Page 240 - 48Fundamentals of Compressible Fluid Mechanics

202CHAPTER 12. EVACUATING/FILING CHAMBERS UNDER EXTERNAL VOLUME CONTROL

Now substituting to the “preferred” variable

(12.9)

#% &

B

L

B

>

L
The analytical solution is applicable only in the case which the flow is choked
L
L

E #%'& >
thorough all the process. The solution is applicable to indirect connection. This
>*

L
L

happen when vacuum is applied outside the tube (a technique used in die casting
>
and injection molding to improve quality by reducing porosity.). In case when the

flow chockless a numerical integration needed to be performed. In the literature, to

create a direct function equation (12.4) is transformed into

(12.10)
B

B
<

with the initial condition of E >*
: E #% & >
>

>
(12.11)

>*

The analytical solution also can be approximated by a simpler equation as

(12.12)
#%'&

The results for numerical evaluation in the case when cylinder is initially at an
> *
atmospheric pressure and outside tube is also at atmospheric pressure are pre-
E
sented in figure 12.2. In this case only some part of the flow is choked (the later
part). The results of a choked case are presented in figure 12.3 in which outside
tube condition is in vacuum. These figures 12.2 and 12.3 demonstrate the impor-
the pressure increases significantly and

verse versa. B B . When #%'& B
tance of the ratio of #%'&
Thus, the question remains how the time ratio can be transfered to param-
B
eters that can the engineer can design in the system.
can be expressed as @
Denoting the area that creates the ratio #%'&

B
B
(12.13)
as the critical area,
provides the needed tool. Thus the exit area,
can be expressed as
@
@

The actual times ratio #%'&

B

B L
L

L
(12.14)

?PO
?PO
?AO
L
L
L
N
N
N
@ L @ L
@ L
*
*
*
* L * L * L
L
L
L