An EKF Based Estimation Scheme for Sedimentation Processes

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Flow Measurement and Instrumentation 21 (2010) 521–530

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Flow Measurement and Instrumentation

journal homepage: www.elsevier.com/locate/flowmeasinst

An EKF based estimation scheme for sedimentation processes in vessels usingEIT-type measurement data

Ahmar Rashid a, Anil Kumar Khambampati a, Bong Seok Kim b, Sin Kim c, Min Jae Kang a,Kyung Youn Kim a,∗

a Department of Electronic Engineering, Jeju National University, Jeju, South Koreab Applied Radiological Science Research Institute, Jeju National University, Jeju, South Koreac Department of Nuclear and Energy Engineering, Jeju National University, Jeju, South Korea

a r t i c l e i n f o

Article history:

Received 31 March 2009

Received in revised form

25 August 2010

Accepted 16 September 2010

Keywords:

Sedimentation

Solids flux theory

State estimation

Electrical impedance tomography

a b s t r a c t

Sedimentation is usually parameterized by settling curves, settling velocities and the concentration

of the constituent layers. The estimation of sedimentation parameters leads to useful information inthe fields of environmental and industrial engineering. This paper presents an extended Kalman filter

(EKF) based dynamic estimation scheme to extract sedimentation parameters from electrical impedancetomography (EIT) measurements obtained across the electrodes attached to the walls of a process vessel.

A state evolutionmodelhas been developedfor three-layer sedimentation based on thesolids flux theoryfor batch sedimentation. The performance of the proposed method has been verified by carrying out

numerical experiments.

© 2010 Elsevier Ltd. All rights reserved.

1. Introduction

Sedimentation is defined as the separation of a suspension

of solid particles into a concentrated slurry and a supernatantliquid, either to concentrate the solid or to clarify the liquid.

There has been considerable research on sedimentation in thelast century as summarized in the literature surveys [1,2]. Several

studies describing the settling phenomenon have been published

[3–12]. The sedimentation can be characterized either as batchsedimentation or continuous sedimentation. If there is no influent

flow into and effluent flow out of the system and the settlingvelocity of the particles at any point depends only on the local

concentration of the particles and the acceleration due to gravity,

the sedimentation is usually referred to as batch sedimentation[6,7]. The sedimentation becomes continuouswhen it is performed

under the influence of external forces, such as a continuous influxof particles into the system [9–11]. Batch sedimentation is used

in the classification of solids, washing, particle size measurement

or mass transfer, solvent extraction, and etc. Its applications arein the fields of waste water treatment and mineral processing,

in chemical, food, pharmaceutical, nuclear and the petroleumindustry,and in themonitoring of sediment transportin the coastal

∗ Corresponding author. Tel.: +82 64 7543664; fax: +82 64 7561745.

E-mail address: [email protected](K.Y. Kim).

zone [7,12]. The results obtained from batch settling tests havebeen effectively used for the design of continuous settlers [10,11].

Although the concentration of the particles varies throughoutthe sediment, most commonly applied mathematical models tothe sedimentation process discretize the settler volume intocompletely stirred, horizontal layers, having sharp interfacialboundaries [13–17]. The number of layers proposed in thesemodels ranges from three to as many as 50. This paper considersthe application of solids flux theory to examine three-layersedimentation [6,17].

The time evolution of the location of the interface boundariesbetween the sedimentation layers is referred to as settling curves,whereas the settling velocities are obtained as derivative esti-

mates of the settling curves. Settling curves and settling veloci-ties are thus often used to parameterize the sedimentation processand the correct determination of these parameters is key to thedesign and modelling of industrial processes [12,18,19]. The tra-ditional methods to determine the settling parameters requiremanual handling of the sedimentation tanks leading to large mea-surement errors, thus limiting their use to laboratory experiments.Optical measurements and image processing techniques providean automatic monitoring of the phenomenon [19,20]. However,these techniques necessitate the use of a transparent tank and can-not be used to monitor the sedimentation of a species in the back-ground of other opaque species. Measurement techniques such asX-ray radiation, ultrasound, microwave and magnetic resonanceimaging can be used under these circumstances. However, these

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522 A. Rashid et al. / Flow Measurement and Instrumentation 21 (2010) 521–530

Electrode

Fig. 1. A schematic diagram of 2D EIT setup to monitor three-layer sedimentation.

