Reaction Coefficient Research Articles

Inverse Analysis of Subgrade Reaction Coefficient of Subway Based on Computer Neural Network

Inverse Analysis of Subgrade Reaction Coefficient of Subway Based on Computer Neural Network

Capture theory models: An overview of their development, experimental verification, and applications to ion-molecule reactions.

Since Arrhenius first proposed an equation to account for the behavior of thermally activated reactions in 1889, significant progress has been made in our understanding of chemical reactivity. A number of capture theory models have been developed over the past several decades to predict the rate coefficients for reactions between ions and molecules-ranging from the Langevin equation (for reactions between ions and non-polar molecules) to more recent fully quantum theories (for reactions at ultracold temperatures). A number of different capture theory methods are discussed, with the key assumptions underpinning each approach clearly set out. The strengths and limitations of these capture theory methods are examined through detailed comparisons between low-temperature experimental measurements and capture theory predictions. Guidance is provided on the selection of an appropriate capture theory method for a given class of ion-molecule reaction and set of experimental conditions-identifying when a capture-based model is likely to provide an accurate prediction. Finally, the impact of capture theories on fields such as astrochemical modeling is noted, with some potential future directions of capture-based approaches outlined.

A shock tube experiment and an improved high-temperature diisopropyl ketone model by Bayesian optimization

Ignition delay times (IDTs) for di-isopropyl ketone (DIPK) were measured at pressures of 6 and 10 atm, equivalence ratios (ϕ) of 0.5, 1.0, and 2, and over a temperature range of 1100 - 1500 K. In the DIPK sub-model proposed by Barari et al., DIPK⇔IC3H7+IC3H7CO (R2) was erroneously written as DIPK⇔IC3H7+ IC4H7O and the consumption paths of dimethyl ketene (DMK) are missing. A DIPK sub-model was proposed on the basis of Barari model by correcting these errors, reassigning the rate coefficients of the DIPK unimolecular decomposition reactions (R1, R2), H-abstraction reactions by O˙H (R3, R4), fuel radical isomerization reaction (R5), and β-scission decomposition reactions (R5-R10) using the analogy method, and adopting the theoretical calculations of the H-abstraction reactions by H˙, C˙H3, and HO˙2 (R11-R16) provided by Allen et al.. The rate coefficients of R1-R10 were globally optimized by the Bayesian optimization algorithm using the experimental IDTs and profiles of DIPK and DMK. The optimized model well predicts not only the IDTs and profiles of DIPK and DMK, but also the laminar flame speeds and the profiles of many intermediates (propyne, propylene, and methyl ketene) and small species (H2, CO, CH4) during the pyrolysis of DIPK. The calculation using the method of Chen et al. shows that no optimized reaction violates the collision limit. The sensitivity and rate of production analysis were used to discuss the DIPK unimolecular decompositions, DIPK H-abstraction reactions, DIPK radicals isomerization and β-scission reactions, and the optimization of the DIPK sub-model.

The role of highly vibrationally excited H2 initiating the nitrogen chemistry

The formation of hydrides by gas-phase reactions between H2 and a heavy element atom is a very selective process. Reactions with ground-state neutral carbon, oxygen, nitrogen, and sulfur atoms are very endoergic and have high energy barriers because the H2 molecule has to be fragmented before a hydride bond is formed. In cold interstellar clouds, these barriers exclude the formation of CH, OH, NH, and SH radicals through hydrogen abstraction reactions. Here we study a very energetically unfavorable process, the reaction of N(4S) atoms with H2 molecules. We calculated the reaction rate coefficient for H2 in different vibrational levels, using quantum methods for v = 0−7 and quasi-classical methods up to v =12; for comparison purposes, we also calculated the rate coefficients of the analogous reaction S (3P)+ H2(v) → SH + H. Owing to the high energy barrier, these rate coefficients increase with v and also with the gas temperature. We implemented the new rates in the Meudon photodissociation region (PDR) code and studied their effect on models with different ultraviolet (UV) illumination conditions. In strongly UV-irradiated dense gas (Orion Bar conditions), the presence of H2 in highly vibrationally excited levels (v ≥ 7) enhances the NH abundance by two orders of magnitude (at the PDR surface) compared to models that use the thermal rate coefficient for reaction N(4S) + H2 → NH + H. The increase in NH column density, N(NH), across the PDR is a factor of ~25. We investigate the excitation and detectability of submillimeter NH rotational emission lines. Being a hydride, NH excitation is very subthermal (Trot ≪ Tk) even in warm and dense gas. We explore existing Herschel/HIFI observations of the Orion Bar and Horsehead PDRs. We report a 3σ emission feature at the ~974 GHz frequency of the NH NJ = 12 − 01 line toward the Bar. The emission level implies N(NH) ≃ 1013 cm−2, which is consistent with PDR models using the new rate coefficients for reactions between N and UV-pumped H2. This formation route dominates over hydrogenation reactions involving the less abundant N+ ion. JWST observations will quantify the amount and reactivity of UV-pumped H2 in many interstellar and circumstellar environments.

