I
W. J. THOMAS and STANISLAW PORTALSKI Chemical Engineering Department, Battersea College of Technology, London, England
Hydrodynamics of
0 0
Countercurrent Flow in Wetted-Wall Columns L I T E R A T U R E METHODS for experimentally determining mass transfer coefficients in wetted-wall columns are significant because they presumably predict interfacial area across which transfer occurs. Capacity coefficients then give values for mass transfer coefficients. However in much of the early work, experimental techniques were inadequate. For a wetted-wall column to be useful in determining mass transfer coefficients, wave characteristics of the liquid film must be defined in both viscous and turbulent conditions. I n this work, these characteristics in liquid films are examined under a wide range of flow conditions, especially the turbulent region.
Equipment A wetted-wall column with a countercurrent air-water system was used (Figure 1). The liquid weir was prepared by grinding the top-edge plane, which was at 90' to thecenter line of the column. A 45 level was carefully ground to produce a knife edge a t the weir. The lower end of the column was belled out uniformly to allow use of a gas inlet tube of the same diameter as the column. Prior to use, the column was degreased and cleaned.
calibrated orifice meter. The column was well mounted, and considerable care taken to ensure that it was truly vertical. Pressure drop across the column was measured with a micromanometer (Figure 2), designed according to Ower (27) and adequately described by him. Considerable thought was given to positioning the pressure taps. Because pressure drop across the column is required, the logical procedure was to place the taps at the top and bottom of the column itself. However, this would undoubtedly disturb both the water film and the air. The purpose of the experiinent was to investigate the nature of wa,ves in the liquid film under different operating conditions; therefore, two alternatives were available-i.e., to include pressure tappings that would not interfere with flow patterns of either
liquid or air, or remove pressure tappings from the column. The first alternative presents considerable experimental difficulty. With present techniques, noninterference could not be ensured; therefore, the second alternative was used. The top tapping is placed just above the liquid level in the top reservoir, and the bottom tapping in the side of the air-calming tube, just below the bottom of the lower reservoir. The recorded pressure drop is therefore the sum of losses over the column, enlargement loss a t the top reservoir, and losses prior to entry into the column. At the same air rates, pressure drop should be constant over those parts of the system which are not wetted by water. I n this case, the difference in pressure loss can he determined a t constant air rate, for a wall just wetted, and when liquid is actually flowing. The tube was
n
Column Dimensions Length of wetted-wall Bore of tube Water temp. Air temp. Air press. Air visc., 2 4 O C. Air density, 30.09 in.Hg, 24' C.
TO 'DRAIN
39.25 in. 0.772 in. 20-220 c. 24' C. 30.09 in. Hg.
NV
C W
NEEDLE -- COCK
w
VALVE
E WAY COCK
0.018 cp.
0.0747 Ib./cu. ft.
The gas inlet tube had a straight length of about 1 foot immediately below the column to act as a calming section. The upper reservoir was large to minimize liquid entrance effects. T h e flow of water from a constant head tank was adjusted by a needle valve. The three-way cock in this line was placed as near to the side of the reservoir as possible. The inlet air was controlled by a needle valve and measured with a
DIFF. PRSSSURe
m
+ I -- - AIR
ORIFICE METER
Figure 1. With this apparatus, pressure drop without ripples of liquid was measured over the wetted-wall column VOL. 50, NO. 7
0
JULY 1 9 5 8
1081
MILLED H l A D BRASSPLATE WITH
~ E D U A T E D VERNI~R
scay
CONNECTION TO TOP OF COLUMN
CONNSCTION TO BOTTOM OF COLUMN
--FINE SCREW
WR4Q
HAIR LINE FOR
liquid holdup. At high liquid rates, the quantity of water held over the weir can be significant compared with the liquid on the wall of the wetted-wall column. This has not been appreciated by previous workers. The procedure used was to stop the flow of water and at the same time destroy the head of water over the weir; the water was allowed to drain for 3 minutes into a measuring cylinder. The amount collected was determined accurately by weighing. Some water is always retained on the wall of the column, and to determine this amount, the wall after draining was dried with a previously weighed piece of dry cotton wool which was then re-weighed. This was done a t each liquid rate.
