Heat Transfer

Assumtions

you are familiar with heat transfer and have read the introduction.
you have read the notations, units and glossary.

1. Equations

heat equation

ρ C_{p} \frac{\partial T}{\partial t} - \nabla \cdot (k \nabla T) = Q, in Ω

$\rho C_p \frac{\partial T}{\partial t} - \nabla \cdot \left( k \nabla T \right) = Q, \quad \text{ in } \Omega$

which is completed with boundary conditions and initial value

at t = 0, T (x, 0) = T_{0} (x)

$\text{at } t=0, \quad T(x,0) = T_0(x)$

1.1. Convective heat transfer

convective heat equation

$\rho C_p \left( \frac{\partial T}{\partial t} + \boldsymbol{u} \cdot \nabla T \right) - \nabla \cdot \left( k \nabla T \right) = Q, \quad \text{ in } \Omega$

1.2. Steady case

steady heat equation

$- \nabla \cdot \left( k \nabla T \right) = Q, \quad \text{ in } \Omega$

steady convective heat equation

$\rho C_p \boldsymbol{u} \cdot \nabla T - \nabla \cdot \left( k \nabla T \right) = Q, \quad \text{ in } \Omega$

1.3. Multi-materials

Given a domain $\Omega \subset \mathbb{R}^d, d=1,2,3$ , $\Omega$ is partitioned into $N_r$ regions $\Omega_i,i=1,\ldots,N_r$ corresponding to different materials (solid or fluid). We consider $\rho_i$ , $C_{p,i}$ and $k_i$ the material properties defined in each regions $\Omega_i$ . We define also $\boldsymbol{n}_i$ the outward unit normal vector associated to the boundary $\partial \Omega_i$ .

$\begin{eqnarray} \rho_i C_{p,i} \frac{\partial T}{\partial t} - \nabla \cdot \left( k_i \nabla T \right) &=& Q, \quad &\text{ in }& \Omega_i,i=1,\ldots,N_r \\ T_{|_{\Omega_i}} &=& T_{|_{\Omega_j}}, \quad &\text{ on }& \partial \Omega_i \cap \Omega_j = \Gamma_{ ij}, \forall i \neq j \\ -k_i \nabla T \cdot \boldsymbol{n}_i &=& k_j \nabla T \cdot \boldsymbol{n}_j, \quad &\text{ on }& \partial \Omega_i \cap \Omega_j = \Gamma_{ ij}, \forall i \neq j \\ \end{eqnarray}$

We assume the operator $\mathcal{L}$ tel que $\mathcal{L} T = \rho_i C_{p,i} \frac{\partial T}{\partial t} - \nabla \cdot \left( k_i \nabla T \right)$ is elliptical.

We multiply $\mathcal{L} u = Q$ by a function test $v \in \mathbf{V}$ and integrates by part on $\Omega_i$ . Which give:

$\rho C_{p,i} \displaystyle \int_{\Omega_i} \frac{\partial T}{\partial t} v - \int_{\Omega_i} \nabla \cdot \left[ k_i \nabla T \right] v = \int_{\Omega_i} Qv, \quad \forall v \in H^1_0(\Omega) \quad for i = 1, \cdots , N_r$

By the formula of Green, we get

$\rho C_{p,i} \displaystyle \int_{\Omega_i} \frac{\partial T}{\partial t} v + \int_{\Omega_i} k_i(y) \nabla T \cdot \nabla v- \int_{\partial \Omega} k_i \nabla T \cdot \boldsymbol{n}_i v = \int_{\Omega_i} Qv \quad \forall v \in \mathbf{V}$

Additivity of the integral, we have

$\sum_{ i=1}^{N_r} \left( \rho C_{p,i} \displaystyle \int_{\Omega_i} \frac{\partial T}{\partial t} v + \int_{\Omega_i} k_i \nabla T \cdot \nabla v- \int_{\partial \Omega_i} k_i \nabla T \cdot \boldsymbol{n}_i v \right) = \sum_{ i=1}^{N_r} \left( \int_{\Omega_i} Qv \right) \forall v \in \mathbf{V}$

Note that

$\bigcup_{ i=1}^{ N } \partial \Omega_i = \bigcup_{ i,j} \Gamma_{ ij} \cup \partial \Omega$

Use the conditions in the interfaces, we get

$\sum_{ i=1}^{N_r} \left( \rho C_{p,i} \displaystyle \int_{\Omega_i} \frac{\partial T}{\partial t} v + \int_{\Omega_i} k_i \nabla T \cdot \nabla v- \int_{\partial \Omega} k_i \nabla T \cdot \boldsymbol{n} v \right) = \sum_{ i=1}^{N_r} \left( \int_{\Omega_i} Qv \right) \forall v \in \mathbf{V}$

