Compressor characteristic processing method

Abstract

The application provides a compressor characteristic processing method, which introduces skeleton characteristic parameters to replace traditional characteristic parameters to represent the characteristics of a compressor, wherein the skeleton characteristic parameters include a work coefficient, a flow coefficient and a loss coefficient, the traditional characteristic parameters include a pressure ratio, a converted flow and an efficiency, a minimum loss point established according to the skeleton characteristic parameters is taken as a skeleton trunk, and branches are established according to the relationship between non-minimum loss points and the skeleton characteristic parameters of the minimum loss point, and the branches are extended to form the characteristics of the compressor in a full speed range, and in the selection of the minimum loss point and the establishment of the relationship between the non-minimum loss points and the skeleton characteristic parameters of the minimum loss point, an optimization algorithm is used to adjust and optimize the position of the minimum loss point. The compressor characteristic processing method can efficiently and high-quality use the characteristics of a small number of working conditions to obtain the characteristics of the remaining working conditions.

Description

Technical Field

[0001] This invention relates to the field of aero-engine performance simulation technology, specifically to a method for processing compressor characteristics. Background Technology

[0002] Currently, the commercial aero-engine industry in China is still in its infancy. Its simulation tools and methods lack technological accumulation and expertise. In particular, the methods for handling compressor characteristics mostly rely on existing commercial software tools, which have limited functionality and do not provide core algorithms and principles, thus failing to meet the demand for high-precision engine performance simulation under all operating conditions.

[0003] Traditional methods for handling compressor component characteristics use pressure ratio π and converted flow rate W. c The characteristics of a compressor are characterized by its efficiency η, such as... Figure 1 and Figure 2 As shown, Figure 1 and Figure 2 The compressor's pressure ratio π and converted flow rate W c Both efficiency η and π were standardized. ds W c,ds and η ds These represent the pressure ratio, converted flow rate, and efficiency at the design point, respectively. Figure 1 and Figure 2 The data on each characteristic curve is the converted speed n corresponding to that curve. c The value of η. In the research and development and testing of aero-engines, characteristic curves at certain speeds are often only available; characteristic curves at other speeds need to be provided through computer simulation and performance calculation. Traditional compressor component characteristic processing methods provide limited characteristic curves due to their small operating range, and the efficiency η exhibits zero or singular points during transitions between compressor, mixer, and turbine states. Figure 2 As shown, this prevents a smooth transition in performance calculations, leading to computational problems.

[0004] In the 1980s, General Electric (GE) developed an extended parametric representation of compressor fans and turbines for NASA. This method, based on the working mechanism of turbomachinery and extensive experimental experience and data, introduces a power coefficient Ψ and a flow coefficient... and loss coefficient Ψ loss Replace pressure ratio π with other skeleton characteristic parameters to calculate flow rate W c Traditional characteristic parameters such as efficiency η are used to characterize the properties of compressors and turbines, while each converted speed n cBelow, that is, the loss coefficient Ψ on each characteristic curve. loss The point with the smallest loss is taken as the backbone point, also called the minimum loss point. First, the different conversion speeds n are determined. c The relationship and trend of the skeleton characteristic parameters at the minimum loss point are then used to calculate the rotational speed n. c The minimum loss point under the given conditions is extended to obtain other converted rotational speeds n. c The minimum loss point is connected to different converted speeds n. c Multiple minimum loss points are formed as follows Figure 3 The dashed lines in the diagram represent the skeleton lines, and then based on each converted rotational speed n... c The skeleton characteristic parameters at the minimum loss point are used to obtain the same converted rotational speed n. c The skeleton characteristic parameters at the non-minimum loss points, and the skeleton characteristic parameters at the minimum and non-minimum loss points, can be converted into traditional characteristic parameters to obtain the compressor characteristics across the entire speed range. This method will be referred to as the skeleton characteristic principle below. Compared to the definition of efficiency η, the loss coefficient Ψ... loss The changes are continuous and without singularities. Therefore, by using the skeleton characteristic principle to process the characteristics of compressor components, the characteristics of compressors can be effectively extended and obtained across the entire speed range based on the characteristic trends of partial speeds, providing a data basis for calculating engine starting performance and windmill performance.

