#686_Ulloa_bozza


	

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PAPER 
 
 
 
 
 

MODELLING OF HYDRATION OF BEAN 
(PHASEOLUS VULGARIS L.): 

EFFECT OF THE LOW-FREQUENCY ULTRASOUND 
 
 
 

L.R. LÓPEZ LÓPEZa, J.A. ULLOA*a,b, P. ROSAS ULLOAb, J.C. RAMÍREZ RAMÍREZc, 
Y. SILVA CARRILLOd and A. QUINTERO RAMOSe 

aPosgrado en Ciencias Biológico Agropecuarias. Unidad Académica de Agricultura, Universidad Autónoma 
de Nayarit, Carretera Tepic-Compostela Km 9, CP 63780 Xalisco, Nayarit, México 

bCentro de Tecnología de Alimentos, Universidad Autónoma de Nayarit, Ciudad de la Cultura Amado 
Nervo, C.P. 63155 Tepic, Nayarit, México 

cUnidad Académica de Medicina Veterinaria y Zootecnia, Universidad Autónoma de Nayarit, Carretera a 
Chapalilla Km 3.5, CP 63700 Compostela, Nayarit, México 

dUnidad Académica de Agricultura, Universidad Autónoma de Nayarit, Carretera Tepic-Compostela Km 9, 
CP 63780 Xalisco, Nayarit, México 

eDepartamento de Investigación y Posgrado, Facultad de Ciencias Químicas, Universidad Autónoma de 
Chihuahua, Circuito Universitario s/n, Campus Universitario No. 2, CP 31240 Chihuahua, Chihuahua, 

México 
*Corresponding author. Tel. +52 3112118851; Fax +52 3112118861 

E-mail address: arulloa5@gmail.com 
 
 
 
 
 

ABSTRACT 
 
Beans of six varieties were soaked in distilled water at 30 °C and exposed to ultrasound at 
powers of 5, 12 or 19 W, in addition to treatment control without ultrasound to attain the 
equilibrium moisture.  From four model studied, the Weibull model presented the best fit 
(R2 0.986-0.999) for the experimental data of the hydration kinetics. Soaking time was 
reduced from 52.6 % to 77.2 %, while the effective diffusivity was increased from 2.25 
times to 3.50 times at 19 W, depending on the bean variety. The ultrasound power 
improved the hydration capability being suitable for possible use in industrial 
applications. 
 
 
 

Keywords: beans, hydration, mathematical modelling, ultrasound, water diffusivity, Weibull´s model 
 



	

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1. INTRODUCTION 
 
Common beans (Phaseolus vulgaris L.) are a grain legumes that belong to the family of 
Fabaceae. They provide an affordable source of protein (16-33%), almost two to three times 
that of cereals), and also are a rich source of dietary fibre, starch, minerals and vitamins 
(MKANDA et al., 2007). Bioactive compounds present in common bean have been 
associated with the prevention and/or regulation of chronic degenerative diseases such as 
obesity, diabetes, coronary heart disease and cancer (CAMPOS-VEGA et al., 2010; PLANS 
et al., 2013). 
On the other hand, the World Health Organization has issued several recommendations 
for reducing overweight, obesity and cardiovascular diseases that also are likely to reduce 
the risk of diet-related diseases, such as type 2 diabetes and obesity. These 
recommendations include the achievement of adequate intakes of non-starch 
polysaccharides through regular consumption of wholegrain cereals, legumes, fruits and 
vegetables (WHO/FAO, 2003). Of the leguminous plants consumed by humans, one of 
those of greatest importance on a worldwide level is the common bean (OLMEDILLA-
ALONSO et al., 2013).  Cooked dry beans along with whole grains have been emphasized 
as the primary nutritional shifts in food intake patterns to a plant-based diet by the 2010 
Dietary Guidelines Advisory Committee (CAMPOS-VEGA et al., 2012).  
Before the cooking step, beans are hydrated in water until maximum weight is reached 
(ULLOA et al., 2016). Hydration capacity is dependent on the ease of water absorption 
through the seed coat to the cotyledons (MKANDA et al., 2007). However, the soaking 
process is a time-consuming step, requiring approximately 12 h at room temperature 
(PIERGIOVANNI, 2011), and many attempts have been directed towards shortening it 
(Abu-Ghannam and McKenna, 1997).   
Ultrasound power is a novel technology in the food industry, and research on its 
application is a rapidly growing field. Ultrasonic waves can cause a rapid series of 
alternative compressions and expansions similar to a sponge when it is squeezed and 
released repeatedly, a phenomenon known as cavitation (CHEMAT et al., 2011). 
Ultrasound cavitation results in the occurrence of microstreaming, which enhances heat 
and mass transfer (CÁRCEL et al., 2012). 
Ultrasound applications were reported to enhances the hydration of chickpeas (YILDIRIM 
et al., 2011), sorghum grains (PATERO and AUGUSTO, 2015), and navy beans (GHAFOOR 
et al., 2014). However, as soaking conditions vary depending on the particular legume or 
food material, it is necessary for practical applications to characterise and optimise these 
conditions. 
Therefore, the objective of the present study was to evaluate the effect of low-frequency 
ultrasound on the kinetics and modeling of the hydration of the six varieties of the most 
consumed beans in Mexico. 
 
