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Bed voidage in annular solid—liquid fluidized beds
Chemical Engineering and Processing. 31 (1992) 221-227
Bed voidage M. Jamialahmadi Department
of Chemical
in annular
221
solid-liquid
fluidized
beds
and H. Miiller-Steinhagen and Materials
Engineering,
The University
of Auckland.
Auckland
(New Zealand)
(Received November 7, 1991; in final form January 23, 1992)
Abstract Accurate prediction of the bed voidage is essential for the calculation of liquid side heat transfer coefficients in fluidized bed heat exchangers. New experimental results for the bed voidage in annuli arc presented. Various published correlations are tabulated and the inadequacy of most of these correlations is demonstrated. The most recently published correlation of Hirata and Bulos in conjunction with the Richardson-Zaki model predicts bed
voidage in annuli with a standard deviation of about 6%.
Introduction
In heat transfer processes where heavy fouling of the heat exchanger surfaces is expected, installation of a fluidized bed heat exchanger may be recommended [l-6]. It has been claimed that the fluidized bed heat exchanger remains totally clean, even under serious fouling conditions. Models available for the prediction of heat transfer coefficients from a wall to a solid-liquid fluidized bed indicate that heat transfer is a strong function of the bed voidage [7]. Therefore, accurate knowledge of bed voidage is crucial in the study of heat transfer and fouling phenomena associated with fluidized bed heat exchangers. Numerous correlations have been proposed in the literature for the velocity-voidage relationship observed during solid-liquid fluidization and during sedimentation. Table 1 summarizes most of the published equations and the conditions for which their application has been recommended. The model suggested by Jean and Fan [25], is based on fluid mechanics and limited to low Reynolds numbers. Richardson and Zaki’s model [14], is still the most widely used correlation for solid&liquid systems. Hirata and Bulos [26] developed an explicit equation for the prediction of bed voidage in solidliquid systems by modifying this model using their own experimental results and most previously published solid-liquid fluidization data. The majority of the presented correlations [8- 13, 15- 17, 23, 251 are purely empirical and apply only over a restricted range of Reynolds numbers or for a specific particle type. Wide discrepancies between the predictions of these models have been observed [22b Furthermore, all these models
0255-2701/92/$5.00
have been developed for tubular flow at ambient temperature; their applicability for annuli and for higher temperatures has not been verified yet. At the beginning of any investigations on heat transfer to fluidized beds it is, therefore, essential to determine which of these correlations can be used to predict the voidage accurately. The objective of the present investigation was to measure the fluidized bed voidage as a function of superficial liquid velocity for conditions similar to those found in test heaters for fouling investigations [27]. Those correlations which cover the entire range of Reynolds number encountered in this investigation were evaluated.
Experimental
equipment
A schematic drawing of the experimental set-up used in this study is shown in Fig. 1. The set-up consists of fluidized bed column, water pump, supply tank, liquid flow meters and control valves. Two annular columns with hydraulic diameters of 1.4 cm and 4.1 cm respectively, and about 60 cm height have been used. The columns, which were made from Perspex, consisted of calming, test and expansion sections. A stainless streel screen fitted above the calming section of the bed supports the solid particles. The liquid passing through the column and the return lines is discharged into the supply tank. A by-pass line is installed for control purposes. Initially, water at 22 “C was used to fluidize glass beads and hemi-cylindrical stainless steel particles, with properties summarized in Table 2.
0
1992 -
Elsevier Sequoia. All rights reserved
222 TABLE
1. Published
correlations
Author
for the bed expansion
of liquid
fluidized
Equation
Steinour
[8]
Lewis et al.
+=s2exp[-4.19(1
--E)]
t
Us, _=$,a= U,
[lo]
2.5( 1 -E)
Us,
[ 1 l]
Hawksley
( - 1 - 39( 1 - e)/64 >
-K=“2exp -Us, = 85.6 U,
[ 121
Jottrand
Lewis and Bowerman
[ 131
Richardson
[ 141
and Zaki
beds Range
[ 91
Brinkman
of applicability
Type of equation
Re, c 0.2; E -Z 0.85
Semitheoretical
Re,<2
Theoretical
1.1
Empirical
0.001 < Re, i 58
Semitheoretical
0.001 < Re, < 0.4
Empirical
2 < Re, i 500
Empirical
Semitheoretical where n n n n n
Ham1
characteristics
= 4.65 + 2OdJD = (4.4 + 18dJD) = (4.4 + 18dJD) = 4.4 Re,$-’ =2.4
3-4.5(1
[I51
u,, ci,=
and Ruth
us, u,
Re, < 0.2 0.2 < Re, < 1 1 500
R~z-O-~~ Re,$-’
-&)“3f4.5(1
-&)5’3-3(1
-&)Z
Re, < 0.2
Theoretical
Re, i 0.5
Semitheoretical
+(I-2.15(1-e))[l-0.75(1-e)“‘]
Re, < 0.4
Semitheoretical
c4,’Ga = 18 Re + 2.7 Re’ 687
0.01 < Re < lo4
Empirical
3 + 2( 1 - &)I’) E3
Loeffler
Oliver
[ 171
Letan
-z
5.7 +&
,
[ 181
Wen and Yu Barnea
[ 161
1-E
and Mizrahi
[19]
4d&, - PJg 3PUZ2
Wen and Fan [Zl]
[221
4.8 = 0.63 + ~ &
Ga
Ga004b3($,--1)=~~3026[l
03W4,
B-A
u
1) =
C$,‘-““[ 1 + 1.5 x 10~%$,3.‘0*]
[231
Empirical
1.5 < Re, < 2200
Semitheoretical
18
Empirical
Ga > 10s
Empirical
=O.O6Re,‘+“’
U& for E =C0.85 for E z 0.85
where B = 0.8~‘.~s B = s2.65
Ganguly
+ 1.13 x 10--1&4”25]
13.9 Re’+’
Us, _s4 14 Z$
and
Al-Dibouni
1
0.687s 1.033
??$ GaO
Garside
-&)I’3
( 1 + 0. I5 Re,O 687)~4.5
U,r -= U,
[20]
E
1 f(l
L, =
where
1.27W Ip,DZ(l - 1.762U +0.95UZ)] U = w t
m
0. I2 < Re, < 43.66
Empirical
223
TABLE 1. (Continued) Author
Equation
Range of applicability
Type of equation
Foscolo et al. [24]
&4 USI u, 4(1 -zE)+c’
Re, i 0.2
Theoretical
USI -= U
0.0777 Re,( 1 + 0.01941 Re,)c4# - 1 0.0388 Re,
0.2 < Re, < 500
VT!
