Simple Analytical Solution Research Articles

Finite element formulation to study thermal stresses in nanoencapsulated phase change materials for energy storage

Nanoencapsulated phase change materials (nePCMs) – which are composed of a core with a phase change material and of a shell that envelopes the core – are currently under research for heat storage applications. Mechanically, one problem encountered in the synthesis of nePCMs is the failure of the shell due to thermal stresses during heating/cooling cycles. Thus, a compromise between shell and core volumes must be found to guarantee both mechanical reliability and heat storage capacity. At present, this compromise is commonly achieved by trial and error experiments or by using simple analytical solutions. On this ground, the current work presents a thermodynamically consistent and three-dimensional finite element (FE) formulation considering both solid and liquid phases to study thermal stresses in nePCMs. Despite the fact that there are several phase change FE formulations in the literature, the main novelty of the present work is its monolithic coupling – no staggered approaches are required – between thermal and mechanical fields. Then, the FE formulation is implemented in a computational code and it is validated against one-dimensional analytical solutions. Finally, the FE model is used to perform a thermal stress analysis for different nePCM geometries and materials to predict their mechanical failure by using Rankine’s criterion.

SPH Analysis of Free Surface Flow over Pivot Weirs

Pivot weirs have been proved efficient to measure, control and stabilize open channel flows in an easy-to-use manner. This work presents a numerical investigation of free surface overflow from full-width rectangular pivot weirs based on the hydrostatic-corrected smoothed particle hydrodynamics (SPH). In verification stage, the draining process of a reservoir-vertical weir system is separately analyzed by activating and deactivating the hydrostatic correction. The SPH results are benchmarked against a simplified analytical solution derived herein. It is shown that the declining reservoir head is more accurately predicted by activating the correction term. The SPH simulations are then validated for five weir inclination angles, ranging from a very mild to vertical configurations. Comparison with relevant experimental data demonstrates satisfactory agreement in each case. It is observed that the location of vena contracta in the overtopping jet shifts further downstream at steeper inclination angles. Moreover, a simple relation among discharge coefficient and weir inclination angle is proposed based on the curve-fitting of SPH results. These two parameters appeared to be negatively correlated so that an increase in the inclination angle poses a decrease in the discharge coefficient and vice-versa. The present analysis can be useful for estimating flow discharge if a simple head measurement is available for a given geometry of the pivot weir.

Revision of archive recovery tests using analytical and numerical methods on thermal water wells in sandstone and fractured carbonate aquifers in the vicinity of Budapest, Hungary

This study offers a reinterpretation of archive aquifer tests, predominantly on the basis of recovery data, from an original datasheet of thermal water wells located in carbonate and sandstone aquifer units in the vicinity of Budapest, Hungary. The study compares the hydraulic conductivity (K) and specific storage (Ss) values derived in the first instance from an aquifer test evaluation. This included an initial application of the classical analytical Cooper and Jacob method. Subsequently, the visual two-zone (VTZ) numerical method was applied, then third, a more complex model, namely, WT software. It was found that the simple analytical solution is not able to represent the field conditions accurately, while in the course of the application of the VTZ model, it proved possible to alter the various hydraulic parameters within reasonable limits to fit the field data. In the case of the VTZ model, the researcher is required to calculate the accuracy of the fitted model separately, while with the WT model, this is automatic, the software seeks out the best fit. In addition to VTZ parameters, the WT model can efficiently incorporate data on up to 500 model layers, water level, and pressure. The optimization of the parameters may be achieved by automatic calibration, improving the accuracy of the numerical results. Recovery tests for 12 wells were numerically simulated to obtain values for vertical and horizontal hydraulic conductivity and specific storage for Triassic and Eocene fractured carbonate and the Upper-Miocene-Pliocene granular sandstone aquifer units. When an analytical solution is applied, only average values could be obtained. The conclusion reached was that the results of the analytical solution can be improved by the use of numerical methods. These methods are able to incorporate basic information on well design, aquifer material and the hydrogeological environment in the course of the evaluation. The revision of the archive recovery data using numerical methods may assist in the quest for better data for numerical flow and transport simulations without the need to perform new tests. In addition, the methods employed here can explain cases in which the original analytical interpretations proved unable to yield reliable data and predictions.

Modelling shear loading of a cantilever with a crack-like defect explicitly including linear parameters

This article considers the influence of shear on a specimen with a crack-like defect (CRLD) as part of the general variational statement of the problem with a distinguished interactive layer. The delta-element fracture criterion is formulated as a generalized energy product (GEP), accounting for the influence of hydrostatic stress and the existence of an external boundary. The GEP convergence with a reduction of the layer thickness in the layers element is shown both in the simplified analytical solution and in the numerical solution by the finite elements method (FEM). The linear size determining the GEP convergence in the layers element is taken for the characteristic size of the finite element conjugate to the physical excision and its continuation. It is shown that, unlike the simplified analytical solution, the solution by FEM allows finding the maximum GEP outside the layer.

