Liquid Metal Flow Research Articles

Investigation on metal vapor characteristics and keyhole/melt pool dynamics in high-power laser welding with side gas flow

High-power laser welding processes involve intense interactions between the laser beam and the base material, resulting in severe welding defects such as humping, spattering, and undercutting. Utilizing side gas flow may provide an effective approach for minimizing these defects. However, the effects of side gas flow on the dynamic behavior and weld formation necessitate further investigation. In this study, an innovative analysis of metal vapor characteristics and keyhole/melt pool dynamics in high-power laser welding with side gas flow is presented, combining experimental and simulation results. A three-dimensional transient fluid flow model was developed, uniquely integrating an adaptive heat source, evaporation-condensation processes, and metal vapor energy attenuation to provide a novel perspective on the effects of side gas flow in high-power laser welding. The results display that side gas flow could suppress plasma, increase thermal input to the melt pool, and intensify the flow of liquid metal toward the pool bottom during high power laser welding, thereby enhancing weld seam penetration. The numerical results indicate that variations in flow rate significantly impact metal vapor morphology, with higher flow rates reducing both the average area and height of the metal vapor and effectively suppressing the size of metal vapor/plasma. When the side gas flow rate is 20 l/min, with the gas flow impact point located at the keyhole opening and the nozzle height set to 3 mm, the side gas flow significantly enhances the depth-to-width ratio of the weld seam in 316L stainless steel laser welding. This work can provide guidance for suppression of weld defects in high-power laser welding of medium-thick steel components, which has important theoretical significance and application value.

Microstructure and Mechanical Properties of a Weld Seam from Magnetron High-Current CO2 Welding

External magnetic field (EMF)-assisted high-current CO2 welding is beneficial for improving the large spatter and poor performance of the welding heat-affected zone for mild steels under high-current welding specifications. In this paper, the droplet transfer behaviors were determined using a high-speed camera on a self-developed magnetically controlled CO2 welding system. Based on these welding specifications, a three-dimensional, transient, multi-energy field coupling welding system model to investigate the mechanism of the droplet and molten pool in EMF-assisted welding was developed. The microstructure and mechanical properties of the welded joint were systematically studied. The results show that the Lorentz force applied by the EMF to twist the droplet decreases the accumulated energy in the short-circuited liquid bridge and changes the liquid metal flow condition, both of which reduce the spatter by 7% but increase the welded joint hardness by 10% and tensile strength by 8%.

The Concept of Using 3D Printing Technology in Ceramic Foundry Filter Manufacturing

AbstractThe article presents the concept of using 3D printing technology in ceramic foundry filter manufacturing. They are a crucial component for obtaining an acceptable quality of nickel superalloys by carrying out the process using the precision casting method. Commonly used filters of this type have a number of disadvantages. They are characterized by irregularity of the filtering structure, high brittleness, lack of resistance to mechanical shocks, and impacts of a stream of liquid metal. All these factors create a risk of introducing the material from the damaged filter into the casting mold, which translates into contamination of the casting alloy, and thus the occurrence of casting defects such as nonmetallic inclusions. The hope for a change in this state of affairs is the use of additive manufacturing technology in their production, which by assumption will allow to obtain a product with a repeatable shape, high mechanical resistance, and a specially designed structure regulating the flow of liquid metal into the mold during casting. The suitability of robocasting and binder jetting technology for producing filtration structures was initially assessed. The results have presented the selection of ceramic powder, as well as the development of the composition of the ceramic paste. The parameters of paste preparation and 3D printing individual processes were described. Moreover, the representative microstructures and basic mechanical properties of samples obtained by both of the compared technologies were presented. Furthermore, the final effect of prototype casting filters with a repeatable shape, manufactured with the use of both technologies, was presented, which were transferred for further technological research in a foundry producing critical aircraft engine parts. The possibilities of using each technology in various applications were also discussed.

