Petroleum University of Technology

Verification of Conduction Heat Transfer Equation in Spherical Coordinates

So many friends of mine asked me about conduction heat equation in spherical coordinates and how to prove

it, since there was not a good resource on the net. So I tried to prove as easiest as possible. Hope to be useful.

We start by selecting a control volume dV, as indicated in the figure:

. .

Also, the following identity is useful:

sin Taylor series is used as well:

Radial Direction

!" ,

$.%&'(&) * +&',) 12 / 0 12 /2 0

!

From Taylor series we have:

23

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23 4

!5 1

Polar Direction

6 !"6

!"6 ,

$.%&'(&) * +&',) 12 / 0 12 /2 0

6 !

From Taylor series we have:

6236 6 6

6 6236 6 4

!5 2

Azimuth Direction

7 !"7

!"7

,

$.%&'(&) * +&',) 12 / 0

7 !

From Taylor series we have:

7237 7 879

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7 7237 879

! 3

Meanwhile we have:

;)

!)

!) 4=>? @ @ 5

1 5

4 !5

4

!5

! @

!)

By dividing both sides by we have:

1

4

!5

1

4

!5

1

!

@

;)

done.

Note that we used Taylor series instead of writing heat flux from the output sides which has long

manipulations.

Alireza Katamipour

Petroleum University of Technology

Abadan Institute of Technology