Rayleigh Line Flows

• Consider the effects of heat transfer on a frictionless compressible flow through a duct, the governing Eq. (41.1), (41.2a), (41.5), (38.8) and (41.6) are valid between any two points "1" and "2".
  
  
• Equation (41.3) requires to be modified in order to account for the heat transferred to the flowing fluid per unit mass, dQ , and we obtain
  
  

  (41.8)

  
  
• So, for frictionless flow of an ideal gas in a constant area duct with heat transfer, we have a situation of six equations and seven unknowns. If all conditions at state "1" are known, there exists infinite number of possible states "2". With an infinite number of possible states "2" for a given state "1", we find if all possible states "2" are plotted on a T- s diagram, The locus of all possible states "2" reachable from state "1" is a continuous curve passing through state "1".
  
  
• Again, the question arises as to how to determine this curve? The simplest way is to assume different values of T₂ . For an assumed value of T₂, the corresponding values of all other properties at "2" and dQ can be determined. The results of these calculations are shown in the figure below. The curve in Fig. 41.4 is called the Rayleigh line.

Fig 41.4 Rayleigh line representation of frictionless flow in a constant area duct with heat transfer

• At the point of maximum temperature (point "c" in Fig. 41.4) , the value of Mach number for an ideal gas is . At the point of maximum entropy( point "b") , the Mach number is unity.
  
  
• On the upper branch of the curve, the flow is always subsonic and it increases monotonically as we proceed to the right along the curve. At every point on the lower branch of the curve, the flow is supersonic, and it decreases monotonically as we move to the right along the curve.
  
  
• Irrespective of the initial Mach number, with heat addition, the flow state proceeds to the right and with heat rejection, the flow state proceeds to the left along the Rayleigh line. For example , Consider a flow which is at an initial state given by 1 on the Rayleigh line in fig. 41.4. If heat is added to the flow, the conditions in the downstream region 2 will move close to point "b". The velocity reduces due to increase in pressure and density, and Ma approaches unity. If dQ is increased to a sufficiently high value, then point "b" will be reached and flow in region 2 will be sonic. The flow is again choked, and any further increase in dQ is not possible without an adjustment of the initial condition. The flow cannot become subsonic by any further increase in dQ .