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Analytical description of the unsteady settling of spherical particles in Stokes and Newton regimes

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Abstract

In most solid processes, the separation of particles is important and designed based on the terminal settling velocity as the characteristically distributed particle property. This settling velocity is determined by distributed particle properties like size, density, and shape. Although the settling of particles driven by an external force (gravitation or centrifugal field) is an unsteady process, the temporal changes of the particle settling velocity are relevant for the microscopic process design. These temporal changes correspond to spatial changes along the path-line. Generally, the particle state of motion is described by solving numerically the force balance equations in fluid dynamic simulations which are highly complex. In this work, algebraic solutions are presented instead, suitable for describing the temporal changes of the settling velocity and the displacement at settling for laminar and turbulent flows around the particle. These equations solely depend upon particle properties like size, density, and shape and are valid independently from the initial velocity. Thus, and at the difference to known publications, it is not assumed that the particles are initially at rest. Finally, the relaxation time at which terminal settling velocity is approximately reached can be clearly identified.

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Abbreviations

A:

Cross-sectional area \((\hbox {m}^{2})\)

\(\hbox {c}_{\mathrm{w}}\) :

Drag coefficient \((-)\)

d:

Diameter (m)

\(\hbox {D}_{\uprho }\) :

Density function \((-)\)

\(\hbox {F}_{\mathrm{A}}\) :

Buoyancy force (N)

\(\hbox {F}_{\mathrm{g}}\) :

Gravity force (N)

\(\hbox {F}_{\mathrm{j}}\) :

Inertia force of the fluid shell (N)

\(\hbox {F}_{\mathrm{T}}\) :

Inertia force of the particle (N)

\(\hbox {F}_{\mathrm{w}}\) :

Drag force (N)

g:

Gravity \((\hbox {m}/\hbox {s}^{2})\)

j:

Volumetric fraction of the fluid shell around the particle

Re:

Reynolds number

s:

Settling distance (m)

t:

Settling time (s)

T:

Absolute temperature (K)

u:

Fluid velocity (m/s)

v:

Particle velocity (m/s)

V:

Volume \((\hbox {m}^{3})\)

0:

Initial state

63, 76, 95, 96:

Characteristic values

f:

Fluid phase

p:

Solid phase, particle phase

r:

Relative

R:

Relaxation

s:

Steady state

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Acknowledgments

This work is part of the priority program SPP 1679 “Dynamische Simulation vernetzter Feststoffprozesse” and is supported financially by the Deutsche Forschungsgemeinschaft (DFG). Interesting discussions with Amir Eshghinejadfard are gratefully acknowledged.

Author information

Authors and Affiliations

  1. Chair of Mechanical Process Engineering, Otto von Guericke University Magdeburg, Universitätsplatz 2, 39106, Magdeburg, Germany

    Hannes Mann, Peter Mueller & Juergen Tomas

  2. NaWiTec, Otto von Guericke University Magdeburg, Universitätsplatz 2, 39106, Magdeburg, Germany

    Thomas Hagemeier

  3. Chair of Fluid Dynamics and Technical Flow, Otto von Guericke University Magdeburg, Universitätsplatz 2, 39106, Magdeburg, Germany

    Christoph Roloff & Dominique Thévenin

Authors
  1. Hannes Mann
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  2. Peter Mueller
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  3. Thomas Hagemeier
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  4. Christoph Roloff
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  5. Dominique Thévenin
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  6. Juergen Tomas
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Corresponding author

Correspondence to Hannes Mann.

Additional information

Translated and extended version of the original research article published in Gloeckner et al. Chem Ing Tech 87(5):644–655 (2015).

The research conducted here is a research performed at the University of Magdeburg using University-owned computers and commercial software.

Appendix: Derivation of particle settling for turbulent flow conditions in Newton-regime

Appendix: Derivation of particle settling for turbulent flow conditions in Newton-regime

1.1 Velocity-time law

From Eq. (27) the velocity-time law is derived. Rearranging gives

$$\begin{aligned} \int \nolimits _{\mathrm{v} = {\mathrm{v}}_0}^\mathrm{v} \frac{\hbox {dv}}{\hbox {v}_\mathrm{s}^2 -\hbox {v}^{2}}=\int \nolimits _{\mathrm{t} = {\mathrm{t}}_0}^\mathrm{t} \frac{1}{\hbox {t}_\mathrm{R} \hbox {v}_\mathrm{s}}\hbox {dt} \end{aligned}$$
(101)

as the equation which has to be solved using the indefinite integral [20]

