Prescribed Heat Flux Research Articles

Cattaneo -Christov heat flux model for magnetohydrodynamic flow in a suspension of dust particles towards a stretching sheet

Abstract In the present paper, the flow of an incompressible, electrically conducting dusty fluid over a stretching sheet is considered. The Cattaneo- Christov heat flux theory is employed to control the thermal boundary layer. The flow equations are transformed into nonlinear ordinary differential equations (NODEs) and which are solved with help of Runge-Kutta 4th order method. Flow equations are examined with respect to boundary conditions namely prescribed wall temperature (PWT) and prescribed heat flux (PHF) cases. In general PWT and PHF boundary conditions are very useful in the industrial as well as manufacturing up and down processes. Impact of the emerging parameters on the dimensionless velocity and temperature as well as friction coefficient and local Nusselt number are examined. We also validated my results with already available literature. It is found that the heat transfer rate of the flow in PWT case is higher than that of PHF case. These results can help us to conclude that for higher heating processes (Heating industries) PWT case and lesser heating processes (Cooling industries) PHF boundary condition is useful.

Numerical study on magnetohydrodynic CNTs-water nanofluids as a micropolar dusty fluid influenced by non-linear thermal radiation and joule heating effect

Numerical study on magnetohydrodynic (MHD) flow and heat transfer of incompressible CNTs-water nanoparticles (single wall carbon and tube (SWCNT) and multiwall cabin nanotube (MWCNT)) as a non-Newtonian micropolar dusty fluid influenced by non-linear thermal radiation and joule heating effect over stretching plate are investigated in this paper. The governing equations have been solved by numerical method. Flow and heat transfer for two cases of prescribed heat flux (PHF) and prescribed surface temperature (PST) are analyzed, As a important result have been mentioned to the reduction of velocity profile due to the Lorentz force by increasing magnetic field parameter, the temperature profile increases by increasing radiation parameter, and MWCNT in comparison to the SWCNT has a greater effect on velocity profile also SWCNT in compared to the MWCNT has a greater effect on temperature profile and heat transfer.

Normal solutions of a boundary-value problem arising in free convection boundary-layer flows in porous media

This paper is concerned with normal solutions of a two-point boundary-value problem of second order which arises in the steady free convection boundary-layer flow over a vertical permeable flat plate being embedded in a saturated porous medium with both prescribed heat flux and suction rate of the plate. We use a very simple argument to prove that there exists a λmin ∈ (1.3782407, 1.4166499) such that this problem has no normal solution for all λ < λmin , a unique normal solution for λ=λmin, and exactly two normal solutions θ1(η) and θ2(η) with θ1(η) < θ2(η) on [0,+∞) for each fixed λ > λmin , which is applied to the reduced boundary-layer system.

Analytical solution of convective heat transfer of a quiescent fluid over a nonlinearly stretching surface using Homotopy Analysis Method

Abstract In this article, an analytical solution of the boundary layer fluid flow and heat transfer of a quiescent viscous fluid over a non-linearly stretching surface is presented. The thermal radiation effects are included in the energy governing equation. Surface velocity and temperature conditions are assumed to be of the power-law form with an exponent of 1/3 for velocity and arbitrary exponent m for surface temperature or heat flux conditions. The system of nonlinear differential equations is solved by Homotopy Analysis Method (HAM) for two cases of Prescribed Surface Temperature (PST) and Prescribed Heat Flux (PHF). The results of this method appear in the form of series expansions, the convergence of which is analyzed carefully. Graphical results are finally presented in order to investigate the influence of Prandtl number (Pr) and thermal radiation on heat transfer phenomena.

Unsteady MHD Slip Flow with Radiative Heat and Mass Transfer over an Inclined Plate Embedded in a Porous Medium

The combined effects diffusion-thermo, chemical reaction, buoyancy forces, radiative heat flux, velocity slip and magnetic field on an unsteady hydromagnetic mixed convective flow of an electrically conducting fluid with heat and mass transfer over an inclined vertical porous plate embedded in a porous medium is studied. The imposed thermal boundary conditions include prescribed uniform plate surface temperature (PST) and prescribed heat flux (PHF). The governing equations are solved analytically with the help of two term perturbation technique. The influence of various thermophysical parameters on the fluid velocity, temperature and species concentration are presented graphically while numerical values of skin friction, Nusselt and Sherwood numbers are presented in tabular form for different values and discussed. A special case of our results show excellent agreement with the earlier results in the literature.

