Chemical Engineering 374

Transcription

1 Chemical Egieerig 374 Fluid Mechaics NoNeoia Fluids Oulie 2 Types ad properies of o-neoia Fluids Pipe flos for o-neoia fluids Velociy profile / flo rae Pressure op Fricio facor Pump poer Rheological Parameers Poer La Fluids

2 No-Neoia Fluids 3 Neoia Bigham Plasic Pseudoplasic Dilia =* =* + y =k* Bigham Plasic 4 3D elasic srucures. Weak solid srucures ha mus be broke Resiss small shear, bu srucure breaks apar ih large shear. The is ~ liear ih du/ Some slurries (coal, grai slurries), seage sludge. Toohpase (o ip) Larger paricles à eak solid srucure à breaks Bigham Plasic =* + y 2

3 Pseudoplasic 5 Mos commo Dissolved or dispersed paricles, like dissolved log chai molecules. Have a radom orieaio i he fluid a res, bu lie up he he fluid is sheared. decreases ih srai rae ops as molecules alig Polymer mels, paper pulp suspesios, pigme suspesios, hair gel, blood, muds, mos slurries Shear-Thiig moor oil Pseudoplasic =K*(-) Dilia 6 Rare Slurries of solid paricles ih barely eough liquid o keep apar. (cor sarch, aer squeezed ou a high shear) A lo srai raes, he fluid ca lubricae solids; a high srai raes, his lubricaio breaks do. icreases ih srai à icreases. Shear hickeig. Dilia =K*(-) 3

4 No-Neoia 7 Time depedece 8 Thixoropic Slurries/soluios of polymers May ko fluids Mos are pseudoplasic Aligable paricles/molecules ih eak bods (H-bodig) Pai Rheopecic Rare Feer ko examples Usually fluids oly sho his behavior uder mild shearig Chages occur ihi he firs 60 sec. for mos processes. Hard o describe Viscoelasic ime ime 4

5 Poer La Fluids 9 Goverig equaios are correc i erms of Expressio for is he model. Called a cosiuive relaio Also have hese for mass ad hea fluxes i hea ad mass rasfer. Neoia flo = dv dy For dilia ad pseudoplasic fluids (mos commo) Poer La = K dv dy > à Dilia < à Psuedoplasic =, K= à Neoia K, are empirical cosas May oher forms Simpler oes have 3 parameers ad give a beer fi, bu are more complex ha poer la form. See Hadou of Book Chaper o Webpage. Lamiar Pipe Flo 0 r+dr r x P x P x+dx r Force Balace: Pressure, sress (P x P x+ x )(2 r r)+(2 xr) r (2 x)(r + r) r+ r =0 Divide 2pDrDx r P x P x+ x x Limi Dx, Dr à 0 = d(r ) = C r + r r (r + r) r+ r r =0 Separae variables ad iegrae ih =0 a r=0 = r 2 5

6 No-Neoia Pipe Flo Mos o-neoia flos are lamiar. Key resuls: (remember, Q is jus volumeric flo rae-vdo) Force balace: Poer la cosiuive relaio Iegrae ih B.C. v=0 a r=r = r 2 dv = K v = 2K / R + r + + Q = Av avg Q is volumeric flo rae Q = 3 8(3 + ) / D 4K 2(3 + ) / D 4K Kieic Eergy Correcio Facor: Momeum Flux Correcio Facor: = 3(3 + )2 ) (5 + 3)(2 + ) = Pressure Drop Lamiar Flo 2 Defie f = 4 2 v2 avg Neoia Force Balace = R 2 Z A 2 R v(r)da 2 No-Neoia v = R2 4 R2 8 r 2 R v = 2K / 2(3 + ) + R + r + / D 4K = 4V avg R f = 64 Re Solve 4 for / ad iser io 2 Iser io for =(some complex expressio) f = 8K 2(3 + )Vavg Vavg 2 6

7 Turbule Flo 3 Defie he fricio facor as before: (Lamiar or Turbule) For urbule flo e had f = f(re, e/d) from dimesioal aalysis. Quesio: Will his ork for o-neoia Flo? Quesio: Wha is he Reyolds umber? No clear defiiio of Re sice is o cosa (depeds o he srai rae dv/, hich depeds o V avg ) Use he same defiiio as he lamiar fricio facor: Re=64/f f = 8K V 2 avg f = 8 V 2 avg 2(3 + )Vavg Re = 8 V 2 avg K = P D/L 2 V avg 2 2(3 + )V avg (Defiiio based o lamiar Neoia, bu used for all regimes) Plo fricio facor versus Re as for Neoia flos, usig he red defiiio of Re. No-Neoia Fricio Facor (Poer La) 4 ffaig = (/4) fdarcy Re = 8 V avg 2 K 2(3 + )V avg 7

8 Rheological Parameers (poer la) 5 Problem: No-Neoia fluid has: Ho o fid K, ad for a give fluid? You eed o measure somehig (ha?) Try a pipe flo D, Q, / Here s ha e ko: dv = K = r 2 dv = K = R 2 D, Q, / à V avg,. The relae hese o K, : Compue (-dv/) from v(r) v = 2K Q = / 3 8(3 + ) 2(3 + ) = K D 4K dv + / D 4K = K dv R + r + / Rheological Parameers (poer la) 6 From v(r), e ge: No = K So a plo of l( ) versus l(-dv/) is liear ih slope, ad iercep l(k). Bu, oe ha (-dv/) ivolves, hich is uko à ha o do? Jus rearrage: dv 2(3 + )Vavg = dv l( )=l(k)+ l( dv/) l( )=l(k)+ l(2(3 + )V avg /) l( )= l(v avg /D)+[l(K)+ l(2(3 + )/)] No, a plo of l( ) versus l(v avg /D) is liear ih slope. Oce is ko, K ca be compued from he iercep (erm i [ ]), or jus compue i aalyically from dv = K ad dv 2(3 + )Vavg = hich give K = (2(3 + )V avg /) 8

9 Recap 7 To compue K, for a o-neoia fluid Measure Q, D, / Compue V avg from Q ad D (area), ha is, Q=A*V avg Compue from Plo l( ) versus l(v avg /D) Fi a lie o he daa (he liear par of he daa) The slope is K is compued from he iercep I: or from K = = R 2 (2(3 + )V avg /) Noe, he uis o K are (kg*s -2 /m) K =exp[i l(2(3 + )/)] Example 8 Give: Diameer Pressure Drop Flo Rae Compue: K, Re Poer hrough a give pipe is as usual, Q*DP = slope = K based o iercep I is 0.63 K =exp[i l(2(3 + )/)] K based o K = (2(3 + )V avg /) is 0.63 ± sblue pois=