techniques are complex, invasive in nature and often too ex-pensive, making it impracticable to use them online. Electricalimpedance tomography (EIT) is a fast and non-invasive measure-ment technique in which an internal image can be reconstructedby imposing an electrical current and measuring voltages acrossthe electrodes attached to the walls of the sedimentation tank[21–23]. This approach is suitable for the online estimation of set-tling parameters (such as settling curves and settling velocities)

without the need to use a transparent settling tank.A major challenge while applying EIT to extract the sedimenta-

tion parameters is its time-varying nature. A dynamic image re-construction algorithm is needed to examine the sedimentationprocess effectively. For dynamic reconstruction, a state evolution-based estimation technique is required. The evolution model,alongwith the observation model, constitutes the state-space represen-tation of the system. In the state-space model, the evolution of state parameters is modelled as a stochastic process. The knowl-edge of the stochastic nature of the state evolution is instrumentalin order to apply EIT for dynamic image reconstruction. A num-ber of state evolution models has been proposed to monitor dy-namic processes using EIT [21–31]. The simplest of these modelsis the so-called random-walk model in which the state param-

eters evolve by a predetermined covariance of the added whitenoise [25–28]. However, application of this model is based on thefact that no prior knowledge of the process is available thereforeit is not considered as an accurate evolution model. For processeswhich evolve with constant velocity or constant acceleration, thekinematic models offer a better solution [21–24]. The kinematicmodels have originated from target tracking problems [32,33]. I n acomparative study, Tossavainen et al. [21] showed that the kine-matic models had better estimation of the settling curves, set-tling velocities and the conductivities of the phase layers than therandom-walk model.Another evolutionmodel which is commonlyused to monitor the flow of a fluid in a pipeline is based on the con-vection diffusion equation [29–31]. Since our final objective is thereal-time monitoring of sedimentation, a model which is simple,computationally inexpensive and incorporates the dynamics of ac-

tual sedimentation phenomenon is needed. A concrete state evolu-tion model based upon the physical aspects of the settling processwould certainly improve the estimation performance of the algo-rithm for a wide range of scenarios. Almost all of the mathemat-ical models that describe layered settling are based on the Kynchtheory of sedimentation, commonly known as the solids flux the-ory for batch sedimentation [6]. It is essentially a kinematical the-ory of sedimentation based on the propagation of sedimentationwaves in the suspension and assumes that the settling velocity of the particles is a function of the concentration alone.

This paper presents a dynamic state estimation scheme toexamine the sedimentation process using two-dimensional elec-tricalimpedancetomography (2D EIT). Themain focus of this paperis to incorporate a priori information into the state evolution equa-

tion based on solids flux theory for batch sedimentation by consid-ering the relationship between the concentration and conductivity

of the constituent layers. The sedimentation model assumes three

layers, i.e., a top liquid layer, a middle slurry layer, and the bottom

compression layer. Since only batch sedimentation is considered,

the concentration of the middle layer is assumed to stay constant

during the course of sedimentation. Extended Kalman filter (EKF)is employed as the dynamic reconstruction algorithm to estimate

the settling curves, settling velocities andconductivities of the con-

stituent layers. Two different scenarios are considered to verify the

performance of the proposed model. The results demonstrate thatthe proposed method has been able to monitor the sedimentation

parameters quite effectively.

The rest of this paper is organized as follows: The three-layersedimentation modelling using solids flux theory is presented in

Section 2. Section 3 gives a brief overview of the EIT forward

problem. Section 4 presents the state estimation formulation of the

inverse solution to the sedimentation monitoring problem using

2D EIT. Section 5 evaluates the performance of the state estimationmodel using extended Kalman filter, and the conclusions are

presented in Section 6.

2. Sedimentation: the layered model

In this section, the model equations for the evolution of the

state parameters are developed. The model assumes three-layer

sedimentation, while a two-layer sedimentation model is derived

as a special case of the three-layer model.

2.1. The three-layers sedimentation model

In the three-layer sedimentation model, the fluids are assumed

to settle into the following layers: the top liquid layer, the middle

dilute slurrylayerand the bottom compression layer [15] as shownin Fig. 1.

The EIT setup in Fig.1 consists of 16 measurement electrodes at-

tached to the walls of the of the sedimentation tank. σ 1, X 1, σ 2, X 2

and σ 3, X 3 are conductivity and concentration of the top, middle,and bottom phase layers, respectively. h1 and h2 are heights of the

interfacial boundaries with respect to the base of the sedimenta-

tion tank.

Among the two interfacial boundaries between these layers,the interface between the liquid layer and the dilute slurry is

usually referred to as sludge blanket. Considering that there is no

influent or effluent flow, the middle layer, along with the sludge

blanket, descends continuously with constant concentration, and

the settling zone is continuously merged into the compressionlayer. Due to the transfer of flux between these two layers, the

compression layer goes through a simultaneousincreasein volume

and concentration. A certain fraction of the flux entering the

compression layer makes up for the difference in concentration

between the two layers, while the other fraction results inincreased concentration for the compression layer [17].



A. Rashid et al. / Flow Measurement and Instrumentation 21 (2010) 521–530 523

Fig. 2. Evolution of the interfacial boundaries and the concentrations of the

constituent layers.

In general, the concentration X 1 of theliquid layer is assumed tobe zero. The concentration X 2 of thedilute slurryis assumedto stayconstant during the course of sedimentation. However, the con-centration of the bottom layer shows a gradual increase towardsthe bottom. Therefore, an average concentration X 3 is calculatedto avoid an unlimited number of compression layers and a correc-tion factor 0 < f c ≤ 1 known as the flux distribution coefficient,is introduced to adapt the approach either to measurement or tothe analytical solution [17].

