Heat Transfer

1. What is Heat Transfer

Heat transfer is defined as the movement of energy due to a difference in temperature. It is characterized by the following mechanisms:

Conduction

Heat conduction takes place through different mechanisms in different media. Theoretically it takes place in a gas through collisions of the molecules; in a fluid through oscillations of each molecule in a “cage” formed by its nearest neighbors; in metals mainly by electrons carrying heat and in other solids by molecular motion which in crystals take the form of lattice vibrations known as phonons. Typical for heat conduction is that the heat flux is proportional to the temperature gradient.

Convection

Heat convection (sometimes called heat advection) takes place through the net displacement of a fluid, which transports the heat content in a fluid through the fluid’s own velocity. The term convection (especially convective cooling and convective heating) also refers to the heat dissipation from a solid surface to a fluid, typically described by a heat transfer coefficient.

Radiation

Heat transfer by radiation takes place through the transport of photons. Participating (or semitransparent) media absorb, emit and scatter photons. Opaque surfaces absorb or reflect them.

2. 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

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

3. 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

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

3.1. Convective heat transfer

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

3.2. Steady case

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

3.3. 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

\[AU = F\]