Wall jet flows of Glauert type: Heat transfer characteristics and the thermal instabilities in analytic closed forms

Heat transfer characteristics of the traditional wall jet flows subject to various thermal boundary conditions including isothermal surface, prescribed temperature, constant heat flux, prescribed heat flux, adiabatic surface and thermally convective surface are documented in analytic closed forms. Heat dissipation has been also included where the similarity energy equation could structurally adjust to this specific term (for the adiabatic case, this term has been necessarily included). In all the studied cases, both the transpiration velocity and moving wall conditions are allowed to exist (where applicable) in such a way, being consistent with the Glauert integral constraint and subject to the context of exponentially decaying wall jet flows.In particular, it is analytically proved (even without having the closed form solutions) that for the Glauert original case, a surface with a prescribed temperature in the form of Tw(x)=mx−14+T∞ serves zero contribution to heat transfer at the wall (the Induced Heat Shield). More precisely, the value −14as the prescribing parameter is the Surface Heat Transfer Stopping Point and below this point, heat transfer phenomenon falls from a usual physical interpretation, expressing spectrums of thermal instabilities through a hyper geometric function. Furthermore, it is argued that a normalized similarity temperature in the form of θ(η)=T−T∞mxn<−14 results in the appearance of an uninterruptable singularity within the heat transfer phenomenon for the Glauert case subject to a rigorous physical ground; and as an immediate practical consequence, there is no physically-valid similarity solution for an adiabatic surface or more precisely, for energy equation with heat dissipation consideration.It should be mentioned that the concept of Induced Heat Shield is discussed in here for the first time (to our knowledge) which may inspire a concealed fact regarding the restrictions of the thermal similarity solutions.Therefore, it is hopeful that heat transfer phenomenon in the Glauert type wall jet flows and the associated physical representations can be better understood by the present research.

MHD VISCOUS NANOFLUID FLOW WITH BASE FLUIDS’ WATER AND KEROSENE IN THE PRESENCE OF A TEMPERATURE GRADIENT DEPENDENT HEAT SINK WITH PRESCRIBED HEAT FLUX

MHD viscous nanofluid flow with viscous dissipation and thermal radiation in the presence of a temperature gradient dependent heat sink is analyzed. Hence, this work mainly deals nano fluids with nanoparticles Cu, Ag, Al, Al2O3 and TiO2 and with base fluids’ water and kerosene. Prescribed heat flux boundary condition is employed on the porous surface. Suitable similarity transformations are introduced for converting nonlinear partial differential equations into the nonlinear ordinary differential equations and then solved by analytically. The influence of various physical parameters over the velocity and temperature of nanofluids Cu-water and Cu-kerosene are examined by using graphs. Skin friction coefficient and Nusselt number of various nanofluids tabulated and analyzed. It is found that skin friction coefficient and heat transfer rate of kerosene based nanofluid is higher than the water based nanofluid in the presence of considered physical effects.

Fe3O4–(CH2OH)2 nanofluid analysis in a porous medium under MHD radiative boundary layer and dusty fluid

In this paper, magnetohydrodynamic (MHD) boundary layer flow and heat transfer of an incompressible Iron (II,III) oxide (Fe3O4)-ethylene glycol ((CH2OH)2) nanoparticle on micropolar fluid with homogeneously suspended dust particles over a stretching sheet in the presence of thermal radiation and Joule heating are investigated. After transforming the partial differential equations (PDEs) governing the problem into the ordinary differential equations (ODEs), Runge–Kutta Fehlberg fourth fifth order numerical method is applied. The main criterion of this paper is to study the effects of different parameters and dimensionless numbers on velocity and temperature distribution in two phases of fluid and dust for two prescribed surface temperature (PST) and prescribed heat flux (PHF) cases. Velocity reduction in both fluid and dust phases is due to the Lorentz force against flow by increasing Hartman number, enhancement of temperature and thickness of thermal boundary layer are due to the increase of radiation parameter and Eckert number for both PST and PHF cases are the most important results. Also, in the final section of this paper, the effect of changes in the value of different parameters on skin friction coefficient and local Nusselt number in (PST and PHF PHF) have been discussed. These results indicated increasing the Hartman parameter (Ha) would cause an increase skin friction coefficient, also increasing the Eckert number would cause an increase local Nusselt number in PST case and reverse effect in PHF case.

