Two homogeneous rods of Length L1 and L2 have the same density and specific heat

Question

Two homogeneous rods of Length L1 and L2 have the same density and specific heats, but their thermal conductivities are K1 and K2, which differ. The two rods are welded together with a weld joint that puts them in perfect contact. The composite rod is laterally insulated and the temperature of the left end is held at 0 degrees, while the right end is held at tau degrees. Assume the system is allowed to come to a steady state where the temperature distribution in the system depends only upon x.

a) Discuss why the temperature and the flux should be continuous at the joint.

b) Determine the steady-state temperature distribution in the composite rod.

Explanation / Answer

SInce two rod are connected in series

so heat rate will be equal in both rod

let the temp at T at middle part

heat rate=K1*A1*T/L1=K2*A2*(tau-T)/L2..........(i)

since heat rate is constant .it means the rate of heat flow is same in both

since specific heat is same for both rods ,the heat flow for mass per unit volume will will same as density is also same so dT/dx will also be same so temperature and fulx will be continuous .

steady state temp at the joint can be found by (i)

T(K1*A1/L1 + K2*A2/L2)=K2*A2*tau

T =(K2*A2*tau)/(K1*A1/L1 + K2*A2/L2)

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