Mathematical Modeling of Drying of Liquid/Solid Slurries in Steady State One-Dimensional Flow

A mathematical model of simultaneous mass, heat and momentum transfer for two-phase flow of a gas and a solid/liquid slurry was developed. The model was applied to calculation of the drying process of coal-water slurry droplets in a gas medium in a steady one-dimensional flow. The model was based on the well-known two-stage drying process for slurry droplets. After the first period of drying, in which the evaporation rate is controlled by the gas phase resistance, the evaporating liquid diffuses through the porous shell (crust) and then, by convection, into the gas medium. Inside the dry external crust of the drop, a wet central core forms, which shrinks as evaporation proceeds. The temperature of the slurry droplet rises. The process ends when the temperature of the dry outer crust reaches the coal ignition temperature in the case of combustion or when the moisture of the particle reaches the final required moisture. The developed model was based on one-dimensional balance equations of mass, energy and momentum for the liquid/solid and gas phases. The system of governing equations was represented by first-order differential equations and solved simultaneously. The numerical solution of the governing equations was obtained using Gear's method. The model permitted calculation     

烟气脱硫喷淋塔的数值模拟

以计算流体力学为基础,在三维坐标系下采用标准k-ε双方程模型求解动量、能量和组分方程,结合浆滴蒸发模型及简化的浆滴脱硫反应模型,以Eu ler-Lagrange方法建立了喷淋塔内烟气脱硫的数值计算模型,模型计算结果与孔华的试验数据符合较好。模型计算结果表明,对于粒径小的喷淋液滴,其烟气脱硫反应和液滴蒸发主要发生在烟气进口附近,而随着液滴粒径的增大,液滴在塔内蒸发和脱硫反应的过程延长。同时,增加烟气温度、降低烟气中SO2的入口质量浓度以及增加液气比均有利于提高脱硫效率。文中模型相对于一维柱塞流模型,能够直观地显示出喷淋塔内的流场、温度场和组分质量浓度场的空间分布。     

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.     

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.     

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.     

A one-dimensional plug-flow model of a counter-current spray drying tower

A one-dimensional numerical model for a detergent slurry drying process in a counter-current spray drying tower is developed for the prediction of the gas and droplet/particle temperature profiles within the tower. The model accommodates droplets/particles over a range of sizes. A semi-empirical slurry droplet drying model is integrated with a counter-current tower simulation based on mass, energy and particulate phase momentum balances in order to calculate the drying rate and the particle residence time within the tower. The coupled first order ordinary differential equations for the two phases are solved numerically using the iterative shooting method in an algorithm developed within MATLAB. The predictions of the numerical model are compared with industrial pilot plant data. The results are found to vary significantly with the specified size distribution of the droplets. Despite the simplicity of the model in ignoring the coalescence, agglomeration, wall deposition and re-entrainment, the model gives reasonable agreement with the experimental data.     

Reply to the Comment by K. Jorgensen et al. to “Analysis of constant rate period of spray drying of slurry”

Previously developed model for spray drying of slurry droplets with high solid content (Liang, 2001) was revised to describe correctly a boundary condition on the outer surface of droplet on basis of the rate of liquid evaporation.     

单颗粒褐煤高温烟气干燥过程数值模拟

褐煤干燥对于提高其品质具有重要意义。为了模拟高温烟气干燥这一高温差、变温差非稳态传热传质过程中褐煤内水分蒸发过程,采用有限体积法建立了一维球坐标系下蒸发界面向内迁移的单颗粒褐煤干燥数学模型,并利用该模型分析了初始烟气温度和颗粒粒径对单个褐煤颗粒干燥特性的影响。模型模拟结果与实验结果对比表明二者变化趋势一致,所建模型能较好地反映出高温烟气干燥过程中褐煤内水分蒸发过程。结果表明,初始烟气温度越高,颗粒粒径越小,蒸发界面向内迁移速度越快,水分脱除越快,干燥时间越短;蒸发界面平均迁移速度均与初始烟气温度和颗粒粒径呈线性关系;在初始烟气温度700℃下,较短的停留时间使得颗粒表面温度未达到挥发分析出温度,本研究中不同粒径褐煤颗粒在干燥过程中基本没有挥发分的析出。     

