Composition Probability Density Function Research Articles

Replacement Strategy for Aging Nuclear Power Instrumentation and Control Equipment Based on Life Margin

The instrumentation and control (I&C) equipment in nuclear power plants gradually ages and becomes obsolete with increased operation time. Its performance deteriorates, and the probability of its failure also increases gradually. The failure of I&C equipment may directly lead to the degradation of the control or protection functions, reducing the reliability and safety margin required by the design. This will hurt the safety and stable operation of nuclear power plants. Therefore, an aging I&C management/replacement strategy is necessary to control and minimize this problem. In this regard, this paper establishes a module lifetime evaluation model described by a composite probability density function for modules composed of multiple components. On this basis, we have developed a multi-objective aging replacement optimization model aimed at high reliability, economy, and feasibility, and propose an equipment aging replacement optimization calculation method based on the linear-weighted discrete state transition algorithm. The procedure is verified by the application of data from nuclear power engineering. The results show that the proposed aging replacement strategy and method can significantly reduce computational difficulty, improve equipment reliability, and lower aging replacement costs.

Study on performance of a relay-assisted UWOC system based on adaptive optics

To improve the performance of underwater wireless optical communication (UWOC) systems, we propose a relay-assisted UWOC system model based on adaptive optics (AO). The closed expressions of the scintillation index, composite channel probability density function, and outage probability of the Gaussian beam before and after AO compensation are derived using the extended Rytov theory and Meijer G-function. The performance variation of an UWOC system with different parameters is analyzed by simulation. The results show that AO correction can compensate for the distorted wavefront and significantly reduce the intensity fluctuation at the receiving end. The proposed system can efficiently alleviate channel fading, improving the outage probability performance of the UWOC system.

Ergodic Capacity Analysis of UAV-Based FSO Links Over Foggy Channels

In this letter, we investigate the ergodic capacity of unmanned aerial vehicle (UAV)-based free space optics (FSO) links over random foggy channel. More specifically, we derive composite probability density function (PDF) and close approximation for the moments of the composite PDF using the statistical model of a UAV-based 3D pointing error and a random foggy channel. With it, we obtain upper bound and asymptotic approximation of the ergodic capacity for the two possible detection techniques of intensity modulation/direct detection (IM/DD) and heterodyne detection at high and low signal-to-noise ratio (SNR) regimes. The numerical results confirm all the presented analytic results via computer-based Monte-Carlo simulations.

Commercial wind turbines modeling using single and composite cumulative probability density functions

As wind turbines more widely used with newer manufactured types and larger electrical power scales, a brief mathematical modelling for these wind turbines operating power curves is needed for optimal site matching selections. In this paper, 24 commercial wind turbines with different ratings and different manufactures are modelled using single cumulative probability density functions modelling equations. A new mean of a composite cumulative probability density function is used for better modelling accuracy. Invasive weed optimization algorithm is used to estimate different models designing parameters. The best cumulative density function model for each wind turbine is reached through comparing the RMSE of each model. Results showed that Weibull-Gamma composite is the best modelling technique for 37.5% of the reached results.

Unified statistical performance of FSO link due to the combined effect of weak turbulence and generalized pointing error with HD and IM/DD

In this paper, we set up the statistical channel model of ship-to-ship (or ship-to-shore) free space optical links considering generalized pointing error and weak turbulence. We combine various pointing error models with weak turbulence and derive the composite probability density functions (PDF) for each case of pointing error model. Also, using the similarity of the composite PDFs, we obtain a unified expression for the composite PDF. Furthermore, we conduct error rate analysis based on both intensity modulation and direct detection (IM/DD) and heterodyne detection (HD). At last, the numerical results confirm that the derived average error rate gives precise prediction on error rate.

