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Interfacial shear stress and momentum transfer in horizontal gas-liquid flow
Chemical Engineering Science, 1966, Vol. 21, pp. 63-75. Pergamon Press Ltd., Oxford. Printed in Great Britain.
Interfacial shear stress and momentum transfer in horizontal gas-liquid flow T. N. SMITH and R. W. F. TAIT Chemical Engineering Department, University of Adelaide, South Australia (Received 21 December 1964; in revisedform 11 March 1965)
Abstract--The stratified flow of an air stream and a water film in a dosed horizontal channel of rectangular section, 16 in. by 2 in., has been investigated for air velocities ranging between 6 and 120 ft/sec and for superficial water rates between 0.000375 and 0.0210 fta/ft sec. The interfacial shear stress has been measured, and has been separated into two components, one resulting from a momentum diffusion as to a smooth boundary, and the other resulting from a distribution'of normal pressure induced in the fluid by the interfacial wave motion. It is concluded that the pressure distribution shear stress at an interface disturbed by steady waves is in agreement with predictions based on the theoretical work of BROOKEBENJAMIN[7] o n flOWpast a wavy boundary. INTRODUCTION THE mechanics of flow of a gas stream and a liquid film moving under the influence of the gas stream are of vital interest in the design of a variety of mass and heat transfer equipment. In this study, the flow of air and a water film in a horizontal channel has been investigated, and the magnitude and origin of the interfacial shear stress have been examined. In particular, the mechanism of m o m e n t u m transfer between the phases has been discussed because of its relation to heat and mass transfer processes. It has been observed by several authors, for example ALVES [1], that the interface in a conduit carrying a gas and a liquid in cocurrent, horizontal, stratified flow may be either smooth, or wavy, or wavy and disperse depending on the flow rates of the two phases. In flow over a fiat, smooth surface, it is well known that m o m e n t u m is transferred from the free flow by both turbulent eddies and molecular diffusion across the boundary layer to a zone of decreasing turbulent intensity close to the boundary. Turbulence is effectively damped at some small distance from the surface, and from this distance m o m e n t u m is transferred towards the boundary by molecular diffusion. When a fluid flows over a surface having protuberances, a pattern of velocity variations is induced in the fluid near the surface as the flow conforms to the boundary profile. In steady flow 63
these velocity changes result in corresponding pressure changes in the fluid so that a pressure distribution stationary relative to the surface is established. Figure l(a) illustrates qualitatively, the velocity and pressure distributions in the flow of an ideal (frictionless) fluid over a surface having a sinusoidal wave profile. In this case, the pressure distribution is in phase with the boundary profile, and exerts no resultant thrust on the surface in the flow direction. In the flow of real fluids, however, the pressure distribution is not, in general, in exact phase with the boundary profile. Figure l(b) shows a pressure distribution which lags the boundary profile by ~r/2. It is readily seen that the action of the distribution upon the profile slopes leads to a resultant stress in the flow direction. This stress derived from the distribution of normal pressure near the boundary is " f o r m drag". A pronounced form drag occurs where there is separation of the fluid boundary layer at protuberances, so producing a large out of phase component in the pressure distribution. Both the m o m e n t u m diffusion and the pressure distribution mechanisms can be expected to contribute to the shear stress at a disturbed gas streamliquid film interface. Where there is dispersion between the phases, the interchange of drops and bubbles introduces an additional m o m e n t u m transfer component.
T. N. SMITHand R. W. F. TAIT
Velocity ÷ changes --
Pressure
÷
changes _
Surface
profile
(a) Pressure changes in phase with surface
profile
Velocity + changes
Pressure changes
~
~
~~
~
f
~
I
~
~
"l" --
Surface
profile
{b) Pressure changes log surface profile by -n'/2
FIG. I.
Velocity and pressure changes in flow over wavy surface.
PREVIOUS AND RELATEDWORK
The gas-liquid interface HANRATTY and ENGEN [2] have given a qualitative description of the gas-liquid interface during their experiments on the cocurrent flow of air and water in a channel of rectangular section, 12 in. wide by 1 in. high. With increasing air velocity, there appeared in sequence the five distinct types of interface patterns; (1) smooth, (2) "two-dimensional" waves with long crests extending the full width of the channel, (3) "squall" waves having short crests and giving the interface a "pebbled" appearance, (4) "roll" waves which formed on the squall surface and travelled at a much greater velocity, and (5) dispersed flow where entrainment of water droplets in the air stream began. The "squall" wave interface, which was a steady pattern and which persisted over a range of air velocities and water rates, was more closely defined by LILLELEHT and HANRATTY [3] w i t h the measure-
ment of wave profilesby alightabsorptiontechnique. For air velocities in the range 12-33 ft/sec and superficial water rates in the range 0.002-0.005 fta/ft see, they found that the root mean square values of wave amplitude and wave frequency varied from 0.0008 to 0.0016 ft and 21 to 26 see -~ respectively. The distribution of wave amplitudes at fixed air and water rates was found to be Gaussian. Measurements of interfacial wave parameters for the flow of an air stream and a water film in the 16 in. wide by 2 in. high channel used in this study have been reported elsewhere [4]. These measurements covered a range of air velocity from 5-120 ft/ sec and of superficial water rate from 0.0003750.0210 fta/ft sec. The wave amplitudes for the steady patterns, the "two-dimensional" and "squall" waves described by HANRATTY and ENGEN [2], ranged up to 0-0025 ft. Wave number (2n/wave length) varied from 50-210 f t - l , and wave velocity ranged from 0-5 to 2.5 ft/sec. The wave amplitudes were found from the changes in the capacitance to earth of a probe in the air stream immediately above the water surface. Adequate resolution of the surface profile was ensured by keeping the size of the probe small in comparison with the interfacial wavelengths. Wave velocities were measured by timing their passages between fixed locations using a stopwatch. The wave numbers were computed from the wave velocities using wave frequencies measured with the capacitance probe apparatus. Further details of these techniques are available in the paper on wave parameters [4].
lnterfacial shear stress HANRATTY and ENGEN [2], VAN ROSStrM [5], and ELLIS and GAY [6] have all reported greater shear stresses for the flow of a gas stream over a disturbed gas-liquid interface than over a smooth, solid surface. Having an adequate description of the interfacial wave profile, an attempt can be made to relate interfacial shear stresses from the gas-liquid system to those developed in the flow of a fluid over a solid surface of comparable profile. The experimental wave parameters reported earlier [4] provide a description of the interfacial profile. The steady waves are nearly sinusoidal and have, typically, 64
Interfacial shear stress and momentum transfer in horizontal gas-liquid flow ratios of wave length and boundary layer thickness to wave amplitude of 40 and 50 respectively. Here the interfacial boundary layer thickness is the distance from the interface to the air stream velocity profile maximum.
laminar sublayer, there can be no profile curvature at this point and the latter effect is absent. The resultant shear stress due to the viscous effect is
Shear stress at a solid boundary
and that due to velocity profile curvature is
"Cv -~ ½ a 2 k 1 6 / a z S o / 3 p a / 3 # 4 / a ( u
The existence of a stationary pressure distribution in non-separated flow over a wavy boundary has been demonstrated theoretically by BROOKE BENJAMIN [7]. Analysing the flow of a fluid over a two-dimensional surface having the form of a small amplitude sine wave, he has shown that the first order perturbations in the flow of the fluid in following the boundary give rise to periodic tangential and normal stresses on the wavy boundary. The periodic tangential stress has no resultant, in the flow direction, over a wave length. However, the normal stress has a component differing in phase from the surface waves by 1t/2 which does have a resultant in the flow direction. The resultant from the normal stress has two components, one based on the effects of viscosity on flow near the boundary, and the other on the velocity profile curvature at the point where the fluid velocity is equal to the wave velocity. The effect due to velocity profile curvature was first shown by MILES [8]. Should the point where the fluid velocity is equal to the wave velocity be inside the
Zc =
a2kap(U--c)
_ c)4~1~2
-4 U"C - ~ qb2,
where a is the wave amplitude, k is the wave number, Zo is the smooth boundary (momentum diffusion) shear stress, p is the fluid density, # is the fluid viscosity, U is the fluid free stream velocity, c is the wave velocity, uc is the normal distance derivative of the fluid velocity u at U = c, and u~ is the second derivative. ~bI is a function which determines the magnitude of the viscous effects near the boundary, while ~b2 gives the effect of velocity profile on the boundary stress. These functions are defined in the Appendix. Measurements of the shear stress at solid surfaces with sinusoidal profiles have been made by STANTON, MARSHALL, and HOUGHTON[9], STREETER[10], MOTZFELD [11], and KONOBEEV and ZrIAVORONKOV [12]. Of these workers, only the last named have studied profiles which approach the geometry of the interracial wave and boundary layer system. The measurements of KONOBEEVand ZHAVORONKOV
Table 1. Comparison o f measured [12] and calculated [7] solid wave friction factors Tube
c
Friction factors 0"~ -- 7o)/p U2
No.
a (mm)
k (mm -1)
dia. (mm)
6
0.325
0.785
15.9
7
0.22
0-785
16.0
8
0.14
0.785
16-3
9
0.36
0.524
15.8
10
0.135
0.524
16.2
16
0.175
0.785
11.1
17
0.10
0"785
11.1
NR~ = 10,000
NR, = 30,000
NRe = 100,000
0.0061 0.0016 0.0027 0"0021 0.0011 0.0012 0.0036 0.0044 0.0006 0.0010 0.0022 0.0009 0.0009 0.0004
0.0041 0.0013 0.0017 0.0011 0.0006 0.0006 0.0022 0.0022 0.0004 0.0004 0.0014 0-0005 0.0004 0.0002
0.0031 0.0006 0.0012 0.0005 0.0006 0.0002 0.0017 0.0009 0.0003 0.0002
Meas. Calc. Meas. Calc. Meas. Calc. Meas. Calc. Meas. Calc. Meas. Calc. Meas. Calc.
