Bingham plastic

Definition

The material is an elastic solid for shear stress ${\displaystyle \tau }$, less than a critical value ${\displaystyle {\tau }_0}$. 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:

${\displaystyle \frac{\mathrm{\partial }u}{\mathrm{\partial }y}=\{\begin{array}{ll}0,& \tau <{\tau }_0\\ \frac{\tau -{\tau }_0}{{\mu }_{\mathrm{\infty }}},& \tau \ge {\tau }_0\end{array}}$

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:

${\displaystyle f=\frac{2h_{\text{f}}gD}{L{V}^2}}$

where:

• ${\displaystyle f}$ is the Darcy friction factor (SI units: dimensionless)
• ${\displaystyle h_{\text{f}}}$ is the frictional head loss (SI units: m)
• ${\displaystyle g}$ is the gravitational acceleration (SI units: m/s²)
• ${\displaystyle D}$ is the pipe diameter (SI units: m)
• ${\displaystyle L}$ is the pipe length (SI units: m)
• ${\displaystyle V}$ 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:

${\displaystyle f_{\text{L}}=\frac{64}{\mathrm{Re}}\left[1+\frac{\mathrm{He}}{6\mathrm{Re}}-\frac{64}{3}\left(\frac{{\mathrm{He}}^4}{f^3{\mathrm{Re}}^7}\right)\right]}$

where:

• ${\displaystyle f_{\text{L}}}$ is the laminar flow Darcy friction factor (SI units: dimensionless)
• ${\displaystyle \mathrm{Re}}$ is the Reynolds number (SI units: dimensionless)
• ${\displaystyle \mathrm{He}}$ is the Hedstrom number (SI units: dimensionless)

The Reynolds number and the Hedstrom number are respectively defined as:

${\displaystyle \mathrm{Re}=\frac{\rho VD}{\mu },}$ and

${\displaystyle \mathrm{He}=\frac{\rho D^2{\tau }_o}{{\mu }^2}}$

where:

• ${\displaystyle \rho }$ is the mass density of fluid (SI units: kg/m³)
• ${\displaystyle \mu }$ is the dynamic viscosity of fluid (SI units: kg/m s)
• ${\displaystyle {\tau }_o}$ is the yield point (yield strength) of fluid (SI units: Pa)

Turbulent flow

Darby and Melson developed an empirical expression that was then refined, and is given by:

${\displaystyle f_{\text{T}}=4\times {10}^a{\mathrm{Re}}^{-0.193}}$

where:

• ${\displaystyle f_{\text{T}}}$ is the turbulent flow friction factor (SI units: dimensionless)
• ${\displaystyle a=-1.47\left[1+0.146e^{-2.9\times {10}^{-5}\mathrm{He}}\right]}$

Note: Darby and Melson's expression is for a Fanning friction factor, and needs to be multiplied by 4 to be used in the friction loss equations located elsewhere on this page.

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:

${\displaystyle f_L=\frac{64}{\mathrm{R}\mathrm{e}}+\frac{64}{\mathrm{R}\mathrm{e}}{\left(\frac{\mathrm{H}\mathrm{e}}{6.2218\mathrm{R}\mathrm{e}}\right)}^{0.958}}$

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:

${\displaystyle f_L=\frac{K_1+{\displaystyle \frac{4K_2}{{\left(K_1+\frac{K_1{K}_2}{K_1^4+3K_2}\right)}^3}}}{1+{\displaystyle \frac{3K_2}{{\left(K_1+\frac{K_1{K}_2}{K_1^4+3K_2}\right)}^4}}}}$

where

${\displaystyle K_1=\frac{16}{\mathrm{R}\mathrm{e}}+\frac{16\mathrm{H}\mathrm{e}}{6{\mathrm{R}\mathrm{e}}^2},}$

and

${\displaystyle K_2=-\frac{16{\mathrm{H}\mathrm{e}}^4}{3{\mathrm{R}\mathrm{e}}^8}.}$

Combined equation for friction factor for all flow regimes

Darby–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:

${\displaystyle f={\left[{f_{\text{L}}}^m+{f_{\text{T}}}^m\right]}^{\frac{1}{m}}}$

where:

${\displaystyle m=1.7+\frac{40000}{\mathrm{Re}}}$

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.