[Balaji2010]Passivity Based Control of Reaction Diffusion Systems

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Passivity based control of reaction diffusion systems: Application to the

vapor recovery reactor in carbothermic aluminum production$

S. Balaji, Vianey Garcia-Osorio, B. Erik Ydstie

Department of Chemical Engineering, Carnegie Mellon University, 5000 Forbes Ave, Pittsburgh, PA 15213, USA

a r t i c l e i n f o

Article history:

Received 20 January 2010

Received in revised form18 May 2010

Accepted 20 May 2010Available online 1 June 2010

Keywords:

Aluminum

Shrinking core model

Gassolid non-catalytic reaction

Moving bed

Inventory control

Passivity

a b s t r a c t

We develop a passivity based controller for a reaction diffusion system. The model used in the

simulation study describes the vapor recovery reactor used in carbothermic aluminum reduction. It

takes into account, non-catalytic gassolid reactions, the moving bed of solid particles and theshrinkage of the unreacted particle core. The reaction diffusion system is solved using the finite element

method. We use passivity based control to adjust the carbon feed and the heat input to achieve the

required conversion and maintain the temperature along the reactor. The efficiency of the reactor is

determined by calculating the extent of vapor recovery and the conversion of carbon particles. The

sensitivity of different parameters such as solid flow rate and column height based on the reactor

performance is also determined. We show that the control scheme based on the inventory balances

performs well under various operating conditions.

& 2010 Elsevier Ltd. All rights reserved.

1. Introduction

The conventional approach for the control of distributed

parameter systems discretizes the partial differential equation

into a large system of ordinary differential equations

through finite difference or finite element method (Ray, 1978;

Christofides, 2001a). Although controllers now can be designed

based on the classical theories like pole placement, linear

quadratic or predictive control, erroneous conclusions may result

concerning the stability, controllability and observability of the

system (Aling et al., 1997; Balas, 1986; Christofides, 2001b). This

is due to the fact that a highly reduced model cannot represent a

DPS accurately. On the other hand, a high order model is

computationally expensive making it unsuitable for real time

controller implementation.

High dimensionality can be avoided using the concepts ofsingular perturbation and inertial manifolds. These methods rely

on time-scale separation and temporal averaging. They can

be applied to systems where the fast dynamics dissipate

(Christofides, 2001b; Brown et al., 1990; Smoller, 1996). Classical

control methods like the Lyapunov method can now be applied

(Christofides and Daoutidis, 1998; Christofides, 1998; El-Farra

et al., 2003). An extension of this method to predictive control can

be seen in Kazantzis and Kravaris (1999). Sliding mode control

has also been proposed to control the DPS systems (Hanczycand Palazoglu, 1995; Sira-Ramirez, 1989). Other approaches

of controlling nonlinear PDE systems include control using

symmetry group representation (Godasi et al., 2002), generalized

invariants (Palazoglu and Karakas, 2000).

In this study, we focus on the control of distributed parameter

systems using passivity based control based on feedback from

spatial rather than temporal averages. In particular, we use

feedback from inventories. The passivity theory can be used as a

guideline to design a control system for large scale, distributed

parameter systems (Desoer and Vidyasagar, 1975). Willems

(1972a,b) developed a systems perspective for passivity and

dissipativity and linked the concept to state space representations.

Byrnes et al. (1991)showed that passivity and Lyapunov stability

is equivalent for a class of feedback systems using geometricmethods. Van der Schaft (1996) developed control methods

linking passivity and L2 stability, whileKrstic et al. (1995)linked

passivity and nonlinear adaptive control.Ydstie and Alonso (1997)

and Alonso and Ydstie (2001) advanced the idea of combining

thermodynamics and passivity and showed that passivity could be

motivated using a storage function related to the available work

and Gibbs tangent plane criteria for phase stability.

The inventory control concept was first formulated and tested

byFarschman et al. (1998). According to this strategy, any process

system can be represented based on inventories. Based on the

balance equation of the inventories, simple control schemes can

be implemented for stable operation of the system. An inventory

Contents lists available atScienceDirect

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

Chemical Engineering Science

0009-2509/$- see front matter& 2010 Elsevier Ltd. All rights reserved.

doi:10.1016/j.ces.2010.05.037

$This work was supported by ALCOA Inc. Corresponding author. Tel.: + 1 412268 2235; fax: + 1 412268 7139.

