Euler Number Research Articles

Influence of Pore Morphology on Permeability through Digital Rock Modeling: New Insights from the Euler Number and Shape Factor

The microscopic pore shape and topology significantly affect fluid transport and occurrence in porous permeable rock. A quantitive characterization of the impact of pore morphology on permeability is currently lacking, which limits the efficient development of underground hydrocarbon resources. This work introduces the Euler number and shape factor to characterize the pore topology and shape of heterogeneous sandstone based on CT imaging. The pore morphology under different pore sizes and the correlation of the Euler number, shape factor, fractal dimension, and surface area are analyzed. Furthermore, a modified Kozeny–Carman equation is established to explain the influence of the Euler number and shape factor on permeability. The results show that with the increase of pore diameter, the Euler number decreases while the shape factor increases. In a connected pore system, the smaller Euler number corresponds to the complex pore network, which leads to the increase in the surface area, shape factor, and fractal dimension. At constant porosity, the shape factor is negatively correlated with permeability, and with increasing Euler number, the heterogeneity of the pore structure increases, resulting in an increase of flow resistance and a decrease of permeability. The results provide a new pore morphology characterization method for digital rock and help to understand the flow mechanism of hydrocarbons in complex pore networks.

Effect of the segmented fin height on the air-side performance of serrated welded spiral fin-and-tube heat exchangers

Serrated welded spiral fin-and-tube heat exchangers (SWSFTHXs) are experimentally investigated. The work primarily focuses on the effects of the segmented fin height ( h s ) with various fin pitches ( f p ) on the air-side performance (ASP). Experimental results show that SWSFTHXs provide a higher air-side heat transfer coefficient than the plain welded spiral fin-and-tube heat exchangers (PWSFTHXs) at the same f p . The h s has a significant effect on the Nusselt number ( Nu ) and Colburn factor ( j ), whereas f p clearly has a greater effect on the friction factor ( f ) and Euler number ( Eu ) than h s . Furthermore, the Nu , j , f , and Eu correlations for PWSFTHXs and SWSFTHXs are also proposed.

The effect of axial cyclone inlet velocity and geometrical dimensions on the flow pattern, performance, and acoustic noise

This article studied the effect of the inlet velocity and various geometric dimensions on the flow pattern, collection efficiency, pressure drop, and acoustic noise of axial cyclones using computational fluid dynamics (CFD). The maximum values of tangential and axial velocity over inlet velocity were obtained for a guide vane revolution of 0.4 and a ratio of the swirler core diameter to the cyclone diameter of 0.3. It was found that the revolution of guide vanes also significantly affected the collection efficiency. In addition, the minimum values of the cut-off Stokes number, Euler number, and the overall sound pressure level (OASPL) were found to be 4.83 × 10−3, 4.014, 22.81 dB, respectively, while the minimal acoustic noise was observed for the guide vanes leading-edge exponent of 1.5. However, an increase in the Reynolds number did not alter the shape of the pressure and velocity contours but affected their magnitudes in the cyclone.

다양한 매개변수에 따른 관내 나노유체의 성능지수

The figure of merit for nanofluids represents the heat transfer performance of nanofluids against basic fluid. In previous studies, only the material properties of nanofluids have been considered for the figure of merit. In this study, the computational thermal flow analyses are performed including not only the material properties but also the geometric shapes and the external effects. The figures of merit are obtained with the four parameters, which are the inlet pressure, the number of return bends, the radius of curvature, and the volume fraction of nanoparticles in the pipes with the return bends. The results show that the three remaining parameters other than the number of return bends act on the figures of merit in a complicated manner. Therefore, a correlation equation with Euler number, Dean number, and Prandtl number associated with the three parameters is derived through the nonlinear regression. These results are expected to be helpful in designing and manufacturing heat exchangers using nanofluids.

