Some Remarks on the Correlation of Bed Expansion in Liquid-Solid Fluidized Beds
Bed expansion of liquid-solid fluidized beds has been correlated empirically in terms of particle Reynolds number and particle Galileo number. T h e proposed correlation is shown to b e applicable from the onset of fluidization to the dilute phase fluidized bed with large voidage. T h e range of Galileo number covered is from 18 to 3 X lo*.
A plot of (N~a/13.9N~e1.4)Ncao.04""~~ -1) US. 9, is shown in Figure 3 for 18 < Nca < 105. The correlation can be expressed by the equation
Empirical correlations of bed expansion in liquid-solid fluidized beds have recently been reported by Ramamurthy and Subbaraju (1973) as
N G/ ~1 8 N ~ , 4t155 NG, < 18
< NGa < N,, > lo5
N,,/13.9NR,14 = $t221
where N,,
18
3N,,/ NRr2 = dt3l 1
= d,,?pp
-
p I ) p l g / p 2and
(1)
dc = [1
-
(2)
lo5
1.21(1 -
The standard deviation for this correlation is about 10%. For Nea > lP, a plot Of us. 9, is shown in Figure 4. The correlation can be expressed by the equation
(3) t)2'J]-'
In this paper the applicability of eq 2 and 3 is exapined. In Figures 1 and 2, data obtained from several investigators (Bena, et al., 1971; Lewis, et al., 1949; Ramamurthy and Subbaraju, 1973; Wilhelm and Kwauk, 1948 (for Figure 1); and Bena, et al., 1971; Ramamurthy and Subbaraju, 1973; Richardson and Zaki, 1954; Wen and Yu, 1966; Wilhelm and Kwauk, 1948 (for Figure 2)) are used to plot N c , / ~ ~ . ~ N RUS., ~Nea . ~ and 3 N ~ a l N ~ eus. ' NGa, using 4, as a parameter. The data of Bena, et al. (1971), plotted in Figure 1 are obtained from the empirical correlation presented in Figure 3 of their original paper. It is seen that the effect of NC, becomes increasingly important as 4f is increased. At @c equal to or less than 2.0, the effect of Nea seems to be negligible since the slope is almost zero. It is also obvious that the influence of Nea is less pronounced in the region of Nca > lo5 than in the region of 18 < NGa < lo5. Since the slope of the lines in Figures 1 and 2 changes approximately linearly with 4,, then one should use the form Ncaa4f+* as a correction factor for the overall correlation, and one can mathematically determine that a = - b = 0.0463 for 18 < Nea < 105 and a = - b = 0.0384 for Nca > 105.
The standard deviation for this correlation is about 6%. In these correlations, the wall effects are small and were consequently neglected. . A comparison of these two correlations, made by constructing a plot of bed voidage, calculated and experimental, is shown in Figures 5-8. Values of t , calculated, were obtained from calculations based on equations 2-5, respectively, and values of t, experimental, were obtained from the experiments of four investigators (Bena, et al., 1971; Lewis, et al., 1949; Ramamurthy and Subbaraju, 1973; Wilhelm and Kwauk, 1948) for 18 < NGa < 1P and five investigators (Bena, et al., 1971; Ramamurthy and Subbaraju, 1973; Richardson and Zaki, 1954; Wen and Yu, 1966; Wilhelm and Kwauk, 1948) for Nc, > 1oJ. It is clear from Figures 5 and 6 that eq 2 and 3 can predict bed expansion from t = 0.65 to t = 1.0, where Nca has only a small effect. However, a tendency to deviate appears as is decreased below 0.65. Once the correction factor is introduced, the calculated and experimental values become consistent as shown in Figures 7 and 8. The small devia-
b
I
\ LEWIS E T A L . ( 1949)
1
R A M A M U R T H Y AND SUBBARAJU (1975) / W I L H E L M AND KWAUK (1948)
1
h N
b
+ + I
I
1
IO
I
I 1 1 1 l
1
I
I
1
I l l
IO'
I 0'
r
p
* +
+# I
I l l
I
+ I
I
I 1 1 1 1
I 0'
Nod
Figure 1. Effect of N,, on the correlation of bed expansion proposed by Ramamurthy and Subbaraju for 18 < NG, 4 1W. 194
Ind. Eng. Chem., Process Des.
