Design of Pressure Vessels

This module should take the user through pressure vessel design application (PVDA). The necessary background is built by the modules “ Mechanical Design Fundamentals” and “Reinforcement Calculations”.

Shape

Process vessels for use in chemical process industry are conceived, designed and fabricated in a fairly small number of shapes. These are generally Shapes of Revolution. The important basic shapes are: cylinder, sphere (or hemi-sphere), ellipsoid (or hemi-ellipsoid), torus, cone (or frustum of a cone) and a flat plate. Using these shapes, several composite pressure vessel shapes can be constituted.

PVDA visualizes a composite vessel shape as a stack of layers, each layer utilizing one of the basic shapes.

Size

The spatial dimensions of each of these shapes can be defined in terms of a few size primitives. For example, a sphere is completely defined by its diameter. A cylinder is completely defined by its diameter and length etc.

The diameter can be given either as “inner diameter” or “outer diameter”. One of them is entered by the user. The other one is calculated after the vessel wall thickness (t) is arrived at suitably. The relationship between the ID and OD is as follows.

OD = ID – 2t.

The values for essential size parameters for each of the shapes comprising the composite vessel shape need to be provided by the user. Connectivity of shapes in adjacent layers allows internal fixation of sizes of some shapes.

Wall Thickness

Design codes provide formulae for calculating the minimum wall thickness for the standard shapes. The application of the formula for any specific shape requires the user to provide design temperature, design pressure, material of construction and its allowable stress at the design

temperature, fabrication and inspection procedure and weld quality factor or weld joint efficiency. The thickness calculated using applicable code formula is the regulation thickness. This much thickness must be available uniformly everywhere in the vessel body.

The regulation thickness needs to be corrected for the corrosion allowance and mill tolerance. The next available commercial plate thickness is then the recommended wall thickness.

Available wall thickness is the thickness left after deducting the mill tolerance and corrosion allowance from the recommended wall thickness. This is obviously more than or at least equal to the regulation thickness. The maximum pressure that the shape can withstand with this available thickness while still complying with the regulation thickness formula is calculated. This is the Maximum Working Pressure (MWP) allowed by the recommended wall thickness.

After fabrication and assembly, the vessel is tested for its pressure integrity. The test if often carried out at a pressure higher than the design pressure as well as the maximum working pressure. It is rated so as to develop in the weakest vessel portion stresses equivalent to the yield stress of the material of construction. The hydrotest pressure is obtained by multiplying the MWP by a factor greater than 1. Logically it is 1-5, the safety factor used in getting allowable stress from material’s yield stress.

Flange Design Calculations

Adjacent vessel shapes (as in adjacent layers of a composite shape) need to be connected. These could be welded on flange connected. PVDA allows the user to specify suitable connection choice. Flanged connection, if selected need to be properly designed.

Various types of flanges and flange surfaces are possible. Choice depends on the service and vessel dimensions. Similarly, a wide choice is available for gasket material. The choice is governed by the service and flange type.

Suitable design formulae lead to calculation of Gasket Circle Diameter, Gasket Width, Bolt Circle Diameter, number of Bolts, Diameter of each Bolt, flange outer diameter and flange thickness. PVDA supports these calculation steps for a given choice of flange type, flange surface, gasket material, gasket thickness and bolt material.

Nozzle Calculations

The openings required for operational and maintenance reasons on various shapes of the vessel are in the form of pipes welded on to the vessel after cutting an appropriately sized opening.

The schedule of the pipe of given nominal bore (NB) is arrived as using the pressure thickness calculation formula for cylinder.

The openings require to be tested for the need of reinforcing pad. Reinforcing pad increases the vessel wall thickness around the opening if necessary by welding a collar around it. The thickness of the collar is designed using area compensation method. The formulae are specific to different shapes on which openings are cut. PVDA supports these calculations.

The nozzles could be of different types. PVDA supports the choice and calculations for specific user-made choice.

Thermal Insulation

A process vessel operating at super-ambient or sub-ambient temperature needs to be insulated to minimize heat egress or ingress respectively. Insulation also is needed as a safety measure so that the skin temperature (exposed surface temperature) of the vessel parts is not very high. The insulation thickness can be calculated on one or more of the three criteria, namely, based on maximum allowable skin temperature, maximum tolerable rate of heat egress/ ingress or economic criteria (considering the capital investment vis-a-vis cost of heat energy lost). PVDA supports the three design criteria.

