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The mesquite pods have a high nutritional value, being a rich source of sugars, proteins, minerals and fibers that can be used as a raw material for the development of a number of food products and technological innovations. The objective of this work was to study the drying kinetics of mesquite pods in a thin layer, 20 mm length and determine the effective diffusion coefficient by adjusting mathematical models that were based on heat transfer and mass fundamentals. For the experiments, we used mature mesquite, with 20% water content (b.u). Drying experiments were performed using temperatures of 50
°
C, 60
°
C, 70
°
C and 80
°
C with drying air speed of 2 m•s
^{-1}
. It was used a convective dryer with fixed bed and upflow air. Mathematical models of Fick, Page, Cavalcanti-Mata, Two Exponential Terms and Henderson & Pabis were used to adjust the experimental data. To calculate the effective diffusivity, flat plate geometry with sample thickness of 6.5 mm was used. Based on the results obtained, it was concluded that the loss of water from mesquite oc
curred during the drying period in decreasing rate; and with increasing drying temperature, total process time was reduced. The diffusion coefficient increased by increasing drying temperature. Mathematical models including theoretical (Fick), semi-theoretical (Page and Cavalcanti-Mata) and empir
ical (Two Exponential Terms) models satisfactorily explain the experimental data of drying mesquite.

Brazil is the country with the greatest potential for research with plant species in the world, because it has the largest and richest biodiversity on the planet distributed in six distinct biomes that have about 55,000 described species [

Mesquite was introduced in Brazil, especially in the Northeast, due to its hardiness and because it has the important characteristic of fruiting in the driest time of the year when natural fodder stocks reach a critical stage, thus providing a great food nutritional value, especially for feeding caprine and bovine [

The pods of mesquite tree are among the oldest foods used by man in the New World. Fruits of mesquite tree are yellow, long, flat pods and usually slightly curved, palatable, aromatic and sweet; it is in pods that is concentrated nutritional value, being a rich source of carbohydrate. The pod is rich in sugars, proteins, minerals and fiber that can be converted into raw material in production of cakes, breads, cookies, drinks, condiments, jams, honeys, puddings, soups, porridge and other tasty and nutritious food [

Because of all this information, mesquite pods have been a potential source of raw material for the development of a range of products and technological innovations that are being studied by researchers from various countries, like mesquite whole meal, produced in Kenya, Brazil, United States, Argentina and Mexico, and Peruvian algarobina, exported to many countries [

Drying is a method of preservation wherein water content and water activity of fruits, vegetables and grains are diminished through mass and heat transfer phenomena, which minimizes chemical, physical, biochemical and microbiological degradation of these products during storage, ensuring their quality and stability [

Mathematical representation of drying process of various agricultural products has, for some time, been studied and used in attempting to predict the phenomena that occur during this process [

Considering the importance of theoretical study of drying process for agricultural products and a limitation of information about the phenomena that occur during the drying of mesquite, the objectives of this study were: to study mesquite drying kinetics with an initial water content of 20% (wet basis), in a thin layer, using temperatures of 50˚C, 60˚C, 70˚C and 80˚C and adjusting mathematical models Fick, Page, Cavalcanti-Mata, Henderson & Pabis and Two Exponential Terms to the experimental data and determine the effective diffusion coefficient using flat plate geometry with a sample with 6.5 mm thickness.

Experiments were performed in Food Engineering Laboratories of the Federal University of Campina Grande, in the city of Campina Grande, PB.

For experiments it was used mature mesquite [Prosopis juliflora (Sw.) DC.], obtained in the city of Serra Branca, PB, with initial moisture content of approximately 20% (wb).

Mesquite samples were received in nylon bags and transported to the site of the experiments, where they were packed in polyethylene film and stored in a horizontal freezer, temperature −50˚C, in order to preserve its characteristics for the study.

Mesquite cleaning was performed with chlorinated water spray at 30 ppm; then rinsing with potable water at room temperature and subsequent drying (with paper towels), allowing thus eliminate of the surface dirt as: earth, debris and impurities of several species that could affect the quality of the product; subsequently, mesquite was subjected to manual cut in order to standardize product at approximately 20 mm length.

