Thrust Action

Engineering Applications of Computational Fluid Mechanics Vol. 6, No. 2, pp. 271–284 (2012)

Received: 1 Sep. 2011; Revised: 4 Jan. 2012; Accepted: 16 Jan. 2012

271

METHODOLOGY FOR THE RESIDUAL AXIAL THRUST EVALUATION IN MULTISTAGE CENTRIFUGAL PUMPS

Simone Salvadori *, Alessandro Marini and Francesco Martelli

Energy Engineering Department “Sergio Stecco”, University of Firenze, Via di S. Marta, 3 50139 Firenze, Italy

* E-Mail: [email protected] (Corresponding Author)

ABSTRACT: One of the most challenging aspects in horizontal pumps design is the evaluation of the residual axial thrust acting on the rotating shaft. The thrust is affected by pump characteristics and working conditions. Solving this problem is easier for a single stage pump than for multistage pumps, even in partially self-balancing opposite impeller configuration. The challenge is then to individuate a procedure that will provide the residual thrust value with a moderate computational effort, dealing with the industrial requests of accuracy and reduced time consumption. A procedure is proposed, which consists in the numerical simulation of each pump component. For each component, the obtained mass-flow/thrust correlations are coupled by using a momentum balance equation used to calculate the axial thrust as a function of the working conditions. The main topic in multistage pump modeling is the leakage flows characterization by means of accurate numerical analysis. Therefore, the cavity flows behavior is investigated and the flow structures individuated. The numerical investigation of the pump’s components provides also a thorough knowledge of fluid dynamic fields. The proposed procedure is able to predict both the direction and the variation of the thrust in a selected range of flow rates, while the value of the thrust is affected by a non-negligible error generated by “real machine” effects.

Keywords: axial thrust, cavity flows, centrifugal pumps, CFD, multistage, momentum balance

1. INTRODUCTION

In a multistage horizontal centrifugal pump, the residual axial thrust is balanced by bearings that can guarantee the mechanical reliability (when properly chosen). Over the pump operating range the main contribution to the axial thrust is due to the impellers flow fields, the leakage flows through wear rings and the pressure distribution that occurs inside the gaps between impeller shrouds and pump stationary walls. A detailed analysis of the origins of multistage pump unbalance has been proposed by Gantar et al. (2002). They underlined the effect of cavity flows and off-design conditions but did not propose a procedure to couple the separated contributions coming from the pump components. The cavity flow behavior is a key parameter for the pressure field evaluation and then for the thrust calculation. From a physical point of view, the flow filed inside the impeller side chambers has been extensively described by the classical contributions by Batchelor (1951) and Stewartson (1953). Many contributions are also available where the leakage flow has been analyzed to provide correlations between the leakage mass-flow, the impeller head, the friction losses and the geometrical parameters. Amongst them it is

worthwhile to remember the works of Denny (1954), Traupel (1958), Worster and Thorne (1959), Daily and Nece (1960), Utz (1972) and more recently Tamm and Stoffel (2002) and Gulich (2003a and b). The main limit of these correlations is that they refer to an “ideal” configuration, which is somehow representative of cavity flows, but neglect two fundamental aspects. The first one is that the geometrical characteristics of side chambers are usually decided by compactness and reliability criteria instead of analyzing disk friction losses. Then, their shape could be very different from the one represented by the correlations, especially considering the front cavity. The second issue is related to the boundary conditions: since cavity and main-flow are coupled by clearances and there are non-linear effects to be considered. Baskharone and Wyman (1999) and Adami et al. (2005) proposed methods based on the numerical modeling to manage with this kind of problem. Thamsen and Bubelach (2011) suggested that the main problem in residual axial thrust evaluation was the non-uniformity of the cavity inlet conditions generated by the impeller/diffuser interaction. The experimental analysis of a single stage pump demonstrated that the flow field and

Engineering Applications of Computational Fluid Mechanics Vol. 6, No. 2 (2012)

