Fourier’s Law

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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 the Fourier’s law. So let us go through the below article to understand what is Fourier’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\[^{\rightarrow }\] = - 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.

Mean heat flux can be represented as:

q = \[\frac{Q}{A}\][W.m\[^{2}\]]

Here, A = heat transfer area.

  • 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[\[\vec{r}\], T(\[\vec{r}\],t)] = \[\frac{\vec{q}_{x}}{\frac{\partial T}{\partial x}}\]

For a lot of materials that are homogeneous, thermal conductivity can be written as k = k (T). The same type of expressions 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):

Q\[_{cond}\] = kA (T1 − T2 / Δx) = −kA (ΔT / Δx)

Now when Δx is zero, the previous equation in differential form can be written as:

Q\[_{cond}\] = −kA (ΔT / Δx)

Furthermore, the 3D form of Fourier’s law is:

q\[^{\rightarrow }\] = - k▽T

After going through what is 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 10-3 [m] = 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 Publicized 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) Homogenous and isotropic material

By now, you must have understood what is Fourier’s law, its derivation, and other related topics. For more materials related to this topic download the Vedantu app today!

FAQ (Frequently Asked Questions)

1. Which Substance Holds the Highest Thermal Conductivity?

Ans. 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 meter per Kelvin; five times more than metals like copper.

2. What Do the Terms Conduction, Convection, and Radiation Mean?

Ans. 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.

3. How to Restrict the Loss of Heat from Radiation?

Ans. 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 windows that contains vacuum or air between its two glass panes.

4. Give Some Day to Day Life Examples of Conduction?

Ans. When you put a pan on a lighted stove, heat transfer takes place from the burner to the utensil. Next, when you immerse a metal spoon in a cup of hot boiling water, it gets heated. Again, if you are holding a piece of chocolate, it will automatically melt due to the heat conducted from your palm.