[0005] However, in the research and development and testing of aero engines, for each calculated rotational speed n... c Often, only a limited number of data points can be obtained to determine the characteristics of the compressor. The above skeleton characteristic principle does not provide an effective method to determine the location of the ideal minimum loss point based on a limited number of data points. At present, a suitable calculation method and tool based on the skeleton characteristic principle has not yet been formed to process the compressor characteristics and ensure the accuracy and effectiveness of performance calculation. Summary of the Invention

[0006] The purpose of this invention is to provide a compressor characteristic processing method that can efficiently and effectively utilize the characteristics of a small number of operating conditions to obtain the characteristics of other operating conditions.

[0007] To achieve the aforementioned objective, a compressor characteristic processing method introduces skeleton characteristic parameters to replace traditional characteristic parameters to characterize the compressor's characteristics. These skeleton characteristic parameters include the power coefficient, flow coefficient, and loss coefficient, while the traditional characteristic parameters include pressure ratio, converted flow rate, and efficiency. The method uses the minimum loss point established based on these skeleton characteristic parameters as the backbone of the skeleton, and establishes branches based on the relationship between non-minimum loss points and the skeleton characteristic parameters of the minimum loss point. This extends and forms the compressor's characteristics across the entire speed range. During the selection of the minimum loss point and the establishment of the relationship between the non-minimum loss points and the skeleton characteristic parameters of the minimum loss point, an optimization algorithm is used to adjust and optimize the position of the minimum loss point.

[0008] In one or more embodiments of the compressor characteristic processing method, the method includes the following steps: S1, obtaining the mechanical related parameters of the compressor and the actual values ​​of the traditional characteristic parameters of multiple data points at a partial converted speed; S2, calculating the work coefficient and the loss coefficient of the multiple data points; S3, setting the data point with the smallest loss coefficient among the multiple data points at each converted speed as the assumed minimum loss point at the corresponding converted speed; S4, adjusting the position of the assumed minimum loss point at each converted speed using an optimization algorithm to obtain the optimized minimum loss point at each converted speed; S5, extending and obtaining extended minimum loss points at other converted speeds based on the relationship and changing trend of the skeleton characteristic parameters of the multiple optimized minimum loss points at the partial converted speeds; S6, obtaining the skeleton characteristic parameters of the corresponding non-minimum loss point at the partial converted speed or the other converted speeds based on the skeleton characteristic parameters of each optimized minimum loss point or the extended minimum loss point; S7, calculating the traditional characteristic parameters based on the skeleton characteristic parameters of the optimized minimum loss point, the extended minimum loss point, and the non-minimum loss point.

[0009] In one or more embodiments of the compressor characteristic processing method, the optimization algorithm employs a particle swarm optimization algorithm. The method for calculating the fitness function of the particle swarm optimization algorithm includes: obtaining the skeleton characteristic parameters of the current assumed minimum loss point; calculating the skeleton characteristic parameters of multiple data points at the corresponding converted speed based on the skeleton characteristic parameters of the current assumed minimum loss point; calculating the traditional characteristic parameters of multiple data points by back-calculating the calculated values ​​of the traditional characteristic parameters based on the skeleton characteristic parameters of the multiple data points; standardizing the calculated values ​​of the traditional characteristic parameters with the actual values ​​of the traditional characteristic parameters obtained in step S1; calculating the deviation between the calculated values ​​and the actual values; and using the sum of the deviations of multiple data points at the same converted speed as the fitness function.

[0010] In one or more embodiments of the compressor characteristic processing method, step S4 further includes manually checking and adjusting the optimized minimum loss point.

[0011] In one or more embodiments of the compressor characteristic processing method, the optimization algorithm employs a swarm intelligence algorithm.

[0012] The compressor characteristic processing method of this invention is based on the principle of skeleton characteristics, establishing a corresponding compressor characteristic conversion algorithm. It also incorporates the characteristics of physical principles and experimental experience. Besides extension (refined extrapolation), it can identify and correct unreasonable data, achieving optimal smoothing of component characteristics, identifying and correcting unreasonable data in test results. It can utilize compressor component characteristics under a limited number of operating conditions to obtain component characteristics under other operating conditions, thus extending the characteristic trends at certain speeds to obtain compressor component characteristics across the entire speed range, providing a data foundation for calculating engine starting performance and turbine performance. In the selection of the minimum loss point and the establishment of the relationship between skeleton characteristic parameters of non-minimum loss points and the minimum loss point, an optimization algorithm is used to adjust and optimize the position of the minimum loss point to obtain a minimum loss point close to the ideal position at each conversion speed. This helps simplify the workflow and workload of compressor characteristic processing, saves manpower, provides high-quality characteristic processing results, improves the efficiency of compressor characteristic processing, ensures the validity of calculation results, and shortens the engine design cycle. Attached Figure Description

