Reduction of Stresses in Cylindrical Pressure Vessels Using Finite Element Analysis

Due to the differential operating pressure of pressure vessels, they are potentially dangerous and accidents involving pressure vessels can be deadly and poses lethal dangers when vessels contents are flammable/explosive, toxic or reactive. Stress induced operating factors (e.g., process-upset, catalyst regeneration) and stress related defects (e.g., fatigue creep, embrittlement, stress corrosion cracking) accounts for approximately 24.4% of reoccurring catastrophic pressure vessels failures in process industries, many of which has resulted in loss of several lives, properties and in some cases preventive measures of evacuation of residents and community enforced ( Sirosh & Niedzwiecki 2008 ). Pressure vessels store large amounts of energy; the higher the operating pressure and the bigger the vessel, the more the energy released in the event of a rupture and consequently the higher the extent of damage or disaster or the danger it poses, (American Petroleum Institute 2001). To prevent stress related vessel rupture and catastrophic failure, it is necessary to identify the main factors that contribute extensively to stress development in pressure vessels and how they can be mitigated. This work presents critical design analysis of stress development using 3D CAD models of cylindrical pressure vessels assembly and finite element engineering simulation of various stress and deformation tests at high temperature and pressure.


Introduction
Due to the differential operating pressure of pressure vessels, they are potentially dangerous and accidents involving pressure vessels can be deadly and poses lethal dangers when vessels contents are flammable/explosive, toxic or reactive. Stress induced operating factors (e.g., process-upset, catalyst regeneration) and stress related defects (e.g., fatigue creep, embrittlement, stress corrosion cracking) accounts for approximately 24.4% of reoccurring catastrophic pressure vessels failures in process industries, many of which has resulted in loss of several lives, properties and in some cases preventive measures of evacuation of residents and community enforced ( Sirosh & Niedzwiecki 2008 ). Pressure vessels store large amounts of energy; the higher the operating pressure -and the bigger the vessel, the more the energy released in the event of a rupture and consequently the higher the extent of damage or disaster or the danger it poses, (American Petroleum Institute 2001). To prevent stress related vessel rupture and catastrophic failure, it is necessary to identify the main factors that contribute extensively to stress development in pressure vessels and how they can be mitigated. This work presents critical design analysis of stress development using 3D CAD models of cylindrical pressure vessels assembly and finite element engineering simulation of various stress and deformation tests at high temperature and pressure.

Rate of pressure vessels accidents
Bulk Transporter (2009) reported that the National Board of Boiler and Pressure Vessel Inspectors in the US recorded the number of accidents involving pressure vessels at an increase of 24% over the course of a year between 1999 and 2000. These statistics include power boilers, steam heating boilers, water heating boilers, and unfired pressure vessels. However, the increased number of accidents was not reflected through to the number of fatalities, as these actually dropped by 33% over this period. By broadening this search, it can be seen that the reporting period of 1992 to 2001 saw a total of 23,338 pressure vessel related accidents which averages at 2,334 accidents per year. Reporting year 2000 saw the highest number of accidents at 2,686 with the lowest at 2,011 in 1998 (National Board of Boiler and Pressure Vessel Inspectors, 2002).
The number of fatalities as a direct result of boiler and pressure vessel accidents has been recorded as 127 over the past 10 years (Air-conditioning, Heating, Refrigeration-The News (2002). During the reported period between 2001 and 2008, the statistics show that the rate of accidents that were directly linked to pressure vessels is not yet on the decline.

Causes of pressure vessel failures
The main causes of failure of a pressure vessel are as follows:

