Large pumping stations with low head play a very important role in some areas in the national economy such as city flood control,agricultural irrigation and drainage,interbasin water transfer,and water environmental improvement. A large pump system is made up of 4 parts,i.e.,inlet conduit,pump impeller (including impeller chamber,hereinafter the same), guide vane,and outlet conduit (see Fig. 1). Among them,the impeller is the core of the pump system,which is used to transfer the energy to the water flow so as to produce the head. The inlet conduit is the transition part between the impeller and the forebay,and the outlet conduit is the transition part between the guide vane and the outlet sump. The guide vane is the adjustment part between the impeller and the outlet conduit.

Fig. 1 Constitution diagram of large pump system with low head

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The flow patterns in the inlet and outlet conduits have a decisive effect on the operations with safety,stability,and high efficiency for the pump set of a large pumping station with low head ^[1] . To satisfy the requirement of engineering applications,it is necessary to study the optimum hydraulic design for the inlet and outlet conduits.

Numerical simulation is the main method to study the hydraulic performance and conduct the optimum hydraulic design for three-dimensional (3D) turbulence flow in the inlet and outlet conduits. The requirement of the flow pattern at the impeller chamber inlet is considered in the optimum hydraulic design for the inlet conduit,and the effect of the flow pattern at the guide vane outlet on the hydraulic performance is considered in the optimum hydraulic design for the outlet conduit. In order to well solve the problems concerned with the boundary condition that are closely related to the numerical simulation for the inlet and outlet conduits of the pump system,their flow patterns are deeply analyzed.

The chord angle of the impeller blade and the velocity parallelogram at the inlet of an axial- flow pump are,respectively,shown in Fig. 2 and Fig. 3,where β₁₀ stands for the design chord angle, v₁₀ ,w₁₀ ,and u stand for the design values of the absolute velocity,the relative velocity,and the implicated velocity,respectively. If the velocity at the inlet section of the impeller chamber is not uniform,or the absolute speed at a certain point on the section is less than (or more than) v₁₀ ,as shown in Fig. 3(a),by v₁₁ (or v₁₂ ),the flow angle β₁₁ (or β₁₂ ) will be less than (or more than) β₁₀ .

Fig. 2 Chord angle of impeller blade

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Fig. 3 Velocity parallelograms at inlet of axial flow pump

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This will cause the flow to hit against the blade front (or back) and bring about head loss in the impeller. Meanwhile,flow separation and vortices will be caused at the back (or front) of the blade so as to lead to the partial negative pressure and cavitation. Similarly,if the velocity at a certain point on the inlet section of the impeller chamber is not vertical to the section,as shown in Fig. 3(b),the velocity parallelogram at the inlet and the flow angle will be changed, which will lead to harmful flow patterns near the impeller inlet.

The effect of an axial-flow pump with low head is more obvious than that of a centrifugal pump with high head because large axial-flow pump is often restricted by the layout condition of the inlet conduit,and the contradiction between the inlet flow pattern and the requirement for the hydraulic design of the pump impeller is rather prominent. Therefore,the inlet flow pattern of the pump with low head needs more attention.

The function of the inlet conduit is to lead the flow from the forebay to the impeller chamber with changing the direction orderly and shrinking evenly and to supply the flow pattern for the hydraulic design to meet the impeller requirement of the impeller chamber inlet so as to ensure the pump operates with stability and high efficiency ^[2] . Harmful inlet flow patterns can not only reduce the pump energy and cavitation performance,but also sometimes cause suction vortex band or underwater vortex band and threaten the stability of the pump operation. The worse the inlet flow pattern is,the more obvious the undesirable effect on the operation condition and hydraulic performance of the pump is. In a large pump system,the outlet section of the inlet conduit is connected directly with the inlet section of the impeller chamber,and its hydraulic design has a great effect on the hydraulic performance of the pump system.

The requirements for the optimum hydraulic design of the inlet conduit may be summarized as follows:

(i) The velocity at the outlet section of the conduit is uniform,and the direction of the flow is vertical to the outlet section.

(ii) The flow changes in the direction orderly,and shrinks even without any harmful flow patterns.

(iii) The main sizes of the conduit are reasonable.

