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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.
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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.
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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
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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.
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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.
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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.
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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.
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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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