CN105550404A - Method for analyzing hydraulic loss of centrifugal pump based on entropy theory - Google Patents

Method for analyzing hydraulic loss of centrifugal pump based on entropy theory Download PDF

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CN105550404A
CN105550404A CN201510890660.6A CN201510890660A CN105550404A CN 105550404 A CN105550404 A CN 105550404A CN 201510890660 A CN201510890660 A CN 201510890660A CN 105550404 A CN105550404 A CN 105550404A
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裴吉
尹庭赟
袁小露
袁寿其
骆寅
司乔瑞
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Jiangsu University
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Abstract

The invention relates to a method for analyzing the hydraulic loss of a centrifugal pump based on an entropy theory. The method comprises the following steps of A, using three-dimensional modeling software for modeling for the computational domain of the centrifugal pump; B, introducing the model of the computational domain obtained in the step A into analysis software for performing mesh dividing; C, performing constant incompressible full flow passage flow field viscous numerical calculation on the computational domain after mesh dividing; D, comparing the lift and efficiency in the numerical calculation result of the step C with the lift and efficiency of the centrifugal pump under design conditions, thus obtaining the mesh dividing with the optimal mesh number; E, based on the mesh dividing with the optimal mesh number obtained in the step D, compiling an entropy generation formula into the post processor of the analysis software by using a programming language, thus obtaining data after entropy analysis; and F, introducing the data after entropy analysis obtained in the step E into data processing software for drawing, and thereby obtaining a corresponding entropy analysis result.

Description

A kind of method based on the hydraulic loss of entropy theory analysis centrifugal pump
Technical field
The present invention relates to the technical field that Flow in Fluid Machinery loss mechanism is relevant, refer more particularly to a kind of method based on the hydraulic loss of entropy theory analysis centrifugal pump.
Background technology
Centrifugal pump is widely used in commercial production every field as a kind of common rotating machinery, because its power consumption total amount is huge, studies its energy response and to realize energy-saving run significant.Literature survey shows mainly hydrodynamically to analyze the research of centrifugal pump energy response, based on experimental test and method for numerical simulation.Numerical simulation is technology maturation in Flow Field Calculation, and comparatively experimental test can provide more detailed internal flow information and extensively be approved.
The size only occurred from hydraulic loss in hydrokinetic angle analysis fluid machinery and position directly perceived not, need to introduce a kind of analytical approach and carry out hydraulic loss in convection cell machinery a situation arises to manifest more intuitively.Inspire in entropy analytical approach successful Application in flowing heat transfer equipment, the present invention uses the hydraulic loss of entropy analysis theories convection cell machine internal to analyze.
For centrifugal water pump, because the viscosity of actuating medium and the existence of eddy stress make the irreversible inside energy of mechanical energy in operational process transform, bring irreversible loss.From the angle analysis of the second law of thermodynamics, this kind of energy loss is the conversion of useful work to idle work form of energy, therefore local entropy product method can be used for the measurement of mechanical power loss and realizes the seizure of pump internal loss, it is quantity of state that entropy produces, can not move by baric flow for single-phase, the Entropy generation of unit mass can provide its transport equation Spurk:
ρ ( ∂ s ∂ t + u ∂ s ∂ x + v ∂ s ∂ y + w ∂ s ∂ z ) = d i v ( q ‾ T ) + φ T + φ θ T 2 - - - ( 1 )
In formula: ρ-density, [Kgm -3]; -heat flow density, [Wm 2]; φ-mechanical energy viscous dissipation, [Wm -3];
φ Θ-entropy produces item, [WKm -3]; U, v, w-x, y, z direction speed, [ms -1]; S-area, [m 2];
X, y, z-x, y, z coordinate, [m]; T-temperature, [K];
Last two of formula (1) the right represents entropy and produces in transport process the mechanism produced, occur as source item and always on the occasion of; Wherein Section 1 is that the entropy produced due to viscous dissipation produces, and Section 2 description is that the entropy that under limited temperature drop, diabatic process produces produces.Unit mass Entropy generation, is also specific entropy, is unique unknown quantity in this formula, is only the function of temperature and pressure.And single-phase can not baric flow move in temperature field and pressure field retrained by the Basic equation group that flows, i.e. continuity equation, the equation of momentum and energy equation.Therefore, the fundamental physical quantity calculated required for entropy product can first be calculated by conventional numeric analogy method, in theory the aftertreatment amount that entropy produces as numerical simulation is processed, entropy can be avoided to produce the directly loaded down with trivial details calculating of transport equation and indirectly obtain entropy product solution comparatively accurately.
