Conjugate Boundary Conditions Research Articles

EFFECT OF JET TO PLATE SPACING ON FILM COOLING PERFORMANCE IN A COMBINED IMPINGEMENT AND FILM COOLING ARRANGEMENT

The influence of jet to plate spacing on conjugate heat transfer from a flat plate with combined impingement and film cooling has been studied both numerically and experimentally. The flow configuration consists of a matrix of impingement holes and multiple staggered rows of cylindrical film holes. One side of the plate is impinged with air jets, while the other side is exposed to hot mainstream as well as the vented air from the film cooling holes. The effect of increasing jet to plate ratio (H/D = 0.4, 1.2, and 2.0) on film cooling flow structure and heat transfer is analyzed under conjugate boundary conditions. The flow and temperature patterns are predicted in the entire computational domain; namely, the plenum chamber, film holes, and mainstream-coolant interaction zone. The mainstream Reynolds number is varied from 49,050 to 130,800 while keeping the jet Reynolds number constant at 825 and the blowing ratio ranging from 0.6 to 1.6. The velocity, turbulence, and temperature distributions at the exit of the film hole together define the film coverage on the interaction surface. The thermochromic liquid crystal (TLC) technique is used to evaluate the surface temperature of the plate and validate the computed values of the surface effectiveness. Both measured and computed effectiveness distributions are found to be comparable. Increase in the H/D values beyond a certain limit causes decrease in film effectiveness on the interaction surface, especially downstream of the film hole. The influence of H/D is noticed for all the three blowing ratios investigated, albeit to varying degree.

Asymptotic Expressions of Fourth Order Sturm-Liouville Operator with Conjugate Conditions

In this paper, it is studied the asymptotic expression of fourth order differential operator with periodic boundary conditions. For this operator, it is also considered conjugate boundary conditions at x=0 which shows discontinuity. For this purpose, firstly asymptotic expression of solutions areobtained. Then by using the the asymptotic formulas of fundamental solutions, asymptotic expression of eigenvalues and eigenfunctions are presented. It is also dealt with the asymptotic expression of same operator with antiperiodic boundary conditions and conjugate conditions

Natural convection in vertical enclosures with conjugate boundary conditions

A numerical investigation and reduced-order modelling of natural convection in a cavity with differentially heated sidewalls is discussed. The effect of conjugate boundary conditions on the cavity's heat transfer and natural flow circulation with varying aspect ratios and Rayleigh numbers is examined. Validation of the canonical differentially heated cavity reveals that a modification to the definition of the Rayleigh number (Ra) is required to reconcile the $Nu\sim Ra^{1/4}$ scaling (where Nu is the Nusselt number) and validity of previously proposed correlations for the heat transfer in vertical enclosures. Dynamic mode decomposition is used to uncover the underlying time-dependent flow structures and the results are compared with stability bifurcation studies in the literature. A flow mode previously associated with smaller aspect ratio cavities is identified as the unstable mode for a larger aspect ratio of 4. The effect of conjugate boundary conditions is scaled based on the ratio of the internal and external boundary layers, wherein higher external Reynolds numbers aid in the heat transfer as the Nusselt number approaches the isothermal limit. A reduced-order theoretical model is proposed to predict the Nusselt number for the conjugate boundary conditions. The performance of conjugate boundary conditions is connected to the flow stability, with the parallel-flow configuration acting in a destabilizing manner while the counterflow configuration has a stabilizing effect and results in the highest heat transfer. We also found that relatively large heat transfer can be achieved with substantially lower external actuation with intermediate aspect ratios, translating to less power in practice.

Fully Coupled Large Eddy Simulation-Conjugate Heat Transfer Analysis of a Ribbed Cooling Passage Using the Immersed Boundary Method

