Determination of the diffusion coefficient of dry mushrooms using the inverse method

Journal of Food Engineering 95 (2009) 1–10

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Journal of Food Engineering journal homepage: www.elsevier.com/locate/jfoodeng

Review

Determination of the diffusion coefficient of dry mushrooms using the inverse method Cristiane Kelly F. da Silva a,*, Zaqueu Ernesto da Silva a, Viviana Cocco Mariani b a b

Laboratório de Energia Solar, Universidade Federal da Paraíba – UFPB, 58051-970 João Pessoa, Paraíba, Brazil Departamento de Engenharia Mecânica, Pontifícia Universidade Católica do Paraná – PUCPR, 80215-901 Curitiba, Paraná, Brazil

a r t i c l e

i n f o

Article history: Received 19 November 2008 Received in revised form 18 March 2009 Accepted 14 April 2009 Available online 22 April 2009 Keywords: Diffusion coefficient Inverse problem Levenberg–Marquardt Differential Evolution Sensitivity study Mass transfer

a b s t r a c t This study determined the diffusion coefficient during hot air drying of mushrooms (Agaricus blazei). Experimental drying kinetics were applied to sliced mushroom at different air temperatures (45, 60, 75, and 80 °C) and air velocities (1, 1.2, 1.75, 2.3, and 2.5 m/s) in order to find an analytical solution to the mass transfer equation for drying using an inverse problem with two distinct optimization techniques: Levenberg–Marquardt and Differential Evolution, in successive trials. The experimental data were also modified by a normal distribution random error. Statistical analyses showed no significant differences between reported and estimated curves. The diffusion coefficients are greater in mushrooms dried at higher temperatures, and those dried at higher drying air velocities. Ó 2009 Elsevier Ltd. All rights reserved.

Contents 1. 2. 3.

4.

5.

6.

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Mass transfer model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Materials and methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1. Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2. Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Inverse problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1. Differential Evolution Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2. Levenberg–Marquardt Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Results and discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1. Sensitivity analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2. Inverse Problem analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Conclusions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1. Introduction The thermal properties of dry foods are of interest to food engineers in modeling drying and dry storage processes and in designing equipment. With the advent of modern computers and * Corresponding author. E-mail addresses: [email protected] (C.K.F. da Silva), [email protected] (Z.E. da Silva), [email protected] (V.C. Mariani). 0260-8774/$ - see front matter Ó 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.jfoodeng.2009.04.009

1 3 3 3 4 4 4 5 5 5 8 9 9 9

numerical methods, nonlinear equations can be accurately solved. However, the reliability of these predictions is directly related to the accuracy with which researchers are able to predict thermophysical properties. Knowledge of the diffusion coefficient of mushrooms is important for researchers who seek to improve drying processes using modeling and simulation methods. The determination of this coefficient requires a series of highly timeconsuming measurements. An alternative is the use of inverse methods to estimate the initial state of the system, unknown

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Nomenclature CR D fm J k L r1, r2, r3 RH S T t v X X0 Xe X Y z

crossover rate diffusion coefficient (m2 s1) mutation factor sensitivity coefficients matrix generation food slice thickness (m) mutually distinct integers relative humidity of drying air (%) objective function temperature (°C) time (s) flow velocity of drying air (m s1) moisture content d.b. (kg kg1) initial moisture content d.b. (kg kg1) equilibrium moisture content d.b. (kg kg1) average moisture content d.b. (kg kg1) measured moisture content (kg kg1) spatial coordinate (m)

parameters, boundary conditions, heat transfer coefficient, thermal properties, the system’s geometry, and, for instance, a temperature or moisture content field (Jarny and Maillet, 1999). Although researchers in food process engineering use inverse methods less frequently, several papers in this area are cited below. The inverse problem to determine heat capacity versus temperature was solved by Zueco et al. (2003). Mendonça et al. (2005) presented an inverse method to estimate the thermal conductivity and volumetric specific heat of spherical fruits during convection heating. Simpson and Cortés (2004) proposed a method to determine three parameters of a model of apparent volumetric specific heat. Zueco et al. (2004) presented an inverse method to determine specific heat during a thawing process. Martins and Silva (2004) determined the specific heat and thermal conductivity of frozen green beans during thawing by measurements and by inverse methods. Bialobrzewski (2006) determined changes in the heat transfer coefficient during celery root drying under natural convection conditions, based on inverse problem formulations. Anderson and Singh (2006) determined heat transfer coefficients during impingement thawing, using a thermal conductivity similar to that of a food product, but with known properties. Monteau (2008) developed an inverse method to estimate the thermal conductivity of bread. Mariani et al. (2008) determined the apparent thermal diffusivity of banana as a function of moisture content and/or average temperature during the drying process. Earlier researches investigating the drying kinetics of mushrooms are discussed next. Krokida et al. (2003) examined the effect of air conditions and characteristic sample size on the drying kinetics of various plant materials (potatoes, carrots, peppers, mushrooms, etc.) during air drying. (Maroulis et al., 1988) used a first-order reaction kinetics model in which the drying constant is a function of the process variables, while the equilibrium moisture content of dried products was fitted to a GAB equation. The drying temperature was the most important factor in the drying rate of all the materials examined, while the effect of air velocity and air humidity was considered lower than that of air temperature. Cao et al. (2003) analysed the drying of Maitake mushrooms using the modified plate drying model. Samples were dried at various air temperatures and relative humidity. Their results indicated that the modified plate drying model for predicting the moisture content and drying rate was reasonably compatible with their experimental results under different drying conditions. Both the

