Rectangular Source Research Articles

The time‐fractional ISPH method for fin circular rotation on MHD bioconvection flow of oxytactic microorganisms and NEPCM within a hexagonal‐shaped cavity

AbstractThis work investigates the bioconvection flow of oxytactic microorganisms and nanoparticle‐enhanced phase change material (NEPCM) within a hexagonal‐shaped cavity containing a rotated cross fin, utilizing the incompressible smoothed particle hydrodynamics (ISPH) method based on a time‐fractional derivative. The study considers the circular rotation of an inner cross fin and the presence of two rectangular heat sources on the plane walls inside the hexagonal cavity. The novelty of this work is its integration of a hexagonal‐shaped cavity with a rotated cross fin and the use of a time‐fractional derivative in the ISPH method to analyze bioconvection flow, offering new insights into the interaction between microorganism motion and heat transfer dynamics in complex geometries. The effects of Darcy, Hartmann, Lewis, Peclet, Rayleigh, and bioconvection Rayleigh numbers on isotherms, heat capacity ratio, microorganism density, velocity fields, and average Nusselt number are analyzed. The key findings demonstrate the significant impact of Rayleigh and bioconvection Rayleigh numbers in enhancing heat distribution and velocity fields, thereby significantly influencing microorganism motion. Due to Lorentz forces, the velocity field decreases by with an increase in the Hartmann number from 0 to 50. The resistance of the nanofluid's velocity becomes apparent as the Darcy number decreases. Increasing Lewis and Péclet numbers cause the microorganisms to shift towards the cavity's boundaries.

The Temperature Dependence of the Parameters of LED Light Source Control Devices Powered by Pulsed Voltage

Ambient temperature has a significant effect on the electrical and luminous parameters of light-emitting diodes (LEDs), which include forward and reverse current, forward voltage, and luminous flux. This paper gives insight into the influence of ambient temperature on the electrical and luminous parameters of LEDs powered by a rectangular pulsed voltage source versus those powered by a constant voltage source. The characteristics of LEDs in LED lighting devices were studied to determine their optimal operating conditions. To this end, rectangular pulse voltages with different pulse filling factors D were considered against the DC voltage source. Characteristics were obtained for the current stabilization mode and for the LED voltage stabilization mode. In both modes, the temperature dependence of the luminous flux, current, voltage, power consumption, and luminous efficiency of the LEDs was studied in the 20 °C to 60 °C range. The optimal LED operating conditions were determined, of which their luminous flux and luminous efficiency are least dependent on ambient temperature. When powered by a rectangular pulse voltage, the LED device drivers’ optimal pulse filling factor and operating frequency were determined.

MIRACLE—a microphone array impulse response dataset for acoustic learning

This work introduces a large dataset comprising impulse responses of spatially distributed sources within a plane parallel to a planar microphone array. The dataset, named MIRACLE, encompasses 856,128 single-channel impulse responses and includes four different measurement scenarios. Three measurement scenarios were conducted under anechoic conditions. The fourth scenario includes an additional specular reflection from a reflective panel. The source positions were obtained by uniformly discretizing a rectangular source plane parallel to the microphone for each scenario. The dataset contains three scenarios with a spatial resolution of 23mm\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$23\\,\ extrm{mm}$$\\end{document} at two different source-plane-to-array distances, as well as a scenario with a resolution of 5mm\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$5\\,\ extrm{mm}$$\\end{document} for the shorter distance. In contrast to existing room impulse response datasets, the accuracy of the provided source location labels is assessed and additional metadata, such as the directivity of the loudspeaker used for excitation, is provided. The MIRACLE dataset can be used as a benchmark for data-driven modelling and interpolation methods as well as for various acoustic machine learning tasks, such as source separation, localization, and characterization. Two timely applications of the dataset are presented in this work: the generation of microphone array data for data-driven source localization and characterization tasks and data-driven model order reduction.

