UNCORRECTED PROOF
ARTICLE IN PRESS
1
Short communication 2
3 Estimation of microwave-assisted drying parameters using
4 adaptive optimization inverse techniques
BJ.L. Pedren˜o-Molina
a,*, J. Monzo´-Cabrera
b, A. Toledo-Moreo
c, D. Sa´nchez-Herna´ndez
b5
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
b 8
Departamento de Teorı´a de la Sen˜al y Radiocomunicaciones, Universidad Polite´cnica de Cartagena,
9 Campus Muralla del Mar, E-30202 Cartagena, Spain
c 10
Departamento de Tecnologı´a Electro´nica, Universidad Polite´cnica de Cartagena,
11 Campus Muralla del Mar, E-30202 Cartagena, Spain
12 Received 31 March 2004; received in revised form 1 June 2004; accepted 1 June 2004
13
14 Abstract
In this work, a parametric adaptive optimization architecture is applied for modelling the direct problem of 15 microwave-assisted drying processes. The proposed architecture, based upon the Levenberg–Marquardt (LM) 16 algorithm, gives a solution for the inverse problem in a complex mathematical model. Experimental results with 17 leather samples in an pre-industrial installation have verified the capabilities of the proposed model for the 18 estimation of optimum non-variable and time-dependent parameters, which are difficult to measure in a real drying 19 process, such as internal evaporation, electric field, and heating-up period. In order to verify the reliability of 20 proposed architecture, some well-known parameters, such as specific heat coefficients, are estimated and 21 contrasted. For parameter and functions estimation, temperature at the centre of the material and moisture content 22 are measured and simultaneously considered. Finally, from analysis of obtained results, some conclusions are 23 extracted with reference to improve the knowledge of internal characterization of microwave-assisted drying 24 processes. 25
D 2004 Elsevier Ltd. All rights reserved. 26
27
0735-1933/$ - see front matterD 2004 Elsevier Ltd. All rights reserved.
doi:10.1016/j.icheatmasstransfer.2004.06.004
B Communicated by Dr. W.J. Minkowycz.
* Corresponding author.
E-mail address: [email protected] (J.L. Pedren˜o-Molina).
International Communications in Heat and Mass Transfer xx (2004) xxx – xxx
www.elsevier.com/locate/ichmt
ICHMT-01250; No of Pages 9
UNCORRECTED PROOF
28 1. Introduction
29 The numerical modelling of microwave-assisted drying processes is a complex task that requires the
30 accurate knowledge of several thermo-physical, dielectric, and boundary condition parameters that
31 appear in the mathematical formulation[1,2]. The experimental estimation of such parameters strongly
32 depends on the considered model and, very often, requires specialised instrumentation, not always
33 available for research activities [3]. The identification of model coefficients from experimental
34 temperature and/or mass transfer data, known as an inverse problem, is an alternative to the experimental
35 determination of those parameters, but it needs the development of specific mathematical and
36 programming techniques[4]. Adaptive optimization algorithms are capable to find optimal parameters
37 for fitting mathematical models to experimental data, when these models are well-posed and
38 dimensioned. Starting from time-domain analysis of temperature distribution and moisture content, an
39 efficient design of the adaptive algorithm can be able to optimize both fixed and time-dependent
40 parameters involved in drying process. The Levenberg–Marquardt (LM) [5] algorithm has been
41 implemented by other authors for simultaneous parameter estimation applied, for example, to
42 experimental data of temperature and moisture contents[6]or to optimize some parameters of Luikov’s
43 model[7]in the construction material-drying processes[8]. On the other hand, optimization algorithms
44 for functions estimation can be found in applications for solving temperature-dependent heat flux in
45 turning tool insert during machining[9], or for specific heat estimation of materials in heat conduction
46 problems[10].
47 In this work, an architecture based on the LM algorithm has been applied to temperature and moisture
48 content measurements in microwave-assisted leather drying processes. Two parameters (the heating-up
49 period and the initial electric field within the sample) and the time-dependent evaporation function of the
50 model described in[11]have been optimized. Then, the estimation of well-known parameters of leather
51 material samples, such as the dry mass and the phase liquid specific heat, has in turn allowed a validation
52 of the proposed architecture and the mathematical model. Results from several experiments with varying
53 operating conditions are also discussed.