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the pressure distributions were not axi-symmetric. This phenomenon is even more important when radial gaps are non negligible. Furthermore, flow recirculations occurring at off design conditions at the impeller exit section are responsible for increased non-uniformities. To overcome all of these problems, the numerical simulation of each pump component has been individuated as a possible solution to the complex problem of axial thrust evaluation. In fact, Computational Fluid Dynamics (CFD) has been demonstrated to be accurate enough to study real cases in the turbo machinery field of interest. Several examples are available in the recent literature for compressors (Zachos et al., 2011), combustion chambers (Marzouk and Huckaby, 2010), turbines (Zhou et al., 2011; Liu and Wang, 2011; Montomoli et al., 2011) and pumps (Sifikhani et al., 2011; Koombua and Pidaparti, 2010). Each contribution to the residual thrust intensity is calculated taking into account the local pressure distribution on the rotating walls. Concerning cavity flows, all of them are analyzed considering the actual geometry and working range with a loosely coupling with the impeller/diffuser stages. The latter are analyzed considering realistic inlet conditions coming from the numerical analysis of the upstream component. The obtained correlations are matched together by means of a momentum balance equation obtained by studying a pump stage. Furthermore, an in-depth study of the fluid dynamic fields inside the single pump components can help to obtain more knowledge of the pump characteristics during the design phase to avoid off-design issues. The proposed approach represents a possible answer to the industrial requests of accuracy and smartness in axial thrust evaluation. The known limits of this method are a simplified approach to the leakage flow evaluation and the loosely coupling of some components. Nevertheless, those approximations will not affect the accuracy of the analysis while the method is applied to a range of working conditions close to the Best Efficiency Point (BEP) (±15%) for centrifugal pumps with small gaps at the impeller outlet diameter.

2. THRUST EVALUATION IN MULTISTAGE PUMPS

Present paper describes the development of a numerical method for the residual axial thrust calculation. The evaluation of the resultant of the forces acting on each single stage is the starting

point to determine the thrust. With reference to the schematic distribution of forces and control volume shown in Fig. 1, this balance can be expressed by Eq. 1:

F

fsF

bsF

inletF

momF

busT

ax (1)

The balance is referred to a control volume containing all the rotating walls. The term Ffs is the global force acting on the front shroud walls while Fbs is relative to the back shroud: both of them are evaluated considering the static pressure distribution along the rotating walls. Finlet is generated by the pressure field at the impeller inlet section and is a function of the local static pressure value. Fmom is the momentum contribution along axial direction and is calculated considering the mean value of inlet velocity. The term Fbus is the pressure integral on the bushing walls. All these terms can be evaluated considering pressure levels, pump geometry and mass-flow conditions. For each pump (at a chosen flow rate) the axial thrust can be calculated once the pressure field inside its components, including leakage cavities, is known. Therefore the multistage pump axial load can be obtained algebraically by adding the single stage contributions, which have been calculated by applying Eq. 2.

2

fs bs inlet inlet

A A

bus bus axinlet

p ndA p ndA p A n

Qn p A n T

A

(2)

In the pump shown in Fig. 2, the flow enters the suction nozzle and passes through the two stages of the first bench. Then a crossover device leads it to the third stage inlet through an annular chamber, thus being responsible for its turning of direction. The fluid evolves in the three stages of the second bench and reaches the discharge nozzle. All the diffusers are characterized by the same geometry except for the ones facing the

Fig. 1 Forces balance for a single stage (the dotted

lines indicate the control volume).


273

Fig. 2 Multistage centrifugal pump (courtesy of

WEIR-Gabbioneta SRL).

Fig. 3 Impeller side chambers.

Fig. 4 Central balancing drum.

crossover. All the impellers are geometrically identical but with different inlet conditions, due to the presence of different upstream components. The opposite impeller configuration helps to balance the axial thrust, but does not guarantee it also in pumps with an even impellers number because of the effect of leakage flows and shroud-stationary walls gaps. To properly choose the thrust bearings, the residual axial load intensity and direction must be evaluated taking into account the contribution of the impeller shroud chambers. The leakage flows and the presence of gaps modify the pump flow rate (and, consequently, the total head) and affect the pressure distribution on the rotating walls. In such a complex machine, three kinds of cavities

are present: front and back impeller shroud cavities and a central drum. Fig. 3 shows the main and leakage flows in the front and back shroud chambers. While the flow inside the front shroud chamber always turns inward in the radial direction, the one inside the back shroud cavity turns according to the local pressure gradient. Even though the mass-flow through the back shroud chamber has a slight influence on the whole pump performance, its contribution to the axial thrust cannot be neglected. The second and the fifth impeller back shroud cavities correspond to the central drum ones in which the leakage flow goes from the second to the first group of impellers (Fig. 4). A lateral balancing drum is located before the first impeller of the second bench and is connected to the suction volute by means of a balancing duct thus keeping the stuffing box pressure very close to the suction pressure.

