KKL Model Research Articles

Free convection of copper–water nanofluid in a porous gap between hot rectangular cylinder and cold circular cylinder under the effect of inclined magnetic field

Natural convection heat transfer of copper–water nanofluid in a porous gap between hot internal rectangular cylinder and cold external circular cylinder under the effect of inclined uniform magnetic field has been investigated. Domain of interest is a porous sector, where horizontal and vertical adiabatic borders are the external circular cylinder radii. Governing equations formulated in dimensionless stream function, vorticity and temperature variables using the Brinkman-extended Darcy model for the porous medium, single-phase nanofluid model with Brinkman correlation for the nanofluid viscosity and Hamilton and Crosser model for the nanofluid thermal conductivity have been solved numerically by the control volume finite element method. Effects of the Rayleigh number, Hartmann number, Darcy number, magnetic field inclination angle, nanoparticles volume fraction, nanoparticles shape factor, nanoparticles material, nanofluid thermal conductivity and dynamic viscosity models and nanofluid electrical conductivity correlation on streamlines, isotherms, local and average Nusselt numbers have been studied. Obtained results have shown the heat transfer enhancement with the Rayleigh number, Darcy number, nanoparticles volume fraction and nanoparticles shape factor, while the heat transfer rate reduces with the Hartmann number and magnetic field inclination angle. At the same time, the average Nusselt number increases at about 16% when nanoparticles volume fraction rises from 0 till 4% for Ra = 105, Ha = 25, while for Ha = 0 one can find the heat transfer rate augmentation at about 9% for the same conditions. In the case of different nanofluid thermal conductivity and dynamic viscosity models, it has been found that KKL model reflects the heat transfer rate reduction with nanoparticles volume fraction, while for the Hamilton–Crosser–Brinkman model, the heat transfer rate increases. Comparison between the Maxwell correlation for the nanofluid electrical conductivity and the base fluid electrical conductivity illustrates an intensification of the convective heat transfer rate for high values of the Rayleigh number (Ra ≥ 104) in the case of Maxwell correlation for the nanofluid electrical conductivity. At the same time, the effect of the nanoparticles volume fraction becomes more significant when nanofluid electrical conductivity is a function of nanoparticles volume fraction.

Simulation of water based nanofluid convective flow inside a porous enclosure via non-equilibrium model

In this article, nanofluid convective flow inside a porous enclosure is simulated by means of two-temperature model. Control volume based finite element method is employed for this purpose. Nanofluid properties are estimated by means of KKL model. Darcy- Boussinesq approximation is utilized for nanofluid flow field. Lorentz forces effects are taken into consideration. Roles of solid-nanofluid interface heat transfer parameter (Nhs), Rayleigh number (Ra), porosity (ε), and Hartmann number (Ha) are examined. Outputs demonstrate that |ψ|max enhance with rise of Nhs but it augments with rise of Ra. Porosity has opposite relationship with temperature gradient.

Entropy generation analysis and heatline visualization of free convection in nanofluid (KKL model-based)-filled cavity including internal active fins using lattice Boltzmann method

Two-dimensional natural convection and entropy generation in a square cavity filled with CuO–water nanofluid is performed. The lattice Boltzmann method is employed to solve the problem numerically. The influences of different Rayleigh numbers 103<Ra<106 and solid volume fractions 0<φ<0.05 on the fluid flow, heat transfer and total/local entropy generation are presented comprehensively. Also, the heatline visualization is employed to identify the heat energy flow. To predict the thermo-physical properties, dynamic viscosity and thermal conductivity, of CuO–water nanofluid, the KKL model is applied to consider the effect of Brownian motion on nanofluid properties. It is concluded that the configurations of active fins have pronounced effect on the fluid flow, heat transfer and entropy generation. Furthermore, the Nusselt number has direct relationship with Rayleigh number and solid volume fraction, and the entropy generation has direct and reverse relationships with Rayleigh number and solid volume fraction, respectively.

Lattice Boltzmann method based on Dual-MRT model for three-dimensional natural convection and entropy generation in CuO–water nanofluid filled cuboid enclosure included with discrete active walls

In the present study, the three-dimensional natural convection and entropy generation in a cuboid enclosure included with various discrete active walls is analyzed using lattice Boltzmann method. The enclosure is filled with CuO–water nanofluid. To predict thermo-physical properties, dynamic viscosity and thermal conductivity, of CuO–water nanofluid, the KKL model is applied to consider the effect of Brownian motion on nanofluid properties. In lattice Boltzmann simulation, two different MRT models are used to solve the problem. The D3Q7-MRT model is used to solve the temperature filed, and the D3Q19 is employed to solve the fluid flow of natural convection within the enclosure. The influences of different Rayleigh numbers 103<Ra<106 and solid volume fractions 0<φ<0.04 and four different arrangements of discrete active walls on the fluid flow, heat transfer, total entropy generation, local heat transfer irreversibility and local fluid friction irreversibility are presented comprehensively.

