Flux Boundary Condition Research Articles

Thermal entry flow problem for Rabinowitsch fluid subject to circular tube and flat channel with uniform heat flux boundary conditions

The heat exchangers, chemical reactors, cooling/heating systems and haemodialysis are specific areas due to which Graetz thermal problem is considered as fundamental flow in the duct along with heat and mass transfer aspects. Owing to its importance we examine the thermal entry flow problem for Rabinowitsch liquid in the pipe and channel ducts along with heat transfer aspects. The heat equation with uniform heat flux conditions is tackled via the superposition principle and separation of variable approach due to the involvement of non-homogeneous boundary conditions. The transformed boundary value problem is solved with the MATLAB algorithm for the calculation of eigenvalues and related eigenfunctions. The special cases for simplified Phan-Thien–Tanner (SPTT) fluid and finite extendable non-linear elastic peterline (FENE-P) fluids are also discussed. For SPTT and FENE-P models the controlling parameter can only take positive values and hence these models can only predict shear-thinning effects. The graphical illustrations of fully developed temperature, local and mean Nusselt numbers are displayed through the main effect brought by the controlling parameter.

An explicit boundary condition-enforced immersed boundary-reconstructed thermal lattice Boltzmann flux solver for thermal–fluid–structure interaction problems with heat flux boundary conditions

For the thermal–fluid–structure interaction (TFSI) problems with moving boundaries, the original implicit boundary condition-enforced immersed boundary method (IBM) by Wang et al. [1] has to generate a large correlation matrix and compute its inversion at each iteration, implying that the virtual memory requirement and computational cost would grow exponentially with the number of Lagrangian points. In this work, we proposed an efficient explicit boundary condition-enforced immersed boundary method for Neumann boundary condition (NBC) to achieve high computational efficiency and maintain similar accuracy as the original implicit IBM [1], which circumvents the needs to assemble a large correlation matrix and inverse it in the original implicit IBM [1] through second-order approximation based on the error analysis using Taylor series expansion. Most importantly, the proposed explicit IBM can efficiently solve practical physics problems with a tremendous amount of Lagrangian points involving Neumann boundary condition. The comparisons of the virtual memory and computational cost between the explicit and implicit IBMs demonstrate that the explicit boundary condition-enforced IBM is not only computational efficient, but also has memory saving performance. The proposed explicit IBM integrated with the reconstructed thermal lattice Boltzmann flux solver (RTLBFS) is validated with some classical benchmarks, and the results show that the proposed explicit IBM can successfully resolve TFSI problems with Neumann boundary condition.

Figures-of-Merit Analysis Using the Thermophysical Properties of Water and Ethylene Glycol Based Reduced Graphene Oxide/Nanodiamond Hybrid Nanofluids

The water and ethylene glycol based stable rGO/nanodiamond hybrid nanofluids were prepared and used for thermophysical properties analysis. The thermophysical properties were measured experimentally at various particle loadings from 0.2% to 1.0% and various temperatures ranging from 20 to 60 °C. From the measured thermophysical properties the figures-of-merit were analyzed by assuming constant heat flux and turbulent boundary conditions by using different models, and also by assuming the hybrid nanofluids flows through a tube. Results indicate that the thermal conductivity of water and ethylene glycol based hybrid nanofluids at Φ = 1.0% vol. is enhanced about 27.87% and 18.8% at 60 °C; viscosity of water and ethylene glycol based hybrid nanofluids at Φ = 1.0% vol. is also enhanced by 72.15% and 86.62% compared to their base fluids at 20 °C. The density is also increased with an increase of particle loadings, whereas, the specific heat is decreased with a rise of particle loadings. Experimental thermophysical properties are fitted into regression equations by using multi linear regression method. The figures-of-merit of all the prepared hybrid nanofluids show its value is less than one under the used particle loadings and temperatures.