RUBBER TUBING,
Figure 2. Pressure drop across the column was measured with this micromanometer
wetted by flooding its surface with water containing 0.1% of Teepol as wetting agent, then cutting the water rate down to a trickle. Thus, with rapid readings, experimental values of pressure drop were obtained over the wet tube without ripples of liquid. Water flow rate was determined by collection over a period and subsequent weighing.
introducing a measuring cylinder between the tank and the column, and collecting the water over a given time. Several measurements were made to ensure that conditions were stable. Liquid holdup a t any liquid rate in still air was determined by a special technique. The three-way cock at the top of the column on the water inlet had a considerably larger bore than the connecting tube. With this cock, water could either enter the weir chamber, or be shut off; also, water in the weir chamber could be drained. I n other words, the head of water over the weir could be destroyed almost instantaneously. This is important when determining the
Holdup
Holdup data, obtained at the various liquid rates were recorded in still air. To do this, the bottom reservoir was removed from the column and a large tank was substituted for water collection. The flow rate was then obtained by
Table 1. Liquid, rate Cc./Min. 1820 1750 1685 1540 1220 1000 795 520 a
Holdup Data
Liquid, Holdupa cc.
Mean Thickness of Liquid Film, m Cm. X lo2
r,
-
Lb./Ft./Hr.
PL
Re, Still Air
41.7 40.0 38.5 37.7 29.1 25.1 23.9 21.4
6.88 6.61 6.37 6.23 4.74 4.09 3.89 3.48
1192 1146 1104 1002 799 655 52 1 341
493 474 456 414 330 271 215 141
1972 1896 1824 1656 1320 1083 860 564
r
Corrected by 0.60 ml. t o allow for water retention on the column wall after draining.
Table II.
Pressure Drop Data
AH
Level of Air
Rate 1 2 3 4 5 6
7 8
1082
2m. M'G. Air Vel. Air Dry Tube, Orifice Ft./Sec. 16.3 14.3 12.5 10.6 8.5 6.6 4.6 2.7
2.65 2.48 2.32 2.14 1.91 1.68
1.40 1.07
Resir. Dry Tube 988 925 865 798 712 627 522 400
AP In. WG. Dry Tube 2.246 1.983 1.737 1.496 1.200 0.962 0.680 0.410
INDUSTRIAL AND ENGINEERING CHEMISTRY
AP In. WG. Wet Tube 2.257 1* 991 1.751 1.506 1.216 0.950 0.683 0.425
Air Rate, G, Lb./Hr. Cu. Ft./Sec. Sq. Ft. x 103 713 667 624 575 514 452 3 76 288
8.61 8.06 7.53 6.94 6.21 5.47 4.56 3.48
Experimental Results
Values of liquid holdup were obtained with still air (Table I), whereas the more interesting condition is with air flowing. However, for the ranges of flow studied, the air appeared to have little effect on
Literature Background Subject Errors in early techniques Correlations proposed, involving plots of JD vs. Reynolds and Schmidt Nos. Reasonable agreement between JD and the friction factor, f, using 2-propanol and water Water films flowing down vertical plates and outside small bore tubes examined Transition flow region suggested between ReL = 25 and 1000 Effect of liquid rate on surface velocity of the liquid and direct effect on turbulence of gas phase in countercurrent flow Liquid surface to av. velocity Ratio is 1.5 Ratio increases from 1.5 to 2.0 up to Re = 80, then remains constant Rippling produces a marked increase in mass transfer rate Wave forms for liquids flowing down polished flat plates measured with a capacitometer and oscillograph Graphical solution to general equation of flow in pipes Nusselt equation Systems of waves examined mathematically Equations for cocurrent annular flow of air-water system Effect of liquid velocity on turbulencein an air stream demonstrated by pressure drop over a wetted-wall column References to recorded pressure drop for a wetted-wall column Thickness of liquid films measured With micrometer With drainage techniques With isotope tracers
(4,19% 34)
COUNTERCURRENT FLOW HYDRODYNAMICS the liquid holdup. Retention of water on the column wall a t each liquid rate was 0.G ml. This amount was added to *he measured quantity of water drained, to give the total holdup. From this and the dimensions of the wetted-wall, mean thicknesses of the liquid films were calculated (Table I). Pressure drop data were obtained at different air rates for the wet and dry tube; however, data for the dry tube were not used in the calculations (Table 11). Under the same conditions of air flow, the pressure drop in a dry and wet tube differ and for determining the friction drag for liquid flow, the wet tube is more suitable. I n other words, skin friction, deduced from the wet tube, probably corresponds more readily to that existing when a liquid flows down a wall. Data were obtained for pressure drop across the system for various air and water rates in countercurrent flow. The experiments were planned to give both viscous and turbulent conditions for water and air. High water rates were especially interesting because few results for liquids at high Reynolds numbers are given in the literature. Values were obtained for various air rates (Table 11) and at constant liquid rates of 320, 800, 1050, 1200, 1420, 1640, 2000, and 2240 cc. per minute. The values are given in a complete set of tables which may be obtained from the American Documentation Institute (33). Only Tables I1 and I11 are included in this report.