E-mail address: [email protected] (B. Erik Ydstie).

Chemical Engineering Science 65 (2010) 47924802
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is an extensive measure of the thermodynamic state of the

system. Some examples of system inventories are total internal

energy (U), total volume (V), total mass (Mi) of speciesi present in

the system. A system is said to be passive, if the inventories

converge to their setpoints and the state variables of the system

converge to their stationary passive state.

Inventory based calculations ignore or overlook the detailed

structure of the dynamics. However, ultimately we obtain a

reduced order model by averaging over space rather than time,resulting in a coarse grained view of the system dynamics. Thus,

the resulting set of equations to be solved will be the model

equations in addition with the macroscopic or integrated

conservation equations. The control design is achieved using

either Lyapunov or passivity design techniques. A description of

this approach and its application to finite dimensional systems

with and without input constraints is given byFarschman et al.

(1998).Ydstie and Jiao (2004)applied inventory and flow control

to a float glass production system and Ruszkowski et al. (2005)

tested this control for a 1D transport reaction process system.

Control of particulate processes described by population balance

equations with the concept of system inventories (Duenas Diez

et al., 2008) is a good example of the inventory based control

applied for an integro-partial differential equation system.

2. Passivity of transport reaction systems

Let S be a convex subset of R+n+2 called the statespace. Let Z

denote an arbitrary point. For example, we can have

Z U,V,M1,. . . , Mnc,XT, where Uis the internal energy, V is the

volume,Mi is mass or moles of chemical component i and Xcan

correspond to the charge, degree of magnetization, area, momen-

tum, etc. In this case, Z is called the Gibbs ensemble.

We assume that there exists a C1 function, S: S/R , called

the entropy such that:

For any positive constant l, SlZ lSZ (S is positivelyhomogeneous of degree one).

For all points Z1,Z2AS and any positive constant l,SlZ1 1lZ2ZlSZ1 1lSZ2 (S is concave).

T @U=@S40 (the temperature is positive).

Fig. 1 shows the entropy function S(Z). Using the C1 property

we define the intensive variables, so that

wT @S

@Z 1

We see from figure that the intensive variables are defined as the

tangent hyperplane to the entropy function. The slope of the

tangent line, w1 defines the intensive variables at the particular

pointZ1. A non-negative function, called a storage function can be

defined now so that for any pair of points Z1, Z2 in S we have

A1Z2 SZ1 wT1Z2Z1SZ2Z0 2

It is easy to verify geometrically that A1(Z2) is non-negative as

shown in Fig. 1. Inequality (2) also follows from Youngs

inequality applied to the concave function S. From Eq. (2) and

the Euler identity for homogeneous functionS(Z) wTZ, we have

A1Z2 wT1Z2SZ2 3

From the Gibbs tangent plane condition, the two points Z1and Z2are in equilibrium if and only ifA1(Z2) 0. It is easy to see that this

condition is equivalent to w1w2. The function A, therefore,

measures the distance from an arbitrary, fixed reference pointS.

A coarse grained transport reaction system is defined locally by

the set of conservation laws for zwritten as partial differential

equations

@rx,tzx,t

@t @fx,t

@x sx,t 4

wherex is the position and tis the time. f(x,t) represents the flux

density andsx,trepresents the density of production.rx,tandz(x,t) are the molar density and local state of the system,

respectively. These variables are connected to the macroscopic

balances

@Zi@t

pit fit

yi hZi 5

via the relationships

fit fLi 1,t fLi,t 6

pt

Z Li 1Li

sx,tdx, ZZ Li 1

Li

rx,tzx,tdx 7

where i 1,2,y,N with N denoting the number of spatial

subdivisions of the system. This is useful in exploiting the

distributed nature of the system with more accuracy. Here, we

have included one dimension only. In the above equations, the

sign is chosen such that it is positive for flow into the system

and negative for flow out of the system. We normally divide

the flux density into orthogonal components (in the sense of

GibbsDuhem) so that

fx,t fconvx,t fdiffx,t 8

where

fconvx,t rx,tzx,tu 9

with u as the center of mass velocity. The division into diffusive

and convective terms can be motivated on physical grounds and

corresponds to a separation into Eulerian and Lagrangian

components.