Greek symbols b vector of unknown parameters Db variation of unknown parameter d relative error of convergence e normal random error k damping parameter r standard deviation of errors in temperature measurements v sensitivity coefficients x random number variable Xm diagonal matrix Subscripts comp computed exp experimental i individual population index

drying constant and surface mass transfer coefficient of the modified plate drying model were expressed as Arrhenius-type functions of temperature. Rodríguez et al. (2005) modeled the drying kinetics of mushrooms under several operational conditions to evaluate the effective diffusion coefficient of moisture. Different ways of microwave-vacuum drying were compared with freeze drying. Their results revealed that a decrement of the applied pressure produces a certain increase in the drying rate allied to lower moisture in the dehydrated end product. Controlling the temperature inside the sample helped to ensure better quality of the dehydrated product. Diffusion coefficients showed a correspondence with product temperature during drying. The microwave-dried samples obtained with moderate power and temperature control of the product showed a degree of quality similar to that obtained by freeze drying. Walde et al. (2006) dehydrated mushrooms by different treatments and various drying systems. The drying rate curves showed a characteristic constant rate period and a falling rate period, except in vacuum drying, where drying rates were very slow. Microwave drying may not be a suitable method for drying mushrooms, since the longer drying times caused charring at the edges. In view of the drying time and quality of products, the fluidized bed drying system was found to be superior. Giri and Prasad (2007) evaluated microwave-vacuum dehydration characteristics of button mushrooms in a microwave oven modified into a drying system. Microwave-vacuum drying reduced the drying time by 70–90% and the dried products had better rehydration characteristics than products subjected to convective air drying. Page’s exponential and empirical model adequately describes microwave-vacuum drying data. A statistical analysis of the drying data showed that the drying rate was highly influenced by microwave power, followed by sample thickness, while system pressure exerted the greatest influence on the rehydration characteristics of dehydrated mushrooms. Jambrak et al. (2007) used ultrasound as a pretreatment method to dry mushrooms, Brussels sprouts and cauliflower in order to reduce the drying time and to understand the effect of ultrasound in the mass transfer process, whose limiting step is diffusivity. The procedures they used were either freeze drying or conventional drying. After the ultrasound treatment, the drying time of all their samples was shorter than that of untreated samples. The rehydration properties of ultrasoundtreated samples were better than those of untreated samples.

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The present study resembles those of Ramallo et al. (2004), Mendonça et al. (2005), and Ruiz-López and Garcia-Alvarado (2007) from certain technical points of view: the geometry is simple and the direct problem is solved analytically. Using Fick’s law, Ramallo et al. (2004) estimated the diffusion coefficients of water and sucrose during the osmotic dehydration of sliced pineapple at different temperatures. The equilibrium water content varied from 34 to 36% for a 60% w/w sucrose solution and was practically independent of temperature. Equilibrium sugar content increased from 45 to 54% as the temperature rose from 30 to 50 °C. Mendonça et al. (2005) estimated the thermal conductivity and volumetric thermal capacity of apples, applying the inverse problem of heat conduction to find an analytical solution to the problem. Ruiz-López and Garcia-Alvarado (2007) proposed an analytical solution for a mass transfer equation for drying considering variable diffusivity. They obtained experimental drying kinetics for sliced mangos at different air temperatures and air velocity to fit their proposed model by means of nonlinear regression with four different degrees of complexity. Their results indicated that the proposed model can accurately reproduce experimental drying kinetics. In the present work, an inverse method is used to estimate the diffusion coefficient. Two optimization methods, Levenberg– Marquardt and Differential Evolution, are implemented and their performance is analysed. Thus, the objectives of this study were to: (1) develop an inverse method to estimate the diffusion coefficient of dry mushrooms; (2) determine the sensitivity coefficients for parameters such as diffusion coefficient and equilibrium moisture; and (3) determine the diffusion coefficient of dry sliced mushrooms as a function of water content using experimental drying kinetics at different air temperatures (45, 60, 75 and 80 °C) and air velocities (1, 1.2, 1.75, 2.3, and 2.5 m/s) and fitting it to an analytical solution of a mass transfer equation. 2. Mass transfer model For the direct problem of mass transfer inside food products subjected to air drying processes, a diffusion model was developed, considering the following assumptions in each case:  the product is represented by the geometrical form of a slice of thickness L;  moisture transfer is predominantly one-dimensional;  the initial moisture content is uniformly distributed throughout the product;  shrinkage is considered negligible;  the diffusion coefficient is considered constant and homogeneous during drying.

