- 2071 Bhadra · 6 marks
Plate with convection and radiation: inside temperature
Short answer
Heat flux from the surface: convection q_conv = 20×(300−20) = 5600 W/m², radiation q_rad = 0.9×5.67e-8×(573.15⁴−293.15⁴) = 5130 W/m², total q = 10730 W/m². Conduction temperature drop ΔT = qL/k = 10730×0.025/50 = 5.36 K. Inside plate temperature = 300 + 5.36 ≈ 305.4 °C.
A 2.5 cm thick plate (k = 50 W/mK) 50 cm by 75 cm is maintained at 300°C. Heat is lost from the plate surface by convection and radiation to the ambient air at 20°C. If the emissivity of the surface is 0.9 and the convection heat transfer coefficient is 20 W/m²K, determine the inside plate temperature. [σ = 5.67 × 10⁻⁸ W/m²K⁴]
Reading of the data. The 300 °C is taken as the temperature of the exposed outer surface, from which heat leaves by convection and radiation. The inside (other) face of the plate is hotter, because heat reaches the surface by conduction through the plate. Heat flow is steady and one-dimensional.
Given: L = 2.5 cm = 0.025 m, k = 50 W/m·K, A = 0.5 × 0.75 = 0.375 m², T_s = 300 °C = 573.15 K, T_∞ = 20 °C = 293.15 K, ε = 0.9, h = 20 W/m²·K, σ = 5.67×10⁻⁸ W/m²·K⁴.
Heat flux leaving the surface
Convection: q_conv = h (T_s − T_∞) = 20 × 280 = 5600 W/m²
Radiation (to surroundings at the air temperature): q_rad = εσ (T_s⁴ − T_∞⁴) = 0.9 × 5.67×10⁻⁸ × (573.15⁴ − 293.15⁴) = 0.9 × 5.67×10⁻⁸ × (1.0791×10¹¹ − 0.0739×10¹¹) = 5130 W/m²
Total: q = 5600 + 5130 = 10730 W/m²
Heat flow rate: Q = qA = 10730 × 0.375 = 4024 W
Temperature drop across the plate
Conduction: q = k (T_i − T_s)/L
T_i − T_s = qL/k = 10730 × 0.025/50 = 5.36 K
Inside temperature
T_i = 300 + 5.36
Answer: inside plate temperature T_i ≈ 305.4 °C.
✓ Answer checked by Claude Opus 5.5 (high effort)
Related questions
- Insulating layer: minimum outside heat transfer coefficient · Fundamental of Thermodynamics & Heat Transfer, chapter 7 · asked 2×
- Insulation layer: minimum outside heat transfer coefficient · Fundamental of Thermodynamics & Heat Transfer, chapter 7 · asked 2×
- Hollow cylinder: heat transfer; temperature at 10 cm · Fundamental of Thermodynamics & Heat Transfer, chapter 7 · asked 2×
- Hollow cylinder: heat transfer and radial temperature · Fundamental of Thermodynamics & Heat Transfer, chapter 7 · asked 2×
- Air nozzle: exit temperature and exit area · Fundamental of Thermodynamics & Heat Transfer, chapter 4
- Heat pump at 50% of ideal: minimum surrounding temperature · Fundamental of Thermodynamics & Heat Transfer, chapter 5
- Laminar forced convection over a heated flat plate · Heat Transfer, chapter 3
- Flat plate collector useful heat and outlet temperature · Energy Resources, chapter 2
- Natural convection from a vertical plate · Heat Transfer, chapter 3
- Newton's law of cooling and convection coefficient · Heat Transfer, chapter 3
All Fundamental of Thermodynamics & Heat Transfer chapter 7 questions →