Probability Density Function Model Research Articles

An Investigation of Rotary Cup Burner Assembly with Three Vehicle-Mounted Cooking Stoves by Numerical Evaluation Method

The adaptability of vehicle-mounted heating systems that include burner and stove remarkably influences the system efficiency, heat flux uniformity, and pollutants emission. In this work, the performance of a rotary cup burner assembly with three different cooking stoves was investigated using ANSYS Fluent software based on five factors of thermal efficiency, heat transfer intensity, heating uniformity, CO emissions, and flue gas outlet temperature. The Eulerian-Lagrangian method was used to perform the diesel spray, and the shear stress transfer k-ω turbulence model and the probability density function model were employed to simulate the turbulent combustion. Based on the simulation results, the performance pentagon of the above five factors was constructed to evaluate the comprehensive performance of the new rotary cup burner system. The rotary cup burner had a good performance when it is used in two staple food stoves and a subsidiary food stove. In staple food stove A, its higher furnace increased the heat exchange area of the vessel, while the higher fireboard of staple food stove B caused a higher heat transfer intensity at the bottom of the vessel. However, the higher fireboard also led to higher CO emissions. In consideration of these two factors, the thermal efficiency of stove A was about 7% higher than that of stove B. Different from the staple food stove, the furnace of subsidiary food stove C had better wrapping to the bottom of the boiler so that it had the highest heat transfer intensity. The obtained performance pentagon shows that the comprehensive adaptability performance of stove A was the best and that of stove B was the worst, which is mainly caused by the height of the fireboard and the shape of the vessel. This research guides the optimization of the heating system and promotes the application of the rotary cup burner.

Optimal swirl number and equivalence ratio for minimizing the soot and NO emissions in turbulent methane combustion

The simultaneous effect of the swirl number (SN) and equivalence ratio on NO and soot emissions in turbulent methane–air combustion was numerically investigated. The realizable k-ε model has been applied for modeling turbulence and eddy dissipation (ED) and probability density function (PDF) models were adopted for chemical reaction. The general gradient diffusion hypothesis (GGDH), high order algebraic model (HOGGDH), and simple eddy diffusivity model have been applied for modeling the turbulent heat flux (THF) vector. The Brooke–Moss model has been employed to predict the soot particle quantities. Comparing the results of the different numerical models (PDF and EDM with algebraic THF) with available experimental data indicated that employing the second-order models significantly leads to the modification of predicting temperature distribution in EDM. However, due to the effect of the PDF model on soot modeling, the PDF model provides more accurate results. The numerical simulations have been performed for various SNs (SN = 0–1.33) and equivalence ratios (0.125–0.75). In all of equivalence ratios the flame temperature and pollutants emission (NO and soot) were strongly affected by the SN. Also, the SN has different effects on NO and soot emission. Finally, for each of NO and soot emission, a correlation relative to effective combustion factors such as SN and equivalence ratio was presented. It is found that by increasing the equivalence ratio, the NO emission increases with a power of 2.3, and soot emission decreases with a power of 8.7.

New Probability Distributions in Astrophysics: X. Truncation and Mass-Luminosity Relationship for the Frèchet Distribution

The Frèchet distribution has aided the modelling of scientific data in many contexts. We demonstrate how it can be adapted to model astrophysical data. We analyze the truncated version of the Frèchet distribution deriving the probability density function (PDF), the distribution function, the average value, the rth moment about the origin, the median, the random generation of values and the maximum likelihood estimator, which allows us to derive the two unknown parameters. This first PDF in the regular and truncated version is then applied to model the mass of the stars. A canonical transformation from the mass to the luminosity allows us to derive a new PDF, which is derived in its regular and truncated version. Finally, we apply this new PDF model on the distribution in luminosity of NGC 2362.

