Shorter Communications Table 1 some results for the different operating conditions are listed. In Fig. 2 the back mixing coefficients are plotted against the superficial gas velocities for the measurements in column II at cocurrent flow. The splitting into two zones occurs up to a gas velocity of about 6cmlsec. This velocity limit corresponds to the transition from bubble flow to coalesced bubble flow which has been described elsewheretl, 81. Similar results were obtained in column I and for countercurrent flow. For countercurrent flow the back mixing coefficients are represented in Fig. 3 for three different liquid velocities. A significant influence of the liquid velocity can not be observed. Recently, Towel1 and Ackermann[4] proposedan empirical correlation for back mixing data in bubble columns where the back mixing coefficient is related to the volumetric flow term: D1’50,$5. On the basis of this relation we found that the following three equations represent our data at a significant level of 95 per cent: In the lower region of the column EL = (1.2kO.12) D1’5v,$5 [cm*/sec],
(3)
in the upper region
EL = (2.4-CO.18) D1%cO~s [cm*/sec],
(4)
and if no splitting into different back mixing zones occurs: EL = (2kO.15) D1%c0.5
[cm%ec].
(5)
Equations (3)-(5) apply to cocurrent current flow of the two phases.
as well as counter-
Institutfiir Technische Chemie Technische Universitiit Berlin Berlin, Germany
W. U. H. Y.
c D EL vG vL x lC
DECKWER GRAESER LANGEMANN SERPEMEN
NOTATION concentration of tracer, g mole/l. column diameter, cm liquid phase back mixing coefficient, cm*/sec superficial gas velocity, cm/set superticial liquid velocity, cm/set length coordinate (distance from source), cm relative gasholdup
REFERENCES EngngSci. 197025 1. HI OHKI Y.and INOUEH.,Chem. PI REITH T., RENKEN S. and ISRAEL B. A., Chem. Engng Sci. 1968 23 619. [31 TOWELL G. D., STRAND C. P. and ACKERMAN G. H., A.I.Ch.E-Znstn Chem. Engrs Joint Meeting Series No. 10, p. 97 1965. G. H., 5th European and 2nd International Symposium on Chemical Reaction 141 TOWELL G. D. and ACKERMAN Engineering, Amsterdam 1972. Preprints p. B 3- 1 Elsevier. PI AOYAMA Y., OGUSHI K., KOIDE K. and KUBOTA H.,J. Chem. EngngJapan 1968 1 158. WI EISSA S. H. and EL-HALWAGI M. M., Ind. Engng Chem. Proc. Des. Dev. 1971 103 1. [71 EISSA S. H. and EL-HALWAGI M. M., Chemie ind. GCnie Chimique 1971104 2080. H., Chemie Inger Techn. 1972 44 697. PI KOLBEL H., BEINHAUER R. and LANGEMANN Chemrcol Engineenng Science,
1973, Vol. 28, pp. 122% 1229.
Pergamon Press.
Printed in Great Britain
Liquid distribution in packed columns (Received 24 January 1972; accepted 6 October 1972) THE MATHEMATICAL description of liquid distribution in gas-liquid random packed columns [ 11has been studied using a diffusion like equation, which for cylindrical coordinates and axial symmetry is:
(1) The build up of liquid wall flow in the column has been taken into account through the boundary conditions used to depict the interaction between the liquid in the packing and the liquid on the wall. Porter and Jones [2] assumed a direct proportionality between the specific liquid rate near the wall and the liquid load on the wall. Stanek and Kolar[3] and Dutkai and Ruckenstein[4] proposed the following absorption-desorption type of rate equation to evaluate the
net cross flow between packing and wall: =k(L(a,z)-K.
W(z)).
