Total solution:
VN+1 + LA = VA + LN
Solute:
Time of contact determines whether Y and X reach equilibrium, but mass balances must hold in any event. We must not forget that the solids are wet. The inert solid might have several times its own weight of solution of composition "Y" adhering to it.
We start at the bottom of the column and include just one contacting
stage:
Going in is the extracting liquid from Stage2 and the feed stream of
solids with adhering liquid.
Coming out is the raffinate from Stage1 and the enriched extracting
liquid. This is the final liquid extracting stream from the column. The
mass balance is:
LA XA + V2 Y2 = VA YA + L1 X1
We know the rates L and V, and we know XA. This leaves three unknowns in one equation (shown bold):
LA XA + V2 Y2 = VA YA + L1 X1
The key is to guess at a value for YA. If equilibrium is reached, YA = X1 or we can consider the stage efficiency to relate YA and X1. In other words, by assuming a value for YA we get Y2 as the only variable.
We now advance to the mass balance for Stage 2, but now we have the values for the streams between Stage1 and Stage2 to use to repeat the same procedure. When we get to the top of the column, there will be an error between the projected calculated value for YB and the actual value for the statement of the problem. This error determines how much correction for the guessed value for YA to iterate until the error is less than some specified criterion.
The above method is based on stating the number of contacting stages available. An alternative is to specify the desired value for XB and to keep incrementing the stages until this value is obtained. Again there must be iteration to reduce the error between the projected calculated value for YB and the actual value for the statement of the problem.
For multi-stage batch systems where fresh solvent free of solute is
used at each stage, the number of contacts can be calculated with the equation
where X/Xo is the ration of solute recovered, N is the number of stages, and alpha = L m / F (liquid volume, distribution ratio divided by solid volume).