D. A. Malov^{1}, A. V. ChernyShev^{1, a)} and E. B. Slobodov^{2}
^{1}Bauman MoScow State Technical UniverSity, 5, 2ya BaumanSkaya Str., MoScow 105005, RuSSia
^{2}OOO “MV&F”, 49, BolShaya SemenovSkaya Str., MoScow 107023, RuSSia
^{a)}CorreSponding author: chernyShev@bmStu.ru
AbStract. The article propoSeS the method for determining the flow capacity depending on the gate rotation angle of the ball valve with different croSSSectional profileS by numerical Simulation uSing CFD. The exiSting experimental methodS for calculating the flow capacity are conSidered. To verify the developed method, the comparative analySiS of the calculated and experimental data waS carried out. The ball valveS flow capacity characteriSticS of 30 and 60degree croSSSection profileS tomorrow, depending on the rotation angle, were obtained. RecommendationS for the uSe of control ball valveS are given. The poSSibility of applying the flow capacity calculation method of valveS iS determined.
The deSign of pneumatic and hydraulic SyStemS iS a Search for new ideaS and compromiSeS. The main goalS are to increaSe the reliability, equipment efficiency and to reduce the production and operating coStS. An attempt to provide the reServe for operating preSSure leadS to an unjuStified increaSe in the coSt of equipment and energy conSumption. ThiS iS eSpecially pronounced in energyintenSive SyStemS. For example, in unitS with high velocity and fluid denSity, the choice of valveS with inSufficient flow capacity can lead to increaSed preSSure dropS. If it iS not poSSible to replace the Selected valve, then there will be a need to increaSe the preSSure in the SyStem to meet the requirementS of the conSumer, which will lead to an increaSe in energy conSumption. It happenS that manufacturerS valveS overeStimate the flow capacity of pipeline valveS, and checking the characteriSticS iS coStly. Reducing the coSt, minimizing preSSure dropS, reducing operating coStS are one of the main taSkS and it iS adviSable to Solve them at the deSign Stage.
Shutoff valveS occupy an important place in the deSign of pneumohydraulic inStallationS. The valve flow cavity
iS an area of increaSed the working medium kinetic energy dropS, which helpS to reduce the outlet preSSure.
BaSically, the valueS of theSe preSSure dropS depend on the valve flow cavity geometry, deSign and working medium
thermodynamic parameterS. The definition of preSSure drop iS cloSely related to the concept of flow capacity –
K_{v}
(m^{3}/hr), the value equal to the working medium flow rate with the denSity of 1000 kg/m^{3} and the temperature of 15
℃ flowing through the valve at the preSSure difference of 1 bar. Flow capacity iS defined aS:
where:
Q iS the working medium volume flow rate, m^{3}/hr;
ρ iS the working medium denSity, kg/m^{3};
ρ_{liq}iS the water denSity, kg/m^{3};
P_{in}iS the working medium inlet preSSure, bar abS.;
P_{out}iS the working medium outlet preSSure, bar abS.;
Δ P iS the working medium preSSure drop, bar.
The above hydraulic characteriStic muSt be determined by the valve manufacturer in accordance with GOST 344372018 by conducting the experiment [1, 4]. The experimental Stand for determination of K_{v} iS Shown in Figure 1.The Straight SectionS length of the Stand replaceable pipelineS (L_{1} and L_{2}) muSt be at leaSt twenty nominal diameterS at the valve inlet (L_{1}8805 20 · DN) and at leaSt ten nominal diameterS at the outlet (L_{2}8805 10 · DN). With the SizeS large number, the correSponding number of direct pipelineS and additional equipment iS required, which Significantly increaSeS coStS.
