Answer:
The time required to reach the center temperature of egg at 70 °c is 0.3 sec.
Explanation:
Given data
Diameter (d) = 5.5 cm = 0.055 m
K = 0.6 [tex]\frac{W}{mk}[/tex]
h= 1400 [tex]\frac{W}{m^{2} K }[/tex]
Initial temperature [tex]T_{i}[/tex] = 4 °c= 277 K
Ambient temperature [tex]T_{o}[/tex] = 97 °c= 370 K
Center temperature T = 70°C = 343 K
Thermal diffusivity α = [tex]0.14[/tex] × [tex]10^{-6} \frac{m^{2} }{s}[/tex]
We know that
[tex]\alpha = \frac{K}{\rho C}[/tex]
[tex]0.14[/tex] × [tex]10^{-6}[/tex] = [tex]\frac{0.6}{\rho C}[/tex]
[tex]\rho C =[/tex] 4.28 × [tex]10^{6} \frac{KJ}{m^{3} K }[/tex]
We know that from the lumped parameter analysis
[tex]\frac{T - T_{o} }{T_{i} - T_{o} } = e (-\frac{ h A}{\rho V C} ) t[/tex]
Put all the values in above equation we get
[tex]\frac{343 - 370 }{277 - 370 } = e (-\frac{ (1400)(6)}{235400} ) t[/tex]
0.29 = 0.965 × t
t = 0.3 sec
Therefore the time required to reach the center temperature of egg at 70 °c is 0.3 sec.
This question involves the concepts of thermal diffusivity, lumped parameter analysis, and thermal conductivity.
It will take "0.26 s" for the center of the egg to reach 70°C.
We will use the formula for the lumped parameter analysis:
[tex]\frac{T-T_o}{T_i-T_o}=e^{\frac{-hA}{\rho VC}}t[/tex]
where,
T = ceneter temperature = 70° C + 273 = 343°C
T₀ = Ambient Temperature = 4°C + 273 = 277 K
T_i = Initial Temperature = 97°C + 273 = 370 K
h = heat transfer coefficient = 1400 W/m².K
A = Area of sphere = 4πr²
t = time interval = ?
V = volume of sphere = [tex]\frac{4}{3}\pi r^3[/tex]
K = thermal conductivity = 0.6 W/m.K
α = thermal diffusivity = 0.14 x 10⁻⁶ m²/s = [tex]\frac{K}{\rho C}=\frac{0.6\ W/m.K}{\rho C}[/tex]
[tex]\frac{1}{\rho C}=\frac{0.14\ x\ 10^{-6}\ m^2/s}{0.6\ W/m.K} = 0.23\ x\ 10^{-6}\ m^2.K/N[/tex]
Therefore,
[tex]\frac{343\ K-277\ K}{370\ K-277\ K}=e^{\frac{-(1400\ W/m^2.k)(4\pi r^2)(0.23\ x\ 10^{-6}\ m^2.K/N)}{(\frac{4}{3}\pi r^3)}}t\\\\0.7096=e^{-(0.000322)(\frac{3}{r})}t\\\\0.7096=e^{-(0.000322)(\frac{3}{0.0275})}t\\\\t=\frac{0.7096}{2.718}\\\\[/tex]
t = 0.26 s
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