Nanofluid heat transfer analysis in a microchannel heat sink (MCHS) under the effect of magnetic field by means of KKL model

The present study investigates heat transfer of nanofluid flow in micro channel heat sink (MCHS) in the presence of a magnetic field. Al2O3-water nanofluid is chosen the best among other nanolfluid and is used as a coolant fluid in solution. The KKL correlation is utilized for calculation of the effective thermal conductivity and viscosity of nanofluid. Modified Darcy equation is applied for porous medium and the two-equation model with thermal dispersion is employed for heat transfer between fluid and solid sections. Since the coupled heat transfer equations of nanofluid and solid phase are nonlinear, the analytical Collocation method (CM) is employed to solve this problem. Effects of the Hartmann number, nanoparticles volume fraction, nanoparticle diameter, porosity, channel aspect ratio on temperature distribution, velocity and Nusselt number are investigated deliberately. Results show that Nusselt number has direct relationship with applying a magnetic field on MCHS.

Simulation of CuO-water nanofluid heat transfer enhancement in presence of melting surface

In this report, impact of melting heat transfer on nanofluid heat transfer enhancement in presence of magnetic field is simulated. Innovative numerical approach is utilized namely CVFEM. KKL model is taken into account to obtain properties of CuO- water nanofluid. Roles of melting parameter (δ), CuO-water volume fraction (ϕ), Hartmann (Ha) and Rayleigh (Ra) numbers are demonstrated in outputs. Results depict that temperature gradient augments with rise of melting parameter and Rayleigh number. Nusselt number detracts with rise of Lorentz forces. At higher concentrations, nanofluid has greater merit to be used for heat transfer enhancement.

Lattice Boltzmann method simulation for CuO-water nanofluid flow in a porous enclosure with hot obstacle

In this research, effect of Lorentz force on nanofluid natural convection inside a porous cavity is reported. Simulations have been done via Lattice Boltzmann Method. KKL model is employed to add Brownian motion effect on nanofluid properties. Impacts of Darcy number (Da), Rayleigh number (Ra), nanofluid volume fraction (ϕ) and Hartmann number (Ha) on heat transfer treatment are demonstrated. Outputs demonstrate that temperature gradient reduces with increase of Hartmann number but it enhances with augment of Darcy number.

Melting heat transfer influence on nanofluid flow inside a cavity in existence of magnetic field

In this research, influence of melting heat transfer on magnetohydrodynamic nanofluid free convection is analyzed by means of CVFEM. KKL model is taken into account to obtain viscosity and thermal conductivity of CuO-water nanofluid. Roles of melting parameter (δ), CuO-water volume fraction (ϕ), Hartmann (Ha) and Rayleigh (Ra) numbers are illustrated in outputs. Results depict that temperature gradient enhances with rise of Rayleigh and melting parameter. Nusselt number decreases with increase of Lorentz forces.

RETRACTED: Mesoscopic simulation of CuO[sbnd]H2O nanofluid in a porous enclosure with elliptic heat source

Abstract In this article, mesoscopic approach has been utilized to investigate magnetic field impact on CuO H 2 O nanofluid free convection inside a porous cavity with elliptic heat source. Simulations have been done via LBM. KKL model is employed to consider Brownian motion impact on nanofluid properties. Influences of Rayleigh number ( Ra ), nanofluid volume fraction ( ϕ ), Hartmann number ( Ha ), Darcy number ( Da ) on heat transfer treatment are demonstrated. Outputs demonstrate that temperature gradient reduces with increase of Ha while it increases with augment of Da , Ra .

RETRACTED: Free convection of CuO–H2O nanofluid in a curved porous enclosure using mesoscopic approach

Abstract In this research, impact of magnetic field on CuO–H 2 O nanofluid free convection inside a curved porous cavity is reported. Simulations have been done via Lattice Boltzmann method (LBM). KKL model is applied to consider Brownian motion impact on nanofluid properties. Impacts of Rayleigh number ( Ra ), Darcy number ( Da ), nanofluid volume fraction ( ϕ ), Hartmann number ( Ha ) on heat transfer treatment are illustrated. Outputs demonstrate that temperature gradient reduces with increase of Ha while it increases with augment of permeability of porous media and buoyancy forces.