Stresses in pressure vessels
Stress is the internal resistance or counterforce of a material to the distorting effects of an external force or load, which depends on the direction of applied load as well as on the plane it acts. According to D. R. Moss (2004) stresses are generally categorised as primary, secondary or peak stresses. Primary stresses are stresses due to pressure (internal or external), mechanical loads and wind which can result in the rupture or total collapse of a pressure vessel, they are the most hazardous. Secondary stresses on the other hand are strain-induced stresses, and can be developed at the junction of major components of a pressure vessel (e.g. radial loads on nozzles) because of stresses caused by relenting load or differential thermal expansion. While Peak stresses are the maximum stress concentration point in addition to the primary and secondary stresses present in a region. Peak stresses are only significant in fatigue conditions and are the sources of fatigue cracks, which are applicable to membrane, bending and shear stresses (Rao. K. R. 2002).
When a thin-walled cylinder is subjected to internal pressure, three mutually perpendicular principal stresses will be set up in the cylinder material, namely the circumferential or hoop stress, the radial stress and the longitudinal stress, (Sharma.S.C .2010). Provided that the ratio of thickness to inside diameter of the cylinder is less than 1/20, it is reasonably accurate to assume that the hoop and longitudinal stresses are constant across the wall thickness, and that the magnitude of the radial stress set up is so small in comparison with the hoop and longitudinal stresses that it can be neglected. This is obviously an approximation since, in practice, it will vary from zero at the outside surface to a value equal to the internal pressure at the inside surface (Hearn.E.J.1998).

Vessel description
The arrangement of a typical pressure vessel is shown in Fig. 2. A typical pressure vessel consists of shell (body of the vessel), closure heads, openings for inspection and instrumentations, and a combination of nozzles for pressure relief or other purpose and supports (Syed U. A. (2009

Finite element model of pressure vessels
In order to proceed with the analysis, three Finite element models were designed and denoted as design case one, two and three. A finite element model consists of boundary conditions, mesh of elements and nodes. Each component of pressure vessel analysed for stress and deformation at the design conditions of 137 MPa and 400 o C for all the cases www.intechopen.com considered. Wind loads and seismic loading were also taken into account. All finite element analyses were run using ANSYS 10.0.
The design parameters taken are as follows: The shell to be used is fully radiographed, hence E is 100% -Monobloc shell assumed.

Model design case one
The pressure vessel model assumed a cylinder with semi ellipsoidal top head and hemispherical bottom head capped. Considering the shell to be monobloc and the design rule that the design pressure P D should not exceed the limit set by article KD-251.1 (division 3 of ASME BPVC section 8) given as: Where Y is the ratio of outer diameter (D o ) to the inner diameter (D i ) of the shell, Y = D o /D i, and ratio of 2 is assumed for safety. Assumed D o = 3048 mm, D i= 1524 mm, Length = 3 x D o and flange thickness = 254 mm. The ellipsoidal top head is considered to be fully radiographed like shell, hence E = 1. Then using given equation: Where K is the stress intensity factor by the equation below: 137 * 3048 * 1 2 * 965 * 1 0.2 * 137 t   Thickness = 220 mm The required thickness of the bottom hemispherical head is normally one-half the thickness of an elliptical or torispherical head for the same design conditions, material, and diameter which will be 110 mm in this case.
For the openings and closures, which are an important requirement for safety, is given as: Where r is the inside radius of nozzle, assuming a 1524 mm nozzle, the required thickness comes out to be 254 mm.
The CAD model of the pressure vessel is shown below: Each model part was meshed with SOLID 45 element with a higher concentration of elements around the nozzles and stress concentration areas. Eight nodes having three degrees of freedom at each node define the element translations in the nodal x, y, and z directions. The element has plasticity, creep, swelling, stress stiffening, large deflection, and large strain capabilities. Subfigure 2 of Fig.3 above shows the meshing of the model parts.

Boundary conditions
Axis symmetric displacement boundary conditions was applied to the two axes to ensure that the body, bottom end of the shell and the head are fully constrained with internal pressure loadings of 137 MPa and temperature of 400 oC applied. The finite analysis results are shown in tables 1 and 2 below. www.intechopen.com

