Main Geometric Analysis of PC-type Single-screw Compressor
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摘要:
为了解决PC型单螺杆压缩机设计中理论输气量、排气压力和耗功的计算问题，对其螺槽容积、扭矩数学模型、封闭螺旋线和排气孔口进行了研究.首先对PC型单螺杆啮合过程中，星轮齿的侵入面积和形心进行了公式推导；其次建立了单螺槽容积、密闭螺槽容积、任意时刻螺槽容积和扭矩的数学模型，给出了封闭螺旋线及排气孔口大小的公式.并且以螺杆直径为117 mm的PC型单螺杆压缩机为例，分析结果表明：利用Pro/E软件建立了单螺槽的实体模型，测量体积发现与计算体积误差在3.771%左右；瞬时扭矩随着进气压力的不同呈现周期性变化；扭矩比值接近1，并且扭矩比与进气压力无关.为PC型单螺杆压缩机的设计和计算提供理论依据.
Abstract:In order to obtain the theoretical displacement, discharge pressure and power consumption when design the PC-type single screw compressor (SSC), the main geometric analysis of PC-type SSC has been carried out in this paper. By analyzing the intermeshing area formed by gate rotor tooth and groove of screw rotor and centroid distance, the mathematical models of groove volume and torque were presented for PC-type SSC. Analysis of a typical PC-type SSC with 117 mm diameter rotor shows that the calculated entire single groove volume has good consistency with that measured by 3D Pro/E software, with percentage error about 3.771%, the periodic variation of instantaneous torque is the same for different inlet pressure levels, the torque ratio, with its value close to 1, is independent of the inlet pressure of SSC and that the power consumption increases with increase of inlet pressure. Results from the established mathematical models of groove volume and torque is of great importance for engineers to design and manufacture the PC-type SSC.
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Keywords:
- single-screw compressor /
- groove volume /
- torque
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20世纪60年代，法国Zimmern等[1-3]提出单螺杆压缩机的概念, 由于单螺杆结构具有受力平衡、低振动、高容积效率等特点[3-5]，被广泛应用于空调制冷、空气压缩机及天然气压缩领域等. 单螺杆结构根据单螺杆和星轮片的形状类型可以分为CP、PP、CC和PC型. PC型单螺杆压缩机不仅具有CP型单螺杆结构的优点，而且具有压比大、流量小、吸气阻力小、无加工局限性等突出优点[6-7]. 在日本，PC型单螺杆压缩机占有很大的市场，并具有继续增长的趋势[6-7].
在单螺杆压缩机设计和计算中，单螺杆螺槽容积是计算理论输气量和功耗的基础，封闭螺旋线和排气孔口的大小是决定进气结束和排气开始的重要部分. 因此，开展单螺杆压缩机主要几何关系的研究对设计加工单螺杆压缩机至关重要.
目前，针对各类型单螺杆结构主要几何关系方面的研究主要集中在CP和CC型方面. 孙光三[8-9]研究了CP型单螺杆与星轮片的主要几何关系，建立了单螺杆螺槽基元容积随着星轮转角变化的数学模型，给出了封闭角的计算公式及齿宽系数相关的经验数据，分析了进气锥角对进气口形状的影响. 王增丽等[10]研究了CP单螺杆压缩机主要几何参数对压缩机性能的影响. 张淑存等[11]对CP型单螺杆中心距系数与啮合副直径比之间的关系进行了数值模拟分析，给出了啮合副直径的取值范围. Yang等[12-15]研究了CP、PP、PC和CC型单螺杆结构的型线方程并建立了相应的数学模型. 汤炎等[6-7]研究了CC型单螺杆压缩机的主要几何关系，建立了CC型单螺杆转子螺槽基元容积随着星轮转角变化的数学模型. 查谦等[16]介绍了PC型单螺杆压缩机的特点、加工技术及相应的机床. 许明文[17]建立了圆柱圆台包络的PC型单螺杆压缩机的型线并给出关键零部件的制造方法. 但是，目前还没有文章研究PC型单螺杆压缩机螺槽容积和扭矩的数学模型及封闭螺旋线和排气孔口公式.
本文针对PC型单螺杆结构的特点，给出了PC型螺杆和星轮结构参数之间的基本关系，推导了PC啮合副运动时，星轮齿的侵入面积和形心公式；进一步建立了单螺槽容积、密封螺槽容积和任意时刻螺槽容积随着星轮转角变化的数学模型和扭矩数学模型；给出了封闭螺旋线和排气孔口大小的公式，为PC型单螺杆压缩机的设计提供理论指导.
1. PC型单螺杆压缩机工作原理
PC型单螺杆压缩机的工作原理如下.
吸气过程：螺杆转子吸气端的螺槽与吸气腔连通，随着螺槽容积的增大而吸气. 当螺槽容积增加到一定程度，即螺槽被星轮齿全部封闭，星轮转到吸气结束角时，结束进气，开始压缩. 如图 1(a)所示.
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图 1 PC型单螺杆压缩机工作原理示意图Figure 1. Working principle diagram of PC-type single-screw compressor压缩过程：随着星轮的转动，螺槽容积不断减小，气体不断被压缩，直到排气开始，即星轮转到排气开始角位置，开始排气. 如图 1(b)所示.
排气过程：星轮转到排气开始角，螺槽与排气口连通，开始排气. 如图 1(c)所示.
2. PC型单螺杆压缩机的基本几何关系
通过螺杆轴线并垂直于星轮中线的平面为主平面. 螺杆与星轮的大部分尺寸都能在主平面上标注出来(如图 2所示). 当Z2与Zg轴重合时，为星轮转角的0度角. 当Z2轴在Zg右侧时，星轮转角为负值；反之为正值.
$$ {i_{12}} = \frac{{{\theta _1}}}{{{\theta _2}}} = \frac{{{Z_2}}}{{{Z_1}}} $$ (1) $$ \gamma = \frac{{360}}{{{Z_2}}} $$ (2) $$ \beta = \frac{\gamma }{2} - \arcsin \left( {\frac{{{e_1}}}{{2{r_2}}}} \right) $$ (3) $$ b = {r_2}\sin \beta $$ (4) $$ {b_1} = b - H\tan \lambda $$ (5) $$ {\delta _1} = \arcsin \left( {\frac{{{b_1}}}{{{r_2}}}} \right) $$ (6) $$ \alpha = \arcsin \left( {\frac{{H\cot \varphi }}{{{r_2}}}} \right) $$ (7) $$ {\alpha _1} = \arcsin \left( {\frac{{H\tan \varphi }}{{{r_2}}}} \right) $$ (8) 式中：i12为P型螺杆与C型星轮啮合传动比；θ1为P型螺杆转角； θ2为C型星轮转角； Z1为P型螺杆槽数； Z2为C型星轮齿数； H为C型星轮齿齿长； b为C型星轮齿齿根部宽； b1为C型星轮齿齿顶部宽； δ1为C型星轮齿齿顶宽对应的齿宽角之半；β为C型星轮齿齿根宽对应的齿宽角之半； e1为P型螺槽肋部最窄距离； α1为螺杆与星轮排气侧啮合角； α为螺杆与星轮进气侧啮合角； $\varphi $ 为Zg与Xs轴之间的夹角；γ为星轮齿的分度角.
3. 螺槽容积和扭矩数学模型
啮合过程中，星轮齿与螺槽始终是吻合的. 星轮齿对螺杆的侵入，由浅入深，再由深到浅，在相对运动中形成封闭的螺槽. 如图 3所示，螺槽体积微元 ${\rm{d}}v = {F_{ci}}{\bar R_{ci}}{\rm{d}}{\theta _1}$ ，因此，星轮齿侵入面积及其形心位置是求螺槽容积和理论排气量的基础；此外，为了进行强度和功率计算，需算出作用在星轮齿上的力，其合力为侵入面积与气体压力之乘积，作用在形心上. 因此，星轮齿侵入螺槽的面积和形心是计算螺槽容积、功率和强度校核的基础. 因此，星轮齿啮入螺槽面积和形心对单螺杆压缩机设计计算至关重要.
