Copper spheres 20 mm in diameter are quenched by being dropped into a tank of water that is maintained at \( T_{\infty} = 280 \, \text{K} \). The spheres may be assumed to be falling at a terminal velocity of \( v_{\infty} = 2.17 \, \text{m/s} \) through the water. Determine the height (\( H \)) of the water tank needed to cool the spheres from an initial uniform temperature of \( T_0 = 360 \, \text{K} \) to a surface temperature of \( T = 310 \, \text{K} \). *Hint*: Estimate \( h \) in order to calculate the Biot number.

Given, Diameter of sphere, D = 20 mm temperature of want in the tank ,T∞ = 280 K terminal velocity, v∞ = 2.17 m/s Initial temperature of the sphere,T0 = 360 K Final temperature of the sphere,Tf = 310 KProperties of the components are as follows: density of copper, ρCu = 8933 kgm3 conductivity of copper, k = 398 Wm·K Heat capacity of copper, Cp,Cu= 387 Jkg·K density of water, ρ = 1000 kgm3 Calculate α as follows: α= kρCp = 398 W/m·K8933 kg/m3×387 J/kg·K = 1.15×10-4 m2/sWrite the expression for force balance as follows: CDπD42ρv∞22 = ρCu-ρgπD36 CDρv∞24 = ρCu-ρgD3 CD = 4ρCu-ρgD3ρv∞2CD = 48933 kg/m3 - 1000 kg/m3×9.81 m/s2×20 mm3×1000 kg/m3×2.17 m/s2×1 m1000 mmCD = 0.44 Calculate the Reynolds number of the particle as follows: Rep = Dvμ = 20 mm×2.17 m/s1.42×10-6 m2/s×1 m1000 mm = 30563 For CD = 0.44 and Rep = 30563 NuD = 439Calculate h as follows: h = NuDkDh = 439×0.582 W/mK20 mm×1000 mm1 mh = 12775 W/m2KCalculate the Bi as follows: Bi = hrkBi = hD2kBi = 12775 W/m2K×20 mm2×398 W/mK×1 m1000 mmBi = 0.32At Bi = 0.32, C1 = 1.0937ξ1 = 0.9472θ = 0.5 Calculate F0 as follows: θ = C1exp-ξ2F00.5 = 1.0937×exp-0.94722×F0 -0.94722×F0 = ln0.5 1.0937F0 = -0.7827-0.8972 F0 = 0.87 Calculate the time taken to fall as follows: t = F0r2αt = F0D/22α = 0.87×20 mm2×1 m1000 mm21.15×10-4 m2/s = 0.76 s Calculate the height of the tank as follows: H = tv∞ = 0.76 s×2.17 m/s =1.65 m Therefore, the required solution is 1.65 m.

Answered: Copper spheres 20 mm in diameter are…

Introduction to Chemical Engineering Thermodynamics

8th Edition

ISBN:9781259696527

Author:J.M. Smith Termodinamica en ingenieria quimica, Hendrick C Van Ness, Michael Abbott, Mark Swihart

Publisher:J.M. Smith Termodinamica en ingenieria quimica, Hendrick C Van Ness, Michael Abbott, Mark Swihart

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Copper spheres 20 mm in diameter are quenched by being dropped into a tank of water that is maintained at \( T_{\infty} = 280 \, \text{K} \). The spheres may be assumed to be falling at a terminal velocity of \( v_{\infty} = 2.17 \, \text{m/s} \) through the water. Determine the height (\( H \)) of the water tank needed to cool the spheres from an initial uniform temperature of \( T_0 = 360 \, \text{K} \) to a surface temperature of \( T = 310 \, \text{K} \). *Hint*: Estimate \( h \) in order to calculate the Biot number.

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