Investigation of heat transfer in square porous-annulus

The current study is focused to analyze the heat transfer characteristics in a porous duct. The mathematical model of heat transfer in a porous duct was solved by converting the governing partial differential equations into a set of algebraic equations with the help of finite element method. A simple three noded triangular element is used to mesh the duct domain. The current problem consists of a square duct with outer walls being exposed to hot temperature T_h, and inner walls subjected to cool temperature T_c. Emphasis is given to investigate the effect of width ratio of cavity on heat and fluid flow characteristics inside the porous medium. The results are reported for various duct width ratios, Rayleigh number etc. It is found that the Nusselt number increases with increase in height of cavity along the vertical walls of duct; however the Nusselt number for certain values of duct ratio oscillates along the width of the porous medium at bottom wall of the cavity.     

Conjugate heat transfer analysis of a heat generating vertical plate

This paper mainly deals with conjugate heat transfer problem pertinent to rectangular fuel element of a nuclear reactor dissipating heat into an upward moving stream of liquid sodium. Introducing boundary layer approximations, the equations governing the flow and thermal fields in the fluid domain are solved simultaneously along with two-dimensional energy equation in the solid domain by satisfying the continuity of temperature and heat flux at the solid–fluid interface. The boundary layer equations are discretized using fully implicit finite difference scheme so as to adopt marching technique solution procedure, while second-order central difference scheme is employed to discretize the energy equation in the solid domain and the resulting system of finite difference equations are solved using Line-by-Line Gauss–Seidel iterative solution procedure. Numerical results are presented for a wide range of parameters such as aspect ratio, A_r, conduction–convection parameter, N_cc, heat generation parameter, Q, and flow Reynolds number, Re. It is concluded that there exist an upper or a lower limiting value of these parameters above or below which the temperature in the fuel element crosses its allowable limit. It is also found that an increase in Re results in considerable increase in overall heat dissipation rate from the fuel element.     

Heat transfer in porous cavity under the influence of radiation and viscous dissipation

Heat transfer under the influence of radiation and viscous dissipation in a square cavity filled with saturated porous medium is analysed. The flow is assumed to follow Darcy law. The governing equations are non-dimensionalised and solved numerically using finite element method. Left vertical surface of the square cavity is maintained at isothermal temperature T_h and right vertical surface at T_c. Results are presented in terms of Nusselt number at hot and cold wall of the cavity for various values of viscous dissipation parameter and radiation parameter. It is seen that the average Nusselt number at hot as well as cold wall increases with increase in radiation parameter.     

Heat generation effects in natural convection inside a porous annulus

The interplay between internal heat generation and externally driven natural convection inside a porous medium annulus is studied in detail using numerical methods. The axisymmetric domain is bounded with adiabatic top and bottom walls and differentially heated side walls sustaining steady natural convection of a fluid with Prandtl number, Pr = 5, through a porous matrix of volumetric porosity, ? = 0.4. The generalized momentum equation with Brinkman–Darcy–Forchheimer terms and the local thermal non-equilibrium based two-energy equation model are solved to determine the flow and the temperature distribution. Beyond a critical heat generation value defined using an internal Rayleigh number, Ra_I,cr^?, the convection transits from unicellular to bicellular mode, as the annulus T_max becomes higher than the fixed hot-wall temperature. The Ra_I,cr^? increases proportionately when the permeability based external Rayleigh number Ra_E^? and the solid–fluid thermal conductivity ratio γ are independently increased. A correlation is proposed to predict the overall annulus Nu as a function of Ra_E^?, Ra_I^?, Da and γ. It predicts the results within ± 20% accuracy.     

