Pressure Vessel Desip.

for High or

Vessel design in accord with the ASME Code requires consideration of the effect of temperature on the maximum stress, but the Code offers no help in determinin~ this effect. This article presents a group of charts with which to find both pressure and thermal stresses.

With the wider use of high-pres­sure processes operating at ex­tremes of temperature, either at the high or at the low end of the temperature spectrum, it has be­come more necessary for the design engineer to pay attention to the combined effects of both pressure and thermal stresses.

Cryogenic Tem.peratures

RAYMOND R. /M.CCARY. )fGu"tt & Co., 1M.

Many vessels are designed to con­form to the ASME Unfired Pres­sure Vessel Code' and compliance with 'Par. UG-22, Item 7, is man­datory as concerns the thermal stress loading. However, although the paragraph stipulates that ves­sel design loadings shall include the effect of temperature gradients on maximum stress in the vessel, the Code does not give any specific iuidance toward this evaluation.

Censequently, chemical engineers

Steady-state thermal stresses under isothermal heat flow '01 inner and outer surfaces of a thick-walled shell; solves Eql. (1) and (2).-~. 1

131 CHEM[CAL ENCINlutlUNC-November 28, 1960

\

PRESSURE VESSEL DESIGN . • •

.: o u

~ on

'" ~ ... a.

cOO

~ 100 -c. l c u

'" >-..c

o -400

+ +

Aluminum ~-"',-.. alloys •

-----~~-:---: o 500 I~OO

Austt tiC stainless sl eels

Corban steels carbon < 0.3 %

1,500 TI ~ Temperature inside vessel, of.

TelJlperature~ependent physical-property factor C for stainless and carbon tee! and aluminum alloys; used in Eqs. (1) and (2).-Fig 2

'u; c.

0:; £; ... 0 CO> U 0

~

::>

'" CO> "0 Vi .~

;; 0;

..:-<Il

!i 'u; c ~ .,; II>

~ II>

.!:! c: GO .,. c: 0 I-

10 4

8

6

4

2

I

1,250,

1,000

1750 I 101~ __ ~ __ ~~~ __ ~ __ ~ __ ~ __ ~~-L __ ~ __ -L __ ~ ___ J

0.1 0.2 0.3 0.4 0.5 0,6 0.7 0.8 0.9 1.0

~ ~ Thickness of vessel wall, inches R Inside radius of shel~ inches

Tangential stresses in the inner surface of a thick-walled vesllel subjected to internal pressure; solves Eq. (G)-Fig. 4

are prone to underestimate the sig­nificance of s,uch stresses in vesilel design, instead leaning heavily on the factor of safety employed. It iR possible for neglect of such ther­mal stresses to lead to conditions similar to the over-temperature hazards in pressure vessels dis­cussed by Rossheim et al.·

Code Limits Allowable Stresses

In addition to requiring that the vessel design be based on the simul­taneous application of the pressure and temperature expected in nor­mal operation, the ASME Unfired Pressure Ves el Code limits the maximum allowable stress in either tension or compression to the values permitted for the materials as listed in Subsection C. It is obvious that this stress limit must be based on the simultaneous application of both the temperature and the pres­sure stresses existing in the vessel to comply with present Code re­quirements. Compliance insures that the material stresses will re­main within the elastic limits of the material of construction.

Although it is possible to design vessels beyond the elastic limits, for conditions beyond the seope of the Code, this requires careful con­sideration of many factors. Among these are transient temperature conditions, cyclic loadings, creep or plastic flow, ductility exhaustion as a result of thermal fatigue. metallurgical phase changes under sustained exposures to temperature, as well as oxidative or corrosive at­tacks from the vessel's contents which change the physical proper­ties of the material of construction of the vessel.

Charts Find Thermal Stresses

In order to simplify the calcula­tions of the combined pressure and thermal stresses as required for compliance with Code require­ments, the author has developed a series of charts which are presented heTe along with examples dealing with the more usual conditions of stress distributions in cylindrical vessels operating under steady-state temperatures.

