Chapter 6, Problem 6.14P

Interpretation Introduction

Interpretation: The rate of flow of the fluid and the total head of the jet 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:-

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

The notations used in equation (1) are:-

P₁ and P₂ are 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_L = Head loss in the section

H = Head developed by the pump

g = Acceleration due to gravity

The flow rate is given as,

  $\text{Q}\text{=}\text{A×v}$   ...... (2)

A = Area of pipe

v = Velocity of water

The motion of the jet here is a projectile motion in which the jet follows a curved path under the action of the gravity.

The motion in horizontal part is in x-direction and the one in vertical part is in y-direction.

The equation of motion in x-direction is,

  $\frac{\text{dx}}{\text{dt}}\text{=}\text{vcosθ}$   ...... (3)

  $\frac{\text{dx}}{\text{dt}}$ = Change in horizontal velocity

v = Velocity of jet

  $\text{θ}$ = Angle of projectile

The equation of motion in y-direction is,

  $\frac{\text{dy}}{\text{dt}}\text{=}\text{vsinθ}\text{-}\text{gt}$   ...... (4)

g = Acceleration due to gravity

t = Time of motion of projectile

  $\frac{\text{dy}}{\text{dt}}$ = Change in vertical velocity