SOLUTION TO CHAPTER 4 EXERCISES: SLURRY TRANSPORT

Transcription

1 SOLUTION TO CHAPTER 4 EXERCISES: SLURRY TRANSPORT EXERCISE 4.1 Sales o a osate slurry ixture are aalyzed i a lab. Te ollowig data describe te relatiosi betwee te sear stress ad te sear rate: 1 Sear Rate,γ& ( sec ) Sear Stress, τ ( Pa ) Te slurry ixture is o-newtoia. I it is cosidered a ower-law slurry, wat is te relatiosi o te iscosity to te sear rate? SOLUTION TO EXERCISE 4.1: First reare a lot o log(sear stress) ersus log (sear rate). τ k & γ logτ log k+ log & γ Sear Rate Sear Stress Log (Sear Stress) Log (Sear Rate) Fro tis lot 1

2 Sloe 0.15 Itercet 1.37 Hece, Sloe 0.15 Itercet log k 1.37 k 3.4 Ns 0.15 / 0.15 τ 3.4 & γ wit γ& i ad τ μa 3.4 & γ & γ sec ad τ i Pa Proble 4.1 Sear Stress Sear Rate Proble 4.1 Log (Sear Stress) Log (Sear Rate)

3 EXERCISE 4.: Veriy equatio 4.7 SOLUTION TO EXERCISE 4.: r z For oe-diesioal, ully-deeloed, laiar low i a ie, te z cooet o te oetu balace siliies to te dieretial equatio 1 O + ( rτ rz ) z r r For a ower-law luid dz τ rz k dr Hece, ΔP k d d O r z + + L r dr dr Here as bee relaced by z Itegratig oce, ΔP + sice te ressure gradiet is costat. L r ΔP C1 d z + k L r dr C1 ust equal to zero sice te elocity gradiet is zero at r 0. Itegratig agai, ΔP r + C z Lk Alyig te boudary coditio tat z 0 at r R z ΔP R r + kl + 1 3

4 Te aerage elocity is equal to R r z R 0 dr Hece ad ΔP R r r R kl ΔP 3+ R R kl ( 3+ ) 1 R 0 D Siliyig wit R 1 ΔPD D 4kL ( ) EXERCISE 4.3: Veriy equatio 4.8 SOLUTION TO EXERCISE 4.3: r z For oe-diesioal, ully-deeloed laiar low i a ie, te z cooet o te oetu balace siliies to te dieretial equatio P 1 O + z r r For ay luid ( rτ ) rz 4

5 Itegratig, Δ Pr L d dr ( rτ ) rz Δ Pr τ rz sice τ rz ust be iite at r 0 L For ie low dz τ rz τ + y μ dr sice d z d r is egatie or ie low Hece, Δ Pr dz τ y + μ L dr Rearragig, Itegratig, Δ Pr τ y dz + Lμ μ dr Δ Pr τ yr + + C 4Lμ μ z Aly te boudary coditio z 0 r R Δ P R τ yr + C 4Lμ μ ( R r ) τ ( r R) ΔP y z + 4Lμ μ Tis elocity roile is alid or R r R. For te lug low regio r R, d z 0 dr. I te lug low regio, te elocity is costat ad ΔPR τrz τ y L 5

6 Te lug low elocity ca be oud by substitutig tis exressio or τ y ito te elocity roile. z ( R R ) PR ( R R) ΔP Δ + 4Lμ Lμ ( R R ) ΔP z or 4μ L 0 r R Bot elocity regios ust be itegrated to deterie te aerage elocity. Itegratig, R R ΔP ΔP τ R R r r+ R r + r R 4μ 0 L R 4μ R L μ r r ( ) d ( y ) ( R) d 4 ΔPR 4 R 1 R 1 + 8μ L 3 R 3 R or Rτ 1 4 4τ 1 τ 0 y y + 4μ 3τ0 3 τ0 were τ 0 is te wall sear stress τ 0 RΔP L EXERCISE 4.4 A slurry beaig as a seudolastic luid is lowig troug a soot roud tube aig a iside diaeter o 5 c at a aerage elocity o 8.5 /s. Te desity o te slurry is 900 kg/ 3 ad its low idex ad cosistecy idex are 0.3 ad k 3.0 Ns 0.3 /. Calculate te ressure dro or a) 50 legt o orizotal ie ad b) 50 legt o ertical ie wit te low oig agaist graity. 6

