Page 223 - 48Fundamentals of Compressible Fluid Mechanics

11.1. GOVERNING EQUATIONS AND ASSUMPTIONS 185

and Fanno flow are used as limiting or bounding cases for the real flow. Addition-
ally, the process in the chamber can be limited or bounded between two limits of
Isentropic process or Isothermal process.
In this analysis, in order to obtain the essence of the process, some simpli-
fied assumptions are made. The assumptions can be relaxed or removed and the
model will be more general. Of course, the payment is by far more complex model
that sometime clutter the physics. First, a model based on Fanno flow model is
constructed. Second, model is studied in which the flow in the tube is Isothermal.
The flow in the tube in many cases is somewhere between the Fanno flow model
to Isothermal flow model. This reality is an additional reason for the construction
of two models in which they can be compared.
Effects such as chemical reactions (or condensation/evaporation) are ne-
glected. There are two suggested itself possibilities to the connection between the
tube to the tank (see the Figure 11.2): one) direct two) through a reduction. The
direct connection is when the tube is connect straight to tank like in a case where
pipe is welded into the tank. The reduction is typical when a ball is filled trough
an one–way valve (filling a baseball ball, also in manufacturing processes). The
second possibility leads itself to an additional parameter that is independent of the
, with the tube area.

The simplest model for gas inside the chamber as a first approximation is
resistance. The first kind connection tied the resistance,
the isotropic model. It is assumed that kinetic change in the chamber is negligible.
(see Figure 11.4). Thus, the stagnation pressure at the tube’s entrance is the same
as the pressure in the chamber.
The mass in the cham-
ber and mass flow out are ex-
Therefore, the pressure in the chamber is equal to the stagnation pressure,
pressed in terms of the cham-
ber variables (see Figure 11.3.
The mass in the tank for perfect 2
gas reads
1

Fig. 11.4: The pressure assumptions in the chamber
(11.1) and tube entrance

And for perfect gas the mass at any given time is
B
E

*
(11.2)

* *

*
and the pressure

difference between the two sides of the tube . The initial condi-
"
The mass flow out is a function of the resistance in tube,

tions in the chamber are B
, and etc. If the mass occupied in the tube is