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半轴壳体左右两面孔加工组合机床的总体设计带CAD图
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702 2013,25(5):702-709 DOI: 10.1016/S1001-6058(13)60415-1 The influence of the flow rate on periodic flow unsteadiness behaviors in a sewage centrifugal pump* PEI Ji (裴吉), YUAN Shou-qi (袁寿其), YUAN Jian-ping (袁建平), WANG Wen-jie (王文杰) Research Center of Fluid Machinery Engineering and Technology, Jiangsu University, Zhenjiang 212013, China, E-mail: (Received September 15, 2012, Revised March 11, 2013) Abstract: To design a single-blade pump with a good performance in a wide operational range and to increase the pump reliability in the multi-conditional hydraulic design process, an understanding of the unsteady flow behaviors as related with the flow rate is very important. However, the traditional design often considers only a single design condition, and the unsteady flow behaviors have not been well studied for single-blade pumps under different conditions. A comparison analysis of the flow unsteadiness behaviors at di- fferent flow rates within the whole flow passage of the pump is carried out in this paper by solving the three-dimensional unsteady Reynolds-averaged Navier-Stokes equations with the Shear Stress Transport (SST) turbulence model. A definition of the unsteadi- ness in the pump is made and applied to analyze the unsteady intensity distributions, and the flow rate effect on the complex unsteady flow in the pump is studied quantitatively while the flow mechanism is also analyzed. The CFD results are validated by experimental data collected at the laboratory. It is shown that a significant flow rate effect on the time-averaged unsteadiness and the turbulence in- tensity distribution can be observed in both rotor and stator domains including the side chamber. The findings would be useful to re- duce the flow unsteadiness and to increase the pump reliability under multi-conditions. Key words: flow rate effect, flow unsteadiness, turbulence intensity, sewage centrifugal pump, multi-conditions Introduction Due to the Rtor-Stator Interaction (RSI), the flow around the impeller of a single-blade centrifugal pump produces a strongly asymmetrical unsteady effect at both Best Efficiency Point (BEP) and other operating points. This phenomenon, therefore, leads to a perio- dical hydrodynamic force acting on the impeller sur- face and results in strong rotor vibrations1. The RSI effect in centrifugal pumps was investigated numeri- cally and experimentally. Feng et al.2-6 investigated the RSI effect in a radial diffuser pump by a compari- son among CFD calculated results, PIV and LDV measurement results, and indicated that the RSI has two kinds of effects on centrifugal pumps with diffu- sers. The first is the downstream effect of the impeller * Project supported by the National Natural Science Foun- dation of China (Grant Nos. 51239005, 51009072), the Na- tional Science and Technology Pillar Program of China (Grant No. 2011BAF14B04). Biography: PEI Ji (1984-), Male, Ph. D. on the stator flow, which is characterized by unsteady effects due to the highly distorted relative impeller flow field and impeller wakes, the second is the up- stream effect of the stator on the impeller flow, which causes unsteady pressures on the relative flow and velocity fluctuations. Studies of single-blade pumps were carried out by using numerical and experimental methods. Benra et al.7 presented an investigation of the periodic un- steady flow in a single-blade pump by using the CFD simulation and PIV measurement methods. The results of transient numerical simulations compare very well with the velocity measurements, but this is only true when all flow details are included into the numerical calculation. Pei et al.8 investigated the flow-induced vibrations of the single-blade sewage water pump im- peller under off-design conditions using numerical and experimental methods. Different strategies of the partitioned fluid-structure interaction simulation for a single-blade pump impeller were studied, and results obtained by one-way and two-way coupling strategies were compared and analyzed9. Pei et al.10 evaluated the transient pressure variation in a single-blade pump 703under multi-conditions by defining the standard devia- tion of the pressure fluctuation in a revolution period. De Souza et al.11 focused on the optimization of single-blade impeller hydraulics from the perspective of both the performance and the solid handling ability by numerical simulations. Yasuyuka et al.12 proposed a method of designing a single-blade centrifugal impe- ller for which the passed particle size is specified based on CFD analysis. De Souza et al.13 addressed the volute design as applied to