The two concentration steps between the three particle con-centrations propagate like wavefronts during the sedimentationprocess considering the principle of mass conservation. The prop-agation velocities v( X 1, X 2) and v( X 2, X 3) for the two interfaces arecalculated as follows

v( X 1, X 2) = −vH ( X 2) (1)

v( X 2, X 3) = f c X 2vH ( X 2)

X 3 − X 2(2)

where vH ( X ) is the settling velocity of the particles due to theirconcentration X and has been described by Vesilind [34] as

vH ( X ) = vH0e−mX (3)

where vH0is the initial settling velocity and m is an empirical

settling parameter. The empirical parameters vH0 and m have beenestimated by many researchers [35–38]. The expression developedby Daiggerand Roper [35] appears more frequently in the literature

vH ( X ) = 187e(−0.148−0.0021SVI) X (4)

where SVI is known as the sludge volume index, and is often usedto quantify the sludge settleability and compaction. SVI is definedas the volume in millilitres occupied by 1 g of a suspension after30 min of settling.

The most popular equation to describe the settling velocityof particles in solid-liquid homogeneous suspensions, usingonly two empirical parameters, is the so-called Richardson–Zakiequation [1,2,8,12]

vH

= vH0

(1 − εL) z (5)

where εL is the liquid fraction and 1 − εL = εs is the solid fractionin the mixture. z is an experimentally determined exponent, isconstant for a particle, and depends upon its Reynolds number Rt ,at vH0

.Let t be the unit time during which the top interface changes

its height from h1 to h′1 calculated as h1 = v( X 1, X 2)t , while

the interface between the bottom two layers changes its heightfrom h2 to h′

2 calculated as h2 = v( X 2, X 3)t . Similarly, theconcentrations of the three layers change by X ′1 − X 1, X ′2 − X 2 and

X ′3 − X 3, respectively. This is shown in Fig. 2. The evolution of thetwo phase boundary interfaces can be expressed as follows

h′1 = h1 − vH ( X 2)t (6)

h′2 = h2 + f c X 2

X 3 − X 2vH ( X 2)t . (7)

Fig. 3. A schematic description of the evolution of the interfacial boundaries and

the concentrations of the constituent layers in a two-layer sedimentation model.

The next step is to calculate the evolution of concentrationprofiles of the sedimentation layers. The concentrations X 1 and X 2stay constant, therefore, X ′1 − X 1 = 0 and X ′2 − X 2 = 0. Since thereis no influent or effluent flow, the total flux entering the bottomlayer should be equal to the total flux leaving the middle layer, i.e.,

X ′3h′2 − X 3h2 = X 2(h1 − h2) − X ′2(h′

1 − h′2). (8)

Inserting the values of h′1 and h′

2 from (6) and (7) into (8) andsetting X ′2 = X 2 we get

X ′3 = X 3h2 + X 2

f c X 2

X 3− X 2+ 1

vH t

h2 + f c X 2

X 3− X 2vH t

(9)

which is the evolution equation for the concentration of thecompression layer.

2.2. The two-layer sedimentation model

The three-layer model description holds before the two interfa-cial boundaries meet. At that point the three-layer sedimentationmodel transforms into a two-layer model: the top layer is assumedto be clear water and the bottom layer is the compression layerwhere the consolidation takes place. (Here we retain our parame-terization of the problem, assuming that the middle layer has dis-appeared.) Fig. 3 gives the schematic description of the two-layermodel.

The propagation velocity v( X 1, X 3) of the boundary interfacewill be calculated as

v( X 1, X 3) =f ( X 3) − f ( X 1)

X 3 − X 1(10)

where f ( X ) is the particle flux due to concentration X , calculatedas f ( X ) = X vH ( X ).

The concentration of X 1 is assumed to be zero, therefore, (10)becomes

v( X 1, X 3) = vH ( X 3). (11)

Now, the evolution of the phase boundary interface and theconcentration profiles can be expressed as follows

h′1 = h1 − vH ( X 3)t . (12)

Since themiddle layer has vanished and there is no exchange of flux between the liquid and the compression layers, the change inthe flux of the compression layer should be zero, i.e.,

X ′3h′1 − X 3h1 = 0. (13)

Inserting (12) into (13), we get

X ′3 =h1

h1 + h1

X 3 (14)

X ′3 =h1

h1 − vH ( X 3)t X 3 (15)

which is the concentration evolution model for the compressionlayer in the case of two-layer sedimentation.




