How can I calculate the cooling time of a steel ball using thermal radiation?

In summary, the conversation is about a physics homework problem involving a steel ball, its density and specific heat capacity, and the time it takes for the ball to cool down to half of its temperature through thermal radiation. The conversation includes equations and an attempt at a solution, with a discrepancy between the calculated result and the correct result. The problem is later identified as an algebra mistake.
  • #1
mstrakl
3
0

Homework Statement


Hi! I need some help by my phisycs homework at universty. Here is the problem:
We have a steel ball, with density of 7800kg/m^3, with speficic heat capacity 460J/kgK. Temperature of the ball is 1700K. How much time does the ball need to cool down to half of it's temperature (850K), if it cools only because of the thermal radiation. Stefan-Boltzman constant is 5,67×10^-8. There is no other radiation on the ball, from the surrounding area.
The ball's mass is 1kg.

Homework Equations




The Attempt at a Solution


I suppose, that the power of radiation (P = j × A = (s.b. constant) × T^4 × A) is decreasing as the temperature drops, so i tried to set up my equaton with differentials:

dQ = Pdt
mc dT = A × (sb const.) × T^4 dt

i integrated the equation, and got this:
mcT^-4dT = A(sb const)dt
-1/3 mcT^-3= A(sb const) × t

integral is from Ta/2 to Ta

and i shrinked the expression and expressed time from the equation

t= 8mc/(Ta)^3×A×(sb. const)

; c= spec. heat capacity
sb const. = stefan boltzman constante = 5,67 × 10^-8
Ta = temp. at start = 1700K

the result i get is 51 secons. The solution should be 313 seconds. What am I doing wrong? Can you help me?
 
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  • #2
mstrakl said:
mcT^-4dT = A(sb const)dt
-1/3 mcT^-3= A(sb const) × t

integral is from Ta/2 to Ta

and i shrinked the expression and expressed time from the equation

t= 8mc/(Ta)^3×A×(sb. const)

Check your algebra... when solving the integral you should have a 7 instead than 8... anyway (using of course 7) I have the right result...
 
  • #3
I did the algebra again, I hope photo sharp enough
... i get the same result, I checked twice.

51,14 seconds.

The correct result should be 313 seconds :S
 

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  • #4
I probably found your problem... when computing
$$ \left(\frac{1}{T^3}\right)_{T/2}^{T}=\frac{1}{T^3/8}-\frac{1}{T^3}=\frac{8}{T^3}-\frac{1}{T^3}=\frac{1}{T^3}(8-1) $$
instead you computed
$$ \frac{1}{T^3/8 -T^3} $$
which is clearly different because then you don't get a factor 1/8...
Using the right formula you should have 313 more or less (I did it before and it was OK) :smile:
 
  • #5
Oh of course... I have found the problem now... Thanks :)
 

Related to How can I calculate the cooling time of a steel ball using thermal radiation?

1. How does cooling by thermal radiation work?

Cooling by thermal radiation is a process in which heat is transferred from one object to another through the emission and absorption of electromagnetic waves. Objects with higher temperatures emit more radiation, and this radiation is absorbed by cooler objects, causing them to cool down.

2. What is the role of thermal radiation in cooling systems?

Thermal radiation plays a crucial role in cooling systems as it allows heat to be transferred from the hot components of a system to the cooler surroundings. This helps to maintain the temperature of the system and prevent overheating.

3. How does the color of an object affect its cooling by thermal radiation?

The color of an object affects its cooling by thermal radiation as different colors have different levels of reflectivity and emissivity. Darker colors tend to absorb more radiation and thus cool down faster, while lighter colors reflect more radiation and may not cool down as quickly.

4. Can thermal radiation be used for cooling in space?

Yes, thermal radiation can be used for cooling in space. In a vacuum, there is no medium for heat to be transferred through conduction or convection, so thermal radiation is the primary method of heat transfer. This is why objects in space, such as spacecraft, have reflective surfaces to prevent overheating from the sun's radiation.

5. How does thermal radiation contribute to the Earth's energy balance?

Thermal radiation plays a significant role in the Earth's energy balance as it is responsible for balancing the incoming solar radiation with the outgoing thermal radiation. This helps to regulate the Earth's temperature and maintain a suitable climate for life to thrive on our planet.

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