Laminar Film Flow down a Right Circular Cone Rlchard L. Zollars and William B. Krantz* Department of Chemical Engineering, University of Colorado, Boulder, Colorado 80302
An asymptotic solution is developed for laminar film flow down a right circular cone. This solution incorporates the effects of surface tension and associated curvature of the film which were not included in a nonsystematic solution available for this flow. The solution developed here includes this ad hoc solution as a limiting case. The results are of particular value to investigators employing the conical wetted-wall column as a laboratory absorber or reactor, and to those interested in studying the stability of this low Reynolds number nonparallel flow.
The solution for the laminar flow of a film down a right circular cone is complicated by the fact that this is a nonparallel decelerating flow. There is need for this basic flow solution since this flow geometry is used in high precision laboratory absorbers and reactors when contact times in the range 0.2 to 1.0 s and large interfacial areas (-80 cm2) are desired. This flow also occurs in commercial equipment such as centrifugal evaporators and venturi absorbers. An explanation of the topographical features emanating from certain geophysical flows, such as that of molten lava down a volcanic cone, would be aided by this basic flow solution. Furthermore, an understanding of this nonparallel film flow may permit us to develop improved models to predict the performance of such contacting devices as trickling filters which involve nonparallel film flows but which are usually analyzed using parallel flow models. The authors' interest in developing a solution to this basic flow stems from an interest in the linear stability problem for this low Reynolds number flow. Indeed only four rigorous solutions to the linear stability problem for nonparallel flows have appeared in the literature. These have been confined to relatively high Reynolds number flows. However, these latter studies indicate that the nonparallel flow effects on the hydrodynamic stability are more pronounced a t lower Reynolds numbers. This brief review clearly establishes the need to investigate an idealized nonparallel flow, such as film flow down a cone, in more detail. A nonsystematic solution for laminar film flow down a cone is given by Bird et al. (1966).This ad hoc solution assumes a semi-parabolic velocity profile for the streamwise velocity component. This then is used in a macroscopic mass balance in order to obtain an equation for the local film thickness. Unfortunately, this solution does not account for any dependence of the velocity profiles or film thickness on the surface tension and associated curvatures of the basic flow. Furthermore, the range of validity of this ad hoc solution is difficult to assess, although intuitively one would expect that it is restricted to very low Reynolds number flows. The purpose of this note is to develop a systematic asymptotic solution for the laminar film flow down a cone incorporating the effects of surface tension associated with the streamwise and lateral curvature of the film. Theoretical Development The coordinate system used here places an x axis along the surface of the cone such that x = 0 is a t the apex of the cone, and a y axis perpendicular to the 1: axis such that y = 0 is at the surface of the cone. A radial coordinate measured from the axis of symmetry of the cone is given in dimensionless form by r = x sin p 6y cos p where p is the
+
apex angle of the cone as measured from the axis of symmetry. The equations of motion corresponding to this coordinate system are given by Millikan (1932). When appropriately nondimensionalized they assume the following form: au au u s i n p 6u cos@ =O (1) ax ay x s i n p 6y cosp
+
-+-+
:
N R , ~u - + u - -
(
+
aP
ii)
a2u
a2u
= -6-++,-62ax ay
:;
ax ay
cos p + cos6 ( 2 ) ax x s i n p 6y cos@ aP a2u a2u N R , (~u ;~+ u a y = - - +63--6a' ay ax2 axay sin p 63--6- s i n p (3) au ax ay x sin p 6y cosp The corresponding dimensionless boundary conditions to be satisfied are given by
+ (6 - - 6 3 -
(
+
)
(
+
)
"I
+(
(" ay
+ 62-)au
u=o
(y=O)
(4)
u=o
(y=O)
(5)
(1 - a2hX2)- 2a2h,
ax
h X 2 au au - - - + NReNW.96 1 - h X 2(ax ) ,a 1 - 6h, tan p - ( x tan p 6h)(l 62hx2)1/2=
au (~-~,)-26-+263aY
[ (1+dsh;E2)3/2
+
+
]
o
( y = h ) (7)
Since this flow involves a free surface, the film thickness is related to the velocity components by the surface kinematic condition given in dimensionless form by
-ah_ -- u ax
u
(Y = h)
In arriving a t the above equations the streamwise and cross-stream velocity components u and u , pressure P , and independent variables x and y were nondimensionalized with the scale factors u,, u c , P,, x c , and y c , respectively. This nondimensionalization introduces the Reynolds number N R e u a y , / u ,Weber number N w e n/puc2yc,and 6 = y c / x c a dimensionless group which is a measure of the cross-stream to streamwise diffusion of vorticity. Since it is the judicious scaling of this problem which permits us to obtain an asymptotic solution, it is worthwhile to summarize these scaling arguments. One relationInd. Eng. Chem., Fundam., Vol. 15, No. 2, 1976
91
ship between the above scale factors is determined by balancing the streamwise gravity forces with the viscous forces; this necessarily scales the equations appropriate to a low Reynolds number flow. We also balance the crossstream gravity forces with the hydrostatic pressure forces. The continuity equation given by eq 1 also yields another relationship between the scale factors. In order to relate the scale factors to the volumetric flow rate Q, we normalize the overall continuity condition given in dimensionless form by
$ = $0
+ 6$1+
(16)
($o)rrrr = 0 ($o)y
($o)yy
(10)
+ ...
