Computation of thermo-solutal convection with Soret-Dufour cross diffusion in a vertical duct containing carbon/metallic nanofluids

Duct flows constitute an important category of modern thermal engineering. Optimizing efficiency has become a significant objective in the 21st century in, for example, heating ventilation and air-conditioning (HVAC), coolant or heat transfer fluid flows in a nuclear power reactor, heat exchanger design etc, and this has been achieved by either new materials (improved thermal insulation properties) constituting the duct walls, novel geometric designs or improved working fluids. Nanotechnology has infiltrated into duct design in parallel with many other fields of mechanical, medical and energy engineering. Motivated by the excellent potential of nanofluids, a subset of materials engineered at the nanoscale, in the present work, a new mathematical model is developed for natural convection in a rectangular vertical duct containing nanofluid. The left and right walls of the duct are maintained at constant and unequal temperatures, while the front and rear walls of the duct are insulated. Thermo-solutal (double-diffusive) natural convection of aqueous nanofluid containing various metallic nanoparticles (e. g. copper, titanium oxide) or carbon-based nanoparticles (e. g. diamond, silicon oxide) is simulated. The Tiwari-Das nanoscale volume fraction model is used in addition to the Brinkman and Maxwell models for defining the properties of the nanofluid. The partial differential conservation equations for mass, momentum and energy are non-dimensionalized via appropriate transformations and the resulting boundary value problem is solved with a second-order accurate implicit finite difference technique employing Southwell-Over-Relaxation (SOR). Mesh independence tests are conducted. Extensive visualization of the solutions for velocity, temperature, nanoparticle concentration (volume fraction) are presented for five different nanoparticles (silicon oxide, diamond, copper, titanium oxide and silver), thermal Grashof number, nanoparticle species (solutal) Grashof number, volume fraction of nanoparticles (i.e. percentage doping), Dufour number, Soret number, Prandtl number, Schmidt number and duct aspect ratio. It is observed that the heat transfer rate (Nusselt number) at both the walls is maximized for diamond nanoparticles and minimized for silicon oxide nanoparticles. Further the heat transfer rate for clear fluid is lower when compared with nanofluid, confirming that nanoparticles achieve the desired thermal enhancement at the boundaries also. The mass transfer at both walls (Sherwood number) however is not significantly influenced by any particular type of nanoparticle, thermal and concentration Grashof number and is depleted with higher values of Dufour, Prandtl, Soret and Schmidt numbers in addition to aspect ratio. However, Sherwood numbers at both the left and right duct walls are substantially boosted with greater solid volume fraction of nanoparticles

Augmentation of heat transfer via nanofluids in duct flows using Fourier-type conditions : theoretical and numerical study

Motivated by developments in thermal duct processing, an investigation is presented to study the behavior of viscous nanoparticle suspensions flowing in a vertical duct subject to Fourier-type conditions. The left wall temperature is kept lower than that of the right wall. Brownian motion and thermophoresis which are invoked via the presence of nanoparticles are incorporated in the study. Numerical solutions with an efficient Runge–Kutta shooting method are also presented at all values of the control parameters. The impact of thermal Grashof number (0≤Λ≤15), Eckert number (0.01≤Ec≤0.04), thermophoresis (0.05≤Nt≤2), and Brownian motion parameters (0.05≤Nt≤2) on the velocity, temperature, and nanoparticle concentration distributions for identical (Bi1=Bi2=10) and differing Biot numbers (Bi1=1,Bi2=10) (at the duct walls) are computed and visualized graphically. With vanishing thermophoresis and Brownian motion parameters, the solutions match exactly with the earlier Newtonian viscous flow computations. Symmetric and asymmetric wall heat conditions are also acknowledged. Intensifying the thermal Grashof number, Eckert number, thermophoresis parameter, and Brownian parameter serve to amplify magnitudes of the velocity and temperature, whereas the nanoparticle concentration field is suppressed. The skin friction and Sherwood number are also computed with various combinations of the flow control parameters. Nusselt number values at the hot duct wall are enhanced with an increase in thermal buoyancy parameter, Eckert number, Brownian motion parameter, and thermophoresis parameter for equal Biot numbers. The opposite trend is computed for different Biot numbers. For any given values of Biot numbers, the mean velocity and bulk temperature are boosted with increase in thermal buoyancy parameter, Eckert number, Brownian motion parameter, and thermophoresis parameter. Hence, it may be inferred that the transport characteristics computed using Fourier-type boundary conditions are substantially different from those based on isothermal boundary conditions in nanofluid duct flows

