Initial Diffusion Coefficient Research Articles

Characteristics of constant pressure dehydrogenation and construction of linear diffusion model for solid hydrogen storage materials: A case study of TiMn1.5 alloy

Solid-state hydrogen storage is considered to be the most promising hydrogen storage technology because of its high volumetric capacity and good security. The process of hydrogenation and dehydrogenation of solid hydrogen storage materials can be characterized by a diffusion kinetic model, which can be used to guide the safe and efficient application of hydrogen storage materials. Currently, experimental study on the dehydrogenation characteristics under constant pressure outer boundary condition has not been carried out methodically, and a simple and reliable diffusion kinetic model under this condition should be established based on the experimental results. In this study, an experimental platform for hydrogen absorption and constant pressure dehydrogenation of solid hydrogen storage materials was independently established. The TiMn1.5 alloy and its interstitial hydrides were taken as the experimental research objects to conduct constant pressure dehydrogenation tests, and the dehydrogenation properties at different temperatures were systematically studied. The results show that the dehydrogenation rate decreases with the increase of testing time. The ultimate capacity of hydrogen release increases first and then decreases with the increasing temperature, reaching the maximum value at 30 °C. It is found that the diffusion coefficient increases linearly with time during constant pressure dehydrogenation process. On this basis, a linear diffusion kinetic model is established. The initial diffusion coefficient D0 calculated by the new model has a quadratic polynomial relationship with temperature, and the increase coefficient α of diffusion coefficient has a linear relationship with temperature. According to the mathematical expressions, the dehydrogenation process of TiMn1.5 alloy at different temperatures can be predicted. The research results can provide theoretical guidance for the safe, efficient and controllable application of hydrogen storage materials.

Analysis of dominant flow in tectonic coal during coalbed methane transport

Diffusion and seepage are the main flow forms of coal seam gas transport, and are one of the key factors in the selection of gas extraction improvement methods. Changes in the physical structure of tectonic coal make gas transport more complex during coalbed methane extraction. In this paper, we develop a multi-field coupled model of methane transport in coal seams, taking into account the effects of tectonics, and theoretically analyze the dominant flow patterns for methane extraction. Then, the evolution of gas dominated flow is analyzed for different initial pressures, initial permeabilities, and initial diffusion coefficients of tectonic and intact coal seams. The results show that the amount of daily methane seepage in tectonic coal increases with the initial pressure of the coal reservoir, but decreases with the initial diffusion coefficient of the coal reservoir. Methane seepage in tectonic coal has a longer control time than in intact coal at different initial pressures, initial permeabilities, and initial diffusion coefficients of the coal reservoir. For different coal reservoir initial pressures, coal reservoir initial permeabilities, and coal reservoir initial diffusion coefficients, the maximum seepage control time for tectonic coal is 20, 17, and 15 times longer than for intact coal, respectively. Finally, the discrepancies of methane dominant flow in tectonic coal and intact coal during methane extraction were analyzed by using the double bottleneck flow model, and methods for methane enhanced extraction in tectonic coal and intact coal were discussed. The results presented in this paper may provide a theoretical reference for the extraction of differentiated gas in coal seams.

The Pore Structure Multifractal Evolution of Vibration-Affected Tectonic Coal and the Gas Diffusion Response Characteristics

Vibrations caused by downhole operations often induce coal and gas outburst accidents in tectonic zone coal seams. To clarify how vibration affects the pore structure, gas desorption, and diffusion capacity of tectonic coal, isothermal adsorption-desorption experiments under different vibration frequencies were carried out. In this study, high-pressure mercury intrusion experiments and low-pressure liquid nitrogen adsorption experiments were conducted to determine the pore structures of tectonic coal before and after vibration. The pore distribution of vibration-affected tectonic coal, including local concentration, heterogeneity, and connectivity, was analyzed using multifractal theory. Further, a correlation analysis was performed between the desorption diffusion characteristic parameters and the pore fractal characteristic parameters to derive the intrinsic relationship between the pore fractal evolution characteristics and the desorption diffusion characteristics. The results showed that the vibration increased the pore volume of the tectonic coal, and the pore volume increased as the vibration frequency increased in the 50 Hz range. The pore structure of the vibration-affected tectonic coal showed multifractal characteristics, and the multifractal parameters affected the gas desorption and diffusion capacity by reflecting the density, uniformity, and connectivity of the pore distribution in the coal. The increases in the desorption amount (Q), initial desorption velocity (V0), initial diffusion coefficient (D0), and initial effective diffusion coefficient (De) of the tectonic coal due to vibration indicated that the gas desorption and diffusion capacity of the tectonic coal were improved at the initial desorption stage. Q, V0, D0, and De had significant positive correlations with pore volume and the Hurst index, and V0, D0, and De had negative correlations with the Hausdorff dimension. To a certain extent, vibration reduced the local density regarding the pore distribution in the coal. As a result, the pore size distribution was more uniform, and the pore connectivity was improved, thereby enhancing the gas desorption and diffusion capacity of the coal.

