Estimate the time required to heat the center of a 1.5-kg roast in a 163°C oven to 77°C. State your assumptions carefully and compare your results with cooking instructions in a standard cookbook.
GIVEN
FIND
The time required to heat the roast
ASSUMPTIONS
SKETCH
With the assumptions listed above, the radius of the spherical roast is given by
Approximate solutions can be used to find the Fourier number. The following parameters (which are listed in
are needed
Using approximate solution we have
For the center r=0, the term becomes indeterminate, so using L Hospital’s rule we get.
For Bi=2.56 for sphere we have
Solving for the time
The Better Homes and Gardens Cookbook recommends cooking a Standing Rib Roast with the oven set at
325°F (163°C) for 27-30 minutes per pound to achieve a center temperature of 170°F (77°C) which is
considered well done.
This calculation yielded 134 minutes for 1.5 kg (3.3 lbs) or 40 minutes per pound. The discrepancy is probably due to inaccuracies in the assumed properties of the roast.
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