We now define an augmented storage function Wso that

Wt

Z Li 1Li

Ax,tdx1

2yiy

i

TyiyiZ0 10

Therefore, the dissipation equality for process systems is given by

(Ruszkowski et al., 2005)

dW

dt

~fl

~wLi 1

Li

Z Li 1

Li

~fT

~X ~wT ~sdxyiyidyi

dt

11

A1

Z1 Z2

Entrop

yS

(Z)

Fig. 1. Entropy function.

S. Balaji et al. / Chemical Engineering Science 65 (2010) 47924802 4793


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From the above equation, we can see that the storage function W

decreases if the right hand side is negative. The term ~fl

~wjLi 1Li

is

due to deviations in boundary conditions, ~fT

~X represents devia-

tions due to convection, diffusion and heat conduction, ~wT ~srepresents deviations due to chemical reaction and power of

compression and yiyidyi=dt is the product of the system input

and system output (inventory control).

In the current study, the focus is on the passivity based control ofa vapor recovery reactor (VRR) used in carbothermic aluminum

production for enhanced performance. A shrinking core model has

been used along with a one-dimensional heterogeneous model to

represent the reactor behavior appropriately. This choice was

motivated by the modeling and experimental studies (Garcia-Osorio

et al., 2001; Fruehan et al., 2004) carried out in the past. In the next

section we show how this process fits into the control theory

developed above.

3. Carbothermic aluminum process: control challenges

The HallHeroult electrolytic process is the only technically and

economically feasible process to produce aluminum today. However,it carries the demerits of high capital cost and high energy

requirements. Thus, with a view of devising a more energy efficient

process, the carbothermic reduction process has gained a significant

interest in the aluminum industries (Motzfeldt et al., 1989; Bruno,

2003; Choate and Green, 2006; Garcia-Osorio, 2003). The main

advantages of the carbothermic process are that it promises to

reduce capital and operating costs by 25% or more. One disadvan-

tage with the carbothermic reduction process is that it is carried out

at very high temperatures 42200 3C (Johansen et al., 2000). The

operational challenges are significant due to the high temperature

and significant amounts of aluminum leave the main reactor in the

form of aluminum and aluminum oxide vapors since the operating

conditions are close to the boiling point of Aluminum 2520 3C.

The aluminum containing vapors cannot be recovered throughsimple cooling since a backward reaction produces a significant

amount of aluminum oxide in the form of molten slag and fine

particles which reduce the process efficiency and its operability.

A counter-current vapor recovery reactor (VRR) with carbon feed has

been proposed to solve these problems. In the VRR, the carbon reacts

with the aluminum compounds in a series of heterogeneous non-

catalytic reactions, forming solid and gas products (refer Fig. 3). The

important solid product is aluminum carbide which is indeed

required in the smelting stage of the aluminum process. The gaseous

product is hot carbon monoxide gas, which can be used for other

purposes like generating electricity or for syn-gas production.

The flow sheet for the aluminum production is based on theReynolds process (Fig. 2, Kibby and Saavedra, 1987). The feed

(Al2O3+C) is heated to around 20003C in the smelting stage where

carbide containing slag and gases are produced based on the reaction:

Al2O3 C-Al4C3Al2O3slag COAl2OAlg 12

The slag is then heated to above 2100 3C, producing a carbon

containing aluminum alloy which floats on the slag and gas as shown

below:

Al4C3Al2O3slag-AlCl COAl2OAlg 13

The aluminum alloy is sent to the purification stage where the carbon

is separated. The amount of aluminum and aluminum sub-oxide

escaping from the smelting stage is quite high and therefore must be

recovered for improved efficiency. As simple cooling is infeasible for

such systems, a vapor recovery reactor is used instead.The VRR is charged with carbonaceous material to produce

aluminum carbide, which is recycled back to the smelting stage

(Fig. 2). The reaction between aluminum vapor and carbon to form

aluminum carbide production is slightly exothermic which leads

to further heating of the charge in the column. However, a slight

increase in temperature is advantageous since it accelerates the

reaction thereby improving production efficiency and operability.

The main control challenges for the process are:

A highly nonlinear systemthermodynamics plays a majorrole in the extent of conversion and the system is a multiphase

reactor system which can be represented appropriately only

with a nonlinear model.

Distributed parametric systemthere is a significant differ-ence in temperature along the system along with the shrinkage

of the carbon particles and hence the conversion and slag

formation vary spatially.

Vapor

Recovery

Reactor

Smelting ProcessGas Fluxing - Purification

-

-

Fig. 2. Simplified diagram of Reynolds process for carbothermic aluminum production.