3. CFD PROCEDURE DESCRIPTION

The full procedure is applied to the described pump designed by WEIR-Gabbioneta SRL (Fig. 2). The evaluation of the component’s performances requires the development of a strategy to manage the following tasks:

The single stage analysis (both impeller and diffuser hydraulic channels);

The simulation of stationary components (volutes and crossover);

The study of the flow conditions inside the front and back shroud impeller side chambers;

The study of the flow conditions inside the central balancing drums;

The final data collection for the residual hydraulic axial thrust calculation.

Several second order accurate 2D/3D steady simulations solving Reynolds Averaged Navier-Stokes (RANS) equations have been carried out using the ANSYS Fluent commercial code. Hybrid meshes have been realized using the commercial tool Centaur by Centaur soft. A second order accurate finite-volume pressure-correction procedure for incompressible flows has been employed with a two-equation k-ε turbulence model with standard wall functions. The choice of the turbulence model is coherent with the near wall mesh quality, which ensures a y+ value higher than 30.The pump has been divided into its main components, which have been analyzed separately to reduce computational


274

costs. The following elements or group of elements have been simulated:

Suction volute;

First impeller;

Each diffuser not facing the crossover device coupled with its downstream stage impeller;

Second diffuser together to crossover device and annular chamber;

Third impeller;

Fifth diffuser and discharge volute;

Impeller side chambers (front and back shrouds);

Central and lateral balancing drums.

The sequence of simulations has been chosen considering each result as a boundary condition for the following CFD analysis. For instance, the impeller boundary conditions have been provided by its upstream stationary component simulation (either diffuser or annular chamber). Finally, all the results have to be combined as indicated in Eq. 2, in order to obtain the axial thrust of each of the two groups of stages, thus helping to understand their reciprocal balancing effects. To study the shroud chambers behavior, the relation between the head across the impeller and the leakage flows should be known. The impeller shroud chambers have been analyzed with 2D axi-symmetric CFD simulations while a 3D model has been used for the central balancing drum. This choice is supported by the results shown by Gantar et al. (2002) for geometries similar to the present one. Flow recirculation in the impeller exit area should not affect leakage flow rate in this case due to the small clearance that separates the main impeller passages from their own shroud chambers as reported by Gantar et al. (2002). As already reported, recently Thamsen and Bubelach (2011) denied this assumption in case of large gaps. In the latter case the proposed approach must be changed to include impeller/cavity interaction and a fully coupled approach under unsteady conditions is suggested. Grid dependence analysis has been performed for all the components. The procedure used to define the optimal mesh is here described for the front shroud only and has been repeated for the other components. The sensibility analysis of the leakage flow and the axial thrust to the spatial resolution has been performed. Grids characterized by a different resolution both nearby and far from the walls have been realized

Fig. 5 Front shroud computational hybrid grids.

(a)

(b)

Fig. 6 Leakage flow (a) and thrust (b) as a function of impeller head for front shroud.

(Fig. 5). Grid 1 contains around 27300 elements, Grid 2 70400 (+258%) and Grid 3 89800 (+379%). In Fig. 6 the leakage mass flow and axial thrust value as a function of the head across the cavity for the front shroud are reported. The shown results are non dimensional with respect to the BEP values. It should be noticed that, despite of the complex geometry of the front shroud


275

cavity, both the leakage mass flow and the axial thrust acting on the rotating wall are a linear function of the impeller head. It can be seen that leakage flow and thrust are underestimated, respectively, by about 22% and 16% by Grid 1 relative to Grid 2, while for Grid 2 and Grid 3 the mismatches are negligible. Therefore, Grid 2 strikes a balance between computational costs and accuracy of results.

