KONA Powder and Particle Journal No. 41 (2024) 58–77/https://doi.org/10.14356/kona.2024010

Review Paper

About Modeling and Optimization of Solid Bowl Centrifuges †Marco Gleiss * and Hermann Nirschl
Institute of Mechanical Process Engineering and Mechanics, Karlsruhe Institute of Technology (KIT), Germany

Many processes involve solid bowl centrifuges as a solid–liquid separation step,
typically used for clarification, thickening, classification, degritting, mechanicaldewatering, and screening. In order to operate solid bowl centrifuges safely, withminimum resource consumption and reduced setup times, modeling and optimization
are necessary steps. This is a challenge due to the complex process behavior, which can
be overcome by developing advanced physical models and process analysis. This review
provides an overview of solid bowl centrifuge applications, their modeling, andaddresses future optimization potentials through digital tools. The impact of dispersed
phase properties such as particle size, shape, surface roughness, structure, composition,and continuous liquid phase is the reason for the lack of generally applicable models.
Laboratory-scale batch sedimentation centrifuges are used to predict material behavior
and develop material functions describing separation-related properties such as
sedimentation, sediment build-up and sediment transport. The combination of material
functions and modeling allows accurate simulation of solid bowl centrifuges fromlaboratory to industrial scale. Since models usually do not cover all influencing
variables, there are often deviations between predictions and the real process behavior.
Gray-box modeling and on-line or in-situ process analytics are tools to improve centrifuge operation.
Keywords: solid bowl centrifuges, material characterization, multiphase flow, process modeling, gray-box modeling, process analytics

1. Introduction

The separation of fine particles from liquids plays animportant role in daily products such as chemicals, pharmaceuticals, healthcare products, ingredients, beverages,
proteins, food or consumer goods, see Table 1 (Anlauf,
2007). Solid–liquid separation is an essential step in theprocessing and handling of products that are often obtainedby precipitation, crystallization, or comminution.
The two main principles of solid–liquid separation are
filtration and centrifugation. Filtration requires a filter medium that retains particles, while centrifugation relies on arotating bowl.
Solid bowl centrifuges are sedimentation centrifuges anduse the physical principle of sedimentation, based on adensity difference between the solid and liquid phases. In
the industry, there are several types of centrifuges that allow the separation of colloids or particles with several
millimeters in size. Typical machines are solid bowl peeler
centrifuges, tubular bowl centrifuges, disk stack centrifuges, or decanter centrifuges.
Fig. 1 shows the categorization of different centrifuge
types in terms of cut size, which is the particle size that has
an equal chance of being in the underflow or overflow. The
†

*

58

Received 14 May 2022; Accepted 8 September 2022
J-STAGE Advance published online 30 June 2023
Corresponding author: Marco Gleiss;
Add: Straße am Forum 8, 76131 Karlsruhe, Germany
E-mail: marco.gleiss@kit.edu
TEL: +49-721-608-42426 FAX: +49-721-608-40405

influencing variables are numerous, and the cut size depends not only on the process conditions and the centrifuge
geometry, but also on the properties of the dispersed andcontinuous phases. Furthermore, the limits can be arbitrarily shifted by suspension pre-treatment such as coagulation or flocculation.
For a spherical particle, in a creeping flow and infinitely
diluted Newtonian fluid, the Stokes’ settling velocity isvalid (Stokes, 1850): ρs  ρl C g x 2
vs 

(1)

18 μl

Here, ρs is the density of solid phase, ρl is the density ofliquid, g is gravity, x is the particle diameter, and μl is the(1)
dynamic
viscosity. The Stokes’ settling velocity can be derived from an equilibrium of forces consisting of buoyancy,
centrifugal, and drag force, neglecting the influence of inertia and assuming small particle Reynolds number Re < 0.1.
For semi-batch or continuous solid bowl centrifuges,
modeling requires the linkage between process conditions,geometric dimensions and material properties (Ambler,1952). One approach to scale-up of solid bowl centrifuges
is Sigma theory, which requires a significant amount ofpilot testing, making the design process time-consuming
and costly.
Increasing computational power and improvements insensor technology have led to the development of morecomprehensive and detailed models and measurement
techniques in recent years (Gleiss et al., 2020; Hammerich

Copyright © 2024 The Authors. Published by Hosokawa Powder Technology Foundation. This is an open
access article under the CC BY license (https://creativecommons.org/licenses/by/4.0/).

Marco Gleiss et al. / KONA Powder and Particle Journal No. 41 (2024) 58–77

Review Paper

Fig. 1 Classification of different centrifuge types depending on the operation mode and applicable particle size range. Adapted from Ref. (Spelter etal., 2010) with permission from Elsevier, Copyright 2010.
Table 1 Overview of different applications of solid bowl centrifuges.

Centrifuge type

Field of application

Tubular bowl centrifuge

Biotechnology a, b
Nanotechnology c, d, eBattery materials f, g

Disc stack centrifuge

Protein separation h
Bio fuels i, j
Pharmaceuticals k, l
Oil industry m

Decanter centrifuge

Food industry n, o
Biotechnology p
Wastewater treatment q, rProtein separation s, t
Minerals processing u, va(Spelter et al., 2010), b(Rathore et al., 2015), c(Kohsakowski et al., 2019),d(Flegler et al., 2016), e(Flegler et al., 2018), f(Sinn et al., 2020), g(Wolf etal., 2021), h(Byrne et al., 2002), i(Najjar and Abu-Shamleh, 2020),
j(Arcigni et al., 2019), k(Kempken et al., 1995), l(Jungbauer, 2013),
m(Khatib et al., 1995), n(Caponio et al., 2014), o(Schubert et al., 2018),
p(Lopes et al., 2013), q(Wakeman, 2007), r(Ginisty et al., 2021), s(Haller etal., 2021), t(Haller and Kulozik, 2019), u(W.F. Leung, 2004), v(Merkl etal., 2012)

et al., 2018). The aim of this review is to provide an overview of the current developments in the field of modelingand optimization of solid bowl centrifuges.
First, the process behavior of solid bowl centrifuges is
explained in more detail. This is followed by a descriptionof the separation properties and their characterization

based on laboratory centrifuges. Finally, different modelingstrategies are discussed.
In this context, process modeling enables the simulationof the process behavior and provides a better understandingof the separation-related behavior of solid bowl centrifuges. Furthermore, this review highlights the capabilities
of dynamic and gray-box modeling. Gray-box modelingcombines physical and data-driven modeling. Direct coupling here means the integration of process analytics tomeasure particle properties on-line or in-situ, since dynamic modeling usually cannot capture all physical effects
with sufficient accuracy.

2. Process behavior of solid bowl centrifuges

The process behavior of solid bowl centrifuges varies
considerably depending on the type of centrifuge and affects the operation. The reasons for this are manifold. The
most important influencing variables can be divided into
three groups: centrifuge geometry, material properties andprocess conditions.

2.1 Batch and semi-batch centrifuges

Analytical and ultracentrifuges have a high degree of
flexibility due to batch operation. The process time can be
adjusted according to the sedimentation behavior, but islimited to small volumes and laboratory scale.
Compared to batch sedimentation centrifuges, semibatch centrifuges have an overflow weir that forms a liquid
pond. This allows continuous clarification of the liquid
phase. Centrifugal acceleration is the driving force forparticle separation. As a result, the accumulated solids forma liquid-saturated sediment on the rotor wall. The sediment59

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build-up influences the flow behavior by shortening theresidence time.
For example, tubular bowl centrifuges are semi-batch
operating machines and have a slim rotor. Conventional
bearing systems provide high relative centrifugal numbersof C = 70,000 with special magnetic bearing technology of
C = 120,000 (Konrath et al., 2016). The relative centrifugal
number is an important value indicating the ratio of centrifugal acceleration to gravity. Due to the high relative centrifugal force, tubular bowl centrifuges are typically used
for separation and classification of colloidal particles, bacteria, or human blood. Disk stack centrifuges operate either
in semi-batch or continuous mode. Manual removal of
sediment from the rotor is required for semi-batch operation, which is commonly used for dilute suspensions.

