Thermal mass flow meter

I am trying to build a thermal mass flowmeter, but I am facing some challenges describing it theoretically. I have an aluminium pipe with a heating element. I am measuring the temperatures of the pipe before and after the heating element. The temperature difference should be in some correlation to the flowrate. However, the temperature difference is different while the flowrate is the same when the temperature of water changes. This means that the transfer of heat to the fluid changes with its temperature.
I found this paper(Laub 1957, MEASURING MASS FLOW with the boundary-layer flowmeter, page 25 of the pdf), where the author describes some theory behind the sensor. However the type of sensor described changes the power of the heater to maintain constant temperature difference and no kind of inlet temperature compensation was mentioned... I modified the equations described in the paper to calculate flowrate.
$$P$$ is the heater input power, $$D$$ is the pipe inner diameter, $$L$$ is the heated segment length, $$k$$ is the fluid thermal conductivity, $$c$$ is the fluid thermal capacity and $$A$$ is the cross sectional area of the heated segment. $$P=hA\Delta T \\ h=2.34\frac{k^{2/3}c^{1/3}}{(DL)^{1/3}}\dot{V}^{1/3} \\ \dot{V}=\sqrt[3]{\frac{P}{0.023\frac{k^{2/3}c^{1/3}}{(DL)^{1/3}}A\Delta T}}$$ How can I add the inlet temperature into the equation?
Thank you for any help.

• The heat transfer rate is directly dependent on the temperature difference between your heating element and the inlet water temperature. In addition, notice that your equation for $h$ is empirical. This is no accident ... heat transfer often involves turbulence, which is STILL a phenomenon that no one can model mathematically. This means that you may well need to take measurements and fit a different equation to your data. Commented Feb 13, 2022 at 1:11
• The equation you gave doesn't seem to match the equation given in the paper. The leading coefficient for the low Re case is not 0.023. Also, the delta T in this equation is the temperature difference between the wall and the mean fluid temperature. Also, there is an additional heat balance equation that you are missing giving the temperature change between inlet and outlet of the heater. Commented Feb 13, 2022 at 1:17