Abstract The study focuses on evaluating fully coupled conjugate heat transfer (CHT) simulation in a ribbed cooling passage with a fully developed flow assumption using large eddy simulation (LES) with the immersed boundary method (IBM-LES-CHT). The IBM-LES and the IBM-CHT frameworks are validated by simulating purely convective heat transfer in the ribbed duct, and a laminar boundary layer flow over a 2D flat plate with heat conduction, respectively. For the main conjugate simulations, a ribbed duct geometry with a blockage ratio of 0.3 is simulated at a bulk Reynolds number of 10,000 with a conjugate boundary condition applied to the rib surface. The nominal Biot number is kept at 1, which is similar to the comparative experiment. It is shown that the time scale disparity between turbulent fluid flow and heat conduction in solid can be overcome by using an artificially high solid thermal diffusivity. While the diffusivity impacts the instantaneous fluctuations in temperature and heat transfer, it has an insignificant effect on the predicted Nusselt number. Comparison between IBM-LES-CHT and iso-flux heat transfer simulations shows that the iso-flux case predicts higher local Nusselt numbers at the back face of the rib. Furthermore, the local Nusselt number augmentation ratio (EF) predicted by IBM-LES-CHT is compared with experiment and another LES conjugate simulation. The present LES calculations predict higher EFs on the leading face of the rib and show a different trend at the trailing face when CHT is activated.

Optimal control of first-order partial differential inclusions in bounded region

The paper is devoted to the Lagrange problem in the bounded region for first-order partial differential inclusions (PDIs). For this, using discretisation method and locally adjoint mappings (LAMs), in the form of Euler–Lagrange type inclusions and conjugate boundary conditions, sufficient optimality conditions are obtained. The transition to a continuous problem with PDIs is possible using a specially proved equivalence theorem. To demonstrate this approach, some semilinear problems and polyhedral optimisation with first-order partial differential inclusions are considered. Furthermore, the numerical results also are provided.

A new hydrodynamic analogy model for the determination of transport phenomena in random packings

A new model for absorption processes in columns equipped with random packings based on the hydrodynamic analogy approach is presented. The packed bed is represented by a bundle of cylindrical channels with the same diameter equal to the hydraulic diameter of the bed. The liquid flow in the real packing is captured by a combination of film, jet and drop flow which are assumed to be the dominating flow patterns in the bed. The film flow is considered as attached to the inner surface of the channels, while jets and drops are falling through the channel center. Each of the flow patterns is modelled independently of one another, in a separate channel type with counter-current gas flow. After certain packing specific lengths of undisturbed flow, each phase is mixed to govern the redirection of the flow and the transition from film to jet and drop flow and vice versa as well as the mixing of gas flows from neighboring packing elements. The fluid dynamics as well as heat and mass transfer are described by the respective rigorous conservation equations. Both phases are related by conjugate boundary conditions at the phase interface. The reactive CO2 absorption into aqueous sodium hydroxide solution is investigated experimentally for two random packings of type 50 mm Pall-Ring and ENVIPAC 2 in an industrial-scale column. The model is validated against the experimentally obtained data.

Dynamic study of ferrodroplet and bubbles merging in ferrofluid by a simplified multiphase lattice Boltzmann method

In this article, the dynamics of ferrodroplet deformation and bubbles merging within the ferrofluid under a uniform magnetic field are studied through numerical simulations. A recently developed simplified multiphase lattice Boltzmann method (SMLBM) is used to capture the hydrodynamic behavior of ferrofluids, the Cahn-Hilliard (CH) equation is employed as the interface tracking algorithm, whose solution is also obtained within the lattice Boltzmann framework and a self-correcting procedure is used to solve the magnetic field along with conjugate boundary conditions for a smooth transition of magnetic permeability at the interface. A series of numerical simulations are performed for a range of magnetic field strength, magnetic susceptibility and Ohnesorge number (Oh), along with a comparison between obtained results and previous experimental and numerical results. The dynamical behavior of ferrodroplets and bubbles merging in ferrofluids is analyzed through the distribution of magnetic field and magnetic flux density, velocity contours, pressure variations, the magnetic energy density and the effect of inertia. Findings of this study may help to explain the mechanism of ferrodroplet deformation and merging of bubbles within the ferrofluid and provide a useful insight into the distribution of flow variables during these deformations and merging phenomena.

