Isentropic Process

The relationship between the pressure and the temperature in the chamber can be approximated as isotropic and therefore

$$\bar{T} = {{T(t)} \over {T(0)}} = \left[ {{{P}(t)}\over {P(0)} } \right] ^{ {k-1}\over {k}} = \bar{P}^{ {k-1}\over {k}}$$ (11.17)

The ratios can be expressed in term of the reduced pressure as followed:

$${\bar{P} \over \bar{T}} = { \bar{P} \over \bar{P}^{k-1 \over k} } = \bar{P}^{1 \over k}$$ (11.18)

and

$${\bar{P} \over \sqrt{\bar{T}}} = \bar{P}^{k +1 \over 2k}$$ (11.19)

So equation (11.13) is simplified into three different forms:

$${d \over d \bar{t}} {\left( \bar{V}{\bar{P}}^ { {1 \over k} } \right)} \pm {\bar{P} ^ {k+1 \over 2k} } \bar{M}(\bar{t}) f[M] = 0$$ (11.20)

$${1 \over k} \bar{P}^{1-k\over k} {d \bar{P} \over d\bar{t}} \bar{... ...ar{V}\over d \bar{t}} \pm {\bar{P} ^ {k+1 \over 2k} } \bar{M}(\bar{t}) f[M] = 0$$ (11.21)

$$\bar{V} {d \bar{P} \over d\bar{t}} + k \bar{P} {d \bar{V}\over d \bar{t}} \pm k {\bar{P} ^ {3k-1 \over 2k} } \bar{M}(\bar{t}) f[M] = 0$$ (11.22)

Equation (11.17) is a general equation for evacuating or filling for isentropic process in the chamber. It should be point out that, in this stage, the model in the tube could be either Fanno flow or Isothermal flow. The situations where the chamber undergoes isentropic process but the flow in the tube is Isothermal are limited. Nevertheless, the application of this model provide some kind of a limit where to expect when some heat transfer occurs. Note the temperature in the tube entrance can be above or below the surrounding temperature. Simplified calculations of the entrance Mach number are described in the advance topics section.