Discussion Forum

Hi,

I would like to check the following Fourier equation: φ =-λ.grad (T) in this simple modele.

My model consists of two concentric cylinders separated by a gas gap. A power density (Q) is placed in the first cylinder. I want to check the Fourier equation in the gas layer (Neon).

Analytically the above equation to calculate returns: T1-T2 = φ / (2 * π * λ * L) * ln (R2/R1) (see page 78 of the document attached INSA).

With: index 1: inner wall of cylinder Neon, index 2: outer wall of cylinder Neon
  R2 = 15 mm, R1 = 14.880 mm, λ (Neon) = 0.054 W (m * K), L = 1 m,
and φ = Q * V (stainless steel inner tube) = 200E5 [W / m ^ 3] * 243.2E-06 m ^ 3 = 4864 W

Analytically obtained T1-T2 = 115 147 K.

COMSOL finds this two temperatures : T1 = 434.75 K and T2 = 319.6 K
So T1-T2 = 115.15 K.

The Fourier equation would then be validated. However, in steady, the radius R1 and R2 are not identical to the cold radius.

According to COMSOL in "deplacement selon Y", R1 = 14.880 + 0.03588 [mm] and R2 = 15 + 0.00575 [mm]. If we integrate these new radius in the analytical equation, we find T1-T2 = 86.11 K.

This means a 30% error between the analytical result and the COMSOL.

I feel that COMSOL does not take into account the movement of the tubes, and so the change in radius. What is the tool that allows to take into account these movements on the final result in temperature?

I checked if the flow were identical to reality. In theory, on the surface of the inner tube of neon was φ = P / S = 4864 / (2 * π * L * R1) = 52024.8W / m ^ 2. And the average over COMSOL normal flow is equal to 51642.1 W / m ^ 2 so 0.7% error.
There is therefore consistent flow.

Thank for answering,

I remain at your disposal for any further information.

Stephane Bendotti