Calculating Heat Loss and Temperature Drop Using Stefan-Boltzmann's Law

[munchy35]

Homework Statement

A ceramic teapot (epsilon(e) = 0.70) and a shiny one (epsilon(e) = 0.10) each hold 0.50 L of tea at 99 degrees C.
a) Estimate the rate of heat loss from each. Consider only radiation, and assume the surroundings are at 20 degrees C.

b.)Estimate the temperature drop after 30 min for each.

Homework Equations

Stefan-Boltzmann's Law

change in Q/change in T = epsilon*Stefan constant*Area (T^4 - T^4)

The Attempt at a Solution

My professor told us to assume that the tea pot is a sphere so we can solve for the area.

so V = 4/3pi R^3 = .05 m^3 R=.2285

A = 4pi R^2 = .6563



Ceramic = (.70)(5.67*10^-8)(.6563)((372^4)-(293^4))
Ceramic = 306.87 W

Shiny = (.1)*5.67*10^-8)(.6563)((372^4)-(293^4))
Shiny = 43.84

Then we're supposed to use two sig figs, which I did,and the program I used keeps telling me I am wrong.

 

[munchy35]

okay so I realized .5 L = .005 m^3

but still I get 66 for ceramic and 9.4 for shiny.

it still says I am wrong

 

[munchy35]

nevermind. i got the right answer.

.5 L = .0005 m^3

gotta work on my conversions!

 

[munchy35]

now i have no idea how to approach part b once i have my answers for part a.

 

[xmonsterx]

hey I am working on the same problem too and i am stuck on part b and i don't know how to even approach this problem. did u figure it out yet?

 

[Oscur]

In the first part you've worked out the *rate* of heat loss, which is energy dissipated over time, right? So you have (with some notation abuse*):

$$H=\frac{E}{T}$$

I may be wrong, but from there, knowing how long the heat was dissipating for (in SECONDS, of course), you should be able to get a change in energy and then temperature...

*Technically this should be a time derivative, as the rate of heat loss is dependent on temperature itself, but...

 

[ideasrule]

Yup, Oscur is right. You won't get an exact answer because H should be the time derivative and not E/T, but that's why the question says "estimate".

 

[xmonsterx]

im still confused. =( so what is the formula to figure out the temperature drop after 30mins? and are we suppose to use derivative to solve this problem? I am so sorry to bother you about this but I am really lost.

are we suppose to use this equation? ---> H = E/T
if so... what is H? E? T?

sooooooo lost.

 

[Oscur]

Ok, basically the rate at which something radiates heat depends on its temperature. This is evident from the form of the Stefan-Boltzmann Law, which has a term in T in it. As a result, to properly calculate the temperature after a given time, you'd need to integrate with respect to time.

As the question says "estimate", however, I'd assume you can just rearrange the equation I gave (in which H is the rate of heat loss, E is a total change in energy and T is a time period, sorry I should have defined my variables and possible the capital T was confusing matters. Call it "t" from now on.)

That way, you have: H*t = E where E is the energy lost by the object. Any more helpful?