Evaluation of Quadrature-based Moment Methods in turbulent premixed combustion

Transported probability density function (PDF) methods are widely used to model turbulent flames characterized by strong turbulence-chemistry interactions. Numerical methods directly resolving the PDF are commonly used, such as the Lagrangian particle or the stochastic fields (SF) approach. However, especially for premixed combustion configurations, characterized by high reaction rates and thin reaction zones, a fine PDF resolution is required, both in physical and in composition space, leading to high numerical costs. An alternative approach to solve a PDF is the method of moments, which has shown to be numerically efficient in a wide range of applications. In this work, two Quadrature-based Moment closures are evaluated in the context of turbulent premixed combustion. The Quadrature-based Moment Methods (QMOM) and the recently developed Extended QMOM (EQMOM) are used in combination with a tabulated chemistry approach to approximate the composition PDF. Both closures are first applied to an established benchmark case for PDF methods, a plug-flow reactor with imperfect mixing, and compared to reference results obtained from Lagrangian particle and SF approaches. Second, a set of turbulent premixed methane-air flames are simulated, varying the Karlovitz number and the turbulent length scale. The turbulent flame speeds obtained are compared with SF reference solutions. Further, spatial resolution requirements for simulating these premixed flames using QMOM are investigated and compared with the requirements of SF. The results demonstrate that both QMOM and EQMOM approaches are well suited to reproduce the turbulent flame properties. Additionally, it is shown that moment methods require lower spatial resolution compared to SF method.

Unified Finite Series Approximation of FSO Performance Over Strong Turbulence Combined With Various Pointing Error Conditions

In this paper, we investigate both the bit error rate (BER) and outage performance of free-space optical (FSO) links over strong turbulence combined with various pointing error conditions. Considering atmospheric turbulence and pointing errors as main factors that deteriorate the quality of an optical link, we obtain a unified finite series approximation of the composite probability density function, which embraces generalized pointing error models. This approximation leads to new unified formulas for the BER and outage capacity of an FSO link, which account for the two possible detection mechanisms of intensity modulation/direct detection and heterodyne detection. Selected simulation results confirm that the newly derived approximations can give precise predictions of both the average BER and the outage capacity of FSO communication that are generally applicable to all environments.

Comparison of Combustion Models for Lifted Hydrogen Flames within RANS Framework

Within the framework of a Reynolds averaged numerical simulation (RANS) methodology for modeling turbulence, a comparative numerical study of turbulent lifted H2/N2 flames is presented. Three different turbulent combustion models, namely, the eddy dissipation model (EDM), the eddy dissipation concept (EDC), and the composition probability density function (PDF) transport model, are considered in the analysis. A wide range of global and detailed combustion reaction mechanisms are investigated. As turbulence model, the Standard k-ε model is used, which delivered a comparatively good accuracy within an initial validation study, performed for a non-reacting H2/N2 jet. The predictions for the lifted H2/N2 flame are compared with the published measurements of other authors, and the relative performance of the turbulent combustion models and combustion reaction mechanisms are assessed. The flame lift-off height is taken as the measure of prediction quality. The results show that the latter depends remarkably on the reaction mechanism and the turbulent combustion model applied. It is observed that a substantially better prediction quality for the whole range of experimentally observed lift-off heights is provided by the PDF model, when applied in combination with a detailed reaction mechanism dedicated for hydrogen combustion.

Performance analysis of FSO coherent BPSK systems over Rician turbulence channel with pointing errors.

