Isentropic Flow

Overview

Engineering reference data for Isentropic Flow in fluid mechanics.

Key Formulas

Reynolds Number

$Re = \frac{\rho v D}{\mu}$

Ratio of inertial to viscous forces — determines flow regime.

Bernoulli's Equation

$P + \frac{1}{2}\rho v^2 + \rho g h = \text{const}$

Conservation of energy for steady, inviscid, incompressible flow.

Continuity Equation

$A_1 v_1 = A_2 v_2$

Conservation of mass for incompressible flow.

Darcy-Weisbach

$\Delta P = f \frac{L}{D} \frac{\rho v^2}{2}$

Pressure drop due to friction in a pipe.

Variables

Symbol	Description	Unit
$Re$	Reynolds number	—
$\rho$	Fluid density	kg/m³
$v$	Flow velocity	m/s
$D$	Characteristic dimension	m
$\mu$	Dynamic viscosity	Pa·s
$P$	Pressure	Pa
$f$	Darcy friction factor	—

Derivation of Isentropic Relations

The condition for isentropic flow (ds = 0) is derived from the general entropy equation for a compressible fluid. For an ideal gas, this simplifies to:

$$ds = c_v \ln\left(\frac{T_2}{T_1}\right) + R \ln\left(\frac{\rho_1}{\rho_2}\right) = c_p \ln\left(\frac{T_2}{T_1}\right) - R \ln\left(\frac{p_2}{p_1}\right) = 0 \tag{1}$$

Using the specific heat ratio $\kappa = \frac{c_p}{c_v}$ and the ideal gas relation $R = c_p - c_v$, equation (1) can be rearranged to establish the fundamental relationships between properties for an isentropic process:

$$\left(\frac{T_2}{T_1}\right)^{\frac{\kappa}{\kappa-1}} = \left(\frac{\rho_2}{\rho_1}\right)^{\kappa} = \frac{p_2}{p_1} \tag{3}$$

From this, two common forms of the isentropic relationship are obtained:

Pressure-Density Relation:

$$\frac{p}{\rho^\kappa} = \text{constant} \tag{4}$$

Pressure-Specific Volume Relation: Using $\rho = 1/v$ (where $v$ is specific volume, m³/kg), this becomes:

$$p v^\kappa = \text{constant} \tag{6}$$

These equations (3, 4, 6) describe the essential state-to-state property relationships for an isentropic process in an ideal gas.

References

• Original Source