Numerical simulation of asymmetric flow of coupled heat and mass transfer in a square cavity

The heat and mass transfer of fluid in a square cavity with a solutal and thermal source is numerically investigated. For different Rayleigh numbers, buoyancy ratio, Soret and Dufour numbers, the bifurcation characteristics of heat and mass transfer in a symmetrical square cavity are studied systematically. The results show that there is a critical Ra_c for onset of bifurcation that changes the fluid flow pattern. When Ra < Ra_c, the streamline, temperature and concentration are symmetrically distributed; when Ra > Ra_c, the transition from a symmetrical state to a stable asymmetric state is observed. The fluid is more prone to bifurcation with increasing of buoyancy. An increment to the Soret and Dufour effects enhances heat transfer symmetry and increases the critical Rayleigh number for breaking symmetry.     

方腔内耦合传热传质非对称流动数值模拟

对带有质热源的方腔内流体传热传质进行数值研究。针对不同Ra、Nc、Sr和Df,探究对称方腔内流体传热传质的分岔特性。结果表明：存在临界Ra_c使流体流动形态发生转变,当Ra<Ra_c时,流体流线、温度场和浓度场对称分布;当Ra>Ra_c时,流体发生偏斜。增大浮升力,流体更易发生分岔现象。增强Soret和Dufour效应可增强传热对称性并增大流体发生分岔的临界Rayleigh数。     

Inverse determination of boundary heat fluxes in a porous enclosure dynamically coupled with thermal transport

The inverse natural convection problem of estimating the heat source profiles in a porous enclosure is proposed in the present work. The physical model for the momentum conservation equation makes use of the Darcy-Brinkman equation, which allows the no-slip boundary condition on a solid wall to be satisfied. An iterative Fletcher-Reeves conjugate gradient method is applied such that the gradient of the cost function is introduced when the appropriate sensitivity and adjoint problems are defined. Particularly, the pressure-based SIMPLE algorithm is adopted to solve the continuum direct, sensitivity and adjoint problems in unification. Effects of thermal Rayleigh number, Darcy number, heat flux profiles, sensor locations and quantity on the accuracy of inverse solutions are investigated with or without the measurement errors. Additionally, the fluid and heat transport structures in the uniform porous layer are analyzed using the streamlines and heatlines, and the heat transfer potential is also explained by the variation of overall Nusselt number. Noise data solutions are regularized by stopping the iterations with the discrepancy principle of Alifanov, before the high frequency components of the random noises are reproduced. The present method solves inverse strong convection problem satisfactorily without any a priori information about the unknown heat flux to be estimated.     

有内热源的多孔层中自然对流的稳定性分析

采用局部非热平衡模型, 通过数值法和Garlerkin近似法, 分析存在均匀内热源和边界浓度梯度时, 有效热导率比、流体和固相间的传热系数、浓度梯度的大小以及内热源在流体与固相内的分布情况对水平多孔层中临界内热源Rayleigh数的影响, 来研究相关参数对自然对流的稳定性的影响, 并得到临界内热源Rayleigh数的表达式。结果表明, 浓度Rayleigh数的增加可以促进自然对流的形成;内热源为正时, 自然对流的形成区域主要位于上半区域;内热源为负时, 自然对流的形成区域位于下半区域, 内热源总是促进自然对流的发生;有效热导率比、流体和固相间的内部传热系数、内热源在流体与固相内的分布情况相互耦合, 影响自然对流的稳定性, 这种影响取决于各参数的范围。     

Lattice Boltzmann simulation of alumina-water nanofluid in a square cavity

A lattice Boltzmann model is developed by coupling the density (D2Q9) and the temperature distribution functions with 9-speed to simulate the convection heat transfer utilizing Al₂O₃-water nanofluids in a square cavity. This model is validated by comparing numerical simulation and experimental results over a wide range of Rayleigh numbers. Numerical results show a satisfactory agreement between them. The effects of Rayleigh number and nanoparticle volume fraction on natural convection heat transfer of nanofluid are investigated in this study. Numerical results indicate that the flow and heat transfer characteristics of Al₂O₃-water nanofluid in the square cavity are more sensitive to viscosity than to thermal conductivity.     

