Steady-state Model Research Articles

MODELING AND SIMULATION OF POST-COMBUSTION CO 2 CAPTURE PROCESS FOR 6.4 MWE BAGASSE FUELED POWER PLANT

The emission of carbon dioxide, a greenhouse gas, contributes to the phenomenon of climate change; the largest source of carbon dioxide emissions are fossil fuel power plants. Post-combustion carbon capture (PCC) technology, which is based on solvents, is currently the most advanced technology for reducing carbon dioxide emissions from power plants. The Post-combustion Process, which uses conventional solvents, has a number of disadvantages, including high energy consumption for regeneration, solvent losses, corrosion of equipment, thermal instability, and environmental impact. Ionic liquids (ILs) and IL-based solvents have produced a novel method of CO 2 capture that is highly efficient, economical, and environmentally friendly. A rate-based steady-state model of the PCC process using the ionic liquid 1-hexyl-3-methylimidazolium glycine ([Hmim] [Gly]) was investigated using Aspen Plus ®. In Aspen Plus ® Electrolyte NRTL method and RK equation of state are used to compute liquid and vapor properties. From the process analysis carried out, it was found that the height and diameter of the absorber packing, the liquid-to-gas ratio and the solvent and flue gas temperatures have a positive effect on the CO 2 capture efficiency. The result showed a maximum CO 2 capture efficiency of 92% at a packed height of 12m and diameter of 0.9m. The best capture efficiency of 98.82% was achieved at an L/G ratio of 4.0.the process simulation study examined five key parameters for ([Hmim] [Gly]) solvent which may reduce the energy and cost required for CO 2 capture. The results show that aqueous ([Hmim] [Gly]) is a promising absorbent for CO 2 capture. Offering a pathway towards more sustainable and efficient carbon capture technologies. This study contributes to the advancement of carbon capture technologies and underscores their significance in mitigating climate change impacts.

Multi input multi output lateral dynamics control of an electric vehicle

Full electric vehicles with multiple and independently controlled powertrains allow for an improvement of vehicle handling capabilities both in steady state and in transient manoeuvres. This paper focuses on active lateral dynamics control of an electric vehicle equipped with 4 in-wheel motors and active rear steering. An integral terminal sliding mode controller (ITSMC) is derived starting from the linearized single track model with the addition of the rear wheel steering angle. The controller has a multi-input multi-output structure and is designed to track vehicle yaw rate and sideslip angle reference quantities through torque vectoring and active rear steering actuation. A novel approach for calculating reference sideslip angle and yaw rate using a logistic function is also presented in this paper. The ITSMC relies on real time knowledge of sideslip angle which cannot be measured in the real vehicle, thus it is estimated through the addition of an extended Kalman filter to the control loop. The performance of the controller is tested with VI-CarRealTime 14 degrees of freedom nonlinear model both for steady state and transient manoeuvres. Simulation results show a good tracking of the reference value with no chattering issues and with an improved behaviour if compared to a sliding mode controller from literature.

Real-time optimization control of variable rotor speed based on Helicopter/ turboshaft engine on-board composite system

In order to adequately leverage the advantages of coaxial compound helicopters (CCH) to enhance overall performance and efficiency through variable rotor speed, and to improve the fuel economy of the composite propulsion system, a real-time optimization control method of variable rotor speed based on helicopter/turboshaft engine on-board composite system is proposed. Firstly, a steady-state model of the CCH required power and an on-board inverse model of the turboshaft engine based on an improved deep neural network propulsion system model (AdamDNN-PSM) are established. On this basis, an on-board adaptive model based on adaptive square root Kalman filter (ASRUKF) estimator is employed to correct the precision of the on-board inverse model and to track the real engine state. Finally, by adopting the feasible sequence quadratic programming (FSQP) method, an optimal speed rotor (OSR) is obtained through real-time optimization control, which reduces fuel consumption during cruising while considering both the geometric and aerodynamic constraints of the rotor. The simulation results demonstrate that the accuracy and real-time performance of the on-board inverse model have been significantly improved. The real-time optimization control method for OSR effectively reduces fuel consumption during cruising and enhances the CCH performance of the entire flight envelope.