Specific entropy is quantity of state is also instantaneous flow, first produces homogenizing process when transport equation carries out by homogenization process during class Reynolds to entropy before numerical solution.Entropy produces can be divided into two: average magnitude part and percent ripple part.
ρ ( ∂ s ‾ ∂ t + u ‾ ∂ s ‾ ∂ t + v ‾ ∂ s ‾ ∂ t + w ‾ ∂ s ‾ ∂ t ) = d i v ( q ‾ T ) - ‾ ρ ( ∂ u ′ s ′ ‾ ∂ x + ∂ v ′ s ′ ‾ ∂ y + ∂ w ′ s ′ ‾ ∂ z ) + φ ‾ T + φ θ ‾ T 2 - - - ( 2 )
In formula (2) be viscous dissipation and the time homogenizing entropy that produces produces, average item and pulsation item can be divided:
φ ‾ T = S p r o . D ‾ + S p r o , D ′ - - - ( 3 )
S p r o , D ‾ = μ T { 2 [ ( ∂ u ‾ ∂ x ) 2 + ( ∂ v ‾ ∂ y ) 2 + ( ∂ w ‾ ∂ z ) 2 ] + ( ∂ u ‾ ∂ y + ∂ v ‾ ∂ x ) 2 + ( ∂ u ‾ ∂ z + ∂ w ‾ ∂ x ) 2 + ( ∂ v ‾ ∂ z + ∂ w ‾ ∂ y ) 2 } - - - ( 4 )
S p r o , D ′ = μ T { 2 [ ( ∂ u ′ ∂ x ) 2 ‾ + ( ∂ v ′ ∂ y ) 2 ‾ + ( ∂ w ′ ∂ z ) 2 ‾ ] + ( ∂ u ′ ∂ y + ∂ v ′ ∂ x ) 2 ‾ + ( ∂ u ′ ∂ z + ∂ w ′ ∂ x ) 2 ‾ + ( ∂ v ′ ∂ z + ∂ w ′ ∂ y ) 2 ‾ } - - - ( 5 )
In formula (2) be that limited temperature drop drives diabatic process and the time homogenizing entropy that produces produces, average item and pulsation item can be divided:
φ θ ‾ T 2 = S p r o , C ‾ + S p r o , C ′ - - - ( 6 )
S p r o , C ‾ = λ T ‾ 2 [ ( ∂ T ‾ ∂ x ) 2 + ( ∂ T ‾ ∂ y ) 2 + ( ∂ T ‾ ∂ z ) 2 ] - - - ( 7 )
S p r o , C ′ = λ T ‾ 2 [ ( ∂ T ′ ∂ x ) 2 ‾ + ( ∂ T ′ ∂ y ) 2 ‾ + ( ∂ T ′ ∂ z ) 2 ‾ ] - - - ( 8 )
Shi Junhua result has occurred that four kinds of entropys produce form, and two mean entropy product items and two entropys of pulsing produce item, are referred to as local entropy productive rate item. the entropy product that directly dissipates, S pro, D'that turbulent dissipation entropy produces, the contribution that average temperature gradient is produced entropy, S pro, C'it is the contribution that fluctuating temperature gradient is produced entropy.Wherein mean entropy produce item can direct solution, and entropy of pulsing produces item and cannot directly calculate.Kock and Herwig research thinks that pulsation entropy product can exist inner link with turbulence model, with tubulence energy dissipative shock wave and thermograde relevant, can obtain:
S p r o , D ′ = ρ ϵ T ‾ - - - ( 9 )
S p r o , C ′ = α t α S p r o , C ‾ - - - ( 10 )
S pro, D'-turbulent flow Entropy generation, [Wm -3k -1]; S pro, C'-fluctuating temperature Entropy generation, [Wm -3k -1]
The entropy produced by average temperature gradient and fluctuating temperature gradient according to the research of Duan Lu produces and can be summed up as one:
S p r o , C = λ e f f T ‾ 2 [ ( ∂ T ‾ ∂ x ) 2 + ( ∂ T ‾ ∂ y ) 2 + ( ∂ T ‾ ∂ z ) 2 ] - - - ( 11 )
Wherein, λ eff=λ+λ t,
Summarize known, local entropy produces that item can pass through formula (4), (9), (11) calculate, and the total Entropy generation in such computational fields can be obtained by the volume integral of local entropy productive rate to computational fields:
ΔS p r o , D ‾ = ∫ V S p r o , D ‾ d V - - - ( 12 )
But Kock result of study shows, all there is stronger wall effect in the Entropy generation of four kinds of forms, and time equal item comparatively obvious.Conventional numeric analogy method can only be separated relatively accurately providing away from wall area, region more close to wall causes local velocity's distribution and Temperature Distribution sharply to change due to the existence of turbulent boundary layer and thermal boundary layer, causes Entropy generation to calculate and occurs peak value and occur departing from true result of separating.So the Entropy generation calculating needs near wall region are given and special concern, otherwise unacceptable mistake will be there is.Zhang Xiang gives near wall entropy and produces the formula calculated, as shown in (15), be called wall Entropy generation, wall Entropy generation is carried out to wall area total Entropy generation that integration just can obtain wall area, and the integral domain therefore in formula (12) will be the calculating core space not comprising wall area.