S. Balaji et al. / Chemical Engineering Science 65 (2010) 479248024794


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Highly integrated with recyclesfromFig. 2, it can be inferredthat there are many recycle streams integrated with the

smelting process (product Al4C3is recycled from both the VRR

and the purification stage).

A multi-phase reaction system which demands complexmodels for accurate prediction of the process.

The available control variables are:

Heat input to the smelting process and the VRR. Carbon feed to the smelting process and the VRR. Alumina feed to the smelting process and the VRR.

In the present work, we concentrate on modeling and control of

VRR only. The proposed control strategy will be extended to the

entire process in a future paper.

4. The vapor recovery reactor

The vapor recovery unit recovers the aluminum gases escaping

from the smelting stage. Carbon particles are fed from the top and

the aluminum vapors enter from the bottom of the reactor(counter-current operation). Almost instantaneous chemical

reaction takes place due to the high temperature. A mass transfer

limited reaction mechanism therefore describes the reaction rate

accurately. It is also assumed that the diffusion of gases into the

solid product layer is the controlling mechanism. This assumption

is based on both experiments carried out at ALCOA and lab

scale experiments in the Department of Materials Science and

Engineering at Carnegie Mellon University (Bruno, 2004; Fruehan

et al., 2004). The mechanism is captured well by postulating

the shrinking core model.

Fig. 3shows a schematic representation of the vapor recovery

reactor.

The main chemical reactions that take place in VRR are

given by

2Al2Og 5Cs2Al4C3s 2COg 14

4Alg 3Cs2Al4C3s 15

5. Model description

In this section, we elucidate the model used to describe the

mass and heat transfer in a non-porous moving bed for chemical

reactions shown in (14) and (15). The modeling approach is based

on the idea that the reactions occur in spherical carbon particles

whose dimensions are negligible when compared with the overall

size of the column. With this assumption, the mass balance

equation for the reactants (in vapor form) is written as

eB@ci@t

vg@ci@z

Deb,i@2ci@z2

si, i Al2O, Al 16

Similarly, the mass balance of the solid reactant (carbon) is

written as

1eB@cs@t

vs@cs@z

DeC@cs@z2

sC 17

The solid concentrationcsis related to the extent of conversion by

cs cs01X 18

Substituting (18) in (17),

@cs0X

@t vs

@cs0X

@z DeC

@2cs0X

@z2 sC 19

The mass balance for the gas product is

eB@cCO@t

vg@CO@z

Deb,CO@2cCO@z2

sAl2O 20

where eB is the bed void fraction, si is the rate of reaction ofspeciesi and Xis the solid conversion. si is negative for reactantsand positive for products. The mixing in axial direction (which is

due to turbulence and presence of packing) is considered in the

model by superposing a dispersion transport mechanism. The flux

associated with this mechanism is described by an expression

analogous to Ficks law for mass transfer. The proportionality

constant is the dispersion coefficient Deb. Froment and Bischoff

(1990) summarized the experimental results concerning the

dispersion coefficient in the axial direction. In these correlations,

the dispersion coefficient is a function of the Peclet number

(based on the particle diameter) and Reynolds number. The

dispersion coefficient in the axial direction is obtained from

Garcia-Osorio and Ydstie (2004).

The energy balance for the solid reactant is given by

1eBCpsrs@Ts@t

vsCpsrs@Ts@z

kfs@2Ts@z2

haTsTg Xn

i

DHisi Q

21

whereQis the energy from an external heat source to maintain

the temperature along the reactor. The energy balance for the

vapor is given by

eBCfrf@Tg@t

vgCfrf@Tg@z

kf@2Tg@z2

haTgTs 0 22

wherea is the specific surface area per unit volume, h is the gas

solid heat transfer coefficient obtained using the RanzMarshall

correlation (Themelis, 1995), DHi is the enthalpy of reaction for

species i and vg is the velocity of the gas stream. In thissimulation, the momentum balance is not solved explicitly. To

compensate for this, the change in velocity is computed with a

corresponding change in the temperature of the gas stream. The

relation is written as

vg vg0TgTg0

23

5.1. Reaction modelproduct layer diffusion controlled

The model takes into account the external transfer of the gas

species onto the surface of the solid particle, the diffusion through

the pores of the solid product layer and the heterogeneous

chemical reaction at the surface of the solid reactant. When the

C, Al4C3

Product Stream

Al2O, Al, CO

Vapors

C

Solid FeedCO

Gas Out

Q

Heat Input

MovingBedSystem

Fig. 3. Vapor recovery reactor.