single-blade impeller pumps. Keays and Meskell14 carried out a numerical study of a single-vanned waste-water pump using the commercial CFD software. However, the unsteady flow field and the turbu- lence behavior caused by the RSI should be analyzed for single-blade pumps under different operational conditions, because the pump is not often running only at the design flow rate. The comparison analysis of flow unsteadiness behaviors under different run- ning conditions is important to gain an insight of the complex flow in the pump and to design a single- blade pump with better performance for a wide range of reliable operations. This paper focuses on a comparison analysis of flow unsteadiness behaviors under different flow rates for a single-blade centrifugal pump in the whole flow passage. The periodic flow unsteadiness results under different conditions are quantitatively compared by defining the unsteady intensity and the turbulence in- tensity in both impeller and volute, which can improve the understanding of the impeller-volute interaction in single-blade pumps as related with the flow rate. Fig.1 Overview of model pump 1. Computational model and method A commercial single-stage single-suction horizo- ntal centrifugal pump with a single-blade impeller is selected as the calculation model. An overview of the pump rotor and the test pump is shown in Fig.1. The design parameters of the pump are listed in Table 1. Three-dimensional, unsteady Reynolds-averaged Navier-Stokes equations are solved using the Shear Stress Transport (SST) turbulence model. The structu- red grids for computational domains are generated using the grid generation tool ICEM-CFD 12.1 and the grid details are shown in Fig.2 with a total number of grid nodes of 2 182 132 for both rotating and sta- tionary domains. The independence of the solutions from the number of grid nodes is checked by simula- ting the flow field with different numbers of grid nodes. The maximum non-dimensional wall distance +y 80 is obtained in the complete flow field. Both the hub and shroud side chambers between the impe- ller and the pump casing are also included in the grid to take the leakage flow effect into account, as shown in Fig.3. The discretization in space is of second-order ac- curacy, and the second-order backward Euler scheme is chosen for the time discretization. The interface between the impeller and the casing is set to the “tran- sient rotor-stator” to capture the transient rotor-stator interaction in the flow, because the relative position between the impeller and the casing changes in each time step with this kind of interface. Two different coordinate systems are utilized for the rotatory and stationary domains, respectively. The inlet boundary condition is set to the total pressure in the stationary frame while the outlet condition is set to the mass flow rate, and all specific values are obtained from la- boratory tests. The smooth wall condition is used for the near wall function. The chosen time step t for the transient simulation is 3.47225104 s for the no- minal rotating speed, corresponding to a changed angle of 3o, therefore, 120 transient results are inclu- ded for one impeller revolution calculation. Within each time step, 10 iterations may be performed and the iteration stops when the maximum residual is less than 103. The convergence criterion for the transient problem is that the result reaches its stable periodicity state, 9 revolutions of the impeller for each opera- tional condition in this case are included. To obtain the stable numerical results, an initial value distribu- tion of the flow parameters as exact as possible is re- quired. To provide this starting solution, a steady cal- culation with the frozen rotor strategy is made. 2. Flow unsteadiness definition The unsteady intensity and the turbulence inten- sity, as the indices of the flow unsteadiness, in both 704 Table 1 Parameters of the pump Design parameters Delivery head des.H 8 m Impeller outer diameter 2D 0.205 m Flow rate des.Q 33.3 l/s Blade width at impeller outlet 2b 0.100 m Rotation speed des.n 1 440 rpm Suction diameter sD 0.100 m impeller and volute domains, are defined according to Ref.2. Fig.2 Grid details Fig.3 Rotor fluid domain view in a meridional plane Fig.4 Velocity triangle Due to the CFD calculation based on URANS equations, the velocity component inside the pump consists of two parts: a time-average component and a periodic component, therefore, the random fluctuating velocity components associated with the unsteady phenomena which are not correlated with the impeller frequency are not considered. Each relative velocity component in the rotating domain and each absolute velocity component in the volute region can be