Natural convection in a heat generating fluid bounded by two horizontal concentric cylinders

Convective flow and heat transfer of a Boussinesq fluid contained between two horizontal concentric cylinders is investigated under the effects of two driving mechanisms – an externally-imposed temperature gradient across the annulus, and a uniform internal heat generation. Numerical results for flow field and temperature distribution are obtained in terms of four dimensionless parameters, namely the radius ratio, R, the Prandtl number, Pr, the Rayleigh number, Ra*, and the ratio, S, between the characteristic temperature induced by internal heating and the applied temperature difference between the boundaries. Depending on the value of S, the flow pattern is made up of either one or two vortices in each half cavity, and heat is transferred into or out of the cavity through the hot wall. In particular, for a certain value of the applied temperature difference, the hot wall apparently acts as a thermally-insulated boundary, the internal heat is completely lost through the cold wall, and the fluid undergoes a transition from a bicellular to a unicellular flow regime.     

Natural convection in a porous medium at high rayleigh numbers — application of prandtl's analogy

The problem of natural convection heat transfer from an isothermal vertical cylinder to a saturated porous medium is solved by the integral method making use of one-seventh profiles for velocity and temperature variation in the boundary layer. Prandtl's analogy is used to obtain the wall heat transfer coefficients. The results obtained for Rayleigh numbers greater than 10⁹ and Prandtl numbers 4 and 0.12 are compared with experimental data from literature show satisfactory agreement. Also an approximate closed form solution is presented for the case of a flat wall to predict local Nusselt numbers.     

Numerical simulation of natural convection in a square enclosure filled with nanofluid using the two-phase Lattice Boltzmann method

Considering interaction forces (gravity and buoyancy force, drag force, interaction potential force, and Brownian force) between nanoparticles and a base fluid, a two-phase Lattice Boltzmann model for natural convection of nanofluid is developed in this work. It is applied to investigate the natural convection in a square enclosure (the left wall is kept at a high constant temperature (T_H), and the top wall is kept at a low constant temperature (T_C)) filled with Al₂O₃/H₂O nanofluid. This model is validated by comparing numerical results with published results, and a satisfactory agreement is shown between them. The effects of different nanoparticle fractions and Rayleigh numbers on natural convection heat transfer of nanofluid are investigated. It is found that the average Nusselt number of the enclosure increases with increasing nanoparticle volume fraction and increases more rapidly at a high Rayleigh number. Also, the effects of forces on nanoparticle volume fraction distribution in the square enclosure are studied in this paper. It is found that the driving force of the temperature difference has the biggest effect on nanoparticle volume fraction distribution. In addition, the effects of interaction forces on flow and heat transfer are investigated. It is found that Brownian force, interaction potential force, and gravity-buoyancy force have positive effects on the enhancement of natural convective heat transfer, while drag force has a negative effect.     

滚塑工艺加热阶段的传热模型的解析解

对从滚塑模具开始受热到其内部粉料开始熔融之间的加热阶段建立一个传热理论模型。在该模型中,忽略模具、内部空气和粉料的温度梯度,即各自只有一个随时间变化的平均温度。将模具的内部分成内部空气区域和粉料区域。内部空气和粉料在与模具发生对流换热的同时,这两者之间通过掺混也在发生对流换热。根据能量守恒原理,对模具、内部空气和粉料分别列出热能平衡方程式,然后通过简化和求解常微分方程组得到了它们平均温度的解析表达式。由此计算所得的内部空气温度与其实验结果吻合的较好,从而证明了此传热模型及其解析解的准确性。     

Analytical modelling of hysteresis heating in rolling contact of rubber covered rollers

Abstract

In a previous paper, an approximate model for the contact between a rubber covered roller and a rigid roller was developed as analytical functional relationships connecting geometric parameters and material properties of the rubber to nip properties such as maximum contact pressure, etc. in a two-dimensional relationship. The results from that development are used in this work, with the objective to provide a means of estimating the temperature rise due to hysteresis heating. In the heat conduction modelling, one-dimensional steady state heat conduction is assumed. The heat source is a power input coming from internal friction developed during the rolling contact. The power input is expressed by the hysteresis in the rubber represented by a loss angle of the material together with the peripheral speed and some parameters taken from the previously developed contact model. The temperature distribution is calculated in accordance with temperature and heat flow boundary conditions.     