65
T. N. Sma-ri and R. W. F. TAn" appear to support the theoretical work of BROOKE BENJAMIN. Table 1 compares values of the wave friction factor taken from KOI~OBEEV and ZHAVORONKOV's paper with values calculated from BROOKE BENJAMIN'S viscous effect. The profile curvature stress is, of course, absent because there is no movement of the wavy boundary. These friction factors are in the form (~t - Zo)/p U 2 where z~ is the total wall stress, and the Reynolds number is in the boundary layer form b~Up/# where b~ is the boundary layer thickness (here, the pipe radius). Only the long wave length, small amplitude results from KONOBEEVand ZHAVORONKOV'S work have been included, because these are the results for profiles comparable with the airwater interfacial wave profiles, and because there is evidence of boundary layer separation in the other results. Indeed, boundary layer separation could account for the relatively poor agreement for Tube No. 6, where there is a combination of shorter wave length and greater amplitude. EXPERIMENTAL Experimental observations were made on an airstream water-film system in a 16 in. wide by 2 in. high horizontal channel. The channel was wide enough for flow in the middle to be treated as two-dimensional, and long enough, 12 ft, for equilibrium flow to be established in the latter half of its length. The apparatus is shown in Fig. 2. Air was drawn into the channel by a fan, and the water film moved through the channel under the influence of the interfacial shear stress.
Fio. 2. Arrangement of experimental channel. (1) Air inlet cone (2) Water inlet piece (3) Channel (4) Stiffening bars (5) Measuring station (6) Water outlet piece (7) Centrifugal fan (8) Head tank (9) Rotameter (10) Sump tank.
A vertically traversing impact tube and a static pressure tapping in the roof of the channel, connected to a micromanometer, were used to find the air layer velocity profile. The air velocity was varied from 5 to 120 ft/sec. The water rate was measured by calibrated rotameters, and was varied from 0.000375 to 0.0210 ft3/ft see (expressed as flow per unit width of channel). The water surface velocity was estimated by dropping small pieces of polyvinyl chloride sheet, about 0.02 in. square and 0.0035 in. thick, onto the water surface and timing their passages between fixed locations with a stopwatch. The surface velocities were estimated to be accurate within ___5 per cent. The water depth was measured by using a vertically traversing pointer, fitted to a micrometer mechanism, within an estimated error of -t-0.0004 ft. Interface and wall shear stresses were found from a balance of shear stress and pressure gradient over the boundary layer, that is the region between the boundary and the air velocity profile maximum. The error in these stress measurements was estimated at + 4 per cent. The methods used for measuring the interfacial wave parameters have already been noted. RESULTS
Interracial pattern In general the interfacial patterns were similar to those described by HANRATTY and ENGEN [2]. In order of appearance with increasing air velocity they were: (1) smooth, (2) two-dimensional waves, with crests extending the full width of the channel, (3) rhombic waves, short-crested waves giving the water surface the "pebbled" appearance described by HANRATTY and ENGEN, (4) unsteady waves, accelerating "roll" type waves, (5) disperse, with penetration of water droplets into the air stream and air bubbles into the water film. The profiles of both the two-dimensional and the rhombic waves were nearly sinusoidal. The amplitudes and frequencies of the two-dimensional waves were quite regular, but those of the rhombic waves varied in an apparently random manner as remarked by LILLELEHTand HANRATTY [3]. Figure 3 shows the areas occupied by the various interface patterns on a water rate-air velocity chart. 66
Interfacial shear stress and momentumtransfer in horizontalgas-liquid flow
210
120
6C 0 x
3C e)
=;
15
,o 7,5
3.75
5-
,'o
z'o Air velocity,
FZG. 3.
go
I 100
fl/sec
Interracial patterns.
There is good reason to believe that the start of interphase dispersion is identifiable with the onset of unsteady waves. It was found impossible to measure the amplitudes of unsteady waves with the capacitance probe because of the continual attachment to the probe tip, and occasionally the supporting rod, of water droplets from the air stream. The attachment of such droplets naturally caused violent pulses in the capacitance bridge circuit, making the instrument inoperable. To avoid the attachment of water droplets it was necessary to move the probe so far from the water surface that the sensitivity of the circuit was reduced to a useless level. Although the concentration of water droplets in the air stream was immeasurably small at the start of unsteady waves, this point must be regarded as the start also of interphase dispersion. In further support of this contention, it was observed that some of the unsteady waves, especially those at the larger water rates, appeared to contain air bubbles, representing dispersion of the gas phase in the liquid.
"roughness" of the gas-liquid interface. The stresses were, therefore, converted to friction factors, and the corresponding boundary layer flows were characterised by Reynolds numbers. In order to be consistent with the notion of the development of shear stress by momentum diffusion to the boundary, the interface friction factor was defined as zJpg(U - u~)2, where ui is the interface velocity. The wall friction factor was defined as z~/pgU 2. The boundary layer Reynolds rnumbers were defined as b~(U-ui)pg/it 9 and bwUpg/l~g respectively, where b~ and b~ are the interface and wall boundary layer thicknesses. In Fig. 4 the interface friction factor is plotted against Reynolds number for all water rates. For comparison the wall friction factors determined in the dry channel are shown as a full line. The dry channel friction factors follow the smooth wall correlation of PgA~DTL [13] very closely up to a Reynolds number of 40,000. Above this Reynolds number the friction factor increases, presumably owing to flow structure effects associated with the geometry of the channel. The differencebetween the "interface" and smooth friction factors generally increased with increasing it
= • ° • • • *
8"0
rotes ftZ/sec
Water
7"C
0 " 0 2 IO 0"0120 0" 0 0 6 0 O" 0 0 . 3 0 0"00150 O" 0 0 O 7 5 0"000375
,, ° .~, 0= =
°0
~
°
m
6.c o
o
o
o
.
o o
= o oo
"G s.o o
=*t
o
o
oo
o
o o
o
4'0 =D'~
||=
o~ °
•= ===
3.0
•
o
•
=l
I
" ~ - + ' =. ".t " ,. : ' .. :~. . "•
°
Interfacial shear stress The thicknesses of the interface and wall boundary layers in any test were, in general, not equal. Consequently the respective shear stresses could not be compared directly in order to evaluate the
2.0
g
,'0
I
~0
Reynolds number
go x l O -3
FZG.4. Interfacefriction factor. 67
100
T. N. SMm-[ and R. W. F.
water rate and Reynolds number, and the point of departure of the interface friction factor from the smooth wall value was earlier, in terms of Reynolds number, with increasing water rate. These trends in interface friction factor follow those in interfacial wave amplitude [4]. It is noteworthy that the friction factors at the two largest water rates reached maxima, then fell with increasing Reynolds number. A similar trend was detected by VAN ROSSUM [5] in his experiments with air over a hydrocarbon oil.
These trends in film depth and surface velocity were followed less closely at the larger water rates.
The #as film The same thickening of the interface boundary layer with the growth of interfacial disturbances as reported by HANRATTY and ENGEN [2] was observed. The velocity profile in the interface boundary layer closely approximated PRANDTL'S logarithmic distribution [13]. b u = U - 2"5 u* I n - . Y
The liquid film The depth of the water film and the water surface velocity are graphed against the air velocity in Fig. 5. The depth ranged from 0.001-0.04 ft and varied approximately inversely with the air velocity, while the surface velocity ranged from 0.09-1.8 ft/ sec and varied nearly directly with the air velocity. Film depth increased approximately as the square root of the water rate, but the rate of increase of surface velocity was less than this. 400 o
o
0 x
300 u u
=" 2oo
u o o
a o
c
o o
E LL
~
u
I00 A
I I0
0 5
Air
I 20 velocity,
%
~
'¢~ tat I 40
ft/see
5"C
0"02 I0 0"0120 ° O" 0 0 6 0 • O" 0 0 5 0 • 0'00150
Water rotes ft / s e c
u~ • .... 2'0
•
0"00075
+
0"000375
~
o ~
~
o
a o •
~
TAIT
The wall friction factors generally showed little difference from those for the flow of air in the dry channel. This is in accord with the observation of ELLIS and GAY [6]. DISCUSSION Interracial shear stress That the shear stress resulting from momentum diffusion is governed by the relative velocity of the interface and the gas stream is demonstrated by the coincidence of the interfacial friction factor, which is defined in terms of relative velocity, and the wall friction factor for smooth interfaces. Extension of the relative velocity argument to the shear stress at a disturbed interface requires that the total interracial shear stress should be divided into two parts and that two friction factors should be calculated. The friction factor for the momentum diffusion stress should be calculated from the interface relative velocity, and the friction factor for the pressure distribution stress should be calculated from the wave relative velocity. Because of the small differences between interface velocity and wave velocity in the tests, less than 1 ft/sec, such an elaboration is not warranted for the comparison of the interfacial shear stresses with those at solid boundaries.
a a
Comparison with stress at a solid boundary
I'C
•
•
; o
t
;; I
I0 Air
FIG. 5.
velocity,
•
-"
+
+
!
l
20
40
ft/sec
Liquid depth and surface velocity.
In those tests where a steady interfacial wave pattern was established, the interfacial shear stress varied in the range 0.04-0.4 lb/ft sec 2. According to its usual definition, 51~/pu* [15], the extent of the air laminar sublayer at a smooth boundary 68
Interfacial shear stress and momentum transfer in horizontal gas-liquid flow
then varied from 0.0011-0.00035 ft for this range of interfacial stresses. Because not all the interfacial stress is derived from m o m e n t u m diffusion at non-smooth boundaries, it may well be more appropriate to use only that part of the stress which is derived from m o m e n t u m diffusion in the calculation of the laminar sublayer thickness. To do so would shorten the above thickness range to 0.0011-0.0006ft. The extent of the laminar sublayer is then of the same order of magnitude as the wave amplitudes measured in the tests. This fact, and the small curvature of the interfacial waves where the length is typically about 50 times the amplitude, would make the prospect of gas boundary layer separation very remote. The distance above the interface at which the air velocity reaches the wave velocity is quite small. For an interfacial shear stress of 0.04 lb/ft sec 2 and an extreme value of wave velocity relative to interface velocity of 1 ft/sec, this distance is 0.0003 ft. This level is within the laminar sublayer, so that there can be no velocity profile curvature. These conclusions suggest that the increased shear stress at the disturbed interface arises from the viscous pressure distribution effect shown theoretically by BROOKE BENJAMIN [7]. Wave friction factors calculated according to the above effect are compared with the experimental values, that is values off~ - f o , for a few tests in Table 2.
