Heat Transfer

1. Notations and Units

Notation Quantity Unit

\(\rho\)

fluid density

\(kg \cdot m^{-3}\)

\(C_p\)

heat capacity at constant pressure

\(J \cdot kg^{-1} \cdot K^{-1}\)

\(T\)

temperature

\(K\)

\(k\)

thermal conductivity

\(W \cdot m^{-1} \cdot K^{-1}\)

\(\boldsymbol{u}\)

fluid velocity

\(m \cdot s^{-1}\)

2. Equations

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

which is completed with boundary conditions and initial value

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

2.1. Convective heat transfer

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

2.2. Steady case

steady heat equation
\[ - \nabla \cdot \left( k \nabla T \right) = Q, \quad \text{ in } \Omega\]
steady convective heat equation
\[\rho C_p \boldsymbol{u} \cdot \nabla T - \nabla \cdot \left( k \nabla T \right) = Q, \quad \text{ in } \Omega\]

2.3. 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

\[AU = F\]