Mathematical modelling of triple diffusion in natural convection flow in a vertical duct with Robin boundary conditions, viscous heating and chemical reaction effects

The triple-diffusive convective flow (thermal diffusion and dual species diffusion) in a viscous fluid flowing within a vertical duct is investigated subject to Robin boundary conditions at the duct walls. Viscous heating and homogenous chemical reaction effects are included. The mass transfer (solutal) buoyancy effects due to concentration gradients of the dispersed components are taken into account using the Boussinesq approximation. Symmetric and asymmetric wall conditions for the temperature are taken into account. The conservation equations are rendered into dimensionless form via suitable transformations and the emerging ordinary differential equations feature a number of dimensionless parameters including thermal Grashof number, two solutal Grashof numbers (one for each of the diffusing components i.e. species 1 and species 2), left and right duct wall thermal Biot numbers, species 1 and species 2 chemical reaction parameters, Brinkman number and temperature difference ratio.These coupled and nonlinear dimensionless conservation equations are solved numerically using theRunge-Kutta shooting method. The solutions obtained numerically are validated with approximate analytical solutions obtained via a regular perturbation method which are valid for small values of Brinkman number.The impact of selected parameters on velocity, temperature and dual species concentration distributions is visualized graphically. Furthermore, the variation of skin friction and Nusselt number with these parameters is also tabulated. The solutions obtained numerically and analytically are found to be equal in the absence of viscous dissipation. However, the deviation is magnified with large values of Brinkman number. In the absence of chemical reaction, the results concur with the earlier computations of Zanchini (1998). Increasing second species solutal Grashof number is observed to decelerate the flow in the left duct half space, to accelerate the flow in the right duct half space and consistently reduce temperatures across the entire duct width. With increasing species 1 chemical reaction parameter the concentration magnitudes are elevated in the left duct half space whereas they are depressed in the right duct half space. A similar response is computed for the influence of species 2 reaction parameter on the concentration profile. Temperatures are strongly enhanced across the duct width with increasing Brinkman number and are symmetric in nature about the channel centerline for the symmetric Biot number case (equal thermal Biot numbers at the left and right walls). These profiles are morphed for the asymmetric Biot number case (equal thermal Biot numbers at the left and right walls). Temperatures descend from the left wall to the right wall, although they are still enhanced with increasing Brinkman number. The simulations are relevant to geochemical transport phenomena, industrial materials processing and thermal duct design

Effects of thermophysical properties on heat transfer at the interface of two immisicible fluids in a vertical duct: numerical study