Re, > 500
-=
Jean and Fan [25]
-= u,
Hirata and Bulos [26]
E
3-4.5(1
=++(l
--)“Sf4.5(1 -@‘3-3(1 3+2(1 -+a
-epk)sRZexp(B(l
-E)2
-Ebb))
Re, < 0.2
Theoretical
As Richardson and Zaki [14]
Empirical
where A = 2.2n + % and
B =2.ln
While most of the experiments have been performed with this set-up, the runs at higher temperatures have been performed with a fluidized bed test section which has been installed in a stainless steel heat transfer fouling test apparatus [27].
Experimental
P
procedure and results
The bed voidage is defined as
flowmeters
V, “‘jTyq$’
Hzdh2 - 4V, Hnd,’
(1)
For a given number of particles, therefore, only the bed height has to be measured. For each particle size and for each column, experiments were performed using 40- 160 g of particles. Expanded bed height data were
drain
Fig. 1. Schematic diagram of experimental set-up.
TABLE 2. Properties of particles
d,(cm)
Mdcm3)
%k
0.31 0.20 0.196
7.9 2.5 7.9
0.45 0.41 0.40
obtained using both increasing and decreasing liquid flow rates to investigate the possibility of hysteresis effects. The bed height for a liquid rate of 0 kg/s is recorded as the static bed height. Static bed voidages were also measured separately for each particle type in both columns. Most experiments were carried out in an arbitrary sequence and some runs repeated to check the reproducibility of the experiments, which proved to be very good. Figure 2 shows the bed voidage measured with stainless steel particles in the two different columns, for superficial liquid velocities ranging from 0 to 50 cm/s. Typical velocity-voidage results for the glass beads are shown in Fig. 3. The general shape of, bed voidage versus superficial liquid velocity curves is characterized by a gradual increase in voidage, followed by a sharp
224
Comparison of Figs. 2 and 3 shows that the hydraulic diameter does not have a significant effect on the bed voidage of dense stainless steel particles while it has a profound effect on light glass particles. This is because the particle Reynolds numbers for stainless steel particles were always greater than 500; in this regime the column diameter is not important any more
a, 0.8 : m ‘2
0.7
B m
06
[141.
Theoretical
Superficial
Fig. 2. Bed voidage steel particles.
#iquid
velocity
as a function
Us1
of superficial
considerations
The equation proposed by Richardson and Zaki [ 141, which is given in detail in Table 1, is the most widely used correlation for solid-liquid systems.
Cm/S
liquid velocity
for
The exponent n is given in Table 1 as a function particle Reynolds number, Re,,
.,L
The Richardson and Zaki model has been criticized because of the discontinuities which exist at the transition points between the different Re, ranges. Furthermore, the Richardson and Zaki model is not a function of the static bed voidage and, therefore, predicts zero voidage at zero liquid velocity. Most recently Hirata and Bulos [26] corrected the above two disadvantages of the Richardson and Zaki model, suggesting the following correlation for the e&imation of the voidage of solid-liquid systems:
0.5
0
5
Superficial
Fig. 3. Bed voidage glass particles.
10
liquid
15
velocity
as a function
20
USI
of superficial
25
cm/s
liquid velocity
of the
& = Epk+ ( 1 for
E&E;,
eB(’
- ?G)
A = 2.2n + f$ increase and a subsequent gradual increase towards an asymptotic value of one. When a fluid is passed slowly upwards through a bed of granular solids, the bed initially remains static. If the velocity is increased, a stage is reached when the particles re-orientate themselves and present a greater crosssectional area to the flow of fluid; this re-adjustment continues until the ‘loosest’ stable arrangement is attained. With further increase, the particles are individually supported by the fluid, the bed becomes fluidized and the bed voidage starts to increase. For velocities greater than the minimum velocity required to initiate fluidization, the bed expands sharply and the particles remain uniformly dispersed in the liquid. The degree of agitation becomes greater, but a sharp interface is maintained between the bed and the clear liquid above the suspension.
(4)
Where (5)
and B = 2.ln
(6)
.spk is the packed bed or static bed voidage, snz is the bed voidage based on the Richardson and Zaki equation and II is the Richardson-Zaki exponent. Hirata and Bulos [26] suggested that the exponent n be calculated from the following continuous empirical equation proposed by Rowe [28] 2( 2.35 + 0.175 Re,0.75) ’ =
(1 +0.175Re:-75)
All correlations given in Table 1, including the Richardson and Zaki model [14], have been presented in a form where the ratio of superficial liquid velocity to terminal velocity of the particles is expressed as a
225
function of the bed voidage, the Reynolds number at the terminal velocity and several physical properties. The prediction of the bed voidage, therefore, necessitates the prediction of the terminal velocity of the particles corrected for the wall effect.