EFFECTIVE SEMANTIC FEATURES TO IMPROVE RETRIEVAL OF LUNG NODULES IN CT SCAN IMAGES

Successful treatment of a patient depends on the accurate determination of the disease type. The advent of big data facilitates the retrieving of medical images and helps physicians in reliable diagnoses using content-based medical image retrieval systems (CBMIR). They consist of a feature extraction module and a distance metric. The extracted textural or deep-based features identify different types of diseases. In the proposed retrieval algorithm, we use the gray level cooccurrence matrix as the common textural characteristics and integrate them with semantic attributes. The semantic features are the geometric characteristics of the tumor that a radiologist employ to distinguish between benign and malignant tumors. These high-level attributes include the Euler number, margin smoothness, and the aspect ratio of the lesion’s size. We used the Minkowski distance measure for computing the similarity of images and applied the proposed algorithm to 200 CT-scan data containing lung lesions obtained from the LIDC database. The types of lesions were benign and malignant. Employing an ablation study, we proved the effectiveness of the semantic feature. The precision of the retrieval results is 93% which is promising compared to recent studies. In the future, we plan to define other kinds of semantic attributes to distinguish stages 1–5 of lung tumors as well.

Quadratic types and the dynamic Euler number of lines on a quintic threefold

We provide a geometric interpretation of the local contribution of a line to the count of lines on a quintic threefold over a field k of characteristic not equal to 2, that is, we define the type of a line on a quintic threefold and show that it coincides with the local index at the corresponding zero of the section of Sym5S⁎→Gr(2,5) defined by the threefold.Furthermore, we define the dynamic Euler number which allows us to compute the A1-Euler number as the sum of local contributions of zeros of a section with non-isolated zeros which deform with a general deformation. As an example we provide a quadratic count of 2875 distinguished lines on the Fermat quintic threefold which computes the dynamic Euler number of Sym5S⁎→Gr(2,5).Combining those two results we get that when k is a field of characteristic not equal to 2 or 5∑Trk(l)/k(Type(l))=1445〈1〉+1430〈−1〉∈GW(k) where the sum runs over the lines on a general quintic threefold.

Determination of Pressure Drop Correlation for Air Flow through Packed Bed of Sinter Particles in Terms of Euler Number

In order to clearly understand the air flow resistance characteristics in vertical tanks for sinter waste heat recovery in the steel industry, experimental research on the air flow pressure drop (FPD) performance in a sinter bed layer (BL) was conducted. Based on a self-made experimental device, the measurement values of air FPD for different experimental conditions were determined firstly, and then the concept of Euler number (Eu) in heat exchangers was introduced into the study of air FPD in BL; the change rules of Eu under different particle diameters were analyzed. Finally, the air FPD correlation in sinter BL was obtained and described in the form of Eu, and the error analysis of obtained air FPD correlation was performed. The results show that, the air FPD increases as a second power relationship with the increase in air superficial velocity when the particle diameter is constant. The decrease amplitude of Eu gradually dwindles when increasing the Reynolds number (Re), and the decrease in the Eu shows a reciprocal relationship with the Re. As the bed geometry factor increases, the FPD coefficient, A, decreases as an exponential relationship, while the FPD coefficient, B, increases as a first power relationship. The obtained air FPD correlation in the form of Eu in the experiment is well compatible with the measurement values, and the mean deviation of obtained correlation is 4.67%, showing good originality.

A new perspective on teaching the natural exponential to engineering students

The natural exponential and logarithm are typically introduced to undergraduate engineering students in a calculus course using the notion of limits. We here present an approach to introduce the natural exponential/logarithm through a novel interpretation of derivatives. This approach does not rely on limits, allowing an early and intuitive introduction of these functions. The question behind our contribution is whether one can introduce derivatives using only polynomials and power series? Motivated by an earlier exposure of engineering students to differential equations, we demonstrate that the natural exponential/logarithm can arise from two common differential equations. Our limit-free approach to derivatives provides an intuitive interpretation of , the Euler number, and an intuitive introduction of time constants in first-order dynamical systems.

Infinitesimal Torelli for elliptic surfaces revisited

In this article we give a new proof for the infinitesimal Torelli theorem for minimal elliptic surfaces without multiple fibers with Euler number at least 24 for nonconstant j -invariant. In the case of constant j -invariant we find a new proof in the case of Euler number at least 72. We also discuss several new counterexamples .