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

In 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     

TRANSIENT BEHAVIOR OF NATURAL CONVECTION INSIDE A VERTICAL CYLINDER

The transient behavior for natural convection heat transfer within a vertical cylindrical enclosure was studied using the SIMPLER numerical algorithm. Initially, the temperature of the internal fluid was uniform. The unsteady state heat transfer was initiated by a temperature step change applied to all boundaries. The boundary temperature was then increased at the same rate as the center temperature until pseudosteady state was achieved. (Pseudosteady state is comparable to steady state convection for a fluid with uniform heat generation.) The transient heat transfer coefficient and the intensity of flow, along with the magnitude and the location of the minimum temperature, mean temperature, and maximum stream function are presented. Temperature and stream function contours are used to show the penetration of heat from the boundary and the development of the flow patterns. The local Nusselt number is also provided as supplementary information. A cylinder with HID = 1 was considered, with the Prandtl number of the fluid equal to 7 and a Rayleigh number of 10⁵. A characteristic length defined as 6 x (volume)/(surface area), which was found to produce the best regression results in our previous studies, was used in defining the Nusselt number and the Rayleigh number. The initial heat transfer rate was found to be correlated by

Nu =0.938 (Dimensionless Time) ^-0567

Oscillatory phenomena were found in support of the prediction made by Patterson and Imberger.     

基于最小二乘法预测的导热反问题求解

根据体表红外热像图获得体内异常热源信息可抽象为一个含有未知内热源的导热反问题,其求解过程需要对计算区域内温度场进行反复计算,对于复杂的三维物理模型反演过程耗时较长。采用粒子群算法用以反演未知参数,并结合最小二乘法对部分粒子位置对应的目标函数值进行预测。在反演过程中,对远离群体的粒子进行位置的重新分配,避免计算资源的浪费。分析不同预测系数对粒子搜索过程的影响,采用了线性递减的预测系数。数值验证结果表明：基于最小二乘法预测的粒子群算法能在保证反演精度的前提下减少导热问题计算次数,缩短反演所需时间。     

Computational Modelling on Heat Transfer Study of Molten Silicon During Multi-crystalline Silicon Growth Process for PV Applications

Multi-crystalline silicon is an important material with advantages of low-production cost and high conversion efficiency for photovoltaic solar cells. Directional solidifi cation has become the main technique for producing mc-Si ingots for solar cell applications. The study is performed in the framework of the incompressible Navier-Stokes equation in the Boussinesq approximation with convection-conduction equations. The computations are carried out in a two dimensional (2D) axisymmetric model by the finite- element method. The influence of the Reynolds numbers, total heat flux and velocity streamline pattern on the silicon melt was simulated and analyzed for various Rayleigh numbers between 10 to 10 ⁶ with the help of a numerical technique. The following key findings are presented in this paper. The velocity field value is increased above 0.02(m/s), heat flux value is increased to 10 ⁴(W/m ²), when the Rayleigh number is increased above 1000. Reynolds numbers are also studied in five parallel horizontal cross-sections of the melt silicon region for various Prandtl numbers at a critical Rayleigh number of 1000. Reynolds numbers are varied between 100 and 10 ⁵ for the Rayleigh numbers between 10 to 10 ⁶. Meanwhile, the melt has high fluctuation when the Prandtl number is increased above 0.01. The flow is converted from laminar to turbulence at a critical Rayleigh number 1000 and Prandtl number 0.01. These results provide important information for controlling the melt fluctuations during the solidification process which are used to increase the average grain size in growing silicon multicrystals and reduce the dislocation density.     

STEADY FILMWISE CONDENSATION ON A FINITE-SIZE HORIZONTAL DISK IN A HOMOGENEOUS POROUS LAYER

This study employs a boundary layer approximation to solve steady filmwise condensation in a homogeneous porous layer near a cold finite-size horizontal disk. Numerical solutions are obtained using the fourth-order Runge-Kutta scheme with a shooting method. A theoretical analysis is performed to predict the influence of suction effects on the rate of condensation heat transfer. The dimensionless average Nusselt number and the thickness of the dimensionless liquid film on the disk surface are examined at various Darcy numbers, Jakob numbers, Prandtl numbers, modified Rayleigh numbers, and suction parameters. Finally, an algebraic correlation function is presented for the average Nusselt number based on the obtained numerical results.     

STEADY FILMWISE CONDENSATION ON A FINITE-SIZE HORIZONTAL DISK IN A HOMOGENEOUS POROUS LAYER

This study employs a boundary layer approximation to solve steady filmwise condensation in a homogeneous porous layer near a cold finite-size horizontal disk. Numerical solutions are obtained using the fourth-order Runge-Kutta scheme with a shooting method. A theoretical analysis is performed to predict the influence of suction effects on the rate of condensation heat transfer. The dimensionless average Nusselt number and the thickness of the dimensionless liquid film on the disk surface are examined at various Darcy numbers, Jakob numbers, Prandtl numbers, modified Rayleigh numbers, and suction parameters. Finally, an algebraic correlation function is presented for the average Nusselt number based on the obtained numerical results.     

