Here, the hydraulic equations for two known non-Newtonian fluids are presented. These are,
- the power law fluid and
- the Bingham plastic fluid.
These equations are to focus on the flow rate $Q$, the Reynolds number $N_{Re}$, and the friction factor $f_F$, for the case of a fluid flowing through a pipe of circular section.
A comment on the friction factors
There are, in fluid mechanics, two different friction factors:
- the Darcy friction factor
- the Fanning friction factor
This is confusing if you are not aware of this fact. The Darcy friction factor (say $f_F$) and the Fanning's (say $f_{FF}$) are related as follows,
You should pay attention when using Darcy's equation for head loss since it is originally and commonly expressed in terms of $f_F$. This means that if you are using $f_{F}$ the head loss can be found from:
In this post, the formulas for the newtonian fluid use the Darcy friction factor $f_F$ while those for power law and plastic Bingham fluids use the Fanning friction factor $f_{FF}$.
Newtonian fluid
The flow rate $Q$ is given by,
$Q=\dfrac{\pi\,R^4}{8\mu}\dfrac{P_0-P_L}{L}$ Eq. (01)
where $R$ is the pipe internal radius and subscripts $0$ and $L$ indicate the end and final of the pipe.
The Reynolds number $N_{Re}$ is given by,
$N_{Re}=\dfrac{D\,V}{\nu}$ Eq. (02)
The friction factor $f_F$,for laminar conditions $N_{Re}<2000$, is given by,
$f_{F-L}=\dfrac{64}{N_{Re}}$ Eq. (03)
for turbulent flow regime $N_{Re}>4000$ by,
$\dfrac{1}{\sqrt{f_{F-T}}}=-2\log_{[10]}\left( \dfrac{\epsilon}{3.72D}+\dfrac{2.51}{N_{Re}\sqrt{f_{F-T}}} \right)$ Eq. (04)
Power law fluid
The flow rate $Q$ is given by,
$Q=\dfrac{\pi\,R^3}{(1/n)+3}\left(\dfrac{\left(\mathbb{P}_0-\mathbb{P}_L\right)R}{2mL}\right)^{1/n}$ Eq. (05)
where $R$ is the pipe internal radius and subscripts $0$ and $L$ indicate the end and final of the pipe. The formula in Eq. (05) was taken from the familiar book Dynamics of Polymeric Liquids by Bird R. B. et al. Also, the velocity given as a function of the pipe radius is,
$v_z=\left( \dfrac{\tau_R}{m} \right)^{1/n}\dfrac{R}{(1/n)+1}\left[ 1-\left( \dfrac{r}{R} \right)^{(1/n)+1} \right]$ Eq. (05a)
where,
$\tau_{rz}=\dfrac{\left( \mathbb{P}_0-\mathbb{P}_L \right)r}{2L}$ Eq. (05b)
and
$\tau_{R}=\tau_{rz}\vert_{r=R}=\dfrac{\left( \mathbb{P}_0-\mathbb{P}_L \right)R}{2L}$ Eq. (05c).
Recall that $\tau_{rz}$ is the shear stress and $\tau_{r}$ is the stress evaluated at the pipe wall. You should be aware that the pressure difference $\mathbb{P}_0-\mathbb{P}_L$ in Eqs. (05) includes the effect of gravity. The Reynolds number $N_{Re}$ is given by,
$N_{Re}=\dfrac{(4n)^{n}\,D^n\,V^{2-n}\rho}{g_c\,m\,(3n+1)^n8^{n-1}}$ Eq. (06)
where $g_c=32.174\,lb_m\,\cdot \,ft/lb_f\, \cdot s^2$ is a units correction factor used in the British units system. The density of the fluid $\rho$ must be given in $lb_m/ft^3$. Equation (06) was taken from the article of Dodge and Metzner [AIChE J 1959 5 (2)].
The friction factor $f_{FF}$, for laminar conditions $N_{Re}<N_{Re-c}$, is given by,
$f_{FF-L}=\dfrac{16}{N_{Re}}$ Eq. (07)
and for turbulent flow $4000<N_{Re}<10^5$ regime by,
$f_{FF-T}=\dfrac{0.0682\,n^{-1/2}}{N_{Re}^{1/(1.87+2.39\,n)}}$ Eq. (08)
and for the transition region $N_{Re-c}<N_{Re}<4000$ by
$f_{FF-Tr}=1.79\times 10^{-4}\exp\left[ -5.24\,n \right]\,N_{Re}^{0.414+0.757\,n}$ Eq. (09)
Finally, the critical Reynolds number $N_{Re-c}$ is defined as,
$N_{Re-c}=2100+875(1-n)$ Eq. (10)
Equations (07-10) were taken from the paper of Darby et al. [Chem. Eng. 1992 99 (9)].
Plastic Bingham fluid
The flow rate $Q$ is given by,
$Q=\dfrac{\pi\,R^3\tau_w}{4\mu_\infty}\left[ 1 - \dfrac{4}{3}\left( \dfrac{\tau_0}{\tau_w}\right)+ \dfrac{1}{3}\left( \dfrac{\tau_0}{\tau_w} \right)^4 \right]$ Eq. (11)
where $R$ is the pipe internal radius, $\mu_\infty$ is called the limiting viscosity and $\tau_0$ is the yield stress. Also, $\tau_w$
$\tau_w=\dfrac{\left(\mathbb{P}_0-\mathbb{P}_L\right)R}{2L}$ Eq. (12)
In Eq. (12) pressures $\mathbb{P}_0$ and $\mathbb{P}_L$ include the hydrostatic contribution in an inclined pipe. Formulas in Eqs. (11-12) were taken from the familiar book Dynamics of Polymeric Liquids by Bird R. B. et al. Subscripts $0$ and $L$ indicate the end and final of the pipe.
The Reynolds $N_{Re}$ and Hedstrom $N_{He}$ numbers are given by,
$N_{Re}=\dfrac{D\,V\, \rho}{\mu_\infty}$ Eq. (13)
$N_{He}=\dfrac{D^2\,\rho\,\tau_0 }{\mu_\infty^2}$ Eq. (14)
In this case the friction factor $f_{FF}$ is given for all flow regimes as,
$f_{FF}=\left( f_{FF-L}^m+f_{FF-T}^m \right)^{1/m}$ Eq. (15)
where,
$m=1.7+\dfrac{40000}{N_{Re}}$ Eq. (16)
and the friction factors for laminar $f_{FF-L}$ and turbulent $f_{FF-T}$ regimes are:
$f_{FF-L}=\dfrac{16}{N_{Re}}\left[ 1+\dfrac{1}{6}\dfrac{N_{He}}{N_{Re}}-\dfrac{1}{3}\dfrac{N_{He}^4}{f_{FF}^3N_{Re}^7} \right]$ Eq. (17)
$f_{FF-T}=\dfrac{10^a}{N_{Re}^{0.193}}$ Eq. (18)
$a=-1.47\left[ 1+0.146\exp\left( -2.9\times 10^{-5}N_{He} \right) \right]$ Eq. (19)
Notice that for plastic Bingham fluids there is no laminar-transition-turbulent regions reported, so that in order to find $f_F$ you must solve numerically Eq. (15). However, you may read the manuscript of Swamee and Aggarwal [J. Pet. Sci. Eng. 2011 76] if you would like to know about an effort to give a critical $N_{Re}$ for these fluids. Equations (13-19) were taken from the paper of Darby et al. [Chem. Eng. 1992 99 (9)].
Watch this video if you do not want to read. Enjoy!