Experimental study on enhancing single-pass air disinfection efficacy for portable disinfection systems with different Far-UVC sources

Two single-pass air disinfection systems utilizing different form factors of Far-UVC sources were designed to achieve indoor air disinfection. The first system, namely the external-module-equipped (EME) system, featured a rectangular UV source module attached to the envelope of the disinfection box chamber. The second system, known as the internal-tube-equipped (ITE) system, incorporated a centrally positioned UV tube within the chamber. To assess their air disinfection efficacy, two bacteria, Salmonella enterica and Staphylococcus epidermidis, and one bacteriophage, MS2 were selected as challenges for the systems operating at a flow rate over 100 m3/h. The baseline EME system inactivated the microbes with the efficacies of 81.4 %, 16.9 %, and 25.1 % for S. enterica, S. epidermidis, and MS2, respectively. In comparison, the baseline ITE system demonstrated higher inactivation efficacies due to a wider irradiated area, with respective efficacies of 94.4 %, 25.8 %, and 37.8 % for the tested microorganisms. Two enhancement measures were implemented by redirecting airflow to the areas with high UV irradiance and enhancing UV reflectance, increasing the inactivation efficacy of the systems by up to 207.0 %. When tested against Staphylococcus epidermidis, compared with an upper-room Far-UVC, the modified EME and ITE systems presented an increase in inactivation rate by 23.0 % and 29.2 % respectively. These results demonstrated the potential of the systems to efficiently and sustainably inactivate a broad spectrum of microbes in future practical applications.

Sizing horizontal metallic inclusions in insulators using lock-in inductive infrared thermography

We present a methodology to size the area and depth of horizontal metallic inclusions of unknown geometry, embedded in insulators, using lock-in inductive thermography. The method is based on averaging the amplitude and phase thermograms along circles concentric with the center of the heated region. We present an analytical calculation of the surface temperature distribution produced by a horizontal circular heat source, taking into account conductive coupling with the air, and we propose to fit the averaged data to this model. We present lock-in thermography data on resin samples containing embedded calibrated circular and rectangular Cu slabs that we excite inductively. The results indicate that it is possible to determine the area and depth of rectangular heat sources with aspect ratio below 1.5 with accuracy better than 10 %.

Experimental study on flame interaction, gas temperature distribution, and radiative heat flux from two rectangular burning propane sources

This work experimentally investigates the flame interaction (flame merging and flame height), gas temperature distribution, and radiative heat flux from two rectangular propane burners by varying total heat release rate (80–360 kW) and burner spacing (0–1.28 m). The results show that the dominant mechanism of air entrainment rate changes from spacing to heat release rate with increasing spacing, making two flames from fully merging to non-merging. A method for predicting the flame height is proposed in terms of air entrainment mechanism, which is proportional to the 2/3 power of the group dimensionless parameter (Q˙total*,Q˙total*=Q˙totalρ0CpT0g1/2LP3/2). With the decrease of spacing, gas temperature in the additional region is distributed in an inverted U-shaped manner in the horizontal direction, reaching its extreme value at the center of the two burners. The vertical gas temperature can be employed to estimate the critical flame merging, which depends on the dimensionless parameters of Q˙total* and S/W. When S/W < 51 Q˙total*, two flames are merged; otherwise, the two flames are separated. For two rectangular fires with different burner spacings, the radiative heat flux received by the external targets is accurately predicted using the Richardson Number (Rix, Rix=gβ(Tf−T0)xU2).

Dynamics of River Flood Waves below Hydropower Dams and Their Relation to Natural Floods

The dynamic behavior of flood waves on rivers is essential to flood prediction. Natural flood waves are complex due to tributary inputs, rainfall variations, and overbank flows, so this study examines hydropower dam releases, which are simpler to analyze because channel effects are isolated. Successive arrival times and heights of peaks along 9 rivers with multiple stream gauges downstream of hydroelectric dams show that flow peaks typically become exponentially lower and wider with distance. The propagation velocity of peaks increases with water depth and channel slope but decreases with downstream distance and greater channel tortuosity. A rich hierarchy of velocities was found. Hydropower pulses progress at or in slight excess of the theoretical celerity, which is faster than the propagation rate of average natural floods, which in turn exceeds the mean velocity of water in the channel, yet the water moves faster than the peaks of record floods. The progressive changes to the height, shape, and velocity of hydropower flow peaks are simulated by the first analytical solution to the convolution integral for a rectangular source pulse that is based on diffusion-advection theory. Available data support some widely held expectations while refuting others. An expanded definition of “water mining” is proposed.