3. Electrical impedance tomography: the forward solver

EIT consists of the forward and inverse problems. The forwardproblem calculates the boundary voltages from the assumedconductivity distribution. In the inverse problem, the internalelectrical conductivity distribution is reconstructed based on theimposed currents and the measured voltages at the electrodesplaced across the body of the domain. A schematic diagram of the

EIT measurement setup has been shown in Fig. 1. The physicalrelationship between the internal conductivity σ ( x, y) and theelectrical potential u ( x, y) on the object Ω ⊂ ℜ2 is governed by apartial differential equation with appropriate boundary conditions(the so-called complete electrode model which considers theshunting effects as well as the contact impedance between theelectrodes and the medium)

∇ · (σ ∇u) = 0, ( x, y) ⊂ Ω (16)

u + zℓσ ∂u

∂ν= U ℓ, ( x, y) ⊂ eℓ, ℓ = 1, 2, . . . , L (17)

∫ eℓ

σ ∂u

∂νdS = I ℓ, ( x, y) ⊂ eℓ, ℓ = 1, 2, . . . , L (18)

σ ∂u∂ν

= 0, ( x, y) ⊂ ∂Ω \ L

ℓ=1

eℓ (19)

where I ℓ is the electrical current injected into the object Ω throughthe ℓ-th electrode eℓ, L is thenumberof electrodes, ν is theoutwardnormal unit vector, zℓ is the contact impedance, and U ℓ is themeasured boundary potential. In order to ensure the existenceand uniqueness of the solution, following two constraints are alsoimposed

L−ℓ=1

I l = 0 and

L−ℓ=1

U ℓ = 0. (20)

The forward problem is solved using the finite element method(FEM) [25,28]. In FEM, the domain is divided into a number of

small triangular elements. The conductivity value inside each finiteelement is assumed to be constant. Let N be the number of nodesinthe finite element mesh. The finite element formulation gives thefollowing system of linear equations [25]

Ab = I (21)

where

A =

B C

CT D

, b =

αβ

and I =

0

I

. (22)

Here, α = (α1, α2, . . . , αN )T ∈ ℜN ×1 and β = (β1, β2, . . . ,

βL−1)T ∈ ℜ(L−1)×1. I = (I 1 − I 2, I 1 − I 3, . . . , I 1 − I L)T ∈ ℜ(L−1)×1

is the reduced current matrix and 0 = (0, . . . , 0)T ∈ ℜN ×1. Thesystem matrix A ∈ ℜ(N +L−1)×(N +L−1) is of the form

B (i, j) =

∫ Ω

σ ∇φi · ∇φ jdΩ +L−

ℓ=1

1

zℓ

∫ eℓ

φiφ jdS,

i, j = 1, 2, . . . , N (23)

C (i, j) = −1

z1

∫ e1

φidS +1

z j+1

∫ e j+1

φidS,

i = 1, 2, . . . , N , j = 1, 2, . . . , L − 1 (24)

C (i, j) =

|e1|

z1

i = j

|e1|

z1

+

e j+1

z j+1

i = j,

i, j = 1, 2, . . . , L − 1 (25)

whereφi

is the two-dimensional first-order basis function and e j

is the area of the electrode j.

4. Sedimentation monitoring: the inverse problem

This section will describe a state estimation formulation of the inverse solution to the sedimentation monitoring problem us-ing electrical impedance tomography (EIT). The state evolutionmodel is developed by the application of the Maxwell–Hewitt rela-tion [39], which establishes a relationship between the conductiv-ity and concentration of the solid particles in a solid-liquid mixtureto the sedimentation model presented in Section 2. This relation-ship was used to estimate the concentration distribution of a givensubstance in a fluid moving in a pipeline using EIT [40].

The sedimentation is parameterized using the locations of in-terfacial boundaries, their velocities and the concentration of itsconstituent layers. Using EIT measurements, EKF provides theestimation of interface locations, interface velocities, and con-ductivities of the sedimentation layers. The concentration of thesedimentation layers can then be determined from the respectiveconductivity estimates.

4.1. The state evolution model

When the only conducting phase in a solid-liquid mixture isthe liquid then the two-dimensional version of Maxwell–Hewittrelation [39] can be used to relate the liquid fraction εL to theconductivity of liquid σ L and that of the mixture σ

1 − εL =1 − σ /σ L

1 + σ /σ L. (26)

If X s is the concentration of the solid in the mixture then thesolid fraction εs = 1 − εL can be expressed in terms of X s, i.e.,εs = X s

ρs+ X s, where ρs is the density of the solid. Solving the above

relations for X s, we get

X s =ρs

2

σ L

σ − 1

. (27)

In the proposed sedimentation model the conductivity of liquidlayer is σ 1 and the conductivities of the two mixture layers areσ 2 and σ 3 respectively, as shown in Fig. 1. So, inserting σ 1 = σ Land replacing X s and σ , first with X 2 and σ 2 and then with X 3 andσ 3, respectively, results in the following relationships between theconcentration and conductivities of sedimentation layers

X 2 =ρs

2

σ 1

σ 2− 1

(28)

X 3 =ρs

2

σ 1

σ 3− 1

. (29)

Inserting the values of X 2 and X 3 from (28) and (29),respectively, into (7) and simplifying

h′2 = h2 + f c σ 3(σ 1 − σ 2)

σ 1(σ 2 − σ 3)vH ( X 2)t . (30)

