Thermal conduction between 3 rods

[j3dwards]

Homework Statement

Rods of copper, brass and steel are welded together to form a Y-shaped figure. The cross-sectional area of each rod is 2.0 cm² . The free end of the copper rod is maintained at 100^◦C, and the free ends of the brass and steel rods at 0 ^◦C. Assume there is no heat loss from the surface of the rods. The lengths of the rods are: copper, 13 cm; brass, 18 cm; steel, 24 cm. The thermal conductivities are: copper, 385 W m⁻¹ K ⁻¹ ; brass, 109 W m⁻¹ K ⁻¹ ; steel, 50.2 W m⁻¹ K ⁻¹
(a)What is the temperature of the junction point?
(b)What is the heat current in each of the three rods?

Homework Equations

H = kA (T_H - T_C)/L

The Attempt at a Solution

(a) I assumed that the heat flow through all 3 rods was the same:

k_c (100 - T)/L_c = k_b (T - 0.0)/L_b = k_s (T - 0.0)/L_s

L_b k_c (100 - T) = L_c k_b]T

And with rearranging:

T = (100L_bk_c)/(L_ck_b + L_bk_c) = 83.0^◦C

Is this correct? Can I just assume that heat flow is the same and ignore the steel rod?

(b) Do i just used: H = kA (T_H - T_C)/L again but for each metal? Because H = dQ/dt?

Copper: dQ/dt = (385)(2 x 10^-4)(100-83)/0.13 = 10.1

 

[Hesch]

(a) I assumed that the heat flow through all 3 rods was the same:

I don't think that's a good assumption.

I'd switch the thermal circuit to an electric circuit ( temperatures = voltages, thermal conductivity = resistors, heat flow = current ).

Use Kirchhoffs current law ( KCL ) to calculate voltage and currents. ( It's only one equation needed ).

 

[rude man]

Is this correct? Can I just assume that heat flow is the same and ignore the steel rod?

Presumably, the junction temperature will be somewhere between 0 and 100C. Does it then make sense that, the ends of the steel rod being at different temperatures, that there be no heat flow thru the steel rod?