Vanadium sulfide decorated at carbon matrix as anode materials for “fast-charging” lithium-ion batteries

Vanadium sulfide is considered a promising electrode material for achieving rapid lithium storage ascribing to its high specific capacity, unique crystal structure, and abundant valence states. Unfortunately, the structural collapse induced by volume expansion during charging and discharging is the key factor affecting rapid lithium storage. This article prepares composite materials with different morphologies through a strategy of decorating vanadium sulfide at carbon matrix. Among them, graphene oxide modified vanadium tetrasulfide composite materials with three-dimensional conductive networks exhibit excellent electrochemical performance. When the current density is 50 mA g−1, the discharging specific capacity reaches 1, 560 mAh g−1. After 2, 500 cycles at a high current density of 10, 000 mA g−1, the specific capacity is about 188 mAh g−1 with charging time of approximately 70 s. GITT testing indicates that the order of magnitude of the logarithm of the initial diffusion coefficient is 10−11 cm2 s−1 owing to the three-dimensional porous conductive network, which not only shortens ion diffusion path but also provides more active sites for lithium-ion storage, rendering it a promising anode material for “fast-charging” lithium-ion batteries.

Analysis of methane diffusion on permeability rebound and recovery in coal reservoirs: Implications for deep coalbed methane-enhanced extraction

A proper understanding of the effect of methane diffusion on coal reservoir permeability rebound and recovery is essential, as coal reservoir permeability is the key parameter influencing the efficiency of coalbed methane migration and computational research on it is lacking. In this paper, the multifield coupling model for methane migration was established. Then, two parameters, the influence coefficient of diffusion on permeability rebound (DPRB) and the influence coefficient of diffusion on permeability recovery (DPRC), were proposed to quantify the effect of methane diffusion on rebound and recovery of coal reservoir permeability. Subsequently, we used COMSOL software to study the variation rules of the coal reservoir permeability rebound time, permeability recovery time, and permeability rebound value, DPRB, and DPRC for different geologic parameters. The results shown that the permeability rebound time and recovery time are proportional to the coal seam initial pressure, but inversely proportional to the initial permeability and initial diffusion coefficient. The rebound value decreases with increasing coal seam initial pressure and initial permeability, but ascends with rising initial diffusion coefficient. DPRB declines with increasing coal seam initial pressure, initial permeability, and initial diffusion coefficient, but they are all greater than 0.7, indicating that methane diffusion has a significant effect on permeability rebound. The DPRC values for different coal seam initial pressures, initial permeabilities, and initial diffusion coefficients are above 0.98, which implies that methane diffusion dominates the permeability recovery process. Finally, a conceptual model was presented to research the mechanism of diffusion influence on rebound and recovery of coal reservoir permeability, and the implications for enhanced drainage of deep coalbed methane were discussed. Therefore, the results of this paper can provide a theoretical foundation for deep coalbed methane-enhanced extraction.

Can initial apparent diffusion coefficient and tumour volume predict future metastases in treatment-naive patients with laryngeal squamous cell carcinoma

Can initial apparent diffusion coefficient and tumour volume predict future metastases in treatment-naive patients with laryngeal squamous cell carcinoma

Comparative analysis of permeability rebound and recovery of tectonic and intact coal: Implications for coalbed methane recovery in tectonic coal reservoirs