Air exit

Fixed bed dryer Temperature control

Air inlet

Fig. 2. Vertical fixed bed dryer used in the experiments.

Based on the above assumptions, the set of equations that defines the mathematical model can be written as:

@X @2X ¼D 2 @t @z

ð1Þ

with initial and boundary conditions,

X ¼ X 0 ; 0 < z < L; t ¼ 0 @X ¼ 0; z ¼ 0; t > 0 @z X ¼ X e ; z ¼ L; t > 0

ð2Þ ð3Þ ð4Þ

where X is the moisture content (kg kg1), D is the diffusion coefficient (m2 s1), and z and t are the independent variables, i.e., position (m) and time (s), respectively. Systems (1)–(4) may have an analytical solution if the above assumptions are valid. This analytical solution for a rectangular geometry is (Crank, 1975):

  1 X  Xe 8 X 1 t 2 2 ¼ 2 exp ð2n þ 1Þ p D X 0  X e p n¼0 ð2n þ 1Þ2 4L2

ð5Þ

where X is the product’s average moisture content along the length of the diffusion path (kg water/kg dry matter). 3. Materials and methods 3.1. Materials The analytical solution to the mass transfer problem was validated by experimental results given by Kurozawa (2005). Fresh

Fig. 1. (a) Mushroom (Agaricus blazei), and (b) sliced mushrooms used in the experiments.

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greenhouse mushrooms (Agaricus blazei) were used with an average moisture content of 88.75% (w.b.), supplied by GAPI – Grupo Agaricus de Piedade, located in the city of Pilar do Sul, state of São Paulo, Brazil. The mushroom samples were approximately 0.05 m in length and 0.03 m in diameter, as illustrated in Fig. 1a. For the convective air drying, the fresh mushroom samples were sliced longitudinally into 0.005 m thick slices, as shown in Fig. 1b. 3.2. Methods The mushrooms were dried on a fixed bed dryer (Fig. 2). Eight mushroom drying experiments were carried out (with a half-thickness of approximately 2.5  103 m) under different conditions of temperature and air velocity, as indicated in Table 1. The moisture content presented in Table 1 was determined from samples weighing 3–5 g placed in aluminum crucibles. The sample/crucible sets were then subjected to a greenhouse method under forced convection at 105 °C for 24 h. After drying, the sample/crucible sets were cooled to ambient temperature in a desiccator and then weighed. The residuals or the difference between the values given by the measurement and the model were used to estimate the quality of the model (Beck and Arnold, 1977). The residuals can be expressed as r = Xexp  Xcomp. Fig. 3 shows residuals versus time plots. When the residuals oscillated around zero, the magnitude of the oscillations had a minimum value of 0.4493 kg kg1 and a maximum of 0.6835 kg kg1, which were of the order of the standard deviation of the measurement errors of 0.3757. This indicates that the measurement was reliable and the analytical solution represented the physical phenomenon. 4. Inverse problem Many optimization methods have been proposed to solve inverse problems. Among the deterministic methods, the Levenberg–Marquardt method has been used successfully in several areas (Kanevce et al., 2005; Mendonça et al., 2005; Mejias et al., 1999). Among the stochastic methods, the Differential Evolution method has been less applied to inverse problems, which include the works of Kanevce et al. (2003), and Mariani et al. (2008). The optimization methods used to estimate the diffusion coefficient depend on the minimization of S, the sum of square of differences between the experimental moisture content and the moisture content computed with a diffusion model

S¼

n X  2 X exp  X comp :

ð6Þ

1

A set of moisture contents was obtained experimentally at discrete times during the unsteady drying process. This set of experimental moisture contents was modified by adding a normal random error to verify the accuracy of the optimization techniques

Table 1 Air drying conditions and moisture contents of mushrooms (Kurozawa, 2005). Test 1 2 3 4 5 6 7 8

T (°C)

v (ms1)

X0 (d.b.) (kg kg1)

Xe (d.b.) (kg kg1)

RH (%)

t (h)

D (m2 s1)

45 75 45 75 40 80 60 60

1.20 1.20 2.30 2.30 1.75 1.75 1.00 2.50

9.8093 9.9688 8.4584 9.2295 9.7160 7.5560 7.3002 7.1137

0.0317 0.0048 0.0265 0.0045 0.0544 0.0051 0.0086 0.0103

25.3 9.9 23.1 9.5 32.7 10.4 13.2 14.5

8 2.5 5 2.5 10 5 7 4

3.88 12.79 6.91 17.5 4.14 14.29 5.56 9.49

Fig. 3. Comparison of measured and theoretical moisture content and related residuals.

in determining the diffusion coefficient. Based on Xexp(z, tj), the solution to the mathematical model, where tj are discrete times within the unsteady period, a new series of moisture contents, called X(z, tj, ej), were obtained by adding a normal random error ej to Xexp (z, tj). Thus