Wave Characteristics in Viscous and Turbulent Flow
The pressure loss across the column, including additional losses already mentioned, may be plotted us. air rate and constant liquid rate (Figure 3). The air rate is plotted directly as pressure drop across the air measuring orifice. Small differences may be significant; therefore Imperial graph paper was used and in Figure 3, which is a reduced replica of the graph, the curves appear as a narrow band. Their characteristics, however, are represented by the band reasonably well. Figure 4 reproduced from the original Imperial graph is area C in Figure 3. At high air rates, slopes of the curves rise more sharply than at lower rates (Figure 3,B). Thus, pressure drop over the column is considerably higher than that for a wet tube. I n Figure 3, B, a definite hump in the curves suggests a change in flow conditions, and in Figure 3,A, the curves cross and their pattern deviates from the remainder of the graph. The profile drag is considered as the
Figure 3. When air rate is plotted directly as pressure loss across the airmeasuring orifice, the curves appear as a narrow band
Level of Air Rate Lw liq. rate, cc./min. W , g./sec./cm. Theor. m, cm. X lo2 Av. vel. of liq. film,ft./sec. urnax.of liq. film,ft./sec. Air vel. relative to av. vel. of liq., ft./sec. Resir relative to av. liq. vel. Air vel. relative to urnax.of liq., ft. sec. Reair relative to urnax. of liq. APT, in. wg. total APD, in. wg. drag A P D , lb./sq. ft. drag r lb./hr./ft. r/ML
Re& = 4 r / p ~ CD+SD
~n.8n.G~
Table 111. 1 320 0.87 2.99 0.95 1.43 3.60 1343
Pressure Drop Data 3 4 5
2
. .. ..
. .. ..
3.43 1279
4.08 3.91 1522 1458 2.350 2.036 0.099 0.045 0.575 0.234 210 87 348 0.312 0.160 158450 71450
. .. ..
. .. ..
. .. ..
. .. ..
3.27 1220
. .. ..
. .. ..
6
7
8 320 0.87 2.99 0.95 1.43
2.63 981
2.35 877
2.02 753
.......... .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..
3.09 1153
2.86 1067
3.75 3.57 3.34 3.11 2.83 2.50 1399 1332 1246 1160 1056 933 1.785 1.517 1.248 0.977 0.690 0.450 0.034 0.011 0.032 0.015 0.007 0.025 0.177 0.057 0.163 0.078 0.036 0.013 210 87 348 0.140 0.053 0.193 0.117 0.076 0.480 54450 17540 5100 23910 10870 39990
.................. .................. .................. VOL. 5 0 , NO. 7
JULY 1958
1083
8 3
E
P a U
A
L Figure 5 . drop
A typical curve obtained when liquid rate is plotted against pressure
Q
Q
I
I
I
I
I
I
I
G Figure
6.
LBS/UR/SQ
FT.
Pressure drop caused by drag increases as air rate, G, increases
IO
i; 0 0
7 6
*
5
N
E
4 v)
v)
w
z
3
Y
z
2
5 U
1
I
Figure
1084
7.
