Surface renewal model, random, Danckwerts

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 ... 

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... 

As previously mentioned, this arises from the consideration that the residence time of a surface element at the interface is very short, so that it is likely that A has never penetrated to the inner edge of the element before it is replaced. Models that limit the depth of the surface element have also been proposed and applied to purely physical mass transfer first — such as the surface rejuvenation model of Danckwerts [1955] and the film penetration model of Toor and Marchello [1958], These were later extended to mass transfer with reaction. Harriott [1962] and Bullin and Dukler [1972] extended these models by assuming that eddies arriving at random times come to within random distances from the interface. This leads to a stochastic formulation of the surface renewal. 

For the Danckwerts model, the random surface renewal analysis, presented in Section 10.5.2, shows that the fraction of the surface with an age between t and t + dt is a function of t = fit) dt and that f(t) = Kq s1 where s is the rate of production of fresh surface per unit total area. 

For the Danckwerts model, the random surface renewal analysis, presented in Section... 

Danckwerts surface renewal theory A conceptual mass transfer model used to describe the transfer of a substance from a liquid to a gas. It assumes that an element of the surface interface comprises a mosaic of elements of various ages. Each element has a random chance of being replaced by another element from the bulk of the liquid. A feamre of the model is that a simple mathematical solution is used for complex cases involving chemical reactions. It was formulated by Peter V. Danckwerts (1916-84). 

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