Where P(C) = (( 0.0017795c
- 0.0573)C + 1.03 15)C
- 1.26222
(15)
R(C) = 0.99947 + 0.01853 sin( 1.848C - 3.14)
(16)
and
Particle terminal velocity corrected for wall effect
C = log,, Ar The terminal velocity of a single particle corrected for the wall effect can be calculated from the following equation proposed by Richardson and Zaki [ 141:
In equation ( 17) Ar is the Archimedes defined by Hartman ef al. [31] as: Ar
log,,
+
=s
=
m
2
d,‘m(pcPI2
(\ d,(Ps- P,lS ‘I2 1
I7
(10)
j
jLDPl
/
As can be seen from equation (IO), the free-falling velocity of the particles is a function of the drag coefficient which is generally given in graphical form [29] as a complex function of the flow conditions. Lydersen [30] fitted these curves for spherical particles to a series of straight lines: for < Re, < 200,000, 500 2.0 < Re, < 500.0, 0.00001 < Re, < 2.0, The above correlations
co = 0.44 C, = 18.5 Re,-0-6 Co = 24.0 Re,-’
and equation
number
which is
PI1
(18)
The terminal velocities predicted using the method suggested by Hartman et al. [31] agree very well with those calculated from the standard equations of Lydersen [30]. For the results which are presented here, the terminal velocities of the particles are, therefore, calculated with the Hartman et al. correlation. Comparison
Therefore, u
=
(8)
( t> Where U, is the terminal velocity or free-fall velocity of spheres in an infinite medium, which can be obtained by equating the viscous drag force to the effective gravitational force:
(17)
of measured
and predicted res$ts
Inspection of the correlations given in Table 1 revealed wide discrepancies between the predictions of several of these correlations and our experimental results for the voidage as a function of velocity. However, excellent agreement between measured and predicted data was obtained with the Hirata and Bulos [26] model in conjunction with the Richardson and Zaki [ 141 exponent, n, which is given in Table 1. Figures 4 and 5 show the comparison between the experimental results and the predictions of the two models suggested by Richardson and Zaki [ 141 and by Hirata and Bulos [26], for glass spheres and stainless For both particles, the steel cylinders, respectively.
(10) give:
for 500.0 < Re, < 200,000
Stainless
(11)
steel
particles
0.9
for 2.0 < Re, < 500.0 Um =
(12)
for 0.00001 < Re, < 2.0
(” ;,p’)g
U, = 0.0556dp2
(13)
Recently, Hartman et al. [31] proposed the following explicit empirical expression for the free-fall velocities of spheres: log,, Re, = P(C) + log,, R(C)
(14)
50
40
0
SupeFficiol
li$id
Fig. 4. Measured and predicted superficial liquid velocity.
veldcqty
U sI
bed voidage
cm/s
as a function
of the
226
Symbols
0.4 0
5uper5ficial
10 15 liquid velocity USI
Fig. 5. Measured and predicted superficial liquid velocity.
bed voidage
20
25
0.5
as a function
of the
Fig.
and
predicted
bed
voidage,
Nomenclature
4 4
Ga g H L 0.5
> R 0.7
Experimental
6. Comparison
1
The authors are indebted to Dipl.-Ing. Bengt Stellingwerf for his assistance in obtaining the experimental data.
CLJ D
T = 22 “C.
0.9
Voidage
Acknowledgements
B C
1
Fig.
of measured
2.52 7.9 79
Additional experimental data have been obtained for the annular fluidized bed heat exchanger at a bulk temperature of 95 “C. These results are compared in Fig. 7 with the predictions of eqn. 4. The general agreement between measured and calculated values is good.
Ar
0.6
7. Comparison
0.8
(9&,3,
T = 95 “C.
A
05
07
Experimental
Richardson and Zaki model underpredicts the experimental results, considerably. For low flow velocity, this model does not predict the static bed voidage. The Hirata and Bulos correlation [26] in conjunction with eqns. (14-18) and the Richardson-Zaki exponent given in Table 1, on the other hand, predicts the experimental results with an average deviation of only about 6%. All measured bed voidages for the investigated solid particles with different densities and sizes are compared with the values predicted by the Hirata and Bulos correlation [26] in Fig. 6. Considering the errors involved in the experimental data and the approximate nature ‘of the correlations itself, the root mean square error of 6% is an excellent indication that the recommended procedure is suitable for the prediction of the bed voidage in annular heat exchangers.
