CN108561195B - Effective control method for vortex cavitation flow in low-temperature liquid expander - Google Patents
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Abstract
The invention discloses an effective control method of vortex cavitation flow in a low-temperature liquid expander, which comprises the steps of researching the vortex cavitation mechanism of the low-temperature liquid expander in consideration of the thermodynamic effect of low-temperature fluid, analyzing the geometric parameter sensitivity of an impeller of the vortex cavitation flow in the low-temperature liquid expander, characterizing the complex vortex cavitation flow in the low-temperature liquid expander, constructing a flow field optimization objective function and a flow field optimization control variable aiming at controlling the vortex cavitation flow, and solving the problem of the vortex cavitation flow optimization control in parallel.
Description
Technical Field
The invention belongs to the fields of low-temperature air separation, low-temperature liquefaction and the like, and relates to an effective control method for vortex cavitation flow in a low-temperature liquid expansion machine.
Background
The cryogenic liquid expander, as a hydraulic machine, is similar to a conventional hydraulic (or hydrodynamic) machine, and cavitation inevitably occurs. The collapse of the cavitation bubbles can generate extremely high local pressure, which causes great impact on the surface material of the structure and generates cavitation corrosion damage; but also induce unit vibration and threaten the stable operation of the liquid expander and even the low-temperature system. Therefore, the method has great significance for effectively inhibiting the vortex cavitation flow of the low-temperature liquid expander.
The cavitation phenomenon refers to a phenomenon that the partial pressure of liquid is lower than the saturated vapor pressure at the corresponding temperature, which leads to the vaporization of the liquid and causes explosive growth and collapse of micro-bubbles. The cavitation phenomenon is widely existed in the fields of hydraulic machinery such as water pumps and water turbines, low-temperature air separation, low-temperature liquefaction and the like. According to the different occurrence positions, four forms of blade-shaped cavitation, gap cavitation, cavity cavitation and local cavitation exist. Generally, cavity cavitation refers to cavitation caused by vortex flow existing in conventional hydraulic machinery (such as a draft tube of a water turbine), the cavity cavitation has high intensity and large occupied space, and is mostly in a shape of a braid, the performance of the hydraulic machinery and the reliability of a unit are directly influenced, and the inducement of the cavitation, namely the vortex flow, is essentially from high-speed rotation of an impeller. As a hydraulic turbine, the impeller rotating at high speed in the low-temperature liquid expander also causes a high-intensity swirling flow and expands to the diffuser pipe downstream thereof, thereby inducing cavitation at the outlet of the impeller and in the diffuser pipe.
In order to inhibit cavitation, some optimization design methods are provided for the impeller design of the conventional hydraulic machine. Patent 201110202524.5 discloses an optimized design method for cavitation erosion resistant centrifugal pump impeller, which uses NSGA-II genetic algorithm as optimization tool to perform multi-objective optimized design for centrifugal pump impeller parameters, thereby improving impeller efficiency and cavitation resistance. Patent 201510679202.8 "a high cavitation resistance centrifugal impeller hydraulic design method" provides a high cavitation resistance centrifugal impeller hydraulic design method, which adopts the method of improving the blade inlet placement angle, the blade thickness distribution, the impeller inlet diameter and the blade inlet width, and improves the cavitation resistance of the centrifugal pump. Patent 201510908837.0 "anti-cavitation axial flow pump impeller design" discloses an anti-cavitation axial flow pump impeller design method, which makes the designed axial flow pump impeller work reliably and have anti-cavitation ability.
Compared with conventional hydraulic machines, the cavitation of the vortex in cryogenic hydraulic machines (such as cryogenic liquid expanders) is more complex. For normal temperature hydromachinery, the thermal effect of the medium is almost negligible, i.e. the influence of temperature on cavitation is negligible. However, due to the remarkable thermodynamic effect of the low-temperature medium, the low-temperature cavitation is very sensitive to temperature change, the latent heat of phase change of the low-temperature medium is large, and the influence factors such as the low-temperature cavitation and the like are all negligible. Essentially, the high speed rotation of the impeller of the cryogenic liquid expander causes high intensity vortical flow at its exit, resulting in local low pressure and temperature rise, directly inducing cavitation. However, the vortex cavitation flow is highly coupled with the low-temperature field, so that the control difficulty is obviously increased. At present, no public data in the aspect is found at home and abroad.
