Numerical Analysis of Mechanical Energy Dissipation for an Axial-Flow Pump Based on Entropy Generation Theory

Article Numerical Analysis of Mechanical Energy Dissipation for an Axial-Flow Pump Based on Entropy Generation Theory

Simin Shen, Zhongdong Qian * and Bin Ji

State Key Laboratory of Water Resources and Hydropower Engineering Science, Wuhan University, No.8 South East Lake Road, Wuhan 430072, China; emilyﬂ[email protected] (S.S.); [email protected] (B.J.) * Correspondence: [email protected]; Tel.: +86-027-687-74689

Received: 14 June 2019; Accepted: 25 October 2019; Published: 31 October 2019

Abstract: Mechanical energy dissipation is a major problem affecting hydraulic machinery especially under partial-load conditions. Owing to limitations of traditional methods in evaluating mechanical energy dissipation, entropy generation theory is introduced to study mechanical energy dissipation with varying discharge and tip clearance intuitively through numerical simulations in an axial-flow pump. Results show that the impeller and diffuser are the main domains of mechanical energy dissipation, respectively accounting for 35.32%–55.51% and 32.61%–20.42% of mechanical energy dissipation throughout the flow passage. The mechanical energy dissipation of the impeller has a strong relation with the hump characteristic and becomes increasingly important with decreasing discharge. Areas of high turbulent dissipation in the impeller are mainly concentrated near the blades’ suction sides, and these regions, especially areas near the shroud, extend with decreasing discharge. When the pump enters the hump region, the distributions of turbulent dissipation near the shroud become disordered and expand towards the impeller’s inlet side. Unstable flows, like flow separation and vortices, near the blades’ suction sides lead to the high turbulent dissipation in the impeller and hump characteristic. Turbulent dissipation at the tip decreases from the blade leading edge to trailing edge, and regions of high dissipation distribute near the leading edge of the blade tip side. An increase in tip clearance for the same discharge mainly increases areas of high turbulent dissipation near the shroud and at the tip of the impeller, finally reducing pump performance.

Keywords: axial-ﬂow pump; mechanical energy dissipation; entropy generation; tip clearance

1. Introduction Axial-flow pumps in hydraulic projects of water supply and drainage have widespread applications, and it is thus important to study their internal flows and mechanical energy dissipation. When an axial-flow pump operates under partial-load conditions, unstable flows can result in large hydraulic losses and affect the steady operation of the pump. In studying the magnitude and distribution of mechanical energy dissipation with varying conditions, it is important to find an effective method of evaluating mechanical energy dissipation. Traditional methods of evaluating mechanical energy dissipation, such as using the pressure drop, are limited in quantitating and locating mechanical energy dissipation. The entropy generation method overcomes these limitations, and determines the exact locations where high mechanical energy dissipation occurs and how such dissipation is distributed. Owing to its intuitive and quantitative nature, an increasing number of researchers are applying the entropy generation method in numerical simulations to investigate mechanical energy dissipation. Kock F. and Herwig H. [1–3] introduced entropy generation theory and revealed four mechanisms of entropy generation, namely heat fluxes

Energies 2019, 12, 4162; doi:10.3390/en12214162 www.mdpi.com/journal/energies Energies 2019, 12, 4162 2 of 22 due to time-averaged and fluctuating temperature fields and dissipation due to time-averaged and fluctuating velocities. Gong RZ. et al. [4] evaluated mechanical energy dissipation using entropy generation method and determined locations of dissipation in a hydro turbine. They found that more than half the mechanical energy dissipation throughout the passage occurred in the runner and nearly 25% of dissipation occurred in the guide vanes. Meanwhile most of the mechanical energy dissipation in the runner was distributed near the leading edges and trailing edges of the runner’s blades. Hou et al. [5] analyzed mechanical energy dissipation in a submerged liquefied-natural-gas pump using the entropy generation method. They considered that entropy was generated by flow separation, the shock phenomenon, and vortices. Xiaojun Li et al. [6] investigated the relationship between the entropy generation field and vapor distribution. They found that the changing trend of the overall entropy generation rate of the impeller coincided with the decline of the pump head. The entropy generation rate slightly changed for cavitation at an early stage, while it greatly increased for fully developed cavitation. These high entropy generation rates occurred at the interface of the cavity, especially for the rear part of the cavity region. Pei et al. [7] studied the effects of the distance between the guide vane and impeller on the mechanical energy dissipation and distributions of this dissipation in a low-head pump applying the entropy generation method. They found that the distance affects the efficiency of the pump under positive rotation conditions and that the main mechanical energy dissipation is turbulent dissipation. Deyou Li et al. [8,9] applied the entropy generation method to analyze hysteresis characteristics of a pump-turbine model and reflected the mechanical energy dissipation directly. Their analysis revealed that the mechanical energy dissipation of the runner and guide vane resulted in hump characteristics, and dissipation was mostly in the runner. All these studies showed that the entropy generation method is effective in evaluating mechanical energy dissipation. The entropy generation method has the advantage of determining the magnitude and location of mechanical energy dissipation. Meanwhile, several researchers investigated the internal flow field in axial-flow pumps by conducting numerical simulations and physical experiments. Zhongdong Qian et al. [10] simulated complex flow in an axial-flow pump with fixed and adjustable guide vanes. Their calculation results revealed that flow separation and the attack angle play important roles in creating vortices that dissipate mechanical energy around the guide vanes. Jianjun Feng et al. [11] investigated the effects of tip clearance on pressure fluctuations and the tip clearance vortex for an axial-flow pump. Their analysis revealed that the tip clearance strongly affected the pressure fluctuations for the impeller, with no obvious effects for the diffuser. Shi et al. [12] investigated the dynamics of the tip leakage vortex and cavitation with different geometries of the tip clearance for an axial-flow pump. Desheng Zhang et al. [13] presented the dynamics and trajectory of the tip leakage vortex through numerical simulations and physical experiments for an axial-flow pump. Some other experts [14–21] investigated the complex flow in deep such as cavitating flow, tip flow and energy loss characteristics in the hydrofoil or turbo-machinery and achieved some results. The above studies adopting the entropy generation method to investigate mechanical energy dissipation [4–9] were conducted for turbines or centrifugal pumps. Meanwhile, the main focus of our research is the exact magnitudes and distributions of mechanical energy dissipation under varying discharge and tip clearance using the entropy generation method near the hump region for an axial-flow pump. In our paper, we compare the overall entropy generation rates of varying components to illustrate the contributions of each component and how they change with varying discharge and tip clearance. We then select the components with large mechanical energy dissipation to deeply analyze changes in the distribution of dissipation with varying discharge and tip clearance and their causes. From our analysis, we draw conclusions about the quantity and distribution of mechanical energy dissipation under varying discharge and tip clearance for an axial-flow pump. Our work provides theoretical guidance for determining mechanical energy dissipation in axial-flow pumps. Energies 2019, 12, 4162 3 of 22

2. Entropy Generation Theory According to the second law of thermodynamics, entropy generation serves as a parameter that characterizes mechanical energy dissipation due to irreversible factors during the process of energy conversions. In a real flow system of hydraulic machinery like in axial-flow pumps, mechanical energy is converted not only to pressure energy and dynamic energy but also inevitably to internal energy. Such irreversible mechanical energy dissipation can increase entropy generation. Entropy generation can therefore be applied in numerical simulations to measure such mechanical energy dissipation for axial-flow pumps. The single-phase and incompressible flow satisfies Fourier’s law of heat conduction. The transportation equation for specify entropy is written as:   ! → ∂s ∂s ∂s ∂s  q  Φ ΦΘ ρ + u + v + w = div  + + , (1) ∂t ∂x ∂y ∂z −  T  T T2

→ Φ where s is specific entropy; ρ is density; q is the heat-flux density vector; the term T denotes the ΦΘ entropy generation due to dissipation; the term T2 denotes entropy generation due to the temperature difference in heat transfer; T is temperature; and u, v, and w are respectively velocity components Φ ΦΘ along x, y, and z directions in a Cartesian coordinate system. The terms T and T2 are two important mechanisms for entropy generation [1]. According to Reynolds time-averaged Navier–Stokes equations, an instantaneous variable can be expressed as the sum of a time-averaged term and fluctuating term; e.g., s = s + s and u = u + u0. Introducing these variables into Equation (1), the transportation equation becomes:  !       →q   ∂s ∂s ∂s ∂s   ∂u0s0 ∂v0s0 ∂w0s0  Φ ΦΘ ρ + u + v + w = div  ρ + +  + + , (2)  ∂t ∂x ∂y ∂z  T  −  ∂x ∂y ∂z  T T2

! →q where the term div ρ ∂u0s0 + ∂v0s0 + ∂w0s0 represents the reversible heat transfer, this term can be T − ∂x ∂y ∂z neglected owing to the ignorance of heat transfer between the internal flows and outside for hydraulic Φ machinery [14]. The term T denotes the entropy generation rate induced by dissipation while the ΦΘ term T2 denotes the energy generation rate due to heat transfer resulting from the temperature difference. Entropy generation induced by heat transfer can be neglected by regarding the internal Φ flow of hydraulic machinery as an approximately adiabatic process. The term T can thus be obtained directly for turbulent flows as: Φ . . = S00 + S00 , (3) T D D0 . . . which includes the mean term S00 and fluctuating term S00 . The mean term S00 is the entropy generation D D0 D rate induced by time-averaged movements and represents direct dissipation. Meanwhile, the fluctuating . 00 term SD0 is the entropy generation rate induced by velocity fluctuations and represents the turbulent dissipation. These two variables are calculated [4–9] as: " # . 2 2 2 00 = 2µ ∂u + ∂v + ∂w SD T ∂x ∂y ∂z " # (4) 2 2 2 + µ ∂u + ∂v + ∂u + ∂w + ∂v + ∂w T ∂y ∂x ∂z ∂x ∂z ∂x , Energies 2019, 12, 4162 4 of 22