Table 2. Wave friction factors Water rate (fta/ft sec) 0"000375 0"00075 0"00150 0"0030 0.0060 0-0120 0"0210
Test Air velocity (ft/sec) 23 "2 37"9 18"9 13"4 26"8 13"4 32"7
Wave friction factor Calculated Measured 0.00054 0"00056 0"0010 0"0053 0"0015 0-0043 0"00036
0'0003 0'0006 0.0004 0.0002 0.0019 0"0004 0"0031
There is fair agreement in some of the tests, but three of them show large variations between the calculated and experimental values. The trends seem to be for an overestimate of the experimental stress at smaller air velocities, by an order of magnitude, and for an underestimate at greater air
velocities, especially for large water rates. This comparison is much less favourable than that in Table l, where the experiments of KONOBEEV and ZHAVORONKOV [12] with regular solid wave profiles seem to confirm BROOKE BENJAMIN'S work. The probable source of the disagreement is in the random variation of the wave parameters over the interface. This variation has two effects. The first follows from the use of the defined average values of wave amplitude and wave number in the calculation. The power product of the average values, a2k 16/a, is patently not the average value of that power product of the random variables over the interface. In general, the average value of the power product is greater than the power product of the averages, with the difference increasing with the spread of the parameter variation. Thus the use of the product a2k 16/3 in the calculation leads to an underestimate of the shear stress. This depression of the calculated shear stresses would be more pronounced for the less regular wave patterns, notably those in the tests at larger water rates and air velocities. It should be mentioned that wave number appears in the functions q~l and ~b2. However, it has very little effect in ~bl, and since ~2 is an attenuation function for waveinduced fluid motion with increasing distance from the interface, the use of the average wave number in it seems to be admissible. The second effect of the random variation of the wave parameters is to interfere with the coherence of the convected air pressure distribution. Possibly this effect is enhanced at smaller wave amplitudes, where the flow oscillations in the air stream are less energetic and more easily disrupted by irregularities in the interfacial waves. This argument could explain the large preponderance of the calculated stresses over the measured ones at low air velocities, just after the transition from a smooth or a small amplitude, two-dimensional wave inter-face. Qualitatively, then, these effects can account for the differences between the experimental shear stresses and those calculated according to the viscous pressure distribution effect shown by BROOKE BENJAMIN [7].
Change in stress with gas velocity The changes in interfacial shear stress with gas 69
T. N. SMITHand R. W. F. velocity and their relation to the general form of the gas-liquid interface are worth comment. To facilitate discussion Fig. 6 showing the difference between the interface friction factors and the smooth wall friction factors, has been drawn.
Water rates f t Z/sec
5"0
t7 0 ' 0 2 1 0 0 " 0 I ?.0 o 0.0060 • • • +
= c=
0"0030 O' 00150 O" 0 0 0 7 5 O' 0 0 0 3 7 5
=,a a
o
m 4.0
o
u
o
a
o
o =
_ 3'C
o
o
o oo m ~ a
o
o
=>
o
=a
•
o
2"C
o~oo
•
o
o ~ ~o
I'O
•
•
m= =m
=
°:.,- ....°°°"i':" "'"
3-2:" •o e
oI
.
•
0 a.a
2
I
I
I
I
5
I0
20
50
Reynolds
number
tOO
x 10 - 3
FIG. 6. Wave friction factor. In general the wave friction factor increases from zero at the onset of waves and continues to increase with increasing gas Reynolds number. In the cases of the larger water rates the wave friction factors reach clear maxima at Reynolds numbers of 50,000 or higher, then begin to decrease. In decreasing, the wave friction factors for the three largest water rates show a tendency to coincide. There are no obvious changes of slope in the curves in Fig. 6 to suggest that the onset of unsteady waves abruptly increases the stress. This is to be expected because these waves first appear infrequently, then increase in number and amplitude with increasing air velocity. Some boundary layer separation is to be expected over the unsteady waves because of their greater amplitudes and sharper
TAIT
peaks. With increasing air velocity, the unsteady wave pressure distribution stress would grow and, presumably, become dominant over the viscous pressure distribution as a source of interfacial shear stress. With the start of dispersion at the gas-liquid interface, an additional stress producing mechanism becomes operative. This is m o m e n t u m exchange between the phases resulting from the movement of droplets and bubbles to and from both phases. The rate of momentum exchange by this mechanism naturally increases as the gas velocity and the extent of dispersion increase. The eventual faU in wave friction factor with increasing gas Reynolds number can be explained by the thinning of the liquid film, and the fall in concentration of dispersed liquid in the gas stream. Thinning of the liquid film increases the wave energy dissipation rate [14] so quieting the interface and reducing the pressure distribution stress, and the rate of increase in dispersion. The increasing air rate then overtakes the rate of increase of dispersion and the concentration of droplets dispersed in the air stream begins to decrease. At extreme air velocities, the liquid film is very thin, and consequently fairly smooth, so that the main stress producing mechanism is the "diffusion" of water droplets across the air stream. The "wave" stress is then simply proportional to the concentration of water in the air stream, subject to a general flow structure trend with Reynolds number. In the limit, the "wave" friction factor decreases to zero when the water concentration reaches zero.
The liquid film The liquid film moves through the channel principally under the influence of the interfacial shear stress. Except at low air velocities and larger water rates where the liquid film thickness became large, the surface gradient and channel pressure gradient stresses are smaller by an order of magnitude [14]. Figure 7 shows the relations between the interfacial shear stress and the liquid film thickness and the liqaid interfacial velocity. Theoretical lines for laminar film flow at rates of 0.00075, 0.0030 and 0.0120 ft3/ft sec under a surface shear stress are shown for comparison. 70
Interfacial shear stress and momentum transfer in horizontal gas-liquid flow 400
(D 20C
'- IOC
i2 2~
I0
20
Interfacial
50
I O0
shear s t r e s s ,
200
500
I000
s e c Z x 10 -3
Ib/ft
2'0
"~"
l'C
.
0'.=
/"
. /
•
w
•.
"
~ 0"2.' " / ~u)
"
/
/
_.~ /
•/ 0'12
/, 10
/1"
j~ , -
*
~
ft2/$ec
."
= 0"0210
~
o 0.006o
*
+
~
*
-- • 0"00075
~,
,
,
,
20
.50
IO0
200
lnterfacial
• 0"0050
~. 0 ' 0 0 1 5 0
shear stress,
Ib/ft
• o.o~375
secZx
500
sition between laminar and turbulent flow structures is about 1500. The Reynolds numbers for the water rates 0.0030 and 0.0060 ft3/ft see are 280 and 560, so that these films should remain in laminar flow. Further, even after the flattening of the velocity profile and dilation of the liquid film, the film velocity and thickness continue to change in direct and reciprocal proportion respectively to the square root of the interfacial shear stress. This behaviour, which is that of a film in laminar flow, is common to all water rates. The evidence then suggests not that a transition to a turbulent flow structure occurs, but that the apparent viscosity of the liquid in the film undergoes a fairly abrupt change. Some apparent viscosities of the liquid film have been calculated from their thicknesses and surface velocities for interfacial shear stresses in the range 0.1-0.2 lb/ft sec 2. The ratios of these apparent viscosities to the true viscosity of the water are given in Table 3.
lO00
Table 3. Apparent viscosity ratios of liquid films
10 -3
FIG. 7. Liquid film depth and surface velocity.
At the three smallest water rates the liquid film is evidently laminar over the full range of observations until the onset of unsteady waves. At larger water rates, the liquid film deviates from theoretical laminar behaviour at decreasing levels of the interfacial shear stress. The points of departure are particularly well marked in the curves for the water rates 0.0030 and 0.0060 fta/ft sec. At interracial stresses of about 0.04 and 0.03 lb/ft see 2 respectively, the rate of increase of surface velocity is arrested and falls temporarily with increasing stress. At the same points the rate of decrease of film thickness also halts. These developments indicate a flattening of the film velocity profile, and, necessarily, an increase in the rate of momentum transfer across the film. Although this is a symptom of turbulence, there are good reasons for believing that there is not a change from laminar flow to turbulent flow in the usually implied sense. The flow of the liquid film is similar to plane Couette flow, that is, the flow of a fluid between two parallel planes in relative motion. According to REICHARDT [16] the critical value of the Reynolds number, Lp/I~, for the tran-
Water rate (fta/ft see) 0"000375 0.00075 0"00150 0.0030 0.0060 0'0120 0"0210
Apparent viscosity ratio 1 1 1"3 1"8 2"1 2"7 2"9
The explanation of the change in apparent viscosity is that the interfacial waves, on their formation, promote the transfer of momentum across the liquid film by the convection associated with the wave motion. The film flow transition points for the water rates 0.00150, 0.0030, 0.0060 ft3/ft see are compared with the observed points for the establishment of interfacial waves (other than small amplitude two-dimensional) in Table 4. The correspondence between the film flow and wave transitions is close. Smaller water rates do not show clear flow transitions, presumably because o f the smaller interfacial disturbances and the stronger attenuation of wave-induced convection with movement away from the interface. In the two largest water rates, it would seem that any film flow 71
T. N. SMrrrIand R. W. F. TAIT transition has been masked by the effects of the surface gradient.
the start of the wall boundary layer is typically 10, the wave motions in the wall boundary layer are negligibly small.
Table 4. Transition air velocities for liquid film flow and interface wave formation Water rate (fts/ft sec)
Flowtransition (ft/sec)
Waveformation (ft/sec)
0.00150 0.0030 0.0060
15 13 12
14 12 11
Momentum transfer The significant mechanisms of interfacial momentum transfer are molecular diffusion and the interchange of drops and bubbles between the phases. The correspondence between the interfacial stresses arising from the presence of waves and those calculated according to BROOKE BENJAMIN'S [7] pressure distribution model seems to be adequate when the arguments made in the discussion of the interracial shear stresses are taken into account. It must be concluded, therefore, that the interfacial stress arising from surface "roughness" arises substantially from the pressure distribution mechanism. Consequently there can be no significant promotion of momentum transfer between the phases by interracial wave motion. Momentum transfer across the separate boundary layers is, however, intensified by the fluid motion resulting from the interfacial wave motion. While this convective motion reaches its greatest amplitude at the interface, there can be no interfacial transfer because the motions of the fluid and the interface conform. There is no means of estimating, from the data collected, the extent of momentum transfer by drop and bubble interchange in the experimental system. However, there is a logical limit to this momentum transfer at any water rate. This is based on complete dispersion of the liquid phase. For illustration of the limit to interchange momentum transfer, suppose that the water rate of 0.0120 ft2/sec is just completely dispersed at an air velocity of 120 ft/sec. The density of the air-water stream is then about 1.5 times that of air and, consequently, the rate of diffusion of momentum to a boundary is also 1.5 times that in an air stream. It is difficult to conceive that the particle interchange friction factor for any other air velocity could exceed that for the air velocity which causes just complete dispersion. At smaller air velocities, dispersion rapidly decreases. At larger air velocities, the water concentration in the air stream falls because of the greater air flow.
