The 1997 ASME Fluids Engineering Division Summer Meeting FEDSM 97 June 6, 1997 FEDSM97 338 THE COMPUTATON OF FLUD-NDUCED FORCES ON CENTRFUGAL MPELLERS ROTATNG AND WHRLNG N A VOLUTE CASNG R.G.K.M. Aarts Department of Mechanical Engineering University of Twente P.O. Box 17 7500 AE Enschede, The Netherlands phone: +31 53 489 557 fax: +31 35 489 3663 Email: r.g.k.m.aarts@wb.utwente.nl J.B. Jonker Department of Mechanical Engineering University of Twente P.O. Box 17 7500 AE Enschede, The Netherlands phone: +31 53 489 591 fax: +31 35 489 3631 Email: j.b.jonker@wb.utwente.nl ABSTRACT A finite element based method has been developed for computing fluid-induced forces on an impeller in a volute casing. Potential flow theory is used assuming irrotational, inviscid and incompressible flow. Both excitation forces and motion dependent forces are calculated. The numerical results are compared with experimental results obtained at the California nstitute of Technology. n two-dimensional and three-dimensional simulations the calculated pump characteristics near the design point are about 0% higher than the experimental curve. This is caused by viscous losses that are not taken into account in our model. The magnitude of the excitation force is predicted well for optimum and high flow rates. At low flow rates the calculated force is too large which is probably related to inaccuracies in the calculated pressure. The motion dependent forces from two-dimensional simulations are in reasonable agreement with the experimental results. The important region of destabilizing tangential forces is predicted fairly well. NOMENCLATURE A Area A h Hydraulic area F Force (on the impeller) H Head Momentum J Angular momentum M (Shaft) moment N number of blades N s Specific speed (Q 1 =(gh) 3 4 ) O Origin Q Volume flow rate T Period c Bernoulli constant e Unit vector in the direction of " f Dimensionless force g Gravitational acceleration n Normal vector p (Static) pressure p 0 Total pressure r Position vector r Outer impeller radius r s Radius of the sliding cut t Time v Absolute velocity w Relative velocity, Circulation " Whirl orbit vector Polar coordinate Density Dimensionless flow rate ' Velocity potential Dimensionless head Angular velocity of the rotation! Angular velocity of the whirl 1 Copyright c 1997 by ASME
Subscripts: 0 Zeroth order 1 First order n Normal component t Tangential component y y 0 P r r s sliding cut 1 NTRODUCTON The fluid-induced forces on the impeller play an important role for the dynamic behavior of the rotor shaft in a pump. These forces originate from the hydrodynamic interaction between the impeller and the volute. Two main fluid-structure interaction mechanisms can be distinguished: (1) the forces from the flow in clearances like in seals and between the shroud and the casing, () the forces from the unsteady flow inside the impeller channels. The latter will be addressed in this paper. n general, forces arise if the pressure distribution is not homogeneous. E.g. at off-design conditions the pressure around the impeller is not constant, which causes a time-varying radial excitation force on the impeller and the pump shaft. Furthermore, vibrations of the impeller disturb the flow field and generate additional so-called motion dependent forces. These forces can initiate unstable subsynchronous impeller motion (see e.g. Verhoeven (1991)). A finite element based method has been developed for computing the fluid-induced forces on an impeller in a volute casing. Results from the simulations are compared with the experimental data from measurements carried out at the California nstitute of Technology (CalTech). n many experiments an impeller X with specific speed N s = 0:57 in a volute A is used (Adkins (1986), Adkins and Brennen (1988)). n Franz and Arndt (1986) measurements are reported in which an impeller Z is used. This is a two-dimensional model of impeller X. The computations reported in this paper focus on impeller Z. THEORETCAL AND NUMERCAL BACKGROUND n this section a condensed description of the theoretical and numerical background of our computational method is given. Details of this model have been published elsewhere (van Essen (1995), Jonker and van Essen (1997))..1 Coordinate systems n our calculations the volute is treated in a fixed coordinate system and the impeller is treated in a relative coordinate system. The relative coordinate system x 0 O 0 y 0 moves with the impeller where the origin O 0 coincides with the impeller axis. The relative motion of the coordinate system x 0 O 0 y 0 in the fixed