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unsteady mt notes - ’TM F453 0 Can-Haw-For Species A in...

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Unformatted text preview: ’TM F453». 0? Can-Haw -For Species A - in a Binary mirhuz MB) ' Can+ Emu: Moor canserwfiwa-Fm MGM” : MassI'n- MusM-r Masséuauld lat-hon wuss 6’0an In 9““ OF ' FA ' mass gwrwlper‘bmfir z” 0 Pa 0%‘E'P a +3 ,3 )u 'i Wfiw ax ay a: 0F accom 9"" i" won/WW 131%“: a AanIWHM' Khml Gt) aci’l 3 *gmaufl = ”'35 m '5' a! a:- R‘ ’H’ we, subsH+u+ez nl Me = - ac <7- 7 1; NA BABE; 4* Xfiwmfi 19°; law gwfihthW as CREE mom a” MOFW Q) My - Danfi + cm W“ _ PM (33 Mb (7;) : GENERAL 12ch a? Coanu'I’rV'E' SW95”? ‘1“- 3%,, (V-CnViD-(v-Dnva) ' RA ’1‘- fur HultimmPunent Syst'vam: replaced by [lfc'jIERJ + R3}. _ ifgm + RB] {ism—[9) was at constant lmnpuraluru armor h: written as {MAII'DI‘ ELI—14 and 15 that must b: occurring, Lhun rA1 {3. RA, Eq. 13.1 1? orir M ismmin (13. I—ZUI maulimus simply Hm HWM‘EUH "usinn 'in .mh‘dfl nr .n’m‘a'unmy Ir.cmunrr-qe'ffliarfun in 11mm“ —2fl| i5 similar [U the firm- iimilarily is lhu IJHHiH fur LIL: and difi'ufiinn prnhlcm}: in .I'uhlcms dcsm'ibcd h}? Fit-Ks nay h'u firund in [1“; JI'tl'JIIfl' fiFa'I iven in §1|$.1 arc summH rimd dinutua. Tim}-r ul'e tabulated [n TethIL: ”3.2-1 w: gjvu 111:: Table: 132—2, the Ilimlfiiun In bu mitten in farms nf I' ra'lhcr trawl}.I th: Jugur'tlhln uf 11w: mean 1:" In M {1 FM 490) we been first uuggcstud by C. H. IE4 {195 U). total mnlm' flux will! respect m «remit; PMS {193M}. : Samir, fixl'nrd‘ University Pmss' The. Equation nf Contlnuity in Curvliinear Cumming“: 559 TABLE ISL-J THE EQUATJUN CIF CGN‘I‘INUITY GIT- .-|1. IN VARIQUS COORDINATE SYSTEMS {Eq- IEI-ID‘; Recraqgrtflnr momfiuun's: a: 55: . w an; . —1+[ A'fi—H'wk .3‘ =H E-‘r Ex: Ey- d: ”l (A) Cyhnn'l'r'mi' rmrrffflmrvs: 711:1 1 Id 1 HRH-“l af‘l' I _' . _ _ g a; _ _ ' _ 5‘ _ 6-“: 1L“! ”Arid-r 6‘0 I 5'; ) _R' {H} .‘prrr'rl'rur' t'mlrdumflr; Er: i r“ l J [ 6-? __.n + __ ,5. i. _ . _. _.. mm. 5” (r3 r3r {r '1'" + ”inf? M m l* M” I?) + F'SLIT-fi-I M_) R" {C} TABLE “5.1—1 m: EQUATION or cummulw or a. mu cowsmm ,. AND *3” [re-,1. IaLm Ri't'l'migun'flr :ererJ'nah's; at I {‘6 ¢ r7“ l m: 17: 3! 2 _ 1 ‘ __‘. _. _' j — _‘f ___E"l_ El II ‘I: { ‘1 :‘I I“ in + h 13': J‘l“ an" F Hy: + 39.3 i R“- (A) {jarfmdrfral (maximum.- .33 1 ( E'n'i . 15:41 “-I a: T — “Jr a; "WTJ 1 a a“ 1 a an:- _ 1 . .4. .1. _ _p. _ ._ r.