CHAPTER 2 INTRODUCTION OF RECIPROCATING COMPRESSOR
2.5 Simulation Results and Remarks
2.5.5 Remarks
The simulation results were initially verified by the experiments in a calorimeter with ASHRAE test conditions (the calorimeter accuracy is 96% for energy efficiency ratio E.E.R).
Table- 2.5.1 reveals both the results for the same testing conditions. The E.E.R. result simulated from the software is 0.9395, which is close to the experiment (E.E.R. = 0.9690).
Unit Simulation Experiment
Inhaled pressure kPa 120 117
Exhaled pressure kPa 1470 1468
Refrigerant mass
flow rate kg/hr 5.1349 5.2290
capacity of
refrigeration kcal/hr 194.6221 189.7000
E.E.R kcal/hr-W 0.9395 0.9690
Table- 2.5.1 Simulated and experimental results
The simulation software is developed for small reciprocating compressor. The dynamic, thermodynamic, fluid dynamics, bearing analysis and valve vibration are included in the software. By using the software, the efficiencies and E.E.R of the reciprocating compressor can be predicted while operating.
The software outputs include the thermal and dynamic forces properties and the forces
exerted on the slider-crank mechanism or other respective effect. When applying a new refrigerant type or carries out a novel element design such as valve or crankshaft, the operating characteristic become a useful reference.
The study debugs and improves the simulation software to meet conformity with the experiments provided by the Energy and Resources Laboratories (ERL) of Industrial Technology Research Institute (ITRI).
By using the software, it can be found that the main factor affecting the compressor efficiencies is the vibration of the valves. Besides, a good lifting of the suction and discharge valves brings a long valve working life and higher compressor efficiencies with good refrigerant condition.
CHAPTER 3
OPITMIZATION MODULE
By extending the simulation software described in the previous chapter, it is expected that an optimization module can construct and append to the software. The module uses the Borland C++ Builder accompanying with OOP (Object-Oriented Programming) concept to construct the Windows GUI (Graphical User Interface), which provides “What you see is what you get” interface that is friendly and simple operating [9]. Users could select various types of design conditions and proceed to simulate different optimization problems in the reciprocating hermetic compressor software. This chapter will make a detail statement about the optimization module.
3.1 Structure of Optimization Module
The section describes the full structure of the optimization module. First of all, the simulation software described in Chapter 2 is considered as an independent simulation module. The optimization module that constructs to link the simulation module, mainly comprises two sub-modules, user interface and optimization solver. The optimization module then must test and verify the linking feasibility between it and the simulation module, furthermore, check and modify errors for developing optimization. After proving the applicability, it is expected the optimization module may be constructed as an independent and easy separating module to apply the software and even be used for other simulation software.
Fig- 3.1.1 shows the flow chart to integrate the simulation and optimization modules making
use of the study. The flow chart mainly includes three parts: user interface, simulation module, and optimization solver. Users initially define and set optimization problems in the user interface sub-module, then make the related arguments as input data and pass it to the simulation module. The simulation module executes the analysis and produces the results called output data. The output data will be transferred to the optimization solver. Through various optimization methods in the optimization solver, new arguments can be generated as new input data, then pass back to the simulation module for next iteration. The details about the procedure presented as following sub-sections.
3.1.1 User Interface
Before commencing optimization, it is necessary to define and formulate the optimization problems. As presented in Fig- 3.1.1, the user interface sub-module help finish this stage. Users just choose what kinds and numbers of design variables, and set the cost function, constraints or bounds in the user interface. The sub-module is friendly and performs easily without coding the software themselves. Besides, it can reduce the some incompatible and time-consuming problems occur in a general manner linking between simulation and optimization. Fig- 3.1.2 to Fig- 3.1.5 shows the schemes of user interface about choosing and setting design variables, constraints, cost function, and involved parameters used in optimization (like convergent criteria). Furthermore, the user interface also can select different optimization method for various optimization problems in this reciprocating compressor simulation software.
3.1.2 Simulation Module
input data can be formed and then transfers to the simulation module. The simulation module is described in Chapter 2, but the prior simulation software provides only interactive form to set or adjust parameters. So the simulation software is recomposed as a batch type mode in this study. That is to say, the modified simulation software can be executed only imports one input data file [10], [11]. As soon as the software has finished, it generates an output data including many results such as efficiencies or performance in the reciprocating compressor.
The output data then sends to optimization solver to proceed with next stage.
3.1.3 Optimization Solver
Referring Fig- 3.1.1 again, from the optimization solver sub-module, the output data is checked whether satisfy the constraint conditions or limitations of defined optimization problem at first. Then it identifies coincidence with the convergent criteria or not. The sub-module procedure will stop if the output data meet the convergent criteria, otherwise using various optimization methods to solve the problem and form a new design variables group as current input data. The current input data then passes back to the simulation module to continue iterations until the final results satisfying the convergent conditions.
