Idea Transcript
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STUDY OF DIFFERENT VISCO-ELASTIC FLUID MODELS
2.1
Introduction: A Bingham plastic is a viscoplastic material which behaves
as a rigid body at low stresses but flows as a viscous fluid at high stress. Bingham[1] proposed its mathematical form. Steffe[23] presented rheological methods in food process. Buckingham[4] demonstrated plastic flow through capillary tubes. Darby and Melson[10] predicted the friction factor for flow of Bingham plastics. Darby et al[11] obtained friction loss in slurry pipes. Swamee and Agrawal[25] obtained explicit equations for laminar flow of Bingham plastic fluids. Its models are commonly used for mud flow in drilling engineering and in the handling of slurries. A common example is toothpaste, which will not be extruded until a certain pressure is applied to the tube. It then is pushed out as a solid plug. In the case of viscous or Newtonian fluid, if the pressure at one end of a pipe is increased this produces a stress on the fluid tending to make it move i.e. shear stress and the volumatric flow rate increases proportionally. However for a Bingham plastic
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stress may be applied but it will not flow until a certain value, the yield stress, is reached beyond this point the flow rate increases steadily with increasing shear stress. Bingham plastic does not exhibit any shear rate, no flow and thus no velocity untill a certain stress is achieved. For the Newtonian fluid the slope is the viscosity which requires the only parameter to describe its flow whereas Bingham plastic requires two parameters, the yield stress and the slope of the line known as the plastic viscosity.
2.2 Friction Factor Formulae: The material is an elastic solid for shear stress , less than a critical value 0 . Once the critical shear stress or yield stress, is exceeded, the material flows in such a way that the shear rate
u 0, y 0 ,
u y
0 0
(2.1)
is directly proportional to the amount by which applied shear stress exceeds the yield stress. It is usually extremely difficult to arrive at exact analytical solution to evaluate the friction factor associated with flow of non-
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Newtonian fluids and hence explicit approximations are used to evaluate it. Once the friction factor has been evaluated the pressure drop may be easily obtained for a given flow by the Dascy-Weisbach equation: f
h f gD LV 2
,
(2.2)
where h f the frictional head loss,
f
=
Darcy friction factor,
L
=
the pipe length
g
=
the gravitational acceleration,
D
=
the pipe diameter,
V
=
the mean fluid velocity.
Laminar Flow An exact description of friction loss for Bingham plastics was fully
developed
laminar
pipe
flow
by
Buckingham[4].
Buckingham-Reiner equation in a dimensionless form reads
The
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64 H e 64 H e4 f L 1 R e 6 R e 3 f 3 Re7
(2.3)
where f
=
the laminar flow friction factor
Re
=
Reynolds number
He
=
Headstrom number
The Reynolds number and Hedstrom numbers are defined as Re
VD ,
D 2 0 He , 2
(2.4)
(2.5)
where be the mass density of the fluid, as the dynamic viscosity of fluid, 0 be the yield point or yield strength of fluid. Turbulent Flow Darby and Melson[10] obtained an empirical relation which was refined by Darby et al[11] as f r 10a Re0.193
(2.6)
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where f r as the turbulent flow friction factor and a 1.47 1 0.146e 2.910
5
He
(2.7)
2.3 Approximations of Buckingham-Reiner Equation: An exact analytical solution of the Buckingham-Reiner equation may not be obtained because it is a fourth order polynomial equation in f. Hence, we use explicit approximations. Swamee-Agrawal[25] approximated which reads 1.143
H 10.67 0.1414 e 64 Re fL 1.16 Re He 1 0.0149 Re Re
He Re
(2.8)
Danish et al[8] presented an explicit method to evaluate the friction factor f by using the Adomian decomposition method. The friction factor contains two terms as K1 fL
1
4K2 K1 K 2 K 1 K14 3K 2 3K 2
K1 K 2 K1 4 K1 3K 2
4
3
(2.9)
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where K1
16 16 H e Re 6 Re2
(2.10)
16 H e4 K2 3Re8
(2.11)
Darby and Melson[10], in view of Churchil[6] and Usagi (1972) obtained a single equation for all flow regimes f f L fT m
m
1 m
(2.12a)
where m 1.7
40000 Re
(2.12b)
Both Swamee-Agrawal equation and the Darby-Melson equation may be combined to give an explicit equation for obtaining the friction factor of Bingham plastic fluids in any regime.
2.4 Concluding Remarks: We have presented friction factor formulae for Bingham plastic fluids. It is obtained that both Swamee-Agrawal equation and Darby-
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Melson equation may be combined to give an explicit equation to
Volume of flow
obtain the friction factor of Bingham plastic fluids in any regime.
Newtonian Liquid
Bingham Plastic Liquid
Yield stress
Shear stress
Figure 1. Bingham Plastic flow as described by Bingham
Bingham Plastic Liquid
Yield stress
Shear stress
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Newtonian Liquid
Yield stress
Shear Rate
Figure 2. Bingham Plastic flow as described currently.
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References (1)
Bingham, E.C. (1916) "An investigation of the Laws of Plastic Flow" U.S. Bureau of Standards Bulletin, 13, 309-353.
(2)
Bingham, E.C. (1922), Fluidity and Plasticity McGraw-Hill (New York) p. 219.
(3)
Bird, R.B., Stewart, W.E., Lighfoot E.N. (1960), Transport Phenomena, Wiley Book Co.; New York.
(4)
Buckingham, E. (1921), "On Plastic Flow through Capillary Tubes", 21: 1154-1156.
(5)
Chhabra,R.P. Richardson J.F., (2008), Non-Newtonian flow and applied rheology, Elsevier.
(6)
Churchill, S.W. (1977), "Friction factor equation spans all fluidflow regimes". Chemical Engineering, Nov. 7, 91-92.
(7)
Churchill, S.W. and Usagi, R.A. (1972), "A general expression for the correlation of rates of transfer and other phenomena". AICHE Journal 18(6), 1121-1128.
(8)
Danish, M. et al (1981): Approximate explicit analytical expressions of friction factor for flow of Bingham fluids in
44
smooth
pipes
Communications
using in
Adomian Nonlinear
decomposition Science
and
method". Numerical
Simulation 16, 239-251. (9)
Darby, R. (1996), Chemical Engineering Fluid Mechanics, Marcel Dekker.
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Fluids,
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