Channel Of Rectangular Cross-section Research Articles

Generation of entropy of turbulent EG-water-Al2O3 hybrid nanofluid flow through a channel of rectangular cross-section

In this work, the problem of the turbulent EG-water-Al2O3 hybrid nanofluids flow through a rectangular channel, kept under constant heat flux, has been solved numerically for different values of the Reynolds number , volume fraction of nanoparticles , effectiveness of nanofluids (EN), diameter of nanoparticle and pumping power . Using the SIMPLE algorithm and the turbulence k–ε model, the governing equations have been discretized, and the FVM has been employed to find the numerical results of the hydrothermal phenomena and the generation of entropy in terms of the average Nusselt numbers , absolute pressure drop , Bejan number (Be), generations of thermal (N thermal), frictional (N frictional) and total entropy (N total). The present study has important applications in the photovoltaic cell/collector used to convert solar radiation into the electrical and thermal energy, and also in the field of biomedical science for analysis of red cell dimensions where the cells are suspended in rectangular channel. It has also been found that with the increase in the value of the diameter of nanoparticle, the formation of thermal entropy increases, while that of the frictional entropy decreases.

Experimental Study on the Effect of Locally Convergent Configurations on the Flow Pattern in PMMA Microchips

The geometries of micro-channel play a key role in forming of digital droplets, and can be real-time or effective controlling methodologies. Local convergence regions are designed in the rectangular cross-section channels on PMMA microchips, in which two-phase coaxial jets are introduced by inserting a syringe needle. The two-phase flow (lubricating oil (continuous phase, flow rate Q c)/deionized water (dispersed phase, flow rate Q d)) is considered. Two geometric control variables, the relative position (needle displacement x) and tapering characteristics (convergence angle α), are naturally adopted to discribe such geometry configurations. The micro-flow under the change of these two parameters is mainly studied in this paper. Four kinds of characteristic flow patterns, namely, sausages, slug, dripping and jetting, are found in the experiment, and their occurring parameters and developing dynamic characteristics are discussed. The experiment shows that the increase of inner needle displacement x can produce higher frequency and finer droplets, which is in consistent with our previous results obtained in round tube experiments and simulations. While increasing the convergence angle α, contrarily, takes opposite effects.

Investigating heat transfer in an upward flow of liquid metal in the mercury facility with a loop of natural circulation

Research of heat transfer is performed in an upward flow of liquid metal in a vertical channel of rectangular cross-section under the condition of one-sided heating and in a vertical pipe in a mercury circuit with a natural circulation loop. Natural circulation loop enables measurements in mixed turbulent convection modes, which are inaccessible on a contour with a forced convection, that is, in the region of small Reynolds (Peclet) numbers of characteristics. Using a two-coordinate probe with a microthermocouple correlation sensor, profiles of averaged velocity and temperature and distribution of a wall temperature are obtained, and heat transfer coefficients are determined. Criterion dependences of Nusselt number on Peclet and Richardson are built. Experimental data on heat transfer are compared with similar data obtained earlier in a loop with a forced flow of mercury.

Vibroextrusion flow of concrete mixtures in a right four-corner pyramidal channel

The purpose of the research was to obtain calculation formulas for describing the flow of concrete mixtures and determining their viscosity directly in the process of vibroextrusion in regular quadrangular pyramidal channels.
 In solving the flow problems, it was taken into account that concrete mixtures are non-Newtonian systems in the conditions of a vibration field, and hydrodynamic theories were used to calculate the processes and rheological characteristics.
 Since the calculation formulas for describing the flow of fluids in a regular quadrangular pyramidal channel are absent and the channel has a square cross section, an analytical dependence was used to characterize the process, which describes the flow of a Newtonian fluid in a rectangular channel of constant cross section. The authors proposed for use simplified formulas for determining the maximum flow rate and flow rate of a Newtonian fluid in a channel of rectangular cross-section.
 The degree of decrease in the flow velocity and the flow rates of a Newtonian fluid in a channel of rectangular cross-section in comparison with the flow between flat parallel plates are analyzed.
 To describe the flow of concrete mixtures in regular quadrangular pyramidal channels, the proposed coefficients for reducing the speed and flow rate, as well as the existing formulas for the flow between flat symmetric stationary walls, which converge, were used.
 The possibility of using the obtained formula for the flow rate in a regular quadrangular pyramidal channel for calculating the viscosity of a concrete mixture during vibration extrusion is shown. To simplify the experimental procedure, a new formula for calculating the viscosity based on the expiration time for a certain amount of concrete mixture has been proposed.
 The formulas obtained are convenient for further mathematical processing and have no restrictions in their application. The proposed method for determining the viscosity expands the possibilities of studying the rheological properties of concrete mixtures in the vibroextrusion of fiber-reinforced concrete mixtures.