[0013] The above and other features, properties and advantages of the present invention will become more apparent from the following description taken in conjunction with the accompanying drawings and embodiments, wherein:

[0014] Figure 1 This is a schematic diagram showing the relationship between the pressure ratio and the converted flow rate of a certain type of compressor.

[0015] Figure 2 This is a schematic diagram showing the relationship between the efficiency and converted flow rate of a certain type of compressor.

[0016] Figure 3 This is a schematic diagram of the frame lines of a certain type of compressor.

[0017] Figure 4 This is a schematic flowchart of the compressor characteristic processing method according to an embodiment of the present invention.

[0018] Figure 5 This is a flowchart illustrating the particle swarm optimization algorithm.

[0019] Figure 6This is a schematic diagram showing the relationship between multiple data points before fitting and the virtual Mach number of the minimum loss point for a certain type of compressor.

[0020] Figure 7 This is a schematic diagram showing the relationship between the loss coefficients of multiple data points before fitting and the minimum loss point of a certain type of compressor.

[0021] Figure 8 This is a schematic diagram showing the relationship between the virtual Mach number of multiple data points after fitting a certain type of compressor and the minimum loss point.

[0022] Figure 9 This is a schematic diagram showing the relationship between the loss coefficients of multiple data points after fitting a certain type of compressor and the loss point with the minimum loss. Detailed Implementation

[0023] The following discloses various implementation methods or embodiments of the described subject matter. To simplify the disclosure, specific examples of the elements and arrangements are described below. These are merely examples and are not intended to limit the scope of protection of the present invention. It should be noted that the accompanying drawings are for illustrative purposes only and are not drawn to scale, and should not be used to limit the actual scope of protection claimed by the present invention. Furthermore, certain features, structures, or characteristics in one or more embodiments of the present invention can be appropriately combined.

[0024] The compressor characteristic processing method of this invention is based on the aforementioned skeleton characteristic principle, and introduces the work coefficient Ψ and flow coefficient. and loss coefficient Ψ loss Replace pressure ratio π with other skeleton characteristic parameters to calculate flow rate W c The compressor characteristics are characterized by traditional characteristic parameters such as efficiency η. The minimum loss point established based on the skeleton characteristic parameters is used as the skeleton backbone, and branches are established by the relationship between the skeleton characteristic parameters of non-minimum loss points and minimum loss points, which are then extended to form the compressor characteristics across the entire speed range.

[0025] During the practical application of the skeleton characteristic principle, the inventors discovered that in the research and development and testing of aero engines, for each calculated rotational speed n... c Often, only a limited number of data points can be obtained, making it difficult to accurately determine the location of the minimum loss point. Multiple adjustments to the assumed minimum loss point are needed to obtain the correct location for each converted rotational speed n. c The minimum loss point is close to the ideal position. During the adjustment process, due to the high degree of uncertainty and the high coupling between parameters, it is necessary to perform global parameter optimization for multiple objectives. This process often takes several weeks, involves trying different directions, and finally selects a more suitable solution as the final processing result. This not only consumes a lot of human resources, but also cannot guarantee the quality of processing.

[0026] Therefore, the compressor characteristic processing method of this invention employs an optimization algorithm to adjust and optimize the position of the minimum loss point during the selection of the minimum loss point and the establishment of the relationship between the skeleton characteristic parameters of non-minimum loss points and the minimum loss point, in order to obtain each converted speed n. c The minimum loss point that is close to the ideal position.