—. _. _ ”1(1- Er{ Er) +r= M: + 3‘3]+R' {3.1 ram ‘TraW'I‘ Phenomm " bd- irdL Situafl'n L-i h-hFoa‘t ml 115 :me In Mmugfiga anxsw I fl-l' (Ufa-Hr 3_C__A+ @963): DAB sing—Cm-QEF act" if 31’“ UsuALLV— we. +ry +o Sf’mp E; fins EQN as now as possmlc based 0" Hm. In [as a: GFflu. Problem P Vs 380”” TWO dl§feren+ problems NZ haw. “‘6 m emd— M Ma firesufl- in alga-eni- Solufims based on al.569— rmgg In -Hu Boundafl/ Candi-hens \- "-—-'-"'—~.—.=' POHDW‘ GM maple. -- - Comma SIMP’I‘FVI'Ij Assvah'ons +9 fig SQImNI£$§ 53:14?- £150 5mm; fl II:" a 'Skadyw —‘>5_C¢ vim u. no conveal-z'an =5 V'CA'UQ. =0 5- Species A 35 very dl'lU-I-c. Q V- cg); =1: 6 Flu): 'inonly onedlrécha‘n(u)$>_§_ {.1 =0 7. 6; Dan r- cons+an+ ' 37‘” L645 kid 3) Some examples..- OD S+eady5+alt 'Hux rm 9a caklys-l' SW‘M‘ I- c Dig-us!» of-a mug Specie +0 0L Isa-Ml scar-F III ‘1 canJ-inum process (Glow-4k In process‘) ' 1: I-I‘II-I a—vau 5"»‘4; Pkuw rxn dd bound | (QMIWIBV‘ M) II comm- A . _ 3% :2 Assumptions: 134 . real-a vhrcooml - ‘ “d- 3ma§°adc =9 afiaa '51 a 3- no mm in bulk @ mac. 3; II. Flux only inz-dl'reol I. -1: mag :0 ’3‘: 37.9! g Q 90b N = - 3.96. + QCNA'I'fiNh ‘2" A 32. 0' fi “7'": ‘3“ Baundavy Canal (5‘ as” m c. -125sz '7‘" 3 CA?) 2: = Q0 I: cab 29:0 N 3 £663 GAB ZIBCAO L- CabZ-l-r' A/h " hi- Examrlcél) Una-end)! DI'EvSIbna‘. km on a alum/9+ 5“”‘“¢- an 50% Boundaries - h . I76 Willem ”W‘Bqag -' ' N ‘ NW yflb Gascon-Fliudfilw YT--""'""' Emzmfl' H F. Ili yl-fl-u-u. *9." (unpaid? . hm A-‘OP mun-PA in carrier 3” RA" KCA I'nrlnhl maflmfiofl A =0» 3+ and“ ”a A LS uerydllwe. (CA0 V6 5mm) ' I. I C E a: W “$3.: %: :32 Assumeflfl'isdilfiemo;h Ufidnl 'm -dll"¢b'hE) +0 maled'oonuedfll) BC's ¢1nrwcm ””51"" '"d w Eh 3:7 r; y a O 4!! “Md EW‘L 3’ I I 4m»: is M . P" EmW‘ 53) A oli'ssom' Wm M he” "9‘“ “I'M-L ‘5 rub'al reach {SJ-cad)" m4; aF AESSoIvth a, Hoe! long will H- 'hh- 10 WWW); dlss'alve? E- I13 Middle man i Assumphéns : sphericd 59mm No Convet'hon . Flux only '1» r—dli’aal’lan Gn-linui-h’ EQN 'nn-spkehcal coord %%= ”mi-a 3:0?) __ EXQIMplfl Tm W+OF Palglx& am at :7'I-nd mm r%'d +0 We. 657.: 0~F~HL£L Solven+t+ha+ 15 in 61 pol maria, film .51th W «mm +WGKVLW " clmm fidugn‘f 13:90wa in Poljw D = 4 K f57cm375 Imhal Com. oFSoluanJr in 91m = rim 13pm (by L00 . Kil ififltifflg‘fi’éfikg I K=~Qnm X I aria. i _ T 3 WW“ ¥ Im Transanr DH? m 13-“: 5 E #0 H0 Convechbn E— mmngfifisbfm no ”in 3* digwm‘ Mass 5‘“ ASSUme US$033?” F @QA '7 ' BNA Occurs only m X*dlred1c¢g\ at BX CHAIM Slaw») g. .0: NA I6 represenkd bes+ b :3: 7 gas. “g"? @) 0d.- x: 51mm (5 rapid afierfo" In +er—Faca candihbns (BourHary Condlfiéns) r Convcdm. 'mass transhr and an ffiUi/r’brl'um dISMbU'I'IbH Coefficienir .