Page 243 - 48Fundamentals of Compressible Fluid Mechanics

12.1. MODEL 205

According to equation (11.5)* is inversely proportional to to area,* . Thus,

@
is canceled and reduced into
@
(12.15)
?AO
N
equation (12.14) the*

?PO
L
N
@ L
@
*

, and the friction
L

Parameters influencing the process are the area ratio,
* L
. From other detailed calculations [?] it was found that the influence

on the pressure development in the cylinder is quite small.
parameter,

The influence is small on the residual air mass in the cylinder but larger on the
of the parameter
Mach number, . The effects of the area ratio, , are studied here since it is
the dominant parameter.
. This parameter rep-
O B
resents the ratio between the filling time and the evacuating time, the time which
U
It is important to point out the significance of the #% &

would be required to evacuate the cylinder for constant mass flow rate at the max-
B
B
imum Mach number when the gas temperature and pressure remain in their initial
, according to

values. This parameter also represents the dimensionless area,
the following equation
Figure 12.4 describes the pressure as a function of the dimensionless time
is almost straight. For

for various values of
. The line that represents

Fig. 12.4: The pressure ratio as a function of the dimensionless time
the pressure increases the volume flow rate of the air until a

large values of
quasi steady state is reached. This quasi steady state is achieved when the volu-

metric air flow rate out is equal to the volume pushed by the piston. The pressure
and the mass flow rate are maintained constant after this state is reached. The
there

pressure in this quasi steady state is a function of
. For small values of
is greater than one the pressure is concave up-

is no steady state stage. When
is less than one the pressure is concave downward as shown

ward and when
in Figures 12.4, which was obtained by an integration of equation (12.9).

12.1.2 Examples
Example 12.1:
Calculate the minimum required vent area for die casting process when the die vol-
)( .

ume is . and
SOLUTION . The required solidification time,* N ?AO $