Buoyancy-induced flow and heat transfer in a porous annulus between concentric horizontal circular and square cylinders

ABSTRACT

This paper reports on natural convection heat transfer in a porous annulus between concentric horizontal circular and square cylinders. The heated inner circular cylinder is maintained at the uniform hot temperature T_h, whereas the cooled outer square duct is held at the uniform cold temperature T_c. A pressure-based collocated finite-volume method is used to numerically investigate the effects on the total heat transfer of Rayleigh number (Ra), Prandtl number (Pr), Darcy number (Da), porosity (?), and annulus aspect ratio (R/L). Results demonstrate that at low Ra values, conduction is the dominant heat transfer mode. Convection contribution to total heat transfer becomes more important beyond a critical Ra value, which decreases with an increase in Da and/or ?. Furthermore, an increase in the enclosure aspect ratio (R/L) leads to an increase in total heat transfer. A similar behavior is obtained with Prandtl number, where predictions indicate higher heat transfer rates at higher Pr values with its effect increasing as Ra increases. Streamlines and isotherms reveal flow separation for some of the reported cases. Limited computations are also performed for natural convection in a porous annulus between two horizontal concentric circular cylinders having the same inner and outer perimeters as the investigated enclosure. Comparison of the predicted average Nusselt number estimates with similar ones obtained in the original enclosure reveals a large percentage difference in values, demonstrating the strong influence of geometry on natural convection in enclosures.     

Effect of viscosity variation on natural convection flow of water–alumina nanofluid in an annulus with internal heat generation

Heat transfer enhancement in a horizontal annulus using the variable viscosity property of an Al₂O₃–water nanofluid is investigated. Two different viscosity models are used to evaluate heat transfer enhancement in the annulus. The base case uses the Pak and Cho model and the Brinkman model for viscosity which take into account the dependence of this property on temperature and nanoparticle volume fraction. The inner surface of the annulus is heated uniformly by a constant heat flux q_w and the outer boundary is kept at a constant temperature T_c. The nanofluid generates heat internally. The governing equations are solved numerically subject to appropriate boundary conditions by a penalty finite‐element method. It is observed that for a fixed Prandtl number Pr = 6.2, Rayleigh number Ra = 10⁴ and solid volume fraction ? = 10%, the average Nusselt number is enhanced by diminishing the heat generation parameter, mean diameter of nanoparticles, and diameter of the inner circle. The mean temperature for the fluids (nanofluid and base fluid) corresponding to the above mentioned parameters is plotted as well. © 2012 Wiley Periodicals, Inc. Heat Trans Asian Res; Published online in Wiley Online Library ( wileyonlinelibrary.com/journal/htj ). DOI 10.1002/htj.21016     

Stability analysis of MHD radiative mixed convective flow in vertical cylindrical annulus: Thermal nonequilibrium approach

In the present study, the influence of the induced magnetic field on the MHD mixed convective electrically conducting fluid flow inside the vertical cylindrical annulus is analyzed numerically. The heat transfer is presumed to be due to a combination of mixed convection and radiation. The stability of the flow is examined when the solid and fluid phases are not in local thermal equilibrium. The governing equations are solved numerically by both finite difference and finite element methods. To control the flow formation rate more accurately the induced magnetic field is also considered in this study. As the magnetic Prandtl number (Pm) and Hartmann number (M) get enhanced, the velocity and induced magnetic fields get retarded in the annulus due to the presence of drag-like force, namely, the Lorentz force. When there is an increase in the mixed convection parameter the induced magnetic field gets enhanced. An increase in radiation parameter tends to decline the fluid temperature and reverse the behavior of the solid temperature. Increment in Pm decreases the wall shear stress near the conducting cylinder. Increasing values of porous, magnetic, and radiation parameters lead to an unstable system with smaller heat transfer coefficient values but the system gets stabilized for larger values of heat transfer coefficient. The results could be used as first-hand information for comprehending and developing the thermal flow phenomenon in porous media. The obtained numerical results are in good accordance with the existing results. Using an artificial neural network, heat transfer characteristics are analyzed through mean square error and regression analysis.     

Effect of the Darcy number on the energy flow and operating conditions of a thermoacoustic porous-medium system

In this paper, a simplified porous medium thermoacoustic system is modeled to observe its energy interaction characteristics and identify its operating conditions mainly as a function of porous medium Darcy number. The governing Darcy–Brinkman momentum equation and energy equation are simplified and linearized by using a first order perturbation analysis. Similar perturbation analysis is usually used to solve the linear thermoacoustic problem in the low Mach number limit. Simplified momentum and energy equations are solved, in the frequency domain, in order to obtain the expressions of the fluctuating velocity (u₁) and temperature (T₁). Time averaged and space averaged heat fluxes and work fluxes are calculated using the expressions of fluctuating velocity and temperature. The effects of the drive ratio (DR), Darcy number (Da_δ), temperature gradient (?T_m), and frequency (f) on the heat flux, work flux, and operating conditions are discussed and graphically presented.     