-: ' . = 6'0 I N W K Cé’t/c’ am: r'h' gnu phase a} Eguilibrium “A ”it ___., q -- - > :— ' ‘0 CG" he. L... (4:) Cd! CA" ( b} c“ ”A q 4"" . 0&3- pg ‘ Coi,..-— kc C? (d) c: (a) Case. (a) Kc l and Rf ‘5' (”Wk-EL: case. (b) K>l 605e,“) K4! case Cd) K“ 05m $9512 finalgfiafié; 58 Making Vamb/cs fDl'mension less 125m?! chamokzrtsh'c Cal/[C EL LeVva d _ l |(u'mr'miole. conc. @an Imevismn gas ____ =. C _ 5"“ Comm U 1;}: am} am. > Y 1 1? 3mg; y CA '— 6" /Kfidfflv'ub%h% . . fl _ IHl'HfiJ cans. Infitm CJWQHS [on \655 :2.) X : ___ Leigh“ L- ‘- CMrac-ki'aSJ-fo 103% scale. C‘FH leTI-Mfl of- Elva) p“ tr Variablc time. dimeflS‘lonless => X __. D L“ "tlme 'INm'Al—Catdbmbgq g Yfl 3, Xn' o NO FLUX BOUNDARY _ = X 5 O CONIim'ofl Ex) Impermeabla. "2-) 2—; O a Burma. PflPiD EXTER'IOKCoNNJECfioN :9 Y: o a) X = rames solvenffi: BDUHDARY 1%,; COND. “a -F1'1m/caW‘ Eur-Shae. caflufifitw Convedrlva FLUX OF A BounmRY :3, *aY : gg. Y5" COMB. “ED _Fi\m/air Suf-Face. |mnz Ei- _- Wit: 5; DAB Can Also 901ve4’14is/Prablem GRAPHI'CALLY Soiuhéns 710 WSW {*dlhiEflSfaflal dig-Fusion (or Conducfibn) In St. é: gfiomfiies (Sal/muffin) haw; hem WW 1}] a VM CW fifgfihicm ¥ormaf Knawfi as G w“ ney- Lari e C hafi‘s Figure. 4.2.2 Gurney—Lyric charts for lrausicnl dil‘funiun in ling: in tin: value of Y flanged oval: tlm voluma. The lflwcr [aJnsl-ab.[hlalnngcylimlunand[c}asphere_1?oreachvalue hm: is the value of Y at I‘m: surface. In the CELEB ufthe. Hint number LIL aset of three lines is shtiwn. Th1: upper Hi = W, the smface is at 1" —- U for all time. an nu Iim: is line is the film: :zrl' ‘Jr' al I11: cent” of the body. 111.: dashed Shawn. {it would he. [he vurliual Y axis.) 19:» cm“ warmer}; film, (Ciajglfifid can: = Roadppnu ma! average 00m: = 52, 97‘ Co = IMPPnM L = Chmacknsht. Le K=Egmrb Dis—Mb egg: h = PM“ = 2mm D215 “/Méqcmz/s ” " 26"“ Cg -'* Gaga/K 3.900 "' 0/ I I u X = H H K'- 03’3“] 165%,) D X10 1: .. ICE-gt B " E's": ,_ .1: 00 (0.2)1 ) "t DAB - f) Usin a . War awn“ QSMC/ / MW (lbw?- g me, 014.. chad (n) (3351311 “flier E" = “D . We; See. )(p Corresponding +0 M = 0.05” f5 /-/ CA” #150 50sz VA/Sfaqpy M2485 TEA-A5125; "' Ali/Amway Nunsx’icmflumvsis a REPRESENT owe REG/0N r/swe 025mm” GRID P035175 am Cane. m EACH 6m: POINT (/5wa A N UM EKiC/Jc HPPKOXMA 770A) F01? DEKIM 7/1/55 Fox AM MUS TEAD Y FROCEgs WE mac Co nc AT EACH 9sz PT AT IMCREMEMML TIME STEPS GIL/EN A momma/mm Co n CENTRAT? a N DISTRIBUTION 36-. Cone. msrkzguflon AT 7.61%: {Is CALCUCHTED BASED CW Coma. mama/7m) AT 1: THE DIk’FEXEMIiAL 50M WE flRE 5’04 wile mafia; = 17%;; FOR 1 - DimenSbnal DiPFUSiaH: ’tfiqaéaTEA = 3%? ASSUME D955 60me WHAT IS THE NuMEfixcfia fi’E‘PEESE’A/fliflbu 0F c3904 $3. 