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Mathematical modelling and optimal control of solar dryers

Mathematical modelling and optimal control of solar dryers Mathematical Modelling of Systems 138 1 -2424/97/0304-297$12.00 1997, Vol. 3, No. 4, pp. 297-322 @Swets & Zeitlinger Mathematical Modelling and Optimal Control of Solar Dryers E. GALLESTEY * AND A.D.B. PA ICE^ ABSTRACT A mathematical model of the solar dryers designed and used in Cuba is presented and discussed. The model consists of a set of ordinary nonlinear differential equations which describe the behaviour of the main parts of the dryer. Using the model, an optimal control problem is for- mulated and solutions within the class of piecewise constant controls are numerically obtained. It is shown that controls which are constant over intervals of 24 hours have the best properties for application to drying plants. Keywords: drying, modelling, optimal control, robust control, solar energy. 1 INTRODUCTION Drying is an important industrial process in the Cuban economy. It is present in such fundamental areas as conservation of agricultural products, conditioning of wood and production of animal food, among others. Traditional dryers work on conventional sources of energy, using large amounts of fuel in maintaining constant internal tem- perature levels. The exploitation costs are very large and too expensive for a small country like Cuba. In the Solar Energy Research Center of Cuba a family of dryers which uses the sun as a source of energy has been designed. The intention was to substitute the conventional energy source by the more ecological and easily available solar energy. The results have been very encouraging. A solar dryer works on the same principle as greenhouses do. It consists of a glass house with black plates under the glass for absorbing the solar radiation. The glass has the property that it does not allow the long wave radiation from the plates to return *Solar Energy Research Center of Cuba, Cuban Academy of Sciences, Havana, Cuba. E-mail: eduardo@mathematik.uni-bremen.de htitute for Dynamical Systems, University of Brernen, Bremen, Germany. E-mail: andrew@mathematik.uni-bremen.de E. GALLESTEY AND A.D.B. PAICE Fig. 1. Solar dryer prototype. to the environment, but at the same time permits solar radiation to enter. The metallic plates form a cabin, inside of which the product to be dried is placed. Fans move the air in order to increase the mass and heat transfer coefficients of in the cabin (See Fig. 1). Maximal temperatures between 5S°C and 65OC are frequently measured within the dryer and drying processes of good quality are obtained. In Fig. 2 graphs representing experiments performed in Cuba with the dryer charged and empty are depicted. The qualitative behaviour of the installation can be easily observed. A more detailed explanation may be found in [I], [2]. Nevertheless, one problem remains unsolved. The air inside the dryer must be periodically exchanged in order to avoid saturation and to extract the evaporated water. Naturally, this causes an energy loss which should be minimized in order to maintain maximum drying efficiency. Thus we arrive at the following optimal control problem: How much air should be extracted at each moment in order to reach the highest efficiency? This is a difficult problem, due to the unpredictable character of the environmental conditions and the strong and complicated correlations between the parameters which determine the dryer behaviour. In this paper we examine various approaches to the solution of this problem, and we evaluate the various approximate solutions obtained in terms of robustness of the achieved performance with respect to changes in the environmental and initial conditions. This paper is organized as follows. First, in Section 2, a mathematical model of the dryer is developed and discussed. It is seen that the output predicted by the model for given environmental conditions closely models that observed in a real dryer. In SOLAR DRYERS 299 Empty Dryer Temperature Charaed D~er Tem~erature Wood Humidity (%) lo 0 0 60 0 100 10 10 160 1;O Time (hr) Fig. 2. Drying of wood. Experiments performed in november 1990 at the Solar Energy Institute, Santiago de Cuba, Cuba. Section 3 some numerical experiments are performed using the dryer model. The aim is to develop an intuition regarding the performance of the dryer given different initial conditions and environmental influences. Section 4 is dedicated to the study of linearizations of the presented model about trajectories with constant controls. This section should increase our understanding of the problem we face and at this point we will be in the position to formulate the full optimal control problem. In Section 5, making use of the obtained results, the optimal control problem is precisely stated and strategy for determining the solution is fixed. Further, piecewise constant controls are studied, and it is shown that from a practical point of view the best possible piecewise constant control is constant over intervals of 24 hours. In Section 6 the goals of the paper are summarized and recommendations for the application of the obtained results are made. 2 THE MATHEMATICAL MODEL 2.1 Differential equations of the model As a first step, a mathematical model for the dryer has been developed. It consists of a set of differential equations which were derived paying attention to the energy and mass balance in the main parts of the dryer. They appear as follows: E. GALLESTEY AND A.D.B. PAlCE Fig. 3. Scheme of the dryer. 1 : upper absorber plate; 2 : east absorber plate; 3 : west absorber plate; 4 : walls; 5 : product; 6 : dryer air Where the following notation has been used: t : Time (hours); Tl : Temperature of the upper absorber plate (OC); SOLARDRYERS T2 : Temperature of the east absorber plate (OC); T3 : Temperature of the west absorber plate (OC); T4 : Temperature of the walls (OC); T5 : Temperature of the product (OC); T6 : Temperature of dryer air (OC); H : Water content in the dryer air (kglkg); A : Water content in the product (kg); Cij : Transfer coefficients, which are positive constants defined by the physical laws governing the process. They are obtained from formulas used by engineers in calculations related to heat transfer. These formulas are well established, which makes us confident in the values of the calculated coefficients, see for example [9]. For all Cij, except C12, C42, CS1, C52, the coefficients are given by where Nu is the Nusselt number, 0.032~e~.~ if Re < lo5 Nu = [ 0.669~eO.~ otherwise pi is the thermal conductivity of the ith component, li is the length of the surface i taken in the direction of the air flow, v,;, and vatr are the cinematic viscosity and the bulk velocity of the air inside the cabin respectively, cpi is the specific heat of component i, and mi is the mass of i. The coefficients Csl and C42 may be calculated by modified versions of (9). In the case of C42, for example, a term should be added which accounts for the thermal conductivity of the composite wall, see [9]. Coefficients C12, CS2 model the only heat transfer process by radiation which has been introduced in the model. They have the form E. GALLESTEY AND A.D.B. PAICE where E, = 0.95 and E, = 0.43 are the emissivity of the upper plate and of the product respectively. The value of E, is a property of the product and was determined experimentally. u is the Stefan Bolzman constant. h : Latent heat of vaporization; c,, , c, : Specific heats of the product and water; md: Dried mass of the product; Q : Heat obtained from the fans; U(t) : Extraction rate (bounded control, 0 5 U(t) 5 U,,,); G1 : Evaporation rate, defined by the equation: a : A slowly time varying positive parameter. It represents the "error" caused by considering the drying process to be pure water evaporation. For the moment we will work with a! a constant, which is true during the first phase of the drying process, and return to this problem later. G2 : Extraction rate of humidity per unity of volume, defined by the equation: where V(t) represents the water contents of the air per unit of volume in the ambient. This is given in the model as: Pus (T) : Saturated vapor pressure. H, : Relative ambient humidity. Sl (t), S2(t), S3(t) : Energy absorbed by the upper, east and west plates respec- tively. For details in their derivation see [5]. Ta(t) : Ambient temperature. A given function of time. SOLAR DRYERS 303 Fl (TI , To), F2 (T2, To), F3 (T3, To) : Heat transfer coefficients through the glass, given as follows (see [3]): where: C = 520(1 - 0.000051@), Ni : Number of sheets of glass in cover i; h,, : Heat transfer coefficient from the glass to the environment by convection; r, : Glass emissivity; pi : Tilt angle of the glass (in degrees); Remark 1: The model is of high order. Nevertheless, models of lower order do not fit the data satisfactorily. For example, a model obtained by unifying equations (1)-(3) in one equation has not been successful. An explanation for that can be found in the fact that the behaviour of the three absorbing plates is very different during the day. The first author has completed a lengthy series of investigations into the modelling of this system, and has come to the conclusion that only systems of higher order may perform as well as this one in modelling the observed behaviour of the dryer. Another possibility would be to replace some equations with algebraic constraints. See Remark 2 in Section 4 for further comments on this approach. It is important to note that: The system is nonlinear in the states. The state equations are affine in the control. 