A computational fluid dynamic method for dense-phase carbon dioxide centrifugal pump

As an advanced approach for large-scale low-carbon utilization of fossil energy, the carbon capture, utilization and storage technology is a promising means to achieve the carbon-neutrality target by 2060. Among the most widely used approaches for carbon storage is the enhanced oil recovery technique, for which the centrifugal pump is used to inject high-pressure dense-phase carbon dioxide (CO2) into the stratum. This work focuses on real fluid effects and inter-stage matching deviation from the incompressibility assumption for a dense-phase CO2 centrifugal pump, of which the inlet condition is close to the fluid critical point. With a thermophysical-state table for CO2 based on the thermophysical-property database of the National Institute of Standards and Technology embedded into a computational fluid dynamics solver, a numerical simulation method for compressible dense-phase CO2 pumps is established. The developed method is validated in the 50kW main compressor for a supercritical CO2 cycle and then applied to the simulation of a multistage dense-phase CO2 pump designed for carbon storage. The stage with the most remarkable fluid compressibility is determined, which provides a priori basis for the follow-up design optimization of the pump. This work is of promising application prospect for design of advanced dense-phase fluid machinery.


Introduction
The carbon capture, utilization and storage (CCUS) technology is an important way to achieve largescale low-carbon utilization of fossil energy and reduce greenhouse gas emissions, which is of important significance for the protection of ecological environment and sustainable development of civilization.The pressurization route for CCUS with a dense-phase carbon dioxide (CO2) pump can lead to remarkably lower total power consumption than that with a supercritical CO2 (SCO2) compressor [1], the process of which in Scheme B can reduce the compression power consumption corresponding to more than half of that of the supercritical compression process in Scheme A, as shown in figure 1.However, the change of working medium density and the deviation of inter-stage matching from the incompressible flow state occur especially when the inlet condition is close to the fluid critical point, which poses challenges to the flow analysis and design of dense-phase CO2 pumps.
Several researchers have conducted computational fluid dynamics (CFD) simulations of SCO2 compressors.Pecnik et al. [2] used a pressure-based CFD solver to numerically simulate the flow in Sandia SCO2 compressor at three rotation speeds.By interpolating the thermophysical properties table, the computation cost is reduced, and the reliability of the numerical analysis method is validated.Hosangadi et al. [3] developed an advanced real-fluid numerical framework for transcritical CO2, and pointed out that the difficulty in achieving a numerical solution increases as the inlet condition approached the critical point.Shao et al. [4] used a density-based CFD solver to simulate the flow in a SCO2 compressor under near-critical point conditions.They found that the constant pressure specific heat (cp) table with a steep ridgeline causes numerical instability, then a quasi-symmetric sampling method is proposed to improve the numerical stability by reducing the strong fluctuation of the cells along the ridgeline in the cp table.As for the numerical simulation of dense-phase CO2 fluid machinery, few studies have been carried out to the best of the authors' knowledge.The main purpose of this work is to develop a CFD method for dense-phase CO2 pumps with fluid compressibility and real-fluid effect taken into account.To deal with the sensitivity of the thermophysical parameters of CO2 to the thermodynamic state, a thermophysical-state table for CO2 based on the thermophysical-property database of the National Institute of Standards and Technology (NIST) is embedded into a commercial CFD solver.The developed method is validated in the 50kW main compressor for a supercritical CO2 cycle.Then, it is applied to the simulation of a multistage dense-phase CO2 pump designed for carbon storage to determine the stage with the most remarkable fluid compressibility.This work is expected to provide reference significance for the development of high-reliability numerical simulation methods for densephase fluid machinery.

Performance parameters of dense-phase CO2 pump
For compressible dense-phase CO2, the polytropic work rather than differential pressure work can accurately describe the work obtained per unit mass of working fluid in the pump, and the polytropic efficiency is more suitable for describing the work capacity [5].Therefore, for the pressurization process of the dense-phase pump, the process-averaged polytropic work and polytropic efficiency are adopted to quantify the performance of the dense-phase pump with the compressibility taken into account.
The process-averaging refers to treating a polytropic process as a process with a constant polytropic exponent, where the polytropic exponent of the entire process is the average value.This method requires knowledge of the thermal state of the inlet and outlet, and the average polytropic process exponent is calculated as: Considering the kinetic energy obtained during the pressurization process, the polytropic work of the pump can be written as: Then, the specific expression for the polytropic work is written as: The head is calculated as , and the polytropic efficiency expression is: It should be noted here that the second term in the right hand side of equation ( 3) is the additional work due to compressibility effect, and thus to achieve a same pressure rise, the polytropic work is higher than the differential pressure work.