ANALYSIS OF THE BIFURCATING DUCT OF AN INLET PARTICLE SEPARATOR IN TRANSONIC FLOW CONDITIONS

Abstract To increase the reliability of turboprop and turboshaft engines in extreme operating conditions, filtering protections such as Inlet Particle Separators (IPS) can be installed at the intake. The flow inside an IPS is highly 3D and unsteady, with fluctuations especially pronounced when transonic conditions are reached. In this paper, we aim at providing the transonic analysis of an industrial IPS designed for aerodynamic lab testing. As a first step, we define the threshold beyond which sonic conditions are reached in the boundary cross sections of the IPS by means of a 1D semi-empirical model. Second, we select a critical configuration, and we perform CFD simulations to analyze the locations and evolution of the transonic flow inside the IPS. Different models are presented, i.e. steady and unsteady Reynolds-Averaged Navier-Stokes, Detached Eddy Simulation and Large Eddy Simulation, characterized by three levels of resolution. The results show the evolution of some transonic shocks: the steady-state model is only providing information on averaged quantities, while the time-resolved simulations offer a more precise overview in the time domain. The analysis in the frequency domain reveals the frequencies of the transonic fluctuations. Finally, we discuss three alternative designs to effectively improve the operating range and mitigate the risks related to the transonic instabilities, comparing the differences in separation efficiency with respect to the baseline case. The results prove that the optimal design choice is given by the trade-off between operating range, pressure losses and separation efficiency.

The active time model of concurrent choice.

The following paper describes a steady-state model of concurrent choice, termed the active time model (ATM). ATM is derived from maximization principles and is characterized by a semi-Markov process. The model proposes that the controlling stimulus in concurrent variable-interval (VI) VI schedules of reinforcement is the time interval since the most recent response, termed here "the active interresponse time" or simply "active time." In the model after a response is generated, it is categorized by a function that relates active times to switch/stay probabilities. In the paper the output of ATM is compared with predictions made by three other models of operant conditioning: melioration, a version of scalar expectancy theory (SET), and momentary maximization. Data sets considered include preferences in multiple-concurrent VI VI schedules, molecular choice patterns, correlations between switching and perseveration, and molar choice proportions. It is shown that ATM can account for all of these data sets, while the other models produce more limited fits. However, rather than argue that ATM is the singular model for concurrent VI VI choice, a consideration of its concept space leads to the conclusion that operant choice is multiply-determined, and that an adaptive viewpoint-one that considers experimental procedures both as selecting mechanisms for animal choice as well as tests of the controlling variables of that choice-is warranted.

Solar radiation and heat sink impact on fluctuating mixed convective flow and heat rate of Darcian nanofluid: Applications in electronic cooling systems

One of the most effective techniques for electronic-cooling devices involves the use of nanofluid, a substance known for its superior heat transfer capabilities. The main challenge in modern production engineering lies in the efficient cooling of electronic devices. Nanofluids have attracted widespread interest in various technical fields due to their exceptional characteristics, which enable effective cooling of electronic devices while improving energy efficiency. In the field of electronic systems, engineers are exploring the potential of nanofluids in a range of applications and phenomena. Choosing the right heat transfer fluid to dissipate heat is crucial in managing electronic component temperatures. This study examines the effects of heat sinks and thermal radiation on the flow of Darcy-Forchheimer nanofluid over a vertical plate, with the aim of enhancing the thermal durability of electronic components. The governing mathematical model has been refined to accurately represent the physical coefficients. To develop a relevant algorithm in the FORTRAN environment, primitive variables are employed in steady-state and oscillatory models, nanomaterial and heat sink. Outputs are presented numerically and graphically using Tecplot 360, revealing that an increase in heat sink capacity leads to a reduction in surface temperature due to the heat sink's ability to absorb excess thermal energy. The use of porous Darcy-Forchheimer material is shown to lower the fluid temperature. The study also shows that the heat transfer rate decreases with an increase in the heat sink coefficient, and the oscillatory frequency of heat transfer reduces with a decrease in the Prandtl number. Decreasing behavior of heat transfer is depicted for maximum thermophoretic factor due to heat sink impact.