Numerical model of a pulsed subcritical streamer microwave discharge for problems of plasma ignition of fuel mixtures in the gas phase

An approximate model for estimating plasma heating and conversion caused by a subcritical streamer microwave discharge has been considered and verified. Ignition occurs in an environment with a pressure of 13 kPa, a temperature of 150 K, there is an external air flow at a speed of up to 500 m/s, an initiator antenna and a flat mirror are used to focus electromagnetic radiation, a stoichiometric mixture of propane-air and pure propane is supplied through the cavity in the antenna. Radiation power is 3 kW. The model is three-stage and semi-empirical. The plasma region and its conductivity are set based on experimental statistics; this is a key feature that reduces the consumption of computing resources. The finite element method is used. At the first stage, the Boltzmann equation for the electron gas in the medium is solved in the zero-dimensional formulation for the given parameters of the external electric field. The distribution functions of the electronic energy are obtained as well as the functions of the reaction coefficients. At the second stage, the Helmholtz equations are solved to obtain the distribution of electromagnetic fields near the antenna-initiator, taking into account the given conducting region “plasma”. Based on the obtained distributions of the electric field, the Joule heating powers and the values of the reaction coefficients are calculated. At the third stage, the Navier-Stokes and transfer equations of various types of particles for a compressible medium are solved; taking into account combustion processes for given local heating and plasma reactions. The results are compared with the data of a physical experiment. The distributions of temperature, composition of the medium, velocity of the medium are considered for given local heating power and additional reactions in the plasma region. A stoichiometric mixture of propane with air and pure propane supplied through the antenna are ignited by plasma under the conditions under consideration: the mixture burns in a small area, propane is oxidized in a thin layer of mixing with air. The temperature fields and composition of the medium are compared with photographs of the flame from the experiment. The numerical study shows that under all the conditions considered the model gives results close to the experiment, but there is an overestimation of the power required for ignition by up to two times. The study of the processes of ignition of gaseous mixtures by a subcritical microwave discharge is of interest for the design of propulsion systems with increased reliability, with the possibility of using hardly flammable mixtures. The proposed model gives approximate estimates, while the requirements for resources and time are significantly reduced compared to classical models.

Theoretical study about the hydrogen abstraction reactions on methyl acetate on combustion conditions

Methyl acetate is considered a prototype molecule to study biodiesel ignition and combustion and is seen as a possible fuel or fuel additive. For this reason, methyl acetate decomposition paths have been investigated in recent years. In the present study, hydrogen abstraction reactions on methyl acetate by OH(2Π), HO2(2A'), H(2Σg), O(3P), and O2(3Σg) were conducted, and the effect of methodology and anharmonic corrections on the rate coefficients were evaluated. The M06-2X and B3LYP-D3 functionals with the cc-pVDZ, cc-pVTZ, aug-cc-pVDZ, and aug-cc-pVTZ basis set were used for methodology evaluation and the CCSD(T)/cc-pVTZ//M06-2X/aug-cc-pVTZ for the rate coefficients calculation and literature comparisons. The rate coefficients were calculated in the range of 250-3500K, including tunneling corrections, methyl-hindered rotations, and anharmonic effects calculated by the VPT2 method. The methodology analysis showed that the B3LYP-D3 functional leads to lower activation energies for all elementary reactions studied, and the double-zeta basis is insufficient to calculate precise rate coefficients. The inclusion of anharmonic corrections consistently lowered the rate coefficients of all elementary reactions studied and changed the Arrhenius plot profile with the temperature. Significant anharmonic effects were observed at higher temperatures, being the reaction with O2(3Σg) the most affected by this correction. Differences superior to 105 cm3 molec-1s-1 in the rate coefficients were observed in some cases.

