230
Ind. Eng. Chem. Fundam., Vol. 17, No. 3, 1978
Literature Cited
Report to Marquette University's Committee on Research, Milwaukee, Wis., 1977. Starling, K. E.. "Fluid Thermodynamic Propertiesfor Light Petroleum Systems", Gulf PublishingCo., Houston, Texas, 1973. Townend, D. T. A,, Bhatt, L. A,, Proc. Roy Soc. London, Ser. A, 134, 502 (1931).
Barker, J. A., Leonard, P. J., Pompe. A,. J. Chem. Phys., 44, 4206 (1966). Bloomer, 0. T., Rao, K. N., lnst. Gas Techno/., Chicago, Res. Bull., 18 (1952). Dawson, P. P., McKetta, J. J., Silberberg, I. H., J. Chem. Eng. Data, 18, 76 (1973). Dawson, P. P., Silberberg, I. H., McKetta, J. J.. J. Chem. Eng. Data, 18, 7 (1973). Gosman, A. L.,McCarthy, R. D., Hust, J. G., Nat. Bur. Stand. (U.S.) Ref. Data, 27 (1949). Greville, T. N. E., SIAMJ. Appl. Math., Nurner. Anal., Ser. B, 1, 53 (1964). Marquardt, D. W., SlAMJ. Appl. Math., 11, 431 (1963). Meter. D. A., "Nonlinear Least Squares (GAUSHAUS)", University of Wisconsin Computing Center, Vol. IV, Rev. B, Sec. 3.22, Madison, Wis., 1966. Rabinovich. V. A., Tokina, L. A,, Berezin, V. M., Teplofiz, Vys. Temp., 8, 789 (1970). Rabinovich, V. A., Tokina, L. A,, Berezin, V. M., Teplofiz. Vys. Temp., 11, 64 (1973). Reamer, H. H., Sage, B. H.. Lacey. W.N., lnd. Eng. Chem., 42, 140 (1950). Rodriguez, L., "Detailed Procedure to Determine LennardJones 12-6 Force Parameters for Pure Substances Based on PVTData of Binary Gas Mixtures",
Department of Mechanical Engineering Marquette University Milwaukee. Wisconsin 53233
Luis Rodriguez
Received for review August 5,1977 Accepted May 1,1978
The author gratefully acknowledges the support provided for this work by Marquette University's Committee on Research. S u p p l e m e n t a r y M a t e r i a l A v a i l a b l e : Detailed procedure to determine Lennard-Jones 12-6 force parameters (69 pages). Ordering information is given on any current masthead page.
Bubble Formation in Non-Newtonian Liquids
The prediction of the volume of bubbles released from a nozzle is of importance to industries employing nonNewtonian liquids. It has been shown in this work that equations of the type V = C(Q2/g)3/5 can be used to predict the volume of bubbles in liquids displayin both shear dependent viscosity and viscoelasticity provided the gas rates are higher than about 0.5 X m 1s.
9
Introduction Several mass-transport operations involve processes where a gas is released from a nozzle and bubbled through a rheologically complex mass. The examples are in the field of fermentation, effluent treatment, polymer production, etc. The kinematics of the motion of the gas bubbles in such masses controls the mass transfer processes. The rise velocity of the swarm of bubbles, for instance, controls not only the gas holdup but also the mass transfer coefficients. The rise velocity, in turn, depends upon the bubble size. It is thus of interest to know the bubble size as a function of the gas properties, liquid properties, and the operational and equipment variables. A considerable body of knowledge exists on the problem of bubble formation in Newtonian fluids (Kumar and Kuloor, 1970). However, the attention paid to the corresponding problem of bubble formation in non-Newtonian fluids has been considerably less. Indeed there are only some scanty experimental data in the literature (Kumar and Kuloor, 1970) where the problem has been examined with some seriousness. The present note is concerned with the aspects of bubble formation in non-Newtonian fluids. '
Background Several models of bubble formation in Newtonian fluids have been presented in the literature. Most of these assume that constant gas flow and constant pressure prevail during the bubble formation. In principle, all the models are based on the balance of forces pertaining to inertia, gravity, surface tension, and viscosity and they predict the bubble size as a function of the geometry of the nozzle, system, and operational variables. For the case of inviscid liquids the model of Davidson and Schuler (1960a) assumes an irrational flow and balances the 0019-7874/78/1017-0230$01.00/0
buoyancy force with inertia. In viscous fluids, the viscous drag term is added by Davidson and Schuler (1960b). The model of Kumar and Kuloor (1967) assumes a two-stage mechanism: an expansion stage followed by a detachment of the bubble. A viscous drag term is added by Kumar and Kuloor (1970) to the first stage for the case of bubble formation in highly viscous liquids. Despite the widely different physical background, both models for the bubble formation in inviscid liquids can be expressed (after some simplification) in the form of
V = C (Q2/g)3/5
(1)
in which the value of the dimensionless coefficient C was found by Davidson and Schuler (1960a) to be 1.387, while Kumar and Kuloor (1967) suggest C = 0.976. Both these values are fairly close to the empirical value of C = 1.722 found by van Krevelen and Hoftyzer (1950). In analyzing the contribution of the viscous drag, one has to consider also the balance of stresses when the gas flow rate becomes very high. In this case it is plausible to assume that the shear and normal stresses, whether in a Newtonian or a non-Newtonian liquid, will have very little influence upon the mode of the bubble formation as compared with the stresses due to inertia. In other words, the model expressed in eq 1 should hold for all types of non-Newtonian liquids if the volumetric gas flow rate is sufficiently high. This point has not been tested in the literature so far and we provide an experimental test of this conjecture in this short note.
