Chapter II Isentropic Flow General Features of Isentropic Flow The flow in pipes and ducts is very often adiabatic. When the duct is short, as it is in nozzles, diffusers, the frictional effects are comparatively small, and the flow may as a first approximat –ion be considered reversible, and, therefore isentropic. The one – dimensional approximation A one – dimensional flow is a flow in which the rate of change of fluid properties normal to the streamline direction is negligibly small compared with the rate of charge along the streamline. If the properties vary over each cross section and we are applying the one – dimensional assumption to the flow in ducts, we in effect deal with certain kinds of average properties for each cross section. Prof. Dr. MOHSEN OSMAN

Consider the isentropic flow of any fluid through a passage of vary- ing cross section. The following physical equations may be written for a control surface extending between the stagnation section and any other section in the channel; I – The first law of Thermodynamics (steady-flow energy equation) Or = stagnation enthalpy This is hold for steady adiabatic flow of any compressible fluid outside the boundary layer. For perfect gases : & From the first law of Thermodynamics : Prof. Dr. MOHSEN OSMAN

1 – For constant – volume process 2 – For constant – pressure process: & Since & II – Second Law of Thermodynamics III – Conservation of Mass Equation (Equation of Continuity) IV –Definition of Mach Number Prof. Dr. MOHSEN OSMAN

Speed of Sound The speed of sound (C) is the rate of propagation of a pressure pulse of infinitesimal strength through a still fluid; it is a thermodynamic property of a fluid.     Apply continuity equation for one dimensional flow (I) Prof. Dr. MOHSEN OSMAN

Apply linear momentum equation in x-direction: (II) From eqns (I) & (II) For isentropic flow as Prof. Dr. MOHSEN OSMAN

For a perfect gas, the equation of state (IV) For reversible adiabatic process Speed of sound = C = (III) For Air: Compare equations (III) & (IV) Prof. Dr. MOHSEN OSMAN

For Liquids & Solids Bulk modulus of elasticity B of the material is defined as : But Differentiate Speed of sound in liquids & solids General Features of Isentropic Flow Apply conservation of mass principle for one-dimensional steady flow: Prof. Dr. MOHSEN OSMAN

Consider SSSF continuity equation Take logarithm of both sides of continuity equation Differentiate both sides of the equation From the second law of thermodynamics, for isentropic flow we have & By combining the first law with the second law, changes of entropy can be related to other state functions. The following equations apply to a process in a one-component system in which gravity, motion, electricity, magnetic, and capillary effects are absent: since for isentropic flow, we have Prof. Dr. MOHSEN OSMAN

Euler’s equation for one-dimensional flow: If variations in height are neglected, then Now eliminate dP and dρ between continuity and speed of sound equation to obtain the following relation between velocity change and area change in Isentropic Duct Flow. Rearrange For subsonic flow M˂ 1 dV˂ 0 when dA˃0 & dV˃0 when dA˂0 Prof. Dr. MOHSEN OSMAN

Prof. Dr. MOHSEN OSMAN Figure 5.2 Nozzle configurations. Figure 5.3 Diffuser configurations. Prof. Dr. MOHSEN OSMAN

For supersonic flow M˃ 1 dV˃0 when dA˃0 & dV˂0 when dA˂0 Since infinite acceleration is physically impossible, the above equation indicates that dV can be finite only when dA=0, that is, a minimum area (throat) or a maximum area (bulge). The throat or converging-diverging section can smoothly accelerate a subsonic flow through sonic to supersonic flow. This is the only way a super-sonic flow can be created by expanding the gas from a stagnant reservoir. Perfect – Gas Relations We can use the perfect–gas and isentropic–flow relations to convert the continuity equation into an algebraic expression involving only area and Mach number, as follows. Equate the mass flux at any section to the mass flux under sonic conditions (which may not actually occur in the duct). Prof. Dr. MOHSEN OSMAN

But from the first law of thermodynamics, we have Or Since then, We have Perfect–gas equation of state Prof. Dr. MOHSEN OSMAN

Then For sonic conditions , substitute M=1 Find relations between Prof. Dr. MOHSEN OSMAN

Similarly, For γ=1.4, this equation takes the numerical form: which is plotted in the figure Also, for γ=1.4, the following numerical versions of the isentropic flow formula are obtained Prof. Dr. MOHSEN OSMAN

Figure 5.1 Property variation with area change. Prof. Dr. MOHSEN OSMAN

The figure shows that the minimum area which can occur in a given isentropic duct flow is the sonic, or critical throat area. Choking For γ=1.4, this reduces to : For Subsonic Flow M≈ For Supersonic Flow M≈ Prof. Dr. MOHSEN OSMAN