Experiments were performed using temperatures of 50˚C, 60˚C, 70˚C and 80˚C, drying air speed of 2 m∙s^{−1}, to constant weight. These parameters were chosen based on the type of raw material, because it is an organic product with sensitive components; as the objective was to preserve maximum nutrients, were not used temperatures above 80˚C; temperatures below 50˚C are not indicated for the study due to very high drying time, thus resulting in high costs and energy consumption.

In drying process was used convective dryer (

The dryer and the electrical resistances were turned on until the desired temperature for each experiment was obtained; any adjustment of temperatures was performed by a temperature controller and a PT100 probe. It was made, also, measures the speeds of air with the aid of the anemometer, and temperature and relative humidity of air, with the aid of thermo-hygrometer.

To determine the kinetics of drying mesquite, weighing were carried out with product in baskets at predetermined intervals until constant weight; all of the experiments were performed in triplicate; experimental data were expressed as a ratio of water content, calculated by Equation (1). Mathematical models of Fick, Page, Cavalcanti-Mata, Two Exponential Terms, and Henderson & Pabis were used to fit to experimental data and are described in

Model | Equation | Reference |
---|---|---|

Fick | [ | |

Page | [ | |

Cavalcanti-Mata | [ | |

Henderson & Pabis | [ | |

Two Exponential Terms | [ |

[

In which:

RX―ratio of water content, dimensionless;

X_{t}―water content at time t, dry basis, bs;

X_{eq}―equilibrium water content, bs;

X_{i}―initial moisture content, bs.

For calculation of the effective diffusivity (D_{ef}) it was used flat plate geometry with a sample with 6.5 mm thick ness, using the Fick model with first term of the series.

For sufficiently long drying times, first term of the series of Fick Equation (n = 0) gives a good estimate solution and Equation (2) can be applied to calculate effective diffusivity.

Linearizing Equation (3.2) and by plotting the graph of ln(RX) versus time, inclination is obtained (K_{0}) according to Equation (3). From the analogy of Page equations, Cavalcanti-Mata and Henderson & Pabis with Fick equation, effective diffusivity was also calculated from Equations (4) and (5) for all drying conditions.

As a criterion to determine the model that best represented the experimental data, it was used coefficient of de termination (R^{2}) and the root mean square error (RMSE), described in Equation (6).

In which:

RMSE―root mean square error;

RX_{pred}―predicted water content ratio;

RX_{exp}―experimental ratio of water content;

N―number of observations.

After drying process, mesquite was packed in flexible film BOPP (biaxially orientated polypropylene) and kept at room temperature of 25˚C ± 3.0˚C.

Drying kinetics and application of mathematical models can be modified by Dring kinetics.

In

Observing

Fick Model | ||||||
---|---|---|---|---|---|---|

Temperature (˚C) | Parameters | |||||

L (m) | D_{ef} (m^{2}∙s^{−1}) | K (s^{−1}) | R^{2} (%) | RMSE | ||

50 | 0.0033 | 0.92 × 10^{−9} | 0.0019 | 99.24 | 0.0238 | |

60 | 0.0033 | 1.22 × 10^{−9} | 0.0028 | 99.77 | 0.0133 | |

70 | 0.0033 | 2.38 × 10^{−9} | 0.0054 | 99.48 | 0.0182 | |

80 | 0.0033 | 2.87 × 10^{−9} | 0.0064 | 98.86 | 0.0268 | |

Page Model | |||||
---|---|---|---|---|---|

Temperature (˚C) | Parameters | ||||

K (s^{−1}) | n | D_{ef} (m^{2}∙s^{−1}) | R^{2} (%) | RMSE | |

50 | 0.0022 | 0.6933 | 0.96 × 10^{−9} | 99.90 | 0.0084 |

60 | 0.0023 | 0.7434 | 1.03 × 10^{−9} | 99.78 | 0.0129 |

70 | 0.0055 | 0.6138 | 2.44 × 10^{−9} | 99.76 | 0.0123 |

80 | 0.0056 | 0.6565 | 2.49 × 10^{−9} | 99.88 | 0.0084 |

Cavalcanti-Mata Model | |||||||||
---|---|---|---|---|---|---|---|---|---|

Temperature (˚C) | Parameters | ||||||||

a_{1} | b | a_{2} | a_{3} | a_{4} | a_{5} | D_{ef} (m^{2}∙s^{−1}) | R^{2} (%) | RMSE | |