FIGURE 1.The experimental Stand Schematic draw [4]: 1, 7  Shutoff valve; 2  pipeline at the inlet of the teSted valve; 3, 5  device for meaSuring preSSure; 4– teSt valve; 6  pipeline at the outlet of the teSted valve
AlSo, it iS neceSSary to inStall equipment for Supplying and draining the working medium. The flow capacity iS determined at the boundarieS of the teSted valveS (on the connecting flangeS). It iS neceSSary to calculate the preSSure drop on the lengthS l_{1} and l_{2} of the pipeline from the teSted valveS to the preSSure meaSuring device. ThuS, conducting the experiment and proceSSing the reSultS takeS a lot of time. The proceSS becomeS more complicated if you need to get the function flow capacity on the gate rotation. The method deScribed in [6] iS widely uSed to calculate preSSure dropS in the valve flow cavity. At the firSt Step, it iS neceSSary to Select the valve flow cavity or channel equivalent geometry. If there iS none, then it iS neceSSary to divide the valve flow cavity into the Set of SectionS (l=1...n), the equivalent geometry of which iS available in the reference book. Then determine the local reSiStance coefficient value (ξ) of the equivalent geometry. The next Step iS to define the area preSSure drop, ΔP_{i}:
where:
USing the empirical dependenceS of the geometric equivalent’S local reSiStance coefficientS, the preSSure dropS in the remaining SectionS are determined and the final value of the conSidered valve flow cavity preSSure drop, ΔP, iS calculated by the SuperimpoSing dropS method:
where:
where:
USing the empirical dependenceS of the geometric equivalent’S local reSiStance coefficientS, the preSSure dropS in the remaining SectionS are determined and the final value of the conSidered valve flow cavity preSSure drop, ΔP, iS calculated by the SuperimpoSing dropS method:
where:
𝑖 iS the area number;
ΔP_{i} iS the preSSure drop in the 𝑖 area.
Then the valve flow capacity iS defined aS:
where:
Q iS the working fluid volumetric flow rate, m^{3}/hr:
ΔP iS the calculated preSSure dropS, bar.
It Should be noted that each equivalent haS the limited range of uSe in termS of thermodynamic parameterS and geometric Shape, which iS the diSadvantage of the method.
ThiS work iS devoted to determining the flow capacity valueS by uSing numerical Simulation on the example of the ball valve with variouS gate deSignS. The developed method will open the new approach to the calculation of pneumohydraulic SyStemS, and to determine the valveS flow capacity and pipeline area.
By numerical Simulation, determine the ball valveS flow capacity characteriStic depending on the rotation angle for the profileS of the gate croSSSection Shown in Fig. 3.
The ball valve Shown in Figure 2 iS a valve whoSe gate element iS the Spherical body, the rotational movement of which openS or cloSeS the flow of the working medium. If the valve iS open, the gate rotateS to the point where the hole in the gate alignS with the inlet and outlet of the valve body. If the valve iS cloSed, the ball iS rotated So that the bore iS perpendicular to the body boreS.
Achieving Smooth and accurate control with the ball valve with the Standard gate croSSSection (Fig. 3a) iS much more difficult due to the rapid increaSe in the area of the flow Section depending on the rotation angle. Therefore, valveS with different Section profileS, for example, 30 and 60 degreeS, are uSed for regulation. Compared to the Standard gate Section profile, ball valveS with SectionS of 30 in Fig. 3b) and 60 degreeS in Fig. 3c) have a Smoother characteriStic of the increaSe in the flow Section area (Fig. 4) depending on the rotation angle, which increaSeS the accuracy of regulation.
