MASS TRANSFER T O FALLING WAVY LIQUID
FILMS IN TURBULENT FLOW SANJOY BANERJEE, D O N A L D S. SCOTT, A N D E D W A R D R H O D E S Dejartment of Chemical Engineering, L'niLersttj of Jt'aterloo, Tt'aterloo, Ontario, Canada
The mass transfer rate a t a free interface can b e related to the rate of viscous dissipation in the turbulent flow near the surface, An order of magnitude estimate for the dissipation for a wavy turbulent liquid film is derived from a consideration of the time-averaged vorticity equation. The liquid phase mass transfer coefficient is then related to the average wave length, amplitude, and speed of the surface disturbances. The predicted values of the transfer coefficients agree well with experimental observations.
there exist considerable experimental data on liquid phase-controlled mass transfer to falling liquid films and other data on fluid mechanical aspects of falling liquid films, there appears to be a marked lack of understanding of these phenomena except a t very lo\\ Reynolds numbers. Since a comprehensive revie\< by Fulford (1964) of the properties of thin liquid films is available: it is not necessary to survey the literature here, except insofar as it directly relates to the present ivork. Liquid films floiving doivn vertical surfaces develop surface instabilities a t all possible Reynolds numbers in the absence of surface active agents (Benjamin, 1957; Yih, 1963). T h e formation of 'ivaves in the film renders the flow pattern so complex that only recently has a mass transfer model (limited to lower R e p o l d s numbers) been developed ivhich gives reasonable agreement it,ith the experimental evidence (Banerjee et ai., 1967). .A factor Fvhich further complicates the anal>& is the very small thickness of falling liquid films. Since waves of ivavelength greater than the film thickness can be expected to have a marked effect on the transport properties, it ivould appear that rvaves of all possible ivavelengths have to be taken into account in attempting to explain experimentally obserired mass transfer coefficients. T h e most commonly reported range of Reynolds numbers for which transition to turbulence occurs is betn.een 800 and 1600. [Reynolds number values used here are based on the use of four times the hydraulic diameter, giving the definition for thin films, -Yne = 4Q 'Y. Many other authors-e.g.> Brauer-have used a Reynolds number one quarter of this value.] .Mthough there is some disagreement on the exact value of the critical Reynolds number: there appears little doubt that falling liquid films can be considered turbulent ivhen the Reynolds number is greater than about 2000 (Fulford, 1964). \Vhile there has been some study of the structure of turbulence near a solid ivall (Black? 1966 ; Einstein and Li, 1956 ; Fage and Toivnend, 1932; Kline and Runstadler: 1959; Mitchell and Hanratty, 1966; Neclclerman, 1961; Popovich, 1966; Popovich and Hummel, 1967; Sternberg, 1963); very little is kno\vn about the effect of a free interface on the nature of a turbulent floiv. Furthermore, the occurrence of Lvaves on a liquid film, by alternately compressing and decompressing it, would undouhtedly affect the turbulence structure. At present, so little is kno\vn about the viscous sublayer that it is not yet possihle to predict the mass transfer rate at high Schmidt numbers from a solid ivall to a turbulent liquid without the use of empirical constants. Therefore it \ ~ o u l dseem even more difficult to arrive at a theory which \vould predict the mass transfer rate at the free interface of a turbulent wavy HILE
liquid film. However, this problem is actually somewhat simpler than that of mass transfer from a solid wall, because surface tension Lvould not damp velocity fluctuations a t a free interface to the same extent as does fluid viscosity near a solid Xvall; and in the absence of external stresses a t the surface, average shear near the surface \vould be very small. There are? at present, t\vo possible approaches to the problem of mass transfer to a turbulent liquid. T h e first is to use a n empirical or semiempirical expression for the eddy diffusivity and then to integrate a n equation like Spalding's (1961) exact differential equation to arrive a t a mean concentration field. T h e other approach is to construct a physical model for the mass transfer process based on a n idealization of the turbulence structure near the interface. T h e first method is useful only if considerable experimental data on mass transfer rates exist, from \vhich the eddy diffusivity can be reliably estimated. If the value of the Schmidt number is high, a meaningful expression for the eddy diffusivity cannot be derived from a knolvledge of the mean velocity field, even if certain simplifying assumptions can be made about the turbulent Schmidt or Prandtl number. This limitation exists because the mass transfer rate at high Schmidt numbers is markedly affected by very small velocity field fluctuations ivliich do not significantly affect momentum transfer near the