Optimized design of steam ejector primary nozzle based on orthogonal test method

This paper proposes an orthogonal test method to optimize the design parameters of a steam ejector. The objective of this method is to improve the cooling system’s efficiency and effectiveness. The L9(33) orthogonal table was established to simultaneously optimize the shape curve of the converging portion of the primary nozzle, the length of the throat, and the angle of the diverging portion, and the entrainment rate and the critical backpressure were used as the performance indexes to formulate 9 groups of test schemes with a total of 108 cases, which were simulated under the same conditions. Based on Computational Fluid Dynamics (CFD) technology, the influence order of the three parameters is derived by polar analysis and the flow field is analyzed at the overall as well as the primary nozzle. The results show that the shape curve of converging portion of the primary nozzle is a sensitive factor affecting the performance of the ejector, which should be emphasized in practical production applications; the Mach number shapes at the primary nozzles of different structures show similarity and overlap at a distance of about 0.03 m from the mixing chamber. The Mach number profiles at different back pressures show want similarity.


Introduction
Solar energy, geothermal energy, industrial waste heat, and recycled environmentally friendly refrigerants are all able to be effectively utilized by the Steam Ejector Refrigeration System (SERS).[1], which plays a significant role in energy conservation and emission reduction.Its advantages are simple structure, high reliability, lower initial as well as operational costs [2].The low efficiency of SERS means it is not widely used in the market [3].
A steam ejector's role in improving plant efficiency as a core unit cannot be overstated [4].It is common for ejectors to be measured by the entrainment rate (ER) and the critical back pressure (CBP) It is possible to make the conclusions drawn from the simulation not limited to specific parameter values making it very generalizable.The entrainment rate is defined as the mass flow rate of the secondary fluid to the primary fluid, and the critical back pressure is defined as the maximum back pressure at which the ejector can be operated stably.Steam ejectors should be designed with high ER and CBP to increase the coefficient of performance, thus enabling SERS to operate more economically.Due to insufficient knowledge of the flow field inside the ejector, there are limitations in rationalizing the design of the ejector and improving the efficiency of the ejector.
The principle of operation is as follows: By the high temperature and pressure of the primary fluid through the Rafael nozzle adiabatic expansion, the potential energy is converted into kinetic energy, due to the pressure at the nozzle outlet is lower than the pressure of the secondary fluid, the secondary fluid is pumped into the mixing chamber, the formation of a single homogeneous mixed fluid after 2 intense mixing.The mixed vapor recovers some of its pressure as it flows through the diffusion chamber and reaches a compliant pressure value at the outlet.
As can be seen from the working process of the ejector, the working principle of the ejector is simple, but the internal flow field is very complex.According to its working principle, it is known that the geometry of the main nozzle has a great influence on the ejector properties.Scholars in various countries have done a lot of research work on how to optimize the structure of the ejector and select the best working conditions, changing the structural dimensions of the ejector, selecting the best working parameters, obtaining the relationship between the structure of the ejector and the optimal working conditions, and then arriving at the structural design of the ejector to improve the improvement of the program and improve its efficiency.The following studies also reveal the reliability and efficiency of computational fluid dynamics techniques in the field of ejector flow analysis and property prediction [5].Over the course of the years many tests have been carried out to characterize the impact of the ejector geometry on its behavior.Pianthong et al. [6] found that the variation of the nozzle exit position determines the behavior of the ejector, and higher entrainment ratios can be obtained when the NXP is farther away from the ejector inlet.Yang et al. [7] simulated and analyzed the effect of various nozzle shapes on the performance of the ejector, and the performance of the ejector using conical and cruciform nozzles was better in comparison.Wang et al. [8] investigated the effect of surface roughness on entrainment rate and showed that the nozzle should be polished as much as possible to increase the entrainment rate.S. Varga et al. [9] S. Varga has showed experimentally that there exists an optimum area ratio for the area ratio of the nozzle to the throat at which the ejector efficiency is relatively high.In the calculation, there are many issues to be considered, such as the treatment of the surge, the effect of various energy losses, etc.If not handled properly, the efficiency of the ejector will be greatly reduced and will not achieve the desired results.
However, most scholars' research on the optimization of the ejector structure has remained at the single-factor analysis.Only a single parameter is studied, and other parameters remain unchanged, which is obviously debatable.Meanwhile, in most cases, only ER is used as a performance metric, while CBP, which is another important metric, has not been studied in detail.In fact, modeling methods that fit the ER values better may not fit the correct CBP values.Moreover, most researchers have focused on studying parameters such as NXP, nozzle shape, etc., without considering additional parameters such as the shape curve of the converging portion, the length of the throat, and the angle of the diverging portion.Therefore, in which a total of 108 cases with 9 bells of different modeling methods are established by CFD simulation method using three-factor, three-level orthogonal test method, this paper aims to establish the order of influence of the three parameters on the behavior of the ejector, and then to optimize the construction of the ejector.

