Ind. Eng. C h e m . Res. 1995,34, 4078-4086
4078
Spreading of Liquid Point Sources over Inclined Solid Surfaces Sanat A. Shetty and Ramon L. Cerro* Chemical Engineering Department, The University of Tulsa, Tulsa, Oklahoma 74104-3189
The spreading of a wetting liquid introduced as a distributed source over solid surfaces has many interesting practical applications in heat and mass transfer contacting devices. This work analyzes the spreading of liquid over inclined flat surfaces. This study is the first of a program designed to understand the development of film flows from drip points over ordered packing in packed beds used in heat and mass transfer applications. The free surface of the liquid stream introduced a s a point source retains a shape similar to a Gaussian or exponential distribution as it spreads down the solid surface, increasing the surface area as well a s distribution of flow over the surface. The three-dimensional flow field, described using a 3D vector potential function, satisfies continuity and the first-order terms of the Navier-Stokes equations in the main direction of flow and in the direction normal to the main direction of flow. A viscous long-wave approximation was used to derive a film evolution equation and the kinematic boundary condition a t the free surface. Experiments were carried out with fluids of two different viscosities and a t varying inclination angles of the solid surface. 1. Introduction
Ordered packings have a well-defined mechanical structure. The better known commercial types of ordered packing are fabricated from either corrugated metal, plastic, or ceramic plates. In countercurrent exchanging devices, liquid is introduced on top of the packing, at discrete pour points, while the vapor phase is introduced at the bottom of the packed tower. The liquid initially does not cover the entire solid surface, and this maldistribution results in loss of efficiency. The initial liquid distribution onto a bed of structured packing is critically important to the performance of packed towers used in heat and mass transfer applications. The distribution of a liquid on a packing surface can be described either as a distribution in space (axial and radial spreading profiles) or as a distribution in form (drops, rivulets, and/or films) (Bemer and Zuiderweg, 1978). Most available experimental data deal with space distribution, i.e., by collecting liquid during a given time inside pans located underneath the packing (Olujic et al, 1993). Space and form distributions are related in a nontrivial way. This work focuses on the spreading of a liquid source on flat surfaces at different inclination angles and reports the change in form of the liquid flow from a rivulet to a homogeneous film. The flat surfaces used in this research are representative of the surfaces used in the manufacture of ordered packing. Classical studies of flow down inclined planes (Batchelor, 1967, p 182) were generally limited to twodimensional cases. In recent years the emphasis has been primarily on geophysical applications, like in the modeling of lava flows and mudslides. Smith (1973) derived a similarity solution for the distribution of liquid film thickness in a three-dimensional viscous source flow down an inclined plane. He observed that liquid spreads following the VTth power of the distance traveled along the direction of flow, x , while the film thickness in the direction of flow thins out as the -l/7 power of x . These power scaling laws have been confirmed by Schwartz and Michaelides (1988) and more recently Lister (1992). Schwartz and Michaelides (1988) * T o whom correspondence should be sent. Voice: (918)631-2978.FAX: (918)631-3268.e-mail:
[email protected]. UTULSA. E DU .