Investigation of heat transfer in dimple-protrusion micro-channel heat sinks using copper oxide nano-additives

The ability of controlling the temperature of a system by using heat transfer and thermodynamics-based technology is known as thermal management. Thermal management is a crucial problem which is to be addressed in the various fields. This study accompanies a numerical work to provide a solution for heat transfer enhancement by studying a straight rectangular cross-section channel. The selected channel is used with and without disturbance in the form of dimple-protrusion with circular & through geometry. Water is employed as the base fluid for identifying flow behavior and heat carrying capacity while the heat flux is applied as a boundary condition. Furthermore, using the same boundary condition, copper oxide nanoparticles with the concentration of 0.10%, 0.15% and 0.20% respectively are added in the channels. Flow conditions are varied from 100 to 900 considering the laminar regime with fully developed flow. Performance enhancement criteria are evaluated by considering the Nusselt number, friction factor, and base cooling. The fully developed velocity profile is maintained at the inlet of test section for all Reynolds numbers. Semi-Implicit Method for Pressure Linked Equations (SIMPLE) pressure-velocity coupling is used with the second-order upwind scheme for discretization and solution approximation. It is observed that circular dimple-protrusion channel is effective for high Reynolds numbers and through dimple protrusion is efficient for intermediate Reynolds numbers. 10% improvement in Nusselt number is observed for circular dimple-protrusion channel with 0.2% CuO concentration by comparing the results of dimple-protrusion channel with straight channel at same nanoparticle concentration. Compared to straight channel with 0% and 6% particles concentration 5% and 7% increase in Stanton number is observed for through disruption case. Dimple channel on the other hand showed 3% and 5% increase. Furthermore, the formation of vortices at high Reynolds number is observed in circular dimple-protrusion channel.

Numerical and Theoretical Study of Performance and Mechanical Behavior of PEM-FC Using Innovative Channel Geometrical Configurations

Proton exchange membrane fuel cell (PEM-FC) aggregation pressure causes extensive strains in cell segments. The compression of each segment takes place through the cell modeling method. In addition, a very heterogeneous compressive load is produced because of the recurrent channel rib design of the dipole plates, so that while high strains are provided below the rib, the domain continues in its initial uncompressed case under the ducts approximate to it. This leads to significant spatial variations in thermal and electrical connections and contact resistances (both in rib–GDL and membrane–GDL interfaces). Variations in heat, charge, and mass transfer rates within the GDL can affect the performance of the fuel cell (FC) and its lifetime. In this paper, two scenarios are considered to verify the performance and lifetime of the PEM-FC using different innovative channel geometries. The first scenario is conducted by adopting a constant channel height (H = 1 mm) for all the differently shaped channels studied. In contrast, the second scenario is conducted by taking a constant channel cross-sectional area (A = 1 mm2) for all the studied channels. Therefore, a computational fluid dynamics model (CFD) for a PEM fuel cell is formed through the assembly of FC to simulate the pressure variations inside it. The simulation results showed that a triangular cross-section channel provided the uniformity of the pressure distribution, with lower deformations and lower mechanical stresses. The analysis helped gain insights into the physical mechanisms that lead to the FC’s durability and identify important parameters under different conditions. The model shows that it can assume the intracellular pressure configuration toward durability and appearance containing limited experimental data. The results also proved that the better cell voltage occurs in the case of the rectangular channel cross-section, and therefore, higher power from the FC, although its durability is much lower compared to the durability of the triangular channel. The results also showed that the rectangular channel cross-section gave higher cell voltages, and therefore, higher power (0.63 W) from the fuel cell, although its durability is much lower compared to the durability of the triangular channel. Therefore, the triangular channel gives better performance compared to other innovative channels.

