Thermal Conductivity - Estimate Temperature

In summary, the problem involves finding the temperature on the inner surface of an insulating case given the temperature on the outer surface and the rate of heat radiation to the surroundings. Using the heat conduction equation, a value for the flux is obtained and then applied to Stefan's Law to find the temperature. The final result is approximately 26.8 degrees Celsius.
  • #1
unscientific
1,734
13

Homework Statement



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Given an insulating case outer surface at 25C, radiates heat to surroundings at 20C. Find temperature inner surface.

Homework Equations





The Attempt at a Solution



The heat is conducted through a thickness of 2.5mm, with a temperature difference of 5oC. Thus ##\frac{\partial T}{\partial z} = 2000##. Using ##k = 0.02##:

We have a value for the flux:

[tex]J = k\frac{\partial T}{\partial z} = 40[/tex]

Using Stefan's Law:

[tex]J = \sigma T^4[/tex]

We get ##T = 163K##, and boy that is one cold case!(-110oC)
 
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  • #2
You seem to think the 5oC difference is between the iPod and the outside of the case.
Looks to me as if that temperature difference is to be calculated from the flux.
Stefan's law works two ways: 25 to 20 should be something like [tex]J = \sigma \left ( T_{\rm case}^4 - T_{\rm environment}^4 \right )[/tex]
 
  • #3
BvU said:
You seem to think the 5oC difference is between the iPod and the outside of the case.
Looks to me as if that temperature difference is to be calculated from the flux.
Stefan's law works two ways: 25 to 20 should be something like [tex]J = \sigma \left ( T_{\rm case}^4 - T_{\rm environment}^4 \right )[/tex]

That's right, because ##J## is the net flux (caused by a temperature gradient). The answer turns out to be ##26.8^o##, which looks right.
 

FAQ: Thermal Conductivity - Estimate Temperature

1. What is thermal conductivity?

Thermal conductivity is a measure of a material's ability to conduct heat. It is the rate at which heat is transferred through a material per unit area, per unit temperature difference.

2. How is thermal conductivity calculated?

Thermal conductivity is calculated by dividing the amount of heat transferred through a material by the product of the cross-sectional area and the temperature gradient.

3. What factors affect thermal conductivity?

The factors that affect thermal conductivity include the type of material, its density, temperature, and the presence of any impurities or imperfections in the material.

4. How can thermal conductivity be measured?

Thermal conductivity can be measured through various experimental methods, such as the hot plate method or the transient hot wire method. It can also be estimated using mathematical models and equations based on the material's properties.

5. Why is estimating temperature using thermal conductivity important?

Estimating temperature using thermal conductivity is important in many scientific and industrial applications, such as designing heating and cooling systems, understanding heat transfer in materials, and predicting the behavior of substances under different thermal conditions.

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