[ \frac{T(t) - T_\infty}{T_i - T_\infty} = \exp\left(-\frac{h A_s}{\rho V c_p} t\right) ] For a sphere: ( A_s/V = 6/D ). [ \frac{100 - 25}{200 - 25} = \exp\left(-\frac{20 \cdot 6}{8933 \cdot 0.02 \cdot 385} t\right) ] [ \frac{75}{175} = 0.4286 = \exp(-0.001744 \cdot t) ] [ \ln(0.4286) = -0.8473 = -0.001744 , t ] [ t \approx 486 , \text{seconds} , (\approx 8.1 , \text{minutes}) ]
Newton’s law of cooling: [ Q = h , A , (T_s - T_\infty) ] [ 600 = h \cdot 0.5 \cdot (80 - 20) ] [ 600 = h \cdot 0.5 \cdot 60 = h \cdot 30 ] [ h = 20 , \text{W/m}^2\text{·K} ] heat transfer example problems
Radiation dominates at high temperatures. Even with a 200 K difference, over 3 kW is transferred. Problem 4: Overall Heat Transfer Coefficient (Conduction + Convection) Scenario: A steam pipe (inner radius ( r_1 = 0.05 , \text{m} ), outer radius ( r_2 = 0.06 , \text{m} )) has ( k = 15 , \text{W/m·K} ). Inside: steam at ( T_{hot} = 200^\circ\text{C} ) with ( h_i = 100 , \text{W/m}^2\text{K} ). Outside: room air at ( T_{cold} = 25^\circ\text{C} ) with ( h_o = 10 , \text{W/m}^2\text{K} ). Find the heat loss per unit length ( Q/L ). Problem 4: Overall Heat Transfer Coefficient (Conduction +
Now heat flux: [ q = \frac{1100 - 50}{0.8334} = \frac{1050}{0.8334} \approx 1260 , \text{W/m}^2 ] Find the heat loss per unit length ( Q/L )