Large Eddy simulation on the Lead-Bismuth Eutectic jet at reactor core outlet

The Lead-Bismuth Eutectic (LBE) Cooled Fast Reactor (LFR) represents a promising Generation-IV technology, anticipated to offer a sustainable, safe, and clean energy solution. LBE fluid from LFR assemblies varies in temperature, causing violent temperature oscillations during the jet mixing that may contribute to structural fatigue, threatening LFR safety. This study focuses on investigating the critical thermal–hydraulic phenomena occurring at the core outlet of the LFR by modeling the assembly head in a realistic reactor configuration. Large Eddy Simulation (LES) is employed to explore the LBE jet characteristics in a three-head model with varying inlet parameters. The analysis includes the fluctuation patterns of both the flow and temperature fields downstream of the assembly head. The results show that the influence area of the assembly head is mainly in the range of about 0–175 mm downstream of the outlet of the assembly head. The temperature fluctuation frequency is the highest near the assembly head, typically within 20 Hz, and the temperature fluctuation frequency in other areas is lower. These findings provide valuable insights for designing LFRs and understanding the flow and heat transfer behavior of liquid metals.

Impact of Mg on the Feeding Ability of Cast Al–Si7–Mg(0_0.2_0.4_0.6) Alloys

The demand for high-performance Al–Si casting alloys is driven by their mechanical properties, making them popular in automotive, aerospace, and engineering industries. These alloys, especially hypoeutectic Al–Si–Mg, offer benefits like high fluidity, low thermal expansion, and good corrosion resistance. Silicon and magnesium primarily define their microstructure and mechanical properties. Silicon enhances fluidity, while magnesium improves strength and fatigue resistance. However, challenges like shrinkage porosity persist during solidification. Understanding solidification feeding regions is crucial, influenced by factors such as chemical composition, solidification characteristics, and casting design. This study investigates magnesium’s influence on feeding ability in hypoeutectic Al–Si7–Mg alloys through experimental tests. Increasing magnesium content from 0% to 0.6% affects the interdendritic and burst feeding regions. This could impact shrinkage porosity formation. The “Sand Hourglass” test results indicate a rise in porosity levels with higher magnesium content, which is linked to the narrowing of interdendritic channels and the formation of magnesium-rich intermetallic compounds. These changes hinder the liquid metal flow, worsening shrinkage porosity. Therefore, magnesium’s role in expanding the interdendritic region is a key factor in developing porosity in cast hypoeutectic Al–Si7–Mg alloys. This study highlights that porosity levels increase from 0% in magnesium-free Al–Si7 to 0.84% in Al–Si7–Mg0.6, underscoring magnesium’s significant impact on the occurrence of porosity in these alloys.

Coupled model for liquid lithium plasma facing components

Numerical analysis provides the design choice and operating window of liquid metal Plasma Facing Components (PFC) concepts. Coupled analysis of boundary plasma together with the surrounding boundary structures is required. To achieve this goal, PPPL is developing a comprehensive multi-physics model for modeling of PFCs in fusion devices. The model includes the fluid-kinetic code SOLPS-ITER and the flow and heat transfer code CFX from ANSYS. SOLPS-ITER was augmented with a liquid metal boundary condition algorithm, allowing direct two-way coupling of the plasma analysis with the two-dimensional analytical slab flow model which includes heat convection in the liquid metal PFC. The target heat flux resulting from this coupled analysis is used as a boundary condition for detailed 3D Computational Fluid Dynamics (CFD) Magneto Hydro Dynamics (MHD) and heat transfer analysis. A new formulation of MHD equations is introduced in the numerical procedure ensuring current conservation of the discretized equations. Results of the 3D analysis are used for final validation of the coupled model. A PFC design where a porous wall is used to stabilize the liquid metal surface, while MHD drive is used to push the liquid metal flow inside the PFC, will be investigated in the regimes where vapor shielding is created for enhanced volumetric plasma heat dissipation.

Accuracy and scalability of incompressible inductionless MHD codes applied to fusion technologies