[0027] The compressor characteristic processing method of this invention is based on the principle of skeleton characteristics, establishing a corresponding compressor characteristic conversion algorithm. It also incorporates the characteristics of physical principles and experimental experience. Besides extension (encrypted extrapolation), it can identify and correct unreasonable data, achieving optimal smoothing of component characteristics and identifying and correcting unreasonable data in test results. It can utilize compressor component characteristics under a small number of operating conditions to obtain component characteristics under other operating conditions, thus extending the characteristic trends at certain speeds to obtain compressor component characteristics across the entire speed range, providing a data foundation for calculating engine starting performance and windmill performance. In the selection of the minimum loss point and the establishment of the relationship between the skeleton characteristic parameters of non-minimum loss points and the minimum loss point, an optimization algorithm is used to adjust and optimize the position of the minimum loss point to obtain the corresponding conversion speed n. c Finding the minimum loss point close to the ideal position helps simplify the workflow and workload of compressor characteristic processing, saves manpower, provides high-quality characteristic processing results, improves the efficiency of compressor characteristic processing, ensures the validity of calculation results, and shortens the engine design cycle.

[0028] The flow chart of the compressor characteristic processing method according to an embodiment of the present invention is as follows: Figure 4 As shown, it includes the following steps:

[0029] S1. Obtain the mechanical parameters of the compressor, and some of the converted speed n. c The actual values ​​of traditional characteristic parameters for multiple data points are shown below, including mechanically relevant parameters such as the first-stage moving blade tip diameter D, the compressor inlet area A, and the design point speed n. ds Traditional characteristic parameters include pressure ratio π and converted flow rate W. c And efficiency η, etc.;

[0030] S2. Calculate the work coefficient for multiple data points. and loss coefficient Ψ loss =Ψ(1-η), where Δh is the enthalpy increase, U is the tangential velocity at the tip of the first-stage compressor blade, and g0 and J are constants. Where ft is feet, lbm is pounds (mass), lbf is pounds (force), s is seconds, and Btu is a unit of heat.

[0031] S3, convert each known rotational speed n from step S1 c Among multiple data points, the loss coefficient Ψ loss The minimum data point is set as the corresponding converted speed n. c The assumed minimum loss point;

[0032] S4. Adjust each converted speed n using an optimized algorithm. c Based on the assumed location of the minimum loss point, obtain the calculated rotational speed n for each speed. c The optimal minimum loss point that is close to the ideal position;

[0033] S5. Calculate the rotational speed n based on the known portion. c The relationship and trend of skeleton characteristic parameters at multiple optimized minimum loss points are analyzed, and other converted rotational speeds n are obtained through interpolation, fitting, or other methods. c The extended minimum loss point;

[0034] S6. Obtain the corresponding partially converted rotational speed n based on the skeleton characteristic parameters of each optimized minimum loss point. c Based on the skeleton characteristic parameters of the non-minimum loss points, other equivalent rotational speeds n are obtained. c Skeleton characteristic parameters of non-minimum loss points;

[0035] S7. Calculate the traditional characteristic parameters of the compressor component based on the skeleton characteristic parameters of the optimized minimum loss point, the extended minimum loss point, and the non-minimum loss point, thereby obtaining the compressor component characteristics of the full speed range characterized by the traditional characteristic parameters.

[0036] Among them, the calculation method for obtaining the skeleton characteristic parameters of non-minimum loss points from the skeleton characteristic parameters of the minimum loss point, based on the method provided in the skeleton characteristic principle, mainly includes: fitting the relationship curve between the skeleton characteristic parameters of the non-minimum loss point and the minimum loss point, for example, Figures 6 to 9 The diagram illustrates the relationship between the skeleton characteristic parameters of multiple known data points of a certain type of compressor, specifically multiple non-minimum loss points and the minimum loss point. Figure 6 This diagram illustrates the relationship between multiple known data points and the virtual Mach number M of the minimum loss point. Figure 6 Each graph in the diagram corresponds to a known converted rotational speed n. c , Figure 6 In the diagram, GH on the horizontal axis represents the work coefficient offset, GH = Ψ - Ψ ML Ψ ML The value of the power coefficient representing the point of minimum loss is indicated by a horizontal axis of 0, while points with a horizontal axis other than 0 represent the corresponding equivalent rotational speed n. c Multiple known data points, i.e. multiple non-minimum loss points; Figure 7 The loss coefficient Ψ between multiple data points and the minimum loss point loss A diagram illustrating the relationship between the two. Figure 7 Each graph in the diagram corresponds to a known converted rotational speed n. c , Figure 7 In the x-coordinate, GH*|GH|=(Ψ-Ψ) ML )*|Ψ-Ψ ML The point with an x-coordinate of 0 is the minimum loss point, and the point with a x-coordinate other than 0 is the corresponding converted rotational speed n. c Multiple known data points, i.e. multiple non-minimum loss points; Figure 8 It is Figure 6 The slope K of the virtual Mach number M relationship curve obtained by fitting the virtual Mach number M relationship curve is... M With converted speed n c Relationship diagram; Figure 9 It is Figure 7 The loss coefficient Ψ loss The loss coefficient Ψ is obtained by fitting the relationship curve. loss The slope K of the relationship curve Ψloss With converted speed n c A schematic diagram of the relationship; through the above fitting calculations, the known conversion speed n can be used. c The fitted curve of the skeleton characteristic parameters at the minimum loss point is extended to obtain other converted rotational speeds n. c The skeleton characteristic parameters of the minimum loss point, and the virtual Mach number M and loss coefficient Ψ of the non-minimum loss point. loss The GH value can be calculated from the slope of the fitted skeleton characteristic parameter curve; therefore, the corresponding converted rotational speed n can be obtained based on the skeleton characteristic parameters at the minimum loss point. c The virtual Mach number M and loss coefficient Ψ at the non-minimum loss point loss Then, the characteristic parameters across the entire speed range can be calculated.