4. STAGE COUPLING APPROACH

The axial thrust on pump shaft is due to the forces balance on the impeller walls; however the correct pressure field evaluation requires also the stationary components simulation to provide the right boundary conditions for the rotating parts analysis. The presence of both stationary and rotating parts in each stage suggests a coupled approach for the diffuser/impeller interaction evaluation, namely to deal with the computation of domains characterized by different periodic conditions because of the different number of vanes/blades. A mixing plane technique has been applied in present study. Diffuser inlet and impeller outlet have been extended in the computational models to avoid errors induced by imposing the boundary conditions on an interface plane exactly corresponding to the real components inlet and outlet sections. The mixing plane approach allows solving separately the rotating and the stationary domains in steady conditions. Data from adjacent zones are tangentially averaged and then imposed as mixed boundary conditions at the interface. This approach removes any unsteadiness deriving from the circumferential variations in the interface plane, thus yielding a steady state result. Nevertheless, this method allows to maintain a non-uniform radial distribution of the variables and then a swirled inlet velocity distribution. Since the impeller eye has a large inlet area and the tangential non-uniformity of the flow at the impeller inlet is negligible with respect to the axial component of the flow, this method provides a realistic evaluation of the stage performance. Also Adami et al. (2005) demonstrated that the approximation of the time-averaged results was quite reasonable especially for the performance parameters evaluation. It must be pointed out that this kind of approach is not accurate for volute pumps. In that case unsteady calculations would be appropriate, although the computational time increase would be a non-negligible factor depending on the flow field complexity and the mesh characteristics.

In the present method, a further hypothesis of the stage kinematics repeatability has been assumed. Therefore, the velocity profile at the impeller exit section has been employed to update the inlet boundary conditions of the diffuser model. The whole process is iterative and can be synthesized in the following main steps:

(a) Imposition of the initial boundary conditions on the diffuser inlet (flow rate and velocity direction) and on the impeller inlet (flow rate) and front shroud leakage inlet (flow rate);

(b) Solution of the RANS equations in both domains;

(c) During the simulation, the CFD solver updates the conditions on the mixing plane averaging the flow field and the static pressure values in tangential direction;

(d) Once the convergence has been reached, the boundary conditions are updated at the interfaces;

(e) The steps from (a) to (d) have to be repeated until convergence is achieved both for velocity profiles and pressure values.

The stage computational hybrid grid is reported in Fig. 7. The spatial discretization consists approximately of 500000 elements both for impeller and diffuser. Inlet and outlet sections of the model are shown as well as the mixing planes. The mean values for the updating of the boundary conditions on the diffuser inlet are obtained on the real interface. Then the velocity vectors are scaled assuming a free vortex distribution and finally imposed at the diffuser inlet. Impeller-diffuser interaction is not limited to the main flow, but also involves the leakage flow that considerably affects both the axial thrust and the efficiency of the whole pump. A complete description of the coupling procedure can be

Fig. 7 Computational grid for steady stage

simulation with mixing plane approach.


276

IMPELLER

Boundary Conditions: Inlet velocity direction (from DIFFUSER EXIT because of repeatability) Balancing flow rate (from SIDE CHAMBER simulation)

DIFFUSERBoundary Condition: Inlet velocity direction (from IMPELLER EXIT because of repeatability) Outlet static pressure (from IMPELLER INLET by mixing plane) Balancing flow rate (from SIDE CHAMBER simulation)

Fig.8 Scheme of complete coupling procedure.

found in Fig. 8. The leakage flow passing through the back shroud chamber is negligible with respect to the main flow rate (less than 0.5% in the range of interest), while the flow passing in the front shroud cavities has been considered (about 1.5%). Therefore, the impeller geometric domain has been modeled including the stage interface with front shroud chamber. Concerning the back shroud it must be underlined that the effect of the leakage flow on the stage performance is neglected while the contribution provided by the pressure field to the axial thrust is very important and is included in the Fbs term of Eq. 1. The numerical study of the impeller shroud chambers has been carried out before the stage analysis, assessing a correlation between the impeller head and the front shroud leakage flow to update the boundary conditions. An iterative cycle on the leakage flow rate has to be performed because the impeller head decreases when the main flow rate increases, while the leakage flow increases with the impeller head. The CFD procedure has also been repeated for different capacities. Stage efficiency and characteristic curve have been obtained and their dimensionless version is shown in Fig. 9. The reference value corresponds to the result obtained at the BEP. The static head has been defined as reported in Eq. 3.