We investigate the performance of coherent free-space optical (FSO) system in terms of bit error rate (BER) evaluation by adopting the modified Rician distribution based coherent channel model which allows taking into consideration the composite effects of both Rician turbulence, including amplitude fluctuation and optical phase distortion, and pointing errors (PEs). By expanding the Rician distribution, a mathematically traceable expression of the probability density function (PDF) for the composite channel is derived in the form of the Meijer-G function. Based on the composite channel PDF, the exact BER expression is obtained, allowing the analysis of BER performance for single-input single-output (SISO) links. This analysis is extended to single-input multi-output (SIMO) links with maximal ratio combining (MRC). With the help of the moment generating function (MGF), the exact BER expression can be simplified into a single integral, facilitating the analysis with high accuracy and reducing calculation complexity. Engineering insights including high-SNR approximated channel PDF, asymptotic BER expression, coding and diversity gains, are investigated and cross-validated for both SISO and SIMO links. Through both analytical and numerical verifications, the impairment due to PEs as well as the effect of modal compensation on the BER performance are discussed in detail, unveiling the fact that their inner relations should be taken into account for optimization. These verify the effectiveness of our models for both SISO and SIMO links with a wide range of different conditions and can be feasibly applied for different types of coherent FSO links.

Numerical analysis of nitrogen oxides in turbulent lifted H2/N2 cabra jet flame issuing into a vitiated coflow

Abstract This paper gives an in-depth insight into NOX (NO, NO2, and N2O) formation of H2/N2 turbulent Cabra jet flame issuing into a hot vitiated coflow. The joint composition probability density function (PDF) was employed to model the combustion and to specify the characteristics of the flame (i.e., scalar variables, concentration of species etc.). The turbulent transport term was modelled by Reynold-Average-Naiver-Stokes (RANS) SSG and molecular mixing was modelled by modified curl model. A combustion mechanism including 13 species and 34 reactions was employed to define the thermochemical state of the flame. The chemical reaction terms were resolved and accelerated by In Situ Adaptive Tabulation (ISAT). The simulation was performed at different equivalence ratios (ER), fuel jet nitrogen content ( Y N 2 , C ), coflow ( T C ) and jet temperatures ( T J ), coflow oxygen ( Y O 2 , C ) and water contents ( Y H 2 O , C ). Results reveal NOX is composed of 30% NO2 and 70% NO in the burner. Reaction rate analysis at different operating points in the ignition kernel demonstrates that N + O H ⇌ N O + H and N O 2 + H ⇌ N O + O H are dominant reactions in NO formation, while N O + H O 2 ⇌ N O 2 + O H is the main reaction in NO2 formation.

Average SEP and channel capacity analysis over Generic/IG composite fading channels: A unified approach

This paper develops an analytical framework to analyse important performance metrics of the wireless communication system over generic fading channels characterized by α−μ/Inverse Gaussian (IG), η−μ/IG, and κ−μ/IG distributions. The important performance metrics, namely, the outage probability (OP), the amount of fading (AOF), the average symbol error probability (SEP) of coherent and non-coherent schemes, the channel capacity under optimal rate adaptation (ORA), channel inversion with fixed rate (CIFR), and truncated CIFR (TCIFR) policy are analysed. Further, low and high signal-to-noise-ratio (SNR) approximations for channel capacity are presented. Furthermore, for asymptotic analysis the approximate composite probability density function (PDF) at the origin was developed and utilized to derive the closed-form expressions for the average SEP of both coherent and non-coherent modulation schemes with maximal ratio combining (MRC), equal gain combining (EGC), and selection combining (SC) diversity schemes. The proposed asymptotic solutions are simple and can be easily used to describe the system behaviour at high SNR. The correctness of the derived results has been verified by comparing them with Monte Carlo simulations.

Validation of Combustion Models for Lifted Hydrogen Flame

Within a Reynolds Averaged Numerical Simulation (RANS) approach for turbulence modelling, a computational investigation of a turbulent lifted H2/N2flame is presented. Various turbulent combustion models are considered including the Eddy Dissipation Model (EDM), the Eddy Dissipation Concept (EDC), and the composition Probability Density Function transport model (PDF) in combination with different detailed and global reaction mechanisms. Turbulence is modelled using the Standard k-ɛ model, which has proven to offer a good accuracy, based on a preceding validation study for an isothermal H2/N2jet. Results are compared with the published measurements for a lifted H2/N2flame, and the relative performance ofthe turbulent combustion models are assessed. It is observed that the prediction quality can vary largely depending on the reaction mechanism and the turbulent combustion model. The best and quite satisfactory agreement with experiments is provided by two detailed reaction mechanisms applied with a PDF model.