Energy Bus-Based Matrix Modeling and Optimal Scheduling for Integrated Energy Systems

Integrated energy systems (IESs) can easily accommodate renewable energy resources (RESs) and improve the utilization efficiency of fossil energy by integrating various energy production, conversion, and storage technologies. However, the coupled multi-energy flows and the uncertainty of RESs bring challenges regarding optimal scheduling. Therefore, this study proposes an energy bus-based matrix-modeling method and a coordinated scheduling strategy for the IES. The matrix-modeling method can be used to formulate the steady- and transient-state balances of the multi-energy flows, and the transient model can clearly express the multi-time-scale characteristics of the different energy flows. The model parameters are fitted with data from experiments and the literature. To address the inherent randomness of the RESs and loads, a coordinated scheduling strategy is designed that contains two components: day-ahead optimization and rolling optimization. Day-ahead optimization uses the system steady-state model and multiple scenarios from the RES and load forecast data to minimize the operation cost while rolling optimization is based on the system’s transient-state model and aims to achieve the optimal real-time scheduling of the energy flows. Finally, a case study is conducted to verify the advantages and effectiveness of the proposed model and optimization method. The results show that stochastic optimization reduces the total daily cost by 1.48% compared to deterministic optimization when considering the prediction errors associated with the RESs and loads, highlighting the stronger adaptability of stochastic optimization to prediction errors. Moreover, rolling optimization based on the system’s transient-state model can reduce the errors between day-ahead scheduling and rolling correction.

Heat Transfer Models and Measurements of Brushless DC Motors for Small UASs

Heat transfer affects a motor’s sizing, its performance, and, ultimately, the overall vehicle’s range and endurance. However, the thermal literature does not have early-stage models for outrunner brushless DC (BLDC) motors found in small unmanned aerial systems (UASs). To address this gap, we have developed a non-dimensional heat transfer model (Nusselt correlation). Parametric experiments of four different-sized BLDC motors under load in Reynolds-matched wind tunnel tests generated data for model correlation. The motors’ aspect ratios (diameter/length) ranged from 0.9 to 1.5. The freestream Reynolds number of the axial flow over the motors ranged from 20,000 to 40,000. The rotational Reynolds number ranged from 10,000 to 20,000. The results showed that aspect ratio had the largest influence on heat transfer, followed by rotational and freestream Reynolds numbers. A steady-state model used the correlation to predict the motor’s ambient temperature differential within 10 K of experimental data. A case study applied the correlation to predict a hypothetical motor’s continuous torque in different environments. The correlation enables conceptual designers to capture thermally-driven trade-offs in early design stages and reduce costly revisions in later stages.

Control and Protection of Medium Voltage Distribution Network Based on Unified Power Flow Controller

The integration of diverse load types, such as large-scale distributed generators and charging stations, presents considerable challenges to distribution networks. The Unified Power Flow Controller (UPFC) provides multiple benefits, including its compact dimensions, cost efficiency, seamless regulation, and swift response. When deployed in distribution networks, the UPFC facilitates the creation of a Distribution Unified Power Flow Controller (DUPFC) system. The DUPFC system enhances medium voltage distribution network regulation but also affects power quality, fault characteristics, and relay protection accuracy. To address these issues, a power flow equivalent circuit model for the medium voltage distribution loop network was developed. The operating principle and steady-state model of the UPFC were investigated, based on distribution loop network parameters and power flow distribution characteristics. A UPFC control strategy was designed to create a suitable DUPFC system for the medium voltage distribution loop network, ensuring coordination between UPFC protection and the distribution system’s protection mechanisms. This system accomplishes rapid fault isolation and automatic load transfer. Finally, a simulation model based on a typical 20 kV distribution loop network structure was selected to validate the effectiveness of the UPFC control strategy and the DUPFC protection system in MATLAB/Simulink.

One-dimensional, multi-fluid model of the plasma-wall transition. II. Negative ions

The plasma-wall transition is investigated by a one-dimensional steady-state multifluid model, which was presented in detail in Part I [T. Gyergyek et al., AIP Adv. 14, 045201 (2024)]. In this work, the plasma-wall transition is analyzed for the case where the plasma consists of singly charged positive ions, electrons, and singly charged negative ions. When the temperature and initial density of the negative ions are varied, a transition between two types of solutions of the model is observed. We call them the low and high solution, with respect to the absolute value of the potential drop. When the density and temperature of the negative ions are above a critical value, the low solution is observed. As the mass of the positive ions increases, these critical values also increase, but only until the ion mass is below about 1000 electron masses. With larger ion masses, the critical density of the negative ions and the temperature no longer change. In the low solution, the potential drop in front of the sheath is determined by the negative ions and is smaller in absolute terms than in the case of the high solution, where the potential drop in front of the sheath is determined by the electrons. If the problem is analyzed on the pre-sheath scale, the transition between the low and high solution is very sharp. However, when the neutrality condition is replaced by the Poisson equation, this transition becomes blurred and the solutions of the model equations exhibit oscillations. The role of the smallness parameter is highlighted. It is shown how the initial electric field is determined. Deviation of the negative ion density profile from the Boltzmann relation is discussed.

