Governing Equation of Time Transient Heat Equation

heat equation is a common example of a parabolic partial differential equation that is used to model heat transfer in solids. It can be written as:

{\partial T \over \partial t} =\alpha \nabla^2T

\rho C_p{\partial T \over \partial t} =k \nabla^2T

Where:

$T$ = temperature

$t$ = time

$α$ = thermal diffusivity (a material property)

$k$ = thermal conductivity

$\rho$ = density

$C_p$ = specific heat

$∇²T$ = the Laplacian of $u$

This equation describes how the temperature of a solid changes over time due to heat conduction. It is often used to study problems such as steady-state heat transfer in a solid, or the transient response of a solid to a sudden change in temperature. There are several different numerical methods that can be used to solve the heat equation, including the finite difference method, the finite element method, and the boundary element method.

Heat Generation, Thermal Convection considerations for Heat Transfer

If you include heat generation, heat flow into and out of the solid, and thermal convection, the heat equation becomes more complex. The resulting equation will typically take the form:

{\partial T \over \partial t} =\alpha \nabla^2 T + q_{gen}

Where:

$q_{gen}$ = internal heat generation $(W/m^3)$

$h$ = convection coefficient $(W/m^2K)$

$T_\infty$ = ambient temperature $(K)$

$q’$ = Heat flow per unit area$(W/m^2)$

The term qgen represents the internal heat generation within the solid, which could come from a chemical reaction or radioactive decay, for example. The term $-h(T – T_\infty)$ represents the heat loss or gain due to thermal convection, where $h$ is the convection coefficient, and $T_\infty$ is the ambient temperature. The term $q’$ represents the heat flow per unit area, through a boundary of the solid, which could come from radiation, conduction or convection.

The full form of the equation describes the heat transfer through the solid in more details, including all possible heat transfer mechanism, such as conduction, convection, and radiation. The equation can be more challenging to solve, and the method of solution will depend on the specific problem and the geometry of the solid.

It’s good to keep in mind that these equation are widely used in heat and mass transfer analysis and numerical methods developed for this purpose can be applied.

Variational Forms of Heat Equation

The finite element method (FEM) is a powerful numerical technique that can be used to solve partial differential equations such as the heat equation. To use the FEM to solve the heat equation, the first step is to discretize the solid into a finite number of elements. Each element is assumed to have a constant temperature over its entire volume, and the temperature is approximated using a set of nodal values at the corners of the element. The temperature at any point within the element can be found by interpolating these nodal values using a set of shape functions.

Once the solid has been discretized, the heat equation can be written in a weak form, also known as a variational form. This is done by multiplying the heat equation by a test function and integrating over the entire domain. The resulting equation is equivalent to the original equation, but it is expressed in terms of the nodal values of the temperature and the gradients of the shape functions. The weak form can be written as:

\int({\partial T\over \partial t}-\alpha \nabla^2T - q_{gen})vdV=0

When integrating the Laplacian of a function $\nabla^2u$ over a domain, integration by parts can be used to convert the double derivative into a gradient. The Laplacian of a function $u$ is defined as the divergence of the gradient of $u$, which is $\nabla^2u=\nabla\cdot(\nabla u)$

To integrate the Laplacian of $T$ over a domain $\Omega$, we use the following steps

1. Integrate over the domain V:

\int_\Omega \nabla^2TvdV = \int_\Omega \nabla \cdot (\nabla T)vdV

2. Apply the divergence theorem to the second term:

\int_\Omega \nabla \cdot (\nabla T)vdV=\int_\Omega  (\nabla T)\cdot vdV-\int_\Omega  (\nabla T)\cdot(\nabla v)dV

3. The first term on the right-hand side is just the dot product of the gradient of $u$ and the $v$, and it is the standard form for the weak form of Laplacian operator.
4. from Divergence theorem,

\int_\Omega  (\nabla u)\cdot vdV= \int_{\partial \Omega}  {\partial (\nabla u)\over \partial n }vdS

4. The second term on the right-hand side represents the surface integral over the boundary of the domain, and it represents the flux of the gradient of u through the boundary. This term can be used as a Neumann boundary condition, representing the normal derivative of the solution on the boundary.

