PRESSURE DROPS FOR PURELY VISCOUS NON-NEWTONIAN FLUID FLOW THROUGH BEDS PACKED WITH MIXED-SIZE SPHERES

Extensive new measurements on pressure drop for the flow of purely viscous non-Newtonian fluids through packed beds made up of binary-and-quaternary size spheres are reported herein. These results have been interpreted using the previously available Sabiri and Comiti capillary model (1995), which has been quite successful in correlating the pressure drop data for the beds of uniform size spherical and nonspherical particles. The resulting predictions are also very good for mixed beds if some results are available for a Newtonian fluid, thereby enabling the evaluation of the tortuosity factor. However, since such data is always not available, an empirical scheme for estimating the tortuosity that allows the prediction of pressure drop with a mean relative error of about 10% is also presented. This sort of accuracy is quite acceptable for process engineering design calculations.     

A Model for Through Drying of Tissue Paper at Constant Pressure Drop and High Drying Intensity

A mathematical model for through drying of paper at constant pressure drop was developed. The model is based on physical properties; hence, basis weight, pressure drop, drying air temperature, pore size distribution, initial gas fraction, and tortuosity are important input parameters to the model. The model was solved for different combinations of the variables basis weight, drying air temperature, and pressure drop corresponding to industrial conditions and the results were compared with data from bench-scale experiments. The simulations show that the drying rate curve is very sensitive to the air flow rate and that correctly modeling the correlation between pressure drop and air flow rate is the most important factor for a successful model for through drying. The model was tuned by adjusting the parameters initial gas fraction and tortuosity in order to give the best possible fit to experimental data. For a given basis weight and pressure drop, different drying air temperatures resulted in relatively constant values of the fitted parameters. This means that the model can well predict the effects of changes in drying air temperature based on a tuning of the model performed at the same basis weight and pressure drop. However, for a given basis weight, an increase in pressure drop yielded fitted parameters that were somewhat different; i.e., a lower initial gas fraction and a higher tortuosity, a change that increases the resistance to air flow. This implies that the correlation between pressure drop and air flow rate in the model does not quite capture the nonlinear relationship shown by the experiments.     

A Model for Through Drying of Tissue Paper at Constant Pressure Drop and High Drying Intensity

A mathematical model for through drying of paper at constant pressure drop was developed. The model is based on physical properties; hence, basis weight, pressure drop, drying air temperature, pore size distribution, initial gas fraction, and tortuosity are important input parameters to the model. The model was solved for different combinations of the variables basis weight, drying air temperature, and pressure drop corresponding to industrial conditions and the results were compared with data from bench-scale experiments. The simulations show that the drying rate curve is very sensitive to the air flow rate and that correctly modeling the correlation between pressure drop and air flow rate is the most important factor for a successful model for through drying. The model was tuned by adjusting the parameters initial gas fraction and tortuosity in order to give the best possible fit to experimental data. For a given basis weight and pressure drop, different drying air temperatures resulted in relatively constant values of the fitted parameters. This means that the model can well predict the effects of changes in drying air temperature based on a tuning of the model performed at the same basis weight and pressure drop. However, for a given basis weight, an increase in pressure drop yielded fitted parameters that were somewhat different; i.e., a lower initial gas fraction and a higher tortuosity, a change that increases the resistance to air flow. This implies that the correlation between pressure drop and air flow rate in the model does not quite capture the nonlinear relationship shown by the experiments.     

A general model for pressure drop prediction across a rotating packed bed

Although rotating beds show promise for intensified separations, the fundamentals of their hydrodynamics are still poorly understood. In many operating conditions, pressure drop in an irrigated bed can be considerably lower than that across a dry bed. Previously published correlations don’t provide good prediction of these phenomena. In this research, a semi-empirical correlation is developed to predict the pressure drop across the rotating packed beds. The results show the agreement of predicted pressure drop with the experimental data is acceptable. This model can also predict the pressure drop unexpected phenomena of higher pressure drop in dry beds than in wet beds.     

