Numerical setup
The axial-flow pump consists of inlet pipe, impeller, information vane, and outlet pipe. Determine 1 exhibits the 4 move sections mannequin of the pump. Desk 1 exhibits the principle attribute information of the pump. This paper primarily research the features of various cascade density on the efficiency of pump, the analysis outcomes generally is a reference for the optimization of axial-flow pump impellers.
3D mannequin of axial-flow pump. (1) Inlet pipe; (2) Impeller; (3) Information vane; (4) Outlet pipe with 60°ellow.
Governing equations and turbulence mannequin
The flowing medium contained in the axial-flow pump is a steady incompressible fluid, and the water move is energized by the rotation of the impeller. Because the temperature of the fluid has no change, the vitality equation might be ignored. Subsequently, the continuity equation lastly solved by CFD is proven within the following formulation: (1) The momentum equation is proven in System (2) under.
The continuity equation is:
$$frac{{partial rho }}{{partial t}}+frac{partial }{{partial {x_{textual content{j}}}}}left( {rho {u_{textual content{j}}}} proper)=0$$
(1)
The conservation of momentum is:
$$frac{{partial rho {u_{textual content{i}}}}}{{partial t}}+frac{partial }{{partial {x_{textual content{j}}}}}left( {rho {u_{textual content{i}}}{u_{textual content{j}}}} proper) – frac{partial }{{partial {x_{textual content{j}}}}}left[ {mu left( {frac{{partial {u_{text{i}}}}}{{partial {x_{text{j}}}}}+frac{{partial {u_{text{j}}}}}{{partial {x_{text{i}}}}}} right)} right]= – frac{{partial p}}{{partial {x_{textual content{i}}}}} – frac{partial }{{partial {x_{textual content{j}}}}}left( {rho overline {{u_{{textual content{i}}}^{{{prime }}}u_{{textual content{j}}}^{{{prime }}}}} } proper)+{S_{textual content{M}}}$$
(2)
Amongst:
$$- rho overline {{u_{{textual content{i}}}^{{{prime }}}u_{{textual content{j}}}^{{{prime }}}}} ={mu _{textual content{t}}}left( {frac{{partial {u_{textual content{i}}}}}{{partial {x_{textual content{j}}}}}+frac{{partial {u_{textual content{j}}}}}{{partial {x_{textual content{i}}}}}} proper) – frac{2}{3}left( {rho ok+{mu _{textual content{t}}}frac{{partial {u_{textual content{ok}}}}}{{partial {x_{textual content{ok}}}}}} proper){delta _{{textual content{ij}}}}$$
(3)
The place ρ represents fluid density, kg/m3; ui and uj characterize the sections of the Reynolds time-average velocity, m/s; xi and xj characterize the elements of the Cartesian-coordinates, m; (:stackrel{-}{p}:)represents the time common stress, Pa; µ represents the dynamic viscosity, Pa· s; (:-rho:stackrel{-}{{u}_{i}^{{prime:}}{u}_{j}^{{prime:}}}:textual content{i}textual content{s}:textual content{t}textual content{h}textual content{e}:)Reynolds stress, Pa; and t is the bodily time, s; (:{mu:}_{textual content{t}}) is the turbulent viscosity, pa·s; SM is the sum of the physique forces, kg/m2·s2; and δij is the “Kronecker perform”.
Close to the wall, the SST ok–ω turbulence mannequin retains the unique ok–ω mannequin, and away from the wall, the SST ok–ω turbulence mannequin applies the ok–ε turbulence mannequin. The mannequin corrects the turbulent viscosity formulation, which may higher switch the shear stress on the wall. On the similar time, this helps to foretell the move of water close to the wall and the separation of the fluid beneath a reverse stress gradient.
Subsequently, on this paper, the SST k-ω turbulence mannequin is chosen to shut the governing equation. Lastly, the Reynolds averaged Navier-Stokes (RANS) equation and the SST k-ω turbulence mannequin are used to simulate and predict the move discipline and efficiency of the pump.