2.2 Continuous centrifuges

Decanter and disk stack centrifuges are continuously
operating machines that have a discharge mechanism forboth the clarified liquid (overflow weir and peeling disk)
and the sediment.
Disk stack centrifuges consist of a conical rotor andconically inclined disks, forming a number of small flow
channel which significantly shortens the settling distance.
As a result, biological particles such as microorganisms,
yeast, mammalian cells or algae can be separated. It is
crucial that these substances have only a small density difference to the liquid phase. This fact makes it necessary tohave a high centrifugal force. Discharge can be either intermittent, pressure controlled or through nozzles. The choiceof discharge mechanism depends on the material properties
and operating conditions. Typically, discharge of materials
in disk stack centrifuges requires a flowable sediment.
The main components of a decanter centrifuge are a feed
tube, a cylindrical conical bowl, and a screw conveyor. The
screw conveyor transports the collected solids to the sediment discharge located in the conical section of the machine. The material behavior of the sediment has asignificant influence on the performance of the centrifuge.
Incompressible sediments exhibit a constant pressure gradient resulting in a constant sediment porosity. The centrifugal force leads to undersaturation of the sediment after it
leaves the liquid pond.
In contrast to the aforementioned behavior, interparticle
forces, which are dominant for finely dispersed particles
influence the sediment formation process in a different way.
Here, the sediment is compressible and high capillary pressures occur within the sediment resulting in saturation of
S = 1. The compressive forces in the sediment lead to anonlinear porosity gradient. Insufficiently compacted compressible sediments tend to flow easily and interfere withsediment transport.
To overcome this issue, a plain disk is welded at theconveyor hub between the feed zone and the conical part

60

Review Paper(Records and Sutherland, 2001). The cake baffle disk has asmaller diameter than the centrifuge bowl and creates a gap
which serves for the pre-compaction of the material. This
ensures a highly compacted non-flowable sediment during
the operation.
Another important function of the cake baffle is to improve separation by preventing turbulence and resuspension of previously separated particles. Furthermore, it ispossible to increase the pond level, which has a positiveeffect on the separation efficiency, especially for fine particles with low settling velocity.

2.3 Flow conditions

The flow conditions in solid bowl centrifuges are crucial
for achieving the desired separation efficiency. In semibatch and continuous solid bowl centrifuges, the liquid
pond moves almost like a rigid-body. In practice, however,
the tangential velocity is smaller than rigid-body motion
due to friction losses or insufficient pre-acceleration of thefeed. One way of describing the real tangential velocity
analytically is a power law that relates the theoretical andactual values as a function of radial position (Gösele, 1968;
Reuter, 1967). When deviations from the rigid-body motion are present, internals such as the screw conveyor,
disks, internals or deflectors improve the pre-acceleration(Romani Fernández et al., 2010).
The axial velocity in solid bowl centrifuges occurs due
to mass conservation between the feed and centrate. In theliterature, two limiting cases are distinguished: a plug flow
and a boundary layer flow. A uniform flow profile acrossthe entire liquid pond characterizes the plug flow (Ambler,
1959; Stahl and Langeloh, 1983). In comparison, theboundary layer flow consists of two different zones: afast-flowing layer near the gas-liquid interface and a stagnant layer below (Leung, 1998). For a particle entering thestagnant layer, the influence of axial velocity is negligible.
Both flow profiles do not accurately represent the real
flow behavior. This has been demonstrated by recent experimental and numerical studies based on Laser Doppler
Anemometry (LDA) and Computational Fluid Dynamics(CFD) (Spelter et al., 2011). The advantage of both methods is the ability to study the flow behavior in a solid bowl
centrifuge in space and time (Romani Fernández et al.,2009).
Fig. 2 (I) illustrates the simulation of a mixture of water(blue) and air (red), its distribution in a vertical solid bowl
centrifuge type STA at a constant flow rate of 150 L/min
and a rotational speed of 2550 rpm. The method investigated here was the Volume-of-Fluid (VOF) method which
considers the physical behavior of immiscible liquids
where surface tension acts on both phases and forms a clear
phase boundary between gas and liquid. Coriolis andcentrifugal forces were included in the conservation of
momentum in the form of volumetric forces (Hirt et al.,

Marco Gleiss et al. / KONA Powder and Particle Journal No. 41 (2024) 58–77

Review Paper

Fig. 2 Left: (I) Contours of volume fraction for air (red) and water (blue) (II) Contours of tangential velocity at 2550 rpm and 140 L/min inlet flow.
(III) Axial velocity of water versus radius for different angular positions (0°, 30°, 45°, 60°) at the height of z = 0.3 m. Operation conditions are
150 L/min and 2550 rpm. Adapted with permission from Ref. (Romani Fernández et al., 2010) under the terms of the CC-BY 4.0 license. Copyright(2010) the Authors, published by Hindawi.

1981). The simulation indicates a sharp gas-liquid interface
and the formation of a liquid pond. Furthermore, the liquid
splits into smaller droplets due the pre-acceleration(Romani Fernández et al., 2010).
A closer look at Fig. 2 (II) depicts a contour plot of thetangential velocity. The increase of tangential velocity occurs due to the previously mentioned rigid-body motion of
the rotor.
The resulting axial flow profile for the investigated vertical solid bowl centrifuge at different angles is shown in
Fig. 2 (III). A detailed analysis of the axial velocity suggests that neither plug flow nor boundary layer flow ispresent. This is due to the absence of slip on the inner wallof the rotor and reduced friction at the gas-liquid interface.
Decanter and disk stack centrifuges have a different flow
behavior compared to vertical solid bowl centrifuges. The
screw conveyor and the disk stack counteract a pure axial
flow. Here, a complex flow behavior is present, which not
only depends on the centrifuge geometry, but also on theflow rate and centrifugal acceleration. In the case of a decanter centrifuge, the screw conveyor and the overflow
weir form a liquid pond. The main flow occurs along theformed helical channel. The mentioned flow direction wasinvestigated by color tracer experiments in a transparent
decanter centrifuge (Faust, 1985). The tracer observation
indicates insufficiently pre-accelerated liquid in the feed
zone, where the liquid is deflected towards the conical part,reverses and flows in the direction of the overflow weir.
In disk stack centrifuges, friction between the disks
dominates which results in hydrodynamic instabilities(Janoske et al., 1999). The flow through the disks is usuallybased on a superposition of tangential, radial, and axial
velocities. This causes the particle to settle on a threedimensional trajectory along the formed disk channel toward the centrate (König et al., 2021). Disk spacing withaxial stands between two disks minimizes the effect of
centripetal force on the particle movement and has a posi-tive effect on the separation efficiency.
Predicting local and temporal changes in flow behavior
is very difficult, and so far, there is no CFD model that cansimulate the flow conditions of an entire disk stack centrifuge. The reasons for this are the operating conditions, thedifficult centrifuge geometry, the high pressure and velocity gradients, and the momentum exchange between thesolid and liquid phases.

2.4 Residence time behavior

Another promising technique to evaluate the flow behavior of solid bowl centrifuges, is the indirect measurement of
the residence time distribution. This method allows to investigate the residence time of fluids or particles in any
kind of semi-batch or continuous apparatus. Fig. 3 illustrates three methods for characterizing the residence time
behavior of a machine. The tracer experiment (Fig. 3 (I)) is
the simplest method with small experimental effort and low
cost for the measurement equipment.
Saturated sodium chloride solution is a tracer that increases the conductivity of the continuous liquid phase after injection, and a sensor at the centrate tracks the change
in conductivity until no change occurs. To avoid phaseseparation during conductivity measurement, the operator
must ensure that the density of the sodium chloride solution
is close to the density of the continuous phase.
The method has been successfully applied to verticalsolid bowl, tubular bowl and decanter centrifuges. Residence time measurements for vertical and tubular bowl
centrifuges show a broad distribution. This distributiondiffers significantly from the behavior calculated by plug orboundary layer flow (Konrath et al., 2015; Romani
Fernández, 2012). In the case of a decanter centrifuge, thevolume flow and the solids content have a significant influence on the residence time distribution (Frost, 2000).
Another method of indirectly measuring flow behavior is
to experimentally characterize system behavior (Fig. 3 (II))

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Review Paper

Table 2 Comparison of hindered settling functions to describe the settling behavior of a concentrated slurry with uniform settling velocity.

