do you realise an annulus is a 2D shape ?

so to make sense of the problem we have to consider a unit length. so your annulus is a section through a pipe, or do you mean a circular tube, a torus ?

so initially everything (air, walls, water) is at the same temp (water temp, which need to be defined). Then you step change the air to 100deg.
so the walls will quickly come up to this temp, of course depending on the thickness of the wall ... (another thing that needs to be dfined). if very thick, then the outer face of the wall (beside the water )will slowly come up the the air temp, the inside face (beside the air) will more quickly come up to air temp.

I guess a key part of the problem is whether the water on the inside of the annulus is connected to the outside water ? Is the temperature of the inside water independent of the outside water ?

Another thing ... how big is the outside water ? "infinite" ?

Of course you realise the issues if the water is heated to 100deg C ...
if the air temp is 99deg C, then the solution will asymptote to 99deg C, if 101dg C then the water near the tube phase changes and then the problem changes !

"Hoffen wir mal, dass alles gut geht !"
General Paulus, Nov 1942, outside Stalingrad after the launch of Operation Uranus.

a circular tube that has a thickness, the outside and the inside of the circular tube isn't connected at all, the thickness could be varied starting from like 3mm with a material of thermal conductivity of 0.15 W/mk, the heat transfer occurring is, convection from the hot air to the inside wall, and from the inside wall to the outside wall by conduction and from the outside wall to the water by convection, but if the temperature of the outside wall is as that of the water (water bath of semi-infinite volume), then there will be no convection between the outside temperature and the water temperature. And initially, only the tube has the water temperature then I introduce the hot air flow through it.


I hope this explained the problem better and what I'm looking for exactly.

ok, with you said "annulus" I didn't take this as a section of a pipe, so you have a pipe with air in it sitting in an "infinite" bath of water, and initially everything is at some temp (what temp will affect the solution). Then you added heated air (what temp makes a difference; yes, you've said "100 deg C") and you want to see temperature profile on the outside surface of the pipe ?

Your really (really !) long sentence confuses me. I imagine that you need to divide the water into different volumes. I think you're saying that you have an infinite supply of air (at some temp); the inside wall of the pipe is the first thing to heat up, then shortly after it the outside wall of the pipe (in contact with the water). I suspect you need to model the pipe in different (radial) zones to detect this, else you have just three zones (the air, the pipe and the water) with heat travelling from the air to the pipe and water until they stabilize (of course, the infinite water will never stabilize).

I suggest you start with a simple problem ... model just three zones ... air, pipe, and water. I imagine you need to set a far boundary for the water ... if infinite I imagine that it'll never change, or change very slowly. I'd start with the initial temp as 20 deg C and 80 deg C air, then when you see something that looks real, play with the temps. I suspect all you'll see is the pipe coming quickly up the temp and then nothing. Then model the near volume of water, say a pipe radius; then more volumes of water ... the model detail you put into the water model the more you'll see heat transfer between the zones. Each successive zone of water should increase in temp (to equal the air temp) and the infinite zone will probably never change. Modelling the pipe in different zones will (should?) slightly refine the heat transfer across the pipe.

"Hoffen wir mal, dass alles gut geht !"
General Paulus, Nov 1942, outside Stalingrad after the launch of Operation Uranus.