Mixed convection of heat generating nanofluid in a lid-driven cavity with uniform and non-uniform heating of bottom wall

Steady state two-dimensional mixed convection in a lid-driven square cavity filled with Cu  –water nanofluid is investigated numerically in the presence of internal heat generation. In the present investigation, bottom wall is uniformly and non-uniformly heated while two vertical walls are fixed and they are thermally insulated. The top wall is moving from left to right at a constant speed. The governing equations are normalized and solved numerically with boundary conditions by finite volume approach using third order accurate upwinding scheme (deferred QUICK). Effects of the pertinent physical parameters are investigated in terms of the flow and temperature fields, as well as Nusselt number distributions. The presented results show that the solid volume fraction plays a significant role on the flow and thermal fields and the Nusselt number distributions for different flow configurations. It is found that Richardson number strongly affect the fluid flow and heat transfer in the cavity. For Ri<1$\mathit{Ri}<1$, the forced convection becomes dominant in the entire cavity, the natural convection relatively weak.     

Investigating the effect of various nanoparticle and base liquid types on the nanofluids heat and fluid flow in a microchannel

In this paper, heat transfer and pressure drop of different nanofluid types in a two-dimensional microchannel is investigated numerically. To do this, an Eulerian–Eulerian two-phase model is used for nanofluid simulation and the governing equations are solved using a finite volume method. Nine different nanoparticles and three different base liquid types (water, ethylene glycol and engine oil) are considered. Heat transfer and pressure drop of different nanofluid types are compared at Re = 100 and 1% volume concentration for different nanoparticles and at constant inlet velocity for different base liquids. Numerical results show an almost equal pressure drop for all the nanoparticles dispersed in water, while, the heat transfer coefficient is highest for water–diamond and is the lowest for water–SiO₂ nanofluids. Also, the pressure drop for water-based nanofluid is very lower than the others and the heat transfer coefficient is the highest for water-based nanofluids.     

Numerical study of dynamic thermal conductivity of nanofluid in the forced convective heat transfer

In this work, forced convective heat transfer of nanofluid in the developing laminar flow (entrance region) in a circular tube is considered. The nanofluid thermal conductivity, as an important parameter, is considered as two parts: static and dynamic part. Simulated results show that the dynamic part of nanofluid thermal conductivity due to the Brownian motion has a minor effect on the heat transfer coefficients, on the other hand, static part of thermal conductivity including nanolayer around nanoparticle has an important role in heat transfer.     

Optimal paths for minimizing entransy dissipation during heat transfer processes with generalized radiative heat transfer law

A common of finite-time heat transfer processes between high- and low-temperature sides with generalized radiative heat transfer law [q ∝ Δ(Tⁿ)] is studied in this paper. In general, the minimization of entropy generation in heat transfer processes is taken as the optimization objective. A new physical quantity, entransy, has been identified as a basis for optimizing heat transfer processes in terms of the analogy between heat and electrical conduction recently. Heat transfer analyses show that the entransy of an object describes its heat transfer ability, as the electrical energy in a capacitor describes its charge transfer ability. Entransy dissipation occurs during heat transfer processes, as a measure of the heat transfer irreversibility with the dissipation related thermal resistance. Under the condition of fixed heat load, the optimal configurations of hot and cold fluid temperatures for minimizing entransy dissipation are derived by using optimal control theory. The condition corresponding to the minimum entransy dissipation strategy with Newtonian heat transfer law (n = 1) is that corresponding to a constant heat flux rate, while the condition corresponding to the minimum entransy dissipation strategy with the linear phenomenological heat transfer law (n = −1) is that corresponding to a constant ratio of hot to cold fluid temperatures. Numerical examples for special cases with Newtonian, linear phenomenological and radiative heat transfer law (n = 4) are provided, and the obtained results are also compared with the conventional strategies of constant heat flux rate and constant hot fluid (reservoir) temperature operations and optimal strategies for minimizing entropy generation. Moreover, the effects of heat load changes on the optimal hot and fluid temperature configurations are also analyzed.     

A new family of unsteady boundary layers over a stretching surface

In this paper, a new family of unsteady boundary layers over a stretching flat surface was proposed and studied. This new class of unsteady boundary layers involves the flows over a constant speed stretching surface from a slot, and the slot is moving at a certain speed. Depending on the slot moving parameter, the flow can be treated as a stretching sheet problem or a shrinking sheet problem. Both the momentum and thermal boundary layers were studied. Under special conditions, the solutions reduce to the unsteady Rayleigh problem and the steady Sakiadis stretching sheet problem. Solutions only exist for a certain range of the slot moving parameter, α. Two solutions are found for −53.55° < α < −45°. There are also two solution branches for the thermal boundary layers at any given Prandtl number in this range. Compared with the upper solution branch, the lower solution branch leads to simultaneous reduction in wall drag and heat transfer rate. The results also show that the motion of the slot greatly affects the wall drag and heat transfer characteristics near the wall and the temperature and velocity distributions in the fluids.     