Photothermal nanobubble nucleation on a plasmonic nanoparticle: A 3D lattice Boltzmann simulation

Nanobubble nucleation by plasmonic heating on a gold spherical nanoparticle under laser illumination is numerically investigated by a 3D liquid-vapor phase change lattice Boltzmann (LB) method. A volumetric heat source is incorporated to the evolution equation of temperature distribution function to simulate photothermal energy conversion effect in a plasmonic gold spherical nanoparticle induced by surface plasma resonance under laser illumination. A halfway bounce-back conjugate boundary condition on nanoparticle/liquid interface for LB simulation, with interfacial thermal resistance taken into consideration, is derived. It is confirmed that nanobubble nucleates on a laser heated hydrophilic gold spherical nanoparticle in a core-shell mode under normal gravity condition. To simulate nanobubble nucleation on a plasmonic nanoparticle in a spherical cap mode, an extraordinary (fictitious) hyper gravity environment would have to be imposed. Effects of nanoparticle surface’s wettability, nanoparticle-liquid interfacial thermal resistance, and nanoparticle’s diameter on nanobubble nucleation under normal gravity are investigated. The size of the nanoparticle is found to have non-monotonic effects on the threshold laser fluence of nanobubble nucleation, with the 60 nm gold nanoparticle having the lowest threshold laser fluence for nucleation. This finding is in agreement with existing experimental results.

Modeling of combustion synthesis in multilayer gasless system

A combustion model for a flat layered composition has been developed, where chemically active layers alternate with inert metallic layers with high thermal conductivity. The heat exchange between the layers was specified by the conjugate boundary conditions. A numerical study of gasless combustion of a multi-layer system with heat-conjugated layers of two types was performed. Optimal layer sizes and parameters of the layer system were obtained to provide the maximum burning rate of the layer package. The effect of increasing the burning rate was found to be associated with heat recovery and an increase in the effective thermal conductivity of the system. The concentration limits of combustion were determined depending on the volume content of the inert element. Replacing the system of inert layers with that of low-calorie mixture layers leads to a model for synthesis of inorganic materials in the "chemical furnace" mode.

Effect of spacing on thermal performance characteristics of Li-ion battery cells

In this article, effect of spacing between the battery cells ($$\bar{W}_{\text{f}}$$) on thermal performance of Li-ion battery cells is investigated in detail. Developing a finite volume method-based numerical code for the present analysis, conjugate boundary condition at the cell and coolant interface is considered. SIMPLE algorithm employed for solving the Navier–Stokes equation is validated with famous benchmark lid-driven cavity problem. The heat generation inside the modern battery cell is uniform in accordance with cell zone. Air being the coolant flows between the channel spacing of the battery cells. Forced laminar flow of coolant and steady-state analysis with operating parameters like heat generation term ($$\bar{S}_{\text{q}}$$), Reynolds number (Re), conduction–convection parameter (ζcc), and aspect ratio (Ar) is analyzed with main focus of $$\bar{W}_{\text{f}}$$. The range of $$\bar{W}_{\text{f}}$$ is from 0.02 to 0.14 varied in steps of 0.02 and Re from 250 to 2000 in step of 250. Coupled heat transfer behavior in terms of maximum temperature and average Nusselt number for these parameters is provided. From the numerical analysis, it is observed that for most of the range of operating parameters, at $$\bar{W}_{\text{f}} = 0.02$$, causes a sudden increase in temperature distribution and rise in maximum temperature above critical limits. Average Nusselt number increased with decrease in $$\bar{W}_{\text{f}}$$ up to 0.04 and below this, it dropped. Spacing of $$\bar{W}_{\text{f}} = 0.04$$ and $$\bar{W}_{\text{f}} = 0.06$$ proved to be an optimal spacing at which average Nusselt number is the highest and the maximum temperature is within the safe limit for parameters considered.

Heat and mass transfer in a polymeric electrolyte membrane-based electrochemical air dehumidification system: Model development and performance analysis

In this paper, a two-dimensional steady-state theoretical model was established, to model the internal transport phenomena in a polymeric electrolyte membrane-based (PEM-based) air dehumidification element. The influences of electrochemical reactions, activation and concentration over-potentials, and Ohmic and electro-osmotic effects were considered. The model was solved by the finite difference method with conjugate boundary conditions. So, with this model, the heat, mass and current transfer through the five layers of the element (diffusion layers, catalyst layers and a PEM) could be described theoretically, as well as the convective heat and mass exchange with adjacent airflows. Compared with the results from previous models, this model showed a much closer trend to the experimental data. The overall error was less than 15%, with an acceptable average error of 8.6%. However, greater deviations were observed under larger airflow conditions, probably due to the assumption of laminar airflow and steady-state heat conduction. Furthermore, by the performance analysis, the maximum moisture gradient was found inside the PEM, so the PEM’s parameters could largely affect the system performance. With the increase in PEM water content, the dehumidification was significantly enhanced, especially when the air humidity was high. If the PEM water content was doubled, the dehumidification rate was increased by 42%. Then, decreasing the PEM thickness also improved the performance. However, the effect became minor if the thickness was less than 100 μm. It was also helpful by increasing the PEM conductivity, although the effect of this variable was relatively small. This study provided theoretical guidance for further system improvement and material preparation for PEM-based dehumidification systems.

