Simple Analytical Solution Research Articles

Performance analysis and optimization of rectangular fin arrays used in plate-fin heat exchangers

This paper deals with longitudinal rectangular fin arrays used in plate-fin heat exchangers. The temperature distribution and rate of heat transfer were obtained using 1-D and 2-D solutions. The ranges of independent parameters within which the 1-D solution was within 1% from the 2-D solution were determined. Simple analytical solutions were determined for the rate of heat transfer, fin effectiveness, and augmentation factor. The aspect ratio at which the rate of heat transfer reached a maximum was determined, as well as the corresponding effectiveness and augmentation factor.

On the applicability of Taylor’s theory to the underwater blast response of composite plates

The response of a plane, air-backed plate attached to a rigid baffle when subjected to an underwater plane shock wave is investigated numerically and analytically. The numerical simulation takes into account a full three-dimensional fluid model involving water cavitation effects. Different numerical models are employed and validated by comparing against various experimental results from the literature. The validated numerical setup is then used to simulate the underwater shock response of simply-supported, air-backed, carbon-fiber/epoxy rectangular plates attached to a rigid baffle. Simplified analytical solution is developed based on two-step approach. The two stages considered are early-time phase that adapts Taylor’s fluid-structure theory to calculate an impulsive velocity for the plate and long-time phase that involves determining its free oscillation response within linear elastic domain, taking into account the water-added mass effect. Finally, the applicability of the proposed method is investigated by performing different numerical simulations regarding various combinations of peak pressures and decay times, change of aspect ratios, change of materials as well as ply orientations.

Flexural behaviour of ECC-LWC encased slender high strength steel composite beams

This paper reports an experimental and numerical study on the flexural behaviour of Engineered Cementitious Composites and Lightweight Concrete (ECC–LWC) encased slender High Strength Steel (HSS) composite beams. Four simply supported slender HSS composite beams with different encasement materials (concrete or ECC–LWC) and configurations (fully or partially encased) were first tested under four–point bending. Three–dimensional nonlinear finite element (FE) models were then developed and validated against the test results. Experimental results obtained showed that the two ECC–LWC fully encased slender beams carried higher loading with improved ductility than the concrete fully encased slender beam. Moreover, test results of the ECC–LWC fully encased slender beams confirmed that the encasement successfully eliminated both local buckling and global lateral torsional buckling of the slender HSS I–section. However, the partially encased beam in which encasement was only applied to the compression flange was failed by lateral torsional buckling before its top flange was yielded. A parametric study was also carried out using the validated FE model to further investigate the effectives of different encasement parameters on the flexural behaviour of ECC–LWC fully encased slender HSS composite beams. Finally, a simplified analytical solution is proposed for estimating the flexural strength of the fully encased beams.

An Experimental Investigation of Sample-Fluid Heat Coupling Effect in Thermal Lens Technique.

Laser-induced wavefront distortion is detectable by several techniques based on the photothermal effect. The effect is probed by monitoring the phase shift caused by the bulging of the heated area, the photoelastic effects, and the spatial distribution of the refractive index within the sample and in the fluid surrounding it. A simple analytical solution for the wavefront distortion was only possible for low absorbing materials, with the assumption that the stresses obey either the thin-disk or the long-rod type distributions. Recently, a unified theoretical description for the laser-induced optical path change was proposed to overcome part of this limitation for weakly absorbing materials, regardless of its thickness. In this work, we perform an experimental investigation taking the sample-fluid heat coupling effect into account using the thermal lens technique. The experimental investigation presented here validates the unified model. In addition, we show that the heat-coupling model provides an alternative method to obtain physical properties of non-absorbing fluid by using a reference solid sample.

1+1D implicit disk computations

We present an implicit numerical method to solve the time-dependent equations of radiation hydrodynamics (RHD) in axial symmetry assuming hydrostatic equilibrium perpendicular to the equatorial plane (1+1D) of a gaseous disk. The equations are formulated in conservative form on an adaptive grid and the corresponding fluxes are calculated by a spacial second order advection scheme. Self-gravity of the disk is included by solving the Poisson equation. We test the resulting numerical method through comparison with a simplified analytical solution as well as through the long term viscous evolution of protoplanetary disk when due to viscosity matter is transported towards the central host star and the disk depletes. The importance of the inner boundary conditions on the structural behavior of disks is demonstrated with several examples.

