Effect Of Internal Heat Generation Research Articles

Effect of internal heat generation on Marangoni convection in a superposed fluid-porous layer with deformable free surface

Linear stability analysis is applied to investigate the effect of internal heat generation on Marangoni convection in a two-layer system comprising an incompressible fluid-saturated porous layer over which lies a layer of the same fluid. The upper free surface is deformable and is subject to a general thermal condition, while the lower boundary is rigid and is fixed at a constant temperature (isothermal) or a constant heat flux. The Beavers-Joseph condition is employed at the interface and the Forchheimer-extended-Darcy equation is employed to describe the flow regime in the porous medium. The linear stability theory and the normal mode analysis are applied and the resulting eigenvalue problem is solved exactly. For both the upper and lower boundaries fixed at a constant heat flux, the analytical asymptotic solution of long wavelength is obtained using regular perturbation technique. It is observed that the critical Marangoni number decreases with an increase in the dimensionless heat source strength. However, an increase of the Bond number and the decrease of the Darcy number will help to slow the process of destabilizing the system. Key words: Marangoni convection, heat generation, porous layer.

Effects of internal heat generation, thermal radiation and buoyancy force on a boundary layer over a vertical plate with a convective surface boundary condition

In this paper we analyse the effects of internal heat generation, thermal radiation and buoyancy force on the laminar boundary layer about a vertical plate in a uniform stream of fluid under a convective surface boundary condition. In the analysis, we assumed that the left surface of the plate is in contact with a hot fluid whilst a stream of cold fluid flows steadily over the right surface; the heat source decays exponentially outwards from the surface of the plate. The similarity variable method was applied to the steady state governing non-linear partial differential equations, which were transformed into a set of coupled non-linear ordinary differential equations and were solved numerically by applying a shooting iteration technique together with a sixth-order Runge–Kutta integration scheme for better accuracy. The effects of the Prandtl number, the local Biot number, the internal heat generation parameter, thermal radiation and the local Grashof number on the velocity and temperature profiles are illustrated and interpreted in physical terms. A comparison with previously published results on similar special cases showed excellent agreement.

Combined effects of internal heat generation and higher order chemical reaction on the non‐darcian forced convective flow of a viscous incompressible fluid with variable viscosity and thermal conductivity over a stretching surface embedded in a porous medium

AbstractIn this paper, we study the combined effects of internal heat generation and higher order chemical reaction on a steady two‐dimensional non‐Darcian forced convective flow of a viscous incompressible fluid with variable dynamic viscosity and thermal conductivity in a fluid saturated porous medium passing over a linear stretching sheet. Using similarity transformations, the governing nonlinear‐coupled partial differential equations are made dimensionless and solved numerically for similarity solutions using very robust computer algebra software Maple 8. The non‐dimensional velocity, temperature and concentration distributions are presented graphically for various pertinent parameters such as relative temperature difference parameter, Darcy number, porosity parameter, reaction rate parameter and the order of the chemical reaction. The variations of Prandtl number and Schmidt number within the boundary layer are also displayed graphically when the fluid dynamic viscosity and thermal conductivity are temperature dependent. From the present numerical computations it is found that Prandtl number as well as Schmidt number must be taken as variables within the flow domain when the fluid's dynamic viscosity and thermal conductivity are variable. In the presence of internal heat generation, dynamic viscosity and thermal conductivity of the fluid are found to be higher than when it is absent. Increasing Darcy number reduces dynamic viscosity as well as thermal conductivity whereas increasing pore size reduces the Schmidt number and increases the Prandtl number within the boundary layer. For higher order reaction the rate of increase in mass transfer function is less compared to the rate of increase for the lower order reaction. © 2011 Canadian Society for Chemical Engineering

Combined effects of internal heat generation and buoyancy force on boundary layer flow over a vertical plate with a convective surface boundary condition

