Chapter 6, Problem 6.5P

Interpretation Introduction

Interpretation: The gage pressure for the given power of the turbine is to be determined by using the energy balance equation.

Concept introduction: The Energy balance across the pipe is given by the Bernoulli’s equation in which the sum of Potential head, Kinetic head and Pressure head is constant. The Bernoulli equation across 2 points of any system is given by:-

  $\frac{{\text{P}}_{\text{1}}}{\text{ρg}}\text{+}\frac{{\text{v}}_{\text{1}}{}^{\text{2}}}{\text{2g}}{\text{+ Z}}_{\text{1}}\text{+}\text{H}\text{=}\frac{{\text{P}}_{\text{2}}}{\text{ρg}}\text{+}\frac{{\text{v}}_{\text{2}}{}^{\text{2}}}{\text{2g}}{\text{+ Z}}_{\text{2}}\text{+}{\text{h}}_{\text{f}}$ ……. (1)

The notations used in equation (1) are:-

P₁and P₂are pressure at inlet and outlet of the pipe

v₁ and v₂ are velocity at inlet and outlet of the pipe

Z₁ and Z₂ are the locations of the liquid.

  $\text{ρ}$ = Density of the water

H = Head developed by the pump

g = Acceleration due to gravity

h_f = Frictional losses

The power delivered by the pump is given as for a given efficiency is,

  $\text{<msup> <mo>P</mo> <mo>′</mo></msup>}\text{=}\text{ηρgHQ}$ ……. (2)

Q = Flow rate of the water

  $\text{<msup> <mo>P</mo> <mo>′</mo></msup>}$ = Power of the pump

  $\text{η}$ = Turbine efficiency

The flow rate is given as,

  $\text{Q}\text{=}\text{A×v}$ ……. (3)

A = Area of pipe

v = Velocity of water