AuthorHindered settling function

Richardson et al. av  vSt 1    RZ

Michaels et al. b v  vSt 1  max 

n

n

(11)
μ

Ekdawi et al. c  l maxv  vSt 1    1  max (12)a(Richardson et al., 1954), b(Michaels et al., 1962), c(Ekdawi etal., 1985)(13)

Fig. 3 Schematic representation of three methods to determine residence time behavior. (I) Tracer experiment, (II) Characterization ofsystem behavior, (III) Numerical simulation. Reprinted with permission from Ref. (Gleiß, 2018) under the terms of the CC-BY 4.0 license.
Copyright: 2018, the Authors, published by KIT scientific publishing.

by examining a step response following a sudden change involumetric flow rate or solids volume fraction during
feeding. Compared to tracer experiments, step response
measurements require considerably more equipment. An
experimental setup consists of two storage tanks for two
feed suspensions with different solids volume fractions anda three-way valve to cause a sudden change in the feed
composition. Time-dependent analysis of the step response
can be performed by time sampling of the centrate flow andgravimetric measurement of the solids volume fraction(Gleiss et al., 2017). A more automated method of measuring step response is to integrate a turbidity sensor on thecentrate for on-line prediction of solids content (Konrath etal., 2014).
The third promising approach to characterize the residence time behavior is CFD simulation (Fig. 3 (III)),
which requires a model to evaluate the flow behavior insolid bowl centrifuges (Gleiß, 2018). Numerical simulations are usually not possible without calibration with experimental data, as CFD relies on many assumptions, suchas turbulence models.

3. Separation behavior

The selection and operation of solid bowl centrifuges
depends not only on the flow conditions, but also on thematerial properties, which are influenced by the particle
and liquid phase properties. Particle size, particle shape,
particle interactions, solids content, densities, and liquid
viscosity play an important role. Moreover, a distinction
must be made between the properties of individual particles, agglomerates and bulk materials (Hammerich et al.,2019a). When evaluating the separation process in batch or62

semi-batch centrifuges, the sedimentation behavior andsediment build-up are important parameters. For continuous centrifuges, sediment transport and discharge must also
be taken into account (Karolis et al., 1986). It is not possible to draw general theoretical conclusions about the separation process in solid bowl centrifuges because theinfluencing variables at the particle level are manifold.

3.1 Settling behavior

Sedimentation is the settling of particles in a surrounding
liquid phase. Typically, sedimentation can be divided into
four areas: free settling, cluster formation, cluster breaking,and hindered settling (Bhatty, 1986). Free settling meansthat the particles do not interfere with each other and settle
under Stokes’ conditions. Moderate increases in solids
content lead to clustering and faster settling. At a criticallevel, the clusters are destroyed by the upward flow, reducing the settling rate. Further increase in solids content leads
to slower settling due to increased momentum exchange.
Theoretical prediction of the described behavior as a function of particle properties is time-consuming and requires
resolved simulations at the particle level (Romaní
Fernández et al., 2013).
Experimental investigation of the influence of particle
and continuous phase properties is therefore more effective. In practice, empirical material functions are commonly used to describe the influence of the solids content
on the settling velocity. A large number of researchers havestudied the settling behavior of monodisperse (Richardson
et al., 1954) and polydisperse particles (Al-Naafá et al.,
1989; Beiser et al., 2004; Ettmayr et al., 2001; Ha et al.,2002). This is summarized in Table 2. One approach todescribe hindered settling is based on (Richardson et al.,1954) and is valid for monodisperse suspensions, where theexponent nRZ depends on the particle Reynolds number.
The approach by Michaels et al. (1962) treats flocculated
systems and considers two correction factors (n, ϕmax) for
hindered settling. The approach by Ekdawi et al. (1985)
takes into account the influence of maximum packing andliquid viscosity.

Marco Gleiss et al. / KONA Powder and Particle Journal No. 41 (2024) 58–77
Table 3 Comparison of different empirical and semi-empirical equations describing compressive yield stress as a function of solids volume
fraction.

AuthorCompressive yield stress

Auzerais et al. a

ps  aBuscall et al. b

ps = a(ϕ – ϕgel)mGreen et al. c  
(14)p p 

Landman et al. d

 
 2
 1
ps  p1 (15)  gel Usher et al. e a max   b    gel  
ps    gel(16) s1  gel 

contacts. The common practice is to fit the material functions to experimental data which are based on batch centrifugation test on a laboratory scale.

4. Material characterization

nmax  

p2

Review Paper

Laboratory and pilot scale trials are essential for the design and optimization of solid bowl centrifuges. However,
due to the amount of material and time required, it is advisable to keep the experimental effort of pilot tests to a minimum. For this purpose, a variety of laboratory centrifuges
are available to measure material-related properties such as
particle size distribution, sedimentation behavior and sediment compressibility. These are described in more detail in
the following sections.


 1

p

k

a(Auzerais et al., 1988), b(Buscall et al., 1988), c(Green et al.,1996), d(Landman et al., 1995), e(Usher et al., 2013)(17)
For dilute suspensions and polydisperse particles, thesettling velocity is not uniform. Rather, a settling velocity
distribution can be observed in the experimental characterization (Balbierer et al., 2019). This makes the development of material functions much more complex, as theinfluence of different particle fractions or species on thesedimentation behavior must be taken into account.

3.2 Sediment formation process

Particles accumulate on the rotor wall after sedimentation and form a liquid-saturated sediment, which allows thetransfer of normal and shear forces due to permanent particle contacts. A saturated sediment behaves either incompressible or compressible while forming.
Incompressible sediments consist mostly of coarser particles x > 10 μm. In this size range, inertial forces dominate, whereas in compressible sediments the influence ofinterparticle forces increases. If the particles are uniformly
distributed in the sediment, incompressible materials havea constant porosity. In contrast, a compressible sedimentexhibits a porosity gradient based on compressive forces inthe sediment. Compressibility modeling combines compressive yield stress and solids volume fraction, wherecompressive yield stress is defined as the integral strengthof a particle network.
Table 3 compares different material functions to modelcompressive yield stress depending on solids volume fraction. All presented models are empirical or semi-empiricalequations which use fit parameters (a, b, k, m, n, p1, and p2)
and characteristic value such as gel point ϕgel or maximum
packing density ϕmax. The gel point determines the transition from a suspension to a sediment (Skinner et al., 2016).
Here, the material behavior changes abruptly, since forces
are transferable in the sediment due to permanent particle

