Experimental study and numerical simulation of internal flow dissipation mechanism of an axial-flow pump under different design parameters

Abstract

This research uses CFD (computational fluid dynamics) to simulate an axial-flow pump and analyzes its internal flow dissipation. The results show that under different operating conditions, with increasing tip cascade density, the pump head gradually increases. With increasing tip cascade density, the pump efficiency gradually decreases under small discharge conditions, and the high-efficiency zone of the axial-flow pump gradually narrows when the pump is operated under large discharge conditions. Under the design operating conditions, the highest efficiency of the axial-flow pump reaches 86.15%. The total entropy generation of the impeller, guide vane, and outlet pipe decreases and then increases with increasing discharge. Meanwhile, the total entropy generation of the impeller is 3.37 W/K, which is the largest among different over-water flow components, accounting for 47%. The turbulence entropy generation (EGTD) ratios of the impeller, guide vane and outlet pipe are 42%, 59%, and 65% under the design operating conditions, respectively. Finally, the results of numerical simulation are reliable as verified by a model test. The research results have implications for improving the efficiency of axial-flow pumps.

Full text

Experimental study and numerical
simulation of internal ow
dissipation mechanism of an axial-
ow pump under dierent design
parameters
Lijian Shi1, Yi Han1, Pengfei Xu2, Yi Sun3, Fengquan Qiao3, Yiyu Chen1, Muzi Xue1 &
Yao Chai1
This research uses CFD (computational uid dynamics) to simulate an axial-ow pump and analyzes its
internal ow dissipation. The results show that under dierent operating conditions, with increasing
tip cascade density, the pump head gradually increases. With increasing tip cascade density, the pump
eciency gradually decreases under small discharge conditions, and the high-eciency zone of the
axial-ow pump gradually narrows when the pump is operated under large discharge conditions. Under
the design operating conditions, the highest eciency of the axial-ow pump reaches 86.15%. The
total entropy generation of the impeller, guide vane, and outlet pipe decreases and then increases with
increasing discharge. Meanwhile, the total entropy generation of the impeller is 3.37 W/K, which is
the largest among dierent over-water ow components, accounting for 47%. The turbulence entropy
generation (EGTD) ratios of the impeller, guide vane and outlet pipe are 42%, 59%, and 65% under the
design operating conditions, respectively. Finally, the results of numerical simulation are reliable as
veried by a model test. The research results have implications for improving the eciency of axial-
ow pumps.
Keywords Axial-ow pump, Cascade density, Entropy generation theory, Hydraulic performance,
Numerical simulation, Model test
List of symbols
D Diameter of the impeller (mm)
Qdes Design discharge (L/s)
NPSHre Required net position suction head (m)
ρ Density (kg/m3)
g Acceleration of gravity (m/s2)
H Pump head (m)
H1 Impeller head(m)
η Pump eciency (%)
η1 Impeller eciency (%)
β Airfoil placement angle (°)
Zi Number of impeller blades
Zg Number of guide vanes
dh Hub ratio
µe Eective dynamic viscosity (pa·s)
µ Dynamic viscosity (pa·s)
µt Turbulent viscosity (pa·s)
T ermodynamic temperature (K)
˙m
Mass discharge rate (kg/s)
1College of Hydraulic Science and Engineering, Yangzhou University, Yangzhou 225000, China. 2Huai’an Water
Resources Survey and Design Institute Co., Ltd, Huai’an 223001, China. 3South-to-North Water Transfer Eastern
Jiangsu Water Resources Co., Ltd, Nanjing 210029, China. email: shilijian@yzu.edu.cn
OPEN
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A Area (m2)
V Volume (m3)
l Chord length of impeller airfoil (m)
t Physical time (s)
t’ Plane 2D cascades distance (m)
σ1 Cascade density at shroud
σ11 Cascade density at hub
Zm Multiple of cascade density
PT
2
Total pressure at pump outlet (Pa)
PT
1
Total pressure at pump inlet (Pa)
k Turbulent kinetic energy (m2/s2)
ε Turbulence dissipation rate (m2/s3)
ω Turbulence eddy current frequency ()
∆h Pipeline hydraulic loss (m)
Sgen Total entropy generation (W/K)
Sgen,
¯
D
(EGDD) Direct entropy generation (W/K)
Sgen,D′
(EGTD) Turbulence entropy generation (W/K)
Sgen,w
(EGWS) Wall entropy generation (W/K)
S¯
D
Direct entropy generation rate (W/K)
SD′
Turbulence dissipation rate (W·m− 3·K− 1)
Sw
Wall dissipation rate (W·m− 3·K− 1)
−→τ
Wall shear stress (Pa)
−→v
Center velocity vector of the rst layer grid near the wall (m/s)
Pk Shear generation of turbulence (kg/m·s3)
SM Momentum source (kg/m2·s2)
δ Identity matrix or Kronecker Delta function
σk3 Constant for k-equation in the SST model
σω2 Constant for ω-equation in the SST model
S Strain tensor (s− 1)
˙
Q
Energy transmission rate (W·m− 3)
Φ Dissipation function (W·m− 3)
Axial-ow pumps are widely chosen because of their large discharge, low head, and high specic rotating speed.
ey are used in elds including irrigation and drainage in agriculture, urban water supply and drainage, water-
jet propulsion for ships, and water transfer in large river basins1–3. CFD has the advantages of low calculation
cost, short calculation time, high precision, safety, and good repeatability. It has worked well in many aspects4–7.
As the central part of the axial-ow pump, the design of the impeller directly impacts the capability of the
pump8–10.
e water ow in an axial pump is complex three-dimensional unsteady turbulence during operation. e
phenomenon of secondary backow and unsteady turbulent ow of the axial-ow pump occur due to the
distortion of the shape of the blade. As a result, the internal energy dissipation of the pump will increase and
the pump eciency will be reduced11–13. As a crucial geometric parameter of the axial-ow pump impeller, the
cascade density directly aects the pump eciency and cavitation capability14,15. e entropy generation method
