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Nuclear Engineering and Design 239 (2009) 2236–2241

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Nuclear Engineering and Design

journa l homepage: www.e lsev ier .com/ locate /nucengdes

tructural evaluation of a piping system subjected to thermal stratification

omnath Chattopadhyayepartment of Engineering, The Pennsylvania State University, College Place, DuBois, PA 15801, United States

r t i c l e i n f o

rticle history:eceived 23 January 2009eceived in revised form 13 July 2009

a b s t r a c t

Piping systems in nuclear power plants are often designed for pressure, mechanical loads originatingfrom deadweight and seismic events and operating thermal transients using the equations in the ASME

ccepted 17 July 2009Boiler and Pressure Vessel Code, Section III. In the last few decades a number of failures in piping haveoccurred due to thermal stratification caused by the mixing of hot and cold fluids under certain lowflow conditions. Such stratified temperature fluid profiles give rise to circumferential metal temperaturegradients through the pipe leading to high stresses causing fatigue damage. A simplified method has beendeveloped in this work to estimate the stresses caused by the circumferential temperature distributionfrom thermal stratification. It has been proposed that the equation for the peak stress in the ASME Section

addi

III piping code include an

. Introduction

Nuclear piping systems (Class 1) are designed according to theules of NB 3600 of the ASME Boiler and Pressure Vessel Code, Sec-ion III (ASME, 2007). The loads producing the stresses originaterom the internal pressure, deadweight, seismic and thermal expan-ion loads and the operating thermal transients. Normally pipingystems are not designed for circumferential temperature variation.owever, during some events, especially under low flow conditions,

ircumferential temperature distribution due to the mixing of hotnd cold fluids (thermal stratification) can lead to significant fatigueailures in piping. The purpose of this work is to estimate the peaktresses due to thermal stratification and to suggest a modificationo the existing design rules of NB 3600 to incorporate such effectsor fatigue evaluation.

The thermal stratification occurs when streams of water at dif-erent temperatures meet. The density of water varies significantly

ith temperature. So when there is a mixing of fluids at differ-nt temperatures in a circular pipe, the warmer fluid, which is lessense, tends to seek the upper portion of the pipe, while the cooleruid remains at the bottom. The effect of the thermal stratificationn the state of stress in the pipe is manifested in two ways: (a)he difference in temperature between the top and bottom of theipe causes greater thermal expansion at the top tending to bow

he pipe. When such bowing is restrained, global bending stressesesult; (b) the interface between the two fluid layers causes a localtress in the pipe due to steep thermal gradient across the pipeection.

E-mail address: sxc72@psu.edu.

029-5493/$ – see front matter © 2009 Elsevier B.V. All rights reserved.oi:10.1016/j.nucengdes.2009.07.017

tional term for thermal stratification.© 2009 Elsevier B.V. All rights reserved.

Such stratified conditions have been observed for the pipingadjacent to the main feed water nozzle for a number of pressurizedwater reactor (PWR) steam generators, and the resulting thermalstresses have caused extensive cracking in those areas (see e.g.USNRC Bulletins 88-08 and 88-11). The thermal stratification pro-duces circumferential temperature gradients in the piping and canlead to very high stresses. These high stresses produce low cyclefatigue damage in piping undermining its structural integrity. Themechanism of stratification has been identified as a source for pipefatigue damage in the United States NRC Bulletin 88-08 and theUnited States NRC Bulletin 88-11. Significant fatigue damage in feedwater lines due to stratification has been reported (Miksch et al.,1985; Thurman et al., 1981). Thermal hydraulic experimental stud-ies have been reported by Kim et al. (1993) in order to establishtemperature loading conditions at the piping surface location forsubsequent fatigue analysis.

