Mathematical Modeling of Drying Kinetics for Apricots: Influence of the External Resistance to Mass Transfer

The drying curves of halved and deseeded apricots obtained during convective drying at different temperatures (from 50 to 90°C) have been examined, and a diffusional model, solved by a finite elements method, has been proposed to simulate the drying kinetics. The importance of taking into account both the internal and the external resistances to mass transfer when modeling the drying curves is discussed. Due to the geometry of halved apricots as a hemisphere losing water only through the flat section, the mass transfer coefficient (k_c) was not correctly estimated through an empirical correlation. Only the identification of this coefficient from the experimental results allowed an accurate simulation, decreasing the mean relative error from 12.3±3.8% when k_c was calculated through the Pasternak and Gauvin correlation to 2.9±1.0% when k_c was identified.     

The Effect of External Mass Transfer Resistance during Drying of Fermented Sausage

The drying mechanism of fermented sausages (sucuks) that were cylindrical rod shaped, 40 cm long and 4 cm diameter, during ripening under natural convection conditions at different temperatures (15 to 30°C) was examined. To simulate the experimental drying curves, three empirical models and a diffusional model assuming negligible external mass transfer resistance were evaluated. The drying rate curves of sucuk samples were also simulated taking into account the influence of the external mass transfer resistance. The equation was solved using the trial-and-error solution algorithm developed in this study and the mass transfer coefficient, k _c , and effective moisture diffusivity, D _eff , were simultaneously determined (1.44 × 10^?8 to 1.93 × 10^?8 m/s and 4.30 × 10^?10 to 6.85 × 10^?10 m²/s, respectively). The proposed model considering the effect of external resistance allowed the accurate simulation of the experimental drying data of sucuks at different temperatures.     

Modeling Hydrodynamics And Mass Transfer Parameters In A Continuous Ozone Bubble Column

A computer model based on the establishment of mass balance equations and on the model of fluids flow “stirred tank in series” was developed in order to calculate the ozone transfer coefficient k_La and kinetic constant k_c of ozone consumption by water. On the basis of experimental data, the correlation for gas holdup ε_g and bubble diameter d_vs, were proposed and used to calculate the specific interfacial area a. The liquid-phase mass transfer coefficient k_L for ozone was evaluated from a and the k_La data.     

From Laboratory Experiments to Design of a Conveyor-Belt Dryer via Mathematical Modeling

Abstract

A conveyor-belt dryer for picrite has been modeled mathematically in this work. The necessary parameters for the system of equations were obtained from regression analysis of thin-layer drying data. The convective drying experiments were carried out at temperatures of 40, 60, 80, and 100°C and air velocities of 0.5 and 1.5 m/sec. To analyze the drying behavior, the drying curves were fitted to different semi-theoretical drying kinetics models such as those of Lewis, Page, Henderson and Pabis, Wang and Singh, and the decay models. The decay function (for second order reactions) gives better results and describes the thin layer drying curves quite well. The effective diffusivity was also determined from the integrated Fick's second law equation and correlated with temperature using an Arrhenius-type model. External heat and mass transfer coefficients were refitted to the empirical correlation using dimensionless numbers (J _h , J _D  = m · Re ⁿ ) and their new coefficients were optimized as a function of temperature. The internal mass transfer coefficient was also correlated as a function of moisture content, air temperature, and velocity.     

Liquid-solid mass transfer in a rotating packed bed reactor with structured foam packing

A rotating packed bed (RPB) reactor has substantially potential for the process intensification of heterogeneous catalytic reactions. However, the scarce knowledge of the liquid–solid mass transfer in the RPB reactor is a barrier for its design and scale-up. In this work, the liquid–solid mass transfer in a RPB reactor installed with structured foam packing was experimentally studied using copper dissolution by potassium dichromate. Effects of rotational speed, liquid and gas volumetric flow rate on the liquid–solid mass transfer coefficient (k_LS) have been investigated. The correlation for predicting k_LS was proposed, and the deviation between the experimental and predicted values was within ± 12%. The liquid–solid volumetric mass transfer coefficient (k_LSa_LS) ranged from 0.04–0.14 1⁻¹, which was approximately 5 times larger than that in the packed bed reactor. This work lays the foundation for modeling of the RPB reactor packed with structured foam packing for heterogeneous catalytic reaction.     