Geometric and operational conditions
The geometry used in the simulations on the basis of the experimental setup designed by Sriveerakul et al [1], which is extensively used for quantitative studies of ejector performance.The main structure of ejector is shown in Table 1.The structure of the steam ejector is schematically shown in Figure 1.

Creation of Meshes and Independence Analysis of Meshes
The numerical study of this work is based on the axial symmetry of the ejector.Using CFD software ICEM to establish the mesh of the steam ejector model as well as the calculation area.Generating meshes requires both efficiency and accuracy.Figure 2 illustrates the mesh delineation of the injector as well as a localized zoomed-in view.Due to the large changes in velocity and pressure gradients in the boundary layer and the mixing region, the density of the mesh in these regions is increased for more accurate simulation.Since the velocity of the secondary fluid at the inlet section of the mixing section is small compared to the working vapor velocity, the lateral inlet of the secondary fluid can be simplified to an annular inlet.Pianthong et al [6] showed through numerical calculations that the simplified 2D model is not significantly different from the 3D model, thus justifying the simplified axisymmetric assumption [10].Mesh independence is investigated with the aim of verifying the mutual independence of the computational results with respect to the number of meshes.
In this paper, three different densities of meshes were chosen with the number of meshes: 153,447, 192,375, 394,300.With all three models, the first near-wall node of all walls is placed in the region y + <1.The operating pressure is 0 Pa.The radial distribution of Mach number (Ma) is given in Figure 3(a) at the inlet of the three lattice throats, and the axial distribution of Mach number along the center axis of the ejector is given in Figure 3b.It can be seen that the simulation results for Ma are essentially the same.The maximum ER values were 0.423, 0.424 and 0.424, respectively.When the number of meshes is larger than 153,447, Ma and ER do not change much, and the Ma distribution is basically the same.Considering the time cost as well as the simulation efficiency, a medium density (192,375) mesh was chosen for further simulation analysis.

Numerical settings
To accommodate the pressure values set for the inlet and outlet, the boundary conditions were set as pressure boundaries.It is assumed that the velocity, temperature, pressure and other parameters in the cross-section are homogeneous one-dimensional flow, so that the effect of the friction resistance of the walls on the flow field is not considered.The wall surface was used without slip and a static wall surface with a standard roughness model was used.Fluid density calculated using ideal gas relations [11][12].The SIMPLE algorithm is selected, the residual convergence criterion is defined as 1e-6, and the monitors for primary and secondary inlet mass flow are set up to detect the mass flow of the working and induced fluids simultaneously.Spatial discretization of turbulent kinetic energy using a second-order windward format.

Turbulence model
The SST model effectively avoids the excessive shear stress caused by the Boussinesq eddy-viscosity model and is more reliable than other models.Xiao et al. [13] discovered that the SST k-ω model predicts the ER as well as the CBP best based on the measured results when a high-density mesh is employed, so the SST k-ω model was chosen to simulate the performance of the steam ejector.

Calculated results
With the aim of analyzing the combined effect of different structural parameters on the function of the steam ejector, we use orthogonal experimental method in order to compare the emulation results under various structures.To ensure the accuracy of the CFD model as well as the boundary conditions, the entrainment rate and CBP are used as the measurements, and the simulation results are compared with the actual measurements in the literature [1] under the same operating conditions.Figure 4 shows a diagram of the experimental setup from the literature [1].This setup allows the correct evaluation of the entrainment ratio of the ejector.The details of the results are listed in Table 2, indicating that the differences among the experimental and simulation results are comparatively small, with the ER and CBP errors ranging between 5.75% and 14%, respectively.Figure 5 shows the calculated results of ER with back pressure Pb.As the backpressure increases, the individual structures show similarity to the experimental values.However, the chosen SST model has a high sensitivity to backpressure gradients, leading to overprediction of recirculation and underestimation of critical backpressure values.Under certain primary and secondary flow pressures, the ejector operation mode is categorised into three regions, clogged flow, non-clogged flow and secondary fluid backflow, based on critical back pressure (CBP) and breakdown back pressure (BBP) thresholds [1].In the blocked flow region, when the backpressure is lower than the critical value, blockage happens to the secondary fluid inside the mixing chamber, leading to the entrainment within the ejector of the same amount of secondary flow, keeping the ER value constant throughout the region.As the mixing of the two streams of steam is a process in which the working steam with the higher velocity and the ejected steam with the lower velocity are gradually mixed completely, the velocities and pressures of the two streams are eventually gradually equalized.At the same time, under these conditions, the pressure distribution does not vary with the outlet pressure until a sudden jump in pressure occurs.This basic characteristic is known as the constant capacity characteristic.In the blockage flow region, the shock wave train that produces the compression effect is found in the throat or diffusion tube section.If the back pressure increases, the shock wave train at the downstream will move upstream and eventually coincide with the first shock wave train.Complex flow situations are more susceptible to structural changes.When the back pressure exceeds the critical back pressure, the secondary fluid in the unclogged flow region is no longer clogged, at this time, the pressure inside the mixing chamber rises, reducing the pressure driving force of the primary and secondary fluids, reducing their mass flow rate, and the ER decreases rapidly as the outlet pressure rises.The shock wave train moves upstream and interferes with the mixing process among the primary and secondary fluids.Further increase in backpressure to reach the breakdown backpressure threshold will result in flow reversal into the secondary inflow port and backflow leading to ejector failure.
From the above analysis, it is clear that the critical pressure value is the optimum outlet pressure value of the ejector at which the ER value is maximized and there is no dissipation loss from the surge.

