CHAPTER 3 OPITMIZATION MODULE
3.3 Cost Function and Constraints Setting
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 (kcal/hr) at the same initial conditions. It is expected to get the result more to promote cooling ability. So this result is also a considering constraint in the problem.
Cost function
There are many results can be the cost function. The E.E.R is a usual index for determining the whole efficiency of the reciprocating compressor. It defines capacity of refrigeration dividing motor input power. The motor stator reaction torque is generated by Motor-Speed-Torque curves of induction motors that show the average torque. The higher E.E.R generally represents the better performance. Due to reducing the complication of optimization, the single objective optimization is used in the problem and the cost function is maximization of E.E.R.
The completed definitions, formulation and setting for the design variables, constraints, cost function are listed in the Table- 4.1.1. The optimum results can be acquired after executing the optimization model stated in Chapter 3. The results and discussions show in the
4.1.2 Results and Discussions
Table-4.1.2 shows the results from the optimization problem. It is found that the suction-valve thickness decreases to lower bound, the diameter of passage and the distance of passage increase to the upper bound could get maximum E.E.R. The decreasing suction-valve thickness for better E.E.R is proved by experiments [18] if considering only this one variable.
In the problem the thinner valve thickness still has better result. The other two design variables are proved when the design bounds are limited in the problem, the large diameter of the passage and distance of diameter arise the cost function E.E.R. The trends for those are reasonable because when increasing these, the pushing force could lift up the valve more for importing more refrigerant.
One constraint could not be satisfied in the problem, the compression efficiency. It illustrates that the needed solution could not be found in the design space for the problem. The problem runs 11 iterations and the related history for these iterations are showed in Fig- 4.1.1 to Fig- 4.1.4. It is known that the designs satisfied the constraints except for the compression efficiency and get the maximum E.E.R. Fig- 4.1.5 shows the constraints sensitivity to design variables in the problem. It can observe the thickness of suction valve influence the volumetric efficiency and capacity of refrigeration more. The distance of passage has greater sensitive effect to compression efficiency. The results are the consultation to manufacture and how to select changes to get best effects without increasing difficulties in practice is depend on experiences and cost to manufacture.
design
variables No. name unit initial
value lower bound upper bound
1 valve thickness m 0.0002 0.0001 0.0004
2 diameter of
valve passage m 0.0056 0.00504 0.00616
3 distance of valve
passage m 0.018 0.015 0.019
constraints No. name initial
value comparison condition
1 volumetric eff. % 65.3 ≧ 70
2 compression eff. % 68.49 ≧ 75
3 capacity of
refrigeration kcal/hr 194.6221 ≧ 200
cost function No. name initial
value requirement
1 EER kcal/hr．
W 0.9395 max
Table- 4.1.1 Formulation of the suction-valve problem
design variables constraints cost function
name unit results name unit results name unit results valve
thickness m 0.0001 volumetric
eff. % 77.07 EER kcal/hr．
W 0.9906 diameter of
valve passage
m 0.00616 compression
eff. % 72.13
Table- 4.1.2 Results of the suction valve optimization
Fig- 4.1.1 History of design variables for the suction-valve problem
Fig- 4.1.2 History of constraints for the suction-valve problem
Fig- 4.1.3 History of cost function for the suction-valve problem
distance of passage
Constraints normalized sensitivity to design variables
volumetric eff. -5.26E-03 -1.00E+00 1.62E-02
compression eff. 1.00E+00 0.00E+00 0.00E+00
capacity of refrigerant 0.00E+00 -1.00E+00 0.00E+00
distance of passage thickness diameter of passage
Fig- 4.1.5 Constraints normalized sensitivity to design variables
4.2 Optimization Problem for Discharge-valve
After the problem stated in 4.1, the characteristics of discharge-valve in the simulation software is another selection to improve the efficiency and performance. Because of higher difference between the cylinder and discharge chamber and shorter time for discharge step, how to design the suitable dimensions and sizes to get benefit is important.
4.2.1 Formulation of Problem
Design variables:
The three design variables, valve thickness, diameter of passage, distance of valve passage for the discharge-valve are using in the optimization problem. The bounds conditions are set by considering the practice in manufacture and consulting with ERL members.
Constraints and cost functions
The volume efficiency, compression efficiency, and capability of refrigeration are also considered as constraints in the problem. The E.E.R is also chosen as the cost function for maximization in the problem.
The definitions, formulation and setting for the design variables, constraints, cost function are listed in the Table- 4.2.1. The results of the optimization problem and discussions show in the next sub-section.
4.2.2 Results and Discussions
Table- 4.2.2 shows the results from the optimization problem. It is found that the
passage and distance of passage increase to the upper bound could get maximum E.E.R. In the problem the thinner valve thickness still has better result. The other two design variables are proved when the design bounds are limited in the problem, the large diameter of the passage and distance of passage arise the cost function E.E.R.