0.4 0.4
0.6
cm/s
$?,,
o’Gloss 0.2 0 Steel 0.196 A: Stee10.310 “‘I’,,“““,“““”
of measured
0.8
0.9
1
voidage
and
predicted
bed
voidage,
Re, Ret Re,
parameter defined in eqn. (5) Archimedes number parameter defined in eqn. (6) parameter defined in eqn. ( 17) drag coefficient column diameter, cm hydraulic diameter, cm equivalent particle diameter, cm Galilei number acceleration due to gravity, m ss2 bed height, cm height of expanded bed, m Richardson-Zaki exponent parameter defined in eqn. ( 15) parameter defined in eqn. ( 16) particle Reynolds number particle Reynolds number at terminal velocity =Re,/exp[S( 1 - E)/~E] Reynolds number defined in ref. 19
227
minimum fluidization velocity, cm s-’ superficial liquid velocity, cm s ’ particle terminal velocity corrected for wall effect, cm ss’ particle terminal velocity in an infinite fluid, cm SC’ volume of liquid, m3 volume of particles, m3 mass of feed, kg bed voidage static bed voidage bed voidage calculated from Richardson-Zaki equation liquid viscosity, g cm-’ s-’ liquid density, g crnd3 solid particle density, g crne3 =[ 1 - 1.21( 1 - E)*/~] -’ dimensionless parameter
References 1 L. P. Hatch and G. G. Weth, Scale control in high temperature distillation utilizing fluidized bed heat exchangers. Research and Development Progress Rept. No. 571, 1910. 2 C. A. Allen and E. S. Grimmett, Liquid fluidized bed heat exchanger design parameters. Department of Energy, Idaho Operations Ofice, contract I-(322)-1570, April 1978. Development of a vertical flash evaporator, 3 D. G. Klaren, Ph.D Thesis, Delft University of Technology, 1975. of’ calcium sulphate scale by a 4 J. A. M. Meijer, Inhibition fluidized bed, Ph.D. Thesis, Delft University of Technology, 1984. Principles and 5 D. G. Klaren, The fluid bed heat exchanger: modes of operation and heat transfer results under severe fouling conditions, Fouling Prev. Res. Dig., 5 ( 1) (1983). and J. Kollbach, New developments in 6 R. Rautenbach fluidized bed heat transfer for preventing fouling, Swiss Chem. 8 (5) (1986) 47 55. 7 J. Kollbach, W. Dahm and R. Rautenbach, Continuous cleaning of heat exchangers with recirculating fluidized bed, Heat Transfer Eng. 8 (4) ( 1987) 26-32. nonflocculated suspen8 J. H. Steinour, Rate of sedimentation: sions of uniform spheres, Ind. Eng. Chem, 36 ( 1944) 618 -624. A calculation of the viscous force exerted by 9 H. C. Brinkman, a flowing fluid on a dense swarm of particles, Appl. Sci. Res. Al (1947) 27-34. of IO W. K. Lewis, E. R. Gilliland and W. Bauer, Characteristics lluidized particles, Ind. Eng. Chem. 41 (1949) 1104- 1117. I1 P. G. W. Hawksley, Some aspects of fluid flow. Paper 7, Institute of Physics and E. Arnold, London, 1951.
12 R. Jottrand, An experimental study of the mechanism of fluidization. J. Appl. Chem., 2 (1952) 17-22. I3 E. W. Lewis and E. W. Bowerman, Fluidization of Solid Particles in Liquids. Chem. Eng. Prog., 48 (1952) 603-61 I. 14 .I. F. Richardson and W. N. Zaki, Sedimentation and Fluidization. Trans. Inst. Chem. Eng. 32 (1954) 35-53. 15 J. Happel, Viscous llow in multiparticle systems: Slow motion of fluids relative to beds of spherical particles and particulate Ruidization and sedimentation of spheres. AIChE. J., 4 (1958) 197-201. 16 A. L. Loelller and B. F. Ruth, Particulate fluidization and sedimentation of spheres. AIChE. J., 5 (1959) 310-315. 17 II. R. Oliver, The sedimentation of suspensions of closely sized spherical particles. Chem. Eng. Sci., 15 (1961) 230-242. I8 (7. Wen and Y. H. Yu, Chem. Eng. Prog. Symp. Ser., 62 (62) (1966) 100-105. I9 E. Barnea and J. Mizrahi, A generalized approach to the fluid dynamics of particulate systems. Part I: General correlation for fluidization and sedimentation in solid multiparticle systems, Chem. Eng. J., 5 (1973) 171- 189. 20 R. Letan, On vertical dispersion two-phase flow, Chem. Eng. Sri., 29 ( 1974) 62 l-624. 21 C. Y. Wen and L. S. Fan, Some remarks on the correlation of bed expansion in liquid-solid fluidized beds, fnd. Eng. Chem. Process Des. Dew, 13 (1974) 194-198. 22 J. Garside and M. R. Al-Dibouni, Velocity-voidage relationships for fluidization and sedimentation in solid-liquid systems, Ind. Eng. Chem. Process Des. Dev., 16 (1977) 206213. 23 LJ. P. Ganguly, Direct method for the prediction of expanded bed height in liquid-solid fluidization. Can. J. Chem. Eng., 58 (1980) 559-563. 24 P. U. Foscolo, L. G. Gibilaro and S. P. Waldram, A unified model for particulate expansion of fluidized beds and flow in fixed porous media, Chem. Eng. Sci., 38 (1983) 1251-1260. 25 R. H. Jean and L. S. Fan, A fluid mechanic-based model for sedimentation and fluidization at low Reynolds numbers, Chem. Eng. Sci., 44 (1989) 353-362. 26 A. Hirata and F. B. Bulos, Predicting bed voidage in solidliquid fluidization, J. Chem. Eng. Japan, 23 (1990) 599-604. 27 R. Blijchl and H. M. Miiller-Steinhagen, Influence of particle size and particle/liquid combination on particulate fouling in heat exchangers, Can. J. Chem. Eng., 68 (4) (1990) 585591. 28 P. N. Rowe, A convenient empirical equation for estimation of the Richardson-Zaki exponent, Chem. Eng. Sci., 24 (1987) 2795-2796. 29 H. Schlichting, Boundary /ayer theory, 6th ed., McGraw-Hill, New York, 1968. 30 A. K. Lydersen, Fluid flow and heat transfer, Wiley, New York, 1989. 31 M. Hartman, V. Havlin, 0. Tronka and W. Carsky, Predicting the free-fall velocity of spheres, Chem. Eng. Sci., 44 (1989) 1743-1745.
