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## Brazilian Journal of Chemical Engineering

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*Print version* ISSN 0104-6632*On-line version* ISSN 1678-4383

### Braz. J. Chem. Eng. vol.17 n.4-7 São Paulo Dec. 2000

#### http://dx.doi.org/10.1590/S0104-66322000000400028

**ADAPTIVE CONTROL OF PASTE DRYING IN SPOUTED BED USING THE GPC ALGORITHM**

**N.A.Corrêa, R.G.Corrêa and J.T.Freire **Chemical Engineering Department, Federal University of São Carlos,

Rod. Washington Luís, Km 235, C.P.676, CEP13565-905, São Carlos - SP, Brazil

E-mail: ronaldo@power.ufscar.br, freire@power.ufscar.br

*(Received: November 22, 1999 ; Accepted: April 6, 2000 )*

Abstract- Spouted bed is an equipment reasonable investigated for drying of many and different materials ranging from granules to pastes and suspensions. This work presents an implementation and tests of an advanced control strategy (GPC) on a spouted bed dryer. Water was used as the ideal paste for tests. A microcomputer PC-486 type was used to maintain humidity and temperature set points of air in the dryer by manipulating electric power of heat exchanger and paste feed flow rate. An instrumentation was set-up with different sensors, interface and final control elements for the process. The tests were carried out with pure water as ideal paste. Stability and performance analysis of the control strategy were accomplished. A stable controller was obtained which encouraged us to investigate this control strategy in the drying of other pastes too.

Keywords: spouted bed dryer, adaptive control, GPC algorithm.

**INTRODUCTION**

Gishler and Mathur (Mathur and Epstein, 1974), for the first time, used the spouted bed as the dryer at Canada in 1954. Since then, several studies using this equipment were published. The continuous drying of pastes and suspensions using spouted bed, with inert particles, became an interesting operation. The number of articles on this issue is growing considerably, mainly concerning food processing, for example, bovine blood, eggs, vegetable extract, tomato pulp, pulps of fruits, yeast, molasses, milk and corn starch (Freire, 1992).

As any other processing unit, a spouted bed dryer must be properly automated to achieve better equipment performance, better product quality, raw material economy, energy economy and safety. Several experimental investigations regarding the control of dryers have already been reported (Seborg *et alli*, 1986), but few concerns to the spouted bed dryer control problems.

Drying is basically a simultaneous heat and mass transfer operation. The dryer performance is strongly affected by conditions and nature of feeding, the gas-solid contact mode, and how energy is introduced into the system. The satisfactory model of a drying process should contain good estimates of drying rate, heat and mass transfer coefficients and solid residence time during drying. These parameters can be estimated from an enormous variety of equations, depending on the equipment and on the operational conditions of drying processes. An intrinsic characteristic of drying processes is their slow dynamics (the transient state is very important in the process description), i.e., once the system is disturbed, the steady-state lags to reach their new values. Besides, drying processes frequently contains long time delays.

Drying models that include heat and mass transfer phenomena, present non-linearity characteristics, because the real process possesses a complex dynamics. Drying process parameters are full of uncertainties due to transfer laws complexity. Therefore, the identification of a drying process is not a simple task, because parameters cannot be easily found, since they are time variant.

Then, how to properly operate (control) a process which changes itself along the time? There are control structures that update itself simultaneously as process changes. Such structures correlate the model parameters recursively in well-defined time intervals. That control type is denominated *adaptive*. Different options of adaptation are available. One mostly used has the parameters updated from optimization based on minimization of deviations between the process variables and the nominal (or reference) model variables, which is being identified.

Based on the exposed above, the objective of this work is implement an adaptive strategy to control spouted bed dryer using pure water as ideal paste of proof. The tests were accomplished using GPC controller for different designs searching the better performance to be used in future drying of another pastes. This paper presents the GPC algorithm for SISO systems, but for MIMO system, like our case, the development is trivial.

