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英文原文英文原文 Design efficiency optimization of one dimensional multi stage axial flow compressor AbstractA model for the optimal design of a multi stage compressor assuming a fixed configuration of the flow path is presented The absolute inlet and exit angles of the rotor the absolute exit angle of the stator and the relative gas densitiesat the inlet and exit stations of the stator of every stage are taken as the design variables Analytical relations of the compressor elemental stage and the multi stage compressor are obtained Numerical examples are provided to illustrate the effects of various parameters on the optimal performance of the multi stage compressor 2007 Elsevier Ltd All rights reserved Keywords Multi stage axial flow compressor Efficiency Analytical relation Optimization 1 Introduction The design of the axial flow compressor is partially an art The lack of accurate prediction influences the design process Until today there are no methods currently available that permit the prediction of the values of these quantities to a sufficient accuracy for a new design Some progresses has been achieved via the application of numerical optimization techniques to single and multi stage axial flow compressor design 1 22 Especially with the development of computational fluid dynamics CFD many more accurate methods of calculating have been presented in many references in which the techniques of CFD have been applied to two and three dimensional optimal designs of axial flow compressors 17 20 However it is still of worthwhile significance to calculate using one dimensional flow theory the optimal design of compressors Boiko 23 presented a detailed mathematical model for the optimal design of single and multi stage axial flow turbines by assuming i a fixed distribution of axial velocities or ii a fixed flow path shape and obtained the corresponding optimized results Using a similar idea Chen et al 22 presented a mathematical model for the optimal design of a single stage axial flow compressor by assuming a fixed distribution of axial velocities In this paper a model for the optimal design of a multi stage axial flow compressor by assuming a fixed flow path shape is presented The absolute inlet and exit angles of the rotor the absolute exit angle of the stator and the relative gas densities at the inlet and exit stations of the stator of each stage are taken as the design variables Analytical relations of the compressor stage are obtained Numerical examples are provided to illustrate the effects of various parameters on the optimal performance of the multi stage compressor 2 Fundamental equations for elemental stage compressor Consider a n stage axial flow compressor see Fig 1 Fig 2 shows the specific enthalpy specific entropy diagram of this compressor For a n stage axial flow compressor there are 2n 1 section stations The stage velocity triangle of an intermediate stage i e jth stage is shown in Fig 3 The corresponding specific enthalpy specific entropy diagram is shown in Fig 4 The performance calculation of multi stage compressor is performed using one dimensional flow theory The analysis begins with the energy and continuity equations and the axial flow velocities of the working fluid and wheel velocities at the different stations in the compressor are not considered as constant that is ij uu ij cc ij where i denotes the ith station and j denotes the jth stage The major assumptions made in the method are as follows The