A Model for the Relative Velocity of a Particle in an Impingement Dryer: Application to Heat Transfer

Abstract

A simplified mathematical model for the relative gas-particle motion in a confined jet impingement dryer is developed. Model predictions based on an unsteady momentum balance are in good agreement with the observed cycling motion of a spherical particle. The model is applied to coriander seeds submerged in a flow field of superheated steam. It is found that relative motion occurs in unsteady turbulent regime, and that steady settling velocity of particles is never achieved. Model results are applied to correlate experimental heat transfer data of an impingement dryer. Experimental Nu numbers compare fairly well with correlations for fluidized systems.     

Modeling of Coriander Seeds Drying in an Impingement Dryer

The aim of this work is to develop a mathematical model to estimate the batch drying curve of coriander seeds in an impingement dryer and to study the axial movement of a seed in a transparent prototype impingement dryer. The apparatus is a horizontal acrylic transparent cylinder with a slight slope to induce the axial and rotational movement of particles. Gas enters tangentially downwards through a narrow slot arranged all along the dryer, flows in a counterclockwise circular motion in the chamber—in crossflow with respect to the solids—and is discharged through an upper lengthwise expansion chamber. As a result of gas drag, the particles advance in a rotational-helicoidal motion between feed and discharge.

Velocity and temperature profiles for gas in 2D turbulent flow were simulated using commercial software from Fluent Inc.^[1,^4] Maximum velocities are shown to be located close the walls; most of the gas is recirculated, and the rest is exhausted. It is assumed that particle trajectories also follow a circular motion near the walls, as observed in the transparent reproduction of the dryer operating with ambient air for small batch of solids and/or a single particle. Air velocities along this trajectory are estimated from the simulated flow field. Particle motion, heating, and drying along this path are described by unsteady momentum, heat, and mass balances when subjected to gas drag and gravity forces.

With respect to the axial trajectory of a coriander seed, for an inlet air velocity of 20 m/s at the slot the average experimental time for a complete circular cycle is 0.18 s and the simulated time is 0.21 s, whereas average experimental residence time is 1.53 s and the simulated time is 0.94 s. Differences between experimental results and simulations are due to air instability, leading to nonhomogeneous air velocity profiles along the equipment. The mathematical model is based on the assumption that air velocity profiles are homogeneous. Experimental observations indicate that the particle does not move along the equipment but sometimes moves backward (or erratically) or spins out advancing, due to an uneven air speed profile, and impacts against the wall. Finally, the drying model gives results that adjust to the batch experimental data, taking into account the deviations found with respect to the axial trajectory from a seed. This is because the model was devised exactly to predict the conduct of the system in batch operation for a particle bed, obtaining results that show the macrocospic response of the equipment (velocity and average temperature of the air). As it happens in this type of phenomenon, the drying rate in the constant period is a function of the adimensional Reynolds number.     

Simulation of Superheated Steam Turbulent Flows and Heat Transfer in an Impingement Dryer

Numerical simulations are presented for two-dimensional flow field and heat transfer characteristics due to a turbulent single slot jet of superheated steam discharging tangentially into a confined cylinder. A finite volumes method was used to solve the equations that describe the problem. Calculations were performed for steady state turbulent flow and unsteady state heat transfer. Constant velocities and superheated steam temperature are imposed at the inflow. Particle properties were assumed to be the same of a cubic particle of carrot. Numerical tests were performed to ensure that the model solutions were “grid independent” and also independent of the turbulence intensity of stated as boundary condition at the inlet slot. The solution procedure developed is fast and that convergence is reached after a few iterations. The results obtained are relevant to flow and heat transfer behaviors of the impingement dryer and it will be useful future studies considering particles interactions.     

UNSTEADY-STATE SIMULATION OF A MDLTI-STAGE CONCURRENT-FLOW MAIZE DRYER

An unsteady - state model of concurrent-flow maize drying was developed. It consists of a set of four partial differential equations (PDE) which are solved by finite differences. The PDE model was verified by comparing it to the steady-state concurrent-flow dryer simulation. The model was used for the development of a process model which in turn is to be employed for the design of an automatic control system of a two-stage concurrent-flow dryer.     

