Exponential Function Research Articles

Theoretical and FEA study on shear performance of novel shear connectors

Shear connectors are essential components for force transfer in composite beams. Compared to conventional welded-bolt connectors, bolted shear connectors offer the advantage of easy removal. However, traditional bolted connectors for steel-concrete composite beams feature threads on the contact surface, reducing the shear cross-sectional area of the bolts. To address this limitation, this paper proposes a novel type of friction-clamping bolt connector, referred to as Connecting Lock Combination Bolts (CLCB11CLCB:Connecting Lock Combination Bolts；). The finite element software ABAQUS was used to develop an analytical model for the CLCB, and a comparative analysis was conducted with traditional connectors. The static performance of the new shear connectors was evaluated, and a shear load capacity formula applicable to CLCB connectors was derived. Additionally, linear and exponential functions were employed to fit the load-slip curve of the CLCB connectors. The results indicate that the shear load capacity of CLCB connectors exceeds that of conventional bolted connectors. Moreover, the bolt diameter and concrete strength are the primary factors influencing the ultimate shear load capacity and stiffness of CLCB connectors. The preload on the bolts enhances the shear stiffness, while parameters such as bolt hole clearance, the height of the lower nutted rod, and the clearance between the upper and lower bolt rods are crucial in ensuring the proper functioning of CLCB connectors. Furthermore, the shear capacity formula provides an accurate prediction of CLCB connector performance with an error margin of less than 6 %, and the load-slip curve model effectively illustrates the load-slip relationship under shear.

Enhanced Safety‐Critical Control of Hypersonic Vehicles With Multiple Constraints

ABSTRACTThis paper proposes an enhanced safety‐critical control scheme for hypersonic vehicles (HSVs) with multiple constraints, where both safety constraints (e.g., actuator saturations and time‐varying angle‐of‐attack [AOA] limitation) and transient performance constraints are considered. In this scheme, the extended exponential control barrier function (EECBF) is constructed by incorporating extended states and robust terms to deal with time‐varying constraint boundaries and uncertainties simultaneously. An adaptive programming for optimal transient performance envelopes is conducted to make a trade‐off between safety and transient performance constraints. It is shown that the proposed scheme for HSVs can confine AOA to time‐varying safety boundaries in the presence of uncertainties while the velocity and height tracking errors can always maintain within the desired transient performance envelopes. Simulations and comparisons are provided to verify the effectiveness of our scheme.

Can the atmospheric boundary layer-based potential evaporation model increase the accuracy of generalized complementary functions?

ABSTRACT Complementary functions could reliably and effectively estimate actual evaporation and have received wide attention in recent years. However, estimating potential evaporation (Epo) greatly influences the accuracy of complementary functions. In this study, we compare the atmospheric boundary layer model (ABL2021) with the Priestly–Taylor model (P-T) and the maximum evaporation model (YR2019) for estimating Epo. Eighty-six flux sites are utilized to fit parameters for three generalized complementary functions, including the sigmoid function (H2018), the polynomial function (B2015), and the exponential function (G2021) with various potential evaporation models. The results suggest that ABL2021 shows the best agreement with the observations at large lake sites. The uncertainties for estimation of actual evaporation induced by different potential evaporation models are larger than different complementary functions. ABL2021 significantly reduces the differences in performance between different complementary functions. It suggests that ABL2021 improves the accuracy of the generalized complementary functions in most cases and can provide a calibration-free method for Epo estimation. G2021 performs better and shows more flexibility than the other generalized complementary functions. Therefore, G2021 combined with ABL2021 shows potential to develop a robust method for estimating actual evaporation based on the complementary principle.

Molecular Clouds in the Outer Milky Way Disk: Sample, Integrated Properties, and Radial Trends with Galactocentric Radius