Experimental Setup and Liquids Used Air was fed into one of the several nozzles (0.025 to 0.762 cm) fitted on circular disk at the bottom of a vertical perspex column of square (diameter 16.5 cm) cross section. The bubbles were allowed to rise freely along the 245-cm length of the 0 1978 American Chemical Society
Ind. Eng. Chem. Fundam., Vol. 17, No. 3, 1978 231
Table I. Properties of Liquids Used solution
density (20 OC) kg/m3 1176.5 1025 1020 1010 1005 1150
70% glycerol 0.75% CMC 0.5% CMC 0.1% CMC 0.05% CMC 0.5% PAA in glycerol 0.1% PAA 0.05% PAA 0.05% PEO 0.1% PEO 0.5% PEO
1008 1003 1000 1003 1005
range of +,
viscosity (20 “C), mPa s
n
K, Pa sn
b
6.46
-
-
-
1.4 0.20 0.18
-
-
0.72 0.76 0.86
-
0.52
5.60
0.815
7.07
0.1-200
-
0.65
0.127
0.83 0.85 0.904 0.899 0.888
0.57 0.509 0.564 0.783 1.56
0.1-200 0.1-200 0.1-200 0.1-200 0.1-200
-
-
3.94
-
.-
-
0.56
-
-
-
-
2.24 1.60 3.01
A, Pa s b
0.744
-
-
S-1
-
1-103 1-103 1-103 -
.....
look I
/a
I
i I
I 51.mc
tneo. . a 3
10-
Figure 1. Bubble volumes in inelastic liquids (gas flow rate between 0.5 X and 60 X m3/s):0,aqueous glycerol; D,0.75%CMC; A , 0.5% CMC; 0,0.1%CMC; U, 0.05% CMC. column. The bubble volume was determined from the gas flow rate and from the frequency of the bubbles which was determined from a film shot by a high-speed (-200 pps) movie camera. The range of gas flow rates was between 0.5 X 10+ and 60 X m3/s. Three different types of liquids were used. The aqueous solution of glycerol represented a Newtonian liquid. Aqueous solutions of carboxymethyl cellulose (CMC) were shear-thinning, while the solutions of polyacrylamid (PAA) and those of poly(ethy1ene oxide) (PEO) showed also a nonzero primary normal stress difference. This is an indication of the viscoelastic behavior which was particularly pronounced with PAA solution in aqueous glycerol. The variable viscosity of all the polymer solutions was interpreted in the form of a power-law type model
K4rl-l
(2) where the apparent viscosity function ha= ha(+)was measured using a Weissenberg Rheogoniometer (R-18) and evaluated as pa = 3Ml2rR3+. The primary normal stress difference N1 which is a manifestation of the viscoelastic behavior was evaluated from the axial thrust T generated between the cone and the plate of the Rheogoniometer by computing N1 = 2T/rR3.The function N1 = N1 (4) was interpolated by an empirical formula ha =
N1 = A+’
(3)
1
I
1 10-1
10-2
I
l
l
10” 2x100
VOlrne thea. )e83
Figure 2. Bubble volumes in viscoelastic liquids (gas flow rate between 0.5 X and 60 X m3/s):0,0.5%PAA in glycerol; ,0.1% PAA; A,0.05% PAA; 0,0.5%PEO; m, 0.1% PEO; A ,0.05% PEO. The values of all these parameters for the liquids used are summarized in Table I.
Results and Their Discussion The comparison of measured bubble volumes with those computed using eq 1is shown in Figure 1 for viscous inelastic fluids and in Figure 2 for viscoelastic fluids. The value of the coefficient C was taken as 0.976. The agreement is very good in the whole range of liquid properties so that it can be concluded that the rheology of the ambient liquid has no influence upon the bubble volumes in the gas flow rates region examined in this work. An earlier work from this laboratory (Acharya et al., 1977, 1978) has shown that the so-called “wave theory” which implies inviscid flow approximation can be successfully applied to the motion of bubbles with terminal rise velocities higher than 20 cmls irrespective of the rheological complexity of the ambient liquids. The data presented in this communication extend the range of phenomena in which the influence of rheology is insignificant. It should be emphasized that the process of the growth of a bubble involves a transient extensional kinematics. Consequently, if the characteristic stretch rate during growth was much higher than the relaxation time of the viscoelastic fluid, then large tensile stresses would be expected to be present.