ΔS p r o , W = ∫ s τ w → · ν p → T ‾ d S - - - ( 15 )
In formula: Δ S pro, W-wall entropy produces, [WK -1]
Therefore, the total entropy in whole system computational fields produces and can calculate, as formula (16); By entropy produce bring irreversible loss is expressed as formula (17).
ΔS p r o = ΔS p r o , D ‾ + ΔS p r o , D ′ + ΔS p r o , C + ΔS p r o , W - - - ( 16 )
I p r o = Σ i I i = Σ i ( T r ΔS p r o , i ) , i = D ‾ , D ′ , C , W - - - ( 17 )
In formula: Δ S pro-volume integral entropy produces, [WK -1]; -directly entropy product, [WK -1];
Δ S pro, D '-turbulent flow entropy produces, [WK -1]; Δ S pro, C-thermal entropy produces, [WK -1];
Δ S pro, W-wall entropy produces, [WK -1]; I pro- damage, [W];
T r-reference temperature, [K];
Summary of the invention
Entropy is produced the theoretical hydraulic loss introducing centrifugal pump and calculates by base of the present invention, and the method is easy to operation, and entropy analysis result well can reflect centrifugal pump interior flow field hydraulic loss situation, the reference that the structural design for follow-up centrifugal pump can provide.
Based on a method for entropy theory analysis centrifugal pump hydraulic loss, comprise the steps:
Steps A) moulding is carried out to the computational fields applying three-dimensional modeling software of centrifugal pump, this computational fields is divided into four parts: induction pipe, impeller, spiral case and outlet;
Step B) moulding of computational fields that steps A obtained imports in ANSYS and carries out stress and strain model: adopt the ICEM in ANSYS to carry out structured grid division to induction pipe in computational fields and outlet; Use Meshing to carry out unstructured grid division to impeller in computational fields and spiral case, be then encrypted the grid of impeller wall and the grid of spiral case wall, the stress and strain model number of this computational fields is no less than 2 groups;
Step C) computational fields that divides step B CEInet lattice carries out permanently full fluid flow on channel tack value can not being pressed to calculate;
Step D) lift under the lift in the result of step C numerical evaluation and efficiency and Centrifugal Pump Design operating mode and efficiency are compared, with lift with efficiency change is not obvious and grid amount is moderate in evaluation criterion, obtain the stress and strain model of Bestgrid number;
Step e) based on the stress and strain model of Bestgrid number that obtains by step D, by coding language, formula 16,17 is compiled in the post processor of analysis software, entropy analyzing and processing is carried out to the interior flow field of computational fields, obtains the data after entropy analyzing and processing;
ΔS p r o = ΔS p r o , D ‾ + ΔS p r o , D ′ + ΔS p r o , C + ΔS p r o , W - - - ( 16 )
I p r o = Σ i I i = Σ i ( T r ΔS p r o , i ) - - - ( 17 )
In formula: Δ S pro-volume integral entropy produces, [WK -1]; -directly entropy product, [WK -1];
Δ S pro, D '-turbulent flow entropy produces, [WK -1]; Δ S pro, C-thermal entropy produces, [WK -1];
Δ spro, W-wall entropy produces, [WK -1]; I pro- damage, [W];
T r-reference temperature, [K];
Step F) draw in the entropy obtained in step e analysis data importing to data processing software, thus obtain corresponding entropy analysis result.