The ok equation and the ω equation are as follows:
$$frac{{partial left( {rho ok} proper)}}{{partial t}}+frac{{partial left( {rho {u_{textual content{i}}}ok} proper)}}{{partial {x_{textual content{i}}}}}=frac{partial }{{partial {x_{textual content{j}}}}}left[ {left( {mu +frac{{{mu _{text{t}}}}}{{{sigma _{k3}}}}} right)frac{{partial k}}{{partial {x_{text{j}}}}}} right]+{p_{textual content{ok}}} – {beta ^{prime}}rho komega$$
(4)
$$frac{{partial left( {rho omega } proper)}}{{partial t}}+frac{{partial left( {rho {u_{textual content{i}}}omega } proper)}}{{partial {x_{textual content{i}}}}}=frac{partial }{{partial {x_{textual content{j}}}}}left[ {left( {mu +frac{{{mu _{text{t}}}}}{{{sigma _{omega 3}}}}} right)frac{{partial omega }}{{partial {x_{text{j}}}}}} right]+2left( {1 – {F_1}} proper)rho frac{1}{{omega {sigma _{omega 2}}}}frac{{partial ok}}{{partial {x_{textual content{j}}}}}frac{{partial omega }}{{partial {x_{textual content{j}}}}}+{alpha _3}frac{omega }{ok}{p_k} – {beta _3}rho {omega ^2}$$
(5)
the place σk3 might be solved utilizing a weighted perform by rewriting the corresponding phrases within the k-ε and k-ω fashions. The perform is as follows:
$$frac{1}{{{sigma _{{textual content{k3}}}}}}={F_1}frac{1}{{{sigma _{{textual content{k1}}}}}}+left( {1 – {F_1}} proper)frac{1}{{{sigma _{{textual content{k2}}}}}}$$
(6)
The eddy viscosity is outlined as:
$${mu _{textual content{t}}}=rho cdot frac{{{alpha _1}}}{{hbox{max} left( {{alpha _1}omega ,S{F_2}} proper)}}$$
(7)
the place S is an pressure price invariant, s-1; and Pok is the turbulence era as a consequence of viscous forces, which is modeled utilizing:
$${P_{textual content{ok}}}={mu _{textual content{t}}}left( {frac{{partial {u_{textual content{i}}}}}{{partial {x_{textual content{j}}}}}+frac{{partial {u_{textual content{j}}}}}{{partial {x_{textual content{i}}}}}} proper)frac{{partial {u_{textual content{i}}}}}{{partial {x_{textual content{j}}}}} – frac{2}{3}frac{{partial {u_{textual content{ok}}}}}{{partial {x_{textual content{ok}}}}}left( {3{mu _{textual content{t}}}frac{{partial {u_{textual content{ok}}}}}{{partial {x_{textual content{ok}}}}}+rho ok} proper)$$
(8)
For incompressible move, (:left(partial:{u}_{textual content{ok}}/partial:{x}_{textual content{ok}}proper)) is small and the second time period on the proper facet of Eq. (8) produces little impact on the era.
The mixing features is the important thing to the success of the strategy. They’re calculated primarily based on the space to the closest floor and on the move variables.
$${F_1}=tanh left( {arg _{1}^{4}} proper)$$
(9)
$${arg _1}=hbox{min} left[ {hbox{max} left( {frac{{sqrt k }}{{{beta ^{prime}}omega y}},frac{{500nu }}{{omega {y^2}}}} right),frac{{4rho {sigma _{varvec{upomega}2}}}}{text{k}}{{C{D_{{text{kw}}}}{y^2}}}} right]$$
(10)
$$C{D_{{textual content{ok}varvec{upomega}}}}=hbox{max} left( {2rho frac{1}{{omega {sigma _{{{varvec{upomega}2}}}}}}frac{{partial ok}}{{partial {x_{textual content{j}}}}}frac{{partial omega }}{{partial {x_{textual content{j}}}}},1.0 occasions {{10}^{ – 20}}} proper)$$
(11)
the place y is the space to the closest wall, m; υ is the kinematic viscosity, m2/s; and ok is the turbulent kinetic vitality, m2/s2.
The SST k-ω turbulence mannequin corrects the eddy viscosity coefficient by modifying its kind as follows:
$${F_2}=tanh left( {arg _{2}^{2}} proper)$$
(12)
$${arg _2}=hbox{max} left( {frac{{2sqrt ok }}{{{beta ^{prime}}omega y}},frac{{500upsilon }}{{omega {y^2}}}} proper)$$
(13)
The information coefficients 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 era concept
There’s all the time a specific amount of mechanical vitality misplaced by means of dissipation and friction within the axial-flow pump, which is transformed into inner vitality and might now not be used, this course of is irreversible, and results in a rise in entropy. Entropy (s) is a state variable, when its transport equation in a single-phase incompressible move is as follows:
$$rho frac{{Ds}}{{Dt}}= – nabla (frac{{vec {q}}}{T})+frac{{{varvec{Phi}}}}{T}+frac{{{{{varvec{Phi}}}_{{varvec{uptheta}}}}}}{{{T^2}}}$$
(14)
the place s is the instantaneous amount, and the instantaneous amount is split into two elements, the common and the fluctuation half, by the Reynolds averaged Navier–Stokes (RANS) turbulence methodology:
$$s=overline {s} +s^{prime}$$
(15)
$$u=overline {u} +{u^{prime}}$$
(16)
The above Formulation (15) and (16) are substituted into System (14) to acquire the next entropy steadiness Eq. (17).
$$rho (frac{{partial overline {s} }}{{partial t}}+overline {U} cdot nabla overline {s} )= – overline {{nabla (frac{{vec {q}}}{T})}} -rho nabla (overline {{U^{prime}s^{prime}}} )+overline {{left( {frac{{{varvec{Phi}}}}{T}} proper)}} +overline {{left( {frac{{{Phi _theta }}}{{{T^2}}}} proper)}}$$
(17)
(:stackrel{-}{{varPhi:}_{{uptheta:}}/{T}^{2}}) don’t calculate (In incompressible fluids, the fluid path of the pump is sort of isothermal); The dissipation entropy yield includes two elements: the direct dissipation price ((:{S}_{stackrel{-}{textual content{D}}})) because of the common move discipline and the turbulence dissipation price ((:{S}_{{textual content{D}}^{{prime:}}})) because of the pulsation velocity.
$$frac{{overline {Phi } }}{T}={S_{bar {{textual content{D}}}}}+{S_{{textual content{D}}^{prime}}}$$
(18)
Based on the Gouy-Stodola theorem, the entropy era price SD might be expressed by the next formulation:
$${S_{textual content{D}}}=frac{{dot {Q}}}{T}=frac{{overline {Phi } }}{T}$$
(19)
the place (:dot{Q}) is the vitality switch price, W·m-3; and Φ is the dissipation perform, W·m-3.