visualizes the location and distribution of dissipation. It is signicant to explore the energy dissipation mechanism
of the ow-passing parts of the axial-ow pumping device under dierent operating conditions16–18. Ma et al.19
examined the inuence of cascade density on the capability of bidirectional axial-ow pumps. When the cascade
density increases, the relative pressure distribution of the blades becomes more uniform, the eciency increases,
and the optimum operating point dris to a small discharge condition. Yu et al.20 researched the changes in the
internal ow characteristics of axial-ow pumps by adapting the angle of attack and cascade density. e results
demonstrated that increasing the cascade density or reducing the inlet angle of attack can improve the pump
eciency and internal ow. Yang et al.21 studied the internal and external characteristics of multiblade axial-
ow pump impellers under dierent cascade density schemes and drew some conclusions. ey concluded that
increasing the cascade density can eectively increase the minimum pressure on the suction side of the impeller
blades then signicantly improve the cavitation performance of the blades. All of the aforementioned scholars
have investigated the eect of cascade density on axial-ow pumps; the dierence lies in the types of pumps
studied and the factors aecting the pumps. Similarly, other scholars have conducted research in a related eld.
Zhao et al.22 optimized the impeller and volute of a multistage double suction centrifugal pump with energy-
saving as the design objective. e results revealed that improving the matching condition between rotor and
stator can improve the pump performance. Osman et al.23 investigated the eects of dierent turbulence models
on the hydraulic performance and internal ow characteristics of a vertical re pump. e results show that SST
k-ω is the best performing turbulence model for such applications. Gu et al.24 utilized a computational uid
dynamics approach to fully analyze a multi-stage centrifugal pump and veried the results with tests. e results
show that the pressure uctuations, eciency and head coecient show periodic variations with the frequency
of the axial oscillation of impeller. Shi et al.25 elaborated the link between the eects of cavitation and the energy
conversion characteristics of helical axial multiphase pump. ey discovered that as the cavitation number σ
became smaller and smaller, the cavitation following a streamline direction on the blade, extended from the
suction side of the blade to the pressure side of the blade. Jiao et al.26 investigated the positive and negative
disconnection processes of a bidirectional full-ow pump through numerical simulations and model tests. And
they discovered that the entropy generation rate of the negative disconnection process was signicantly larger
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than that of the positive disconnection process. Deng et al.27 found that the entropy generation in the main ow
regions was critical in hydraulic loss, and that turbulent dissipation in the main ow regions of the impeller and
volute casing play a dominant role in pump eciency. e near-wall entropy generation of the impeller was
positively correlated with the ow velocity but had little eect on the volute casing. Pei et al.28 employed the
Rortex analysis method and Entropy analysis parameters to study the energy dissipation mechanism of a double-
suction centrifugal pump. e results showed that the local shear plays a decisive role in energy dissipation in the
pump. In related research on the cascade density of axial-ow pumps, the cascade density changes monotonously
and linearly from the shroud to the hub. Simultaneously, the variation of cascade density is essential for the
analysis of the impact of pump performance under a wide range of operating conditions.
ese related studies illustrate the eect of cascade density on pumps, but there is still a lack of combining
experiments to explore the eect of dierent cascade density schemes on the performance of axial-ow pumps.
e purpose of this paper is to investigate the eects of dierent cascade density schemes on the energy and
cavitation characteristics of the axial-ow pump, and to improve the eciency of axial-ow pumps. Based on
the abovementioned research results, this paper intends to optimize the design of the axial-ow pump impeller.
Meanwhile, combined with computational uid simulation and experiments, the functions of cascade density
on the hydraulic performance of pump as well as the internal energy dissipation loss of the optimized scheme
are mainly analyzed.
Numerical calculations
Numerical setup
e axial-ow pump consists of inlet pipe, impeller, guide vane, and outlet pipe. Figure1 shows the four ow
sections model of the pump. Table1 shows the main characteristic data of the pump. is paper mainly studies
the functions of dierent cascade density on the performance of pump, the research results can be a reference for
the optimization of axial-ow pump impellers.
Governing equations and turbulence model
e owing medium inside the axial-ow pump is a continuous incompressible uid, and the water ow is
energized by the rotation of the impeller. Since the temperature of the uid has no change, the energy equation
can be ignored. erefore, the continuity equation nally solved by CFD is shown in the following formula: (1)
e momentum equation is shown in Formula (2) below.
e continuity equation is:
Parameters Valu e
Design discharge Qdes 360L/s
Rotating speed n1450 r/min
Impeller diameter D300mm
Blades number of impeller Zi4
Impeller tip clearance 0.15mm
Hub ratio dh0.433
Blades number of guide vanes Zg7
Inlet pipe inlet diameter 350mm
Blade angle of impeller 0°
NPSHre ≤ 7m
Tab le 1. Main data of the axial-ow pump.
Fig. 1. 3D model of axial-ow pump. (1) Inlet pipe; (2) Impeller; (3) Guide vane; (4) Outlet pipe with 60°ellow.
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∂ρ
∂t
+
∂
∂xj
(ρuj
)=0
(1)
e conservation of momentum is:
∂ρu
i
∂t +
∂
∂x
j
(ρuiuj)
−∂
∂x
j
µ
∂u
i
∂x
j
+
∂u
j
∂x
i
=
−∂p
∂x
i−∂
∂x
j
ρu′
iu′
j

