Expansion vessel. Pressurized equipment design. PED.

H Research Edition. ISSN 2532-5612. Publication 2020-21,06 Division edition Group HTNET

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With the publication 2020-21.06 we dealt with the design of an expansion vessel.

After identifying the normative references that regulate the pressure equipment,

then the PED directive (2014/68 / EU), with reference to the project data that

classify the pressure equipment to category IV (Annex II of the PED directive) we

chose as a design method, the DBF analytical method envisaged by the EN 13445

standard that the PED directive has adopted. Having identified the mechanical

characteristics of the material used for the construction of the vessel (yield point,

traction limit and other mechanical properties), we calculated the safety factor

based on the analysis of the loads that the pressure equipment will have to support

under operating conditions (both normal and exceptional). Since 6 nozzles were

connected to the expansion tank, it was necessary to check the reinforcement of

the nozzle which confirms the request for reinforcement for the nozzles connected

to the casing, with a larger diameter. Assuming three types of (periodic) oscillating

loads, we carry out the fatigue test in all the load hypotheses considered. The

checks and analyzes carried out on the results and calculations have demonstrated

the required design safety even in exceptional cases such as temperature and

pressure peaks, confirming the elastic regime (even if at the limit) with a

temperature more than double the operating temperature (from 220 ° C at 450 ° C)

in combination with a peak pressure equal to twice the operating pressure (from 25

bar to 50 bar). With the intention of achieving certainty of the results obtained, it

was appropriate to size the container using an alternative calculation method to

Electronic copy available at: https://ssrn.com/abstract=3555693

Expansion vessel.

Pressurized equipment design. PED. Author: engineer Vito Gnazzo

Publisher: Htnet edition


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compare the results; for this purpose we used the calculation model provided by

ASME VIII, which provides for the calculation of the pressure necessary for the

complete plasticization of the section according to the Tresca criterion and which

having provided, the same results as the model provided for by the EN 13445 DBF

standard (model calculation based on the hypothesis of not exceeding the yield

point of the material, therefore in the hypothesis of linearity determined by Hooke's

law) it allows us to believe that the certainty of the results has been reached, aimed

at operating safety of the pressure equipment. The project calculation ends with the

Elasto Plastic Stress Analysis: deformation analysis according to the DBA Method

(ASME VIII Div. 2) which allows us to calculate the plastic deformation, the collapse

limit of the material and to carry out a study of the deformations at the elastic field

limit and in the plastic field by Navier's theory (Claude-Louis-Marie Navier) which

applies the constitutional laws of materials known as Hooke's law for the elastic

field and the constitutional laws of Ramberg-Osgood material for the field plastic.

Keywords: PED, directive 2014/68/UE, pressure vessel, expansion vessel, DBF, EN 13445, safety factor, fatigue limit factor, Shigley, Juvinall, Pressure Equipment Directive, nozzle, nozzles, verification of nozzle reinforcement, fatigue , Wöhler diagram SN, Ramberg-Osgood, Hooke.

Author: engineering Vito Gnazzo. Publication of 2020/02/06 number 06 for magazine H Research Edition No 2020-21.


Publication list

H Research Edition Italian: publication 06 magazine number 2020-21

EXPANSION TANK Design of pressure equipment. PED. publication 2020-21 06

Cha. 1 – Calculation for the sizing of a pressure vessel…………………………………………………………………………pag 5

1. 1 Introduction pag 5 1. 2 P ED Directive, Pressure Equipment Directive, standards for

the design of pressure vessels pag 6 1. 3 Sizing methods used in the project pag 6 1. 4 Main standards used for the design

of pressure vessels pag 7

Cha. 2 ‐ Sizing of the pressure vessel in the side view shown below (expansion tank)………………………………………………… pag 9

2. 1 Realization material pag 10 2. 2 Apparatus category classification pag 11 2. 3 Safety factor calculation pag 12 2. 4 Calculation of the allowable tension pag 15 2. 5 Pressure calculation pag 16 2. 6 Cylindrical mantle sizing pag 18 2. 7 Dimensionamento fondi sferici pag 22

Cha. 3 ‐ Actions of the weight of the fluid and the weight of the container………………………………………………………………………………………..…..… pag 24

3. 2 Stress calculation pag 27


Publication list

Cha. 4 – Calculation of reinforcement area required for the presence of nozzle connected to the container……………………………………… pag 31

4. 1 Geometric limits of the reinforcement areas pag 31 4. 2 Check reinforcement areas pag 34

4. 2.1 Check nozzles, mantle – Nozzle A3, A4 pag 34 4. 2.2 Calculation of reinforcement areas – Nozzle A3, A4 pag 38

4. 3 Check nozzle for valve and pressure gauge

mantle – Nozzle A5, A6 pag 42 4. 4 Check nozzle funds – Nozzle A1, A2 pag 45

Cha. 5 – Fatigue stress analysis ………………………………………………………………………… pag 52

5. 1 Pressure vessel sizing results:

expansion tank pag 53 5. 2 Oscillating loads pag 54

Cha. 6 – Analysis of calculations with combination of loads ……………………………… pag 63

6. 1 Analysis of the calculation with alternative methods pag 65 6. 2 Elasto Plastic Stress Analysis: deformation analysis,

Method DBA (ASME VIII Div. 2) pag 69 6. 3 Plastic analysis pag 77

VII – SOURCES FOR RESEARCH …………………………..…..………..…..pag. 83

VIII – BIBLIOGRAPHY…………………………..…………………..…..pag. 84

Author: engineering Vito Gnazzo date: 2020/02/06



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Expansion vessel.

Design of pressure equipment. PED.

Calculation for the sizing of a pressure vessel 1-INTRODUCTION In the dimension design of the pressure vessel, the most common load conditions should be considered:

Internal and external pressure, own weight, actions transmitted by the weight of any equipment: machines, internals, actions transmitted by the container motion,

Supports, cyclical and dynamic actions produced by variations in pressure and temperature, wind, snow, Impulsive actions, such as those due to "blow" (is a hydraulic phenomenon that occurs in a conduit when a flow of liquid moving in it is abruptly stopped by the sudden closing of a valve or when a closed conduit in Under pressure is opened suddenly),

Temperature gradients and differential thermal expansion, pressurization test.

It is, however, the responsibility of the manufacturer to identify all actions that may be expected to take place during the operating life, which may be relevant for safety,

Including those resulting from any reasonably foreseeable misuse of the equipment itself.

Often the pressure component includes welded joints whose presence tends to reduce the permissible tension values of the component itself by means of a coefficient called "efficiency" of the weld. The welding efficiency generally depends on the following factors: welding type (pp head, corner angle, etc.), NDE or NDD ("Non Destructive Examination" or "Non Destructive Testing" US, X-ray, etc.) After welding, the thicknesses of the welded parts, the working temperatures, the type of base material, must be evaluated. “The operating conditions determine the safety coefficient to be taken and



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may cause variations in structural calculations with variations”. 1.2 - Directive PED, Pressure Equipment Directive, regulations for the design of pressure vessels The PED Directive refers to harmonized standards (eg EN 13445) for the design and construction of pressure equipment; It is understood that PED's safety requirements are deemed to be satisfied if it complies with the design and construction requirements of EN 13445.

Category I: for less dangerous equipment, EC certification is subject to the "self-certification" of the manufacturer.

Category II: CE certification is obligatory through a notified body which, without entering into the merit of the design, carries out production monitoring in the manner selected by the manufacturer;

Category III: CE certification is obligatory through a notified body. If the manufacturer has not certified his quality system, including the design, the in-depth testing of the prototype to be certified by CE is also foreseen;

Category IV: requires the highest level of control of design and production.

1.3 - Sizing methods used in the project

Design by Formulas (DBF): Sizing and checking of the container are based on pre-packed formulas (formulas) designed to cover, with adequate safety coefficients, all the main situations encountered in the design of a pressure vessel; When formulas are usually based on simple or semi-empirical models that are not very accurate, so the security coefficients tend to be higher.

Design by Analysis (DBA): sizing and checking the pressure



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vessel are based on accurate analysis of the actual tension state, usually only available with Finite Element Method (FEM) bases. The DBA approach is necessary for cases not covered by DBF relationship, but is also used as an alternative to the latter, unless simple analytical models are sufficient. By providing the DBA approach more accurate analysis, so the security coefficients used tend to be lower.

