Fourier’s Law

It is well known that the conduction of heat occurs when the molecules of matter vibrate or agitate, causing them to transmit energy to the molecules in their immediate vicinity. As the molecules collide with one another, heat energy is transferred from a higher temperature area to a lower temperature area. Fourier’s law is followed in this procedure. In order to better understand Fourier’s law. It is also referred to as the law of thermal conduction equations or the law of thermal conductivity, depending on how it is expressed. There are certain concepts that students should learn and not understand in order to understand this law index, such as Newton’s law of cooling, Ohm’s law, heat transfer, change of state, specific heat capacity, and temperature measurement.

What is Fourier’s Law, and how does it work?

In thermal conduction, Fourier’s law states that the rate of heat transfer through a material is proportional to the negative gradient in temperature and the area (perpendicular to the gradient) of the surface through which the heat flows.

Heat transfer processes can be quantified using rate equations, which are mathematical representations of the processes. In order to understand the rate equation for conduction (a mode of heat transfer), we must first understand Fourier’s law of thermal conduction. Specifically, it states that the rate of heat transfer across a substance is directly proportional to the negative gradient in temperature and area, at 90 degrees to that gradient, in which the heat transfer occurs.

Fourier’s Law is expressed in a differentiable form

The differential form of Fourier’s law is as follows:

q = – k▽T

Where,

q is the local heat flux density in W.m²

k is the conductivity of the material in W.m^-1.K^-1

▽T is the temperature gradient in K.m^-1

In one-dimensional form:

 q_x=-k ∂T ⁄ ∂x 

Integral form

Fourier’s law Integral form is as follows:

∂Q ⁄ ∂t=-k∯_sΔ T. dS 

Where,

is the amount of heat transferred per unit time

dS is the surface area element

When the same equation is given in the differential form, which is the basis of heat equation derivation:

 Q Δt ⁄ =-kA(ΔT ⁄ Δx) 

Where,

A is the area of the cross-sectional surface

ΔT is the temperature difference between the endpoints

Δx is the distance between two ends

Fourier’s law in terms of conductance

Δ Q ⁄ Δt=UA(-ΔT )

Where,

U is the conductance

In a substance, the thermal conductivity (k) is nothing more than the proportionality constant that was obtained through the expression. When energy transfer occurs rapidly through the process of conduction in a body, that body is referred to as an excellent thermal conductor. Furthermore, it has a statistically significant value of k.

In order to find the solution of Fourier’s law, the relationship between geometry, temperature difference, and thermal conductivity of the material must first be established and then tested. In the year 1822, Joseph Fourier published his conclusion, which stated that “the heat flux resulting from thermal conduction is proportional to the magnitude of the temperature gradient and opposite to it in sign.”

Heat flux

The rate of heat transfer per unit area normal to the direction of heat transfer is referred to as the heat flux. It is also referred to as heat flux density in some circles. Because it is a vector quantity, it has both a magnitude and a direction associated with it.

Thermal Conductivity

Thermal conductivity is a property of a solid body that can be used to quantify the characteristics of heat transfer in that body. It is important to remember that Fourier’s law applies to all states of matter, whether they are solid, liquid, or gaseous. As a result, it can be defined in terms of both liquids and gases.

In solids and liquids, the thermal conductivity of a maximum number of substances varies with temperature, and in gases, the thermal conductivity varies with pressure.

k=k_r→,T(_r→,t)=q_x→/(∂T/∂x)

Thermal conductivity can be written as k = k for a large number of materials that are homogeneous in nature (T). Thermal conductivities in the y (negative) and z (negative) directions are both expressed using the same type of expression, which is shown below. Thermal conductivity, on the other hand, is not dependent on the path of heat transfer in the case of an isotropic substance.

k_x = k_y = k_z = k

It can be deduced from the above expression that as thermal conductivity and temperature difference increase, conduction heat flux increases as well. In general, the thermal conductivity of a solid substance is more significant than that of a liquid or gas substance. This is due to the difference in intermolecular spaces between the two molecules.

Fourier’s Law Derivation

The temperature difference between two points in a short distance is denoted by the letters T1 and T2. The distance is denoted by x, the area is denoted by A, and the conductivity of the material is denoted by k. It is therefore possible to construct the following equation in a single dimension:

Q _cond = kA (T1 − T2 / Δx) = −kA (ΔT / Δx)

When x is equal to zero, the differential form of the previous equation can be written as follows:

Q _cond = −kA (ΔT / Δx)

Furthermore, the three-dimensional form of Fourier’s law is as follows:

q→=−k∇T

Conclusion

It is also known as the law of heat conduction, and it is primarily defined as follows: the rate at which heat transfers through a material is considered to be proportional to the negative gradient present in the temperature and to the area of the material that is at right angles to the gradient through which the heat flows are both proportional to the negative gradient present in the temperature. The integral form and the differential form are both equivalent representations of this law. This law is considered to be in an empirical relationship, which means that it is based on observational information.