Chapter 5, Problem 5.15P

Interpretation Introduction

Interpretation: The theoretical power required by the fan is to be determined.

Concept introduction: The power required by the fan is the amount of energy required to increase the pressure for a given amount of fluid. The power required is calculated as,

  $\text{P}\text{=}\text{ρHQ}$ .......... (1)

The notations used here are,

  $\text{ρ}$ = Density of the fluid

P = Power developed by the fan

Q = Flow rate by the duct

H = Total head developed by the fan

The density of the fluid can be given as,

  $\text{ρ}\text{=}\frac{\text{PM}}{\text{RT}}\mathrm{.......}(2)$

In the equation (2), notations used are,

P is the pressure at which gas enters

M is the molecular weight of gas

R is the Universal Gas constant

T is the temperature at which gas enters

The Reynolds number is given as,

  $\text{Re = }\frac{\text{ρ<mover accent="true"><mo> v </mo><mo>¯</mo></mover> D}}{\text{μ}}$ .......... (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 friction, there are some frictional losses in the duct and the head developed due to this friction is frictional head.

The head developed due to this friction is given as,

  ${\text{h}}_{\text{f}}\text{=}\frac{{\text{2fL<mover accent="true"><mo> v </mo><mo>¯</mo></mover>}}^{\text{2}}}{\text{D}}$ .......... (4)

f = Fanning friction factor

h_f = Frictional head developed

L = Length of the duct

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.