outlet inletp pH

g

(3)

As observed in the stage simulation, the impeller performances strongly depend on the inlet conditions. To simulate the first and the third

impeller, the suction volute and the crossover device have been studied by a stand-alone approach as well. For these components, correlations between the evolving flow rate and the pressure variation have been obtained. Furthermore, velocity distributions of the suction volute and crossover device have been tangentially averaged at the interface plane and then imposed on the downstream impeller inlet. This method is similar to mixing plane one, but neglects both the steady and unsteady interactions. The importance of considering realistic boundary conditions for the impeller inlet is evidenced by Fig. 10, where an example of the pressure field and streamlines in the crossover are reported for the BEP. As can be seen, mechanical support devices provide non-uniform conditions at the crossover exit/impeller inlet section, which can be estimated in yaw angle values up to 15° in off-design conditions.

Non Dimensional Stage Head and Efficiency

0.0

0.2

0.4

0.6

0.8

1.0

1.2

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4flow rate / flow rate @ BEP [-]

he

ad

/ h

ea

d @

BE

P [

-]

0.0

0.2

0.4

0.6

0.8

1.0

1.2

eff

/ e

ff @

BE

P [

-]

Stage Head

Stage Efficiency

Fig.9 Non dimensional stage head and efficiency.


277

Fig. 10 Pressure field and streamlines in crossover.

Non Dimensional Impeller Head

0.70

0.75

0.80

0.85

0.90

0.95

1.00

1.05

1.10

0.70 0.80 0.90 1.00 1.10 1.20 1.30 1.40

flow rate / flow rate @ BEP [-]

hea

d /

hea

d o

f fi

rst

imp

elle

r @

BE

P [

-]

First Impeller

Third Impeller

Stage Impeller

Fig. 11 Dimensionless head for simulated impellers.

Non Dimensional Impeller Efficiency

0.90

0.95

1.00

1.05

1.10

0.70 0.80 0.90 1.00 1.10 1.20 1.30 1.40flow rate / flow rate @ BEP [-]

hea

d /

hea

d o

f fi

rst

imp

elle

r @

BE

P [

-]

First Impeller

Third Impeller

Stage Impeller

Fig. 12 Dimensionless efficiency for simulated

impellers.

Impellers performance and efficiency curves have been also plotted (Figs. 11 and 12). The shown results are non dimensional with respect to the value obtained for the first impeller at the pump BEP. The first impeller shows the highest head for every flow rate due to the fact that the flow enters the first impeller axially, that is at the design conditions. Instead, the third impeller has lower values of head that get further and further from the first impeller ones since the flow rate grows. In fact, the flow entering the third impeller has not been straightened by the diffuser but directly comes from the annular chamber where only structural elements are present (Fig. 10). The other impellers’ head is a linear function of the flow rate and has halfway values between the first and the third ones. The efficiency curves confirm good performances for the first impeller and worst values for the third one. However the third impeller efficiency is higher than expected at low flow rates, probably due to the leakage flow effect. The results confirm a high dependence of impeller performance on its inlet flow field distribution.

5. SIDE CHAMBERS AND BALANCING DRUM ANALYSIS

Impeller shroud chambers have been studied by means of a 2D axi-symmetric approach. To get an exhaustive leakage characterization, the CFD analysis has been performed for different heads across the cavity, calculating the corresponding leakage mass-flow and the contribution to the axial thrust. Cavity flows are therefore solved at different operating points in a preliminary phase and do not require to be calculated at every update of the stage coupling boundary conditions. The results have been employed to assess correlations between the head across the chambers, the