3.1 星轮齿啮入螺槽面积和形心
当一个星轮齿从开始啮入螺槽到完全脱离螺槽，与螺槽相交的面积形状会出现7种情况，如图 4(a)~图 4(g)所示，并且图形的形状均是由图中直线P3P4、P4P6、P5P6分别和P1P2、P1P7之间的组合面积叠加而成. 因此，在求星轮齿啮入螺槽面积时，首先求出这几条直线的方程，如式(9)~(13)所示. 然后求出直线P3P4、P4P6、P5P6分别和P1P2、P1P7组成的各部分面积和形心坐标，如式(14)~(31)所示. 最后求出这7部分星轮齿啮入螺槽的面积和形心坐标，如表 1所示；形心距如图 5所示，可以由式(32)求出.
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图 4 星轮齿啮入螺槽所形成的相交形状Figure 4. Intersection part between gate rotor tooth and screw groove表 1 星轮齿啮入面积和形心坐标Table 1. Intermeshing tooth area and centroid coordinates序号 星轮齿啮入面积和形心 积分上下限取值 1 $\begin{array}{l} {F_{{\rm{c}}1}} = {F_1} + {F_2}\\ {\overline X _{{\rm{c}}1}} = ({F_1}{\overline X _1} + {F_2}{\overline X _2})/{F_{{\rm{c}}1}}\\ {\overline Z _{{\rm{c}}1}} = ({F_1}{\overline Z _1} + {F_2}{\overline Z _2})/{F_{{\rm{c}}1}} \end{array}$ $\begin{array}{l} {t_1} = ( - H + {C_z} - {r_1})/\sin \varphi + {C_y}\cot \varphi \tan {\theta _2})/\cot \varphi \\ {t_2} = {t_3} = 0.5b\\ {t_4} = ( - {C_2} + ({C_z} - {r_1})/\sin \varphi + {C_y}\cot \varphi \tan {\theta _2})/({C_1} + \cot \varphi /cos \; {\theta _2}) \end{array}$ 2 $\begin{array}{*{20}{l}} {{F_{{\rm{c2}}}} = {F_3} + {F_1} + {F_2}}\\ {{{\bar X}_{{\rm{c2}}}} = ({F_3}{{\bar X}_3} + {F_1}{{\bar X}_1} + {F_2}{{\bar X}_2})/{F_{{\rm{c2}}}}}\\ {{{\bar Z}_{{\rm{c2}}}} = ({F_3}{{\overline Z }_3} + {F_1}{{\bar Z}_1} + {F_2}{{\bar Z}_2})/{F_{{\rm{c2}}}}} \end{array}$ $\begin{array}{*{20}{l}} {{t_5} = ( - {C_2} + {C_z} - {r_1})/\sin \varphi + {C_y}\cot \varphi \tan {\theta _2})/( - {C_1} + \cot \varphi /{\rm{cos}}\;{\theta _2})}\\ \begin{array}{l} {t_6} = {t_1} = - 0.5b\\ {t_2} = {t_3} = 0.5b \end{array}\\ {{t_4} = ( - {C_2} + ({C_z} - {r_1})/\sin \varphi + {C_y}\cot \varphi \tan {\theta _2})/({C_1} + \cot \varphi /{\rm{cos}}\;{\theta _2})} \end{array}$ 3 $\begin{array}{*{20}{l}} {{F_{{\rm{c3}}}} = {F_3} + {F_1} + {F_2} + {F_4}}\\ {{{\bar X}_{{\rm{c3}}}} = ({F_3}{{\bar X}_3} + {F_1}{{\bar X}_1} + {F_2}{{\bar X}_2} + {F_4}{{\bar X}_4})/{F_{{\rm{c3}}}}}\\ {{{\bar Z}_{{\rm{c3}}}} = ({F_3}{{\bar Z}_3} + {F_1}{{\bar Z}_1} + {F_2}{{\bar Z}_2} + {F_4}{{\bar Z}_4})/{F_{{\rm{c3}}}}} \end{array}$ $\begin{array}{*{20}{l}} {{t_5} = ( - {C_2} + {C_z} - {r_1})/\sin \varphi + {C_y}\cot \varphi \tan {\theta _2})/( - {C_1} + \cot \varphi /{\rm{cos}}\;{\theta _2})}\\ {\begin{array}{*{20}{l}} {{t_6} = {t_1} = - 0.5b}\\ {{t_2} = {t_3} = 0.5b}\\ {{t_4} = {t_7} = - {r_2}\sin {\theta _2}} \end{array}}\\ {{t_8} = ( - {C_2} - {C_y}\tan \varphi \tan {\theta _2})/({C_1} - \tan \varphi /{\rm{cos}}\;{\theta _2})} \end{array}$ 4 $\begin{array}{*{20}{l}} {{F_{{\rm{c4}}}} = {F_3} + {F_1} + {F_5} + {F_4}}\\ {{{\bar X}_{{\rm{c4}}}} = ({F_3}{{\bar X}_3} + {F_1}{{\bar X}_1} + {F_5}{{\bar X}_5} + {F_4}{{\bar X}_4})/{F_{{\rm{c4}}}}}\\ {{{\bar Z}_{{\rm{c4}}}} = ({F_3}{{\bar Z}_3} + {F_1}{{\bar Z}_1} + {F_5}{{\bar Z}_5} + {F_4}{{\bar Z}_4})/{F_{{\rm{c4}}}}} \end{array}$ $\begin{array}{*{20}{l}} {{t_5} = ( - {C_2} + {C_z} - {r_1})/\sin \varphi + {C_y}\cot \varphi \tan {\theta _2})/( - {C_1} + \cot \varphi /{\rm{cos}}\;{\theta _2})}\\ {\begin{array}{*{20}{l}} {{t_6} = {t_1} = - 0.5b}\\ {{t_2} = {t_9} = - {r_2}\sin {\theta _2}}\\ {{t_{10}} = {t_7} = 0.5b} \end{array}}\\ {{t_8} = ( - {C_2} - {C_y}\tan \varphi \tan {\theta _2})/({C_1} - \tan \varphi /{\rm{cos}}\;{\theta _2})} \end{array}$ 5 $\begin{array}{*{20}{l}} {{F_{{\rm{c5}}}} = {F_3} + {F_6} + {F_5} + {F_4}}\\ {{{\bar X}_{{\rm{c5}}}} = ({F_3}{{\bar X}_3} + {F_6}{{\bar X}_6} + {F_5}{{\bar X}_5} + {F_4}{{\bar X}_4})/{F_{{\rm{c5}}}}}\\ {{{\bar Z}_{{\rm{c5}}}} = ({F_3}{{\bar Z}_3} + {F_6}{{\bar Z}_6} + {F_5}{{\bar Z}_5} + {F_4}{{\bar Z}_4})/{F_{{\rm{c5}}}}} \end{array}$ $\begin{array}{*{20}{l}} {{t_5} = ( - {C_2} + {C_z} - {r_1})/\sin \varphi + {C_y}\cot \varphi \tan {\theta _2})/( - {C_1} + \cot \varphi /{\rm{cos}}\;{\theta _2})}\\ \begin{array}{l} {t_6} = {t_{11}} = - {r_2}\sin {\theta _2}\\ {t_{12}} = {t_9} = - 0.5b\\ {t_{10}} = {t_7} = 0.5b \end{array}\\ {{t_8} = ( - {C_2} - {C_y}\tan \varphi \tan {\theta _2})/({C_1} - \tan \varphi /{\rm{cos}}\;{\theta _2})} \end{array}$ 6 $\begin{array}{*{20}{l}} {{F_{{\rm{c6}}}} = {F_6} + {F_5} + {F_4}}\\ {{{\bar X}_{{\rm{c6}}}} = ({F_6}{{\bar X}_6} + {F_5}{{\bar