Heat transfer through two metal bars

[kidsmoker]

Homework Statement

A steel bar 10cm long is welded end to end to a copper bar 20cm long. Both bars are insulated perfectly on their sides. Each bar has a square cross section, 2cm on a side. The free end of the steel bar is maintained at 100 celsius and the free end of the copper bar is maintained at 0 celsius. Find the temperature at the junction of the two bars.

The attempt at a solution

The problem is a worked example so I can see the method for solving it. But I don't understand why they take the heat currents through the two bars to be equal? Would someone be able to explain it to me? :)

Thanks!

 

[tiny-tim]

… maintained … maintained …

I don't understand why they take the heat currents through the two bars to be equal?

Hi kidsmoker!

Read every word in exam questions …

the word "maintained" tells you that the system has reached equilibrium …

so heat in must equal heat out at all points.

 

[baba89]

Heat transfer through two metal bars can be explained using the principle of thermal conductivity. In this case, the two bars are made of different materials, steel and copper, which have different thermal conductivities. This means that they have different abilities to transfer heat.

When the free end of the steel bar is maintained at 100 celsius and the free end of the copper bar is maintained at 0 celsius, there will be a temperature gradient along the length of each bar. This means that the temperature will gradually decrease from 100 celsius to 0 celsius in the steel bar, and from 0 celsius to 100 celsius in the copper bar.

However, at the junction where the two bars are welded together, the temperature must be the same for both bars. This is because heat will flow from the hotter steel bar to the colder copper bar in order to achieve thermal equilibrium. This transfer of heat will continue until the temperatures at the junction are equal.

In order for the temperatures at the junction to be equal, the heat currents through both bars must also be equal. This is because the rate of heat transfer is directly proportional to the temperature difference between the two ends of the bar. Therefore, if the heat currents through both bars are equal, the temperature difference at the junction will also be equal, resulting in the same temperature at the junction for both bars.

In summary, the heat currents through the two bars are taken to be equal because this ensures that the temperatures at the junction will also be equal, allowing for thermal equilibrium to be achieved. This is an important consideration when analyzing heat transfer through different materials.