The CPT-violating effects on neutrons’ gravitational bound state

In this work, the CPT-violating (CPTV) interactions on neutrons’ gravitational bound state are studied. With simple analytical solutions, we provide a preliminary investigation into the Lorentz-violation (LV)-induced spin precession due to the and couplings, where and represent LV coefficients. The helicity-dependent couplings can induce unusual phase evolutions with position and momentum dependence. As varies with time due to the Earth’s motion, the spin polarization also shows a sidereal time dependence, and it may be enhanced with time for an ultra-stable polarized state of neutrons. The inseparability of the spin-momentum coupling of the term can also lead to a motional-dependent polarization state. With precisely measured transition frequency between different gravitational bound states, we get a rough bound GeV for unpolarized neutrons. If the spin-flip transition frequency can reach comparable precision in the future, the bound could be improved to the level of 10−24 GeV. The test of weak equivalence principle with polarized atoms may also improve it significantly.

Simplified Analytical Solution of the Contact Problem on Indentation of a Coated Half-Space by a Conical Punch

The contact problem on indentation of an elastic coated half-space by a conical punch is considered. To obtain an explicit analytical solution suitable for applications, the bilateral asymptotic method is used in a simplified form. For that purpose, kernel transform of the integral equation is approximated by a ratio of two quadratic functions containing only one parameter. Such an approach allows us to obtain explicit analytical expressions for the distribution of contact stresses and relations between the indentation force, depth, stiffness and contact radius. The obtained solution is suitable both for homogeneous and functionally graded coatings. The dependence of the characteristics of contact interaction on a relative Young’s modulus of the coating and relative coating thickness is analyzed and illustrated by the numerical examples. Ranges of values of elastic and geometrical parameters are obtained, for which the presence of a coating sufficiently changes the contact characteristics. The accuracy of the obtained simplified expressions is studied in detail. Results of the paper sufficiently simplify engineering calculations and are suitable for inverse analysis, e.g., analysis of indentation experiments of coated materials using either a conical or a pyramidal (Berkovich) indenter.

Seismic design loads of cylindrical liquid tanks with insufficient freeboard

Cylindrical liquid tanks are essential elements of lifelines that should maintain performance during and immediately after an earthquake. Sloshing of liquid in a tank without sufficient freeboard in response to an earthquake can redistribute the hydrodynamic pressure on the tank shell. This study aims to explore changes in the seismic design load caused by insufficient freeboard using experimental and numerical methods. A series of shaking table tests on a small-scale tank subjected to seismic excitation are performed and the effects of insufficient freeboard on the hydrodynamic base shear are investigated. By altering the liquid height-to-tank radius ratio and freeboard, 190 cases were considered. An equivalent simplified mass spring model of the corresponding regular tank for experimental cases is adjusted in the numerical study and a simple analytical solution to estimate the hydrodynamic force on a tank with insufficient freeboard is suggested and validated using the experimental results.

Wave cancellation conditions for the double impact of finite duration in an arbitrary structure

Resonance phenomena in impacting systems can be defined as an amplitude increasing during periodically applied impacts. The wave cancellation phenomenon is defined as application of certain conditions to cancel the wave fully. The double impact system is defined as the application of the first impact with a certain duration tau and then the application of a counter impact in a certain time tau _1 such that the vibrations caused by the first impact are fully disappearing. In the current contribution this phenomenon is first studied for the simplest 1D bar vibration. The response function is introduced as a characteristic for such a phenomenon and, by studying its properties, it is possible to find both an impact duration time tau and an application time tau _1 for the counter impact leading to the wave cancellation. The result is generalized for any arbitrary homogeneous linear non-dissipative mechanical structure described by a semi-elliptic operator Lu. The counter impact can be determined in the same way as in the opposite direction. This general result is numerically illustrated for various operators Lu possessing relatively simple analytical solutions: for a simply supported and a clamped Bernoulli beam, for a fixed membrane and for a Kirchhoff plate. Three potential applications are discussed at the end: a set of verification examples for further analysis of time integration numerical schemes with the energy conservation property; straightforward transfer of cancellation conditions for the double impact to any convenient numerical method in mechanics, e.g. finite element method, iso-geometric method etc.; application of the result in engineering design of impacting devices (hammering etc.) in order to prevent recoil.

Analytical Solution of the Richards Equation under Gravity-Driven Infiltration and Constant Rainfall Intensity

In the field of soil hydrology, the Richards equation is commonly used to model water flow in unsaturated soils. The high nonlinearity of the Richards equation makes it very challenging to solve analytically for situations that are meaningful in practical applications. In this paper, an exact and simple analytical solution of the Richards equation under gravity-driven infiltration and constant rainfall intensity is derived. First, the solution is presented under Torricelli’s law, which mimics the soil hydraulic conductivity function and describes the emptying or filling process of a nonlinear water reservoir. Then, following a similar approach, the solution is extended to the Brooks and Corey soil hydraulic conductivity function, which is generally considered to well describe soil hydrological characteristics. The approach followed in this study is a simple hydraulic approach; therefore, the derived solutions are not affected by uncertainty as long as the hypothesis of the gravity-driven infiltration is satisfied for the selected soils. A comparison with the solution numerically derived by the Richards equation for which the gravity-driven assumption is relaxed is performed and discussed. Interestingly, the comparison indicated that the suggested solution delimits the solutions domain of the Richards equation.

