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Chapter 5: Problem 134

Water enters a pump impeller radially, It leaves the impeller with a tangential component of absolute velocity of \(10 \mathrm{m} / \mathrm{s}\). The impeller exit diameter is \(60 \mathrm{mm}\), and the impeller speed is \(1800 \mathrm{rpm}\). If the stagnation pressure rise across the impeller is 45 kpa, determine the loss of available energy across the impeller and the hydraulic efficiency of the pump.

Short Answer

Expert verified
The detailed calculations based on the steps will provide the loss of available energy across the impeller and the hydraulic efficiency of the pump.

Step by step solution

01

Calculation of impeller tip speed

The impeller speed is given in revolutions per minute. This should be converted into radian per second, because all the other speeds are in meter per second. Convert the impeller speed from rpm to radian per second by the formula \(\omega = \frac{2\pi N}{60}\) where N = 1800 rpm. Now, to find the impeller tip speed (U), multiply the impeller's angular velocity (obtained in the conversion) by the radius of the impeller: \(U = \omega \times r\). The radius r can be calculated from the given diameter.
02

Calculation of fluid absolute exit angle

We can find the fluid absolute exit angle (\( \beta_2\)) by applying the formula \( \tan(\beta_2) = \frac{V}{U}\) where V is the tangential component of absolute velocity (10 m/s), and U is the impeller tip speed calculated in Step 1.
03

Calculate the loss of available energy across the impeller

The energy equation can be used to calculate the loss of available energy. It can be found by the formula \( \Delta H_{L} = \Delta H_{sto} - U V \cos(\beta_2)\) where \( \Delta H_{sto}\) is the stagnation pressure rise across the impeller (given as 45 kPa, convert to m by dividing by density of water and acceleration due to gravity), while \( UV \cos(\beta_2)\) is the useful component of the energy imparted to the fluid.
04

Determine the hydraulic efficiency of the pump

The hydraulic efficiency (\( \eta_H \)) can be calculated using the formula \( \eta_H = \frac{U V \cos(\beta_2)}{\Delta H_{sto}}\). It represents the ratio between the useful energy imparted to the fluid and the total energy provided by the rotor.

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Most popular questions from this chapter

An inward flow radial turbine (see Fig. P5.136) involves a nozzle angle, \(a_{1},\) of \(60^{\circ}\) and an inlet rotor tip speed, \(U_{1},\) of 30 ft \(/ \mathrm{s}\). The ratio of rotor inlet to outlet diameters is \(2.0 .\) The radial component of velocity remains constant at 20 ft/s through the rotor, and the flow leaving the rotor at section (2) is without angular momentum. If the flowing fluid is water and the stagnation pressure drop across the rotor is 16 psi, determine the loss of available energy across the rotor and the hydraulic efficiency involved.

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