Spreading of a Liquid Point Source over a Complex Surface

626

Ind. Eng. Chem. Res. 1998, 37, 626-635

Spreading of a Liquid Point Source over a Complex Surface Sanat Shetty† and Ramon L. Cerro*,‡ Chemical Engineering Department, The University of Tulsa, Tulsa, Oklahoma 74104

The flow of thin viscous films over complex surfaces is relevant to the description of heat- and mass-transfer processes over ordered packings. This work focuses on the gravity-induced bulk spreading of a liquid point source over periodic surfaces. This simulates the flow from a drip point over ordered packings in packed beds. Experimental observations indicate that in the absence of contact lines, for corrugations normal to the vertical, the liquid film adopts a shape similar to a Gaussian distribution as it spreads down the solid surface. For this type of flow, a viscous long-wave-type approximation was used to derive a film evolution equation. The film evolution equation retains all three possible driving forces, i.e., components of gravity normal and tangent to the solid surface and capillarity. Agreement between experimental and predicted film thicknesses is seen to be quite good. Experimental data are given also on flow over complex surfaces with corrugations inclined with respect to the vertical. 1. Introduction The initial liquid distribution onto a bed of structured packings is critically important to the performance of packed towers used in heat- and mass-transfer applications. In countercurrent exchanging devices, liquid is introduced at the 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 entirely cover the solid surface, and this maldistribution results in a loss of efficiency. Though considerable progress has been made in industrial applications and performance evaluations of structured packings, relatively little experimental effort has been devoted to the understanding of the sources and extent of liquid maldistribution in a packed bed with structured packings. A uniform liquid distribution is imperative for developing the full efficiency of the packing, and this is more so for the high-performance ordered packings, since the inherently fixed geometry of structured packings makes it difficult to correct uneven liquid flows. The importance of understanding the initial liquid distribution onto a bed of structured packings has been recognized by packing manufacturers and large production companies, who have put in considerable effort in developing and testing liquid distributors (Perry et al., 1990). Pilot-plant-scale and even full-scale experiments can remove some of the uncertainty in the design of liquid distributors. These experiments, however, are not designed to deal with the fundamentals of wetting and spreading of liquids over complex surfaces. Ordered packings have a very well-defined structure, which includes the macro- and microstructures. The macrostructure in the form of overall channel geometry configures the mixing cells for bulk flow of the vapor phase. It is important to understand the liquid distribution of the point source as it flows down over the packing surface to be able to predict the spatial distri* To whom correspondence should be addressed. † Present address: FINTUBE, 4150 S. Elwood, Tulsa, OK 74107. ‡ Present address: Department of Chemical and Materials Engineering, University of Alabama in Huntsville, Huntsville, AL 35899.

bution of the liquid film. This work deals with the bulk spreading of a point source of liquid over a model test surface which simulates the macrostructure of the ordered packing. Previous to this work (Shetty and Cerro, 1993, 1995), film evolution equations were developed to predict flow profiles for film flows over complex surfaces and the spreading of a point source over an inclined flat surface. There are two ways to describe liquid distribution in packed beds: (1) in form (drops, rivulets, films) and (2) in space (axial and radial profiles) (Bemer and Zuiderweg, 1978). Packing efficiency as related to the effective mass-transfer area is dependent on form as well as on spatial distribution. Most research on liquid distribution has been done on the spatial distribution of liquid in commercial or pilot-plant towers. Spatial distributions in random packings have been modeled using a process similar to random walk, which can be in turn modeled using scalar diffusion equations (Cihla and Schmidt, 1957). Le Goff and Lespinasse (1962) modeled the liquid as flowing down preferred paths. Porter (1968) reconciled these two using a probabilistic model using a frequency distribution to generate diffusion type equations for liquid flow. Albright (1984) used a random walk simulator to show that packings have a natural frequency of distribution. If the distributor can achieve a better distribution than this, the flow will quickly degrade toward it; on the other hand, if the initial liquid distribution is not as good as the natural frequency, a large column length will be needed to achieve it. Hoek et al. (1986) used this concept to characterize the liquid distribution requirements of random and ordered packings. Since ordered packings quickly develop a good distribution only in one direction, i.e., parallel to the layers of corrugated sheets, the anisotropy has to be corrected by stacking successive layers at a 90° angle with respect to each other. More recently experiments have been carried out to study the liquid spreading by injecting a single jet of liquid at the center of one packing element, so that the spreading between sheets could be studied (Stikkelman et al., 1989; Olujic et al., 1993). Olujic et al. (1993) observed that, for a packing with holes or slits, the liquid will stay on the sheet on which it is introduced and is

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Ind. Eng. Chem. Res., Vol. 37, No. 2, 1998 627

