JULIO / DICIEMBRE 2020 - VOLUMEN 30 (2)
/ ISSN electrónico: 2215-2652
Esta obra está bajo una Licencia de Creative Commons. Reconocimiento - No Comercial - Compartir Igual 4.0 Internacional
DOI 10.15517/ri.v30i2.39588
Ingeniería 30 (2): 63-76, julio-diciembre, 2020. ISSN: 2215-2652. San José, Costa Rica
Modelling Matambú bean (Phaseolus vulgaris) hydration kinetics
using an automated digital image analysis
Modelado de la cinética de hidratación del frijol Matambú (Phaseolus
vulgaris) utilizando un análisis automatizado de imágenes digitales
Ovidio Valerio Cubillo
Docente e Investigador
Universidad de Costa Rica, San José, Costa Rica
ovidio.valerio@ucr.ac.cr
ORCID: https://orcid.org/0000-0003-2299-0968
Guilllermo Vargas Elías
Docente e Investigador
Universidad de Costa Rica, San José, Costa Rica
guillermo.vargaselias@ucr.ac.cr
ORCID: https://orcid.org/0000-0001-8562-0062
Luis Barboza Barquero
Profesor adjunto e Investigador CIGRAS
Universidad de Costa Rica, San José, Costa Rica
luisorlando.barboza@ucr.ac.cr
ORCID: https://orcid.org/0000-0002-4140-6598
Recibido: 8 de noviembre 2019 Aceptado: 27 de marzo 2020
_________________________________________________________
Abstract
Common bean is one of the most consumed crops worldwide and of great importance for food security
in Central America. In Costa Rica, there is not much information regarding the physical properties of the
relatively new variety of Matambú bean. The aim of this study was to determine the speed in which Matambú
bean (Phaseolus vulgaris) expands after grain soaking. An automated digital image processing pipeline
for measuring individual grain volume and simultaneously retaining its integrity was validated and used.
The previously described pipeline uses a mirror at 45 degrees so that a superior and lateral image could be
taken and the 3 perpendicular diameters that compose a triaxial ellipsoid could be measured from grains.
The geometric distances where calibrated with known geometries with high correlations (R
2
=0.999) and
the approximated volume was validated with a pycnometer with an average difference of 16 %. Volumetric
expansion was evaluated with Peleg and sigmoid models for water temperatures of 20-70 oC. The equations
kinetic parameters were described as functions of temperature using the Arrhenius equation with an activation
energy of 22.410 kJ mol
-1
. The obtained results are potentially useful for future studies on food properties,
grain quality, process, and packaging design.
Keywords:
Digital image processing; Food properties; volumetric expansion; activation energy
Esta obra está bajo una Licencia de Creative Commons. Reconocimiento - No Comercial - Compartir Igual 4.0 Internacional
DOI 10.15517/ri.v30i2.39588
Ingeniería 30 (2): 63-76, julio-diciembre, 2020. ISSN: 2215-2652. San José, Costa Rica
Resumen
El frijol común es uno de los cultivos más consumidos en todo el mundo y es de gran importancia para la
seguridad alimentaria en América Central. En Costa Rica no hay mucha información sobre las propiedades
físicas de la variedad relativamente nueva de frijol Matambú. El objetivo de este estudio fue determinar
la velocidad a la cual el frijol Matambú (Phaseolus vulgaris) se expande después del remojo. Se validó y
utilizó un sistema de procesamiento de imágenes digitales automatizado para medir el volumen de granos
individuales reteniendo su integridad. El sistema descrito anteriormente utiliza un espejo a 45 grados para que
se pueda tomar una imagen superior y lateral para medir los 3 diámetros perpendiculares que componen un
elipsoide triaxial a partir de granos. Las distancias geométricas se calibraron con geometrías conocidas con
altas correlaciones (R
2
= 0.999) y el volumen aproximado se validó con un picnómetro con una diferencia
promedio de 16 %. La expansión volumétrica se evaluó con modelos Peleg y sigmoides para temperaturas del
agua de 20-70 oC. Los parámetros cinéticos de las ecuaciones se describieron como funciones de temperatura
utilizando la ecuación de Arrhenius con una energía de activación de 22.410 kJ mol
-1
. Los resultados obtenidos
son potencialmente útiles para futuros estudios sobre las propiedades de los alimentos, la calidad del grano,
el proceso y el diseño del empaque.