1.4 - Main standards used for the design of pressure vessels • ASME VIII

Div. 1 – Approach DBF Div. 2 – Approccio DBF+DBA

• EN 13445 – Approccio DBF+DBA Verification criteria for structural safety: The different standards differ from each other in the verification criteria

adopted for the calculation of the equivalent sigma eq defined as id

in this calculation: ASME VIII - Div. 1 (DBF): Lamé criterion (max. Normal tension)

eq 22

used for slightly ductile materials. ASME VIII - Div. 2 (DBF): Tresca criterion (max. Tangential tension)

2maxji

max



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Where eq max2

which in this publication is id. Criterion

used for ductile materials. ASME VIII - Div. 2 (DBA): Von Mises criterion (distortion energy)

eq )(3 223

213

212323121

23

22

21

Where with 03

you get eq 212

221 3)(

, wich with respect to the Tresca criterion, an excess approximation is used in the

calculation of eq. The Von Mises test criterion is used for ductile

materials.. EN13445 (DBF + DBA): Tresca criterion (max. Tangential tension)

2maxji

max

Where eq max2

, in this design calculation, is id. Criterion

used for ductile materials..



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2 – Sizing of the pressure vessel in the side view below (expansion vessel).

Nozzle: A1 process connection, A2 state level connection to float, A3 visual level indicator

upper connection, A4 visual level lower connection, A5 pressure gauge connection, A6 connection for emergency valve.

The container has a cylindrical configuration with an internal diameter

of D = 320 mm and is closed with hemispherical bottoms, it is placed on

two saddles A and B separated by a distance LAB = 400 mm and comes

out a length L0 = 160 mm on both sides. Contains steam at a constant

pressure of 25 bar with a maximum operating temperature of 220 °C;

being for this temperature 23.193 bar the vapor pressure, it appears to

be mainly liquid. The density of the fluid is ρ = 0.840 kg/dm3 (p = 25bar,

t = 220 °C). The project calculation is carried out to size the thickness

of the mantle and the substrates assuming that the degree of safety is

at least 2 as calculated in the following sub paragraphs. The study of



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the expansion vessel is also carried out with the hypothesis of

oscillating loads assuming the following three hypotheses of periodic

loads: 1) Pmax = 25 bar; Pmin = 20 bar; 2) Pmax = 25 bar; Pmin = 15

bar; 3) Pmax = 25 bar; Pmin = 5 bar; with an average oscillation of 5

minutes.

2.1 - REALIZATION MATERIAL

Material used for the realization: steel P355NH (for references,

annexes according to UNI EN 10028); tensile tensile strength and yield

strength at a temperature of 220 °C

R

= 465 2mm

N

S = 275 2mm

N

The amm must be the last value before the break therefore at the

maximum equal to R.

At a temperature of 20 °C they appear to be 2630490 mmNr and

2345355 mmNs . The minimum values for the project calculations

will be assumed.



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2.2 - Apparatus category classification

The PED ("Pressure Equipment Directive") which refers, for the design and

construction of pressure equipment, to the Harmonized Standards (eg EN 13445),

means that the safety requirements of the PED are considered satisfied if the

requirements are met of design and construction of EN 13445. For this purpose it

establishes the following categories of pressure equipment and for each one of

them a different level of safety:

• Category I: in which the less dangerous equipment falls, CE certification is

provided on the basis of the manufacturer's "self-certification".

• Category II: CE certification is compulsory through a notified body, which,

without going into the merits of the design, provides for carrying out production

surveillance, in the manner chosen by the manufacturer;

• Category III: CE certification is mandatory through a notified body. If the

manufacturer has not also certified its quality system, including the design, in-

depth tests are also carried out on the prototype to be certified CE;

• Category IV: the highest level of control of design and production is required.

Through Annex II, table 2, of the same PED directive (2014/68 / EU) and for the

type of gas that affects the expansion vessel, subject of this calculation, it is

possible to identify the classification category of the expansion vessel ( table

below):

TABLE 2 of Annex II of the directive (UE) 2014/68



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With reference to the project data, from the table above, the expansion

vessel is classified in category IV (5800 bar • l).



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2.3 - SAFETY COEFFICIENT CALCULATION

The following considerations are made from a load analysis based on

the information collected (supplied by the manufacturer):

st

t

C

K

where is it

K is the stress concentration factor (notch effect), takes into

account uncertainties of various origins and stC is the collaboration

coefficient (departure from the behavior of materials hypothesized by

Hooke's linear law):

For the project he is hired:

98,0stC

1K

96,1 where is it:

4321 with

1 which considers any material defects and inaccurate values of its

characteristics (assumes values between 1 and 1.15). It is assumed equal to

1,15.

2 which considers uncertainties in determining the loads acting with

particular reference to impulsive loads and impacts as well as dynamic

pressure, water hammer. It assumes different values for fragile



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materials (normally between 1.5 and 2) and ductile materials (normally

between 1.1 and 1.3). With this coefficient the loads due to the

dynamic pressure, the water hammer, reaction forces and reaction

moments caused by supports, connections, pipes etc. are considered,

therefore it is assumed equal to 1.5 (dynamic pressure, water hammer ,

piping reactions, load peaks).

3 which considers errors due to the simplifications of the stress

calculation, for example in the ideal schematization of the constraints

(absence of friction) can vary between 1 and 1.5 (fragile materials) and

1 ÷ 1.1 (ductile materials). It is assumed equal to 1.

4 is the actual safety margin introduced to keep the system sufficiently

far from the failure conditions. Its value depends on the importance of

the component under consideration: it will be much higher the more

vital it is to avoid its failure (for example, cableway attacks, bridges,

lifts, biomechanical prostheses, ...). Normally the value varies between

1.1 and 1.5. With this coefficient we consider the increase due to the

calculation model adopted which compared to the DBA model (for

example: FEM finite element method, analytical model based on

formulas of lesser approximation) requires higher safety coefficients

and therefore is assumed to be equal to 1,14; the approximation of the

model adopted could, in some cases, reduce the states of tension that

are actually present in the sections; to consider that the analytical

calculation formulas adopted in this project assume an increase that

considers the above mentioned case studies.

96,114,115,115,1



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298,0

96,11

st

t

C

K

Increases due to erosion conditions, decomposition of unstable fluids are

excluded from the analysis carried out on loads and the stresses due to

traffic e winds are negligible; considered the effects that could cause any

malfunctions of the equipment provided for in paragraph 2.11.2 of Annex I

of the PED directive (nozzle A6) which limit any pressure peaks to 10% of

the maximum pressure (approximately 28 bar) and increases in

temperature which remain would cause a reduction of the elastic limit of

the material; the analysis of the loads and the related combination can be

found in the last paragraph.

2.4 - Calculation of the allowable tension.

Since the degree of safety is not less than 2, the safety coefficient γ is

equal to 2, so the admissible tension is:

amms

R mm

N

25,137

2

275

With reference to Annex 1 of Directive (EU) 2014/68, paragraph 7, which

establishes wich for ferritic steel, including normalized steel (rolled steel), and

excluding fine grain steels and steels who have undergone a special heat



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treatment, the amm cannot exceed the minimum between 32 of teR and 125

of 20mR , the value of 25,137 mmNamm is the correct choice, where

teR elasticity limit, indicates the value at the calculation temperature, as

appropriate:

— the maximum scrolling limit for a material with a minimum and

maximum scrolling limit,

— of the conventional elastic limit, equal to 1.0%, for austenitic steel and

unalloyed aluminum,

— of the conventional elasticity limit equal, in other cases, to 0.2%..

20mR indicates the minimum value of the tensile strength at 20 ° C;

indeed

2220 275 mmNRes

22220 5,137183

3

2mmNmmNRes

220 490 mmNRms

2220 5,137204

12

5mmNmmNRmr

It should also be considered that the steel used is fine-grained, hot rolled.

2.5 - Pressure calculation

The point of greatest stress is identified.