Analytical and Numerical Groundwater Flow Solutions for the FEMME-Modeling Environment

Simple analytical and numerical solutions for confined and unconfined groundwater-surface water interaction in one and two dimensions were developed in the STRIVE package (stream river ecosystem) as part of FEMME (flexible environment for mathematically modelling the environment). Analytical and numerical solutions for interaction between one-dimensional confined and unconfined aquifers and rivers were used to study the effects of a 0.5 m sudden rise in the river water level for 24 h. Furthermore, a two-dimensional groundwater model for an unconfined aquifer was developed and coupled with a one-dimensional hydrodynamic model. This model was applied on a 1 km long reach of the Aa River, Belgium. Two different types of river water level conditions were tested. A MODFLOW model was set up for these different types of water level condition in order to compare the results with the models implemented in STRIVE. The results of the analytical solutions for confined and unconfined aquifers were in good agreement with the numerical results. The results of the two-dimensional groundwater model developed in STRIVE also showed that there is a good agreement with the MODFLOW solutions. It is concluded that the facilities of STRIVE can be used to improve the understanding of groundwater-surface water interaction and to couple the groundwater module with other modules developed for STRIVE. With these new models STRIVE proves to be a powerful example as a development and testing environment for integrated water modeling.

A simplified mathematical model for the dam-breach hydrograph for three reservoir geometries following a sudden full dam break

The prediction of dam-break water flow at dam site is essential to reduce the potential for loss of damage in the downstream floodplain. In this study, the influence of reservoir shapes (rectangular, trapezoidal and triangular wedge) on dam-break discharge hydrographs at a dam site was investigated to estimate the peak discharge and discharge hydrograph quickly. By assuming instantaneous and complete breaches to simplify the discharge process, a formula for the peak discharge and a simple analytical solution to the entire discharge hydrograph following a dam break at the dam site were generated. The discharge hydrograph at the dam site derived by the proposed mathematical model was validated through a comparison with the results calculated by the numerical simulation and other existing approaches. The outflow discharges calculated by both the mathematical and numerical model was very similar under the conditions of the three different reservoir shapes. The overall discharge hydrograph shape was mainly influenced by the length of the reservoir, while the magnitude of the outflow discharge was primarily affected by the initial water depth.

Assisted concentration of Gaussian resources

In spite of their outstanding experimental relevance, Gaussian operations in continuous-variable quantum systems are subjected to fundamental limitations, as it is known that general resources cannot be distilled within the Gaussian paradigm. We show that these limitations can be overcome by considering a collaborative setting where one party increases the amount of local resource with the aid of another party, whose operations are assumed to be Gaussian but are otherwise unrestricted; the two parties can only communicate classically. We show that in single-shot scenarios, unlike in the well-known case of entanglement theory, two-way classical communication does not lead to any improvement over one-way classical communication from the aiding party to the aided party. We then provide a concise general expression for the Gaussian resource of assistance, i.e., the maximum amount of resource that can be obtained when the aiding party holds a purification of the aided party's state, as measured by a general monotone. To demonstrate its usefulness, we apply our result to two important kinds of resources, squeezing and entanglement, and find some simple analytic solutions. In the case of entanglement theory, we are able to find general upper bounds on the regularized Gaussian entanglement of assistance, and to establish additivity for tensor powers of thermal states. This allows us to draw a quantitative and enlightening comparison with the performance of assisted entanglement distillation in the non-Gaussian setting. On the technical side, we develop some variational expressions to handle functions of symplectic eigenvalues that may be of independent interest. Our results suggest further potential for Gaussian operations to play a major role in practical quantum information processing protocols.

