Page 239 - 48Fundamentals of Compressible Fluid Mechanics

12.1. MODEL 201

Equation (12.2) can be non–dimensionlassed as

(12.2)

The governing equation (11.10) that was developed in the previous chapter
> >* E
(11) obtained the form as
>*
(12.3)

where > > > * N > * @ > are two different charac-
. Notice that in this case that there

>
?PO

. The

is associated with the ratio of the volume and the tube
> *
?PO
>*
is as-
N
?PO
@
>*
* *
sociated with the imposed time on the system (in this case the elapsed time of the
N
@
and the “maximum” time,*

teristic times: the “characteristic” time,*
piston stroke). N
?AO
first characteristic time,*
Equation (12.3) is an nonlinear first order differential equation and can be
characteristics (see equation (11.5)). The second characteristic time,*
rearranged as follows
(12.4)

>
Equation (12.4) is can be solved > E
B

>*
isn’t function of the time.
>
<
B
: E #% & >
>

>*
The solution of equation (12.4) can be obtained by transforming and by
. The reduced Pres-
only when the flow is chocked In which case
Utilizing this definition and there implication
reduce equation (12.4)
> and therefore

>
introducing a new variable
(12.5)

sure derivative,

>

where B And equation (12.5) can be further simplified as

>*
E

>*
(12.6)

E E
B
#%'& >

Equation (12.6) can be integrated to obtain >* >*

E E E

(12.7)

or in a different form E L L L L L E >*
L
L
E
L
(12.8)

L
L

L
L
L L
E
>*
L
L