TY - JOUR
AU - Lin,, Jiyou
AB - Abstract This paper analyzes rotor profile and meshing law of twin-screw expander. Elementary area is calculated by meshing sequence method, and elementary volume is solved by integration. The influence of geometric characteristics such as torsion angle and internal expansion angle of male rotor on the thermal performance of twin-screw expander is studied. The results show the following: reasonably reducing internal expansion angle and positive rotor torsion angle can increase isentropic adiabatic power of twin-screw expander. Taking isentropic adiabatic efficiency as optimization goal, sequential quadratic programming algorithm is used to optimize the structural parameters of twin-screw expander, and the influence of leakage on its functional force is analyzed. Considering leakage effect, actual shaft power of twin-screw expander is 16.7 kW, the isentropic adiabatic power under ideal conditions is 21.3 kW, and the adiabatic efficiency is 78.71%. 1 INTRODUCTION Twin-screw expander has the advantages of high efficiency, good stability and strong adaptability, and has broad application prospects in the field of energy recovery [1]. The research on the factors and laws affecting thermal performance of twin-screw expander is an important prerequisite for improving the operating efficiency of organic Rankine cycle and improving the recovery capacity of the system [2]. Research on rotor profile is an important basis for studying performance of twin-screw expanders. At present, there are few studies on the expander rotor profile at home and abroad, and the research focuses on the profile line of the compressor. [3] optimized rotor profile and compressor housing size by constructing a rotor profile mathematical model. Ma et al. [4] designed a new type of screw compressor rotor profile and optimized the design of rotor profile. Wu and Fong [5] proposed that the performance of the screw compressor and the pros and cons of rotor profile can be judged according to the shape of the seal line when the male and female rotors mesh. Although internal working process of the compressor and the expander is reversed, the law of rotor meshing and the thermal characteristics of the two types of equipment will vary with the angle of rotation. At present, experimental methods are used to study the thermal performance of expanders under a specific working medium and specific working conditions to obtain a fitting empirical correlation [6]. Yeqiang Zhang [7] of Beijing University of Technology conducted a performance experiment on a single-screw expander to study the influence of factors such as screw speed, expansion ratio and back pressure on the performance of a single-screw expander. Starting from the geometric parameters of the expander, the research on the overall thermal performance parameters of the expander is still rare, and there is no performance optimization method based on structural parameters. This paper constructs a mathematical model of rotor profile of twin-screw expander and the model is used to study the influence of geometric features on the thermal performance of twin-screw expander. In this paper, the multivariate function non-linear programming method is used to optimize the design of isentropic adiabatic efficiency, and the influence of leakage on the thermal performance of twin-screw expander is analyzed. 2 RESEARCH ON GEOMETRIC CHARACTERISTICS OF TWIN-SCREW EXPANDER 2.1 Introduction to geometric feature parameters The geometric characteristics of twin-screw expander mainly include area among the teeth, elementary volume, area utilization coefficient and torsion angle coefficient. Wherein, elementary volume is a single inter-tooth volume enclosed by male and female rotors and the chassis. Inter-tooth area is the projection of elementary volume on rotor end face. Area utilization factor of twin-screw expander Cn1 characterizes the extent of utilization of the total area within rotor diameter range, which is Cn1=Z1(S1+S2)D12 (1) Cφ=V0Vmax (2) Vmax=S1L+S2L (3) L=τ1360T1 (4) The study of the geometric characteristics of twin-screw expander needs to first determine area between the teeth and the elementary volume, then determine area utilization factor and torsion angle coefficient, and finally effect of geometric characteristics on the thermal performance of a twin-screw expander. 