Bingham plastic
Encyclopedia
A Bingham plastic is a viscoplastic material that behaves as a rigid body at low stresses but flows as a viscous
fluid
at high stress. It is named after Eugene C. Bingham
who proposed its mathematical form.
It is used as a common mathematical model
of mud
flow in offshore 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.
) and the volumetric flow rate increases proportionally. However for a Bingham Plastic fluid (in blue), stress can 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. This is roughly the way in which Bingham presented his observation, in an experimental study of paints. These properties allow a Bingham plastic to have a textured surface with peaks and ridges instead of a featureless surface like a Newtonian fluid.
Figure 2 shows the way in which it is normally presented currently. The graph shows shear stress
on the vertical axis and shear rate on the horizontal one. (Volumetric flow rate depends on the size of the pipe, shear rate is a measure of how the velocity changes with distance. It is proportional to flow rate, but does not depend on pipe size.) As before, the Newtonian fluid flows and gives a shear rate for any finite value of shear stress. However, the Bingham Plastic again does not exhibit any shear rate (no flow and thus no velocity) until a certain stress is achieved. For the Newtonian fluid the slope of this line is the viscosity
, which is the only parameter needed to describe its flow. By contrast the Bingham Plastic requires two parameters, the yield stress and the slope of the line, known as the plastic viscosity.
The physical reason for this behaviour is that the liquid contains particles (e.g. clay) or large molecules (e.g. polymers) which have some kind of interaction, creating a weak solid structure, formerly known as a false body, and a certain amount of stress is required to break this structure. Once the structure has been broken, the particles move with the liquid under viscous forces. If the stress is removed, the particles associate again.
τ, less than a critical value
. Once the critical shear stress
(or "yield stress
") is exceeded, the material flows in such a way that the shear rate, ∂u/∂y (as defined in the article on viscosity
), is directly proportional to the amount by which the applied shear stress exceeds the yield stress:
where:
where:
The Reynolds number and the Hedstrom number are respectively defined as:
, and
,
where:
where:
The Swamee-Aggarwal equation is given by:
where:
, and
where:
Both Swamee-Aggarwal equation and the Darby-Melson equation can be combined to give an explicit equation for determining the friction factor of Bingham plastic fluids in any regime. Relative roughness is not a parameter in any of the equations because the friction factor of Bingham plastic fluids is not sensitive to pipe roughness.
Viscosity
Viscosity is a measure of the resistance of a fluid which is being deformed by either shear or tensile stress. In everyday terms , viscosity is "thickness" or "internal friction". Thus, water is "thin", having a lower viscosity, while honey is "thick", having a higher viscosity...
fluid
Fluid
In physics, a fluid is a substance that continually deforms under an applied shear stress. Fluids are a subset of the phases of matter and include liquids, gases, plasmas and, to some extent, plastic solids....
at high stress. It is named after Eugene C. Bingham
Eugene C. Bingham
Eugene Cook Bingham was a professor and head of the Department of Chemistry at Lafayette College. Bingham made many contributions to rheology, a term he is credited with introducing. He was a pioneer in both its theory and practice...
who proposed its mathematical form.
It is used as a common mathematical model
Mathematical model
A mathematical model is a description of a system using mathematical concepts and language. The process of developing a mathematical model is termed mathematical modeling. Mathematical models are used not only in the natural sciences and engineering disciplines A mathematical model is a...
of mud
Mud
Mud is a mixture of water and some combination of soil, silt, and clay. Ancient mud deposits harden over geological time to form sedimentary rock such as shale or mudstone . When geological deposits of mud are formed in estuaries the resultant layers are termed bay muds...
flow in offshore engineering
Offshore construction
Offshore construction is the installation of structures and facilities in a marine environment, usually for the production and transmission of electricity, oil, gas and other resources....
, and in the handling of slurries
Slurry
A slurry is, in general, a thick suspension of solids in a liquid.-Examples of slurries:Examples of slurries include:* Lahars* A mixture of water and cement to form concrete* A mixture of water, gelling agent, and oxidizers used as an explosive...
. A common example is toothpaste
Toothpaste
Toothpaste is a paste or gel dentifrice used with a toothbrush as an accessory to clean and maintain the aesthetics and health of teeth. Toothpaste is used to promote oral hygiene: it serves as an abrasive that aids in removing the dental plaque and food from the teeth, assists in suppressing...
, which will not be extruded until a certain pressure
Pressure
Pressure is the force per unit area applied in a direction perpendicular to the surface of an object. Gauge pressure is the pressure relative to the local atmospheric or ambient pressure.- Definition :...
is applied to the tube. It then is pushed out as a solid plug.
Explanation
Figure 1 shows a graph of the behaviour of an ordinary viscous (or Newtonian) fluid in red, for example in a pipe. If the pressure at one end of a pipe is increased this produces a stress on the fluid tending to make it move (called the shear stressShear stress
A shear stress, denoted \tau\, , is defined as the component of stress coplanar with a material cross section. Shear stress arises from the force vector component parallel to the cross section...
) and the volumetric flow rate increases proportionally. However for a Bingham Plastic fluid (in blue), stress can 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. This is roughly the way in which Bingham presented his observation, in an experimental study of paints. These properties allow a Bingham plastic to have a textured surface with peaks and ridges instead of a featureless surface like a Newtonian fluid.