expre- ssed as follows: ( , ,) =( , ) +( , ,)uuuWx yWx yWx y (1) ( , ,) =( , )+( , ,)rrrW x yW x yW x y (2) ( , ,) =( , )+( , ,)uuuCx yCx yCx y (3) ( , ,) =( , )+( , ,)rrrCx yCx yCx y (4) where uW represents the relative circumferential velo- city component, rW represents the relative radial velo- city component, uC and rC represent the absolute cir- cumferential velocity component and the absolute ra- dial velocity, respectively, and they can be calculated in cylindrical systems according to the azimuth angle from the absolute velocity components in x-axis and y-axis directions. Here the meridional component mC is assumed to be equal to the radial component rC . uW, rW, uC and rC represent the time-averaged com- ponents, obtained from all results in one impeller re- volution period. uW, rW, uC and rC represent the periodic components caused by different relative impeller circumferential positions to the volute tongue. The velocity components can be shown in a typical velocity triangle of centrifugal pumps, as in Fig.4. The relative velocity components can be obtained from Eqs.(5) and (6) ( , ,) =( , ,)( , )uuWx yCx yU x y (5) ( , ,) =( , ,)rrW x yCx y (6) where t is the impeller position at the instant time t , and T is the period of the pump rotation. The time-averaged velocity components can be calculated from Eqs.(7) through (10) 70501( , ) =( , ,)dTuutWx yWx ytT (7) 01( , ) =( , ,)dTrrtW x yW x ytT (8) 01( , ) =( , ,)dTuutCx yCx ytT (9) 01( , ) =( , ,)dTrrtCx yCx ytT (10) Fig.5 Definition of angle parameters and monitor points In order to examine quantitatively the unsteadi- ness of the impeller relative flow, the impeller un- steady intensity uI is defined in Eq.(11). uI is calcu- lated by the root mean square of two periodic compo- nents normalized by the impeller tip speed 2U . And the time-averaged unsteady intensity uI is calculated according to Eq.(12) considering results of 120 time steps in one impeller revolution, and the examined points in the rotor region are in the rotating frame of reference. The stator unsteady intensity uS is defined by the root mean square of two absolute periodic velocity components in Eq.(13), and the time-avera- ged unsteady intensity uS is calculated according to Eq.(14). 2221( , ,)+( , ,)2( , ,) =uruWx yWx yIx yU (11) 01( , ) =( , ,)dTuutIx yIx ytT (12) 2221( , ,)+( , ,)2( , ,) =uruCx yCx ySx yU (13) 01( , ) =( , ,)dTuutSx ySx ytT (14) Fig.6 The comparison of experimental and numerical pressure fluctuation results on the casing Fig.7 Relative velocity vector distribution at midspan for =Q 33l/s, =0o 706 Fig.8 Comparison of multi-condition results of time-averaged rotor unsteadiness at midspan The turbulence intensity uT is defined in Eqs.(15) and (16) based on the turbulence kinetic energy ( ,K x ,)y, and the isotropic assumption is used to calcu- late the turbulence kinetic energy with consideration of two components. The time-averaged turbulence in- tensity uT in one period is calculated according to Eq.(17) 2( , ,) =( , , ,)3K x yK x y z (15) 2( , ,)( , ,) =uK x yT x yU (16) 01( , ) =( , ,)dTuutT x yT x ytT (17) 3. Results of comparison For analyzing the flow unsteadiness under va- rious conditions, two coordinate systems are defined, the stationary coordinate frame marked as x, y and the rotating coordinate frame marked as , , as shown in Fig.5. The rotating position of the impeller is indicated by defining the rotating angle between the positive axis of the rotating coordinate frame and the positive y axis of the stationary coordinate frame in the clockwise direction, and =0o means that the trailing edge of the impeller is at the top posi- tion. The angle of the circumferential position in the stationary coordinate is defined as , and =0o when = 0 mx and y0 m. The circumferential posi- tion angle increases in the clockwise direction in the x, y coordinate system, as shown in Fig.5. Experimental data are collected for the model pump in the laboratory of the Institute of Turboma- chinery at University of Duisburg-Essen, Germany, to verify the accuracy of the calculation. The transient pressure results at the transient pressure measurement point, shown in Fig.5, on the casing by both CFD and experimental methods are qualitatively compared under multi-conditions, as shown in Fig.6, and the numerical results are validated. Figure 7 shows the relative velocity vector distri- bution at the midspan under the design condition (= 33l/s)Q when =0o from an unsteady simula- tion result, and the legend represents the velocity scale in the calculation domain. The stable flow patterns can be observed in the volute domain, and the velocity is relatively small. Therefore, the kinetic energy of the flow is transformed into the pressure energy very well in this