UNCORRECTED PROOF
ARTICLE IN PRESS
1
2 A novel predictive architecture for microwave-assisted
3 drying processes based on neural networks
BJ.L. Pedren˜o-Molina
a,T, J. Monzo´-Cabrera
a, 4 A. Toledo-Moreo
b, D. Sa´nchez-Herna´ndez
a5
a 6
Departamento de Tecnologı´as de la Informacio´n y las Comunicaciones, Universidad Polite´cnica de Cartagena,
7 Campus Muralla del Mar, E-30202 Cartagena, Spain
bDepartamento de Tecnologı´a Electro´nica, Universidad Polite´cnica de Cartagena, Campus Muralla del Mar, 8
9 E-30202 Cartagena, Spain
10 Abstract 11
In this contribution, a novel learning architecture based on the interconnection of two different learning-based 12 neural networks has been used to both predict temperature and drying curves and solve inverse modelling 13 equations in microwave-assisted drying processes. In this way, a neural model that combines the accuracy of neural 14 networks based on Radial Basis Functions (RBF) and the algebraic capabilities of the matrix polynomial structures 15 is presented and validated. The architecture has been trained by temperature (Tc(t)) and moisture content (Xt(t)) 16 curves, which have been generated by a previously validated drying model. The results show that the neural model 17 is able to very accurately predict both kind of curves for any combination of the considered input variables (electric 18 field and air temperature) provided that an appropriate training process is performed. The proposed configuration 19 also permits the solution of the inverse problem in the drying process by finding the optimal value for the electric 20 field. This provides Tc(t) or Xt(t) curves that fit to a desired drying condition in a specific time slot. 21
D 2005 Published by Elsevier Ltd. 22
23 Keywords: Predictive system; Neural network modelling; Microwave-heating applications
24
0735-1933/$ - see front matterD 2005 Published by Elsevier Ltd.
doi:10.1016/j.icheatmasstransfer.2005.05.001
BCommunicated by J.W. Rose and A. Briggs.
TCorresponding author.
E-mail address: [email protected] (J.L. Pedren˜o-Molina).
International Communications in Heat and Mass Transfer xx (2005) xxx – xxx
www.elsevier.com/locate/ichmt
ICHMT-01347; No. of Pages 8
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25 1. Introduction
26 The conventional analysis of heat and mass transfer processes is mainly based on the physical
27 modelling of these phenomena, by finding and solving the differential equations that describe them. In
28 this way, many authors have published very different mathematical models which produce good results
29 under several constraints as described in[1]. Yet, the key shortcomings for this kind of models are the
30 need of an accurate characterization of the model coefficients, particularly temperature and/or moisture
31 content dependency [1,2]. In contrast, the neural-network-based architectures give solutions to the
32 appearance of equations restraints when the operating environment is unknown. Since this kind of
33 structures is learning based, the rest of the process or environment parameters are no longer necessary
34 when the mapping between the input variables of the process and the output functions is known.
35 Although the employment of neural networks allows for the determination of parameter optimization
36 problems[4], they are commonplace for the identification and prediction of dynamic processes. Thus, a
37 fuzzy neural algorithm is employed in [5] to control wood drying processes and a back-propagation
38 neural structure is developed for the precise modelling of rice drying in[6]. On the other hand, most
39 neural network structures applied to drying processes involve solutions in which the output of the
40 network is a unique solution [7] for an input set of one-dimensional parameters, rather than a time-
41 dependent function.
42 In this work, however, the proposed model is able to learn and predict complete drying curves, such as
43 sample temperature Tc(t) or moisture content Xt(t) evolution, from only two numerical input parameters.
44 The proposed neural structure, based on Radial Basis Functions (RBF) neural networks and polynomial
45 learning structures, has been trained from several temperature and drying curves which have been
46 obtained from a previously validated microwave-assisted drying model[2,3].