coordinate system xoy can be split into two contributions: The rotation of the impeller with the angular velocity and the motion of the whirl orbit vector " (Fig. 1). n this paper we only discuss a plane circular whirl motion with radius " and constant speed!, O " O 0 Figure 1: The inertial and rotating coordinate systems for a whirling impeller. The sliding cut at radius r s is the interface between rotor and stator part. so and the time derivative _" is t x x 0 " = "e; (1) _" = "!e; () in which e is a unit vector in the direction of ". n the impeller region it is often convenient to use the fluid velocity w relative to the the moving coordinate system x 0 O 0 y 0. n a point P it is related to the absolute velocity v by v = w +r+"!e; (3) where r is the position vector of P in x 0 O 0 y 0.. Potential flow equations n pumps with well-designed impellers and backward curved blades there is a broad range of operating conditions in which no flow separation occurs and the flow may be treated as inviscid and incompressible. Assuming irrotational flow at the inlet the velocity v is the gradient of a velocity potential ' v = r': (4) The continuity equation can then be written as rr'=0: (5) n a two-dimensional flow field it is possible to account for a varying axial width b by using rbr'=0; (6) provided rb is small. The pressure p is obtained from the unsteady Bernoulli equation @' + 1 v v + p = c(t); (7) Copyright c 1997 by ASME
where is the density and c is the Bernoulli constant which only depends on the time t. n the impeller region Eq. (7) is rewritten as @' 0 + 1 v v, ( r + "!e)v+ p =c(t); (8) where the superscript 0 indicates that the time derivative is taken in the relative coordinate system. Slit lines are present to make the geometry of impeller and volute singly-connected. The potential jumps across these slit lines correspond to circulations e.g. around a blade (combined with its wake) or around the whole impeller. These circulations are defined over any closed curve S,= S vds: (9) The circulation, i around blade i of the impeller is not determined uniquely in the solution of the potential ' from Eq. (5) or Eq. (6), unless extra restrictions are imposed upon the flow. These restrictions follow from the so-called Kutta condition which requires that the relative velocities of the fluid at the trailing edges are tangential to the blades. n unsteady flow the blade circulations change and vorticity is shed of. We distinguish two numerical methods. n the quasi-steady approach the circulation around the blades is determined in order to satisfy the Kutta condition at each instant. n the unsteady approach vortex wakes are implemented and the vorticity that is shed of the blades is transported along wake curves. By imposing continuous pressure and normal velocity across the wake curves, expressions for the transport equations of the vorticity along these wakes can be derived (Jonker and van Essen (1997)). n the computations the impeller and volute region are connected at a sliding cut (Fig. 1). This sliding cut consists of two coinciding circles. The rotation of the impeller is implemented by altering the connections between the circles. Thus the meshes in both regions do not have to be modified to account for the continuously changing impeller-volute configuration. The wake curves are confined to the impeller region and are truncated near the sliding cut. The vorticity that reaches the end of the wake curve is removed..3 Perturbation method n the case of a whirling impeller the sliding cut is generally not a fixed circle in the stator region. This complicates the coupling between both regions. A solution has been found by using a regular perturbation analysis. For small amplitudes of the whirl motion, the flow field is written as a series of powers of "=r v = v 0 + " v r 1 + O ( " ) ; (10) r where r is the outer impeller radius. The subscripts 0 and 1 denote the zeroth order and first order velocities, respectively. The CV Figure : Control volume for the calculation of the impeller forces and shaft momentum. zeroth order velocity is associated with the centric rotation of the impeller, i.e. " =0. The first order velocity is the perturbation due to the whirling motion. Similarly, other quantities like the pressure are splitted p = p 0 + " p 1 + O ( " ) (11) r r and substituted in the equations mentioned earlier in this section. Equating equal powers of "=r gives systems of zeroth order and first order equations which both can be solved for a centric position of the impeller (Jonker and van Essen (1997)). The excitation