Now we are ready to formulate our state estimation model tomonitor sedimentation using EIT. Let us use x to denote the setof state parameters required to be estimated. Within the discretetime state estimation framework, the parameter x is treated as astochastic nonlinear state estimation problem. The correspondingstate evolution and measurement equations for x will be definedas

xk+1 = f k( xk) + wk (31)

V k+1 = U k+1( xk+1) + vk+1 (32)

where the subscript k is the state index, f k is the non-linear state

transition model and U k is the non-linear observation model. wkand vk denote the process and measurement noise, respectively




and are assumed to be zero-mean Gaussian-distributed noise. Theparameter vector xk can be defined as

xk = (h1k, h2k

, vHk, σ 1k

, σ 2k, σ 3k

)T ∈ ℜM×1, for M = 6 (33)

where h1kand h2k

are the heights of the phase boundaryinterfaces, vHk

is the settling velocity, and σ 1k, σ 2k

and σ 3kare the

conductivities of the sedimentation layers, respectively, at state

index k. Using the Eqs. (6) and (30), the component-wise stateevolution equations for h1kand h2k

can be written as

h1k+1= h1k

− vHkt (34)

h2k+1= h2k

+ f cσ 3k

(σ 1k− σ 2k

)

σ 1k(σ 2k

− σ 3k)

vHkt . (35)

The random-walk evolution model is used for the rest of theparameters (vHk

, σ 1k, σ 2k

, σ 3k). Notice that since the concentration

of themiddle layer is constant, vHkis constant, hence the evolution

model for the top interface is essentially a kinematic model.Finally, as the two interface boundaries meet each other, the

three-layer model transforms into a two-layer model. From thatpoint onwards, it becomes a four-parameter estimation problem,i.e.,

xk = (h1k, vHk

, σ 1k, σ 3k

)T ∈ ℜM×1, for M = 4. (36)

Once again, the parameterization of the inverse problem isretained, only subtracting the parameters related to the middlelayer from the state vector.

4.2. Extended Kalman filter formulation

Recursive Kalman filter is a widely used estimation algorithmwhen the state evolution and observation equations are linear.Since the sedimentation state parameters evolve nonlinearly andthe observation equation is also nonlinear, suboptimal estimatescanbe obtained by using linearapproximations. Themost commonapproach for this purpose is the extended Kalman filtering in

which linearization is obtained with respect to the previous stateestimate. The linearized state evolution is obtained as

φk =∂ f k

∂ x

x= xk|k

(37)

such that we have Eq. (38) given in Box I.Similarly, the linearized observation model is obtained as

V k+1 = V k+1|k + J k+1

xk+1 − xk+1|k

+ vk+1

+ higher order terms (39)

where V k+1|k = U k+1( xk+1|k) is the previous measurement esti-

mate and the Jacobian J k+1 ∈ ℜL×M is defined as

J k+1 =

∂U k+1

∂ x x= xk+1|k

. (40)

If we define the pseudo-measurement yk+1 ∈ ℜL×1

yk+1 ≡ V k+1 − U k+1( xk+1|k) + J k+1 xk+1|k

= J k+1( xk+1|k) xk+1 + vk+1 (41)

where vk ∈ ℜL×1 is the pseudo-measurement noise

vk+1 = vk+1 + higher order terms (42)

then, the complete set of equation for the extended Kalman filteris as follows

Time Update:

xk+1|k = f k( xk|k) (43)

C k+1|k = φkC k|kφT k + Q . (44)

Fig.4. A schematicdiagramof thechangein theareaof theuppertriangularregion

of an FEM element Ωm, with respect to the change in the p-th interface location,

h p = h′ p − h p.

Measurement Update:

K k+1 = C k+1|k J T k+1[ J k+1C k+1|k J T

k+1 + R]−1 (45)

C k+1|k+1 = (I N − K k+1 J k+1)C k+1|k (46)

xk+1|k+1 = xk+1|k + K k+1[ yk+1 − J k+1 xk+1|k] (47)

where, C k|k ∈ ℜM×M is the error covariance matrix, Q ∈ ℜM×M and

R ∈ ℜL

×L

are process noise and measurement noise covariancesrespectively, K k ∈ ℜM×L is the Kalman gain and I N ∈ ℜM×M is an

identity matrix.