50 | 0.5064 | 0.0022 | 0.6861 | 0.4834 | 0.7282 | −0.0075 | 0.95 × 10^{−9} | 99.92 | 0.0076 |

60 | 0.5048 | 0.0024 | 0.7432 | 0.5006 | 0.7433 | −0.0285 | 1.04 × 10^{−9} | 99.88 | 0.0096 |

70 | 0.5337 | 0.0034 | 0.5652 | 0.5337 | 0.5652 | −0.0703 | 1.51 × 10^{−9} | 99.94 | 0.0061 |

80 | 0.5124 | 0.0041 | 0.6316 | 0.5124 | 0.6316 | −0.0252 | 1.81 × 10^{−9} | 99.94 | 0.0063 |

Henderson & Pabis Model | |||||
---|---|---|---|---|---|

Temperature (˚C) | Parameters | ||||

a | K (s^{−1}) | D_{ef} (m^{2}∙s^{−1}) | R² (%) | RMSE | |

50 | 0.8732 | 0.0036 | 1.60 × 10^{−9} | 98.07 | 0.0381 |

60 | 0.8838 | 0.0049 | 2.10 × 10^{−9} | 99.13 | 0.0256 |

70 | 0.8118 | 0.0065 | 2.80 × 10^{−9} | 96.94 | 0.0441 |

80 | 0.8476 | 0.0071 | 3.14 × 10^{−9} | 97.53 | 0.0394 |

Two Exponential Terms Model | ||||||
---|---|---|---|---|---|---|

Temperature (˚C) | Parameters | |||||

a | K_{0} | b | K_{1} | R^{2} (%) | RMSE | |

50 | 0.6118 | 0.0023 | 0.3479 | 0.0413 | 99.79 | 0.0124 |

60 | 0.1812 | 0.0036 | 0.7848 | 0.0442 | 99.84 | 0.0110 |

70 | 0.6550 | 0.0048 | 0.3129 | 0.0559 | 99.81 | 0.0111 |

80 | 0.3386 | 0.0054 | 0.6313 | 0.0679 | 99.81 | 0.0110 |

content, which complies with studies with drying various products: rice [

Still analyzing

It is noticed that mesquite, with initial moisture content of 20% (b.u) led 1230, 990, 780 and 660 minutes to reach equilibrium moisture content: 7.0%, 6.5%, 6.0% and 5.5% (b.u) when subjected to drying process at temperatures of 50˚C, 60˚C, 70˚C and 80˚C, respectively. It is observed trend in the values of equilibrium water content when it decreases with increasing temperature, a fact that can be explained by the increase of the vapor partial pressure difference between drying air and product, so that increases mass transfer rate; this upgrade of energy to the process results in a higher excitation of water molecules, increasing the distance between them and reducing their attractive forces. This kind of behavior is typical of many agricultural products and has been observed in beans drying study [

Diffusion model (2nd Law of Fick) was adjusted to experimental data using the first term of the series, therefore, from that term, diffusion coefficient and adjusting the curves showed very similar values.

It can be seen in ^{−9} m^{2}∙s^{−1}. According to [^{−11} to 10^{−9} m^{2}∙s^{−1}. [^{−11} to 10.42 × 10^{−11} m^{2}∙s^{−1} to temperatures of 30˚C and 70˚C, respectively. [^{−10} to 41.48 × 10^{−10} m^{2}∙s^{−1} to temperatures of 45˚C, 60˚C, 75˚C, 90˚C and 105˚C, respectively.

