Effect of ultrasound treatment on the hydration kinetics and cooking times of dry beans (Phaseolus vulgaris)

Abstract

The effect of ultrasound on the hydration kinetics and cooking times of six bean varieties was studied. Samples of beans were exposed to ultrasound treatments (10, 20 and 30 min; 40 KHz, 130 W) at 30°C in addition to control treatments. In most cases, the observed maximum moisture content of the dry beans (1.16 ± 0.03 to 1.22 ± 0.02 g/g) after soaking did not vary with the bean variety and ultrasound exposure time. Effective diffusivity significantly increased (P < 0.05) up to 45 times, and the time to obtain the equilibrium moisture content was reduced by 17.6–58.8%, with respect to control treatments, when treated with ultrasound for 30 min. The percentage reduction in cooking time ranged from 5.4 to 43.0. Ultrasound treatment significantly reduced the soaking and cooking times of the bean varieties studied, but the extent of such reduction depended on the bean variety and ultrasound exposure time.

Se estudió el efecto del ultrasonido en las cinéticas de hidratación y tiempos de cocción de seis variedades de frijol. Las muestras de frijol se expusieron a tres tratamientos de ultrasonido (10, 20 y 30 min; 40 KHz 130 W) a temperatura ambiente (30°C), además de los tratamientos control sin ultrasonido. En la mayoría de los casos, el contenido de humedad máximo observado de los frijoles (1,16 ± 0,03 to 1,22 ± 0,02 g/g) después del remojo no mostró variación con la variedad de frijol y tiempo de exposición al ultrasonido. En todos los casos, las difusividades efectivas se incrementaron significativamente (P < 0,05) hasta 45 veces, mientras que el tiempo para conseguir los contenidos de humedad de equilibrio disminuyeron del 17,6 al 58,8% con respecto a los tratamientos control, cuando se aplicó el tratamiento de ultrasonido de 30 min. El porcentaje de reducción del tiempo de cocción varió de 5,4 a 43,0. El ultrasonido redujo significativamente los tiempos de remojo y cocción, pero tales reducciones dependieron de la variedad de frijol y tiempo de exposición al ultrasonido.

Keywords:

• ultrasound treatment
• dry beans; soaking
• cooking
• hydration kinetics

Palabras clave:

• tratamiento de ultrasonido
• frijol
• remojo
• cocción
• cinética de hidratación

1. Introduction

The common bean (together with corn) constitutes the staple diet for most Mexicans and has been used in the preparation of traditional dishes and meals for many years. Beans are a good source of carbohydrates, protein, dietary fibre (primarily insoluble fibre), vitamins and some minerals (to a lesser extent), and the hull contains flavonoids that act as antioxidants (Gallegos-Infante et al., 2010).

Clinical studies have shown that regular consumption of beans helps decrease the incidence and multiplicity of colon cancer, gastrointestinal tract alterations, cardiovascular diseases and diabetes (López et al., 2013). The physiological effects of dry bean consumption may be due to the presence of abundant phytochemicals, including polyphenols, which possess both anticarcinogenic and antioxidant properties (Xu & Chang, 2011). Compared to other carbohydrate sources, beans have a low glycaemic index, which may help control blood glucose levels in diabetics and people with other chronic degenerative diseases (Silva-Cristobal, Osorio-Díaz, Tovar, & Bello-Pérez, 2010).

Furthermore, beans contain several antinutritional factors that interfere with the bioavailability of nutrients and include inhibitors such as trypsin, chymotrypsin and amylase as well as phytic acid, flatulence-producing oligosaccharides, saponins and lectin. However, the contents of all of these antinutritional factors can be reduced or eliminated by certain culinary practices, such as discarding the soaking water before cooking or the use of a sodium bicarbonate or citric acid soaking solution prior to cooking heat treatment (Fernandes, Nishida, & Da CostaProença, 2010).

Bean hydration prior to cooking has been reported to have an important role in determining the cooking time and appearance as well as the extent of protein denaturation and starch gelatinisation of legumes (Güzel & Sayar, 2012). However, the soaking process is a time-consuming step, and many attempts have been directed towards shortening it. As soaking conditions vary depending on the particular legume, it is necessary for practical applications to characterise and optimise these conditions (Yildirim, Öner, & Bayram, 2010).

Power ultrasound is a novel technology in the food industry, and research on its application is a rapidly growing field. Its new uses include the following: pasteurisation, sterilisation, generation of emulsions, disruption of cells, promotion of chemical reactions, inhibition of enzymes, tenderisation of meat and modification of crystallisation (Chandrapala, Oliver, Kentish, & Ashokkumar, 2012). 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, Huma, & Khan, 2011). Ultrasound cavitation results in the occurrence of microstreaming, which enhances heat and mass transfer (Cárcel, García-Pérez, Benedito, & Mulet, 2012).

Ultrasound applications were reported to promote oligosaccharide leaching in legumes (Han & Baik, 2006), reduce the cooking time of rice (Wambura, Yang, & Wang, 2008), decrease the soaking time of chickpeas (Yildirim, Öner, & Bayram, 2011) and soften and decrease the cooking time of chickpeas (Yildirim, Öner, & Bayram, 2013). However, there are no studies on ultrasonic applications to decrease the soaking and cooking time of beans.

The aim of the present work was to study the effect of ultrasound treatment on the hydration kinetics and cooking times of the main six dry bean varieties consumed in Mexico.