(2)
In both cases, the solutions of Eq. (l), predict the same asymptotic or equilibrium value for the amount of liquid flowing down the wall, when packing depth equals infinity: &=A
(3)
1+K.. 2
In a previous work [5], it was shown that Eq. (2) is of little practical use for correlation and prediction of the equilibrium wall flow. This is due to the fact that the parameter K is a
1225
function of column diameter, liquid rate and packing size. On the basis of a simple model, which states that the liquid on the wall comes from the peripheral area of the packed column cross section, a new equation was proposed that proved successful to correlate all known equilibrium wall flow experimental data: I$,, = 400 ((
previously by Porter and Templeman[6]. It was found that the variation of wall flow with packing depth was more uniform at short packing depths, if the initial layer of packing remained unchanged. Therefore the experimental procedure was modified to allow for the variation of packing height without disturbing the top layer of packing. The liquid distributation was measured in this way at three liquid rates: 5000, 20000 and 50000 Kgr/m*hr. The packings used were ceramic Raschig rings of 1.5, 2-O and 3*0cm. The column diameters were 12 and 20 cm. Typical results of variation of liquid wall flow with packing depth are shown in Fig. 1. Values of equilibrium wall flow were also obtained for other
(4)
where 5 is an empirical wall parameter, independent column diameter. For ceramic Raschig rings: 5. pT62= 6.7, L-O.151
of
(5)
This correlation holds over the following ranges: L = 480050000 Kgr/m*hr, packing diameter l-3 cm and column diameter: 533-30 cm. The values of wall flow vs. packed height can be used to check the validity of the above boundary conditions. In this regard Porter and Jones found that their solution overestimates the flow of liquid on the wall. The solution of Dutkai and Ruckenstein is more realistic. With regard to the effect of column diameter on the rate of build up of wall flow, both solutions indicate that the packing depth required to reach a given fraction of the equilibrium wall flow increases with column diameter. However, the experimental data of Porter and Templeman [6] over a wide range of column diameters (533-30.0cm) failed to show any definite effect of the column diameter on the rate of build up of wall flow. The same conclusion can be drawn from the data of Dutkai and Ruckenstein. In this work a different radial boundary condition is derived on the basis of the elementary process of liquid transfer between a packing piece and the wall. Experiments of wall flow vs. packed height for the case of initial uniform liquid distribution were performed to verify the proposed model.
The
EXPERIMENTAL experimental equipment
60-
20
Packed height, Fig.
60
40
cm
1. Experimental and predicted
values of wall flow. l d=1*5cm, D=l2cm and L=5OOOKgr/m*hr, n d= 2.Ocm, D = 20 cm and L = 20900 Kgr/m%r; v d = 3.0 cm, D = 20 cm and L = 20000 Kgr/m*hr.
RESULTS is similar to that used
Table 1. Equilibrium values of liquid distribution between packing and wall d
D
(cm) 1.5 1.5 2.0 2.0 2.0 2.0 2.5 2.5 3.0 3.0 3.0 3.0
12 20 12 14.7 20.0 25.0 14.7 25.0 12.0 14.7 20.0 25.0
Mean Values:
Lo = 5000 Kgr/m%r W, WdI, (Kgrlm .hr) 95.5 111.7 90.0 93.0 97.5 112.0 100.0 125.0 101.0 105.0 122.0 119.0 106.0
15.7 12.6 10.1 8.5 7.2 7.8 7.8 7.6 9.5 7.1 7.3 5.8
L,, = 2OtXKlKgr/m*hr We/I, W,
(Kgrlm .hr) 306.0 360.0 312.0 262.0 326.0 305.0 328.0 364.0 345.0 278.0 366.0 376.0 328.0
9.4 8.4 7.3 4.6 5.4 4.6 5.7 4-9 6.8 3.5 4.4 4.2
L,, = 50000
W,
Kgr/m*hr WC/I,
W&m. hr) 663-o 645 675 615 735 887 850 665 773 815 795 752
7.2 5.2 5.5 4.2 4.7 5.7 5.8 3.2 4.9 4.7 3.6 3.1
740-o
The 95 per cent confidence limits for the mean values are within k7.5 per cent of the reported data.