Further, in steps A, 3D sculpting software is Creo or Proe.
Further, the Fluent in ANSYS is adopted to carry out permanently full fluid flow on channel tack value can not being pressed to calculate comprise to step C computational fields: to select Reynolds turbulence model and Standard law of wall to computational fields, MRF model is adopted to be coupled to induction pipe, impeller, spiral case and outlet, according to SIMPLEC algorithm, couple solution is carried out to the pressure in the flow field of computational fields and speed, energy equation is added to computational fields, PRESTO! Form carries out discrete to the pressure term of energy equation; Upstreame scheme carries out discrete to the convective term of energy equation; Using Second-Order Central Difference form to energy equation all the other every carry out discrete; Then to the import given speed boundary condition of induction pipe, setting tubulence energy and hydraulic diameter; The outlet of outlet adopts free discharge boundary condition; Solid wall surface adopts without slip boundary condition; The outlet of water inlet pipe and the import of impeller, the outlet of impeller and the import of spiral case adopt Interface, setting numerical value convergence residual values; Finally carry out computing.
Further, in step B, stress and strain model is 120W, 150W, 177W, 200W grid amount.
Further, in step D, the stress and strain model of Bestgrid number is 177W.
Further, tubulence energy and hydraulic diameter are set as 3% and 150mm respectively.
Further, its character numerical value convergence residual error is set as 1e-4.
Beneficial effect of the present invention:
1. entropy is produced the theoretical hydraulic loss calculating introducing centrifugal pump by the present invention, and the method is easy to operation, and entropy analysis result well can reflect centrifugal pump interior flow field hydraulic loss situation, for the structural design of follow-up centrifugal pump can provide reference.
2. pair stress and strain model number is no less than two groups, can after numerical evaluation, carry out the analysis of grid independence, choose the grid number that Bestgrid divides, carry out aftertreatment, it is moderate that the stress and strain model of Bestgrid number is that 177W mono-aspect meets lift and efficiency and grid amount, on the other hand, save finishing time, raise the efficiency.
3. in the present invention, by adding energy equation, be applied in discrete processes by energy equation, thus be the importing that follow-up entropy produces equation, the analysis that entropy produces data result provides the foundation.
4. using the calculating of Entropy generation item as an aftertreatment pattern, the transport equation avoiding complicated loaded down with trivial details entropy calculates, and brings great simplification.
Accompanying drawing explanation
Fig. 1 is centrifugal pump computational fields schematic diagram.
Fig. 2 is centrifugal pump stress and strain model figure.
Fig. 3 is the scale map being lost in total losses of each computational fields of centrifugal pump.
Fig. 4 is that entropy dissimilar under each operating mode of centrifugal pump produces composition diagram.
Reference numeral: 1-induction pipe; 2-impeller; 3-spiral case; 4-outlet.