Numerical investigation on thermal-hydraulic performance of a spiral plate heat exchanger

The thermal-hydraulic performance of counter-current spiral plate heat exchanger (SPHE) under laminar flow regime is numerically investigated by adopting computational fluid dynamics (CFD) and field synergy principle in this paper. The effects of number of turns of spiral and Reynolds number on the average Nusselt number, heat transfer effectiveness, and Euler number are evaluated. The evidence shows that stronger convective heat transfer takes place at inner turns and areas around inner walls of channel. The heat transfer effectiveness increases with reducing increment and Euler number increases with increasing increment as the number of turns increases. When Reynolds number increases, both of heat transfer effectiveness and Euler number decrease due to large increments in mass flow rate. Consequently, an increase in number of turns of spiral at higher Reynolds numbers results in a greater benefit and a lower increment in cost. The numerical results in current research are also compared with the published article. It is found that the effect of thermal resistance of walls plays a vital role in evaluating the relation of heat transfer effectiveness and number of transfer units (NTU) at higher Biot numbers.

Ganglia dynamics during imbibition and drainage processes in nanoporous systems

We study the gas–water transient imbibition and drainage processes in two-dimensional nanoporous media using our recently developed lattice Boltzmann model. To describe the microscopic molecular interactions, the model employs a pseudopotential that correlates the local density and interaction strength to perform simulation at a mesoscopic scale. The primary interest is ganglia dynamics in the nanoporous media affected by fluid and geometrical properties of the porous structure. We performed sensitivity analyses on the fluid and rock characteristics such as the Euler number, gas–water interfacial area, water film area, capillary pressure, pore size distribution, specific surface area, and wettability. The simulation results revealed the fingering nature of the nonwetting phase. In the imbibition process, the flow pathway of water results in isolated and trapped gas bubble clusters because of the strong attraction between water and solid surfaces. In the drainage process, the pressure difference between the gas phase and the water phase depends on both the capillary pressure and the disjoining pressure due to the presence of water film. Pore topography and specific surface area control the continuity of the fluid phases in the imbibition process. In nonwet systems, the water phase starts fingering in the nanoporous system. The present work elucidates the microscopic ganglia dynamics of gas–water two-phase flow in nanoporous media. The microscopic scale details will help establish the macroscopic flow equation to accurately predict two-phase flow in shale gas, tight oil, and caprock seals.

SO(5) Landau model and 4D quantum Hall effect in the SO(4) monopole background

We investigate the $SO(5)$ Landau problem in the $SO(4)$ monopole gauge field background by applying the techniques of the non-linear realization of quantum field theory. The $SO(4)$ monopole carries two topological invariants, the second Chern number and a generalized Euler number, specified by the $SU(2)$ monopole and anti-monopole indices, $I_+$ and $I_-$. The energy levels of the $SO(5)$ Landau problem are grouped into $\text{Min}(I_+, I_-) +1$ sectors, each of which holds Landau levels. In the $n$-sector, $N$th Landau level eigenstates constitute the $SO(5)$ irreducible representation with $(p,q)_5=(N+I_+ + I_--n, N+n)_5$ whose function form is obtained from the $SO(5)$ non-linear realization matrix. In the $n=0$ sector, the emergent quantum geometry of the lowest Landau level is identified as the fuzzy four-sphere with radius being proportional to the difference between $I_+$ and $I_-$. The Laughlin-like wavefunction is constructed by imposing the $SO(5)$ lowest Landau level projection to the many-body wavefunction made of the Slater determinant. We also analyze the relativistic version of the $SO(5)$ Landau model to demonstrate the Atiyah-Singer index theorem in the $SO(4)$ gauge field configuration.

Impact of the training loss in deep learning-based CT reconstruction of bone microarchitecture.

