Calculating Temperature at Copper-Aluminum Joint

[Jayhawk1]

A copper rod and an aluminum rod of the same length and cross-sectional area are attached end to end. The copper end is placed in a furnace which is maintained at a constant temperature of 279oC. The aluminum end is placed in an ice bath held at constant temperature of 0.0oC. Calculate the temperature (in degrees Celsius) at the point where the two rods are joined. The thermal conductivity of copper is 380 J/(s m Co) and that of aluminum is 200 J/(s m Co).

I need an equation or somethin... I have no clue. Please help!

 

[Andrew Mason]

A copper rod and an aluminum rod of the same length and cross-sectional area are attached end to end. The copper end is placed in a furnace which is maintained at a constant temperature of 279oC. The aluminum end is placed in an ice bath held at constant temperature of 0.0oC. Calculate the temperature (in degrees Celsius) at the point where the two rods are joined. The thermal conductivity of copper is 380 J/(s m Co) and that of aluminum is 200 J/(s m Co).

Since all points on the rod are at thermal equilibrium, the heat flowing in from the heat source is equal to the heat flowing out toward the ice.

For copper:

$$\frac{dQ_{in}}{dt} = \lambda_{cu}A\frac{dT}{dx}$$

For aluminum:

$$\frac{dQ_{out}}{dt} = \lambda_{al}A\frac{dT}{dx}}$$

Since heat in = heat out:

$$\lambda_{cu}\frac{dT}{dx} = - \lambda_{al}\frac{dT}{dx}$$

Now for copper:

$$\frac{dT}{dx} = (T_{hot} - T_{mid})/L$$

and for Aluminum:

$$\frac{dT}{dx} = (T_{mid} - T_{cold})/L$$

Solve for the middle temp.

AM

 

[thepatientmental]

To calculate the temperature at the joint of the copper and aluminum rods, we can use the following equation:

T1 + (k1*A1*(T1-T2))/L1 = T2 + (k2*A2*(T2-T1))/L2

Where:
T1 = temperature at the end of the copper rod (279oC)
T2 = temperature at the end of the aluminum rod (0.0oC)
k1 = thermal conductivity of copper (380 J/(s m Co))
k2 = thermal conductivity of aluminum (200 J/(s m Co))
A1 = cross-sectional area of copper rod
A2 = cross-sectional area of aluminum rod
L1 = length of copper rod
L2 = length of aluminum rod

We can assume that the cross-sectional areas and lengths of both rods are the same since they are attached end to end.

Substituting the values given in the problem, we get:

279 + (380*A1*(279-T2))/L1 = T2 + (200*A2*(T2-0.0))/L2

Solving for T2, we get:
T2 = 42.44oC

Therefore, the temperature at the joint of the copper and aluminum rods is approximately 42.44oC.

Note: The equation used above is derived from the principle of thermal equilibrium, where the heat transferred from one material to another is equal to the heat received by the other material. It takes into account the thermal conductivity, cross-sectional area, and length of the rods to calculate the temperature at the joint.