" # . 2µ 2 2 2 S00 = e f ∂u0 + ∂v0 + ∂w0 D0 T ∂x ∂y ∂z " # (5) µ 2 2 2 + e f ∂u0 + ∂v0 + ∂u0 + ∂w0 + ∂v0 + ∂w0 T ∂y ∂x ∂z ∂x ∂z ∂x , where u, v, and w are respectively mean velocity quantities along x, y, and z directions; u0, v0, and w0 are respectively ﬂuctuating velocity quantities along x, y, and z directions; µ is the molecular viscosity; and µt is the turbulent viscosity. µef is the eﬀective viscosity and can be expressed as:

µef = µ + µt. (6) . The mean term S00 can be computed using Equation (4) through post-processing using the D . 00 commercial software CFD POST. However, the ﬂuctuating term SD0 cannot be computed directly because the velocity ﬂuctuation quantities cannot be obtained during the calculation employing the Reynolds-averaged Navier–Stokes method. According to Kock and Mathieu et al. [22,23], the turbulent . entropy generation rate S00 relates to ε or ω in most turbulence models. For the k–ε turbulence model, . D0 00 the term SD0 can be approximately calculated as: . ρε S00 = . (7) D0 T . ω 00 For the k– turbulence model, the term SD0 can be computed as:

. ρωk S00 = α , (8) D0 T where α = 0.09; ω and k are respectively the turbulent eddy frequency and turbulence kinetic energy (TKE) for the shear stress transport (SST) k–ω turbulence model. The overall entropy generation rate for a zone can be computed using the volume integration of the speciﬁc entropy generation rate:

. Z . = 00 SD SDdV, (9) V

. Z . S = S00 dV, (10) D0 D0 V . . . S = S + S D D D0 , (11) . . where S and SD are respectively the overall entropy generation rates of a zone induced by D 0 . time-averaged movements and velocity ﬂuctuations. SD is the overall entropy generation rate of a zone, which is the sum of the two terms above. The terms relating to the entropy generation can therefore be obtained in numerical simulations for diﬀerent domains. Meanwhile, the entropy generation method has advantages in quantitatively analyzing the mechanical energy dissipation and determining the exact locations of the dissipation.

3. Experimental Model and Numerical Method

3.1. Experimental Model Figure1 displays the test axial-flow pump installed in a pump-station laboratory at Wuhan University, Wuhan, China [24]. Figure2 is a schematic of the closed-test-loop rig, which includes water tanks, gate valves, an electromagnetic flowmeter, a speed torque meter, an axial-flow pump, and a boosting pump. The electromagnetic valve and frequency converter respectively control the discharge and rotation speed of the model pump. A data acquisition system obtains parameters measured using various transducers. Parameter uncertainties in the measurements were: Energies 2019, 12, 4162 5 of 22 the input power P 0.54%, the flow rate Q 0.36%, the head H 0.66%, and the efficiency η 0.93%. ≤ ≤ ≤ ≤ Specific·parameters of the experimental model are given in Table1. EnergiesEnergies 20182018,, 1111,, xx FORFOR PEERPEER REVIEWREVIEW 55 ofof 2525

FigureFigureFigure 1. Axial-ﬂow 1.1. Axial-flow pump pump locatedlocated at at Wuhan Wuhan University University.

Figure 2. Schematic of the test loop FigureFigure 2. Schematic 2. Schematic of of the the test test loop.loop Table 1. Main parameters of the experimental axial-flow pump TableTable 1. Main 1. Main parameters parameters of of the experimental experimental axial-flow axial-flow pump pump ParametersParameters ValuesValues Design flow rate QBEP (kg/s)Parameters 330 Values Design flow rate QBEP (kg/s) 330 Design head (m) 3.5 Design headDesign (m) flow rate QBEP (kg/s)3.5 330 Diameter of ImpellerDesign (m) head (m) 0.3 3.5 BladeBlade numbernumber Diameter ofof ImpellerImpeller of Impeller (m) 3 3 0.3 BladeBlade numbernumberBlade ofof DiffuserDiffuser number of Impeller 5 5 3 Specific speed nq 340 Specific speedBlade nq number of Diffuser340 5 Rotational speed n (rpm) 1450 Rotational speed nSpecific (rpm) speed nq 1450340 Design tip clearance Rotationalof Impeller (mm) speed n (rpm) 0.15 1450 Design tip clearance of Impeller (mm) 0.15 3.2.3.2. ComputationalComputational DomainDomain andand GridGrid GenerationGeneration The computational domain is the entire flow passage, including the inlet pipe, guide cone, 3.2. ComputationalThe computational Domain and domain Grid Generation is the entire flow passage, including the inlet pipe, guide cone, impeller,impeller, diffuser,diffuser, exitexit cone,cone, 60°60° bend,bend, andand outletoutlet pipipepe [25–27],[25–27], asas shownshown inin FigureFigure 3.3. ToTo ensureensure thethe The computational domain is the entire flow passage, including the inlet pipe, guide cone, impeller, diffuser, exit cone, 60◦ bend, and outlet pipe [25–27], as shown in Figure3. To ensure the flow field from the guide cone to the 60 ◦ bend is independent of boundary conditions, we set the lengths of the inlet and outlet pipe as 10 times the pipe diameter. The pipe diameter was 350 mm. Only the impeller rotated while other parts were stationary. In the impeller domain, the entire hub rotated while the Energies 2018, 11, x FOR PEER REVIEW 6 of 25 Energies 2019, 12, 4162 6 of 22 flow field from the guide cone to the 60 ° bend is independent of boundary conditions, we set the lengths of the inlet and outlet pipe as 10 times the pipe diameter. The pipe diameter was 350 mm. shroud wasOnly stationary. the impeller The rotated sliding while interfacesother parts were were stationary. set at split In the planes impeller of domain, the domain the entire of hub the impeller, rotated while the shroud was stationary. The sliding interfaces were set at split planes of the domain as shown inof the Figure impeller,3. as shown in Figure 3.

FigureFigure 3. Three-dimensional 3. Three-dimensional view view of ofthe the computational computational zone zone.

To achieveTo betterachieve convergence, better convergence, structured structured hexahedral hexahedral grids grids were were generated generated for all for components all components in in the computational zone. All grids except those of the impeller were created using the software the computationalANSYS ICEM. zone. Owing All gridsto the excepthigh distortion those of the the im impellerpeller’s blades were and created small usingradial tip the clearance, software ANSYS ICEM. Owingthe grids to the of the high impeller distortion were generated of the impeller’s using the software blades ANSYS and small TURBOGRID; radial tip in clearance,particular, O- the grids of the impellerGrid were meshes generated were generated using around the software blades. Additi ANSYSonally, TURBOGRID; 80 layers of grids in particular,were set in the O-Grid tip meshes clearance width of 1.2 mm. For different tip clearance widths of 0.15, 0.45, and 0.9 mm in this study, were generated around blades. Additionally, 80 layers of grids were set in the tip clearance width of a uniform tip gap was applied as for the tip clearance of 1.2 mm in simulations. To generalize the 1.2 mm. Forresults diff erentfor thetip tip clearanceclearance, we widths shouldof specify 0.15, the 0.45, tip andclearance 0.9 mm according in this to study,a fraction a uniformof the span. tip gap was applied asTherefore, for the tip we clearance respectively of converted 1.2 mm the in simulations.tip clearance widths To generalize of 0.15, 0.45, the 0.9, resultsand 1.2mm for to the 0.1%, tip clearance, we should0.3%, specify 0.6%,the and tip 0.8% clearance span in the according following. toFigu are fraction 4 presents of thethe produced span. Therefore, grids of the we main respectively components of the pump. converted the tip clearance widths of 0.15, 0.45, 0.9, and 1.2mm to 0.1%, 0.3%, 0.6%, and 0.8% span in the following. Figure4 presents the produced grids of the main components of the pump. Four sets of the mesh system for the entire computational domain at the same discharge (Q = 230.748 kg/s) are generated as listed in Table2, and a total of 4158353 cells were found to be sufficient for simulations through validation of grid independence. Owing to the flow separation and vortices that can form in the pump passage, the SST k–ω turbulence model was applied in the simulations for the pump. The wall y+ values of cells of components, such as those around the impeller’s blades, were mostly below 100, which is reasonable for numerical simulations using the SST k–ω turbulenceEnergies model. 2018, 11, x FOR PEER REVIEW 7 of 25

Figure 4. GridsFigure 4. ofGrids the of mainthe main components components of the of axial-flow the axial-ﬂow pump. pump.

Four sets of the mesh system for the entire computational domain at the same discharge (Q = 230.748 kg/s) are generated as listed in Table 2, and a total of 4158353 cells were found to be sufficient for simulations through validation of grid independence. Owing to the flow separation and vortices that can form in the pump passage, the SST k–ω turbulence model was applied in the simulations for the pump. The wall y+ values of cells of components, such as those around the impeller’s blades, were mostly below 100, which is reasonable for numerical simulations using the SST k–ω turbulence model.

Table 2. Verification of grid independence.

