(Solution) - Estimate the time required to heat the center of a -(2025 Original AI-Free Solution)

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Estimate the time required to heat the center of a 1.5-kg roast in a 163°C over to 77°C. State your assumptions carefully and compare your results with cooking instructions in a standard cookbook.
GIVEN
A roast in an oven
Mass of the roast (m) = 1.5 kg
Oven temperature (T?) = 163°C
Final temperature of the roast?s center (Tf) = 77°C
ASSUMPTIONS
The shape of the roast can be approximated by a sphere
The roast temperature is initially uniform at (To) = 20°C
The properties of the roast are approximately those of water
Thermal conductivity (k) = 0.5 W/(m K)
Density (?) = 1000 kg/m3
Specific heat = 4000 J/(kg K)
A uniform heat transfer coefficient of (hc) = 18 W/(m2 K) exists between the roast and the oven air (midline of the range for free convection given in Table1.4.)