Bed voidage M. Jamialahmadi Department
of Chemical
in annular
221
solid-liquid
fluidized
beds
and H. Miiller-Steinhagen and Materials
Engineering,
The University
of Auckland.
Auckland
(New Zealand)
(Received November 7, 1991; in final form January 23, 1992)
Abstract Accurate prediction of the bed voidage is essential for the calculation of liquid side heat transfer coefficients in fluidized bed heat exchangers. New experimental results for the bed voidage in annuli arc presented. Various published correlations are tabulated and the inadequacy of most of these correlations is demonstrated. The most recently published correlation of Hirata and Bulos in conjunction with the Richardson-Zaki model predicts bed
voidage in annuli with a standard deviation of about 6%.
Introduction
In heat transfer processes where heavy fouling of the heat exchanger surfaces is expected, installation of a fluidized bed heat exchanger may be recommended [l-6]. It has been claimed that the fluidized bed heat exchanger remains totally clean, even under serious fouling conditions. Models available for the prediction of heat transfer coefficients from a wall to a solid-liquid fluidized bed indicate that heat transfer is a strong function of the bed voidage [7]. Therefore, accurate knowledge of bed voidage is crucial in the study of heat transfer and fouling phenomena associated with fluidized bed heat exchangers. Numerous correlations have been proposed in the literature for the velocity-voidage relationship observed during solid-liquid fluidization and during sedimentation. Table 1 summarizes most of the published equations and the conditions for which their application has been recommended. The model suggested by Jean and Fan [25], is based on fluid mechanics and limited to low Reynolds numbers. Richardson and Zaki’s model [14], is still the most widely used correlation for solid&liquid systems. Hirata and Bulos [26] developed an explicit equation for the prediction of bed voidage in solidliquid systems by modifying this model using their own experimental results and most previously published solid-liquid fluidization data. The majority of the presented correlations [8- 13, 15- 17, 23, 251 are purely empirical and apply only over a restricted range of Reynolds numbers or for a specific particle type. Wide discrepancies between the predictions of these models have been observed [22b Furthermore, all these models
0255-2701/92/$5.00
have been developed for tubular flow at ambient temperature; their applicability for annuli and for higher temperatures has not been verified yet. At the beginning of any investigations on heat transfer to fluidized beds it is, therefore, essential to determine which of these correlations can be used to predict the voidage accurately. The objective of the present investigation was to measure the fluidized bed voidage as a function of superficial liquid velocity for conditions similar to those found in test heaters for fouling investigations [27]. Those correlations which cover the entire range of Reynolds number encountered in this investigation were evaluated.
Experimental
equipment
A schematic drawing of the experimental set-up used in this study is shown in Fig. 1. The set-up consists of fluidized bed column, water pump, supply tank, liquid flow meters and control valves. Two annular columns with hydraulic diameters of 1.4 cm and 4.1 cm respectively, and about 60 cm height have been used. The columns, which were made from Perspex, consisted of calming, test and expansion sections. A stainless streel screen fitted above the calming section of the bed supports the solid particles. The liquid passing through the column and the return lines is discharged into the supply tank. A by-pass line is installed for control purposes. Initially, water at 22 “C was used to fluidize glass beads and hemi-cylindrical stainless steel particles, with properties summarized in Table 2.
0
1992 -
Elsevier Sequoia. All rights reserved
222 TABLE
1. Published
correlations
Author
for the bed expansion
of liquid
fluidized
Equation
Steinour
[8]
Lewis et al.
+=s2exp[-4.19(1
--E)]
t
Us, _=$,a= U,
[lo]
2.5( 1 -E)
Us,
[ 1 l]
Hawksley
( - 1 - 39( 1 - e)/64 >
-K=“2exp -Us, = 85.6 U,
[ 121
Jottrand
Lewis and Bowerman
[ 131
Richardson
[ 141
and Zaki
beds Range
[ 91
Brinkman
of applicability
Type of equation
Re, c 0.2; E -Z 0.85
Semitheoretical
Re,<2
Theoretical
1.1
Empirical
0.001 < Re, i 58
Semitheoretical
0.001 < Re, < 0.4
Empirical
2 < Re, i 500
Empirical
Semitheoretical where n n n n n
Ham1
characteristics
= 4.65 + 2OdJD = (4.4 + 18dJD) = (4.4 + 18dJD) = 4.4 Re,$-’ =2.4
3-4.5(1
[I51
u,, ci,=
and Ruth
us, u,
Re, < 0.2 0.2 < Re, < 1 1
R~z-O-~~ Re,$-’
-&)“3f4.5(1
-&)5’3-3(1
-&)Z
Re, < 0.2
Theoretical
Re, i 0.5
Semitheoretical
+(I-2.15(1-e))[l-0.75(1-e)“‘]
Re, < 0.4
Semitheoretical
c4,’Ga = 18 Re + 2.7 Re’ 687
0.01 < Re < lo4
Empirical
3 + 2( 1 - &)I’) E3
Loeffler
Oliver
[ 171
Letan
-z
5.7 +&
,
[ 181
Wen and Yu Barnea
[ 161
1-E
and Mizrahi
[19]
4d&, - PJg 3PUZ2
Wen and Fan [Zl]
[221
4.8 = 0.63 + ~ &
Ga
Ga004b3($,--1)=~~3026[l
03W4,
B-A
u
1) =
C$,‘-““[ 1 + 1.5 x 10~%$,3.‘0*]
[231
Empirical
1.5 < Re, < 2200
Semitheoretical
18
Empirical
Ga > 10s
Empirical
=O.O6Re,‘+“’
U& for E =C0.85 for E z 0.85
where B = 0.8~‘.~s B = s2.65
Ganguly
+ 1.13 x 10--1&4”25]
13.9 Re’+’
Us, _s4 14 Z$
and
Al-Dibouni
1
0.687s 1.033
??$ GaO
Garside
-&)I’3
( 1 + 0. I5 Re,O 687)~4.5
U,r -= U,
[20]
E
1 f(l
L, =
where
1.27W Ip,DZ(l - 1.762U +0.95UZ)] U = w t
m
0. I2 < Re, < 43.66
Empirical
223
TABLE 1. (Continued) Author
Equation
Range of applicability
Type of equation
Foscolo et al. [24]
&4 USI u, 4(1 -zE)+c’
Re, i 0.2
Theoretical
USI -= U
0.0777 Re,( 1 + 0.01941 Re,)c4# - 1 0.0388 Re,
0.2 < Re, < 500
VT!
