Showing papers in "Chemical Engineering Research & Design in 2001"

Design and Optimization of Fully Thermally Coupled Distillation Columns: Part 1: Preliminary Design and Optimization Methodology

Abstract: The design of a fully thermally coupled distillation column, or its thermodynamically equivalent arrangement, the dividing wall distillation column, is more complex than conventional arrangements because of the greater number of degrees of freedom. All of these degrees of freedom must be initialized before rigorous simulation can be performed. The distribution of stages in the various sections of the column, the reflux ratio, vapour and liquid splits on either side of the fully thermally coupled columns and feed condition must all be initialized. Yet these are important degrees of freedom that all interact with each other in the design. A new approach to the design of fully thermally coupled columns is proposed in this paper. The procedure uses the equilibrium stage composition concept developed for the design of azeotropic distillation systems 1 . The method is semi-rigorous in nature, providing an initial design that is very close to the results of rigorous simulation. The approach then allows the degrees of freedom to be optimized simultaneously and an optimized initial design established for rigorous simulation. A case study has been used to demonstrate the application of the new method.

On the Use of Linear Models for the Design of Water Utilization Systems in Process Plants with a Single Contaminant

Abstract: This paper addresses the optimum design of water utilization systems when a single contaminant is present. The application of the necessary conditions of optimality allows an LP or MILP formulation depending on the objective function of choice. Several examples are presented to illustrate the proposed methodology and to point out that several alternative solutions are available.

Study of Bubble Formation Under Constant Flow Conditions

Abstract: Bubble size is one of the key parameters in the design of two-phase gas-liquid bubble column reactors. Accurate knowledge of this parameter is essential for the prediction of gas holdup, heat and mass transfer coefficients. The previousfindings, particularly with respect to the infiuence of orifice size and physical properties of the liquid phase on bubble size, are often of contradictory nature. In this paper, extensive new experimental results are presented for regions where published data are insufficient. The suitability of artificial neural networks for identification of the process variables and modeling is evaluated. The Radial Basis Function (RBF) neural network architecture was used successfully to generate a nonlinear correlation for the prediction of bubble diameter. This correlation predicts the present data and the control data of other investigators with excellent accuracy.

FLOW VISUALIZATION IN STIRRED VESSELS A Review of Experimental Techniques

Abstract: Stirred vessels are being used not only in chemical processes for simple contacting or blending operations, but also in novel configurations and processes, as in mineral processing and/or wastewater treatment, with specific requirements, like low shear or regions in the vessel with high and low turbulence levels. The techniques that are available for the study of the flow patterns induced by the various types of agitators, e.g., classical pressure or velocity measurements with Pitot tubes or hot-wire anemometers, and novel ones like laser Doppler velocimetry, laser-induced fluorescence and particle image velocimetry are reviewed and their usefulness for particular situations is discussed.

Numerical Simulation of the Two-Phase Flow in an Axially Stirred Vessel

Abstract: The two-phase flow field in an axially stirred vessel has been calculated using the commercially available CFD-code CFX4. The multi-fluid approach was used with separate k and e equations solved for each phase. The results were compared with experimental phase-Doppler anemometry data. Particular attention was given to the distribution of slip velocities in the tank. Qualitative agreement between calculations and experimental datawas obtained for the axial component of the slip and for the turbulence kinetic energy difference between the phases. The radial and tangential components of the slip vector were greatly under-predicted in the calculations. The drag was found to be most important of the investigated inter-phase momentum transfer terms. Four different drag models were tested and found to give very similar results. Finally, the importance of the slip velocity as well as the particle velocity for the mass transfer in stirred tanks was discussed.

Experiments and CFD predictions of solid particle distribution in a vessel agitated with four pitched blade turbines

Abstract: The distribution of solid particles in a high aspect-ratio baffled tank agitated with four 45° pitched blade turbines (PBT) was investigated using both experimental measurements and CFD simulations. Dilute suspensions of glass beads in water and moderately viscous liquids were considered. The measurement of axial particle concentration profiles was conducted by means of a light attenuation technique. Fully predictive simulations of solid-liquid suspensions were performed using a Sliding-Grid approach coupled with the Eulerian-Eulerian Two Fluid Model and the ‘homogeneous’ two-phase k-ɛ turbulence model. The simulated particle axial concentration profiles were compared with the experimental data and good agreement was found.