Xðz; tj ; ej Þ ¼ X exp ðz; t j Þ þ ej

ð7Þ

where ej = xjr, xj is the normally distributed random number, with zero mean and a standard deviation of 0.05. The values of xj were produced by a random number generator. The Differential Evolution and Levenberg–Marquardt methods were implemented and used as the search technique. A brief description is given below of the two numerical techniques, whose efficiency and reliability in the solution of inverse problems in heat and mass transfer engineering have already been proven. 4.1. Differential Evolution Method Differential Evolution (DE) is an evolutionary algorithm proposed by Storn (1995), and Storn and Price (1995). While DE shares similarities with other evolutionary algorithms (EA), it differs significantly in that the information about distance and direction of the current population is used to guide the search process. DE uses the differences between randomly selected vectors (individuals) as the source of random variations for a third vector (individual), referred to as the target vector. Trial solutions are generated by adding weighted difference vectors to the target vector. This process is referred to as the mutation operator where the target vector is mutated. A crossover step is then applied to produce an offspring, which is only accepted if it improves the fitness of the parent individual. The basic DE algorithm is described in greater detail below with reference to the three evolution operators: mutation, crossover, and selection. Mutation is an operation that adds a vector differential to a population vector of individuals, according to the following equation:

  zi ðk þ 1Þ ¼ xi;r1 ðkÞ þ fm  xi;r2 ðkÞ  xi;r3 ðkÞ

ð8Þ

where i = 1, 2,..., N is the individual’s index of population, k is the generation, xi ðkÞ ¼ ½xi1 ðkÞ; xi2 ðkÞ; . . . ; xin ðkÞT stands for the position of the i-th individual of population of N real-valued n-dimensional

C.K.F. da Silva et al. / Journal of Food Engineering 95 (2009) 1–10

vectors, zi ðkÞ ¼ ½zi1 ðkÞ; zi2 ðkÞ; . . . ; zin ðkÞT stands for the position of the i-th individual of a mutant vector; r1, r2 and r3 are mutually different integers and also different from the running index, i, randomly selected with uniform distribution from the set f1; 2; . . . i  1; i þ 1; . . . ; Ng, and fm > 0 controls the amplification of the difference between two individuals so as to avoid search stagnation, and is usually taken from the range [0.1, 1]. Crossover applied to the population is employed to generate a trial vector by replacing certain parameters of the target vector with the corresponding parameters of a randomly generated donor vector. For each vector, zi(k+1), an index rnbr() 2 {1,2,...,n} is randomly chosen using uniform distribution, and a trial vector, ui ðk þ 1Þ ¼ ½ui1 ðk þ 1Þ; ui2 ðk þ 1Þ; . . . ; uin ðk þ 1ÞT , is generated with

 uij ðk þ 1Þ ¼

zij ðk þ 1Þ; se randbðjÞ 6 CR ou j ¼ rnbrðiÞ; xij ðkÞ; se randbðjÞ > CR ou j–rnbrðiÞ;



ui ðk þ 1Þ; if f ðuðk þ 1ÞÞ 6 f ðxi ðkÞÞ; xi ðkÞ; otherwise

to the Gauss method (Yang and Gao, 2007). The iterative procedure starts with an initial guess, b0, and at each step the vector b is modified until:

   ðkþ1Þ ðkÞ   bi  bi   < d;  ðkÞ  bi  þ n

for i ¼ 1; 2; 3 . . .

ð12Þ

where d is a small number that must be chosen by the investigator (typically 103) and n (<1010). The LM method is quite a robust and stable estimation procedure whose main advantage is a good rate of convergence (Fguiri et al., 2007). 5. Results and discussion

ð9Þ

In the above equations, randb(j) is the j-th evaluation of a uniform random number generation with [0,1] and CR is in the range [0,1]. Selection is the procedure of producing better offspring. To decide whether or not the vector ui(k+1) should be a member of the population comprising the next generation, it is compared with the corresponding vector xi(k). Thus, if f denotes the objective function under minimization, then

xi ðk þ 1Þ ¼

5

ð10Þ

In this case, the cost of each trial vector ui(k+1) is compared with that of its parent target vector xi(k). If the cost, f, of the target vector xi(k) is lower than that of the trial vector, the target is allowed to advance to the next generation. Otherwise, the target vector is replaced by a trial vector in the next generation. Storn (1996) and Storn and Price (1997) proposed ten different strategies for DE based on the individual being disturbed, the number of individuals used in the mutation process and the type of crossover used. The strategy described above is known as DE/ rand/1, meaning that the target vector is randomly selected, and only one difference vector is used. This strategy is considered here.