4.3. Calculation of Jacobian

The calculation of the Jacobian is paramount to the correct

application of EKF to solve this problem. The complete description

of the Jacobian formulation can be found in the Refs. [22,28]. This

section only presents a brief description of the Jacobian. In EIT,

the voltages are usually measured only at some of the electrodes

(specific to the applied current pattern). For P injected current

patterns, the voltages U ∈ ℜE×P at E measurement electrodes can

thus be obtained as [28]

U = RT I = MA−1 I (48)

where R = A−1MT ∈ ℜ(N +L−1)×E is a pseudo-resistance matrix and

M ∈ ℜE×(N +L−1) is an extended measurement matrix. The Jacobian

with respect to the i-th parameter of the state vector x can then be

calculated as follows

∂U

∂ xi

= −MA−1 ∂ A

∂ xi

A−1 I = −RT ∂ A

∂ xi

b = −RT 1

∂B

∂ xi

α (49)

where R1 = R(1 : N , :) ∈ ℜN ×E . In the proposed sedimentation

monitoring model, the Jacobian hasto be calculated with respect to

the phase interfaces and the conductivities. It means only ∂B

∂h p( p =

1, 2) and ∂ B

∂σ q(q = 1, 2, 3) has to be calculated

∂B

∂h p

=−

m|Ωm⊂supp(φiφ j)

(σ u − σ l)

|Ωm|

∂ ˜ Au

∂h p

∫ Ωm

∇φi.∇φ jdΩ (50)

∂B

∂σ q=

∫ supp(ϕiϕ j)∩ ˜ Aq

∇φi.∇φ jdΩ (51)

where supp(φiφ j) expresses the part of the domain Ω , with both

the basis functions φi and φ j to be non-zero. Near the phase

interfaces,the integrals of ∇φi·∇φ j are calculated as area-weighted

averages of the phase conductivities σ u and σ l of the upper and

lower parts, respectively, of the triangular elements Ωm split by

the p-th phase interface crossing through them.|∂ ˜ Au|∂h p

corresponds




∂ f k

∂ x=

1 0 −t 0 0 0

0 1 f c t σ 3(σ 1 − σ 2)

σ 1(σ 2 − σ 3) f c vH t

σ 3σ 2

σ 21 (σ 2 − σ 3) f c vH t

σ 3(σ 3 − σ 1)

σ 1(σ 2 − σ 3)2f c vH t

σ 2(σ 1 − σ 2)

σ 1(σ 2 − σ 3)2

0 0 1 0 0 00 0 0 1 0 00 0 0 0 1 0

0 0 0 0 0 1

(38)

Box I.

a

b

Fig. 5. Estimated interface heights and settling velocity for scenario 1 with 1%

zero-mean Gaussian-distributed noise. (a) Evolution of true and estimated mean

interface height. (b) Evolution of true and estimated mean settling velocity. The

error bars show the standard deviation s of the estimated parameters at each

measurement instance.

to the change in the area of the upper regions of the split triangleswith respect to a fractional change in the interface location h p =

h′ p − h p. Considering a simple scenario, as shown in Fig. 4,

|∂ ˜ Au|∂ h p

can

be calculated as

|∂ ˜ Au|

∂h p

= limh p→0

( xR − xL + xR + xR − xL − xL)

2h p

× h p = xR − xL (52)

since xR → 0 and xL → 0 as h p → 0.

For the implementation of the Jacobian with respect to the

conductivities, it is only required to compute the integral of the

basis functions of the triangular elements in the area of interest,

i.e., the elements lying between the interfaces, and the elements

through which the interfaces are crossing.

a

b

Fig. 6. Estimated conductivityand concentrationfor scenario1 with1% zero-mean

Gaussian-distributed noise. (a) Evolution of true and mean estimated conductivity

of the top, middle and bottom layers, respectively. (b) Evolution of true and mean

estimated concentrationof the bottom and middlelayers,shown as top and bottom

curves, respectively. The error bars show the standard deviation s of the estimated

parameters at each measurement instance.

5. Results and discussion

The performance of the state estimation model has been

verified by performing a set of numerical experiments. For the

numerical experiments a sedimentation tank with a 4 m of height

and 3 m of width was considered. The EIT setup consisted of a

total of 16 electrodes, with eight of them attached to both sides

of the walls of the sedimentation tank. In order to measure the

voltage across the electrodes, the so-called complete electrode

model with an effective contact impedance of the electrodes set

to 0.01 m2 was used [25]. The trigonometric current injection

protocol was used such that the current was applied to all the

16 electrodes simultaneously and the resulting voltages were

measured from all the electrodes. With 16 electrodes, eight cosine

and seven sine current patterns are possible. Out of all thepossible

trigonometric current patterns, only the first two modes of cosineand sine current patterns were used alternatively considering




a

b






that they are more sensitive compared to the other modes [28].This resulted in reasonable performance even with an incompleteset of current patterns. The time between two measurementswas two minutes. The settling curves, settling velocities andconductivities of the constituent layers were obtained by repeatingthe current injections and voltage measurements several timesduring sedimentation.

The mesh used to solve the forward problem consisted of 7472 triangular elements with 3865 nodes. The mesh used for theinverse problem was composed of 1868 triangular elements with999 nodes. The different mesh structure for the forward and theinverse solvers was used to avoid the so called inverse crime [41].

In EIT, the voltage measurements are often noisy in nature.Therefore, zero-mean Gaussian noise with standard deviationof 0.01% of the maximum computed voltage and zero-meanGaussian noise with 1% and 2% standard deviations, respectively,relative to the corresponding measured voltage was added to thesimulated voltage data. Similarly, zero-mean Gaussian noise with0.2% standard deviation relative to the simulated conductivitiesand the heights of the phase interfaces was added. The latter(process) noise was added to give random spatial variation to thesimulated parameters.