Calculation method for thermophysical property of CO2
Due to the intense change of CO2 physical properties near the critical point, small temperature or pressure changes will lead to sharp changes in specific constant pressure heat capacity, density and other physical properties.To accurately predict the dense-phase CO2 flow within the CFD framework, it is necessary to introduce a reasonable physical property model, including the equation of state and transport properties.The thermal equation of state developed by Span and Wagner [6] has been used to interpolate the thermodynamic properties for near-critical point simulations due to its high accuracy [7,8].The core of the high-accuracy Helmholtz free energy-type SW equation is to calculate the thermal physical properties of CO2 by using the auxiliary function, which can calculate the physical parameters from the triple point to the temperature of 1100K and the pressure of 800MPa.The dimensionless Helmholtz free energy consists of two parts, namely, the ideal part and the residual part.The Vesovic equation [9] has good prediction accuracy for CO2 viscosity and thermal conductivity and covers the thermal state range of most engineering applications.The temperature range is from 200K to 1500K for viscosity and from 200K to 1,000K for thermal conductivity, respectively, and the pressure range is from 0 to 100MPa, all satisfying the requirements of the present work.

CFD method
To conduct the numerical simulations, the commercial CFD solver ANSYS CFX is employed to solve the Reynolds-averaged Navier-Stokes (RANS) equations.Due to the unclosed nature of the RANS equations, the turbulence model is introduced for closure of the momentum and energy equations.The Its main advantage is that it considers turbulent shear stress and does not cause excessive prediction of eddy viscosity.For the simulation of fluid machinery near the CO2 critical point, the SST model has been widely used due to its ability to accurately predict the flow in the rotating domain and capture the flow separation characteristics near the wall [12][13][14].The second-order high-resolution scheme is used to discretize the convection term, and the other diffusion and source terms are treated with the secondorder central scheme.
To integrate the calculation method for thermophysical property of CO2 into the CFD solver, the real gas property (RGP) table is generated by using the program package RGP-Gen-master developed by the Department of Mechanical and Aerospace Engineering of Carleton University, Canada.The program package can call the working fluid physical property model from the multipurpose NIST REFPROP 10 [15] database, which employs SW equation of state and Vesovic equation.

Validation of Numerical Simulation Method
Due to the lack of openly published experiment data of dense-phase CO2 pumps, the Sandia National Laboratories SCO2 compressor [16] is adopted for validation of the developed numerical analysis method.Its inlet condition is near the critical point, and the compressibility is even stronger than that of the dense-phase pump flow.Therefore, the simulation results of the compressor can be used to assess the effectiveness and accuracy of the numerical simulation method in the follow-up densephase pump.The geometric model of the impeller is reconstructed, with the inlet and outlet sections extended to avoid the influence of non-uniform flow at near the boundaries on the prediction accuracy.The original impeller model and the reconstructed geometric model are shown in figure 2. The CFD solver ANSYS CFX is employed to solve the RANS equations.Total inlet temperature and pressure, inlet direction, outlet mass flowrate, solid wall with adiabatic non-slip wall, and symmetry surface with periodic boundary conditions are set as boundary conditions.Turbulence intensity of 5% with zero velocity angle is prescribed at the inlet and the reference pressure is set to zero.The aerodynamic performance parameters and pressure ratio of the impeller inlet and outlet are calculated by using the weighted average of mass flowrate.The multiblock structured grid is generated to discretize the computational domain.To accurately calculate the boundary layer flow with the SST turbulence model, the grid cell hight at the wall is limited to 1 in wall units.The grid details are shown in figure 3. The isentropic efficiency of the simulation with four computational grids at the design point conditions is compared for the grid-independence verification.From figure 4, the results show that the efficiency tends to be constant after the third grid.Therefore, the third grid is selected for subsequent simulation, with a grid cell number of about 820,000.The flows in the compressor at the design speed of 75,000 r• min -1 and an off-design speed of 50,000 r• min -1 are simulated.The comparison between the pressure ratio and isentropic efficiency obtained at the design point is shown in table 1.According to the results, the relative error between the pressure ratio and isentropic efficiency obtained from CFD simulation and experimental data is within 2%, indicating that it has high simulation accuracy for the design point.The comparison between the pressure ratio and isentropic efficiency simulated at 50,000 r• min -1 and experimental data is shown in figure 5.The highest efficiency point at this speed is near the left side of 2.5 kg• s -1 , and the calculated mass flowrate is slightly higher than the experimental data.The simulation errors of the pressure ratio and isentropic efficiency are about 2.22% and 5.15%, respectively.The results indicate that the deviation between the off-design speed CFD simulation performance and experimental data is significantly greater than the design speed deviation, and the deviation is significant at small and medium flowrates, that is, for simulation results near surge conditions, the deviation is significant.

