Project ForgeA very large metal plate of thickness d and thermal conductivity k is cooled symmetrically on both faces by air at T∞=300K with heat transfer coefficient h; the plate's centerline temperature is TP. In which of the following case(s) can the lumped parameter model be used to study the heat transfer in the plate? (A) h=10 W/m²K, k=100 W/mK, d=1mm. (B) h=100 W/m²K, k=100 W/mK, d=1m. (C) h=100 W/m²K, k=1000 W/mK, d=1mm. (D) h=1000 W/m²K, k=1 W/mK, d=1m.
- h=10 Wm⁻²K⁻¹, k=100 Wm⁻¹K⁻¹, d=1 mm, TP=350 K
- Bh=100 Wm⁻²K⁻¹, k=100 Wm⁻¹K⁻¹, d=1 m, TP=325 K
- h=100 Wm⁻²K⁻¹, k=1000 Wm⁻¹K⁻¹, d=1 mm, TP=325 K
- Dh=1000 Wm⁻²K⁻¹, k=1 Wm⁻¹K⁻¹, d=1 m, TP=350 K
Explanation
The lumped-capacitance model requires Biot number Bi=h·Lc/k≪1, with characteristic length Lc=d/2 for a plate cooled symmetrically on both faces. (A): Bi=10×0.0005/100=5×10⁻⁵ — valid. (B): Bi=100×0.5/100=0.5 — too large, NOT valid. (C): Bi=100×0.0005/1000=5×10⁻⁵ — valid. (D): Bi=1000×0.5/1=500 — far too large, NOT valid.
Source: GATE ME 2023 Official Paper (2023)
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