Modeling of Drying Curves for Some Foodstuffs Using a Kinetic Equation of High Order

ABSTRACT

Non-linear regression was used to fit a high order kinetic model to drying curves for mango and cassava. For comparison purposes, the same experimental data were also adjusted by non-lines regression to a model based on exponential series, and to another exponential model by simple linear regression. The high order model fit the experimental data with higher precision (r2 Al.98) than the other exponential series models. The proposed higher order model is mathematically simple as compared to the exponential series, and could be used for simulation of drying processes.     

Kaffir Lime Leaf (Citrus hystric DC.) Drying Using Tray and Heat Pump Dehumidified Drying

Desorption isotherms of mature kaffir lime leaves are determined and a nonlinear regression program was applied to the experimental data to fit with any of the four moisture sorption isotherm models. It was found that the modified Halsey model could fit the best. Tray and heat pump–dehumidified drying of kaffir lime leaves were conducted and it was found that the modified Page model was the most effective one. The drying constant was fitted to drying air temperature using the Arrhenius model. Effective moisture diffusivities were determined using the drying data. Heat pump–dehumidified drying reduced drying time and provided dried kaffir lime leaves with higher amount of citronellal than tray drying.     

Analysis of the Drying Kinetics of Brachiaria brizantha (Hochst. Stapf) Grass Seeds at Different Drying Modes

Forage grass seeds have a high economical importance in the Brazilian bovine cattle breeding and seed drying is a fundamental stage of processing to guarantee their stability and allow their storage for long periods. In this context, the objective of this work was to develop a pilot-scale belt dryer, which operates also as a fixed and fluidized bed. Brachiaria brizantha seeds dehydration was analyzed under different air velocities and temperatures. Experimental data of moisture content variation along the drying time was successfully fit to a one-term exponential model. The experimental drying rate points were calculated by approximating the derivatives to finite differences. Its behavior was accomplished fitting curves of the drying rate versus water content and time. Finally, fissure, germination, and vigor rates were analyzed as a function of the drying conditions so that the experimental conditions combine the best process efficiency with the best physiological quality maintenance.     

Analysis of the Drying Kinetics of Brachiaria brizantha (Hochst. Stapf) Grass Seeds at Different Drying Modes

Forage grass seeds have a high economical importance in the Brazilian bovine cattle breeding and seed drying is a fundamental stage of processing to guarantee their stability and allow their storage for long periods. In this context, the objective of this work was to develop a pilot-scale belt dryer, which operates also as a fixed and fluidized bed. Brachiaria brizantha seeds dehydration was analyzed under different air velocities and temperatures. Experimental data of moisture content variation along the drying time was successfully fit to a one-term exponential model. The experimental drying rate points were calculated by approximating the derivatives to finite differences. Its behavior was accomplished fitting curves of the drying rate versus water content and time. Finally, fissure, germination, and vigor rates were analyzed as a function of the drying conditions so that the experimental conditions combine the best process efficiency with the best physiological quality maintenance.     

Mathematical modelling of wheat drying by fractional order and assessment of transport properties

The purpose of this study is to evaluate both the temperature and the initial moisture content of the material in mathematical models of drying. For this, empirical lumped parameter models were fitted based on experimental data of moisture over time. Furthermore, a new semi-empirical drying kinetics model was applied. This model was developed using the generalization of arbitrary order of the Lewis equation obtained through the Laplace transform. After performing the fit, the fractional order model for drying wheat seeds as a temperature function was generalized. Distributed parameter models were also fitted to evaluate the influence of initial moisture content on drying kinetics and to estimate the moisture profile along the position inside the seed. It was verified that the fractional order model presented statistical results similar to models with a higher number of constants, being used to generalize the kinetic drying model for the three wheat cultivars. Generalized models showed better fits for the 3 cultivars with first-degree function, and the maximum global deviation was 10%, 15%, and 20% for the cultivars BRS–Atobá, BRS–Jacana, and BRS–Sanhaço, respectively. In addition, the distribution of moisture content inside the seed was verified by the distributed parameter model, which predicted the experimental data with an overall deviation of around 10%.     

