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CH-307 EQIPMENT DESIGN Report on Types, applications and design of vessels of Non-Circular crossection. By Radhesh Sirohiya Enroll no-09112048 Vivek Pandey Enroll no-09112047 Voggu Vikas Reddy Enroll no-09112031 B.Tech 3 rd Year Chemical Engineering
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Page 1: Report on Non Circular Pressure Vessels

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CH-307 EQIPMENT DESIGN

Report on

Types, applications and design of

vessels of Non-Circular crossection.

By

Radhesh Sirohiya Enroll no-09112048

Vivek Pandey Enroll no-09112047

Voggu Vikas Reddy Enroll no-09112031

B.Tech 3rd

Year Chemical Engineering

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Acknowledgement

We would like to acknowledge and extend our heartfelt gratitude to all those who

have directly or indirectly helped us and made the completion of this Project possible

especially Sh. Vimal C. Srivastava, -- for his vital encouragement and suppor,for his

assistance, constant reminders and much needed motivation and help and inspiration

he extended to us.

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Introduction

The aim of the report is to set down, in readily available fashion, the

fundamental theory needed for the design of some typical pressure

vessels of non-circular cross section.

Of these the most common are the rectangular section tanks. They are

often used as bulk storage containers or as baths in the treatment of

metals and fibres and surface coating processes etc. For this reason

vessels of this type have been given special attention. Other shapes are

also included by reference rather than by a worked example.

In explanation of the underlying theory a number of fully worked out

examples are given showing the procedure which may be adopted

when preparing the necessary calculation sheets for different types of

vessels.

There are no national or international standards or codes of practice

that will cover all of the types. Here ASME VIII, Div.1, Appendix 13

probably offers the best guidance on a number of different designs.

Unfortunately, in its present form, it is rather cumbersome and

requires considerable time for proper understanding and assimilation.

Another useful source of information on rectangular tanks can be

found in the Theory and Practical Design of Bunkers. Rectangular

section headers are also covered by the Swedish Pressure Vessels

Code, British Standard BS I 113: 19694 and the Italian Standard

ANCC-VSR Collection Section VSR IS: 1978. The last two

references need to be viewed with considerable reservations as they

appear to contain a number of discrepancies which are inconsistent

with the fundamental theory.

Where there is no relevant code, the procedure outlined here follows

the same logic, based on fundamental engineering theory as used in

the codes and should therefore be equally acceptable. Such procedure

should be regarded as evidence of good modern day general

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engineering practice in this field. It is hoped that it will promote a

better understanding of the problems associated with such vessels

which are often either ignored or not given the consideration and

attention which they deserve. Such tanks can be quite complex in their

detailed design and unawareness on the part of the designer and/or

fabricator, to appreciate the various aspects can lead to costly

ramifications later on. These tanks although they appear to be very

simple indeed, can nonetheless cause considerable embarrassment if

not assessed adequately at the outset.

Additional guidance is given on square/rectangular ducting. Normally

such ducting is restricted to 20 psig (0·138 N/mm2). However, the

procedure outlined in this article has no limitation per se.

Comparison of the rectangular vessels with the equivalent size

cylindrical (circular cross-section) vessels indicates that the former are

rather inefficient. Cylindrical vessels will sustain considerably higher

pressures, for the same wall thicknesses and size, see Fig. 20.

However, practical consideration will often force the designer to select

a rectangular shape as the best available option.

The fundamental theory is applicable to both external and internal

pressure. Worked examples given in the text refer to internal pressure

for the simple reason that, for the external pressure application,

considerable gaps still exist in the knowledge of the allowable

compressive stress levels which will not cause buckling or plastic

collapse in rectangular and other noncircular tanks. In such instances it

should be possible to use the design data contained in the British

Standard BS 449: 1969 for checking the main stiffeners and beams.

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Fundamental theory for rectangular

section pressure vessels

Figure 1 shows the basic geometry of the rectangular vessel with

sharp corners and which is subjected to a uniform pressure of p.