This is how we can integrate by parts the Laplacian operator in the weak form. The final result is ∫∇u.(∇v) dV – ∫∂(∇u)/∂n . v dS. The first term is the standard form for the weak form of Laplacian operator and the second term can be used as a Neumann boundary condition.

Where: u = temperature (unknown function) v = test function (arbitrary function)

The above equation is true for any test function v. The next step is to discretize the above equation using finite element method and to form the set of linear algebraic equation. Then the above set of equation can be solved using numerical linear algebra techniques such as the Gaussian elimination or iterative methods like Conjugate gradient or GMRES.

The above is a brief description of the finite element method for solving the heat equation with additional heat source term, convection and heat flow. The specifics of the implementation will depend on the specific problem being solved, including the geometry of the solid and the boundary conditions that must be imposed.

To derive the bilinear or linear form for each term, we need to approximate the temperature u and the test function v using a set of shape functions. In the case of the finite element method, the shape functions are usually chosen to be polynomials of a certain degree, such as linear or quadratic.

For the temporal term, ∂u/∂t we can use constant shape functions for example: ∫(∂u/∂t) v dV = ∫ (u1 – u0) * (v1 – v0) / Δt dV where u1 and u0 are temperature values at two consecutive time steps, v1 and v0 are the test function values at two consecutive time steps.

For the Laplacian term α ∇²u, ∫ α ∇²u v dV = α ∫ (∂²u/∂x² + ∂²u/∂y² + ∂²u/∂z²) v dV and for a two dimensional problem it can be simplified as ∫ α ∇²u v dV = α ∫ (∂²u/∂x² + ∂²u/∂y²) v dV.

By substituting the shape functions into the above equation, we can obtain a bilinear or linear form of the Laplacian term.

For the heat generation term, qgen , ∫qgen v dV = ∫qgen v dV , as it is a constant term and it can be directly multiplied with the shape function

For the convection term, -h(u – T∞), ∫-h(u – T∞)v dV = -h ∫ (u – T∞)v dV .

For the heat flow term q’ ∫ q’v dV = ∫ q’ v dS .

Note that the above expressions are for a 2D problem and for 3D problem you need to take the derivatives in all three direction x,y and z. Also, keep in mind that the form of the equation will depend on the specific type of shape functions used in the analysis and the boundary conditions of the problem.

In the case of anisotropic conduction, the thermal conductivity tensor can be represented as:

λ = [λxx λxy λxz λyx λyy λyz λzx λzy λzz]

Where λxx, λyy, λzz are the thermal conductivity coefficients in the x, y, and z directions, respectively, and λxy, λxz, λyz, λyx, λzx, λzy are the off-diagonal elements that represent the cross-coupling effects between different directions.

To incorporate anisotropic conduction into the heat equation, we need to replace the Laplacian term with the divergence of the heat flux. The heat flux is given by:

q = -λ ∇u

Where λ is the thermal conductivity tensor and ∇u is the gradient of the temperature. The divergence of the heat flux is given by:

∇ . q = – ∇ . (λ ∇u) = -λ ∇²u – ∇λ ∇u

Therefore, the heat equation with anisotropic conduction becomes:

∂u/∂t = – ∇ . (λ ∇u) + qgen – h(u – T∞) + q’

By substituting the shape functions into the above equation, we can obtain a bilinear or linear form of the anisotropic conduction term. The resulting form will be a bit more complex than the isotropic conduction case, as it will depend on both the nodal values of the temperature and the thermal conductivity tensor, as well as the gradient of the shape functions

It’s important to note that this equation holds for 3D cases, and for 2D cases the terms of the thermal conductivity tensor will be reduced and for the one-dimensional problem, the thermal conductivity tensor will be a scalar.