Machine learning assisted modeling of mixing timescale for LES/PDF of high-Karlovitz turbulent premixed combustion

Accurate modeling of mixing in the transported probability density function (PDF) method remains a great challenge, especially for turbulent premixed combustion under extreme conditions such as high Karlovitz number Ka. Recently, a power-law based mixing timescale model was developed (Zhang et al., Proceedings of the Combustion Institute, 2021, 38(2): 2917–2927) for the large-eddy simulations (LES)/PDF modeling of high-Ka number turbulent premixed flames. It is found in this work that the power-law mixing timescale model is highly sensitive to the model parameters. It is thus critically needed to develop accurate calibration of these model parameters. The empirical specification of the model parameters developed in Zhang et al. is found to be inadequate for accurate modeling of the mixing timescale. Machine learning is introduced as an attractive alternative in this work for the specification of the model parameters. A high-Ka number DNS jet flame is used as the training and validation of the machine learning models. The choices of the input parameters are discussed and compared for the machine learning models. The effect of differential molecular diffusion on mixing is examined by including the effect of the Lewis number in the training of the machine learning models. The performance of different machine learning algorithms is compared for the specification of the mixing model parameters. Overall, excellent performance of the machine learning models is observed for assisting the mixing modeling. The feasibility, interpretability, applicability, generality, and portability of using machine learning are discussed in general to provide a perspective on applying data-driven machine learning for turbulent combustion modeling studies.

Spatiotemporal just noticeable difference modeling with heterogeneous temporal visual features

Developing accurate Just-noticeable difference (JND) models are challenged by complicated HVS characteristics and nonstationary features of video sequence. Great efforts have been devoted to JND modeling, and inspiring performance improvements are witnessed in the literature, especially spatial JND models. However, there are not only urgent requirement but also technical potentiality for improving temporal JND models fully accounting for the temporal perception characteristics. In terms of temporal JND modeling, there are two challenges, one is how to extract perceptual feature parameters of source video, and the other is how to quantitatively characterize the interaction relationship between feature parameters and HVS characteristics? Firstly, this work extracts content-aware temporal feature parameters having predominate impacts on vision perception, including motion (foreground/background), pixel-correspondence duration and inter-frame residue fluctuation intensity along temporal trajectory, and investigates the HVS responses to these four heterogeneous feature parameters. Secondly, this work proposes respective probability density functions (PDF) in the perception sense to quantitatively depict the attention and suppression perception responses of feature parameters, accounting for the temporal perception characteristics. Using these PDF models, we fuse the heterogeneous feature parameters from the viewpoint of uniform dimension,i.e. self-information measured visual attention and information entropy measured masking uncertainty, achieving heterogeneous parameter homogenization. Thirdly, with self-information and entropy results, this work then proposes a temporal weight model, by striking the balance between visual attention and masking suppression, to adjust the spatial JND threshold, and then develops the improved spatiotemporal JND model. Intensive simulation results verity the effectiveness of the proposed spatiotemporal JND profile, with competitive model accuracy compared with the-state-of-the-art candidate models.

A new maximum entropy method for estimation of multimodal probability density function

The high-precision estimation of a multimodal probability density function is a difficult problem in many engineering fields. We propose a new method to improve the estimation accuracy based on the fractional moment-based maximum entropy method with a nonlinear transformation and a multi-peak recognition method. For the translation parameters in the nonlinear transformation, three approaches, such as sample-based least square polynomial fitting technique, sample-based kernel density estimation and classical maximum entropy method, are presented to determine the parameters. By adjusting the translation parameter, the valley of the probability density function curve can be translated to the position with a larger slope, and the distance between adjacent peaks is enlarged to avoid the wrong fitting form of the probability density function curve with multiple peaks. After the parameters of the transformation are obtained, the fractional moment-based maximum entropy method is applied to predict the probability density function of the transformed performance function. Two numerical examples are used to verify the accuracy and stability of the proposed method. Two engineering examples are introduced to illustrate the applicability and efficiency of the proposed method in the real-life engineering setting. It is concluded that the proposed method uses fewer moments with less additional calculation costs, and has good computational efficiency and applicability for the modeling of multimodal probability density functions, compared with the classical fractional moment-based maximum entropy method.

Impact of Pointing Errors on the Performances of Double Rician FSO Channels

In this paper we will propose new analytically traceable probability density function (PDF) model for free space optics (FSO) turbulence, obtained as a generalization of double Ricean turbulence model, that encompasses both large-scale and small-scale turbulence eddy effects along by taking into account performance decreasing influence of misalignment introduced through boresight pointing error model. Consequently, after delivering the closed-form expressions for the newly introduced double FSO model, we obtain the analytical expressions for the bit error rate (BER) performance for the Double Rician distribution affected by misalignment. Numerical results will show the impact of system parameters on FSO link performance and we will provide full performance analysis. © 2021.