Where

L = the longer span

h = the shorter span

l1= second moment of area of the beam BCB about its neutral axis

l2 = second moment of area of the beam BAB about its neutral axis

Due to symmetry about axes AA and CC it will be convenient to

analyse one quadrant only of the cross-section shown. This quadrant

is in equilibrium under the action of the loads and moments indicated

in Fig. 2.

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Clearly from the balance of the horizontal and vertical forces acting

on the quadrant we obtain

Tc =

which represents the direct tensile load in member BC, and

TA =

the tensile load in member AB respectively. In evaluating these

tensile forces the thicknesses t1 and t2 are considered to be negligible

in comparison with dimensions such as L and h, i.e.

(

effectively equals to L/2 .

In any member of a structure subjected to bending the total strain

energy is given by

U = ∫

...............(1)

where M is the bending moment at any point on the member caused

by the combined effect of the imposed loads and the supporting forces

and moments, whether statically determinate or not.

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Two further postulates (Castigliano's theory) help to solve the

problem. These are:

(i) The partial differential coefficient of the strain energy in a

structure with respect to a load F acting on a structure, is

equivalent to the displacement of F along its line of action,

i.e.

= ∫

= ᵟ ............. (2)

(ii) The partial differential coefficient of the strain energy with

respect to a moment acting on a structure is equivalent to

the angle through which that portion of the structure

rotates when the moment is applied

= ∫

= ᵠ .............(3)

When, as in this case, the support of the structure (point A) does not

give way under the action of the loading, then there is no deformation

of the structure at this point of support and the two expressions just

quoted can be equated to zero.

By setting down the equation for moments along AB and BC and by

considering the strain energy due to bending (by integrating along AB

and BC respectively) it can be shown that the moments at the three

important points A, B and C become for a general case

MA

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(

) ..................(4)

MB

(

) ................(5)

Mc=

(

) ...................(6)

Where

K=

and

Notice that

MB=

-

(

)

where the first term denotes the bending moment at mid span for a

simply supported beam BCB under the action of uniform load p. For a

uniform wall thickness throughout, the parameter

K=

which is the same as P and the three moment expressions simplify to

the following

MA

(

) ................(7)

MB

(

) ................(8)

Mc=

(

) ..............(9)

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where once again

the ratio of shorter to longer spans. By substituting specific values for

the parameter (i.e. = 0,0·1,0·2, ... etc. up to = 1) we can express

the three moments in a very much simplified form

MA= 22 pLA .....................(10)

MB= 22 pLB .....................(11)

MC= 22 pLc .....................(12)

where A, B and c are the three new parameters which, for uniform

wall thickness throughout, are dependent on ratio only.

The plots for these three parameters are shown in Fig. 3, where after

simplification these can be written as

A=

.....................(13)

B=

.....................(14)

c=

.....................(13)

Identical plots to those shown in Figure 3 were obtained from the

equations given in ASME VIII, Appendix 13 and the Swedish

Pressure Vessels codel indicating that these are also based on

fundamental engineering theory.

When comparing the above equations with the corresponding

parameters given in these two codes it must be borne in mind that the

latter3 specifies the two spans as 2m x 2n so that the relevant

constants a will dint-r factor of four, since M reference ( l) and M = a

3pm reference (3). Thus for consistency a1L2 must be equal to a3 m2.

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As L2 == 4m2 hence this factor of is contained in the parameter ex.1

in all the expressions for moment given as M = apm 2Hence it can be

seen that this approach presented in this article will satisfy both

ASME VIII,

Div.1 and the Swedish Pressure Vessels code for the plain rectangular

vessels but refer has still to be made to these codes for the allowable

design stress and the weld factor where necessary. From the plots

shown in Fig 3 the moment distribution curve along each member can

be quite easily obtained by the following method.