The above can be substituted into the equations of motion and corresponding boundary conditions in terms of the stream function. Retaining only the zeroth-order terms in 6 yields the following differential equation and associated boundary conditions to be solved for $0:
(Y = 0 )
(18)
0
(Y = 0 )
(19)
=0
(Y = h )
(20)
+ x sin p cos p = 0
($o)yyr
(17)
=0
($OL =
The preceding constitute four relations between the five scale factors. They can be used to express four of the scale factors in terms of the scale factor x , L which will remain unspecified since there is no characteristic length in the streamwise direction. The use of an unspecified length scale factor in defining suitable parameters for perturbation expansion solutions is discussed by Van Dyke (1964). These scale factors then assume the following form:
62$2
(y = h )
(21)
The subscript x and y’s denote differentiation with respect to these variables. Note that eq 17 does not contain any derivatives with respect to x . The x dependence of $0 enters only through the tangential and normal stress boundary conditions given by eq 20 and 21 since both of these conditions are evaluated at h which is a function of x . Retaining first-order terms in 6 yields the following set of equations to be solved for $1:
u, = (--)113Q2g a2uL2
Q
u, = -
2nL2 Qp3g2u
pc =
1/3
(L)
The boundary conditions given by eq 4 and 5 are the noslip and no-flow conditions a t the solid surface. The boundary conditions given by eq 6 and 7 are the tangential and normal force balances at the free surface. Note that in the latter, the first and second terms in brackets are the streamwise and lateral curvature effects, respectively. Hence these terms introduce the surface tension effect as indicated by the presence of the Weber number. Note that we have not attempted to satisfy any boundary conditions for specified values of x . This proves to be no limitation since we will solve this system of equations via a perturbation scheme. The resulting zeroth-order set of equations will not contain any derivatives with respect to x and hence will not require any boundary conditions for this independent variable. Thus this is a singular perturbation scheme akin to that used in boundary layer problems. Indeed it is fortunate that we do not need to prescribe boundary conditions in x since it is not obvious which conditions we should choose to satisfy here. Since this is a two-dimensional flow it is convenient to define a stream function $ such that
Equations 2 through 8 can be recast in terms of $. The resulting equations will not be given here as they are quite cumbersome. Indeed the resulting equation of motion in terms of the stream function involves some 52 terms most of which are higher order terms not considered to the order which this equation is solved via the perturbation scheme described below. The interested reader desiring more details of this solution is referred to Zollars (1974). For low Reynolds number film flows the parameter 6 will be small. This suggests that the problem can be solved via a perturbation expansion in this parameter of the form 92
Ind. Eng. Chem., Fundam., Vol. 15, No. 2, 1976
6($l)yy
6 cos p
--
6y cos p
x sin (3
(Y = h ) (25)
6 cos p
--
x sin p
($o)yy
+ x sin p cos p - 6 x sin2 ph,
Note that the effect of the lateral curvature appears in eq 26 whereas the effect of streamwise curvature will not appear until terms of order J2 are retained. This becomes obvious when the terms contained in brackets in eq 7 are expanded in a Taylor series about 6 = 0. It would appear from eq 7 that the streamwise curvature term is proportional to N w e P and hence is of order 6 if N w , = O(l/S). However, in converting to the stream function formulation one eliminates P - P, from eq 7 by differentiating eq 7 with respect to x and substituting for 6P/6x from eq 2. Hence it is seen that 6P/6x is of order 1/6 relative to the other terms in the resulting form of eq 7. In arriving at eq 17 through 26 several ordering arguments were necessary. These are N R , = O(1); l / r = O(1); and N w e = O(l/6). The symbol 0 (“big oh”) used here connotes for example in the case N R =~ O(1) that N R has ~ an ~ fact can be smaller upper bound of order unity; N R in without violating the ordering arguments used in retaining and casting out terms in arriving at eq 17 through 26. The
first of these ordering arguments will restrict the maximum volumetric flow rate for a given set of fluid properties. The second will restrict the solution from describing the flow near the apex of the cone; this does not prove to be a very limiting restriction in most practical cases as will be shown later. The third will restrict the solution from describing flows which have very large Weber numbers. Note that the kinematic surface condition given by eq 8 indicates that h is of the form h = ho
+ 6hl+ 62h2+ . . .
