Tools › Fluids and Heat Transfer › Conduction › Thermal Stress and Conduction › Example
Steel bolt in an aluminum sleeve heated 65 °C, a single-pane window, a copper rod fin and a thermocouple bead: thermal stress, conduction network, fin and lumped transient (Hibbeler 4.12, Çengel 3-2, Incropera 3.8 and 5.1)
| Tool | Thermal Stress and Conduction |
| Source | Hibbeler, R. C. (2014). Mechanics of Materials, 9th ed. Pearson Prentice Hall, Upper Saddle River, NJ. Example 4.12; Çengel, Y. A. (2003). Heat Transfer: A Practical Approach, 2nd ed. McGraw-Hill, New York. Example 3-2; Incropera, F. P., DeWitt, D. P., Bergman, T. L., and Lavine, A. S. (2007). Fundamentals of Heat and Mass Transfer, 6th ed. John Wiley and Sons, Hoboken, NJ. Examples 3.8 and 5.1. |
| Level | Undergraduate |
Problem
Four textbook cases on one page. (1) An A-36 steel bolt (400 mm², E 200 GPa, α 12 µm/m/°C) inside a 2014-T6 aluminum sleeve (600 mm², 73.1 GPa, 23 µm/m/°C) is snug at 15 °C; find the force in bolt and sleeve at 80 °C. (2) A 0.8 m × 1.5 m window of 8-mm glass (k 0.78 W/m·K) separates 20 °C room air (h 10 W/m²·K) from −10 °C outdoor air (h 40); find the heat loss and the inner glass temperature. (3) A 5-mm copper rod (k 398) at a 100 °C wall in 25 °C air (h 100) is long enough to be an infinite fin; find m and the heat it removes. (4) A thermocouple bead (ρ 8500 kg/m³, c 400 J/kg·K, k 20 W/m·K) of 0.706 mm diameter in a gas with h 400 W/m²·K has a 1-s time constant; how long from 25 °C to 199 °C in a 200 °C stream?
Inputs to Enter
- Thermal stress, parallel members: dT 65 °C; member 1 E 200 GPa, α 12, A 400 mm², L 150 mm, n 1; member 2 E 73.1 GPa, α 23, A 600 mm², n 1; P 0
- Conduction, plane wall: A 1.2 m²; one layer 8 mm, k 0.78; h_i 10, T_i 20 °C; h_o 40, T_o −10 °C
- Fin: pin, infinite tip, k 398, h 100, T_b 100 °C, fluid 25 °C, D 5 mm (L 200 mm for the plot)
- Transient: sphere r 0.353 mm, ρ 8500 kg/m³, c 0.4 kJ/(kg·K), k 20, h 400, T_i 25 °C, fluid 200 °C, t 5 s, target 199 °C, 10 series terms
Answers
- Bolt F = A E (δ/L − α dT) = +20.26 kN (book 20.3 kN), sleeve −20.26 kN; bolt stress 50.64 MPa, sleeve −33.76 MPa; common elongation 0.1550 mm
- Window: R = 0.08333 + 0.008547 + 0.02083 = 0.11271 K/W; q = 30/0.11271 = 266.2 W (book 266 W); inner glass −2.18 °C (book −2.2 °C), outer −4.45 °C; U 7.39 W/(m²·K)
- Copper rod: m = √(4h/(kD)) = 14.18 m⁻¹ (book 14.2), q_f = √(h P k A_c) θ_b = 8.31 W (book 8.3 W), ε_f 56.4, infinite-fin length 2.65/m = 0.187 m (book 0.19 m)
- Thermocouple: Bi = h(r/3)/k = 2.35 × 10⁻³, τ = ρ c (r/3)/h = 1.000 s, t = τ ln(175/1) = 5.17 s (book 5.2 s); the sphere series gives Bi 0.00706, Fo 236, ζ₁ 0.1454, C₁ 1.0021 and 198.8 °C at 5 s, the same as the lumped value
Notes
The tool's default inputs are these four cases; press Submit. Every printed number is checked in test_thermal_stress.php together with Hibbeler Examples 4.10 and 4.11, Beer Concept Application 2.6, Timoshenko's bimetal curvature, Çengel Examples 3-3, 3-8, 3-9 and 4-3, Holman Example 2-2, Incropera Examples 3.5, 3.9 and 5.3 and Tables 3.4 and 5.1, and against scipy (References/Thermal-Stress-Conduction/thermal_check.py).