The E.E.R. seems can get better results comparing to the previous problem (1.0975 >
0.9906). However, volumetric efficiency and compression efficiency seems have different trends comparing to previous problem. The new design variables group in previous problem can inhale more refrigerant to get higher capacity of refrigeration (230.6542 kcal/hr·W), so the volumetric efficiency is higher (77.07%). However, the compression efficiency is just 72.13%
and causing the lower E.E.R value (0.9903). Nevertheless, in this problem the capacity of refrigerant is just 220.46 kcal/hr·W, but can get higher E.E.R value (1.0975) due to its higher compression efficiency (80.58%). These conditions can also verified that a compromise between volumetric efficiency and compression efficiency.
All constraints could be satisfied in the problem. The problem also runs 6 iterations and the related history for these iterations are showed in Fig- 4.2.1 to Fig- 4.2.4. It is known that the designs after iteration 6 are feasible and designs before it had some violation of constraints.
The normalized sensitivity for 3 design variables to cost function shows in Fig- 4.2.5, found that the diameter of the passage is more sensitivity to E.E.R in this problem. It also can be a reference when deciding modification of dimension of the suction-valve by the designer.
Fig- 4.2.6 shows the normalized constraints sensitivity to design variables. It is found that the diameter of discharge-valve is more sensitive to three constraints, the second is thickness of discharge-valve, the distance of distance of discharge-valve affects least to constraints in the problem. It could be a reference for inaccuracy of manufacture in practical.
design
valve passage m 0.0045 0.00405 0.00495
3 distance of valve
passage m 0.018 0.015 0.02
constraints No. name initial
value comparison condition
1 volumetric eff. % 65.3 ≧ 70
2 compression eff. % 68.49 ≧ 75
3 capacity of
refrigeration kcal/hr 194.6221 ≧ 200 cost
function No. name initial
value requirement
1 EER kcal/hr．
W 0.9395 max
Table- 4.2.1 Formulation of the discharge-valve problem
design variables constraints cost fucntion
name unit results name unit results name unit results valve
thickness m 0.0001 volumetric eff. % 73.66 EER kcal/hr．W 1.0975 diameter of
Table- 4.2.2 Results of the discharge valve optimization
Fig- 4.2.1 History of design variables for the discharge-valve problem
Fig- 4.2.2 History of constraints for the discharge-valve problem
Fig- 4.2.3 History of cost function for the discharge-valve problem
Fig- 4.2.4 History of constraint violation and convergence parameter
Fig- 4.2.5 Cost function normalized sensitivity to design variables
Constraints normalized sensitivity to design variables
volumetric eff.
compression eff.
capacity of refrigerant
volumetric eff. 1.46E-03 6.95E-01 7.19E-01
compression eff. 0.00E+00 -4.26E-01 -9.05E-01
capacity of refrigerant 0.00E+00 0.00E+00 1.00E+00
distance of passage thickness diameter of passage
Fig- 4.2.6 Constraints normalized sensitivity to design variables
4.3 Synthetic Optimization of Suction and Discharge-valves
After discussing the suction-valve and discharge-valve optimizations respectively, It is expected that considering both two valves at the same time, what’s the relations and meaning of the results. The integrated optimization for these two components is developed in this section.
4.3.1 Formulation of Synthetic Optimization Problem
Design variables:
The design variables defined in the problem include the six parameters that discussing in the previous two optimization problems. In the mean time, all their bounds are identical using in it.
Constraints and Cost function:
In order to comparing the previous results in section 4.1 and section 4.2, the constraint conditions and cost function are use the same setting from the previous two optimization problems.
The detail definitions, formulations and related setting conditions of this problem are listed in the Table- 4.3.1.
4.3.2 Results and Discussions
Table- 4.3.2 shows the optimum results. The distances of suction-valve and
thickness of suction-valve and discharge-valve decrease and the diameter of discharge-valve passage increase slightly can meet the convergent requirement, the E.E.R is 1.0165. It is deduced that this design variables group is a local maximum result for the cost function.
Besides, all the constraint conditions are satisfied in the problem. So the design variables group is a feasible solution, Fig- 4.3.1 to Fig- 4.3.4 show some other history of the related arguments in the problem.