High accuracy barrier heights, enthalpies, and rate coefficients for chemical reactions

Quantitative chemical reaction data, including activation energies and reaction rates, are crucial for developing detailed kinetic mechanisms and accurately predicting reaction outcomes. However, such data are often difficult to find, and high-quality datasets are especially rare. Here, we use CCSD(T)-F12a/cc-pVDZ-F12//ωB97X-D3/def2-TZVP to obtain high-quality single point calculations for nearly 22,000 unique stable species and transition states. We report the results from these quantum chemistry calculations and extract the barrier heights and reaction enthalpies to create a kinetics dataset of nearly 12,000 gas-phase reactions. These reactions involve H, C, N, and O, contain up to seven heavy atoms, and have cleaned atom-mapped SMILES. Our higher-accuracy coupled-cluster barrier heights differ significantly (RMSE of ∼5 kcal mol−1) relative to those calculated at ωB97X-D3/def2-TZVP. We also report accurate transition state theory rate coefficients {k}_{infty }(T) between 300 K and 2000 K and the corresponding Arrhenius parameters for a subset of rigid reactions. We believe this data will accelerate development of automated and reliable methods for quantitative reaction prediction.

Diffusion and reaction in a two-dimensional multilayer body: Analytical solution and imaginary eigenvalue analysis

Heat and mass transfer in multilayer bodies occur commonly in a number of engineering problems, such as thermal management, manufacturing and reacting systems. Much of the past literature on theoretical analysis of multilayer diffusion-reaction problems is based on the assumption of one-dimensional transport. Given the geometrical complexity of practical engineering systems, two- or three-dimensional analysis may be needed for improved accuracy. This work presents theoretical analysis of coupled diffusion and reaction occurring in a general, two-dimensional multilayer body. The transient temperature distribution is written in the form of an infinite series, and the eigenequation for this problem is derived. Adiabatic or isothermal conditions along the side walls are accounted for by appropriate choice of eigenvalues in that direction. Results are shown to be in good agreement with numerical simulations. It is shown that the temperature distribution may converge or diverge, depending on the specific values of various problem parameters, such as reaction coefficients and diffusivities in each layer, as well as the nature of the boundary conditions in both directions. An analysis for the existence of imaginary eigenvalues, which are related to divergence at large times is presented. The theoretical model is used to predict the limits of design parameters to prevent thermal runaway in a two-layer Li-ion cell. This work improves upon the limited past work on multi-dimensional multilayer transport by accounting for the reaction term, and by identifying the possibility of imaginary eigenvalues in such problems. Due to the close relationship between imaginary eigenvalues and thermal runaway, these results may contribute towards improved safety and reliability of practical engineering systems.

Investigation of the combustion process of activated fuel in an automotive diesel engine

BACKGROUND: Intensive development of the transport sphere and the extension of the fleet of tractors and cars, as well as the tightening of standards and requirements in the field of exhaust gas emissions have resulted in increased requirements for the combustion process. This problem can be solved by improving the working process in the engine by preheating the fuel to 300 C in the high pressure fuel supply system. Under external thermal influence, the conditions of mixture formation are improved, an increase in the rate of preflame reactions and a positive change in the dynamics of the combustion process are observed. When potential energy of hydrocarbon molecules increases in the activated complex system, energy is redistributed among active molecules. Along with this, the rate coefficient of a chemical reaction increases due to the concentration of foci of hydrocarbon molecules that have reached the energy barrier.
 The object of research is the process of diesel fuel combustion at preliminary external influence on it.
 AIMS: The aim is to study the combustion process of a motor-tractor diesel engine upon fuel activation. Development of a scheme of individual phases of the activated diesel fuel combustion of depending on the temperature at heating.
 METHODS: Theoretical study and analysis of the combustion process of thermally preprepared fuel for an automotive diesel engine. Theoretical determination of the dependence of the rate of formation and concentration of toxic components and dispersed particles on the rate constant of direct and reverse chemical reactions, as well as on the activation temperature of fuel at emission in exhaust gases.
 RESULTS: The actual coefficient of molecular change of the working mixture increases when diesel fuel is activated. The rate of formation of total nitrogen oxides and hydrocarbons increases by 1.79% and 3.66%. The proportion of unburned carbon in the combustion process on activated fuel varies within (24) %.
 CONCLUSIONS: Theoretically, a scheme for changing individual phases of the activated fuel combustion process by time and temperature has been developed. Fuel heating accelerates the pre-flame preparation in the liquid phase, shortens the duration of the rapid gorenje phase. The concentration of toxic indicators depends on the constant of the chemical reaction rate and the activation temperature of the fuel.