$$\begin{aligned} \int \frac{\hbox {dx}}{\hbox {Q}}=\int \frac{\hbox {dx}}{\hbox {a}^{2}-\hbox {x}^{2}}=\left\{ {{\begin{array}{l} {\left[ {\frac{1}{2\hbox {a}}\hbox {ln}\frac{\hbox {a}+\hbox {x}}{\hbox {a}-\hbox {x}}} \right] _{\left| \mathrm{x} \right| <a}} \\ {\left[ {\frac{1}{2\hbox {a}}\hbox {ln}\frac{\hbox {x}+\hbox {a}}{\hbox {x}-\hbox {a}}} \right] _{\left| \mathrm{x} \right| >a}} \\ \end{array}}} \right. \end{aligned}$$
(102)

which is valid for \(\hbox {v}_{\mathrm{s}} \ne 0\). After rigorous examination, it can be shown that

$$\begin{aligned} \int \frac{\hbox {dx}}{\hbox {Q}}=\left[ {\frac{1}{2\hbox {a}}\hbox {ln}\frac{\hbox {a}+\hbox {x}}{\hbox {a}-\hbox {x}}} \right] _{\mathrm{x} = \mathrm{x}_0}^\mathrm{x} =\frac{1}{2\hbox {a}}\hbox {ln}\frac{\frac{\hbox {a}+\hbox {x}}{\hbox {a}-\hbox {x}}}{\frac{\hbox {a}+\hbox {x}_0 }{\hbox {a}-\hbox {x}_0}} \end{aligned}$$
(103)

and

$$\begin{aligned} \int \frac{\hbox {dx}}{\hbox {Q}}= & {} \left[ {\frac{1}{2\hbox {a}}\hbox {ln}\frac{\hbox {x}+\hbox {a}}{\hbox {x}-\hbox {a}}} \right] _{\mathrm{x}= \mathrm{x}_0}^\mathrm{x} =\frac{1}{2\hbox {a}}\hbox {ln}\frac{\frac{\hbox {x}+\hbox {a}}{\hbox {x}-\hbox {a}}}{\frac{\hbox {x}_0 +\hbox {a}}{\hbox {x}_0 -\hbox {a}}} \end{aligned}$$
(104)
$$\begin{aligned} \int \frac{\hbox {dx}}{\hbox {Q}}= & {} \frac{1}{2\hbox {a}}\hbox {ln}\frac{-\left( {\frac{\hbox {a}+\hbox {x}}{\hbox {a}-\hbox {x}}} \right) }{-\left( {\frac{\hbox {a}+\hbox {x}_0}{\hbox {a}-\hbox {x}_0}} \right) }=\frac{1}{2\hbox {a}}\hbox {ln}\frac{\frac{\hbox {a}+\hbox {x}}{\hbox {a}-\hbox {x}}}{\frac{\hbox {a}+\hbox {x}_0 }{\hbox {a}-\hbox {x}_0}} \end{aligned}$$
(105)

are equal. Therefore, Eq. (113) describes the particles’ state of motion.