54 2. Theoretical study
55 2.1. Heat and mass transfer analysis
56 The mathematical model employed on the study of microwave-assisted drying of leather has been
57 previously reported and validated in[11]although the most important equations are reproduced here for
58 a better understanding of the parameter estimation procedure. In this case, the differential equations that
59 rule the conversion of electromagnetic energy into heat and mass transfer are provided in Eqs. (1–7). See
60 key to symbols in the nomenclature section.
qsDHvð ÞX BX
Bt ¼ g tð Þ2pd f eoeV Xð ÞE2ð ÞX ð1Þ
61 E Xð Þ ¼ Eo
ffiffiffiffiffiffiffiffiffiffiffiffi eW Xð Þ
p ð2Þ
UNCORRECTED PROOF
62 g tð Þ ¼
gx
thd t tbth
gx tzth
ð3Þ
63 qscp
BTc
Bt ¼ 2pd foeoeW Xð ÞjE Xð Þj2þ kt
B2Tc
Bx2 qdevDHev
BX Bt
ð4Þ
64
kt
BTc
Bx ¼ htðTs TairÞ ð1 evÞdms
A DHev
BX Bt
: ð5Þ
65 6667 In these equations, the thermal conductivity (kt) and the specific heat (cp), are assumed to be
68 dependent on the moisture content with a linear relationship,
kt ¼ ktsþ ktwdX ð6Þ
cp¼ cpsþ cpwdX : ð7Þ 69
70 71 72 2.2. Temperature and moisture response analysis for adaptive architecture design
73 The identification of the most important parameters and functions implemented into the mathematical
74 model is based on the analysis and characterization of the probed temperature time-dependent response
75 at the centre of the material Tc(k) and of the moisture content X(k). These data have been obtained from
76 real experiments with different conditions for airflow temperature (Tair) and microwave power level ( P).
77 Hence, by considering experimental results, the selection of the initial electric field (E0) and heating-up
78 period (th) parameters, together with the internal evaporation function ev(k), have been selected as goals
79 of the proposed adaptive architecture. The inherent difficulty of these parameters to be measured or
80 interpreted from the experimental responses of temperature or moisture content data, has made an
81 accurate estimation quite difficult and validation techniques absolutely essentials for this work.
82 Moreover, the simultaneous estimation of cps and cpw has permitted an additional validation of the
83 proposed algorithm.
84 2.3. Description of the adaptive model
85 The proposed adaptive scheme fits the Tc(k) and X(k) outputs from direct mathematical model to
86 experimental results by optimizing the vector of non-variable parameters [E0, th, cps,cpw], described in
87 Eqs. (1–3, 7) and the natural evolution of ev(k) function that produces the best adjustment for the
88 solution of direct model. In[11], the expression for ev(k) function was defined as in Eq. (8), wherea, b,
89 c, and d are constants which are estimated by means of empirical data. These constants do not always
90 ensure that ev(k) keeps below one, its maximum theoretical value, as it depends on Tc(k), and the
91 theoretical limit is ensured by the minimal value in Eq. (8).
evð Þ ¼ mink
"
1; aþ
Tcð Þ bk c
d#
: ð8Þ
92 93
UNCORRECTED PROOF
94 However, the upper limitation for this expression, justifying maximum level of internal evaporation
95 which could be physically reached by the described process, introduces a restriction to the real evolution
96 of ev(k). Due to this, the previously mentioned model was not able to predict the experimental
97 temperature variations for some trials. One of the advantages of the application of the LM optimization
98 algorithm to the mathematical model, when it is well dimensioned, is to eliminate restrictions to internal
99 functions. Thus, by taking into account the exponential evolution of Eq. (8) and the obtained results, a
100 third-order polynomial equation has been defined and tested for ev(k). Consequently, the considered new
101 expression for this function is given by Eq. (9), whereAi parameters are adaptively estimated for each
102 different process.
evð Þ ¼ Ak 3k3þ A2k2þ A1kþ A0: ð9Þ
103 104105 By considering Eq. (9), where ev(k) depends on the sampling period k, together with E0, th, cps, and
106 cpwparameters, this algorithm generates the vector wY given by Eq. (10)to be optimized by the proposed
107 architecture from temperature and moisture content measurements.
wY¼ t E0;th;cps;cpw;A3;A2;A1;A0b: ð10Þ 108 109110 InFig. 1, the designed architecture is presented. Differences in drying behaviours have been obtained
111 by means of variation of two main input parameters to the process: the airflow temperature (Tair) and the
112 microwave power level ( P).
