Calculating Temperature at Copper-Aluminum Joint

In summary: IT, in summary, the temperature at the point where the copper and aluminum rods are joined can be calculated by setting the heat flow equations for copper and aluminum equal to each other and solving for the middle temperature. The thermal conductivity of copper and aluminum, as well as the temperatures of the heat source and ice bath, are needed for this calculation.
  • #1
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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!
 
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  • #2
Jayhawk1 said:
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:

[tex]\frac{dQ_{in}}{dt} = \lambda_{cu}A\frac{dT}{dx}[/tex]

For aluminum:

[tex]\frac{dQ_{out}}{dt} = \lambda_{al}A\frac{dT}{dx}}[/tex]

Since heat in = heat out:

[tex]\lambda_{cu}\frac{dT}{dx} = - \lambda_{al}\frac{dT}{dx}[/tex]

Now for copper:

[tex]\frac{dT}{dx} = (T_{hot} - T_{mid})/L[/tex]

and for Aluminum:

[tex]\frac{dT}{dx} = (T_{mid} - T_{cold})/L[/tex]

Solve for the middle temp.

AM
 
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  • #3


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.
 

1. How do I calculate the temperature at a copper-aluminum joint?

To calculate the temperature at a copper-aluminum joint, you will need to use the heat transfer equation, which takes into account the thermal conductivity and heat capacity of both materials, as well as the temperature difference between them. This equation can be found in most thermodynamics textbooks or online calculators.

2. What factors affect the temperature at a copper-aluminum joint?

The temperature at a copper-aluminum joint is affected by several factors, including the thermal conductivity and heat capacity of both materials, the temperature difference between them, and the surface area of the joint. Any external heat sources or cooling methods may also impact the temperature at the joint.

3. How accurate are temperature calculations for copper-aluminum joints?

The accuracy of temperature calculations for copper-aluminum joints depends on the accuracy of the input parameters, such as thermal conductivity and heat capacity values, as well as the assumptions made in the calculation. It is important to use reliable data and make realistic assumptions to ensure accurate results.

4. Can the temperature at a copper-aluminum joint be measured directly?

Yes, the temperature at a copper-aluminum joint can be measured directly using a thermometer or a thermal imaging device. However, this may not be practical for all situations, especially in industrial settings where joints may be difficult to access or in high-temperature environments.

5. How can I minimize temperature differences at copper-aluminum joints?

To minimize temperature differences at copper-aluminum joints, it is important to choose materials with similar thermal properties, such as thermal conductivity and heat capacity, and to ensure proper contact between the two materials. Using thermal interface materials, such as thermal paste or pads, can also help improve heat transfer and minimize temperature differences at joints.

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