Disclosure of Invention
The invention aims to overcome the defects of the prior art and provides an effective control method for vortex cavitation flow in a low-temperature liquid expander, which can effectively improve the performance and the operation reliability of the low-temperature liquid expander.
In order to achieve the purpose, the effective control method of the vortex cavitation flow in the low-temperature liquid expander comprises the steps of low-temperature liquid expander vortex cavitation mechanism research considering the low-temperature fluid thermodynamic effect, impeller geometric parameter sensitivity analysis of the vortex cavitation flow in the low-temperature liquid expander, characterization of complex vortex cavitation flow in the low-temperature liquid expander, construction of a flow field optimization objective function and a flow field optimization control variable aiming at controlling the vortex cavitation flow, and parallel solving of a vortex cavitation flow optimization control problem.
The specific process of the low-temperature liquid expander vortex cavitation mechanism research considering the low-temperature fluid thermodynamic effect comprises the following steps: a Rayleigh-Plesset cavitation model is adopted to study the internal cavitation flow of the low-temperature liquid expander, and the Rayleigh-Plesset cavitation model is combined with a numerical model of the whole liquid expander to simulate the vortex cavitation flow of the liquid expander.
The Rayleigh-Plesset cavitation model includes a volume fraction control Equation that treats cavitation as a two-phase three-component system, a mixed phase mass, momentum, and energy Equation based on the assumption that each component has the same velocity, and a Rayleigh-Plesset Equation for predicting vaporization rate, cavitation bubble generation, and cavitation bubble destruction.
The method specifically comprises the following steps of:
changing geometric parameters of the impellers to obtain impellers of different shapes; carrying out geometric modeling, meshing and cavitation flow numerical simulation and analysis on each impeller to determine 7 impeller geometric parameters most sensitive to vortex cavitation, wherein the seven impeller geometric parameters comprise an included angle alpha between the outer end surface of the inducer and a radial surface1Included angle alpha between inner end surface of inducer and radial surface3Maximum wrap angle theta of blade center parabola at average radius of outer end surface of inducer in circumferential directionOMBlade angle at the blade top of the outer end surface of the inducer
Blade angle at average radius of outer end surface of inducer
Radius R at blade root of impeller outlet1And the radius R at the blade top of the outlet of the impeller2。
The characterization expression of the complex vortex cavitation flow in the low-temperature liquid expansion machine comprises the characterization expression of the vortex flow in the low-temperature liquid expansion machine and the characterization expression of the vortex cavitation flow in the low-temperature liquid expansion machine;
characterizing the swirl flow in the cryogenic liquid expander by using the total pressure loss coefficient zeta of the cryogenic liquid expander, wherein,
wherein, PtIs the total pressure, P, of the cryogenic liquid expandert＝p+0.5ρ(u2+v2+w2) Q is the flow of the cryogenic liquid expander, AinThe inlet area of the cryogenic liquid expander.
The geometric parameter sensitivity analysis of the impeller of the vortex cavitation flow in the low-temperature liquid expander shows that the average pressure of the impeller outlet of the low-temperature liquid expander influences the vortex cavitation flow in the low-temperature liquid expander, so that the average pressure of the impeller outlet is nondimensionalized
Characterizing the vortical cavitation flow in a cryogenic liquid expander, wherein,
wherein the content of the first and second substances,
for the initially designed impeller exit mean static pressure, Pa'veAnd designing the outlet average static pressure of the impeller for the optimization process candidate.