Palabras clave:
Procesamiento de imágenes digitales; propiedades de los alimentos; expansión volumétrica; energía de
activación
Ingeniería 30 (2): 63-76, julio-diciembre, 2020. ISSN: 2215-2652. San José, Costa Rica DOI 10.15517/ri.v30i2.39588
65
1. INTRODUCTION
The common bean (Phaseolus vulgaris) is one of the crops with the longest history in the
Americas and of the highest consumption worldwide especially in Latin America, Africa and Asia
(Arias, Valverde, Fonseca & Melara, 2010). The plant has undergone a process of domestication
where genotypes have been selected to grow in stressful environments and meet necessary nutritional
demands (Rao, 2014). In Costa Rica, bean varieties have been breed with tolerance to disease and
droughts from which the Matambú species is one of the most recent in 2013 (Hernández, Chaves,
Araya & Beebe, 2018). Because of how recent and localized these developments are, information
regarding its physical properties for post-harvest processing is scarce. Soaking is widely used as a
pretreatment of cereals and legumes before other processing systems. In grains, soaking is used to
hydrate a grain evenly in order to generate the necessary water for starch gelatinization and protein
denaturation during cooking (Zanella-Díaz, Mújica-Paz, Soto-Caballero, Welti-Chanes & Valdez-
Fragoso, 2014). It is also a very important step before processes such as fermentation (Egounlety
& Aworh, 2003). A way to optimize the previously described processes is to describe physical
changes in a biological material as a time dependent function.
Several models have been developed to describe the rate of grain soaking. Peleg´s equation
(eq. 1) corresponds to the most used model to describe hydration kinetics for food products. This is
an empirical, two parameters equation for moisture absorption curves for materials exposed to the
atmosphere for short times before the moisture level approaches equilibrium levels with medium
(Peleg, 1988):
(1)
This model has been used to describe products such as sorghum (Kashiri, Kashaninejad & Aghajani,
2010), soy (Quicazán, Caicedo & Cuenca, 2012), rice, and barley (Cardoso, Ascheri & Carvalho,
2014). The Peleg model despite being the most used model cannot describe an initial lag phase,
observed during hydration of some dry grains, which has been corrected by the formulation of a
sigmoid equation (eq. 2) (Kaptso et al., 2008). This is a three-parameter model where the delay
time which describes a slow absorption phase and the moisture at equilibrium are considered.
(2)
In his work for imbibed adzuki bean (Vigna angularis), Oliveira et al. (2013) showed that
parameters k and have a temperature dependence considering the activation energy for which
the Arrhenius equation was used. This equation represents the dependence of the rate of a chemical
reaction with respect to temperature for a rst order chemical reaction. The variation of a parameter
P is presented in equation 3, which is the modied Arrhenius equation.
VALERIO, VARGAS Y BARBOZA: Modelling Matambú bean (Phaseolus vulgaris) hydration kinetics...
66
(3)
The previously described equation has been used for products such as red bean (Phaseolus
vulgaris L.), chickpea (Cicerarietinum L.) (Shafaei, Masoumi & Roshan, 2016), cowpea (Vigna
unguiculata), and Bambara groundnuts (Voandzeia subterranea) (Kaptso et al., 2008).
All the presented equations have been used to describe changes in moisture or mass, but studies
that have modeled volumetric expansion as a function of time are limited in grains as it would
generally require a time consuming individual analysis involved in traditional measuring methods.
Most of the methods listed in the references are based in bulk analysis, disregarding individual grain
variation. Because of this, the aim of this work is to describe bean volumetric expansion with a
practical and accessible methodology through automated image processing. The method presented
in this paper offers great reliability on the resulting data as it sheds light on many variables that are
normally ignored.
2. MATERIALS AND METHODS
2.1 Location and plant material
This work was performed at the facilities of the Research Center of Grains and Seeds (CIGRAS)
at the University of Costa Rica. Matambú bean samples (Phaseolus vulgaris) with 12.35 ± 0.78 %
moisture (wet basis) were harvested in January 2016 at the experimental station Baudrit Fabio, Ala-
juela, Costa Rica. Prior to use, the product was classied using circular screens with perforations
sizes of 4.37 and 4.76 mm. This created a homogenous sample that was manually cleaned, sealed
in plastic bags, and stored in a dry place before using.
2.2 Image acquisition and analysis
The image acquisition was performed using a D7100 Nikon digital camera with a focal aperture
of f/8, 5EV exposure, resolution of 6000 4000 pixels, and mounted on a rail, which maintains a
constant distance from the observer to the object. All photographic equipment is conned in a cham-
ber to avoid incidence of light from the outside with only white lights that are perpendicular to the
line of sight. The action of the camera is controlled from a computer Intel Core i5 with a processor
power of 3.00 GHz and 5.00 GB of RAM, using Camera Control Pro. Each grain was placed on a
plate covered by blue germination paper (Hoffman Manufacturing) to create a contrasting back-
ground. A mirror at 45 degrees was used so that a superior and lateral image could be taken and the
3 perpendicular diameters (a, b, c) that compose a triaxial ellipsoid could be measured. A scalimeter
was used to associate pixel value to SI units. The volume could then be calculated indirectly (eq. 4).
