Transient Analytical Solution Research Articles

Transient temperature analysis and engineering application of steel shell immersed tunnel protected by fireproof board

Based on the Shenzhen-Zhongshan Link, the transient analytical solution of the steel shell immersed tunnel under the protection of fireproof board was deduced and verified through fire test. Then, the variation of the maximum temperature of the bottom steel shell with the thickness and thermal conductivity of the fireproof board was analyzed and suggestions on the selection of fireproof boards were given. Finally, a fitting formula for the maximum temperature of bottom steel shell was proposed. The results show that: (1) When the thermal contact resistance of the steel-concrete interface and the thermal resistance of cavity are 0.01 m2 °C/W and 0.02 m2 °C/W respectively under the case of 25 mm thick fireproof board, the variation of the analytical solution is consistent with test results, and the peak value is similar. (2) Based on the three-dimensional diagram of d -λ-T, with 300 °C as the fire resistance limit of structure, the range of the fireproof board parameters is obtained: when λ reaches 0.14 and 0.4 respectively, d shall not be less than 10 mm and 28.38 mm (3) The scatter points obtained from analysis were fitted using quadratic polynomials and linear functions respectively. Considering the accuracy and simplicity, the linear fitting formula is given as T = 275.66–7.41*d+644.6*λ.

Transient analytical solution for coupled water–gas transport in unsaturated soil cover of landfill

A new analytical solution for coupled water–gas transport in unsaturated landfill cover system is proposed, which can consider transient diffusive-advective transport of gas under transient water transport condition. The proposed analytical solution is first verified against the laboratory measurements and numerical simulations. Using the proposed solution, parametric studies are conducted on the coupled transient behaviors of water–gas transport. The results show that ignoring the coupling effect can lead to significant error in gas transport results under various conditions, e.g., the surface gas emission flux is underestimated by about 28.3%∼79.8% due to ignoring the spatial and temporal variations of water content induced by transient water transport. The influencing parameters (i.e., initial saturation degree, rainfall patterns, and evaporation rate) all have significant impact on the coupled water–gas transport behaviors. The magnitude of the error induced by ignoring the coupling effect increases with increasing initial saturation degree and increasing evaporation rate (e.g., for initial saturation degree increasing from 0.35 to 0.95, the accumulated percolation at cover bottom increases by 10.9 times and the gas emission flux at cover surface decreases by 64.7%). The results highlight that it’s imperative to consider the effect of transient water transport when one needs more accurate assessment of gas transport results in the cover. This analytical solution can also be utilized to aid landfill cover design and to verify other numerical models.

Theoretical and numerical study on transient acoustic wave propagation across ice layers in the Arctic Ocean.

The study of transient acoustic wave propagation across the Arctic Ocean ice layer provides theoretical guidance for the design of trans-ice acoustic communication systems. In this study, the Arctic Ocean was modeled as an ice-water composite structure, where the ice and water are regarded as an elastic solid and liquid, respectively. An analytical transient solution for acoustic wave propagation in this structure was derived using the eigenfunction expansion method. Further, the numerical procedures were presented and used to analyze the acoustic wave propagation characteristics across the ice layer. The results show that waveforms corresponding to the radial displacements are more severely distorted than the axial displacements. The amplitudes of the radial and axial displacements decreased rapidly with increasing propagation distance. The ice thickness had a greater impact on the radial displacement than axial displacement; the thicker the ice, the greater the distortion for both radial and axial displacements.

Prediction of oxygen distribution in the waste mass from an aeration well in bioreactor landfills.

The key to upgrade the efficiency of aerobic remediation of landfills is to determine the distribution characteristics of oxygen concentration during aerobic ventilation. This study discusses the distribution law of oxygen concentration with time and radial distance based on a single-well aeration test at an old landfill site. The transient analytical solution of the radial oxygen concentration distribution was deduced using the gas continuity equation and approximation of calculus and logarithmic functions. Oxygen concentration data from the field monitoring were compared with the results predicted by the analytical solution. The results indicated that the oxygen concentration initially increased and then decreased with prolonged aeration time. With an increase in radial distance, the oxygen concentration rapidly declined, followed by a gradual decrease. The influence radius of the aeration well increased slightly when the aeration pressure increased from 2 to 20 kPa. The field test data agreed with the analytical solution prediction results, preliminarily verifying the reliability of the oxygen concentration prediction model. Results from this study provide a basis of guidelines for the design, operation and maintenance management of a landfill aerobic restoration project.