[

δ[x,y] ) δ0[x] exp -

Figure 1. Schematic showing the introduction position of the point source.

transported along the channel walls (using the upper and lower sides) to the end of the channel. This results in a large lateral component to the movement of liquid inside the bed. For a packing with a closed surface, the liquid film is forced to flow over the corrugation ridges, making use of the crossing point at which two adjacent packing sheets touch, to provide the channels of the adjacent sheet with liquid. This continuous splitting of a liquid stream reduces the extent of lateral transport but improves the quality of liquid distribution on the small scale. The results obtained by these experiments, however, do not reflect the flow structure for areas smaller than the catching pans used to collect the liquid in these experiments. Extensive experimental data have been obtained for the spreading of a liquid point source introduced on top of a sheet of ordered packing with horizontal corrugations. A film evolution equation based on the viscous long-wave approximation was used to develop a model for flow and dispersion. Part of the data have been obtained for spreading of a point source over the macrostructure with corrugations inclined at a 45° angle with respect to the vertical. 2. Theory Consider the flow of a point source of liquid introduced at the top of a periodic surface (Figure 1). The liquid is introduced at a point O at the top of the surface just before the first corrugation. A Cartesian coordinate system is defined such that the main direction of flow is X. The Y-axis is along the cross-slope direction, i.e., along the width of the solid surface. The Z-axis is normal to the X- and Y-axes and also normal to the average position of the solid surface. The position of the solid surface is described by a periodic function which describes the alternating half-circles. R is the amplitude of the macrostructure and (xc, zc) are the center points of the half-circles.

zs ) zc ( xR2 - (x - xc)2

(1)

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 defined as y ) 0. The cross-sectional shape of the free surface can be described, within experimental error, by the following normal probability distribution function

]

y2 2m[x]2

(2)

Here δ0[x] is the maximum film thickness at the centerline, y ) 0. The function m[x] 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 was shown that, in the absence of contact lines, the normal distribution is a more accurate expression to describe the film thickness profile (Shetty and Cerro, 1995). The film thickness distribution is symmetric about the location of the maximum at the centerline and 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 f ∞. At any cross section, for a constant value of the variable x, the film thickness profile described by eq 2 has two characteristic lengths determining its shape: (1) the peak height, δ0[x] ) O (δ0[x)0]), and (2) the standard deviation, m[x] ) O (m[x)0]). 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 affects only the film thickness profiles very near this origin. We choose the first “crest” of the solid surface or the position of maximum solid surface amplitude as the origin. In most of the flow field, the representation of the film profile given by eq 2 is a very accurate similarity transformation subject to the constraint that the peak height is much smaller than the standard deviation. A small parameter, R, can be defined such that

R)

δ0[x)0] 2m[x)0]

,1

(3)

For a liquid film flowing over the macrostructure of an ordered packing, there is a local equilibrium between gravity and viscous forces (Shetty and Cerro, 1993). The liquid film, therefore, behaves as if it were flowing down a flat surface of continuously varying inclinations. The average velocity in the X-direction, for any value of -π/2 < θ[X] < π/2, must therefore be of the same order of magnitude as the average velocity of a viscous falling film flowing under the effect of gravity on a vertical surface. This velocity, referred to as Nusselt’s velocity, is given by

U[x] )

Fg δ [x]2 3µ 0i

(4)

Assume that the velocities in the x and y directions are of the order of Nusselt’s velocity. Coordinates x and y are scaled by the width of the dispersion at x ) 0, i.e., 2mi, z is scaled by the peak film thickness at x ) 0, i.e., δ0i.

u ) UU

x ) 2miX ν ) UV

δ ) δ0iδ* y ) 2miY

z ) δ0iZ t ) t*(2mi)/U

Using continuity it is possible to show that

w ) RUW

(5)

U, V, and W and u, v, and w are the dimensionless and

628 Ind. Eng. Chem. Res., Vol. 37, No. 2, 1998

dimensional components of velocity in the x, y, and z directions, respectively. In a liquid film pressure is determined by the hydrostatic component and by capillary action at the free surface. The capillary contribution to the pressure field is a function of the mean curvature of the free surface. The definition of the mean curvature is dominated by the second derivatives of the film thickness with respect to the x and y coordinates. As a consequence, the dimensionless expression for the mean curvature is scaled with respect to the peak height and the standard deviation

2H )

2HR 2mi

(6)

where H and H are the dimensionless and dimensional mean curvatures of the free surface. The hydrostatic component of the pressure term is proportional to the film thickness. The vector components of the pressure gradient are then given by

R3 (2H ) ) 0 n.T - n NCa

0i

∂p Uµ ∂P ) ∂y δ 2 ∂Y 0i

∂p Uµ ∂P ) ∂z δ 2R ∂Z

(7)