We present a catalog of 32,162 12CO molecular clouds that covers the northern outer Galactic plane (l = [15°, 165°] and l = [195°, 230°], and b = [−5.°25, +5.°25]). The catalog was produced using a DBSCAN algorithm applied to the Milky Way Imaging Scroll Painting project data. We systematically analyze both the integrated properties and scaling relationships of these molecular clouds across different Galactocentric radii (8 < R GC < 26 kpc). An interesting finding is that each cloud property’s mean, median, and maximum values generally decrease systematically with increasing R GC, approximated quantitatively by exponential functions. The mass and size spectra for the entire sample show power-law indices of γ = −2.00 ± 0.13 and an upper mass limit of M 0 = (4.5 ± 2.8) × 106 M ⊙, and γ = −3.68 ± 0.48 with an upper size limit of R 0 = 144 ± 80 pc, respectively. These spectra exhibit steeper slopes and reduced upper limits as R GC increases. The derived scaling relations are M=12.00Reff2.41±0.003 , σv=0.25Reff0.66±0.003 , and α vir = 37.2M −0.40 ± 0.002 for the whole outer Galaxy clouds. These scaling relations show slightly steeper correlations observed in more distant locations and notably deviate from Larson’s relations. We find that 44.3% of the molecular mass resides in gravitationally bound structures, a proportion significantly higher than systematic studies from the CfA survey and external galaxies. Moreover, comparisons among different R GC subsamples also reveal a decrease in the amount of bound mass with increasing R GC. The scale length of the molecular disk is estimated to be approximately 2 kpc. These results, based on a constant CO conversion factor of XCO=2.0×1020cm−2Kkms−1−1 , may be slightly altered by variations in X CO. In summary, our findings provide robust evidence for the influence of Galactic evolution on molecular cloud properties, at least in the outer Galaxy region studied.

АЛГОРИТМИ ПРОГРАМНОЇ РЕАЛІЗАЦІЇ ЦИФРОВИХ СТАТИСТИЧНО ОПТИМАЛЬНИХ РЕГУЛЯТОРІВ АСИНХРОННИХ ЕЛЕКТРОПРИВОДІВ ІЗ СТОХАСТИЧНИМИ НАВАНТАЖЕННЯМИ

The paper is devoted to the development of algorithms for the programm implementation of digital statistically optimal regulators of electric drives, which are under the influence of dynamic loads that change according to stochastic laws. On the basis of continuous transfer functions of statistically optimal regulators for similar loads, which can be approximated by exponential and exponential-cosine correlation functions, the following algorithms were obtained in the form of recurrent difference equations, which can be implemented by software. Recommendations have been developed for the accuracy of determining the parameters of the corresponding digital regulators. Ref. 11, fig. 5.

Dynamic description of soliton solutions of nonlinear schrödinger equation with higher order dispersion and nonlinear terms

Abstract The article presents an analytical solution for the higher-order nonlinear Schrödinger equation (NLSE), which describes the propagation of short light pulses in monomode optical fibers. Various traveling wave solutions are obtained using the generalized exponential rational function method, a technique with substantial applications in physics and mathematics. Additionally, the parameters leading to the occurrence of optical bright and multipeak solitons in this medium are provided along with their formation conditions. The derived solutions are graphically displayed to enhance the understanding of the model’s physical phenomena. This approach is credible, potent, and successful in solving a wide variety of different models of this kind that arise in the applied sciences. Its robustness, strength, and efficiency make it suitable for addressing various higher-order nonlinear problems in current research fields, extending beyond the models encountered in the applied sciences.

Assessing the impact of rice-wheat-maize residue decomposition rate and nutrient dynamics of residue and soil using different placement method in the IGP of India

The rice-wheat cropping system (RWCS) provides the world’s population with staple foods, and it is crucial to maintain global food demand and security. Food systems are a complex ecosystem and sustain many feedback mechanisms. Crop residue management is one of those feedback mechanisms that was assessed under conservation agriculture, and a decomposition study was analyzed for the rice-wheat cropping system using rice, wheat, and maize crop residue for decomposition rate and nutrients release under agricultural practices (zero till, raised beds). Different zero tillage techniques in Samastipur demonstrated an accelerated decomposition trend, which was especially noticeable in the straw from wheat and rice. At the same time, permanent bed systems showed a relatively larger residue mass, especially in the case of wheat and maize straw. Permanent bed systems (wheat-maize system), particularly those using wheat straw, held the largest amounts of residue mass when the total residue mass throughout the sites was taken into account. Samastipur showed higher nutrient release for all the rice residue in wheat, rice residue in maize, and wheat residue in rice except maize residue in rice as compared to the Karnal sites. Decomposition kinetics, modeled via a first-order exponential decay function, showed high correlations (R2: 0.941 to 0.996) across treatment methods. The research underscores the significant effect of agroecological factors on residue decomposition and nutrient release, irrespective of residue type, highlighting the importance of tailored residue management practices for enhanced nutrient cycling and sustainability. These findings contribute to the optimization of residue management strategies in RWCS, promoting sustainable agriculture practices in the face of climate change and increasing food security demands.