This optimization solver “Most” [12] is a multifunctional optimization system tool and provides many methods for different optimization problems. A Sequential Quadratic Programming (SQP) method is selected to deal with single objective optimization problems for continuous variables for accuracy, reliability and efficiency in the optimization module and described below:
SQP method uses Karush-Kuhn-Tucker (KKT) conditions as a basis and can divided three stages [13]. The first is using BFGS (Broyden-Fletcher-Goldfarb-Shanno) method to update the approximate second information for the Lagrange function (like Hessian matrix for keeping the Hessian approximation positive definite). After calculating Hessian of the
Lagrange function, QP(Quadratic Programming) sub-problem can be solved at each iteration to get the solution as a search direction for next stage. The third are the determination of step size and descent function calculation. Using the search direction generated before, a step size can be resolved to minimize the descent function where is defined by Pshenichy and Danilin [14].
Besides SQP method, the modified branch-and-bound algorithm which converts discontinuous design space into a continuous one by dropping discontinuous restrictions is used to solve discrete optimization problems [15]. For multi-objective optimization, the solver also provides Compromise Programming, Goal Programming and the Surrogate Worth Trade-off method for designer to find the best compromise solution [12], [16], [17].
3.1.4 Considerations
Consulting Fig- 3.1.1 and previous description, it is found that some advantages in the optimization module. These advantages state bellow:
1. If the design consideration of the optimization problem changes, the user interface can easily set and complete the formulation, moreover, establish required setting conditions in the problem for the convenience of developing simulation and optimization.
2. The optimization solver provides several algorithms for dealing with different types of optimization problems. The related algorithmic parameters used in different optimization methods are decided appropriately and automatically in the module.
3. Due to the optimization are separated distinctly, the optimization module can even utilize in other analysis simulation programs readily.
Fig- 3.1.1 Optimization procedure
Formulate:
(1). Design variables
(2). Cost function to be minimized (3). Constraints must be satisfied
Setting:
(1). Selecting variables
(2). Setting up cost function (3). Deciding restrained conditions 1. User Interface
Does the design satisfy convergence criteria?
Change the design using an optimization method (various optimum solvers)
No
Fig- 3.1.2 Design variables setting page
Fig- 3.1.3 Constraints setting page
Fig- 3.1.4 Cost function setting page
Fig- 3.1.5 Related parameters setting page
3.2 Design Variables Setting In User Interface
For an optimization problem, how to set and decide the parameters as design variables is a major consideration. It involves many requirements or conditions in various fields. For the hermitic reciprocating compressor, the simulation explained in Chapter 2 combines valve dynamic, thermal simulation, mechanism simulation, all have input parameters in respective parts. By user interface, all these parameters can be selected arbitrarily when formulating optimization problems. All the parameters from different parts are explained as following sub-sections, and the discussion about those in the optimization problems are illustrated in Chapter 4.
3.2.1 Numerical Simulation Parameters
The setting page for numerical simulation parameters using in the optimization module are presented in Fig- 3.2.1. Users can easily mark or cancel parameters they need in defined optimization problems. It includes such as refrigerant properties of inlet and outlet of the compressor, motor rotational speed, numerical step-size, etc. Table- 3.2.1 lists the 11 parameters and its default values applying in numerical simulation pages of the optimization module.
3.2.2 Slider-Crank Parameters Setting
The parameters of slider-crank mechanism are setting here and the related geometries and sizes are constructed in this page showed in Fig- 3.2.2. The decisions in this part may affect the behaviors in the slider-crank mechanism results. If users want to focus optimization
11 parameters can be defined or marked in this page and their initial values.
3.2.3 Eccentric Shaft and Mass Redistribution Setting
The Fig- 3.2.3 shows setting page about mass redistribution and shaft in user interface, it has 7 parameters including positions of the shifting counterweight. It is also a consideration if users have requirement for optimization problems concerning eccentric shaft distance or position, or the positions of the shifting counterweight. The 7 parameters and its default values in the setting pages are described in Table- 3.2.3.
3.2.4 Bearing Parameters Setting
The four bearing properties and dimensions and initial values are listed Table- 3.2.4 Influences in bearing properties and lubrication conditions are important factors for mechanical power loss. In the user interface, bearing properties can be chose and defined as design variables for optimization. Fig- 3.2.4 shows the setting pages for the four bearings between slider-crank and crankshaft.