Quasi-site-specific multivariate probability distribution model for sparse, incomplete, and three-dimensional spatially varying soil data

ABSTRACT In a previous work, the first two authors proposed a data-driven method that can construct a site-specific multivariate probability density function model for soil properties using sparse, incomplete, and spatially variable site investigation data. The spatial variability was limited to the depth direction (horizontal variability was not considered). This data-driven method is referred to as GPR-MUSIC-X. In the current paper, two improvements with respect to GPR-MUSIC-X are made. First, the one-dimensional spatial variability considered by GPR-MUSIC-X is extended to three-dimensional spatial variability (denoted by GPR-MUSIC-3X). Second, a hierarchical Bayesian model (HBM) is adopted to learn the cross-correlation (correlation among different soil parameters) behaviour of generic sites in a soil database accounting for site differences (or uniqueness), and the learning outcome is incorporated into GPR-MUSIC-3X. The resulting model is a quasi-site-specific model (denoted by HBM-MUSIC-3X) because it not only is based on site-specific data but also is informed by the soil database in a manner sensitive to site uniqueness. A case history is used to illustrate the effectiveness of the proposed HBM-MUSIC-3X.

A new approach to maximize the wood production in the sustainable management of Amazon forest

• Key message We presented a method to determine the optimal wood production in Amazon forest. The database that feeds the procedure is faster to obtain when compared to other methods. The simulations resulted in higher wood volume production when compared to the current management system. Moreover, it avoids overexploiting several species which could occur due to felling trees before maturity. • Context Currently in Brazil, generalized forest law rules the management of the entire set of timber species and typologies. However, it is important to consider dynamics specificity of each species to ensure management sustainability, especially for the Amazon forest with its wide floristic diversity. • Aims To develop a procedure to determine which logging diameter would achieve optimal wood production by species, using Apuleia leiocarpa (Vogel) J.F. Macbr., Erisma uncinatum Warm., Hymenolobium excelsum Ducke, and Trattinnickia burserifolia Mart. as studying case. • Methods Two main methodologies of analysis by species were combined: probability density function (PDF) and growth modeling. The growth models were used to derive the volume increment curves at the individual tree level. To detect the points of maximum annual increment in volume at the population tree level, we used PDF with adjusted growth equations. • Results The population maximum annual volumetric increments occurred in smaller diameters compared to that of the individual level. When combining shorter cutting cycles with the population biological rotation point considered the minimum felling diameter (MFD), we observed higher annual increments in volume than that achieved using the Brazilian law criteria (MFD = 50 cm) or other MFD tested. • Conclusion The procedure proposed may be used by forest managers and forest law-makers, aiming to maximize sustainable wood production in the Amazon forest.

Modeling and Performance Optimization of Unmanned Aerial Vehicle Channels in Urban Emergency Management

With the development of smart cities, the use of unmanned aerial vehicles (UAVs) for interactive information exchange between air and ground can provide effective support for the deployment of emergency work. However, the existing UAV air-to-ground channels often use a single channel model. Considering that the density and distribution of obstructions on information transmission paths at different heights are different, only using a single channel model greatly affects the reliability of communications. Aiming at addressing the different channel characteristics of air-to-ground channels at different heights, a height-based adaptive SUUL-SULA channel model is proposed in this paper. Firstly, in the ultra-low altitude environment, the influence of large-scale fading and small-scale fading on the envelope of the received signal is discussed based on the classic LOO model, and the probability density function and bit error rate model of the received signal are derived. Secondly, a SULA channel model based on Jakes’ model is proposed in the low-altitude environment. The uniform circular array beamforming technology is adopted to realize the design of the Doppler frequency shift compensation algorithm. Finally, the simulation results show that the SUUL-SULA model effectively reduces the bit error rate of the system and improves the reliability of communication. Therefore, this model can provide effective physical support for the application of UAV in smart city emergency management.

