Creator:Raymond R. MacCary
Date Created:August 21, 1961
Place Created:
Keywords:pressure vessels,cryogenic temperatures
Context:article publicated by the ASME
**************************************************
Vessel design in accord with the ASME Code requires consideration of
the effect of
f »
temperature on the maximum stress, but the Code offers no help in determining this effect. This article presents a group of charts with which to find both pressure and thermal stresses.
With the wider use of high-pres-sure 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.
Many vessels are designed to conform to the ASME Unfired Pressure Vessel Code1 and compliance with Tar. 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 guidance toward this evaluation. Consequently, chemical engineers
Pressure Vessel Design for High or
Cryogenic Temperatures
RAYMOND R. MACCARY, MalUtt & Co., Inc.
Steady-state thermal stresses under isothermal heat flow on inner and outer surfaces of a thick-walled shell; solves Eqs. (1) and (2).—Fig. 1
Chemicat. Engineering—November 28, i960
PRESSURE VESSEL DESIGN
u -400	0	500	1,000	1,500
T, = Temperature inside vessel, °F.
Temperature-dependent physical-property factor C for stainless	and carbon uteels an<i aluminum alloys; used in Eqs. (1) and (2).—Fig 2
Carbon steeis
carbon < 0.3%
. 1
Austenitic stainless ' steels
f: i
are prone to underestimate the significance of such stresses in vessel design, instead leaning heavily on the factor of safety employed. It is 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 Vessel 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.
Thickness of vessel wall, inches
R Inside radius of shell, inches
Tangential stresses in the inner surface of a thick-walled vessel subjected to internal pressure; solves Eq. (6)—Fig. 4
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 hep along with examples dealing with the more usual conditions of stress distributions in cylindrical vessels operating under steady-state temperatures.
132
November 28, I960—Chemical Engineering
Dimenfionless geometric factors Yi and F, for cylindrical shells of varying t/R ratio; used In Eqs. (1) and (2).—Fig. 8
of the shell only, since the governing design stress can usually be determined by these values, as Examples 1 and 2 will show:
where G = a E/[2 (1 - and
S{T. #1)	= CYi&T	(1)
<S<r. H)	- CYt&T	(2)
S(T. SI)	" S(Tt »1)	(3)
S(T. Zl)	" Sit. fc)	(4)
S<r. £i>	= S(T B) = 0	(5)

Y,
r_i__
In (l+</«) [ In (1+1/A) "
1 -
1
(1 +</*)»
(1+V
2- — 1 '»)' - l J
Other terms are as given in the table of nomenclature on page 136.
Temperature stresses become significantly large mainly in thick-walled vessels where there can be a temperature difference between the inner and outer surfaces of the vessel shell. Therefore, the charts are set up to aid in designing vessels of wall thickness anywhere in the thickness-to-radius range t/R of 0.1 to TXT. "
Calculation of the thermal stresses due to a temperature gradient across the vessel wall is based on the simplifying assumption that there is steady heat flow through the wall. For thick-walled vessels, this leads to a logarithmic temperature gradient, as discussed in Refs. S, 4 and 5. The thermal stress equations. as derived through the theory irf elasticity for regions other than the vessel ends, may be conveniently simplified and expressed in the form of single terms of the geometric parameter t/R to make their use »«rs tractable.
The following relationships give the v&luas of the thermal stresses for the inside and outside surfaces
t Thickness of vessel woll, inches R Inside rodius of shell,inches
Tangential stresses in the outer surface of a thick-walled vessel subjected to internal pressure; solves Eq. (7).—Fig. 5
i. hemic< Encineuinc—November 28, 1960
13 3
PRESSURE VESSEL DESGN
The importance of determining an accurate temperature difference A3" at operating conditions will be obvious from these relationships which bring out its direct effect on thermal stress. Insulation and ambient conditions play a controlling role on vessel design. In addition, the temperature dependent properties of the modulus of elasticity and the coefficient of expansion assume significance at the extremes
of operating temperatures as will be evident from Fig. 2. Data on the thermal properties of the metals most commonly used in vessel 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-
peratures of the shell gives an accurate measure of the shell strains. The coefficients of expansion vary only slightly within the range of XT, so the inside temperature of the shell may be considered, within a practical degree of accuracy, as 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 modulus of elasticity in tension and that in torsional rigidity.
Find Pressure Stresses Too
The similar relationships for the stresses in the vessel wall due to internal pressure alone can also be expressed in terms of the same geometric parameter t/R used for thermal expansion. Here the formulas are based on the Lame equations as indicated in Refs. 3, 4 or 5:
-p[1+W/2«)«~i ] (6)
StfMa -	(7)
S(P. ZD Sip zi)
-^Wsj'-i] (8)
SlP. - - P	(9)
Sw. bi, - 0	(10)
The tangential pressure stress given by Eq. (6) is equivalent to the design formula of Par. UA-2 in Appendix I of the ASME Un-fired Pressure Vessel Code. The Code also allows the use of the modified membrane shell formula of Par. UG-27c when thickness of the shell does not exceed 0.885SE. This alternate equation may be used in the calculations for the tangential pressure stress without significantly changing the results.