Choice of insulating material is dependent on the services conditions as well as economics. PVDA supports choice of insulation from among a list available in the data base as well as a user-defined insulation.

Reinforcement Calculations

Theoretical Basis

Consider an infinite, flat, uniformly thick plate of a metal subjected to tensile load along one direction. The load is such that it develops tensile stresses all along its skin as shown

Consider now an infinitesimally small, circular cross-section hole punctured in this plate. The radius of the hole is ‘a’.

Things change drastically because of this ‘opening’ made into a plate. An easy visualization of this is offered by considering the induced stresses as pathways for transmission of load across the plate. These pathways are disrupted due to the cutting of a hole into the whole.

One thus expects a uniform stress pattern to be disrupted and stress “intensification” along the edges of the hole aligned to the stress direction and also a stress “rarification” in the vicinity of the hole in the transverse direction.

Theory provides a quantitative feel of the phenomenon. A formula to calculate stress levels at a point defined by polar coordinates ( ) as shown in the figure is as follows.

r

a

=

Substitution of various values of and in the formula gives a quantitative feel of the stress intensification around the opening.

Important positions are the 12 O’clock and 6 O’clock positions as well as 3 O’clock and 9 O’clock positions. From symmetry considerations, one can home in on 12 O’clock ( ) and 3 O’clock ( ) positions for further study. Let us see what is the situation on the edge of the hole ( ) along these directions and also as one moves away from the edge, one radius at a time (i.e. at ). 5 4 3 2 1

1 2 3 4 5

The calculations and the stress profiles offer important insights into the implications of making an opening.

As we travel along the 3 O’clock position, we see that stresses at the edge have intensified to 3 times their value before making an opening. Stress increases to 3 . This intensification attenuates very fast and the stress is 1.21 even as one moves a radius away from the edge of the hole. It reduces further and there is no intensification of any consequence beyond

.

The situation along the 12 O’clock axis is even more interesting. The tangential position along this axis is transverse to the original stress lines. There were no stresses along this direction initially. Cutting of a hole, however, induced a compressive stress at the edge of the opening. The compressive stress reverses within one radius, becomes tensile and then dies down fast.

Allowable stress for pressure vessel design is often derived by reducing the yield stress of the MoC at design temperature by a factor of safety. Most commonly recommended factor of safety is 1.5

Sa =

This, when coupled with the observed stress intensification around the opening, indicates the engineering unacceptability of stress intensification. For example, let the plate be stretched initially such that the tensile stresses reach the allowable level for the MoC. When the hole is punctured, a stress intensification factor of 3 would mean that the stresses would reach a level of 3 Sa or 2 Sy. The plate would thus yield plastically and deform around the opening. This may not be acceptable. Something therefore needs to be done around the opening to keep the intensified stresses within the allowable as much as possible.

One of the possibilities is to opt for a thicker plate (preferably thrice as thick as the requirement to keep stresses within allowable prior to cutting an opening). This would be uneconomical. Keeping in mind that the stress intensification attenuates with a circle of double the radius of the opening and fall below engineering safely margines, one therefore considers the need to provide a ‘collar’ or reinforcing pad to strengthen the stress carrying cross-section of the plate locally.

We thus carry from the theory two points.1) Something needs to be done because the stress intensification

might take the stresses beyond engineering safety margines.2) This ‘something’ needs to be done only within a circle of double

the diameter of the opening.

What about the assumptions?

Let us now revisit some of the assumptions behind the theory giving us the formula that led to above conclusions.

One of the assumptions was regarding the plate being flat. Pressure vessels and their closures are essentially not so. However the dimensions of the shape on which an opening for the nozzle is made is much larger as compared to the nozzle diameter. The nozzle thus sees a reasonably flat surface around it, if not a perfectly flat one. We therefore persume that the assumption is not that restrictive as to make the theory inapplicable in practical situations.

Another assumption was regarding the infinite expanse of the plate. Our vessels are of finite dimensions. However, as the stress intensification attenuates within few radiuses from the hole, whether the plate exits beyond that or not is not of much concern. This assumption is, therefore, not considered to be very restrictive.