232 Ind. Eng. Chem. Fundam., Vol. 17, No. 3, 1978
The contribution of the viscous stresses and these large extensional stresses will be included in the resistance to deformation due to the internal stresses. However, the agreement of the inviscous fluid model with the experimental data indicates that the contribution of the internal stresses is quite negligible in relation to the inertial stresses in the range studied. The practical implications of the present observations are obvious. It would appear that the traditionally used equations such as eq 1could be safely used in practice for predicting the bubble sizes in highly viscous non-Newtonian fluids provided the flow rates are of the order of m31s per one nozzle opening. Such a case might arise, for instance, in the prediction of bubble sizes in aerated non-Newtonian fermentation broths used in the production of antibiotics such as penicillin or streptomycin.
Nomenclature A = material parameter, eq 3 b = material parameter, eq 3 C = coefficient, eq 1 g = acceleration due to gravity K = material parameter, eq 2 M = torque due to the shearing of the liquid sample kept between the plate and the cone in the Rheogoniometer n = material parameter, eq 2 N1 = primary normal stress difference
Q = volumetric gas flow rate R = radius of the cone-and-plate setup T = axial thrust between the cone and the plate of the Rheogoniometer due to the shearing of the liquid sample V = volumeof bubble Greek Symbols
i. = shear rate K~
= apparent viscosity
Literature Cited Acharya, A., Mashelkar, R. A., Ulbrecht, J., Chem. Eng. Sci., 32,863 (1977). Acharya, A., Mashelkar, R. A., Ulbrecht, J., Can. J. Chem. Eng., to be published, 1978. Davidson, J. F., Schuler, B. 0. G., Trans. Inst. Chem. Eng., 38, 335 (1960a). Davidson, J. F.. Schuler, B. O.G., Trans. Inst. Chem. Eng., 38, 144 (1960b). Kurnar, R.,Kuloor, N. R., CHEMTECH, 19, 733 (1967). Kurnar, R., Kuloor, N. R.,Adv. Chem. fng., 8, 256 (1970). van Krevelen, D. W., Hottyzer, P. J., Chem. Eng. Prog., 46, 29 (1950).
Department of Chemical Engineering University of Salford Salford M5 4 WT, England
Arunima Acharya R. A. Mashelkar Jaromir J. Ulbrecht*l
Received for review October 31, 1977 Accepted May 8, 1978
Correspondence regarding this paper should be sent to this author at the Department of Chemical Engineering, State University of New York at Buffalo, Amherst, N.Y. 14260.
Note on the Computation of Component Fugacity Coefficients
In the computation of vapor-liquid equilibrium a method proposed by Joffe leads to substantial simplification of the arithmetic. The method is restricted to pseudocritical temperatures which depend on composition only. However, one wishes sometimes to regard pseudocritical temperature as dependent also on the system temperature. In that case the relation used by Joffe for evaluating d In p/dT, is no longer valid. It is now shown that this relation can still be applied in a formal sense, if the pseudocritical temperature is taken at the current system temperature and its variation with temperature is ignored. Thus Joffe's algorithm is given a wider range of applicability.
Generalized equations of state employing the three-parameter corresponding states principle have in recent years reached a degree of perfection which makes them eminently suitable for vapor-liquid equilibrium computation. This note is concerned with such equations of state which are explicit in reduced pressure P, or compressibility factor 2, and in which the other variables are reduced temperature T,, reduced volume v,, and the acentric factor w . When the critical (or for mixtures pseudocritical) parameters are T,, P,, and v,, then the reduced properties are by definition
T , = TIT,; P , = PIP,;
v, = vlv,
(1)
The pseudocritical pressure as a rule is derived from the other mixture parameters through
P , = RT,Z,Iu,
(2)
In the calculation of vapor-liquid equilibrium the equation of state and its associated combination rules for the pseudocritical parameters and for the w of the mixture is used to compute the individual component fugacity coefficients pi in both phases. The fundamental relation for this is 0019-7874/78/1017-0232$01.00/0
In
rpi =
RT
v
1( E ) ani
V,T,nj
- g] d V - In 2 V
(3)
(at constant T ) .For a large and complicated equation of state this leads to a lengthy formula for In cpi and consequent high computational expense. An attractive shortcut method is based on the observation by Joffe (1948) that In cpi can be treated as a partial property , value for the total mixture, so that of In p ~the a In (PM a In ( a ~ (4) l n p i = l n m + - - ax
dxj
This requires only In (PM and its derivatives with respect to composition to be evaluated, thus giving a substantial saving in computation time. The shortcut method was demonstrated by Joffe (1976, 1977), and was also successfully applied by Plocker (1977). Since In (OM is available as a function of T,, P,, and w , which themselves are functions of composition, finding the differentials of In ( a in~ eq 4 involves evaluation of (a In +daTr)p,,w. For this it is customary to use the relation 0 1978 American Chemical Society
