MULTILAYER VESSELS

DEPARTMENT OF CHEMICAL ENGINEERING EQUIPMENT DESIGN

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TABLE OF CONTENTSS.NO.1 2 3 4 5 6 7 8 9 10 11 12

TITLEAbstract Theory Multilayer vessel with shrink fitted head Determination of hoop stresses Determination of shrinkage stresses Determination of interferences required in shrinkfitted vessels Thermal expansion for shrink-fitting Multilayer construction using weld shrinkage Ribbon and wire wound vessels Theory of ribbon and wire winding Thin shells wound at constant tension References

PAGE NO.3 3 9 14 14 15 17 18 22 24 24 25

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ABSTRACTMultilayer pressure vessel is designed to work under high pressure condition. Optimization of thickness of each layer in multilayer vessel is carried out by Genetic Algorithm and then stress distribution is analyzed under optimum shrinkfit condition. The fatigue life is calculated for shrink-fit multilayer vessel. Thickness of each vessel is considered as design variable and objective function is maximum hoop stress through-out the thickness at the given working pressure. Multilayer vessel is assumed to be constructed by insertion of different vessels with zero interference and zero clearance such that interface pressure at the mating surfaces is equal to the pressure generated at the same surface due to interference fit. The mathematical model is derived from basic governing equation of thick cylinder. The appropriate boundary conditions are applied to each successive layer. Effect of number of shells on the maximum value of hoop stress is analyzed. Apart from this, effect of overall thickness of pressure vessel on the effectiveness of multi-layering is brought into focus. Stress distribution and fatigue life for the obtained thickness of each vessel from Genetic Algorithm is nearer to that obtained from Lagranges multiplier method.

THEORY:- SOME CONSTRUCTION METHODS AND ADVANTAGESThe trend in chemical production is toward increasing capacities of process plants and units by higher operating pressures and temperatures and by the use of larger equipment. Each of these trends requires thicker equipment walls. High pressure equipment diameter often reaches 4-5 m, with wall thicknesses up to 300 ram, compared to 800-1200 mm which was the limit in the chemical industry earlier. The high cost of such unique equipment results in the development of process units without spare equipment which greatly increases the requirements for its reliability and service life. At first thick-walled shells were fabricated as single forgings. This required large steel-melting, forging-pressing, and metal-cutting equipment. The high cost of such equipment and the requirement for unique and expensive technological equipment restricted the production of thick-walled equipment. In addition, this technology did not guarantee equipment with the required design3

parameters. The development of welding technique permitted the fabrication of larger forged-welded and pressed- welded vessels, in effect removing the technological limitation on vessel length. In spite of the reduction of the cost of these vessels their fabrication still required expensive equipment (for forging rings or bending thick plates). Increased vessel diameter was retarded by the limitation of obtaining thick roiled plate of consistent high quality throughout its cross section and the complication of bending the plate. The thick plates and low ratio of beading radius to thickness, which is characteristic of high pressure vessels, necessitated hot rolling of shells. This required the construction of heat-treating furnaces and larger roiling equipment. Concepts which were accepted in the field of artillery design in the second half of the last century played an important role in the development of larger high pressure vessels. At that time Russian engineers developed the theory of multilayer barrels for large caliber guns made of several cylindrical shells with each consecutive shell being stretch-fitted over the previous one. About 40 years ago A. O. Smith (USA) started test fabrication of multilayer vessels which featured concentric layers in the vessel wails. The experience was utilized in the United States in developing a variety of processes operating at high pressure. The method developed in the United States was also applied in a number of other countries. The production of multilayer vessels was developed in the Federal German Republic using a different technique: winding of a special rolledprofiled tape in the form of a helix. Longitudinal stresses caused by internal pressure are transmitted without welding by mechanical contact between layers. This is achieved by tongues and grooves which fit into each other during the fabrication of the cylindrical portion of the vessels. The vessels are harmonized by the use of a central tube. Wound multilayer vessels produced by this method played an important part of the development of processes for fixed nitrogen production. However, this vessel-fabrication technology was found to be inadequate due to the poor matching of tongues and grooves during the tape winding.