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Shorter Communications sizes of packing and column diameters and the results are reported in Table 1. Liquid dispersion coefficients, 9, were determined for 2-O and 3-Ocm ceramic Raschig rings and the experimental results are plotted against packing diameter in Fig. 2, together with data from other workers. The liquid dispersion coefficients were measured following the experimental procedure proposed by Porter et ol. [7]. E
u
I
Values of W, and WJI’, are reported in Table 1 and W, is always several times greater than Fe for the different experimental conditions covered during this work. The effect of curvature is also dedcted from the dam of Table 1 being the ratio W./I, greater for the smaller packings. The values of W, are almost the same irrespective of the packing size or column diameter, for a given L,,. Therefore, W, is a characteristic value that is mainly a property of the column wall or periphery 151. Formulation of the radial boundary condition Let us consider that nT, is the total number of wetted contact points with the wall of the packing per unit area of column wall; out of this number n, corresponds to pieces that are oriented in positions such that cross flow will take place from the Raschig ring to the wall and np corresponds to packing pieces that take liquid away from the column wall. From the experiments of Sheridan and Donald[8], the liquid flow to the wall at a suitable contact point is proportional to the liquid load on the corresponding packing unit. Then the cross flow in g/set, for each of the n, contact points is:
Jti = m.
Packing diameter.
The liquid flow on the wall is not uniformly distributed, it may flow in rivulets; if the local liquid load on the wall is ascribed to each of the n,, contact point, the reverse flow to the packing per point is:
cm
Fig. 2. Liquid dispersion coefficients for ceramic Raschig rings. A This work, l Dutkai and Ruckenstein [4] and 0 Porter et al. [7]. MECHANISM OF LIQUID ACCUMULATION ON THE COLUMN WALL In the case of uniform initial liquid distribution the build up of the liquid wall flow starts right at the top layer of the packing, where conveniently oriented packing pieces in contact with the wall transfer to it a proportion of the liquid flow that is wetting them. The mechanism of this cross flow was studied by Sheridan and Donald[8], their work shows that “the orincioal ohenomenon controlling flow from a packing un& to an adjacent wall is the shape of the liquid meniscus beneath the contact point” [9]. They found that the shape of this meniscus is dependent upon the irrigation intensity and of the difference in curvature of the surfaces in contact. For instance, if a sphere and a wall are in contact and both are simultaneously irrigated, the cross flow is always directed from the sphere to the wall. If a Raschig ring is touching the column wall, the flow of liquid to the wall will be favoured for most of the possible positions of its axis. This behaviour can be easily verified experimentally. The trend of the liquid to flow down the wall in a packed column is illustrated by comparing the liquid load on both the packing and the wall, when the final liquid distribution in the column is reached. The liquid load is defined as the liquid flow per unit of perimeter and for the equilibrium conditions is obtained as follows: forthepacking for the wall
Ie = &(I -$%cJ _
rc,
IP+!-$
(8)
Therefore, at a given packing depth, the net flow to the wall per unit area of column wall is: J(z) = ~~J~-~jJ,,=~r(z)n,-~‘,w(z)n,
1
1
(10)
Where I(z) is the average value of the liquid load on the peripheral zone and W(z) is the mean intensity of the liquid load on the wall. When the liquid linal distribution in the column is reached the net exchange of liquid between packing and wall will be zero, thus
(11) If the mechanism for the cross flow in both directions is-the same a reasonable simplification is obtained taken fw = fp = f, then
and using this relationship in Eq. (10): J(Z) =fk
(13)
with the use of the obvious equality I’(z)/Ie = L(z, a)/L,, Eq. (13) may be further simplified considering that J_.(z, a)& is close to unity. This assumption is reasonable because the maximum value that this ratio can attain, at the top of the packing, is l/(1 -da,), that for most practical situations is less than two. Furthermore, it can be seen from the experimental data of Dutkai and Ruckenstein [4, lo], of liquid flow
1227
Shorter Communications in the peripheral zone of the column, that this ratio drops to its limiting value of one at very short distances from the top of the packing. Then, defining f=f. np the cross flow per unit area is: J(z) =f.
(We--W(z)).
(14)
In this equation the driving force for the cross flow is expressed only in terms of the liquid load on the wall and the characteristic value of wall flow, W,, appears explicitly. The variation of liquid load on the wall can be obtained by direct integration of the differential form of Eq. (14): J(z)dz=dW=f.
(If’,-W)
.dz
(15)
that with W = 0 at z = 0, is:
(w/we) = ($d&,) = 1 -exp (-fz).