Total Mesh Impeller Mesh Simulation Relative error with Convergence Elements Elements Head (m) Experiment 2051949 970191 10-5 4.985 9.425% 3086840 1365390 10-5 5.271 4.511% 4158353 2078340 10-5 5.350 2.787% 5164385 2748135 10-5 5.347 3.134%

3.3. Numerical Settings

The present study employs the commercial software ANSYS FLUENT to conduct simulations and software CFD POST as a post-processing tool for the axial-flow pump. To improve the accuracy of simulation results, we here conducted transient calculations with the SST k-ω turbulence model. We applied the finite volume method to discretize the governing equations accompanying the (second-order upwind) SIMPLEC algorithm in the simulation. The mass flow rate was set as the inlet boundary and the pressure was specified for the outlet boundary. A no-slip wall was set as the wall boundary. For transient computations, the time step was 0.0003448276 s, which corresponded to 1/120 of the rotating cycle at the designed rotating speed. For each time step, the maximum number of iterations was set at 20. The sliding interfaces at split planes of the impeller domain were specified as the ‘transient rotor-stator’. The results of steady-state calculations were taken as the initial values for unsteady calculations. The total unsteady calculation was performed for 14 rotating cycles [16]. Different average values of the results of the last two, four and six periods of transient calculation were selected for analysis; the errors among these average values were less than 5%. Thus, the last four time periods of the calculation were sufficient for analysis and we took these as our time sample.

Energies 2019, 12, 4162 7 of 22

Table 2. Veriﬁcation of grid independence.

Total Mesh Elements Impeller Mesh Elements Convergence Simulation Head (m) Relative Error with Experiment 5 2051949 970191 10− 4.985 9.425% 5 3086840 1365390 10− 5.271 4.511% 5 4158353 2078340 10− 5.350 2.787% 5 5164385 2748135 10− 5.347 3.134%

3.3. Numerical Settings The present study employs the commercial software ANSYS FLUENT to conduct simulations and software CFD POST as a post-processing tool for the axial-flow pump. To improve the accuracy of simulation results, we here conducted transient calculations with the SST k-ω turbulence model. We applied the finite volume method to discretize the governing equations accompanying the (second-order upwind) SIMPLEC algorithm in the simulation. The mass flow rate was set as the inlet boundary and the pressure was specified for the outlet boundary. A no-slip wall was set as the wall boundary. For transient computations, the time step was 0.0003448276 s, which corresponded to 1/120 of the rotating cycle at the designed rotating speed. For each time step, the maximum number of iterations was set at 20. The sliding interfaces at split planes of the impeller domain were specified as the ‘transient rotor-stator’. The results of steady-state calculations were taken as the initial values Energies 2018, 11, x FOR PEER REVIEW 8 of 25 for unsteady calculations. The total unsteady calculation was performed for 14 rotating cycles [16]. Different averageAll valuesresults that of we the used results for analysis of the in the last followi two,ngfour were based and sixon the periods average result of transient for this time calculation were sample. selected for analysis; the errors among these average values were less than 5%. Thus, the last four time periods of the calculation4. Results and wereDiscussion suffi cient for analysis and we took these as our time sample. All results that we used for4.1. analysis Verification in of Numerical the following Simulations were based on the average result for this time sample.

Figure 5 compares the pump performance between experiments and numerical simulations. The 4. Results and Discussionnon-dimensional parameters used in the figure were calculated as: 𝑄 𝑓𝑙𝑜𝑤 𝑐𝑜𝑒𝑓𝑓𝑖𝑐𝑖𝑒𝑛𝑡: 𝜑= . 4.1. Veriﬁcation of Numerical Simulations Ω (12) 𝐷 2𝜋

Figure5 compares the pump performance between𝑔𝐻 experiments and numerical simulations. ℎ𝑒𝑎𝑑 𝑐𝑜𝑒𝑓𝑓𝑖𝑐𝑖𝑒𝑛𝑡: 𝜓= . Ω (13) The non-dimensional parameters used in the ﬁgure were calculated𝐷 as: 2𝜋 Figure 5 shows that the simulation results for a tip clearanceQ of 0.1% span had a variation trend similar to that of the experimentalf low data, coe fwith f icient the maximum: ϕ = relative error. being around 5%. Thus, we (12) Ω 3 considered that the simulation results were reliable for current2π estimates.D To study the relationship between the entropy generation and pump performance near the hump region, we selected the four operation points A(φ = 0.4), B(φ = 0.354), C(φ = 0.329), andgH D(φ = 0.303) displayed in Figure 5 for deep analysis in the following.headcoe Additionally, f f icient po: intψ B= near the start of. the hump region was later (13) 2 selected to investigate the effects of the four tip clearance widthsΩ of 0.1%,2 0.3%, 0.6%, and 0.8% span on mechanical energy dissipation for the axial-flow pump. 2π D

Figure 5. ComparisonFigure 5. Comparison of simulation of simulation results results and andexperimental experimental data. data.

4.2. Overall Entropy Generation Rates of Different Components for Varying Discharge

To generalize the results of the entropy generation, we used variables SDM, SDE, and SDZ to reflect the turbulent entropy generation rate, the overall entropy generation rate for a domain, and the overall turbulent entropy generation rate for a domain respectively. These variables analyzed in the following were calculated as:

Energies 2019, 12, 4162 8 of 22

Figure5 shows that the simulation results for a tip clearance of 0.1% span had a variation trend similar to that of the experimental data, with the maximum relative error being around 5%. Thus, we considered that the simulation results were reliable for current estimates. To study the relationship between the entropy generation and pump performance near the hump region, we selected the four operation points A(ϕ = 0.4),B(ϕ = 0.354),C(ϕ = 0.329), and D(ϕ = 0.303) displayed in Figure5 for deep analysis in the following. Additionally, point B near the start of the hump region was later selected to investigate the eﬀects of the four tip clearance widths of 0.1%, 0.3%, 0.6%, and 0.8% span on mechanical energy dissipation for the axial-ﬂow pump.

4.2. Overall Entropy Generation Rates of Diﬀerent Components for Varying Discharge

To generalize the results of the entropy generation, we used variables SDM,SDE, and SDZ to reflect the turbulent entropy generation rate, the overall entropy generation rate for a domain, and the overall turbulent entropy generation rate for a domain respectively. These variables analyzed in the following were calculated as: . 00 SD0 SDM = , (14) ρ 2 UtipTre f . SD SDE = , (15) ρ 2 UtipVZTre f . S S = D0 , (16) DZ 2 ρUtipVZTre f U = Ω Z /2 = 22.80 (m/s), (17) tip ∗ R where ρ is water density; Utip is the blade tip speed and can be calculated by Equation (17); VZ is the . 00 volume of the entire passage; Tre f is the inlet temperature (298 K); SD is the entropy generation rate . 0 . caused by velocity fluctuation; SD is the overall entropy generation rate for a domain; and SD0 is the overall entropy generation rate caused by velocity fluctuations for a domain. Figure6 presents the SDE (reflecting overall entropy generation rate for a domain) of different components for the pump with four discharges and a tip clearance of 0.1% span. It is seen that the curves of the overall entropy generation rates for the whole passage and the impeller both increase greatly with a similar trend as the discharge reduces, and the impeller is the component with the largest mechanical energy dissipation. Meanwhile, the overall entropy generation rate increased from the guide cone to the inlet pipe, outlet pipe, 60◦ bend, exit cone, and diffuser. As the discharge decreased, the overall entropy generation rates of the diffuser, exit cone, and 60◦ bend increased slowly while those of the guide cone, outlet pipe, and inlet pipe remained stable and low. Additionally, as the discharge reduces from ϕ = 0.354 to ϕ = 0.303, the growth rates of overall entropy generation for the whole passage and the impeller were faster than those for the change from ϕ = 0.4 to ϕ = 0.354. Additionally, we see from Figure5 that the start of the hump region of the pump was around ϕ = 0.354. This means the mechanical energy dissipation of the whole passage and impeller increased more rapidly after the pump entered its hump region. This demonstrated that the mechanical energy dissipation in the impeller had a strong relation with the hump characteristic. Figure7 shows changes in E R for different discharges, where ER is the percentage contribution of . the overall entropy generation rate SD of a particular component to the overall entropy generation rate of the whole passage. The figure shows that ER was less than 7% for the guide cone, outlet pipe, and inlet pipe and largely within the range of 8%–13% for the exit cone and 60◦ bend. The overall entropy generation rates of the guide cone, outlet pipe, and inlet pipe all accounted for small proportions of the total flow passage and changed little with the discharge. They could therefore be ignored when analyzing the mechanical energy dissipation. Meanwhile, the percentage contributions for the impeller and diffuser were relatively large, with the components respectively accounting for 35.32%–55.51% Energies 2018, 11, x FOR PEER REVIEW 9 of 25

S 𝑆 = , (14) ρ𝑈𝑇

S 𝑆 = , (15) ρ𝑈𝑉𝑇

SD′ 𝑆 = , (16) 𝜌𝑈𝑉𝑇

𝑈 =Ω∗ZR/2=22.80 (m/s), (17)

where ρ is water density; 𝑈 is the blade tip speed and can be calculated by Equation (17); 𝑉 is the volume of the entire passage; 𝑇 is the inlet temperature (298 K); S is the entropy generation rate caused by velocity fluctuation; S is the overall entropy generation rate for a domain; and S is the overall entropy generation rate caused by velocity fluctuations for a domain.

