It is not entirely clear from the question, where the instrument readings take place. The following equations might need to be adapted in order to reflect the sensor positions.
The basic equation for the mass-flow $\dot{m}$ is:
$\dot{m} = A \cdot \rho \cdot v$
$A$, $\rho$, and $v$ need to be evaluated or measured at the same location.
The ideal gas law can be used here but be aware the static values for temperature $T$ and pressure $p$ have to be used here:
$\rho = \frac{p_\mathrm{s}}{RT_\mathrm{s}}$
The specific gas constant for air is something around $287.058\frac{J}{kgK}$. It changes with temperature but this does not seem to be the problem right now.
Usually the problem is now to compute $T_\mathrm{s}$.
The following equations use an isentropic process to compute first the Mach-Number and then the static Temperature. Step 1 is to solve the equation for $M$:
$\frac{p_\mathrm{s}}{p_\mathrm{t}} = \left( 1 + \frac{\gamma - 1}{2} M^2\right)^{\frac{\gamma}{\gamma-1}}$
Here $\gamma$ is the isentropic coefficient which is around $1.4$ for air depending on the humidity. $p_\mathrm{t}$ and $p_\mathrm{s}$ are the static and total pressure. Their relation is:
$p_\mathrm{t} = p_\mathrm{s} + p_\mathrm{d}$
$p_\mathrm{d}$ is often calculated with $\frac{1}{2}\rho u^2$ this simplification creates an error in the density $\rho$ of 5% for a Mach-Number $M=0.3$. So it is possible with the result of step 1 to check if this was the source of the problem.
With the calculation of the Mach-Number the isentropic relation of total and static temperature can be used to calculate the static temperature in step 2:
$\frac{T_\mathrm{s}}{T_\mathrm{t}} = \left( 1 + \frac{\gamma - 1}{2} M^2\right)^{-1}$
Here the total temperature $T_\mathrm{t}$ is the temperature of the gas in the pressure supply.
The final step 3 is the calculation of the velocity $v$. This will be done using the equation for the Mach-Number:
$M = \frac{u}{a}$
Here the only unknown is $a$ (the speed of sound) which can be computed using the following relation:
$a = \sqrt{\gamma R T_\mathrm{s}}$
Doing these calculations by hand is laborious but it's easy to set up a script or sheet to do these calculations. There are three constants:
- Through flow area, $A$
- Specific gas constant, $R$
- Isentropic coefficient, $\gamma$ (sometimes its $\kappa$)
And there are three measurement values:
- Total pressure, $p_\mathrm{t}$
- Static pressure, $p_\mathrm{s}$
- Total temperature, $T_\mathrm{t}$