13.2 Isentropic Flow

Isentropic Flow

Isentropic flow definition and assumptions

Isentropic flow describes an idealized gas flow that is both reversible and adiabatic. "Adiabatic" means no heat is exchanged with the surroundings, and "reversible" means no entropy is generated by friction, viscosity, or turbulence. Together, these conditions keep entropy constant throughout the flow, which is where the name "isentropic" comes from.

Beyond the reversible-adiabatic requirement, the standard assumptions are:

• Steady, one-dimensional flow of an ideal gas with constant specific heats ($c_p$ and $c_v$ don't change with temperature)
• No shaft work from turbines or compressors
• Gravity effects are negligible

These assumptions sound restrictive, but they apply surprisingly well to real flows through nozzles and diffusers where friction losses are small and the process happens fast enough that heat transfer is minimal.

Derivation of isentropic flow relations

The isentropic relations come from combining the first law of thermodynamics (conservation of energy) with the ideal gas law and the definition of an isentropic process ($Pv^{\gamma} = \text{const}$, where $\gamma = c_p / c_v$ is the specific heat ratio).

The three core relations link pressure $P$, density $\rho$, and temperature $T$ between any two points (1 and 2) in the flow:

• Pressure–density–temperature ratio:

$\frac{P_2}{P_1} = \left(\frac{\rho_2}{\rho_1}\right)^{\gamma} = \left(\frac{T_2}{T_1}\right)^{\frac{\gamma}{\gamma-1}}$

• Density ratio (equivalent forms):

$\frac{\rho_2}{\rho_1} = \left(\frac{P_2}{P_1}\right)^{\frac{1}{\gamma}} = \left(\frac{T_2}{T_1}\right)^{\frac{1}{\gamma-1}}$

• Temperature ratio (equivalent forms):

$\frac{T_2}{T_1} = \left(\frac{P_2}{P_1}\right)^{\frac{\gamma-1}{\gamma}} = \left(\frac{\rho_2}{\rho_1}\right)^{\gamma-1}$

Notice these are not three independent equations. They're all rearrangements of the same underlying relation $P \propto \rho^{\gamma}$. If you know any one ratio (say $P_2/P_1$), you can find the other two.

Calculation of isentropic flow properties

Using the relations above in practice follows a straightforward pattern:

1. Identify known quantities at a reference point (often the stagnation state, where velocity is zero). You need at least two independent properties (for example, $P_0$ and $T_0$).
2. Determine the property ratio between the reference point and the point of interest. This usually comes from a Mach number relation or a given pressure/temperature measurement.
3. Apply the isentropic relation to find the unknown property.

For example, if you know the pressure and temperature at a point, you can find density directly from the ideal gas law:

$\rho = \frac{P}{RT}$

where $R$ is the specific gas constant for the working fluid (for air, $R \approx 287 \, \text{J/(kg·K)}$).

This approach lets you map out pressure, density, and temperature throughout a nozzle or diffuser without needing to solve the full Navier-Stokes equations.

Nozzle behavior in isentropic flow

How a nozzle affects the flow depends on its geometry and whether the flow is subsonic or supersonic.

Converging nozzles:

1. As the cross-sectional area decreases, flow velocity increases.
2. Pressure, density, and temperature all decrease along the flow direction.
3. The maximum velocity is reached at the throat (the point of minimum area). For a purely converging nozzle, the throat is the exit, and the flow there can reach at most Mach 1.

Diverging sections (downstream of a throat):

The behavior here depends on the pressure ratio across the nozzle:

1. Subsonic flow (back pressure is not low enough to reach the critical ratio): velocity decreases in the diverging section while pressure, density, and temperature increase. The diverging section acts as a diffuser.
2. Supersonic flow (back pressure is low enough): velocity continues to increase past the throat while pressure, density, and temperature continue to decrease. This is how converging-diverging (de Laval) nozzles accelerate flow to supersonic speeds.

The transition between these regimes is governed by the critical pressure ratio:

$\frac{P^*}{P_0} = \left(\frac{2}{\gamma+1}\right)^{\frac{\gamma}{\gamma-1}}$

Here $P^*$ is the pressure at the throat when the flow is exactly sonic (Mach 1), and $P_0$ is the stagnation pressure. For air ($\gamma = 1.4$), this ratio works out to approximately 0.528. If the back pressure drops below $P^*$, the throat becomes choked at Mach 1 and the mass flow rate reaches its maximum for that stagnation condition.