Effect of Axial Agitator Configuration (Up-Pumping, Down-Pumping, Reverse Rotation) on Flow Patterns Generated in Stirred Vessels

Abstract: Single phase turbulent flow in a tank stirred with two different axial impellers, a pitched blade turbine (PBT) and a Mixel TT (MTT), has been studied using Laser Doppler Velocimetry. The effect of the agitator configuration, i.e. up-pumping, down-pumping and reverse rotation, on the turbulent flow field, as well as power, circulation and pumping numbers has been investigated. An agitation index for each configuration was also determined. In the down-pumping mode, the impellers induced one circulation loop and the upper part of the tank was poorly mixed. When up-pumping, two circulation loops are formed, the second in the upper vessel. The PBT pumping upwards was observed to have a lower flow number and to consume more power than down-pumping, however, the agitation index and circulation efficiencies were notably higher. The MTT has been shown to circulate liquid more efficiently in the up-pumping configuration than in the other two modes. Only small effects of the MTT configuration on the power number, flow number and pumping effectiveness have been observed.

Scaling up Bubble Column Reactors with the Aid of CFD

Abstract: Bubble column reactors, used widely in industry, often have large column diameters (up to 6 m) and are operated at high superficial gas velocities (in the range of 0.1 to 0.4 ms −1 ) in the churn-turbulent flow regime. Experimental work on bubble column hydrodynamics is usually carried out on a scale smaller than 0.3 m, at superficial gas velocities lower than 0.25 ms −1 . The extrapolation of data obtained in such laboratory scale units to the commercial scale reactors requires a systematic approach based on the understanding of the scaling principles of bubble dynamics and of the behaviour of two-phase dispersions in large scale columns. We discuss a multi-tiered approach to bubble column reactor scale up, relying on a combination of experiments, backed by Computational Fluid Dynamics (CFD) simulations for physical understanding. This approach consists of the following steps: (a) description of single bubble morphology and rise dynamics (in this case both experiments and Volume-of-Fluid (VOF) simulations are used); (b) modelling of bubble-bubble interactions, with experiments and VOF simulations as aids; (c) description of behaviour of bubble swarms and the development of the proper interfacial momentum exchange relations between the bubbles and the liquid; and (d) CFD simulations in the Eulerian framework for extrapolation of laboratory scale information to large-scale commercial reactors.

Flow boiling of refrigerant R141B in small tubes

Abstract: Small circular and non-circular tubes are widely encountered in compact evaporators and condensers. This paper presents an experimental study of two-phase flow and heat transfer of refrigerant R141b, in small tubes. Four circular tubes with diameters of 1.1, 1.8, 2.8, 3.6 mm and one square tube of 2×2mm 2 were used in the test programme. The parameter ranges were mass flux 50 ∼ 3500kg m −2 s −1 , heat flux 1 ∼ 300kWm −2 , and inlet pressure 1 ∼ 3 bar resulting in mean boiling heat transfer coefficients of 0.1 ∼ 10 kW m −2 C −1 . It was found that local heat transfer coefficients are not only a strong function of heat flux but also a function of vapour quality and a weaker function of mass flux for the all the small tubes tested, showing that both nucleate boiling and convective evaporation occur in small tubes. The mean heat transfer coefficient was found to be is primarily a function of the heat flux, rather than the mass flux. Comparison with data in the literature shows that a general flow map developed from adiabatic two-phase flow tests can provide guidance for prediction of flow regimes in heated tubes. This study provides useful design data for two-phase flow and heat transfer in small tubes and channels.

The Implications and Challenges of Enhanced Heat Transfer for the Chemical Process Industries

Abstract: This paper considers the many techniques that have been developed to enhance convective heat transfer, with the emphasis on heat exchangers found in the chemical process industries. A summary is given of the techniques that are effective for the various modes of heat transfer. Of particular interest currently is compound enhancement that involves the simultaneous application of several techniques, so as to produce an enhancement that is larger than the individual techniques operating separately. There are many experimental and analytical challenges that relate to obtaining and applying data. The current advanced enhancement represents third generation heat transfer technology. This technology will find increasing application in all industries.

