Thermal Conduction Homework: Find Rod's Conductivity

In summary, the problem involves a rod with insulated sides immersed in boiling water at 100C and a water/ice mixture at 0C. The rod has a uniform cross-sectional area of 4.04 cm^2 and length 91cm. Under steady state conditions, the rod conducts heat that melts 1.0g of ice every 34 seconds. To find the thermal conductivity, the equation H=dQ/dt=k*Area(TempChange)(1/length) is used, along with the heat of fusion of water (3.34*10^5 J/kg). The correct answer is 220 W/m*K. The mistake in the solution was not converting units properly and not accounting for the rate at which mass
  • #1
Joshb60796
62
0

Homework Statement


A rod, with sides insulated to prevent heat loss, has one end immersed in boiling water at 100C and the other end immersed in a water/ice mixture at 0C. The rod has uniform cross-sectional area of 4.04 cm^2 and length 91cm. Under steady state conditions, heat conducted by the rod melts the ice at a rate of 1.0g every 34 seconds. What is the thermal conductivity of the rod?


Homework Equations


H=dQ/dt=k*Area(TempChange)(1/length)
Heat of Fusion of water is 3.34*10^5 J/kg


The Attempt at a Solution


(3.34*10^5*91)/(34seconds*100C*4.04cm^2) = 2200

My answer key says 220 W/m*K, I've tried converting 91cm to .91m and 4.04cm^2 to .000404m^2 and I get the same 2200 answer. I think I'm making a conversion error but I'm not sure, please advice, thank you.
 
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  • #2
have you realized that it is 1x10^-3 kg melted in 34secs
 
  • #3
Your work is off by a factor of 1000 because you forgot to multiply the heat of fusion of water (J/kg) by the rate at which mass is melting (0.001 kg). Only 334 J of heat is being transferred in 34 sec, NOT 334,000 J.
 
  • #4
hmm I believe I follow what you are saying. I recalculated with 334J and my answer comes to 2.20 but that's still not what the answer key says. Am I just dense or is maybe the key incorrect?
 
  • #5
wait, maybe the 2.20 is in W/cm*K and the 220 on the key is W/m*K? ...nevermind, that would be backwards
 
  • #6
I don't know what to tell you, because it's just arithmetic at this point. You are messing it up somewhere and just need to be meticulous and get it right.
 
  • #7
Joshb60796 said:

Homework Statement


A rod, with sides insulated to prevent heat loss, has one end immersed in boiling water at 100C and the other end immersed in a water/ice mixture at 0C. The rod has uniform cross-sectional area of 4.04 cm^2 and length 91cm. Under steady state conditions, heat conducted by the rod melts the ice at a rate of 1.0g every 34 seconds. What is the thermal conductivity of the rod?


Homework Equations


H=dQ/dt=k*Area(TempChange)(1/length)
Heat of Fusion of water is 3.34*10^5 J/kg


The Attempt at a Solution


(3.34*10^5*91)/(34seconds*100C*4.04cm^2) = 2200

My answer key says 220 W/m*K, I've tried converting 91cm to .91m and 4.04cm^2 to .000404m^2 and I get the same 2200 answer. I think I'm making a conversion error but I'm not sure, please advice, thank you.
(334*0.91)/(34seconds*100C*.000404) = 221
 
  • #8
Thank you Chester, I'd been messing with this over and over and apparently it's like trying to grammar check your own novel, I never tried both converting to meters, and fixing my gram and kilogram mistake. Thank you so much :) I don't know how I missed it now.
 

1. What is thermal conductivity and how is it measured?

Thermal conductivity is the measure of a material's ability to conduct heat. It is measured in watts per meter Kelvin (W/mK) and is determined by measuring the rate of heat transfer through a material of known thickness and temperature difference.

2. What is the purpose of finding a rod's thermal conductivity?

The purpose of finding a rod's thermal conductivity is to understand how well the material will transfer heat. This information is important in various fields such as engineering, materials science, and thermodynamics.

3. How is thermal conductivity affected by different materials?

Thermal conductivity is affected by a material's physical properties such as density, specific heat, and molecular structure. Generally, materials with higher density and specific heat will have higher thermal conductivity.

4. How does temperature affect thermal conductivity?

In most materials, thermal conductivity increases with an increase in temperature. However, some materials, such as metals, may experience a decrease in thermal conductivity at very high temperatures due to changes in their atomic structure.

5. What factors can affect the accuracy of measuring a rod's thermal conductivity?

The accuracy of measuring a rod's thermal conductivity can be affected by factors such as the material's surface roughness, moisture content, and impurities. The measurement method and equipment used can also impact the accuracy of the results.

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