interface. Consequently, mass transfer coefficients can be predicted by the first method only if considerable empirical knoivledge of the mass transfer rates in very similar situations is available. T h e second method attempts to relate mass transfer coefficients to certain hydrodynamic parameters which in certain cases may reduce to a single parameter called "the surface reneival rate"-e.g., Danckwerts (1 951). Hojvever, the values of such hydrodynamic parameters have not yet been determined rvithout recourse to observations of mass transfer rates. Certainly if a theoretical basis for determining surface reneival rates in terms of easily measured hydrodynamic quantities could he found, the second method ~vouldbe preferable to the first. Furthermore, the method utilizing a knowledge of the eddy diffusivity has to be severely modified if the mean velocity gradient vanishes a t the interface, as in the case of turbulent falling liquid films. This difficulty, coupled lvith the physically unrealistic analogy upon which the eddy diffusivity concept is based, severely limits the fundamental importance of the first method in the case considered here. [Phillips (1967) has shown from a completely different viewpoint that there may be reasons to believe that a n eddy viscosity coefficient can be used in regions of lois shear.] T h e object of this paper is to put forward a hypothesis which may be used to estimate the quantities required for the
development of a physical model. T h e results obtained from the application of this hypothesis are then applied to the particular case of a turbulent wavy liquid film to obtain estimates of mass transfer coefficients.
The Physical Model
The time scale of eddies in a turbulent fluid is proportional where f is the wave number and E(&)is the to 1/[k3E(&)]1/Z, usual three-dimensional spectrum-see, for example, To\\ nsend (1956). The viscous clissipation is proportional to k2E(k) and reaches a maximum for large f , corresponding to a scale of eddy sizes where most of the viscous dissipation occurs. The quantity f 3 E ( i )comes to a ver) sharp maximum a t slightly higher values of than the previous group, corresponding to the viscous cutoff for the spectrum, or, in other words, this maximum occurs for the smallest eddy sizes. I t is thus evident that the smallest eddiw in a turbulent fluid have associated with them time scales ibhich are markedly smaller than those of larger eddies If a general curvilinear coordinate system (XI, x ~ x, 3 ) is chosen with x 2 directed into the liquid film, the mean velocity field can be Icritten V ( x g ) and a time characteristic of the strain rate is (V,s)-I (Corrsin, 1958; Lumley and Panofsky, 1964). Thus the “criiical” wave number. k,,, a t which the eddy structure is expectEd to be isotropic is given by
Figure 1. Idealized diagram of small eddy structure near interface
to determine the concentration distribution of a solute in such a velocity field. T h e details of Lamont’s Lvork are not given here, but a n idealized situation is considered, which allows the derivation of a result very nearly the same as that obtained by him. Consider the flow situation shoivn in Figure 1. The streamlines illustrated are physically unrealistic, but convey a n idea of the eddy structure. Provided the thickness of the diffusion layer is much less than the length scale, 1, the concentration field is described by the equation,
with the boundary conditions
i? = O a t x z = 0 E = 1 atxz+
i?=latxl=O Near the interface of a turbulent falling liquid film V,Bwill be very small and isotropy can be expected even a t fairly small wave numbers. However. surface tension will probably give rise to anisotropy a t small \Lave numbers and the value of gCT will be considerably iicreased. Even if this is the case and therefore leads to a conservatively high estimate for k,,, it is very likely that a t the highest ~ a v numbersisotropy e can be assumed. T h e foregoing arguments refer primarily to the eddy structure “near” the interface and therefore it is of interest to give a more precis: definition of “near.” This is done below. Obviously, right at the interface, turbulence will be completely damped. Consider the eddy structure near an interface. T h e eddies within which the flo\v is dominated by viscous effects-i.e., the smallest eddies-have a characteristic time which is much shorter than that of larger eddies. Consequently, the small eddies swept near the interface by the larger eddies reside near the interface for a time which is much longer than the small eddy time scale. I t is therefore a reasonable hypothesis to consider the mass transfer rate as being governed by diffusion into the smallest eddies. This will be true if it is assumed that the Schmidt number is I,O large that the distance that the solute diffuses in the time characteristic of the smallest eddies is small compared to the length scales of these eddies. If 7 is the time scale of the smallest eddy, then the distance the solute diffuses in time T is approximatvly 4 I t is shown below that the
