Answer:
Explanation:
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Answer:
Average convection heat transfer coefficient, [tex]\dot{Q_{ave} } = 832.42 W[/tex]
time taken for the process, [tex]\triangle t = 37.9 min[/tex]
Explanation:
The average convection heat transfer rate is calculated using the formula:
[tex]\dot{Q_{ave} } = h A_{s} (T_s - T_{\infty} )[/tex]
The surface area of the steel ball is given by :
[tex]A_{s} = \pi D^{2} \\A_{s} = \pi * 0.21^{2} \\A_{s} = 0.139 m^2[/tex]
Free stream temperature, [tex]T_{\infty} = 30^{0} C[/tex]
Initial temperature of the ball, T₁ = 350°C
Final temperature of the ball, T₂ = 250°C
Average surface temperature of the ball:
[tex]T_s = \frac{T_1 + T_2}{2} \\T_s = \frac{350 + 250}{2}\\T_s = 300^{0} C[/tex]
Velocity of air, V = 6 m/s
Diameter of the ball, D = 0.21 m
Viscosity, v = 1.608 * 10⁻⁵ m²/s
Reynold number Re can be calculated by using the formula: [tex]Re = \frac{VD}{v}[/tex]
[tex]Re = \frac{6 * 0.21}{1.608 * 10^{-5} }[/tex]
Re = 78358.21
The Nusselt number can be calculated by using the equation:
[tex]Nu = 2 + (0.4Re^{0.5} + 0.06Re^{0.67} ) (Pr^{0.4}) (\frac{\mu_{\infty}}{\mu_s})^{0.25} \\Nu = 2 + (0.4*78358.21^{0.5} + 0.06*78358.21^{0.67} ) (0.7282^{0.4}) (\frac{1.872*10^{-5}}{2.934*10^{-5}})^{0.25}[/tex]
Nu = 179.95
The heat transfer coefficient can be calculated using the formula:
[tex]h = \frac{k* Nu}{D} \\h = \frac{0.02588* 179.95}{0.21}\\h = 22.18 W/m^2 k[/tex]
[tex]\dot{Q_{ave} } = h A_{s} (T_s - T_{\infty} )[/tex]
[tex]\dot{Q_{ave} } = 22.18 * 0.139 (300 -30 )\\\dot{Q_{ave} } = 832.42 W[/tex]
The time taken for the process, [tex]\triangle t = \frac{Q_{total} }{\dot{Q_{ave} }}[/tex]
[tex]Q_{total} = mc_{p} (T_1 - T_2)[/tex]
Volume of the steel ball, [tex]V = \frac{\pi * D^3 }{6}[/tex]
[tex]V = \frac{\pi * 0.21^3 }{6}[/tex]
V = 0.0049 m³
Density of steel, [tex]\rho = 8055 kg/m^{3}[/tex]
Mass of the steel, [tex]m = \rho V[/tex]
m = 8055*0.0049
m = 39.47 kg
Total rate of heat transfer: [tex]Q_{total} = mc_{p} (T_1 - T_2)[/tex]
Specific heat capacity of steel ball, [tex]c_p[/tex] = 480 J/kg
[tex]Q_{total} = 39.47 *480 (350 - 250)[/tex]
[tex]Q_{total} = 1894560 J[/tex]
[tex]\triangle t = \frac{Q_{total} }{\dot{Q_{ave} }}[/tex]
[tex]\triangle t = 1894560/832.42\\\triangle t = 2275.97s\\\triangle t = 2275.97/60\\\triangle t = 37.9 min[/tex]