47 2. Theoretical study
48 2.1. Neural architecture design
49 From previous contributions concerning microwave-assisted drying processes [2,3,8], one can
50 conclude that Xt(t) varies smoothly along the different drying stages while Tc(t) shows steeper changes
51 during dehydration. Additionally, both curves are highly dependent on the electric field strength (E) and
52 the air-flow temperature (Tair). It is for this reason that both E and Tairhave been chosen as the input
53 parameters for the neural network model.
54 In order to design a neural architecture which is able to predict both Xt(t) or Tc(t) from E and Tair, it is
55 necessary to consider neural network models based on learning algorithms mainly focused on the
56 solution of non-linear problems. In this case, the same neural architecture will be used to learn and
57 predict Xt(t) or Tc(t) regardless of their time evolution characteristics, as illustrated in Fig. 1. Two
58 different levels can be distinguished in this architecture, where wjk are the neuron weights of the first
59 network for the kth trial, WW is the matrix associated to the second network and y(t) is a general
60 purpose time-dependent curve that represents either Xt(t) or Tc(t). In the first level of the proposed
61 architecture, the output function length, t = [0, T 1], is divided into M time intervals in order to project the T points of y(t) onto M neurons (M b T) whose weights have been properly learned. If we assume that 62
63 Zi (i = [1, 2]) are the input parameters of the drying process (E and Tairrespectively), the relationship
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64 between Zi and y(t) within each time interval will be learned by the RBF neural network during the
65 training stage and then, it is later assigned to a neuron weight wj ( j = [1, M]), where M is the time
66 interval number. The second architecture level projects the M weights of the RBF neurons onto E and
67 Tair. This projection is carried out by other network whose structure depends basically on the
68 relationship between each wk and Zi. In this case, a third degree polynomial model has been employed,
69 which provides sufficient final precision for the MY2 projection in each interval as discussed further
70 on. This second level permits the generation of appropriate RBF neuron weights from whatever value
71 for E and Tair.
72 2.2. RBF neural networks for drying processes identification
73 RBFs are supervised neural networks[9]whose structure provides a solution to the local interpolation
74 of non-linear functions. This is the case of the generic function y(t) considered in this work since Xt(t) or
75 Tc(t) do not have a linear behaviour. Although the neurons activation in the RBF model is carried out by
76 radial functions, the RBF model has a linear expression for the estimation (y˜(t)) of y(t). Therefore, for
77 each pair of input variables [E, Tair]k corresponding to the kth trial, the estimation of y˜k(t) is given by
78 Eqs. (1) and (2),
˜yykð Þ ¼t XM
j¼1
wjk/jð Þt ð1Þ
79 /jð Þ ¼ expt ðt lj2
r2j
!
ð2Þ
80 81 where /j(t) is the jth Gaussian radial function, lj and rj are the centre and standard deviation of /j(t),
82 wjk is the weight value associated to /j(t) for the kth trial (k = [1, N]), and N is the number of trials
83 during the learning period.
k
Tair
E
Z={ , } Model [2,3] Numerical
wjk
E Tair
)
~ tyk( ) (t yk
) , , , (E3E2 ETair f
V=
WW Φ(t)
Level 2 Level 1 -
+ Hk
Hk
Fig. 1. General scheme of the proposed neural model.
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84 The estimation of wjkis carried out by using the gradient descent algorithm[10]to minimize the cost
85 function described in Eq. (3)
Hk ¼ XT
t¼ 0
ykð Þ ˜yyt kð Þt
ð Þ2 ð3Þ
86 8788 As a conclusion, the application of the RBF neural network to the estimation of y˜k(t) permits to
89 obtain, after the training stage, the optimal values for wjkfrom curves generated by the validated drying
90 model [2,3]. In the prediction stage, Eq. (1) supplies the approximation of yk(t) for Tc(t) or Xt(t).
91 Transforming Eq. (1)into a matrix notation,
˜yykð Þ ¼ Wt k FðððtÞÞÞT ð4Þ
92 93 with Wk being the 1 M dimension vector containing the RBF neuron weights for the kth trial, and
%(t)Tthe vector whose elements are the M Gaussian functions, which are independent of k, as described 94 95 in Eq. (2).