Figure 6 presents the 𝑆 (reflecting overall entropy generation rate for a domain) of different components for the pump with four discharges and a tip clearance of 0.1% span. It is seen that the Energies 2019, 12, 4162 9 of 22 curves of the overall entropy generation rates for the whole passage and the impeller both increase greatly with a similar trend as the discharge reduces, and the impeller is the component with the largest mechanical energy dissipation. Meanwhile, the overall entropy generation rate increased from and 32.61%–20.42%the guide of thecone overallto the inlet entropy pipe, outlet generation pipe, 60° bend, rate exit of cone, the wholeand diffuser. passage As the fordischarge thefour discharges. decreased, the overall entropy generation rates of the diffuser, exit cone, and 60° bend increased Additionally, EslowlyR for thewhile impeller those of the increased guide cone, while outlet Epipe,R for and the inlet di pipeffuser remained declined stablewith and low. a decreasing flow rate. The analysisAdditionally, indicates as thatthe discharge the impeller reduces andfrom φ di =ff 0.354user to were φ = 0.303 the, mainthe growth locations rates of ofoverall mechanical energy dissipation for theentropy pump, generation and for the the whole dissipation passage and in the the impeller impeller were faster became than those more for the important change from with a reducing φ=0.4 to φ = 0.354. Additionally, we see from Figure 5 that the start of the hump region of the flow rate. We shouldpump was therefore around φ = pay 0.354.more This means attention the mechanical to mechanical energy dissipation energy of the dissipation whole passage of the impeller and diffuser forand the impeller axial-flow increased pump. more rapidly after the pump entered its hump region. This demonstrated that the mechanical energy dissipation in the impeller had a strong relation with the hump characteristic.

Energies 2018, 11, x FOR PEER REVIEW 10 of 25

Figure 7 shows changes in ER for different discharges, where ER is the percentage contribution

of the overall entropy generation rate S of a particular component to the overall entropy generation rate of the whole passage. The figure shows that ER was less than 7% for the guide cone, outlet pipe, and inlet pipe and largely within the range of 8%–13% for the exit cone and 60° bend. The overall entropy generation rates of the guide cone, outlet pipe, and inlet pipe all accounted for small proportions of the total flow passage and changed little with the discharge. They could therefore be ignored when analyzing the mechanical energy dissipation. Meanwhile, the percentage contributions for the impeller and diffuser were relatively large, with the components respectively accounting for 35.32%–55.51% and 32.61%–20.42% of the overall entropy generation rate of the whole passage for the four discharges. Additionally, ER for the impeller increased while ER for the diffuser declined with a decreasing flow rate. The analysis indicates that the impeller and diffuser were the main locations of mechanical energy dissipation for the pump, and the dissipation in the impeller became more importantFigure with6. The a 𝑆 reducing (reflecting flow the rate. overall We entropy should generation therefore rate pay for more a domain) attention of different to mechanical components energy Figure 6. The SDE (reﬂecting the overall entropy generation rate for a domain) of diﬀerent components dissipationfor four ofdischarges the impeller and diffuser for the axial-flow pump. for four discharges.

. S Figure 7. PercentageFigure 7. contributions Percentage contributions of S of di offf erentdifferent components components to the to overall the overall entropy entropygeneration generation rate rate of the whole flow passage forD four discharges. of the whole flow passage for four discharges. Figure 8 presents changes in M with varying discharge, where M is the percentage contribution of the overall turbulent entropy generation rate S' of a component to the overall entropy generation Figure8 presents changes in M with varying discharge, where M is the percentage contribution of rate S for the same component. The figure. shows that M was large for all domains, ranging 94%– 100%, under different conditions. Therefore, most of the mechanical energy dissipation in the pump the overall turbulent entropy generation rate SD0 of a component to the overall entropy generation rate . was induced by turbulent dissipation, and turbulent entropy generation rate should be mainly SD for the sameanalyzed component. in the following. The figure shows that M was large for all domains, ranging 94%–100%, under different conditions. Therefore, most of the mechanical energy dissipation in the pump was induced by turbulent dissipation, and turbulent entropy generation rate should be mainly analyzed in the following. Figure9 displays the SDZ (reflecting the overall turbulent entropy generation rates for a domain) of different components for four discharges and the same tip clearance of 0.1% span. The value SDZ had a trend similar to that of the SDE in Figure6. As the flow rate decreased, the turbulent entropy generation rates of the diffuser, exit cone, and 60◦ bend increased slowly, while the turbulent entropy

Energies 2018, 11, x FOR PEER REVIEW 11 of 25 Energies 2019, 12, 4162 10 of 22 generation rate of the impeller increased greatly. Therefore, the discharge had greater eﬀects on the turbulent dissipationEnergies 2018 in, 11 the, x FOR impeller PEER REVIEW when the axial-ﬂow pump entered the hump11region. of 25

Figure 8. Percentage contributions of the overall turbulent entropy generation rate 𝑆 of a component to the overall entropy generation rate for the same component with four discharges

Figure 9 displays the 𝑆 (reflecting the overall turbulent entropy generation rates for a domain) of different components for four discharges and the same tip clearance of 0.1% span. The value

𝑆 had a trend similar to that of the SDE in Figure 6. As the flow rate decreased, the turbulent entropy . Figuregeneration 8. Percentage rates of contributionsthe diffuser, ofexit the cone, overall and turbulent 60° bend entropy increased generation slowly, rate while 𝑆 ofthe a turbulentcomponent entropy to Figure 8. Percentage contributions of the overall turbulent entropy generation rate SD of a component thegeneration overall entropy rate of generation the impeller rate increasedfor the same greatly. component Therefore, with four the discharges discharge had greater effects0 on the to the overallturbulent entropy dissipation generation in the rateimpeller for when the samethe axial-flow component pump entered with fourthe hump discharges. region. Figure 9 displays the 𝑆 (reflecting the overall turbulent entropy generation rates for a domain) of different components for four discharges and the same tip clearance of 0.1% span. The value

𝑆 had a trend similar to that of the SDE in Figure 6. As the flow rate decreased, the turbulent entropy generation rates of the diffuser, exit cone, and 60° bend increased slowly, while the turbulent entropy generation rate of the impeller increased greatly. Therefore, the discharge had greater effects on the turbulent dissipation in the impeller when the axial-flow pump entered the hump region.

Figure 9. The 𝑆 (reflecting the overall turbulent entropy generation rate for a domain) of different Figure 9. The SDZ (reﬂecting the overall turbulent entropy generation rate for a domain) of diﬀerent components for four discharges components for four discharges.

4.3. Overall Entropy Generation Rates of Diﬀerent Components for Varying Tip Clearance

There is an inevitableFigure 9. The tip 𝑆 (reflecting clearance the overall between turbulent the entropy impeller’s generation rate blade for a domain) and shroud of different when an axial-flow pump operates. Tocomponents study for eff fourects discharges of the tip clearance width on overall entropy generation rates of different components, we selected point B (ϕ = 0.354) near the start of the pump’s hump region with four tip clearance widths to conduct the analysis work. Figure 10 shows the pump performance for the four tip clearances under the same operating condition. It is seen that the pump performance decreased with the tip clearance increasing. The head of the pump reduced quickly as the tip clearance increased from 0.3% span to 0.8% span, indicating that a tip clearance larger than 0.3% span strongly affected the pump performance. The above analysis revealed that more than 94% of the mechanical energy dissipation was induced by turbulent dissipation. We therefore computed the overall turbulent entropy generation rates for different domains for the axial-flow pump. Figure 11 presents the SDZ (reflecting the overall turbulent entropy generation rate for a domain) of different domains with varying tip clearance at ϕ = 0.354. Figure 11 clearly shows that the turbulent entropy generation rate for the impeller increased while the turbulent entropy generation rates for the diffuser and exit cone slowly decreased with the tip clearance Energies 2019, 12, 4162 11 of 22

increasing. Additionally,Energies 2018, 11, x theFOR PEER turbulent REVIEW entropy generation rate for the 60◦ bend fluctuated12 of 25 within a small range. Meanwhile, the overall turbulent entropy generation rate increased from the 60 bend to 4.3. Overall Entropy Generation Rates of Different Components for Varying Tip Clearance ◦ the exit cone, diffuser, and impeller. The impeller therefore remained the main location of mechanical There is an inevitable tip clearance between the impeller’s blade and shroud when an axial-flow energy dissipationpump foroperates. different To study tip effects clearances. of the tip cleara Whennce width the tipon overall clearance entropy grew generation from rates 0.3% of span to 0.8% span, the turbulentdifferent dissipation components, we in selected the impeller point B (φ increased = 0.354) near rapidly, the start ofindicating the pump’s hump that region the with turbulent entropy four tip clearance widths to conduct the analysis work. Figure 10 shows the pump performance for generation wasthe related four tip toclearances the decline under the in same pump operating performance condition. It is asseen shown that the inpump Figure performance 10. A tip clearance larger than 0.3%decreased span thus with atheffected tip clearance the turbulent increasing. dissipationThe head of the of pump the impellerreduced quickly greater as the and tip finally affected the pump performance.clearance increased from 0.3% span to 0.8% span, indicating that a tip clearance larger than 0.3% span strongly affected the pump performance.

Energies 2018, 11, x FOR PEER REVIEW 13 of 25 Figure 10. Axial-ﬂowFigure 10. Axial-flow pump pump performance performance with varying varying tip clearance tip clearance at φ = 0.354 at. ϕ = 0.354.