3.2.5 The Control Volume Parameters Setting
In the previous simulation software of the reciprocating compressor it divides six control volumes for simulation. All related parameters are introduced in the optimization module. The user interface could make them as design variables in accordance with various optimization problems. Descriptions about individual control volume as follows:
Chambers size setting
As show in Fig- 3.2.5, this setting page defines the related size or thermal parameters of suction and discharge chamber. In addition, the cylinder parameters could be set in it. Table- 3.2.5 expresses the parameters and its initial value for simulation. Though the related parameters in chambers and cylinder are universal for industrial use, the optimization module also provides them the feasibility as design variables for developing optimization in the reciprocating compressor.
Mufflers parameters setting
The suction and discharge mufflers are dimensioned and selected in this setting page showed in Fig- 3.2.6. The parameters and its default values are listed in Table- 3.2.6.
Suction plenum and passages conditions parameters setting
The parameters about suction plenum and channels are displayed in Fig- 3.2.7. This setting page provides various parameters in suction plenum and passages conditions for users to formulate optimization problems. Table- 3.2.7 displays the parameters and its initial values.
3.2.6 Suction and Discharge Reed Valves Setting
The sizes, geometry, and physical properties of the suction and discharge valves are indicated and chose in this setting page. Due to [6], [7] and previous studies, it is known that the reed valve behaviors take a key role because of requirements for durability, stiffness, and operating life in the reciprocating compressor. Therefore, this study takes great efforts to proceeding design optimization of the valve characteristics and these results and discussions are illustrated in Chapter 4.
The suction and discharge reed valve setting page are showed in Fig- 3.2.8, Fig-
Fig- 3.2.1 Numerical simulation parameters page
Fig- 3.2.2 Slider-crank parameters page
Fig- 3.2.3 Eccentric shaft and mass redistribution page
Fig- 3.2.4 Bearing parameters setting page
Fig- 3.2.5 Chambers & cylinder parameters setting page
Fig- 3.2.6 Mufflers parameters setting page
Fig- 3.2.7 Suction plenum & passages setting page
Fig- 3.2.8 Suction valve setting page
Fig- 3.2.9 Discharge valve setting page
Default value Unit
Refrigerant properties
Pressure 120 kPa
Table- 3.2.1 Numerical simulation parameters list
slider-crank mechanism
parameters default Unit parameters default Unit R 0.007515 m piston weight 4.2648E-02 kg
shaft and mass redistribution parameters parameters default Unit
Dbb 0.045 m
lcsr 0.000814551 m4
Table- 3.2.3 Shaft and mass redistribution parameters list
bearing parameters Unit clearance ratio 700
slider-crank related bearing parameters
A length 0.014 m
A radius 0.008 m
B length 0.014 m
B radius 0.008 m
eccentric shaft related bearing parameters
C length 0.014 m
C radius 0.008 m
D length 0.014 m
D radius 0.008 m
Table- 3.2.4 Bearing parameters list
Table- 3.2.5 Chamber and cylinder parameters list
suction muffler discharge muffler
parameters default Unit parameters default Unit
volume 9.00E-05 m3 volume 9.00E-05 m3
Table- 3.2.6 The mufflers parameters list
suction chamber discharge chamber cylinder
parameters default Unit parameters default Unit parameters default Unit volume 4.53E-06 m3 volume 4.53E-06 m3 thickness 0.01 m
suction plenum passages areas
parameters default Unit parameters default Unit
volume 0.003145926 m3
suction plunem
thickness 0.002 m compressor inlet
section area 1.2566E-05 m2 thermal
conductivity coefficient
14.9 J/m2·℃ compressor outlet
section area 1.2566E-05 m2
Table- 3.2.7 Plenum and passages parameters list
suction valve discharge valve
parameters default Unit parameters default Unit
shape shape
center of circle 0.018 center of circle 0.018
0 0
simulating parameters
simulating parameters
damping ratio 0.75 damping ratio 0.75
density 7850 kg/m3 density 7850 kg/m3
Young's module 2.10E+11 Pa Young's module 2.10E+11 Pa
physical parameters physical
parameters
radius 0.0028 m reducing orifice
radius 0.0028 m
valve passage
diameter 0.004 m valve passage
diameter 0.004 m
Limiting lift 0.002 m limiting lift 0.001 m
3.3 Cost Function and Constraints Setting
In consideration of optimization problems, the construction of the mathematic formulation is a key point for solving the problem well. Correct formulation can express the design problem accurately. For this reason, the formulation should exactly choose design parameters as variables, cost function and constraints, furthermore, combine those to image the substance of the formulation in the optimization problem. Section 3.2 has explained the parameters decision, the cost function and constraints setting constructed in user interface will interpret in this section.