An accurate semi-empirical model for PMT pulse signal analysis.

The energy information of pulse signals is significantly important for applications such as computed tomography (CT), positron emission tomography (PET), and research on defects in condensed matter. Time-over-threshold (TOT) and multi-voltage threshold (MVT) are commonly used digitization methods in sampling pulse signal. However, both approaches rely on a mathematical model of the pulse signal to derive energy information. This study proposes a semi-empirical mathematical model for pulse signals formation process in scintillation crystal-coupled photomultiplier tube(PMT) probes, by utilizing the CR-RC shaping method. This mathematical model accurately describes output of the PMT pulse signals. This study analyzes a substantial dataset of pulse signals, comparing the performance of the newly designed mathematical model with that of the double exponential function in terms of their ability to fit pulse signals. The results indicate that the mathematical model developed herein achieves an average R2 of 0.9255, significantly surpassing the 0.9155 of the double exponential function, thereby demonstrating its superior fitting efficacy.

Security analysis and adaptive false data injection against multi-sensor fusion localization for autonomous driving

Multi-sensor Fusion (MSF) algorithms are critical components in modern autonomous driving systems, particularly in localization and AI-powered perception modules, which play a vital role in ensuring vehicle safety. The Error-State Kalman Filter (ESKF), specifically employed for localization fusion, is widely recognized for its robustness and accuracy in MSF implementations. While existing studies have demonstrated the vulnerability of ESKF to sensor spoofing attacks, these works have primarily focused on a black-box implementation, leading to an insufficient security analysis. Specifically, due to the lack of theoretical guidance in previous methods, these studies have consistently relied on exponential functions to fit attack sequences across all scenarios. As a result, the attacker had to explore an extensive parameter space to identify effective attack sequences, lacking the ability to adaptively generate optimal ones. This paper aims to fill this crucial gap by conducting a thorough security analysis of the ESKF model and presenting a simple approach for modeling injection errors in these systems. By utilizing this error modeling, we introduce a new attack strategy that employs constrained optimization to reduce the energy needed to reach the same deviation target, guaranteeing that the attack is both efficient and effective. This method increases the stealthiness of the attack, making it harder to detect. Unlike previous methods, our approach can dynamically produce nearly perfect injection signals without requiring multiple attempts to find the best parameter combination in different scenarios. Through extensive simulations and real-world experiments, we demonstrate the superiority of our method compared to state-of-the-art attack strategies. Our results indicate that our approach requires significantly less injection energy to achieve the same deviation target. Additionally, we validate the practical applicability and impact of our method through end-to-end testing on an AI-powered autonomous driving system.

Estimating the Lifetime Parameters of the Odd-Generalized-Exponential–Inverse-Weibull Distribution Using Progressive First-Failure Censoring: A Methodology with an Application

This paper investigates statistical methods for estimating unknown lifetime parameters using a progressive first-failure censoring dataset. The failure mode’s lifetime distribution is modeled by the odd-generalized-exponential–inverse-Weibull distribution. Maximum-likelihood estimators for the model parameters, including the survival, hazard, and inverse hazard rate functions, are obtained, though they lack closed-form expressions. The Newton–Raphson method is used to compute these estimations. Confidence intervals for the parameters are approximated via the normal distribution of the maximum-likelihood estimation. The Fisher information matrix is derived using the missing information principle, and the delta method is applied to approximate the confidence intervals for the survival, hazard rate, and inverse hazard rate functions. Bayes estimators were calculated with the squared error, linear exponential, and general entropy loss functions, utilizing independent gamma distributions for informative priors. Markov-chain Monte Carlo sampling provides the highest-posterior-density credible intervals and Bayesian point estimates for the parameters and reliability characteristics. This study evaluates these methods through Monte Carlo simulations, comparing Bayes and maximum-likelihood estimates based on mean squared errors for point estimates, average interval widths, and coverage probabilities for interval estimators. A real dataset is also analyzed to illustrate the proposed methods.