In any section of the cylinder, at point C the pressure turns out to be:



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MPabarpc 5,2250000025

In point D the pressure due to the weight of the fluid must be added to the

above pressure:

02,181,98402500000ghpp CD

Mpa508,2

MpapD 508,2 being

211

mm

NMPa

then 2

508,2mm

NpD



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2.6 - Cylindrical mantle sizing

In a thin-walled pressure vessel, tensions can be calculated with the

following relationships

s

rpm 2

s

rpn

pr

The calculation will be made by referring to the existing pressure on the

bottom, or the pressure Dp, using the Tresca method as a verification

criterion for structural safety, we obtain:

minmax id

In this dimensioning, the minimum tension is r while the maximum

tension is m

ps

rpp

s

rpid

)(

for the calculation of the thickness, the structural stability equation is

used



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ammE max

The coefficient E represents the efficiency of the welds which normally

takes the following values:

E=1 for complete radiographic control (100%)

E=0,85 for spot radiographic control (10%)

Begin id max replacing

ammps

rp

ps

rpamm

from which the thickness that turns out to be is obtained:

pE

rps

amm

being the radius half the diameter equal to 160 mm, replacing is obtained



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mms 97,2508,25,1371

160508,2

if a coefficient of 0.85 (instead of 1) is adopted for the welds, the

thickness becomes:

mms 51,3508,25,13785,0

160508,2

therefore in the first calculation analysis the choice of a thickness equal

to 2.97 mm (or 3.51 mm for E=0.85) is adequate. From the application of

Directive 2014/29/EU that repeals Directive 2009/105/EC, rationalization

of EC Directive 404/1987, adopted in Italy with Legislative Decree number

82 of 19 May 2016, for the possible presence, to make the container, of

longitudinal welds made with a non-automatic welding process, the

thickness is multiplied by the coefficient 1.15; therefore with a

E=1 the thickness must be at least 3,51 mm

E=0,85 the thickness must be at least 4,04 mm

Having, the pressure vessel a product value maximum operating pressure

for the volume greater than 5800 bar. liter, category IV, can be carried out

following a project in which the thickness of the parts under pressure is

calculated in application of directive 2014/68 / EU (adopted in Italy with



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law decree number 26 of 15 February 2016) which excludes the possibility

of applying the experimental method (without calculation) which would

determine the thickness so as to allow the container to withstand, at room

temperature, a pressure equal to at least 5 times the maximum working

pressure, with a permanent circumferential deformation of no more than

1%. It should be noted that the structural calculation project, an

obligation imposed by the directive, carried out by an authorized

technician (qualified engineer) allows to obtain a considerable saving

of material in compliance with the maximum safety requirements

necessary.

mmSE 51,31

mmSE 05,485,0



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2.7 - Spherical seeds bottom sizing In this case the relations for calculating the tension, are:

s

rpm 2

s

rpn 2

pr

By applying the Tresca criterion for the verification of structural safety,

we obtain:

ps

rpp

s

rpid

2

)(2

From which we obtain the thickness which for E=1 is equal to

)508,25,1371(2

160508,2

)(2 pE

rpS

amm

mm49,1

while for E=0.85 it results

)508,25,13785,0(2

160508,2

)(2 pE

rpS

amm

mm75,1



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Considering the possible presence of longitudinal welds made with a non-

automatic welding process, the thickness is multiplied by the coefficient

1.15; therefore

E=1 the thickness must be at least 1,71 mm

E=0,85 the thickness must be at least 2,01 mm

So in the first analysis of calculation the thickness of the funds turns out

to be

mmSE 71,11

mmSE 01,285,0 Note: In the realization of pressure vessels with non-thin walls, in the absence of structural constraints, the hemispherical bottom is chosen because it involves a reduction of metal used since it can have a thinner thickness than the thickness of the shell, up to half; a choice that is not used for thin-walled pressure vessels where the same thickness is used for both the shell and the hemispherical bottoms.



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3 – ACTIONS OF THE WEIGHT OF THE FLUID AND THE PRESSURE VESSEL.

The weights of the fluid and the shell operate as a continuous load q

which, however, is constant in the cylindrical area, but variable in the

two bottoms.

The volume of a spherical segment, proportional to the weight, is given

by the relationship

)3

(2 hrhV if

the trend of which is shown in the graph.

Assuming the bottoms as cylindrical, an increase in design safety is

obtained as well as a simplification of the calculations, so that the tank

becomes a cylinder with a length Lt=720 mm with an internal diameter

Di=320 mm and an external diameter De=328 mm.

The volume of the fluid turns out to be:



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322 232,0720,0320,014,3 mLrV tif

The weight relative is:

gmQ ff gVf

gLrQ tif2

N93,47681,9720,0160,0840 2

dividing the total weight of the fluid by the total length gives the

equivalent continuous load:

mm

N

m

N

L

Qq

t

ff 662,040,662

720,0

93,476

The mantle weight is:

tim LsrV 2

gLsrQ timm 2'



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N91,21281,9720,0004,0160,027500

The total weight of the 6 nozzles must be added to the weight thus

calculated (ref. Nozzle table, in the following subparagraphs) which from

the tables EN 1092 is 11 kg in total, therefore

NQQQ nozzlemm 82,32081,91191,212'

the total load is the sum of the two weights:

NQQQ mft 79875,79782,32093,476

dividing the total weight by the total length gives the equivalent

continuous load:

mm

N

L

Qq

t

t 11,1720

798

The system for the purposes of structural calculation is schematized as a

beam supported with a continuous load, the supports of which are

connected to the mantle by saddles; the scheme adopted for the

structural calculation allows an increase in safety compared to the

hypothesis of supports connected to the bottoms, since the tension state

due to the internal pressure is halved compared to the mantle:



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The diagram of the moment turns out to be:

3.2 - Stress calculation

The most stressed section is identified by calculating the moment where

it reaches its maximum intensity, in sections A and B and in the center

line that we call E.

In sections A and B, for the symmetry of the system, the moments

assume the same intensity:



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Nmm

LqM A

B 140802

1601,1

221

In section E it is calculated:

NmmLR

LqM AEA

EE 7920200

2

798

2

3601,1

2

221

To calculate the maximum tension indicated with maxf

, the modulus

of constant bending resistance in the whole section is calculated:

444

44

204615364

360364

322

6464 mmD

DD

We

ie

f

Since the maximum bending moment in section E in module, it is possible

to calculate the maximum tension maxf

through the calculation below:

2max 039,0204615

7920

mm

N

W

M

F

Ef



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The stress state in the section is represented in the red figure, where in

point A it generates a maximum compression of value maxf and in point

B a traction of value maxf.

By now calculating the tension n, the maximum tension generated by

the fluid pressure is obtained:

236,100

4

160508,2

mm

N

s

prn

Therefore in point B there is a maximum tension mt (cylindrical mantle

sigma) calculated as follows:

2max 100039,036,100mm

Nfnmt

As predicted by the size of the expansion vessel and therefore quantity of

fluid (steam) contained therein, the in the shell is approximately equal

to the generated by the load pressure, resulting in the calculation

being negligible as the generated by the weight of the fluid and

expansion tank.



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The sizing project of the pressure vessel is considered completed

because applying the Tresca criterion

mtrmtid minmax

the resulting ideal tension mtid is lower than the sliding tension,

the limit beyond which the plastic sliding starts, i.e. plastic deformation

preceding the breakage; the limit sigma has been indicated with sigma R

defined as breakdown. The thickness S equal to 4 mm is the thickness

for which the pressure vessel with cylindrical walls and

hemispherical bottoms complies with the design safety requirements

(operating conditions, maximum operating pressure, maximum and

minimum operating temperature, ...).

If they had been detected oscillating loads then proceed with the fatigue check and eventually increase the thickness to reach the required project duration (unlimited duration).



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4 - CALCULATION OF REINFORCEMENT AREA REQUIRED FOR THE PRESENCE OF NOZZLES CONNECTED TO THE PRESSURE VESSEL

Considering that by introducing an opening on the side casing of the

container or on bottoms, material is eliminated and the flow of tensions

through this material is interrupted causing a concentration of

tangential tension around the hole generated for the positioning of the

nozzles (nozzle), it is necessary replace the missing material with other

material, close enough to the opening to be affected by the flow of

tensions. The replacement material can be provided by a surplus of

vessel thickness, compared to the minimum required, or by special

reinforcements located in the opening area.

Therefore we proceed

checking the geometric limits within which the material that can

withstand the stresses must be contained,

the calculation of the values of the area necessary for the

transmission of the tensions in the opening area

to calculate the area values deriving from the material already

available, in the thickness of the vessel and the reinforcement

any additional reinforcement arrangements.

4.1 - Geometric limits of the reinforcement areas

Size limits for validity of following rules (otherwise additional rules):

• D<60 in. (1524 mm), d<0.5D o 20 in (508 mm)

• D>60 in. (1524 mm), d<0.33D o 40 in. (1016 mm)

where d internal diameter of the nozzle, D internal diameter of the container. Since the diameter of the container (expansion tank) is 360 mm, the material removal can have a maximum diameter of 180 mm. The nozzles to be connected to the container are listed below (and do not require additional models or interventions for connection with respect to the design and calculation choices present here °)



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__________________________________ (°)When the above relationships are not verified, it is certainly possible, as well as necessary, to intervene by increasing the size of the container.