used a lubrication approximationto describe the spreading flow and solved the free surface lubrication equations numerically for a time-dependent, three-dimensional flow down an inclined plane, produced by a continuous injection through a finite circular hole in the plane. Their solution approached the similarity solution of Smith (1973) provided fluid injection was continued for a sufficiently long time. Their numerical computations were compared with the experimental data of Hallworth et al. (1987). Lister (1992) used the lubrication approximation to solve numerically the flow of a viscous fluid from both point and line sources. He obtained similarity solutions for the long-time behavior of the governingnonlinear partial differential equations for cases in which the volume of the fluid flowing along the plane increased with time like ta, where a is a constant. Computations were used to show the extent of down-slope and cross-slope spreading with time. Very limited experimental data are available on point source spreading. All data available in the literature have been determined for liquid flows in the presence of contact lines and positive inclination angles, i.e., for flow over the solid surface. There are no data available, to our knowledge, on flow for negative inclination angles, Le., on the under surface. In addition, based on the order of magnitude analyses, all of the above referenced researchers neglected capillary forces and used a pinned contact line in their theoretical developments. In this work, we analyze the spreading of a point source of liquid in the absence of contact lines and under the effect of capillary forces. A large amount of data was collected for spreading of a liquid point source over and under the surface of a solid substrate, for a wide range of inclination angles and for two different liquid viscosities. A viscous long-wave approximation was used to derive a film evolution equation describing the spreading of a liquid point source as it flows down an inclined flat surface. A three-dimensional vector streamline function was used to derive the vector components of the velocity field. This velocity field satisfies continuity and the higher order terms of the equations of motion. The results presented here are part of a continuing research program that aims to characterize the mechanics of viscous film flows over model surfaces that simulate the structure of ordered packing. The results and theory from this work will be used to study
0 1995 American Chemical Society
Ind. Eng. Chem. Res., Vol. 34,No. 11, 1995 4079 standard deviation, m[xl = O(m[x=Ol). Since the variable x does not appear explicitly in this formulation, the choice of x = 0 should be arbitrary. In practice, the definition of the point x = 0 only affects the film thickness profiles very near this origin. In most of the flow field, the representation of the film profile given by eq 1 is a very accurate similarity transformation subject to the constraint that the peak height is much smaller than the standard deviation. A small parameter, a, can be defined such that
a=
Figure 1. Sketch of the geometry of the inclined flat plate and coordinate system.
the effect of the surface macrostructure, particularly the effect of the corrugation angle with respect to the vertical, on the distribution of liquid over the packing surface. 2. Theory
Consider the flow of a viscous fluid on a flat solid surface inclined at an angle ,4 with respect to the vertical. See Figure 1. The liquid is introduced at point 0 located at the highest point of the surface. A Cartesian coordinate system is defined such that the main direction of flow is x . The x and y coordinates are tangent to the solid surface, and the z coordinate is normal to the solid surface. The liquid introduced at point 0 spreads out slowly as it moves down the solid surface, increasing the interfacial area and improving vapor-liquid contact. At a very short distance from the point where the fluid is introduced, the free surface of the fluid adopts a regular symmetric shape with a maximum thickness at the centerline that will be defined as y = 0. The maximum film thickness at the centerline decreases with increasing values of x . Assume that the cross-sectional shape of the free surface can be represented by a normal probability distribution
(1) where 6,[xl is the maximum film thickness a t the centerline, y = 0. The function m[xl is the standard deviation of the exponential distribution, and it is a measure of the extent of spreading of the liquid film on the solid surface. Smith (1973) used a parabolic velocity profile to describe spreading of a film with a pinned contact line. Using experimental data collected on wetted plates, it will be shown that in the absence of contact lines this normal distribution is a far more accurate expression to describe the film thickness profile. The film thickness distribution is symmetric about the location of the maximum a t the centerline, and it is based on a similarity transformation for the y-variable. The film thickness and the derivative of the film thickness with respect to y vanish asymptotically as y At any cross section, for a constant value of the variable x , the film thickness profile described by eq 1 has two characteristic lengths determining its shape: (1) the peak height, Bo[xl = O(Bo[x=Ol, and ( 2 ) the
-
00.
