Stress assessment in piping under synthetic thermal loads emulating turbulent fluid mixing

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Stress Assessment in Piping under Synthetic Thermal Loads Emulating Turbulent Fluid

Mixing

Oriol Costa Garrido, Samir El Shawish, Leon Cizelj

Abstract

Thermal fatigue assessment of pipes due to turbulent fluid mixing in T-junctions is a

rather difficult task because of the existing uncertainties and variability of induced thermal

stresses. In these cases, thermal stresses arise on three-dimensional pipe structures due to

complex thermal loads, known as thermal striping, acting at the fluid-wall interface. A

recently developed approach for the generation of space-continuous and time-dependent

temperature fields has been employed in this paper to reproduce fluid temperature fields of a

case study from the literature. The paper aims to deliver a detailed study of the three-

dimensional structural response of piping under the complex thermal loads arising in fluid

mixing in T-junctions.

Results of three-dimensional thermo-mechanical analyses show that fluctuations of

surface temperatures and stresses are highly linearly correlated. Also, surface stress

fluctuations, in axial and hoop directions, are almost equi-biaxial. These findings,

representative on cross sections away from system boundaries, are moreover supported by the

sensitivity analysis of Fourier and Biot numbers and by the comparison with standard one-

dimensional analyses. Agreement between one- and three-dimensional results is found for a

wide range of studied parameters. The study also comprises the effects of global thermo-

mechanical loading on the surface stress state. Implemented mechanical boundary conditions

develop more realistic overall system deformation and promotes non- equibiaxial stresses.

Keywords

Turbulent fluid mixing; heat transfer analyses; thermal stress fluctuations; spectral analysis;

global thermo-mechanical loading; mechanical boundary conditions.

Highlights

- Generation of complex space-continuous and time-dependent temperature fields.

- 1D and 3D thermo-mechanical analyses of pipes under complex surface thermal loads.

- Surface temperatures and stress fluctuations are highly linearly correlated.

- 1D and 3D results agree for a wide range of Fourier and Biot numbers.

*Manuscript

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- Global thermo-mechanical loading promotes non-equibiaxial stress state.

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Nomenclature

t Time � Simulated time �, � Non-dimensional axial � ∈ ��0, �⁄ � and circumferential � ∈ ��0,2��� coordinates ����, �� Field of amplitudes of harmonic n

���, ��� Non-dimensional wave numbers in axial and circumferential directions � Main flow velocity normalized with pipe inner radius

� Integer defining the range for ��� distribution: ��� ∈ ����, � n Harmonic number �� Phase of harmonic n �� Discrete angular frequency ��, �� Discrete frequency and Nyquist frequency N Number of temperature readings Δ , Δ� Time and frequency intervals l, �, ! Length, inner and outer radius of the pipe Radial coordinate of the pipe wall, ∈ ��� , ! � h Heat transfer coefficient "#, $% Fourier and Biot numbers & Thermal diffusivity ', () Density and specific heat of pipe wall material

�, * Thermal conductivity and expansion of pipe wall material +, , Young modulus and Poisson’s ratio of pipe wall material -.��, �, � Two-dimensional and time-dependent fluid temperature ".��, �, � Two-dimensional and time-dependent fluid temperature fluctuation -./01��, �� Variance field of fluid temperature fluctuations -.2304��, �� Field of fluid mean temperatures

-.145��, �� Field of fluid temperature ranges maxima

-.126 Root mean square of fluid temperature fluctuations at certain position -6�, �, �, � Time-dependent temperature fields of the pipe wall "6�, �, �, � Time-dependent temperature fluctuation fields of the pipe wall -6126�� Radial profile of root mean square temperature fluctuations of the pipe wall 7�!�899 Convective surface heat flux :1, :;, :< Radial, hoop and axial stresses :126�� Radial profile of root mean square stress fluctuations of the pipe wall :20= Maximum stress at the surface under fully clamped surface expansion Sf Scale factor to input data r(F-z) Linear correlation factor between temperature and axial stress fluctuations r(F-�) Linear correlation factor between temperature and hoop stress fluctuations r(�-z) Linear correlation factor between axial and hoop stress fluctuations

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1. Introduction

Thermal loads generated by turbulently mixing fluids at different temperatures in T-

junction piping are known to be the cause of thermal fatigue in the surrounding material. Steel

components in existing nuclear power plants (NPPs) have experienced cracking due to

thermal fatigue induced by this type of fluid phenomenon and, in several cases, the cracks

propagated through wall, provoking primary water leakage within the containment

(NEA/CSNI, 2005, 2012). These incidents proved that thermal fatigue has important

implications on structural integrity of plant components and, subsequently, on nuclear safety.

Various research projects have been dedicated to the understanding of thermal fatigue

in general, which include the particular case of turbulent fluid mixing. Clear examples are the

European THERFAT project (Metzner and Wilke, 2005), the international project coordinated

by IAEA under the framework of fast reactor technologies (IAEA, 2002), the American

research project (EPRI, 2003) and the Japanese research project (Fukuda et al., 2003). The

multidisciplinary nature of thermal fatigue assessment forced the involved areas of research to

progress in parallel.

Firstly, characterization of thermal loads acting on structural components and the flow

patterns that develop downstream of the T-junction have been possible by means of

experimental facilities (Kamide et al., 2009; Smith et al., 2013). In particular, the Vattenfall

benchmark facility (NEA/CSNI, 2011) was initially conceived for the development and

validation of computational fluid dynamic (CFD) codes, mathematical approaches and

turbulence models for the reliable simulation of fluid mixing phenomena. At the present

moment, CFD with large-eddy simulations (LES) scheme give accurate thermal loads acting

on the surrounding structure when simulations are performed considering adiabatic

surrounding walls (Kuczaj et al., 2010). In these cases, heat transfer between fluid and

structure is modeled employing a heat transfer coefficient approach (Chapuliot et al., 2005).

In parallel to the validation of these advanced computational tools, mechanical response of the

material to thermal loads has been predicted analytically using one dimensional (1D) methods

(Kasahara et al., 2002), leading to the European Procedure for Assessment of High Cycle

Thermal Fatigue (Dahlberg et al., 2007). The procedure provides screening criteria of

temperature difference between the mixing fluids for the requirement of fatigue analyses,

usual values of heat transfer coefficients between fluid and structure as well as suitable

correction factors to be applied on the derived stresses using the proposed sinusoidal (SIN)

method. In the SIN method the fluid temperature is anticipated at a single point and assumed

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to vary sinusoidally with time at a given frequency and amplitude. Then, thermal, mechanical

and fatigue analyses are performed for different temperature parameters assuming that the

pipe wall temperature varies only in radial direction. However, it is known that 1D methods

intrinsically omit the global response of the structure.

The second field of research tries to quantify heat transfer between fluid and structure,

i.e., to predict pipe wall temperature fluctuations. By means of novel sensors in experimental

facilities, the fluid and structure temperatures are captured simultaneously (Fontes et al.,

2009; Kimura et al., 2009) and transfer functions are derived in the frequency domain.

Computer simulation tools are also available which allow obtaining structural temperature

fields by computing the fluid thermo-hydraulics, the fluid-wall heat convection and wall heat

conduction simultaneously (Kloeren and Laurien, 2011; Kuhn et al., 2010). These tools

employ LES with conjugate heat transfer (CHT) and give very promising results. However,

they are under development and the required computer resources and time are excessively

elevated. Nevertheless, the validation of this kind of computer codes will be possible through

ongoing international projects which aim to determine proper thermal loads induced on

piping; see (Kuschewski et al., 2013; Miyoshi et al., 2012) and references therein.