Abelian Higgs model with charge conjugate boundary conditions

The abelian Higgs model is studied on the lattice with charge conjugate boundary conditions. A locally gauge invariant operator for the charged scalar field is constructed and the charged scalar particle mass is calculated in the Coulomb phase of the lattice model. Agreement is found with the mass calculated in Coulomb gauge. The gauge invariant scalar field operator is used to calculate the Higgs boson mass in the Higgs region and to show that the charged particle disappears from the spectrum in the confined regime.

Sedimentation of two non-magnetic particles in magnetic fluid

Magnetic fluid is a stable suspension of solid phase magnetic particles of diameter about 10 nm in a nonmagnetic carrier fluid like water or alcohol. Nowadays, the magnetic fluid is widely used in industry areas such as sealing, damping, lubricating, sound regulation, heat dissipation, and MHD beneficiation. Researchers have paid great attention to the behaviors of non-magnetic particles (NPs) in the magnetic field because magnetic fluid containing NPs can form different microstructures, which are easily controlled by applying a magnetic field. In order to appropriately use the properties of magnetic fluid in industry, it is necessary to study the interaction among NPs in detail. In this paper, a multi-physical numerical model is employed to investigate the sedimentation of two NPs in magnetic fluid subjected to an applied magnetic field. The magnetic fluid flow is simulated by lattice Boltzmann method, and magneto hydrodynamics is calculated with a self-correcting procedure of a Poisson equation solver, which enables the Ohm's law to satisfy its conservation law. A dipole force model is used to obtain the dipole interaction force between particles. In addition, as the permeability of the magnetic fluid is quite different from those of the NPs and magnetic fluid, correctly establishing the conjugate boundary condition of the magnetic intensity at the interface between the particles and surrounding fluid is a key because it affects the magnetic induction in the fluid-structure interaction area. A smooth transition scheme of the conjugate boundary condition for magnetic intensity at the interface between the particles and surrounding fluid is used in this work. The aim of this work is to investigate sedimentation of two NPs in magnetized magnetic fluid. By changing the ratio of magnetic permeability and the magnetic parameter, it is found that altering the ratio of magnetic permeability is more effective to change the trajectories of NPs, while changing the magnetic parameter can just give rise to a slight transform of particle trajectories. This can provide good theoretical support for the application of magnetic fluid in industry area, because the results in the present simulation can quantitatively analyze the controlling of the movement of NPs.

Conjugate boundary condition, hidden particles and gauge-Higgs inflation

We propose an idea that hidden particles can be separated according to gauge quantum numbers from the visible ones by the difference of boundary conditions (BCs) on extra dimensions. We formulate five-dimensional (5D) gauge theories yielding conjugate BCs besides ordinary ones on S1/Z2, and examine physical implications concerning hidden particles on an extension of the Standard Model (SM) coexisting different types of BCs. A model with conjugate boundary conditions is applied on a gauge-Higgs inflation scenario.

Numerical simulations of conjugate convection combined with surface thermal radiation using an Immersed-Boundary Method

Dynamic and thermal interaction problems involving fluids and solids were studied through a finite volume-based Navier-Stokes solver, combined with immersed-boundary techniques and the net radiation method. Source terms were included in the momentum and energy equations to enforce the non-slip condition and the conjugate boundary condition including the radiative heat exchange. Code validation was performed through the simulation of two cases from the literature: conjugate natural convection in a square cavity with a conducting side wall; and a cubical cavity with conducting walls and a heat source. The accuracy of the methodology and the validation of the inclusion of moving bodies into the simulation was performed via a theoretical case

Non-Isothermal Moulding of Composite Products with Impregnation of the Porous Layer

The non-isothermal filling of a flat mould cavity by a Newtonian fluid with simultaneous two-dimensional impregnation of the porous (reinforcing) layer is investigated. The flow in the cavity is described by inertialess Navier–Stokes equations, the flow in the porous layer by Darcy's law, and the flow in the region adjacent to the ‘liquid–porous layer’ boundary by the Brinkman equation. Viscosity is considered to be temperature dependent. The temperature fields in the channel region and in the permeable layer are interrelated by conjugate boundary conditions of the fourth kind.