ABERs of LDPC-Coded Multi-Hop FSO Over Double GG Fading Channels With Pointing Error and Path Loss

The average bit error rate (ABER) performances of low-density parity-check (LDPC)-coded multi-hop free-space optical (FSO) communication system with decode-and-forward strategy over the aggregated double generalized gamma (double GG) fading channels for plane and spherical waves are investigated. In particular, on the basis of the unified closed-from expressions of the composite probability density function and cumulative distribution function in terms of the Meijer’s G function, the analytical expression of the ABER for this multi-hop FSO system is derived by generalized Gauss-Laguerre quadrature rule and verified by Monte Carlo simulation. Furthermore, LDPC codes are adopted to improve the system performance. The results show that the ABER performance over double GG distribution is mainly limited by both pointing error and path loss for these two transmission beams, whereas the fading can be significantly improved by LDPC codes, and the coding gain improvement becomes more apparent by decreasing the normalized beam-width, increasing the normalized jitter and hop numbers. This work is beneficial for the FSO system design.

Application of the PDF transport model to nonreacting jets using an adaptive Monte Carlo method

ABSTRACTA nonreacting methane turbulent jet flame is simulated using a composition Probability Density Function (PDF) transport model combined with a parabolic flow model closed using a k–ε turbulence model. The model is validated using the measurements of Birch et al. The turbulent concentration field of a methane jet, J. Fluid Mech., vol. 88, pp. 431–449, 1978. For the most part, the agreement between the model predictions and measured PDFs, mean mixture fraction, Root Mean Squared (RMS) mixture fraction, and higher moments is very good, except where jet intermittency effects are important.Although the validation of the model is encouraging, the requirement for large sample sizes limits how the model implemented with the standard Monte Carlo method can be taken forward. To address this issue, the Monte Carlo method is modified such that the number of particles in a control volume is adapted based on a criterion developed from the central limit theorem. The adaptive Monte Carlo method requires approximately 20% of the runtime of the standard Monte Carlo method.

LES/PDF for premixed combustion in the DNS limit

We investigated the behaviour of the composition probability density function (PDF) model equations used in a large-eddy simulation (LES) of turbulent combustion in the direct numerical simulation (DNS) limit; that is, in the limit of the LES resolution length scale Δ (and the numerical mesh spacing h) being small compared to the smallest flow length scale, so that the resolution is sufficient to perform a DNS. The correct behaviour of a PDF model in the DNS limit is that the resolved composition fields satisfy the DNS equations, and there are no residual fluctuations (i.e. the PDF is everywhere a delta function). In the DNS limit, the treatment of molecular diffusion in the PDF equations is crucial, and both the ‘random-walk’ and ‘mean-drift’ models for molecular diffusion are investigated. Two test cases are considered, both of premixed laminar flames (of thickness δL). We examine the solutions of the model PDF equations for these test cases as functions of Δ/δL and h/δL. Each of the two PDF models has advantages and disadvantages. The mean-drift model behaves correctly in the DNS limit, but it is more difficult to implement and computationally more expensive. The random-walk model does not have the correct behaviour in the DNS limit in that it produces non-zero residual fluctuations. However, if the specified mixing rate Ω normalised by the reaction timescale τc is sufficiently large (Ωτc ≳ 1), then the residual fluctuations are less than 10% and the observed flame speed and thickness are close to their laminar values. Away from the DNS limit (i.e. h/δL ≳ 1), the observed flame thickness scales with the mesh spacing h, and the flame speed scales with Ωh. For this case it is possible to construct a non-general specification of the mixing rate Ω such that the flame speed matches the laminar flame speed.

Free-space communications over exponentiated Weibull turbulence channels with nonzero boresight pointing errors.