In structural mechanics, the derivation of wall stresses from the induced temperature

fields is performed with computer codes that employ finite-element (FE) solvers. These are

necessary to obtain appropriate structural responses emerging from the complex three-

dimensional (3D) thermo-mechanical loading as a consequence of large-scale flow

instabilities of different frequencies and global deformation of the system. In the literature one

can find few examples of mechanical analyses performed with thermal loads derived from

CFD-LES-CHT simulations. However, these state-of-the-art and computationally demanding

tools simulate one specific experimental case with stress levels that are usually scaled up to

crack initiation levels (Kamaya and Nakamura, 2011) or, when more realistic cases are

simulated, the study of the structural response is limited, most probably, due to the complexity

of the overall computer simulation and uncertainties in the involved phenomena (Hannink and

Blom, 2011; Niffenegger et al., 2013). Nevertheless, a comparison of thermal stresses

obtained with different approaches has been performed by Blom et al. (2007).

Certainly, necessary and most important input to 3D thermo-mechanical analyses of

T-junction piping are space-continuous surface thermal loads which cannot be obtained in

experimental facilities. Hence, in support to experiments and in parallel to the development of

CFD codes, a novel approach has been recently developed for the generation of two-

dimensional (2D), continuous and time-dependent temperature fields while reproducing fluid

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temperature statistics from experimental readings at discrete near-wall locations (Costa

Garrido et al., 2014). The approach has been proven to deliver credible surface thermal loads

while reproducing discrete experimental input, in a fast and reliable fashion. In this paper, the

approach is therefore employed to generate complex thermal fields with input data from the

Vattenfal benchmark facility.

The main purpose of this paper is to study the 3D structural response of pipes under

complex thermal loads arising in T-junctions where fluids at different temperatures

turbulently mix. In addition to this, the effect of mechanical boundary conditions on the

global thermo-mechanical state of the 3D structural system is also analyzed and main findings

are presented in the Results section. In the Sensitivity analyses section, different values of

heat transfer related parameters are studied with the aim to support and evaluate the scope of

the main findings given the Results section. This section also highlights differences between

3D and standard 1D (through thickness) pipe models by comparing the obtained results under

equivalent surface temperatures. To the authors’ knowledge this is the first attempt to

systematically evaluate the differences of wall stress fluctuations obtained with 3D and 1D

methods. At the end, a summary of main findings can be found in the Conclusions.

2. Generation of Thermal Loads: Demonstration Case

Surface temperature fields are generated employing a novel approach proposed in

Costa Garrido et al. (2014). The strength of the approach is to deliver space-continuous and

time-dependent surface thermal loads from discrete data at field locations. The modeled fields

reasonably reproduce the attributes of flow patterns and thermal load fields in a

statistical/stochastic sense, in fluid mixing circumstances. The underlying concept relies on

the approximation of the fluid temperature fluctuations with linear superposition of plane

waves while reproducing the fluid temperature statistics from experimental readings or CFD

simulations at discrete near-wall locations. In particular, the approach requires, and perfectly

reproduces by design, the mean and normalized power spectral density (PSD) of fluid

temperatures near the pipe surface at several field locations. For the demonstration case given

in this paper, first two moments of the temperature histories -.��, �, � are employed as input

data, i.e., the mean and variance:

-.2304��, �� ? @

A B -.��, �, �C AD ,

".��, �, � ? -.��, �, � � -.2304��, ��, (1)

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-./01��, �� ? @A B ".��, �, �EC A

D ,

and the PSD profile is assumed from turbulent theory (Hinze, 1975). The adopted PSD profile

was shown to be reasonable when compared to experimentally obtained PSD in Hannink and

Timperi (2011). In any case, main conclusions of this study are not supposed to be affected by

this choice. The simulated fluid temperature fluctuation fields ".��, �, � are approximated in

time domain ∈ �0, � with the expression:

".��, �, � ? ∑ ����, �� cosJ�� K ���� K ���� K ��LMNO@�P@ , (2)

where ����, �� stands for the position dependent amplitude of a specific harmonic, �� ?2��� is the angular frequency, ��� and ��� are wave numbers in axial and circumferential

directions and �� is the phase. N temperature readings are assumed to be taken at a constant

time interval Q and the frequency domain is discretized from 0 to Nyquist frequency

�� ? 1 �2Q �⁄ as �� ? �S � 1� �TQ �⁄ for S=1,2,…,N/2+1. Eq. (2) allows relating the

amplitudes �� directly to the power spectral density (Costa Garrido et al., 2014). The wave

numbers and the phases, however, are determined on physical reasoning, as described below.

The fields are defined in the near wall region of the pipe’s inner surface downstream of a T-

junction where two fluids at different temperatures mix together. The 2D space is delimited

by the inner perimeter of the pipe and its length (l). Both dimensions are normalized by the

inner radius of the main pipe (�) leading to dimensionless coordinates (�,�) where � ∈��0, �⁄ � and � ∈ ��0,2���. The 2D fields are characterized by the (also dimensionless) wave

numbers ��� and ���. Wave numbers ��� define axial velocities of hot and cold spots within

temperature fluctuation fields, which are set constant and equal to the main flow velocity �

(Blom et al., 2007; Hannink and Blom, 2011):

��� ? � EUVW8 . (3)

For consistency � has the units of ��. As there is no net velocity expected in circumferential

direction, the average wave number ��� should be zero. Moreover, ��� ensure the

circumferential symmetry of our domain, ".��, 0, � ? ".��, 2�, � and were shown to

influence the size and shape of the hot and cold spots within the fluctuation fields. In this

respect, ��� are chosen here from a standard uniform distribution of integers ����, � defined

by parameter �. The difficulty of the approach resides in the selection of the phases �� which

are initially chosen randomly from a uniform distribution ��0,2��. The phases undergo a

minimization process in order to ensure that physical values of the generated temperatures are

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kept within a prescribed tolerance of the limiting, cold and hot, mixing fluids temperatures

(Costa Garrido et al., 2014). Nevertheless, the approach assures that the prescribed statistics

and PSD profiles are maintained at all field locations.

Fig. 1. (a) Sketch of the Vattenfall benchmark facility. (b) Input mean temperature

field with experimental data locations marked with black dots. (c) Input variance field of fluid temperature fluctuations. Crosses with labels indicate locations used to present results

throughout the paper. Applied scale factor Sf=6.66 to the original data from Westin et al. (2008). Grid values in (b) and (c) are interpolated linearly from experimental locations.

Values at z=0 are assumed here to generate complete temperature field.

