Some transformations for natural convection on a vertical flat plate embedded in porous media with prescribed wall temperature

This study analyzes transformation for boundary layer equations for two-dimensional steady natural convection along a vertical flat plate embedded in porous media. Three different boundary conditions (uniform, power law, and exponential variation), which result in different wall temperature, are identified and processed. Based on the distance along flat plate, governing equations after transformation divide the flow pattern into three subregions. These three subregions characterize the distinct relationship among patterns for pure fluid flow, non-Darcian flow expressed in nonsimilar equations, and pure porous flow. It is also found that similarity solution exists for the whole flow region as the wall temperature distribution is in linear variation and the inertia resistance is without consideration.     

MHD free-convection flow in open-ended vertical concentric porous annuli

Analytical solutions for fully developed MHD natural-convection flow in open-ended vertical concentric porous annuli are presented. Four fundamental boundary conditions have been investigated and the corresponding fundamental solutions are obtained. These four fundamental boundary conditions are obtained by combining each of the two conditions of having one boundary maintained at uniform heat flux or at uniform wall temperature with each of the conditions that the opposite boundary is kept isothermal at the inlet fluid temperature or adiabatic. Expressions for the flow and heat-transfer parameters are given for each case. These fundamental solutions may be used to obtain solutions satisfying more general thermal boundary conditions.     

Mixed convection boundary layer flow on a horizontal plate in a uniform stream

A mixed convection boundary layer on a horizontal plate for uniform wall temperature/uniform heat flux is investigated using a computer extension of the perturbation series. The first 17 terms for the uniform wall temperature case and the first ten terms for the uniform heat flux case are computed for a Prandtl number σ = 0.72. The direct expansion is transformed by a Euler transform and other techniques. The results for buoyancy aiding or opposing the main flow are presented. The present work predicts the result to two digit accuracy for the entire domain of the streamwise coordinate. For uniform wall temperature, the maximum error is 5.983% for skin friction and 1.072% for heat transfer. For uniform heat flux, the maximum error is 6.9% for skin friction and 1.9% for wall temperature.     

A way to explain the thermal boundary effects on laminar convection through a square duct

A way using the reformulation of the energy conservation equation in terms of heat flux to explain the thermal boundary effects on laminar convective heat transfer through a square duct is presented. For a laminar convection through a square duct, it explains that on the wall surface, the velocity is zero, but convection occurs for uniform wall heat flux (UWHF) boundary in the developing region due to the velocity gradient term; for uniform wall temperature (UWT) boundary, only diffusion process occurs on the wall surface because both velocity and velocity gradient do not contribute to convection; for UWHF, the largest term of the gradient of velocity components (the main flow velocity) on the wall surface takes a role in the convection of the heat flux normal to the wall surface, and this role exists in the fully developed region. Therefore, a stronger convection process occurs for UWHF than for UWT on the wall surface. The thermal boundary effects on the laminar convection inside the flow are also detailed.     

Laminar free-convection in glycerol with variable physical properties adjacent to a vertical plate with uniform heat flux

The steady laminar boundary layer flow of glycerol along a vertical stationary plate with uniform heat flux is studied in this paper. The density, thermal conductivity and heat capacity of this liquid are linear functions of temperature but dynamic viscosity is a strong, almost exponential, function of temperature. The results are obtained with the numerical solution of the boundary layer equations. Both upward flow (plate heating) and downward flow (plate cooling) is considered. The variation of μ with temperature has significant influence on wall heat transfer and much stronger influence on wall shear stress. It was also found that the similarity exponent, which is equal to 0.20 for the classical problem with constant properties, is lower than 0.20 in the upward flow and higher than 0.20 in the downward flow.     