TABLE NOZZLE

Pressure vessel bottoms: A1 ATTACK ON THE PROCESS UNI

EN1092-1 DN25, PN25

A2 FLOAT STATUS LEVEL ATTACK

UNI EN1092-1 DN50, PN25

Pressure vessel mantle: A3

UPPER ATTACK VISUAL LEVEL INDICATOR UNI EN1092-1 DN50, PN25

A4

LOWER ATTACK VISUAL LEVEL INDICATOR UNI EN1092-1 DN50, PN25

A5 CONNECTION FOR PRESSURE GAUGE / PRESSURE SWITCH

1/2" GF

A6 CONNECTION FOR EMERGENCY VALVE

1/2" GF

Schemes and geometric data nozzle (nozzles) to be connected



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EN 1092-1

PN=25

Therefore the following material removal will be carried out: on the bottoms a 33.7mm hole and a 60.3mm hole on the casing there are two 60.3mm holes and two holes for pressure gauge and emergency valve whose dimensions require a 30mm hole.



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4.2 - VERIFICATION OF REINFORCEMENT AREAS (verification to the reinforcement) Below we will refer to a coefficient E = 0.85 (efficiency of the welds made with a non-automatic procedure) therefore, with reference to the calculations of the sub paragraphs above, the same thickness of construction of the expansion vessel is chosen both for the funds and for

the mantle and equal to mms 00,4 resulting, therefore, for the mantle

an mmsr 51,3 and for the funds an mmsr 01,2 . Area required for the design load (pressure 25 bar):

4.2.1 - VERIFICATION THE NOZZLE FOR MANTLE - NOZZLE A3, A4 The first check will be carried out for the larger nozzle on the mantle (for the A3 and A4 nozzles of the same DN 50 and PN 50: reference "nozzle table"), one of the nozzles which in the specific case corresponds to one of the two nozzles to be connected on the cylindrical mantle:



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35

230,19151,35,54 mmsdA rr

where d1 is provided by the table EN 1092-1 (d1 in correspondence of the DN 50 nozzle, subtracting the ring of thickness s=sn)

pressure vessel available area (not stressed by design pressure):

))(2(1 rssdXA available penetration area (not stressed by design pressure)

)(22 rnn ssYA that in the hypothesis of different material between the mantle and the nozzles, it must be multiplied by the ratio of

the respective yield stresses )()(

vesselnozzle

amm

amm

The European standard EN 13445 establishes that when



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OKAAAr 21 when instead, it happens

ENTREINFORCEM21 AAAr In the specific case with reference to the European standard EN 13445, for the mantle it happens that

)90.25.2,00.45.2min()5.2,5.2min( nssY

mm25,7

where ns it is obtained from the table above EN 1092-1, that is the

thickness s of height 1h (reference to the nozzle scheme that precedes the table EN 1092-1) equal to 2.90mm. While always with reference to the European standard EN 13445, X assumes the following value:

)90.200.42

5.54,5.54max()

2,max( nssd

dX

mm5,54

Therefore the calculation of areas 1A and 2A turns out to be:

21 71,26)51,300,4()5,545,542( mmA

)90,2(25,722 rnSA



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For the calculation of rnS it is necessary to know the minimum thickness stressed by the load (pressure P=25 bar) of the project which in the hypothesis of the same material between the pressure vessel and the nozzle )5,137( 2mmNamm and with E=0.85 turns out to be:

pE

rps

ammrn

being the radius half the diameter equal to 27.25 mm, substituting, we

obtain

mmsrn 60,0508,25,13785,0

25.27508,2

mmSrn 60,0

Therefore 2A becomes 2

2 35,33)60,09,2(25,72 mmA

The sum of 1A and 2A turns out to be

35,3371,2630,191 21 AAAr

ENTREINFORCEM06,60



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Therefore, it is necessary to supply material to be carried out, for example, by

means of electrode or continuous wire welding to the extent calculated below:

4.2.2 - CALCULATION OF REINFORCEMENT AREA - NOZZLE A3, A4

The reinforcement area required in application of the European standard

EN 13445 can be determined as follows:

reinforcement area required Are

221 24,13106,60030,191 mmAAAA rre

rererenre ssYssdXA )(2)22( post



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39

nsdXB 22 you gets

rererere ssYBsA )(2 and therefore

0)2(2 2 rerere AsYBs from which it is possible to derive

4

8)2(

4

)2( 2re

re

AYBYBs

calculating the value of B e

substituting is obtained

mmB 7,489,225,545,542

4

24,1318)25,7270,48(

4

)25,7270,48( 2

res

mmsre 23,2

In the specific case, a weld must be made that allows the supply of

material distributed both on the side of the mantle and on the side of the

nozzle for the entire diameter of the nozzle, with an average thickness of

about 3.3 mm per side; the thickness must extend for about 14 mm per

side. Using, for example, a welding with adequate feed speed (cm / min)

carried out by means of a fuse electrode which melting supplies the

required filler metal; (certified that for the circumferential and longitudinal



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assemblies it is the obligation of the manufacturer to guarantee the

realization through full penetration welds or with welds of equivalent

efficiency both for simple and not simple containers such as this

expansion tank). The supply of material can also be distributed mainly on

the mantle, in which case the thickness of the intake must be increased

by the extent of which the thickness of the intake distributed on the side

of the nozzle is reduced. The extension of the supply must remain

constrained to the required reinforcement area and therefore 14 mm being

the estimate of the longitudinal area in which the increase in the stress

state is reduced; to consider that the increase in tensions around the hole

of the nozzle is about three times the tensions of the surrounding areas of

the mantle.

It is interesting to calculate the thickness through the notch effect K which

can be determined from the standardized tables such as the graph below

which appears to be approximately 2.7 ÷ 2.8 and using the relationship for

the thin shells of the upper paragraphs:



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Considering that a tension accumulation in the section equal to about

three times the tension due to the working load corresponds to a loading

pressure of about 75 bar, it is obtained that:

mmpE

rps

amm

23,95,75,1371

1605,7

1E is adopted in the calculation of res because it is considered for

comparison

the increase in thickness made only on the coat.

The thickness increase is obtained by subtracting the design thickness

from s (paragraph 2.6),

mmsre 26,697,223,9 .

By distributing the intake of material calculated by checking the

reinforcement to the nozzle only on the mantle, you gets an increase in

thickness of approximately

mmsre 76,6)6,09,2(23,22

The greater thickness obtained with the nozzle verification calculation

method is due to the processing with which the supply of material is

carried out, therefore it is not an increase in thickness obtained during the

lamination of the mantle but an increase obtained through processes such

as welds that they cannot be considered free from imperfections thus

requiring an increase in thickness (increase).



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4.3 - NOZZLE VERIFICATION FOR VALVE AND CAPE GAUGE - NOZZLE A5, A6 The reinforcement verification for the connection of the nozzle ends with

the calculations for connections A5 and A6 with reference to the "nozzle

table". These are 2 nozzles with DN equal to ½ "and for a hole of 30 mm.

The thickness Sn is 8mm e:

214,4951,314 mmsdA rr

where d is provided by the difference between the diameter of the hole and the thickness of the nozzle; with reference to the thickness s for the

latter and in determining rnS , the thickness of the thread, the propeller must be considered).

Pressure vessel available area (not stressed by design pressure):



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vesselnozzle

amm

amm

The European standard EN 13445 establishes that when

OKAAAr 21 when it happens

ENTREINFORCEM21 AAAr In the specific case with reference to the European standard EN 13445, for the mantle it happens that

)00.65.2,00.45.2min()5.2,5.2min( nssY

mm00,10

where ns is 6mm, obtained from the EN 1092-1 tables.

