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.
Highlights
A frequency quantum interpretation of the surface renewal model of mass transfer Chanchal Mondal† and Siddharth G
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
In contrast with the earlier model of Higbie [3] which had assumed that all surface elements had the same residence time at the gas–liquid interface, Danckwerts derived an exponential age distribution by using the postulate that all surface elements, irrespective of their individual ages, had equal probability of being replaced by fresh elements arriving from the bulk liquid
Summary
The surface (assumed to be of unit area) of a turbulent liquid is visualized as being populated by a mosaic of chaotic eddies or liquid elements. As tp → ∞, it reduces to the well-known, steady-state age distribution Se−St, which was proposed by Danckwerts [2] In this case, it is assumed that there are liquid elements, visualized as being ‘blue’, that are already present on the surface at tp = 0 when the process starts and ‘red’ elements begin displacing the blue elements by the mechanism of surface renewal. It can be observed that, as Stp → 0, i.e. as S → 0 (low surface renewal) or as tp → 0 (near the start of the process), which implies that St → 0, equation (2.11) reduces to equation (2.19), i.e. the Danckwerts distribution (Case 1) reduces to the uniform distribution This is analogous to the reduction of the Planck formula to the classical Rayleigh–Jeans formula in the limit of low frequencies in the energy spectrum radiated by a blackbody [1]. The cumulative age distribution corresponding to equation (2.23) is expressed by t tp
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