+S
M
(2)
Among:
−
ρu′
iu′
j=µt
∂u
i
∂x
j
+
∂u
j
∂x
i−2
3

ρk +µt
∂u
k
∂x
k
δ
ij
(3)
Where ρ represents uid density, kg/m3; ui and uj represent the sections of the Reynolds time-average velocity,
m/s; xi and xj represent the components of the Cartesian-coordinates, m;
−
p
represents the time average pressure,
Pa; µ represents the dynamic viscosity, Pa· s;
−
ρ
−
u′
i
u′
j
is the Reynolds stress, Pa; and t is the physical time, s;
µt
is the turbulent viscosity, pa·s; SM is the sum of the body forces, kg/m2·s2; and δij is the “Kronecker function”.
Near the wall, the SST k-ω turbulence model retains the original k-ω model, and away from the wall, the SST
k-ω turbulence model applies the k-ε turbulence model. e model corrects the turbulent viscosity formula,
which can better transfer the shear stress at the wall. At the same time, this helps to predict the ow of water near
the wall and the separation of the uid under a reverse pressure gradient.
erefore, in this paper, the SST k-ω turbulence model is chosen to close the governing equation. Finally, the
Reynolds averaged Navier-Stokes (RANS) equation and the SST k-ω turbulence model are used to simulate and
predict the ow eld and performance of the pump.
e k equation and the ω equation are as follows:
∂(ρk)
∂t +
∂(ρu
i
k)
∂x
i
=
∂
∂x
j
µ+
µ
t
σ
k3∂k
∂x
j
+pk−β′ρkω (4)
∂
(
ρω
)
∂t
+
∂
(
ρu
i
ω
)
∂xi
=
∂
∂xj
µ+
µ
t
σω3∂ω
∂xj
+ 2 (1
−
F1)ρ1
ωσω2
∂k
∂xj
∂ω
∂xj
+α3
ω
k
pk
−
β3ρω
2
(5)
where σk3 can be solved using a weighted function by rewriting the corresponding terms in the k-ε and k-ω
models. e function is as follows:
1
σk3
=F1
1
σk1
+ (1
−
F1)
1
σk2
(6)
e eddy viscosity is dened as:
µ
t=ρ
·α
1
max (α1ω, SF2)
(7)
where S is an strain rate invariant, s-1; and Pk is the turbulence generation due to viscous forces, which is modeled
using:
P
k=µt
∂u
i
∂xj
+
∂u
j
∂xi∂u
i
∂xj−2
3
∂u
k
∂xk
3µt
∂u
k
∂xk
+ρk

(8)
For incompressible ow,
(∂u
k/∂ xk)
is small and the second term on the right side of Eq.(8) produces little
eect on the generation.
e blending functions is the key to the success of the method. ey are calculated based on the distance to
the nearest surface and on the ow variables.
F1
= tanh

arg
4
1
(9)
arg
1=min

max
√
k
β′ωy,500ν
ωy2

,4ρσω2
kCDkwy2

(10)
CD
kω=max

2ρ1
ωσω2
∂k
∂xj
∂ω
∂xj
,1.0
×
10−20

(11)
where y is the distance to the nearest wall, m; υ is the kinematic viscosity, m2/s; and k is the turbulent kinetic
energy, m2/s2.
e SST k-ω turbulence model corrects the eddy viscosity coecient by modifying its form as follows:
F2
= tanh

arg
2
2
(12)
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arg
2=max

2
√
k
β′ωy,500υ
ωy2

(13)
e data coecients for the equations are as follows:β’ = 0.09; σk1 = 1.176; σk2 = 1.0; σω3 = 2; σω2 = 1.1682;
α3 = 0.44; β3 = 0.0828; α1 = 5/9.
Entropy generation theory
ere is always a certain amount of mechanical energy lost through dissipation and friction in the axial-ow
pump, which is converted into internal energy and can no longer be used, this process is irreversible, and
leads to an increase in entropy. Entropy (s) is a state variable, when its transport equation in a single-phase
incompressible ow is as follows:
ρDs
Dt
=
−∇
(
q
T
)+
Φ
T
+
Φ
θ
T
2 (14)
where s is the instantaneous quantity, and the instantaneous quantity is divided into two parts, the average and
the uctuation part, by the Reynolds averaged Navier–Stokes (RANS) turbulence method:
s=s+s′
(15)
u=u+u′
(16)
e above Formulas (15) and (16) are substituted into Formula (14) to obtain the following entropy balance
Eq.(17).
ρ
(∂s
∂t
+U
·∇
s)=
−∇
(q
T
)
−
ρ
∇
(U′s′)+

Φ
T
+

Φθ
T
2

(17)
−
Φθ/T 2
do not calculate (In incompressible uids, the uid path of the pump is almost isothermal); e
dissipation entropy yield comprises two parts: the direct dissipation rate (
S−
D
) due to the average ow eld and
the turbulence dissipation rate (
SD′
) due to the pulsation velocity.
Φ
T
=S¯
D+SD′ (18)
According to the Gouy-Stodola theorem, the entropy generation rate SD can be expressed by the following
formula:
S
D=
˙
Q
T
=
Φ
T
(19)
where
˙
Q
is the energy transfer rate, W·m-3; and Φ is the dissipation function, W·m-3.
erefore, the principle of entropy generation is essentially consistent with the Gouy-Stodola theorem.
S
D=2µeff
T

