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##

The ``Simple'' General Case

Balloon Problem
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 container 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 assumption isn't relaxed here.

Any continuous positive monotonic function
can be expressed into a polynomial function.
However, as first approximation and simplified approach can
be done by a single term with a different power as

When

can be any positive value including zero, 0
.
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 reduced pressure are linear^{11.6}
Value of
above it will Convex and below it concave.

Notice that when
equation (11.49)
reduced to equation (11.43).

After carrying-out differentiation results

Again, similarly as before, variables are separated and integrated as
follows

Carrying-out the integration for the initial part if exit results in

The linear condition are obtain when

That is just bellow 1 (

) for

.

** Next:** Advance Topics
** Up:** Rapid evacuating of a
** Previous:** Simple Semi Rigid Chamber
** Index**
Created by:Genick Bar-Meir, Ph.D.

On:
2007-11-21