Computed tomography (CT) is a technique of choice to image bone structure at different scales. Methods to enhance the quality of degraded reconstructions obtained from low-dose CT data have shown impressive results recently, especially in the realm of supervised deep learning. As the choice of the loss function affects the reconstruction quality, it is necessary to focus on the way neural networks evaluate the correspondence between predicted and target images during the training stage. This is even more true in the case of bone microarchitecture imaging at high spatial resolution where both the quantitative analysis of bone mineral density (BMD) and bone microstructure is essential for assessing diseases such as osteoporosis. Our aim is thus to evaluate the quality of reconstruction on key metrics for diagnosis depending on the loss function that has been used for training the neural network. We compare and analyze volumes that are reconstructed with neural networks trained with pixelwise, structural, and adversarial loss functions or with a combination of them. We perform realistic simulations of various low-dose acquisitions of bone microarchitecture. Our comparative study is performed with metrics that have an interest regarding the diagnosis of bone diseases. We therefore focus on bone-specific metrics such as bone volume and the total volume (BV and TV), resolution, connectivity assessed with the Euler number, and quantitative analysis of BMD to evaluate the quality of reconstruction obtained with networks trained with the different loss functions. We find that using norm as the pixelwise loss is the best choice compared to or no pixelwise loss since it improves resolution without deteriorating other metrics. Visual Geometry Group (VGG) perceptual loss, especially when combined with an adversarial loss, allows to better retrieve topological and morphological parameters of bone microarchitecture compared to Structural SIMilarity (SSIM) index. This however leads to a decreased resolution performance. The adversarial loss enhances the reconstruction performance in terms of BMD distribution accuracy. In order to retrieve the quantitative and structural characteristics of bone microarchitecture that are essential for postreconstruction diagnosis, our results suggest to use norm as part of the loss function. Then, trade-offs should be made depending on the application: VGG perceptual loss improves accuracy in terms of connectivity at the cost of a deteriorated resolution, and adversarial losses help better retrieve BMD distribution while significantly increasing the training time.

Performance characteristics of a small scale cyclone separator operated in different flow regimes

The flow pattern, cyclone pressure drop and particle penetration through a sampling cyclone have been studied at a wide range of flow rate 0.22–7.54 LPM using the LES simulations that have been validated based on experimental penetration data. The cyclone performance has been described by three dimensionless characteristics, the Euler number Euoutstat, cut-size ψ50 and slope of the transformed penetration curve κ1. Three main flow regimes and four sub-regimes have been revealed. The effects of the flow rate (Reynolds number) on the dimensionless cyclone performance characteristics have been described and a one-term power series model has been proposed. Additionally, the effect of the aspiration efficiency on the cyclone cut-size has been determined.

Proof of some conjectural congruences modulo p 3

In this paper, we mainly prove the following conjectures of Z.-H. Sun [New congruences involving Apéry-like numbers, preprint (2020). Available at arXiv:2004.07172v2]: Let p>3 be a prime. Then where , and stands for the nth Euler number.

Effect of rheology and solids concentration on hydrocyclones performance: A study involving the design variables of an optimized hydrocyclone

Despite several advances in the field of separations in hydrocyclones, achieving high separation efficiency in these devices when operating with concentrated non-Newtonian suspensions is still a challenge, and any improvement in this aspect can be a significant contribution. Thus, the objective of this study was to evaluate the behavior of a hydrocyclone operating with concentrated pseudoplastic suspensions. The device used in this study was the Maximum Overall Efficiency Hydrocyclone (MOEH), which is the result of a geometric optimization study developed by our research group whose objective was to obtain a hydrocyclone geometry that led to high separation efficiencies. The polymer carboxymethyl cellulose (CMC) was used as a rheological modulator at different concentrations. Changing the concentration of CMC in the suspension from 0.2 wt% to 1.0 wt%, it was observed reductions of 27% and 23% for the Euler number and overall efficiency of the MOEH hydrocyclone, respectively. When operating with dilute pseudoplastic suspensions (CCMC < 0.6 wt%), the drop in the separation performance could be mitigated with appropriate increments in the dimensions of the underflow diameter and reductions in the vortex finder length. Above a concentration of 0.8 wt% CMC the hydrocyclone operated as a stream splitter, with reduced efficiency of less than 2%, and the modifications in the design variables were not sufficient to reverse the detrimental effect of the increased slurry viscosity. The results obtained in this study show the relevant effect of non-Newtonian fluids in the functioning of hydrocyclones and the importance of the correct choice of geometric configurations and operating conditions to reduce the impact of the rheology in the separation performance.