2 aX‘“ Comma fillowinj example, 1.5+ Deriva-Hve. a) Per b SLOPE of Cum: 5: W. b can BE Avmximrep as i d :3 £831: Cfib'cfifl- $32$§3 X AX: A)! plfiem fiflag fith= CHC'Cab E) FORWARD . DIFFERENCE $933 CAl+éQfi = CAc'CAa-I =>CQI$aI x am: 313x d1 rem ax: 3% (M) H 95%” égiLlflLExA' = CAa+CAc' iCAb -—-————*M (M) ‘IIIIA'I' Is THE NUMERICAL REPRESEWIIIIoII 0F 31L ? ? ? 5.3,. can the. represenI-ed Using Backward Forward or CenIraI DrFFeI-ence. schem es Shawn .~g~ eviouSI “g n. i- I 1|. I m I! “I THE FINITE DIFFERENCE II) I315 CASE Is ESTIMATED FOR A SET var In SPACF AT DIFFERENT “TIME IIITEIIVAIS W fl m f ' fly? fiflfjfigbfifim %W}ifi+fid raffle +9.9. {AWL 1"} FQRWARD D rencg‘ SGheme shalt be 03w .2RESULT :D fit (CA:+'+ 04:; 1M #:21)%__)C;l‘ 5X?" f ‘- CALIE I If’IrIIc.aI‘.ilIIIpIIIIII IIII’ILngAd-“Him“Lfii JI,In’wIiIIIIILII, USE ”Poo/MW 1C0? "Numerical” Salmon £199. : magical at 3" Inihel Cond'liién" CA = .1000 PPM 19”” X 4"" ““0 Boardary Candi {16m fl : 0 mi K: 'ZCM 3X Boundary Common " CAI : CAei = 0 art K:OCM K The method of Iinee [MULJ is a general technique For the solution of partial dif— ' ferential equations that has been introduced in Prohleru 3.9. This method uti- lizee ordinaryr differential equations for the time derivative and finite differences on the Spatial derivativearTl-ie finite difierence elements for this problem are shown in Figure 7—11, where the interior of the slab has been divided into N = 8 intervals mvolving N + 1 = 9 nodes. . . f .Elcre _. _ Pflesmlfi» BIL-'1) r‘lo'l Cmdd' I 5-- io: = 1}. {LI-15' cm.- 6 EEK-posed Surface * l I | lEiounderzvr Conditions: I l l (a) & (l3) CA1 is main: tamed at constant value CM oLunia J a r l' I I I‘ I I i | I I I I I I lNo Maee Flux B If]? I in 0W wards Chase 12‘: ha dE’SCri be if: 091;? a Cami-ml d1 me. ~F0rmula .55? (15 Gun ordinary denuéfiw WEE Use SWUIJraI/Leous DiSjCe/Lewal 5%” SO) [/2 r“ . 7 fnih'afi conchhbms EV 'mJa'uornadeS . Values gar ConS-Pavrfs Ax) DAB) K - Final Value 199:" 41m (Hugs) PDIJYMATH Results TJJIEH} Unsteady-slate Mass Transfer in it Slab 0:40.200: RETF.E.130 (30!:le ted values of the DEE! varia'hlu Variable. iniLiel value mjmme] value maximal Heine fing‘: value 0 0 _ 0 1.DE+Db i_0fl+05 :02 0.000 4.2233-05 0.002 4.2233-05 003 0.002 0.2030—05 0.002 0-2000—05 000 0.002 1.2022-00 0.002 £-2025~0§ 0.05 0.002 1.03E=-00 0.003 1.03E-04 000 0.002 1.?003—04 0-002 5.0000—04 an? 0 002 1.9003-04 0.003 1.0990u04 000 0.002 2.1215—01 0.00? 2.1210-04 DAB 4.03—00 4.00—0? 