The isentropic flow function is a useful and straightforward way to determine the massflow rate of a compressible fluid if other of the fluid's basic properties are known. In general, the massflow rate of a fluid through a cross-sectional area $A$ is equal to
$\dot{m}=\rho VA$.
Now, if the fluid is compressible and the Ideal Gas Law applies, then
$\dot{m}=\rho VA=\left(\frac{P}{RT}\right)(M\sqrt{\gamma RT})A=PAM\sqrt{\frac{\gamma}{RT}}$.
Both the stagnation temperature and stagnation pressure are preferred flow variables to their static counterparts, so the above equation can be rewritten as
$\dot{m}=P_0 \left(\frac{P}{P_0}\right)AM\sqrt{\frac{\gamma (T_0/T)}{R(T_0)}}$,
and the stagnation properties (as well as the through-flow area) can be moved to the LHS of the equation:
$\frac{\dot{m}\sqrt{T_0}}{P_0 A}=\left(\frac{P}{P_0}\right)M\sqrt{\frac{\gamma}{R}\left(\frac{T_0}{T}\right)}$
If the flow is isentropic (as we are assuming), we know that
$\frac{P}{P_0}=\left(\frac{P_0}{P}\right)^{-1}=\left(\frac{T_0}{T}\right)^\frac{\gamma}{1-\gamma}$,
which gives us
$\frac{\dot{m}\sqrt{T_0}}{P_0 A}=M\sqrt{\frac{\gamma}{R}}\left(\frac{T_0}{T}\right)^{\frac{1}{2}+\frac{\gamma}{1-\gamma}}=M\sqrt{\frac{\gamma}{R}}\left(\frac{T_0}{T}\right)^{\frac{1+\gamma}{2(1-\gamma)}}$.
Again invoking our assumption of isentropic flow, we know that the stagnation temperature ratio is related to the local Mach number by the following equation:
$\frac{T_0}{T}=1+\frac{\gamma-1}{2}M^2$
which, when plugged into the previously derived expression gives us the isentropic flow function $FF_T$:
$FF_T=\frac{\dot{m}\sqrt{T_0}}{P_0 A}=M\sqrt{\frac{\gamma}{R}}\left(1+\frac{\gamma-1}{2}M^2\right)^{\frac{1+\gamma}{2(1-\gamma)}}$
To compute the massflow rate we simply rearrange the isentropic flow function relation...
$\boxed{\dot{m}=P_0 AM\sqrt{\frac{\gamma}{RT_0}}\left(1+\frac{\gamma-1}{2}M^2\right)^{\frac{1+\gamma}{2(1-\gamma)}}}$.
The local volumetric flow rate $\dot{V}$ can be found by dividing the local massflow rate by the static density:
$\boxed{\dot{V}=\frac{\dot{m}}{\rho}=MA\sqrt{\gamma RT_0}\left(1+\frac{\gamma-1}{2}M^2\right)^{-\frac{1}{2}}}$
Stagnation temperature can be found via a minimally-invasive thermocouple or RTD, while the Mach number can be derived from pitot-static flow measurements. If implemented correctly, they will give an accurate calculation of the flow rates without altering the system at all. The actual area will probably need to be modified to account for the effect of viscous boundary layers, in which case $A$ will be replaced with $A_e$, the "effective" through-flow area. 707