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Momentum Transfer

Momentum Transfer. Jul-Dec 2006 Instructor: Dr. S. Ramanathan Office: CHL 210 Email: srinivar@iitm.ac.in Class Notes: http://www.che.iitm.ac.in/~srinivar. Overview. Background & Motivation Course Syllabus What will be covered and what will not be Examples Goals & Pre-requisites

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Momentum Transfer

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  1. Momentum Transfer Jul-Dec 2006 Instructor: Dr. S. Ramanathan Office: CHL 210 Email: srinivar@iitm.ac.in Class Notes: http://www.che.iitm.ac.in/~srinivar

  2. Overview • Background & Motivation • Course Syllabus • What will be covered and what will not be • Examples • Goals & Pre-requisites • Evaluation • Tentative Schedule • Text Books / References

  3. Transport Phenomena Reaction Kinetics Chemical Engineering Momentum Mass Heat

  4. Background : • Most of the momentum transfer equations are similar to heat and mass transfers • Momentum transfer: Focus is on fluids Heat and Mass Transfer: Also include solid Heat Transfer: Radiation (no corresponding phenomena in momentum and mass transfer) • Similarities in problems will be discussed as appropriate

  5. Motivation Momentum Transfer: Fluid Mechanics Understanding Lab Results Design Manufacturing (Production/ Maintenance) Troubleshooting To do these things, how much do I have to know

  6. Course Syllabus: What will be covered? And to what extent? Fundamentals (ideal cases) Some applications (more realistic, but not very) • Most real-life issues, ==> kinetics & heat/mass/momentum transfer together • Analytical solutions not possible in many cases What will not be covered? • Compressible , supersonic flows • Only limited exposure to non-newtonian fluids • Computational Fluid Dynamics (CFD) • limited exposure to Perturbation methods • ...and so on

  7. Course Syllabus: • Statics: • To refresh the basics Dynamics: • Mass Balance • Momentum Balance (Linear & Angular) • Energy Balance • Frictional losses • Boundary layer theory • Flow past/through

  8. Examples • Pumps, Turbines • Heat Exchangers, Distillation column • Fluidized or Fixed bed reactors • CVD reactors (micro electronics) • Artificial blood vessels (Bio)

  9. Examples • Production of Sulfuric Acid • used in fertilizers, car batteries etc

  10. Examples • Monsanto Process • Pump air (N2+O2) and burn Sulfur • Provide large area of catalyst • “Scrub” with water • Store the sulfuric acid • For a given production (ton per day), • What is the pump capacity needed? • Design and operation of reactor • How to measure the flow rate? • What if something goes wrong? How to detect it and how to respond? (Detection of leak through chemical sensor, pressure sensor etc)

  11. Goals: • Understanding and approaching problems which involve Momentum Transfer • ==> Pumps, flow through pipes • ==> Separation (filtration, adsorption etc) • More emphasize on application and less on proof • Also prepare for future courses • Momentum Transfer Lab • Transport phenomena Calculus (PDE), Complex Variables Little bit of programming Final Exam - 40 Quizzes - 2 * 20 = 40 and Project/Assignment -20

  12. Tentative Schedule Quiz-1 Quiz-2

  13. Text Books / References • Class Notes / Slides • Slides will be on the internal server • Text: Fundamentals of Momentum, Heat and Mass Transfer by Welty, Wicks , Wilson & Rorrer (4th edition) John Wiley & Sons • Reference: Transport Phenomena by Bird, Stewart and Lightfoot, edition, McGraw Hill • Fluid Mechanics and its applications by Gupta & Gupta • Other sources referenced will be mentioned in the class

  14. Statics

  15. Statics • Fluid: changes shape continuously when a tangential force is applied • Pressure at any point in a stationary fluid is same in all directions • Pressure vs Distance • Consider only gravity effects • ie. Ignore electromagnetic, chemical (eg.osmosis) and other forces

  16. Po = atm h r P bottom = Po + r g h Statics • Constant Density • (eg Liquids) Application: Manometer

  17. Approx air temp vs height 80 Height (km) 50 10 -120 -60 0 Temp (C) Fig from “Introduction to Fluid Mechanics” by Fox & McDonald, page 53 Statics • Variable Density • eg Gases

