ENHANCEMENTS TO THE EUROPEAN PRESSURE VESSEL CODES: A FOCUS ON CONICAL COMPONENTS UNDER INTERNAL PRESSURE

International Journal of Recent Advances in Mechanical Engineering (IJMECH), Vol.12, No.4, November 2023
DOI:10.14810/ijmech.2023.12402 11
ENHANCEMENTS TO THE EUROPEAN PRESSURE
VESSEL CODES: A FOCUS ON CONICAL
COMPONENTS UNDER INTERNAL PRESSURE
Isaak Dassa and Dimitrios Mertzis
My Company Projects, Olimpiou Diamanti 20, GR 54626, Thessaloniki, Greece
ABSTRACT
This paper presents an analysis of conical components in pressure vessels emphasizing their significance
when utilized to transition between different diameters or slopes. It aims to propose enhancements to the
European Pressure Vessel Codes concerning conical components under internal pressure focused on
improving safety, reliability, and compliance with industry regulations. The influence of cone apex angles,
shell geometries and design pressure on cone plate thickness calculations is presented. Cylinder diameters
between 1000 and 2500 mm and cone angles between 30° and 60° are inputs used to quantify the
difference in cone thickness for a design pressure between 10 and 200 barg. The proposed method leads to
increased cone thickness compared to AD2000 up to 2.2 mm for the selected range of applications
resulting in safer pressure vessel design. Moreover, the proposed approach leads to thinner cones (up to
22 mm for studied range) resulting in significant material savings during manufacturing.
KEYWORDS
Pressure vessels, internal pressure, cone junction, EN13445, AD2000, cone thickness optimization
1. INTRODUCTION
Pressure vessels are containers designed to hold gases or liquids at different pressures. Due to
their importance in many fields like petrochemical, aerospace, and nuclear industries, pressure
vessel design has been an active area of research for many decades. One key aspect of pressure
vessel design is the optimization of conical components, which are crucial elements included in
many types of pressure vessels. The optimization objective of pressure vessel design problems
involving conical components is typically to minimize the total cost while ensuring structural
safety and meeting other requirements such as weight, strength, and durability.
Computing pressure vessels is a complex undertaking, comprising numerous mathematical
expressions, empirical equations, diagrams and tables. These formulae are collated in reference
manuals titled "Codes" that were initially assembled by engineering societies from nations with
established mechanical engineering background. The Codes offer not only the calculation
formulas but also additional technical directives for vessel construction including proper material
selection and necessary thermal treatments to alleviate stress (annealing) to fabricate safer
vessels.
There are a total of five main available codes, ASME VIII – USA [1], EN13445 –European
Union [2], AD2000 – Germany [3], PD5500 – United Kingdom [4], which prevail in the pressure
vessel market. In general, the calculation codes for pressure vessels of European countries AD
Merkblatt (Germany), Codap (France), Ispesl VSR (Italy), and BS5500 (UK) were reissued in
2000 (under the names AD2000, CODAP2000, PD5500) to incorporate the requirements of the

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newly established at the time European legislation on pressure vessels PED 97/23/EC [5] (which
was reissued in 2014 as PED 2014/68/EU [6]).
The pressure vessel code AD2000, which has a long history of use, remains valid and is favored
by designers due to its streamlined calculation techniques that permit manual design. Its
implementation also enables the creation of industry-leading lightweight vessels as a result of the
reduced safety margins; however, this necessitates skilled mechanical engineers and limits user
accessibility. In conjunction with German DIN standards, it offers solutions for automated
production line manufacturing processes. Furthermore, there are stringent material quality
demands associated with its utilization.
The European Standard EN13445 had replaced the pressure vessel design regulations of Britain,
but after Brexit in 2020, the latter was reinstated. This standard is renowned for its exceptional
quality and comprehensively addresses issues that other codes fail to cover such as evaluating
fatigue in nozzle joints due to connected pipe loads, determining backing ring dimensions for
floating head heat exchanger tube sheets, and calculating horizontal support structures (saddle)
for vessels. The use of PD5500 was once restricted to materials conforming only to British
standards, limiting its global application. However, the 2000 revision replaced older codes with
European standards and enabled English foundries to collaborate with their European
counterparts for steel production toward "European unification." This modification facilitated the
worldwide construction of vessels using PD5500. It is noteworthy that PD5500-compliant vessels
conform to the thickness standards equivalent to those of AD2000, albeit with marginally lower
material quality specifications. This renders it a cost-efficient option but with slightly diminished
quality expectations.
The pressure vessel design code of the European Union, commonly known as the "Euro-code," is
a highly advanced and comprehensive standard that employs sophisticated methods and
equations. The use of this code results in vessels with exceptional specifications and demands
wall thicknesses within the range of ASME and AD2000/PD5500. One significant advantage it
offers is its full compliance with P.E.D legislation standards, which are harmonized. However,
owing to its rigorous quality requirements, constructing vessels using this code may incur higher
costs than those manufactured under ASME regulations.
Conical components are crucial elements that play a significant role in many types of pressure
vessels, and their optimization is essential to ensure structural safety, meet industry requirements,
and minimize costs. Conical components can be found in pressure vessels where a change in
diameter or slope is required, such as at the junction of two cylindrical sections or at the
transition between a cylindrical section and a dished end. Furthermore, conical components can
also be used to minimize the overall weight of a pressure vessel without sacrificing its strength or
integrity. The utilization of conical components can enhance the efficiency of pressure vessels by
minimizing turbulence and pressure drop. This facilitates an even fluid flow transition that
diminishes potential erosion and corrosion on the walls of said vessel. Thorough design and
analysis must be conducted to guarantee the safety and dependability of such a component, as
they are vulnerable to considerable strain concentrations and high stresses. Conical components
must endure diverse loading circumstances like thermal stresses, internal pressures, or external
pressures. Incorporating pressure vessel design codes is vital to ensure reliable performance,
safety, and compliance with industry regulations.
The purpose of this paper is to suggest enhancements to the European Pressure Vessel Codes
concerning conical components under internal pressure. The focus of the proposed improvements
will be on enhancing safety and reliability and ensuring compliance with industry regulations.
The current study will examine the existing European codes and standards that regulate the