회분식 공정-저장조 망구조의 최적설계를 위한 해석적인 방법

This study overcomes the limitation of the classical lot sizing method developed on the basis of the single product and single stage assumption. The superstructure of the plant considered consists of a network of serially and/or parallelly interlinked processes and storage units. The processes involve chemical reactions with multiple feedstock materials and multiple products. The objective function for optimization is minimizing the total cost composed of setup and inventory holding costs as well as the capital costs of constructing processes and storage units. And the relative optimality of a certain process can be computed to help the sequence decision of optimizing works. The novel Periodic Square Wave (PSW) model was used to analyze production and inventory system. The advantage of the PSW model comes from the fact that the model provides a set of simple analytic solutions in spite of a detailed description of the material flow between processes and storage units. The resulting simple analytic solution can greatly enhance the proper and quick investment decision for plant design and operaton problem confronted in diverse economic situations.

Analytical solution to temperature-induced deformation of suspension bridges

Varying ambient temperature may cause significant deformation of long-span suspension bridges. This study investigates the mechanisms of the temperature-induced mid-span deflection and tower-top horizontal displacement of suspension bridges, and formulates the general analytical solutions to the thermal responses of ground-anchored suspension bridges. The temperature-induced mid-span deflection can be determined from the superposition of deflections caused by each of the separate temperature changes of the main-span cable, side-span cables, and towers, while the tower-top horizontal displacement is a combination of the thermal effects of the side-span cables and towers. It is found that the cable temperature plays the dominant role in the mid-span deflection and tower-top horizontal displacement. Moreover, the temperature effective length is proposed and a unified formula of the temperature-induced bridge responses is obtained. The accuracy of the proposed formulas is verified through the field monitoring data of the 2132-m-long Tsing Ma Bridge at Hong Kong. The present study not only provides a simple, general, and unified analytical solution to the temperature-induced deformation of long-span suspension bridges, but also assists engineers in understanding the mechanisms of thermal behaviours of such bridges.

Constructing spherically symmetric Einstein\u2013Dirac systems with multiple spinors: Ansatz, wormholes and other analytical solutions

In this paper we present a detailed calculation of an Ansatz that allows to obtain spherically symmetric Einstein–Dirac configurations in d-dimensions. We show that this is possible by combining 2^{lfloor frac{d-2}{2} rfloor } Dirac fields, making use of the properties of the angular dependence of the spinors in a spherical background. By applying this Ansatz, we investigate some simple analytical solutions. One of them is a regular wormhole supported by the Dirac fields. Other solutions include a pathological black hole and a naked singularity. We analyze the domain of existence and properties of all these solutions.

Accurate and Practical Energy Detection over α-μ Fading Channels

In this study, a novel and exact closed-form expression for detection probability of energy detection (ED) in terms of Meijer’s G-function over α-μ generalized fading channels was derived. It is more accurate and practical than the existing exact expressions and has wide application prospects in the performance evaluations in various areas of wireless communications, especially in the wireless sensor network (WSN) and the cognitive radio network (CRN). Furthermore, an exact and simple analytical solution for the sample size meeting the desired detection performance in terms of the probability mass function of a Poisson distribution was also solved. Simulations verified the detection performance and accuracy of our derived expressions with a small sample size compared to the existing exact expressions and approximations.

A new theoretical approach for the performance simulation of multijunction solar cells

AbstractA new theoretical approach is proposed for the performance simulation of multijunction (MJ) solar cells, starting from the weakness and strength of the Hovel model and of the transfer matrix method for describing the propagation of electromagnetic waves inside the solar cell structure. It is based on the scattering matrix method (SMM) and on a simplified generation function that allow describing with good accuracy the propagation of electromagnetic waves in the solar cell device, preserving, at the same time, the possibility of getting simple analytical solutions of the continuity equations. The numerical stability of the new theoretical approach is first demonstrated on triple junction InGaP/InGaAs/Ge solar cells, in which the Ge substrate is considered as the last layer (layer N) and then as the N‐1 layer. Further, the new theoretical approach is applied to simulate the performance of thin quadruple junction (QJ) InGaP/InGaAs/SiGeSn/Ge solar cells, in two‐ and three‐terminal configurations. Efficiency values of up to 45.1% and 44.9%, respectively, have been simulated at 1000× concentration, by considering the MJ limited by the InGaAs subcell. Finally, it is estimated that the QJ InGaP/InGaAs/SiGeSn/Ge solar cell has the potential to reach efficiencies over 50% by assuming proper antireflective coatings.