**THE SPOUTED BED**

A spouted bed is normally a cylindrical vessel, with a conical base, loaded with particles. The cone is truncated to form a nozzle where warm air is fed in. The air feed flow rate should be enough to *spout* the particles. Spouted bed geometry allows particles to possess a cyclical movement, facilitating batch or continuous operation. Depending on the drying process, the particles can be grains or seeds, inert spheres or any other geometry, tablets, candies and many others. Impelled by the fluid, the particles travel the spout channel, decelerate forming a source and then they drop in the annular region (the region between spout and bed wall). The particles in annular region present very dense packing and slow descending movement. Most of the particles move downwards to conical base and move upwards again after going into the spout channel encounter the spout, then establishing a cyclical movement. The characteristic behavior of a spouted bed is expressed by a relationship between bed pressure drop (D*P*) and superficial fluid velocity or volumetric flow of air (*Q*). This behavior is obtained increasing the fluid flow rate gradually, starting from fixed bed, up to obtain a stable spout; and decreasing the fluid flow rate until spout collapse. One important operation point is achieved when spout flow rate is minimum, i. e., the minimum condition for a stable spout. A smaller flow rate would end the spout an undesirable fixed bed operation would be started. The movement of particles is essential to increase heat and mass transfer area of paste and to detach the dry powder (product) from inert particles. It is always difficult to operate the spouted bed equipment exactly at the minimum spout flow rate , because any disturbance can extinguish the spout. Common disturbances are paste feed temperature and paste feed flow rate, air temperature and air flow rate. Concerning the spout collapse, it is a common procedure to operate the spouted bed with a flow rate value larger than the minimum spout flow rate. Evidence of spout sensitivity in presence of paste can be found in Cunha *et alli* (1998).

**THE GPC ALGORITHM**

The control algorithm GPC (Generalized Predictive Control) was developed by Clarke *et alli* (1987) and it consists in an advanced strategy of predictive and adaptive control. Figure 1 shows a simplified block diagram of GPC. It belongs to the class of the "self-tuning" controllers, and it is feasible in the treatment of:

(a) Non-minimum phase or unstable systems;

(b) Systems with inconstant or unknown dead time and process gains;

(c) Unknown order.

The spouted bed dryer presents one of these problems at least, if not all of them.

Adaptive controllers were already used in several engineering fields in countless applications, from military (control of ships, airplanes and missiles - Aström, 1983) to medical purposes (arterial blood pressure control by vasodilator flow rate regulation using GPC in tests with anesthetized dogs - Timmons *et alli*, 1997). Specifically in chemical processes, there are several papers of adaptive control in experiments and simulations. Seborg *et alli* (1986) made a survey of several works and they reported adaptive control applications in furnaces, boilers, grinders, distillation columns, dryers, nuclear reactors, and many others.

Like most of the predictive controllers with internal model (IMC - Garcia and Morari, 1982), GPC minimizes an objective function. The objective function has the following form:

(1) |

The model to be incorporated in the predictive control follows the discrete linear structure:

(2) |

*A* and *B* are polynomials with order *na* and *nb* in shift operator domain *q ^{-1}*, and D is the differential; i. e.,

(3) |

(4) |

(5) |

The term *q ^{-k}* refers to a dead-time of response (dead-time =

*k*). The model given by Eq. (2) is called as CARIMA (Controlled Auto-Regressive Integrating Moving-Average).

Considering the prediction horizon *R*, Eq. (1) can be rearranged into the following form:

(6) |

The minimization of Eq. (6) with respect to D** u** leads to the following solution:

(7) |

To decrease the computational effort, it is worth to have Eq. (7) over-determined with *L* <*R*, i. e. ** G_{1}** is a

*R*x

*L*matrix derived from

**(**

*G**R*x

*R*matrix).

Aström and Wittenmark (1989) present some rules to design *n*, *R* and *L*:

*n*: if dead time *k* is known, *n*=*k*. If not, *n*=1.

*R*: it is selected in such way that *R*x*T* is equal to the rise time of a step response (*T* is the sampling interval).

*L*: for simple systems, *L*=1 yields good results. For more complex systems *L* should be, at least, equal to the number of unstable or weakly damped poles.

To turn the GPC adaptive, it is necessary to estimate recursively (at each sampling interval) the components of the ** G_{1} **matrix and consequently the

*A*and

*B*polynomials. Recursive least square estimation was used (RLS - Aström and Wittenmark, 1989), or

(8) |

(9) |

with

(10) |

(11) |

where **q** is the vector containing the *A* and *B* coefficients, **f** is the process variables vector, ** P** is the covariance matrix and

*Y*is a weight called forgetting factor (0<

*Y*£1).

