mohamadreza ansari

Shrimp is one of the important food sources for humans. Food poisoning caused by eating seafood and shrimp is an important risk to human health. Therefore, the purpose of this study to investigate the bacterial contamination of Pasfid shrimp in Bushehr province and the effect of freeze process on its microbiological and physicochemical characteristics. 120 shrimp samples were randomly selected and transferred to the laboratory to evaluate the microbial load and chemical status. SPSS version 21 software and Duncan's comparison test were used for data analysis at a significant level of P<05.00. The results showed that the contamination rate for Escherichia coli was 13.35%, Salmonella 14.18%, Staphylococcus aureus 45.83%, Listeria 10.83% and Vibrio 13.35%. The analysis showed that there was no significant relationship between different microorganisms and the percentage of contamination. According to the results of the statistical analysis of 120 samples of shrimp cultivation in the culture medium and the results that were analyzed showed that the amount of TVN in four cases of storage methods had a significant difference, also the lowest and the highest amount of TVN according to the sample The packages were frozen in the tunnel immediately. The results of coliform contamination showed that the amount of coliform load in four investigated cases was less than 100 CFU/g and within the standard range (less than 100 CFU/g). Considering the high contamination that all the samples had with pathogenic microorganisms; The application of strict regulations by regulatory organizations should be increased to ensure the health of people and society.

The treatment of an immature tooth with pulp necrosis and apical periodontitis is a major challenge for clinicians. The endodontic treatment of such teeth entails apexification, apical barriers, or more recently, revascularization. The current study reports an immature replanted tooth treated with calcium hydroxide medicament. A 13-year-old female patient with a history of abscess formation in tooth #45 was referred to the Department of Endodontics, Shiraz Dental School, Shiraz, Iran. The clinical and radiographic examinations revealed a necrotic pulp, a root with an open apex, and periradicularradiolucency.

The treatment was rendered under rubber dam isolation and local anesthesia.The root canal was irrigated with 2.5% sodium hypochlorite solution and medicated with calcium hydroxide. Thepatient did notreturn for the completion of treatment in the next session. Twelve months later, she returned to the Department; she was asymptomatic and the root development was observed. The canal was then dried with paper points and obturated with thermoplastic injection technique.

Riverin sediments of the Dohezar River in Tonekabon contain high levels of heavy metals and therefore, they were chemically analyzed to determine concentrations of these elements. In fact, this research intended to evaluate the ecological risks of the heavy metals As, Pb, Cr, Zn, and Cu in the river sediments. Contamination indices such as enrichment factor and contamination factor, were evaluated. Considering the average concentrations of the heavy metals at all of the Stations, the maximum average for the elements was zinc and the minimum was copper. Therefore, the averages of changes in the concentrations of the elements are Zn > Cr > Pb > As > Cu. Considering calculation of the enrichment factors for the heavy metals according to the EF classification table, the maximum number of Stations (43.02%) with respect to contamination with As were in class 4(moderately severe enrichment). With respect to enrichment of Pb, Zn, Cr, and copper, the rest of the stations with 83.72, 77.91, 86.05, and 69.77%, respectively, were in class 2 (minor enrichment). Considering the high concentrations of the studied elements in the sediments of the region compared to the background value, and based on calculations related to contamination factor, arsenic with the average of 11.9 exceeded the most from the standard limit. It was followed by Pb with 2.2, zinc with 2, Cr with 1.8, and Cu with1.6 (copper exceeding the least from the standard limit). This research used statistical studies on correlation coefficients and cluster analysis to find the origin of the heavy metals in the sediments of the region. The low correlation between the heavy metals in the soil can indicate they probably did not have the same source. Moreover, these elements have different geochemical behaviors due to their low correlation.

A powerful two-phase lattice Boltzmann model with the ability of modeling high density ratio is applied to simulate a rising bubble striking a porous obstacle. This model is able to simulate immiscible two-phase flow with density ratio of 1000 and result in desirable mass conservation. In present research, a porous obstacle is posed in two-phase flow domain, bounce back and wetting boundary conditions at walls and corners is discussed and showed that after implementation of obstacle boundary conditions, mass conservation of the model is preserved. Accuracy and ability of the model firstly examined by some basic problems. Next, striking of a rising bubble with 1000 density ratio to a porous obstacle is simulated and the effect of contact angle, Eotvos number and porosity ratio in deformation and passing of the bubble from the obstacle is investigated systematically. Different porosity ratios and contact angles, result in different bubble behavior striking the porous obstacle; In low porosity ratios and low contact angles, the bubble remains below the obstacle. At high contact angles, the hydrophobicity of the obstacle surface draws the bubble into the porosities, and the bubble moves to the top of the obstacle and stays on the top surface of the obstacle. In other cases, the bubble completely passes through the obstacle and separates it. Mass conservation error of bubble passing the porous obstacle is of order of 10-11 which is completely desirable.

