One Dimensional Heat Conduction

Introduction

Heat conduction in a certain direction has a specific rate that is in proportion with the temperature gradient. The temperature gradient can be explained as the change rate of temperature with that distance in a certain direction. Heat conduction is called to be steady when the temperature is constant or does not vary from time to time. Heat conduction can also be explained as a thermal energy flow or transfer that takes place from those particles that have more energy in a given medium to those particles that have less energy. Therefore, in one-dimensional heat flow, the condition on temperature is its difference in only one direction which eventually causes the heat transfer. A larger rate of heat transfer happens when the temperature difference is also larger.

One Dimensional heat Conduction

The phenomena of heat transfer can happen in all directions and it is certainly not restricted. When the heat flow does happen from one medium to three other directions, it is three-dimensional. Similarly, when from a certain medium the heat transfer takes place only towards two different directions, it’s two-dimensional. Finally, when from one medium the heat transfer happens only in one direction, it’s one-dimensional. Therefore, as the dimensions reduce the heat transfer changes the temperature significantly. This is because when from a certain medium the heat is diffused to two different directions, the temperature change will not be affected as much as if the transfer would take place only in one direction.

Therefore, in the case of one-dimensional heat conduction, the difference in temperature lies only in one direction and not multiple directions, necessitating the heat flow in one direction only.

For instance, when an egg is put in boiling water, the egg has a temperature way lower than the temperature of the boiling water. The heat transfer starts from the boiling water to the egg (in one direction, i.e., towards the egg), this is one-dimensional heat conduction or heat transfer.

Fourier’s Law

Fourier’s law of heat conduction expresses that the rate of conduction of heat or flow of heat is proportional to the area that is normal the heat transfer’s direction and the temperature difference across those mediums, while it is inversely proportional to the other medium’s distance in that specific direction.

One dimensional heat flow equation or one-dimensional heat conduction equation as per Fourier’s law stands in the following differential form:-

Q_Cond= -kAdT/dx

Where,

k= Material’s thermal conductivity,

dT/dx= Temperature gradient.

One Dimensional Heat Conduction Equation

The change and balance of heat elements can be expressed through:-

(Heat conduction rate at x) + (rate of heat conduction at x +change in x) + heat generation rate inside the element) = (Change rate of the element’s energy content)

The one-dimensional heat conduction equation for cylinder, plane wall, and sphere can be expressed in the following manner:-

1rn ∂rrn k∂T∂r+ egen=ρc∂T∂t

Where,

n = 0 (in case of a plane wall); n=1 (in case of a cylinder); n=2 (in case of a sphere)

In case of heat transfer in a plane wall, r has to be replaced by x.

Conclusion

The topic of one-dimensional heat conduction has been explained thoroughly and in a lucid manner. The introduction section introduces one towards heat conduction and its scopes. The aspects of steady and unsteady heat conduction have been looked into in the context of one-dimensional heat flow. Moreover, the theoretical differences between one-dimensional heat flow with two and three-dimensional heat flows have been established. In one-dimensional heat flow, the condition on temperature has been explained. One dimensional heat flow equation and one-dimensional heat conduction equations have been mentioned. The FAQs section attempts to address the most probable queries that might arise. Moreover, this section makes the understanding of the topic of one-dimensional heat conduction much easier. FAQs sections contain additional information for this purpose.