al Flows (continued) So ar we have talked about internal lows ideal lows (Poiseuille low in a tube) real lows (turbulent low in a tube) Strategy or handling real lows: How did we arrive at correlations? What do we do with the correlations? Dimensional analysis and data correlations non-dimensionalize ideal low; use to guide exts on similar non-ideal lows; take data; develo emirical correlations rom data use in MEB; calculate ressure-dro low-rate relations Emirical data correlations riction actor ( P) versus (Q) in a ie grahical correlations (low in a ie) correlation equations (low in a ie) laminar turbulent turbulent 6 0.079 4.0 log 0 0.5 0.4 < 00 4000 0 4000 rom Denn, Process Fluid Mechanics, Prentice-Hall 980; 5 5 (Geankolis rd ed)
al Flows (continued) Other internal lows: rough ies - need an additional dimensionless grou k - characteristic size o the surace roughness k - relative roughness (dimensionless roughness) D 4.0 log k 4.67 + D 0 +.8 k Colebrook correlation (>4000) al Flows (continued) Surace Roughness or Various Materials Material k (mm) Drawn tubing (brass,lead, glass, etc.).5x0 - Commercial steel or wrought iron 0.05 Ashalted cast iron 0. Galvanized iron 0.5 Cast iron 0.46 Wood stave 0.-.9 Concrete 0.- Riveted steel 0.9-9 rom Denn, Process Fluid Mechanics, Prentice-Hall 980; 46
al Flows (continued) Other internal lows: low through noncircular conduits Emirically, it is ound that vs. correlations or circular conduits matches the data or noncircular conduits i D is relaced with equivalent hydraulic diameter D H. 4(cross sectional area) D 4 ( wetted erimeter) H R H Equivalent hydraulic diameter Hydraulic radius al Flows (continued) Flow Through Noncircular Conduits Flow through equilateral triangular conduit and calculated with D H solid lines are or circular ies rom Denn, Process Fluid Mechanics, Prentice-Hall 980; 48 Note: or some shaes the correlation is somewhat dierent than the circular ie correlation; see Perry s Handbook
Non-Circular Crosssections have alication in the new ield o microluidics Chemical & Engineering News, 0 Set 007, 4 4
al Flows (continued) Other internal lows: entry low in ies low through a contraction low through an exansion low through a Venturi meter low through a butterly valve etc. see Perry s Handbook al Flows (continued) Now, we will talk about external lows ideal lows (low around a shere) real lows (turbulent low around a shere, other obstacles) Strategy or handling real lows: How did we arrive at correlations? What do we do with the correlations? Dimensional analysis and data correlations non-dimensionalize ideal low; use to guide exts on similar non-ideal lows; take data; develo emirical correlations rom data calculate drag - suericial velocity relations 5
(equivalent to shere alling through a liquid) z (r,θ,φ) Steady low o an incomressible, Newtonian luid around a shere Creeing Flow θ y sherical coordinates symmetry in the φ dir g calculate v and drag orce on shere neglect inertia ustream v z v low vr v v θ 0 rθφ Eqn o Continuity: g cosθ g g sinθ 0 r r rθφ P P( r, θ) ( r v ) sin r v θ + θ 0 r sinθ θ Steady low o an incomressible, Newtonian luid around a shere Creeing Flow Eqn o Motion: v ρ + v v P + µ v + ρ g t steady state neglect inertia SOLVE BC: no sli at shere surace BC: velocity goes to ar rom shere v 6
SOLUTION: Creeing Flow around a shere P µ v R P0 ρgr cosθ R r τ µ [ v + ( v) ] T v v v cosθ R r 4 all the stresses can be calculated rom v R + cosθ r R sinθ 4 r 0 0 R r rθφ Bird, Stewart, Lightoot, Transort Phenomena, Wiley, 960, 57; comlete solution in Denn SOLUTION: Creeing Flow around a shere What is the total z-direction orce on the shere? total vector orce on shere total z- direction orce on the shere F kˆ ππ [ rˆ ( τ PI )] r 0 0 F vector stress on a µscoic surace o unit normal rˆ total stress at a oint in the luid R integrate over the entire shere surace R sinθ dθ dφ evaluate at the surace o the shere 7
Force on a shere (creeing low limit) comes rom ressure comes rom shear stresses kˆ 4 F Fz πr ρg + πµ Rv + 4πµ Rv buoyant orce orm drag riction drag Bird, Stewart, Lightoot, Transort Phenomena, Wiley, 960, 59 stationary terms (0 when v0) Stokes law: kinetic orce kinetic terms F kin 6πµ Rv Nondimensionalize eqns o change: v t * * * * * * * + v v P + Nondimensionalize eqn or F kin : * * v + g Fr * Steady low o an incomressible, Newtonian luid around a shere Turbulent Flow deine dimensionless kinetic orce conclude () or C D C D () C D drag coeicient Fz, kinetic πd ρv 4 take data, lot, develo correlations 8
take data, lot, develo correlations Laminar low: C D πd 4 Stokes law ( 6πµ Rv ) ρv 4 Steady low o an incomressible, Newtonian luid around a shere Turbulent Flow Turbulent low: C D Calculate CD rom terminal velocity o a alling shere (see BSL 8; Denn 56) ( ρ ρ) shere ρ 4 Dg v all measurable quantities grahical correlation Steady low o an incomressible, Newtonian luid around a shere 4 McCabe et al., Unit Os o Chem Eng, 5th edition, 47 9
correlation equations Steady low o an incomressible, Newtonian luid around a shere laminar turbulent 4 8.5 0.60 < 0.0 500 turbulent 0.44 500 00,000 BSL, 94 use correlations in engineering ractice article settling entrained drolets in distillation columns article searators dro coalescence (See Denn, BSL, Perry s) al Flows (continued) Other external lows: rough sheres objects o other shaes lows ast walls airlane light 0
al Flows (continued) Now, we ve done two classes o real lows: internal lows (low in a conduit) external low (around obstacles) We can aly the techniques we have learned to more comlex engineering lows. We will discuss two examles briely:. Flow through acked beds. Fluidized beds ion exchange columns acked bed reactors acked distillation columns iltration low through soil (environmental issues, enhanced oil recovery) luidized bed reactors Flow through Packed Beds voids solids solids voids x -sectional area voids ε x -section o bed solids x -sectional area solid ε x -section o bed I the hydraulic diameter D H concet works or this low, crosssection then we already know () rom ie low.