While always with reference to the European standard EN 13445, X assumes the following value:

)00.600.42

14,14max()

2,max( nssd

dX

mm17



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21 8,9)51,300,4()14172( mmA

)00,6(00,1022 rnSA

For the calculation of rnS it is necessary to know the minimum thickness stressed by the load (pressure P = 25 bar) of the project which in the hypothesis of the same material between the pressure vessel and the nozzle )5,137( 2mmNamm and with E = 0.85 turns out to be:

pE

rps

ammrn

being the radius half the diameter equal to 7 mm, replacing is obtained,

next

mmsrn 15,0508,25,13785,0

00.7508,2

mmSrn 15,0

Therefore 2A becomes

22 20,117)14,000,6(00,102 mmA




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20,1178,914,49 21 AAAr

OK 00,127

4.4 - VERIFICATION, NOZZLE ON FONDI - NOZZLE A1, A2 The second check will be carried out for the nozzle of smaller size on the bottom, the nozzle A1 (nozzle PN 50 and DN 25) of the "nozzle table" The area loaded by the design pressure result to be:

229,5701,25,28 mmsdA rr

where d is provided by the table EN 1092-1 (d1 at the DN 50 nozzle)



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Since it is a thin-walled thickness, the thickness of the bottoms is chosen equal to the thickness of the coat and therefore equal to 4mm=s. Pressure vessel available area (not stressed by the design pressure):




vesselnozzle

amm

amm

With reference to the following verification conditions

OKAAAr 21

ENTREINFORCEM21 AAAr the technician proceeds with the calculation of Y and X:

mmssY n 50,6)60.25.2,00.45.2min()5.2,5.2min(

where ns equal to 2.60mm is obtained from the table above EN 1092-1,

that is the value of the thickness s of height 1h (reference to the nozzle scheme that precedes the table EN 1092-1). While X takes the following value:

)60.200.42

5.28,5.28max()

2,max( nssd

dX



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mm5,28


21 72,56)01,200,4()5,285,282( mmA

)60,2(50,622 rnSA

For the calculation of rnS it is necessary to know the minimum thickness stressed by the load (pressure P = 25 bar) of the project which in the hypothesis of the same material between the pressure vessel and the nozzle )5,137( 2mmNamm and with E = 0.85 is result to be:

pE

rps

ammrn

being the radius half the diameter equal to 14.25 mm, replacing is

obtained

mmsrn 31,0508,25,13785,0

25.14508,2

mmSrn 31,0

Therefore 2A it becomes 2

2 77,29)31,060.2(50,62 mmA



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77,2972,56875,49 21 AAAr

)(49,86 OK

___________________________

(°)If the manufacturer had chosen to use the thickness for the spherical seeds bottoms equal to 2.01mm (therefore 2mm), it is easy to verify that it is necessary to make reinforcement on the nozzle connection; it is also easy to verify that to avoid reinforcement, just increase the bottom thickness by one millimeter (3mm). However for thin-walled containers it is equaled to the thickness of the coat and therefore to 4mm.

Nozzle A2. To complete the check of the nozzles on the bottoms it is necessary to carry out the reinforcement calculation for the nozzle A2 with PN 50 and nominal diameter DN equal to 50, then the twice the nozzle A1 (reference "nozzle table"). The area loaded by the design pressure is:

255,10901,250,54 mmsdA rr

where d is provided by the table EN 1092-1 (d1 at the DN 50 nozzle) As for the calculation of the previous point: since it is a thin-walled thickness, the thickness of the bottoms is chosen equal to the thickness of the coat and therefore equal to 4mm=s. Pressure vessel available area (not stressed by the design pressure):



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vesselnozzle

amm

amm

With reference to the following verification conditions

OKAAAr 21

ENTREINFORCEM21 AAAr il tecnico procede con il calcolo di Y e Y:

mmssY n 25,7)90,25.2,00.45.2min()5.2,5.2min(

where ns is obtained from the table above EN 1092-1, i.e. the thickness

s of height 1h (nozzle scheme reference preceding the table EN 1092-1) equal to 2.90mm. While X takes the following value:

)90,200.42

5.54,5.54max()

2,max( nssd

dX

mm5,54



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21 46,108)01,200,4()5,545,542( mmA

)90,2(25,722 rnSA

Per il calcolo di rnS è necessario conoscere lo spessore minimo sollecitato dal carico (pressione P = 25 bar) del progetto che nell'ipotesi dello stesso materiale tra il recipiente a pressione e l'ugello )5,137( 2mmNamm e con E = 0,85 risulta essere:

pE

rps

ammrn

essendo il raggio metà del diametro pari a 27,25 mm, si ottiene la

sostituzione

mmsrn 60,0508,25,13785,0

25.27508,2

mmSrn 60,0

Therefore 2A it becomes 2

2 35,33)60,090,2(25,72 mmA




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35,3346,108375,95 21 AAAr

)(81,141 OK

___________________________

(°) Anche in questo caso se il costruttore scegliesse di non utilizzare lo stesso spessore del mantello, per i

fondi semisferici, ma lo spessore di 2,01mm (quindi 2mm) si verifica facilmente la necessità di apportare

materiale sul collegamento del nozzle; e come nel calcolo precedente basta aumentare lo spessore del

fondo di un millimetro (3mm) per eliminare la necessita di apportare rinforzo. Tuttavia trattandosi di un

recipiente a pareti sottili lo spessore dei fondi semisferici viene scelto uguale allo spessore del mantello e

quindi pari a 4mm.



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5 - FATIGUE STRESS ANALUSIA The trend of the loads that urge the expansion tank is shown below, in the event of

constant loads, therefore, in the absence of oscillating loads:

In the hypothesis of constant loads, the expansion vessel is not stressed by any

variable load, therefore with periodic oscillations such as symmetrical or

asymmetrical cyclic loads and therefore it is not necessary to carry out any study on

the tension state that could cause fatigue failure, not there are periodic variations of



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53

the loads and therefore of the tension state generated which may affect the sizing.

In this case, the calculation ends with the following table.

5.1 - RESULTS PRESSURE VESSEL DIMENSIONING: EXPANSION TANK With reference to the calculations and related analyzes performed, the following

values are adequate for the sizing of the expansion tank:

Cylindrical coat thickness = 4 mm

Hemispherical bottom thickness = 4 mm

The A1 and A2 nozzle connections do not require reinforcement.

A5 and A6 nozzle connections do not require reinforcement.

Reinforcement to be applied to the A3 and A4 nozzles connected on the shell:

• reinforcement area = 184.65 mm2

• reinforcement thickness to be evolved for the entire circumference of the

connection = 2.23 mm

• thickness length on both sides = approx. 14.7 mm.

Steel used for the construction of the expansion tank: P355NH steel (for

references, annexes according to UNI EN 10028).

Coefficient of welds made with non-automatic procedure E = 0.85 (spot checks:

10%).

Increase in vessel thickness for non-automatic welding process 1.15



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Type of welds required: carried out by qualified personnel and at total depth for the

surfaces in contact to be welded and for the nozzles with a reinforcing area with a

meltable electrode or with the addition of material in the measurements indicated

above.

Adopted safety factor 2 (calculation report details)

Verification of fatigue resistance not required since the action of periodically

oscillating loads is not detected.

Further information is provided by the technician in the calculation report and

related project drawings of the expansion tank.

5.2 - OSCILLATING LOADS

If we assume variable loads with periodic oscillation it is necessary to carry out the

fatigue verification; therefore we assume the existence of a load as shown below:

The following 3 loading hypotheses are assumed:



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55

1) Pmax=25 bar; Pmin= 20 bar;

2) Pmax=25 bar; Pmin=15 bar;


for an average oscillation of every 5 minutes.

With reference to the mantle of the expansion vessel, further checks on the bottoms

being unnecessary being the manufacturer's choice, for the latter, the same

thickness of the shell and considering that the stress state in the hemispherical

bottoms is about half of the shell,

the oscillating loads of the three hypotheses above provide for the thickness of

4mm the following values of max and min

)1 2max 00,100mm

N

2min 00,80mm

N

)2 2max 00,100mm

N

2min 00,60mm

N

)3 2max 00,100mm

N

2min 00,20mm

N

In application of the EN 13445 standard, fatigue verification is carried out using the

following calculation formulas:

kP

PK

P

PdS

maxmax

)1( dove K

k1

where is the design tension or the equivalent corresponding to the maximum

load. It is obvious that dS it is also equal to



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maxmax

minmax

max

minmax )1( kP

Pk

P

PPdS

dSkk )()1( minmaxmaxmax

min

Before proceeding with the determination of the life of the expansion vessel using

the SN curve of Wöhler, it is necessary to calculate the modifying factor of the

fatigue limit of a material, K, which is defined in the following ways (with reference

to the design data provided by the manufacturer, i.e. analysis of loads, shape

details and surface finishes):

Shigley ka - surface finish, kb - dimensions, kc – gradient, type of tension, reliability kd - operating temperature, ke - other causes kf - carving factor

where fedcba KKKKKKK

Juvinall Cs - surface finish, Cg - dimensions / gradient of tension, Cl - type of tension kf - carving factor

where flqs KCCCK

Using Shigley's method, we will assume the following values:



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ka = 1 (fine surface polishing) , kb = 1 (being the thickness less than 8mm), kc = 0,868 (essendo trazione per un'affidabilità di circa il 95%), kd = 1 (the operating temperature being lower than 350 ° C) ke = 0,97 (residual tensions) kf = 1 (no carvings) resulting in a value of K (capital K) equal to

842,0197,01868,011 K and of k (lowercase k) equal to

19,11

kk

1) FIRST CALCULATION HYPOTHESIS

1) Pmax=25 bar; Pmin= 20 bar;

)1 2max 00,100mm

N

2min 00,80mm

N

applying the Tresca criterion, the maximum equivalent turns out to be

250,102

mm

N

from which, it turns out that



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MpaKP

PdS 40,2450,10219,1

25

5

max

from the following Wöhler S-N diagram we obtain the number of probabilistic cycles

before breaking due to fatigue.