I
FL= E90
ReL = 1160 m = 4 ~ 2 2x I d 2 c m . I
I
I
I
l
l
1
A critical point occurs when viscous flow changes to turbulent flow
INDUSTRIAL AND ENGINEERING CHEMISTRY
difference between total pressure drop with water and air flowing, and pressure drop for the wet tube; this can be read directly from Figure 3. A set of eight curves were drawn showing the effect of liquid rate, ReL, o n , pressure drop for constant values of air rate. All have similar shapes and are reproducible from the tables (33). I n Figure 5, which is a typical example of these curves, the pressure drop a t A corresponds to the wet tube case with no waves present-i.e., only skin friction. The increase in the total pressure drop with ReL from A to B is consistent with conditions in the viscous range of liquid flow, the waves being comparatively large. The decrease beyond B suggests a change in flow conditions. I t i s reasonable to assume that beyond B the transition region of flow is entered which persists to C where again a change in flow conditions leads to turbulence. In all the curves of this type, B occurs at about ReL = 900 to 'i000,and C at ReL = 1300 to 1500. B probably marks the end of the viscous range, and the onset of a partly turbulent condition coincides with the breakup of large liquid waves which exist in the viscous range. The smaller waves would result in decreased pressure drop because of drag; thus, the total pressure drop would decrease and continue to decrease with increasing turbulence until a fully turbulent condition is reached at C. Various workers have reported the onset of turbulence at ReL values between 1000 and 1200. Friedman and Miller (73), and Jackson (27) have shown that waves exist a t ReL values as low as q0. Dukler and Bergelin (70) demonstrated photographically that the liquid waves are larger in the viscous range than when turbulence commences. These observations substantiate this work which in fact shows that turbulence begins earlier than previously stated-i.e., at a value of Re& = 1080 ( 9 ) , and the transition region extends up to a comparatively high value of Re& = 1500 before a fully turbulent condition is reached. The breakup of large waves occurring a t the end of the viscous range at all air rates, is substantially independent of the air rate within the ranges used. The liquid flow is therefore critical in this region, and the curves present a quantitative basis for studying the effects of waves. Eight graphs showing pressure drop caused by drag us. air rate were drawn from the experimental values, each for a constant liquid rate. All curves had the same characteristic shape, as shown in Figure 6. Points A , B, C, and D occurred at the same values of G at different liquid levels. Commencing with B, as the air rate, G, is increased, so the pressure drop caused by drag increases and reaches a maximum at C. B corresponds approximately to G = 600 and Reai, = 1200 to 2000 (relative to the
COUNTERCURRENT FLOW HYDRODYNAMICS liquid). C corresponds to G = 800 and Reai, = 1500 to 2300 (relative to the liquid). The air flow is therefore in the viscous range, and favors formation of large waves in the liquid, which because of drag, results in increased pressure drop with increased air rate. Beyond C,there is a change in conditions resulting in a decreased AP,,, which reaches a minimum a t D. Further increase in the air rate then produces a rapid rise in AP,,,,. C must mark the end of the range for viscous air flow. The transition region of flow, leading to turbulence, exists between C and D, and after D air flow is fully turbulent. The partly turbulent condition of the air between C and D enhances the effect of change in liquid wave form resulting in a decrease in the friction drag. Beyond D, the pressure drop due to drag increases with air rate much more rapidly than previously, and the general effect of turbulent drag is evident with a consistent wave character. Between A and B at very low air rates, certain anomalous results were obtained, because the true condition was masked either by end effects of liquid flow on the air, or by other disturbances. Thickness and Critical Flow in the liquid Film
LIQUID RATE
CC'S/m,
For viscous flow (72)
200
For the thickness of a viscous film (26) (3)
u
By knowing the values of W ,pL, and p~ a theoretical value of m can be calculated: using Equation 3 with this m value, u~,,.and urn,,. may be determined. I t is impossible to predict theoretically with confidence the values of rn, us.,., and urnax. for a turbulent film. Both us.,. and urnax,are greater for turbulent than for viscous flow; however, these have been determined assuming viscous conditions even in the turbulent region. This is not exact but within the ranges considered, the error is not likely to be appreciable. Jackson, Johnson, and Ceaglske (79) found that up to ReL = 80, umx./u,,. increased to a value of 2.0, remaining constant thereafter. Unfortunately, the range of operation did not extend into the turbulent region. As a result of later work (27) Jackson stated that with wave motion, a parabolic velocity gradient gives way to a linear gradient ( ~ ~ . / u ~ . ,1.5 . , and 2.0, respectively).