2. Materials and methods

2.1. Plant materials

Bean (Phaseolus vulgaris) seeds from six varieties, which are highly consumed in México and classified as most preferred (Azufrado, Mayacoba, Flor de Mayo and Negro Jamapa) and preferred (Garbancillo and Pinto), were used for this study and were obtained from the Mercado de Abastos, located in Tepic, Nayarit, Mexico. Samples were stored in hermetically sealed bags at room temperature in a dark room. These food legumes were separated from broken, small and split seeds, as well as from dust and other excessive materials, and they were cleaned and size-graded manually. The physical properties, including the size, shape and colour, of the dry bean samples were analysed according to the methods reported by Kaptso et al. (2008) and are summarised in Table 1. The chemical composition of the dry bean samples according to the Association of Official Analytical Chemists methods (AOAC, 1995) is shown in Table 2.

Table 1. Several physical properties of dry beans.

Tabla 1. Varias propiedades físicas de los frijoles.

	Dry bean variety
	Azufrado	Mayacoba	Flor de Mayo	Negro Jamapa	Garbancillo	Pinto
Weight 100 seeds (g)	23.5 ± 0.21^e	46.2 ± 0.13^a	29.2 ± 0.14^b	17.5 ± 0.10^f	27.0 ± 0.28^c	26.4 ± 0.11^d
Length (mm)	10.91 ± 0.65^b	12.77 ± 0.69^a	12.90 ± 0.63^a	9.90 ± 0.80^c	10.37 ± 0.68^d	12.83 ± 0.67^a
Width (mm)	6.83 ± 0.37^b	7.30 ± 0.57^a	7.32 ± 0.33^a	6.34 ± 0.55^c	7.04 ± 0.43^d	7.44 ± 0.37^a
Thickness (mm)	4.58 ± 0.35^b	6.23 ± 0.44^c	5.03 ± 0.45^a	4.21 ± 0.44^d	5.35 ± 0.53^e	5.02 ± 0.38^a
Sphericity	0.64 ± 0.02^a	0.65 ± 0.05^a	0.61 ± 0.02^b	0.65 ± 0.04^a	0.70 ± 0.02^c	0.61 ± 0.02^b
Geometric diameter (mm)	6.98 ± 0.44^b	8.31 ± 0.48^c	7.80 ± 0.46^a	6.61 ± 0.58^d	7.31 ± 0.41^e	7.73 ± 0.46^a
Colour
L*	54.33 ± 3.13^a	63.87 ± 3.80^b	91.23 ± 3.40^c	84.50 ± 0.35^d	60.82 ± 3.46^e	96.18 ± 3.04^f
a*	2.01 ± 1.42^a	−1.59 ± 2.25^c	6.34 ± 1.59^d	−0.21 ± 0.34^b	2.50 ± 1.77^a	−0.48 ± 0.76^b
b*	35.95 ± 3.47^a	29.5 ± 5.14^b	−4.21 ± 2.81^c	−9.4 ± 0.13^d	25.51 ± 2.18^e	1.30 ± 1.55^f

Note: values are means ± standard deviation of three replicates. Means with the same superscript along the lines are not significantly different at P < 0.05.

Nota: Valores promedio ± desviación estándar de tres repeticiones. Los valores promedio con la misma letra a través de las líneas no son significativamente diferentes a P < 0,05.

Table 2. Chemical composition of dry beans.

Tabla 2. Composición química del frijol.

Dry bean variety	Moisture (g/kg)	Protein (g/kg, N x 6.25)	Fat (g/kg)	Ash (g/kg)	Total carbohydrates (g/kg)
Azufrado	117.0 ± 0.9	235.3 ± 1.0	21.4 ± 0.5	47.0 ± 0.9	579.3 ± 1.5
Mayacoba	118.9 ± 1.0	234.5 ± 0.9	16.5 ± 0.4	44.4 ± 0.7	585.7 ± 1.9
Pinto	116.9 ± 0.8	238.2 ± 1.1	6.3 ± 0.3	40.7 ± 0.6	597.9 ± 1.6
Peruano	113.7 ± 1.1	237.4 ± 0.8	46.8 ± 0.4	40.5 ± 0.7	561.6 ± 1.8
Flor de Mayo	115.3 ± 0.9	233.6 ± 1.0	10.9 ± 0.3	41.5 ± 0.5	598.7 ± 1.5
Negro Jamapa	117.6 ± 0.7	238.2 ± 0.9	14.7 ± 0.4	44.6 ± 0.6	584.9 ± 1.7

Note: values are means ± standard deviation of three replicates.

Nota: Valores promedio ± desviación estándar de tres repeticiones.

2.2. Ultrasound treatment

Each bean sample (10 g) was placed in separate 500 mL beakers, and 400 mL of distilled water was added into each sample at room temperature (30°C). For ultrasound treatment, a 40 kHz ultrasound bath (Branson, Model MTH-3510; a power of 130 W; a tank capacity of 5 L; internal dimensions of 290 × 150 × 150 mm; an acoustic energy density of 0.026 W/cm³) was used. Beakers containing the bean samples of each variety were placed directly into the ultrasound bath to receive the ultrasound treatment for 10, 20 or 30 min. The temperature increase during the experiments was not significant (less than 2°C). For control treatments, beakers of 500 mL with bean samples (10 g) of each variety with 400 mL of distilled water were placed in a water bath at 30°C.

2.3. Water absorption determination

After ultrasound treatment, the bean samples were removed from the ultrasonic bath and kept at 30°C. Water absorption of the beans was determined according to the Leal-Oliveira et al. (2013) method at regular intervals of 30 min. All soaking tests were performed in triplicate, and the results were recorded as a moisture percentage of dry matter.

2.4. Modelling of water absorption

Three mathematical models, namely Peleg’s, first order, and sigmoid models, were utilised in the present study. Peleg’s kinetic model applies to weight gain during hydration (Jideani & Mpotokwana, 2009) and assumes the form:

(1)

where W(t) is the moisture content of the seeds at time t, W₀ is the initial moisture content of the unsoaked seeds, k₁ is the Peleg rate constant and k₂ is the Peleg capacity constant. According to this model, the equilibrium moisture, W_e (i.e. when t → ∞), is given by:

(2)

In the first-order (exponential) hydration model (Cox, Gupta, & Abu-Ghannam, 2012), the following equation is used:

(3)

where W(t) is the moisture content at time t, W₀ is the initial moisture content, W_e is the equilibrium moisture content and k (1/min) is the constant rate of hydration.