67
(4)
In this experiment, c corresponds to the minor diameter observed from the lateral view given
by the mirror while a and b describe the mayor and minor diameters from the superior image.
Digital pictures were processed by a set of macros produced in ImageJ 1.50b (Schneider, Rasband
& Eliceiri, 2012). The photographs were contrasted and then transformed to a black and white
image where the beans were analyzed, using a relative L*a*b color scale to automatically separate
the grains from their background. Consequently, each grain is studied as a particle that can be
computationally approximated to an ellipse (Figure 1).
Figure 1. Digital image processing workow: (A) camera photograph, (B) Cropped area of study,
(C) Image binary conversion, (D) Image analysis
2.3 Calibration
The equipment was calibrated using an image that was generated in AutoCAD 2013 printed
in 1: 1 scale of 9 ellipses and a sphere with different diameter values and centroid tilt angles. The
printed imaged was analyzed using the digital image processing method and the ellipses parameters
that were produced from ImageJ were compared with the real values from the original picture. As
the macros try to compare individual shapes to virtual ellipses, the correlation between the theore-
tical and virtual ellipse parameters would display the precision and exactitude of the digital image
processing method. A validation was done for 100 grains in which a digital Vernier caliper, image
processing and pycnometer measurements where compared. An individual analysis was performed
with consecutive use of a Vernier caliper, image processing, and pycnometer for each grain. The
VALERIO, VARGAS Y BARBOZA: Modelling Matambú bean (Phaseolus vulgaris) hydration kinetics...
68
results were compared using tests for the signicance of the Pearson product-moment correlation
coefcient and one-way ANOVA test with post-hoc Tukey test.
2.4 Determination of volumetric expansion during soaking
Once the computer digital image processing system was calibrated, this was used to calculate
the initial and nal volumes of beans during expansion. Separate grains were hydrated in divided
petri dishes and were supercially dried, measured and then removed. Individual measurements
of grain with 10 repetitions were made before and after pre-dened time intervals. Increasing time
lapses were used to calculate the volumetric expansion ratio for each sample, up to a total soaking
time which ranged from 1100 to 45 min. This was done for water temperatures of 20, 30, 40, 50,
60 and 70 oC that were thermostatically controlled. The volumetric expansion ratio was expressed
as the coefcient between the nal and initial volumes per individual grain.
Matambú bean volumetric expansion kinetics were evaluated using Peleg´s equation and the
sigmoidal model. For the models that appropriately described the experimental data, their parameters
were modelled as a function of the process temperature. The criteria of model selection were the
use of the Pearson Product Moment correlation coefcient and consistency regarding the process
boundary conditions.
Additionally, the maximum volume expansion ratio was compared across different tempera-
tures, based on samples of 20 grains that were exposed extended time periods and covering up to
24 hours.
3. RESULTS AND DISCUSSION
3.1 Validation of Digital Image Processing (DIP) equipment
The error caused by direct measurement of distance within the equipment used to determine
the diameters of the approximate ellipse to beans was studied and found to be low (Figure 2). For
this, high correlation coefcients were found (Table 1) where the Pearson coefcient showed a
signicant correlation (p <0.01). No differences were found when comparing scalimeter values for
the real image and the reected by the mirror (p< 0.05) for 10 measurements of 1 mm, as such it
can be assumed that that the virtual and real image are the same for the 45-degree position.
A signicant correlation (p< 0.05) was also found when comparing the diameters a, b, and c for
the Vernier caliper and the DIP methods (Figure 3). Despite having a signicant correlation (p<
0.05) (Figure 4) and no signicant differences between the average values (p< 0.05), the volume
measurements displayed an error when using the triaxial ellipsoid as an approximation for the
shape of Matambú beans. The average error for both the Vernier caliper and digital image pro-
cessing remains as a 16 % lower than the reference value provided by the pycnometer (Table 2).
An explanation can be that the flattened ellipsoid shape does not consider the size generated by
the curvature of the grain as was observed for barley (Walker & Panozzo, 2012).
