Heat conduction equation in cylindrical coordinates

3 min readJan 27, 2017

Heat conduction equation of cylindrical coordinates

What is the equation for cylindrical coordinates?

We have already seen the derivation of heat conduction equation for Cartesian coordinates. Now, consider a cylindrical differential element as shown in the figure.

We can write down the equation in Cylindrical Coordinates by making TWO simple modifications in the heat conduction equation for Cartesian coordinates.

a. Replace (x, y, z) by (r, φ, θ)

b. Use factors

and modify the equation in Cartesian coordinates. The terms in the numerators go inside the bracket with k, while the denominators go in the denominator outside the bracket.

The differential heat conduction equation in Cartesian Coordinates is given below,

Now, applying the two modifications mentioned above:

Hence,

Special cases

(a) Steady state

Steady state refers to a stable condition that does not change over time. Time variation of temperature with respect to time is zero.

(b) Uniform properties

If the material is homogeneous and isentropic, the thermal conductivity of the material would be constant.
{Comment: What do you mean by homogeneous and isentropic material? The term homogenous means that the values of physical properties of a material do not vary with position within the body of the material. E.g., The value of thermal conductivity at position (r1, φ1, z1) will be same as those at some other position (r2, φ2, z2). The term isentropic means, the value of physical properties at a point in different directions will be same. That is to say kr=kφ=kz at a point.}