ENGG 199 Reacting Flows Spring Lecture 2b Blending of Viscous, Non-Newtonian Fluids

Transcription

1 ENGG 199 Reacting Flows Spring 2006 Lecture 2b Blending of Viscous, Non-Newtonian Fluids Copyright 2000, A.. Etchells, R..Grenville & R.D. LaRoche All rights reserved.

2 Re-Cap In turbulent regime, viscosity does not influence: Power input. Flow. Blend time. In transitional regime, viscosity influences: Flow. Blend time. In laminar regime, viscosity influences: Power input. Flow. Blend time (for turbines - not for helical ribbons). Need to know viscosity. ENGG 199 Lecture 2b Slide 2

3 Newtonian Fluids: Newtonian Fluids Ratio of shear stress to shear rate is constant. Constant of proportionality is the fluid s viscosity. Viscosity will be uniform throughout stirred tank: Shear rates are high in the impeller zone. Low elsewhere. ENGG 199 Lecture 2b Slide 3

4 Non-Newtonian Fluids: Non-Newtonian Fluids Ratio of shear stress to shear rate is not constant. Non-linear relationship. Measure shear stress over range of shear rates in Viscometer. Regression of data - Power Law: n Apparent viscosity at given shear rate: A n (n 1) ENGG 199 Lecture 2b Slide 4

5 ield Stress Fluids More extreme non-newtonian behavior. Fluid will not move until a certain level of shear stress is exerted. No motion -No Shear Rate. Shear stress at which motion starts is called the ield Stress. Two models for data: A n A n ENGG 199 Lecture 2b Slide 5

6 Design Problems Power consumption by impellers in laminar regime: P Dimensionless blend time in the transitional regime: N Motion of ield Stress fluids: ENGG 199 Lecture 2b Slide 6

7 Power Consumption in the Laminar Regime ork of Metzner et al. (Late 50 s - Early 60 s). Measure power consumption in the laminar regime with Newtonian fluids. Get values of P for impellers tested. P P N 2 D 3 Measure power consumption in non-newtonian fluids. Value of P allows apparent viscosity to be estimated. A P P N 2 D 3 ENGG 199 Lecture 2b Slide 7

8 Power Consumption in the Laminar Regime Having measured the power law constants, and n: Found that shear rate is proportional to impeller speed. A k S N 1 n 1 For turbines: 10 ks 14 For helical ribbons: k S c D ENGG 199 Lecture 2b Slide 8

9 In the laminar regime: Power Calculation P P A N 2 D 3 A ( k S N) ( n 1) P P k ( n S 1) N ( n 1) D 3 Check for Newtonian case. hat happens when n = 1? ENGG 199 Lecture 2b Slide 9

10 Blending in the Transitional Regime Blending rate for vessel is determined by local blending rate in region of baffle. hat is the shear rate and apparent viscosity here? Several papers published using Metzner s method: Does not work! Gives shear rate in impeller region - not wall / baffle. Does not account for processes in bulk of vessel. Metzner s method only valid in laminar regime. Need to estimate shear rate for: all / baffle region. Transitional regime. ENGG 199 Lecture 2b Slide 10

11 Force Balance Impeller exerts force on fluid causing it to move. all and baffles exert equal and opposite force. From Transport Phenomena: Tq S R v r ds A Rp Baff da Assume that the shear rate at the wall does not vary with position: v r ENGG 199 Lecture 2b Slide 11

12 Substituting equations: Tq Force Balance Making assumptions about the fluid velocity at the baffles and integrating: 1 Tq ( v ) T Having measured and n: S RdS A Rp Baff da 1 n ( n 1) ENGG 199 Lecture 2b Slide 12

13 Results for Pitched Blade Turbine - RMS ENGG 199 Lecture 2b Slide 13

14 Correlation - Pitched Blade Turbine ENGG 199 Lecture 2b Slide 14

15 Correlation - All Impellers ENGG 199 Lecture 2b Slide 15

16 Blend Time Calculation Use wall viscosity to calculate Re and Fo. Plot: N versus Re Po 1/ 3 Re versus 1 / Fo Compare with Newtonian data. Excellent agreement. Implications for impeller selection? ENGG 199 Lecture 2b Slide 16

17 Impeller Selection In the transitional regime: Po 1/ 3 Re 186 Fo 1 2 / 3 T D 2 / 3 T 2 / 3 For Newtonian fluids, no effect of impeller type. For non-newtonian fluids, high torque impeller (high Po) will give shorter blend time for equal power input. ENGG 199 Lecture 2b Slide 17

18 hy? High torque per unit volume gives high shear stress (and rate) at wall: Tq 3 T Tq / T 3 1 n Tq 3 T 1 n A ( n 1) Tq 3 T ( n 1) n Results in low apparent viscosity at the wall. ENGG 199 Lecture 2b Slide 18

19 ield Stress Fluids Important class of fluids for industry. Fluid must experience certain shear stress before it will move: A n A n Typical yield stress is Pa. At low shear rates, apparent viscosity is very high. ENGG 199 Lecture 2b Slide 19

20 Shear Stress - Shear Rate Models These models are simply fits of data to a model. No physical basis. A yield stress fluid may be considered a shear-thinning fluid with very low n. A shear-thinning fluid may be considered a yield stress fluid with a low yield stress. Need to design mixers that work and yield stress model allows us to do this. ENGG 199 Lecture 2b Slide 20

21 Observations A cavern of moving fluid is created around the impeller. A the boundary, the shear stress generated by the impeller is equal to the fluid s yield stress. As impeller speed increases, cavern size increases. Until cavern reaches wall: H C xd C Once cavern reaches wall, cavern grows axially. ENGG 199 Lecture 2b Slide 21

22 Hirata & Aoshima, ChERD, ENGG 199 Lecture 2b Slide 22

23 Cavern Diameter Prediction of Cavern Diameter: D D C PoRe Re Minimum speed when cavern reaches vessel wall: hen D C then T N 2 D 2 N C 2 T 1.32 D Po D 3 2 ENGG 199 Lecture 2b Slide 23

24 Growth of Cavern Once cavern reaches vessel wall, it grows axially: H T C Need to compare power and speed of single impeller design versus multiple impellers (intersecting caverns). x N N C Value of x dependent on impeller type: Rushton turbine: x = 0.40 Paddle: x = 0.45 Pitched blade turbine: x = 0.55 Propeller: x = 0.75 ENGG 199 Lecture 2b Slide 24

25 Shear-thinning vs. ield stress Correlations For yield stress fluids, when cavern reaches wall: 3 T Po 2 2 N D D Po N 3 T 2 D 5 Tq 3 T For shear-thinning fluids: Tq 3 T ENGG 199 Lecture 2b Slide 25