2.2 Meshing sequence method for calculating the area between teeth The mathematical description of twin-screw expander rotor profile includes dynamic and static coordinate system of male and female rotor, the envelope condition and the equations of the teeth of male and female rotors. In this paper, the unilateral asymmetric cycloidal-pin-tooth arc-shaped rotor profile is taken as the research object, and inter-tooth area of twin-screw expander is solved. This paper proposes the meshing sequence method to calculate inter-dental encroachment area of each micro-vehicle corner, and area between the teeth at each meshing moment is solved by the difference between the maximum inter-tooth area and the inter-tooth interfering area. The principle of the meshing sequence method is tooth curve equations of male and female rotors represent the angles at which the corresponding points on the male and female rotor curves need to be rotated when they mesh with each other. Through transformation of the dynamic and static coordinates, the coordinates of each point on tooth curve of each corner can be obtained. The curve formed by these points and area surrounded by the top circle of male and female rotors is area between rotor teeth at the moment of the corner. The specific steps are the following: Determine tooth curve and the tip circle that surrounds area between the teeth at a certain moment. Then, the coordinates of each point are determined by the mathematical expression of male and female rotor tooth curves and the addendum circle which surround elementary area at this moment. Through the dynamic and static coordinate transformation formulas, the coordinates of the points of male and female rotor teeth and the tip circle that participate in the encroachment area between the envelope teeth can be obtained. The dynamic and static coordinate transformation formulas is {x′1=x1cosφi−y1sinφiy′1=x1sinφi+y1cosφi (5) After the coordinates of each point are known, the trapezoidal calculation method is used to calculate the intrusion area between the teeth of male and female rotors and tooth envelope of the crest. The trapezoidal calculation method is S(φi)=∑j=1n[xj(φi)−xj−1(φi)]⋅[yj(φi)+yj−1(φi)]2 (6) Area between the teeth at each meshing moment can be obtained by the difference between the maximum inter-tooth area and inter-tooth encroachment area obtained by the meshing sequence method. Among them, a schematic diagram of the meshing of tooth curve BA and HG is shown in Figure 1. Figure 1. Open in new tabDownload slide Meshing diagram of tooth curve BA and HG. Figure 1. Open in new tabDownload slide Meshing diagram of tooth curve BA and HG. 2.3 Integral calculation method of primitive volume For the calculation of the maximum elementary volume, it is necessary to clarify the variation of area between the teeth and the elementary volume. The specific change law is: the first phase (0≤φ≤φ1) with the increase of the encroachment of male and female rotors, area between the teeth projected on the suction end face gradually increases elementary volume. When male rotor is rotated through φ1 angle, the teeth of female rotor completely sweep across inter-tooth area of male rotor on the suction end face. The second phase (φ1≤φ≤τ1) indicates from the end of the first phase to the time when male rotor turns a torsion angle τ1 of male rotor. At this phase, with the advancement to the exhaust end face, area between the teeth remains unchanged, and elementary volume is continuously increased. The third phase (τ1≤φ≤τ1+φ1) indicates that from the end of the second stage to the time when male and female rotors are gradually released from engagement. Similar to the first phase, male rotor turned over the angle of τ1+φ1 ⁠, and area occupied by the teeth is gradually reduced, and the increase in elementary volume is maximized. The basic dimensions of rotor are shown in Table 1. The change rule of elementary area at each stage is shown in Figure 2. The maximum elementary volume can be found by integrating inter-tooth area with the angle of male rotor: V0=V0(φ)=T12π∫0φS(φ)dφ (7) Table 1. Design parameter table. τ1 ϕ1 D1 D2 λ L R A Z1 Z2 ps 270 124.5 0.084 0.0768 1.5 0.12 0.0164 0.064 0.004 0.006 1630 τ1 ϕ1 D1 D2 λ L R A Z1 Z2 ps 270 124.5 0.084 0.0768 1.5 0.12 0.0164 0.064 0.004 0.006 1630 Note: The angle is radians, and the pressure unit is kPa. Open in new