UNSTEADY-STATE SIMULATION OF A MDLTI-STAGE CONCURRENT-FLOW MAIZE DRYER

An unsteady - state model of concurrent-flow maize drying was developed. It consists of a set of four partial differential equations (PDE) which are solved by finite differences. The PDE model was verified by comparing it to the steady-state concurrent-flow dryer simulation. The model was used for the development of a process model which in turn is to be employed for the design of an automatic control system of a two-stage concurrent-flow dryer.     

PAPER DRYING: A STRATEGY FOR HIGHER MACHINE SPEED. II. IMPINGEMENT AIR DRYING FOR HYBRID DRYER SECTIONS

For printing and heavier grades, combining cylinder and impingement air drying into a multiple technique dryer section can enable higher productivity through higher machine speed. The large differences in local moisture content and temperature across the sheet which develop quickly under high intensity impingement drying provide the potential for reducing drying time by sheet reversal between impingement drying cylinders. This advantage was determined experimentally under low intensity impingement drying conditions. Use of the micro-scale based McGill dryer simulator for determining the advantage from sheet reversal between impingement drying cylinders was demonstrated for both laboratory and industrial impingement drying intensities. For completing the drying of 205 g/m² linerboard from 0.3 to 0.5 kg/kg dry under 400°C air jets of Re 20000, drying time is about 30% less with sheet reversal between two impingement drying cylinders than for a single, larger diameter cylinder. This extensively validated dryer simulator enables determining advantageous design specifications and operating conditions for hybrid dryer sections involving combinations of cylinder and impingement air drying, a concept with potential to become common industrial practice.     

Motions of a sphere in a time-dependent stokes flow: A generalization of Faxen’s law

A general solution of the unsteady Stokes equation in spherical coordinates is derived for flow in the exterior of a sphere, and then applied to study the arbitrary unsteady motion of a rigid sphere in an unbounded single fluid domain which is undergoing a time-dependent mean flow. Calculation of the hydrodynamic force and torque on the sphere leads to a generalization of the Faxen’s law to time-dependent flow fields which satisfy the unsteady Stokes equation. For illustrative purposes, we consider the relative motion of gas bubbles which undergo very rapid oscillations so that the generalized Faxen’s law derived for a solid sphere can be applied. We also demonstrate that our results reduce to those of Faxen for the steady flow limit.     

Numerical Study of Heat Transfer and Fluid Flow in Multiple turbulent Impinging Jets

ABSTRACT

Impingement flows have been studied extensively for various geometries and configurations, but because of the complexity of the turbulent flow and its strong dependence on the geometry of the flow, further investigation is required to identify the suitable model for specific cases. This paper presents a study of various k–E turbulence models in order to identify the best model for an array of multiple confined impinging slot jets, with exhaust ports in the confinement surface located symmetrically between adjacent jets. Such a configuration is used in a novel drum dryer for black liquor. The “High Reynolds number” turbulence models including the standard k–E model fail to predict heat transfer to impingement surface accurately although they do predict the flow field reasonably well. On the other hand, the “Low Reynolds number” models yield considerably better results for both fluid flow and heat transfer. All computed results are compared with experimental data reponed in the literature. This work was motivated by the need to select an optimal multiple impinging jet configuration for a novel drum dryer for Kraft black liquor. It is also pertinent to impingement dryers for paper, films, textiles etc.     

Optimal Control of the Primary Drying Stage of Freeze Drying of Solutions in Vials Using Variational Calculus

The problem of operating freeze drying of pharmaceutical products in vials placed in trays of a freeze dryer to remove free water (in frozen state) at a minimum time was formulated as an optimal control problem. Two different types of freeze dryer designs were considered. In type I freeze dryer design, upper and lower plate temperatures were controlled together, while in type II freeze dryer design, upper and lower plate temperatures were controlled independently. The heat input to the material being dried and the drying chamber pressure were considered as control variables. Constraints were placed on the system state variables by the melting and scorch temperatures during primary drying stage. Necessary conditions of optimality for the primary drying stage of freeze drying process in vials are derived and presented. Furthermore, an approach for constructing the optimal control policies that would minimize the drying time for the primary drying stage was given. In order to analyze optimal control policy for the primary drying stage of the freeze-drying process in vials, a rigorous multi-dimensional unsteady state mathematical model was used. The theoretical approach presented in this work was applied in the freeze drying of skim milk. Significant reductions in the drying times of primary drying stage of freeze drying process in vials were obtained, as compared to the drying times obtained from conventional operational policies.     

