Combined signal flow diagram for figures 17.3 and 17.5
The equation for the inert gas valve is:
Similarly, the equation for the vent valve is:
where wIG = inert gas flow, lbm/sec we - vent flow, lbm/ sec Pq = column pressure, lbf/ft2 Pig = inert gas supply pressure, lbf/ft2 PR = pressure downstream of vent valve, lbf/ft2 6C = controller output signal
These equations may be combined into the partial signal flow diagram of Figure 17.7. Note that the valve gains, dwIG/ddc and dwe/dd„ are both assumed to be positive; reverse action of the inert gas valve is obtained by the - 1 term. If we can assume that PjG and PR are sufficiently constant, we may reduce Figure 17.7 to the form of Figure 17.8. The term Cc is the acoustic capacitance of the column, vapor line to condenser, and the condenser.
For the case where reboiler steam is flow or flow-ratio controlled we can now combine Figures 17.8 and 17.3 into the signal flow diagram of Figure 17.9. Note the pressure feedback on wc through '6Tcp/dPcp, and the addition of the pressure measurement, KmGm(s), and the controller, KcGc(s) ■ Figure 17.9 can be reduced to the form of Figure 17.10 where:
From equation (17.6) we can see that open-loop pressure dynamics are essentially first order. Since in most cases the inert gas bleed and the vent flow are fairly small, the valve gains, dwIG/ddc and dwc/ddc, tend to be small. Together with the first-order dynamics, this commonly leads to large controller gains (small proportional bands) and control valve saturation for fairly small disturbances.
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