Finding thermal conductivity with conduction and convection.

[miniradman]

Homework Statement

Determine the thermal conductivity of a metal (assume linear heat distribution at steady-state, and well insulated), given,

T_hot = 96.8
T_cold = 29.5

There is also:
13.7 W being pumped in at one end.
Water convecting heat away on the other end.

Homework Equations

Fourier's Law:
$q = -kA\frac{∂T}{∂x}$

The Attempt at a Solution

Hello, I was wondering about whether or not the water convecting heat will change the value of q in any way? I thought to use fourier's law, all you needed was the amount of energy going into one end and the temperature difference. I assumed that because the system is at steady-state (and is well insulated, the amount of heat going in will be the same as what is being convected out by the water. Hence q = 13.7W

However I'm getting a value of 13ish for thermal conductivity, and this material is suppose to be Aluminium so that's way off.

I got the equation:

$k=-\frac{qΔx}{A((Tcold)-(Thot))}$

which looks correct to me, however its been a long day...

ps.

I used the following Python code to computer the sum.

dx = 0.03
A = 4.504 * (10**(-4))
q = 13.7
T2 = 100.1
T3 = 97.9
T6 = 28.5
T7 = 26.5

def thermalConductivity():
    k = -((q*dx)/(A*(((T6+((T6-T7)/2))-(T3-((T2-T3)/2))))))
    return k

print(thermalConductivity())

 

[maajdl]

You should tell us the dimensions of the piece of metal with the proper units.
If you are "a way off", this is likely to be a problem of proper use of units.
You should also confirm us that the temperatures your indicate are the two end of the metal piece.
In addition, you should explain you Python: we do not clearly identify all the data:
- you indicate 4 temperatures, while 2 only are mentioned in the statement of the problem
- none of these 4 temperatures match the two temperature given in the statement of the problem
- length and cross section seem to be dx and A in the Python code but are not shown in the statement of the problem and are given no units
- the formula you use is very surprising, since it is based on 4 temperatures, and you should explain why you used it

A more careful statement of the problem will probably solve your problem

 

[miniradman]

Hello, maadj
Thanks for the response.

- you indicate 4 temperatures, while 2 only are mentioned in the statement of the problem

I used an approximation method to obtain the values for Thot and Tcold. Where T2 and T3 and are temperatures close to the position of Thot and T6 and T7 are temperatures close to Tcold. I worked them out prior to using the equation I have given (did do a calculation by hand, and my values agree with the python one).

length and cross section seem to be dx and A in the Python code but are not shown in the statement of the problem and are given no units

So:
dx = thickness of metal sample = 30mm = 0.03m
A = area of the cross sectional face with diameter 25mm = 4.504 * (10^-4)m

 

[maajdl]

Why didn't you write simply:

k = -q /( (Tcold-Thot)/dx ) / A = 13.7 W/m/°C approximatively

which is what you also obtained!

Why do you think that would be wrong?

 

[miniradman]

I seem to be error prone with my manual calculations, also, I needed to calculate the thermal conductivities of two other metals, at different temperatures. Which is why I wrote the program as well.

I know that is wrong because it seems like I'm a whole order of magnitude from a literature value of 205.0 W/m K

 

[maajdl]

I still don't understand why the T6, T7, T2, T3 in your program.
What is exactly the statement of the problem and what are the actual data?
And was it explicitly said that this metal was aluminium?
What was exactly written about this metal, was is said that it is a metal?

Is the problem theoretical, or is it the analysis of a lab experience?
If it is an experience, what is the setup and what are the conditions?

Could you be crystal-clear about the statement of the problem?
Thanks