The open-loop transfer function of a fluid-flow system can be written as

where Ñ‚ = RC, R is a constant equivalent to the resistance offered by the orifice so that 1/R = ½kH0-1/2 and C = the cross-sectional area of the tank. Since ΔH = R ΔQ2, we have the following for the transfer function relating the head to the input change:

For a closed-loop feedback system, a float-level sensor and valve may be used as shown in Figure P4.1. Assuming the float is a negligible mass, the valve is controlled so that a reduction in the flow rate, ΔQ1, is proportional to an increase in head, ΔH, or ΔQ1 = -KΔH. Draw a closed-loop flow graph or block diagram. Determine and compare the open-loop and closed-loop systems for
(a) Sensitivity to changes in the equivalent coefficient R and the feedback coefficient K,
(b) The ability to reduce the effects of a disturbance in the level ΔH(s), and
(c) The steady-state error of the level (head) for a step change of the input ΔQ1(s).

(s) AQi(s) RCs 1 GI(s) FIGURE P4.1 Tank level control.