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Next: The Impulse Function Up: Isentropic Tables Previous: Isentropic Isothermal Flow Nozzle Index General RelationshipIn this section, the other extreme case model where the heat transfer to the gas is perfect, (e.g. Eckert number is very small) is presented. Again in reality the heat transfer is somewhere in between the two extremes. So, knowing the two limits provides a tool to examine where the reality should be expected. The perfect gas model is again assumed (later more complex models can be assumed and constructed in a future versions). In isothermal process the perfect gas model readsSubstituting equation (4.76) into the momentum equation^{4.6} 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). From the Figure 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)). It should be emphasized that the stagnation pressure decrees. It is convenient to find expression for the ratio of the initial stagnation pressure (the stagnation pressure before entering the nozzle) to the pressure at the throat. Utilizing equation (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 |