A comprehensive theoretical and numerical investigation is presented for two fluids with different physical properties. The effects of buoyancy and viscous heating are addressed. Non-isothermal wall conditions are applied at the walls. The front and rear walls of the duct are perfectly insulated. Numerical solutions for the reduced non-dimensional Navier-Stokes equations and coupled energy conservation equation are obtained using a finite difference method with second-order accuracy. Opting suitable conditions at the interface the two different solutions for two different fluids are extracted. The effects of Grashof number (thermal buoyancy parameter), viscosity ratio, thermal conductivity ratio, Eckert number (dissipation parameter), Prandtl number and duct aspect ratios (for the two immiscible fluid regions) on the flow field are visualized graphically. The value of the average Nusselt number is also tabulated for the two-fluid model. A grid-independence study is conducted. The solutions obtained by the numerical code are also validated by comparing with the benchmark solutions of the one fluid model and also with the simpler solutions of two fluid models available in the literature. Promoting Grashof number, Eckert number, Prandtl number and upper region aspect ratio (i.e. simultaneous decrease lower region aspect ratio) the Nusselt number increases at the left wall and decreases at the right wall in both the regions. However, the converse effect is computed with greater values of ratio of conductivity and viscosity. With increasing viscosity ratio parameter significant flow acceleration is induced in the upper half region of the duct whereas deceleration is caused at the bottom of the duct. Prescribing different values of aspect ratios in the upper and lower duct regions is found to generate a noticeable movement of the interface. The computations show that percentage changes in y 0 Nu = (heat transfer rate at the left wall of the duct) are 19.3334, 19.9350, 19.9423, 19.9965, 20.1926% in correspondence with a change in Grashof number from 5, 10, 20, 50, to 100 respectively. Percentage changes in Nusselt number are 19.9102, 19.9547, 19.9999, 20.0451, 20.0901% for values of Prandtl number of 0.01, 0.5, 1.0, 1.5, 2 respectively. The simulations are relevant to crystal growth technologies, buoyancy-driven fires in atria and geophysical convection

Convective fluid flow and heat transfer in a vertical rectangular duct containing a horizontal porous medium and fluid layer

Purpose-A numerical analysis is presented to investigate thermally and hydrodynamically fully developed convection in a duct of rectangular cross-section containing a porous medium and fluid layer. Design/methodology/approach-The Darcy-Brinkman-Forchheimer flow model is adopted. A finite difference method of second-order accuracy with the Southwell-OverRelaxation Method (SORM) is deployed to solve the non-dimensional momentum and energy conservation equations under physically robust boundary conditions. Findings-It is found that the presence of porous structure, and different immiscible fluids exert a significant impact in controlling the flow. Graphical results for the influence of the governing parameters i.e. Grashof number, Darcy number, porous media inertia parameter, Brinkman number and ratios of viscosities, thermal expansion and thermal conductivity parameters on the velocity and temperature fields are presented. The volumetric flow rate, skin friction and rate of heat transfer at the left and right walls of the duct are also provided in tabular form. The numerical solutions obtained are validated with the published work and excellent agreement is attained. Originality/value-To the authors best knowledge this work original in developing the numerical code using FORTRAN to assess the fluid properties for immiscible fluids. The study is relevant to geothermal energy systems, thermal insulation systems, resin flow modeling for liquid composite molding processes and hybrid solar collectors

Modeling the onset of thermosolutal convective instability in a non-Newtonian nanofluid-saturated porous medium layer

The onset of double-diffusive (thermosolutal) convection in horizontal porous layer saturated with an incompressible couple stress nanofluid saturated is studied with thermal conductivity and viscosity dependent on the nanoparticle volume fraction. To represent the momentum equation for porous media, a modified Darcy-Maxwell nanofluid model incorporating the effects of Brownian motion and thermophoresis has been used. The thermal energy equation includes regular diffusion and cross diffusion (Soret thermo-diffusion and Dufour diffusothermal) terms. A linear stability analysis depends on the normal mode technique and the onset criterion for stationary and oscillatory convection is derived analytically. The nonlinear theory based on the representation of the Fourier series method is applied to capture the behavior of heat and mass transfer. It is found that the couple stress parameter enhances the stability of the system in both the stationary and oscillatory convection modes. The viscosity ratio and conductivity ratio both enhance heat and mass transfer. Transient Nusselt number is found to be oscillatory when time is small. However, when time becomes very large, all the three transient Nusselt number values approach to their steady state values

Unsteady squeezing flow of a magnetized nano-lubricant between parallel disks with Robin boundary conditions