Trailing Vortices of Rushton Turbine: PIV Measurements and CFD Simulations with Snapshot Approach

Abstract: Understanding fluid dynamic characteristics of trailing vortices behind impeller blades and the capability to computationally simulate these vortices is essential for reliable design and scale-up of stirred reactors. In this paper, trailing vortices behind the blades of a standard Rushton turbine were studied using particle image velocimetry (PIV). Angle resolved and angle averaged flow fields near the impeller blades were measured and the structure of trailing vortices was studied in detail. A computational snapshot approach of Ranade and Dommeti was extended and used to simulate flow generated by the Rushton turbine in baffled stirred vessels. The approach was implemented using the commercial CFD code, FLUENT (of Fluent Inc, USA). Two turbulence models, namely, standard k – ɛ model and renormalization group version (RNG) of k – ɛ model were used for simulating the flow in stirred vessels. Predicted results were compared with the angle resolved PIV measurements to examine whether the computational model captures the flow structures around impeller blades. Predicted results were also compared with the angle averaged PIV data. Predicted gross flow characteristics like pumping number were also compared with the present and previously published experimental data. The results and conclusions drawn from this study will have important implications for extending the applicability of CFD models for simulating flow near impeller blades.

The effect of size, location and pumping direction of pitched blade turbine impellers on flow patterns: lda measurements and cfd predictions

Abstract: A combined LDA-CFD (MRF method) study has been undertaken of the turbulent flow associated with two sizes of 6-bladed 45° pitch blade turbine at three clearances, either pumping up or down. Good quantitative agreement for mean velocity vectors has been obtained but that for RMS values was poor. The mean discharge angle is more vertical: (a) for the smaller impeller at all clearances; (b) for both sizes when an impeller approaches a surface towards which it is pumping.

Experimental Analysis of Floc Size Distribution and Hydrodynamics in a Jar-Test

Abstract: The present work concerns coagulation-flocculation in drinking water treatment units. This study focuses on the link between hydrodynamics and floe size distribution. Experimental analysis of both floe size distributions and local hydrodynamics are performed in a jar-test vessel. In this study, water quality (pH) and coagulant (both type and dose) are fixed. Floe size distributions are analysed by image processing. The temporal evolution of the floe size distributions is determined from a statistical analysis of experimental data. It leads to basic information on floe agglomeration kinetics. The velocity field is analysed using Particle Image Velocimetry (PIV). The statistical analysis of the velocity field leads to the determination of the average velocity field and the average velocity gradients. In addition, instantaneous velocity fields and associated instantaneous velocity gradients are determined. It is shown to be important to locate their maximum values. The analysis highlights the close relationship between the characteristic size of the floes and hydrodynamics, that can be expressed in terms of local dissipation of turbulence or in terms of velocity gradient.

Impeller Geometry Effect on Velocity and Solids Suspension

Abstract: The effect of varying impeller geometrical parameters, including impeller type, number of impeller blades, blade pitch angle and blade thickness, on the turbulent velocity fields in mixing tanks has been studied through LDV measurements. Pitch bladed turbines and disc turbines were used in the experiments. It was found that universal time-mean velocity profiles exist for both pitch bladed turbines and disc turbines. Flow number correlations based on power number for both pitch bladed turbines and disc turbines are presented. It was found that there is a link between impeller pumping capacity and the S parameter in the Zwietering correlation, and this can be expressed as NQ × S = K, where NQ is the impeller flow number and K is a non-dimensional constant independent of the solid/liquid material property and impeller geometry.

Applications of the Turbulent Entrainment Assumption to Immiscible Gas-Liquid and Liquid-Liquid Systems

Abstract: The entrainment assumption, originally developed to describe mixing of miscible fluid pairs, has been used successfully in previous investigations to predict the liquid inflow velocity at the boundary of gas or vapour jets submerged in liquids. In this paper, the power of the entrainment assumption is further revealed by the application of the Ricou-Spalding form of the assumption to the breakup of drops and jets in liquid and gas surroundings, and to liquid entrainment from initially stratified two-phase regions.

Assessment of Large Eddy Simulations for Agitated Flows

Abstract: Large eddy simulations (LES) on the single-phase flow driven by a pitched blade impeller in a baffled stirred tank reactor were performed. The geometry, and operation conditions (defined in terms of Re = 7,300) were chosen to comply with the experimental work by Schafer et al . As the turbulence in stirred tanks is strongly off-equilibrium, no straightforward criteria with respect to spatial and temporal resolution of the simulations can be formulated, and experimental validation becomes of prime importance. In this study, the influence of the resolution, and the specific choice of the subgrid-scale model (standard Smagorinsky model, and structure function model) on the simulated flow field were investigated.