Diffusivity is understood as a physical phenomenon in which water migrates from center to surface material, and this parameter strongly influenced by drying temperature, as observed in this study, where: with increasing heat transfer rate occasioned by raising drying temperature, it was noticed a higher diffusion coefficient of mesquite. [^{−10} m^{2}∙s^{−1} to crambe seed, using temperatures from 30˚C to 70˚C with an initial moisture content of 22% (b.u).

It is noticed that the value of diffusivity drying temperature of 70˚C is about twice the value found for temperature of 60˚C; this numerical difference to the order of magnitude of 10^{−9} is justified because it is a biological material, wherein drying phenomenon cannot be generalized because they may have intrinsic characteristics that differentiate them from each other.

In

In

It is observed in

Still analyzing

In literature it appears that Page model is quite successfully used in describing drying kinetics of various agricultural products, as several authors, including [^{−1}, that this model expresses satisfactorily the drying phenomena, obtaining coefficient determination of above 99% and medium∙square deviation of less than 0.05.

It can be seen in _{1}”, “a_{2}”, “a_{3}”, “a_{4}” and “a_{5}”, used in mathematical adjustments presented, in general, random behavior; however, Cavalcanti-Mata model represents experimental data in order exceeding 99% (correlation coefficients) and their medium∙squared deviations are smaller than 0.01 at all temperatures.

In Cavalcanti-Mata equation, the value of the parameter “b” is the effective diffusivity as well, since equation derived from model Fick to 2 terms of the series with time correction by means of potential coefficients a_{2} and a4. It is necessary to ask what the effective diffusivity should be considered or would be correct. It is noted that all coefficients are in the same order of magnitude. It is observed also that in Cavalcanti-Mata model, diffusivity data are more consistent because it is not found to exist values that are twice each other in drying function of temperature and there is a proportionality between these values, as observed in equidistance between drying curves.

In

The parameter “a” of Henderson & Pabis model in case of mesquite pod varied from 0.8118 to 0.8838 between drying temperatures of 50˚C and 80˚C, noting a random behavior. [^{−1}, with values for parameter “a” ranging from 0.849 to 0.882.

In

It is observed in _{0}” and “K_{1}” from two-term exponential model increased when temperature of drying increased from 50˚C to 80˚C; parameters “a” and “b” of the same model have randomized behavior for all drying temperatures of mesquite. [_{1}” with increasing of drying temperature, studying the kinetics of mesquite grain.

In

Based on the results of drying mesquite, it is concluded that:

The drying mesquite occurred during the decreasing rate period, and with increasing drying temperature, there was a reduction in total process time, reaching the fastest equilibrium water content ranging from 7.0 5.5% to temperatures of 50˚C to 80˚C, respectively;

Diffusion coefficients, calculated from Fick model increased with increasing of drying temperature of 50˚C to 80˚C, with values of 0.92 to 2.87 × 10^{−9} m^{2}∙s^{−1};

Mathematical models―Theoretical (Fick) and semi theoretical (Page and Cavalcanti-Mata) models satisfactorily represent the experimental data of mesquite drying at temperatures of 50˚C, 60˚C, 70˚C and 80˚C, with larger determining coefficients and 99% medium∙squared deviation less than 0.05 except for Fick model at temperature of 80˚C, where coefficient of determination was 98.86%;

The empirical model of Two Terms Exponential also provided good fit to the experimental data, showing coefficients of determination higher than 99% and lower medium∙square deviation 0.05;

Among the models studied, the model of Henderson & Pabis was presented the lowest determination coefficients ranging from 96.94% (70˚C) to 99.13% (60˚C).

Rennan Pereira de Gusmão,Thaisa Abrantes Souza Gusmão,Mário Eduardo Rangel,Moreira Cavalcanti-Mata,Maria Elita Martins Duarte, (2016) Mathematical Modeling and Determination of Effective Diffusivity of Mesquite during Convective Drying. American Journal of Plant Sciences,07,814-823. doi: 10.4236/ajps.2016.76076