tab Table 1. Design parameter table. τ1 ϕ1 D1 D2 λ L R A Z1 Z2 ps 270 124.5 0.084 0.0768 1.5 0.12 0.0164 0.064 0.004 0.006 1630 τ1 ϕ1 D1 D2 λ L R A Z1 Z2 ps 270 124.5 0.084 0.0768 1.5 0.12 0.0164 0.064 0.004 0.006 1630 Note: The angle is radians, and the pressure unit is kPa. Open in new tab Figure 2. Open in new tabDownload slide The relationship between the inter-dental area occupied by rotor and the male rotor torsion angle. Figure 2. Open in new tabDownload slide The relationship between the inter-dental area occupied by rotor and the male rotor torsion angle. The variation of elementary volume obtained by integration is shown in Figure 3. During the engagement of male and female rotor teeth of twin-screw expander from the beginning to the disengagement, male rotor turned over the angle of τ1+φ1 ⁠. During this time, elementary volume increases from zero to the maximum, completing the inspiration and expansion process. At the same time, elementary volume on the other side is reduced from the maximum value to zero, completing the exhaust process. Figure 3. Open in new tabDownload slide The relationship between the element volume and the male rotor torsion angle. Figure 3. Open in new tabDownload slide The relationship between the element volume and the male rotor torsion angle. 3 ANALYSIS OF THERMAL PERFORMANCE OF TWIN-SCREW EXPANDER 3.1 Twin-screw expander thermal performance The thermal performance of twin-screw expander mainly includes internal expansion ratio, volumetric flow rate, and isentropic adiabatic power. In order to facilitate the analysis of the thermal performance of twin-screw expander and ignore some minor factors, this paper makes the following assumptions: The thermal process of twin-screw expander is an ideal process, and the working fluid is an ideal gas. Calculation condition meets the pressure and temperature values in the design conditions. That is, internal and external expansion ratios are the same. Internal expansion ratio εi means that the ratio of the pressure at the end of the working fluid intake to the instantaneous pressure before the start of the exhaust is εi=pspd=(V0Vi)k=εvk (8) εv=V0Vi=Cφτ1τ1−φc+φ1−AaS (9) Theoretical volumetric flow rate qvt is the sum of rotor’s volume per unit time per unit time, which is qvt=CφCn1nT1D12 (10) The expanding gas can be treated as an ideal gas, so isentropic adiabatic power of twin-screw expander Pad is Pad=kk−1psqvt[(pspd)k−1k−1]=kk−1psqvt[εik−1k−1] (11) 3.2 Analysis of influence of geometric characteristics on thermal performance of expander For fixed-line twin-screw expanders, the relationship between the number of teeth of male and female rotors, male and female rotor outer diameter, the radius of the teeth, and the nominal diameter have been determined. Therefore, the key structural parameters affecting the geometric features are lead, rotor diameter, male rotor torsion angle and internal expansion angle. Through calculation and analysis, it can be concluded that area utilization coefficient of twin-screw expander does not change with the change of rotor diameter and rotor torsion angle, and is always a constant; At the same time, the magnitude of torsion angle factor is also independent of lead and rotor diameter. However, it has a monotonous relationship with the twist angle of male rotor, as shown in Figure 4. The specific mathematical description of torsion angle and torsion angle of male rotor can be obtained by formula fitting. The correlation coefficient of the fitting result R2 is 0.99, and the fitting result is close to the real result. At the same time, by calculation, when torsion angle of male rotor is reduced from 270° to 240° under the same conditions, the maximum elementary volume of the twin-screw expander is reduced by 12.9%. Therefore, torsion angle of male rotor has a great influence on the geometric characteristics of twin-screw expander. Figure 4. Open in new tabDownload slide The influence of male rotor torsion angle on torsional angle coefficient. Figure 4. Open in new tabDownload slide The influence of male rotor torsion angle on torsional angle coefficient. Figures 5 and 6 show the influence of lead on the thermal performance. The volume flow and lead of twin-screw expander show a monotonically increasing relationship. The change in the isotropic thermal power of lead is caused by the change of the volume flow, so