Validation of a Method of Determination of Apparent Diffusivity Versus Composition in Solids

Abstract

A method is presented for computing the values of apparent diffusivity in solids with respect to the concentration of the diffusing substance (water or sodium chloride). This method does not require any assumption upon the mathematical relationship between diffusivity and concentration. It can be applied to experimental measurements of local concentration versus position within the solid (profiles) with relatively few measurements (circa 10) and a mathematical smoothing of the experimental data by using an artificial neural network model. The method was first validated on simulated data obtained by using a constant diffusivity value and on experimental profiles when the relation between diffusivity and concentration was given. It was then applied to original experimental moisture profiles obtained by putting gelatin gels with different initial moisture contents into contact for up to 14 days. The method was also successfully applied to five sets of experimental moisture and sodium chloride profiles taken from the literature and obtained from different food products. Apparent diffusivities calculated by our method were found in agreement with those obtained by authors using different numerical methods to compute the diffusivity values.     

Effective Moisture Diffusivity Estimation from Drying Data. A Comparison Between Various Methods of Analysis

Moisture diffusivity is the most crucial property in drying calculations. Literature data are scarce due to the variation of both experimental measurement techniques and methods of analysis. The effect of using different methods of analysis on the same experimental drying data is examined in this work. Detailed and simplified mathematical models, incorporating moisture diffusivity as model parameter, are applied. It is proved, that significant differences in the calculated values of moisture diffusivity result when different models are used, and probably these differences explain the variation in literature data. Thus, the adoption of a standardised methodology will be of great importance in moisture diffusivity evaluation.

The above findings resulted from the application of four alternative models on the drying data of three common food materials, potato, carrot and apple. A typical pilot plant scale dryer with controlled drying air conditions was used for the experiments. The moisture content dependence of the diffusion coefficient was proved significant at the last drying stage, while the temperature dependence followed the well known Arrhenius relation. The effects of considering external mass transfer and volume shrinkage during drying, were also investigated.     

Effective Moisture Diffusivity Estimation from Drying Data. A Comparison Between Various Methods of Analysis

ABSTRACT

Moisture diffusivity is the most crucial property in drying calculations. Literature data are scarce due to the variation of both experimental measurement techniques and methods of analysis. The effect of using different methods of analysis on the same experimental drying data is examined in this work. Detailed and simplified mathematical models, incorporating moisture diffusivity as model parameter, are applied. It is proved, that significant differences in the calculated values of moisture diffusivity result when different models are used, and probably these differences explain the variation in literature data. Thus, the adoption of a standardised methodology will be of great importance in moisture diffusivity evaluation.

The above findings resulted from the application of four alternative models on the drying data of three common food materials, potato, carrot and apple. A typical pilot plant scale dryer with controlled drying air conditions was used for the experiments. The moisture content dependence of the diffusion coefficient was proved significant at the last drying stage, while the temperature dependence followed the well known Arrhenius relation. The effects of considering external mass transfer and volume shrinkage during drying, were also investigated.     

Convective drying of osmotically dehydrated apples described by three-dimensional numerical solution of the diffusion equation with analysis of water effective diffusivity spatial distribution

Abstract

This study presents two liquid diffusion models to represent the convective drying of apple, osmotically dehydrated in sucrose solution, cut into parallelepiped-shaped pieces. Model 1 considered water diffusivity and the volume of the slices with constant values. Model 2 considered water effective diffusivity and the dimensions of the slices as variable. The numerical solution of the three-dimensional diffusion equation in Cartesian coordinates was obtained through the finite volume method, with a fully implicit formulation and boundary condition of the third kind. Process parameters were estimated by an optimizer using experimental data. A spatial distribution analysis was carried out for water effective diffusivity and moisture content in the apple slices. The results showed that the concentration of the osmotic solution used in the pretreatment influenced the drying process and that the mathematical model that considered a variable diffusivity and shrinkage was more suitable to describe the experimental data.     

DRYING OF ONION: EFFECTS OF PRETREATMENT ON MOISTURE TRANSPORT

ABSTRACT

This study investigated the drying of osmosed and fresh onions. Onion slices (0.8 × 0.8 × 0.15 cm) soaked in sodium chloride solutions (10 and 15% w/w) for 60 min at 22°C were submitted to air drying. The experimental kinetics data obtained were employed to determine effective diffusivity, using a mass transfer model based on Fick's law of diffusion applied to thin slabs. The results show those samples soaked in the 10% NaCl solution had faster drying rates and larger moisture diffusion coefficients. The drying time of onions can be reduced to less than half by introducing an hour of osmotic dehydration in a salt solution. The dried previously osmosed samples presented a more natural coloration than the untreated ones did.     

