By Radoslav Paulen, Miroslav Fikar
This research concentrates on a common optimization of a selected category of membrane separation approaches: these regarding batch diafiltration. present practices are defined and operational advancements in keeping with optimum keep an eye on concept are urged. the 1st a part of the publication introduces the idea of membrane techniques, optimum regulate and dynamic optimization. Separation difficulties are outlined and mathematical types of batch membrane procedures derived. The regulate concept makes a speciality of difficulties of dynamic optimization from a chemical-engineering standpoint. Analytical and numerical equipment that may be exploited to regard difficulties of optimum keep an eye on for membrane procedures are defined. the second one a part of the textual content builds in this theoretical foundation to set up options for membrane versions of accelerating complexity. each one bankruptcy starts off with a derivation of optimum operation and keeps with case stories exemplifying a number of facets of the keep an eye on difficulties into consideration. The authors paintings their manner from the restricting flux version via more and more generalized types to suggest an easy numerical method of the final case of optimum operation for batch diafiltration tactics. Researchers drawn to the modelling of batch procedures or within the capability business functions of optimum keep an eye on idea will locate this monograph a worthwhile resource of thought, guideline and ideas.
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Extra info for Optimal Operation of Batch Membrane Processes
In comparison with continuous processes, batch operations allow to use membranes with reduced area in order to reach the target product quality, that usually leads to smaller space requirement and lower investment costs . Moreover, batch processing is particularly suited for applications where the process liquor is manufactured in batches or lots, before any subsequent separation is undertaken. A schematic diagram of a discontinuous membrane diafiltration process is shown in Fig. 16. Considering a process liquor with two solutes, the general purpose of such batch plant can be summarised as to increase the macro-solute concentration from c1,0 to c1,f , and to reduce the micro-solute concentration from c2,0 to c2,f .
6c) 32 2 Optimal Control Problem • Point inequality constraint Gc = g (x, u, p, tc ) , Fc = 0, Jc ≤ 0. 6d) The presented canonical forms of constraint functions can be adjoined to the cost functional J using a vector of Lagrange multipliers ν ∈ Rn c to form an augmented functional J¯ J¯ = J + nc νc Jc . 8) c=1 Equivalently, this functional can be written as tf J¯ = G¯ + t0 with G¯ = G + nc νc Gc , and F¯ = F + c=1 nc νc Fc . 3 Process Model In principle, the process model represents an additional set of equality constraints since in general it consists of a set of algebraic, differential, and/or functional equations which, if satisfied, give input–output or inner mathematical description of phenomena taking place in the observed system.
A more efficient procedures exist for cleaning of membranes. These include physical, chemical, and biological operations. 2 Mathematical Modelling of Membrane Processes Prediction of the membrane flux is a key element to modelling of membrane processes which further enables effective design of operation for studied membrane process or system. 1) where q is volumetric (solvent) flux, A represents (effective) membrane area, c stands for the molar concentration of diffusing mass, D denotes a so-called diffusivity coefficient, and x represents a position in space.