Reduced osmotic modulus

The expression in brackets represents the interparticle interactions. It gives the contribution by which the true molar mass is modified to yield the measurable apparent molar mass M pp(c), which is a function of the concentration. A dimensionless quantity can be obtained by multiplying Eq. (78b) with which will be called the reduced osmotic modulus... 

Fig.32a,b. Plot of the reduced osmotic modulus MJM (c) as a function of reduced concentration X = = cIc A2 I. The solid lines represent theoretical curves for the... 

Fig. 35. Plot of the reduced osmotic modulus from anhydride cured linear and cross-linked phenylglycidylethers [165,173-175]...

In a diagram where the reduced osmotic modulus is plotted as a function of the parameter X, a universal curve is found for linear flexible chain molecules independent of the molar mass. Macromolecules of different architecture, for instance star-like, random branched or stiff macromolecules, deviate from this curve in a characteristic way. Figure 5.6 displays this for PMAC dissolved in benzene [15]. 

Figure 5.6 Plot of the reduced osmotic modulus (M /RT) (dn/dc) j versus the parameter X = A M c for two fractions of PMAC in benzene at 20°C. The theoretical curves for hard spheres and flexible chains are drawn as full lines [20].

The thermodynamic approach does not make explicit the effects of concentration at the membrane. A good deal of the analysis of concentration polarisation given for ultrafiltration also applies to reverse osmosis. The control of the boundary layer is just as important. The main effects of concentration polarisation in this case are, however, a reduced value of solvent permeation rate as a result of an increased osmotic pressure at the membrane surface given in equation 8.37, and a decrease in solute rejection given in equation 8.38. In many applications it is usual to pretreat feeds in order to remove colloidal material before reverse osmosis. The components which must then be retained by reverse osmosis have higher diffusion coefficients than those encountered in ultrafiltration. Hence, the polarisation modulus given in equation 8.14 is lower, and the concentration of solutes at the membrane seldom results in the formation of a gel. For the case of turbulent flow the Dittus-Boelter correlation may be used, as was the case for ultrafiltration giving a polarisation modulus of ... 

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