Overall heat transfer coefficient for free convection exchanger

[Engn_sam]

I am trying to design a heat exchanger for a stratified hot water tank which means that the heat transfer on that side of the exchanger will be totally dependent on free convection. I plan on using a shell and tube set-up but I have no idea on how to calculate the required mass flow rate from the hot side since the mass flow from the cold side is supposed to be governed by free convection. If I could find a way to calculate the overall heat transfer coefficient in that situation, I would be able to calculate the required mass flow rate.
Normally, I would just set the mass flow rate and boundary temperatures (T_in and T_out) and use that to calculate the required heat transfer rate (Q) using

Q = mass flow * specific heat capacity * ΔT​

then use the Q and the boundary temperatures of the other side to calculate the mass flow rate of the other side.
I got lost after realising that the only quantities I can set are the inlet and out let temperatures from the side and the inlet temperature from the hot water source, Since the mass flow of the tank side is solely determined by free convection, I need a way to estimate the product of overall heat transfer coefficient that the temperature difference in the tank can sustain; the temperature difference determines the density differences so I assume it would also determine the buoyancy which determines, in conjunction with the heat capacity, the max heat rate that can be sustained.

thank you

 

[insightful]

If I could find a way to calculate the overall heat transfer coefficient [U₀] in that situation, I would be able to calculate the required mass flow rate.

This is very difficult. With water both inside and outside the tubes, a reasonable first guess for U₀ is 1000 W/(m²K).

 

[Engn_sam]

This is very difficult. With water both inside and outside the tubes, a reasonable first guess for U₀ is 1000 W/(m²K).

Yh it is, there are already solutions out there but they are all terribly confusing and I was just hoping someone on here would know of some sort of relationship between the UA (U being overall heat transfer coefficient and A being heat transfer area) product to inlet and outlet temperature. That way I can assume the U and solve for the A of the heat exchanger while also using it to get the mass flow of the hot side.