Effects of the crack geometric features on the probability density of spherical healing agent particles in concrete

Accurately assessing and predicting the probability distribution characteristics of the healing agent particle on the crack surface is the prerequisite for the design of self-healing concrete. To accurately evaluate the probability distribution features, probability density function models considering the geometric characteristics of cracks were established for different healing agent particles and verified by a large-scale sample size test. The results showed that the distribution of the healing agent particle in concrete was consistent with a Binomial distribution. Poisson approximate was applicable to small size and large added quantity of healing agent particles while Normal approximate was applicable to large-size and low-dosing quantity healing agent particles. Aggregates would have an impact on the geometric characteristics of cracks, increasing the amount of healing agent particles distributed on the crack surface and the variance. To guarantee the healing effect, a calculation method for the theoretical dosage of the healing agent particle with a 95% confidence level was established. The research work provides a basis for the quantitative design of self-healing concrete.

Study on Lifted Flame Stabilization Under Different Background Pressures

Abstract A numerical experimental investigation is presented for a steady methane lifted flame and a nonreaction jet flow in a co-flow of hot combustion products from lean premixed air-hydrogen combustion. The main objective has been to analyze the dependence of methane jet flame stability on the background pressure: a pressurized vitiated co-flow burner (PVCB) has been used to study the methane lifted flame and nonreaction jet flow under different background pressures (1–1.5 bars). The lifted flame is characterized by a liftoff height, which has been measured with a high-speed camera, and a central jet flow defined by the jet velocity, which has been measured by means of a high-sensitivity Schlieren imaging system. The experimental results show that the liftoff height decreases for an increment in the background pressure (from 1 to 1.5 bar at 1073 K) and in the co-flow temperature (from 1058 K to 1118 K at 1 bar). The standard deviation of the liftoff height also reduces for an increase in either the background pressure or the co-flow temperature, which indicates that the liftoff height is more stable at higher background pressures and co-flow temperatures. As far as the experimental tests on the nonreaction jet flow is concerned, the jet velocity becomes extinct faster as the background pressure rises, which is consistent with the decrease in the liftoff height as the background pressure grows. The evolution of the jet velocity has been proved to be another important factor that affects the liftoff height under different background pressures (physical factor), in addition to the fuel autoignition delay (chemical factor). The simulation data led with a Reynolds-averaged Navier–Stokes (RANS)/probability density function (PDF) model show that an increment in the background pressure makes the temperatures increase and induces a brighter yellow part of lifted flame, which leads to more soot production. This proves that the flame is not completely premixed. On the other hand, the Schlieren images of the non-reaction jet flow highlight that the flame is partially premixed, since the edge of the jet is not well defined, as the jet penetration increases with time. The liftoff height values of the flame in the numerical simulations were found to be generally higher than those measured in the corresponding experiments. This discrepancy was caused by an appreciable radiation heat loss at the thermocouple. A correlation was therefore developed for the thermocouple temperature measurement in order to correct the inaccuracy.

A spatially constrained asymmetric Gaussian mixture model for image segmentation

The finite Gaussian mixture model (GMM) is a flexible and powerful tool for addressing many computer vision and pattern recognition problems. The Gaussian distribution is a probability distribution that is symmetric with respect to the mean. However, in many segmentation applications, the observed data obey an asymmetric distribution. Furthermore, the GMM is sensitive to imaging noise. To alleviate these issues, a new finite anisotropic asymmetric normal mixture model is presented in this paper. Note that GMM is a degraded case of our proposed model. First, the proposed model employs anisotropic spatial information to reduce the effect of imaging noise while preserving the details, such as edges, corners and slim structure objects. Second, the anisotropic spatial information is coupled into the skew normal distribution to fit the observed data obeying an asymmetric distribution. Then the modeling and estimation of the object intensity probability density function are proposed by using the anisotropic skew normal mixture model. The proposed model not only has the capability to fit the observed data obeying a non-symmetric distribution, but also can reduce the effect of noise while preserving the objects details. Finally, expectation maximization (EM) algorithm is adopted to estimate the model parameters in order to maximize the log-likelihood function. The experiment results on synthetic images and natural grayscale images demonstrate the superior performance of the proposed model compared with other state-of-the-art segmentation methods.