The following sigmoid mathematical model (Kaptso et al., 2008) was also tested:

(4)

where W(t) is the moisture content at time t, W_e is the equilibrium moisture content and k (1/min) is the constant rate of hydration. The soaking time, τ (min), is defined as the time needed to attain half saturation (50%) of the seeds.

Fitting procedures and hydration rate constants were determined from the results by minimisation of the sum of quadratic differences between observed and model-/equation-predicted values using Newton’s method available in the Solver for Excel spreadsheet (Phomkong, Soponronnarit, & Thammarutwasik, 2010).

The goodness of fit of the tested mathematical models to the experimental data was evaluated from the coefficient of determination (R²), sum square error (SSE), root mean square error (RMSE) and the chi-square (χ²) between the predicted and experimental values. The lowest values of SSE, RMSE and χ² in addition to the highest values of R² (≈1.0) are considered as optimum criteria to evaluate the fit quality of the models used (Cox et al., 2012).

2.5. Determination of the effective diffusivity (D_eff)

To eliminate the effect of the seed radius on the constant rate of diffusion, the effective diffusivity was calculated using the analytical solution of the one dimensional Fick’s law of diffusion with constant moisture diffusivity for a sphere (Kaptso et al., 2008). In this respect, the following equation was used: k = π²D_eff∕r²; where D_eff (m²/s) is the effective diffusivity; k (1/s) is the constant rate of diffusion for the model described above with the best fit quality and r (m) is the seed radius of the dry bean, which is half of the geometric diameter.

2.6. Cooking time determination

The soaked beans from each ultrasonic treatment were cooked by heating at 98 ± 1°C in a hot plate in 500 mL beakers with 350 mL of distilled water. The cooking water was heated to 98°C and then the beans were added. The cooking time was sensorially determined according to the method reported by Wani, Sogi, and Gill (2013).

2.7. Moisture content

The moisture content was determined in triplicate by the Association of Official Analytical Chemists (AOAC, 1995) method.

2.8. Statistical analysis

The results of physical characteristics and chemical composition were expressed as the mean ± standard deviation. One-way analysis of variance (ANOVA) and mean comparison analyses were conducted using the Statgraphics Plus 5.0 software (Statistical Graphics Corp., Rockville, MD, USA) to test the significant differences between treatments at P ≤ 0.05.

3. Results and discussion

3.1. Kinetics of water absorption

Understanding water absorption in legumes during soaking is of practical importance because it affects subsequent processing operations and the quality of the final product. The kinetics of water absorption of the dry bean seeds are shown in Figure 1. In all cases, water absorption increased with the duration of soaking. However, the behaviour of the hydration kinetics among the dry bean varieties and soaking treatments was different. When 30 min of ultrasound treatment was applied, the water absorption curves for the control treatments of the Azufrado, Mayacoba and Garbancillo varieties were sigmoid in shape, and the Flor de Mayo variety exhibited the typical hyperbolic curve. A sigmoid shape in kinetics indicates that the water absorption curve exhibits a lag phase, which is followed by a phase of the time-linear relationship water absorption and a phase with a slower rate of absorption. This type of kinetics has been observed in seeds of cowpea and common bean (Kaptso et al., 2008; Piergiovanni, 2011). The typical hyperbolic form is characterised by an initial high rate of water absorption, followed by slower absorption in later stages. This type of kinetics has been reported by many authors for many food stuffs (Abu-Ghannam, 1998; Yildirim et al., 2011). The form of the absorption curve may reflect the degree of shell hardness in the seeds (Kaptso et al., 2008). From this perspective, the dry beans of the Azufrado, Mayacoba and Garbancillo varieties presented more hard shell features than the dry beans of the Flor de Mayo, Negro Jamapa and Pinto varieties. Slow hydration may be due to hemicellulose and pentosans in the seed coat, which hinder the penetration of water, as well as to the quick moisture absorption of the middle lamella in legumes lacking seed coats (Zamindar, Baghekhandan, Nasirpour, & Sheikhzeinoddin, 2013). The change in shape of the water absorption curves of the Azufrado, Mayacoba and Garbancillo dry bean varieties, which changed from sigmoid by soaking without ultrasound exposure to hyperbolic by soaking for 30 min with ultrasound exposure, was of interest. This change suggested that the application of ultrasound for soaking had an effect similar to increasing the soaking temperature, thus overcoming the defect of low water absorption observed in the seeds (Kaptso et al., 2008).

Figure 1. Varietal and ultrasound treatment differences in the changes in moisture content of Azufrado (a), Mayacoba (b), Flor de Mayo (c), Negro Jamapa (d), Garbancillo (e) and Pinto (f) seeds during soaking. Solid lines indicate predictive plots for non-linear regression of Peleg’s model.

Figura 1. Diferencia de las variedades y tratamientos de ultrasonido en los cambios en contenido de humedad de las semillas de Azufrado (a), Mayacoba (b), Flor de Mayo (c), Negro Jamapa (d), Garbancillo (e) y Pinto (f) durante el remojo. Las líneas continuas indican las curvas predictivas por regresión no lineal del modelo de Peleg.

The parameters generated from the application of the different mathematical models (sigmoid, first-order and Peleg’s models) describing the kinetics of water absorption by the dry beans are presented in Tables 3–5. In most cases, the R² values were high with low values of SEE, RMSE and χ² suggesting that the models clearly described the water absorption in the different dry bean varieties. The R² values ranged from 0.8404 to 0.9993 taking into account all the soaking treatments, dry bean varieties and the three mathematical models (Tables 3–5). However, the first-order kinetic model had the lowest R² values among the three models, especially for soaking without ultrasound treatment.