3.3.1 Cost Function Setting
The final results (performances or efficiencies) show in the reciprocating compressor simulation software will be stated as the selections of the cost or objective function in optimization problems. The related figure, Fig- 3.3.1 shows the wasted work of various parts of the reciprocating compressor system from simulation, and refrigerant mass flow rate, E.E.R.
other performance indices. The users can select different kinds of results or indices as cost function in user interface.
The statement of the performance indices are illustrated as below:
1. Volumetric efficiency:
[realflowrate idealflowrate] (%) 2. Compressor efficiency:
[idealcompression work realcompression work] (%) 3. Mechanical Efficiency:
[realcompression work totalmechanical work] (%) 4. Motor efficiency:
[bearingoutput work motor output work] (%) 5. Flow rate of refrigerant:
[volumetricefficiency×idealmaximumflowrate] (kg/hr) 6. Capacity of refrigeration:
[
flowrateof refrigerant × enthalpy difference]
(kcal/hr) 7. Energy efficiency ratio(EER):[capacity of refrigeration motor input work]
The users initially plan to decide what the cost function for its problem is and then selects it in the user interface module to do the consequent simulation and optimization. At present, the optimization module adopts the single objective function to reduce the complexity of the problem, and the module also can accomplish discrete and the multi-objective functions optimization problem.
Fig- 3.3.1 Cost function selecting page
3.3.2 Constraints Setting
Design constraints are used to limit the range of design variables and can divide into explicit and implicit constraints, boundary and characteristic constraints, and equal or unequal constraints. In the user interface module, when deciding the design variables for an optimization problem, the users also can set their boundary constraints (lower bound and upper bound) at the same time. As show in Fig- 3.3.2, an example, the numerical simulation variables setting page can select the design variables and identify its ranges.
The constraint conditions just set the boundary constraints in the optimization module presently, but in fact the accurate problem formulation should consider many other kinds of constraints like characteristic or implicit constraints to get the highly believable results in simulation and optimization. Thus, how to define the related constraints from various design variables in an optimization problem is an important issue to discuss. The statements for this are presented in the next Chapter.
Fig- 3.3.2 Constraints selecting parameters list
CHAPTER 4
OPTIMIZATION OF THE REED VALVES
From the description in previous Chapters, it is found that the valve characteristics play the critical role for efficiencies and performance of the hermetic reciprocating compressors.
So these characteristics can be used to produce some design problems for optimization. This Chapter will take detailed statement for results and discussions about the optimization problems.
4.1 Optimization Problem for Suction-valve
The paper [18] considers the affections of E.E.R and refrigerant mass flow rate for different suction-valve thickness. The reason for choosing suction-valve thickness as a variable is that there are actual sizes can carry out experiments for verification. So in the section the parameters that can be modified in practical manufacture to do experiments are initially selected as design variables in the optimization problem.
4.1.1 Formulation of Problem
The mathematic model of the suction-valve constructed in the simulated software applying the assumed method and deriving by Lagrangian approach. Chapter 2 has a simple introduction. The details could refer the Appendix [A] .
The physical characteristics in the suction-valve are considered in the problem. The reasons for selecting these design variables, constraint conditions and cost function are
Design variables:
1. valve thickness (tvalve)
Due to the mathematic model, it is known that different valve-thickness will affect the cross-section and moment of inertia of the valve, more over to mass, stiffness. So if too stiff the valve plate causes over-compression and delay of closing. However the inessential fluctuation of the valve plate produced because of too flexible valve [7].
Besides, the valve thickness is considered as a design variable for the problem.
2. diameter of valve passage (ds)
The influence of effective force area and effective flow area of valve is upon the change for diameter of valve passage. However, the experimental coefficient β using for correcting pushing force (Appendix [A] ) is also affected by diameter of valve passage. So it is also a design variable in the problem.
3. distance of valve passage (l1)
Let reviewing Fig- 2.3.1 again, the l1 is defined for the parameter. It represents the position that the center of the diameter of passage located to and due to symmetry of valve, the length l1 is considering only one direction.
Therefore, these three variables are highly nonlinear and have coupled influence to the pushing force, and respectively have effect for other calculations. So how to decide the sizes from their relations to promote the efficiency is the target of the optimization problem.
Constraints:
1. compression efficiency
The compression efficiency changes with the effective force area of valve [19]. It is expected to arise from the increasing effective force area to get the better efficiency, and
forms a constraint in the problem.
2. volume efficiency
If the designer hopes to obtain high volume efficiency, he may limit the size and number of valves, however, may tend to lower area and decrease the compression efficiency [19]. As a general rule, high volume efficiency and high compression efficiency (low power requirement) do not go together. It must get a compromise between the two constraints.
3. capacity of refrigeration
This result depends on the testing conditions of the import and export refrigerant.
The experimental results show that the capacity of refrigeration approximates to 190
The experimental results show that the capacity of refrigeration approximates to 190