In the first hypothesis we have an infinite life of the expansion vessel, therefore

being able to work for an unlimited number of cycles.



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2) CALCULATION HYPOTHESIS


)2 2max 00,100mm

N

2min 00,60mm

N


250,102

mm

N

from which, it turns out that

MpaKP

PdS 79,4850,10219,1

25

10

max

In the second load hypothesis, by reading the number of cycles N in Wöhler's S-N

diagram, we obtain:

with a probability of breaking at 32%, a life of approximately 3.06E+6 cycles is

found

with a 40% probability of breakage it results in a life of approximately 5.73E+6

cycles

with a probability of breakage greater than 40%, an unlimited life results.



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60

From an analysis of the calculations of the upper sub paragraphs results in an

unlimited life if the manufacturer chooses the thickness of 6 mm; in this case there

would also be a reduction in the supply of material for reinforcement, a reduction of

about 60%, easily deducible from the analysis of the calculations relating to the

verification of the reinforcement for the nozzles connected on the mantle; no

reinforcement is required for the other nozzles connected to the expansion tank. As

for spherical seeds bottoms, there is an unlimited life even with a thickness of 4

mm.



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3) CALCULATION HYPOTHESIS


)3 2max 00,100mm

N

2min 00,20mm

N


250,102

mm

N

from which it results that

MpaKP

PdS 58,9750,10219,1

25

20

max

In the third load hypothesis, from reading the number of cycles N in Wöhler's S-N

diagram, we have a limited life which, based on the probability of failure, assumes

the following maximum cycle numbers:

with a probability of breakage of 32%, a life of approximately 7.04E+4 cycles

results,


cycles,

with a probability of breakdown of 56%, a life of approximately 4.60E+5 cycles

results,


cycles,

with a probability of breakdown of 71%, a life of approximately 7.30E+5 cycles

results,

with a failure probability of 80%, a life of approximately 8.71E+5 cycles results,



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cycles,

with a probability of breakage greater than 90%, an unlimited life results.

From an analysis of the calculations made in the previous sub paragraphs would

require a thickness of 10 mm to obtain an unlimited number of cycles with a

probability of breaking of 40%; in this case the analysis of the calculations for the

verification of the reinforcement to the nozzles shows the possibility of not adding

reinforcement material with reference to the nozzles connected to the mantle, not

requiring additions for the other nozzles even with lower thicknesses (4mm). In this

case the manufacturer could choose a different thickness for the spherical seeds

bases, or a thickness equal to 5 mm.



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6 - CALCULATION ANALYSIS WITH COMBINATION OF LOADS

From the analysis of the loads that will solicit the expansion tank, only the changes

in operating pressure and temperature are relevant at the end of the design.

The trend of the elastic limit with the temperature can be represented by the graph

below:

Therefore an increase in temperature of more than double the operating

temperature (from 220 to 450 °C) would reduce the elastic limit by 40%.

Considering that the maximum yield point for the material used, at 220 °C, is 293

MPa and that at double the temperature would be 207 MPa it would still be higher

than the stress state of the mantle (maximum stress section) MPamt 36,100

considering the constant pressure at operating pressure also in application of the

other loads considered with the safety factor of 2.

If a pressure peak equal to twice the operating pressure is assumed with a

consequent increase in temperature of at least 100 °C it would happen that the



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64

elastic limit of the material would assume the value of 266 MPa, the stress state in

the section of maximum stress would become MPamt 00,200 allowing the

pressurized equipment to remain in operation in the elastic range with a safety

coefficient of 1.33 (considering that the safety coefficient considers the other load

conditions that can occur with less probability than the operating loads).

If an increase in temperature is assumed with a total peak of 450 °C combined with

the pressure peak of 50 bar, the stress state in the shell would increase, to 200

MPa and the elastic limit would drop to 207 MPa which in the absence of other

loads the the apparatus would remain in the elastic field, otherwise it would arrive at

the plastic field while remaining below the breaking limit. To be considered that this

last case is highly unlikely since the pressure equipment connected to a system

that operates at a pressure of 25 bar and with a temperature below 450 °C being

the operating temperature of the expansion vessel equal to 220 °C.

In order to guarantee further safety in the initial determination of the elastic limit, the

additional safety of considering the yield s at the temperature of 220 ° C equal to

275 MPa instead of 293 MPa as shown in the above graph has been adopted, in

order to consider any frequent increases of temperature with respect to the

operating temperature of 220 ° C.



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6.1 - ANALYSIS OF CALCULATIONS WITH ALTERNATIVE METHODS

In order to obtain the certainty of the result, the designer, in addition to simulating

and combining the presence of peak loads and assumed regulatory obligations as

enshrined in the PED directive itself, performs further checks on the results

obtained by adopting different calculation models with the intention to make a

comparison. In this alternative calculation analysis, the ASME VIII, Division II

calculation model is adopted, which provides for the calculation of the pressure

necessary for the complete plasticization of the section, according to the Tresca

criterion.

Considering the equation of radial equilibrium from the Lamé problem and the

plasticization condition according to Tresca (valid for each r) the following system of

differential equations is defined

srrrr

rrrr dr

dr

C

from which, it is obtained

rdr

d srr

with the following boundary conditions

0)(

)(.

sR

PRCC

rr

rr

By solving the integral, you get



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66

Crr srr )ln()(

from which applying the boundary conditions

0)( sRrr pRrr )(

risult

sR

rr srr ln)(

therefore, it is also true that

sR

RR srr ln)(

that

s

P

sR

R

ln

and that

s

P

eR

sR

from which it is possible to derive t in the following expression

1s

P

eRs

that with sm ES it results to be

1ES

P

meRs



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67

valid for any thickness or for any ratio Rs

.

The maximum plasticization of the section, for the construction material at a

temperature of 220 ° C, according to the calculation method ASME VIII - Division 2

and in application of paragraph 7.1.2 of Annex I of the PED directive (2014/68/UE)

must not exceed the following value:

5,1,

4,2min

5,1,

4,2min SRy

m

UTSS

MPa75,19343.196,75.193min

being the ammm MPaS 5,137 value calculated above with a safety factor

of 2, less than 193.75 MPa, it is suitable for determining the minimum thickness

using this calculation method:

mmes 46,31160 5,13785.0

5,2

since it is not necessary to carry out further checks with reference to the spherical

seeds bottoms where the stress state is about half of the stress state generated in

the mantle and being easily verifiable that the minimum thickness required for the

nozzles is identical to the thickness calculated with the method envisaged by the

EN standard 13445 (DBF) for the thin thicknesses used in the paragraph relating to

the verification of reinforcement for nozzle, in fact it turns out:

mmmmesA rn 31.0308,0125,141 5,13785.0

5,2



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68

mmmmesA rn 60.0589,0125,272 5,13785.0

5,2

mmmmesAA rn 60.0589,01250,274,3 500,137850.0

500,2

mmmmesAA rn 15.0151,01000,76,5 500,137850.0

500,2

Therefore, considering the choice of the thickness equal to 4 mm for the mantle and

compared with the result obtained through the alternative calculation method, equal

to 3.46 mm, the certainty that the designer requires to guarantee the conditions of

maximum safety is considered achieved, considered the operating conditions of the

expansion vessel.



