021010 sec9_1-18

# 021010 sec9_1-18 - Scaling Arguments The Fourier Number Turkey Cooking Woman's Home Companion Cook Book Weight 6-10 10-16 18-25 n weight Tune/Unit

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Scaling Arguments — The Fourier Number Turkey Cooking Woman's Home Companion Cook Book Weight n weight Tune/Unit Weight in time/weight 6-10 (8) 2.1 20-25 3.1 10-16 (13) 2.6 18-20 2.95 18-25 (21) 3.0 15-18 2.8 Suppose the whole family is getting together, and we needed to cook a 30-pound turkey. How long should we cook it? says 13 nt nM   tM Why should a 30-pounder takes in 30 = 3.4 ~ 14 min/lb? 9-1

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Turkey Cooking Why do bigger turkeys take less time to cook per pound? Turkeys are spheres? — 1st approximation We know that  2 0 , s s TT r ef R      , everything here just depends on dimensionless or "scaled" radius and time. Basically, to cook a turkey, we need to make sure that the interior temperature reaches a certain value for a certain time; hence: Since in each case, 0 , a  , we should have the same dimensionless time, 0 0 2 t R 2 0 R Now this gives us the real time, which depends on R 2 of the turkey, assuming all turkeys have the same α : The mass of turkey (sphere) is: 13 3 or M M RR 2 23 M M     so the actual time it takes for a turkey to cook goes like M . If, as in the cookbook, we want to determine the time/mass, we have to: per pound M M MM or M 1/3 . nt nM  Hence, the theoretical justification of our graph, without any tough PDE's to solve. Often in engineering analysis, the dimensionless Fourier number, τ , is sufficient to scale the problem. 9-2
ChE 120B Important Numbers on Heat Transfer  2 p kt CL - Fourier number — dimensionless time ratio of convective heat transfer Biot number conductive heat transfer hL k When to do a complete analysis — when not to. First, always do the simplest scaling relationship. Transient problem – ID, [0 2 _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 2 , ,C T y=0 y=L 0 p xb TT x tx   in dimensionless form 0 When to use tabulated data/scaling: Note that for transient problems in dimensionless form, the solutions are functions of 2 0 , char. dim. surface surface t   Char. Dim x/L, 0 / rr , etc. And Biot# = .dim. h k char So, all problems are similar and can be used to scale up when needed. What is important in the system geometry, the Fourier number and the Biot number. 9-3

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ChE 120B Hamburger Cooking Scaling For Biot# >> 1, τ = α t/L 2 governs the time it takes for a given dimensionless temperature change. Imagine – McDonalds is planning a 1/2-pounder to replace the 1/4-pounder. To use the same buns, the diameter of the burger is the same but the patty is thicker, however, the centerline temperature of patty must be at the sample temperature as before to prevent intestinal problems due to raw meat. Using the same cooking surface, how much longer will the 1/2-pounder take to cook?
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## This note was uploaded on 03/22/2011 for the course CHE 120B taught by Professor Zasadinski during the Winter '10 term at UCSB.

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021010 sec9_1-18 - Scaling Arguments The Fourier Number Turkey Cooking Woman's Home Companion Cook Book Weight 6-10 10-16 18-25 n weight Tune/Unit

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