$$\begin{aligned}&\frac{1}{2\hbox {v}_\mathrm{s}}\left[ {\hbox {ln}\frac{\hbox {v}_\mathrm{s} +\hbox {v}}{\hbox {v}_\mathrm{s} -\hbox {v}}} \right] _{\mathrm{v} = \mathrm{v}_0}^\mathrm{v} =\frac{1}{\hbox {t}_\mathrm{R} \hbox {v}_\mathrm{s} }\left[ \hbox {t} \right] _{\mathrm{t} = \mathrm{t}_0}^\mathrm{t} \end{aligned}$$
(106)
$$\begin{aligned}&\hbox {ln}\frac{\frac{\hbox {v}_\mathrm{s} +\hbox {v}}{\hbox {v}_\mathrm{s} -\hbox {v}}}{\frac{\hbox {v}_\mathrm{s} +\hbox {v}_0}{\hbox {v}_\mathrm{s} -\hbox {v}_0}}=\frac{2}{\hbox {t}_\mathrm{R}}\left( {\hbox {t}-\hbox {t}_0} \right) \end{aligned}$$
(107)
$$\begin{aligned}&\frac{\frac{\hbox {v}_\mathrm{s} +\hbox {v}}{\hbox {v}_\mathrm{s} -\hbox {v}}}{\frac{\hbox {v}_\mathrm{s} +\hbox {v}_0}{\hbox {v}_\mathrm{s} -\hbox {v}_0}}=\exp \left( {\frac{2}{\hbox {t}_\mathrm{R}}\left( {\hbox {t}-\hbox {t}_0} \right) } \right) \end{aligned}$$
(108)
$$\begin{aligned}&\hbox {v}_\mathrm{s} +\hbox {v}=\left( {\hbox {v}_\mathrm{s} -\hbox {v}} \right) \left( {\frac{\hbox {v}_\mathrm{s} +\hbox {v}_0}{\hbox {v}_\mathrm{s} -\hbox {v}_0}\exp \left( {\frac{2}{\hbox {t}_\mathrm{R}}\left( {\hbox {t}-\hbox {t}_0} \right) } \right) } \right) \end{aligned}$$
(109)
$$\begin{aligned}&\hbox {v}\left( {\frac{\hbox {v}_\mathrm{s} +\hbox {v}_0}{\hbox {v}_\mathrm{s} -\hbox {v}_0}\exp \left( {\frac{2}{\hbox {t}_\mathrm{R}}\left( {\hbox {t}-\hbox {t}_0} \right) } \right) +1} \right) \nonumber \\&\quad =\hbox {v}_\mathrm{s} \left( {\frac{\hbox {v}_\mathrm{s} +\hbox {v}_0}{\hbox {v}_\mathrm{s} -\hbox {v}_0}\exp \left( {\frac{2}{\hbox {t}_\mathrm{R}}\left( {\hbox {t}-\hbox {t}_0} \right) } \right) -1} \right) \end{aligned}$$
(110)
$$\begin{aligned}&\hbox {v}=\hbox {v}_\mathrm{s} \frac{\left( {\frac{\hbox {v}_\mathrm{s} +\hbox {v}_0}{\hbox {v}_\mathrm{s} -\hbox {v}_0}\exp \left( {\frac{2}{\hbox {t}_\mathrm{R}}\left( {\hbox {t}-\hbox {t}_0} \right) } \right) -1} \right) }{\left( {\frac{\hbox {v}_\mathrm{s} +\hbox {v}_0 }{\hbox {v}_\mathrm{s} -\hbox {v}_0}\exp \left( {\frac{2}{\hbox {t}_\mathrm{R}}\left( {\hbox {t}-\hbox {t}_0} \right) } \right) +1} \right) } \end{aligned}$$
(111)
$$\begin{aligned}&\hbox {v}=\hbox {v}_\mathrm{s} \frac{\left( {\frac{\left( {\hbox {v}_\mathrm{s} +\hbox {v}_{0}} \right) \exp \left( {\frac{2}{\hbox {t}_\mathrm{R}}\left( {\hbox {t}-\hbox {t}_0} \right) } \right) -\left( {\hbox {v}_\mathrm{s} -\hbox {v}_0} \right) }{\hbox {v}_\mathrm{s} -\hbox {v}_0}} \right) }{\left( {\frac{\left( {\hbox {v}_\mathrm{s} +\hbox {v}_{0}} \right) \exp \left( {\frac{2}{\hbox {t}_\mathrm{R}}\left( {\hbox {t}-\hbox {t}_0} \right) } \right) +\left( {\hbox {v}_\mathrm{s} -\hbox {v}_0} \right) }{\hbox {v}_\mathrm{s} -\hbox {v}_0}} \right) } \end{aligned}$$
(112)
$$\begin{aligned}&\hbox {v}=\hbox {v}_\mathrm{s} \frac{\left( {\hbox {v}_\mathrm{s} +\hbox {v}_{0}} \right) \exp \left( {\frac{2}{\hbox {t}_\mathrm{R} }\left( {\hbox {t}-\hbox {t}_0} \right) } \right) -\hbox {v}_\mathrm{s} +\hbox {v}_0}{\left( {\hbox {v}_\mathrm{s} +\hbox {v}_{0}} \right) \exp \left( {\frac{2}{\hbox {t}_\mathrm{R}}\left( {\hbox {t}-\hbox {t}_0} \right) } \right) +\hbox {v}_\mathrm{s} -\hbox {v}_0} \end{aligned}$$
(113)

In case of simple initial conditions \(\hbox {t}_{0} = 0\) and \(\hbox {v}_{0} = 0\) (particle at rest), Eq. (116) is the simplified expression that has been shown by Brennen [13].