Liquid-Liquid Dispersion in an SMX-Sulzer Static Mixer

Abstract: Break-up processes have been investigated in an SMX-Sulzer static mixer. Oil/water and water/oil dispersions have been studied over a range of Reynolds, Weber and viscosity numbers. A semiempirical model has been developed for the prediction of mean drop size by equating the disruptive force acting upon the drop to the cohesive forces due to surface and viscous resistance to breakage. To establish the model, the liquid–liquid dispersions are assumed to be pseudo-fluids and the energy dissipation rate is calculated from pressure drop measurements expressed through a capillary model. Experimental data are correlated with Reynolds and Weber numbers and are analysed in order to show the influence of surface and viscous resistance on the breakage process.

Design and Optimization of Fully Thermally Coupled Distillation Columns: Part 2: Application of Dividing Wall Columns in Retrofit

Abstract: This paper addresses the application of dividing wall columns in retrofit. It emphasizes the need to take maximum advantage of the existing hardware with minimum capital outlay. Based on this study, several practical issues associated with the application of the dividing wall column in retrofit have been identified and as a result, its thermodynamically equivalent arrangements, such as the prefractionator arrangement and the Petyluk column, are often recommended instead. A case study involving the improvement of energy efficiency and capacity expansion of the NGL separation train has been illustrated to demonstrate the analysis involved.

Kinetic Study of Zinc Oxide Reduction by Methane

Abstract: Zinc oxide reduction by methane can be an alternative method for zinc production Methane is more reactive than coke (the common reducing agent), thus, the operating temperature can be considerably decreased Therefore, this method can omit or simplify the troublesome splash zinc condenser of the usual processes Moreover, the gaseous products of this method can be used as synthesis gas for the petrochemical industries In this work, the kinetics of zinc oxide reduction with methane was studied by thermogravimetry, and simultaneous and online gas analysis Then by applying a suitable model, kinetic parameters were determined These parameters are essential for the design of pilot or industrial plants

Saturated Flow Boiling of Water in a Narrow Channel: Experimental Investigation of Local Phenomena

Abstract: This paper describes some observations of flow boiling of water at near-atmospheric pressure in a single vertical channel of rectangular cross-section 2 mm × l mm, heated on three sides over a length of 248 mm, with a window on the fourth side. Measurements have been made of the fluctuating wall temperatures and pressures at several stations along the channel, with simultaneous high-speed video recordings of the flow patterns at a mass flux of 134 kg m −2 s −1 and heat fluxes of 50–100 kW m −2 . The frequency spectra of the fluctuations depend on the compressibility of the inlet flow to the channel. It is shown that the processes of initiation of boiling and of heat transfer differ from flow boiling in large channels. The acceleration of liquid slugs by confined bubbles causes pressure pulses to propagate the full length of the channel. Heat transfer is distributed in time between convection and nucleate boiling. The experimental limitations are discussed.

Effect of Pipe Diameter on the Performance of Drag-Reducing Polymers in Annular Gas-Liquid Flows

Abstract: In a recent paper, Al-Sarkhi and Hanratty examined the influence of a drag-reducing polymer on air-water annular flows in a 9.53 cm pipeline. This paper describes similar studies in a 2.54 cm pipe, for which the annular flow has different characteristics and for which the friction factors assume larger values in air-water systems. Both studies show a maximum drag reduction that is accompanied (in most cases) by a change to a stratified flow for which the concentration of drops in the gas phase is zero or close to zero, and for which the friction factor is about 1.5 times what would be observed for air flowing alone. The amount of drag reduction, therefore, depends on the magnitude of the friction factor for the air-water flow. Drag reductions up to 63% were observed in the 2.54 cm pipe. This is to be compared with the 48% previously achieved in the 9.54 cm pipe. Larger concentrations of polymer are needed to obtain maximum drag reduction in the smaller pipe.