Moisture Diffusivity in Rice Components During Absorption and Desorption

Moisture diffusivity values of different rice kernel components, namely endosperm, bran and husk, are required to solve mathematical models describing absorption and desorption processes. In addition to the rice variety and temperature, the moisture diffusivity also depends on its instantaneous moisture content or water activity (a_w) and whether rice is absorbing or desorbing moisture. This research was undertaken to determine moisture diffusivity values of rough rice components in different a_w ranges during absorption and desorption. Experiments were performed to measure sorption rates of different rice forms, including white rice, brown rice, and rough rice kernels. Mathematical models were developed to predict their moisture distribution during moisture sorption processes. These models were solved by finite element method using Comsol Multiphysics® simulation program. Moisture diffusivity values of different rice components were calculated and found to be different during absorption and desorption. Diffusivity of rice endosperm was higher during desorption than absorption at a_w higher than 0.20 and increased with an increase in a_w in 0.20–0.80 a_w range. Diffusivity of bran remained almost the same with a_w while diffusivity of husk decreased with an increase in a_w. Results obtained in this research demonstrated that the moisture diffusivity of different rice components varies significantly with the change in water activity or moisture and should be accounted in the mathematical models.     

A transient mathematical model for maximum respiration activity and oxygen diffusion coefficient estimation in non-steady-state biofilms

A transient mathematical model was established in order to evaluate oxygen diffusivity in non-steady-state biofilms. A submerged fixed bed biofilm system with efficient medium recirculation was investigated for p-toluenesulphonic acid degradation by Comamonas testosteroni T-2 in a multi-species biofilm. Static mixer elements (Sulzer Chemtech Ltd, Switzerland) were used as a support matrix for biofilm formation. Biofilm respiration was estimated using the dynamic gassing-out oxygen uptake method. Based on the dissolved oxygen concentration profiles, the oxygen diffusion coefficient and the maximum respiration activity were calculated. The values of the dissolved oxygen diffusion coefficient varied with biolfilm development and values reported here (2×10⁻¹⁰–1.2×10⁻⁹ m² s⁻¹) are in good agreement with literature data. Calculated oxygen consumption rates fit well with values obtained in respirometry tests with washed out biofilms. The knowledge of diffusivity changes in biofilms is particularly important for removal capacity estimation and appropriate reactor design. © 1998 Society of Chemical Industry     

Effective Diffusivity and Evaporative Cooling in Convective Drying of Food Material

This article presents mathematical models to simulate coupled heat and mass transfer during convective drying of food materials using three different effective diffusivities: shrinkage dependent, temperature dependent, and the average of those two. Engineering simulation software COMSOL Multiphysics was utilized to simulate the model in 2D and 3D. The simulation results were compared with experimental data. It is found that the temperature-dependent effective diffusivity model predicts the moisture content more accurately at the initial stage of the drying, whereas the shrinkage-dependent effective diffusivity model is better for the final stage of the drying. The model with shrinkage-dependent effective diffusivity shows evaporative cooling phenomena at the initial stage of drying. This phenomenon was investigated and explained. Three-dimensional temperature and moisture profiles show that even when the surface is dry, the inside of the sample may still contain a large amount of moisture. Therefore, the drying process should be dealt with carefully; otherwise, microbial spoilage may start from the center of the dried food. A parametric investigation was conducted after validation of the model.     

Effects of material and environmental parameters on chloride penetration profiles in concrete structures

On the basis of the transport mechanism of chloride ion, a prediction model of chloride penetration into concrete structures has been developed. The model includes the diffusion of chloride and its dependences on temperature, age, relative humidity, chloride binding and chloride convection by moisture transport. The experimental program has been set up to verify the model developed in the present study. Several series of concrete specimens were immersed in 3.5% chloride solutions for 15 weeks, and the chloride profiles of the specimens were measured and compared to the predicted chloride profiles. In addition, field measurements have been also conducted. From 10-year-old bridge piers, the chloride profiles in concrete under tidal zone were measured and compared with the predicted chloride profiles. The effects of chloride binding, relative humidity, temperature, exposure condition, and age-dependence on the chloride penetration in concrete were clarified from the present analyses. It was found from the present study that all these variables affect greatly the chloride penetration profiles in concrete. The comparison of the laboratory and field test data with the present theory confirms that the proposed model can be realistically used to predict the penetration of chloride ions into concrete structures under sea environments. Further, these results may be efficiently used for the realistic assessment and design for durability of concrete structures.     

