Comparative calculations with a mathematical model designed by the authors, which takes into consideration energy transfer from gas flowing through a given channel to gas which penetrates this channel from an adjacent channel, as well as a model which omits this phenomenon, respectively, were made for the process of separating gas mixtures carried out with an inert sweep gas in the fourend capillary membrane module. Calculations were made for the process of biogas separation using a PMSP polymer membrane, relative to helium as the sweep gas. It was demonstrated that omitting the energy transfer in the mathematical model might lead to obtaining results which indicate that the capacity of the process expressed by the value of feed flux subjected to separation is by several percent higher than in reality.
The paper presents the mathematical modelling of selected isothermal separation processes of gaseous mixtures, taking place in plants using membranes, in particular nonporous polymer membranes. The modelling concerns membrane modules consisting of two channels - the feeding and the permeate channels. Different shapes of the channels cross-section were taken into account. Consideration was given to co-current and counter-current flows, for feeding and permeate streams, respectively, flowing together with the inert gas receiving permeate. In the proposed mathematical model it was considered that pressure of gas changes along the length of flow channels was the result of both - the drop of pressure connected with flow resistance, and energy transfer by molecules of gas flowing in a given channel to molecules which penetrate this channel from the adjacent channel. The literature on membrane technology takes into account only the drop of pressure connected with flow resistance. Consideration given to energy transfer by molecules of gas flowing in a given channel to molecules which penetrate this channel from the adjacent channel constitute the essential novelty in the current study. The paper also presents results of calculations obtained by means of a computer program which used equations of the derived model. Physicochemical data concerning separation of the CO2/CH4 mixture with He as the sweep gas and data concerning properties of the membrane made of PDMS were assumed for calculations.
In this study the authors minimise the total process cost for the heating of solid particles in a horizontal fluidised bed by an optimal choice of the inlet heating gas temperature profile and the total gas flow. Solid particles flowed along the apparatus and were heated by a hot gas entering from the bottom of the fluidised apparatus. The hydrodynamics of the fluidised bed is described by a two-phase Kunii - Levenspiel model. We assumed that the gas was flowing only vertically, whereas solid particles were flowing horizontally and because of dispersion they could be additionally mixed up in the same direction. The mixing rate was described by the axial dispersion coefficient. As any economic values of variables describing analysing process are subject to local and time fluctuations, the accepted objective function describes the total cost of the process expressed in exergy units. The continuous optimisation algorithm of the Maximum Principle was used for calculations. A mathematical model of the process, including boundary conditions in a form convenient for optimisation, was derived and presented. The optimization results are presented as an optimal profile of inlet gas temperature. The influence of heat transfer kinetics and dispersion coefficients on optimal runs of the heating process is discussed. Results of this discussion constitute a novelty in comparison to information presented in current literature.
The mathematical model of postproduction suspension concentration by microfiltration has been developed. This model describes a process conducted in a batch system with membrane washing by reverse flow of permeate. The model considerations concern filtration pseudocycles consisting of the filtration period and the membrane washing period. The balances of continuous phase volume, dispersed phase mass and energy, for each period of pseudocycle respectively, have been presented.