Optimization of the Drying Process of Carrot (Daucus carota v. Nantes) on the Basis of Quality Criteria

The drying curves and the degradation kinetics of three different quality attributes (total carotenoids (TC) and total polyphenols (TP) contents and antioxidant activity (AA)) of carrots during drying at different temperatures (from 40 to 90°C) have been experimentally evaluated and modeled. A diffusional model taking into account the solid shrinkage and both the external and internal water transfer resistances was used to accurately represent the water transfer in carrot during drying (average mean relative error (MRE) of 3.3 ± 0.6%). The effective moisture diffusivity was found to follow the Arrhenius relationship (E_a = 76.0 kJ/mol) and the mass transfer coefficient a linear dependence with air temperature. The Weibull model was used to satisfactorily simulate the degradation kinetics of the three quality attributes considered (average MRE of 2.8 ± 1.2% for TC content, 5.7 ± 1.0% for TP content, and 3.6 ± 1.8% for AA); these were the E_a of 52.7 kJ/mol for TC; 22.1 kJ/mol for TP; and 27.5 kJ/mol for AA kinetics. By using the proposed models, the estimated optimum drying temperature to best retain the total carotenoids content ranged between ca. 42–46°C; meanwhile, in order to maintain the TP content and the antioxidant activity at the highest levels, the drying needed to be carried out at temperatures of ca. 60–75°C. These results indicate that the TC retention is more influenced by the drying temperature, while the TP and AA retentions are more sensitive to drying time exposure. However, it was possible to establish a global optimum air temperature which ranged between 52.6 and 57.7°C, decreasing the TC, TP, and AA retentions by less than 2.2% from their respective optimal values.     

Characterization of isotherms and thin-layer drying of red kidney beans,Part II: Three-dimensional finite element models to estimate transient mass and heat transfer coefficients and water diffusivity

Three-dimensional finite element models with consideration of shrinkage and irregular shape were developed to estimate the relationships among the transient heat and mass transfer coefficients, the transient water diffusivity, and the temperature and moisture content of the red kidney beans being dried under different drying conditions. An equation was developed to calculate the transient mass transfer coefficient using the measured time–moisture content data. This calculated transient mass transfer coefficient was further used to calculate the transient heat transfer coefficient. To verify the predicted temperature on the surface of the red kidney beans, surface temperature was measured using a handhold infrared thermometer. These measured temperature and time–moisture content data were used to determine the transient water diffusivity using the least square method when the red kidney bean kernel experienced a shrinkage during drying. Strong relationship among the transient heat and mass transfer coefficients, the water diffusivity, and the ratio of the transient heat and mass transfer coefficients was revealed. This relationship could be used to predict temperature and moisture content of the red kidney beans during the entire drying period. The Lewis number?=?27, and the ratio of the transient heat over mass transfer coefficients was 10765?J?m^?3?k^?1 at 30 and 40°C, and 10729?J?m^?3?k^?1 at 50°C. Shrinkage did not significantly influence the value of the estimated transient water diffusivity.     

Experimental Study of the Heat and Mass Transfer During Drying in a Fluidized Bed Dryer

ABSTRACT

Drying curves obtained in a pilot-scale fluidized bed dryer using biological source solids (sawdust, soya and fish meal) were used to estimate the parameters involved in heat and mass transfer phenomenas: heat transfer coefficient and moisture diffusivity coefficient. Parameters involved in mass transfer were estimated from drying models based on diffusional mechanisms and others that in addition consider internal and external resistance to the mass transfer. The estimate ef ective diffusivity coefficient was between 2x10-11 to lx10 (m2/s) for the considered products. Heat transfer coefficient was estimated from drying data points in the constant drying rate period when the external resistance to the mass transfer controls the process.     

Gas‐Liquid Mass Transfer in Fibrous Bed Reactor with Counter‐Current Liquid Recycle

In this work, the gas‐liquid mass transfer in a lab‐scale fibrous bed reactor with liquid recycle was studied. The volumetric gas‐liquid mass transfer coefficient, k_La, is determined over a range of the superficial liquid velocity (0.0042–0.0126 m.s^–1), gas velocity (0.006–0.021 m.s^–1), surface tension (35–72 mN/m), and viscosity (1–6 mPa.s). Increasing fluid velocities and viscosity, and decreasing interfacial tension, the volumetric oxygen transfer coefficient increased. In contrast to the case of co‐current flow, the effect of gas superficial velocity was found to be more significant than the liquid superficial velocity. This behavior is explained by variation of the coalescing gas fraction and the reduction in bubble size. A correlation for k_La is proposed. The predicted values deviate within ± 15 % from the experimental values, thus, implying that the equation can be used to predict gas‐liquid mass transfer rates in fibrous bed recycle bioreactors.     

