- Email: [email protected]
Determination of diffusion and convective transfer coefficients in food drying revisited: A new methodological approach
b i o s y s t e m s e n g i n e e r i n g 1 6 2 ( 2 0 1 7 ) 3 0 e3 9
Available online at www.sciencedirect.com
ScienceDirect journal homepage: www.elsevier.com/locate/issn/15375110
Research Paper
Determination of diffusion and convective transfer coefficients in food drying revisited: A new methodological approach Francisco J. Arranz a, Tatiana Jimenez-Ariza b, Belen Diezma c, Eva C. Correa d,* Grupo de Sistemas Complejos, Departamento de Ingenierı´a Agroforestal, ETSI Agronomica Alimentaria y de Biosistemas, Universidad Politecnica de Madrid, Av. Puerta de Hierro 2, 28040 Madrid, Spain b sicas e Ingenierı´a, Universidad San Buenaventura, Carrera 8H # 172 e 20, Bogota , Facultad de Ciencias Ba Colombia c Laboratorio de Propiedades Fı´sicas y Tecnicas Avanzadas en Agroalimentacion (LPT_TAGRALIA), Departamento de Ingenierı´a Agroforestal, ETSI Agronomica Alimentaria y de Biosistemas, Universidad Politecnica de Madrid, Av. Puerta de Hierro 2, 28040 Madrid, Spain d Laboratorio de Propiedades Fı´sicas y Tecnicas Avanzadas en Agroalimentacion (LPF_TAGRALIA), Departamento de Quı´mica y Tecnologı´a de Alimentos, ETSI Agronomica Alimentaria y de Biosistemas, Universidad Politecnica de Madrid, Av. Puerta de Hierro 2, 28040 Madrid, Spain a
article info
It is usual to determine diffusion and convective transfer coefficients using classical
Article history:
approximate relationships involving dimensionless numbers. These approximate relation-
Received 3 January 2017
ships were developed in the past, in order to avoid the difficulties associated with calculations
Received in revised form
of large series expansions and its corresponding expansion coefficients given by transcen-
9 July 2017
dental equations. However, the development of improved computing techniques has
Accepted 24 July 2017
removed these difficulties, such that currently such calculations can be easily performed. An iterative methodology is proposed that takes advantage of current computational capabilities, avoiding to use approximate relationships. The proposed methodology is applied to
Keywords:
generated data and also to experimental data from carrot drying. Additionally, a MATLAB®
Mass Biot number
implementation of the proposed iterative methodology, along with the input data files cor-
Mass Fourier number
responding to the results presented in this paper, is provided as supplementary material.
Iterative numerical methodology
© 2017 IAgrE. Published by Elsevier Ltd. All rights reserved.
Carrot
1.
Introduction
In the development of suitable models for complex processes involving foods, which are applicable to a broad set of external
* Corresponding author. E-mail address: [email protected] (E.C. Correa). http://dx.doi.org/10.1016/j.biosystemseng.2017.07.005 1537-5110/© 2017 IAgrE. Published by Elsevier Ltd. All rights reserved.
conditions, the correct and accurate determination of the physical parameters involved is a central issue. Especially important in food engineering are heat and mass transfer processes, which are characterised by the corresponding diffusion and convective transfer coefficients.
b i o s y s t e m s e n g i n e e r i n g 1 6 2 ( 2 0 1 7 ) 3 0 e3 9
Nomenclature Bi D Di F Fo M M M0 Meq M[ SD Sh T a b h hi [ [y
Biot number Effective diffusion coefficient (m2 s1) Effective diffusion coefficient calculated at the i-th iteration (m2 s1) Mass flow (g [water] (m2 s1) Dimensionless time (Fourier number) Dry basis moisture (g [water] g1 [dry matter]) Position averaged moisture (g [water] g1 [dry matter]) Initial uniform moisture (g [water] g1 [dry matter]) Moisture at equilibrium (g [water] g1 [dry matter]) Moisture at food sample-air interface (g [water] g1 [dry matter]) Generalised significant figures in diffusion coefficient Generalised significant figures in mass transfer coefficient Sample temperature ( C) Intercept in the fitting to the straight line y ¼ a þ bt Slope in the fitting to the straight line y ¼ a þ bt (s1) Convective mass transfer coefficient (m2 s1) Convective mass transfer coefficient calculated at the i-th iteration (m2 s1) Half thickness of the sample (mm) Half width of the sample (mm)
It is noteworthy that for industrial purposes, as is pointed out by Zhao et al. (2014), foods are usually dried to remove the moisture up to a certain level at which microbial spoilage and deteriorative chemical reactions are minimised, and accurate determination of diffusion and convective transfer coefficients is required in order to facilitate accurate designs. Different methods have been described in the literature to obtain experimental values for these coefficients. Zhao et al. (2014) used a trial-and-error method based on a comparison of analytic and experimental drying curves, tuning the value of the diffusion coefficient until the matching of both curves is reached approximately. In similar fashion, Białobrzewski (2007) obtained both diffusion and convective transfer coefficients by comparing numerical (as opposed to analytical) and experimental drying curves. Note that analytical solutions are available only for some simple systems, whilst numerical solutions are available for any system. Conversely, analytical solutions have arbitrarily high accuracy, but the accuracy of numerical solutions is limited by numerical approximation, especially in the case of partial differential equations. Also, in order to obtain the convective transfer coefficient, the reaction engineering approach has been applied by considering drying as a competitive process between condensation and evaporation (Compaore et al., 2017). However, the most widely used method is the fitting of the experimental drying curves, in logarithmic form, to a straight line. For simple one-dimensional geometries, such as for slices or cylinders, by assuming the first term approximation in
[z t ti v w0 wdm weq wi x yi
Di F FN mn r s2D s2h s2i s2w
31
Half height of the sample (mm) Time (s) The i-th time measured in the drying process (s) Air velocity induced by the extractor fan (m s1) Initial sample weight in the drying process (g) Sample dry matter weight (g [dry matter]) Sample weight at equilibrium in the drying process (g) Sample weight at each time ti in the drying process (g) Position across thin layer sample thickness (mm) The i-th ordinate value corresponding to the average dimensionless relative moisture at each time ti in the drying process Convergence of first term approximation for the i-th iteration Average dimensionless relative moisture The N terms series expansion approximation to the average dimensionless relative moisture The n-th positive root of the characteristic equation mntanmn ¼ Bi Sample dry matter density (g [dry matter] m3 ) Variance of the diffusion coefficient (m2 s1) Variance of the mass transfer coefficient (m s1) Variance corresponding to the i-th ordinate value yi in the drying process Variance corresponding to the instrumental error of the weighing device (g)
the series expansion solution, diffusion coefficient can be obtained from the corresponding slope (Karim & Hawlader, 2005; Mghazlia et al., 2017; Sampaio et al., 2017; Srikiatden & Roberts, 2006). Additionally, by using approximate relationships involving dimensionless numbers, in particular the Biot number, convective transfer coefficient is also obtained , Cruz, & (Dhalsamant, Tripathy, & Shrivastava, 2017; Guine Mendes, 2014; Tripathy & Kumar, 2009). Note that, in order to obtain the correct values of the coefficients, it is important that the experimental conditions correspond to the mathematical conditions under which the analytical solution was obtained. However, it is not common to find in the literature details explaining the drying experimental methodology which could fulfil such conditions. Moreover, it is important to determine the accuracy of the resulting coefficients and the corresponding significant figures. Note that the instrumental error of the experimental data will propagate in the calculations when the coefficients are obtained, such that the significant figures in the initial experimental data will decrease (or at best remain the same) in the calculated coefficients. In this regard, as is well known, the calculability of diffusion and convective transfer coefficients depends on the Biot number, therefore this issue should be analysed in any method for the determination of diffusion and convective transfer coefficients. In drying experiments, since the mass diffusion coefficient (and also convective mass transfer coefficient) depends on temperature, in order to calculate the mass diffusion
32
b i o s y s t e m s e n g i n e e r i n g 1 6 2 ( 2 0 1 7 ) 3 0 e3 9
coefficient (and convective mass transfer coefficient) by fitting to an analytical solution of mass diffusion equation where the diffusion coefficient (and convective mass transfer coefficient) is assumed to be constant, it is mandatory to maintain the sample with uniform temperature in its whole volume. In this regard, as was shown by Srikiatden and Roberts (2006) when studying potatoes and carrot drying, when the initial temperature of the sample and target temperature differ, the maintenance of surrounding air temperature at target temperature is not sufficient to achieve uniform temperature, with temperature gradients occurring inside the sample. It is necessary firstly to heat the sample, using a microwave device which leads to uniform temperature, up to target temperature, and then maintain the surrounding air temperature at the target temperature throughout the drying assay. The likely reason for wrong determination of the mass diffusion coefficient is due to existence of temperature gradients inside the sample, as was pointed out in the early literature (Chen & Johnson, 1969; Chirife, 1983; King, 1968; Vaccarezza, Lombardi, & Chirife, 1974). Additionally, aside from temperature, other conditions must be controlled in order to correctly obtain mass diffusion and convective mass transfer coefficients. Thereby, sample volume is usually assumed constant in obtaining analytical solutions of diffusion equation, so that shrinkage should be controlled in food samples (e.g., by taking drying data over the initial period, when shrinkage is negligible). Besides, the asymptotic moisture value (i.e., sample moisture at infinite time) is also usually assumed constant, so that the relative humidity of the surrounding air should be maintained invariable. On the other hand, in the fitting of the corresponding analytical solution to obtain mass diffusion and convective mass transfer coefficients, some approximations are usually made. Firstly, in the infinite series expansion of the analytical solution, the first term approximation is taken, that is, solution is approximated by taking only the first term in series expansion. As is well known, this is a very accurate approximation only for large enough time (Crank, 1975; Luikov, 1968). Time threshold is well defined in terms of dimensionless time, but dimensionless time depends on the diffusion coefficient, so that some iterative process should be used. Secondly, the logarithmic form of first term approximation leads to a straight line, so that the fitting process is straightforward. Nevertheless, direct calculation of diffusion and convective transfer coefficients from fitting intercept and slope involves the solving of a transcendental equation, so that different approximate relationships involving dimensionless numbers were developed in order to avoid this difficulty (Luikov, 1968; Pflug & Blaisdell, 1963). Despite that these difficulties being easily avoidable, due to the development of computers and numerical methods, the approximate classical schemes are et al., 2014; Karim & Hawlader, 2005; still used (Guine Srikiatden & Roberts, 2006; Tripathy & Kumar, 2009). In order to determine diffusion and convective transfer coefficients from drying data, an iterative methodology is presented that takes advantage of current numerical and computational capabilities, avoiding to use the approximate classical schemes, and numerically solving the corresponding transcendental equations. As an additional feature, from the range of obtained parameters involved in this numerical
solving, it can be identified the mismatching of experimental conditions and mathematical conditions, and so, the unsuitability of used data. As will be explained in detail below, experimental data are first transformed to an asymptotic linear form and then fitted to a straight line. The mathematical conditions under which the analytical solution is obtained lead to a very narrow range for the intercept of this asymptotic straight line. So that, a fitting intercept value out of range implies that experimental data have been obtained under experimental conditions that differs from mathematical conditions, hence is unsuitable for the determination of diffusion and convective transfer coefficients. The proposed methodology is applied to generated data, in order to test its behaviour, and also to experimental data from carrot drying at different temperatures, including an unsuitable data set corresponding to non-isothermal drying, included in order to illustrate the detection of the unsuitability of data set. Additionally, a MATLAB® implementation of the proposed iterative methodology, along with the input data files corresponding to the results presented in the paper, is provided as supplementary material.