100 90 80 70 n -
o
60
Y.
v N
u d
VI
50 Po
u" 50
20
IO
LlQUiD RATE
CC's/&
Figure 8. These curves confirm the theory of wave formation from viscous to turbulent flow as deduced from Figure 4 VOL. 50, NO. 7
JULY 1958
1085
DIRECTION
stant air rare. This minimum occurs at a liquid rate of about 1400 cc. per minute and ReL = 1500. The drag force then increases with increasing liquid rate in the fully turbulent range. When CD.SDG’ us. liquid rate at constant air rate is plotted (Figure 9), the curves have maxima and minima at the same values of liquid rate as in Figure 8. These two figures confirm the theory of wave form change from viscous to turbulent flow as deduced from Figure 4.
OF LlQUID
I
(m-y)x L.i,
t
Shear Force Distribution in a Turbulent Film AP (m-y,X
DlRECTlOX OF
Figure 9. Element of liquid film extending from a solid plate or tube wall to the gas-liquid interface A. B.
The shaded a r e a is the element of liquid film Forces acting on the film in countercurrent operation
The assumption of a linear gradient for the bulk of the film was inconsistent with observed film thicknesses. Therefore, because definite information was lacking, U , , ~ , ~ U , , was ~ taken as 1.5. The thickness of the liquid film was calculated from the experimental results of holdup. Figure 7 shows a critical value of m and liquid rate (m = 0.0422 cm.; ReL = 1160). This critical point must correspond to the change from viscous to turbulent flow. The value of Re& = 1160 obtained experimentally is in good agreement with the value, ReL = 1080, predicted theoretically by Dukler and Bergelin (70). Fallah, Hunter, and Nash (72) obtained a critical value of m = 0.05 cm. and ReL = 2000, using a drainage technique. However, they ignored the marked effect from volume of liquid at the weir and included this in their liquid holdup. Jackson (27) determined liquid film thickness using an isotope tracer technique. I n his graphs of film thickness us. the liquid flow rate, the sharp break associated with a change from viscous to turbulent conditions is not shown; however, the scale of the plot may be responsible for this. I t also appears that in certain cases, conditions were in the turbulent range. Drag Force and Wave Characteristics
The drag force in a wetted-wall column operating with a countercurrent system can be F = CD.SD.L.~TY:AP,
(4)
where AP,, the stagnation pressure (ZS), is given by
1086
(5)
Combining Equations 4 and 5 F
=
C D , S D ,&)
pB
2.Y
.L . 2 ~ r
(6)
also, F = A P D . T Y= ~ Ro.2ar.L
(7)
From Equations 6 and 7
No theoretical method exists for predicting CD and SD under any condition of liquid flow. To do this, characteristics in the liquid film would have to be theoretically predicted. In hydraulic problems, CD.SO are usually determined from experiments, and a similar procedure can be applied here. Cn.SD can be evaluated by determining the values of G, L , r, and p o experimentally. APD =
APtatai
-
A P s k i n fiiotian
For single phase vertical gas flow in a tube of radius r, the force relationship is R,.2.rrr.L = AP.rr2 - L,p,.ar2
(11)
so that R u = ?(% 2 L -
(12)
p,)
.4ssuming that the gas-liquid interface acts like a solid wall relative to the gas, it follows that
(9)
From the value of APskln obtained from experiments on the wet tube, and total pressure drop, also obtained experimentally, APD may be calculated. Liquid rate is predominant in determining the liquid wave pattern. Therefore, to examine further conditions in the film, CD. SD and CD S D , G2were plotted us. jiquid rate for constant air rate values (Figures 8 and 9). In Figure 8, the curves are similar to those in Figure 4. The maxima correspond to the beginning of the transition range (Re& = 900). This is followed by a breakup of the large viscous waves and a decrease to a minimum value of APD and consequently of C D S Da t con-
INDUSTRIAL AND ENGINEERING CHEMISTRY
Because of the complex and random nature of the wave at a gas-liquid interface, mathematical equations that take into account the wave form are difficult to develop. Until more is learned about waves experimentally, a theoretical treatment which includes wave formation will probably not be successful. I t would be helpful if equations defining the turbulent flow were developed for the “plane case.” In practice the interface will generally not be planar for Reynolds numbers greater than 20 in a water film. Consider an element of liquid film extending from a solid plate or tube wall to the gas-liquid interface. In Figure 9, the shaded portion represents a vertical slice of this element. The forces acting on this slice of film in countercurrent operation are shown in Figure 9,B. Summing these forces, we have R , . X . L . = ( m - y ) . X . L . p r ,- R c . X . L . - AP.(m - y ) . X (10)
Substituting in Equation 10 for Ri, and eliminating X.L., R, = ( m
-YXPL
- AP/L)
-
(r+m)($
-
P,)
(14)
This equation defines the shear distribution in the liquid film. Velocity Distribution in a Turbulent Film
The shear force in the liquid film is related to the velocity gradient by the equation, R, =
g
.12 (du,/dy)2
(15)
COUNTERCURRENT FLOW H Y D R O D Y N A M I C S The velocity distribution equation can now be obtained by integration of Equation 21. The integrated equation is
considered here as part of the laminar layer. Then LT
=
2Tr.