isentropic adiabatic power and lead also show a monotonically increasing relationship. The influence of rotor diameter on thermal performance can also be obtained from Figures 5 and 6. Because the diameter and lead of rotor affect the thermal performance of twin-screw expander by changing the size of internal working cavity of the expander, effect of rotor diameter on the thermal performance of the expander is similar to lead. At the same time, it is found by data fitting that the volume flow and isentropic adiabatic power of twin-screw expander are proportional to the cube of rotor diameter. The correlation coefficient of the fitting result R2 is 0.99, and the fitting result is close to the real result. Figure 5. Open in new tabDownload slide The influence of lead and rotor diameter on volume flow. Figure 5. Open in new tabDownload slide The influence of lead and rotor diameter on volume flow. Figure 6. Open in new tabDownload slide The influence of lead and rotor diameter on isentropic adiabatic power. Figure 6. Open in new tabDownload slide The influence of lead and rotor diameter on isentropic adiabatic power. The variation law of torsion angle and internal expansion ratio of male rotor is shown in Figure 7. It can be seen that internal expansion ratio is monotonically decreasing with torsion angle of male rotor. However, if torsion angle of male rotor is too small, the suction opening will be too small, which makes it difficult to design the structure. From Figure 7, the influence of the expansion angle on the thermal performance of another geometric characteristic parameter affecting internal expansion ratio can also be obtained. Internal expansion ratio is in a monotonically increasing relationship with internal expansion angle. The increase of internal expansion angle can increase internal expansion ratio of twin-screw expander, but excessively increasing internal expansion angle often causes difficulties in the structural design of the suction opening, and also increases the resistance loss when the gas flows through the suction opening. Therefore, internal expansion angle is not too large. From Figure 8, it can be concluded that the volume flow rate increases with the increase of torsion angle of male rotor. This is the result of an increase in torsion angle of male rotor resulting in an increase in the torsion angle factor. From Figure 9, it can be concluded that isentropic adiabatic power of twin-screw expander is monotonously decreasing with respect to torsion angle of male rotor. The influence of torsion angle coefficient on the increase of isentropic adiabatic power is less than the influence of internal expansion ratio on isentropic adiabatic power reduction. At the same time, it can be concluded from the figure that internal expansion angle is positively correlated with isentropic adiabatic power of twin-screw expander. Figure 7. Open in new tabDownload slide The influence of male rotor torsion angle and internal expansion angle on internal expansion ratio. Figure 7. Open in new tabDownload slide The influence of male rotor torsion angle and internal expansion angle on internal expansion ratio. Figure 8. Open in new tabDownload slide The influence of male rotor torsion angle on volume flow. Figure 8. Open in new tabDownload slide The influence of male rotor torsion angle on volume flow. Figure 9. Open in new tabDownload slide The influence of male rotor torsion angle and internal expansion angle on isentropic adiabatic power. Figure 9. Open in new tabDownload slide The influence of male rotor torsion angle and internal expansion angle on isentropic adiabatic power. 3.3 Optimization of rotor profile structure parameters In this paper, isentropic adiabatic power is taken as the objective function, and theoretical volumetric flow rate is used as the constraint condition. Lead, male rotor torsion angle and internal expansion angle are selected as the optimization variables, because their influence on isentropic adiabatic power are similar. The non-linear programming method of multivariate function is used to optimize rotor profile of twin-screw expander. In this paper, the sequential quadratic programming algorithm is used to solve the above constrained non-linear optimization problem [8]. The objective function, constraints, and optimization interval are as follows: maxPad=kk−1Psqvt[(Cφτ1τ1−φc+φ1−AaS)k−1−1] (12) {qvt=CφCnn1T0D12=0.342110≤T0≤130150≤φc≤200240≤τ1≤300 (13) The results of optimizing twin-screw expander with a rotor diameter of 0.08 m, a rotational speed of 2500 rad/min, and an suction pressure of 1630 kPa are shown in Table 2. Table 2. Structural parameter optimization results. Lead T0(m) Male rotor torsion angle τ1(°) Internal expansion angle φc(°) Isentropic adiabatic power Pad(kW) 118 247 150 21.3 Lead T0(m) Male rotor torsion angle τ1(°) Internal expansion angle φc(°) Isentropic adiabatic power Pad(kW) 118 247 150 21.3 Open in new tab Table 2. Structural parameter optimization results. Lead T0(m) Male rotor torsion angle τ1(°) Internal expansion angle φc(°) Isentropic adiabatic power Pad(kW) 118 247 150 21.3 Lead T0(m) Male rotor torsion angle τ1(°) Internal expansion angle φc(°) Isentropic adiabatic power Pad(kW) 118 247 150 21.3 Open in new tab The results show that reasonable reduction of internal expansion angle and the positive rotor torsion angle can increase isentropic adiabatic power of twin-screw expander, but it will bring difficulties to the structural design of the suction opening and increase the resistance loss of the gas flow. That torsion angle of male rotor is too small will affect the working length of the expander, resulting in incomplete expansion of actual expansion process of the expander. Increasing lead and rotor diameter of twin-screw expander can increase the volumetric flow and power of the expander, but it will increase the manufacturing cost. 4 IMPACT OF LEAKAGE ON TWIN-SCREW EXPANDERS Twin-screw expander is a multi-cavity structure, and the operation of each elementary volume is the same. In this paper, a primitive volume is used as control body for research. Ignoring the process of meshing male and female rotors of twin-screw expander, control body is regarded as a working cavity surrounded by elementary volume and the wall of the casing, as shown in Figure 10. Figure 10. Open in new tabDownload slide Schematic diagram of control body. Figure 10. Open in new tabDownload slide Schematic diagram of control body. Due to leakage during actual operation of the expander, it can be seen that the mass change amount of the inflow and outflow control body suction and exhaust orifices and leakage mass leakage amount leaking into and out of control body satisfy the formula: dm=dm1−dm2=dma−dmb (14) Leakage methods that have a large impact on the performance of the expander include: leakage caused by the gap between the top of the female and male rotors and the wall of the casing and leakage caused by the leaking triangle. It is considered that leakage formed at the suction end face, the exhaust end face, contact line and a part of leakage triangle is an external leak, which does not affect the pressure change of the working cavity of twin-screw expander. Therefore, this paper only considers leakage caused by the gap between the top of male and female rotors and the wall surface of the casing [9]. An ideal gas nozzle model is adopted for controlling leakage passage in the body, and the mass change generated by leakage passage satisfies the ideal gas isentropic flow nozzle formula [10]: m0=Apρ02kRTα(k−1)(1−(pβpα)kk−1) (15) Use R134a as the working medium, and assume that the working medium is always the ideal gas during operation: Does not consider the pressure pulsation effect; Does not consider the heat exchange between the working gas and the body. Solve actual process of the process considering effects of leakage under the initial conditions described in Section 2.3. Calculate the thermal performance of actual process according to the state of suction and exhaust of the expander, and compare it with the thermal performance under ideal conditions in Section 2.3. The calculation results are shown in Table 3. Table 3. Thermal performance table. Calculation method Pd/kPa Td/K εi Pad/kW Ideal process 557.22 301.43 2.97 21.3 Actual process 594.62 302.05 2.74 16.7 Calculation method Pd/kPa Td/K εi Pad/kW Ideal process 557.22 301.43 2.97 21.3 Actual process 594.62 302.05 2.74 16.7 Open in new tab Table 3. Thermal performance table. Calculation method Pd/kPa Td/K εi Pad/kW Ideal process 557.22 301.43 2.97 21.3 Actual process 594.62 302.05 2.74 16.7 Calculation method Pd/kPa Td/K εi Pad/kW Ideal process 557.22 301.43 2.97 21.3 Actual process 594.62 302.05 2.74 16.7 Open in new tab Actual