c- = - c - c 0 cm
T h e local mass transfer coefficient can be derived from the solutions to the above equation as 1
and the spatially averaged mass transfer coefficient becomes
Lamont’s calculations give for the average mass transfer coefficient
T h e result obtained is very similar to that from the surface renewal model of Danckwerts (1351), providing that the surface reneival rate is given by
The length scales of the smallest eddies are about 1 = earlier assumption is therefore justified if l/qd= is largethat is, if the Schmidt number is of the order of 1000 or more. Diffusion into Eddies with Wholly Viscous Flow. This problem was considered independently by Lamont (1966), who solved the Savier-Stokes equations for small Reynolds numbers using appropi-iate boundary conditions a t the free interface and considering a spatially periodic solution for the velocity field. H e then used the resulting velocity distributions
I t is of interest to estimate a t what distance from the surface, Equation 1 can be expected to hold-Le., what is meant by “near” the interface-and to Xvhat distance from the interface surface tension would be expected to influence the eddy structure. Harriott (1962) sholved that the expression for the mass transfer coefficient predicted by the surface renewal theory is accurate when the group s/dFr is less than 0.5, lvhere R is VOL. 7
the average distance to within which eddies approach the interface. Therefore, when discussing the eddy structure near the interface, "near" is a distance R from the interface such that 8 , ' d z < 0.5. An approximate value of R for a falling 10-5 sq. cm. water film can be easily obtained. Consider D per second, 7 low2second, then R l o p 4to cm. I t is assumed that right a t the interface there is no turbulent motion in the usual three-dimensional sense. \Vith increasing distance from the interface, motions of increasing length and velocity scales appear. Thus, it follows that the smallest eddies will appear closest to the interface. I t is necessary to estimate the distance, d, from the interface, over Ivhich these eddies may be expected to occur. A conservatively high value ofdcan be obtained by considering that, as an eddy approaches the interface, its kinetic energy is transformed into the configurational energy of the surface (the value of d estimated in this \ \ ay would be too high because viscous dissipation has been neglected). The kinetic energy contained in an eddy of length scale I and velocity scale u is approximately p13u2. An eddy of length scale I can distort the interface such that the capillary pressure can a t most be about a i l acting in the immediate vicinity of the interface over an area of the order of 12. Therefore, ( g / l ) I2d N pI3v2 or d N p12v2/a. For water, u 75 dynes per cm., u 1 cm. per second, and I to 10-3cm., so t h a t d - 10-6cm. Since R > d, it is possible to apply Equation 1 to the falling rvater films being considered here. I t is probable that Equation l can also be applied to other liquids, although it may be necessary to check the relative magnitudes of R and d in each case. The length and velocity scales of the eddies lvhich provide the main contribution to the viscous dissipation of energy can be written as
u = (ue)l/4
(Batchelor, 1947, 1953). I t is not necessary for the Reynolds number of the largest eddies to be very high for Equation 3 to hold; in fact, it is only necessary that the Reynolds number be very much greater than unity. These scales are obtained from the first similarity hypothesis of Kolmogorov, which implies that the probability distributions defining the turbulence structure for these eddies are uniquely determined by e, the viscous dissipation of energy per unit mass, and u, the kinematic viscosity. Thus Equation 1 can be written as
(4) e occurs to the one-quarter power in the equation for k L and reasonable estimations of g L can be made from a knowledge of the order of magnitude of e . Equation 4 is one of the main results of the present paper, since it is generally applicable to problems involving transport processes into a turbulent liquid with a free interface. LYhen there are external stresses on the liquid surface, it is more difficult to accept Equation 4, since it depends on the local isotropy of the eddies for which viscous dissipation is appreciable. However, it may serve as a first approximation which may sometimes be adequate. The viscous dissipation of energy may be calculated from the exact relation (Batchelor, 1953)
An example of the use of this method is given below, where the theory developed is applied to turbulent falling liquid films. Calculation of Mass Transfer Coefficient A theory due to Phillips (1961) is used to calculate the viscous dissipation of energy. Since it is not immediately obvious that the theory is applicable to turbulent wavy liquid films, a brief outline of the method is given. Consider the time-averaged vorticity equation in Cartesian coordinates where the velocity derivatives given form the rate of strain tensor.