0 The control is bounded from above and below. E. GALLESTEY AND A.D.B. PAICE alpha 1\ Equlibrium Humidity 100 Pmduct Humidity (%) Fig. 4. Approximate behaviour of the coefficient a during the drying process. 2.2 Assumptions made about the physical process In order to derive equations (1)-(8), the following assumptions have been made: 0 Assumption 1: All the transversal sections are equivalent. Assumption 2: There is a strong turbulent convection of the air. This assumption allows us to assume the homogeneity of the air at every moment and permits us to consider that the heat and mass transfers process inside the dryer are realized only by convection. Only one radiation coefficient is taken into account, the heat transfer between the product and the upper plate. Assumption 3: The product surface is always wet. That means that the dryer process occurs slowly enough to permit the use of the simplified expression (15) for the evaporation velocity, or more precisely, to take a! as a constant close to 1. For design purposes this assumption is not very important, due to the generality of the cases which must be taken into account, however for optimal control purposes this assumption may be false and can lead to incorrect results. The parameter a! depends in a very complicated way on the product itself, the product humidity, and the drying process conditions. That is why, in designing a controller for a real process, a! has to be considered a time-varying hardly measurable parameter. Its approximate behaviour during the drying process can be seen in Fig. 4. SOLAR DRYERS 305 Time (hr) Time (hr) 50 I I 1.5, 0 5 10 15 20 Time (hr) Time (hr) ..\\1 0 5 10 15 20 Time (hr) Time (hr) Fig. 5. Examples of graphs obtained with the model. 2.3 Typical outputs of the model All the main parts of the dryer and their interactions are reflected in the model. It offers complete information about the temperatures of the main parts of the dryer and the water content in the air and product. It also permits the observation of quantities which are hard to measure such as the energy losses through the distinct parts, the evaporation velocity, etc. Typical graphs obtained with the model are shown in Fig. 5. 2.4 Model validation The model has been tested and its ability to predict the dryer dynamics has been proved. The experiments were made in March-July 1994 in Santiago de Las Vegas, Ciudad de la Habana, Cuba, using water for modeling the product (a, = 1). Graphs which represent the model predictions and the measured data can be found in Figs. 6 and 7. As can be seen, the model follows the measured data almost perfectly. It has to be pointed out that a better agreement between the modeled and measured data could be obtained if more detailed measurements of the solar radiation were available at the location where the experiments were carried on. The model a function of time (the functions Si), which requires the instantaneous irradiance as is difficult and expensive to obtain. This fact made the use of certain interpolation procedures necessary in order to obtain approximations of the real Si, which, due to space considerations, are not commented on here. 306 E. GALLESTEY AND A.D.B. PAICE 6 40 +g & 40 5 30 020 5 10 15 20 25 30 35 40 45 Time (hr) Time (hr) Fig. 6. Comparisons between the exoeriment verformed on Aoril4th and 5th (0) and calculations 5 10 15 20 25 30 35 40 45 Time (hr) 5 10 15 20 25 30 35 40 45 Time (hr) Fig. 7. Comparisons between the experiment performed on July 4th and 5th (0) and calculations (-) At this moment the mathematical model is being used in designing and optimizing new dryers at the Solar Energy Research Center in Cuba. It is inserted in a computer user interface, written by the first author, which allows a comfortable simulation process. SOLAR DRYERS 3 SIMULATION STUDIES Before trying to implement some of the methods of Control Theory, we wish to get an idea of the dryer responses resulting from controls which seem reasonable from the point of view of the physical process. We are going to look for the different dynamics caused by changes in the parameters, initial states conditions, environmental conditions, etc. This will improve our knowledge about the dryer and should permit a better orientation in the problem. This goal will be achieved by using the simulation program Gnans [7], which offers all the necessary facilities. The simulations were performed using the standard solar radiation, corresponding to a location in Havana. The solar radiations curves were derived using methods developed at the Solar Energy Research Center, which give trustworthy average month values for the solar radiation. See [5] for details. In order to fix a configuration during the simulations, parameters which define the dryer and its environment were chosen. Some of the most important are listed in the Table 1. Unless otherwise specified, these are the parameters used. In the same table the initial conditions are depicted. Table 1. Dryer parameters and initial conditions used during the simulations Parameters Initial Conditions Parameter Value Units State Value Units length 1 rn final time 24 hour width 3 rn TI 33 O C height 2 rn T2 33 OC roof angle 10 grades T3 33 O C wall height 1 rn T4 33 OC month April Ts 33 OC extraction 60 rn3/hr T6 33 O C initial humidity 80 % H 0.02 kglkg product specific heat 5.1 k J/ kg/OC A 70 kg 3.1 Changes in the initial conditions Not all the initial conditions have the same impact on the system behaviour. We know from the physics of the process that the initial conditions in the states which do not have large storage capacities have little influence on the long term dryer behaviour. This is the case for the temperatures of the plates and the air TI, T2, T3, T6, (it is known that the plates and the air have a very small heat capacity), and the humidity of the air inside the dryer H (air exchange with the environment makes any initial 308 E. GALLESTEY AND A.D.B. PAICE 0.5 1 1.5 2 Time (hr) Time( hr) Fig. 8. Comparisons between the effects of changes in the initial conditions of TI, T2, T3, T6 and H. Each curve is obtained by changing the corresponding initial condition, while leaving the others unchanged. In the last graph ten curves have been drawn. humidity value within the dryer converge to a physically realizable value). This means that these states are completely determined by the dynamics of the process. Some example calculations illustrate the above comments, Fig. 8. As can be seen no matter which initial values these states have, they converge quickly to a trajectory fixed by the dynamics of the system, without any strong influence on the dryer performance. More important appear to be the initial conditions of T4, T5 and A. Their effect lasts longer and in the case of T5 even seems to be important during the first hours. See Fig. 9. The most important conclusion of this section is that slow and fast dynamics coexist in the system (two time scales). This fact should be taken into account in the future numerical treatment of the problem. It can thus be concluded that, except for the water content of the product, the states of the dryer depend mainly on the environmental conditions within the last few hours. Because of the fact that the environmental influences are well predictable only for the next 24 hours, and the fact that the product water content can be measured only once or twice a day, we are going to study the dryer and its optimization only in periods of 24 hours. 3.2 Changes in the environmental conditions The environmental conditions play a fundamental role in the dryer behaviour. As can be appreciated from the results depicted in Fig. 10 any changes in the ambient, which '0 SOLARDRYERS 309 "0 5 10 15 20 Time (hr) 1.5, - '" d t--..'l 9 5 , ' o----- I I 0 5 10 15 20 0 5 10 15 20 Time (hr) Time (hr) .- - - _ _ ziOm d 5 0 5 10 15 20 Time (hr) Time (hr) Fig. 9. Comparisons between the effects of changes in the initial conditions of T4, T5 and A. are modeled in the system through the functions T, (ambient temperature), V (ambient absolute humidity), and So (radiation level on the horizontal surface), leads to changes in the amount of evaporated water on that day. This effect can be most clearly observed in the dependence of the evaporation on the solar energy, which seems to be, as may be expected, the fundamental parameter. 