The specific process of constructing the flow field optimization objective function and the flow field optimization control variable aiming at controlling the vortex cavitation flow comprises the following steps:
using non-dimensionalised impeller outlet mean pressure
Linear combination with total pressure loss coefficient zeta of cryogenic liquid expanderAn objective function for optimally controlling the vortex cavitation of the warm liquid expander, wherein the constructed objective function for optimally controlling the vortex cavitation of the warm liquid expander is as follows:
eff0and eff' is the isentropic efficiency, delta, of the whole machine in the initial design and optimization processes, respectivelyeffAmplitude of downward floating for efficiency, C1And C2Respectively represent Pa'veAnd a weight between ζ, wherein,
the flow field optimization control variable comprises an included angle alpha between the outer end surface of the inducer and the radial surface1Included angle alpha between inner end surface of inducer and radial surface3Maximum wrap angle theta of blade center parabola at average radius of outer end surface of inducer in circumferential directionOMBlade angle at the blade top of the outer end surface of the inducer
Blade angle at average radius of outer end surface of inducer
Radius R at blade root of impeller outlet1And the radius R at the blade top of the outlet of the impeller2。
The specific process of parallel solving the vortex cavitation flow optimization control problem is as follows: combining a low-temperature expansion machine vortex cavitation flow characterization method with an optimization method based on adaptive sampling to construct a low-temperature liquid expansion machine vortex cavitation flow optimization control method, wherein the low-temperature liquid expansion machine vortex cavitation flow optimization control method comprises a Crigin model initialization module, an adaptive sampling optimization module and an automatic sample analysis module;
the cramThe specific working process of the gold model initialization module is as follows: determining an optimization variable alpha1,α3,θ0M,
R1,R2By using DOE experimental design to optimize the variable alpha1,α3,θ0M,
R1,R2Selecting a plurality of experimental samples in the variation range, and performing flow field CFD numerical simulation to obtain
And maximum and minimum values of ζ, then
And substitution of maximum and minimum values of ζ
To determine C1And C2；
The adaptive sampling optimization module comprises the following steps: constructing and updating a proxy model; solving an EI auxiliary function by combining an optimization algorithm with a Krigin proxy model to obtain a new impeller design, and then updating the proxy model by using the new impeller design;
the automatic sample analysis module comprises the following steps: obtaining a new optimization variable alpha1,α3,θ0M,
R1,R2Then the new optimization variable alpha is added1,α3,θ0M,
R1,R2Converting into impeller three-dimensional blades, performing iterative computation on the flow field by using a flow field solver, and performing iterative computation according to the flow field solverJudging whether convergence is required under the convergence condition, acquiring information of a flow field when convergence is required, and utilizing the information
Calculating an objective function value to obtain an optimized impeller blade profile; when not converging, then classify the current optimization variable into the malformed solution, and then use the penalty function to generate a larger optimization variable α1,α3,θ0M,
R1,R2The range of variation of (a).
The invention has the following beneficial effects:
the effective control method for the vortex cavitation flow in the low-temperature liquid expander can effectively control the vortex flow caused by the rotation of the impeller, inhibit the cavitation induced by the vortex flow, avoid the vibration and shutdown of the low-temperature liquid expander unit and the shutdown of the air separation liquefaction device induced by the vortex cavitation and effectively improve the performance and the operation reliability of the low-temperature liquid expander by controlling the vortex cavitation flow in the low-temperature liquid expander.
Drawings
FIG. 1 is a schematic illustration of a parameterization of the impeller shape;
FIG. 2 is a block diagram of the flow of the vortex cavitation flow optimization control of the present invention.
Detailed Description
The invention is described in further detail below with reference to the accompanying drawings:
referring to fig. 1, the effective control method of the vortex cavitation flow in the cryogenic liquid expander according to the present invention includes a study on a vortex cavitation mechanism of the cryogenic liquid expander considering a thermodynamic effect of a cryogenic fluid, an analysis on geometric parameter sensitivity of an impeller of the vortex cavitation flow in the cryogenic liquid expander, a characterization of a complex vortex cavitation flow in the cryogenic liquid expander, a construction of a flow field optimization objective function and a flow field optimization control variable for the purpose of controlling the vortex cavitation flow, and a parallel solution of a vortex cavitation flow optimization control problem.
1. The specific process of the low-temperature liquid expander vortex cavitation mechanism research considering the low-temperature fluid thermodynamic effect comprises the following steps:
the method is characterized in that a Rayleigh-Plesset cavitation model is adopted to research the cavitation flow in the low-temperature liquid expander, the Rayleigh-Plesset cavitation model is combined with a numerical model of the whole liquid expander to simulate the vortex cavitation value of the low-temperature liquid expander, and in order to consider the thermodynamic benefit of the low-temperature fluid, the saturated vapor pressure and the surface tension are particularly expressed as functions changing along with the temperature; in the iterative calculation process of the flow field, the saturated vapor pressure and the surface tension are updated in real time along with the change of the temperature field; mechanism research shows that the swirl flow in the low-temperature liquid expander originates from the tail edge of a high-speed rotating impeller and expands to a diffuser pipe along with the main flow, so that local static pressure in an impeller outlet and the downstream diffuser pipe is reduced, the temperature is increased, and further cavitation is induced.