2.
Materials and methods
2.1. Determination of diffusion and convective transfer coefficients 2.1.1.
Mathematical model
It is assumed that, for a thin layer sample, food drying process is well described by means of the one-dimensional diffusion equation vM v vM ¼ D ; vt vx vx
(1)
with uniform initial condition Mðx; t ¼ 0Þ ¼ M0 ;
(2)
and boundary conditions
vM ¼ 0; vx x¼0
(3)
and
vM h ¼ M[ Meq ; vx x¼[ D
(4)
where M is the dry basis moisture, t is the time, x is the position across thin layer sample thickness (with origin x ¼ 0 at internal centre), [ is the sample half thickness, and D and h are the physical drying parameters, namely, effective diffusion and convective mass transfer coefficients, respectively. Besides, M[ ¼ M(x ¼ [, t) is the moisture at food sample-air interface, and Meq ¼ Mðx ¼ [; t/∞Þ is the corresponding moisture at equilibrium. Note that symmetric external conditions, i.e. the same air conditions at both sides of the thin layer, have been assumed. Since initial condition in Eq. (2) is also symmetric, the whole system is symmetric, so that the mass flow F ¼ rD(vM/vx) given by Fick's law, where r is the dry matter density, will be null at the internal centre, leading to the boundary condition in Eq. (3).
b i o s y s t e m s e n g i n e e r i n g 1 6 2 ( 2 0 1 7 ) 3 0 e3 9
Moreover, if food sample temperature is maintained constant during drying process, then the diffusion coefficient D can be assumed to be constant. Additionally, if the air velocity is constant, the convective transfer coefficient h can also be assumed constant. Finally, if the relative humidity of air is maintained constant, then the moisture at equilibrium Meq will be constant. Under these additional conditions, the solution of diffusion equation in Eq. (1) under initial condition in Eq. (2), and boundary conditions in Eqs. (3) and (4), is given by the following series expansion (Crank, 1975; Luikov, 1968) ∞ x Mðx; tÞ Meq X 2sinmn Dt ¼ cos mn exp m2n 2 ; [ M0 Meq mn þ sinmn cosmn [ n¼1 (5) being mn the positive roots of the equation mn tanmn ¼
h[ ; D
(6)
ordered such that 0 < mn < mnþ1. The experimentally measurable overall moisture content M of the sample is obtained by averaging over the whole sample volume MðtÞ ¼
1 [
Z[ Mðx; tÞdx;
(7)
0
so that, the overall content moisture corresponding to series expansion in Eq. (5) will be 2 ∞ MðtÞ Meq X 2sin mn Dt exp m2n 2 : ¼ M0 Meq mn ðmn þ sinmn cosmn Þ [ n¼1
FðFoÞ ¼
n¼1
2Bi 2 exp m2n Fo ; m2n Bi þ Bi þ m2n
(10)
If diffusion process is much faster than convective transfer process (external problem), then Bi/0 and the positive roots of Eq. (10) are given by mn ¼ (n 1)p. Conversely, if convective transfer process is much faster than diffusion process (internal problem), then Bi/∞ and the corresponding roots are given by mn ¼ ð2n 1Þp=2. So that, when both processes are simultaneously relevant (0 < Bi < ∞), the positive roots of Eq. (10) are bounded by the two extreme cases, that is,
2.1.2.
(12)
In other words, for high enough values of the Fourier number, moisture time evolution FðFo; BiÞ is well described by the first term approximation F1 ðFo; BiÞ, that expressed in logarithmic form 2 2sin m1 m2 D 12 t; (13) logFzlog m1 ðm1 þ sinm1 cosm1 Þ [ takes the form of a straight line y ¼ a þ bt with intercept a ¼ log
2 2sin m1 ; m1 ðm1 þ sinm1 cosm1 Þ
(14)
and slope m2 D b ¼ 12 : [
(9)
where F ¼ ½MðtÞ Meq =ðM0 Meq Þ is the average dimensionless relative moisture, Fo ¼ Dt/[2 is the dimensionless time known as Fourier number, and Bi ¼ h[/D is the Biot number, which determines the relative importance between diffusion and convective transfer processes. Notice that, in dimensionless form, the coefficients mn are defined as the positive roots of the equation
ðn 1Þp < mn < ð2n 1Þp=2:
lim FðFo; BiÞ ¼ F1 ðFo; BiÞ:
Fo/∞
(15)
So that, by taking a set of drying data {w0,w1,…,weq}, the corresponding ordinate values can be calculated1
2
mn tanmn ¼ Bi:
FðFo; BiÞ in Eq. (9) must be calculated with sufficient accuracy, i.e, by taking a large enough number of terms N in the series expansion leading to a high accuracy approximation FN ðFo; BiÞ. For this purpose, the corresponding first N roots of the transcendental equation in Eq. (10) need to be calculated with high accuracy, too. However, since the roots are clearly bounded by the inequalities in Eq. (11), there exist different numerical algorithms that are able to calculate them very efficiently. Specifically, the algorithm of Forsythe, Malcolm, and Moler (1977) have been used, which is implemented in the MATLAB® function fzero. Moreover, it is well known that the series expansion in Eq. (9) converges rapidly and monotonically for high enough values (depending on Biot number value) of the Fourier number (Crank, 1975; Luikov, 1968), so that, as time increases, moisture curve tends asymptotically to the first term in the series, that is
(8)
As is well known, the solutions of diffusion equation can be expressed in terms of dimensionless magnitudes (Crank, 1975; Luikov, 1968). In this way, the series expansion in Eq. (8) can be expressed as follows ∞ X
33
(11)
Iterative numerical methodology
In order to obtain the diffusion and convective transfer coefficients from drying data, the dimensionless moisture
yi ¼ log
wi weq ; w0 weq
(16)
where w0 is the initial sample weight at t0 ¼ 0, wi is the sample weight at each time ti(i ¼ 1,2,…), and weq is the sample weight at equilibrium. Then, intercept a and slope b can be obtained by means of the least squares fitting to a straight line of the data pairs {ti,yi}(i ¼ 1,2,…). It should be noted that, by using the logarithm function in the ordinate values definition yi, the corresponding variances s2i will increase with time, so that, a weighted least squares fitting should be used in the obtaining of slope and intercept values. Indeed, assuming the same variance s2w for all weight data wi, which is determined by the weighing device used (instrumental error), the variances for the ordinate values yi(i ¼ 1,2,…) are given by2 1
Note that, in order to calculate the average dimensionless relative moisture F from drying data, only weight data w are needed, and no average moisture values M should be calculated, M M w w since M0i Meq ¼ w0i weq : eq eq 2 Given a function f(x1,…,xN) of variables xk with variances s2k ðk ¼ 1; …; NÞ, error propagation theory provides the following 2 P vf s2k : formula for the variance s2 of the function f.s2 ¼ N k¼1 vxk
34
s2i
b i o s y s t e m s e n g i n e e r i n g 1 6 2 ( 2 0 1 7 ) 3 0 e3 9
¼
2s2w
1 wi weq
2 þ
wi weq
1
! 1 þ 2 : w0 weq w0 weq (17)
The weighted least squares fitting was carried out by using the algorithm of Strang (1986), which is implemented in the MATLAB® function lscov. After fitting process, the first root value m1 is calculated from Eq. (14). Notice that, as in the calculation of the roots of Eq. (10) discussed above, it is a transcendental equation, so that the algorithm of Forsythe et al. (1977) has also been used. Then, diffusion coefficient D is calculated from Eq. (15), convective transfer coefficient h is calculated from Eq. (6), and Biot number Bi from Eq. (10). However, the coefficients thus obtained could be wrong, since the first term approximation F1 ðFo; BiÞ is valid for high enough values (depending on the Biot number value) of the Fourier number, and both Fourier and Biot numbers depend on diffusion and convective transfer coefficients. So that, in order to ensure the correctness of the coefficients calculated, an iterative process is mandatory. Thus, iteratively, the (i1)-th data pair will be deleted and the corresponding coefficients Di and hi recalculated by fitting of remaining data pairs, until the convergence of first term approximation within data accuracy, namely FN ðFoi ; Bii Þ F1 ðFoi ; Bii Þ si ;
(18)
where the values of Fourier and Biot numbers are recalculated at each iteration from fitting coefficients Di and hi. When the condition in Eq. (18) is fulfilled, the corresponding coefficients Di and hi can be considered the correct coefficients. Observe that, in order to verify the convergence of first term approximation to exact values, we have used a high accuracy approximation FN ðFo; BiÞ that can be considered as exact within data accuracy. Finally, notice that by applying the inequalities in Eq. (11) to the first root m1, the intercept a and the slope b of the fitted straight line must necessarily fulfil the following conditions log
8 < a < 0; p2
(19)
and b < 0:
(20)
So that, the non-fulfilment of conditions above, when fitting a given data set, implies that the experimental setup breaks the model assumptions, and then the fitting coefficients obtained will be wrong. In this regard, condition in Eq. (19) is particularly relevant since it establishes a very narrow range, a2(0.21, 0), for checking the suitability of experimental data set. A MATLAB® implementation of the proposed iterative methodology, along with the input data files corresponding to the results presented in this paper, is available in the supplementary material.
2.2.
Experimental setup
The carrot samples were obtained from entire carrots (Daucus carota L. cv. Nantes) purchased from local market, which were
manually water washed, peeled, and lengthwise sliced with a kitchen slicer (Cooking, CASA International nv, Itegem, Antwerpen, Belgium) in order to obtain constant thickness. The samples were sliced from central part of carrots, and eventually cut into rectangular pieces. The dimensions of each carrot sample were measured by means of a digital caliper gauge (DigitCal® Classic 05.30035, Swiss Instruments Ltd., Mississauga, Ontario, Canada). Before drying process, each carrot sample was heated in a sticos Taurus S.L., microwave oven (PTMW 700, Electrodome rida, Spain), with the aim of achieving a uniform Oliana, Le temperature in the whole sample. With the purpose of determine diffusion and convective transfer coefficients at different temperatures, distinct heating periods were considered to reach temperatures around 40, 50, 60, and 70 C. In order to avoid the start of drying process throughout the heating period, a small bowl with water was introduced into microwave oven along with each sample. After heating, sample temperature was measured in each case by using a type T thermocouple thermometer (HH2001TC, Omega Engineering Inc., Stamford, CT, USA), giving temperatures of 40, 55, 64, and 72 C. Before each sample was introduced into the weighing device, it was sandwiched between two metal grids in order to keep the slice shape. The setpoint temperature, corresponding to each measured temperature at which the parameters will be estimated, was established in each case in the weighing device (AMB 300 moisture balance, Inscale Measurement Technology Ltd., Bexhill, East Sussex, UK), and the sample weight recording every 20 s was started. As shown in Fig. 1, the programmable weighing device comprises a chamber, with the weighing pan inside, equipped with two halogen heaters, and also a RS-232 interface for connection to a computer, so that weighing data are automatically recorded in a file. The instrumental error of the weighing device, which is required by the proposed iterative methodology in Eq. (17), is sw ¼ 0.001 g. Additionally, at the top of the chamber, there is a ventilation grill. Observe that, in order to achieve the required symmetric external conditions (i.e., the same air conditions at both sample sides), the sample was raised on four feet above the sample pan. Air temperature and also air relative humidity were monitored by means of an electronic logger (iButton® Hygro, CA, chron™ DS1923, Maxim Integrated Products Inc., San Jose USA) located into the weighing device chamber. In order to prevent the increase of relative humidity in the sample surroundings, an extractor fan (EVERFLOW F128025BU, SilverStone Technology Co. Ltd., New Taipei, Zhonghe, Taiwan) was mounted on the ventilation grill of the weighing device chamber [see Fig. 1(a)]. Extracted air velocity was measured by using a multifunctional precision manometer with Prandtl's € hler Messgera € te Kehrgera € te GmbH, Bad tube (DC 100pro, Wo Wu¨nnenberg, Paderborn, Germany). Lastly, each carrot sample was entered into a drying oven (Conterm Poupinel, JP SELECTA S.A., Abrera, Barcelona, Spain) at 105 C until steady weight was achieved, and thereafter sample dry weight was determined with a precision balance (ADP 720/L, Adam Equipment Co. Ltd., Kingston, Milton Keynes, UK).
b i o s y s t e m s e n g i n e e r i n g 1 6 2 ( 2 0 1 7 ) 3 0 e3 9
35
Fig. 1 e (a) Overview of the weighing device with the mounted extractor fan. (b) Section view of the weighing device with the sample raised on four feet above the sample pan. (c) Detail of the sample sandwiched between two metal grids with the four feet.
