Conversion between Power and Flow rate of Fan taking into consideration the geometry of the fan I need to find the relation between two properties of a fan, that is between power and flow-rate. I expect such a formula to depend on the pressure and on the geometry of the fan. 
I have found in here: https://www.axair-fans.co.uk/news/applications/understanding-basic-fan-laws/  a relation between rotational velocity of a fan with each of pressure, power and flow rate. Then following those relations, a relation between power and flow rate could be build easily. But this relation is unuseful since I don't need to convert between two states of each same property. 
By the way, the kind of fan is one with slots or holes. I just need the method of building such a relation no matter how the fan looks like then I could apply that to the fan I have. 
Thank you
 A: I calculated the flow parameters through the oval opening of the air purifier Dyson pure cool . I wondered how the air flow is created through the opening when blowing a jet through a thin slot around the perimeter of the oval. I compiled a mathematical model and wrote code for Mathematica 12. Figure 1 shows the flow lines and the magnitude of speed when blowing jets on the perimeter of the oval (shows one section across the oval).

The second question that interested me was how the stream penetrates the filter. Here is a simple example where a stream is filtered through a thin channel system. Figure 2 shows the magnitude of velocity (left), the velocity profile at the outlet (in the center) and the longitudinal component of velocity in a  flow behind the filter.

Finally, the third question is how air is distributed through the ventilation system. Fig. 3 shows a ventilation duct with several outputs: at the top, the distribution of velocity magnitude is shown, in the center these are velocity profiles in each branch, and below is the pressure distribution in several sections.

To maintain air circulation in the system, it is necessary to install a fan of a certain power, which is able to develop the required pressure and provide the required air flow. The standard formula relating fan power ($P_e$), air flow rate ($q_v$), pressure drop ($P_r$) and system efficiency ($\eta $) is
$$P_e=\frac {q_v P_r}{\eta}$$
The international standard uses specific fan power (SFP):
$$SFP=\frac {\sum P_e}{q_v}$$
The efficiency of converting electrical power to pressure and air flow rate depends on the type of fan:
$$\eta =\frac {P_{out}}{P_{in}}=\eta _{vsd} \eta _{motor} \eta _{transf} \eta _{fan}$$
$P_{out}$ is useful output of the fan in [W]=$\Delta P_rq_v$, $\Delta P_r$ is pressure rise in [Pa], $q_v$ is flow rate in $m^3/s$;
$P_{in}$ is power input [W];
$\eta _{vsd}$ is variable speed drive efficiency;
$\eta _{motor}$ is motor efficiency; 
$\eta _{transf}$ is power transfer efficiency;
$\eta _{fan}$ fan aerodynamic efficiency.    
Blade speed and pressure distribution (my calculations for viscous flow) 
 
Fan efficiency depends on flow and diameter. In fig. Figure 5 shows the dependence of the efficiency of five types of fans on $q_v$ and diameter.

In most cases, the pressure in the system decreases with increasing flow rate, but the efficiency increases, so the power increases with increasing flow rate. Typical dependencies are shown in Fig. 6. for a real fan (left) and in theory (right)

The theory that I am developing is very simple. It is assumed that the pressure drop in the system is related to the flow rate by a quadratic dependence:
$$\Delta p=p_0(1-aq_v^2)$$
In fig. 7 shows the dependence of fan pressure on flow rate for several types of fans.

There are two types of system efficiency - with reaching a constant as in Fig. 5 or with a local maximum as in Fig. 6:
$$\eta =\frac {kq_v}{1+bq_v+cq_v^2}$$
Then the electric power depends on the flow rate as
$$P_e=\frac { q_v\Delta p}{\eta}=(p_0/k)(1-aq_v^2)(1+bq_v+cq_v^2)$$
Here the parameters $a, b, c, p_0/k$ depend on the geometry of the fan.This dependence is shown in Fig. 6. Qualitatively, this dependence is consistent with experiment.
A: the power delivered by the fan blades to the air moving through it will be equal to the mass flow rate of air through the fan times the pressure difference across the fan. it will also be equal to the power delivered by the motor times the efficiency of the fan. 
the power delivered by the motor to the fan will be equal to the torque developed by the motor times its RPM. it will also be equal to the power delivered by the AC mains times the efficiency of the motor. 
the power delivered by the AC mains to the motor will be equal to the current flow through the motor times the mains voltage. 
A: From the link you posted you can derive the equation $$\frac{P_2}{P_1}=\left(\frac{\dot V_2}{\dot V_1}\right)^3$$ where I have changed their notation slightly. I will use $P$ for power and $p$ for pressure, and $\dot V$ for the volumetric flow rate and $V$ for volume and $v$ for flow velocity, and $A$ for area.
As you said, you do not want to compare two points, so we can collect terms to get $$P=k \ \dot V^3$$ which is your desired relationship between power and volumetric flow rate.
Now, $k$ is a characteristic of the fan, and must be calculated or measured at some given power level for the fan. Ideally, a smaller $k$ will mean less power for a given volumetric flow rate, so we would like a fan that minimizes $k$. Generally this will be difficult to calculate theoretically, but it should be something that can be determined by the specifications for any given fan. 
However, since you are specifically interested in the geometry, we can do a little rearranging to get a sense of how the geometry may affect things: $$k=\frac{P}{\dot V^3}=\frac{p\ \dot V}{\dot V^3}=\frac{p}{\dot V^2}=\frac{p}{(A\ v)^2}$$
So, to minimize $k$ you want to have a low pressure and high velocity over a large area. However, realistically pressure and velocity and area are all related, so you cannot usually just increase one without affecting the others. However, this does give you a general idea that increasing the area will reduce the power for a fixed flow rate.