Re, > 500
-=
Jean and Fan [25]
-= u,
Hirata and Bulos [26]
E
3-4.5(1
=++(l
--)“Sf4.5(1 -@‘3-3(1 3+2(1 -+a
-epk)sRZexp(B(l
-E)2
-Ebb))
Re, < 0.2
Theoretical
As Richardson and Zaki [14]
Empirical
where A = 2.2n + % and
B =2.ln
While most of the experiments have been performed with this set-up, the runs at higher temperatures have been performed with a fluidized bed test section which has been installed in a stainless steel heat transfer fouling test apparatus [27].
Experimental
P
procedure and results
The bed voidage is defined as
flowmeters
V, “‘jTyq$’
Hzdh2 - 4V, Hnd,’
(1)
For a given number of particles, therefore, only the bed height has to be measured. For each particle size and for each column, experiments were performed using 40- 160 g of particles. Expanded bed height data were
drain
Fig. 1. Schematic diagram of experimental set-up.
TABLE 2. Properties of particles
d,(cm)
Mdcm3)
%k
0.31 0.20 0.196
7.9 2.5 7.9
0.45 0.41 0.40
obtained using both increasing and decreasing liquid flow rates to investigate the possibility of hysteresis effects. The bed height for a liquid rate of 0 kg/s is recorded as the static bed height. Static bed voidages were also measured separately for each particle type in both columns. Most experiments were carried out in an arbitrary sequence and some runs repeated to check the reproducibility of the experiments, which proved to be very good. Figure 2 shows the bed voidage measured with stainless steel particles in the two different columns, for superficial liquid velocities ranging from 0 to 50 cm/s. Typical velocity-voidage results for the glass beads are shown in Fig. 3. The general shape of, bed voidage versus superficial liquid velocity curves is characterized by a gradual increase in voidage, followed by a sharp
224
Comparison of Figs. 2 and 3 shows that the hydraulic diameter does not have a significant effect on the bed voidage of dense stainless steel particles while it has a profound effect on light glass particles. This is because the particle Reynolds numbers for stainless steel particles were always greater than 500; in this regime the column diameter is not important any more
a, 0.8 : m ‘2
0.7
B m
06
[141.
Theoretical
Superficial
Fig. 2. Bed voidage steel particles.
#iquid
velocity
as a function
Us1
of superficial
considerations
The equation proposed by Richardson and Zaki [ 141, which is given in detail in Table 1, is the most widely used correlation for solid-liquid systems.
Cm/S
liquid velocity
for
The exponent n is given in Table 1 as a function particle Reynolds number, Re,,
.,L
The Richardson and Zaki model has been criticized because of the discontinuities which exist at the transition points between the different Re, ranges. Furthermore, the Richardson and Zaki model is not a function of the static bed voidage and, therefore, predicts zero voidage at zero liquid velocity. Most recently Hirata and Bulos [26] corrected the above two disadvantages of the Richardson and Zaki model, suggesting the following correlation for the e&imation of the voidage of solid-liquid systems:
0.5
0
5
Superficial
Fig. 3. Bed voidage glass particles.
10
liquid
15
velocity
as a function
20
USI
of superficial
25
cm/s
liquid velocity
of the
& = Epk+ ( 1 for
E&E;,
eB(’
- ?G)
A = 2.2n + f$ increase and a subsequent gradual increase towards an asymptotic value of one. When a fluid is passed slowly upwards through a bed of granular solids, the bed initially remains static. If the velocity is increased, a stage is reached when the particles re-orientate themselves and present a greater crosssectional area to the flow of fluid; this re-adjustment continues until the ‘loosest’ stable arrangement is attained. With further increase, the particles are individually supported by the fluid, the bed becomes fluidized and the bed voidage starts to increase. For velocities greater than the minimum velocity required to initiate fluidization, the bed expands sharply and the particles remain uniformly dispersed in the liquid. The degree of agitation becomes greater, but a sharp interface is maintained between the bed and the clear liquid above the suspension.
(4)
Where (5)
and B = 2.ln
(6)
.spk is the packed bed or static bed voidage, snz is the bed voidage based on the Richardson and Zaki equation and II is the Richardson-Zaki exponent. Hirata and Bulos [26] suggested that the exponent n be calculated from the following continuous empirical equation proposed by Rowe [28] 2( 2.35 + 0.175 Re,0.75) ’ =
(1 +0.175Re:-75)
All correlations given in Table 1, including the Richardson and Zaki model [14], have been presented in a form where the ratio of superficial liquid velocity to terminal velocity of the particles is expressed as a
225
function of the bed voidage, the Reynolds number at the terminal velocity and several physical properties. The prediction of the bed voidage, therefore, necessitates the prediction of the terminal velocity of the particles corrected for the wall effect.