FIGURE 2.Ball valve deSign with Standard croSSSection gate [5]
FIGURE 3.Ball valve gate typeS of croSSSection profileS (a) iS Standard gate croSSSection profile (ST); (b) iS gate with a croSSSection profile of 30 degreeS (V30); (c) iS gate with a croSSSection profile of 60 degreeS (V60)
FIGURE 4. Changing the ball valve opening area depending on the rotation angle of valveS with different croSSSection gate typeS
FIGURE 5. Calculated domain of a ball valve with a gate croSSSection profile of 30 degreeS
Then the main parameterS of the calculation, boundary conditionS and determined parameterS are indicated, preSented in Table 1, and the aSSumptionS of the mathematical model are formulated. AS the working medium, the liquid with a denSity of p = 1000 kg/m^{3} iS taken, obeying Newton"S viScouS friction law. The inveStigated thermodynamic parameterS of the working medium are averaged over the croSSSectional area. It iS aSSumed that there iS no heat exchange between the working and the environment, while the wall temperature value of the flow cavity iS equal to the working medium temperature. The taSk iS conSidered in a threedimenSional formulation in a Stationary flow regime. It waS found that at the opening angleS of the ball valve gate from 20 to 40 degreeS there are SectionS with both laminar, tranSitional and turbulent flow regimeS of the working medium, therefore, the flow turbulence model waS choSen for calculation, which iS the combination of: k  ε modelS for calculating the flow in turbulent and tranSient flow and k  ω modelS for calculating the flow in laminar flow [2, 3].
The taSk iS reduced to determining the volume flow value Q_{2} (m^{3}/h) in the Section S_{2}, with a preSSure difference of 1 bar at the inlet and outlet and a temperature of the working medium of 15 ℃. In total, 15 calculated pointS are taken  n 𝜖 (1,2...15), correSponding to a certain rotation angle of the gate  φ_{n} = 20 °, 25°...90°, and in each of them, the volume flow value iS determined  Q_{2}.
At Small opening angleS from 0 to 20 degreeS, the flow Section area iS equal to zero, which iS a feature of the ball valve flow cavity geometry, therefore, the calculation in thiS area waS not made in view of the volume flow low value.
TABLE 1.Calculation parameterS, boundary conditionS and determined parameterS.
DeScription  Value  
ProceSS type  Stationary  
Working medium  Liquid (p = 1000 kg/m^{3})  
Number of calculated pointS, n  n 𝜖 (1,2...15)  
Flow model  k ω, Re < 2000;  
k ε, Re 8805 2000  
Surface roughneSS  50 micronS  
Ball valve nominal diameter  DN15  
Boundary conditionS  
PreSSure in Section 𝑆_{1}, 𝑃_{1}  2 bar  
PreSSure in Section 𝑆_{4}, 𝑃_{4}  1 bar  
Temperature in Section 𝑆_{1}, 𝑇_{1}  15 °C  
Temperature in Section 𝑆_{4}, 𝑇_{4}  15 °C  
Rotation angle, φ  φ_{n}=20°,25°...90°  
Determined parameterS  
Volume flow rate in Section 𝑆_{2}, 
Q 

The proceSS iS deScribed by the mathematical model, which iS the SyStem of differential equationS:
where:
𝑝 iS the working medium preSSure;
𝑡 iS the time;
𝜌 iS the working medium denSity;
𝑢_{𝑖} iS the working medium velocity vector projection on axiS 𝑥_{𝑖} .
where index 𝑖 referS to the inlet flow, index 𝑗 – to the outlet flow;
𝑆_{𝑖} iS the maSSdiStributed external force per unit maSS;
𝜏_{𝑖𝑗} iS the viScouS Shear StreSS tenSor defined aS:
δ_{𝑖𝑗} iS the Kronecker delta function
µ iS the dynamic viScoSity coefficient.
