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.GIVENA roast in an ovenMass of the roast (m) = 1.5 kgOven temperature (T??) = 163?CFinal temperature of the roast??s center (Tf) = 77?CASSUMPTIONSThe shape of the roast can be approximated by a sphereThe roast temperature is initially uniform at (To) = 20?CThe properties of the roast are approximately those of waterThermal conductivity (k) = 0.5 W/(m K)Density (ρ) = 1000 kg/m3Specific 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.)

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T- 163°C Oven Roast m = 1.5 kg T.=20°C Convection Heat Transfer Coefficient W/m? K Btu/h ft? °F Fluid Air, free convection Superheated steam or air, forced convection Oil, forced convection Water, forced convection Water, boiling Steam, condensing 1-5 6-30 30-300 5-50 60-1,800 10-300 300-18,000 50-3,000 3,000-60,000 6,000-120,000 500-10,000 1,000-20,000