Subsequently, the precept of entropy era is basically in keeping with the Gouy-Stodola theorem.
$${S_{overline {{textual content{D}}} }}=frac{{2{mu _{{textual content{eff}}}}}}{T}left[ {{{left( {frac{{partial overline {u} }}{{partial x}}} right)}^2}+{{left( {frac{{partial overline {u} }}{{partial y}}} right)}^2}+{{left( {frac{{partial overline {u} }}{{partial z}}} right)}^2}} right]+frac{{{mu _{{textual content{eff}}}}}}{T}left[ {{{left( {frac{{partial overline {u} }}{{partial y}}+frac{{partial overline {v} }}{{partial x}}} right)}^2}+{{left( {frac{{partial overline {u} }}{{partial z}}+frac{{partial overline {w} }}{{partial x}}} right)}^2}+{{left( {frac{{partial overline {v} }}{{partial z}}+frac{{partial overline {w} }}{{partial y}}} right)}^2}} right]$$
(20)
$${S_{{textual content{D}}^{prime}}}=frac{{2{mu _{{textual content{eff}}}}}}{T}left[ {{{left( {frac{{partial u^{prime}}}{{partial x}}} right)}^2}+{{left( {frac{{partial v^{prime}}}{{partial y}}} right)}^2}+{{left( {frac{{partial w^{prime}}}{{partial z}}} right)}^2}} right]+frac{{{mu _{e{textual content{ff}}}}}}{T}left[ {{{left( {frac{{partial u^{prime}}}{{partial y}}+frac{{partial v^{prime}}}{{partial x}}} right)}^2}+{{left( {frac{{partial u^{prime}}}{{partial z}}+frac{{partial w^{prime}}}{{partial x}}} right)}^2}+{{left( {frac{{partial v^{prime}}}{{partial z}}+frac{{partial w^{prime}}}{{partial y}}} right)}^2}} right]$$
(21)
$${mu _{e{textual content{ff}}}}=mu +{mu _{textual content{t}}}$$
(22)
the place µeff is the efficient dynamic viscosity, pa·s ; µ is the dynamic viscosity, pa·s; and µt is the turbulent viscosity, pa·s.(::stackrel{-}{u}), (:stackrel{-}{v}), and (:stackrel{-}{w}), respectively, characterize the sections of the time-averaged velocity within the x, y, and z instructions, m/s.(:{u}^{{prime:}}), (:{v}^{{prime:}}), and (:{w}^{{prime:}}), respectively, characterize the elements of the pulsating velocity within the x, y, and z instructions, m/s.
The pulsation velocity element just isn’t out there when utilizing the Reynolds imply methodology. Based mostly on Kock et al.29 and Mathieu et al.30, (:{S}_{{textual content{D}}^{{prime:}}}) in Eq. (21) above might be calculated by turbulence mannequin. The (:{S}_{{textual content{D}}^{{prime:}}}) of the SST k-ω turbulence mannequin might be given by:
$${S_{{textual content{D}}^{prime}}}={beta ^{prime}}frac{{rho omega ok}}{T}$$
(23)
the placeβ’ is an empirical fixed within the SST k-ω mannequin, roughly equal to 0.0931; ok is the turbulent kinetic vitality, m2/s2; and ω is the turbulence eddy frequency, s-1.
Contemplating the excessive velocity gradient on the wall, further turbulent dissipation losses are unavoidable. The wall entropy era is calculated as follows:
$${S_{textual content{W}}}=frac{{vec {tau } cdot vec {v}}}{T}$$
(24)
the place(:overrightarrow{:tau:}) is the wall shear stress, Pa; and (:overrightarrow{v}) is the middle velocity vector of the primary layer mesh on the wall, m/s.
The entropy era vitality dissipation of every half might be obtained by quantity integration of every native entropy era price. The wall entropy era might be obtained by floor integration of the wall entropy era price. The entropy era equation for every half is as follows:
$${S_{{textual content{gen,}}overline {{textual content{D}}} }}=int_{V} {{S_{overline {{textual content{D}}} }}} dV$$
(25)
$${S_{{textual content{gen,D}}^{prime}}}=int_{V} {{S_{{textual content{D}}^{prime}}}} dV$$
(26)
$${S_{{textual content{gen,W}}}}=int_{A} {{S_{textual content{W}}}} dA$$
(27)
the place(:{:S}_{textual content{g}textual content{e}textual content{n},stackrel{-}{textual content{D}}}:)(EGDD) is the direct entropy era brought on by the time-average velocity;(:{:S}_{textual content{gen},{textual content{D}}^{{prime:}}})(EGTD) is the turbulent entropy era brought on by the pulsating velocity; and (:{S}_{textual content{g}textual content{e}n,textual content{W}})(EGWS) is the wall entropy era brought on by the wall velocity gradient.
In conclusion, the whole entropy era ((:{S}_{textual content{g}textual content{e}textual content{n}})) of the convective move discipline is the same as the sum of EGDD, EGTD, and EGWS.
$${S_{{textual content{gen}}}}={S_{{textual content{gen,}}overline {{textual content{D}}} }}+{S_{{textual content{gen,D}}^{prime}}}+{S_{{textual content{gen,W}}}}$$
(28)
Contemplating that the temperature of the fluid within the axial-flow pump stays fixed in the course of the move course of, System (29) determines the hydraulic loss calculated based on the entropy era System (28).
$${h_{{textual content{gen}}}}=frac{{T cdot {S_{{textual content{gen}}}}}}{{mathop mlimits^{ cdot } g}}$$
(29)
the place, (:dot{m}) is the mass discharge price of the pump, L/s; and T is the temperature, Okay.