The production of vorticity by the mean velocity is zero near the surface, and if the diffusion of vorticity from the bottom boundary is neglected, then in Equation 5 the production of vorticity by the oscillatory rate of strain ( e i j ) , and the mean rate of strain ( E i j ) , associated with the wave motion, as well as the production by the random rate of strain ( e i j ) T associated with the turbulence, is balanced by the dissipation of vorticity through viscosity. Consider the average wave profile to be 9 a cos k ( x - i t ) , \vhere a is the average wave amplitude and cis the average wave velocity; then ( e i j ) , , which is proportional to the spatial velocity derivatives, is of order ak2c. If there is no slip at the bottom boundary ( E i j ) uis -a2k3c (from Longuet-Higgins, 1951) very near the surface-Le., for ekd + 1.0. There is no restriction to deep or shallow water for the estimation of ( J ! ? , ~ ) ~ . The generation of vorticity due to vortex stretching by the existing turbulent structure-i.e., (Eij)T-near the surface is of the same order. If the eddies responsible for _viscous dissipation are isotropic, 2 u wiV2wi ,- uu2 1-2 where - I,
their length )l/4. Since-e = -vu2, then - scale, is equal to ( u 3 / e -ulP2 N (w2)1'2 and it follows that 2 u wtV2wi (u2)3/2. Equating the dissipation of vorticity to the generation by the mean rate of strain and using E = v w2, we obtain c v a4kec2. The covariance between the vorticity field and the cyclic rate of strain (eij), is very small, since the net strain over a cycle is zero and the vorticity increment is more closely correlated to the strain than to the rate of strain, and therefore ( e i j ) , can be neglected. T h e mass transfer coefficient is then obtained as
A proportionality constant, which according to the previous theory must be nearly equal to unity, may be included on the right-hand side of Equation 6 but in view of the scattered nature of the existing data, there is little to be gained from its inclusion a t this stage. I n this derivation no account has been taken of external stresses on the liquid surface. I n the case where there is a wind stress (or some other stress) on the surface, the production of vorticity by the mean velocity profile near the surface has to be considered. Equation 6 is strictly valid only \Then there is a constant driving force. However, even if the driving force changes because of transfer of solute into the liquid film, Equation 6 is approximately true if a reasonable average driving force is used. Comparison with Experiment To obtain values for the liquid phase mass transfer coefficient from Equation 6, it is necessary to use information about average values of wave amplitude, wave number, and wave velocity.
All waves must be taken into consideration in computing the average values of the amplitude, length, and wave speed. Few data exist which take into account all the waves, which makes it difficult to obtain reliable values. Fulford (1962) measured wavelengths by photographing vertical falling water films and then averaging the wavelength over several vertical lines. His values may be high, because some of the smallest waves were not easily discernible although the trend of his data indicates that wavelengths appear to approach a value of 1 cm. for large Reynolds number. I t is known from linearized stability analysis that the wavelength of the most highly amplified wave is about 1 cm. (Anshus and Goren, 1966). This value appears to vary only very slowly with Reynolds number. Tailby and Portalski (1962) also found that the wavelength a t wave inception is constant a t about 1 cm. for a very large range of Reynolds numbers. Until more detailed data on wavelengths are available, a n average wavelength of 1 cm. over the entire range of Reynolds numbers is therefore used as a reasonable approximation. Information concerning wave amplitudes for water films can be obtained from Brauer’s (1956) paper. Brauer measured the frequency,f, a t which a particular film thickness, J , occurred and obtained a dimensionless plot for the quantities,
6.0 X 10-4
Few data exist for values of the wave velocity. Fulford (1962) indicates that for turbulent falling water films, the smoothed curve for the ratio of the wave velocity, c, to the average film velocity, C, is
T h e average velocity of the liquid film can be written as
where 6 , the average film thickness, has been found from Brauer’s data to be b = 0.02005 vzI3
Other correlations for the average film thickness are available but differ very little from Equation 11. T h e mean velocity is then
Q = 12.5 Y
~ N/~ ~~ 7 / 1 5
and it follows from Equation 10 that for water a t ordinary temperatures the choice of dimensicnless group containing J to be used depending on whether the film thickness was greater or less than J , ~ the ~ ~ thickness , which occurred with the maximum frequency, f m s x . Brauer also found that in the turbulent region aT, the film thickness for the deepest wave troughs, was constant a t about 0.3 mm. with Reynolds number. H e obtained a rather complex curve for 8B, the thickness of the highest wave crests. His curve indicated that there was some form of transition in the wave structure over a Reynolds number range of about 2000 to 3000. For the purpose of the present analysis the best straight line was placed through the data on a log-log plot of 6 B us. ,ITRe.T h e data are represented with a maximum spread of +20% between Reynolds numbers of 1200 and 7000 by the equation
= 0.01 AfrRe2/5cm.