A permeable squeezed flow analysis of Maxwell fluid near a sensor surface with radiation and chemical reaction

The goal of present work is to explore the two dimensional, unsteady and incompressible squeezed Maxwell fluid flows between infinite parallel plates in a permeable medium. Flows in porous channels/tubes are of particular relevance in a variety of biological and mechanical engineering applications. Sensor surfaces are employed to detect germs and help to diagnose diseases in medical. The computational aspects of thermal radiation and heat generation for thermal boundary layer are investigated. In important engineering activities such as drying, temperature regulation, and moisture distribution, chemical reactions affect heat and mass movement. Mass transfer affected by chemical reaction is analyzed. For porous medium, Darcy law is applied and the governing equations are simplified by boundary layer approach. The similarity transformation is applied to transform the governing equations into ODEs which are solved by BVP4C method. The effects of porosity parameter, permeable velocity and Maxwell number on velocity are studied numerically and graphically. The impacts of heat radiation, Prandtl number and heat generation on temperature field are addressed. The influence of Schmidt number and chemical reaction on concentration field is examined. Velocity of the fluid drop for the improvement in porosity parameter, permeable velocity and Maxwell number. Thermal radiation and heat generation increase the temperature whereas mass diffusivity fall by increasing chemical reaction coefficient and the Schmidt number. Effects of Maxwell parameter, heat generation and chemical reaction on physical quantities are studied.

Gas-Phase Ozone Reaction Kinetics of C5-C8 Unsaturated Alcohols of Biogenic Interest.

Unsaturated alcohols are volatile organic compounds (VOCs) that characterize the emissions of plants. Changes in climate together with related increases of biotic and abiotic stresses are expected to increase these emissions in the future. Ozonolysis is one of the oxidation pathways that control the fate of unsaturated alcohols in the atmosphere. The rate coefficients of the gas-phase O3 reaction with seven C5-C8 unsaturated alcohols were determined at 296 K using both absolute and relative kinetic methods. The following rate coefficients (cm3 molecule-1 s-1) were obtained using the absolute method: (1.1 ± 0.2) × 10-16 for cis-2-penten-1-ol, (1.2 ± 0.2) × 10-16 for trans-2-hexen-1-ol, (6.4 ± 1.0) × 10-17 for trans-3-hexen-1-ol, (5.8 ± 0.9) × 10-17 for cis-3-hexen-1-ol, (2.0 ± 0.3) × 10-17 for 1-octen-3-ol, and (8.4 ± 1.3) × 10-17 for trans-2-octen-1-ol. The following rate coefficients (cm3 molecule-1 s-1) were obtained using the relative method: (1.27 ± 0.11) × 10-16 for trans-2-hexen-1-ol, (5.01 ± 0.30) × 10-17 for trans-3-hexen-1-ol, (4.13 ± 0.34) × 10-17 for cis-3-hexen-1-ol, and (1.40 ± 0.12) × 10-16 for trans-4-hexen-1-ol. Alkenols display high reactivities with ozone with lifetimes in the hour range. Rate coefficients show a strong and complex dependence on the structure of the alkenol, particularly the relative position of the OH group toward the C═C double bond. The results are discussed and compared to both the available literature data and four structure-activity relationship (SAR) methods.

AN ATTEMPT TO UNIFY EVALUATION OF GROUND DEFORMATION MODULUS USED FOR COEFFICIENT OF HORIZONTAL SUBGRADE REACTION EVALUATION OF PILES

In this paper, deformation modulus in an evaluation method of coefficient of subgrade reaction were evaluated. As a result, the variation of deformation modulus obtained from the pressuremeter tests can be suppressed its variation by considering the strain dependency of the ground rigidity. Furthermore, the relationships between the deformation modulus obtained from the pressuremeter test and the PS logging can be approximated proportionally, and the relationships between the deformation modulus and the SPT N-value are approximated by a power approximation formula with an intercept, which enable to unify evaluation of the deformation modulus.