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chemical reaction at the interface is almost instantaneous, the

resistance offered by the reaction is negligible when compared with

that of the diffusion through the product layer. Thus, the overall rate

is controlled by the diffusion of the reactants through the product

layer. The rate of diffusion is obtained by making pseudo-steady-

state approximation (Bischoff, 1963; Luss, 1968). That is, the rate of

accumulation of the gas species in the product layer is negligible

when compared with the diffusive fluxes (Szekely et al., 1976).

Further modeling assumptions are:

The size of the particle is considered constant during thecourse of the reaction.

Sintering is neglected. The reactions are bi-directional i.e. reversible.

The reaction rate per particle for Eqs. (14) and (15) is given by the

shrinking core model as reported byGarcia-Osorio et al. (2001)for

the aluminum, carbon system.

4pr2De,Al2OdcAl2Odr

4pDe,Al2 O

1rc

1r0

cAl2O cCOKE,Al2O

KE,Al2O1 KE,Al2O

4pr2crCbAl2O

drcdt

24

4pr2De,AldcAldr

4pDe,Al

1rc

1r0

cAl 1KE,Al

4pr2crC

bAl

drcdt

25

whereDe,Al and De,Al2O are the effective diffusivities of aluminum

and aluminum sub-oxide vapors, respectively, in the product

layer. KE,Al2O and KE,Al are the equilibrium reaction constants for

reactions (14) and (15), respectively, rc is the radius of the

unreacted carbon particle andr0is the total radius of the particle.

Therefore, the reaction rate of each chemical species derived

for Eqs. (24) and (25) is (Smith, 1981):

sAl2 O31eBDeb,Al2O

r30cAl2O

cCOKE,Al2O

1

rc

1

r0

1 KE,Al2O1 KE,Al2O

26

sAl31eBDeb,Al

r30cAl

1

KE,Al

1

rc

1

r0

127

sCO sAl2O 28

The effective diffusivity for the product layer was calculated by

Fruehan et al. (2004) through experiments. The reaction rate for

the solid particle is determined by

RC dnCdt

d

dtrC

4

3pr3c 4prCr

2c

drcdt

29

Therefore, the reaction rate of an unreacted carbon particle is

obtained by combining Eqs. (24) and (25) with Eq. (29) ( Missen

et al., 1999):

Deb,AlbAl cAl 1

KE,Al

1

rc

1

r0

1

Deb,Al2ObAl2O cAl2O cCOKE,Al2O

1

rc

1

r0

1 KE,Al2O1 KE,Al2O

30

The relation between the radius of the unreacted carbon particle

(rc) and the initial particle radius (r0) is

rc r01X1=3 31

The initial and boundary conditions are as follows:

cAlz,0 0, cAl2 Oz,0 0, cCOz,0 cCO0

Tgz,

0 Tg0,

Tsz,

0 Ts0,

X 0

cAl0,t cAl0 ,dcAldz

z L

0

cAl2O0,t cAl2O0 ,dcAl2Odz

z L

0

cCO0,t cCO0 ,dcCOdz

z L

0

Tg0,t Tbottom,dTgdz

z L

0

TsL,t Ttop,dTsdz

z 0

0

XL,t 0,dX

dt

z 0

0 32

The initial conditions are established such that only pure carbon

particles are present inside the column at time t0. The boundary

conditions at the column end (zL) assume that the gas product

stream is removed immediately from the unit (Amundson, 1956).

The equilibrium constants were calculated using the database

FACT (Pelton and Degterov, 1999; Pelton and Bale, 1999)

developed by Pelton and Degterov for the Al2O3Al4C3 system.Eqs. (16)(22) with reaction rates corresponding to Eqs. (26)(28)

now correspond to the course grained system (4).

6. Method of solution

The model consists of six PDEs with the physical parameters

like density, viscosity as a function of the state variables. The

reaction rate is non-linear resulting in a set of partial differential

equations with a high degree of non-linearity. The proposed

equations are solved by finite elements method using the

Multiphysics modeling software COMSOL (2005). In this work,

the equations are implemented and solved on a one-dimensional

domain. Mesh refinement is carried out until the solution reachedconvergence. Four hundred and eighty mesh points were used

with 5766 degrees of freedom to solve the equations with

required accuracy. A transient analysis with direct linear system

solver (UMFPACK) is used to solve the PDEs simultaneously.

