A Single-Tank Liquid-Level System with a Pump Consider the single-tank liquid-level system shown in Figure 7.11, where a pump is connected to the bottom of the tank through a valve of linear resistance R. The inlet to the pump is open to the atmosphere, and the pressure of the fluid increases by Δp

Question

A Single-Tank Liquid-Level System with a Pump

Consider the single-tank liquid-level system shown in Figure 7.11, where a pump is connected to the bottom of the tank through a valve of linear resistance [latex]R[/latex]. The inlet to the pump is open to the atmosphere, and the pressure of the fluid increases by [latex]\Delta p[/latex] when crossing the pump. Derive the differential equation relating the liquid height [latex]h[/latex] and the volume flow rate [latex]q_{o}[/latex] at the outlet. The tank's cross-sectional area [latex]A[/latex] is constant. The density [latex]\rho[/latex] of the liquid is constant.

Accepted Answer

We begin by applying the law of conservation of mass to the tank,

[latex]\frac{\mathrm{d} m}{\mathrm{~d} t}=q_{\mathrm{mi}}-q_{\mathrm{mo}}.[/latex]

The fluid mass inside the tank is [latex]\rho A h[/latex]. For constant fluid density and constant cross-sectional area,

[latex]\frac{\mathrm{d} m}{\mathrm{~d} t}=\rho A \frac{\mathrm{d} h}{\mathrm{~d} t}.[/latex]

The mass flow rate into the tank is

[latex]q_{\mathrm{mi}}=\frac{p_{1}-p_{2}}{R}.[/latex]

where [latex]p_{1}=p_{\mathrm{a}}+\Delta p[/latex] and [latex]p_{2}=p_{\mathrm{a}}+\rho g h[/latex], which is equal to the hydrostatic pressure at the bottom of the tank. Thus,

[latex]q_{\mathrm{mi}}=\frac{p-\rho g h}{R}.[/latex]

The mass flow rate out of the tank can be expressed in terms of the volume flow rate [latex]q_{0}[/latex] as

[latex]q_{\mathrm{mo}}=\rho q_{\mathrm{o}} .[/latex]

Substituting these expressions into the law of conservation of mass gives

[latex]\rho A \frac{\mathrm{d} h}{\mathrm{~d} t}=\frac{p-\rho g h}{R}-\rho q_{\mathrm{o}}.[/latex]

Rearranging the equation gives

[latex]\rho A \frac{\mathrm{d} h}{\mathrm{~d} t}+\frac{\rho g}{R} h-\frac{p}{R}=-\rho q_{0}.[/latex]

For a liquid-level system with two or more tanks, we apply the law of conservation of mass to each tank.

Question 7.4: A Single-Tank Liquid-Level System with a Pump Consider the s...

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