In this paper, we present analytical expressions for the performance of urban free-space optical (FSO) communication systems under the combined influence of atmospheric turbulence- and misalignment-induced fading (pointing errors). The atmospheric turbulence channel is modeled by the exponentiated Weibull (EW) distribution that can accurately describe the probability density function (PDF) of the irradiance fluctuations associated with a transmitted Gaussian-beam wave and a finite-sized receiving aperture. The nonzero boresight pointing error PDF model, which is recently proposed for considering the effects of both boresight and jitter, is adopted in analysis. We derive a novel expression for the composite PDF in terms of a convergent double series involving a Meijer's G-function. Based on the statistical results mentioned above, exact expressions for the average bit error rate of on-off keying modulation scheme and the outage probability are developed. To provide more insight, we also perform an asymptotic error rate analysis at high average signal-to-noise ratio. Our analytical results indicate that the diversity gain for the zero boresight case is determined only by the ratio between the equivalent beamwidth at the receiver and the jitter standard deviation, while for the nonzero boresight case, the diversity gain is related to the ratio of the equivalent beamwidth to the jitter variance as well as the parameter of the EW distribution.

A Detailed Validation Study of Multi-Environment Eulerian Probability Density Function Transport Method for Modeling Turbulent Nonpremixed Combustion

The current work involves the validation of presumed shape multi-environment Eulerian probability density function (PDF) transport method (MEPDF) using direct quadrature method of moments (DQMOM)-interaction by exchange with mean (IEM) approach for modeling turbulence chemistry interactions in nonpremixed combustion problems. The joint composition PDF is represented as a collection of finite number of Delta functions. The PDF shape is resolved by solving the governing transport equations for probability of occurrence of each environment and probability-weighted mass fraction of species and enthalpy in Eulerian frame for each environment. A generic implementation of the MEPDF approach is carried out for an arbitrary number of environments. In the current work, the MEPDF approach is used for a series of problems to validate each component of MEPDF approach in an isolated manner as well as their combined effect. First of all, a nonreactive turbulent mixing problem with two different Reynolds numbers (Re = 7000 and 11,900) is used for validation of the mixing and correction terms appear in the MEPDF approach. The second problem studied is a diffusion flame with infinitely fast chemistry having an analytical solution. The reaction component is validated by considering a 1D premixed laminar flame. In order to validate the combined effect of mixing and turbulence chemistry interactions, two different turbulent nonpremixed problems using global one-step chemistry are used. The first reactive problem used is H2 combustion (DLR Flame H3), while the second reactive validation case is a pilot-stabilized CH4 flame. The current predictions for all validation problems are compared with experimental data or published results. The study is further extended by modeling a turbulent nonpremixed H2 combustion using finite-rate chemistry effects and radiative heat transfer. The current model predictions for different flame lengths as well as minor species are compared with experimental data. The current model gave excellent predictions of minor species like OH. The differences in the current predictions with experimental data are discussed.

Free-Space Optical Communication with Nonzero Boresight Pointing Errors

The performance of free-space optical (FSO) communication systems is compromised by atmospheric fading and pointing errors. The pointing errors are widely considered as a combination of two components: boresight and jitter. A statistical model is investigated for pointing errors with nonzero boresight by taking into account the laser beamwidth, detector aperture size, and jitter variance. A novel closed-form probability density function (PDF) is derived for this new nonzero boresight pointing error model. Furthermore, we obtain closed-form PDF for the composite lognormal turbulence channels and finite series approximate PDF for the composite Gamma-Gamma turbulence channels, which is suitable for terrestrial FSO applications impaired by building sway. We conduct error rate analysis of on-off keying signaling with intensity modulation and direct detection over the lognormal and Gamma-Gamma fading channels. Asymptotic error rate analysis and outage probability of such a system are also presented based on the derived composite PDFs. It is shown that the boresight can only affect the coding gain, while the diversity order is determined by the atmospheric fading effect as well as the pointing error effect.