Thermal stratification by itself is generally not a concern forfatigue failure. However, if a mechanism for cycling exists, fatiguecracking can occur. The sources of thermal cycling are due to (a)changes in interface level, (b) changes in temperature and (c) tur-bulence penetration. Under certain thermal-hydraulic conditions,the interface between the hot and cold layers can become turbu-lent leading to high frequency cycling over a narrow range in thevertical profile of the pipe (EPRI TASCS, 1994). This phenomenon iscalled thermal striping and has been reported for PWR feed waternozzles (Thurman et al., 1981), where the interface level betweenhot and cold flowing fluids oscillates rapidly with periods rang-

ing from 0.1 to 10 s and is thus exposed to a rapidly varying fluidtemperature. This produces fluctuating surface stresses, which aretypically small in magnitude, but the number of cycles is so largethat these stresses could contribute significantly to fatigue crackinitiation.

S. Chattopadhyay / Nuclear Engineering a

Nomenclature

A pipe cross-sectional areaC1 secondary stress index for membrane (hoop) stress

(peak)C2 secondary stress index for bending stress (peak)C3 secondary stress index for thermal stress (peak)C ′

3 secondary stress index for thermal stress (primaryplus secondary)

D0 pipe outside diameterE elastic modulus of the pipe materialEab average cold elastic modulus between sides ‘a’ and

‘b’ of the discontinuityf0(x) displacement functionf1(x) displacement functionI moment of inertia of the pipe cross-sectionK1 local stress index for membrane (hoop) stress (peak)K2 local stress index for bending stress (peak)K3 local stress index for thermal stress (peak)Kstrat stratification intensification factorMi applied bending moment to the pipeMT effective bending moment due to circumferential

temperature gradientP0 applied internal pressure in the pipePT effective axial force due to circumferential temper-

ature gradientRm mean pipe radiusSm allowable design stress intensity for the pipe mate-

rialSp peak stress intensity in the pipeT temperature at a particular location of the pipeTa average temperature of side ‘a’Tb average temperature of side ‘b’T1, T2,. . ., T13 temperatures at various circumferential pipe

locationsTm1, Tm2,. . ., Tm12 mean temperatures at various circumfer-

ential pipe segments�T1 linear component of the temperature gradient�T2 nonlinear component of the temperature gradient�Tstrat effective stratification temperature gradientt thickness of the pipeu axial displacementx coordinate in the axial directiony coordinate in the horizontal directionz coordinate in the vertical direction˛ coefficient of thermal expansion for the pipe mate-

rial˛a coefficient of thermal expansion for side ‘a’˛b coefficient of thermal expansion for side ‘b’εxx axial elastic strain in pipe

spmgstsm

t

due to differential rate of heating between regions of unequal wallthickness. A gross discontinuity may also be present at a bimetal-lic welded joint where different coefficients of thermal expansionexist. The thickness gradients as well as the gross discontinuitystresses are addressed in the piping design allowable for peak stress

� pipe material Poisson’s ratio�xx axial elastic stress in pipe

Jhung and Choi (2008) have investigated the effect of thermaltratification on the structural integrity of a nuclear power plantressurizer surge pipeline. They have developed finite elementodels of the surge line using several element types available in a

eneral purpose structural analysis program. They have performedtress analysis to determine the response characteristics for various

ypes of top-to-bottom temperature differentials due to thermaltratification and have concluded that thermal stratification is aajor contributor to the fatigue life of the surge line.

Bain et al. (1992) have concluded that the circumferentialemperature distribution is critical to the analysis of thermal strat-

nd Design 239 (2009) 2236–2241 2237

ification and has a significant effect on piping local stresses andbending moments. They have used a semicircular two-dimensionalmodel with 18 circumferential elements and 3 elements throughthe pipe thickness to model the pipe and have obtained the out-side wall temperatures by applying fluid temperature profile andfilm coefficient at the pipe inside wall. The fluid temperature dis-tribution includes step changes, step changes with a linear interfaceregion and linear distribution. For each of these cases they analyti-cally compute correction factors to be applied to the top to bottomtemperature difference as well as to the resulting bending momentsto compute stresses. The maximum value of the correction factorwas found to be 1.18.

