Call the concentration of the taint in the cream x, and in the steam y, both as mass fractions,
From the condition that, at equilibrium, the concentration of the taint in the steam is 10 times that in the cream:
and in particular, 10x1 = y1
Now, y1 the concentration of taint in the steam leaving the stage is also the concentration in the output steam
The incoming steam concentration = y2 = 0 as there is no taint in the entering steam.
The taint concentration in the entering cream is xa = 8 ppm.
The mass ratio of stream flows is 1 of cream to 0.75 of steam and if no steam is condensed this ratio will be preserved through the stage.
1/0.75 = 1.33 is the ratio of cream flow rate to steam flow rate = L/V.
Applying eqn. (9.6) to the one stage n = 1,
which is the concentration of the taint in the leaving cream, having been reduced from 8 ppm.
Based on the step-by-step method of calculation, it was suggested by McCabe and Thiele (1925) that the operating and equilibrium relationships could very conveniently be combined in a single graph called a McCabe-Thiele plot
The essential feature of their method is that whereas the equilibrium line is plotted directly, xn against yn, the operating relationships are plotted as xn against yn+1. Inspection of eqn. (9.5) shows that it gives yn+1 in terms of xn and the graph of this is called the operating line. In the special case of eqn. (9.6), the operating line is a straight line whose slope is L/V and whose intercept on the y-axis is (ya - xaL/V).
Considering any stage in the process, it might be for example the first stage, we have the value of y from given or overall conditions. Proceeding at constant y to the equilibrium line we can then read off the corresponding value of x, which is x1. From x1 we proceed at constant x across to the operating line at which the intercept gives the value of y2. Then the process can be repeated for y2 to x2, then to y3, and so on. Drawing horizontal and vertical lines to show this, as in the Fig. 9.4, a step pattern is traced out on the graph. Each step represents a stage in the process at which contact is provided between the streams, and the equilibrium attained. Proceeding step-by-step, it is simple to insert sufficient steps to move to a required final concentration in one of the streams, and so to be able to count the number of stages of contact needed to obtain this required separation. [Fig. 9.4 both illustrates the general process, with two stages, and also gives numerical data to solve later Example (9.5) of a two-stage steam stripping/gas absorption process].
On the graph of Fig. 9.4 are shown the operating line, plotting xn against yn+1, and the equilibrium line in which xn, is plotted against yn. Starting from one terminal condition on the operating line, the stage contact steps are drawn in until the desired other terminal concentrations are reached. Each of the numbered horoizontal lines represents one stage.This procedure is further explained in Example 9.5.
Contact-Equilibrium Processes - Part 2: APPLICATIONS > GAS ABSORPTION
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