基于变径基圆渐开线涡旋压缩机的几何模型及优化研究
Geometrical Model and Optimization of Scroll Compressor Based on Involute of Circle with Variable Radii
责任编辑: 王一凡
收稿日期: 2022-04-27 修回日期: 2022-06-6 接受日期: 2022-06-17
基金资助: |
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Received: 2022-04-27 Revised: 2022-06-6 Accepted: 2022-06-17
作者简介 About authors
彭斌(1976-),教授,博士生导师,从事涡旋机械、低温余热发电技术研究;E-mail:
针对现有变径基圆涡旋压缩机几何模型的定量分析及优化研究不够完善的问题,给出了变径基圆涡旋压缩机从吸气过程开始直到排气过程结束的完整工作腔容积变化规律.比较了格林定理法、图解法及法向等距法的容积计算差异性,分析了产生计算误差的原因,验证了几何模型的正确性.推导了齿面积利用率的计算公式,构建了行程容积及齿面积利用率的灵敏度模型,得出基圆半径变化系数及内侧型线发生角的影响尤为显著的结论.建立了一种考虑非线性约束且以行程容积与齿面积利用率为优化目标的多目标优化分析模型.结果表明:多目标遗传算法(NSGA-II)比多目标粒子群(MOPSO)算法综合性能更佳,且相较于优化前,某款汽车空调用变径基圆涡旋压缩机的行程容积提升了6.5%,齿面积利用率降低了4.27%.该研究能为变径基圆涡旋压缩机的优化设计及定量研究提供进一步的理论参考.
关键词:
Considering the problems that quantitative analysis and optimization of the geometrical model of scroll compressor created from involute of circle with variable radii are incomplete, the changing trend of the working chamber volume from the beginning of the suction process to the end of the discharge process is given. The volume calculation differences of Green’s theorem method, the graphical method, and the normal isometric method are compared, the reasons for calculation errors are analyzed, and the correctness of the geometrical model is verified. The calculation formula of teeth area utilization rate is deduced, and the sensitivity models of suction volume and teeth area utilization rate are constructed. It is concluded that the influence of the base circle radius variation coefficient and the occurrence angle of the inner profile are particularly significant. A multi-objective optimization analysis model considering nonlinear constraints and taking the suction volume and teeth area utilization rate as the optimization goals is established. The results show that the multi-objective genetic algorithm NSGA-II has a better comprehensive performance than the multi-objective particle swarm algorithm MOPSO and the suction volume of the scroll compressor created from the involute of circle with variable radii for an automobile air conditioner has been increased by 6.5% and the teeth area utilization rate has been reduced by 4.27%. This paper can provide further theoretical reference for the optimal design and quantitative research of the scroll compressor constructed from involute of circle with variable radii.
Keywords:
本文引用格式
彭斌, 刘慧鑫, 陶耀辉.
PENG Bin, LIU Huixin, TAO Yaohui.
以上对涡旋压缩机的研究主要集中在等壁厚圆渐开线涡旋类型且关于变径基圆涡旋压缩机几何模型的研究,存在着模型不完善、不精确等问题.同时对变径基圆涡旋压缩机的优化研究考虑的约束较少[13].因此,本文在完善变径基圆涡旋压缩机几何模型的基础上,尝试构建一种考虑约束条件的多目标优化分析模型,为变径基圆涡旋压缩机的优化研究及定量分析提供进一步的理论参考.
1 变径基圆涡旋压缩机的几何模型
1.1 变径基圆涡旋压缩机涡旋齿的几何模型
表1 基本几何结构参数
Tab.1
参数 | 符号 | 取值 |
---|---|---|
初始基圆半径 | a0/mm | 3.675 |
基圆半径变化系数 | δ0/(mm·rad-1) | -0.05 |
离散的多方指数 | δ1 | 1 |
回转半径 | Rob/mm | 4.361 7 |
内侧型线发生角 | αin/rad | 1.430 44 |
外侧型线发生角 | αou/rad | 0 |
内外侧型线展角 | φin, φou/rad | — |
修正展角 | Φ/rad | 0.6π |
终端渐开线展角 | φe/rad | 6π |
动涡旋齿外侧型线方程:
式中:
动涡旋齿内侧型线方程:
图1
图1
动静涡旋齿几何模型
Fig.1
Schematic diagram of geometrical model of orbiting and fixed scroll teeth
图2
图2
双圆弧齿头修正原理及齿头面积计算模型
Fig.2
Modification principle and teeth head area calculation model of double-arc modification
修正齿轴向投影面积为
式中:
1.2 变径基圆涡旋压缩机3种工作腔容积求解方法比较分析
式中: xf,in, yf,in为静涡旋齿内侧型线方程.
