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Chapter 12: Problem 36

The system resistance for a pipeline is given by \(\Delta p_{\mathrm{sys}}=2.0 Q^{2}\) where \(\Delta p_{\text {ys }}\) is the pressure rise required of a pump to deliver the flow rate \(Q\) through the piping system. A pump has the pressure-rise-flow characteristic given by \(\Delta p_{p}=30.0-3.0 Q^{2} .\) In both curves, \(\Delta p\) is in \(\mathrm{kPa}\) and \(Q\) is in \(\mathrm{m}^{3} / \mathrm{s}\), Find the pump input power if this punp is placed in this piping system and the pump overall eff ciency is \(90 \%\)

Short Answer

Expert verified
The input power required by the pump is approximately 27.358 kW.

Step by step solution

01

Establish Relationship between Pump and System

To start solving this problem, we need to equate the given system and pump characteristics. The common variable between the two equations is \(Q\) (flow rate). This gives an equation like this: \(2.0 Q^{2} = 30.0 - 3.0 Q^{2}\)
02

Solve for Flow Rate Q

After simplifying, the equation from Step 1 becomes \(5.0 Q^{2} = 30.0\), which can be solved for \(Q\) (flow rate) by doing some algebra. Solving this quadratic equation gives \(Q = \sqrt{30.0 / 5.0} = 2.45 \, m^{3}/s\)
03

Find the Pressure Rise

Now we can substitute \(Q = 2.45 \, m^{3}/s\) into the pump's characteristic equation to find the pressure rise. This gives \(\Delta p_{p} = 30.0 - 3.0 \times (2.45)^{2} = 10.05 \, kPa\)
04

Calculate Power Delivered by Pump

Using equation for power delivered by pump, \(P = \Delta p_{p} \times Q\), substituting values from Step 2 and 3 gives: \(P = 10.05 \times 10^{3} \times 2.45 = 24.6225 \, kW\)
05

Calculate the Pump Input Power

Now let's consider the pump's efficiency (\(90\% = 0.9\)), we can find the pump's input power using equation \(\text{Input Power} = \text{Delivered Power} / \text{Efficiency}\). Substituting known values: \(\text{Input Power} = 24.6225 / 0.9 = 27.358 \, kW\)

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Most popular questions from this chapter

A model fan with wheel diameter 32 in. is tested at a speed of \(1750 \mathrm{rpm}\). The test fluid is air with density \(0.075 \mathrm{lbm} / \mathrm{ft}\). At its \(\mathrm{BEP}\), the fan produces \(8000 \mathrm{ft}^{3} / \mathrm{min}\) at total pressure rise of 8 in. \(\mathrm{H}_{2} \mathrm{O}\). A geometrically similar fan is to handle \(200,000 \mathrm{ft}^{3} / \mathrm{min}\) of flue gas with density \(0.050 \mathrm{lbm} / \mathrm{ft}^{3}\) and 30 in. \(\mathrm{H}_{2} \mathrm{O}\) total pressure rise. Determine the required size and speed of the flue gas fan. Note any assumptions and/or limitations.

A centrifugal pump having an impeller diameter of \(1 \mathrm{m}\) is to be constructed so that it will supply a head rise of 200 mat a flowrate of \(4.1 \mathrm{m}^{3} / \mathrm{s}\) of water when operating at a speed of \(1200 \mathrm{rpm} .\) To study the characteristizs of this pump, a \(1 / 5\) scale, geometrizally similar model operated at the same speed is to be tested in the laboratory. Determine the required model discharge and head rise. Assume that both model and prototype operate with the same effciency (and therefore the same flow coefficient)

A certain axial-flow pump has a specific speed of \(N_{x}=5.0\) If the pump is expected to deliver 3000 gpm when operating against a 15 -ft head, at what speed (rpm) should the pump be run? Draw a sketch of the p.ump impeller (front and side views).

Water to run a Pelton wheel is supplied by a penstcck of length \(\ell\) and diameter \(D\) with a friction factor \(f\). If the only losses associated with the flow in the penstock are due to pipe friction. show that the maximum power output of the turbine occurs when the nozzle diameter, \(D_{1},\) is given by \(D_{1}=D /(2 f \ell / D)^{1 / 4}\)

A prototype fan has a \(20-f\) diameter, an inlet pressure of 14.40 psia, an inlet temperature of \(70^{\circ} \mathrm{F},\) and a speed of 90 rpm. \(\mathrm{A}\) \(\frac{1}{10}-\operatorname{scal} \mathrm{e}\) model of the fan has the same inlet pressure and emperature. an inlet power of \(1.24 \mathrm{hp}\), a flow rate of \(220 \mathrm{ft}^{3} / \mathrm{min}\), and a speed of \(1800 \mathrm{rpm} .\) Find the corresponding input power and flow rate of the prototype fan. Neglect Reynolds number effects.
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