113 The learning process in the proposed algorithm is based on the simultaneous minimization of the error
114 that measures the discrepancy between the desired (experimental) and current (estimated) values of Tc(k)
115 and X(k), using the Levenberg–Marquardt method. In this method, the change D wð YÞ in the adjustable
116 parameters vector wðYÞ is obtained by solving:
adD wY¼ 1
2 rYErr ð11Þ
117 118 where Err is an error index. The elements of a matrix are given by Eq. (12),
aij ¼ ð1 þ kdijÞ X
k
BTcð wY;kÞ
Bwi dBTcðwY;kÞ Bwj
þ FdX
k
BX wðY;kÞ
Bwi dBX wðY;kÞ Bwj
!
ð12Þ 119 120 where k is an index representing the sample number, F is a weighting factor, and d is the so-called
121 Kronecker impulse function. The k variable in Eq. (12) is a parameter that is updated every period. If it is
122 very small, a matrix becomes an approximation to the Hessian, and the method behaves as the inverse-
) (w Err
Real platform for microwave- assisted drying
Mathematical model equations(1-7)
Parameters w
+ - ) (
) (
k T
k X
cm m
Updating w Tair
P
E0,th,cps,cpw,ev(k)
) , (
) , (
k w T
k w X
c
Fig. 1. Proposed scheme for optimization of parameters and internal functions of mathematical model.
UNCORRECTED PROOF
123 Hessian method. If k>>1, the method gets closer to the steepest descent method. In this application,
124 TcðwY;kÞ and X wðY;kÞ are the curves of temperature and moisture content obtained after executing the
125 model with the current parameters, while Tcm(k) and Xm(k) are the experimentally measured evolution of
126 both magnitudes. To obtain the error index, the two curves are subtracted and the resulting values are
127 squared and summed, as shown in Eq. (13).
ErrðwYÞ ¼ X
k
Tcmð Þ Tk c
wY;k 2
þ Fd X
k
Xmð Þ Xk
wY;k 2
: ð13Þ
128 129130 The partial derivatives appearing in Eq. (12) are estimated in this work by incremental
131 approximations. Starting from initial random values, both a and rYErr are evaluated, and solving Eq.
132 (11)a correction for the values ofwY is obtained as:
wYV¼ wYþ D wY ð14Þ
133 134 with wYVbeing the updated value for parameter vector wY. In the LM algorithm each iteration reduces the
135 error until the predetermined error level is reached or a local minimum is found.
136 2.4. Experimental setup
137 In order to validate the model and as a requirement for the designed architecture, an experimental
138 setup has been employed. The microwave drier used in all the tests has been previously described in
139 [11], although it should be remarked that it consisted of a multimode rectangular cavity with mode
140 stirrers, which ensured a uniform spatial field distribution within the sample during the whole process.