Table 3. Water absorption characteristics of dry beans by the effect of ultrasound exposure time following the sigmoid model.

Tabla 3. Características de absorción de agua del frijol por efecto del tiempo de exposición a ultrasonido siguiendo el modelo Sigmoide.

Time (min)	Kinetic parameters	Dry bean variety
		Azufrado	Mayacoba	Flor de Mayo	Negro Jamapa	Garbancillo	Pinto
0 (control)	Wobs.eq (g/g H2O db)	1.21 ± 0.01^a	1.22 ± 0.02^a	1.21 ± 0.02^a	1.20 ± 0.03^a	1.22 ± 0.02^a	1.22 ± 0.01^a
	Weq (g /g H2O db)	1.20 ± 0.02^a	1.15 ± 0.03^abc	1.19 ± 0.02^ab	1.14 ± 0.02^c	1.13 ± 0.01^c	1.16 ± 0.02^ab
	k (10^–2/min)	1.26 ± 0.04^b	1.45 ± 0.06^a	1.07 ± 0.04^c	0.69 ± 0.04^e	0.72 ± 0.03^e	0.81 ± 0.02^d
	τ (min)	179.7 ± 3.1^d	165.6 ± 2.5^e	153.3 ± 2.8^f	276.0 ± 3.3^b	343.8 ± 4.1^a	213.7 ± 2.8^c
	SSE	0.0016	0.0058	0.0257	0.0207	0.0478	0.0260
	RMSE	0.0097	0.0211	0.0378	0.0294	0.0446	0.0380
	χ^2	0.0001	0.0005	0.0016	0.0009	0.0021	0.0014
	R^2	0.9993	0.9968	0.9878	0.9926	0.9846	0.9884
10	Wobs.eq (g/g H2O db)	1.20 ± 0.02^a	1.22 ± 0.01^a	1.20 ± 0.02^a	1.20 ± 0.03^a	1.21 ± 0.02^a	1.21 ± 0.02^a
	Weq (g/H2O g db)	1.18 ± 0.02^a	1.18 ± 0.02^a	1.19 ± 0.02^a	1.10 ± 0.03^b	1.16 ± 0.03^ab	1.20 ± 0.02^a
	k (10^–2/min)	1.17 ± 0.04^c	1.55 ± 0.06^a	1.46 ± 0.06^a	0.73 ± 0.07^e	0.94 ± 0.05^d	1.26 ± 0.05^b
	τ (min)	174.0 ± 2.5^b	119.7 ± 3.2^d	100.3 ± 2.9^e	243.5 ± 3.3^a	164.4 ± 2.3^c	116.0 ± 2.5^d
	SSE	0.0040	0.0066	0.0171	0.0284	0.0185	0.0178
	RMSE	0.0153	0.0234	0.0337	0.0376	0.0320	0.0344
	χ^2	0.0002	0.0006	0.0013	0.0015	0.0011	0.0011
	R^2	0.9982	0.9957	0.9895	0.9860	0.9905	0.9905
20	Wobs.eq (g/g H2O db)	1.20 ± 0.02^a	1.21 ± 0.02^a	1.10 ± 0.02^b	1.16 ± 0.03^a	1.20 ± 0.02^a	1.20 ± 0.03^a
	Weq (g/g H2O db)	1.19 ± 0.02^a	1.19 ± 0.01^a	1.15 ± 0.02^ab	1.11 ± 0.02^b	1.16 ± 0.02^a	1.19 ± 0.02^a
	k (10^–2/min)	1.44 ± 0.05^b	1.90 ± 0.04^a	1.15 ± 0.05^c	0.91 ± 0.03^e	0.99 ± 0.05^d	1.38 ± 0.04^b
	τ (min)	97.4 ± 3.5^d	87.0 ± 2.8^e	105.7 ± 3.6^c	178.7 ± 3.8^a	152.0 ± 3.7^b	86.4 ± 2.9^e
	SSE	0.0273	0.0094	0.0434	0.0158	0.0198	0.0376
	RMSE	0.0413	0.0292	0.0578	0.0297	0.0341	0.0500
	χ^2	0.0019	0.0006	0.0039	0.0009	0.0013	0.0028
	R^2	0.9836	0.9932	0.9618	0.9912	0.9890	0.9742
30	Wobs.eq (g H2O/g db)	1.21 ± 0.02^a	1.20 ± 0.02^a	1.21 ± 0.02^a	1.19 ± 0.03^a	1.17 ± 0.03^a	1.22 ± 0.02^a
	Weq (g H2O/g db)k	1.20 ± 0.02^a	1.20 ± 0.02^a	1.21 ± 0.03^a	1.16 ± 0.02^ab	1.15 ± 0.02^b	1.21 ± 0.03^a
	(10^–2/min)	1.57 ± 0.05^c	2.66 ± 0.04^b	8.08 ± 0.07^a	1.05 ± 0.03^f	1.19 ± 0.06^e	1.34 ± 0.05^d
	τ (min)	74.5 ± 3.2^d	46.3 ± 2.5^e	21.9 ± 2.4^f	121.8 ± 5.2^a	100.5 ± 4.2^b	83.4 ± 3.8^c
	SEE	0.0527	0.0461	0.0521	0.0289	0.0356	0.0514
	RMSE	0.0637	0.0679	0.0807	0.0425	0.0487	0.0606
	χ^2	0.0040	0.0057	0.0086	0.0020	0.0023	0.0042
	R^2	0.9630	0.9570	0.9431	0.9809	0.9742	0.9621

Notes: values are means ± standard deviation of three experiments. The values with the same letter along the lines are not significantly different (P < 0.05). k is the constant rate of hydration, τ is the half saturation time, W_eq and W_obs.eq are the predicted and observed equilibrium moisture content, respectively. SEE is the standard error of estimation, RMSE is the root mean square error, R² is the coefficient of determination and χ² is the chi-square.