3.3 Simulations with constant controls Constant controls are of special interest because they are easy to implement in practice and relatively easy to study from a mathematical point of view. A detailed study of the optimal constant controls should increase our understanding of the problem and allow a better formulation of the mathematical problem. The most straightforward and natural dryer performance index is the total daily evaporation and in this section we will understand as optimal the constant controls which maximize this performance. We want to study in this section how this dryer performance can be improved by using constant air extractions. Optimal constant controls corresponding to different initial and environmental conditions were calculated. Attention was also paid to the interrelation between the parameter a from (7) and the optimal constant controls. Dependence on environmental conditions In Fig. 11 we can observe the dependence of the evaporation on the constant controls. For small extraction values the dependence is strong. But after a certain E. GALLESTEY AND A.D.B. PAICE , /.1 L 5 0 --- 0 5 10 15 20 Time hr T~me (hr) q , 4 e25 2 20 __-- . , I' 150 5 10 15 20 0 5 10 15 20 Time (hr) Time (hr) 2 5 '0 5 10 15 20 Time (hr) Time (hr) Fig. 10. Effec Evaporation Month Evaporation 0 50 100 Extraction Month Better Extraction Month Maximal Evaporation ::I 246810 246810 month month Fig. 11. Dependence of the optimal constant control on the environment. value, the curves become very flat and, although the optimal control exists and it strongly depends on the month, larger values than the optimal do not substantially affect the performance of the dryer. SOLAR DRYERS 311 20 30 40 50 extraction lnitial Conditions on T4 I<, g 55 .- 0 50 100 20 30 40 50 extraction Initial Conditions on T5 10, 0 1 70 0 80 100 120 0 50 100 extraction lnitial Conditions on A Fig. 12. Dependence of the optimal constant control on the initial conditions. Left: Evaporation vs. extraction for each initial condition, while the others initial conditions remain unchanged. Right: Best optimal control for each initial condition. Dependence of the constant control on initial conditions As shown in Section 3.1, the only initial conditions able to significantly influence the dryer dynamics are those of T4, T5 and A. In Fig. 12 we examine the dependence of the optimal constant controls on these initial conditions. It can be observed that the same situation as in the above paragraph is found: The optimal controls are very different for each initial condition, but the improvements in the performance from one control to another are very small. Dependence of the constant control on the parameter a In order to test the sensitivity of the value of the optimal constant control on the parameter a from (7) different values of a were used and the optimal control for each of them was calculated. The results of the calculations are summarized in Fig. 13. In the graphs it is easy to see the close dependence of the control on a. This characteristic is accentuated when alpha takes values between 0.1 and 0.6, which corresponds to the last days of the drying process (See Fig. 4). An optimal control strategy for the last days of drying should take into account this fact. 3.4 Summary of the simulations results Through the simulation study of the model the following characteristics of our problem have proven to be important: E. GALLESTEY AND A.D.B. PAICE Best Extraction (m"3Ihr) Daily Evaporation (kg) so1 8 0 50 100 extraction (m"3hr) Fig. 13. Dependence of the optimal constant control on the parameter a. Left: Curves with constant a. Right: Optimal Extraction for each a. Fast and slow dynamics are present in the model. The slow dynamics have a time constant of the order of 2 hours, which is short in comparison to the run time of the process, which is at least 5 days. The optimization problem can be regarded as a finite horizon problem. We may choose a final time, which we call tf. tf will be chosen as 24 hours, since the environmental influences are approximately predictable with this period. The system is mainly influenced by the environment, which cannot be a priori determined. The optimal controls obtained have to be analyzed with regard to robustness. The second most significant source of perturbations in the model is the parameter a. 4 LINEARIZATION OF THE MODEL The degree of stiffness of the model will play a fundamental role in the method to be used in solving the optimal control problem. In trying to learn more about it, a SOLAR DRYERS 313 model obtained by linearizing the equations (1) to (8) about trajectories with nominal constant controls has been derived and tested. In this paragraph the following notations will be used. xo : The standard initial conditions of the model used before, taken from Table 1. u*: A given nominal constant control. x*(t) = 4(t, xo, u*): Solution of the full system defined by the equations (1) to (8) corresponding to the control u* and initial condition xo. S(t, x*(t), u*): Right sides of the equations from (1) to (8 ) evaluated at the nominal trajectory x*(t) and the control u*. u(t) = u;(t) - u*: The difference between the actual control uz(t) and the nominal control u* . xl(t) = @(t, xo, u) : Vector of states of the linearized system. It is a linear approximation of the deviation of the state x (t) = 4 (t, xo, uz) from the nominal state x*(t) = #(t, xo, u*) and is a solution of the following linear system of equations. as A(t) = -(t, x*, u*), ax as ~(t) = -(t, au x*, u*), XL = XL(~, XO, uz, u*) = x*(t) + @(xO, uz - u*): This will be called the "solution of the linearized system". It was found that the matrices A(t) and B(t) appearing in (22) are slowly time varying and that the linear system (22) is a good approximation of the full dryer model. A comparison of the behaviour of the linearized model with that of the full model is given in Fig. 14. The four graphs compare the trajectories x* = 4(t, xo, 40) generated by the full model for u* = 40m3/ hr from one side, and the solution of the linearized system xL(t, XO, 40,60). AS may be seen, there are no significant differences between the solutions. This increases confidence in conclusions made on the basis of studies of the linearized model. By examining the eigenvalues of A, we can quantify how ill conditioned our problem is. The condition number (Ih,,,/h,i, I) of the linearized system as a function of time, together with the maximal and the minimal eigenvalues is depicted in Fig. 15. E. GALLESTEY AND A.D.B. PAICE - U=40 ... Linearization 0 5 10 15 20 Time Fig. 14. Comparisons between the solution to the linearized model and the of full one for u:(t) = 40. The linearization was made about u*(r) = 60. Only the real parts of the eigenvalues were drawn, the imaginary parts are of order It can be seen that the condition number reaches large values ( lo7) which implies that an attempt to use techniques such as Pontryagin's Maximum Principle will lead to numerical difficulties which are hard to avoid. Remark 2: A method to avoid the ill conditionedness of the problem is to reduce the order of the system by converting the fast dynamics present in equations (1)-(3) and (7) into algebraic constraints. Nevertheless, this leads to a complex system of time varying differential algebraic equations (see for example the expressions in (18)) which would be very difficult to solve. It is believed that the gain in "well conditionedness" of the model is not enough, at least at this stage of the investigation, to justify the difficulties in finding solutions of such a reduced model. 5 THE OPTIMAL CONTROL PROBLEM At this point we are in a position to define our optimal control problem. Of course, we should take advantage of what we have learned about the dynamics of the dryer during the simulation experiments. The plant is already designed, it is believed in the best possible way, although this could also be a matter of study, and the only control available to influence the process is to extract the air within the dryer. SOLAR DRYERS 0 5 10 15 20 25 Time (hr) Fig. 15. Largest and smallest eigenvalues of the matrix A as a function of time. 5.1 Optimal control problem formulation The behaviour of the optimal constant controls indicates that the range of values in which the optimal controls may take on can be easily obtained in the practice, see [4]. Taking this into account, we do not penalize large control values in the performance index. Thus the following simple expression has been chosen. where A is the same as in equation (8). This corresponds to the requirement of obtaining the maximum possible evaporation on the selected time interval [0, tf]. As the control is to be implemented by a fan, we restrict ourselves to considering the class of controls to be piecewise continuous. The optimal control problem may now be formulated as follows: Find the control U* : [0, tf] -+ R such that J(U*) 5 J(U) forall other piecewise continuous U : [0, tf] -+ R. J(U) is taken from equation (23),and is subject to constraints corresponding to equations (I), ...,( 21) with initial conditions taken from Table I. Problems of this general class have been studied for many years. A classical approach is to apply the Pontryagin Maximum Principle or the Bellrnan Equation, for an introduction see for example [6]. 