Transient Analysis of a Finite Queueing System with Bulk Arrivals in IoT-Based Edge Computing Systems

Queueing models can be used for making decisions about the resources required to provide high quality service. In this paper, a finite capacity single server queueing model with bulk arrivals is studied in IoT-based edge computing systems. The transient analysis of the model is carried out and the transient analytical solution to the system is derived with a group of recursive coefficients by using the ordinary differential equations (ODEs) technique. From which the steady-state probabilities are solved. Then, some performance metrics of interest are derived along with numerical results. Although the paper is initiated from the IoT based edge computing platform, the proposed system modeling and analysis method can be extended to more general situations such as telecommunication, manufacturing, transportation, and many other areas that are closely related to people’s daily lives.

Transient and Quasi-Steady-State Analytical Methods for Simulating a Vertical Gas Flow in a Landfill with Layered Municipal Solid Waste

The exploration of gas pressure and its distribution in landfills is a principal concern for the design and management of municipal solid waste (MSW) landfills. A one-dimensional analytical model was proposed to simulate a vertical gas flow in a landfill with layered MSWs. The transient analytical solution was obtained by the method of superposition and eigenfunction expansion and verified by a semi-analytical solution and numerical simulation. According to the results of a parametric analysis by the transient analytical solution, a vertical gas flow in landfills can be simplified to a quasi-steady-state flow. A quasi-steady-state analytical solution for simulating a vertical gas flow in a landfill with layered MSW is proposed. The quasi-steady-state analytical solution showed good agreement with the transient solution. Both temporal and spatial variations of the gas generation rate in MSW landfills were considered in the transient and quasi-steady-state analytical methods. A comparison between the proposed quasi-steady-state solution and the previously described steady-state solution showed that using the average gas generation rate or the gas generation rate corresponding to the average age of the MSW layer in the steady-state solution resulted in an error in the estimation of the gas pressure in landfills. The proposed solutions are reliable and can provide a reference for the design, management, and subsequent restoration of landfills.

MTPS Analytical Temperature and Heat Flux Solution With Thermal Contact Resistance

Abstract An exact transient analytical solution of the temperature fields and heat fluxes for the modified transient plane source (MTPS) method is presented. The MTPS sensor is made of a thin disk with a heater/sensor and a heated guard ring on its surface, providing a one-dimensional heat flow to a semi-infinite sample in contact with its surface. The MTPS sensor measures the thermal effusivity of the sample. Two different solutions are derived, one is in the form of an infinite series with no thermal resistance, and the other is in integral form and includes thermal resistance between the MTPS sensor surface and the measured sample. The theoretical model is supplemented by experimental results.

A Porothermoelastic Model Considering the Dynamic Temperature-Perturbation Boundary Effect for Borehole Stability in Fractured Porous Rocks

Summary Conventional drilling design tends to inappropriately predict the mud density required for borehole stability of deep fractured porous rocks, such as shale, tight sandstone, and hot dry rock, because it is treated as a single-porosity case and even introduces the influence of weak plane to cover the effect on the fracture system. When the external loadings are applied, fractured porous rocks naturally display two different poromechanical responses of the matrix system and the fracture system considering respective hydraulic and mechanical properties. Besides, the constant temperature difference between the drilling mud and formation rock is often chosen as a boundary condition to solve the temperature balance equation, and thus the incorrect prediction of temperature variations of fractured rock further leads to inappropriate evaluation of the pore pressure and stress fields considering the thermo-hydromechanical (THM) coupling (porothermoelastic model), since a dynamic temperature-perturbation boundary condition related to the temperature at the borehole wall actually accounts for the circulating effect of the drilling mud. Therefore, this paper first uses the API RP 13D (1995) model in combination with the circulating temperature-fields model of Raymond (Raymond 1969) to obtain a set of fully transient analytical solutions to circulating temperature fields, including the four types of temperature inside the drilling pipe, borehole annulus, at the borehole wall, and formation. Furthermore, under local thermal equilibrium (LTE) condition, one considers the dynamic temperature-perturbation boundary condition and provides semianalytical porothermoelastic solutions to the field variables around a vertical borehole subjected to nonhydrostatic stresses in fractured porous rock with dual porosity and dual permeability. The solutions for field variables are obtained in line with the plane strain assumption. The variables include displacements, stresses, and two pore pressures of the matrix system and fracture system. The model is verified by the analytical solutions in the case of a porous medium with a single-porosity one under LTE condition. The main results show that the dual-porosity medium displays a higher borehole instability potential than the single-porosity one. This increasingly cooling effect increases the higher risk of the tensile transverse fracturing when the constant temperature-perturbation boundary condition chooses a smaller temperature difference than that of the dynamic case at a later time. The drilling mud-pressure window narrows with increasing time when the coupled porothermoelastic model is considered. It suggests that the drilling engineer takes into consideration the dynamic temperature-perturbation boundary effect, fracture spacing, and fracture width into the predrilling design of the time-dependent safe mud-pressure window (SMPW).