Abstract It is well-known that magnetohydrodynamics (MHD) dominates the dynamic of the liquid metal flows inside the breeding blankets (BB) of future nuclear fusion plants by magnetic confinement. MHD is a multiphysics phenomenon involving both electromagnetism and incompressible fluid mechanics. From the computational point of view, the simulation of MHD flows in fusion relevant conditions entails a significant challenge. Indeed, due to the shape of the induced electrical currents inside the bulk of the fluid, high spatial resolutions are needed to capture the large gradients found in boundary layers and 3D effects. Besides, solving the equations accurately typically requires very small time steps for the transient algorithms. Over the past few decades, some parallel MHD codes have been developed with success to simulate complex flows in increasingly realistic geometries. Among them, the MHD tools of commercial CFD platforms have attracted attention due to their relatively soft learning curve. Most of these codes are based on the so called ϕ-formulation which, by applying the divergence free condition of the current density to the Ohms law, reduces the electromagnetic part of the problem to a single Poisson equation. As a downside, the approach segregates the fluid and electromagnetic problem. In practice, this establishes important limits to the mesh element size, to the mesh quality and to the time-step needed to obtain accurate and stable solutions that maintains charge conservation at a discrete level. In this work, these limits are explored for the commercial platform ANSYS-Fluent using a test geometry under different conditions. As an alternative, a new code based on Finite Element Methods (FEM) is introduced as well. This open-source code, called GridapMHD (https://github.com/gridapapps/GridapMHD.jl), aims at solving the full set of MHD equations using a monolithic approach. GridapMHD is still in early stages of development but it has already shown promising results.

Interaction of inertia and magnetic force in the liquid metal flow past a magnetic obstacle

In this paper, we present a numerical study of the three-dimensional behavior of a liquid metal flow in an insulating rectangular duct of narrow cross section past a localized magnetic field (i.e., a magnetic obstacle) produced by two parallel square magnets arranged externally on the walls of the duct. A series of simulations are conducted focused mainly on describing the interplay between inertial and magnetic forces in a wide range of interaction parameters (1.8<N<48) by varying the Reynolds number while the Hartmann number is kept fixed (Ha = 75). The analyzed configuration coincides with that studied experimentally by Domínguez et al. [“Experimental and theoretical study of the dynamics of wakes generated by magnetic obstacles,” Magnetohydrodynamics 51(2), 215–224 (2015)] and, as a first step, experimental data from local variables (streamwise velocity component) and global parameters (oscillation frequency and kinetic energy of the wake) are consistently replicated by the numerical model. Furthermore, to complement the flow phenomenology, the transition to different flow structures as the interaction parameter varies is explored. It is found that when the magnetic forces predominate over inertia, stationary vortex patterns with two, four, and six vortices appear while, unlike the hydrodynamic flow past a bluff body, the increase in inertial effects leads to a reduction in the number of vortices and eventually to their disappearance, reaching a state in which the magnetic obstacle becomes imperceptible to the flow. The existence of a critical value of the interaction parameter that maximizes the kinetic energy of the wake is confirmed numerically and corroborated from the experimental data.

Моделирование гидродинамики жидкого металла в ячейке МГД-сепаратора

Ensuring the purity of metals and alloys is a key problem in both the metallurgical and nuclear industries. The difference in the electrical conductivity of the molten metal and impurity particles allows their separation from each other in a non-contact way using an electromagnetic force. In addition to the implementation of phase separation, such force inevitably sets the liquid metal in motion, changing the velocity distribution in the separation cells. This effect is most pronounced in channels of rectangular geometry and therefore should be studied extensively. In this paper, we numerically study the hydrodynamics in a separation cell in the presence of partitions of various configurations and in the presence or absence of an external weighting electromagnetic force with the aim to determine the cell geometry providing the most efficient separation and accumulation of impurity particles in it during the cyclic pumping of liquid metal. It has been found that the intense force action leads to the occurrence of parasitic vortices in the channel, which reduce the efficiency of the separation process. The initial flow velocity does not practically affect the topology of the flow in the channel, whereas the height and configuration of the baffles inside the channel, together with the magnitude of the external weighting force have a significant effect on the flow of liquid metal. It is shown that in the zones between the baffles, the parasitic vortices that could potentially facilitate the washout of impurity from this space are not formed. Based on the results of numerical study the optimum configuration of the MHD-separator cell is selected. It corresponds to the mode with the most effective separation according to four parameters: partition height, magnitude of the external weighting electromagnetic force, flow velocity, and topology of intermediate partitions.

Porosity suppression mechanism analysis in narrow-gap oscillating laser-MIG hybrid welding of aluminum alloys based on keyhole stability and molten pool flow behavior

In this study, the keyhole dynamic behavior and molten pool flow were investigated to elucidate the effects of oscillating laser on porosity in narrow-gap oscillating laser-MIG hybrid welding of aluminum alloys. A weld with a porosity of 0.08% was produced at the laser oscillating frequency of 600 Hz and diameter of 1 mm, while 2.27% without oscillating laser. Compared to non-oscillating laser, keyhole opening size increased by about 2.3 times and keyhole depth fluctuation was suppressed. The forces acted on the interface between keyhole wall and molten pool became more balanced. The increased vapor recoil pressure overcame the combined closure effect of gravity and molten pool pressure. While applying oscillating laser, the vortex in molten pool was eliminated and the liquid metal flow was more stable. The combined impact force caused by droplets transfer and liquid metal convection due to constraining effect of narrow gap was buffered, which had less effect on molten pool and keyhole stability. The liquid metal tended to flow from the molten pool inside to surface, which favored the escape of bubbles from the molten pool.