Electrodissolution of aluminium thin film microband electrodes

This paper discusses the electrodissolution of aluminium thin films as microband electrodes (length = 5 × 10^–3 m) in terms of mass transfer determined by voltammetry and a.c.-impedance techniques as a function of bandwidth (20 to 2000 nm) in 0.1m NaOH solution. The anodic polarization curves of the aluminium microband electrodes show that current density is enhanced with decreasing bandwidth. The ac impedance response suggests that a steady-state diffusion layer appears the more markedly, the smaller the bandwidth. The anodic polarization curves are analysed on the basis of the combined Butler-Volmer high field approximation and the semi-cylindrical diffusion field approximation. As a result of the analysis, the electrodissolution proceeds by a mixed kinetic-mass transfer controlled reaction. The analysis also makes it possible to distinguish the semi-cylindrical diffusive mass transfer contribution to the electrodissolution from the kinetic contribution, i.e., mass transfer index linearly diminishes with decreasing bandwidth. The increased current density is attributable to the decreased mass transfer contribution, i.e., the more predominant semi-cylindrical diffusive mass transfer as compared to laminar diffusive mass transfer.Nomenclature k _a anodic kinetic constant - k _c cathodic kinetic constant - F Faraday constant - _s kinetic transfer coefficient for anodic reaction - _c kinetic transfer coefficient for cathodic reaction - c ^s surface concentration - V anodic polarization - D diffusion coefficient - d diffusion layer thickness - z number of electrons transferred - l length of microband electrode - w bandwidth of microband electrode - r radius of cylinder     

Analysis of the Shrinkage Effect on Mass Transfer During Convective Drying of Sawdust/Sludge Mixtures

Convective drying of wastewater sludges and sawdust/sludge mixtures was studied. The first part of this work was an experimental study performed in a cross-flow convective dryer using 500 g of wet material extruded through a disk with circular dies of 12 mm. The results showed that the sawdust addition has a positive impact on the drying process from a mass ratio of 2/8, on a dry basis, with observed drying rates higher than the original sludge. The second part of this work consisted of developing a drying model in order to identify the internal diffusion coefficient and convective mass transfer coefficient from the experimental data. A comparison was made between fitted drying curves, well represented by the Newton's model, and the analytical solutions of the diffusion equation applied to a finite cylinder. Variations of dimensional characteristics, such as the volume and exchange surface of the sample bed, were obtained by X-ray tomography. This technique allowed us to confirm that shrinkage, which is an important phenomenon occurring during sludge and sawdust/sludge mixture drying, must be taken into account. The results showed that both the internal diffusion coefficient and convective mass transfer coefficient were affected by mixing and sawdust addition. The internal diffusion coefficient changed from 7.77 × 10^?9 m²/s for the original sludge to 7.01 × 10^?9 m²/s for the mixed sludge and then increased to 8.35 × 10^?9 m²/s for the mixture of a mass ratio of 4/6. The convective mass transfer coefficient changed from 9.70 × 10^?8 m/s for the original sludge to 8.67 × 10^?8 m/s for the mixed sludge and then increased to 12.09 × 10^?8 m/s for the mixture of a mass ratio of 4/6. These results confirmed that sawdust addition was beneficial to the sludge drying process as the mass transfer efficiency between the air and material increased. Reinforcing the texture of sludge by adding sawdust can increase the drying rate and decrease the drying time, and then the heat energy supply will be reduced significantly. The study also showed that neglecting shrinkage phenomenon resulted in an overestimation for the internal diffusion coefficient for the convective drying of sludges and sawdust/sludge mixtures.     

