REUVEN DOBRY' and R. K. FINN' University of Illinois, Urbana, 111.
Mass Transfer to a Cylinder at Low Reynolds Numbers Basic information from this study
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..
h Is applied directly to design of dust filters h Confirms the analogy between heat and mass transfer S E \ . E R A L investiqators (73. 13) have studied the transfer of heat from cylindrical wires to fluids in transverse laminar flow, and still others (4. 27) have attempted to measure the drag forces acting on a cylinder in this region-i.e.. a t Reynolds numbers less than 1. No corresponding studies of mass transfer have been made. however. A knowledge of the diffusional transport of mass is of more than academic interest. For example. it may be used to predict the efficiency with which a bed of cylindrical fibers can remove submicron particles from a n air stream. ivhere diffusion is the principal mechanism of collection. Plir-borne particles of unit density and 0.02 to 0.12 micron in diameter are characterized by Schmidt numbers ranging from 1000 to 3O:OOO. Because such high values of the Schmidt number are also attained in the diffusion of molecules or ions through liquids. data concerning mass transfer in condensed systems can be applied directly to the diffusion of aerosol particles. T h e relationship betLveen the collection efficiency of a single fiber. 7 0 . and the Nusselt number for mass transfer is a simple one ( 3 ) .
Moreover, from the efficiency of a single fiber it is possible to calculate the performance of a whole mat of fibers if certain simplifving assumptions are made SherIvood and Pigford (78) have reviebved most of the literature on mass transfer to cylinders. Powell (75, 76) studied the evaporation of water into air; Lohrisch (72) observed the absorption of water vapor on caustic cylinders and ammonia on cylinders wetted with phosphoric acid; and Linton and Sherwood ( 7 7) measured the rate of solution in water of cast cvlinders of organic acids. None of the above work was done at Reynolds numbers Present address, School of Chemical and Metallurgical Engineering, Cornel1 University, Ithaca, N. Y .
1 540
IoLier than 400. probabh because of limitations in the- experimental techniques. I t is not practical. for example, to make a \vetted or cast cylinder of sufficiently small size to reach the laminar region. If the cylinder is not small, the flow rates must be so loiv that density gradients and thermal convection upset the floiv pattern. T\vo ne\v methods of study have been suggested ( 3 ) , ivhich circumvent the above limitations. These are: 1 . Measurement of the diffusionlimited rate of absorption of dye by a cylindrical fiber 2. Measurement of the diffusionlimited electrical current discharging at a cylindrical microelectrode T h e second of these methods proved to be generally superior and so was used to collect the data reported here.
Theory Langmuir ( 9 ) developed the first theory of mass transfer to a cylinder in connection with the problem of aerosol filtration, basing his derivation on Lamb's equation (8) for the velocity field around a single cylinder. Langmuir assumed that diffusion occurs only during the time that fluid passes from 0 = x '6 to 0 = 5 x '6. where 0 = 0 is taken in the upstream direction and the origin is at the center of the collector. His approximate, somewhat arbitrary. solution can be expressed as
f ; ( R ) 2 1.3 R'
C = 2(2 - InI
(3)
and
The function of R appearing in brackets in Equation 2 is rather cumbersome, but over short ranges in R it can be approximated by a simple exponential function ( 3 ) . For values of R between 0.083 and 0.745
INDUSTRIAL AND ENGINEERING CHEMISTRY
(5)
This relationship applies to a region ivhich is of interest in aerosol filtration namely a t Reynolds numbers from 0.0.3 to 0.4 and Schmidt numbers from 1000 to 28.000. Combination of Equations 2 to 5 results in
Another approach to the prediction of mass transfer has been proposed recently by Friedlander (.5) Lvho applied a boundary-layer concept. Although this method of attack is appropriate for flow at high Reynolds numbers. Friedlander adapted it for usc at loiv Reynolds numbers by incorporating the laminar stream function of Tomotika and Aoi (79). T h e latter is a more accurate description of the flow field than that given by Lamb. .4s in the Langmuir treatment. certain arbitrary assumptions Lvere made by Friedlander, and the resulting equation
is limited to flow \vhere the "boundary layer" is thin. It applies only for Reynolds numbers less than lo-" and Schmidt numbers greater than lo5. a region not encountered in aerosol work. In a separate calculation, Friedlander estimated that at a Reynolds number of 10-' and Schmidt numbers greater than 100 ,Vhu =
where
ig
0.557 (.Vpejl
(8)
BetLveen the limits of applicability of Equations 7 and 8 the Nusselt number \vas obtained by interpolation. Still another ivay to predict the rate of mass transfer is to make use of the analog)- to heat transfer. McAdams (73) has correlated the extensive heat transfer data of Davis (2) by the equation .Vsu
0.86 . V R ~ " , ~ ~ . V ~ (~9~) . ~
which is only approximate because it covers a \vide range of Reynolds numbers (from 0.1 to 200). Piret and co-
tvorkers ( 7 1 ) studied heat transmission from fine wires to Lvater and found that their data fit the following equation for Reynolds numbers from 0.08 to 10
surface is substantially zero.) T h e quantitative expression for the so-called "diffusion current" is
T O BOTTOM
RESEiiVOlR
(11)
.Vsu = 0.965 ~VR.O~**.VP~O~~ (10) S o attempt xias made by those who collected data on heat transfer to give a theoretical interpretation of the results. Nevertheless? mass transfer may be predicted by simply replacing the dimensionless groups for heat transfer Lvith corresponding ones for mass transfer. Thus the Prandtl number becomes the Schmidt number, etc. I n designing fibrous filters. use has been made only of the Langmuir theory. \.slues of single-fiber efficiency. as calculated by the theory, are combined with such bed parameters as void fraction and fiber orientation to yield a n over-all efficiency for the filter. I n reviewing the diffusion-controlled collection of aerosols. Chen ( 7 ) has found a lack of agreement between predicted and observed filtration efficiencies, but it is not clear \
