CHAPTER 4 OPTIMIZATION OF THE REED VALVES
4.3 Synthetic Optimization of Suction and Discharge-valves
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
[1]
Huang, J. S., “Simulation of Reciprocating Compressor Performance,” Master Thesis, National Chiao Tung University, in Chinese, 2001.[2] 簡聰海， “數值分析使用 C 語言，” 全華科技圖書股份有限公司， 二版一刷， 2002。
[3]
Hsiung, M. D., “Integration of the Reciprocating Compressor Performance Simulation,”Master Thesis, National Chiao Tung University, 2002.
[4]
Gatecliff, G. W., “Analytic Analysis of the Forced Vibration of the Sprung Mass of a Reciprocating Hermetic Compressor,” Proceedings of the 1972 International Compressor Engineering Conference at Purdue, 1972.[5] Marriott, L. W., “Motion of the Sprung Mass and Housing of a Reciprocating Hermetic Compressor during Starting and stopping,” Proceedings of the 2000 International Compressor Engineering Conference at Purdue, pp. 823-830, 2000.
[6] Ooi, K. T., Chai, G. B., and Kwek, E. C., “A Simple Valve Model to Study the Performance of a Small Compressor,” Proceedings of the 1992 International Compressor Engineering Conference at Purdue, p. 147, 1992.
[7] Cheng, S. H., Hu, Y. Z. Robert., “Nonlinear Vibration Analysis of Reed Valves,”
Proceedings of the 2000 International Compressor Engineering Conference at Purdue, pp. 437-441, 2000.
[8] Leonard, M., “Computational Methods in Structure Dynamics,” Sijthoff & Noordhoff, International Publishers, Alphen ann den Rijn, The Netherlands, 1980.
[9] 李勁， “精通 C++ Builder 6，” 文魁資訊股份有限公司， 初版二刷， 2003。
[10] 陳正凱， “C++ 函式庫精華錄，” 金禾資訊股份有限公司， 初版五刷， 2004。
[11] 古頤榛， “C++ 全方位學習，” 碁峰資訊股份有限公司， 2002。
[12] Tseng, C. H., “Most 1.1” and “Most 1.1 Manual,” Technical Report, National Chiao Tung University, January 1996.
[13] Arora, J. S., “Introduction to Optimization Design,” Elsevier Academic Press, 2004.
[14] Psheenichny, B. N., Danilin, Y. M., “Numerical Methods in Extremal Problems (2nd ed.),” Moscow: Mir Publishers, 1982.
[15] Tseng, C. H., Wang, L. W. and Ling, S. F., “Enhancing Branch and Bound Method For Structural Optimization,” ASCE, Journal of Structural Engineering, 121, 5, pp. 831-837, 1995.
[16] Rao, S. S., Venkayya, V. B. and Khot, N. S., “Optimization of Actively Controlled Structures Using Goal Programming Techniques,” Int. J. for Num. Methods in Engrg, 26, pp. 183-197, 1988.
[17] Tseng, C. H., Lu, T. W. and Wang, L. W., “Multiobjective Optimization with Non-continuous Design Variables,” Journal of CSME, 13, 6, pp. 547-560, 1992.
[18] Yu, P. Y., Hsiao, T. L., Cheng, Y. C., and Chang, Y. C., “Performance Estimation of Hermetic Reciprocating Compressor with Computer Model,” International Compressor Engineering Conference at Purdue, C021, pp. 3 -7, July 2004.
[19] Bloch, H. P., “A Practical Guide to Compressor Technology,” McGraw-Hill, Inc., New York, 1995.
Appendix [A]
Symbols of the reed valve analysis:
)
C2 The equation of the secondary valve line segment E Young’s modulus of the valve
f(t) Generalized force l Length of the valve
l1 Distance of the applied force related to the coordinate I(x) moment of inertia of the valve cross-sectional
r s Radius of the suction orifice ps Pressure in the suction chamber p Pressure in the cylinder
q(t) Time-dependent generalized coordinate
valve
t Thickness of the valve t)
(x,
u Z-direction displacement of the valve w n The natural frequency
Mathematical Formulation of Reed Valve [1]
The Lagrangian approach is used to derive the governing equation of motion for a cantilever type valve. The Kinetic energy (T), potential energy (U), and the work (W), of external load could be obtained with the application of the Lagrangian’s equation.
The assumed-modes method [2] is used to simplify the governing equations of the reed valve vibration. Fig-A.1 shows the valve displacement relation. Since the natural boundary conditions will be automatically accounted in the kinetic and potential energy, its conditions are not particularly considered here. The valve displacement, u(x, t), are as following:
q(t)
φ(x) : Valve admissible function (or shape function) that can be obtained form the free vibration analysis. The first mode consideration is sufficient for the function. The admissible function [2] satisfied the boundary conditions depicted as: q(t) : The time-dependent generalized coordinate
Fig-A.1 Suction valve displacement, u(x, t) t)
the extern load, are expressed as following: Then, applying the Lagrange’s equation in [2], W
q
( & , substituting
the expression for the work of the extern load, the Kinetic energy, and potential energy into the energy expression, and adding the damping effect yields the governing equation of the valve vibration below. The pushing force coefficient, β [12][13], shown in Fig-A.2 is obtained from experiment and is related with the valve displacement, u(x, t). Now, a damping effect, ξ is added to the equation (A-6), it becomes:
# The valve motion equation above is only suitable before the valve impact the seat. The valve velocity while vibration is affected by the collision and the velocity after impacting the
seat can be simply obtained by multiplying a reflecting factor, Cr. The seat is considered as a rigid body and the stiffness is much higher than the valve. Therefore, the fully valve motion equation can be derived by the equation (A-11) and the following equation.
q C
q&after =− r& (A-15)
Fig-A.2 pushing force coefficient β
h: valve lift
r: diameter of passage
Reference
[1] Cheng, S. H., Hu, Y. Z. Robert., “Nonlinear Vibration Analysis of Reed Valves,”
Proceedings of the 2000 International Compressor Engineering Conference at Purdue, p.437-441, 2000.
[2] Leonard, M., “Computational Methods in Structure Dynamics, “Sijthoff & Noordhoff, Alphen aan den Rijn,” The Netherlands, 1980.
[3] Ferreira, R. T. S., Driessen, J. L., “Analysis of the Influence of Valve Geometric Parameters on the Effective Flow and Force Areas,” Proceedings of the 1986 international compressor engineering conference at Purdue, p.632-640, 1986.
[4] Deschamps, C. J., Ferreria, R. T. S., and Prata, A.T., “The Effective Flow and Force Areas in Compressor Valve,” Proceedings of the 1988 international compressor engineering conference at Purdue, 1988.