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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
610
(8) 2.1
2025
3.1
1016
(13) 2.6
1820
2.95
1825
(21) 3.0
1518
2.8
Suppose the whole family is getting together, and we needed to cook a 30pound
turkey. How long should we cook it?
says
13
nt
nM
tM
Why should a 30pounder takes in
30 = 3.4 ~ 14 min/lb?
91
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View Full Document 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.
92
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.
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View Full Document 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 1/2pounder to replace the 1/4pounder. 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 1/2pounder 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.
 Winter '10
 Zasadinski

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