operating process of twin-screw expander has a shaft power of 16.7 kW, which is less than the ideal process isentropic adiabatic power of 21.3 kW. Therefore, under the condition that leakage exerts a functional force on twin-screw expander, the adiabatic efficiency is 78.71%. The relationship between the working chamber pressure and elementary volume under actual process of leakage is compared with the change law under ideal conditions, as shown in Figure 11. The results show the following: actual operation process is consistent with the ideal process change law. During the expansion phase, the ideal and actual operating process pressure decreases as the working chamber increases, and the pressure in each elementary volume of actual process is slightly larger than the ideal operating process. This is caused by a mass leak in actual process. Figure 11. Open in new tabDownload slide Diagram of actual process and ideal process. Figure 11. Open in new tabDownload slide Diagram of actual process and ideal process. 5 RESULTS In this paper, elementary volume is calculated by the meshing sequence method, and the influence of the geometrical parameters of twin-screw expander on the thermal performance is analyzed. Considering leakage effect comprehensively, the isotropy adiabatic efficiency is taken as the objective function, and twin-screw expander is optimized: Mathematical modeling of a twin-screw expander with a single-sided asymmetric cycloidal-pin-tooth arc-shaped rotor profile using the meshing principle and elementary area is calculated by the meshing sequence method. The influence of geometric characteristic parameters on the thermal performance of twin-screw expander is analyzed. Reasonable reduction of internal expansion angle and positive rotor torsion angle can improve isentropic adiabatic efficiency. At the same time, it will bring difficulties to the structural design of the suction opening and increase the resistance loss of the gas flow; increasing lead and rotor diameter of twin-screw expander can increase the volumetric flow and power of the expander, but the manufacturing cost will increase. Taking isentropic adiabatic efficiency as the objective function, the sequential quadratic programming algorithm is used to optimize the design. The optimal isentropic adiabatic power under ideal conditions is 21.3 kW. Taking R134a as the working medium and considering leakage effect, the shaft power of twin-screw expander is 16.7 kW and the adiabatic efficiency is 78.71%. About the Author Chenghu Zhang (1980–), male, Associate Professor, Doctor who is mainly engaged in research in the field of ‘thermal energy and thermals’. List for symbols Symbol Meaning Dimension Cn1 area utilization factor Z1 number of teeth of male rotor S1 male rotor elementary area m2 S2 female rotor elementary area m2 D1 male rotor outer diameter m Cφ torsion angle coefficient V0 actual maximum elementary volume m3 Vmax theoretical maximum elementary volume m3 L effective working length m τ1 male rotor torsion angle ° T1 lead m x1 male and female rotor tooth curve abscissa m y1 male and female rotor tooth curve ordinate m x′1 abscissa of male and female rotor tooth curve after rotor rotates by φi angle m y′1 ordinate of male and female rotor tooth curve after the rotor rotates by φi angle m S(ϕi) elementary area after rotor has turned φi angle m2 j jth point xj abscissa of the jth point m yj ordinate of the jth point φ male rotor angle ° φ1 first stage angle ° D2 female rotor outer diameter m λ transmission ratio between male and female rotors R tooth height radius m A center distance m Z2 number of teeth of female rotor εi internal expansion ratio ps gas pressure when elementary volume is disconnected from the suction port kPa pd gas pressure when elementary volume is connected to the suction port kPa k isentropic index of gas Vi elementary volume at the beginning of the expansion process m3 εv content ratio φc internal expansion angle ° S elementary area m2 Aa integral value of elementary area to the corner when the angle of male rotor is φa m2 qvt volume flow m3 n speed of male rotor r/min Pad isentropic adiabatic power kW dm amount of change in the volume of control body kg dm1 leakage into control body mass dm2 leakage out control body mass dma flow into control body from the suction port kg dmb outflow control body mass from the exhaust