∂u
∂x
2
+

∂u
∂y
2
+

∂u
∂z
2
+µeff
T

∂u
∂y +∂v
∂x
2
+

∂u
∂z +∂w
∂x
2
+

∂v
∂z +∂w
∂y
2
(20)
S
D′=2µeff
T

∂u′
∂x

2
+

∂v′
∂y

2
+

∂w′
∂z

2

+µeff
T

∂u′
∂y +∂v′
∂x

2
+

∂u′
∂z +∂w′
∂x

2
+

∂v′
∂z +∂w′
∂y

2

(21)
µeff=µ+µt
(22)
where µe is the eective dynamic viscosity, pa·s ; µ is the dynamic viscosity, pa·s; and µt is the turbulent viscosity,
pa·s.
−
u
,
−
v
, and
−
w
, respectively, represent the sections of the time-averaged velocity in the x, y, and z directions,
m/s.
u′
,
v′
, and
w′
, respectively, represent the components of the pulsating velocity in the x, y, and z directions,
m/s.
e pulsation velocity component is not available when using the Reynolds mean method. Based on Kock et
al.29 and Mathieu et al.30,
SD′
in Eq.(21) above can be calculated by turbulence model. e
SD′
of the SST k-ω
turbulence model can be given by:
S
D′=β′
ρωk
T
(23)
whereβ’ is an empirical constant in the SST k-ω model, approximately equal to 0.0931; k is the turbulent kinetic
energy, m2/s2; and ω is the turbulence eddy frequency, s-1.
Considering the high velocity gradient at the wall, additional turbulent dissipation losses are unavoidable.
e wall entropy generation is calculated as follows:
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S
W=
τ ·v
T
(24)
where
−→τ
is the wall shear stress, Pa; and
−→v
is the center velocity vector of the rst layer mesh at the wall, m/s.
e entropy generation energy dissipation of each part can be obtained by volume integration of each local
entropy generation rate. e wall entropy generation can be obtained by surface integration of the wall entropy
generation rate. e entropy generation equation for each part is as follows:
S
gen,D =
V
SDdV (25)
S
gen,D′=
V
SD′dV (26)
S
gen,W =
A
SW
dA
(27)
where
Sgen,
−
D
(EGDD) is the direct entropy generation caused by the time-average velocity;
Sgen,D′
(EGTD) is the
turbulent entropy generation caused by the pulsating velocity; and
Sgen,W
(EGWS) is the wall entropy generation
caused by the wall velocity gradient.
In conclusion, the total entropy generation (
Sgen
) of the convective ow eld is equal to the sum of EGDD,
EGTD, and EGWS.
S
gen
=Sgen,D +Sgen,D′+Sgen,W
(28)
Considering that the temperature of the uid in the axial-ow pump remains constant during the ow process,
Formula (29) determines the hydraulic loss calculated according to the entropy generation Formula (28).
h
gen =
T·S
gen
·
mg
(29)
where,
˙m
is the mass discharge rate of the pump, L/s; and T is the temperature, K.
Mesh independence and convergence analysis
e impeller and guide vane are structured meshed in the turbo-grid to meet the quality requirements. e inlet
pipe with the water guide cone and the outlet pipe with the motor sha are structured in ICEM CFD and the
mesh quality is above 0.4, which is good quality. Figure2 below shows the grid drawing of the impeller, guide
vane, inlet pipe, and outlet pipe.
Verication of grid independence is intended to reduce or eliminate the inuence of the number and size of
grids on the calculation results. e impeller is a rotating part, and the other parts are stationary, so it is necessary
to verify the number of the impeller grids to ensure the accuracy of the calculation. Under the design operating
conditions, eight dierent impeller grid number schemes are utilized to evaluate impeller grid independence.
Figure3 demonstrates that as the number of impeller grids increases from 651,152 to 710,444, the change of
pump eciency slows down signicantly and the eciency values growth rate is less than 2%. erefore, the
total number of impeller grids is approximately 651,152, and the grid count for the entire axial-ow pump is
approximately 4.5million.
e grid quality has a signicant impact on the results of numerical simulations. It is important to ensure
grid convergence to strike a balance between computational resources and the numerical calculations accuracy.
e GCI (Grid Convergence Index) criterion of the Richardson extrapolation method is used to verify the
convergence of the mesh. Finally, three groups of ne mesh N1 = 4,511,756, medium mesh N2 = 1,316,776, and
coarse mesh N3 = 573,075 are used to research the mesh convergence of pump. e grid renement coecients
are all greater than 1.3. en, the eciency parameters of the pump are analyzed by discretization error analysis.
Refer to the GCI calculation program of Yang et al.32. Table2 shows the calculation results. e convergence
index GCI21 is 0.154%, and the discretization error is small. erefore, the nal grid number is 4,511,756.
Numerical settings
e discretization of governing equations adopts a nite volume method which is based on nite elements, and
the convective terms are implemented in a high-resolution format. e ow eld solution uses a fully implicit
multi-grid coupled solution technique, coupling the continuity and momentum equations. Table3 below shows
the boundary conditions set for the computational domain of the pump. e maximum number of iteration
steps is set to 2000 in the solver control.
Design of dierent axial-ow pump impeller schemes
To investigate the eect of dierent cascade densities on the axial-ow pump, based on the selected airfoil, the
airfoil placement angle, the maximum airfoil camber ratio, and the maximum airfoil thickness ratio are kept
constant. By changing the tip cascade density, the eect of dierent cascade density on the performance of the
axial-ow pump is studied.
In designing the impeller, the impeller blades are divided into eleven airfoil sections. R is the distance from
each airfoil section to the center of rotation of the impeller. Figure4a demonstrates the cross-sectional schematic
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diagram of the impeller airfoil. According to formulas (30) and (31), when each airfoil Section R is determined,
the cascade distances t’ and r(i) remain unchanged. Figure4b shows a schematic diagram of the two-dimensional
leaf channel.
t
′=
2πR
Z
(30)
r
(i)=
R
(i)
D
(31)