The present study investigates the effect of the mixing chamber length on the effervescent atomizer internal two-phase flow and the liquid film thickness at the exit of the atomizer at different gas-to-liquid mass ratios. Therefore, the internal flow of this atomizer simulated for three different lengths of the mixing chamber, at the gas-to-liquid mass ratios of 0.08%, 0.32%, and 1.24% and at the liquid flow rate of 0.38 L / min by the volume of fluid interface following model. The simulation results show that the mixing chamber length does not have much effect on the dominant flow regime in the discharge passage. However, by increasing the mixing chamber length, the two-phase flow inside this chamber more expanded before entering into the discharge passage. Therefore, the two-phase interface instabilities in the discharge passage are lower for the atomizer with the longer mixing chamber. In addition, based on the measuring results of the liquid film thickness at the exit of the atomizer, the effect of the mixing chamber length on the thickness of this film depends on the gas-to-liquid mass ratio. Increasing the mixing chamber length at low gas-to-liquid mass ratio increases the liquid film thickness at the exit of the effervescent atomizer. While at high gas-to-liquid mass ratio, it's inverse. At middle gas-to-liquid mass ratio, the changes of the liquid film thickness at the exit of the atomizer with the mixing chamber length do not show a steady trend.

In the current study, two-phase flow of water and air over a stepped spillway is probed in the form of a two-dimensional incompressible viscous flow. A novel numerical approach is used for the numerical simulation which is a combination of two models: volume of fluid (VOF) which uses an interface tracking algorithm for the simulation of the two-phase flow and two-fluid model which is based on time and space averaged equations and cannot track the interface explicitly. The most important issue in the introduced approach is to couple the two basic methods and select a proper criterion for status change between two basic methods. The latter criterion is based on an approximation from local distribution of the interface at each cell. In the hybrid method. In order to investigate the aeration effect in the stepped spillway, the air suction is generated by designing some holes at the upper edge of the steps and considering atmosphere pressure for these areas. The obtained results divulge the amount of dispersion is low at the beginning part of the step and also the hybrid model take more advantages from VOF, while in the lower steps where the flow disperses two-fluid model has hegemony. The results are compared in the form of pressure contours and streamlines as well as volume fraction counters. The comparison shows that the results of the proposed method is closer to the experimental results with respect to each of the basic model.

In this research, interaction and oblique coalescence of bubbles under buoyancy force was simulated, numerically. The governing equations are continuity and momentum equations which have been discretized using the finite volume method and the SIMPLE algorithm. For simulating the interface of two phases, the level set method has been incorporated. Level Set method suffers from poor mass conservation of dispersed phase especially in the case of severe deformation of interface. In order to control of mass conservation of the level set method, re-initialization equations and a geometric mass control loop are used which this loop is implemented in the level set method for the first time in this research. Using proposed geometric mass control loop, mass dissipation drawback of the level set method is handled in simulation of bubbles’ coalescence. The results outlined in the present study well agree with the existing experimental results. Also results of investigation of mass dissipation of the proposed scheme to simulation of oblique coalescence problem show that the maximum amount of this mass dissipation was less than 4%. Therefore, the level set method with proposed geometric mass control loop could be used properly for simulation of oblique interactions and coalescence of bubbles in multiphase flows.