Using the implicit Euler method for the time term:

$\frac{\partial T}{\partial t} (t^{ k+1}) \approx \frac{ T (t^{ k+1}) - T(t^k)}{ dt} \quad \forall t^k \in \mathbb{ R^+} \text{ et } k \in \mathbb{N}$

Denoting $T^k = T(t^k)$ , we write the formula in $t^{ k+1}$ , we obtain:

$\sum_{ i=1}^{N_r} \left( \rho C_{p,i} \displaystyle \int_{\Omega_i} \frac{ T^{k+1}}{dt} v + \int_{\Omega_i} k_i \nabla T^{k+1} \cdot \nabla v - \int_{\partial \Omega} k_i \nabla T^{k+1} \cdot \boldsymbol{n} v \right) = \sum_{ i=1}^{N_r} \left( \int_{\Omega_i} \frac{T^{k}}{dt} v + \int_{\Omega_i} Qv \right) \quad \forall v \in \mathbf{V}$

So, the weak wording becomes:

The weak formulation

$\text{ On cherche } T \in \mathbf{H} \text{ telle que:} \\ a(T^{k+1}, v) = l(v) \quad \forall v \in \mathbf{V} \\ \text{ and} \quad a(T^{k+1}, v) = \sum_{ i=1}^{N_r} \left( \rho C_{p,i} \displaystyle \int_{\Omega_i} \frac{ T^{k+1}}{dt} v + \int_{\Omega_i} k_i \nabla T^{k+1} \cdot \nabla v - \int_{\partial \Omega} k_i \nabla T^{k+1} \cdot \boldsymbol{n} v \right) \\ l(v) = \sum_{ i=1}^{N_r} \left( \int_{\Omega_i} \frac{T^{k}}{dt} v + \int_{\Omega_i} Qv \right)$

So we have $a(u_{k+1},v)$ a continuous bilinear form coercive in $v \in \mathbf{V}$ and $l(\phi)$ a continuous linear form . We are in a Hilbert space, so we have all the conditions for the application of the Lax-Milgram theorem. So this problem is well posed.

Correct approximation:

We use the Galerkin approximation method:

Let $\{ \mathcal{T}_h \}$ a family of meshes of $:\Omega$ .

Let $\{ \mathcal{K}, P, \sum \}$ a finite element of Lagrange of reference of the degree $k \geq 1$ .

Let $P^k_{c,h}$ the conforming approximation space defined by

$P^k_{ c,h} = \{ v \in C^0(\Omega), \forall \mathcal{K} \in \mathcal{T}_h, v|_{\mathcal{K}} \in \mathbb{P}_k(\mathcal{K}) \}$

To obtain a conformal approximation in V, we add the boundary conditions

$V_h = P^k_{c,h} \cap V$

Discrete problem is written:

Problème discrète

$\text{ Find } T_h \in V_h \text{ such that} \\ a(T_h, v_h) = l(v_h) \quad \forall v_h \in V_h$

Let $\{ \varphi_1, \varphi_2, ..., \varphi_N \}$ the base of $V_h$ . An element $T_h \in V_h$ is written as

$T_h = \sum^{N}_{l=1} T_l \varphi_l$

Using $v$ as a basic function of $V_h$ , our problem becomes

$\sum_{ i=1}^{N_r} \left( \rho C_{p,i} \displaystyle \int_{\Omega_{i}} \sum_{ l=1}^N T^{k+1}_l \frac{ \varphi_l }{dt} \varphi_j + \int_{\Omega_i} k_i \sum_{ l=1}^N T^{k+1}_l \nabla \varphi_l \cdot \nabla \varphi_j - \int_{\partial \Omega} k_i \sum_{ l=1}^N T^{k+1}_l \nabla \varphi_l \cdot \boldsymbol{n} \varphi_j \right) = \sum_{ i=1}^{N_r} \left( \int_{\Omega_i} \sum_{ l=1}^N T^{k}_l \frac{ \varphi_l }{dt} \varphi_j + \int_{\Omega_i} Q \varphi_j \right)$

The variational problem of approximation is then equivalent to a linear system

Algebraic problem

$\text{Determine } T_l \text{ satisfying} \\ \sum_{ l=1}^N a(\varphi_l, \varphi_j) T^{k+1}_l = l(\varphi_j) \forall j = 1, \cdots , N$

Introduce

$A = (a(\varphi_i , \varphi_j)), \quad 1 \leq i,j \leq N , \\ U^{k+1} = (T_1^{k+1}, T_2^{k+1}, ..., T_N^{k+1}) \in \mathbb{R}^{N}, \\ F = (l(\varphi_1), l(\varphi_2), ..., l(\varphi_N)) \in \mathbb{R}^{N}$

We write the system in matrix form

$AU = F$