132 November 28, 1960-CHEMlcAL ENCINEEIUNC

ra

hermal

Dim ioal geometric factors Y. and Y. for cylindrical shells of varying t l R Ntio; used in Eqa. (1) and (2).-Fig. S

of the ah 11 only, since the govern­ing design stress can usually be determined by these vl\lues, as Ex­ample, 1 and 2 will show.

SIT. ") "" CY,~ T (1 ) SIT. ft) = ' Y%~ T (2 ) S('I' . • I) co S(7. " ) (3) SrI'. tS) ., . ·( T . f, ) (4)

8(T. JlI) = SIT lIS) "" 0 (5)

4 .-'" Q. --~ s;. ... "0

'" u 2 0 -...

::> on ... "'0 ' ;;; '5 0

~'03 N ..

cL <J'I 8

~ .. c 6 ~ ... .. ~

CA I '0

4

~ -c .. CI' C CI ~

2

where C - a E/[2 (1 - 1' ) ] and

Y. = [ In (l~t/R) -;-=- ~ _I -- J ' ) H Rl'

. [1 2 ] h = In (l+t , H) - (I -r(/ R)'=-1

Other terms are a.s given in the table of nomenci&ture on paie 136.

Solves EQ. ( 7)

+

2,500 I

2,000

_ 1 ,5~

' ,250]

1,000

. r

10Z~ __ ~_ _~ __ ~I __ ~ __ J-__ J-__ J-__ ~.~5~ 0. 2 03 04 05 0.6 0.7 08 09 10 01

, Thic ness of vessel wall , inches -: R InSIde rodius of shell.lnches

r es Ta entia} litre!! .. in the outer surface of a thick-waned vessel aubjected to out.8lde surfaces internal pressure; aolves Eq. (7).-Fig. 6

.... "-'UI.U'Ic-Novem~r 28, 1960 133

\.

PRESSURE VESSEl DESIGN . • .

The importance of determining an accurate temperature difference AT at operating conditions will be obvious from these relationships which bring out its direct effect on thennal lItl"ess. Insulation and am­bient conditions play a controlling role on vessel design. In addition, the temperature dependent proper­ties of the modulus of ela ticity and the coefficient of expansion as­sume significance at the extremes

6 '

4

'" n . N .... a: -<fl

"0 2 c: 0

-N

0: <fl

~ in lei c: .... ,,;

8 ., -'" '" <U

'" 6 ..

! 4[ ,

of operating temperatures as will be evident from Fig. 2. Data on the thermal properties of the metals most commonly used in veRsel con­struction will be found in Refs. 6 through 11.

Since the thermal stress equa­tions are based on the temperature difference existing at the operating temperature, the average of the instantaneous coefficients of expan­sion at the inside and outside tem-

, ~ --1-~, i J ~ - i

Solves Eq. (8)

I

r~ I

--1 500 I

I ",l __ _ 0 1 C 2 03 0 4 0.50.6070.809 1.0

I _ Thickness 0' vesse l Ilia d, nches - - Ins d~ rad ius of she ll,l'lches

Lon~tudinal shell stresSE'S, assumed constant. throughout the thickness, in thick­walled vessels subjected to internal pressure; solves Eq. (8).-Fig. 6

peratures of the shell ive an ac­curate measure of the shell strains. The coefficients of expansion vary only slightly within the range of AT, so the inside temperature of the shell may be considered, within a practical degree of accuracy, ag suitable for determining the in­stantaneous coefficient of thermal expansion. Poisson's ratio remains essentially constant at both low and high temperatures as indicated by the constancy of the relationships between the modulu~ of elasticity in tension and that in torsional rigidity.

