Simulate Heat Propagation in Vertical Water Column I have a vertical cylinder with water, along the height of that cylinder I have n temperature sensors. I would like to simulate the change in temperature measured by those sensors. Heat is going to be propagated by conduction and also convection (since cold water is heavier it will end up in the bottom of the cylinder).
Can anyone point me in the right direction? Which equations should I take into account?
P.S. I am assuming the cylinder is completely isolated from the outside world and the walls of the cylinder don't transfer heat.
 A: Since you anticipate having both convective heat transfer generated by fluid motion due to differential buoyancy as well as diffusive heat transfer (aka conduction), you are going to have to solve both a momentum and an energy equation. Assuming the kind of parameter range typical for water as the fluid under reasonable pressures and temperatures, it's likely that the Boussinesq aproximation will be sufficient for your purposes. Let us know if you need information beyond what's in the Wikipedia article.
A: You don't have enough information to tell, but the situation you're describing is called free convection, when the only force acting on the water is the buoyancy created by a temperature difference. For a non-trivial solution, I need to assume that there is some initial unstable temperature imbalance. Namely, that the hot starts at the bottom. Similar situations can be created by heat transfer from the cylinder, but you stated explicitly that this is not the case.
A analytical solution would involve calculating a bunch of non-dimensional constants, with the end goal of characterizing the heat transfer of the system with the Nusselt number. This number is key to understanding how your system is going to behave. The other constants are Reynold's Number, $$Re = \frac{density \cdot characteristic \space velocity \cdot characteristic \space speed }{dynamic \space viscosity},$$ Prandtl number, $$ Pr = \frac{specific \space heat \cdot viscosity}{thermal \space conductivity}, $$ for free convection, there an extra one called the Grashof number $$Gr = \frac{g \cdot \beta (T_s - T_{\inf}) \cdot L^3}{\nu^2}, $$ $\beta$ tells you how much a given fluid expands when it's heated, $\nu$ is dynamic visocity, L is charactaristic lenth. The last two can be combined to for the Rayleigh number $$ Ra = Gr \cdot Pr$$ If you match all these numbers with the right equation (the closest one I can find is free convection between two symmetrically heated, isothermal, vertical plates), you will get an equation for the Nusselt number. $$ Nu = \frac {1}{24} Ra \frac {S}{L} (1 - exp(\frac {-35}{Ra \frac {S}{L}}))^{3/4}.$$ This empirical relationship should be reasonably close to your situation.
That's the analytical solution, but that all seems a litle bit off topic considering your question. If you were looking for someone to steer you in the right direction, I think a study of what all these numbers mean would be appropriate.
If all of that seems a little overwhelming and you have a big processor on your computer, you could always just simulate the system numerically. This would probably be the closest to answering your question, because you have already discretized the system with your n measurement devices.