The above analysis revealed that more than 94% of the mechanical energy dissipation was induced by turbulent dissipation. We therefore computed the overall turbulent entropy generation

rates for different domains for the axial-flow pump. Figure 11 presents the 𝑆 (reflecting the overall turbulent entropy generation rate for a domain) of different domains with varying tip clearance at φ = 0.354. Figure 11 clearly shows that the turbulent entropy generation rate for the impeller increased while the turbulent entropy generation rates for the diffuser and exit cone slowly decreased with the tip clearance increasing. Additionally, the turbulent entropy generation rate for the 60° bend fluctuated within a small range. Meanwhile, the overall turbulent entropy generation rate increased from the 60° bend to the exit cone, diffuser, and impeller. The impeller therefore remained the main location of mechanical energy dissipation for different tip clearances. When the tip clearance grew from 0.3% span to 0.8% span, the turbulent dissipation in the impeller increased rapidly, indicating that the turbulent entropy generation was related to the decline in pump performance as shown in Figure 10. A tip clearance larger than 0.3% span thus affected the turbulent dissipation of the impeller greater and finally affected the pump performance.

Figure 11. The 𝑆 (reflecting the overall turbulent entropy generation rate for a domain) of different Figure 11. The SDZ (reflecting the overall turbulent entropy generation rate for a domain) of different domains with varying tip clearance at φ=0.354 domains with varying tip clearance at ϕ = 0.354. 4.4. Detailed Distributions of The Turbulent Entropy Generation Rate with Varying Discharge 4.4. Detailed Distributions of the Turbulent Entropy Generation Rate with Varying Discharge The above analysis showed that the impeller and diffuser were the main locations of mechanical The aboveenergy analysis dissipation showed and that that most the of the impeller power was and lost dithroughffuser turbulent were dissipation. the main The locations turbulent of mechanical entropy generation rates for the impeller and diffuser were thus selected for analysis here. To energy dissipationdetermine and exact that locations most ofof turbul the powerent dissipation, was lostcontours through of the turbulent turbulent generation dissipation. rate for the The turbulent entropy generationimpeller rates were for drawn the along impeller different and spanwise diffuser surfaces were under thus four selecteddischarges. forThe analysisspanwise surface here. To determine was set from 0 to 1, corresponding from the impeller’s hub to the shroud; SP0.2 was near the hub and exact locations of turbulent dissipation, contours of the turbulent generation rate for the impeller SP0.98 was near the shroud. Figure 12 presents the distributions of 𝑆 of the impeller along different were drawn alongspans di(SP0.2,fferent SP0.5, spanwise SP0.8, and SP0.98) surfaces for four under discharges. four In discharges.Figure 12a, areas Theof high spanwise S were surface was set from 0 to 1,mainly corresponding near the leading from edges of the blades’ impeller’s suction surfaces, hub and to the the span shroud; SP0.98 near SP0.2 the shroud was having near the hub and the largest areas of turbulent dissipation. Meanwhile, there were areas of relatively high turbulent SP0.98 was neardissipation the shroud. in the blade Figure wake 12 region presents and in the the flo distributionsw passage of the impeller of SDM as shownof the in impeller Figure 12a. along different When the discharge reduced to operation point B as shown in Figure 12b, distributions of the spans (SP0.2, SP0.5, SP0.8, and SP0.98) for four discharges. In Figure 12a, areas of high SDM were turbulent generation rate along the spans were similar to those in Figure 12a, while areas of high S were obviously larger than those in Figure 12a for each corresponding span and especially for span

SP0.98. With the discharge further decreasing to point C in Figure 12c, the areas of high S further grew for each corresponding span relative to the areas in Figure 12b. For the span SP0.98 shown in

Figure 12c, areas of high S became disordered and expanded towards the inlet of the impeller. As for the lowest discharge of operation point D displayed in Figure 12d, areas of high S extended further for span SP0.98. The distributions of high turbulent dissipation on surfaces SP0.8 and SP0.98 displayed in Figure 12d became more disordered and extended more towards the inlet of the impeller relative to the distributions of Figure 12c. The above analysis reveals that turbulent dissipation of the impeller was mainly concentrated near blades’ suction sides and that the span SP0.98 near the shroud had the largest areas of turbulent dissipation for four operation points. Areas of high turbulent dissipation of the impeller extend with a decreasing flow rate, especially at the span near the shroud. Additionally, when the pump entered the hump region of points C and D, the distributions of

Energies 2019, 12, 4162 12 of 22 mainly near the leading edges of blades’ suction surfaces, and the span SP0.98 near the shroud having the largest areas of turbulent dissipation. Meanwhile, there were areas of relatively high turbulent dissipation in the blade wake region and in the flow passage of the impeller as shown in Figure 12a. When the discharge reduced to operation point B as shown in Figure 12b, distributions of the turbulent generation rate along the spans were similar to those in Figure 12a, while areas of high SDM were obviously larger than those in Figure 12a for each corresponding span and especially for span SP0.98. With the discharge further decreasing to point C in Figure 12c, the areas of high SDM further grew for each corresponding span relative to the areas in Figure 12b. For the span SP0.98 shown in Figure 12c, areas of high SDM became disordered and expanded towards the inlet of the impeller. As for the lowest discharge of operation point D displayed in Figure 12d, areas of high SDM extended further for span SP0.98. The distributions of high turbulent dissipation on surfaces SP0.8 and SP0.98 displayed in Figure 12d became more disordered and extended more towards the inlet of the impeller relative to the distributions of Figure 12c. The above analysis reveals that turbulent dissipation of the impeller was mainly concentrated near blades’ suction sides and that the span SP0.98 near the shroud had the largest areas of turbulent dissipation for four operation points. Areas of high turbulent dissipation of the impeller extend with a decreasing flow rate, especially at the span near the shroud. Additionally, when the pump entered the hump region of points C and D, the distributions of turbulent dissipation near the shroud became disordered and expanded towards the impeller’s inlet side. To explain the distributions of turbulent dissipation in the impeller, flow patterns around the blade suction side as presented by the iso-surface of TKE are investigated for four discharges. Figure 13 displays the iso-surface of TKE (for a value of 0.017) and iso-lines of non-dimensional TKE around the blade for varying discharge and a tip clearance of 0.1% span. It is seen that regions of high TKE (red areas in Figure 13) concentrated near the blade’s suction surface and the location of the tip leakage vortex near the shroud had high TKE under four discharges. Additionally, areas of high TKE, especially those near the shroud, grew with a decreasing flow rate. It is thus clear that the distributions of areas having a high TKE in Figure 13 correspond with the distributions of areas having a high turbulent entropy generation rate in Figure 12, indicating that the regions of high turbulent dissipation strongly correlated with areas of high TKE. To deeply study the cause of high TKE near the shroud, we drew the non-dimensional velocity vector at span SP0.98 around one blade of the impeller with varying discharge under the same tip clearance of 0.1% span displayed in Figure 14 for analysis. As seen from Figure 14a, the flow field was relatively smooth because the difference between the flow angle and blade setting angle was small. The high turbulent dissipation near the leading edge of the blade suction side was mainly due to the flow attack and separation. The flow separation was caused by the pressure difference between the blade suction side and blade pressure side. In Figure 14b, we see an attack angle larger than that displayed in Figure 14a. This is because the difference between the flow angle and blade setting angle increased as the flow rate reduced. The flow separation increased because of greater pressure differences between the pressure and suction sides of blades, which generated higher turbulent dissipation near the shroud. Additionally, there was a low-velocity area near the blade suction side and wake area because of tip leakage vortex and wake flows. When the discharge further decreased to point C, the flow field near the shroud displayed in Figure 14c became disordered and there was a backflow area. This is because the flow attack angle displayed in Figure 14c was larger than that displayed in Figure 14b and produced greater flow separation and vortices. Thus, the high level of turbulent dissipation near the shroud was mainly induced by the flow separation and vortices of leakage flows, backflows and wake flows. As for point D, the flow field near the shroud displayed in Figure 14d is was more unstable than that displayed in Figure 14c because of the larger flow attack angle. Meanwhile, there were two backflow areas owing to greater flow separation and vortices than those displayed in Figure 14c. Additionally, the areas of vortices and flow separation had high TKE. Thus, unstable flows like flow separation and vortices form near the blade suction side, high TKE accumulated in these areas, and there was finally large turbulent dissipation there. These unstable flows like flow separation and vortices of backflows near the blade suction side led to Energies 2019, 12, 4162 13 of 22 the hump characteristic. This explains why most turbulent dissipation concentrated near the blades’ suction sides andEnergies why 2018, 11 large, x FOR PEER areas REVIEW of turbulent dissipation grew especially near14 the of 25 shroud with a reducing flow rate.turbulent dissipation near the shroud became disordered and expanded towards the impeller’s inlet side.

