Mathematical modelling of thermal conductivity for nanofluid considering interfacial nano-layer

Maxwell’s classical model for predicting effective thermal conductivity of colloidal solution predicts the thermal conductivity of nanofluids quite satisfactorily. However, Maxwell’s model does not consider the effect of interfacial layer, Brownian motion of nano-particle and nanoparticle aggregation. In this paper, the effect of interfacial layer on thermal conductivity is considered. A simple expression has been derived to determine thermal conductivity of nanofluid considering interfacial layer formed on the nano particles. The thermal conductivity of the interfacial layer has been precisely determined and results are found to be closer to the experimental values, hence, further improving the results of classical Maxwell model.     

Effective Thermal Conductivity Modelling for Closed-Cell Porous Media with Analytical Shape Factors

This paper presents a fully analytical model for the effective thermal conductivity of two-phase porous media with two-/three-dimensional closed cells, applicable to honeycombs and closed-cell foams. The present model combines an existing analytical expression derived based on the Laplace heat conduction equation with an analytical shape factor which corrects the deviation caused from a non-circular (or non-spherical) pore inclusion. Results demonstrate the validity of the present model capable of analytically estimating the effective thermal conductivity of closed-cell porous media. The simple yet accurate model provides the physical mechanisms of how effective thermal conductivity depends upon the shape of pores.     

Correlation equations for optimum design of annular fins with temperature dependent thermal conductivity

Correlation equations for optimum design of annular fins with temperature-dependent thermal conductivity are obtained in the present work. The nonlinear fin equation which is associated with variable thermal conductivity condition is solved by Adomian decomposition method that provides an analytical solution in the form of an infinite power series. The optimum radii ratio of an annular fin which maximizes the heat transfer rate has been found as a function of Biot number and the fin volume for a given thermal conductivity parameter describing the variation of the thermal conductivity. The fin volume is fixed to obtain the dimensionless geometrical parameters of the fin with maximum heat transfer rate. The data from the present solutions is correlated for a suitable range of Biot number and the fin volume. The simple correlation equations presented in this work can assist for thermal design engineers for optimum design of annular fins with temperature-dependent thermal conductivity.     

Effective thermal conductivity of dispersed materials

Measurements have been made by the comparison method for the effective thermal conductivity of dispersed materials which consist of substances with different thermal conductivities. The applicability of existing predicting formulae is discussed in detail as comparing their predicted values with the present data. A new predicting formula is proposed through analyzing the experimental data, the numerical results, and also the data obtained with the electrolytic-bath. It is found that the proposed formula has a wider range of applicability than that of previously reported ones.     

Creeping-flow rotation of a slip spheroid about its axis of revolution

The problem of rotation of a rigid spheroidal particle about its axis of revolution in a viscous fluid is studied analytically and numerically in the steady limit of negligible Reynolds number. The fluid is allowed to slip at the surface of the particle. The general solution for the fluid velocity in prolate and oblate spheroidal coordinates can be expressed in an infinite-series form of separation of variables. The slip boundary condition on the surface of the rotating particle is applied to this general solution to determine the unknown coefficients of the leading orders, which can be numerical results obtained from a boundary collocation method or explicit formulas derived analytically. The torque exerted on the spheroidal particle by the fluid is evaluated for various values of the slip parameter and aspect ratio of the particle. The agreement between our hydrodynamic torque results and the available analytical solutions in the limiting cases is good. It is found that the torque exerted on the rotating spheroid normalized by that on a sphere with radius equal to the equatorial radius of the spheroid increases monotonically with an increase in the axial-to-radial aspect ratio for a no-slip or finite-slip spheroid and vanishes for a perfectly slip spheroid. For a spheroid with a specified aspect ratio, the torque is a monotonically decreasing function of the slip capability of the particle.     

The evolution of the brabender farinograph

The importance of the consistency of wheat flour dough in baking is defined. This is normally determined with the aid of a recording dough mixer.Such a mixer was first patented byHogarth, a Scottish miller in 1889 but proved inadequate. About 40 years laterHankóczy a Hungarian cereal chemist initiated the line of recording dough mixers which are used today.His andBrabender's work are reviewed which led from a factory mixer fitted with an ammeter, via the early torque recording laboratory instruments to the modern Farinograph. The basic unit is described in some detail.Slides of the various attachments are shown which allow application to grinding and high speed mixing both on a batch and continuous basis.The use of the system for the testing of cereal products, fats, rubber, filler and plastics is described.The theoretical significance of the results obtained is briefly discussed. The lecture is illustrated by 14 slides.Paper presented at the Conference on Experimental Rheology, University of Bradford, April 17–19, 1968. — Original paper unpublished.     

