Heat - Transfer Example Problems

For steady-state 1D conduction without heat generation:

The outside air convection is the bottleneck. Insulating the pipe would dramatically reduce heat loss. Problem 5: Lumped Capacitance – Transient Cooling Scenario: A copper sphere (diameter ( D = 0.02 , \text{m} )) at ( T_i = 200^\circ\text{C} ) is suddenly placed in air at ( T_\infty = 25^\circ\text{C} ) with ( h = 20 , \text{W/m}^2\text{K} ). Copper properties: ( \rho = 8933 , \text{kg/m}^3 ), ( c_p = 385 , \text{J/kg·K} ), ( k = 401 , \text{W/m·K} ). Check if lumped capacitance is valid. If yes, find the time to reach ( 100^\circ\text{C} ). heat transfer example problems

[ R_{conv,i} = \frac{1}{100 \cdot 2\pi \cdot 0.05} = \frac{1}{31.416} = 0.03183 , \text{m·K/W} ] For steady-state 1D conduction without heat generation: The

[ Q = 5.67 \times 10^{-8} \cdot 5.44 \times 10^{10} = 5.67 \times 544 = 3084 , \text{W} ] Copper properties: ( \rho = 8933 , \text{kg/m}^3

[ R_{conv,o} = \frac{1}{10 \cdot 2\pi \cdot 0.06} = \frac{1}{3.7699} = 0.2653 , \text{m·K/W} ]