Unites and nomenclature
thermal conductivity ()
density ()
specific heat ( )
Diffusivity ( )
Mass ()
Heat flux ()
Emissivity ()
Stefan-Boltzmann constant ()
distance ()
Temperature (
Environment temperature ()
Heat transfer coefficient ()
2D heat transfer with convection and radiation
In this problem we model a 2D problem where heat is lost from a bar. The bar is sufficiently long as to ignore heat transfer in the third dimension. We assume an isothermal initial temperature and model the cooling of the component to room temperature. Note that when including radiation, it is convenient to model the problem in units of Kelvin opposed to Celsius.
These notes provide the partial differential equations of interest, and both explicit and implicit finite difference schemes. Use the information to derive the finite difference schemes.
The notes provide an example for the top surface and top left corner. You need to repeat the process for the other surfaces and corners.
Interior regions
Heat is transferred within the interior of the bar through condition. For three dimensions;
1
Let us utilise an effective heat capacity that includes latent heat contributions, simplifying Equation 1 to
2
We can then ignore heat transfer in the z direction
3
We can expand the derivative to and rearrange to obtain
4
Where .
We can discretize the temperature field and describe the temperature at individual locations. Let and describe the temperature at the nodes at locations and in the current and future time step, as illustrated in Figure 1, which shows the neighbouring nodes.
Figure 1: Temperature discretization of interior nodes considering the current or future time step.
An explicit scheme is obtained if we consider the spatial derivatives of the temperature field in the current time step. If we consider the spatial derivatives of temperature in the future time step, we derive an implicit finite difference scheme. Explicit and implicit finite difference schemes are shown below;
Explicit
Implicit
5
6
7
8
9
10
11
12
We can apply finite difference with the following one sided forward fist derivative for the temporal derivative
13
Substitute Equation 13 for the temporal derivative and either the explicit or implicit finite difference approximations for the spatial derivatives shown Equations 5-12 into Equation 4 and derive your finite difference scheme.
Boundary regions – the top edge
We are modelling a case where we can ignore heat fluxes acting in the z direction. Consider the heat flowing through the volume highlighted in red in Figure 2. Note the dimensions of the control volume.
Figure 2: Thermal fluxes acting upon the top surface considering either the current temperature field or the temperature field in the future time step.
We can calculate the rate of heat transfer through balancing the fluxes acting at the location with the rate at which heat can flow
14
Where refers to the mass of the control volume and is given by
15
The size of the control volume in the z direction is given by which is also reffered to by in your notes.
An explicit approximation of the fluxes is given in Equations 16 to 19.
16
17
18
19
Equations 20 to 23 provide the implicit approximations.
20
21
22
23
Substitute either the implicit or explicit approximations for the heat fluxes in addition to the forward first derivative finite difference equation shown in Equation into Equation 14 and derive your finite difference scheme.
If you want to use the in-built Matlab matrix operation functions, you may wish to use an explicit approximation of radiation and an implicit approximation of the other fluxes;
24
This way you can avoid part of the A matrix being to the power of 4.
Boundary regions – top left corner
Consider the heat flowing through the volume highlighted in red in Figure 3. Note the dimensions of the control volume.
Figure 3: Thermal fluxes acting upon the top left corner considering either the current temperature field or the temperature field in the future time step.
The change in temperature at the top corner is determined using Equation 14, however for the corner node the mass of the control volume is given by
15
Explicit approximations for the thermal fluxes are
16
17
18
19
The implicit approximations are given by
16
17
18
19
Substitute either the implicit or explicit approximations for the thermal fluxes, and the forward fist derivative scheme shown in Equation 13 into Equation 14 and derive your finite difference scheme.
Again, if you want to use the Matlab in-built matrix functions to solve , you may wish to use an explicit description of the radiation term with an implicit description of the other fluxes.
17
18