volute under the design condition. Because the circumferential velocity of the domain is zero, the relative velocity is equal to the absolute velocity in the volute. In the rotor domain, a significantly unbalanced velocity distribution can be observed because of the asymmetrical blade shape. A relatively low velocity appears near the blade pressure side and the impeller eye position. A relatively large velocity appears near the blade suction side and at the circumferential area of the blade outlet, and the maximum velocity can be found at the outermost position of the impeller. The almost same phenomenon can be observed at each examined flow rate. However, the relative velocity distribution results are obtained at a specified time, the velocity field is also fluctuating with time which is called the unsteadiness in this paper, and this pheno- menon cannot be shown in a single figure. To study the intensity distribution of the flow fluctuation and the mechanism of the unsteadiness phenomenon, a number of pictures of flow distributions at each time point or at a number of monitor points at each position containing the time-history results should be obtained to describe the phenomenon in the whole flow passage, which is not efficient and even possible. Therefore, applying the time-averaged unsteadiness intensity coe- fficients defined in this paper to analyze the unsteadi- ness mechanism in the pump is a better choice as with the coefficients one can use a single value at each lo- cation to describe the unsteadiness intensity with con- sideration of the whole time-history fluctuating results for an impeller revolution. Figure 8 shows the comparison of multi-condi- tion results of the time-averaged rotor unsteadiness 707 Fig.9 Comparison of multi-condition results of time-averaged volute unsteadiness at midspan Fig.10 Comparison of multi-condition results of time-averaged turbulence intensity in the impeller at midspan coefficients at the midspan. For each flow rate, rela- tively large values can be observed near the impeller outlet and near the blade pressure side, and the values increase when the flow rate increases. At three exami- ned flow rates, the maximum uI result can be obse- rved at the trailing edge close to the pressure side of the blade for = 42 l/sQ. Because under a large flow rate condition, more power is required from the blade by the fluid while more kinetic energy is transferred to the fluid, and the wake flow with a relatively high velocity near the blade trailing edge will have a stro- nger interaction downstream with the volute tongue, and a strong velocity fluctuation will appear. At a small flow rate, less power is required from the blade, and a weak interaction between the fluids in the rotor and in the stator is observed, which means a weak periodic velocity fluctuation. Figure 9 shows a comparison of multi-condition results of the time-averaged volute unsteadiness coe- fficients at the midspan. It is shown that large values uS0.035 can be only found under the off-design conditions, in the volute channel at a low flow rate and in the area near volute tongue at a large flow rate, and no such values can be found under the design con- dition. The reason is that, at a large flow rate, because of the strong interaction between the blade and the tongue as shown in Fig.8, the strong velocity fluctua- tion can be found near the tongue area. At a small flow rate, although a weak rotor stator interaction causes a weak velocity fluctuation near the tongue area, an unstable flow in the volute channel can be ob- served with unstable velocity fluctuations because the volute is designed at the design flow rate, and with less fluid in the channel, no smooth flow can be obtai- ned, and a strong unsteady flow exists. Under the de- sign condition, a best matching between the impeller and the volute can be obtained. Although the flow un- steadiness is not the smallest in the impeller, a relati- vely smooth and stable flow in the volute can be found, and finally a good compromise between the flow in the impeller and the volute can be obtained as compared with off design conditions. Figure 10 shows the comparison of multi-condi- tion results of the time-averaged turbulence intensity in the impeller at the midspan. uT represents the flow random fluctuation intensity with consideration of the cumulative effect of the flow results in an impeller re- volution. At each flow rate, relatively large values can be observed in the area near the impeller outlet and in the area near the blade pressure side, and the values increase when the flow rate decreases, because at a large flow rate, much more energy of the rotating blade is transferred to the fluid in a relatively strong periodic flow by