Modeling of Diffusion in Capillary Porous Materials During the Drying Process

ABSTRACT

This paper presents a model of heterogenous diffusion in capillary porous materials during the process of drying. The governing heat and mass transfer equations have been established using the liquid as well as vapor flow. Two models have been presented. Model 1 does not consider the heat conduction while the model 2 has been established by considering the conduction. The developed models and the numerical solutions of the resulting differential equations can take into account the moisture and temperature dependent thermophysical properties of the product. All equations have been established in spherical coordinates but the programme written for the purpose of calculations can be used for other geometries also. Numerical calculations have been performed for gas concrete and tiles using model 1, while model 2 has been used for gas concrete only because of the lack of data for thermophysical properties of the tile. For gas concrete it was seen that conduction has only marginal effect on the drying process and the numerical predictions of the drying process were reasonably accurate.     

Two-Fluid, Two-Dimensional Model for Pneumatic Drying

Pneumatic drying is a widely used process in the chemical industries and includes simultaneous conveying and heat and mass transfer between the particles and the heat gas. The increase in the use of this unit operation requires reliable mathematical models to predict processes in the industrial facilities. In the present study a Two-Fluid model has been used for modeling the flow of particulate materials through pneumatic dryer. The model was solved for a two-dimensional steady-state condition and considering axial and radial profiles for the flow variables. A two-stage drying process was implemented. In the first drying stage, heat transfer controls evaporation from the saturated outer surface of the particle to the surrounding gas. At the second stage, the particles were assumed to have a wet core and a dry outer crust; the evaporation process of the liquid from a particle is assumed to be governed by diffusion through the particle crust and by convection into the gas medium. As evaporation proceeds, the wet core shrinks while the particle dries. The numerical procedure includes discretization of calculation domain into torus-shaped final volumes, solving conservation equations by implementation of the SIMPLE (Semi-Implicit Method for Pressure-Linked Equations) algorithm and controls over coupling of phases by IPSA (Interphase Slip Algorithm). The developed model was applied to simulate a drying process of wet PVC particles in a large-scale pneumatic dryer and to a drying process of wet sand in a laboratory-scale pneumatic dryer. The numerical solutions are compared successfully with the results of independent numerical and experimental investigations. Following the model validation, the two-dimensional distributions of the flow characteristics were examined.     

Two-Fluid,Two-Dimensional Model for Pneumatic Drying

Abstract

Pneumatic drying is a widely used process in the chemical industries and includes simultaneous conveying and heat and mass transfer between the particles and the heat gas. The increase in the use of this unit operation requires reliable mathematical models to predict processes in the industrial facilities. In the present study a Two-Fluid model has been used for modeling the flow of particulate materials through pneumatic dryer. The model was solved for a two-dimensional steady-state condition and considering axial and radial profiles for the flow variables. A two-stage drying process was implemented. In the first drying stage, heat transfer controls evaporation from the saturated outer surface of the particle to the surrounding gas. At the second stage, the particles were assumed to have a wet core and a dry outer crust; the evaporation process of the liquid from a particle is assumed to be governed by diffusion through the particle crust and by convection into the gas medium. As evaporation proceeds, the wet core shrinks while the particle dries. The numerical procedure includes discretization of calculation domain into torus-shaped final volumes, solving conservation equations by implementation of the SIMPLE (Semi-Implicit Method for Pressure-Linked Equations) algorithm and controls over coupling of phases by IPSA (Interphase Slip Algorithm). The developed model was applied to simulate a drying process of wet PVC particles in a large-scale pneumatic dryer and to a drying process of wet sand in a laboratory-scale pneumatic dryer. The numerical solutions are compared successfully with the results of independent numerical and experimental investigations. Following the model validation, the two-dimensional distributions of the flow characteristics were examined.     

A comprehensive review on modeling of pneumatic and flash drying

Pneumatic conveying drying (PCD) is a widely used process in the industries and is a combination of heat and mass transfer and pneumatic handling technology. Drying processes consume large amounts of energy and, therefore, reduction in operating cost will be extremely beneficial for the industry. Many studies have been conducted to model and optimize the pneumatic drying. This review article focuses on the different strategies used in the literature to model pneumatic drying processes. An analysis is provided for the different mathematical modeling and its components such as balance and complementary equations and modeling assumptions. Two-fluid theory, Eulerian granular, and the discrete element method are reviewed as well as gas–solid flow modeling methods. In addition, the numerical methods and the main studied parameters in the field of pneumatic drying are investigated. To this end, heat and mass transfer coefficients, gas and dispersed phase properties are reviewed.     