3.
Results and discussion
3.1.
Determination of coefficients from generated data
In order to test the behaviour of the proposed iterative methodology, the series expansion in Eq. (9) with original coefficients D, h, and other conditions described in Table 1, was used to generate 18 sample weight data {w1, w2,…,w18} along drying process. By taking N ¼ 14 terms in series expansion, an accuracy of 0.001 g in the worst case Fo ¼ 0 was achieved. Gaussian noise was introduced into generated data, so that the final data set with weighing accuracy sw ¼ 0.01 g was obtained. Pseudorandom values for Gaussian noise were calculated by using the algorithms of Marsaglia (Marsaglia & Tsang, 2000; Marsaglia & Zaman, 1991) which are implemented in the MATLAB® function randn. The successive iterations corresponding to applying the proposed iterative methodology to generated data have been depicted in panels (a)e(g) of Fig. 2. At each iteration, the first data point is deleted, and the remaining data fitted to a straight line leading to the corresponding values of diffusion Di and convective transfer hi coefficients. Note that, in the first iterations, the original fitting data points and the corresponding exact points obtained from Eq. (9) with the fitting coefficients Di and hi differ significantly. As iterative process progresses both data sets get closer, and finally, when convergence is reached at i ¼ 7, they match within weighing
Table 1 e Original and calculated diffusion and convective transfer coefficients in the determination of coefficients from generated data. Note that calculated coefficients correspond to those in Fig. 2(g) rounded to two significant figures. Coefficients Original Calculated
D (m2 s1)
h (m s1)
Bi
4.76$1011 4.8$1011
2.74$107 2.7$107
11.5 11
Data generation conditions: [ ¼ 2 mm, w0 ¼ 6.47 g, weq ¼ 0.77 g, sw ¼ 0.01 g.ti ¼ {0.5, 1, 1.5,…,9 h}.
accuracy sw. Notice that convergence condition in Eq. (18) is graphically represented in panel (h) of Fig. 2. It is worth noting that we have verified in a heuristic manner that, as is shown in Fig. 2 and summarised in Table 1 for the present case, diffusion and convective transfer coefficients are obtained roughly with one significant figure less than the weighing values. Thus, the generated data set starts with w1 ¼ 5.86 g and ends with w18 ¼ 3.01 g, corresponding to three significant figures, and diffusion and convective transfer coefficients are obtained with two significant figures. The number of significant figures obtained depends on the Biot number, as well as the features of the experimental setup, mainly the accuracy of the weighing data. In order to establish the number of significant figures obtained in the calculation of diffusion and convective transfer coefficients under typical conditions in drying experiments, we have carried out several series of simulations by generating weighing data and fitting generated data to obtain the coefficients, in a similar way to the example above. Sample half thickness [ ¼ 2 mm, weighing device accuracy sw ¼ 0.001 g with 4 significant figures, and range 109 1012 m2 s1 for the order of magnitude of the diffusion coefficient are assumed as typical conditions in drying experiments. Additionally, for each Biot number Bi ¼ {0.001,0.01,0.1,1,10,100,1000} a set of 10 diffusion coefficients are randomly obtained, and for each random diffusion coefficient 10 sets of weighing data are generated at Fourier numbers Fo ¼ {0,0.1,0.2,…,1}. By comparing original diffusion and convective transfer coefficients with those calculated from generated data sets, the value of the generalised number of significant figures SX is obtained. The generalised number of significant figures in X is defined by means of the logarithmic difference SX ¼ 1 þ log10 ðXÞ log10 ðsX Þ;
(21)
where X ¼ D, h is the value of the corresponding coefficient at a given Biot number, and sX is the standard deviation of the coefficient from the different simulations (10 simulations in our case) at that Biot number. Notice that, for the same Biot number but different random diffusion coefficient, somewhat different values of SX will be obtained, which will be
36
b i o s y s t e m s e n g i n e e r i n g 1 6 2 ( 2 0 1 7 ) 3 0 e3 9
Fig. 2 e Iterative process for obtaining diffusion and convective transfer coefficients applied to generated data described in Table 1. At each iteration (i ¼ 1,2,…,7), the logarithm of average dimensionless relative moisture F versus Fourier number Fo has been depicted for fitting data (open circles), fitted straight line corresponding to first term approximation F1 ðFoi ; Bii Þ (thick line), exact drying curve corresponding to high accuracy approximation FN ðFoi ; Bii Þ (thin line), and the points over exact curve corresponding to Fourier numbers of fitting data (filled circles). (a) Initial fitting with all data. (b)e(g) Consecutive fitting iterations, where the first datum is deleted at each iteration until reaching convergence. Fitting coefficients Di (10¡11 m2 s¡1) and hi (10¡7 m s¡1) and also c each iteration are indicated. (h) Decimal logarithm of the difference Di ¼ FN ðFoi ; Bii Þ F1 ðFoi ; Bii Þ at convergence. Values corresponding to deleted data points (open circles) and used fitting data (filled circles) are depicted. Data error at convergence s7 is represented in dotted line.
statistically characterised by the corresponding mean and standard deviation. The results of these simulations are shown in Fig. 3, where the mean values and the standard deviation values of the generalised significant figures corresponding to each generated data set are represented as filled circles and error bars, respectively. Observe that, as expected, optimal results for both coefficients jointly are obtained in the range 1 Bi 10. Moreover, as Biot number increases the accuracy of diffusion coefficient also increases, whilst the accuracy of convective transfer coefficient falls quickly. Even so, at Bi ¼ 100 the accuracy of transfer coefficient is not too low. On the other hand, as Biot number decreases the accuracy of diffusion coefficient falls quickly, and the accuracy of transfer coefficient also falls, although it decreases slower than the diffusion coefficient accuracy. This is not a surprising behaviour, since the transfer coefficient value is calculated through Eq. (6) by using the previously obtained diffusion coefficient. However, until Bi ¼ 0.01 the accuracy of calculated transfer coefficient is very good. The iterative process shown in Fig. 2 can be reproduced, and additional details obtained, by running the MATLAB® program with the input data file corresponding to generated data available in the supplementary material.
3.2. data
Determination of coefficients from experimental
The proposed iterative methodology has been applied to experimental data corresponding to carrot drying
Fig. 3 e Generalised significant figures, SD and Sh, obtained in the calculation of diffusion (thick line) and convective transfer (thin line) coefficients, respectively, as a function of the Biot number Bi for different simulations performed under typical conditions in drying experiments (see text for details).
b i o s y s t e m s e n g i n e e r i n g 1 6 2 ( 2 0 1 7 ) 3 0 e3 9
experiments. In order to establish the temperature dependence of diffusion and convective transfer coefficients, isothermal drying at different temperatures was performed. The final fitting at convergence is depicted for each temperature in Fig. 4 (top panels) along with the corresponding graphically represented convergence conditions (bottom panels). The obtained diffusion and convective transfer coefficients are summarised in Table 2 in addition to some sample parameters as Biot number Bi, thickness 2[, width 2[y, height 2[z, initial weight w0, equilibrium weight weq, and dry matter weight wdm. Note that, in this case, weighing data have four significant figures, so that three significant figures have been assumed for the obtained coefficients. By means of simulations as those depicted in Fig. 3, the accuracy of the obtained diffusion and convective transfer coefficients has been estimated, namely sD ¼ 0.03$109 (m2 s1) and sh ¼ 0.01$106 (m s1), respectively. Since the main goal of this section is to illustrate the application of the proposed methodology, for the sake of clarity only one data set at each temperature has been considered. However, in order to obtain a better estimation of the coefficients accuracy, accounting correctly for the experimental error, several data sets at each temperature should be regarded, and the corresponding error calculated in standard manner. As is expected, both diffusion coefficient and convective transfer coefficient mostly grow as temperature increases. Nevertheless, as is shown in Table 2, diffusion coefficient decreases very slightly from the value at T ¼ 55 C and the value at T ¼ 64 C. We assume that this effect is due to the
37
different drying behaviour of core and cortex parts in carrots and the slightly different core-cortex ratio in each sample. Notice that, in order to achieve the one-dimensional behaviour for the thin layer samples, it was not possible to separate core and cortex parts. In this regard, Srikiatden and Roberts (2006) obtained slightly higher values of the diffusion coefficient for core tissue than for cortex tissue in carrots (D. carota L. cv. Red Core Chantenay) cut into cylindrical samples. Specifically, they found at temperatures of 40, 50, 60, and 70 C, the values of 1.01, 1.32, 1.90, and 2.34 (109 m2 s1) for core tissue, and values of 0.76, 0.92, 1.06, and 1.36 (109 m2 s1) for cortex tissue, respectively. Taking into account that carrot samples of Srikiatden and Roberts correspond to a different cultivated variety than those used in the present work, it can be considered that their results and our results are in good agreement. Additionally, Markowski (1997) found the value h ¼ 1.37$107 (m s1) for convective transfer coefficient of carrots (cv. not specified) at T ¼ 60 C and natural convection. Considering that our results correspond to forced convection, albeit rather at a low air velocity, it seems reasonable that our results are somewhat higher. Moreover, it is important to highlight that, since an analytical solution of the diffusion equation under certain conditions is used, in order to obtain physical coefficients of carrots rather than unphysical best fit parameters, these conditions should be approximated in drying experiments, as much as possible. Thus, one-dimensional behaviour for the thin layer samples is achieved by taking thin layer samples with width and height one order of magnitude greater than
Fig. 4 e Iterative process for obtaining diffusion and convective transfer coefficients applied to experimental data from carrot slices drying summarised in Table 2. For different temperatures, the logarithm of average dimensionless relative moisture F versus the Fourier number Fo has been depicted for fitting data (open circles), fitted straight line corresponding to first term approximation F1 ðFoi ; Bii Þ (thick line), exact drying curve corresponding to high accuracy approximation FN ðFoi ; Bii Þ (thin line), and the points over exact curve corresponding to Fourier numbers of fitting data (filled circles). (a)e(d) Final fitting at convergence for temperatures T ¼ 40, 55, 64, 72 C, respectively. (e)e(h) Decimal logarithm of the difference Di ¼ FN ðFoi ; Bii Þ F1 ðFoi ; Bii Þ at convergence corresponding to temperatures T ¼ 40, 55, 64, 72 C, respectively. Values corresponding to deleted data points (open circles) and used fitting data (filled circles) are depicted. Data error at convergence for each temperature si (i ¼ 39,34,78,18) is represented in dotted line. Fitting coefficients Di (10¡9 m2 s¡1) and hi (10¡6 m s¡1) and also the Biot number calculated at convergence are indicated.