278

leakage flow rate and the axial thrust on the rotating walls. The obtained equations are consistent with the correlations proposed by Stirling (1982), Denny (1954) and Worster and Thorne (1959), currently in use at WEIR-Gabbioneta SRL. The flow inside the cavity moves outward in radial direction at the rotating wall and inwards at the casing wall. The central core rotating speed is lower than the impeller one (Batchelor, 1951; Stewartson, 1953).The rotating velocity inside the chambers depends on peripheral conditions, namely on the impeller rotating speed and on the head across the cavity itself. Due to pump geometry, head across the front shroud cavity corresponds to impeller head, while the leakage flow in the back shroud chamber depends on the pressure rise in the diffuser (Fig. 3). Therefore, leakage mass-flow and its contribution to the axial thrust depend on the pump working conditions. Results from the front shroud chamber analysis have been discussed in a previous section. In Fig. 13a the back shroud impeller chamber grid is shown. Its leakage flow rate and its contribution to the residual axial thrust as a function of the diffuser head are visible in Fig. 15. Both leakage flow rate and axial load are a linear function of the diffuser head. Considering the central balancing drum, the head across the cavity depends on the working conditions in the second group of impellers and in the crossover device. The computational grid is reported in Figs. 13b and 14, where a section of the mesh is presented, while the obtained results are visible in Fig. 16. All the correlations can be expressed as linear function of the head across the balancing drums. The two contributions to the thrust have opposite behaviors. The thrust on the second impeller is nearly constant in the range of interest while high variations on the fifth impeller can be observed. In fact, for a variation of the head of 40% the axial thrust almost decreases at the same percentage. The most interesting conclusion is that looking at the contributions of each chamber, the central balancing drum is the key element for residual axial thrust intensity. Changing this component allows to control the thrust.

6. SIDE CHAMBERS FLOW FIELD ANALYSIS

Head across the cavity results in a centripetal flow with inwards structures at the stationary wall and outwards ones at the rotating walls. Local tangential velocity vθ divided by the impeller

Fig. 13 Back shroud (a) and central drum (b)

computational hybrid grids.

Fig. 14 Detail of computational grid of central

balancing drum along section A-A.

Diffuser Head vs Leakage and Thrust for Back Shroud

0.6

0.7

0.8

0.9

1.0

1.1

1.2

0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1diffuser head / diffuser head @ BEP [-]

leak

age

/ le

akag

e @

BE

P [

-]

0.90

0.95

1.00

1.05

1.10

thru

st /

th

rust

@ B

EP

[-]

Leakage Flow

Thrust

Fig. 15 D Leakage flow and thrust as a function of

impeller head for back shroud.

Head vs Leakage Flow and Thrusts for Central Drum

0.80

0.90

1.00

1.10

1.20

0.7 0.8 0.9 1.0 1.1head (central drum) / head (central drum) @ BEP [-]

leak

age

/ lea

kag

e @

BE

P

-1.2

-1.1

-1.0

-0.9

-0.8

-0.7

-0.6

thru

st /

thru

st @

BE

P

Leakage Flow

Thrust Fifth Impeller

Thrust Second Impeller

Fig. 16 Leakage flow and thrust as a function of head

across the central drum.


279

velocity u (Eq. 4) has been reported in Fig. 17 for the front and back shroud cavities together with the streamlines. Repeating calculations showed that peripheral velocity with respect to the impeller one depends on the mass-flow condition through the cavity.

loc locr

ru u

(4)

The dimensionless rotating velocity distribution in the middle of the back cavity is reported in Fig. 18 at different flow rates. Fluid rotating speed inside the cavity presents a typical value for dimensionless radius ranging from 0.55 to 1. The typical characteristics of the Batchelor flow (Batchelor, 1951) can be observed, namely, the fluid rotates as a solid body between two boundary layers. For the internal regions, the distribution called Stewartson flow (Stewartson, 1953) is encountered, with the classic “vortex” distribution. Increasing the mass-flow through the cavity, the presence of the Stewartson distribution is increased, while the value of constant rotating velocity of the core flow is reduced. This agrees with the results of Debuchy et al. (1998). The conclusion was that the Batchelor type flow can be observed at low mass-flow rates, and far from the periphery because of the influence of the inlet conditions. It was also found that the flow structure near the axis is strongly affected by a