0i

Under these definitions, the equations of motion can be written in dimensionless form by substitution of eqs 5-7 into the continuity equation and the three components of the linear momentum balance. The three dimensionless components of velocity and the pressure term are expanded as power functions of the small parameter R. Substituting the above equations into the dimensionless form of the equations of motion and discarding all terms affected by powers of R results in a zero-order approximation of the equations of motion:

(8)

∂P(0) ∂2U(0) + NSt cos θ[X] + ∂X ∂Z2

(9)

∂P(0) ∂2U(0) + ∂Y ∂Z2

(10)

∂P(0) - RNSt sin θ[X] ∂Z

(11)

0)-

at Z ) Zs

(13)

[x

(14)

Here, δ*[X,Y] ) δ[x,y]/δ0i is the dimensionless film thickness and NCa ) µU /σ 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

∂U(0) ∂V(0) ) )0 ∂Z ∂Z

at Z ) Zs + δ*[X,Y]

(15)

while the normal component of the free surface boundary condition recovers the Young-Laplace equation

P(0) +

R3 (2H ) ) 0 NCa

at Z ) Zs + δ*[X,Y] (16)

Despite being multiplied by the capillary term remains because the Capillary number is a small number and the curvature of the free surface may be large near the location of the liquid source. The component of momentum normal to the surface (eq 11) is integrated across the film thickness and used to obtain an expression for the pressure field upon introduction of the normal stress boundary condition:

P(0) ) (δ*[X,Y] + Zs(X) - Z)RNSt sin θ[X] -

Here, NRe ) U δ0i/ν is the Reynolds number and NSt ) δ0i2g/Uν is the inverse Stokes number. θ is the angle of the tangent to the solid surface at a surface location. This angle can be defined as a function of the geometrical parameters of the solid surface.

θ[x] ) arctan(dzs/dx) ) arctan

at Z ) Zs + δ*[X,Y]

R3,

∂U(0) ∂V(0) ∂W(0) + )0 ∂X ∂Y ∂Z

0)-

U)V)W)0

and continuity of stresses at the free surface

∂p Uµ ∂P ) ∂x δ 2 ∂X

0)-

magnitude analysis encompasses the larger order terms of the equations of motion. The body force term, RNSt sin θ[X], remains despite being multiplied by R. This analysis is consistent with that of Kheshgi’s (1984) and is justified on the basis that gravity is the main spreading force for all inclination angles, θ > 0. This analysis is similar to the long-wave expansion first used by Benney (1966) to analyze two-dimensional long waves on liquid films. The three forces that drive the flow remain in the zero-order expansion, i.e., components of gravity tangent and normal to the solid surface and capillarity. These terms are balanced by the viscous shear force. Inclusion of higher order terms of R will account for the effect of inertia and normal viscous stresses. The boundary conditions needed to integrate eqs 8-11 are the no-slip condition at the solid surface

((xs - xc)

]

R2 - (xs - xc)2 (12)

The system of equations obtained by this order of

R3NCa-1(2H *) (17) Equation 17 is substituted into eqs 9 and 10, which are then solved for the two components of the velocity field. For a Cartesian system, the velocity components are, for the x-direction

(

U ) NSt cos θ[X] - RNSt sin θ[X]

)

∂δ* + ∂X

(Z - Zs(X))2 R3 ∂2H * [(Z - Zs(X))]δ*[X,Y] (18) NCa ∂X 2 and for the y-direction


(

V ) -RNSt sin θ[X]

[

)

R3 ∂2H * ∂δ* + × ∂Y NCa ∂Y

]

(Z - Zs(X))2 (Z - Zs(X))δ*[X,Y] (19) 2

The velocity profiles for U and V are parabolic, highlighting the close relationship with Nusselt’s velocity profile. 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 15. The component of velocity normal to the solid surface, W[X,Y,Z], can be calculated knowing U[X,Y,Z] and V[X,Y,Z] using the continuity equation. The expression for W[X,Y,Z], being very long, is not given here. The free surface of the spreading film is a three-dimensional surface where the principal radii of curvature, in general, do not coincide with any of the coordinate directions. However, using the parametric representation of the free surface

R B ) bx ı + by j +B k δ(x,y) the mean curvature can be represented as (Scriven, 1981)

(1 + δy2)δxx - 2δxδyδxy + (1 + δx2)δyy

2H )

(1 + δx2 + δy2)3/2

Q ) Uδ0i2mi

∫-∞∫Z

Zs+δ[x,y]

[( ) () s

() π 2

π 2

(23)

x(2c1) δ30i

The second relationship is developed using the kinematic boundary condition at the free surface:

U

Ca

4 RNSt sin θδ* 0[X] ∂m*[X] 24 ∂X

0.5

]

RNSt sin θδ*0[X]3m[X]* ∂Ys[X] (21) 3 ∂X

0.5

(22)

s

s

(24)

which gives

5R3δ*0[X]4 ∂m[x]* 8N m[x]*2 ∂X

4 RNSt cos θδ* 0[X] m*[X] ∂θ 16 ∂X

6 0

∫ZZ +δ*[X,Y]VB(0) dZ

0.5

π 6

2mi )

∂δ*[X,Y] ) -∇ B ∂t*

R3δ*0[X]3 ∂δ* 0[X] 2NCam*[X] ∂X

() () () ()

6 0

The length scale for the x- and y-axes can now be defined as

3 NSt cos θ δ* 0[X] m[X]* 3

0.5

π 2

π 2

U[X,Y] dZ dY )

NSt sin θδ*0[X]3m[x]* ∂δ0[X]* + 6 ∂X

0.5

1/2 2

(20)

0.5

π 2

3 1 c1 ) [g 3νQ cos β(π) ] δ (x) δ (x)

This approximation, strictly, is valid only when the derivatives of m*[X], Ys[X], δ*[X], and θ[X] are negligible. In practice, however, the leading term of eq 21 is much larger than the neglected terms, for all values of X. Using this approximation, the expression for the film thickness profile given by eq 2, becomes

0.5

π 6

Uδ0i2mi

2m[x]2 )

2 *6 δ*[X,Y] ) δ* 0[X] exp[-2Y δ0 [X]]

At steady state, the flow rate at a cross surface defined for X ) constant is independent of X. Using eq 18 for the velocity along the primary direction of flow and neglecting higher order derivatives and higher powers of derivatives, the following expression for the flow rate is obtained: ∞

determined in order to completely specify the film thickness profiles. For the given solid surface geometry and a fluid of given properties, the liquid flow rate is the only variable that can be adjusted during the experiments. To use these independent experimental variables to evaluate the characteristic functions, functional relationships must be developed between m*[X], δ*[X], and Q. One of these relationships is obtained from the theoretical expression for the flow rate (eq 21). Using the leading term in eq 21, i.e., neglecting all terms being multiplied by the small parameter, δ, and writing in dimensional form, the function m[X] can be approximated as follows:

Substitution of experimental values of the peak film thickness, standard deviation of the film thickness distribution in eq 21, allows one to estimate flow rates in good agreement with the experimental flow rate. Equation 21 has two unknown variables, the peak film thickness, δ*0[X] and the standard deviation of the distribution, m[X]*. These two unknowns must be

∂U ∂V +V - W ) 0 evaluated at Z ) Zs + ∂X ∂Y δ*[X,Y] (25)

Equations 24 and 25 are the mathematical expressions for the kinematic boundary condition. These equations are usually referred to as the film evolution equations because they describe changes in film thickness with position along one of the main axes of flow. Substitution of eq 22 into eq 24 eliminates m[X] and reduces it to a nonlinear ordinary differential equation in one variable, δ*0[X]. 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 be obtained by integrating the film evolution equation at Y ) 0, i.e., at the centerline where the profile is symmetric. The resulting equation is still extremely long and would be unmanageable to conventional solution methods. An order of magnitude analysis was therefore done to retain only the important terms. This was done by primarily neglecting higher powers of R, higher order derivatives, and higher powers of the derivatives. This gives


∂δ* 0 2 2 *3 12NSt cos θδ*2 0 + 6NSt cos θδ0 ∂X ∂θ ∂θ 8RNSt cos θδ*3 - 8RNSt2 cos2 θδ*4 + 0 0 ∂X ∂X ∂θ 2 - 12RNSt2 cos θδ*3 3RNSt2δ*4 0 sin θ 0 × ∂X

[

∂Ys ∂θ ∂Ys + 11R2NSt2 cos θδ*4 ) 0 sin θ ∂X ∂X ∂X 192 3 *16 R δ0 + 16RNSt sin θδ*10 0 Ca ∂Ys ∂θ (26) 3NSt2 cos θ sin θδ*4 - 6NSt cos θδ*2 0 0 ∂X ∂X sin θ

(

)