INFLUENCE OF ROUGHTY OF A RECTANGULAR OPEN CHANNEL ON FLOW SPEEDS AND WATER FLOW

<p>This paper presents an analysis of water flow and flow velocity in an open channel of rectangular cross section as a function of channel roughness, ie Manning roughness coefficient. Three different cases of finishing the open channel of rectangular cross-section were analyzed, namely the concrete channel, the channel with smoothed cement mortar and the channel in very poor condition with shore erosion overgrown with sedges and large stones at the bottom of the channel. Analyzes were performed on a rectangular channel 3.0 m wide and with water depths in the channel from 0.5 m to 1.5 m with a step of 0.1 m, for steady flow. The longitudinal fall of the bottom of the rectangular channel in all analyzed cases was 1%.u urađene na pravougaonom kanalu širine 3,0 m i sa dubinama vode u kanalu od 0,5 m do 1,5 m sa korakom od 0,1 m, za ustaljeno tečenje. Podužni pad dna pravougaonog kanala u svim analiziranim slučajevima je iznosio 1%. The paper showed that higher Mannig roughness coefficients give lower water flow in the canal. The importance of the paper is reflected in the fact that for each of the three analyzed treatments of the rectangular channel, a quadratic function of water velocity and flow depending on the water depth in the channel was obtained. During the hydraulic calculation, the type of lining of the open rectangular channel should be taken into account in order for the dimensioned channel to be the most favorable in terms of maximum throughput.</p>

Influence of wettability and initial size on the merging dynamics of droplet within a Y-shaped bifurcating channel

Present study numerically investigates the merging dynamics of two identical dispersed droplets within a Y-shaped bifurcating channel of rectangular cross-section. A finite element method based solver is adopted to analyze the flow dynamics within the bifurcating channel for different wall wettability conditions, initial size of the dispersed droplets and for a fixed Capillary number. The merging dynamics is described with the help of morpho-dynamic evolution along with the temporal evolution of wetted area of the droplets. It reveals that the daughter droplet generated from the mating of identical parent droplets at the bifurcating tip has the affinity to stick at the junction for hydrophilic and neutral wettability configurations of the walls, thus generating longer size droplet. On the contrary, for hydrophobic wettability of the walls, the parent droplets merge to each other bypassing the bifurcating junction while mating and subsequently entrapping of droplets takes place within the daughter droplet. Moreover, the influence of wall wettability is insignificant when the initial sizes of the parent droplets are smaller.

On the structure of the orthotropic 3D permeability tensor of an anisotropic porous body in heat and mass transfer problems

The structure of a 3D permeability tensor with orthotropic properties is proposed to describe various heat and mass transfer processes in anisotropic porous media. This structure of the permeability tensor has advantages over the monoclinic and triclinic because of the minimum number of elements to be determined, which greatly simplifies the formulation, conduct and processing of the corresponding experiments. In addition, the confirmation of the operability of such a structure opens up new opportunities for the artificial creation of porous materials with specified properties and the required anisotropy architecture on 3D printers. Using the Jacobians of rotations in the Cartesian reference system, the matrix of rotations is found as a result of successive multiplication of Jacobians by different angles around the axes of the base coordinate system. The final form of the tensor is identified as the product of the orthotropic tensor on the left and right by the turn matrix and the transposed turn matrix, respectively. This representation allowed us to calculate the inverse permeability tensor and synthesize a mathematical 3D model of Darcy-Brinkman type of unidirectional flow in a porous channel of rectangular cross-section on the example of hydrodynamic filtration of a Newtonian fluid. It is shown that such linearization does not level the property of anisotropy in a porous medium, while maintaining three-dimensionality.

Investigating the Effects of Bed Roughness on Incipient Motion in Rigid Boundary Channels with Developed Hybrid Geno-Fuzzy versus Neuro-Fuzzy Models