Sampling Errors in Observed Gravity Wave Momentum Fluxes from Vertical and Tilted Profiles

Observations from radiosondes or from vertically pointing remote sensing profilers are often used to estimate the vertical flux of momentum due to gravity waves. For planar, monochromatic waves, these vertically integrated fluxes are equal to the phase averaged flux and equivalent to the horizontal averaging used to deduce momentum flux from aircraft data or in numerical models. Using a simple analytical solution for two-dimensional hydrostatic gravity waves over an isolated ridge, it is shown that this equivalence does not hold for mountain waves. For a vertical profile, the vertically integrated flux estimate is proportional to the horizontally integrated flux and decays with increasing distance of the profile location from the mountain. For tilted profiles, such as those obtained from radiosonde ascents, there is a further sampling error that increases as the trajectory extends beyond the localised wave field. The same sampling issues are seen when the effects of the Coriolis force on the gravity waves are taken into account. The conclusion of this work is that caution must be taken when using radiosondes or other vertical profiles to deduce mountain wave momentum fluxes.

Analysis of static and dynamic parameters of the wooden window glass frame

An integral part of each the window is its frame. This frame is a support of the glass panel. It is a part of window used for any load distribution to the other parts of buildings. Investigation of parameters of this member acting in the interaction system (glass-frame) is important for numerical modelling and simplified analytical solution. The main aim of the paper is to present experimental investigation of the static and dynamic parameters results of usual wooden and plastic windows.

Diffraction of Acoustic Waves on a Half-Plane in the Case of Impedance Boundary Conditions. Applications to Sound Barriers

A method is proposed and the simple analytical solutions are obtained for the diffraction problems of acoustic waves on a half-plane with the impedance boundary condition on its one side and the Neumann or Dirichlet condition on the opposite side. It is shown that the properties of these solutions are essential for the optimal design of sound barriers with sound-absorbing walls.

The Influx-Management Envelope Considering Real-Fluid Behavior

Summary The influx-management envelope (IME), Culen et al. (2016), is a decision-making tool for how to deal with an influx during managed-pressure-drilling (MPD) operations that offers a substantial improvement over the traditional MPD well-control matrix (WCM). However, in the case study by Gabaldon et al. (2017), it has been found that the simplified analytical solution introduced by Culen et al. (2016) makes the IME inaccurate for many real-world applications. This paper extends the original IME that considers the gas migration in the annulus as a single bubble, without making the simplifying assumptions that are required to make the equations explicit and analytically solvable. The underlying equations that are required to develop the IME are derived from first principles, and it is shown that using realistic equations of state (EOSs) for gas, and a single-bubble-type model, the equations for the IME can be numerically solved yielding less conservative limits. The proposed approach is significantly faster than constructing the IME through high-fidelity simulations. The proposed method allows for the calculation of a peak circulating pressure at the surface and the maximum weak-point pressure for a given kick size and initial shut-in pressure, as well as a kick envelope with respect to formation limits. This makes the contributions of this paper also relevant for traditional well-control situations. The effect of considering various simplifying assumptions on the resulting IME is studied, and different scenarios that compare the results of the proposed approach with that of the original single-bubble equation for the IME (Culen et al. 2016) are presented.

High‐strength steel wires containing corrosion pits: stress analysis and critical distance based fatigue life estimation

AbstractThe present paper deals with the problem of assessing the fatigue lifetime of high‐strength steel wires containing corrosion pits, with this investigation being based on a large number of experimental data selected from the technical literature. To evaluate the stress concentration phenomena characterising corroded metallic wires, according to the state‐of‐the‐art knowledge, pits were modelled either as semi‐ellipsoidal cavities or as hemispherical notches. The stress concentration factors, Kt, associated with these simplified pit geometries were calculated numerically by solving numerous three‐dimensional FE models. Subsequently, the Kt values being determined according to this standard numerical procedure were post‐processed systematically to derive simple analytical solutions suitable for estimating, in situations of engineering interest, the stress concentration factors associated with pitting corrosion. Finally, after making some assumptions to derive the necessary fatigue properties, the Theory of Critical Distances was used in the form of the Point Method and the Line Method to reanalyse the literature data being collected. This systematic validation exercise allowed us to prove that the Theory of Critical Distances is successful also in assessing the fatigue lifetime of high‐strength metallic cables containing corrosion pits, with the obtained estimates falling within an error factor of 3. Therefore, as far as wires weakened by corrosion pits are concerned, it was demonstrated that the Theory of Critical Distances can be used to post‐process the local linear‐elastic stress fields when they are not only determined numerically but also estimated by using those standard analytical solutions which are strictly valid solely for conventional notches.