(16)
The simple form of Eq. (16) allows the determination off directly from experimental values 4, vs. z, for the case of initial uniform liquid distribution. On the other hand the radial boundary condition proposed by Dutkai and Ruckenstein[14] leads to an infinite series solution for W(z) that precludes a direct determination of the wall factor k. The radial flow of liquid in the packing is proportional to (X./&), then the flow to the wall is: (17) From Eqs. (15)-(17) it is obtained the value of (aL/ar) in the packing at r = a. Thus the boundary condition at the wall is: aL
f&.
(ar>r=a=-
Lo. n .f. exp (-fi)
2.9
(18)
Ocr
(19)
This equation together with: =L#)
L(0.r)
$z,O)=o
osz
(19’)
are the set of initial and boundary conditions used for the solution of Eq. (1). The solution in terms of one dimensionless parameter B = a/(a*f) and two dimensionless variables R andZ is as follows:
+9,,
1 (Rz l+
II
G-G-
l
Equation (21) allows a direct comparison of experimental and theoretical values of liquid distribution. For the computation of liquid distribution from Eqs. (20) and (2 l), values of the wall flow parameter f and the liquid dispersiop coefficient L%are needed. Porter et al. [7] and Kolar and Stanek[l l] reported that at moderate liquid rates the liquid dispersion coefficient is independent of the specific flow rate in a packed column operating without countercurrent gas flow. Dutkai and Ruckensein[lO] studied the effect of gas rate up to the flooding region and found that “the gas velocity has no important effect up to values of the order of about 70 per cent of the flooding velocity”. They reported values of 9 for several sizes of Raschig rings in the flooding region that were, as an average, 35 per cent higher than those obtained without gas flow. The effect of packing diameter upon 9 can be obtained using the random walk model, for the dispersion of liquid in the packing, that predicts: B.z=fs.A=.
The following assumptions are usually made regarding s and A[12]: s = zld A=d which replaced back into Eq. (22) render a direct proportionality between _Q and d. This proportionality will hold in as much laminar film flow of liquid in the packing prevails. The values of 9 vs. d for Raschig rings that are reported in Fig. 2, were obtained for conditions of moderate liquid rates and without countercurrent gas flow. The value off, for a given operating condition, is obtained from wall flow vs. packed height experimental values. Using Eq. ( 16) it is found numerically the value off that minimizes the square deviation of predicted and experimental values. The theoretical values of wall flow vs. packed height are drawn as full curves in Fig. 1. The effect of packing diameter, liquid rate and column diameter upon the rate of build up of wall flow can be studied from the influence of these variables upon the wall flow parameter J Values off for the experimental conditions covered in this work are reported in Table 2. The values off for Raschig rings of 1.5 and 3.0 cm obtained in 12 cm and 20 cm diameter columns are similar,
1) exp (-Z) Table 2. Experimental values off and +W
mJdSnR)(exp (-Z) -PSn* exp (-PWZ)) +lz PSnZ . .MSn) (PSnZ- 1) I=,
(20)
where Sn are the roots of J1 (Sn) = 0. The proportion of total flow rate appearing within a radius R that limits the central packing zone is obtained by integration of Eq. (20):
$+4_
LI+ (RZ
1
D= d (cm) 1.5
exp(-Z)
2.0
* 2. JI(SnR) (exp (-Z) -_PSnZexp (-/3Sn*Z)) /3R. SrPJo(Sn) @SnZ- 1) I. It=,
3.0
~-8/3-l
0
(22)
>
+z
(21) 1228
L (Kgrlm’hr) 5000 20000 5OOOo 5000 20000 5OoOO 5000 20000 5OoOo
dJWe 0.636 0,510 0442 0600 0.520 0.450 0.673 0,575 0.515
12cm
D=20cm
f
dJ,
f
0.065 0.080 0.122 0.041 0.061 0.085 0.022 0.039 0.048
0.470 0.360 0.258 0.390 0.326 0.294 0486 0.366 0.318
0.056 0.063 0.122 0.070 0.100 0.120 0.023 0.043 0~080