Find Pressure Stresses Too

The similar relationships for the !ltresses in the vessel wall due to internal pressure alone can also be expressed in terms of the same geo­metric parameter tl R used for ther­mal expan ion. Here the formulas are based on the Lame equations as indicated in Refs. 3, 4 or 5:

SIP . ' 11 = P [1+ ( l -H/~)I-l] (6)

S(/' o Is) -l' [11 +1/~)'-l] (7)

S IP. Z I) = S.p %1)

.. P [ ( H -I ~? \I-l ] (S)

S CPo RI) = - P (9J SIP , Bt) ~ 0 (1 0 )

The tangential pressnre stres:! given by Eq. (6) is equivalent t o the design formula of Par. UA-2 in Appendix I of the ASME Un­fired Pressure Ve~sel Code. The Code also allows the use of the modified membrane shell fonnula of Par. UG-27c when thickness of the shell does not exceed O.885SE. This alternate equation may be used in the calculations for the tangen­tial pressure stress without sig­nificantly changing the results.

Eqs. (l) and (2) are solved by Fig. 1. The physical properties fac­tor C and the dimensionlcss shape factors Y1 and Y. are found from Figs. 2 and 3. F.qs. (6), (7) and (8~ are solved directly in terms of tlR and the internal pressure P by means of Figs. 4, 5 and 6. The use of these equations is illustrated in Examples 1 and 2. Here, having com put e d independently the

lH November 28, 1960-CREMICAl. ENCtml'!1UNC

, ....

t-~~2: ' I " ... ' .1 .. . . ' ~. : .1. .. .1 . -' ..

;-m C!lMTl!'w'l-t 0 sn,AI o o

r...... 10

.,..,. I . ,.

•

PRESSURE VESSEL DESIGN • • •

stresses due to pressure and to tem­perature, we sum them algebrai­cally to give the resultant tensile or compressive stress distribution in the vessel wall. It will be noted in both examples that t e problem is solved twice, once for tlR = 0.83 and again for tlR = 0.50. In each case it will be obllerved that a thick­ness ratip of tlR = 0.33 could han­dle the job without exceeding the allowable maximum stress if in­ternal pressure alone where pro­ducing wall stresses. The addition of temperature stress, however, in each case makes it necessary to go to a tlR of 0.5 to maintain the com­bined stresses below the allowable limit permitted by the Code.

amples 1 and 2 have indicated the stress values for a given tempera­ture difference t:.T, it is advisable to recheclc the computation of the tem­perature gradient based on the tinal thickness selected for the shell.

In applications where the a.llow­able working stresses are based on creep limitations of the materials used for vessel construction, some relief of the thermal stresses may be expected to occur under operat­ing condition!!. However, the recog­nition of relaxation of this sort has not as yet become a part of the Code, although such plastic flows may be within the elastic limits.

At the right of each example is a graph of the stress pattern through the vessel wall, with tensile stresses shown as positive and compressive stresses negative.

Now Try Examples

c

B

p

Example 1 has been selected to R show that the summation of the

S'I',ru tangential stresses at the outside of the shell governs the design. In Ex- S'''.'S) ample 2 the flow of heat is from the outside to the inside of the shell and 8'I'.nJ AT is considered as negative, there- S

" .W by reversing the sign of the ther-mal stresses. In this case, the sum-mation of the stresses at the inside S(P.n.} of the shell dictates the design. Al-though longitudinal and radial S'P."} stresses normally do not govern de- 8,r.ru sign, their values are included to complete the stress pattern. S'!'.")

Such stress patterns a.pply only S to the steady-state conditions of the

,r.Il}

vessel. Transient conditions, as the S ,r.nl vessel reaches a steady-state level, warrant pecial attention. An in- S

,r hi dication of the stress pattern may be obtained by selecting several S''1'''' levels of pressure and temperature other than the final steady-state ' ~ level, and employing the corre- • sponding allowable stresses and T, thermal properties. These indica-tions of stress pattern should not be I1T construed as a substitute for a more rigorous analysis under transient temperature conditions, but it may serve to guide the engineer toward Y. further investigations, Y