The outlet section of the inlet conduit is divided into a certain number of cells,according to the requirements of the hydraulic design of the axial-flow pump impeller for the inlet flow field. Two objective functions are introduced ^[3] . One is the uniformity function of the velocity distribution V u . It is expressed by

Click Browse formula

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where u_ti stands for the transverse velocity of each unit in the outlet section.

The above objective functions provide the quantitative indexes to evaluate the quality of the inlet flow field for the optimum hydraulic design of the inlet conduit in a large pump system. To satisfy the requirement of the hydraulic design for the axial-flow pump impeller,we choose V_u = 100% and θ = 90°. For practical engineering design,the goal of the optimum hydraulic design for the inlet conduit is to make the values of the objective function of the outlet flow field in the inlet conduit close to the ideal values as much as possible.

In a large pump system,the outlet section of the inlet conduit is connected closely with the inlet section of the impeller chamber. It is used to be thought that the flow at the outlet of the inlet conduit is much near the impeller so that it follows the impeller rotation to be turned, causing the so-called “pre-swirl” flow. This phenomenon has a great effect on the boundary condition in the numerical simulation for the flow pattern in the inlet conduit. In this paper, the model test method is used to research this problem.

Vertical axial-flow pump system is widely applied in pumping station projects. The flow in this type of pump system has to be turned 90° to enter the impeller chamber. In order to research the flow patterns at the axial-flow pump impeller chamber inlet under the same operation condition as the actual one,a typical vertical axial-flow pump system consisted of an elbow inlet conduit and a siphon outlet conduit is chosen to be the research object,and the ZBM791-100 axial-flow pump model is used as the model pump in the pump system.

The schematic diagram and photo of the model test device for the vertical pump system are shown in Fig. 4 and Fig. 5,respectively. To visually observe and record the flow pattern at the impeller chamber inlet,the inlet and outlet conduits are made of transparent plexiglass ^[4] ,and red threads are pasted on the inner wall of the inlet conduit. The location of the red thread section is shown in Fig. 6. A net,which is used to paste red threads,is formed by nylon threads at the inlet section of the impeller chamber. The red thread locations at the section is shown in Fig. 7,where solid lines stand for walls of the inlet conduit and guide cap,dotted lines stand for the nylon threads,and black spots stand for the locations at which the red threads are pasted. The test discharge is controlled by adjusting the opening degree of the gate valve.

Fig. 4 Sketch of model test device with vertical axial-flow pump system

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Fig. 5 Photo of model test device with vertical axial-flow pump system

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Fig. 6 Locations of red thread section

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Fig. 7 Red thread locations at section

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(i) Normal operation condition

The test discharge is made with Q = 42L/s by adjusting the opening degree of the gate valve in the normal operating condition for the pump. From Fig. 8,we can see that the red threads on both the outer and middle circles point at the axial direction of the pump. As the red threads are near the guide cap,the red threads on the inner circle point at the direction parallel to the generatrix of the guide cap. It is indicated by the test results that in the normal operation condition,the flow in front of the impeller chamber of an axial-flow pump is steady, and there is no “pre-swirl”,which means that the flow does not rotate as the impeller does.

Fig. 8 Flow pattern at impeller chamber inlet (Q = 42L/s)

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(ii) Small discharge operation condition

The test discharge is made with Q = 20L/s by reducing the opening degree of the gate valve. It can be observed that the red threads on the outer circle are obviously deflected and rocked. The deflecting direction is the same as the rotation direction of the impeller,the direction of the red threads on the inner circle is still parallel to the generatrix of the guide cap,and the red threads on the middle circle are swung back and forth uncertainly. The flow pattern in front of the impeller chamber inlet under the small discharge operation condition is shown in Fig. 9. The test results indicate that in this condition,the pre-swirl in the same direction as the rotation direction of the impeller appears in the flow in the outer circle of the impeller chamber inlet section,the flow in the inner circle flows to the impeller chamber along the outer wall of the guide cap,while the flow in the middle circle is in an unsteady motion state between the above two states.

