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(Wm; ' 5 Chemical Engineering 374 Fluid Mechanics
Fall 201 1 Scaling  Recall: Nature does not “know” about our units.
— kilogram, pounds
— meters, miles
 seconds, hours  Nature only knows about fundamental dimensions, and
relative sizes with those dimensions. — When we assign values to those dimensions or combinations of
those dimensions, we do so relative to our own contrived scales. — Most problems have more natural scales than our units (1 m).  What is the most natural length scale for flow in a room?
' Then measure length in “room” units ) x/L mom i BYU  How long to cook? — Trial and error? — Approximate the geometry and properties and
solve a heat transfer problem? — Solve a complex numerical solution?
 Cookbook Instructions call for ~20 min per lb.  Does this make sense? Why or why not? 5;qu WWW 0C 6 CO “If you roast at 325°F
(160°C), subtract 2
minutes or so per pound.
If the roast is
refrigerated just before going into the
oven, add 2 or 3 minutes
per' pound. “...You thought this was
going to be a simple
answer, didn’ tyou? “ r 62.. airbags Roast
6L _ 85:2 61,2 3:2 Ts
3T 027" T
4— = —Q 0
(9t 31.2
a = i (And1lC, 2 BCs) ._
PCP L  Unsteady heat equation
— Assuming constant thermal conductivity
— 3 dimensional, Cartesian coordinates  Dimensional homogeneity: each term has the
same dimensions and the same units.
— k—) J/s*m*K; p‘) kg/m3; cp)J/kg*K; a)m2/s  Nondimensionalize, then scale the equation. {Bvuj What does the heat equation say?  We don’ t want to solve this equation, just examine it. Ta Surface Center Nondimensionalize Select reference quantities: Tref, L, 17 Make a direct substitution.
Simplify: divide through by Tref/t = 42—: (
Now all terms are nondimensional. The group (XI/L2 is called a dimensionless group and is the Fourier number: a ratio of the physical time to the
characteristic diffusion time. Now scale the equation: — Cengel and Boles call this
“normalization”  If the terms in (...) are 0(1), then
since the LHS is 0(1), the term in [...]
must be 0(1) also. Then what to
choose for "c, L? n I I — L is just the domain size. Img' _ 2 intrinsic reference
_ Then tL /a quantities so that each term ’  in the dimensional equations
 These are the characteristic scales of Transform info The Prodm
the problem_ of a constant factor which
closely estimates the term' 5
 Rather than measure length in units order of magnitude and a ,
u n . drmensronless factor of unit
of meters , nature prefers units of order of magnitude" roast srze . Lin and Sega1938  Now what does this say about cooking a roast?
 The timescale scales as L2  But mass scales as L3 12 r
10 —
2 8; —M"(2l3)
7' oc L g 6 —M“(1)
3 i: 4
m 0‘ L 7' oc m2/3 2
l o
LOle/3 oz 4681012 . . . , Mass
 So cooking time scales With m2/3, not With m  We have not had to solve anything, but we know something valuable
about the problem.  if you know the time for one roast, you can extrapolate to another.
o Relationships among parameters! 5 BYU  Nondimensionalization also reduces the number of
parameters by showing that they are not independent,
but come in characteristic groups. — Conduction problem: vary the Fourier number alone, not alpha
and L separately. — Book gives simple equation of motion, where initial position,
velocity and gravity parameters are collapsed into one paramter,
the Foude number.  The Fourier number was the ratio of timescales, the Froude
number is the ratio of forces. — Many others: Friction factor, Drag coefficient, Knudsen number,
Mach number, REYNOLDS number. o Nonpremixed Flames Oxidizer Products Fuel Largescale jet ﬂame simulations: 12 weeks on 10,000 processors Ethylene combustion with complex chemistry: 19 species and 167 chemical reactions Speed setup with a simpler 1step chemical reaction.
Reaction characteristics? ' ,_ Beauty and betrayal in this equation T ,/ 771811: = H2 CHI CNDON AR was CUM} canon
H um C!" III H14": ICIDM cm
0 an In!!! CHZCHO um m C3010 m cm 61H! cam M! M M2!"
0" 00 C2!“ can w AI scam
N20 602 CINE 900!“ CM “H60 CHI H02 NCO CING AC!!!“ 04"“! COMO“ .03"!
“ID! CMZO “CEO cClHl “H01 C5". CHICCHZ
C CNZOII CHICO C4011 CJHT 05H! CHICHO
CH CH3!) NCCOH HZCAD 50”! 650150 CIHJCHD (I = A 1"0 IIMEIR'D)
REACTIONS A 0 E 1.20"] a) 02m
2.0mm @ 0".“ 1.20911 1 .0 0.0
SMEH‘I 1.0 0.0 3. 0*“! <9 "*0" 3.01904 2.1 0200.0
4.9000! 0 OH'KIZ 1.00E01l 0.0 0.0
5. WHO! G) OH‘HOZ SSSEOOI 2.0 «mm 0.090! m H+CO 7. MN! <=> MHCO
I. OOCHIZ (=) HIOED
B.WCHH (I) HOHCO 5.7054! 0.0 0.0
0.006013 0.0 0.0
LBDEHS 00 0.0
1.50E913 0.0 0.0 457. C4017?" C9 WM] 1.80911 0.0 0.0 050‘ “H1901 <3) “0104*” 1.00591! 0.0 0.0 lSlC‘HTQHOI <=> CH2090HHC3H5 2.00E4‘3 00 0.0 ‘SDJMHY‘HCO (K) CJHH‘CD 0.00E¢13 0.0 0.0 001. “"795”! 3> CMIGOCHJ LIOEHS 0.0 0.0 ‘62. QHMCZHJ <0 “H? 7.035030 4.5 “220.0
I ‘03. cmmspm <=> C‘Hﬂl‘ﬂ) 1.50E9l3 0.0 0.0 02H4 + 302 —> 2C02 + 2H20 % = _Ic(T)lC21‘1410'1[0’4’11'6 Problem: Very Stiff Explicit ODE integration — Very small timestep sizes required.
— Takes too many steps.
— Why? Apply Characteristic Timescale _ A[C2H4]mar [02/14] 5 — A T = ﬁ = [C2H410'9
H—ﬁ—‘HC t” [C'21‘lalo'1 77111:: As [02H4] becomes small. 1: becomes small.
The small reaction order is the culprit. Fix: As [C2H4] becomes small, increase reaction order. [C2H4]0.9[1—exp(—800Yb2m)] 021114 + 302 —» 2002 + 2H20 " [C2H4]0.1 _) [02H4]0.1+0.9 exD(—800YC2H¢) .
10‘ 1o" 10‘ 10" 1o" 10“ 10°
Ethylene Concentration oce 10"
10"
g 102
K
g 10"
.E
a 10*
E 106
Z 10‘
1
10‘710"10'51n41c'310'210" 10°
Ethylene concentration ...
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This note was uploaded on 03/11/2012 for the course CHE 374 taught by Professor Davidlignell during the Fall '12 term at Brigham Young University, Hawaii.
 Fall '12
 DavidLignell

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