316 E. GALLESTEY AND A.D.B. PAICE In the first approach, it is necessary to derive the associated Hamiltonian system, where the adjoint system satisfies a terminal condition. This would require a numerical treatment of the problem as an analytic solution is unattainable. In our case, difficul- ties in this approach were encountered, due to the stiffness present in the system of differential equations. Another approach is to approximate the class of allowable controls by piecewise constant controls with fixed switching time intervals and solve the associated optimal control problem. This has the advantage of converting an optimization problem over an infinite dimensional Banach space into an optimization problem involving only a finite number of parameters. The experience collected in studying the system response to various environmental and initial conditions, and the study of the linearized system suggests that before attempting a difficult classical method, suboptimal controls should be considered. Here a suboptimal control is understood to be the control derived by considering the optimal control problem over a restricted class of controls, which is chosen to reflect the characteristics of the controller in the real system. This should at least clarify whether or not the improvement of the performance index through the optimal controls is of sufficient significance to justify the amount work required in finding them. 5.2 Optimal piecewise constant controls We now investigate the system's behaviour obtained by considering piecewise constant controls. We proceed as follows: The time interval of interest [0, tf] is divided into m equal subintervals, and during each of these subintervals a constant valued control is applied to the system. More explicitly, we define a cost function G, : lRm t lR, Gn,(un') = J(U) = A(tf), (cf. equation (23)) where u E lRm is a vector with components u; equal to the value of the extraction on the interval i and U is a control defined in by the following expression: Our optimization problem is now to find the minimum of the function Gm(.) subject to the restrictions 0 5 ui 5 Urn,, , i = 1 . . . m, and to compare the solutions for different m. In order to solve this problem, a program has been written in C++ which calculates the solutions to this problem making use of the facilities of the NAG Fortran Library [8], see Fig. 16. The suboptimal controls and the pe8ormance index It is easy to see that any physically implementable control may be approximated by the control defined by u E lRm, for large enough m. Thus, although the optimal piece- SOLARDRYERS Suboptimal Piecewise Constant Controls Fig. 16. Piecewise suboptimal controls. wise constant control may not in general lie in our class of controls, from a practical point of view it may be sufficiently well approximated by the control associated with u* E IRnl, where G,(u*) is the minimum. By increasing rn, we obtain controls closer to the optimal and can evaluate the improvements in the performance of the dryer. Due to limitations of the minimization algorithm, which appear when rn becomes very large , we were able to increase rn only up to 480 which is equivalent to intervals of 3 min of real time. From a practical point of view, we believe this is sufficient to determine an implementable control which approximates the optimal. In Fig. 16 the controls which minimize the functions G, corresponding to m = 1 (intervals of 24 hours, corresponding to constant controls), rn = 24 (intervals of 1 hour) and rn = 240 (intervals of 6 minutes) are depicted. They were calculated for a standard day in April, with the same parameters as used previously (see Table 1). In Fig. 17 we see the increments in performance for these three controls. We find that there is not a strong difference between the values of the performance index achieved by the different controls. Robustness of the suboptimal controls Another argument for choosing the control for the dryer should be the robustness of the control with respect to uncertainties in the environmental and initial conditions. We do not expect invariance of the dryer behaviour with respect to the environmental conditions, however, we want to implement the control which best "desensitizes" the dryer performance to uncertainties in weather conditions (cloudiness, median ambient E. GALLESTEY AND A.D.B. PAICE Daily Evaporation Time (hr) Fig. 17. Performance for each of the controls. temperature and median ambient relative humidity) and in the humidity of the product (the parameter a). We have calculated the sensitivity of the performance index with respect to these factors for each of the three selected suboptimal controls. The sensitivity to aparameter p is understood to be the partial derivative of the performance index with respect to p evaluated at a given value p,. For example Fig. 18 plots the performance index E,) against the.cloudiness (kh), and the sensitivity 3 against kh, while (evaporation, all other parameters were held constant at the standard values. These derivatives were obtained by using numerical differentiation routines included in the NAG Fortran Library. The results can be seen in the series of Figures 18, 19,20 and 21. It can be appreciated that in three cases (cloudiness, the parameter a and ambient humidity) the moduli of the derivatives are smaller for the constant control than for the other more sophisticated suboptimal controls. Only with respect to the ambient temperature is the contrary situation observed, albeit with very small values for the derivative moduli. That means that the constant controls are at least as robust as the others. At the same time, it should be pointed out that the observed improvements of the performance index do not encourage an attempt to implement controls more complicated than the constant ones. SOLARDRYERS Daily Evaporation Sensitivity of the Evaporation 12.57 0 0.5 1 cloudiness (kh) Fig. 18. Evaporation and sensitivity with respect to the cloudiness (kh). Daily Evaporation Sensitivity of the Evaporation 9I 0.5 1 alpha alpha Fig. 19. Evaporation and sensitivity with respect to cr. E. GALLESTEY AND A.D.B. PAICE Fig. 20. Evaporation and sensitivity with respect to the ambient humidity. Daily Evaporation Sensitivity of the Evaporation Fig. 21. Evaporation and sensitivity with respect to the ambient temperature. SOLARDRYERS 32 1 6 CONCLUSIONS At this point, we are in a position to give final recommendations for the control of solar dryers. It has been found that constant controls over a 24 hour period unite the charac- teristics desired of a controller in optimal way: the achievement of high levels of performance, simplicity in implementation and the best possible robustness. The con- trols may be calculated for a variety of possible environmental conditions and then put in an appropriate form at the disposition of the industry, for example in tables de- termining the recommended control corresponding to given weather conditions, water content of the product and a given product (estimation of the parameter a). In this paper the following goals have been achieved: 0 An exhaustive mathematical model of solar drying plants has been presented and validated. With the aid of this model, it has been shown how to obtain controls which optimize the efficiency of these plants. 0 Programs have been developed which make the calculation of the above men- tioned controls for each new configuration highly efficient, making possible the immediate application of the obtained recommendations in future dryer models. ACKNOWLEDGEMENTS We thank Prof. Dr. Luis BCrriz PCrez of the Solar Energy Research Center of Cuba and Prof. Dr. Diederich Hinrichsen of the University of Bremen for the many fruitful discussions during the development of this work. The Deutscher Akademischer Austauschdienst is gratefully acknowledged for the financial support given to the first author. REFERENCES 1. L. BCmz and D. Degas.: Secador Solar de Madera. Editorial Comision Nacional de Energia, Havana, (1990). 2. L. BCrriz and D. Degas.: Secador Solar MultipropBsito. Editorial Comision Nacional de Energia, Havana, (1 990). 3. J. Duffie and W. Beckman.: Solar Engineering of thermal processes. John Wiley & Sohn, New York, (1991). 4. H. P. Garg.: Advances in Solar Energy Technology. D.Reide1 Publishing Company, (1987). 5. M. A. Guerra and M. R. Vazquez.: Manual de Radiacion Solar para la Republica de Cuba. Editorial Academia de Ciencias de Cuba, Habana, (1992). 6. F. L. Lewis and V. L. Syrmos.: Optimal Control. John Wiley & Sons, New York, (1995). E. GALLESTEY AND A.D.B. PAICE 7. B. Mirtensson.: Gnans: A program for stochastic and deterministic dynamical systems, reference manual. Report 300, Institute for Dynamical Systems. Bremen Univer- sity, Bremen, (1993). 8. Mark 16 NAG Fortran Library Users' Guide.: Numerical Algorithm Group. Inc. Oxford, (1994). 9. M. Mikheyev. Fundamentals of Heat Transfer. Mir Editions, Soviet Union, (1978). http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Mathematical Modelling of Systems Taylor & Francis