4.1 Particle size distribution

The particle size is an imperative parameter for the evaluation and design of solid bowl centrifuges. Since the particles are not uniformly distributed, it is helpful to measurea particle size distribution (PSD).
Since most particles are non-spherical, the concept ofequivalent diameter is applied to calculate particle size.
The equivalent diameter is a reference value to the real
particle diameter and depends on the measurement technique. Examples are the Sauter’, the Feret’, and the Stokes’
diameter. For solid bowl centrifuges, it is desirable to use
sedimentation-based methods for particle size analysis. All
sedimentation methods have in common that the determination of the equivalent diameter is based on the Stokes’
velocity.
Typically, sedimentation methods are divided into incremental or cumulative methods. For the incremental method,
three main principles are conceivable: the pipette, thephoto-, and the X-ray sedimentation method (Hayakawa etal., 1998). The cumulative method also differentiates threemeasuring principles: the manometer centrifuge, the sedimentation balance, and the imbalance centrifuge (Yoshida
et al., 2001).
Two analytical centrifuges based on the incremental
method are CPS disk centrifuge and LUMiSizer® analytical centrifuge (Detloff et al., 2007; Neumann et al., 2013).
The CPS disk centrifuge uses the well known principleof differential centrifugal sedimentation (DCS). The particle size is analyzed by applying a density gradient in therotating disk. After injection of a small amount of suspension, the particles migrate in radial direction depending on
their individual settling velocity. Particles are counted by a
CCD sensor which measures the signal of a monochromatic laser beam.
In contrast to the DCS, the measuring principle of the
LUMiSizer analytical centrifuge is the spatial and temporalmeasurement of absorbance in a rectangular cuvette(Detloff et al., 2007). Both methods use Stokes’ settling
velocity to calculate the equivalent particle diameter. This

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20

C=21

18

Sediment height (mm )supposes additional physical parameters such as the solid
density, liquid density, liquid viscosity, and the refractive
index of both phases.
There are other physical principles to measure particle
size or shape distributions, such as laser diffraction, ultrasonic spectroscopy, X-ray scattering or dynamic light scattering. Details on the measuring principles can be found in(Frank et al., 2022).

Review Paper

C=42
C=212

16

C=302

14

C=214612
10
8Real�ve
centrifugal
number

6
42

4.2 Settling behavior measurement

The use of measuring cylinders in the Earth’s gravitational field is one way to measure the sedimentation rate.
For concentrated suspensions, experiments usually show aclear phase boundary between the clarified liquid and thesedimenting solids, indicating hindered settling (Lester etal., 2005). Modern instruments measure the settling velocity automatically (Horozov et al., 2004). To reduce measuring time, the settling behavior of finely dispersed particles
can be determined using analytical centrifuges (Detloff etal., 2007). According to the measuring principle of particle
size measurement, the temporal and spatial variation ofabsorbance along the radial position of a cuvette in the
LUMiSizer is used to calculate the settling velocity distribution from experimental data.

4.3 Sediment characterization

Batch sedimenting centrifuges are also suitable for theexperimental characterization of sediment formation. There
are several methods that differ in the amount of measurement required and the complexity of data post-processing.
The simplest method is to centrifuge a sample at constant
speed and measure the solids volume fraction gravimetrically. The volume-averaged compressive yield stress canbe derived from the analysis of the sediment at different
speeds. However, information about the porosity gradientin the sediment is lost by this method. Alternatively, twodifferent measurement setups in combination with batch
sedimenting centrifuges allow to measure the local solids
volume fraction as a function of sediment height.
The centrifuged sediment can be divided into a defined
number of layers after centrifuging. The solid volume fraction of each layer is then measured gravimetrically. A disadvantage of this method is the destruction of the sediment.
Non-invasive X-ray measurements are another promising technique for the analysis of local sediment structures.
Examples for measuring cake structures with X-rays are
the LUMiReader X-Ray or micro-computed tomography(μCT) (Peth, 2010). Both measurement principles are limited to the Earth’s gravitational field, which requires a twostep measurement procedure based on centrifugation andsubsequent X-ray measurement (Sobisch et al., 2016).
Fig. 4 illustrates the experimental result of the mentioned two-step procedure for solids volume fraction depending on sediment height. The suspension studied was a

64

0

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

Solids volume fraction (-)

Fig. 4 Spatial distribution of solids volume fraction in a limestone
sediment as a function of centrifugal acceleration. The measurement
requires a two-step measuring procedure. First, suspension is centrifuged with the LUMiSizer analytical centrifuge at a speed. Subsequently, the sediment structure is investigated with X-Ray measurementwith the LUMiReader X-Ray.

mixture of limestone and water, which was first centrifuged
and then measured in the LUMiReader X-Ray.
Closer examination of the measured data serves to study
the influence of the relative centrifugal number on thecompressibility of the formed sediment. Two effects areobservable. First, the solids volume fraction increases toward the bottom of the cell. Second, the solids volume
fraction of the sediment shifts to a higher value with increasing centrifugal acceleration.
However, this method is only applicable to sedimentsthat exhibit plastic behavior. This means that the sedimentdoes not tend to elastic recovery after reducing the centrifugal force (Hammerich et al., 2019b). Typically, biologicalsystems tend to behave elastically due to the existing cellstructure (Usher et al., 2013).
Additionally, the sediment transport in decanter centrifuges leads to a superposition of shear yield and normal
stress (Erk et al., 2004). Shear stress moves and rearranges
the particles in the sediment, resulting in denser packing(Hammerich et al., 2020; Radel et al., 2022).
To study the influence of shear yield stress on the sediment compressibility, several measurement principles canbe found in the literature. One method is to measure theeffect of shear in a vane-under-compressional-loading rheometer (Höfgen et al., 2020). Here, a cylinder at the top of
the rheometer produces a normal stress on the sediment,
while at the same time a vane rotates at a constant angularvelocity and introduces a shear yield stress. Furthermore,
modified ring shear cells are suitable to analyze sedimentswith a saturation of S = 1. This involves the modification of
the ring shear cell by incorporating semi-permeable membranes at the top and bottom to allow the liquid to escapeduring the experiment (Erk et al., 2004; Hammerich et al.,2020).

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5. Modeling of solid bowl centrifuges

During process development, accurate estimation ofcentrifuge type and size with the lowest possible productand resource consumption is essential. It is also importantto optimize existing solid bowl centrifuges to increase
productivity and minimize energy consumption and maintenance.
The following section provides an overview of different
methods for predicting centrifuge size and process behavior. A variety of approaches can be found in the literature.
This review focuses on the scale-up using analytical equations such as sigma theory, process modeling for real-time
simulation, CFD simulations of solid bowl centrifuges, andthe extension of process modeling to gray-box models toimprove the predictive accuracy of process models(Menesklou et al., 2021a).

5.1 Steady-state models

The first published theory to describe clarification insolid bowl centrifuges is the Sigma theory (Ambler, 1952).
Sigma theory is based on several assumptions. These include spherical particles in a creeping flow, uniform flow,
and Stokes settling. Also, the approach neglects insufficient
pre-acceleration or turbulences, resulting in the assumptionof uniform distribution of the solid at the feed. Furthermore, hydrodynamic interactions between solid and liquid
which are present for high solids content and the sedimentbuild-up are also not considered (Leung, 1998).
The basis of the original Sigma theory is a shallow pondwith (rw/rb ≥ 0.75). Here, rw is the weir radius and rb is thebowl radius. Extensions are available for deep pond solidbowl centrifuges (rw/rb ≤ 0.65) (Records and Sutherland,2001). To predict the clarification, the theory assumes thata solid bowl centrifuge removes the cut size with a probability of 50 %. With the above assumption, the flow rate ofa solid bowl centrifuge can be calculated by the product ofsettling velocity in the earth’s gravity field and the Σ-value
which is defined as an equivalent clarification area:

Qf  2 Σ vg

(2)

Looking more closely at Table 4, the Σ-value considersthe(2)
influence of the centrifuge geometry by integrating theweir radius, the bowl radius, and the centrifuge length.
Additionally, the Σ-value takes the angular velocity into
account. At this point it should be noted, that there are further modifications of the Sigma theory, which can be foundin the literature (Ambler, 1952; 1959). Due to the previously mentioned assumptions underlying the Sigma theory,
it is not reasonable to use this approach for designing thecapacity of a solid bowl centrifuge. Instead, Sigma theory
is applied to transfer for a geometrically similar machine toa larger scale with known experimental pilot scale data. For
tubular bowl and decanter centrifuges, the radial dimension
is defined by the bowl radius and the weir radius. For disc

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Table 4 Overview of Σ-value for different types of solid bowl centrifuges. According to Ref. (Ambler, 1961).