Furuhashi et al. (2008) have obtained evaluation charts for tem-perature distribution and thermal stresses in cylindrical vessels forthermal stratification. In their formulation the radius of the shellis large compared to the thickness thereby leading to a model ofa flat plate exposed to a flowing fluid on one face with the otherface insulated. They found the thermal stress to depend on theratio of the temperature attenuation coefficient to the shell materialattenuation coefficient, as well as the stratification profile.

The structural response due to stratification and striping hasbeen addressed from the standpoint of fracture mechanics in anumber of studies. Lee and Song (1993) have investigated thebehavior of a small crack located at the thermal striping zone ina thermally stratified pipe using the finite element method. Theyhave concluded that the stress intensity factor depends on the oscil-lating frequency and the heat transfer coefficient. Jones (2003)assumed a small crack located in the thermal stratification zoneand found the crack growth to depend on the oscillation frequency,material properties and crack depth.

In this study, an analytical technique has been employed toevaluate the stresses due to thermal stratification. The effect ofthermal striping has not been considered. The existing ASME pipingdesign equations are first discussed. This is followed by a procedurefor calculating thermal stresses in a pipe due to circumferentialtemperature gradient. The numerical solution is an approximateone that employs typical piping dimensions for a PWR plant, andmakes use of standardized fluid temperature profiles as measuredby thermocouples placed around the circumferential locations inthe piping in a number of power plants. The peak stress due tothrough thickness temperature variation has not been explicitlyconsidered, but conservatively a value of 2.0 is used. The result-ing stress intensification factor is applied to the calculated thermalstress and then used for fatigue evaluation. This procedure is rec-ommended to be used in the ASME Code for fatigue evaluation.

2. ASME Code piping design criteria for thermal stresses

In piping systems the fluid temperature variation gives rise to atemperature gradient occurring through the thickness of the pipewall. The temperature variations also cause differences in aver-age wall temperatures at locations of gross piping discontinuities

Fig. 1. Coordinate system for the pipe.

2 ering a

i

S

b

C

Hoi

Ftf

238 S. Chattopadhyay / Nuclear Engine

ntensity range (ASME, 2007), which states,

p = K1C1P0D0

2t+ K2C2

MiD0

2I+ K3

2(1 − �)E˛|�T1|

+ K3C3Eab|˛aTa − ˛bTb| + E˛

(1 − �)|�T2| (1)

Sp is used for fatigue evaluation. Excluding the effects of thermalending the following requirement must be satisfied:

P D M D

1

0 0

2t+ C2

i 0

2I+ C ′

3Eab|˛aTa − ˛bTb| ≤ 3Sm (2)

ere K1, K2, K3 are the local stress indices, C1, C2, C3, C ′3 are the sec-

ndary stress indices for the component under consideration, P0s the applied internal pressure, Mi the applied bending moment,

ig. 2. (a) Stratified temperature distribution for Profile 1 (from ASME-81-PVT-3). (b) Stemperature distribution for Profile 3 (Thurman et al., 1981). (d) Stratified temperature disor Profile 5 (Thurman et al., 1981). (f) Stratified temperature distribution for Profile 6 (Th

nd Design 239 (2009) 2236–2241

D0 is the pipe OD, I is the moment of inertia, t is the pipe thick-ness, E the elastic modulus, ˛ the coefficient of thermal expansion,� the Poisson’s ratio material. �T1 and �T2 are the linear andnonlinear components used to approximate the temperature gra-dient across a pipe wall. The stress that occurs at locations ofgross discontinuity is represented by Eab|˛aTa − ˛bTb|, where Eabis the average cold modulus of elasticity between sides ‘a’ and ‘b’ ofthe discontinuity. ˛a, ˛b are the coefficients of thermal expansionand Ta, Tb are the average temperatures at sides ‘a’ and ‘b’ of the

discontinuity.