13
manufacturing of conical components under internal pressure, identify any shortcomings or gaps
in these regulations, and propose practical solutions to improve their effectiveness.
2. LITERATURE REVIEW
Designing conical components can be challenging due to their non-uniform shape, and requires
advanced analysis methods such as finite element analysis or analytical methods. There has been
a significant amount of research conducted on conical components in pressure vessel design, with
a focus on improving their safety, reliability, and efficiency. Over time, the design techniques for
conical components in pressure vessels have undergone evolution and diversification to ensure
their safety and efficiency. The earlier practices relied on classical analytical methods such as
Lame's [7], Barlow's [8], and thin-shell theories [9] that were based on fundamental principles of
stress and strain offering simplicity but with limitations in addressing complex geometries or
real-life conditions. Hence empirical and semi-empirical approaches emerged by incorporating
experimental data to refine design calculations.
With the advent of powerful computational tools, Finite Element Analysis (FEA) and other
numerical methods have become increasingly popular for analyzing conical components, offering
greater accuracy and the ability to model intricate geometries and diverse loading scenarios.
When comparing design methodologies, each approach has its advantages and limitations.
Classical methods provide a straightforward understanding of the underlying mechanics, while
empirical methods can bridge gaps in the analytical models. FEA and numerical methods offer
precision and flexibility but require computational resources and expertise. Factors influencing
the choice of design method include the complexity of the component, available resources, and
project-specific requirements. Existing literature provides valuable insights into best practices
and recommendations for selecting the most suitable method for a given application, ultimately
enhancing the safety and performance of pressure vessels.
The European Pressure Vessel Codes, including EN 13445 and PD5500, provide guidelines for
designing pressure vessels with conical components. These codes outline specific requirements
and limitations concerning conical sections, including geometrical specifications such as the
minimum thickness, radius of curvature, and angle. The European Pressure Vessel Codes offer
direction and standards for the creation and evaluation of conical elements, with particular
attention given to the specifications surrounding geometrical and stress analysis. The recent
enhancements in these codes have emphasized more precise directives for stress analysis while
also striving for greater accuracy and uniformity in design computations. Enhancements made to
conical component design and examination may lead to pressure vessel designs that are safer,
more effective, as well as economically advantageous. The code sections focused on cone
components in EN13445-3 are 7.6 and 8.6, in AD2000 are B2, and in PD5500 sections 3.5.3 and
3.6.3.
While the European Pressure Vessel Codes provide valuable guidance for designing conical
components, there are still limitations and gaps in existing research and codes. One limitation is
that the codes may not adequately address complex geometries or loading scenarios.
Additionally, there may be variations in the level of detail and precision provided by different
codes, leading to inconsistencies in design approaches and potential safety hazards.
The following studies suggest that there may be errors in the pressure vessel design code
EN13445. Hadley and Garwood [10] found that the design rules of EN 13445 Annex B and BSI
PD 5500 Appendix D do not support the EN approach, suggesting an urgent need for further
examination. Sandsträm et al. [11] found that the models used as a basis for EN 13445 seem
somewhat conservative, in particular for thinner gauges. Neither of these papers explicitly