96 2.3. Interpolation network for curves prediction
97 In order to obtain information about the dependence order of each RBF neuron with respect to the
98 input variables of the process and to design the second level network, the Wkobtained in each trial have
99 been projected onto [E, Tair]k, which are the input variables used to generate yk(t). Taking into account
100 all the trials used in the learning stage, this projection results in the WW matrix. This matrix contains the
101 mapping between E and Tairand the contribution of each Gaussian function to the generation of y˜(t). In
102 order to establish the dependence of each neuron versus E and Tair variations, a two-dimensional
103 projection of all the M neuron weights has been generated. Fig. 2shows, for both Xt(t) and Tc(t), the
104 projection for the fourth neuron of the weight matrix. The rest of the M 1 neuron weights present very
105 similar behaviours with respect to E and Tair. From the surface analyses inFig. 2, one can conclude that
106 there is a third-order dependence of the weights in the RBF network versus E in both Xt(t) and Tc(t)
Fig. 2. 3D projection of the 4th RBF weight over E and Tairfor estimated (a) Xt(t) and (b) Tc(t).
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107 curves. At the same time, these surfaces also show a linear dependence with respect to Tair, which
108 justifies the selection of the polynomial structure of the second architecture level.
109 By following the scheme in Fig. 1, applying the polynomial model to the RBF neuron weights and
110 considering the matrix expression in Eq. (4), y˜(t) can be generated, for each pair of inputs [E, Tair], by
111 means of
yðt;Tair;EÞ ¼ wjdAðt;li;rÞT ¼Y
VðE;TairÞdWWdAðt;li;rÞT ð5Þ
112 113 where theY
V components are represented inFig. 1, and the matrix WW is learned by following the linear 114 regression algorithm for the R and S matrixes shown in Eqs. (6) and (7).
S¼
Tair1 E1
Tair2 E2
v v
TairN EN
3 77
5 R¼
w11 w12 : : : w1M
w21 w22 : : : w2M
v v v
wN 1 wN 2 : : : wN M
3 77 5 2
66 4 2
66
4 ð6Þ
WW¼ ðSTdSÞ1dSTdR ð7Þ 115
116 117 118 2.4. Inverse calculation for electric field intensity estimation
119 One of the main advantages of using the proposed neural configuration, based on interconnected RBF
120 neural networks and polynomial matrix equations, is the possibility to readily solve the inverse problem.
121 In this case, this implies the estimation of the optimal value of one of the input variables of the process
122 from desired output results in terms of temperature or moisture content. This is particularly important for
123 microwave-assisted drying processes since it provides the initial drying configuration that ensures a final
124 drying level to be reached in a previously established time. In this work, we have fixed Tairand a target
125 value (K) for the sample temperature (Tc0) or the moisture content (Xt0), which has to be reached within
126 t0 seconds. With these conditions, the proposed model is able to estimate the optimal value for E that
127 generates the Tc(t) or Xt(t) curves that fit to the desired points [Tc0; t0] or [Xt0; t0]. By particularizing
128 expression (5) for a specific t0 and Tair, Eq. (8) is obtained.