An interesting modification of the technology for multilayer vessel fabrication with concentric layers was developed in the Polish Peoples' Republic. This technique reversed the order of assembling the layers used by A. O. Smith: fabrication is started from the outside layer and progresses toward the inside instead of starting from the inner layer. This results in a tighter assembled4

cylindrical casing and a more favorable distribution of residual stresses. Jacques produced multilayer vessels using the "Multiwall" method. These vessels are fabricated of pre calibrated and preheated shells with 30-50 mm thick walls which are slipped over each other to form the desired wall thickness. The use of thick sheet steel, which is required for these vessels, reduces the advantages of this type of multilayer construction. A new construction and fabrication technology for multilayer vessels was developed: the vessels walls were made by spiral winding of long, wide steel ribbons. The vessel was made leak tight by the internal welded seam while the required vessel strength was obtained by using the correct number of spiral windings. The individual multilayer shells could be welded together forming a cylindrical vessel of any required length. The rapid development of the economy and the increased requirement for thick-walled vessels by the chemical industry after the war forced resumption of fabrication of multilayer vessels and continued investigation and improvement of this type of design. The design of spirally wound vessels was significantly improved by the introduction of a central tube on which wide ribbons of steel are wound directly from coils: these vessels are now called coiled. Other technological and design improvements were introduced. The advantages of coiled multilayer vessels, proposed in the USSR, were evaluated abroad and a Japanese firm has started fabrication. At the present time the use of multilayer walls is universally accepted as economical due to the following advantages. Multilayer wall construction removes the limitations previously imposed in designing thick-walled vessels: high pressure vessels with practically unlimited wall thickness can be fabricated per- mitting the use of larger diameter vessels. The reliability of high pressure multilayer vessels is higher than that of solid wall vessels due to the layered construction and the use of thin steel with better and more predictable properties. A multilayer wall can localize cracks or other defects in one layer. Cracks in solid wall vessels propagate through the entire wall thickness and destruction are practically instantaneous in case of a failure. The energy liberated in a very short period of time creates dangerous explosive conditions. In a multilayer wall, where the metal is purposely separated, a crack formed in one layer cannot propagate across the gap and must be initiated anew on the other side of this gap. This inhibits and slows down crack propagation. The cost of multilayer vessels is lower due to reduced metal requirements, the use of simpler equipment, and a reduction in labor and fabricating time. The inner surface of a multilayer vessel can be made to meet any requirement by selecting5

the proper metal for the inner layer. This eliminates the necessity for fabricating the entire vessel of expensive special steel or the use of complicated and expensive methods of providing a protective layer inside the vessel. Less productive area is required for fabricating multilayer vessels. Large rolling, pressing, and heat extracting equipment is made available for other purposes. Welded vessels with solid walls thicker than 35 mm require high temperature tempering to relieve residual stresses. Solid wall or bimetallic wall vessels for hydrogen-containing atmospheres are fabricated of high-alloy hydrogen-resistant alloys which require high temperature heat treatment after welding. Large furnaces, long periods for heating, soaking, and cooling, and high energy consumption are required for large vessels. Multilayer vessels are fabricated of less expensive, lower alloy steel which does not require high temperature heat treatment after welding. In spite of the many advantages of multilayer over solid wall vessels, the justification for their use is determined primarily by the wall thickness. The choice of design and fabrication technology of multilayer vessels (walls made of concentric layers or by coiling) depends to a large extent on the strength requirement of the vessel and the reliability of multi- layer wall fabrication.