T h e mean amplitude was obtained by finding the mean crest height and the mean trough height, subtracting, and dividing the result by 2. By graphical integration, the mean of the ratio of crest height to maximum crest height was found to be
and the mean ratio of trough heights found in the same way was
where = denotes agreement to about +lo%. The mean trough height is subtracted from the mean crest height and divided by 2, giving the mean amplitude as
Brauer found that y j m a x varied from 0.5 to 0.65 mm. over a large range of Reynolds numbers, with a n average of about 0.6 mm. Substituting this average value for yjmax, as well as 8~ = 0.3 mm., and, substituting Equation 7 into Equation 8 yields
c = 9.64 X lo-*
If the appropriate values for the wavelength, wave amplitude (from Equation 9), and wave speed (from Equation 12) are substituted into Equation 6 and the value Y = 0.01 sq. cm. per second is assumed for water, the mass transfer coefficient is obtained as
I n Figure 2, data for the mass transfer coefficient from experiments by various workers (Emmert and Pigford, 1954; Kamei and Oishi, 1955; Miller, 1948; Sherwood and Pigford, 1954) have been plotted, together \zith a solid line representing Equation 13. All the data shown apply to turbulent falling water films in the absence of gas flow when a sparingly soluble gas is being absorbed. Conclusions T h e mass transfer coefficients predicted by the proposed model agree well with the experimental data Lvith a maximum spread of about +40y0 and - 3070 for Reynolds numbers ranging from 1500 to 8000. This range of deviation is well tvithin the scatter of the experimental results available and of the uncertainties in the correlations used in deriving Equation 13, I t is possible, of course, that compensating errors may have been made in the assumptions for various values of the average wave properties, although as far as possible a consistent viewpoint has been adopted throughout. T h e method, given earlier, of calculating the surface renewal rate appears to be of value, especially if there is a dominant eddy size and frequency. When the turbulent flow is less easily characterized, some meaningful average of the eddy properties would have to be taken. The importance of understanding the role of the wave parameters in predicting mass transfer rates to flowing turbulent films is emphasized by the encouraging agreement between theory and experiment achieved in this model. VOL. 7
COLUMN LENGTH 3.73 f t
also required in the Reynolds number range 800 to 1600 which defines the transition region between laminar and turbulent wavy flow.
8. PIGFORD (1954)
absorption at 25O C 71 ? I 25OC [email protected]
KAME1 8 OlSHl (1955) absorDtion at 8.5OC ,, 14' C 25OC
We express gratitude to G. D. Fulford, Department of Chemical Engineering, University of Waterloo, for his help in providing data and many useful suggestions. One of us (S. B.) had many valuable discussions with E. 0. Moeck and P. E. Tremblay, Atomic Energy of Canada, Ltd., Chalk River.
,, ,, 35OC ,, 50' C
$ 1 I ,
MILLER ( 1 9 4 8 ) A absorption at 54) F ,, 54OF
2.4 f t
3.8 f t
or concentration, g. moles /cc. = interfacial concentration, g moles/cc. c, = bulk concentration, g. moles/cc. c, = dimensionless concentration d = distance from the interface a t which an eddy motion of velocity scale u and length scale I can be expected, cm. D - = molecular diffusivity, sq. cm./sec. E ( k ) = three-dimensional spectrum, cc./sec.z = frequency with which a film thickness occurs f fm,, = maximum frequency R = average distance of approach of eddies to interface, cm. = average wave number-Le., 2n/wavelength-cm.-1 k L = average mass transfer coefficient, cm./sec. k,, = local mass transfer coefficient: cm./sec. / = length scale of eddies in viscous dissipation range, cm. iVRe = Reynolds number, 4Q/v. Q = flow rate per unit wetted perimeter, cc./(sec. cm.) s = surface renewal rate? set.-' u l , u = velocity scale of eddies in viscous dissipation range, cm./sec. = average velocity in liquid film, cm./sec. = film thickness measured from wall, cm. y yfmaa = film thickness which occurs with maximum frequency, c
0 / *
6000 Eo00 10,000
Re Figure 2. -From
k / d E vs.