141 The microwave power levels used in the tests were 500, 750, and 920 W and the frequency was 2.450
142 MHz. The oven had a fan that provided a forced airflow, which temperature was controlled and varied
143 between 30 and 47 8C with the aid of some resistances. Thus, the setup allowed for a variation of
144 external conditions in these drying tests. The leather samples were 2-mm-thick Nubuck-ND type hides,
t1.1 Table 1
Initial conditions for the tests and results for parameters vector estimation wð ÞY
Test 1 2 3 4 5 6
t1.2
Initial conditions Tair(8C) 30,90 30,97 32,33 45,20 46,20 45,90
t1.3
P (W) 500 750 920 750 500 920
t1.4
X0 0,9058 0,9194 0,9295 0,7978 0,8602 0,9122
t1.5
ms(Kg) 0,1040 0,0930 0,0950 0,0910 0,1030 0,0980
t1.6
Estimated parameters E (V) 2.652,2 2.972,1 3.463,9 2.957,8 2.660,4 3.075,3 t1.7
th(s) 29,84 28,06 36,33 30,71 34,57 29,67
t1.8
cps(J/Kg 8C) 1.607,4 1.614,2 1.581,1 1.609,2 1.595,5 1.649,1 t1.9
cpw(J/Kg 8C) 4.189,1 4.189,3 4135,1 4.186,6 4.177,8 4.254,6 t1.10
A3 4,2109 5,5109 6,6108 8,8109 2,4109 2,3108 t1.11
A2 2,4106 1,0105 3,0105 7,9106 5,2106 2,3105 t1.12
A1 1,8105 0,004 0,002 0,002 0,003 0,007
t1.13
A0 0,762 0,517 0,791 0,791 0,575 0,292
t1.14
UNCORRECTED PROOF
145 with a surface 3535 cm2. The average dry mass was around 100 g while the initial moisture content
146 was around 1 (dry basis).
147 3. Results and discussion
148 For the estimation of the physical parameters, six different experiments were carried out. The
149 proposed adaptive algorithm was applied to temperature values measured at the centre of the body
150 Tcm(k) and moisture content probing Xm(k). The acquisition interval for all the experiments varied
151 between 360 and 800 s, with a sampling rate of 1 s. The initial conditions for the tests and the obtained
152 results are shown inTable 1. Employed material was leather which known values for cpsand cpwwere
153 1.600 and 4.180 (J/Kg 8C), respectively[11].
Fig. 2. Measured and estimated temperature and internal evaporation function.
UNCORRECTED PROOF
154 For these experiments, the F ponderation factor was set to 10 and the initial value for k parameter to
155 0.01. All the parameters in vector wY have been estimated without restrictions. Nevertheless, relative
156 mean error percentage for parameters which can be compared to reported values [11], such as cps and
157 cpw, were about 0.6% and 0.2%, respectively. InFig. 2, both the estimated Tc(k) and the measured Tcm(k)
158 temperature and the internal evaporation function ev(k) are illustrated for the six different tests. The
159 results have been obtained from six experiments and the consideration of the estimated parameters
160 represented inTable 1. By taking into account the absence of restrictions in this architecture, the close
161 agreement between Tc(k) and Tcm(k), and the similarity between reported and estimated values for cps
162 and cpw, then the estimation for the E0, th, A3, A2, A1, A0 parameters of the weight vector can be also
163 considered accurate.
164 From the behaviour of the different microwave-assisted drying tests described in Fig. 2, it can be
165 observed that the absence of restrictions for the natural evolution of ev(k) function supposes an excellent
166 technique for obtaining extremely accurate estimated values for Tcm(k). In fact, although no restrictions
167 were imposed on ev(k), this function keeps below one (model limit) most of the time. The reason for
Fig. 3. Estimated thversus Tcm(k) and Xm(k) curves.
UNCORRECTED PROOF
168 ev(k) exceeding this limit can be based on the fact that temperature was measured at discrete locations
169 along the hide. Thus, in these points it is possible that local evaporation values exceeds the average
170 drying rate due to microwave energy absorption rates higher than the average microwave power
171 absorption in the leather sample as reported in[11]. It can also be observed that the temperature at the
172 centre decreases when ev(k) increases. This indicates that internal evaporation plays a major role in
173 temperature and moisture content profiles when modelling microwave-assisted drying processes. In
174 Fig. 3, estimated th values are depicted versus Tcm(k) and Xm(k) curves, in which only the first 250
175 samples have been drawn for a better threpresentation. Obtained results for thare in good agreement to
176 measured data of temperature and moisture content. In fact, this parameter seems to properly delimit the
177 heating-up period and the constant drying rate period for Tcm(k) and Xm(k) curves.
178 As results inTable 1andFig. 3show, the dependence of the th parameter on the external conditions
179 for this mathematical model cannot be easily determined. One of the advantages of the proposed
180 architecture is to give an estimated value for this parameter, in order to define the different stages of
181 microwave-assisted drying.