Thermal instability analysis in magneto-hybrid nanofluid layer between rough surfaces with variable gravity and space-dependent heat source

The onset of thermal instability with hybrid nanoparticles Al2O3–Cu in nanofluid is investigated with the combined effects of variable gravity, variable heat source and Lorentz force in a porous medium. Its applications are in many areas like chemical engineering, geophysics, astrophysics, etc. Based on literature, many gravity variations are assumed in the present analysis, along with a space-dependent heat source/sink parameter that varies along the width of the channel. The finite difference-based Lobatto IIIa method has been applied to solve the system under no nanoparticle flux and rough boundary conditions, and approximate analytical results are generated for some special cases. The roughness parameters ([Formula: see text]), Darcy number (Da), gravity variation function ([Formula: see text]), and Chandrasekhar number (Q) delay the onset of convection and thus stabilize the system. It is also observed that an increase in the Lewis number Le, power index in variable heat source, and the nanoparticle Rayleigh concentration number Rn decreases the critical Rayleigh number [Formula: see text] which destabilizes the system, and increases the critical wave number, which enlarges the convection cell size. In the case of exponential variation ([Formula: see text]) in the gravity variation parameter, the system becomes stabilized due to a delay in the onset of convection. In addition, we have considered multilayer perceptron-artificial neural network (MLP-ANN) computation to predict the critical Rayleigh number as per function of important controlling parameters. The data set of 625 observations is chosen keeping 70% for testing, 15% for training and 15% for validation using efficient Levenberg−Marquardt back propagation algorithm with optimal accuracy measures i.e., root mean square deviation (RMSE), root mean relative error (RMRE) and [Formula: see text] (coefficient of determination). Finally, the regression plots are drawn that correlate, target and output data.

Shear stress regulation in cylindrical arteries through medial growth and nitric oxide release

The mechanisms employed by blood vessels in order to adapt to their hemodynamic environment are important for our general understanding of disease and development. Changes in arterial geometry are generally induced by two effects: vasodilation and/or constriction; and growth and remodeling ("G &R"). The first can occur over short periods of a few minutes, while the second usually occurs over timescales of weeks or months. The free radical Nitric oxide (NO) is one of the few biological signaling molecules that is gaseous. When smooth muscle cells internalize NO, they lengthen and ultimately induce a relaxation of the artery. Platelet-Derived Growth Factor (PDGF) is a growth factor released by smooth muscle cells and platelets that regulates cell growth and division. In this paper we present a single-layered, axisymmetric hyperelastic model for a deforming, growing artery in which the opening angle is regulated by NO and growth is induced by PDGF. Our model describes vasodilation and G &R in a long cylindrical artery regulated by a steady-state Poiseuille flow. The transport of NO released by the endothelium is governed by a diffusion equation with a shear-stress dependent flux boundary condition. Arterial opening angle is assumed to be a Hill function of the wall-averaged NO concentration. We find that both growth and NO help to regulate shear stress with respect to the flow rate, but regulation through growth occurs only at large times. In contrast, regulation through NO is immediate but can only occur as long as the opening angle is able to continually decrease as a function of flow rate. Our model is calibrated using experimental data from ligated, control, and anastomosed carotid arteries of adult and weanling rabbits. Our results generate shear stress/flow rate and lumen radius/flow rate curves that agree with experimental data from control and NO-inhibited rabbit carotid arteries.

Experimental investigation of heat transfer and pressure losses across staggered Body Centered cubic arrays fabricated by Laser Powder Bed Fusion

Truss type lattice structures are regarded as a promising approach to design compact heat exchangers that are characterized also by good mechanical performances. The design of these structures can be tailored to satisfy multifunctional requirements and manufactured by exploiting the capabilities of additive manufacturing. This study reports on the experimental characterization of the aerothermal behaviour of new Body-Centred Cubic (BCC) lattice stagger arrays in a rectangular channel, which differ by the truss diameter. The heat exchangers are made from AlSi10Mg by means of Laser Powder Bed Fusion, and the geometric conformity and surface roughness of the internal features were characterized through optical and non-destructive methods to precisely define the boundary conditions at which the aero-thermal performances are provided. By so doing, some marginal defects caused by inherent limitations of the technology were observed, and relevant information in the perspective of designing new structures on a smaller characteristic scale were collected. The samples were then tested on a dedicated test rig by applying uniform heat flux boundary conditions up to a Reynolds number of 20′000. The experimental rig and methodology were firstly tested on a smooth channel and the results were found to be consistent with the literature. The BCC structure with the largest diameter performs the best in terms of heat transfer in view of the higher wetted surface. On the other hand, as expected, the friction factor shows the opposite trend. However, the thermal performance factor does not differ much among the different configurations, with slightly better performances observed for the larger diameter truss. Finally, correlations were found to predict both aero and thermal performances of the considered structure by introducing a modified formulation for the friction factor, Nusselt and Reynolds numbers.