Numerical simulation of MHD mixed convection in alumina–water nanofluid filled square porous cavity using KKL model: Effects of non-linear thermal radiation and inclined magnetic field

This study examines the influence of non-linear thermal radiation and inclined magnetic field on mixed convection in a square porous cavity. Here, the cavity is saturated with alumina–water nanofluid. Darcy-Brinkman-Forchheimer-extended model has been adopted to formulate governing differential equations. After transforming equations to dimensionless form, these are solved by utilizing weighted residual Galerkin finite element method. Latest Koo-Kleinstreuer-Lee (KKL) model is employed for the evaluation of effective thermal conductivity and dynamic viscosity of nanofluid. Effect of pertinent parameters in specific ranges such as Richardson number (0.01 ≤ Ri ≤ 100), radiation parameter (0 ≤ Rd ≤ 5), temperature ratio parameter (1.1 ≤ Nr ≤ 1.4), inclined magnetic field parameter (0°≤ γ ≤ 90°), Darcy number (10−6 ≤ Da ≤ 10−3), porosity parameter (0.2 ≤ ϵ ≤ 0.8) and volume fraction of solid particles (0 ≤ ϕ ≤ 0.04) has been studied and presented in the form of streamlines, isotherms and plots. Moreover, kinetic energy and average temperature have also been taken into account for better understanding the flow philosophy. It is found that radiation parameter, temperature ratio parameter, Darcy number and porosity parameter augment heat transfer, kinetic energy and average temperature while inclined magnetic field parameter in selected aforementioned range declines heat transfer due to hot bottom wall.

Lattice Boltzmann method simulation for MHD non-Darcy nanofluid free convection

Magnetohydrodynamic nanofluid convective flow in cubic porous enclosure is reported. Lattice Boltzmann Method is selected as mesoscopic approach. Brownian motion impact is taken into account via KKL model. Roles of Darcy number (Da), Hartmann number (Ha), Rayleigh number (Ra), and Al2O3 volume fraction (ϕ) are presented. Outputs are illustrated in forms of velocity contours, isokinetic energy, streamlines, isotherms and Nusselt number. Results indicate that temperature gradient over the hot surface augments with rise of Darcy numbers and ϕ while it reduces with augment of Lorentz forces. Nusselt number enhances with increase of buoyancy forces and permeability of porous media. Nanofluid motion enhances with augment of ϕ,Da,Ra while it decreases with augment of Ha.

Influence of viscous dissipation on a copper oxide nanofluid in an oblique channel: Implementation of the KKL model

This paper aims to study the flow of a nanofluid in the presence of viscous dissipation in an oblique channel (nonparallel plane walls). For thermal conductivity of the nanofluid, the KKL model is utilized. Water is taken as the base fluid and it is assumed to be containing the solid nanoparticles of copper oxide. The appropriate set of partial differential equations is transformed into a self-similar system with the help of feasible similarity transformations. The solution of the model is obtained analytically and to ensure the validity of analytical solutions, numerically one is also calculated. The homotopy analysis method (HAM) and the Runge-Kutta numerical method (coupled with shooting techniques) have been employed for the said purpose. The influence of the different flow parameters in the model on velocity, thermal field, skin friction coefficient and local rate of heat transfer has been discussed with the help of graphs. Furthermore, graphical comparison between the local rate of heat transfer in regular fluids and nanofluids has been made which shows that in case of nanofluids, heat transfer is rapid as compared to regular fluids.

Numerical investigation of nanofluid melting heat transfer between two pipes

CuO-water nanofluid melting heat transfer between two circular cylinders is studied in existence of horizontal magnetic field. Effect of various shapes of nanoparticles on nanofluid properties is considered. The KKL model is utilized for estimating viscosity of nanofluid. Runge-Kutta integration scheme is selected for solving ODEs. Numerical procedures are examined for various active parameters namely shapes of nanoparticle, melting parameter (δ), Hartmann number (Ha), Eckert number (Ec) and Reynolds number (Re). Results show that temperature gradient increases with increase in Hartmann number but it decreases with increase in melting parameter. Velocity reduces with rise of Lorentz forces and melting parameter but it increases with increase in Reynolds number.

