What I try to model?

I try to create a heat transfer model for a cryostat. The heat transfer mechanisms between the layers are solid conduction, radiation and internal heating.

The graphic below shows the cryostat with its several layers and respective temperatures. The red rods represent mechanical and thus thermical connection between them. The space between the layers is evacuated.
Cryostat model

What does the thermodynamic model look like?

In the simplified model the layers are connected via 1-dimensional rods (i.e. 1-d heat conduction) with the heat conduction fluxes $\dot{q}_{12}$, $\dot{q}_{23}$, $\dot{q}_{34}$, ..., between the layers. Radiation and internal heating are represented as additional heat flux point sources $\dot{q}^*_{1}$, $\dot{q}^*_{2}$, $\dot{q}^*_{3}$, ... , which are introduced at each layer (and are assumed to be constant and known). The graphic below shows the 1-dimensional model.

Simplified heat transfer model

What is the model supposed to do?

The temperatures ($T_1, T_2$, ...) and additional heat flux sources ($\dot{q}^*_{1}$, $\dot{q}^*_{2}$, ...) at each layer shall be defined by me.

The model then shall deliver the resulting heat fluxes between each layer ($\dot{q}_{12}$, $\dot{q}_{23}$, ...).

(This is interesting as $\dot{q}_{12}$, $\dot{q}_{23}$,... represent the needed cooling power of a cryocooler connecting two consecutive layers.)

What equations do I use?

(1) $\dot{q}_{12} = -k \cdot \frac{\partial T}{\partial x}$

(2) $ \frac{\partial}{\partial x} ( k \cdot \frac{\partial T}{\partial x}) + \dot{q}_{\text{source}} = \rho c \frac{\partial T}{\partial t} $

I seek a stationary solution, hence the right hand sight of (2) is zero. We assume $k=\text{const.}$ for each individual rod.

$$ k \cdot \frac{\partial^2 T}{\partial x^2} + \dot{q}_{\text{source}} = 0 $$

Integration delivers: $$ \frac{\partial^2 T}{\partial x^2} = -\frac{\dot{q}_{\text{source}}}{k} $$

$$ \frac{\partial T}{\partial x} = -\frac{\dot{q}_{\text{source}}}{k} x + C_1 $$

$$ T(x) = -\frac{\dot{q}_{\text{source}}}{k} x^2 + C_1 x + C_2 $$

Boundary conditions:

  1. $T(x_1) = T_1$,
    $T(x_2) = T_2$,

  2. $-k \frac{\partial T(x_1)}{\partial x} = \dot{q}^*_{1} $,
    $-k \frac{\partial T(x_2)}{\partial x} = \dot{q}^*_{2} $,

What is my problem in modelling it?

I don't know how to go on from here. For each connecting rod, there are 4 boundary conditions (2 temperatures and 2 heat fluxes at both ends), but only 2 integration constants ($C_1$ and $C_2$).

I'm not sure where the additional heat flux terms ($\dot{q}^*_{1}$, $\dot{q}^*_{2}$, ...) have to be taken into account in the formulas. Is it correct that they come up as boundary conditions, or should they be taken into account as the internal heat production term $\dot{q}_{\text{source}}$? If not, is $\dot{q}_{\text{source}} = 0$?

Additionally, when we look at, for example, node 2: How should the inward heat flux $\dot{q}_{12}$, the outward heat flux $\dot{q}_{23}$ and the additional heat flux $\dot{q}^*_{2}$ be linked in a formula?

Lastly, is the underlying assumption of stationarity adequate to describe the given problem?

I'd be thankful for any idea and advice.


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