Page 225 - 48Fundamentals of Compressible Fluid Mechanics

11.2. GENERAL MODEL AND NON-DIMENSIONED 187

and utilizing the definition of characteristic time, equation (11.5), and substituting
into equation (11.8) yields

(11.9)

>

Note that equation (11.9) can be modified by introducing additional param-
> >
>

3
%

. For cases, where the process time
>
>
%
is important parameter equation (11.9) transformed to
>*
*
?AO
(11.10)
N
eter which referred to as external time,*

?PO
when and * * * N > > > .

@
>

It is more convenient to deal with the stagnation pressure then the actual
> *

>

pressure at the entrance to the tube. Utilizing the equations developed in Chapter
%
?PO
4 between the stagnation condition, denoted without subscript, and condition in a
> > >
>

N
are all are function of* in this case. And where* * *
tube denoted with subscript 1. The ratio of is substituted by

(11.11)

It is convenient to denote >
E

>

>
>
(11.12)

Note that is a function of the time. Utilizing the definitions (11.11) and sub-

E

stituting equation (11.12) into equation (11.9) to be transformed into

(11.13)

> >*

Equation (11.13) is a first order nonlinear differential equation that can be solved

*
>

for different initial conditions. At this stage, the author isn’t aware that is a general
> %

>
4
solution for this equation . Nevertheless, many numerical methods are available to
>
>
%
>*
solve this equation.
11.2.1 Isentropic process
The relationship between the pressure and the temperature in the chamber can be
approximated as isotropic and therefore
(11.14)

* *

3 This notation is used in many industrial processes where time of process referred to sometime as

the maximum time. >
>
4 To those mathematically included, find the general solution for this equation.