Modeling of Coriander Seeds Drying in an Impingement Dryer

The aim of this work is to develop a mathematical model to estimate the batch drying curve of coriander seeds in an impingement dryer and to study the axial movement of a seed in a transparent prototype impingement dryer. The apparatus is a horizontal acrylic transparent cylinder with a slight slope to induce the axial and rotational movement of particles. Gas enters tangentially downwards through a narrow slot arranged all along the dryer, flows in a counterclockwise circular motion in the chamber—in crossflow with respect to the solids—and is discharged through an upper lengthwise expansion chamber. As a result of gas drag, the particles advance in a rotational-helicoidal motion between feed and discharge.

Velocity and temperature profiles for gas in 2D turbulent flow were simulated using commercial software from Fluent Inc.^{[ 1} Quispe , J.F. ; Canales , E.R. ; Bórquez , R.M. Simulation of turbulent flows in an impingement dryer by an extended κ-ε model . Computer Methods in Applied Mechanics and Engineering 2000 , 190 , 625 – 637 .[Crossref] , [Google Scholar], ⁴ Soto , V. ; Bórquez , R.M. Simulation of superheated steam turbulence flows and heat transfer in an impingement dryer . Drying Technology 2003 , 21 ( 2 ), 311 – 328 .[Taylor & Francis Online], [Web of Science ®] , [Google Scholar] ^] Maximum velocities are shown to be located close the walls; most of the gas is recirculated, and the rest is exhausted. It is assumed that particle trajectories also follow a circular motion near the walls, as observed in the transparent reproduction of the dryer operating with ambient air for small batch of solids and/or a single particle. Air velocities along this trajectory are estimated from the simulated flow field. Particle motion, heating, and drying along this path are described by unsteady momentum, heat, and mass balances when subjected to gas drag and gravity forces.

With respect to the axial trajectory of a coriander seed, for an inlet air velocity of 20 m/s at the slot the average experimental time for a complete circular cycle is 0.18 s and the simulated time is 0.21 s, whereas average experimental residence time is 1.53 s and the simulated time is 0.94 s. Differences between experimental results and simulations are due to air instability, leading to nonhomogeneous air velocity profiles along the equipment. The mathematical model is based on the assumption that air velocity profiles are homogeneous. Experimental observations indicate that the particle does not move along the equipment but sometimes moves backward (or erratically) or spins out advancing, due to an uneven air speed profile, and impacts against the wall. Finally, the drying model gives results that adjust to the batch experimental data, taking into account the deviations found with respect to the axial trajectory from a seed. This is because the model was devised exactly to predict the conduct of the system in batch operation for a particle bed, obtaining results that show the macrocospic response of the equipment (velocity and average temperature of the air). As it happens in this type of phenomenon, the drying rate in the constant period is a function of the adimensional Reynolds number.     

Optimal Control of the Primary Drying Stage of Freeze Drying of Solutions in Vials Using Variational Calculus

Abstract

The problem of operating freeze drying of pharmaceutical products in vials placed in trays of a freeze dryer to remove free water (in frozen state) at a minimum time was formulated as an optimal control problem. Two different types of freeze dryer designs were considered. In type I freeze dryer design, upper and lower plate temperatures were controlled together, while in type II freeze dryer design, upper and lower plate temperatures were controlled independently. The heat input to the material being dried and the drying chamber pressure were considered as control variables. Constraints were placed on the system state variables by the melting and scorch temperatures during primary drying stage. Necessary conditions of optimality for the primary drying stage of freeze drying process in vials are derived and presented. Furthermore, an approach for constructing the optimal control policies that would minimize the drying time for the primary drying stage was given. In order to analyze optimal control policy for the primary drying stage of the freeze-drying process in vials, a rigorous multi-dimensional unsteady state mathematical model was used. The theoretical approach presented in this work was applied in the freeze drying of skim milk. Significant reductions in the drying times of primary drying stage of freeze drying process in vials were obtained, as compared to the drying times obtained from conventional operational policies.     