[ [Gu,.dy + LJo
If the liquid film thickness is small compared with the radius of the tube, that is r and, ( m - y ) . A P / L if ( r - m ) and ( r / 2 ) . pg are negligible, then (m
- Y ) P L - (r/z)(AJ'/L)
= (PL/S).~2.(duu/dY)2( 1 7 )
For a single phase flow in smooth pipes, the results of Prandtl and Von KBrrnkn are well summarized by Bakhmeteff (7). These theories assume that liquid velocity a t any point in the fluid is a function of the local shear stress. As a result, thickness of the laminar layer will depend on shear stress at the laminar layer. For single phase flow in a pipe, a core of turbulent fluid may possibly mask the conditions existing near the solid wall ; therefore, principles valid for full pipes, may lead to error when applied to films. This is especially true for countercurrent gas-liquid film flow where boundary conditions are predominant. However, experimental data on film flow is inadequate to elaborate on differences between the two cases. Therefore, it is necessary to assume that the mixing length is the same for a liquid film as for single phase flow in a pipe; and that shear stress, a t the boundary of a laminar layer in a liquid film, bears the same relationship to layer thickness as does wall shear for single phase flow in a pipe. Accepting these assumptions, the equations of Nikuradse (25) relating laminar layer thickness to kinematic viscosity and wall shear for single phase flow in smooth pipes, may be applied to films. The Nikuradse equations are
[g(m- Q)1"2
Y
0.4
-
[2(m 48 (rn
Y2 u ) + 16 ( m -
+ . . .1 + c
y3
- a)3
u)2
+
If it is assumed that the velocitv , gradient is constant in the laminar layer then u = k.y. and
(k)(y).A limiting
where a = condition is that (m-y)pL
> Ri for converAt the .boundary of the laminar subgency of the expansion of (1 - A)"' m-a layer, ug = k.6. The rate of flow in the I n other words the gravity term must be turbulent layer is greater than the interfacial shear. Equation 22 is similar to the velocity distribuLW,. = Z i r r u u . dy (28) tion equation developed by Prandtl for full pipes. The higher order terms may The value for uy is obtained from Equabe neglected in Equation 22, so that the tion 24 and is substituted in Equation equation giving the velocity distribution 28. Hence in the turbulent region of the film becomes [uli =
[g(m
-
& . y)]"'x
Integration of Equation 29 gives Lturb. = 2 ~ r . AX
0.4
Y
[2(m
-
& . F)
[m(ln.m
- 1) - a(ln6 ma 48B2 + ?j3
(mq + +%)I
+
- 1) (30)
The solutions to Equations 27 and 28 can now be obtained bv determining m and 6 . As indicated. '8 can be obutained from the Nikuradse (25) equation T o simplify this equation, let
Cg(,
-L 2P2 0.4
A =
.