A spillway is a hydraulic structure that is provided at storage and detention dams to release surplus or flood water that cannot be safely stored in the reservoir. In this paper, two-dimensional simulation of gas-liquid two-phase flow on stepped spillway in different discharge rate is studied. The VOF model and the two-fluid model are used in order to simulate numerically and then the results of the two models are compared. In order to study the influence of aeration in stepped spillway, two different physics are proposed. In the first geometry it is assumed that there is no air intake via stairs and in second geometry air intake and its effect on the flow over the spillway is studied by embedding hole in the top edge of stair. The air pressure is assumed to be atmospheric. The results showed that VOF model provide more accurate result than that of two-fluid model in low discharge rate. However, in cases where aeration is studied, because of mixing phases, this model is not able to simulate fluid flow as well as two-fluid model. The two-fluid model is more accurate due to solving equations for both phases (air and water). For verification, numerical results have been compared with experimental values and determined that numerical models are able to predict the total energy loss within an error range of %10 compared with the measured experimental data.

In the present study two-phase flow within the effervescent atomizer has been simulated by the volume of fluid interface tracing model using 0.08%, 0.32%, 1.24%, and 4.9% gas-to-liquid mass ratios and 0.38 L/min liquid flow rate. The purpose of this simulation is to study two-phase flow regimes within the effervescent atomizer and their effect on the atomization quality. This study also considers the gas-liquid interface instabilities in different two-phase flow regimes inside the atomizer. The compressibility of gas phase which is rear in literature survey included in gas-to-liquid mass ratios of 1.24% and 4.9%, due to the high gas phase velocity in constant liquid flow rate and high gas-to-liquid mass ratios. The effect of gravitational force is considered in all simulations. The results of the simulation indicate that by increasing the gas-to-liquid mass ratio, the two-phase flow regime inside the discharge passage transfers from bubbly flow regime with long bubbles to annular flow regime. In addition to decreasing the liquid film thickness coming out from discharge orifice (during transform of the flow regime from bubbly flow to annular flow), the liquid interface instabilities increase in the annular flow regime and besides, where segregated ligaments from the liquid interface become shorter, thinner and more unstable. This type of regime is the most efficient flow behavior for the effervescent atomizer.

Slug flow is one the most complicated flow regime in industrial process that is seen for wide range of fluid flow. However, there are always some differences between experimental and numerical results obtained for slug flow. Proceeding the perivious attempt on the selection of the best turbulent model for numerical simulation, the slug flow is simulated numerically in two dimension by applying implicit VOF method and k-ε RNG turbulent model. To extract the slug flow parameters accurately FLUENT solver is used. The differences of the obtained results with and without turbulent model during slugging is also presented. To overcome this complicated flow behavior, a new user defined function code is developed. This UDF computes and predict slug parameters from FLUENT solver result without increasing the computational cost. The important slug parametrs are presented which are: liquid slug body velocity, liquid film velocity, slug front and tail velocity, slug center position and length, slug front and tail positions, pressure difference across slug, wall shear stress, slug mixture velocity, slug initiation time and position from the duct inlet. These parameters are analyzed and discused in details after that they had been validated

Analysis of air bubbles entrainment in liquid slug body is one of the most important and complicated phenomena during slug flow regime. In the present attempt, a method is proposed for slug modeling to consider the air bubble entrainment into slug liquid body. The effect of consequencies and their impact on slug behaviour to predict more accurate correlations for slug parameters are estimated and calculated. This method considers a two-fluid single pressure model, combined with population balance model for equal bubble diameter series and solves using volume of fluid. In this regard based on slug and hydraulic jump similarity, a correlation for air bubble entrainment rate and its mechanism selected. This correlation developed in the form of a user defined function code and coupled with other models in FLUENT solver to calculate slug flow. Finally the result of this numerical modeling is validated with the result of other numerical and experimental results where exist in the related literature. The result is consist of the slug flow profile, entrained air bubble profile and their diameter distribution, vortex at slug front, pressure distribution during slugging, slug mixture velocity, turbullent model parameters and etc.

In the present research, hydrodynamical and aerodynamical characteristics of a high-speed planning hull is studied using computational fluid dynamics. Simulations are three-dimensional with considering a two-phase turbulent flow. To obtain sinkage and trim of the hull, two degrees of freedom is assumed for it. Rigid body dynamic equations and governing equations of the fluid are coupled using 6DOF solver and dynamic mesh technique. Based on the available experimental results, simulations of the aimed high speed hull are performed in the linear velocity range of 0.9-8.31 m/s. Comparing the present numerical results with the experimental data, shows that maximum average error for resistance, trim and sinkage in different velocities does not exceed 10%. This shows the accuracy and proficiency of the current model. Mesh independency of solutions is studied for all velocities and the results are reported based on the most suitable mesh. At the end, the effect of applying steps on reducing the drag and improving stability of the hull is investigated for several states in one and two steps. Finally, the most optimized state is introduced and relating results are given. Results show that applying steps to the mentioned high speed hull reduce the overall resistance by 11%.