38
b i o s y s t e m s e n g i n e e r i n g 1 6 2 ( 2 0 1 7 ) 3 0 e3 9
Table 2 e Diffusion and convective transfer coefficients obtained for carrot slices and additional sample parameters. T ( C) 40 55 64 72
D (m2 s1) 10
7.27$10 1.55$109 1.53$109 2.95$109
h (m s1)
Bi
2[ (mm)
2[y (mm)
2[z (mm)
w0 (g)
weq (g)
wdm (g)
6.54$107 1.20$106 1.29$106 2.65$106
1.28 1.45 1.21 1.76
2.85 3.75 2.87 3.92
32.80 27.40 29.51 23.43
61.17 62.48 62.96 52.13
4.795 5.450 4.986 3.986
0.662 0.890 0.733 0.479
0.593 0.845 0.702 0.456
Obtained coefficients accuracy: sD ¼ 0.03$109 m2 s1, sh ¼ 0.01$106 m s1. Weighing device accuracy: sw ¼ 0.001 g. Air velocity: v ¼ 0.71 m s1.
thickness, as is shown in Table 2. Homogeneous initial moisture M0 can be assumed for fresh carrots preserved in moisture insulated atmosphere. Additionally, diffusion D and convective transfer h coefficients must be constant. Since both coefficients depend on temperature, carrot sample temperature should be uniform and constant. Sample temperature homogeneity is achieved by using a microwave device in the heating process, and it is kept constant by controlling surrounding air temperature, which was monitored by means of an electronic logger. Notice that convective transfer coefficient depends also on surrounding air velocity and carrot sample shape, both conditions being kept constant as is explained below. Besides, sample moisture at equilibrium Meq must also be constant, which is achieved by controlling relative humidity of surrounding air. In order to avoid the relative humidity increase due to drying process, an extractor fan at very low velocity was used (generating constant air velocity), and relative humidity was monitored through an electronic logger. Finally, shape and size (volume geometry in diffusion equation) must be constant along drying process. Carrot slices were sandwiched between two metal grids in order to keep the slice shape. Nevertheless, shrinkage is an unavoidable effect in carrot drying, which changes size and also shape, albeit most shrinkage occurs at the initial stages of drying process, since it is strongly correlated with the moisture decrease. However, the proposed methodology discards initial data, where most shrinkage occurs. Additionally, a data set within a narrow moisture range has been considered, so that shrinkage can be neglected. Notice that, as verified, weighing data corresponding to a nonnegligible shrinking period lead to a convex (instead of concave) drying curve, as in the case of non-isothermal drying described below, being detected by the proposed methodology as unsuitable data. The sample temperature homogeneity is a remarkable condition in order to obtain correct diffusion and convective transfer coefficients from drying data fitting. As it has been shown by Srikiatden and Roberts (2006) in drying experiments with carrots and also with potatoes, convective hot air drying with constant air temperature leads to non-isothermal drying, since a temperature gradient is induced inside samples, giving experimental drying curves that do not fit the corresponding analytical solution of diffusion equation with constant diffusion and transfer coefficients. In this regard, an example of checking of suitability of data set provided by the proposed iterative methodology in Eq. (19) is shown in Fig. 5. Drying data in Fig. 5 correspond to a carrot slice sample initially at room temperature (~20 C) that is drying at constant air temperature of 65 C. Observe that the drying curve given by experimental data is convex (negative second derivative), whilst the
Fig. 5 e Iterative process applied to unsuitable (nonisothermal) experimental data from carrot slices drying. The logarithm of average dimensionless relative moisture F versus time t has been depicted for fitting data (open circles), and for the corresponding fitted straight line logðFÞ ¼ a þ bt (thick line). The unsuitability of data set is detect at first step of process, since fitting intercept a is out of range, i.e., a;ð0:21; 0Þ.
analytical solution on which our methodology is based is a concave curve (positive second derivative). As a consequence of this feature, fitting intercept a is out of the required range (0.21, 0) in Eq. (19), being a positive value a > 0, and so detecting the unsuitability of data set. The iterative process corresponding to each experimental data set at different temperatures shown in Fig. 4, as well as that corresponding to the unsuitable data set, can be reproduced, and additional details obtained, by running the MATLAB® program with the corresponding input data file available in the supplementary material.
4.
Conclusions
Physical drying parameters, namely, mass diffusion and convective mass transfer coefficients, have a significant relevance on proper determination of optimal drying conditions in convective hot air food drying, in order to achieve the aims of
b i o s y s t e m s e n g i n e e r i n g 1 6 2 ( 2 0 1 7 ) 3 0 e3 9
food industry. In this paper, a new methodological approach has been presented for the determination of diffusion and convective transfer coefficients from drying data, which takes advantage of current numerical and computational capabilities versus the classical scheme, based on approximate relationships involving dimensionless numbers, that is usually applied. As in classical schemes, data fitting to asymptotic first term in series expansion solution is used, but in our method the Fourier number threshold is iteratively calculated, since the Fourier number depends on diffusion coefficient. Also, by using standard numerical methods, involved transcendental equations are accurately solved, so that series expansion solution can be obtained with arbitrarily high accuracy, and also diffusion and convective transfer coefficients can be directly obtained from fitting parameters. As an additional feature, from the range of obtained parameters involved in the numerical solving, it can be identified the unsuitability of used data, avoiding calculation of wrong coefficients. The proposed methodology was tested by applying to generated data, resulting in obtaining diffusion and convective transfer coefficients, for Biot numbers roughly within the range 1e10, with only one significant figure less than the weighing data values. Additionally, the new methodological approach has been applied to experimental data from carrot drying at different temperatures, showing an adequate behaviour and giving reasonable results. Also, our method was applied to nonisothermal drying data, showing the feature of automatic detection of unsuitable data. Finally, it is noteworthy that although our method has been specified for the infinite plate one-dimensional diffusion system, the extension of the proposed iterative methodology to other typical one-dimensional diffusion systems with analytical solution, as those corresponding to cylindrical or spherical geometries, is straightforward. Notice that, in these cases, the same diffusion coefficient should be obtained for the different geometries (assuming that the diffusion coefficient is a physical property of food tissue), but different convective transfer coefficients will be obtained for each geometry, since it is assumed that the convective transfer coefficient depends on solid shape, among others factors.
Acknowledgments The funding of this work has been covered by Comunidad de Madrid and European Union through S2013/ABI-2747 (TAVSCM) project.
Appendix A. Supplementary data Supplementary data related to this article can be found at http://dx.doi.org/10.1016/j.biosystemseng.2017.07.005.
references
Białobrzewski, I. (2007). Determination of the mass transfer coefficient during hot-air-drying of celery root. Journal of Food Engineering, 78(4), 13881396.
39
Chen, C. S., & Johnson, W. H. (1969). Kinetics of moisture movement in hygroscopic materials. I. Theoretical considerations of drying phenomena. Transactions of the ASAE, 12(1), 109e113. Chirife, J. M. (1983). Fundamentals of the drying mechanism during air dehydration of foods. In A. S. Mujumdar (Ed.), Advances in drying (Vol. 2). New York, US: Hemisphere Publishing Corp. Compaore, A., Dissa, A. O., Rogaume, Y., Putranto, A., Chen, X. D., Mangindaan, D., et al. (2017). Application of the reaction engineering approach (REA) for modeling of the convective drying of onion. Drying Technology, 35(4), 500e508. Crank, J. (1975). The mathematics of diffusion (2nd ed.). London, UK: Oxford University Press. Dhalsamant, K., Tripathy, P. P., & Shrivastava, S. L. (2017). Moisture transfer modeling during solar drying of potato cylinders considering shrinkage. International Journal of Green Energy, 14(2), 184e195. Forsythe, G. E., Malcolm, M. A., & Moler, C. B. (1977). Computer methods for mathematical computations. Englewood Cliffs, US: Prentice-Hall. , R. P. F., Cruz, A. C., & Mendes, M. (2014). Convective drying Guine of apples: Kinetic study, evaluation of mass transfer properties and data analysis using artificial neural networks. International Journal of Food Engineering, 10(2), 281e299. Karim, M. A., & Hawlader, M. N. A. (2005). Mathematical modelling and experimental investigation of tropical fruits drying. International Journal of Heat and Mass Transfer, 48(23e24), 4914e4925. King, C. J. (1968). Rates of moisture sorption and desorption in porous dried foodstuffs. Food Technology, 22(4), 165e171. Luikov, A. V. (1968). Analytical heat diffusion theory. New York, US: Academic Press. Markowski, M. (1997). Air drying of vegetables: Evaluation of mass transfer coefficient. Journal of Food Engineering, 34(1), 55e62. Marsaglia, G., & Tsang, W. W. (2000). The ziggurat method for generating random variables. Journal of Statistical Software, 5(8), 1e7. Marsaglia, G., & Zaman, A. (1991). A new class of random number generators. Annals of Applied Probability, 1(3), 462e480. Mghazlia, S., Ouhammoua, M., Hidara, N., Lahninea, L., Idlimamb, A., & Mahrouz, M. (2017). Drying characteristics and kinetics solar drying of Moroccan rosemary leaves. Renewable Energy, 108, 303e310. Pflug, I. J., & Blaisdell, J. L. (1963). Methods of analysis of precooling data. ASHRAE Journal, 5(11), 33e41. Sampaio, R. M., Neto, J. P. M., Perez, V. H., Marcos, S. K., Boizan, M. A., & Da Silva, L. R. (2017). Mathematical modeling of drying kinetics of persimmon fruits (Diospyros kaki cv. Fuyu). Journal of Food Processing and Preservation, 41(1), e12789. Srikiatden, J., & Roberts, J. S. (2006). Measuring moisture diffusivity of potato and carrot (core and cortex) during convective hot air and isothermal drying. Journal of Food Engineering, 74(1), 143e152. Strang, G. (1986). Introduction to applied mathematics. Wellesley, US: Wellesley-Cambridge Press. Tripathy, P. P., & Kumar, S. (2009). A methodology for determination of temperature dependent mass transfer coefficients from drying kinetics: Application to solar drying. Journal of Food Engineering, 90(2), 212e218. Vaccarezza, L. M., Lombardi, J. L., & Chirife, J. (1974). Kinetics of moisture movement during air drying of sugar beet root. International Journal of Food Science & Technology, 9(3), 317e327. Zhao, D., An, K., Ding, S., Liu, L., Xu, Z., & Wang, Z. (2014). Twostage intermittent microwave coupled with hot-air drying of carrot slices: Drying kinetics and physical quality. Food and Bioprocess Technology, 7(8), 2308e2318.