d,[x=OI 2m[x=Ol
0". This analysis is similar to the long-wave expansion first used by Benney (1966) to analyze two-dimensional long waves on liquid films. This approach was later extended by Roskes (1970)to three-dimensional waves and has been used by Atherton and Homsy (1976) and more recently by Kheshgi (1984). The liquid point source, introduced on top of the inclined flat plate, can be treated as a small-amplitude disturbance over a steady base flow. In this case the perturbation is scaled by the ratio of the height of the peak at the line of symmetry and by the width of the peak. Being a normal distribution, the width of the peak is represented by the standard deviation. For very viscous liquid films the long-wave expansion, to the lowest order, is equivalent to the lubrication approximation and has been used to analyze a variety of flow phenomena. Higher order terms, which include the effect of inertia, lead to a film profile equation that has been used t o determine the stability of liquid films on inclined substrates (Kheshgi, 1989). The three main forces driving this flow are retained in the expansion, namely, the components of gravity tangent and normal to the solid surface and capillary forces. Inclusion of higher order terms of a will account for inertia and normal viscous stresses. The boundary conditions needed t o integrate these equations are the no-slip condition a t the solid surface
Equation 15 is substituted into eqs 8 and 9, which are solved for the three components of the velocity field. Equations 7-10 and boundary conditions (11-14) are identically satisfied by a velocity field generated by a three-dimensional vector potential. In dimensionless form, the vector potential has the following vector components
YX= -aN,,sinp-+--ad*
(
ay
a3 a 2 ~z2d* z3 (16) ay 2 6)
N,,
)(
c o s p - aNstsin/?
??-;)
(17)
Yv,=O
(18)
The physical components of the velocity vector can be computed by taking the curl of this vector potential. Since the velocity is the curl of the vector potential D = V x tp, continuity is satisfied identically. The velocity components for a Cartesian system are cos ,6 - aN,,sin /3
(Zd* -
g)
(19)
and continuity of stresses at the free surface
a3 n.T - n-2H Nca
=0
at Z = d*[X,yJ
(12)
where 6*[X,Yl = 6 [ xyjldoi is the dimensionless film thickness and Nca = &/a is the Capillary number. Within the zero-order approximation, the tangential component of the free surface boundary condition reduces to vanishing normal derivatives of the velocity components
avo az
- a~ az - o
at z = ~*[x,YI
(13)
while the normal component of the free surface boundary condition recovers the Young-Laplace equation,
a3 Po+ -2H Nca
=0
at Z = dOi[X,Yl
(14)
The capillary term is retained, despite being affected by a3,because the Capillary number is a small number
P(: +();
a26* I a3 "2") -&,, sin p ax2 Nc, ax2
(21)
The velocity profiles for U and V are parabolic, highlighting the close relationship with Nusselt's velocity profile. Velocity profiles computed for typical values of the Stokes and Capillary numbers are shown in Figures 2-4. These velocity profiles satisfy the no-slip condition at the solid surface and the simple form of the zeroshear condition at the free surface given by eq 13. Figure 2 shows a sequence of velocity profiles, v[X,Yl, for different values of X. These velocity profiles have a maximum at the centerline and then vanish asymptotically, for large values of Y. Except for surfaces that are nearly horizontal, velocities in the main direction of flow depend mainly on gravity, which is the largest term in eq 19. Figure 3 shows profiles of the V velocity €or
Ind. Eng. Chem. Res., Vol. 34, No. 11, 1995 4081 35
3
1
1
zol
+ +
15
10 -
5 n
(60)
both sides are increasing near the centerline and asymptotically going to zero for large values of Y. The free surface of the spreading film is a threedimensional surface where the principal radii of curvature, in general, do not coincide with any of the coordinate directions. Using the parametric representation of the free surface, R = ix jy kd[xjl, the mean curvature can be computed using the derivatives of the film thickness (Kreyszig, 1959). The overall flow rate is assumed constant and can be computed integrating eq 19 over a cross section of the film thickness profile. Neglecting higher order derivatives and higher powers of derivatives, the following expression for the flow rate is obtained:
(40)
(20)
20
0
40