Finite element analysis of natural convection in a triangular enclosure: Effects of various thermal boundary conditions

The phenomena of natural convection in a right-angled triangular enclosure is studied numerically. A penalty finite element analysis with bi-quadratic elements is used for solving the Navier-Stokes and energy balance equations. The detailed study is carried out in two cases depending on various thermal boundary conditions:

(a)

Vertical wall is uniformly or linearly heated while inclined wall is cold isothermal.

(b)

Inclined wall is uniformly or linearly heated while vertical wall is cold isothermal.

In all the cases horizontal bottom wall is adiabatic and the aspect ratio of the lengths of base and height is considered to be one. The present numerical procedure adopted in this investigation yields consistent performance over a wide range of parameters of Rayleigh number Ra(10³?Ra?10⁵) and Prandtl number Pr(0.07?Pr?1000) in all the cases mentioned above. Numerical results are presented in terms of stream functions, temperature profiles and Nusselt numbers. It has been found that at low Rayleigh numbers (Ra?10⁴), the isotherms are almost parallel near the bottom portion of the triangular enclosure while at Ra=10⁵, the isotherms are more distorted. This is because the heat transfer is primarily due to conduction for lower values of Rayleigh number. As Rayleigh number increases, there is a change from conduction dominant region to convection dominant region, and the critical Rayleigh number corresponding to on-set of convection is obtained. It has been shown that the average Nusselt number for vertical wall is times that of the inclined wall as the bottom wall is adiabatic and that verifies the thermal equilibrium of the system for all case studies.     

COMPREHENSIVE, THEORETICALLY BASED, CORRELATING EQUATIONS FOR FREE CONVECTION FROM ISOTHERMAL SPHERES

Correlating equations are developed for the local and mean Nusselt number for free convection from an isothermal sphere as a function of the Rayleigh and Prandtl numbers. These expressions are based primarily on theoretical solutions for limiting cases, and hence are presumed to be more reliable than purely empirical correlations. The predictions of the proposed expressions are, however, validated by comparisons with prior experimental data. The expressions for the mean Nusselt number are shown to be applicable for all Ra and Pr. The expressions for the local Nusselt number are limited in applicability to the laminar boundary layer regime. The same equations are applicable to mass transfer and to combined heat and mass transfer in terms of the Sherwood, Schmidt and appropriately modified Rayleigh numbers.     

Laminar mixed convection in the thermal entrance region of horizontal rectangular channels with uniform heat input axially and uniform wall temperature circumferentially

This paper presents a numerical study on laminar mixed convection in the thermal entrance region of horizontal rectangular channels with uniform heat input axially and uniform wall temperature circumferentially. A relatively novel numerical method of solution is developed to obtain the developing velocity and temperature fields. The values of Prandtl number are 0.7 and 7.2, corresponding to air and water, respectively. The values of Rayleigh number are 0, 10⁴, 3 × 10⁴ and 10⁵. The channel aspect ratios considered are 0.2, 0.5, 1, 2 and 5. Variations in local friction factor ratios and local Nusselt numbers are presented. It is found that the circumferential boundary condition of uniform wall temperature significantly increases the value of local Nusselt number as compared to that found in earlier works under the boundary condition of uniform wall heat flux. But the boundary condition effect on the friction factor is shown to be comparatively minor. The asymptotic solutions at z → are compared to the existing numerical data with good agreement.     

Rayleigh–Benard convection in a chemically active gas in the chemical equilibrium state

The Rayleigh–Benard convection in a chemically active gas in the chemical equilibrium state is numerically studied in the Boussinesq approximation. A flat layer with isothermal horizontal boundaries free from shear stresses is considered. Thermodynamic parameters of the gas (hydrogen–oxygen mixture) are calculated by the previously proposed model of chemical equilibrium. It is shown that the allowance for recombination and dissociation processes leads to the emergence of an additional factor at the Rayleigh number. An expression for the growth rate of infinitesimal perturbations and a relation for the critical Rayleigh number as a function of temperature are derived. It is found that the neutral curves consist of the upper (instability due to heating from below) and lower (instability due to heating from above) branches. Results calculated for a nonlinear steady mode are reported.