Heat Equation – Heat Conduction Equation
In previous sections, we have dealt especially with one-dimensional steady-state heat transfer, which can be characterized by the Fourier’s law of heat conduction. But its applicability is very limited. This law assumes steady-state heat transfer through a planar body (note that, Fourier’s law can be derived also for cylindrical and spherical coordinates), without heat sources. It is simply the rate equation in this heat transfer mode, where the temperature gradient is known.
But a major problem in most conduction analyses is to determine the temperature field in a medium resulting from conditions imposed on its boundaries. In engineering, we have to solve heat transfer problems involving different geometries and different conditions such as a cylindrical nuclear fuel element, which involves internal heat source or the wall of a spherical containment. These problems are more complex than the planar analyses we did in previous sections. Therefore these problems will be the subject of this section, in which the heat conduction equation will be introduced and solved.
General Heat Conduction Equation
The heat conduction equation is a partial differential equation that describes the distribution of heat (or the temperature field) in a given body over time. Detailed knowledge of the temperature field is very important in thermal conduction through materials. Once this temperature distribution is known, the conduction heat flux at any point in the material or on its surface may be computed from Fourier’s law.
The heat equation is derived from Fourier’s law and conservation of energy. The Fourier’s law states that the time rate of heat transfer through a material is proportional to the negative gradient in the temperature and to the area, at right angles to that gradient, through which the heat flows.
A change in internal energy per unit volume in the material, ΔQ, is proportional to the change in temperature, Δu. That is:
∆Q = ρ.cp.∆T
Using these two equation we can derive the general heat conduction equation:
This equation is also known as the Fourier-Biot equation, and provides the basic tool for heat conduction analysis. From its solution, we can obtain the temperature field as a function of time.
In words, the heat conduction equation states 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 equal the rate of change of thermal energy stored within the volume.
Additional simplifications of the general form of the heat equation are often possible. For example, under steady-state conditions, there can be no change in the amount of energy storage (∂T/∂t = 0).
One-dimensional Heat Equation
One of most powerful assumptions is that the special case of one-dimensional heat transfer in the x-direction. In this case the derivatives with respect to y and z drop out and the equations above reduce to (Cartesian coordinates):
Heat Conduction in Cylindrical and Spherical Coordinates
In engineering, there are plenty of problems, that cannot be solved in cartesian coordinates. Cylindrical and spherical systems are very common in thermal and especially in power engineering. The heat equation may also be expressed in cylindrical and spherical coordinates. The general heat conduction equation in cylindrical coordinates can be obtained from an energy balance on a volume element in cylindrical coordinates and using the Laplace operator, Δ, in the cylindrical and spherical form.
Obtaining analytical solutions to these differential equations requires a knowledge of the solution techniques of partial differential equations, which is beyond the scope of this text. On the other hand, there are many simplifications and assumptions, that can be applied to these equations and that lead to very important results. In the next section we limit our consideration to one-dimensional steady-state cases with constant thermal conductivity, since they result in ordinary differential equations.
Boundary and Initial Conditions
As for another differential equation, the solution is given by boundary and initial conditions. With regard to the boundary conditions, there are several common possibilities that are simply expressed in mathematical form.
Because the heat equation is second order in the spatial coordinates, to describe a heat transfer problem completely, two boundary conditions must be given for each direction of the coordinate system along which heat transfer is significant. Therefore, we need to specify four boundary conditions for two-dimensional problems, and six boundary conditions for three-dimensional problems.
Four kinds of boundary conditions commonly encountered in heat transfer are summarized in following section:
Conduction with Heat Generation
In the preceding section we considered thermal conduction problems without internal heat sources. For these problems the temperature distribution in a medium was determined solely by conditions at the boundaries of the medium. But in engineering we can often meet a problem, in which internal heat sources are significant and determines the temperature distribution together with boundary conditions.
In nuclear engineering, these problems are of the highest importance, since most the heat generated in nuclear fuel is released inside the fuel pellets and the temperature distribution is determined primarily by heat generation distribution. Note that, as can be seen from the description of the individual components of the total energy energy released during the fission reaction, there is significant amount of energy generated outside the nuclear fuel (outside fuel rods). Especially the kinetic energy of prompt neutrons is largely generated in the coolant (moderator). This phenomena needs to be included in the nuclear calculations.
For LWR, it is generally accepted that about 2.5% of total energy is recovered in the moderator. This fraction of energy depends on the materials, their arrangement within the reactor, and thus on the reactor type.
Note that, heat generation is a volumetric phenomenon. That is, it occurs throughout the body of a medium. Therefore, the rate of heat generation in a medium is usually specified per unit volume and is denoted by gV [W/m3].
The temperature distribution and accordingly the heat flux is primarily determined by:
- Geometry and boundary conditions. Different geometry leads to completely different temperature field.
- Heat generation rate. The temperature drop through the body will increase with increased heat generation.
- Thermal conductivity of the medium. Higher thermal conductivity will lead to lower temperature drop.