In present study, the comprehension of different types of bedforms generated by different bed roughness, in rectangular and circular cross-sectional rigid boundary channels and the resistance generated by the bed roughness against the flow are evaluated. Moreover, the critical condition of sediment motion under bed roughness influence is investigated. This study presents novel contributions in solving engineering problems that suffer from the lack of knowledge on the incipient motion of sediment particles in rigid boundary channels under bed roughness effects. Furthermore, soft computing and evolutionary computation methods have been combined to develop a new novel predictive model. The Evolutionary GENOFIS also aims at improving the ANFIS tool depends on Sugeno Fuzzy Inference System. The advantages of the proposed hybrid GENOFIS approach over the ANFIS tool is its novel ability to tackle the incipient prediction problem by using less fuzzy based rules and to represent the consequent part of the model as a constant, linear or non-linear function as well as with possible simultaneous combinations of them. For validation and comparison purposes, model performances of hybrid GENOFIS, ANFIS and data-based linear regression approaches are evaluated via experimental data partitioned for testing purposes. The model results are compared through corresponding RMSE and CE values. It is found that the hybrid GENOFIS model results outperformed the ANFIS and linear regression models. Consequently, novel experimental data-driven incipient motion formula and the hybrid GENOFIS approach are proposed to modeling complex incipient motion problems that are full of uncertainty and complication.

Flexural wave-based soft attractor walls for trapping microparticles and cells.

Acoustic manipulation of microparticles and cells, called acoustophoresis, inside microfluidic systems has significant potential in biomedical applications. In particular, using acoustic radiation force to push microscopic objects toward the wall surfaces has an important role in enhancing immunoassays, particle sensors, and recently microrobotics. In this paper, we report a flexural-wave based acoustofluidic system for trapping micron-sized particles and cells at the soft wall boundaries. By exciting a standard microscope glass slide (1 mm thick) at its resonance frequencies <200 kHz, we show the wall-trapping action in sub-millimeter-size rectangular and circular cross-sectional channels. For such low-frequency excitation, the acoustic wavelength can range from 10–150 times the microchannel width, enabling a wide design space for choosing the channel width and position on the substrate. Using the system-level acousto-structural simulations, we confirm the acoustophoretic motion of particles near the walls, which is governed by the competing acoustic radiation and streaming forces. Finally, we investigate the performance of the wall-trapping acoustofluidic setup in attracting the motile cells, such as Chlamydomonas reinhardtii microalgae, toward the soft boundaries. Furthermore, the rotation of microalgae at the sidewalls and trap-escape events under pulsed ultrasound are demonstrated. The flexural-wave driven acoustofluidic system described here provides a biocompatible, versatile, and label-free approach to attract particles and cells toward the soft walls.

A computational model for predicting filtration performance of 3D-magnetic filters under different channel geometries, particle sizes and flow conditions

High-gradient magnetic separation (HGMS) is an appropriate technique for filtration of ultra-fine and weakly magnetic or even nonmagnetic particles by coagulation methods. The run time for the commercial 3D-codes in this field takes hours or days for computing such a 3D-complex physics. This paper presents a low time-consuming numerical model for parametric investigation of the performance of 3D-HGMS filters for filtration of a specific fluid which flows within various rectangular and square cross-sectional channels and has not studied up to now. The performance of the 3D-HGMS filters for filtration of fully developed flow is modeled by post-processing the 2D-simulation results of COMSOL Multiphysics software. This model is validated with the available experimental results with a maximum relative error of 15 %. The results of parametric study demonstrate that the capture efficiency is inversely related to the Reynolds number and the vertical spacing between the rods and is directly related to the particle diameter and channel aspect ratio. In addition, the performance of the 3D-HGMS filter with the triangular arrangement of the rods is better than that with the rectangular matrix arrangement. Furthermore, the total amount of particles captured on the rod matrix is obtained for particles with Gaussian distribution of diameter size at the channel inlet under different matrix geometries, particle sizes, and flow Reynolds numbers.

Measurement of the flow behavior index of Newtonian and shear-thinning fluids via analysis of the flow velocity characteristics in a mini-channel

An in-situ measurement technique to determine the rheology of a fluid based on the experimentally measured velocity profile of a flow in a mini-channel is introduced. The velocity profiles of a Newtonian and different shear-thinning fluids along a rectangular channel were measured using shadowgraph particle image velocimetry (PIV). Deionized water and different concentrations of a polyacrylamide solution were used as Newtonian and shear-thinning fluids, respectively and were studied at different Reynolds numbers. The flow indices of the fluids were determined by comparing the experimental velocity profile measurements with developed theory that takes into account the non-Newtonian nature of the fluids rheology. The results indicated that the non-Newtonian behavior of the shear-thinning fluid intensified at lower Reynolds numbers and it behaved more as a Newtonian fluid as the Reynolds number increased. A comparison between the power law index determined from experimental monitoring of the velocity profile at different Reynolds numbers and measurements from a rheometer reflected good agreement. The results from the study validate the new approach of the rheology measurement of Newtonian and non-Newtonian flows through straight, rectangular cross-section channels. The proposed approach can be further utilized using other methods such as X-ray PIV to characterize the rheology of non-transparent fluids and in general, for all non-Newtonian fluids.