(a)

(b)

Fig. 17 Non-dimensional local rotational velocity distribution inside front/back impeller chamber.

weak stream, which enhances the level of the core-swirl ratio near the axis. Pressure distribution on the back and front shroud of the impeller is qualitatively reported in Fig. 19. The pressure distribution inside is highly influenced by the flow conditions. In Fig. 20, the actual radial distribution of static pressure inside the back shroud cavity is reported. A different value of leakage flow rate divided by the leakage flow rate at BEP corresponds to each curve. All the curves are plotted considering the same reference pressure at the impeller exit. Let us consider now the same distribution divided by a term expressing the local centrifugal force (Eq. 5), as reported for different mass-flow in Fig. 21. It should be underlined that loc expresses the local fluid rotational speed in front and back impeller chambers and r is the local radial coordinate. It can be observed that all curves are coincident except between r/R=0.45 and r/R=0.55, where the outlet of the back shroud chamber is located.

2 2'loc

pr

p

(5)

Fig. 18 Dimensionless rotating velocity inside the

back impeller cavity at different mass-flow (referred to the leakage mass-flow at BEP).

Fig. 19 Leakage contributions to axial thrust.


280

Fig. 20 Radial distribution of static pressure inside

back shroud chamber at different mass-flow rates.

Fig. 21 Radial distribution of dimensionless pressure

inside back shroud chamber at different mass flow rates.

7. AXIAL THRUST: RESULTS AND DISCUSSION

To calculate the axial thrust acting on the whole pump, the contribution of every stage has been separately evaluated according to Eq. 2. Concerning the cavities (the Ffs and Fbs terms), the contribution to the thrust is calculated by adding the real corresponding reference pressure to the pressure integral on the rotating wall:

, ,

,

ax cavity exit impeller

A A

exit impeller

A A

T p ndA p ndA

p ndA pndA

(6)

The area is referred to both the front and back impeller shroud chambers. Similarly the axial thrust is calculated for central and lateral balancing drums. If directed from the second to the first bench a force is considered positive. Each stage contribution to the axial thrust for the design flow rate is reported in Fig. 22, highlighting the opposite effects of impeller shroud chambers. The values are non dimensional with respect to the absolute value of the axial thrust at BEP. It can be noticed that, even for the opposite alignment of the two groups of stages, the global thrust is not null. The impeller back shroud contributions are higher than the corresponding one from the impeller front shroud. Furthermore, the second bench has a higher number of stages and the opposite configuration of the groups is not

sufficient to balance the global thrust. The residual thrust is negative in the chosen reference system and it means that the shaft is in compression. It can be also verified that the greater contribution is due to the central balancing drum where the flow is driven by pump head and by wear rings clearances eventually modified by mechanical wear. Current result is consistent with design assumptions usually considered by the industry. In fact, when two benches are present, the residual axial thrust is considered to be generated by the uneven pump stage, when existing. In our case the fifth stage provides a non-dimensional contribution of 0.98, which means that the residual axial thrust is totally generated by the last stage. For a proper design of the thrust bearings, it is still required to know the maximum thrust in pump operating range. It can be also underlined that the contribution of each component to the residual thrust is one or two orders of magnitude higher than the final value. Then, the use of the proposed procedure for bearing dimensioning requires a careful evaluation of the accuracy of the CFD simulation and an experimental validation of the procedure. Analyses have been carried out between the 80% and the 115% of the BEP flow rate as reported in Fig. 23. Experimental study has been performed by WEIR-Gabbioneta SRL using load cells. Axial thrust values are non-dimensional with respect to the absolute value obtained experimentally for the BEP condition.


281

Fig. 22 Discrete residual thrust evaluation at BEP flow rate.

Fig. 23 Residual axial thrust between 80% and 115% of BEP flow rates.