Again, the capillary term multiplied by R3 has been retained using the argument that the Capillary number is very small and may be an important driving force. All of the symbolic computations leading up to eq 26 were done using Mathematica on an HP7000 workstation. 3. Experimental Setup The experimental setup is shown schematically in Figure 2. This setup was described in detail in an earlier paper (Shetty and Cerro, 1995). Only the more important aspects/differences will be described here. The test surface is shown here with the corrugations inclined at a 45° with respect to the vertical. Most of the tests, however, were done with corrugations normal to the vertical line. The test surface has a period of 1 in. and an amplitude of 0.25 in. The test surface is made of stainless steel and is 0.2032 m wide and 0.432 m long. This length includes 16 periods of the half-circle pattern which enables complete development of the flow. The test surface is mounted on a custom-made supporting frame, which is then mounted on a tubular bench. The inclination of the entire test surface could be changed with respect to the vertical over a full 90°, as shown in Figure 2. A small liquid jet flowing out of a glass tip was deposited on the top of the first period of the solid surface at a single drip point to create an initial mal distribution much like the initial distribution of a liquid in industrial packed beds. The liquid used in these experiments was silicone oil (poly(dimethylsiloxane)) of kinematic viscosity 0.1 St. Silicone oil was used since it wets the stainless steel 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 was used to recycle the liquid back to the overhead tank. Liquid flow rates were measured by weighing liquid collected over a fixed interval of time. Liquid flow rates were varied to encompass the range of flow rates per drip point typically found in commercial units. A very thin needle mounted on a x-y precision translator was used to detect the position of the free surface and the solid surface. The translator ensemble was mounted on a tubular bench kept parallel to the test surface such that the needle could be placed normal to the test surface at any point on the test surface. The

entire experimental assembly was mounted on an Oriel optical table. The back of the test surface and one end of the needle were connected to a laboratory ohmmeter. Due to the large difference in electrical conductivity between the solid surface and silicone oil, it was possible to detect the position of the solid surface when the needle touched it. However, since the conductivity of silicone oil is very low, the position of the free surface was detected with a combination of side-lighting and a video camera zoomed in on the liquid surface. Experimental measurements are assumed to be good down to 5 mm. These measurements were done at several values of x conveniently distributed along the length of the plate. 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 error in the position of two consecutive solid surface maxima was found to be no greater than 0.3°. 4. Results and Discussion Figures 3-6 show the intermediate stages in the development of the stable wetting profile of the point source of liquid introduced onto the macrostructure kept with its corrugations horizontal. There are a variety of phenomena taking place during these intermediate stages of spreading. These phenomena may be the roots of some peculiar start-up problems of industrial packed distillation and absorption columns. To understand these figures, it is important to remember that the liquid wets the test surface, and at the begining of the experiment, the surface is initially covered with a very thin layer of liquid. During the initial stages of the experiment the liquid drips down from one crest to another under the influence of gravity, rather than following the surface contour as shown in Figure 3. Notice the liquid bridges joining the crests of the solid surface even when there are no contact lines present. After a few seconds, the liquid bridges move into the valleys of the solid surface and the thread of liquid follows the contour of the surface as shown in Figure 4. The flow patterns depicted in Figures 3 and 4 are two possible solutions of the equations of motion, suggesting a bifurcation behavior even at these low Reynolds numbers. The presence of the bridges is facilitated by the initial conditions of the experiment, but this flow loses stability and evolves toward the contour flow. The contour flow shown in Figure 4 reminds one of the teapot effect first described by Reiner (1956). Slowly, the main driving force in the cross-slope direction, g sin[θ[X]]∆δ, acts upon the liquid thread and the liquid spreads for positive inclination angles of the solid surface. For positive inclination angles the component of the gravity force normal to the solid surface drives the liquid away from the centerline, while for negative inclination angles it brings it back toward the center of symmetry (Shetty and Cerro, 1995). This sequence of effects, promoting and hindering spreading, gives rise to the wine glass pattern seen in Figure 5. Notice the movement of the liquid away from the center of symmetry when the liquid flows over the corrugation and toward the middle and even reverting to a thread, when the liquid flows under the corrugation. A corrugated surface, as a result of this phenomenon, spreads slower than a flat vertical surface.


Figure 2. Schematic of the experimental setup.

Figure 3. Stage 1 in the development of a stable wetting profile for point source spreading over the complex surface.




Eventually, the width of the wine glass pattern increases as shown in Figure 6 and will encompass the entire width of the periodic surface. In actual packing configurations, this effect could be faster due to the effect of the points of contact between adjacent sheets of packing. Figure 7 shows the free surface profiles for one period of the solid surface at different positions across the width of the solid surface. The free surface profiles have

already evolved from a jumping thread to a wine glass pattern that follows the contour of the solid surface closely. There is not a uniform film thickness across the width of the solid surface but a maximum at the center of symmetry and a lateral film thickness asymptotically approaching zero on either side. The difference in film thickness between the center and the sides is more evident on the underside of the solid surface, for negative inclination angles and in the valley, when the


Figure 7. Free surface profiles over a given period at different positions across the solid surface width. y (cm): 9, -0.8; [, -0.3; f, 0; O, 0.3; ], 0.8.