Notas: Valores promedio ± desviación estándar de tres experimentos. Los valores con la misma letra a través de las líneas no son significativamente diferentes (P < 0,05). k es la constante de de velocidad de hidratación, τ es el tiempo medio de saturación. W_eq and W_obs.eq son los contenidos de humedad de equilibrio predichos y observados, respectivamente. SEE es el error estándar de estimación, RMSE es la raíz del error cuadrático medio, R² es el coeficiente de determinación y χ² es chi-cuadrada.

Table 4. Water absorption characteristics of dry beans by the effect of ultrasound exposure time following the first-order model.

Tabla 4. Características de absorción de agua del frijol por efecto del tiempo de exposición a ultrasonido siguiendo el modelo de Primer Orden.

Time (min)	Kinetic parameters	Dry bean variety
		Azufrado	Mayacoba	Flor de Mayo	Negro Jamapa	Garbancillo	Pinto
0 (control)	Weq (g/g H2O db)	1.21 ± 0.04^a	1.22 ± 0.03^a	1.21 ± 0.02^a	1.21 ± 0.05^a	1.22 ± 0.04^a	1.22 ± 0.03^a
	k (10^–2/min)	0.44 ± 0.03^a	0.45 ± 0.02^a	0.49 ± 0.03^a	0.27 ± 0.03^c	0.21 ± 0.02^d	0.35 ± 0.03^b
	SSE	0.2340	0.2167	0.0527	0.2222	0.4975	0.1495
	RMSE	0.1140	0.1291	0.0541	0.0962	0.1439	0.0864
	χ^2	0.0137	0.0166	0.0031	0.0096	0.0216	0.0078
	R^2	0.9121	0.8810	0.9751	0.9208	0.8404	0.9335
10	Weq (g/g H2O db)	1.21 ± 0.02^a	1.22 ± 0.03^a	1.19 ± 0.02^a	1.20 ± 0.02^a	1.20 ± 0.01^a	1.21 ± 0.02^a
	k (10^–2/min)	0.44 ± 0.01^c	0.64 ± 0.02^b	0.73 ± 0.02^a	0.30 ± 0.03^d	0.45 ± 0.02^c	0.64 ± 0.01^b
	SSE	0.1631	0.0854	0.0169	0.1420	0.0687	0.0442
	RMSE	0.0979	0.0843	0.0336	0.0864	0.0617	0.0509
	χ^2	0.0101	0.0077	0.0012	0.0074	0.0040	0.0027
	R^2	0.9281	0.9209	0.9896	0.9302	0.9648	0.9765
20	Weq (g /g H2O db)	1.20 ± 0.01^a	1.21 ± 0.02^a	1.10 ± 0.02^c	1.17 ± 0.01^b	1.20 ± 0.02^a	1.20 ± 0.02^a
	k (10^–2/min)	0.74 ± 0.02^c	0.87 ± 0.02^a	0.82 ± 0.02^b	0.41 ± 0.02^e	0.48 ± 0.02^d	0.81 ± 0.01^b
	SSE	0.0068	0.0342	0.0140	0.0758	0.0579	0.0193
	RMSE	0.0206	0.0558	0.0328	0.0648	0.0583	0.0358
	χ^2	0.0004	0.0034	0.0011	0.0044	0.0036	0.0013
	R^2	0.9959	0.9754	0.9876	0.9583	0.9678	0.9867
30	Weq (g/g H2O db)	1.21 ± 0.02^a	1.20 ± 0.02^a	1.21 ± 0.02^a	1.20 ± 0.01^a	1.18 ± 0.02^a	1.22 ± 0.01^a
	k (10^–2/min)	0.92 ± 0.03^c	1.53 ± 0.02^b	3.15 ± 0.01^a	0.58 ± 0.03^f	0.69 ± 0.02^e	0.82 ± 0.02^d
	SSE	0.0046	0.0074	0.0153	0.0179	0.0174	0.0178
	RMSE	0.0175	0.0272	0.0438	0.0335	0.0341	0.0357
	χ^2	0.0003	0.0008	0.0021	0.0011	0.0012	0.0013
	R^2	0.9967	0.9930	0.9832	0.9881	0.9873	0.9868

Notes: values are mean ± standard deviation of three experiments. The values with the same letter 0 along the lines are not significantly different (P < 0.05). k is the constant of rehydration rate and W_eq is the predicted equilibrium moisture content. SSE is the sum square error, RMSE is the root mean square error, R² is the coefficient of determination and χ² is the chi-square.

Notas: Valores promedio ± desviación estándar de tres experimentos. Los valores con la misma letra a través de las líneas no son significativamente diferentes (P < 0,05). k es la constante de de velocidad de hidratación. W_eq es el contenido de humedad de equilibrio predicho. SEE es el error estándar de estimación, RMSE es la raíz del error cuadrático medio, R² es el coeficiente de determinación y χ² es chi-cuadrada.

Table 5. Water absorption characteristics of dry beans by the effect of ultrasound exposure time following Peleg´s model.

Tabla 5. Características de absorción de agua del frijol por efecto del tiempo de exposición a ultrasonido siguiendo del modelo de Peleg.

