Heat conduction equation is defined as the differential equation through which we can define the conduction in any shape of body and we can calculate the temperature at any point in a medium in any situation like steady flow, transient flow, one dimension flow, three dimension flow etc. It is governing equation of conduction. There are various shapes of body such as rectangular shape, cylindrical shape, spherical shape and combine shape.
Heat Conduction Equation
he 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.
Heat conduction equation for rectangular coordinate
When we want to define conduction in a wall or at a point in
a body which is represent by rectangular coordinate then we have to use
rectangular heat conduction equation which is given by:
Where, T = Temperature at specified point, e = Heat generated per unit volume, C = Specific heat of medium material, ρ = Density of medium material, k = Thermal conductivity of medium material, This equation may reduce in various forms in various
conditions. Some of them are given below:
- For constant thermal conductivity:
- For steady state ( ∂T / ∂t = 0 ):
- For no heat generation ( e = 0 )
Heat conduction equation for cylindrical coordinates
When we want to define conduction in cylindrical body then we have to use cylindrical heat conduction equation which is given by:
Heat conduction equation for spherical coordinates
When we want to define conduction in spherical body then we have to use spherical heat conduction equation which is given by
All these heat conduction differential equations are solved by the use of two boundary conditions. The mathematical expression of thermal condition at the boundary is known as boundary condition. Some of boundary conditions for one dimension heat conduction are given below. When the temperature of both end are known.
When the heat enter and heat rejected by the surface or heat flux are known.
When the boundary is insulated at one end, then heat transfer at that end becomes equal to zero.
When the surface is exposed into environment then heat transfer by conduction equal to heat transfer by convection.
Where – sign is taken for heat rejection.
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 via way of means of situations on 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.
or 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.