Figure 2 shows the way in which it is normally presented currently. The graph shows shear stress
Shear stress
A shear stress, denoted \tau\, , is defined as the component of stress coplanar with a material cross section. Shear stress arises from the force vector component parallel to the cross section...
on the vertical axis and shear rate on the horizontal one. (Volumetric flow rate depends on the size of the pipe, shear rate is a measure of how the velocity changes with distance. It is proportional to flow rate, but does not depend on pipe size.) As before, the Newtonian fluid flows and gives a shear rate for any finite value of shear stress. However, the Bingham Plastic again does not exhibit any shear rate (no flow and thus no velocity) until a certain stress is achieved. For the Newtonian fluid the slope of this line is the viscosity
Viscosity
Viscosity is a measure of the resistance of a fluid which is being deformed by either shear or tensile stress. In everyday terms , viscosity is "thickness" or "internal friction". Thus, water is "thin", having a lower viscosity, while honey is "thick", having a higher viscosity...
, which is the only parameter needed to describe its flow. By contrast the Bingham Plastic requires two parameters, the yield stress and the slope of the line, known as the plastic viscosity.
The physical reason for this behaviour is that the liquid contains particles (e.g. clay) or large molecules (e.g. polymers) which have some kind of interaction, creating a weak solid structure, formerly known as a false body, and a certain amount of stress is required to break this structure. Once the structure has been broken, the particles move with the liquid under viscous forces. If the stress is removed, the particles associate again.
Definition
The material is rigid for shear stressShear stress
A shear stress, denoted \tau\, , is defined as the component of stress coplanar with a material cross section. Shear stress arises from the force vector component parallel to the cross section...
τ, less than a critical value
. Once the critical shear stressShear stress
A shear stress, denoted \tau\, , is defined as the component of stress coplanar with a material cross section. Shear stress arises from the force vector component parallel to the cross section...
(or "yield stress
Yield (engineering)
The yield strength or yield point of a material is defined in engineering and materials science as the stress at which a material begins to deform plastically. Prior to the yield point the material will deform elastically and will return to its original shape when the applied stress is removed...
") is exceeded, the material flows in such a way that the shear rate, ∂u/∂y (as defined in the article on viscosity
Viscosity
Viscosity is a measure of the resistance of a fluid which is being deformed by either shear or tensile stress. In everyday terms , viscosity is "thickness" or "internal friction". Thus, water is "thin", having a lower viscosity, while honey is "thick", having a higher viscosity...
), is directly proportional to the amount by which the applied shear stress exceeds the yield stress:
Friction Factor Formulae
In fluid flow, it is a common problem to calculate the pressure drop in an established piping network. Once the friction factor, f, is known, it becomes easier to handle different pipe-flow problems, viz. calculating the pressure drop for evaluating pumping costs or to find the flow-rate in a piping network for a given pressure drop. It is usually extremely difficult to arrive at exact analytical solution to calculate the friction factor associated with flow of non-Newtonian fluids and therefore explicit approximations are used to calculate it. Once the friction factor has been calculated the pressure drop can be easily determined for a given flow by the Darcy–Weisbach equation:where:
-
is the frictional head loss (SI units: m) -
is the friction factor (SI units: Dimensionless) -
is the pipe length (SI units: m) -
is the gravitational acceleration (SI units: m/s²) -
is the pipe diameter (SI units: m) -
is the mean fluid velocity (SI units: m/s)
Laminar flow
An exact description of friction loss for Bingham plastics in fully developed laminar pipe flow was first published by Buckingham. His expression, the Buckingham-Reiner equation, can be written in a dimensionless form as follows:where:
-
is the laminar flow friction factor (SI units: Dimensionless) -
is the Reynolds number (SI units: Dimensionless) -
is the Hedstrom number (SI units: Dimensionless)
The Reynolds number and the Hedstrom number are respectively defined as:
, and,where:
-
is the mass density of fluid (SI units: kg/m3) -
is the kinematic viscosityViscosityViscosity is a measure of the resistance of a fluid which is being deformed by either shear or tensile stress. In everyday terms , viscosity is "thickness" or "internal friction". Thus, water is "thin", having a lower viscosity, while honey is "thick", having a higher viscosity...
of fluid (SI units: m²/s)
Turbulent flow
Darby and Melson developed an empirical expression that determines the friction factor for turbulent-flow regime of Bingham plastic fluids, and is given by:where:
-
is the turbulent flow friction factor (SI units: Dimensionless) -
Approximations of the Buckingham-Reiner equation
Although an exact analytical solution of the Buckingham-Reiner equation can be obtained because it is a fourth order polynomial equation in f, due to complexity of the solution it is rarely employed. Therefore, researchers have tried to develop explicit approximations for the Buckingham-Reiner equation.Swamee-Aggarwal Equation
The Swamee Aggarwal equation is used to solve directly for the Darcy–Weisbach friction factor f for laminar flow of Bingham plastic fluids. It is an approximation of the implicit Buckingham-Reiner equation, but the discrepancy from experimental data is well within the accuracy of the data.The Swamee-Aggarwal equation is given by:
Danish-Kumar Solution
Danish et al. have provided an explicit procedure to calculate the friction factor f by using the Adomian decomposition method. The friction factor containing two terms through this method is given as:where:
, andDarby-Melson Equation
In 1981, Darby and Melson, using the approach of Churchill and of Churchill and Usagi, developed an expression to get a single friction factor equation valid for all flow regimes:where:
Both Swamee-Aggarwal equation and the Darby-Melson equation can be combined to give an explicit equation for determining the friction factor of Bingham plastic fluids in any regime. Relative roughness is not a parameter in any of the equations because the friction factor of Bingham plastic fluids is not sensitive to pipe roughness.