7. Results (open loopbase case)

Table 1 shows a list of parameters used for the base case

simulation. The simulation results showing the conversion of Al,

Al2O vapors and carbon particles until the system reaches steady-

state are shown in this section. The various lines in Figs. 47

represent the profiles at different time instances. Fig. 4shows the

Table 1

Model parameters (base case).

Parameter Parameter name Nominal value

D Diameter of the column 0.15 m

De,i Effective diffusivity of species i in the

solid product layer

0.8 104 m2/s

vg Gas velocity 1.28 m/s

vs Solid velocity 7 104 m/s

Z Height of the column 0.30 m

r0 Radius of the pellet 0.0125 m

Tbottom Temperature at the bottom of the

vapor recovery unit

2230K

B Bed porosity 0.25

s Solid density 2267 kg/m3



6/11

normalized concentration of Al vapors with respect to the

normalized column height at various time instances. The

steady-state profile is clearly seen in the figure for the operating

conditions specified inTable 1.Fig. 5represents the concentration

profile of the aluminum sub-oxide (Al2O) along the reactor.

Comparing the concentrations of aluminum and aluminum sub-

oxide at steady-state we observe that the conversion of aluminum

sub-oxide is less when compared with that of aluminum.

Therefore, the percentage recovery of aluminum vapors ishigher than that of the aluminum sub-oxide vapors.

Figs. 6 and 7show the gas temperature and the solid particles

temperature along the reactor, respectively. In the energy balance

three heating mechanisms are considered: (a) heat exchange

between the gas and the solid, (b) heat generation due to the

exothermic reaction and (c) heat from an external source. For the

base case simulations, the energy from the external heat source is

not considered and equated to zero. However, it is one of the

manipulated variables for the proposed control strategy which

will be discussed later (Section 10). The profiles show that the

heat transfer coefficient is high enough to heat the solid particles

to a required high temperature. Also, due to the exothermicFig. 4. Normalized Al vapor concentration vs normalized column height.

Fig. 5. Normalized Al2O vapor concentration vs normalized column height.

Fig. 6. Normalized gas temperature vs normalized column height.

Fig. 7. Normalized carbon temperature vs normalized column height.

0 200 400

600 800 10001200

14001600

00.2

0.40.6

0.81

0

0.2

0.4

0.6

0.8

0.9

NormalizedColumnHeight(z*)

CarbonConversion

(X%)

Time(s)

Fig. 8. Plot of carbon conversion with respect to normalized column height and

time.



7/11


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9. Inventory control

Farschman et al. (1998)used the macroscopic balance

@v

@t fm,z,d pm,z,d 35

to derive control structures for chemical processes. We have

indicated that the flux and production variables depend on

vectors of manipulated variables m, state variables z and

disturbances d. We furthermore note that Eq. (35) is a positive

system since the state consists of elements that are non-negative.

All states of the system are stabilized (in the sense of Lyapunov) if

the total energy and the total mass are bounded. We now have the

following result.

Let vn denote a time-varying reference. Then, the synthetic

input and output pair as given below is passive:

u fp dv

dt

e vv 36

LetV 12vvTvv. The vectorsu andeare related in a passive

manner which can be easily verified. By differentiating Vwe get

_V eT _e eT _v _v eTu 37

The inputoutput pair is called synthetic since it does not

necessarily correspond to what is directly measured and manipu-

lated in the real process.

A control strategy which ensures that an inventory asympto-

tically tracks a desired set point is called inventory control. It

follows from the stability of feedback systems and Eq. (36), that

inventory control is input/output stable when we use feedback-

feedforward control in the form

u fm,z,d ^pm,z,d dv

dt Ce 38

In this expression f denotes the estimate of the net flow and ^p

denotes the estimate of the net production. Furthermore, let~f ff and ~pp ^p. The operatorC(e), which maps errors into

synthetic controls, should be strictly input passive (see Appendix).

We can then use the stability of the feedback systems to show

that the gain from the error d ~f ~p to set point errors is given

by inverse of the L2 gain of the operator C. The stability result

developed (Ruszkowski et al., 2005) requires that the mapping^fm

,

z,

dis invertible with respect to the manipulated variable m.

The control system converges to zero error if the uncertainty can

be modeled as a constant offset.