Ingeniería 30 (2): 63-76, julio-diciembre, 2020. ISSN: 2215-2652. San José, Costa Rica DOI 10.15517/ri.v30i2.39588
69
Figure 2. Comparison of the centroid tilt angle, minor and major
diameters of the theoretical ellipses versus experimental data
Table 1. Theoretical versus experimental linear regression constants for digital
image analysis of drawn ellipses with known parameters (Figure 2)
Variable Equation f(x) R
2
Major Diameter 0.069+ 1.008x 0.999
Minor diameter -0.006+ 1.011x 0.999
Centroid Tilt Angle 4.467+ 0.982x 0.999
Table 2. Analysis of Vernier and Digital Image Processing (DIP)
indirect volume measurement accuracy using pycnometer as reference
Instrument Volume (mL) Error (decimal)
Vernier 0.131 A ± 0,090 15.9 A ± 12,9
DIP 0.123 A ± 0,032 15.8 A ± 10,9
Pycnometer 0.140 A ±0,032 N/A
* Means with a common letter are not signicantly different for different measurement methods (p> 0.05)
VALERIO, VARGAS Y BARBOZA: Modelling Matambú bean (Phaseolus vulgaris) hydration kinetics...
70
Figure 3. Diameter measurements a, b and c of Vernier caliper versus Digital Image Processing method
Figure 4. Measurement of volumes of pycnometer versus Digital Image Processing method
Ingeniería 30 (2): 63-76, julio-diciembre, 2020. ISSN: 2215-2652. San José, Costa Rica DOI 10.15517/ri.v30i2.39588
71
3.2 Models for volumetric expansion ratio
It was found that, after the grains gain their maximum mass and reach equilibrium, there are
no signicant differences for the average values in the volumetric expansion ratio (p> 0.05) and
the temperature range studied (Table 3). It is determined that for the studied bean variety, there is
an equilibrium point of 2.427, which is a constant calculated from the average of the maximum
volumetric expansion ratios of the analyzed temperatures. With this consideration, Peleg´s equation
for volume ratio did not show acceptable results due to the impossibility of forming an equilibrium
point (Figure 5.A, R
2
> 0.9). This effect is corrected on the sigmoidal model ( Figure 5.B, R
2
> 0.9).
Table 4 and 5 show the regression constants for the described equations.
Figure 5. Volumetric expansion for (A) Peleg and (B) sigmoid equations, as well as parameter variations
with temperature for (C) Peleg and (D) sigmoid models
VALERIO, VARGAS Y BARBOZA: Modelling Matambú bean (Phaseolus vulgaris) hydration kinetics...
72
Table 3. Volumetric expansion equilibrium values for the
temperature range studied
T(
o
C) V
eq
20 2,339 ± 0,247
30 2,539 ± 0,447
40 2,558 ± 0,295
50 2,346 ± 0,387
60 2,443 ± 0,151
70 2,337 ± 0.44
Total Average 2,427
Table 4. Regression constants for Peleg´s equation
Temperature (
o
C) Parameter Coefcient
Standard
Error
t-value P-value R
2
Residuals
20
k
1
92,947 20,192 4,603 <0,0001
0,900 0,380
k
2
0,514 0,042 12,235 <0,0001
30
k
1
131,173 26,868 4,882 <0,0001
0,972 0,087
k
2
0,470 0,038 12.39 <0,0001
40
k
1
36,374 11,481 3,168 0,003
0,977 0,053
k
2
0,570 0,051 11,171 <0,0001
50
k
1
46,716 14,713 3,175 0,003
0,984 0,028
k
2
0,429 0.08 5,378 <0,0001
60
k
1
25,208 7,378 3,417 0,001
0,969 0,073
k
2
0,504 0,047 10,694 <0,0001
70
k
1
29,549 9,180 3,219 0,002
0,969 0,068
0,450 0,066 6,879 <0,0001
Ingeniería 30 (2): 63-76, julio-diciembre, 2020. ISSN: 2215-2652. San José, Costa Rica DOI 10.15517/ri.v30i2.39588
73
Table 5. Regression constants for sigmoid equation
Temperature (
o
C) Parameter Coefcient
Standard
Error
t-value P-value R
2
Residuals
20
M
eq
2.432 0.024 99.906 <0,0001
0.959 0.156
(min)
80 8.047 9.942 <0,0001
K 0.007 0 16.406 <0,0001
30
M
eq
2.432 0.024 99.906 <0,0001
0.984 0.051
(min)
34.226 5.763 5.94 <0,0001
k 0.009 0.001 14.05 <0,0001
40
M
eq
2.432 0.024 99.906 <0,0001
0.968 0.074
(min)
10.235 4.023 2.544 0.014
k 0.019 0.002 10.355 <0,0001
50
M
eq
2.432 0.024 99.906 <0,0001
0.994 0.011
(min)
15.641 3.453 4.529 <0,0001
k 0.024 0.002 10.144 <0,0001
60
M
eq
2.432 0.024 99.906 <0,0001
0.997 0.007
(min)
11.547 2.226 5.188 <0,0001
k 0.036 0.004 9.19 <0,0001
70
M
eq
2.432 0.024 99.906 <0,0001
0.986 0.031
(min)
11.259 2.348 4.795 <0,0001
0.034 0.003 10.638 <0,0001
The variation of both Peleg (Figure 5.C) and sigmoidal equation´s parameters (Figure 5.D) was
studied for the temperature range of 20-70
o
C. Only on the second model a clear tendency is
found. The kinetic parameter increases proportionally with temperature while the delay time
decays and remains as a constant due to the lack of any sigmoidal form on the expansion kinetics.