Yet another assumption was regarding the infinitesimal dimension of the hole. The assumption was necessary to ensure that the hole remained circular inspite of stress intensification. Practically sized holes would actual deform and attain an over shape. This deformation actually helps redistribution and alleviation of stresses. Stress levels in the case of finite sized openings are thus likely to be more benign than what the theory predicts. Theory thus offers more alarming estimates. Use of the theory for practically sized nozzles is therefore acceptable.

The important limitation was regarding the unidirectional force inducing unidirectional stresses in the plate. In practical situation, we have a 2-D scenario. For example, a cylinder pressurized from inside or outside experiences stresses in circumferential direction (Hoop’s stresses) as well as axial direction. A sphere and other shapes as well, have stresses in 2 orthogonal directions.

Stress intensification can be quantified using superposition. Effect of stresses in one direction is superimposed on the same calculated for stresses in the other direction.

Calculations show that stress intensification is actually less in the case of cylinders and spheres. The order of severity of stress intensification is

flat plate cylinder sphere

(High Low)

Area Compensation Method

The need for the provision of a reinforcing pad around an opening is ascertained and the pad thickness is arrived at using the area compensation method stipulated by the codes. It is applicable to a cylindrical nozzle provided on any shape of a vessel or a closure.

The premise on which the area compensation method is based is very simple. It identifies the load bearing metal cross-sectional area which is lost due to the act of making an opening. It attempts to compensate this area loss by providing extra thickness in the affected vicinity of the hole.

It is important to get a correct picture of the area that is purported to be lost due to an opening. Consider the flat plate again. Let it be stretched in one direction such that the stresses are just equal to the allowable stress. Let the plate thickness be ‘t’ everywhere. We now contemplate to remove a circular area of diameter ‘d’ in a lane of width ‘d’ as shown below.

d

The load bearing metal cross-section that would be lost because of removing a disc of diameter ‘d’ is clearly not the area of the circle. Instead it is a rectangle of width ‘d’ and thickness ‘t’.

t t d

This area can be returned back to the plate by welding a disc of thickness ‘t’ of outer diameter ‘2d’ and inner diameter ‘d’. This would provide an extra area of d t / 2 on either side of the lost area ‘d t’ as shown.

2d

d t t

This in essence is the concept of area compensation. The actual calculations are somewhat more elaborate and incorporate the decision steps leading to the wall thickness calculations wherein the regulation thickness gets corrected for corrosion/erosion allowance and mill tolerance on plate thickness before the next available commercial thickness is recommended.

Let us consider a cylindrical vessel/pipe of outer diameter subjected to an internal design pressure of . Let the corrosion allowance be

and mill tolerance . Let the recommended plate thickness be .

Let a nozzle (or branch connection) of OD , ID and nominal thickness ( ) be required to be provided on this vessel/pipe (header). Let the mill tolerance be . Corrosion allowance and design pressure would be and as for header. This is so because the vessel and nozzle face identical service conditions.

It helps to consider the steps that go in recommending the header and branch thickness. Regulation thickness is calculated, corrosion allowance is added, mill tolerance is provided and the next higher commercial thickness is recommended. There is often an extra thickness available in the header design. This amounts to an extra area available in the affected zone to handle stress intensification. Compensation area can take advantage of this discount. Often, this extra area available is more than the area lost. No extra area by way of reinforcing pad is required in this case. The nozzle is then said to be ‘self compensating’.

The nozzle thickness calculations go through a similar sequence. There is thus some extra thickness (and hence area) available in the nozzle itself. It is believed that this extra thickness available in the nozzle up to a height of above the header OD can be accounted for in the area available.

If the nozzle is protruding inside the header, its portion up to a depth of is also considered as providing extra area to handle stress intensification.

A reinforcing pad is provided only if the area ‘lost’ due to cutting an opening is more than the area ‘available’ (due to over design) in the header portion, nozzle portion above the header and the nozzle portion inside the header. This area accounting has several nuances further to try and avoid provision of a reinforcing pad.

Consider the area that is lost. As seen earlier, it is a rectangle of width equal to the diameter of the hole and height equal to the ‘thickness’. Each term requires to be qualified further.