当φou=φe时有:
因此,单个吸气腔轴向投影面积计算公式为
压缩腔投影面积随主轴转角θ的变化规律为
图3
图3
不同容积求解方法计算结果对比
Fig.3
Comparison of calculation results of different working chamber volume solving methods
由图3可见,利用3种方法求解的压缩腔轴向投影面积变化规律高度一致.对3个不同转角位置(如图3中1, 2, 3)所示的曲线放大观察,格林定理法与图解法的误差较小而法向等距与其两者的误差相对较大,在这3个位置处格林定理法与图解法的误差分别为 0.036%、0.055%及 0.042 3%,主要原因是作图时的测量误差;法向等距法与图解法的误差分别为0.106%、0.128%及 0.078 4%.两者产生差异的原因是格林定理法的计算过程是直接利用生成变径基圆涡旋齿的型线参数方程进行代入求解,因此,误差相对较小;而法向等距法进行了几何等效处理,在平移组成封闭工作腔的内外型线时会造成微小的面积偏移误差,格林定理法虽然精度较高,但由于组成封闭工作腔的型线类型在不断变化,对于吸气过程及后续压缩过程面积的推导极其抽象复杂[17].相反,法向等距法由于其简单易懂且精度满足要求,广泛应用于变壁厚涡旋压缩机的几何模型快速建立中.
表2 单腔轴向投影面积
Tab.2
1.3 完整工作腔容积的变化规律
文献[15]中利用法向等距法给出了变径基圆涡旋压缩机压缩腔及排气腔容积的计算公式,对相应公式进行坐标变换及利用法向等距法推导出吸气腔的容积V0,计算公式如下.
当0≤θ≤2π时,
式中:H为齿高;L0为基线长,
行程容积为
式中:V0(2π)为θ=2π时V0的函数值.
理论内压缩比为
式中:
图4
图4
工作腔容积及其变化率变化规律
Fig.4
Evolution and derivative of working chamber volume with orbiting angle
1.4 齿面积利用率计算模型
图5
动、静涡旋齿啮合所需最小盘径[3]:
动、静涡旋齿面积:
式中:Sm为齿头采用双圆弧修正的修正齿轴向投影面积.
因此,齿面积利用率为
不同初始基圆半径a0对应的齿面积利用率随基圆半径变化系数δ0的变化如图5(b)所示.从图中可知,随着
2 型线参数对齿面积利用率及行程容积的影响
为了研究各型线参数对变径基圆涡旋压缩机几何性能的影响程度大小,为后续的优化分析确定最佳的决策变量,定义行程容积与齿面积利用率的灵敏度模型为
式中:
图6
3 多目标优化分析模型的建立与求解
基于建立的变径基圆涡旋压缩机几何模型,对某款汽车空调用变基圆半径渐开线涡旋压缩机[18]几何结构参数及性能进行优化分析,比较不同优化算法求解性能指标的差异.
3.1 目标函数及决策变量的确定
从上述分析知,
3.2 约束条件的确定
(1) 变径基圆涡旋压缩机常用于汽车空调等领域[11].因此,动静涡旋齿的最小啮合盘径定为
(2) 采用法向等距法生成涡旋齿,回转半径大小会影响齿厚大小及内外型线位置.因此,将其定为
(3) 为了使得优化分析有意义,须保证优化分析之后的内压缩比不小于优化前的内压缩比.因此,将其定为
(4) 考虑到变径基圆涡旋压缩机在φe处涡旋齿的强度及刚度,将该处齿厚t(φe)定为
(5) 为了使得涡旋压缩机的壁厚变化规律适应工质在工作腔内压力、温度向中心腔靠近逐渐增大的变化规律,将基圆半径变化系数定为
3.3 求解结果及比较分析
多目标优化算法与传统的线性加权法相比,能够把可供决策者选取的多组解以非劣解集的形式可视化,更具优势.多目标遗传算法(NSGA-II)及多目标粒子群(MOPSO)算法是在传统遗传算法及粒子群算法的基础上,加入了非支配排序算子,超体积(HV)、反世代距离(IGD)、计算时间(T)能够衡量算法的优劣,HV值越大,IGD值越小,时间越短,算法越优秀.表3所示为以上两种算法求解标准测试函数BT1的相关性能指标对比.从表3可知,在相同种群数量及迭代次数的条件下,NSGA-II在计算时间、HV及IGD指标均优于MOPSO.图7所示为多目标优化算法NSGA-II及MOPSO求解标准测试函数BT1(f1, f2为其两个目标函数)的结果.由图可见,MOPSO求解[22]获得的pareto前沿均匀性及准确性均差于NSGA-II对应的求解结果.因此,对于测试函数BT1, NSGA-II更具优势.图8(a)所示为只考虑δ0,αin上下边界约束条件的求解结果,由图可知,对于较少的决策变量,MOPSO与NSGA-II能解得差异极小的非劣解集,但MOPSO对应解的分布均匀性弱于NSGA-II, 且计算时间为12.70 s,多于11.11 s.图8(b)为考虑非线性约束条件的求解结果,其计算时间为25.56 s,虽计算效率弱于图8(a)中的结果,但求解的非劣解集更精确,范围更小,决策者选取优秀的解也更容易.从图8(b)中选取一组非劣解及其对应的型线参数与优化前作对比,如表4所示.
