Question 7.4.9: Liquid-Level System with an Orifice Consider the liquid-leve...
Liquid-Level System with an Orifice
Consider the liquid-level system with an orifice, treated in Example 7.3.2. The model is
A \frac{d h}{d t} = q_{vi} − q_{v0} = q_{vi} − C_{d} A_{o} \sqrt{2gh}Consider the case where A = 2 ft² and C_{d} A_{o} \sqrt{2g} = 6. Estimate the system’s time constant for two cases: (i) the inflow rate is held constant at q_{vi} = 12 ft^{3}/sec and (ii) the inflow rate is heldconstant at q_{vi} = 24 ft^{3}/sec.
Substituting the given values, we obtain
2 \frac{d h}{d t} = q_{vi} − q_{v0} = q_{vi} − 6 \sqrt{h} (1)
When the inflow rate is held constant at the value q_{ve}, the liquid height reaches an equilibrium value h_e that can be found from the preceding equation by setting dh/dt equal to zero. This gives 36 h_{e} = q^{2}_{ve}.
The two cases of interest to us are (i) h_{e} = (12)^{2}/36 = 4 ft and (ii) h_{e} = (24)^{2}/36 = 16 ft. Figure 7.4.11 is a plot of the flow rate 6\sqrt{h} through the orifice as a function of the height h.
The two points corresponding to h_{e} = 4 and h_{e} = 16 are indicated on the plot.
In Figure 7.4.11 two straight lines are shown, each passing through one of the points of interest (h_{e} = 4 and h_{e} = 16), and having a slope equal to the slope of the curve at that point.
The general equation for these lines is
q_{vo} = 6 \sqrt{h} = 6 \sqrt{h_{e}} + \left(\frac{d q_{vo}}{dh} \right)_{e} (h − h_{e}) = 6 \sqrt{h_{e}} + 3 h^{−1/2}_{e} (h − h_{e})
Substitute this into equation (1) to obtain
2 \frac{d h}{d t} = q_{vi} − 6 \sqrt{h_{e}} − 3h^{−1/2}_{e} (h − h_{e}) = q_{vi} − 3 \sqrt{h_{e}} − 3h^{−1/2}_{e} h
The time constant of this linearized model is 2 \sqrt{h_{e}}/3, and is 4/3 sec for h_{e} = 4 and is 8/3 sec for h_{e} = 16. Thus, if the input rate q_{vi} is changed slightly from its equilibrium value of q_{vi} = 12, the liquid height will take about 4(4/3) or 16/3 sec to reach its new height.
If the input rate q_{vi} is changed slightly from its value of q_{vi} = 24 , the liquid height will take about 4(8/3) or 32/3 seconds to reach its new height.
Note that the model’s time constant depends on the particular equilibrium solution chosen for the linearization. Because the straight line is an approximation to the 6 \sqrt{h} curve, we cannot use the linearized models to make predictions about the system’s behavior far from the equilibrium point. However, despite this limitation, a linearized model is useful for designing a flow control system to keep the height near some desired value. If the control system works properly, the height will stay near the equilibrium value, and the linearized model will be accurate.
