The convection coefficient for flow over a solid sphere may be determined by submerging the sphere, which is initially at 25 °C, into the flow, which is at 75 °C, and measuring its surface temperature at some time during the transient heating process. If the sphere has a diameter of 0.1 m, a thermal conductivity of 22 W/(m·K), and a thermal diffusivity of 0.40 ×10-5 m2/s, at what time will a surface temperature of 60 ºC be recorded if the convection coefficient is 300 W/(m2·K)?

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rejkjavik rejkjavik · Answer 1 · 2019-10-17T07:17:57+02:00

Answer:

t = 59.37 s

Explanation:

Given data:

thermal diffusivity [tex]= \alpha = \frac{k}{\rho c_p} =0.40\times 10^{-0.5}[/tex]

theraml conductivity = k = 22 W/m.K

h = 300 W/ m^2.K

[tex]T_i[/tex] = 25 degree C = 298 k

[tex]T_o[/tex] = 60 degree C = 333 k

[tex]T_{\infty} [/tex]= 75 degree C = 348 L

diameter d = 0.1 m

characteristics length Lc = r/3 = = 0.0166

[tex]Bi = \frac{hLc}{K} = \frac{300\times 0.0166}{22} = 0.226[/tex]

[tex]\tau = \frac{\alpha t}{lc^2} = \frac{0.4\times 10^{-5}\times t}{0.0166^2}[/tex]

[tex]\tau = 0.036 t[/tex]

[tex]\frac{T_o -T_{\infty}}{T_i -T_{\infty}} = Ae^[\lambda^2 \tau}[/tex]

at Bi = 0.226

Ai = 0.982

[tex]\lambda = 0.876[/tex]

[tex]\frac{333348}{298-348} = 0.982e^{-0.879^2 0.036t}[/tex]

[tex]0.3 = 0.982 e^{-0.2t}[/tex]

[tex]0.305 = e^{-0.2t}[/tex]

-1.187 = - 0.02t

t = 59.37 s