Figure 8. Cross-slope film thickness profiles. x (cm): 9, 32.5; [, 34.1; f, 34.68, 0, 36.78; ], 37.11; *, 37.69; O, 38.43.

center of the film grows at the expense of the surrounding regions. The recurring effect of gravity for positive and negative angles is evident again in Figure 8 where the liquid film thickness profiles in the direction normal to flow are shown at different positions along the solid surface. The values of the variable x account for positions along the wavy solid surface, as indicated on the upper right side of the figure. The centerline at y ) 0 corresponds to the center of symmetry of the liquid source. Here x ) 35.15 is at the peak of the solid surface. As the film moves along the solid surface, the film thickness varies in a complex manner which is a combination of film flow over the periodic surface and spreading over and under the solid surface. For all periods of the solid surface, the minimum film thicknesses are at the crest of the solid surface and the maximum thicknesses are at the valleys. Notice that the areas under the film profiles are different at different positions. The overall flow rate, however, is the same for all positions because the average velocities change with position. It is clearly seen that a film profile spreads out on the top side of the surface but retraces to the center, on the surface underside. The logarithm of the film thickness normalized using the thickness at the centerline is shown as a function of y2 for different values of x in Figure 9. The figure shows, within experimental error, straight lines of different slopes for each value of x. The slope of these

Figure 9. ln(δ/δ0) vs y2. x (cm): 9, 32.5; f, 36.48; +, 37.69; O, 38.43.

Figure 10. Normalized cross-slope film thickness profiles. x (cm): 0, 36.78; ], 37.11; *, 37.69; O, 38.43.

lines is -1/(2m2[x]). Figure 10 shows the same experimental data shown in Figure 9 but plotted as the dimensionless film thickness δ*(X) versus the dimensionless normal surface coordinate, y* ) [y2/(2m[x]2]0.5. The variance of the normal distribution, m[x], was calculated using the slopes of the lines in Figure 9. The fact that all data collapse into a single curve demonstrates that the normal distribution is indeed a similarity transformation that can be used to represent the film thickness profiles. Figures 11 and 12 show the film thickness profiles along the direction of flow for different positions across the solid surface (i.e., different values of y) for two different flow rates. The film thicknesses shown in Figures 11 and 12 are normal film thickness values, that is, measured in a direction that is normal to the solid surface at the point of measurement. Experimental normal film thicknesses necessary for comparison with the theoretical values of the normal film thickness, are calculated from experimental data using an algorithm that has been described in detail elsewhere (Shetty and Cerro, 1993). For low liquid flow rates the centerline film thickness profile is distinctly double-periodic, as was seen for film flow over the macrostructure of ordered packings (Shetty and Cerro, 1993). The peaks of the normal film thickness profiles coincide with the positions on the solid surface where the inclination angle is more removed from the vertical. The maximum value of the normal film thickness of a spreading film slowly


Figure 11. Normal film thickness profiles along the direction of flow at different positions across the width of the solid surface. Q ) 1.45 mL/min. y (cm): 9, -2.5; ], -1; O, 0; 0, 1; [, 2.5.

Figure 13. Effect of liquid flow rate on normal film thickness profiles. Q (mL/min): 9, 0.49; ], 1.45; f, 4.16; 0, 6.98; [, 9.04; O, 12.6.

spreading of a liquid point source over the macrostructure of the solid surface. At the limit as θ f 0, i.e., for a point source spreading over a vertical flat surface, the two dominant terms are the viscous and capillary forces. For NSt ) 3.0, this equation reduces to (Shetty and Cerro, 1995):

δ*14 0 +

Figure 12. Normal film thickness profiles along the direction of flow at different positions across the width of the solid surface. Q ) 12.96 mL/min. y (cm): 9, -2.7; [, -0.7; O, 0; 0, 0.8; ], 2.8.

decreases steadily as the film moves downstream. Increasing the flow rate results in a film thickness profile where the difference between the thickness of the peaks and valleys is less pronounced (Figure 12). This effect is more evident at positions away from the centerline. The effect of flow rates on the maximum film thickness is explored in Figure 13, which shows the centerline film thickness at different positions along the surface and for different flow rates. For the relatively viscous liquid used in this experimental work, the film thickness profiles stabilize very quickly; i.e., the film thickness profiles on successive periods are nearly identical after only the first three or four periods of the solid surfaces. This effect is more pronounced for small flow rates. The thickness of the film centerline reaches an asymptotic value after a few cycles of the solid surface, but the maximum thickness at lines adjacent to the centerline (i.e., y ) (1) also reaches an asymptotic maximum but of smaller size. This indicates that, after a few cycles of the solid surface, the film thickness profiles go through a stable sequence of wider and narrower film profiles but spreading essentially stops. This situation is very different from spreading on an inclined surface, where the pull of gravity keeps spreading the film toward the sides, indefinitely. The film evolution equation (26) describes the evolution of the centerline film thickness profile for the