force F 0 is calculated from the zeroth order problem. A straightforward method is to integrate the pressure force on the surfaces of the blades. However, at off-design conditions the flow at the inlet of the impeller is not along the blades and discontinuities in the velocity may exist in the calculations. These discontinuities affect the pressure according to Eq. (8). Consequently, the force is not computed accurately, especially in computations with thin blades. A method which minimizes the effects of the velocities and pressures at the leading edges uses a control volume attached to the impeller as shown in Fig.. Applying the law of conservation of momentum gives for the total force F c on the fluid in the control volume F c = D Dt ; (1) where is the momentum of the fluid inside the control volume = Z CV v da: (13) Using Reynolds transport equation (Shames (198)) the material time derivative of in Eq. (1) can be expressed in the local time derivative D Dt = @ CS + v(w n)ds; (14) where CS is the surface of CV. The force F c is composed of the impeller force F and the pressure force on the inlet and outlet surfaces CS io of the control volume F c =,F, pnds: (15) CS io 3 Copyright c 1997 by ASME n
Substituting Eqs. (14) and (15) into Eq. (1) and taking into account that w n =0at the blades gives F =, @ CS, v(w n)ds, pnds: (16) io CS io The excitation force F 0 is obtained by substituting the zeroth order solution into Eq. (16). Analogously, using the conservation of angular momentum J z = Z CV (r v) z da (17) an expression for the shaft momentum can be derived M =, @J z CS, (r v) z (w n)ds io, CS io p(r n) z ds: (18) For a control volume bounded by cylinders centered at the z axis the third term is zero. The motion dependent forces are derived from the first order solution. n the presentation of the results we consider two components of the first order forces. These normal and tangential components F 1n and F 1t are relative to the whirl orbit as shown in Fig. 3. The tangential component F 1t is of special importance for the rotor dynamics. The motion dependent force F 1 has a component in the direction of the whirl motion _", iff 1t and the ratio!= have equal signs. n that case the motion dependent force has a destabilizing effect on the rotor motion. For two-dimensional calculations a set of computer programs are developed to solve the unsteady zeroth and first potential flow equations in two dimensions (THWCOMP) andfor postprocessing (THWPOST). For the three-dimensional simulations the software package COMPASS is available. This package is developed at our University (van Esch et al. (1995)). t y F 1t F 1n computes the three-dimensional flow very efficiently by using a substructuring technique and an implicit treatment of the Kutta conditions..4 Pressure calculation n all simulations special attention should be paid to the computation of the pressure with Eqs. (7) and (8). The force in Eq. (16) depends strongly on the pressure differences at the surface of the control volume. Especially the computation of the time derivative may introduce errors. n unsteady flow the blade circulations change and vorticity is transported along the wake curves. f vorticity is removed at the end of a wake curve the total circulation of the blade and the wake changes and the potential jump across its accompanying slit line changes. That leads to different time derivatives @'= on both sides of the slit line and after substitution in the Bernoulli equation a discontinuous pressure is found. To avoid these discontinuities two kind of time steps are commonly used. During a backward time step the potential jumps across slit lines are kept constant and the time derivatives are calculated. During the next time step these jumps are adjusted. All pressures are continuous using this algorithm. However, significantly different pressures are found if time derivatives are computed using all time steps. For highest accuracy a time average force is computed avoiding time derivatives. Using Eq. (7) with c(t) = 0the total pressure p 0 = p + 1 (19) v in the stator region can be written as p 0 =, @' : (0) The zeroth order flow is periodic with period T = 1 (1) N where N is the number of blades. The difference in the velocity potential ' between any two points outside the sliding cut is a continuous function of time and is also periodic. As a result, the time averaged total pressure in period T O "!t x p 0 =, @' () in both points should be the same. 1 The average total pressure difference across the impeller p 0 is related to the average shaft momentum according to p 0 = M Q : (3) Figure 3: The components F 1n and F 1t of the first order force normal and tangential to the whirl orbit. 1 Note that the control volume is in the impeller region where the function '(t) in a fixed point is not continuous due to the presence of wakes, blades and slit lines. This mathematically complicates the computation of @'=. 4 Copyright c 1997 by ASME