Mesh independence and convergence evaluation
The impeller and information vane are structured meshed within the turbo-grid to fulfill the standard necessities. The inlet pipe with the water information cone and the outlet pipe with the motor shaft are structured in ICEM CFD and the mesh high quality is above 0.4, which is sweet high quality. Determine 2 under exhibits the grid drawing of the impeller, information vane, inlet pipe, and outlet pipe.
Grid mannequin of every part of the axial-flow pump.
Verification of grid independence is meant to scale back or eradicate the affect of the quantity and measurement of grids on the calculation outcomes. The impeller is a rotating half, and the opposite elements are stationary, so it’s essential to confirm the variety of the impeller grids to make sure the accuracy of the calculation. Underneath the design working situations, eight completely different impeller grid quantity schemes are utilized to guage impeller grid independence. Determine 3 demonstrates that because the variety of impeller grids will increase from 651,152 to 710,444, the change of pump effectivity slows down considerably and the effectivity values progress price is lower than 2%. Subsequently, the whole variety of impeller grids is roughly 651,152, and the grid depend for all the axial-flow pump is roughly 4.5 million.
Verification evaluation of impeller grid independence.
The grid high quality has a big influence on the outcomes of numerical simulations. It is very important guarantee grid convergence to strike a steadiness between computational assets and the numerical calculations accuracy. The GCI (Grid Convergence Index) criterion of the Richardson extrapolation methodology is used to confirm the convergence of the mesh. Lastly, three teams of high-quality mesh N1 = 4,511,756, medium mesh N2 = 1,316,776, and coarse mesh N3 = 573,075 are used to analysis the mesh convergence of pump. The grid refinement coefficients are all larger than 1.3. Then, the effectivity parameters of the pump are analyzed by discretization error evaluation. Discuss with the GCI calculation program of Yang et al.32. Desk 2 exhibits the calculation outcomes. The convergence index GCI21 is 0.154%, and the discretization error is small. Subsequently, the ultimate grid quantity is 4,511,756.
Numerical settings
The discretization of governing equations adopts a finite quantity methodology which relies on finite components, and the convective phrases are carried out in a high-resolution format. The move discipline resolution makes use of a totally implicit multi-grid coupled resolution approach, coupling the continuity and momentum equations. Desk 3 under exhibits the boundary situations set for the computational area of the pump. The utmost variety of iteration steps is ready to 2000 within the solver management.
Design of various axial-flow pump impeller schemes
To analyze the impact of various cascade densities on the axial-flow pump, primarily based on the chosen airfoil, the airfoil placement angle, the utmost airfoil camber ratio, and the utmost airfoil thickness ratio are saved fixed. By altering the tip cascade density, the impact of various cascade density on the efficiency of the axial-flow pump is studied.
In designing the impeller, the impeller blades are divided into eleven airfoil sections. R is the space from every airfoil part to the middle of rotation of the impeller. Determine 4a demonstrates the cross-sectional schematic diagram of the impeller airfoil. Based on formulation (30) and (31), when every airfoil Part R is set, the cascade distances t’ and r(i) stay unchanged. Determine 4b exhibits a schematic diagram of the two-dimensional leaf channel.
$${t^{{prime }}}=frac{{2pi R}}{Z}$$
(30)
$${r_{left( {textual content{i}} proper)}}=frac{{{R_{left( {textual content{i}} proper)}}}}{D}$$
(31)
the place i = 1, 2, 3…11, 1 represents the shroud, and 11 represents the hub.
Axial-flow pump airfoil cross-section and 2D cascade schematic diagram.
The cascade density of the airfoil part between the impeller hub and the shroud adopts the legislation of equal energy, so the cascade density of every airfoil part might be managed by the 2 parameters σ1 and Zm, exactly based on the next formulation: (32), (33), (34), and (35). To discover the affect of σ1 on the efficiency of the axial-flow pump, Zm=1.433 is saved fixed. By altering σ1, the impact of various cascade densities on the efficiency of the pump is studied.
$${lambda _{textual content{2}}}=frac{{{d_{textual content{h}}} cdot left( {{Z_{textual content{m}}} – 1} proper) cdot {sigma _{textual content{1}}}}}{{1 – {d_{textual content{h}}}}}$$
(32)
$${lambda _{textual content{1}}}={sigma _{textual content{1}}} – lambda {}_{{textual content{2}}}$$
(33)
$${sigma _{left( {textual content{i}} proper)}}={lambda _{textual content{1}}}+frac{{{lambda _{textual content{2}}}}}{{{r_{left( {textual content{i}} proper)}}}}$$
(34)
$${Z_{textual content{m}}}=frac{{{sigma _{{textual content{11}}}}}}{{{sigma _{textual content{1}}}}}$$
(35)
the place Zm is the a number of of cascade density, σ1 is the cascade density on the shroud, σ11 is the cascade density on the hub, σ(i) is the cascade density of every airfoil part of the blade, and dh is the hub ratio.
The values of σ1 are 0.7, 0.75, 0.8, 0.85, 0.9, 0.95, and 1.0, for a complete of seven schemes, amongst which the σ1 = 1.0, Zm=1.433 scheme is the preliminary scheme (OS) in the course of the check; Desk 4 demonstrates the cascade density for various schemes. Desk 5 shows the sectional information of every airfoil beneath completely different schemes.
