Lecture_15_notes - MW ‘ "‘ D}HM;o-=«‘...

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Unformatted text preview: MW ‘ "‘ D}HM;o-=«‘ IL\Na'/§I‘§ 4,”) S641..g awry.) -. vbimwr‘vfiks’} grauPs —' 65:4"? M “ion at: 3M W {on 4” "26 an 61/ waJn‘rrw- J . j— + QC! t-() (:3 kg ’ [M A” ‘36 4 w? 5 ;’( Q) 707M mud" MM #4 Sam-q uwfylg _'——P winemslbké/ Ho-«Uc‘a'l/r ® Nan—DimmW—“u T57 ~: 6’7“ 137 62W;.-47 Wk m 5m.“ u-«.‘}; at Tana-4 ‘ “a {0? 99M (.57 Qiu5041tw4> M Arm Tl' — w 5W?“ .. d2? _ D5 ‘ : r )4 {:a v :a ; gag” . v "t 1?. 20 ( Baal; ‘ u 7 l 0,0 / mevxé 2 , 1—,, I 20 I vb M $ 31 m :9 9 <; (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 t-L /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 Sega|1938 - 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 Large-scale jet flame simulations: 1-2 weeks on 10,000 processors Ethylene combustion with complex chemistry: 19 species and 167 chemical reactions Speed setup with a simpler 1-step chemical reaction. Reaction characteristics? ' ,_ Beauty and betrayal in this equation T ,/ 771811: = H2 CHI CNDON AR was CUM} canon H um C!" III H14": I-CIDM 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 IIM-EIR'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> CM-IGOCHJ 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.

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Lecture_15_notes - MW ‘ "‘ D}HM;o-=«‘...

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