The moisture content distribution depends on z, t and on other physical parameters, particularly:

X ¼ Fðz; t; X 0 ; X e ; DÞ:

ð13Þ

Sensitivity equations were set up using differentiation of the model equation with respect to the diffusion coefficient, D, and equilibrium moisture content, Xe. For comparison of sensitivity coefficients which do not have the same units, we used the reduced sensitivity coefficients obtained by multiplying the original sensitivity coefficients by their own parameter. The sensitivity coefficients were defined as:

@X ; @X e @X ¼D : @D

vXe ¼ X e

ð14Þ

vD

ð15Þ

In Eqs. (14) and (15), the sensitivity coefficients represent the variation in the variable state due to a change in the value of an unknown parameter. In other words, they ascertain the influence of the parameter on the moisture content distribution. Sensitivity coefficients are therefore important for estimating parameters (Beck and Arnold, 1977). The following section illustrates the parametric sensitivity analysis as presented in Niliot and Lefèvre (2004).

4.2. Levenberg–Marquardt Method The Levenberg–Marquardt (LM) Method was derived by modifying the ordinary least squares norm and is a combination of the Gauss and Steepest Descent methods. Based on the criterion of ordinary least squares, the iterative formula has the following expression (Press et al., 1990)



h i1 bðkþ1Þ ¼ bðkÞ þ ðJ k ÞT J k þ kk Xkm ðJ k ÞT Y  Xðbk Þ

ð11Þ

where k is the number of iterations, k is a positive scalar called damping parameter, Xm is a diagonal matrix, and J is the sensitivity coefficient matrix defined as JðbÞ ¼ @X T ðbÞ=@b. The purpose of the matrix term kk Xkm in Eq. (11) is to damp oscillations and instabilities due to the ill-conditioned character of the problem, by making its components larger than those of JTJ, if necessary. The damping parameter is set large in the beginning of the region around the initial guess used for the exact parameters. With this approach, the matrix JTJ is not required to be non-singular at the beginning of iterations and the Levenberg–Marquardt method tends to the steepest descent method, that is, a fairly small step is taken in the negative gradient direction. The parameter kk is then gradually reduced as the iteration procedure advances to the solution of the parameter estimation problem, and then the Levenberg–Marquardt method tends

5.1. Sensitivity analysis Fig. 4a and b show the evolution of the reduced sensitivity coefficient to the equilibrium moisture content, Xe, as a function of time for each test performed in this study. Note that the sensitivity coefficients increased over time until stability was reached. The curves indicate that the variation in the sensitivity coefficient is more significant when the process is performed at low values of drying temperature, i.e., the mass transfer increases along with increasing temperature. Fig. 4a and b also present the effects of air velocity on the sensitivity coefficient. Note that procedures performed with low air velocity lead to higher sensibility coefficients. Physically, this indicates that processes carried out at low temperature and air velocity lead to higher equilibrium moisture content, and the equilibrium condition is reached over a longer interval of time. Therefore, depending on the desired end product, experiments carried out at low temperatures and air velocities are not feasible. El-Aouar et al. (2003) demonstrated that increasing the air temperature and air velocity reduces the drying time of the end product. Fig. 5a and b illustrate the evolution of the sensitivity coefficient to the diffusion coefficient, D, over time. An analysis of the curves of the sensitivity coefficient reveals the influence of the air drying temperature and velocity on the process. Note that the sensitivity

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C.K.F. da Silva et al. / Journal of Food Engineering 95 (2009) 1–10

0.06

0.04

0.05 0.03

1: T=45°C, v=1.20 m/s 2: T=75°C, v=1.20 m/s 3: T=45°C, v=2.30 m/s 4: T=75°C, v=2.30 m/s

0.02

χ Xe

χ Xe

0.04 5: T=40°C, v=1.75 m/s 6: T=80°C, v=1.75 m/s 7: T=60°C, v=1.00 m/s 8: T=60°C, v=2.50 m/s

0.03

0.02 0.01 0.01

0

0 0

1

2

3

4

5

6

7

0

8

1

2

3

4

5

6

Time (h)

Time (h)

(a)

(b)

7

8

10

9

0

0

-0.5

-0.5

-1

-1

-1.5

-1.5

χD

χD

Fig. 4. Sensitivity coefficient for Xe in each test. (a) Tests 1–4, (b) Tests 5–8.

-2

-2 1: T=45°C, v=1.20 m/s 2: T=75°C, v=1.20 m/s 3: T=45°C, v=2.30 m/s 4: T=75°C, v=2.30 m/s

-2.5

5: T=40°C, v=1.75 m/s 6: T=80°C, v=1.75 m/s 7: T=60°C, v=1.00 m/s 8: T=60°C, v=2.50 m/s

-2.5

-3

-3 0

1

2

3

4

5

6

7

0

8

1

2

3

4

5

Time (h)

Time (h)

(a)

(b)

6

7

8

9

10

Fig. 5. Sensitivity coefficient for D in each test. (a) Tests 1–4, (b) Tests 5–8.