3-1
and
uy = A.1n.y
-
The value of A', the constant, varies from 5 to 30 over the transition range of flow. Nikuradse quotes a value of 11.6 from the point of intersection of the extrapolated viscous and turbulent region curves on a U + and y + plot. From Equations 15 to 17 if a further AP L
simplification is made that - (m is negligible, then
where A = coefficient of friction in a smooth pipe and X = 4f. The mixing length is I = 0.4~. Substituting for 1 in Equation 17,
(26)
_I
(22)
Equation 23 now is and
[mu,dyl
JS
This is the velocity distribution equation in the turbulent region of the liquid film. Rate of Flow of the liquid Film
Rs = ( m - 6
) p ~
2
( y-
(m
p,)
- 8)
=
- B ) P L - Rs
(32)
I n this way, 6 may be evaluated from Equations 31 and 32. T o evaluate U8,
The quantity of liquid flowing in a turbulent film on a tube wall is given by L T where LT = 2nr
which on rearrangement becomes
&L
m u Y .dy
(25)
The flow is made up by a contribution from the laminar sublayer and the turbulent zone. The buffer layer is
Integrating this equation
Using Equations 31, 32, and 34, and if m, p, Ri and p L are known, then 6 and ug VOL. 5 0 , NO. 7
JULY 1 9 5 8
1087
can be calculated. These values can then be substituted in Equations 27 and 30 to give values for L L ~and ~ . Lturb., respectively. By carrying out experiments where a wetting agent is added to water, the planar case can be approximated and the values of Lturb.found experimentally and theoretically. These values should agree for the truly planar case; but, if waves are present the values will differ, and this difference will be a measure of the extent of wave formation.
Nomenclature
C CD d
F
f Summary
An improved experimental method, using a drainage technique in a wettedwall column, gives more accurate values of the mean thicknesses of moving water films. A critical point exists between viscous and turbulent flow, where rn = 0.0422 cm. and ReL = 1160. Experimental results of pressure drop determination across a wetted-wall column with an air-water countercurrent system are given. Different curves obtained by plotting pressure drop against flow conditions show definite characteristics which can be quantitatively interpreted. These curves show that a viscous flow condition may be associated with large waves, and that these waves begin to break up into smaller waves at an ReL = 900 to 1000. This breakup occurs a t lower Re& values than previously stated for the beginning of the transition region. Further, the lower pressure drop associated with smaller waves continues to decrease throughout the transition range. When full turbulence is reached the pressure drop increases. The transition region extends up to ReL = 1500 in some cases; this value is higher than previously stated. Curves summarizing the effect of air rate at constant liquid rates, on the pressure drop caused by drag are analyzed. The Reynolds numbers of the air flow are calculated relative to the liquid. With air flow in the viscous range, the pressure drop caused by large liquid waves is enhanced. I n the transition range of air flow, the partly turbulent character of the air enhances the breakup of waves occurring in the transition flow range of the liquid, and gives a further decrease in friction drag. In the turbulent air range the pressure drop due to drag increases more rapidly with increasing Re& than at lower air rates. Thus, air enhances the effects caused by liquid flow, but the wave character of the liquid is primarily governed by liquid flow. Using equations derived for shear stress and velocity distribution in the film and using experimentally determined values of film thicknesses and pressure drops, the quantity flow of liquid in the laminar sub-layer and turbulent layer can be calculated.
1088
G g k
L LN.
LT 1
m N AP APD APT AP,
RD Ri R, Reai, ReL
integration constant dragcoefficient diameter of inner tube, ft. drag force, lb. friction factor mass velocity of air, lb./hr. sq. ft. = gravitational constant, ft./hr.2 = mass transfer coefficient = length of wetted wall, ft. = liquid rate, cc./min. &rb. = volume rate of liquid flow in laminar and turbulent region, respectively, cu. ft./hr. = total volume rate of liquid flow, cu. ft./hr. = Prandtl mixing length, ft. = thickness of liquid film, ft. = Nikuradse constant = pressure drop, lb./sq. ft. = pressure drop caused by profile drag, lb./sq. ft. = total pressure drop, lb. sq. ft. = stagnation pressure drop, lb./ sq. ft. = shear force caused by profile drag, lb./sq. ft. = shear force at the gas-liquid interface, lb./sq. ft. = shear force at a distance y from the tube wall or boundary, lb./sq. ft. = Reynolds number of air flow = Reynolds number of liquid film - 417 _
= = = = = =
r
= inner radius of tube, ft. =
projected area for drag, sq. ft.
u ~ ~ .= ( ~average ~ ~ ) air velocity, ft./sec. urnax,= maximum velocity of the liquid
u* u+
film, ft./sec. = velocity in the axial direction at y ft./sec. = friction velocity, ft./sec. = universal velocity parameter, =
-
UY U*
W
= liquid rate per unit periphery,
X y+
r
= liquid film width, ft. = universal distance parameter = liquid flow rate per unit pe-
6
= thickness of laminar layer, ft.
g./sec. cm.
riphery, lb./ft. hr. coefficient of friction
1
=
fiL
= liquid viscosity, lb./ft. hr. = kinematic viscosity of the liquid,
vL
sq. ft./hr. gas and liquid density, respectively, lb./cu. ft. = shear a t pipe wall, lb./cu. ft.
pe, pL = T,
References Bakhmeteff, B. A,, “The Mechanics of Turbulent Flow.” Calvert, S., Williams, B., A.1.Ch.E. Journal 1, 78 (1955). Chilton, T. H., Colburn, A. P., I N D . ENG.CHEM. 26,1183 (1934).