Large frictional forces, which increase exponentially with the contact angle, are developed in moving a flexible element along a cylindrical surface. Friction is further increased by radial forces pressing each layer against the next. The friction forces which try to keep each layer from moving relative to the neigh- boring layers also act on both surfaces of the layer. Coiled multilayer walls tightly wound under these conditions approach the state of solid wall construction and the displacement between layers with reduction of gaps and compaction of the wall will be of a local, random nature and of insignificant magnitude occurring only in a few spots. It is important to have a flawless start and finish of the coiling in fabricating vessels by the coil method. This does not represent any great difficulty. If there are no appreciable stresses at both ends of the coil the strength of such a vessel will not be lower than that of a concentric layer vessel fabricated with equal care. The difference in the strength of walls fabricated by the two methods becomes less significant as the number of layers increases. The stresses in multilayer shells of the two types in the elastic stress range differ significantly due to local deformation during compaction of the walls and the elimination of6

deviations from ideal theoretical flexure. However the scatter of stresses in the initial stage of loading a multilayer vessel does not have any significant effect on the pressure which causes vessel failure. Compaction of the layers and variation of their flexure under the effect of internal pressure is accompanied by the development of localized in- significant, and therefore safe, plastic deformations. Non-uniformity of tangential stresses, which are used to calculate the vessel strength, increases with higher external to internal diameter ratios for multilayer wall vessels just as it does for solid wall vessels. However increased stress on the inner surface of the vessel is maintained only during the period of elastic operation of the vessel. After plastic deformation spreads from the inner vessel surface to the entire wall thickness the stresses are equalized and the excess stress on the inner layers which had existed causes only a small difference in the values of plastic metal deformation in the inner and outer wall layers. Since the increase in plastic deformation of the inner layers constitutes only a small fraction of one percent, while vessel rupture occurs only after deformation equal to several percent, the effect of thick walls in vessels is not as pronounced as the elastic range design calculations indicate. Actually failure of multilayer vessels can be caused by nonuniformity of strength properties of metal or some stress and de- formation concentrators on the outer surface of the vessel. This has been occasionally observed in tests. Let us assume that in a coiled vessel only the inner and outer layers are actively loaded, since the intermediate layers serve primarily as filling which transmits the load from the inner to the outer layers. The intermediate layers could perform this function if they were cut into sections along the circumference. In fact the radial displacement of the coils is accompanied by the corresponding tangential deformation similar to that in a vessel with concentric wall layers. While properly designed and well fabricated thick-walled multilayer vessels of both types are identical from the strength standpoint, coiled vessels have great technological advantages over concentric layer vessels. Concentric layer vessels have weld seams in each layer, it is difficult to control the quality of seams of small cross section welded on top of a thick wall section which has been previously assembled and welded. It is impossible to inspect the seams after assembly which reduces the reliability of the vessels. Coiled vessels can be fabricated without any welding inside the wall thickness. This reduces the amount of welding required and increases the reliability of the vessel. One long steel ribbon is used in fabricating a multilayer coiled shell. The number of components required for fabricating a shell7

with concentric layers is at least equal to the number of layers; all components are different. It is impossible to predict the size of shells which will be required since, even if the shells fit tightly, the tolerance in actual metal thickness is cumulative. The fabrication of coiled multilayer vessels utilizes the achievements of metallurgy more effectively. A wide ribbon which is produced in continuous rolling mills, coiled in sections of the required length, can be used. The width of the ribbon (1500- 2350 mm) and its thickness (2-12 mm) are well suited to multilayer vessel fabrication. The fabrication of coiled multilayer shells requires considerably less time, labor, and fabricating space. No prefabrication (specifically rolling), repeated pulling of the shells over the preceding one, welding of the longitudinal seams, and surface finishing of these seams flush with the base metal are required. Winding of coiled shells can be performed continuously to the required wall thickness using relatively simple equipment.

Coiled shells can be fabricated with stricter adherence to the desired conditions, i.e., ribbon tension, temperature, etc. Tension caused by shrinkage of weld seam metal in concentric layer shells introduces an unpredictable factor which is undesirable in welded pressure vessels. Welding of seams in each layer in concentric layer construction causes some surface deformation which prevents the layers from touching and, consequently, decreases the density of the multilayer shell. Comparative tests of two vessels, coiled and concentric layers showed that coiled vessels have a denser winding. Local gaps between layers during circumferential butt-welding of shell sections are conducive to cracking. The coiling method makes more effective use of recently developed thermally toughened steels. In concentric layer vessel fabrication each layer has at least one weld seam which is a weak spot (heat reflected weld zone) reducing the strength of the entire multilayer part. Thus comparison of the two main methods for fabricating multilayer vessels shows that coiling has significant advantages.

