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I need to estimate the time it would take for the inside of a hollow cylinder to reach a limit temperature (detailed data below), in order to adjust the wall thickness and choose an insulating material to delay heating as much as possible. Being a total beginner in thermodynamics, I don't understand how to proceed, what modes of heat transmission to use.

The situation is as follows:

I have a hollow cylinder with internal diameter D1, external diameter D2 and height h. The cylinder is immersed in oil at a constant temperature T2 and has an initial internal temperature T1. The oil is stationary relative to the cylinder.

I therefore tried to make an initial estimate of the time it would take to heat up by means of a heat flow calculation, but I find a result that I find totally inconsistent.

Data :

D1 = 35mm ; D2 = 40mm ; h = 60mm

T1 = 40°C ; T2 = 90°C

I've done the calculation for walls covered with aerogel (it's not realistic, but it's a first estimate), thermal conductivity λ=0.03W.K^(-1).m^(-1), thickness e = 7mm.

The surface area of my cylinder is therefore :

S=2πrh=2π×0.02×0.06=0.00753 m2

For the flux, I get :

ϕ=((λ.ΔT)/e)*S=((0.03×(90-40))/0.07)*0.00753=1.61567 W

To obtain the heating time, I divide the flow by the volume of air in the cylinder multiplied by the heat capacity of the air (1.256 kJ.m-3).

Volume of air in the cylinder (removing the volume taken up by the aerogel):

V=π*(r^2)*h=π*(0.02-0.007×2)^2×0.06=6.7858e-6

So the time I find is :

t=(cp×V)/ϕ=(1256×6.7858e-6)/1.61567=0.005 s

I don't think this is the right way to do it, should heat transfer be calculated in all modes (conduction, conduction, radiation)?

An additional problem that I haven't yet taken into account, but which is very important, is that the cylinder contains an electronic circuit that also heats up. By insulating the cylinder, the temperature stabilizes at around 40°C.

Thank you very much for your help