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Governing Equations and Assumptions

The process of filing or evacuating a semi flexible (semi rigid) chamber through a tube is very common in engineering. semi rigid chamber For example, most car today equipped with an airbag. For instance, the models in this Chapter are suitable for study of the filling the airbag or filling bicycle with air. airbag The analysis is extended to include a semi rigid tank. The term semi rigid tank referred to a tank that the volume is either completely rigid or is a function of the chamber's pressure.

As it was shown in this book the most appropriate model for the flow in the tube for a relatively fast situation is Fanno Flow. The Isothermal model is more appropriate for cases where the tube is relatively long in-which a significant heat transfer occurs keeping the temperature almost constant. As it was shown in Chapter (9) the resistance, $\frac{4fL}{D}$ , should be larger than . Yet Isothermal flow model is used as the limiting case.

**Figure 11.2:** A schematic of two possible connections of the tube to a single chamber

**Figure 11.3:** A schematic of the control volumes used in this model

The Rayleigh flow model requires that a constant heat transfer supplied either by chemical reactions or otherwise. This author isn't familiar with situations in which Rayleigh flow model is applicable. And therefore, at this stage, no discussion is offered here.

Fanno flow model is the most appropriate in the case where the filling and evacuating is relatively fast. In case the filling is relatively slow (long $\frac{4fL}{D}$ than the Isothermal flow is appropriate model. Yet as it was stated before, here Isothermal flow and Fanno flow are used as limiting or bounding cases for the real flow. Additionally, 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 simplified 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. semirigid tank!limits

Effects such as chemical reactions (or condensation/evaporation) are neglected. 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 resistance. The first kind connection tied the resistance, $\frac{4fL}{D}$ , with the tube area.

The simplest model for gas inside the chamber as a first approximation is the isotropic model. It is assumed that kinetic change in the chamber is negligible. Therefore, the pressure in the chamber is equal to the stagnation pressure, $P\approx P_0$ (see Figure (11.4)). Thus, the stagnation pressure at the tube's entrance is the same as the pressure in the chamber.

**Figure:** The pressure assumptions

The mass in the chamber and mass flow out are expressed in terms of the chamber variables (see Figure 11.3. The mass in the tank for perfect gas reads

$\displaystyle {d m \over dt } - \dot{m}_{out} = 0$

(11.1)

And for perfect gas the mass at any given time is

$\displaystyle m = { P(t) V (t) \over R T (t)}$

(11.2)

The mass flow out is a function of the resistance in tube, $\frac{4fL}{D}$ and the pressure difference between the two sides of the tube $\dot{m}_{out} (\frac{4fL}{D}, P_1/P_2)$ . The initial conditions in the chamber are

and etc. If the mass occupied in the tube is neglected (only for filling process) the most general equation ideal gas (11.1) reads

$\displaystyle { d\over dt} \overbrace{\left( {PV \over RT} \right)}^{m} \pm \ov... ... \overbrace{c_1 M_1(\frac{4fL}{D}, {P_2 \over P_1})}^{U}}^{{\dot{m}}_{out}} = 0$

(11.3)

When the plus sign is for filling process and the negative sign is for evacuating process.

Next: General Model and Non-dimensioned Up: Evacuating SemiRigid Chambers Previous: Evacuating SemiRigid Chambers Index

Created by:Genick Bar-Meir, Ph.D.
On: 2007-11-21