Entropy generation analysis for convective heat transfer of nanofluids in tree-shaped network flowing channels

In this paper, convective heat transfer processes of nanofluids flowing through tree-shaped network flowing channels with prescribed heat flux on channel wall are investigated. The entropy generation rate equations of nanofluids flowing through the tree-shaped network channels are deduced; the interaction between volume fractions of nanoparticles and entropy generation rate is investigated. For H-shaped tree network flowing channels, the recurrence formulas of total entropy generation rate and the order number of flowing channels are deduced, and the numerical examples are presented. A new dimensionless number is proposed to describe the entropy generation properties of nanofluids. Analytical solutions of entropy generation rate distribution of any order of H-shaped construct of flowing channel using nanofluids are obtained.

Study of velocity and temperature distributions in boundary layer flow of fourth grade fluid over an exponential stretching sheet

This study deals with the investigation of boundary layer flow of a fourth grade fluid and heat transfer over an exponential stretching sheet. For analyzing two heating processes, namely, (i) prescribed surface temperature (PST), and (ii) prescribed heat flux (PHF), the temperature distribution in a fluid has been considered. The suitable transformations associated with the velocity components and temperature, have been employed for reducing the nonlinear model equation to a system of ordinary differential equations. The flow and temperature fields are revealed by solving these reduced nonlinear equations through an effective analytical method. The important findings in this analysis are to observe the effects of viscoelastic, cross-viscous, third grade fluid, and fourth grade fluid parameters on the constructed analytical expression for velocity profile. Likewise, the heat transfer properties are studied for Prandtl and Eckert numbers.

Experimental study of mixed convection heat transfer in a vertical channel with a one-sided semicylindrical constriction with prescribed heat flux

An experimental study of mixed convection heat transfer is carried out in a vertical channel with a one-sided semicylindrical constriction with prescribed heat flux while the other bounding walls are insulated and adiabatic. The semicylinder is placed horizontally at the mid-plane with a blockage ratio (BR, ratio between the semicylinder diameter and the thickness of the rectangular section) of 0.3 and a semicylinder aspect ratio (AR, ratio between the length and diameter of the semicylinder) of 6. The effect of opposing buoyancy on the flow and thermal behavior is analyzed for fixed Prandtl number of Pr = 7, Reynolds number based on semicylinder diameter of 20 ≤ Re ≤ 350 and buoyancy strength or modified Richardson number, Ri* = Gr*/Re2, from 20 to 350. For relatively large values of Ri*, flow visualization images and thermal analysis confirm the presence of a complex three-dimensional (3D) two-vortex structure with two recirculation bubbles present at the forebody and rear of the semicylinder. Surface temperature distributions and averaged Nusselt number at different Reynolds and modified Richardson numbers have been obtained. The results show that variation of the local temperature distributions with angular position and spanwise location become evident, and their relation to the presence of a complex 3D vortex structure that develops close to the semicylindrical constriction has been studied and discussed in detail. Moreover, empirical correlations for the overall Nusselt number are obtained using both Re and Gr* and Re and Ri* as the controlling parameters.

Temperature Distribution in Flux Channels with Discrete Contact Boundary Conditions

ABSTRACTAn analytical approach for the thermal behavior of two-dimensional rectangular flux channels with arbitrary boundary conditions on the source plane is presented. The boundary condition along the source plane can be a combination of the first kind boundary condition (Dirichlet or prescribed temperature) and the second kind boundary condition (Neumann or prescribed heat flux). To model the boundary conditions along the source plane, the method of least squares is used. The proposed solution is in the form of Fourier series expansion and can be applied to both symmetrical and non-symmetrical channels. This method is more general than other approaches and there is no need to use equivalent heat flux distributions to model isothermal heat sources. The general approach for obtaining the multidimensional temperature profile in flux channels and the advantages of the least-square method is discussed. The proposed solution can be used to calculate the temperature at any specified point in the flux channel. Two case studies are presented. The first case study is a flux channel with five discretely specified contact temperatures along the source plane. The second case study has both of the first kind and second kind boundary conditions on the source plane. The analytical results for both systems are compared with finite element method using a commercial software package. It is shown that the proposed approach can precisely model the temperature profile over the flux channel.