Conjugate conduction-forced convection heat transfer analysis of a rectangular nuclear fuel element with non-uniform volumetric energy generation

The main objective of this paper is to present a comparative study of uniform and non-uniform volumetric energy generation in a rectangular nuclear fuel element washed by upward moving stream of liquid sodium. Employing finite difference schemes, the boundary layer equations governing the flow and thermal fields in the fluid domain are solved simultaneously with two-dimensional energy equation in the solid domain by satisfying the continuity of temperature and heat flux at the solid–fluid interface. Numerical results are presented for a wide range of aspect ratio, A_r, conduction–convection parameter, N_cc, total energy generation parameter, Q_t, and flow Reynolds number, Re_H. It is concluded that for the same total energy generation, a somewhat realistic non-uniform volumetric energy generation puts greater restriction on the thermal power generation as compared to the idealistic uniform volumetric energy generation. Further, it is found that despite the total energy generation being the same for two cases, the non-uniform volumetric energy generation within the fuel element results in considerably higher energy dissipation rate.     

The influence of a magnetic field on the operating conditions of a single plate thermoacoustic system

We model a single-plate thermoacoustic system under the action of a transverse magnetic force. Simplified momentum and energy equations are solved, in the frequency domain, in order to obtain the expressions of the fluctuating velocity (u₁) and temperature (T₁). Time averaged and space averaged heat fluxes and work fluxes are calculated. The effects of the drive ratio (DR), Hartmann number (Ha_δ), temperature gradient (∇T_m), and frequency (f) on the heat flux, work flux, and operating conditions are discussed and graphically presented.     

Thermal analysis of natural convection and radiation in porous fins

In this study, the effects of radiation and convection heat transfer in porous media are considered. The geometry considered is that of a rectangular profile fin. The porous fin allows the flow to infiltrate through it and solid-fluid interaction takes place. This study is performed using Darcy's model to formulate heat transfer equation. To study the thermal performance, three types of cases are considered viz. long fin, finite length fin with insulated tip and finite length fin with tip exposed. The theory section addresses the derived governing equation. The effects of the porosity parameter S_h, radiation parameter G and temperature ratio C_T on the dimensionless temperature distribution and heat transfer rate are discussed. The results suggest that the radiation transfers more heat than a similar model without radiation.     

Buoyant flow in a vertical fluid saturated porous annulus: The Brinkman model

Laminar, steady and parallel buoyant flow in a concentric vertical annulus filled with a porous medium is studied. The Darcy–Brinkman model and the Oberbeck–Boussinesq approximation are employed. The governing equations are written in a dimensionless form and solved analytically by means of the variation of constants method. In order to investigate the onset of flow reversal, both at the inner and at the outer boundary, a dimensionless reference temperature θ₀ is introduced. Moreover, this quantity is related to the usual Gr/Re parameter. The limiting cases of Darcy flow and clear fluid are investigated as well.     

Thermal conductivity variation on natural convection flow of water–alumina nanofluid in an annulus

This work is focused on the numerical modeling of steady laminar natural convection flow in an annulus filled with water–alumina nanofluid. The inner surface of the annulus is heated uniformly by a uniform heat flux q and the outer boundary is kept at a constant temperature T_c. Two thermal conductivity models namely, the Chon et al. model and the Maxwell Garnett model, are used to evaluate the heat transfer enhancement in the annulus. The governing equations are solved numerically subject to appropriate boundary conditions by a penalty finite-element method. A parametric study is conducted and a selective set of graphical results is presented and discussed to illustrate the effects of the presence of nanoparticles, the Prandtl number and the Grashof number on the flow and heat transfer characteristics for both nanofluid models. It is found that significant heat transfer enhancement can be obtained due to the presence of nanoparticles and that this is accentuated by increasing the nanoparticles volume fraction and Prandtl number at moderate and large Grashof number using both models. However, for the Chon et al. model the greatest heat transfer rate is obtained.     