Effect of Magnetic Field on the Developing Thermal Field in a Duct Filled with Porous Media under Local Thermal Non-Equilibrium with a Nonlinear Flow Model

In this article, the numerical study of the influence of a magnetic field at the laminar forced convection in a thermally developing region coming under the influence of local thermal non-equilibrium (LTNE) of parallel plate channels completely immersed in the porous material is investigated. Constant wall heat flux boundary conditions are applied to the walls of the channel. In the nonlinear flow model, the Darcy-Brinkman-Forchheimer equation governs the flow field in the porous region, which is assumed to be unidirectional. The system is defined by certain well-known parameters, these being Darcy number (Da), thermal conductivity ratio (κ), Forchheimer number (F), Hartmann number (M), and Biot number (Bi). Numerical solutions have been obtained by applying a successive accelerated replacement (SAR) scheme. Exact solutions for the dimensionless temperature and the fully developed Nusselt number in the absence of the Forchheimer number (F = 0), for the fully developed thermal field, are obtained for the linear flow model, the Darcy-Brinkman model. Plots are given for the dimensionless temperature profiles in the fluid and solid phases, wall temperature, as well as the local Nusselt number at the parallel plate channel, which has been displayed. The effect of the magnetic field and the thermal conductivity ratio has a significant effect on the local Nusselt number.

Random fractional partial differential equations and solutions for water movement in soils: Theory and applications

AbstractThis paper analyses a set of random fractional partial differential equations (rfPDEs) for water movement in soils. The rfPDEs for both rigid and swelling soils are solved for both a random flux boundary condition (BC), and random concentration BC. Solutions from a random flux BC are presented for the large‐time and small‐time situations with the large‐time solution as a very simple method for determining the flux through the surface of the soil. The equation of cumulative infiltration is presented with random parameters of the rfPDE subject to a random concentration BC. The simulations using the results of the rfPDE for the two types of BCs yielded encouraging and stable results based on two sets of field data: the first set of the data was measurements at a single site while the second set was from 26 measurements in a small catchment. The results suggest that the presented procedures are very useful methods for the interpolation, extrapolation, and prediction of hydrological variables and parameters such as water content, hydraulic conductivity or the flux through the surface of the soil. The methodologies presented in this paper are able to reveal and reproduce the realistic hydrological processes in nature which are often stochastic and random.

Numerical simulation of cathode-spot crater formation and deuterium desorption process on hydrogen titanium cathode with flux boundary conditions and different impregnation degree

In this paper, the crater formation process of single cathode spot on hydrogen titanium electrode is modeled. In this model, combined with the deuterium diffusion equation, the flux boundary condition was considered to calculate the cathode spot desorption rate with different impregnation degree, and the deformation process of the cathode spot crater was simulated by a hydrodynamic model. The simulation results show that the current, size, and depth of a single cathode spot crater tend to decrease with increasing impregnation degree, which is consistent with many related experimental results. The simulation result shows that the desorption of deuterium in a single cathode spot crater mainly occurs in the ignition stage, and the positions of desorption are successively distributed in the liquid metal flow area on the side wall of the crater and the high temperature area in the center of the crater. The desorption rate of deuterium drops rapidly after cathode spot quenching. At the same time, the increase of arc current and impregnation degree can improve the deuterium desorption rate of a single cathode spot.

