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Next: Relationships for Small Mach Up: Stagnation State for Ideal Previous: Stagnation State for Ideal   Index

General Relationship

It is assumed that the flow is one-dimensional. Figure (4.1) describes a gas flow through a converging-diverging nozzle. It has been found that a theoretical state known as the stagnation state is very useful in simplifying the solution and treatment of the flow. The stagnation state is a theoretical state in which the flow is brought into a complete motionless condition in isentropic process without other forces (e.g. gravity force). Several properties that can be represented by this theoretical process which include temperature, pressure, and density et cetera and denoted by the subscript ``0 .''

First, the stagnation temperature is calculated. The energy conservation can be written as

Perfect gas is an ideal gas with a constant heat capacity, $ C_p$ . For perfect gas equation (4.1) is simplified into
Again it is common to denote $ T_{0}$ as the stagnation temperature. Recalling from thermodynamic the relationship for perfect gas
and denoting $ k \equiv C_{p} \div C_{v}$ then the thermodynamics relationship obtains the form
and where $ R$ is a specific constant. Dividing equation (4.2) by $ (C_p T)$ yields
Now, substituting $ c^2 = {kRT}$ or $ T = c^2 /k R$ equation (4.5) changes into
By utilizing the definition of $ k$ by equation (4.4) and inserting it into equation (4.6) yields

It very useful to convert equation (4.6) into a dimensionless form and denote Mach number as the ratio of velocity to speed of sound as

Mach number Inserting the definition of Mach number (4.8) into equation (4.7) reads

Figure: Perfect gas flows through a tube
\begin{figure}\centerline{\includegraphics {cont/variableArea/tube}}

The usefulness of Mach number and equation (4.9) can be demonstrated by this following simple example. In this example a gas flows through a tube (see Figure 4.2) of any shape can be expressed as a function of only the stagnation temperature as opposed to the function of the temperatures and velocities.

The definition of the stagnation state provides the advantage of compact writing. For example, writing the energy equation for the tube shown in Figure (4.2) can be reduced to

The ratio of stagnation pressure to the static pressure can be expressed as the function of the temperature ratio because of the isentropic relationship as

In the same manner the relationship for the density ratio is
A new useful definition is introduced for the case when M=1 and denoted by superscript ``$ *$ .'' The special case of ratio of the star values to stagnation values are dependent only on the heat ratio as the following:

Figure: The stagnation properties as a function of the Mach number, k=1.4
Image stagnation

next up previous index
Next: Relationships for Small Mach Up: Stagnation State for Ideal Previous: Stagnation State for Ideal   Index
Created by:Genick Bar-Meir, Ph.D.
On: 2007-11-21