• Although the calculations of Ex-

Nomenclature Temperature-dependent phys­ical properties factor = I1E I [%(1 - I')J; see Fig. 2. Modulus of elaaticity, psi. at T,_ Intern 1 pressure, p 1. Inside radius of vessel shell, in. Tanlential prel8ur atre s at inside surface of shell, psi. Tangential pressure atress at outaide surface of shell. pai. Lon&,itudinal pressure strNs at inside surface of shell, psi. Longitudinal pressure stress at outside surface of shell, psi Radial pressure strt"s at in­side surface of hell, psi. Radial pressure stress at out­side surface of shell, psi. Tangential thermal strt'ss at inside surface of shell, psi. Tsngential thermal stress at outside surface of shell, psi. Longitudinal thermal stress at inside surface of shell, psi. Longitudinal thermal stre88 at outside surface of shell, psi. Radial thermal stress at m­side surface of shell, psi. Radial thermal tr s at out­side lurface or shell, p i. Shell thicknes , in. Inside surface temperature of shell, OF. Out8ide surface temperature of ahell, OF. TeDlperature dilJerence be­tween inner and outer sur· faces of shell; positive when heat flow is outward, negative wben heat flow is inward. Dimenaionlelll geometric shape factor, from Fl&'. 3. Dimensionless pometric shape factor, from Fig. 3.

•

Ref rl!'DC

1. ASME Unfired PretI!!!ure V Rules ot Conlltructfon. PubL by Am ot Meoh. Engre., New York. 18511.

2. Ros.helm, D. B., .t ol....p Un! Veaael Overtemperature Huarda," .A.SME paper 6&-A·3U.

a. Timoabenlto, 8., and J. N. Ooodter, "Theory ot E1a.stfclty," 2nd Ed. , MoGraw­HlI! Book Co., Inc., New York, 11151.

4. "Chemical Engineera' Handbook." 3rd Ed., J. H. Perry, Ed I tori McGraw-HUl Book Co., Inc., 'ew York, 9110.

5. Con:1i~.. W., "Hlgb P aure Technolo 'Y,' Chap. IJ McGraw-mil Book Co., Ino., New York, ;95 .

6. "COde tor Preuure PipIng," ABA D31.1-1»55, T,.blea II. •. Pub!. by Amer. Boo. ot Mech. E~., New York, un.

7. "Reeumll ot HI&'h Temperature In­,·estIgt.Uona Q)lIducte<l 'During 1 e. -60," publ. by Tlmk n Roll r Dea.r1na Co., Can­ton, OhJo.

8. Furman, D. E., Thermal ExpansIon Characteristics ot Salnl_ Ste Is Be­tween -300 F. 'and !LOOO F., J. Jhta'-, Apr. 1950; TraM, .tSMB, Vol. 188.

9. Chelton. D. B., and D. B. Mann. "Cryogenlc Date BOOk," Nat. Bur. of Std." WADe Tech. Rept. U-S, ltar, 1 Ii , FtC'. 104.

10. "Cryo~eD1o Appllcatlona of Alcoa Aluminum," pubL by Alumloum Co. ot AmerIca.

11. McClintock, R. Il0l .. and H. P. Glb­bona, "MeclIanlca.l PropertJea ot Struc­tural MaterIal. at Low T aturea a Compllatloo From the Literature." • at. Bur. of Stel •. Monograph 13, Jane 1960, Refs. p. 199.

Meet the Author

RA YMOND R. IACCARY is project 6 gitleering 71Ia7tag r for alLet & Co., Inc., of Pittsburgh, Pc., trill" II has OV6T-all direction 01 }Woe SIt pl t design, engineering, conetructi@, op­erational .tartups and ma' tcna e procedures.

Readen will recall hi8 recent ar­ticle (CHEM. ENG" Oct. 17, 19 0) 011

pressure ves.el de.ign for economfl. In 1949, he co-authored a.m of th rile artirle. ilt this fMgad~ on 1Itssel duigft for eztremfl P1'naures,

Mr. Macca11l attended Cooper Un­ion 17utituu of TechnolollY where he earn.d hi8 Bachelor's in 711 cha wi engineering in 19 .. 0.

136 November 28, 1960-CuEMrCAL ENelNE c