Investigation of thermal energy exchange potential of a gravitational water vortex

The conversion of hydel energy of the water vortex formed under gravity is well known in the form of gravitational water vortex turbines; however, the thermal energy exchange potential of the gravitational water vortex flow (GWVF) is yet to be explored. Heat transfer investigation of GWVF is important because the natural gravity being the sole driving force can considerably reduce the pumping power requirement. The present study is the first of its kind to investigate hydro-thermal characteristics of the gravitational water vortex heat exchanger (GWVHE) using an in-house developed experimental test rig. The proposed heat exchanger involves a spiral channel of rectangular cross-section constructed around a cylindrical basin generating a GWVF. For various inlet mass flow rates and temperature combinations, energy balance between the two water streams as well as Nusselt number correlations are determined. Experimentally, a reasonable energy agreement has been achieved with a maximum loss of 30% among the two different inlet temperature tested conditions. Moreover, maximum rise in cold side temperature for the tested conditions was 6K, whereas for the hot side the measure in temperature drop was 8K. A numerical simulation has also been conducted to virtually predict the performance of the designed heat exchanger. The simulation process has shown an improved energy balance with a maximum loss of 10%. A comparison of the experimental and numerical results shows that a GWVF has the potential to effectively exchange the heat between the two fluid streams moving under gravity. The performance of the proposed GWVHE with a pipe in pipe heat exchanger (PPHE) with parallel flow configuration (without vortex) has also been modeled and compared. For the same conditions, the maximum difference in temperature drop and gain between GWVHE and PPHE is 5K and 2.5K, respectively. The present study may act as a benchmark for the new class of GWVF based heat exchangers.

Viscoplastic fingering in rectangular channels.

We experimentally study the viscous fingering problem of viscoplastic fluids in channels of rectangular cross section. We find that a yield stress-dependent capillary number (Ca^{*}) and an aspect ratio-dependent Bond number (Bo^{*}) can classify the finger shape into ramified and unified fingering patterns, and the finger flow regime into yield stress, viscosity, and aspect ratio-buoyancy-dominated regimes. For these regimes, we provide the transition boundaries using Ca^{*} and Bo^{*} and propose simple relations to predict the finger width, for a wide range of flow parameters, versus the capillary number, the channel aspect ratio, and the rheology of the viscoplastic fluid.

Nanoparticles phenomenon for the thermal management of wavy flow of a Carreau fluid through a three-dimensional channel

A hybrid nanofluid phenomenon is considered involving nanoparticles since such particles are potential medication transportation devices in biomedical applications. Moreover, the expansion in heat transfer can be accomplished by refining either flow mechanism with geometry annoyance or thermal conductivity of the liquid itself. This motivation encourages the authors to discuss the three-dimensional study to analyze the peristaltic transport of Carreau nanofluid in a cross section of rectangular channel. This investigation may be helpful in physiology, chemical industries and biomedical apparatus, especially for the dismissal of cancerous cells where the heat convection would be followed by nanoparticles through three-dimensional tube/duct. In the current analysis, the constitutive equations are managed under the assumptions of long wave length and low Reynolds number assumptions. After making use of some suitable dimensionless quantities, we gathered the partial differential equations in nonlinear coupled form which are then handled by the homotopy perturbation method. The variational occurrence of all emerging parameters affecting the flow is analyzed through graphical treatment. Velocity distribution is also plotted for three dimensions. Trapping scheme is also presented through streamlines which implies the flow phenomenon through circulating empty bolus traveling toward the flow. A comparative analysis is also made to differentiate between the behavior of Newtonian and Carreau fluid model. On the other hand, the pumping rate of 3D rectangular channel is also contrasted with a 3D square duct. It is analyzed that peristaltic pumping rate is enhanced with the growing values of aspect ratio, power law index and local nanoparticles Grashof number; however, the inverse results are faced with Brownian motion factor and Weissenberg number. It is also visualized that pumping rate in 3D rectangular enclosure is higher than a square duct geometry.