Develop., Vol.
13, No. 2, 1974
-
+ -I0'
t
\
I
l o r-- *
-
/ u > 'h I
I
I
1
R A M A M U R T H Y AND SUBBARAJU ( 1 9 7 3 ) RICHARDSON AND ZAKI ( 1 9 5 4 ) W I L H E L H A N D KWAUK WE: ( 1 9 4AND 8 ) YU
I
1
1
1
1
( 1 976:
l
,
N0o
Figure 2. Effect of N'a on the correlation of bed expansion proposed by Ramamurthy and Subbaraju for N G a > 105.
s 0
BENA
LEWIS
o
0
E T AL
119711
E T AL
119491
BENA
J
RAM4YURTnY 4ND 5UBB4RAJU 119731
RAYAMURTHV 4ND SUBBARAJU 119731
WlLnELY
AND
KWAUK
10
Figure 3. Correlation of bed expansion in liquid-solid fluidized beds for 18 < N,, < 105.
119711
R I C H I R D S D N AND I4Kl ( 1 9 5 4 )
11940)
+e
E T AL
D
a
WEN AND
YU
0
WILMELY ,19461
4ND
119661 KW4UK
10
0, Figure 4. Correlation of bed expansion in liquid-solid fluidized beds for N,, > 105. Ind. Eng. Chern., Process Des. Develop., Vol. 13, No. 2, 1974
195
03 03
,
6
I
I
04
0 5
I
1
I
or
O B
0 9
E ,
IXILIIYLNTAL
Figure 5. Comparison of the calculated values from Ramamurthy and Subbaraju's equation and experimental values of the bed voidage for 18 < N G 105.
E
"1'''1
"""'
'
'
' ' ''''"1
,
, 1 1 1 1 1 1 1
'
'""7
I
1
RICHARDSON A U D Z A K I l I 9 6 4 1 WEN
AND Y U I l 9 6 ( l l
W I L H E L Y AND K W b U X I 1 9 4 1 1
0 4 t
03
os
-1
/ 04
as
06
c ,
00
a7
a*
Lo
t x ~ 1 1 t u t n r ~ L
Figure 6. Comparison of the calculated values from Ramamurthy and gubbaraju's equation and experimental values of the bed voidage for N G >~ 105.
I
,,,,,,I
Io=
I
1 1 1 1 1 1 1 1
IO'
Llll$@
10.
NO.
LEWIS E T A L
Figure 9. Relationship between NG,, NRe, and fluidized beds for 18 < NGa < 105.
I19491
in liquid-solid
R b M A M U R T H Y AND SUBBARAJU I 1 9 7 3 1
09
W I L H E L M A N D KWAUK 1 1 9 4 8 1
t
05-
-
03 03
1
1
0 4
0 5
I 0 6
E ,
I
I
I
I
07
OB
09
I O
EXPERIMENTAL
Figure 7. Comparison of the calculated values from the equation of this work and experimental values of the bed voidage for 18 < N~~ < 105. 196
c
Ind. Eng. Chem., Process
Des. Develop., Vol. 13, NO. 2,1974
tion of the data points from the line indicates an improved prediction by the new correlations. Since eq 4 and 5 are rather complicated, a trial and error procedure is normally required for the prediction of t ; however, for convenience, plots of N R us. ~ Nc, with t as a parameter have been constructed and are presented in Figures 9 and 10. These figures can be used to compute bed expansion without going through a trial and error procedure. Extending eq 4 and 5 to the minimum fluidization point, and assuming I#I~ = 6.31 (emf = 0.42), we have
(NRe)mf = 0.00134N,,,3"890 18
< NG,l
lo5
10'
(6)
(5)
The minimum fluidization correlation equation of Bena, et al. (1971), for NG, < lo5 is
the point on onset of fluidization. The correlations of the other investigators (Ramamurthy and Subbaraju, 1973), while adequate over certain ranges, are not as accurate as the equations presented in this study. Acknowledgment This work is sponsored by the National Science Foundation Grant No. GK-10977.