Infinite families of Hecke-Rogers type series and their truncated representations

Recently, Andrews and Merca studied the truncated version of Euler's pentagonal number theorem. Their work opened up a new study on truncated series. Since then, many truncated versions of theta series have been discovered. Very recently, Wang and Yee generalized the study on truncated series to Hecke-Rogers type series and obtained three new truncated Hecke-Rogers series. Inspired by these works, we not only prove several infinite families of Hecke-Rogers type series and their truncated versions, but also establish some inequalities on some interesting partition functions in this paper. In particular, we generalize some results due to Wang and Yee, and He. Our work mainly relies on some formulas on the evaluation of certain terminating basic hypergeometric ϕ23 series.

Influence of Blade Fracture on the Flow of Rotor-Stator Systems with Centrifugal Superposed Flow

Rotor-stator cavities are often found in turbomachinery; they supply cold air that is bled from the compressor to the turbine blades. The pressure of the outlet of a rotor-stator cavity is axisymmetric under normal circumstances. However, its pressure would be non-axisymmetric in the event of blade fracture. The impact of blade fracture on a rotor-stator cavity with centrifugal superposed flow is studied in this paper. The Euler number E, the rotational Reynolds number Reφ, and the low-pressure zone range θ are investigated and, for the first time, with the non-axisymmetrical boundary conditions employing numerical simulation. The results of the numerical calculations show that after turbine blade fracture, the velocity is more affected in the downstream region at a high radius, especially when the Reφ is large. As for the distribution of the mass flow rate, there may be a critical θc at which the other blades are least affected. The θc would increase as the Reφ or the E increase, and the θc≅0.2 when Cw=10,137, Reφ=5.12×105, and 0.2≤E≤0.4. In addition, the thrust coefficient increases as the E or the θ increases, and the increase in the thrust coefficient does not exceed 4% when the E=0.2 and the θ=0.1 in this paper. However, the moment coefficient on the rotating shaft is almost independent of the E and the θ. An increase in the Reφ will reduce the effect of turbine blade fracture on the thrust and moment coefficients, when the Reφ is small.

Comparative analysis of proficiencies of various textures and geometric features in breast mass classification using k-nearest neighbor

This paper introduces a comparative analysis of the proficiencies of various textures and geometric features in the diagnosis of breast masses on mammograms. An improved machine learning-based framework was developed for this study. The proposed system was tested using 106 full field digital mammography images from the INbreast dataset, containing a total of 115 breast mass lesions. The proficiencies of individual and various combinations of computed textures and geometric features were investigated by evaluating their contributions towards attaining higher classification accuracies. Four state-of-the-art filter-based feature selection algorithms (Relief-F, Pearson correlation coefficient, neighborhood component analysis, and term variance) were employed to select the top 20 most discriminative features. The Relief-F algorithm outperformed other feature selection algorithms in terms of classification results by reporting 85.2% accuracy, 82.0% sensitivity, and 88.0% specificity. A set of nine most discriminative features were then selected, out of the earlier mentioned 20 features obtained using Relief-F, as a result of further simulations. The classification performances of six state-of-the-art machine learning classifiers, namely k-nearest neighbor (k-NN), support vector machine, decision tree, Naive Bayes, random forest, and ensemble tree, were investigated, and the obtained results revealed that the best classification results (accuracy = 90.4%, sensitivity = 92.0%, specificity = 88.0%) were obtained for the k-NN classifier with the number of neighbors having k = 5 and squared inverse distance weight. The key findings include the identification of the nine most discriminative features, that is, FD26 (Fourier Descriptor), Euler number, solidity, mean, FD14, FD13, periodicity, skewness, and contrast out of a pool of 125 texture and geometric features. The proposed results revealed that the selected nine features can be used for the classification of breast masses in mammograms.

Truncated Series with Nonnegative Coefficients from the Jacobi Triple Product

Andrews and Merca investigated a truncated version of Euler's pentagonal number theorem and showed that the coefficients of the truncated series are nonnegative. They also considered the truncated series arising from Jacobi's triple product identity, and they conjectured that its coefficients are nonnegative. This conjecture was posed by Guo and Zeng independently and confirmed by Mao and Yee using different approaches. In this paper, we provide a new combinatorial proof of their nonnegativity result related to Euler's pentagonal number theorem. Meanwhile, we find an analogous result for a truncated series arising from Jacobi's triple product identity in a different manner.