INDIRECT THERMAL DRYING OF LIGNITE: DESIGN ASPECTS OF A ROTARY DRYER

ABSTRACT

An investigation of the thermal drying of lignite has been carried out, by using an indirect heat pilot rotary drum. The process aims at the production of dry lignite and clean steam as part of a gasification procedure. Both flighted and bare drum modes have been employed. Temperature profiles along the dryer length, the amount of evaporation (moisture conversion) and the solids residence time distribution (RTD) were measured. A non-isothermal model was tested under three different regimes of solids flow. Model integration, by taking account of experimental amount of evaporation at dryer exit and temperature profiles along the dryer length, has been utilized in the validation of drying kinetics and heat transfer correlations. Model predictions compare satisfactorily with the operating data of an indirect heat industrial lignite dryer. Overall heat transfer coefficients of the pilot rotary dryer were found to agree well with those reported for direct heat dryers.     

Drying of Fish Press-Cake with Superheated Steam in a Pilot Plant Impingement System

Drying tests for pine sawdust and mackerel press-cake with hot air and with superheated steam were carried out in a pilot impingement cylindrical dryer. Wet particles move axially along the dryer, adjusting the inclination of the equipment, whereas hot gases circulate in cross-flow against the particles, forming a corotational impingement front. Feed rate and residence time of solids were studied experimentally as a function of dryer inclination for hot air and superheated steam as drying media. Drying rates and heat and mass transfer coefficients were found to increase at shorter residence times and higher gas temperatures. Dried mackerel press-cake with superheated steam resulted in a product with high moisture removal and very low losses of the valuable omega-3 fatty acids.     

On the unsteady forces during the motion of a sedimenting particle

We consider the unsteady motion of a sedimenting rigid spherical particle in order to examine the relative strength of the hydrodynamical forces acting on particles in fluid flows. The relative strength of the forces on all stages of the particle motion is a major concern for closing constitutive equations describing the more complex motion of particulate flows such as fluidised beds. The formulation results in a first order nonlinear integro-differential equation in terms of the instantaneous velocity of the sphere. This equation is made dimensionless and the particle Reynolds number and the fluid-particle density ratio are identified as the relevant physical parameters describing the particle motion. We obtain analytical solutions for the limits of small density ratios and small Reynolds number. In addition, a numerical solution is used for arbitrary values of the density ratio. The results show that the motion of spherical particles is significantly affected by the unsteady drag dominated by the memory Basset force on the early stages of the motion and on the approach to the steady state (terminal velocity). The present calculations indicate that the unsteady hydrodynamic drags might become of the same order of magnitude of the dominant viscous drag for flows with moderate particle-fluid density ratio. Therefore, unsteady drags should be taken into account on modelling multiphase particulate flows with moderate density ratio.     

Modeling a multiple-zone air impingement dryer

A complete model for the simulation of the drying process of a binary system in a modular air impingement dryer is presented. The underlying equations and the numerical scheme were chosen such that large-scale parameter studies could be performed efficiently with an ordinary PC. The influence of the operating parameters on the drying progress and the energy efficiency was investigated, considering a wide range of technically important constraints such as proper web handling, the risk of explosion or the generation of bubbles in the coating.     

Drying of Fish Press-Cake with Superheated Steam in a Pilot Plant Impingement System

Drying tests for pine sawdust and mackerel press-cake with hot air and with superheated steam were carried out in a pilot impingement cylindrical dryer. Wet particles move axially along the dryer, adjusting the inclination of the equipment, whereas hot gases circulate in cross-flow against the particles, forming a corotational impingement front. Feed rate and residence time of solids were studied experimentally as a function of dryer inclination for hot air and superheated steam as drying media. Drying rates and heat and mass transfer coefficients were found to increase at shorter residence times and higher gas temperatures. Dried mackerel press-cake with superheated steam resulted in a product with high moisture removal and very low losses of the valuable omega-3 fatty acids.     