The Two-Dimensional Direct and Inverse Scattering Problems with Generalized Oblique Derivative Boundary Condition

Consider the two-dimensional exterior problem for the Helmholtz equation with a generalized oblique derivative boundary condition, which arises in some new scattering problems such as the scattering of tidal waves by islands under suitable assumptions. Different from the well-studied scattering problems with Dirichlet, Neumann, and impedance boundary conditions, the boundary condition in our new model involves a linear combination of the normal and tangential derivatives of the wave field with complex coefficient. Compared with the classical scattering models, the tangential derivative on the obstacle boundary leads to some essential differences such as the symmetric property of the Green function and the reciprocity principle of the scattering data. Both the direct and inverse scattering problems are studied in this paper. First, based on the Lax--Phillips method, we show the unique solvability of the direct scattering problem and the analyticity of the solution on the wave number. Second, we investigate the Green function and the far-field pattern for our scattering model, which play a key role in the corresponding inverse scattering problems. It is found that the symmetry of the Green function and the reciprocity relations need new characterizations in terms of the other scattering problem with conjugate boundary condition in our case. Finally, we study two inverse scattering problems for which the uniqueness results are established using our new results for the direct scattering problem.

Heat transfer in a large triangular-roof enclosure based on the second law analysis

Application of entropy generation minimization method for the optimization of natural convection in triangular-roof enclosures can be helpful in enhancing the thermodynamic efficiency of the system. In this paper, the numerical solution of the entropy production was performed due to natural convection of laminar air flow in a large isosceles triangular-roof enclosure with different boundary conditions such as the greenhouses conjugate boundary conditions. The simulation results showed that the entropy generation number in an enclosure with partially heated from bottom wall center, is less than an enclosure with partially heated from bottom wall corner. Also, in a large triangular-roof greenhouse, the entropy generation number decreases with increasing of the greenhouse aspect ratio (L/H).

Conjugate points for fractional differential equations

Abstract Let b > 0. Let 1 < α ≤ 2. The theory of u 0-positive operators with respect to a cone in a Banach space is applied to study the conjugate boundary value problem for Riemann-Liouville fractional linear differential equations D 0+α u + λp(t)u = 0, 0 < t < b, satisfying the conjugate boundary conditions u(0) = u(b) = 0. The first extremal point, or conjugate point, of the conjugate boundary value problem is defined and criteria are established to characterize the conjugate point. As an application, a fixed point theorem is applied to give sufficient conditions for existence of a solution of a related boundary value problem for a nonlinear fractional differential equation.

Elastic stresses in an encapsulated gas bubble

The main objective of the present work is to study the elastic behaviour and the non-linear radial oscillations of a thin spherical shell encapsulating a perfect gas under the influence of an external acoustic pressure field. Our basic motivation is to derive a full elastic description of the above conjugate fluid mechanics problem. The mathematical model is reduced to solve a modified Rayleigh–Plesset equation to predict the external radius together with a Cauchy equation needed to predict the deformations and elastic stresses of the spherical thin shell. We impose conjugate boundary and initial conditions to predict the radius, elastic deformations and stresses and the resulting governing equations are solved numerically. Using dimensionless variables, we introduce, among others, three characteristic parameters that dictate the dynamic response of the encapsulated bubble: β that represents the competition between acoustic forces and induced stresses into the elastic solid, πx is the ratio between the equilibrium pressure and the elasticity modulus of the shell and βA defines the competition between the characteristic pressures of the system given lines below. The numerical predictions reveal that this last dimensionless parameter βA regulates clearly a pronounced cavitation limit and a non-linear oscillating behaviour. The most relevant physical variables of the problem such as the external radius of the shell, the elastic deformations and stresses are obtained as functions of time and different values of the above dimensionless parameters.