 
2. MATERIALS AND METHODS 
 
2.1. Material 
 
Common bean seeds from six varieties, which are highly consumed in Mexico 
(RODRÍGUEZ-LICEA et al., 2010) and classified as most preferred (Azufrado, Mayacoba, 
Pinto, Peruano Bola, Flor de Mayo and Negro Jamapa), were used for this study and were 
obtained from the Mercado de Abastos, located in Tepic, Nayarit, Mexico. These food 
legumes were separated from broken, small and split. Seeds were cleaned and size-graded 



	

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manually. Samples were stored in hermetically sealed bags inside closed plastic jars at 
room temperature (25 °C) in a dark room. 
 
2.2. Chemical and morphological characterization of the bean seeds 
 
The chemical composition (moisture, protein, fat, ash, and carbohydrates) of the beans was 
determined following the official methods of the AOAC (2002). For the morphological 
characterization, 100 beans of each variety were selected, and the weight (we), length (l), 
width (w) and depth (d) were measured to determine the geometric mean diameter (Gm), 
arithmetic mean diameter (Am), square mean diameter (Sm) and the radius of an equivalent 
sphere (r) according to the methods reported by GAFHOOR et al. (2014). 
 

2.3. Seed coat content 
 
To determine the percentage of the coat of the seeds, the TIZAZU and EMIRE (2010) 
method with some modifications was used. Thirty bean seeds were soaked in distilled 
water for 8 h, and the coat was removed manually, separating it from the cotyledon. The 
seed coats collected were dried in an oven at 60° C for 24 h, followed cooling in a 
desiccator. It was then weighed and the percentage of seed coat was calculated. 
 
2.4. Ultrasound soaking treatments 
 
A sample of 5 g of beans was used in each one of the three replicates. The bean samples 
were soaked in a 100-mL beaker with 50 mL of distilled water and exposed to ultrasound 
at powers of 5, 12 or 19 W (20 kHz, 30 °C), using a GE-130 ultrasonic processor (Cole 
Parmer, Vernon Hills, Connecticut, Illinois) provided of a titanium probe of 6 mm. At 
specific time intervals (10 min for ultrasound treatments and 30 min for control, until 
stabilization), the seeds were removed from the water, drained, superficially blotted with 
absorbent paper, weighed and returned to the water. 
 
2.5. Modeling the kinetics of hydration 
 
For fitting the moisture uptake of soaked bean, four models were used to estimate the 
parameters associated with each model. The list of the models, and the respective 
equations used in this study, is presented in Table 1. The best fitted model was determined 
by the highest coefficient of determination (R2) and the lowest values of the root mean 
square error (RMSE) and chi-square (χ2) amongst the predicted and experimental results 
(COX et al., 2012). 
	
2.6. Effective diffusivity (Deff) 
 
Deff was determined according to the method reported by KAPTSO et al. (2008) with the 
following equation: 
 
 𝑐 = !

!!!""
!!

 Eq. 1 
 
Where Deff (m2/s) is the effective diffusivity, c (s-1) is the rate constant of the water 
absorption of the mathematical model with the best fit and r (m) is the value of radius of 
the equivalent sphere. 
	



	

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Table 1. Models used to describe moisture content and rate of moisture uptake by effect of ultrasound. 
 