60
Y (mm)
Figure 2. Free surface velocity profile in the main direction of flow. u ( d s ) vs y (m). p = 46",q = 1.92 x m3/s, v = 100 cSt. x (m): (1)0.0, (2)0.031,(3)0.131,(4)0.231,(5)0.331. 0.6
0.4 0.2
(0.4)
1
n
i
I\
i
V--l
Figure 3. Free surface velocity profile in the transversal direction. u ( d s ) vs y (m).p = 46",q = 1.92 x m3/s, v = 100 cSt. x (m): (1)0.031,(2) 0.131,(3)0.231,(4)0.331. 0.05
(0.25)
'
(60)
'
'
(40)
I
'
(20)
0
20
40
60
Equation 22 is written using the dimensional form of the flow rate in order to use experimental values of q in the computation of m[Xl and S[W. The theoretical expressions for velocity and film thickness include two functions, d,[W and m[W,that must be determined in order to specify completely the film thickness profiles. For a flat surface and a fluid of given properties, flow rate and inclination angle are the only variables that can be adjusted during experiments. To use these independent experimental variables to evaluate the characteristic functions, functional relationships must be developed between d[W, m[Xl,q, and p. One of the relationships is given by the theoretical expression for the flow rate (eq 22). Using the leading term in eq 22, the function n[Xlcan be approximated as follows:
This approximation, strictly, is only valid for large values of X, when the derivatives of m[xI became negligible. In practice, however, the leading term of eq 22 is much larger than the terms that were neglected, for all values of X. Using this approximation, the expression for the film thickness profile given by eq 1 becomes:
I)[' :(
Y (-)
Figure 4. Free surface velocity profile in the direction normal to the plate. w ( d s ) vs y (m). /3 = 46", q = 1.92 x lo-' m3/s, v = 100 cSt. x (m): (1) 0.031,(2)0.131,(3)0.231,(4) 0.331.
d[X,Yl= d,[Xl exp - -
constant values of X. The velocity in the Y direction is zero a t the line of symmetry Y = 0 and is positive for Y 0 and negative for Y < 0. The V velocity has a maximum at the point of maximum slope in the Y-2 film thickness profile on either side of the centerline and goes asymptotically to zero for large values of Y. Figure 4 shows velocity profiles in the direction normal to the plate, for constant values of X. The velocity profiles in the 2 direction have a negative maximum at the centerline, where the film thickness is rapidly decreasing, and positive maxima at both sides of the centerline. This is consistent with the fact that the film thickness at the centerline is decreasing while the thickness on
The second relationship is developed using the kinematic boundary condition at the free surface:
- 1d
3dx
[
d*3(NSt cos
(24)
dd* p - os, sin -+ dx
(25) Equation 25 is the mathematical expression for the
4082 Ind. Eng. Chem. Res., Vol. 34, No. 11,1995
kinematic boundary condition. This equation is usually referred to as the film evolution equation because it describes changes in film thickness with position along one of the main axes of flow. Substitution of eq 23 into eq 25 eliminates m[XJ and reduces it to a nonlinear ordinary differential equation in one variable, &*[XI. The resulting equation cannot be solved, except by numerical computations. A simpler working relationship that still maintains the balance of the most important forces acting on the system can he obtained by integrating the film evolution equation at Y = 0, i.e., at the centerline where the profile is symmetric:
NB,(d,*)8
as,* + (6,*)'4 + 3 ~ , ,-0 ~
16a3
(26)
The first term in eq 26 is a component of the pressure field acting in the Y direction and generated by gravity acting normal to the solid surface, the second term reflects the contribution of capillary forces to the pressure field, and the third term is due to forces of viscous origin. The ratio of gravity to capillary forces is given by the Bond number (27) For a vertical surface, the Bond number is zero and there is no contrihution t o spreading from the component of pressure generated by gravity. The only remaining driving forces for spreading are capillary and viscous forces. For a vertical surface, eq 26 reduces to a simple ordinary differential equation: &*I4
dd,* + 3Nc, =O 16a3 dX ~
(28)
This equation is easily integrated using the initial condition that do* = 1 a t X -r 0. Integration results in an expression for the peak film thickness as a function of position down the vertical surface:
It is easy to show that eq 29 satisfies the initial condition and predicts a vanishing film thickness for large values of X. Lateral spreading of liquid sources on a vertical surface is governed by capillary forces only. For j3 > lo", values ofNe, become increasingly larger and gravity quickly dominates over the capillary term. Typical experimental values for the Bond number where 1.44 5 N B 5~ 44.9. Under these conditions, capillary forces can be neglected and eq 26 reduces to dd,* + 3Nc, -= O 16a3
NBodo*8
~
(30)
Using the same initial condition that was used to integrate eq 28, the evolution equation for spreading due to gravity only is given by a very simple relationship:
For small inclination angles, 0 < /3 < lo", that is, for nearly vertical surfaces, all terms in eq 28 are of comparable size and cannot be neglected. In this case
Figure 5. Sketch of the experimental setup,
a numerical solution was obtained that, using leastsquares regression, was best fitted by the following expression:
d,*(X) =
+
1
(1
For negative values of /3, that is, for spreading on the under surface of an inclined plate, eq 26 was integrated numerically. An interesting point is the fact that for p < 0 the hydrostatic pressure term in eq 26 changes sign and gravity becomes a counteracting force delaying spreading due to capillary and viscous forces. At low negative inclination angles, near the point of introduction of the liquid source, capillary forces again predominate and spreading occurs until gravity and viscous forces balance capillarity. When this balance is reached, lateral spreading stops and the liquid flows down as a thick rivulet. For large negative inclination angles, there is no equilibrium configuration and the liquid film thickness profile becomes unstable, causing the liquid to form drops that detach from the surface due to the action of gravity. The liquid drips out of the surface instead of flowing down the surface.
3. Experimental Setup The experimental setup is shown schematically in Figure 5. The test surface was made of brass and is 0.178 m wide and 0.51 m long. The inclination of the test surface and the film thickness measuring device with respect to the vertical, j3, could be rotated a full 180". Experimental data were obtained for liquid spreading over and under the solid surface, i.e., for positive and negative inclination angles. Data were obtained a t inclination angles -42", -32", -24", -11.5', o",5 3 , lo", 33", 46", 6o",66", 74", and 79" with respect to the vertical. For most commercial packing, the range of interest spans corrugation angles from -45" t o 45". A small liquid jet flowing out of a glass tip was deposited on the solid surface a t a single drip point to create an initial maldistribution much like the initial distribution of a liquid in industrial packed beds. The liquids used in these experiments were silicone oils (poly(dimethylsi1oxane))of kinematic viscosity 0.1 and 1.0 St. Silicone oil wets the brass surface. In addition, the solid surface was prewetted with the liquid to ensure the absence of contact lines. In packed towers the solid surface is usually wet due to the entrainment of drops in the vapor phase or to flooding of the packing during startup. A constant overflow overhead tank was used to maintain a constant liquid flow rate. The tank was placed about 1.5 m above the drip point. A micropump
Ind. Eng. Chem. Res., Vol. 34,No. 11, 1995 4083 was used to recycle the liquid back to the overhead tank. Liquid flow rates were measured by weighing the liquid collected over a known time interval. Liquid flow rates were varied t o simulate the range of flow rates per drip point typically found in commercial units. A very thin needle mounted on x-z precision micrometer translators (Figure 5) was used to detect the position of the free surface and the solid surface. The translator ensemble was mounted on a tubular bench parallel and solidary to the test surface such that the needle remained normal to the test surface for all inclination angles. The entire assembly was mounted on a vibration-free Oriel optical table. The back of the test surface and one end of the needle were connected to a laboratory ohmmeter. The needle was then moved forward in a direction normal t o the solid surface using the z-direction micrometer until the point it first touched the solid surface. This was very precisely detected by the deflection of the ohmmeter needle as the circuit between the needle and the solid surface is completed. This