Time (min)	Kinetic parameters	Dry bean variety
		Azufrado	Mayacoba	Flor de Mayo	Negro Jamapa	Garbancillo	Pinto
0 (control)	Weq (g/g H2O db)	1.29 ± 0.04^a	1.22 ± 0.03^ab	1.23 ± 0.03^ab	1.25 ± 0.03^ab	1.18 ± 0.04^b	1.23 ± 0.03^ab
	k1 (10^2 min g/g H2O db)	3.00 ± 0.08^e	3.25 ± 0.07^d	2.16 ± 0.09^f	5.34 ± 0.12^b	6.49 ± 0.15^a	3.78 ± 0.04^c
	k2 (g/gH2O db)	0.25 ± 0.01^b	0.00 ± 0.00^e	0.47 ± 0.04^a	0.11 ± 0.01^d	0.00 ± 0.00^e	0.22 ± 0.02^c
	SSE	0.0779	0.0307	0.0139	0.0090	0.1046	0.0195
	RMSE	0.0658	0.0486	0.0277	0.0194	0.0660	0.0312
	χ^2	0.0048	0.0027	0.0008	0.0004	0.0047	0.0010
	R^2	0.9707	0.9831	0.9934	0.9967	0.9664	0.9913
10	Weq (g/g H2O db)	1.26 ± 0.02^a	1.25 ± 0.03^ab	1.23 ± 0.02^abc	1.19 ± 0.02^c	1.22 ± 0.01^bc	1.25 ± 0.02^ab
	k1 (10^2 min g/g H2O db)	2.98 ± 0.05^b	2.01 ± 0.07^d	1.25 ± 0.07^f	4.88 ± 0.08^a	2.53 ± 0.09^c	1.48 ± 0.07^e
	k2 (g/g H2O db)	0.24 ± 0.01^c	0.26 ± 0.02^c	0.60 ± 0.02^a	0.07 ± 0.03^d	0.40 ± 0.02^b	0.57 ± 0.01^a
	SSE	0.0336	0.0065	0.0077	0.0040	0.0030	0.0214
	RMSE	0.0444	0.0232	0.0227	0.0142	0.0130	0.0355
	χ^2	0.0022	0.0006	0.0005	0.0002	0.0001	0.0014
	R^2	0.9851	0.9958	0.9952	0.9980	0.9976	0.9886
20	Weq (g/g H2O db)	1.22 ± 0.01^a	1.25 ± 0.02^a	1.11 ± 0.02^c	1.17 ± 0.01^b	1.21 ± 0.02^a	1.21 ± 0.02^a
	k1 (10^2 min g/g H2O db)	1.14 ± 0.03^c	1.21 ± 0.04^c	1.20 ± 0.04^c	3.09 ± 0.06^a	2.33 ± 0.07^b	1.00 ± 0.04^d
	k2 (g/g H2O db)	0.65 ± 0.01^a	0.48 ± 0.02^b	0.66 ± 0.02^a	0.34 ± 0.02^c	0.42 ± 0.02^b	0.67 ± 0.03^a
	SSE	0.0072	0.0051	0.0077	0.0027	0.0054	0.0114
	RMSE	0.0212	0.0216	0.0227	0.0124	0.0179	0.0276
	χ^2	0.0005	0.0005	0.0005	0.0001	0.0003	0.0008
	R^2	0.9956	0.9962	0.9945	0.9984	0.9969	0.9921
30	Weq (g/g H2O db)	1.21 ± 0.02^a	1.21 ± 0.02^a	1.19 ± 0.02^a	1.18 ± 0.01^ab	1.17 ± 0.02^b	1.21 ± 0.01^a
	k1 (10^2 min g/g H2O db)	0.77 ± 0.08^d	0.45 ± 0.07^e	0.15 ± 0.05^f	1.59 ± 0.04^a	1.27 ± 0.06^b	0.91 ± 0.04^c
	k 2(g/gH2O db)	0.73 ± 0.02^b	0.74 ± 0.03^b	0.85 ± 0.02^a	0.58 ± 0.03^d	0.64 ± 0.03^c	0.68 ± 0.03^c
	SSE	0.0009	0.0025	0.0006	0.0050	0.0058	0.0081
	RMSE	0.0081	0.0158	0.0090	0.0178	0.0197	0.0240
	χ^2	0.0007	0.0003	0.0001	0.0003	0.0004	0.0006
	R^2	0.9993	0.9976	0.9992	0.9966	0.9957	0.9940

Notes: values are means ± standard deviation of three experiments. The values with the same letter along the lines are not significantly different (P < 0.05). k₁ and k₂ are Peleg´s constants and W_eq is the predicted equilibrium moisture content. SSE is the sum square error, RMSE is the root mean square error, R² is the coefficient of determination and χ² is the chi-square.

Notas: Valores promedio ± desviación estándar de tres experimentos. Los valores con la misma letra a través de las líneas no son significativamente diferentes (P < 0,05). k₁ and k₂ son las constantes de Peleg. W_eq es el contenido de humedad de equilibrio predicho. SEE es el error estándar de estimación, RMSE es la raíz del error cuadrático medio, R² es el coeficiente de determinación y χ² es chi-cuadrada.

In general, the examination of the parameters obtained from the models applied showed that the kinetic rate constant k of the sigmoid and first-order models increased with increasing ultrasound time. The k₁ parameter for Peleg´s model decreased as the ultrasound time was increased from 0 (control treatment) to 30 min, and the k₂ parameter increased. According to the results obtained in this study, the effect of ultrasound treatment was analogous to the effect of increasing the temperature during soaking. A study with Shiitake mushroom (Lentinus edodes) has shown that increasing the temperature of immersion water from 30 to 50°C reduces the k₁ value of Peleg´s model by 74.2% (García-Segovia, Andrés-Bello, & Martínez-Monzó, 2011). Another study with edible Irish brown seaweed has shown that the k₁ value of Peleg´s model is reduced by 66.6% when the hydration water temperature is increased from 20 to 60°C (Cox et al., 2012). In our study, the reduction of the k₁ value in Peleg´s model resulting from a 30 min ultrasound treatment ranged from 70.2% for the Negro Jamapa bean variety to 93.0% for the Flor de Mayo bean variety. Solomon (2007) suggested that k₁ may represent the water absorption rate in the early phases of the hydration process and that k₂ is related to the maximum capacity of water absorption or to the equilibrium moisture content.

