Study on critical speed of rotation in the multistage high speed centrifugal pumps rotors

Transcription

1 Internationa Journa of Heat and echnoogy Vo. 36, No. 1, March, 218, pp Journa homepage: Study on critica speed of rotation in the mutistage high speed centrifuga pumps rotors Yabin ian *, Anjie Hu Schoo of Civi Engineering and Architecture, Southwest University of Science and echnoogy, Mianyang 6211, China Corresponding Author Emai: Received: 17 August 217 Accepted: 1 December 217 Keywords: critica speed of rotation, fuid-structure interaction, mutistage centrifuga pump, rotor dynamics. ABSRAC Apart from hydrauic performance, the critica rotate speed in wet operating condition (the iquid ubricant state) is an important parameter in the designing of the mutistage high speed centrifuga pumps rotors. In this paper, the finite eement dynamic mode of the rotor in dry state (without ubricate) is anaysed by the ubricant fuid appied to the disk and cyinder parts, and a resistance equation of the infuence of the fuid is proposed based on the anaysis. his resistance is then couped into the dry state finite eement dynamic mode to obtain that of the wet state, which is considered as the fuid-soid couped interaction. With this mode, the study further studies the infuence of severa factors on the inherent frequency of the rotor, and simuates the bending and torsion formation of the physica mode. Simuation resuts show that, when the fuid s infuence is considered, the inherent frequency of the rotor is very different from that of the dry state, and hence this infuence cannot be negected in the designing of the rotor. hese resuts are aso verified by corresponding experiments. 1. INRODUCION hroughout the main body of your paper, pease foow these prescribed settings: 1) the font is mosty imes New Roman; 2) amost a the words are typed in 1 points except; 3) each ine throughout the paper is singe-spaced; 4) in most cases, 1 pts spacing sha be eft above and beow any heading, tite, caption, formua equation, figure and tabe. Actuay, as mentioned in the abstract section, it wi be rather easy to foow these rues as ong as you just repace the content here without modifying the form. High speed mutistage centrifuga pump has been extensivey appied to petroeum, chemica industry, eectric power, stee and other important industry areas. As an important industria equipment, the pump units are getting arger and the corresponding speed is getting higher as the singe-machine capacity increases. he pump s rotating speed now can easiy exceed the first critica speed of the pump rotor, and can even approach the second and third critica speed. If the pump amost operates at critica speeds, the system wi be unstabe due to the sympathetic vibration, which wi pose a formidabe chaenge to the steady operation of the pump. o avoid the probem of high speed operation, the critica speeds of the pump rotor must be precisey cacuated when the high speed mutistage centrifuga pump is designed. he critica speeds of genera rotor system are aways cacuated based on the dynamic characteristics of pump shaft and the oi fim bearing. However, the dynamic characteristics of the rotor system are aso infuenced by the fuid (ubricating oi) within the system. As the operating speed getting higher, the effects of the fuid on the system become more obvious. Hence, to guarantee the computationa precision, the fuidstructure interaction between the rotor system and the fuid must be considered in this case. [1] inite eement dynamic mode has ong been appied to the study of the rotor dynamic simuation since the 197s, and has been graduay deveoped in these years.[2] Compared with the origina modes which ony considered the inertia movement s infuence, the atest modes are more cosed to the engineer practice, since the factors such as rotationa inertia, gyroscopic coupe, axia oad, interna and externa anticorrosion, shearing deformation, bearing spring, foundation easticity, and so on are a discussed in these modes. Moreover, the infuence of the siding bearing and seaing ubricate oi has been widey appied to the research of sma circuation gap structure, which greaty improved the finite eement dynamic mode. [3, 4] However, some fata factors are sti negected in these modes. One of them is the probem of fuid-structure interaction of the circuating fow within a arge gap. [5, 6] As we a know, the rotor system is immersed in the ubricating oi, which highy infuences the motion of the rotor, especiay when the rotating speed is very high. Large deviation wi appear if this factor is negected. Athough the fuid-structure interaction is introduced in many commercia structura dynamics software, such as ADINA [7], ANSYS [8] etc., detaied modes for a certain rotor system sti need to be estabished. Schoars have done a ot of work to precisey predict the dynamic characteristics of the rotor system. Simiar to the oi fim bearing simuation, there are some schoars such as ian Yabin [9], Antunes [1, 11], Sun Qiguo [12] studied the dynamic coefficient of the eccentric rotor when the fuid machinery soaked in the circuation fuid in the big gap. However, the fuid-structure interaction mechanism in the rotor system is much different from the oi fim bearing, the dynamic coefficient is not appropriate for describing the dynamic characteristics of the rotate system. Schoars [13] aso simuated the fuid-structure interaction by introducing the mass effects of the fuid. However, this method negects the viscosity resistance of the fuid, which increases as the rotate speed goes up. 31