Experimental and theoretical investigation of molecular diffusion coefficient of propylene in NMP

In this study, the absorption of propylene in N-methyl pyrolidone (NMP) was experimentally performed at three different temperatures (276.15, 293.15, and 328.15 K) using the pressure decay method and as a result, the equilibrium data, Henry's law constants, and kinetic data were reported. It was shown that the solubility and diffusivity are two important factors affecting the kinetic behavior of the system. This absorption system was mathematically modeled using Fick's second law accompanied by a time dependent boundary condition. An analytical method followed by numerical optimization was used to estimate the diffusion coefficient of propylene in NMP at different operating temperatures. The results demonstrated that the calculated diffusion coefficient obeys an Arrhenius type model. The resulting mathematical model was applied to calculate the number of absorbed moles of the gas. It shows a deviation of about 10% in comparison with the experimental measurements. Furthermore, the time dependent concentration profile along the liquid depth was also predicted.     

DIFFUSION-CONTROLLED ADSORPTIVE UPTAKE OF COAL AND PETROLEUM ASPHALTENES IN A NiMo/Al2O3 HYDROTREATING CATALYST

In this work, hindered diffusion of one coal and two petroleum asphaltenes was studied by adsorptive uptake in THF from a bath surrounding a NiMo/Al₂O₃ catalyst. A mathematical model for the adsorption-diffusion of asphaltenes was developed. The model parameters were obtained by simulating the experimental data with the model solution. Several asphaltene fractions were defined via SEC (Size Exclusion Chromatography), with the molecular weight of each fraction being determined by its elution characteristics. It was found that both the coal and petroleum asphaltenes have very broad molecular weight distributions; however, the molecular weights of the coal asphaltenes (50-2000) were much smaller than those of the petroleum asphaltenes (200-30000). The uptake rates for asphaltene fractions with different molecular weights varied, depending on their diffusion rates and adsorption constants. Simulation results showed that even though the properties of coal and petroleum asphaltenes were quite different, the values of model parameters for the fractions of the three asphaltenes had the same trend and could be estimated by same numerical expressions; with increasing molecular weight of the fraction, the adsorption constant monotonically increased, and the effective diffusivity decreased. The experimental uptake data for the three asphaltenes as a function of molecular weight were well represented by same mathematical model.     

存在浓度峰的氯离子分布曲线及其扩散参数的计算方法

干湿交替下氯离子分布曲线中可能存在氯离子浓度峰。本文主要研究了不同水灰比对氯离子浓度峰的影响,以及适合这种氯离子分布曲线计算扩散参数的方法。实验结果显示,峰值氯离子浓度( Cmax )随水灰比呈双曲线关系增大,而浓度峰出现的位置与水灰比呈较好线性增长关系。此外,去掉氯离子增大阶段并以浓度峰出现的深度为零点拟合氯离子下降阶段的方法较为适合被应用于出现浓度峰的氯离子分布曲线中来获得扩散参数。     

Determination of appropriate effective diffusivity for different food materials

Effective diffusivity is the most important key parameter needed in the analysis, design, and optimization of heat and mass transfer during food drying process. In general, two types of effective diffusivities are used to develop the mathematical modeling of food drying, namely, moisture-dependent effective diffusivity (MDED) and temperature-dependent effective diffusivity (TDED). However, no study has extensively investigated which effective diffusivity is more accurate in predicting drying kinetics. The main goal of this study is to determine the appropriate effective diffusivity for predicting the drying kinetics. Drying models were developed for different fruits and vegetables based on moisture-dependent and temperature-dependent effective diffusivities. COMSOL Multiphysics, a finite element-based engineering simulation software is used to solve the coupled heat and mass transfer equations. 3D moisture profiles were developed to investigate the spatial moisture distribution during drying. Extensive experimental investigation on five types of fruits and vegetables was conducted and results were compared with the simulated results. The experiments were repeated thrice, and the average of the moisture content at each value was used for constructing the drying curves. Close agreement between experimental and simulated results validates the models developed. It was observed that the moisture profile and temperature profile in case of MDED were more closely fitted with the experimental results. For all fruits and vegetables, the moisture ratio with MDED was significantly lower than moisture ratio with TDED. This finding confirms that the MDED is more accurate for predicting kinetics in food drying. Moreover, the moisture ratio of apple was lowest whereas pear showed the highest moisture ratio. On the other hand, carrot showed a considerably lower moisture ratio compared to potato.     