3

2.5

Test 1: D Test 1:Xe

2

Test 2: D Test 2:Xe

1.5

Test 3: D Test 3:Xe

1

Test 4: D Test 4:Xe

0.5 0 1

-0.5

2

3

4

Time (h)

-1

5

6

7

Test 5: D Test 5: Xe

2.5

8

Reduced Sensitivity Coefficients

Reduced Sensitivity Coefficients

3

2

Test 6: D Test 6: Xe

1.5

Test 7: D Test 7: Xe

1

Test 8: D Test 8: Xe

0.5 0 1

-0.5

2

3

4

5

Time (h)

6

-1

-1.5

-1.5

-2

-2

-2.5

-2.5 -3

-3

(a)

(b)

Fig. 6. Sensitivity coefficients for Xe and D in all drying tests: (a) Tests 1–4, (b) Tests 5–8.

7

8

9

10

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coefficients show an initially declining and negative evolution at the beginning of drying, after which they steadily increase. Considering the behavior of the curves in absolute values, it can be seen that the sensitivity coefficient reaches its maximum value at higher air drying temperatures and air velocity. Physically, this indicates that the diffusion coefficient can be determined more accurately if the process takes place at higher temperatures. This finding is in agreement with the results reported by El-Aouar et al. (2003), who showed that increasing both air temperature and velocity favored the mass transfer process, and hence, increased the effective water diffusivity of fresh papayas, which varied from 1.72  109 m2 s1 to 4.78  109 m2 s1. The results in Fig. 6 compare the reduced sensitivity coefficients to the equilibrium moisture content, Xe, and to the diffusion coefficient, D. The main purpose of this figure is to verify their linear interdependence. Thus, Fig. 6 indicates that:  The parameters are linearly independent, i.e., the sensitivity coefficients vary in different ways.

The above analysis demonstrates that, in the present study, it was impossible to estimate two parameters simultaneously in only one experiment.

10

10 Test 1: LM Test 1: ED Test 1: Kurozawa (2005) Test 2: LM Test 2: ED Test 2: Kurozawa (2005)

8 7 6 5 4

8 7 6 5 4

3

3

2

2

1

1 0

0 0

1

2

3

4

5

6

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9

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10

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6

Time (h)

Time (h)

(a)

(b)

7

8

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10

8 7

5: LM 5: ED 5: Kurozawa (2005) 6: LM 6: ED 6: Kurozawa (2005)

Test 7:LM Test 7:ED Test 7:Kurozawa Test 8:LM Test 8:ED Test 8:Kurozawa

9 8

Moisture Content (kg/kg)

Test Test Test Test Test Test

9

Moisture Content (kg/kg)

Test 3: LM Test 3: ED Test 3: Kurozawa Test 4: LM Test 4: ED Test 4: Kurozawa

9

Moisture Content (kg/kg)

9

Moisture Content (kg/kg)

 Sensitivity to the equilibrium moisture content, Xe, is close to zero during the drying process. Therefore, it is impossible to estimate this parameter, since low sensitivity and variations in this parameter in the same process imply deviations that are indistinguishable from the theoretical model.  Sensitivity to the diffusion coefficient, D, in absolute values, reaches a maximum, declining thereafter until the end of the drying process. The sensitivity of this parameter is high, indicating that, moment to moment, the moisture content is highly dependent on the diffusion coefficient. In terms of parameter estimation, this indicates that minor changes in the value of this parameter affect the model.  The curves of the reduced sensitivity coefficients show that the highest value in absolute terms is reached in the initial hours of the drying process.

6 5 4 3

7 6 5 4 3

2

2

1

1

0

0 0

1

2

3

4

5

6

Time (h)

(c)

7

8

9

10

0

1

2

3

4

5

Time (h)

6

7

8

(d)

Fig. 7. Experimental and predicted drying kinetics (without the inclusion of measurement errors) of mushrooms in different drying conditions. (a) Tests 1 and 2, (b) Tests 3 and 4, (c) Tests 5 and 6, (d) Tests 7 and 8.

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affected the drying curves, decreasing the drying time of the samples. As expected, the air velocities also affected the drying curves, albeit to a lesser extent than air temperature. This finding is congruent with the results obtained by Krokida et al. (2003), who reported that the drying temperature was the most important factor affecting the drying rate in all the materials they tested, while the effect of air velocity was found to be lower than that of air temperature. Table 2 shows the diffusion coefficients obtained from adjusting the two optimization methods to all the drying conditions, as well as the respective coefficient of determination, R2. The highest diffusion coefficient was found in test 4 (approximately 17.80  1010 m2 s1) and the lowest in test 1 (approximately 4.084  1010 m2 s1). Therefore, the rise in temperature caused this coefficient to increase, probably due to the lower theoretical moisture content gradient in the product. Thus, a comparison of the results of tests 1 and 3, or 2 and 4 indicates that increasing the air velocity led to higher diffusion coefficient values. The diffusion coefficients obtained in tests 3 and 4 were approximately 25% higher than those calculated from tests 1 and 2, respectively. The kinetic energy of water molecules in food increases with rising temperatures and drying air velocities, reducing the