The aim of the present work is to examine the impact of magnetized nanoparticles (NPs) in enhancement of heat transport in a tribological system subjected to convective type heating (Robin) boundary conditions. The regime examined comprises the squeezing transition of a magnetic (smart) Newtonian nanolubricant between two analogous disks under an axial magnetism. The lower disk is permeable whereas the upper disk is solid. The mechanisms of haphazard motion of NPs and thermophoresis are simulated. The non-dimensional problem is solved numerically using a finite difference method in the MATLAB bvp4c solver based on Lobotto quadrature, to scrutinize the significance of thermophoresis parameter, squeezing number, Hartmann number, Prandtl number and Brownian motion parameter on velocity, temperature, nanoparticle concentration, Nusselt number, factor of friction and Sherwood number distributions. The obtained results for the friction factor are validated against previously published results. It is found that friction factor at the disk increases with intensity in applied magnetic field. The haphazard (Brownian) motion of nanoparticles causes an enhancement in thermal field. Suction and injection are found to induce different effects on transport characteristics depending on the specification of equal or unequal Biot numbers at the disks. The main quantitative outcome is that, unequal Biot numbers produce significant cooling of the regime for both cases of disk suction or injection, indicating that Robin boundary conditions yield substantial deviation from conventional thermal boundary conditions. Higher thermophoretic parameter also elevates temperatures in the regime. The nanoparticles concentration at the disk is boosted with higher values of Brownian motion parameter. The response of temperature is similar in both suction and injection cases; however, this tendency is quite opposite for nanoparticle concentrations. In the core zone, the resistive magnetic body force dominates and this manifests in a significant reduction in velocity i.e. damping. The heat buildup in squeeze films (which can lead to corrosion and degradation of surfaces) can be successfully removed with magnetic nanoparticles leading to prolonged serviceability of lubrication systems and the need for less maintenance

Perturbation and numerical study of double-diffusive dissipative reactive convective flow in an open vertical duct containing a non-darcy porous medium with robin boundary conditions

A mathematical model for thermosolutal convection flow in an open two-dimensional vertical channel containing a porous medium saturated with reactive Newtonian fluid is developed and studied. Robin boundary conditions are prescribed, and a first-order homogenous chemical reaction is considered. The Darcy–Forchheimer model is used to simulate both the first- and second-order porous mediums’ drag effects. For the general non-Darcy-case, a numerical solution is presented using the Runge–Kutta quadrature and a shooting method. The influences of thermal (0≤λ1≤15) and solute Grashof numbers (0≤λ2≤20) , Biot numbers (1≤Bi1≤10,Bi2=10) , Brinkman number (0≤Br≤0.5) , first-order chemical reaction parameter (2≤α≤8) , porous medium parameter (2≤σ≤8) and Forchheimer (inertial drag) parameter (0≤I≤12) on the evolutions of velocity, temperature and concentration (species) distributions are visualized graphically. Nusselt number and skin friction at the walls are also computed for specific values of selected parameters. The study is relevant to the analysis of geothermal energy systems with chemical reaction

MIXED CONVECTION OF A COMPOSITE POROUS MEDIUM IN A VERTICAL CHANNEL WITH ASYMMETRIC WALL HEATING CONDITIONS

[[abstract]]The fully developed laminar mixed convection flow of a composite porous medium in a vertical channel is presented. The flow is modeled using a Brinkman model. The viscous and Darcy dissipation terms are also included in the energy equation. Three types of thermal boundary conditions, such as isothermal-isothermal, isoflux-isothermal, and isothermal-isoflux, for the left-right walls of the channel are presented. The coupled nonlinear governing equations are solved using a regular perturbation method. The effects of various parameters on the flow, such as porous parameter, ratio of Grashof number to Reynolds number, width ratio, viscosity ratio, and thermal conductivity ratio, are discussed.[[note]]SC

Fully developed magneto convection flow in a vertical rectangular duct

[[abstract]]An analysis is performed to study the MHD free convection flow in a vertical rectangular duct for laminar and fully developed regime taking into consideration the effects of Ohmic heating and viscous dissipation. Numerical solutions are found using finite difference method of second-order accuracy. The effects of various physical parameters such as Hartmann number, aspect ratio, buoyancy parameter and circuit parameter are presented graphically. It is found that as Hartmann number, buoyancy parameter and aspect ratio increase, the upward and downward flow rates are increased for open circuit but decrease for short circuit.[[note]]SC