Transient electromagnetic perspective technology in the ultra-long coal mining face

The ultra-long coal mining face has the advantages of high recovery efficiency and low production cost; however, the widening of the coal face poses the difficulty of detecting the hidden geological bodies inside the coal face, and using conventional geophysical methods gives rise to imprecise detection results. Therefore, this paper proposes a transient electromagnetic perspective technology based on the conventional mining transient electromagnetic method. A long rectangular loop is used as the transmitting source in one coal mining tunnel of the coal mining face, and it is received in the opposite coal mining tunnel. This method is characterized by a long detection distance and high construction efficiency. Firstly, the paper used numerical simulation to investigate the spread of the transient electromagnetic field from the long rectangular loop source in both a uniform full-space model and a full-space laminated model. This showed the transient electromagnetic field propagation law and the effect of the coal seam thickness. Then, we analyzed the transient electromagnetic perspective signal characteristics of typical geological structures and gave the all-time apparent resistivity calculation method. Conclusively, the feasibility of this method is ultimately confirmed by the analysis of the measured data. Also, the research results provide a new technological approach to detecting hidden geological structures in the ultra-long coal mining faces.

Experimental study and newly proposed correlations for maximum ceiling gas temperature rise in tunnel fires with natural ventilation

As a critical factor in tunnel fire safety, previous research on model predictions of the maximum ceiling gas temperature rise under natural ventilation has not demonstrated consistent results. These discrepancies are commonly attributed to variations in tunnel dimensions and materials. To address this issue, an experimental investigation utilizing heptane pool fires with different dimensions was conducted in a model-scale tunnel. The results show that the burning rate increases with an increasing aspect ratio of the fire source for the selected diameter range of 0.078–0.169 m. The flame length was characterized using a modified diameter of the fire source considering the air entrainment, and found that the non-dimensional flame length is 2.4 times the 2/5th power of the non-dimensional heat release rate. Finally, incorporating the concept of flame virtual origin, two different forms of correlations were developed to quantify the maximum ceiling gas temperature rise under different impingement conditions of the fire plume, and the coefficients were determined as 0.065 and 0.215 by numerous experimental results from different studies. The newly proposed models demonstrate good applicability to the fire sources with different dimensions, especially for the rectangular fire source with a large aspect ratio, thus solving the problem of insufficient accuracy of classical models under this condition.

Ground deformation due to natural resource extraction in the Western Canada Sedimentary Basin

The study links multiple natural resource extraction activities in the Western Canada Sedimentary Basin (WCSB) with observations of ground deformation. Sentinel-1, RADARSAT-2, and RADARSAT Constellation Mission (RCM) Synthetic Aperture Radar (SAR) data acquired as early as 2015 were used to compute deformation maps and ground deformation time series. Ground deformation produced by induced earthquakes (2015 Mw 4.6 in British Columbia and 2022/23 Mw 5.1/4.6 in Alberta; and 2017/18 Mw 5.0/4.2 slow slip events in British Columbia) were mapped with individual interferograms. Long-lasting ground deformation due to Steam-Assisted Gravity Drainage (SAGD) and Cyclic Steam Stimulation (CSS) enhanced oil recovery and coal and potash underground mining were mapped with the Multidimensional Small Baseline Subset (MSBAS) time series method. When both ascending and descending data were available, MSBAS was used to compute two-dimensional (2D) deformation time series and rates. It was observed that seismic and aseismic ground deformation due to hydraulic fracturing and saltwater injection in WCSB had a similar surface expression (paired anomaly), spatial extent (5-10 km) and orientation (striking NW-SE), suggesting that triggering processes might be technogenic. In the case of hydraulic fracturing, ground deformation appeared soon after the beginning of the operation, while not with saltwater injection, as it took years. The source mechanism of the 2015 Mw 4.6 earthquake was obtained through inverse modelling of the two descending interferograms. The rectangular source model corresponded to a fault at 1.98 ± 0.09 km depth, with 126° ± 7° strike, 14° ± 2° dip, 72° ± 7° rake, 1.61 ± 0.24 km width, 2.54 ± 0.23 km length, and 0.19 ± 0.07 m slip. In the case of in-situ enhanced oil recovery, ground deformation varied significantly depending on the technique. Continuous uplift observed at the SAGD site had a rate of only about 0.05 m/year, while reversible uplift and subsidence at CSS sites exceeded 0.2 m/month, even though wells in both cases were located at approximately similar depths (380 m for SAGD vs 520 m for CSS). In the case of underground mining, the spatial extent of subsidence depended on the extraction depth (200-500 m for coal vs 1000 m for potash). When a mining operation was conducted in an area with topographic relief, horizontal deformation was observed in addition to subsidence. Various observed deformation signals may precede more consequential deformation events.