[0037] Optionally, the optimization algorithm in step S4 adopts a swarm intelligence algorithm, such as a genetic algorithm, a particle swarm optimization algorithm, or an artificial fish swarm algorithm, etc., so as to realize the automated optimization process of global optimum in the optimization and adjustment of multi-solution problems, which can greatly improve the efficiency of compressor characteristic processing, and save processing time and ensure processing quality while ensuring the formation of component characteristics across the entire speed range.

[0038] Optionally, the minimum loss point calculated by the optimization algorithm in step S4 can be manually checked and adjusted. For example, check whether the minimum loss point was not obtained due to the limitations of the pre-set boundary conditions of the optimization algorithm, including the range of values ​​for the minimum loss point. Check whether the fitting curves of the skeleton characteristic parameters of the minimum loss point, extended minimum loss point, and non-minimum loss point with the converted speed are smooth and conform to the trend of the compressor characteristics. Check whether the calculated values ​​of the traditional characteristic parameters calculated from the skeleton characteristic parameters of the minimum loss point, extended minimum loss point, and non-minimum loss point are within a certain range, such as no more than 3%. For unreasonable optimization results, adjust the optimization settings and recalculate or make manual adjustments to ensure that the calculation results of the optimization algorithm meet the requirements.

[0039] In one embodiment of the compressor characteristic processing method of the present invention, the optimization algorithm in step S4 adopts the particle swarm optimization algorithm, and the process of the particle swarm optimization algorithm is as follows: Figure 5 As shown, it includes:

[0040] 1. Determine technical specifications

[0041] 2. Initialize the particle swarm;

[0042] 3. Calculate the fitness of the particles, that is, calculate the value of the fitness function;

[0043] 4. Determine whether pbest, gbest, or nbest meets the update conditions. If the update conditions are met, proceed to step 5. If the update conditions are not met, proceed to step 3 again. Here, pbest is the best position found by each particle at present, i.e., the individual best particle; gbest is the best position found by all particles at present, i.e., the global best particle; and nbest is the best position found by each of the current particle's neighbors, i.e., the neighborhood best particle.

[0044] 5. Update pbest, gbest, and nbest, and determine whether the stopping condition has been met. If the stopping condition has not been met, repeat step 3. If the stopping condition has been met, such as reaching the preset maximum step size (maximum number of iterations) or convergence accuracy, output the calculation result, i.e., the optimal solution.

[0045] The technical indicators in step 1 include fitness function, stopping conditions, etc. The technical indicators of the particle swarm optimization algorithm differ for different optimization objects. In one embodiment of the compressor characteristic processing method of the present invention, the calculation method of the fitness function includes the following steps:

[0046] a) Obtain the skeleton characteristic parameters of the current assumed minimum loss point, including the work coefficient. Flow coefficient Loss coefficient Ψ loss =Ψ(1-η) and the virtual Mach number M, where C x Let M be the axial velocity of the airflow at the compressor inlet. The virtual Mach number M can be calculated using the following formula:

[0047]

[0048] Among them, W cmax The maximum converted flow rate is given by r, where r is the adiabatic index.

