Dynamic analysis of the mechanical seals of the rotor of the labyrinth screw pump

Transcription

1 IOP Conferene Series: Materials Siene and Engineering PAPER OPEN ACCESS Dynami analysis of the mehanial seals of the rotor of the labyrinth srew pump To ite this artile: A Y Lebedev et al 17 IOP Conf. Ser.: Mater. Si. Eng View the artile online for updates and enhanements. Related ontent - Dynami analysis of nonlinear behaviour in inertial atuators M Dal Borgo, M Ghandhi Tehrani and S J Elliott - A mehanial seal for very high pressures D Anson - Dynami analysis of entrifugal mahines rotors supported on ball bearings by ombined appliation of 3D and beam finite element models I V Pavlenko, V I Simonovskiy and M M Demianenko This ontent was downloaded from IP address on 17/11/18 at 16:3

2 (HERVICON+PUMPS-17) IOP Conf. Series: Materials Siene and Engineering (17) 135 doi:1.188/ x/33/1/135 Dynami analysis of the mehanial seals of the rotor of the labyrinth srew pump A Y Lebedev 1, P M Andrenko 1 and A L Grigoriev 1 1 National Tehnial University "Kharkiv Polytehni Institute", Ukraine, Kharkiv andrenko1947@gmail.om Abstrat. A mathematial model of the work of the mehanial seal with smooth rings made from ast tungsten arbide in the ondition of liquid frition is drawn up. A speial feature of this model is the allowane for the thermal expansion of a liquid in the gap between the rings; this effet ating in the onuntion with the fritional fores reates additional pressure and lift whih in its turn depends on the width of the gap and the speed of sliding. The developed model displays the proesses of separation, transportation and heat removal in the ompation elements and also the resistane to axial movement of the ring arising in the gap aused by the pumping effet and the frition in the flowing liquid; the inertia of this fluid is taken into aount by the mass redution method. The linearization of the model is performed and the dynami harateristis of the transient proesses and the fored osillations of the devie are obtained. The onditions imposed on the parameters of the mehanial seal are formulated to provide a regime of the liquid frition, whih minimizes the wear. 1. The introdution and the statement of task The labyrinth srew pump (LSP, sealing assembly is shown in Figure 1 [1]) is installed in an oil well at a depth of about 1 km at the lower end of the pipeline, through whih the vane pumps lift the oil to the surfae of the earth; up to the speified mark, it flows under the influene of layer s pressure. (а) Figure 1. Mehanial seal of the rotor of the LSP: (a) design; (b) sheme. 1 outlet hamber; spring; 3,4 lips of rings; 5 shaft; 6 spring support. (b) Content from this work may be used under the terms of the Creative Commons Attribution 3. liene. Any further distribution of this work must maintain attribution to the author(s) and the title of the work, ournal itation and DOI. Published under liene by Ltd 1