(a) For members BCB, span L First draw to a suitable scale the

free end moment distribution curve BCB which is given by

the standard equation

Mxb=

................(16)

where x is the distance from point B. (Distance B-B to represent the

span length L.) Refer back to Fig. 3 to obtain the relevant bending

moment at the corners, M. = a.pL2. Draw a new "zero bending

moment" axis 0-0 at a distance equal to M. below the original datum

line BB (as shown in Fig. 4a). The resultant sketch will give the

complete bending moment distribution diagram for the longer span

length L. Moment at any point along BB is then simply given by the

vertical intercept, either above or below

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the new datum line 0-0 as the case may be. In this particular instance

Fig. 4 shows the moment distribution curve which is applicable to the

following three geometries: (i) datum line 0-0 for a rectangular header

whose hlL ratio equals 0·5, (ii) datum line 0-0 for square header, i.e. h

= L and (iii) for built in beam where h = o. The points of contraflexure

and also shown for these cases. This sort of information could prove

useful when the decision has to be made on the best location of the

welded seam or any other outside attachment.

(b) For members BAB, span h

Similar procedure to that described above can be used to obtain the

moment distribution diagram for the shorter span. The only difference

in procedure is that the initial free end bending moment curve is now

given by the equation

Mxa = ph2

1

h

xx

2

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=2

2

t

pb

where x is the distance from point B (towards A this time). The new

distance BB should now represent, to the same scale as above, the

shorter span h.

The basic engineering theory and the above procedure indicate that

each member of a rectangular section vessel can be treated as an

initially simply supported (free end) beam uniformly loaded along its

entire length which is then subjected to the end moments MB , the

latter determined from Fig. 3. This approach will be useful for

calculating the central deflection of the members. This is illustrated in

Figs. 4(b) and (c) and the plot for the central deflection of the longer

span L is given in Fig. 5. So far we have dealt essentially with

uniform wall rectangular vessels. The preceding basic theory is

equally applicable to rectangular vessels which have peripheral

stiffeners spaced along the length of the vessel as shown in Fig.6. In

such cases we have to check not only the strength of the stiffeners but

also the stress levels in the wall panels between these stiffeners.

The strength of the peripheral stiffeners can be determined by the

method described above, as for the plain rectangular vessels, by

substituting ps for the uniform pressure load p used in the preceding

analysis. Equations (4), (5) and (6) can be used directly for a general

case where the second moments of area of the stiffeners I and 11 and

the wall thicknesses I2 of the two main sides are different.

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For uniform wall and stiffener sections Equations (10), (11) and (12)

and Fig. 3 become once more applicable provided p’s is substituted

for p in the relevant equations.

The wall panels between the stiffeners can be treated as rectangular

panels fixed (built-in) at all four edges and subjected to a uniform

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pressure load p over the entire area. Reference (7) covers this

particular case and gives the maximum bending stress, which occurs

at the centre of the long edges, as

=2

2

5.0t

pb or

2

2

5.0t

pb

where the value of depends solely on the ratio of the two sides a/h, h

is the width or the shorter span and t is the panel plate thickness. Fig.

7 gives the plot for the variable for various a/b ratios. Notice that for

alb values above 2·15 the parameter = 0·5, giving

This represents the same situation that occurs for a built-in beam of

span b. Here the end moment

= 2

2

2

2

5.012

6

t

pL

t

pLM

and the plate section modules for ac unit width strip

Hence the bending stress at the built in edge

= 2

2

2

2

5.012

6

t

pL

t

pLM

p = 2375.010

5.25.1

cmkg

i.e. the same as above. This confirms that for wall panels whose alb

ratio exceeds 2·15 we can treat the central portion of such panels as a

fixed-in beam of span equal to the width of the panel.

One further detail which will require consideration is the solution for

the corner wall panels, whether the corner occurs between the main

side panels or between the side panels and the flat ends which may

have transverse stiffeners. Such details can be dealt with by

evaluating the bending moments and tensile loads shown in Fig. 8.