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6.2 – Elasto Plastic Stress Analysis: deformation analysis, method DBA

(ASME VIII Div. 2)

With reference to the Protection Against Local Failure verification, in the Elasto

Plastica analysis the admissible limit deformation is given by the following function,

which takes into account the effects of the triaxiality of the tension state:

3

1

311

2 e

sl I

m

LuL e

with

)(3211

I

and the equivalent tension according to Von Mises given by

2

)()()( 231

232

221

e

that, in ordinary operating conditions,

n 1 , r 2 , m 3 , risult

2

)()()( 222mnmrrn

e



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70

MPa06,89

MPaI 97,1471

From the following table it is possible to calculate the admissible limit plastic

deformation before the collapse of the material



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71

from which, it turns out:

3

1

06,893

97,147

245,01

2,2

245,0 eL

166,0L .

By repeating the calculations it is easy to verify that up to the e

, calculated with

the Von Mises criterion, close to the collapse stress state, the value of the limiting

plastic deformation (collapse) remains approximately the same value:

166,0L

for higher values it would have no meaning since the structure collapsed.

By conducting a complete elasto-plastic analysis in large displacements for the

verification against plastic collapse, it is necessary to impose the verification of the

following relationship:

Lcfpeq

where in addition to the above defined L remain to be identified:

peq= plastic deformation accumulated in the considered point

cf= plastic deformation possibly accumulated during the forming process



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72

The cf deformation is due to the possible presence of residual plastic

deformations linked to previous cold plastic deformation processes and can only be

omitted if the product undergoes stress relieving heat treatments before installation;

hot forming processes may also be negligible.

In the determination of peq , the areas in which the stress state undergoes

accumulations of plastic deformations are considered which for the present

expansion tank and with reference to the verification of the reinforcement for nozzle

(upper paragraphs) caused a supply of material for reinforcement on the mantle,

while for spherical seeds substrates, the choice of thickness equal to the thickness

of the mantle in order to increase the yield strength in said most stressed areas. For

the connection areas between nozzle and bottoms it was not necessary to make

reinforcement being the thickness of the nozzles and the greater thickness of the

bottoms sufficient to support the localized increase in tensions.

Therefore an analysis of the deformations at the limit of the elastic field is carried

out considering the tension state n , m , r , determined by the yield

stress that in the absence of other stresses and at the operating temperature of

220 °C is reached by a pressure peak less than 69 bar = 6,90 MPa:

MPan 00,275

MPam 50,137

MPar 90,6



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The analysis of deformations at the limit of the elastic field is carried out by applying

Hooke's law for the constitutional bond of the material which developed through

Navier's theory provides the following relationships:

)(

1mrnn E

)(1

mnrr E

)(1

rnmm E

It must be said that in this type of analysis a plane stress state is assumed which

can be used for reduced longitudinal lengths, however it can also be used for

longer lengths since the results are affected by approximation by excess, in fact in

the above equations by setting m equal to zero it is easy to verify that the sum of

the three deformations (modules) would be 00222,0 greater than the

deformation calculated below assuming a three-component stress state for an

isotropic material, whose value turns out to be 00211,0 . Despite the

previous observation, the hypothesis of a state of plane tension is not used below,

since it is not necessary to carry out further increases in the calculations and a

more precise study of the deformations is necessary both in the elastic field and in

the precise plastic field. Once the three deformations at the limit of the elastic head

have been determined, the technician proceeds to determine the variation of the

thickness and then to the relative e .



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Being Young's modulus equal to GPaE 215190 and considering that the

operating temperature causes a variation of about 95% (indicative graphic

reference above), for the material used the value of GPaE 197 already

reduced is adequate to consider the thermal effects at the operating temperature

equal to C220 ; the Poisson coefficient assumes values between

3,028,0 which for the material used is adequate for 28,0 . With

reference to the characteristics of the above material, the deformation is:

ndeformatio ntialcircumfere mantle00119,0

)50,13728,090,628,000,275(10197

13

n

therefore under the assumed limit load conditions the elastic limit deformation of the

mantle in the circumferential direction (increase in the circumference) is

approximately:



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75

mmrl nn 20,128,6.16000119,02

elongation which causes a variation in the radius of

mmrr n 190,016000119,0

)(reductionn deformatio thicknessmantle000621,0

)50,13728,000,27528,090,6(10197

13

r

the elastic radial deformation of the mantle (elastic reduction of the thickness) is

approx:

mmsl rr 0025,04000621,0

ndeformatio allongitudin mantle000297,0

)90,628,000,27528,050,137(10197

13

m

the elastic longitudinal limit deformation of the mantle (elongation of the mantle) is

approx:

mmll mm 120,0400000297,0

The conservation of the material that forms the expansion vessel implies that

0)( mrnmsn lslllll



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76

and allows you to calculate s , the total thickness reduction, as follows:

mmllls mrn 313,0

Therefore the deformation of the thickness at the limit of the elastic field turns out to

be:

etot

s

s

0783,04

313,0

0

The plastic deformation limit frist before breaking, 166,0L , determines a

possible reduction of the thickness of no more than mmrp 664,0 that

the accumulation of deformation peq can cause.

In the elastic analysis conducted, the condition required by the Elastic Stress

Analysis of ASME VIII, Div. 2 (DBA method), it is verified.:

S4321 or ammrmn 4 than replacing

50,137490,650,13700,275

5506,405



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6.3 – Plastic analysis

Considering that during the elastic deformation there are increases in the stress

state due to rigid connections such as the pipes that connect the expansion tank to

the system, it is possible to calculate the plastic deformation generated in the

elastic limit condition of the load, also caused by increases in the stress state in

presence of material defects; therefore multiple increases in the stress state are

assumed starting from 20% beyond the elastic limit (yield point of the material) at

the operating temperature of 220 ° C, to evaluate the plastic deformation of the coat

thickness, resulting from the accumulation of deformations plastics in the three

main directions of tension.

1) First peak load 20% greater than the elastic limit load

MPan 00,330

MPam 00,165

MPap 28,8

Using the Ramberg-Osgood law as a constitutional bond in the plastic field:

ntp K or

n

pt K

1

and therefore the total deformation turns

out to be

n

pep

et H

1

where H and n for low carbon hot rolled steel assume the following values:



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H = stress coefficient assumed equal to 965 MPa

n = Hardening exponent assumed equal to 0,14

The calculating the plastic deformation in the direction of maximum tension

pn risult:

00166,000,965

00,33000119,0

14,01

p

nenn

the circumferential deformation, considering the conservation of the mass that

forms the container, results in a reduction in thickness equal to mm266,0 of

which mm0751,0 is plastic.

The calculating the plastic deformation in the longitudinal direction with

pm risult

000300,000,965

00,165000297,0

14,01

p

memm

longitudinal deformation (considering mass conservation) results in a reduction in

thickness equal to mm120,0 of which mm00133,0 is plastic.

In the radial direction only elastic deformation results, therefore

000621,0 em

pm

emm

the radial deformation (considering mass conservation) results in a reduction in

thickness equal to mm0025,0 .



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79

Therefore the total reduction of the thickness, sum of the elastic reduction plus the

plastic reduction turns out to be tsmm 390,0 of which

psmm 076,0 is plastic; the corresponding deformation turns out to be:

166,00975,04

390,0

0

Lt

t s

s

while 'p it results to be

019,04

076,0

0

'

s

spp



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2) Second peak load 30% greater than the elastic limit load

MPan 389

MPam 50,194

MPap 97,8

in determining the n the reduction in thickness of the vessel due to the previous

plastic deformation is considered.

The calculating the plastic deformation in the direction of maximum tension

pn results:

00271,000,965

00,38900119,0

14,01

p

nenn

the circumferential deformation, considering the conservation of the mass that

forms the container, results in a reduction in thickness equal to mm434,0 of

which mm243,0 is plastic.

The calculating the plastic deformation in the longitudinal direction with

pm results

000308,000,965

50,194000297,0

14,01

p

memm



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81

longitudinal deformation (considering mass conservation) results in a reduction in

thickness equal to mm123,0 of which mm00423,0 is plastic.

