Heat flow problem- what did I do wrong?

In summary: In this case, you would need to use the thermal conductivities and temperatures of both copper and steel, as well as the cross-sectional areas. Set the heat fluxes equal to each other and solve for \Delta x. This will give you the length of the steel section.
  • #1
frasifrasi
276
0
A long rod, insulated to prevent heat loss along its sides, is in perfect thermal contact with boiling water (at atmospheric pressure) at one end and with an ice-water mixture at the other View Figure . The rod consists of 1.00-m section of copper (one end in steam) joined end-to-end to a length L2 of steel (one end in ice). Both sections of the rod have cross-section areas of 4.00 cm^2. The temperature of the copper-steel junction is 65.0 deg C after a steady state has been set up.

View Figure at http://session.masteringphysics.com/problemAsset/1042082/4/YF-17-70.jpg


1. How much heat per second flows from the steam bath to the ice-water mixture?

2. What is the length L2 of the steel section?

Now, here is my solution:

---------------------------------------------------------------------------------

∇q = 0
Because there is only heat flux along the rod this simplifies to:
dq/dx = 0 => q = constant
along the rod.

Heat flux is defined as
q = k·dT/dx
k is thermal conductivity

For constant q and k you find:
q = - k·ΔT/Δx

So the heat flow through the rod is
Q = A·q = -A·k·ΔT/Δx

The heat flow is constant , That means heat flow through copper section as through steel section as through the whole rod.

1.
consider copper section:
k = 400 W/Km for copper

Hence:
Q = -A·k·ΔT/Δx
= -4.00×10-4m² · 400W/Km · (65°C - 100°C) / 1m
= 5.6W

I put in 5.6, and it told me it was the wrong answer.

What did i do wrong?

Thanks.
 
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  • #2
Steel and copper have different thermal conductivities, so the temperature differentials will be different. The boiling water at 1 atm is at 100°C, which is sat temp at 1 atm, and the ice is at 0°C.

The interface is at 65°C, and the heat flux on both sides is equal, but the dT/dx is not.

Find expressions for the heat flux in the copper and steel at the interface and set them equal.
 
  • #3
How would you do that? I am not good with heat flux.

Please help me get started.
 
Last edited:
  • #4
How would I set up the expression?
 
  • #5
Q = A·q = -A·k·ΔT/Δx is correct.

Heat flux = q = -k ΔT/Δx. One knows the ΔT's and the length of copper.

So q (copper) = q (steel), and the k's are different.

The 5.6 seems correct, but for 1 sec the heat (energy) should be J. W is J/s, which is power.

Does the answer input require units?
 
Last edited:
  • #6
It requires W. I put in 5.6 and got "try again."

I don't know what to do. Shouldn't I add both q's?
 
  • #7
Please! help me like a comdemned animal.
 
  • #8
In order to have steady-state, what goes in one end must go out the other, i.e. the rate of heat transfer is constant along the length of the bars. The formula one used seems correct.
 
  • #9
Thermal Conductivity

Looks like you used a Thermal Conductivity value of 400 which is close to the value that wikipedia has. However, my book (and yours if same book p 664) uses the value of 385.0 for the thermal conductivity of copper.

...so you did it correctly. However, the answer using a thermal conductivity of 385 would be 5.39W.
 
  • #10
Length of L_2

How do you actually find the length?

Thanks in advance
 
  • #11
vanillacreme said:
How do you actually find the length?

Thanks in advance
One would use the heat flux equation and solve for [itex]\Delta x[/itex].
 

1. Why is my heat flow problem not balancing?

There could be several reasons for this. One common mistake is using incorrect units in your calculations. Make sure all values are in the correct units and be consistent throughout your calculations. Another possibility is that you have missed a term in your equations or made a calculation error. Double check your work and make sure all equations are correctly applied.

2. Why is my calculated heat transfer rate different from the expected value?

There are a few potential reasons for this discrepancy. One possibility is that your assumptions or boundary conditions are not accurate. Make sure all assumptions are reasonable and all boundary conditions are correctly applied. Another possibility is that there may be errors in your data or measurements. Check your data and measurements for accuracy and precision.

3. Why is my heat flow problem giving me a negative value?

In some cases, a negative value for heat transfer rate may be expected and correct. However, if you are not expecting a negative value, it could be due to incorrect sign conventions or a mistake in your calculations. Make sure to carefully follow the sign conventions for your problem and check your calculations for errors.

4. Can I use the same equations and methods for all heat flow problems?

No, different heat flow problems may require different equations and methods. It is important to carefully read and understand the problem statement and choose the appropriate equations and methods for that specific problem. Additionally, some problems may have simplifying assumptions that can affect the equations and methods used.

5. How can I improve my understanding of heat flow problems?

To improve your understanding of heat flow problems, it is important to have a strong foundation in thermodynamics and heat transfer principles. Practice solving different types of heat flow problems and make sure to understand the underlying concepts and equations. You can also seek help from a mentor or tutor, or use online resources and textbooks for additional learning and practice.

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