Fully transient analytical solution for solute transport in 1D deforming saturated porous media considering nonlinear compressibility and permeability

In this study, a series of analytical solutions for coupled consolidation and solute transport in 1D saturated deformable porous media considering nonlinear compressibility and permeability relationships is presented. The proposed analytical solutions consider the combined effects of consolidation, diffusion, advection, dispersion, sorption and degradation. The proposed analytical solutions are successfully validated against the experimental results of consolidation-induced solute transport tests and verified against the CST3 numerical model. The results in model verification indicate that the proposed analytical solution shows good efficiency and accuracy and the assumptions taken in deriving the analytical solution of consolidation governing equation have a limited impact on the accuracy of the proposed analytical solution. Using the verified analytical solution, a series of simulations is conducted to investigate the effects of consolidation and solute transport mechanisms (i.e., advection, dispersion, sorption and degradation) on trichloroethylene (TCE) transport through the compacted clay liner (CCL) system. The results indicate that, regardless of which solute transport mechanism is considered, the consolidation of CCL can substantially affect the TCE transport through the CCL liner system. The advection induced by the leachate head and the sorption of CCL increase the effect of consolidation on TCE transport through the CCL liner system, whereas the dispersion of TCE decreases this effect. Moreover, degradation (both the degradation of dissolved and sorbed TCE masses) can barely affect the effect of consolidation on TCE transport.

Transient modeling and analysis of fractional-order resonant very high frequency boost converter

PurposeThis paper aims to present a simplified method to analyze the transient characteristics of a fractional-order very high frequency (VHF) resonant boost converter. The transient analytical solutions of state variables obtained by this method could be used as a guide for parameter design and circuit optimization.Design/methodology/approachThe VHF converter is decoupled into a simplified equivalent circuit model and described by the differential equation. The solution of the simplified equivalent circuit model is taken as the main oscillation component of the transient state variable. And the equivalent small parameter method (ESPM) and Kalman filter technology are used to solve the differential equation of the converter to obtain the steady-state ripple component. Then, by superimposing the abovementioned two parts, the approximate transient analytical solution can be acquired. Finally, the influence of the fractional order of the energy storage elements on the transient process of the converter is discussed.FindingsThe results from the proposed method agree well with those from simulations, which indicates that the proposed method can effectively analyze the transient characteristic of the fractional-order VHF converter, and the analytical solution derived from the proposed mathematical model shows sufficient accuracy.Originality/valueThis paper proposes for the first time a method to analyze the transient characteristics of a fractional-order VHF resonant boost converter. By combining the main oscillated solution derived from the simplified equivalent circuit model with the steady-state solution based on ESPM, this method can greatly reduce the computation amount to estimate the transient solution. In addition, the discussion on the order of fractional calculus of energy storage components can provide an auxiliary guidance for the selection of circuit parameters and the study of stability.

Analytical transient analysis of temporal boundary value problems using the d'Alembert formula.

Temporal boundary value problems (TBVPs) provide the foundation for analyzing electromagnetic wave propagation in time-varying media. In this paper, we point out that TBVPs fall into the category of unbounded initial value problems, which have traveling wave solutions. By dividing the entire time frame into several subdomains and applying the d'Alembert formula, the transient expressions for waves propagating through temporal boundaries can be evaluated analytically. Moreover, unlike their spatial analogs, TBVPs are subject to causality. Therefore, the resulting analytical transient solutions resulting from the d'Alembert formula are unique to temporal systems.