182 4. Conclusions
183 In this paper, an optimization architecture has been proposed for solving inverse problems in the
184 microwave-assisted drying processes of laminar materials. The solution is sought by estimating non-
185 variable and time-dependent parameters of the considered mathematical model. Experimental measure-
186 ments of temperature and moisture content feedback to the system in order to improve the convergence
187 of unknown parameters of the model, such as the heating-up period or the initial internal electric field.
188 Therefore, four main drying parameters and the internal evaporation function–difficult to be measured in
189 practice–have been simultaneously estimated in this work. The interpretation of different time dependent
190 responses of the ev(k) function for each experiment, which have been generated as a result of applying
191 this adaptive algorithm, has permitted the establishment of an accurate relationship between the
192 temperature and the internal evaporation involved in microwave-assisted drying processes. Likewise,
193 small errors obtained from the comparison with well-known parameters of the considered material, such
194 as specific heat coefficients of water and dry material, have also allowed for proper validation of the
195 proposed algorithm. Finally, this architecture provides a general methodology for solving inverse
196 problems in microwave drying when the mathematical model is well-posed and dimensioned.
197 Nomenclature
198 A Total area of the sample (m2)
199 cps Dry material specific heat (J/kg 8C)
200 cpw Liquid phase specific heat (J/kg 8C)
201 ev Internal evaporation function (dimensionless)
202 E Electric field absolute value (V/m)
203 E0 Initial value for electric field evolution (V/m)
204 f Frequency (Hz)
205 ht Convective heat transfer coefficient (W/m2 8C)
206 k Discrete time index
UNCORRECTED PROOF
207 kts Dry material thermal conductivity (W/m 8C)
208 ktw Liquid phase thermal conductivity (W/m 8C)
209 ms Dry mass of leather sample (kg)
210 P Power supply (watts)
211 Tc Centre temperature of the sample (8C)
212 Tcm Measured centre temperature of the sample (8C)
213 Ts Surface temperature of the sample (8C)
214 Tair Air temperature in the microwave cavity (8C)
215 t Time (s)
216 th Heating-up period delimiting time (s)
217 x x Coordinate value (m)
218 X Moisture content (dry basis)
219 Xm Measured moisture content (dry basis)
220 X0 Initial moisture content (dry basis)
221 DHevLatent heat of evaporation (J/kg)
222 eV Dielectric constant (dimensionless)
eU Dielectric loss factor (dimensionless) 223
224 qs Density of leather samples (kg/m3)
g Moisture evaporation efficiency 225
226
227 References
228
229 [1] G.V. Barbosa-Ca´novas, H. Vega-Mercado, Dehydration of Foods, Chapman & Hall, New York, 1996.
230 [2] A.S. Mujumdar, et al., Handbook of Industrial Drying, vol. 1, Marcel Dekker, New York, 1995.
231 [3] L.B. Dantas, H.R.B. Orlande, R.M. Cotta, Int. J. Thermal Science 41 (2002) 217.
232 [4] S.K. Kim, B.S. Jung, H.J. Kim, W.I. Lee, Experimental Thermal and Fluid Science 27 (2003) 697.
233 [5] D.W. Marquardt, Journal of the Society for Industrial and Applied Mathematics 11 (1963) 431.
234 [6] L.B. Dantas, H.R.B. Orlande, R.M. Cotta, International Journal of Heat and Mass Transfer 46 (2003) 1587.
235 [7] A.V. Luikov, Heat and Mass Transfer in Capillary-Porous Bodies, Pergamon Press, Oxford, 1966.
236 [8] Z.B. Maroulis, C.T. Kiranouidis, D. Marinos-Kouris, Journal of Food Engineering 14 (1991) 241.
237 [9] S. Kim, International Communications in Heat and Mass Transfer 28 (4) (2001) 537.
238 [10] P.J. Bendeich, J.M. Barry, W. Payten, Applied Mathematical Modelling 27 (2003) 337.
239 [11] J. Monzo´-Cabrera, et al., Microwave and Optical Technology Letters 32 (2002) 465.
240
