Subtopic: Transient heat transfer in simple geometry
Question Statement:
A plate shaped frozen food has a thickness of 20 mm, average thermal conductivity of 2.58 W m⁻¹ °C⁻¹, density of 1080 kg m⁻³, specific heat of 3550 J kg⁻¹ °C⁻¹, and a uniform temperature of −20 °C. The food material is suddenly immersed in a well stirred hot water maintained at a constant temperature of 90 °C. The heat transfer coefficient between the food material and hot water is 25 W m⁻² °C⁻¹. The time required for the centre temperature of the food material to reach 30 °C in minutes is
Options:
(A) 3.9
(B) 7.9
(C) 15.5
(D) 31.0
Official Answer Key:
C
Step-by-Step Solution & Explanation
This is a transient (lumped/Heisler-chart) conduction problem for a slab heated symmetrically from both faces. Half-thickness L = 0.010 m. Thermal diffusivity α = k/(ρCp) = 2.58/(1080×3550) = 6.73×10⁻⁷ m²/s. Biot number Bi = hL/k = 25×0.010/2.58 = 0.0969 (small, so the one-term Heisler-chart approximation for a slab centre temperature is appropriate). Dimensionless temperature: θ = (T−T∞)/(Ti−T∞) = (30−90)/(−20−90) = (−60)/(−110) = 0.545. Using the one-term slab solution with the corresponding eigenvalue for Bi≈0.10 (λ₁≈0.311, A₁≈1.0161): θ = A₁exp(−λ₁²Fo) ⇒ 0.545 = 1.0161·exp(−0.0968·Fo) ⇒ Fo = −ln(0.545/1.0161)/0.0968 ≈ 6.44. Time t = Fo·L²/α = 6.44×(0.010²)/6.73×10⁻⁷ ≈ 957 s ≈ 15.9 ≈ 15.5 min, matching option (C) (small variation depending on the precision of the Heisler chart coefficients used).
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