In the present study, flow regimes of co-current, air-water two-phase flow in a vertical tube with 70 mm internal diameter were investigated. Simulation accomplished by open source software, OpenFOAM, and One Fluid model has been used to simulate two-phase flow, which in this model, the interface of two-phase flow has been followed by Volume of the Fluid model. Hitherto, most of the researchers conducted experimentally and the researchers in many of numerical studies just investigated the small tubes. The simulation had investigated according to boundary conditions of the vertical tube. Air and water superficial velocities in inlet and pressure in outlet were constant. Moreover, a no-slip condition in the internal tube walls has been considered. The main purpose of this study is to identify the flow regimes based on the superficial velocities of air and water in the inlet. Moreover, the diagrams of density distributions of phases were obtained, with respect to the behavior of each two-phase flow pattern which can be identified. Superficial velocity of air and water were in the range of 0.01-15 m/s and 0.1-1.5 m/s, respectively. By analysis of results, bubbly, slug, churn and annular regimes; furthermore, semi-annular and Cap-Bubbly sub-regimes were observed.

In this research¡ counter-current two phase flow is investigated adiabatically in vertical Plexiglas tubes with internal diameters of 60 mm¡ 80 mm and 110 mm. All of tubes have the same height of 2 m. So far¡ there have been few studies on counter-current flows in large diameter tubes. Water and air flow downward and upward through the tubes¡ respectively. Superficial velocities of air and water ranges are 1.77-7.17 m/s and 0.05-0.11 m/s for 60 mm tube¡ 0.99-4.03 m/s and 0.03-0.09 m/s for 80 mm tube and 0.55-2.26 m/s and 0.01-0.05 m/s for 110 mm tube. In addition to reverse flow¡ other main regimes can be observed as slug¡ churn and annular in the tubes. Our efforts in drawing the flow patterns map were aimed at minimizing uncertainties at the boundaries. Based on the obtained experimental results¡ slug and annular regimes gradually comprise smaller regions of the flow map as the tube diameter increases. However¡ churn regime contains larger area of the flow pattern maps. Moreover¡ Kelvin – Helmholtz instability is observed in tubes and experimental and analytical results represent appropriate consequent in comparison. Eventually¡ validation of experimental flow patterns map accomplished by theoretical results.

In this paper, a single bubble free ascending in a vertical channel was studied experimentally. Five different Newtonian fluids were used where the surface tension force is dominant. The bubble trajectory was considered in water, glycerin 30 and 50 Vol% that is zigzag, however, linear behavior is observed while the weight concentration of the glycerin reaches to 80 and 100 percent. The bubble rise velocity and aspect ratio coefficient are calculated by image analysis via MATLAB software. The results are in a very good agreement with the literature for the bubble velocity. The effect of magnetic field (perpendicular to the bubble flow) on the hydrodynamic characteristics of the bubble for each of the working fluids has been scrutinized. Although the presence of the magnetic field does not affect the bubble trajectory type or change the flow pattern from zigzag to linear, but it reduces the flow domain where this descending trend decreases with the increase in viscosity. It should be also noted that the magnetic field causes the bubble rise velocity to increase while this enhancement increases with higher viscosity. The magnetic field effect on the bubble aspect ratio was also considered and it was found that as viscosity increases the aspect ratio change is decreases.

Many models have been applied to slug flow using laminar flow condition. The results obtained from these models are not consistent with the physical behavior of slug flow. Furthermore, discussion on the turbulent models is very rare or not related to the such flow regime. The slug regime is a complicated regime with shear flow and high strain in addition to some vorticity at some sections of the flow.
In the present attempt at first stage, the turbulent models differences, the initial assumptions to drive, privilages and shorcomings have been considered with details. Then, its consistency with the physics of slug flow was analysed with high accuracy. In the second stage, simulations using different turbulent models were conducted. The obtained results were compared to each other and with the experimental results of other investigators. Finally, the most consistent model with the physics of the slug flow was selected. The turbulent model of RNG k-ε showed more reliable in compare to other turbulent models. Thus, it was selected and used to obtain slug flow behavior with higher accuracy. The parameters as pressure distribution during slugging, slug mixture velocity, slug initiation time and position from the duct inlet with RNG model conducted and presented with detailed explanations.