CFD simulation of bubbly flow in a long coaxial heat exchanger

The numerical investigation of a two-phase bubbly flow inside a long vertical counterflow heat exchanger has been performed in this study. The heat transfer was running from the steam moving inside the tubing to the two-phase bubbly water in the annulus. A two-dimensional steady-state heat transfer and fluid flow model are based on the coupling of a probability density function (PDF) coupled with a k-epsilon turbulence model is determined to be the best computation technique in this large-scale case. The k-epsilon model was used to analyze the carrier fluid (water or steam) and the PDF model to calculate bubble flow. The authors desired to receive practical results of the heat transfer process rather than paying attention to the formation of bubbles, their coalescence, tracking flow regimes, and other turbulence-related features that are useless in industrial applications. The article explored two cases: short (800 m) and long heat exchanger (8000 m). Several thermodynamic parameters were on track: flow velocities, pressure, temperature, and concentration of the bubbles in the radial direction. Einstein's correction equation adjusted the viscosity of the bubbles. Solution converged with non-uniform grid spacing in the longitudinal and radial directions. The results gave satisfactory conclusions about the validity of the obtained data with theoretical and experimental results. This article opens a new way on computational fluid dynamics (CFD) application to a large-scale simulation of long vertical or horizontal pipelines and coaxial heat exchangers with the coupled bubbly flow and heat transfer applicable to geothermal and petroleum industry problems.

Vibrational fatigue failure prediction of a brake caliper used for railway vehicles based on frequency domain method

A brake caliper used for railway vehicles was introduced in this paper. In order to predict the fatigue failure under random vibration condition, firstly a dynamic model was established and proved by comparing the results of modal simulation and modal test. The weak points of the structure were identified by taking random response analysis, and the corresponding PSD (power spectral density) spectrums of stress were obtained. Furthermore three different PDF (probability density function) models of stress amplitude distribution were employed to describe the distributions of stress at weak points, then the fatigue damages were calculated based on Miner’s linear cumulative damage theory and the fatigue failures were predicted. Finally, an accelerated failure test was designed and carried out to verify the validity of the predictions proposed by three models. The result shows that, the Dirlik’s model and Steinberg’s model can provide more exact results and be suitable for vibration fatigue failure prediction of brake caliper.

A framework for fatigue life prediction of materials under the multi-level cyclic loading

Fatigue life prediction of materials under the cyclic loading is still a particularly challenging task remains to be resolved. This study aims to develop a generic framework for fatigue life prediction under the multi-level cyclic (MLC) loading with consideration of the loading sequence effects. Firstly, a conditional probability density function (PDF) model is presented to quantify the fatigue life distributions of materials under any constant amplitude cyclic (CAC) loading. Subsequently, a fatigue damage accumulation model is proposed from the perspective of cumulative failure probability, which is capable of accounting for the nonlinear characteristics of damage accumulation and the loading sequence effects. Finally, a generic framework for fatigue life prediction of materials under the MLC loading is developed based on the proposed models. The fatigue life data of fiber reinforced composites under the two-level and three-level cyclic loading are utilized to verify the proposed framework. The results show that the predicted fatigue lives agree well with the fatigue test data of fiber reinforced composites. Moreover, the proposed framework can be easily extended to calculate fatigue life of materials under any MLC loading.

Numerical investigation on the optimized arrangement for high-temperature corrosion after low NOx transformation

In response to high-temperature corrosion after low NOx transformation with separate over fire air (SOFA), combined arrangement of corner and wall tangential firing burners, as well as anti-tangential and deflecting air technology are studied. Numerical simulation with realizable $$k-\varepsilon$$ two equation and Non-premixed combustion including probability density function model (PDF) is carefully adopted. Result conveys that application of combined arrangement can remarkably reduce the tangent circle in the furnace, lowering the probability that coal particles might strike the furnace wall. Also, lower CO and temperature near the wall would greatly decelerate the high-temperature corrosion. Further studies conclusively show that the performance has no linear relation to the angle in deviation cases. If properly arranged, combined burners layouts behave better than deviation (including anti-tangential) methods in restraint of high-temperature corrosion. All the technologies play effective roles in NOx reduction that the best-performing anti-tangential case 4 has a 50.7% drop from original case. Remission of high-temperature corrosion will have great economic and security importance.