3NCa ∂δ* 0 )0 16R3 ∂X

(27)

For a corrugated surface, a similar approximation consists of using the leading viscous term (i.e., the first term on the left-hand side of eq 26) and the term for the primary driving force in the cross-slope direction (i.e., the second term on the right-hand side of eq 26) to give *2 16R sin θ[X]δ*10 0 + 12 cos θ[X]δ0

∂δ* 0 )0 ∂X

(28)

Capillary forces make a negligible contribution to the spreading of a point source over the macrostructure of a packing surface. Neglecting the capillary term in eq 26 allowed it to be solved numerically for the test case of R ) 0.088. The theoretical film thickness profile for the centerline, δ*0(x), computed using eq 26 is compared with experimental film thickness profiles in Figure 14. The minimum value at the valley of the first period of the experimental film thickness is significantly larger than the computed values, but agreement becomes significantly better for successive periods of the solid surface. Notice that the x-axis in this plot is the x-coordinate normalized by the solid surface period (l and not 2mi) to make it easier to distinguish the peaks and valleys of the solid surface. In industrial practice, packing corrugations are inclined with respect to the vertical at angles 30° < β < 60°. A set of data was also taken for the spreading of a point source over a complex corrugated packing surface with corrugations inclined at β ) 45° with respect to the vertical. The angle of inclination was measured as positive to the left-hand side, and the liquid was introduced at the right edge of the test surface as shown in Figure 2. As in fully developed film flow (Shetty and Cerro, 1997), the liquid flows along the direction of the corrugations and spreads over the corrugation ridges in its generally downward flow. This succession of flows


Figure 14. Comparison of theoretical and experimental normal film thickness profiles for point source spreading over the complex surface. R ) 0.088; O, theory; 9, experimental.

Figure 15. Locii of points of maximum distance travelled by liquid film for point source spreading over the complex surface placed inclined. Q (mL/min): O, 4.67; 0, 8.25.

along the channels and over the ridges effectively decreases the angle of flow with respect to the horizontal axis. The liquid no longer spreads toward the right side edge of the packing surface because this would essentially require the liquid to flow upward against the gravity forces. In fact, the right edge of the region where the liquid film is detectable, moves away from the vertical line traced from the liquid source. Figure 15 shows the locus of the points were the right side edge of the liquid film becomes undetectable. These experiments were done at two different flow rates to show the fact that inertial forces are negligible. Notice that the line representing this movement away from the source has a constant inclination solely determined by surface geometry. In packing surfaces this asymmetric movement is counterbalanced by the presence of another corrugated sheet with inclinations at a negative value of β. Figure 16 shows the film thickness profiles along the length of the surface for different positions across the width of the surface. Figure 17 shows the same experimental data as a surface in a three-dimensional plot, making it easier to visualize the location of the film thickness maxima. The measurements shown in Figure 17 are on a square grid oriented in the same direction as the test surface. The asymmetry of the film thickness profiles is more evident here. Notice the shape of the

Figure 16. Normal film thickness profiles for point source spreading over the complex surface placed inclined. Y (cm): O, 18.64; 9, 12.14; f, 6.84; 0, 4.14; ], 2.44; *, 2.14.

Figure 17. Surface plot of film thickness profiles for point source spreading over the complex surface placed inclined.

film at the right edge as it abruptly goes from a maximum to nearly zero. Computation of these profiles would require a nonsymmetric film thickness profile, substantially different from eq 2 used in the computation of the flow over horizontal corrugations. 5. Concluding Remarks Experimental data have been obtained for the spreading of a liquid point source over the macrostructure of a type of ordered packing with corrugations horizontal and at a 45° angle with respect to the vertical. For horizontal corrugations, the thickness profiles in the direction normal to flow are best described by a Gaussian distribution. Thus, flows spreading over a complex surface are a linear combination of two base flows, that is, the film flow over a periodic surface and the flow of a point source spreading over and under an inclined flat surface. For these thin films, the linear combination is possible due to the absence of inertial effects. A film evolution based on the long-wave approach was developed and validated by comparing it with experimental data. The most important conclusion is that dispersion depends strongly on the corrugation angle and on the angle of inclinations of corrugations with respect to the vertical. Films are spread out on the upper side of the corrugated surface but tend to revert back to rivulets


when flowing on the underside of the solid surface. The overall effect of this sequence of positive and negative spreading is that the film reaches a stable distribution and spreading stops after a few solid surface cycles. It must be pointed out that in this analysis the perturbations introduced by the points of contact between adjacent corrugated sheets are not taken into account; thus, it is possible that they may have an important role in spreading. Corrugations inclined with respect to the vertical create a degree of asymmetry on the film profiles that prevents its theoretical analysis. Spreading takes place toward the downhill side of the corrugations and is virtually nil to the other side. This is simply due to the fact that the film will not spread against gravity.