Let us consider a dynamic system represented as

dx

dt fx gx,u,d, x0 x0

y hx 39

The model equations (16)(31) used in this study can be easily

represented as the above dynamic system. If the inventories of thesystem are denoted by Z, then the inventory balance for the

proposed dynamic system is (Ruszkowski et al., 2005):

dZx

dt fy,u,d px 40

where,px @Z=@xfp; and fy,u,d @Z=@xgp

p is the production rate, pn is the production rate at steady-

state and f is the supply rate.

Based on this, many control schemes like onoff control, PID

control, gain scheduling, nonlinear control can be implemented

based on the objective function, constraints and performance

criteria. For example, if a proportional-integral control is

formulated, the control law would be

fy,u,d pp KcZZ

KctI

Z t

0ZZdt

41

whereKcis the controller gain, tIis the integral time constant andZ is the inventory.

10. VRR control study

The objective is to control the total carbon holdup (Mc, carbon

inventory) and the total internal energy (Uc) of the system. The

manipulated variables are carbon feed and the amount of heat

energy supplied from an external heat source (Q). Writing a

simple balance equation for carbon in the reactor and implement-

ing the inventory control gives

JcinJcoutKc,1McM

c

Kc,1tI,1

Z t0

McMcdt 42

whereJinc is the total flux of carbon entering the system and Jout

c is

the total flux of carbon leaving the system. As per the simulations,

the effect of integral action to control the carbon inventory is

almost negligible. However, for completeness, the integral term is

included in the equation. Similarly, inventory balance equation for

the total internal energy is given by

minh

in Q mout

houtKc,2UcUc

Kc,2tI,2

Z t0

UcUcdt 43

From simulation results, the second controller to maintain the

energy inventories by changing the amount of heat Qsupplied to

the system results in offsets with a simple proportional control.Thus, integral action becomes inevitable for the second controller.

Both the controllers are implemented simultaneously as a given

disturbance will affect both the inventories.

The resulting equation for the manipulated variablescarbon

feed and the external heat source is directly implemented into the

boundary condition and the model equations, respectively. The

corrective action governed by the control strategy is a simple

expression (Eqs. (42) and (43)) with integral terms built in it. By

incorporating such expressions in the model equations (which is a

system of PDEs) results in an integro-partial differential equation

system. The control action is calculated and implemented under

the common platform of solving PDEs in COMSOL Multiphysics.

The control action is initially tested with changes in set points

for both the carbon hold up and the internal energy of the system.

Fig. 10. Normalized steady-state carbon temperature for varying column height.



9/11

InFig. 11, the step changes made in the carbon hold up reference

profile and the corresponding change in the carbon inventory is

shown in the first subplot. A closer view of the change at a

particular step change is shown in the second subplot. The

corresponding change in manipulated variable is shown in Fig. 12.

Similarly, the change in reference value for the energy inventory

and the equivalent changes made to the system by the controller

are given inFig. 13. The heat supplied by the external source is

modified to match the energy inventory to the proposed set point(Fig. 14).

Velocity of the vapor stream is believed to be a disturbance

variable. Thus, the controller is tested for a change in the inlet

velocity of the aluminum vapors. The set point for energy

inventory (1.5e7 J) and the carbon holdup (0.33mol) are now

fixed. The change in vapor velocity and the respective variations

seen in the inventories and the manipulated variables are shown

in Figs. 1517. From the results obtained, it can be inferred that

the proposed control strategy performs satisfactorily in

maintaining the inventories of the system at the specified level.

Thus, based on the reactor configuration and the operating

0 500 1000 1500 2000 2500 3000

0.35

0.4

0.45

0.5

CarbonHoldupinVRR(m

ol)

1500 1550 1600 1650 1700 17500.34

0.35

0.36

0.37

0.38

0.39

Carbon

HoldupinVRR(mol)

Reference profileActual profile

Reference profiledata2

Time (s)

Time (s)

Fig. 11. Set point changes in total carbon holdup and the corresponding inventory

variations.

Fig. 12. Change in manipulated variable

carbon feed.

Fig. 13. Set point changes in total internal energy of VRR and the corresponding

inventory variations.

Fig. 14. Change in manipulated variableexternal heat source.

Fig. 15. Series of step changes in the vapor velocityas a disturbance to the

system.



