Penetration theory random surface renewal

DatKkwerts theory cf penetration with random surface renewal modifies diis picture by proposing an infinite range of ages for elements of the surface. The probability of an element of surface being rqrlaced by a fresh eddy is considered to be independem of the age of that element. Danckwerts introduced this modification by defining a surfiKe age distribution function, (r), such that the fraction of surface with ages between r and r -l- is 6(r)dir. If die probability of replacement of a surface element is independem of its age, Danckwetts showed that k = (sD), where s is the fractional rate of surface renewal. 

Penetration theory can also be applied to turbulent conditions by assuming the turbulence spectrum to consist of large eddies, capable of surface renewal, and small eddies responsible for the presence of eddy diffusivity The small eddies are damped when an element of liquid reaches the interface so that, during its residence time there, mass transfer occurs in accordance with the assumptions of the penetration theory If all the eddies stay at the interface for the same interval of time we talk about penetration theory with regular surface renewal or the Higbie model If there is random distribution of residence times with an age-independent fractional rate of surface renewal, s, the term penetration theory with random surface renewal, or the Danckwerts nK)del, is employed In the case of the Higbie model, the mass transfer coefficient is the same as that given by eqn (18). For the Danckwerts model it takes the form... 

In surface renewal models the liquid surface is assumed to consist of a mosaic of elements with different age at the surface. The rate of absorption at the surface is then an average of the rates of absorption in each element, weighted with respect to a distribution function (t)—see Eq. 6.2-5. Under this heading of surface renewal theory we will also occasionally mention results of Higbie s [23] so-called penetration-theory, which can be considered as a special case in which every element is exposed to the gas for the same length of time before being replaced. The main emphasis of this section is on the Danckwerts [24] approach using the distribution function for completely random replacement of surface elements ... 

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