Centrifuge type

Sigma value

Batch sedimenting centrifuges

ΣTubular bowl or decanter
centrifuges (shallow pond)

Σ

(18)

ω 2V
 2r 2 
4.6 log  rw  rb 

πlω 2
g

rb 2  rw 2 
 2r 2 ln  2 b 2  rb  rw Tubular bowl or decanter
centrifuges (deep pond)πlω 2 rb 2  rw 2 2g
r2
ln  b 2 (19) rw 

Disc stack centrifugesΣΣ

2πnω 2
rdo 3  rdi 3 cot θ3g(20)(21) is defined by r r
stack centrifuges the radial dimensiondo, diand θ, which are the outer radius, inner radius and inclination angle of the disc channel.
In terms of practice, this means that pilot tests are essential for the selection and scale-up of solid bowl centrifuges.
For the description of two geometrically similar machines
the following equation is valid:
Qf 2
Σ
 2
Qf1
Σ1

(3)

The integration of correction factors ξ1 and ξ2 allows totransfer for non-geometrically similar machines:
(3)
Qf 2
ξ Σ
 2 2
(4)
Qf1
ξ1 Σ1
Another mathematical model for the scale-up of decanter centrifuges is the g-volume, which is a modified(4) of the sigma theory. Similar to the Sigma theory, theformg-volume also allows the comparison of geometrically different decanter centrifuges. The g-volume is the ability of adecanter to clarify a suspension, but is not a measure of theavailable surface area as in the Sigma theory, but of theavailable centrifuge volume.
Therefore, the g-volume approach considers the influence of sediment build-up on the clarification process(Stahl, 2004). The dimensionless Leung number has beendeveloped for spin tube, disk, tubular bowl, and decanter
centrifuges. Leung number depends on the properties ofsuspended particles, liquid and design parameters (Leung,2004).

5.2 CFD simulation

CFD is based on the conservation of mass and momentum, also known as the Navier-Stokes equations. The Navier-Stokes equations can only be solved using
numerical methods such as the finite volume method. This

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requires a computational grid. The advantage of CFD is theability to simulate temporal and spatial changes in velocity
and pressure. The optimization of solid bowl centrifuges is
an interesting task in terms of process behavior and energy
consumption. However, due to the presence of several
phases (gas, liquid and solid), the simulation of the material
behavior is very challenging. In addition, high pressure andvelocity gradients lead to very small time steps.
When simulating multiphase flows, CFD distinguishes
between the Euler-Euler and Euler-Lagrange methods. The
Euler-Euler method treats the continuous and disperse
phases as interacting continua. A volume-averaged dragforce, allows to model the momentum exchange between
several phases. In contrast, the Euler-Lagrange method
uses a Euler model to estimate the behavior of the continuous phase. The dispersed phase is modeled by Newton’s
equation of motion. The implementation of a drag force isessential to calculate the momentum exchange between
discrete particles and the liquid phase.
One problem with the computational time of the EulerLagrange method is that sediments in solid bowl centrifuges consist of billions of particles. Simulating this large
number of particles is not possible with today’s computing
power.
One promising method to solve this issue is to reduce thedomain size. For solid bowl centrifuges, this means that
only specific parts of the machine such as the infeed wheel
or discharge zone are simulated (Bell et al., 2016;
Törnblom, 2018).
A suitable method for simulating the velocity and pressure field of gas-liquid flows without taking into accountthe influence of particles is the Volume-of-Fluid (VOF)
method (Hirt et al., 1981). VOF is mostly used to simulate
gas-liquid interfaces by incorporating the surface tension

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as volume force into conservation of momentum (Romani
Fernández et al., 2009). The VOF approach is well suitedfor the optimization process of solid bowl centrifuges withthe target of minimized friction loss of the rotating liquid
pond to ensure rigid-body motion (Romani Fernández etal., 2010). Furthermore, the coupling of CFD with the discrete element method (DEM) is also used to study the
temporal change of sediment build-up (Romani Fernández
et al., 2013). DEM uses additional forces for particle andparticle–wall interactions by considering nonlinear contact
models.
Another method to study the process of sedimentbuild-up in solid bowl centrifuges is based on an extended
mixture model, as this involves significantly lower computational costs compared to Euler-Euler or Euler-Lagrange
models (Hammerich et al., 2018).
The extended mixture model does not solve the mass andmomentum conservation for each phase separately, but
considers the suspension as a mixed phase. The algorithm
is based on different rheological models for the suspension
and the sediment. Power laws are suitable to consider the
influence of solids volume fraction on suspension viscosity(Quemada, 1977). Whereas sediments in many cases havea yield point which depends for compressible materials on
the solid volume fraction (Erk et al., 2004).
Fig. 5 (I) shows the simulation of the sediment formation process for a constant time step in a tubular bowl centrifuge for two scenarios: a cohesive non-flowable (1.) anda pasty flowable material (2.) (Hammerich et al., 2019a).
The rheological behavior of the sediment has a significant
influence on the sediment distribution. For easily flowablematerials, shear forces at the sediment surface result in asediment transport and a uniform distribution of theaccumulated solids. For cohesive materials, the friction

Fig. 5 (I) Contour plot of axial velocity and sediment solids volume fraction in a tubular bowl centrifuge type Z11 for two materials with different
rheological behavior. 1. Cohesive non-flowable sediment. 2. Pasty, flowable sediment. (II) Residence time distribution F as a function of flow numberτ for three cases: empty machine (a), sediment formation process of a cohesive sediment (b) and a pasty sediment (c). Adapted with permission from
Ref. (Hammerich et al., 2019a). Copyright 2019, Wiley.

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within the sediment is high and a slope angle has formed,
leading to higher axial dispersion and back-mixing in theflow domain.
Additionally, Fig. 5 (II) compares the residence time
distribution F depending on the flow number τ for threedifferent scenarios. The flow number is the ratio of elapsed
time to mean residence time and allows to compare theresidence time distribution of different centrifuges and operating conditions.
Further investigations are devoted to the flow field in
disk stack centrifuges. For biological systems, the shear
stress in the feed and discharge zone is crucial to keep thestress on microorganisms low and the vitality high, such as
in mammalian cell cultures (Shekhawat et al., 2018). To
predict the separation behavior of a disk stack centrifuge,detailed information about the flow and particle trajectories
within two disks is necessary. A suitable simulation method
is direct numerical simulation (DNS) which can be used to
predict flow instabilities that significantly affect particle
trajectories (Janoske et al., 1999b). Furthermore, CFD is apromising method to compare different centrifuge types
such as disk stack and tubular bowl centrifuges(Esmaeilnejad-Ahranjani et al., 2022).
The complex geometry of a decanter centrifuge with therotating screw conveyor and the cylindrical-conical bowl
makes it difficult to study the process behavior. Few studies
have focused on olive oil and polymer production (Zhu etal., 2013; 2020). Current studies assume stationary rotation
and apply a multi reference frame (MRF) (Zhu et al.,2020). MRF is a stationary approach to treat rotation of asystem and does not consider the transient nature of therelative motion between the screw conveyor and the centrifuge bowl.
In summary, some numerical studies are available. However, some simplifications have been made here in the areaof turbulence modeling and momentum exchange between
solid and liquid.

5.3 Process modeling

In contrast to CFD, the goal of process modeling is tohave a simplified model that can predict the dynamic behavior with minimal computational effort. Ideally, processmodels should be faster than real-time simulation to enablefurther applications such as advanced process control (Sinnet al., 2020).

5.3.1 Batch sedimenting centrifuges

A widely used approach to simulate the sedimentation-consolidation process for batch sedimenting centrifuges is based on a 1D advection-diffusion equation(Bürger et al., 2001; Garrido et al., 2003). These 1D models
assume highly concentrated suspensions where hinderedsettling is present to neglect the influence of segregation(Kynch, 1952).