3.2. Saturation moisture content

In most cases, the observed maximum moisture content of the soaked beans (1.16 ± 0.03 to 1.22 ± 0.02 g/g) did not show variation with the variety and exposure time to ultrasound (Table 3).

Generally, the predicted saturation moisture contents obtained by the proposed first-order model were similar to the experimental saturation moisture contents (Table 4). In the case of the sigmoid model, the predicted saturation moisture contents were lower than the observed moisture contents (Table 3), but the predicted saturation moisture contents were higher than the observed moisture contents in Peleg’s model in some cases (Table 5). In addition, the second constant of Peleg´s model (k₂), which has been shown to be linked to the maximum water absorption capacity of seeds, varied significantly according to the bean variety in this work (P < 0.05), thereby agreeing with previous reports (Solomon, 2007).

3.3. Constant rate of hydration

Each of the models produced a constant rate of hydration that varied significantly with the seed variety and ultrasound treatment. The sigmoid (Table 3) and exponential (Table 4) constant rates of hydration, namely k and k₁, respectively, were shown to systematically increase with increasing ultrasound time. The values of the constant rates of hydration of such models were in the same order of magnitude, which are in agreement with the values of the exponential constant rate of hydration reported by Abu-Ghannam (1998) for kidney beans, who showed that the constant rate, k, varied from 1.50 × 10^–2 at 20°C to 3.18 × 10^–2/min at 40°C. In our study, the sigmoid constant rate, k, increased with a longer ultrasound treatment time analogous to the soaking temperature reported for kidney beans. For the Flor de Mayo dry bean, the k value varied from 1.07 × 10^–2 (control treatment) to 7.01 × 10^–2/min when the ultrasound time was 30 min, and the k value varied from 0.35 × 10^–2 to 0.81 × 10^–2/min under the same respective conditions for the Pinto dry bean. In this study, the sigmoid constant rates were in the same order of magnitude but higher than the exponential constant rate, which is in agreement with the results reported by Kaptso et al. (2008) for cowpea and Bambara groundnut seeds.

3.4. D_eff changes with ultrasound treatment

The D_eff variation of the different dry bean varieties with ultrasound treatment, using the k-values of the sigmoid model, is shown in Table 6, which verifies the increase of D_eff when ultrasound treatment was applied. D_eff of the dry beans for the control treatments varied in the following order: Flor de Mayo > Azufrado > Pinto > Mayacoba > Garbancillo > Negro Jamapa. However, when a 30 min ultrasound treatment was applied, the order of variation was modified as follows: Mayacoba > Pinto > Azufrado > Garbancillo >Pinto > Negro Jamapa. In the case of the Azufrado, Flor de Mayo, Mayacoba and Pinto bean varieties, the increase of D_eff due to the 30 min ultrasound treatment was one order of magnitude. The ultrasound treatment increased the water diffusion of dry beans during soaking due to an increase in the mass diffusion rate (Bhaskaracharya, Kentish, & Ashokkumar, 2009), which may be attributed to ultrasound affecting mass transfer through the cellular wall and the membrane at the tissue level, e.g. microstirring, reducing boundary layers, sponge effects and structural changes (Schössler, Thomas, & Knorr, 2012). For soaking at 20°C, the D_eff values in chickpea were 1.40 × 10^–10, 1.70 × 10^–10 and 2.01 × 10^–10 m²/s for non-ultrasound, 25 kHz 100 W ultrasound and 25 kHz 300 W ultrasound treatments, respectively (Yildirim et al., 2011). In our study, D_eff values changed from 2.60 × 10^–10 to 3.24 × 10^–09, 1.72 × 10^–10 to 7.75 × 10^–09, 2.76 × 10^–10 to 2.08 × 10^–09, 1.29 × 10^–10 to 1.94 × 10^–10, 1.63 × 10^–10 to 2.70 × 10^–10 and 2.05 × 10^–10 to 3.38 × 10^–9 m²/s for Azufrado, Mayacoba, Flor de Mayo, Negro Jamapa, Garbancillo and Pinto bean varieties, respectively, when the samples were left untreated or treated with ultrasound for 30 min.

Table 6. Varietal difference of dry beans in the diffusion coefficient of hydration by the effect of ultrasound exposure time.

Tabla 6. Diferencia varietal del frijol en el coeficiente de hidratación por efecto del tiempo de exposición a ultrasonido.

Dry bean variety	Ultrasound exposure time (min)	Diffusion coefficient (m^2 s^−1)
Azufrado	0 (control)	2.60 × 10^–10 ± 7.2 × 10^–11
	10	2.42 × 10^–10 ± 3.2 × 10^–11
	20	2.96 × 10^–10 ± 2.2 × 10^–11
	30	3.24 × 10^–09 ± 1.0 × 10^–12
Mayacoba	0 (control)	1.72 × 10^–10 ± 4.2 × 10^–11
	10	4.53 × 10^–10 ± 1.8 × 10^–11
	20	5.50 × 10^–10 ± 1.5 × 10^–11
	30	7.75 × 10^–09 ± 0.9 × 10^–11
Flor de Mayo	0 (control)	2.76 × 10^–10 ± 3.2 × 10^–11
	10	3.77 × 10^–10 ± 1.2 × 10^–11
	20	2.96 × 10^–10 ± 1.0 × 10^–11
	30	2.08 × 10^–09 ± 0.8 × 10^–10
Negro Jamapa	0 (control)	1.29 × 10^–10 ± 6.2 × 10^–11
	10	1.35 × 10^–10 ± 3.2 × 10^–11
	20	1.68 × 10^–10 ± 2.0 × 10^–11
	30	1.94 × 10^–10 ± 0.9 × 10^–11
Garbancillo	0 (control)	1.63 × 10^–10 ± 2.2 × 10^–11
	10	2.13 × 10^–10 ± 1.2 × 10^–11
	20	2.24 × 10^–10 ± 0.8 × 10^–11
	30	2.70 × 10^–10 ± 0.5 × 10^–11
Pinto	0 (control)	2.05 × 10^–10 ± 2.2 × 10^–11
	10	3.18 × 10^–10 ± 1.2 × 10^–11
	20	3.49 × 10^–9 ± 0.8 × 10^–10
	30	3.38 × 10^–9 ± 0.2 × 10^–10