al Flows (continued) More Comlex Alications I: Flow through Packed Beds What is ressure-dro versus low rate or low through an unconsolidated bed o monodiserse sherical articles? D shere diameter low or or irregular articles: D volume o articles 6 surace area o articles a v We will choose to model the low resistance as low through tortuous conduits with equivalent hydraulic diameter D H 4R H. al Flows (continued) Hagen-Poiseuille equation: average velocity in the interstitial regions v ( P P ) L D µ L 0 Flow through Packed Beds We will choose to model the low resistance as low through tortuous conduits with equivalent hydraulic diameter D H 4R H. BUT, what are D H and average velocity in terms o things we know about the bed? suericial velocity Q v x -sectional area o voids Q v0 x -section o entire bed x -sectional area voids v 0 v x -section o bed v v 0 ε v ε void raction
al Flows (continued) Flow through Packed Beds BUT, what is D H in terms o things we know about the bed? D H RH 4 volume available or low total wetted surace volume o voids volume o bed ε wetted surace av( ε) volume o bed D ε 6( ε) rom Denn, Process Fluid Mechanics, Prentice-Hall 980; 69 article surace volume o articles volume o articles volume o bed D H D ε ( ε) al Flows (continued) Flow through Packed Beds Now, ut it all together... analogous to or or ies we write: v 0 ε D D ε v H ( ε) v L 4 ρv0 ( P P ) L D 0 ( P P ) L D µ L 0 ρv0 D ( P P ) 0 L L rom Denn, Process Fluid Mechanics, Prentice-Hall 980; 69
al Flows (continued) Flow through Packed Beds Now, ut it all together... 0D v 0 ρv ε ε 6µ ( ε) rom Denn, Process Fluid Mechanics, Prentice-Hall 980; 69 µ ρv ( ε) 7 ε ( ) 0D ε Now we ollow convention and deine this as / and this as 7 al Flows (continued) Flow through Packed Beds When we check this relationshi with exerimental data we ind that a better it can be obtained with, 50 +. 75 7 Ergun Equation A data correlation or ressure-dro/low rate data or low through acked beds. rom Denn, Process Fluid Mechanics, Prentice-Hall 980; 69 ρv0d µ ( ε ) ε ( ε ) 4
Flow through Packed Beds 50 +. 75 rom Denn, Process Fluid Mechanics, Prentice-Hall 980; 709; original source Ergun, Chem Eng. Progr., 48, 9 (95). al Flows (continued) Flow through Packed Beds What did we do? We assumed the same unctional orm or P and Q as laminar ie low with, hydraulic diameter substituted or diameter hydraulic diameter exressed in measureables resulting unctional orm was it to exerimental data (new and deined or this system) scaling was validated by the it to the exerimental data we have obtained a correlation that will allow us to do design calculations on acked beds 5
al Flows (continued) More Comlex Alications II: Fluidized beds Can we use the Ergun equation (or ressure dro versus low rate in a acked bed) to calculate the minimum suericial velocity at which a bed becomes luidized? In a luidized bed reactor, the low rate o the gas is adjusted to overcome the orce o gravity and luidize a bed o articles; in this state heat and mass transer is good due to the chaotic motion. The P vs Q relationshi can come rom the Ergun eqn at small 50 +. 75 neglect low v al Flows (continued) More Comlex Alications II: Fluidized beds Now we erorm a orce balance on the bed: When the orces balance, inciient luidization gravity net eect o gravity and buoyancy is: ( ρ ρ)( ε)alg bed volume ( ε)al ressure (Ergun eqn) P A buoyancy 6
al Flows (continued) More Comlex Alications II: Fluidized beds When the orces balance, inciient luidization eliminate P; solve or v 0 ( ρ ρ)( )ALg P A ε 50 v 0 ( ρ ρ) gd 50µ ε ( ε) velocity at the oint o inciient luidization al Flows SUMMARY IDEAL FLOWS internal lows (Poiseuille low in a ie) external low (low around a shere) µscoic balances REAL FLOWS internal lows ( vs ) external low (C D vs ) nondimensionalization REAL ENGINEERING UNIT OPERATIONS internal lows (ies, uming) external low (acked beds, luidized bed reactors) aly engineering aroximations using reasonable concets and correlations obtained rom exeriments. 7