Effective thermal conductivity of heterogeneous multi-component materials: an SPH implementation

Modeling heat transfer and fluid flow in materials with complicated micro-structures is a major challenge to numerical methods due to their multiscale and multiphysics nature. A relatively novel numerical technique—the meshless smoothed particle hydrodynamics (SPH) method has the potential of making a significant contribution to this research field. In the present SPH modeling effort, a 2D modeling system is devised for the prediction of the effective thermal conductivity in heterogeneous materials containing two or three different components. The microscopic component configuration inside the materials is constructed in the SPH methodology by randomly assigning particles as a certain component to meet the required macroscopic composition. For heterogeneous two-component materials, the effective thermal conductivity predicted by the modified effective medium theory model with the so-called “flexible” factor f equal to 4.5 agrees well with the SPH data. On the basis of a simple “step-process” concept, the effective thermal conductivity of a heterogeneous multi-component material can be derived from the corresponding “degenerate” materials which consist of fewer components.     

Effect of an inhomogeneous interphase zone on the bulk modulus and conductivity of a particulate composite

A model is presented of a particulate composite containing spherical inclusions, each of which are surrounded by a localized region in which the elastic moduli vary smoothly with radius. This region may represent an interphase zone in a composite, or the transition zone around an aggregate particle in concrete, for example. An exact solution is derived for the displacements and stresses around a single inclusion in an infinite matrix, subjected to a far-field hydrostatic compression, and is then used to derive an approximate expression for the effective bulk modulus of a material containing a random dispersion of these inclusions. The analogous conductivity (thermal, electrical, etc.) problem is then discussed, and it is shown that the expression for the normalized effective conductivity corresponds exactly to that for the normalized effective bulk modulus, if the Poisson ratios of both phases are set to zero.     

Effective thermal conductivity of foam concretes: Homogenization schemes vs experimental data and FEM simulations

This paper focuses on the prediction of thermal conductivity of foam concretes. Various analytical forms for their effective thermal conductivity according to their porosity p (air voids volume) have been first derived based on five well-known Mean-Field Homogenization (MFH) schemes. These predictions were found to be very close for low porosities but move away gradually with increasing porosity. Thus, in order to determine the best homogenization scheme predicting the effective thermal conductivity of foam concretes, MFH predictions were confronted with experimental data obtained on nineteen foam concretes and also with numerical results obtained from 3d Finite Element Method (FEM) simulations conducted on an idealized foam concrete. These comparisons have shown that the normalized effective thermal conductivity of foam concrete is closely framed by the power law (1- p)^3/2 given by the Differential scheme and by the hyperbolic law (1- p)/(1+ p/2) given by the Mori-Tanaka scheme.     

Experimental and Numerical Modeling of the Thermal Conductivity of High-Viscosity Hydrocarbon Systems

A mathematical model is developed to numerically predict the heating of heavy hydrocarbon systems. A comparative analysis of numerical and experimental data is performed. It is found that the thermal conductivity of a hydrocarbon system under study heated from an initial temperature of 24°C to 100°C increases by a factor of 40 and, with allowance for free convection, an additional substantial (up to 16 times) increase in heat transfer due to enhanced effective thermal conductivity is observed. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 46, No. 6, pp. 96–102, November–December, 2005.     

A Geometrical Pore Model for Estimating the Microscopical Pore Geometry of Soil with Infiltration Measurements

This paper presents a simple geometrical pore model designed to relate characteristic pore radii of the porous network of soils with macroscopic infiltration parameters. The model composed of a stack of spherical hollow elements is described with two radii values: the pore access radius and the actual pore radius. The model was compared to cylindrical pore models and its mathematical consistency was assessed. Soil sorptivity S and the second parameter A of the Philip infiltration equation (1957), have been determined by numerically simulated infiltration. A diagram and an empirical relation have been set in order to relate the pore access and pore radii to the infiltration parameters S and A. The consistency of the model was validated by comparing the predicted sorptivity and hydraulic conductivity values, with the widely used unsaturated soil hydraulic functions (van Genuchten, 1980). The model showed good agreement with experimental infiltration data, and it is therefore concluded that the use of a model with two radii improves the relation between microscopic pore size and macroscopic infiltration parameters.     