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MULTILAYER VESSEL WITH SHRINK-FITTED HEADThe relationships which follow were first presented by H. L. Cox in 1936. The theory as developed is based upon the assumption that the maximum combined stresses (hoop-stress plus shrinkage stress) existing at the inner surface of each of the several shells will attain a certain identical value. It is assumed that both the internal and external diameters are known and that number of shells is to be a minimum. It is also assumed that the combined cylinder is fabricated by shrinking each successive shell from the inside outwards and that after each shell is shrunk-on, the outside diameter is machined to size before the next cylinder is shrunk-on to the inner shell or shells. Therefore the designer must determine the number of shells and their radii plus the interference (the amount by which the outside diameter exceeds the inside diameter of the next shell ). The relationship for designing such a vessel may be derived as follows :

d i , d 2 ,....d n = diameter of successive intershell surfaces, inches pi = internal pressure, pounds per inch po = external pressure, pounds per square inch p1 , p 2 ,.. pn 1 = successive interface pressures with pi and po acting

f q = hoop stress set up at the inside of each shell with pi andpo acting, pounds per square inch (to be same for all

shells)' p1 , p1 ,......... p1 = interface shell pressure that exists when pi = po 2 n

According to the Lame thick-walled-vessel theory, considering the rth 1 shell and using the sign convention gives:

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fr ! a But

b r2

!a

4b d2

f r ! pr

Therefore pr !

b r2

a!

4b a d2 b rr21 ! a 4b d r21

(1)

And pr 1 ! f r 1 ! a

(2)

Fig. 1: Diagram of a multilayer shell showing notation used in derivation

f q ! ft !

b rr2

a!

4b d r2

a

(3)

Therefore

p r f q ! 2 apr 1d r 1 f q d r2 ! a(d r21 d r2 ) Dividing through by d r2 gives:

(4)

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p r 1d r212 dr

f r ! a(

d r21 d r2

1)

(5)

Let

d r 1 ! K r 1 dr

(6)

Substituting Eq. 6 into Eq. 5 give: pr 1 K r21 f q ! a ( K r21 1) Dividing Eq. 7 by Eq.4 gives: pr 1K r21 f q pr f q Let Cr 1 ! !2 K r 1 1 2

(7)

(8)

2 K r21 1 K r21

(9)

It follows from Eq. 8 that

Cr 1 p r 1 pr ! (1 Cr 1 ) f qAnd similarly

(10)

Cr pr p r 1 ! (1 Cr ) f qNow if r=1, If r=2, If r=3, If r=4,

(11)

C1 p1 p0 ! (1 C1 ) f q C2 p2 p1 ! (1 C2 ) f q C3 p3 p 2 ! (1 C3 ) f q Cn po pn 1 ! (1 Cn ) f q

This series can be represented by:(C1C2 C3 .......C n ) p 0 pi ! (1 C1C2 ....C n ) f q

(12)

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Let

C1C 2C3 .......Cn ! F

(13)

Eq. 12 then becomes:

Fpo pi ! (1 F ) f qSolving for f q gives: fq ! pi Fpo ( p p ) ( Fpo po ) ! i o F 1 F 1

(14)

Therefore

f q ! po (

pi po ) F 1

(15)( pi po ) obviously

The smallest value of f q for a given pressure difference

exists when F has a maximum value. The method of Lagrange multipliers may be used for determining such a constrained maximum. This method indicates that maximum value of F exists whenK1 ! K 2 ! K 3 ...... ! K n ! K

(16)

Substituting Eq. 16 gives: K! d d r 1 d1 d 2 d 3 ! ! ! ! ......... n dn di d 1 d 2 d n 1 (17)