- - - From Pigford's (1 941) equation (SherwoDd and Pigford, 1959) for
Kamei and Oishi's system a t
= average wave amplitude, cm. = average film thickness, cm. = average wave velocity cm./sec.
A model for the eddy structure a t a free interface has been proposed. The mean shear stress a t the interface is assumed to vanish and it is shown that the mass transfer process is likely to be controlled by small eddies for which viscous dissipation is important. An expression for the mass transfer coefficient in terms of the viscous dissipation near the surface has been obtained. By making use of the root mean square vorticity near the surface and appl+g a theory proposed by Phillips, the viscous dissipation near the surface has been found as a function of wavelength, amplitude, and velocity. Using various data for water films from the literature to obtain expressions for the average values of the wave number, amplitude, and velocity, an equation has been obtained for the transfer coefficient as a function of Reynolds number. There is good agreement bet\veen theory and published experimental data for absorption of sparingly soluble gases into falling Tvavy turbulent films. The study was restricted to turbulent flow in the absence of lvind stresses on the liquid surface. Further analytical work would be required before applications to cocurrent or countercurrent gas-liquid flows could be considered. Further work is 26
= film thickness a t highest wave crests, cm. = film thickness a t deepest wave troughs, cm.
= viscous dissipation of energy per unit mass, ergs/g.
= average lvavelength, cm. = kinematic viscosity, sq. cm./sec.
sec. v 7
= surface tension, dynes/cm. = time scale of eddies in viscous dissipation range, sec.
= vorticity, set.-'
literature Cited Anshus, B. E., Goren, S.L., A.I.Ch.E. J . 12, 1004 (1966). Banerjee, S., Rhodes, E., Scott, D. S., Chem. Eng. Sci. 22, 43 (1967). Batchelor, G. K., Proc. Cambridge Phil. Soc. 43, 533 (1947). Batchelor, G. K., "The Theory of Homogeneous Turbulence,"
Cambridge University Press, Cambridge, 1953. Benjamin, T. B., J . Fluid*Mech. 2, 554 (1957). Black, T. J., Heat Transfer and Fluid Mechanics Institute, Santa Clara, paper 21, p. 366, 1966. Brauer, H., V D I Forschungs. 457 (1956). Corrsin, S., National Advisory Committee for Aeronautics, NACA R M 5 8 B l l (1958). Danckwerts, P. V., Ind. Eng. Chem. 43, 1470 (1951). Einstein, H. A., Li, H., Proc. A m . SOC.Cicil Engrs. 82, paper 945 (1956). Emmert, R. E., Pigford, R. L., Chem. Eng. Progr. 50, 87 (1954). Fage, h., Townend, H. C. H., Proc. Roy. Soc. London 135A, 636 (1932). Fulford, G. D., Adoan. Chem. Eng. 5 , 151-236 (1964). Fulford, G. D., Ph.D. thesis, Birmingham, 1962.
Harriott, P., Chem. Eng. Scz. 17, 149 (1962). Kaniei, S.,Oishi, J., M t m . Fac. En?.,Kjoto C n 1 ~17, . 277 (1955). Kline, S. J., Runst‘idler, P. LV.,J. A p p l . .tfech. 26, 166 (1959). Laniont, J. C., Ph.D. th’:sis, British’colunibia, 1966. Longuet-Higgins, M. S Phil. T1-ans. Roy. Soc. London A245, 535 (1353). Luriiley, J . L., Panofsky, €1. :\., “ T h e Structure of Atmospheric Turbulence,” p. 86, Interscience, New York, 1964. Miller, E. G., S.B. thesis, Uni\-ersity of Delaware, 1948. Mitchell, J . E., Hanratty, T. J., J . F/uidAWrch.26, 199 (1966). Neddmiian, K. M., Chem. Eng. Sci.16, 120 (1961). Phillips, 0. M., J . Fluid.2llech. 27, 131 (1967). Phillius. 0. M.. J . G ~ o b h v sRrs. . 66. 2889 (1961). Pigfoid, K. L., b11.D. ihesis, Univeikty of‘Illinois, 1941. Popo\-ich, A . T., Ph.D. thesis, Toronto, 1966. Popovich, A . T., Iluminel, R . L., Cheni. E1747. Sci.22, 21 (1967). Slierwood, T. K., Pigford, K. L., “.Absorption and Extraction.” 2nd ed., p. 267, McGraw-Hill. New York, 1954.