The matrix ** G** contains the values of the closed loop discrete response

**to a step (or a pulse) on the input variables**

*y***. To evaluate its components based on the**

*u**A*and

*B*parameters, it is necessary to use the Diophantine equation (Clarke

*et alli*, 1987):

(12) |

This equation is used recursively in the following form:

(13) |

(14) |

The ** G** elements are given by:

(15) |

When deviations are null, covariance matrix ** P** grows infinitely (wind-up). To avoid this, a variable forgetting factor

*Y*was used (Ydstie

*et alli*, 1985):

(16) |

Where *N _{0}* is the memory length arbitrarily defined. It should be a positive number with typical values of 10

^{4}.

**EXPERIMENTAL SET-UP AND OPERATION**

Figure 2 shows a scheme of the experimental set-up which presents a spouted bed and several peripheral devices.

To start the spout, a motor frequency w was chosen. This frequency was then applied in the blower's (WEG 5500W) motor, which changes its speed rotation and, consequently, the air flow rate *Q*. This frequency was modulated by a frequency inverter (Siemens, MD550). The air flow rate *Q* was measured by a double orifice plate connected to a pressure transducer (Smar, LD 300). The *Q* - w loop can be closed using a controller (Smar, CS500) with a PI (proportional-integral) mode to maintain *Q* constant. However, this work kept this control loop open. Experiments of this PI control loop to improve the operation of spouted beds can be found in Corrêa *et alli* (1999).

The bed pressure drop D*P* was measured by another transducer LD 300. The air humidity in the output (*X _{s}*) was measured by a polymeric capacitive meter (Vaisala, HMD20UB). The inlet air humidity (

*X*) was evaluated by a wet-bulb/dry-bulb psychrometer. All temperatures (

_{e}*T*,

_{s}*T*,

_{bs}*T*,

_{a}*T*) were measured by type J thermocouples (Salcas). The electrical power supplied to the inlet fluid heat exchanger

_{au}*POT*and the paste feed flow rate

*W*were established by a power module or thyristor (Therma, TH6021A/25) and a diaphragm pump (Wallace-Thiernam, Premia P75 MEGA), respectively. All transducers and control final elements signals were in the range of 4-20mA standard type (mV for thermocouples). These signals were sent to a PC-AT486 computer using an AD/DA converter interface (Lynx, CAD 1236-D), after previous treatment in a signal conditioner (Lynx, MS1000) at each 1s. The spouted bed is built in inox steel and it has a conical cylindrical geometry (diameter=15cm, air feed hole diameter =3.5cm, height =67.5cm, load height=23.5cm, load=3600g of glass particles as inert with diameter=0.36cm). Warm air was supplied with a flow rate in the range of 0 – 2.0 m

^{3}/min.

The computer program for the experiments was written in C (Microsoft Quick C^{â} ) and it includes four basic blocks: 1) Input signals reading and output signals sending. 2) Graphic monitoring. 3) Data storage in disk. 4) Identification and control (GPC) algorithm.

The sampling interval for acquisition and monitoring was 1s. The GPC algorithm was elaborated and tested in simulations using MATLAB^{â} before being implemented in C.

Specifically, block 4 consists of a sequencing algorithm that contains two parts:

i) Model parameters pre-adjustment and covariance matrix ** P** adaptation. During 3600s two independent sequences of random frequency signals (PRBS - Pseudo Random Binary Signal) were applied on

*W*and

*POT*with time intervals of

*TA*. A random value of +/-0.25g/s was added to the nominal operational value of

*W*=0.25g/s at each

*TA*time interval and a random value of +/-0.3kW was added to the nominal operational value of

*POT*=1.5kW. For the matrix

**adjustment it was started with**

*P***=diag(10**

*P*^{4}).

ii) Implementation of the GPC control. The identification is continued with *TA*. Manipulated variables *W* and *POT* are updated and sent to the system at each time interval of *TC*.

The drying in the spouted bed was accomplished with the objective of testing the implementation of controller designs for start-up, set point tracking and disturbance rejection. Therefore, the control variables were the outlet air humidity (*X _{s}*) and the air temperature at the exit (

*T*-

_{s}^{o}C). The manipulated variables were the paste feed flow rate (

*W*- g/s) and the electrical power supplied to the inlet fluid heat exchanger (

*POT*- kW).

