NUMERICAL AND EXPERIMENTAL ANALYSIS OF THE STRENGTH OF TANKS DEDICATED TO HOT UTILITY WATER

The focus of this paper are experimental and numerical strength tests of domestic hot water storage tanks. The tests involved the verification of the minimum wall thickness for the assumed operating parameters while meeting all safety standards. The authors presented numerical and experimental analyses for the verification of strength parameters of axial cylindrical tanks due to the lack of methodological guidelines for this type of equipment. In order to verify the conducted theoretical considerations and calculations, experimental tests of samples of front welds produced with austenitic steel as well as a pressure test for the whole tank were conducted using a research test stand. * SZEL-TECH Szeliga Grzegorz, Wojska Polskiego Street 3, 39-300 Mielec, Poland, e-mail: p.balon@szel-tech.pl, e-mail: bartek.kielbasa@gmail.com ** Rzeszów University of Technology, The Faculty of Mechanical Engineering and Aeronautics, Department of Mechanical Engineering, Powstańców Warszawy Avenue 9, 35-959 Rzeszów, Poland, e-mail: erejman@prz.edu.pl *** AGH University of Science and Technology, Faculty of Mechanical Engineering and Robotics, Department of Manufacturing Systems, Mickiewicza Avenue 30-B4, 30-059 Kraków, Poland, e-mail: szostak@imir.agh.edu.pl **** Rzeszów University of Technology, The Faculty of Mechanical Engineering and Aeronautics, Department of Thermodynamics, Powstańców Warszawy Avenue 9, 35-959 Rzeszów, Poland, e-mail:robsmusz@prz.edu.pl


INTRODUCTION
A pressure vessel is a reservoir manufactured to contain fluid (liquid or gas) at a pressure substantially different from the ambient.Cylindrical pressure vessels have widespread industrial applications and have become a type of equipment widely used in industry and everyday life.They are used in power plants, nuclear reactors, chemical processing reactors, and food industries.They appear in industrial compressed air receivers and domestic hot water storage tanks.Other application areas of pressure vessels are recompression chambers, distillation towers, autoclaves, oil refineries, petrochemical plants, vehicle airbrake reservoirs, and storage tanks for liquefied gases such as ammonia, propane, butane, and LPG, etc.They often perform under extreme pressure and temperature conditions (Li, Sheng & Zhang, 2012;Lakshmi Devi & Hari Shankar, 2016).Fundamental loads acting in the tanks are internal hydrostatic pressure and internal uniform pressure.The design of the pressure vessels is mainly related to their strength and the strength study mainly includes stress concentration analysis in the neighborhood of the head and cylindrical shell joint and in the heads of the tank.Pressure vessels can be closed at the ends by different shapes of the heads from flat plates to hemispherical domes.At the junction between a cylindrical shell and a vessel head there is discontinuity of meridional curvatures.These discontinuities of curvatures disturb the membrane stress state and have significant influence on the strength of the structure.This issue has been analyzed in books (Ziółko, 1986;Harvey, 2000) and standards (PN-EN 1993-1-6:2007, 2007; PN-EN 1993-4-2:2009, 2009).One of the most important topics in design is optimal shaping of the entire structure of the tanks.In many works the problem of the optimal tank shape was analyzed (Ventsel & Krauthammer, 2001;Błachut & Magnucki, 2008;Lewiński & Magnucki, 2010, 2012) and special attention was paid to the junction between the vessel and its head because this region is usually subject to considerable stress concentration due to the edge effect (Krużelecki & Proszowski, 2012).
The aim of this work was to describe the strength performance of a cylindrical vertical domestic hot water storage tank.Numerical calculations and experimental strength tests were made to verify the wall thickness for assumed work parameters while meeting all safety standards.
The correct assessment of strength properties requires both numerical and experimental testing of the tank, including its critical nodes, e.g.welded joints.In the first stage of strength tests, tensile tests of samples, tests of mechanical parameters of material and tests of welded joints were conducted.Then an analytical estimation of peripheral and radial stress values was carried out.In the second stage, numerical analysis of the model's real object mapping was performed using the MES software.In the third stage, in order to assess the strength of the structure, strain measurements were done in the shell of the tank subjected to pressure.
The main purpose of the experimental tests was to obtain the data to assess the strength of the tank subjected to the internal pressure load.Assessment of strength properties requires experimental tests of both: tank and critical nodes, e.g.welded joints.In the first stage of strength tests, tensile tests of welded butt joints were carried out.In the second stage the structural behavior of a cylindrical tank subjected to internal pressure was investigated.