3 (HERVICON+PUMPS-17) IOP Conf. Series: Materials Siene and Engineering (17) 135 doi:1.188/ x/33/1/135 The oil in the avity of the LSP has the temperature of 1 C and the pressure of about 1 MPa. It is known [] that there is a lot of gas whih is dissolved in the layer s oil and during the lifting along the well due to the pressure drop, about /3 of its mass is released in the form of bubbles and thus the oil beomes a two-phase medium. The task of LSP is to rush the bubbles and mix them with the liquid, whih is neessary for a reliable operation of the vane pumps. During the operation the LSP raises the oil pressure by. MPa, and this differene ats on the mehanial seal of the output hamber, shown in Figure 1 The presene of the pressure drop, as well as the speifi physial properties of the working body and a very high purity of the sealing surfaes (roughness of the rings made of the wear-resistant alloy of two tungsten arbides is only.3 miro mill.), reate the prerequisites for the maintaining the liquid frition regime in the gap what minimizes the wear. The task of the researh was to determine the ompation parameters whih ensure this mode.. The features of the sealed fluid For every 1 m inrease in depth, the temperature of the earth's rok rises by 3 degrees. The depth of the layer from whih oil is extrated reahes about 3 km and the pressure in the layer is lose to 3 MPa. When the oil is being lifted along the well its temperature hanges slightly and the pressure dereases threefold whih is a result of the release of a large part of the dissolved hydroarbon gas and nitrogen into the gas phase. In this ase the initial visosity of the liquid phase inreases by times. The rotation of the rotor leads to a separation of the oil in the outlet hamber in terms of density: the water whih is present in the layer s oil is pressed to the walls of the hamber and the gas phase remains in the enter. As a result the liquid hydroarbon frations are fed to the ompat but their visosity does not remain onstant and varies within rather wide limits. It is generally aepted that the average kinemati visosity of the degassed and separated oil at a temperature of 1 C is about 4 St, but in the layer s oil there is a lot of gas is dissolved the visosity is muh lower; besides, some types of the oil have a higher visosity []. In the layer, the visosity of oil (in St) obeys the Mayer equation the visosity of the same oil at a temperature of t 3.5 lay ln(.8)/ ln(.8) 93/(73 ) (1) C ; t lay 1 C Figure. Changing the index K depending on the visosity of the layer s oil and temperature: K 1 t lay 1 C, K t lay 7 C, K t lay 1 C. Differentiating the both sides of equality (1) with respet to the temperature the oeffiient of hange (index) of the visosity is obtained: 1 K d dtlay tlay / 3.5 (1.8/ ) ln(.8)/(73 ) In order to take into aount the partial degassing during the lifting of the oil through the well and its separation in the pump, the visosity in the right side of this formula should be redued by times. As a result, the dependenies presented in Figure are obtained. The kinemati visosity of oil in

4 (HERVICON+PUMPS-17) IOP Conf. Series: Materials Siene and Engineering (17) 135 doi:1.188/ x/33/1/135 the pump varies within the limits of St; for further analysis an average value med St will be taken. 3. The hanges in the temperature of the liquid and the material of the ring along the length of the gap It was noted in the work [3] that a radial temperature drop leads to a oniity of the gap, whih in its turn entails a hange in the lift fore. To alulate the oniity, the heat ondution problem for the radial setion of the ring is solved (Figure 3). The release of the heat is aused by the operation of the fluid frition fore and the heat flux density q is desribed by the formula q x R x h h x () ( ) (( ) ) /( ( )), is the density and kinemati visosity of the oil; R, is the mean radius and the angular veloity of the rotation of the ring; h is the gap s height on the outer radius of the ring; h is the hange of the gap; x is the radial oordinate, direted along the flow of the fluid; x[, H ] ; h () ; H is the width of the ring. The task, after negleting the radial ontration of the ring and the flow of heat from the pump shaft, takes the form of: T ; T / x ; / ; / ; /.5 x T x T T y T y q xh y (3) yhy and its solution was obtained by Fourier's method of separation of variables [4] in the form of a series T x y H A H H y H x H (4) (, ) [ sh( / )] h( / )os[ (1 / )],1,... Y the separation onstants and the Fourier oeffiients A are defined by the equalities: tg H, 1 1 H ( ) os( (1 / )) A H q x x H dx ; H 1 [.] is Laplae operator; T T / x T / y ; is the oeffiient of heat transfer to the liquid in the output hamber of the LSP (its temperature in the model (3) is onsidered zero); is the average thermal ondutivity of the material of the ring (in the alulation the thermal -1-1 ondutivity of the ast tungsten arbide 9.5 Wm K is used); H Y is the height of the ring. r 3

