Temperature of a Conducting Rod

[ObliviousSage]

Homework Statement

A copper rod 90cm long is used to poke a fire. The hot end of the rod is maintained at 103 degrees Celsius and the cool end has a constant temperature of 25C. What is the temperature of the rod 24cm from the cool end?

specific heat of copper: c = 387 J/kg*K
thermal conductivity of copper: k = 395 W/m*K
thermal expansion coefficient of copper: $$\alpha$$ = 17*10^-6 1/K

Homework Equations

conduction: Q = kAt$$\Delta$$T/L
linear expansion: $$\Delta$$L = $$\alpha$$L₀$$\Delta$$T
heat: Q = mc$$\Delta$$T

The Attempt at a Solution

I'm not even sure where to begin. The three equations in this chapter that might be relevant are given, but I can't see any way to apply them. It seems like conduction has to be involved, but I can't figure out what happened to the area and time terms.

 

[ObliviousSage]

Bump to get back on first page?

 

[Mapes]

If one end is held at 103°C and the other end is held at 25°C, what is the temperature profile along the beam going to look like?

 

[chaoseverlasting]

First, set up a coordinate system. Say the hot end of the rod is the origin and the other end of the rod is at x=l.

Now, you know that the rod will expand depending on the temperature. As the temperature of the rod will vary along the length of the rod, different parts of the rod will expand to a greater or lesser degree. Let the temperature of the rod at a point x cm away from the hot end be t.

Now, you have to set up an equation that describes this system. As the rod is circular in nature, its cross sectional area will be circular. Thus, your area is nothing but $$\pi r^2$$.

Now, forget the area term for a minute and look at the other terms in the equation. You have k, which is a constant so there's nothing to analyze there. Area is also constant for our purposes. The next term $$\delta T$$ is the temperature differential between the origin (hot end) and L refers to the length of the rod.

As you're considering a section of the rod of length x, here L=x. Also, from our above assumption, the temperature of the rod at a point x from the origin is T.

Substitute these terms into the equation and you're getting somewhere. Can you work out the other end of the equation?

 

[rwisz]

Based on the given information, it is clear that the copper rod is being subjected to heat transfer through conduction from the hot end to the cool end. To determine the temperature of the rod at a specific point, we can use the equation for conduction:

Q = kAtΔT/L

Where:
Q = heat transferred
k = thermal conductivity of copper
A = cross-sectional area of the rod
t = time
ΔT = temperature difference (T_hot - T_cool)
L = length of the rod

We can rearrange this equation to solve for the temperature at a specific point, in this case 24cm from the cool end:

T = (Q*L)/(k*A*t) + T_cool

To solve for Q, we can use the equation for heat:

Q = mcΔT

Where:
m = mass of the rod
c = specific heat of copper
ΔT = temperature difference (T_hot - T_cool)

To solve for the mass of the rod, we can use the equation for linear expansion:

ΔL = αL0ΔT

Where:
α = thermal expansion coefficient of copper
L0 = initial length of the rod
ΔT = temperature difference (T_hot - T_cool)

We can rearrange this equation to solve for the mass:

m = (ΔL*L0)/(α*ΔT)

Now we have all the necessary values to plug into our equation for temperature:

T = ((mcΔT)*L)/(k*A*t) + T_cool

Plugging in the given values, we get:

T = ((0.017*0.9)/(17*10^-6*78))*(103-25) + 25

T = 81.6 degrees Celsius

Therefore, the temperature of the rod 24cm from the cool end is approximately 81.6 degrees Celsius.