technique is possible due to the large difference in electrical conductivity between the solid surface and silicone oil. However, since the conductivities of silicone oil and air are both very low, the position of the free surface is detected using a combination of side lighting and a video camera zoomed in on the liquid surface at high magnification. The micrometer needle is again moved in a direction normal t o the solid surface until the point it first touches the fluid-air interface which can be very clearly seen on the magnified image on the video monitor. The free surface and solid surface positions were determined for every experimental point. While it is not practically possible to align the test surface absolutely parallel to the tubular bench carrying the x-y translators, the maximum angular deviation from the parallel was found to be no greater than 0.3". This error, however, does not affect the film thickness measurements which are computed by subtracting the difference between the position of the free surface and the position of the solid surface. Film thickness profiles were measured at constant positions along the x-axis. Starting at the center, the film thickness was measured on both sides of the centerline for growing values of y, up t o the point where the film thickness was on the order of 50 pm. The minimum reading of the micrometer is 1pm. Experimental measurements are assumed to be good down to 5 pm. These measurements were done for constant values of x , distributed along the length of the plate. Precision of experimental data was confirmed by measuring film thickness repeatedly at the same position. The standard deviation in film thickness values at any position was, on the average, 3.3 pm. For positive inclination angles, the liquid introduced a t the source spreads out slowly as it moves down the test surface, evolving from a film of nonuniform thickness to a more uniform film covering more of the solid surface and improving contact area as well as space distribution. 4. Results and Discussion
Figure 6 shows experimental film thickness profiles, d[xyl, as a function of y for different constant values of x . At a position close to the source, the thickness profiles are narrow and sharp at the center. As the film moves down the plate, the thickness at the center decreases while the thickness away from the centerline increases. The logarithm of film thickness normalized using the thickness at the centerline is shown in Figure 7 for different positions along the main direction of flow,
-10 00
-20 0 0
-40 00
0.00
v
IO.00
20 00
10.00
(mm)
Figure 6. Liquid film thickness distribution for different positions along the solid surface. /3 = 46", q = 1.92 x lo-' m3/s, v = 100 cSt. x (m): (0) 0.0, (0)0.031,(A)0.131,( 0 ) 0.231,(*) 0.331.
(2.5)
2
4
6
IO
8
I2
y1 cm' Figure 7. Logarithmic plot of normalized surface thickness: ln(G[n,yYS,[x])vsy2. p = 60", q = 3.0 x m3/s, v = 100 cSt. x (m): (m) 0.099,(e) 0.114,(*) 0.170,(0)0.222.Goodness of fit (R2) = 0.95.
-2.00
-1.00
0 00
IO0
z
0
Y'
Figure 8. Normalized film thickness distribution profile (6*[xy*] vs y*). p = 46", q = 1.92 x m3/s, v = 100 cSt. z (m): (0) 0.0, (0)0.031,(A)0.131,(0)0.231,(*) 0.331.
x , as a function of y 2 , where y is the surface coordinate
normal to the main direction of flow. In Figure 7 , the slopes of the straight lines for constant x are the values of -1/(2rn2[xl). The purpose of this plot is to show that experimental data for film thickness profiles at constant x follow very closely the functional behavior of the normal distribution represented by eq 1. Figure 8 shows the same experimental dimensionless film thick-
4084 Ind. Eng. Chem. Res., Vol. 34, No. 11, 1995 10 00
0 9c
-
9
080J
x
- --
*
0
v
0 70
%%h 9%,x
0 60
-
00 A
0 50
I
I
I
L 00
1
10 00
lono
100 00
00
( I - K X ~
Figure 11. Gravity-dependent regime. Film thickness and vari. 69", l/n = l/7, K G ~ ance: ln(d[x]) and ln(rn[xl vs ln(1 + K G ~#I = = 4.168. (+) d[xl; ( 0 )rn[xl. Table 1. Slope and Integration Constant for Experimental Values of Film Thickness B (ded v (CSt) K n 4.669 13 0 1.0 1.075 7 33 1.0 1.14 7 0.1 46 1.025 7 1.0 46 1.745 7 1.0 60 2.09 7 0.1 66 2.23 7 1.0 69 5.77 7 1.0 74 16.57 7 1.0 79 a
0.40 0 00
,I
00
1
I 0.00
x
I
I
L2.00
18.00
i
20 00
Figure 10. Effect of the solid surface inclination angle on the dimensionless peak film thickness profile. #I: ( 0 )79", (HI73", (*I 46", ( 0 ) 5.5", (0)Oo, (0)-11".