Kinetic studies on radio-selenium uptake by ion exchange resin

Kinetic studies on the sorption of radioactive Se (IV) ion on Amberlite MB9L have been carried out under a wide range of temperature, pH, initial concentration, and resin weight. Analyses of the experimental data by two theoretical models commonly used to explain the ion exchange kinetics and shrinking core model (SCM), and homogeneous diffusion model (HDM) against exponential diffusion decay model (EDM) show a greater preference of EDM over HDM and SCM at all reaction conditions. The obtained regression coefficient values in case of HDM and SCM are in general not high enough to give a satisfactory fit for the experimental data. The estimated values for EDM parameters provides a good satisfactory fit for the yielded adsorption results as the corresponding linear regression values are high. The sorption process is found to be accompanied with fast film diffusion and slow diffusion between the reacted layer and the shrinking un-reacted core within the ion exchange resin. It also shows that ion exchange as the controlling step offers the lowest probability due to the highest obtained diffusion parameters over the normal tested reaction conditions.     

超细矿物掺合料对高强混凝土干燥收缩影响

开展了微珠、超细矿粉及硅灰三种超细矿物掺合料对高强混凝土干燥收缩变化规律的研究,采用干燥收缩模型对试验结果进行拟合,并预测其收缩终值.结果表明:微珠和超细矿粉均能减小高强混凝土干燥收缩,且随掺量增加干燥收缩减小越明显;而硅灰的加入会增大高强混凝土的干燥收缩,其掺量越高,混凝土干燥收缩程度越大.采用指数方程得到的预测曲线随着龄期的发展与实测收缩值接近,拟合优度均达到99％以上.     

A fractional kinetic model for drying of cement-based porous materials

In this work, we attempt to characterize the drying phenomena of cement-based porous materials (CBPMs) using a fractional kinetic model that is represented by a function for the kinetics of complex systems and characterized by a stretched exponential and/or power-law function with two parameters: the order index n and the fractional time index α. The fractional kinetic model recovers the classic Lewis and Page drying models as well as the pseudo first-order and pseudo second-order adsorption/desorption models with appropriate n and α values and boundary conditions. The fractional kinetic and classic drying models (i.e., the Lewis, Henderson and Pabis, Page, and logarithmic models) were used to interpret the experimental drying data for cement pastes and mortars. Rearrangement of the fractional kinetic correlation generates a linear log ?log plot. The results showed that the values obtained using the fractional kinetic model and Page models were in better agreement with the experimental data than the values obtained using the other selected models. The results may also suggest that drying of CBPMs is more than a diffusion-controlled process.     

Mathematical modeling and influence of ultrasonic pretreatment on microwave vacuum drying kinetics of lotus (Nelumbo nucifera Gaertn.) seeds

The people of Southeast Asia often use lotus as a highly sought-after food source. Here, the effects of ultrasonic pretreatment on the drying kinetics of lotus (Nelumbo nucifera Gaertn.) seeds under microwave vacuum drying were investigated. The best fit model to predict the drying kinetics was also proposed. Lotus seeds were subjected to ultrasonic pretreatment at frequencies of 20, 35, and 80?kHz and power intensity of 0.75 and 1.50?W/g for 10?min using an ultrasonic bath and then to microwave vacuum drying. Five different mathematical models were fitted to the experimental data and a newly proposed model was selected based on model with highest regression coefficient (R²), lowest root mean square error (RMSE), sum square error (SSE), and chi-square (χ²), respectively. Time-domain nuclear magnetic resonance and field scanning electron microscope were used to describe the water state of ultrasonic samples and examine microstructure, respectively. The results showed that ultrasonic pretreatment performed at a relatively low frequency and relatively high power intensity had a positive effect on reducing the drying time (6.25–31.25%) during microwave vacuum drying because of the redistribution of water and the formation of microchannels. In parallel studies, the new model showed the best fit to the drying curve.     

Drying Kinetics Prediction of Solid Waste Using Semi‐Empirical and Artificial Neural Network Models

The drying process of organic solid waste is investigated, based on an experimental study involving its drying kinetics. The experiments were conducted in a thin‐layer fixed‐bed dryer under various operational conditions. The problem of selecting the best fit for solid waste moisture content as a function of time is addressed as well, using artificial neural network (ANN) models and four well‐known drying kinetics correlations commonly applied to biological materials. According to the statistical analysis employed, the simulations showed good results for the ANN, and the Overhults model provided optimum agreement with experimental data among all other models evaluated. Empirical correlations between the Overhults model parameters and the drying operational conditions using nonlinear regression techniques were determined.     