As an indicator of the process, the kinetic parameter was described according to the Arrhenius
modied equation (eq. 5, R
2
=0.963):
VALERIO, VARGAS Y BARBOZA: Modelling Matambú bean (Phaseolus vulgaris) hydration kinetics...
74
(5)
While the delay time was described with an exponential decay equation (eq. 6, R
2
=0,920):
(6)
Which both combined produce the general equation (eq. 7):
(7)
The obtained results serve as a tool to predict volumetric expansion in a way that can be used
for food process design and grains technology. More importantly, it is a basis that will be used
in future studies to describe mass expansion using more complex and sophisticated models that
require knowledge of water concentration, volume, and supercial area. The presented results
might not be as accurate as 3D reconstructions methods (Roussel, Geiger, Fischbach, Jahnke &
Scharr, 2016) as it remains dependent of geometric approximations; however, these are neces-
sary as they are requirements for an statistical shape generalization for grains.
3. CONCLUSIONES
A tool based on image analysis (DIP) to determine the dimensions of beans in the three orthogonal
positions was developed. The results are statistically similar to measurements made with Vernier and
with the advantage of eliminating the bias produced by the operator. This provides an innovative,
low cost methodology that enables an analysis that otherwise would be impractical. The operational
time of the described method for 10 grains requires seconds of computer camera control and data
processing whereas a Vernier and a pycnometer would require, respectively, 5 minutes and half
an hour to provide the same amount of results. Despite that the system presented in this paper is
accurate, the geometric assumption that the Matambú bean has an ellipsoid shape has a difference
or error of 16 %. However, the previously described error is not signicantly different according
to ANAVA tests. This occurs due to the variance that tends to exist in biological materials, but it
does not invalidate the notion that the attened ellipsoid shape does not consider the size generated
by the curvature of the grain.
Peleg’s equation is not a valid model to describe beans volumetric expansion while the
sigmoid equation provides a good general description of the process, despite not having an initial
low absorption phase. Temperature is not a factor on the volume kinetics equilibrium point, which
allows for the sigmoid model to be used as a tool to predict volume change and support food process
Ingeniería 30 (2): 63-76, julio-diciembre, 2020. ISSN: 2215-2652. San José, Costa Rica DOI 10.15517/ri.v30i2.39588
75
design and quality control (Shaur Rahman, 2005). The volumetric expansion curves produced
in this paper can now be used as a starting point to understand more complex processes used for
grain quality. Soaking can reduce cooking time as it is dependent on bean humidity and variety
from which even color remains an important distinction (Kinyanjui et al., 2017). Determining the
specic point in the volume kinetics, which can optimize any post-harvest process, is a task for
future studies. As this paper focuses on studying a relatively recent variety of bean that is important
for Costa Rican culture, the previous results are uniquely important.
NOMENCLATURE
a,b,c major diameters (mm)
w.b. wet basis
E
a
activation energy (kJ mol
-1
)
k kinetic constant sigmoid (s
-1
, min
-1
, h
-1
)
k
1
absorption rate Peleg (w.b.
-1
h)
k
2
absorption capacity Peleg (w.b.
-1
)
M(t) moisture at time t (w.b)
M
0
initial moisture content (w.b)
M
eq
equilibrium moisture content (w.b)
M(t) moisture at time t
P pre-exponential factor
R universal gas constant (8.314 kJ mol
-1
K
-1
)
R
2
determination coefcient
t time (s, min, h)
T temperature (
o
C, K)
delay time sigmoid (s, min, h)
V volume (m
3
)
V
0
initial volumetric expansion ratio
V
eq
equilibrium volumetric expansion ratio
V(t) volumetric expansion ratio at time t
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