Numerical setup
The axial-flow pump consists of inlet pipe, impeller, information vane, and outlet pipe. Determine 1 exhibits the 4 move sections mannequin of the pump. Desk 1 exhibits the principle attribute information of the pump. This paper primarily research the features of various cascade density on the efficiency of pump, the analysis outcomes generally is a reference for the optimization of axial-flow pump impellers.
3D mannequin of axial-flow pump. (1) Inlet pipe; (2) Impeller; (3) Information vane; (4) Outlet pipe with 60°ellow.
Governing equations and turbulence mannequin
The flowing medium contained in the axial-flow pump is a steady incompressible fluid, and the water move is energized by the rotation of the impeller. Because the temperature of the fluid has no change, the vitality equation might be ignored. Subsequently, the continuity equation lastly solved by CFD is proven within the following formulation: (1) The momentum equation is proven in System (2) under.
The continuity equation is:
$$frac{{partial rho }}{{partial t}}+frac{partial }{{partial {x_{textual content{j}}}}}left( {rho {u_{textual content{j}}}} proper)=0$$
(1)
The conservation of momentum is:
$$frac{{partial rho {u_{textual content{i}}}}}{{partial t}}+frac{partial }{{partial {x_{textual content{j}}}}}left( {rho {u_{textual content{i}}}{u_{textual content{j}}}} proper) – frac{partial }{{partial {x_{textual content{j}}}}}left[ {mu left( {frac{{partial {u_{text{i}}}}}{{partial {x_{text{j}}}}}+frac{{partial {u_{text{j}}}}}{{partial {x_{text{i}}}}}} right)} right]= – frac{{partial p}}{{partial {x_{textual content{i}}}}} – frac{partial }{{partial {x_{textual content{j}}}}}left( {rho overline {{u_{{textual content{i}}}^{{{prime }}}u_{{textual content{j}}}^{{{prime }}}}} } proper)+{S_{textual content{M}}}$$
(2)
Amongst:
$$- rho overline {{u_{{textual content{i}}}^{{{prime }}}u_{{textual content{j}}}^{{{prime }}}}} ={mu _{textual content{t}}}left( {frac{{partial {u_{textual content{i}}}}}{{partial {x_{textual content{j}}}}}+frac{{partial {u_{textual content{j}}}}}{{partial {x_{textual content{i}}}}}} proper) – frac{2}{3}left( {rho ok+{mu _{textual content{t}}}frac{{partial {u_{textual content{ok}}}}}{{partial {x_{textual content{ok}}}}}} proper){delta _{{textual content{ij}}}}$$
(3)
The place ρ represents fluid density, kg/m3; ui and uj characterize the sections of the Reynolds time-average velocity, m/s; xi and xj characterize the elements of the Cartesian-coordinates, m; (:stackrel{-}{p}:)represents the time common stress, Pa; µ represents the dynamic viscosity, Pa· s; (:-rho:stackrel{-}{{u}_{i}^{{prime:}}{u}_{j}^{{prime:}}}:textual content{i}textual content{s}:textual content{t}textual content{h}textual content{e}:)Reynolds stress, Pa; and t is the bodily time, s; (:{mu:}_{textual content{t}}) is the turbulent viscosity, pa·s; SM is the sum of the physique forces, kg/m2·s2; and δij is the “Kronecker perform”.
Close to the wall, the SST ok–ω turbulence mannequin retains the unique ok–ω mannequin, and away from the wall, the SST ok–ω turbulence mannequin applies the ok–ε turbulence mannequin. The mannequin corrects the turbulent viscosity formulation, which may higher switch the shear stress on the wall. On the similar time, this helps to foretell the move of water close to the wall and the separation of the fluid beneath a reverse stress gradient.
Subsequently, on this paper, the SST k-ω turbulence mannequin is chosen to shut the governing equation. Lastly, the Reynolds averaged Navier-Stokes (RANS) equation and the SST k-ω turbulence mannequin are used to simulate and predict the move discipline and efficiency of the pump.