Where P(C) = (( 0.0017795c
- 0.0573)C + 1.03 15)C
- 1.26222
(15)
R(C) = 0.99947 + 0.01853 sin( 1.848C - 3.14)
(16)
and
Particle terminal velocity corrected for wall effect
C = log,, Ar The terminal velocity of a single particle corrected for the wall effect can be calculated from the following equation proposed by Richardson and Zaki [ 141:
In equation ( 17) Ar is the Archimedes defined by Hartman ef al. [31] as: Ar
log,,
+
=s
=
m
2
d,‘m(pcPI2
(\ d,(Ps- P,lS ‘I2 1
I7
(10)
j
jLDPl
/
As can be seen from equation (IO), the free-falling velocity of the particles is a function of the drag coefficient which is generally given in graphical form [29] as a complex function of the flow conditions. Lydersen [30] fitted these curves for spherical particles to a series of straight lines: for < Re, < 200,000, 500 2.0 < Re, < 500.0, 0.00001 < Re, < 2.0, The above correlations
co = 0.44 C, = 18.5 Re,-0-6 Co = 24.0 Re,-’
and equation
number
which is
PI1
(18)
The terminal velocities predicted using the method suggested by Hartman et al. [31] agree very well with those calculated from the standard equations of Lydersen [30]. For the results which are presented here, the terminal velocities of the particles are, therefore, calculated with the Hartman et al. correlation. Comparison
Therefore, u
=
(8)
( t> Where U, is the terminal velocity or free-fall velocity of spheres in an infinite medium, which can be obtained by equating the viscous drag force to the effective gravitational force:
(17)
of measured
and predicted res$ts
Inspection of the correlations given in Table 1 revealed wide discrepancies between the predictions of several of these correlations and our experimental results for the voidage as a function of velocity. However, excellent agreement between measured and predicted data was obtained with the Hirata and Bulos [26] model in conjunction with the Richardson and Zaki [ 141 exponent, n, which is given in Table 1. Figures 4 and 5 show the comparison between the experimental results and the predictions of the two models suggested by Richardson and Zaki [ 141 and by Hirata and Bulos [26], for glass spheres and stainless For both particles, the steel cylinders, respectively.
(10) give:
for 500.0 < Re, < 200,000
Stainless
(11)
steel
particles
0.9
for 2.0 < Re, < 500.0 Um =
(12)
for 0.00001 < Re, < 2.0
(” ;,p’)g
U, = 0.0556dp2
(13)
Recently, Hartman et al. [31] proposed the following explicit empirical expression for the free-fall velocities of spheres: log,, Re, = P(C) + log,, R(C)
(14)
50
40
0
SupeFficiol
li$id
Fig. 4. Measured and predicted superficial liquid velocity.
veldcqty
U sI
bed voidage
cm/s
as a function
of the
226
Symbols
0.4 0
5uper5ficial
10 15 liquid velocity USI
Fig. 5. Measured and predicted superficial liquid velocity.
bed voidage
20
25
0.5
as a function
of the
Fig.
and
predicted
bed
voidage,
Nomenclature
4 4
Ga g H L 0.5
> R 0.7
Experimental
6. Comparison
1
The authors are indebted to Dipl.-Ing. Bengt Stellingwerf for his assistance in obtaining the experimental data.
CLJ D
T = 22 “C.
0.9
Voidage
Acknowledgements
B C
1
Fig.
of measured
2.52 7.9 79
Additional experimental data have been obtained for the annular fluidized bed heat exchanger at a bulk temperature of 95 “C. These results are compared in Fig. 7 with the predictions of eqn. 4. The general agreement between measured and calculated values is good.
Ar
0.6
7. Comparison
0.8
(9&,3,
T = 95 “C.
A
05
07
Experimental
Richardson and Zaki model underpredicts the experimental results, considerably. For low flow velocity, this model does not predict the static bed voidage. The Hirata and Bulos correlation [26] in conjunction with eqns. (14-18) and the Richardson-Zaki exponent given in Table 1, on the other hand, predicts the experimental results with an average deviation of only about 6%. All measured bed voidages for the investigated solid particles with different densities and sizes are compared with the values predicted by the Hirata and Bulos correlation [26] in Fig. 6. Considering the errors involved in the experimental data and the approximate nature ‘of the correlations itself, the root mean square error of 6% is an excellent indication that the recommended procedure is suitable for the prediction of the bed voidage in annular heat exchangers.