AbstractThis paper considers the effect of buoyancy force and internal heat generation on laminar thermal boundary layer over a vertical plate with a convective surface boundary condition. We assumed that left surface of the plate is in contact with a hot fluid while a stream of cold fluid flows steadily over the right surface with a heat source that decays exponentially. Using a similarity variable, the steady state governing non‐linear partial differential equations have been transformed into a set of coupled non‐linear ordinary differential equations, which are solved numerically by applying shooting iteration technique together with fourth order Runge–Kutta integration scheme. The effects of Prandtl number, local Biot number, the internal heat generation parameter and the local Grashof number on the velocity and temperature profiles are illustrated and interpreted in physical terms. A comparison with previously published results on special case of the problem shows excellent agreement. From our results, an overshoot of fluid velocity within the boundary layer is observed due to combined effect of buoyancy force and internal heat generation, in addition, internal heat generation causes thickening of thermal boundary layer. © 2011 Canadian Society for Chemical Engineering

Effect of internal heat generation on the onset of Brinkman–Benard convection in a ferrofluid saturated porous layer

The effect of internal heat generation on the criterion for the onset of convection in a horizontal ferrofluid saturated porous layer in the presence of a uniform magnetic field is studied using the Brinkman–Lapwood extended Darcy flow model with fluid viscosity different from effective viscosity. The lower boundary is taken to be rigid – isothermal and the upper surface is free and subject to the general thermal condition. The resulting eigenvalue problem is solved numerically using the Galerkin method with the Rayleigh number as the eigenvalue. The stability of the system is found to be dependent on the dimensionless internal heat source strength N s , Darcy number Da, viscosity ratio Λ , Biot number Bi, magnetic Rayleigh number, R m and the nonlinearity of fluid magnetization parameter M 3. The effect of increase in the value of R m , N s and M 3 is to hasten, while increase in the value of Bi, Λ and Da −1 is to delay the onset of thermomagnetic convection in a ferrofluid saturated porous layer. Further, increase in the value of Bi, Da −1 and N s as well as decrease in Λ , R m and M 3 is to diminish the dimension of convection cells.

Onset of Benard-Marangoni Ferroconvection with Internal Heat Generation

The effect of internal heat generation on the onset of Benard-Marangoni convection in a horizontal ferrofluid layer heated from below in the presence of a uniform vertical magnetic field is studied. The lower boundary is rigid and the upper free boundary is assumed to be flat and undeformable. Both the boundaries are considered to be perfectly insulated to temperature perturbations and the resulting eigenvalue problem is solved numerically using the Galerkin technique as well as analytically by regular perturbation technique with wave number as a perturbation parameter. It is observed that the analytical results agree well with those obtained numerically. It is noted that the combined effect of magnetic Rayleigh number and dimensionless internal heat source strength is to reinforce together and to hasten the onset of Benard-Marangoni ferroconvection compared to their presence in isolation. In addition, the nonlinearity of fluid magnetization is found to have no influence on the criterion for the onset of ferroconvection. Some existing results are reproduced as particular cases from the present study.

HEAT-TRANSFER ANALYSIS OF MHD FLOWDUE TO A PERMEABLE SHRINKING SHEET EMBEDDED IN A POROUS MEDIUM WITH INTERNAL HEAT GENERATION

The Adomian decomposition method with Pade´ approximants is employed in this study as a simple, non-perturbative alternative to solve the magnetohydrodynamic (MHD) boundary-layer flow due to a permeable shrinking sheet embedded in a porous medium. The effects of internal heat generation in the flow are considered. The effects of suction on skin friction and heat-transfer rate at different Pade´ approximants are tabulated, while the effects of various parameters on the flow velocity and temperature are discussed and presented graphically. © 2010 by Begell House, Inc.

On the onset of natural convection in differentially heated shallow fluid layers with internal heat generation

This paper reports the results of an analytical and numerical investigation to determine the effect of internal heat generation on the onset of convection, in a differentially heated shallow fluid layer. The case with the bottom plate at a temperature higher than the top plate mimics the classical Rayleigh Benard convection. However, internal heat generation adds a new dimension to the problem. Linear stability analysis is first carried out for the case of an infinitely wide cavity. The effect of aspect ratio on the onset of convection is studied by solving the full Navier–Stokes equations and the equation of energy and observing the temperature contours. A bisection algorithm is used for an accurate prediction of the onset. The numerical results are used to plot the stability curves for eight different aspect ratios. A general correlation is developed to determine the onset of convection in a differentially heated cavity for various aspect ratios. For an aspect ratio of 10, it is seen that the cavity approaches the limit of an infinite cavity. Analytical results obtained by using linear stability analysis agree very well with the “full” CFD simulations, for the above aspect ratio.