[0049] b) Based on the skeleton characteristic parameters of the assumed minimum loss point, the corresponding converted rotational speed n can be calculated using the method provided in the aforementioned skeleton characteristic principle. c The skeleton characteristic parameters of multiple known data points, i.e. multiple known non-minimum loss points;

[0050] c) Based on the skeleton characteristic parameters of the multiple known data points obtained in the previous step, the calculated values ​​of the traditional characteristic parameters of the multiple known data points are derived by reverse calculation.

[0051] d) Standardize the calculated values ​​of the traditional characteristic parameters of multiple known data points with the actual values ​​of the traditional characteristic parameters of multiple known data points obtained in step S1.

[0052] e) Compare the calculated and actual values ​​of the traditional characteristic parameters of multiple known data points under the same converted speed, and take the distance between the two closest points as the single-point deviation, and use the sum of the deviations as the fitness function.

[0053] While the present invention has been disclosed above with reference to preferred embodiments, it is not intended to limit the invention. Any variations and modifications can be made by those skilled in the art without departing from the spirit and scope of the invention. Therefore, any modifications, equivalent changes, and alterations made to the above embodiments based on the technical essence of the present invention, without departing from the scope of the invention, fall within the protection scope defined by the claims of the present invention.

Claims

1. A compressor characteristic processing method, which characterizes the compressor characteristics by introducing skeleton characteristic parameters instead of traditional characteristic parameters. The skeleton characteristic parameters include a power coefficient, a flow coefficient, and a loss coefficient. The traditional characteristic parameters include a pressure ratio, a converted flow rate, and an efficiency. The method uses the minimum loss point established based on the skeleton characteristic parameters as the skeleton backbone, and establishes branches based on the relationship between non-minimum loss points and the skeleton characteristic parameters of the minimum loss point, extending and forming the compressor characteristics across the entire speed range. The method is characterized by... In the process of selecting the minimum loss point and establishing the relationship between the skeleton characteristic parameters of the non-minimum loss point and the minimum loss point, an optimization algorithm is used to adjust and optimize the position of the minimum loss point; Specifically, the following steps are included: S1. Obtain the mechanical parameters of the compressor, as well as the actual values ​​of the conventional characteristic parameters at multiple data points under some converted speeds; S2. Calculate the work coefficient and the loss coefficient for the plurality of data points; S3. Among the multiple data points under each converted speed, the data point with the smallest loss coefficient is set as the assumed minimum loss point under the corresponding converted speed; S4. Use an optimization algorithm to adjust the position of the assumed minimum loss point under each of the converted speeds to obtain the optimized minimum loss point under each of the converted speeds; S5. Based on the relationship and changing trend of the skeleton characteristic parameters of the multiple optimized minimum loss points under the partial converted speed, the extended minimum loss points under other converted speeds are obtained. S6. Based on the skeleton characteristic parameters of each optimized minimum loss point or extended minimum loss point, obtain the skeleton characteristic parameters of the corresponding partial converted speed or the non-minimum loss point under other converted speeds; S7. Calculate the traditional characteristic parameters based on the skeleton characteristic parameters of the optimized minimum loss point, the extended minimum loss point, and the non-minimum loss point.

2. The compressor characteristic processing method as described in claim 1, characterized in that, The optimization algorithm employs particle swarm optimization, and the method for calculating the fitness function of the particle swarm optimization algorithm includes: Obtain the skeleton characteristic parameters of the current assumed minimum loss point; Based on the skeleton characteristic parameters of the current assumed minimum loss point, calculate the skeleton characteristic parameters of multiple data points at the corresponding converted speed. Based on the skeleton characteristic parameters of multiple data points, the calculated values ​​of the traditional characteristic parameters of multiple data points are calculated in reverse. The calculated values ​​of the traditional characteristic parameters are standardized with the actual values ​​of the traditional characteristic parameters obtained in step S1. The deviation between the calculated value and the actual value is calculated, and the sum of the deviations of multiple data points under the same converted speed is used as the fitness function.

3. The compressor characteristic processing method as described in claim 1 or 2, characterized in that, Step S4 further includes manually checking and adjusting the optimized minimum loss point.

4. The compressor characteristic processing method as described in claim 1, characterized in that, The optimization algorithm employs a swarm intelligence algorithm.

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