14
addresses the question "Does the pressure vessel design code EN13445 contain any errors
according to the literature review?", but they both suggest that there may be errors in the code.
Kemper [12] discusses the importance of understanding the underlying theories of engineering
codes and numerical modeling in forensic engineering. The text discusses a criminal negligence
case involving the misapplication of the ASME Pressure Vessel Code in an initial forensic
analysis. Through reverse engineering and basic material science knowledge, flaws were revealed
in the original analysis, highlighting the importance of understanding engineering codes and
numerical modeling principles. A subsequent accurate FEA report was successfully used in court,
demonstrating that these methods can be applied to other engineering codes and standards for
reliable analyses in forensic investigations. Finally, Prieto et al. [13], evaluate ASME PV codes
through examining errors from production procedures and measurement systems, while
proposing a decision tool based on significance levels. The study suggests avoiding adjusting
pressure units due to large error accumulation, and only using length conversion for parts up to
125 mm in size.
More specific research on cone-cylinder junctions under internal pressure included a
comprehensive review published by Pietraszkiewicz and Konopińska [14], in which several
junctions, including cone-cylinder intersections have been studied. Under internal pressure, cone-
cylinder junctions experience compressive stresses near their joining point. This makes them
prone to asymmetric or axisymmetric buckling failure. Zhao and Teng [15, 16, 17] presented
experimental results on buckling of cone-cylinder intersections under internal pressure and the
failure behavior of cone-cylinder-skirt-ring junctions in steel silos under simulated bulk solid
loading. Teng [18], focuses on the examination of finite-element bifurcation behavior and
strength across different shell geometries at cone-cylinder intersections. The authors, highly
active in this topic, have developed equations for buckling strength that, along with their
previously established axisymmetric collapse strength equations, aim to specify the scope of shell
geometries whose resilience is regulated by asymmetric buckling. Both studies referenced
suggest that there may be inadequacies or flaws in pressure vessel design codes such as EN13445
and ASME PV codes. An effective strategy to enhance the rigidity of the junction is to locally
augment the thickness of the wall in proximity to its intersection. Another option would be
incorporating stiffeners at said intersection, which can effectively bolster its overall strength.
Khalili and Showkati [19], experimentally and numerically studied the buckling behavior of
stiffened cone-cylinder junctions subjected to internal pressure showing that the buckling mode
and load resulting from non-linear analysis are compatible with that of experimental results.
Zamani et al. [20] present an elastic solution for the truncated cone-cylinder shell intersection
under internal pressure. A power series method is used to obtain the general solution for cone
equations. The impact of cone apex angle on stress distribution in conical and cylindrical parts is
studied, as well as how intersection and boundary locations affect circumferential and
longitudinal stresses quantitatively.
The influence of cone apex angles, shell geometries, and thickness-to-diameter ratios on stress
distribution and buckling behavior is a key research finding in the study of cone-cylinder shell
junctions in pressure vessels. These factors play a crucial role in determining the structural
performance and safety of such vessels.
1. Cone apex angles: The apex angle of a cone affects stress distribution in its conical and
cylindrical sections. Different angles can result in varying concentrations of stress at the
junction, influencing overall strength and stability.
2. Shell geometries: The geometries of cylindrical and conical shells, including dimensions and
curvatures, significantly affect stress distribution and buckling in cone-cylinder junctions.

15
Larger diameter or longer cylindrical sections may yield distinct stress concentrations and
buckling behavior than smaller diameters or shorter sections.
3. Thickness-to-diameter ratios: The ratio of shell thickness to diameter significantly impacts
stress distribution and buckling behavior in cone-cylinder shell junctions. Variations in this
ratio can alter structural stiffness, resistance to buckling, and overall stability of pressure
vessels.
Engineers may improve the safety, dependability, and structural proficiency of pressure vessels
by comprehending how these factors impact stress distribution and buckling behavior in cone-
cylinder shell junctions. Additionally, this knowledge can guide the creation of codes and criteria
for designing and evaluating pressure vessels that incorporate cone-cylinder connections.
3. METHODOLOGY
When designing pressure vessels with conical components, the German Pressure Vessel Code
AD2000 is commonly used as a basis for calculation. The PV code utilizes a specific formula
incorporating the diameter DK to calculate thickness values in conical components subjected to
internal pressure conditions. It should be noted that this methodology has been leveraged in other
European Union codes regarding cone verifications as well. In particular, per paragraph AD2000
B2 - 8.1.2 [3]:
(1)
The thickness of the junction between the cone and cylinder is determined by calculating it both
within and outside of the taper region. To calculate the cone's thickness outside of the taper areas,
the characteristic cone diameter DK is utilized. The shell's taper area is defined as x1 in length,
while that of a cone's length is represented by x2 within which an area for this purpose can be
established (see
Figure 1 for convergent and divergent cones).
Figure 1. Convergent cone junction geometry with (middle) and without knuckle (left) and divergent cone
junction geometry (right) according to AD2000 [3]
To ensure structural integrity, it is critical to maintain the thickness of the junction (s1) along
lengths x1 and x2, or 1.4 times the length x2 for smaller (divergent) junctions. The iterative process
is used to determine the appropriate s1 which is generally thicker than that of adjacent shell and

