Chapter 6, Problem 6.7P

Interpretation Introduction

Interpretation: The length of the pipe for air at the sonic velocity is to be determined in an adiabatic flow.

Concept introduction: The length between the starting and the end point of a pipe is known as the maximum length of the pipe. The maximum length for a given Mach number is given as,

  $\frac{{\text{<mover accent="true"><mo>f </mo><mo>¯</mo></mover> L}}_{\text{max}}}{{\text{r}}_{\text{H}}}\text{=}\frac{\text{1}}{\text{γ}}\left(\frac{\text{1}}{{\text{Ma}}_{\text{a}}{}^{\text{2}}}\text{-}\text{1}\text{-}\frac{\text{γ+1}}{\text{2}}\text{ln}\frac{\text{2}\left\{\text{1+}\left[\text{(γ-1)/2}\right]{\text{Ma}}_{\text{a}}{}^{\text{2}}\right\}}{{\text{Ma}}_{\text{a}}{}^{\text{2}}\text{(γ+1)}}\right)$ ....... (1)

The notations used here are,

  ${\text{Ma}}_{\text{a}}$ = Mach number at the entrance of the pipe

  ${\text{L}}_{\max }$ = Maximum length of the pipe

  ${\text{r}}_{\text{H}}$ = Hydraulic radius of pipe = $\frac{\text{D}}{\text{4}}$

  $\text{γ}$ = Heat capacity ratio

  $\text{<mover accent="true"><mo>f </mo><mo>¯</mo></mover>}$ = Average friction factor

The Reynolds number is given as,

  $\text{Re = }\frac{\text{ρuD}}{\text{μ}}$ ....... (2)

The notations used here are,

u = Fluid speed

D = Diameter of pipe

  $\text{μ}$ = Viscosity of the fluid

  $\text{ρ}$ = Density of fluid

The Mach number is given as,

  $\text{Ma}\text{=}\frac{\text{u}}{\text{a}}\text{=}\frac{\text{Speed}\text{of}\text{fluid}}{\text{Speed}\text{of}\text{sound}}$ ....... (3)

The speed of sound is given as,

  $\text{a}\text{=}\sqrt{\frac{\text{γRT}}{\text{M}}}$ ....... (4)

M = Molecular weight of the air

R = Universal gas constant

T = Temperature of the air

The density of air can be given as,

  $\text{ρ}\text{=}\frac{\text{PM}}{\text{RT}}\mathrm{.......}\text{(5)}$

In the equation (5), notations used are,

P is the pressure at which air enters

M is the molecular weight of air

R is the Universal Gas constant

T is the temperature at which air enters