Fig. 9 Flow pattern at impeller chamber inlet (Q = 20L/s)

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The discharge of the model pump under the small discharge operation condition is about 50% discharge in the optimal operation condition,and is identical with that of “the saddle- shaped area” of the model pump performance curve. It can be speculated that in the small discharge condition,the “secondary backflow” appears in the model pump,and it extends from the impeller chamber to the front of its inlet,showing as a “pre-swirl” in front of the impeller chamber inlet. There are already a few studies on the flow in pumps of the small discharge operation condition ^[5] . Cheng et al. ^[5] showed that in small discharge operation conditions,a backflow area could be seen obviously at the head of the suction side of the blade outer edge in axial-flow pumps and at the tail of the suction side near the blade root,and there was a larger backflow area in the inlet which was extended to the inner of the blade groove. Liang et al. ^[6] found that in small discharge operation conditions,the relative flow angle at the impeller inlet was small,and the flow rushed to the pressure side,forming larger flow separation at the suction side. This phenomenon is the most obvious at the impeller edge. Lin et al. ^[7] pointed out that when the discharge of axial-flow pump decreased,the angle between the flow relative velocity and the circumferential velocity decreased. In this case,if the blade angle is not changed,the flow attack angle will be increased; if the attack angle is increased to a certain extent,the flow separation is caused at the airfoil surface. These show that the “pre-swirl” at the impeller chamber inlet of the axial-flow pump in a small discharge operation condition represents the internal characteristics of the “secondary backflow” in the pump,and the saddle-shaped area in the performance curve of pump represents the external characteristics.

It is necessary for the optimum hydraulic design of inlet conduits to solve the Reynolds- averaged Navier-Stokes equations for a number of flow fields of inlet conduit schemes with different 3D geometric shapes ^{[8, 9]} and to carry out 3D turbulent flow numerical simulations. Therefore,the problem for the boundary conditions of the flow field simulation must be well solved.

(i) Inlet boundary of computational flow field

The inlet section of the conduit is submerged into a certain depth below the forebay water level,and the flow gets through the inlet section at a certain slope angle. The velocity distri- bution at the section is not known until the flow field computation is finished so that the inlet section of the computational flow field extends from the inlet section of the conduit inlet to an enough distance in the forebay. This inlet section is perpendicular to the flow direction and the inflow velocity here can be regarded to be uniform. The discharge of the inlet flow field is a known condition (i.e.,the pump design flow). Therefore,the velocity-inlet boundary condition can be applied to the inlet flow field at the inlet section.

(ii) Outlet boundary of computational flow field

The optimum hydraulic design of the conduit is carried out under the design condition,and there is no “pre-swirl” at the inlet of the impeller chamber of an axial-flow pump in a normal operation. Therefore,the outflow boundary condition can be applied to the inlet flow field at the outlet section. In order to apply this boundary condition more exactly,the outlet must be extended as far as twice pipe diameters along the flow direction.

(iii) Solid wall boundary

The boundary conditions of the solid wall in the computational inlet flow field,including the forebay bottom,the inlet conduit wall,and the guide cap wall in front of the impeller chamber inlet,are treated based on the solid wall law ^[10] .

(iv) Water surface

The inlet flow field includes a part of the forebay,on the surface of which the velocity and the turbulent kinetic energy of the surface are treated as those at symmetry,and the shear stress caused by the wind and heat exchange with atmosphere is neglected ^[10] .

The region and boundary conditions of the flow field numerical computation for an elbow inlet conduit in the vertical pump system are shown in Fig. 10.

Fig. 10 Region and boundary conditions of flow field numerical computation for elbow inlet conduit

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Based on the 3D flow field numerical simulation for the elbow inlet conduit,the velocity distribution uniformity and the average angle of the flow at the conduit outlet section are 98.3% and 88.6°,respectively,under the design discharge. The flow fields on the upper,middle,lower profiles and near the conduit surface are shown in Fig. 11. In the straight part of the inlet conduit,the flow pattern is smooth,and the velocity is gradually increased as the area of the conduit section is gradually decreased; after the flow comes into the curved part of the conduit, the velocity is turned and accelerated rapidly; under the effect of the centrifugal force,the velocity close to the inside conduit is greater than that close to the outside conduit,and this phenomenon is consistent with the principle of uniform velocity moment. The flow separation in the curved part cannot be found because the flow turns 90° with rapidly lateral contraction. In the conic part of the conduit,a larger velocity has appeared close to the outside conduit at the beginning due to the strong effect of the inertial force. Near the conduit outlet,the flow tends to have a uniform distribution and to be perpendicular to the outlet section by the adjusting in the conic part ^[3] .