Note: values are means ± standard deviation of three experiments.

Nota: Valores promedio ± desviación estándar de 3 experimentos.

3.5. Effect of ultrasound on soaking time

The soaking time, τ (min), defined as the time needed to attain half saturation (50%) of the seeds according to the sigmoid model used in this study, was greatly reduced by the ultrasound treatment (Table 3). The τ reductions were 105.2, 119.3, 131.9, 154.2, 243.3 and 130.3 min for the Azufrado, Mayacoba, Flor de Mayo, Negro Jamapa, Garbancillo and Pinto bean varieties, respectively. The τ reductions increased with increasing ultrasound treatment times. As previously discussed, the power ultrasound enhances the mass transfer in processes where diffusion takes place. Power ultrasound introduces pressure variation at solid/liquid interfaces and, therefore, increases the moisture absorption rate. Acoustic energy also causes oscillating velocities and microstreaming at the interfaces, which may affect the diffusion boundary layer. Furthermore, ultrasonic waves produce a rapid series of alternating contractions and expansions (sponge effect) of the material in which they are travelling; this alternating stress creates microscopic channels that may ease moisture gain, thereby reducing the soaking time (Yildirim et al., 2010, 2011). In addition, acoustic waves may produce cavitations of water molecules inside the solid matrix, which may be beneficial for gaining strongly attached moisture (Yildirim et al., 2013).

The ultrasound treatment of dry beans during soaking significantly (P < 0.05) decreased the time to obtain equilibrium moisture content (which defines the soaking time) in all studied ultrasound treatments (10–30 min) compared to soaking without ultrasound treatment. The times (min) to obtain the equilibrium moisture contents for the control treatments of the Azufrado, Mayacoba, Flor de Mayo, Negro Jamapa, Garbancillo and Pinto dry bean varieties were 510, 360, 510, 690, 690 and 570 (Figure 1), respectively, which were reduced by 17.6, 25.0, 58.8, 31.8, 39.1 and 31.6% when a 30 min ultrasound treatment was applied.

3.6. Effect of ultrasound treatment on cooking time

The results of the ultrasound treatment on the cooking time of dry beans in this study are shown in Table 7. These results demonstrated that ultrasound treatment reduced the cooking time. The percentage of cooking time reduction ranged from 5.4 to 43.0% depending on the ultrasound treatment and bean variety. The highest reduction of cooking time was observed for the Negro Jamapa variety (reductions of 36.9 and 43.0% for 10 and 30 min ultrasound treatments, respectively), and the lowest reduction of cooking time was observed for the Mayacoba variety (reductions of 5.4 and 21.6% for 10 and 30 min ultrasound treatments, respectively).

Table 7. Cooking time of dry beans by the effect of ultrasound exposure time.

Tabla 7. Tiempo de cocción del frijol por efecto del tiempo de exposición a ultrasonido.

	Ultrasound exposure time (min)
Variety	0 (control)	10	20	30
Azufrado	94 ± 3^a	83 ± 2^b	80 ± 2^b	73 ± 3^c
Mayacoba	37 ± 2^a	35 ± 2^a	31 ± 2^b	29 ± 2^b
Flor de Mayo	60 ± 3^a	55 ± 2^b	54 ± 3^b	47 ± 2^c
Negro Jamapa	65 ± 3^a	41 ± 2^b	38 ± 2^b	37 ± 2^b
Garbancillo	51 ± 2^a	44 ± 2^b	41 ± 2^bc	39 ± 2^c
Pinto	45 ± 2^a	39 ± 2^b	35 ± 2^b	29 ± 2^c

Note: values are means ± standard deviation of three experiments. Means with the same superscript along the lines are not significantly different at P < 0.05.

Nota: Valores promedio ± desviación estándar de tres experimentos. Las valores promedios con la misma letra en las líneas no son significativamente diferentes a P < 0,05.

Some studies have demonstrated the ultrasound effect on the reduction of cooking times of rice (Wambura et al., 2008) and chickpea (Yildirim et al., 2011). Moreover, Janve et al. (2013) recently reported that power ultrasound has the potential to accelerate the nixtamalisation process by reducing the steeping time from 20 h in the traditional nixtamalisation to 1 h in the power ultrasound-assisted nixtamalisation, which is in agreement with the results obtained in this study.

4. Conclusions

The application of ultrasound significantly reduced the soaking and cooking times of six of the main bean varieties consumed in México, but the extent of such reductions depended on the bean variety and ultrasound exposure time. In most cases, the observed maximum moisture content of the dry beans by soaking did not show variation with the bean variety and ultrasound exposure time. The ultrasound treatments of dry beans during soaking significantly decreased the time to reach equilibrium moisture content. However, further studies should be performed to confirm that ultrasound treatment lowers the energy requirement for the soaking and the cooking times of beans.

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

The authors wish to thank Promep-SEP-México for providing funds to support this research through grant [grant number 2649-UAN-CA-6].