Chapter 5, Problem 5.12P

Interpretation Introduction

Interpretation: The pressure drop in the power plant and the power required is to be determined.

Concept introduction: The change in pressure across a pipe due to smoothness or roughness is known as the pressure drop. The pressure drop is given as,

  ${\text{P}}_{\text{1}}\text{-}{\text{P}}_{\text{2}}\text{=}{\text{ρh}}_{\text{f}}$ ....... (1)

The notations used here are,

  ${\text{h}}_{\text{f}}$ = Friction head loss

  $\text{ρ}$ = Density of the fluid

The power for the system is given as,

  $\text{P}\text{=}{\text{ρh}}_{\text{f}}\text{Q}$ ....... (2)

The notations used here are,

P = Power developed by the duct

Q = Flow rate by the duct

The Reynolds number is given as,

  $\text{Re = }\frac{\text{ρ<mover accent="true"><mo>V</mo><mo>¯</mo></mover><mo>D</mo>}}{\text{μ}}$ ....... (3)

The notations used here are,

  $\text{<mover accent="true"><mo>V</mo><mo>¯</mo></mover>}$ = Average velocity of the fluid

D = Diameter of duct

  $\text{μ}$ = Viscosity of the fluid

  $\text{ρ}$ = Density of fluid

Due to presence of elbow, valves, etc, there are some losses in the pipe which are counted in the energy balance equation. The head losses in the valve and elbow is given as,

  ${\text{h}}_{\text{f}}\text{=}\left(\frac{\text{4fL}}{\text{D}}\text{+}{\text{K}}_{\text{f}}\right)\frac{{\text{<mover accent="true"><mo>V</mo><mo>¯</mo></mover>}}^{\text{2}}}{\text{2g}}$ ....... (4)

f = Fanning friction factor

K_f = Loss coefficient in pipe fittings

L = Length of the duct

g = Acceleration due to gravity

The roughness parameter is given as, $\frac{\text{k}}{\text{D}}$ where k is roughness of the pipe. It shows how much friction is present in the pipe and with the help of the roughness parameter, f can be calculated from the friction factor chart.