10/11

conditions, the inventory of the system can be changed such that

the required conversion is attained. Also, the maximum

temperature of the system can be controlled by controlling the

energy inventory of the system.

11. Conclusions

A distributed parametric model that describes the mass and

heat transfer in a non-porous moving bed is developed in this

work. The model can be employed to simulate the VRR operating in

transient regime. The model takes into account the external

transfer of the gaseous species onto the surface of the solid

particle, the diffusion through the pores of the solid product layer

and the heterogeneous chemical reaction at the surface of the solid

reactant. The reactions are mass transfer limited and hence a

shrinking core model is used to describe the reaction rate. A

sensitivity study of the VRR has been performed using the model.

The analysis reveals that the Al2O and Al gas conversions increase

with increasing charcoal feed rate, while solid conversion

decreases. For the same gas flow and solid feed rate, taller column

leads to higher conversions. A control study has been carried out

based on the passivity-based inventory control law. The total

carbon hold up in the system and the total internal energy of the

system are taken as controlled variables. The vapor velocity is

considered as the disturbance variable with the carbon feed and

the heat from an external heat source as the manipulated variables.

From the results obtained, the proposed control performs well in

controlling the inventories of the system through which other

crucial parameters like the extent of conversion and the maximum

temperature attained in the reactor can be controlled.

Nomenclature

b stoichiometric coefficient

c concentration, mol/m3

Cpf heat capacity of fluid, J/kg/K

Cps heat capacity of solid, J/kg/K

cS 0 initial concentration of solid (carbon), mol/m3

Deb dispersion coefficient in axial direction, m2/s

h heat transfer coefficient, W/m2/K

KE equilibrium constant

kf thermal conductivity of fluid, W/m/K

kfs thermal conductivity of solid, W/m/K

Nu Nusselt numberPr Prandtl number

r0 radius of the pellet, m

rc radius of the unreacted core pellet, m

Re Reynolds number

t time, s

T0 initial temperature, K

vg superficial gas velocity, m/s

vs solid velocity, m/s

X extent of carbon conversion

z axial coordinate, m

DH heat of reaction, J/mol

eB void fractionrf fluid density, kg/m

3

rs solid density, kg/m3s reaction rate, mol/(m3 s)mf fluid viscosity, kg/ms

Appendix A. Passivity theory

Passivity theory integrates the effects of input and output

variables in the analysis of stability and dynamic state evolution.

Its applications extend to both linear and non-linear systems with

intuitive and relatively simple results as passive systems are easy

to control. Simple control strategies like PI or PID can be devised

that stabilize the system at a specific operating point.

Let us consider a system with input u and output y. Suppose

that there exists a nonnegative storage (C1) functionW(t), such that

dW

dt rfpvvbJxJ22

The system is

1. Passive ifb 0.

2. Strictly input passive ifx u and b40.

3. Strictly output passive ifx y and b40.

4. Strictly state passive ifx represents the state and b 0.

The notation J J2denotes theL2norm for a function defined on a

domain O such that for any square integrable vector x we have

JxJ22 ZO

xTxdOo1

0 500 1000 1500 2000 2500 30001.47

1.48

1.49

1.5

1.51

1.52

x 107

TotalInternalEnergyinVRR(J)

0 500 1000 1500 2000 2500 30002

2.5

3

3.5

4

4.5

HeatSupplied(Q,

inJ)

Reference ProfileActual Profile

Time (s)

x 106

Time (s)

Fig. 16. Closed loop simulation for the energy inventory.

0 500 1000 1500 2000 2500 30001

0.5

0

0.5

1

1.5

CarbonHoldupinVRR(mol)

0 500 1000 1500 2000 2500 30000.45

0.455

0.46

0.465

0.47

0.475

0.48

Ca

rbonFeed

Reference ProfileActual Profile

Time (s)

Time (s)

Fig. 17. Closed loop simulation for the total carbon holdup.



11/11

O is defined by the semiclosed set 0,1 0,L and subsets

thereof.

Willems (1972b)proved that ifWhas a strong local minimum

then there is an intimate connection between dissipativity and

Lyapunov stability. In this case, W is convex and it is also a

Lyapunov function. Byrnes et al. (1991) developed these ideas

further and established conditions for stabilizability of nonlinear,

finite dimensional systems using passivity. It was shown that a

finite dimensional system is stabilizable if it is feedbackequivalent to a passive system. These results suggest a close

relation between passivity and the methods of irreversible

thermodynamics.

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