working fluid flows stably relative to the vanes stators and rotors which rotate at a fixed speed The working fluid is compressible non viscous and adiabatic The mass flow rate of the working fluid is constant The compression process is homogeneous in the working fluid The absolute outlet angle of the working fluid in jth stage is equal to the absolute inlet angle of the working fluid in j 1 th stage The effects of intake and outlet piping are neglected The specific enthalpies at every station are as follows j 2 2ji2j i 1 2iihc 1 j 2 2j 11i2j 1 i 1 2iihc 2 The total profile losses of the jth stage rotor and the stator are calculated as follows 2 2 2 rjrj2j 12j 12j 12j 12j 12j 12j 1rj 2 2 hwGFuGctgF 3 2 22 rjsj 2j2j2j2jsj 2 1 2hcGFctg 4 Where ri is the total profile loss coefficient of jth stage rotor blade and sj is that of jth stage stator blade Fig 1 Flow path of a n stage axial flow compresso r Fig 2 Enthalpy entropy diagram of a n stage compressor Fig 3 Velocity triangle of an intermediate stage Fig 4 Enthalpy entropy diagram of an intermediate stage The blade profile loss coefficients ri and sj are functions of parameters of the working fluid and blade geometry They can be calculated using various methods and are considered to be constants When ri and sj are functions of the parameters of the working fluid and blade geometry the loss coefficients can be calculated using the method of Ref 24 which was employed and described in Ref 21 The optimization problem can be solved using the iterative method 1 First select the original values of ri and sj and then calculate the parameters of the stage 2 Secondly calculate the values of ri and sj and repeat the first step until the differences between the calculated values and the original ones are small enough The work required by the jth stage is j2j u 2j2j 1 u 2j 12j2j2j 12j 1 2j2j2j 12j 1 GG hu cucu ctguctg FF 5 The work required by the jth rotor is 2222 2j 12j2j2j 1 rj 22 wwuu h 6 The degree of reaction of the jth stage compressor is defined as rjj hh Hence one has u 2j 222 a 2j2j2j 1 j a 2j2j2j 1 11 1 2 kctgctg kkctgctg 7 Where u i k a i 12kin are the velocity coefficients and they are defined as a ia ia 111ii kccFF and u ii1 kuu The constraint conditions can be obtained from the energy balance equation for the one dimensional flow j 2 2 2j 112ji2j2j2j i 1 1 20AiihGFctg 8 j 2 2 2j12j 1i2j 12j 12j 1 i 1 1 20AiihGFctg 9 3 Mathematical model for the behaviour of the multi stage compressor The compression work required by each stage is j 1hjn The total compression work required by the multi stage compressor is n cj j 1 hh The stagnation isentropic enthalpy rise of every stage is s j h The sum of the stagnation isentropic enthalpy rise of each stage is n s j j 1h while the stagnation isentropic enthalpy rise of the multi stage compressor is sc h One has n s jzsc j 1 1 hh The stagnation isentropic efficiency of the multi stage axial flow compressor is n scsccsci i 1 hhhh 10 The total energy balance of a n stage compressor gives nnn 2n 1jZscjrsj j 1j 1j 1 10Ahhhh 11 Eq 11 can be rewritten as 122 1 0jActg 22323 0Actgctg 2j 122j22j 0Actgctg 2j22j 122j 1 0Actgctg 12 2n22n 122n 1 0jn Actgctg 2n 122n 122n 1sc 0Actgctgh For convenience in order to make the constraints dimensionless some parameters are defined 2 22222 j111jjj1 211 1cictgy fctg 13 2 uj111ujjjj1 2 1 1 j u cik ctgy fctg 14 2 222 j11uj1 21 1uikctg 15 2 2 2 j1j1j uj1j1 2 2 2wiciu ciui 16 Where 1 1 are the aerodynamic functions and 11 ca where ais the stagnation sound