Research on Heat Transfer Characteristics and Typical Parameters Sensitivity Analysis of the Deep Borehole Heat Exchanger Combined With the Geothermal Wells

Abstract Based on abundant hydrothermal geothermal resources at the depth of 1000–2000 m formation in the basin of the BoHai Bay, the deep borehole heat exchanger (DBHE) combined with the geothermal wells is proposed. According to the modified thermal resistance and capacity model (MTRCM), the heat transfer models inside and outside borehole are established. The transient analytical solutions which are the vertical temperature profiles in the inlet (outlet) pipe and the grout of the DBHE and the corresponding dimensionless form are obtained by deducing and solving the heat transfer models inside the borehole. The mathematical model and the analytical solutions are validated by the experimental data and existing studied data. This paper utilizes respectively the Matlab2012 and the Feflow7.1 to solve the heat transfer models inside and outside the DBHE. The sensitivity analysis is performed to examine the influence of typical parameters on the DBHE heat transfer characteristics, including the quantity of geothermal water exploitation, the well distance between the pumping well and the DBHE, the DBHE inlet temperature, the DBHE depth, and the flowrate of circulating water. Under the action of geothermal wells, the heat transfer mechanism is changed in the thermal reservoir, and the DBHE heat transfer capacity can effectively enhance while the quantity of geothermal water exploitation increases and the well distance decreases. However, with the change of the quantity of geothermal water exploitation, the growth rate of the DBHE heat transfer capacity reduces and the sensitivity of the change of the typical parameters on the DBHE heat transfer performance reduces.

An Enhanced Numerical One-Dimensional Transient Fuel Rod Code for Light Water Reactors.(Dept.M)

In this paper a new enhanced simple numerical fuel rod code is developed, which is used to calculate the transient one-dimensional fuel temperature behavior in Light Water Reactors. The proposed code is then applied to determine the transient temperature behavior of fuel due to sudden rise in heat generation rate. The obtained results are then compared with that obtained from an analytical transient solution. The steady state values obtained from both solutions are then compared with that obtained from the corresponding closed form solution of this problem. It is proved that the numerical code is more powerful and accurate.

Artificial aquifer recharge and pumping: transient analytical solutions for hydraulic head and impact on streamflow rate based on the spatial superposition method

The behaviour of transient groundwater mounds in response to infiltration from surface basins has been studied for decades, but some common settings still lack analytical solutions. It has been shown that applying mathematical integration to the line-sink solution developed by Hantush in the 1960s for pumping an unconfined aquifer, considering recharge over the surface of a defined area, is identical to his solution for groundwater mounding below a rectangular basin. This implies that the superposition principle can, generally, be used to directly include pumping wells, as well as aquifer boundaries, to derive a unique solution. Moreover, other line-sink solutions can be used with a spatial superposition method for addressing a variety of hydrogeological settings, provided the behaviour of the relevant partial differential equations is linear. Based on this principle and on existing line-sink solutions, several analytical solutions are proposed that are able to consider a rectangular recharging area and a pumping well in an unconfined aquifer: (1) near a stream, (2) between a stream and a no-flow boundary, with and without the influence of natural recharge, (3) near a stream that partially penetrates an aquifer and (4) for a multi-layer aquifer. For cases including streams, transient solutions of the impact on streamflow rate are also presented. The proposed analytical solutions will be of practical use for managed aquifer recharge, in particular the design of structures for artificially recharging an aquifer, possibly pumped by one or several wells.

Heat Transfer Analysis of Flat Heat Pipe With Enhanced Microchannel Shape

Flat heat pipe has many excellent characteristics such as high thermal conductivity, reversibility of heat flow direction and so on. In this paper, a flat heat pipe model with multiple microchannels is established, and the effects of different microchannel shapes and different materials on the heat transfer efficiency of flat heat pipe are studied. Then, based on the general rectangular and inverted trapezoidal microchannel shape, the enhancement is carried out. That is to fill a number of materials in the bottom of the microchannel to change the bottom shape of the microchannel. In the steady-state thermal simulation of flat heat pipe, the axial temperature of the flat heat pipe with enhanced microchannel shape is lower than that of the flat heat pipe with general microchannel, which shows that the enhanced microchannel shape is helpful to the heat dissipation of flat heat pipe. The fluid simulation results show that the enhanced rectangular microchannel shape can accelerate the flow velocity of working fluid. Therefore, the flat heat pipe with enhanced microchannel shape has better heat transfer performance. In order to obtain the local temperature of flat heat pipe quickly, the transient analytical solution is provided. In this paper, the temperature field calculated by using transient analytical solution is compared with that simulated by COMSOL. The results show that the maximum error between the results of discrete heat source model and continuous heat source model calculated by transient solution and COMSOL simulation results is less than 2%.

Analytic transient solutions of a cylindrical heat equation

The principles of superposition and separation of variables are used here in order to investigate the analytical solutions of a certain transient heat conduction equation. The structure of the transient temperature appropriations and the heat-transfer distributions are summed up for a straight mix of the results by means of the Fourier-Bessel arrangement of the exponential type for the investigated partial differential equation.