7 SOLUTION TO EXERCISE 4.4: First ceck i te low is laiar or turbulet Re trasitio 0.3 ( 1.9) Retrasitio 345 Re at low coditios, Re Te rictio actor 1.7 kg s 3.0 Re > Retrasitio turbulet is gie by ( ) log Re Ns Solig or wit Re ad 0.3, yields a) te ressure dro or orizotal low is te L Δ D ρ A V kg 50 Δ ( 0.006) 900 ( 8.5 ) s 7

8 kn Δ 338 b) te ressure dro or ertical low is gie by or Δ +Δ z ρ g L Δ ρg + ρgδ z ρ + ρgδz D kg 50 kg Δ s s kn Δ 779 EXERCISE 4.5 Te cocetratio o a water-based slurry sale is to be oud by dryig te slurry i a oe. Deterie te slurry weigt cocetratio gie te ollowig data: Weigt o cotaier lus dry solids 0.31kg Weigt o cotaier lus slurry 0.48kg Weigt o cotaier 0.1kg Deterie te desity o te slurry i te solid seciic graity is 3.0. SOLUTION TO EXERCISE 4.5: Weigt o dry solids kg Weigt o slurry kg Cocetratio o solids by weigt Te desity o te slurry is oud by 8

9 ρ kg kg ρ kg EXERCISE 4.6 A coal-water slurry as a seciic graity o 1.3. I te seciic graity o coal is 1.65, wat is te weigt ercet o coal i te slurry? Wat is te olue ercet coal? SOLUTION TO EXERCISE 4.6: C w ( 1 C ) 1 Cw w + ρ ρ ρ s ( 1 ) 1 C w C w Solig or, C w ass coal C w ass slurry Cwρ Volue ractio coal ρs Volue ractio coal 0.46 EXERCISE 4.7 Te ollowig reology test results were obtaied or a ieral slurry cotaiig 60 ercet solids by weigt. Wic reological odel describes te slurry ad wat are te aroriate reological roerties or tis slurry? 9

10 Rate o Sear (1/s) Sear Stress (Pa) SOLUTION TO EXERCISE 4.7: Te sear stress at zero sear rate is 4.0 Pa. Hece, tis slurry exibits yield stress equal to 4.0 Pa. I order to deterie weter te slurry beaes as a Biga luid or i it ollows te Herscel-Bulkley odel, we eed to lot τ τ ersus sear rate. τ τ y (Pa) Sear Rate (1/s) y A lot o o y τ τ ersus sear rate o a aritetric scale is ot liear. Howeer, a lot y τ τ ersus sear rate o a log-log scale is liear (te data or zero sear rate is excluded) k & τ τ γ y l τ τ y l k + l & γ 10

11 l ( τ τ y ) lγ& Sear Rate Sloe 0.81 Itercet -1.6 l k Ns k EXERCISE 4.8 A ud slurry is draied ro a tak troug a 50 t. log orizotal lastic ose. Te ose as a ellitical cross-sectio, wit a ajor axis o 4 ices ad a ior axis o ices. Te oe ed o te ose is 10 eet below te leel i te tak. Te ud is a Biga lastic wit a yield stress o 100 dyes/c, a lastic iscosity o 50c, ad a desity o 1.4 g/c 3. a) At wat elocity will water drai ro te ose? b) At wat elocity will te ud drai ro te ose? SOLUTION TO EXERCISE 4.8: Alyig te odiied Beroulli equatio to te syste betwee oits 1 ad, 11