It is to be noted here that the piping design code has no provisionfor treating the cases where circumferential temperature gradientsare encountered. This work provides a typical example where sucheffects are explicitly investigated.

ratified temperature distribution for Profile 2 (Thurman et al., 1981). (c) Stratifiedtribution for Profile 4 (Thurman et al., 1981). (e) Stratified temperature distributionurman et al., 1981).

S. Chattopadhyay / Nuclear Engineering and Design 239 (2009) 2236–2241 2239

(Cont

3d

pbtslmtpbo

T

Atw

u

T

ε

Tt

�

Tch∫

Fig. 2.

. Stresses due to non-axisymmetric temperatureistribution

The basis of the solution stems from the assumption that theroblem is statically determinate and free of external loads. Theasis for calculating the stresses due to circumferential tempera-ure gradient is based on Bernoulli–Euler assumption that the planeections which are plane and perpendicular to the beam axis beforeoading remain so after loading and the effect of lateral contraction

ay be neglected. The Bernoulli–Euler assumption requires thathe axial displacement be a linear function of the coordinates in thelane of the cross section. For this problem, the temperature distri-ution in the pipe due to stratification is assumed to be dependentnly on the vertical distance z (see Fig. 1),

= T(z) (3)

ssuming the plane sections to remain plane and perpendicular tohe axis before and after bending, the axial displacement may beritten as,

= f0(x) + zf1(x) (4)

he axial strain is given by,

xx = ∂u

∂x= f ′

0(x) + zf ′1(x) (5)

he axial strain is therefore a linear function of the distance fromhe neutral axis.

The axial stress is therefore,

xx = E(εxx − ˛T) = E[f ′0(x) + zf ′

1(x) − ˛T) (6)

he functions f0(x) and f1(x) must be determined from equilibrium

onsiderations. Considering the equations of static equilibrium, weave for force equilibrium,

�xxdA = 0 (7)

inued )

And for moment equilibrium,∫�xxzdA = 0 (8)

Eqs. (7) and (8) assume that the problem is statically determinateand free of external loads. This is a reasonably valid assumption forregions remote from the support locations of the pipe. Boley andWeiner (1960) further contend that smoother the axial temperaturedistribution, the more accurate is the elementary theory.

Substituting �xx from Eq. (6) into Eqs. (7) and (8), we have,

f ′0(x)

∫dA + f ′

1(x)

∫zdA =

∫˛TdA (9)

f ′0(x)

∫zdA + f ′

1(x)

∫z2dA =

∫˛zTdA (10)

Since z is measured from the neutral axis, we have,∫zdA = 0 (11)

Therefore using Eqs. (9) and (10) we obtain,

f ′0(x) =

∫˛TdA∫

dA= PT

EA(12)

and,

f ′1(x) =

∫˛zTdA∫z2dA

= MT

EI(13)

Therefore the equation for the axial stress takes the form,

�xx = PT

A+ MT z

I− E˛T (14)

where,

PT = E˛

∫TdA (15)

2240 S. Chattopadhyay / Nuclear Engineering and Design 239 (2009) 2236–2241

Table 1Specified temperature (◦C) at the inner surface of the pipe (location along pipe circumference: location 1—top, location 13—bottom).

Profile 1 2 3 4 5 6 7 8 9 10 11 12 13

1 182 169 132 92 68 55 47 43 40 39 38 37 372 284 283 282 279 274 265 250 226 188 142 106 88 823 864 595 236 62

a

M

4

tms

l∫

∫

wcu

�

Dt

�

5

pad

fim

T12) +T9)

2 − T

− T8

From Fig. 4, the maximum range of stresses occurs at the topof pipe and equals 72–(−124) = 196 MPa (based on Profiles 2 and1) which becomes the peak stress amplitude if a magnificationfactor of 2 is conservatively used to account for through thick-ness variation and other constraining effects. For the alternating

215 209 190 156 11671 70 69 66 63

270 269 265 257 244 2256 253 244 227 201 1

nd,

T = E˛

∫zTdA (16)

. Discretization of integrals

Let us consider a pipe of mean radius Rm and thickness t. Let theemperature be specified at locations spaced 15 degrees apart. This

eans there are 12 segments for which the temperature data arepecified over half the pipe circumference.

