CE427 - CHEMICAL ENGINEERING LABORATORY III FALL 2005 MATHEMATICAL MODELLING OF TANK DRAINING

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1 CE47 - CEMICA ENGINEERING ABORATORY III FA 005 MATEMATICA MODEING OF TANK DRAINING Ojvs: Dvlop r mml modls o vryng omplxy o prd m rqurd o drn vrl ylndrl nk nd ompr modls w xprmnl d. Sysm: Two nks lod n 11 Jrvs v dmrs o ns nd 4.0 ns. T wo nks v x lns r rpll w 0.49 n dmr us vng drn lngs. T lngs vll r pproxmly 6, 1, 4, 36, 48, 60, nd 7 ns long. Tr r wo yps o onron ssmls pld on nks. T lrgr-dmr nk s srp-dgd onron nd smllr-dmr nk s roundd onron. T nks n lld w wr usng pump nd plugs or drn lns r vll. A prssur rnsdur, lod nk x, rnsms d o PC runnng w progrm w rords wr lvl s unon o m s nk drns. Modl dvlopmn I nk sysm s oprd s sown n gur low, low n, F, s sun o mnn lvl n nk ns wl nk s ng drnd roug pp o lng. F

2 Modl vl 1: In smpls ppro o prolm, Brnoull quon wou ron n wrn or suon dsrd ov oosng son sur o lqud n nk nd son pon o dsrg o drn pp. Sn prssur o sons s mospr, p nd p r qul, nd p p =. A sur o lqud, vloy!! s nglgl, nd rm / s droppd. Tus, rmnng rms n Brnoull quon wou ron r: wr g=lron o r ll =lng o drn pp = lqud lvl. g ( + ) = (1) From Eqn. (1), n sn vloy dsrg wll vry w g o lqud n nk. I low, F, s soppd nd nk s llowd o drn, n unsdy s mrl ln on nk nd x pp gvs: d S =! S () d (no mnus sgn s ndd us drvv s ngv) S nd S r rosssonl rs o nk nd drn pp, rspvly. Susuon or rom Eqn.(1) no Eqn.() (qus-sdy s pproxmon), rrrngmn, nd ngron yld: D 1! d = "! d g + 0 d (3) wr =lux (or drn) m = nl lqud lvl = nl lqud lvl d= drn u dmr D= nk dmr Tus m rqurd o drn nk rom n nl lqud lvl o nl lqud lvl s prdd y Modl 1 o :

3 D = { } (4) d g + 1 Modl lvl : Ovously, Modl 1 dos no oun or s o sold oundrs o nk nd drn pp. T Brnoull quon s xndd o oun or xsn o lud ron y ddng rm,, rprsnng ll ron gnrd pr un mss o lud ours wn sons nd. Tror, ddng s rm o Eqn. (1) would yld: g ( + ) = + (5) I n rgud mjor sour o ron loss n our nk drnng sysm s suddn onron o ross son rom nk o drn pp. T ron loss rom suddn onron s proporonl o vloy d n drn pp nd s gvn y xprsson! = K (6) wr K s lld onron loss on. T vlu o onron loss on vrs w gomry o onron. Modl s dvlopd y prodng s n Modl 1 o on: D (1 + K ) = { } (7) d g + Modl lvl 3: Wl Modl ook no oun mjor sour o ron loss, ron loss du o suddn onron o ross son s low nrs drn pp s no only sour o ron loss n our sysm. Modl 3 nluds ron loss du o ron wn wll o drn pp nd lud srm ( ron wn lud nd wll o nk s ngld). In s s, ron rm n Brnoull quon w ron (Eqn. (5)) would nlud, dnong skn ron: s g ( + ) = + + s (8) 3

4 wr = 4 s d = Fnnng ron or d = drn pp dmr T loss ons nd Fnnng ron or r dsussd n Cpr 5 o MC, Sm, nd rro (M,S,&) nd s xpd xprmnr wll onsul x or urr normon. Prodng s n dvlopmn o prvous modls would yld: 4 D (1 + K + ) d = { } (9) d g 3 + (In ngron ld o Eqn. (9), ws ssumd ron or,, s onsn rougou drnng pross. Expln wy on n ssum s s vld) Commns nd Ims o Consdr Drn Tm Msurmns: For msurmns o drn m, nlly ll nks o ovrlow ln wl pluggng drn u. Tn rmov plug wl smulnously srng d quson sysm o rord drn m. 1) Us o nks nd ls 5 drn us. On ls r rdngs or u. ) Bus s dul o drmn xly wn nk s mpy, rord m o drn o onssn dp (1 or ns) ov oom o nk. 3) Drmn lux ms prdd y ll r modls. No or modl 3 you wll nd o sm ron or. To do s, you wll nd o lul vrg vloy o wr lvng u sd on volum o wr n nk gnnng nd ow long ook or o r nl lqud lvl. 4) ow do msurd vlus ompr o os prdd? W modl mor urly prdd drn m? Inlud unrny lms or your msurd drn ms. 5) Plo drn ms s unon o lng o drn pp lng. Inlud modls prdons. 4

5 6) Dos ngng pp lng sgnnly lr lux m msurd? Dos dpnd on onron yp? 7) W urr rnmns would you suggs or mml modl? Rrns 1. MC, Sm, nd rro, Un Oprons o Cml Engnrng, 6 Edon, MGrw- ll, Prry, R.. nd D.W. Grn (ds.), Cml Engnr s ndook, 7 d., MGrw-ll, Pr-l omwork or Tnk Drnng Exprmn- o ompld ndvdully 1. Sow sps o on Eqn.(7), gnnng w Eqn.() nd Eqn.(5).. Fnd n xprsson or vlu or onron loss on or srp-dgd onron (s rrn). 3. Rp prolm or roundd onron (s rrn). 4. Durng your xprmn, you ound smll nk n l(d=4 n.) ks 3. sonds o drn wr 5 C rom lqud lvl o 77 ns o lqud lvl o ns roug 4 n drn pp (d=0.49 n). Bsd on your d, w s vrg vloy dsrg o drn pp durng drnng pross? 5. Drmn Fnnng ron or or xprmnl d n prolm 4 ssumng drn pp s smoo. 5