Heat conduction equation in spherical coordinates
Heat conduction equation in spherical coordinates
What is the equation for spherical coordinates?
We have already seen the derivation of heat conduction equation for Cartesian coordinates. Now, consider a Spherical element as shown in the figure:
We can write down the equation in Spherical 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 factor
and modify the equation in Cartesian coordinates. The terms in the nominator go inside the bracket with k, while the denominator go in the denominator outside the bracket.
The differential heat conduction equation in Cartesian Coordinates is given below,
Now, applying 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 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, 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 that 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.}
(c ) No heat generation within the element
In case, when there is no heat generation within the material, the differential conduction equation will become,
(d) One-dimensional form of equation
If heat conduction in any one direction is in dominance over heat conduction in other directions,