The near constancy, for any one liquid rate, of the wave-induced increase in apparent viscosity in the liquid film is a consequence of the thinning of the film with increasing interfacial shear stress. Owing to the increasing viscous damping in the thinning film, the rate of growth of wave amplitude is slowed, and the penetration of the wave motion into the liquid film is reduced. Because of the free surface of the liquid film, and of the presence of waves at that surface, the similarity between the film flow and plane Couette flow is not exact. Therefore, it is not suggested that a film Reynolds number less than 1,500 is a sufficient condition for the indefinite maintenance of laminar flow with increasing air velocity.
The 9as film The change in the velocity profile in the interface gas boundary layer with interfacial disturbances is due to the same cause as is the flattening of the liquid layer velocity profile, convection of momentum across the boundary layer by wave motion in the fluid. In the air film, this mechanism of momentum transfer is additional to the molecular diffusion and turbulent transport processes. There is the possibility of secondary effects on momentum transfer, by modification of the turbulent flow structure of the fluid. For interfacial waves of small amplitude, however, it seems that such effects should be small in comparison with the primary effect. This explanation also accounts for the insignificant effect of the interfacial disturbances on the dry wall boundary layer. Since the amplitude of the fluid motion due to the interfacial waves decreases with the distance from the interface approximately as exp(-klYl) [7] and the value of kly[ at 72
Interfacial shear stress and momentum transfer in horizontal gas-liquid flow There are similar limits to the particle interchange m o m e n t u m transfer for the o t h e r water rates. I n general, the limits for smaller water rates w o u l d be d i s p r o p o r t i o n a t e l y smaller, because o f the greater air velocities necessary to disperse the thinner, m o r e stable liquid films.
are quite i n d e p e n d e n t , to the first o r d e r at least, o f the interfacial waves. This is n o t to d e n y a m e c h a n i c a l d e p e n d e n c e o f the extent o f dispersion, between the phases, a n d o f the transfer resulting f r o m t h a t dispersion, on the degree o f interfacial disturbance. M o m e n t u m transfer across the films o n either side o f the interface is, however, e n h a n c e d b y the convection i n d u c e d b y the interfacial wave m o t i o n . I n the case o f a t u r b u l e n t film, where there is a thin l a m i n a r sublayer at the interface, the effect o n the overall diffusion resistance o f the film is insignificant. But for a l a m i n a r film the resistance is substantially reduced.
CONCLUSION T h e shear stress at the interface o f a gas stream a n d a n e n t r a i n e d liquid film is m a d e u p o f w h a t m a y be r e g a r d e d as two quite distinct c o m p o n e n t s , one o f which acts on the interfacial surface, while the o t h e r acts o n the interfacial disturbances. T h e shear stress c o m p o n e n t acting on the surface is generated b y m o m e n t u m transfer processes. A s with the flow o f a fluid p a s t a s m o o t h b o u n d a r y , there is m o l e c u l a r diffusion across the b o u n d a r y l a m i n a r sublayer, resulting in a flux o f m o m e n t u m to the b o u n d a r y . W h e r e there is dispersion between the phases, a n a d d i t i o n a l m o m e n t u m transfer occurs b y interchange o f particles between the fluid streams. B o t h m o m e n t u m transfer processes d e p e n d o n the relative velocity o f the fluid streams. T h e stress c o m p o n e n t acting on the interfacial disturbances is the result o f a d i s t r i b u t i o n o f n o r m a l pressure in the gas stream. P r o v i d e d t h a t the interfacial waves c a n be a d e q u a t e l y defined, the theoretical w o r k o f BROOE:E BENJAMIN [7] c a n be used to estimate the pressure d i s t r i b u t i o n shear stress for a n y flow in the interface b o u n d a r y layer. F o r small wave amplitudes, however, the theoretical stresses are smaller t h a n the e x p e r i m e n t a l stresses b y a n o r d e r o f m a g n i t u d e . This difference m i g h t be explained b y the interference o f r a n d o m variat i o n in wave p a r a m e t e r s , a n d p o s s i b l y o f t u r b u l e n t flow fluctuations, with the f o r m a t i o n o f c o h e r e n t pressure distributions. The i n t e r p h a s e m o m e n t u m transfer m e c h a n i s m s
NOTATION a Wave amplitude b Film thickness b~ Interface gas boundary layer thickness bw Wall gas boundary layer thickness c Wave velocity fi Interface friction factor f0 Smooth boundary friction factor k Wave number Pv Periodic normal pressure from viscous effects u Velocity uc Velocity at level u = c m Interface velocity u* Friction velocity x Longitudinal distance parameter y Normal distance parameter Zo Parameter in Appendix L Liquid volume flow per unit width U Maximum in gas stream velocity profile /z Fluid viscosity /zg Gas viscosity /9 Fluid density pu Gas density "re Wave shear stress from profile curvature effects ~-~ Interfacial shear stress 70 Smooth boundary shear stress "r~v Periodic shear stress from periodic normal pressure "ro Wave shear stress from viscous effects •rw Wall shear stress q~l Function in Appendix ¢2 Function in Appendix
REFERENCES [1] [2] [3] [4] [5] [6] [7]
ALVESG. E., Chem. Enon# Proyr. 1954 50 449. HANRATTYT. J. and ENGENJ. M., A. L Ch.E. Jl., 1957 3 229. LILLELErrrL. U. and HANRATTV T. J., A. L Ch. E. Jl., 1961 7 548. SMrra T. N. and TArt, R. W. F., Aust. J. appl. ScL, 1964 15 247. VAN ROSSUMJ. J., Chem. Enong Science, 1959 11 35. ELLISS. R. M. and GAY B., Trans. Inst. Chem. Engrs., 1949 37 206. BROOKEBENJAMrNT,, J. Fluid. Mech., 1959 6 161. 73
T. N. SMITH and R. W. F. TAlT [8] [9] [10] [11] [12] [13] [14] [15] [16]
MILES J. W., J. Fluid Mech., 1957 3 185. STAr,TON T. E., MARSHALLD., and HOUOnTON R., Proe. R. Soe., 1932 A137 283. STI~E'rER V. J., Trans. Am. Soe. ely. En#rs., 1936 101 681. MOTZr~LD J. A., Z. Angew. Math. Mech., 1937 17 193. KONOBEEVV. I. and Za-IAVORONKOVN. M., Int. Jl. Chem. Engng 1962 2 431. Pr.ANDTL L., Z. Vet. dt. Ing., 1933 77 105. SMrrrI T. N., Ph.D. Thesis, University of Adelaide, 1964. y o n K A Z A N T., Trans. Am. Sci. Mech. Engrs. 1939 61 705. RFaCHARDTH., Z. An#ew Math. Mech., 1956 36 26.
where
APPENDIX
Z o = kl/apl/a#I/aCZo2/a Stress at a solid wavy boundary
Values of 41 tabulated by BROOKE BENJAMIN are For the periodic normal stress component given below in graphical form. lagging the boundary profile phase by n/2, BROOKE 2"0 BENJAMIN'S [7] equation (7.43) gives Pv = - a k l a / a z o 5/3p4/3~t4]3( U - ¢ ) 4 4 1 4 2
cos(kx - 1t/2). The periodic shear stress due to this component acting on the boundary y = a cos k x is Tpv = - a k P v
sin
kx,
"e: ~°o ~
which integrated, and averaged over a wave length becomes
-
i
4
]2
or in friction factor form, -
FIo. 1.
I
2 3 Zo Values of 41.
The function ¢2 accounts for the effects of the fluid velocity profile on the boundary stress, and is defined
To = ~a2kl6/aTo 5/3p4/a[g4/3(U -- C)z!'4142 '
.~v/p(U
I t
o
C)2 = ½a2kl6/afo ~/ap-g/a~t4/a
42 =
( U - c)-4/3¢~42.
The function ¢1 determines the magnitude of the viscous effects near the boundary, and is d e f i n e d by BROOKE BENJAMIN as
o \U--
c]
exp(-kly) dy
In evaluating this integral a logarithmic velocity profile, such that
has been assumed.
¢1 = ¢(Z0)
R r s u m r - - L ' a u t e u r exprrimente l'~coulement stratifi6 d'un courant d'air et d'un film d'eau darts un canal horizontal ferm6 de section rectangulaire: 16 × 2 inch pour des vitesses de gaz comprises entre 6 et 120 ft/sec, et pour des vitesses superficielles de liquide comprises entre 0,000375 et 0,0210 fta/ft]sec. La rrsistance [interfaciale a 6t6 mesurre e t a 6t6 drcomposre en 2 616ments, Fun rrsultant de la force de diffusion relative tt la srparation lisse, l'autre rrsultant de la rrpartition de la pression normale produite clans le fluide par un mouvement ondulatoire interfacial. On en conclue que la rrsistance tt la distribution de la pression ~ l'interface perturbre par les ondes stationnaires, est en accord avec les prrdictions fondres sur la throrie de BROOKE BENJAMIN, relative h l'rcoulement dans le voisinage de la srparation ondulante. Zusammenfassung--Untersucht wird die laminare StrSmung von Luft oder eines Wasserfilms durch einen horizontal liegenden rechteckigen Kanal (16 × 2 in.), wobei die StrSrnungsgeschwindigkeit der Luft zwischen 6 und 120 ft/sec und die Durchstitze yon Wasser zwischen 10,000375 und 0,0210 ft3/ft sec liegen. Die Schersparmungen an der Phasengrenze werden gemessen und in 'zwei Komponenten zerlegt, deren eine auf dem Impulsiibergang an die glatte Wand und deren andere auf dem Druckgeftille
74
Interfacial shear stress and momentum transfer in horizontal gas-liquid flow im strOmenden Medium (das durch Wellenbewegungen an der Phasengrenze entsteht) beruht. Die Scherspannung dutch den Druckgradienten an einer dutch station~ire Wellen gestOrten Phasengrenze stimmt mit der theoretischen Voraussage von B~OOKE BSN~AMIN[7] ffir Str6mungen fiber gewellte Fl~ichen tiberein.
