Predictive control with multiobjective optimization: Application to a sludge drying operation

Computers and Chemical Engineering 78 (2015) 70–78

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Predictive control with multiobjective optimization: Application to a sludge drying operation A. Fuentes ∗ , J.P. Ploteau, P. Glouannec Univ. Bretagne-Sud, EA 4250, LIMATB, F-56100 Lorient, France

a r t i c l e

i n f o

Article history: Received 18 June 2014 Received in revised form 8 April 2015 Accepted 15 April 2015 Available online 23 April 2015 Keywords: Offline predictive control Multiobjective optimization Multiphysics modeling Operating input parameters

a b s t r a c t The main objective of this study is to develop an offline tuning of the operating input parameters for a sludge drying operation, by using multiobjective optimization techniques combined with a predictive control method. The manipulated variables concerned are the temperature and the relative humidity of the drying air (Tair , RHair ). The optimal time for the reversal operation of the product is also investigated. The optimization procedure is coupled to a one-dimensional numerical model that allows the simulation of moisture content and temperature field evolutions in the product during the drying step. A genetic algorithm is used to identify the two manipulated variables, at each step time, by minimizing simultaneously three objective functions over a finite horizon. These objective functions are linked to penalties concerning the heating and dehumidifying of the outside air used for the drying stage and to a global moisture content gap relative to a drying target. First, the heat and mass transfer model is validated for the drying step of a plate sample of sludge, with a reversal operation. Afterwards, the optimization procedure is carried out, and the results are discussed in terms of an energetic analysis. © 2015 Elsevier Ltd. All rights reserved.

1. Introduction For processes already in operation or for studies focused only on products, the energetic efficiency of the drying stage is often directly linked to the control procedure of the operating input parameters (Banga et al., 2001; Zorrilla et al., 2003; Dufour, 2007). As far as drying processes are concerned, many authors use the response surface method, obtained by design of experiments, in order to realize afterwards a tuning of the optimal parameters through the use of a desirability function (Karimi et al., 2012; Silva et al., 2013; Erbay et al., 2014; Patil et al., 2014). The response surface method leads to linking the outputs (observables) of a process to the inputs (manipulated variables) with the help of a first or a second order polynomial function. However, the difficulties of using this method appear when strong nonlinearities are present. Furthermore, no discontinuities in the evolution of the observables should occur within the field of the study concerned. Process input parameters could also be determined with the use of determinist or heuristic optimization methods. Barttfeld et al. (2006) have proposed an optimization of a drying process of a liquid film on a substrate. The manipulated variables (temperature and air velocity) are identified according to four scenarios. The goal

∗ Corresponding author. Tel.: +33 2 97 87 45 43. E-mail address: [email protected] (A. Fuentes). http://dx.doi.org/10.1016/j.compchemeng.2015.04.017 0098-1354/© 2015 Elsevier Ltd. All rights reserved.

of the first two scenarios is to obtain a film with a certain final solvent content by minimizing the total heat consumption with a fixed production rate or by maximizing the production rate. The other two scenarios concern minimizing the final solvent mass in the film or maximizing the total flow of evaporated solvent. Each scenario, represented by an objective function, is examined separately (mono-objective optimization). Nevertheless, the parameter estimation procedure requires, most of the time, the optimization of simultaneously possibly conflicting multiple objectives. There are different ways to approach a multiobjective optimization problem. The most common one is to transform a multiobjective into a mono-objective optimization problem by making a single composite objective function as the weighted sum of the objectives. However, authors mention the difficulties of this transition, notably to obtain optimal points from non-convex objective functions (Marler and Arora, 2004). The general approach is to determine an entire Pareto optimal solution set. Multiobjective optimizations are often conducted with the help of a Genetic Algorithm (GA), which provides a Pareto surface from a manipulated population. This metaheuristic algorithm considers some elementary operations (selection, crossover, mutation) that are widely described in the literature (Konak et al., 2006). Several studies have used a GA as a multiobjective optimization solver in order to identify the operating input parameters of thermal or chemical processes (Yuzgec et al., 2006; Arbiza et al., 2008; Sendín et al., 2010; Liu and Sun, 2013).