Convective thermal conditions and the thermophoretic effect's impacts on a hybrid nanofluid over a moving thin needle

This research focuses on the heat source/sink, chemical reaction, and thermophoretic particle deposition in the influence of hybrid nanofluid over a moving thin needle subjected to a magnetic field. Using the appropriate transformations, a group of nonlinear partial differential equations may be converted to ordinary differential equations. Additionally, with the aid of computational software, the RKF-45 approach is used for the numerical assessment, as well as the shooting operation. It should be mentioned that the results' approval demonstrates a strong association with the previous findings. The resulting graphs mainly explain the fundamental characteristics of hybrid nanofluids and nanofluids, as well as the consequences of different restrictions. An increase in needle size enhances the velocity profile, temperature profile, and concentration profile. The radial and axial velocity profiles are reduced when the magnetic constraint is increased, whereas the thermal and concentration patterns are reversed. Improved heat source-sink as well as Biot number values will enhance the thermal profile. The concentration profiles will decrease due to reaction rate restrictions and thermophoretic limits. The inclusion of solid volume fraction reduces surface drag forces while increasing the rate of mass transfer. In most circumstances, hybrid nanofluid plays a prominent role than nanofluid.     

Lie symmetry group analysis of magnetic field effects on free convective flow of a nanofluid over a semi-infinite stretching sheet

We investigate the steady two-dimensional flow of an incompressible water based nanofluid over a linearly semi-infinite stretching sheet in the presence of magnetic field numerically. The basic boundary layer equations for momentum and heat transfer are non-linear partial differential equations. Lie symmetry group transformations are used to convert the boundary layer equations into non-linear ordinary differential equations. The dimensionless governing equations for this investigation are solved numerically using Nachtsheim–Swigert shooting iteration technique together with fourth order Runge–Kutta integration scheme. Effects of the nanoparticle volume fraction ϕ, magnetic parameter M, Prandtl number Pr on the velocity and the temperature profiles are presented graphically and examined for different metallic and non-metallic nanoparticles. The skin friction coefficient and the local Nusselt number are also discussed for different nanoparticles.     

Application of a lumped model to solids with linearly temperature-dependent thermal conductivity

The application of a simple lumped model to unsteady cooling (or heating) processes in solids involving heat convection is limited by the value of the Biot number, Bi. For Bi < 0.1, assuming constant thermal properties, the lumped model approximates the exact solutions with only a small error. In this paper we study the lumped model for a 1-D rectangular solid, when thermal conductivity depends linearly on temperature, a type of dependence very common in metals and alloys at a wide range of working temperatures. From the study, new limits for the Biot are deduced as a function of a sole dimensionless parameter defined from the extreme values of thermal conductivity. The Biot limits depend on the thermal process (heating or cooling) and on the type of temperature dependence—positive or negative.     

Numerical analysis of the thermal behaviour of a shell-and-tube heat storage unit using phase change materials

This work presents a numerical study of a latent heat storage unit (LHSU) consisting of a shell-and-tube. The shell space is filled with two phase change materials (PCMs), P116 and n-octadecane, with different melting temperatures (50 °C and 27.7 °C, respectively). A heat transfer fluid (HTF: water) flows by forced convection through the inner tube, and transfers the heat to PCMs. In order to compare the thermal performances of the latent heat storage unit using two phase change materials (LHSU2) and a single PCM (LHSU1), a mathematical model based on the conservation energy equations was developed and validated with experimental data. Several numerical investigations were conducted in order to examine the impact of the key parameters: the HTF inlet temperature (ranges from 50 to 60 °C), the mass flow rate of the HTF and the proportion mass of PCMs, on the thermal performances of the latent heat storage units using two PCMs and a single PCM, during charging process (melting). This parametric study provides guidelines for system thermal performance and design optimization.     

Computation of melting with natural convection inside a rectangular enclosure heated by discrete protruding heat sources

This paper presents the results of a numerical investigation of the heat transfer by natural convection during the melting of a phase change material (PCM, n-eicosane with melting point of 36 °C) contained in a rectangular enclosure. This latest is heated by three discrete protruding heat sources (simulating electronic components) placed on one of its vertical walls. The power generated by heat sources is dissipated in PCM. The advantage of using this cooling scheme is that the PCMs are able to absorb high amount of heat generated by the heat sources, without acting the fan during the charging process (melting of the PCM). The thermal behavior and thermal performance of the proposed PCM based-heat sink are numerically investigated by developing a mathematical model based on the mass, momentum and energy conservation equations. The obtained numerical results show the impact of various key parameters on the cooling capacity of the PCM-based heat sink. Correlations encompassing a wide range of parameters were developed in terms of the dimensionless secured operating time (time required by one of the electronic components before reaching its critical temperature, T_cr ∼ 75 °C) and the corresponding liquid fraction, using the asymptotic computational fluid dynamics (ACFD) technique.     

