Project ForgeSteady two-dimensional heat conduction takes place in a body with insulated sides except two surfaces P and Q maintained at fixed temperatures. The temperature gradient ∂T/∂x at surface Q (effective width 2m) is 10 K/m. The body's cross-section narrows to an effective width of 1m at surface P. Constant thermal conductivity 0.1 W/m.K. The values of ∂T/∂x and ∂T/∂y at surface P are
- ∂T/∂x=20 K/m, ∂T/∂y=0 K/m
- B∂T/∂x=0 K/m, ∂T/∂y=10 K/m
- C∂T/∂x=10 K/m, ∂T/∂y=10 K/m
- D∂T/∂x=0 K/m, ∂T/∂y=20 K/m
Explanation
With all other boundaries insulated, the total heat flow rate must be identical through every cross-section. At Q: q=-k(∂T/∂x)Q·AQ=-k(10)(2). At P (width 1m, same flow direction as the normal to P): -k(∂T/∂x)P·(1)=-k(10)(2) ⟹ (∂T/∂x)P=20 K/m. Since P is itself a prescribed-temperature boundary with its outward normal along x, and no flow crosses the insulated sides, ∂T/∂y=0 at P.
Source: GATE ME 2008 (2008)
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