A comparison of Eq. 16 with Eq. 19 indicates that

C1 ! C 2 ! C3 ! .......C n ! CSubstituting Eq. 18 into Eq. 13 gives:

(18)

F ! C1C2C3 .........C n ! C nSubstituting Eq. 19 into Eq. 15 gives:

(19)

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f q ! po (

pi po Cn 1

)

(20)

Where po = external pressure on vessel, pounds per square inchpi = internal pressure in vessel, pounds per square inch

n = total number of shells in vessel wall with thicknesses satisfying Eq. 17

fq

= combined

stress at interface of each shell (pressure stresses

superimposed on shrinkage stress) C= constant as defined by Eq. 9 and Eq. 6

A comparison of EQ. 16 and Eq.18 indicates that Eq. 9 may be written as: C! Where K !( d d d r 1 d ) ! ( r 2 ) ! ( r 3 ) ! ..........( n ) dr d r 1 dr 2 d n 1 (22) 2K 2 1 K2 (21)

Therefore it follows thatC !n

2 n K 2n (1 K 2 ) n

(23)

Substituting Eq. 23 into Eq. 20 gives:fq ! [ (1 K 2 ) n ( pi po ) 2 n K 2 n (1 K 2 ) n ] po

(24)

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Eq. 24 is the general relationship for determining the maximum combined stresses at the interface of the concentric shells and at the inside surface of the innermost shell. The combined stresses hoop) at these locations all have the same numerical value. This equation for usual case where the external pressure is zero (gauge), po ! 0 , reduces to:fq ! pi (1 K 2 ) n 2 n K 2 n (1 K 2 ) n

(25)

DETERMINATION OF THE HOOP STRESSES AT THE OUTER SURACES OF MULTILAYER VESSELSThe individual shells may be treated in accordance with Lames theory if the pressures at the interfaces are computed by use of Eq.26. The hoop stresses may be computed by

ft !

2 pi d i2 po d o 2 d o d i2

2 d i2 d o

d2

[

pi po2 d o d i2

]

as an alternative to the use of Eq. 25

DETERMINATION OF SHRINKAGE STRESSESThe shrinkage stresses may be readily determined by the method of superposition. The stress variation, assuming the vessel is a monobloc shell having the same inside diameter and outside diameter as the multilayer vessel, may be determined in each by use of Lames Theory.

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The stress variation in each of the shells can be determined by the methods presented in the previous section. The shrinkage stresses are obtained by subtraction with proper allowance for signs.

DETERMINATION OF INTERFERENCES REQUIRED IN SHRINKFITTED VESSELSThe necessary total difference in diameters (interference) may be calculated by the relationships developed by Cox. The general relationship is EU r 1 ! n [C n r ( K 2 r 1) 2C n ]( pi po ) (27) 2n dr (C 1)( K 1) where U r = difference in diameters, inchesd r = outside diameter of the rth shell, inches

E C n r

= modulus of elasticity of shell material, pounds per square inch = 2K 2 1 K2 (from Eq. 21)

= total number of shells = interface number, numbering outward

K!

OD ratio ID

(from Eq. 22)

pi = internal pressure, pounds per square inch gauge

p o = external pressure, pounds per square inch gaugeSubstituting for C in Eq. 27 by means of Eq. 21 gives:15

EU r 2 n r K 2 n 2 r [( K 2 1) r ( K 2r 1) 2 r 1 K 2 r ]( pi po ) ! dr ( K 2 r 1)[2 n K 2 n 1( K 2 1) n ]

(28)

For a multilayer shrink-fitted vessel fabricated from two shells, n=2 and r=1. Therefore

EU1 2K 2 ! [ 2 ]( pi po ) d1 3K 1

(29)

