# Shell balances in cylindrical coordinates

I have a question regarding writing a shell balance for a cylindrical system with transport in one direction (in any area of transport phenomena). When we set up the conservation equation(say steady state), we multiply the flux and the area at the surfaces of our control volume and plug them into the eqn. Afterwards I believe that we divide the resulting equation by the volume of the control volume before taking the limit as DelX,DelZ or DelR approaches 0. In cartesian coordinates we divide this by delXdelYdelZ, but why in cylindrical CV do we divide by 2.pi.DelR.L instead of 2.pi.R.DelR.L (which is the volume of our CV)?

Chestermiller
Mentor
I have a question regarding writing a shell balance for a cylindrical system with transport in one direction (in any area of transport phenomena). When we set up the conservation equation(say steady state), we multiply the flux and the area at the surfaces of our control volume and plug them into the eqn. Afterwards I believe that we divide the resulting equation by the volume of the control volume before taking the limit as DelX,DelZ or DelR approaches 0. In cartesian coordinates we divide this by delXdelYdelZ, but why in cylindrical CV do we divide by 2.pi.DelR.L instead of 2.pi.R.DelR.L (which is the volume of our CV)?
Of course, you could also divide by ##2\pi r \Delta r L##. That won't change the final result. But, the authors must of thought it was more convenient and aesthetically pleasing to do it their way. Both ways are right.

Of course, you could also divide by ##2\pi r \Delta r L##. That won't change the final result. But, the authors must of thought it was more convenient and aesthetically pleasing to do it their way. Both ways are right.
Yeah but it changes the final answer you get from integrating the resulting ODE.

Chestermiller
Mentor
Yeah but it changes the final answer you get from integrating the resulting ODE.
Please show me how you think the answer will be different. Start out by showing me the two versions of the ODE that you get.