Start-up presents the following steps: 1) Turn on the blowers motor and the air heater. 2) Establish the spout of the inert particles. 3) Keep the *POT* at a fixed value and wait about 40min in order to obtain a steady value of *T _{s}* (for

*POT*=1.5kW,

*T*=75

_{s}^{o}C). 4) Turn on the diaphragm pump with

*W*<1g/s of pure water. Wait about 15min for steady-state re-establishment. 5) Initiate the program to accomplish model pre-fitting based on water drying. 6) After pre-fitting, wait set points to settle down (

*T*=60

_{s}^{o}C and

*X*=0.02). 7) Change set points to test the control performance.

_{s}

**RESULTS AND DISCUSSION**

The linear model used in the identification has the following form:

(17) |

(18) |

The model order was chosen based on some simplifications and observations: *X _{s}* response is very fast (order of minutes) in comparison with

*T*(order of hours) for disturbances in

_{s}*W*, and therefore

**=**

*A***0**in Eq. (17); besides

*X*is almost insensible when

_{s}*POT*changes. Therefore

**B**=

**0**(Eq. 17) for

*POT*, since the bed does not collapse.

Figure 3 presents the start-up procedure test for one of the three GPC controller designs. Variable *X _{s}* tracked tightly the set point (

*X*=0.02), but variable

_{s}*T*showed oscillations around its set point (

_{s}*T*=60

_{s}^{o}C). Besides,

*POT*saturated periodically in its maximum value.

Figure 4, regarding controller test with a different design, also shows the *POT* saturation and the *T _{s}* response with damped oscillation. The larger value of

*TA*(360s) allowed the controller's pre-fitting with more complete information regarding

*T*response arising from excitations (PRBS). A sluggish control was observed due to the larger value of

_{s}*TC*. A disturbance occurred, as exhibited in Figure 4, due to the total consumption of water in the feed tank. After filling this tank (after 80s), the control was recovered.

The value of *f* was increased to limit even more the manipulated variables and to avoid saturation. *TC*=15s was employed to accomplish a faster actuation. Thus, a new controller was designed, which performance is presented in Figure 5. For this controller design, it was observed a better performance and larger stability. The responses settled down more quickly with soft actuation without manipulated variables saturation. Regarding linear model fitting, Figures 3, 4 and 5 show the convergence of some parameters (all relative to the *X _{s}* response, and an example of each,

*A*and

*B*, relative to the

*T*response).

_{s}

At the first instants the parameters were adjusted abruptly. Such behavior could be attributed to the forgetting factor *Y* be always very close to unity. Under this condition the parameters vary quickly and the old data are discarded fast.

Figure 6 presents a set point change test with the last designed controller (refer to the Figure 5). Plots show the performance after the pre-fitting stage. The *T _{s}* response tracked the first set point (

*T*=60

_{s}^{o}C) and even after the change to the second set point (

*T*=65

_{s}^{o}C) it fell down. This behavior was the result of the water flow rate increase in the feed, starting from

*t*=4900s to force

*X*track from set point

_{s}*X*=0.02 until the set point

_{s}*X*=0.03. After 7000s, the manipulated variable

_{s}*POT*arose to compensate the energy wasted properly, and forcing

*T*to track the new set point (

_{s}*T*=65

_{s}^{o}C).

**CONCLUSIONS**

As observed in the responses obtained from the different experiments, it can be concluded that controller GPC had good robustness.

The GPC controller with the last tuning procedure showed satisfactory performance. The controller implemented was able to track set point changes and to reject disturbances with a stable closed loop behavior.

The manipulated variables must be weighted strongly (high *f* values). Thus they would be inside the real operational limits of the equipment (without saturation). Furthermore, the abrupt movements of the atuactors would be avoided.

In agreement with the model parameters convergence during the identification, the introduction of a variable forgetting factor is feasible.

For the system in question, the control interval should be short (15s) and the adaptation interval should be long (6 min).