In this analysis, temperature fields are generated using the input parameters (mean and

variance) from the Vattenfall benchmark facility given in Westin et al. (2008). The simulated

fields consists of N=512 time intervals with time step Δ =0.02 s giving a simulated time

�=10.22 s, maximum frequency of ��=25 Hz and a frequency interval �=0.0977 Hz. The

main flow velocity is set to �=0.8/0.07 s-1 from the experimental bulk velocity and normalized

by the pipe inner radius �=0.07 m. The temperature fields are simulated within a 0.05

tolerance of the limiting fluid temperatures. The temperature difference of the incoming fluids

in the Vattenfall benchmark facility was 15 °C. However, the screening criteria performed in

the European Procedure for Assessment of High Cycle Thermal Fatigue (Dahlberg et al.,

2007) stipulates a minimum temperature difference of 80 °C (in stainless steels and water

coolant) for the requirement of fatigue analyses. Therefore, in the present study, a scale factor

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(Sf) of 6.66 has been applied to the Vattenfall mean temperatures and variances in order to

increase the temperature difference up to nearly 100 °C. A schematic description of the

Vattenfall facility as well as the input data used for the generation of temperature fields are

presented in Fig. 1. Note, in Fig. 1c, the points located 4 diameters downstream and

circumferentially at � equal to 0 (2�), � 2⁄ , � and 3� 2⁄ . They are named D4 TOP, D4 LEFT,

D4 BOTTOM and D4 RIGHT, respectively, and will be used throughout this study to present

main results. This cross section contains the locations with highest variance of fluid

temperature fluctuations (D4 RIGHT and D4 LEFT), and is located at reasonable distance

from boundary conditions at z = 0 and l/ri.

Fig. 2. Computed distribution of temperature range maxima from the generated

thermal loads for the presented case with �=2.

The results presented here stand for the case with wave numbers in circumferential

direction, ���, limited by parameter �=2 (Costa Garrido et al., 2014). We note that the mean

temperatures and PSDs at grid locations within the generated temperature fields are

independent on the choice of � and �. Fig. 2 shows the distribution of temperature range

maxima computed from the generated thermal loads as:

-.145��, �� ? max\�-.��, �, � � min\�-.��, �, � . (4)

As one would expect, the regions with higher temperature ranges match those with higher

variance, shown in Fig. 1c.

In Fig. 3 the simulated fluctuation fields and the computed temperature fields are

shown at two different times. In the fluctuation fields one can observe hot and cold spots

representing the maxima and minima of Eq. (2). The temperature fields are obtained with the

addition of the mean temperature field, Fig. 1b, to the simulated fluctuations. The fluid

temperatures pattern is consistent with “tongue” behavior which breaks into hot spots (Kuczaj

et al., 2010). However, these hot spots emerging from the broken tongue should not be

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confused with the hot/cold spots within the fluctuation fields. Note the point D4 RIGHT,

located in the bordering region between the cold/hot flows. This location has the highest

variance of the temperature fluctuations, Fig. 1c.

Fig. 3. Time progression of simulated fluid temperature fluctuations (left) and fluid

temperature fields (right). The fluid temperatures are calculated by addition of the mean temperature field, Fig. 1b, to the simulated fluid fluctuations. Labels mark the region with

“tongue” behavior and location with highest variance.

3. Results

Heat transfer and mechanical analyses of a pipe wall subjected to the simulated

thermal load fields are presented in this section. In particular, surface thermal stresses

developed due to time dependent temperature gradients in the wall are studied in time and

frequency domain. All simulations have been performed using the finite element solver

ABAQUS (Simulia, 2012).

3.1. Thermal analyses of pipe wall

Analyses presented in this study have been performed on the straight and downstream

part of the Vattenfall T-junction, Fig. 1a, with a length of 1.4 m and considering Stainless

Steel AISI 304L for the pipe material with wall thickness of 9.6 mm. The material properties

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used in the analyses are given in Table 1. The insertion hole of the vertical pipe into the main

pipe has been omitted in all the analyses.

Table 1: Material properties of the pipe

Symbol Property Material SS AISI 304L

' Density (kg/m3) 7900

() Specific Heat (J/kgK) 493.0

� Thermal conductivity (W/mK) 15.29

* Thermal expansion (1/K) 15.67e-6

+ Young modulus (GPa) 193

, Poisson's ratio (-) 0.3

The pipe wall has been meshed using 87,040 hexahedral quadratic elements of type

DC3D20 (Simulia, 2012). The mesh comprises 64 elements equally distributed in

circumferential direction and 68 elements in axial direction with higher density at axial cross

section D4. Approximate element edge size near D4 cross section is 8 mm. The wall thickness

contains 20 elements with denser mesh (bias 10) towards the inner surface. The smallest

element thickness in radial direction is 0.12 mm.

A convection boundary condition has been set at the inner surface of the pipe. The

heat flux on the surface (7�!�899 ) due to convection is linearly correlated to solid (-6���) and

fluid (-.) temperature difference:

7�!�899 ? �_�-6��� � -.�, (5)

and must be satisfied at each time increment. In Eq. (5), the heat transfer coefficient, _, is a

key parameter in this type of analyses. The convection phenomenon, coupled with the time-

dependent heat conduction in the wall, acts as a filter between the fluid and surface

temperatures. Recent studies of coupled CFD analyses with conjugate heat transfer of fluid

mixing in T-junctions have shown values of heat transfer coefficient varying between 3,000

and 7,000 W/m2K (Hannink and Blom, 2011). In Dahlberg et al. (2007) a recommended value

is 15,000 W/m2K. Therefore, this value has been adopted here as a conservative assumption.

Thermal analyses consist of two steps. The pipe temperature is set initially at the

temperature of the main flow, i.e., 100 °C considering the applied Sf. In the first step the pipe

reaches a steady-state temperature distribution with the fluid mean temperature field as the

boundary condition, Fig. 1b. The second step consists of a transient analysis of time length

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10.22 s with a time increment of 0.02s. These values are equal to those used for the simulated

fluid temperature fields shown in the previous section which are now used as sink temperature

in the convection boundary condition. Moreover, in the second step, the FILM subroutine

(Simulia, 2012) is used to import the time-dependent fluid temperature fields during the

analysis which are interpolated at surface nodal locations.

Fig. 4. Temperature fluctuations (left) and temperature fields (right) of the pipe’s inner

surface as a result of heat transfer analyses. The temperature fluctuations are computed from the wall temperatures. Results presented at same times as in the fluid (Fig. 3).

Fig. 4-right shows the temperature distributions of the pipe’s inner surface resulting

from the heat transfer analyses at same times as in the fluid (see Fig. 3-right). Clear

attenuation of the calculated surface temperatures can be observed when compared to the fluid

temperatures. Further, pipe wall temperature fluctuations have been computed from the wall

temperatures obtained in the heat transfer analysis. The temperature fluctuations of the pipe’s

inner surface are represented in Fig. 4-left. Hot and cold spots are also originated at the inner

surface and resemble those of the fluid temperature fluctuations (Fig. 3-left). Surface

temperature fluctuations that resemble hot and cold spots traveling along the pipe surface was

initially pointed out by Hannink and Blom (2011).

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Fig. 5. Computed distribution of temperature range maxima at the pipe’s inner surface.

The distribution of surface temperature range maxima has also been computed and can

be seen in Fig. 5. The maximum temperature range occurs at the position with higher variance

of the fluid temperature fluctuation, i.e., D4 RIGHT location. The fluid has a maximum

temperature range at this location of almost 90 °C, Fig. 2, while at the pipe surface it reduces

to slightly above 40 °C.

Fig. 6. Filtering effect between fluid (-.) and pipe surface (-6���) temperatures.

Temperature histories, distributions and power spectral densities (PSD) are shown for 4 circumferential positions (LEFT, TOP, BOTTOM and RIGHT) at cross section located 4

diameters downstream. In legend, fluid and surface temperature ranges in (°C).