Double-Cell Prefabricated Utility Tunnel Composed of Groove-Shaped Elements: An Extended Study of Stiffness Reduction Method

Based on the semi-rigid characteristics of a double-cell prefabricated utility tunnel composed of groove-shaped elements (PUTCGE), this paper proposes an extended study of a stiffness reduction method for PUTCGEs by introducing the stiffness reduction and bending moment adjustment coefficients of the sidewalls. The effects of the joint stiffness ratio, the aspect ratio of width to height, the coefficient of subgrade reaction, and the embedment depth of the utility tunnel on the stiffness reduction and bending moment adjustment coefficients were investigated, and calculation formulas were established. The results showed that the stiffness reduction coefficient increased with the increase in the joint stiffness ratio, while the bending moment adjustment coefficient decreased with the increase in the joint stiffness ratio. The stiffness reduction coefficient increased with the decrease in the aspect ratio of width to height, while the effect of the aspect ratio of width to height on the bending moment adjustment coefficient was found to be very small. It was also found that neither the coefficient of subgrade reaction nor the embedment depth had a significant effect on the stiffness reduction and bending moment adjustment coefficients. The stiffness reduction method for the PUTCGE was validated through comparison with a full-scale physical model test. The proposed method can provide a reference for designing PUTCGEs for underground construction.

Rate Coefficients for OH + NO (+N2) in the Fall-off Regime and the Impact of Water Vapor

The termolecular, association reaction between OH and NO is a source of nitrous acid (HONO), an important atmospheric trace gas. Rate coefficients for the title reaction as recommended by evaluation panels differ substantially at the temperatures and pressures that prevail in the Earth’s boundary layer where the reaction is in the fall-off regime between low- and high-pressure limiting rate coefficients. Using pulsed laser methods for generation and detection of OH, we have reinvestigated the kinetics of the title reaction at pressures of 22–743 Torr (1 Torr = 1.333 hPa) and temperatures (273, 298, and 333 K) in pure N2 and in N2–H2O bath gases. In situ optical absorption measurements were used to rule out any bias due to NO2 or HONO impurities. Our rate coefficients (k1) in N2 bath gas are parametrized in terms of low-pressure (k0) and high-pressure (k∞) rate coefficients and a fall-off parameter (FC) with k1,0N2 = 7.24 × 10–31 (T/300 K)−2.17 cm6 molecule–2 s–1, k1,∞ = 3.3 × 10–12 (T/300 K)−0.3 cm3 molecule–1 s–1, and FC = 0.53. Used with the “Troe” expression for termolecular reactions, these parameters accurately reproduce the current data in the fall-off regime and also capture literature rate coefficients at extrapolated temperatures. The presence of water vapor was found to enhance the rate coefficients of the title reaction significantly. The low-pressure limiting rate coefficient in H2O bath gas is a factor 5–6 larger than in N2, at room temperature (k1,0H2O = 4.55 × 10–30 (T/300 K)−4.85 cm6 molecule–2 s–1) indicating that H2O is much more efficient in quenching the association complex HONO* through collisional energy transfer. Based on measurements in N2–H2O mixtures, a parametrization of k1 including both N2 and H2O as third-body quenchers was derived. Neglecting the effect of H2O results, e.g., in an underestimation of k1 by >10% in the tropical boundary layer.

Derivation of multiple but finite number of imaginary eigenvalues for a two-layer diffusion-reaction problem

Diffusion-reaction problems occur commonly in heat and mass transfer analysis. In the particular case of a multilayer body under certain conditions, such problems are known to result in imaginary eigenvalues that indicate divergence in the temperature/concentration field at large times. For small values of the reaction coefficient, past work has derived conditions in which an imaginary eigenvalue may exist. This work generalizes this result by showing that more than one, but not infinite imaginary eigenvalues may exist under certain conditions in a one-dimensional two-layer diffusion-reaction problem. A systematic investigation of the eigenequation in the imaginary space is carried out in order to derive a closed form expression for the number of imaginary eigenvalues for any given problem. It is shown that, for other parameters being fixed, the larger the values of the reaction coefficients, the greater is the number of imaginary eigenvalues. However, unlike real eigenvalues, it is shown that the number of imaginary eigenvalues is never infinite. While presented for a two-layer body, similar techniques may be applied for a more general multilayer body, although the derivation of explicit equations may be more challenging. Besides its theoretical importance, accounting for imaginary eigenvalues in the problem discussed here is clearly important to ensure accuracy of the mathematical model. The present work contributes towards this by systematically identifying the number and nature of imaginary eigenvalues that occur in multilayer diffusion-reaction heat and mass transfer problems.

