Conduction through a spherical shell.

Click For Summary
SUMMARY

The discussion focuses on solving the heat conduction problem through a spherical shell with a thickness of 5 cm and a radius of 0.5 m. The initial formula used is Q=KA(T2-T1)/L, where K represents thermal conductivity. The user is guided to derive the differential area element in polar coordinates, dA = rdθ * rsinθ dφ, and to express the heat transfer as dQ = rdθ * rsinθ dφ * K(T1-T2)/0.05. While a full integration is suggested, it is noted that an exact solution requires accounting for the varying area between the inner and outer surfaces of the shell.

PREREQUISITES
  • Understanding of heat conduction principles
  • Familiarity with polar coordinates in calculus
  • Knowledge of thermal conductivity (K) and its application
  • Basic integration techniques
NEXT STEPS
  • Study the derivation of heat conduction equations in spherical coordinates
  • Learn about the impact of varying surface areas on heat transfer
  • Explore numerical methods for solving differential equations in thermal analysis
  • Investigate advanced topics in heat transfer, such as transient conduction
USEFUL FOR

Students and professionals in thermal engineering, physicists, and anyone involved in solving heat transfer problems in spherical geometries.

kristen32123
Messages
1
Reaction score
0
How do I solve for the temperature at a certain time for conduction through a spherical shell? I have a shell 5 cm thick with a radius of 0.5m. I am starting with Q=KA(T2-T1)/L and I am completely lost as to where to go with it.

Thanks!
 
Physics news on Phys.org
Imagine the sphere has lines of latitude and longitude drawn on it. The (differential element of ) area of a rectangle defined by these lines is ( working in polar coords r, \theta, \phi)

dA = rd\theta*rsin\theta d\phi

so, using your formula

dQ = rd\theta*rsin\theta d\phi*K(T_1-T_2)/0.05

Now integrate out d\theta, d\phi between 0 and 2*pi, and that gives the formula for the whole shell.

Of course you don't actually need to integrate - just think about it and the answer is obvious.

This solution is not exact because we're ignoring the difference in area btween the outer and inner surfaces ...

To solve exactly one needs a differential dL of thickness. Not a trivial problem.
 
Last edited:

Similar threads

Electric potential due to shell containing a charge at an offset outside
  • · Replies 5 ·
Replies
5
Views
1K
Spherical conducting shell enclosing a non-conducting core
  • · Replies 5 ·
Replies
5
Views
771
Two concentric conducting spherical shells and resistor in between
  • · Replies 44 ·
2
Replies
44
Views
5K
Charged sphere and charged conducting shell
  • · Replies 9 ·
Replies
9
Views
3K
Electric field of spherical shell
  • · Replies 2 ·
Replies
2
Views
2K
Conductive and grounded shells
  • · Replies 9 ·
Replies
9
Views
2K
Electric field inside non-conducting spherical shell
  • · Replies 5 ·
Replies
5
Views
5K
Find the electric field inside and outside of a spherical shell superposition
  • · Replies 12 ·
Replies
12
Views
3K
How Does Charging a Conducting Shell Affect Potential Difference?
  • · Replies 7 ·
Replies
7
Views
2K
Spherical conductor shell problem
  • · Replies 19 ·
Replies
19
Views
4K