Temperature filtering effects due to coupling of heat convection and conduction in the

wall can be further seen in Fig. 6. Time histories of fluid (-.) and pipe surface (-6���) temperatures are shown at 4 circumferential positions at the cross section located 4 diameters

downstream, i.e., D4 LEFT, TOP, BOTTOM and RIGHT presented in Fig. 1c. Spectral

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analyses of the fluctuations as well as the distribution of fluid and solid temperatures are

presented in Fig. 6. The amplitudes of temperature fluctuations in solid are reduced

significantly. Moreover, high frequency temperature fluctuations are more attenuated than

low frequency ones. This same effect was observed in the literature from experimental

measurements (Fontes et al., 2009; Kimura et al., 2009). However, from CFD simulations

with conjugate heat transfer this effect seems to be less acute (Hannink, 2008; Kloeren and

Laurien, 2011). Note the imposed theoretical fluid PSD profile (Costa Garrido et al., 2014)

which is reproduced at all field locations.

3.2. Mechanical analyses of pipe wall

Time-dependent thermal fields of the pipe wall obtained in heat transfer analyses are

used here as a source of thermal stresses. Such one-way or uncoupled mechanical analyses

were also used elsewhere (Chapuliot et al., 2005; Hannink, 2008; Hannink and Blom, 2011;

Kamaya and Nakamura, 2011) since it is assumed that structural displacements do not affect

either the flow condition or wall thermal fields. The mesh used in mechanical analyses is the

same as the one described in the previous section. Hexahedral quadratic elements of the type

C3D20 (Simulia, 2012) have been used to perform mechanical analyses.

3.2.1. Stress fluctuations at inner surface

Thermal fatigue is induced by stress fluctuations, which develop due to the variation

of temperature gradients inside the structure. Temperature gradients arise due to localized

temperature fluctuations in the structure. Therefore, in terms of fatigue assessment,

consideration of structural temperature fluctuations as a source of stress fluctuations may be

sufficient. This reasoning has been applied in recent studies which try to define more realistic

thermal boundary conditions in 1D methods, by means of complex CFD-LES-CHT

simulations (Hannink and Blom, 2011). In this way, mechanical boundary conditions,

explained in detail in the next section, do not produce any relevant effects on stresses (the

pipe wall is assumed stress free at 0 °C). Fig. 7 shows the axial and Mises stress fluctuations

of the pipe’s inner surface, at same times as in Fig. 3 and Fig. 4. Clear correspondence can be

observed between cold/hot spots in the fluid/solid temperature fluctuations (Fig. 3 and Fig. 4,

left) and surface stress concentrations.

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Fig. 7. Axial and mises stress fluctuations of the pipe’s inner surface as a result of

mechanical analyses using wall temperature fluctuations as thermal loads. The pipe wall is assumed stress free at 0 °C. Results presented at same times as in Fig. 3 and Fig. 4. Cold/hot spots in the fluid/solid temperature fluctuation fields (Fig. 3 and Fig. 4, left) match surface

stress concentrations.

Surface-localized wall temperature fluctuations induce an almost equi-biaxial stress

state at the surface. This can be seen in Fig. 8 and Fig. 9 at the 4 circumferential locations D4

LEFT, TOP, BOTTOM and RIGHT. Specifically, Fig. 8 shows, for each of the locations, time

histories of surface temperature fluctuations compared to the induced, axial and hoop, stress

fluctuations. Positive surface temperature fluctuations generate compressive stresses while

negative fluctuations cause tensile stresses. It can be also appreciated that both stresses

closely follow the temperature fluctuations. This is emphasized by the linear correlation

factors between fluctuations and axial stresses r(F-z) and hoop stresses r(F-θ). It can be noted

that correlation factors are close to -1 at all locations, however, the correlation between

surface temperature fluctuations and hoop stresses is slightly lower than for axial stresses.

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Fig. 8. Comparison between time histories of surface temperature fluctuations and,

axial and hoop, stress fluctuations. Results presented for the circumferential positions (LEFT, TOP, BOTTOM and RIGHT) at cross section located 4 diameters downstream. Stresses follow temperature fluctuations. In legends: linear correlation factors between surface

temperature and, axial r(F-z) and hoop r(F-θ), stress fluctuations.

The almost equi-biaxial stress state of the pipe surface can be further derived by

correlating axial and hoop stress intensities. This is shown in Fig. 9 at same locations as in

Fig. 8. It can be observed that linear correlation between stress fluctuations r(θ-z) is rather

close to one at all locations. Fig. 9 also includes axial stress distributions that, to some extent,

match distribution of surface temperatures in Fig. 6.

Fig. 9. Correlation between axial and hoop temperature fluctuations as thermal loads

(LEFT, TOP, BOTTOM and RIGHTlegends: computed stress ranges (in MPa)

axial

The study follows with a spectral analysis of surface stresses.

temperature and stress fluctuations in frequency domain. One can observe that the

spectral densities (PSDs) of temperature

Recall that the linear correlation obtained between temperature fluctuations and

in time domain is lower than with axial stresses

power level fluctuations of hoop stresses

the stresses’ PSDs at 5 Hz are two orders of magnitude lowe

This agrees with the behavior of stresses

(Blom et al., 2007; Chapuliot et al., 2005

approaches.

orrelation between axial and hoop surface stress fluctuations using wall as thermal loads. Results presented for the circumferential positions

LEFT, TOP, BOTTOM and RIGHT) at cross section located 4 diameters downstream(in MPa) and linear correlation factors between hoop and

axial stress fluctuations, r(θ-z).

The study follows with a spectral analysis of surface stresses. Fig. 10 shows surface

stress fluctuations in frequency domain. One can observe that the power

temperature fluctuations and stresses follow very similar trend.

obtained between temperature fluctuations and hoop stress

is lower than with axial stresses (see Fig. 8). This seems to be reflected in

hoop stresses in the low frequency ranges up to 1Hz. Moreover,

two orders of magnitude lower than for the lowest frequency

the behavior of stresses in the frequency domain from previous studies

Chapuliot et al., 2005; Kasahara et al., 2002) which used more theoretical

using wall

circumferential positions diameters downstream. In

hoop and

surface

power

similar trend.

hoop stresses

be reflected in

low frequency ranges up to 1Hz. Moreover,

r the lowest frequency.

from previous studies

which used more theoretical

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Fig. 10. PSDs of surface temperature and stress fluctuations at the circumferential

positions (LEFT, TOP, BOTTOM and RIGHT) at cross section located 4 diameters downstream. PSDs of fluctuations and stresses follow very similar trend.

Distributions of axial stress range maxima induced at the inner surface can be seen in

Fig. 11. Regions with higher stress ranges are found to match those with greater variance of

the fluid temperature fluctuation, matching the observation with the temperature ranges (see

Fig. 2 and Fig. 5). This same behavior has also been found in recent studies by Kamaya and

Nakamura (2011).

Fig. 11. Distribution of axial stress range maxima at the pipe’s inner surface. Higher

stress ranges occur at locations with higher variance of the fluid temperature fluctuation.

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3.2.2. Mechanical boundary conditions and their effect on surface thermal stresses

Wall temperature fluctuations generate stress fluctuations over a mean stress state.