Development of a Microwave–Vacuum-Based Dehydration Technique for Fresh and Microwave–Osmotic (MWODS) Pretreated Whole Cranberries (Vaccinium macrocarpon)

A bench scale microwave–vacuum (MWV) dryer was developed using a modified consumer-grade microwave oven. MWV dehydration was first tested as a standalone method on whole frozen–thawed berries. Subsequently, a new combination drying technique was developed employing microwave osmotic dehydration under continuous-flow medium-spray (MWODS) conditions together with MWV as a secondary drying operation. Fresh (frozen–thawed) and MWODS pretreated berries were dried under a range of MWV treatments employing continuous and decreasing microwave power settings (duty cycles). Initial microwave power density for all treatments was approximately 10.2 W/g and magnetron power-on and power-off times varied from 3 to 15 s and 27 to 15 s, respectively. Drying times to 20% (db) were recorded and energy consumption was calculated according to the total magnetron power-on time where overall it was found that drying times and energy consumption decreased with increasing MWV process intensity, where drying times for all MWV treatments were significantly shorter than those of conventional air drying. Drying kinetics were fit using two models (exponential and Page's empirical model), where Page's model better fit the experimental data. The quality of the berries was monitored visually through evidence of scorching in order to screen treatments and establish upper limits of treatment intensity for further studies.     

PARAMETER ESTIMATION OF FICK'S LAW DRYING EQUATION

Distributed parameter drying models such as the Fick's law diffusion model, unlike the lumped parameter model of van Meel whose parameters can be easily estimated by regression, suffer from the difficulty in estimating the parameters of the models quantitatively with accuracy. In the past they were estimated by visual inspection of the theoretical drying curves which fit the experimental drying curve best In this work, a quantitative parameter estimation technique originally suggested by Chavent, is developed by minimizing the integrated squares of error between theoretical and experimental curves over the drying lime (the criterion) subjected to the constraints that the theoretical curve is governed by the constant diffusivity Fick's taw diffusion equation (the constraint). Although the estimation of Fick's law constant diffusivity can be done by using the analytical solution developed by Crank, the use of the Fick's law model here is simply to demonstrate the utility of the proposed technique which can be used in more complex distributed models. The optimization problem is to solve for the adjoint equation for which the value of the Fick's law diffusivity minimizes the criterion. The Lagrangian derivative is solved by using a discrete derivative of the criterion. The theoretical curves are generated by using simple explicit (FSE) and modified Crank-Nicholson (FCR) algorithms The drying of oil palm kernels are used as a case study. Ii is found that the estimated diffusivities of moisture in oil palm kernels range from 0 5 to 5.0 × 10^-10 m²sol;s which are comparable with published data. It is also found that the estimated diffusivity is dependent on the initial moisture content.     

FITTING CASSAVA DRYING KINETIC WITH A HIGH ORDER EQUATION

The experimental drying curves with air flow rates of cassava parallelepipeds were analyzed at different and thickneses to obtain an empirical model for the cassava drying kinetics. Experimental results were modeled with first and higher order kinetic models. The kinetic parameters were fitted by non-linear regression to experimental drying curves with a power law model. The parameters of the power law model were statistically significant in a non-lineal joint interval confidence of 99%. The effect of temperature, air velocity and sample thickness on drying kinetic results, showed a consistent behavior with respect to drying theory.     

Mathematical Modeling of Drying Processes of Selected Fruits and Vegetables

This study investigates the behavior of fruit and vegetable samples during drying. The experimental data are fitted to several different thin-layer drying models. Regression analysis is used to determine model parameters, while statistical indicators serve to evaluate the goodness of fit. The power function model gives the best fit for all examined samples. Based on this model, different drying and heat storage technologies can be combined to ensure that the required residual moisture content of an agricultural product is reached. It is demonstrated on the case of a specific Togolese processing plant that under favorable conditions, fossil fuel consumption can be decreased by 33 %.     

A New Variable Diffusion Drying Model for the Second Falling Rate Period of Paddy Dried in a Rapid Bin Dryer