2 o precisey predict the critica speed of the pump rotor, in this paper, an improved fuid-structure interaction deaing method is appied to the finite eement mode. With this mode, the effects of the fuid on critica speeds of the pump rotor is further numericay discussed [14, 15]. 2. MAHEMAICAL MODELS 2.1 Dynamic mode of the dry mutistage centrifuga pumps rotors he mode of dry condition particuary refers to the rotor working condition without considering the fuid s infuence. In this case the rotor system vibration equation is given by Mu( t) Ku( t) (1) where, M is the mass, K is the stiffness matrix, u (t) and u(t) are the finite eement node acceeration and dispacement respectivey. he soution of the above equation can be considered as a harmonic function, hence the foowing equation shoud be satisfied: det K 2 M (2) With this equation, a seria combination of λi and ωi2 can be obtained. Each ωi2 refers to a feature vector {ф i}, which represents a certain format of free vibration. he inherent frequency can then be given by f i i 2 (3) Where f i refers to the ith inherent frequency. 2.2 Gyroscopic effect he gyroscopic effect on the rotor system can be expressed by the agrange function, which is given by u Mu u Gu u Ku Lu, u Where u represents n-dimensiona dispacement vector, and Gu is the gyroscopic term, in which G is a antisymmetric matrix. he dynamic equation of the rotor system can then be written as Mu Gu Ku f1 t where f 1 (t) is the externa force. he above equation is a second order differentia equation. o sove it, the soution of the corresponding homogeneous equation must be obtained first. With the soution of the homogeneous equation, the soution of the Eq. (5) can then be obtained by Duhame integration as (4) (5) 2.3 uid-structure interaction force he fuid-structure interaction force is infuenced by the shape of the immersed structure and its vibration ways. or some typica vibration cases, the force can be cacuated by some given equations. In this paper, the movement of rotor system can be regarded as the vibration of discs and cyinder in the fuid. he corresponding interaction forces are given as foows. (1) ransationa and corner vibration of discs he vibration of a disc can be both transationa and corner vibration. If a disc with radius R does the transationa vibration in the direction parae to its axis in the fuid, the interaction force between the fuid is given by m x c x (7) da da where m da = 3πR 2 ηρ 2ω, c da = 3πR(1 + 2R 3δ ), ρ and η represents the density and the viscosity of the fuid, and is the vibration frequency respectivey, δ = 2η ωρ. In the corner vibration cases, the fuid resistance moment on the disc is given by M i g da da (8) where i da = 3π 8 R4 ηρ 2R (1 + ), g 2ω 3δ da = 3π 8 ηr3 (1 + R ). δ (2) ransationa and corner vibration of cyinders Cyinders aso do the transationa and corner vibration. If the cyinders do the transationa vibration in the direction parae to its axis in the fuid, the fuid resistance of the transation on the unit ength can be written as foows: q m x c x sa sa (9) where m sa = 6πR ηρ 2ω (1 + R 3δ ), c sa = 6πη(1 + R δ ) (1) In the cases of corner vibration, the resistance moment of fuid to the rotation centre corner is given by 2 dm m c s ds sa sa where, m sa and c sa are aso given by Eq. (9). 2.4 he fuid-soid couping mode (11) o numericay anayze the infuence of fuid on the rotor, these inter-action forces or moments above shoud be couped with the motion equations. (1) he equation of motion disc According to the Eqs. (7) and (8), the noda forces caused by fuid strength in the disk can be written as Mu Gu Ku (6) 32