0.03—01 0.00—07 deltax 0.025 0.025 0.025 0.025 059 0.000 2.1023—04 0.0030004 0.1020 00 000 0 0 0 0 K 1 1 1 1 LOH 0.000 0 0.00? 0 905 3310;: I'HHFH} Dilferenliel equetiene asentered by the user I 1 1 .dlcfilfldlt} = DAE‘ECAE -2*W4-EA1 00503:”? | :11 0003000} = DAE*[BM+2‘EA3+C-A2]Jdeltax02 |__'-] d{fifi.4lfd{t} u DAEl'lCAfi-‘E'flmmflawelmflz r 4 .' dlfififila’dll} = DAB-{em-z-ehemmwmmez r01 0(000 F00} r- DAB‘lCALZ'CAfi-rcfifi Fdeflaxfi '. I: J 0{CA?}Id{t} = mafia—00000000001502 H | 00000001 = DASH:CM-TCABl-Cenmmlax“? Exellel: 00.00.0000 00 animal! by the user 1 l 1 DAB = 4 000? L;-'. ; denim: 0.025 ,r ‘1] GAE: a lfll==01|lhen[2.De-EjeiseH-fi‘CAB-CAT-yaj -u 000=00 | E] K = 1 [01 001 -- llfl==0}then{2_00-3}e1se{CAOfK] Independent 1.0.1 Hal-Ha wrleble neme: l initial value... .0 final value 100000 Freels'len Step 132090055. h = 0.000001 Truncatlen errer telemnee. 005' = 0.000001 General number of differentlal equetlerle: T number of explicit: equatiens: EI- El'eta file-.k‘nuneteady diffuelen p0] 1 £1153 £13354 El "DE-4 -1.|.|IlE--‘|- ZINE-'3 nmm I3 EIDEIJ INE‘II ZINE-1- JflDE4 4IIIIJIZ-‘i Useful Finite Difference Appreximatiens Table 11-1 FII'ET-Drdar Finne [Infererm Apprmmtm Dlfieranee First Order Farmula Furw Md. Difference L‘C- _ {I fur iflxI} : M {p.41 Fis'et Derivative dr ix- Backward Difference 1 -,} _ .1: } for -d flip" '- r AI "fl “I {la-'2) First. Derivative- d’: 3“ [Perl-ward Difference ,: - I _ a, I +2“. d {if i =. quitf’ fif!x£+1]+f{xfi+1}{A'3] 5113mm! Derivative #1? I 333:1 Backward Difference ,1 _ '1 ._ *1” if“: 1 u “I" J————“‘ ' 1 ' ”F 1“ 3’ {A41 $94"an Derivative eh"? I $.11 Table A42 Semndflrdm Flmlu Dillemnm Amunmauum Dfll‘arenee Second Order Fern-ruin Furwurd Diflhrenre 'J-flxI-r 4, #13:“ I} _ (”I I II} “II-SI d 1"” #flx } - ———-——. Fired. Derivntwu :1“ L EM DcIntml [Hfierenee d . _ I'll-1',- + I]. _ Ifh-I. l3 A-fi f0]. I LIE—{114'} _ 1 { ) First Tim-Native 1 Eur. lienkward ELIE-whee :I’ - ”fl” III-HI _ 11" + fix: _:J 'Ffil‘ [1' II — .—————— {flu-'3'} Wit-Ht DerIvum-C f a 313-1: Table A—2 Second-Order Fume Di‘flmeme hwmknal‘bne- Difference seemed Order meula —_————l F ' 1-11 D'i'f' . . I . .. I IIIIrIrII-wa 1 erenee “firm-j : 2ft1.I.:-5f{1l+lj+r1f{x”3:411”}; [A-fl] 59::an Derivative dx? ' I it: L‘entra'l Difference 1 . . . - [x . :-1{.r-}+flr_ : 1101' . . ifE-r." =. [f I -r 1 I 1]- 1 1 {AI-9] beeund TJenvatlve rfl" ' M- ' d 'J' . 1- . . Eifliwu l ifl‘erenee d___flx_3 = Enam— fiflxid I} +4313: 2* . {{ng- —'.+J [IA-1n] Second Derivative d5: ‘ it?" ...
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