X}_5} + {F_4}{{\bar X}_4})/{F_{{\rm{c5}}}}}\\ {{{\bar Z}_{{\rm{c6}}}} = ({F_6}{{\bar Z}_6} + {F_5}{{\bar Z}_5} + {F_4}{{\bar Z}_4})/{F_{{\rm{c5}}}}} \end{array}$ $\begin{array}{*{20}{l}} {{t_{11}} = ( - {C_2} - {C_y}\cot \varphi \tan {\theta _2})/( - {C_1} - \cot \varphi /{\rm{cos}}\;{\theta _2})}\\ \begin{array}{l} {t_{12}} = {t_9} = - 0.5b\\ {t_{10}} = {t_7} = 0.5b \end{array}\\ {{t_8} = ( - {C_2} - {C_y}\tan \varphi \tan {\theta _2})/({C_1} - \tan \varphi /{\rm{cos}}\;{\theta _2})} \end{array}$ 7 $\begin{array}{*{20}{l}} {{F_{{\rm{c7}}}} = {F_6} + {F_5}}\\ {{{\bar X}_{{\rm{c7}}}} = ({F_6}{{\bar X}_6} + {F_5}{{\bar X}_5})/{F_{{\rm{c6}}}}}\\ {{{\bar Z}_{{\rm{c7}}}} = ({F_6}{{\bar Z}_6} + {F_5}{{\bar Z}_5})/{F_{{\rm{c6}}}}} \end{array}$ $\begin{array}{*{20}{l}} {{t_{11}} = ( - {C_2} - {C_y}\cot \varphi \tan {\theta _2})/( - {C_1} - \cot \varphi /{\rm{cos}}\;{\theta _2})}\\ {{t_{12}} = {t_9} = - 0.5b}\\ {{t_{10}} = ( - h + {C_y}\tan {\theta _2}\tan \varphi ){\rm{cos}}\;{\theta _2}/\tan \varphi } \end{array}$ 直线P1P2方程：
$$ {Z_2} = {X_2}\frac{{\tan \varphi }}{{\cos {\theta _2}}} - {C_y}\tan \varphi \tan {\theta _2} $$ (9) 直线P1P7方程：
$$ {Z_2} = - {X_2}\frac{{\cot \varphi }}{{\cos {\theta _2}}} + \frac{{{C_x} - {r_1}}}{{\sin \varphi }} + {C_y}\cot \varphi \tan {\theta _2} $$ (10) 直线P3P4方程：
$$ {Z_2} = {{\rm{C}}_1}{{\rm{X}}_2} + {C_2} $$ (11) 式中： ${{\rm{C}}_1} = \frac{{ - h}}{{b - {b_1}}},{{\rm{C}}_2} = \frac{{hb}}{{b - {b_1}}}$ .
直线P4P6方程：
$$ {Z_2} = h $$ (12) 直线P5P6方程：
$$ {Z_2} = - {C_1}{X_2} + {C_2} $$ (13) 利用数值积分公式，直线P3P4、P4P6、P5P6分别和P1P2、P1P7组成的面积公式为
$$ \begin{array}{*{20}{c}} {{F_1} = \int_{{t_1}}^{{t_2}} {\left( {{P_4}{P_6} - {P_1}{P_7}} \right){\rm{d}}{X_2}} = }\\ {\left[ {\left( {h - \frac{{{C_x} - {r_1}}}{{\sin \varphi }} - {C_y}\cot \varphi \tan {\theta _2}} \right){X_2} - \frac{{\cot \varphi }}{{2\cos {\theta _2}}}X_2^2} \right]_{{t_1}}^{{t_2}}} \end{array} $$ (14) $$ \begin{array}{*{20}{c}} {{F_2} = \int_{{t_3}}^{{t_4}} {\left( {{P_3}{P_4} - {P_1}{P_7}} \right){\rm{d}}{X_2}} = }\\ {\left[ {\left( {{C_2} - \frac{{{C_x} - {r_1}}}{{\sin \varphi }} - {C_y}\cot \varphi \tan {\theta _2}} \right){X_2} - } \right.}\\ {\left. {\frac{1}{2}\left( {{C_1} + \frac{{\cot \varphi }}{{\cos {\theta _2}}}} \right)X_2^2} \right]_{{t_3}}^{{t_4}}} \end{array} $$ (15) $$ \begin{array}{*{20}{c}} {{F_3} = \int_{{t_5}}^{{t_6}} {\left( {{P_5}{P_6} - {P_1}{P_7}} \right){\rm{d}}{X_2}} = }\\ {\left[ {\left( {{C_2} - \frac{{{C_x} - {r_1}}}{{\sin \varphi }} - {C_y}\cot \varphi \tan {\theta _2}} \right){X_2} - } \right.}\\ {\left. {\frac{1}{2}\left( { - {C_1} + \frac{{\cot \varphi }}{{\cos {\theta _2}}}} \right)X_2^2} \right]_{{t_5}}^{{t_6}}} \end{array} $$ (16) $$ \begin{array}{*{20}{c}} {{F_4} = \int_{{t_7}}^{{t_8}} {\left( {{P_3}{P_4} - {P_1}{P_2}} \right){\rm{d}}{X_2}} = }\\ {\left[ {\left( {{C_2} + {C_y}\tan \varphi \tan {\theta _2}} \right){X_2} - \frac{1}{2}\left( {{C_1} - \frac{{\tan \varphi }}{{\cos {\theta _2}}}} \right)X_2^2} \right]_{{t_7}}^{{t_8}}} \end{array} $$ (17) $$ \begin{array}{*{20}{c}} {{F_5} = \int_{{t_9}}^{{t_{10}}} {\left( {{P_4}{P_6} - {P_1}{P_2}} \right){\rm{d}}{X_2}} = }\\ {\left[ {\left( {h + {C_y}\tan \varphi \tan {\theta _2}} \right){X_2} - \frac{{\tan \varphi }}{{2\cos {\theta _2}}}X_2^2} \right]_{{t_9}}^{{t_{10}}}} \end{array} $$ (18) $$ \begin{array}{*{20}{c}} {{F_6} = \int_{{t_{11}}}^{{t_{12}}} {\left( {{P_5}{P_6} - {P_1}{P_2}} \right){\rm{d}}{X_2}} = }\\ {\left[ {\left( {{C_2} + {C_y}\tan \varphi \tan {\theta _2}} \right){X_2} - \frac{1}{2}\left( { - {C_1} + \frac{{\tan \varphi }}{{\cos {\theta _2}}}} \right)X_2^2} \right]_{{t_{11}}}^{{t_{12}}}} \end{array} $$ (19) 根据静距公式，直线P4P6、P3P4、P5P6和P1P7、P1P2分别组成的面积形心坐标公式为
