Chapter 8, Problem 8.12P

(a)

Interpretation Introduction

Interpretation:

The work done per mole of a gas for a single stage adiabatic compressor is to be determined.

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.

Work done by the gas for a compressor is given as,

  $\text{W}\text{=}\frac{\text{<msup><mo>P </mo><mo>′</mo></msup>}}{\text{<mover accent="true"><mo>m </mo><mo>˙</mo></mover>}}$ ..... (1)

  $\text{<msup><mo>P </mo><mo>′</mo></msup>}$ = Power of the gas in Btu/min

  $\text{<mover accent="true"><mo>m </mo><mo>˙</mo></mover>}$ = Mass flow rate of the gas in lb mol/min

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

  $\text{γ}$ = Heat capacity ratio

  $\text{η}$ = Stage efficiency

Heat capacity ratio is calculated as,

  $\text{γ}\text{=}\frac{{\text{C}}_{\text{p}}}{{\text{C}}_{\text{p}}\text{-}\text{R}}$ ..... (3)

C_p = Specific heat of gas

R = Universal gas constant = 1.99 $\text{cal/mol}\text{.°C}$

(b)

Interpretation Introduction

Interpretation:

The work done per mole of a gas for a 2 stage adiabatic compressor is to be determined.

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.

Work done by the gas for a compressor is given as,

  $\text{W}\text{=}\frac{\text{<msup><mo>P </mo><mo>′</mo></msup>}}{\text{<mover accent="true"><mo>m </mo><mo>˙</mo></mover>}}$ ..... (1)

  $\text{<msup><mo>P </mo><mo>′</mo></msup>}$ = Power of the gas in Btu/min

  $\text{<mover accent="true"><mo>m </mo><mo>˙</mo></mover>}$ = Mass flow rate of the gas in lb mol/min

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

  $\text{γ}$ = Heat capacity ratio

  $\text{η}$ = Stage efficiency

Heat capacity ratio is calculated as,

  $\text{γ}\text{=}\frac{{\text{C}}_{\text{p}}}{{\text{C}}_{\text{p}}\text{-}\text{R}}$ ..... (3)

C_p = Specific heat of gas

R = Universal gas constant = 1.99 $\text{cal/mol}\text{.°C}$