A High Step-Down SiC-Based T-Type LLC Resonant Converter for Spacecraft Power Processing Unit

A spacecraft power processing unit (PPU) is utilized to convert power from solar arrays or electric batteries to the payload, including electric propulsion, communication equipment, and scientific instruments. Currently, a high-voltage converter is widely applied to the spacecraft PPU to improve power density and save launch weight. However, the high voltage level poses challenges such as high step-down ratios and high power losses. To achieve less conduction loss, a SiC-based T-type three-level (TL) LLC resonant converter is proposed. To further broaden the gain range and achieve high step-down ratios, a variable frequency and adjustable phase-shift (VFAPS) modulation scheme is proposed. Meanwhile, the steady-state time-domain model is established to elaborate the operation principles and boundary conditions for soft switching. Furthermore, the optimal resonant element design considerations have been elaborated to achieve wider gain range and facilitate easier soft switching. Furthermore, the numerical solutions for switching frequency and phase shift (PS) angle under each specific input could be figured out. Finally, the effectiveness of this theoretical analysis is demonstrated via a 500-W experimental prototype with 650∼950-V input and constant output of 48-V/11-A.

Test Method for Rapid Prediction of Steady-State Temperature of Outer Rings of Bearings under Grease Lubrication Conditions

Temperature has a great influence on the stability of bearing performance. For the study of bearing steady-state temperature, this paper proposes a test method to quickly predict the steady-state temperature of the outer ring of a bearing, which solves the problems in traditional theoretical calculations and simulation analysis methods such as the large number of calculations, complex models, and large errors. Firstly, a mathematical prediction model is established according to the bearing temperature-rise law; then, a bearing steady-state temperature detection device is designed; and finally, the prediction model parameters are solved according to the experimental results, and experimental verification is carried out. It is shown that the prediction model has high accuracy under different load and speed conditions, and the error between the predicted steady-state temperature and the tested steady-state temperature is less than 0.7 °C. This prediction method reduces the single test time of the same speed to 60 min, which greatly improves the efficiency of the temperature detection test. The steady-state temperature model has important theoretical significance in guiding the study of the limiting speed of bearings.

Development of multiple input supply based modified SEPIC DC–DC converter for efficient management of DC microgrid

The development of DC microgrids is reliant on multi-input converters, which offer several advantages, including enhanced DC power generation and consumption efficiency, simplified quality, and stability. This paper describes the development of a multiple input supply based modified SEPIC DC–DC Converter for efficient management of DC microgrid that is powered by two DC sources. Here Multi-Input SEPIC converter offers both versatility in handling output voltage ranges and efficiency in power flow, even under challenging operating conditions like lower duty cycle values. These features contribute to the converter's effectiveness in managing power within a DC microgrid. In this configuration, the DC sources can supply energy to the load together or separately, depending on how the power switches operate. The detailed working states with equivalent circuit diagrams and theoretical waveforms, under steady-state conditions, are shown along with the current direction equations. This paper also demonstrates the typical analysis of large-signal, small-signal, steady-state modeling techniques and detailed design equations. The proposed configuration is validated through the conceptual examination using theoretical and comprehensive MATLAB simulation results. Detailed performance analysis has been done for different cases with various duty ratios. Finally, to show the competitiveness, the multi-input SEPIC topology is compared with similar recent converters.