Transported PDF modeling of pulverized coal jet flames

A transported composition probability density function (PDF) method is developed for pulverized coal combustion. A consistent hybrid Lagrangian particle/Eulerian mesh algorithm is used to solve the modeled PDF transport equation for the gas phase, with finite-rate gas-phase chemistry. The model includes k–ε turbulence, gradient transport for scalars, and a Euclidean minimum spanning tree (EMST) mixing model. A separate Lagrangian description is used to solve for the coal particle phase, including particle tracking, coal devolatilization and surface reaction models. Interphase coupling models are developed to handle the interaction between the gas phase and the solid phase. Radiative heat transfer is modeled by a P1 model for a gray absorbing emitting and scattering gas–particle system. Two independent laboratory-scale pulverized coal jet flames (“flame A” and “flame B”) are simulated using the new model. For flame A, the baseline model reproduces the measured mean and rms particle axial velocity reasonably well. Some discrepancies are found in particle temperature and gas-phase concentrations, which may in part be due to the uncertainties in the experimental data. Sensitivities of model results to coal-related model variations, turbulence–chemistry interactions, different interphase coupling strategies, and finite-rate chemistry are explored to establish sensitivities and to determine which aspects of the models are most important. The same model is applied to a second flame (flame B), the only change being in parameters related to the different coal composition. It is found that experimental standoff heights cannot be reproduced for three different stoichiometric ratios using a single model. Time scales for chemical reactions, devolatilization and turbulence are extracted and compared, to study the level of turbulence–chemistry–particle interactions in flame A and to test the popular assumption of equilibrium chemistry in coal combustion modeling. Mixture-fraction statistics for flame B are explored to test assumptions that have been proposed for mixture-fraction-based coal models. While the usual assumption of Beta distributions is found to be appropriate, assumptions of statistical independence of two mixture fractions are not valid.

Simulations of transient n-heptane and n-dodecane spray flames under engine-relevant conditions using a transported PDF method

Time-dependent Reynolds-averaged CFD is performed for transient turbulent spray flames in a high-pressure, constant-volume chamber for two single-component fuels using skeletal chemical mechanisms. The simulations span a range of initial pressures, temperatures and compositions that correspond to conventional and advanced (e.g., low-temperature) compression-ignition engine combustion. The objectives are to establish the extent to which turbulent fluctuations in composition and temperature influence ignition delays and lift-off lengths and turbulent flame structure under engine-relevant conditions, and to provide insight into turbulence-chemistry interactions. This is done by comparing results from a model that accounts for turbulent fluctuations using a transported composition probability density function (PDF) method with those from a model that ignores the influence of turbulent fluctuations on local mean reaction rates (a locally well-stirred reactor – WSR – model). For robust diesel combustion conditions, the WSR and PDF computed ignition delays and lift-off lengths are close to each other, and both are in good agreement with experiment. For lower initial temperatures, ignition delays and lift-off lengths from the two models are significantly different, and the results from the PDF model are in better agreement with experiment. The differences are especially striking for n-dodecane. There the PDF-model computed ignition delays and lift-off lengths are within 10% of measured values for initial temperatures of 900K and higher (for 22.8kg/m3 density, 15% oxygen), while the WSR model predicts an ignition delay that is three times the measured value at 900K. At an initial temperature of 800K, the WSR model fails to ignite, whereas the PDF model computed ignition delay and lift-off length are within 30% of the measured values. In all cases, the WSR and PDF models produce significantly different turbulent flame structures, and the differences increase with decreasing initial temperature and oxygen level. The WSR model produces a thin laminar-like flame, while the PDF model gives a broadened turbulent flame brush that is qualitatively more consistent with what is expected for these highly turbulent flames and what is observed experimentally. Thus, while it may be possible to reproduce some global ignition characteristics using a WSR model (depending on the choice of chemical mechanism), turbulent fluctuations play an increasingly important role at lower initial temperatures and oxygen levels.