“Failure envelops for foundation subjected to inclined and eccentric loading considering steady state and transient flow conditions in unsaturated soils”

In this paper, an analytical method and semi-empirical equations are presented for determining the bearing capacity failure envelopes for foundations subjected to inclined or eccentric loads in unsaturated soils considering the influence of steady and transient state flow conditions. Firstly, the analytical method is proposed extending the slip line theory coupled with analytical solutions for different matric suction profiles. The failure envelopes in the V - H (Vertical - Horizontal) and V - M (Vertical - Moment) loading spaces for two fine-grained soils were investigated by conducting parametric studies using the proposed method. The influence of the ground water table depth variation, internal friction angles, surface flux boundary conditions and different infiltration durations were considered. Based on the parametric studies results, semi-empirical equations were proposed for the failure envelopes in the two loading spaces. The validity of the proposed failure envelope equations is examined by providing comparisons with the proposed analytical method and published results from the literature. Good agreement is found between the fitted failure envelopes and that obtained from the analytical method and published studies.

Theoretical investigation on heat leakage distribution between vapor and liquid in liquid hydrogen tanks

As a key factor affecting thermal behaviors of liquid hydrogen (LH2) tanks, heat leakage plays an important role in accurate prediction of pressure build-up for safe storage and transportation of LH2. Uniform heat flux between vapor and liquid in LH2 tanks is widely adopted as thermal boundary condition in predicting pressure build-up process. However, a distribution of heat flux between vapor and liquid was observed during the self-pressurization process in the experimental test. In light of this, an analytically theoretical model of revealing the energy exchange process among the vapor, liquid and inner wall is proposed to investigate the heat leakage distribution ratio (HDR) between vapor and liquid in LH2 tanks. The feasibility of the model is validated by the experimental results from NASA. In the whole self-pressurization process of 25,000 s, HDR reduces from 0.803 to 0.235 under a liquid fill ratio of 90% and a total heat leakage of 71.3 W. The results show that the existence of inner wall and different thermal properties between the vapor and liquid make the heat leakage flux non-uniformly distributed into the vapor and liquid. And the geometric structure of tank, thermal properties and initial states of the vapor and liquid have a significant effect on HDR. When coupling the model with thermal multi-zone model, the relative error in pressure prediction is reduced by 61.8% against experimental results. Benefiting from the coupled model, the relative error in pressure prediction caused by the uniform heat flux boundary condition reduces from 90.16% to 8.15%. The present work establishes theoretical foundation on analyzing heat leakage distribution between the vapor and liquid for LH2 tanks, and provides useful guidance on modifying boundary conditions in accurately predicting thermal behaviors of LH2 tanks.

Thermodynamic analysis for peristaltic flow of Carreau-Yasuda magneto nanofluid through a porous medium

Thermodynamic analysis for mixed convective peristaltic transportation of Carreau–Yasuda (CY) magneto nanofluid is presented. The assumed nanofluid is a mixture of graphene nano-powder and ethylene glycol. The Carreau–Yasuda model is considered to represent the non-Newtonian features of nanofluids. A mathematical model is developed under the effects of a porous medium, variable thermal conductivity, Brownian motion, viscous dissipation, magnetic field, thermophoresis, and Ohmic heating assumptions. Mass and heat transfer aspects are also considered. The flow and heat/mass transfer equations were made using Buongiorno’s model of nanofluids. No-slip and zero mass flux boundary conditions are entertained for velocity, thermal, and concentration profiles. A numerical solution via built-in technique is obtained for a coupled system of equations. Graphical outcomes predict that due to Ohmic heating characteristics temperature profile increases. The heat transfer rate at the wall decreases for higher thermal conductivity parameter. The irreversible rate can be controlled by maintaining Darcy’s number. The concentration of nanomaterials increases when the thermophoresis parameter is improved. Fluid velocity is slowed down by enhancing the Hartman number and it improves by enhancing the thermal Grashof number.

