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Next: Rayleigh Flow Tables Up: Rayleigh Flow Previous: Introduction   Index

Governing Equation

The energy balance on the control volume reads

the momentum balance reads
The mass conservation reads
Equation of state
There are four equations with four unknowns, if the upstream conditions are known (or downstream conditions are known). Thus, a solution can be obtained. One can notice that equations (10.2), (10.3) and (10.4) are similar to the equations that were solved for the shock wave.
The equation of state (10.4) can further assist in obtaining the temperature ratio as
The density ratio can be expressed in terms of mass conservation as
Substituting equations (10.5) and (10.7) into equation (10.6) yields
Transferring the temperature ratio to the left hand side and squaring the results gives

Figure 10.2: The temperature entropy diagram for Rayleigh line
The Rayleigh line exhibits two possible maximums one for $ dT/ds = 0 $ and for $ ds /dT =0$ . The second maximum can be expressed as $ dT/ds = \infty$ . The second law is used to find the expression for the derivative.

Let the initial condition $ M_1$ , and $ s_1$ be constant and the variable parameters are $ M_2$ , and $ s_2$ . A derivative of equation (10.11) results in
Taking the derivative of equation (10.12) and letting the variable parameters be $ T_2$ , and $ M_2$ results in
Combining equations (10.12) and (10.13) by eliminating $ dM$ results in
On T-s diagram a family of curves can be drawn for a given constant. Yet for every curve, several observations can be generalized. The derivative is equal to zero when $ 1 - kM^2 = 0$ or $ M = 1 /\sqrt{k}$ or when $ M \rightarrow 0$ . The derivative is equal to infinity, $ dT/ds = \infty$ when $ M=1$ . From thermodynamics, increase of heating results in increase of entropy. And cooling results in reduction of entropy. Hence, when cooling is applied to a tube the velocity decreases and when heating is applied the velocity increases. At peculiar point of $ M = 1 /\sqrt{k}$ when additional heat is applied the temperature decreases. The derivative is negative, $ dT/ds < 0$ , yet note this point is not the choking point. The choking occurs only when $ M=1$ because it violates the second law. The transition to supersonic flow occurs when the area changes, somewhat similarly to Fanno flow. Yet, choking can be explained by the fact that increase of energy must be accompanied by increase of entropy. But the entropy of supersonic flow is lower (see Figure (10.2)) and therefore it is not possible (the maximum entropy at $ M=1$ .).

It is convenient to refer to the value of $ M=1$ . These values are referred to as the ``star''10.1values. The equation (10.5) can be written between choking point and any point on the curve.

The temperature ratio is

The stagnation pressure ratio reads

next up previous index
Next: Rayleigh Flow Tables Up: Rayleigh Flow Previous: Introduction   Index
Created by:Genick Bar-Meir, Ph.D.
On: 2007-11-21