Heat transfer problem - conduction in a cylinder

To-Ti=C_1*ln(ro)+C_2-C_1*ln(ri)-C_2C_1=(To-Ti)/ln(ro/ri)C_2=TiT(r)=(To-Ti)/ln(ro/ri)*ln(r)+TiIn summary, the temperature distribution equation for a cylinder with no convection and constant temperature across its length is T(r)=(To-Ti)/ln(ro/ri)*ln(r)+Ti. To find the temperature at r=a, plug in the value for a in the equation.
  • #1
ana111790
42
0

Homework Statement


The following cylinder has a temperature inside Ti and temperature outside To. Using the general equation for heat conduction in a cylinder, write the temperature distribution equation as a function of the radius T(r). What is the temperature midway at r=a? (Take the heat conductivity = k, and length of cylinder is L).
Assume no convection and constant temperature across the length of the cylinder.

[PLAIN]http://img40.imageshack.us/img40/7344/radiusproblem.jpg

Homework Equations



Fourier's Law in cylindrical coordinates: q''= -k (dT/dr)

The Attempt at a Solution


Boundary conditions:
r=ri, T=Ti
r=ro, T=To

So integrating Fourier's equation with these boundary points I get:
To-Ti= -roq'' ln(ro/ri)

I think this gives the temperature difference though, not the distribution and I also have the q'' (flux term) still in the equation as an unknown. How would I find the temperature distribution and T(r=a)?
 
Last edited by a moderator:
Physics news on Phys.org
  • #2
Steady-state, no heat generation for cylinder: d/dr(r*dT/dr)=0

integrate twice with respect to r: T(r)=C_1*ln(r)+C_2
 

1. What is heat transfer problem - conduction in a cylinder?

Heat transfer problem - conduction in a cylinder refers to the study of how heat is transferred through a cylindrical object, such as a rod or pipe, due to differences in temperature. This is typically analyzed using the Fourier heat conduction equation, which takes into account parameters such as the thermal conductivity of the material and the temperature gradient.

2. What is the difference between conduction, convection, and radiation?

Conduction, convection, and radiation are three different mechanisms by which heat can be transferred. Conduction is the transfer of heat through a solid material, while convection is the transfer of heat through a fluid (such as air or water). Radiation is the transfer of heat through electromagnetic waves. In the case of conduction in a cylinder, heat is transferred through the solid material of the cylinder.

3. How is the temperature distribution in a cylinder affected by conduction?

The temperature distribution in a cylinder is affected by conduction because heat is transferred from areas of higher temperature to areas of lower temperature. This results in a temperature gradient within the cylinder, with the temperature being highest at the source of heat and decreasing as it moves towards the cooler end.

4. What factors influence the rate of heat conduction in a cylinder?

The rate of heat conduction in a cylinder is influenced by several factors, including the thermal conductivity of the material, the temperature difference between the two ends of the cylinder, the length and cross-sectional area of the cylinder, and the boundary conditions at the ends of the cylinder (e.g. insulated or exposed to external heat source).

5. What are some real-life applications of conduction in a cylinder?

Conduction in a cylinder has many practical applications, such as in the transfer of heat in heating and cooling systems, in cooking appliances like ovens and stovetops, and in industrial processes like metal casting. It is also important to understand for the design and optimization of thermal insulation materials and systems.

Similar threads

Heat transfer between objects by conduction
    • Engineering and Comp Sci Homework Help
Replies
13
Views
1K
Spec'ing the power of heater for an atypical heat exchanger problem
    • Engineering and Comp Sci Homework Help
Replies
22
Views
1K
Heat transfer problem for a long rectangular bar
    • Engineering and Comp Sci Homework Help
Replies
4
Views
1K
Heat and Mass Transfer: Heat Out
    • Engineering and Comp Sci Homework Help
Replies
1
Views
3K
1-D Heat Distribution in Long Cylinder
    • Engineering and Comp Sci Homework Help
Replies
1
Views
1K
Engineering Heat transfer from composite pipe -- Is my answer right?
    • Engineering and Comp Sci Homework Help
Replies
1
Views
942
Logic applied to making isothermal assumption
    • Engineering and Comp Sci Homework Help
Replies
1
Views
2K
I Warm-up time for a hollow cylinder
    • Thermodynamics
Replies
5
Views
1K
Heat Loss Through a Short Cylinder
    • Engineering and Comp Sci Homework Help
Replies
1
Views
1K
Heat and Mass Transfer Problem
    • Engineering and Comp Sci Homework Help
Replies
7
Views
2K

Hot Threads

Recent Insights

Back
Top