Viscous dissipation and thermoconvective instabilities in a horizontal porous channel heated from below

A linear stability analysis of the basic uniform flow in a horizontal porous channel with a rectangular cross section is carried out. The thermal boundary conditions at the impermeable channel walls are: uniform incoming heat flux at the bottom wall, uniform temperature at the top wall, adiabatic lateral walls. Thermoconvective instabilities are caused by the incoming heat flux at the bottom wall and by the internal viscous heating. Linear stability against transverse or longitudinal roll disturbances is investigated either analytically by a power series formulation and numerically by a fourth order Runge-Kutta method. The special cases of a negligible effect of viscous dissipation and of a vanishing incoming heat flux at the bottom wall are discussed. The analysis of these special cases reveals that each possible cause of the convective rolls, bottom heating and viscous heating, can be the unique cause of the instability under appropriate conditions. In all the cases examined, transverse rolls form the preferred mode of instability.     

Heat Transfer Characteristics of Laminar Flow in Internally Finned Tubes under Various Boundary Conditions

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Flow due to a porous stretching/shrinking sheet with thermal radiation and mass transpiration

This study investigates the behavior of carbon nanotubes (CNT) approaching an unsteady flow of a Newtonian fluid over a stagnation point on a stretching surface employing porous media. It flows when the liquid begins to move with the progression of time. Heat exchange with the environment has an impact on the flow. The implicitly limited component technique is used to solve the nondimensional partial differential equation with an associated boundary layer, which is an unstable system. Analytically, the solutions, as well as the required boundary conditions, are obtained. The effects of mass transpiration, volume fraction, and heat radiation on Newtonian fluid flow through porous media are explored. Single- and multi-walled CNTs are used as well as water, as base fluids in the experiment. The impact of thermal radiation and heat source/sink is shown in the energy equation, which is solved under four different cases: uniform heat flux case, constant wall temperature case, general power-law wall heat flux case, and general power-law wall temperature case. By supplying distinct physical characteristics, a theoretical analysis of the existence and nonexistence of unique and dual solutions may be explored. These physical parameters determine the velocity distribution and temperature distribution. Prescribed surface temperature (PST) and prescribed wall heat flux (PHF) heat transfer solutions can be written using confluent hypergeometric equations, and generic power-law PST and PHF situations can also be expressed using confluent hypergeometric equations. The graphical representations assist in the discussion of the current study's findings.     

Fully developed laminar flow and heat transfer in smooth trapezoidal microchannel

Numerical analysis of fully developed laminar slip flow and heat transfer in trapezoidal micro-channels has been studied with uniform wall heat flux boundary conditions. Through coordinate transformation, the governing equations are transformed from physical plane to computational domain, and the resulting equations are solved by a finite-difference scheme. The influences of velocity slip and temperature jump on friction coefficient and Nusselt number are investigated in detail. The calculation also shows that the aspect ratio and base angle have significant effect on flow and heat transfer in trapezoidal micro-channel.     

基于燃烧与水动力耦合模型的锅炉蒸汽管超温特性研究

提出一种基于燃烧与水动力耦合模型的锅炉蒸汽管壁温度数值模拟方法,对某660 MW超临界切圆燃烧锅炉壁温进行了计算分析。以均匀外壁温为边界条件,利用Fluent软件模拟了煤粉气固流动、燃烧和辐射等过程,获得了炉内不同位置受热管的传热热流。再以热流分布为边界,采用MATLAB软件建立了工质流动及气-壁-汽换热方程组,Fluent软件重新计算的壁温边界。通过编写模型间的网格映射函数,实现壁温的耦合计算。研究表明:壁温计算值与实测值的最大相对误差在2%以内;炉膛出口残余旋转使水平烟道左侧和右上方热流较大,高温再热器和末级过热器的外壁温沿炉宽方向呈双峰分布;高温再热器整级受热管出口壁温的峰谷差值远高于末级过热器,实际运行中应特别注意高温再热器靠烟道左侧管屏外圈管子向火侧弯头处的超温。     

Transient effects in natural convection cooling of vertical parallel plates

This work presents results from a numerical study of transient natural convection between vertical parallel plates. Two boundary conditions – uniform wall temperature and uniform heat flux – are considered. Results presented include the rate of heat transfer for uniform wall temperature and the maximum wall temperature for uniform heat flux. Also presented are simple correlations to calculate the minimum heat transfer and the maximum wall temperature during the transient period. It is found that for uniform wall temperature the ratio of the minimum heat transfer to the steady state heat transfer decreases with length of the channel, and for uniform heat flux the maximum transient temperature has a maximum of about 9% over the steady state temperature.     