$$\begin{aligned} \hbox {v}= & {} \hbox {v}_\mathrm{s} \frac{\left( {\hbox {v}_\mathrm{s} +0} \right) \exp \left( {\frac{2}{\hbox {t}_\mathrm{R}}\left( {\hbox {t}-0} \right) } \right) -\hbox {v}_\mathrm{s} +0}{\left( {\hbox {v}_\mathrm{s} +0} \right) \exp \left( {\frac{2}{\hbox {t}_\mathrm{R}}\left( {\hbox {t}-0} \right) } \right) +\hbox {v}_\mathrm{s} -0} \end{aligned}$$
(114)
$$\begin{aligned} \hbox {v}= & {} \hbox {v}_\mathrm{s} \frac{\exp \left( {\frac{2\hbox {t}}{\hbox {t}_\mathrm{R}}} \right) -1}{\exp \left( {\frac{2\hbox {t}}{\hbox {t}_\mathrm{R}}} \right) +1} \end{aligned}$$
(115)
$$\begin{aligned} \hbox {v}= & {} \hbox {v}_\mathrm{s} \tanh \left( {\frac{\hbox {t}}{\hbox {t}_\mathrm{R}}} \right) \end{aligned}$$
(116)

1.2 Distance-time law

Based on Eqs. (33) or (116), the distance-time law in the Newton regime is derived. Therefore,

$$\begin{aligned} \frac{\hbox {ds}}{\hbox {dt}}= & {} \hbox {v}_\mathrm{s} \frac{\left( {\hbox {v}_\mathrm{s} +\hbox {v}_{0}} \right) \exp \left( {\frac{2}{\hbox {t}_\mathrm{R}}\left( {\hbox {t}-\hbox {t}_0} \right) } \right) -\hbox {v}_\mathrm{s} +\hbox {v}_0 +\left( {\hbox {v}_\mathrm{s} -\hbox {v}_0} \right) -(\hbox {v}_\mathrm{s} -\hbox {v}_0 )}{\left( {\hbox {v}_\mathrm{s} +\hbox {v}_{0}} \right) \exp \left( {\frac{2}{\hbox {t}_\mathrm{R}}\left( {\hbox {t}-\hbox {t}_0} \right) } \right) +\hbox {v}_\mathrm{s} -\hbox {v}_0} \end{aligned}$$
(117)
$$\begin{aligned} \int \nolimits _{\mathrm{s} = {\mathrm{s}}_0}^\mathrm{s} \hbox {ds}= & {} \hbox {v}_\mathrm{s} \int \nolimits _{\mathrm{t} = \mathrm{t}_0 }^\mathrm{t} \left( \frac{\left( {\hbox {v}_\mathrm{s} +\hbox {v}_{0}} \right) \exp \left( \frac{2}{\hbox {t}_\mathrm{R}}\left( {\hbox {t}-\hbox {t}_0} \right) \right) +\hbox {v}_\mathrm{s} -\hbox {v}_0 }{\left( {\hbox {v}_\mathrm{s} +\hbox {v}_{0}} \right) \exp \left( {\frac{2}{\hbox {t}_\mathrm{R}}\left( {\hbox {t}-\hbox {t}_0} \right) } \right) +\hbox {v}_\mathrm{s} -\hbox {v}_0}\right. \nonumber \\&\left. -\,2\frac{\hbox {v}_\mathrm{s} -\hbox {v}_0}{\left( {\hbox {v}_\mathrm{s} +\hbox {v}_{0}} \right) \exp \left( {\frac{2}{\hbox {t}_\mathrm{R}}\left( {\hbox {t}-\hbox {t}_0} \right) } \right) +\hbox {v}_\mathrm{s} -\hbox {v}_0} \right) \hbox {dt} \end{aligned}$$
(118)
$$\begin{aligned} \int \nolimits _{\mathrm{s} = {\mathrm{s}}_0}^\mathrm{s} \hbox {ds}=\hbox {v}_\mathrm{s} \left( {\int \nolimits _{\mathrm{t} = {\mathrm{t}}_0}^\mathrm{t} \hbox {dt}-2\left( {\hbox {v}_\mathrm{s} -\hbox {v}_0} \right) \int \nolimits _{\mathrm{t} = {\mathrm{t}}_0}^\mathrm{t} \left( {\frac{1}{\frac{\hbox {v}_\mathrm{s} +\hbox {v}_{0}}{\exp \left( {\frac{2}{\hbox {t}_\mathrm{R}}\hbox {t}_0} \right) }\exp \left( {\frac{2}{\hbox {t}_\mathrm{R}}\hbox {t}} \right) +\hbox {v}_\mathrm{s} -\hbox {v}_0}} \right) \hbox {dt}} \right) \end{aligned}$$
(119)

needs to be solved. Usage of the indefinite integral [20]

$$\begin{aligned} 2\hbox {b}\int \frac{1}{\hbox {b}+\hbox {c}\exp \left( {\hbox {ax}} \right) }\hbox {dx}= & {} 2\hbox {b}\left( {\frac{\hbox {x}}{\hbox {b}}-\frac{1}{\hbox {ab}}\ln \left( {\hbox {b}+\hbox {c}\exp \left( {\hbox {ax}} \right) } \right) } \right) \nonumber \\= & {} 2\hbox {x}-\frac{2}{\hbox {a}}\ln \left( {\hbox {b}+\hbox {c}\exp \left( {\hbox {ax}} \right) } \right) \end{aligned}$$
(120)

and offers a possibility to solve the right term of Eq. (119). Hence, Eq. (124) is the general distance-time law.