Isentropic efficiency
The performance validation results of the Sandia SCO2 compressor show that the relative error is within 2% at design point and is within about 5% at the off-design point, and the effectiveness of the numerical method is validated.

Numerical Simulation of Dense-phase CO2 Pump
The developed numerical simulation method is applied to the flow simulation of an 11-stage densephase pump design prototype designed for the CCUS industry.The last 10 stages are designed as repeating stages, while the first stage has a different configuration.Some important design parameters are shown as table 2, including geometry and operating parameters.From geometry parameters, it can be seen that the inlet section of the first stage is bigger than that of the repeating stages, and the number of blades of first stage is less, in which way the cavitation on the inlet of the first stage could be avoided to the designer's knowledge.To reduce the computation cost, the whole flow in the pump is solved in a separate-stage solution mode.Two sets of geometric models are constructed for the first stage and the other repeating stages, respectively.First, the geometry module in ANSYS is used to extract the fluid domain of the first and repeating models, and then ANSYS MESH is employed to generate the unstructured grids.The regions with obvious geometric characteristics and the boundary layer are refined.The grid cell hight at the wall is limited to 1 in wall units to ensure the accurate use of SST k   .The grid generation and details of boundary layer refinement of the two models are shown in figure 6.
To include the pressure and temperature status of the multi-stage pump pressurization process, the thermophysical properties RGP table is set as: pressure range 6 to 25Mpa, temperature range 260 to 310K, and table resolution is 100.For the setting of boundary conditions, the first and repeating stages are specifically expanded in table 3 and table 4.  Before taking the performance prediction, grid-independence verification is conducted as shown in figure 7. The results show that the pressure ratio and polytropic efficiency tends to be constant after the third grid for both models.The grid with about 760,000 cells and about 630,000 cells are used for the first stage and the repeating stages, respectively.

Performance prediction
According to the separate-stage solution mode, the CFD prediction of multi-stage dense-phase pumps is carried out, and six operating points corresponding to the mass flowrate from small to large are selected, namely 40, 45, 50, 55, 60, and 65 kg• s -1 .The method of weighted average of mass flowrate is adopted to obtain the thermal state parameters of the cross-section in post-processing, where the pressure ratio is determined as the total pressure ratio of the inlet and outlet cross-section, and the polytropic work and polytropic efficiency are determined according to equation ( 3) and ( 4), respectively.The pressure ratio, head, temperature and total enthalpy ratio, density ratio, and polytropic efficiency performance curves of the entire machine are obtained as shown in figure 8. and total enthalpy ratio, (d) density ratio, (e) polytropic efficiency.By analyzing the performance curve of the entire machine, it can be seen that the simulated peakefficiency point is around 50 kg• s -1 on the right side, corresponding to a head of about 1353m, and polytropic efficiency of about 81.09%.The pressure ratio and head decrease with the increase of mass flowrate, and the decrease in large flowrate becomes more severe, resulting in blockage conditions; The temperature ratio and total enthalpy show a decreasing trend with the increase of mass flowrate, and this trend remains basically unchanged.The reason for this is that these two ratios can approximately reflect the size of total power consumption, so they are close to the law of total power consumption changing with flowrate, and basically show a downward sloping straight line in the figure; The density ratio first slightly increases and then decreases with the increase of flowrate, and decreases faster under high flow conditions; It can also be recognized that the flow characteristics of the CO2 dense-phase pump are similar to those of conventional pumps.
Although the density and temperature increase during the pressurization process, there is little change.At the simulated peak-efficiency of 50 kg• s -1 , the temperature and density of the entire machine only increase by 3.61% and 2.51%, respectively.In addition, the simulated peak-efficiency point is close to the design mass flowrate of 52.08 kg• s -1 of the multi-stage pump.The simulated total outlet pressure is 21.809 MPa, slightly higher than the target pressure of 20 MPa, indicating that the multi-stage pump design is basically reasonable and can meet the background requirements.