Eqs. (1) and (2) are solved by Fig. 1. The physical properties factor C and the dimensionless shape factors Y, and Y, are found from Figs. 2 and 3. Eqs. (6), (7) and (8) are solved directly in terms of t/R 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 computed independently the
_ Thickness o' vessel moll,inches R Inside rodius of shell,inches
Longitudinal shell stresses, assumed constant throughout the thickness, in thick-walled vessels subjected to internal pressure; solves Eq. (8).—Fig. 6
134
November 28, 1960—Ckemicai. ENC.wrr.wNc
Example 1: High Pressure and High Temperature
Conditions—Internal pressure P = 1,250 psi. Inside temperature T, =r 950 F„ with AT at 25 F. Shell material it, carbon steel. SA-201 Gr. A. Joint effi<ieney = 1. Maximum allowable stress (tensile or compressive) = 4,500 psl. at S50 F. (Code table UC8-2S). Note (♦) marks governing design stress.
Stress & Eq. Fig. No. </„—0.33 VV^O.
Due to internal P	V-X.T"		• .,-
S(p, 6)) Eq. (6)	4	+ 4,460	+ 3,250
S(P, e,> Eq. (7)	5	+ 3,210	+ 2,000
Due to AT			
S(T, e,) Eq. (»	1,2,3	-3,120	-3,220
S(T.ej) Eq (2)	1. 2, 3	+ 2,560	+ 2,450
Stress Summation			
	4 + 1	1-1,340	+30 *
s<p,6s) + .Scr.es>	5 + 1	+ 5,770*	+ 4,450*
Due to internal P			
S<p,z^ Eq..(8J-	6	+ 1,605	+ 1,000
S<p. zt) Eq. (8)	6	+ 1.605	+ 1,000
Due to AT			
S(T. Zj) Eq. (3)	K 2.3	—3,120	-3.220 <
S(T.Z.) Eq. (4)	1,2,3	+ 2,560	+ 2,450
Stress Summation			
s<p, ZO +S<T. Zx)	6+1	-2,515	-2,220
S(p. zt) +S(T»Z3)	6+1	+ 4,165	+ 3,450
Due to internal P			
S(P,Rl)= - P Eq. (9)		—1,250	-1,250
S<p.Rj)~ 0 Eq. (10)		0	0
Due to AT			
S(T. R,) == 0 Eq. (5)		0	0
•(T.Ki) = 0«q. (»)		0	0
Stress Summation	* 1 1		
S(P.Rt) " S(T RO		—1,250	-1,250
S(P,R»> +S(T Ml)		0	
Example 2: High Pressure and Low Temperature
Condition*—Internal pressure P = 3,0W psi. Inside temperature T» = —250 F„ wit* AT -50 F. Shell material is stainless steel, SA-240 Type 316 L. Joint efficiency — L Maximum allowable stress (tensile or compressive) as 17,500 psi. (Code limits stress to value at —20 E.). Note (*) marks governing design stress
Stress & Eq.
Due to internal P			
Str.e,) Eq. (6)	? 4	+17,300	+ 10,714
S(P,6s> Eq. (7)	5	4 14,300	+ 7,714
Due to AT			
S<T.e,> Eq. (1)	1,2,3	b 6.500	+ 6,740
S(T, e,) Eq. (2)	1,2,3	- 5,780	- 5,550
Stress Summation			
Scp.eo + S<T.9,)	4+1	+ 23,800	+ 17,454
S(P, 01) +S(T.ei)	'S:3-1	+ 8,520	+ 2,164
Due to internal P			
S<p, Ti) Eq. (8)	6	4 7,150	+ 3,857
S(p. z2) Eq. (8)	6	4- 7,150	+ 3,857
Due to AT			
S(T Z,) Eq. (3)	1.2.3	4 6,500	+ 6.740
S<T, z-i Eq. (4)	1,2,3	- 5,780	- 5.550
Stress Summation			-1—~
S(P, *,) +S(T.	6+1	413,650	+10,597
S<p, *,) + S(T, St)	6+1	4 1,370	— 1,693
Due to internal P			
S(»,*o= PEq. (9)		- 3400	- 3.000
Srp.H,)= OEq. (10)		0	0
Due to AT			
S(T.R,)= 0 Eq. (5)		0	0
SrT, *,)— 0 Bq. (5)		0	°
Strew Summation			
S(P,R,) + ®<*. *i)		- ifiOQ	3,000
s<* «») +S,t, m%)		0	•
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 the problem is solved twice, once for t/R = 0.83 and again for t/R = 0.50. In each case it will be observed that a thickness ratip of t/R = 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 t/R of 0.5 to maintain the combined stresses below the allowable limit permitted by the Code.