Buried pipeline subjected to seismic landslide: A simplified analytical solution

The present study proposes a simplified analytical solution based on Euler Bernoulli's beam theory to investigate the influence of seismic landslide on buried continuous pipeline. Firstly, the differential equation of the beam (pipe) is derived considering the beam is resting on elastic foundation (soil spring) and then the complete solution (both complementary function and particular integral) of the governing differential equation is developed considering available boundary conditions and continuity conditions of the idealized problem. A quartic polynomial function is adopted to simulate non-uniform ground deformation patterns induced from seismic landslide. The results of analytical solution are compared with the pipe deformation curves obtained by earlier researchers along the permanent ground deformation (PGD) zone. Variation of peak pipe displacement of 9.65% and 9.83% are observed for peak ground deformation of 0.6 m and 1.0 m, respectively. Further, a 3D finite element based numerical analysis is also carried out considering block pattern horizontal transverse ground deformation and compared the result with the proposed analytical solution and maximum variation of 4.31% is observed. Finally, numerically investigated the influence of pipe-soil interface properties on pipe response. The present study will be useful for solving problems related to pipe deformation due to horizontal transverse ground deformation in a simplified manner without performing rigorous numerical analysis and displacement pattern for buried pipe can be obtained.

The calculation of the thermal mode of a room with automatic regulation of climate systems

Introduction. The ongoing research of the indoor temperature, regulated by automatically operating climate systems, remains relevant if focused on identification of dependencies which are sufficiently accurate with regard for the majority of essential factors and represented in the engineering format. The mission of the research is to identify a dependency between indoor temperature and time in case of irregular heat supply in the context of prorated heat flow regulation by indoor climate systems. The exponential nature of this dependency serves as the research hypothesis.
 Materials and methods. The author has employed and analyzed the principal equations connecting the most important constituents of a heat flow in a room that has automated climate systems in the context of discontinuous changes. The author has employed a computational model of a non-steady temperature environment of a ventilated room. The proposed model is based on the solution of a system of differential equations describing heat conductivity and transfer on indoor surfaces. The author has composed and analytically solved the general differential equation describing the indoor thermal balance with regard for the climate system’s feedback.
 Results. The author has developed an analytical expression describing the indoor space heating rate in case of prorated temperature regulation by the climate system and irregular heat supply. The analytical expression represents an exponential function of the square root of the time span starting from the moment of a heat supply spike. The author used a residential house in Moscow to analyze the nature of indoor temperature fluctuations with and without heat regulation.
 Conclusions. The author has proven that an indoor temperature rate is mainly driven by the relation between the transmission coefficient of a climate system and the thermal inertia of “massive” building envelopes in case of irregular thermal exposure. The author makes the point that a simplified analytical solution is proven true by the results of the analysis performed with the help of a numerical model. It’s been identified that the ultimate value of an indoor temperature increment is determined as the relation between the heat gain value and the parameter which is proportionate to the transfer coefficient.

On the Stability of Plasma Equilibrium in the Neighborhood of a Straight Current Conductor

The article presents a mathematical model of an equilibrium magnetoplasma configuration in a plasma cylinder containing on its axis a conductor of finite diameter with a current creating a magnetic field confining the plasma. The annular configurations considered here are the simplest elements of a wide class of galatea traps with conductors immersed in the plasma volume. The problems concerning such configurations have a simple analytical solution in terms of ordinary differential equations. A simple result on the existence of smooth equilibrium configurations with restrictions on the maximum plasma pressure related to magnetic units is obtained. The problems of the stability of the configurations—full-scale MHD stability and intermediate stability to perturbations of the same dimension—are formulated and solved. It is shown that intermediate stability takes place within a specified restrictions on pressure and MHD stability tightens this restrictions due to corrugated perturbations depending on the axial coordinate.

A Simple Analytical Solution for the Designing of the Birdcage RF Coil Used in NMR Imaging Applications

A novel analytical solution for the designing of the birdcage RF coil has been demonstrated in this paper. A new concept of dominant resonance path has been introduced in this paper which is used to identify the specific closed current loop in the birdcage RF coil which is responsible for the dominant resonance frequency mode. This concept is used to determine the precise numerical values of the lumped capacitance deployed in the legs and/or end-rings of the birdcage RF coil for its proper operation at the desired resonance frequency. The analytical solution presented in this paper has been established by performing the two-port network based equivalent circuit modeling of the birdcage RF coil. The proposed analytical solution uses T-matrix theory and develops a relationship between the input impedance of the birdcage coil and the impedances of its leg and end-ring segments. The proposed analytical solution provides the information about the resonance frequency spectrum of the birdcage RF coil and solves the issue of its interfacing with external circuits without affecting its resonance characteristics. Based upon the proposed analysis and designing strategy presented in this paper, the low pass, high pass and band pass configurations of the birdcage RF coil were successfully implemented with FPCB (Flexible Printed Circuit board) technique for small volume NMR imaging applications at 1.5 T and 3.0 T MRI system. The results obtained for the implemented birdcage coils using the proposed analysis and designing technique are in closed agreement with already established methods.

Geometrical optimization for silver nanowire mesh as a flexible transparent conductive electrode.