Experience with Experimental Standards for Measurements of Various Parameters in Stirred Tanks: A Comparative Test

Abstract: Stirred tanks are used for several operations in industrial practice. Numerous scientific papers have been presented in literature dealing with experimental results on these applications. Comparisons and valuations of these data often fail because geometric parameters, experimental conditions, and measurement techniques differ notably. herefore, it can be observed that correlations derived on the basis of these experiments often show large discrepancies. In a cooperative test of nine German working groups different experiments were carried out in stirred tanks under completely standardized conditions. Thereby, common measurement techniques were examined in their reliability.

Influence of Gas Flow Rate on the Structure of Trailing Vortices of a Rushton Turbine: PIV Measurements and CFD Simulations

Abstract: Trailing vortices behind rotating impeller blades play crucial role in determining gas accumulation behind them. The gas accumulation behind blades affects the pumping and power dissipation capacity of the impeller and thus significantly affects the performance of gas–liquid stirred reactors. Understanding fluid dynamic characteristics of these trailing vortices and capability to computationally simulate these vortices is, therefore, essential for reliable design and scale-up of stirred reactors. In this paper, we have used particle image velocimetry (PIV) technique and CFD model based on computational snapshot approach for systematically studying influence of gas flow rate on structure of trailing vortices behind blades of a Rushton turbine. PIV measurements were carried out in a standard, fully baffled stirred vessel (H/T= 1) with a flat bottom. Vessel diameter was 0.4 m. A six bladed standard Rushton turbine was placed at one third of liquid height with a ring sparger. Four baffles of 1/10 T width were placed at equal spacing. Tap water was used as a medium in the vessel. Measurements were carried out at five different gas flow rates to vary the dimensionless gas flow number in the range of 0.01 to 0.06. Both, angle resolved and angle averaged flow fields near the impeller blades were measured. The structure of trailing vortices in presence of gas was studied in detail. A Eulerian–Eulerian, two fluid model was used to simulate dispersed gas–liquid flow in stirred vessel. A computational snapshot approach was used to simulate impeller rotation. The computational model was implemented using the commercial CFD code, FLUENT (of Fluent Inc., USA) with the help of user defined subroutines. The computational model was used to simulate flow in stirred vessel operating under conditions used in the experiments. The results of this study will have important implications for extending the applicability of CFD models for simulating multiphase stirred reactors.

The Effect of Swirl on Flow Stability in Spray Dryers

Abstract: The effect of varying the amount of inlet air swirl on the stability of the flow patterns in a small-scale, co-current spray dryer has been investigated. The objective of this work was better understanding of the effect of the vane angle on the flow patterns and to determine whether a particular vane angle provided superior performance. The dryer studied was a cylinder on cone unit, with a drying chamber 0.8 m in diameter and 1.61 m tall, and fitted with adjustable swirl vanes tightly surrounding a Delavan GA1 two-fluid atomiser. Swirl vane angles between 0° and 45°, in 5° increments, were investigated using a complementary combination of flow visualisation and laser Doppler velocimetry (LDV) techniques. No single swirl vane angle resulted in behaviour that was clearly steady throughout the dryer, but a swirl vane angle of about 25° was considered to be an appropriate selection. This vane angle, corresponding to a Swirl number of approximately 0.45, gave an observable degree of stability in much of the flow domain and good air-spray mixing without excessive spreading of the spray cloud and wall deposition. The introduction of spray had a significant effect on the flow behaviour, so that air-only studies did not adequately represent the flow conditions with spray.

Contact angles of droplets during spread and recoil after impinging on a heated surface

Abstract: The contact angle of a droplet impinging upon a hot surface undergoes changes as the drop spreads and recoils. The motion of the liquid and the effect of evaporation from the edges of the drop affect the contact angle. The changes in the contact angle during spreading affect the spreading characteristics of the droplet upon impact. However, the models available in literature for the maximum spreading ratio (maximum spread diameter divided by the initial droplet diameter) do not include this effect. In addition, the actual area of contact of a droplet is needed in the heat transfer studies. The present work reports an experimental study conducted to characterize the contact angle variation as a function of the heater surface finish, heater material, and heater surface temperature prior to impact. It is seen that the dynamic advancing contact angle extends beyond the equilibrium advancing and receding contact angles during the motion of the interface. Since the droplet spreading is influenced by the dynamic advancing contact angle, it is proposed to use the dynamic contact angle measurements in the available models for maximum spreading ratio. The experimental results obtained on the maximum spreading diameter indicate the validity of this approach.

