Chapter 6, Problem 6.3P

Interpretation Introduction

Interpretation: The rate of the temperature increase in the reservoir 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:-

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

H = Head developed by the pump

g = Acceleration due to gravity

The amount of heat required is given as,

  $\text{Q}\text{=}\text{ρVc}\frac{\text{dT}}{\text{dt}}$   ……. (1)

Q = Heat required

V = Volume of the reservoir

  $\text{ρ}$ = Density of fluid

c = Specific heat of the fluid

  $\frac{\text{dT}}{\text{dt}}$ = Change in temperature with time

The change in heat due to head developed is given as,

  ${\text{Q}}_{\text{1}}\text{=}\text{mH}$   ……. (2)

m = Mass flow rate of the fluid

The mass flow rate from momentum equation is,

  $\text{m}\text{=}\text{ρAv}$   ……. (3)

A = Area of the pipe

v = Velocity of the fluid