velocity and 2 1 2 1 1 ai kk jj11uj j fFFlk l is the relative area jj1 y is the relative density where l is the height of the blade and a 11 cu is flow coefficient Introducing the isentropic coefficient used by Boiko 23 one has 1 i j j j i 1 exp ss R 17 Where k k 1 iisi ii 18 Therefore the constraint conditions can be rewritten as j 11 u 2i2iu 2i 12i 1k 11 k 2j 12j2j 1 2 i 1 2i2i2i 12i 1 1 2 1 1 1 kctgkctg Ay y fyfctg 2 2j2 11 222 2j2j1 1 1 10 1 ctg y fctg 19 j 11 u 2i2iu 2i 12i 1k 11 k 2j2j 12j 2 i 1 2i2i2i 12i 1 1 2 1 1 1 kctgkctg Ay y fyfctg 2 2j 12 11 222 2j 12j 11 1 1 10 1 ctg yfctg 20 n u 2i2iu 2i 12i 1 2n 11zSC i 1 2i2i2i 12i 1 1 kctgkctg A y fyf 1 2 22 n 12i 1 u 2i 1ri 22 i 1 2i 12i 12i 12i 1 1 2 ctg k yfyf 1 22 n 2 2isi 22 i 1 2i2i 1 10 2 ctg y f 21 and the stagnation isentropic efficiency of the multi stage axial flow compressor can be rewritten as n scsc1u 2i2i2i2iu 2i 12i 12i 12i 1 i 1 kctgy fkctgyf 22 Where 2 scsc1 hu is isentropic work coefficient of the multi stage The isentropic work coefficient of each stage is defined as 2 sisi2i 1 hu Now the optimization problem is to search the optimal values of i aand i yfor finding the maximum value of the objective function sc under the constraints of Eqs 19 21 4 Solution procedure Once the system variables the objective function and the constraints are defined a suitable method has to be adopted to determine the values of the design variables that maximize the objective function while satisfying the given constraints The present optimization model is a non linear programming procedure with Table 1Relative areas for the stations Station i 1234567 Relative area i f 10 9360 8860 8090 7290 7010 647 Table 2Original and optimal design plans 参数上限下限原始数据最佳数据 s 0 732 s 0 732 s 0 732 s 0 6 0 59 0 59 0 49 0 59 1 549080 589172 685874 911666 5570 2 359049 5045 0045 0045 00 3 549084 133876 343177 5568 2003 4 359049 5045 0045 0045 00 5 549066 41159 708069 058255 7046 6 359049 541845 0045 0046 6157 7 549089 9990 0090 9989 6147 2 y 031 0891 04591 09131 093 3 y 031 1481 14741 15491 0798 4 y031 4241 39701 39001 2624 5 y031 4241 41171 41981 2624 6 y031 5651 53721 60911 3345 7 y 031 6181 63381 66711 4450 sopt 0 90200 90500 90740 8955 5 Numerical example In the calculations 1 u i k 1 330 um s 1 288TK 1 4k n 3 R 286 96 J kg K 0 04 z 0 025 rj and0 02 sj are set The relative areas at every station are listed in Table 1 It should be pointed out that there will be some influence on the relation of the optimization objective with these dimensionless parameters if are functions of the working fluid parameters and geometry parameters of the flow path configuration However the relation obtained will not change qualitatively For a 3 stage compressor there are 13 design variables and 7 constraint conditions Besides the lower and upper limit value constraints of the 13 design variables should also be considered in the calculations The lower and upper limits of the optimization variables the original design plan and the optimization results for different flow coefficients and work coefficients are listed in Table 2 It can be seen that the optimization procedure is effective and practical The calculations show that the optimal stagnation isentropic efficiency sopt is an increasing function of the work coefficient and a decreasing function of the flow coefficient The effect of the work coefficient on the optimal stagnation isentropic efficiency is larger than that of the flow coefficient Also for various values你of the flow coefficients and work