An Introduction to Fourier’s Law
It is known that the conduction of heat takes place when the molecules of matter vibrate or agitate and transmit energy to the adjacent molecules. As the neighbouring molecules collide, heat energy is transferred from a higher temperature area to a lower one. This process abides by Fourier’s law. So let us go through the article below to understand what is Fourier’s law. Fourier’s law is also called the law of thermal conduction equations or the law of thermal conductivity. This law is taught in schools in the NCERT book that follow the CBSE curriculum. In order to understand this law index, there are certain concepts that students should learn and not understand like Newton's law of cooling, ohm's law, heat transfer, change of state, specific heat capacity, measurement of temperature.
Fourier’s law is also known as the law of heat conduction, it mainly states that the heat transfer rate through a material is considered to be proportional to the negative gradient present in the temperature, as well as to the area which is at right angles to the gradient, through which the heat flows. This law has two equivalent forms- integral form and differential form. This law is considered to be in an empirical relationship that is based on observation.
If the thermal conductivity of a solid is greater than the ability to conduct heat through it will also be greater. However, if the heat conductivity of the solid is lower than the ability to conduct heat through it will also be lower. The rate at which heat conduction takes place requires certain factors for the medium through which it takes place like the geometry, thickness, material, the temperature difference across the medium also matters.
Key Concepts of the Topic
Key concepts whose knowledge is necessary for the understanding of Fourier’s law-
Temperature and heat
Measurement of temperature
Ideal-gas equation and absolute temperature
Thermal expansion
Specific heat capacity
Calorimetry
Change of state
Heat transfer
Newton’s law of cooling
Ohm’s law
What is Fourier’s Law?
“Fourier’s law of thermal conduction states that the rate of heat transfer through a material is proportional to the negative gradient in the temperature and the area (perpendicular to the gradient) of the surface through which the heat flows.”
Processes of heat transfer are measurable in the form of rate equations. The rate equation for conduction (a mode of heat transfer) is deduced based on Fourier’s law of thermal conduction. It says that the rate of transfer of heat across a substance is directly proportional to the negative gradient in temperature and area, at 90 degrees to that gradient, in which flow of heat occurs.
The differential form of Fourier’s law can be represented as:
q = - k▽T
where,
∇T is temperature gradient (K. m-1)
k is the conductivity of the materials (W. m-1. K-1)
q is the heat flux density vector (W. m-2)
Thermal conductivity (k or λ) of a substance is nothing but the proportionality constant acquired in the expression. A body in which energy transfer occurs rapidly by the process of conduction is considered an excellent thermal conductor. Also, it has a significant value of k.
To find the solution of Fourier’s law, the relationship of geometry, temperature difference, and thermal conductivity of the material is derived. Joseph Fourier first introduced this law in the year 1822 and concluded: “the heat flux resulting from thermal conduction is proportional to the magnitude of the temperature gradient and opposite to it in sign”.
Heat flux
Heat flux is the heat transfer rate per unit area normal to the direction of heat transfer. It can also be referred to as heat flux density. Since it is a vector quantity, it has both magnitude and direction.
Thermal Conductivity
Features of heat transfer of a solid body can be quantified by the virtue of its property, thermal conductivity. You must remember that Fourier’s law is applicable to all states of matter, be it solid, liquid, or gas. Hence, it can be defined for liquids and gases as well.
The thermal conductivity of a maximum number of solids and liquids varies with temperature, and for gases, it depends on pressure.
\[k = k r^{\rightarrow},T(r^{\rightarrow},t) = \frac{q_x^\rightarrow}{\frac{\partial T}{\partial x}}\]
For a lot of materials that are homogeneous, thermal conductivity can be written as k = k (T). The same type of expression is related to thermal conductivities in y (negative) and z (negative) directions. However, for an isotropic substance, thermal conductivity is not dependent on the path of heat transfer.
kx = ky = kz = k
From the above expression, it can be said that when thermal conductivity and temperature difference increases, conduction heat flux also increases. Generally, a solid substance’s thermal conductivity is more significant than a liquid and gas. This is because of the difference in intermolecular spaces.
Did You Know?
Diamond is the hardest material and has the highest thermal conductivity.
