Derivation of Heat Equation

Heat Equation

Heat is a form of energy that transfers from one medium to the other mediums and it usually travels from the hotter region to the colder region of the conductor. There are ways to transfer the heat according to the medium of the conductor. Heat is transferred in solids by the process of conduction, in liquids and gases by the process of convection, and electromagnetic waves in the form of the radiation of heat. The heat equation in one dimension is a partial differential equation that describes how the distribution of heat evolves over the period of time in a solid medium, as it spontaneously flows from higher temperature to the lower temperature that will be the special case of the diffusion.

Heat Equation Derivation

Derivation of the heat equation in one dimension can be explained by considering a rod of infinite length. The heat equation for the given rod will be a parabolic partial differential equation, which describes the distribution of heat in a rod over the period of time. As the heat energy is transferred from the hooter region of the conductor to the lower region of the conductor.

The form of the equation is given as:

\[\frac{∂u}{∂t}\] = α[\[\frac{∂^{2}u}{∂x^{2}}\] + \[\frac{∂^{2}u}{∂y^{2}}\] + \[\frac{∂^{2}u}{∂z^{2}}\]]

where, α is a real coefficient of the equation which represents the diffusivity of the given medium.

Derivation of the Heat Equation in One Dimension

The amount of heat energy required to raise the temperature of  the given rod by ∂T degrees is 

CM. ∂T, which is  known as the specific heat of the conductor,

Where,

C -  positive physical constant of heat determined by the conductor
M - the mass of the conductor

The rate at which heat energy transferred in the surface of the conductor is directly proportional to the surface area and the temperature gradient at the surface of the conductor and this constant of proportionality is known as the thermal conductivity of heat which is denoted by K

Consider a rod of finite length with cross-sectional area A and mass density ρ.

The temperature gradient of the function is given as

\[\frac{∂T}{∂x}\](x + dx,t)

The rate at which the heat energy transferred from the right end of the given rod is given as

KA\[\frac{∂T}{∂x}\](x + dx,t)

The rate at which the heat energy transferred from the left end is given as

KA\[\frac{∂T}{∂x}\](x , t)

As the temperature gradients are positive from both ends then the temperature of the conductor must increase.

As the heat flows from the hot region to a cold region of the given rod, heat energy should enter from the right end of the rod and transferred to the left end of the rod.

So the equation as per the condition is given as

KA\[\frac{∂T}{∂x}\](x + dx,t) - KA\[\frac{∂T}{∂x}\](x , t)dt where it is the time period.

Now the temperature change in the given rod is can be written as

\[\frac{∂T}{∂x}\](x , t)dt

The mass of the rod will be

Density = mass/volume

  ρ = M/A.dx

M =  ρA.dx

Now, the heat equation can be written as

CρAdx\[\frac{∂T}{∂x}\](x , t)dt = KA[\[\frac{∂T}{∂x}\](x + dx,t) - \[\frac{∂T}{∂x}\](x , t)]dt

Dividing both sides of the above equation by dx and dt and taking limits of it dx and it ->0, then CρA\[\frac{∂T}{∂x}\](x , t) = KA\[\frac{∂^{2}T}{∂x^{2}}\](x , t)  

The equation will be,

\[\frac{∂T}{∂x}\](x , t) = α\[^{2}\]\[\frac{∂^{2}T}{∂x^{2}}\](x , t)

Where,

α\[^{2}\] = \[\frac{K}{Cρ}\]

is the thermal diffusivity of the given rod. 

Hence the above-derived equation is the Heat equation in one dimension.

There are so many other ways to derive the heat equation. However, here it is the easiest approach.  In detail, we can divide the condition of the constant in three cases post which we will check the condition in which, the temperature decreases, as time increases. It is the phenomena of the heat or any form of energy that they will lose energy while traveling from one medium to the other.  Ultimately after the integration, we will get the same equation of the heat in one dimension.

Application of the Heat Equation

• The heat equation is used to modify the automobile engines, as it tells you about the specific heat of the conductor which gives you the idea about the rate of heat absorption by the engine and capacity to hold the heat.

• The most common use in the medical field is when the patient gets relief from pain with the help of the hot water bag. In that case, the heat gets transferred from the hotter region to the colder region.