Available online at www.sciencedirect.com
ScienceDirect journal homepage: www.elsevier.com/locate/issn/15375110
Research Paper
Determination of diffusion and convective transfer coefficients in food drying revisited: A new methodological approach Francisco J. Arranz a, Tatiana Jimenez-Ariza b, Belen Diezma c, Eva C. Correa d,* Grupo de Sistemas Complejos, Departamento de Ingenierı´a Agroforestal, ETSI Agronomica Alimentaria y de Biosistemas, Universidad Politecnica de Madrid, Av. Puerta de Hierro 2, 28040 Madrid, Spain b sicas e Ingenierı´a, Universidad San Buenaventura, Carrera 8H # 172 e 20, Bogota , Facultad de Ciencias Ba Colombia c Laboratorio de Propiedades Fı´sicas y Tecnicas Avanzadas en Agroalimentacion (LPT_TAGRALIA), Departamento de Ingenierı´a Agroforestal, ETSI Agronomica Alimentaria y de Biosistemas, Universidad Politecnica de Madrid, Av. Puerta de Hierro 2, 28040 Madrid, Spain d Laboratorio de Propiedades Fı´sicas y Tecnicas Avanzadas en Agroalimentacion (LPF_TAGRALIA), Departamento de Quı´mica y Tecnologı´a de Alimentos, ETSI Agronomica Alimentaria y de Biosistemas, Universidad Politecnica de Madrid, Av. Puerta de Hierro 2, 28040 Madrid, Spain a
article info
It is usual to determine diffusion and convective transfer coefficients using classical
Article history:
approximate relationships involving dimensionless numbers. These approximate relation-
Received 3 January 2017
ships were developed in the past, in order to avoid the difficulties associated with calculations
Received in revised form
of large series expansions and its corresponding expansion coefficients given by transcen-
9 July 2017
dental equations. However, the development of improved computing techniques has
Accepted 24 July 2017
removed these difficulties, such that currently such calculations can be easily performed. An iterative methodology is proposed that takes advantage of current computational capabilities, avoiding to use approximate relationships. The proposed methodology is applied to
Keywords:
generated data and also to experimental data from carrot drying. Additionally, a MATLAB®
Mass Biot number
implementation of the proposed iterative methodology, along with the input data files cor-
Mass Fourier number
responding to the results presented in this paper, is provided as supplementary material.
Iterative numerical methodology
© 2017 IAgrE. Published by Elsevier Ltd. All rights reserved.
Carrot
1.
Introduction
In the development of suitable models for complex processes involving foods, which are applicable to a broad set of external
* Corresponding author. E-mail address: [email protected] (E.C. Correa). http://dx.doi.org/10.1016/j.biosystemseng.2017.07.005 1537-5110/© 2017 IAgrE. Published by Elsevier Ltd. All rights reserved.
conditions, the correct and accurate determination of the physical parameters involved is a central issue. Especially important in food engineering are heat and mass transfer processes, which are characterised by the corresponding diffusion and convective transfer coefficients.
b i o s y s t e m s e n g i n e e r i n g 1 6 2 ( 2 0 1 7 ) 3 0 e3 9
Nomenclature Bi D Di F Fo M M M0 Meq M[ SD Sh T a b h hi [ [y
Biot number Effective diffusion coefficient (m2 s1) Effective diffusion coefficient calculated at the i-th iteration (m2 s1) Mass flow (g [water] (m2 s1) Dimensionless time (Fourier number) Dry basis moisture (g [water] g1 [dry matter]) Position averaged moisture (g [water] g1 [dry matter]) Initial uniform moisture (g [water] g1 [dry matter]) Moisture at equilibrium (g [water] g1 [dry matter]) Moisture at food sample-air interface (g [water] g1 [dry matter]) Generalised significant figures in diffusion coefficient Generalised significant figures in mass transfer coefficient Sample temperature ( C) Intercept in the fitting to the straight line y ¼ a þ bt Slope in the fitting to the straight line y ¼ a þ bt (s1) Convective mass transfer coefficient (m2 s1) Convective mass transfer coefficient calculated at the i-th iteration (m2 s1) Half thickness of the sample (mm) Half width of the sample (mm)
It is noteworthy that for industrial purposes, as is pointed out by Zhao et al. (2014), foods are usually dried to remove the moisture up to a certain level at which microbial spoilage and deteriorative chemical reactions are minimised, and accurate determination of diffusion and convective transfer coefficients is required in order to facilitate accurate designs. Different methods have been described in the literature to obtain experimental values for these coefficients. Zhao et al. (2014) used a trial-and-error method based on a comparison of analytic and experimental drying curves, tuning the value of the diffusion coefficient until the matching of both curves is reached approximately. In similar fashion, Białobrzewski (2007) obtained both diffusion and convective transfer coefficients by comparing numerical (as opposed to analytical) and experimental drying curves. Note that analytical solutions are available only for some simple systems, whilst numerical solutions are available for any system. Conversely, analytical solutions have arbitrarily high accuracy, but the accuracy of numerical solutions is limited by numerical approximation, especially in the case of partial differential equations. Also, in order to obtain the convective transfer coefficient, the reaction engineering approach has been applied by considering drying as a competitive process between condensation and evaporation (Compaore et al., 2017). However, the most widely used method is the fitting of the experimental drying curves, in logarithmic form, to a straight line. For simple one-dimensional geometries, such as for slices or cylinders, by assuming the first term approximation in
[z t ti v w0 wdm weq wi x yi
Di F FN mn r s2D s2h s2i s2w
31
Half height of the sample (mm) Time (s) The i-th time measured in the drying process (s) Air velocity induced by the extractor fan (m s1) Initial sample weight in the drying process (g) Sample dry matter weight (g [dry matter]) Sample weight at equilibrium in the drying process (g) Sample weight at each time ti in the drying process (g) Position across thin layer sample thickness (mm) The i-th ordinate value corresponding to the average dimensionless relative moisture at each time ti in the drying process Convergence of first term approximation for the i-th iteration Average dimensionless relative moisture The N terms series expansion approximation to the average dimensionless relative moisture The n-th positive root of the characteristic equation mntanmn ¼ Bi Sample dry matter density (g [dry matter] m3 ) Variance of the diffusion coefficient (m2 s1) Variance of the mass transfer coefficient (m s1) Variance corresponding to the i-th ordinate value yi in the drying process Variance corresponding to the instrumental error of the weighing device (g)
the series expansion solution, diffusion coefficient can be obtained from the corresponding slope (Karim & Hawlader, 2005; Mghazlia et al., 2017; Sampaio et al., 2017; Srikiatden & Roberts, 2006). Additionally, by using approximate relationships involving dimensionless numbers, in particular the Biot number, convective transfer coefficient is also obtained , Cruz, & (Dhalsamant, Tripathy, & Shrivastava, 2017; Guine Mendes, 2014; Tripathy & Kumar, 2009). Note that, in order to obtain the correct values of the coefficients, it is important that the experimental conditions correspond to the mathematical conditions under which the analytical solution was obtained. However, it is not common to find in the literature details explaining the drying experimental methodology which could fulfil such conditions. Moreover, it is important to determine the accuracy of the resulting coefficients and the corresponding significant figures. Note that the instrumental error of the experimental data will propagate in the calculations when the coefficients are obtained, such that the significant figures in the initial experimental data will decrease (or at best remain the same) in the calculated coefficients. In this regard, as is well known, the calculability of diffusion and convective transfer coefficients depends on the Biot number, therefore this issue should be analysed in any method for the determination of diffusion and convective transfer coefficients. In drying experiments, since the mass diffusion coefficient (and also convective mass transfer coefficient) depends on temperature, in order to calculate the mass diffusion
32
b i o s y s t e m s e n g i n e e r i n g 1 6 2 ( 2 0 1 7 ) 3 0 e3 9
coefficient (and convective mass transfer coefficient) by fitting to an analytical solution of mass diffusion equation where the diffusion coefficient (and convective mass transfer coefficient) is assumed to be constant, it is mandatory to maintain the sample with uniform temperature in its whole volume. In this regard, as was shown by Srikiatden and Roberts (2006) when studying potatoes and carrot drying, when the initial temperature of the sample and target temperature differ, the maintenance of surrounding air temperature at target temperature is not sufficient to achieve uniform temperature, with temperature gradients occurring inside the sample. It is necessary firstly to heat the sample, using a microwave device which leads to uniform temperature, up to target temperature, and then maintain the surrounding air temperature at the target temperature throughout the drying assay. The likely reason for wrong determination of the mass diffusion coefficient is due to existence of temperature gradients inside the sample, as was pointed out in the early literature (Chen & Johnson, 1969; Chirife, 1983; King, 1968; Vaccarezza, Lombardi, & Chirife, 1974). Additionally, aside from temperature, other conditions must be controlled in order to correctly obtain mass diffusion and convective mass transfer coefficients. Thereby, sample volume is usually assumed constant in obtaining analytical solutions of diffusion equation, so that shrinkage should be controlled in food samples (e.g., by taking drying data over the initial period, when shrinkage is negligible). Besides, the asymptotic moisture value (i.e., sample moisture at infinite time) is also usually assumed constant, so that the relative humidity of the surrounding air should be maintained invariable. On the other hand, in the fitting of the corresponding analytical solution to obtain mass diffusion and convective mass transfer coefficients, some approximations are usually made. Firstly, in the infinite series expansion of the analytical solution, the first term approximation is taken, that is, solution is approximated by taking only the first term in series expansion. As is well known, this is a very accurate approximation only for large enough time (Crank, 1975; Luikov, 1968). Time threshold is well defined in terms of dimensionless time, but dimensionless time depends on the diffusion coefficient, so that some iterative process should be used. Secondly, the logarithmic form of first term approximation leads to a straight line, so that the fitting process is straightforward. Nevertheless, direct calculation of diffusion and convective transfer coefficients from fitting intercept and slope involves the solving of a transcendental equation, so that different approximate relationships involving dimensionless numbers were developed in order to avoid this difficulty (Luikov, 1968; Pflug & Blaisdell, 1963). Despite that these difficulties being easily avoidable, due to the development of computers and numerical methods, the approximate classical schemes are et al., 2014; Karim & Hawlader, 2005; still used (Guine Srikiatden & Roberts, 2006; Tripathy & Kumar, 2009). In order to determine diffusion and convective transfer coefficients from drying data, an iterative methodology is presented that takes advantage of current numerical and computational capabilities, avoiding to use the approximate classical schemes, and numerically solving the corresponding transcendental equations. As an additional feature, from the range of obtained parameters involved in this numerical
solving, it can be identified the mismatching of experimental conditions and mathematical conditions, and so, the unsuitability of used data. As will be explained in detail below, experimental data are first transformed to an asymptotic linear form and then fitted to a straight line. The mathematical conditions under which the analytical solution is obtained lead to a very narrow range for the intercept of this asymptotic straight line. So that, a fitting intercept value out of range implies that experimental data have been obtained under experimental conditions that differs from mathematical conditions, hence is unsuitable for the determination of diffusion and convective transfer coefficients. The proposed methodology is applied to generated data, in order to test its behaviour, and also to experimental data from carrot drying at different temperatures, including an unsuitable data set corresponding to non-isothermal drying, included in order to illustrate the detection of the unsuitability of data set. Additionally, a MATLAB® implementation of the proposed iterative methodology, along with the input data files corresponding to the results presented in the paper, is provided as supplementary material.