A. Fuentes et al. / Computers and Chemical Engineering 78 (2015) 70–78

In addition, model predictive control (MPC) is a relatively recent technique for the optimal control of processes; by its simplicity, it has created a real interest in the industrial field (Morari and Leeb, 1999; Qin and Badgwellb, 2003). This control method belongs to the class of model-based control strategies. The principle of this method is to control the operating input parameters of a process in order to follow a reference trajectory closely, while taking into account the future behavior of the system. The tuning of the operating input parameters is based on the resolution of an optimization problem, and the future behavior of the system is obtained with the help of a numerical prediction model. In the context of drying processes, researchers have shown an interest for tuning the optimal operating input parameters by using a predictive control method. Thus, Dufour et al. (2003) used this method for drying a coat of paint with an infrared source. In their study, the issue is of Single Input Single Output (SISO) type. The main objective is to follow an imposed temperature kinetic for the paint through determining the optimal command policy of the infrared emitter. Temmerman et al. (2009) have proposed an optimization for a pasta drying process by using a predictive control strategy. The optimization is conducted with a single objective function related to the pursuit of a drying kinetic target. The resolution of the underlying optimization problem is carried out with a deterministic algorithm (LevenbergMarquard). The manipulated variables for the process optimization are the temperature and the relative humidity of the drying air. Laabidi et al. (2008) have worked on the development of a predictive control strategy with multicriteria optimization for a nonlinear multimodel system, in order to realize a set point tracking. A GA is used for the resolution of the multicriteria optimization problem. The procedure provided an optimal control law of the system. The authors emphasize that the predictive control method associated with a multiobjective optimization problem enabled them to take into consideration some perturbations due to noise or load variations of the system more efficiently. Thus, in the context of drying operations, few studies deal with the development of a predictive control procedure based on a multicriteria optimization problem coupled with a nonlinear multiphysics model. The operating input parameters tuning procedure proposed in the present study is based on a predictive control strategy with receding horizon combined with a nonlinear multiphysics model and a genetic algorithm for solving the multiobjective optimization problem. The optimization procedure proposed is applied

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to a field that arouses interest among researchers: sludge drying (Vaxelaire and Cézac, 2004; Bennamoun et al., 2013). In addition, modeling the reversal operation during the drying step is also rarely investigated in the literature. The optimization procedure proposed in this study can also determine the ideal moment for the reversal operation of the product. In this context, an effort is focused on the development of a numerical model that allows taking into account the reversal operation on the one hand, and enables an accurate prediction of the water content evolution of the product according to the control signal matched (Tair , RHair ), on the other hand. First, the experimental setup and the heat and mass transfer model used are presented. The second part of the paper presents the optimization procedure and the objective functions characterizing the multiobjective aspect of the optimization problem concerned. A model validation with a reversal operation is carried out preliminary to the optimization phase, in order to prove its suitability. Then, two optimization cases are presented: one with a basic multiobjective optimization solved with a genetic algorithm, and another one whereby the predictive control method is added to this optimization procedure.

2. Formulation of the drying problem The dynamic optimization procedure is applied to the thermal drying, at low temperature, of an agricultural sludge. A plane flat configuration is used for the sample, with convection and conduction exchanges on each face (Fig. 1). The laboratory dryer used allows adjusting the thermo-aeraulic conditions of the drying air applied on the upper face of the sample. A temperature can be imposed at the bottom of the sample (Tb ) by using a contact-heating device. A mass loss measurement is used to obtain the evolution of the average moisture content in dry base X. This apparatus has been presented in a previous study (Louarn et al., 2014). The goal of our study is to determine several pairs of variable optimal operating input parameters (air temperature and relative humidity) during the drying cycle. In order to accelerate the drying step, the ideal moment for the reversal operation is also looked for. The reversal operation involves manually extracting the product from the crucible and turning it over (by inversion of the upper and lower face) before reintroducing it into the crucible in order to continue

Fig. 1. Summary of the proposed procedure.

