Steady state heat flow: radiation and conduction

[Sudikshya Pant]

Homework Statement

One end of a solid cylindrical copper rod 0.200 m long and 0.0250 m in radius is inserted into a large block of solid hydrogen at its melting temperature, 13.84 K. The other end is blackened and exposed to thermal radiation from surrounding walls at 500.0 K. (Some telescopes in space employ a similar setup. A solid refrigerant keeps the telescope very cold—required for proper operation— even though it is exposed to direct sunlight.) The sides of the rod are insulated, so no energy is lost or gained except at the ends of the rod. (a) When equilibrium is reached, what is the temperature of the blackened end? The thermal conductivity of copper at temperatures
near 20 K is 1670 W/m^2 K. b) At what rate (in kg>h) does the solid hydrogen melt?

Homework Equations

H = Aes(T^4-T_s^4)H =dQ/dt =kA(T^4-Tc^4)/L

The Attempt at a Solution

I tried to solve it by taking the equilibrium temperature as T and equated the two equations above thinking that the rate at which the radiation is absorbed by the black end equals to the rate at which the heat is conducted from that end to the colder end. But that gives me wrong answer. I can't seen to understand the equilibrium and thus, have to admit don't understand what to do and why I should do it. Thus, please help me solve as well as understand this problem. Thank you.

 

[mjc123]

Have another look at your heat conduction equation. Does it involve T⁴ terms?

 

[Sudikshya Pant]

Have another look at your heat conduction equation. Does it involve T⁴ terms?

No. I solved it without using the power four. Sorry, for the error here. (I can't even edit it now, can I?). Actually, I even kept the answer of T in the above equations but they don't seem to give equal values. Thus, I came to a conclusion that my method is wrong. The answer is 14.26K.

 

[mjc123]

Your approach seems to be correct - I get 14.26 that way - but as you haven't shown your working it's not possible to identify where the mistake is.
Simplifying hint: As T << 500, approximate (500⁴ - T⁴) by 500⁴.

 

[Sudikshya Pant]

I would like to know what value did you use for 'e' of copper. Am I wrong to use 0.3 for it?

Wait...that's where I went wrong. I should use 1 for "black" end of copper right?

 

[mjc123]

I used 1 and got the right answer.