3 Q M u C u 1d da 1d da 1d Q M u C u 2 d da 2 d da 2 d where, M da = ( m da i da ), M da = ( c da g da ) he motion equations of the disc are given as foows: M u G u Q Q d 1d d 2 d 1d 1d (12) M u G u Q Q d 2 d d 1d 2 d 2 d (13) with the Eq. (12), the Eq. (13) can be rewritten as M u G u C u Q d 1d d 2 d d 1d 1d M u G u C u Q d 2 d d 1d d 2 d 2 d (14) where, M d = M d + M da, C d = C da. (2) he motion equation of shaft unit 1) he noda forces caused by fuid resistance According to the principes of virtua work and notes x = [N]{U 1s }, the foowing equations can be obtained: (15) U Q xd U N d 1S 1S 1S and Q N d 1S (16) With notes d=qds, the above equation can be rewritten as Q M u C u K u 1S sa 1s sa 1s sa 1s K sa (17) where, M sa = m sa [N] Nds, C sa = c sa [N] Nds, = k sa [N] Nds, the symmetry force is then given by Q M u C u K u 2 S sa 2 s sa 2 s sa 2 s (18) 2) he noda forces caused by fuid drag torque Simiary, according to the principes of virtua work and notes θ = N u 1s, the foowing equations can be obtained: (19) M u Q dm u N dm 1s 1s 1s and M Q N dm 1s (2) with the Eqs. (13) and (14), we can get M M M M Q M u C u K u 1s sa 1s sa 1s sa 1s (21) where, M sa c sa [N ] N s 2 ds, K sa the symmetry force is aso given by M M M M Q M u C u K u 2 s sa 2 s sa 2 s sa 2 s = m sa [N ] N s 2 ds, C sa = = k sa [N ] N s 2 ds (22) 3) the shaft unit equations of motion considering the fuid. M M u G u K u Q Q Q s 1s s 2 s s 1s 1s 1s 1s M M u G u K u Q Q Q s 2 s s 1s s 2 s 2 s 1s 2 s (23) taking the equation (16), (17), (2), (21) into (23), and rearranging, we can get M u G u C u K u Q s 1s s 2 s s 1s s 1s 1s M u G u C u K u Q s 2 s s 1s s 2 s s 2 s 2 S (24) where, M s = M s + M sa + M M sa, C s = C sa + C M sa, K s = K sa + K M sa. (3) he motion equations of overa system he motion equations of the overa system considering the fuid shoud have the foowing forms: M U C U GU K U Q M U GU C U K U Q (25) Or M U + C U + K U = Q where M = ( M 1 M 1 ) is the system inertia matrix, C = ( C 1 G 1 G 1 C ) is the damping matrix, and K = ( K 1 1 K ) is the 1 stiffness matrix. Other matrixes definitions are given as foows: M 1 is formatted by the shaft unit inertia matrix M s and the disc ce inertia matrix M d, C 1 is formatted by the shaft unit damping matrix C s and the disc unit damping matrix C j, and K 1 is formatted by the shaft unit eement stiffness matrix K s and the disc stiffness matrix K j. 3. HE INIE ELEMENMODELIN ANSYS he geometry mode can be carried out in the finite eement software ANSYS. he rotor system in this paper can be treated as a beam. here are mainy three kinds of beam eements in the software: Beam 3, Beam4, and Beam 188. Beam3 refers to the 2D beam eements, hence it can ony be appied to 2D probem. Beam 4 and Beam 188 are both 3D beam eements, and the shape of the section need to be defined. However, the rotor system in the present work has a singe shape of cyinder, and the appication of the above methods compicate the issue. o simpify the mode, this paper appied the pipe eement 33