$$ \begin{array}{*{20}{c}} {{F_1}{{\bar X}_1} = \int_{{t_1}}^{{t_2}} {\left( {{P_4}{P_6} - {P_1}{P_7}} \right){X_2}{\rm{d}}{X_2}} = }\\ {\left[ {\frac{1}{2}\left( {h - \frac{{{C_x} - {r_1}}}{{\sin \varphi }} - {C_y}\cot \varphi \tan {\theta _2}} \right)X_2^2 - \frac{{\cot \varphi }}{{3\cos {\theta _2}}}X_2^3} \right]_{{t_1}}^{{t_2}}} \end{array} $$ (20) $$ \begin{array}{*{20}{c}} {{F_2}{{\bar X}_2} = \int_{{t_3}}^{{t_4}} {\left( {{P_3}{P_4} - {P_1}{P_7}} \right){X_2}{\rm{d}}{X_2}} = }\\ {\left[ {\frac{1}{2}\left( {{C_2} - \frac{{{C_x} - {r_1}}}{{\sin \varphi }} - {C_y}\cot \varphi \tan {\theta _2}} \right)X_1^2 - } \right.}\\ {\left. {\frac{1}{3}\left( {{C_1} + \frac{{\cot \varphi }}{{2\cos {\theta _2}}}} \right)X_2^2} \right]_{{t_3}}^{{t_4}}} \end{array} $$ (21) $$ \begin{array}{*{20}{c}} {{F_3}{{\bar X}_3} = \int_{{t_5}}^{{t_6}} {\left( {{P_5}{P_6} - {P_1}{P_7}} \right){X_2}{\rm{d}}{X_2}} = }\\ {\left[ {\frac{1}{2}\left( {{C_2} - \frac{{{C_x} - {r_1}}}{{\sin \varphi }} - {C_y}\cot \varphi \tan {\theta _2}} \right)X_2^2 - } \right.}\\ {\left. {\frac{1}{3}\left( { - {C_1} + \frac{{\cot \varphi }}{{2\cos {\theta _2}}}} \right)X_2^2} \right]_{{t_5}}^{{t_6}}} \end{array} $$ (22) $$ \begin{array}{*{20}{c}} {{F_4}{{\bar X}_4} = \int_{{t_7}}^{{t_8}} {\left( {{P_3}{P_4} - {P_1}{P_2}} \right){X_2}{\rm{d}}{X_2}} = }\\ {\left[ {\frac{1}{2}\left( {{C_2} + {C_y}\tan \varphi \tan {\theta _2}} \right)X_2^2 - \frac{1}{2}\left( {{C_1} - \frac{{\tan \varphi }}{{\cos {\theta _2}}}} \right)X_2^3} \right]_{{t_7}}^{{t_8}}} \end{array} $$ (23) $$ \begin{array}{*{20}{c}} {{F_5}{{\bar X}_5} = \int_{{t_9}}^{{t_{10}}} {\left( {{P_4}{P_6} - {P_1}{P_2}} \right){X_2}{\rm{d}}{X_2}} = }\\ {\left[ {\frac{1}{2}\left( {h + {C_y}\tan \varphi \tan {\theta _2}} \right)X_2^2 - \frac{{\tan \varphi }}{{3\cos {\theta _2}}}X_2^3} \right]_{{t_9}}^{{t_{10}}}} \end{array} $$ (24) $$ \begin{array}{*{20}{c}} {{F_6}{{\bar X}_6} = \int_{{t_{11}}}^{{t_{12}}} {\left( {{P_5}{P_6} - {P_1}{P_2}} \right){X_2}{\rm{d}}{X_2}} = }\\ {\left[ {\frac{1}{2}\left( {{C_2} + {C_y}\tan \varphi \tan {\theta _2}} \right)X_2^2 - } \right.}\\ {\left. {\frac{1}{3}\left( { - {C_1} + \frac{{\tan \varphi }}{{\cos {\theta _2}}}} \right)X_2^3} \right]_{{t_{11}}}^{{t_{12}}}} \end{array} $$ (25) $$ \begin{array}{*{20}{c}} {{F_1}{{\bar Z}_1} = }\\ {\int_{{t_1}}^{{t_2}} {\left( {{P_4}{P_6} - {P_1}{P_7}} \right)\left( {{P_1}{P_7} + \frac{1}{2}\left( {{P_4}{P_6} - {P_1}{P_7}} \right)} \right){\rm{d}}{X_2}} = }\\ {\frac{1}{2}\left[ {\left( {{h^2} - {{\left( {\frac{{{C_x} - {r_1}}}{{\sin \varphi }} + {C_y}\cot \varphi \tan {\theta _2}} \right)}^2}} \right){X_2} + } \right.}\\ {\left. {\frac{{\cot \varphi }}{{\cos {\theta _2}}}\left( {\frac{{{C_x} - {r_1}}}{{\sin \varphi }} + {C_y}\cot \varphi \tan {\theta _2}} \right)X_2^2 - \frac{{{{\cot }^2}\varphi }}{{3{{\cos }^2}{\theta _2}}}X_2^3} \right]_{{t_1}}^{{t_2}}} \end{array} $$ (26) $$ \begin{array}{*{20}{c}} {{F_2}{{\bar Z}_2} = }\\ {\int_{{t_3}}^{{t_4}} {\left( {{P_3}{P_4} - {P_1}{P_7}} \right)\left( {{P_1}{P_7} + \frac{1}{2}\left( {{P_3}{P_4} - {P_1}{P_7}} \right)} \right){\rm{d}}{X_2}} = }\\ {\frac{1}{2}\left[ {\left( {C_2^2 - {{\left( {\frac{{{C_x} - {r_1}}}{{\sin \varphi }} + {C_y}\cot \varphi \tan {\theta _2}} \right)}^2}} \right){X_2} + } \right.}\\ {\left( {{C_1}{C_2} + \frac{{\cot \varphi }}{{\cos {\theta _2}}}\left( {\frac{{{C_x} - {r_1}}}{{\sin \varphi }} + {C_y}\cot \varphi \tan {\theta _2}} \right)} \right)X_2^2 - }\\ {\left. {\frac{1}{3}\left( {C_1^2 + \frac{{{{\cot }^2}\varphi }}{{{{\cos }^2}{\theta _2}}}} \right)X_2^3} \right]_{{t_3}}^{{t_4}}} \end{array} $$ (27) $$ \begin{array}{*{20}{c}} {{F_3}{{\bar Z}_3} = }\\ {\int_{{t_5}}^{{t_6}} {\left( {{P_5}{P_6} - {P_1}{P_7}} \right)\left( {{P_1}{P_7} + \frac{1}{2}\left( {{P_5}{P_6} - {P_1}{P_7}} \right)} \right){\rm{d}}{X_2}} = }\\ {\frac{1}{2}\left[ {\left( {C_2^2 - {{\left( {\frac{{{C_x} - {r_1}}}{{\sin \varphi }} + {C_y}\cot \varphi \tan {\theta _2}} \right)}^2}} \right){X_2} - } \right.}\\ {\left( {{C_1}{C_2} + \frac{{\cot \varphi }}{{\cos {\theta _2}}}\left( {\frac{{{C_x} - {r_1}}}{{\sin \varphi }} + {C_y}\cot \varphi \tan {\theta _2}} \right)} \right)X_2^2 - }\\ {\left. {\frac{1}{3}\left( {C_1^2 + \frac{{{{\cot }^2}\varphi }}{{{{\cos }^2}{\theta _2}}}} \right)X_2^3} \right]_{{t_5}}^{{t_6}}} \end{array} $$ (28) $$ \begin{array}{*{20}{c}} {{F_4}{{\bar Z}_4} = }\\ {\int_{{t_7}}^{{t_8}} {\left( {{P_3}{P_4} - {P_1}{P_2}} \right)\left( {{P_1}{P_2} + \frac{1}{2}\left( {{P_3}{P_4} - {P_1}{P_2}} \right)} \right){\rm{d}}{X_2}} = }\\ {\frac{1}{2}\left[ {\left( {C_2^2 - {{\left( {{C_y}\tan \varphi \tan {\theta _2}} \right)}^2}} \right){X_2} + \left( {{C_1}{C_2} + } \right.