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Enclosed Rectangular Tank with Flat Bolted Top Cover with Gasket

Seal Fig. 16 shows the essential details of the proposed tank. The

tank is to contain liquid of specific density of 1·0 up to a maximum

depth of 1·2 m. The "gas space" pressure == 0·069 N/mm 2 (IO

psig). Because of the presence of the flanged joint the cross-section of

the tank cannot be considered as an integral entity. In this instance the

cover and the tank have to be treated independently.

(i) Consider the cross-section of the tank itself. The vertical stiffeners

on the side wall can be considered as beams built-in at the lower ends

and simply supported at the flange face joint level and loaded as

shown in Fig. 17. The pressure at the bottom of the tank will be given

by P2 == 0.069 + 0.012 - 0·081 N/mm 2 The pitch between the

stiffeners = 440 mm. Thus the loading per unit length of span will be

PI = 0·069 x 440 = 30·36 Newtons per unit length of span AB,

and P2 == 0.012 x 440 = 5·28 Newtons per unit Of span AB.

The two loading conditions shown above will produce the following

bending moments at

base (point B)

assuming that the triangular load distributton is over the entire

length of 1·5 m instead of 1·2 m. This will give slightly conservative

results. Thus the combined bending

moment at base will be

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M = MI + M2

=(30·36/8) X 15002 + 5·28 X 1500

2 - 9330750 N.mm

Hence the bending stress at this location is 125.08 N/mml

In addition there will be a tensile load equal to = (P2 L)/2

acting on each stiffener. Therefore

Ωd =(81 * 440 * 1500 )/(2*23.4*100)= 1142 N/mm1

So that the total tensile stress

Ωt = 125.08 + 11·42 = 136·50 N/mm1.

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Fig 17

Hence the combined tensile stress level Ω=124.34 N/mm2.The above

calculations indicate once again that the plate panels and the stiffener

sections are adequate for the loading conditions specified. Now that

the overall design check has been made on the main components the

designer can carry out detailed calculations on other features

particular to his case, such as ftangedjoint details and the adequacy of

supporting sections. Note that if this enclosed tank had been of

integral construction the maximum combined tensile stress

level would only be 101 N/mm2 .Compare this with the

136·50 N/mm2 calculated at the base of the vertical stiffner.

Fig 18

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The information presented here is based on the basic theory contained

in Reference (8) by combining the two separate loading conditions for

panels L and h respectively. Basic information on critical moments

and tensile loads is also given for

(a) rectangular vessels with radiused corners-see Fig. 9;

(b) elliptical vessels-see Fig. 10;

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(c) long vessels of uniform thickness-see Figs. II and 12. Table 1 in

the Appendix gives some basic equations for the simple geometries

and loading systems considered in this article.

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WORKED EXAMPLES

1. Open top rectangular tank with continuous horizontal wall

stiffeners

Figure 13 shows the essential details of one such tank measuring

5500 x 2000 x 2500 mm deep. The tank is to contain liquid of specific

density 1·5. It is to be supported on beam members forming part of

the general plant structure.

The tank is to be built from 6mm thick plate material of 432 N/mm2

ultimate strength. The corrosion effect on the plate thickness is

considered to be negligible during the useful life of the tank.

I t would normally require several attempts to establish the optimum

size of the stiffener and their respective spacing. The following check

will deal with the tank as shown in Fig. 13 in order to demonstrate the

design method rather than the final choice.

The pressure distribution on the tank walls will be linear and as

shown in Fig. 14. The pressure at the bottom of the tank due to the 2·5

m head of liquid of specific density of 1·5 will be

p = 2375.010

5.25.1

cmkg

(as 10 m head of water is equivalent to 1 atii or 1 kg/cm2 pressure), or

in Newtons per mm2 this pressure is equivalent to

2

5605001032.352.2

2

1 2

1p

= 476.15 Newtons per mm of span

364.05500

2000

= 0.0368 N/mm2 or 3·68 x 10

-2 N/mm

2

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(A) Check on Stiffeners

(i) Considering the first stiffener from the bottom, namely SI. It is

fabricated from 200 x 100 x 8 mm rectangular hollow section whose

properties are as follows

1xx= moment of inertia = 2269 em4

Zxx = elastic modulus= 227 cm3

A= sectional area = 45.1 cm2

Applications of non- circular vessels.

