Derive a relationship between solvent concentration, c^S, and time to control solvent addition rate, in order to maintain constant supersaturation in a cooled, batch crystallizer with no seeding. Use (17-101) for solubility as a function of solvent content and apply the solute mass balance in (2) of Example 17.21.

Example 17.21.

Dilution batch crystallization is often performed at a constant rate of diluent addition, k_m, given by

In terms of km, the batch solution volume, V{t}, which includes solvent plus diluent volumes, varies according to

where the initial batch volume, V0, is free of diluent. General expressions for both the time rate of change of target species solubility, cs, in (17-89) and of batch solution volume, V{t}, in (2) at a constant rate of diluent addition are useful to predict outcomes of dilution batch crystallization. One of these outcomes is the time required to obtain a target crystal size, which is examined in Example 17.22. It is necessary to differentiate (17-89) and (2) to obtain general expressions for the time rate of change of target species solubility and of batch solution volume. A typical rate for (1) is 10-6 L of diluent per L of solvent per second. A value of 10 L of solvent plus diluent per L of diluent for kd in (17-89) is common. What are the magnitudes of the general expressions for these typical values of diluent addition rate and solubility proportionality constant, if the initial solvent volume is 1 L?

dca dt km (1)