The objectives of this article is to propose a new drying model for the second falling rate period known as the variable diffusion controlled period that follows after the first falling rate period and to propose a new method to determine the second critical moisture content that separates these two periods. Experimental work on paddy drying at minimum fluidization velocity was carried out in a rapid bin dryer. The effects of operating temperatures (60-120°C) and bed depths (2-6 cm) on the paddy drying characteristics were investigated. It was found that the normalized drying rate of paddy was proportional to the normalized moisture content in the first falling rate period but in the second falling rate period, the normalized drying rate of the material varies exponentially with the normalized moisture content. The different relationship between the normalized drying rate and the normalized moisture content in the first and second falling rate periods indicate that two different mechanism of moisture transport are at work. The new exponential model of the second falling rate period and the linear model of the first falling rate period were found to fit the experimental data very well. Derivation from variable diffusion equation shows that the linear model is the result of constant diffusion coefficient whereas the new exponential model is the result of linear diffusion coefficient. This also implies that the first falling rate period is a constant diffusion controlled period and the second falling rate period is a variable diffusion controlled period. In addition, drying kinetics data of a drying process that fits the exponential model over a very slow drying period will show that the drying process is under the effect of a linear diffusion coefficient. It was also found that the proposed new method to determine the second critical moisture content that distinguishes between the first and second falling rate periods by using a sudden change in the value of the drying rate gradient to a much lower value at that point is more rigorous and yet simpler than the method of determining the specific location of the receding drying boundary since it is based on the behavior of the actual drying kinetic data.     

DRYING KINETICS FOR SOME FRUITS: PREDICTING OF POROSITY AND COLOR DURING DEHYDRATION.

ABSTRACT

Drying kinetics of four fruits (prune, quince, fig and strawberry) were studied by using a simple mass transfer mathematical model involving a characteristic parameter (K) as a function of process variables. The model was tested with data produced in a laboratory air dryer, using non-linear regression analysis. The investigation involved three values of sample thickness (5, 10, 15mm) and three different air temperatures (50, 60, 70?°C).

The parameters of the model were found to be greatly affected by sample thickness and air temperature. The effect of moisture content on the porosity of three fruits (namely avocado, prune and strawberry) was also investigated. A simple mathematical model was used to correlate porosity with moisture content. It was found that porosity increased with decreasing moisture content.

Samples of three fruits (avocado, prune and strawberry) were investigated to estimate color changes during conventional drying at 70?°C. A first order kinetic model was fitted to experimental data with great success. It is found that the color of avocado and strawberry change while the color of prune remains the same.

Furthermore, the water sorption isotherms of three fruits (avocado, prune and strawberry) at 25° temperature were determined experimentally and the parameters at the GAB equation were evaluated by means of non-linear regression analysis. The use of the above equation produced a very good fit.     

DRYING KINETICS FOR SOME FRUITS: PREDICTING OF POROSITY AND COLOR DURING DEHYDRATION.

Drying kinetics of four fruits (prune, quince, fig and strawberry) were studied by using a simple mass transfer mathematical model involving a characteristic parameter (K) as a function of process variables. The model was tested with data produced in a laboratory air dryer, using non-linear regression analysis. The investigation involved three values of sample thickness (5, 10, 15mm) and three different air temperatures (50, 60, 70 °C).

The parameters of the model were found to be greatly affected by sample thickness and air temperature. The effect of moisture content on the porosity of three fruits (namely avocado, prune and strawberry) was also investigated. A simple mathematical model was used to correlate porosity with moisture content. It was found that porosity increased with decreasing moisture content.

Samples of three fruits (avocado, prune and strawberry) were investigated to estimate color changes during conventional drying at 70 °C. A first order kinetic model was fitted to experimental data with great success. It is found that the color of avocado and strawberry change while the color of prune remains the same.

Furthermore, the water sorption isotherms of three fruits (avocado, prune and strawberry) at 25° temperature were determined experimentally and the parameters at the GAB equation were evaluated by means of non-linear regression analysis. The use of the above equation produced a very good fit.     

Effect of two stage, tray and heat pump assisted-dehumidified drying on drying characteristics and qualities of dried ginger

Desorption isotherms for sliced gingers have been measured. A non-linear regression programme was used to fit four moisture sorption isotherm models to the experimental data. The Modified Halsey and Modified Oswin models gave the best fit for X_e = f(RH_e, T) and RH_e = f(X_e, T), respectively. Tray and heat pump dehumidified drying incorporated by single and two stage drying were conducted. It was found that the modified Page model was the most effective. The drying constant was fitted to drying air temperature using the Arrhenius model. Effective moisture diffusivities were determined using the drying data. The heat pump dehumidified drying incorporated by the two stage drying could reduce the drying time at 40 °C by 59.32% and increase 6-gingerol content by 6%. Quality evaluation by 6-gingerol content, rehydration ratio and ΔE* showed the best quality for dried sliced gingers in the heat pump dehumidified drying incorporated by the two stage drying at 40 °C.