The Heat Diffusion Equation

It is important to know the temperature distribution as it allows us to see how temperature varies with position inside a medium. Once the distribution is known, Fourier’s law can be used to calculate the conduction heat flux at any point in the medium or on its surface or any other important quantities of interest.

In a solid, temperature distribution allows study of structural integrity by determining thermal stress, expansions and deflections. Temperature distribution can also help to optimize the thickness of an insulating materials, determine the compatibility of special coatings or adhesives used within the material.

In order to determine the temperature distribution, we can follow the same methodology used in the chapter on Relationship to the First Law of Thermodynamics (see here) of applying energy conservation. In this case, we:

• define a differential control volume,
• identify the relevant energy transfer processes
• introduce the appropriate rate equations

All this results in a differential equation whose solution for a set of boundary conditions, provides the temperature distribution in the medium.

Let us consider a homogeneous medium within which there is no advection (bulk motion) and the temperature distribution T(x, y, z) is expressed in Cartesian coordinates. We then define an infinitesimally small differential control volume dx·dy·dz as shown below:

Formulate the first law at an instant of time, followed by considering the energy processes relevant to this control volume. If there is no motion or with uniform motion, there are no changes in the mechanical energy and no work is being done on the system. Thus, only thermal forms of energy need to be considered. If there are temperature gradients, heat transfer through conduction will occur across each of the control surfaces. The conduction heat rates perpendicular to each control surfaces at the x-, y- and z-coordinate locations are expressed by the terms q_x, q_y, and q_z, respectively. Using Taylor series expansion and neglecting higher-order terms, we can express the conduction heat rates at the opposite surfaces as:

Equation 1a, states that the x-component of the heat transfer rate at x + dx is equal to the value of this component at x plus the amount by which is changes with respect to x times dx.

Within the medium there may also be an energy source term associated with the rate of thermal energy generation (E_g). This term is represented as:

E_g = q dx dy dz                 (2)

where q is the rate at which energy is generated per unit volume of the medium (W/m³).

Additionally, changes may occur in the amount of the internal thermal energy stored by the material in the control volume. If the material is not experiencing a change in phase, latent energy effects are not relevant, and the energy storage term may be expressed as:

E_st = ρC_p(∂T / ∂t) dx dy dz                    (3)

where ρC_p(∂T / ∂t) is the time rate of change of the sensible (thermal) energy of the medium per unit volume

The terms E_g and E_st represent different physical processes. The energy generation term (E_g) is an indicator of some energy conversion process involving thermal energy on one hand and some other form of energy like chemical, electrical, or nuclear, on the other. The term is positive (a source) if thermal energy is being generated in the material at the expense of some other energy form; it is negative (a sink) if thermal energy is being consumed. In contrast, the energy storage term (E_st) refers to the rate of change of thermal energy stored by the matter.

On a rate basis, the general form of the conservation of energy requirement is:

E_in + E_g – E_out  = E_st

Hence, recognizing that the conduction rates constitute the energy inflow E_in and outflow E_out, and substituting Equations 2 and 3, we obtain:

Substituting from Equations 1:

The conduction heat rates in an isotropic material may be evaluated using Fourier’s law:

where each heat flux component of Equation 4 has been multiplied by the appropriate control surface (differential) area to obtain the heat transfer rate. Substituting Equations 6 into Equation 5 and dividing out the dimensions of the control volume (dx dy dz), we obtain: 

Equation 7 is the general Cartesian coordinates form of the heat diffusion equation. This equation is referred to as the heat equation and is the basic tool for heat conduction analysis and its solution helps us obtain the temperature distribution T(x, y, z) as a function of time. The heat equation describes the conservation of energy.  In simple words, Equation 7 means that at any point in the medium, the net rate of energy transfer by conduction into a unit volume plus the volumetric rate of thermal energy generation must be equal to the rate of change of thermal energy stored within the volume.

In equation 7, the term ∂(k∂T/∂x)/∂x is related to the net conduction heat flux into the control volume for the x-coordinate direction. That is, multiplying by dx we get:

The flux expressions in the y- and z- directions are similar to equation 8.

It is often possible to simplify Equation 7. For example, if the thermal conductivity is constant, the heat equation is:

where α = k/ρc_p is the thermal diffusivity.

Additional simplifications of the general form of the heat equation are also possible. For example, under steady-state conditions, there can be no change in the amount of energy storage; hence Equation 7 reduces to:

If the heat transfer is one-dimensional (e.g., in the x-direction) and there is no energy generation, Equation 10 reduces to:

This shows that, under steady-state, one-dimensional conditions with no energy generation, the heat flux is a constant in the direction of transfer.

The heat equation may also be expressed in cylindrical and spherical coordinates.

i) Cylindrical Coordinates

The differential control volume for cylindrical coordinates is as shown below:

When we express Equation 3 from The Conduction Rate Equation blog entry (see here) in cylindrical coordinates, the general form of the heat flux vector and hence of Fourier’s Law is:

Where equations 13 represent the heat flux components in the radial (r), circumferential (ϕ), and axial directions (z), respectively.

Applying an energy balance to the cylindrical coordinate’s differential control volume, the following general form of the heat equation is obtained:

ii) Spherical Coordinates

The differential control volume for cylindrical coordinates is as shown below:

The general form of the heat flux vector and Fourier’s law in spherical coordinates is:

Equations 16 are the heat flux components in the radial (r), polar (θ), and azimuthal directions (ϕ), respectively.

Applying an energy balance to the spherical coordinate’s differential control volume, the following general form of the heat equation is obtained: