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Simple mass balance controllers for continuous sedimentation Fernando Betancourt a,∗ , Fernando Concha a , Daniel Sbárbaro b a b

Departamento de Ingeniería Metalúrgica, Facultad de Ingeniería, Universidad de Concepción, Concepción, Chile Departamento de Ingeniería Eléctrica, Facultad de Ingeniería, Universidad de Concepción, Concepción, Chile

a r t i c l e

i n f o

Article history: Received 21 August 2012 Received in revised form 12 March 2013 Accepted 18 March 2013 Available online 3 April 2013 Keywords: Continuous sedimentation Clariﬁer–thickener Steady state Control PI

a b s t r a c t The rational use of water in the mineral processing industry has become an important issue due to the geographical location of many plants. The increase of capacity in many copper concentrators has lead to an increased effort for recovering the maximum amount of water in the solid–liquid separation process. Thickeners work continuously to produce a concentrated underﬂow and a water overﬂow free from particulate matter. The behavior of many processes can be represented by a set of intensive and extensive variables. In this case, practice has shown that standard feedback control based on intensive variables has not been very easy to tune and effective in providing consistent operations. In many plants, thickeners operate with poor standards, with high dosages of ﬂocculants, overﬂows with high ﬁne particles contents and highly variable underﬂows. This work presents a novel nonlinear PI controller which is able to stabilize thickener operation using a simple control structure. An internationally accepted model and calibration using plant data is used to illustrate the design methodology and the level of performance attained by the controllers. The analysis of the results points out the improved performance by using extensive variables. In addition some guidelines concerning controllers tuning are also provided. © 2013 Elsevier Ltd. All rights reserved.

1. Introduction Thickener control represents some challenges due to their long time constants, poor instrumentation, nonlinaerities and disturbances. In order to deal with these issues, the use of expert control systems based on fuzzy rules have been suggested by several authors (Schoenbrun, Hales, & Bedell, 2002), to control simultaneously the level of interface and the underﬂow concentration variables, but no evaluation has been reported in the literature. On the other hand, structures based on standard PI controllers can be found in some concentrators, but these have never been in operation due to the lack of proper tuning and the inherent nonlinear behavior of the process. In the literature, it can also be found a control structure using PI controllers and a high selector to deal with the control of level and underﬂow concentration at the same time (Gupta & Yan, 2006). A comparison between these approaches has been carried out in Segovia, Concha, and Sbárbaro (2011). Since level and underﬂow concentration are intensive variables the closed loop expressions are complex and difﬁcult to analyze. Thus tuning these controllers has proved a difﬁcult task. The use of extensive variables, however, has been recognized to provide simple and effective control laws (Farschman, Viswanath, & Ydstie, 1998; Georgiou, Caston, & Georgakis, 1987). This fact

∗ Corresponding author. Tel.: +56 412204202. E-mail addresses: [email protected] (F. Betancourt), [email protected] (F. Concha), [email protected] (D. Sbárbaro). 0098-1354/$ – see front matter © 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.compchemeng.2013.03.014

has also been acknowledged in some previous work on thickener ˜ & Low, 1980), a concontrol. In the patent (Furness, Quinonez, trol structure based on inventories is proposed but no account on the control of the underﬂow concentration is given. Diehl (2008b) has suggested the use of a proportional control for controlling the inventory with time variant saturating output for satisfying both control objectives. Unfortunately the use of only a proportional controller require a feedforward term in order to get rid of the steady state error inherent to this structure. Recently predictive controller (Cortés & Cerda, 2010) has been also used to stabilize the operation of thickeners. Even though all these strategies have demonstrated to be useful none of them provide tuning guidelines neither formal stability analysis. Both issues are very important from a practical point of view. In this work, based on a macroscopic model, we propose an alternative method based on a nonlinear proportional plus integral (PI) controller which provides more ﬂexibility both in terms of tuning and stability analysis. To simplify the analysis, the control of the sedimentation process is only considered, without taking into account the effect of ﬂocculants. In order to test the proposed control structures spatially onedimensional models are used. These models go back the kinematic theory of Kynch (1952), which describes the batch settling of an ideal suspension of small, equal-sized rigid spheres in a viscous ﬂuid by the conservation law t + fbk ()z = 0,

F. Betancourt et al. / Computers and Chemical Engineering 54 (2013) 34–43

35

where is the solid volume fraction as a function of the depth z and time t. The characteristics of the solid are described by the Kynch batch ﬂux density function fbk . Numerous studies (see, Concha & Bürger, 2002, 2003 for historical overviews) and extensive use in industry made this theory a powerful tool to model and design settling equipments (Concha & Barrientos, 1993). The extension of the kinematic Kynch’s theory to clariﬁer–thickener units leads to a conservation law that depends discontinuosly on the spatial variable z. This discontinuity is caused by the split of the suspension inlet into upwards and downwards bulk ﬂows into the clariﬁcation and thickening zones (see Fig. 1). In real-world applications, many suspensions consist of small ﬂocs, i.e., they are ﬂoculated. These mixtures include, for example, inorganic slurries such as tailings from mineral processing, which are ﬂocculated artiﬁcially in order to enhance settling rates, as well as biological sludges in wastewater treatment. They form compressible sediments, which are characterized by curved isoconcentration lines in cylindrical settling columns, and can not be predicted by the kinematic theory of Kynch. A model considering this phenomenon is provided by the sedimentation–consolidation processes (Bustos, Concha, Bürger, & Tory, 1999). The governing equation is given by a quasi-linear strongly degenerate parabolic equation, which degenerates into a ﬁrst-order hyperbolic conservation law when the concentration is less than a critical concentration or also called gel point c , where c is the concentration at which ﬂocs star to touch each other. In Bürger, Karlsen, and Towers (2005a), an entension of the Kynch theory considering the compression phenomenon was developed by studying the well-posedness of weak entropy solutions for conservation laws with discontinuous coefﬁcients. Moreover, also in Bürger et al. (2005a) the authors performed a steady state analysis and gave a reliable numerical method for computing approximate solutions. This paper is organized as follows: Section 2 describes the phenomenological model. Section 3 describes the control problem and the proposed control strategies. Section 4 provides some simulations results for illustrating the main features of these strategies. Finally, in Section 5 some ﬁnal remarks are given and future research is outlined.

in four zones: the thickening zone (0 < z < zR ), the clariﬁcation zone (zL < z < 0), the underﬂow zone (z > zR ) and the overﬂow zone (z < zL ). The unit is continuosly fed at z = 0 with a volume feed rate QF of concentration F . The underﬂow is characterized by the volume rate QR and its concentration D and the overﬂow by the volume rate QL and its concentration E (see Fig. 1). Along this work, we will assume that E = 0.

2. Mathematical model

fbk () =

Troughout this section we follow Bürger et al. (2005a). The height z denotes a downwards increasing depth variable, and it is assumed that all ﬂow variables depend only on z and time t. We consider that the volume solid concentration remains constant at each cross sectional area. The clariﬁer–thickener is split

2.1. Balance equations Since we consider a unit with constant cross-sectional area S, then it is possible to replace the volume ﬂows Q(·) by velocity ﬂows q(·) : = Q(·)/S. According to Bürger et al. (2005a) the conservation of mass yields t + (q(t) + (1 − )vr )z = 0,

where q(t) is the bulk velocity of the suspension and vr is the relative velocity between solid and liquid phases. The kinematic sedimentation theory of Kynch (1952) is based on assuming that vr = vr (). He postulated that the relative velocity vr is a variable given by a Constitutive Relation reﬂecting the Principle of Objectivity since vr is a velocity difference. In general, the relative velocity vr is given in term of the Kynch ﬂux function fbk (). We use here

vr =

fbk () , (1 − )

which gives us t + (q(t) + fbk ())z = 0.

(2.2)

The function fbk () is usually assumed to be piecewise differentiable with fbk () = 0 for ≤ 0 or max ≤ , where max is the (0) > maximum solids concentration, fbk () > 0 for 0 < < max , fbk ( 0 and fbk max ) ≤ 0. In this paper the Richardson-Zaki expression Richardson and Zaki (1954) for the ﬂux density function is employed

v∞ (1 − )C for0 ≤ ≤ 1, 0

otherwise,

with max = 1, v∞ > 0 is the settling velocity of a single particle in an unbounded ﬂuid and C ≥ 1 is an experimental coefﬁcient. Flux functions with more than one inﬂexion point are treated in Bürger, Concha, Karlsen, and Narváez (2006).

feed volume rate: Q F solids volume fraction: φF

overflow volume rate: Q L solids volume fraction: φE

qL

qR

(2.1)

zL

overflow level

0

feed level

zR

discharge level

clarification zone

thickening zone

discharge volume rate: Q R solids volume fraction: φD Fig. 1. Clariﬁer–thickener unit.

z

36

F. Betancourt et al. / Computers and Chemical Engineering 54 (2013) 34–43

The sedimentation–consolidation theory outlined in Bustos et al. (1999), which includes sediment compressibility, leads to the expression

vr =

fbk () (1 − )

1−

e () ∂ g ∂z

,

(2.3)

where is the solid–ﬂuid density difference, g is the acceleration of gravity and e () is the effective solid stress function, which corresponds to the second constitutive expression that characterized the suspension. This function satisﬁes e () ≥ 0 and

e () :=

de () d

=0

for ≤ c ,

>0

for > c .

(2.4)

e () =

for ≤ c ,

0 ˛ exp(ˇ)

(2.5)

for > c ,

fbk ()e () ; g

A() :=

a()d,

(2.6)

0

t + (q(t) + fbk ())z = A()zz .

(2.7)

We remark that Eq. (2.7) is a ﬁrst-order hyperbolic conservation law for < c and second order parabolic for > c . Moreover, the location of the type-interface = c is not known a priori and is part of the solution. 2.2. The clariﬁer–thickener model In early attempts (Bustos & Concha, 1992; Bustos, Paiva, & Wendland, 1996) the sedimentation problem was studied as an initial-boundary value problem for (2.7), where the feed and discharge mechanisms were represented by boundary conditions. In order to avoid boundary conditions, Chancelier, Cohen De Lara, and Pacard (1994) and Diehl (1996) proposed to deﬁne the bulk velocities qR and qL by q(z, t) =

qR (t)

forz > 0,

qL (t)

forz < 0.

(2.8)

In the clariﬁer–thickener model (Bürger et al., 2005a), (2.7) is used with q(t) = qR (t) for z > 0 and q(t) = qL (t) for z < 0. Furthermore it is assumed that in the overfow and underﬂow zone, the solid material is transported with the same velocity as the liquid. The feed is incorporated by adding a singular source term to right hand side of (2.7). Then, the problem can be written as

∂ ∂ ∂ A() 1 (z) , z ∈ R, t > 0 + f ((z), ) = ∂t ∂z ∂z ∂z

(2.9)

(z, 0) = 0 (z),

(2.10)

where f ((z), ) := 1 (z)fbk () + 2 (z)( − F ),

:= (1 , 2 ),

(2.11)

with

1 (z) :=

1 for zL < z < zR , 0

otherwise.

;

2 (z) :=

t > 0,

m(0) = m0 ≥ 0,

(2.13) (2.14)

zR

m(t) =

(t, z)dz. zL

We remark that along this work it is assumed E = 0. 2.4. Steady states

the following governing equation is obtained

For controlling the unit, the total mass of solids m will be used as an internal controlled variable. Let m0 be the mass of the clariﬁer–thickener unit at t = 0. From the global mass balance we have

and

where ˛ and ˇ can be found through experiments (Garrido, Bürger, & Concha, 2000). Replacing (2.3) into (2.2) and deﬁning a() :=

2.3. The macroscopic model

dm(t) = sQ F (t)F (t) − sQ R (t)D (t), dt

Here we use the empirical formula

For the model with variable cross sectional area we refer to Bürger, Damasceno, and Karlsen (2004). The well-posedness of (2.9)–(2.12) is not a trivial task. Existence and uniqueness was proved in Bürger et al. (2005a) in the sense of weak entropy solutions by developing a theory on strongly degenerate parabolic equations with discontinuous functions.

qL

for z < 0,

qR

for z > 0. (2.12)

The steady states proﬁles, = (z), are deﬁned by the ordinary differential equation obtained from (2.9) by setting to zero the time derivative, considering the jump and entropy jump conditions at the discontinuities of the solution and the discontinuities of the ﬂux at z = zL , z = 0 and z = zR (Bürger & Narváez, 2007). Depending on the sediment level height attained by integration we ﬁnd in practice (plants) two modes of operation: conventional and of high capacity. The conventional mode of operation is characterized by a zero overﬂow concentration and by a sediment entirely located below the feed level. In this mode of operation, there are three distinguishable regions. The compression region, which is located between the hindered settling zone and the discharge level, and is characterized by a concentration proﬁle that joins continuosly the value D with the critical value c at the sediment level z = zc , which separates the compression region from the hindered settling region. In the hindered settling region, the concentration assumes a constant value called, conjugate concentration l < c (see, Bürger & Narváez, 2007 for a deﬁnition). Finally, the clear liquid region comprises the clariﬁcation and overﬂow zones (see Fig. 2 left). In the high rate mode of operation, unlike the conventional mode, the sediment level is supposed to be located above the feed level. In this mode all solids are located within the compression zone and there is no hindered settling zone above. The concentration proﬁles decrease from D at z = zR to the critical value c at the interface z = zc , which divides the compression region from the clear liquid region and marks a jump from c to 0 (see Fig. 2 right). In Bürger et al. (2005a), it was showed that steady states can be uniquely determined by imposing the values of F , D , E and qF . However, there is no rule that permit to know a priori if the tuple (F , D , E , qF ) can yield to a reachable steady state. Then, a numerical computation must be used to determine if (F , D , E , qF ) produce a real steady state or not. This can be performed by integrating the time-independent version of (2.9) (for example by a Runge-Kutta method) taking into account the restrictions given by the Rankine-Hugoniot condition, entropy jump condition and the entropy inequality (Bürger & Narváez, 2007). Fig. 3 shows typical concentration proﬁles for a CT unit in steady state. These restrictions made it possible to calculate the physical relevant solution, since solutions of (2.9) are not in general unique. Then, also the solid mass accumulated in the unit m is unique in

F. Betancourt et al. / Computers and Chemical Engineering 54 (2013) 34–43

overflow volume rate: QL = QR − QF ≤ 0

feed volume rate: Q F ≥ 0 solids volume fraction: φF

φF , Q F ≥ 0 QL ≤ 0

overflow zone zL

overflow level φ=0

clear liquid region (φ = 0 )

clarification zone 0

sediment level (z = z c )

z = zc

feed level

hindered settling region (0 < φ ≤ φc )

φc < φ ≤ φD

thickening zone

compression region (φc < φ ≤ φD )

zR discharge volume rate: Q R ≥ 0 solids volume fraction: φD

37

discharge level

discharge zone φD , Q R ≥ 0 z

Fig. 2. Clariﬁer–thickener unit operated in conventional mode (left) and in high capacity mode (right).

steady state for a given tuple (F , D , E , qF ). This is a big difference in comparison with the model without compression, where to get uniqueness in the total mass the height of sediment must be impose Diehl (2008b). This behavior can be explained by the governing equation. The addition of the compression, which is reﬂected in the parabolic right hand side of (2.9), gives “stability” to the concentration proﬁle since, in the compression region, it must be a monotone continuous function of z in the steady state. In the case without compression, only discrete concentration values can be found at steady state in the unit Diehl (2001). 3. Control objective and strategies Since its invention, the clariﬁer–thickener has been used to separate the solids from liquid in a suspension. However the purpose of the unit can vary depending on in what industrial process is used. Following Diehl (2008b) (see also Diehl, 2001, 2005, 2006, 2008a), the clariﬁer–thickener must • produce a low (preferably zero) overﬂow concentration; • produce a discharge concentration over a minimun value and below a maximum value; • be robust to variations in the feed variables. The physical input variables are the feed concentration F and the feed volume ﬂow QF . In practice, the manipulated variable is the discharge ﬂux QR , which can be modiﬁed through the use of a pump with a Variable Frecuency Drive (VFD). A design requirement for the unit is to produce a discharge concentration below a maximum value * , which is given by hydraulic transport conditions. Nevertheless, it is also important to produce a discharge concentration D close to this value in order to reduce the effort in the next stage of the process, e.g., ﬁltration and drying. Then, a

natural controlled variable in this framework should be the discharge concentration D . On the other hand, it is also important to keep under control the total mass accumulated in the clariﬁer–thickener due to inventory strategies. A “desired” (optimal) state of operation can be deﬁned from the conditions of the above description and the characteristics of the steady states. We note that the next deﬁnition does not distinguish between conventional or high rate mode of operation. Deﬁnition. A clariﬁer–thickener unit operates in optimal conditions if: (1) it produces a zero overﬂow concentration (2) it produces a discharge concentration D below a maximum value * (3) there is a jump concentration at the concentration value c Remark. Along this work we assume that the unit is initially operated in optimal conditions and since it is not possible to determine a priori the feasibility of a steady state, it is considered that in any case for a given pair (F , qF ), the tuple (F , D < * , E = 0, qF ) gives a reachable steady state. It can be proved that it is true for a region in the space of variables (Diehl, 2012). Following the ideas of Diehl (2008b), three Nonlinear Proportional Integral regulators for the clariﬁer–thickener unit are proposed. 3.1. PI strategy In this case the objetive is to keep under control the solid mass in the unit assuming that the discharge concentration is known. Let QF , F be the feed variables. Let msp be the desired mass, where the superindex sp means set point. The following controller is introduced (see Fig. 4) QR (t) = KP

m(t) − msp s D (t)

t

+ KI

(m( ) − msp )d 0

.