where i = 1, 2, 3…11, 1 represents the shroud, and 11 represents the hub.
Fig. 3. Verication analysis of impeller grid independence.
Fig. 2. Grid model of each section of the axial-ow pump.
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e cascade density of the airfoil section between the impeller hub and the shroud adopts the law of equal
strength, so the cascade density of each airfoil section can be controlled by the two parameters σ1 and Zm,
precisely according to the following formulas: (32), (33), (34), and (35). To explore the inuence of σ1 on the
performance of the axial-ow pump, Zm=1.433 is kept constant. By changing σ1, the eect of dierent cascade
densities on the performance of the pump is studied.
λ
2=dh
·
(Zm
−
1)
·
σ1
1−dh
(32)
Fig. 4. Axial-ow pump airfoil cross-section and 2D cascade schematic diagram.
Location Boundary condition
Inlet of pump Total pressure
outlet of pump Mass discharge
Solid wall No-slip wall
Interfaces on both sides of impeller in stabilization calculation Stage
Other computing domain interfaces None
Method of mesh connection GGI
Downstream velocity limit Stage average velocity
Impeller shroud velocity Counter-rotating wall
Residual type RMS
Convergence accuracy 10−5
Tab le 3. Boundary condition settings.
Parameter ф=Eciency
N1, N2, N34,511,756, 1,316,776, 573,075
r21,r32 1.507, 1.320
ф1, ф2, ф385.116%, 84.945%, 84.463%
p3.647
ϕ21
ext
,
ϕ32
ext
85.22%, 85.21%
e21
a
,
e32
a
0.2%, 0.56%
e21
ext
,
e32
ext
0.12%, 0.31%
GCI21
fine
,
GCI32
fine
0.154%, 0.22%
Tab le 2. Calculation of mesh discretization error.
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λ1=σ1−λ2
(33)
σ
(i)=λ1+
λ
2
r
(i)
(34)
Z
m=
σ11
σ1
(35)
where Zm is the multiple of cascade density, σ1 is the cascade density at the shroud, σ11 is the cascade density at
the hub, σ(i) is the cascade density of each airfoil section of the blade, and dh is the hub ratio.
e values of σ1 are 0.7, 0.75, 0.8, 0.85, 0.9, 0.95, and 1.0, for a total of seven schemes, among which the
σ1 = 1.0, Zm=1.433 scheme is the initial scheme (OS) during the test; Table4 demonstrates the cascade density
for dierent schemes. Table5 displays the sectional data of each airfoil under dierent schemes.
Experiment test and validation
e experiment was conducted to verify the external characteristics of the pump. Figure5 shows the test
installation, including the inlet tank, pressure outlet tank, voltage stabilizing rectier, electromagnetic ow
meter, and control valve. Utilize CNC machining to produce the brass material impeller for the axial-ow pump,
and through welding techniques, use the welded steel material to construct the guide vane. Figure6 depicts a
model of the main components of the axial-ow pump. Figure7 is a comprehensive characteristic curve diagram
of the pump. Aer the experimental system uncertainty calculation, the calculation results meet the accuracy
requirements of the pump model and pumping device model acceptance test protocol (SL 140–2006), indicating
that the experimental results are reliable.
Figure8 compares the model test and numerical simulation results of the pump. e impeller used in the
axial-ow pump test is the impeller data of the initial scheme (OS). By comparing the data of experiment and
initial protocol (OS), it can be found that the eciency curve of the numerical simulation is in good agreement
with the experiment under the large discharge condition. An unavoidable error in the small discharge condition
may be due to the particular disturbance at the inlet of the pump. Under the design operating conditions, the
experiment test eciency is 84.22%, and the head is 6.076m. In comparison, numerical simulation calculates an
eciency of 84.24% with a head of 5.95m. e eciency error is 0.024%, and the head error is 2.07%, both of
which fall within the acceptable error range and satisfy the calculation accuracy requirement. Furthermore, by
comparing the scheme (S4) and the initial scheme (OS), the pump eciency increases, the eciency of scheme
(S4) is 0.99% higher than that of scheme (OS) under the design discharge condition and 3.87% higher than
r(i) β(°) gmax/l δmax/l
1.0 15.81 0.027 0.025
0.937 18.18 0.028 0.031
0.874 20.56 0.030 0.036
0.811 23.02 0.033 0.043
0.748 25.63 0.036 0.050
0.685 28.47 0.040 0.058
0.622 31.64 0.046 0.067
0.559 35.28 0.052 0.077
0.496 39.62 0.058 0.089
0.433 44.97 0.065 0.010
0.367 51.3 0.073 0.012
Tab le 5. Sectional data of each airfoil under dierent schemes.
Scheme
σ(i)
Section
I II III IV V VI VII VIII IX X XI
S1 0.7 0.71 0.73 0.74 0.76 0.78 0.81 0.84 0.88 0.93 1.00
S2 0.75 0.76 0.78 0.79 0.81 0.84 0.86 0.90 0.94 1.00 1.07
S3 0.8 0.81 0.83 0.85 0.87 0.89 0.92 0.96 1.00 1.06 1.15
S4 0.85 0.86 0.88 0.90 0.92 0.95 0.98 1.02 1.07 1.13 1.22
S5 0.9 0.92 0.93 0.95 0.98 1.00 1.04 1.08 1.13 1.20 1.29
S6 0.95 0.97 0.98 1.01 1.03 1.06 1.09 1.14 1.19 1.26 1.36
OS 1.0 1.02 1.04 1.06 1.08 1.12 1.15 1.20 1.25 1.33 1.43
Tab le 4. Cascade density of each airfoil section under dierent schemes.
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that of scheme (OS) under the large discharge condition. And the range of the high-eciency zone expands
by optimizing the density of the tip cascade. Finally, comparing numerical simulation and test results further
demonstrates the dependability of numerical simulation and tests.
Results and discussions
Comparison of hydraulic characteristic under dierent schemes
Figure9a, b show the head-discharge and eciency-discharge characteristic curves of the axial-ow pump of
dierent schemes. is indicates that the variation range of the head between dierent σ1 values is larger for
small discharge conditions (Q ≤ 340L/s) than for large discharge conditions (Q ≥ 380L/s), indicating that the
variation of σ1 aects the axial-ow pump more under small discharge conditions than under large discharge
conditions. Simultaneously, under small discharge conditions, with an increase in σ1, the pump head gradually