Model Comparisons of Flow and Chemical Kinetic Mechanisms for Methane–Air Combustion for Engineering Applications

The reasonably accurate numerical simulation of methane–air combustion is important for engineering purposes. In the present work, the validations of sub-models were carried out on a laboratory-scale turbulent jet flame, Sandia Flame D, in comparison with experimental data. The eddy dissipation concept (EDC), which assumes that the molecular mixing and subsequent combustion occur in the fine structures, was used for the turbulence–chemistry interaction. The standard k-ε model (SKE) with the standard or the changed model constant C1ε, the realizable k-ε model (RKE), the shear-stress transport k-ω model (SST), and the Reynolds stress model (RSM) were compared with the detailed chemical kinetic mechanism of GRI-Mech 3.0. Different reaction treatments for the methane–air combustion were also validated with the available experimental data from the literature. In general, there were good agreements between predictions and measurements, which gave a good indication of the adequacy and accuracy of the method and its further applications for industry-scale turbulent combustion simulations. The differences between predictions and measured data might have come from the simplifications of the boundary settings, the turbulence model, the turbulence–reaction interaction, and the radiation heat transfer model. For engineering predictions of methane–air combustion, the mixture fraction probability density function (PDF) model for the partially premixed combustion with RSM is recommended due to its relatively low simulation expenses, acceptable accuracy predictions, and quantitatively good agreement with the experiments.

A fuzzy preference-based Dempster-Shafer evidence theory for decision fusion

Dempster-Shafer evidence theory (D-S) is an effective instrument for merging the collected pieces of basic probability assignment (BPA), and it exhibits superiority in achieving robustness of soft computing and decision making in an uncertain and imprecise environment. However, the determination of BPA is still uncertain, and merely applying evidence theory can sometimes lead to counterintuitive results when lines of evidence conflict. In this paper, a novel BPA generation method for binary problems called as the base algorithm is designed based on the kernel density estimation to construct the probability density function models, using the pairwise learning method to establish binary classification pairs. By means of the new BPA generation method, a new decision-making algorithm based on D-S evidence theory, fuzzy preference relation and nondominance criterion is effectively designed. The strength of the proposed method is presented in applying pairwise learning, which transforms the original complex problem into simple subproblems. With this process, the complexity of the problem to be solved is greatly reduced, which increases the feasibility for industrial applications. Furthermore, the fuzzy computing technique is used to aggregate the output for each single subproblem, and the nondominance degree of each class is determined from the fuzzy preference relation matrix, which can be directly used for the determination of the input instance. Based on several industrial-based classification experiments, the proposed BPA generation method and decision-making algorithm present the effectiveness and improvement in terms of precision and Cohen’s kappa.

Stochastic and spectra contents of detonation initiated by compressible turbulent thermodynamic fluctuations

A canonical system that contains the basic elements of detonation initiated by compressible turbulence thermodynamic fluctuations was proposed by Towery et al. [“Detonation initiation by compressible turbulence thermodynamic fluctuations,” Combust. Flame 213, 172–183 (2020)] and successfully tested using direct numerical simulation (DNS). In the present study, the DNS dataset is used to assess the applicability of a few compressible turbulence theories to the problem of detonation and to investigate some statistical aspects of the problem and the spectra of the turbulence kinetic energy (TKE) and reactive scalar fields within the context of detonating and deflagrating compressible flows. It has been found that scaling of the dilatational and solenoidal components of compressible turbulence as reported in the literature does not apply to the case of detonation or deflagration. Also, the detonating cases were found to have significantly lower Karlovitz numbers compared to the nondetonating ones and to conform to the Bray–Moss–Libby premixed combustion (bimodal probability density function) model for intense turbulence and large Damkhöler number. The nondetonating cases show distributed pdfs. The stronger combustion in the detonating cases shows effects on the spectra of the TKE and reactive scalars, particularly when the wavenumber is scaled with the Kolmogorov length and the Schmidt number.