Chapter 6, Problem 6.7P

Interpretation Introduction

Interpretation: The volume rate of the flow of air in the duct and the horsepower output of the fan needs to be determined.

Concept introduction: The Energy balance across the fan is given by Bernoulli’s equation in which the sum of potential energy, kinetic energy and pressure energy is constant. The Bernoulli equation across 2 points of any system is given by:-

  $\frac{{\text{P}}_{\text{1}}}{\text{ρ}}\text{+}\frac{{\text{v}}_{\text{1}}{}^{\text{2}}}{\text{2}}{\text{+ gZ}}_{\text{1}}\text{=}\frac{{\text{P}}_{\text{2}}}{\text{ρ}}\text{+}\frac{{\text{v}}_{\text{2}}{}^{\text{2}}}{\text{2}}{\text{+ gZ}}_{\text{2}}$ ........ (1)

Here, P₁ and P₂ are pressures at 2 points of the system in Pascal.

v₁ and v₂ are velocities at 2 points of the system in m/s.

Z₁ and Z₂ are the locations of the inlet and outlet of the air.

  $\begin{array}{l}\text{ρ}\text{=}\text{Density}\text{of}\text{air}\text{=}\text{1}\text{.22}{\text{kg/m}}^{\text{3}}\\ \text{g}\text{=}\text{Acceleration}\text{due}\text{to}\text{gravity}\text{= }\text{9}\text{.8}{\text{m/s}}^{\text{2}}\end{array}$

The volumetric flow rate in the air duct is given as,

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

Q = Volumetric flow rate of air

A = Area of duct

v = Velocity of air at the outlet

  $\begin{array}{l}{\text{P}}_{\text{1}}\text{-}{\text{P}}_{\text{2}}\text{=}{\text{ρ}}_{\text{w}}\text{gh}\\ \text{Here,}\\ \text{h}\text{=}\text{Reading of the differential manometer }\\ {\text{ρ}}_{\text{w}}\text{=}\text{Density}\text{of}\text{water}\end{array}$ ........ (3)

The power output is given as,

  $\begin{array}{l}\text{<mover accent="true"><mo>P</mo><mo>˙</mo></mover>}\text{=}\text{ρQ}\frac{{\text{v}}_{\text{2}}{}^{\text{2}}}{\text{2}}\\ \text{<mover accent="true"><mo>P</mo><mo>˙</mo></mover>}\text{=}\text{Power}\text{output}\\ {\text{v}}_{\text{2}}\text{=}\text{Output}\text{velocity}\text{of}\text{air}\end{array}$ ........ (4)