Closed Form Solutions For Mixed Convection With Magnetohydrodynamic Effect in a Vertical Porous Annulus Surrounding an Electric Cable

Mixed convection Darcy flow in a vertical porous annulus around a straight electric cable is investigated. It is assumed that the flow is fully developed and parallel. Moreover, the Boussinesq approximation is used. The magnetic field with a steady electric current in the cable is radially varying according to the Biot–Savart law. Two flow regimes are investigated. The first is mixed convection with negligible effects of internal heat generation due to Joule heating and viscous dissipation. The second is forced convection with important effects of heat generation. In these two special cases, closed form expressions of the velocity profile and of the temperature profile, as well as of the flow rate and the Nusselt number, are obtained. The main features of these solutions are discussed.

Magnetic field and internal heat generation effects on the free convection in a rectangular cavity filled with a porous medium

A numerical investigation of the steady magnetohydrodynamics free convection in a rectangular cavity filled with a fluid-saturated porous medium and with internal heat generation has been performed. A uniform magnetic field, inclined at an angle γ with respect to the horizontal plane, is externally imposed. The values of the governing parameters are the inclined angle γ = 0, π/6, π/4 and π/2, Hartmann number Ha = 0, 1, 5, 10 and 50, Rayleigh number Ra = 10, 100, 10 3 and 10 5, and the aspect ratio a = 0.01, 0.2, 0.5 and 1 (square cavity). It is shown that the intensity of the core convection is considerably affected by the considered parameters. It is also found that the local Nusselt number Nu Y decreases on the bottom wall as γ increases (magnetic field changes its direction from the horizontal to the vertical direction) and vice versa for the top wall of the cavity. The reported results are in good agreement with the available published work in the literature.

Hydromagnetic three-dimensional free convective heat transfer overa stretching surface embedded in a non-Darcian porous medium inthe presence of heat generation or absorption

A general analysis is developed to study fluid-flow and heat-transfer characteristics for steady, three-dimensional flow over a linearly stretching porous vertical surface embedded in a non-Darcian medium. A generalized flow model is used in the present study to include the effects of the macroscopic viscous term and the microscopic inertial force. The flow is subjected to a uniform transverse magnetic field normal to the plate. The effect of internal heat generation or absorption is also considered. The governing three-dimensional partial-differential equations for the present case are transformed into ordinary-differential equations using three-dimensional similarity variables. The resulting equations are solved numerically. Velocity distribution, temperature distribution, surface shear stresses, and wall-heat transfer rate are computed for various values of the Prandtl number, magnetic field parameter, inverse Darcy number, porous-medium inertia coefficient, heat generation/absorption coefficient, and mass-transfer coefficient.PACS No.: 44.30.+v

MHD free-convection flow of a non-Newtonian power-law fluid at a stretching surface with a uniform free-stream

The momentum and heat transfer in laminar boundary layer of a non-Newtonian power-law fluid flowing past a stretching surface with uniform surface heat flux and uniform free-stream, are analyzed. The sheet is linearly stretched in the presence of a uniform transverse magnetic. The effect of internal heat generation or absorption is also considered. The governing similar equations are solved numerically for several values of the power-law viscosity index and the non-Newtonian Prandtl number of the fluid. A discussion is provided for the effect of the magnetic field and heat source/sink parameters on the velocity and temperature fields. Friction coefficients and Nusselt number are calculated for the mentioned parameters.

Hall effects on MHD free convection flow of a non-newtonian power-law fluid at a stretching surface

The momentum and heat transfer in laminar boundary layer of a non-Newtonian power-law flow fluid flowing past a stretching surface with uniform surface heat flux, are analyzed. The sheet is linearly stretched in the presence of a uniform strong transverse magnetic field and the Hall effects are taken into account. The effect of internal heat generation or absorption is also considered. The governing similar equations are solved numerically for several values of the power-law viscosity index and the non-Newtonian Prandtl number of the fluid. A discussion is provided for the effect of the magnetic field, Hall and heat source/sink parameters on the velocity and temperature fields. Friction coefficients and Nusselt number are calculated for the mentioned parameters

Combined effect of internal heat generation and magnetic field on free convection and mass transfer flow in a micropolar fluid with constant suction

This paper studied the problem of free convection with mass transfer flow for a micropolar fluid bounded by a vertical infinite surface with an exponentially decaying heat generation, under the action of a transverse magnetic field. Numerical calculations are carried out for the various parameters entering into the problem. The numerical values of the skin friction, the wall couple stress, the rate of heat transfer, the concentration gradient at the wall and the boundary layer thickness are shown graphically as functions of the magnetic field parameter.

