Second, either the ideal square root or appropriated power law model presumes turbulent flow. For your orifice meter testing, you need to be sure that the flow rate remains in the turbulent region.
Third, turbulence-induced fluctuations on the dP signal create noise on the measured F (F appears measured because it is displayed, but F is actually calculated by the calibration equation and the dP sensor signal). The flow turbulence is greater at higher flow rates, but the square root functionality attenuates in the high F range. Accordingly, at low F, as F increases, the measurement noise will increase, but at high F, as F increases, the noise will decrease. Precision is reported as some measure of variability, But this value of precision would change with the flow rate value. You would need to characterize precision at several F values.
Fourth, you could quantify precision by experimental replicate testing or by propagating uncertainty. Use propagation of uncertainty when you are seeking to estimate the errors related to the pressure- and temperature-induced influences on fluid density—measured F. But you will need models that relate pressure and temperature to each source of variation.
Fifth, understand both aspects of measurement error: precision and accuracy. Both can be reported as rms values. Accuracy is measured by closeness to the true value. To report accuracy, you need to compare the "measured" F to a true value of F. To do this, calibrate the orifice meter, then compare the average of the noisy measured F value to a known standard. Even if accuracy is perfect at some flow rate, because the simple calibration equation will not exactly capture nature’s behavior over the entire range, the measured F will not represent the truth over the entire range. Accuracy will change with F. In contrast to accuracy, precision is related to repeatability, replicate variability or noise-induced fluctuations. To report precision, take many samples at one constant flow rate value and calculate the standard deviation of the replicate measurements.
Finally, after you have quantified either accuracy or precision, something will change. Someone will adjust the noise damping feature of the dP cell or the noise filter on the signal or calculated value, which will change the precision and lag associated with the measurement, or the orifice will erode, corrode, etc., or someone will adjust an upstream feature (thermowell insertion, bypass, etc.) which will shift the F to dP relation within the devices, making the calibration equation inaccurate.
With best practices for calibration, installation and maintenance, handbooks report orifice flowmeter error in the 3% to 5% range. They also indicate that 10% error is not to be unexpected. Accordingly, a major concern about flow rate measurement error might justify consideration of an alternate flowmeter type.
R. Russell Rhinehart