Inverse heat conduction problem in a thin circular plate and its thermal deflection

An inverse problem of transient heat conduction in a thin finite circular plate with the given temperature distribution on the interior surface of a thin circular plate being a function of both time and position has been solved with the help of integral transform technique and also determine the thermal deflection on the outer curved surface of a thin circular plate defined as 0 ? r ? a, 0 ? z ? h. The results, obtained in the series form in terms of Bessel’s functions, are illustrated numerically.     

Thermal dispersion driven by the spontaneous imbibition process

In this work, we have theoretically analyzed the thermal dispersion process under the influence of the spontaneous imbibition of a liquid trapped in a capillary element, considering the presence of a uniform temperature gradient. The capillary element is represented by a porous medium which is initially found at temperature T₀ and pressure P₀. Suddenly, the lower part of the porous medium touches a liquid reservoir at temperature T_l and pressure P₀. This contact between both phases, in turn causes spontaneously the imbibition process. Using a one-dimensional formulation of the average conservation laws, we derive the corresponding nondimensional momentum and energy equations. The numerical solutions permit us to evaluate the position and velocity of the imbibition front as well as the temperature profiles and Nusselt numbers. The above results are shown by taking into account the influence of three dimensionless parameters: the ratio of the characteristic thermal time to the characteristic imbibition time, β, the ratio of the hydrostatic head of the imbibed liquid to the characteristic pressure difference for the imbibition front, α, and the ratio of the dispersive thermal diffusivity to the effective thermal diffusivity of the medium, Ω. The predictions show that temperature profiles and the heat transfer process are strongly dependent on thermal dispersion effects, indicating a clear deviation in comparison with the case of Ω = 0 that represents the absence of the thermal dispersion.     

A spatial-fractional thermal transport model for nanofluid in porous media

In this paper, a spatial fractional-order thermal transport equation with the Caputo derivative is proposed to describe convective heat transfer of nanofluids within disordered porous media in boundary layer flow. This equation arises naturally when the effect of anomalous migration of nanoparticles on heat transfer is considered. The numerical results show that local Nusselt numbers of four different kinds of nanofluids are all inversely proportional to the fractional derivative exponent β. Based on this finding, it is concluded that the anomalous diffusion of nanoparticles improves the convective heat transfer of nanofluids and the space fractional thermal transport equation may serve as a candidate model for studying nanofluids. Additionally, the effects of other involved physical parameters on temperature distribution and Nusselt number are presented and analyzed.     

Numerical study of melting in an enclosure with discrete protruding heat sources

In order to explore the capability of a solid–liquid phase change material (PCM) for cooling electronic or heat storage applications, melting of a PCM in a vertical rectangular enclosure was studied. Three protruding generating heat sources are attached on one of the vertical walls of the enclosure, and generating heat at a constant and uniform volumetric rate. The horizontal walls are adiabatic. The power generated in heat sources is dissipated in PCM (n-eicosane with the melting temperature, T_m = 36 °C) that filled the rectangular enclosure. The advantage of using PCM is that it is able to absorb high amount of heat generated by heat sources due to its relatively high energy density. To investigate the thermal behaviour and thermal performance of the proposed system, a mathematical model based on the mass, momentum and energy conservation equations was developed. The governing equations are next discretised using a control volume approach in a staggered mesh and a pressure correction equation method is employed for the pressure–velocity coupling. The PCM energy equation is solved using the enthalpy method. The solid regions (wall and heat sources) are treated as fluid regions with infinite viscosity and the thermal coupling between solid and fluid regions is taken into account using the harmonic mean of the thermal conductivity method. The dimensionless independent parameters that govern the thermal behaviour of the system were next identified. After validating the proposed mathematical model against experimental data, a numerical investigation was next conducted in order to examine the thermal behaviour of the system by analyzing the flow structure and the heat transfer during the melting process, for a given values of governing parameters.     

A Variational Approach to Discontinuous Problems with Critical Sobolev Exponents

We employ variational techniques to study the existence and multiplicity of positive solutions of semilinear equations of the form − Δu = λh(x)H(u − a)u^q + u^2* − 1 in R^N, where λ, a > 0 are parameters, h(x) is both nonnegative and integrable on R^N, H is the Heaviside function, 2* is the critical Sobolev exponent, and 0 ≤ q < 2* − 1. We obtain existence, multiplicity and regularity of solutions by distinguishing the cases 0 ≤ q ≤ 1 and 1 < q < 2* − 1.     