Table 2 Diffusion coefficient and standard deviation obtained by each optimization method for all the drying tests. Simulations without errors added to the measurements. Tests

Levenberg–Marquardt

Differential Evolution

D (m2 s1) 1 2 3 4 5 6 7 8

R2 10

14

3.932  10 ± 5.333  10 12.850  1010 ± 2.312  1013 10 ± 1.219  1013 6.849  10 17.560  1010 ± 3.828  1013 4.071  1010 ± 5.608  1014 14.290  1010 ± 3.519  1013 5.591  1010 ± 1.099  1013 9.460  1010 ± 2.177  1013

D (m2 s1)

0.997381 0.998334 0.995909 0.998013 0.987843 0.953263 0.997886 0.997037

R2 10

4.084  10 13.180  1010 6.976  1010 17.800  1010 4.171  1010 14.580  1010 5.994  1010 9.639  1010

0.997145 0.998377 0.995649 0.997915 0.986284 0.948666 0.997201 0.996839

5.2. Inverse Problem analysis Fig. 7 shows experimental and simulated drying curves (without the inclusion of measurement errors) of mushrooms vs. drying time at drying air temperatures ranging from 45 to 80 °C, and air velocities ranging from 1 to 2.5 m s1. The simulation was done with a diffusion model using two optimization methods to solve the inverse problem. As can be observed in Fig. 7, air temperature

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10 Test 1:LM Test 1:ED Test 1:Kurozawa (2005) Test 2:LM Test 2:ED Test 2:Kurozawa (2005)

Moisture Content (kg/kg)

8

Test 3:LM Test 3:ED Test 3:Kurozawa (2005) Test 4:LM Test 4:ED Test 4:Kurozawa (2005)

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Moisture Content (kg/kg)

9

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(b)

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10 Test 5:LM Test 5:ED Test 5:Kurozawa (2005) Test 6:LM Test 6:ED Test 6:Kurozawa (2005)

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Test 7:LM Test 7:ED Test 7:Kurozawa (2005) Test 8:LM Test 8:ED Test 8:Kurozawa (2005)

9 8

Moisture Content (kg/kg)

9

Moisture Content (kg/kg)

4

Time (h)

7 6 5 4 3

7 6 5 4 3

2

2

1

1

0

0 0

1

2

3

4

5

Time (h)

(c)

6

7

8

9

10

0

1

2

3

4

Time (h)

5

6

7

(d)

Fig. 8. Experimental and predicted drying kinetics (including measurement errors) of mushrooms in different drying conditions. (a) Tests 1 and 2, (b) Tests 3 and 4, (c) Tests 5 and 6, (d) Tests 7 and 8.

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Table 3 Diffusion coefficient and standard deviation obtained by each optimization method for all the drying tests. Simulations for 30 estimates of experiments obtained with r = 0.05. Tests

Levenberg–Marquardt 2

D(m s 1 2 3 4 5 6 7 8

1

Differential Evolution 2

rD

) 10

3.327  10 12.325  1010 5.877  1010 16.867  1010 3.158  1010 13.599  1010 5.172  1010 8.690  1010

R 12

2.918  10 2.641  1011 5.496  1012 3.604  1011 5.026  1012 3.661  1011 2.144  1012 4.399  1011

0.997740 0.998250 0.997014 0.998232 0.996010 0.963517 0.998258 0.997592

attractive forces, which, in turn, increases the release of water from the food. The diffusion coefficient represents the water removal rate in the diffusion process occurring inside the product; hence, the values of this property are higher in processes carried out at higher levels of operational conditions. The R2 values demonstrate that the two optimization methods yielded a similar performance under the various experimental conditions applied here. These values also indicate that test 2 presented the best fitting. To validate the two proposed optimization methods, the set of experiments was modified, i.e., the experiments were simulated with Gaussian distribution with zero mean and a standard deviation of 0.05 in the 30 simulations. Fig. 8 shows the moisture content estimated by the two optimization methods as well as the experimental moisture content, indicating that stable results were obtained. Thus, the Levenberg–Marquardt and Differential Evolution methods predicted the experimental curves accurately, providing a good estimation of the diffusion coefficient, mainly at higher temperatures, as indicated in Table 3, which lists the average values and standard deviation of the diffusion coefficient. These results indicate that the diffusion coefficients obtained in tests 2, 4, 6 and 6 showed a higher standard deviation than the other tests, confirming the sensitivity analysis depicted in Fig. 4, which indicates that the sensitivity of the diffusion coefficient increases in processes conducted at higher temperatures. Some of the results listed in Table 2 are very similar to the experimental results without the inclusion of errors given in Table 1, confirming that the optimization methods (Differential Evolution and Levenberg–Marquardt) provided an adequate solution, despite the noise present in the experimental results. Our results confirm that the diffusion coefficient can be estimated accurately by both optimization methods. 6. Conclusions The inverse problem of the determination of the diffusion coefficient was solved using the Levenberg–Marquardt and Differential Evolution methods. The diffusion model with a constant diffusion coefficient and without the assumption of shrinkage adequately represented the mushroom drying process. The Levenberg– Marquardt and Differential Evolution optimization methods were both validated statistically and applied to eight drying experiments, showing good performance. To analyse the consequences of imprecise knowledge of the other model parameters, the sensitivity to parameters was investigated, revealing low sensitivity to the equilibrium moisture content and high sensitivity to the diffusion coefficient. Hence, the moisture content is highly dependent on the diffusion coefficient. Note that the two optimization methods accurately estimated the diffusion coefficient in all the tests. When the set of experiments was simulated using Gaussian distribution with zero mean and a standard deviation of 0.05, the diffusion coefficients estimated by the two optimization methods were similar to those obtained previously. Thus, the inclusion of errors in