Similarly values for other values of r and n can be find out using Eq. 28

SIMLIFIED RELATIONSHIPSThe general relationships for multilayer-vessel construction have been presented in the previous sections according to the method developed by Cox. The use of these relationships is somewhat involved; also, for practical reasons most shrink-fitted multilayer vessels consist of two shells. Therefore it is desirable to work with simplified relations for the case of two shell construction in which the external pressure is zero. For this condition the combined shrinkage stress and pressure stress at the inner surface of each shell ( f q ) is given by Eq. 25fq ! pi (1 K 2 ) n 2 n K 2 n (1 K 2 ) n

This equation may be written as: 1 K 2 n ( ) pi 2K 2 fq ! 1 K 2 n 1 ( ) 2K 2 Eq. 21 C is defined as: C! 2K 2 1 K216

(30)

Substituting into Eq. 30 gives: fq ! ( ) Cn 1 pi (31)

Solving this equation for C gives :

C!n(

pi

fq

1)

(32)

This relationship gives the value of C for n number of shells with pi (internal working pressure) and f q (allowable stress) specified. The K value given by Eq. 21 is K ! (C 2C ) (33)

An examination of Eq. 32 and 33 indicates that when n=2, pi ! 3 f q , C=2 and

K ! g .Thus, when the working pressure is three times the allowable stress, the theoretical required thickness approaches infinity. This situation may be compared with that of the Lames equation for a monobloc shell, in which the theoretical required shell thickness approaches infinity when the working pressure approaches the allowable stress of the material. Thus, the use of multilayer-vessel theory theoretically permits a threefold extension of the design range, two-shell construction being assumed. A greater extension may be obtained by using more than two shells. The practical limits of two-shell construction occur in the range in which the working pressure is from one half to twice the allowable working stress.

THERMAL EXPANSION FOR SHRINK-FITTINGIn order to shrink-fit successive shells upon one another, the outer shell must be expanded by heating. It may then be slipped over the inner shell or shells and allowed to cool. The expansion must be sufficient to overcome the required

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interference after cooling and also to provide some clearance for easy assembly. The diametric enlargement of the outer shell resulting from heating is: (d ! E ((t )d where (d = increase in shell diameter, inches (34)

d = inside diameter of shell, inches (t = temperature of heating minus room temperature degree FahrenheitE = coefficient of thermal expansion

MULTILAYER CONSTRUCTION USING WELD SHRINKAGEIf a narrow longitudinal band of from one tenth to one twentieth of the circumference of the outer shell were heated to 10 to 20 times this temperature difference, the same effect would be produced by cooling. In the cooling of a longitudinal welded joint such an effect can be produced. Therefore a possible convenient method of prestressing would take advantage of the shrinkage stresses in the longitudinal welded joints of shells. For transverse shrinkage of butt welds the following relationship holds:

s ! 0.1716

a 0.0121w t

(35)

Where s=transverse shrinkage, inches a = cross-sectional area of weld, square inches t = plate thickness, inches w= average width of weld, inches Substituting into Eq. 35 the dimensions for a single V butt weld of a -in. plate gives shrinkage values in the order of magnitude of 1/8-in. This shrinkage is greater than necessary to prestress successive shells to the desired amount.18

Therefore some of the shrinkage must be absorbed by the provision of a clearance between successive shells at the time of welding. At this procedure depends upon shop technique, it is not possible to compute the final stresses that exist upon completion of the shrinkage stresses. In the technique used by the A. O. Smith Company an inner shell having a thickness usually greater than in. and often in. is first fabricated. This shell is not perforated and serves to contain the fluid to be held under pressure in the vessel. Subsequent shells usually in. in thickness are progressively wrapped around the inner shell, tightened mechanically, and welded longitudinally. These subsequent shells are perforated with small holes for venting. Cylindrical rings are inserted at both ends of the inner shell during these operations to maintain a true cylindrical shape. The welds are staggered around the circumference to minimize localization of any excessive stresses at or near the welded joints and are ground flush prior to the adding of subsequent layers. After the vessel shell has been built up to the desired thickness with successive layers, the ends of the built-up shell are machined for the welding groove for the attachment of a formed head. In the final vessel 12 or more successive layers are often used. A number of these vessels were tested to destruction. The test data obtained have provided some valuable information concerning this method of fabrication. Whereas monobloc vessels which fail under high pressure service often fragment, the multilayer vessels do not fail in such manner. Considerable deformation occurs prior to failure, and when a leak develops in the inner shell, fluid escapes through the perforations in the outer shells. This provides a system of venting which gives warning of possible rupture. In the tests to destruction some of the vessels were stress relieved and others were not. One vessel which was not stress relieved withstood 8% greater stress than the corresponding vessel which was stress relieved. This indicates the desirability of not stress relieving, in order to retain the compressive stresses developed in fabrication.