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A. Fuentes et al. / Computers and Chemical Engineering 78 (2015) 70–78

the drying step. To perform the reversal operation successfully, the product must be in a solid state for the handling process, and the lower face must not be in a sticky state. The drying stage is conducted until 90% of the initial water content is reduced. The determination of the optimal command law is obtained by solving a multiobjective optimization problem. One of the optimization criteria is linked to the pursuit of a drying kinetic target established during a drying operation with the highest air temperature and lowest relative humidity (CDG ) (Fig. 1). The other two criteria that have to be minimized are related to the costs for heating (CTair ) and dehumidifying (Cw ) the outside air before introducing it into the dryer.

fluid mechanic correlation (Incropera and DeWitt, 2002) and the radiative part hr is estimated from linearization of the radiative heat flux (Kutz, 2003), according to the following assumptions: equality between walls and drying air temperature and weak temperature gap between product surface and wall temperature.



(X)

∂T  = hg(v,T ) (Tair − T(z=e,t) ) − ∂z z=e

The sludge is static inside the dryer and disposed in a 200 mm × 200 mm crucible. For this study, the thickness of the sludge is fixed at 3 cm. Velocity, temperature, and relative humidity of the drying air are measured and controlled during the drying step. A hygrometer and a thermocouple are used to regulate drying air conditions. In parallel, a mass measure and a pyrometer are used to follow respectively the evolution of the global water content and the surface temperature during the experiment. An embedded thermocouple in the crucible is used to regulate the temperature imposed on the bottom of the product.

−s Dl(X,T )

∂X  ∂z z=e



Lv

(5)

The air/sludge boundary condition for mass transfer (6) can be written as a function of the difference between the vapor pressure at the surface and the vapor pressure in the air (7).



−s Deff(X,T )

2.1. Experimental setup and protocol





Fm = km(v)

∂X  = Fm ∂z z=e

Mv (a P sat − RHair Pvsat ) air(T ) RTfilm w(X) v surf(T )

(6)

(7)

At the bottom of the sludge sample, a Dirichlet’s condition (Tf ) and a null mass flux are imposed (8). z = 0 : {T (0, t) = Tf ; Fm = 0}

(8)

In order to obtain the drying kinetic, the unsteady-state computation of the multiphysics model is realized with Comsol® . 3. Optimization procedure

2.2. Heat and mass transfer model The heat and mass transfer and their couplings are included in a macroscopic model. The small size of the sample allows retaining the assumption of a uniform temperature and water content on the surface, which simplifies the modeling to one-dimensional geometry. The formulation of the governing Eqs. (1) and (2) was carried out according to the following assumptions: - Three phases are considered: solid, liquid, and vapor - The shrinkage and thermomigration effects are neglected - The effective diffusion coefficient Deff (3) are temperature and local water content dependent (Louarn et al., 2014) - The initial temperature and moisture distributions inside the solid are supposed to be uniform (4) The endothermic effect due to water evaporation is taken into account in the heat transfer Eq. (1). The specific heat capacity evolves according to a mixing law between the capacities of water and dry solid. Effective thermal conductivity is a function of water content X. The combination of the mass conservation equations in both the liquid and gas phases leads to (2), where X represents the water (liquid + vapor) content in dry basis. The effective diffusion coefficient (Deff ) includes the liquid and vapor diffusion mechanisms inside the sludge matrix (3). D and Dv were obtained from an identification procedure presented in a previous study (Zaknoune et al., 2013). (X) cpeq(X)

∂X ∂ − ∂z ∂t

∂T ∂ + ∂t ∂z





−(X)

∂X Deff(X,T ) ∂z



∂T ∂z



=

∂ ∂z



s Dv(X,T )

∂X ∂z



Lv

(1)