Fig. 11 Flow fields for elbow inlet conduit

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In a large pump system with low head,the inlet section of the outlet conduit is connected with the outlet section of the guide vane. Therefore,the boundary condition of the inlet section of the outlet conduit depends on the flow pattern at the outlet of the guide vane. The effect of the outlet conduit is to lead the flow from the outlet of the guide vane to the outlet sump with changing the direction orderly and diffusing gently so that the kinetic energy of the flow can be recovered as much as possible in this process. The velocity is decreased,and the kinetic energy is recovered in a way of enlarging the section area of the outlet conduit gradually. The average velocity at the inlet of the conduit is about 4.0 m/s,and that at the outlet of the conduit is generally controlled below 1.5 m/s. For the pump system with low head,it is usually controlled below 1.0 m/s.

The type of the outlet conduit in a large pumping station is variable. In each conduit, however,the flow diffusion is a common characteristic. The head loss produced in the flow diffusion process is closely affected by the flow pattern in the conduit. Under the diffusion condition,vortex is easily formed due to the flow separation,and the effect of this harmful flow pattern is particularly notable for the conduit head loss. Therefore,the vortex in the conduit must be avoided as far as possible.

The requirements of the conduit for optimum hydraulic design may be summarized as follows:

(i) The kinetic energy is recovered,and the conduit head loss (outlet head loss included) is decreased as low as possible;

(ii) The flow changes in the direction orderly,and diffuses gently without any harmful flow pattern;

(iii) The main sizes of the conduit are reasonable.

The total head loss of the conduit is the quantitative index to evaluate its hydraulic perfor- mance. The effect of recovering the kinetic energy of the flow for the conduit can be expressed as follows:

(i) The kinetic energy loss at the conduit outlet is small.

(ii) The head loss in the conduit is small.

Assume ξ to be the head loss coefficient for the conduit. Then,the total head loss of the conduit ∆h oc may be expressed as

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The objective function of the optimum hydraulic design for the conduit may be written as

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The energy is given to the flow with the rotating impeller having a high speed in an axial- flow pump. Therefore,the flow from the impeller chamber is provided with a larger tangential velocity. A guide vane is installed behind the impeller chamber in order to adjust the tangential velocity into the axial velocity and to recover the flow rotational kinetic energy ^{[11, 12]} . It was revealed that the pump efficiency with guide vane was higher in the range between 2% and 3% than that without guide vane ^[13] . Theoretically,only for the guide vane with an infinite number of blades,can the tangential velocity at the outlet of guide vane be eliminated completely. Actually,in order to decrease the head loss of the guide vane itself,the blade number is limited (5-7 pieces in general). The outflow from the guide vane still has a certain tangential velocity, whose rotational direction is the same as that of the impeller. Consequently,the flow flows into the conduit in a spiral way at last. The velocity triangles at the outlet of the impeller, inlet,and outlet of the guide vane are shown in Fig. 12,where v 4 ,v 4u ,and v 4m are the absolute velocity,the tangential velocity,and the axial velocity of the flow at the outlet of guide vane, respectively.

Fig. 12 Velocity triangles at outlet of impeller and inlet and outlet of guide vane

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The outlet conduit in the pump system is used to accept the outflow from the guide vane, at the outlet of which the flow patterns determine the boundary conditions of the outlet con- duit. Without any question,the velocity circulation will inevitably affect the flow pattern and hydraulic performance of the conduit.

The effect of the flow velocity circulation at the outlet of the guide vane of an axial-flow pump system on the hydraulic performance of the outlet conduit is obvious. The quantitative research about the effect is very few at present. This is mainly because that the circulation is hard to be measured. In order to solve this problem,the swirl meter is designed and used.