75
Interfacial shear stress and momentum transfer in horizontal gas-liquid flow T. N. SMITH and R. W. F. TAIT Chemical Engineering Department, University of Adelaide, South Australia (Received 21 December 1964; in revisedform 11 March 1965)
Abstract--The stratified flow of an air stream and a water film in a dosed horizontal channel of rectangular section, 16 in. by 2 in., has been investigated for air velocities ranging between 6 and 120 ft/sec and for superficial water rates between 0.000375 and 0.0210 fta/ft sec. The interfacial shear stress has been measured, and has been separated into two components, one resulting from a momentum diffusion as to a smooth boundary, and the other resulting from a distribution'of normal pressure induced in the fluid by the interfacial wave motion. It is concluded that the pressure distribution shear stress at an interface disturbed by steady waves is in agreement with predictions based on the theoretical work of BROOKEBENJAMIN[7] o n flOWpast a wavy boundary. INTRODUCTION THE mechanics of flow of a gas stream and a liquid film moving under the influence of the gas stream are of vital interest in the design of a variety of mass and heat transfer equipment. In this study, the flow of air and a water film in a horizontal channel has been investigated, and the magnitude and origin of the interfacial shear stress have been examined. In particular, the mechanism of m o m e n t u m transfer between the phases has been discussed because of its relation to heat and mass transfer processes. It has been observed by several authors, for example ALVES [1], that the interface in a conduit carrying a gas and a liquid in cocurrent, horizontal, stratified flow may be either smooth, or wavy, or wavy and disperse depending on the flow rates of the two phases. In flow over a fiat, smooth surface, it is well known that m o m e n t u m is transferred from the free flow by both turbulent eddies and molecular diffusion across the boundary layer to a zone of decreasing turbulent intensity close to the boundary. Turbulence is effectively damped at some small distance from the surface, and from this distance m o m e n t u m is transferred towards the boundary by molecular diffusion. When a fluid flows over a surface having protuberances, a pattern of velocity variations is induced in the fluid near the surface as the flow conforms to the boundary profile. In steady flow 63
these velocity changes result in corresponding pressure changes in the fluid so that a pressure distribution stationary relative to the surface is established. Figure l(a) illustrates qualitatively, the velocity and pressure distributions in the flow of an ideal (frictionless) fluid over a surface having a sinusoidal wave profile. In this case, the pressure distribution is in phase with the boundary profile, and exerts no resultant thrust on the surface in the flow direction. In the flow of real fluids, however, the pressure distribution is not, in general, in exact phase with the boundary profile. Figure l(b) shows a pressure distribution which lags the boundary profile by ~r/2. It is readily seen that the action of the distribution upon the profile slopes leads to a resultant stress in the flow direction. This stress derived from the distribution of normal pressure near the boundary is " f o r m drag". A pronounced form drag occurs where there is separation of the fluid boundary layer at protuberances, so producing a large out of phase component in the pressure distribution. Both the m o m e n t u m diffusion and the pressure distribution mechanisms can be expected to contribute to the shear stress at a disturbed gas streamliquid film interface. Where there is dispersion between the phases, the interchange of drops and bubbles introduces an additional m o m e n t u m transfer component.
T. N. SMITHand R. W. F. TAIT
Velocity ÷ changes --
Pressure
÷
changes _
Surface
profile
(a) Pressure changes in phase with surface
profile
Velocity + changes
Pressure changes
~
~
~~
~
f
~
I
~
~
"l" --
Surface
profile
{b) Pressure changes log surface profile by -n'/2
FIG. I.
Velocity and pressure changes in flow over wavy surface.
PREVIOUS AND RELATEDWORK
The gas-liquid interface HANRATTY and ENGEN [2] have given a qualitative description of the gas-liquid interface during their experiments on the cocurrent flow of air and water in a channel of rectangular section, 12 in. wide by 1 in. high. With increasing air velocity, there appeared in sequence the five distinct types of interface patterns; (1) smooth, (2) "two-dimensional" waves with long crests extending the full width of the channel, (3) "squall" waves having short crests and giving the interface a "pebbled" appearance, (4) "roll" waves which formed on the squall surface and travelled at a much greater velocity, and (5) dispersed flow where entrainment of water droplets in the air stream began. The "squall" wave interface, which was a steady pattern and which persisted over a range of air velocities and water rates, was more closely defined by LILLELEHT and HANRATTY [3] w i t h the measure-
ment of wave profilesby alightabsorptiontechnique. For air velocities in the range 12-33 ft/sec and superficial water rates in the range 0.002-0.005 fta/ft see, they found that the root mean square values of wave amplitude and wave frequency varied from 0.0008 to 0.0016 ft and 21 to 26 see -~ respectively. The distribution of wave amplitudes at fixed air and water rates was found to be Gaussian. Measurements of interfacial wave parameters for the flow of an air stream and a water film in the 16 in. wide by 2 in. high channel used in this study have been reported elsewhere [4]. These measurements covered a range of air velocity from 5-120 ft/ sec and of superficial water rate from 0.0003750.0210 fta/ft sec. The wave amplitudes for the steady patterns, the "two-dimensional" and "squall" waves described by HANRATTY and ENGEN [2], ranged up to 0-0025 ft. Wave number (2n/wave length) varied from 50-210 f t - l , and wave velocity ranged from 0-5 to 2.5 ft/sec. The wave amplitudes were found from the changes in the capacitance to earth of a probe in the air stream immediately above the water surface. Adequate resolution of the surface profile was ensured by keeping the size of the probe small in comparison with the interfacial wavelengths. Wave velocities were measured by timing their passages between fixed locations using a stopwatch. The wave numbers were computed from the wave velocities using wave frequencies measured with the capacitance probe apparatus. Further details of these techniques are available in the paper on wave parameters [4].
lnterfacial shear stress HANRATTY and ENGEN [2], VAN ROSStrM [5], and ELLIS and GAY [6] have all reported greater shear stresses for the flow of a gas stream over a disturbed gas-liquid interface than over a smooth, solid surface. Having an adequate description of the interfacial wave profile, an attempt can be made to relate interfacial shear stresses from the gas-liquid system to those developed in the flow of a fluid over a solid surface of comparable profile. The experimental wave parameters reported earlier [4] provide a description of the interfacial profile. The steady waves are nearly sinusoidal and have, typically, 64
Interfacial shear stress and momentum transfer in horizontal gas-liquid flow ratios of wave length and boundary layer thickness to wave amplitude of 40 and 50 respectively. Here the interfacial boundary layer thickness is the distance from the interface to the air stream velocity profile maximum.
laminar sublayer, there can be no profile curvature at this point and the latter effect is absent. The resultant shear stress due to the viscous effect is
Shear stress at a solid boundary
and that due to velocity profile curvature is
"Cv -~ ½ a 2 k 1 6 / a z S o / 3 p a / 3 # 4 / a ( u
The existence of a stationary pressure distribution in non-separated flow over a wavy boundary has been demonstrated theoretically by BROOKE BENJAMIN [7]. Analysing the flow of a fluid over a two-dimensional surface having the form of a small amplitude sine wave, he has shown that the first order perturbations in the flow of the fluid in following the boundary give rise to periodic tangential and normal stresses on the wavy boundary. The periodic tangential stress has no resultant, in the flow direction, over a wave length. However, the normal stress has a component differing in phase from the surface waves by 1t/2 which does have a resultant in the flow direction. The resultant from the normal stress has two components, one based on the effects of viscosity on flow near the boundary, and the other on the velocity profile curvature at the point where the fluid velocity is equal to the wave velocity. The effect due to velocity profile curvature was first shown by MILES [8]. Should the point where the fluid velocity is equal to the wave velocity be inside the
Zc =
a2kap(U--c)
_ c)4~1~2
-4 U"C - ~ qb2,
where a is the wave amplitude, k is the wave number, Zo is the smooth boundary (momentum diffusion) shear stress, p is the fluid density, # is the fluid viscosity, U is the fluid free stream velocity, c is the wave velocity, uc is the normal distance derivative of the fluid velocity u at U = c, and u~ is the second derivative. ~bI is a function which determines the magnitude of the viscous effects near the boundary, while ~b2 gives the effect of velocity profile on the boundary stress. These functions are defined in the Appendix. Measurements of the shear stress at solid surfaces with sinusoidal profiles have been made by STANTON, MARSHALL, and HOUGHTON[9], STREETER[10], MOTZFELD [11], and KONOBEEV and ZrIAVORONKOV [12]. Of these workers, only the last named have studied profiles which approach the geometry of the interracial wave and boundary layer system. The measurements of KONOBEEVand ZHAVORONKOV
Table 1. Comparison o f measured [12] and calculated [7] solid wave friction factors Tube
c
Friction factors 0"~ -- 7o)/p U2
No.
a (mm)
k (mm -1)
dia. (mm)
6
0.325
0.785
15.9
7
0.22
0-785
16.0
8
0.14
0.785
16-3
9
0.36
0.524
15.8
10
0.135
0.524
16.2
16
0.175
0.785
11.1
17
0.10
0"785
11.1
NR~ = 10,000
NR, = 30,000
NRe = 100,000
0.0061 0.0016 0.0027 0"0021 0.0011 0.0012 0.0036 0.0044 0.0006 0.0010 0.0022 0.0009 0.0009 0.0004
0.0041 0.0013 0.0017 0.0011 0.0006 0.0006 0.0022 0.0022 0.0004 0.0004 0.0014 0-0005 0.0004 0.0002
0.0031 0.0006 0.0012 0.0005 0.0006 0.0002 0.0017 0.0009 0.0003 0.0002
Meas. Calc. Meas. Calc. Meas. Calc. Meas. Calc. Meas. Calc. Meas. Calc. Meas. Calc.