Radio Continuum Halos of Seven Nearby Large Galaxies Using uGMRT

We present the results of deep radio observations of seven nearby large galaxies observed using the upgraded Giant Metrewave Radio Telescope (uGMRT) 0.3–0.5 GHz receivers with an angular resolution of ∼10″. The achieved sensitivities of these observations range from ≈15 to 50 μJy beam−1, which is a factor of ≈3–4 lower than the previous observations at these frequencies. For two galaxies (NGC 3344 and NGC 3627) with moderate inclination angles, significant diffuse emissions are seen for the first time. The detected radio halos in the vertical direction are significantly larger in our 0.4 GHz maps compared to the observations at ∼1.5 GHz for four nearly edge-on galaxies—NGC 3623, NGC 4096, NGC 4594, and NGC 4631. For these four galaxies, significantly larger halos are also detected along the galaxy disk. For NGC 3623 and NGC 4594, we could detect elongated radio disks that were not seen before. We also present new uGMRT images of NGC 3344 and NGC 3623 at 1.3 GHz and a new VLA image of NGC 3627 at 1.5 GHz. We fitted an exponential function to the flux densities along different cross-cuts and found a significantly wider distribution at the 0.4 GHz uGMRT images compared to the high-frequency images at ∼1.5 GHz. Using maps at 0.144, 0.4, and ∼1.5 GHz, we made spectral index maps of the seven sample galaxies and found a steepening of the spectrum up to a value of ∼−1.5 in the halo regions of the galaxies.

Hybrid energy storage lifespan optimization based on an enhanced fuel-cell degradation model and meta-heuristic algorithm

Enhancing the supply of power in terms of availability, dependability, and security are the current goals of the electricity industry. Incorporating renewable energy sources (RES) into the electrical grid to address the issue of clean energy scarcity has consequently garnered a great deal of attention. However, the utilization of hybrid energy storage systems (HESS) is required due to the high degree of electricity supply instability of RES paired with variations in energy demand levels. HESS still faces issues with lifespan deterioration and the need to increase electricity availability. This research, therefore, suggests an improved polymer electrolyte membrane fuel cell (PEMFC) degradation model and the study proposed the multi-objective fire dragonfly algorithm (MOFDA) technique. This strategy integrates the dragonfly algorithm and firefly algorithm, and the proposed equation incorporates the ameliorating factor as an exponential function for enhancement to deal with the system’s instability, which causes the HESS to age quickly. The comparison of the HESS with the proposed PEMFC model and the HESS model shows a huge improvement in the parameters that affect the lifespan predicted period. The research delves into the efficacy of a proposed technique that optimizes six benchmark trusses functions using MOFDA, a multi-objective algorithm. MOFDA’s performance is compared to that of seven other competitive multi-objective algorithms. The Friedman test indicates that each algorithm yields different results, and the MOFDA algorithm is the best overall solution, with both p-values being less than the significance value of α = 0.06. To ensure a fair evaluation, the population size and maximum number of iterations are both set to 100 and 200, respectively. Each multi-objective metaheuristic algorithm independently runs 30 times. The system is modeled and evaluated using MATLAB and Simulink to assess the performance of the study.

On the embedding between the variable Lebesgue space Lp(⋅)(Ω) and the Orlicz space L(log⁡L)α(Ω)

We give a sharp sufficient condition on the distribution function, |{x∈Ω:p(x)≤1+λ}|, λ>0, of the exponent function p(⋅):Ω→[1,∞) that implies the embedding of the variable Lebesgue space Lp(⋅)(Ω) into the Orlicz space L(log⁡L)α(Ω), α>0, where Ω is an open set with finite Lebesgue measure. As applications of our results, we first give conditions that imply the strong differentiation of integrals of functions in Lp(⋅)((0,1)n), n>1. We then consider the integrability of the maximal function on variable Lebesgue spaces, where the exponent function p(⋅) approaches 1 in value on some part of the domain. This result is an improvement of the result in [6].

Determination Method of Reaction Rate and Mass Transfer Parameters in Cathode Catalyst Layer in PEFC

