Film‐penetration model for mass and heat transfer

Abstract

The two major models of the mechanism of mass and heat transfer between two phases are the film theory (16) and the penetration theory (2, 3). The film theory assumes that there is a region in which steady state molecular transfer is controlling; the penetration theory assumes that the interface is continuously replaced by eddies and that unsteady state molecular transfer into the eddies controls the transfer in this region.There are three classes of problems to which these theories have been applied : (1) transfer between a solid and a fluid in turbulent flow; (2) transfer between two fluids, at least one of which is in turbulent flow; and, (3) transfer between two fluids in an apparatus of discontinuous geometry such as packed column. In flow in a packed column the concept of surface renewal by eddies is replaced by that of a laminar liquid which mixes at discontinuities in the packing (2). If the flow is laminar except a t the points of mixing, neither the film theory nor the analysis developed below will apply to the liquid phase.In the absence of any other resistances the film theory predicts a first-power dependence of the transfer rate on the diffusivity or thermal conductivity, and the penetration theory predicts a square root dependence. Danckwerts (3) has shown that neither the film nor penetration theory is completely valid for a packed column, and Hanratty (6) points out that at high Schmidt numbers the penetration theory gives better results than the film theory.The purpose of this paper is to show that the film and penetration theories are not separate, unrelated concepts but rather are limiting cases of a more general model and that the two theories, rather than being mutually exclusive, are actually complementary. THEORETICAL DEVELOPMENTThe boundary layer extending from the front end of a flat plate or from the inlet to a conduit will be considered. A laminar zone is assumed to exist which is bounded by a turbulent region, and heat or mass is being transferred between the surface and the fluid. For a short distance along the plate the transfer must be by an unsteady state mechanism, for the penetration of the diffusing substance will take a finite contact time (corresponding to a finite distance along the surface) to reach the edge of the film. Some distance along the surface the transfer will reach its steady value and there will be no inore accumulation in the film. The short distance corresponds to the penetration theory, the long distance to the film theory, and for intermediate distances the transfer process has the characteristics of both mechanisms.The transfer between a gas and stirred liquid, which as postulated by Danckwerts has its surface randomly replaced by eddies of fresh fluid from the bulk of the liquid, will be considered. If the eddies remain in the surface a short period of time, each element may be assumed to absorb matter or heat at the interface by unsteady state conduction. As the life of an element increases, the penetration into the element increases and again, after a lo...

Reaeration in open-channel flow

“…The film-penetration model was developed by Dobbins (1956) and later by Toor and Marchello (1958). Certain modifications have been discussed in the literature (see, for example, Toor, 1963, andHarriott, 1962), but essentially, the film-penetration theory consists of assuming molecular diffusion into laminar surface elements of differential area and thickness, L/, which exist on top of a turbulent liquid which has uniform solute concentration, CL-The ages of the surface elements are assumed to be distributed according to Danckwerts' (1951) Marchello and Toor (1963) presented a modification of the film-penetration model in which the surface layer is assumed not to be replaced but to be instantaneously mixed at times which are distributed according to the age-distribution function of Danckwerts (1951).…”

Evaporation from porous surfaces into turbulent airflows: Coupling eddy characteristics with pore scale vapor diffusion

“…The original SR theory inspired a class of related models based on various heuristic depictions of the near-wall turbulent fluid flows. Some of the well-known variants of the SR theory include film-penetration models [Toor and Marchello, 1958;Brusset et al, 1973], periodic growth-breakdown models [Einstein and Li, 1958;Ruckenstein, 1958;Meek and Baer, 1970;Pinczewski and Sideman, 1974], random surface renewal models [Hanratty, 1956;Fortuin and Klijn, 1982;Fortuin et al, 1992], and surface rejuvenation models [Harriott, 1962;Bullin and Dukler, 1972;Thomas et al, 1975;Loughlin et al, 1985]. These eddy renewal models assume that the replacement of individual fluid elements near a surface may be represented as a stochastic process driven by a turbulent flow field away from the surface.…”

A Non-Wetting Packed Bed Gas Scrubber

“…Whereas the former assumes the same exposure time for each fluid contact with the interface, the latter employs a wide spectrum of exposure times and averages the varying degrees of penetration. Toor and Marchello (1958) described the Film-Penetration Theory in which the Film and Penetration Theories were shown to be limiting cases of this more general model.…”