Fourier’s Law Derivation
Consider T1 and T2 to be the temperature difference through a short distance of an area. Here the distance is Δx and the area is denoted as A and k is the material’s conductivity. Henceforth, the following equation can be formed (in one dimension):
Qcond = kA (T1 − T2 / Δx) = −kA (ΔT / Δx)
Now when Δx is zero, the previous equation in differential form can be written as:
Qcond = −kA (ΔT / Δx)
Furthermore, the 3D form of Fourier’s law is:
\[q^{\rightarrow} = -k\nabla T\]
After going through Fourier's law and related topics, next take a look at a solved example of heat loss.
Numerical Example Showing Loss of Heat through Windows
One of the major reasons for heat loss from a house is through its windows. Find the heat flux rate from a glass window having an area 1.5 m x 1.0 m and width 3.00 mm, provided the temperatures of outer and inner surfaces are 13.0 degrees Celsius and 14.0 degrees Celsius.
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Answer:
Over here, temperatures of the surfaces are given based on the conditions of the inner and outer parts of the house. Hence, the flow of heat takes place from a higher temperature inside the house to lower temperatures outside.
By using Fourier’s law equation, the following can be assumed:
Thermal conductivity of glass k = 0.96 W / m.K
Then,
Heat flux q = 0.96 W/m.K x 1 K / 3.0 x 10-3m = 320 W / m2
Net loss of heat from the window:
qloss = q . A = 320 x 1.5 x 1.0 = 480 W
Do It Yourself
1. The Fourier Theory Was Published in Which Year?
(a) 1830 (b) 1845 (c) 1824 (d) 1822
2. Determine the Wrong Assumption Made on Fourier’s Law.
(a) No heat is generated internally
(b) Non-linear temperature profile
(c) Heat conduction in steady-state
(d) Homogeneous and isotropic material
Conclusion
By now, you must have understood what is Fourier’s law, its derivation, and other related topics. This makes it easy to learn further concepts.
FAQs on Fourier’s Law
1. How to derive Fourier’s law?
Consider T1 and T2 to be the temperature difference through a short distance of an area. In this equation, the distance Δx is donated as A, and the area is donated as k which is the material’s conductivity. Therefore, the equation is as follows-
Qcond = kA (T1 − T2 / Δx) = −kA (ΔT / Δx)
Here we can see that, Δx is zero, therefore, the previous equation in differential form can be written as:
Qcond = −kA (ΔT / Δx)
In 3D for Fourier’s law can be written as-
\[q^{\rightarrow} = -k\nabla T\]
This is how Fourier law or the law of thermal conductivity is derived.
2. What are the important topics that are needed to understand Fourier’s law?
The important topics related to Fourier’s law are as follows-
Temperature and heat
Measurement of temperature
Ideal-gas equation and absolute temperature
Thermal expansion
Specific heat capacity
Calorimetry
Change of state
Heat transfer
Newton’s law of cooling
Ohm’s law
Students are advised to learn and understand these various concepts as it is only after gaining knowledge of these, can a student learn about the things taught further in classes such as Fourier’s law or the law of heat conduction or the law of thermal conductivity. These topics are prescribed by the CBSE and students wanting to get a good score in their class 12 board examination are advised to get a good grasp of these concepts of physics.
3. Which Substance Holds the Highest Thermal Conductivity?
Besides graphene and graphite, diamond is also considered to be a good thermal conductor at room temperature. It has a thermal conductivity greater than 2,000 watts per metre per Kelvin; five times more than metals like copper.
4. What Do the Terms Conduction, Convection, and Radiation Mean?
When thermal energy transfer takes place through direct contact, it is termed as conduction. When thermal energy is transferred during the movement of a gas or liquid, it is called convection. When the transfer of thermal energy occurs because of thermal emission, it is known as radiation.
5. How to Restrict the Loss of Heat from Radiation?
Simple techniques like fitting curtains, carpets, and draught excluders can be done to reduce the loss of heat. You can also install reflective foil in the room walls or on it. Moreover, you can prevent heat loss from windows with the help of double glazing. These are a unique type of window that contains vacuum or air between its two glass panes.