Experimental Study and Mathematical Modeling of Green Pea Drying in a Spouted Bed

In this study, green pea drying is investigated experimentally in a laboratory-scale spouted bed dryer. A mathematical model is also developed to investigate the effect of operating conditions on the performance of the system. The effect of operating parameters such as inlet air temperature, particle size, and flow rate of the drying air on the performance of the dryer are studied experimentally. In order to build the process model, it is necessary to analyze the transport in both solid and gas phases. A complete set of equations with no adjustable parameters is derived for existing zones in the spouted bed dryer in order to predict variations in the temperature and moisture content of the solid and gas phases with time for batch drying conditions. Model results are compared with corresponding experimental data. Agreement between the model results and experimental data is good.     

Model-based multivariable control of the drying of a thin sheet of fibres in a continuous infrared dryer

We developed an Internal Model Control (IMC) algorithm for drying a thin textile fabric, continuously passing through an electric infrared dryer, based on a reduced linear model of the drying dynamics. This model relates the controlled variables, the humidity and temperature of the web at the dryer outlet, to the manipulated variables, the electrical power supplied to the sources and the web speed through the dryer, and also to changes in the initial humidity of the web at the dryer inlet. The control algorithm was first tested by simulation using the model in regulation mode, and in set-point tracking mode, to vary the manipulated variables so as to maintain the controlled variables at their respective set-points when the inlet web temperature and humidity were changed. The performance under simulated operational conditions was compared to that of a conventional feedback proportional-integral (PI) controller coupled with a feedforward control. The IMC controller was then tested directly in regulation mode using a pilot scale infrared dryer, acting simultaneously on the manipulated variables, the emitter power and the web speed, to control the fabric temperature and humidity at the dryer outlet. The experimental results were compared with those from the above feedback-feedforward controller, on the pilot scale dryer. The results have indicated that the closed-loop stability of the process is assured simply by choosing a stable IMC controller. Also, such a controller does not require the design of specific compensators for the strong interactions between variables of the drying process.     

STEAM DRYING OF WOOD CHIPS IN PNEUMATIC CONVEYING DRYERS

ABSTRACT

A model for a pneumatic conveying dryer is presented. Although the main emphasis is put on superheated steam drying of wood chips, it can be used for other porous materials as well

The model includes a comprehensive two-dimensional model for the drying of single wood chips which accounts for the main physical mechanisms occurring in wood during drying. The external drying conditions in a pneumatic conveying dryer were calculated by applying the mass, heat and momentum equations for each incremental step in dryer length. A plug flow assumption was made for the dryer model and the single particle and dryer models were solved in an iterative manner. The non-spherical nature of wood chips were accounted for by measuring the drag and heat transfer coefficients

Model calculations illustrate the complex interactions between steam, particles and walls which occur in a flash dryer. The drying rate varies in a very complex manner through the dryer. The internal resistance to mass transfer becomes very important in The drying of less permeable wood species such as spruce. Two effects were observed as the particle size was increased: firstly the heat transfer rate decreased, and secondly the residence time increased. To some extent, these effects compensate for each other, however, the net result is that larger chips have a higher final moisture content.     

Modeling and simulation of the drying of thin sheets in a continuous infrared dryer

The model gives the temperature and moisture distributions of the air, and of the moist sheet, as a function of time and distance in the dryer. The influence of the sheet's velocity and that of the radiant energy on the dryer performance as well as the effect of the moisture content of the entering sheet have been studied. A set of 27 experiments was carried out using the infrared dryer in order to calibrate the model. In these, the following three variables each had three operational levels: web velocity, initial web moisture and heating power. The model predictions agreed very well with the experimental data. Model predictions using arithmetic averages for the parameters, and parameters correlated with operational variables, are also presented and discussed.