The ok equation and the ω equation are as follows:
$$frac{{partial left( {rho ok} proper)}}{{partial t}}+frac{{partial left( {rho {u_{textual content{i}}}ok} proper)}}{{partial {x_{textual content{i}}}}}=frac{partial }{{partial {x_{textual content{j}}}}}left[ {left( {mu +frac{{{mu _{text{t}}}}}{{{sigma _{k3}}}}} right)frac{{partial k}}{{partial {x_{text{j}}}}}} right]+{p_{textual content{ok}}} – {beta ^{prime}}rho komega$$
(4)
$$frac{{partial left( {rho omega } proper)}}{{partial t}}+frac{{partial left( {rho {u_{textual content{i}}}omega } proper)}}{{partial {x_{textual content{i}}}}}=frac{partial }{{partial {x_{textual content{j}}}}}left[ {left( {mu +frac{{{mu _{text{t}}}}}{{{sigma _{omega 3}}}}} right)frac{{partial omega }}{{partial {x_{text{j}}}}}} right]+2left( {1 – {F_1}} proper)rho frac{1}{{omega {sigma _{omega 2}}}}frac{{partial ok}}{{partial {x_{textual content{j}}}}}frac{{partial omega }}{{partial {x_{textual content{j}}}}}+{alpha _3}frac{omega }{ok}{p_k} – {beta _3}rho {omega ^2}$$
(5)
the place σk3 might be solved utilizing a weighted perform by rewriting the corresponding phrases within the k-ε and k-ω fashions. The perform is as follows:
$$frac{1}{{{sigma _{{textual content{k3}}}}}}={F_1}frac{1}{{{sigma _{{textual content{k1}}}}}}+left( {1 – {F_1}} proper)frac{1}{{{sigma _{{textual content{k2}}}}}}$$
(6)
The eddy viscosity is outlined as:
$${mu _{textual content{t}}}=rho cdot frac{{{alpha _1}}}{{hbox{max} left( {{alpha _1}omega ,S{F_2}} proper)}}$$
(7)
the place S is an pressure price invariant, s-1; and Pok is the turbulence era as a consequence of viscous forces, which is modeled utilizing:
$${P_{textual content{ok}}}={mu _{textual content{t}}}left( {frac{{partial {u_{textual content{i}}}}}{{partial {x_{textual content{j}}}}}+frac{{partial {u_{textual content{j}}}}}{{partial {x_{textual content{i}}}}}} proper)frac{{partial {u_{textual content{i}}}}}{{partial {x_{textual content{j}}}}} – frac{2}{3}frac{{partial {u_{textual content{ok}}}}}{{partial {x_{textual content{ok}}}}}left( {3{mu _{textual content{t}}}frac{{partial {u_{textual content{ok}}}}}{{partial {x_{textual content{ok}}}}}+rho ok} proper)$$
(8)
For incompressible move, (:left(partial:{u}_{textual content{ok}}/partial:{x}_{textual content{ok}}proper)) is small and the second time period on the proper facet of Eq. (8) produces little impact on the era.
The mixing features is the important thing to the success of the strategy. They’re calculated primarily based on the space to the closest floor and on the move variables.
$${F_1}=tanh left( {arg _{1}^{4}} proper)$$
(9)
$${arg _1}=hbox{min} left[ {hbox{max} left( {frac{{sqrt k }}{{{beta ^{prime}}omega y}},frac{{500nu }}{{omega {y^2}}}} right),frac{{4rho {sigma _{varvec{upomega}2}}}}{text{k}}{{C{D_{{text{kw}}}}{y^2}}}} right]$$
(10)
$$C{D_{{textual content{ok}varvec{upomega}}}}=hbox{max} left( {2rho frac{1}{{omega {sigma _{{{varvec{upomega}2}}}}}}frac{{partial ok}}{{partial {x_{textual content{j}}}}}frac{{partial omega }}{{partial {x_{textual content{j}}}}},1.0 occasions {{10}^{ – 20}}} proper)$$
(11)
the place y is the space to the closest wall, m; υ is the kinematic viscosity, m2/s; and ok is the turbulent kinetic vitality, m2/s2.
The SST k-ω turbulence mannequin corrects the eddy viscosity coefficient by modifying its kind as follows:
$${F_2}=tanh left( {arg _{2}^{2}} proper)$$
(12)
$${arg _2}=hbox{max} left( {frac{{2sqrt ok }}{{{beta ^{prime}}omega y}},frac{{500upsilon }}{{omega {y^2}}}} proper)$$
(13)
The information coefficients 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 era concept
There’s all the time a specific amount of mechanical vitality misplaced by means of dissipation and friction within the axial-flow pump, which is transformed into inner vitality and might now not be used, this course of is irreversible, and results in a rise in entropy. Entropy (s) is a state variable, when its transport equation in a single-phase incompressible move is as follows:
$$rho frac{{Ds}}{{Dt}}= – nabla (frac{{vec {q}}}{T})+frac{{{varvec{Phi}}}}{T}+frac{{{{{varvec{Phi}}}_{{varvec{uptheta}}}}}}{{{T^2}}}$$
(14)
the place s is the instantaneous amount, and the instantaneous amount is split into two elements, the common and the fluctuation half, by the Reynolds averaged Navier–Stokes (RANS) turbulence methodology:
$$s=overline {s} +s^{prime}$$
(15)
$$u=overline {u} +{u^{prime}}$$
(16)
The above Formulation (15) and (16) are substituted into System (14) to acquire the next entropy steadiness Eq. (17).
$$rho (frac{{partial overline {s} }}{{partial t}}+overline {U} cdot nabla overline {s} )= – overline {{nabla (frac{{vec {q}}}{T})}} -rho nabla (overline {{U^{prime}s^{prime}}} )+overline {{left( {frac{{{varvec{Phi}}}}{T}} proper)}} +overline {{left( {frac{{{Phi _theta }}}{{{T^2}}}} proper)}}$$
(17)
(:stackrel{-}{{varPhi:}_{{uptheta:}}/{T}^{2}}) don’t calculate (In incompressible fluids, the fluid path of the pump is sort of isothermal); The dissipation entropy yield includes two elements: the direct dissipation price ((:{S}_{stackrel{-}{textual content{D}}})) because of the common move discipline and the turbulence dissipation price ((:{S}_{{textual content{D}}^{{prime:}}})) because of the pulsation velocity.
$$frac{{overline {Phi } }}{T}={S_{bar {{textual content{D}}}}}+{S_{{textual content{D}}^{prime}}}$$
(18)
Based on the Gouy-Stodola theorem, the entropy era price SD might be expressed by the next formulation:
$${S_{textual content{D}}}=frac{{dot {Q}}}{T}=frac{{overline {Phi } }}{T}$$
(19)
the place (:dot{Q}) is the vitality switch price, W·m-3; and Φ is the dissipation perform, W·m-3.
