2522
Ind. Eng. Chem. Res. 1987,26, 2522-2528
Interface Interaction in Two-Phase Gas-Non-Newtonian Liquid Flow C. H.Oht and R. Mahalingam* Department of Chemical Engineering, Washington State University, Pullman, Washington 99164-2710
A theoretical and experimental investigation of two-phase flow of gas-non-Newtonian liquid systems was carried out under horizontal and isothermal flow conditions, in order to evaluate some of the interfacial effects which govern the momentum transfer in such systems. By considering an interaction to exist a t the interface between the phases, an analytical model was developed for wavelength and pressure drop parameters in two-phase wavy and annular regimes and compared with experimental measurements. Experimentation was carried out in a two-phase flow system using air and aqueous solutions of Superloid (0.1%,0.25%, and 0.5% w/w) and containing three different types of surface active agents in solution. The measurements involved pressure drop, holdup and wave characteristics. The model developed incorporates into it the wave motion a t the interface, thus improving upon the pressure drop predictions. The deviations, however, appear to increase with increasing pseudoplasticity; this is explained through calculations on equilibrium entrainment and interface roughness. It is also observed that the measured wavelength is influenced by the pseudoplasticity of the liquid only to a limited extent. Similarly, the role of surfactants in stabilizing the waves also appears to be limited.
Most of the research in two-phase flow appears concentrated on gas-Newtonian liquid systems, despite the increasing industrial importance of situations wherein the liquid phase is non-Newtonian: for instance, biochemical broths, polymer solutions and melts, drilling muds, and slurries. Momentum transfer in such two-phase gasnon-Newtonian liquids is, at present, practically unpredictable, and a few models developed (Oliver and Young Hoon, 1968; Mahalingam and Valle, 1972; Mahalingam, 1980; Eisenberg and Weinberger, 1979; Deshpande and Bishop, 1983) do not adequately account for the interaction occurring at the interface between the gas and the liquid. Thus, there is considerable impetus for examining the two-phase flow behavior involving non-Newtonian liquids, taking into consideration the interfacial interaction, and this has served as the primary motivation for the current work. Even in gas-Newtonian liquid flow, phenomenological theories for the creation of a particular interface or the various interactions occurring at the interface are not yet developed. Development of correlations for various flow patterns and the identification of several interfacial actions, such as interfacial waviness and resultant roughness, droplet detachment and resultant entrainment, droplet redeposition, etc., however, appear to be progressing satisfactorily, albeit slowly. Current experimentation in the field appears to be directed toward evaluating various second- and third-order parameters in these flow systems, which can, in turn, explain the observed first-order variables such as pressure drop and phase holdup. Physical Model and Governing Equations The system considered is one of cocurrent flow, with a thin film of non-Newtonian fluid in laminar motion and a simultaneously flowing turbulent air phase, both in the horizontal plane, as shown in Figure 1. On the assumption that the film thickness is small compared with any length scale in the flow direction, two-dimensional forms of the ‘Present address: EG&G Idaho, Inc., Idaho Falls, ID 83415. 0888-5885/87/2626-2522$01.50/0
equations of continuity and motion are analyzed. The surface which is initially smooth at low flow rates of air changes to a wavy one as the flow rate of air is increased. As the air velocity increases, the wave surface is disturbed, eventually resulting in rippling surfaces. The “Leveque approximation” is used and the following assumptions are made: (a) small wave film thickness, (b) two-dimensional wave flow, (c) no entrainment, and (d) no radial pressure gradient at the interface. The method of analysis is an extension of Kapitza’s (1948) integral momentum technique and of Davis’s (1969). The equations of motion are
The continuity equation is -au* + - = av* ax* ay*
0
(3)
or
The boundary condition is a2a*
at y = a * (5) ax*2 The integration of (2) and substitution of (5) into the integrated equation give an expression for p*: a2a* p* = p*g - u -- pg(y* - a*) (6) ax*2 Next, the coordinate x* is transformed into a moving coordinate t* for u*, a*, and p*. Surface waves are not assumed to be appreciably damped due to surface elasticity; hence, E* = x* - wt* (7) p=pg-u-
0 1987 American Chemical Society
Ind. Eng. Chem. Res., Vol. 26, No. 12, 1987 2523 V
t Y
t I
a
COMPRESSED AIR FROM MAINS
a'
The power-law model for pseudoplastic liquid (shear thinning) is T
= k?"
(10) By consideration of wave motion, a similar velocity profile is applied locally; Le.,
Using dimensionless variables, u = u*/iio u = u*/iio a = a*/ao
T*aol/s
T
=kiiol/s
Sla2s+2$29-1+' + S2a28+1a'+2s + S3as+2p?s-1~' +
+ + + + Sga8+la'p+ S9a2rS-'-r']+ R*
S4as+2JB-1V?sS6a*+1arFp S6a2r2s-1T' ii0
u
@
PRESSURE GAUGE
ph MICRO METERING
@
PRESSURE TRANSDUCER
W
MANOMETER
V N E
ON-OFF V A L E
j, i
\
I
(12)
k,
soln concn,
temp,
wt%
O C
n
Nsn/m2
N/m
0.10 0.25 0.50
19 20 18
0.93 0.84 0.59
0.031 0.058 0.472
0.065 0.055 0.052
Q,
A mass balance is taken around a control volume defined by two vertical planes normal to the flow. If one plane is located a t the position where the film thickness is a, and the second is at the position where the average film thickness is ao*,then a material balance can be written for unit width of the test section such that the control volume moves at the wave velocity. There is no mass transfer across the interface or across the solid wall. Thus, pa*(w - ii*) = pao*(w - iio)
z = w/iio
ii = ' s a u dy a o
Integrating the velocity profile, eq 14, we obtain
The interfacial shear stress is eliminated from (12) by equating (13) and (15) to give T
=
(:)I"[
z
-
% (i)as+lp]l" + -
a$
(16)
+
is eliminated because z can be measured and can be calculated by the pressure drop equation for the gas phase. For small amplitude waves, (17) a = 1 + 4(*) T
W
Q LIQUID PUMP
Figure 2. Schematic diagram of experimental setup.
C; = C;*/ao
for large L, substituting for u*, &*lay*, and au*/aJ/*, and averaging (9) in the y direction by integrating between y = 0 and y = a, we obtain
-[S,as+lp-l+'
i
ROTAMETER
Table I. Rheological and Surface Tension Properties of Aqueous Solutions of Superloid
where +* = ap,/aC;*, the pressure gradient in the gas phase. In the absence of waves, the velocity distribution is a function of the pressure gradient, $, in the conduit as well as of the interfacial shear exerted by the turbulent gas phase on the liquid, i.e.,
iio = Jaou* dy*
0
e ON
(8)
The substitution of (4)-(7) into (1) gives
y = y*/ao
1
LEGEND
LIQUID FEED TANK 2 . HOLD-UP M S S E L 3 GAS DISENGAGEMENT TANK 4. M N T 5 . Y-MIXER 1.
x * Figure 1. Schematic description of the model under consideration.
where 4, the wave shape factor, is expected to be periodic and 141