Online Flow Measurement of Liquid Metal Solutions Based on Impact Force Sequences: Modeling Analysis, Simulation, and Validation of Experimental Results.

Aiming at the existing high-temperature liquid metal flow online accurate measurement by the metal melt characteristics, installation space, and high-temperature environment adaptability limitations, this paper innovatively puts forward a soft measurement method based on the impact force generated in the fluid flow process as an observational variable series. Fluid mechanics theory and simulation software are used to analyze and verify the feasibility of the impact force as an observable variable to measure the flow rate, followed by the construction of the CNN-LSTM-CNN-Double (CLCD) flow measurement model of impact force and flow rate based on the parameters of the learning rate and the number of training times, and finally the construction of a test platform for the flow measurement, and the validity of the method is verified through actual operation.

Two-Field Excitation for Contactless Inductive Flow Tomography.

Contactless inductive flow tomography (CIFT) is a flow measurement technique allowing for visualization of the global flow in electrically conducting fluids. The method is based on the principle of induction by motion: very weak induced magnetic fields arise from the fluid motion under the influence of a primary excitation magnetic field and can be measured precisely outside of the fluid volume. The structure of the causative flow field can be reconstructed from the induced magnetic field values by solving the according linear inverse problem using appropriate regularization methods. The concurrent use of more than one excitation magnetic field is necessary to fully reconstruct three-dimensional liquid metal flows. In our laboratory demonstrator experiment, we impose two excitation magnetic fields perpendicular to each other to a mechanically driven flow of the liquid metal alloy GaInSn. In the first approach, the excitation fields are multiplexed. Here, the temporal resolution of the measurement needs to be kept as high as possible. Consecutive application by multiplexing enables determining the flow structure in the liquid with a temporal resolution down to 3 s with the existing equipment. In another approach, we concurrently apply two sinusoidal excitation fields with different frequencies. The signals are disentangled on the basis of the lock-in principle, enabling a successful reconstruction of the liquid metal flow.

Numerical Calculations of Liquid-Metal Magnetohydrodynamic Flows in Rectangular Ducts with Sudden Contractions

For the design of the liquid-metal blanket in a fusion reactor, numerical calculations have been carried out on liquid-metal magnetohydrodynamic flows in rectangular ducts with sudden contractions. Conservation equations of fluid mass and fluid momentum and the Poisson equation for electrical potential have been solved numerically. The numerical calculations have been conducted for a Hartmann number of ~10 000; a Reynolds number of ~10 000; and contraction ratios (CRs) of 2, 3, and 4. The pressure loss through the contraction has been estimated by the loss coefficient ζ divided by the interaction parameter N, i.e. ζ/N. The loss coefficient ζ/N through the contraction parallel to the magnetic field is much larger than that through the corresponding contraction perpendicular to the magnetic field. The loss coefficient ζ/N increases consistently with the CR and does not change very much with N. While ζ/N also does not change very much with the wall conductance ratio for the contraction parallel to the magnetic field, ζ/N increases gradually with the wall conductance ratio for the contraction perpendicular to the magnetic field.

Optimization of lithium vapor box divertor evaporator location on NSTX-U using SOLPS-ITER

