Lecture 1 - Drag & Terminal Velocity.pdf

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1. ChE 3203/4221 Particle Technology
Forces on Submerged Particles
Drag & Terminal Velocity
1
Safat Anam
Lecturer
Dept. of ChE, RUET

2. Concept of:
Boundary layer growth and separation
Drag force on submerged particles
Total forces on submerged particles
Creeping flow around a spherical particles
Navier stokes equation
Terminal Velocity
2

3. Concept of Boundary Layer Growth and Separation
Ref: Foust, Chapter-13
3

4. 4
Theboundarylayerseparatesfromtheobstacle'ssurfacetoformavortex-filled
wake.Thisphenomenonisknownasboundarylayerseparation.
Outsidetheboundarylayer,andthewake,theflowpatternisirrotationaland
essentiallyinviscid.

5. In the Laminar regime:
B.L Thickness:
Point stress:
Mean stress:
In the turbulent regime:
B.L Thickness:
Point stress:
Mean stress:
5

6. Nomenclature:
6

7. Transition point:
Themeanstressforthe
laminarportionoftheB.L
thatliesbetweeny=0
toy=y
c:
Average stress
over the plate:
7

8. Types of drag
Aforceisexertedonthesolidbythefluidwhilemovingthroughit.Thisforceisa
combinationof-1.BLdrag(orskindragorskinfriction)and2.Formdrag.
[Significantfrictional
lossoccurbecauseof
accelerationand
decelerationwhenthe
fluidchangespathto
passaroundasolid
bodysetintheflow
path.]
8

9. (Boundary layer
growth begins at
the stagnation
point and
continues over the
entire surface)
(Attheexactcenter
ofthebodythefluid
willhavezero
velocity.Thisis
knownasstagnation
point)
(The tangential
stress on the body
arising from
transfer of
momentum
originating in the
slowing down of
the boundary
layer is the skin
friction)
(ThefluidoutsidetheB.Lis
subjectedtoaccelerationdue
inparttochangeinpath.As
thefluidisdivertedinpath
topassaroundthebody,a
forceisexerteduponthe
bodybythefluid)
9

10. Skinfrictionorskindrag:Thetangential
stressonthebodyarisingfromtransferof
momentumoriginatingintheslowing
downoftheboundarylayeristheskin
friction.
FormdragorPressuredrag:The
summationofallforcesonthebodydue
toaccelerationanddeceleration
constitutestheformdragofthebody.
Formdragarisesfromthepressurefield
aroundthebodycausedbythebodyshape
andthedisplacementeffectofthe
boundarylayeraswellasseparationof
theflowfromthatbodyshapebecauseof
viscouseffects.
10

11. Skin drag and Form drag
Thetransferofmomentumresultedinatangentialstressordragona
smoothsurfacethatwasorientedparalleltotheflowdirection.This
phenomenonistraditionallycalledskinfrictionorskindrag.
Ifanysurfaceisincontactwithafluidandarelativemotionexists
betweenthefluidandthesurface,skinfrictionwillexistbetweenthesurface
andthefluid.
Inadditiontoskinfriction,significantfrictionallossesoccurbecauseof
accelerationanddecelerationofthefluid.Theaccelerativeeffectsoccur
whenthefluidchangespathtopassaroundasolidbodysetintheflowpath.
Thisphenomenoniscalledformdrag.
11

12. Ref: Fluid Mechanics R.K
Bansal, Chapter-14
12

13. 13

14. 14

15. 15

16. 16

17. 17

18. The drag co efficient ( C
D) is defined as:
Correlationsareusuallypresentedgraphicallyonlogarithmicplotsof
C
DasafunctionofN
Re
18

19. (C
Dvs.N
Replotfor
particlesofdifferent
shapes)
AboveN
Re=10
5
theboundarylayerisconsideredturbulentforallshapesandtheaccelerative
effectspredominate.Itisinthisregionthatthedragdiagramcanberepresentedby:C
D=const
Ref: Foust, Chapter-13
19

20. 20

21. 21
What is it mean?

22. 22

23. 23

24. 24

25. 25

26. 26

27. 27

28. 28

29. ConsidertheflowofanincompressiblefluidaboutasolidsphereofradiusRand
diameterD.Thefluid,withdensityρandviscosityμ,approachesthefixedsphere
verticallyupwardinthezdirectionwithauniformvelocityv
ꭃ,.Forthisproblem,
"creepingflow"meansthattheReynoldsnumberRe=Dvρ/μislessthanabout0.1.
Thisflowregimeischaracterizedbytheabsenceofeddyformationdownstream
fromthesphere.
29
Transport Phenomena –
Bird & Stewart, 2
nd
ed
(Chapter 2, Section 2.6)
Read & Understand the
complete derivation

30. 30

31. 31

32. Normal force:
Tangential force:
Total forces on the spherical particle:
32

33. 33
Solvethisgivensetofequationstoobtainthetotalforceonasphericalparticles
subjectedtocreepingflow

34. 34

35. 35
Navier Stokes Equation:

36. 36

37. 37
Terminal Velocity of Particles
Ref: Foust, Chapter-22

38. 38

39. 39

40. 40

41. 41

42. 42

43. 43

44. 44

45. 45

46. 46

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48. 48

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53. 53

54. 54
Foust Exercise Problem: 22.1, 22.3, 22.4, 22.5, 22.6