Model and reference Equation 

Peleg (PELEG, 1988) M! = M! +
!

!!!!!!
           (2) 

Sigmoid (LEAL-OLIVEIRA et al., 2013) M! =
!!

!!!"# !!(!!!)
           (3) 

First order (GHAFOOR et al., 2014) MR = 1 − exp (−k!t)       (4) 

Weibull (ZURA et al., 2013) MR = 1 − exp − !
!

!
    (5) 

 
MR is the rate of moisture uptake, and is given by the equation: 𝑀𝑅 = !!!!! 

!!!!!
 

where M0 is the initial moisture content of the bean, Mt is the moisture content of bean at time t, and Me is the 
final moisture content at equilibrium. 
t is the hydration duration (in min), and the variables c1, c2, k1, k, τ, α and β are the coefficients used in 
nonlinear regression analysis with the various models. 
 
 
2.7. Statistical analysis 
 
One-way analysis of variance and Tukey tests were performed to determine the difference 
between the values of seed morphological traits (we, l, w, d, Gm, Am, Sm and r), chemical 
composition (moisture, proteins, fats, ashes and total carbohydrates), %Cs and the kinetic 
parameters of the soaking treatments (Me and Deff) of the different varieties of beans, using 
Statgraphics Plus 5.0 statistical package (Statistical Graphics Corp., MD, USA). 
Significance level were tested at P < 0.05. 
 
 
3. RESULTS AND DISCUSSION 
 
3.1. Chemical and morphological characterization of the bean seeds 
 
Table 2 shows the results of we, l, w, d, Gm, Am, Sm and r of the six bean varieties studied. 
According to the obtained results, the Azufrado bean variety presented the lowest values 
in the majority of the morphological characteristics studied, and the Mayocoba and Flor de 
Mayo varieties presented the highest values. Although Flor de Mayo and Azufrado beans 
had the highest r (0.41 cm) and the lowest r (0.30 cm), respectively, the majority of the 
varieties of bean used in this study presented a higher r value in comparison with the 
Navy bean (GHAFOOR et al., 2014). 
Table 3 shows the chemical composition of the six varieties of beans studied. The values of 
the proximal composition of the six bean varieties of this study are similar to those 
reported by WANI et al. (2015) for other bean varieties, with the exception of the moisture 
content for the Mayocoba variety, which was the highest, and the fat content for the Pinto 
variety, which was the lowest. 
 
3.2. Kinetics of water absorption 
 
The water absorption kinetics of the different varieties of bean studied are shown in Fig. 1. 
The time to attain the Me ranged from 330 min to 660 min depending of the bean variety, 
being the shortest time for Mayacoba and longest time for Negro Jamapa. According to 
this finding, exposure to ultrasound reduced the soaking time of the bean; however, at a 
higher power of ultrasound, the time lapse was even lower. From the six different varieties 
of bean studied, after of the ultrasound treatments at 5W, 10 W and 19 W, the shortest and 
longest soaking times were 173 min and 280 min, 108 and 360 min, and 123 min and 313 



	

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min, respectively, corresponding to Mayacoba and Peruano Bola, Mayacoba and Negro 
Jamapa, and Pinto and Negro Jamapa. 
 
 
Table 2. Characteristics of the six varieties of common bean (Phaseolus vulgaris) seeds. 
	

Parameters 
Bean varieties 

Mayocoba Azufrado Pinto Peruano Bola Flor de Mayo Negro Jamapa 
we (g) 0.39±0.03a 0.25±0.02d 0.36±0.05b 0.32±0.04c 0.35±0.06b 0.24±0.04d 
l (cm) 1.29±0.06a 1.08±0.06c 1.25±0.08b 1.03±0.06d 1.25±0.01b 0.96±0.07e 
w (cm) 0.70±0.04cd 0.66±0.03e 0.73±0.05b 0.72±0.03bc 0.76±0.05a 0.69±0.08d 
d (cm) 0.59±0.05a 0.47±0.04d 0.56±0.05b 0.61±0.03a 0.52±0.04c 0.53±0.04c 
Gm (cm) 0.82±0.04ª 0.70±0.03

d 0.79±.0.04ab 0.76±0.03c 0.79±0.06b 0.71±0.04d 
Am (cm) 0.86±0.04ª 0.74±0.03

d 0.85±0.05ab 0.78±0.03c 0.84±0.06b 0.73±0.04b 
Sm (cm) 0.41±0.01

d 0.37±0.01e 0.41±0.01d 0.84±0.02b 0.87±0.04a 0.80±0.03c 
r (cm) 0.35±0.01d 0.30±0.00e 0.34±0.02d 0.39±0.01b 0.41±0.02a 0.37±0.02c 