Commercial fusion reactors will be faced with extremely high divertor target heat fluxes that will require mitigation. Simulations of detachment in an NSTX-U scenario projected to have 92 MW m−2 unmitigated peak target heat flux are presented, which reaches sub-10 MW m−2 target heat flux using a highly dissipating lithium vapor box divertor design. The lithium vapor box is a detached divertor design which employs lithium vapor evaporation and condensation to contain lithium below the X-point. Previous SOLPS modeling has indicated a lithium vapor box can reduce the heat flux down to 10 MW m−2 via simultaneous evaporation from the Private Flux Region (PFR) and the Common Flux Region (CFR) sides of the vapor box. It is found here that PFR evaporation has improved access to the separatrix leading to significantly more efficient power dissipation than CFR evaporation. Simulations of target evaporation with an evaporation distribution that is self-consistent with the temperature of a Capillary Porous System with Fast flowing liquid lithium could reach nLi / ne∼ 0.025–0.030 at the Last Closed Flux Surface (LCFS) depending on the liquid metal flow speeds and lithium sputtering yield, while PFR-side evaporation can reach acceptable heat fluxes with nLi / ne∼ 0.038 at the LCFS. However, PFR evaporator performance can be improved if the target is allowed to be hot enough such that it reflects lithium, reaching nLi / ne∼ 0.028 and reducing required lithium evaporation. Ultimately PFR evaporation and target evaporation are found to have similar ability to produce acceptable heat flux solutions with minimal upstream concentration.

Study on the liquid metal flow transitions behind a circular cylinder under the axial magnetic field

We study the magnetohydrodynamic flow around a circular cylinder confined in a rectangular duct. In this configuration, both the circular cylinder and the walls of the rectangular duct are electrically insulating, while the magnetic field aligns with the axial direction of the cylinder. The experimental measurements are performed by controlling two parameters Re and N (N is the ratio of electromagnetic forces to inertial forces) in the ranges of (180–722) and (0.8–264), respectively. Utilizing the electrical potential method, we employ both movable and wall probes to obtain the local flow velocities in the wake of the cylinder. By analyzing the space correlation of the signals obtained from the wall probes, a distinct transition in the flow behavior is observed, transitioning from a three-dimensional state to a quasi-two-dimensional (Q2D) state when the external magnetic field reaches a sufficient strength (N > 3). Additionally, the Q2D state allows for a further subdivision based on the scaling relationship Re/Ha∼(0.41–0.42) in the stability map, thereby distinguishing between steady and unsteady flow states, which is consistent with findings from previous studies.

Development of liquid metal targets for high-repetition intense laser and pulsed-power discharge applications

In this study, we investigate the response of linear liquid metal flow to a magnetic field when used as a load for high-repetition pulsed-power discharges and as a sheet for a laser target. When a magnetic field was applied perpendicular to the liquid metal flow, the flow deformed while maintaining its thickness. This approach may reduce current and enhance light sources in Z-pinch and alternative X-pinch discharges. However, low-frequency fluctuations were more prominent when the magnetic field was applied parallel to the liquid metal flow in sheet form. These findings suggest that combining liquid metal flow with a magnetic field could effectively facilitate high-repetition pulsed-power discharges and laser operations.

Infiltration and permeability of porous ceramics simulated by random networks

A network computational model for a 3D ceramic structure is developed. The model is applied to study the impact of geometric and material parameters of structure on the liquid metal flow through random porous ceramic medium in pressure infiltration processes. The characteristic geometric features of the ceramic structure favorable for liquid metal flow during the infiltration process are determined. The method of structural approximation and constructive homogenization are applied, and the discrete stationary Stokes equations on random graphs are considered. This approach gives a robust algorithm to determine the macroscopic permeability K of interpenetrating phases. The dependencies of K on the distribution of connections (windows) between the cells (inclusions) are derived. The numerical simulations demonstrate that the permeability K does not depend on the scaled distribution sizes of windows. This implies that K is proportional to the mean value of the window areas. The considered model takes into account a random complex structure of 3D ceramic. Hence, it complements the previous study on the local transport properties of tubes (windows) connecting the cells.

Temperature‐dependent viscosity effects on heat transfer characteristics of grade three fluid in electromagnetohydrodynamic flow between large parallel plates maintained at uniform temperatures