increases, and the eciency gradually decreases. is is because the longer the cord at the impeller shroud, the
more the impeller will work and the higher the head will be. However, the corresponding friction area between
the blade and the uid will increase, resulting in a decrease in the pump eciency. In general, for the axial-ow
pump, if Zm remains constant, although increasing σ1 can increase the pump head, it leads to a decrease in the
pump eciency and narrows the high-eciency area of pump operating conditions. ere σ1 should not be too
large and should be kept within reasonable limits.
e density multiplier of the impeller cascade of the pump remains unchanged. As the σ1 decreases, the twist
degree of the axial-ow pump blade decreases, the maximum displacement of the tip increases, and the maximum
structural stress reduces. erefore, reducing the tip cascade density improves the structural performance of the
pump and ensures the ecient, safe, and stable operation of the pump station.
Figure10a, c, show the head-discharge and eciency-discharge characteristic curves, respectively, of the
impeller for dierent schemes. e gure shows that the larger σ1 is under dierent discharge conditions, the
greater the slope of the impeller’s head-discharge characteristic curve is. e change in σ1 has a signicant
eect on the impeller head under small discharge conditions, whereas it has no eect under the large discharge
Fig. 6. Model pump.
Fig. 5. Diagram of test pump. (1) inlet tank, (2) axial-ow pump, (3) pressure outlet tank, (4) bifurcated tank,
(5) regulating gate valve, (6) voltage stabilizer rectier, (7) electromagnetic ow meter, (8) control gate valve,
(9) auxiliary pumping device, (10) dierential pressure gauge.
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conditions. In contrast, under small discharge conditions, σ1 has little inuence on the impeller eciency. Under
large discharge conditions, as σ1 increases, the impeller eciency decreases gradually.
Figure 10b is the relationship curve between the head of the impeller and the head of the initial scheme
minus ΔH1 and σ1 under dierent discharge conditions. is reveals that with the gradual decrease in σ1, the
head drop ΔH1 of the impeller is greater. Similarly, a dierent σ1 has little inuence on the impeller under large
discharge conditions. e magnitude of the inuence is greater under small discharge conditions. Figure10d is
the relationship curve between the impeller eciency under dierent schemes and the eciency of the initial
scheme minus Δη1 and σ1 under dierent discharge conditions. is illustrates that Δη1 uctuates under small
discharge conditions; however, the design operating condition has almost no eect. Under large discharge
conditions, the greater the increase in σ1 is, the greater the eciency drop Δη1 of the impeller.
Figure11 illustrates the relationship curve between the hydraulic loss of the pipeline and discharge. Under
dierent schemes, the hydraulic loss in the pipelines decreases and then increases with increasing discharge, and
there is a minimum value near the design operation conditions. e hydraulic loss of pipelines under dierent
schemes is the same, under large discharge conditions. However, the hydraulic loss also increases gradually with
the σ1 increasing under the small discharge conditions, which is consistent with the result that the larger σ1 is for
axial-ow pump under small discharge conditions, the lower eciency is.
Comparison of cavitation performance under dierent schemes
is paper selects S1, S4, the initial scheme (OS), and the suction-side pressure of the blade to predict the
NPSHre (NPSH required) of the pump under the design operating conditions. e formula for the calculation
is as follows:
NPSH
re =
P
0
ρg −P
min
ρg
+0.
24
(36)
where P0 is the inlet total pressure, 1atm, and Pmin is the suction side minimum pressure value of the blade.
According to many tests on axial-ow pumps, the cavitation of impellers oen occurs near the shroud on the
suction surface of the blade and close to 10–20% of the inlet width. erefore, in CFD, the suction side span of
Fig. 7. Comprehensive characteristic curve of axial-ow pump.
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the blade is taken as = 0.9, and the minimum pressure of the blade width at the airfoil section; is approximately
10–20% away from the blade inlet.
Figure12 depicts the NPSHre of the impeller for dierent schemes. It reveals that along the streamlined
direction, the suction side pressure of the impeller decreases rstly and then increases when the span is between
0.1 and 0.75. e maximum NPSHre for the pump in S1 is 7.36m, the NPSHre for the pump in S4 is 6.55m, and
the minimum NPSHre for the pump in the OS is 6.1m. e smaller the necessary NPSHre of the impeller is, the
greater the cavitation margin and the lower the probability that the pump will cavitate. Both the OS and S4 can
satisfy the requirements.
Fig. 9. Performance curve of the axial-ow pump under dierent schemes.
Fig. 8. Comparison of experimental and numerical simulations.
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Figure13 is the suction side pressure cloud diagram of the pump blade. is demonstrates that as σ1 increases,
the pressure gradient at the blade suction side becomes more uniform, the range of the low-pressure region
decreases, and the range of the inlet low-pressure region at the impeller suction side close to the blade suction
side accounts for approximately 20% of the blade width, which is agreement with the results reected in Fig.8.
Entropy generation analysis of the axial-ow pump
In this section, three typical working conditions are selected in S4: small discharge conditions (Q = 280 L/s),
design operation conditions (Q = 360L/s), and large discharge conditions (Q = 420L/s) for an axial-ow pump.
Entropy generation in axial-ow pumps can be divided into local and overall entropy generation.
Analysis of local entropy generation