Analysis of Load Transfer Law of Short Pile-Net Composite Foundation for High-Speed Railway

Although pile-net composite foundation has been widely used in the construction of high-speed railway, load transfer characteristics of its structural system during the filling of roadbed are seldom studied, so field studies are conducted in the Gan-Long (Ganzhou-Longyan) high-speed railway DK86 + 998.0–DK87 + 191.2 test section, aiming to explore the bearing characteristics of a short pile-net composite foundation over low to medium compressible soil. The study encompasses a field measurement of Earth pressure and pore water pressure which lasted almost two years, and a systematic analysis of variation law of soil pressure between piles and their tops and that of pore pressure in reinforcement zone and retaining layer in medium and low compressibility soil area. The results suggest that the soil in the test section bears the characteristic of a low water content, low porosity ratio, and high liquid limit, with compression factor being approximately 0.25, and test values of the natural soil pressure and the lateral distribution are close to the theoretical values. Soil pressure at the top of the piles is higher than that between the piles by approximately 2.4 times because of the differences in pile-net rigidity. Besides, the soil pressure of the pile-net composite foundation follows the pattern of “jagged” in the transverse direction of the roadbed. The filling load of the composite foundation soil arch is 80 kPa and the composite foundation reaction coefficient is 6.4 kN/m3 when the filling is stable. The pile-soil stress ratio at the shoulder and the center of the line is 3.2 and 2.8, respectively. The change of the hole pressure at the pile end point is larger than that at the reinforcement area, and the side verification short pile can effectively transmit the additional stress of the upper part to the pile end soil layer.

A hybrid high-order method for Sobolev equation with convection-dominated term

We consider a new hybrid high-order method for the Sobolev equation with convection-dominated term. We adopt the Pk polynomials to approximate the primary unknown on mesh elements and faces where k is a given positive integer. The stability and high accuracy are obtained by using reconstruction to formulate the gradient term and convective term and by adding a suitable stabilization term. We analyze the semi-discrete formulation of the hybrid high-order method and use Crank-Nicolson difference in time to get the corresponding fully-discrete scheme. We prove stability and convergence of the approximate solution in energy norm. The energy errors hold irrespective of the reciprocal of dispersion, diffusion and reaction coefficients. Numerical results underline the efficiency of the proposed methods and our analytical findings.

Electrochemical and Kinetic studies for the Degradation of Salicylaldehyde in Non –Aqueous Media

In this study, the electro-oxidation of salicylaldehyde (SA) was investigated in non aqueous media at different scan rate on a platinum electrode by eco friendly cyclic voltammetry (CV) technique. Kinetic and electrochemical parameters were evaluated and reported in terms of standard heterogeneous rate constant (k0 , 6.115×102 s-1), anodic electron transfer rate constant (kox, 1.540×103 s-1), electron transfer coefficient of reaction (α, 0.197), formal potential (E0,1.989 V), and linearity under the influence of scan rate.

Settlements and Subgrade Reactions of Surface Raft Foundations Subjected to Vertically Uniform Load

The settlements and corresponding soil reactions of a surface raft foundation subjected to vertically uniform load at the sites of granular or cohesive soils have been qualitatively introduced to engineers in the past. This study intends to verify the foundation load-response mechanism using three-dimensional finite-element analyses. The Mohr–Coulomb soil model was used to simulate the nonlinear effects of the granular and cohesive soils. The coefficients of subgrade reactions of the soil were back-calculated and compared to those obtained from the rigid foundation model. For all the square surface rafts studied, non-uniform settlements with the largest value at the center can be found. For the smallest rafts located in clay, the soil reactions were found to be rather consistent. For larger rafts in clay, the soil reactions would be changed according to the soil stiffness and raft dimensions. For surface rafts in sand, the effects of soil stiffness on soil reactions appeared to be insignificant. The soil reactions were generally higher at the center and varied with raft dimensions. For the smallest raft in sand, large soil reactions can be found at the edge. The coefficients of subgrade reactions were found to be more consistent with the smallest values occurring at the center. The results of this study indicate that the load-response mechanism of the raft foundation is rather complicated. Such load-response mechanisms are strongly affected by the relative rigidity of the foundation depending on soil stiffness and foundation dimensions.

Maximum principle for time-fractional parabolic equations with a reaction coefficient of arbitrary sign

We consider time-fractional parabolic equations with a Caputo time derivative of order α∈(0,1). For such equations, we give an elementary proof of the weak maximum principle under no assumptions on the sign of the reaction coefficient. This proof is also extended for weak solutions, as well as for various types of boundary conditions and variable-coefficient variable-order multiterm time-fractional parabolic equations.