Model-based predictive control (MBPC) is a process control methodology that uses a knowledge model to predict the process output at future discrete instants (k), over a specified prediction horizon (HPred ). The knowledge model used in this study has been detailed above (Section 2.2). The future manipulated variables sequence (Tair(k) and RHair(k) ) over a specified control horizon (HCmd ) is obtained by the resolution of a multiobjective optimization problem. As often highlighted in other studies (Konak et al., 2006), genetic algorithms are well designed for the resolution of a multiobjective optimization problem. Typically, a multiobjective problem has no single global solution, and it is often necessary to determine a set of optimal points in the sense of Pareto. With the genetic algorithm, a Pareto optimal population is obtained if there is no other individual that improves all the objective functions simultaneously. Before solving a multiobjective optimization problem with the use of a GA, it is necessary to define the size of the genetic string used for each individual. In our case, the length of the genetic string is equal to the number of manipulated variables considered (two in this study, Tair and RHair ) multiplied by the number of prediction times (HPred ) (9). Each manipulated variable coded in the genetic string corresponds to an operating input value applied during a Sample Time (ST). Moreover, they are bounded by the technical limits of the dryer and by the external conditions (10). However, as the control signal is manipulated only within the control horizon in MBPC, the operating input parameters remain constant afterwards (11). [Tair(k=1) , Tair(k=2) , . . ., Tair(k=HPred ) , RHair(k=1) , RHair(k=2) , . . ., RH air(k=HPred ) ]

=0

Deff(X,T ) = Dl(X,T ) + Dv(X,T ) t = 0 : {X(z,0) = Xini ; T(z,0) = Tini }

(2)



(3)

Tair ext ≤ Tair ≤ Tair max

(4)

RHair min ≤ RHair ≤ RHair ext

For the upper thermal boundary condition (5), hg is a global exchange coefficient that takes into consideration the convective and radiative heat transfer. The convective part hc is issued from

  Tair(k) = Tair(H )  Cmd if (k > HCmd ) →   RHair(k) = RHair(HCmd )

(9)

(10)

(11)

A. Fuentes et al. / Computers and Chemical Engineering 78 (2015) 70–78

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Fig. 3. Example of selected points for the CDG objective function calculus.

Fig. 2. Optimization flowchart.

(7). Thus, Fm remains non-null up to the end of the drying step. Consequently, Xt(k) must be always different and lower than Xi(k) . CDG =

The GA starts with the generation of an initial population of individuals (Fig. 2 – Step 0), which are composed of a random selection set of genes. Each individual of the population thus generated is applied as an input condition of the multiphysics model (Fig. 2 – Step 1). After numerical computations, an evolution of the average moisture content of the product is obtained for each subject of the population during the prediction horizon considered. The values of operating input parameters and average water contents obtained for each individual are then used to calculate the objective functions (Fig. 2 – Step 2), often named fitness functions. GA stopping criteria, based on a Pareto optimal population or a maximum generation reached, are then tested (Fig. 2 – Step 3). If the exit criteria of the GA are not reached, a new population is generated with the use of the genetic algorithm operators (Fig. 2 – Step 4), and this population is afterwards reintroduced as input data of the multiphysics model. The procedure is iterated until it is possible to obtain a Pareto optimal population. Once the stopping criteria of the GA are reached, an individual is selected on the Pareto surface using the weighted global criterion method (Marler and Arora, 2004). The retained individual is the one that obtained the smallest weighted sum of the objectives on the Pareto surface (16). Different weightings could be applied to objectives for this selection, based on various strategies (e.g. less energy consumption or a shorter drying time). Once an individual is selected on the Pareto surface, the predictive control strategy leads to retain only the first control action (k = 1) coded in the gene of the individual. Afterwards, the horizons are moved toward the future, with a defined ST, and the optimization procedure is restarted at the initial step (Fig. 2 – Step 0). Three dimensionless objective functions, bounded between 0 and 1, are considered for the optimization problem. The first criterion, calculated with Eq. (12), takes into consideration an average moisture content gap relative to a drying target (Fig. 3). The subscripts i and f correspond respectively to the initial and the final water content of the sludge issued from the optimization, and the subscript t corresponds to the target water content. The drying target, evaluated by computation, corresponds to the highest drying rate obtained by the use of the highest temperature and the lowest relative humidity of the drying air during the entire drying step. Furthermore, the drying step is always stopped before reaching the vapor pressure equilibrium between air and sludge surface