  18. B 10 cm A Water 25 cm Hg Example rMercury = 13,600 kg/m3 Pbm. 2.13 PA-PB=? B’ PB’-PB= r1 * g * h1 PA-PB’ = r2 * g * h2 -r1 * g * h2 Actually used for flow rate measurement

  19. Example Pbm 2.22 z Practical depth for a suited diver is ~ 180 m What is the error in assuming density is constant?

  20. Example Coin on water: Surface Tension F Indication of force between liquid-metal vs liquid/liquid

  21. Statics • Acceleration due to other forces • eg centrifuge, accelerating vehicle • In centrifuge, usually g is negligible compared to a • Otherwise use vector algebra to add g and a

  22. Example: Centrifuge r a • To separate materials based on density difference • in case gravity is insufficient (for reasonable separation) • Acceleration expressed as N times “g” • Typically acceleration >> g • Ignore gravity effects

  23. Example: Slow rotation For lesser acceleration h1 At z=h1, r=0, P = Patm On the surface, P = Patm Equation of free surface

  24. Conservation of Mass • In any control volume • Mass flux in - mass flux out = Mass accumulation rate. • If (mass in) is taken as -ve, then • Accumulation rate + Flux(out -in) =0 S Vol V-velocity n-normal vector

  25. Conservation of Mass • Reynold’s Theorem (generalization) • For a property B (Mass, for example) • and corresponding b (per unit mass) Rate of change (system) = Flux+ Accumulation See Transport Phenomena, by Bird Stewart Lightfoot for an analogy

  26. Reynold’s Transport Thm • B = Mass • ==> b =1 • DB/Dt =0; Eqn of Conservation of Mass • B = Momentum • ==> b = velocity • Momentum Eqn • B = Angular Momentum • ==> b = r x v (Angular Momentum Eqn) • etc..

  27. Mass conservation • Simplifications • Steady State : (gas or liquid) • d/dt =0 • Mass in = Mass out • For liquids (Volume in = Volume out) • Constant density & fixed control Volume: • d/dt (V) =0 • Volume in = Volume out • True even for unsteady state

  28. Examples • Pbm. 4.8, 4.5, 4.12, 4.18, 4.11,4.9 4.15, 4.20, 4.22, 4.21, 4.24

  29. Examples Pbm. 4.18, steady state V1 V2 d1 = d2 = 2 cm Q1 = 0.0013 m3/s V2 = 2.1 m/s A3 = 100 * (p 1e-3*1e-3/4) There are 100 holes of 1 mm dia in the shower V3

  30. Examples Pbm. 4.8 A1 V2,a2 A2 V1,a1 Area =A, Velocity =V, Acclrn = a. Find V2, A2 V1 (t) A1 = V2 (t) A2 a1A1 = a2A2

  31. Examples Pbm. 4.5, steady state V/2 m/s 0.5 m Long, 0.1 m R V m/s 6 m/s

  32. X Y Examples Pbm. 4.11 V2 r1 r2 Vw V1 Consider stationary control volume

  33. X Y Examples Pbm. 4.11 V2 r1 r2 Vw V1 Consider control volume moving @ Vw

  34. Examples Pbm. 4.9, one dimension, steady flow

  35. r Examples Pbm. 4.12 R

  36. Examples Pbm. 4.15 Steady flow liquid film thickness is “h” width “into the paper” is W V0 h Y X

  37. 0 Examples Constant Velocity V Varying thickness “b” Infinitely long plate (in one direction) Exit velocity is (a) flat or (b) parabolic Pbm. 4.14 V b 2L Mass Flux Y direction Accmln rate Consider unit depth for control Vol

  38. Velocity of outgoing fluid = V(y) Examples Pbm. 4.14 V b For a flat profile, V(y) = constant, say Vavg 2L Y direction For a parabolic profile,

  39. 0 Examples d1= 2cm, d2=0.8 mm Pbm. 4.21 d2 d1 V How fast should the plunger move (ie find V) (a) if there is no leakage (b) if leakage between tube and plunger is 10% of needle flow Mass Flux Accmln rate

  40. Qn: What is the flow rate across the Horizontal surface? Examples Pbm. 4.13 Vx =V0 V0 0 Height=6d d V0 Vx =V0 V0

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