As the inlet part of pump, the inlet pipe is used to provide good inow conditions for the inlet of the axial-ow
pump. Figure14 represents the ratio of entropy generation under dierent working conditions of the inlet pipe.
Under each operating condition of the inlet pipe, EGWS accounts for approximately 70% and EGTD accounts
for 30%, indicating that the inlet pipe is primarily aected by wall entropy generation. However, the percentage
of EGDD for each operating condition is approximately equal to 0%, which is negligible and therefore not
reected in the gure. e wall shear force and shear velocity signicantly impact the inlet pipe.
e pump impeller is a rotating part that is entirely dierent from the rest of the stationary parts. It is
meaningful to study the proportion of internal entropy generation and the distribution position of high entropy
generation. Figure15 shows the ratio of entropy generation of the impeller under dierent operation conditions.
As discharge increases, the EGTD in the impeller rstly decreases and then increases, but the EGWS increases
rstly and then decreases. e EGWS reaches a maximum of 57%, and the EGTD reaches a minimum of 42%
under design operating conditions. EGTD and EGWS are nearly the same. is illustrates that although the
EGTD due to the strong impeller rotation accounts for a particular proportion, the local loss caused by the wall
shear force is also a more nonnegligible part of the impeller loss.
Figure16 represents the EGWS at the impeller hub. e entropy generation on the surface of the hub is
mainly distributed at the upper end under small discharge conditions (Q = 280 L/s). In contrast, under the
Fig. 10. Eect of dierent σ1 values on the performance of the impeller.
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Fig. 12. NPSHre of the axial-ow pump under dierent schemes.
Fig. 11. Change curves of pipeline hydraulic loss under dierent schemes.
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design and large discharge conditions, most entropy generation occurs in the center of the hub. is could be
due to the local high entropy generation caused by the secondary return ow at the outlet of impeller hub under
small discharge conditions. Figure17 depicts the EGWS at the impeller shroud, and the EGWS at the impeller
shroud is the smallest under design operation condition. In general, when the pump is operated under the
Fig. 14. Entropy generation ratio diagram of the inlet pipe.
Fig. 13. Pressure contour diagram of the blade suction side under dierent schemes.
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design conditions, the Sgen produced by the impeller is the smallest, the functional ability of the impeller is the
strongest, and the pump eciency is the highest.
e function of the guide vane is to eliminate the velocity cycling of the water ow at the impeller outlet and
to convert the velocity energy into pressure energy. Figure18 shows the ratio of entropy generation of the guide
Fig. 16. Distribution of entropy generation of the impeller hub.
Fig. 15. Entropy generation ratio diagram of the impeller.
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vane under dierent working conditions. e EGTD of the guide vane occupies the central part under each
discharge condition, which suggests that the guide vane reduces the turbulence and vortex inside the pump. As
discharge increases, the EGTD of guide vane decreases rstly and then increases, and the EGWS increases rst
and then decreases. Under design operating conditions, EGWS reaches a maximum of 40%, and EGTD reaches a
minimum of 59%. Figures19 and 20 demonstrate that under small discharge conditions, the entropy generation
of the guide vane is mainly distributed on the blade pressure side. e suction side is where the majority of
guide vane blade entropy generation occurs under the design and large discharge operating conditions. Under
dierent operating conditions, the high entropy generation of guide vane is largely distributed at the front end
of guide vane inlet, which may be because of the inuence of static and dynamic interference at the junction of
the impeller and the guide vane.
Fig. 18. Entropy generation ratio diagram of the guide vane.
Fig. 17. Distribution of entropy generation of the impeller shroud.
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e outlet pipe is the outlet part of the pump, and its function is to further recover the velocity circulation at
the guide vane outlet. Figure21 shows the ratio of entropy generation under dierent working conditions of the
outlet pipe. e EGTD of outlet pipe under dierent operating conditions also occupies a signicant proportion,
which indirectly shows the ability of the outlet pipe to maximize the recovery of water kinetic energy at the outlet
of the guide vane. With increasing discharge, the EGTD of the outlet pipe rst decreases and then increases,
and the EGWS rstly increases and then decreases. Under design operating conditions, the EGWS reaches a
maximum of 35%, and EGTD reaches a minimum of 65%. e percentage of EGDD is approximately equal to
0% at all operating conditions. Figure22 reveals that the high entropy generation area is largely concentrated
on the inlet of outlet pipe. is is because the ow is kept circulating at a certain velocity aer passing through
the guide vane. Aer owing into outlet pipe, the strong rotation of the water ow driven by the rotation of the
motor sha concentrates on the inlet section of the pipe.
Analysis of total entropy generation
Figure23a, b depict the total entropy generation (Sgen) and the proportion of entropy generation of each over-
water ow component for dierent operating conditions of S4. e Sgen of the inlet pipe increases as the discharge
increases, but the proportion is negligible and can be ignored. e Sgen of the impeller, guide vane, and outlet
pipe decreases rst with increasing discharge. It increases aer being small, and there is a minimum value under