Critical Exponents of Quasilinear Parabolic Equations

In this paper we study the critical exponents of the Cauchy problem in Rⁿ of the quasilinear singular parabolic equations: u_t = div(|∇u|^m − 1∇u) + t^s|x|^σu^p, with non-negative initial data. Here s ≥ 0, (n − 1)/(n + 1) < m < 1, p > 1 and σ > n(1 − m) − (1 + m + 2s). We prove that p_c ≡ m + (1 + m + 2s + σ)/n > 1 is the critical exponent. That is, if 1 < p ≤ p_c then every non-trivial solution blows up in finite time, but for p > p_c, a small positive global solution exists.     

Spatial-resolution,lumped-capacitance thermal model for cylindrical Li-ion batteries under high Biot number conditions

The time-efficient yet accurate thermal modeling of the battery cells for electric and hybrid electric vehicles is essential improving the performance, safety, and lifetime of the battery system. This paper presents a spatial-resolution, lumped-capacitance (LC) thermal model for cylindrical battery cells under high Biot number (Bi ? 1) conditions where the classical LC thermal model is generally inapplicable because of a significant temperature variation in the cell volume. The spatial-resolution LC model was formulated using zero- and first-order Hermite integral approximations. For model validation, a one-dimensional, transient analytical (exact) solution using Green functions was obtained for a cylindrical Li-ion battery cell with uniform volumetric battery heat generation of Joule and entropic heating under convective cooling boundary conditions. It was found from the comparison of the results that the spatial-resolution LC thermal model can accurately and quickly predicts the cell temperatures (core, skin and area-averaged) under various dynamic battery duty cycles even for high Biot numbers due to highly convective conditions such as liquid cooling.     

Numerical investigation of natural convection in a rectangular enclosure due to partial heating and cooling at vertical walls

A comprehensive numerical investigation on the natural convection in a rectangular enclosure is presented. The flow is induced due to the constant partial heating at lower half of the left vertical wall and partial cooling at upper half of the right vertical wall along with rest walls are adiabatic. In this investigation the Special attention is given to understand the effect of aspect ratio and heat source intensity i.e. Rayleigh number, Ra, on the fluid flow configuration as well as on the local and average heat transfer rates. The range of Rayleigh (Ra) and aspect ratio (A) is taken [10³, 10⁶] and [0.5, 4] respectively. The results are presented in terms of stream function (ψ), temperature (θ) and heat transfer rates (local Nusselt numbers Nu_L, and average Nusselt numbers Nu). The numerical experiments show that increasing of Ra implies the enhancement of thermal buoyancy force, which in turn increases the thermal convection in the cavity. As a result, the local as well as average heat transfer rate is expected to increase. The local transfer rate (Nu_L) is increases in the small region near the left vertical wall of the left wall of the cavity and after that it is decreases in the middle portion of heated region. And, it start to increase near to the middle point of left wall. It is also observed that the local heat transfer is increases as increases the aspect ratio. The average heat transfer rate (Nu) is increases as the aspect ratio A increases from 0.5 to 1 and beyond that it is decreases smoothly. It is also found that the heat transfer rate attains its maximum value at aspect ratio one.     

On the Hermitian positive definite solutions of nonlinear matrix equation X + A∗XA = Q

Nonlinear matrix equation X^s + A∗X^−tA = Q, where A, Q are n × n complex matrices with Q Hermitian positive definite, has widely applied background. In this paper, we consider the Hermitian positive definite solutions of this matrix equation with two cases: s ? 1, 0 < t ? 1 and 0 < s ? 1, t ? 1. We derive necessary conditions and sufficient conditions for the existence of Hermitian positive definite solutions for the matrix equation and obtain some properties of the solutions. We also propose iterative methods for obtaining the extremal Hermitian positive definite solution of the matrix equation. Finally, we give some numerical examples to show the efficiency of the proposed iterative methods.     

Flow and heat transfer of a nanofluid over a nonlinearly stretching sheet: A numerical study

Steady, laminar boundary fluid flow which results from the non-linear stretching of a flat surface in a nanofluid has been investigated numerically. The model used for the nanofluid incorporates the effects of Brownian motion and thermophoresis. The resulting non-linear governing equations with associated boundary conditions are solved using variational finite element method (FEM) with a local non-similar transformation. The influence of Brownian motion number (Nb), thermophoresis number (Nt), stretching parameter (n) and Lewis number (Le) on the temperature and nanoparticle concentration profiles are shown graphically. The impact of physical parameters on rate of heat transfer (−θ′(0)) and mass transfer (−?′(0)) is shown in tabulated form. Some of results have also been compared with explicit finite difference method (FDM). Excellent validation of the present numerical results has been achieved with the earlier nonlinearly stretching sheet problem of Cortell [16] for local Nusselt number without taking the effect of Brownian motion and thermophoresis.