s D (t)

Q F (t)

msp

±

PI

÷

Q R (t)

φF (t)

CT φD (t)

Fig. 3. Steady states concentration proﬁles for a CT with d = 100m, zR = 1.5m, zL = −0.5m, fbk = 0.0001(1 − )5 m/s, e () = 5.35 exp(17.9) Pa, = 1650 kg/m3 and F = 0.155.

(3.1)

Fig. 4. Block diagram for the control system (3.1).

m(t)

38

F. Betancourt et al. / Computers and Chemical Engineering 54 (2013) 34–43

Then, the following controller is introduced

QR (t) =

m(t) − msp (t)

sp QR (t)+KP

s D (t)

t 0

+KI

(m( ) − msp ( ))d

s D (t)

,

(3.8) Replacing (3.8) in (2.13) yields dm(t) dt

Fig. 5. Block diagram for the control system (3.6).

sp

m( )d

(m( ) − m ( ))d

,

t > 0.

0

t

t

−KI

The closed loop equation is (see (2.13)) dm(t) = QF (t)s F (t) − KP m(t) − KI dt

sp

= QF (t)s F (t) − QR (t)s D (t) − KP (m(t) − msp (t))

sp

,

t > 0, (3.2)

0

where the deviation variable m(t) := m(t) − msp is deﬁned. Eq. (3.2) can be written in the system form, x (t) = Ax(t) + bQF (t)F (t),

sp

The solid balance in steady state QF (t)F (t) = QR (t)D , gives us

dm(t) sp sp = QR (t)s (D − D (t)) − KP m(t) − msp (t) dt

t

(m( ) − msp ( ))d

− KI

(3.3)

,

t > 0.

(3.9)

0

with x(t) :=

⎛

⎞

m(t)

⎝

t

m( )d

⎠ , A :=

−KP

−KI

1

0

Eq. (3.9) can be written in the system form,

, b :=

1 0

.

(3.4)

1,2

x (t) = Ax(t) + c(t)D (t) + b(t),

m(t) =

0

If KP and KI are chosen so that the eigenvalues of matrix A are in the left hand side of the complex plane, then if QF and F remain constants it is easy to demonstrate that m(t) → 0 as t→ ∞. In addition, the control parameters can be used to shape the transient of the closed loop system. The eigenvalues depend on the values of KP and KI as follows KP =− ± 2

KP2 − 4KI

(3.5)

2

and the natural frequency will be just ωn =

factor = KP /(2

where

(3.11)

⎞

m(t)

⎝

x(t) :=

t

(m( ) − msp ( ))d

⎠,

(3.12)

0

A :=

−KP

−KI

1

0

KI and the damping

KI ).

0 x(t)

1

⎛

(3.10)

c(t) :=

sp

−QR (t)s

, b(t) :=

.

sp

sp

KP msp (t) + QR (t)D (t)s −msp (t)

,

(3.13)

0 3.2. PI strategy with disturbance feedforward For controlling the solid mass in the unit and knowing F , QF and D , alternatively, the following controller can be introduced QF (t)F (t) QR (t) = + KP D (t)

m(t) − msp s D (t)

t + KI

0

(m( ) − msp )d s D (t)

.

(3.6) With the same arguments as before, it is easy to check that the expression for m and its primitive is given by x(t) = exp(At)x(0).

(3.7)

The convergence of this controller is direct from the sign of the real part of the eigenvalues of A. A block diagram of control system (3.6) is shown in Fig. 5. 3.3. PI strategy with reference feedforward In this controller, we want to hold the discharge concentration D manipulating the volumetric discharge QR through a mass sp sp sp controller. Given QF , F and D , we deﬁne QR (t) as QR (t) = sp QF (t)F (t)/D . Keeping this in mind, it is possible to obtain a unique steady state if it is exists, and we assume that is the case. Integratsp ing the steady state corresponding to the tuple (QF , F , D ) we ﬁnd sp sp sp the total mass m (t) = m (QF (t), F (t), D ).