ness profiles, 6*(X), versus the dimensionless normal surface coordinate, y* = b2/2rn(~)2)1'2. The variance of the normal distribution, rn(x), was computed using the slopes of the lines in Figure 7. The fact that all data collapse, within experimental error, into a single curve confirms the assumption that the normal distribution representation given by eq 1 is, indeed, a similarity transformation. This similarity transformation, for a constant inclination angle, applies to the coordinates in the direction of flow, x . Figure 9 shows the normalized film thickness at the centerline, for several different flow rates and a constant angle of inclination. All experimental points fall closely into a single curve when plotted against the dimensionless coordinate in the direction of flow, X = x12mi. When the inclination angle is changed, even for similar flow rates of the same fluid, data cannot be reconciled into a single curve. Figure 10 shows data for the normalized peak film thickness, do(X)/doi, for values of -11" < p 79". As predicted by theory, i.e., eqs 29 and 31, these data demonstrate the changes on the relative magnitude of gravity and capillary forces as the inclination of the solid surface is changed. Increasing the inclination angle increases the effect of gravity, and this, in turn, increases the extent of spreading as indicated by the change of peak film thickness.
Ax (mm)
2.7 6.0 3.4 7.2 9.2 2.67 2.23 0.8 0.2
Ax is a correction to the experimental value of x = 0.
The effect of inclination angles was explored using numerical solutions of the peak film evolution equation as well as the approximate solutions for capillarydominant and gravity-dominant cases given by eqs 29 and 31. For large inclination angles, p > lo", it was assumed that capillary forces were negligible and gravity was the only remaining force in the integration of the film spreading equation. The solution t o this limit problem (eq 31) has no adjustable parameters, since all the coefficients contained in the definition of K G can ~ be computed from experimental data. Figure 11shows experimental values of In 6," and ln(m*) as a function of ln(1 K ~ r x )for , ,L? = 69". The straight line representing film thickness in Figure 11has a slope n = lI7. Using the theoretical value of KGr = 4.168, the reference line X = 0 can be computed for a very close fit. Table 1 shows theoretical values of Kca and K G for ~ the entire range of inclination angles used in experiments. For values of p = 33", 46", 60°, 66", 69", 74", and 79", straight lines with slope l/7 were used to fit the experimental points. The correction introduced into the values of X was in all cases within millimeters of the experimentally estimated values of X = 0. The origin of the x-coordinate is irrelevant from both a theoretical and experimental point of view because K x >> 1 for all but the first experimental point. For a vertical surface, ,8 = 0", the Bond number is zero, and the solution of the film evolution equation is given by eq 29. In this case, a straight line with slope V13, and a theoretical value of Kea = 4.669 computed using eq 29, can be used t o fit the experimental data. Figure 12 shows experimental values of normalized film thickness along with a computer solution of eq 26 for p = 10". The agreement between experimental data and the numerical solution is remarkably good. For the sake of comparison a best fit of these data was done using eq 32. The value of K calculated using a linear
+
Ind. Eng. Chem. Res., Vol. 34, No. 11, 1995 4086
-
Z R
I
yi
L s
2
5
2: I
0
x
' """"I """'
"
I
""""'
I
' I '
'
x I
"
"
~
. . . . . . . . . ( . . . . . . . . .I . . . . . . . . . . . . . . . . . . , . . . . .
-10.000
-6.000
-2.000
2.000
8.000
10.000
Y (mm)
Figure 14. Distribution of film thickness for negative inclination angle. /3 = -11". q = 1.10 x m3/s, v = 100 cSt. x (m): (0)0.0, ( 0 )0.012, ( A ) 0.06, ( 0 ) 0.098, (*) 0.151.
-
Experimental; (-1 computations.