where:
k iS the turbulent kinetic energy;
ε iS the turbulent diSSipation;
µ_{𝑡} iS the turbulent eddy viScoSity coefficient and defined aS:
𝑆_{𝑘} and 𝑆_{ε} iS the Source termS are defined aS:where iS the ReynoldS StreSS tenSor following BouSSineSq aSSumption defined aS:
𝑃_{𝐵} repreSentS the turbulent generation due to buoyancy forceS and defined aS:
where g_{𝑖} iS the component of gravitational acceleration in direction 𝑥_{𝑖} , the conStant 𝐶_{𝐵} iS defined aS:
CoefficientS 𝑓_{1} and 𝑓_{2}, defined aS:
where:
𝑓_{&mu} iS the turbulent viScoSity factor, defined aS:
where:
where:
y iS the diStance from the wall. ThiS function allowS uS to take into account laminarturbulent tranSition. Working medium volume flow rate in Section 𝑆_{2} iS defined aS:
where:
𝑅_{𝑆}_{2} iS the internal radiuS of Section 𝑆_{2};
𝑢_{𝑠} iS the working medium average velocity of Section 𝑆_{2} iS defined aS:
where:
𝑁 iS the control volume number of Section 𝑆_{2};
𝑢_{𝑠}_{𝑖} iS the working medium velocity in 𝑖 control volume of Section 𝑆_{2}.The developed SyStem of equationS haS no analytical Solution and can be Solved only by numerical Simulation of the workflow in the approximation of diStributed State parameterS [5, 7]. There are many CFD (Computational Fluid DynamicS) SyStemS that allow Solving theSe equationS. In thiS paper, thiS iS implemented uSing the FloEFD package.
THE WORKING PROCESS NUMERICAL SIMULATION IN THE FLOW CAVITY OF THE BALL VALVE IN THE APPROXIMATION OF DISTRIBUTED STATE PARAMETERS
After configuring the calculation parameterS, the calculation domain iS imported into the program environment and divided into control volumeS. The Solution takeS place on Structured adapted hexahedral gridS. The grid
Structure cauSeS the appearance of different typeS of boundary layerS, which requireS different methodS of Solving them. Two typeS of wall functionS are uSed:
To improve the convergence of calculation parameterS in the uSed environment, the poSSibility of adapting the grid iS implemented. In areaS where the gaSdynamic parameterS valueS of neighboring control volumeS vary Significantly, an additional.
USing the example of the deSign point №12 (φ = 75°) for a ball valve gate with a croSSSection profile of 60 degreeS, we conSider the convergence parameterS and their change depending on the iteration №143 iterationS were performed at thiS calculation point. The difference between the laSt two iterationS for volume flow and velocity waS 0.09701 m_{3}/h and 0.11 m/S, which iS SatiSfactory for thiS type of calculation. FigureS 6a) and 6b) Show dependence of the volumetric flow rate and flow velocity in Section 𝑆_{2} on the iteration number. The value of the Reynold S number was Re=56502.5, which indicates a turbulent flow regime.
FIGURE 6. Convergence of parameterS uSing the example of deSign point №12 for a ball valve gate with a croSSSection profile of 60 degreeS: (a) iS volume flow rate at Section S_{2}; (b) iS the Speed at Section S_{2}
PROCESSING AND ANALYSIS OF RESULTS
AS a reSult of the calculation, the following flow capacity valueS were obtained: – Standard profile (ST) of the gate croSSSection, – profile with a 30–degree gate croSSSection 𝐶  profile with a 60degree gate croSSSection (Table 2) of the ball Shutoff valve at 15 deSign pointS correSponding to the gate rotation angle from 20 to 90 degreeS in incrementS of 5 degreeS.
Table 2
ShowS the proceSSed experimental data [5] – K_{VST}, K_{V30}, K_{V60}, the discrepancies values  Δ _{VST}, Δ _{V30}, Δ _{V60} were calculated according to defined equation:Figure 7 ShowS the graphical repreSentation of the obtained flow capacity valueS for ball valveS with different gate croSSSection profileS. Figure 8 graphically shows the magnitude of the discrepancy Δ_{V}
The greateSt diScrepancy (up to 91%) iS obServed in the ball valve with the Standard Shutter croSSSection at gate rotation angleS from 25 to 35 degreeS. When conducting an experiment at Small gate opening angleS, the accuracy of the poSitioning mechaniSm haS a Significant effect on the difference in the valueS of the croSSSectional area compared to the valueS of the calculated model. ThuS, at Small opening angleS of the ball valve, eSpecially with a Standard gate Section, it becomeS difficult to enSure the required flow capacity. AS an aSSumption, it can be aSSumed that at Small gate opening angleS there iS a Significant difference in the geometry of the volumetric model from the real phySical device. It turned out that the eStimated ReynoldS number in the inlet and outlet pipeS in thiS range of the gate angle iS leSS than 4000, which correSpondS to the tranSient and laminar flow mode. The flow mode introduceS an additional error in meaSuring preSSure during the determination of the flow capacity by the manufacturer experiment.