Models for prediction of hydrodynamic characteristics of gas–solid tapered and mini-tapered fluidized beds

Prediction of hydrodynamic characteristics is a prerequisite in the design and operation of tapered and mini-tapered fluidized beds. This paper has been focused on the development of generalized models for prediction of minimum fluidization velocity and maximum pressure drop in gas–solid tapered and mini-tapered fluidized beds. The empirical correlations were developed based on dimensionless analysis of empirical data. These correlations have the ability to predict the minimum fluidization velocity and maximum pressure drop in both tapered and cylindrical beds (the beds with tapered angle of zero). The empirical data were collected from tapered beds with different cone angles for various particles. The predicting capability of correlations has been discussed. Predicted values of minimum fluidization velocity and maximum pressure drop by the proposed models compared well with the empirical data. The effects of tapered angle are also discussed.     

Pressure drop across packed beds in the low Knudsen number regime

The solutions of Happel and Kuwabara describing flow through systems of multiple spheres have been compared with available pressure drop data to determine the applicability of cell models for analyzing flow in packed beds. The comparisons indicate that these solutions describe the flow with reasonable accuracy. Of the two solutions, the Kuwabara flow field has been found to give better agreement between predictions and experimental results. In order to describe flow in beds operating at low Knudsen numbers the Kuwabara solution has been extended by allowing for gas slippage at the collecting surfaces. The resulting solution is presented as a correction to the Basset-Millikan expression and accounts for the presence of neighboring spheres in low Knudsen number flows. Using this extended solution, the pressure drop across packed beds is predicted as a function of Knudsen number and packing density.     

A study of particle shape and size effects on hydrodynamic parameters of trickle beds

Uniform-spherical and cylindrical-extrudate particles are employed to study air-water downflow in a packed bed of 14 cm i.d. The effect of particle shape, neglected in the literature so far, is shown to be very significant. A packed bed of extrudates displays significantly greater global dynamic liquid holdup h_d and pressure drop, as well as a trickling-to-pulsing transition boundary at higher gas flow rates, compared to beds of spheres of comparable size. Moreover, packed extrudates exhibit a significant increase of holdup, h_d, in the axial flow direction, a trend reported for the first time as there are no similar data available in the literature; on the contrary beds of spherical particles are characterized by practically constant h_d in the axial direction. Although an explanation for this h_d axial variation is not obvious, one might attribute it to the anisotropy and non-uniformity of interstitial voids of packed cylindrical particles. For beds of uniform spheres, in the diameter range examined (3-6 mm), the effect of size on both dynamic holdup and pressure drop, although quite pronounced, is not as significant as the effect of particle shape. An extensive survey of literature data, obtained with similar spherical particles, suggests that small bed diameters have an appreciable influence on trickling-to-pulsing transition boundary. Comparisons are reported with literature methods for predicting the measured parameters; discrepancies between data and predictions may be partly due to the inadequacy of a single “equivalent” diameter to represent both shape and size of non-spherical particles; predictive methods performing best are also identified.     

Numerical and Experimental Study of Catalyst Loading and Body Effects on a Gas‐Liquid Trickle‐Flow Bed

The influence of tortuosity and fluid volume fractions on trickle‐flow bed performance was analyzed. Hydrodynamics of the gas‐liquid downward flow through trickle beds, filled with industrial trilobe catalysts, were investigated experimentally and numerically. The pressure drop and liquid holdup were measured at different gas and liquid velocities and in two different loading methods, namely, sock and dense catalyst loading. The effect of sharp corners on hydrodynamic parameters was considered in a bed with rectangular cross section. The reactor was simulated, considering a three‐phase model, appropriate porosity function, and interfacial forces based on the Eulerian‐Eulerian approach. Computational fluid dynamics (CFD) simulation results for pressure drop and liquid holdup agreed well with experimental data. Finally, the velocity distribution in two types of loading and the effect of bed geometry in CFD results demonstrated that pressure drop and liquid holdup were reduced compared to a cylindrical one due to high voidage at sharp corners.     