Drying at Sub-zero Temperatures: Case Study on In-store Drying of Grains

Drying using ambient subzero temperatures is of potential interest for thermosensitive products. Existing theoretical drying models have been used to predict the response of the system to different aeration systems. The model is based on enthalpy balances and includes water freezing and deposition of water on the surface of the commodity. It uses thermophysical properties of the commodity (i.e., maize in this study) and ambient weather data collected from northeastern China. Water within the grain is modelled as bound, free or frozen. The physical state of water under subzero temperatures has been investigated using a differential scanning calorimeter and nuclear magnetic resonance spectrometry. It has been established that the quantity of bound water was around 17%. Thermophysical properties characterizing the drying behavior of maize kernels cv. Huangmo 417, the most common variety grown in northeastern China, were determined under a wide range of moisture contents and drying temperatures. Those were: particle and bulk density, porosity, thermal conductivity, specific heat, thin layer drying, and sorption isotherms. It could be established that the thermal conductivity and specific heat were strongly dependent on temperature and relative humidity and that the sorption isotherms followed the 5-term Guggenheim-Anderson-de-Boer model. The industrial-scale in-store drying experiments in northeastern China have demonstrated the feasibility of in-store drying under subzero conditions. Advantages in terms of reduced susceptibility of maize to mould formation have been established, resulting in improved quality and financial returns to the processor.     

A Thermo-Hydro-Mechanics Bidirectional Coupling Mathematical Model for Drying of Biological Porous Medium

Based on Fickian diffusion theory, Fourier's law of heat conduction and thermoelasticity mechanics, a thermo-hydro-mechanics bidirectional coupling mathematical model has been developed to simulate the hot air convective drying of biological porous media. The transient model, composed of a system of partial differential equations, was solved by finite difference methods. The numerical results were compared with available experimental data obtained during the drying of potatoes. The numerical results obtained using the mathematical model were in good agreement with the experimental data. Numerical simulations of the drying curve variations and the spatio-temporal distributions of moisture, temperature, and drying stresses and strains were evaluated.     

ELECTRICAL-RESISTANCE ELEMENT ARRAYS OPERATING AT HIGH VOLTAGES FOR HEATING AND DRYING IN INDUSTRIAL APPLICATIONS

Abstract

Drying using ambient subzero temperatures is of potential interest for thermosensitive products. Existing theoretical drying models have been used to predict the response of the system to different aeration systems. The model is based on enthalpy balances and includes water freezing and deposition of water on the surface of the commodity. It uses thermophysical properties of the commodity (i.e., maize in this study) and ambient weather data collected from northeastern China. Water within the grain is modelled as bound, free or frozen. The physical state of water under subzero temperatures has been investigated using a differential scanning calorimeter and nuclear magnetic resonance spectrometry. It has been established that the quantity of bound water was around 17%. Thermophysical properties characterizing the drying behavior of maize kernels cv. Huangmo 417, the most common variety grown in northeastern China, were determined under a wide range of moisture contents and drying temperatures. Those were: particle and bulk density, porosity, thermal conductivity, specific heat, thin layer drying, and sorption isotherms. It could be established that the thermal conductivity and specific heat were strongly dependent on temperature and relative humidity and that the sorption isotherms followed the 5-term Guggenheim-Anderson-de-Boer model. The industrial-scale in-store drying experiments in northeastern China have demonstrated the feasibility of in-store drying under subzero conditions. Advantages in terms of reduced susceptibility of maize to mould formation have been established, resulting in improved quality and financial returns to the processor.     

多孔介质对流干燥机理及其模型

在对现有的多孔介质对流干燥传热、传质模型归类分析的基础上,从介质内部热湿迁移机制出发,建立了能较完善、较准确地描述多孔介质在恒速段及降速段热质传递规律的“三耦合-六场量”混合理论模型.同时针对干燥问题数值模拟中的移动边界问题,提出了一种迭代修正的思想,并发展了相应的数值计算方法.对砖的干燥模拟计算结果表明,本文的模型较其他模型具有更好的精确性.     