Determination of the normal anisotropy coefficient R
A vessel is a device that consists of two main parts: the bottoms and the shell.While the shaping of the shell does not cause any technological problems, forming the bottoms is a serious problem.This is particularly important when shaping products with large diameters requiring expensive and heavy dies (Bałon & Świątoniowski, 2016a(Bałon & Świątoniowski, , 2016b;;Bałon, Świątoniowski, Szostak & Kiełbasa, 2016).For this reason, the properties of the sheet material must be very precisely defined for the description of the stamping process.One of the most important parameters is the normal anisotropy coefficient R. Therefore, research on this parameter has been carried out.
The normal anisotropy R ratio by Lankford is defined as the ratio of transverse strain increments during uniaxial stretching.Assigning the main directions for flat stress state of indices: 1direction of rolling, 2direction perpendicular to the rolling direction in the plate plane, 3normal direction to the plate surface and determied by deij the tensor components of the plastic strain increment, this coefficient -for the sample cut at an angle α to the direction of rollingwhich we can express in the form of a quotient (Bałon & Świątoniowski, 2016a(Bałon & Świątoniowski, , 2016b;;Bałon & Świątoniowski, 2013).
and respectively: and  90 =  11  33 (2) In practice, with homogeneous and proportional deformations, in determining the anisotropy coefficients, in the place of deij increments, the final values of real strains can be used, so the final formula will take the form (Dyrektywa 97/23/ WE, 1997;Rozporządzenie Rady Ministrów, 2002): (3) where: b0, b1widths of the sample before and after deformation, g0, g1thicknesses of the sample before and after deformation.
The measurement results allow us to state that the value of the steel yield point determined at 100 °C with a disproportionate elongation Rp0,2 [MPa] varies from 276.1 MPa (for the 0° direction samples) to 283.5 MPa (for the 90° direction samples), which indicates no significant influence of the directivity of the material structure on its properties after rolling.This is also confirmed by the results relating to the value of immediate tensile strength Rm -548.8MPa (for the 0° direction samples) and 551.3 MPa (for the 90° direction samples).In turn, the damage elongation measured after assembling both parts of the sample reaches 53%.
Measurements of the strains of the samples during thier uniaxial stretching were used to determine the normal anisotropy coefficient R. The mean Rsr value was calculated using the dependence: where: R0, R45, R90values of normal anisotropy coefficients for the direction according to the direction of rolling, directed at an angle of 45° and transverse to the direction of rolling The average value of the normal anisotropy coefficient in a steel sheet of DIN 1.4541 is Rst = 0.9832.Therefore, the value of the anisotropy coefficient for the sample cut at an angle can be described by the formula: where: b0, b1measured lengths of the sample before and after elongation.
On the basis of the conducted tests, it was found that the tested steel sheet DIN 1.4541 does not show normal anisotropy to a degree that can significantly affect the course of the pressing process.In the most general case, during stamping of the bottom from a flat disc, in the flange zone, the material is subjected to radial tensile stresses and peripheral compressive stresses, and thus the product of main stresses fulfills the unevenness s11s22 < 0.
In this case, according to the condition of metal transitioning into the plastic state of Mises-Hill, the absolute decreases, and the necesssity for plasticizing stress values |s11| and |s22| could be expected only at R > 1.
The bottoming operation is a cold process, hence the deformation of the material formed in subsequent phases leadsthrough dislocations in the crystal latticeto the strengthening of the blank material.This phenomenon must be taken into account when determining the process parameters.

Isotropic hardening model
In the stamping process, the kinematic model better reflects the physical aspect of the proces than the isotropic model.In the considered case of stamping, in which the load increases continuously up to the maximum value, there is no need to take into account stress hysteresis and therefore, the sufficient accuracy is ensured by a much simpler isotropic model.The nonlinear material with isotropic hardening is defined by the value Rp0,2 and the value Rm, while the section between Rp0,2 and Rm is defined by the tangent of the angle of inclination of the α curve.Transformation of the stiffness matrix takes place once strain or stresse values do not exceed the value specified by Rp0,2 and above this value each time at successive iterations.
Hardening curve of the material determined with extrapolation according to the method of Krupkowsky-Gesetz: where: n = 0.31500, φ0 = 0.00055, K = 2442.194.