16
cone regions outside the taper zones. Assuming that junction thickness s1 is known (which needs
to apply both at the cylindrical and the conical part of the junction), then the taper distances x1
and x2 are calculated based on equations (2)-(4) of AD2000 B2 – par. 8.1.1 [3].
(2)
(3)
(4)
While for the thickness of the conical shell outside the taper area is given through equation (5)
[3].
(5)
Where DK is the characteristic diameter, p is the design pressure, K is the material yield stress, S
is a safety factor, v is a joint coefficient, φ is the cone angle and c1, c2 are mill under-tolerance
and corrosion allowance respectively. Overall, the most important size to define in order to obtain
the best estimation for cone thickness sg, is the equivalent cone diameter DK. Though the current
equation for DK is technically correct, there are certain nuances that should be taken into account
to further enhance its precision. In pressure vessel design, engineers typically rely on calculation
formulas that are dependent on either the inside or outside diameter of the component. This
allows for a determination of thickness based on known measurements. However, slight
variations in geometry necessitate adjustments to required thickness calculations depending on
which basis of design is chosen. For example, assuming a conical structure with an axial length
of 800 mm and a material thickness of 30 mm, this cone is hypothetically linked to two distinct
cylindrical shells with outside diameters measuring 1000 mm and 1800 mm respectively. Both
cylindrical components maintain a uniform material thickness of 30 mm. If the design strategy is
based on matching the outside diameter of the cone (see Figure. 2) at the junction with the
outside diameter of the cylindrical shell, it would lead to an inward extension of the cone
thickness. By mapping the geometry of such an arrangement, we ascertain an internal diameter at
the median point of the cone height, often perceived as a DK calculation diameter, yielding an
internal diameter of approximately 1332.918 mm. Adopting the same conical structure with an
axial length of 800 mm and a thickness of 30 mm, we consider its attachment to cylindrical shells
of adjusted internal diameters. The first cylinder internal diameter is 940 mm, derived from the
original external diameter of 1000 mm less twice the material thickness (1000-2*30 mm). The
second cylinder, on the other hand, has an internal diameter of 1740 mm, computed similarly as
the original 1800mm diameter less twice the material thickness (1800-2*30 mm). In this case,
each cylinder maintains the consistent material thickness of 30 mm.
In an alternative design strategy focused on matching the inside diameter of the cone with the
inside diameter of the cylindrical shell (
Figure 3), the cone thickness would need to extend externally. Consequently, based on the
geometric illustration of such an arrangement, we achieve an internal diameter at the midpoint of
the cone height (potentially considered a DK calculation diameter). This configuration yields an
internal diameter of roughly 1340 mm. This comprehensive examination and comparison of
design strategies based on outside and inside diameters, provide critical insights into the
implications of design decisions on the geometric and structural characteristics of the cone-
cylinder junction.

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Figure. 2. Characteristic diameter DK (dimension in parenthesis) at the cone median axial length when
calculating based on the outside diameter. For the same shell thickness, the cone extends 3.54 mm to the
inside of the vessel
Figure 3. Equivalent diameter DK (dimension in parenthesis) at the cone median axial length when
calculating based on the inside diameter. For the same shell thickness, the cone extends 3.54 mm towards
the outside of the vessel
The AD2000 equation (1) serves as an instructive tool for determining the characteristic diameter
in the context of a convergent cone, characterized by a large junction. However, for the case of a
divergent cone, distinguished by a smaller junction, the equation's applicability is not specified.
Consequently, engineers are required to engage in a more hands-on approach, manually sketching
the geometrical representation of the cone to ascertain the accurate DK value, thereby reflecting
the necessity for more adaptive methodologies in certain engineering scenarios.
In the context of engineering design, cones are typically characterized by their primary
dimensions namely small and large diameters, along with their lengths. From this geometric
information, the conical angle φ can be derived. AD2000 [3] formulas operate under the

18
assumption that this angle φ is pre-established. However, the determination of the angle φ can be
complex under certain conditions. In the case of a cone without knuckles and with a known
length, the task of finding the angle φ can be relatively straightforward. Conversely, the process
becomes more intricate when the cone features knuckles. This can occur at the larger or smaller
junction, or even at both junctions simultaneously.
The presence of knuckles introduces an additional layer of complexity in the computation of the
conical angle φ, thereby necessitating a more detailed and careful approach. Therefore, the
accurate determination of this angle under these conditions poses a substantial challenge to
engineers and implicates the equivalent diameter DK calculation (which involves finding the
correct angle φ).
The AD2000 equation typically provides the characteristic diameter under new, un-corroded
conditions. Nevertheless, many engineers factor in anticipated corrosion into their calculations,
adopting a more pragmatic and safety-oriented perspective. Corrosion allowance is commonly
considered to reduce the thickness of the vessel from the interior, thereby enlarging the
"corroded" internal diameters compared to their "new" counterparts. Therefore, the incorporation
of the "corroded" diameter in these computations usually results in a greater thickness
requirement. This cautious methodology, while perhaps conservative, is inherently safety-focused
and ensures the resilience and longevity of the vessel under corrosive conditions.
In the context of this analysis, the input data required are an integral part of the methodological
approach and they are primarily focused on the dimensions and characteristics of the pressure
vessel cones and their junctions with the adjacent cylindrical shells.
Figure 4. Illustration of cone junction geometry symbols used in this methodology
The first key parameter is Da, which is the outside diameter of the large cylindrical shell at the
junction with the cone. The next input parameter is Sl,af, representing the thickness of the large
cylindrical shell at the cone junction after forming, which can be calculated as the selected
nominal thickness of the cone junction Sl minus a forming allowance c1. This parameter may be

