alt="equation"/>
1.3.5.3 Stagnation Properties
Stagnation properties are those thermodynamic properties a flowing compressible fluid would possess if it were brought to rest adiabatically and reversibly, i.e. isentropically, and without heat and work transfer. The stagnation state is a convenient hypothetical state that simplifies many of the equations involving flow by taking account of the kinetic energy terms in the steady flow energy equation implicitly.
The stagnation enthalpy ht is the enthalpy that a gas stream of enthalpy h and velocity C would possess when brought to rest adiabatically and without work transfer. The energy equation thus becomes
(1.63)
For a perfect gas, h = cpT and the corresponding stagnation temperature Tt is
(1.64)
Applying the concept of stagnation properties to an adiabatic compression, the energy Eq. (1.60) becomes
Rearranging, we get
(1.65)
Temperature‐measuring devices such as thermometers and thermocouples in reality measure the stagnation temperature of the flow and not the static temperature. Thus, introduction of stagnation temperatures simplifies solving the energy equation by eliminating the kinetic energy term and the need to measure flow velocity.
The stagnation pressure pt is defined as the pressure the gas stream would possess if the gas were brought to rest adiabatically and reversibly. Using Eqs. (1.48) and (1.64), pt can be written as
(1.66)
1.3.5.4 Isentropic Flow
Examples include flow in ducts, nozzles, and diffusers without heat transfer and work being done. Knowing
we obtain
Also,
Combining the last two equations, we get
(1.67)
From Eqs. (1.48) and (1.67)
(1.68)
The pressure ratio for Ma = 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 pt > 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
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
or, rearranging,
(1.71)
Now
Also