Thermally developing flow and heat transfer in rectangular microchannels of different aspect ratios

Laminar convective heat transfer in the entrance region of microchannels of rectangular cross-section is investigated under circumferentially uniform wall temperature and axially uniform wall heat flux thermal boundary conditions. Three-dimensional numerical simulations were performed for laminar thermally developing flow in microchannels of different aspect ratios. Based on the temperature and heat flux distributions obtained, both the local and average Nusselt numbers are presented graphically as a function of the dimensionless axial distance and channel aspect ratio. Generalized correlations, useful for the design and optimization of microchannel heat sinks and other microfluidic devices, are proposed for predicting Nusselt numbers. The proposed correlations are compared with other conventional correlations and with available experimental data, and show very good agreement.     

Analysis of flow and heat transfer in evaporator porous wicking structure of a flat heat pipe

With a specified pressure distribution, an analytical investigation was conducted to explore the flow and heat transfer characteristics in an evaporator porous wicking structure of a flat heat pipe. The boundary effect on the flow rate is more significant than the inertia, and both the boundary and inertia effects exert very little influence on fluid layer thickness and velocity distribution. The bottom of the porous layer is at a quite uniform temperature, and the heat flux is almost normal to the solid boundary. © 2008 Wiley Periodicals, Inc. Heat Trans Asian Res; Published online in Wiley InterScience ( www.interscience.wiley.com ). DOI 10.1002/htj.20237     

Buoyant Darcy flow driven by a horizontal temperature gradient: A linear stability analysis

The linear stability of a fluid saturated porous layer bounded by two parallel impermeable plane walls is investigated. The lower wall is subject to a uniform heat flux, while the upper wall is subject to a linearly varying temperature in a horizontal direction. Two parameters govern the onset of convection in the porous layer: the vertical Darcy–Rayleigh number, and the horizontal Darcy–Rayleigh number. The objective of this study is to obtain the onset conditions for the instability of the basic parallel flow in the layer. The governing balance equations are written in a dimensionless form and solved on assuming oblique roll disturbances, arbitrarily oriented in the horizontal plane. Mathematically, this leads to a system of two ordinary differential equations to be solved as an eigenvalue problem. The solution, carried out numerically, provides the neutral stability condition. The numerical solution is performed by employing a procedure based on the sixth-order Runge–Kutta method and on the shooting method for satisfying the boundary conditions at the upper boundary wall.     

NON-DARCIAN FORCED CONVECTION ANALYSIS IN AN ANNULUS PARTIALLY FILLED WITH A POROUS MATERIAL

Numerical solutions are presented for fully developed forced convection in concentric annuli partially filled with a porous medium. The porous medium is attached at the inner cylinder, which is maintained at uniform heat flux or at uniform wall temperature while the outer cylinder is adiabatic. The Brinkman-Forchheimer-extended Darcy model was used to model the flow inside the porous medium. The dependence of the fluid flow and heat transfer on several parameters of the problem is thoroughly documented. The inertia coefficient at which the inertial effects reduce the flow rate by 5% is determined as a function of the Darcy number for various thicknesses of the porous substrate. It is also shown that a critical thickness at which the value of the Nusselt number reaches a minimum does not exist if the effective thermal conductivity of the fluid-saturated porous medium is much higher than the fluid conductivity.     