AbstractHeat transfer characteristics in the electromagnetohydrodynamic flow of a third‐grade fluid between parallel plates have already been examined by researchers, but studies on the effect of temperature‐dependent viscosity on velocity, temperature, and heat transfer coefficients are very few. The present study investigates the heat transfer behavior of a third‐grade fluid flow between large parallel plates, taking temperature‐dependent viscosity when magnetic and electric fields are imposed externally. The plate walls are subjected to uniform temperatures. Consideration of temperature‐dependent viscosity results in the formulation of the phenomenon by nonlinear, coupled differential equations, which are solved by applying the least‐square method (LSM). Solving coupled, nonlinear equations by the LSM requires a great deal of modification in the implementation process. The obtained results in terms of dimensionless velocity are validated with the results of earlier studies and are also in a close match with the results of the Adomian decomposition method of the current study. Results indicate that for a higher value of the viscosity parameter, at first, velocity increases with an increase in the non‐Newtonian parameter. Beyond a certain value of the non‐Newtonian parameter, velocity starts reducing. Temperature, at all points of any cross‐section, increases up to a certain non‐Newtonian parameter and then reduces with the rise in the same parameter. For higher values of the viscosity parameter and low non‐Newtonian parameters, the upper plate requires an enhanced rate of cooling. Whereas, for higher values of the non‐Newtonian parameter at higher viscosity parameters, a lower heating rate is required. The results of the mathematical model can serve to be useful in the field of liquid metal flows in metallurgical industry, micropumps, medical and biological sectors, and microelectromechanical applications.

A novel Halbach array permanent magnet flowmeter for liquid metal flow measurement. Part Ⅱ: Impact of velocity distribution on measurement signals and optimization design for electrodes

In this paper, the principles of Halbach array permanent magnet flowmeter (HA-PMFM) in reducing the sensitivity of measurement signals to velocity distribution by using arc-shaped electrodes and multi-electrodes, respectively, are analyzed. The impact laws of disturbance amplitudes of three velocity distribution forms (waterfall, skewed, and blockage types) on the measurement signals acquired by HA-PMFM with point electrodes, arc-shaped electrodes, and multiple electrodes, respectively, were investigated by COMSOL Multiphysics software. The results show that, in comparison to HA-PMFM with point electrodes, HA-PMFM with arc-shaped electrodes mitigates the unevenness of the weight function from 0.68 to 0.12. Under strong velocity distribution disturbances at constant flow velocity, optimized arc-shaped electrodes can reduce the measurement error from a maximum of 15 % to within 7 %, and multi- electrodes can reduce the measurement error from a maximum of 15 % to within 5 % compared to the point electrodes. However, when the uniform incoming flow rate is varied, the optimized arc-shaped electrodes can reduce the measurement error from 2 % to within 1 %, and the multi-electrodes can reduce the error from 2 % to within 1.5 %. This study offers valuable insights for enhancing the measurement accuracy of HA-PMFM in engineering applications and provides guidance for the design of liquid metal electromagnetic flowmeters in various fields, particularly for those compact systems using liquid metal as flow medium in which the flowmeter has to be installed on the flow passage with strongly distorted velocity distribution.

SIMMER code two-fluid model well-posedness and stability for simulating liquid metal (lead–lithium eutectic), water vapor, and non-condensable gases multi-component multi-phase flows

Liquid metals are employed as a coolant in liquid metal fast reactors (LMFR) and are considered breeders and coolants for future power-producing fusion reactors. The interaction of liquid metals with other coolants, such as air and water, is one of the possible occurrences in liquid metal-cooled reactors. Although the SIMMER-III computer code is currently in the validation and verification phase, it is a prospective candidate code for correctly investigating possible accidents. This study aims to determine the stability and well-posedness of the SIMMER-III code Eulerian-Eulerian two-fluid model (TFM) under any such accident scenario. This work also considers the influence of the virtual mass force and diffusion forces in momentum on TFM stability in accelerated liquid metal, steam, and non-condensable gaseous flows throughout all flow regimes. The characteristics method is used to assess the ill-posed nature of TFM for all types of accidents and accident scenarios in a hypothetical simplified system model with several components (Lead–Lithium, non-condensable gases, and water vapor). It has been discovered that the analysis findings vary from the air and water two-component two-phase flows (characteristics roots spectrum and error growth rate patterns) and are particularly sensitive to the diffusion and virtual mass coefficients. This is because liquid metals have a higher density than liquid water and steam, resulting in strong virtual mass forces and weak diffusion forces in liquid-metal and gas two-phase flows. Because of the extensive range of possible interactions between fluxes of different components, producing an accurate representation of diffusion in multi-component mixtures is difficult. Because of this, it is strongly suggested that the virtual mass coefficient and diffusion coefficients be handled more accurately for these kinds of flows. The values of these coefficients significantly affect how accurate proposed TFM predictions are, but there hasn't been much research on how to estimate them. The study also sheds light on the model's accuracy and highlights the areas where the model's predictions will be mathematically trustworthy.