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

$$h +{U^2 \over 2} = h_0$$ (4.1)

Perfect gas is an ideal gas with a constant heat capacity, $C_p$ . For perfect gas equation (4.1) is simplified into

$$C_{p} T + {U^2 \over 2} = C_{p} T_{0}$$ (4.2)

Again it is common to denote $T_{0}$ as the stagnation temperature. Recalling from thermodynamic the relationship for perfect gas

$$R = C_{p} -C_{v}$$ (4.3)

and denoting $k \equiv C_{p} \div C_{v}$ then the thermodynamics relationship obtains the form

$$C_{p} = {k R \over k -1 }$$ (4.4)

and where $R$ is a specific constant. Dividing equation (4.2) by $(C_p T)$ yields

$$1 + { U ^{2} \over 2 C_{p} T} = { T_{0} \over T}$$ (4.5)

Now, substituting $c^2 = {kRT}$ or $T = c^2 /k R$ equation (4.5) changes into

$$1 + {k R U ^{2} \over 2 C_{p} c^{2} } = { T_{0} \over T}$$ (4.6)

By utilizing the definition of $k$ by equation (4.4) and inserting it into equation (4.6) yields

$$1 + {{k -1} \over 2} \; { U ^{2} \over c^{2} } = { T_{0} \over T}$$ (4.7)

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

$$M \equiv {U \over c}$$ (4.8)

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

$${T_0 \over T} = 1 + { k -1 \over 2 } M^{2}$$ (4.9)

Figure: Perfect gas flows through a 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

$$\dot{Q} = C_{p} ({T_{0}}_{B} - {T_{0}}_{A}) \dot{m}$$ (4.10)

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

$${P_0 \over P } = \left( { T_0 \over T} \right) ^ {k \over k -1} = \left( 1 + { k -1 \over 2 } M^{2} \right)^ {k \over k -1}$$ (4.11)

In the same manner the relationship for the density ratio is

$${\rho_0 \over \rho } = \left( { T_0 \over T} \right) ^ {1 \over k -1} = \left( 1 + { k -1 \over 2 } M^{2} \right)^ {1 \over k -1}$$ (4.12)

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:

$${T^{*} \over T_0} = {{c^{*}}^2 \over {c_0}^2} = {2 \over k+1}$$ (4.13)

$${P^{*} \over P_0} = \left(2 \over k+1 \right)^{k \over k-1}$$ (4.14)

$${\rho^{*} \over \rho_0} = \left(2 \over k+1 \right)^{1 \over k-1}$$ (4.15)

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