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Essential material properties are the flux density function, the effective solid stress, the density of solid and liquid phase, and the liquid viscosity (Garrido et al., 2000). A
major limitation of 1D models is that they are not suitablefor capturing the physical behavior caused by internal disks
in disk stack centrifuges, sediment transport in decanter
centrifuges, and particle classification for low concentrated
suspensions in tubular bowl centrifuges. Models that consider the geometry and the influence of the particle size
distribution are necessary to describe the above process
behavior.

5.3.2 Tubular bowl centrifuges

The modeling of the technical separation in tubular bowl
centrifuges is complex, since the flow conditions and material properties have a significant influence on the process
behavior. One promising strategy to simulate the process
behavior of tubular bowl centrifuges is a pseudo-2D compartment approach. The idea of compartmentalization is to
distribute the particles along the axial length of the rotor
based on the calculated sedimentation path, see Fig. 6 (II)(Spelter et al., 2013). Compared to steady-state models
such as the Sigma theory, the pseudo-2D compartment
model predicts the temporal change of sediment build-up
and the degree of filling. The degree of filling is the ratio of
sediment to centrifuge volume and provides information
about the current filling status. One limitation of thepseudo-2D compartment model is the modeling of the particle motion which is based on the assumption of a plug
flow neglecting the influence of the turbulences and backmixing (Spelter et al., 2013).
The extension of pseudo-2D to 2D compartment modeling addresses the previously discussed limitation by accounting for nonideal flow behavior by interconnecting adefined number of compartments to approximate the residence time behavior of tubular bowl centrifuges (Gleiss etal., 2020). Fig. 6 illustrates the differences between the twopresented algorithms. The 2D compartment model divides
the tubular bowl centrifuge into two zones: the sedimentation and sediment zone. Both zones consider different material streams and material functions to capture the physicalbehavior of the suspension and sediment. The upper zone is
the sedimentation zone. Compartments of the sedimentation zone consist of one incoming and two outgoingstreams. The feature of the algorithm is to calculate theparticle classification along the rotor for a predefined number of particle classes. A more detailed description of thealgorithm can be found in (Gleiss et al., 2020).
The sediment zone is located below the sedimentation
zone to calculate the sediment build-up for incompressible
and compressible materials. A compartment of the sediment zone consists of one ingoing stream which represents
the separated solids from the sedimentation zone. The relationship between compressive yield stress and solids

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Fig. 6 (I) Schematic representation of a tubular bowl centrifuge. (II)Pseudo 2D-compartment modeling according to Ref. (Spelter et al.,2013) with the discretized sediment zone. (III) 2D-compartment modeling of a tubular bowl centrifuge based on the interconnection of compartments which are based on the sedimentation and sediment zone
(Gleiss et al., 2020).

volume fraction is used to compute sediment compressibility which can be measured in batch sedimenting centrifuges as mentioned earlier in Section 4.3.
It should be noted that the sediment influences the separation process, as the available free flow cross-section is
reduced and results in a shorter residence time.
Fig. 7 depicts the comparison between simulation andexperiment for the sediment distribution of colloidal silica
particles at a flow rate of 0.2 L/min and two different centrifugal numbers of C = 19,200 (I) and C = 38,500 (II) fora tubular bowl centrifuge type Z11 from CEPA. The flow
direction is from left to right. As already mentioned, particles accumulate at the inner wall of the rotor. Due to thepolydispersity of the particle system and different settling
velocities, the sediment accumulation is not uniform, but
has a constant slope along the axial length of the tubular
bowl centrifuge. A closer look at the relative centrifugal
numbers C = 19,200 and C = 38,500 shows different timeintervals for the filling process. For C = 38,500, the rotor
fills faster compared to C = 19,200, because the centrifugal
force leads to faster settling rates and therefore faster solids
accumulation. Analytical equations such as the Sigma theory (see Section 5.1) do not allow to estimate temporalchanges in solid bowl centrifuges. However, this is neces-

68

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Fig. 7 Comparison between simulation and experiment for sediment
distribution along the axial length Lax of the rotor for colloidal silica
particles at a constant flow rate of 0.2 L/min. (I) C = 19,200, (II)C = 38500.

sary to achieve a constant separation performance, especially for tubular bowl centrifuges.

5.3.3. Decanter centrifuges

The modeling of decanter centrifuges requires the consideration of sediment transport as one of the most important influencing variables during operation. As mentioned
earlier, the material properties of the sediment are critical,
and a distinction must be made between incompressible
and compressible sediments. For incompressible materials,
friction between the material, the screw conveyor and thecentrifuge bowl is dominant. Friction in the conical part ofthe machine has the highest impact on the screw torque(Bell et al., 2014; Reif and Stahl, 1989). After the material
leaves the liquid pond, undersaturation occurs which increases the frictional forces. The required power consumption can be determined from four components: feed
acceleration, product transport, windage and power transmission losses (Bell et al., 2014).
In contrast, compressible sediments have a totally different transport behavior. The pasty consistency leads to considerable difficulties during sediment transport when
applying unfavorable operating conditions. This can lead tototal failure of the machine. However, so far only simplified material functions and models can be found which
represent the rheological behavior of compressible
sediments with good accuracy (Karolis et al., 1986). As

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previously described, the material has a yield point depending on solids volume fraction (Erk et al., 2004; Mladenchev,
2007).
For decanter centrifuges, the concept of 2D compartment modeling is also a suitable approach to calculate theprocess behavior (Gleiss et al., 2017). However, the approach needs to be extended to the process behavior of decanter centrifuges. Fig. 8 (I) shows the structure of a 2D
compartment model for countercurrent decanter centrifuges. In contrast to tubular bowl centrifuges, the simulation domain is an unrolled screw channel which is formedby the screw conveyor and the overflow weir. Furthermore,
Fig. 8 (II) illustrates the detailed structure of a single compartment with the inlet and outlet flows (Menesklou et al.,2020). The 2D compartment model for decanter centrifuges considers two additional material streams in the sediment zone to model the sediment transport by screw

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conveyor system (Menesklou et al., 2020). The applicability of the 2D compartment approach for scale-up purposes
was tested using different decanter centrifuges. Fig. 9 (I)depicts the comparison between simulation and experiment
for the solids mass fraction in the centrate for three countercurrent decanter centrifuges of different sizes, which are
not geometrically similar. With increasing speed, a higher
centrifugal force occurs and the solids mass fraction of thecentrate decreases. Another application of 2D compartment
modeling is to simulate the solids mass fraction of the sediment, see Fig. 9 (II) (Menesklou et al., 2021b). The material functions were determined at laboratory scale andintegrated into the 2D compartment model. The successful
simulation of different machine sizes is a first step awayfrom many pilot tests towards a predictive design andscale-up starting from laboratory tests to significantly
reduce the number of pilot tests.

Fig. 8 (I) Schematic representation of a countercurrent decanter centrifuge and unrolled helical screw channel which is the basis of the 2Dcompartment model. (II) Ingoing and outgoing streams for the sedimentation and sediment zone of one compartment in the 2D compartment approach.
Adapted from Ref. (Menesklou et al., 2020) with permission from Elsevier, Copyright 2020.

Fig. 9 (I) Comparison between simulation and experiment for solids mass fraction of the centrate for three countercurrent decanter centrifuges ofdifferent size which are not geometric similar. (II) Comparison between simulation and experiment for the mean solids mass fraction of the concentrate
for three countercurrent decanter centrifuges of different size which are not geometric similar. The investigated product was finely dispersed limestone.
Reprinted from Ref. (Menesklou et al., 2021b) under the terms of the CC-BY 4.0 license. Copyright: (2021) The Authors, published by MDPI.

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A further application of a decanter centrifuge is the particle classification, where the local flow behavior is crucial
for accurately predicting the particle size distribution in thecentrate. While the 2D compartment model assumes a uniform flow profile and ideal back-mixing, extensions of theapproach take into account the local flow profile and theinfluence of screw conveyor motion on the flow (Bai et al.,2021a).

