Non-Newtonian flow in annulus
Input(s)
r: Inner Radius (ft)
R: Outer Radius ( \(f t\) )
n: Power Law Constant 2 (dimensionless)
\(\kappa\): Dilatational Viscosity \((\mathrm{cP})\)
vo: Input Velocity (ft/s)
\(\rho\): Fluid Density \(\left(\mathrm{lb} / \mathrm{ft}^{3}\right)\)
Output(s)
\(v_{z}\): Output Velocity (ft/s)
w: Mass Flow Rate (lb/s)
Formula(s)
\[
\begin{gathered}
\mathrm{v}_{\mathrm{z}}=\mathrm{v}_{\mathrm{o}} *\left(\frac{\left(\left(\frac{\mathrm{r}}{\mathrm{R}}\right)^{1-\left(\frac{1}{\mathrm{n}}\right)}\right)-1}{\left(\kappa^{1-\left(\frac{1}{\mathrm{n}}\right)}\right)-1}\right) \\
\mathrm{W}=\left(\frac{2 * \pi * \mathrm{R}^{2} * \rho * \mathrm{v}_{\mathrm{o}}}{\left(\kappa^{1-\left(\frac{1}{\mathrm{n}}\right)}\right)-1}\right) *\left(\frac{1-\left(\kappa^{3-\left(\frac{1}{\mathrm{n}}\right)}\right)}{3-\left(\frac{1}{\mathrm{n}}\right)}-\left(\frac{1-\kappa^{2}}{2}\right)\right)
\end{gathered}
\]
Reference(s)
Bird, R.B., Stewart, W.E. and Lightfoot, E.N. (2002). Transport Phenomena (Second Ed.). John Wiley & Sons, Chapter: 8, Page: 258.