Such vessels have wide application as component parts of air-cooled

heat exchangers, duct work, special piping, extrusion chambers and

specialty vessels used in laundry and hospital service and heat transfer

applications.

A pressure vessel having a non-circular axial cross-section to be

used either as an aircraft engine duct or as an aircraft fuel tank. The

use of pressure vessels is particularly important with respect to

aircraft engine ducts and fuel tanks since an engine duct or fuel tank

having an elliptical-shaped axial cross-section is aerodynamically less

protrusive than an equivalent circular cross-sectional vessel. Certain

problems, however, result from the vessel's non-circular axial cross-

sectional configuration. Where the internal pressure forces in a

circular pressure vessel create simple "hoop" tension forces in the

walls of a "circular" pressure vessel, the walls of a "non-circular"

pressure vessel become subject to varying bending moments around

the periphery. The bending moments result from the non-circular

section tending to become round under the internal pressure, with the

bending moments being maximum at the nose and tending to "open

up" the nose curvature, reducing to zero, then reversing the bending

direction at the top and bottom centerlines.

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In the conventional approach to dealing with these bending loads,

internal or external frames have been added to the non-circular

pressure vessel. These frames are generally added around the entire

surface periphery of the pressure vessel, and are closely-spaced along

the entire length, thereby greatly increasing the weight and cost of the

pressure vessel. An additional disadvantage of using these frames is

the deflection of the shell's skin between each of the frame members

in the presence of pressure This skin deflection increases aerodynamic

drag, and thereby reduces propulsion efficiency, thus again

necessitating close spacing of the frames leading to increased weight

and cost.

An object of the present invention is to provide a non-circular

pressure vessel that avoids the use of support frames and their

corresponding disadvantages. An additional objective of the present

invention is to provide a non-circular pressure vessel with essentially

constant compression and tension loads around the periphery despite

the varying bending moments created by the non-circular cross-

sectional axial configuration. A further objective is to provide a non-

circular pressure vessel that is able to carry fuselage axial

compression and bending loads resulting from its attachment to the

fuselage. A further objective of the present invention is to provide a

non-circular pressure vessel that has minimal weight and maximum

aerodynamic smoothness, and which may be constructed with

minimal cost and with improved reliability.

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Conclusions

The basic engineering theory outlined in this article shows how we

can check the design of a number of non-circular cross-section

pressure vessels. The worked examples demonstrate how we can

represent various details of a rectangular tank by replacing them with

simplified geometries which can subsequently be evaluated by the

fundamental engineering theory. We must endeavour to make each

theoretical representation as close to the real component as possible.

The closer the approximation between model and actual detail the

higher the allowable design stress levels we can adopt.

The simplification procedure and the degree of sophistication needed

will depend on how arduous will be the intended duty, on the

confidence of our knowledge of the material properties and other

factors. I t is Important to realise the implications of the

simplifications and assumptions which have been made. If the

theoretical model, or the simplified geometry, is too far detached from

the real detail the design calculations may become invalid.

In other cases we can compensate for any simplifications by using

much lower design stress levels or by ensuring that the theoretical

representation is conservative. Personal experience and knowledge of

the fundamental engineering theory will dictate the course of the

appropriate action. This approach is certainly not recommended for

the beginners. If you are one then seek advice.

It is hoped that by outlining some of the salient features of the non-

circular pressure vessels this article would prove useful to the

designers and fabricators alike, and that it would, in some small way,

lead to fewer failures of the type normally classified' as due to poor or

inadequate design.