Chapter 8, Problem 8.4P

Interpretation Introduction

Interpretation: The number of stages required by the compressor for given compression ratio is to be determined along with the shaft work and brake horsepower.

Concept introduction: A compressor is a device that increases the pressure of a gas by decreasing its volume. A stage in a compressor is the number of times the gas is compressed. The number of stages in a compressor is given by the compression ratio,

Compression ratio = $\frac{{\text{P}}_{\text{b}}}{{\text{P}}_{\text{a}}}$ = ${\left(\frac{{\text{P}}_{\text{b}}}{{\text{P}}_{\text{a}}}\right)}^{\frac{\text{1}}{\text{n}}}$

  ${\text{P}}_{\text{a}}$ = Inlet Pressure of the gas

  ${\text{P}}_{\text{b}}$ = Outlet Pressure of the gas

For number of stages the value of ${\left(\frac{{\text{P}}_{\text{b}}}{{\text{P}}_{\text{a}}}\right)}^{\frac{\text{1}}{\text{n}}}$ should be nearly constant.

n is the number of stages.

The shaft work is the work done by the compressor in the number of stages given and it is given as,

  ${\text{W}}_{\text{pr}}\text{=}\frac{{\text{P}}_{\text{a}}\text{γ}}{{\text{(γ-1)ρ}}_{\text{a}}}\left[{\left(\frac{{\text{P}}_{\text{b}}}{{\text{P}}_{\text{a}}}\right)}^{\text{1-}\frac{\text{1}}{\text{γ}}}\text{-1}\right]\mathrm{.......}\text{(1)}$

W_pr = Shaft work

  $\text{γ}\text{=}$ Compressibility ratio

The brake horsepower is given as,

  ${\text{P}}_{\text{B}}\text{=}\frac{\text{1}\text{.304}\text{×}{\text{10}}^{\text{-4}}{\text{T}}_{\text{a}}{\text{q}}_{\text{0}}}{\text{η}}\frac{\text{γ}}{\text{γ-1}}\left[{\left(\frac{{\text{P}}_{\text{b}}}{{\text{P}}_{\text{a}}}\right)}^{\text{1-}\frac{\text{1}}{\text{γ}}}\text{-1}\right]\mathrm{.......}(2)$

T_a= Inlet temperature in $\text{°R}$

  ${\text{q}}_{\text{0}}$ = Volume of compressed gas in std ft³/min

P_B = Brake horsepower