We would like our design to be functional right through the service life. Corrosion would have caused increase of the nozzle ID (which is the size of the opening also) to over this period. This is therefore considered as the design basis for the diameter of the opening to be used in reinforcement calculations. As a consequence, the affected area on the header extends to a circle of diameter . Reinforcing pad, if at all provided, will have this as its OD.

The ‘thickness’ to be used in calculating area lost is also important. What is indeed lost is the regulation thickness. The rest which comprised of the allowances, tolerances and extra is not consequence here. Regulation thickness would have helped keep the stresses at allowable level. This thickness is what is ‘missed’ as an opening is made.

Couple of other points are also very important. The opening for the nozzle is unlikely to be located on an existing weld joint of the header or its vicinity. A weld or an opening is a weakness in the structure and fabrication wisdom would dictate that these should not occur simultaneously. If this is so, then the regulation thickness for the header should be calculated using Weld Joint Efficiency value as 1 in the appropriate regulation thickness formula for the header shape. The regulation thickness thus may not be imported directly from previous calculations done at the time of header design. Note that this consideration reduces the value of regulation thickness, thereby lowering the estimate of area lost.

Another point is regarding the choice of the formula to be used for the regulation thickness itself. It should be the code formula for a shape ‘seen’ by the nozzle. It may not make difference if the nozzle is placed on a sphere, hemisphere, cylinder, flat plate or an ellipsoidal closure. For a dished (torispherical) closure or a cone housing a nozzle, it does make a difference.

If the nozzle is on the ‘crown’ of a dished closure, the shape around it is actually a sphere with diameter double that of the vessel. While designing the closure, formula pertaining to the dished closure would have been used. While calculating regulation thickness to be used in calculating area lost, one should use formula for a sphere instead. Note that this consideration also reduces the value of regulation thickness, thereby lowering the estimate of area lost.

Similar is the case for nozzle on a cone. The thickness of the cone is arrived at using the base diameter of the cone. As one moves towards the tip of the cone, the regulation thickness requirement decreases and extra thickness increases. To avail of this extra thickness in reinforcement calculation, one should calculate the regulation thickness afresh using cone diameter at a level corresponding to the center of the opening. Note that this consideration also reduces the value of regulation thickness, thereby lowering the estimate of area lost. In fact, a properly located nozzle on a cone can often be made ‘self compensating’.

Let us now put together the balance sheet of the load bearing metal area affected due to an opening.

Area Lost

Where is the regulation thickness recalculated with above discussed consideration.

Area Available

From Header:

is the thickness that may not be available as per mill tolerance. If M% is the mill tolerance for header, then

From Nozzle portion outside the header:

( )

From nozzle portion protruding inside the header:

The last expression needs some clarification. The protruding portion of the nozzle is subject to same pressure on either side of its wall. The differential pressure on this wall is thus zero. There is thus no regulation thickness requirement. At the same time, corrosion is eating into this wall from inside as well as outside. Corrosion over the expected service life is thus twice the corrosion allowance.

The participating heights of the nozzle, and , (participating in sharing the extra stresses) are given as follows.

Note that for a non-protruding nozzle, . The regulation thickness of the nozzle, , is imported directly from its previous calculations done for deciding nozzle thickness. No correction for weld joint efficiency is required in this calculation as the entire nozzle with its seam welding (if any) is in the affected area.

The balance sheet attempts to hammer down the estimate of area lost. The area available is estimated by looking for as much area available in the vicinity as possible. In fact, even ‘weldment’ area in the affected rectangle is accounted for in area available if such estimates are available.

weldment area

A reinforcement pad is provided if area deficit is greater than zero.

The deficit area is provided in the affected zone welding a reinforcing pad of thickness given as

The formula is self explanatory in view of the discussions above and the figure.

Although not explicitly stated, it is pressumed that the reinforcing pad is of the same material as that of the header/nozzle. This would normally be the case, as welding together dissimilar metal could lead to galvanic corrosion. However, considering the fact that the pad is not exposed to the corrosive process fluid, if a dissimilar material is chosen for the pad for economic considerations, an appropriate correction to the pad thickness is called for.

Codes recommend on upward revision of the thickness if pad material allowable stress ( ) is lower than that of the header/nozzle (Sa). Logically, the revision is as follows.

Codes however do not allow a downward revision for a stronger pad material. The calculation formula should thus be

For Fabrication consideration, a pad thickness of less than 5mm is not recommended.