thus indicating that the rate of approach of wall flow to equilibrium conditions is independent of column diameter. For the 2.0 packing the values in the column of 20 cm are about 50 per cent larger than those of the column of 12 cm. With regard to the influence of packing size. It can be seen from Table 2, that for the 12 cm dia. column f decreases with packing size. A similar conclusion may be obtained in the 20 cm dia. column, with the exception of the data for 2.0cm Raschig rings that are again abnormally high. The value off for a given packing size and column diameter increases with the specific liquid rate. The variation off with the specific liquid rate and packing diameter are consistent with the proposed model, that states that f is proportional to the number of wetted contact points per unit area of column wall. This number increases with the liquid rate in a packed column and decreases with packing size. The variation of f with liquid rate is slight therefore it is a good approximation to assume that f is constant in the solution ofEq. (1). A value of z = l/f gives the packing depth required to reach 63.2 per cent of &,c The values of z = l/f range from 8.0 cm for d = l-5 cm and L = 50000 Ker/mzhr-45 cm for d = 3.0 cm and L = 5000 Kgr/m%r. Therefore the equilibrium wall flow is reached in very short depths of packing. The limit of Eq. (20) for the packing depth going to infmity is simply: (1 -I&). Therefore the solution of the differential equation with the proposed boundary condition at the wall does not establish a relationship between equilibrium wall flow and column diameter. The values of $J~ are independently obtained from Eqs. (4) and (5). The distribution of liquid in a packed column with wall effect depends on the value of the dimensionless parameter p = g/a’f As 9 increases andfdecreases with packing size, the value of p is strongly dependent of packing diameter. The extreme values of /3 foundin this work are 0.02 and 0. IS. The experimental finding of starvation of liquid flow in the peripheral zone of the packing, found at low values of p,
is due to the fact that the radial transfer of liquid by dispersion in the packing is slower than the cross flow of liquid to the wall. ESTEBAN A. BRIGNOLE GUSTAV0 ZACHARONEK JORGE MANGOSIO Planta Photo de Ingenieria Quimica Universidad National de1 Sur Bahia Blanca Argentina a 2 D d k K a” R’ Z W s
NOTATION column radius, L total surface area of packing per unit volume, L-’ liquid spreading coefficient, L column diameter, L packing nominal diameter, L wall parameter of Eq. (2) equilibrium wall factor of Eq. (2), L-r specific liquid rate, M/LzT liquid flow rate, M/T radial position, L r/a, dimensionless radial position fz,dimensionless packing height liquid load on the wall, M/LT number of vertical displacements in the bed
Greek symbols B g/a% dimensionless
5 wall factor & percentage A magnitude ponding I liquid load
parameter of Eqs. (20) and (21) of Eq. (4) of wall flow of horizontal displacement in the bed corresto one vertical displacement on the packing, M/LT
Subscripts 0 conditions at the top of the packing
e
equilibrium conditions at z --+ ~0
REFERENCES [II CIHLA Z. and SCHMIDT O., Colln Czech. Chem. Commun. 1957 22 896. PI PORTER K. E. and JONES M. C., Trans. Znstn Chem. Engrs 1963 41240. [31 STANEK V. and KOLAR V., Colln Czech. Chem. Commun. 1965 30 1054. E., Chem. Engng Sci. 1968 23 1365. [41 DUTKAI E. and RUCKENSTEIN [51 BRIGNOLE E. and MANGOSIO J., Lat. Am. J. Chem. Engng Appl. Chem. 19712 99. WI PORTER K. E. and TEMPLEMAN J. J., Trans. Instn Chem. Engrs 1968 46 86. J. J., Trais Instn Chem. Engrs 1968 46 74. 171 PORTER K. E., BARNETT V. D. andTEMPLEMAN PI SHERIDAN M. B. and DONALD M. B.. 2nd. Chemist 1959 35 439. 191 SHERIDAN M. B. and DONALD M. B., lnd. Chemist 1959 35 487. E., Chem. Engng Sci. 1970 24 433. UOI DUTKAI E. and RUCKENSTEIN [Ill STANEK V. and KOLAR V., Colln Czech. Chem. Commun. 1968 33 2636. [W TOUR R. S. and LERMAN F., Trans. Am. Instn Chem. Engrs 1939 35 719.
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