0 ‘21
11 1
33 3
22 2
44 4
55 5
I
I
a p r e e i c t i o n s of eqoarion 30 b p r e d c t i o n s ny B i r d e t al. 1 A3: 1 7 8 , A 2 = 3 31 2 A 3 : 3390.A2:1.70x104
+
(27)
This then permits us to solve eq 17 and 22 for $0 and $1, respectively. The resulting solution for $ is given by V L
+cos2P
+N
(y+--”)
ho2Y2 +xsin2P(ho), 4 12
(x6
R sin ~ cos2 (3 -- ho2y5- y 7 I 2h05y2) 90 1260 45
The above equation permits us to determine u and u at any point along the cone. The terms ho and hl are the zerothand first-order terms, respectively, in the solution for h. The latter is obtained by substituting eq 28 into the kinematic surface condition to yield a first-order differential equation for h. The integration of this equation is straightforward; the resulting integration constant is datermined via the normalized overall continuity condition given by eq 9. The resulting equation for h is given by
where K=(
sin /3 cos P
Discussion It is interesting to note that the asymptotic solution given by eq 28 and 29 reduces to that presented by Bird et al. if only terms of zero order in 6 .are retained. This then establishes this ad hoc solution on a rigorous basis. Furthermore, it represents an improvement on the latter solution in that the first-order terms in 6 include the effect of the lateral curvature of this flow. If eq 28 and 29 are recast in dimensional form the unspecified scale factor L necessarily will cancel out. For purposes of generalizing these results and presenting them in graphical form it is convenient to define a new length scale ~. length scale factor is suggested factor ( Q u / 2 ~ g ) l / This when eq 29 is recast in dimensional form. When this length scale factor is used to nondimensionalize all lengths appearing in the dimensional form of eq 29, this equation assumes the following form:
where A2 = Qg1/3/2au513 and A3 = ~ ~ / p u ~ / ~ This g ’ / ~ choice . of dimensionless groups is convenient since it isolates those
0
40
x.
120
“60
200
Figure 1. Dimensionless film thickness vs. dimensionless distance along a 30° cone showing velocity profiles a t two locations.
parameters characterizing the flow into A2 and those characterizing the fluid into As. Typical results are given in Figure 1 which shows a plot of the dimensionless film thickness ha vs. dimensionless axial distance x * for both the solution given by eq 30 denoted by the “a” curves and that presented by Bird et al. denoted by the “b” curves. The upper set of curves denoted by l a and l b are for A3 = 1.78, a value typical of light mineral oils; the lower set of curves denoted by 2a and 2b are for A3 = 3390, the value for water. The value of Q for both sets of curves is 5 cm3/s; this corresponds to a value of A2 = 3.31 for the light mineral oils and A2 = 1.70 X lo4 for water. The apex angle of the cone P = 30’. The u velocity profiles a t two locations are superimposed on these curves. In the case of the light mineral oils these locations are a t XI equal 10.4 and 36.4 corresponding to dimensional distances of 2 and 7 cm, respectively. In the case of water they are at x * equal to 37.5 and 131 corresponding to dimensional distances of 2 and 7 cm, respectively. The velocity profiles are plotted on a relative basis; that is, all velocities were scaled relative to the maximum velocity, namely the surface velocity for curve 2a at the upstream location which was us = 46.2 cm/s. This velocity is repraented by the horizontal reference line at the upstream location of curve 2a. The velocities within the film or a t other locations for either of the two fluids then can be determined by comparing the appropriate horizontal length indicated by the velocity profiles to the length of the reference line. Note that the curves in Figure 1 are not extended to small values of x * because of our ordering argument l / r = O(1). These curves have been arbitrarily terminated a t upstream locations corresponding to 1.5 cm for the light mineral oil and 2.2 cm for water. In order to quantitatively assess the range of validity of the solution developed here it is necessary to estimate the undetermined scale factor L . Physically L corresponds to the length over which the streamwise velocity component undergoes a characteristic change of uc. A first guess at an appropriate L might be the local value of Xd, that is, the distance from the apex a t which the film thickness and velocity profiles are to be determined. Using this estimate of L and the criterion that “small 6” corresponds to 6 = 0(0.1), one finds that our ordering argument l / r = o(1)is satisfied for X d I 4.04 cm for the mineral oil curve in Figure 1 and for X d I 1.12 cm for the water curve. However, the fact that the zeroth- and first-order solutions (i.e., curves “a” and “b”) agree nearly identically a t X d = 7 cm indicates that X d is a very conservative estimate of L . Thus the solution developed here Ind. Eng. Chem., Fundam., Vol. 15, No. 2 , 1976