The first term in Eq. (18) vanishes due to the periodicity, so we can write for a cylindrical control volume M =, (r v) z (w n)ds: (4) CS io Combining Eqs. (19), (), (3) and (4) gives the average pressure p outside the impeller region p =, (r v) z (w n)ds, 1 ; (5) Q v CS io which can be computed without using time derivatives. This p is substituted in e.g. Eq. (16) to compute an average force F. nlet mpeller slit Sliding cut Blade slit Wakes curve Control volume Outlet 3 TWO-DMENSONAL CALCULATONS WTH THE CALTECH GEOMETRY Franz and Arndt (1986) report results from experiments with the two-dimensional centrifugal impeller Z in a single volute A. The impeller has a constant axial width and it consists of five logarithmic spiral blades with a blade angle of 5 o. The thickness of each blade is one-eighth of the total spacing between the pressure sides of two successive blades. Figure 4 illustrates the configuration. The volute is modeled from the drawings (Adkins and Brennen (1988)). A simplified cross-section is used as shown in Fig. 4. t consist of a diverging part with an angle d =0 o on both sides up to a radius r c. The part for radii between this cutoff radius r c and the outer radius r o is a converging section with a constant angle c on both sides. Two quantities are considered as functions of the angle from the tongue : The cross-sectional area Z A = b(r)dr; (6) and the hydraulic area Z b(r) A h = dr; (7) r at a number of positions in the volute. The radii r c and r o are calculated as functions of to get an agreement of A and A h between our model and the actual geometry. With c = 56:4 o values of r o () are found that are very close to the actual outer radii. The dimensionless pump characteristic is shown in Fig. 5. The flow coefficient and the head coefficient are defined as Q = (8) r b; = gh : (9) r For most values of the computed head is larger than the measured head. This is due to the use of an inviscid flow model in = 310 o d Figure 4: CalTech impeller Z in volute A: The twodimensional computational domain and the axial width b of volute A at 310 o from the tongue. The thin curve in graph is the actual width according to Adkins (1986). The thick line is the approximation used in the calculations. The dashed curve in the volute in graph is at the position of the cut-off radius r c in graph. which dissipative effects in the fluid are neglected. At off-design conditions the quasi-steady approach gives a smaller head. t was found that the changes in the blade circulations of the rotating impeller are smaller if wakes are taken into account. Apparently this gives a more realistic characteristic. The magnitude of the zeroth order excitation force is shown in Fig. 5. The force is scaled as F f = r 3b (30) The magnitude of the force is in qualitative agreement with the measurements. At low flow rates the force is overestimated. This is probably related to errors in the computed pressure. Experimentally optimal flow is found at = 0:086. The minimum value of f 0 found in the calculations for =0:093 indicates a r c c r o 5 Copyright c 1997 by ASME
0.8 5.0 0.6 f 1n 4.0 3.0 =0:086 0.4.0 0. 1.0-1.0-1.0-0.5 0.5 1.0!= Experimental data (Adkins (1986)) Unsteady computations 3 Quasi-unsteady computations 4 Experimental data (Adkins (1986)) Unsteady computations 3 Quasi-unsteady computations 4 f 0 0.15 0.10 f 1t.0 1.0 =0:086 5-1.0 0 Figure 5: CalTech impeller Z in volute A: the pump characteristic and the magnitude of the average zeroth order fluid force acting on the impeller. -.0-1.0-0.5 0.5 1.0!= Figure 6: The normal and tangential component of the first order fluid force acting on the impeller for = 0:086. higher flow rate for optimal conditions. Extended research revealed that the excitation forces are very sensitive to the shape of the volute (van Essen (1995)). n Fig. 6 the motion-dependent forces are shown for optimum flow. n each graph experimental results and forces computed with the quasi-steady and with the unsteady approach are shown. For positive whirl ratios (!