Both experiments and CFD indicate that in the range considered for the investigated pump, the maximum thrust is obtained at the lowest flow rates corresponding to the highest value of pump head. Second order polynomials fit the data with high accuracy: the value of R2 is 1.000 for the proposed method and 0.979 for the experiments. These results suggest a linear relation between the residual thrust and the pump head in the selected range of flow rates. The numerical method also succeed in the evaluation of both the residual thrust direction and its reduction at the higher flow rates. The discrepancies between the experimental and numerical data are not negligible. To quantify the inaccuracy, the difference between the data with

respect to the experimental value is evaluated as follows:

, .100axax CFD

ax

T TTd

(7)

The trend of this curve is linear, as demonstrated by the R2 value of 0.998 for a linear law. This unexpected result suggests that a key phenomenon is missing in the numerical model and that this phenomenon can be represented by a linear function of the flow rate. It must be pointed out that no experimental uncertainties are available and hence it is not possible to ascertain the actual difference between the numerical data and the experiments. Further studies are necessary to individuate the flow feature to be accounted for.


282

8. CONCLUSIONS

A computational study of a horizontal multistage pump has been carried out, with the residual axial thrust evaluation as the main target. The contribution of each single pump component to the axial load has been estimated by a CFD investigation of its internal flow and pressure field. Then, all the computational results all collected and a methodology to get the residual axial thrust of the whole pump is developed. The suggested method is flexible and the obtained results are consistent with the results given by standard tools in use in industry. Furthermore, each component is studied considering realistic boundary conditions by means of a loosely coupled approach with the upstream/downstream components. Impellers and diffusers have been coupled and analyzed by applying a mixing plane approach. The procedure described by the authors allows to analyze also the interaction between the main flow and the leakage flow, providing accurate evaluation of stage performances. Comparisons between stage curves evidenced the effect of considering different inlet conditions and leakage flows. Leakage flows through wear rings of shroud chambers have been simulated separately with 2D axi-symmetric models. The obtained results in terms of correlations are consistent with the literature information and represent the key feature for the thrust evaluation, especially when considering the central balancing drums. Flow fields inside the cavities show the typical structures of rotating cores. The main limit of this approach is that the tangential non-uniformity generated by the blade-row interaction and the flow recirculation is neglected, thus providing ideal curves only. It must be pointed out that experimental correlations, which are often obtained in controlled environments, also have the same limits. A more detailed analysis of the effect of non-uniform inlet conditions for cavity flows is hence necessary. Coming to the thrust curve, the selected method suggests a proper dimensioning of bearings that can guarantee mechanical reliability. The trend of the curve is well recognized as well as the direction of the thrust over the whole range, which is one of the points of interest for the industrial application of the selected method. It could be concluded that the selected method would be very useful to individuate trends when modifying pump’s crucial components, such as the balancing drum or the cavity flows’ shape and

dimensions (during the design phase) or assessing wear rings decreasing performances. The main limit of the numerical approach is its poor accuracy when applied to pumps with relatively wide gaps for leakage flows. Furthermore, some assumptions are invalid when evaluating off-design conditions. Nevertheless, the force balance does not fail in the listed cases and it is possible to perform fully coupled simulations of the impeller/diffuser/ cavity region of flow at a higher computational cost, if necessary. The leakage flows should be studied by means of fully coupled approach and the effect of the statistical distribution of the tolerances could be considered as well.

NOMENCLATURE

A passage area [m2] BEP best efficiency point (pump design

point) CFD computational fluid dynamics d discrepancy between experimental and

numerical data [%] F force [N] g gravitational acceleration [m/s2] H head [m] n normal vector [-] p pressure [Pa] Q flow rate [mc/s] r radial coordinate [m] R impeller radius [m] T thrust [N] u blade velocity [m/s] v velocity [m/s] y+ distance of the first cell center from the

solid wall [-] Subscripts and Superscripts ax axial bs back shroud bus bushing cav cavity fs front shroud inlet impeller inlet section loc local mom momentum outlet impeller outlet section Greek letters ρ density [kg/mc] tangential local fluid rotational speed in front and


283

back impeller chambers [rad/s] rotating speed [rad/s]

ACKNOWLEDGEMENTS

The authors are grateful to Dr P. Adami and Dr S. Della Gatta from the Energy Engineering Department of the University of Florence, and Ing. G. Marenco, Ing. A. Piva and Ing. L. Bertolazzi from WEIR-Gabbioneta SRL for their valuable suggestions and support during the development of this work.

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