4 mode PIPE 16 instead, which considered the rotor eements as soid pipes. Besides, the vane whee was modeed by the mass eements and its rotationa inertia was added on the eements. Other geometric detais, such as rounding, gaskets, keyways and pores were negected since they have itte infuence in the simuation. Stainess stee H41 (1Cr13) has been chosen as the materia of the rotor, the properties of which are given as foows: poisson ratio of.3, density of 785 Kg/m3, and young moduus of 2 GPa. 4. ANALYSIS O HE ROOR DYNAMICS With the proposed mode, the critica speeds of the rotor are numericay anayzed in the present work. o iustrate the infuence of fuid (oi) on the system, this paper separatey simuates the system performance with and without fuid. he dry condition and wet condition are adopted here to represent the cases of simuation resuts with and without fuid s effects. he operating parameters of the high speed 1 stages centrifuga pump are given as foows: the rated speed is 75 r/min, the shaft diameter is 85mm, the impeer is 215mm, the chief of the spinde is about 2353mm and the diameter of the support is 7mm. he proposed mode is appied to ANSYS by programming the parametric designing anguage (APDL), which is contained in ANSYS software. he gyro effects are aso considered in the mode since the rote speed is rather high for the rotor. By chosen the numerica mode of each infuentia factors, this paper numericay anayzes these infuences on the rotor vibration and critica speeds separatey. 4.1 Infuence of support stiffness on the critica speed he infuence of the support stiffness on the critica speed is given in tabe 1. As we can see from this tabe, the simuated critica speed increased with the support stiffness. he corresponding vibration shape is given in ig. 1. As we can see in this figure, when the eastic support is taken into consideration in the mode, the rigidity of the rotor decreased, which ead to the decrease of the critica speed. he vaue of the critica speed is about 5.6E+5N/m to 5.6E+7N/m in this case. Since the infuence of the critica speed is so obvious, great errors may be made if it is negected in the simuation. Ony in the case of the rigidity support, such infuence can be negected. Stiffness of 5.6E+5N/m Stiffness of 5.6E+6N/m Stiffness of 5.6E+7N/m Stiffness of 5.6E+8N/m Stiffness of 5.6E+9N/m Stiffness of 5.6E+1N/m igure 1. vibration shape with different support stiffness abe 1. he inherent frequency with different support stiffness Support stiffness 5.6E+5 5.6E+6 5.6E+7 5.6E+8 5.6E+9 5.6E+1 1st order frequency nd order frequency rd order frequency th order frequency Infuence of gyroscopic effect on the inherent frequency he infuence of the gyroscopic effect on the critica speed is given in tabe 2. It can be seen from this tabe that the critica rotor speed is reduced by the backward-precession whie added by the forward-precession. With the increase of the rotor speed, the infuence of the precession gets more obvious. he infuence of the gyroscopic effect on the vibration type is given by the figures 2-4. It can be seen from these figures that the gyroscopic has itte infuence on the vibration type of the rotor. 34

5 abe 2. he inherent frequency with gyroscopic effect Rotor speed r/min Direction backward forward backward forward backward forward backward forward 1st order frequency nd order frequency rd order frequency igure 2. he first order vibration type of the forward and backward precession igure 3. he second order vibration type of the forward and backward precession igure 4. he third order vibration type of the forward and backward precession 4.3 Infuence of fuid-structure interaction on the inherent frequency On the base of the gyroscopic effect mode, the infuence of fuid-structure interaction on the critica speed is given in tabe 3. It can be seen from this tabe that the critica speed of each order is reduced. he reason of the decrease is that additiona mass is added on the rotor by the fuid. However, the infuence of the fuid on the speed is very imited since the additiona mass is quite sma, and this effect woud be more obvious if the additiona mass gets arger. igs. 5-7 give the infuence of the fuid on the vibration type, it can be seen from these figures that the vibration types change itte compared with the cases without fuid-structure interaction. abe 3. he inherent frequency with the fuid-structure interaction 1st order frequency 2nd order frequency 3rd order frequency backward forward backward forward backward forward

6 igure 5. he first order vibration type of the forward backward precession with the fuid-structure and interaction igure 6. he first order vibration type of the forward and backward precession with the fuid-structure interaction igure 7. he first order vibration type of the forward and backward precession with the fuid-structure interaction 4.4 Infuence of choma support stiffness on the inherent frequency he infuence of choma support stiffness on the critica speed is given in tabe 4. abe 4. he inherent frequency with the choma stiffness Choma stiffness 1.5E+5 N/m 1.5E+6 N/m 1.5E+7 N/m 1.5E+8 N/m 1st order backward forward nd order backward forward (torsion mode) 3rd order backward forward It can be seen from this tabe that the vibration frequency of the rotor is added with the choma support stiffness. he effect of the choma on the rotor is simiar to the effect of the support when its stiffness is constant, and this effect can be used to enhance the stabiity of the rotor, since it can increase the corresponding inherent frequency. igs shows the corresponding vibration type under the effects of the choma support. It can be seen from these figures that the vibration type changes itte when the stiffness of the choma increased from 1.5E+5N/m to 1.5E+6 N/m, whie it changes obviousy when the stiffness gets to 1.5E+7 N/m. he reason of these resuts is that the high stiffness choma can be viewed as an additiona support which can restrain the transformation of the rotor. 36