A prediction model for the effective thermal conductivity of nanofluids considering agglomeration and the radial distribution function of nanoparticles

Conventional heat transfer fluids usually have low thermal conductivity, limiting their efficiency in many applications. Many experiments have shown that adding nanosize solid particles to conventional fluids can greatly enhance their thermal conductivity. To explain this anomalous phenomenon, many theoretical investigations have been conducted in recent years. Some of this research has indicated that the particle agglomeration effect that commonly occurs in nanofluids should play an important role in such enhancement of the thermal conductivity, while some have shown that the enhancement of the effective thermal conductivity might be accounted for by the structure of nanofluids, which can be described using the radial distribution function of particles. However, theoretical predictions from these studies are not in very good agreement with experimental results. This paper proposes a prediction model for the effective thermal conductivity of nanofluids, considering both the agglomeration effect and the radial distribution function of nanoparticles. The resulting theoretical predictions for several sets of nanofluids are highly consistent with experimental data.     

Finite element analysis of the axially symmetric motion of an incompressible viscous fluid in a spherical annulus

This paper presents results obtained by employing a modified Galerkin finite element method to analyse the steady state flow of a fluid contained between two concentric, rotating spheres. The spheres are assumed to be rigid and the cavity region between the spheres is filled with an incompressible, viscous, Newtonian fluid. The inner sphere is constrained to rotate about a vertical axis with a prescribed angular velocity, while the outer sphere is fixed. Results for the circumferential function Ω, streamfunction ψ, vorticity function ζ and inner boundary torque T₁ are presented for Reynolds numbers Re ? 2000 and radius ratios 0.1 ? α ? 0.9. The method proved effective for obtaining results for a wide range of radius ratios (0.1 ? α ? 0.9) and Reynolds numbers (0 ? Re ? 2000). Previous investigators who employed the finite difference method experienced difficulties in obtaining results for cases with radius ratios α ? 0.2, except for small Reynolds numbers (Re ? 100). Results for Ω, Ψ, ζ and T₁ obtained in this study for radius ratios 0.8 ≤ α ≤ 0.9 verified the development of Taylor vortices reported by other investigators. The research indicates that the method may be useful for analysing other non-linear fluid flow problems.     

A Fractal Approach to Model Soil Structure and to Calculate Thermal Conductivity of Soils

Heat transport in soils depends on the spatial arrangement of solids, ice, air and water. In this study, we present a modified fractal approach to model the pore structure of soils and to describe its influence on the thermal conductivity. Three different fractal generators were sequentially applied to characterize a wide range of particle- and pore-size distributions. The given porosity and particle-size distribution of a clay, clay loam, silt loam and loamy sand were successfully modeled. The thermal conductivity of the fractal soil model was calculated using a network of resistors. We applied a renormalization approach to include the effects of smaller scale structures. The predictions were compared with the empirical Johansen' model (Johansen, 1975), that postulates a simple linear relationship between ice content and thermal conductivity. For high ice-saturated conditions, the calculated thermal conductivity agrees well with the empirical model. To describe partial ice saturation, we assumed that some pores were coated by ice films enclosing the air-filled center. In addition, we introduced a reduced heat exchange coefficient of the particles for unsaturated conditions. The ice-saturated and -unsaturated thermal conductivity calculated with this approach was very similar to that estimated by the empirical model. The variation of the thermal conductivities for different spatial arrangements of pores and particles in the prefractals were determined. Extreme values deviate more than 50% from the mean values.     

Comparison of geometry dependent resistance models with conventional models for estimation of effective thermal conductivity of two-phase materials

The geometry dependent resistance models are used to estimate the effective thermal conductivity of two-phase materials based on the unit cell approach. The algebraic equations are derived based on isotherm approach for various geometries. The effective thermal conductivity of the above models are found and compared with experimental data with a minimum and maximum deviation of ±3.976 and ±19.55%, respectively. The present models are good agreement with experimental results.     

