer coefficients [10]. 4.
CONCLUSIONS
Detailed study on beds of equilateral and non-equilateral solid cylinders has revealed the similarity between their qualitative behaviour and that of beds of spherical particles. The findings of this work have highlighted the inadequacy of the commonly used Ergun correlations. As a general rule, decreasing the particle size reduces the mean voidage of the bed and, thereby, increases the pressure drop across it. A predictive correlation for pressure drop in packed beds incorporating the size of solid cylindrical particles and their proportion to the tube diameter has been developed. It has been shown that the predictive correlation developed for equilateral solid cylinders can also be useful to determine the pressure drop across beds of non-equilateral cylinders for the range of tube to particle diameter ratios considered. In addition to this, the effect of particle shape on the energy losses has also been investigated. It has been found that mean voidage is not sufficient to characterise the structure of the bed and additional parameters, such as tortuosity, are required to explain the pressure drop characteristics of beds packed with different shapes. Acknowledgements--The authors are very grateful to the financial support provided by the University of Leeds (AK) and the Ministry of Higher Education of Iran (AV).
REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9. 10.
S. Ergun, Chem. Engng Prog. 48, 89 (1952). D. Handley and P. J. Heggs, Trans. Inst. Chem. Engrs 46, 251 (1968). F. MacDonald, M. S. El-Sayed, K. Movand and F. A. L. Dullien, Ind. Engng Chem. Fundam. 18, 199 (1979). E. A. Foumeny, F. Benyahia, J. A. Castro, H. A. Moallemi and S. Roshani, Int. J. Heat Mass Transfer 36, 536 (1993). Design information for packed bed towers. The Norton Company, U.S.A. (1977). E. A. Foumeny and S. Roshani, Chem. Engng Sci. 46, 2363 (1991). Y. Cohen and A. B. Metzner, A. L Chem. J. 27, 705 (1981). S. Roshani, Ph.D. Thesis, University of Leeds (1990). D. W. Marquardt, J. Soc. Ind. Appl. Math. 11, 431 (1963). E. A. Foumeny, P. J. Heggs and J. Ma, Heat Exchange Engineering: Design Aspects o f Fixed Beds, Vol. 6 (in press).