 Xt (k) − Xf (k)

HPred

1 HPred

k=1

Xt (k) − Xi (k)

(12)

The other two criteria penalize the heating (13) and the dehumidifying (14) of the outside air. Tair ext and wair ext are respectively the temperature and the specific humidity of the outside air. wair min and Tair max are respectively the lowest moisture content and the highest temperature of the drying air. The penalty dehumidification Cw is only evaluated if the specific humidity of the drying air is lower than outside (15). CTair =

 T (k) − T air air ext

HPred

1 HPred

k=1

Tair max − Tair ext

(13)

Cw (k)

 

1  wair ext − wair (k) HPred

Cw =

HPred

k=1

wair ext − wair min

If wair (k) ≥ wair ext : Cw (k) = 0

(14) (15)

The selection of the individual on the Pareto surface is based on the weakest weighting sum of objectives (16), with ˛, ˇ and , the positive weighting coefficients linked to the desired optimization strategy. min(˛CDG + ˇCTair + Cw )

(16)

In this study, the one-dimensional heat and mass transfer induces a water content gradient inside the product during drying. The aim of the reversal operation is to expose the bottom face of the sludge, where higher water content than the upper face is obtained, to the drying air condition. Thus, to simulate the reversal operation with the numerical model, the boundary conditions (5)–(7) and (8) are permuted. Two criteria are previously fixed, in order to define the ideal moment to realize the operation. Firstly, the bottom of the sludge must not be in a sticky phase (17), and secondly the evaporated mass flux (Fm ) must be decreasing (18) and below a prescribed value (19). Fm mini was established experimentally and correspond to a weak value of Fm  (18). X(0, t) ≤ Xsticky

(17)

d(Fm ) ≤0 dt

(18)

 Fm =

Fm ≤ Fm mini

(19)

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A. Fuentes et al. / Computers and Chemical Engineering 78 (2015) 70–78

Table 1 Initial and operating conditions.

Tsurf [°C] 60

22 ◦ C 1 m/s 25 ◦ C 70% ≈12 gwater /kgDA 0.125

Tini = Vair = Tair ext = RHair ext = wair ext = Xfinal =

Initial temperature of the sludge Air velocity in dryer Outside temperature Outside relative humidity Outside specific humidity Targetable final moisture content

Tsurf (Model)

Tsurf (Exp)

50 40 30

The resolution of the optimization problem is achieved with the Global Optimization Toolbox of Matlab® .

20 10

4. Results and discussion

0 0

The multiphysics model presented in (Section 2.2) has already been validated, with the same product, for a drying case without reversal operation (Louarn et al., 2014). One of the aims of this study is to assess the ability of the model to simulate a drying step with a reversal operation for fixed drying air conditions (Section 4.1) and variable drying air conditions obtained by the optimization procedure (Section 4.2.1). The confrontations between numerical and experimental results are based on the overall water content, evaporation mass flux, and surface temperature. The optimization procedure is first carried out with short horizons (HPred = HCmd = ST) (Section 4.2.1) and second with larger horizons (Section 4.2.2). Then, an energetic analysis is proposed in order to compare both optimization methods (Section 4.3). Initial and operating conditions used for this study are shown in Table 1. 4.1. Model validation with reversal operation For this experiment, the temperature and the relative humidity of the drying air are fixed respectively at 50 ◦ C and 11%. In these conditions, the reversal operation is performed after 10 h. The bottom temperature before the reversal operation is imposed at 60 ◦ C, and afterwards reduced to 40 ◦ C. The comparison between experimental and numerical drying kinetics shows the ability of the model to predict the global water content evolution of the sludge (Fig. 4). The computed surface temperature is also in agreement with the measured one (Fig. 5). The maximal gap observed is about 5 ◦ C, just after the reversal operation. Although the mass evaporation flux from the experiment is slightly noisy, it is possible to observe a good fit between simulation and experiment (Fig. 6). Moreover, we observe the advantage of making a reversal operation, notably given the presence of a mass flux peak that contributes to accelerate the