Subsequently, the precept of entropy era is basically in keeping with the Gouy-Stodola theorem.
$${S_{overline {{textual content{D}}} }}=frac{{2{mu _{{textual content{eff}}}}}}{T}left[ {{{left( {frac{{partial overline {u} }}{{partial x}}} right)}^2}+{{left( {frac{{partial overline {u} }}{{partial y}}} right)}^2}+{{left( {frac{{partial overline {u} }}{{partial z}}} right)}^2}} right]+frac{{{mu _{{textual content{eff}}}}}}{T}left[ {{{left( {frac{{partial overline {u} }}{{partial y}}+frac{{partial overline {v} }}{{partial x}}} right)}^2}+{{left( {frac{{partial overline {u} }}{{partial z}}+frac{{partial overline {w} }}{{partial x}}} right)}^2}+{{left( {frac{{partial overline {v} }}{{partial z}}+frac{{partial overline {w} }}{{partial y}}} right)}^2}} right]$$
(20)
$${S_{{textual content{D}}^{prime}}}=frac{{2{mu _{{textual content{eff}}}}}}{T}left[ {{{left( {frac{{partial u^{prime}}}{{partial x}}} right)}^2}+{{left( {frac{{partial v^{prime}}}{{partial y}}} right)}^2}+{{left( {frac{{partial w^{prime}}}{{partial z}}} right)}^2}} right]+frac{{{mu _{e{textual content{ff}}}}}}{T}left[ {{{left( {frac{{partial u^{prime}}}{{partial y}}+frac{{partial v^{prime}}}{{partial x}}} right)}^2}+{{left( {frac{{partial u^{prime}}}{{partial z}}+frac{{partial w^{prime}}}{{partial x}}} right)}^2}+{{left( {frac{{partial v^{prime}}}{{partial z}}+frac{{partial w^{prime}}}{{partial y}}} right)}^2}} right]$$
(21)
$${mu _{e{textual content{ff}}}}=mu +{mu _{textual content{t}}}$$
(22)
the place µeff is the efficient dynamic viscosity, pa·s ; µ is the dynamic viscosity, pa·s; and µt is the turbulent viscosity, pa·s.(::stackrel{-}{u}), (:stackrel{-}{v}), and (:stackrel{-}{w}), respectively, characterize the sections of the time-averaged velocity within the x, y, and z instructions, m/s.(:{u}^{{prime:}}), (:{v}^{{prime:}}), and (:{w}^{{prime:}}), respectively, characterize the elements of the pulsating velocity within the x, y, and z instructions, m/s.
The pulsation velocity element just isn’t out there when utilizing the Reynolds imply methodology. Based mostly on Kock et al.29 and Mathieu et al.30, (:{S}_{{textual content{D}}^{{prime:}}}) in Eq. (21) above might be calculated by turbulence mannequin. The (:{S}_{{textual content{D}}^{{prime:}}}) of the SST k-ω turbulence mannequin might be given by:
$${S_{{textual content{D}}^{prime}}}={beta ^{prime}}frac{{rho omega ok}}{T}$$
(23)
the placeβ’ is an empirical fixed within the SST k-ω mannequin, roughly equal to 0.0931; ok is the turbulent kinetic vitality, m2/s2; and ω is the turbulence eddy frequency, s-1.
Contemplating the excessive velocity gradient on the wall, further turbulent dissipation losses are unavoidable. The wall entropy era is calculated as follows:
$${S_{textual content{W}}}=frac{{vec {tau } cdot vec {v}}}{T}$$
(24)
the place(:overrightarrow{:tau:}) is the wall shear stress, Pa; and (:overrightarrow{v}) is the middle velocity vector of the primary layer mesh on the wall, m/s.
The entropy era vitality dissipation of every half might be obtained by quantity integration of every native entropy era price. The wall entropy era might be obtained by floor integration of the wall entropy era price. The entropy era equation for every half is as follows:
$${S_{{textual content{gen,}}overline {{textual content{D}}} }}=int_{V} {{S_{overline {{textual content{D}}} }}} dV$$
(25)
$${S_{{textual content{gen,D}}^{prime}}}=int_{V} {{S_{{textual content{D}}^{prime}}}} dV$$
(26)
$${S_{{textual content{gen,W}}}}=int_{A} {{S_{textual content{W}}}} dA$$
(27)
the place(:{:S}_{textual content{g}textual content{e}textual content{n},stackrel{-}{textual content{D}}}:)(EGDD) is the direct entropy era brought on by the time-average velocity;(:{:S}_{textual content{gen},{textual content{D}}^{{prime:}}})(EGTD) is the turbulent entropy era brought on by the pulsating velocity; and (:{S}_{textual content{g}textual content{e}n,textual content{W}})(EGWS) is the wall entropy era brought on by the wall velocity gradient.
In conclusion, the whole entropy era ((:{S}_{textual content{g}textual content{e}textual content{n}})) of the convective move discipline is the same as the sum of EGDD, EGTD, and EGWS.
$${S_{{textual content{gen}}}}={S_{{textual content{gen,}}overline {{textual content{D}}} }}+{S_{{textual content{gen,D}}^{prime}}}+{S_{{textual content{gen,W}}}}$$
(28)
Contemplating that the temperature of the fluid within the axial-flow pump stays fixed in the course of the move course of, System (29) determines the hydraulic loss calculated based on the entropy era System (28).
$${h_{{textual content{gen}}}}=frac{{T cdot {S_{{textual content{gen}}}}}}{{mathop mlimits^{ cdot } g}}$$
(29)
the place, (:dot{m}) is the mass discharge price of the pump, L/s; and T is the temperature, Okay.