.1;
,
,
,,,,,
,
, , , , , ,,,
, ,
, C:E;R P ;
Nomenclature a = correlation constant, dimensionless b = correlation constant, dimensionless d , = particle diameter, L g = gravitational acceleration, LO-2 NG, = Galileo number, dp3(pp- p l ) p g / p 2 , dimensionless NRe = Reynolds number, d p u p l / p ,dimensionless (NRe)mf = Reynolds number a t incipient fluidization point, dpUmfPI/p, dimensionless u = superficial velocity of fluid through beds, LO-1 umf = minimum fluidization velocity, LO-l
1
10 10
IO6
IO’
100
NGP
Figure 10. Relationship between NGa,N R ~and , fluidized beds for NGa> 105.
c
in liquid-solid
Under the conditions lo5 > Nc, >> 19, the denominator of this equation can be simplified such that the equation becomes ( A’,,)mi = 0.00138N(;,0s9“ (8) This equation is practically identical with eq 6. In addition, for Nca > 105,the Bena, et al., correlation is ( ICrKe)n,f =
0.03865N(;c,U’”OL
(9)
Examining eq 7 and 9 more closely, one notes that only a small difference exists between the coefficients while the exponents are equal. Once again, the resemblance between the two equations is very close. Reviewing the conclusions drawn previously, it is evident that the present correlation can be applicable from
Greek Letters t = bed voidage, dimensionless e m f = bed voidage at incipient fluidization point, dimensionless de = [l - 1.21(1 - t ) 2 ’ 3 ] - 1 , dimensionless p = viscosity of the fluid, M L - I T - l p1 = density of the fluid, M L - 3 p , = density of the particle, ML-3 Literature Cited Bena, J., Havalda, I., Matas, J., Collect. Czech. Chem. Comrnun., 36, 3563 (1971). Lewis, W. K., Gilliland, E. R., Bauer, W. C., lnd. Eng. Chem., 41, 1104 (1949). Ramamurthy, K., Subbaraju. K., lnd. Eng. Chem., Process Des. Develop., 12, 184 (1973). Richardson, J. F., Zaki, W. N.. Trans. lnst. Chem. Eng., 32,35 (1954). Wen, C. Y., Yu, Y. H., Chem. Eng. Prog. Symp. Ser., 62, 100 (1966). Wilhelm, R. H., Kwauk, M., Chem. €ng..Progr., 44, 201 (1948).
D e p a r t m e n t of C h e m i c a l Engineering West Virginia University Morgantown, West Virginia 26506
C.
Y. Wen* L. S. Fan
Receioed for reuieu! July 12, 1973 Accepted December 5 , 1973
CORRESPONDENCE Some Remarks to Marina’s Modification of the NRTL Equation
Sir: Recently Marina and Tassios (1973) have proposed a modification of the NRTL equation (Renon and Prausnitz, 1968) which consists of substituting for the adjustable parameter CY the value -1. The mentioned authors deduced, on the basis of analysis of the measured and calculated values of activity coefficients, that this value could substitute rather ambiguous rules for the estimation of this parameter (Renon and Prausnitz, 1968) or its relatively complicated optimalization from the measured equilibrium data. It is very probable that the proposed value will be sufficient for a number of the homogeneous and heterogeneous systems, but, on the other hand, the use of any particular value of cy restricts, in our opinion, the applicability of the NRTL equation. A part of our former calculations extended into the region of negative cy values is illustrated in
Figure 1. By using the values in this figure, one can judge the limits of the applicability of the NRTL equation with a certain value of cy dependent on ( and (G11)xo. The parameter l denotes - 0.51, where X O corresponds to the compositions at which the quality G11 = 0.43429 x a2(g/RT)/dX12 reaches its minimum value (G1l)xo. The symbols are defined as follows: g is the molar Gibbs energy, T the absolute temperature, XIthe mole fraction, and R the gas constant. The critical isotherm is therefore characterized by (G1l)xo = 0 and X o = X,,where X, is the composition corresponding to the critical point. The analogous values for an ideal symmetrical system are (G11)xo = 1.73’72andXo = 0.5, i e . , t; = 0. The limits of the applicability of the NRTL equation are also determined by the following facts. Let us consider, for example, CY = 0.5; in this case the NRTL equation
1x0
Ind. Eng. C h e m . , Process Des. Develop., Vol. 13, No. 2,1974
197