} \right.}\\ {\left. {\left. {\frac{{{C_y}{{\tan }^2}\varphi \tan {\theta _2}}}{{\cos {\theta _2}}}} \right)X_2^2 + \frac{1}{3}\left( {C_1^2 - \frac{{{{\tan }^2}\varphi }}{{{{\cos }^2}{\theta _2}}}} \right)X_2^3} \right]_{{t_7}}^{{t_8}}} \end{array} $$ (29) $$ \begin{array}{*{20}{c}} {{F_5}{{\bar Z}_5} = }\\ {\int_{{t_9}}^{{t_{10}}} {\left( {{P_4}{P_6} - {P_1}{P_2}} \right)\left( {{P_1}{P_2} + \frac{1}{2}\left( {{P_4}{P_6} - {P_1}{P_2}} \right)} \right){\rm{d}}{X_2}} = }\\ {\frac{1}{2}\left[ {\left( {{h^2} - {{\left( {{C_y}\tan \varphi \tan {\theta _2}} \right)}^2}} \right){X_2} + } \right.}\\ {\left. {\frac{{{C_y}{{\tan }^2}\varphi \tan {\theta _2}}}{{\cos {\theta _2}}}X_2^2 - \frac{{{{\tan }^2}\varphi }}{{3{{\cos }^2}{\theta _2}}}X_2^3} \right]_{{t_9}}^{{t_{10}}}} \end{array} $$ (30) $$ \begin{array}{*{20}{c}} {{F_6}{{\bar Z}_6} = }\\ {\int_{{t_{11}}}^{{t_{12}}} {\left( {{P_5}{P_6} - {P_1}{P_2}} \right)\left( {{P_1}{P_2} + \frac{1}{2}\left( {{P_5}{P_6} - {P_1}{P_2}} \right)} \right){\rm{d}}{X_2}} = }\\ {\frac{1}{2}\left[ {\left( {C_2^2 - {{\left( {{C_y}\tan \varphi \tan {\theta _2}} \right)}^2}} \right){X_2} + \left( {{C_1}{C_2} + } \right.} \right.}\\ {\left. {\left. {\frac{{{C_y}{{\tan }^2}\varphi \tan {\theta _2}}}{{\cos {\theta _2}}}} \right)X_2^2 - \frac{1}{3}\left( {C_1^2 + \frac{{{{\tan }^2}\varphi }}{{{{\cos }^2}{\theta _2}}}} \right)X_2^3} \right]_{{t_{11}}}^{{t_{12}}}} \end{array} $$ (31) 如图 5所示，形心到螺杆轴的距离公式为
$$ {{\bar R}_{ci}} = \left( {\sqrt {C_x^2 + C_y^2} - {{\bar Z}_{ci}}} \right)\sin \varphi - \left( {{r_2}\sin {\theta _2} + {{\bar X}_{ci}}} \right)\cos \varphi $$ (32) 3.2 螺槽容积和扭矩数学模型
单螺槽容积是指整个螺槽的容积，考虑了星轮齿完全啮入螺槽之前的那部分容积，单螺槽容积同样分为7部分计算，然后求和得到整个螺槽的容积
$$ {V_{{\rm{whole}}}} = {i_{12}}\sum\limits_{i = 1}^7 {\int_{{\theta _{{\rm{c}}i{\rm{s}}}}}^{{\theta _{{\rm{c}}i{\rm{e}}}}} {{F_{{\rm{c}}i}}{{\bar R}_{{\rm{c}}i}}{\rm{d}}{\theta _2}} } $$ (33) 式中：θcis为第i部分的星轮开始角；θcie为第i部分星轮结束角.
密闭螺槽容积是指星轮齿恰好完全啮入螺槽时的螺槽容积，记为封闭螺槽容积Vsealed. 公式为
$$ {V_{{\rm{sealed}}}} = {i_{12}}\sum\limits_{i = 6}^7 {\int_{{\theta _{{\rm{c}}i{\rm{s}}}}}^{{\theta _{{\rm{c}}i{\rm{e}}}}} {{F_{{\rm{c}}i}}{{\bar R}_{{\rm{c}}i}}{\rm{d}}{\theta _2}} } $$ (34) 若求任意时刻的螺槽容积Vany，只需知道对应时刻星轮的转角θ，看对应时刻星轮转角属于哪个区域，把对应区域的积分下限换为θ即可. 比如θc6s≤θ<θc6e, 则对应的螺槽容积为
$$ {V_{{\rm{any}}}} = {i_{12}}\left( {\int_\theta ^{{\theta _{{\rm{c}}6{\rm{e}}}}} {{F_{{\rm{c}}i}}{{\bar R}_{{\rm{c}}i}}{\rm{d}}{\theta _2}} + \int_{{\theta _{{\rm{c}}7{\rm{s}}}}}^{{\theta _{{\rm{c}}7{\rm{e}}}}} {{F_{{\rm{c}}i}}{{\bar R}_{{\rm{c}}i}}{\rm{d}}{\theta _2}} } \right) $$ (35) 由于PC型单螺杆压缩机一个工作过程内同时完成2次压缩2次排气，因此，理论排气量可以利用式
$$ \dot V = 2n{z_1}{V_{\rm{s}}} $$ (36) 计算. 假定气体为绝热压缩过程，槽中的气体压力按公式
$$ p = {p_{{\rm{in}}}}{\left( {\frac{{{V_{\rm{s}}}}}{{{V_{\rm{d}}}}}} \right)^\kappa } $$ (37) 计算. 式中：pin代表进气压力；Vs代表吸气结束时的螺槽容积；Vd代表排气开始时的螺槽容积；κ代表气体绝热指数.
根据压缩机工作过程，瞬时扭矩的计算分为相应的3部分来计算. 假设整个压缩过程为绝热压缩，排气背压等于排气开始的压力，并且假设作用在星轮片下表面的压力为排气压力，则作用在一个星轮齿上的瞬时扭矩计算公式为
$$ T = \left\{ \begin{array}{l} \left( {{p_{{\rm{out}}}} - {p_{{\rm{in}}}}} \right){F_{{\rm{c}}i}}{{\bar R}_{{\rm{c}}i}},\;\;\;\;\;\;\;\;\;\;\;\;{\theta _{{\rm{cls}}}} \le {\theta _2} \le {\theta _{{\rm{se}}}}\\ \left( {{p_{{\rm{out}}}} - p\left( {{V_{{\rm{ce}}}},{s_{{\rm{in}}}}} \right)} \right){F_{{\rm{c}}i}}{{\bar R}_{{\rm{c}}i}},\;\;{\theta _{{\rm{se}}}} < {\theta _2} \le {\theta _{{\rm{ds}}}}\\ \left( {{p_{{\rm{out}}}} - {p_{{\rm{out}}}}} \right){F_{{\rm{c}}i}}{{\bar R}_{{\rm{c}}i}},\;\;\;\;\;\;\;\;\;\;\;{\theta _{{\rm{ds}}}} < {\theta _2} \le {\theta _{{\rm{c7e}}}} \end{array} \right. $$ (38) 在螺杆与星轮啮合过程中，每次啮合可能会有2、3个齿与螺槽啮合. 由于星轮齿是均布的，因此任意转角时，PC压缩机的瞬时扭矩TS为
$$ {T_{\rm{S}}} = T\left( {{\theta _2}} \right) + T\left( {{\theta _2} + \gamma } \right) + T\left( {{\theta _2} + 2\gamma } \right) $$ (39) 平均扭矩 $\overline T $ 模型公式为
$$ \bar T = \frac{{\int_{{t_1}}^{{t_2}} {{T_{\rm{S}}}{\rm{d}}t} }}{{{t_2} + {t_1}}} $$ (40) 扭矩比λtr是螺杆转子不同转角下对应的理论瞬时扭矩与压缩机的平均扭矩比值. 扭矩比反映单螺杆压缩机是否可以平稳运行. 如果转矩比越接近于1，表明单螺杆压缩机运行平稳. 扭矩比公式为
$$ {\lambda _{{\rm{tr}}}} = \frac{{{T_{\rm{S}}}}}{{\bar T}} $$ (41) 4. 封闭螺旋线和排气孔口
PC型单螺杆压缩机靠轴向吸气且总是在螺槽最深处吸气，吸气阻力很小. 但是，此种压缩机也可以开径向吸气孔口，其目的是减少单螺杆齿顶与气缸内壁之间的摩擦损失.