65
T. N. Sma-ri and R. W. F. TAn" appear to support the theoretical work of BROOKE BENJAMIN. Table 1 compares values of the wave friction factor taken from KOI~OBEEV and ZHAVORONKOV's paper with values calculated from BROOKE BENJAMIN'S viscous effect. The profile curvature stress is, of course, absent because there is no movement of the wavy boundary. These friction factors are in the form (~t - Zo)/p U 2 where z~ is the total wall stress, and the Reynolds number is in the boundary layer form b~Up/# where b~ is the boundary layer thickness (here, the pipe radius). Only the long wave length, small amplitude results from KONOBEEVand ZHAVORONKOV'S work have been included, because these are the results for profiles comparable with the airwater interfacial wave profiles, and because there is evidence of boundary layer separation in the other results. Indeed, boundary layer separation could account for the relatively poor agreement for Tube No. 6, where there is a combination of shorter wave length and greater amplitude. EXPERIMENTAL Experimental observations were made on an airstream water-film system in a 16 in. wide by 2 in. high horizontal channel. The channel was wide enough for flow in the middle to be treated as two-dimensional, and long enough, 12 ft, for equilibrium flow to be established in the latter half of its length. The apparatus is shown in Fig. 2. Air was drawn into the channel by a fan, and the water film moved through the channel under the influence of the interfacial shear stress.
Fio. 2. Arrangement of experimental channel. (1) Air inlet cone (2) Water inlet piece (3) Channel (4) Stiffening bars (5) Measuring station (6) Water outlet piece (7) Centrifugal fan (8) Head tank (9) Rotameter (10) Sump tank.
A vertically traversing impact tube and a static pressure tapping in the roof of the channel, connected to a micromanometer, were used to find the air layer velocity profile. The air velocity was varied from 5 to 120 ft/sec. The water rate was measured by calibrated rotameters, and was varied from 0.000375 to 0.0210 ft3/ft see (expressed as flow per unit width of channel). The water surface velocity was estimated by dropping small pieces of polyvinyl chloride sheet, about 0.02 in. square and 0.0035 in. thick, onto the water surface and timing their passages between fixed locations with a stopwatch. The surface velocities were estimated to be accurate within ___5 per cent. The water depth was measured by using a vertically traversing pointer, fitted to a micrometer mechanism, within an estimated error of -t-0.0004 ft. Interface and wall shear stresses were found from a balance of shear stress and pressure gradient over the boundary layer, that is the region between the boundary and the air velocity profile maximum. The error in these stress measurements was estimated at + 4 per cent. The methods used for measuring the interfacial wave parameters have already been noted. RESULTS
Interracial pattern In general the interfacial patterns were similar to those described by HANRATTY and ENGEN [2]. In order of appearance with increasing air velocity they were: (1) smooth, (2) two-dimensional waves, with crests extending the full width of the channel, (3) rhombic waves, short-crested waves giving the water surface the "pebbled" appearance described by HANRATTY and ENGEN, (4) unsteady waves, accelerating "roll" type waves, (5) disperse, with penetration of water droplets into the air stream and air bubbles into the water film. The profiles of both the two-dimensional and the rhombic waves were nearly sinusoidal. The amplitudes and frequencies of the two-dimensional waves were quite regular, but those of the rhombic waves varied in an apparently random manner as remarked by LILLELEHTand HANRATTY [3]. Figure 3 shows the areas occupied by the various interface patterns on a water rate-air velocity chart. 66
Interfacial shear stress and momentumtransfer in horizontalgas-liquid flow
210
120
6C 0 x
3C e)
=;
15
,o 7,5
3.75
5-
,'o
z'o Air velocity,
FZG. 3.
go
I 100
fl/sec
Interracial patterns.
There is good reason to believe that the start of interphase dispersion is identifiable with the onset of unsteady waves. It was found impossible to measure the amplitudes of unsteady waves with the capacitance probe because of the continual attachment to the probe tip, and occasionally the supporting rod, of water droplets from the air stream. The attachment of such droplets naturally caused violent pulses in the capacitance bridge circuit, making the instrument inoperable. To avoid the attachment of water droplets it was necessary to move the probe so far from the water surface that the sensitivity of the circuit was reduced to a useless level. Although the concentration of water droplets in the air stream was immeasurably small at the start of unsteady waves, this point must be regarded as the start also of interphase dispersion. In further support of this contention, it was observed that some of the unsteady waves, especially those at the larger water rates, appeared to contain air bubbles, representing dispersion of the gas phase in the liquid.
"roughness" of the gas-liquid interface. The stresses were, therefore, converted to friction factors, and the corresponding boundary layer flows were characterised by Reynolds numbers. In order to be consistent with the notion of the development of shear stress by momentum diffusion to the boundary, the interface friction factor was defined as zJpg(U - u~)2, where ui is the interface velocity. The wall friction factor was defined as z~/pgU 2. The boundary layer Reynolds rnumbers were defined as b~(U-ui)pg/it 9 and bwUpg/l~g respectively, where b~ and b~ are the interface and wall boundary layer thicknesses. In Fig. 4 the interface friction factor is plotted against Reynolds number for all water rates. For comparison the wall friction factors determined in the dry channel are shown as a full line. The dry channel friction factors follow the smooth wall correlation of PgA~DTL [13] very closely up to a Reynolds number of 40,000. Above this Reynolds number the friction factor increases, presumably owing to flow structure effects associated with the geometry of the channel. The differencebetween the "interface" and smooth friction factors generally increased with increasing it
= • ° • • • *
8"0
rotes ftZ/sec
Water
7"C
0 " 0 2 IO 0"0120 0" 0 0 6 0 O" 0 0 . 3 0 0"00150 O" 0 0 O 7 5 0"000375
,, ° .~, 0= =
°0
~
°
m
6.c o
o
o
o
.
o o
= o oo
"G s.o o
=*t
o
o
oo
o
o o
o
4'0 =D'~
||=
o~ °
•= ===
3.0
•
o
•
=l
I
" ~ - + ' =. ".t " ,. : ' .. :~. . "•
°
Interfacial shear stress The thicknesses of the interface and wall boundary layers in any test were, in general, not equal. Consequently the respective shear stresses could not be compared directly in order to evaluate the
2.0
g
,'0
I
~0
Reynolds number
go x l O -3
FZG.4. Interfacefriction factor. 67
100
T. N. SMm-[ and R. W. F.
water rate and Reynolds number, and the point of departure of the interface friction factor from the smooth wall value was earlier, in terms of Reynolds number, with increasing water rate. These trends in interface friction factor follow those in interfacial wave amplitude [4]. It is noteworthy that the friction factors at the two largest water rates reached maxima, then fell with increasing Reynolds number. A similar trend was detected by VAN ROSSUM [5] in his experiments with air over a hydrocarbon oil.
These trends in film depth and surface velocity were followed less closely at the larger water rates.
The #as film The same thickening of the interface boundary layer with the growth of interfacial disturbances as reported by HANRATTY and ENGEN [2] was observed. The velocity profile in the interface boundary layer closely approximated PRANDTL'S logarithmic distribution [13]. b u = U - 2"5 u* I n - . Y
The liquid film The depth of the water film and the water surface velocity are graphed against the air velocity in Fig. 5. The depth ranged from 0.001-0.04 ft and varied approximately inversely with the air velocity, while the surface velocity ranged from 0.09-1.8 ft/ sec and varied nearly directly with the air velocity. Film depth increased approximately as the square root of the water rate, but the rate of increase of surface velocity was less than this. 400 o
o
0 x
300 u u
=" 2oo
u o o
a o
c
o o
E LL
~
u
I00 A
I I0
0 5
Air
I 20 velocity,
%
~
'¢~ tat I 40
ft/see
5"C
0"02 I0 0"0120 ° O" 0 0 6 0 • O" 0 0 5 0 • 0'00150
Water rotes ft / s e c
u~ • .... 2'0
•
0"00075
+
0"000375
~
o ~
~
o
a o •
~
TAIT
The wall friction factors generally showed little difference from those for the flow of air in the dry channel. This is in accord with the observation of ELLIS and GAY [6]. DISCUSSION Interracial shear stress That the shear stress resulting from momentum diffusion is governed by the relative velocity of the interface and the gas stream is demonstrated by the coincidence of the interfacial friction factor, which is defined in terms of relative velocity, and the wall friction factor for smooth interfaces. Extension of the relative velocity argument to the shear stress at a disturbed interface requires that the total interracial shear stress should be divided into two parts and that two friction factors should be calculated. The friction factor for the momentum diffusion stress should be calculated from the interface relative velocity, and the friction factor for the pressure distribution stress should be calculated from the wave relative velocity. Because of the small differences between interface velocity and wave velocity in the tests, less than 1 ft/sec, such an elaboration is not warranted for the comparison of the interfacial shear stresses with those at solid boundaries.
a a
Comparison with stress at a solid boundary
I'C
•
•
; o
t
;; I
I0 Air
FIG. 5.
velocity,
•
-"
+
+
!
l
20
40
ft/sec
Liquid depth and surface velocity.
In those tests where a steady interfacial wave pattern was established, the interfacial shear stress varied in the range 0.04-0.4 lb/ft sec 2. According to its usual definition, 51~/pu* [15], the extent of the air laminar sublayer at a smooth boundary 68
Interfacial shear stress and momentum transfer in horizontal gas-liquid flow
then varied from 0.0011-0.00035 ft for this range of interfacial stresses. Because not all the interfacial stress is derived from m o m e n t u m diffusion at non-smooth boundaries, it may well be more appropriate to use only that part of the stress which is derived from m o m e n t u m diffusion in the calculation of the laminar sublayer thickness. To do so would shorten the above thickness range to 0.0011-0.0006ft. The extent of the laminar sublayer is then of the same order of magnitude as the wave amplitudes measured in the tests. This fact, and the small curvature of the interfacial waves where the length is typically about 50 times the amplitude, would make the prospect of gas boundary layer separation very remote. The distance above the interface at which the air velocity reaches the wave velocity is quite small. For an interfacial shear stress of 0.04 lb/ft sec 2 and an extreme value of wave velocity relative to interface velocity of 1 ft/sec, this distance is 0.0003 ft. This level is within the laminar sublayer, so that there can be no velocity profile curvature. These conclusions suggest that the increased shear stress at the disturbed interface arises from the viscous pressure distribution effect shown theoretically by BROOKE BENJAMIN [7]. Wave friction factors calculated according to the above effect are compared with the experimental values, that is values off~ - f o , for a few tests in Table 2.