The permeability rebound and recovery of coal reservoirs is one of the key factors affecting the efficient recovery of coalbed methane (CBM). Most studies focus on the permeability rebound and recovery of intact coal reservoirs. Conversely, the permeability rebound characteristics of tectonic coal have only been rarely investigated. In this paper, an improved fully coupled mathematical model for methane transport of tectonic coal and intact coal seam was established. Then, the phenomenon of permeability rebound and recovery was analyzed theoretically, and the permeability rebound time, rebound value and recovery time were proposed to compare the difference in permeability evolution. In addition, the permeability rebound characteristics of tectonic and intact coal were evaluated for different geological parameters of coal reservoirs. The results indicate that the rebound time and recovery time of both intact and tectonic coal increase with the increase of initial gas pressure. As the initial permeability increases, the permeability rebound time of intact and tectonic coal show a decreasing trend. The permeability rebound time of intact coal decreases with increasing initial diffusion coefficient, while that of tectonic coal shows the opposite trend. Furthermore, the change in permeability recovery time for tectonic coal is 6.59 times larger than for intact coal, indicating that the effect of permeability rebound phenomenon is more significant during gas extraction in tectonic coal reservoirs. Finally, a conceptual model was proposed to explain the differential mechanism of permeability rebound between tectonic and intact coal, and its implications for CBM recovery in tectonic coal reservoirs was discussed. Therefore, the results presented in this paper can provide a theoretical basis for the efficient development of CBM in tectonic coal reservoirs.

Numerical Analysis of Permeability Rebound and Recovery during Coalbed Methane Extraction and Its Implications for Enhanced Coalbed Methane Extraction through Injection of Superheated Water.

Permeability rebound and recovery are pivotal in determining the efficacy of coalbed methane (CBM) extraction and the impact of superheated water injection during thermally enhanced CBM extraction. Existing research predominantly focuses on the roles of effective stress and methane desorption shrinkage, often neglecting the critical influence of the temperature. Therefore, our study introduces a mathematical model incorporating heat-fluid-solid coupling and a permeability evolution model considering temperature variations. The model was used to analyze the phenomenon of permeability rebound and recovery during CBM extraction and the effects of various factors on it. The results show that the permeability rebound and recovery time increase with initial gas pressure but decrease with initial diffusion coefficient and permeability. Initial coal seam temperature has little effect on the permeability rebound time, which increases the recovery time. The permeability rebound value rises with the initial diffusion coefficient but falls with the initial gas pressure and permeability, and the coal seam temperature has little impact on it. In addition, whether to consider the temperature on the permeability evolution is compared. The results reveal that temperature impact causes an elevation in permeability rebound, prolongs rebound and recovery time, and reduces postrecovery permeability ratio compared to the scenario without temperature influence. Inspired by the law of permeability evolution, this paper discusses the impact of injection pressure and temperature on the effectiveness of superheated water injection in the initial stage of enhanced CBM recovery engineering practice. The findings offer valuable insights into selecting optimal injection parameters tailored to various coal seams.

Study on the influence of stress constraint conditions on multi-scale gas emission characteristics in in-situ coal

Gas drainage in boreholes is characterized as an essential means to control abnormal gas emissions during mining. Unfortunately, its drainage efficiency is terribly restricted by high in-situ stress in deep underground coal seams. Gas drainage involves gas emission under stress constraint conditions (SCCs), as well as multi-scale gas migration (gas desorption, diffusion, and seepage). Nevertheless, how SCCs influence gas emission behaviors remains unclear. Considering this status quo, this paper focuses on researching He and CH4 emissions under different SCCs, and analyzes the influence of SCCs on gas emission by integrating the initial gas diffusion coefficient and the diffusion attenuation coefficient. In addition, a novel method for calculating equivalent diffusion and seepage pore radii by the T2 spectrum of nuclear magnetic resonance was proposed. Finally, according to the multi-mechanism gas emission model that tracks the dynamic evolutions of the equivalent diffusion and seepage pore radii, the influences of equivalent pore radius and pressure on the conductance of different diffusion mechanisms were further quantitatively analyzed. The research in this paper is expected to provide a theoretical reference for efficient gas drainage in deep coal seams.

Diffusion coefficient and tortuosity: Brownian Motion

The objective of this research is to determine the chloride ion diffusion through Brownian motion using images. The research also aims at the search for diffusional physical tortuosity due to Brownian motion to understand the transport flow of the chloride ion. Through micro-CT scanned images of concrete samples, a 3D reconstruction is performed for the representative element and define its microscopic properties. The implementation of the Brownian motion of the chloride ion particles in the porous network of the representative element considered the initial diffusion coefficient defined by the Stokes-Einstein equation. The diffusion coefficient is an important parameter for predicting the depth of attack of the chloride ion and its displacement due to Brownian motion helps to define the tortuosity of this diffusive transport. The research used computational modeling with MATLAB® software. The results show diffusion due to Brownian motion and tortuousness.