CONVECTWE DRYING MODEL OF SOUTHERN PINE

ABSTRACT

A transient one dimensional first principles model is developed for the drying of a porous material (wood is used as an example) that includes both heat and mass transfer. Heat transfer by conduction and convection, mass transfer by binary gas diffusion, pressure-driven bulk flow in the gas and liquid, and diffusion of bound water are included in the analysis. The diffusive mass transfer terms are modeled using a Fickian approach, while the bulk flow is modeled assuming Darcian flow. Depending on the state (pendular or funicular) of the moisture in the wood, appropriate terms are considered in the development of the governing mass equations. The results provide distributions within the material of each moisture phase (vapor, liquid, and bound), temperature, and total pressure. Information regarding the drying rate and evaporation rate is also presented. Average distributions are obtained as a function of time, and compared with experimental data from the literature. It is observed that the total pressure within the material can be considerably above one atmosphere during the drying process.     

AN INTEGRAL MODEL FOR DRYING OF HYGROSCOPIC AND NONHYGROSCOPIC MATERIALS WITH DIELECTRIC HEATING

Heat and mass transport phenomena in drying assisted by microwave or radio-frequency dielectric heating are analyzed. When drying at temperatures near boiling point or with high temperature gradients, the effect of the gas phase pressure gradient on moisture transfer within the solid can be important. The governing heat and mass transfer equations, including consideration of internal heat generation and the effect of the gas phase pressure gradient, are derived and solved in a one-dimensional system using an integral method. The integral model has been used to simulate dielectrically-enhanced convective drying of beds of polymer pellets, glass beads and alumina spheres with flow over the bed surface. Model predictions of drying rates and temperatures agree well with experimental data for these cases.

The model provides a relatively fast and efficient way to simulate drying behavior with dielectric heating, and may be useful in design and optimization of dielectrically-enhanced convective drying processes.     

Modeling and simulation of isothermal process of natural gas pipeline drying with dehydrated air

Abstract

Ventilative drying is an essential drying method to protect natural gas pipeline from corrosion and prevent the formation of hydrate. A mathematical model describing the problem of pipelines drying with dehydrated air is developed based on the fundamental principle of mass, momentum, and species conservation in present work. To enhance the accuracy of the model, a new correlation of the evaporation rate of water in pipeline is introduced by taking interfacial resistance into account. Finite volume method (FVM) is employed to discretize the governing equations with first-order upwind scheme for mass and momentum equations and QUICK scheme for species equations. To reduce calculation load, a new solution strategy of semi-implicit method for S_m-linked equations is proposed to solve the nonlinear algebraic equations after the discretization of mathematical model. The results indicate that average absolute deviation between experimental data and the calculations are 8.8% and 8.2% for two cases, respectively. By analysis of dynamic process of the ventilative drying, it is found that the drying process takes place from the ends toward the center of the pipeline. In addition, the results also demonstrate that the nonuniform distribution of the liquid holdup may reduce the service efficiency of dehydrated air at the late period of drying process. Thus, we suggest that the drying process shifts to soak-test stage when the water vapor at the outlet reaches a minimum set value. Finally, a number of influential factors on the drying process are studied.     

RADIO FREQUENCY VACUUM DRYING OF WOOD. I. MATHEMATICAL MODEL

A one-dimensional mathematical model to describe the transport phenomena during continuous radio frequency/vacuum (RF/V) drying of thick lumber was developed from general conservation equations. When drying at temperatures near the boiling point, as in RF/V drying, the effect of the gas phase pressure gradient on moisture transfer within the solid can be very important. The controlling resistances and transport mechanisms are discussed in detail. In addition, capillary transport in RF/V drying is discussed and its effect is compared with convective drying. The model provides a relatively fast and efficient way to simulate vacuum drying behavior assisted by dielectric heating. As an example, the governing heat and mass transfer equations, including consideration of internal heat generation and the effect of gas phase pressure gradient, are derived and solved in a one-dimensional system using a finite volume method. The effect of changes of the most important parameters on the predictions of the model is also presented.     

