Fundamentals of Heat Engines. Jamil Ghojel

      Combining the last two equations, we get

(1.67)

      From Eqs. (1.48) and (1.67)

(1.68)

      The pressure ratio for M_a = 1 at γ = 1.4 is equal to 1.893. This is the critical pressure ratio for air. To achieve supersonic flow, the stagnation pressure needs to be such that p_t > 1.893p.

      1.3.5.5 Speed Parameter

      It was shown in Section 1.2 that dimensional analysis and similitude can be used to derive functional representations of complex flow systems, such as compressors, using a reduced number of nondimensional groups of properties. Among the groups discussed were the velocity parameter and mass flow parameter (Eq. 1.39a). To find a physical interpretation of these seemingly arbitrary combinations of physical properties, consider first the ratio :

      Combining this equation with Eq. (1.67) we obtain

(1.69)

      Now, for a given compressor blade design and impeller tip speed U, C = f(U) and U = f(ND); hence, from Eq. (1.69) for the compressor inlet conditions

      The compressor speed parameter is a function of the flow Mach number at the inlet (flight Mach number for a turbojet engine) and thermodynamic properties of the fluid. For a given gas with known thermodynamic properties and a compressor of fixed size,

      (1.70)

      Hence, the nondimensional speed parameter is directly proportional to the Mach number.

      1.3.5.6 Mass Flow Parameter

The mass flow rate and density of a fluid are and ρ = p/RT. Combining these equations with the equation for the Mach number, we obtain

      or, rearranging,

(1.71)

      Now

      Also

      hence

      Combining this equation with Eqs. (1.67), (1.68), and (1.71) we obtain

      Finally, for the compressor inlet conditions

(1.72)

      Rearranging Eq. (1.72) yields

      (1.73)

      For a compressor of fixed size and constant fluid properties, the flow parameter is a function of the Mach number:

      (1.74)

      1.3.5.7 Applications of the Energy Equation

Simple nozzle or diffuser. A nozzle is a device that increases the velocity of a gas or vapour at the expense of pressure (Figure 1.11a). A diffuser is a device that increases the pressure of a gas or vapour at the expense of velocity (Figure 1.11b).

Figure 1.11 Schematic diagrams of a (a) nozzle; (b) diffuser.

Figure 1.12 The reciprocating internal combustion engine as a steady‐flow system.

      There is no work or heat transfer and negligible potential energy change in both systems. If the input kinetic energy is of considerable value (the gas is approaching the nozzle in Figure

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