19
obtained through an iterative procedure or it can be approximately equal to the selected nominal
thickness after accounting for the forming allowance.
The third parameter Ri is the internal knuckle radius at the large junction. This parameter is set to
zero if no knuckle exists at the junction. The outside diameter of the small cylindrical shell at the
cone junction is indicated as da. Its thickness after forming is sl,af and it can be obtained in a
similar manner as Sl,af, with the same considerations applying for the forming allowance c1. We
also consider ra, the external knuckle radius at the small junction. This value is also set to zero in
cases where no knuckle exists. This parameter is typically the sum of the forming diameter Ri and
Sl,af. The cone thickness after forming, denoted by Sg,af, is selected initially by the user. Finally,
the total length of the cone is represented by L, and c2 stands for the corrosion allowance. The
output data from our analysis can be succinctly described through a series of equations. Each
equation corresponds to a specific parameter and is formulated based on the corresponding input
data.
Initially, size A, which is the distance between the knuckle centers is calculated as:
(6)
In case of no knuckles on the large and small junction, the second and the fourth term of equation
(6) are eliminated respectively. Following this, the internal diameters at the large and small
junctions are computed respectively:
(7)
These internal diameters can be adjusted to take into account corrosion.
(8)
The corroded internal knuckle radius at the large junction is also accounted for:
(9)
The conical angle φ is found by addition of φ1 and φ2:
(10)
(11)
Where the first and second term in the arcsin function in equation (10) are eliminated in case of
no knuckle at the large and small junction respectively. Following the determination of the cone
angle, the values X2 and x2 are found according to equation (3) for the large and small junction
respectively. Before proceeding, x2 is increased by 40% to ensure structural integrity as
mentioned in the previous section.
Next, the lengths La, the pivot lengths L1, L2, and their sum Li are calculated so that the
characteristic diameters are found.

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(12)
(13)
(14)
(15)
(16)
(17)
The cone junction design, which corresponds to the inside diameter approach, utilizes
comparable input data as the outside diameter-based design discussed earlier. However, instead
of using external diameters Da and da for small and large cylindrical shells respectively, internal
diameters Di and di are used in this case.
(18)
(19)
,
(20)
(21)
For the design based on the inside diameter, the outside diameters are obviously calculated
through equations (7) and the rest of the output parameters up to DK and dK are calculated through
equations (8) to (17). The two set of formulas analysed in this section are the basis of this study
and, according to the authors, significant improvement can be achieved during conical
components design. Initially, potential corrosive processes are taken into account, thereby
enhancing safety and realism. Another noteworthy aspect of the proposed approach lies in its
ingenuity to accurately compute DK even when knuckles are present at either junction of the cone.
As such, this method can be applied across a wide range of scenarios without compromising on
practicality or versatility.
Table 1. Input parameters for cone plate thickness determination in the selected case studies
Input parameter Symbol Value Units
Design Pressure p 9.6 barg
Design Temperature Ts 300 °C
Material EN10028-2: P355GH, ≤16 t ≤ 40 mm
Material yield stress K 225 MPa
Safety factor S 1.5 -
Joint coefficient v 0.85 -
Mill under-tolerance c1 0.25 mm
The above AD2000 refinement method is illustrated through three case studies, the input
parameters and results of which are shown in the Results section. The first case study assumes no
corrosion allowance, while the design is based on inside diameter (Di, di), the second case study

21
is also based on inside diameter however corrosion is assumed according to AD2000. The third
case study bases the design on the outside diameter, and DK is assumed in new condition.
To quantify the effect of the difference in DK in the resulting cone thickness, and consequently
rigidity and safety), a pressure vessel scenario is assumed and the three cases are compared. The
process and engineering parameters adopted are listed in Table 1. Finally, the calculated
characteristic diameter DK derived through the methodological approach presented in this study is
comparatively analysed with the corresponding characteristic diameter DK determined through
equations 7.6-6 and 7.6-8 as prescribed in the EN13445 standard [2]. It is noted that equation 7.6-
8 (equation 23) leads to highly increased diameters resulting in excessive material use.
(22)
(23)
Figure 5. Geometry depiction of cone junctions without (A) and with (B) knuckles according to
EN13445 [2]
4. RESULTS AND DISCUSSION
In the realm of engineering, the precise calculation of DK, the diameter at the junction of conical
and cylindrical components, is crucial for the design and safety of pressure vessels. The AD2000
standard, a widely accepted guideline for pressure vessel design, provides equations for
calculating DK. However, these equations may not always yield the most accurate results,
particularly for large junctions. Therefore, it is essential to refine these calculations to ensure the
structural integrity and safety of the pressure vessels.
The refinement of DK in relation to the AD2000 DK equation is shown through three different
case studies (Case study 1a, 1b and 2). The considerations are made for the large junction, as the
AD2000 DK formula for small junction is not available.
Case 1: Base design on inside diameter, (a) without and (b) with corrosion allowance
In the first case, the base design is based on the inside diameter, and no corrosion allowance is
considered. The engineering input for the large junction includes an inside diameter (Di) of 3000
mm and an outside diameter (Da) of 3100 mm. The thickness at the large junction (Sl,af) is 50 mm,
and the radius of the large junction (Ri) is 300 mm. For the small junction, the inside diameter (di)
is 1000 mm, the outside diameter (da) is 1070 mm, and the thickness (sl,af) is 35 mm. The radius
of the small junction (ra) is 150 mm. The angle φ is 52.033°, the thickness of the gasket (sg,af) is
20 mm, and the length of the junction is 1000 mm. Using these parameters, the calculated DK is