yðt0;EÞ ¼Y
VðEÞdWWdAðt0ÞT ¼ K ð8Þ
129 130131 Solving Eq. (8) for the variable E, a solution for the inverse problem can be reached provided that the
132 desired target point [K, t0] belongs to the learned range for E and Tair. Thus, the solution for the optimal
133 electric field, considering the form of the vectorY
V fromFig. 1, will be obtained by solving the third 134 degree equation given by,
aE3þ bE2þ cE þ d ¼ K ð9Þ
135 a¼ A6ð Þt0
b¼ A3ð Þ þ Tt0 airA5ð Þt0
c¼ A1ð Þ þ Tt0 airA4ð Þt0
d¼ A7ð Þ þ Tt0 airA2ð Þt0
g
ð10ÞUNCORRECTED PROOF
136 137 where the coefficients Ak(t0) are calculated as:
Akðt0Þ ¼ XM
j¼1
WWkjdUkð Þt0 ð11Þ
138 139 140 3. Results and discussion
The diagram inFig. 1and Eqs. (1)–(11) have been used both for curves’ learning and prediction and to solve the 141 inverse problem in microwave-assisted drying processes. The temperature and drying curves for the learning and 142 validation procedures have been provided by the drying model described in [2,3]. During the learning stage, 143 random values for the input parameters E and Tairhave been generated and different curves were obtained. A 144 moisture content of 0.912 (dry basis) and sample temperature of 26 8C have been used as initial boundary 145 conditions for all the simulations. For other initial simulation parameters, the readers should refer to[2,3]. The time 146 interval for training and testing the neural model was set to T = 600 s. For all the Gaussian functions of the RBF 147 neural network li= Td i/M (i a [1, M]) and r = 3T/M. The learning intervals for each input parameter (E and Tair) 148 are [1246.6; 3630.8] and [30.30; 69.84], respectively. Moreover, it must be pointed out that the proposed neural 149 model has been trained separately for the identification of Xt(t) or Tc(t) curves. 150
time (sec.)
0 100 200 300 400 500 600 700
Moisture content Xt (dry basis)
-0,6 -0,4 -0,2 0,0 0,2 0,4 0,6 0,8 1,0 1,2 1,4 1,6
Xt provided by [2,3]
2 Trials 4 Trials 6 Trials 8 Trials 10 Trials
Tair = 45 °C E = 2500 v/m M = 10 RBF neurons
time (sec.)
0 100 200 300 400 500 600 700
Moisture content Xt (dry basis)
-0,5 0,0 0,5 1,0 1,5
Xt provided by [2,3]
Xt estimated with 1 RBF neuron Xt estimated with 2 RBF neurons Xt estimated with 10 RBF neurons
Tair = 45 °C E = 2500 v/m TRIALS = 10
a) b)
Fig. 3. Predictive behaviour of the proposed model for estimation of Xt(t) from E and Tairinputs.
time (sec.)
0 100 200 300 400 500 600 700
Centre temperature of the sample Tc (°C) 0 50 100 150 200 250 300
Tc from [2,3]
4 Trials 6 Trials 10 Trials 15 Trials
Tair = 50 °C E = 3500 v/m M = 12
time (sec.)
0 100 200 300 400 500 600 700
Centre temperature of the sample Tc (°C) 30 40 50 60 70 80 90 100 110
Tc provided by [2,3]
Tc from 2 neurons Tc from 4 neurons Tc from 6 neurons Tc from 8 neurons Tc from 12 neurons
Tair = 50 °C E = 3500 v/m TRIALS = 15
a) b)
Fig. 4. Predictive behaviour of the proposed model for estimation of Tc(t) from inputs E and Tair.
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The model performance for the prediction of Xt(t) from the input parameters (E and Tair) has been evaluated as a 151 function of the number of trials and RBF neurons. For this purpose, different neuron weight matrixes and training 152 curves have been autonomously generated and learned.Fig. 3a shows the neural model prediction of Xt(t) as a 153 function of the number of trials whileFig. 3b illustrates the neural model estimation of Xt(t) versus the neuron 154 weights’ number. From this figure, one can conclude that the higher the trials and neurons number, the more 155 accurate performance for the neural model, which is normally expected. Yet,Fig. 3also shows that 10 trials for the 156 neural network training and 10 neuron weights are sufficient to precisely predict the temporal evolution of Xt(t), 157 and that the model is much more sensitive to trial amount than to neurons number for smooth curves such as Xt(t). 158 The same assessment has been carried out for to the other involved curve: Tc(t).Fig. 4a shows the behaviour of 159 the temperature neural model prediction versus the number of training trials for the RBF neural network, whileFig. 160 4b represents the prediction convergence of this model versus the number of RBF neuron weights. FromFig. 