Mathematical modelling and optimal control of solar dryers

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Mathematical Modelling of Systems 138 1 -2424/97/0304-297$12.00 1997, Vol. 3, No. 4, pp. 297-322 @Swets & Zeitlinger Mathematical Modelling and Optimal Control of Solar Dryers E. GALLESTEY * AND A.D.B. PA ICE^ ABSTRACT A mathematical model of the solar dryers designed and used in Cuba is presented and discussed. The model consists of a set of ordinary nonlinear differential equations which describe the behaviour of the main parts of the dryer. Using the model, an optimal control problem is for- mulated and solutions within the class of piecewise constant controls are numerically obtained. It is shown that controls which are constant over intervals of 24 hours have the best properties for application to drying plants. Keywords: drying, modelling, optimal control, robust control, solar energy. 1 INTRODUCTION Drying is an important industrial process in the Cuban economy. It is present in such fundamental areas as conservation of agricultural products, conditioning of wood and production of animal food, among others. Traditional dryers work on conventional sources of energy, using large amounts of fuel in maintaining constant internal tem- perature levels. The exploitation costs are very large and too expensive for a small country like Cuba. In the Solar Energy Research Center of Cuba a family of dryers which uses the sun as a source of energy has been designed. The intention was to substitute the conventional energy source by the more ecological and easily available solar energy. The results have been very encouraging. A solar dryer works on the same principle as greenhouses do. It consists of a glass house with black plates under the glass for absorbing the solar radiation. The glass has the property that it does not allow the long wave radiation from the plates to return *Solar Energy Research Center of Cuba, Cuban Academy of Sciences, Havana, Cuba. E-mail: eduardo@mathematik.uni-bremen.de htitute for Dynamical Systems, University of Brernen, Bremen, Germany. E-mail: andrew@mathematik.uni-bremen.de E. GALLESTEY AND A.D.B. PAICE Fig. 1. Solar dryer prototype. to the environment, but at the same time permits solar radiation to enter. The metallic plates form a cabin, inside of which the product to be dried is placed. Fans move the air in order to increase the mass and heat transfer coefficients of in the cabin (See Fig. 1). Maximal temperatures between 5S°C and 65OC are frequently measured within the dryer and drying processes of good quality are obtained. In Fig. 2 graphs representing experiments performed in Cuba with the dryer charged and empty are depicted. The qualitative behaviour of the installation can be easily observed. A more detailed explanation may be found in [I], [2]. Nevertheless, one problem remains unsolved. The air inside the dryer must be periodically exchanged in order to avoid saturation and to extract the evaporated water. Naturally, this causes an energy loss which should be minimized in order to maintain maximum drying efficiency. Thus we arrive at the following optimal control problem: How much air should be extracted at each moment in order to reach the highest efficiency? This is a difficult problem, due to the unpredictable character of the environmental conditions and the strong and complicated correlations between the parameters which determine the dryer behaviour. In this paper we examine various approaches to the solution of this problem, and we evaluate the various approximate solutions obtained in terms of robustness of the achieved performance with respect to changes in the environmental and initial conditions. This paper is organized as follows. First, in Section 2, a mathematical model of the dryer is developed and discussed. It is seen that the output predicted by the model for given environmental conditions closely models that observed in a real dryer. In SOLAR DRYERS 299 Empty Dryer Temperature Charaed D~er Tem~erature Wood Humidity (%) lo 0 0 60 0 100 10 10 160 1;O Time (hr) Fig. 2. Drying of wood. Experiments performed in november 1990 at the Solar Energy Institute, Santiago de Cuba, Cuba. Section 3 some numerical experiments are performed using the dryer model. The aim is to develop an intuition regarding the performance of the dryer given different initial conditions and environmental influences. Section 4 is dedicated to the study of linearizations of the presented model about trajectories with constant controls. This section should increase our understanding of the problem we face and at this point we will be in the position to formulate the full optimal control problem. In Section 5, making use of the obtained results, the optimal control problem is precisely stated and strategy for determining the solution is fixed. Further, piecewise constant controls are studied, and it is shown that from a practical point of view the best possible piecewise constant control is constant over intervals of 24 hours. In Section 6 the goals of the paper are summarized and recommendations for the application of the obtained results are made. 2 THE MATHEMATICAL MODEL 2.1 Differential equations of the model As a first step, a mathematical model for the dryer has been developed. It consists of a set of differential equations which were derived paying attention to the energy and mass balance in the main parts of the dryer. They appear as follows: E. GALLESTEY AND A.D.B. PAlCE Fig. 3. Scheme of the dryer. 1 : upper absorber plate; 2 : east absorber plate; 3 : west absorber plate; 4 : walls; 5 : product; 6 : dryer air Where the following notation has been used: t : Time (hours); Tl : Temperature of the upper absorber plate (OC); SOLARDRYERS T2 : Temperature of the east absorber plate (OC); T3 : Temperature of the west absorber plate (OC); T4 : Temperature of the walls (OC); T5 : Temperature of the product (OC); T6 : Temperature of dryer air (OC); H : Water content in the dryer air (kglkg); A : Water content in the product (kg); Cij : Transfer coefficients, which are positive constants defined by the physical laws governing the process. They are obtained from formulas used by engineers in calculations related to heat transfer. These formulas are well established, which makes us confident in the values of the calculated coefficients, see for example [9]. For all Cij, except C12, C42, CS1, C52, the coefficients are given by where Nu is the Nusselt number, 0.032~e~.~ if Re < lo5 Nu = [ 0.669~eO.~ otherwise pi is the thermal conductivity of the ith component, li is the length of the surface i taken in the direction of the air flow, v,;, and vatr are the cinematic viscosity and the bulk velocity of the air inside the cabin respectively, cpi is the specific heat of component i, and mi is the mass of i. The coefficients Csl and C42 may be calculated by modified versions of (9). In the case of C42, for example, a term should be added which accounts for the thermal conductivity of the composite wall, see [9]. Coefficients C12, CS2 model the only heat transfer process by radiation which has been introduced in the model. They have the form E. GALLESTEY AND A.D.B. PAICE where E, = 0.95 and E, = 0.43 are the emissivity of the upper plate and of the product respectively. The value of E, is a property of the product and was determined experimentally. u is the Stefan Bolzman constant. h : Latent heat of vaporization; c,, , c, : Specific heats of the product and water; md: Dried mass of the product; Q : Heat obtained from the fans; U(t) : Extraction rate (bounded control, 0 5 U(t) 5 U,,,); G1 : Evaporation rate, defined by the equation: a : A slowly time varying positive parameter. It represents the "error" caused by considering the drying process to be pure water evaporation. For the moment we will work with a! a constant, which is true during the first phase of the drying process, and return to this problem later. G2 : Extraction rate of humidity per unity of volume, defined by the equation: where V(t) represents the water contents of the air per unit of volume in the ambient. This is given in the model as: Pus (T) : Saturated vapor pressure. H, : Relative ambient humidity. Sl (t), S2(t), S3(t) : Energy absorbed by the upper, east and west plates respec- tively. For details in their derivation see [5]. Ta(t) : Ambient temperature. A given function of time. SOLAR DRYERS 303 Fl (TI , To), F2 (T2, To), F3 (T3, To) : Heat transfer coefficients through the glass, given as follows (see [3]): where: C = 520(1 - 0.000051@), Ni : Number of sheets of glass in cover i; h,, : Heat transfer coefficient from the glass to the environment by convection; r, : Glass emissivity; pi : Tilt angle of the glass (in degrees); Remark 1: The model is of high order. Nevertheless, models of lower order do not fit the data satisfactorily. For example, a model obtained by unifying equations (1)-(3) in one equation has not been successful. An explanation for that can be found in the fact that the behaviour of the three absorbing plates is very different during the day. The first author has completed a lengthy series of investigations into the modelling of this system, and has come to the conclusion that only systems of higher order may perform as well as this one in modelling the observed behaviour of the dryer. Another possibility would be to replace some equations with algebraic constraints. See Remark 2 in Section 4 for further comments on this approach. It is important to note that: The system is nonlinear in the states. The state equations are affine in the control. 