$$\begin{aligned}&\left[ \hbox {s} \right] _{\mathrm{s} = \mathrm{s}_0}^\mathrm{s} =\hbox {v}_\mathrm{s} \left( {\left[ \hbox {t} \right] _{\mathrm{t}= \mathrm{t}_0}^\mathrm{t} -\left[ {2\hbox {t}-\frac{2}{\left( {\frac{2}{\hbox {t}_\mathrm{R}}} \right) }\ln \left( {\frac{\hbox {v}_\mathrm{s} +\hbox {v}_{0}}{\exp \left( {\frac{2}{\hbox {t}_\mathrm{R}}\hbox {t}_0} \right) }\exp \left( {\frac{2}{\hbox {t}_\mathrm{R}}\hbox {t}} \right) +\hbox {v}_\mathrm{s} -\hbox {v}_0} \right) } \right] _{\mathrm{t} = \mathrm{t}_0}^\mathrm{t}} \right) \end{aligned}$$
(121)
$$\begin{aligned}&\hbox {s}-\hbox {s}_0 =\hbox {v}_\mathrm{s} \left( {\hbox {t}-\hbox {t}_0 -2\left( {\hbox {t}-\hbox {t}_0} \right) +\hbox {t}_\mathrm{R} \hbox {ln}\frac{\left( {\hbox {v}_\mathrm{s} +\hbox {v}_{0}} \right) \exp \left( {\frac{2}{\hbox {t}_\mathrm{R}}\left( {\hbox {t}-\hbox {t}_0} \right) } \right) +\hbox {v}_\mathrm{s} -\hbox {v}_0}{\left( {\hbox {v}_\mathrm{s} +\hbox {v}_{0}} \right) \exp \left( {\frac{2}{\hbox {t}_\mathrm{R}}\left( {\hbox {t}_0 -\hbox {t}_0} \right) } \right) +\hbox {v}_\mathrm{s} -\hbox {v}_0}} \right) \end{aligned}$$
(122)
$$\begin{aligned}&\hbox {s}-\hbox {s}_0 =\hbox {v}_\mathrm{s} \left( {-\left( {\hbox {t}-\hbox {t}_0} \right) +\hbox {t}_\mathrm{R} \hbox {ln}\frac{\left( {\hbox {v}_\mathrm{s} +\hbox {v}_{0}} \right) \exp \left( {\frac{2}{\hbox {t}_\mathrm{R}}\left( {\hbox {t}-\hbox {t}_0} \right) } \right) +\hbox {v}_\mathrm{s} -\hbox {v}_0}{\hbox {v}_\mathrm{s} +\hbox {v}_{0} +\hbox {v}_\mathrm{s} -\hbox {v}_0}} \right) \end{aligned}$$
(123)
$$\begin{aligned}&\hbox {s}=\hbox {s}_0 +\hbox {v}_\mathrm{s} \left( {-\left( {\hbox {t}-\hbox {t}_0} \right) +\hbox {t}_\mathrm{R} \hbox {ln}\frac{\left( {\hbox {v}_\mathrm{s} +\hbox {v}_{0}} \right) \exp \left( {\frac{2}{\hbox {t}_\mathrm{R}}\left( {\hbox {t}-\hbox {t}_0} \right) } \right) +\hbox {v}_\mathrm{s} -\hbox {v}_0}{2\hbox {v}_\mathrm{s}}} \right) \end{aligned}$$
(124)

For \(\hbox {s}_{0} = 0, \hbox {t}_{0} = 0\) and \(\hbox {v}_{0} = 0\), Eq. (125) results.

$$\begin{aligned} \hbox {s}=\hbox {v}_\mathrm{s} \left( {\hbox {t}_\mathrm{R} \hbox {ln}\frac{\hbox {exp}\left( {\frac{2}{\hbox {t}_\mathrm{R} }\hbox {t}} \right) +1}{2}-\hbox {t}} \right) \end{aligned}$$
(125)

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Mann, H., Mueller, P., Hagemeier, T. et al. Analytical description of the unsteady settling of spherical particles in Stokes and Newton regimes. Granular Matter 17, 629–644 (2015). https://doi.org/10.1007/s10035-015-0584-y

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