Page 233 - 48Fundamentals of Compressible Fluid Mechanics

11.4. RAPID EVACUATING OF A RIGID TANK 195

Carrying differentiation result in

(11.44)

Similarly as before, the variables are separated as

E
>

>

>
>

>*
(11.45)

B
The equation (11.50) integration obtains the form >

>
>
* !
(11.46)

The physical meaning that the pressure remains larger thorough evacuating pro-

E

E

cess, as results in faster reduction of the gas from the chamber.

>
>* !
>
11.4.5 The “Simple” General Case
The relationship between the pressure and the volume from the physical point of
view must be monotonous. Further, the relation must be also positive, increase of
the pressure results in increase of the volume (as results of Hook’s law. After all, in
the known situations to this author pressure increase results in volume decrease
(at least for ideal gas.).
In this analysis and previous analysis the initial effect of the chamber con-
tainer inertia is neglected. The analysis is based only on the mass conservation
and if unsteady effects are required more terms (physical quantities) have taken
into account. Further, it is assumed the ideal gas applied to the gas and this as-
sumption isn’t relaxed here.
Any continuous positive monotonic function can be expressed into a poly-
nomial function. However, as first approximation and simplified approach can be
done by a single term with a different power as
(11.47)

When can be any positive value including zero, . The physical meaning of

* ,
is that the tank is rigid. In reality the value of lays between zero to one. When
is approaching to zero the chamber is approaches to a rigid tank and vis versa
when the the chamber is flexible like a balloon.
(

(
There isn’t a real critical value to . Yet, it is convenient for engineers to
(

further study the point where the relationship between the reduced time and the
(
6
reduced pressure are linear Value of above it will Convex and and below it
(
concave. (
(
6 Some suggested this border point as infinite evocation to infinite time for evacuation etc. This
undersigned is not aware situation where this indeed play important role. Therefore, is waiting to find
such conditions before calling it as critical condition.