5

10

15

20

25

30

t [h] Fig. 5. Surface temperatures.

drying stage of the product. This is explained by the presence of a wet surface to ambient air after the reversal operation. All these observations show that the multiphysics model is able to predict correctly the thermal and hydric behavior of the sludge during a drying step with a reversal operation. 4.2. Optimal conditions The optimization procedure developed in this study allows tuning the operating input parameters (Tair , RHair ) every two hours (ST = 2 h) and the ideal moment for the reversal operation. A small sample time (ST) allows to find a compromise between total computational time and accurate description of the command policy. It should be noted that the optimization tool presented here takes into account the product only, independently of the technical resources that realize the drying air conditions. Hence, the selection of the individual on the Pareto surface is realized by giving equal importance to each criterion (˛ = ˇ =  = 1/3) (16). The drying kinetic target, used for the calculus of the cost function CDG , is established by considering Tair max and RHair min . The conditions used for optimization and simulation of the drying step are detailed in Table 2. 4.2.1. Optimal conditions from basic optimization The optimization procedure, conducted with the shortest horizons (HPred = HCmd = ST = 2 h), has led to the following command policy (Fig. 7), with a turnaround operation provided at the 16th hour (Fig. 8). First, we note that the moment of the reversal operation is different for the optimized conditions (16 h) compared to Fm

Xsludge

1

1 Xsludge (Model)

0,9

Xsludge (Exp)

Fm (Model)

0,9

0,8

0,8

0,7

0,7

0,6

0,6

0,5

0,5

0,4

0,4

0,3

0,3

0,2

0,2

0,1

0,1

Fm (Exp)

0

0 0

5

10

15

20

t [h] Fig. 4. Drying kinetics.

25

30

0

5

10

15

20

t [h] Fig. 6. Mass fluxes.

25

30

A. Fuentes et al. / Computers and Chemical Engineering 78 (2015) 70–78 Table 2 Operating and optimization conditions. 50 ◦ C 60 ◦ C 25 ◦ C 10% 70% 2h 1/3

Tb = Tair max = Tair min = RHair min = RHair max = ST= ˛ = ˇ = =

75

Xsludge 1

Imposed temperature at the bottom of the product Highest air temperature Lowest air temperature Lowest air relative humidity Higher air relative humidity Sample Time (ST) Weighting coefficients for the selection of the individual on the Pareto surface

Xsludge (Model) Xsludge (Exp)

0,9 0,8 0,7 0,6 0,5 0,4 0,3

Tair [°C]

RHair [%]

60

60

50

50

40

40

30

30

20

20

10

10

0,2 0,1 0 0

5

10

15

20

25

30

t [h] Fig. 9. Drying kinetics: experimental and computed.

Fm 1

Tair (Opt) RHair (Opt)

Tair (Control) RHair (Control)

0 5

10

15

20

Fm (Exp)

0,8

0 0

Fm (Model)

0,9

25

0,7

30

t [h]

0,6

Fig. 7. Optimal set points (Tair opt , RHair opt ) and applied control set points (Tair control , RHair control ).

0,5 0,4 0,3

the target conditions (14 h). The necessary criteria to authorize the reversal operation (17)–(19) are reached earlier with the target, due to more favorable drying conditions. Otherwise, a strong fluctuation of the optimal air temperature is observed, notably between the 10th and the 16th hour (Fig. 7). The poor anticipation of the drying gap evolution, in relation to the target, induced a fluctuation of the manipulated variables from one time step to the other. Afterwards, an experimental test is realized with the results from the optimization phase. Three levels are considered for the temperature and the relative humidity of the drying air (Fig. 7). These set points are then applied to the dryer as well as to the simulation. By comparing the experimental and numerical drying kinetics, we observe that the evolution of the average moisture content of the product is quite correctly predicted (Fig. 9). Nevertheless, a slight difference appears, justified by the difficulty for the dryer to follow the experimental set point modifications. As in the preliminary case (Section 4.1), we observe experimentally and numerically a consequent mass flux peak due to the reversal Xsludge