Orthogonal tests
Orthogonal test method is an important branch of statistics, which has been widely used in industrial production and experiments [14].Traditional single-factor analysis has a large number of tests, high test costs, and does not consider the interaction among multiple factors, and systematic errors will occur under many sites.In this article, the orthogonal test method is adopted, and the basic idea is to select representative factors and levels for testing in comprehensive test sites, thus reducing the number of tests and the consumption of human, material and financial resources.Therefore, it is a fast and economical method of experimental design and one of the most important programmes for the study of multifactorial and multilevel problems to date.
Orthogonal tests generally use orthogonal tables to analyse the measurements.The influence of the three factors was analyzed with CFD simulations (the shape curve of the converging portion, throat length and angle of the diffusion section) on the performance of the ejector.To satisfy the requirement of arbitrary orthogonal tables, each factor corresponds to 3 levels, and the L9(3 3 ) orthogonal experimental factorials and the corresponding orthogonal experimental tables are shown in Tables 3  and 4.
The simulation results are plotted in Table 5. Changes in nozzle structure result in changes in flow resistance and therefore changes in working vapor flow rate.Overall, the working fluid flow rate of the conical shrinkage section is significantly smaller than that of the curved shrinkage section.
Table 6 shows the entrainment rate analysis table and Table 7 shows the critical back pressure analysis table.Based on the R-value, the factors which influence the ER of the steam ejector follow the order A (shape curve of the converging portion) > B (length of the throat) > C (diverging portion angle).According to Table 7, the factors affecting the critical back pressure of the steam ejector in the order of A (shape curve of the converging portion) > C (diverging portion angle) = B (length of the throat).Based on this structure, the shape curve of the converging portion is the main consideration.Changing the shape profile of the convergent section leads to a change in the strength of the nonequilibrium condensation, which in turn leads to a change in the flow energy loss [15].In practical production applications, the shape curve of the main nozzle should be prioritized and a multifactorial analysis should be performed to analyze the effect of any factor on the ejector's performance.(c) (d)

Conclusion
Since there are many structural parameters affecting the performance of steam ejectors, the optimal design of an ejector is a multi-parameter, multi-objective problem.The following conclusions are drawn from the orthogonal test method as well as from the study of the flow details of different structures of ejectors: (1) Comparatively, the main factor affecting the ER as well as the CBP of a steam ejector is the shape profile of the shrinkage section of the main nozzle.When the entrainment rate is used as an indicator, the optimisation results are better when the shape curve is straight, while the optimisation results are better when the critical back pressure is used as an indicator for the quadratic curve.The curved surface of the inner wall of the nozzle should be designed to minimize pressure loss.(2) The variation of nozzle structure produces different flow resistance and hence the working steam flow rate varies accordingly.Overall, the working fluid flow rate of the conical shrinkage section is significantly smaller than that of the curved shrinkage section.(3) As the back pressure increases, the second shock train merges with the first shock train, increasing the complexity of the flow.Numerical results for non-obstructed flows are more susceptible to structural changes than obstructed flows.(4) The variation of Mach number at the main nozzle shows similarity as the pressure increases.
Despite the different structures, the Mach numbers coincide at about 0.03 m from the mixing chamber inlet.

Figure 3 .
Figure 3. Mach number variations at different meshes: (a) at the throat inlet, (b) along the ejector.

Figure 5 .
Figure 5.Comparison of simulation results and actual measurements for different structures of ejectors.

Figure 6 .
Figure 6.Models of ejectors of different configurations and Mach numbers along the centerline of the main nozzle: (a) overall Mach number at 3.0 kPa, (b) main nozzle Mach number at 3.0 kPa, (c) overall Mach number at 5.4 kPa, (d) main nozzle Mach number at 5.4 kPa.

Table 1 .
Primary geometries of the ejectors.

Table 2 .
Comparison verification of simulated and measured results.

Table 5 .
Data Simulation Results.

Table 6 .
Analysis of entrainment rates.

Table 7 .
Analysis of critical back pressure.