System (3.3) can be written in system form as shown in Fig. 6. In order to address the convergence to the desired result, some knowledge about the relation between D (t) and m(t) is needed. If the relationship satisﬁes the following inequality D (t) L2 C m(t) L2 ,

(3.14)

then it is possible to demonstrate the stability and convergence of the closed loop provided that the gain of the composed system satisﬁes the small gain theorem (Desoer & Lin, 1985). Thus, it would be possible to ﬁnd a small controller gains that render the closed loop stable, of course at expenses of slower transient responses. In steady state (see Fig. 7), we can see that there exists a static gain satisfying the condition (2.12). 4. Numerical examples For the simulations we use the numerical scheme proposed in Bürger, Karlsen, and Towers (2005b). For the sake of completeness a description is given. Let z be the mesh parameter of the spatial variable z and deﬁne zj : = jz, and discretize the vector by j+1/2 : = (zj+1/2 ) and the initial datum by Uj0 := u0 (zj+1/2 ) with zj+1/2 = zj + z/2. For n > 0 the numerical approximation is deﬁned through the formula Ujn+1 =

n , U n ) − h( n n Ujn − (h( j+1/2 , Uj+1 j−1/2 , Uj , Uj−1 )) j

n n (A(Uj+1 ) − A(Ujn )) − 1,j−1/2 (A(Ujn ) − A(Uj−1 ))) + ( z 1,j+1/2

F. Betancourt et al. / Computers and Chemical Engineering 54 (2013) 34–43

39

Q F (t)

φF (t)

Q sp R (t) φsp D

msp (t)

SS

±

÷

PI

+

Q R (t)

m(t)

CT φD (t)

Q F (t)

φF (t) Fig. 6. Block diagram for the control system (3.8).

considered. The solid is determined by the constitutive relations (Becker, 1982)

fbk () e ()

=

0.0006025(1 − )12.59 0

otherwise, for ≤ 0.2,

5.35 exp(17.9)

for > 0.2.

0

=

for0 ≤ ≤ 1,

and its density is 2650 [kg/m3 ]. The liquid considered here is water. Along this section it is assumed that the unit is initially operated in optimal conditions and in steady state for the input variables QF = 700[t/h], F = 0.15, and D = 0.28 which gives us an initial mass of solids m0 of 886.6[t]. Fig. 7. Total solid mass acummulated in the unit m as a function of the discharge concentration D in steady state computed numerically.

where : = t/z and

h(, u, v) :=

1 f (, u) + f (, v) − 2

v

|fu (, )|d u

is the Engquist–Osher numerical ﬂux. For the simulations it is considered a discretization with z = 0.05 and t = 2000z2 . We notice that this choice respect the CFL stability condition for the numerical scheme and then, the convergence to the entropy weak solution is guaranteed (Bürger et al., 2005a). For all the numerical examples a clariﬁer–thickener of 48[m] of diameter, 4[m] in height with zR = 3[m] and zL = −1[m] will be

4.1. Example 1 The performance of controller (3.1) is illustrated in Figs. 8 and 9 by simulating different scenarios. In both examples the controller is connected at t = 0+ with the unit initially in steady state. Since the controller does not consider the feed variables, it takes a long time to stabilize the unit because it must “accumulate” error to get an acceptable corrected action. In Fig. 8, at t = 5 ×105 s a step change in the set point of mass is introduced. The new set point is msp = 1.2m0 . We see that the controller is able to get the expected result in 105 s. In Fig. 9, the unit is affected by a step change in the value of F from 0.15 to 0.17. The controller is able to compensate the effect of the disturbance. The value of KI is 2 × 10−9 in the two cases. The selection of the gains KP and KI was done considering (3.3) (to obtain

6

x 10

m[kg]

2

KP=5.34 E−5 K =4 E−5 P

1

K =2.67 E−5 P

0

1

2

3

4

5

6

7

8

9

−4

10 5

x 10

t[s]

q R [m/s]

x 10

K =5.34 E−5 P

1

K =4 E−5 P

K =2.67 E−5 P

0 0

1

2

3

4

5

6

7

8

9

5

t[s]

φD

10 x 10

0.3

K =5.34 E−5

0.2

KP=4 E−5

0.1 0

P

KP=2.67 E−5 1

2

3

4

5

6

7

8

9

t[s] Fig. 8. Example 1: Dynamic behavior of the clariﬁer–thickener under a step change in the set point of mass for controller (3.1).