4, it 161 can be concluded, unlike that for Xt(t), that 15 trials during the learning stage and 12 RBF neurons are sufficient to 162 obtain a good prediction of the temporal evolution for Tc(t) in the considered drying process. Again the neural 163 model seems to be very sensitive to the training process as expected fromFig. 3. Nevertheless, it is clear fromFig. 164 4b that the temperature prediction is more sensitive to the number of neurons than in the case of moisture content. 165 This may be due to the fact that temperature time evolution is less linear than moisture content evolution which 166 implies the need of more slots for the considered time interval t a [0, T]. 167
Finally, the capabilities of the proposed model to give accurate solutions for the inverse problem are evaluated 168 and analysed. Eqs. (8)–(11) have been applied to solve the inverse problem in the drying process to find the 169 optimal value of E that matches the desired target point [Tc0; t0] or [Xt0; t0]. In order to test the resolution of 170 inverse problems, the temperature (Tc0) and moisture content (Xt0) targets have been kept constant and different 171 values for t0have been considered, and the results are shown in Fig. 5. 172
InFig. 5a, the optimum value for E, the moisture content target objective (0.5 dry basis), the drying curves for 173 both the drying model [2,3] and the neural network architecture are shown. Likewise, Fig. 5b illustrates the 174 temperature target (50 8C), the optimum value for E and the temperature curves provided by the drying and the 175 neural model. AsFig. 5 shows, the matching error at the targets [Tc0; t0] or [Xt0; t0] is negligible, while the 176 predictive identification of the drying and temperature curves is very precise. 177178
179 4. Conclusions
180 In this paper, a model based on artificial intelligence has been applied to the design of a novel
181 architecture which interconnects the adaptive characteristics of the RBF (Radial Basis Function) neural
time (sec.)
0 100 200 300 400 500 600 700
Humidity contentXt (dry basis)
-0,2 0,0 0,2 0,4 0,6 0,8 1,0
Xt (t) from [2,3]
Xt1NN(t ; t0=200) Xt2NN(t ; t0=250) Xt3NN(t ; t0=300) Xt4NN(t ; t0=350) 0,5
E1 = 3.183,1 (0,08%) E2 = 2.799,1 (0,21%) E3 = 2.519,1 (1,06%) E4 = 2.312,1 (1,14%)
INVERSE PROBLEM E1 E2 E3 E4
time (sec.)
0 100 200 300 400 500 600 700
Sample Temperature Tc(t) (°C)
20 40 60 80 100 120
Tc (t) from [2,3]
Tc1NN (t ; t0 = 20) Tc2NN (t ; t0 = 30) Tc3NN (t ; t0 = 50) Tc4NN (t ; t0 = 100) Tc5NN (t ; t0 = 150)
50 E1 = 3.643,0 (0,63%)
E2 = 3.049,6 (0,02%) E3 = 2.524,6 (0,78%) E4 = 2.002,6 (0,71%) E5 = 1.690,6 (0,98%) INVERSE PROBLEM
E1
E2
E5 E3 E4
a) b)
Fig. 5. Proposed model applications for estimation of (a) Xt(t) and (b) Tc(t), fitting [Tc0, t0] and [Xt0, t0].
UNCORRECTED PROOF
182 networks with the algebraic tools of the polynomial matrix equations. This neural architecture has been
183 applied to model and predict the sample temperature and moisture content evolution in microwave-
184 assisted drying processes during a long time interval, where the air temperature and the electric field are
185 considered as the input parameters of the process. Additionally, the use of RBF and polynomial networks
186 has allowed the resolution of the inverse problem by finding the optimal value of the microwave electric
187 field that forces Tc(t) or Xt(t) to reach a target value, Tc0 or Xt0, at a desired instant t0. The obtained
188 results for both the temperature and moisture content prediction and the inverse calculation show that
189 this modelling technique is very precise provided that a proper training process and neuron number
190 estimation is carried out. The main advantage of the proposed learning-based model is to provide a
191 closed solution for the described inverse problem, which is difficult to solve by the conventional drying
192 models based on differential equations. Although this neural architecture has been tested on a very
193 particular microwave-assisted drying model, the obtained conclusions can be readily extended to other
194 drying models or techniques due to the adaptive capabilities of neural networks.
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