1 law Stokes’ A written report submitted to Engr. Ramer P. Bautista Submitted by Alvarez, Lanna Marie L. Blay, Nathalie Kristine E.. Dimaano, Roy Albert C. Galicia, James Manipol, Mark Glenn G. Ordonio, Mark Angelo A. (ChE 147 - T) Department of Chemical Engineering College of Engineering and Agro-Industrial Technology University of the Philippines Los Baños March 2012 In partial fulfillment of the requirements for the undergraduate subject ChE 147 (Applications of Fluid Dynamics in Chemical Engineering)
Transcript
1
law
Stokes’
A written re
port submitte
d to
Engr. Ramer P. Bautist
a
Submitted by
Alvarez, Lanna M
arie L.
Blay, Nathalie Krist
ine E..
Dimaano, Roy Albert C
.
Galicia, James
Manipol, Mark Glenn G.
Ordonio, Mark Angelo A.
(ChE 147 - T)
Department o
f Chemical Engineering
College of Engineering and Agro-Industri
al Technology
University of th
e Philippines Los B
años
March 2012
In p
artia
l fulfillm
ent o
f the
req
uire
me
nts fo
r the
un
de
rgra
dua
te su
bje
ct
ChE 1
47 (A
pp
lica
tions o
f Fluid
Dyna
mic
s in C
he
mic
al En
gin
ee
ring
)
2
law
Stok
es’
Drag Coefficient
The friction factor fF, defined as the ratio of the shear stress to the product of the velocity head and density, is
shown to be useful in treating fluid flow through conduits. An analogous factor, the drag coefficient CD, is used
for immersed solids. Consider a smooth sphere immersed in a flowing fluid and at a distance from the solid
boundary of the stream sufficient for the approaching stream to be in potential flow, as shown in Fig. 1.
Figure 1. Flow past immersed sphere.
The drag coefficient CD is defined as the ratio of total drag to the same product of the velocity head and densi-
ty, or
(1)
where FD is the total drag (force acting on the solid), AP is the projected area of the solid normal to the flow,
(π/4)Dp2 for a sphere, ρ is the density of the stream, and uo is the velocity of approaching stream (assumed con-
stant over the projected area). This equation is important wherever momentum transfer at a fluid-flow bounda-
ry must be examined. Thus, it may be applied to the design of particle-separation equipment as well as to that
of piping systems (Foust et al, 1980).
For particles having shapes other than spherical, it is necessary to specify the size and geometric form of the
body and its orientation with respect to the direction of flow of the fluid. One major dimension is chosen as the
characteristic length, and other important dimensions are given as ratios to the chosen one. Such ratios are
called shape factors (McCabe et al, 1985).
From dimensional analysis, the drag coefficient of a smooth solid in an incompressible fluid depends upon a
Reynolds number and the necessary shape factors. The Reynolds number for a particle in a fluid is defined as
(2)
where D0 is the characteristic length.
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Stok
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Drag coefficients for compressible fluids increase with increase in the Mach number when the latter becomes
more than about 0.6 (McCabe et al, 1985). Coefficients in supersonic flow are generally greater than in subson-
ic flow.
Figure 2. Drag coefficients for spheres, disks, and cylinders.
Fig. 2 shows the relationship of CD and NRe, p for spheres, long cylinders, and disks. However, the curves that
show these relationships are valid only for a maintained orientation for which the axis of the cylinder and the
face of the disk are perpendicular to the direction of flow (McCabe et al, 1985). If, for example, a disk or cylin-
der is moving by gravity or centrifugal force through a quiescent fluid, it will twist and turn as it moves freely
through the fluid.
Stokes’ Law
George Gabriel Stokes, famous for his work describing the motion of a sphere through viscous fluids, was an
Irish-born mathematician who spent much of his life working with fluid properties. This directed to the devel-
opment of Stokes' Law in the 1840s. This equation shows the force needed to move a small sphere through a
continuous, quiescent fluid at a certain velocity. It is based primarily on the radius of the sphere and the vis-
cosity of the fluid.
From the complex nature of drag, it is not surprising that the variation of CD with NRe, p is more complicated
than that of fF with NRe, p. The variations in slope of the curves of CD vs. NRe, p at different Reynolds numbers
are the results of the interplay of the various factors that control form drag and wall drag. Their effects can be
followed by discussing the case of the sphere.
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Stok
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For low Reynolds numbers, the drag force for a sphere conforms to a theoretical equation called Stokes’ law,
written as
(3)
The drag coefficient predicted by Stokes’ law is
(4)
In theory, Stokes’ law is valid only when NRe, p is considerably less than unity. Practically, as shown by the left-
hand portion of the graph of Fig. 2, Equations (3) and (4) may be used with small error for all Reynolds num-
ber less than 1. At low velocities at which the law is valid, the sphere moves through the fluid by deforming it.
The wall shear is the result of the viscous forces only, and inertial forces are negligible. The motion of the
sphere affects the fluid at considerable sentences from the body, and if there is a solid within 20 or 30 diame-
ters of the sphere, Stokes’ law must be corrected for the wall effect. Referred as creeping flow, this flow treated
by the law is valuable for calculating the resistance of small particles, such as dusts or fogs, moving through
gases or liquids of low viscosity, or for the motion of larger particles through highly viscous liquids.
Figure 3. Flow past single sphere, showing separation and wake formation: (a) laminar flow in boundary lay-
er; (b) turbulent flow in boundary layer; B: stagnation point, C: separation point.
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Stok
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As the Reynolds number is increased to 10 or above, well beyond the range of Stokes’ law, separation occurs
at a point just forward of the equatorial plane, as shown in Fig. 3a, and a wake, covering the entire rear hemi-
sphere, is formed. Wakes, backwater zones of strongly decelerated fluid, are characterized by a large friction
loss, developing a large form drag. In a wake, the angular velocity of the vortices (thus the kinetic energy of
rotation), is large. By Bernoulli principle, the pressure in the wake is less than that in the separated boundary
layer; a suction develops in the wake, and the component of the pressure vector acts in the direction of flow.
The pressure drag, and hence the total drag, is large, much greater than if Stokes’ law still applied.
At moderate Reynolds number, the vortices disengage from the wake in a regular fashion, forming in the
downstream fluid a series of moving vortices known as a ‘vortex street.’ However, at Reynolds number above
about 2,500, vortices are no longer shed from the wake, a stable boundary forms, originating at the apex point
B in Fig. 3. The boundary layer grows and separates, flowing freely around the wake after separation. The drag
coefficient is nearly constant, as shown in Fig. 2, for spheres and cylinders it increases slightly with the Reyn-
olds number. As the Reynolds number is increased, transition to turbulence takes place, first in the free bound-
ary layer and then in the boundary layer still attached to the front hemisphere of the sphere. When turbulence
occurs in the latter, the separation point moves towards the rear of the body and the wake shrinks, as shown in
Fig. 3b. Both friction and drag decrease, and the remarkable drop in drag coefficient from 0.45 to 0.10 at a
Reynolds number of about 250,000 is a result of the shift in separation point when the boundary layer attached
to the sphere becomes turbulent. At Reynolds number above 300,000, the drag coefficient is nearly constant.
The Reynolds number at which the attached boundary layer becomes turbulent is called the critical Reynolds
number for drag. The curve for spheres shown in Fig. 2 applies only when the fluid approaching the sphere is
non-turbulent or when the sphere is moving through a stationary fluid. If the approaching fluid is turbulent, the
critical Reynolds number is sensitive to the scale of turbulence and becomes smaller as the scale increases. For
example, if the scale of turbulence, defined as , is 2 per cent, the critical Reynolds number is
about 140,000. One method of measuring the scale of turbulence is to determine the critical Reynolds number
and use a known correlation between the two quantities.
The curve of CD vs. NRe for an infinitely long cylinder normal to the flow is much like that of a sphere, but at
low Reynolds number, CD does not vary inversely with NRe because of the two-dimensional character of the
flow around the cylinder. For short cylinders, such as catalyst pellets, the drag coefficient falls between the
values for spheres and long cylinders and varies inversely with the Reynolds number at very low Reynolds
numbers. Disks do not show the drop in drag coefficient at a critical Reynolds number, because once the sepa-
ration occurs at the edge of the disk, the separated steam does not return to the back of the disk and the wake
does not shrink when the boundary layer becomes turbulent. Bodies which show this type of behaviour are
called bluffed bodies. For a disk the drag coefficient CD is approximately unity at Reynolds numbers above
2,000.
The drag coefficients for irregularly shaped particles such as coal or sand appear to be about the same as for
spheres of the same nominal size at Reynolds numbers less than 50. However the curve of CD vs. NRe levels out
at NRe ≈ 100, and the values of CD are 2 to 3 times those for spheres in the range NRe = 500 - 3,000 (McCabe et
al, 1985).
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es’
Terminal Settling Velocity
Consider a small spherical body, falling freely due to gravity in a viscous medium.
Figure 4. Forces actiong on a body during free-fall.
The various forces acting on the body are:
Weight of the body, acting downwards;
Viscous drag FV, acting upwards (opposing motion of the body); and
Upthrust or buoyant force FT of liquids, equal to weight of the displaced liquid.
When the velocity of a falling body increases and continues to increase until the accelerating and resisting forc-
es are balanced, the velocity of that particle remains constant during the remainder time of fall (Foust et al,
1980). That constant velocity is the terminal velocity, or
(5)
where ut is the terminal velocity and ρs and ρ are the solid and liquid densities, respectively.
Settling particles may undergo fluctuating motions owing to vortex shedding, among other factors. Oscillation
is enhanced with increasing separation between the mass and geometric centers of the particle. Variations in
mean velocity are usually less than 10 per cent. The drag force on a particle fixed in space with fluid moving is
somewhat lower than the drag force on a particle freely settling in a stationary fluid at the same relative veloci-
ty.
Terminal velocity independent of the drag coefficient can also be calculated in laminar flow. The resisting
force due to fluid friction acting on a sphere when the relative motion produces laminar flow has been shown
using Stokes law with this new equation
(6)
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Stok
es’
This is another statement of the Stokes’ law, and is applicable to the fall of spherical particles in laminar flow.
For example, it is used for calculating viscosity using a falling-ball viscometer. A ball of known diameter falls
through a fluid of unknown viscosity in a tube. The time of fall between two points is measured, and by Equa-
tion (6), the viscosity can be determined.
When an object falls from rest, its velocity u(t) is
(7)
where b, which is the drag coefficient. u(t) asymptotically approaches the terminal velocity ut = mg/b. For a
certain b, heavier objects fall faster. Stokes’ law assumes a low velocity of a small falling particle. Now the
drag, Stokes drag has a coefficient b equal to b = 6ðrç. This is the coefficient used in equation (4). The deriva-
tion of b is easy for the parameters r and ç, but the problem is the factor 6ð. A factor 2ð is caused by a pressure
effect and a factor 4ð by friction.
Applications
Stokes’ law has many applications in science, such as in earth science where measurement of the setting time
gives the radius of soil particles. In air, the same theory can be used to explain why small water droplets (or ice
crystals) can remain suspended in air (as clouds) until they grow to a critical size and start falling as rain (or
snow and hail). It also explains why the speed of a raindrop is less than a freely falling body with constant ve-
locity, from the height of clouds. The same law helps a man coming down with the help of a parachute, to slow
down.
Figure 5. The falling-ball viscometer.
Stokes' law is the basis of the falling-ball viscometer, in which the fluid is stationary in a vertical glass tube. A
sphere of known size and density is allowed to descend through the liquid. If correctly selected, it reaches ter-
minal velocity, which can be measured by the time it takes to pass two marks on the tube. Electronic sensing
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law
Stok
es’
can be used for opaque fluids. Knowing the terminal velocity, the size and density of the sphere, and the densi-
ty of the liquid, Stokes' law can be used to calculate the viscosity of the fluid. A series of steel ball bearings of
different diameters are normally used in the classic experiment to improve the accuracy of the calculation. The
school experiment uses glycerine as the fluid, and the technique is used industrially to check the viscosity of
fluids used in processes. Several school experiments often involve varying the temperature and/or concentra-
tion of the substances used in order to demonstrate the effects this has on the viscosity. Industrial methods in-
clude many different oils, and polymer liquids such as solutions.
Figure 6. Diagram of a hematocrit tube.
In medicine, a well-known application is the precipitation of blood cells. After setting, on the bottom of a hem-
atocrit tube are the red cells, the erythrocytes, since they are large and have the highest density. In the middle
are the white cells, the leucocytes despite their often larger volume. However, they are less dense and especial-
ly less smooth, which slows their speed of setting. On top, and hardly visible, is a thin band of the much small-
er platelets, the thrombocytes. The relative height of the band (cylinder) with the red cells is the hematocrit.
Although the red cells are not spherical and the medium is not large at all (a narrow hematocrit microtube), the
process still behaves rather well according to Stokes’ law. Red cells can clotter to money rolls, which set faster.
Another important application is the process of centrifugation of a biochemical sample. The centrifuge is used
to shorten substantially the setting time. In this way proteins and even smaller particles can be harvested, such
as radio nucleotides (enrichment of certain isotopes of uranium in an ultracentrifuge). With centrifugation, the
same equations hold, but the force of gravity g should be replaced by the centrifugal acceleration a, or
(8)
where f is the number of rotations/s and R the radius of the centrifuge (the distance of the bottom of the tube to
the center). In biochemistry, a can easily reach 104 g and in physics even 106 g.
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Sample Problems
1. Calculate the equivalent radius of a red blood cell given the following parameters: hematocrit
(assume 0.45 L/L), settling velocity (0.003 m/hour = 0.83∙10-6 m/s), density of red cells (1120
kg/m3) and plasma (1000 (kg/m3).
Solution:
Using Equation (6) gives a radius equal to 3.5 μm. Actually, the red cell is disk-shaped with a
radius of about 3.75 μm and a thickness of 2 μm.
2. Consider a small sphere with radius r = 1 μm moving through water at a velocity u of 10 µm/s.
Using 10-3 as the dynamic viscosity of water in SI units, find the drag force.
Solution:
Using Equation (6) gives a drag force of 0.2 pN. This is about the drag force that a bacterium
experiences as it swims through water.
3. The viscosity of glycerin is temperature dependent, being 1.49 Pa·s at 20oC and 0.95 Pa·sec at
25oC. A ball of radius 1.2mm, weighing 0.05 grams, was dropped through glycerine at 23oC at
a speed of 0.022 m/s. Find the drag force.
Solution:
At 23oC, assume viscosity to be at 1.17 Pa·s, just taking a linear interpolation.
Using Equation (6) gives
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es’
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References
(n. a.). (n. d.). Stokes’ Law. Retrieved March 14, 2012 from http://en.wikipedia.org/wiki/Stokes'_law
(n. a.). (n. d.). Stokes’ Law. Retrieved March 17, 2012 from http://www.tutorvista.com/content/physics/physics
-iii/solids-and-fluids/stokes-law.php
(n. a.). (n. d.). Stokes’ law and hematocrit. Retrieved March 17, 2012 from http://onderwijs1.amc.nl/medfysica/
doc/StokesLawHematocrit.htm
(n.a.). (1998) Stokes’ Law. Retrieved March 17, 2012 from http://www.cord.edu/faculty/ulnessd/legacy/
fall1998/sonja/stokes.htm
Foust, A. S., Wenzel, L. A., Clump, C. W., Maus, L., and Andersen, L. B. (1980). Principles of Unit Opera-
tions (2nd ed.). Singapore: John Wiley & Sons.
Fowler, M. (2006). Dropping the Ball (Slowly). Retrieved March 14, 2012 from http://