Analysis of flow and heat transfer in water based nanofluid due to magnetic field in a porous enclosure with constant heat flux using CVFEM

Impact of Lorentz forces on CuO–water nanofluid flow in a permeable enclosure is presented by means of CVFEM. Darcy’s law is applied for porous media. In order to predict properties of nanofluid, KKL model has been utilized. Important parameters are inclination angle (ξ=0∘ to 90°), CuO–water volume fraction (ϕ=0 and 0.04), Hartmann (Ha=0 to 20) and Rayleigh (Ra=102,250 and 103) numbers for porous medium. A formula for Nuave is provided. Results demonstrated that Nusselt number detracts with enhancement of ξ,Ha. Heat transfer augmentation detracts with rise of buoyancy forces but it enhances with rise of inclination angle and Hartmann number.

Nanofluid heat transfer between two pipes considering Brownian motion using AGM

Nanofluid flow between two circular cylinders is studied in existence of magnetic field. KKL model is applied for nanofluid. Thermal radiation effect has been considered in energy equation. AGM is selected for solving ODEs. Semi analytical procedures are examined for various active parameters namely; aspect ratio, Hartmann number, Eckert number and Reynolds number. Results indicate that temperature gradient enhances with rise of Ha, Ec and η but it reduces with augment of Re. Velocity reduces with rise of Lorentz forces but it augments with rise of Reynolds number.

CuO–Water Nanofluid Magnetohydrodynamic Natural Convection inside a Sinusoidal Annulus in Presence of Melting Heat Transfer

Impact of nanofluid natural convection due to magnetic field in existence of melting heat transfer is simulated using CVFEM in this research. KKL model is taken into account to obtain properties of CuO–H2O nanofluid. Roles of melting parameter (δ), CuO–H2O volume fraction (ϕ), Hartmann number (Ha), and Rayleigh (Ra) number are depicted in outputs. Results depict that temperature gradient improves with rise of Rayleigh number and melting parameter. Nusselt number detracts with rise of Ha. At the end, a comparison as a limiting case of the considered problem with the existing studies is made and found in good agreement.

CuO-water nanofluid free convection in a porous cavity considering Darcy law

The natural convection of a CuO-water nanofluid in a permeable cavity is simulated using the Darcy law. The Brownian motion impact on the properties of the nanofluid is taken into account using the KKL model. The effect of Lorentz forces on the nanofluid hydrothermal behavior are considered. The control volume based finite element method is applied to solve the final equations. Roles of CuO-water volume fraction ( $\phi$ ), Rayleigh (Ra) and Hartmann (Ha) numbers for a porous medium are reported. Results show that the Nusselt number decreases with increasing Ha but it increases with increasing $\phi$ , Ra.

MHD forced convection and entropy generation of CuO-water nanofluid in a microchannel considering slip velocity and temperature jump

Flow field, heat transfer and entropy generation of forced convection of CuO-water nanofluid is investigated in a parallel plate microchannel in the presence of magnetic field. Two vertical micromixers are attached on the hot walls of the microchannel. To consider the effect of the Brownian motion of the nanoparticles, the KKL model is utilized to estimate thermal conductivity of the nanofluid. The governing equations, which are accompanied with the slip velocity and temperature jump boundary conditions, are solved by the finite volume method (FVM) and SIMPLER algorithm. The study is conducted for the Reynolds numbers in the range of 10 < Re < 100, Hartmann numbers in the range of 0 < Ha < 40, Knudsen numbers ranging of 0 < Kn < 0.1 and volume fraction of nanoparticles ranging of 0 < φ < 0.04. The results show that when the Hartmann or Reynolds numbers, or the volume fraction of nanoparticles increase, the average Nusselt number and the total entropy generation rate increase. Furthermore, when Knudsen number increases, the total entropy generation rate decreases.

Instability of quasicrystalline order in the local Kalugin-Kitaev-Levitov model

In 1978 Alexander and McTague noticed that a very simple Landau theory of crystallization led naturally to the consideration of structures that we would now describe as icosahedral quasicrystals. They concluded, however, that a body-centered-cubic (bcc) crystal would be the most favored phase. Shortly after the experimental discovery of such quasicrystals, Kalugin, Kitaev, and Levitov (KKL) argued that a very natural extension of the Alexander-McTague model could easily stabilize the icosahedral quasicrystal over the bcc crystal. Both investigations considered only the quadratic and cubic terms in the bulk free energy. It has been pointed out, however, that if the free energy with a local quartic term is directly minimized, then the quasicrystal is never a global minimum. We show here, in addition, that in the presence of a local quartic term the quasicrystal is not even metastable in the KKL model, being unstable against a collapse to the bcc crystal.