= > 0) the normal forces are in reasonable agreement with the experimental results. For negative whirl ratios the calculated forces are too small. The influence of the unsteady wakes is small. Apparently the added mass that is related to the normal force is not changed if wakes are present. The calculated tangential forces without unsteady wakes are too large. Better agreement is found if the unsteady wakes are included. The important region of destabilizing tangential forces is predicted fairly well. At off-design conditions similar results are found, although the calculated normal forces deviate more from the experimental results at higher flow rates. More detailed research showed that the first order forces are, in contrast to the zeroth order forces, not very sensitive to the shape of the volute (Jonker and van Essen (1997)). However, small changes in f 1t may affect the size of the region of destabilizing forces and it is possible to decrease this width in Fig. 6 with a different volute design. 4 THREE-DMENSONAL CALCULATONS WTH THE CALTECH GEOMETRY For three-dimensional calculations impeller Z and volute A are modeled in a similar way as is discussed in the previous section. The domain and the axial width of Fig. 4 are combined in the three-dimensional geometry shown in Fig. 7. Shaft momentum and excitation forces are computed using a control volume slightly larger than the blade region. Figure 8 shows the results in comparison with the results from the twodimensional simulations. n all simulations wakes have been taken into account. Clearly both the momentum and forces are in good agreement from both simulations. Apparently, Eq. (6) may be used in the two-dimensional simulations although rb is rather large in the volute. 6 Copyright c 1997 by ASME
0.8 0.6 0.4 0. Figure 7: Three-dimensional model of the CalTech impellers Z in volute A. The the hub and blade surfaces in the impeller and the lower half of the volute are shown. Experimental data (Adkins (1986)) D unsteady computations 3 3D unsteady computations + 5 DSCUSSON The calculated pump characteristic is near the design point about 0% higher than the experimental curve. This is caused by viscous losses that are not taken into account in our model. The magnitude of the excitation force is predicted well for optimum and high flow rates. At low flow rates the calculated force is too large. The motion dependent forces are in reasonable agreement with the experimental results. The normal component is too large for a negative whirl speed ratio. The range of whirl speed ratios where the tangential component may destabilize the motion is predicted fairly well. Future work will deal with the computation of the first order flow in three-dimensional simulations in mixed-flow impellers, including impeller X. Also work is undertaken to improve the accuracy of the pressure calculation. ACKNOWLEDGMENTS The research reported in this paper has been financially supported by the European Community in the scope of the BRTE-EURAM program under contract BRE-CT94-0945 (APHRODTE). REFERENCES Adkins, D. and Brennen, C. (1988). Analyses of hydrodynamic radial forces on centrifugal pump impellers. ASME J. Fl. Engineering, Vol. 110(), pp. 0 8. Adkins, D. R. (1986). Analyses of hydrodynamic forces on centrifugal pump impellers. PhD thesis, California nstitute of Technology, Pasadena (CA). Report Number 00.3. Esch, B. van, Kruyt, N., and Jonker, J. (1995). An efficient method for computing three-dimensional potential flows in hydraulic turbomachines. n Ninth nternational Conference on Finite Elements in Fluids New Trends and Applications, Venice, taly, pages 633 644. f 0 0.15 0.10 5 0 Figure 8: Results from three-dimensional calculations (+) on CalTech impeller Z in volute A in comparison with results from two-dimensional calculations and experimental data. Graph shows the pump characteristic and graph shows the magnitude of the average zeroth order fluid force. Essen, T. van (1995). Fluid-nduced mpeller Forces in Centrifugal Pumps, Finite Element Calculations of Unsteady Potential Flow in Centrifugal Pumps. PhD thesis, University of Twente. Franz, R. and Arndt, N. (1986). Measurements of hydrodynamic forces on a two-dimensional impeller and a modified centrifugal pump. CalTech, Report No. E49.4. Jonker, J. and Essen, T. van (1997). A finite element perturbation method for calculating fluid induced forces on a whirling centrifugal impeller. nt. J. Num. Methods in Engineering, Vol. 40(), pp. 69 94. Shames,. H. (198). Mechanics of fluids. McGraw-Hill Book Company, nd edition. Verhoeven, J. (1991). Rotor dynamics of centrifugal pumps, A matter of fluid forces. The Shock and Vibration Digest, Vol. 3(8), pp. 3 17. 7 Copyright c 1997 by ASME