0,2 0,1 0 0

5

10

15

t [h]

20

25

30

Fig. 10. Mass fluxes: experimental and computed.

operation (Fig. 10). A good agreement concerning surface temperatures is also found (Fig. 11). All these observations show the relevance of the multiphysics model used in the reversal operation case with variable drying air conditions. 4.2.2. Optimal conditions with predictive control The results presented here come from an optimization phase with command and prediction horizons respectively equal to 6 h (HCmd ) and 10 h (HPred ). The identified air temperature values rise progressively (Fig. 12). It is noteworthy that the maximal temperature (Tmax ), fixed for the drying target, is reached a few times only. In Tsurf [°C]

1

60

Xsludge (Target)

0,9

Tsurf (Model)

Xsludge (Opt)

0,8

Tsurf (Exp)

50

0,7 40

0,6 0,5

30

0,4 20

0,3 0,2

10

0,1 0

0

0

5

10

15

20

25

t [h] Fig. 8. Target and optimal drying kinetics.

30

0

5

10

15

20

25

30

t [h] Fig. 11. Surface temperatures: experimental and computed.

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A. Fuentes et al. / Computers and Chemical Engineering 78 (2015) 70–78

Tair [°C]

RHair [%]

60

60

50

50

40

40

Tair (Opt with MBPC) RHair (Opt with MBPC)

30

30

20

20

10

10

0

0 0

5

10

15

t [h]

20

25

30

Fig. 15. Bottom water contents.

Fig. 12. Optimal set points (Tair opt , RHair opt ).

Xsludge

1

Xsludge (Target) Xsludge (Opt with MBPC)

0,9 0,8 0,7 0,6 0,5 0,4 0,3 0,2 0,1 0 0

5

10

15

t [h]

20

25

30

however, a slight difference noticed on the evolution of the average moisture content (Fig. 13). This gap is related to the drying air conditions only, due to an identical temperature imposed at the bottom of the sludge in both cases. The surface temperature obtained for the target progressively increases, whereas that obtained with the optimization procedure fluctuates in relation to the variation of the drying air temperature (Fig. 16). Overall, the temperature level is higher for the target. A decrease of the surface temperature is observed just after the reversal operation, simultaneously with the appearance of the second mass flow peak. The necessary heat to evaporate the water on the product surface, following the turnaround operation, is taken on the product. This phenomenon punctually decreases the surface temperature during the 16th hour.

Fig. 13. Drying kinetics.

4.3. Energetic analysis parallel, the relative humidity values decrease, but the limit value used for the drying target (10%) is never reached. By comparison with the basic optimization case (Section 4.2.1), with short horizons, a prediction horizon of 10 h has reduced the command oscillations (Fig. 12) while achieving a better pursuit of the target (Fig. 13). In these optimal conditions, the goal sludge water content can be obtained faster (25 h) than in the case with short horizons (30 h). Evolutions of the evaporation mass fluxes between the target and optimization case with predictive control are quite similar (Fig. 14), except concerning the first peak (t = 2 h). The value of the first peak observed for the target conditions is explained by the initial drying air conditions at higher temperature and at lower relative humidity than the optimization case. Concerning the reversal operation, the fixed conditions (17)–(19) are reached at the same time (t = 14 h). The bottom water contents are similar in the target and the optimization case with predictive control (Fig. 15), with,

The conductive and convective heat fluxes are compared for the three drying cases previously examined. For both optimization cases, the conductive fluxes are higher than the target, owing to a lower overall temperature of the sludge (Fig. 17). At the reversal moment, a conductive peak is systematically observed. This phenomenon is due to a bottom temperature (previously surfacic) lower than the imposed temperature (Tb ). Globally, the conductive fluxes issued from optimization are almost always positive, in contrast to the target for which they become negative as early as the 18th hour of drying. A negative value of the conductive flux means that the product becomes locally warmer than the imposed bottom temperature. For the basic optimization case, with short horizons, the convective flux before the reversal operation (t ≤ 16 h) is lower, due to the low level of the drying air temperature (Fig. 18). In contrast,

Fig. 14. Evaporation mass fluxes.

