I am trying to design a ducted axial fan. The fan will be pulling air through a filter, and ejecting out into the atmosphere (For a CO2 direct air capture application). However I want to start with the physical parameters (blade diameter/hub diameter/airfoil profile/AOA/rpm etc). While varying these parameters I am trying to predict flow rate and static pressure. I think I understand how to determine the static pressure based on the thrust provided by the blades, and thus generate a pressure-rpm curve, but what I don't understand is how to determine volume flow rate theoretically. Is it simply a matter of a blade velocity diagram (where I would only know the tangential velocity for a given rpm)? Or is there a way of finding flow rate/axial velocity based on the pressure? I can not find an answer to this in the usual equations for fan design. I am essentially trying to find the axial velocity and static pressure provided by a fan based purely on the physical characteristics of the fan. I intend on validating the design using CFD afterwards. Thank you.
For unducted fans a relatively simple very fast program to estimate the performance of a fan or rotor is at https://au.mathworks.com/matlabcentral/fileexchange/21994-analysis-of-a-rotor-blade-system-using-blade-element-momentum-theory. It is designed for helicopters without blade twist or variable chords but could probably be modified to accommodate twist. The output array out1 contains Thrust Generated, Power Required, Figure of Merit, so divide the thrust by the disc area to get pressure, and power/thrust is an upper bound of the average flow speed.
However for ducted fans the effect of the duct and blade tips interacting is far more complex and there is no simple solution. here's a plot of the Figure of Merit vs rpm for 3 tip clearances and an open rotor. This was taken from https://www.sciengine.com/SCTS/doi/10.1007/s11431-021-2110-x;JSESSIONID=7905c56c-4cdb-4bd9-9cc6-e640d30beae2
Flow rate = V_induced * Area of the duct (at the rotor plane)
V_induced can be calculated using Momentum (or combined Blade Element Momentum Theory, BEMT), for open rotors. (this reference uses V0 notation, and gives a method to iteratively find induced (in-rotor-plane) velocity.)
Effect of the duct can be approximated using empirical methods. [I came across this fantastic paper about testing ducted fans]
Be warned that your "filter" will be affecting the results dominantly. Of course, you can compare your various fan designs (parameter sets).
For validation, you'll probably find the optimum parameters after building and testing a few (or several) fans.