Pressurization characteristics
In order to obtain the pressurization characteristics of dense-phase pump, the stage with the highest density change within the dense-phase pump is identified, and the power capacity of each stage is evaluated, taking the 50 kg• s -1 working condition as an example, the distribution of density changes and head at each stage are shown in figure 9.It can be seen that the highest density change occurs at the second stage, with a relative change of about 0.25%.In addition, the head of the repeating stages is roughly the same, slightly higher than that of the first stage, indicating that the power capacity of the first stage impeller is slightly lower than that of the repeating stages.

Similarity characteristics
When the geometric size ratio and speed ratio of the real pump and the model are within a certain range, according to the equivalent similarity law, the flowrate and head of the two pumps satisfy: Based on the combination parameters of centrifugal compressors and the concept of general performance curves, taking the scaling ratio L =1 m into account because of reserching same model, the combined flowrate and combined head of the pump are given as follows: Within a certain range of speed changes, the efficiency of the pump remains approximately unchanged, and the general performance curves should also overlap.Therefore, five speeds are selected for simulation using the first stage as an example to verified the similarity law.The combined head and flowrate can be obtained by equation ( 7) and ( 8), respectively, to get the combined head curve.Then, the first stage head, combined head, and polytropic efficiency general performance curves at variable speed are shown in figure 10, respectively.(c) polytropic efficiency It can be seen that the combined head curve of the first stage under variable speed basically coincides with the efficiency curve, satisfying the similarity law.In fact, after obtaining the general performance curve, the polytropic efficiency and combined head can be obtained at any speed and flowrate near the design speed range, providing great convenience for obtaining performance at variable speed.
Additionally, the inlet condition of the dense-phase pump may change in actual operation, especially due to temperature changes caused by seasonal change.To explore the similar characteristics of dense-phase pumps under different inlet temperatures, the inlet temperatures of the first stage are changed to 10 ℃, 17 ℃, 24 ℃, and 31 ℃, and general performance curves at variable temperature are created together with the original operating conditions of 3 ℃, as shown in figure 11.From figure 11, it can be seen that the combined head curves of the first stage basically coincide under variable temperature, while the polytropic efficiency slightly increases with the increase of inlet temperature, basically meeting the similarity law.The reason why the analysis efficiency increases with the increase of temperature is that with the increase of inlet temperature, the flow state approaches the critical temperature of CO2, the compressibility of dense-phase CO2 fluid further improves, and the polytropic efficiency increases to a certain extent.In addition, it is not difficult to find that the two sets of general performance curves for the first stage variable speed and variable inlet temperature actually overlap, indicating that variable speed or inlet temperature can only be achieved by using a set of general performance curves.By utilizing this characteristic, the conversion of the entire unit's variable operating condition performance can be greatly simplified.

Flow analysis
Different flow conditions 1, 3, and 6 are selected for flow analysis for the first and sixth stages (represented by the repeating stages).The distribution of static pressure on the meridional surface, 90% blade height velocity, and S3 section of the blade leading edge under different flow conditions of first stage are shown in figures 12, 13 and 14, respectively.
As shown in figure 12, the outlet pressure of the impeller gradually decreases with the increase of flowrate.From figure 13, under low flow conditions, there is a differential positive incidence angle at the blade inlet, and there is flow separation on the suction surface, indicating poor flow conditions.As the flowrate increases, this situation gradually improves.This characteristic is in line with the theoretical understanding of turbomachinery.According to figure 14, as the flowrate increases, the low-pressure area on the suction surface and the high-pressure area on the pressure surface gradually diffuse towards the blade cover side, and the blade load is more uniform along the blade height direction.The reason is that the first stage inlet blade width is longer, and the main flow area is close to the blade disc side under small and medium flowrates.Under high flowrate conditions, the flow distribution along the blade height is more uniform.The simulated flow results of the sixth stage are analyzed, and the static pressure distribution on the meridian surface is similar to that of the first stage.The distribution of 90% blade high speed of the sixth stage and the static pressure distribution on the S3 section of the blade leading edge are given below, as shown in figure 15 and figure 16 respectively.
Compared with figure 15 and figure 13, it can be seen that corresponding to three different flow conditions, the inlet speed of the sixth stage blade is higher than that of the first stage impeller.This is because the blade inlet section of the repeating stage is smaller than that of the first stage, and the inlet speed of the repeating stage impellers is higher under the condition of small density change, so as to keep the volume flow basically unchanged.The law of positive incidence angle generated by its small flowrate is similar to that of the first stage.Comparing figure 16 and figure 14, under different flow conditions, the low-pressure area and high-pressure area of the suction surface of the sixth stage are evenly distributed along the blade height, which is different from the results of the first stage.The reason is that the inlet blade width of the repeating stage is smaller, and the flow velocity in the leading edge direction is also more uniform under low flow conditions, making the flow at different blade heights more uniform than that of the first stage.