5 (HERVICON+PUMPS-17) IOP Conf. Series: Materials Siene and Engineering (17) 135 doi:1.188/ x/33/1/135 x y q H Figure 3. The setion and the deformation of the ring and the heat flow in the gap. Figure 4. The graphs of the oeffiients of the proportionality for the temperatures. Integrating the series (4) over the height of the ring the following data is obtained: H Y 1 T( x, y) dy.5h os( / ) A,1,... x H (5) and, it is typial, that this result does not depend on the height of the ring H Y. The hange in the temperature of the liquid in the gap (i.e., the temperature of the hannel wall) is determined by the equality ( ).5 1 [ th( / )] 1 T (1 os [ (1 / )]) l x H A HY H x H (6),1,... and when the ratio HY.5H is weakly dependent on the height H Y. The analysis showed that in the sums (4) - (6) one an onfine the first term; then all the thermal harateristis of the ring are proportional to one temperature T : T Kp( a) T ; T Ks( a) T ; T Kn( a) T ; grad T Kp( a) T / H r l in x T.5 qmed H / r is the alulated value of the temperature obtained under the ondition of the transfer of the heat with density q R h from the inner radius of the ring to the outer med ( ( med ) / ) radius; Rmed R.5 H ; T r, Tl are the radial temperature differenes of the ring and the liquid; T in is the temperature of the material in the initial setion of the ring; the orresponding thermal elongations in the inlet Hin and the outlet setions Hout are determined by the formulas H T H; H T H, (7) in in out out 6-1 is the oeffiient of thermal linear expansion of the ast tungsten arbide, K ; the dependene of the oeffiients Ks, Kp, Kn on the parameter is given in the graphs of Figure 4. With a good organization of the heat removal from the rings, the values are loated in the interval In [3], the restrition on the gradient of the expansion of the material of the ring is indiated; it is , that is why, the ondition Kp( a) T 6 1 must be satisfied. From this point, as well as from the graphs in Figure 4, it follows that for a ring from the ast tungsten arbide, the value Tl of 4

6 (HERVICON+PUMPS-17) IOP Conf. Series: Materials Siene and Engineering (17) 135 doi:1.188/ x/33/1/135 the rise in the temperature of the liquid in the hannel should not exeed 6.5K. 4. The hange in ring s temperature in the transient proess If the gap or other parameters used in the right side of the equation () are being hanged during the time t, then a stationary task (3) is transformed into a non-stationary task: T T / t ; T / x ;... ; T / y.5 q ( x, t ) x, is the speifi heat and the density of the material of the rings. Applying the Laplae transformation and the Fourier method, the following equalities are obtained: H Y yhy T x y s dy H A s H s x H 1 1 (,, ).5 ( )( ) os[ (1 / )],1,... p 1 ( ) H (, ) os[ (1 / )] A s H q x s x H dx s is the Laplae variable; as a result for thermal deformations Hin, Hout the solutions for images in the form of generalized Fourier series are obtained: in.5 ( )( ) os, out.5 ( )( ) H H A s H s H H A s H s 1 ( ) H (, ) os( / ) A s H q x s x H dx, 1 Returning to the originals and using the first terms of the series (the others are small), the previous proportionalities are obtained H ( t) K H T ( t); H ( t) K H T ( t) (8) in in out out and the differential equation for the design temperature T () t : H dt ( t)/ dt T ( t).5 q ( t) H ; 1 q ( t) ( R ( t)) / h ( t) (9) med 5. The ondition of the stati equilibrium of the sealing ring If the gap exeeds the size of the roughness by 3 times or more then the liquid frition regime is observed. Here, the rotating ring does not have a ontat with the stationary ring and its equilibrium is ahieved when the oppositely direted fores are equal Fspr ( h) Fp ( h) Fgr the fore of the spring Fspr ( h) Fspr. zspr h, the fore of gravity F gr (in whih Arhimedes' law is taken into aount, as well as the third part of the mass of the moving oils of the spring) and the fluid pressure fore in the gap H Fp ( h ) ( R x) p( x) dx (1) h is the initial learane on the outer radius R of the ring; z spr the oeffiient of the spring s stiffness; x the radial oordinate of the setion; px ( ) the diagram of the oil overpressure in the gap, whih satisfies the known differential equation [5] dp / dx 1 oilu / h or if the flow veloity u passes to the mass flow m ( R x) hu, 5