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ANALYSIS OF TEST DATAFig. 2 shows a plot of the induced compressive hoop stresses at the inside surface of the inner shell as determined by circumferential strain measurements during the progressive wrapping of successive shells. The vessel used in this text had an inside diameter of 48 in. The vessel wall consisted of an inner shell in. thick wrapped with 32 concentric layers each in. in thickness. The curve gives an indication of the cumulative effect of the shrinkage of successive layers on the inner shell. One method for analyzing these data involves the prediction of the experimental curve by use of simplified theoretical relationships. Consider the inner

Fig. 2

shell and the first wrapped layers as an inner shell under external pressure and an outer shell under internal pressure, respectively. The interface pressure is common to both shells. A summation of forces about a symmetrical plane can be made according to the membrane theory. The forces in the two shells are equal and opposite, or Fi ! f i t i l Fi ! f1t1l

(for inner core) (for first wrapped layer)

Equating and solving for f i gives: ( f i ) n !1 ! f1 t1 ! incremental stress induced in inner core by first wrapped layer ti20

but ( f i ) n ! 0 ! 0

therefore (f i ! ( f i ) n !1 ( f i ) n ! 0 ! f i t1 ti

Assuming that there is uniform stress addition across and considering that the inner core and the inner core and the first layer are a unit, we may treat that the second wrapped layer in like manner. The incremental stress included in the inner core plus the first layer by a second wrapped layer is: ( f i )n ! 2 ! f 2 ( t2 ) ti t1 (36)t1 ! t 2 ! t 3 ! t n

( f i ) n !3 ! f 3 (

t3 ) ti t1 t 2

For uniform thickness of successive shells, Therefore ( ( f i ) n ! f n ( tn ) ti (n 1)t n

(37)

The stress in the nth wrapped shell is produced by inducing strain resulting from weld shrinkage. The total free weld shrinkage is a function of weld joint dimensions. If the same plate thickness and weld is used in each successive weld, the free weld shrinkage will be a constant. Part of this free weld shrinkage is used in overcoming clearance between layers, and part in compressing the inner layers elastically. The major portion of the free weld shrinkage develops elastic strain and resultant stress in the layer itself. Assuming for purposes of simplification that each clearance in the free weld shrinkage, we find that the total circumferential strain contributing to stress development will remain the same. The unit strain will be proportional to the diameter. The analysis of the stresses was based on thin wall theory. As the number of shells increases, the wall becomes progressively thicker and the thin-wall analysis should be accomplished by application of Lames21

analysis for thick-wall vessels. The wall is under an external pressure because of the weld shrinkage.

RIBBON- AND WIRE-WOUND VESSELSThis technique of winding cylindrical shells subjected to high internal pressure is old and has long been used for reinforcing gun barrels.

(a)Wire and Flat-ribbon Windings.These vessels are used only for absorbing hoop and radial stress and offer no restraint to axial load. An inner monobloc shell must be used having a minimum thickness sufficient to absorb the axial internal pressure load. In the process of

Fig. 3: Typical section of a vessel with wire or flat-ribbon windings

winding, the inner shell may be considered to behave as a vessel under an external pressure induced by the winding. This pressure at the interface between shell and wire windings also acts as an internal pressure on the wire windings. An internal pressure in the vessel induces hoop-tension stresses in both the inner monobloc shell and the outer windings. Thus, under the operating conditions the inner monobloc shell may be considered to have both internal and external pressures, and the windings to have induced stresses resulting from winding tension and internal pressure.22