Table 2. Wave friction factors Water rate (fta/ft sec) 0"000375 0"00075 0"00150 0"0030 0.0060 0-0120 0"0210
Test Air velocity (ft/sec) 23 "2 37"9 18"9 13"4 26"8 13"4 32"7
Wave friction factor Calculated Measured 0.00054 0"00056 0"0010 0"0053 0"0015 0-0043 0"00036
0'0003 0'0006 0.0004 0.0002 0.0019 0"0004 0"0031
There is fair agreement in some of the tests, but three of them show large variations between the calculated and experimental values. The trends seem to be for an overestimate of the experimental stress at smaller air velocities, by an order of magnitude, and for an underestimate at greater air
velocities, especially for large water rates. This comparison is much less favourable than that in Table l, where the experiments of KONOBEEV and ZHAVORONKOV [12] with regular solid wave profiles seem to confirm BROOKE BENJAMIN'S work. The probable source of the disagreement is in the random variation of the wave parameters over the interface. This variation has two effects. The first follows from the use of the defined average values of wave amplitude and wave number in the calculation. The power product of the average values, a2k 16/a, is patently not the average value of that power product of the random variables over the interface. In general, the average value of the power product is greater than the power product of the averages, with the difference increasing with the spread of the parameter variation. Thus the use of the product a2k 16/3 in the calculation leads to an underestimate of the shear stress. This depression of the calculated shear stresses would be more pronounced for the less regular wave patterns, notably those in the tests at larger water rates and air velocities. It should be mentioned that wave number appears in the functions q~l and ~b2. However, it has very little effect in ~bl, and since ~2 is an attenuation function for waveinduced fluid motion with increasing distance from the interface, the use of the average wave number in it seems to be admissible. The second effect of the random variation of the wave parameters is to interfere with the coherence of the convected air pressure distribution. Possibly this effect is enhanced at smaller wave amplitudes, where the flow oscillations in the air stream are less energetic and more easily disrupted by irregularities in the interfacial waves. This argument could explain the large preponderance of the calculated stresses over the measured ones at low air velocities, just after the transition from a smooth or a small amplitude, two-dimensional wave inter-face. Qualitatively, then, these effects can account for the differences between the experimental shear stresses and those calculated according to the viscous pressure distribution effect shown by BROOKE BENJAMIN [7].
Change in stress with gas velocity The changes in interfacial shear stress with gas 69
T. N. SMITHand R. W. F. velocity and their relation to the general form of the gas-liquid interface are worth comment. To facilitate discussion Fig. 6 showing the difference between the interface friction factors and the smooth wall friction factors, has been drawn.
Water rates f t Z/sec
5"0
t7 0 ' 0 2 1 0 0 " 0 I ?.0 o 0.0060 • • • +
= c=
0"0030 O' 00150 O" 0 0 0 7 5 O' 0 0 0 3 7 5
=,a a
o
m 4.0
o
u
o
a
o
o =
_ 3'C
o
o
o oo m ~ a
o
o
=>
o
=a
•
o
2"C
o~oo
•
o
o ~ ~o
I'O
•
•
m= =m
=
°:.,- ....°°°"i':" "'"
3-2:" •o e
oI
.
•
0 a.a
2
I
I
I
I
5
I0
20
50
Reynolds
number
tOO
x 10 - 3
FIG. 6. Wave friction factor. In general the wave friction factor increases from zero at the onset of waves and continues to increase with increasing gas Reynolds number. In the cases of the larger water rates the wave friction factors reach clear maxima at Reynolds numbers of 50,000 or higher, then begin to decrease. In decreasing, the wave friction factors for the three largest water rates show a tendency to coincide. There are no obvious changes of slope in the curves in Fig. 6 to suggest that the onset of unsteady waves abruptly increases the stress. This is to be expected because these waves first appear infrequently, then increase in number and amplitude with increasing air velocity. Some boundary layer separation is to be expected over the unsteady waves because of their greater amplitudes and sharper
TAIT
peaks. With increasing air velocity, the unsteady wave pressure distribution stress would grow and, presumably, become dominant over the viscous pressure distribution as a source of interfacial shear stress. With the start of dispersion at the gas-liquid interface, an additional stress producing mechanism becomes operative. This is m o m e n t u m exchange between the phases resulting from the movement of droplets and bubbles to and from both phases. The rate of momentum exchange by this mechanism naturally increases as the gas velocity and the extent of dispersion increase. The eventual faU in wave friction factor with increasing gas Reynolds number can be explained by the thinning of the liquid film, and the fall in concentration of dispersed liquid in the gas stream. Thinning of the liquid film increases the wave energy dissipation rate [14] so quieting the interface and reducing the pressure distribution stress, and the rate of increase in dispersion. The increasing air rate then overtakes the rate of increase of dispersion and the concentration of droplets dispersed in the air stream begins to decrease. At extreme air velocities, the liquid film is very thin, and consequently fairly smooth, so that the main stress producing mechanism is the "diffusion" of water droplets across the air stream. The "wave" stress is then simply proportional to the concentration of water in the air stream, subject to a general flow structure trend with Reynolds number. In the limit, the "wave" friction factor decreases to zero when the water concentration reaches zero.
The liquid film The liquid film moves through the channel principally under the influence of the interfacial shear stress. Except at low air velocities and larger water rates where the liquid film thickness became large, the surface gradient and channel pressure gradient stresses are smaller by an order of magnitude [14]. Figure 7 shows the relations between the interfacial shear stress and the liquid film thickness and the liqaid interfacial velocity. Theoretical lines for laminar film flow at rates of 0.00075, 0.0030 and 0.0120 ft3/ft sec under a surface shear stress are shown for comparison. 70
Interfacial shear stress and momentum transfer in horizontal gas-liquid flow 400
(D 20C
'- IOC
i2 2~
I0
20
Interfacial
50
I O0
shear s t r e s s ,
200
500
I000
s e c Z x 10 -3
Ib/ft
2'0
"~"
l'C
.
0'.=
/"
. /
•
w
•.
"
~ 0"2.' " / ~u)
"
/
/
_.~ /
•/ 0'12
/, 10
/1"
j~ , -
*
~
ft2/$ec
."
= 0"0210
~
o 0.006o
*
+
~
*
-- • 0"00075
~,
,
,
,
20
.50
IO0
200
lnterfacial
• 0"0050
~. 0 ' 0 0 1 5 0
shear stress,
Ib/ft
• o.o~375
secZx
500
sition between laminar and turbulent flow structures is about 1500. The Reynolds numbers for the water rates 0.0030 and 0.0060 ft3/ft see are 280 and 560, so that these films should remain in laminar flow. Further, even after the flattening of the velocity profile and dilation of the liquid film, the film velocity and thickness continue to change in direct and reciprocal proportion respectively to the square root of the interfacial shear stress. This behaviour, which is that of a film in laminar flow, is common to all water rates. The evidence then suggests not that a transition to a turbulent flow structure occurs, but that the apparent viscosity of the liquid in the film undergoes a fairly abrupt change. Some apparent viscosities of the liquid film have been calculated from their thicknesses and surface velocities for interfacial shear stresses in the range 0.1-0.2 lb/ft sec 2. The ratios of these apparent viscosities to the true viscosity of the water are given in Table 3.
lO00
Table 3. Apparent viscosity ratios of liquid films
10 -3
FIG. 7. Liquid film depth and surface velocity.
At the three smallest water rates the liquid film is evidently laminar over the full range of observations until the onset of unsteady waves. At larger water rates, the liquid film deviates from theoretical laminar behaviour at decreasing levels of the interfacial shear stress. The points of departure are particularly well marked in the curves for the water rates 0.0030 and 0.0060 fta/ft sec. At interracial stresses of about 0.04 and 0.03 lb/ft see 2 respectively, the rate of increase of surface velocity is arrested and falls temporarily with increasing stress. At the same points the rate of decrease of film thickness also halts. These developments indicate a flattening of the film velocity profile, and, necessarily, an increase in the rate of momentum transfer across the film. Although this is a symptom of turbulence, there are good reasons for believing that there is not a change from laminar flow to turbulent flow in the usually implied sense. The flow of the liquid film is similar to plane Couette flow, that is, the flow of a fluid between two parallel planes in relative motion. According to REICHARDT [16] the critical value of the Reynolds number, Lp/I~, for the tran-
Water rate (fta/ft see) 0"000375 0.00075 0"00150 0.0030 0.0060 0'0120 0"0210
Apparent viscosity ratio 1 1 1"3 1"8 2"1 2"7 2"9
The explanation of the change in apparent viscosity is that the interfacial waves, on their formation, promote the transfer of momentum across the liquid film by the convection associated with the wave motion. The film flow transition points for the water rates 0.00150, 0.0030, 0.0060 ft3/ft see are compared with the observed points for the establishment of interfacial waves (other than small amplitude two-dimensional) in Table 4. The correspondence between the film flow and wave transitions is close. Smaller water rates do not show clear flow transitions, presumably because o f the smaller interfacial disturbances and the stronger attenuation of wave-induced convection with movement away from the interface. In the two largest water rates, it would seem that any film flow 71
T. N. SMrrrIand R. W. F. TAIT transition has been masked by the effects of the surface gradient.
the start of the wall boundary layer is typically 10, the wave motions in the wall boundary layer are negligibly small.
Table 4. Transition air velocities for liquid film flow and interface wave formation Water rate (fts/ft sec)
Flowtransition (ft/sec)
Waveformation (ft/sec)
0.00150 0.0030 0.0060
15 13 12
14 12 11
Momentum transfer The significant mechanisms of interfacial momentum transfer are molecular diffusion and the interchange of drops and bubbles between the phases. The correspondence between the interfacial stresses arising from the presence of waves and those calculated according to BROOKE BENJAMIN'S [7] pressure distribution model seems to be adequate when the arguments made in the discussion of the interracial shear stresses are taken into account. It must be concluded, therefore, that the interfacial stress arising from surface "roughness" arises substantially from the pressure distribution mechanism. Consequently there can be no significant promotion of momentum transfer between the phases by interracial wave motion. Momentum transfer across the separate boundary layers is, however, intensified by the fluid motion resulting from the interfacial wave motion. While this convective motion reaches its greatest amplitude at the interface, there can be no interfacial transfer because the motions of the fluid and the interface conform. There is no means of estimating, from the data collected, the extent of momentum transfer by drop and bubble interchange in the experimental system. However, there is a logical limit to this momentum transfer at any water rate. This is based on complete dispersion of the liquid phase. For illustration of the limit to interchange momentum transfer, suppose that the water rate of 0.0120 ft2/sec is just completely dispersed at an air velocity of 120 ft/sec. The density of the air-water stream is then about 1.5 times that of air and, consequently, the rate of diffusion of momentum to a boundary is also 1.5 times that in an air stream. It is difficult to conceive that the particle interchange friction factor for any other air velocity could exceed that for the air velocity which causes just complete dispersion. At smaller air velocities, dispersion rapidly decreases. At larger air velocities, the water concentration in the air stream falls because of the greater air flow.