7 (HERVICON+PUMPS-17) IOP Conf. Series: Materials Siene and Engineering (17) 135 doi:1.188/ x/33/1/135 dp / dx [6 m /( Rh )] [ ( x)/( ( x) ( x))] (11) 3 3 r h r ( R x) / R, h ( h h) / h, / the relative values of the radius, the gap and the kinemati visosity in the setion x ; is the value of the visosity at the inlet to the seal. For the differential equation (11), the boundary value task with boundary onditions is solved p() p, ph ( ), p is the preset pressure drop on the slits, from whih the orretion for the entrifugal fore of Coriolis is subtrated pc.5medvt. med ; t. med V is an average tangential -1 veloity of fluid in the gap (in the investigated LSP at the operating mode Vò. ñð 5.5m ); med is its average density. Then the result of the solution is substituted into the integral (1). If it is assumed that the gap and the visosity along the length of the gap do not hange, then integration gives the value of the fore F ( h ) f k p / p R R f R is the area of the ring; and the orretion kr 1.5 H / Rmed takes into aount the taper, aused by the annular form of the gap; it is present in all further results. It is taken into aount that the values ( x) and hx ( ) vary with the hange in the temperature: ( x) 1 K ( Tl( x)/ Tl) ; h( x) 1 p( H / h ) kh ( Tl ( x)/ Tl )) k Kp( a) / Ks( a ).8 ; T T ( H ) H l l K is the visosity index at t 1 C (Figure ). lay And the amendment ( x) differs slightly from 1, therefore 3 ( x) (1 Tl( x)/ Tl) ; l H, 3 ( x)/( h( x)) ( x) T ( ( H / h ) k K /3) The use of the funtion ( x ) in equation (11), and then in the integral (1), leads to the following result: F ( h ) f k ( p / ) [1.45 /(1.4 )] (1) p R R In partiular, if 1, i.e. the taper of the gap leads to an almost omplete overlap of the flow near the inner radius of the ring, then Fp( h ) fr p and this is the maximum tearing fore whih an be obtained during the liquid frition regime. However, if in the operating mode of the LSP this result is obtained (due to the orresponding tightening of the spring), this an lead to overheating of the sealing ring. For the test ompation, the onnetion between the visosity, the outer gap h and the maximum oil heating temperature in the gap is determined by the equation: Tl T Ks a R H h k H T (13) l ( )[ ( med) /( / )]/( H l ) hereof it is easy to express the height of the gap through the values of the visosity and the temperature: h Ks a R H T k H T ( )[ ( med) / ]/ l H l Then the formula (1) was used, whih was rewritten in the following form: F ( h ) f k ( p / )[1 F] p R R 6

8 (HERVICON+PUMPS-17) IOP Conf. Series: Materials Siene and Engineering (17) 135 doi:1.188/ x/33/1/135 F T A ; A [.9( H k / h K /6)]/(1.8 H k T ) (14) l T T H H l The oeffiient F indiates the fration of the repulsive fore whih was aquired after the ring was heated. In Figure 5 (a), the graphs orresponding to the dependenes (13), (14) for the parameters of the analyzed ompation are plotted at the maximum permissible value 6.5 Ñ, and also at 5C and 8C. With a visosity med St, the gap h is.65 μm, and at the inner radius it dereases to.45 μm,.5 μm and.4 μm aordingly. If the tightening of the spring is seleted and remains unhanged then the oeffiient F is not hanged. As we an see, for equal values F an inrease in the visosity leads to an inrease in the steady temperature drop and following this rule the tightening must be made at the highest visosity, F 1 and it is diffiult to make suh a fine adustment. But if the visosity flutuation takes a short time, then, as we shall show later the temperature differene will also vary little. Therefore the tightening of the spring should be alulated based on the average visosity of the oil. T l (а) Figure 5. The oeffiients F of inrease in the lift fore and the initial gaps hi, μm with temperature differene 8C (urves 1), 6.5 C (urves ) and 5C (urves 3) and the oil temperature: (a) 1 C, (b) C. In the formula (14), two fators affet the hange in the lift fore: the thermal expansion of the rings inreases this fore and the derease in visosity with the inreasing of the temperature redues on the ontrary. With the visosity med St, the first fator is stronger than the seond one by an order of magnitude, but at high visosities (due to the inrease in the gap) this gap is redued. If the temperature is signifiantly redued (Figure 5 (b)), then with inreased visosities 6 St, it is going to be diffiult to adust the seal to work in liquid frition mode. 6. The modeling the movement of the ring in the transient modes As it follows from (9) the time of the thermal inertia of the rings is T H 1.6 s, but when the ring moves the stronger fators whih make slower the transient proesses are ating. It is onsidered to observe the one dimensional non stationary motion of a the visous inompressible fluid along the gap of the variable and the varying height h( x, t ), desribed by the well-known equations (here S ( R x) h( x) this is the area of the annular setion of the gap): m t um x S p x m h S t u S x m x / / / 1 / ; ( ) / ( ) / / The veloity of the flow is small, so the onvetive terms are not onsidered: (b) 7