The swirl meter is consisted of 4 pieces of even and straight blades in parallel with the pipe axis,and the blades may revolve freely round the axis (Fig. 13 and Fig. 14). The blades will synchronously rotate with the flow due to the action of the tangential velocity. Its angular velocity is namely the average angular velocity of the flow. A glistening piece is stuck to the blade side,and a photoelectric sensor is installed at a relevant position on the pipe wall. The sensor is connected with its secondary instrument and emits a beam. When the blade with the glistening piece rotates via the sensor,the beam will be reflected by the glistening piece. This optical signal may be received by the sensor,and thus the light pulse is produced. The instrument connected with the sensor may number the light pulses within per unit time,by which the rotational speed of the swirl meter n will be worked out.

Fig. 13 Sketch of structure for swirl meter

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Fig. 14 Photo of swirl meter

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(i) Calculation for the average angular velocity

Based on the rotational speed n,the average angular velocity ω can be calculated by

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(ii) Calculation for the average tangential velocity

Assume r to be the radius of an arbitrary point at the outlet section of the guide vane. The tangential velocity v t of the flow at the point can be written as

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If R₁ and R₂ denote,respectively,the radius of the outlet section and the radius of the hub at the outlet section,then the tangential kinetic energy E_t of the water with ∆Z thickness is

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If v_t and r denote the average tangential velocity and its corresponding average radius of the outlet section in the sense of energy,respectively,then

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(iii) Calculation for the average circulation

The calculation for the flow velocity circulation at the outlet section of the guide vane needs to be integrated in the section,which is a toroidal multiply connected domain. L₁ and L₂ are the outer and inner ring boundary lines of the domain,respectively,and their integral route directions are opposite ^[14] . Considering the consistent requirement of the rotate direction of the flow at the outlet section with that of the pump impeller,it is defined in this paper that the circulation direction coincident with the impeller is plus so as to make the calculation result of the circulation be plus. The circulation Γ at the outlet section is defined by

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According to (12),the following formula can be obtained:

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The flow velocity circulation at the guide vane outlet section of the axial flow pump model ZBM791-100 has been tested by the help of the swirl meter. The rotational speed of the measured swirl meter n is 270 r/min under the design condition. The radius of the outlet section R₁ and the radius of the hub at the outlet section R₂ are 0.075m and 0.025m,respectively. According to (5) and (13),the flow average circulation at the guide vane outlet section is calculated to be 0.944 m²/s.

According to the above test results,the effect of the guide vane on the eliminating circulation at the impeller outlet section can be analyzed. Based on (14),which is the theoretical head for a vane pump,the circulation Γ₂ at the guide vane inlet section (i.e.,the impeller outlet section) can be calculated by

Click Browse formula

Taking the axial flow pump model ZBM791-100 for example,the pump head is 4.8 m under the design condition when the rotational speed of the impeller is 1450 r/min. The flow velocity circulation at the guide vane inlet section is 1.95 m²/s. Thus,the effect of the guide vane on the eliminating circulation may be described as a radio of 0.51 between the circulation at the outlet and the inlet sections of the guide vane.

For a outlet conduit with a certain shape,the head loss is not the least for the flow with no circulation under the condition of the same discharge ^[15] . Keeping the same design discharge, the model tests for the relationship between the flow circulation at the outlet section and the head loss of the siphon and low hump outlet conduit are completed. The test results indicate that the relationships between the circulations at the inlet section and the head losses of the outlet conduits are open cubic curves with the upper direction,which means that there are optimal circulations for the losses ^[16] . The optimal circulations for the siphon and the low hump conduit are 0.972 m²/s and 1.308 m²/s,respectively.

At large bending or sharp diffusion in the outlet conduit,the harmful flow,such as separa- tion and vortex flow,will appear in the outlet conduit because of the inertia effect if the flow flows into the outlet conduit without circulation. If the flow has a certain circulation,it will be affected by a centrifugal force and forced to move with stronger rotation along the circumfer- ential direction of the conduit section ^[11] . The separation trend of the flow is restricted by the circumferential motion,and the vortex area is prevented from being generated or enlarged so that the flow pattern is improved in the conduit. The larger the circulation is,the better the effect to improve the flow pattern is. On the other hand,the loss of the tangential kinetic energy may also be caused by the velocity circulation. The larger the circulation is,the larger the loss of the tangential kinetic energy is. Assume the flow circulation to be increased gradually from zero. Then,the head loss will be decreased gradually because of the improvement of the flow pattern. As the circulation is increased continuously,the loss of the tangential kinetic energy of the flow will be increased,but the decrease trend of the conduit head loss is maintained. When the circulation is increased to such a value that the increased value of the tangential kinetic energy loss is larger than the decreased value of the conduit head loss because of the improvement of the flow pattern,and the conduit head loss will begin to be increased. The optimal circulation of the flow occurs at the conversion point between the above two conditions.