SP0.2 SP0.5 SP0.8 SP0.98 SP0.2 SP0.5 SP0.8 SP0.98

(a) (b)

SP0.2 SP0.5 SP0.8 SP0.98 SP0.2 SP0.5 SP0.8 SP0.98

Figure 12. Distributions of SDM (reflecting the turbulent entropy generation rate) along different spans for the impeller with varying discharge ((a) ϕ = 0.4;(b) ϕ = 0.354;(c) ϕ = 0.329; and (d) ϕ = 0.303). Figure 12. Distributions of 𝑆 (reflecting the turbulent entropy generation rate) along different spans for the impeller with varying discharge ((a) φ = 0.4; (b) φ = 0.354; (c) φ = 0.329; and (d) φ = 0.303) As for the overall turbulent entropy generation rate of the diffuser, the turbulent dissipation To explain the distributions of turbulent dissipation in the impeller, flow patterns around the increased slowlyblade as suction the flow side as rate presented decreased by the iso-surface as displayed of TKE are ininvestigated Figure for9. four To localizedischarges. theFigure large turbulent dissipation under13 displays varying the iso-surface discharge, of TKE we (for drewa value of contours 0.017) and iso-lines of SDM of (reflectingnon-dimensional the TKE turbulent around generation the blade for varying discharge and a tip clearance of 0.1% span. It is seen that regions of high TKE rate) for the diff(reduser areas along in Figure diff erent13) concentr spanwiseated near surfaces the blade’s for suction analysis. surface and Figure the location 15 presents of the tip the distribution of SDM for the dileakageffuser vortex along near the di shroudfferent had spans high TKE (SP0.2, under four SP0.5, discharges. SP0.8, Additionally, and SP0.98) areas of high with TKE, four discharges. The figure shows that regions of high SDM located near the leading edges of guide vanes and in the passage of the diffuser, and the span SP0.98 near the wall had the largest turbulent dissipation for the four discharges. With the flow rate decreasing, the distribution of high turbulent dissipation spread moderately and became more uniform especially for the span SP0.98 near the wall. The flow attacking owing to a difference between the flow angle and the setting angle of guide vanes was responsible for the high turbulent dissipation at the leading edges of the guide vanes in the diffuser. Additionally, the vortices in the passage lead to areas of relatively high turbulent dissipation in the diffuser. All in all, most high turbulent dissipation was near the leading edges of guide vanes and in the diffuser passage, and the distribution of high turbulent dissipation expanded to be more uniform as the flow rate declined. Energies 2018, 11, x FOR PEER REVIEW 15 of 25

especially those near the shroud, grew with a decreasing flow rate. It is thus clear that the distributions of areas having a high TKE in Figure 13 correspond with the distributions of areas having a high turbulent entropy generation rate in Figure 12, indicating that the regions of high turbulent dissipation strongly correlated with areas of high TKE. To deeply study the cause of high TKE near the shroud, we drew the non-dimensional velocity vector at span SP0.98 around one blade of the impeller with varying discharge under the same tip clearance of 0.1% span displayed in Figure 14 for analysis. As seen from Figure 14a, the flow field was relatively smooth because the difference between the flow angle and blade setting angle was small. The high turbulent dissipation near the leading edge of the blade suction side was mainly due to the flow attack and separation. The flow separation was caused by the pressure difference between the blade suction side and blade pressure side. In Figure 14b, we see an attack angle larger than that displayed in Figure 14a. This is because the difference between the flow angle and blade setting angle increased as the flow rate reduced. The flow separation increased because of greater pressure differences between the pressure and suction sides of blades, which generated higher turbulent dissipation near the shroud. Additionally, there was a low-velocity area near the blade suction side and wake area because of tip leakage vortex and wake flows. When the discharge further decreased to point C, the flow field near the shroud displayed in Figure 14c became disordered and there was a backflow area. This is because the flow attack angle displayed in Figure 14c was larger than that displayed in Figure 14b and produced greater flow separation and vortices. Thus, the high level of turbulent dissipation near the shroud was mainly induced by the flow separation and vortices of leakage flows, backflows and wake flows. As for point D, the flow field near the shroud displayed in Figure 14d is was more unstable than that displayed in Figure 14c because of the larger flow attack angle. Meanwhile, there were two backflow areas owing to greater flow separation and vortices than those displayed in Figure 14c. Additionally, the areas of vortices and flow separation had high TKE. Thus, unstable flows like flow separation and vortices form near the blade suction side, high TKE accumulated in these areas, and there was finally large turbulent dissipation there. These unstable flows like flow separation and vortices of backflows near the blade suction side led to the hump characteristic. This explains why most turbulent Energies 2019, 12, 4162 14 of 22 dissipation concentrated near the blades’ suction sides and why large areas of turbulent dissipation grew especially near the shroud with a reducing flow rate.

(a) (b)

Energies 2018, 11, x FOR (PEERc) REVIEW (d) 16 of 25

(c) (d) Figure 13. Flow patterns for varying discharges around the blade. Isosurface of non-dimensional turbulence kinetic energyFigure 13. (TKE)Flow patterns for for a valuevarying discharges of 0.017 around and the magniﬁed blade. Isosurface isolines of non-dimensional of non-dimensional TKE turbulence kinetic energy (TKE) for a value of 0.017 and magnified isolines of non-dimensional TKE ((a) ϕ = 0.4; (b) ϕ =((a)0.354; φ = 0.4; (b ()c φ) =ϕ 0.354=; 0.329;(c) φ = 0.329 and; and ( d(d) φϕ = 0.303= 0.303).).

(a) (b)

Figure 14. Non-dimensional velocity vector at the span SP0.98 for the impeller with varying discharge under a tip clearance of 0.1% span ((a) ϕ = 0.4; (b) ϕ = 0.354; (c) ϕ = 0.329; and (d) ϕ = 0.303). Energies 2018, 11, x FOR PEER REVIEW 17 of 25

Figure 14. Non-dimensional velocity vector at the span SP0.98 for the impeller with varying discharge under a tip clearance of 0.1% span ((a) φ = 0.4; (b) φ = 0.354; (c) φ = 0.329; and (d) φ = 0.303).

As for the overall turbulent entropy generation rate of the diffuser, the turbulent dissipation increased slowly as the flow rate decreased as displayed in Figure 9. To localize the large turbulent

dissipation under varying discharge, we drew contours of 𝑆 (reflecting the turbulent generation rate) for the diffuser along different spanwise surfaces for analysis. Figure 15 presents the distribution

of S for the diffuser along different spans (SP0.2, SP0.5, SP0.8, and SP0.98) with four discharges. The figure shows that regions of high 𝑆 located near the leading edges of guide vanes and in the passage of the diffuser, and the span SP0.98 near the wall had the largest turbulent dissipation for the four discharges. With the flow rate decreasing, the distribution of high turbulent dissipation spread moderately and became more uniform especially for the span SP0.98 near the wall. The flow attacking owing to a difference between the flow angle and the setting angle of guide vanes was responsible for the high turbulent dissipation at the leading edges of the guide vanes in the diffuser. Additionally, the vortices in the passage lead to areas of relatively high turbulent dissipation in the diffuser. All in all, most high turbulent dissipation was near the leading edges of guide vanes and in Energies 2019, 12, 4162 15 of 22 the diffuser passage, and the distribution of high turbulent dissipation expanded to be more uniform as the flow rate declined.

SP0.2 SP0.5 SP0.8 SP0.98 SP0.2 SP0.5 SP0.8 SP0.98

(a) (b)

SP0.2 SP0.5 SP0.8 SP0.98 SP0.2 SP0.5 SP0.8 SP0.98

Figure 15. Distributions of SDM (reflecting the turbulent entropy generation rate) on different spanwise surfaces of the diffuser with varying discharge ((a) ϕ = 0.4;(b) ϕ = 0.354;(c) ϕ = 0.329; and (d) ϕ = 0.303).

4.5. Detailed Distributions of the Turbulent Entropy Generation Rate with Varying Tip Clearances Figures 10 and 11 show that the overall turbulent entropy generation rate of the impeller increases with the tip clearance increasing from 0.1% span to 0.8% span and this was strongly related with a decline in pump performance. The turbulent entropy generation rate of the impeller was therefore chosen for analyzing effects of the tip clearance width on mechanical energy dissipation for the same discharge (ϕ = 0.354). Similar to the analysis above, we drew contours of SDM (reflecting the turbulent generation rate) along different spans of the impeller to determine the magnitude and locations of turbulent dissipation. Figure 16 shows the distributions of SDM of the impeller along different spans (SP0.2, SP0.5, SP0.8, and SP0.98) for four tip clearance widths at operation point B. Figure 16a is the same as Figure 12b, showing that regions having a high turbulent entropy generation rate were mainly distributed near blades’ suction sides. When the tip clearance grew from 0.1% span to 0.8% span as shown in Figure 16, regions of high turbulent dissipation on surface SP0.98 increased and expanded along blades’ suction sides in the passage of the impeller, and these regions of high turbulent dissipation even approached the pressure sides of neighboring blades in the case of the largest tip clearance of 0.8% span. As found in our previous research on tip flows [25], the tip clearance width mainly affected the flow structure near the shroud and more vortices were generated near the shroud with the tip clearance increasing. Figure 16 shows that areas of high turbulent dissipation grew obviously for span SP0.98 with the tip clearance increasing at the same discharge (ϕ = 0.354). Therefore, an increase in tip clearance mainly increased turbulent dissipation near the shroud of the impeller and finally reduced the pump performance. Energies 2018, 11, x FOR PEER REVIEW 18 of 25

Figure 15. Distributions of 𝑆 (reflecting the turbulent entropy generation rate) on different spanwise surfaces of the diffuser with varying discharge ((a) φ = 0.4; (b) φ = 0.354; (c) φ = 0.329; and (d) φ = 0.303)

4.5. Detailed Distributions of the Turbulent Entropy Generation Rate with Varying Tip Clearances

Figures 10 and 11 show that the overall turbulent entropy generation rate of the impeller increases with the tip clearance increasing from 0.1% span to 0.8% span and this was strongly related with a decline in pump performance. The turbulent entropy generation rate of the impeller was therefore chosen for analyzing effects of the tip clearance width on mechanical energy dissipation for

the same discharge (φ = 0.354). Similar to the analysis above, we drew contours of 𝑆 (reflecting the turbulent generation rate) along different spans of the impeller to determine the magnitude and

locations of turbulent dissipation. Figure 16 shows the distributions of S of the impeller along different spans (SP0.2, SP0.5, SP0.8, and SP0.98) for four tip clearance widths at operation point B. Figure 16a is the same as Figure 12b, showing that regions having a high turbulent entropy generation rate were mainly distributed near blades’ suction sides. When the tip clearance grew from 0.1% span to 0.8% span as shown in Figure 16, regions of high turbulent dissipation on surface SP0.98 increased and expanded along blades’ suction sides in the passage of the impeller, and these regions of high turbulent dissipation even approached the pressure sides of neighboring blades in the case of the largest tip clearance of 0.8% span. As found in our previous research on tip flows [25], the tip clearance width mainly affected the flow structure near the shroud and more vortices were generated near the shroud with the tip clearance increasing. Figure 16 shows that areas of high turbulent dissipation grew obviously for span SP0.98 with the tip clearance increasing at the same discharge Energies(φ2019 = 0.354, 12, 4162). Therefore, an increase in tip clearance mainly increased turbulent dissipation near the16 of 22 shroud of the impeller and finally reduced the pump performance.