Two test scenarios were considered and EKF was used to verifythe performance of the proposed model. The first scenario focusedon the three-layer model in which the sedimentation parameterswere estimated before the two interfaces met each other. The

second scenario considered the case where the three-layer modeltransformed into the two-layer model.

a

b

Fig.8. Estimatedconductivity and concentrationfor scenario1 with2% zero-mean

Gaussian-distributed noise. (a) Evolution of true and mean estimated conductivity

of the top, middle and bottom layers, respectively. (b) Evolution of true and mean

estimatedconcentration of the bottom and middle layers, shown as topand bottom

curves, respectively. The error bars show the standard deviation s of the estimatedparameters at each measurement instance.

In order to verify the stability of the proposed algorithm MonteCarlo type simulations of n = 25 runs (each run with a differentnoise seed) was performed for the two scenarios. A quantitativeanalysis of the algorithm has been carried out by plotting the mean

x =∑n

i=1

xi

n

and standard deviation s =

1

n−1

∑n

i=1( xi − x)2

of the estimated parameters in the graphical format.

5.1. Test scenario 1

This test scenario was generated by assuming the initial con-centrations for the middle sedimentation and bottom compressionlayers to be X 2 = 400 g/l and X 3 = 800 g/l. The heights of thephase interfaces were initialized to h1 = 3.7 m and h2 = 0.5 m,

respectively. The sedimentation parameters, just discussed, wereevolved over a period of 60 min, using the evolution model pre-sented in Section 2. The true conductivity of the liquid layer wasassumed to be 0.5 mS/m. The true conductivities of the remainingtwo layers at each measurement step were generated by solving(28) and (29) for σ 2 and σ 3, respectively. The liquid is assumed tobe water while the solid is assumed to be wet sand with densityρs = 1900 g/l. The simulated scenario corresponds to the settlingof natural beach sediments at high concentrations [12].

In order to solve the inverse problem, an initial guess for all thestate parameters was required. The initial estimates for the stateparameters were

x0|0 = (3.8, 0.4, 0, σ best, σ best, 1.01σ best)T .

The best homogenous conductivity σ best was used asa commoninitial guess for the conductivities by estimating the best fitting




a

b






global conductivity value to the voltage measurements at stateindex k = 1. This estimate can be obtained by using the methodof least squares [21]. However, the initial conductivity of thebottom layer is chosen as 1.01σ best to avoid an undefined valuewhile using (32). Since no prior knowledge of the flux distributioncoefficient f c value was available, f c = 0.7 was arbitrarily selectedfor the inverse solver throughout the simulations. The state noisecovariance and the error covariance were initialized as

Q = diag[2 × 10−4, 2 × 10−4, 2 × 10−4, 8 × 10−4,

10−5, 3 × 10−4]

C 0|0 = diag[10−4, 10−4, 10−5, 10−4, 10−4, 10−4]

while the measurement noise covariance matrix was chosen asR = 5 × 10−1I L and R = 6 × 10−1I L for 1% and 2% measurementnoise cases respectively.

Figs. 5 and 6 show the estimated sedimentation parametersfor the first scenario with 1% zero-mean Gaussian-distributedmeasurement noise added relative to the true voltage data, whileFigs. 7 and 8 show the estimated results for this scenario with 2%zero-mean Gaussian-distributed measurement noise. The plottedresults show the mean and standard deviation of the estimatedparameters at each measurement instance. The mean is plottedwith a solid line whereas the standard deviation is plotted usingthe error bars.

The estimated results, both for 1% and 2% noise cases, showthat EKF is successful in estimating the state parameters with

reasonable accuracy. However, there is a transition period beforethe estimated parameters converge to their respective true values.

a

b

Fig. 10. Estimated conductivity and concentration for scenario 2 with 1% zero-

mean Gaussian-distributed noise. (a) Evolution of true and mean estimated

conductivityof thetop, middleand bottom layers, respectively.(b) Evolutionof true

and mean estimated concentration of the bottom and middle layers, shown as top

and bottom curves,respectively. Theerror barsshow thestandarddeviations ofthe

estimated parameters at each measurement instance.

The convergence time varies fordifferent parameters. Forexample,the conductivity of themiddlelayershowsvery quick convergence,whereas a stable estimate of the top and bottom layer conductivityis obtained shortly before half of the simulation time. This isbecause the best homogeneous value of the conductivity, whichis chosen as theinitial guess, is incidentally closer to the true valueof the middle layer’s conductivity and far from the true upperand bottom layer’s conductivity. This variation in the transitionperiod is quite predictable since EKF is very sensitive to the initialguess. The settling velocity estimate is poorer compared to otherestimated parameters. This is also expected as the random walkmodel is used to estimate the settling velocity and also Jacobianwith respect to the velocity is not available. The estimation curvesfor the phase interfaces and the concentration show reasonableconvergence as well.

