We have two devices, A and B, with different pipe diameters, lengths, and bend angles through which water flows. We want to compare the differences in flow rate and pressure drop between device A and device B by inserting an orifice plate at specific positions in each device and measuring the flow rate and differential pressure. A pressure gauge C and a flow meter are installed before the orifice plate, and a pressure gauge D is installed after the orifice plate. The pipe diameter at the location where the orifice plate is installed differs between device A and device B, but the diameter of the orifice plate itself is the same. By measuring the difference in pressure values before and after the orifice plate (C - D), we can calculate the differential pressure (P) and determine the absolute flow rate value Q (in L/min) from the flow meter. The measurement positions' heights are the same for both device A and device B.

As much as possible, it is acceptable to approach it from an ideal standpoint.

Now, here are the questions: Can the relationship between the differential pressure and flow rate be expressed as Q = kP using Bernoulli's theorem? Here, k represents a device-specific proportionality constant. To compare devices A and B, should we compare the differential pressure using a common flow rate reference instead of comparing P? What can we understand from this differential pressure comparison? For example, if P in device A is smaller than P in device B, can we conclude that device B has a higher pipe friction resistance and device A has a more favorable pipe structure for flow? If we change the orifice diameter in the same device, I believe that P would vary even with the same Q. In the case of the same device, is it acceptable to compare the differences in Q based on P? Water's density () changes with temperature. Can we incorporate this density variation into the equation Q = kP? I would like to know the relationship between Q, P, and , including the calculation process leading to that relationship.