Geostatistical and multitemporal InSAR method and elastic deformation model for detailed characterizations of topographic change and magma reservoirs: Application to the Tangkuban Parahu Volcano, Indonesia

This study proposes the entire detection of the surface deformation of a volcanic area using interferometric synthetic aperture radar (InSAR) with geostatistics to specify the locations and shapes of the sources causing the deformation. We selected the Tangkuban Parahu area, West Java, Indonesia, as our study site and used a Sentinel-1 dataset. In this study, we combine persistent scatterer and small baseline subset InSAR with a geostatistical technique, i.e., turning bands simulation. This combination can compensate for undetected displacement pixels and decompose the line-of-sight displacement into vertical and horizontal east–west displacements using both the ascending and descending mode results and the least-squares method. This two-dimensional geostatistics and multitemporal InSAR approach is referred to as 2D GMT-InSAR. A detailed displacement pattern is revealed by 2D GMT-InSAR for undetected pixels in low-coherence areas. Then, the vertical displacement pattern in the study period from 2014 to 2019, mainly uplift with local subsidence, is reproduced via the joint use of the Mogi and Okada models for circular and rectangular parallelepiped sources of the deformation, respectively. Via a Bayesian inversion, one circular source, equivalent to a magma reservoir, and one dyke source are identified. The displacement pattern is likely caused by pressure increases in the magma reservoir, associated with the volcanic activation, and in the dyke, associated with the gas and fluid flows from the reservoir, which induced a phreatic explosion in July 2019. A connection of the dyke with the old Sunda volcanic system is found; this new finding is a result of the caldera collapse during the Sunda volcanic eruption.

Influence of magnetic field on MHD mixed convection in lid-driven cavity with heated wavy bottom surface

This study investigates the influence of a rectangular heat source on magnetohydrodynamic hybrid convection flow in a lid-driven cavity. The effects of various parameters, such as the heat source size, magnetic field strength, and heat absorption/generation, are analyzed. The results show that increasing the heat source size decreases the average Nusselt number along the heated wall. The average Nusselt number also decreases with higher magnetic field strength and heat generation, while it increases with heat absorption. The major finding is to apply an important technique the Galerkin weighted residual technique of the finite element (FE) method to solve the non-dimensional equations and the associated boundary conditions. The isotherms are used to show the temperature distribution in a domain. Streamline present the flow field in the enclosure. However, it is easy to realize the direction and intensity of the heat transfer particularly in convection problems which the path of heat flux is perpendicular and the isotherm due to convection effect. Thus, the purpose of this research is to study the results of mixed convection. The effects of location and height of the partitions are considered for the various Richardson numbers. Fluid flow field, thermal field and heat transfer are presented through the streamlines and isotherms, respectively. Results are substantiated relating to the published work.

Thermographic Estimation of the Area and Depth of Buried Heat Sources for Nondestructive Characterization of Horizontal Defects

We present a methodology to estimate quantitatively the area and depth of horizontal defects that generate heat in non-destructive tests such as burst vibrothermography or inductive thermography, without previous knowledge of the shape of the heat source. The method is based on extracting the temporal evolution of the temperature at the centre of the heated region, together with the thermogram obtained at the end of the excitation. The temperature displayed in this thermogram is averaged in circumferences concentric with the centre of the heated region to obtain an averaged radial profile which is fitted, together with the temporal evolution of the temperature, to a circular heat source model. By fitting synthetic data corresponding to rectangular heat sources with added noise, we analyse the accuracy of the method to retrieve the area and depth of the heat source for different depths and aspect ratios. Experimental results show that the method is able to estimate the area and depth of heat sources with aspect ratios below 1/1.5 with accuracy of about 10%.