Optimal shape design for packaging containing heating elements by inverse heat transfer method

The present study is concerned with the determination of the optimal shape for a package containing multiple heating elements. The optimization tool has been developed based on the inverse heat transfer (IHT) approach, incorporating a direct problem solver, a numerical grid generator, a direct-differentiation sensitivity analyzer, and the conjugate gradient method. Shape design that leads to a specified outer surface temperature distribution is predicted by the approach. In this study, the effects of internal heat generation on optimal shapes of the packagings have also been evaluated. Several practical cases with various imbedded heating elements and thermal conditions are studied. Results show that the approach provides an efficient computer-aided design scheme for the shape profile determination.

Hall Effects on Magnetohydrodynamic Free-Convection Flow at a Stretching Surface with a Uniform Free-Stream

Free convection flow of a conducting fluid near an isothermal sheet with a uniform free stream of constant velocity and temperature is investigated. The sheet is linearly stretched in the presence of a uniform strong transverse magnetic field and the Hall effects are taken into account. The effect of internal heat generation or absorption is also considered. The boundary layer equations governing the problem under consideration have been transformed into a system of ordinary differential equations, which are solved numerically. Velocity and temperature functions are shown on graphs and the numerical values of functions affecting the shear stress and the rate of heat transfer are entered in a table. The effects of the magnetic field, the Hall and the heat source/sink parameters on these functions are discussed. The results are compared with those known from the literature.

Effects of throughflow and internal heat generation on the onset of convection in a fluid layer

The throughflow and internal heat generation effects on the onset of convection in an infinite horizontal fluid layer are investigated. The boundaries are considered to be rigid (however permeable) and perfectly conducting. The resulting eigenvalue problem is solved by using the Galerkin method, and the effects of various parameters in the stability results are analyzed. The results indicate that the stability of the system is significantly affected by both throughflow and internal heat generation in the fluid layer. The Prandtl number comes into play due to the presence of throughflow and it has a profound effect on the stability of the system. It is found that, in the presence of internal heating, throughflow in one direction supresses convection while throughflow in the other direction encourages it.

Hall and Ion-Slip Effects on MHD Free Convective Heat Generating Flow Past a Semi-infinite Vertical Flat Plate

The object of the present work is to study the effects of Hall and ion-slip currents on free convection flow past a semi-infinite vertical flat plate for the case of a strong magnetic field imposed perpendicularly to the plate. The effect of internal heat generation or absorption is also considered. The derived fundamental equations on the assumption of small magnetic Reynolds number are solved numerically. The velocity profiles, temperature profiles, the skin friction components on the plate and heat transfer coefficient are computed and discussed in detail for various values of the magnetic field parameter, Hall parameter, ion-slip parameter and heat source/sink parameter.

Novel model for evolution of normal zones in composite superconductors

This paper presents a numerical analysis of the evolution of normal zones in a composite superconductor based on a novel heat conduction equation which takes into account the finite speed of thermal wave propagation and the inertia of the internal heat source. The results are compared with those obtained from the classical parabolic heat conduction equation. It is shown that appreciable differences can occur between the two models, in qualitative as well as quantitative terms. For large times these differences do not disappear because of the effect of internal heat generation.

The Effects of Internal Heat Generation, Anisotropy, and Base Temperature Nonuniformity on Heat Transfer from a Two-Dimensional Rectangular Fin

Abstract A finite-element approach is used to study the two-dimensional performance of a straight fin of rectangular profile operating in a convecting environment. The analysis includes the effects of internal heat generation, anisotropy of fin material, base temperature elevation (depression), and unequal heat transfer coefficients for tip and main convection surfaces. Numerical results are presented showing the impact of these effects on the heat removal capability of the fin. The graphical information provided is believed to be useful for predicting the performance of a two-dimensional rectangular fin under more general and realistic conditions.