EXPERIMENTAL RESEARCH OF FRONT WELDS
Steel 1.4541 (EN designation X6CrNiTi18-10), which was used in fabrication of the tank belongs to the largest austenitic stainless steels group with high corrosion resistance, and can be welded in all dimensions without becoming susceptible to intergranular corrosion.The chemical composition of the steel 1.4541 shown in Table 1.The analysis concerns the mechanical properties of the TIG welding butt joints with argon shielding.Test samples were prepared from 2.0 mm thick sheets of steel 1.4541.
In accordance with standards, a tensile test was carried out on a certified strength machine and the results of the strength tests performed on welded joints made of steel 1.4541 under quasi-static conditions are shown in Table 2.The tests show that destruction and broken areas of all samples occurred outside the heat affected zone and the tensile strength of the joint was higher than the base material.Furthermore, ultimate tensile strength (UTS) of the steel 1.4541 samples was higher than normative values of the UTS (normative UTS of the steel 1.4541: 520 MPa).
In the next step, the experimental tests on an internally pressurized tank were carried out.The subject of the experimental test was a cylindrical pressure vessel with a mean diameter of the cylindrical part equal 480 mm, made of steel X6CrNiTi18-10.The wall thickness of the cylindrical part of the tank and the top bottom head walls is the same and equal 2 mm and the main geometric dimensions of the head geometry are shown in Fig. 2. The heads of the tank were manufactured by spinning and the shape of the top and middle surface of the head was defined by spline curve.The bottoms were manufactured according to the DIN 28013 norm (Warunki Urzędu Dozoru Technicznego WUDT/UC/2003, 2005; Bałon, Świątoniowski, Szostak & Kiełbasa, 2016).

Fig. 2. Head tank geometry
For the joining of samples, the classic TIG method and the method with argon bluing on the side of the root of the weld were used.In the case of stretching samples made of X6CrNiTi18-1 steel made with the TIG method without argon bluing from the side of the ridge, the sample was destroyed in the cross-section of the weld, and for welding with the bluingoutside the weld area (Bałon & Świątoniowski, 2014).
Based on the above tests, it can be concluded that, with regard to the same welded material, the TIG welding technology using argon blown from the root side of the weld significantly improves the quality of the weld surface as well as its tensile strength.The average value of tensile strength of the connection was Rm = 630.6MPa.
For this reason, the TIG method was proposed for the welding of the vessel, with argon bluing on the ridge side (Bałon, Świątoniowski & Szostak, 2015;Bałon & Świątoniowski, 2014).

ANALYTICAL AND NUMERICAL STREGTH CALCULATION OF THE VESSEL
The designed tank is dedicated to storing water with a temperature of up to 100 °C and a working pressure P1 = 6 bar equal to the pressure of water feeding from the network.Nominal, design and trial pressures were adopted in accordance with the regulatory literature in force in the European Union and Poland.
The basis for the design of the tanks is Ps pressure which is the maximum allowable pressure specified by the producer for which the device has been designed (Dyrektywa 97/23/WE, 1997; Rozporządzenie Ministra Gospodarki, 2005;PN-EN 13445-1, 2014;PN-EN 13445-3, 2014).
The following assumptions were made for the calculation of the vessel: Due to the negligibly low hydrostatic pressure, which is approx.1.8% of the working pressure, its influence on the design pressure was not taken into account.
Design pressure (PN-EN 13445-1, 2014): Base on ( PN-EN 10131:2006, 2006), the design pressure Pd should meet the condition: = 1.5   = 11.25  (9) Test pressure: The test pressure Ptest should be: = 1.43   = 10.72  (10) The higher pressure from Pd and Ptest: should be taken as the design pressure: For the assumed bottom geometry and pressure of Pd = 11.25 bar, numerical calculations of the bottom using the COMSOL program were performed.The maximum reduced stresses reached the value of 210 MPa, which is lower than the material yield point of about 260 MPa, so the structure is safe.In the case of vessel design according to (Rozporządzenie Ministra Gospodarki, Pracy i Polityki Społecznej, 2003), it is necessary to determine the design temperature T, which affects the strength properties of the vessel material.It is determined with dependencies: where: TCmaximum temperature of the medium.
According to (PN-EN 13445-1, 2014), the permissible stress fd is the criterion stress for assessing the force of the vessel material.Using the design method consistent with the recognized engineering practice, and having the material from which the vessel will be made, in accordance with EN 10088-3: 2005, the strength properties of the tank material were adopted as follows: = 540  (15) It should be stated that the material parameters obtained during the tests significantly exceed the values given by the standard.Due to the fact that the designed tank works at a temperature of T = 70 °C, it can be estimated on the basis of the EN 10088 norm that the appropriate mechanical properties for the tested steel are: The permissible stresses fd are determined on the basis of chapter 6.4 according to (PN-EN 1993-1-6:2007, 2007).
= max [(145,3); min(181,7) ; ( 165)] (20) The f stresses in the structure should satisfy the following conditio: Considering that the assessment of weld properties was done by testing a welded metal sample and ther are conditions for visual assessment of the weld in the structure, z = 0.85 was accepted for the calculations.
For the internal diameter of the tank of De = 480 mm, the theoretical wall thickness of the tank was determined as follows:

Calculation of the tank wall thickness with regard to operating conditions
The nominal sheet thickness en is determined by the formula: where: Ccorrosion allowance, δeallowance for tolerances of rolled sheets (lower tolerance deviation), δmwall thickness allowance due to additional pressure stresses (δm = 0) was assumed.
According to American data for the X6CrNiTi18-1 steel (marking according to AISI -304), the reduction in wall thickness for a working water environment for a year is (with research being carried out for 15 years): Assuming τ = 15 years as the expected tank life, the following was obtained: = 0.14 mm according to EN10131:2006 (29) = 1.92 + 0,14 + 0.00111 = 2.061  (30) Due to the fact that the axial stresses are twice as small, the wall thickness considered in the axial direction will be smaller.The proposed solution assumes the thickness of the sheet as en = 2 mm.
For the proposed solution, the main stresses will be: = 92.17 < 181.7  (the condition is met) Taking into account that the lowest acceptable stress value, which is significantly lower than experimentally determined, was used for calculations, the sheet thickness of e = 2 mm can be considered as sufficient.

EXPERIMENTAL RESEARCH
The main goal of the research was to obtain the data needed for the experimental assessment of the force of the vessel material at a certain value of its internal pressure load.The object undergoing testing was a pressure vessel with an average diameter of the cylindrical part of Dsr = 480 mm, made of X6CrNiTi18-10 steel.The thickness of the bottom walls and the cylinder itself is the same and equals 2 mm.Bottoms of cylindrical tanks usually have ellipsoidal or toroidal-spherical shapes and are characterized by their connection to the cylindrical surface of the tank in the axial plane of the cross-section at the place of the greatest curvature.This results in the disturbance of the membrane state of internal forces in the structure (Fig. 6).There is transverse force and a bending moment which cause changes in stress distribution, both in the bottom part and in the cylindrical part of the tank.Therefore, in order to select the most intensive areas of the tank, auxiliary calculations were made using the finite element method (Fig. 6).
The greatest stresses, both circumferential and longitudinal, occur in the region at the very bottom of the bottom.Then longitudinal stresses decrease, reaching negative values near the bottom connection with the cylindrical part.The circumferential stresses reach the second local maximum near the connection to the cylindrical part.
In order to determine the stresses at the characteristic points of the structure, strain gauges were glued on them.Figure 7 shows the location of the strain gauges on the bottom.For circumferential and longitudinal strain measurements, biaxial tensiometers of the type TF 3-2x / 120 from Tenmex were used.
Additionally, two strain gauges were installed on the cylindrical part.One was installed at a distance of 10 mm from the weld, and the second halfway up the cylindrical part of the tank.In addition, in order to verify the measurements, in the same configuration as for the top end, strain gauges were installed on the very bottom of the bottom.In order to compensate for the influence of temperature on the measurement results, strain gauges were mounted on the unloaded part of the tank.
For circumferential and longitudinal strain measurements, seven biaxial tensiometers of the type TF 3-2x / 120 from Tenmex were glued on to the head of the tank.In order to compensate for the influence of temperature on the unloaded steel X6CrNiTi18-10 plate single-axis strain gauges were mounted.Additionally, two strain gauges were installed on the cylindrical part.To measure the pressure in the tank, a piezoelectric transducer was used.The strain gauges and pressure transducer were integrated with the Catman measuring system using the four Hotminger Spider-8 amplifiers.During the test, data from 26 measurement channels were recorded.The pressure test was carried out as follows.After calibrating the measurement channels and stabilizing the indications of the strain gauges, the appropriate pressure in the tank was developed by the water pump.