Influence of the Addition of Dietary Fiber on the Drying Curves and Microstructure of a Dry Fermented Sausage (Sobrassada)

The formulation of a dry fermented sausage has been modified by the addition of carrot dietary fiber (CDF; 3, 6, 9, and 12% [w/w]), and the influence of this change on the drying curves and food microstructure has been studied. The CDF content influenced the initial moisture content as well as the drying rate. A diffusion model taking into account the change in the product formulation has been proposed to simulate the drying curves. A constant mass transfer coefficient of 2.53 × 10^?8 m/s was obtained and the effective water diffusivity varied exponentially with the CDF content from 0.99 × 10^?11 m²/s (0% CDF) to 2.08 × 10^?11 m²/s (12% CDF). The simulation of the drying curves was satisfactory (mean relative error of 0.5 ± 0.1%). No differences in the microstructure related to the proteolytic process were found among samples with different CDF contents.     

Influence of Pressure on the Hydrodynamics and Mass Transfer Parameters of an Agitated Bubble Reactor

The present study deals with the pressure effects on the hydrodynamic flow and mass transfer within an agitated bubble reactor operated at pressures between 10⁵ and 100 × 10⁵ Pa. In order to clarify the flow behavior within the reactor, liquid phase residence time distributions (RTD) for different operating pressures and gas velocities ranging between 0.005 m/s and 0.03 m/s are determined experimentally by the tracer method for which a KCl solution is used as a tracer. The result of the analysis of the liquid‐phase RTD curves justifies the tank‐in‐series model flow for the operating pressure range. Good agreement is obtained between theoretical and experimental results assuming the reactor is operating as perfectly mixed. Two parameters characterizing the mass transfer are identified and investigated in respect to pressure: the gas‐liquid interfacial area and volumetric liquid‐side mass transfer coefficient. The chemical absorption method is used. For a given gas mass flow rate, the interfacial area as well as the volumetric liquid mass transfer coefficient decrease with increasing operating pressure. However, for a given pressure, a and k_La increase with increasing gas mass flow rates. The mass transfer coefficient k_L is independent of pressure.     

Evaluation of Mass Transfer Resistances from Drying Experiments

A new approach to experimental evaluation of mass transfer resistances from drying experiments is proposed. A composite model of ginseng root mass transfer, based on one-dimensional treatment of diffusive and convective resistances as additive components of radial mass transfer, was developed. Mass transfer resistance was evaluated from a linear relationship between measured flux and thermodynamic driving force. Partitioning of mass transfer resistance into diffusive (core and skin) and convective (air boundary layer) resistances was done by modification of boundary conditions: (a) high (3 m/s) and low (1 m/s) air velocity; (b) skin removal. Total radial mass transfer resistance was evaluated as (146 ± 6) ? 10⁶ s/m at 38°C, significantly decreasing to (48 ± 1.5) ? 10⁶ s/m at 50°C. Boundary resistance was evaluated as (54 ± 5) ? 10⁶ s/m at 38°C and (26 ± 3) ? 10⁶ s/m at 50°C in the entire range of moisture contents. Core and skin resistances were both moisture dependent: core resistance increased from initial value of (6 ± 1) ? 10⁶ s/m to (61 ± 6) ? 10⁶ s/m toward the end of drying, whereas skin resistance decreased from initial value of (92 ± 5) ? 10⁶ s/m to (25 ± 5) ? 10⁶ s/m at the endpoint of drying. However, the sum of core and skin resistances, which represents composite diffusive resistance of intact ginseng root, was constant and independent of moisture content: (91 ± 4.6) ? 10⁶ s/m at 38°C and (22 ± 1.6) ? 10⁶ s/m at 50°C. The relationship between mass transfer resistance R and drying rate factor k = 1/RC was used for verification of the composite model.     

Improvement of Convective Drying of Carrot by Applying Power Ultrasound—Influence of Mass Load Density

Power ultrasound is considered to be a novel and promising technology with which to improve heat and mass transfer phenomena in drying processes. The aim of this work was to contribute to the knowledge of ultrasound application to air drying by addressing the influence of mass load density on the ultrasonically assisted air drying of carrot. Drying kinetics of carrot cubes were carried out (in triplicate) with or without power ultrasound application (75 W, 21.7 kHz) at 40°C, 1 m/s, and several mass load densities: 12, 24, 36, 42, 48, 60, 72, 84, 96, 108, and 120 kg/m³. The experimental results showed a significant (p < 0.05) influence of both factors, mass load density and power ultrasound application, on drying kinetics. As expected, the increase of mass load density did not affect the effective moisture diffusivity (D_e, m²/s) but produced a reduction of the mass transfer coefficient (k, kg water/m²/s). This was explained by considering perturbations in the air flow through the drying chamber thus creating preferential pathways and, as a consequence, increasing external mass transfer resistance. On the other hand, it was found that the power ultrasound application increased the mass transfer coefficient and the effective moisture diffusivity regardless of the mass load density used. However, the influence of power ultrasound was not significant at the highest mass load densities tested (108 and 120 kg/m³), which may be explained from the high ratio (acoustic energy/sample mass) found under those experimental conditions. Therefore, the application of ultrasound was considered as a useful technology with which to improve the convective drying, although its effects may be reduced at high mass load densities.     