IntroductionIt is important to optimize the cathode catalyst layer (CCL) because the oxygen reduction reaction (ORR) in the CCL is sluggish. Our previous study showed that the mass transfer resistance in the CCL can be quantified by measuring the ORR rate dependency on the oxygen partial pressure at fixed temperature[1,2]. In this study, polarization curves were measured and analyzed at varied oxygen partial pressure and temperature. The objective of this study is to establish the quantification methods of mass transfer and reaction rate parameters, which explain the behavior of polarization curves at various conditions.ExperimentalA cell whose active area was 20 mm × 20 mm was used in experiments. The membrane electrode assembly (MEA, Eiwa Corp.) consists of proton exchange membrane (PEM, Chemours NafionTM, NR-212) and two catalyst layers of each side of the PEM. Platinum catalyst supported on Ketjen Black (Pt/C weight ratio: 1.0) and a NafionTM ionomer (ionomer/carbon weight ratio: 1.0) consists of the catalyst layer. Oxygen and nitrogen were supplied to the cathode in order to vary the oxygen partial pressure. H2, N2, and O2 were humidified in bubblers to keep the inlet relative humidity (RH) at 0.8. The cell temperature was varied at 50, 60, 70, and 80 °C. The Oxygen mole fraction at the cathode was also varied at 1.5, 3, 6, 10, and 21 % on a dry basis.Results and DiscussionPolarization curves at varied oxygen partial pressure, p Oc and temperature are shown in Figs. 1–4. Since the reaction and mass transfer rate are improved at higher temperature, the current density at fixed cathode electromotive force, E cm increases when temperature increases as far as RH is identical. Arrhenius plot at the E cm of 0.2 V is shown in Fig. 5. Since the reaction rate is proportional to the oxygen partial pressure[3,4], the current divided by the oxygen partial pressure on the vertical axis in Fig. 5 is proportional to the observed reaction rate constant. Fig. 5 shows that the apparent activation energy decreases as temperature increases at the p Oc range of 2.5 to 7.7 kPa. This result indicates that the reaction rate saturates when temperature increases due to the effect of oxygen permeation resistance of the ionomer layer. The observed reaction rate constant decreases at higher p Oc , particularly at 15.9 kPa, due to the effect of proton transfer resistance. Fig 6. shows the apparent activation energies plotted against E cm . Both apparent activation energies of the ORR did not depend on E cm . Tafel slope, b c should be the function of temperature, T as expressed in Eq. (1): b c=RT/4aF, (1)where R, a, and F is the gas constant, transfer coefficient, and Faraday constant, respectively. The ORR rate constant, k vc ° is the exponential function of the inverse of the temperature. Considering Eq. (1), k vc ° should be expressed as follows: k vc° = k vc0exp(-E/RT-E c/b c) = k vc0exp(-(E+4aFE c)/RT), (2)where E is the activation energy. Although the apparent activation energy of the ORR should increase when E cm increases, the measurement results were independent of E cm , especially below 0.65 V. It indicates that the Tafel slope does not depend on temperature. The apparent activation energies of the ORR and of the oxygen permeance through the ionomer layer were obtained as 52 kJ/mol and 15 kJ/mol, respectively.Reaction rate constant, effective oxygen diffusion coefficient, effective proton conductivity, and oxygen permeance through the ionomer layer can be determined by applying the dimensionless isothermal 1D cathode model[1, 2, 4] to the obtained polarization curves.ConclusionsThe polarization curves at varied oxygen partial pressure and temperature were measured and analyzed. As the activation energy of the reaction is higher than that of diffusion and permeation, the observed activation energy in the Arrhenius plot at varied cathode electromotive force decreased when temperature increases. The observed activation energy of the ORR did not depend on the cathode electromotive force.AcknowledgementsThis work was supported by the FC-Platform Program: Development of design-for-purpose numerical simulators for attaining long life and high performance project (FY 2020–2023) conducted by the New Energy and Industrial Technology Development Organization (NEDO), Japan. This work was supported by Japan Society for the Promotion of Science (JSPS) Grants-in-Aid for Scientific Research (KAKENHI) Grant Number 22KJ1899.

Integration of smart buildings with high penetration of storage systems in isolated 100 % renewable microgrids

The operation of smart buildings is an important feature in energy-transition programs. Nowadays, smart buildings typically use renewable energy resources and energy storage systems to feed their energy demands. In this research, a bi-level stochastic model is put forward for integration of smart buildings with high penetration of storage systems in isolated 100 % renewable microgrids. In the first level, each smart building solves its operational planning model and submits its desired power exchange to the microgrid; then, in the second level, microgrid operator solves its operational planning model, considering all requests from smart buildings. Each smart building performs as a prosumer having an electric water heater, air conditioner, photovoltaic modules, storage system and electric vehicles. A dynamic demand response (DR) program is used in which microgrid operator awards incentive for shift-down of demand of a building at times and scenarios in which demand of the building is higher than its average demand. The incentives to be paid for shift-down and shift-up of non-thermal demand of buildings are exponential functions of the difference between demand and average demand of the building. The findings approve the efficiency of the proposed bi-level methodology; the results confirm that thanks to storage systems, renewable energy resources and demand response program, the operation of the smart buildings is profitable; moreover, the microgrid operator exploits the responsiveness of smart buildings through DR program to decrease its operational planning cost. The results show that storage systems increase the daily profit of the smart buildings by 95 %. The results also show that DR and building-to-grid capability respectively increase the daily profit of the smart buildings by 43.8 % and 164 %.