The Rayleigh-Plesset cavitation model includes a volume fraction control Equation that treats cavitation as a two-phase three-component system, a mixed phase mass, momentum, and energy Equation based on the assumption that each component has the same velocity, and a Rayleigh-Plesset Equation for predicting vaporization rate, cavitation bubble generation, and cavitation bubble destruction.
2. The method specifically comprises the following steps of analyzing the geometric parameter sensitivity of the impeller of the vortex cavitation flow in the low liquid expansion machine:
referring to fig. 1, the geometric parameters of the impeller are changed to obtain impellers with different shapes; carrying out geometric modeling, meshing and cavitation flow numerical simulation and analysis on each impeller to determine 7 impeller geometric parameters most sensitive to vortex cavitation, wherein the seven impeller geometric parameters comprise an included angle alpha between the outer end surface of the inducer and a radial surface1Included angle alpha between inner end surface of inducer and radial surface3Maximum wrap angle theta of blade center parabola at average radius of outer end surface of inducer in circumferential directionOMBlade angle at the blade top of the outer end surface of the inducer
Blade angle at average radius of outer end surface of inducer
Radius R at blade root of impeller outlet1And the radius R at the blade top of the outlet of the impeller2。
3. The characterization expression of the complex vortex cavitation flow in the low-temperature liquid expansion machine comprises the characterization expression of the vortex flow in the low-temperature liquid expansion machine and the characterization expression of the vortex cavitation flow in the low-temperature liquid expansion machine;
characterizing the swirl flow in the cryogenic liquid expander by using the total pressure loss coefficient zeta of the cryogenic liquid expander, wherein,
wherein, PtIs the total pressure, P, of the cryogenic liquid expandert＝p+0.5ρ(u2+v2+w2) Q is the flow of the cryogenic liquid expander, AinThe inlet area of the cryogenic liquid expander.
The geometric parameter sensitivity analysis of the impeller of the vortex cavitation flow in the low-temperature liquid expander shows that the average pressure of the impeller outlet obviously influences the internal cavitation flow of the liquid expander, the size of the average pressure not only reflects the internal cavitation characteristic of the impeller, but also reflects the cavitation characteristic in a diffuser pipe at the downstream of the impeller, so that the average pressure of the impeller outlet is nondimensionalized
Characterizing the vortical cavitation flow in a cryogenic liquid expander, wherein,
wherein the content of the first and second substances,
impeller Outlet average static pressure, P ', for initial design'aveFor setting candidates in the optimization processAnd (4) calculating the average static pressure at the outlet of the impeller.
4. The specific process of constructing the flow field optimization objective function and the flow field optimization control variable aiming at controlling the vortex cavitation flow comprises the following steps:
using non-dimensionalised impeller outlet mean pressure
And linearly combining with the total pressure loss coefficient zeta of the low-temperature liquid expander to construct an objective function for optimally controlling the vortex cavitation of the low-temperature liquid expander, wherein the zeta represents the flow loss caused by the vortex flow at the outlet of the impeller and the average pressure at the outlet of the dimensionless impeller
Characterizing cavitation flow, and constructing a target function of vortex cavitation optimization control of the low-temperature liquid expander, wherein the target function is as follows:
the above objective function is taken as an optimization target, the impeller vortex cavitation flow can be effectively inhibited, wherein eff0And eff' is the isentropic efficiency, delta, of the whole machine in the initial design and optimization processes, respectivelyeff(optionally 2%) is the magnitude of the downward efficiency float to prevent overall expander performance degradation during flow field optimization, C1And C2Are respectively P'aveAnd a weight between ζ, wherein,
before optimization begins, N experimental samples are determined from the experimental design. Obtaining flow data and related information by numerical simulation, and establishing an initial agent model; obtaining the sample of the experiment by statistical analysis
And ζ, and then the parameter C1And C2。
The flow field optimization control variable comprises an included angle alpha between the outer end surface of the inducer and the radial surface1Included angle alpha between inner end surface of inducer and radial surface3Maximum wrap angle theta of blade center parabola at average radius of outer end surface of inducer in circumferential directionOMBlade angle at the blade top of the outer end surface of the inducer
Blade angle at average radius of outer end surface of inducer
Radius R at blade root of impeller outlet1And the radius R at the blade top of the outlet of the impeller2。
5. The specific process of parallel solving the vortex cavitation flow optimization control problem is as follows:
combining a low-temperature expansion machine vortex cavitation flow characterization method with an optimization method based on adaptive sampling to construct a low-temperature liquid expansion machine vortex cavitation flow optimization control method, referring to fig. 2, wherein the low-temperature liquid expansion machine vortex cavitation flow optimization control method comprises a kriging model initialization module, an adaptive sampling optimization module and a sample automatic analysis module;
the specific working process of the kriging model initialization module is as follows: determining an optimization variable alpha1,α3,θ0M,
R1,R2By using DOE experimental design to optimize the variable alpha1,α3,θ0M,
R1,R2Selecting a plurality of experimental samples in the variation range, simulating by a flow field CFD numerical value to obtain a corresponding objective function value, and establishing an initial objective function value on the basisThe agent model is obtained by analyzing and counting D0E test samples
And maximum and minimum values of ζ, then
And substitution of maximum and minimum values of ζ
To determine C1And C2；
The adaptive sampling optimization module comprises the following steps: constructing and updating an agent model, namely replacing time-consuming CFD calculation with the agent model to complete the evaluation of candidate design and accelerate the optimization process; an optimization algorithm is combined with the kriging proxy model, an EI auxiliary function is solved to obtain a new impeller design, and then the proxy model is updated by using the new impeller design;
the automatic sample analysis module comprises the following steps: obtaining a new optimization variable alpha1,α3,θ0M,
R1,R2Then the new optimization variable alpha is added1,α3,θ0M,
R1,R2Converting into impeller three-dimensional blades, performing iterative calculation on the flow field by using a flow field solver, judging whether the convergence is performed or not according to convergence conditions, acquiring the information of the flow field when the convergence is performed, and then utilizing the flow field solver to perform iterative calculation on the flow field
Calculating an objective function value to obtain an optimized impeller blade profile; when not converging, then classify the current optimization variable into the malformed solution, and then use the penalty function to generate a larger optimization variable α1,α3,θ0M,
R1,R2The range of variation of (a).
And continuously interacting the main program of the optimization algorithm with the automatic sample analysis module, updating the Kriging model in real time, capturing the characteristic information of the optimization problem, correcting the optimization path in real time, searching along the direction of the global solution until the optimization condition is reached, stopping searching and outputting the optimized impeller blade profile.
Claims (1)
1. An effective control method of vortex cavitation flow in a low-temperature liquid expander is characterized by comprising the steps of researching the vortex cavitation mechanism of the low-temperature liquid expander considering the thermodynamic effect of low-temperature fluid, analyzing the geometric parameter sensitivity of an impeller of the vortex cavitation flow in the low-temperature liquid expander, characterizing the complex vortex cavitation flow in the low-temperature liquid expander, constructing a flow field optimization objective function and a flow field optimization control variable aiming at controlling the vortex cavitation flow, and solving the problem of the vortex cavitation flow optimization control in parallel;
the specific process of the low-temperature liquid expander vortex cavitation mechanism research considering the low-temperature fluid thermodynamic effect comprises the following steps: researching internal cavitation flow of the low-temperature liquid expander by adopting a Rayleigh-Plesset cavitation model, and combining the Rayleigh-Plesset cavitation model with a numerical model of the whole liquid expander to simulate the vortex cavitation flow of the liquid expander;
the Rayleigh-Plesset cavitation model comprises a volume fraction control equation for regarding cavitation as a two-phase three-component system, a mixed phase mass, momentum and energy equation based on the assumption that all components have the same speed, and a Rayleigh-Plesset equation for predicting gasification rate, cavitation generation and cavitation destruction;
the method specifically comprises the following steps of:
changing geometric parameters of the impellers to obtain impellers of different shapes; geometric modeling, meshing and cavitation flow numerical simulation and division are carried out on each impellerAnalyzing to determine 7 impeller geometric parameters which are most sensitive to the vortex cavitation, wherein the 7 impeller geometric parameters comprise an included angle alpha between the outer end surface of the inducer and a radial surface1Included angle alpha between inner end surface of inducer and radial surface3Maximum wrap angle theta of blade center parabola at average radius of outer end surface of inducer in circumferential directionOMBlade angle at the blade top of the outer end surface of the inducer
Blade angle at average radius of outer end surface of inducer
Radius R at blade root of impeller outlet1And the radius R at the blade top of the outlet of the impeller2；