7 (a) 1st order (b) 2nd order (c) 3rd order igure 8. he first 3 orders vibration type of the forward and backward precession with the choma support stiffness of 1.5E+5 N/m (a) 1st order (b) 2nd order (c) 3rd order igure 9. he first 3 orders vibration type of the forward and backward precession with the choma support stiffness of 1.5E+6 N/m (a) 1st order (b) 2nd order (c) 3rd order igure 1. he first 3 orders vibration type of the forward and backward precession with the choma support stiffness of 1.5E+7 N/m (a) 1st order (b) 2nd order (c) 3rd order igure 11. he first 3 orders vibration type of the forward and backward precession with the choma support stiffness of 1.5E+8 N/m 37

8 4.5 Infuence of additiona axia stress on the inherent frequency he infuence of additiona axia stress on the critica speed is given in tabe 5. It can be seen from this tabe that the inherent frequency aso increased obviousy under the infuence of the axia stress. he corresponding vibration types are given in ig. 12, from which it can be found that the vibration type changes a ot after the axia stress is considered, and the bending vibration decreased and critica speed increased, which is beneficia to the stabiity of the rotor. abe 5. he inherent frequency with the choma stiffness 1st order 2nd order 3rd order (a) 1st order (b)2nd order (c)3rd order igure 12. he vibration types with the additiona axia stress In this section, the paper has numericay discussed the 5 main factors infuencing the inherent frequency of the rotor. rom these simuation resuts, it can be found that the infuences of these factors are different, among these factors, the fuid-structure interaction, choma support, and axia stress have obvious effects on the operating character of the rotor: the inherent frequency of the wet state is obviousy higher than the dry state; the choma support enhances the inherent frequency most when its stiffness is high; the axia stress infuences not ony the inherent frequency of the rotor but aso its corresponding vibration type. he infuences of these three factors are a good to the stabiity of the rotor, since they can enhance the stiffness of the rotor and restrain the deformation of the rotor 5. EXPERIMEN VERIICAION o verify the simuation resuts, the corresponding experiments are aso carried out in this paper. he 3 8 3/4-1stg HSB high speed mutistage centrifuga pump is chosen as the experiment object. And the dynamic signa acquisition system is made up of KD15 piezoeectric buit-in circuit (ICP) acceeration sensor, CYB11 type pressure transmitter, charge ampifier, pressure sensor preconditioning, and U328 coector. Each bade rotor is set with 4 coection points, and the samping time is 3 min. he experiment resuts are given by the tabe 6. abe 6. Experimenta tested frequencies of the rotor system 1st order 2nd order 3rd order 4th order It can be seen from this tabe that, compared with the simuation resuts, the critica rotor speed is a itte ower in the experiment, whie the inherent frequency is a itte higher. his is mainy because the infuencing factors of the experiment are more than that of simuation. A in a, the experiment resuts are cose to that of simuation, which verifies the simuation resuts. 6. CONCLUSIONS his paper proposes a resistance equation of the infuence of fuid on the high speed mutistage centrifuga pump, and appies the equation to the simuation of the inherent frequency. In the simuation, severa factors infuences are compared and discussed in detai. inay, it verifies the simuation with the corresponding experiments. oowing concusions can be drawn based on the resuts: (1) By integrating the roe of fuid into the motion equation of the whoe system to get the fuid-soid couping finite eement mode of the rotor dynamics, the dynamics anaysis of a rea high speed mutistage centrifuga pump rotor system can be done by the computer program with the finite eement mode. hese anayses can verify the theory that fuid has an impact on dynamics performance of transversa subsystem. (2) Simuation resuts show that the differences of the critica speed between dry and wet states in mutistage centrifuga pump are obvious. Hence the effect of fuid-soid couping on dynamic characteristics must be considered in cacuation process. (3) he dynamics anaysis of high speed mutistage centrifuga pump rotor shows that the effect of fuid on impeer can improve the stabiity of the rotor. (4) he experiment resuts show a good consistence with the simuation resuts, which verifies the reiabiity of the present mode. ACKNOWLEDGMENS Project(13zx7145) supported by Research oundation for PHD of Southwest University of Science and echnoogy(13zx7145&11zx716). Mianyang, appied technoogy research and deveopment project funding (14G - 9-4, 15zd218) 38

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