A frequency quantum interpretation of the surface renewal model of mass transfer

Abstract

The surface of a turbulent liquid is visualized as consisting of a large number of chaotic eddies or liquid elements. Assuming that surface elements of a particular age have renewal frequencies that are integral multiples of a fundamental frequency quantum, and further assuming that the renewal frequency distribution is of the Boltzmann type, performing a population balance for these elements leads to the Danckwerts surface age distribution. The basic quantum is what has been traditionally called the rate of surface renewal. The Higbie surface age distribution follows if the renewal frequency distribution of such elements is assumed to be continuous. Four age distributions, which reflect different start-up conditions of the absorption process, are then used to analyse transient physical gas absorption into a large volume of liquid, assuming negligible gas-side mass-transfer resistance. The first two are different versions of the Danckwerts model, the third one is based on the uniform and Higbie distributions, while the fourth one is a mixed distribution. For the four cases, theoretical expressions are derived for the rates of gas absorption and dissolved-gas transfer to the bulk liquid. Under transient conditions, these two rates are not equal and have an inverse relationship. However, with the progress of absorption towards steady state, they approach one another. Assuming steady-state conditions, the conventional one-parameter Danckwerts age distribution is generalized to a two-parameter age distribution. Like the two-parameter logarithmic normal distribution, this distribution can also capture the bell-shaped nature of the distribution of the ages of surface elements observed experimentally in air–sea gas and heat exchange. Estimates of the liquid-side mass-transfer coefficient made using these two distributions for the absorption of hydrogen and oxygen in water are very close to one another and are comparable to experimental values reported in the literature.

A surface renewal model for unsteady-state mass transfer using the generalized Danckwerts age distribution function

“…Using equations (A.8) and (A.15) in the appendix (i.e. electronic supplementary material, S1) of the manuscript of Mondal & Chatterjee [ 29 ] and generalizing them to unsteady-state conditions yields the fundamental differential equation that describes the unsteady-state GD age distribution function f ( t , t p ): where t is the age of a surface element and t p is the time elapsed since the commencement of the mass or heat transfer process (i.e. the process or ‘clock’ time).…”

“…Recently, Mondal & Chatterjee [ 29 ] have given a deeper theoretical understanding of the Danckwerts surface renewal model by invoking the concept of a frequency quantum, which is analogous to the energy quantum postulate which Planck used to derive his famous formula for the energy density of black-body radiation. By assuming that surface elements of a particular age have renewal frequencies that are integral multiples of this fundamental frequency quantum, and by further assuming a Boltzmann-type distribution for the renewal frequency, they showed that a population balance for these elements led to the Danckwerts surface age distribution.…”

“…For all four cases they derived explicit mathematical expressions for the rates of physical gas absorption at the gas–liquid interface and dissolved-gas transfer to the bulk liquid from the surface. Mondal & Chatterjee [ 29 ] showed that, under unsteady-state conditions, these two rates were not equal but had an inverse relationship, and only with the progress of absorption towards steady state did they approach one another.…”

“…Like the two-parameter LN distribution, the GD distribution was also able to capture the bell-shaped nature of experimentally measured surface age distribution, which, as mentioned earlier, has been observed in air–sea gas and heat exchange. Mondal & Chatterjee [ 29 ] used the LN and GD distributions to calculate the steady-state liquid-side mass-transfer coefficient for the absorption of hydrogen and oxygen in water at three different wind speeds as used by Garbe et al [ 30 ] in their experiments on air–water heat exchange on the Heidelberg Aeolotron. They found that, for both distributions, estimates of the mass-transfer coefficient were very close to one another and comparable to the experimental values reported by Hutchinson & Sherwood [ 33 ] who investigated the absorption of eight different pure gases at 25°C in a stirred flask containing water whose surface was exposed to the gas.…”

“…The chief objective of the present work is to extend the two-parameter GD age distribution beyond the steady-state form derived by Mondal & Chatterjee [ 29 ], to unsteady-state conditions. This distribution is then used to analyse the unsteady-state absorption of a gas into a large volume of liquid, assuming negligible gas-side mass-transfer resistance.…”