Top-Tray and Overhead System Composition Dynamics

We assume further that the vapor from the top tray is totally condensed. We can then write the following transfer function relating the vapor composition yr and reflux composition Xr  

The term Lr s)/Oc ) relates reflux flow to controller output signal for the moment there are no implications as to whedier the manipulation is simply a valve, the set point to the secondary flow controller, or die ratio in a ratio-control system. 

Rippin and Lamb point out that if we provide a perfect feedforward control system to adjust Lr so that feed changes in flow and composition do not change jVr, then the transmission Xr s) is broken and there is no feedback of Xr(s) down the column. It is possible to accomplish the same thing by making l/xp sufficiently smaller than the resonant frequency of the closed-loop composition-control system. This has been shown experimentally by Aikmam to be true of a plant column. The mathematical explanation is simple. 

This is the mixing time constant of the top tray. Then  

Signal flow diagram for top tray and overhead system 

---