Denoting �� = 15◦ = �/12 and �A = tRm �� = (�/12) tRm, the fol-owing approximations to the integrals result:

TdA = �Rmt

6

12∑1

Tmi (17)

TzdA = �R2mt

6

12∑1

Tmi sin �mi (18)

here Tm1, Tm2,. . ., Tm12, are the mean temperatures at variousircumferential pipe segments. The expression for the axial stresssing Eq. (14) (the maximum value) can therefore be written as:

xx = E˛

12

12∑i=1

Tmi + E˛

6sin �mi

12∑i=1

Tmi sin �mi − E˛T (19)

enoting T the average temperature across the pipe cross section,he maximum axial stress is now given by,

xx = E˛

6sin �mi

12∑i=1

Tmi sin �mi + E˛(Tave − T) (20)

. Numerical example

The numerical example uses data for a typical piping for a PWRlant (Thurman et al., 1981) for which the geometric parameters

�xx = E˛

6

[0.4957(T1 − T13) + 0.9576(T2 −+0.7011(T4 − T10) + 0.4958(T5 −

�xx = E˛ �Tstrat

�Tstrat = 16

[0.4957(T1 − T13) + 0.9576(T

+0.4958(T5 − T9) + 0.2566(T6

nd the material properties are: geometric parameters: outsideiameter D0 = 0.4064 m, thickness t = 0.02145 m.

Material properties: Young’s modulus E = 192 GPa, and coef-cient of thermal expansion ˛ = 12 × 10−6/K. For simplicity allaterial properties are assumed to be independent of temperature.

66 54 47 43 41 40 3956 52 48 45 42 41 40

191 148 110 85 70 62 59120 90 70 58 50 47 46

6. Stratification temperature profiles

As indicated by Thurman et al. (1981) during normal operation anumber of temperature measurements were taken around the pipecircumference both for the inside and outside surfaces. Analysisof the test data indicated that the distribution could be groupedinto six basic profiles corresponding to different levels of interfacebetween the hot and cold fluids. The profiles are indicated in Fig. 2athrough f where the solid lines represent the fluid temperaturesand the temperatures marked by diamond are the calculated metaltemperatures on the outside surface.

The numerical values of the temperatures specified at the innersurface of the pipe for Profiles 1 through 6 are indicated in Table 1.

Based on the numerical values of temperatures in Table 1, themean temperatures at various circumferential pipe segments arecalculated and shown in Fig. 3.

The maximum axial stress from Eq. (20) with the proposed dis-cretization scheme can now be written as:

0.8586(T3 − T11)+ 0.2566(T6 − T8)

]+ E˛

(12∑

ii=1

Tm1 − T1

)(21)

Eq. (21) can be formally written for �xx in terms of �Tstrat expressedas:

12) + 0.8586(T3 − T11) + 0.7011(T4 − T10)

)] +12∑i=1

Tm1 − T1

(22)

The stress distributions for Profiles 1 through 6 have been calculatedusing Eq. (22) and are shown in Fig. 4.

Fig. 3. Temperature input to the approximate numerical model.

S. Chattopadhyay / Nuclear Engineering a

s3Tptre

7

sp

twtttbp

mo

Fig. 4. Stress distribution across the pipe diameter for Profiles 1 and 6.

tress amplitude 196 MPa of the allowable number of cycles is about0,000 using design fatigue curve of carbon steel from ASME (2007).he plant data in the work of Thurman et al. (1981) indicates a com-arable number of stratification temperature excursions. This leadso a significant fatigue usage factor at the top of the pipe, which cor-elates with the fatigue cracks observed at this location (Thurmant al., 1981).

