Chapter 6, Problem 6.28P

Interpretation Introduction

Interpretation: The discharge velocity of the air − water mixture is to be determined by using the energy balance equation.

Concept introduction: The Energy balance 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:-

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

The notations used in equation (1) are:-

P₁and P₂are pressure at inlet and outlet

v₁ and v₂ are velocity at inlet and outlet

Z₁ and Z₂ are the locations of the fluid.

  $\text{ρ}$ = Density of the fluid

H = Head developed

g = Acceleration due to gravity

The mass balance equation state that the mass flow in is equal to the mass flow out.

  ${\text{<mover accent="true"> <mo>m</mo> <mo>˙</mo></mover>}}_{\text{M}}\text{=}{\text{<mover accent="true"> <mo>m</mo> <mo>˙</mo></mover>}}_{\text{A}}\text{+}{\text{<mover accent="true"> <mo>m</mo> <mo>˙</mo></mover>}}_{\text{W}}$ ...... (2)

  ${\text{<mover accent="true"> <mo>m</mo> <mo>˙</mo></mover>}}_{\text{M}}$ = Mass flow rate of the mixture

  ${\text{<mover accent="true"> <mo>m</mo> <mo>˙</mo></mover>}}_{\text{A}}$ = Mass flow rate of the air

  ${\text{<mover accent="true"> <mo>m</mo> <mo>˙</mo></mover>}}_{\text{W}}$ = Mass flow rate of the water

The mass flow rate is given as,

  $\text{<mover accent="true"> <mo>m</mo> <mo>˙</mo></mover>}\text{=}\text{ρAv}$ ...... (3)

A = Area of the pump