This mean stress state, on the other hand, arises from the global thermo-mechanical loading of

the structure. Presented 3D pipe model comprises only a portion of the whole piping system

and its global deformation and loading highly depends on the considered boundary conditions

(BCs). Thus, the aim is to develop mechanical BCs to be applied on the pipe section

considered and that would resemble, at a low computational cost, the omitted parts of the

installation.

In this section, mechanical analyses are performed with wall temperature fields

obtained in the heat transfer analyses. Two types of mechanical BCs are considered and their

influence on the global thermo-mechanical loading of the structure is studied. In this case, the

pipe wall is stress free at 100 °C. The first BC, named BC-Soft, is shown in Fig. 12a. The

nodes in the pipe’s cross section at z=0 are kept on plane (axial displacement Uz=0). In this

cross section, the outer surface node at the top of the pipe is pinned to avoid free body

movement. The end cross section z=l/ri is free. Fig. 12a shows the Mises stress distribution at

the inner pipe surface (deformation scale factor 30), at simulation time 6.6s (same as Fig. 4-

right). A pipe bending is clearly seen due to the existing circumferential temperature gradients

between top (hot tongue) and bottom (cold). The BC at z=0 assumes that omitted upstream

piping is completely stiff in axial direction while the BC at the end of the pipe is too soft since

there is no modeling of downstream piping. This free-end BC allows free bending of the pipe.

Away from z=0, stresses prove to be equal to those arising with a completely free pipe (not

shown here).

The second BC that has been considered, named BC-Hard, is shown in Fig. 12b. The

wall rigidity induced by the omitted parts of the installation, i.e., a T-junction (inlet and

branch pipes) and outlet pipe, have been simulated with the addition of beams. The lengths of

the inlet and outlet beams are assumed equal to the 3D pipe length (1.4 m) and the branch

beam length to 0.5 m. The beam components are meshed with 50 (inlet, outlet) and 20

(branch) linear elements of type BC31 (Simulia, 2012). The beams’ implementation assumes

a pipe cross-section equal to that of the 3D pipe and they are stress free also at 100 °C. The

temperature during the simulation is kept at 100 °C for the inlet beam and set to 200 and 150

°C for the branch and outlet beams, respectively. The boundary condition applied to the outer

nodes of the beams suppresses all degrees of freedom except for the outlet beam which can

move in the axial direction. Further, the beam’s nodes located at the 3D pipe’s boundaries are

20

kinematically coupled with the respective cross sections of the 3D pipe. No heat conduction is

assumed between beams and the 3D pipe. Fig. 12b shows the Mises stress distribution at the

inner surface (deformation scale factor 300) at same time as in Fig. 12a. Comparing the

deformation states from both BCs in Fig. 12 it can be observed that with BC-Hard, cross

section z=0 is able to slightly rotate around the X axis while the outlet beam brings the

stiffness from omitted piping downstream, i.e., avoiding free bending. Note also the negative

displacement of the system in Y direction which is induced by the temperature increase of the

inlet beam. The resulting 3D pipe’s deformation together with the beams’ deformations

proves to be a reasonable overall system’s deformation. The modeling approach also has

shown to be computationally inexpensive.

Fig. 12. Effects of mechanical boundary conditions (BCs) on the deformation and

stress state. Mises stress distribution of the pipe’s inner surface at time 6.6s for (a) BC-Soft (scale factor 30) and (b) BC-Hard (scale factor 300). Boundary conditions are described in

text. The system is stress free at 100 °C. Beam temperatures during analyses are set to 100 °C for inlet, 200 °C for branch and 150 °C for outlet.

The effect of BCs on surface stress states is shown in Fig. 13 at 4 circumferential

locations D4 LEFT, TOP, BOTTOM and RIGHT. For the two boundary conditions

considered, hoop and axial stresses are correlated in the same way as in Fig. 9. In fact, both

stress fluctuations remain unchanged under the application of BCs. Note that a global thermo-

mechanical state affects mainly the axial loading of the structure since hoop stresses remain

basically unchanged. The variation in axial stresses is indicated in Fig. 13 by the computed

values of their mean in each location. It can be observed that LEFT and RIGHT locations are

not affected by the BCs. In TOP location the axial stresses become more compressive; in fact

with BC-Hard axial stresses shift to fluctuate under compressive values. On the other hand,

the situation is reversed in the BOTTOM location where the stresses become always tensile

with BC-Hard. This stress state could make crack initiation and propagation more likely at

this location. Furthermore, the almost equi-biaxial stress state is now lost. In the presented

21

case study, however, these two locations, TOP and BOTTOM where BCs have a clear effect

on the mean stress state, turn out to undergo relatively low stress ranges.

Validation of BC-Hard modeling technique has been performed by repeating the same

analyses with half and double length of all beams and also with constant beam temperatures

throughout the simulation (100 °C). In all the cases, results showed a very small difference,

±5 MPa, of mean stress levels. This implies that, for the system considered here, the main

contributor to mean stress state of the structure is the actual mean temperature distribution.

Note that different mean temperature distribution, for instance generated by different flow

pattern condition, would affect the overall deformation of the system. The indicated mean

stresses in Fig. 13 have also proved to increase, in this elastic regime, linearly with the fluids

temperature difference.

Fig. 13. Correlation between axial and hoop surface stresses using wall temperature as

thermal loads. Results presented for the two BCs considered, BC-Soft and BC-Hard, in circumferential positions (left, top, bottom and right) at cross section located 4 diameters

downstream. In legends within brackets: mean axial stresses (:�2304) in MPa. Arrows indicate that axial stresses become always tensile with BC-Hard in D4 BOTTOM, and compressive, in

D4 TOP.

4. Sensitivity analyses

The results given in Section 3.2 can be summarized as follows:

22

• Axial and hoop stress fluctuations at the pipe’s inner surface are almost linearly correlated to surface temperature fluctuations (r(F-z) and r(F-θ) are close to -1 in Fig. 8).

• Axial and hoop stress fluctuations at the pipe’s inner surface are similar in amplitude (equi-biaxiliaty; r(θ-z) is close to 1 in Fig. 9).

• Previous two points yield very similar PSD profiles of temperature and stress fluctuations. (Fig. 10).

• Pipe’s global thermo-mechanical loading is highly influenced by the mean temperature field and boundary conditions. The surface stress state is not necessarily equi-biaxial and may turn out to fluctuate in tensile/compressive values (Fig. 13).

These results agree with recent studies in the literature. Recommended works include

Kamaya and Nakamura (2011), Hannink (2008), Niffenegger et al. (2013) and Chapuliot et al.

(2005). Other related works that try to simplify these complex studies with 1D (through-

thickness) methods are Blom et al. (2007), Hannink and Blom (2011) and Hannink and

Timperi (2011). This section will focus on a detailed study of the first two points listed above,

i.e., an investigation of wall stress fluctuations with emphasis on the surface as potential

location of crack initiation. The study also includes 1D analyses with equivalent surface

thermal loads in order to compare wall stress fluctuations evolving from 3D and 1D models.