A Photonic Spectrum Analyzer for Waveguide-Coupled Sources in the Terahertz Range

Signals in the terahertz range are mainly wireless, waveguide-bound or on-chip signals. To measure their spectral power, linewidth and harmonics, usually spectrum analyzers are used. Here we present a photonic spectrum analyzer for waveguide-bound signals for the frequency range from 450 GHz to 1.05 THz with the possibility to extend its operation to on-chip signals. Therefore, a photonically generated local oscillator frequency is used to downconvert the signal under test within a photoconductor. A Vivaldi antenna feeds the signal from a dielectric waveguide to the photoconductor. This allows the measurement of any signal that can be transported via a dielectric waveguide. We show details on the transition from a rectangular metallic hollow waveguide to a dielectric waveguide and spectral measurements of a rectangular metallic hollow waveguide source. The same transition can also be used to attach metallic waveguide-coupled ground-source-ground probes in order to characterize on-chip sources.

Experimental studies on sidewall pool fire for assessing green façades hazard

Fire hazard assessment on green façade used to be carried out by a window plume. Large amount of fuel was burnt. A small pool fire placed adjacent to the façade can give similar effect on assessing the hazardous consequences. Fire plume characteristics provide important information of fire hazards adjacent to façade. The average flame height of rectangular source fire located against different sidewall materials were investigated in this study.Burning tests were conducted with rectangular burners of different dimensions with adjacent walls of different thermal properties, including steel plate and calcium silicate board. Experimental results of fire source with sidewall were compared with those in open space. Due to sidewall blockage, a deep necking-in structure was observed at the flame base and entrained air for combustion. Effect of aspect ratio of the fire pool and effect of heat loss through a sidewall on the average flame height were investigated. Convection lost would decrease the temperature of the fuel-air mixture and should not be ignored. On average, the mean flame height of the fire source with steel plate was 1.08 times that with calcium silicate board. A model for estimating the average flame height was developed by modifying the mirror model.

A thermo-metallurgical-mechanical model for microstructure evolution in laser-assisted robotic roller forming of ultrahigh strength martensitic steel

Laser-assisted robotic roller forming (LRRF) apparatus and process were developed to bend a plate to form a straight channel for ultrahigh strength steel MS1300. Since the thermal processing during the roller forming impacts the microstructure and mechanical behavior of the steel, an integrated thermo-metallurgical-mechanical finite element simulation considering the heat source, phase transformation and material constitutive models was established. A rectangular laser source was devised to homogenize the temperature around the bending corner and a new surface heat source model was proposed and validated. The phase transformation model accounting for the austenitization process, austenite decomposition and tempering was embedded in the finite element model through self-developed user subroutines. The predicted microstructure evolution and the phase distribution were consistent with experimental microstructure characterization. More specifically, it is found that tempering dominates at the inner layer of the bend, resulting in two different phases, i.e., the original and tempered martensitic phases after the LRRF process. The outer layer of the bend, however, goes through austenitization, quenching, and tempering processes, resulting in a combination of fresh martensite, a small amount of tempered martensite and retained austenite phases.

Air Kerma Calculation in Diagnostic Medical Imaging Devices Using Group Method of Data Handling Network

The air kerma, which is the amount of energy given off by a radioactive substance, is essential for medical specialists who use radiation to diagnose cancer problems. The amount of energy that a photon has when it hits something can be described as the air kerma (the amount of energy that was deposited in the air when the photon passed through it). Radiation beam intensity is represented by this value. Hospital X-ray equipment has to account for the heel effect, which means that the borders of the picture obtain a lesser radiation dosage than the center, and that air kerma is not symmetrical. The voltage of the X-ray machine can also affect the uniformity of the radiation. This work presents a model-based approach to predict air kerma at various locations inside the radiation field of medical imaging instruments, making use of just a small number of measurements. Group Method of Data Handling (GMDH) neural networks are suggested for this purpose. Firstly, a medical X-ray tube was modeled using Monte Carlo N Particle (MCNP) code simulation algorithm. X-ray tubes and detectors make up medical X-ray CT imaging systems. An X-ray tube's electron filament, thin wire, and metal target produce a picture of the electrons' target. A small rectangular electron source modeled electron filaments. An electron source target was a thin, 19,290 kg/m3 tungsten cube in a tubular hoover chamber. The electron source-object axis of the simulation object is 20° from the vertical. For most medical X-ray imaging applications, the kerma of the air was calculated at a variety of discrete locations within the conical X-ray beam, providing an accurate data set for network training. Various locations were taken into account in the aforementioned voltages inside the radiation field as the input of the GMDH network. For diagnostic radiology applications, the trained GMDH model could determine the air kerma at any location in the X-ray field of view and for a wide range of X-ray tube voltages with a Mean Relative Error (MRE) of less than 0.25%. This study yielded the following results: (1) The heel effect is included when calculating air kerma. (2) Computing the air kerma using an artificial neural network trained with minimal data. (3) An artificial neural network quickly and reliably calculated air kerma. (4) Figuring out the air kerma for the operating voltage of medical tubes. The high accuracy of the trained neural network in determining air kerma guarantees the usability of the presented method in operational conditions.