5.3.4 Disk stack centrifuges

The process modeling of disc stack centrifuges is currently limited to the use of analytical equations and does
not consider process dynamics. Existing models are steadystate and divide disk stack centrifuges into different zones:
the pre-acceleration zone, the disk stack zone and the resuspension zone (König et al., 2020). However, it is also conceivable to apply the 2D-compartment model also to disk
stack centrifuges. This would have the advantage of a uniform modeling strategy for different solid bowl centrifuges
which would improve comparability and thus the selectionof the optimal centrifuge type especially for industries
where no centrifuge experts are not available.

5.4 Gray-box modeling

The idea of gray-box modeling is to combine white-box
and black-box models. White-box models represent thephysical laws, i.e. conservation of mass, momentum, andenergy. An example of a white-box model is the 2D compartment model introduced in Section 5.3. Black-box
models are mostly data-driven algorithms (e.g. support
vector machines, linear, nonlinear regression, or neuralnetworks) that are trained and validated with experimental
data (Xiong et al., 2002). Data acquisition can be accomplished by integrating off-line, on-line, or in-situ measurement techniques to determine particle properties at theprocess. On-line and in-situ measurement has tremendous
potential for the safe and cost-effective operation of solidbowl centrifuges and is necessary to collect data for graybox models. Gray-box models are suitable for applications

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where white-box models are not accurate enough. For solidbowl centrifuges in particular, there is a complex relationship between centrifuge geometry, material properties andprocess conditions that has not been adequately captured
by white-box models. At the same time, data-driven algorithms require a huge amount of experimental data which isoften not available due to the costs of measurement equipment. Furthermore, black-box models are not extrapolatable and are difficult to adapt to new conditions without
enough new experimental data.
Two basic structures of a gray-box model are the paralleland serial structures which are compared in Fig. 10. The
advantage of the parallel structure is that the white-box andblack-box models can be easily combined into a gray-box
model. If the white-box model has a poor accuracy theblack-box model takes over to simulate the process. A disadvantage of the parallel structure is that both models do
not work simultaneously. Depending on the accuracy of theprediction, either the white-box or the black-box model isused for the calculation (Menesklou et al., 2021a). In thecase of a white-box model with a poor accuracy, the graybox model will be a pure black-box model, which needs ahuge amount of data.
An improvement to the aforementioned drawback is theserial structure which uses the black-box model to be able
to improve the uncertainties of the white-box model(Nazemzadeh et al., 2021). For example, it is conceivable
to use the black-box model to predict the influence of flow
conditions in a serial gray-box model structure or to correct
separation-related properties, such as the hindered settlingfunction, through on-line learning algorithms. However,this is challenging task and currently part of the research in
the field of particle technology.
Fig. 11 shows experimental results of the solids massfraction in the centrate of a pilot scale decanter centrifuge
and simulation data based on a developed gray-box model
based on a parallel structure. In this study, the white box
model is the previously discussed 2D compartment modeland the black box model corresponds to a neural network

Fig. 10 Comparison between the two main structures of a gray-box model which is typically based on a deterministic and data-driven model. (I)Parallel structure (II) Serial structure.

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Fig. 11 Comparison between white-box and gray-box modeling for theinvestigated centrate mass fraction of a pilot-scale decanter centrifuge
as a function of flow rate and confidence interval at constant speed. The
model system is a limestone-water suspension. Reprinted from Ref.
(Menesklou et al., 2021a) under the terms of the CC-BY 4.0 license.
Copyright 2021, the Authors, published by MDPI.

with one input, hidden and output layer. The training procedure of the neural network consists of a training and a test
phase. The training data includes pilot test data for material
properties measured at the feed, centrate and concentrate
and the relevant process conditions.
Another important point for the application of a graybox model with parallel structure is the confidence interval
necessary to allow the intervention of the black-box model
when the white-box model has a poor accuracy. For thecase presented in Fig. 11, there is a deviation of the whitebox model at higher speeds. The smaller the confidence
interval is set, the earlier the correction of the white-box
model prediction by the black-box model occurs. As mentioned earlier, this allows the prediction of process behavior for decanter centrifuges to be improved by integrating
experimental data.

6. Process analytics

Process monitoring is playing an increasingly importantrole in optimizing the operation of solid bowl centrifuges toachieve a higher product quality and shorter setup times. To
avoid reject rates or exceeding tolerance limits, quality
control is moving from off-line to on-line, or in-situ applications. On-line measurements typically use a bypass toguide the material into the measurement system. In contrast
to off-line measurements, on-line detection requires both asampling system and often a dilution station to measureparticle properties with high accuracy. The following section summarizes several measurement methods for on-line
and in-situ characterization of particle and bulk properties
that have been applied to solid bowl centrifuges.

6.1 On-line measurement of particle properties

In tubular bowl centrifuges, the flow conditions changeover time due to the accumulated solids. To maintain consistent product characteristics, the rotor speed or flow rate
must be adjusted during the centrifugation process. How-

Fig. 12 Comparison of sensor data and gravimetric measurement forthe product loss as a function of time for colloidal silica for the noncontrolled case (n constant) and the controlled case (n controlled). For atubular bowl centrifuge type Z11. Reprinted from Ref. (Konrath et al.,2014) with permission from Elsevier. Copyright 2014.

ever, in order to change the rotor speed, it is essential tomonitor the particle properties at the centrate. For dilute
suspensions, light scattering sensors are suitable to determine the solids volume fraction on-line (Konrath et al.,2014). To integrate such a system into the process, calibration with the initial suspension is required. Typically, a dilution series is necessary, which can be characterized by
off-line measurements.
Fig. 12 illustrates the comparison of the product loss forthe controlled (n controlled) and the non-controlled (nconstant) scenario for a tubular bowl centrifuge type Z11.
The controller design is based on the step response after
changing solids volume fraction at the feed. A closer lookat Fig. 12 reveals a fundamentally different process behavior for the non-controlled and the controlled case.
While the product loss increases with time for n =constant, since the machine fills continuously, the productloss remains constant for n = controlled. In this case, thedecreasing residence time is compensated by increasing therotational speed. Light scattering sensors can also be used
to detect turbidity in the centrate of decanter and disk stackcentrifuges to monitor centrate clarification and control theprocess to maintain product quality (Leung, 2007).
Another measurement system for on-line analysis of total suspended solids is UV-Vis spectroscopy. Unlike lightscattering sensors, UV-Vis spectroscopy does not operate ata constant wavelength, but uses the full range of ultraviolet
and visible light to produce an absorption spectrum. UVVis spectroscopy also requires extensive calibration basedon a dilution series for the measured suspension. A
promising technique to correlate the measured UV-Vis datawith the solids volume fraction is the use of multivariate
regression which is typically based on generalized ordinary
least squares (Winkler et al., 2020). Another application of
on-line UV-Vis spectroscopy is to detect the composition of71

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binary mixtures (Winkler et al., 2021). One example is theon-line measurement of polymethyl methacrylate (PMMA)
and zinc oxide (ZnO) in an aqueous system. Both particle
systems consist of submicron particles. For composition
prediction, both components must scatter a UV-Vis signal
at different wavelengths. By directly coupling UV-Vis datawith product-related properties such as the product loss, theUV-Vis sensor acts as a soft sensor that directly outputs thequality of a classification, or fractionation process (Winkler
et al., 2021).
Fig. 13(a) shows the comparison between off-line andon-line measurement of the temporal change of the productloss for the fractionation of PMMA and ZnO. The results
indicate a good agreement between reference values andthe prediction also for the variation of the centrifugal number. Additionally, Fig. 13(b) illustrates the prediction of thetemporal change of the product loss during the classification of a suspension consisting of PMMA and water.
Another method of recording the change in solids volume fraction in the centrate over time is to use a measuring
cell with a light source and a camera system. The measured
data can be evaluated using a convolutional neural network(CNN). Prediction requires a pre-trained CNN that has
been trained with experimental data (Bai et al., 2021b). The
presented deep learning algorithm is a hybrid neural network based on a CNN and a short-term memory network(LSTM). The LSTM helps to ensure that previous data is
not forgotten, when searching for the correct gradientsduring the new training period (Bai et al., 2021b). The

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measuring system can be applied to a low concentrated
suspension with a solids mass fraction of up to 0.5 wt %,which allows to detect the clarification in the centrate. Online measurement of particle size distribution is challenging, but there are several methods such as laser diffraction,
ultrasonic extinction, and X-ray. The practicality of an instrument depends on the particle properties, the solids volume fraction, and the process itself. If the measurement
time is less than the residence time, the instrument can beused directly for process control after successful calibration. However, if the measurement time is longer than theresidence time, the direct coupling of process control andmeasurement is challenging due to the different timescales. Furthermore, most particle size measurement techniques are only suitable for a defined particle size range,
making experiments irreplaceable.