Second-Order Slip Flow of Magneto-Nanofluids Along a Stretching Cylinder with Prescribed Heat Flux

Second-Order Slip Flow of Magneto-Nanofluids Along a Stretching Cylinder with Prescribed Heat Flux

A NURBS-based scaled boundary finite element method for the analysis of heat conduction problems with heat fluxes and temperatures on side-faces

The scaled boundary finite element method (SBFEM) combined with isogeometric analysis (IGA) is proposed to solve the two-dimensional steady-state heat conduction problems in complex geometries. The main benefit of SBFEM is that the spatial dimension of analyzed domain is reduced by one and the solution is analytical in the radial direction. In this method, only the boundary of the computational domain requires discretization with finite elements leading to the reduction of computational efforts. However, SBFEM suffers from the finite element method related drawbacks. In the case of the complex geometric shapes, a large number of elements are necessary to obtain the exact representation of geometry in finite element method. Isogeometric analysis is a novel numerical technique based on the non-uniform rational B-splines (NURBS), where the geometry can be exactly represented. Moreover, this technique yields superior numerical accuracy, efficiency and convergence property in comparison to finite element method. In the proposed method, the segments of domain boundary with complex geometries are described with NURBS basis functions in IGA, while the straight segments of boundary are represented with polynomial basis functions as in the conventional SBFEM. Thus, the present approach combines the advantages of both SBFEM and IGA. The heat conduction problems of complex geometry can be more effectively handled with the proposed method considering the prescribed heat fluxes and temperatures on side-faces. The accuracy and efficiency of the proposed formulation are demonstrated by modeling five numerical examples involving the complicated geometry.

Magnetohydrodynamic three-dimensional nonlinear convective flow of viscoelastic nanofluid with heat and mass flux conditions

The present research focuses on three-dimensional nonlinear convective flow of viscoelastic nanofluid. Here, the flow is generated due to stretching of a impermeable surface. The phenomenon of heat transport is analyzed by considering thermal radiation and prescribed heat flux condition. Nanofluid model comprises of Brownian motion and thermophoresis. An electrically conducting fluid is accounted due to consideration of an applied magnetic field. The dimensionless variables are introduced for the conversion of partial differential equations into sets of ordinary differential systems. The transformed expressions are explored through homotopic algorithm. Behavior of different dimensionless parameters on the non-dimensional velocities, temperature and concentration are scrutinized graphically. The values of skin friction coefficients, Nusselt and Sherwood numbers are also calculated and elaborated. It is visualized that the heat transfer rate increases with Prandtl number and radiation parameter is higher.

Radiative entropy generation in a gray absorbing, emitting, and scattering planar medium at radiative equilibrium

Radiative entropy generation through a gray absorbing, emitting, and scattering planar medium at radiative equilibrium with diffuse-gray walls is investigated. The radiative transfer equation and radiative entropy generation equations are solved using discrete ordinates method. Components of the radiative entropy generation are considered for two different boundary conditions: two walls are at a prescribed temperature and mixed boundary conditions, which one wall is at a prescribed temperature and the other is at a prescribed heat flux. The effect of wall emissivities, optical thickness, single scattering albedo, and anisotropic-scattering factor on the entropy generation is attentively investigated. The results reveal that entropy generation in the system mainly arises from irreversible radiative transfer at wall with lower temperature. Total entropy generation rate for the system with prescribed temperature at walls remarkably increases as wall emissivity increases; conversely, for system with mixed boundary conditions, total entropy generation rate slightly decreases. Furthermore, as the optical thickness increases, total entropy generation rate remarkably decreases for the system with prescribed temperature at walls; nevertheless, for the system with mixed boundary conditions, total entropy generation rate increases. The variation of single scattering albedo does not considerably affect total entropy generation rate. This parametric analysis demonstrates that the optical thickness and wall emissivities have a significant effect on the entropy generation in the system at radiative equilibrium. Considering the parameters affecting radiative entropy generation significantly, provides an opportunity to optimally design or increase overall performance and efficiency by applying entropy minimization techniques for the systems at radiative equilibrium.