10 5

x 10

40

F. Betancourt et al. / Computers and Chemical Engineering 54 (2013) 34–43 6

x 10 m[kg]

2

K =5.34 E−5 P

K =4 E−5 P

1

K =2.67 E−5 P

0

1

2

3

4

5 t[s]

6

7

8

9

10 5

x 10

−4

K =5.34 E−5 P

1

K =4 E−5 P

R

q [m/s]

x 10

KP=2.67 E−5 0 0

1

2

3

4

5 t[s]

6

7

8

9

5

0.3

KP=5.34 E−5

0.2

KP=4 E−5

D

φ

10 x 10

K =2.67 E−5 P

0.1 0

1

2

3

4

5 t[s]

6

7

8

9

10 5

x 10

Fig. 9. Example 1: Dynamic behavior of the clariﬁer–thickener under a step change in the feed concentration F for controller (3.1).

0.3 0.25 0.2 φ

complex eigenvalues) and from trial and error to keep controlled the overshoot. The same gains will be used in Examples 4.2 and 4.3. For this example we also include the concentration proﬁle in a 3-D plot (Fig. 10) to illustrate the complete behavior of the unit under a step change in the feed concentration with KP = 4 ×10−5 .

0.15 0.1

4.2. Example 2

0.05

We simulate the same scenarios as in Example 4.1 but with controller (3.6). The results are shown in Figs. 11 and 12. This controller reacts in a better way than controller (3.1) because of the feedforward term. Accordingly to Eq. (3.3) one can expect some classical behavior of the system, however the mass in the unit is calculated numerically integrating (2.9), then deviations from these values are expected. In Fig. 13, we compare the performance of controller (3.6) and (3.1). At t = 0+ the unit is in steady state and both controllers are

0 10 8 6 x 10

4

5

2 0

t[s]

−1

2

1

0 z[m]

Fig. 10. Example 1: Concentration proﬁles of the clariﬁer–thickener under a step change in the feed concentration for controller (3.1).

6

m[kg]

x 10

K =5.34 E−5

1.1

P

K =4 E−5

1

P

K =2.67 E−5 P

0.9 0

1

2

3

4

5 t[s]

6

7

8

9

10 5

x 10

−5

K =5.34 E−5 P

5.5

K =4 E−5 P

R

q [m/s]

x 10

K =2.67 E−5 5 0

P

1

2

3

4

5 t[s]

6

7

8

9

10 5

x 10

0.3 K =5.34 E−5 K =4 E−5

0.28

KP=2.67 E−5

P

φ

D

P

0.29

0

1

2

3

4

3

5 t[s]

6

7

8

9

Fig. 11. Example 2: Dynamic behavior of the clariﬁer–thickener under a step change in the set point of mass for controller (3.6).

10 5

x 10

F. Betancourt et al. / Computers and Chemical Engineering 54 (2013) 34–43

41

5

m[kg]

8.9

x 10

KP=5.34 E−5 K =4 E−5

8.85

P

KP=2.67 E−5 8.8 0

1

2

3

4

5 t[s]

6

7

8

9

5

−5

qR [m/s]

7

x 10

K =5.34 E−5 P

6.5

KP=4 E−5

6

KP=2.67 E−5

0

1

2

3

4

5 t[s]

6

7

8

9

10 5

x 10 KP=5.34 E−5

0.28

K =4 E−5

D

φ

10 x 10

P

0.27

KP=2.67 E−5

0.26 0

1

2

3

4

5 t[s]

6

7

8

9

10 5

x 10

Fig. 12. Example 2: Dynamic behavior of the clariﬁer–thickener under a step change in the feed concentration F for controller (3.6).

2

K =4 E−5; K =2 E−9

x 106

P

I

Controller 3.1 Controller 3.6

1.8

connected and at t = 105 a step change in the set point of mass is introduced. It can be seen that after the stabilization of controller (3.1), both follow the same behavior.

m[kg]

1.6 1.4

4.3. Example 3

1.2 1 0.8 0.6

0

1

2

3

4

5

6

7

8

t[s]

9

10 x 10 5

Fig. 13. Example 2: Dynamic behavior of the clariﬁer–thickener under a step change in the set point of mass for controller (3.6) and (3.1).