DRYING OF CLAY. II RHEOLOGICAL MODELISATION AND SIMULATION OF PHYSICAL PHENOMENA

The present study proposes the development of a complete mathematical modelling transfer phenomena involving at the same time heat, mass and momentum transfer during the drying process of clay. Clay is a generic example of colloid materials forming particulate gels. That can be considered as bi-constituent, homogeneous, isotropic, and highly deformable. The model was numerically solved by the finite difference method and validated by comparison of the numerical results with a previous set of experiments data. The simulation has allowed the determination of spatio-temporal evolution within the solid of different variables: temperature fields, moisture contents, displacement, deformation and stresses. The parametric sensibility has been analyzed in the case of thermophysical properties and the external heat transfer coefficient. Various values of external conditions have been analyzed.     

Modelling of batch fluidised bed drying of pharmaceutical granules

This paper presents a mathematical model based on a three-phase theory, which is used to describe the mass and heat transfer between the gas and solids phases in a batch fluidised bed dryer. In the model, it is assumed that the dilute phase (i.e., bubble) is plug flow while the interstitial gas and the solid particles are considered as being perfectly mixed. The thermal conductivity of wet particles is modelled using a serial and parallel circuit. The moisture diffusion in wet particles was simulated using a numerical finite volume method. Applying a simplified lumped model to a single solid particle, the heat and mass transfer between the interstitial gas and solid phase is taken into account during the whole drying process as three drying rate periods: warming-up, constant rate and falling-rate. The effects of the process parameters, such as particle size, gas velocity, inlet gas temperature and relative humidity, on the moisture content of solids in the bed have been studied by numerical computation using this model. The results are in good agreement with experimental data of heat and mass transfer in fluidised bed dryers. The model will be employed for online simulation of a fluidised bed dryer and for online control.     

DRYING OF CLAY. II RHEOLOGICAL MODELISATION AND SIMULATION OF PHYSICAL PHENOMENA

ABSTRACT

The present study proposes the development of a complete mathematical modelling transfer phenomena involving at the same time heat, mass and momentum transfer during the drying process of clay. Clay is a generic example of colloid materials forming particulate gels. That can be considered as bi-constituent, homogeneous, isotropic, and highly deformable. The model was numerically solved by the finite difference method and validated by comparison of the numerical results with a previous set of experiments data. The simulation has allowed the determination of spatio-temporal evolution within the solid of different variables: temperature fields, moisture contents, displacement, deformation and stresses. The parametric sensibility has been analyzed in the case of thermophysical properties and the external heat transfer coefficient. Various values of external conditions have been analyzed.     

Numerical modelling of heat transfer processes with supplementary data

Mathematical models of heat transfer processes are usually studied using formulation that is unique in the mathematical sense. For example, in solving initial-boundary value problems in conduction heat transfer, the mathematical model consists of the governing equation and initial and boundary conditions. An extension of such an approach by introducing the supplementary information (or data) concerning the process has been proposed here as a method for verifying the accuracy of the model equations. This means that the mathematical model consists of more equations than unknowns which leads in consequence to a finite set of probable solutions. A criterion for choosing the most probable solution has been proposed. Special attention has been paid to numerical formulation. Computational methods have been derived using the Lagrange multipliers. Theoretical considerations have been illustrated by computing the temperature distribution inside a laboratory combustion chamber.     

流化床氛围下多孔物料干燥传热传质的数值模拟

用有限差分法数值求解一个热、质传递耦合模型,理论研究多孔物料流化床干燥过程。方程离散采用全隐格式的控制容积方法,三对角矩阵法（TDMA）用来求解线性方程组。选用球形的苹果丁作为多孔物料。在典型操作条件下,通过分析温度、饱和度和压力的分布侧形,讨论了物料内部的热、质传递机理。在对比条件下,考察了气体入口温度、气速和床面积因子对干燥过程的影响。结果表明：干燥过程受气、固相间的耦合传热传质的影响十分明显,干燥时间随气体入口温度和气速的提高而减少;随床面积因子的增大而增加。     

瓷质砖湿坯对流干燥过程的传热传质研究

引用建立于Whitaker的体积平均方程和Darcy定律基础止的多孔介质内部热质传递的等效耦合扩散模型，寻出一组关于液体饱和度、温度和气相压力的新支配方程，应用该方程组对瓷质砖坯体干燥过程进行了数值分析和实验测定。在平均含湿饱和度的变化方面，数值解与实验结果十分吻合。还改变影响坯体干燥过程的一些因素进行计算机模拟计算，通过改变这些因素的大小来考察计算结果，以期获得某些定性或定量的结论，从而用以指导实际生产过程。     