22
2215.126 mm both for the classical AD2000 approach and the refined method proposed in this
study. This result is confirmed when the junction is sketched, with RK (the radius at the junction)
being half that of DK.
Table 2: Input parameters of case study 1a, base design on inside diameter with no corrosion allowance.
The DK calculated is 2215.126 mm
Parameter Value Unit Parameter Value Unit
Di 3000
mm
Da 3100
mm
Sl,af 50 Ri 300
Ri 300 Sl,af 50
di 1000 da 1070
sl,af 35 sl,af 35
ra 150 Angle φ 52.033 °
sg,af 20
L 1000
DK (refined) 2215.126
DK (AD2000) 2215.126
mm
DK (EN13445) 2542.124
Similarly assuming the same geometry as case 1a but also considering corrosion allowance (c2) at
6 mm, the characteristic diameter (DK) calculated with this study is 2256.810 mm while with the
AD2000 equation, the diameter DK is smaller nearly 7.4 mm (2249.427 mm) while for the
EN13445 code DK is 2547.107.
Case 2: Base design on outside diameter, DK in new condition (or as per AD2000 equation)
In the second case, the base design is based on the outside diameter. The engineering input for the
large junction includes an outside diameter (Da) of 3100 mm. The thickness at the large junction
(Sl,af) is 50 mm, and the radius of the large junction (Ri) is 300 mm. For the small junction, the
outside diameter (da) is 1070 mm, and the thickness (sl,af) is 35 mm. The radius of the small
junction (ra) is 150 mm. The angle φ is 52.85°, the thickness of the gasket (sg,af) is 20 mm, and the
length of the junction is 1000 mm. Using these parameters, the calculated DK is 2233.136 mm
when using classic engineering input. However, when the AD2000 equation is used, the
calculated DK is lower, at 2196.906 mm. This discrepancy can be attributed to the different
methods used in this study and the AD2000 equation.
These case studies highlight the importance of refining the calculation of diameter DK in relation
to the EN13445 and the AD2000 DK equation, the latter particularly for large junctions. The base
design, whether it is based on the inside or outside diameter, and the consideration of factors such
as corrosion allowance, can significantly impact the calculated DK. For the above examples, and
according to the input parameters (process and geometry) shown in Table 1, plate thickness
differences occur which are described in
In the third case study, we examine the configuration of a conical component with knuckles,
where the design is based on the outside diameter. This case is particularly noteworthy due to the
significant difference observed in the resulting DK and thickness values. One of the key variables
in this case study is the angle of the conical component, which is adjusted by varying the length
of the cone. A matrix of different angles is considered, and for each angle, the corresponding DK
diameter is calculated. This approach allows for a comprehensive analysis of the relationship
between the angle of the conical component and the resulting DK diameter. It provides valuable
insights into the optimization of the design of conical components, contributing to the

23
enhancement of the structural safety, efficiency, and cost-effectiveness of pressure vessels. The
resulting characteristic diameters for a range of large junction diameters and angles is presented
in the following three tables.
Table 4.
As, expected, since the characteristic diameter and the cone angle are the major geometrical
properties that are used by Equation (5), the resulting thickness for the first case (design based on
internal diameter without corrosion) is identical. Differences occur for the other two examples
which, since the cone angle is the same, are entirely dependent on the characteristic diameter
(DK).
Figure 6: Base design on inside diameter with no corrosion allowance geometry drawing showing
the dimensions provided in Table 2 and the resulting equivalent radius RK (1107.56 mm).
Table 3. Input parameters of case study 2, base design on outside diameter without corrosion
Parameter Value Unit Parameter Value Unit
Da 3100
mm
Da 3100
mm
Sl,af 50 Ri 300
Ri 300 Sl,af 50
da 1070 da 1070
sl,af 35 sl,af 35
ra 150 Angle φ 52.85 °
sg,af 20
L 1000
DK (study) 2233.136
DK (AD2000) 2196.906
mm
DK (EN13445) 2529.603
In the third case study, we examine the configuration of a conical component with knuckles,
where the design is based on the outside diameter. This case is particularly noteworthy due to the

24
significant difference observed in the resulting DK and thickness values. One of the key variables
in this case study is the angle of the conical component, which is adjusted by varying the length
of the cone. A matrix of different angles is considered, and for each angle, the corresponding DK
diameter is calculated. This approach allows for a comprehensive analysis of the relationship
between the angle of the conical component and the resulting DK diameter. It provides valuable
insights into the optimization of the design of conical components, contributing to the
enhancement of the structural safety, efficiency, and cost-effectiveness of pressure vessels. The
resulting characteristic diameters for a range of large junction diameters and angles is presented
in the following three tables.
Table 4. Difference in the calculated cone plate thickness calculated for the selected case studies between
the proposed methodology and the AD2000 PV code. Input parameters as in Table 1.
Case study
φ c2 DK sg Diff.
(°) (mm) (mm) (mm) (%)
Refined equations
1a
52.033
0
2215.126 13.75
0
AD2000 2215.126 13.75
Refined equations
1b 6
2256.81 20.01
0.225
AD2000 2249.427 19.96
Refined equations
2 52.855 0
2233.136 14.12
1.619
AD2000 2196.906 13.9
Table 5. Characteristic diameter (DK) for a range of large cylinder junction external diameters and cone
angles according to the proposed approach
Da / φ 30° 40° 50° 60°
1000
mm
808.20 732.53 644.16 538.81
1250 1044.16 963.37 869.21 756.81
1500 1281.41 1195.95 1096.58 977.94
1750 1519.72 1430.02 1325.85 1201.27
2000 1758.87 1665.19 1556.54 1426.46
2250 1998.67 1901.26 1788.31 1653.32
2500 2239.09 2138.07 2021.17 1881.20
Table 6. Diameter DK for cylinder external diameters and cone angles according to AD2000
Da / φ 30° 40° 50° 60°
1000
mm
790.88 717.21 631.30 528.82
1250 1026.84 948.05 856.36 746.81
1500 1264.09 1180.63 1083.73 967.94
1750 1502.41 1414.70 1312.99 1991.27
2000 1741.55 1649.87 1543.69 1416.46
2250 1981.35 1885.94 1775.46 1643.32
2500 2221.77 2122.75 2008.32 1871.21
Table 7. Diameter DK for cylinder external diameters and cone angles according to EN13445
Da / φ 30° 40° 50° 60°
1000
mm
875.34 823.51 762.063 689.39
1250 1118.32 1063.93 999.60 923.39
1500 1361.94 1305.21 1238.27 1158.96
1750 1606.09 1547.252 1477.90 1395.62