Embodiment
Below in conjunction with accompanying drawing, the present invention will be further described:
Steps A) apply Creo 3D sculpting software moulding is carried out to the computational fields of centrifugal pump.As shown in Figure 1, computational fields is divided into four parts: induction pipe 1, impeller 2, spiral case 3 and outlet 4;
Step B) computational fields that steps A obtained imports in ANSYS software and carries out stress and strain model, as shown in Figure 2, adopts the ICEM in ANSYS to carry out structured grid division to induction pipe in computational fields 1 and outlet 4; Meshing is used to carry out unstructured grid division to impeller in computational fields 2 and spiral case 3; Simultaneously for meeting Standard law of wall to Y +requirement, be encrypted the grid of impeller 2 wall and the grid of spiral case 3 wall, stress and strain model is 120W, 150W, 177W, 200W grid amount;
Step C) adopt the Fluent in ANSYS to step C computational fields carry out permanent can not press full fluid flow on channel tack value calculate comprise: Reynolds turbulence model and Standard law of wall are selected to computational fields, MRF model is adopted to be coupled to induction pipe 1, impeller 2, spiral case 3 and outlet 4, according to SIMPLEC algorithm, couple solution is carried out to the pressure in the flow field of computational fields and speed, energy equation is added to computational fields, PRESTO! Form carries out discrete to the pressure term of energy equation; Upstreame scheme carries out discrete to the convective term of energy equation; Using Second-Order Central Difference form to energy equation all the other every carry out discrete; Then to the import given speed boundary condition of induction pipe 1, tubulence energy and hydraulic diameter are set as 3% and 150mm respectively; The outlet of outlet 4 adopts free discharge boundary condition; Solid wall surface adopts without slip boundary condition; The outlet of water inlet pipe 1 and the import of impeller 2, the outlet of impeller 2 and the import of spiral case 4 adopt Interface, and numerical value convergence residual error is set as 1e-4, finally carries out computing;
Step D) lift under the lift in the result of step C numerical evaluation and efficiency and Centrifugal Pump Design operating mode and efficiency are compared, with lift with efficiency change is not obvious and grid amount is moderate in evaluation criterion, the stress and strain model obtaining Bestgrid number is the grid amount of 177W;
Step e) based on the grid amount of 177W that obtains by step D, by the UDF coding language in ANSYS, formula 16,17 is compiled in the post processor of analysis software, entropy analyzing and processing is carried out to the interior flow field of computational fields, obtains the data after entropy analyzing and processing;
ΔS p r o = ΔS p r o , D ‾ + ΔS p r o , D ′ + ΔS p r o , C + ΔS p r o , W - - - ( 16 )
I p r o = Σ i I i = Σ i ( T r ΔS p r o , i ) - - - ( 17 )
In formula: Δ S pro-volume integral entropy produces, [WK -1]; -directly entropy product, [WK -1];
Δ S pro, D'-turbulent flow entropy produces, [WK -1]; Δ S pro, C-thermal entropy produces, [WK -1];
Δ S pro, W-wall entropy produces, [WK -1]; I pro- damage, [W];
T r-reference temperature, [K];
Step F) entropy obtained in step e analyzed in data importing to data processing software and draws, can obtain under different flow operating mode, in centrifugal pump computational fields each several part entropy analysis result and centrifugal pump, dissimilar entropy produces composition.
Under Fig. 3 represents different flow operating mode, each parts internal entropy of induction pipe 1, impeller 2, spiral case 3 and outlet 4 produces the proportionate relationship be lost in the loss of pump total entropy product.It is less that entropy as can be seen from Figure 3 in different flow operating mode lower inlet pipe 1 and outlet 4 produces loss ratio, and the entropy in impeller 2 and spiral case 3 produces loss large percentage, be respectively 30% and about 60%, illustrate that impeller 2 and spiral case 3 are main places that pump irreversible factor occurs, should pay close attention during design more.Along with the increase of flow, induction pipe 1 and outlet 4 internal entropy produce to lose to be increased comparatively slowly, and the entropy of spiral case 3 produces loss to be increased to some extent, and the loss of the entropy in impeller 2 region product presents reduction trend.
Under Fig. 4 represents different flow operating mode, entropy dissimilar in centrifugal pump computational fields produces composition, and the entropy product ratio produced by thermograde is less and ignore.As can be seen from Figure 4, the entropy product proportion of the generation that directly dissipates is less, under different flow operating mode, only have 0.54% ~ 3.48%; Turbulent flow entropy produces and produces proportion comparatively greatly with wall viscosity entropy, and be respectively 47% ~ 55% and 44% ~ 52%, the main cause of irreversible hydraulic loss when therefore centrifugal pump runs is turbulent dissipation and wall viscous friction.Design discharge 200m 3under/h, turbulent flow entropy product produces proportion with wall entropy and is respectively 47.91% and 48.61%; Along with the increase of flow, the variation tendency of turbulent flow entropy output now first less rear increase, wall entropy produces then slowly to be increased always, but the rapid drawdown occurred under maximum flow.
Note: concrete implementation model is the centrifugal pump of IS series, and model is IS150-125-250, its design discharge 200m 3/ h, lift 20m, rotating speed 1450r/min, the number of blade 6, and under design conditions, its hydraulic efficiency can reach 92%.
Described embodiment is preferred embodiment of the present invention; but the present invention is not limited to above-mentioned embodiment; when not deviating from flesh and blood of the present invention, any apparent improvement that those skilled in the art can make, replacement or modification all belong to protection scope of the present invention.