A New Approach to Transient Vibration Analysis of Two-Dimensional Beam Structures at Medium and High Frequencies

Abstract Transient analysis of medium-frequency (mid-frequency) and high-frequency vibrations plays an important role in the research and development of complex structures in aerospace, automobile, civil, mechanical, and ship engineering. Low-frequency analysis tools, like the finite element methods, do not work well for mid- and high-frequency problems because they require a huge number of degrees-of-freedom and consequently costly computation, and are sensitive to material properties and boundary conditions. High-frequency analysis tools, such as the statistical energy analysis (SEA) and its variations, are unsuitable for midfrequency problems because they describe the vibrational behaviors of multibody structures in a global manner and cannot provide detailed local information about displacements and internal forces. In this paper, a new method, which is called the augmented distributed transfer function method (DTFM), is proposed for transient vibration analysis of two-dimensional beam structures at medium and high frequencies. Without the need for discretization and numerical integration, the augmented DTFM consistently delivers analytical transient solutions from low to high-frequency regions. A unique feature of the proposed method is that it can provide local information about system response, such as the displacements and internal forces of a structure, at any point and in any frequency region. Additionally, the proposed method provides a platform for model reduction, by which, a balance of efficiency and accuracy in mid- and high-frequency analyses can be achieved. The proposed method is demonstrated in numerical examples.

Total enthalpy-based lattice Boltzmann simulations of melting in paraffin/metal foam composite phase change materials

Phase change materials (PCM) have become a popular choice for building thermal management due to their low cost, chemical stability and high energy density. Though, their low thermal conductivity is a limiting factor in their use. To overcome this limitation, there has been considerable interest in the application of highly conductive substrates such as metal foams. These offer a potential to increase the thermal performance of PCM and to broaden their area of application. However, the influence of micro structured properties on melting is not completely understood and difficult to explore experimentally. In this study, a lattice Boltzmann method (LBM) based on the two relaxation time (TRT) collision scheme for the simulation of melting and conjugate heat transfer is proposed, validated and applied to melting in three-dimensional (3D) structures of composite PCM-metal foam latent heat storages. The model and its implementation is validated against the analytical transient solution of the Stefan problem, proving superlinear grid convergence and close agreement for a large range of lattice relaxation times and Stefan numbers. The interfacial diffusion is found to be effectively limited by leveraging a TRT collision scheme. Very close accordance to measurements and simulation results obtained with other methods is shown for the validation case of melting gallium including natural convection in 2D and 3D. Subsequently, the melting of paraffin in two complex metal foam geometries is simulated. The present simulations successfully describe the multi-domain heat transfer in 3D, where the thermal conductivity of the foam is more than 1000 times larger than that of the paraffin. The predicted progression of the melting front and the influence of the different foam’s specific surface area are in close agreement to earlier simulations.

Transient ellipsoidal combustion model for a porous burner in microgravity

The current study develops a transient combustion model formulated in oblate ellipsoidal coordinates to analyze the behavior of non-buoyant burner-generated diffusion flames. The combustion model is axially symmetric and considers a porous gas-fueled burner called the Burning Rate Emulator (BRE), which is idealized as an ellipsoidal disk. An approximate analytical transient solution for the flame shape and heat transfer to the surface of the burner is generated as a product of the exact steady-state result and the asymptotic transient result that becomes exact far from the burner. Microgravity BRE experiments conducted at NASA Glenn's 5.18-s Zero Gravity Research Facility indicated the evolution of an approximately ellipsoidal flame moving away from the burner with steady state not achieved during the 5-second test period. The microgravity experimental results are shown to be in good agreement with the mathematical model, which can help predict the flame behavior beyond the duration of the test.

Fully transient analytical solution for degradable organic contaminant transport through GMB/GCL/AL composite liners

In this study, analytical solution for degradable organic contaminant transport through a composite liner consisting of a geomembrane (GMB) layer, a geosynthetic clay liner (GCL) and an attenuation layer (AL) is derived by the separation of variables method. The transient contaminant transport in the whole composite liner can be well described avoiding some weird phenomena in existing analytical solutions. The results of parametric study show that GCL has significant effect on improving the barrier efficiency especially for scenarios with high leachate head. The biodegradation and adsorption in GCL have significant influence on the contaminant transport through the composite liner when the half-life of contaminant in GCL is less than 5 years. Otherwise, the effect can be neglected.