Tissue evolution of Al0.67Cu0.33 alloy during melting and solidification by molecular dynamics simulation

To analyze the various factors affecting the solidification organization of Al-Cu alloys, the melting and solidification processes of Al0.67Cu0.33 alloy were simulated by the embedded atomic method (EAM) potential. It is found that the diffusion coefficient of the system keeps rising with the increasing initial temperature and relaxation time. It is a positive correlation between initial temperature, relaxation time and diffusion coefficient. The RDF analysis results are shown that as the cooling rate decreases, the crystal content of the solidified tissue increases and crystal structure characteristics of the system become more obvious. At the cooling rate of 1 × 1010 K/s, the percentage of system crystal compositions reaches 78.4%. The system shows obvious crystal structure characteristics. However, when the cooling rate was 1 × 1014 K/s, the crystalline content of the system is only 0.3% and shows obvious non-crystalline characteristics. It provides a theoretical basis for improving the properties and optimizing the microstructure of Al-Cu alloy.

How pore structure evolves in shale gas extraction: A new fractal model

The reservoir can be divided into four components with different properties: inorganic matrix, hydraulic fracture, natural fracture and kerogen according to the internal structure characteristics and the natural gas migration mode after hydraulic fracturing. Previous studies usually focus on the relationship between reservoir porosity and permeability, but ignore the influence of reservoir microstructure on gas production. The microstructure of different reservoir components can be described by corresponding fractal dimensions. The structure of fractures can be characterized by fractal dimension of fractures and the fractal dimension of pores can characterize the pore structure of inorganic matrix. Considering the complexity of mass transfer between components and structural deformation, in order to describe the evolution of reservoir microstructure and the influence of various factors on gas production, a fully coupled multi-scale fractal model is established. To verify that the model is reliable, the simulation results contrast with the conclusions of field tests. The effects of adsorption parameters, initial diffusion coefficient and compressibility coefficient on gas production and pore structure evolution were studied. The results show that:(1) when the adsorption constant decreases, the variation of the fractal dimension of pore structure becomes more obvious, and the shale reservoir's gas production and pore structure are most sensitive to the volume constant; (2) the deformation of pore structure increases with the increase of effective diffusion coefficient; (3) the change of compressibility of a single medium only has a major impact on the evolution of its own structure.

Predicting the life cycle durability and ecological performance of cementitious coatings with a fuzzy‐stochastics–based approach

AbstractCementitious coating effectively prevents chloride‐induced corrosion in offshore reinforced concrete (RC) structures. Strain hardening cementitious composites (SHCC) is a class of advanced cementitious composites with superior ductility and durability. Due to the lack of long‐term test data, the current probability‐based design guidelines may not be applicable to SHCC coating. This paper develops a fuzzy‐stochastic–based approach to assess the durability performance of coatings using cement mortar and engineered cementitious composites (ECC), a specific type of SHCC. The age factor and the mean of chloride diffusion coefficient are modeled as fuzzy variables, and the chloride diffusion coefficient follows Gaussian distribution. Based on limited test data and subjective opinion, membership functions are proposed for the mean of initial chloride diffusion coefficient and the age factor of cement and ECC. To achieve equivalent durability performance of cement mortar coating, the amount of uncracked and cracked ECC coating are estimated to be around 33% and 25%, respectively. Based on equivalent durability performance, life cycle assessment (LCA) is conducted to compare the ecological performance of the two coatings. A case study, based on a 25‐m‐long simply supported box girder in Shenzhen Bay, is given to elaborate the proposed approach and the evaluation results of durability performance and ecological impacts.