Conclusions
In the present work, a numerical simulation method for dense-phase CO2 centrifugal pump with the compressibility and real-fluid effect taken into account is developed in order to deal with the change in fluid density and inter-stage matching of multistage pumps.The developed method is validated in the 50kW main compressor for a supercritical CO2 cycle in the Sandia National Laboratory and then applied to the simulation of an 11-stage dense-phase CO2 pump designed for carbon capture, utilization and storage.The main conclusions are as follows:  The validation results in the SCO2 compressor show that the predicted static pressure ratio and isentropic efficiency under design and off-design conditions are both in good agreement with the experimental data, validating the effectiveness of the developed numerical simulation method for compressibility and real-fluid effect.
 The numerical simulation results of the 11-stage dense-phase CO2 pump show that the simulated peak-efficiency of the pump is around 50 kg• s -1 and close to the design mass flowrate, and the total outlet pressure meets the design requirements.The second stage has the most remarkable density change, and the first stage has slightly lower head than that of the repeating stages, which provides a priori basis for the follow-up design optimization of the pump.The performance of the first stage at 2500-3500rpm rotation speeds and 3-31 ℃ inlet temperatures generally satisfies the similarity law. The detailed flowfield analysis for the first and sixth stages of the dense-phase CO2 pump shows that a positive inlet incidence angle occurs at low flowrates.Additionally, due to the smaller inlet throughflow area, the inflow profile along the spanwise direction of the rest repeating stages is more uniform than that of the first stage.

Figure 1 .
Figure 1.Transcritical CO2 pressurization scheme from 1 bar to 150 bar [1].As for the numerical simulation of dense-phase CO2 fluid machinery, few studies have been carried out to the best of the authors' knowledge.The main purpose of this work is to develop a CFD method for dense-phase CO2 pumps with fluid compressibility and real-fluid effect taken into account.To deal with the sensitivity of the thermophysical parameters of CO2 to the thermodynamic state, a thermophysical-state table for CO2 based on the thermophysical-property database of the National Institute of Standards and Technology (NIST) is embedded into a commercial CFD solver.The developed method is validated in the 50kW main compressor for a supercritical CO2 cycle.Then, it is applied to the simulation of a multistage dense-phase CO2 pump designed for carbon storage to determine the stage with the most remarkable fluid compressibility.This work is expected to provide reference significance for the development of high-reliability numerical simulation methods for densephase fluid machinery.
two-equation model and is developed based on the model proposed by Wilcox [10].Initially, Wilcox's k   model is highly sensitive to freestream conditions.Menter [11] added a mixing factor to the near wall k   model and the external k   model to solve this problem.

Figure 2 .
Figure 2. Sandia compressor model: (a) original impeller, (b) reconstructed impeller.The CFD solver ANSYS CFX is employed to solve the RANS equations.Total inlet temperature and pressure, inlet direction, outlet mass flowrate, solid wall with adiabatic non-slip wall, and symmetry surface with periodic boundary conditions are set as boundary conditions.Turbulence intensity of 5% with zero velocity angle is prescribed at the inlet and the reference pressure is set to zero.The aerodynamic performance parameters and pressure ratio of the impeller inlet and outlet are calculated by using the weighted average of mass flowrate.The multiblock structured grid is generated to discretize the computational domain.To accurately calculate the boundary layer flow with the SST turbulence model, the grid cell hight at the wall is limited to 1 in wall units.The grid details are shown in figure3.