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
Example 1 has been selected to show that the summation of the tangential stresses at the outside of the shell governs the design. In Example 2 the flow of heat is from the outside to the inside of the shell and A7* is considered as negative, thereby reversing the sign of the thermal stresses. In this case, the summation of the stresses at the inside of the shell dictates the design. Although longitudinal and radial stresses normally do not govern design, their values are included to complete the stress pattern.
Such stress patterns apply only to the steady-state conditions of the vessel. Transient conditions, as the vessel reaches a steady-state level, warrant special attention. An indication of the stress pattern may be obtained by selecting several levels of pressure and temperature other than the final steady-state level, and employing the corresponding allowable stresses and thermal properties. These indications of stress pattern should not be construed as a substitute for a more rigorous analysis under transient temperature conditions, but it may serve to guide the engineer toward further investigations.
Although the calculations of Ex-
amples 1 and 2 have indicated the stress values for a given temperature difference aT, it is advisable to recheck the computation of the temperature gradient based on the final thickness selected for the shell.
In applications where the allowable 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 conditions. 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.
Nomenclature C	Temperature-dependent phys-
ical properties factor = a El [2(1 — m)]; see Fig. 2. Modulus of elasticity, psi. at T»
Internal pressure, psi. Inside radius of vessel shell, in.
Tangential pressure stress at inside surface of shell, psi. Tangential pressure stress at outside surface of shell, psi. Longitudinal pressure stress at inside surface of shell, psi. Longitudinal pressure stress at outside surface of shell, psL
Radial pressure stress at inside surface of shell, psi. Radial pressure stress at outside surface of shell, psi. Tangential thermal stress at inside surface of shell, psi. Tangential thermal stress at outside surface of shell, psi. Longitudinal thermal stress at inside surface of shell, psi. Longitudinal thermal stress at outside surface of shell, psi.
Radial thermal stress at inside surface of shell, psi. Radial thermal stress at outside surface of shell, psi. Shell thickness, in. Inside surface temperature of shell, «F.
Outside surface temperature of shell, °F.
Temperature difference between inner and outer surfaces of shell; positive when heat flow is outward, negative when heat flow is inward. Yi	Dimensionlesa geometric
shape factor, from Fig. 3. }',	Dimensionless geometric
shape factor, from Fig. 3.
•	Instantaneous coefficient of
expansion, in./in./°F. st T,. m	Poisson'g ratio, 8.3 for steels,
0.33 for AI alloys.
References
1.	ASME Unflred Pressure Vessel Code,
Rules of Construction. Publ. by Amer. Soe. of Mach. Engrs., New York, 1969.
2.	Kosshelm, D. B., at at, "Pressure Vessel Overtemperature Hazards," ASME paper 59-A-319.
3.	Timoshenko, a, and J. N. Ooodler, "Theory of Elasticity " 2nd Ed.. MoQraw-hlll Book Co., Inc., New York. 1981.
4.	"Chemical Engineers' Handbook,- 3rd Ed., J. H. Perry, Editor, McQraw-HUI Book Co.. Inc., New York. 1950.
6. Comings, E. W„ "High Pressure Technology,' Chap 6, McGraw-Hill Book Co., Inc., New York, I »SS.
6.	Code for Pressure Piping," ABA BJ1.1-1955. Tables It, 21. Publ. by Amer. Soc. of Mseh. Engrs., New York, 1955.
7.	"Resume of High Temperature Investigations Conducted I>urlng 1948-50," publ. by Tlmken Roller Bearing Co., Canton, Ohio.
8.	Furman, D. B., Thermal Expansion Characteristics of Stainless Steels Between —300 F. 'and 1,000 F., J. Metal*. Apr. 1950; Trant. ASUS, Vol. 188.
9.	Chelton, d. B„ and D. B. Mann, "Cryogenic Date Book," Nat. Bur. of Stds., WADC Tech. Kept. 59-8, Mar. 1969, Fig. 104.
10.	"Cryogenic Applications of Alooa Aluminum," PubL by Aluminum Co. of America.
11.	McCllntoclc. R. M.. and H. P. Gibbons, "Mechanical Properties of Structural Materials at Low Temperatures, a CompllaUon From the Literature." Nat. Bur. of Stds. Monograph 13, Jane 1960, Refs. p. 199.
Meet the Author
RAYMOND R. MACCARY is project engineering manager for Mallet & Co., Inc., of Pittsburgh, Pa., where he has over-all direction of process plant design, engineering, construction, operational startups and maintenance procedures.
Readers will recall his recent article (Cbem. Eng., Oct. 17, 1990) on pressure vessel design for economy. In 19i9, he co-authored a series of three articles in this magazine on vessel design for extreme pressures.
Mr. Maccary attended Cooper Union Institute of Technology where he earned his Bachelor's in mechanical engineering in 1940.
E
P R
Sirm S/r.tu
Sir, n>
S(r.n) Sir, ti) 8ir.nl
S(T.n> SlT.ll) S(T,tl>
StT.tl)
StT.Mt) t
J*.
T,
AT
136
November 28, 1960—Chemical Encineexikc