9 (HERVICON+PUMPS-17) IOP Conf. Series: Materials Siene and Engineering (17) 135 doi:1.188/ x/33/1/135 p/ x 1 m/( S h ) u; m/ x ( h/ h) S And the aeleration u, by virtue of the non-ompressibility of the liquid is onsidered to be proportional to the axial aeleration of the ring: The integration of the equation u[( x.5 H) / h] d h / dt dp / dx [( x.5 H)/ h] d h/ dt under the boundary onditions p(), ph ( ) and substitution of the result into the integral (1) leads to the appearane of inertia fore F m d h dt redued to the ring, in red /, red m (1/3) f k h (.5 H / h) red oil R R m is the mass of fluid in the gap, Note that the mass m red inreases in the proportion to the redution of the gap and in the working modes of the ompation exeeds the mass m R of the moving ring (to whih, aording to the known rule, the inertial mass of the spring s turns is shown) in several times. The equation p / x 1 m/( S h ) oinides with (11), but the magnitude of the mass m flow due to non-stationarity of the proess for different ross setions an be different. Note that on the righthand side of the seond equation the density of the liquid varies with its temperature, and oil [ ( H / h)( k T ( x)/ T K )] T oil H l l in l -1 is the oeffiient of thermal expansion of oil,.1k, and h(, t) dh / dt is the speed of the axial movement of the ring; Kin Kn( a)/ Kp( a) (Figure 4). In order to determine the value m the seond equation was integrated: m( x) m() [ ( z) h( z)/ h( z)] S( z) dz x then this funtion was inserted into the equation (11) and the integral (1) using the established proedure. As a result the following formula for detahment strength was obtained: F f k P A T m d h dt m h dh dt m h B T p R R( / )[1 T l ] red / 1 red ( / ) / 1 red ( / ) T l B K ( ( H / h) k ) ; K T T T oil H 1/15 K /1. in Taking into aount the fores of the spring and the inertia of the ring, the equation of axial osillations is obtained: ( mr mred ) d h / dt (1 / h ) mred dh / dt zsprh frkr ( p / )[1 AT Tl ] (1 / h ) mred B Tl Fgr Fspr. zspr Kin H Tl. T The last term on the right-hand side of (15), whih takes into aount the deformation of the spring aused by the thermal expansion of the sealing rings an be negleted. Note that the mass m red inreases in the proportion to the hange in the gap and on the workers modes of ompation exeeds the mass m R in several times. The varying part of the fore depends on the temperature and the rate of the hange of the oil (15) 8