It is necessary for optimum hydraulic design of outlet conduits to solve the Reynolds- averaged Navier-Stokes equations for the flow fields of a number of outlet conduit schemes with different 3D geometric shapes and to carry out 3D turbulent flow numerical simulations ^{[8, 9]} . Therefore,the problem related to the boundary conditions has to be well solved.

(i) Inlet boundary for computational flow field

In order to apply the inlet boundary condition accurately,the inlet section of the compu- tational flow field of the outlet conduit is extended with a distance of two times of the pipe diameter upstream from its inlet section,where the distribution of the flow velocity may be considered to be uniform. The computational discharge is the design value for a single pump. Therefore,the velocity inlet boundary condition at the inlet section can be applied to the com- putational flow field. At the same time,because the flow from the guide vane has residual circulation,a certain angular speed needs to be be set at the section. Its direction is the same as that of the impeller rotation,and its value may be measured by the swirl meter which is introduced above. Because the pump model adopted by the pumping station is usually con- firmed after the pump equipment tender has been finished,when the optimum hydraulic design for the outlet conduit is being carried out at the preliminary design stage,the existing model test datum can be referenced and the circulation of a similar pump model can be applied. Ac- cording to the measurement and comparison results of the circulations from the guide vane of the excellence axial-flow pump models used in common at home,their velocity circulations at the design condition have not many differences.

(ii) Outlet boundary for computational flow field

The outlet section of the computational flow field for the outlet conduit is set in the outlet sump with an enough distance from the outlet section of the conduit. The outlet section is perpendicular to the flow direction,where the outflow boundary condition may be applied.

(iii) Solid wall boundary

The boundary conditions of the solid wall in the computational flow field,including the outlet sump bottom,the outlet conduit wall,and the guide cap wall behind the guide vane outlet,are treated based on the solid wall law ^[10] .

(iv) Water surface

The outlet flow field includes a part of the outlet sump,on the surface of which the velocity and the turbulent kinetic energy of the surface are treated as those at symmetry,with neglecting the shear stress caused by the wind and heat exchange with atmosphere ^[10] .

The region and boundary conditions of the flow field numerical calculation for the siphon outlet conduit of a vertical pump system are shown in Fig. 15.

Fig. 15 Region and boundaries of flow field numerical calculation for siphon outlet conduit

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Based on the numerical calculation results of the 3D flow field,the head loss of the design discharge for the siphon outlet conduit is 0.349 m. The flow field of the conduit is shown in Fig. 16,from which it may be seen that the flow at the inlet section of the conduit has a certain circulation so that the flow can flow into the conduit helically; the height of each section in the upward segment of the siphon outlet conduit decreases gradually,while the width of each section increases gradually so that the velocity in the upward segment decreases gradually; because the turning angle and diffusion of the segment are gentler,the separation flow does not occur in the segment; affected by both the flow inertia and the circulation,the flow field of the left and right sides in the downward segment is asymmetry; in the downward segment,the direction of the main flow leans to the upside and the left side (along the flow direction),while a local vortex occurs in the downside and the right side (along the flow direction) ^[17] .

Fig. 16 Flow field in siphon outlet conduit

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(i) Under the normal operation conditions,there is essentially no pre-swirl flow at the inlet of the impeller chamber of an axial-flow pump system,based on which the boundary condition at the inlet conduit outlet may be defined.

(ii) The flow at the outlet section of the guide vane of an axial-flow pump system has a certain residual circulation,which has a great effect on the hydraulic performance of the outlet conduit,and there is an optimum circulation for making the hydraulic loss of the outlet conduit to be the least. Therefore,it is strongly suggested to design the guide vane according to the optimum circulation.

(iii) The residual circulation at the outlet section needs to be considered for the inlet bound- ary condition of the outlet conduit,and the value of the circulation may be measured by the model test designed specially.