6.2 In-situ measurement of slurry properties

The in-situ measurement of particle properties in solidbowl centrifuges is very challenging due to the high centrifugal acceleration and requires costly sensors, but provides detailed information about the local physicalbehavior. Several aspects are necessary for the applicationof in-situ sensors in solid bowl centrifuges. First, gaps areneeded inside the rotor to install the sensors. Therefore, it isnecessary to calculate the influence of the gaps on the mechanical strength of the rotor. Second, signal transmission
requires a transmitter, e.g., Wi-Fi, to transmit the measured
signal to the receiver.

(a)

(b)

(c)

Fig. 13 (a) Comparison between soft-sensor prediction and reference for the temporal change of product loss for a binary mixture of polydisperse
PMMA and ZnO particles for a tubular bowl centrifuge type Z11. (b) Comparison between soft-sensor prediction and reference for the temporal changeof product loss P PMMA for a polydisperse PMMA suspension at a constant relative centrifugal number C (c) Temporal change of relative centrifugal
number for the two illustrated experiments (a) and (b). Here, the solid line (expF) represents the centrifugal number for the case of a binary mixture(a) and the dashed line (expC) the PMMA system (b). Reprinted from Ref. (Winkler et al., 2021) under the terms of the CC-BY 4.0 license. Copyright2021, the Authors, published by MDPI.

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rotor of a tubular bowl centrifuge type GLE from CEPA.
The rotor consists of a glass fiber reinforced plastic (GRP).
The adapted rotor is necessary for the undisturbed MRI
measurements of sediment distribution. The analysis of
sediment involves a contrast agent such as carbon black(CB). CB generates a white contrast during MRI measurement, see Fig. 14 (II). A closer look at Fig. 14 (II) depicts
that the temporal filling process starts from the right (inflow) to the left (centrate). The axial flow and the simultaneous classification of the particles along the rotor explain
the non-uniform sediment build-up according to the previously mentioned CFD results and process simulation.
In summary, in-situ measurement methods provide very
accurate insight into the physical behavior of solid bowl
centrifuges, but experiments are time-consuming andequipment costs are high, limiting their use to research.

7. Conclusions
Fig. 14 (I) glass fiber reinforced plastic (GBP) tube and original rotorof tubular bowl centrifuge GLE from CEPA GmbH. (II) MRI analysis ofrotor with longitudinal and lateral cross section. Flow direction fromright to left. Adapted with permission from Ref. (Stahl et al., 2008).
Copyright 2008, Wiley.

Wireless electrical resistance (WERD) is a technique toinvestigate sediment thickness at high relative centrifugal
number (Prayitno et al., 2020). The measurement system
records the change in electrical conductivity of the liquid
phase in-situ by integrating multiple pairs of electrodes
along the cylindrical conical bowl. The measured data canbe used to indirectly calculate the solids volume fraction in
the sediment.
An extension of this method is the Periodic Segmentation Technique in Wireless Electrical Resistance Detector(psWERD), which enables in-situ prediction of material
functions such as the hindered settling function for decanter centrifuges. The periodic segmentation technique isnecessary due to the influence of relative motion between
the screw conveyor and the centrifuge bowl. The screw
conveyor disturbs the measurement signal, due to the different material and is filtered by the periodic segmentation
technique (Prayitno et al., 2022).
Another promising non-invasive and in-situ analysis of
the sediment distribution is the Magnetic ResonanceImaging (MRI), which is an imaging technique based on
the physical principle of nuclear magnetic resonance(Hardy, 2011). This is done by placing the rotor of thecentrifuge in the tube of the tomograph.
In contrast to WERD or psWERD, the size of the tomography limits the application of MRI to small solid bowl
centrifuges. An example of MRI is the study of sediment
distribution in tubular bowl centrifuges (Spelter et al.,2013; Stahl et al., 2008). Fig. 14 (I) presents an adapted

This work presented an overview of research in the fieldof modeling and optimization of solid bowl centrifuges.
Modeling on different time and length scales helps to better
understand the physical behavior of solid bowl centrifuges
and optimize their operation. One of the major challenges
in the selection and development of suitable models arises
from the influence of distributed particle properties.
Conventional steady-state methods for the design ofsolid bowl centrifuges are based on several assumptions
and are not predictive without several pilot scale experiments. The transfer from pilot to industrial scale is usually
only possible for geometrically similar machines in combination with decades of experience of the manufacturers. A
counter idea to conventional methods is to apply processmodeling and to consider material functions for separationrelated properties such as sedimentation, sediment build-up,
and sediment transport. The measurement of the material
properties is not carried out on the pilot but on a smaller
laboratory scale, with well-established laboratory centrifuges. Furthermore, CFD helps to gain a better understanding of the usually inaccessible parts of a solid bowl
centrifuge. At the same time, CFD can be used to optimize
different centrifuge geometries, such as the feed accelerator
or other parts of the machine.
However, since any form of modeling is based on assumptions and therefore does not completely reproduce thephysics, there will always be deviations. This is wheregray-box models have great potential to successively improve the modeling of solid bowl centrifuges. Data-driven
methods such as supervised learning and neural networks
allow an intensive analysis of process data.
Nevertheless, this requires the measurement of particle
properties on-line or in-situ by integrating process analytics. Various measurement techniques such as turbidity or
UV-Vis sensors, camera systems, laser diffraction, ultrasonicextinction and X-ray are available. However, due to the73Marco Gleiss et al. / KONA Powder and Particle Journal No. 41 (2024) 58–77

high cost of measuring equipment, individual testing of theparticle system is necessary.
In summary, useful tools for the modeling and optimization of solid bowl centrifuges are available and their application has already been demonstrated in research. The task
now is to further improve the algorithms and measurement
methods. With further developments and the interconnection of gray-box models, process analytics, and advanced
process control, the overriding goal of autonomous centrifugation will be possible in the future.

Acknowledgments

The authors gratefully acknowledge the funding by
German Research Foundation (DFG) in the priority program SPP 1679 “Dynamic simulation of interconnected
solids processes” and the Federal Ministry of Economic
Affairs and Climate Action in the program IGF—Industrial
Collective Research.

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Authors’ Short Biographies
Marco Gleiss
Marco Gleiss has studied Chemical Engineering at the Karlsruhe Institute of Technology (KIT)and subsequently became a doctoral student at the Institute of Mechanical Process Engineering and
Mechanics (KIT), where he got his Ph.D in Chemical Engineering and became Senior scientist. His
research interest is in the field of solid–liquid separation, on the digitization of filtration and centrifugation, the development of digital twins, and autonomous processes.

Hermann Nirschl
Hermann Nirschl received his Ph.D. in fluid mechanics from the Technical University in Munich
in 1994. In the years between 1997 and 2003 he was responsible for process engineering developments for the 3M company. From 2003 on he is Professor for Mechanical Process Engineering at the
KIT. The focus of the research is on particle technology with a special emphasis on separation processes, numerical simulation and the development of particle characterization technologies.

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