Axisymmetrical Case

In this section, we pass the Heat Equation in axisymmetrical coordinates.

1. Transient Case

If you want more explanation see Axisymetric Coordinates.

We note \(\Omega_c^{axis}\) (respectively \(\Gamma_c^{axis}\), \(\Gamma_{Nc}^{axis}\) and \(\Gamma_{Rc}^{axis}\)) the representation of \(\Omega_c\) (respectively \(\Gamma_c\), \(\Gamma_{Nc}\) and \(\Gamma_{Rc}\)) in axisymmetric coordinates.

We note \(u = \begin{pmatrix} u_r \\ u_{\theta} \\ u_z \end{pmatrix}_{cyl}\) the coordinates of \(u \in \mathbb{R}^3\) in cylindrical base.

We note \(\mathbf{n}^{axis} = \begin{pmatrix} n^{axis}_r \\ n^{axis}_z \end{pmatrix}_{cyl}\) the exterior normal of \(\Gamma_c^{axis}\) on \(\Omega_c^{axis}\).

1.1. Differential Formulation

We suppose the geometry, the parameters and \(T\) are independant by \(\theta\) (of cylindric coordinates \((r,\theta,z)\)). The temperature becomes \(T(x,y,z) = T(r,z)\)

The Laplacian exprims in axisymmetrical coordinates :

\[\begin{eqnarray*} \Delta T &=& \frac{\partial^2 T}{\partial r^2} + \frac{1}{r} \frac{\partial T}{\partial r} + \frac{1}{r^2} \frac{\partial^2 T}{\partial \theta^2} + \frac{\partial^2 T}{\partial z^2} \\ &=& \frac{\partial^2 T}{\partial r^2} + \frac{1}{r} \frac{\partial T}{\partial r} + \frac{\partial^2 T}{\partial z^2} \end{eqnarray*}\]

The equation Heat Equation becomes in axisymmetric coordinates :

Differential Formulation of Static Heat Equation in axisymmetric coordinates
\[\text{(Heat Axis)} \left\{ \begin{matrix} \rho C_p \, \frac{\partial T}{\partial t} - k \Delta T = Q \text{ on } \Omega_c^{axis} \\ \frac{\partial T}{\partial \mathbf{n}^{axis}} = 0 \text{ on } \Gamma_{Nc}^{axis} \\ - k \, \frac{\partial T}{\partial \mathbf{n}^{axis}} = h \, \left( T - T_c \right) \text{ on } \Gamma_{Rc}^{axis} \\ \end{matrix} \right.\]

With \(\Delta T = \frac{\partial^2 T}{\partial r^2} + \frac{\partial^2 T}{\partial z^2}\)

1.2. Weak Formulation

Like for Heat Equation, we have :

\[\small{ \int_{\Omega_c^{axis}}{ \rho C_p \, \frac{\partial T}{\partial t} \, \phi \, dxdydz } + \int_{\Omega_c^{axis}}{ k \, \nabla T \cdot \nabla \phi \, dxdydz } = \int_{\Omega_c^{axis}}{ Q \, \phi \, dxdydz } - \int_{\Gamma_{Rc}^{axis}}{ h \, (T-T_c) \, \phi \, d\Gamma } }\]

We change the variables in axisymmetric coordinates (see Formula of Changement of Variables) :

Weak Formulation of Heat Equation in axisymmetric coordinates
\[\small{ \text{(Weak Heat Axis)} \int_{\Omega_c^{axis}}{ \rho C_p \, \frac{\partial T}{\partial t} \, \phi \, r \, drdz } + \int_{\Omega_c^{axis}}{ k \, \tilde{\nabla} T \cdot \tilde{\nabla} \phi \, r \, drdz } = \int_{\Omega_c^{axis}}{ Q \, \phi \, r \, drdz } - \int_{\Gamma_{Rc}^{axis}}{ h \, (T-T_c) \, \phi \, r \, d\Gamma } }\]

With \(\tilde{\nabla} = \begin{pmatrix} \frac{\partial}{\partial r} \\ \frac{\partial}{\partial z} \end{pmatrix}\)

1.3. Time Discretization

We discretize in time with time step \(\Delta t\). We note \(f^n(\mathbf{x}) = f(n\Delta t, \mathbf{x})\).

If, we know \(T^n\), the equations become :

Time Discretization of Heat Equation in axisymetric coordinates
\[\small{ \begin{eqnarray*} \int_{\Omega_c^{axis}}{ \rho C_p \, \frac{T^{n+1}}{\Delta t} \, \phi \, r \, drdz } + \int_{\Omega_c^{axis}}{ k \, \tilde{\nabla} T^{n+1} \cdot \tilde{\nabla} \phi \, r \, drdz } = \int_{\Omega_c^{axis}}{ Q \, \phi \, r \, drdz } - \int_{\Gamma_{Rc}^{axis}}{ h \, (T^{n+1} - T_c) \, \phi \, r \, d\Gamma } + \int_{\Omega_c^{axis}}{ \rho C_p \, \frac{T^n}{\Delta t} \, \phi \, r \, drdz } \end{eqnarray*} }\]

2. Stationary Case

This section presents the AV-Formulation for the stationary regime as a special case of MQS. We assume that all parameters are time independent.

2.1. Differential Formulation

Differential Formulation of Static Heat Equation in axisymmetric coordinates
\[\text{(Heat Static Axis)} \left\{ \begin{matrix} \rho C_p \, \frac{\partial T}{\partial t} - k \Delta T = Q \text{ on } \Omega_c^{axis} \\ \frac{\partial T}{\partial \mathbf{n}^{axis}} = 0 \text{ on } \Gamma_{Nc}^{axis} \\ - k \, \frac{\partial T}{\partial \mathbf{n}^{axis}} = h \, \left( T - T_c \right) \text{ on } \Gamma_{Rc}^{axis} \\ \end{matrix} \right.\]

2.2. Weak Formulation

Weak Formulation of Static Heat Equation in axisymmetric coordinates
\[\small{ \text{(Weak Heat Static Axis)} \int_{\Omega_c^{axis}}{ \rho C_p \, \frac{\partial T}{\partial t} \, \phi \, r \, drdz } + \int_{\Omega_c^{axis}}{ k \, \tilde{\nabla} T \cdot \tilde{\nabla} \phi \, r \, drdz } = \int_{\Omega_c^{axis}}{ Q \, \phi \, r \, drdz } - \int_{\Gamma_{Rc}^{axis}}{ h \, (T-T_c) \, \phi \, r \, d\Gamma } }\]