(b) Interlocking ribbon Winding.The principle involved in this design consists of interlocking the windings by means of grooved profiles to permit the winding to carry a portion of axial load. Special lathes are required for wrapping .these lathes can be of the same type used in machining monobloc-vessels shells except that the load carriage must be modified to accommodate the reel of tape, the tape heating and cooling equipment, and a profile hack up roll The below figure shows a sectional drawing of a design for high pressure vessel for coal hydrogenation. This vessel was designed to be fabricated of a low chrome vanadium steel similar to SAE6115 suitable for being heat treated during wrapping. The steel has a yield strength of about 90000 psi and ultimate strength of about 130,000 psi.

Fig. 4: Wickelofen flanges

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THEORY OF RIBBON AND WIRE WINDINGRibbon and wire-wound vessels may be fabricated with either a thin or thick shell (inner core plus windings).thick shells with windings are used obviously because of the greater strength resulting from wound construction with monobloc construction. The reason for the use of wound thin shells is not so apparent. One example of the use of such shells is the case in which the inner core must be fabricated of a noncorrosive or ductile material that does not have sufficient strength to resist the tensile (hoop) loads produced by the internal pressure.

THIN SHELLS WOUND AT CONSTANT TENSIONThe simplest application of ribbon or wire windings involves the use of a thin shell onto which is wound flat ribbon or wire of the same material of construction as the shell with constant tension in the wire during winding as the ribbon or wire is wound onto the shell, it causes a circumferential compressive stress to develop in the shell. In such an application an internal shell must be designed of sufficient thickness to resist the axial load resulting from the internal pressure. The axial stress is only half of the hoop stress according to membrane theory. Therefore the inner shell needs to be about half as thick as the corresponding monobloc shell. The necessary additional shell material to absorb the hoop stress is made of wire winding the inner core may be considered to behave as thin walled vessel if the operating pressure is moderate. The compressive stress induced in the inner shell may be determined by taking a summation of forces about a diametric plane with no internal pressure in the vessel. At this plane the total tensile force in the wires is equal and opposite in sign to the compressive forces in the shell.

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((f i ) n ! f n

tn t i (n 1)l n

(38)

But for ribbon and square-wire windings,

( fi ) p !

pi d i pi d i ! 2l 2li 2nl n

(39)

And for round wire,

( fi ) p !

pi d i 2l i nrwT

(40)

f i and f n may be determined by simultaneous solution.pi d i ( d i t i ) Ei 2nt n 2t i (d i 2t i nt n ) En pi d i (d 2t i nt n ) E n 2nt n i 2t i ( d i t i ) Ei

( fn ) p !

(41)

( fi ) p !

(42)

To calculate the stress in the shell under the influence of internal pressure, Eq. (42) may be substituted for the second term in Eq. (40). To calculate maximum stress in the winding under the influence of pressure Eq. (41) may be substituted in Eq. (39). Equations for square and round wire winding on a shell of dissimilar metal corresponding to Equations (41) and (42) may be derived.

REFERENCES1. N. V. Kalakutskii, Investigation of Internal Stresses in Cast Iron and Steel [in Russian], St. Peters- burg (1882)25

2. 3.

4. 5.

6.

V. Gadolin, Theory of Cannon Reinforced by Hoops [in Russian], Artilleriiskii Zhurnal, No. 12, St. Petersburg (1861). I. Rychkov and N. V. Ashmarin, Multilayer Thick-walled High Pressure Vessels [in Russian], Amerkianskaya Tekhnika Promyshlennost, No. 2 (1946). Yu. Shirenbek, Brenstoff-Chemie, 23, (1950). I.Prkhal, Large Welded Two Layer High Pressure Vessels and ForgedWelded [Russian translation], Chekhoslovatskaya Tyazhelaya Promyshlennost, No. 12 (1962). L.E. Brownell and E. H. Young, Process Equipment Design, (1959).

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