Claims (7)

1., based on a method for entropy theory analysis centrifugal pump hydraulic loss, it is characterized in that comprising the steps:
Steps A) moulding is carried out to the computational fields applying three-dimensional modeling software of centrifugal pump, this computational fields is divided into four parts: induction pipe (1), impeller (2), spiral case (3) and outlet (4);
Step B) moulding of computational fields that steps A obtained imports in ANSYS and carries out stress and strain model: adopt the ICEM in ANSYS to carry out structured grid division to induction pipe (1) and outlet (4); Meshing is used to carry out unstructured grid division to impeller in computational fields (2) and spiral case (3), then be encrypted the grid of impeller (2) wall and the grid of spiral case (3) wall, the stress and strain model number of this computational fields is no less than 2 groups;
Step C) computational fields that divides step B CEInet lattice carries out permanently full fluid flow on channel tack value can not being pressed to calculate;
Step D) lift under the lift in the result of step C numerical evaluation and efficiency and Centrifugal Pump Design operating mode and efficiency are compared, with lift with efficiency change is not obvious and grid amount is moderate in evaluation criterion, obtain the stress and strain model of standard compliant grid number;
Step e) based on the stress and strain model of standard compliant grid number that obtains by step D, by coding language, entropy being produced formula compiles in the post processor of analysis software, entropy analyzing and processing is carried out to the interior flow field of computational fields, obtains the data after entropy analyzing and processing;
ΔS p r o = ΔS p r o , D ‾ + ΔS p r o , D ′ + ΔS p r o , C + ΔS p r o , W
I p r o = Σ i I i = Σ i ( T r ΔS p r o , i )
In formula: Δ S pro-volume integral entropy produces, [WK -1]; -directly entropy product, [WK -1];
Δ S pro, D '-turbulent flow entropy produces, [WK -1]; Δ S pro, C-thermal entropy produces, [WK -1];
Δ S pro, W-wall entropy produces, [WK -1]; I pro- damage, [W];
T r-reference temperature, [K]; i = D ‾ , D ′ , C , W .
Step F) draw in the entropy obtained in step e analysis data importing to data processing software, thus obtain corresponding entropy analysis result.
2. a kind of method based on the hydraulic loss of entropy theory analysis centrifugal pump according to claim 1, it is characterized in that, in steps A, 3D sculpting software is Creo or Proe.
3. a kind of method based on the hydraulic loss of entropy theory analysis centrifugal pump according to claim 1, it is characterized in that, the Fluent in ANSYS is adopted to carry out permanently full fluid flow on channel tack value can not being pressed to calculate comprise to step C computational fields: to select Reynolds turbulence model and Standard law of wall to computational fields, adopt MRF model to induction pipe (1), impeller (2), spiral case (3) and outlet (4) are coupled, according to SIMPLEC algorithm, couple solution is carried out to the pressure in the flow field of computational fields and speed, energy equation is added to computational fields, form carries out discrete to the pressure term of energy equation, upstreame scheme carries out discrete to the convective term of energy equation, Using Second-Order Central Difference form to energy equation all the other every carry out discrete, then to the import given speed boundary condition of induction pipe (1), setting tubulence energy and hydraulic diameter, the outlet of outlet (4) adopts free discharge boundary condition, solid wall surface adopts without slip boundary condition, the outlet of water inlet pipe (1) and the import of impeller (2), the outlet of impeller (2) and the import of spiral case (4) adopt Interface boundary condition, setting numerical value convergence residual values, finally carry out computing.
4. a kind of method based on the hydraulic loss of entropy theory analysis centrifugal pump according to claim 1, is characterized in that, in step B, stress and strain model is 120W, 150W, 177W, 200W grid amount.
5. a kind of method based on the hydraulic loss of entropy theory analysis centrifugal pump according to claim 1, is characterized in that, in step D, the stress and strain model of the standard compliant grid number obtained is 177W.
6. a kind of method based on the hydraulic loss of entropy theory analysis centrifugal pump according to claim 3, it is characterized in that, tubulence energy and hydraulic diameter are set as 3% and 150mm respectively.
7. a kind of method based on the hydraulic loss of entropy theory analysis centrifugal pump according to claim 3, is characterized in that, numerical value convergence residual error is set as 1e-4.
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