T fica v fRm8T

V 2(-AP) So (-AP)So where f is the absolute viscosity of filtrate; c, the mass of cake per unit volume of filtrate collected; a, the specific cake resistance; -AP, the pressure drop across the cake and filter; So, the filter area; and Rm, the filter resistance. In RO, c is the solute collected on the membrane (in the concentration boundary layer) per unit volume of permeate; and Rm, the resistance of the membrane. All the other parameters have similar meanings as explained earlier in Chapter 7.

The volume flux F is V/±So. Using this and solving the above equation for V/±So = F,

tSo fic aV + 2SofRm

Initially neglecting the resistance of the solute in the concentration boundary layer, fic aV in the denominator of the first factor on the right-side of the equation may be set to zero, producing

Now, considering the resistance of the solute, designate the combined effect of compressibility, membrane resistance Rm, and solute resistance as am . Analogous to cake filtration, call am as specific membrane resistance. Hence, tSo Ham am = amo(-AP )s (8.11)

where s is an index of membrane and boundary layer compressibility. When s is equal to zero, am is equal to amo, the constant of proportionality of the equation.

Calling the pressure in the feed side as Pf, the net pressure Pfn acting on the membrane in the feed side is

where nf is the osmotic pressure in the feed side. Also, calling the pressure in the permeate side as Pp, the net pressure Ppn acting on the membrane in the permeate side is

where np is the osmotic pressure in the permeate side. Thus,

-AP = Pfn - Ppn = (Pf - nf) - (Pp - Kp) = (Pf - Pp) - (n - Kp) (8.14) and the flux F is

= -7- [(Pf - Pp) - (n - np)] = a (-AP)1-s (8.15) pamo(-A P) Vamo

Table 8.1 shows osmotic pressure values of various solutes. Some generalizations may be made from this table. For example, comparing the osmotic pressures of 1,000 mg/L of NaCl and 1,000 mg/L of Na2SO4, the former has about 1.8 times that of the osmotic pressure of the latter. In solution for the same masses, NaCl yields about 1.6 times more particles than Na2SO4. From this it may be concluded that osmotic pressure is a function of the number of particles in solution. Comparing the 1,000 mg/L concentrations of Na2SO4 and MgSO4, the osmotic pressure of the former is about to 1.4 times that of the latter. In solution Na2SO4 yields about 1.3 more particles than MgSO4. The same conclusions will be drawn if other comparisons are made; therefore, osmotic pressure depends on the number of particles in solution. From this finding, osmotic pressure is, therefore, additive.

Determination of amo and s. The straight-line form of Equation (8.15) is ln(pF) = (1 - 5)ln(-AP) - lnamo (8.16)

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