Mesh independence and convergence evaluation
The impeller and information vane are structured meshed within the turbo-grid to fulfill the standard necessities. The inlet pipe with the water information cone and the outlet pipe with the motor shaft are structured in ICEM CFD and the mesh high quality is above 0.4, which is sweet high quality. Determine 2 under exhibits the grid drawing of the impeller, information vane, inlet pipe, and outlet pipe.
Grid mannequin of every part of the axial-flow pump.
Verification of grid independence is meant to scale back or eradicate the affect of the quantity and measurement of grids on the calculation outcomes. The impeller is a rotating half, and the opposite elements are stationary, so it’s essential to confirm the variety of the impeller grids to make sure the accuracy of the calculation. Underneath the design working situations, eight completely different impeller grid quantity schemes are utilized to guage impeller grid independence. Determine 3 demonstrates that because the variety of impeller grids will increase from 651,152 to 710,444, the change of pump effectivity slows down considerably and the effectivity values progress price is lower than 2%. Subsequently, the whole variety of impeller grids is roughly 651,152, and the grid depend for all the axial-flow pump is roughly 4.5 million.
Verification evaluation of impeller grid independence.
The grid high quality has a big influence on the outcomes of numerical simulations. It is very important guarantee grid convergence to strike a steadiness between computational assets and the numerical calculations accuracy. The GCI (Grid Convergence Index) criterion of the Richardson extrapolation methodology is used to confirm the convergence of the mesh. Lastly, three teams of high-quality mesh N1 = 4,511,756, medium mesh N2 = 1,316,776, and coarse mesh N3 = 573,075 are used to analysis the mesh convergence of pump. The grid refinement coefficients are all larger than 1.3. Then, the effectivity parameters of the pump are analyzed by discretization error evaluation. Discuss with the GCI calculation program of Yang et al.32. Desk 2 exhibits the calculation outcomes. The convergence index GCI21 is 0.154%, and the discretization error is small. Subsequently, the ultimate grid quantity is 4,511,756.
Numerical settings
The discretization of governing equations adopts a finite quantity methodology which relies on finite components, and the convective phrases are carried out in a high-resolution format. The move discipline resolution makes use of a totally implicit multi-grid coupled resolution approach, coupling the continuity and momentum equations. Desk 3 under exhibits the boundary situations set for the computational area of the pump. The utmost variety of iteration steps is ready to 2000 within the solver management.
Design of various axial-flow pump impeller schemes
To analyze the impact of various cascade densities on the axial-flow pump, primarily based on the chosen airfoil, the airfoil placement angle, the utmost airfoil camber ratio, and the utmost airfoil thickness ratio are saved fixed. By altering the tip cascade density, the impact of various cascade density on the efficiency of the axial-flow pump is studied.
In designing the impeller, the impeller blades are divided into eleven airfoil sections. R is the space from every airfoil part to the middle of rotation of the impeller. Determine 4a demonstrates the cross-sectional schematic diagram of the impeller airfoil. Based on formulation (30) and (31), when every airfoil Part R is set, the cascade distances t’ and r(i) stay unchanged. Determine 4b exhibits a schematic diagram of the two-dimensional leaf channel.
$${t^{{prime }}}=frac{{2pi R}}{Z}$$
(30)
$${r_{left( {textual content{i}} proper)}}=frac{{{R_{left( {textual content{i}} proper)}}}}{D}$$
(31)
the place i = 1, 2, 3…11, 1 represents the shroud, and 11 represents the hub.
Axial-flow pump airfoil cross-section and 2D cascade schematic diagram.
The cascade density of the airfoil part between the impeller hub and the shroud adopts the legislation of equal energy, so the cascade density of every airfoil part might be managed by the 2 parameters σ1 and Zm, exactly based on the next formulation: (32), (33), (34), and (35). To discover the affect of σ1 on the efficiency of the axial-flow pump, Zm=1.433 is saved fixed. By altering σ1, the impact of various cascade densities on the efficiency of the pump is studied.
$${lambda _{textual content{2}}}=frac{{{d_{textual content{h}}} cdot left( {{Z_{textual content{m}}} – 1} proper) cdot {sigma _{textual content{1}}}}}{{1 – {d_{textual content{h}}}}}$$
(32)
$${lambda _{textual content{1}}}={sigma _{textual content{1}}} – lambda {}_{{textual content{2}}}$$
(33)
$${sigma _{left( {textual content{i}} proper)}}={lambda _{textual content{1}}}+frac{{{lambda _{textual content{2}}}}}{{{r_{left( {textual content{i}} proper)}}}}$$
(34)
$${Z_{textual content{m}}}=frac{{{sigma _{{textual content{11}}}}}}{{{sigma _{textual content{1}}}}}$$
(35)
the place Zm is the a number of of cascade density, σ1 is the cascade density on the shroud, σ11 is the cascade density on the hub, σ(i) is the cascade density of every airfoil part of the blade, and dh is the hub ratio.
The values of σ1 are 0.7, 0.75, 0.8, 0.85, 0.9, 0.95, and 1.0, for a complete of seven schemes, amongst which the σ1 = 1.0, Zm=1.433 scheme is the preliminary scheme (OS) in the course of the check; Desk 4 demonstrates the cascade density for various schemes. Desk 5 shows the sectional information of every airfoil beneath completely different schemes.