Periodic free convection from a vertical plate in a saturated porous medium, non-equilibrium model

The problem of the free convection from a vertical heated plate in a porous medium is investigated numerically in the present paper. The effect of the sinusoidal plate temperature oscillation on the free convection from the plate is studied using the non-equilibrium model, i.e., porous solid matrix and saturated fluid are not necessary to be at same temperature locally. Non-dimensionalization of the two-dimensional transient laminar boundary layer equations results in three parameters: (1) H, heat transfer coefficient parameter, (2) K_r, thermal conductivity ratio parameter, and (3) λ, thermal diffusivity ratio. Two additional parameters arise from the plate temperature oscillation condition which are the non-dimensional amplitude (ε) and frequency (Ω). The fully implicit finite difference method is used to solve the system of equations. The numerical results are presented for 0 ? H ? 10, 0 ? K_r ? 10, 0.001 ? λ ? 10 with the plate temperature oscillation parameters 0 ? Ω ? 10 and 0 ? ε ? 0.5. The results show that the thermal conductivity ratio parameter is the most important parameter. It is found also that increasing the amplitude and the frequency of the oscillating surface temperature will decrease the free convection heat transfer from the plate for any values of the other parameters.     

Finite element simulation of hydromagnetic convective flow in an obstructed cavity

The hydromagnetic natural convective flow and heat transfer characteristics in a square cavity with a solid circular heated obstacle located at the center have been investigated numerically. The left vertical surface of the cavity is uniformly heated of temperature T_c and other three surfaces are adiabatic. The obstacle consists of constant heat T_h. Under all circumstances the condition T_h > T_c is maintained. The physical problem is represented mathematically by sets of governing equations and the developed mathematical model is solved by employing Galerkin weighted residual finite element simulation. The behavior of the fluid in the ranges of Prandtl number (0.073-2.73), Hartmann number (0-50) and Joule heating parameter (1-7) is explained in details. It is found that the flow and temperature fields are strongly dependent on the above stated parameters for the ranges considered. The variation of the average Nusselt number (Nu) for various Prandtl number (Pr) is also presented.     

Soret effect on the boundary layer flow regime in a vertical porous enclosure subject to horizontal heat and mass fluxes

The combined thermo- and double-diffusive convection in a vertical tall porous cavity subject to horizontal heat and mass fluxes was investigated analytically and numerically using the Darcy model with the Boussinesq approximation. The investigation focused on the effect of Soret diffusion on the boundary layer flow regime. The governing parameters were the thermal Rayleigh number, R_T, the Lewis number, Le, the buoyancy ratio, N, the Soret parameter, M, which characterized the Soret effect, and the aspect ratio of the enclosure, A_r. The results demonstrated the existence of a boundary layer flow solution for which the Soret parameter had a strong effect on the heat and mass transfer characteristics. For M ≠ 1 and M ≠ −1/Le, the profiles of the vertical velocity component, v, temperature, T, and solute concentration, S, exhibited boundary layer behaviors at high Rayleigh numbers. Furthermore, as R_T increased, the temperature and solute concentration became vertically and linearly stratified in the core region of the enclosure. The thermo-diffusion effect on the boundary layer thickness, δ, was discussed for a wide range of the governing parameters. It was demonstrated analytically that the thickness of the boundary layer could either increase or decrease when the Soret parameter was varied, depending on the sign of the buoyancy ratio. The effect of R_T on the fluid flow properties and heat and mass transfer characteristics was also investigated.     