Analysis on the Effects of Different Receiver Structures and Porous Parameters on the Volumetric Effects and Heat Transfer Performance of Porous Volumetric Solar Receiver

The volumetric solar receiver is an important heat transfer component of the concentrated solar power (CSP) system. Moreover, in order to improve the absorption of concentrated solar radiation, the porous media are widely used in volumetric solar receiver. In recent years, many studies were concerned with the effects of porosity, pore number density and size, Reynolds number, and Darcy number on heat and flow performance in volumetric solar receiver. However, there are few studies on the effects of structure type and geometric parameters on the volumetric effects and the radiation characteristics of a porous volumetric solar receiver with nonuniform heat flux boundary condition. In this contribution, in order to analyze the effects of structures on the volumetric effects and heat transfer performance, we design six types of porous volumetric receiver structures, whereas the volume of different structures keeps consistent. Then, we apply the local non-thermal-equilibrium model and also consider the effect of structure shape on the concentrating solar radiation transport characteristic. Furthermore, we apply the Gaussian distribution model (GDM) to simulate the actual nonuniform heat flux boundary condition. The result shows that drum-type structure (RT-III) has the best heat transfer performance among the six structures; e.g., the outlet average temperature of fluid is up to 851 K when the pore size and porosity of porous media are 2 mm and 0.7, respectively, which is 41 K increased than the scaled cone-type structure. In addition, the pore diameter has less influence on the outlet temperature of fluid, but it has greater influence on the pressure drop. The contribution can provide a reference for this type of solar receiver design and reconstruction.

Thermal Convection of an Ellis Fluid Saturating a Porous Layer with Constant Heat Flux Boundary Conditions

The present work analyzes the thermal instability of mixed convection in a horizontal porous channel that is saturated by a shear-thinning fluid following Ellis’ rheology. The fluid layer is heated from below by a constant heat flux and cooled from above by the same heat flux. The instability of such a system is investigated by means of a small-disturbances analysis and the resulting eigenvalue problem is solved numerically by means of a shooting method. It is demonstrated that the most unstable modes on the instability threshold are those with infinite wavelength and an analytical expression for such conditions is derived from an asymptotic analysis. Results show that the non-Newtonian character of the fluid has a destabilizing role.

Numerical Study on the Heat Transfer Characteristics of Cu-Water and TiO2-Water Nanofluid in a Circular Horizontal Tube

A numerical simulation of convective heat transfer coefficient (hconv) was studied with Cu-Water and TiO2-Water nanofluids flowing through a circular tube subjected to uniform wall heat flux boundary conditions under laminar and turbulent regimes. Four different concentrations of nanofluids (ɸ = 0.5, 1, 1.5 and 2%) were used for the analysis and the Reynolds number (Re) was varied from laminar (500 to 2000) to turbulent flow regime (5000 to 20,000). The dependence of hconv on Re and ɸ was investigated using a single-phase Newtonian approach. In comparison to base fluid, average hconv enhancements of 10.4% and 7.3% were noted, respectively, for the maximum concentration (ɸ = 2%) and Re = 2000 for Cu-Water and TiO2—water nanofluids in the laminar regime. For the same ɸ under the turbulent regime (Re = 20,000), the enhancements were noted to be 14.6% and 13.2% for both the nanofluids, respectively. The random motion (Brownian motion) and heat diffusion (thermophoresis) by nanosized particles are the two major slip mechanisms that have more influence on the enhancement of hconv. In addition, the Nusselt number (Nu) of the present work was validated for water with the Shah and Dittus Boelter equation and found to have good agreement for both the regimes.

Experimental evaluation of fluid connectivity in two-phase flow in porous media

In this work, we provide a physically-consistent modeling approach for two-phase porous media flow, by including percolating interfacial area and saturation as state variables. For this purpose, we combine two continuum theories for two-phase flow which have been individually proven to be conditionally valid. This means the potential use of the connected-to-the-flow interfacial area as a state variable is tested utilizing time-resolved microfluidic experiments, for various flux boundary conditions. Moreover, we observe and study a linear relation between the percolating saturation and interfacial area, which is persistent for the tested boundary conditions. In our microfluidic experiments, we employ optical microscopy to perform cyclic immiscible displacement experiments. Our results show that a continuum model, where capillary pressure, saturation , and specific interfacial area of the clusters connected to the flow are considered, is closer to a universal description of two-phase flow than the common approaches, where the only state variable is saturation.

