Chapter 6, Problem 6.29P

Interpretation Introduction

Interpretation: The exit velocity and the pressure drop at point-1 are 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:-

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

The notations used in equation (1) are:-

P₁ and P₂ are 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_L = Head loss in the section

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

The control volume around the mixing is given as,

  $\sum {\text{F}}_{\text{y}}\text{=}{\displaystyle \underset{\text{C}\text{.S}\text{.}}{\iint }{\text{v}}_{\text{y}}\text{ρ(<mover accent="true"><mo>v</mo><mo>→</mo></mover>}\text{.<mover accent="true"><mo>n</mo><mo>→</mo></mover> )dA}}$   ...... (4)

F_y = Force on the body and is given as,

  $\sum {\text{F}}_{\text{y}}=\text{ΔPA}$   ...... (5)

  $\Delta \text{P}$ = Pressure drop in the fluid