2.
Materials and methods
2.1. Determination of diffusion and convective transfer coefficients 2.1.1.
Mathematical model
It is assumed that, for a thin layer sample, food drying process is well described by means of the one-dimensional diffusion equation vM v vM ¼ D ; vt vx vx
(1)
with uniform initial condition Mðx; t ¼ 0Þ ¼ M0 ;
(2)
and boundary conditions
vM ¼ 0; vx x¼0
(3)
and
vM h ¼ M[ Meq ; vx x¼[ D
(4)
where M is the dry basis moisture, t is the time, x is the position across thin layer sample thickness (with origin x ¼ 0 at internal centre), [ is the sample half thickness, and D and h are the physical drying parameters, namely, effective diffusion and convective mass transfer coefficients, respectively. Besides, M[ ¼ M(x ¼ [, t) is the moisture at food sample-air interface, and Meq ¼ Mðx ¼ [; t/∞Þ is the corresponding moisture at equilibrium. Note that symmetric external conditions, i.e. the same air conditions at both sides of the thin layer, have been assumed. Since initial condition in Eq. (2) is also symmetric, the whole system is symmetric, so that the mass flow F ¼ rD(vM/vx) given by Fick's law, where r is the dry matter density, will be null at the internal centre, leading to the boundary condition in Eq. (3).
b i o s y s t e m s e n g i n e e r i n g 1 6 2 ( 2 0 1 7 ) 3 0 e3 9
Moreover, if food sample temperature is maintained constant during drying process, then the diffusion coefficient D can be assumed to be constant. Additionally, if the air velocity is constant, the convective transfer coefficient h can also be assumed constant. Finally, if the relative humidity of air is maintained constant, then the moisture at equilibrium Meq will be constant. Under these additional conditions, the solution of diffusion equation in Eq. (1) under initial condition in Eq. (2), and boundary conditions in Eqs. (3) and (4), is given by the following series expansion (Crank, 1975; Luikov, 1968) ∞ x Mðx; tÞ Meq X 2sinmn Dt ¼ cos mn exp m2n 2 ; [ M0 Meq mn þ sinmn cosmn [ n¼1 (5) being mn the positive roots of the equation mn tanmn ¼
h[ ; D
(6)
ordered such that 0 < mn < mnþ1. The experimentally measurable overall moisture content M of the sample is obtained by averaging over the whole sample volume MðtÞ ¼
1 [
Z[ Mðx; tÞdx;
(7)
0
so that, the overall content moisture corresponding to series expansion in Eq. (5) will be 2 ∞ MðtÞ Meq X 2sin mn Dt exp m2n 2 : ¼ M0 Meq mn ðmn þ sinmn cosmn Þ [ n¼1
FðFoÞ ¼
n¼1
2Bi 2 exp m2n Fo ; m2n Bi þ Bi þ m2n
(10)
If diffusion process is much faster than convective transfer process (external problem), then Bi/0 and the positive roots of Eq. (10) are given by mn ¼ (n 1)p. Conversely, if convective transfer process is much faster than diffusion process (internal problem), then Bi/∞ and the corresponding roots are given by mn ¼ ð2n 1Þp=2. So that, when both processes are simultaneously relevant (0 < Bi < ∞), the positive roots of Eq. (10) are bounded by the two extreme cases, that is,
2.1.2.
(12)
In other words, for high enough values of the Fourier number, moisture time evolution FðFo; BiÞ is well described by the first term approximation F1 ðFo; BiÞ, that expressed in logarithmic form 2 2sin m1 m2 D 12 t; (13) logFzlog m1 ðm1 þ sinm1 cosm1 Þ [ takes the form of a straight line y ¼ a þ bt with intercept a ¼ log
2 2sin m1 ; m1 ðm1 þ sinm1 cosm1 Þ
(14)
and slope m2 D b ¼ 12 : [
(9)
where F ¼ ½MðtÞ Meq =ðM0 Meq Þ is the average dimensionless relative moisture, Fo ¼ Dt/[2 is the dimensionless time known as Fourier number, and Bi ¼ h[/D is the Biot number, which determines the relative importance between diffusion and convective transfer processes. Notice that, in dimensionless form, the coefficients mn are defined as the positive roots of the equation
ðn 1Þp < mn < ð2n 1Þp=2:
lim FðFo; BiÞ ¼ F1 ðFo; BiÞ:
Fo/∞
(15)
So that, by taking a set of drying data {w0,w1,…,weq}, the corresponding ordinate values can be calculated1
2
mn tanmn ¼ Bi:
FðFo; BiÞ in Eq. (9) must be calculated with sufficient accuracy, i.e, by taking a large enough number of terms N in the series expansion leading to a high accuracy approximation FN ðFo; BiÞ. For this purpose, the corresponding first N roots of the transcendental equation in Eq. (10) need to be calculated with high accuracy, too. However, since the roots are clearly bounded by the inequalities in Eq. (11), there exist different numerical algorithms that are able to calculate them very efficiently. Specifically, the algorithm of Forsythe, Malcolm, and Moler (1977) have been used, which is implemented in the MATLAB® function fzero. Moreover, it is well known that the series expansion in Eq. (9) converges rapidly and monotonically for high enough values (depending on Biot number value) of the Fourier number (Crank, 1975; Luikov, 1968), so that, as time increases, moisture curve tends asymptotically to the first term in the series, that is
(8)
As is well known, the solutions of diffusion equation can be expressed in terms of dimensionless magnitudes (Crank, 1975; Luikov, 1968). In this way, the series expansion in Eq. (8) can be expressed as follows ∞ X
33
(11)
Iterative numerical methodology
In order to obtain the diffusion and convective transfer coefficients from drying data, the dimensionless moisture
yi ¼ log
wi weq ; w0 weq
(16)
where w0 is the initial sample weight at t0 ¼ 0, wi is the sample weight at each time ti(i ¼ 1,2,…), and weq is the sample weight at equilibrium. Then, intercept a and slope b can be obtained by means of the least squares fitting to a straight line of the data pairs {ti,yi}(i ¼ 1,2,…). It should be noted that, by using the logarithm function in the ordinate values definition yi, the corresponding variances s2i will increase with time, so that, a weighted least squares fitting should be used in the obtaining of slope and intercept values. Indeed, assuming the same variance s2w for all weight data wi, which is determined by the weighing device used (instrumental error), the variances for the ordinate values yi(i ¼ 1,2,…) are given by2 1
Note that, in order to calculate the average dimensionless relative moisture F from drying data, only weight data w are needed, and no average moisture values M should be calculated, M M w w since M0i Meq ¼ w0i weq : eq eq 2 Given a function f(x1,…,xN) of variables xk with variances s2k ðk ¼ 1; …; NÞ, error propagation theory provides the following 2 P vf s2k : formula for the variance s2 of the function f.s2 ¼ N k¼1 vxk
34
s2i
b i o s y s t e m s e n g i n e e r i n g 1 6 2 ( 2 0 1 7 ) 3 0 e3 9
¼
2s2w
1 wi weq
2 þ
wi weq
1
! 1 þ 2 : w0 weq w0 weq (17)
The weighted least squares fitting was carried out by using the algorithm of Strang (1986), which is implemented in the MATLAB® function lscov. After fitting process, the first root value m1 is calculated from Eq. (14). Notice that, as in the calculation of the roots of Eq. (10) discussed above, it is a transcendental equation, so that the algorithm of Forsythe et al. (1977) has also been used. Then, diffusion coefficient D is calculated from Eq. (15), convective transfer coefficient h is calculated from Eq. (6), and Biot number Bi from Eq. (10). However, the coefficients thus obtained could be wrong, since the first term approximation F1 ðFo; BiÞ is valid for high enough values (depending on the Biot number value) of the Fourier number, and both Fourier and Biot numbers depend on diffusion and convective transfer coefficients. So that, in order to ensure the correctness of the coefficients calculated, an iterative process is mandatory. Thus, iteratively, the (i1)-th data pair will be deleted and the corresponding coefficients Di and hi recalculated by fitting of remaining data pairs, until the convergence of first term approximation within data accuracy, namely FN ðFoi ; Bii Þ F1 ðFoi ; Bii Þ si ;
(18)
where the values of Fourier and Biot numbers are recalculated at each iteration from fitting coefficients Di and hi. When the condition in Eq. (18) is fulfilled, the corresponding coefficients Di and hi can be considered the correct coefficients. Observe that, in order to verify the convergence of first term approximation to exact values, we have used a high accuracy approximation FN ðFo; BiÞ that can be considered as exact within data accuracy. Finally, notice that by applying the inequalities in Eq. (11) to the first root m1, the intercept a and the slope b of the fitted straight line must necessarily fulfil the following conditions log
8 < a < 0; p2
(19)
and b < 0:
(20)
So that, the non-fulfilment of conditions above, when fitting a given data set, implies that the experimental setup breaks the model assumptions, and then the fitting coefficients obtained will be wrong. In this regard, condition in Eq. (19) is particularly relevant since it establishes a very narrow range, a2(0.21, 0), for checking the suitability of experimental data set. A MATLAB® implementation of the proposed iterative methodology, along with the input data files corresponding to the results presented in this paper, is available in the supplementary material.
2.2.
Experimental setup
The carrot samples were obtained from entire carrots (Daucus carota L. cv. Nantes) purchased from local market, which were
manually water washed, peeled, and lengthwise sliced with a kitchen slicer (Cooking, CASA International nv, Itegem, Antwerpen, Belgium) in order to obtain constant thickness. The samples were sliced from central part of carrots, and eventually cut into rectangular pieces. The dimensions of each carrot sample were measured by means of a digital caliper gauge (DigitCal® Classic 05.30035, Swiss Instruments Ltd., Mississauga, Ontario, Canada). Before drying process, each carrot sample was heated in a sticos Taurus S.L., microwave oven (PTMW 700, Electrodome rida, Spain), with the aim of achieving a uniform Oliana, Le temperature in the whole sample. With the purpose of determine diffusion and convective transfer coefficients at different temperatures, distinct heating periods were considered to reach temperatures around 40, 50, 60, and 70 C. In order to avoid the start of drying process throughout the heating period, a small bowl with water was introduced into microwave oven along with each sample. After heating, sample temperature was measured in each case by using a type T thermocouple thermometer (HH2001TC, Omega Engineering Inc., Stamford, CT, USA), giving temperatures of 40, 55, 64, and 72 C. Before each sample was introduced into the weighing device, it was sandwiched between two metal grids in order to keep the slice shape. The setpoint temperature, corresponding to each measured temperature at which the parameters will be estimated, was established in each case in the weighing device (AMB 300 moisture balance, Inscale Measurement Technology Ltd., Bexhill, East Sussex, UK), and the sample weight recording every 20 s was started. As shown in Fig. 1, the programmable weighing device comprises a chamber, with the weighing pan inside, equipped with two halogen heaters, and also a RS-232 interface for connection to a computer, so that weighing data are automatically recorded in a file. The instrumental error of the weighing device, which is required by the proposed iterative methodology in Eq. (17), is sw ¼ 0.001 g. Additionally, at the top of the chamber, there is a ventilation grill. Observe that, in order to achieve the required symmetric external conditions (i.e., the same air conditions at both sample sides), the sample was raised on four feet above the sample pan. Air temperature and also air relative humidity were monitored by means of an electronic logger (iButton® Hygro, CA, chron™ DS1923, Maxim Integrated Products Inc., San Jose USA) located into the weighing device chamber. In order to prevent the increase of relative humidity in the sample surroundings, an extractor fan (EVERFLOW F128025BU, SilverStone Technology Co. Ltd., New Taipei, Zhonghe, Taiwan) was mounted on the ventilation grill of the weighing device chamber [see Fig. 1(a)]. Extracted air velocity was measured by using a multifunctional precision manometer with Prandtl's € hler Messgera € te Kehrgera € te GmbH, Bad tube (DC 100pro, Wo Wu¨nnenberg, Paderborn, Germany). Lastly, each carrot sample was entered into a drying oven (Conterm Poupinel, JP SELECTA S.A., Abrera, Barcelona, Spain) at 105 C until steady weight was achieved, and thereafter sample dry weight was determined with a precision balance (ADP 720/L, Adam Equipment Co. Ltd., Kingston, Milton Keynes, UK).
b i o s y s t e m s e n g i n e e r i n g 1 6 2 ( 2 0 1 7 ) 3 0 e3 9
35
Fig. 1 e (a) Overview of the weighing device with the mounted extractor fan. (b) Section view of the weighing device with the sample raised on four feet above the sample pan. (c) Detail of the sample sandwiched between two metal grids with the four feet.