orifice kg m0 nozzle mass flow kg/s Ap nozzle throat area m2 ρ0 nozzle fluid density kg/m3 Tα nozzle high pressure zone temperature K pα nozzle high pressure zone pressure kPa pβ nozzle pressure in the low pressure zone kPa Symbol Meaning Dimension Cn1 area utilization factor Z1 number of teeth of male rotor S1 male rotor elementary area m2 S2 female rotor elementary area m2 D1 male rotor outer diameter m Cφ torsion angle coefficient V0 actual maximum elementary volume m3 Vmax theoretical maximum elementary volume m3 L effective working length m τ1 male rotor torsion angle ° T1 lead m x1 male and female rotor tooth curve abscissa m y1 male and female rotor tooth curve ordinate m x′1 abscissa of male and female rotor tooth curve after rotor rotates by φi angle m y′1 ordinate of male and female rotor tooth curve after the rotor rotates by φi angle m S(ϕi) elementary area after rotor has turned φi angle m2 j jth point xj abscissa of the jth point m yj ordinate of the jth point φ male rotor angle ° φ1 first stage angle ° D2 female rotor outer diameter m λ transmission ratio between male and female rotors R tooth height radius m A center distance m Z2 number of teeth of female rotor εi internal expansion ratio ps gas pressure when elementary volume is disconnected from the suction port kPa pd gas pressure when elementary volume is connected to the suction port kPa k isentropic index of gas Vi elementary volume at the beginning of the expansion process m3 εv content ratio φc internal expansion angle ° S elementary area m2 Aa integral value of elementary area to the corner when the angle of male rotor is φa m2 qvt volume flow m3 n speed of male rotor r/min Pad isentropic adiabatic power kW dm amount of change in the volume of control body kg dm1 leakage into control body mass dm2 leakage out control body mass dma flow into control body from the suction port kg dmb outflow control body mass from the exhaust orifice kg m0 nozzle mass flow kg/s Ap nozzle throat area m2 ρ0 nozzle fluid density kg/m3 Tα nozzle high pressure zone temperature K pα nozzle high pressure zone pressure kPa pβ nozzle pressure in the low pressure zone kPa Open in new tab Symbol Meaning Dimension Cn1 area utilization factor Z1 number of teeth of male rotor S1 male rotor elementary area m2 S2 female rotor elementary area m2 D1 male rotor outer diameter m Cφ torsion angle coefficient V0 actual maximum elementary volume m3 Vmax theoretical maximum elementary volume m3 L effective working length m τ1 male rotor torsion angle ° T1 lead m x1 male and female rotor tooth curve abscissa m y1 male and female rotor tooth curve ordinate m x′1 abscissa of male and female rotor tooth curve after rotor rotates by φi angle m y′1 ordinate of male and female rotor tooth curve after the rotor rotates by φi angle m S(ϕi) elementary area after rotor has turned φi angle m2 j jth point xj abscissa of the jth point m yj ordinate of the jth point φ male rotor angle ° φ1 first stage angle ° D2 female rotor outer diameter m λ transmission ratio between male and female rotors R tooth height radius m A center distance m Z2 number of teeth of female rotor εi internal expansion ratio ps gas pressure when elementary volume is disconnected from the suction port kPa pd gas pressure when elementary volume is connected to the suction port kPa k isentropic index of gas Vi elementary volume at the beginning of the expansion process m3 εv content ratio φc internal expansion angle ° S elementary area m2 Aa integral value of elementary area to the corner when the angle of male rotor is φa m2 qvt volume flow m3 n speed of male rotor r/min Pad isentropic adiabatic power kW dm amount of change in the volume of control body kg dm1 leakage into control body mass dm2 leakage out control body mass dma flow into control body from the suction port kg dmb outflow control body mass from the exhaust orifice kg m0 nozzle mass flow kg/s Ap nozzle throat area m2 ρ0 nozzle fluid density kg/m3 Tα nozzle high pressure zone temperature K pα nozzle high pressure zone pressure kPa pβ nozzle pressure in the low pressure zone kPa Symbol Meaning Dimension Cn1 area utilization factor Z1 number of teeth of male rotor S1 male rotor elementary area m2 S2 female rotor elementary area m2 D1 male rotor outer diameter m Cφ torsion angle coefficient V0 actual maximum elementary volume m3 Vmax theoretical maximum elementary