In this example the behavior of controller (3.8) is analyzed under a step change in the disturbance F and a step change in the set point of discharge concentration D . In Fig. 15 an step change in sp sp the discharge concentration from D = 0.28 to D = 0.3 is studied. Step change in feed concentration F is shown in Fig. 14 keeping sp constant D . The values of KI and KP are the same as in previous examples. From these simulations can be clearly seen that the proposed controller can cope with disturbances quite effectively and changes in the set point of discharge concentration. It must be noted that this controller is able to keep under control the discharge

6

m[kg]

x 10 1.2

KP=5.34 E−5

1

KP=2.67 E−5

K =4 E−5 P

0

1

2

3

4

5 t[s]

6

7

8

9

5

−5

qR [m/s]

6

x 10

K =5.34 E−5 P

5.5

KP=4 E−5

5 0

KP=2.67 E−5 1

2

3

4

5 t[s]

6

7

8

10 5

KP=5.34 E−5

D

K =4 E−5 P

0.29

KP=2.67 E−5

0.28 0

9

x 10

0.3 φ

10 x 10

1

2

3

4

5 t[s]

6

7

8

9

10 5

x 10

Fig. 14. Example 3: Dynamic behavior of the clariﬁer–thickener under a step change in the set point of discharge concentration D for controller (3.8).

42

F. Betancourt et al. / Computers and Chemical Engineering 54 (2013) 34–43 6

x 10

K =5.34 E−5

m[kg]

1.1

P

K =4 E−5

1

P

K =2.67 E−5 P

0.9 0

1

2

3

4

5 t[s]

6

7

8

9

10 5

x 10

−5

R

q [m/s]

x 10 6.6 6.4 6.2 6 5.8 5.6 0

K =5.34 E−5 P

K =4 E−5 P

K =2.67 E−5 P

1

2

3

4

5 t[s]

6

7

8

9

10 5

x 10 K =5.34 E−5

φD

P

KP=4 E−5

0.28

K =2.67 E−5 P

0.275 0

1

2

3

4

5 t[s]

6

7

8

9

10 5

x 10

Fig. 15. Example 3: Dynamic behavior of the clariﬁer–thickener under a step change in the feed concentration F for controller (3.8).

concentration, nevertheless the mass in the unit changes with the perturbations and the corresponding steady state. 5. Conclusions The design of controllers for thickener based on a macroscopic model, representing a mass balance, leads to simple control laws compared with the ones that may be designed using the full microscopic model. The control laws presented in this work are based on the idea of linearizing the closed loop dynamic, and they can easily be tuned to satisfy some transient speciﬁcations. In addition, the stability of the closed loop system can be analyzed using standard tools. Three different controllers have been proposed with different level of input information. Simulation examples have highlighted these characteristics and demonstrated the effectiveness of the proposed approach illustrating the relation between information and performance. The main shortcomings of these controllers are the lack of dynamic control over some internal variables such as the sediment level, the absence of saturating bounds for the discharge volumetric ﬂow and the assumption of constant properties of the solid. Future work considers the analysis of the effect of ﬂocculants in the performance of these controllers as well as the control of internal variables as the sediment level. The saturating bounds over the control variable (QR ), can be easily taked into account, but as the controller has integral action some anti-windup measure must be implemented. Moreover, the calculation of the lower bound is a difﬁcult task since the calculation of steady states must be done numerically and can not be determined a priori from the input variables and constitutive relations. Acknowledgments The authors acknowledge prof. Raimund Bürger for reading and commenting on the manuscript. FB acknowledges support by Conicyt Programm “Inserción de Capital Humano en la Academia” 791100041. FC and DS acknowledge support by Proyecto Innova 08 CM01-17, AMIRA project P-996 and Conicyt. References Bürger, R., & Narváez, A. (2007). Steady-state, control, and capacity calculations for ﬂocculated suspensions in clariﬁer–thickeners. International Journal of Mineral Processing, 84, 274–298.

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F. Betancourt et al. / Computers and Chemical Engineering 54 (2013) 34–43 ˜ Furness, K. M., Quinonez, M., & Low, S. T. (1980). Thickener control system. US Patent 4226714, October 7. Garrido, P., Bürger, R., & Concha, F. (2000). Settling velocities of particulate systems: 11. Comparison of the phenomenological sedimentation–consolidation model with published experimental results. International Journal of Mineral Processing, 60, 213–227. Georgiou, A., Caston, L., & Georgakis, Ch. (1987). On the dynamic properties of the extensive variable control structures. Chemical Engineering Communications, 60, 119–144. Gupta, A., & Yan, D. (2006). Mineral processing design and operation: An introduction. Elsevier.

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