the RSI effect, and less energy of fluid is spent in the turbulent random fluctuation. At a small flow rate, the periodic flow fluctuation is relati- vely weak with an unexpected impeller-volute mat- ching as shown in Fig.8 while a chaotic flow with more unstable vortexes will be produced, therefore, more residual energy will be spent in the turbulent velocity random fluctuation. Under the design condi- tion, relatively moderate distribution results are obtai- ned. 708 Fig.11 Comparison of multi-condition results of time-averaged turbulence intensity in the volute at midspan Fig.12 Comparison of multi-condition results of time-averaged unsteadiness on the cross section of side chambers Fig.13 Comparison of multi-condition results of time-averaged turbulence intensity on the cross section of side chambers Figure 11 shows the comparison of multi-condi- tion results of time-averaged turbulence intensity in the volute at the midspan. A high turbulent flow can be seen near the tongue for an impeller revolution under each condition, and the area is bigger under the off-design conditions. The reason is that, at large and small flow rates, the flow in the volute is not smooth, the volute tongue will have a great influence on the flow velocity and much energy losses are caused. Therefore, the flow in the area becomes quite chaotic and varies strongly with time. Figure 12 shows the comparison of multi-condi- tion results of uI on the cross section of the side cha- mbers, and the position of the cross section is shown in Fig.3. The values increase with the increase of the flow rate, because the source of the periodic flow phe- nomenon is in the impeller and volute channels, the flow in the side chamber is affected by the periodical unsteady flow in the main channel. The periodic un- steadiness intensity increases with the increase of the flow rate, as shown in Fig.8, therefore, the influence on the side chamber flow becomes significant with the increase of the flow rate. Figure 13 shows the comparison of multi-condi- tion results of uT on the cross section of the side cha- mbers, and the values increase with the increase of the flow rate. The reason is that, at a large flow rate, a higher flow velocity in the pump channel is obtained, and more energy can be transferred to the side cha- mber fluid. Therefore, a part of energy can make the flow in the side chamber chaotic and make the flow relatively more turbulent in the impeller revolution. Although a relatively large uT can be found in the im- peller channel at a small flow rate, as shown in Fig.10, the impeller flow can be found to have a small influe- nce on the side chamber flow under this condition. 4. Conclusions To consider the flow rate effect, the behavior of the unsteady flow in the pump is analyzed from a new angle of view by defining the flow unsteadiness coe- 709fficients for a single-blade centrifugal pump within the whole flow passage. The CFD results are validated by experimental data collected at the laboratory. The un- steadiness distribution results under multi-conditions are studied while the flow mechanism behind that is analyzed. The following conclusions are drawn: (1) For the flow in the impeller channel, a strong velocity fluctuation will appear at a large flow rate while a weak periodic velocity fluctuation is observed at a small flow rate. (2) A strong velocity fluctuation can be found at a large flow rate near the tongue area in the volute, and at a small flow rate, an unstable flow can be obse- rved in the volute spiral channel with strong velocity fluctuations. (3) The extents of the periodic velocity fluctua- tion and the turbulent random fluctuation increase with the increase of the flow rate in the side chamber channels. (4) A comparison analysis can reveal the beha- viors of the flow unsteadiness in detail as related with the flow rate, and the findings would be useful to re- duce the obvious flow unsteadiness and to increase the pump reliability when the pump is running under mul- ti-operational conditions. Acknowledgement The authors would like to thank Prof. Benra F.-K. and Dr. Dohmen H. J. from University of Duisburg- Essen in Germany for their kind help and guidance on the research work and for providing the model of the pump. References 1 BENRA F.-K. Numerical and experimental investiga- tion on the flow induced oscillations of a single-blade pump impellerJ. Journal of Fluids Engineering, 2006, 128(4): 783-793. 2 FENG J., BENRA F.-K. and DOHMEN H. J. Investiga- tion of periodically unsteady flow in a radial pump by CFD simulations and LDV measurementsJ. Journal of Turbomachinery, 2011, 133(1): 011004. 3 FENG J., BENRA F.-K. and DOHMEN H. J. Investiga
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