Measurement of temperature-dependent thermal conductivity and viscosity of TiO2-water nanofluids

Nanofluid is an innovative heat transfer fluid with superior potential for enhancing the heat transfer performance of conventional fluids. Many attempts have been made to investigate its thermal conductivity and viscosity, which are important thermophysical properties. No definitive agreements have emerged, however, about these properties. This article reports the thermal conductivity and dynamic viscosity of nanofluids experimentally. TiO₂ nanoparticles dispersed in water with volume concentration of 0.2–2 vol.% are used in the present study. A transient hot-wire apparatus is used for measuring the thermal conductivity of nanofluids whereas the Bohlin rotational rheometer (Malvern Instrument) is used to measure the viscosity of nanofluids. The data are collected for temperatures ranging from 15 °C to 35 °C. The results show that the measured viscosity and thermal conductivity of nanofluids increased as the particle concentrations increased and are higher than the values of the base liquids. Furthermore, thermal conductivity of nanofluids increased with increasing nanofluid temperatures and, conversely, the viscosity of nanofluids decreased with increasing temperature of nanofluids. Moreover, the measured thermal conductivity and viscosity of nanofluids are quite different from the predicted values from the existing correlations and the data reported by other researchers. Finally, new thermophysical correlations are proposed for predicting the thermal conductivity and viscosity of nanofluids.     

Heat- and mass-transport in aqueous silica nanofluids

Using the transient hot wire and pulsed field gradient nuclear magnetic resonance methods we determined the thermal conductivity and the solvent self-diffusion coefficient (SDC) in aqueous suspensions of quasi-monodisperse spherical silica nanoparticles. The thermal conductivity was found to increase at higher volume fraction of nanoparticles in accordance with the effective medium theory albeit with a smaller slope. On the other hand, the SDC was found to decrease with nanoparticle volume fraction faster than predicted by the effective medium theory. These deviations can be explained by the presence of an interfacial heat-transfer resistance and water retention by the nanoparticles, respectively. We found no evidence for anomalous enhancement in the transport properties of nanofluids reported earlier by other groups.     

Thermal Analysis of Natural Convection Porous Fins

This work introduces a simple method of analysis to study the performance of porous fins in a natural convection environment. The method is based on using energy balance and Darcy’s model to formulate the heat transfer equation. The thermal performance of porous fins is then studied for three types of fins: long fin, finite-length fin with insulated tip and a finite-length fin with tip exposed to a known convection coefficient. It is found from the analysis that the effect of different design and operating parameters such as: Ra number, Da number, thermal conductivity ratio, Kr and length thickness ratio on the temperature distribution along the fin is grouped into one newly defined parameter called S_H. The effect of the variation of S_H on the porous fin thermal performance is established. The effect of varying the fin length and thermal conductivity ratio on the heat transfer rate from the fin is investigated and compared with that for a solid fin at certain conditions. It is found that the heat transfer rate from porous fin could exceed that of a solid fin. It is also found that increasing the fin length and effective thermal conductivity enhances the heat transfer from the fin up certain limit, where a further increase in these parameters adds no improvement to the fin performance. On Leave from Jordan University of Science and Technology, Irbid-Jordan     

空心材料导热性能预测研究

本文建立了一种预测空心材料导热性的方法。研究了空心材料的导热性。用柱形空心材料分析了体分比和孔洞的排列方式对整体材料导热性的影响，用圆柱形、方柱形空心材料和含裂纹材料，分析了空心形状对材料导热性的影响。同他人的实验结果和某些现有的理论模型比较表明，本文方法是有效的。本文的结果能够很好地解释实验结果。     

Effect of temperature on the effective thermal conductivity of n-tetradecane-based nanofluids containing copper nanoparticles

Nanofluids were prepared by dispersing Cu nanoparticles(~20nm) in n-tetradecane by a two-step method.The effective thermal conductivity was measured for various nanoparticle volume fractions(0.0001-0.02) and temperatures(306.22-452.66 K).The experimental data compares well with the Jang and Choi model.The thermal conductivity enhancement was lower above 391.06 K than for that between306.22 and 360.77 K.The interfacial thermal resistance increased with increasing temperature.The effective thermal conductivity enhancement was greater than that obtained with a more viscous fluid as the base media at 452.66 K because of nanoconvection induced by nanoparticle Brownian motion at high temperature.