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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
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3. Concept of Boundary Layer Growth and Separation
Ref: Foust, Chapter-13
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4. 4
The boundary layer separates from the obstacle's surface to form a vortex-filled
wake . This phenomenon is known as boundary layer separation.
Outside the boundary layer, and the wake, the flow pattern is irrotational and
essentially inviscid.

5. In the Laminar regime:
B.L Thickness:
Point stress:
Mean stress:
In the turbulent regime:
B.L Thickness:
Point stress:
Mean stress:
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6. Nomenclature:
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7. Transition point:
The mean stress for the
laminar portion of the B.L
that lies between y = 0
to y = yc :
Average stress
over the plate:
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8. Types of drag
A force is exerted on the solid by the fluid while moving through it. This force is a
combination of - 1. BL drag (or skin drag or skin friction) and 2. Form drag.
[Significant frictional
loss occur because of
acceleration and
deceleration when the
fluid changes path to
pass around a solid
body set in the flow
path.]
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9. (Boundary layer
growth begins at
the stagnation
point and
continues over the
entire surface)
(At the exact center
of the body the fluid
will have zero
velocity. This is
known as stagnation
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)
(The fluid outside the B.L is
subjected to acceleration due
in part to change in path. As
the fluid is diverted in path
to pass around the body, a
force is exerted upon the
body by the fluid)
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10. Skin friction or skin drag: The tangential
stress on the body arising from transfer of
momentum originating in the slowing
down of the boundary layer is the skin
friction.
Form drag or Pressure drag: The
summation of all forces on the body due
to acceleration and deceleration
constitutes the form drag of the body.
Form drag arises from the pressure field
around the body caused by the body shape
and the displacement effect of the
boundary layer as well as separation of
the flow from that body shape because of
viscous effects. 10

11. Skin drag and Form drag
 The transfer of momentum resulted in a tangential stress or drag on a
smooth surface that was oriented parallel to the flow direction . This
phenomenon is traditionally called skin friction or skin drag.
 If any surface is in contact with a fluid and a relative motion exists
between the fluid and the surface, skin friction will exist between the surface
and the fluid .
 In addition to skin friction , significant frictional losses occur because of
acceleration and deceleration of the fluid . The accelerative effects occur
when the fluid changes path to pass around a solid body set in the flow path .
This phenomenon is called form drag.
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12. Ref: Fluid Mechanics R.K
Bansal, Chapter-14
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18. The drag co efficient ( CD ) is defined as:
Correlations are usually presented graphically on logarithmic plots of
CD as a function of NRe
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19. (CD vs. NRe plot for
particles of different
shapes)
Above NRe = 105 the boundary layer is considered turbulent for all shapes and the accelerative
effects predominate. It is in this region that the drag diagram can be represented by: CD = const
Ref: Foust, Chapter-13
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What is it mean?

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29. Consider the flow of an incompressible fluid about a solid sphere of radius R and
diameter D. The fluid, with density ρ and viscosity μ, approaches the fixed sphere
vertically upward in the z direction with a uniform velocity vꭃ,. For this problem,
"creeping flow" means that the Reynolds number Re = Dvρ/μ is less than about 0.1.
This flow regime is characterized by the absence of eddy formation downstream
from the sphere.
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Transport Phenomena –
Bird & Stewart, 2nd ed
(Chapter 2, Section 2.6)
Read & Understand the
complete derivation

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32. Normal force:
Tangential force:
Total forces on the spherical particle:
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Solve this given set of equations to obtain the total force on a spherical particles
subjected to creeping flow

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Navier Stokes Equation:

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Terminal Velocity of Particles
Ref: Foust, Chapter-22

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