Yes, your observation is correct. There is a relation between fluid flow rate and heat transfer.

The flow rate is determined by Reynold's number (Re). If Re < 2300 then flow remains laminar; if 2300< Re < 4000 then it's in transition state and when Re > 4000 then it's turbulent. In the case of turbulent flow (assuming incompressible, steady-state and fully developed), there is an additional mechanism for heat transfer provided by radial direction due to eddy currents. Thus the rate of heat transfer increases for high Re. On the other hand, the transfer of heat is only in the transverse direction in the case of laminar flow (assuming the tube to be uniform which does not have any obstructions and curvature).

Nusselt number (Nu) is primarily used to correlate heat transfer in the laminar flow where Nu is the function of Re and Prandtl number (Pr). The Pr is the ratio of the amount of mass transfer to the heat or energy transfer. For the gases, the ratio is in the neighbourhood of 1 which means heat transfer is directly proportional to fluid flow rate. For water, Pr is around 5 which means the mass transfer is 5 times faster than heat transfer. For highly viscous oil, Pr is around 100.

In various studies, it's found that heat transfer is most efficient in the thermal entrance region (boundary layer) where the viscosity of the fluid is different from the viscosity of the fluid at the axis.

Well, there is a huge literature available on this topic. I have tried my best to give you a glimpse of the same. I hope it will be useful for you.

For laminar flow the relationship between flow rate and viscosity is called the Poiseuille equation : https://en.wikipedia.org/wiki/Hagen%E2%80%93Poiseuille_equation

This equation contains the physical property, dynamic viscosity. The dynamic viscosity of water as a function can be looked up: e.g. https://www.engineeringtoolbox.com/water-dynamic-kinematic-viscosity-d_596.html

or if you want an equation try page 11 of this

https://resources.saylor.org/wwwresources/archived/site/wp-content/uploads/2011/04/Viscosity.pdf

There is another factor apart from viscosity: density of the liquid as a function of temperature.

If you just consider Bernoulli's equation (u2=2(p1-p0)/rho) and assume a fixed pressure difference, you'll notice that the velocity increases as density (rho) decreases. This leads to a higher volumetric flow rate (u~rho-1/2) but to a lower mass flow rate (rho·u~rho+1/2).

In practice, things can be a bit more complicated because changes in density and viscosity might both lead to changes in velocity profile and turbulence levels, but since both viscosity and density decrease as temperature increases, the tendency will be towards higher velocities and higher volumetric flow rates.

The influence of density mentioned above is mainly relevant for turbulent flows. The Hagen-Poisseuille (for purely laminar flow) equation does not include density; it depends only on viscosity.

Finally, as a last thought: As temperature goes up, so does vapor pressure. Which means that cavitation becomes more likely, especially near the inlet to the pipe. Occurence of cavitation could lead to a lower effective cross section or lower effective density and thus reduce flow rate. The exact effect will depend on the specific details of the situation.

Concerning the density of water, the special detail should be kept in mind that (at 1 atmosphere) density actually increases between 0°C and 4°C and only then starts to decrease as temperature increases. The effect of temperature on is not particularly large, e.g. there are many oils for which density (and viscosity) decrease to a much larger extent as temperature goes up.

Unless you are limited to laminar flow, you'll not be able to find "neat analytical solutions" to your question.

But you can start by collecting data about your liquid: How do viscosity and density depend on temperature?

For laminar flow, use the Hagen-Posseuille equation; for turbulent flow, use the well-known power law as a first step, with tabulated values for the exponent (1/n) and the friction factor fD. If you put all this together, you'll develop a feeling for how the temperature dependency of viscosity and density affect flow rate. When building your model, be careful as the tabulated data will be given as a function of Reynolds number, which depends both on your inputs (dynamic viscosity, density) but also on velocity, which is the output (result) you are looking for.

Unfortunately, there are no easy answers once turbulence sets in. The amount of effort needed increases with the accuracy you are looking for. How exactly temperature affects flow rate will depend on details you have not specifed, e.g. the length of the pipe, the roughness of its walls and the shape of the pipe inlet.

I do not agree with Leick when he says:

There is another factor apart from viscosity: density of the liquid as a function of temperature.

Morelife Zibayiwa is talking about water.

Now, from 80 to 0°C the dynamic viscosity coefficient changes with a factor of about 473% while the density by a factor of less than 3%.

Which is the parameter that mostly affects the Reynolds number which is crucial in the determination of the Darcy friction factor that regulates the flow rate of water flowing through a pipe of uniform diameter?

Think it over!

Regards

Giovanni Maria Carlomagno I am really thankful to this point of view which add on to the emphasis almost every professional is indicating: the significance of Reynolds number to the flow. In this regard; let me further highlight my objective over this research. I would like to come up with a collective formula which relates the Hall effect sensor voltage frequency to the function of flow rate, in which flow rate is determined by some physical variables such as liquid temperature, pressure, medium friction and of course viscosity and density. I thank you

Your formula depends on if your flow is laminar, transitional or fully turbulent and if your pipe is smooth or rough. In fully turbulent flow the friction factor is independent on Reynolds number for rough pipes. Look to the Moody diagram.

Regards

Great!!! Thank you for the insights, I think I am better equipped to carry out the lab work. I will publish the data from a number of experiments, conclusions will be better made here. I thank you