Natural convection from a vertical cone in a porous medium due to the combined effects of heat and mass diffusion with non-uniform wall temperature/concentration or heat/mass flux and suction/injection

An analysis has been carried out to study the non-Darcy natural convention flow of Newtonian fluids on a vertical cone embedded in a saturated porous medium with power-law variation of the wall temperature/concentration or heat/mass flux and suction/injection with the streamwise distance x. Both non-similar and self-similar solutions have been obtained. The effects of non-Darcy parameter, ratio of the buoyancy forces due to mass and heat diffusion, variation of wall temperature/concentration or heat/mass flux and suction/injection on the Nusselt and Sherwood numbers have been studied.     

The effect of upper surface conditions on convection in a shallow cavity with differentially heated end-walls

The effect of upper surface boundary conditions on the flow structure in shallow cavities with differentially heated end-walls is examined. Matched asymptotic solutions, valid for small cavity aspect ratios are presented for the following cases; uniform shear stress with zero heat flux, uniform heat flux with zero shear stress, and heat flux linearly dependent on surface temperature with zero shear stress. It is shown that these changes in surface boundary conditions have an important influence on the temperature and flow structure within the cavity.     

Melting effect on mixed convective heat transfer with aiding and opposing external flows from the vertical plate in a liquid-saturated porous medium

In this paper, melting effect on mixed convective heat transfer from a porous vertical plate with uniform wall temperature in the liquid-saturated porous medium with aiding and opposing external flows is numerically examined at steady state. The resulting boundary value problems (BVPs) are comprehensively solved by Runge–Kutta–Gill method and Newton’s iteration for similarity solutions. As shown in the results, for aiding and opposing external flows, it is all found that the rate of convective heat transfer at the interface of solid and liquid phases is reduced with increasing melting strength. Additionally, the melting phenomenon decreases the thermal boundary layer regions of mixed convection in a porous medium. With melting effect, the heat transfer rate is also shown to be asymptotically approaching the forced or free convection as the value of Gr/Re approaches the limits of zero and infinity for aiding external flow; and the criteria for pure forced and mixed convection from an isothermal vertical flat plate in porous media with aiding and opposing external flows are established in melting process.     

Extended Graetz problem including streamwise conduction and viscous dissipation in microchannel

Extended Graetz problem in microchannel is analyzed by using eigenfunction expansion to solve the energy equation. The hydrodynamically developed flow is assumed to enter the microchannel with uniform temperature or uniform heat flux boundary condition. The effects of velocity and temperature jump boundary condition on the microchannel wall, streamwise conduction and viscous dissipation are all included. From the temperature field obtained, the local Nusselt number distributions are shown as the dimensionless parameters (Peclet number, Knudsen number, Brinkman number) vary. The fully developed Nusselt number for each boundary condition is obtained also in terms of these parameters.     

Natural convection heat transfer from an inclined wavy plate in a bidisperse porous medium

This work studies the natural convection heat transfer from an inclined wavy plate in a bidisperse porous medium with uniform wall temperature. The two-velocity two-temperature formulation is used to derive the governing equations of this system. The Prandtl coordinate transformation is used to transform the wavy surface into a regular plane, and the obtained equations are then simplified further by the order-of-magnitude analysis to give the boundary layer equations. The cubic spline collocation method is used to solve the boundary layer governing equations. The effects of dimensionless amplitude, angle of inclination, inter-phase heat transfer parameter, modified thermal conductivity ratio, and permeability ratio on the heat transfer and flow characteristics are studied. Increasing the modified thermal conductivity ratio and the permeability ratio can effectively enhance the natural convection heat transfer of the inclined plate in bidisperse porous media. Moreover, the thermal non-equilibrium effects are significant for low values of the inter-phase heat transfer parameter. As the dimensionless amplitude increases, both the fluctuations of the local Nusselt number for the f-phase and the p-phase with the streamwise coordinate are enhanced.