0.4 0.4
0.6
cm/s
$?,,
o’Gloss 0.2 0 Steel 0.196 A: Stee10.310 “‘I’,,“““,“““”
of measured
0.8
0.9
1
voidage
and
predicted
bed
voidage,
Re, Ret Re,
parameter defined in eqn. (5) Archimedes number parameter defined in eqn. (6) parameter defined in eqn. ( 17) drag coefficient column diameter, cm hydraulic diameter, cm equivalent particle diameter, cm Galilei number acceleration due to gravity, m ss2 bed height, cm height of expanded bed, m Richardson-Zaki exponent parameter defined in eqn. ( 15) parameter defined in eqn. ( 16) particle Reynolds number particle Reynolds number at terminal velocity =Re,/exp[S( 1 - E)/~E] Reynolds number defined in ref. 19
227
minimum fluidization velocity, cm s-’ superficial liquid velocity, cm s ’ particle terminal velocity corrected for wall effect, cm ss’ particle terminal velocity in an infinite fluid, cm SC’ volume of liquid, m3 volume of particles, m3 mass of feed, kg bed voidage static bed voidage bed voidage calculated from Richardson-Zaki equation liquid viscosity, g cm-’ s-’ liquid density, g crnd3 solid particle density, g crne3 =[ 1 - 1.21( 1 - E)*/~] -’ dimensionless parameter
References 1 L. P. Hatch and G. G. Weth, Scale control in high temperature distillation utilizing fluidized bed heat exchangers. Research and Development Progress Rept. No. 571, 1910. 2 C. A. Allen and E. S. Grimmett, Liquid fluidized bed heat exchanger design parameters. Department of Energy, Idaho Operations Ofice, contract I-(322)-1570, April 1978. Development of a vertical flash evaporator, 3 D. G. Klaren, Ph.D Thesis, Delft University of Technology, 1975. of’ calcium sulphate scale by a 4 J. A. M. Meijer, Inhibition fluidized bed, Ph.D. Thesis, Delft University of Technology, 1984. Principles and 5 D. G. Klaren, The fluid bed heat exchanger: modes of operation and heat transfer results under severe fouling conditions, Fouling Prev. Res. Dig., 5 ( 1) (1983). and J. Kollbach, New developments in 6 R. Rautenbach fluidized bed heat transfer for preventing fouling, Swiss Chem. 8 (5) (1986) 47 55. 7 J. Kollbach, W. Dahm and R. Rautenbach, Continuous cleaning of heat exchangers with recirculating fluidized bed, Heat Transfer Eng. 8 (4) ( 1987) 26-32. nonflocculated suspen8 J. H. Steinour, Rate of sedimentation: sions of uniform spheres, Ind. Eng. Chem, 36 ( 1944) 618 -624. A calculation of the viscous force exerted by 9 H. C. Brinkman, a flowing fluid on a dense swarm of particles, Appl. Sci. Res. Al (1947) 27-34. of IO W. K. Lewis, E. R. Gilliland and W. Bauer, Characteristics lluidized particles, Ind. Eng. Chem. 41 (1949) 1104- 1117. I1 P. G. W. Hawksley, Some aspects of fluid flow. Paper 7, Institute of Physics and E. Arnold, London, 1951.
12 R. Jottrand, An experimental study of the mechanism of fluidization. J. Appl. Chem., 2 (1952) 17-22. I3 E. W. Lewis and E. W. Bowerman, Fluidization of Solid Particles in Liquids. Chem. Eng. Prog., 48 (1952) 603-61 I. 14 .I. F. Richardson and W. N. Zaki, Sedimentation and Fluidization. Trans. Inst. Chem. Eng. 32 (1954) 35-53. 15 J. Happel, Viscous llow in multiparticle systems: Slow motion of fluids relative to beds of spherical particles and particulate Ruidization and sedimentation of spheres. AIChE. J., 4 (1958) 197-201. 16 A. L. Loelller and B. F. Ruth, Particulate fluidization and sedimentation of spheres. AIChE. J., 5 (1959) 310-315. 17 II. R. Oliver, The sedimentation of suspensions of closely sized spherical particles. Chem. Eng. Sci., 15 (1961) 230-242. I8 (7. Wen and Y. H. Yu, Chem. Eng. Prog. Symp. Ser., 62 (62) (1966) 100-105. I9 E. Barnea and J. Mizrahi, A generalized approach to the fluid dynamics of particulate systems. Part I: General correlation for fluidization and sedimentation in solid multiparticle systems, Chem. Eng. J., 5 (1973) 171- 189. 20 R. Letan, On vertical dispersion two-phase flow, Chem. Eng. Sri., 29 ( 1974) 62 l-624. 21 C. Y. Wen and L. S. Fan, Some remarks on the correlation of bed expansion in liquid-solid fluidized beds, fnd. Eng. Chem. Process Des. Dew, 13 (1974) 194-198. 22 J. Garside and M. R. Al-Dibouni, Velocity-voidage relationships for fluidization and sedimentation in solid-liquid systems, Ind. Eng. Chem. Process Des. Dev., 16 (1977) 206213. 23 LJ. P. Ganguly, Direct method for the prediction of expanded bed height in liquid-solid fluidization. Can. J. Chem. Eng., 58 (1980) 559-563. 24 P. U. Foscolo, L. G. Gibilaro and S. P. Waldram, A unified model for particulate expansion of fluidized beds and flow in fixed porous media, Chem. Eng. Sci., 38 (1983) 1251-1260. 25 R. H. Jean and L. S. Fan, A fluid mechanic-based model for sedimentation and fluidization at low Reynolds numbers, Chem. Eng. Sci., 44 (1989) 353-362. 26 A. Hirata and F. B. Bulos, Predicting bed voidage in solidliquid fluidization, J. Chem. Eng. Japan, 23 (1990) 599-604. 27 R. Blijchl and H. M. Miiller-Steinhagen, Influence of particle size and particle/liquid combination on particulate fouling in heat exchangers, Can. J. Chem. Eng., 68 (4) (1990) 585591. 28 P. N. Rowe, A convenient empirical equation for estimation of the Richardson-Zaki exponent, Chem. Eng. Sci., 24 (1987) 2795-2796. 29 H. Schlichting, Boundary /ayer theory, 6th ed., McGraw-Hill, New York, 1968. 30 A. K. Lydersen, Fluid flow and heat transfer, Wiley, New York, 1989. 31 M. Hartman, V. Havlin, 0. Tronka and W. Carsky, Predicting the free-fall velocity of spheres, Chem. Eng. Sci., 44 (1989) 1743-1745.