25
2000 1850.67 1789.84 1718.25 1633.21
2250 2095.57 2032.87 1959.13 1871.65
2500 2340.79 2276.28 2200.56 2110.58
Figure 7. The effect of design pressure on the difference in the cone plate thickness calculation between the
proposed method and pressure vessel Code AD2000. K = 225 MPa, S = 1.5, c1 = 0.3 mm, c2 = 1 mm, v =
0.85.
It is clear that since two established PV codes calculate vastly different characteristic diameters,
experienced engineers would follow AD2000 since it results in smaller cone plate thickness, in
cases where EN13445 is not mandatory.
Figure 7 presents the relationship between design pressure and the absolute thickness difference
when comparing the outcomes of a new proposed method and the conventional AD2000 code for
pressure vessels. Both the proposed method and AD2000 produce the same thickness difference
irrespectively of the cylinder diameter and the cone angle. EN13445 on the contrary, results in
thickness differences which are a function of cone angle and cylinder diameter, thus an arbitrary
representative diameter (2000 mm) and angle (50°) are depicted in the graph of
Figure 7. A clear, linear correlation is apparent: as the design pressure increases, the absolute
thickness difference consistently rises across the entire pressure range examined, from 20 to 200
bar.
This analysis provides evidence that at higher design pressures, the discrepancy between the
thickness measurements of the two methods becomes more significant. This relationship holds
true regardless of the pressure level, illustrating a reliable trend that can be used for anticipatory
planning and decision-making. These findings underscore the substantial influence that design
pressure has on the absolute difference in thickness estimations between the two methods.
Therefore, when high design pressure is part of the vessel's specifications, users of the new
proposed method should expect greater variations from the AD2000 code's thickness. This
knowledge is crucial for appropriate resource allocation, ensuring safety standards, and
optimizing design parameters when using the proposed method under varying design pressures.

26
Figure 8. Effect of cone angle on the difference in the cone plate thickness calculation between the
proposed method and pressure vessel Code AD2000, p = 50 bar, K = 225 MPa, S = 1.5, c1 = 0.3 mm, c2 = 1
mm, v = 0.85.
Figure 8 establishes a correlation between cone angle, cone diameter, and the difference in
thickness between the proposed method and the traditional AD2000 code. As the cone angle
widens, there is a consistent decrease in the thickness difference across all examined diameters,
ranging from 1000 up to 2500 mm.
This pattern reveals that cones with larger angles (60°) yield smaller discrepancies in thickness
measurements when compared to cones with smaller angles (30°), irrespective of the calculation
method used. Notably, the impact of the cone angle on the thickness difference is more
significant for larger diameters.
Similarly, the refined method proposed in this study is compared to the results from EN13445
through Equations (5) and (23). A comprehensive comparison between the cone plate thickness
calculated using the refined method and the DK equation (23) from the EN13445 code for the
different cone diameters and angles selected (
Figure 9). At a broader level, the data reflects a consistent pattern: the refined method
consistently returns a thickness lower than the EN13445 code, indicated by the negative
percentage difference. This variance implies that the refined method leads to a more economical
design without sacrificing the safety requirements stipulated by the code. As the cone angle
increases from 30° to 60°, the discrepancy in thickness calculation becomes more pronounced,
with the maximum divergence reaching -20.7% for a cone with a diameter of 1000 mm at 60°
cone angles. This suggests that the refined method significantly deviates from the EN13445 code,
particularly for larger cone angles, which could lead to substantial material savings.