Sherwood, T. K., Pigford, R. L., “Xdsorption and F,xtraction,” 2nd ed., p. 266, McGraw-Hill, New York, 1959. Spalding.. D. B., International Developments in Heat Transfer, ASMl:/I. Mech. E. 11, 439 (1961). Sternberg, J., J . Fluid .Wech. 1 3 , 241 (1963). Tailby, S. R., Portalski, S . , Trans. Inst. Chem. Engrs. London 40, 114 (1962). Townsend, A . A , , “The Structure of Turbulent Shear Flow,” p. 10, Cambridge University Press, Cambridge, 1956. Yih, C. S., Phys. Fluids 6, 321 (1963). RECEIVED for review October 27, 1966 ACC~PTE Septeniber D 5, 1967 \Vork supported by the National Research Council of Canada and Atomic Energy of Canada.
EXCESS PRESSURE DROP IN LAMINAR FLOW THROUGH SUDDEN CONTRACTION Newtonian Liquia’s G l A N N l A S T A R I T A A N D G U I D O G R E C O Istztuto di Elrtitochimica, l,,‘niversity of -YapIes, .Vapler, rial)
The Couette and Hagenbach contributions to the excess pressure drop in laminar flow through sudden contractions are discussed. Values reported in the literature are theoretical, and have been obtained on the basis of assumptions which are seriously violated in the case of a sharp-edged contraction. Experimental data show that the excess pressure drop is much larger than predicted b y published analyses, over the entire range of Reynolds numbers.
HE excess pressure drop (above the distributed losses due Tto \vel1 developed flow) \vhich is encountered in the entrance to a circular tL be under conditions of laminar flow is of interest in tlie analysis of capillary viscometer data. T h e latter are particularly important in the rheological characterization of non-Seivtonian fluids, a n d thus a correct prediction of the excess pressure drop for such fluids is particularly important. Moreover, there has recently been some hint in the literature (Astarita a n d hfetzner, 1966) that anomalous values of the excess pressure drop could be encountered in the case of viscoelastic liquids: antl that these might be used as a tool for measuring elastic properties of such liquicls. As a starting point for investigating these phenomena. it has proved necessary to evaluate the excess pressure drop in the case of Ne\vtonian liquids: a problem which is not without interest in its o\vn right. Surprisingly enough, experimental data on this sutiject are very scarce, ivhile published theoretical analyses. though complex and mathematically elegant, rest on physically un1,varranted assumptions. This \vork reports the results of a n experimental investigation carried out for the simplest possible geometry-Le., a sharp-edged sudden contraction connecting two circular tubes of different diameters. T h e results s h ~that \ ~ usually accepted correlations largely underestimate the excess pressure drop. T h e present Lvork is th’e first part of a more complete investigation ; the second part will deal with non-Newtonian liquids, both purely viscous and viscoelastic.
State of the Art
T h e excess pressure drop in laminar f l o \ ~through a sudden contraction connecting two circular pipes of different diameters, Ap? can be defined as tlie difyerence among the actual pressure drop between a section far upstream and one far do\vnstream of the contraction, a n d the pressure drop \vhich \vould exist if only distributed losses relative to \vel1 developed flow in both tubes \vere present. T h e problem of predicting Ap has been studied since the middle of tlie last century (Poiseuille, 1846), yet u p to no\\- there seems to have been little improvement over the tliroretical results of Boussinesq (1 890, 1891) and the experimental data of Knilihs (1897). An extensive review of existing kno\vledge has been given recently by Holmes (1967), ivho proposes the equation: Ap
where C is the average velocity in the do\\nstreani tube, and K and K‘ are t\ro constants. E; antl /;‘/‘Re are often referred to as the Hagenbach a n d the Couette correction, respecti\,ely. Equation 1 has a form which accommodates all the published correlations, with the exception of a re\\- Lvhich predict I; and K’ to be (weak) functions of the Kcynolds number. Boussinesq’s (1890, 1891) value for the Hagerit-racli correction, I; = 2.24, is in practice as good as any obtainecl later; three of the most recent xvorks on the subject (Holmes, 1967; VOL. 7