Probabilistic analysis of corrosion initiation in existing reinforced concrete structures with imprecise random field

Reasonable assessment of corrosion initiation probability in existing reinforced concrete structures is an important premise for ensuring the safety and durability of such structures. In this paper, a simplified dual time-dependent chloride diffusion model considering the load effect is established and verified by comparing its results with the experimental data. Then, a modelling method of imprecise random fields is developed. In addition, a novel probabilistic analysis method of corrosion initiation in existing RC structures with imprecise random fields is proposed. The case analysis results show that the corrosion resistance of existing structures will be overestimated without considering the effect of imprecise random fields. In comparison, the upper bound of cumulative probability of corrosion initiation is 2.29 times of that using the existing deterministic method in this case. Parametric analyses show that the imprecise random field of concrete cover (Cd) has the greatest effect on the cumulative probability of corrosion initiation, followed by initial surface chloride concentration (Cs) and chloride diffusion coefficient (Da). Compared with the interval coefficient of variation and correlation length, the cumulative probability of corrosion initiation is more sensitive to the interval mean, especially the interval mean of Cd. Additionally, the effect of interval correlation length of Cs is sequentially greater than that of Cd and Da.

The role of internally incorporated nano-silica in recycled aggregate concrete: Modification of transport properties

This paper conducted an experimental investigation on the compressive strength, transport properties behavior and its microstructure of recycled aggregate concrete (RAC) internally modified with nano-SiO2 (NS). The capillary water absorption and chloride penetration resistance of RAC incorporating 0, 30%, 50% and 100% weight replacement of RCAs (namely, R0, R30, R50 and R100) with different NS dosages (0, 1%, 2% and 3% of cementitious materials) were experimentally investigated. The coupled action of NS and RCAs on the resistance to water absorption and chloride penetration into concrete were further discussed. Mercury intrusion porosimetry (MIP) was used to determine the pore structure of mortar in RAC and scanning electronic microscopy-energy dispersive spectrum (SEM-EDS) was selected to characterize microstructural properties of ITZs and mortar. The results indicate that the optimum dosage of NS was determined to be 3%. The 28-day compressive strength of R100 with 3% NS addition was nearly 1.15 time that of R100 without NS addition, and the initial sorptivity and chloride diffusion coefficient of R100 with 3% NS addition was 23.1% and 37.10% lower, respectively, than that of R100 without NS addition. This phenomenon can be attributed to filling effect, nucleation effect and pozzolanic effect of NS. The coupled action of the RCA content and NS dosages has a slightly influence on the sorptivity and chloride diffusion coefficient. The initial sorptivity of R100 with 3% NS addition was about 1.47 time as much as that of R0 without NS addition. The pore structure of specimens is refined by NS according to the obtained MIP results, and a proportion of harmful pores and less harmful pores are transformed into harmless pores. In addition, the microscale analyses suggest that NS promises an optimization of ITZ structures and compacts the mortar matrix, thus improving the mechanical and durability-related properties of RAC.

Kinetic characteristics of desorption and diffusion in raw coal and tectonic coal and their influence on coal and gas outburst

The basic physical parameters and gas desorption characteristics of raw coal and tectonic coal composite seams in northern China were tested, and the gas diffusion kinetic characteristics of raw coal and tectonic coal were investigated. The difference in gas hazard between the raw coal seam and tectonic coal seam was analysed based on this. The results found that: under the same conditions, the adsorption capacity, initial gas discharge rate, accumulated gas desorption volume, initial gas desorption rate and effective diffusion coefficient of tectonic coal are higher than those of raw coal, while Protodyakonov’s coefficient of tectonic coal is much smaller than that of raw coal. This, together with the lower permeability of tectonic coal seam, easily leads to the formation of a gas enrichment zone in tectonic coal seam easily and triggers coal and gas outburst when the external stress environment changes. Under the same conditions, raw coal is more sensitive to changes in initial gas adsorption equilibrium pressure, and the development of cleats and fractures of raw coal is higher than that of tectonic coal, which makes the gas in the area of raw coal easier to extract than that in the area of tectonic coal. When carrying out gas control in the composite coal seam of raw coal and tectonic coal, it is necessary to consider the gas desorption and diffusion characteristics in the raw coal and tectonic coal and to develop precise coal seam gas control measures based on the principles of safety and economy. The results of this study are of great relevance to the prevention and control of coal and gas outburst disasters in raw coal and tectonic coal seams to ensure the safe and efficient mining of coal resources.