既然有径向孔口，就要计算单螺杆转子的理论封闭螺旋线. 理论封闭螺旋线如图 6所示，是指星轮齿后侧边与螺杆外圆的交线，即直线P5P6与P1P2的交点X51在YSOSZS形成的轨迹. 所以封闭螺旋线公式为
$$ \left\{ \begin{array}{l} {Z_{565}} = \left( {{X_{56}}\cos {\theta _2} - {r_2}\cos \beta \sin {\theta _2}} \right)\sin \varphi - \\ \;\;\;\;\;\;\;\;\left( { - {C_1}{X_{56}} + {C_2}} \right)\cos \varphi + {C_z}\\ s = {i_{12}}{r_1}\left( {{\theta _2} - {\alpha _1}} \right)\\ {\alpha _1} \le {\theta _2} \le \alpha + {\beta _1} \end{array} \right. $$ (42) 式中， ${X_{56}} = \frac{{ - {C_2} - {C_y}\cot \varphi \tan {\theta _2}}}{{ - {C_1} - \cot \varphi /cos{\theta _2}}},{\alpha _1} = \arcsin (\frac{{{r_1}\sin \varphi - 0.5{b_1}}}{{{r_2}}})$ .
为获得需求的排气压力，需合理设计排气孔口，如图 6所示. PC型单螺杆压缩机排气口位置可根据已知的压力与星轮转角关系曲线来确定排气开始角θds. 当星轮齿后侧脱离螺槽时，相应的星轮转角为θ2=α+β1，这就是排气结束角，即排气孔口终端的转角位置. 若令星轮转角θ2=θds时，此时排气开始，则排气口形状公式为
$$ \left\{ \begin{array}{l} {Z_{34S}} = \left( { - {r_2}\cos \beta \sin {\theta _2} + {X_{34}}\cos {\theta _2}} \right)\sin \varphi - \\ \;\;\;\;\;\;\;\;\left( {{C_1}{X_{34}} + {C_2}} \right)\cos \varphi + {C_z}\\ s = {i_{12}}{r_1}\left( {{\theta _2} - {\theta _{{\rm{ds}}}}} \right)\\ {\alpha _{{\rm{ds}}}} \le {\theta _2} \le \alpha + {\beta _1} \end{array} \right. $$ (43) 式中 ${X_{34}} = \frac{{\left( { - {C_2} - {C_y}\cot \varphi \tan {\theta _2}} \right)}}{{\left( { - {C_1} - \cot \varphi /cos{\theta _2}} \right)}}$ .
5. 结果分析
基于上述PC型单螺杆压缩机的螺槽容积数学模型，以单螺杆直径117 mm的PC型单螺杆压缩机为例，分析螺槽容积导数、星轮吸气结束角以后的螺槽容积、螺槽中气体压力及理论内容积比随着星轮转角的变化，并且用三维软件Pro/E画出整个单螺槽，验证整个单螺槽容积模型；基于扭矩数学模型，分析了设计内容积比为5时的瞬时扭矩和扭矩比随着螺杆转角的变化. 单螺杆直径117 mm的PC型单螺杆压缩机的具体结构参数如表 2所示.
表 2 PC型单螺杆压缩机结构参数Table 2. Structure parameters of PC-type SSC参数 数值 啮合传动比 17/6 螺杆直径/mm 117 星轮直径/mm 100 Zg轴与Ys轴的夹角 $\varphi $ /(°) 30 齿轮齿侧面倾斜角λ/(°) 0 星轮齿齿长/mm 25 通过数值积分，上述直径117 mm单螺杆整个单螺槽容积是19 567.78 mm3. 利用三维软件对整个单螺槽进行建模，如图 7所示，通过用软件分析测量模型显示整个单螺槽容积为18 856.7 mm3. 整个螺槽容积计算值与三维软件测量的螺槽容积相比，误差为3.771%. 所以文中螺槽容积的数学模型可以用来预测整个螺槽容积、密闭螺槽容积、槽中气体的压力变化. 117 mm的最大密闭容积为10 777.657 mm3，所以PC型单螺杆压缩机适用于微小气量领域.
如图 8所示，螺槽容积的导数随着星轮转角增加先增加后减小，几乎成对称趋势. 如图 9所示，当星轮转过吸气结束角时，螺槽容积随着星轮转角的增加而减小. 如图 10所示，螺杆直径为117 mm的单级PC单螺杆压缩机的内容积比最大可以达到33左右，说明了PC型单螺杆压缩机具有大压比的特点. 如图 11所示，螺槽中气体压力随着星轮转角的增加而增加，但当增加到排气开始角时PC型单螺杆压缩机开始排气，如果不考虑背压损失，如图 11所示，螺槽中气体压力保持不变，为一条直线. 并且图中显示随着进气压力的增加，螺槽中气体压力也是增加的. 综合图 9、11所示，设计PC型单螺杆压缩机时，根据用户需求背压，根据式(37)确定排气压力对应的螺槽容积Vd，然后反求排气开始角，最后根据式(43)确定排气孔口位置，如图 6所示.
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图 8 螺槽容积导数随着星轮转角的变化Figure 8. Differential coefficient of groove volume versus rotary angle of gate rotor![]()
图 9 吸气结束以后，螺槽容积随着星轮转角的变化Figure 9. Groove volume after suction end versus rotary angle of gate rotor![]()
图 11 螺槽气体压力随着星轮转角的变化Figure 11. Pressure in groove volume versus rotary angle of gate rotor假设设计内容积比为5，进气状态为设定压力下的饱和蒸汽，如图 12所示，单螺杆压缩机扭矩每隔60°重复出现. 在一个周期中，无论进口压力等于多少，相对的瞬时扭矩最大值和最小值均分别出现在单螺杆转子转过的相同角度附近. 从图中还可以看出，单螺杆压缩机的扭矩随着进口压力的增加而增加.
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图 12 PC型单螺杆压缩机扭矩随着螺杆转角的变化Figure 12. Torque of PC-type single-screw compressor versus rotary angle of single-screw rotor如图 13所示，PC型单螺杆压缩机的扭矩比与进气压力无关，并且在等于1的附近浮动，浮动值为±0.06左右，表明PC型单螺杆压缩机运行平稳.
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图 13 PC型单螺杆压缩机扭矩比随着单螺杆转角的变化Figure 13. Torque ratio of PC-type single-screw compressor versus rotary angle of single-screw rotor6. 结论
1) 给出了PC型单螺杆压缩机结构的基本关系、星轮齿啮入面积和形心坐标公式，在此基础上，进一步给出了整个单螺槽容积、密闭螺槽容积和任意时刻螺槽容积的数学模型以及扭矩数学模型，这些公式是计算理论输气量、螺槽中气体压力和功耗的基础，是设计符合用户需求的依据. 给出了排气孔口和封闭螺旋线的公式.
2) 以螺杆直径117 mm的PC型单螺杆压缩机为例，分析了螺槽容积、内容积比和排气压力随着星轮转角的变化，发现内容积比的理论极限可以达到33左右，说明PC型单螺杆压缩机适用于高压比场合.
3) 根据扭矩数学模型，发现PC型单螺杆瞬时扭矩和扭矩比随着螺杆转角呈现周期性变化，扭矩比在1附近浮动，与进气压力无关，表明PC型单螺杆压缩机可以稳定运行.