TWO-FLUID MODEL FOR PNEUMATIC DRYING OF PARTICULATE MATERIALS

The Two-Fluid model has been used for modeling the flow of particulate materials through pneumatic dryer. The model was solved for a one-dimensional steady-state condition and was applied to the drying process of wet PVC particles in a large-scale pneumatic dryer and to the drying process of wet sand in a laboratory-scale pneumatic dryer. A two-stage drying process was implemented. In the first drying stage, heat transfer controls evaporation from the saturated outer surface of the particle to the surrounding gas. At the second stage, the particles were assumed to have a wet core and a dry outer crust; the evaporation process of the liquid from a particle assumed to be governed by diffusion through the particle crust and by convection into the gas medium. As evaporation proceeds, the wet core shrinks while the particle dries. The drying process is assumed to stop when the moisture content of a particle falls to a predefined value or when the particle riches the exit of the pneumatic dryer. Our developed model was solved numerically and two operating conditions, adiabatic and given pneumatic dryer wall temperature, were simulated. Comparison between the prediction of the numerical models of Rocha and DryPak, (Pakowski, 1996), which were presented by Silva and Correa (1998), with the prediction of our numerical simulation reviled better agreements with DryPak then with the models of Rocha. The results of the developed model were also compared with experimental results of Baeyens et al. (1995) and Rocha.     

TWO-FLUID MODEL FOR PNEUMATIC DRYING OF PARTICULATE MATERIALS

The Two-Fluid model has been used for modeling the flow of particulate materials through pneumatic dryer. The model was solved for a one-dimensional steady-state condition and was applied to the drying process of wet PVC particles in a large-scale pneumatic dryer and to the drying process of wet sand in a laboratory-scale pneumatic dryer. A two-stage drying process was implemented. In the first drying stage, heat transfer controls evaporation from the saturated outer surface of the particle to the surrounding gas. At the second stage, the particles were assumed to have a wet core and a dry outer crust; the evaporation process of the liquid from a particle assumed to be governed by diffusion through the particle crust and by convection into the gas medium. As evaporation proceeds, the wet core shrinks while the particle dries. The drying process is assumed to stop when the moisture content of a particle falls to a predefined value or when the particle riches the exit of the pneumatic dryer. Our developed model was solved numerically and two operating conditions, adiabatic and given pneumatic dryer wall temperature, were simulated. Comparison between the prediction of the numerical models of Rocha and DryPak, (Pakowski, 1996), which were presented by Silva and Correa (1998), with the prediction of our numerical simulation reviled better agreements with DryPak then with the models of Rocha. The results of the developed model were also compared with experimental results of Baeyens et al. (1995) and Rocha.     

A Simulation Model for Heat Pump Dryer Plants for Fruits and Roots.

Dryer design requires food properties, drying rate and mass-heat transfer coefficients. These values change continuously during drying due to changes in food fractions, particularly the water fraction. The high energy demand and costs allied to inefficient devices, creates a great need for new processing equipment. Along these guide-lines, several heat pump drying research projects were established at the Norwegian University of Science and Technology. The heat pump dryer provides high quality final product as its drying conditions can be controlled. Its efficiency and non-polluting operation come from closed air-refrigerant circuits and from its ability to fully recover the latent heat of moist air as it exits the drying chamber. Most of the above features are quite the opposite of the conventional dryer characteristics. Several experiments were made on heat pump drying of fruits and roots at temperatures from -22.5 to 40°C to obtain data and correlations on thermophysical properties, specific enthalpy and rehydration. Also, tests were done on drying rate, moisture content, drying constant, effective mass diffusivity and heat and mass transfer equations. The next important phase is the development of a simulation model to predict the performance and characteristics of the heat pump dryer plant. The objectives of the present work are to develop and lo test a heat pump dryer simulation model. The simulation provides results on the characteristics of both plant and components which are integrated by heat and mass transfer equations. The program has menus with click-on icons, input and output pop-up dialogue boxes. The usual commands such as, file-open, file-save, edit-delete are available in this program simply called Hpdryer. The model contains moist air psychrometric. natural and conventional refrigerant property libraries. Ammonia is a time-tested, self-alarming and natural refrigerant. It has been used extensively in the past, and it has better thermodynamic and transport properties than halocarbons. Safety is easily attained by design and its restrictive standards have helped increase its use in several countries. There are 36 ammonia installations in Norway and in the United Kingdom, including a drying plant. Ammonia has zero Odp, zero Gwp and the recent R&D has led to viable small-sale heat pump plants. Ammonia and dichlarodifluoromethane refrigerants were used in the test cases simulated by Hpdryer madel.     

