Project ForgeConsider a rod of uniform thermal conductivity whose one end (x=0) is insulated and the other end (x=L) is exposed to flow of air at temperature T∞ with convective heat transfer coefficient h. The cylindrical surface of the rod is insulated so that the heat transfer is strictly along the axis of the rod. The rate of internal heat generation per unit volume inside the rod is given as q̇=cos(2πx/L). The steady state temperature at the mid-location of the rod is given as T_A. What will be the temperature at the same location, if the convective heat transfer coefficient increases to 2h?
- AT_A + q̇L/(2h)
- B2T_A
- T_A
- DT_A(1 − q̇L/4πh) + (q̇L/4πh)T∞
Explanation
Total heat generated over the rod, ∫₀^L cos(2πx/L)dx = 0 (a full period integrates to zero). Since x=0 is insulated, ALL generated heat must exit through x=L at steady state — but that net generation is zero, so the flux at x=L is always zero, regardless of h. The convective BC, −k(dT/dx)|_L = h(T(L)−T∞), then forces T(L)=T∞ for ANY h≠0. Solving the ODE confirms the entire temperature profile T(x)=T∞+(L²/4π²k)[cos(2πx/L)−1] is completely independent of h. So T at the midpoint remains T_A unchanged.
Source: GATE ME 2022 Session 1 (ME-1) (2022)
This question is part of a 15-day revision cycle — sign in and Project Forge will bring it back the day before you'd forget it.