θ: angle of tangent to the solid surface with the vertical, rad ν: kinematic viscosity, m2/s F: density, kg/m3 σ: surface tension coefficient, N/m ψ: three-dimensional vector potential function 2H: dimensionless mean curvature

Acknowledgment

Literature Cited

The early stages of this research were supported by Grant CTS-8912784 of the National Science Foundation, Division of Chemical and Thermal Systems, Fluid, Particulate & Hydraulic Systems Program.

Albright, M. A. Packed tower distributors tested. Hydrocarbon Process. 1984, 173. Bemer, G. G.; Zuiderweg, F. J. Radial liquid spread and maldistributions in packed columns under different wetting conditions. Chem. Eng. Sci. 1978, 33, 1637. Benney, D. J. Long Waves on liquid films. J. Math. Phys. 1966, 45, 150. Cihla, Z.; Schmidt, O. Collect. Czech., Chem. Commun. 1957, 22, 896. Hoek, P. J.; Wesselingh, J. A.; Zuiderweg, F. J. Small scale and large scale liquid maldistrubution in packed columns. Chem. Eng. Res. Dev. 1986, 64, 431. Kheshgi, H. S. The motion of viscous liquid films. Ph.D. Dissertation, The University of Minnesota, Minneapolis, MN, 1984. Le Goff, P.; Lespinasse, B. Rev. Inst. Fr. Pet. 1962, 17, 21. Olujic, Z.; Stoter, F.; de Graauw, J. Liquid distribution performance of structured packings with different surface characteristics Presented at the AIChE Spring National Meeting, Houston, TX, 1993. Perry, D.; Nutter, D. E.; Hale, A. Liquid Distribution for Optimum Packing Performance. Chem. Eng. Prog. 1990, Jan, 30-35. Porter, K. E. Liquid flow in packed columns. Part I: The rivulet model. Trans. Inst. Chem. Eng. (London) 1968, 46, T69. Reiner, M. The Teapot Effect. Phys. Today (Sept), 1956, 9, 1620. Scriven, L. E. Surface Geometry for Capillarity, ChEn 8104 Course Handout. The University of Minnesota, Minneapolis, MN, 1981. Shetty, S. A.; Cerro, R. L. Flow of a Thin Film Over a Periodic Surface Int. J. Multiphase Flow 1993, 19, No. 6, 1013. Shetty, S. A.; Cerro, R. L. Spreading of Liquid Point Sources Over Inclined Solid Surfaces Ind. Eng. Chem. Res. 1995, 34, No. 11, 4078. Smith, P. C. A similarity solution for slow viscous flow down an inclined plane. J. Fluid. Mech. 1973, 58, 275-288. Stikkelman, R.; de Graauw, J.; Olujic, Z.; Wesselingh, H. A study of gas and liquid distributions in structured packings. Chem. Eng. Technol. 1989, 12, 445.

Nomenclature c1: dimensional constant defined in eq 22 g: acceleration due to gravity, m/s2 gx, gy: acceleration due to gravity in the X- and Zdirections, m/s2 2H: mean curvature, 1/m m[x]: standard deviation of the film thickness distribution, m n: normal vector NCa: Capillary number, mU/σ NRe: Reynolds number, FUδ/µ NSt: inverse Stokes number, δ2g/Uu p: pressure, Pa P: dimensionless pressure Q: total flow rate, m3/s R: amplitude of the solid surface, m t: time, s t*: dimensionless time T: stress tensor u, v, w: dimensional velocity components, m/s U, V, W: dimensionless velocity components U : Nusselt’s velocity, m/s x, y, z: coordinate system X, Y, Z: dimensionless coordinate system Greek Letters R: perturbation parameter, δ0/2m β: angle of inclination of corrugations with respect to gravity, rad δ: film thickness, m δ*: dimensionless film thickness ∇: gradient operator µ: dynamic viscosity, Pa‚s

Subscripts c: center 0: peak film thickness 0i: peak film thickness at x ) 0 s: solid surface Superscript (n): nth-order perturbation term, n ) 0, 1, 2

Received for review April 22, 1997 Revised manuscript received October 27, 1997 Accepted October 28, 1997 IE970291T

Spreading of a Liquid Point Source over a Complex Surface

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