Experimental Study of the Heat and Mass Transfer During Drying in a Fluidized Bed Dryer

Drying curves obtained in a pilot-scale fluidized bed dryer using biological source solids (sawdust, soya and fish meal) were used to estimate the parameters involved in heat and mass transfer phenomenas: heat transfer coefficient and moisture diffusivity coefficient. Parameters involved in mass transfer were estimated from drying models based on diffusional mechanisms and others that in addition consider internal and external resistance to the mass transfer. The estimate ef ective diffusivity coefficient was between 2x10-11 to lx10 (m2/s) for the considered products. Heat transfer coefficient was estimated from drying data points in the constant drying rate period when the external resistance to the mass transfer controls the process.     

Drying paper by impinging jets of superheated steam. Part 1: Constant drying rate in superheated steam

Over a wide range of operating conditions, the drying of paper by impinging jets of superheated steam proceeds by a constant rate period followed by a falling rate period. The constant drying rate, investigated here in the jet temperature range 150 ≤ T_j/ ≤ 465°C and over jet Reynolds numbers of 1000 ≤ Re_j ≤ 12000, is predicted within + 12% by a heat transfer expression using Martin's (1977) correlation for the heat transfer coefficient corrected for mass transfer by the Couette flow approximation factor, and a property ratio to account for the large temperature difference between the jet and the paper.     

A theoretical unsteady‐state model for kL of bubbles based on the framework of wide energy spectrum

An unsteady‐state model for predicting mass‐transfer coefficient k_L of bubbles with mobile surface was developed for turbulent gas‐liquid dispersions. This model was derived from an unsteady‐state convection and diffusion equation through a characteristics method. Unlike the previous work, this model considered the contributions of the amount of fluid structures of different scales (i.e., eddies) existing in the turbulent flows, the frequency of eddies arriving at the surface, the deformation and oscillation of bubbles to mass transfer. This model was based on the framework of wide energy spectrum and can account for the role of eddies of different sizes in mass transfer. Thus the assumption adopted by the previous models that mass transfer was controlled by eddies of certain sizes is no more needed. The overall k_L predicted by the proposed model showed a better agreement with the reported experimental data. © 2015 American Institute of Chemical Engineers AIChE J, 62: 1007–1022, 2016     

Experimental Study of Oxygen Mass Transfer Coefficient in Bubble Column with High Temperature and High Pressure

The volumetric gas‐liquid mass transfer coefficient, k_Lα, for oxygen was studied by using the dynamic method in slurry bubble column reactors with high temperature and high pressure. The effects of temperature, pressure, superficial gas velocity and solids concentration on the mass transfer coefficient are systemically discussed. Experimental results show that the gas‐liquid mass transfer coefficient increases with the increase in pressure, temperature, and superficial gas velocity, and decreases with the increase in solids concentration. Moreover, k_Lα values in a large bubble column are slightly higher than those in a small one at certain operating conditions. According to the analysis of experimental data, an empirical correlation is obtained to calculate the values of the oxygen volumetric mass transfer coefficient for a water‐quartz sand system in two bubble columns with different diameter at high temperature and high pressure.     

Prediction of liquid film mass transfer coefficients in packed columns using liquid holdup

A unique characteristic linear dimension (d), defined as the cube root of the specific liquid holdup (h_sp) in the packed column, was used to correlate successfully the liquid film mass transfer coefficient k_La for gas absorption-desorption for sparingly soluble gases in liquids below loading. To produce this simple, dimensionless correlation, k_La data reported in literature were used, covering a wide range of physical properties of liquids, packings and operating conditions. This new approach showed operating holdup as an important factor in gas liquid mass transfer.