Numerical Simulations of Gas Flow Patterns Within a Tall-Form Spray Dryer

Abstract: Numerical simulations of the air flow patterns within a small scale tall-form countercurrent spray dryer have been performed. The simulations were performed using CFX 4.3, a finite volume based, computational fluid dynamics package. This study represents the first application of the Very Large Eddy Simulation (VLES) approach to the simulation of spray dryers. They have been performed in order to gain a more detailed understanding of the flow patterns and their stability in this design of dryer, which is commonly used in countercurrent drying applications, such as the drying of detergents. Limited validation of the simulations was achieved through comparison against qualitative experimental flow pattern information. It was found that by altering the angle of the inlet air streams into the dryer, the nature of the flow within the dryer could be significantly altered. In the majority of the cases simulated, large transients developed in the flow, the nature of these transients being critically dependent on the inlet conditions. The existence of such transients would be detrimental to actual spray dryer performance, however the flow patterns can be stabilised by introducing a large amount of swirl into the chamber.

The Transient Response of Granular Flows in an Inclined Rotating Cylinder

Abstract: This paper reports the results of an experimental and theoretical study to investigate the transient response of the granular flow through a laboratory scale inclined rotating cylinder to large step changes in one of three variables: (i) mass feed rate, (ii) rotation speed or (iii) axis inclination. Experimental measurements are reported for a range of operating conditions, sizes of step and a number of cylinder geometries. A mechanistic model for the transient response is derived based on a published steady state model. The dynamic model parameters are the cylinder radius, length, discharge dam height, axis inclination and rotation speed, the granular feed rate, and the bulk density and dynamic angle of repose of the granular material. The model has no adjustable parameters, making it useful for scale-up. The model takes the form of a non-linear partial differential equation, equivalent to a one dimensional unsteady diffusion equation with variable coefficients, and solution has been obtained numerically. Good agreement is found between the model and experiment both in the range of cases considered in the current study, and also for published experimental work, for which the cylinder size and granular material properties differ substantially from those of this work.

Modelling of an Industrial Fluid Catalytic Cracking Unit Using Neural Networks

Abstract: An artificial neural network (ANN) model for determining the steady-state behaviour of an industrial Fluid Catalytic Cracking (FCC) unit is presented in this paper. Industrial data from a Greek petroleum refinery were used to develop, train and check the model. FCC is one of the most important oil refinery processes. Due to its complexity the modelling of the FCC poses a great challenge. The proposed model is capable of predicting the volume percent of conversion based on six input variables. This work is focused on determining the optimum architecture of the ANN, in order to gain good generalization properties. The results show that the ANN is able to accurately predict the measured data. The prediction errors in both training and validation data sets are almost the same, indicating the capabilities of the model to accurately generalize when presented with unseen data. The neural model developed is also compared to an existing non-linear statistical model. The comparison shows that the neural model is superior to the statistical model.

The Split of Horizontal Stratified Flow at a Large Diameter T-Junction

Abstract: We are pleased to contribute this paper to the Special Geoff Hewitt Issue. This paper is particularly apposite as it was he who encouraged the large-scale work of which this is the latest part. He also encouraged the first studies on T-junctions by BJA and the late Peter Whalley.Phase split data have been obtained for a horizontal stratified flow approaching an equal diameter (0.127 m diameter) junction. In addition, the shape and position of the interface of the flow approaching the junction have been measured. The information has been combined to effect improvements to models to predict the phase split. The improved model gives good predictions for larger and smaller diameter T-junctions. Further improvements, particularly to remove empiricism, are required.

On the Criteria of Nucleate Pool Boiling Enhancement by Surfactant Addition to Water

Abstract: By testing against collected data from twelve surfactant additives, criteria for a given surfactant as a competent additive in enhancing nucleate boiling heat transfer of water is proposed in this work. As a result of the criteria, high solubility, high equilibrium surface tension depression, and low equilibrium contact angle depression are required. A surfactant is incapable of enhancing boiling of water, in which it dissolves sparsely. For a highly soluble surfactant, boiling heat transfer enhancement by its addition is enhanced by the depression of equilibrium surface tension but suppressed by the depression of equilibrium contact angle. These two effects counterbalance each other and result in different degrees of boiling enhancement. Surfactant additives can be categorized accordingly.