D(m2 s1)

R2

rD 10

4.075  10 13.012  1010 7.117  1010 17.517  1010 4.159  1010 14.377  1010 5.967  1010 9.553  1010

12

5.1656  10 2.059  1011 1.0178  1018 2.805  1011 5.418  1012 2.308  1011 8.192  1012 1.512  1011

0.997162 0.998356 0.995321 0.998030 0.986476 0.951901 0.997261 0.996937

measured values has only a negligible effect on the solution. The best fitting was obtained using higher temperatures in the diffusion model. The highest value obtained in this study for the mushroom diffusion coefficient was approximately 17  1010 m2 s1, while the lowest was approximately 4  1010 m2 s1. Because of the effective nature of the diffusivity derived from the proposed kinetic model and the variability of the operational conditions applied in the drying tests, a single diffusion coefficient value for all the drying experiments could not be considered. Note that the diffusion coefficient of test 1 was lower than that of tests 2 and 3, in which temperature and velocity, respectively, were higher. Therefore, higher diffusion coefficients were obtained under higher operational conditions. Increased temperature and velocity were the factors that led to the greatest increases in the drying constant and decreased the equilibrium moisture content of the mushrooms. Acknowledgments The authors would like to thank CAPES (Brazil) for the scholarship granted to the first author, as well as CNPq (Brazil) (processes: 568221/2008-7, 474408/2008-6, and 302786/2008-2) for its financial support of this work. References Anderson, B.A., Singh, R.P., 2006. Effective heat transfer coefficient measurement during air impingement thawing using an inverse method. International Journal of Refrigeration 29 (2), 281–293. Beck, J.V., Arnold, K.J., 1977. Parameter Estimation in Engineering and Science. John Wiley & Sons, New York. Bialobrzewski, I., 2006. Determination of the heat transfer coefficient by inverse problem formulation during celery root drying. Journal of Food Engineering 74 (3), 383–391. Cao, W., Nishiyama, Y., Koide, S., 2003. Thin-layer drying of Maitake mushroom analysed with a simplified model. Biosystems Engineering 85 (3), 331–337. Crank, J., 1975. The Mathematics of Diffusion, second ed. Clarendon Press, Oxford. El-Aouar, A.A., Azoubel, P.M., Murr, F.E.X., 2003. Drying Kinetics of Fresh and Osmotically Pré-Treated Papaya (Carica papaya L.). Journal of Food Engineering 59 (1), 85–91. Fguiri, A., Daouas, N., Borjini, N., 2007. Experimental inverse analysis for the determination of boundary conditions in the parallel hot wire technique. Experimental Thermal and Fluid Science 31 (3), 209–220. Giri, S.K., Prasad, S., 2007. Drying kinetics and rehydration characteristics of microwave-vacuum and convective hot-air dried mushrooms. Journal of Food Engineering 78 (2), 512–521. Jambrak, A.R., Mason, T.J., Paniwnyk, L., Lelas, V., 2007. Accelerated drying of button mushrooms, Brussels sprouts and cauliflower by applying power ultrasound and its rehydration properties. Journal of Food Engineering 81 (1), 88–97. Jarny, Y., Maillet, D., 1999. Problèmes inverses et estimation de grandeurs en thermique, Métrologie thermique et techniques inverses, Ecole d’hiver Metti’99, Odeillo – Font – Romeu. Presses Universitaires de perpignan 1, 50. Kanevce, G.H., Kanevce, L.P., Dulikravich, G.S., Colaço, M.J., 2003. An inverse method for drying at high mass transfer Biot number. In: Proceedings of the ASME Summer Heat Transfer Conference, Las Vegas, USA. Kanevce, G.H., Kanevce, L.P., Mitrevski, V.B., Dulikravich, G.S., Orlande, H.R.B., 2005. Inverse approaches to drying of bodies with significant shrinkage effects. In: Proceedings of the 5th International Conference on Inverse Problems in Engineering: Theory and Practice, Cambridge, UK. Krokida, M.K., Karathanos, V.T., Maroulis, Z.B., Marinos-Kouris, D., 2003. Drying kinetics of some vegetables. Journal of Food Engineering 59 (4), 391–403.

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