However, it may exists other local maximum results for the problem. Here, trying to take others as initial design variables group for the problem. After testing several times the best results shows in Table- 4.3.3. It is found that the capacity of refrigeration increase enormously and the volumetric efficiency and compression efficiency are also rise more. Besides, the E.E.R value promotes to 1.2448 and the design variables group could be a good reference if the design results can meet the manufacturing limit.
design
thickness m 0.0002 0.0001 0.0004
2
diameter of suction-valve
passage
m 0.0056 0.00504 0.00616
3
distance of suction-valve
passage
m 0.018 0.015 0.19
4 discharge-valve
thickness m 0.0002 0.0001 0.0003
5
diameter of discharge-valve
passage
m 0.0045 0.00405 0.00495
6
distance of discharge-valve
passage
m 0.018 0.015 0.02
constraints No. name initial
value comparison condition
1 volumetric eff. % 65.3 ≧ 70
2 compression eff. % 68.49 ≧ 75
3 capacity of
refrigeration kcal/hr 194.6221 ≧ 200 cost
fucntion No. name initial
value requirement
1 EER kcal/hr．
W 0.9395 max
Table- 4.3.1 Formulation of synthetic problem
design variables constraints cost fucntion name unit results name unit results name unit results
suction-valve
thickness m 0.00025 volumetric
eff. % 72.47 EER kcal/hr
．W 1.0165
diameter of suction-valve
passage
m 0.0056 compression
eff. % 73.60
thickness m 0.0002
diameter of
Table- 4.3.2 Results of the synthetic optimization
design variables constraints cost fucntion name unit results name unit results name unit results
suction-valve
thickness m 0.0001 volumetric
eff. % 94.02 EER kcal/hr．
W 1.2448
diameter of suction-valve
passage
m 0.00616 compression
eff. % 90.55
thickness m 0.0001
diameter of
Table- 4.3.3 Results of the best synthetic optimization
Fig- 4.3.1 History of design variables for the synthetic optimization problem
volumetric efficiency
compression efficiency
capacity of refrigeration
Fig- 4.3.2 History of constraints for synthetic optimization problem
Fig- 4.3.3 History of cost function for the synthetic optimization problem
Fig- 4.3.4 History of constraint violation and convergence parameter
CHAPTER 5
CONCLUSIONS AND FURTHER WORKS
5.1 Conclusions
Debugging and improvement of the comprehensive performance simulation software, which the operating condition is below 500 W, developed before is carried out in the study. It has conformity with the experiments provided by the Energy and Resources Laboratories (ERL) of Industrial Technology Research Institute (ITRI).
The optimization module is constructed and developed in the simulation software. It is constructed by using Borland C++ Builder, and based on objective-oriented programming (OOP) concept of programming design with window-based GUI interface. It helps to reduce times for doing experiments of the actual reciprocating compressor are complicated and time-consuming. The conclusions of the thesis can be summarized as bellow:
1. The simulation software of the hermetic reciprocating compressor combines thermal dynamic, mechanism, valve dynamic, bearing analysis are improved, and recomposed for developing design optimization.
2. The optimization module constructed here includes two sub-modules, user interface and optimization solver. The user interface provides very friendly and easily operating interactive interface to help end user formulate and set up different optimization problems.
When the problems change, the user can easily transfer it including cost function, design variables, and constraint conditions by the user interface and proceed to analyze and develop design optimization of the reciprocating compressor. Besides, it also contains
3. The optimization solver is applied in the module. The solver automatically provides suitable algorithmic parameters to match up its methods, help to lower complication for using it by the user. It also provides functions to deal with continuous, discrete and multi-objective optimization problems.
4. By the optimization module, end-user can handle a great quantity of optimization problems regardless of numbers or types of design variables, constraints and cost function.
5. The optimal results are provided in the study for discussing affections of suction-valve and discharge-valve to energy efficiency ration, E.E.R. It could be a consultation and provides a tendency to improve efficiency when proceeding actual experiments.
5.2 Further Works
By integrating the simulation software greatly to predict the behavior and performances of the hermetic reciprocating compressor, and linking the optimization module to the software, the designer can develop and study it more conveniently and quickly to save the manpower and cost. There are several considerations and issues should study and discuss in the future as follows:
(1). Verification of the optimization results with experiments:
The optimization results in the study should be verified by experiments to prove the feasibility and reliability.
(2). Continuing developing design optimization:
The study just discuss affections of some valve properties to efficiency of the reciprocating compressor in the software, the other design considerations, like material properties of valve, cylinder characteristics and so on, possibly have influence for simulation, should study from now on. Besides, the discrete and multi-objective optimization problems also can be investigated in later studies.
(3). Different refrigerant type
The software uses the HFC-134a as the refrigerant. With the trend of using natural refrigerants in the compressors, whether the simulation software can introduce other refrigerant such as CO2, or introduce one without causing greenhouse effect such as R-600a, forms an issue in the further research.
(4). Debugging the software
The software uses many experimental formulas. Once the dimension or size changes, the formulas are suitable for use or not are very important, must correct and prove its applicability.
References
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[4]
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