10 (HERVICON+PUMPS-17) IOP Conf. Series: Materials Siene and Engineering (17) 135 doi:1.188/ x/33/1/135 temperature, whih is determined from the equation (9). Equations (9), (16) are supplemented by an equation desribing the angular osillations () t of the sealing ring relatively to the shaft: d / dt f ( / h) R d / dt z f ( / h) R (16) m. tors R med spr. tors R med m. tors, z spr. tors is the moment of the inertia of the ring and the torsional rigidity of the spring and the formula (9) for the heat flux is refined: q ( t) ( R ( ( t) d / dt)) / h ( t) med Thus, the dynamis of the ompation is desribed by a system of three nonlinear differential equations and is of the 5th order. 7. The frequeny analysis and the model simplifiation The linearization of the system of equations (9), (15), (16) and the analysis of the frequenies of its free osillations showed that in the transient ondensation proess there are damped periodi and aperiodi osillations. The periodi angular vibrations of the rotating ring have a frequeny of about 1 Hz and the attenuation derement that inreases as the gap dereases; these quantities are mainly determined by the oeffiients of equation (16). The aperiodi proess of hanging the radial temperature differene in the gap has a time onstant T 1.6 s whih is mainly determined by the oeffiients of equation (9). The axial movements of the ring have two time onstants: (1 m / m ) h /(1 ) h /(1 ) 1 s, 1( m / z ) / h s h.1 R red h. red spr As one an see, the first proess is too short-lived and it is not neessary to take it into aount when modeling it. To eliminate this proess it is suffiient to exlude the fore of inertia from equation (15). As a result, the equation of the motion has the first order: dh / dt [ h /(1 mred )] frkr( p / )[1 AT Tl ] Fgr Fspr. zsprh B T T l (17) 8. The results of solving the test problems Figures 6-9 show the results of the alulation of transient proesses aused by the various fators. The hange in values is shown in a relative form: h h( t)/ h(), ( t)/ (), T T ( t)/ T (), Fd Fd( t)/ F l l spr. Fd is the sum of the ative fores applied to the ring (without aounting the fritional fore); the argument t as it was before designates the time and is measured in seonds. In Figure 8, the values h and T are shown in the perentages and the frations of a perent. Analyzing the graphs, one an ome to the onlusion that the fritional fores effetively dampen the ation of the perturbations so that the integrity remains. The leakage of oil through the end gap does -1 not exeed.5 Gh what is onsidered permissible for this devie. 9

11 (HERVICON+PUMPS-17) IOP Conf. Series: Materials Siene and Engineering (17) 135 doi:1.188/ x/33/1/135 (a) (b) Figure 6. Flutuation of visosity (in times): (a) into the large, (b) into the smaller side. (a) (b) Figure 7. Flutuation of the pressure drop (in times): (a) into the large, (b) into the smaller side. (a) (b) Figure 8. Fored osillations with a frequeny of 5 Hz: (a) axial vibration of the housing with an amplitude of. mm, (b) unevenness of the pump feed.5%. The value h is shown as a perentage, and T in tenths of a perent (Ppm). Time t is ounted from the beginning of the ause of the osillations. 1

12 (HERVICON+PUMPS-17) IOP Conf. Series: Materials Siene and Engineering (17) 135 doi:1.188/ x/33/1/135 Figure 9. Simulation of power failure with a duration of 6 s. Moreover, there is no signifiant overheating of the rings in these proesses. 9. Conlusion In order to ensure a stable operation of the mehanial seal in the liquid frition mode the designer needs to reate an additional hydrodynami fore whih tends to inrease the gap between the rings by the inreasing of the speed of the shaft or the pressing fore. The authors have left the traditional view whih says that the ause of this fore is a roughness or and undulation of the surfaes, and there are explanations about this ase. The maximum size of irregularities on the ground surfae of a ring made from ast tungsten arbide does not exeed.3 μm, whih is an order of magnitude smaller than the working gap. In addition, ast tungsten arbide this is 96% (by weight) wolfram and its high firmness prevents an inrease in roughness during the usage what ours on the ontrary with the seals whih made from soft omposite materials. It turned out that the appearane and the hange of the additional lifting fore an be explained by the uneven temperature deformation of the rings whih lead to the appearane of the taper of the gap. The lak of empiriism in the formula of the lifting fore made it possible to explain the stati and the dynamis of the ompation from a unified position. The alulated equations reflet the fators known from the literature but in a losed form they are onsidered and used for the first time. Further on, the authors plan to speify the model: take into aount the heat flowing from the shaft as well as the regular plasti deformations of the sealing surfaes arising during operation. Then they will look for the ways to identify the model but if to take into aount the extreme values of oil pressure and temperature it will be diffiult to do this. Referenes [1] Andrenko P M and Lebedev A Y 17. Labyrinth srew pumps (Kharkiv: Panov) p 156 [] Eygelson A S and Shaykhali D M Calulation of density and visosity of layer oil by data of surfae degassing. (Mosow: Geology of oil and gas) 11 [3] Mayer E 1978 Mehanial seals (Mosow: Mehanial Engineering) p 88 [4] Farlow S 1983 Partial differential equations for researhers and engineers (Mosow: Mir) p 381 [5] Loitsyansky L G 1978 Mehanis of liquids and gases (Mosow: Nauka) p