Figure 4‐11. Primary cuts for the two‐column distillation unit distillation unit. In order to construct a control volume that joins the primary cuts of streams #3 and #4, we have two choices. One choice is to enclose the second column by joining the primary cuts of streams #3, #4 and #5, while the second choice is illustrated in Figure 4‐12 where we have shown Control Volumes I and II. If Control Volume II were constructed so that it joined streams #3, #4 and #5, it would not be connected to the single source of the necessary information, i.e., the mass flow rate of stream #1. In that case, the information about stream #5
would cancel in the balance equations and we would not be able to determine the mass flow rate in stream #5.
Since the data are given in terms of mass fractions and the mass flow rate of stream #1, the appropriate macroscopic balance is given by Eq. 4‐7. For steady-state conditions in the absence of chemical reactions, the three species mass balances are given by
Since convective effects will dominate at the entrances and exits of the two
Figure 4‐12. Control volumes for two‐column distillation unit control volumes, we can neglect diffusive effects and express the three mass balances in the form
Here we have represented the fluxes in terms of the mass fractions since the stream compositions are given in terms of mass fractions that are constrained by Constraints:
Before attempting to determine the flow rates in streams 2, 3, 4 and 5, we need to perform a degree of freedom analysis to be certain that the problem is well-posed.
We begin the analysis with Control Volume II, and as our first step in the degree of freedom analysis we list the process variables as