PS y Y 
After the finite element analysis of design case one, large stress was found around the nozzle junction of the cylindrical shell and also found stress developed at the bottom and top head. Ways of mitigating the stress is addressed in the model design case two. The design parameters are kept the same as in Design case one except for the temperature rating tensile strength of the material that is 806 MPa. The diametric ratio Y assumed is 1.5 derived from equation (1): Outer and inner diameter assumed were 3048 mm and 2030 mm respectively. Reconsidering the nozzle thickness in design case two, the value of S is taken as 806 MPa using equation (4) To mitigate the stress discovered during the finite element analysis of design case one, reinforcement pad is introduced to lower the stresses around the junction of shell and nozzle of the pressure vessel model. The total cross-sectional area of reinforcement A required in any given plane for a vessel under internal pressure should be not less than: Where d = diameter in the given plane of the finished opening, (mm).
t r = minimum thickness which meets the requirements of KD-230 in the absence of the opening (mm).
And F is calculated from the graph in non-mandatory appendix H of section 8 division 3 of ASME BPVC to be 1.  Table 4. Summary of finite element results

Case three design model
In design case 3, the shortcomings of both design cases 1 and 2 were considered from design point of view to lower the stresses to an acceptable range. Reinforcement pad is used along with the use of diametric ratio of 2.0, i.e. Y = 2 as calculated in design case 1. Outer diameter of 3048 mm and the inner diameter of 1524 mm with length of cylindrical shell being three times the outer diameter for structure stability. Nozzle thickness kept same as design case 2. Fig. 4. Design model for case three

Element type
Each component of the pressure vessel is simplified using symmetry. Solid 3D Element was used to mesh the models with fine mesh around the nozzle and shell junction. Pressure and temperature of 137 MPa and 400 o C applied on the internal surface of the components. The results obtained from finite analysis are given below:

Discussion
Finite element analysis result in design case one was compared with the allowable stress intensities of ASME BVPC Division 3 and it was found that the stresses in the cylindrical shell exceeded the value of the allowable stress with almost 70 %. Also, the result shows higher stress development along the nozzle length away from the shell and at the bottom semi spherical head. However, the stresses developed in ellipsoidal enclosures head has a very less margin than the allowable stress of about 4%, which can be improved by using better thickness to diameter ratio.
In the design case two model, the shell thickness was reduced and thickness of the nozzle increased as well as the addition of reinforcement pad of high alloy steel SA-705 grade XM-13 around the nozzle to shell junction. The finite element analysis shows that less stresses were developed in the shell wall than the allowable and there was a reduction of stress from 2552 MPa in design case one to 780 MPa in design case two; about 70% drop in stresses. However, despite the decline in the stresses to 780 MPa, it is still higher than the allowable stress by the ASME Code. Finally in the design case three, the cylindrical shell was modified and the reinforcement pad introduced in design case two retained to reduce stresses around the nozzle to shell junction and a skirt length of 254 mm was added around the enclosure heads. This is necessary due to the result of the analysis obtained in the design cases one and two which showed that highest stresses were concentrated on the lower end of the heads where there was formation of joins to the shell. The finite element analysis shows that with the addition of the skirt, there was lesser development of stresses in the lower ends of the heads. The table below shows the comparison of the stresses obtained from finite element analysis and the stresses allowed by the ASME Code.

Conclusion
Three main factors are seen to contribute extensively to the development of stresses in pressure vessels. These are thickness, nozzle positions and the joints of the enclosure heads. From the model design cases used in this research, it could be seen that as the thickness of pressure vessel increases, the stresses decreases, however this is not a viable solution due to cost. Nozzles though are safety relief devices and important component of pressure vessels comes with its own disadvantages of increasing weak areas and stress concentration. However, this was mitigated by use of high alloy reinforcement pads as applied in the design case two and three of this work. The high strength reinforcement pad used has a chemical composition of titanium 0.4 to 1.20%; hollow disc shaped with rectangular section can also reduce the stresses concentration around the nozzle. Finite Element Analysis represents a numerical technique for finding approximate solutions to partial differential equations as well as integral equations, permitting the numerical analysis of complex structures based on their material properties. This book presents 20 different chapters in the application of Finite Elements, ranging from Biomedical Engineering to Manufacturing Industry and Industrial Developments. It has been written at a level suitable for use in a graduate course on applications of finite element modelling and analysis (mechanical, civil and biomedical engineering studies, for instance), without excluding its use by researchers or professional engineers interested in the field, seeking to gain a deeper understanding concerning Finite Element Analysis.