0 The control is bounded from above and below. E. GALLESTEY AND A.D.B. PAICE alpha 1\ Equlibrium Humidity 100 Pmduct Humidity (%) Fig. 4. Approximate behaviour of the coefficient a during the drying process. 2.2 Assumptions made about the physical process In order to derive equations (1)-(8), the following assumptions have been made: 0 Assumption 1: All the transversal sections are equivalent. Assumption 2: There is a strong turbulent convection of the air. This assumption allows us to assume the homogeneity of the air at every moment and permits us to consider that the heat and mass transfers process inside the dryer are realized only by convection. Only one radiation coefficient is taken into account, the heat transfer between the product and the upper plate. Assumption 3: The product surface is always wet. That means that the dryer process occurs slowly enough to permit the use of the simplified expression (15) for the evaporation velocity, or more precisely, to take a! as a constant close to 1. For design purposes this assumption is not very important, due to the generality of the cases which must be taken into account, however for optimal control purposes this assumption may be false and can lead to incorrect results. The parameter a! depends in a very complicated way on the product itself, the product humidity, and the drying process conditions. That is why, in designing a controller for a real process, a! has to be considered a time-varying hardly measurable parameter. Its approximate behaviour during the drying process can be seen in Fig. 4. SOLAR DRYERS 305 Time (hr) Time (hr) 50 I I 1.5, 0 5 10 15 20 Time (hr) Time (hr) ..\\1 0 5 10 15 20 Time (hr) Time (hr) Fig. 5. Examples of graphs obtained with the model. 2.3 Typical outputs of the model All the main parts of the dryer and their interactions are reflected in the model. It offers complete information about the temperatures of the main parts of the dryer and the water content in the air and product. It also permits the observation of quantities which are hard to measure such as the energy losses through the distinct parts, the evaporation velocity, etc. Typical graphs obtained with the model are shown in Fig. 5. 2.4 Model validation The model has been tested and its ability to predict the dryer dynamics has been proved. The experiments were made in March-July 1994 in Santiago de Las Vegas, Ciudad de la Habana, Cuba, using water for modeling the product (a, = 1). Graphs which represent the model predictions and the measured data can be found in Figs. 6 and 7. As can be seen, the model follows the measured data almost perfectly. It has to be pointed out that a better agreement between the modeled and measured data could be obtained if more detailed measurements of the solar radiation were available at the location where the experiments were carried on. The model a function of time (the functions Si), which requires the instantaneous irradiance as is difficult and expensive to obtain. This fact made the use of certain interpolation procedures necessary in order to obtain approximations of the real Si, which, due to space considerations, are not commented on here. 306 E. GALLESTEY AND A.D.B. PAICE 6 40 +g & 40 5 30 020 5 10 15 20 25 30 35 40 45 Time (hr) Time (hr) Fig. 6. Comparisons between the exoeriment verformed on Aoril4th and 5th (0) and calculations 5 10 15 20 25 30 35 40 45 Time (hr) 5 10 15 20 25 30 35 40 45 Time (hr) Fig. 7. Comparisons between the experiment performed on July 4th and 5th (0) and calculations (-) At this moment the mathematical model is being used in designing and optimizing new dryers at the Solar Energy Research Center in Cuba. It is inserted in a computer user interface, written by the first author, which allows a comfortable simulation process. SOLAR DRYERS 3 SIMULATION STUDIES Before trying to implement some of the methods of Control Theory, we wish to get an idea of the dryer responses resulting from controls which seem reasonable from the point of view of the physical process. We are going to look for the different dynamics caused by changes in the parameters, initial states conditions, environmental conditions, etc. This will improve our knowledge about the dryer and should permit a better orientation in the problem. This goal will be achieved by using the simulation program Gnans [7], which offers all the necessary facilities. The simulations were performed using the standard solar radiation, corresponding to a location in Havana. The solar radiations curves were derived using methods developed at the Solar Energy Research Center, which give trustworthy average month values for the solar radiation. See [5] for details. In order to fix a configuration during the simulations, parameters which define the dryer and its environment were chosen. Some of the most important are listed in the Table 1. Unless otherwise specified, these are the parameters used. In the same table the initial conditions are depicted. Table 1. Dryer parameters and initial conditions used during the simulations Parameters Initial Conditions Parameter Value Units State Value Units length 1 rn final time 24 hour width 3 rn TI 33 O C height 2 rn T2 33 OC roof angle 10 grades T3 33 O C wall height 1 rn T4 33 OC month April Ts 33 OC extraction 60 rn3/hr T6 33 O C initial humidity 80 % H 0.02 kglkg product specific heat 5.1 k J/ kg/OC A 70 kg 3.1 Changes in the initial conditions Not all the initial conditions have the same impact on the system behaviour. We know from the physics of the process that the initial conditions in the states which do not have large storage capacities have little influence on the long term dryer behaviour. This is the case for the temperatures of the plates and the air TI, T2, T3, T6, (it is known that the plates and the air have a very small heat capacity), and the humidity of the air inside the dryer H (air exchange with the environment makes any initial 308 E. GALLESTEY AND A.D.B. PAICE 0.5 1 1.5 2 Time (hr) Time( hr) Fig. 8. Comparisons between the effects of changes in the initial conditions of TI, T2, T3, T6 and H. Each curve is obtained by changing the corresponding initial condition, while leaving the others unchanged. In the last graph ten curves have been drawn. humidity value within the dryer converge to a physically realizable value). This means that these states are completely determined by the dynamics of the process. Some example calculations illustrate the above comments, Fig. 8. As can be seen no matter which initial values these states have, they converge quickly to a trajectory fixed by the dynamics of the system, without any strong influence on the dryer performance. More important appear to be the initial conditions of T4, T5 and A. Their effect lasts longer and in the case of T5 even seems to be important during the first hours. See Fig. 9. The most important conclusion of this section is that slow and fast dynamics coexist in the system (two time scales). This fact should be taken into account in the future numerical treatment of the problem. It can thus be concluded that, except for the water content of the product, the states of the dryer depend mainly on the environmental conditions within the last few hours. Because of the fact that the environmental influences are well predictable only for the next 24 hours, and the fact that the product water content can be measured only once or twice a day, we are going to study the dryer and its optimization only in periods of 24 hours. 3.2 Changes in the environmental conditions The environmental conditions play a fundamental role in the dryer behaviour. As can be appreciated from the results depicted in Fig. 10 any changes in the ambient, which '0 SOLARDRYERS 309 "0 5 10 15 20 Time (hr) 1.5, - '" d t--..'l 9 5 , ' o----- I I 0 5 10 15 20 0 5 10 15 20 Time (hr) Time (hr) .- - - _ _ ziOm d 5 0 5 10 15 20 Time (hr) Time (hr) Fig. 9. Comparisons between the effects of changes in the initial conditions of T4, T5 and A. are modeled in the system through the functions T, (ambient temperature), V (ambient absolute humidity), and So (radiation level on the horizontal surface), leads to changes in the amount of evaporated water on that day. This effect can be most clearly observed in the dependence of the evaporation on the solar energy, which seems to be, as may be expected, the fundamental parameter. 