The results for the 2% noise case show a relatively slowerconvergence for some of the sedimentation parameters. Thestandard deviation of all the estimated parameters is higher for the2% noise. This is quite expected since the higher EIT measurementnoiseoften leadsto deterioratedperformance of the reconstructionalgorithm. Also note that the performance of EKF is dependent onthe choice of coefficients of noise covariance matrices, C 0|0, Q andR, respectively. The values of the aforementioned coefficients havebeen chosen using a trial and error approach.

5.2. Test scenario 2

This test scenario was generated to verify the performanceof the method when the two phase boundaries meet each other

marking the transition of the three-layer sedimentation modelinto a two-layer model. The initial concentrations for the middle




a

b


zero-mean Gaussian-distributed noise. (a) Evolution of true and estimated meaninterface height. (b) Evolution of true and estimated mean settling velocity. The



sedimentation andbottomcompression layers were assumed to be X 2 = 250 g/l and X 3 = 1000 g/l, respectively. The initial heightsof the phase boundaries were chosen as h1 = 2.5 m and h2 =1 m, respectively. The total time for this scenario was also 60 min;however, the interfaces meet each other earlier as compared tothe previous scenario. The initial guess for the estimated stateparameters was chosen as follows

x0|0 = (2.6, 0.9, 0, σ best, σ best, 1.01σ best)T .

A common initial guess for the upper two conductivities wascalculated using the least squares method, with a slightly differentvalue for the third conductivity as already explained. The state

noise covariance and the error covariance were initialized asQ = diag[2 × 10−3, 10−3, 10−3, 5 × 10−4, 10−4, 4 × 10−4]

C 0|0 = diag[10−4, 10−4, 10−4, 10−4, 10−4, 10−4].

The measurement noise covariance matrix for this scenario waschosen as R = 6 × 10−1I L and R = 7 × 10−1I L for the simulationswith 1% and 2% measurement noise cases, respectively.

Figs. 9 and 10 show the reconstructed results for the secondscenario with 1% measurement noise while Figs. 11 and 12 showthe reconstructed results for this scenario with 2% measurementnoise.

The effect of 1% and 2% measurement noise on theperformanceof the algorithm, in terms of the deviation of the estimated param-eters from the respective true values as well as from the mean es-

timated values, has been explained in the previous scenario. Theestimated results for the interface, after the sedimentation layers

a

b

Fig. 12. Estimated conductivity and concentration for scenario 2 with 2% zero-

mean Gaussian-distributed noise. (a) Evolution of true and mean estimated

conductivityof thetop, middleand bottom layers, respectively. (b)Evolution of true

and mean estimated concentration of the bottom and middle layers, shown as top

and bottom curves,respectively. Theerror barsshow thestandarddeviations oftheestimated parameters at each measurement instance.

merge intothe compression layer, show good convergence besides

a transition period in the beginning. This transition period is evi-

dent in the form of a deviation of the estimated parameters from

their respective true values. The transition period is due to the fact

that the state parameter x transforms from a six parameter vec-

tor to a four parameter vector. The concentration curves show a

reasonable reconstruction. The fluctuations in the concentrationcurves are due to the deviations in the estimated conductivities.

Also, notice that as the three-layer model transforms into the two-

layer model, conductivity and concentration curves for the middle

layer cease to exist.The settling velocity also drops down suddenly,

as the sedimentation is dominated by the compression afterwards.

The later process is generally slower than the former.

In general, the small difference between the true and estimated

values of sedimentation parameters can be due to various reasons.

One reason is that the EKF uses the Taylor series expansion for the

linearization of measurement equation and the higher order terms

of the Taylor series are neglected while calculating the Jacobian.

Other possible reasons include the use of random-walk evolution

model for some of the parameters, measurement errors (included

as zero-mean Gaussian-distributed noise added to the simulated

measured voltages), modelling errors (included as process noisewhile generating the true data), and suboptimal tuning of the EKF

parameters.

6. Conclusions

A dynamic estimation scheme to analyze the sedimentationprocess using simulated EIT voltage measurements across the




electrodes attached to the side walls of a vessel is presented. A

state evolution model is formulated based on the solids flux theoryfor batch sedimentation. Extended Kalman filter is employed as aninverse solver and the state vector is composed of settling curves,settling velocities and conductivities of the constituent layers. The

robustness of the proposed algorithm has beenverified by carryingout Monte Carlo type simulations for two different scenariosby adding variable zero-mean Gaussian-distributed measurementnoise. The results show a promising performance of the method.

The proposed state evolution model has currently been for-mulated for a three layer batch sedimentation scenario whichassumes a constant concentration for the middle layer. Introduc-tion of model equations related to the continuous sedimentation

and adapting them to cater for an unlimited number of compres-sion layers can be considered for future work.

Acknowledgement

This work was supported by the research grant of Jeju NationalUniversity in 2009.

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An EKF Based Estimation Scheme for Sedimentation Processes

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