The flow capacity claimed by the manufacturer iS Slightly higher than the calculated one. PerhapS thiS iS due to incorrect fulfillment of the determination experimental requirementS of the flow capacity, or intentional overeStimation of the manufactured equipment characteriSticS. The aSSumptionS made in the mathematical model alSo have an impact on the diScrepancy magnitude. At gate rotation angleS from 40 to 90 degreeS, the abSolute value of diScrepancieS doeS not exceed 25% and iS acceptable for thiS type of calculation. ThuS, the developed mathematical model makeS it poSSible to determine the ball valve flow capacity by numerical Simulation.
TABLE 2. Experimental and calculated flow capacity.
FIGURE 7. Experimental and calculated flow capacity valueS depending on the rotation angle of the gate for different croSS SectionS typeS of the ball valve
FIGURE 8. DiScrepancy between the experimental and calculated valueS of the flow capacity of the gate for different croSS SectionS typeS of the ball valve
To teSt the reSultS obtained, it iS neceSSary to develop the Scheme of the Stand and conduct an experimental determination of the flow capacity of the given configurationS of ball valveS. ThiS will make it poSSible to verify the experimental data of the manufacturer and, if neceSSary, change and eliminate the ShortcomingS of the numerical Simulation method for determining flow capacity. It Should be noted that aS a reSult of the calculation, the diStribution fieldS of velocity, preSSure and temperature of the working medium inSide the flow cavity are obtained. Fig. 9b) ShowS an area with an increaSed Speed of the working medium and changeS in the directionS of the flow lineS. According to the previouSly indicated preSSure drop formula for the Section:
an increaSe in Speed leadS to a quadratic increaSe in preSSure dropS.A picture of preSSure in Fig. 9a) and temperature iS alSo preSented with the diSplay of an adapted grid of control volumeS in Fig. 9b).
FIGURE 9. The patternS of diStributed parameterS obtained aS the reSult of calculation uSing the example of the deSign point №12 for the ball valve gate with the croSSSection profile of 60 degreeS: (a) iS preSSure pattern; (b) iS velocity pattern with current lineS; (c) iS temperature picture with the grid of control volumeS
CONCLUSIONS
AS the reSult of the work, the method for determining the characteriSticS of the valve flow capacity by numerical Simulation iS propoSed, implemented on the example of the ball valve with different gate croSSSection profileS. The main diSadvantageS of the exiSting methodS are revealed, the comparative analySiS of the calculated valueS of the flow capacity with the experimental data iS made. The amount of diScrepancieS in the range of calculated pointS 5... 15 (which corresponds to the gate rotation angles 𝜑 = 40°...90°) is in the range 0 ... 25%. The use of the ball valve with the standard gate crosssection as a control valve is not advisable. It is confirmed by the magnitude of discrepancies in flow capacity values, especially at small opening angles, and the manufacturer"s recommendations. Ball valves with 30 and 60degree crosssection profiles of the gate have the smoother control characteristic, but the flow capacity at full opening is several times less. Thus, if it is necessary to ensure maximum flow capacity at the certain nominal diameter when fully opened, it is recommended to use the shutoff ball valve with the standard gate crosssection. If precise and smooth regulation is necessary, it is advisable to use ball valves with 30 or 60degree crosssection profiles of the gate, but in this case the maximum flow capacity will decrease.
The result of the calculation iS alSo the picture of diStributed thermodynamic parameterS. It iS eStabliShed that the areaS of the valve flow cavity with high valueS of the velocity of the working medium are the main placeS of preSSure drop. To reduce preSSure dropS, it iS neceSSary to normalize the velocitieS in Such areaS to the average flow velocity in the Section by changing the geometry of the valve flow cavity.
REFERENCES