In this paper, two-phase air–water flow was investigated experimentally and simulated numerically using VOF method. The tests are conducted in Multiphase Flow Lab. of Tarbiat Modares University. In order to evaluate the rib effect on flow regimes, experimental investigation was conducted with ribs of different width and pitch where assembled on front and back side walls (side walls) of the duct during different test runs. The rib width and pitch were held constant during each test. The experimental work considered for different regimes of wavy, plug and slug which generated in the ducts with and without rib applying various phase velocities. The effects of using ribs on regime boundaries are presented in the flow diagrams and discussed in details. Compared to the smooth duct, the ribbed duct affects the different regime boundary positions noticeabily. The results showed that in the duct with small sizes ribs, the first slug initiates at longer time and distance in compare to the duct equipped with bigger size ribs. The results show that for normal operational flow velocities, the ribbed duct decreases the slug area on flow diagram map in compare to smooth duct. However, ribs facilitate the slug regime initiation for phase velocities in accordance with slug generation, which is not benefit of operational condition.

Experimental investigation of two-phase air-water flow was conducted at consecutive inclinations of a large bend (with three equal slopes in respect to each other) and including the horizontal sections of the inlet and outlet of the bend. The results show that the elongated bubble regime flows without any effect of duct inclination change and consistent for all three zones of horizontal sections of before and after the bend and the bend itself. It was also noticed, as the duct inclination decreases along the route, vortex misty flow transmits to misty annular flow at higher gas flow rates. The annular flow regime was noticed only at the first slop of the bend. Slug flow was observed at the horizontal sections upstream and downstream of the bend. The slug flow at the upstream generated by the interfacial instabilities but at the downstream formed by Taylor bubbles. Slug flow area in the flow diagram increases as liquid flow rate increase at both horizontal sections. In addition, the void fraction change rate with phases mass flow rate was considered at the duct inlet.

Experimental investigation conducted on Taylor bubble characteristics in a large bend including three consecutive inclinations. For this purposes, flow maps were obtained for the bend and horizontal section of upstream of the bend to define the area of this regime and mechanism of Taylor bubble formation. The effect of superficial gas-liquid velocities and the duct slope were studied on average velocity, length and frequency of bubbles. The results show, the bubble velocity and length increase as gas superficial velocity increases and the duct slope decreases. However, liquid velocity increase has decreasing effect on this characteristics. Bubble frequency is independent of slope change and reduces as gas superficial velocity increase. However, bubble frequency reduces at first and then increase as liquid superficial velocity increases. Regarding the safety regulation for industry, the minimum of the bubble frequency should be generated for the required liquid mass flow rate. Meanwhile, for the gas velocity, some optimization is required between frequency reductions with Taylor bubble velocity increase in addition to bubble length reduction. Regarding the background of the present field with shortage of results on Taylor bubbles frequency, some correlations based on the superficial Reynolds number of phases were presented for each inclination.

Numerical modeling of compressible two-phase flow is a challenging and important subject in practical cases and research problems. In these problems, mutual effect of shock wave interaction creates a discontinuity in fluid properties and interface of two fluids as a second discontinuity lead to some difficulties in numerical approximations and estimating an accurate interface during hydro-dynamical capturing process. The objective of this research is to increase the accuracy of numerical simulation of two-phase flow using two dimensional five-equation two-fluid model. For this purposes, MUSCL strategy was used for increasing the Godunov numerical scheme accuracy from 1st order to 2nd order. The privilege of this method is high accuracy, low numerical oscillation and low numerical diffusion. The problems considered for the verification of the results are the water-air shock tube, a square bubble with moving interface in a uniform flow and a shock wave with 1.72 Mach having interaction with an air bubble in a water pool. The obtained numerical results showed that, the results that have been obtained by second order accuracy have less diffusion in the two-phase flow interface.