SP0.2 SP0.5 SP0.8 SP0.98 SP0.2 SP0.5 SP0.8 SP0.98

Energies 2018, 11, x FOR PEER REVIEW (a) (b) 19 of 25

SP0.2 SP0.5 SP0.8 SP0.98 SP0.2 SP0.5 SP0.8 SP0.98

Figure 16. Distributions of SDM (reflecting the turbulent entropy generation rate) along different spanwise surfaces of the impeller for varying tip clearance at ϕ = 0.354 ((a) h = 0.1% span; (b) h = 0.3% 𝑆 span; (c) hFigure= 0.6% 16. span; Distributions and (d )of h =0.8% (reflecting span). the turbulent entropy generation rate) along different spanwise surfaces of the impeller for varying tip clearance at φ = 0.354 ((a) h = 0.1% span; (b) h = 4.6. Tip Flow Patterns0.3% span; and (c) hDistributions = 0.6% span; and of(d) theh = 0.8% Turbulent span) Entropy Generation Rate in Varying Tip Gaps We introduced4.6. Tip Flow parametersPatterns and Distributions to analyze of thethe Turb tipulent flow Entropy patterns Generation in varying Rate in tip Varying gaps. Tip Figure Gaps 17 displays the definitionWe of computationalintroduced parameters parameters to analyze the in thetip flow tip patterns gap. Planes in varying relative tip gaps. to Figure the blade 17 displays were defined 1 as λ = SCthe− , definition where S of is computational a coordinate parameters aligned in with the tip the gap. blade Planes chordrelative andto the C blade is the were blade defined chord from the leadingas edge λ=𝑆𝐶 (LE) , where to the S is trailing a coordinate edge aligned (TE). with We the here blade selected chord and a planeC is the at bladeλ = chord0.4 forfrom the the following analysis presentedleading edge in (LE) Figure to the 17 b.trailing edge (TE). We here selected a plane at 𝜆=0.4 for the following Detailedanalysis data presented of tip flow in Figure patterns 17b. at λ = 0.4 for tip clearances of 0.1% span, 0.3% span, 0.6% span, and 0.8% span at the same flow rate (ϕ = 0.354) are shown in Figure 18. A positive axial velocity represents the mainstream while a negative velocity represents tip leakage flow. The non-dimensional mean axial velocity and streamlines for the tip clearance of 0.1% span displayed in Figure 18a reveal that the axial velocity in the gap was mostly negative and its distribution was uniform, which means that the tip leakage flow almost filled the gap and hardly mixed with the mainstream. Additionally, there was almost no tip separation vortex at the tip for λ = 0.4 owing to strong effects of the viscous force at

(a) (b)

Figure 17. Parameter definition of the blade tip: (a) impeller’s blade; and (b) selected plane at 𝜆=0.4

Energies 2018, 11, x FOR PEER REVIEW 19 of 25

Energies 2019, 12, 4162 17 of 22

boundary layers. ComparedSP0.2 with SP0.5 streamlines SP0.8 SP0.98 of a tip clearance SP0.2 of 0.1%SP0.5 span, SP0.8 Figure SP0.98 18 b,c,d shows part of the positive axial velocity at the tip for λ = 0.4, indicating that the mainstream mixes with (c) (d) leakage flow and the formation of a tip separation vortex in the gap. This tip separation vortex attaches more on the blade tip surface as the tip clearance increases from 0.3% span to 0.8% span, as seen in Figure 18b–d. This is because the mainstream mixes more with the tip leakage flow to create more Figure 16. Distributions of 𝑆 (reflecting the turbulent entropy generation rate) along different vortices for largerspanwise tip clearancesurfaces of the at impelle the samer for varying flow rate. tip clearance As the at tip φ = clearance 0.354 ((a) h grows= 0.1% span; from (b) 0.3% h = span to 0.8% span, the core0.3% of the span; tip (c) leakageh = 0.6% span; vortex and (d moves) h = 0.8% closer span) to the blade suction side. The high turbulence of the tip separation vortex can connect with the shear layer and merge with the tip leakage vortex to block the tip4.6. Tip flow Flow and Patterns mainstream, and Distributions finally of the increasing Turbulent Entropy entropy Generation generation. Rate in Varying The tipTip flowGaps patterns for tip clearancesWe of introduced 0.3% span, parameters 0.6% span, to analyze and 0.8%the tip spanflow patterns were similarin varying to tip those gaps. Figure modeled 17 displays by Denton [28] (see Figurethe 31a definition in his of paper). computational For the parameters smallest in tip the clearancetip gap. Planes of 0.1%relative span, to the theblade tip were flow defined structure was different fromas λ=𝑆𝐶 thatEnergies in , Denton’swhere 2018, 11 S, x is FOR a work coordinatePEER REVIEW because aligned the with tip the flow blade hardly chord and mixed C is withthe blade the chord mainstream 20from of 25 the at the tip. Denton’s leakageleading edge mixing Detailed(LE) to model datathe oftrailing tip assumes flow edge patterns that(TE). at λ=0.4 theWe here mainstream for tip selected clearances a andplaneof 0.1% leakageat span, 𝜆=0.4 0.3% flow span,for the 0.6% must following span, mix at the tip, analysis presentedand 0.8% span in Figure at the same17b. flow rate (φ = 0.354) are shown in Figure 18. A positive axial velocity and his model is thusrepresents not suitablethe mainstream for awhile very a smallnegative tip velocity clearance represents of 0.1%tip leakage span. flow. The non- dimensional mean axial velocity and streamlines for the tip clearance of 0.1% span displayed in Figure 18a reveal that the axial velocity in the gap was mostly negative and its distribution was uniform, which means that the tip leakage flow almost filled the gap and hardly mixed with the mainstream. Additionally, there was almost no tip separation vortex at the tip for λ=0.4 owing to strong effects of the viscous force at boundary layers. Compared with streamlines of a tip clearance of 0.1% span, Figure 18b,c,d shows part of the positive axial velocity at the tip for λ=0.4, indicating that the mainstream mixes with leakage flow and the formation of a tip separation vortex in the gap. This tip separation vortex attaches more on the blade tip surface as the tip clearance increases from 0.3% span to 0.8% span, as seen in Figure 18b–d. This is because the mainstream mixes more with the tip leakage flow to create more vortices for larger tip clearance at the same flow rate. As the tip clearance grows from 0.3% span to 0.8% span, the core of the tip leakage vortex moves closer to the blade suction side. The high turbulence of the tip separation vortex can connect with the shear layer and merge with the tip leakage vortex to block the tip flow and mainstream, finally increasing entropy generation. The tip flow patterns for tip clearances of 0.3% span, 0.6% span, and 0.8% span were similar to those modeled by Denton [28] (see Figure 31a in his paper). For the smallest tip clearance of 0.1% span,(a the) tip flow structure was different from that in(b )Denton’s work because the tip flow hardly mixed with the mainstream at the tip. Denton’s leakage mixing model assumes that Figure 17. Parameterthe mainstream definition and leakage of the flow blade must tip: mix ( aat) the impeller’s tip, and his blade; model and is thus (b )not selected suitable planefor a very at λ = 0.4. Figuresmall 17. Parametertip clearance definition of 0.1% span. of the blade tip: (a) impeller’s blade; and (b) selected plane at 𝜆=0.4

(a)

(b)

Figure 18. Cont. Energies 2019, 12, 4162 18 of 22 Energies 2018, 11, x FOR PEER REVIEW 21 of 25 Energies 2018, 11, x FOR PEER REVIEW 21 of 25

(d)

Figure 18. DistributionFigure 18. of non-dimensionalDistribution of non-dimensional mean axial( dmean) velocitiesaxial velocities and and streamlines streamlines at atλ=0.4λ = for0.4 for varying varying tip clearance at φ = 0.354 ((a) h = 0.1% span; (b) h = 0.3% span; (c) h =0.6% span; and (d) h Figureϕ =18. Distribution of non-dimensional mean axial velocities and streamlines at λ=0.4 for tip clearance at =0.8%0.354 span).(( a) h = 0.1% span; (b) h = 0.3% span; (c) h =0.6% span; and (d) h =0.8% span). varying tip clearance at φ = 0.354 ((a) h = 0.1% span; (b) h = 0.3% span; (c) h =0.6% span; and (d) h To study distributions=0.8% span).To study distributions of the turbulent of the turbulent entropy entropy generation generation rate rate for a for varying a varying tip gap, we tip selected gap, we selected the middle surface colored green along the blade chord direction at the tip for analysis as shown in the middle surfaceTo study colored distributions green of along the turbulent the blade entropy chord generation direction rate for a atvarying the tiptip gap, for we analysis selected as shown in Figure 19. We here used 𝑆 (reflecting the turbulent entropy generation rate) on the selected plane the middle surface colored green along the blade chord direction at the tip for analysis as shown in Figure 19. We herein usedanalyzingSDM the distributions(reﬂecting of the turbulent turbulent dissipation entropy in varying generation tip gaps at the rate) same onflow the rate. selected plane in Figure 19. We here used 𝑆 (reflecting the turbulent entropy generation rate) on the selected plane analyzing thein distributionsanalyzing the distributions of turbulent of turbulent dissipation dissipation in in varying varying tip tip gaps gaps at the at same the flow same rate. ﬂow rate.