Improved Fractional Differences with Kernels of Delta Mittag–Leffler and Exponential Functions

Improved Fractional Differences with Kernels of Delta Mittag–Leffler and Exponential Functions

A Lindemann–Weierstrass theorem for 𝐸-functions

Abstract 𝐸-functions were introduced by Siegel in 1929 to generalize Diophantine properties of the exponential function. After developments of Siegel’s methods by Shidlovskii, Nesterenko and André, Beukers proved in 2006 an optimal result on the algebraic independence of the values of 𝐸-functions which generalizes the Lindemann–Weierstrass theorem. Since then, it seems that no general result was stated concerning the relations between the values of a single 𝐸-function. We prove that André’s theory of 𝐸-operators and Beuker’s result lead to a Lindemann–Weierstrass theorem for 𝐸-functions in its “linear independence” formulation. As a consequence, we show that all transcendental values at algebraic arguments of an entire hypergeometric function are linearly independent over Q ̄ \overline{\mathbb{Q}} .

Torsion and Extension of Functionally Graded Mooney–Rivlin Cylinders

We analytically study finite torsional and extensional deformations of rubberlike material circular cylinders with the two material moduli in the Mooney–Rivlin relation assumed to be continuous functions of the undeformed radius. It is shown that under null resultant axial load on the end faces the cylinder length increases upon twisting. Furthermore, when the two moduli are affine functions of the radius the inhomogeneity parameters can be found to have the maximum shear stress occur at a pre-determined interior point. Whereas the radial stress is finite at the center of a cross-section of a homogeneous material cylinder, it may have large values for an inhomogeneous material cylinder. The closed-form solutions provided herein for the two moduli having affine, power-law and exponential functions of the radius should benefit numerical analysts verify their algorithms and engineers design soft material robots for improving their performance under torsional loads.

Aspect ratio-dependent etching in silicon using XeF2: experimental investigation and comparative analysis with dry etching methods

Abstract Silicon machining plays a crucial role in shaping three-dimensional structures for Micro-Electro-Mechanical Systems (MEMS) applications. This study investigates Aspect Ratio Dependent Etching (ARDE) across various silicon etching processes, with a particular focus on Xenon Difluoride (XeF2) etching, in comparison to Reactive Ion Etching (RIE) and Deep Reactive Ion Etching (DRIE).By exploring different etching parameters, the study highlights the presence of ARDE in both plasma and non-plasma etching processes. Additionally, it is demonstrated that ARDE can be modeled by a saturating exponential function through experimental adjustment of parameters, enabling the prediction of etching profiles.

Experimental study on the effect of dual fire sources elevation on flame merging characteristics in road tunnels with natural ventilation

A series of experiments was carried out using a 1:10 small-scale model tunnel to investigate the effect of different fire source elevations on the flame merging characteristics of dual fire sources under natural ventilation. CFD numerical simulation software was used to analyze the impact of fire source elevation on flame merging characteristics and air entrainment from the perspectives of lateral velocity, longitudinal velocity, vertical velocity and air entrainment rates. The effects of fire source spacing and heat release rate on the probability of flame merging and the height of the flame merging point under different fire source elevations were investigated. It is found that as the fire source spacing S increases, the flame tilt angle θ first increases and then stabilizes for the first time. With a further increase in S, θ increases again and stabilizes for the second time. The first stabilization is mainly controlled by the interaction between the fire sources, while the second stabilization is primarily influenced by natural ventilation from both ends of the tunnel. As the fire source elevation increases, the flame merging probability decreases with the addition of air entrainment between the fire sources. What is more, when the fire source elevation exceeds 8 cm, the effect of the fire source elevation on the flame merging probability is no longer obvious. The prediction model of flame merging probability under different fire source elevations is proposed. Meanwhile, the dimensionless fire source elevation h/D and the modified dimensionless heat release rate Q̇DS∗ are used to express the dimensionless height of flame merging point Zm/Hef. The piecewise function of Zm/Hef and Q̇DS∗ is obtained. WhenQ̇DS∗≤88,h/D≤8, the relationship between Zm/Hef and Q̇DS∗ could be described with exponential function and whenhD>0.8, the fire source elevation has little impact on Zm/Hef. WhenQ̇DS∗>8, Zm/Hef is approximately constant and increases with fire source elevation.