Turbulent Heat Transfer Characteristics of a W-Baffled Channel Flow - Heat Transfer Aspect

In this work, the thermal behavior of a turbulent forced-convection flow of air in a rectangular cross section channel with attached W-shaped obstacles is investigated. The continuity, momentum and energy equations employed to control the heat and velocity in the computational domain. The turbulence model of k-ε is employed to simulate the turbulence effects. The finite volume method with SIMPLE algorithm is employed as the solution method. The results are reported temperature, local and average Nusselt numbers, and mean velocity contours. The subject is relevant and important for industrial applications.

A Study of Travel Time for Different Open Channels

An increase in developmental activities such as afforestation, paved surfaces and construction of buildings and other structures leads to an increase in surface runoff and peak discharge from the watershed. These all cause a decrease in detention storage and surface depression and thus diminish the concentration time that flow will take and distributes flow to the adjoining stream quickly rather than that would have taken before development or urbanization. Despite the importance of time of concentration, planners and engineers are often puzzled by different profiles of the channels and their equation available in the literature without knowing the accuracy of each formula. In this paper, kinematic wave theory integrated with the Manning’s equation has been applied for the comparative assessment of the performance of the various cross-sectional channels. The result of different channel profiles toward travel time of flow has been matched for nine channel profiles. Of the nine channel profiles, it was found that the deep rectangular cross-sectional channel possesses the highest time of travel. Therefore, the use of deep rectangular channel yields lesser watershed runoff. The parabolic channel with more depth yields the lesser time of travel; therefore, the use of parabolic channel profile yields larger watershed runoff.

Transient Conjugate Heat Transfer Numerical Simulation in Superfluid Helium

Computational simulations of superfluid helium are needed in order to improve the cooling system design of superconducting magnets in particle accelerators and to achieve a better understanding of the transient phenomena during magnet quenches. A conjugate heat transfer numerical model based on the C++ toolbox OpenFOAM [1] is implemented to three-dimensional case studies involving superfluid helium and heating sources. The governing equations of the solver are modified according to the Kitamura’s model [2], a simplified version of the two-fluid model developed by Khalatnikov [3]. This simplified model is based on the assumption that the thermo-mechanical effect term and the Gorter-Mellink mutual friction term prevail on the others in the superfluid component momentum equation. Simulations are performed with the thermal conductivity function of superfluid helium both from theory [4] and the formulation used by Sato [5], who normalized the function according to a different conductive heat flux exponential coefficient determined from data analysis. An empirical calculation of the Kapitza conductance [6] is adopted in order to simulate the thermal resistance at the interface between helium and solids. Steady-state and transient simulations are compared to experimental data available in the literature. For such purpose, data are used from Van Sciver’s experiment in a helical coil [7] and a rectangular cross-section channel experiment conducted at CEA Paris-Saclay. The experiments comprised heaters and multiple temperature probes situated at different locations to track the temperature distribution and evolution of the superfluid helium state.

Suction Vortices in a Pump Sump—Their Origin, Formation, and Dynamics

Abstract The origin, formation mechanism, and dynamics of suction vortices in a pump sump have been clarified by large eddy simulation (LES) applied to two different computational models. The first one is a pump-sump model with uniform flow entering a water channel of rectangular cross section and a vertical suction (outlet) pipe installed at its downstream end. LES with different wall boundary conditions have revealed that the origin of a submerged vortex is the mean shear of the approaching boundary layers that develop on the bottom and side walls of the sump. Detailed investigations have revealed that deviation of the mean flow triggers conversion of the vorticity axis to the vertical direction. The local acceleration of the vertical flow stretches the aforementioned vertical vortex, which results in the formation of a submerged vortex. The second one is a simplified computational model composed of a paraboloid of revolution and aims to accurately simulate the stretch of the viscous core of a submerged vortex that has appeared under the suction pipe of the pump-sump model. The differences between the models, especially predictions of the minimum pressure, imply that cavitation could have been initiated in the viscous core, if it had been taken into account, as is observed in the pump-sump experiment at the same condition. Parametric studies with different initial swirl numbers from 0.12 to 16.3 have clarified the behavior of the submerged vortex. It was found that a strong submerged vortex appears only at a relatively small range of the swirl numbers from 1.25 to 3.