Structure of critical perturbations in a horizontal layer of melted water with the prescribed heat flux at the boundaries

The paper deals with the investigation of critical perturbations structure in a horizontal layer of fluid with temperature inversion of density. The lower boundary of the layer is rigid and upper boundary is free and undeformable. The constant vertical heat flux is imposed at both boundaries. The thermocapillarity, evaporation and radiation are neglected. The temperature dependence of the density is assumed to be quadratic. The coordinate of the density inversion point which shows the position of the plane of maximal density in the fluid layer in the conductive state is used for the description of density inversion effect. The influence of the location of density inversion point on structure of critical perturbations is studied. It is shown that, depending on the location of the inversion point, velocity profile of longwave critical perturbations can have two-floor or three-floor structure. Stability to finite-wavelength perturbations is studied for the entire range of their existence. The asymptotic formulas for critical values of the Rayleigh number and wave number are obtained, the structure of finite-wavelength critical perturbations .is determined.

Three-dimensional flow of nanofluid with heat and mass flux boundary conditions

This article presents magnetohydrodynamic three-dimensional flow of nanofluid over an exponentially stretching surface with prescribed heat flux (PHF) and prescribed concentration flux (PCF) conditions. An exponentially stretching surface is used to generate the flow. Effects of Brownian motion, thermophoresis, viscous dissipation and Joule heating are also taken into account. Boundary layer and small magnetic Reynolds number assumptions are considered in the mathematical modeling. The conversion from nonlinear partial differential equations to nonlinear ordinary differential equations is taken into place by using suitable transformations. The resulting nonlinear system is solved via convergent homotopic approach. Graphs are sketched in order to analyze that how the temperature and concentration profiles are affected by distinct physical flow parameters. Skin-friction coefficients and local Nusselt and Sherwood numbers are also computed and analyzed.

Comparative Assessment of the Finite Difference, Finite Element, and Finite Volume Methods for a Benchmark One-Dimensional Steady-State Heat Conduction Problem

The finite difference (FD), finite element (FE), and finite volume (FV) methods are critically assessed by comparing the solutions produced by the three methods for a simple one-dimensional steady-state heat conduction problem with heat generation. Three issues are assessed: (1) accuracy of temperature, (2) accuracy of heat flux, and (3) satisfaction of global energy conservation. It is found that if the order of accuracy of the numerical discretization schemes is the same (central difference for FD and FV, linear basis functions for FE), the accuracy of the temperature produced by the three methods is similar, except close to the boundaries where the FV method outshines the other two methods. Consequently, the FV method is found to predict more accurate heat fluxes at the boundaries compared to the other two methods and is found to be the only method that guarantees both local and global conservation of energy irrespective of mesh size. The FD and FE methods both violate energy conservation, and the degree to which energy conservation is violated is found to be mesh size dependent. Furthermore, it is shown that in the case of prescribed heat flux (Neumann) and Newton cooling (Robin) boundary conditions, the accuracy of the FD method depends in large part on how the boundary condition is implemented. If the boundary condition and the governing equation are both satisfied at the boundary, the predicted temperatures are more accurate than in the case where only the boundary condition is satisfied.

Hydromagnetic nanofluid flow past a stretching cylinder embedded in non-Darcian Forchheimer porous media

The present article presents the hydromagnetic nanofluid flow past a stretching cylinder embedded in non-Darcian Forchheimer porous media by using Buongiorno’s mathematical model (Buongiorno in J Heat Transf 128:240–250, 2006; Nadeem et al. in J Taiwan Inst Chem Eng 45:121, 2014, Nadeem et al. Appl Nanosci 4:625–631, 2014). Thermal radiation via Roseland’s approximation (Akbar et al. in Chin J Aeronaut 26:1389–1397, 2013; Nadeem and Haq in J Aerosp Eng 28:04014061, 2012), Brownian motion, thermophoresis and Joule heating effects are also considered. To explore thermal characteristics, prescribed heat flux and prescribed mass flux boundary conditions are deployed. Governing flow problem consists of PDEs in the cylindrical form, which are converted into system of nonlinear ODEs by applying applicable similarity transforms. ODEs are tackled by RK–Fehlberg fourth–fifth-order numerical integration scheme with shooting algorithm. Impact of numerous involving physical parameters on flow features like temperature distribution, velocity distribution, Sherwood number, local Nusselt number and skin friction coefficient is shown through graphs and tables.