INDUSTRIAL AND ENGINEERING CHEMISTRY
(27)
PL
sD
u,
(4) Chwang, C. T., thesis, MIT, 1928; I N D . ENG.CHEM. 26,428 (1934). (5) Claasen, H., Zentr. Zuckerind. 26, 497 (1918). (6) Cooper, C. M., Drew, T. B., McAdams, W. H., IND. ENG.CHEM. 26, 428 (1934). (7) Cooper, C . M., W’illey, G. S., Zbid., 26, 430 (1934). (8) Coulson, J. M., Richardson, 5. F., “Chemical Engineering,” Pergamon Press, London, 1954. (9) Dukler, A. E,, Ph.D. thesis, Univ. of Delaware, June 1951. (10) Dukler, A. E., Bergelin, 0. P., Chem. Eng. Progr. 48, 557 (1952). (11) Emmert, R. E., Pigford, R. L., Ibid.. 50.87 (1954). Fallah: J.’. Hunter. T. G.. Nash.
(28) (29) (30) (31)
Gilliland, E. R., Ibid., 26, 516 (1934). Greenewalt, C. H., Ibid., 18, 1291 (1926). Grimley, S. S., Ph.D. thesis, University of London 1947; Trans. Inst. Chem. Engrs. 23, 228 (1945). Higbie, R., Trans. A m . Inst. Chem. Engrs. 31, 365 (1935). Hopf, L., Ann. Physik 32, 777 (1910). Jackson, M. L., Johnson, R . T., Ceaglske, N. H., Proc. Midwestern Conf. Fluid Dynamics, 1950, J. W. Edwards, Ann Arbor, Mich., 1951. Jackson, M. L., Ceaglske, N. H., I X D . ENG.&EM. 42, 1188 (1950). Jackson, M. L., A.I.Ch.E. Journal 1, 231 (1955). Kapitsa, P., Zhur. E h @ . i. Teoret. Fiz. 18, 3 (1948). von Kbrmfm. Th.. Trans. A m . SOC. Mech. Engri. 61, 705 (1939). Kirkbride, c. G., I N D . EM. CHEhf. 26, 425 (1934). Nikuradse, J., J . Ferschungsheft, V. D. I., No. 361 (1933). Nusselt, W., Z. Ver. deut. Zng. 60, 541 (1916). Ower, E“,“The Measurement of Air Flow,” Chapman 8r. Hall, London, 1949. Prandtl, L., “Essentials of Fluid Dynamics,” Blackie & Son, London, 1952. Radford, B. A., M.S. thesis, Univ. of Alberta, 1949. Schoktlisch, A., Sitzber. Akad. Wiss. Wien. Math-naturw. Kl. 129, IIA, 895 11920). Sherwdod, T. K., IXD.ENG. CHEM. 33, 425 (1941); Ibid., 42, 2077 (1 \ _95n) __”,.
(32) Sherwood, T. K., Pigford, R . L., “Absorption and Extraction,” 2nd ed., McGraw-Hill, New York, 1952. (33) Thomas, W. J., Portalski, S., American Documentation Institute, No. 5598, Washington 25, D. C. (34) Zimmerman, 0. T., Lavine, I., “Chem. Eng. Lab. Equipment,” Ind. Research Services, Dover, N. H., 1943 RECEIVED for review November 2, 1956 ACCEPTED January 3, 1958 Material supplementary to this article has been deposited as document No. 5598 with the AD1 Auxiliary Publications Project, Photoduplication Service, Library of Congress, Washington 25, D. C. A copy may be secured by citing the document number and by remitting $2.50 for photoprints or $1.75 for 35-mm. microfilm. Advance payment is required. Make checks or money orders payable to Chief, Photoduplication Service, Library of Congress.