Effect of coal particle size distribution on packed bed pressure drop and gas flow distribution

This paper focuses on the role of coal particle size distribution on pressure drop and gas flow distribution through packed coal beds. This fundamental knowledge is helpful in better understanding the operational behaviour of fixed bed dry bottom gasifiers. The Sasol synfuels plants in South Africa use 80 such gasifiers to convert more than 26 million tons of coal per annum to synthesis gas, and ensuring stable operation is of primary importance to ensure high synthesis gas production rates and gasifier availability. Pressure drop measurements on laboratory scale equipment were conducted to investigate the effect of particle size distribution on packed bed pressure drop. The well-known Ergun equation for pressure drop does not accommodate the effect of size distribution on pressure drop. A novel approach was followed to model pressure drop through simulated coal bed structures using Computational Fluid Dynamics (CFD). The coal bed structures were simulated by assuming that the coal particles are represented by randomised convex polyhedra in three-dimensional space. The computational space was divided into polyhedra using statistical Voronoi tessellation technique, which have been shown to be versatile in modelling problems in many fields, e.g. filtration, molecular physics, metallurgy, geology, forestry and astrophysics. This approach of flow modelling through packed coal beds is able to accommodate size distribution effects on pressure drop and gas flow distribution. The modelling work shows large deviations from plug flow with broad size distributions. The lowest bed pressure drop with the closest approximation to plug flow is obtained with the narrowest particle size distribution. Low gas flow rates are also beneficial for reducing excessive channel flow. Combustion profiles for different particle size distributions were studied using a pilot scale combustor. The combustion profiles provide confirmation of the CFD modelling results, namely that narrow particle size distributions and low gas flow rates reduce channel burning. Excessive channel burning was observed for broad particle size distributions, and is enhanced by high gas flow rates. The experimental and modelling work which was conducted, clearly indicate that narrow coal particle size distributions are desirable for optimum gas flow distribution and lowest packed bed pressure drop.     

The generalized Lévêque equation and its practical use for the prediction of heat and mass transfer rates from pressure drop

Based on the generalized Lévêque equation (GLE) a new type of analogy between pressure drop and heat transfer has been discovered, that may be used in the cross corrugated channels of chevron type plate heat exchangers, in packed beds, in tube bundles, in crossed rod matrices or in many other spacewise periodic arrangements.Experimental data on heat transfer in tube bundles in crossflow, both inline, and staggered arrangements, had been recently tested in greater detail. Using an empirical correlation for pressure drop in these arrangements from the literature that has been successfully tested against a large number of experimental pressure drop data, heat transfer data collected earlier could be very well represented from the pressure drop correlation and the GLE. The data for staggered bundles have been shown to be in better agreement with this new method, than with the existing empirical heat and mass transfer correlations. Somewhat larger deviations for inline tube bundles had been found at lower Reynolds numbers. Here a simple and physically reasonable correction function of Re is presented, which leads to a better agreement for the inline bundles, too.Additionally, it can be shown for a number of literature data on tube bundles and on crossed rod matrices that the agreement with the GLE prediction is even better if original pressure drop data from the same sources are available in place of a pressure drop correlation.The method results in reasonable heat or mass transfer predictions from frictional pressure drop, which may be widely used in chemical engineering applications.     

Using Artificial Neural Network to Predict the Pressure Drop in a Rotating Packed Bed

Although rotating beds are good equipments for intensified separations and multiphase reactions, but the fundamentals of its hydrodynamics are still unknown. In the wide range of operating conditions, the pressure drop across an irrigated bed is significantly lower than dry bed. In this regard, an approach based on artificial intelligence, that is, artificial neural network (ANN) has been proposed for prediction of the pressure drop across the rotating packed beds (RPB). The experimental data sets used as input data (280 data points) were divided into training and testing subsets. The training data set has been used to develop the ANN model while the testing data set was used to validate the performance of the trained ANN model. The results of the predicted pressure drop values with the experimental values show a good agreement between the prediction and experimental results regarding to some statistical parameters, for example (AARD% = 4.70, MSE = 2.0 × 10^?5 and R² = 0.9994). The designed ANN model can estimate the pressure drop in the countercurrent flow rotating packed bed with unexpected phenomena for higher pressure drop in dry bed than in wet bed. Also, the designed ANN model has been able to predict the pressure drop in a wet bed with the good accuracy with experimental.     