93
probably can be applied even closer to the apex than shown in Figure 1. This then substantiates our earlier claim that the ordering argument llr = O(1) does not prove to be a very limiting restriction. Figure 1 indicates that the first-order terms in 6 can represent a significant improvement over the zeroth-order solution presented by Bird et al. For example, in the case of water the zeroth-order solution overpredicts h* by 19% and underpredicts u* by 16% at the upstream location. These deviations would of course be even larger if curves 2a and 2b were extended further toward the apex as appears to be justified based on the preceding discussion. The two solutions are seen to come in closer agreement as we move progressively down the cone. Parameter studies indicated that the two solutions agree more closely for small values of A2 and A 3 and large values of p. Conclusions An improved solution for film flow down a right circular cone has been developed via a perturbation technique. This solution includes the solution presented by Bird et al. as a limiting case. The latter solution can result in significant error in predicting h, u, and u when the dimensionless groups A2 and A3 are large or when the cone apex angle (3 is small. Furthermore, Zollars (1974) has found that even when the two solutions are within close agreement, the first-order terms in 6 contained in the solution presented here can have a significant effect on the stability of this flow. The solution presented here should be of particular value to those investigators utilizing the conical wetted-wall column as a laboratory absorber or reactor. In particular, eq 28 permits us to determine both u and u. The latter velocity component sometimes is ignored in analyzing data for such devices; that is, the absorption is viewed as diffusive mass transfer into a nonuniform flow. The presence of this transverse velocity component could significantly increase the mass transfer. Ignoring its presence in some cases could result in overestimating the mass transfer coefficients determined from such devices. The perturbation solution presented here could be carried out to higher order terms in 6 thus extending its range of applicability. It is also possible to solve the perturbation scheme using the ordering argument Nwe = O ( l / P ) . This would result in including the streamwise curvature terms with the first-order terms in 6. Furthermore, it is worthwhile to mention that the solution technique employed here can be extended to other nonparallel flows such as film flow over a sphere or the condensate film flow in vertical condensers. Acknowledgment A portion of the work presented in this paper was completed while one of the authors (W.B.K) was a Fulbright Lecturer in the Department of Chemical Engineering a t Istanbul Technical University in Turkey. This author grate-
94
Ind. Eng. Chem., Fundam., Vol. 15, No. 2, 1976
fully acknowledges this support from the Fulbright-Hays Program and the cooperation he received from the faculty and staff a t Istanbul Technical University. The authors also wish to thank Dr. Byron E. Anshus of the Chevron Research Company and Professor Stephen Whitaker of the University of California a t Davis for their helpful comments concerning this paper. Nomenclature A2 = Qg1/'/2au'/' A3 = ~ / p u ~ ! ~ g ~ / ' g = acceleration of gravity h = local film thickness nondimensionalized with respect to Y c h* = local film thickness nondimensionalized with respect to ( Q u / B i ~ g ) ~ / ~ L = unspecified scale factor in streamwise direction N R = ~ Reynolds number u a c / u N w e = Weber number u/py,uC2 P = pressure P, = (Qp3g2~/2~L)1/3 P , = ambient pressure Q = volumetric flow rate r = radial coordinate nondimensionalized with respect to xc
u = streamwise velocity component nondimensionalized with respect to uc uc = (Q2g/4a2uL2)1/3
u, = surface velocity u = cross-stream velocity component nondimensionalized
with respect to u, = Q/2rL2 x = streamwise coordinate nondimensionalized with respect to x , x * = streamwise coordinate nondimensionalized with respect to ( Q 1 ~ / 2 i ~ g ) l / ~ x, = L X d = dimensional streamwise coordinate y = cross-stream coordinate nondimensionalized with respect to y , y , = (Qv/2i~gL)l/~ U,
Greek Letters @ = apex angle of cone as measured from axis of symmetry 6 = YC/X, u = kinematic viscosity p = density u = surface tension J. = stream function nondimensionalized with respect to UCYC
Literature Cited Bird, R . B., Stewart, W. E.,Lightfoot, E. N.. "Transport Phenomena", p 121, Wiley. New York, N.Y., 1966. Millikan, C. B.. Trans. Am. Soc. Mech. Eng., 54, 29 (1932). Van Dyke, M., "Perturbation Methods in Fluid Mechanics", pp 124-132, Academic Press, New York, N.Y.. 1964. Zollars, R . L., Ph.D. Thesis, University of Colorado, Boulder, Colo., 1974.
Received for reuiew April 29, 1975 Accepted January 29,1976