A Simulation Model for Heat Pump Dryer Plants for Fruits and Roots.

ABSTRACT

Dryer design requires food properties, drying rate and mass-heat transfer coefficients. These values change continuously during drying due to changes in food fractions, particularly the water fraction. The high energy demand and costs allied to inefficient devices, creates a great need for new processing equipment. Along these guide-lines, several heat pump drying research projects were established at the Norwegian University of Science and Technology. The heat pump dryer provides high quality final product as its drying conditions can be controlled. Its efficiency and non-polluting operation come from closed air-refrigerant circuits and from its ability to fully recover the latent heat of moist air as it exits the drying chamber. Most of the above features are quite the opposite of the conventional dryer characteristics. Several experiments were made on heat pump drying of fruits and roots at temperatures from -22.5 to 40°C to obtain data and correlations on thermophysical properties, specific enthalpy and rehydration. Also, tests were done on drying rate, moisture content, drying constant, effective mass diffusivity and heat and mass transfer equations. The next important phase is the development of a simulation model to predict the performance and characteristics of the heat pump dryer plant. The objectives of the present work are to develop and lo test a heat pump dryer simulation model. The simulation provides results on the characteristics of both plant and components which are integrated by heat and mass transfer equations. The program has menus with click-on icons, input and output pop-up dialogue boxes. The usual commands such as, file-open, file-save, edit-delete are available in this program simply called Hpdryer. The model contains moist air psychrometric. natural and conventional refrigerant property libraries. Ammonia is a time-tested, self-alarming and natural refrigerant. It has been used extensively in the past, and it has better thermodynamic and transport properties than halocarbons. Safety is easily attained by design and its restrictive standards have helped increase its use in several countries. There are 36 ammonia installations in Norway and in the United Kingdom, including a drying plant. Ammonia has zero Odp, zero Gwp and the recent R&D has led to viable small-sale heat pump plants. Ammonia and dichlarodifluoromethane refrigerants were used in the test cases simulated by Hpdryer madel.     

THE DEVELOPMENT AND VALIDATION OF A SYSTEM MODEL FOR A COUNTERCURRENT CASCADING ROTARY DRYER

An overall system model for a countercurrent rotary dryer has been developed with the ullimale aim of assessing controller pairings in these dryers. This model is based on heat and mass balances within dryer regions combined with two subsidiary models, one describing the equipment (which determines particle transport and heat transfer)and the other describing the behaviour of the material (the drying kinetics). Six partial differential equations have been set up to evaluate six state variables: solids moisture content, solids temperature, gas humidity, gas temperature, solids holdup and gas holdup as functions of time and rotary dryer length. A control-volume method has been used to reduce the six partial differential equations with respect to time and the length of the rotary dryer to six ordinary differential equations in time.

The drying model has been implemented in the SPEEDUP flowsheeting package (with FORTRAN subroutines) The model has been validated by fifteen experiments-in a pilot scale countercurrent-flow rotary dryer (0.2m in diameter and 2m in length)     

A COMPREHENSIVE MATHEMATICAL AND NUMERICAL MODELING OF DEEP-BED GRAIN DRYING

This paper deals with comprehensive mathematical and numerical modeling of deep-bed grain drying. In order to build the process model, it is necessary to analyze the transport in both grain and gas phases. Experimental works were carried out for a layer of grain bed in order to validate the models. The models consider momentum, energy, and mass conservation within grain and drying air phase. The two-dimensional dynamic equations of energy and mass conservation are solved numerically by finite-difference method (FDM) and utilizing alternating direction implicit algorithm within grain and drying air phase, while momentum conservation are solved by finite difference method by utilizing Semi-Implicit Method for Pressure-Linked Equations (SIMPLE) algorithm. Furthermore, the models will be applied in consideration with developing and designing dryer in order to simulate humidity and temperature profiles of the drying gas together with moisture content and temperature of grain across dryer in term of the dryer performance. The simulations show that the models can be used to predict the dynamic drying characteristic profiles as well as the superficial velocity of drying air phase across dryer.