Hydrodynamics of trickle bed reactors at high pressure: Two-phase flow model for pressure drop and liquid holdup, formulation and experimental validation

A large experimental database has been established at IFP on the same experimental setup to measure simultaneously pressure drop and liquid holdup in packed bed reactor operated in trickle for a large range of operating conditions. The varying parameters are liquid viscosity and density, gas density, bed particle shape and size. The range for gas density range is particularly large (from 1.3 to ), thanks to the use of dense gas to simulate very high pressure conditions. This data bank has been first used to compare the prediction accuracy of the different models from the literature. Finally, the mechanistic model proposed by Attou et al. [1999. Modelling of the hydrodynamics of the cocurrent gas-liquid trickle flow through a trickle-bed reactor. Chemical Engineering Science 54, 785-802] has been improved by adding a new formulation for liquid film tortuosity in two-phase flow conditions. This model has been validated over the whole data range and the accuracy has been checked with data external to the data bank. The prediction accuracy is significantly increased when compared with the best available models for pressure drop and liquid retention in trickle flow reactors.     

CENTRIFUGAL FLUIDIZATION: EFFECTS OF PARTICLE DENSITY AND SIZE DISTRIBUTION

Based on the model of the centrifugal fluidized bed as a homogeneous viscous rotating fluid, the equation for conservation of momentum in the tangential direction is solved to obtain expressions for bed pressure drop and the radial variations of tangential velocity within the bed region. Comparisons with available data on bed pressure drop indicate that the fluidized bed rotates as a rigid body. Experimental data are also presented on minimum fluidization velocity and bed pressure drop with different particle densities and size distributions, verifying the validity of the available models for packed bed and fluidized bed pressure drop and minimum fluidization velocity in a centrifugal fluidized system.     

CENTRIFUGAL FLUIDIZATION: EFFECTS OF PARTICLE DENSITY AND SIZE DISTRIBUTION

Based on the model of the centrifugal fluidized bed as a homogeneous viscous rotating fluid, the equation for conservation of momentum in the tangential direction is solved to obtain expressions for bed pressure drop and the radial variations of tangential velocity within the bed region. Comparisons with available data on bed pressure drop indicate that the fluidized bed rotates as a rigid body. Experimental data are also presented on minimum fluidization velocity and bed pressure drop with different particle densities and size distributions, verifying the validity of the available models for packed bed and fluidized bed pressure drop and minimum fluidization velocity in a centrifugal fluidized system.     

CFD modelling of liquid fluidized beds in slugging mode

Computational Fluid Dynamics (CFD) modelling has been used to simulate a liquid fluidized bed of lead shot in slugging mode. Simulations have been performed using a commercial code, CFX4.4. The kinetic model for granular flow, which is already available in CFX, has been used during this study. 2D time-dependent simulations have been carried out at different water velocities. Simulated aspects of fluidization such as voidage profiles, slug formation, pressure drop and pressure fluctuations have been analysed. The fluid-bed pressure drop was found to be greater than the theoretical one at all velocities, in agreement with experimental observations reported for fully slugging fluidized beds. Power spectral density analysis of the pressure signal was used to investigate the development of the flow pattern and the structure of the fluid-bed with increasing fluidizing velocity. A comparison between experimental and simulated results is also reported.     