3.
Results and discussion
3.1.
Determination of coefficients from generated data
In order to test the behaviour of the proposed iterative methodology, the series expansion in Eq. (9) with original coefficients D, h, and other conditions described in Table 1, was used to generate 18 sample weight data {w1, w2,…,w18} along drying process. By taking N ¼ 14 terms in series expansion, an accuracy of 0.001 g in the worst case Fo ¼ 0 was achieved. Gaussian noise was introduced into generated data, so that the final data set with weighing accuracy sw ¼ 0.01 g was obtained. Pseudorandom values for Gaussian noise were calculated by using the algorithms of Marsaglia (Marsaglia & Tsang, 2000; Marsaglia & Zaman, 1991) which are implemented in the MATLAB® function randn. The successive iterations corresponding to applying the proposed iterative methodology to generated data have been depicted in panels (a)e(g) of Fig. 2. At each iteration, the first data point is deleted, and the remaining data fitted to a straight line leading to the corresponding values of diffusion Di and convective transfer hi coefficients. Note that, in the first iterations, the original fitting data points and the corresponding exact points obtained from Eq. (9) with the fitting coefficients Di and hi differ significantly. As iterative process progresses both data sets get closer, and finally, when convergence is reached at i ¼ 7, they match within weighing
Table 1 e Original and calculated diffusion and convective transfer coefficients in the determination of coefficients from generated data. Note that calculated coefficients correspond to those in Fig. 2(g) rounded to two significant figures. Coefficients Original Calculated
D (m2 s1)
h (m s1)
Bi
4.76$1011 4.8$1011
2.74$107 2.7$107
11.5 11
Data generation conditions: [ ¼ 2 mm, w0 ¼ 6.47 g, weq ¼ 0.77 g, sw ¼ 0.01 g.ti ¼ {0.5, 1, 1.5,…,9 h}.
accuracy sw. Notice that convergence condition in Eq. (18) is graphically represented in panel (h) of Fig. 2. It is worth noting that we have verified in a heuristic manner that, as is shown in Fig. 2 and summarised in Table 1 for the present case, diffusion and convective transfer coefficients are obtained roughly with one significant figure less than the weighing values. Thus, the generated data set starts with w1 ¼ 5.86 g and ends with w18 ¼ 3.01 g, corresponding to three significant figures, and diffusion and convective transfer coefficients are obtained with two significant figures. The number of significant figures obtained depends on the Biot number, as well as the features of the experimental setup, mainly the accuracy of the weighing data. In order to establish the number of significant figures obtained in the calculation of diffusion and convective transfer coefficients under typical conditions in drying experiments, we have carried out several series of simulations by generating weighing data and fitting generated data to obtain the coefficients, in a similar way to the example above. Sample half thickness [ ¼ 2 mm, weighing device accuracy sw ¼ 0.001 g with 4 significant figures, and range 109 1012 m2 s1 for the order of magnitude of the diffusion coefficient are assumed as typical conditions in drying experiments. Additionally, for each Biot number Bi ¼ {0.001,0.01,0.1,1,10,100,1000} a set of 10 diffusion coefficients are randomly obtained, and for each random diffusion coefficient 10 sets of weighing data are generated at Fourier numbers Fo ¼ {0,0.1,0.2,…,1}. By comparing original diffusion and convective transfer coefficients with those calculated from generated data sets, the value of the generalised number of significant figures SX is obtained. The generalised number of significant figures in X is defined by means of the logarithmic difference SX ¼ 1 þ log10 ðXÞ log10 ðsX Þ;
(21)
where X ¼ D, h is the value of the corresponding coefficient at a given Biot number, and sX is the standard deviation of the coefficient from the different simulations (10 simulations in our case) at that Biot number. Notice that, for the same Biot number but different random diffusion coefficient, somewhat different values of SX will be obtained, which will be
36
b i o s y s t e m s e n g i n e e r i n g 1 6 2 ( 2 0 1 7 ) 3 0 e3 9
Fig. 2 e Iterative process for obtaining diffusion and convective transfer coefficients applied to generated data described in Table 1. At each iteration (i ¼ 1,2,…,7), the logarithm of average dimensionless relative moisture F versus Fourier number Fo has been depicted for fitting data (open circles), fitted straight line corresponding to first term approximation F1 ðFoi ; Bii Þ (thick line), exact drying curve corresponding to high accuracy approximation FN ðFoi ; Bii Þ (thin line), and the points over exact curve corresponding to Fourier numbers of fitting data (filled circles). (a) Initial fitting with all data. (b)e(g) Consecutive fitting iterations, where the first datum is deleted at each iteration until reaching convergence. Fitting coefficients Di (10¡11 m2 s¡1) and hi (10¡7 m s¡1) and also c each iteration are indicated. (h) Decimal logarithm of the difference Di ¼ FN ðFoi ; Bii Þ F1 ðFoi ; Bii Þ at convergence. Values corresponding to deleted data points (open circles) and used fitting data (filled circles) are depicted. Data error at convergence s7 is represented in dotted line.
statistically characterised by the corresponding mean and standard deviation. The results of these simulations are shown in Fig. 3, where the mean values and the standard deviation values of the generalised significant figures corresponding to each generated data set are represented as filled circles and error bars, respectively. Observe that, as expected, optimal results for both coefficients jointly are obtained in the range 1 Bi 10. Moreover, as Biot number increases the accuracy of diffusion coefficient also increases, whilst the accuracy of convective transfer coefficient falls quickly. Even so, at Bi ¼ 100 the accuracy of transfer coefficient is not too low. On the other hand, as Biot number decreases the accuracy of diffusion coefficient falls quickly, and the accuracy of transfer coefficient also falls, although it decreases slower than the diffusion coefficient accuracy. This is not a surprising behaviour, since the transfer coefficient value is calculated through Eq. (6) by using the previously obtained diffusion coefficient. However, until Bi ¼ 0.01 the accuracy of calculated transfer coefficient is very good. The iterative process shown in Fig. 2 can be reproduced, and additional details obtained, by running the MATLAB® program with the input data file corresponding to generated data available in the supplementary material.
3.2. data
Determination of coefficients from experimental
The proposed iterative methodology has been applied to experimental data corresponding to carrot drying
Fig. 3 e Generalised significant figures, SD and Sh, obtained in the calculation of diffusion (thick line) and convective transfer (thin line) coefficients, respectively, as a function of the Biot number Bi for different simulations performed under typical conditions in drying experiments (see text for details).
b i o s y s t e m s e n g i n e e r i n g 1 6 2 ( 2 0 1 7 ) 3 0 e3 9
experiments. In order to establish the temperature dependence of diffusion and convective transfer coefficients, isothermal drying at different temperatures was performed. The final fitting at convergence is depicted for each temperature in Fig. 4 (top panels) along with the corresponding graphically represented convergence conditions (bottom panels). The obtained diffusion and convective transfer coefficients are summarised in Table 2 in addition to some sample parameters as Biot number Bi, thickness 2[, width 2[y, height 2[z, initial weight w0, equilibrium weight weq, and dry matter weight wdm. Note that, in this case, weighing data have four significant figures, so that three significant figures have been assumed for the obtained coefficients. By means of simulations as those depicted in Fig. 3, the accuracy of the obtained diffusion and convective transfer coefficients has been estimated, namely sD ¼ 0.03$109 (m2 s1) and sh ¼ 0.01$106 (m s1), respectively. Since the main goal of this section is to illustrate the application of the proposed methodology, for the sake of clarity only one data set at each temperature has been considered. However, in order to obtain a better estimation of the coefficients accuracy, accounting correctly for the experimental error, several data sets at each temperature should be regarded, and the corresponding error calculated in standard manner. As is expected, both diffusion coefficient and convective transfer coefficient mostly grow as temperature increases. Nevertheless, as is shown in Table 2, diffusion coefficient decreases very slightly from the value at T ¼ 55 C and the value at T ¼ 64 C. We assume that this effect is due to the
37
different drying behaviour of core and cortex parts in carrots and the slightly different core-cortex ratio in each sample. Notice that, in order to achieve the one-dimensional behaviour for the thin layer samples, it was not possible to separate core and cortex parts. In this regard, Srikiatden and Roberts (2006) obtained slightly higher values of the diffusion coefficient for core tissue than for cortex tissue in carrots (D. carota L. cv. Red Core Chantenay) cut into cylindrical samples. Specifically, they found at temperatures of 40, 50, 60, and 70 C, the values of 1.01, 1.32, 1.90, and 2.34 (109 m2 s1) for core tissue, and values of 0.76, 0.92, 1.06, and 1.36 (109 m2 s1) for cortex tissue, respectively. Taking into account that carrot samples of Srikiatden and Roberts correspond to a different cultivated variety than those used in the present work, it can be considered that their results and our results are in good agreement. Additionally, Markowski (1997) found the value h ¼ 1.37$107 (m s1) for convective transfer coefficient of carrots (cv. not specified) at T ¼ 60 C and natural convection. Considering that our results correspond to forced convection, albeit rather at a low air velocity, it seems reasonable that our results are somewhat higher. Moreover, it is important to highlight that, since an analytical solution of the diffusion equation under certain conditions is used, in order to obtain physical coefficients of carrots rather than unphysical best fit parameters, these conditions should be approximated in drying experiments, as much as possible. Thus, one-dimensional behaviour for the thin layer samples is achieved by taking thin layer samples with width and height one order of magnitude greater than
Fig. 4 e Iterative process for obtaining diffusion and convective transfer coefficients applied to experimental data from carrot slices drying summarised in Table 2. For different temperatures, the logarithm of average dimensionless relative moisture F versus the Fourier number Fo has been depicted for fitting data (open circles), fitted straight line corresponding to first term approximation F1 ðFoi ; Bii Þ (thick line), exact drying curve corresponding to high accuracy approximation FN ðFoi ; Bii Þ (thin line), and the points over exact curve corresponding to Fourier numbers of fitting data (filled circles). (a)e(d) Final fitting at convergence for temperatures T ¼ 40, 55, 64, 72 C, respectively. (e)e(h) Decimal logarithm of the difference Di ¼ FN ðFoi ; Bii Þ F1 ðFoi ; Bii Þ at convergence corresponding to temperatures T ¼ 40, 55, 64, 72 C, respectively. Values corresponding to deleted data points (open circles) and used fitting data (filled circles) are depicted. Data error at convergence for each temperature si (i ¼ 39,34,78,18) is represented in dotted line. Fitting coefficients Di (10¡9 m2 s¡1) and hi (10¡6 m s¡1) and also the Biot number calculated at convergence are indicated.