An optimization method of rough surface adapted small rectangular LD laser source based on combination of Powell prism and cylindrical lenses

A combination optimization method was proposed to design a small-size rectangular laser spot and decrease the measurement error caused by the uneven intensity distribution of LD laser source. First, the intensity distribution of LD was analyzed and the combination lens structure of Powell prism and cylindrical lenses was proposed. The equations were established to calculate the profile parameters of lenses. And an illuminance loss model δΔED±R was built to determine the values of laser spot size, working reference and range. Secondly, the specific design process of our method was given, the 3D model was established and simulated in ZEMAX to further improve the imaging uniformity. Finally, the collimation, uniformity and location deviation of the spot were discussed. The average uniformity was 95.626%, and the difference of location deviation on shot peened surfaces was 0.004 μm in x direction and 0.121 μm in y direction when compared with the white ceramic surface.

Spatial position recognition method of semi-transparent and flexible workpieces: A machine vision based on red light assisted

In the automatic sorting process, overlapping translucent and flexible workpieces on the conveyor belt, blurring the imaging edge features of translucent and flexible workpieces is a challenge to locate the upper and lower workpieces spatially, we propose a method for locating translucent and flexible workpieces spatially under the overlapping environment in conjunction with the most common automatic sorting of translucent and flexible workpieces such as infusion tube drip buckets. Firstly, we propose a rectangular surface light source based on 650 nm band and monocular CCD for imaging translucent workpieces such as infusion tube drip buckets and optimize the imaging parameters. Secondly, we study a feature matching recognition algorithm for flexible workpieces that are prone to deformation, construct a mapping relationship between the position of overlapping layers and imaging quality of translucent and flexible workpieces such as infusion tube drip buckets based on clarity and information entropy, and establish The mapping relationship between the position of the overlapping layers and the imaging quality of translucent and flexible workpieces such as infusion tube drip buckets is constructed based on clarity and information entropy, and a local spatial coordinate conversion model is established. Finally, the spatial positioning coordinates of overlapping and non-overlapping translucent and flexible workpieces in the local coordinate system are identified, and the results show that the imaging method and theory can be effectively applied to the identification of overlapping and spatial positioning coordinates in the automatic sorting of translucent workpieces such as infusion tube drip buckets.

Experimental study on flame merging of two parallel rectangular fire sources with ground effect

This paper investigates the flame merging process of two parallel rectangular fire sources with ground effect. The experiments were carried out with different burner sizes (aspect ratio w/d = 1–3), heat release rates per unit area Q″ and burner spacings s. It is found that the flame interaction is weak when w/d = 1 and the flame merging process could be further divided into four states. The relationship between the flame merging probability P and the dimensionless burner spacing s/Zf0 (Zf0 is the flame height of one single fire) could be described by the Boltzmann function. The coefficient a and b of the function have definite physical significance and increase with w/d. An exponential function correlates a, b and w/d, and a semi-empirical model of P is established. A correction factor C constructed with P and As=0/d (As=0: short side length of the overall fire source when s = 0) is proposed to correct the dimensionless heat release rate Q˙d*. Thus, the characteristic dimension d of one single fire could be applied to the whole flame merging process. A semi-empirical formula is developed to describe the relationship between the dimensionless flame height L/d and CQ˙d*.