Heat and mass transfer analysis for boundary layer stagnation-point flow towards a heated porous stretching sheet with heat absorption/generation and suction/blowing

A similarity analysis is performed to investigate the structure of the boundary layer stagnation-point flow and heat transfer over a stretching sheet in a porous medium subject to suction/blowing and in the presence of internal heat generation/absorption. A scaling group of transformations is applied to get the invariants. Using the invariants, a third and a second order ordinary differential equations corresponding to the momentum and energy equations are obtained respectively. Boundary layer velocity and temperature profiles are determined numerically for various values of the ratio of free stream velocity and stretching velocity, the permeability parameter, suction/blowing parameter, heat source/sink parameter, Prandtl number. It is found that the horizontal velocity increases with the increasing value of the ratio of the free stream velocity (ax) and the stretching velocity (cx). The temperature decreases in this case. At a particular point of the porous stretching sheet, the non-dimensional fluid velocity decreases with the increase of the permeability of the porous medium and also with the increasing suction parameter when the free stream velocity is less than stretching velocity whereas fluid velocity increases with the increasing injection parameter. But when the free stream velocity is greater than the stretching velocity the opposite behaviour of horizontal velocity is noticed. The dimensionless temperature at a point of the sheet decreases due to suction but increases due to injection. The temperature at a point is found to decrease with the increasing Prandtl number.     

Effect of viscous dissipation and radiation on natural convection in a porous medium embedded within vertical annulus

The effect of viscous dissipation and thermal radiation on natural convection in a porous medium embedded within a vertical annular cylinder is investigated. The inner surface of the cylinder is maintained at an isothermal temperature $T_w$ and the outer surface is maintained at ambient temperature $T_{\infty }$. The fluid is assumed to obey the Darcy law. Finite element method is used to solve the partial differential equations governing the fluid flow and heat transfer behavior. The study is focused to investigate the combined effect of viscous dissipation and radiation. Results are presented for different values of the viscous dissipation parameter, radiation parameter, radius ratio, aspect ratio and Rayleigh number. It is observed that the viscous dissipation parameter reduces the average Nusselt number at hot surface. However, the average Nusselt number increases at the cold surface due to increased viscous dissipation parameter.     

Thermodynamic equilibrium analysis of steam methane reforming based on a conjugate solution of material balance and law action mass equations with the detailed energy balance

Thermodynamic equilibrium analysis of the steam methane reforming (SMR) process to synthesis gas was studied. For this purpose, the system equations of the material balance and the equations of law mass action were solved by dichotomy method. The investigation was performed for a wide range of operational conditions such as a temperature, pressure, and inlet steam-to-methane ratio. The results obtained, with the help of developed algorithms, were compared with the results obtained via different commercial and open-source programs. All results are in excellent agreement. The operational conditions for the probable formation of carbon were determined. It was established that for the temperature range above 1100 K, the probability of carbon formation is absent for steam-to-methane ratio above units. In order to determine the amount of heat supplied per 1 mol to the reformer, the heat balance equation was obtained to achieve a targeted degree of methane conversion. With the help of the heat balance equation, it was established that the resulting transformation of substances in the steam methane reformer can be presented as a sequential heating of feed streams of methane and steam from the inlet temperature T₁ to the outlet temperature T₂, heat for SMR reaction at the temperature T₂, and heat for transformation of the part of the produced carbon monoxide (CO) via the water-gas shift (WGS) reaction at the temperature T₂. The presented algorithm of thermodynamic analysis gives an appearance of the dependence of the product composition and the amount of required heat from operating conditions such as the temperature, pressure, and steam-to-methane ratio.     

Heat and fluid flow characteristics of liquid sodium flowing past a nuclear fuel element with non-uniform energy generation

The prime objective of the present study is to analyze numerically the steady state fluid flow and heat transfer characteristics of liquid sodium as a coolant flowing past over a rectangular nuclear fuel element having non-uniform volumetric energy generation. Accordingly, employing stream function-vorticity formulation and using finite difference schemes, the equations governing the flow and thermal fields in the coolant are solved simultaneously with energy equation for the fuel element by satisfying the conditions of continuity of temperature and heat flux at the solid–fluid interface. Keeping Prandtl number Pr = 0.005 for liquid sodium as constant, numerical results are presented and discussed for a wide range of aspect ratio A_r, conduction–convection parameter N_cc, total energy generation parameter Q_t and Reynolds number Re_H. It is concluded that the rate of heat dissipation from the fuel element to the coolant is independent of A_r, N_cc and Q_t, whereas it increases in proportion to the increase in Re_H. It is also found that for a given material of the fuel element, there is an upper limiting value of N_cc and Re_H beyond which decrease in coolant temperature is negligibly small.