Stagnation properties and Mach number relations

The stagnation (total) state is the state reached when a flowing fluid is brought to rest adiabatically (and isentropically for the pressure). The static properties $T$, $p$ are those measured moving with the fluid.

Stagnation temperature

From the steady flow energy equation for adiabatic deceleration to rest:

$$h_0 = h + \frac{C^2}{2} \;\Rightarrow\; c_p T_0 = c_p T + \frac{C^2}{2}$$ $$T_0 = T + \frac{C^2}{2c_p}$$

Using $c_p = \dfrac{\gamma R}{\gamma - 1}$ and $a^2 = \gamma R T$, so $C^2 = Ma^2 \gamma R T$:

$$\boxed{\frac{T_0}{T} = 1 + \frac{\gamma - 1}{2}Ma^2}$$

Stagnation pressure

For isentropic deceleration, $p_0/p = (T_0/T)^{\gamma/(\gamma-1)}$:

$$\boxed{\frac{p_0}{p} = \left(1 + \frac{\gamma - 1}{2}Ma^2\right)^{\gamma/(\gamma-1)}}$$

At $Ma = 1$ (critical state): $T^*/T_0 = 2/(\gamma + 1) = 0.833$ and $p^*/p_0 = 0.528$ for air ($\gamma = 1.4$).

Points to note

• In adiabatic flow without work, $T_0$ stays constant along a nozzle even with friction; $p_0$ falls because of friction (entropy rise).
• In a compressor or turbine, $T_0$ changes by the work done, so $w = c_p \Delta T_0$.
• For small Mach number, $p_0 \approx p + \tfrac12\rho C^2$.

Why used

1. The kinetic energy is large in gas turbines ($Ma$ of 0.3 to 1 or more), so static and total values differ greatly.
2. A thermocouple or pitot tube measures nearly total values, and performance (efficiency, pressure ratio) is defined on a total basis, e.g. $\eta_{tt}$.
3. Total conditions are independent of the flow velocity, so they give a common reference at each engine station (inlet, compressor exit, turbine exit).