12 or, 0 + +Δ z+δ g z z1+ g were L g D ad z z Need to deterie te ydraulic diaeter o te ie wit te ellitical cross-sectio: D 4 cross-sectioal area wetted erieter a b D π ( 4)( πab) ( a + b ) D ( 4)( i)( 1 i).53 i i + 1 i Pluggig i ubers (SI uits), s s () 1

13 Solutio Procedure: 1) Calculate He ) Guess elocity 3) Calculate Re 4) Fid ro Figure 6. 5) Ceck goerig equatio () 6) I goerig equatio is ot satisied, guess a ew elocity kg 1400 ( ) 10 ρ D τ He s kg 3 y s μ kg He 3,153 Solutio ia iteratie rocedure or water: kg ( ) 3.65 ρd s 3.65 s Re.3 10 μ kg s 5 5 (Re.3 10 ;soot tube) Solutio ia iteratie rocedure or ud: kg ( ) s 3. Re s kg 0.05 s 3 He 3 4 (Re ;.3 10 )

14 EXERCISE 4.9 A coal slurry is oud to beae as a ower-law luid wit a low idex 0.3, a seciic graity 1.5, ad a aaret iscosity o 70c at a sear rate 100s -1. a) Wat oluetric low rate o tis luid would be required to reac turbulet low i a 1/ i. I.D. soot ie wic is 15 t. log? b) Wat is te ressure dro (i Pa) i te ie uder tese coditios? SOLUTION TO EXERCISE 4.9: a) First, calculate Re trasitio or Re trasitio ( 0.3) ( ) 0.3 Re 340 trasitio Also, eed to calculate k, te cosistecy idex μ a k & 1 γ kg k s s kg k 1.76 s Alyig equatio 4.19, te aerage elocity i te soot ie ca be oud. Re 8ρ D k ( 6+ ) kg ( ) kg s Solig or 14

15 1.61 s Te oluetric low rate Q is te i π s 1 i Q (Cross-Sectioal Area, A C) s 3 b) L Δ ρ D Re kg 4.57 Δ ( 0.007) s Δ Pa EXERCISE 4.10 A ud slurry is draiig ro te botto o a large tak troug a 1 log ertical ie tat is 1 c I.D. Te oe ed o te ie is 4 below te leel i te tak. Te ud beaes as a Biga lastic wit a yield stress o 10 N/, a aaret iscosity o 0.04 kg/ s, ad a desity o 1500 kg/ 3. At wat elocity will te ud slurry drai ro te ose? 15

16 SOLUTION TO EXERCISE 4.10: Alyig te odiied Beroulli equatio to te syste aboe z z1+ g Assuig te low is laiar, te ead loss due to ie rictio is gie by ρ g 3 Alyig equatio 4.30, 3μ L 16τ yl + D 3D ρ g kg N ( )( 1 ) s ( 0.01 ) kg s

17 Pluggig tis ead loss back ito te odiied Beroulli equatio, Rearragig, ± (oter solutio yields a egatie elocity) s Ceck origial assutio to see i low is laiar kg ( 0.01 ) μ kg 0.04 s 3 ρd s Re 1300 Flow is laiar ad origial assutio is correct. EXERCISE 4.11 A ud slurry is draiig i laiar low ro te botto o a large tak troug a 5 log orizotal ie tat is 1 c iside diaeter. Te oe ed o te ie is 5 below te leel i te tak. Te ud is a Biga lastic wit a yield stress o 15 N/, a aaret iscosity o 0.06 kg/s, ad a desity o 000 kg/ 3. At wat elocity will te ud slurry drai ro te ose? 17

18 SOLUTION TO EXERCISE 4.11: Alyig te odiied Beroulli equatio to te syste aboe z z1+ g Assuig te low is laiar, te ead loss due to ie rictio is gie by 3μ L 16τ L + y 3 D 3D ρ g ρ g kg N ( )( 5 ) s ( 0.01 ) kg s

19 Pluggig tis ead loss back ito te odiied Beroulli equatio, Rearragig, s ± (oter solutio yields a egatie elocity) s Ceck origial assutio to see i low is laiar Re ρ D μ kg s kg 0.06 s Re 00 low is laiar 19