Figure 4 .
Figure 4. Grid-independence verification at design point.The flows in the compressor at the design speed of 75,000 r• min -1 and an off-design speed of 50,000 r• min -1 are simulated.The comparison between the pressure ratio and isentropic efficiency obtained at the design point is shown in table1.According to the results, the relative error between the pressure ratio and isentropic efficiency obtained from CFD simulation and experimental data is within 2%, indicating that it has high simulation accuracy for the design point.The comparison between the pressure ratio and isentropic efficiency simulated at 50,000 r• min -1 and experimental data is shown in figure5.The highest efficiency point at this speed is near the left side of 2.5 kg• s -1 , and the calculated mass flowrate is slightly higher than the experimental data.The simulation errors of the pressure ratio and isentropic efficiency are about 2.22% and 5.15%, respectively.The results indicate that the deviation between the off-design speed CFD simulation performance and experimental data is significantly greater than the design speed deviation, and the deviation is significant at small and medium flowrates, that is, for simulation results near surge conditions, the deviation is significant.

Figure 8 .
Figure 8. Performance curve of entire machine: (a) pressure ratio, (b) head, (c) temperatureand total enthalpy ratio, (d) density ratio, (e) polytropic efficiency.By analyzing the performance curve of the entire machine, it can be seen that the simulated peakefficiency point is around 50 kg• s -1 on the right side, corresponding to a head of about 1353m, and polytropic efficiency of about 81.09%.The pressure ratio and head decrease with the increase of mass flowrate, and the decrease in large flowrate becomes more severe, resulting in blockage conditions; The temperature ratio and total enthalpy show a decreasing trend with the increase of mass flowrate, and this trend remains basically unchanged.The reason for this is that these two ratios can approximately reflect the size of total power consumption, so they are close to the law of total power consumption changing with flowrate, and basically show a downward sloping straight line in the

Figure 9 .
Figure 9. Pressurization characteristics: (a) density changes, (b) head distribution.It can be seen that the highest density change occurs at the second stage, with a relative change of about 0.25%.In addition, the head of the repeating stages is roughly the same, slightly higher than that of the first stage, indicating that the power capacity of the first stage impeller is slightly lower than that of the repeating stages.

Figure 10 .
Figure 10.General performance curves at variable speed: (a) head, (b) combined head,(c) polytropic efficiency It can be seen that the combined head curve of the first stage under variable speed basically coincides with the efficiency curve, satisfying the similarity law.In fact, after obtaining the general performance curve, the polytropic efficiency and combined head can be obtained at any speed and flowrate near the design speed range, providing great convenience for obtaining performance at variable speed.Additionally, the inlet condition of the dense-phase pump may change in actual operation, especially due to temperature changes caused by seasonal change.To explore the similar

Figure 11 .
Figure 11.General performance curves at variable temperature: (a) head, (b) combined head, (c) polytropic efficiency.From figure11, it can be seen that the combined head curves of the first stage basically coincide under variable temperature, while the polytropic efficiency slightly increases with the increase of inlet temperature, basically meeting the similarity law.The reason why the analysis efficiency increases with the increase of temperature is that with the increase of inlet temperature, the flow state approaches the critical temperature of CO2, the compressibility of dense-phase CO2 fluid further improves, and the polytropic efficiency increases to a certain extent.In addition, it is not difficult to find that the two sets of general performance curves for the first stage variable speed and variable inlet temperature actually overlap, indicating that variable speed or inlet temperature can only be achieved by using a set of general performance curves.By utilizing this characteristic, the conversion of the entire unit's variable operating condition performance can be greatly simplified.

Figure 14 .
Figure 14.Static pressure on S3 section of the leading edge: (a) 40 kg• s -1 , (b) 50 kg• s -1 , (c) 65 kg• s -1 .The simulated flow results of the sixth stage are analyzed, and the static pressure distribution on the meridian surface is similar to that of the first stage.The distribution of 90% blade high speed of the

Table 1 .
CFD simulation and experimental data comparison of Sandia compressor at design point.

Table 2 .
Design parameters of the 11-stage dense-phase pump.

Table 3 .
Boundary conditions of the first stage.

Table 4 .
Boundary conditions of the repeating stages.