COMPARISON BETWEEN SINGLE AND DOUBLE INTEGRAL TRANSFORMATION SOLUTIONS OF HEAT CONDUCTION IN SOLID-STATE ELECTRONICS

The design of modern electronic devices has been dealing with challenges on thermal control. In this work, it is proposed two different ways of modeling the temperature field in Solid State Electronics (SSE) using integral transforms, with several heat generations in the domain of the microchip and external convection. Two proposed approaches solve the heat conduction formulation on the SSE using the Classical Integral Transform Technique (CITT): One performing a single transformation (CITT-ST) and the other performing a double transformation (CITT-DT). Both methodologies are compared and achieved similar results. The simpler analytical solution by CITT-DT contrasts with a complex and cumbersome analytical manipulation of CITT-ST. The results show that CITT-ST is more efficient to obtain the solution, requiring a lower truncation order, for the problem of heat conduction in Solid State Electronics even though it has a more complex formulation.

An Analytical Solution for Lithium-Ion Batteries Cooling

Cooling of the lithium-Ion batteries is an important issue in electric vehicles industries. This paper introduces a simple analytical solution for the battery cooling by considering different parameters that affect the air and liquid cooling of lithium-Ion batteries. It was assumed that the cell dimension is 204mm*129mm.

Analytical solutions of transient heat conduction in multilayered slabs and application to thermal analysis of landfills

The study of transient heat conduction in multilayered slabs is widely used in various engineering fields. In this paper, the transient heat conduction in multilayered slabs with general boundary conditions and arbitrary heat generations is analysed. The boundary conditions are general and include various combinations of Dirichlet, Neumann or Robin boundary conditions at either surface. Moreover, arbitrary heat generations in the slabs are taken into account. The solutions are derived by basic methods, including the superposition method, separation variable method and orthogonal expansion method. The simplified double-layered analytical solution is validated by a numerical method and applied to predicting the temporal and spatial distribution of the temperature inside a landfill. It indicates the ability of the proposed analytical solutions for solving the wide range of applied transient heat conduction problems.

Model Predictive Control Tuning by Inverse Matching for a Wave Energy Converter

This paper investigates the application of a method to find the cost function or the weight matrices to be used in model predictive control (MPC) such that the MPC has the same performance as a predesigned linear controller in state-feedback form when constraints are not active. This is potentially useful when a successful linear controller already exists and it is necessary to incorporate the constraint-handling capabilities of MPC. This is the case for a wave energy converter (WEC), where the maximum power transfer law is well-understood. In addition to solutions based on numerical optimization, a simple analytical solution is also derived for cases with a short prediction horizon. These methods are applied for the control of an empirically-based WEC model. The results show that the MPC can be successfully tuned to follow an existing linear control law and to comply with both input and state constraints, such as actuator force and actuator stroke.

Analytical solution to estimate the point of application of resultant passive earth thrust against unsaturated retaining structures

ABSTRACT This paper describes a simplified analytical solution for computing the magnitude and the point of application of the resultant passive earth thrust, as well as the passive earth pressure distribution against inclined walls retaining unsaturated soils under different steady flow conditions. The present analysis was performed based on the assumptions consistent with the Coulomb approach within the framework of the limit equilibrium method. Possible modifications of induced negative pore pressure or matric suction within the unsaturated zone above the water table have been considered depending upon the steady flow conditions, soil types, and groundwater table positions. Results of the parametric study showed considerable alterations in the passive earth pressure of unsaturated soils due to different parameters and highlight the importance of incorporating suction stress changes directly in the stability analysis of earth-retaining structures.

Localized Conical Edge Modes of Higher Orders in Photonic Liquid Crystals

Most studies of the localized edge (EM) and defect (DM) modes in cholesteric liquid crystals (CLC) are related to the localized modes in a collinear geometry, i.e., for the case of light propagation along the spiral axis. It is due to the fact that all photonic effects in CLC are most pronounced just for a collinear geometry, and also partially due to the fact that a simple exact analytic solution of the Maxwell equations is known for a collinear geometry, whereas for a non-collinear geometry, there is no exact analytic solution of the Maxwell equations and a theoretical description of the experimental data becomes more complicated. It is why in papers related to the localized modes in CLC for a non-collinear geometry and observing phenomena similar to the case of a collinear geometry, their interpretation is not so clear. Recently, an analytical theory of the conical modes (CEM) related to a first order of light diffraction was developed in the framework of the two-wave dynamic diffraction theory approximation ensuring the results accuracy of order of δ, the CLC dielectric anisotropy. The corresponding experimental results are reasonably well described by this theory, however, some numerical problems related to the CEM polarization properties remain. In the present paper, an analytical theory of a second order diffraction CEM is presented with results that are qualitatively similar to the results for a first order diffraction order CEM and have the accuracy of order of δ2, i.e., practically exact. In particular, second order diffraction CEM polarization properties are related to the linear σ and π polarizations. The known experimental results on the CEM are discussed and optimal conditions for the second order diffraction CEM observations are formulated.