Surface age distribution

If, on the other hand, it is assumed that contact times for the individual fluid elements vary at random, an exponential surface age distribution characterized by a fractional rate of renewal s may be used (34). This approach is caUed surface renewal theory and results in... 

The average transfer rate according to the Higbie model for surface age distribution then becomes ... 

In the Danekwens model of mass transfer it is assumed that the fractional rate of surface renewal s is constant and independent of surface age. Under such conditions the expression for the surface age distribution function is = If the fractional rate of surface renewal were proportional to surface age (say s — bt. where b is a constant), show that the surface age distribution function would then assume the form ... 

Surface renewal theory (Danckwerts, 1951) proposes that there is an infinite range of ages for elements of the surface and the surface age distribution function (t) can be expressed as... 

In the Danckwerts model of mass transfer it is assumed that the fractional rate of surface renewal, s is constant and independent of surface age. Under such conditions the expression for the surface age distribution function is s st. 

If the fractional rate of surface renewal were proportional to surface age (say, s = bt, where b is a constant), show that the surface age distribution function would then assume the form ... 

The classic penetration model of Higbie (1935) is based on the assumption that all the fluid elements reside at the interface for the same length of time. The surface age distribution for this model is... 

Figure 9.3. Surface age distribution according to Danckwerts the elements undergo random surface renewal at frequency s.

The Danckwerts surface age distribution (Eq. 9.1.6) leads to an average value of the mass transfer coefficient given by... 

Further retaxation of the assumptions of the film aed penetration theories was suggested by Danckwerta1 who viewed the process as one of transient one-dimensioual diffusion to packets or elements of fluid that reside at the phase interface for varying periods of time. Therefore, the model is that of the penetration theory with a distribution of conract times, The surface age distribution p(t) is defined such that 4(i)tb is the fraction of surface that has seslded at the interface for a time between I and t + di. The mass transfer flua for the entire surface is obtained by integration or the instantaneous flux over all exposure times ... 

Whereas Higbie assumed a constant renewal time, Dankwert s (1951) extension of the penetration theory employs a wide spectrum of eddy contact times. These eddy-like fluid packets are assumed to remain in contact with the interface for variable times from zero to infinity. He assumed a surface-age distribution function, which skews the contact times to small values it is... 

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