 
Values are given as means±standard deviation (n = 100). Different superscript letters in the same row 
indicate significant differences (P < 0.05).  
we: weight; l: length; w: width; d: depth; Gm: geometric mean diameter; Am: arithmetic mean diameter; Sm: 
square mean diameter; r: radius of an equivalent sphere. 
 
 
Table 3. Seed proximate composition of the six varieties of common bean (Phaseolus vulgaris). 
	

Component 
Bean varieties 

Mayocoba Azufrado Pinto Peruano Bola Flor de mayo Negro Jamapa 
Moisture (%) 12.8 ±0.1a 11.2±0.1bc 11.0±0.4bcd 11.4±0.3b 9.5±0.1e 8.9±0.1f 
Fat (%) 1.6±0.2bc 2.1±0.2b 0.6±0.1d 4.7±0.5a 1.0±0.1cd 1.5± 0.1bcd 
Ash (%) 4.5±0.1a 4.8±0.2a 4.1±0.1a 4.5±0.1a 4.0±0.2a 4.6±0.1a 
Protein (%, N x 6.25) 23.5± 0.5a 23.6±0.1a 23.9±0.5a 23.8±0.4a 23.2±0.2a 23.9±0.5a 
Total carbohydrates 
(%) 57.6±0.5 58.3±0.2 60.4±0.5 55.6±0.5 62.3±0.2 61.1±0.5 

 
Values are given as means±standard deviation (n = 3). Different superscript letters in the same row indicate 
significant differences (P < 0.05). 
 
 
On the other hand, the Azufrado and Peruano Bola bean varieties presented an initial lag 
phase or lateness (period with a low water absorption rate) for the 5 W ultrasound 
treatment and the control treatment, whereas the Pinto, Flor de Mayo and Negro Jamapa 
bean varieties presented initial lag phase for the ultrasound treatments of 5 W and 12 W, 
as well as the control treatment.  
The behavior of the hydration kinetics of this study has been observed in the conventional 
soaking (25-55° C) of common bean by PIERGIOVANNI (2011), who classified the bean 
varieties into three groups as a function of the rate of hydration (fast, intermediate and 
slow); the fast and intermediate bean varieties did not present an initial lag phase, but the 
slow hydration bean varieties did have an initial lag phase. According to this classification, 
the Mayocoba bean variety can be considered to have a rapid hydration rate, in contrast to 
the rest of the bean varieties studied, even though with the application of the ultrasound 
power, especially at 19 W, the initial lag phase of the bean hydration of the varieties of 



	

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Azufrado, Pinto, Peruano Bola and Flor de Mayo was reduced. KAPTSO et al. (2008) 
reported that the application of ultrasound for soaking has an effect similar to increasing 
the soaking temperature, thus overcoming the defect of low water absorption observed in 
the seeds. 
 
 

 
 
 
Figure 1. Water absorption kinetic of (a) Mayocoba, (b) Azufrado, (c) Pinto, (d) Peruano Bola, (e) Flor de 
Mayo and (f) Negro Jamapa bean varieties during soaking at different ultrasound powers. Solid lines 
represent the corresponding Weibull model fitted to the experimental data. 
 
 
 
The statistical parameters generated from the application of the different mathematical 
models (Weibull, Peleg, First Order, and Sigmoid) describing the kinetics of water 
absorption for the bean varieties are presented in Table 4. In general, the Sigmoid model 



	

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presented higher values of R2 (0.998) and lower χ2 (0.000) and RSME (0.019) than the Peleg 
and First Order models (Table 4); in addition, Sigmoid model can describe the initial lag 
phase followed by a high rate absorption phase and, finally, a stationary phase (LEAL-
OLIVEIRA et al., 2013). However, the Weibull model presented the best fit, giving the 
highest values of R2 (0.986-0.999) and the lowest of χ2 (0.000-0.002) and RSME (0.013-0.044), 
in accordance with MARABI et al. (2003), who found that such model is one of the best for 
describing the kinetics of food hydration. 