In the radial direction only elastic deformation results, therefore

000621,0 em

pm

emm

In the radial direction only elastic deformation results, therefore mm0025,0 .

the total thickness reduction is obtained as the sum of the elastic reduction plus the

plastic reduction, which turns out to be tsmm 560,0 of which

psmm 247,0 is plastic; the corresponding deformation is:

140,04

560,0

0

s

stt

The admissible limit deformation being verified by the following report

Lcfpeq

where cf is assumed to be zero, peq remains to be calculated given the sum

of the total deformation generated by the second peak plus the residual plastic

deformation generated by the first load peak:



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82

166,0160,0140,0019,0' tp

while p it results to be

062,04

247,0

0

''

s

s pp

Under such conditions the container reaches the plastic collapse conditions,

furthermore pressure peaks would cause the certain collapse.

******** FINE CALCOLI DI PROGETTO **********



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VII - Sources for research:

Norma UNI EN 13445 – Pressure vessel | ASME VIII div.2 - Rules for

Construction of Pressure Vessels - Alternative Rules, The American Society

of Mechanical Engineers, New York, U.S.A. | UNI EN 13445, Unfired

pressure vessels, Background to the rules in Part 3, Design -

copy | DIRECTIVE (UE) 2014/68, PED | Legislative Decree number 26 of 15

February 2016 | Notes from the Mechanical Engineering Course, Porf. ing. A.

De Iorio, University of Engineering, Federico II, Naples, Italy | MASTER

DEGREE COURSE IN MECHANICAL ENGINEERING, MACHINE

CONSTRUCTION COURSE, University IN SUPREMEA DIGNITATIS 1343

| Steel P355NH | A - UNI EN 1092-1, flanges | B - UNI EN 1092-1, flanges |



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VIII - BIBLIOGRAPHY. For research, we referred to the quotes below.

PUBLICATION 2020-21,06 1) Standard UNI EN 13445 - Pressure vessels. The standard applies to containers that are not exposed to flame and subjected to a maximum permissible pressure greater than 0.5 Bar. The standard defines the requirements for design, materials, manufacture, inspection and tests. UNI EN 13445 is harmonized standard for Directive 2014/68 / EU "PED". 2) ASME VIII div. 2 - Rules for Construction of Pressure Vessels - Alternative Rules, The American Society of Mechanical Engineers, New York, U.S.A. This division of Section VIII provides the requirements applicable to the design, manufacture, inspection, testing and certification of pressure vessels operating at internal or external pressures above 15 psig. Such ships can be fired or not burned. This pressure can be obtained from an external source or by applying heat from a direct or indirect source, or any combination of it. These rules provide an alternative to the minimum requirements for pressure vessels under the rules of division 1. In comparison, the requirements of Division 1, Division 2 on materials, design and non-destructive testing are more stringent; however, higher stress values which intensify the values are allowed. The rules of division 2 concern only ships that must be installed in a fixed place for a specific service where the control of operation and maintenance is maintained during the useful life of the ship by the user who prepares or has the design specifications prepared. . These rules may also apply to human-occupied pressure vessels in general in the diving sector. The rules relating to the use of the UME and UV ASME product certification marks are also included. 3) UNI EN 13445, Unfired pressure vessels, Background to the rules in Part 3, Design. The European standard EN 13445 "Pressure tanks not powered" provides a precedent in this after 10 years of discussions between experts, a European consensus has been reached in the field of pressure equipment. Part 3 which was prepared by a group of expert European leaders under the guidance of Dr. Fernando LIDONNICI, Sant'Ambrogio, (Milan, Italy), represents a great progress in European technical convergence. The adoption of the first release of the EN 13445 standard in May 2002 was the first step in a continuous process of development and improvement. This new standard benefits from the contribution of the whole European experience; as such, it includes innovative capabilities and provides solutions for modern subjects. The CEN design rules promote limit analysis and design by analysis - Direct path. Design has strategic importance for the future and competitiveness of the pressure equipment industry. The optimal design offers substantial advantages such as: thickness reduction and damage control in service with increased safety and drastic maintenance costs reduction. The goal of this book is to explain the background of these rules, to help the industry apply them most effectively. It was initiated by EPERC, the European Pressure Research Equipment Council, and a contract was awarded by CEN, the European Standardization Committee, with the support of the European Commission.



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4) DIRECTIVE 2014/68 / EU OF THE EUROPEAN PARLIAMENT AND OF THE COUNCIL of 15 May 2014 concerning the harmonization of the laws of the Member States relating to the making available on the market of pressure equipment; (recast), (Text with EEA relevance). 5) Legislative Decree number 26 of 15 February 2016. Implementation of Directive 2014/68 / EU of the European Parliament and of the Council of 15 May 2014 on the harmonization of the laws of the Member States relating to the making available on the market of pressure equipment ( recast). 6) Notes from the Mechanical Engineering Course, Porf. ing. A. De Iorio, University of Engineering, Federico II, Naples, Italy. 7) MASTER DEGREE COURSE IN MECHANICAL ENGINEERING, MACHINE CONSTRUCTION COURSE, University IN SUPREMEA DIGNITATIS 1343. 8) Steel P355NH. Steel P355NH. HOT ROLLED COIL STEEL SHEETS Quality: P355NH. Steel for pressure vessels. Quality P355NH. EN10028-3 standard. W. Nr. 1.0565. Rolling state N - Normalized. Brief description: Fine grain structural steel for pressure vessels. Good weldability and good hot and cold forming properties in the normalized lamination state. Usage applications Pressure vessels, boilers, steam boilers, pressure pipes, heat exchangers and compressors. 9) A - UNI EN 1092-1, flanges: FLANGE UNI 3, FLANGE EN 1092.1, ASME B16.5, SPECTACLE BLIND, LONG COLLAR FLANGES, ASME B16.47 A-B, B.S. 3293, AWWA C207, ASME B 16.36 ORIFICE FLANGES, API SPEC. 6 B - 6 BX, DIMENSIONS. 10) B - UNI EN 1092-1, flanges: Bolts for UNI flanges, Flat flanges UNI EN 1092-1 PN6 (EX UNI 2276-67 PN 6), Flat flanges UNI EN 1092-1 PN10 (EX UNI 2277-67 PN10), Flat flanges UNI EN 1092-1 PN16 (EX UNI 2278-67 PN16), Flat flanges UNI EN 1092-1 PN25 / PN40 (EX UNI 6083-67 PN25 - UNI 6084-67 PN40), Collar flanges UNI EN 1092-1 PN6 ( EX UNI 2280- 67 PN6), Collar flanges UNI EN 1092-1 PN10 (EX UNI 2281- 67 PN10), Collar flanges UNI EN 1092-1 PN16 (EX UNI 2282- 67 PN16), Collar flanges UNI EN 1092 -1 PN25 (EX UNI 2283- 67 PN25), Collar flanges UNI EN 1092-1 PN40 (EX UNI 2284- 67 PN 40), Collar flanges UNI EN 1092-1 PN64 / PN100 (EX UNI 2285-67 PN64 - UNI 2286-67 PN100), Blind flanges UNI EN 1092-1 PN6 (EX UNI 6091- 67 PN6), Blind flanges UNI EN 1092-1 PN10 (EX UNI 6092- 67 PN10), Blind flanges UNI EN 1092-1 PN16 (EX UNI 6093- 67 PN16), Blind flanges UNI EN 1092-1 PN25 / PN40 (EX UNI 6094-67 PN25 - UNI 6095-67 PN40), Blind flanges UNI EN 1092-1 PN63 / PN100 (EX UNI 6096-67 PN64 - UNI 6097- 67 PN100), Free flanges UNI EN 1092-1 PN6 (EX UNI 6088 PN6), Free flanges UNI EN 1092-1 PN10 / PN16 (EX UNI 6089-67 PN10 - UNI 6090-67 PN16), Threaded collar flanges UNI EN 1092-1 PN6 / PN16 (EX UNI 2253/54 PN6 / PN16), Aluminum flanges UNI 6089 DIN 2642, Sealing surface of flanges UNI EN 1092-1, Type "E" with mouth (EX UNI 2225 SM), Type "F" recessed (EX UNI 2225 SF), Flange sealing surface UNI EN 1092-1, Type "C" with tongue (EX UNI 2226 SM), Type "D" with groove (EX UNI 2226 SF), Surfaces of step seal of flanges UNI EN 1092-1 (EX UNI 2229-67), tolerances flanges UNI EN 1092-1, flanges ANSI class 150 LB, flanges ANSI class 150 LB, flanges ANSI class 300 LB, flanges ANSI class 300 LB, ANSI flanges class 600 LB, ANSI flanges class 600



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LB, Approximate weights of the flanges ANSI B 16.5, Finishing contact faces, Machining tolerances ANSI.


Expansion vessel. Pressurized equipment design. PED.

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