4.1. Sensitivity analysis of wall temperature fluctuations

Localized surface thermal stresses originate when the expansion or contraction of

surface material is precluded, to some extent, by neighboring volumes of material at different

temperatures. Hence, it is expected that surface stresses depend on existing temperature

gradients in the pipe wall, stronger the temperature gradients, higher the thermal stress. A

sensitivity analysis is therefore performed to study the influence of heat transfer related

parameters on thermal stresses’ behavior using same fluid temperature fields shown in

Section 2. Furthermore, the same analyses are also performed considering radial temperature

gradients only using a 1D pipe wall model. Finally, an assessment of 1D analyses versus

complete 3D ones is performed by comparing wall thermal-thickness results with equivalent

fluid temperature histories on the surface. The explicit finite difference equations for the 1D

time-dependent heat diffusion through the pipe wall thickness, convective heat transfer at the

inner surface and adiabatic outer surface were derived in a previous paper by the same authors

and have been used here in 1D analyses (Costa Garrido et al., 2013). Studies presented in this

section compare wall thickness results at point D4 RIGHT. For 1D analyses the fluid

temperature signal given in Fig. 6 has been used at this location. In order to ensure the

23

stability of the 1D numerical approach (in thermal analyses), simulation time increment is

reduced accordingly and fluid temperature behavior is assumed linear within the initial time

increment Δ =0.02s.

Time-dependent heat diffusion in a solid with convection boundary condition, Eq. (5),

is characterized by two non-dimensional numbers, i.e., Fourier number (Fo) and Biot number

(Bi) which are defined as (Incropera and DeWitt, 1996):

"# ? a\bcdN , $% ? ecd

f . (6)

The Bi number is found in convection problems and is equal to the ratio of heat transfer

resistance inside the solid to the thermal resistance of the boundary layer. The Fo number is a

dimensionless time which characterizes the heat conduction through the media and is defined

as the ratio of heat diffusivity rate to the rate of heat storage. & ? � ()'⁄ , in Fo number

definition, stands for the thermal diffusivity defined with thermal conductivity (�), specific

heat (()) and density ('). _, in the expression of the Bi number, is the heat transfer

coefficient. The time increment Δ =0.02s and pipe thickness 9.6mm are used here to define

the characteristic time ̃ and characteristic length hd, respectively. Fig. 14 shows 9 different

cases, in Bi-Fo space, considered in the sensitivity analysis. Note that the reference case, c1,

corresponds to the results shown in Section 3.2.1 with pipe material properties given in Table

1 and heat transfer coefficient equal to 15,000 W/m2K. It has to be pointed out that in order to

observe relevant differences between the two approaches compared in this section (1D and 3D

pipe wall models), a large parameter space of Bi-Fo numbers has been investigated even

though extreme values considered are non-physical in practical terms.

24

Fig. 14. Cases considered in the sensitivity analysis of heat transfer related parameters.

Symbols with labels mark 9 different sets of Fo and Bi numbers; c1 corresponds to case presented in Section 3.2.1. Sketches in right-side summarize wall thermal-thickness results

from Fig. 15.

Changes in wall thermal diffusivity induced by different Fo and Bi numbers result in

variations of wall thermal-thickness. A thermally-thin wall, for instance, undergoes similar

temperature fluctuations throughout the thickness. Hereon, a root mean square (rms) of

temperature fluctuations is used as a measure of fluctuations, and it is defined as the square

root of variance defined in Eq. (1). Fig. 15 presents the radial profiles of wall temperature

fluctuations, -6126��, normalized with the inner surface value, -6126���, for the 9 cases

considered. Such normalization allows comparing wall thermal-thickness among different

cases. In Fig. 15 lines and symbols denote results from 3D and 1D analyses respectively. Fig.

15a presents results of cases closer to the reference case (c1) and Fig. 15b results of more

distant cases (cases can be located with the use of Fig. 14). The results show that Fo number

has a bigger influence on the profile of normalized fluctuations. Flat profiles, which indicate

thermally-thin walls, develop for high Fo numbers. Conversely, low Fo numbers yield

temperature fluctuations localized at the surface, i.e., thermally-thick walls. The influence of

Bi number on the profile is, on the other hand, lower. These observations, summarized in Fig.

14, can be seen by comparing profiles of c6-c1-c8 (with same Bi) and c9-c1-c7 (with same

Fo). Fig. 15 also shows for almost all cases a perfect agreement between wall fluctuations

derived from 3D and 1D thermal analyses. However, the 1D results in c6 and c2 are slightly

higher than in 3D, therefore indicating that radial heat diffusion in 3D is slightly lower in

these two cases. This behavior is clearly observed in Fig. 15b for c3. Lower radial heat

diffusion in 3D analyses is given by the presence of hoop and axial heat fluxes. The heat, in

25

this case, flows in the other two directions and influences wall thermal-thickness. Moreover,

by monitoring wall heat fluxes it has been observed that radial ones decrease by 2 orders of

magnitude from c1 to c3, becoming of the same order of magnitude as axial and hoop heat

fluxes. However, in all other cases, radial heat fluxes prevail and dominate the thermal

behavior of the wall.

Fig. 15. Radial profiles of temperature fluctuations, -6126��, normalized with the

inner surface value, -6126���. Results given for 9 considered cases with different Fo and Bi numbers defined in Fig. 14. (a) cases near c1. (b) cases far away from c1. Lines and symbols denote results from 3D and 1D analyses respectively. See Fig. 16 for corresponding surface

temperature fluctuations values.

Fig. 16 shows inner surface temperature fluctuations, -6126���, normalized with the

fluid value (-.126�D4RIGHT� = 16.81 °C). Results are distributed according to the wall

thermal-thickness observed in Fig. 15: from thermally-thick (left) to thermally-thin (right) in

the horizontal axis of Fig. 16. Bars and dots represent 3D and 1D analyses results,

respectively. The studied cases span the whole range of possibilities: from practically non-

existing fluctuations in c5 to fully following the fluid in c3. Note that both, Bi and Fo,

numbers contribute to surface temperature fluctuations

influence observed when comparing the

same Bi. Nevertheless, these results also indicate that

correlate with surface temperature fluctuations

differences between 3D and 1D temperature fluctuations.

profile is observed in Fig. 15, 1D surface fluctuations

3D ones.

Fig. 16. Inner surface temperature fluctuationsvalue (-.126�D4RIGHT� = 16.81 °C)3D and 1D analyses results, respectively

wall thermal-thickness observed in

The special case c2 requires further

number indicating a very low heat capacity

are rapidly diffused through the wall. In the 1D model and with adiabatic outer surface, the

inner surface must then follow the fluid temperature. In the 3D model, however, heat

flow in other two directions so that this effect is less acute. Thus,

Fig. 16 tend to follow the fluid more closely in 1D than in 3D. This effect is

observable in c3 due to the very high

surface temperatures in both, 1D and 3D models.

It is also worth to mention the fact

actually increases the wall thermal-thickness, i.e. lowers the profile of temperature

fluctuations through thickness (see c7

has to be noted that, in absolute terms, cases

temperature fluctuations than cases with

profiles with surface values are lower.

contribute to surface temperature fluctuations. However, Bi number has a bigger

when comparing the results for c9-c1-c7 with same Fo and c8-c1-c6 with

these results also indicate that wall thermal-thickness does not

correlate with surface temperature fluctuations. Moreover, all cases except c2 do not show

differences between 3D and 1D temperature fluctuations. In case c2, where a flat normalized

1D surface fluctuations present a 50% increase with respect to

urface temperature fluctuations, -6126���, normalized with the fluid

C). All cases are defined in Fig. 14. Bars and dots represent 3D and 1D analyses results, respectively. Results distributed in horizontal axis according to

thickness observed in Fig. 15: from thermally-thick (left) to thermally(right).

