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Feedback    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 (4.116)

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 (4.117)

The definition of pressure specific heat is and second derivative is Maxwell relation hence, (4.118)

First, the differential of enthalpy is calculated for real gas equation of state as (4.119)

Equations (4.27) and (3.19) are combined to form (4.120)

The mechanical energy equation can be expressed as (4.121)

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 (4.122)

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 (4.123)

Substituting relation for stagnation density (3.19) results (4.124)

For the integration results in (4.125)

For the integration becomes (4.126)

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 (4.127)

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 (4.128)

For the case of  (4.129)

The Mach number can be obtained by utilizing equation (3.34) to defined the Mach number as (4.130)

Integrating equation (4.120) when results (4.131)

To carryout the integration of equation (4.131) looks at Bernnolli's equation which is (4.132)

After integration of the velocity (4.133)

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). (4.134)

When or when  (4.135)

The mass flow rate for the real gas  (4.136)

And for  (4.137)

Fliegner's number in this case is (4.138)

Fliegner's number for is (4.139)

The critical ratio of the pressure is (4.140)

When or more generally when this is a ratio approach (4.141)

To obtain the relationship between the temperature and pressure, equation (4.131) can be integrated (4.142)

The power of the pressure ratio is approaching when z approaches 1. Note that (4.143)

The Mach number at every point at the nozzle can be expressed as (4.144)

For the Mach number is (4.145)

The pressure ratio at any point can be expressed as a function of the Mach number as (4.146)

for  (4.147)

The critical temperature is given by (4.148)

and for  (4.149)

The mass flow rate as a function of the Mach number is (4.150)

For the case of the mass flow rate is (4.151) 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