Compressible Flow Free Textbook

Next: The ``Simple'' General Case Up: Rapid evacuating of a Previous: The Isothermal Process Index

Simple Semi Rigid Chamber

A simple relation of semi rigid chamber when the volume of the chamber is linearly related to the pressure as

$\displaystyle V(t) = a P(t)$

(11.47)

where

is a constant that represent the physics. This situation occurs at least in small ranges for airbag balloon etc. The physical explanation when it occurs beyond the scope of this book. Nevertheless, a general solution is easily can be obtained similarly to rigid tank. Substituting equation (11.42) into yields

$\displaystyle {d \over d \bar{t}} \left( \bar{P}^{1 + k \over k} \right) - \bar{P}^{k +1 \over 2k} \bar{M} f[M] = 0$

(11.48)

Carrying differentiation result in

$\displaystyle {1 + k \over k} \bar{P}^{1 \over k} {d \bar{P} \over d \bar{t}} - \bar{P}^{k +1 \over 2k} \bar{M} f[M] = 0$

(11.49)

Similarly as before, the variables are separated as

$\displaystyle \int_0^{\bar{t}} dt = {k \over 1+k} \int _{1}^{\bar{P}} {\bar{P}^{k - 1 \over 2k} d\bar{P} \over \bar{M} f[M]}$

(11.50)

The equation (11.45) integrated to obtain the form

$\displaystyle \bar{t} = {2k^2 \over \bar{M} f[M] (3k - 1) (1+k)} \left[ 1 - \bar{P}^{3k -1 \over 2k} \right]$

(11.51)

The physical meaning that the pressure remains larger thorough evacuating process, as results in faster reduction of the gas from the chamber.

Next: The ``Simple'' General Case Up: Rapid evacuating of a Previous: The Isothermal Process Index

Created by:Genick Bar-Meir, Ph.D.
On: 2007-11-21