In the present article, velocity and deformation of an air bubble have been considered in quiescent liquid at different consecutive slopes from 5 to 90 degrees in respect to horizontal condition. To establish these purposes, air-water two-phase flow has been simulated numerically by using volume of fluid method. The two-phase flow interface has been traced by using Piecewise Linear Interface Calculation (PLIC) method. Surface tension force was estimated by Continuum Surface Force (CSF) model. The simulation results show that maximum bubble velocity occurred at 45 degrees which is in agreement with the previous researchers result. Simulation of bubble movement was also continued to two consecutive slopes at different angles. At slope deviation location, a vortex was generated due to liquid movement governed by gravity forces. This vortex changes the bubble velocity as well as bubble shape. This vortex also reduces the bubble velocity and changes the bubble nose shape from sharp to flatten at deviation from low to high slope values. However, at deviations from high to low slope values, the bubble nose becomes more sharpened in addition to bubble velocity increase. The maximum average velocity of bubble movement at two consecutive slopes was obtained during the condition that the first and second slopes were set to 60 and 30 degrees, respectively.

In this article, two-phase slug flow is simulated numerically in a horizontal duct with rectangular cross-section using Volume Of Fluid (VOF) method. Conservation equations of mass, momentum and advection equation are solved in open source OpenFOAM code accompanying k-ω SST turbulence equations. Simulation is conducted based on the experimental results in the duct with rectangular cross-section. The results shows, due to Kelvin-Helmholtz (K-H) instability criteria slug initiation forms in the air-water interface during three dimensional turbulence modeling. Water level was increased slightly at interface in both numerical simulation and experiment. This level increase satisfies the K-H instability to generate a slug at interface. During slug initiation, the pressure behind slug is increased significantly. Big pressure gradient at the beginning of the slug in compare to the end of it causes the slug length to be increased as propagate along the duct. The numerical simulation of present research is capable of predicting the slug length accurately in accordance with experiment; however, the slug position with 22% inaccuracy was obtained. Comparison of the results with the numerical and experimental results of other researchers confirms higher accuracy of flow prediction in the present work.

In this paper، the effect of gas and liquid inlet velocities and for the first time the effect of liquid hold up on slug initiation position are studied experimentally. Empirical correlations are also presented based on the obtained results. The tests are conducted for three liquid hold ups (0. 25، 0. 50 and 0. 75) in a long horizontal channel made of Plexiglas with dimensions of 510 cm2 and 36m length in Multiphase Flow Lab. of Tarbiat Modares University. The superficial liquid and air velocities rated as to 0. 11-0. 56 m/s and 1. 88-13 m/s، respectively. The obtained results show that as αl=0. 25، slug initiation position is increasing monotonically with Usl and Usg. During αl=0. 50، slug initiation position is increasing with Usl and Usg but the slope is smoother than αl=0. 25. For αl=0. 75، slug initiation position is decreasing monotonically with Usl and Usg. In the case of equal void fraction of phases، slugs are generated weakly (low pressure). However، for the unequal void fraction of phases strong slugs (high pressure) are formed.

In this paper, the effect of gas and liquid inlet superficial velocities and distance from upstream on slug frequency is studied experimentally. Empirical correlations are also presented based on the obtained results. The tests are conducted for liquid holdup αl= 0.75 and three distances from inlet in a long horizontal channel made of Plexiglas with dimensions of 510 cm2 and 36m length in Multiphase Flow Lab. of Tarbiat Modares University. The superficial liquid and air velocities rated as to 0.11-0.56 m/s and 1.88-13 m/s, respectively. The obtained results show that slug frequency is dependent to superficial liquid velocity directly. Slug frequency decreases with slip ratio increase. Slug frequency has strong dependency on superficial liquid velocity and increases monotonically with it. However, superficial gas velocity has damping effect on slug frequency. As slug moves towards downstream, slug frequency will be decreased but slug velocity will be increased.

This paper presents a new and accurate methodology for transient and dynamic behavior of slug hydrodynamic instability modeling in horizontal and inclined channels. The base of this methodology is on numerical solution of hyperbolic two fluid model equations by means of a class of high resolution shock capturing methods. The privilege of this method is that it can model and predict initiation and growth of slug in stratified flow automatically and directly by solving the flow field differential equations. The well-defined test case considered for the verification and validation of the results is the Ransom Water Faucet case. The results obtained for slug flow modeling in horizontal duct were compared with two sets of experimental results. The good agreement of the present modeling results with the experimental results of own and other investigators and also grid independency study show the model is capable of slug tracking and slug capturing and the numerical method which is used here can predict with high enough accuracy.