Pressure test
The pressure test was carried out as follows.After calibrating the measurement channels and stabilizing the indications from the strain gauges, the appropriate pressure value in the tank was forced by means of the pump and its constant value was maintained so as to obtain the determined conditions.This state was maintained for 15 seconds to be able to register a sufficiently large sample of measurement data.The measurement data was recorded and saved every 20 ms.At the same time, the tightness of the system was controlled, i.e., checks were made for the presence of water leaks.The pressure was measured in increments of 1 bar.The tests were carried out at a constant temperature of 24 °C.
In the pressure range from 1 to 9 bar, there was no slow pressure drop when pumping stopped.However, for higher pressures, its slow decline was noticeable, despite the pump's cessation.No leakage of water from the installation was found during this time.The pressure drop was caused by exceeding the yield point in the zone with the highest stresses, which caused the start of the "flow" of the material and the increase of the volume of the tank.This process deepened with increasing pressure, i.e. the gradient of pressure drop increased with increasing pressure level.This meant that an ever-larger material zone reached the limit of plasticity.The pressure test was carried out up to 30.9 bar.A further increase in pressure was not possible due to insufficient pump capacity, i.e. the plastic flow phenomenon was already so intense that it was not possible to achieve higher pressure values.
At the conclusion of the test, the tank was inspected and no leaks were found.All welded connections had been leak-proof and were undamaged.The degree of deformation, which is the effect of plastic deformation can be noticed by comparing the shape of the bottoms before and after the pressure test (Fig. 9).At a distance of about 170 mm, circumferential (tensile) stresses reach the local minimum, then increase rapidly and change the sign.
The highest compressive (peripheral) stresses occur along the radius of the bottom passage into the cylindrical part.It is the effect of combining the ellipse with the straight part forming the cylindrical part of the tank.There is a transverse force and a bending moment that causes stress accumulation, both in the bottom part and in the cylindrical part of the tank.
Longitudinal stresses are positive (stretching) up to the point of 230 mm, where they change the sign.Along with the increase of the radius there is a slight decrease to the point with the coordinate of 70 mm.Then they grow again up to 220 mm, where they reach their maximum value.
As in the case of circumferential stresses, there is stress accumulation in the transition zone in the cylindrical part.
As you can see, this area has a decisive impact on the strength of the structure.

CONCLUSIONS
The conducted numerical and experimental studies show that the FEM method gives results inflated in relation to the analytical calculations, although it indicates the directions where the maximum stress increases should be expected.
The analytical method recommended by the relevant standards allows for the safe design of tanks and the maintenance of good safety ranges.
The material properties of steel given by standards are smaller than the actual values.When designing, for minimal safety factors, you must have your own material base, created as a result of tests in certified laboratories.
In the construction of tanks made of austenitic steel, exceeding the yield point is not synonymous with the destruction of the structure.For large stress values (exceeding the yield point) significant deformations and displacements occur, but the structure remains sealed.
Austenitic steels show good weldability, and the selection of a suitable welding technology ensures adequate weld strength and preservation of its integrity even when the parent material is destroyed.

Fig. 4 .
Fig. 4. A state of reduced stresses in the bottom according to DIN 28013 loaded with the design pressure of Pd = 11.25 bar, e = 2 mmthickness of the wall (COSMOL) = Pd = 11.25 MPadesign pressure, erequired wall thickness, zconnector factor (Dyrektywa 97/23/WE, 1997): z = 1when the tested object together with the weld is subjected to destructive and non-destructive tests, z = 0.85when the object with the weld is subjected to random nondestructive tests, z = 0.7for non-destructive visual tests.

Fig. 5 .
Fig. 5.The view of the vessel prepared for research

Fig. 7 .
Fig. 7. View of tested tank with strain gauges

Fig. 9 .
Fig. 9. Changes in the shape of the tank bottom at an increase of pressure up to 30 bar