Effect of pressure, Mg content and AP size on the microscale flame structure of Mg-based propellant

The Mg-based propellant is considered to be a promising fuel for underwater propulsion, but the mechanism of the change of the burning rate is unclear. The Mg-based propellant was divided into two parts on the basis of the BDP multi-flame model: large-size oxidizer particle and pseudo-propellant consisting of fine oxidizer and other components, and the micro steady-state combustion model of Mg-based propellant was established. Quantitatively analysis on the gas-phase multi-flame structure was performed, which focused on the effect of pressure, Mg content, and oxidizer particle size on the burning rate. This model was verified by the results of combustion experiments on ammonium perchlorate (AP) /hydroxyl‑terminated polybutadiene (HTPB) composite propellant and Mg-based propellant, and the mean absolute percentage error between the burning rate obtained from the simulation and the experimental measurements was less than 5%. Results indicated that the flame structure of Mg-based propellant exhibited typical BDP flame characteristic. As pressure increased from 1 MPa to 4 MPa, the premixed AP flame and the preliminary diffusion flame were closer to the burning surface, which enhanced the heat feedback. And accordingly, the burning rate raised from 5.93 mm/s to 9.79 mm/s. Mg content mainly affect the combustion process of the pseudo-propellant. The premixed flame above the pseudo-propellant gradually disappeared as the Mg content increased from 15% to 63%, which led the burning rate to decrease from 17.45 mm/s to 7.83 mm/s. AP particle size affected the effectiveness of the preliminary diffusion flame on the burning surface and resulted in a change in the burning rate. These findings can provide theoretical guidance for the design of Mg-based propellant formulations.

The contradiction between lithospheric thermal and seismic structures reveals a nonsteady thermal state – A case study from Northeast China

The nature of the lithosphere, as the outermost layer of the solid Earth, has long been a focus of the Earth sciences. Many methods have been used to study the properties and structure of the lithosphere, including gravity, magnetism, electricity, seismic, and geothermics. Since the 1970 s, a coupled heat-flow heat-production steady-state model has been employed to study the thermal structure of the continental lithosphere and has helped researchers explain many geothermal observations. However, the steady-state model does not work well in some areas. Taking Northeast China as a case study, we systematically calculated the lithospheric thermal structure, compared the result with seismic research, and concluded that the lithospheric thermal state is nonsteady in the southern Songliao Basin, the southern Greater Xing’an Range and the Changbai Volcano. The lithosphere in southern Songliao Basin features a hot top and a cold bottom. The lithosphere in southern Greater Xing’an Range and the Changbai Volcano feature a cold top and a hot bottom. Our numerical simulation has demonstrated that the response of surface heat flow to thermal disturbance at the bottom of the lithosphere often lags by more than 20 Myr, resulting in a significant disparity between the thermal structure calculated based on surface heat flow and the velocity structure of seismic waves. In addition, this study of the thermal structure in Northeast China shows that the temperature of the Moho in Songliao Basin is very high (>600 °C), which may be an important cause of the lack of Cenozoic volcanism in the Songliao Basin.

Smart management of pressure regulating stations to maximize hydrogen injection in a gas distribution network

Gas distribution networks traditionally maintain constant pressure setpoints at regulating stations, ensuring a consistent supply to end-users with a high safety margin. However, the prospect of hydrogen injection in the gas networks necessitates rethinking operational strategies. This study examines the environmental and fluid-dynamic implications of implementing a smart pressure management approach in regulating stations within a distribution network with a localized hydrogen injection. The research introduces a novel simulation-based optimization algorithm, incorporating a steady-state fluid-dynamic model with hydrogen tracking, used to optimize the pressure setpoints in the stations. The algorithm is applied to a two-pressure levels looped gas network with multiple gas supply nodes and a single hydrogen injection. The test case results reveal that utilizing dynamic pressure setpoints leads to a 22% increase in hydrogen penetration compared to standard operation. Therefore, this study highlights that smart pressure modulation in a gas network with hydrogen injection can unlock a latent decarbonization potential, while simultaneously adhering to safety limits.

Evolution of structure and spectroscopic properties of a new 1,3-diacetylpyrene polymorph with temperature and pressure.

A new polymorph of 1,3-diacetylpyrene has been obtained from its melt and thoroughly characterized using single-crystal X-ray diffraction, steady-state UV-Vis spectroscopy and periodic density functional theory calculations. Experimental studies covered the temperature range from 90 to 390 K and the pressure range from atmospheric to 4.08 GPa. Optimal sample placement in a diamond anvil cell according to our previously presented methodology ensured over 80% data coverage up to 0.8 Å for a monoclinic sample. Unrestrained Hirshfeld atom refinement of the high-pressure crystal structures was successful and anharmonic behavior of carbonyl oxygen atoms was observed. Unlike the previously characterized polymorph, the structure of 2°AP-β is based on infinite π-stacks of antiparallel 2°AP molecules. 2°AP-β displays piezochromism and piezofluorochromism which are directly related to the variation in interplanar distances within the π-stacking. The importance of weak intermolecular interactions is reflected in the substantial negative thermal expansion coefficient of -55.8 (57) MK-1 in the direction of C-H...O interactions.