	

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Table 4. Statistical parameters of the mathematical models fitted to the kinetics of hydration of six bean varieties. 
 

Model Ultrasound power (W) 
Mayocoba Azufrado Pinto Peruano Bola Flor de Mayo Negro Jamapa 

R2 χ2 RSME R2 χ2 RSME R2 χ2 RSME R2 χ2 RSME R2 χ2 RSME R2 χ2 RSME 
Sigmoidal Control 0.961 0.003 0.050 0.997 0.001 0.023 0.997 0.001 0.027 0.989 0.002 0.035 0.991 0.002 0.036 0.993 0.001 0.016 
 5 0.970 0.002 0.046 0.998 0.000 0.019 0.996 0.001 0.024 0.996 0.001 0.022 0.987 0.002 0.039 0.996 0.001 0.020 
 12 0.983 0.002 0.040 0.996 0.001 0.024 0.998 0.000 0.019 0.994 0.001 0.029 0.997 0.001 0.020 0.995 0.001 0.025 
 19 0.985 0.002 0.036 0.996 0.001 0.024 0.997 0.001 0.023 0.985 0.002 0.040 0.992 0.001 0.034 0.995 0.001 0.028 
 
First Order 

 
Control 0.989 0.001 0.026 0.971 0.006 0.070 0.991 0.002 0.046 0.961 0.006 0.070 0.959 0.006 0.076 0.906 0.016 0.123 

 5 0.990 0.001 0.026 0.921 0.013 0.110 0.940 0.009 0.091 0.949 0.007 0.084 0.981 0.002 0.046 0.933 0.010 0.095 

 12 0.966 0.004 0.056 0.929 0.013 0.108 0.922 0.014 0.112 0.943 0.009 0.092 0.966 0.005 0.069 0.911 0.014 0.113 
 19 0.991 0.001 0.028 0.949 0.009 0.090 0.951 0.009 0.087 0.956 0.007 0.080 0.977 0.004 0.059 0.942 0.008 0.087 
 
Peleg 

 
Control 0.994 0.000 0.020 0.977 0.004 0.062 0.994 0.002 0.038 0.973 0.004 0.059 0.962 0.006 0.075 0.959 0.007 0.081 

 5 0.979 0.002 0.038 0.961 0.007 0.077 0.979 0.003 0.053 0.972 0.004 0.062 0.986 0.002 0.039 0.988 0.002 0.039 
 12 0.948 0.006 0.070 0.975 0.005 0.064 0.971 0.005 0.067 0.959 0.007 0.078 0.986 0.002 0.0433 0.987 0.002 0.041 
 19 0.979 0.002 0.044 0.980 0.004 0.056 0.979 0.004 0.057 0.970 0.005 0.059 0.993 0.001 0.031 0.993 0.001 0.031 
 
Weibull 

 
Control 0.995 0.000 0.017 0.996 0.001 0.022 0.999 0.000 0.013 0.997 0.000 0.0171 0.995 0.001 0.025 0.998 0.001 0.018 

 5 0.991 0.001 0.025 0.997 0.001 0.022 0.998 0.000 0.016 0.999 0.001 0.010 0.997 0.001 0.018 0.991 0.001 0.033 

 12 0.986 0.002 0.036 0.998 0.000 0.019 0.999 0.000 0.014 0.996 0.001 0.0213 0.996 0.001 0.021 0.991 0.002 0.035 
 19 0.993 0.001 0.025 0.999 0.000 0.015 0.999 0.000 0.013 0.997 0.000 0.0163 0.993 0.001 0.033 0.984 0.002 0.044 

 



	