The near constancy, for any one liquid rate, of the wave-induced increase in apparent viscosity in the liquid film is a consequence of the thinning of the film with increasing interfacial shear stress. Owing to the increasing viscous damping in the thinning film, the rate of growth of wave amplitude is slowed, and the penetration of the wave motion into the liquid film is reduced. Because of the free surface of the liquid film, and of the presence of waves at that surface, the similarity between the film flow and plane Couette flow is not exact. Therefore, it is not suggested that a film Reynolds number less than 1,500 is a sufficient condition for the indefinite maintenance of laminar flow with increasing air velocity.
The 9as film The change in the velocity profile in the interface gas boundary layer with interfacial disturbances is due to the same cause as is the flattening of the liquid layer velocity profile, convection of momentum across the boundary layer by wave motion in the fluid. In the air film, this mechanism of momentum transfer is additional to the molecular diffusion and turbulent transport processes. There is the possibility of secondary effects on momentum transfer, by modification of the turbulent flow structure of the fluid. For interfacial waves of small amplitude, however, it seems that such effects should be small in comparison with the primary effect. This explanation also accounts for the insignificant effect of the interfacial disturbances on the dry wall boundary layer. Since the amplitude of the fluid motion due to the interfacial waves decreases with the distance from the interface approximately as exp(-klYl) [7] and the value of kly[ at 72
Interfacial shear stress and momentum transfer in horizontal gas-liquid flow There are similar limits to the particle interchange m o m e n t u m transfer for the o t h e r water rates. I n general, the limits for smaller water rates w o u l d be d i s p r o p o r t i o n a t e l y smaller, because o f the greater air velocities necessary to disperse the thinner, m o r e stable liquid films.
are quite i n d e p e n d e n t , to the first o r d e r at least, o f the interfacial waves. This is n o t to d e n y a m e c h a n i c a l d e p e n d e n c e o f the extent o f dispersion, between the phases, a n d o f the transfer resulting f r o m t h a t dispersion, on the degree o f interfacial disturbance. M o m e n t u m transfer across the films o n either side o f the interface is, however, e n h a n c e d b y the convection i n d u c e d b y the interfacial wave m o t i o n . I n the case o f a t u r b u l e n t film, where there is a thin l a m i n a r sublayer at the interface, the effect o n the overall diffusion resistance o f the film is insignificant. But for a l a m i n a r film the resistance is substantially reduced.
CONCLUSION T h e shear stress at the interface o f a gas stream a n d a n e n t r a i n e d liquid film is m a d e u p o f w h a t m a y be r e g a r d e d as two quite distinct c o m p o n e n t s , one o f which acts on the interfacial surface, while the o t h e r acts o n the interfacial disturbances. T h e shear stress c o m p o n e n t acting on the surface is generated b y m o m e n t u m transfer processes. A s with the flow o f a fluid p a s t a s m o o t h b o u n d a r y , there is m o l e c u l a r diffusion across the b o u n d a r y l a m i n a r sublayer, resulting in a flux o f m o m e n t u m to the b o u n d a r y . W h e r e there is dispersion between the phases, a n a d d i t i o n a l m o m e n t u m transfer occurs b y interchange o f particles between the fluid streams. B o t h m o m e n t u m transfer processes d e p e n d o n the relative velocity o f the fluid streams. T h e stress c o m p o n e n t acting on the interfacial disturbances is the result o f a d i s t r i b u t i o n o f n o r m a l pressure in the gas stream. P r o v i d e d t h a t the interfacial waves c a n be a d e q u a t e l y defined, the theoretical w o r k o f BROOE:E BENJAMIN [7] c a n be used to estimate the pressure d i s t r i b u t i o n shear stress for a n y flow in the interface b o u n d a r y layer. F o r small wave amplitudes, however, the theoretical stresses are smaller t h a n the e x p e r i m e n t a l stresses b y a n o r d e r o f m a g n i t u d e . This difference m i g h t be explained b y the interference o f r a n d o m variat i o n in wave p a r a m e t e r s , a n d p o s s i b l y o f t u r b u l e n t flow fluctuations, with the f o r m a t i o n o f c o h e r e n t pressure distributions. The i n t e r p h a s e m o m e n t u m transfer m e c h a n i s m s
NOTATION a Wave amplitude b Film thickness b~ Interface gas boundary layer thickness bw Wall gas boundary layer thickness c Wave velocity fi Interface friction factor f0 Smooth boundary friction factor k Wave number Pv Periodic normal pressure from viscous effects u Velocity uc Velocity at level u = c m Interface velocity u* Friction velocity x Longitudinal distance parameter y Normal distance parameter Zo Parameter in Appendix L Liquid volume flow per unit width U Maximum in gas stream velocity profile /z Fluid viscosity /zg Gas viscosity /9 Fluid density pu Gas density "re Wave shear stress from profile curvature effects ~-~ Interfacial shear stress 70 Smooth boundary shear stress "r~v Periodic shear stress from periodic normal pressure "ro Wave shear stress from viscous effects •rw Wall shear stress q~l Function in Appendix ¢2 Function in Appendix
REFERENCES [1] [2] [3] [4] [5] [6] [7]
ALVESG. E., Chem. Enon# Proyr. 1954 50 449. HANRATTYT. J. and ENGENJ. M., A. L Ch.E. Jl., 1957 3 229. LILLELErrrL. U. and HANRATTV T. J., A. L Ch. E. Jl., 1961 7 548. SMrra T. N. and TArt, R. W. F., Aust. J. appl. ScL, 1964 15 247. VAN ROSSUMJ. J., Chem. Enong Science, 1959 11 35. ELLISS. R. M. and GAY B., Trans. Inst. Chem. Engrs., 1949 37 206. BROOKEBENJAMrNT,, J. Fluid. Mech., 1959 6 161. 73
T. N. SMITH and R. W. F. TAlT [8] [9] [10] [11] [12] [13] [14] [15] [16]
MILES J. W., J. Fluid Mech., 1957 3 185. STAr,TON T. E., MARSHALLD., and HOUOnTON R., Proe. R. Soe., 1932 A137 283. STI~E'rER V. J., Trans. Am. Soe. ely. En#rs., 1936 101 681. MOTZr~LD J. A., Z. Angew. Math. Mech., 1937 17 193. KONOBEEVV. I. and Za-IAVORONKOVN. M., Int. Jl. Chem. Engng 1962 2 431. Pr.ANDTL L., Z. Vet. dt. Ing., 1933 77 105. SMrrrI T. N., Ph.D. Thesis, University of Adelaide, 1964. y o n K A Z A N T., Trans. Am. Sci. Mech. Engrs. 1939 61 705. RFaCHARDTH., Z. An#ew Math. Mech., 1956 36 26.
where
APPENDIX
Z o = kl/apl/a#I/aCZo2/a Stress at a solid wavy boundary
Values of 41 tabulated by BROOKE BENJAMIN are For the periodic normal stress component given below in graphical form. lagging the boundary profile phase by n/2, BROOKE 2"0 BENJAMIN'S [7] equation (7.43) gives Pv = - a k l a / a z o 5/3p4/3~t4]3( U - ¢ ) 4 4 1 4 2
cos(kx - 1t/2). The periodic shear stress due to this component acting on the boundary y = a cos k x is Tpv = - a k P v
sin
kx,
"e: ~°o ~
which integrated, and averaged over a wave length becomes
-
i
4
]2
or in friction factor form, -
FIo. 1.
I
2 3 Zo Values of 41.
The function ¢2 accounts for the effects of the fluid velocity profile on the boundary stress, and is defined
To = ~a2kl6/aTo 5/3p4/a[g4/3(U -- C)z!'4142 '
.~v/p(U
I t
o
C)2 = ½a2kl6/afo ~/ap-g/a~t4/a
42 =
( U - c)-4/3¢~42.
The function ¢1 determines the magnitude of the viscous effects near the boundary, and is d e f i n e d by BROOKE BENJAMIN as
o \U--
c]
exp(-kly) dy
In evaluating this integral a logarithmic velocity profile, such that
has been assumed.
¢1 = ¢(Z0)
R r s u m r - - L ' a u t e u r exprrimente l'~coulement stratifi6 d'un courant d'air et d'un film d'eau darts un canal horizontal ferm6 de section rectangulaire: 16 × 2 inch pour des vitesses de gaz comprises entre 6 et 120 ft/sec, et pour des vitesses superficielles de liquide comprises entre 0,000375 et 0,0210 fta/ft]sec. La rrsistance [interfaciale a 6t6 mesurre e t a 6t6 drcomposre en 2 616ments, Fun rrsultant de la force de diffusion relative tt la srparation lisse, l'autre rrsultant de la rrpartition de la pression normale produite clans le fluide par un mouvement ondulatoire interfacial. On en conclue que la rrsistance tt la distribution de la pression ~ l'interface perturbre par les ondes stationnaires, est en accord avec les prrdictions fondres sur la throrie de BROOKE BENJAMIN, relative h l'rcoulement dans le voisinage de la srparation ondulante. Zusammenfassung--Untersucht wird die laminare StrSmung von Luft oder eines Wasserfilms durch einen horizontal liegenden rechteckigen Kanal (16 × 2 in.), wobei die StrSrnungsgeschwindigkeit der Luft zwischen 6 und 120 ft/sec und die Durchstitze yon Wasser zwischen 10,000375 und 0,0210 ft3/ft sec liegen. Die Schersparmungen an der Phasengrenze werden gemessen und in 'zwei Komponenten zerlegt, deren eine auf dem Impulsiibergang an die glatte Wand und deren andere auf dem Druckgeftille
74
Interfacial shear stress and momentum transfer in horizontal gas-liquid flow im strOmenden Medium (das durch Wellenbewegungen an der Phasengrenze entsteht) beruht. Die Scherspannung dutch den Druckgradienten an einer dutch station~ire Wellen gestOrten Phasengrenze stimmt mit der theoretischen Voraussage von B~OOKE BSN~AMIN[7] ffir Str6mungen fiber gewellte Fl~ichen tiberein.
75