图7
图8
表3 性能指标对比
Tab.3
算法名称 | 种群大小 | 迭代次数 | T/s | HV | IGD |
---|---|---|---|---|---|
NSGA-II | 200 | 800 | 4.81 | 0.712 4 | 0.008 032 |
MOPSO | 200 | 800 | 5.19 | 0 | 3.888 300 |
由表4可知,优化分析之后的优化型在满足内压比的条件下,较优化前,行程容积增大了6.95%,齿面积利用率下降了4.26%,证明了本优化分析模型的实用性.同时,在本优化分析模型及测试函数上,NSGA-II综合性能均优于MOPSO.
表4 优化型与优化前几何性能对比
Tab.4
名称 | δ0 | αin/rad | Vs×105/m3 | ζ | ε |
---|---|---|---|---|---|
优化前 | -0.020 0 | 1.317 8 | 2.316 | 0.276 7 | 4.315 6 |
优化型 | -0.014 3 | 1.222 0 | 2.477 | 0.264 9 | 4.343 6 |
4 结语
完善了变径基圆涡旋压缩机的几何模型,给出了其完整工作过程的容积变化规律,讨论了格林定理法、图解法及法向等距法计算结果的差异性,3种方法的计算结果误差小,均可用于变径基圆涡旋压缩机几何模型的求解,但法向等距法更简单易懂.建立了变径基圆涡旋压缩机齿面积利用率的计算模型,分析了各几何结构参数对行程容积及齿面积利用率的影响程度,发现基圆半径变化系数及内侧型线发生角的影响尤为显著.构建了一种以Vs与ζ为优化目标的多目标优化分析模型,优化分析之后的性能均优于优化前,且优化算法NSGA-II较MOPSO更具优势,该模型能为变径基圆涡旋压缩机的优化研究提供进一步的理论参考.
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DOI:10.3901/JME.2020.23.118
[本文引用: 1]
A new type of variable thickness scroll compressor line composed of circular involutes with different base circle radii is proposed, which is composed of a circular involute I+ circular involute II+ circular involute I. The method of generating the profile is discussed, the general equations of the profile are given, and a series of geometric models of the circular involute variable thickness scroll compressor are established. Aiming at the established geometric model, the influence of the control coefficients <i>θ</i>, <i>φ</i><sup>*</sup> and <i>R<sub>or</sub></i> on the geometric performance of the variable thickness scroll compressor is analyzed. Based on the circular involute Ⅰ, a geometric model of variable thickness scroll compressors with circular involute Ⅰ + higher-order curve + circular involute Ⅰ is constructed, and the geometric performance of two types of variable thickness scroll compressors is comprehensively analyzed. The results show that:<i>θ</i> taking the median value <i>θ</i><sub><i>M</i></sub>,<i>φ</i><sup>*</sup> taking the value of <i>φ</i><sub>end</sub>-2(<i>n</i><sub>1</sub>+1)∏, and <i>R<sub>or</sub></i> taking the larger, the corresponding geometric performance is better. Higher-order curve variable thickness scroll compressors are similar to the circular involute variable thickness scroll compressors corresponding to the median value <i>θ</i><sub><i>M</i></sub>, and can be substituted for each other.A new type of variable thickness scroll compressor line composed of circular involutes with different base circle radii is proposed, which is composed of a circular involute I+ circular involute II+ circular involute I. The method of generating the profile is discussed, the general equations of the profile are given, and a series of geometric models of the circular involute variable thickness scroll compressor are established. Aiming at the established geometric model, the influence of the control coefficients <i>θ</i>, <i>φ</i><sup>*</sup> and <i>R<sub>or</sub></i> on the geometric performance of the variable thickness scroll compressor is analyzed. Based on the circular involute Ⅰ, a geometric model of variable thickness scroll compressors with circular involute Ⅰ + higher-order curve + circular involute Ⅰ is constructed, and the geometric performance of two types of variable thickness scroll compressors is comprehensively analyzed. The results show that:<i>θ</i> taking the median value <i>θ</i><sub><i>M</i></sub>,<i>φ</i><sup>*</sup> taking the value of <i>φ</i><sub>end</sub>-2(<i>n</i><sub>1</sub>+1)π, and <i>R<sub>or</sub></i> taking the larger, the corresponding geometric performance is better. Higher-order curve variable thickness scroll compressors are similar to the circular involute variable thickness scroll compressors corresponding to the median value <i>θ</i><sub><i>M</i></sub>, and can be substituted for each other.
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