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.
FAQs on Derivation of Heat Equation
1. What is the one-dimensional heat equation and what does it describe?
The one-dimensional heat equation is a fundamental partial differential equation (PDE) that describes how temperature is distributed and changes over time within a long, thin object, like a metal rod. It is mathematically expressed as ∂u/∂t = k * ∂²u/∂x². This equation helps in predicting the temperature u(x,t) at any position 'x' at any given time 't', based on the material's properties.
2. Why is the heat equation classified as a partial differential equation?
The heat equation is classified as a partial differential equation (PDE) because the function it describes, temperature (u), depends on more than one independent variable—specifically, position (x) and time (t). The equation involves partial derivatives with respect to both variables (∂/∂t and ∂²/∂x²), linking the rate of change of temperature in time to its curvature in space. An ordinary differential equation (ODE), by contrast, only involves derivatives with respect to a single independent variable.
3. What do the key variables and constants in the heat equation represent in physics?
In the standard one-dimensional heat equation, ∂u/∂t = k * ∂²u/∂x², each term has a specific physical meaning:
- u(x,t): Represents the temperature at a specific position 'x' along the object at a given time 't'.
- ∂u/∂t: Represents the rate of change of temperature with respect to time at a fixed point.
- ∂²u/∂x²: Represents the second partial derivative of temperature with respect to position. It describes the concavity of the temperature profile and is related to how heat spreads out.
- k: Represents the thermal diffusivity of the material. It is a constant that measures how quickly heat propagates through a medium.
4. What are the key physical assumptions made when deriving the one-dimensional heat equation?
The derivation of the one-dimensional heat equation relies on several important simplifying assumptions to make the model work. These include:
- The object (e.g., a rod) is perfectly insulated along its sides, ensuring that heat flows strictly in one dimension (along the x-axis).
- The material of the object is homogeneous and isotropic, meaning its properties like density (ρ), specific heat capacity (c), and thermal conductivity (K) are uniform throughout and the same in all directions.
- There are no internal sources or sinks of heat within the material itself.
- The physical properties of the material, such as thermal conductivity and specific heat capacity, do not change with temperature.
5. How does the concept of a temperature gradient form the basis for deriving the heat equation?
The concept of a temperature gradient is central to the derivation via Fourier's Law of Heat Conduction. This law states that the rate of heat flow across a surface is proportional to the negative of the temperature gradient (the rate of change of temperature with respect to position, ∂u/∂x). A steeper gradient means a faster heat flow. The derivation combines this law with the principle of conservation of energy (heat entering a section minus heat leaving it equals the change in internal energy) to establish the relationship between the time-derivative and the spatial second-derivative of temperature, resulting in the heat equation.
6. What is the importance of the heat equation in real-world applications beyond the physics classroom?
The heat equation is a powerful tool used to model and predict heat transfer in numerous practical scenarios. Its key applications include:
- Engineering: For designing efficient cooling systems in electronics (like heat sinks for processors), car engines, and industrial machinery.
- Material Science: To analyse thermal stresses in materials during processes like welding, casting, and quenching to prevent cracking or failure.
- Geophysics: In modelling the flow of heat within the Earth's crust and mantle to understand geological phenomena.
- Building and Construction: To design effective insulation for homes and buildings to maintain comfortable temperatures and improve energy efficiency.