PREDICTION OF MINIMUM SPOUTING VELOCITY IN TWO-DIMENSIONAL FLAT BOTTOM SPOUTED BEDS

Spouted beds have been used in industry for operations such as drying, catalytic reactions, and granulation. Conventional cylindrical spouted beds suffer from the disadvantage of scaleup. Two-dimensional beds have been proposed by other authors as a solution for this problem. Minimum spouting velocity has been studied for such two-dimensional beds. A force balance model has been developed to predict the minimum spouting velocity and the maximum pressure drop. Effect of porosity on minimum spouting velocity and maximum pressure drop has been studied using the model. The predictions are in good agreement with the experiments as well as with the experimental results of other investigators.     

Cavity formation and breakup forces in conical spouted beds

A method has been developed to deduce a “breakup” force in a packed bed based on measured pressure drop and internal cavity size hysteresis data in a conical spouted bed. The pressure drop over the vertical jet is estimated by the Eulerian-Eulerian two-fluid model using the commercial Fluent software. The pressure drop over the packed bed is extracted from the measured total pressure drop following a flow rate descending process in the spouted bed, while the “breakup” force is determined from the combination of measured total pressure drop and internal jet height following the flow ascending process, the simulated pressure drop over the gas jet and the pressure drop over the loosely packed upper bed section. Such a proposed method can be applied in the future to develop a generalized expression for the “breakup” force in spouted beds and other packed bed systems where a vertical fluid jet is issued into the packed particles.     

Prediction of critical fluidization velocity and maximum bed pressure drop for binary mixture of regular particles in gas–solid tapered fluidized beds

The problems associated with conventional (cylindrical) fluidized beds, viz., fluidization of wider size range of particles, entrainment of particles and limitation of fluidization velocity could be overcome by using tapered fluidized beds. Limited work has been carried out to study the hydrodynamics of single materials with uniform size particles in tapered beds. In the present work, an attempt has been made to study the hydrodynamic characteristics of binary mixtures of homogeneous and heterogeneous regular particles (glass bead and sago) in tapered fluidized beds having different tapered angles. Correlations have been developed for critical fluidization velocity and maximum bed pressure drop for gas–solid tapered fluidized beds for binary mixtures of regular particles. Model predictions were compared with experimental data, which were in good agreement.     

SINGLE FLUID FLOW IN POROUS MEDIA

A comprehensive study on single fluid flow in porous media is carried out. The volume averaging technique is applied to derive the governing flow equations. Additional terms appear in the averaged governed equations related to porosity ε, tortuosity τ, shear factor F and hydraulic dispersivity D _h. These four parameters are uniquely contained in the volume averaged Navier-Stokes equation and not all of them are independent. The tortuosity can be related to porosity through the Brudgemann equation, for example, for unconsolidated porous media.

The shear factor models are reviewed and some new results are obtained concerning high porosity cases and for turbulent flows. It is known that there are four regions of flow in porous media: pre-Darcy's flow, Darcy's flow, Forchheimer flow and turbulent flow. The transitions between these regions arc smooth. The first region, the pre-Darcy's flow region represents the surface-interactive flows and hence is strongly dependent on the porous media and the flowing fluid. The other flow regions are governed by the flow strength of inertia. For Darcy's flow, the pressure gradient is found to be proportional to the flow rate. The Forchheimer flow, however, is identified by a strong inertia! effects and the pressure gradient is a parabolic function of flow rate. Turbulent flow is unstable and unsteady flow characterized by chaotic flow patterns. The pressure drop is slightly lower than that predicted using the laminar flow equation.

The hydraulic dispersivity is a property of the porous media. It may be considered as the connectivity of the pores in a porous medium. It characterizes the dispersion of mementum, heat and mass transfer. In this paper, only the dispersion of momentum is studied.

Single fluid flow through cylindrical beds of fibrous mats and spherical particles has been used to show how to solve the single fluid flow problems in porous media utilizing the knowledge developed in this communication. Both the pressure drop and axial flow velocity profiles are computed using the developed shear factor and hydraulic dispersion models. Both the predicted velocity profile and pressure drop compare fairly well with the published experimental data.