Chapter 6, Problem 6.25P

Interpretation Introduction

Interpretation: The amount of heat added to a fluid is to be calculated when there is a change in internal energy of the system followed by other energy changes.

Concept introduction: The Energy flow in the fluid across a control surface is given by the following equation,

  $\frac{\partial \text{Q}}{\text{dt}}\text{=}{\displaystyle \underset{\text{C}\text{.S}\text{.}}{\iint }\text{(e+}\frac{\text{P}}{\text{ρ}}}\text{)ρ(v}\text{.n)dA }\mathrm{.......}\text{ (1)}$

The total heat is thus given as the sum of internal energy, kinetic energy, potential energy and pressure energy of the system.

The equation (1) can be written after simplifying as,

  $\text{Q = <mover accent="true"><mo>m</mo><mo>˙</mo></mover>}\left[{\text{(u}}_{\text{2}}{\text{-u}}_{\text{1}}\text{)+}\frac{{\text{(P}}_{\text{2}}{\text{-P}}_{\text{1}}\text{)}}{\text{ρ}}\text{+}\frac{{\text{(v}}_{\text{2}}{}^{\text{2}}{\text{-v}}_{\text{1}}{}^{\text{2}}\text{)}}{\text{2}}{\text{+g(Z}}_{\text{2}}{\text{-Z}}_{\text{1}}\text{)}\right]$   ............ (2)

Here,

P₂- P₁ is change in Pressure of the fluid

u₂- u₁is change in Internal energy of the fluid

v₂² − v₁² is change in velocity of the fluid

Z₂ − Z₁ is change in position of the fluid

   m is the mass flow rate of the fluid