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图 1 PC型单螺杆压缩机工作原理示意图
Figure 1. Working principle diagram of PC-type single-screw compressor
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图 4 星轮齿啮入螺槽所形成的相交形状
Figure 4. Intersection part between gate rotor tooth and screw groove
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图 8 螺槽容积导数随着星轮转角的变化
Figure 8. Differential coefficient of groove volume versus rotary angle of gate rotor
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图 9 吸气结束以后，螺槽容积随着星轮转角的变化
Figure 9. Groove volume after suction end versus rotary angle of gate rotor
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图 11 螺槽气体压力随着星轮转角的变化
Figure 11. Pressure in groove volume versus rotary angle of gate rotor
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图 12 PC型单螺杆压缩机扭矩随着螺杆转角的变化
Figure 12. Torque of PC-type single-screw compressor versus rotary angle of single-screw rotor
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图 13 PC型单螺杆压缩机扭矩比随着单螺杆转角的变化
Figure 13. Torque ratio of PC-type single-screw compressor versus rotary angle of single-screw rotor
表 1 星轮齿啮入面积和形心坐标
Table 1 Intermeshing tooth area and centroid coordinates
序号 星轮齿啮入面积和形心 积分上下限取值 1 $\begin{array}{l} {F_{{\rm{c}}1}} = {F_1} + {F_2}\\ {\overline X _{{\rm{c}}1}} = ({F_1}{\overline X _1} + {F_2}{\overline X _2})/{F_{{\rm{c}}1}}\\ {\overline Z _{{\rm{c}}1}} = ({F_1}{\overline Z _1} + {F_2}{\overline Z _2})/{F_{{\rm{c}}1}} \end{array}$ $\begin{array}{l} {t_1} = ( - H + {C_z} - {r_1})/\sin \varphi + {C_y}\cot \varphi \tan {\theta _2})/\cot \varphi \\ {t_2} = {t_3} = 0.5b\\ {t_4} = ( - {C_2} + ({C_z} - {r_1})/\sin \varphi + {C_y}\cot \varphi \tan {\theta _2})/({C_1} + \cot \varphi /cos \; {\theta _2}) \end{array}$ 2 $\begin{array}{*{20}{l}} {{F_{{\rm{c2}}}} = {F_3} + {F_1} + {F_2}}\\ {{{\bar X}_{{\rm{c2}}}} = ({F_3}{{\bar X}_3} + {F_1}{{\bar X}_1} + {F_2}{{\bar X}_2})/{F_{{\rm{c2}}}}}\\ {{{\bar Z}_{{\rm{c2}}}} = ({F_3}{{\overline Z }_3} + {F_1}{{\bar Z}_1} + {F_2}{{\bar Z}_2})/{F_{{\rm{c2}}}}} \end{array}$ $\begin{array}{*{20}{l}} {{t_5} = ( - {C_2} + {C_z} - {r_1})/\sin \varphi + {C_y}\cot \varphi \tan {\theta _2})/( - {C_1} + \cot \varphi /{\rm{cos}}\;{\theta _2})}\\ \begin{array}{l} {t_6} = {t_1} = - 0.5b\\ {t_2} = {t_3} = 0.5b \end{array}\\ {{t_4} = ( - {C_2} + ({C_z} - {r_1})/\sin \varphi + {C_y}\cot \varphi \tan {\theta _2})/({C_1} + \cot \varphi /{\rm{cos}}\;{\theta _2})} \end{array}$ 3 $\begin{array}{*{20}{l}} {{F_{{\rm{c3}}}} = {F_3} + {F_1} + {F_2} + {F_4}}\\ {{{\bar X}_{{\rm{c3}}}} = ({F_3}{{\bar X}_3} + {F_1}{{\bar X}_1} + {F_2}{{\bar X}_2} + {F_4}{{\bar X}_4})/{F_{{\rm{c3}}}}}\\ {{{\bar Z}_{{\rm{c3}}}} = ({F_3}{{\bar Z}_3} + {F_1}{{\bar Z}_1} + {F_2}{{\bar Z}_2} + {F_4}{{\bar Z}_4})/{F_{{\rm{c3}}}}} \end{array}$ $\begin{array}{*{20}{l}} {{t_5} = ( - {C_2} + {C_z} - {r_1})/\sin \varphi + {C_y}\cot \varphi \tan {\theta _2})/( - {C_1} + \cot \varphi /{\rm{cos}}\;{\theta _2})}\\ {\begin{array}{*{20}{l}} {{t_6} = {t_1} = - 0.5b}\\ {{t_2} = {t_3} = 0.5b}\\ {{t_4} = {t_7} = - {r_2}\sin {\theta _2}} \end{array}}\\ {{t_8} = ( - {C_2} - {C_y}\tan \varphi \tan {\theta _2})/({C_1} - \tan \varphi /{\rm{cos}}\;{\theta _2})} \end{array}$ 4 $\begin{array}{*{20}{l}} {{F_{{\rm{c4}}}} = {F_3} + {F_1} + {F_5} + {F_4}}\\ {{{\bar X}_{{\rm{c4}}}} = ({F_3}{{\bar X}_3} + {F_1}{{\bar X}_1} + {F_5}{{\bar X}_5} + {F_4}{{\bar X}_4})/{F_{{\rm{c4}}}}}\\ {{{\bar Z}_{{\rm{c4}}}} = ({F_3}{{\bar Z}_3} + {F_1}{{\bar Z}_1} + {F_5}{{\bar Z}_5} + {F_4}{{\bar Z}_4})/{F_{{\rm{c4}}}}} \end{array}$ $\begin{array}{*{20}{l}} {{t_5} = ( - {C_2} + {C_z} - {r_1})/\sin \varphi + {C_y}\cot \varphi \tan {\theta _2})/( - {C_1} + \cot \varphi /{\rm{cos}}\;{\theta _2})}\\ {\begin{array}{*{20}{l}} {{t_6} = {t_1} = - 0.5b}\\ {{t_2} = {t_9} = - {r_2}\sin {\theta _2}}\\ {{t_{10}} = {t_7} = 0.5b} \end{array}}\\ {{t_8} = ( - {C_2} - {C_y}\tan \varphi \tan {\theta _2})/({C_1} - \tan \varphi /{\rm{cos}}\;{\theta _2})} \end{array}$ 5 $\begin{array}{*{20}{l}} {{F_{{\rm{c5}}}} = {F_3} + {F_6} + {F_5} + {F_4}}\\ {{{\bar X}_{{\rm{c5}}}} = ({F_3}{{\bar X}_3} + {F_6}{{\bar X}_6} + {F_5}{{\bar X}_5} + {F_4}{{\bar X}_4})/{F_{{\rm{c5}}}}}\\ {{{\bar Z}_{{\rm{c5}}}} = ({F_3}{{\bar Z}_3} + {F_6}{{\bar Z}_6} + {F_5}{{\bar Z}_5} + {F_4}{{\bar Z}_4})/{F_{{\rm{c5}}}}} \end{array}$ $\begin{array}{*{20}{l}} {{t_5} = ( - {C_2} + {C_z} - {r_1})/\sin \varphi + {C_y}\cot \varphi \tan {\theta _2})/( - {C_1} + \cot \varphi /{\rm{cos}}\;{\theta _2})}\\ \begin{array}{l} {t_6} = {t_{11}} = - {r_2}\sin {\theta _2}\\ {t_{12}} = {t_9} = - 0.5b\\ {t_{10}} = {t_7} = 0.5b \end{array}\\ {{t_8} = ( - {C_2} - {C_y}\tan \varphi \tan {\theta _2})/({C_1} - \tan \varphi /{\rm{cos}}\;{\theta _2})} \end{array}$ 6 $\begin{array}{*{20}{l}} {{F_{{\rm{c6}}}} = {F_6} + {F_5} + {F_4}}\\ {{{\bar X}_{{\rm{c6}}}} = ({F_6}{{\bar X}_6} + {F_5}{{\bar X}_5} + {F_4}{{\bar X}_4})/{F_{{\rm{c5}}}}}\\ {{{\bar Z}_{{\rm{c6}}}} = ({F_6}{{\bar Z}_6} + {F_5}{{\bar Z}_5} + {F_4}{{\bar Z}_4})/{F_{{\rm{c5}}}}} \end{array}$ $\begin{array}{*{20}{l}} {{t_{11}} = ( - {C_2} - {C_y}\cot \varphi \tan {\theta _2})/( - {C_1} - \cot \varphi /{\rm{cos}}\;{\theta _2})}\\ \begin{array}{l} {t_{12}} = {t_9} = - 0.5b\\ {t_{10}} = {t_7} = 0.5b \end{array}\\ {{t_8} = ( - {C_2} - {C_y}\tan \varphi \tan {\theta _2})/({C_1} - \tan \varphi /{\rm{cos}}\;{\theta _2})} \end{array}$ 7 $\begin{array}{*{20}{l}} {{F_{{\rm{c7}}}} = {F_6} + {F_5}}\\ {{{\bar X}_{{\rm{c7}}}} = ({F_6}{{\bar X}_6} + {F_5}{{\bar X}_5})/{F_{{\rm{c6}}}}}\\ {{{\bar Z}_{{\rm{c7}}}} = ({F_6}{{\bar Z}_6} + {F_5}{{\bar Z}_5})/{F_{{\rm{c6}}}}} \end{array}$ $\begin{array}{*{20}{l}} {{t_{11}} = ( - {C_2} - {C_y}\cot \varphi \tan {\theta _2})/( - {C_1} - \cot \varphi /{\rm{cos}}\;{\theta _2})}\\ {{t_{12}} = {t_9} = - 0.5b}\\ {{t_{10}} = ( - h + {C_y}\tan {\theta _2}\tan \varphi ){\rm{cos}}\;{\theta _2}/\tan \varphi } \end{array}$ 
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表 2 PC型单螺杆压缩机结构参数
Table 2 Structure parameters of PC-type SSC
参数 数值 啮合传动比 17/6 螺杆直径/mm 117 星轮直径/mm 100 Zg轴与Ys轴的夹角 $\varphi $ /(°) 30 齿轮齿侧面倾斜角λ/(°) 0 星轮齿齿长/mm 25
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[1] ZIMMERN B. Worm rotary compressors with liquid joints: US 3180565 A[P]. 1965-04-27.
[2] ZIMMERN B, PATEL G C. Design and operating characteristics of the Zimmern single screw compressor[J]. Japanese Journal of Applied Physics, 1974, 13(12): 411-414. http://industry.wanfangdata.com.cn/dl/Detail/NSTLHY?id=NSTLHY_NSTL_HYCC0211231025
[3] ZIMMERN B. Rotary interengaging worm and worm wheel with specific tooth shape: US 3932077 A[P]. 1976-01-13.
[4] WANG W, WU Y T, MA C F, et al. Experimental study on the performance of single screw expanders by gap adjustment[J]. Energy, 2013, 62(6): 379-384. http://www.doc88.com/p-9035334634182.html
[5] ZHANG Y Q, WU Y T, XIA G D, et al. Development and experimental study on organic Rankine cycle system with single-screw expander for waste heat recovery from exhaust of diesel engine[J]. Energy, 2014, 77: 499-508. doi: 10.1016/j.energy.2014.09.034
[6] 汤炎, 金光熹. CC型单螺杆压缩机的设计基础[J]. 流体机械, 1990(4): 21-28. http://www.cnki.com.cn/Article/CJFDTOTAL-LTJX199004004.htm TANG Y, JIN G X. The design basis of CC-type single-screw compressor[J]. Fluid Machinery, 1990(4): 21-28. (in Chinese) http://www.cnki.com.cn/Article/CJFDTOTAL-LTJX199004004.htm
[7] 汤炎, 金光熹. CC型单螺杆压缩机主要几何关系的研究[J]. 西安交通大学学报, 1989(3): 7-15 http://www.cnki.com.cn/Article/CJFDTOTAL-XAJT198903001.htm TANG Y, JIN G X. Study of some basic geometric problems of a CC-type single-screw compressor[J]. Journal of Xi'an Jiaotong University, 1989(3): 7-15. (in Chinese) http://www.cnki.com.cn/Article/CJFDTOTAL-XAJT198903001.htm
[8] 孙光三. 单螺杆压缩机主要几何关系的研究[J]. 西安交通大学学报, 1981(5): 24-35. http://www.cnki.com.cn/Article/CJFDTOTAL-XAJT198903001.htm SUN G S. The study of some basic geometric problems of the monoscrew compressors[J]. Journal of Xi'an Jiaotong University, 1981(5): 24-35. (in Chinese) http://www.cnki.com.cn/Article/CJFDTOTAL-XAJT198903001.htm
[9] SUN G. The investigation of some basic geometric problems of the single screw compressor[J]. China Quarterly, 1988, 173(173): 246-247.
[10] 王增丽, 冯全科. 单螺杆压缩机几何参数对压缩机性能的影响[J]. 流体机械, 2014(11): 34-37. doi: 10.3969/j.issn.1005-0329.2014.11.007 WANG Z L, FENG Q K. Effect of geometry parameters on performance of single screw compressor[J]. Fluid Machinery, 2014(11): 34-37. (in Chinese) doi: 10.3969/j.issn.1005-0329.2014.11.007
[11] 张淑存, 余心源, 金光熹. 单螺杆压缩机啮合副主要几何关系研究[J]. 压缩机技术, 2006(6): 21-23. http://www.cnki.com.cn/Article/CJFDTOTAL-YSJJ200606006.htm ZHANG S C, YU X Y, JIN G X. Research of the essential geometric relations of engagement pair for single screw compressor[J]. Compressor Technology, 2006(6): 21-23. (in Chinese) http://www.cnki.com.cn/Article/CJFDTOTAL-YSJJ200606006.htm
[12] YANG S C. A mathematical model of the rotor profile of the single-screw compressor[J]. Proceedings of the Institution of Mechanical Engineers Part C, 2002, 216(3): 343-351. doi: 10.1243/0954406021525052
[13] YANG S C. A mathematical model of a CC-type single-screw compressor[J]. Proceedings of the Institution of Mechanical Engineers Part C, 2004, 218(4): 437-448. doi: 10.1177/095440620421800408
[14] YANG S C. Profile generation and analysis for a PP-type single-screw compressor[J]. The International Journal of Advanced Manufacturing Technology, 2006, 30(9): 789-796. doi: 10.1007/s00170-005-0132-8
[15] YANG S C, HUANG T H, LAI C H. Modeling and manufacture of a PC-type single-screw compressor rotor[J]. Arabian Journal for Science and Engineering, 2014, 39(5): 4221-4229. doi: 10.1007/s13369-014-1078-7
[16] 查谦, 方宜荣. PC型单螺杆压缩机[J]. 压缩机技术, 1995(2): 6-11. http://cdmd.cnki.com.cn/Article/CDMD-10530-1013301518.htm ZHA Q, FANG Y R. PC-type single-screw compressor[J]. Compressor Technology, 1995(2): 6-11. (in Chinese) http://cdmd.cnki.com.cn/Article/CDMD-10530-1013301518.htm
[17] 许明文. PC型单螺杆压缩机型线设计及关键零部件制造实验研究[D]. 湘潭: 湘潭大学, 2012. XU M W. Study on profile generation and manufacture for critical components of the PC-type single screw compressor[D]. Xiangtan: Xiangtan University, 2012. (in Chinese)
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