27
Figure 9. Effect of cone angle on the difference in the cone plate thickness calculation between the
proposed method and pressure vessel Code EN13445, p = 50 bar, K = 225 MPa, S = 1.5, c1 = 0.3 mm, c2 =
1 mm, v = 0.85.
Regarding the cone diameter, the discrepancy in thickness calculation decreases as the diameter
increases, indicating a trend towards convergence with the EN13445 code. For example, at 30°,
the difference reduces from -7.1% for a 1000mm cylinder to -4.2% for a 2500mm cylinder cone.
This implies that the discrepancy between the two methods decreases as the large cylinder
junction diameter becomes larger.
Figure 10 provides insights into the relationship between design pressure, junction diameter, and
the variance in thickness when comparing the new proposed method and the traditional AD2000
code for pressure vessels. With the increase in design pressure, there is a clear and consistent
uptick in the thickness difference across all cone diameters examined, ranging from 1000 up to
2500 mm. This data reveals that at higher design pressures, the deviation in thickness
measurements from the two methods is more pronounced, regardless of the large junction
diameter. This effect is consistent for all diameters studied, and the increased discrepancies at
higher pressures are more prominent for smaller diameters. These findings suggest that design
pressure is a significant factor influencing the difference in thickness estimations between the two
methods.
Overall, EN13445 provides a solid set of equations for cone geometries. Based on bibliography
however [21] and the original concept of the Code which is based on the East German rules for
pressure vessels there are some discrepancies in formulas (which remain unchanged since the
first issue in PrEN13445-3: 1999) and there are also a few points (recognized by the Committee)
that should be improved.

28
Figure 10. Effect of design pressure and DK on the difference in the cone plate thickness calculation
between the proposed method and pressure vessel Code AD2000, φ = 30°, K = 225 MPa, S = 1.5, c1 = 0.3
mm, c2 = 1 mm, v = 0.85.
Modifying the EN13445 equation 7.6-8 (Equation (23)) to match the proposed equation (24)
leads to characteristic diameters DK much closer to the ones calculated either through AD2000 or
the refined approach proposed in this study. With the suggested modification the cone plate
thickness difference between the refined method proposed and EN13445 is - 0.9% on average
compared to the current - 9.6% for the presented diameter and cone angles [2].
(24)
In conclusion, these findings suggest that the proposed refined method might offer more
economical solutions compared to EN13445, and safer vessels compared to AD2000 especially
for designs with larger cone angles and smaller diameters. Nonetheless, the benefits should be
carefully weighed against other factors such as safety, manufacturing complexities, and the
acceptability of deviation from established codes. It is important to note that while the proposed
method may yield more efficient designs, it remains crucial to maintain compliance with safety
regulations, and each design should undergo careful scrutiny and testing before implementation.
5. CONCLUSIONS
It is recognized that the AD2000 formula for determining the characteristic diameter DK is
generally applied with a design strategy oriented towards the inside diameter, while
simultaneously incorporating the outside diameter of the junction as an input variable. However,
the precision of the cone thickness calculation equation (5) warrants careful attention. Any slight
deviations in the calculated diameter can significantly influence the required thickness of the
cone structure in the design. Hence, an optimized and refined DK formula is instrumental to
ensure accurate predictions, thereby effectively dictating the appropriate cone thicknesses and
thereby maintaining the structural integrity and operational efficiency of the overall system. In
this study, refined methodologies for the estimation of the characteristic diameter, a crucial
parameter in the design of pressure vessel cones under internal pressure, are introduced. This
technique is tailored for both types of cone transitions as per AD2000 B2 standards, covering
both convergent and divergent cones. Crucially, the developed equations consider a corrosion

29
allowance, hence ensuring a more realistic and safe vessel design when corrosive processes are
anticipated. Moreover, this innovative approach facilitates the calculation of DK even in the
presence of knuckles at either junction of the cone, thus augmenting the practicality and
versatility of the analysis.
ACKNOWLEDGEMENTS
This research has been co‐financed by the European Union and Greek national funds through the
Operational Program Competitiveness, Entrepreneurship and Innovation, under the call
RESEARCH – CREATE – INNOVATE (project code: T2EDK-04885).
REFERENCSE
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[2] European Standards, "Standards EN 13445:2021 for Unfired pressure vessels all parts," European
Standards, 2021.
[3] AD 2000-MERKBLATT, "A 403, 2021 Edition, June 2021 - Pressure vessel equipment - Equipment
for detecting and limiting pressure and temperature," AD 2000-MERKBLATT, 2021.
[4] BSI, "PD 5500:2021 Specification for unfired fusion welded pressure vessels," BSI, 2021.
[5] European Parliament and of the Council, "Directive 97/23/EC of the European Parliament and of the
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[10] I. Hadley and S. Garwood, "Prevention of brittle fracture in pressure vessels: A review of the design
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30
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AUTHORS
Isaak Dassa holds a Mechanical Engineering Diploma from Aristotle University of
Thessaloniki.. His research spans fundamental and applied areas in petrochemical
construction, including pressure vessels, heat exchangers, towers, and boilers, in
accordance with regulations and codes. Isaak has expertise in computational design of
pressure vessels, heat exchangers, towers, and boilers, adhering to regulations and codes.
Dimitrios Mertzis holds a PhD and a Mechanical Engineering Diploma from Aristotle
University of Thessaloniki. His research lies in the field of biomass and bioenergy,
thermochemical process design and prototyping. He has performed two post-doctoral
reseearches in the fields of catalytic reactions and modelling. Dr. Mertzis has supervised
numerous theses in the Mechanical Engineering Dept. of Aristotle Univeristy of
Thessaloniki where he was an adjunct lecturer beteween 2017-2019.

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