Finding temperature at junction of rods

Question

Let's say I have three tubs, from left to right Tub 1, Tub 2, Tub 3.

Tubs 1 and 3 are at 50 degrees Celsius and Tub 2 is at 10 degrees Celsius.

Tubs 1 and 2 are connected by a copper rod, and tubs 2 and 3 are connected by a brass rod.

So it looks like this:

(50) --(copper)-- (10) --(brass)-- (50)

I want to find the temperature of the middle tub after a long time (assuming no heat loss to surroundings). I want to use conservation of energy and $H = (kA/L) \Delta t$, but I'm just not sure how to compute this or set it up.

Would I do $ (T-10) = (k_{Cu} A_{Cu}/L_{Cu})(50-T) +(k_{Br} A_{Br}/L_{Br})(50-T) $, then solve for T. I'm not sure how to treat this.

Answer 1

Heat continues to flow between the tubs through the conductors, from hotter areas to colder ones, until the temperature difference between them is (including the connectors) zero, i.e. they are all at the same temperature $T_e$ (Zeroth Law of thermodynamics): full thermodynamic equilibrium has been reached and there's no more heat flow.

Because the system is adiabatic ($\Sigma Q=0$), we can write (initial state v. end state):

$$m_1cT_1+m_2cT_2+m_3cT_3=m_1cT_e+m_2cT_e+m_3cT_e$$

Assuming $m_1c=m_2c=m_3c$ ($m$ is water mass, $c$ water specific heat),

we get simply:

$$T_e=\frac{T_1+T_2+T_3}{3}$$

The nature of the connectors does not affect the end result, only the time needed to reach full thermodynamic equilibrium.

Answer 2

Assuming the tubs are equally full of water (each mass M) then it is just delta T of cold tub x M = delta T of the warm tubs x 2M. Where delta T is T -10 for the cold an 50-T for the hot ones. It does not matter what material connects the tubs in the long term, but the pipe with the highest conductivity will just cool that tub a little faster but eventually it will all even out.

The formulas you had with k A over L x delta T are for heat transfer rates but because the tubs are changing T you need a diff eqn to get the T as a function of time AND the flows are NOT equal, I don't think you use these equations for this type of question.

If you were keeping tub 1 at 50C forever and tub 2 at 10C forever, then you could calculate the heat transfer rate with your formulas!