The special case c2 requires further discussion. c2 (and also c3), has a very high

very low heat capacity (storage). Therefore, surface temperature changes

are rapidly diffused through the wall. In the 1D model and with adiabatic outer surface, the

inner surface must then follow the fluid temperature. In the 3D model, however, heat may

this effect is less acute. Thus, c2 surface fluctuations

tend to follow the fluid more closely in 1D than in 3D. This effect is however not

n c3 due to the very high Bi number which, together with very high Fo, enforces

1D and 3D models.

the fact that for a given Fo, an increase of Bi number

thickness, i.e. lowers the profile of temperature

c7-c1-c9 cases, c3-c2 and c4-c5 in Fig. 15). First of all it

oted that, in absolute terms, cases with higher Bi (for given Fo) deliver higher

cases with lower Bi (see Fig. 16). However, the normalized

are lower. It can be concluded that an increase of boundary layer

has a bigger

c6 with

do not show

where a flat normalized

0% increase with respect to

, normalized with the fluid Bars and dots represent

according to thick (left) to thermally-thin

very high Fo

. Therefore, surface temperature changes

are rapidly diffused through the wall. In the 1D model and with adiabatic outer surface, the

may

surface fluctuations in

not

, enforces

number

). First of all it

deliver higher

). However, the normalized

It can be concluded that an increase of boundary layer

27

conductivity not only increases the wall fluctuations but also its thermal thickness. For the

two extreme cases, c3 and c2, a slight increase of wall thermal-thickness in 3D with respect to

1D results can be observed (lower 3D profiles than 1D in Fig. 15-b). This indicates an

enhancement of heat diffusion in circumferential and axial directions versus the radial one.

This can be understood from the fact that at high Bi and in 3D, surface temperatures are

higher everywhere, therefore, increasing also circumferential and axial heat fluxes and, as

described above for case c3, lowering the actual profiles.

Finally, c5 has no temperature fluctuations at the surface because of very low Bi and

Fo numbers. This gives a high thermal resistance of the boundary layer with respect to the

solid media and a high heat capacity (storage). Although the normalized profile through

thickness for c5 shows very thermally-think wall, the absolute values are almost 0.

Fig. 17. PSDs comparison between fluid and 3D surface temperature fluctuations. All

cases are defined in Fig. 14.

A comparison between spectral analyses for fluid and 3D surface temperature

fluctuations is shown in Fig. 17. The theoretical PSD imposed in the fluid is perfectly

reproduced by c3, Fig. 17a. Very similar PSD profiles to c1 are obtained for c2 and c4 with

small differences at high frequencies. Also, levels of PSD for cases closer to the reference c1,

Fig. 17b, are noticed to be distributed in the same way as surface temperature fluctuations, in

Fig. 16. Note that PSD of c5 is not visible in the ranges considered in Fig. 17 due to low level

of its temperature fluctuations. Anyhow, these spectral analyses’ results are expected since the

28

area bellow normalized PSD corresponds to the variance of temperature fluctuations,

therefore, to square of -6126���. 4.2. Sensitivity analysis of stress fluctuations

In this section, stress fluctuations evolving in 3D pipe are compared to 1D ones where

only radial temperature gradients are assumed to be present in the pipe wall. The 1D thermal

stresses for a long hollow circular cylinder, in cylindrical coordinates, are given in Noda et al.

(2003):

:1�, � ? pq@rs t� @

uN B "6�′, �′C′uuw K uNruwN

uN�uxNruwN� B "6�′, �′C′uxuw y,

:;�, � ? pq@rsz @

uN B "6�′, �′C′uuw K uNOuwN

uN�uxNruwN�B "6�′, �′C′uxuw � "6�, �{,

:<�, � ? pq@rsz Es|

uxNruwN B "6�′, �′C′uxuw � "6�, �{,

(7)

where � and ! denote inner and outer radius respectively, * thermal expansion coefficient, +

Young modulus, , Poisson’s ratio, "6�, � represent radial temperature fluctuations of pipe

wall (solid) and ,}, in the axial stress definition, is equal to 1 and , for free-end pipe (~< ? ~D)

and fixed-end pipe (~< ? 0) conditions, respectively. It is worth to point out that 1D hoop and

axial stresses at the surface simplify to:

:���, � ? pq@rsz E

uxNruwN B "6�′, �′C′uxuw � "6�� , �{,

:���, � ? pq@rsz Es|

uxNruwN B "6�′, �′C′uxuw � "6��, �{. (8)

Therefore, for free-end pipe condition, surface axial stresses are found to be equal to hoop

stresses, i.e., ,}=1 in Eq. (8) and meaning perfect equi-biaxial surface stress fluctuations.

The correlation between surface stresses and temperature fluctuations depends on the

time-dependent integral in Eq. (8). The value of this integral proves to be low since surface

stresses and temperature fluctuations are almost linearly correlated and, therefore, correlation

between hoop and axial stresses should also be high for fixed-end conditions (,}=,). Further,

considering fixed-end conditions, the correlation between fluctuations and axial stresses will

be higher than with hoop stresses since the integral is, in this case, multiplied by additional

Poisson’s factor. These statements agree with results presented in Fig. 8 to Fig. 10 for 3D

surface stresses of the reference case (c1). However, it is not obvious which (1D) boundary

29

condition better resembles 3D stress behavior and under which conditions. Thus, it is of

interest also to assess the effects of 1D boundary conditions.

The aim of this section is to study the stress fluctuations’ behavior for different

thermal states of the pipe wall and to assess differences between 3D and 1D models. The

influence of different Bi and Fo numbers on wall stresses, in 3D and 1D, is studied at point

D4 RIGHT. The hoop stress fluctuations are first studied since they are representative of both,

free and fixed, 1D boundary conditions. Then, the correlation of axial and hoop stresses at the

surface is compared for 3D and 1D analyses employing fixed boundary conditions. Finally,

the radial profiles of axial stress fluctuations are studied for 3D and 1D analyses employing

the two boundary conditions.

Fig. 18 presents the radial profiles of hoop stress fluctuations, :;126��, normalized

with the inner surface value, :;126���, for selected cases. The wall thermal-thickness

represented by normalized -6126�� in Fig. 15 has a clear correspondence with normalized

:;126�� in Fig. 18. However, in Fig. 18 local minima of hoop stress fluctuations appear as the

wall becomes thermally-thinner. These are clearly visible for c2, c3 and c6 cases. The minima

from 3D and 1D results match for c6 but 1D stress profiles are over predicted at the outer

surface. The minima no longer match for c2 and c3 cases and the 1D profiles are clearly over-

predicted. However note that absolute values of 1D fluctuations in c2 and c3 cases is are

actually very low, as will be shown next.

Fig. 18. Radial profiles of hoop stress fluctuations :;126��, normalized with the inner

surface value, :;126���. Results given for selected cases defined in Fig. 14. Lines and symbols denote results from 3D and 1D analyses respectively. See Fig. 19 for corresponding

surface stress fluctuation values.

Fig. 19 shows inner surface hoop stress fluctuations, :;126���, with results distributed

according to wall thermal-thickness in the horizontal axis, as in Fig. 16. Bars and dots

represent, also here, 3D and 1D analyses results, respectively.

one can observe a considerable overlap between the two plots, except for c2 and c3 cases.