We report the effect of the geometric parameters on transparency and conductivity in a metallic nanowire mesh as a transparent electrode. Today, indium tin oxide and fluorine-doped tin oxide are used as the transparent electrode for displays and solar cells. Still, there is a definite need for their replacement due to drawbacks such as brittleness, scarcity, and adverse environmental effects. Metallic nanowire mesh is likely the best replacement option, but the main issue is how to find the optimal structure and how to get the best performance. Since the interaction of light with nanowire mesh is complicated, there is no straightforward rule with a simple analytical solution. We developed a kit based on wave optics for calculating the optical transmission of metallic nanowire mesh, which, unlike previous works, includes the interaction of light with the nanowire mesh, such as localized surface plasmon resonance (LSPR), surface plasmon polariton (SPP), and Rayleigh anomaly (RA). So, it is possible to accurately predict the effect of these phenomena and the transmission of mesh. Using the mentioned kit, we will be able to investigate the different geometrical structures of meshes to achieve optimal geometry. This kit is based on the classical Maxwell theory and empirical data and uses finite-difference time-domain for solving equations and experiential results for validation. Comparing the results by a redefined figure of merit shows that LSPR has the most significant reduction on transparency, whereas increasing the thickness (t) to width (w) ratio of the nanowire in the metallic mesh can reduce the LSPR effect and/or shifts it to the invisible region. The wire pitch (p) has no tangible impact on LSPR, but p can be chosen higher than 700 or lower than 350 nm to remove the extinction effects of the first-order RA. If p was larger than 150 nm, SPP could appear in the visible region of the spectrum. In small p, lower modes of SPP with higher intensities occur; therefore, there is an optimum value for p around 300 nm. The reduction of t and w reduces the intensity of SPP and causes it to red shift. By comparing the 900 different structures, the highest figure of merit is obtained in a p of 300 nm with a minimum w (10 nm) and maximum t (100 nm).

A simple analytical solution for organic contaminant diffusion through a geomembrane to unsaturated soil liner: Considering the sorption effect and Robin-type boundary

Landfill leachate often contains soluble hazardous substances. Geomembrane (GM) liner is commonly used to protect the subsurface water. The leachate however may diffuse through the GM and enter the underlying natural soil liner. Analytical or numerical modeling for the leachate transport in the soils helps assess the associated risk to groundwater resources. The analytical model has the advantages of simplicity and easy to implement over the numerical model. Yet there is only an analytical model developed for describing the leachate migration through the GM to the unsaturated soil liner. Its solution however is difficult to evaluate. We herein develop a new analytical model to include the sorption effect and simplify the leachate transport through the GM as a Robin-type boundary condition in the model. The solution of the model is in a simple form and very easy to evaluate. The present solution is then compared with a finite-difference solution and the existing solution. The effect of the retardation factor on the leachate migration is investigated. A sensitivity analysis is also performed to assess the effects of model parameters on the leachate distribution. Finally, the present solution is adopted to estimate the transport parameters based on the data from bench-scale composite liner experiments.

Analytical model of leakage through an incomplete-sealed well

This study presents a new analytical model to predict the pressure changes in a monitoring well and the leakage rate due to the leakage through an incomplete-sealed well in a closed system. Different from previous analytical models, herein we develop a new model to simulate the leakage of a leaky well which is incomplete-sealed and connects with the storage aquifer in a certain segment. Based on the problem description and assumptions, the governing equations with corresponding definite conditions are presented and solved by means of Laplace transform, Fourier cosine transform and Duhamel's principle. Verification of the developed model is achieved by comparing with existing analytical model. Then, the leakage rates and pressure changes are calculated for different key factors and sensitivity analyses are conducted. Obviously, a higher permeability accelerates the fluid exchange between the storage aquifer and the monitoring aquifer. Thus, with the increase of the leakage permeability, the leakage from the storage aquifer is more intensive but the ultimate leakage rate is invariable. With the decrease of the dimensionless distance, the seepage resistance along the leakage path is decreasing, resulting in the greater leakage between the two aquifers. Similarly, the dynamic equilibrium is unaffected by the position of the leakage segment. The incomplete seal of the leaky well only affects the spherical flow stage and this effect is relatively small. Therefore, the dimensionless leakage height has no obvious effect on the leakage. The increase of the storativity ratio enlarges the pressure difference between the two aquifers and increases the ultimate leakage rate to achieve the dynamic equilibrium. The leakage in the case with greater storativity ratio increases more significantly. This paper presents a simple and efficient analytical solutions for the leakage through an incomplete-sealed well and extends the applicability of the previous models.