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Figure 1 also shows the kinetics of bean hydration fitted to the Weibull model where α is a 
scale parameter and β a shape parameter. The scale parameter defines the rate of moisture 
uptake process (α is the reciprocal of the process rate constant) and represents the time 
needed to accomplish the approximately 63% of the moisture uptake process, while the 
shape parameter is a behavior index, which depends on the process mechanism.  
The kinetic parameters generated from the application of the different mathematical 
models for evaluating the hydration kinetics of beans by effect of the ultrasound treatment 
are shown in Table 5. According to the results obtained, the constants of hydration rate, k 
and k1, of the Sigmoidal and First Order Models, respectively, were increased for effect of 
ultrasound power from 1.49 to 4.04 times and from 1.93 to 3.56 times in comparison with 
the control treatment, depending of the bean variety. Regarding Peleg’s model, the first 
constant, c1, which has been shown to be linked to the hydration was reduced for all bean 
varieties by effect of ultrasound in comparison with the control treatment. Therefore, the 
application of ultrasound and its increasing of power level improved the hydration 
properties such as reflected in the rate constants of the Sigmoidal, First Order and Peleg 
models. 
In relation to the α parameter of Weibull model, its value decreased with increasing 
ultrasound power for all the bean varieties (Table 5), in agreement with the reported 
results by GHAFOOR et al. (2014) for Navy bean.  
On the other hand, the values of kinetic parameter of hydration β of Weibull model (Table 
5) for the soaking treatments of the different varieties of beans were high (0.834-1.995). 
According to MACHADO et al. (1999), the Weibull model predicts the initial lag phase of 
the hydration kinetics when β is greater than 1, as it was observed for the different 
treatments and bean varieties in this study, except for the control treatment in the 
Mayocoba variety. 
 
3.3. Seed coat content 
 
The Flor de Mayo and Mayacoba bean varieties presented the highest %Cs (9.85±0.62 and 
8.82±0.62, respectively), followed by the Negro Jamapa (8.65±0.02), Peruano Bola 
(7.92±0.36), Azufrado (7.56±0.19) and Pinto (7.48±0.50) varieties bean. In this study, the 
varieties of bean with the first and third highest %Cs (Flor de Mayo and Negro Jamapa) 
required a higher soaking lapse to attain Me in comparison with the fourth and fifth 
highest %Cs (Peruano Bola and Azufrado). However, the time to attain Me of the Pinto 
variety with a %Cs lower than the Mayacoba variety was longer (Fig. 1).  
On the other hand, the varieties of bean with a higher %Cs presented a higher coat rupture 
by effect of the ultrasound (Table 6). In the varieties of Azufrado, Pinto, Peruano Bola, Flor 
de Mayo and Negro Jamapa, the percentage of seeds with a higher percentage of coat 
rupture increased as the ultrasound power was higher (5, 12 and 19 W). The Negro 
Jamapa and Flor de Mayo varieties presented coat rupture in all soaking treatments, 
including the control. The results suggest that the percentage of the coat rupture depended 
on bean variety and the ultrasound power used in the soaking process, agreeing with the 
report of PAN et al. (2010). 
 



	

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Table 5. Kinetic parameters for the models fitted to the hydration kinetics data of six bean varieties. 
 