3.3 Simulations with constant controls Constant controls are of special interest because they are easy to implement in practice and relatively easy to study from a mathematical point of view. A detailed study of the optimal constant controls should increase our understanding of the problem and allow a better formulation of the mathematical problem. The most straightforward and natural dryer performance index is the total daily evaporation and in this section we will understand as optimal the constant controls which maximize this performance. We want to study in this section how this dryer performance can be improved by using constant air extractions. Optimal constant controls corresponding to different initial and environmental conditions were calculated. Attention was also paid to the interrelation between the parameter a from (7) and the optimal constant controls. Dependence on environmental conditions In Fig. 11 we can observe the dependence of the evaporation on the constant controls. For small extraction values the dependence is strong. But after a certain E. GALLESTEY AND A.D.B. PAICE , /.1 L 5 0 --- 0 5 10 15 20 Time hr T~me (hr) q , 4 e25 2 20 __-- . , I' 150 5 10 15 20 0 5 10 15 20 Time (hr) Time (hr) 2 5 '0 5 10 15 20 Time (hr) Time (hr) Fig. 10. Effec Evaporation Month Evaporation 0 50 100 Extraction Month Better Extraction Month Maximal Evaporation ::I 246810 246810 month month Fig. 11. Dependence of the optimal constant control on the environment. value, the curves become very flat and, although the optimal control exists and it strongly depends on the month, larger values than the optimal do not substantially affect the performance of the dryer. SOLAR DRYERS 311 20 30 40 50 extraction lnitial Conditions on T4 I<, g 55 .- 0 50 100 20 30 40 50 extraction Initial Conditions on T5 10, 0 1 70 0 80 100 120 0 50 100 extraction lnitial Conditions on A Fig. 12. Dependence of the optimal constant control on the initial conditions. Left: Evaporation vs. extraction for each initial condition, while the others initial conditions remain unchanged. Right: Best optimal control for each initial condition. Dependence of the constant control on initial conditions As shown in Section 3.1, the only initial conditions able to significantly influence the dryer dynamics are those of T4, T5 and A. In Fig. 12 we examine the dependence of the optimal constant controls on these initial conditions. It can be observed that the same situation as in the above paragraph is found: The optimal controls are very different for each initial condition, but the improvements in the performance from one control to another are very small. Dependence of the constant control on the parameter a In order to test the sensitivity of the value of the optimal constant control on the parameter a from (7) different values of a were used and the optimal control for each of them was calculated. The results of the calculations are summarized in Fig. 13. In the graphs it is easy to see the close dependence of the control on a. This characteristic is accentuated when alpha takes values between 0.1 and 0.6, which corresponds to the last days of the drying process (See Fig. 4). An optimal control strategy for the last days of drying should take into account this fact. 3.4 Summary of the simulations results Through the simulation study of the model the following characteristics of our problem have proven to be important: E. GALLESTEY AND A.D.B. PAICE Best Extraction (m"3Ihr) Daily Evaporation (kg) so1 8 0 50 100 extraction (m"3hr) Fig. 13. Dependence of the optimal constant control on the parameter a. Left: Curves with constant a. Right: Optimal Extraction for each a. Fast and slow dynamics are present in the model. The slow dynamics have a time constant of the order of 2 hours, which is short in comparison to the run time of the process, which is at least 5 days. The optimization problem can be regarded as a finite horizon problem. We may choose a final time, which we call tf. tf will be chosen as 24 hours, since the environmental influences are approximately predictable with this period. The system is mainly influenced by the environment, which cannot be a priori determined. The optimal controls obtained have to be analyzed with regard to robustness. The second most significant source of perturbations in the model is the parameter a. 4 LINEARIZATION OF THE MODEL The degree of stiffness of the model will play a fundamental role in the method to be used in solving the optimal control problem. In trying to learn more about it, a SOLAR DRYERS 313 model obtained by linearizing the equations (1) to (8) about trajectories with nominal constant controls has been derived and tested. In this paragraph the following notations will be used. xo : The standard initial conditions of the model used before, taken from Table 1. u*: A given nominal constant control. x*(t) = 4(t, xo, u*): Solution of the full system defined by the equations (1) to (8) corresponding to the control u* and initial condition xo. S(t, x*(t), u*): Right sides of the equations from (1) to (8 ) evaluated at the nominal trajectory x*(t) and the control u*. u(t) = u;(t) - u*: The difference between the actual control uz(t) and the nominal control u* . xl(t) = @(t, xo, u) : Vector of states of the linearized system. It is a linear approximation of the deviation of the state x (t) = 4 (t, xo, uz) from the nominal state x*(t) = #(t, xo, u*) and is a solution of the following linear system of equations. as A(t) = -(t, x*, u*), ax as ~(t) = -(t, au x*, u*), XL = XL(~, XO, uz, u*) = x*(t) + @(xO, uz - u*): This will be called the "solution of the linearized system". It was found that the matrices A(t) and B(t) appearing in (22) are slowly time varying and that the linear system (22) is a good approximation of the full dryer model. A comparison of the behaviour of the linearized model with that of the full model is given in Fig. 14. The four graphs compare the trajectories x* = 4(t, xo, 40) generated by the full model for u* = 40m3/ hr from one side, and the solution of the linearized system xL(t, XO, 40,60). AS may be seen, there are no significant differences between the solutions. This increases confidence in conclusions made on the basis of studies of the linearized model. By examining the eigenvalues of A, we can quantify how ill conditioned our problem is. The condition number (Ih,,,/h,i, I) of the linearized system as a function of time, together with the maximal and the minimal eigenvalues is depicted in Fig. 15. E. GALLESTEY AND A.D.B. PAICE - U=40 ... Linearization 0 5 10 15 20 Time Fig. 14. Comparisons between the solution to the linearized model and the of full one for u:(t) = 40. The linearization was made about u*(r) = 60. Only the real parts of the eigenvalues were drawn, the imaginary parts are of order It can be seen that the condition number reaches large values ( lo7) which implies that an attempt to use techniques such as Pontryagin's Maximum Principle will lead to numerical difficulties which are hard to avoid. Remark 2: A method to avoid the ill conditionedness of the problem is to reduce the order of the system by converting the fast dynamics present in equations (1)-(3) and (7) into algebraic constraints. Nevertheless, this leads to a complex system of time varying differential algebraic equations (see for example the expressions in (18)) which would be very difficult to solve. It is believed that the gain in "well conditionedness" of the model is not enough, at least at this stage of the investigation, to justify the difficulties in finding solutions of such a reduced model. 5 THE OPTIMAL CONTROL PROBLEM At this point we are in a position to define our optimal control problem. Of course, we should take advantage of what we have learned about the dynamics of the dryer during the simulation experiments. The plant is already designed, it is believed in the best possible way, although this could also be a matter of study, and the only control available to influence the process is to extract the air within the dryer. SOLAR DRYERS 0 5 10 15 20 25 Time (hr) Fig. 15. Largest and smallest eigenvalues of the matrix A as a function of time. 5.1 Optimal control problem formulation The behaviour of the optimal constant controls indicates that the range of values in which the optimal controls may take on can be easily obtained in the practice, see [4]. Taking this into account, we do not penalize large control values in the performance index. Thus the following simple expression has been chosen. where A is the same as in equation (8). This corresponds to the requirement of obtaining the maximum possible evaporation on the selected time interval [0, tf]. As the control is to be implemented by a fan, we restrict ourselves to considering the class of controls to be piecewise continuous. The optimal control problem may now be formulated as follows: Find the control U* : [0, tf] -+ R such that J(U*) 5 J(U) forall other piecewise continuous U : [0, tf] -+ R. J(U) is taken from equation (23),and is subject to constraints corresponding to equations (I), ...,( 21) with initial conditions taken from Table I. Problems of this general class have been studied for many years. A classical approach is to apply the Pontryagin Maximum Principle or the Bellrnan Equation, for an introduction see for example [6]. 