Fig. 16. Surface temperature.

A. Fuentes et al. / Computers and Chemical Engineering 78 (2015) 70–78

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Table 3 Energetic indicators. Numerical results

 t=TDT (Tair (t) − Text )dt t=0 t=TDT

Total drying time (TDT)

t=0

(wext − wair (t))dt

Basic optimization case

Optimization with predictive control

Target

30 h (+28%)

25.5 h (+8%)

23.5 h

644 ◦ C h (−22%)

696 ◦ C h (−15%)

823 ◦ C h

74.5 (gwater /kgDA ) h (+114%)

2.8 (gwater /kgDA ) h (−92%)

34.9 (gwater /kgDA ) h

Fig. 19. Total heat fluxes.

Fig. 17. Conductive heat fluxes at the bottom of the product.

the conductive flux is three times greater than the target until the reversal moment. Even if the total flux after the reversal operation (16 h < t ≤ 24 h) appears to be higher than the target (Fig. 19), the drying gap previously generated cannot be made up (Fig. 8). The lack of anticipation of this optimization method does not enable following the drying target closely. The analysis of the convective fluxes from the optimization with a predictive control shows that apart from the first peak, the levels reached are quite similar to the target (Fig. 18). During the reversal operation, the convective fluxes are almost the same. The conductive fluxes are also very close to the target (Fig. 17). Despite the first peak, the sum of the fluxes absorbed by the sludge is quite similar in the target and the optimization case with predictive control (Fig. 19). Consequently, the identified air temperature policy first leads to a convective flux that limits the drying gap toward the target and second, avoids overheating the product in order to restrict negatives values of the conductive flux. The time required to obtain the final water content is 23.5 h for the drying target. The drying takes an extra 2 h for the optimization with predictive control and an extra 6.5 h for the basic optimization (Table 3). The calculus of an integral, on the drying time, of the temperature difference (Tair − Text ) can provide an indicator bound

Fig. 18. Convective heat fluxes at the surface of the product.

to the required energy for air heating. A lower value of this energetic criterion 696 ◦ C h for the predictive control case can be obtained instead of 823 ◦ C h for the target. This result shows that a minimum energetic optimization of about 15% can be realized. The final cost of dehumidifying is also estimated with the calculus of an integral, by taking into account only the specific humidity values that are lower than the outside hygrometric conditions (wair ext ). The results of the optimization with predictive control show that the drying stage with almost no dehumidification of the outside air could be possible. In this way, a strong reduction of the energetic cost could also be realized. In contrast, the basic optimization case leads to a great air dehumidification (+114%) at the end of the drying stage, in order to catch up the generated drying gap with the target. From this criterion, it appears worthwhile to perform optimization with a sufficient prediction horizon.

5. Conclusion In the present study, an optimization procedure is proposed and applied to a sludge drying operation. The procedure is based on a predictive control strategy combined with a multiobjective optimization problem solved with a genetic algorithm. A heat and mass transfer model, also validated in this study, is coupled to the procedure as a prediction model. For the drying operation concerned, the dynamic optimization procedure has led to an offline determination of the optimal operating input parameters (Tair , RHair ), together with the ideal moment for the reversal operation. The determined command policies show significant advantages over constant operating input parameters, with notably a reduction of the costs linked to the drying operation. Moreover, for the particular case under investigation, it has been shown that the energy consumption for heating the drying air can be reduced up to 15% and the dehumidification is strongly reduced. When comparing both proposed optimization methods, we observe that the optimization with predictive control improves the quality of the target tracking. The proposed control algorithm can be easily adapted to other batch processes. In addition, on-going work is considering a faithful experimental implementation of the optimal operating input parameters.

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