Model Soaking treatment 
Bean varieties 

Mayocoba Azufrado Pinto Peruano Bola Flor de Mayo Negro Jamapa 

Sigmoidal Control k = 5.351x10
-2 

τ = 26.906 
k = 1.740x10-2 
τ = 104.356 

k = 1.261x10-2 
τ = 131.756 

k = 2.034x10-2 
τ = 87.458 

k = 1.803x10-2 
τ = 114.714 

k = 1.111x10-2 
τ = 225.728 

 5 W k = 5.433x10
-2 

τ = 24.634 
k = 2.666x10-2 
τ = 84.062 

k = 1.900x10-2 
τ = 104.513 

k = 2.571x10-2 
τ = 76.631 

k = 2.267x10-2 
τ = 71.294 

k = 1.436x10-2 
τ = 139.061 

 12 W 
k = 11.354x10-2 
τ = 16.368 

k = 3.920x10-2 
τ = 54.383 

k = 2.681x10-2 
τ = 82.217 

k = 3.739x10-2 
τ = 56.192 

k = 2.754x10-2 
τ = 65.193 

k = 1.595x10-2 
τ = 143.056 

 19 W k = 15.757x10
-2 

τ = 10.395 
k = 5.020x10-2 
τ = 39.387 

k = 5.100x10-2 
τ = 38.667 

k = 4.524x10-2 
τ = 43.394 

k = 3.540x10-2 
τ = 46.346 

k = 1.659x10-2 
τ = 110.453 

First Order Control k1 = 2.458x10
-2 k1 = 0.750x10

-2 k1 = 0.576x10
-2 k1 =0.8445x10

-2 k1 = 0.694x10
-2 k1 = 3.678x10

-3 
 5 W k1 = 2.711x10

-2 k1 = 0.945x10
-2 k1 = 0.742x10

-2 k1 = 1.051x10
-2 k1 = 1.060x10

-2 k1 = 5.729x10
-3 

 12 W k1 = 4.425x10
-2 k1 = 1.460x10

-2 k1 = 0.965x10
-2 k1 = 1.422x10

-2 k1 = 1.219x10
-2 k1 = 5.575x10

-3 
 19 W k1 = 6.848x10

-2 k1 = 2.000x10
-2 k1 = 2.050x10

-2 k1 = 1.808x10
-2 k1 = 1.703x10

-2 k1 = 7.121x10
-3 

 
Peleg Control 

c1 = 23.619 
c2 = 0.959 

c1 = 123.560 
c2 = 0.568 

c1 = 160.676 
c2 = 0.569 

c1 = 100.512 
c2 = 0.626 

c1 = 123.575 
c2 =0.558 

c1 = 329.984 
c2 = 0.321 

 5 W c1 = 30.503 c2 = 0.884 
c1 = 123.423 

c2 = 0.374 
c1 = 156.812 
c2 = 0.3657 

c1 = 102.118 
c2 = 0.471 

c1 = 81.568 
c2 = 0.564 

c1 = 223.034 
c2 = 0.274 

 12 W c1 = 18.995 c2 = 0.818 
c1 = 79.673 
c2 = 0.328 

c1 = 123.713 
c2 = 0.3115 

c1 = 69.255 
c2 = 0.505 

c1 = 79.689 
c2 = 0.468 

c1 = 234.070 
c2 = 0.159 

 19 W c1 = 10.078 c2 = 0.868 
c1 = 54.001 
c2 = 0.402 

c1 = 51.482 
c2 = 0.4301 

c1 = 54.229 
c2 = 0.506 

c1 = 52.746 
c2 = 0.475 

c1 = 163.072 
c2 = 0.297 

 
Weibull Control 

α = 42.481 
β = 0.834 

α = 141.833 
β = 1.493 

α = 181.544 
β = 1.336 

α =124.212 
β = 1.372 

α = 150.984 
β = 1.534 

α = 286.092 
β = 1.995 

 5 W α = 37.404 β = 1.076 
α = 110.515 
β = 1.838 

α = 139.957 
β = 1.607 

α = 102.955 
β = 1.593 

α = 96.963 
β = 1.259 

α = 182.245 
β = 1.6121 

 12 W α = 23.207 β = 1.514 
α = 71.718 
β = 1.732 

α = 107.840 
β = 1.788 

α = 74.790 
β = 1.660 

α = 86.939 
β = 1.425 

α = 183.437 
β = 1.880 

 19 W α = 15.019 β = 1.159 
α = 52.644 
β = 1.599 

α = 51.570 
β = 1.588 

α = 58.377 
β = 1.545 

α = 62.154 
β = 1.287 

α = 146.326 
β = 1.483 

 
 



	

Ital. J. Food Sci., vol 29, 2017 - 298 

3.4. Equilibrium moisture content (Me) 
 
The varieties of beans studied presented a Me that oscillated from 107.18 to 133.99 % d.b. 
(Table 6). The Me in the soaking control treatments increased for all the bean varieties, 
except for Peruano Bola (Table 6). The effect caused by the ultrasound power in the bean 
hydration is similar to the effect caused by the increase of temperature for the 
conventional soaking of bean (SHAFAEI et al., 2014), sesame seeds (KHAZAEI and 
MOHAMMADI, 2009), rice (CHEEVITSOPON and NOOMHORM, 2011), and Botswana 
Bambara (JIDEANI and MPOTOKWANA, 2009). 
 

 
Table 6. Effect of soaking treatment on the hydration properties of beans at 30 °C. 
 

