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Consider Figure 17.4, which shows a liquid of specific volume vl (m3/kg) being pumped from a vessel at pressure pl (Pa) and height zl (m) up to a vessel at pressure P4 and height z4. The conductance between the suction vessel and the pump (including pipe entrance effects) is CLI (mE), while the conductance between pump and discharge vessel is Ct.2 (m2). The pump is situated at a height ze (m), and is rotating at a speed N (rps). The pressure at pump inlet is PPI (Pa), and is PP2 (Pa) at pump outlet.
Applying equation (4.8 l) to the pipe connecting the suction vessel to the pump, the mass flow is given by:

Similarly, the flow from the pump exit to the discharge vessel is given by:

Here the specific volume, vl, has been retained because the temperature rise across the pump is usually small. Squaring and adding the last two equations produces:

or

where CL is the conductance characterizing the complete length of pipework between the suction and discharge vessel, given by

(cf. equation (4.82)), and Ape is the pressure rise across the pump

Equation (17.50) implies that, as far as the calculation of flow is concerned, it does not matter where the pump is situated in the line between the suction and discharge vessels. [Note that it does mattter in practice, and the pump will normally be situated as near the suction vessel as possible, so as to avoid the pump's inlet pressure falling below the vapour pressure of the liquid, when cavitation will reduce pumping efficiency and cause damage to the pump.]

The head developed across the pump at speed, N, is given by equation (17.28), and the corresponding pressure difference is:

where No is the design speed (rps) and the volume flow, Q is related to the mass flow, W, by:

Combining equations(17.50), (17.53) and (17.54) gives the following equation in volume flow rate, Q:

Solving equation (17.55) for a general, nonlinear function, fl,=, of head versus flow will require iteration.
However, the function may be represented by a loworder polynomial

where ai are constant coefficients. Sufficient accuracy is often obtained using a either a second-order or a third-order representation, allowing equation (17.55) to be re-expressed as:

Equation (17.57) may be solved either iteratively or, because it is a cubic, analytically. If a second-order expression for the pump characteristic, f e l , is sufficient then a3 = 0, and equation (17.57) becomes a quadratic, the solution of which is particularly easy.