CONVECTWE DRYING MODEL OF SOUTHERN PINE

A transient one dimensional first principles model is developed for the drying of a porous material (wood is used as an example) that includes both heat and mass transfer. Heat transfer by conduction and convection, mass transfer by binary gas diffusion, pressure-driven bulk flow in the gas and liquid, and diffusion of bound water are included in the analysis. The diffusive mass transfer terms are modeled using a Fickian approach, while the bulk flow is modeled assuming Darcian flow. Depending on the state (pendular or funicular) of the moisture in the wood, appropriate terms are considered in the development of the governing mass equations. The results provide distributions within the material of each moisture phase (vapor, liquid, and bound), temperature, and total pressure. Information regarding the drying rate and evaporation rate is also presented. Average distributions are obtained as a function of time, and compared with experimental data from the literature. It is observed that the total pressure within the material can be considerably above one atmosphere during the drying process.     

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。     

Convective Drying Kinetics of Single Droplets of Aqueous Glucose

The convective drying kinetics of single droplets of aqueous glucose was measured using a single droplet drying rig. The effects of air temperature and velocity were evaluated. It was found that the droplet of aqueous glucose shrank uniformly, retaining a nearly spherical shape during drying. The normalized volume (d/d₀)³ of the droplet decreased linearly with its moisture content. A constant-drying-rate-like period occurred when the moisture content of the droplet was higher than an amount of about 1.0 kg kg^?1 dry solid. The diameter of the droplet decreased sharply due to the evaporation of water, while its temperature remained at a wet-bulb-like temperature in this period. When the moisture content of the droplet was lower than the above-mentioned value, the drying transferred to a falling-drying-rate-period, during which the temperature of the droplet rose quickly and approached the air temperature as drying continued. The effect of air temperature on the drying of single droplets of aqueous glucose was more pronounced when compared with that of air velocity.     

某些无机盐溶液滴粒的蒸发干燥

本文研究了某些常见无机盐的不同浓度水溶液滴粒在高温气流中的蒸发干燥.发现大多数试样滴粒的干燥过程包括(1)湿球温度下的恒速干燥;(2)降速干燥;(3)沸点下恒速干燥;(4)第二次降速干燥四个阶段,而不是两个阶段.这种四阶段干燥机理可以作为对任何含溶解固体滴粒蒸发干燥过程的完整的一般化描述.研究了某些因素对干燥过程特性的影响;测定了各干燥阶段转变时的临界湿含量.     

AN INTEGRAL MODEL FOR DRYING OF HYGROSCOPIC AND NONHYGROSCOPIC MATERIALS WITH DIELECTRIC HEATING

Heat and mass transport phenomena in drying assisted by microwave or radio-frequency dielectric heating are analyzed. When drying at temperatures near boiling point or with high temperature gradients, the effect of the gas phase pressure gradient on moisture transfer within the solid can be important. The governing heat and mass transfer equations, including consideration of internal heat generation and the effect of the gas phase pressure gradient, are derived and solved in a one-dimensional system using an integral method. The integral model has been used to simulate dielectrically-enhanced convective drying of beds of polymer pellets, glass beads and alumina spheres with flow over the bed surface. Model predictions of drying rates and temperatures agree well with experimental data for these cases.

The model provides a relatively fast and efficient way to simulate drying behavior with dielectric heating, and may be useful in design and optimization of dielectrically-enhanced convective drying processes.     

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.     

Modeling conventional drying of wood: Inclusion of a moving evaporation interface

A one-dimensional mathematical model is presented that accounts for a moving evaporation interface in simulating the coupled heat and mass transfer during convective drying of wood. In the model proposed, the only mechanism considered in water transport within wood is diffusion. Additionally, the transport of moisture is dominated by the gradient of the moisture content. The controlling equations were established from Whitaker’s volume averaging laws and solved numerically with the finite volume method. The simulation results for the density of vapor and the volume rate of evaporation indicate that the migrating moisture was mainly in the form of gas under conditions of lower moisture content. The evaporation interface moved at approximately constant speed and the evaporation rate of the interface decreased with time. Finally, the core temperature and average moisture content in wood were successfully simulated.