Thermal, Mechanical and Swelling Studies of Semi-Interpenetrating Polymer Network Composed of Natural Rubber and Polyacrylamide

Semi-interpenetrating polymer network (IPN) composed of two polymers, namely natural rubber (NR) and polyacrylamide (PAAm) were prepared by crosslinking with glutaraldehyde. The NR/PAAm semi-IPN films were developed by varying the concentration of crosslinking agents and the polymer blend ratios. The impact of preparation parameters on the mechanical properties was studied. The semi-IPN was characterized by FTIR, DTG, TG and SEM. Swelling studies were done by using toluene as solvent. The kinetics and mechanism of swelling were investigated and the initial rate of swelling, swelling constant, swelling exponent, equilibrium swelling and diffusion coefficient were also determined.

Manifestations of static and dynamic heterogeneity in single molecule translational measurements in glassy systems.

Rotational-translational decoupling in systems near Tg, in which translational diffusion is apparently enhanced relative to rotation, has been observed in ensemble and single molecule experiments and has been linked to dynamic heterogeneity. Here, simulations of single molecules experiencing homogeneous diffusion and static and dynamic heterogeneous diffusion are performed to clarify the contributions of heterogeneity to such enhanced translational diffusion. Results show that time-limited trajectories broaden the distribution of diffusion coefficients in the presence of homogeneous diffusion but not when physically reasonable degrees of static heterogeneity are present. When dynamic heterogeneity is introduced, measured diffusion coefficients uniformly increase relative to input diffusion coefficients, and the widths of output distributions decrease, providing support for the idea that dynamic heterogeneity can drive apparent translational enhancement. Among simulations with dynamic heterogeneity, when the frequency of dynamic exchange is correlated with the initial diffusion coefficient, the measured diffusion coefficient behavior as a function of observation time matches that seen experimentally, the only set of simulations explored in which this occurs. Taken together with experimental results, this suggests that enhanced translational diffusion in glassy systems occurs through dynamic exchange consistent with wide underlying distributions of diffusion coefficients and exchange coupled to local spatiotemporal dynamics.

Gas desorption law of granular coal in negative-pressure environment and calculation of gas loss during negative-pressure sampling

• The gas desorption law at different AEPs and NP conditions was revealed. • The variation law of the diffusion coefficient was obtained by inversion. • The method of gas loss in the process of negative pressure sampling is analyzed. To explore the gas desorption law of granular coal during negative-pressure (NP) slag discharge sampling, atmospheric-pressure (AP) desorption, BP desorption and negative-to-atmospheric pressure (NP-AP) desorption experiments were performed. Based on the experimental results, the variation laws of the gas diffusion coefficient in the NP and NP-AP environments were analyzed through numerical simulation, and the variation laws of gas desorption in the processes of NP desorption and AP desorption were comparatively analyzed. The results show that the gas diffusion coefficient under the NP condition conforms to a power function. At the same adsorption equilibrium pressure (AEP), the initial diffusion coefficient monotonically increases with the rise of NP, that is, the NP environment facilitates gas desorption. At the same desorption environment pressure, the diffusion coefficient is smaller and varies more stably under a higher AEP. Within the same NP sampling time, a higher AEP corresponds to a larger volume of cumulative gas desorption, and a higher NP corresponds to a greater gas loss in the sampling process. This proves that shortening the sampling time can reduce the gas loss at the NP stage and weaken the influence on the desorption at AP. During NP sampling, the logarithmic method is more accurate than the t method in terms of gas loss calculation, its relative error ranging from −56.25% to 16.45%. The relative error is pretty small in the high NP environment and decreases with the passage of the sampling time. Based on the linear relationship between the environmental NP value and the calculation error, the gas loss calculation formulas corrected by error analysis were obtained. The research results can serve as a guideline for further mastering the NP sampling technology.

Regularization Cuckoo Search Algorithm for Multi-Parameter Optimization of the Multi-Laminated Controlled Release System

The multi-laminated controlled release system has been widely used in the modern pharmaceutical industry because of its simple structure and safety, which can control the drug release behavior by regulating its internal parameters. To achieve three desired drug release behaviors of constant rate, linear reduction, and nonlinearity, which can be controlled by the optimization of the controllable parameters in the controlled release system, such as initial concentration and diffusion coefficient, a new regularization cuckoo search algorithm based on the cuckoo search algorithm and Tikhonov regularization method is proposed. The numerical examples confirm the stability, effectiveness, and feasibility of the proposed method. The effect of the number of model layers on the optimization results is also analyzed.