Variety/Property 
Soaking treatments 

Control 5 W  12 W 19 W 
Mayocoba     
Me (% g/g d.b.) 110.08b 107.18c 111.46ab 113.98a 
Rupture of coat (%) 0.0a 0.0a 0.0a 0.0a 
Deff (x 10

-10 m2/s) 4.845c 5.501c 8.852b 13.717a 

Azufrado     
Me (% g/g d.b.) 124.47b 124.32b 128.60a 128.38a 
Rupture of coat (%) 0.0b 0.0b 0.0b 5.0a 

Deff (x 10
-10 m2/s) 0.380d 0.488c 0.752b 1.026a 

Pinto     
Me (% g/g d.b.) 123.72b 124.46b 127.24a 127.69a 
Rupture of coat (%) 0.0b 0.0b 0.0b 16.6a 
Deff (x 10

-10 m2/s) 1.097d 1.413c 1.839b 3.847a 

Peruano Bola     

Me (% g/g d.b.) 124.47ª 125.08a 127.18a 126.96a 
Rupture of coat (%) 0.0b 0.0b 0.0b 1.5a 
Deff (x 10

-10 m2/s) 2.135b 2.616b 4.168a 4.633a 

Flor de mayo     
Me (% g/g d.b.) 125.96b 125.59b 128.39ab 133.99a 
Rupture of coat (%) 15.5c 29.0b 40.0a 49.0a 
Deff (x 10

-10 m2/s) 1.967c 3.056b 3.414b 5.025a 

Negro Jamapa     
Me (% g/g d.b.) 123.07c 124.68bc 126.22ab 128.05a 
Rupture of coat (%) 1.6b 10.0a 13.3a 21.6a 
Deff (x 10

-10 m2/s) 0.711c 1.287b 1.275b 1.60a 

 
Values are given as means±standard deviation (n = 3). Different superscript letters in the same row indicate 
significant differences (P < 0.05).  
 
 
3.5. Effect of ultrasound on the effective diffusivity (Deff) 
 
To calculate Deff (Eq. 6), the inverse of α was used (1/α) as the rate constant of water 
absorption. Table 6 shows the effects of the ultrasound treatments during the bean soaking 
process on Deff for the distinct studied varieties. 
 



	

Ital. J. Food Sci., vol 29, 2017 - 299 

The soaking treatment with ultrasound at 19 W obtained the highest values of Deff  for all 
bean varieties. The increase of Deff observed in this study as the ultrasound power was 
increased is congruent with a study on chickpeas, where the value of Deff of the soaking 
control treatment at 30 °C was 1.87 x 1010 m2/s and it was increased with the application of 
ultrasound to values of 2.10 x 1010 and 2.62 x 1010 (m2/s) for the soaking treatments at 25 
kHz and 100 W and 25 kHz and 300 W, respectively (YILDIRIM et al., 2011).  
The increase of the absorption of water during the assisted soaking process with 
ultrasound is due to the formation of microscopic channels in the grains, which reduce the 
internal resistance to mass transference (FUENTE-BLANCO et al., 2006). 
In this study we observed that increasing the ultrasound power in the soaking treatments 
of the beans generated a proportional increase on the values of Deff (Fig. 2), in similarity to 
the proportional increasing of the water absorption during soaking of bean by effect of 
increase of temperature (SHAFAEI et al., 2014). 
 

 

 
 
Figure 2. Effect of low frequency ultrasound power (20 kHz) on the Deff in the soaking of the Mayocoba (a), 
Azufrado (b), Pinto (c), Peruano Bola (d), Flor de mayo (e) and Negro Jamapa (f) bean varieties. 
 
 



	

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4. CONCLUSIONS 
 
The application of ultrasound reduced the lapse of soaking in the six varieties of beans 
studied, although the reduction depended of the bean variety and the ultrasound power. 
The soaking treatment at 19 W presented the highest values of Deff, Me and the least time to 
attain Me. However, this ultrasound treatment also presented coat rupture of the beans, in 
different proportion, except for the Mayocoba variety. Of the models used, the Weibull 
model presented the best fit for the experimental data of the hydration kinetics, in the 
different soaking treatments and for the majority of the bean varieties. 
 
 
ACKNOWLEDGEMENTS 
 
During the devolvement of this study, financial support was obtained from the Consejo Nacional de Ciencia y 
Tecnología (CONACYT) with scholarship number 550780. 
 
 
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Paper Received November 17, 2016  Accepted December 31, 2016