Systematic mathematical model was featured for designing the installations and operational conditions in multi-stage feed and breed reverse osmosis (RO) in order to economize the cost for concentration of cheese whey under limiting stable volume flux (Jvlim)without fouling. The total RO cost for removing one m3 of permeate was divided into twoterms; a changeable cost (C
θ) which varies depending upon flux level, and the othercost. Formulae to calculate the costs for RO operation were derived. Relationships among factors in RO operation were studied by introducing equations for membrane transport, concentration polarization and Jvlim. The systematic mathematical model, which was also introduced in the present study, was applied to calculate C
θ from the values of concentration factor and the recirculation flow rate, and consequently this enabled to conceive the optimumplant design and operation mode which minimize the cost for RO by monitoring concentration factors and recirculation flow rates on every stage in RO operation. The minimum cost for RO was estimated regarding four typical experimental cases, in which the values of Jvlim had been determined, on concentation of cheese whey of 100m
3 by three fold in 20h by means of the three-stage feed and breed RO plant equipped with plate-and-frame Arlamo-26 modules and a FL-190 membrane of 0.90 in salt rejection rate. The RO cost was minimized when the whey of pH5.8 was concentrated at 40°C after standing for one hour at 50°C. The total cost for RO, to remove 1m3 of the permeate, was estimated to be 1140yen which was allotted to itemized costs of 4% for energy for high-pressure feed pump, 7% for energy for recirculation pump, 11% for membrane replacement, 13% for solids loss, 1% for waste water treatment, 2% for cleaning, 8% for the pretreatment of whey to adjust pH with CO
2, 16% for labour and 38% for depreciation and interest for investment. Changeable costs for the every stage were capable to be expressed as a function of recirculation flow rate, and then the values for recirculation flow rate and concentration factor, which were to minimize the total RO cost, were estimated by simplified computation.
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