A kinetics-based approach to the steady-state and impedance response of photoelectrodes

A kinetics-based analytical model is developed to describe the steady-state and impedance response of photoelectrodes in the case where charge carrier build-up modifies the potential distribution across the semiconductor-electrolyte junction. The predictions of the steady-state model are compared with numerical calculations. It is shown that carrier build up should accelerate interfacial hole transfer by increasing the potential drop across the Helmholtz layer. This effect may explain observations of higher reaction orders with respect to hole concentration during photoelectrochemical oxygen evolution. The analytical model is extended to photoelectrochemical impedance spectroscopy (PEIS). The calculations indicate that interpretation of PEIS measurements becomes more complicated if high illumination intensities are used. The modeling of the analytically calculated PEIS response using equivalent circuits is discussed with the aim of understanding the physical significance of the resistive and capacitive elements.

Modeling, Simulation, and Membrane Wetting Estimation in Gas-Liquid Contacting Processes Including Shell-Side Reaction: Biogas Upgrading Using DEA Solution.

The basic principles of a steady-state mass transfer model and the resistance-in-series film model are assessed with the aid of a series of experiments in a gas-liquid contact membrane mini-module (3 M Liqui-Cel MM-1.7 × 5.5) using an aqueous solution of diethanolamine (DEA) of 0.25 M (mol/L) for biogas upgrading. Experimental data show that CO2 removal may exceed 67% and reach 100% in combination with the highest possible recovery of CH4 when employing biogas flow rates in the range of 2.8 × 10-5 - 3.6 × 10-5 m3/s and solvent flow rates within 0.47 × 10-5 - 0.58 × 10-5 m3/s. For the experimental data set, a correlation has been developed, effectively interpolating CO2 removal with the gas and liquid flow rates. The wetting values calculated are concentrated close to each other for the same liquid flow rate without considerably depending on the gas flow rate, especially when applying the Hikita-Yun (reaction rate-shell-side correlation) compared with the Hikita-Costello pair. Furthermore, the calculated wetting diminishes with increasing liquid flow rate, a result that is consistent with previous modeling attempts and relevant literature indications. The assumption of enhanced mass transfer in the liquid-filled part of the membrane pores due to the reaction is scrutinized, leading to objectionable computational wetting values. It is shown that for a concentration of DEA equal to 0.25 M the Hatta numbers and the enhancement factors are not equal in the whole reaction path; thus, the choice of the shell-side correlation has an appreciable impact on the overall analysis, especially for the determination of the wetting values.

Numerical prediction of mean flow and acoustic field of a supersonic impinging jet

The three-dimensional turbulent mean flow and acoustic field of a supersonic jet impinging on a solid plate is studied computationally using the general purpose CFD code Ansys Fluent. A pressure-based coupled solver formulation with the second order weighted central-upwind spatial discretization is applied to compute transient solutions. Cold and hot jet thermal conditions are considered. Mean flow characteristics are investigated by a steady-state modeling approach. Acoustic radiation of impingement tones is simulated using a transient time-domain formulation. The effects of turbulence in steady-state are modeled by the SST k-ω turbulence model. The Wall-Modeled Large-Eddy Simulation (WMLES) model is applied to compute transient solutions. The near-wall mesh on the impingement plate is fine enough to resolve the viscosity-affected near-wall region all the way to the laminar sublayer. Nozzle-to-plate distance is parameterized in the model for automatic re-generation of the mesh and results. Steady-state predictions of hover lift loss and mean jet velocity distributions are compared with experimental data, and favorable agreement is reported. The transient solution reproduces the mechanism of impingement tone generation by the interaction of large scale vortical structures with the impingement plate. The acoustic near-field is directly resolved by Computational Aeroacoustics (CAA) to accurately propagate impingement tone waves to near-field microphone locations. Calculated impingement tone frequencies and sound pressure levels agree with experimental values.