the design operating conditions. Meanwhile, under the design operating condition, the entropy generation of
impeller is the largest among all the over-water ow components, which is 3.37W/K, accounting for 47% of the
total entropy generation, which is nearly half. is indicates that under the design conditions, impeller is the
central area of energy dissipation of the pump, which also suggests that the workability of impeller is enhanced.
e axial-ow pump has the highest eciency.
Hydraulic loss comparison between entropy generation and pressure dierence
e hydraulic loss of each over-water ow section is calculated utilizing the pressure dierence method, as
shown in Formula (30). R is dened as the ratio of the hydraulic loss calculated using the entropy generation
method to the hydraulic loss calculated using the dierential pressure method, and Formula (42) is used to
calculate R:
h
p=
p
in
−p
out
ρg
(37)
where, Pin is the total pressure at the pump inlet, Pa, and Pout is the total pressure at the pump outlet, Pa.
R
=
h
pro
hp
(38)
is paper selects S4 to explore the relationship between the R of each over-water ow section and each discharge
condition, and Fig.24 displays the results. Under each discharge condition, the R of the water inlet pipe is 1.3,
Fig. 20. Entropy generation distribution of the guide vane shroud.
Fig. 19. Entropy generation distribution of the guide vane hub and blades.
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Fig. 22. Distribution of entropy generation inside the outlet pipe.
Fig. 21. Entropy generation ratio diagram of the outlet pipe.
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the R of impeller and guide vane domain increases rstly and then decreases with increasing discharge, and the
R of the impeller domain uctuates between 0.595 and 0.686. e guide vane domain ratio uctuates between
0.5 and 0.64, and the R of outlet pipe uctuates between 0.83 and 1.1.
e R of impeller to guide vane domain and outlet pipe uctuates and is small because the high-velocity
rotating water ow inside the impeller domain retains high rotational strength aer owing into guide vane.
Likewise, the rotating ow of water aer guide vane enters outlet pipe causes a specic disturbance. In general,
the hydraulic loss calculated by entropy generation method is highly consistent with which calculated by
dierential pressure method, and it is reliable within a specic error range.
Conclusions
is study compares the results of numerical simulation and experimental validation and therefore veries
the reliability of numerical simulation. e eects of dierent tip cascade density on the axial-ow pump
performance and internal ow dissipation mechanism are studied and the following conclusions are drawn:
(1) e highest eciency of the axial-ow pump reached 86.15% for all optimization schemes. e pump head
increases gradually with an increase in σ1 under the full discharge condition, while the eciency gradually
decreases. e variation range of large discharge rate is less obvious than that of small discharge rate. e
high-eciency zone of the pump widens as σ1 decreases. ere is a minimum value of the hydraulic loss of
pipelines near the design operation condition.
(2) e pressure gradient on the blade suction side turns increasingly uniform as σ1 increases under the design
operating conditions. e size of low-pressure area gradually decreases, NPSHre of the axial-ow pump
gradually decreases, and the cavitation performance of the pump gradually improves as σ1 increases. e
minimum NPSHre of the impeller for the scheme 4 is 6.55m.
Fig. 24. Ratio R of each ow-passing component.
Fig. 23. Total entropy generation and proportion of each ow-passing component.
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(3) e EGDD is minimal and negligible in each over-water ow section. In the axial-ow pump impeller,
EGTD and EGWS are close under each discharge condition. However, in the guide vane and outlet pipe,
EGTD is greater than EGWS. e Sgen of the impeller, guide vane, and outlet pipe rstly decreases and then
increases with increasing discharge. e impeller has the highest entropy generation among the over-water
ow components under the design operating conditions, accounting for 47% of the Sgen. Impeller is the
central area of energy dissipation. e hydraulic loss results of entropy generation are consistent with those
of pressure dierence calculation.
Data availability
Data is provided within the manuscript or supplementary information les.
Received: 26 April 2024; Accepted: 6 November 2024
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Scientic Reports | (2024) 14:27619 21
| https://doi.org/10.1038/s41598-024-79101-y
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Author contributions
Conceptualization, L.S. and Y.H.; methodology, P.X. and Y.S.; soware, Y.H. and F. Q.; validation, L.S.; formal
analysis, P.X. and Y.S.; data curation, Y.C.and Y.H.; writing-original dra preparation, Y.C., Y.C. and Y.H.; writ-
ing- review and editing, L.S., Y.C. and Y.H.; visualization, Y.C. and M.X.; supervision, L.S. and P.X.; project
administration, L.S.; funding acquisition, L.S.All authors have reviewed the manuscript and agreed to the pub-
lished version of the manuscript.
Funding
is paper was supported by the National Natural Science Foundation of China (No.52209116, No.52276041);
the Scientic and Technological Research and Development Program of South-to-North Water Transfer in
Jiangsu Province (No.JSNSBD202201); Yangzhou Science and Technology Plan Project City-School Coopera-
tion Project (No.YZ2022178); Open Research Subject of Key Laboratory of Fluid Machinery and Engineering
(Xihua University), Sichuan Province (No. LTDL-2022004); Yangzhou University’s “Youth and Blue Project”
funding program.
Declarations
Competing interests
e authors declare no competing interests.
Additional information
Correspondence and requests for materials should be addressed to L.S.
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