coefficients the optimal absolute exit angle of the last stage always approaches90 6 Conclusion In this paper the efficiency optimization of a multi stage axial flow compressor for a fixed flow shape has been studied using one dimensional flow theory The universal characteristic relation of the compressor be haviour is obtained Numerical examples are presented The results can provide some guidance as to the performance analysis and optimization of the multi stage compressor This is a preliminary study It will be necessary to use multi objective numerical optimization techniquesand artificial neural network algorithms for practical compressor optimization 中文译文 中文译文 一维多级轴流压缩机性能的解析优化一维多级轴流压缩机性能的解析优化 摘要对多级压缩机的优化设计模型 本文假设固定的流道形状以入口和出口的动叶绝对角 度 静叶的绝对角度和静叶及每一级的入口和出口的相对气体密度作为设计变量 得到压缩 机基元级的基本方程和多级压缩机的解析关系 用数值实例来说明多级压缩机的各种参数对 最优性能的影响 关键词轴流压缩机效率分析关系优化 1引言引言 轴流式压缩机的设计是工艺技术的一部分 如果缺乏准确的预测将影响设计过程 至今 还没有公认的方法可使新的设计参数达到一个足够精确的值 通过应用一些已经取得新进展 的数值优化技术 以完成单级和多级轴流式压缩机的设计 计算流体动力学 CFD 和许多 更准确的方法特别是发展计算的CFD技术 已经应用到许多轴流式压缩机的平面和三维优化 设计 它仍然是使用一维流体力学理论用数值实例来计算压缩机的最佳设计 Boiko通过以 下假设提出了详细的数学模型用以优化设计单级和多级轴流涡轮 1 固定的轴向均匀速度 分布 2 固定流动路径的形状分布 并获得了理想的优化结果 陈林根等人也采用了类似 的想法 通过假设一个固定的轴向速度分布的优化设计提出了设计单级轴流式压缩机一种数 学模型 在本文中为优化设计多级轴流压缩机的模型 提出了假设一个固定的流道形状 以 入口和出口的动叶绝对角度 静叶的绝对角度和静叶及每一级的入口和出口的相对气体密度 作为设计变量 分析压缩机的每个阶段之间的关系 用数值实例来说明多级压缩机的各种参 数对最优性能的影响 2基元级的基本方程基元级的基本方程 考虑图1所示由n级组成的轴流压缩机 其某一压缩过程焓熵图和中间级的速度三角形 见图2和图3 相应的中间级的具体焓熵图如图4 按一维理论作级的性能计算 按一般情况 列出轴流压缩机中气体流动的能量方程和连续方程 工作流体和叶轮的速度 在不同级的轴 向流速不为常数 即考虑 ij uu ij cc ij 时的能量和流量方程 在下列假定下分析 轴流压缩机的工作 相对于稳定回转的动叶 静叶和导向叶片机构 气体流动是稳定的 流体是可压缩 无黏性和不导热的 通过级的流体质量流量为定值 在实际工质的情况下 压缩过程是均匀的 本级出口绝对气流角为下一级进口角绝对气流角 忽略进出口管道的影响 在每一级的具体焓如下 j 2 2ji2j i 1 2iihc 1 j 2 2j 11i2j 1 i 1 2iihc 2 第j阶段的动叶和静叶的焓值损失总额计算如下 2 2 2 rjrj2j 12j 12j 12j 12j 12j 12j 1rj 2 2 hwGFuGctgF 3 2 22 rjsj 2j2j2j2jsj 2 1 2hcGFctg 4 其中 ri 是第j阶段动叶叶片轮廓总损失系数 sj 是第j阶段静叶叶片轮廓总损 失的系数 图1n级轴流式压缩机的流量路径 叶片轮廓损失系数 ri 和 sj 是工作流体和叶片的几何功能参数 它们可以使用各种方法 及视作常量来计算 当 ri 和 sj 看做工作流体和叶片的几何功能参数时 可以使用Ref迭代 的方法来计算损失系数 使用迭代方法解决计算损失系数 1 选择 ri 和 sj 初始值 然后计算各级的参数 2 计算的 ri sj 值 重复第一步 直到计算值和原值之间的差异足够小 第j阶段理论所需计算得 j2j u 2j2j 1 u 2j 12j2j2j 12j 1 2j2j2j 12j 1 GG hu cucu ctguctg FF 5 第j阶段实际所需计算得 图2n级压缩机的焓熵图 图3 中间级的速度三角形 图4 中间级的焓熵图 2222 2j 12j2j2j 1 rj 22 wwuu h 6 基元级反应度定义为 rjj hh 因此有 u 2j 222 a 2j2j2j 1 j a 2j2j2j 1 11 1 2 kctgctg kkctgctg 7 在这里 u i k a i 12kin 视作速度系数 它们的计算为 a ia ia 111ii kccFF 和 u ii1 kuu j 2 2 2j 112ji2j2j2j i 1 1 20AiihGFctg 8 j 2 2 2j12j 1i2j 12j 12j 1 i 1 1 20AiihGFctg 9 3级组的数学模型级组的数学模型 压缩机各级的比压缩功为 j 1hjn 则总的比耗功为 n cj j 1 hh 各级的滞止等 熵能量头为 s j h 则级组各级滞止等熵比压缩功总和为 n s j j 1h 级组等熵比压缩功为 sc h 则 n s jzsc j 1 1 hh 为压缩机的重热系数 根据定义 多级压缩机通流部分滞止等熵效率 为 n scsccsci i 1 hhhh 10 求解确定各级能量头的分配 nnn 2n 1jZscjrsj j 1j 1j 1 10Ahhhh 11 方程式 11 同样可以写作 122 1 0jActg 22323 0Actgctg 2j 122j22j 0Actgctg 2j22j 122j 1 0Actgctg 12 2n22n 122n 1 0jn Actgctg 2n 122n 122n 1sc 0Actgctgh 出于方便 一些参数简化约束计算做了如下定义 2 22222 j111jjj1 211 1cictgy fctg 13 2 uj111ujjjj1 2 1 1 j u cik ctgy fctg 14 2 222 j11uj1 21 1uikctg 15 2 2 2 j1j1j uj1j1 2 2 2wiciu ciui 16 这里 1 1 是气动力函数 11 ca 在这里的 a是滞止声速相对应的 2 1 2 1 1 ai kk 且 jj11uj j fFFlk l 是相对面积 jj1 y 是相对密度 l 是叶片高 a 11 cu 是流量系数 通过Boiko的论文引入等熵线系数 一个是 1 i j j j i 1 exp ss R 17 这里 k k 1 iisi ii 18 因此约束条件也可写作 j 11 u 2i2iu 2i 12i 1k 11 k 2j 12j2j 1 2 i 1 2i2i2i 12i 1 1 2 1 1 1 kctgkctg Ay y fyfctg 2 2j2 11 222 2j2j1 1 1 10 1 ctg y fctg 19 j 11 u 2i2iu 2i 12i 1k 11 k 2j2j 12j 2 i 1 2i2i2i 12i 1 1 2 1 1 1 kctgkctg Ay y fyfctg 2 2j 12 11 222 2j 12j 11 1 1 10 1 ctg yfctg 20 n u 2i2iu 2i 12i 1 2n 11zSC i 1 2i2i2i 12i 1 1 kctgkctg A y fyf 1 2 22 n 12i 1 u 2i 1ri 22 i