MEASUREMENT OF WATER DIFFUSIVITY IN AQUEOUS LITHIUM BROMIDE AND LITHIUM CHLORIDE SOLUTIONS

A relatively simple method was employed for measurement of water diffusivity in aqueous lithium bromide and lithium chloride solutions. The twin bulb apparatus used for these measurements was developed using an analogy between this apparatus and the conventional diaphragm cell apparatus. Tritiated water (TOH) was used as a tracer for these experiments because of its chemical similarity and proximity to the molecular weight of water. High tracer activity used at the beginning of the experiments allowed the use of relatively shorter time duration for each experiment (s;≈ 20 h) and a quasi-steady state equation to calculate the diffusivity from the observed tracer activity data

Initially, the water diffusivity in lithium bromide solutions for concentrations varying from 0.5 M to 3 M (22.1 weight percent) was measured to obtain a comparison with published values. The lithium bromide concentration was further varied from 3 M to 11 M (57.4 weight percent) to obtain data in the concentration range usually employed in absorption cooling applications, which is near the solubility limit. The water diffusivity was found to vary with lithium bromide concentration, and to have a maximum of 16.7 × 10^-10m²/s at 4 M. These diffusivity values were also compared with those obtained using an equation proposed by Rosevaere etal. (1941) for nonideal solutions. The water diffusivity in lithium chloride solutions was measured at concentrations of 7 M (25.7 weight percent) and 8 M (28.87 weight percent) to obtain a comparison with published values. The lithium choride concentration was further varied from 8 M to 13 M (43.24) weight percent) to gather data beyond those of other researchers. Similar to the trend of water diffusivity in lithium bromide solutions, water diffusivity in lithium chloride solutions was also found to vary with concentration with a minimum of 6.2 × 10^-10m²/s near the solubility limit.     

Mass Transfer in the Celery Slice: Effects of Temperature, Moisture Content, and Density on Water Diffusivity

Statistical tests were applied to determine the effects of temperature, moisture content, density, and porosity of material on the effective moisture diffusion coefficient during convective drying of root celery. In biological materials with colloidal capillary-porous structure (like root celery), which shrink considerably during drying and show high heterogeneity, the effective water diffusion coefficient depends not only on material temperature and moisture content, but also on its density. It was found that statistical tests can be applied to predict which independent variables should describe the water diffusivity in colloidal capillary-porous materials. A mathematical model of the effective water diffusion coefficient in root celery was formulated as Arhenius-type equation with moisture content of the raw material, its temperature and density as independent variables.     

A Simple Data Processing Approach for Drying Kinetics Experiments

A simple mathematical approach is proposed to be applied to drying kinetics raw data processing. The data collected in a drying experiment of powder cork under constant air drying conditions served as case study to present the methodology. Two functions (linear and third-degree polynomial) were used to fit solid moisture content in the constant drying rate and the falling rate periods. The drying rate curve was obtained by differentiation and the time at which the drying rate period's transition occurs was determined iteratively until virtually continuous functions were achieved. The critical moisture content was easily identified and two falling drying rate periods were detected.

The powder cork moisture decrease was also used to test several semiempirical models available in the literature. The Logarithmic, Midilli, and Page Modified I models were the ones that revealed the best correlations performance. When the methodology proposed was applied using these models, the critical moisture content was underpredicted.

The effective moisture diffusivity and the activation energy were also obtained for powder cork after the proposed mathematical approach has been applied on the raw data obtained in experiments performed at different air drying temperatures.     

Mass Transfer in the Celery Slice: Effects of Temperature,Moisture Content,and Density on Water Diffusivity

Abstract

Statistical tests were applied to determine the effects of temperature, moisture content, density, and porosity of material on the effective moisture diffusion coefficient during convective drying of root celery. In biological materials with colloidal capillary-porous structure (like root celery), which shrink considerably during drying and show high heterogeneity, the effective water diffusion coefficient depends not only on material temperature and moisture content, but also on its density. It was found that statistical tests can be applied to predict which independent variables should describe the water diffusivity in colloidal capillary-porous materials. A mathematical model of the effective water diffusion coefficient in root celery was formulated as Arhenius-type equation with moisture content of the raw material, its temperature and density as independent variables.