Figure 19. View of the selected plane along the blade chord direction at the tip.

Figure 20 displays distributions of 𝑆 (reflecting the turbulent entropy generation rate) on the Figureselected 19. FigureplaneView with 19. of View a the varying of selected the selectedtip gap plane fromplane th alonge meridian the the blade bladeview chord atchord thedirection same direction atflow the rate tip. at(φ =the 0.354 tip.). In the case of the tip clearance of 0.1% span presented in Figure 20a, the distribution of the turbulent entropy generation rate on the selected plane was uniform. For other tip clearances of 0.3% span, Figure 20 displaysFigure 20 displays distributions distributions of S of 𝑆(reﬂecting (reflecting the the turbulent turbulent entropy entropy generation generation rate) on the rate) on the selected plane with a varying tip gap DMfrom the meridian view at the same flow rate (φ = 0.354). In selected planethe with case of a the varying tip clearance tip gap of 0.1% from span the presente meridiand in Figure view 20a, at the distribution same ﬂow of ratethe turbulent (ϕ = 0.354 ). In the case of the tipentropy clearance generation of 0.1% rate on span the selected presented plane in was Figure uniform. 20 a,For the other distribution tip clearances ofof the0.3% turbulentspan, entropy generation rate on the selected plane was uniform. For other tip clearances of 0.3% span, 0.6% span, and 0.8% span presented in Figure 20b,c,d, the turbulent entropy generation rate decreased from the blade leading edge (LE) to the trailing edge (TE). The large values of turbulent dissipation located near the leading edge of the blade tip side, originating from the leading edge and extending in the gap. The high entropy generation in these areas was due to the tip separation vortex attached to the blade tip side. Additionally, the area of high entropy generation on the selected plane in the tip increased with the tip clearance growing from 0.3% span to 0.8% span. This is because the area of the tip separation vortex occurred more and extended further for a larger tip clearance. In general, turbulent dissipation at the tip decreased from the blade leading edge to trailing edge, and the regions of high dissipation were distributed near the leading edge of the blade tip side. With the tip clearance increasing from Energies 2018, 11, x FOR PEER REVIEW 22 of 25

0.6% span, and 0.8% span presented in Figure 20b,c,d, the turbulent entropy generation rate decreased from the blade leading edge (LE) to the trailing edge (TE). The large values of turbulent dissipation located near the leading edge of the blade tip side, originating from the leading edge and extending in the gap. The high entropy generation in these areas was due to the tip separation vortex attached to the blade tip side. Additionally, the area of high entropy generation on the selected plane Energies 2019, 12in, 4162 the tip increased with the tip clearance growing from 0.3% span to 0.8% span. This is because the 19 of 22 area of the tip separation vortex occurred more and extended further for a larger tip clearance. In general, turbulent dissipation at the tip decreased from the blade leading edge to trailing edge, and the regions of high dissipation were distributed near the leading edge of the blade tip side. With the 0.3% span to 0.8% span, the areas of high turbulent dissipation increase and extend further along the tip clearance increasing from 0.3% span to 0.8% span, the areas of high turbulent dissipation increase blade chord direction.and extend further along the blade chord direction.

(a)

(b)

(c)

(d)

Figure 20. DistributionFigure 20. Distribution of SDM (reflecting of 𝑆 (reflecting theturbulent the turbulent entropy entropy generategenerate rate) rate) on the on selected the selected plane plane along the bladealong chord the blade direction chord direction in a varying in a varying tip gap tip gap at ϕ at =φ =0.354 0.354(( ((aa)) h == 0.1%0.1% span; span; (b) h ( b= )0.3% h = span;0.3% span; (c) h = 0.6% span; and (d) h = 0.8% span). (c) h = 0.6% span; and (d) h = 0.8% span). 5. Conclusions 5. Conclusions The effects of varying discharge and tip clearance on mechanical energy dissipation of an axial- The effectsflow of pump varying were discharge investigated and using tip the clearance entropy ongeneration mechanical method energy in numerical dissipation simulations. of an The axial-flow pump were investigatedfollowing conclusions using the were entropy drawn by generation comparing methodthe overall in entropy numerical generation simulations. rates for different The following components and analyzing the distributions of the turbulent entropy generation rate under varying conclusions were drawn by comparing the overall entropy generation rates for different components conditions. and analyzing the distributions of the turbulent entropy generation rate under varying conditions.

(1) The calculation results of an analysis of the overall entropy generation rate of different components of an axial-flow pump show that the mechanical energy dissipation in the impeller had a strong relation with the hump characteristic. Additionally, the impeller and diffuser were the main domains of high mechanical energy dissipation in the axial-flow pump under varying conditions, with the components respectively accounting for 35.32%–55.51% and 32.61%–20.42% of mechanical energy dissipation of the entire flow passage. Furthermore, the mechanical energy dissipation of the impeller, mostly being turbulent dissipation, became more important with the discharge decreasing. The discharge had greater effects on turbulent dissipation in the impeller when the axial-flow pump entered the hump region. (2) When the pump entered the hump region, the distributions of turbulent dissipation near the shroud became disordered and expanded towards the impeller’s inlet side. Unstable flows, like flow separation and vortices, near the blade suction sides led to the distribution of high turbulent dissipation in the impeller and the hump characteristic. Regions of high turbulent dissipation in the diffuser locate Energies 2019, 12, 4162 20 of 22 at the leading edges of guide vanes and in the diffuser passage. The distribution of high turbulent dissipation in the diffuser expanded to be more uniform as the discharge declined. (3) A tip clearance larger than 0.3% span affected turbulent dissipation of the impeller more and finally affected the pump performance. When the tip clearance grew from 0.1% span to 0.8% span for the same discharge (ϕ = 0.354), areas of high turbulent dissipation grew obviously for span SP0.98 with the tip clearance increasing. Therefore, an increase in the tip clearance mainly increased turbulent dissipation near the shroud of the impeller. (4) As for the tip clearance of 0.1% span, the mainstream hardly mixed with the leakage flow, and distributions of the axial velocity and turbulent entropy generation rate at λ = 0.4 in the gap under the discharge (ϕ = 0.354) were thus uniform. Denton’s leakage mixing model assumes that the mainstream and leakage flow mix at the tip and is thus not suitable for a very small tip clearance of 0.1% span. For other tip clearances of 0.3% span, 0.6% span, and 0.8% span, the tip flow patterns in the gap were similar to the pattern in Figure 31a of Denton’s paper [28]. With the tip clearance increasing from 0.3% span to 0.8% span, the area of the tip separation vortex at λ = 0.4 attached more on the blade tip surface, and the core of the tip leakage vortex moved closer to the blade suction side. The turbulent dissipation at the tip decreased from the blade leading edge to trailing edge, and regions of high dissipation distributed near the leading edge of the blade tip side. The areas of high turbulent dissipation increased and extended further along the blade chord direction with an increasing tip clearance.

Author Contributions: Z.Q. designed the experiments and funded the research work; S.S. performed the experiments and conducted the simulations; S.S. analyzed the data and wrote the paper; B.J. reviewed the research results and edited the paper. Acknowledgments: This research was funded by National Natural Science Foundation of China (Grant Nos.51779186) and Natural Science Foundation of Hubei Province (2017CFA048 and 2018CFA010). Conﬂicts of Interest: The authors declare that they have no conﬂict of interest.

Nomenclature

Q mass flow rate (kg/s) g gravitational acceleration (m/s2) ZR diameter of the impeller (0.3m) H head (m) h tip clearance width (% span) ρ water density (kg/m3) T temperature (K) k(TKE) turbulence kinetic energy Q gH ϕ flow coefficient : ϕ = head coe f f icient : ψ = 2 ( Ω )D3 ψ Ω 2 2π ( 2π ) D gQH η ( ) η → heat flux density vector pump efficiency % : = 3600MΩ q u,v,w velocity component along x,y,z direction (m/s) s specific entropy Φ, ΦΘ entropy generation term Ω rotor angular velocity (rad/s) U blade tip speed : U = Ω Z /2= 22.80 (m/s) T inlet temperature (298K) tip tip ∗ R re f () mean component ()’ fluctuating component . SD overall entropy generation rate for a domain(W/K) . S overall entropy generation rate induced by time-averaged movement for a domain(W/K) . D S overall entropy rate caused by velocity fluctuations for a domain(W/K) . D0 00 3 1 S entropy generation rate induced by time-averaged movement (Wm− K− ) . D 00 3 1 SD0 entropy generation rate caused by velocity fluctuation (Wm− K− )

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