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Next: The Impulse Function Up: Isentropic Tables Previous: Isentropic Isothermal Flow Nozzle Index
Substituting equation (4.76) into the momentum equation4.6 yields
Integration of equation (4.77) yields the Bernoulli's equation for ideal gas in isothermal process which reads
Thus, the velocity at point 2 becomes
The velocity at point 2 for stagnation point, reads
Or in explicit terms of the stagnation properties the velocity is
Transform from equation (4.78) to a dimensionless form becomes
Simplifying equation (4.82) yields
Or in terms of the pressure ratio equation (4.83) reads
As oppose to the adiabatic case ( ) in the isothermal flow the stagnation temperature ratio can be expressed
Utilizing conservation of the mass to yield
Combining equation (4.86) and equation (4.84) yields
The change in the stagnation pressure can be expressed as
The critical point, at this stage, is unknown (at what Mach number the nozzle is choked is unknown) so there are two possibilities: the choking point or M=1 to normalize the equation. Here the critical point defined as the point where M=1 so results can be compared to the adiabatic case and denoted by star. Again it has to emphasis that this critical point is not really related to physical critical point but it is arbitrary definition. The true critical point is when flow is choked and the relationship between two will be presented.
The critical pressure ratio can be obtained from (4.84) to read
Equation (4.87) is reduced to obtained the critical area ratio writes
Similarly the stagnation temperature reads
Finally, the critical stagnation pressure reads
The maximum value of stagnation pressure ratio is obtained when at which is
For specific heat ratio of , this maximum value is about two. It can be noted that the stagnation pressure is monotonically reduced during this process.
Of course in isothermal process . All these equations are plotted in Figure (4.6).4.3 it can be observed that minimum of the curve isn't on M=1 . The minimum of the curve is when area is minimum and at the point where the flow is choked. It should be noted that the stagnation temperature is not constant as in the adiabatic case and the critical point is the only one constant.
The mathematical procedure to find the minimum is simply taking the derivative and equating to zero as following
Equation (4.94) simplified to
It can be noticed that a similar results are obtained for adiabatic flow. The velocity at the throat of isothermal model is smaller by a factor of . Thus, dividing the critical adiabatic velocity by results in
On the other hand, the pressure loss in adiabatic flow is milder as can be seen in Figure (4.7(a)). 4.89) the following relationship can be obtained
Notice that the critical pressure is independent of the specific heat ratio, , as opposed to the adiabatic case. It also has to be emphasized that the stagnation values of the isothermal model are not constant. Again, the heat transfer is expressed as
It can be noticed that temperature in the isothermal model is constant while temperature in the adiabatic model can be expressed as a function of the stagnation temperature. The initial stagnation temperatures are almost the same and can be canceled out to obtain
By utilizing equation (4.100) the velocity ratio was obtained and is plotted in Figure (4.7(b)).
Thus, using the isentropic model results in under prediction of the actual results for the velocity in the supersonic branch. While, the isentropic for the subsonic branch will be over prediction. The prediction of the Mach number are similarly shown in Figure (4.7(b)).
Two other ratios need to be examined: temperature and pressure. The initial stagnation temperature is denoted as . The temperature ratio of can be obtained via the isentropic model as
While the temperature ratio of the isothermal model is constant and equal to one (1). The pressure ratio for the isentropic model is
and for the isothermal process the stagnation pressure varies and has to be taken into account as the following:
where is an arbitrary point on the nozzle. Using equations (4.88) and the isentropic relationship, the sought ratio is provided.
Figure (4.8) shows that the range between the predicted temperatures of the two models is very large, while the range between the predicted pressure by the two models is relatively small. The meaning of this analysis is that transferred heat affects the temperature to a larger degree but the effect on the pressure is much less significant.
To demonstrate the relativity of the approach advocated in this book consider the following example.
Next: The Impulse Function Up: Isentropic Tables Previous: Isentropic Isothermal Flow Nozzle Index Created by:Genick Bar-Meir, Ph.D.
On: 2007-11-21 include("aboutPottoProject.php"); ?>