Derivation of Energy Equation in Cylindrical Coordinates

[clmohn]

So, I have searched the globe to find the derivation of the conservation of energy equation in 3D cylindrical coordinates. I have looked in about every heat transfer, thermodynamic, thermofluid books in the library. I have spent about 4 hours on the internet trying to find something, but no luck.
It seems like EVERY textbook I have looked through has the cartesian derivation and when I look at the problem sets at the end of the chapter what do I see? "Derive the control volume three dimensional energy equation in cylindrical coordinates." It's as if the publisher refused to print the cylindrical derivation. They just threw the end result in cylindrical coordinated in the appendix and call it a day.
I know it is a very lengthy and relatively complex derivation, but come on.

My question is, does anyone have a source for the derivation of the energy equation in cylindrical coordinates? I would truly appreciate it if you could link one.
Thank you for your time and reading my frustrations. =)

Chris

 

[lalbatros]

Probaly all you need is there:

http://en.wikipedia.org/wiki/Nabla_in_cylindrical_and_spherical_coordinates

Just observe how the divergence comes in the energy conservation law.

 

[clmohn]

Finite elemet

Thanks,
But, I am more looking for a finite element differential of the energy equation in cylindrical coordinates...

The free body is set up like a differential slice of pie (3d) with energy flux and work flux in and out of the control volume.

Thanks,
chris

 

[lalbatros]

Let's call (r,t,z) the cylindrical coordinates and d their witdh on and element.
Let's call (R,T,Z) the heat (energy) flux for these directions (in W/m²).
Let's cal Q the source of heat in the volume.

By simple geometric inspection, the net balance of heat flowing along the radial surfaces is:

br = R(r+dr,,) (r+dr) dt dz - R(r,,) r dt dz

You can get similar expressions for the other directions.
For the total heat balance you get then:

br + bt + bz = Q dr rdt dz

where the rhs is the total source in the volume element.

 

[dhar666]

Hi to Everyone.

I need also "derivation of energy equation in spherical coordinates" but i also could'nt find anything. Is there any information about this by explaining how to derive the equation by term by term?

 

[minger]

The fundamental equation is the same. The main difference is the Laplacian operator. Look for derivations on that, because the only difference is 1/r or sin(theta) terms. Those come from simple trig relations to find distances.

So, just follow along with the cartesian derivation, but keep a sketch of a cylindrical/spherical element. When you need to find an area, instead of delx*dely, you'll have something else.

 

[dhar666]

Thank you very much for your help.