Isentropic flow is a key concept in fluid mechanics, describing ideal gas behavior without heat transfer or friction. It's crucial for understanding how pressure, density, and temperature change in various flow situations, from nozzles to diffusers.

By applying conservation of energy and ideal gas laws, we can predict flow properties at different points. This helps engineers design efficient systems and analyze complex flow scenarios, making isentropic flow a fundamental tool in fluid dynamics.

Isentropic Flow

Isentropic flow definition and assumptions

• Occurs without heat transfer or friction involves a reversible adiabatic process (no heat exchange with surroundings) and no dissipative effects (viscosity or turbulence)
• Assumes steady, one-dimensional flow of an ideal gas with constant specific heats
• No shaft work (turbines or compressors) or heat transfer occurs during the flow process
• Neglects the effects of gravity on the flow

Derivation of isentropic flow relations

• Derived from conservation of energy and ideal gas law principles
• Pressure ratio $\frac{P_2}{P_1} = \left(\frac{\rho_2}{\rho_1}\right)^{\gamma} = \left(\frac{T_2}{T_1}\right)^{\frac{\gamma}{\gamma-1}}$ relates pressure $P$, density $\rho$, temperature $T$, and specific heat ratio $\gamma$ between two points
• Density ratio $\frac{\rho_2}{\rho_1} = \left(\frac{P_2}{P_1}\right)^{\frac{1}{\gamma}} = \left(\frac{T_2}{T_1}\right)^{\frac{1}{\gamma-1}}$ expresses the relationship between density, pressure, and temperature
• Temperature ratio $\frac{T_2}{T_1} = \left(\frac{P_2}{P_1}\right)^{\frac{\gamma-1}{\gamma}} = \left(\frac{\rho_2}{\rho_1}\right)^{\gamma-1}$ connects temperature changes to pressure and density variations

Calculation of isentropic flow properties

• Isentropic relations enable calculation of pressure, density, and temperature at different flow points
• With two known properties at a point, the third can be found (if pressure and temperature are given, density is $\rho = \frac{P}{RT}$ where $R$ is the specific gas constant)
• Allows determination of flow conditions throughout an isentropic process (nozzles, diffusers)

Nozzle behavior in isentropic flow

• Converging nozzles:
  1. Flow velocity increases as cross-sectional area decreases
  2. Pressure, density, and temperature decrease along flow direction
  3. Maximum velocity reached at nozzle throat (minimum area)
• Diverging nozzles:
  1. Flow behavior depends on pressure ratio across nozzle
  2. Subsonic flow (pressure ratio < critical value): velocity decreases, pressure, density, and temperature increase along flow
  3. Supersonic flow (pressure ratio > critical value): velocity increases, pressure, density, and temperature decrease along flow
• Critical pressure ratio $\frac{P^}{P_0} = \left(\frac{2}{\gamma+1}\right)^{\frac{\gamma}{\gamma-1}}$ determines flow regime, $P^$ is throat pressure, $P_0$ is stagnation pressure