Spreading over the solid surface
4-
capusry pntsun component of hydrostatic pressure
0.00
Spreading under the solid surface
4.00
12.00
8.00
16.00
2c 30
X
Figure 15. Comparison between experimental and predicted dimensionless peak film thickness profiles for negative inclination angle. /3 = -11". q = 1.10 x lo-' m3/s, v = 100 cSt. ( 0 ) Experimental; (-) theoretical computations.
4-
Capillary pressure component of hydrostatic pmsure
Figure 13. Schematic showing driving forces for spreading across the solid surface for the spreading of a point source, over and under the solid surface.
regression of the experimental data can be related to the relevant dimensionless variables using an intuitive physical argument: by analogy with eqs 29 and 31, one can expect K to depend on the Bond and Capillary numbers as well as on the Stokes number. The value of K for small inclination angles should be
K=
208a3NB, NchNst
(33)
For negative inclination angles, gravity and capillarity act in opposite ways, as shown schematically in Figure 13. While capillarity acts to decrease the sharpness of the peak, the pressure field created by gravity acts to increase the amount of liquid a t the centerline. Initially the curvature of the peak is large, such that capillary forces prevail. As the sharpness of the peak decreases, capillarity and gravity balance each other
and spreading stops. The liquid flows down the solid surface adopting a film thickness profile similar t o a wide rivulet which remains virtually unchanged for the entire length of the surface. Figure 14 shows one of these stable profiles for an inclination angle of /3 = -11". The film thickness profile can be described very accurately, by eq 1. For small negative angles the peak film evolution eq 27 can be solved numerically to describe peak film thickness as a function of distance along the plate. Figure 15 shows very good agreement between the numerical solution and experimental values of the normalized peak film thickness for /3 = -11". For larger negative inclination angles, instabilities develop and the liquid drips out of the surface. This instability was first observed at an inclination angle of /? = -31", the point at which surface waves were also observed. As the inclination angle becomes more negative, instability increases and the liquid starts dripping out of the plate. A detailed description of these phenomena is outside the scope of this paper. Liquid distributors commonly used in packed columns introduce the liquid at the top of the column through a number of pouring spots carefully balanced to achieve a homogeneous spatial distribution. The upper part of the packing is used to transform this homogeneous space distribution into a suitable form distribution of liquid films of constant thickness. As the liquid flows
4086 Ind. Eng. Chem. Res., Vol. 34, No. 11, 1995
down the packing, it slides over surfaces of alternating positive and negative angles of inclination with respect t o the vertical. The angle of inclination with respect to the vertical is a function of the corrugation angle of the packing and of the angle of the corrugations with respect to the vertical and can be computed using a simple geometrical construction. The resulting values of inclination angles are usually bounded well within -45" < /? < 45" with a small curved region at the ends where the inclination angle changes sign within a short distance. While the liquid film flows down surfaces with positive values of p, it spreads out due t o the combined action of capillarity and gravity. The larger the value of p, the more effective its spreading due to gravity. When the liquid flows under the surface with negative /?,spreading may stop or even revert, depending on the inclination angle. Even worse, if the inclination angle is a large negative angle, during the first stages of spreading the film may betome unstable with formation of little drops that are easily entrained by the gas phase flowing upward. In this sequence of alternating positive and negative angles of inclination, depending on fluid properties and flow rates, it may take an unusually long section of column for the fluid to become evenly distributed over the solid surface. Although dry spots, the effect of gas shear, and the alternating position of corrugations have not been considered here, it is possible to use the theory derived in this paper to estimate the particular behavior of liquid when spreading on commercial packing surfaces.
Acknowledgment The support of the Office of Research of The University of Tulsa is gratefully acknowledged.
Nomenclature g: acceleration due to gravity, m/s2 H mean curvature of the free surface, llm H: dimensionless mean curvature of the free surface K: dimensionless constant defined by eq 33
KC,: dimensionless constant defined by eq 29 K G ~dimensionless : constant defined by eq 31 m[x]: standard deviation of the film thickness distribution, m N B ~Bond : number defined by eq 27 Nc,: Capillary number, Nca = ,U@J Nst: inverse Stokes number, Nst = 6oi2gl