38
b i o s y s t e m s e n g i n e e r i n g 1 6 2 ( 2 0 1 7 ) 3 0 e3 9
Table 2 e Diffusion and convective transfer coefficients obtained for carrot slices and additional sample parameters. T ( C) 40 55 64 72
D (m2 s1) 10
7.27$10 1.55$109 1.53$109 2.95$109
h (m s1)
Bi
2[ (mm)
2[y (mm)
2[z (mm)
w0 (g)
weq (g)
wdm (g)
6.54$107 1.20$106 1.29$106 2.65$106
1.28 1.45 1.21 1.76
2.85 3.75 2.87 3.92
32.80 27.40 29.51 23.43
61.17 62.48 62.96 52.13
4.795 5.450 4.986 3.986
0.662 0.890 0.733 0.479
0.593 0.845 0.702 0.456
Obtained coefficients accuracy: sD ¼ 0.03$109 m2 s1, sh ¼ 0.01$106 m s1. Weighing device accuracy: sw ¼ 0.001 g. Air velocity: v ¼ 0.71 m s1.
thickness, as is shown in Table 2. Homogeneous initial moisture M0 can be assumed for fresh carrots preserved in moisture insulated atmosphere. Additionally, diffusion D and convective transfer h coefficients must be constant. Since both coefficients depend on temperature, carrot sample temperature should be uniform and constant. Sample temperature homogeneity is achieved by using a microwave device in the heating process, and it is kept constant by controlling surrounding air temperature, which was monitored by means of an electronic logger. Notice that convective transfer coefficient depends also on surrounding air velocity and carrot sample shape, both conditions being kept constant as is explained below. Besides, sample moisture at equilibrium Meq must also be constant, which is achieved by controlling relative humidity of surrounding air. In order to avoid the relative humidity increase due to drying process, an extractor fan at very low velocity was used (generating constant air velocity), and relative humidity was monitored through an electronic logger. Finally, shape and size (volume geometry in diffusion equation) must be constant along drying process. Carrot slices were sandwiched between two metal grids in order to keep the slice shape. Nevertheless, shrinkage is an unavoidable effect in carrot drying, which changes size and also shape, albeit most shrinkage occurs at the initial stages of drying process, since it is strongly correlated with the moisture decrease. However, the proposed methodology discards initial data, where most shrinkage occurs. Additionally, a data set within a narrow moisture range has been considered, so that shrinkage can be neglected. Notice that, as verified, weighing data corresponding to a nonnegligible shrinking period lead to a convex (instead of concave) drying curve, as in the case of non-isothermal drying described below, being detected by the proposed methodology as unsuitable data. The sample temperature homogeneity is a remarkable condition in order to obtain correct diffusion and convective transfer coefficients from drying data fitting. As it has been shown by Srikiatden and Roberts (2006) in drying experiments with carrots and also with potatoes, convective hot air drying with constant air temperature leads to non-isothermal drying, since a temperature gradient is induced inside samples, giving experimental drying curves that do not fit the corresponding analytical solution of diffusion equation with constant diffusion and transfer coefficients. In this regard, an example of checking of suitability of data set provided by the proposed iterative methodology in Eq. (19) is shown in Fig. 5. Drying data in Fig. 5 correspond to a carrot slice sample initially at room temperature (~20 C) that is drying at constant air temperature of 65 C. Observe that the drying curve given by experimental data is convex (negative second derivative), whilst the
Fig. 5 e Iterative process applied to unsuitable (nonisothermal) experimental data from carrot slices drying. The logarithm of average dimensionless relative moisture F versus time t has been depicted for fitting data (open circles), and for the corresponding fitted straight line logðFÞ ¼ a þ bt (thick line). The unsuitability of data set is detect at first step of process, since fitting intercept a is out of range, i.e., a;ð0:21; 0Þ.
analytical solution on which our methodology is based is a concave curve (positive second derivative). As a consequence of this feature, fitting intercept a is out of the required range (0.21, 0) in Eq. (19), being a positive value a > 0, and so detecting the unsuitability of data set. The iterative process corresponding to each experimental data set at different temperatures shown in Fig. 4, as well as that corresponding to the unsuitable data set, can be reproduced, and additional details obtained, by running the MATLAB® program with the corresponding input data file available in the supplementary material.
4.
Conclusions
Physical drying parameters, namely, mass diffusion and convective mass transfer coefficients, have a significant relevance on proper determination of optimal drying conditions in convective hot air food drying, in order to achieve the aims of
b i o s y s t e m s e n g i n e e r i n g 1 6 2 ( 2 0 1 7 ) 3 0 e3 9
food industry. In this paper, a new methodological approach has been presented for the determination of diffusion and convective transfer coefficients from drying data, which takes advantage of current numerical and computational capabilities versus the classical scheme, based on approximate relationships involving dimensionless numbers, that is usually applied. As in classical schemes, data fitting to asymptotic first term in series expansion solution is used, but in our method the Fourier number threshold is iteratively calculated, since the Fourier number depends on diffusion coefficient. Also, by using standard numerical methods, involved transcendental equations are accurately solved, so that series expansion solution can be obtained with arbitrarily high accuracy, and also diffusion and convective transfer coefficients can be directly obtained from fitting parameters. As an additional feature, from the range of obtained parameters involved in the numerical solving, it can be identified the unsuitability of used data, avoiding calculation of wrong coefficients. The proposed methodology was tested by applying to generated data, resulting in obtaining diffusion and convective transfer coefficients, for Biot numbers roughly within the range 1e10, with only one significant figure less than the weighing data values. Additionally, the new methodological approach has been applied to experimental data from carrot drying at different temperatures, showing an adequate behaviour and giving reasonable results. Also, our method was applied to nonisothermal drying data, showing the feature of automatic detection of unsuitable data. Finally, it is noteworthy that although our method has been specified for the infinite plate one-dimensional diffusion system, the extension of the proposed iterative methodology to other typical one-dimensional diffusion systems with analytical solution, as those corresponding to cylindrical or spherical geometries, is straightforward. Notice that, in these cases, the same diffusion coefficient should be obtained for the different geometries (assuming that the diffusion coefficient is a physical property of food tissue), but different convective transfer coefficients will be obtained for each geometry, since it is assumed that the convective transfer coefficient depends on solid shape, among others factors.
Acknowledgments The funding of this work has been covered by Comunidad de Madrid and European Union through S2013/ABI-2747 (TAVSCM) project.
Appendix A. Supplementary data Supplementary data related to this article can be found at http://dx.doi.org/10.1016/j.biosystemseng.2017.07.005.
references
Białobrzewski, I. (2007). Determination of the mass transfer coefficient during hot-air-drying of celery root. Journal of Food Engineering, 78(4), 13881396.
39
Chen, C. S., & Johnson, W. H. (1969). Kinetics of moisture movement in hygroscopic materials. I. Theoretical considerations of drying phenomena. Transactions of the ASAE, 12(1), 109e113. Chirife, J. M. (1983). Fundamentals of the drying mechanism during air dehydration of foods. In A. S. Mujumdar (Ed.), Advances in drying (Vol. 2). New York, US: Hemisphere Publishing Corp. Compaore, A., Dissa, A. O., Rogaume, Y., Putranto, A., Chen, X. D., Mangindaan, D., et al. (2017). Application of the reaction engineering approach (REA) for modeling of the convective drying of onion. Drying Technology, 35(4), 500e508. Crank, J. (1975). The mathematics of diffusion (2nd ed.). London, UK: Oxford University Press. Dhalsamant, K., Tripathy, P. P., & Shrivastava, S. L. (2017). Moisture transfer modeling during solar drying of potato cylinders considering shrinkage. International Journal of Green Energy, 14(2), 184e195. Forsythe, G. E., Malcolm, M. A., & Moler, C. B. (1977). Computer methods for mathematical computations. Englewood Cliffs, US: Prentice-Hall. , R. P. F., Cruz, A. C., & Mendes, M. (2014). Convective drying Guine of apples: Kinetic study, evaluation of mass transfer properties and data analysis using artificial neural networks. International Journal of Food Engineering, 10(2), 281e299. Karim, M. A., & Hawlader, M. N. A. (2005). Mathematical modelling and experimental investigation of tropical fruits drying. International Journal of Heat and Mass Transfer, 48(23e24), 4914e4925. King, C. J. (1968). Rates of moisture sorption and desorption in porous dried foodstuffs. Food Technology, 22(4), 165e171. Luikov, A. V. (1968). Analytical heat diffusion theory. New York, US: Academic Press. Markowski, M. (1997). Air drying of vegetables: Evaluation of mass transfer coefficient. Journal of Food Engineering, 34(1), 55e62. Marsaglia, G., & Tsang, W. W. (2000). The ziggurat method for generating random variables. Journal of Statistical Software, 5(8), 1e7. Marsaglia, G., & Zaman, A. (1991). A new class of random number generators. Annals of Applied Probability, 1(3), 462e480. Mghazlia, S., Ouhammoua, M., Hidara, N., Lahninea, L., Idlimamb, A., & Mahrouz, M. (2017). Drying characteristics and kinetics solar drying of Moroccan rosemary leaves. Renewable Energy, 108, 303e310. Pflug, I. J., & Blaisdell, J. L. (1963). Methods of analysis of precooling data. ASHRAE Journal, 5(11), 33e41. Sampaio, R. M., Neto, J. P. M., Perez, V. H., Marcos, S. K., Boizan, M. A., & Da Silva, L. R. (2017). Mathematical modeling of drying kinetics of persimmon fruits (Diospyros kaki cv. Fuyu). Journal of Food Processing and Preservation, 41(1), e12789. Srikiatden, J., & Roberts, J. S. (2006). Measuring moisture diffusivity of potato and carrot (core and cortex) during convective hot air and isothermal drying. Journal of Food Engineering, 74(1), 143e152. Strang, G. (1986). Introduction to applied mathematics. Wellesley, US: Wellesley-Cambridge Press. Tripathy, P. P., & Kumar, S. (2009). A methodology for determination of temperature dependent mass transfer coefficients from drying kinetics: Application to solar drying. Journal of Food Engineering, 90(2), 212e218. Vaccarezza, L. M., Lombardi, J. L., & Chirife, J. (1974). Kinetics of moisture movement during air drying of sugar beet root. International Journal of Food Science & Technology, 9(3), 317e327. Zhao, D., An, K., Ding, S., Liu, L., Xu, Z., & Wang, Z. (2014). Twostage intermittent microwave coupled with hot-air drying of carrot slices: Drying kinetics and physical quality. Food and Bioprocess Technology, 7(8), 2308e2318.