. Conclusions

In this study a semi-analytical method has been used to calculatetresses due to circumferential temperature gradients in the piperoduced by thermal stratification.

The peak stress in the ASME Code Eq. (1) should have an addi-ional term equal to Kstrat E˛�Tstrat evaluated from Eq. (22) alongith a stress intensification factor, Kstrat to account for through

hickness temperature variation and other constraining effects. Forhis problem this factor is conservatively taken as 2.0. Accordingly,he magnitudes of the stresses due to thermal stratification have

een determined to be quite significant and have the potential toroduce low cycle fatigue failure in pipelines.

The analytical approach treats the problem as a statically deter-inate system that is free of external loads. Furthermore the effect

f lateral contraction as influenced by the Poisson’s ratio is ignored.

nd Design 239 (2009) 2236–2241 2241

The solutions, albeit simplified, exhibit the same trend as the finiteelement solution results of Thurman et al. (1981). One signifi-cant correction factor that should be applied to the solution stemsfrom the end conditions which have not been considered in thiswork. The other correction comes from the Poisson’s ratio and theexact solution will deviate more from the approximate solution forgreater values of Poisson’s ratio for the pipe material. Finally thereis a possible contribution from the through thickness temperaturevariation, which is also tied to the Poisson’s ratio. A blanket cor-rection factor of 2.0 is suggested in this work to account for allthese effects. This seems reasonable based on the correction fac-tors reported by Bain et al. (1992) and those reported by Furuhashiet al. (2008).

References

ASME Boiler and Pressure Vessel Code, 2007. Section III, Nuclear Power Components.American Society of Mechanical Engineers, New York.

Bain, R.A., Collins, S.O., Testa, M.F., 1992. New insights into thermal stratification offeedwater piping in PWR plants. In: PVP-Vol. 235, Design and Analysis of PressureVessels Piping and Components. American Society of Mechanical Engineers, NewYork.

Boley, B.A., Weiner, J.H., 1960. Theory of Thermal Stresses. John Wiley and Sons, NewYork.

EPRI Report TR-103581, 1994. Thermal Stratification, Cycling and Striping (TASCS).Electric Power Research Institute, Palo Alto, CA.

Furuhashi, I., Kawasaki, N., Kasahara, N., 2008. Evaluation charts of thermal stressesin cylindrical vessels induced by thermal stratification of contained fluid. Journalof Computational Science and Technology 2 (4), 547–558.

Jhung, M.J., Choi, Y.W., 2008. Surge line stresses due to thermal stratification. NuclearEngineering and Technology 40 (3), 239–250.

Jones, I.S., 2003. Small edge crack in a semi-infinite solid subjected to thermal strat-ification. Theoretical and Applied Fracture Mechanics 39, 7–21.

Kim, J.H., Roidt, R.M., Deardoff, A.F., 1993. Thermal stratification and reactor pipingintegrity. Nuclear Engineering and Design 139, 83–95.

Lee, J.B., Song, J.H., 1993. A study of a crack located at the striping zone in a thermallystratified pipe. Nuclear Engineering and Design 143, 229–237.

Miksch, M., Lenz, E., Lohberg, R., 1985. Loading conditions in horizontal feedwaterlines of LWR’s influenced by thermal shock and thermal stratification effects.Nuclear Engineering and Design 137, 387–404.

US Nuclear Regulatory Commission Bulletin 88-08, Thermal stresses in piping con-

nected to reactor coolant systems.

US Nuclear Regulatory Commission Bulletin 88-11, Pressurizer surge line thermalstratification.

Thurman, A.L., Mahlab, M.S., Boylstein, R.E., 1981. 3-D Finite Element Analysis for theInvestigation of Feedwater Line Cracking in PWR Steam Generators, ASME-81-PVP-3.