316 E. GALLESTEY AND A.D.B. PAICE In the first approach, it is necessary to derive the associated Hamiltonian system, where the adjoint system satisfies a terminal condition. This would require a numerical treatment of the problem as an analytic solution is unattainable. In our case, difficul- ties in this approach were encountered, due to the stiffness present in the system of differential equations. Another approach is to approximate the class of allowable controls by piecewise constant controls with fixed switching time intervals and solve the associated optimal control problem. This has the advantage of converting an optimization problem over an infinite dimensional Banach space into an optimization problem involving only a finite number of parameters. The experience collected in studying the system response to various environmental and initial conditions, and the study of the linearized system suggests that before attempting a difficult classical method, suboptimal controls should be considered. Here a suboptimal control is understood to be the control derived by considering the optimal control problem over a restricted class of controls, which is chosen to reflect the characteristics of the controller in the real system. This should at least clarify whether or not the improvement of the performance index through the optimal controls is of sufficient significance to justify the amount work required in finding them. 5.2 Optimal piecewise constant controls We now investigate the system's behaviour obtained by considering piecewise constant controls. We proceed as follows: The time interval of interest [0, tf] is divided into m equal subintervals, and during each of these subintervals a constant valued control is applied to the system. More explicitly, we define a cost function G, : lRm t lR, Gn,(un') = J(U) = A(tf), (cf. equation (23)) where u E lRm is a vector with components u; equal to the value of the extraction on the interval i and U is a control defined in by the following expression: Our optimization problem is now to find the minimum of the function Gm(.) subject to the restrictions 0 5 ui 5 Urn,, , i = 1 . . . m, and to compare the solutions for different m. In order to solve this problem, a program has been written in C++ which calculates the solutions to this problem making use of the facilities of the NAG Fortran Library [8], see Fig. 16. The suboptimal controls and the pe8ormance index It is easy to see that any physically implementable control may be approximated by the control defined by u E lRm, for large enough m. Thus, although the optimal piece- SOLARDRYERS Suboptimal Piecewise Constant Controls Fig. 16. Piecewise suboptimal controls. wise constant control may not in general lie in our class of controls, from a practical point of view it may be sufficiently well approximated by the control associated with u* E IRnl, where G,(u*) is the minimum. By increasing rn, we obtain controls closer to the optimal and can evaluate the improvements in the performance of the dryer. Due to limitations of the minimization algorithm, which appear when rn becomes very large , we were able to increase rn only up to 480 which is equivalent to intervals of 3 min of real time. From a practical point of view, we believe this is sufficient to determine an implementable control which approximates the optimal. In Fig. 16 the controls which minimize the functions G, corresponding to m = 1 (intervals of 24 hours, corresponding to constant controls), rn = 24 (intervals of 1 hour) and rn = 240 (intervals of 6 minutes) are depicted. They were calculated for a standard day in April, with the same parameters as used previously (see Table 1). In Fig. 17 we see the increments in performance for these three controls. We find that there is not a strong difference between the values of the performance index achieved by the different controls. Robustness of the suboptimal controls Another argument for choosing the control for the dryer should be the robustness of the control with respect to uncertainties in the environmental and initial conditions. We do not expect invariance of the dryer behaviour with respect to the environmental conditions, however, we want to implement the control which best "desensitizes" the dryer performance to uncertainties in weather conditions (cloudiness, median ambient E. GALLESTEY AND A.D.B. PAICE Daily Evaporation Time (hr) Fig. 17. Performance for each of the controls. temperature and median ambient relative humidity) and in the humidity of the product (the parameter a). We have calculated the sensitivity of the performance index with respect to these factors for each of the three selected suboptimal controls. The sensitivity to aparameter p is understood to be the partial derivative of the performance index with respect to p evaluated at a given value p,. For example Fig. 18 plots the performance index E,) against the.cloudiness (kh), and the sensitivity 3 against kh, while (evaporation, all other parameters were held constant at the standard values. These derivatives were obtained by using numerical differentiation routines included in the NAG Fortran Library. The results can be seen in the series of Figures 18, 19,20 and 21. It can be appreciated that in three cases (cloudiness, the parameter a and ambient humidity) the moduli of the derivatives are smaller for the constant control than for the other more sophisticated suboptimal controls. Only with respect to the ambient temperature is the contrary situation observed, albeit with very small values for the derivative moduli. That means that the constant controls are at least as robust as the others. At the same time, it should be pointed out that the observed improvements of the performance index do not encourage an attempt to implement controls more complicated than the constant ones. SOLARDRYERS Daily Evaporation Sensitivity of the Evaporation 12.57 0 0.5 1 cloudiness (kh) Fig. 18. Evaporation and sensitivity with respect to the cloudiness (kh). Daily Evaporation Sensitivity of the Evaporation 9I 0.5 1 alpha alpha Fig. 19. Evaporation and sensitivity with respect to cr. E. GALLESTEY AND A.D.B. PAICE Fig. 20. Evaporation and sensitivity with respect to the ambient humidity. Daily Evaporation Sensitivity of the Evaporation Fig. 21. Evaporation and sensitivity with respect to the ambient temperature. SOLARDRYERS 32 1 6 CONCLUSIONS At this point, we are in a position to give final recommendations for the control of solar dryers. It has been found that constant controls over a 24 hour period unite the charac- teristics desired of a controller in optimal way: the achievement of high levels of performance, simplicity in implementation and the best possible robustness. The con- trols may be calculated for a variety of possible environmental conditions and then put in an appropriate form at the disposition of the industry, for example in tables de- termining the recommended control corresponding to given weather conditions, water content of the product and a given product (estimation of the parameter a). In this paper the following goals have been achieved: 0 An exhaustive mathematical model of solar drying plants has been presented and validated. With the aid of this model, it has been shown how to obtain controls which optimize the efficiency of these plants. 0 Programs have been developed which make the calculation of the above men- tioned controls for each new configuration highly efficient, making possible the immediate application of the obtained recommendations in future dryer models. ACKNOWLEDGEMENTS We thank Prof. Dr. Luis BCrriz PCrez of the Solar Energy Research Center of Cuba and Prof. Dr. Diederich Hinrichsen of the University of Bremen for the many fruitful discussions during the development of this work. The Deutscher Akademischer Austauschdienst is gratefully acknowledged for the financial support given to the first author. REFERENCES 1. L. BCmz and D. Degas.: Secador Solar de Madera. Editorial Comision Nacional de Energia, Havana, (1990). 2. L. BCrriz and D. Degas.: Secador Solar MultipropBsito. Editorial Comision Nacional de Energia, Havana, (1 990). 3. J. Duffie and W. Beckman.: Solar Engineering of thermal processes. John Wiley & Sohn, New York, (1991). 4. H. P. Garg.: Advances in Solar Energy Technology. D.Reide1 Publishing Company, (1987). 5. M. A. Guerra and M. R. Vazquez.: Manual de Radiacion Solar para la Republica de Cuba. Editorial Academia de Ciencias de Cuba, Habana, (1992). 6. F. L. Lewis and V. L. Syrmos.: Optimal Control. John Wiley & Sons, New York, (1995). E. GALLESTEY AND A.D.B. PAICE 7. B. Mirtensson.: Gnans: A program for stochastic and deterministic dynamical systems, reference manual. Report 300, Institute for Dynamical Systems. Bremen Univer- sity, Bremen, (1993). 8. Mark 16 NAG Fortran Library Users' Guide.: Numerical Algorithm Group. Inc. Oxford, (1994). 9. M. Mikheyev. Fundamentals of Heat Transfer. Mir Editions, Soviet Union, (1978).

Journal

Mathematical Modelling of Systems – Taylor & Francis

Published: Jan 1, 1997

Keywords: drying; modelling; optimal control; robust control; solar energy

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