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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 $ ds =0$ and utilizing the equation of the state $ dh = dP/\rho$ . The enthalpy is a function of the temperature and pressure thus, $ h = h (T, P)$ and full differential is
The definition of pressure specific heat is $ C_p \equiv
{\partial h \over \partial T } $ 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 $ k$ . 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 $ n > 1$ the integration results in
For $ n=1$ 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 $ \rho^{*}$ is the density at the throat (assuming the chocking condition) and $ A^{*}$ is the cross area of the throat. Thus, the mass flow rate in our properties
For the case of $ n=1$
The Mach number can be obtained by utilizing equation (3.34) to defined the Mach number as
Integrating equation (4.120) when $ ds =0$ 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 $ n=1$ or when $ n \rightarrow 1 $

The mass flow rate for the real gas $ \dot{m} = \rho^{*} U^{*} A^{*}$

And for $ n=1$

Fliegner's number in this case is

Fliegner's number for $ n=1$ is

The critical ratio of the pressure is
When $ n=1$ or more generally when $ n \rightarrow 1 $ 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 $ k-1 \over k$ when z approaches 1. Note that

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

For $ n=1$ the Mach number is
The pressure ratio at any point can be expressed as a function of the Mach number as
for $ n=1$
The critical temperature is given by
and for $ n=1$

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

For the case of $ n=1$ the mass flow rate is

A design is required that at a specific point the Mach number shou...
...ferent from the point?
You can assume that $k=1.405$.

  1. The solution is simplified by using Potto-GDC for $ M=2.61$ 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

    $\displaystyle P_0 = {P_0 \over P} P = {2.61 \over 0.04943} \sim 52.802 [Bar]

    The stagnation temperature is

    $\displaystyle T_0 = {T_0 \over T} T = {300 \over 0.42027} \sim 713.82K

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

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