volume m3 L effective working length m τ1 male rotor torsion angle ° T1 lead m x1 male and female rotor tooth curve abscissa m y1 male and female rotor tooth curve ordinate m x′1 abscissa of male and female rotor tooth curve after rotor rotates by φi angle m y′1 ordinate of male and female rotor tooth curve after the rotor rotates by φi angle m S(ϕi) elementary area after rotor has turned φi angle m2 j jth point xj abscissa of the jth point m yj ordinate of the jth point φ male rotor angle ° φ1 first stage angle ° D2 female rotor outer diameter m λ transmission ratio between male and female rotors R tooth height radius m A center distance m Z2 number of teeth of female rotor εi internal expansion ratio ps gas pressure when elementary volume is disconnected from the suction port kPa pd gas pressure when elementary volume is connected to the suction port kPa k isentropic index of gas Vi elementary volume at the beginning of the expansion process m3 εv content ratio φc internal expansion angle ° S elementary area m2 Aa integral value of elementary area to the corner when the angle of male rotor is φa m2 qvt volume flow m3 n speed of male rotor r/min Pad isentropic adiabatic power kW dm amount of change in the volume of control body kg dm1 leakage into control body mass dm2 leakage out control body mass dma flow into control body from the suction port kg dmb outflow control body mass from the exhaust orifice kg m0 nozzle mass flow kg/s Ap nozzle throat area m2 ρ0 nozzle fluid density kg/m3 Tα nozzle high pressure zone temperature K pα nozzle high pressure zone pressure kPa pβ nozzle pressure in the low pressure zone kPa Open in new tab REFERENCES 1 Smith IK , Stosic N , Kovacevic A . Expanders for power recovery . Power Recovery Low Grade Heat Means Screw Expanders 2014 ; 12 : 1 – 12 . doi:https://doi.org/10.1533/9781782421900.1 . WorldCat 2 Tang H , Wu H , Wang X , et al. Performance study of a twin-screw expander used in a geothermal organic rankine cycle power generator . Energy 2015 ; 90 : 631 – 42 . doi:https://doi.org/10.1016/j.energy.2015.07.093 . Google Scholar Crossref Search ADS WorldCat 3 Stosic N , Smith LK , Kovacevic A , et al. Geometry of screw compressor rotors and their tools . J Zhejiang Univ-Sci A 2011 ; 12 : 310 – 26 . doi:https://doi.org/10.1631/jzus.A1000393 . Google Scholar Crossref Search ADS WorldCat 4 Ma Y , Zeng X , Chang G . Optimized design for rotor profile of screw compressor . Compressor Technol 2004 ; 5 : 15 – 7 . doi:https://doi.org/10.3969/j.issn.1006-2971.2004.05.005 . WorldCat 5 Wu Y , Fong Y . Improved rotor profiling based on the arbitrary sealing line for twin-screw compressors . Mechanism and Machine Theory 2008 ; 43 : 695 – 711 . doi:https://doi.org/10.1016/j.mechmachtheory.2007.05.008 . Google Scholar Crossref Search ADS WorldCat 6 Hu F , Zhang Z , Chen W , et al. Experimental investigation on the performance of a twin-screw expander used in an ORC system . Energy Procedia 2017 ; 110 : 210 – 5 . doi:https://doi.org/10.1016/j.egypro.2017.03.129 . Google Scholar Crossref Search ADS WorldCat 7 Yeqiang Z 2015 . ‘Study on the performance of orcsystem with a single-screw expander.’ PhD Diss. Beijing University of Technology. 8 Du JF 2015 . ‘Tooth contact pattern analysis and design technology research of cycloid hypoid gears.’ PhD Diss., Northwestern Polytechnical University. 9 Qi Y , Yu Y . Numerical simulation and analysis of leakage process in twin-screw expander . J Shanghai Jiao Tong Univ 2016 ; 04 : 496 – 501 . doi:https://doi.org/10.16183/j.cnki.jsjtu.2016.04.003 . WorldCat 10 Tang YQ , Yu YF . Design and optimization of the low-temperature waste-heat dual-cycle power generation system . J Chin Soc Power Eng 2012 ; 2 : 487 – 93 . doi:https://doi.org/10.3969/j.issn.1674-7607.2012.06.013 . WorldCat © The Author(s) (2019). Published by Oxford University Press. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted reuse, distribution, and reproduction in any medium, provided the original work is properly cited.
TI - Research on structural parameter optimization and thermal performance of twin-screw expander
JF - International Journal of Low-Carbon Technologies
DO - 10.1093/ijlct/ctz026
DA - 2019-08-31
UR - https://www.deepdyve.com/lp/oxford-university-press/research-on-structural-parameter-optimization-and-thermal-performance-fnY69YnoD1
SP - 386
VL - 14
IS - 3
DP - DeepDyve
ER -