**NOMENCLATURE**

A, B | polynomials of parameters a and b |

, AB | matrices of parameters a and b |

a, _{j}b _{j} | polynomials parameters |

a(i,j), | parameters as matrix elements |

b(i,j) | |

e, | deviations or errors |

e | vector of e () |

E, _{j}F _{j} | polynomials of the Diophantine equation |

f | weighting factor for the actions Du; element of the polynomial F |

G _{j} | polynomial of the discrete response to a step in manipulated variable |

G | matrix (RxR)of G polynomial components_{j} |

G_{1} | over-determined matrix (RxL) derived from G |

g _{j} | component of polynomial G |

i | integers (i=1,... ,L) |

j | integers (j=n,... ,R) |

J | cost or objective function |

k | delay or dead time (s) |

L | control horizon |

N | initial instant of prediction |

N _{0} | memory length |

P | covariance matrix |

POT | electrical power supplied to the heat exchanger to warm the inlet air (kW) |

D P | pressure drop across the bed (mmHg) |

q ^{j} | shift operator |

Q | air flow admitted in the bed (m^{3}/min) |

R | prediction horizon |

t | current instant |

T, _{a}T _{au,} | ambient temperature, ambient humid bulb , |

T,_{bs}T, _{s} | temperature, air temperature at the exit (at the humidity meter position) and temperature at the exit, respectively (^{o}C) |

TA, TC | sampling intervals for adaptation and control, respectively |

u | manipulated variable |

D u | manipulated variable movement or action |

D u | vector of Du(Du=[Du(t+1-k)...Du(t+R-k)]^{T}) |

W | paste feed flow rate (g/s) |

X, _{e}X _{s} | bed inlet and bed outlet air humidity (mass fraction), respectively |

y, , y _{sp} | control variable, model response and set point, respectively |

y | vector of y ()y= [y(t+1)...y(t=R)]^{T} |

vector of (=[ (t)...)_{R}(t)]^{T} | |

y_{sp} | vector of set-pointsm (y)_{sp}=[y_{sp}(t+1)...y_{sp}(t+R)]^{T} |

f | vector of A and B parameters |

q | vector of process variables |

w | frequency applied in the blowers motor to manipulate the air flow rate (Hz) |

Y | forgetting factor |

**ACKNOWLEDGEMENTS**

The authors would like to thank the financial support provided by CNPq (under grant: 142849/97-9) and PRONEX (under grant: 41.96.0897.00).

**REFERENCES**

Aström, K.J., Theory and Application of Adaptive Control – A Survey, Automatica, Vol. 19, No. 5, pp. 471-486 (1983). [ Links ]

Aström, K.J. and Wittenmark, B., Adaptive Control, Addison-Wesley Pub. Co., Lund, Sweden, (1989). [ Links ]

Clarke, D.W.; Mohtadi, C.; Tuffs, P.S., Generalized Predictive Control - Part I. The Basic Algorithm, Automatica, Vol. 23, No. 2, pp. 137-148 (1987). [ Links ]

Corrêa, N.A.; Freire, J.T.; Corrêa, R.G., Improving Operability of Spouted Beds Using a Simple Optimizing Control Structure, Brazilian Journal of Chemical Engineering, Vol. 16, No. 4, pp. 359-368 (1999). [ Links ]

Cunha, F.O.; Spitzner Neto, P.I.; Freire, J.T., Paste Drying Investigations: Influence of Paste, Proceedings of the Brazilian Congress of Particulate Systems - XXVI ENEMP, Teresópolis, Brazil, pp. 323-329 (1998) (*in portuguese*). [ Links ]

Freire, J.T., Paste Drying in Spouted Bed. In: Freire, J.T. and Sartori, D.J.M. Special Topics in Drying, Vol. 1, Chap. 2, pp. 42-85, EDUFSCar, São Carlos, Brazil (1992) (*in portuguese*). [ Links ]

Garcia, C.E. & Morari, M., Internal Model Control. 1. Unifying Review and Some New Results, Ind. Eng. Chem. Process Des. Dev., Vol. 21, No. 2, pp. 308-323 (1982). [ Links ]

Mathur, K.B. & Epstein, N., Spouted Beds, Academic Press, N.Y., (1974). [ Links ]

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Timmons, W.D.; Chizeck, H.J.; Casas, F.; Chankong, V.; Katona, P.G., Parameter-Constrained Adaptive Control, Ind. Eng. Chem. Res., Vol. 36, No. 11, pp. 4894-4905 (1997). [ Links ]

Ydstie, B.E.; Kershenbaum, L.S.; Sargent, R.W.H., Theory and Application of a Self-Tuning Controller, AIChE Journal, Vol. 31, No. 11, pp. 1771-1780 (1985). [ Links ]