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Next: Normal Shock Up: Isentropic Flow Previous: Isothermal Table   Index

# The effects of Real Gases

To obtained expressions for non-ideal gas it is communally done by reusing the ideal gas model and introducing a new variable which is a function of the gas properties like the critical pressure and critical temperature. Thus, a real gas equation can be expressed in equation (3.19). Differentiating equation (3.19) and dividing by equation (3.19) yields

Again, Gibb's equation (4.27) is reused to related the entropy change to the change in thermodynamics properties and applied on non-ideal gas. Since and utilizing the equation of the state . The enthalpy is a function of the temperature and pressure thus, and full differential is

The definition of pressure specific heat is and second derivative is Maxwell relation hence,

First, the differential of enthalpy is calculated for real gas equation of state as

Equations (4.27) and (3.19) are combined to form

The mechanical energy equation can be expressed as

At the stagnation the definition requires that the velocity is zero. To carry the integration of the right hand side the relationship between the pressure and the density has to be defined. The following power relationship is assumed

Notice, that for perfect gas the n is substituted by . With integration of equation (4.121) when using relationship which is defined in equation (4.122) results

Substituting relation for stagnation density (3.19) results

For the integration results in

For the integration becomes

It must be noted that n is a function of the critical temperature and critical pressure. The mass flow rate is regardless to equation of state as following

Where is the density at the throat (assuming the chocking condition) and is the cross area of the throat. Thus, the mass flow rate in our properties

For the case of

The Mach number can be obtained by utilizing equation (3.34) to defined the Mach number as

Integrating equation (4.120) when results

To carryout the integration of equation (4.131) looks at Bernnolli's equation which is

After integration of the velocity

It was shown in Chapter (3) that (3.33) is applicable for some ranges of relative temperature and pressure (relative to critical temperature and pressure and not the stagnation conditions).

When or when

The mass flow rate for the real gas

And for

Fliegner's number in this case is

Fliegner's number for is

The critical ratio of the pressure is

When or more generally when this is a ratio approach

To obtain the relationship between the temperature and pressure, equation (4.131) can be integrated

The power of the pressure ratio is approaching when z approaches 1. Note that

The Mach number at every point at the nozzle can be expressed as

For the Mach number is

The pressure ratio at any point can be expressed as a function of the Mach number as

for

The critical temperature is given by

and for

The mass flow rate as a function of the Mach number is

For the case of the mass flow rate is

Solution

1. The solution is simplified by using Potto-GDC for the results are

Isentropic Flow Input: M k = 1.4
M T/T0 ρ/ρ0 A/A* P/P0 PAR F/F*
2.61 0.423295 0.116575 2.92339 0.0493458 0.144257 0.633345
2. The stagnation pressure is obtained from

The stagnation temperature is

3. Of course, the stagnation pressure is constant for isentropic flow.

Next: Normal Shock Up: Isentropic Flow Previous: Isothermal Table   Index
Created by:Genick Bar-Meir, Ph.D.
On: 2007-11-21