The characterization expression of the complex vortex cavitation flow in the low-temperature liquid expansion machine comprises the characterization expression of the vortex flow in the low-temperature liquid expansion machine and the characterization expression of the vortex cavitation flow in the low-temperature liquid expansion machine;
characterizing the swirl flow in the cryogenic liquid expander by using the total pressure loss coefficient zeta of the cryogenic liquid expander, wherein,
wherein, PtIs the total pressure, P, of the cryogenic liquid expandert＝p+0.5ρ(u2+v2+w2) Q is the flow of the cryogenic liquid expander, AinThe inlet area of the cryogenic liquid expander;
the geometric parameter sensitivity analysis of the impeller of the vortex cavitation flow in the low-temperature liquid expander shows that the average pressure of the impeller outlet of the low-temperature liquid expander influences the vortex cavitation flow in the low-temperature liquid expander, so that the average pressure of the impeller outlet is nondimensionalized
And carrying out characterization expression of the vortex cavitation flow in the low-temperature liquid expander, wherein,
wherein the content of the first and second substances,
for the initially designed impeller exit mean static pressure, Pa'veDesigning the outlet average static pressure of the impeller for candidate design in the optimization process;
the specific process of constructing the flow field optimization objective function and the flow field optimization control variable aiming at controlling the vortex cavitation flow comprises the following steps:
using non-dimensionalised impeller outlet mean pressure
And linearly combining with the total pressure loss coefficient zeta of the low-temperature liquid expander to construct an objective function of the vortex cavitation optimization control of the low-temperature liquid expander, wherein the constructed objective function of the vortex cavitation optimization control of the low-temperature liquid expander is as follows:
eff0And eff' is the isentropic efficiency, delta, of the whole machine in the initial design and optimization processes, respectivelyeffAmplitude of downward floating for efficiency, C1And C2Respectively represent
And a weight between ζ, wherein,
the flow field optimization control variable comprises an included angle alpha between the outer end surface of the inducer and the radial surface1Included angle alpha between inner end surface of inducer and radial surface3Maximum wrap angle theta of blade center parabola at average radius of outer end surface of inducer in circumferential directionOMBlade angle at the blade top of the outer end surface of the inducer
Blade angle at average radius of outer end surface of inducer
Radius R at blade root of impeller outlet1And the radius R at the blade top of the outlet of the impeller2(ii) a N is an experimental sample;
the specific process of parallel solving the vortex cavitation flow optimization control problem is as follows: combining a low-temperature expansion machine vortex cavitation flow characterization method with an optimization method based on adaptive sampling to construct a low-temperature liquid expansion machine vortex cavitation flow optimization control method, wherein the low-temperature liquid expansion machine vortex cavitation flow optimization control method comprises a Crigin model initialization module, an adaptive sampling optimization module and an automatic sample analysis module;
the specific working process of the kriging model initialization module is as follows: determining flow field optimization control variable alpha1,α3,θ0M,
R1,R2By using DOE experimental design to optimize the control variable alpha in the flow field1,α3,θ0M,
R1,R2Selecting a plurality of experimental samples in the variation range, and performing flow field CFD numerical simulation to obtain
And maximum and minimum values of ζ, then
And substitution of maximum and minimum values of ζ
To determine C1And C2；
The adaptive sampling optimization module comprises the following steps: constructing and updating a kriging proxy model; solving an EI auxiliary function by combining an optimization algorithm with a Krigin proxy model to obtain a new impeller design, and then updating the proxy model by using the new impeller design;
the automatic sample analysis module comprises the following steps: obtaining new flow field optimization control variable alpha1,α3,θ0M,
R1,R2Then optimizing the control variable alpha according to the new flow field1,α3,θ0M,
R1,R2Generating corresponding impeller three-dimensional blades, performing iterative calculation on the flow field by adopting a flow field solver, judging whether the convergence is performed or not according to convergence conditions, acquiring the information of the flow field when the convergence is performed, and then utilizing the convergence condition to perform iterative calculation on the flow field
Calculating an objective function value to obtain an optimized impeller blade profile; when the flow field optimization control variable is not converged, classifying the current flow field optimization control variable into a malformed solution, and then generating a larger flow field optimization control variable alpha by utilizing a penalty function1,α3,θ0M,
R1,R2The range of variation of (a).
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