Numerical simulation of film boiling heat transfer in the presence of a uniform electric field using front tracking

Film boiling has many applications in various industries, such as industrial furnaces, heat exchangers in nuclear and fossil power plants. In this paper, film boiling on a horizontal plate in the presence of a uniform electric field is simulated in two dimensions by a finite difference/front tracking method with constant heat flux and constant wall temperature boundary conditions. Applying an electric field is an active method that is widely used to enhance heat transfer. Simulation of this phenomenon is of great importance in industry to predict the growth and movement of steam bubbles produced. The electric field is applied by an electric potential difference to the upper and lower boundaries. It is assumed that the working fluid is a completely dielectric fluid. The electric force is a volumetric force that is added directly to the Navier-Stokes equations. The present numerical method is validated by simulation of film boiling in the presence and absence of electric field and comparing with available experimental and numerical results. The effect of the electric field on the Nusselt number for constant heat flux and constant wall temperature boundary conditions are investigated. The effect of dimensionless numbers is also studied. Generally, the Nusselt number increases by applying an electric field. The bubble growth time, bubble thickness and vapor film thickness are reduced by applying an electric field. Small bubbles are formed on the surface with increasing electric field for constant heat flux boundary condition. The number of bubble growth sites increased with increasing electric field in the constant wall temperature boundary condition, or in other words, the instability wavelength decreased.

On phase-field equations of Penrose–Fife type with the conserved order parameter under flux boundary condition: Global-in-time solvability and uniform boundedness

We study an initial boundary value problem for a system of conserved phase-field model proposed by Penrose and Fife ([13,14]) under flux boundary condition for the temperature in higher space dimensions. In contrast to other works related to this problem, we are concerned with the above mentioned problem under the correct form of the flux boundary condition for the temperature, which is obliged to overcome additional difficulties in mathematical treatment. We prove that the initial boundary value problem for such a system admits a unique global-in-time strong solution in Sobolev–Slobodetskiĭ spaces. Moreover, it is shown that the temperature keeps positive and both temperature and order parameter are uniformly bounded in two dimensions through the time evolution.

A Hilliges-Weidlich-type scheme for a one-dimensional scalar conservation law with nonlocal flux

<abstract><p>The simulation model proposed in [M. Hilliges and W. Weidlich. A phenomenological model for dynamic traffic flow in networks. <italic>Transportation Research Part B: Methodological</italic>, <bold>29</bold> (6): 407–431, 1995] can be understood as a simple method for approximating solutions of scalar conservation laws whose flux is of density times velocity type, where the density and velocity factors are evaluated on neighboring cells. The resulting scheme is monotone and converges to the unique entropy solution of the underlying problem. The same idea is applied to devise a numerical scheme for a class of one-dimensional scalar conservation laws with nonlocal flux and initial and boundary conditions. Uniqueness of entropy solutions to the nonlocal model follows from the Lipschitz continuous dependence of a solution on initial and boundary data. By various uniform estimates, namely a maximum principle and bounded variation estimates, along with a discrete entropy inequality, the sequence of approximate solutions is shown to converge to an entropy weak solution of the nonlocal problem. The improved accuracy of the proposed scheme in comparison to schemes based on the Lax-Friedrichs flux is illustrated by numerical examples. A second-order scheme based on MUSCL methods is presented.</p></abstract>

Numerical solutions to two-dimensional fourth order parabolic thin film equations using the Parabolic Monge-Ampere method

<abstract><p>This article presents the Parabolic-Monge-Ampere (PMA) method for numerical solutions of two-dimensional fourth-order parabolic thin film equations with constant flux boundary conditions. We track the PMA technique, which employs special functions to acclimate and force the mesh moving associated with the physical PDE representing the thin liquid film equation. The accuracy and convergence of the PMA approach are investigated numerically using a one two-dimensional problem. Comparing the results of this method to the uniform mesh finite difference scheme, the computing effort is reduced.</p></abstract>