Diafiltration ultrafiltration cycle

Due to the complication of diafiltration, determination of the optimum ultrafiltration cycle normally requires time consuming experimental work or tedious calculations. 

A comprehensive mathematical analysis of batch ultrafiltration coupled with diafiltration is presented. The time cycle of the ultrafiltration-diafiltration has been correlated with the volume initially charged, percent of solute recovered, membrane area and flux. The optimum diafiltration volumes which result in the minimum cycle time or the minimum membrane area were solved for in terms of the operating conditions. 

In this paper, complete mathematical formulations for correlating the time cycles with other operating conditions are presented. The optimum diafiltration cycle (in terms of volume fraction), and the total cycle time are solved as functions of membrane area, flux, initial volume and recovery. Convenient charts, which can be used as a guide in designing or modifying an ultrafiltration process, are provided. 

From inspection of Equation (5), it can be seen that the total time cycle is the sum of ultrafiltration and the dia-filtration cycles with the diafiltration cycle given by the second term on the right-hand side of the equation. 

Equation (14) is an implicit algebraic equation of the optimum relative diafiltration volume, Ud. It can be solved numerically by any one of a number of methods, e. g. Newton, Raphson Technique, (Lapidus, 1962). Once the value of Ud is determined, the optimum time cycles of the ultrafiltration and diafiltration stages, Tu and Td, can be calculated readily from Eqs. (12) and (5). 

The optimum time cycle and the relative diafiltration volume in the ultrafiltration-diafiltration process can be expressed as a function of three variables, P, Q, and R. P and Q are simple functions of the initial volume, membrane area, and flux (P = mA/Vo, Q = bA/Vo), and R is the solute recovery. From these, the time cycle and relative diafiltration volume (Vd/Vo) can be solved at various values of m, b, Vo, A, and R (m and b are respectively the slope and intercept of the flux, J = m In Vo/V + b). At a fixed recovery, the optimum time cycle and the relative diafiltration volume become functions of only two variables P and Q. Thus, the optimum operating condition can be simply plotted as function of P and Q. These plots, providing convenient and sufficient information, can be used as a guide in the design and operation of the ultrafiltration process. 

Td = Time cycle of diafiltration phase Tu = Time cycle of ultrafiltration phase U = Volume fraction remained (V/Vo)... 

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