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Rayleigh Curve

As we know, Rayleigh flow is called as the flow with heat addition. The curve or plot or state chart dealing with heat addition is called Rayleigh curve. Lets derive the expression for this curve in p-v and h-s chart.

For 1D, it can be expressed as pA + ρv2A = const, called as Impulse function or Thrust function.

We can draw a p-v diagram for flow with heat addition or Rayleigh flow like Hugoniot curve, using momentum equation as,

This equation is the equation for straight line on p-v diagram where k2 corresponds to slope which eventually is the mass flow rate. Hence, slope of the line joining any point, corresponding to initial state and final state on Rayleigh curve, represents mass fluxes or mass flow rates. This fact is same as that observed for Hugoniot curve.

Slope of this Rayleigh line can be calculated as

p1 + v1k2 = p2 + v2k2

Such curve represented by points 1, 2, 3 and 4 for flow with heat addition is as shown in Figure 1 along with the isentropic and isothermal line on p-v chart

For better understanding, consider the process of heat addition in subsonic flow. Suppose given conditions are described by point 1 in Fig. 1. Change in thermodynamic states of the flow in the process of heat addition is shown in the figure by a straight line. Slope of this straight line is proportional to the mass flow rate and is given by Eq. 1. Here point 2 represents the conditions after certainly amount of heat addition. Point 2 essentially has lesser pressure and higher specific volume as that of point 1 since expansion of the subsonic flow takes place due to heat addition. Increase in temperature can also be observed here for the subsonic flow. Sufficient amount of heat addition would lead to reach point 3 from initial conditions 1. We can clearly see in this figure (1) that the Rayleigh line is tangent to an isotherm at point 3, hence the temperature given by the corresponding isotherm is the maximum attainable temperature by adding heat in the given subsonic flow of initial conditions 1. Conditions represented by point 4 become possible by further addition of heat. It can also be seen here that Rayleigh line is tangent to an isentropic at point 4, hence point 4 represents the maximum entropy point or sonic point. Reduction in temperature in the process 3-4 is clearly evident in the presence of heat addition. Therefore there are two critical points in Rayleigh curve, one of which corresponds to maximum enthalpy or temperature and other corresponds to maximum entropy or total temperature or total enthalpy.

We can as well use h-s diagram to explain the heat addition process in the same subsonic flow as shown in Fig. 2.

Fig.2 Rankine curve in h-s chart