This indicates that surface temperature fluctuations (the second term of Eq.

stress) contribute much more to the hoop stress fluctuations than the radial temper

fluctuations, i.e., the wall thermal-thickness (the first term of Eq.

Therefore, again almost a linear correlation between surface tempe

stress fluctuations is expected in almost all considered cases (except c2 and c3).

In Fig. 19 the case c5, with zero

fluctuations as expected. However, c3 with highest temperature fluctuations undergoes

3D stress fluctuations than the reference case c1

analyses. Also small 1D stresses are obta

thin cases, as observed for c2 and c3 in 1D results (

effectively preclude volume changes and thus induce thermal stresses.

generation of surface stress fluctuations in 3D

the next section. It can be anticipated

proximity of material volumes at different temperature

temperature fluctuations levels but stress fluctuations are lower in c1.

obtained for c7 which shares same Fo

Fig. 19. Inner surface hoop stress14. Bars and dots represent 3D and 1D analyses results, respectively. Results distributed in

horizontal axis according to wall thermal(left) to thermally

represent, also here, 3D and 1D analyses results, respectively. Comparing Fig. 19 and Fig.

a considerable overlap between the two plots, except for c2 and c3 cases.

cates that surface temperature fluctuations (the second term of Eq. (8) for 1D hoop

stress) contribute much more to the hoop stress fluctuations than the radial temperature

thickness (the first term of Eq. (8) for 1D hoop stress).

Therefore, again almost a linear correlation between surface temperature fluctuations and

stress fluctuations is expected in almost all considered cases (except c2 and c3).

zero temperature fluctuations, presents also zero stress

. However, c3 with highest temperature fluctuations undergoes

reference case c1, and almost zero stresses arise in 1D

1D stresses are obtained for c2. It is worth to point out that in thermal

cases, as observed for c2 and c3 in 1D results (Fig. 15), there is no wall material that can

effectively preclude volume changes and thus induce thermal stresses. The explanation for the

generation of surface stress fluctuations in 3D analyses for c2 and c3 cases is given in detail

It can be anticipated here that nonzero stresses are caused by the laterial

material volumes at different temperatures. Moreover, c1 and c4, share

vels but stress fluctuations are lower in c1. Maximum stresses are

Fo number with c1 but one order of magnitude higher

hoop stress fluctuations, :;126���. All cases are defined in

. Bars and dots represent 3D and 1D analyses results, respectively. Results distributed in horizontal axis according to wall thermal-thickness observed in Fig. 15: from thermally

(left) to thermally-thin (right).

Fig. 16

a considerable overlap between the two plots, except for c2 and c3 cases.

for 1D hoop

ature

for 1D hoop stress).

rature fluctuations and

stress

. However, c3 with highest temperature fluctuations undergoes lower

It is worth to point out that in thermally-

), there is no wall material that can

xplanation for the

in detail in

laterial

Maximum stresses are

order of magnitude higher Bi.

All cases are defined in Fig. . Bars and dots represent 3D and 1D analyses results, respectively. Results distributed in

: from thermally-thick

31

4.2.1. Finding a relation between surface stresses and wall thermal-thickness

The analysis of surface stresses is performed next. A surface stress ratio is here

defined as the hoop stress fluctuations at the surface, :;126���, normalized with a case-

specific maximum stress:

:20= ? pq������uw�@rs , (9)

calculated with -6126��� values in Fig. 16. This maximum stress arises from thermally

induced volume change of surface material layer with fully clamped in-plane directions

movement but free in the surface’s normal direction. Higher surface temperature fluctuations

are usually assumed to generate higher thermal stresses. However this statement omits the

thermal state of the component and, as shown in results given in Fig. 16 and Fig. 19, this is

not necessarily always true. Thus, comparing computed surface stresses to the maximum

surface stress in fully clamped conditions, i.e., surface stress ratio, will reflect the influence of

wall thermal-thickness on stress levels since the effect of the actual loading at the surface is

suppressed with the normalization. Further, also the linear correlation factors between hoop

and axial stress fluctuations, r(θ-z), at the surface have been computed for all the cases

considered, Fig. 14. It is worth to recall that r(θ-z) is also a measure of stress equi-biaxiality

since mean stress effects have been eliminated. The correlations prove to be, in general, quite

high. In order to properly observe relevant effects, they are subtracted from the unity and

plotted in a log scale where 0 represents perfect equi-biaxial stress state.

The results of surface stress ratios and stresses’ correlations have been summarized in

Fig. 20. As in Fig. 16 and Fig. 19, results are distributed according to wall thermal-thickness

observed in Fig. 15: from thermally-thick (left) to thermally-thin (right) in the horizontal axis.

Bars and dots represent 3D and 1D analyses results, respectively. Both results, for surface

stress ratios and stress equi-biaxiality, prove to correlate with the wall thermal thickness.

32

Fig. 20. Top: Results of surface stress ratios (left vertical axis) defined as surface hoop

stress fluctuations, :;126���, normalized with case-specific maximum stress :20=, Eq. (9), and linear correlation factors between hoop and axial stress fluctuations at the surface r(θ-z) (right vertical axis) after being subtracted from the unity. Bars and dots represent 3D and 1D analyses results, respectively. Results distributed in horizontal axis according to wall thermal-

thickness. Down: Sketches summarizing wall thermal-thickness behavior of the cases.

Values of surface stress ratios are represented in left vertical axis of Fig. 20. Results

closer to 1 are those with very thermally-thick wall, i.e., when the temperature and stress

fluctuations are concentrated very close to the surface. As the wall becomes thermally-

thinner, no material volume is available to fully restrict the volume change of the surface

layer, hence, developing lower stress levels. 1D stresses are in general over predicted with

respect to 3D ones, except for c2 and c3 cases. This is also shown in Fig. 19 and the

explanation is given below. Although no apparent thermal differences turn out from 1D and

3D analyses for c1, c7, c8 and c9 (see Fig. 15 and Fig. 16), 1D surface stress fluctuations may

be up to 10% higher than 3D ones. This value represents the over-conservatism in 1D

methods due to combined effects from thermal and mechanical analyses. Also note that even

though c5 reaches fully clamped stress levels because of very surface localized fluctuations

33

(see Fig. 15b), stresses are really low since temperature fluctuations are almost 0 (see Fig. 16

and Fig. 19).

Before moving to the stress correlations, also given in Fig. 20, it is worth to comment

on the very thermally-thin wall cases c2 and c3 for which 3D stress fluctuations are higher

than 1D ones. These two are special cases where the effect of hoop and axial temperature

gradients are indeed relevant. Fig. 21 compares the 3D pipe wall thermal response between c1

and c3 cases, at specific time, when a fluid hot spot passes the point D4 RIGHT. The

fluctuations are very surface localized in c1, therefore, radial heat fluxes are predominant and

thermal stresses are mainly induced by above surface material volumes which restrict the

surface expansion. However, in c3 practically the whole wall thickness fluctuates and radial

heat fluxes are of same order than circumferential and axial heat fluxes. The expansion of the

wall thickness, with the shape of the fluid hot spot, is now restricted by adjacent axial and

circumferential material volumes. These are the 3D stress fluctuations that cannot be

reproduced with 1D models. In Fig. 20, 1D results for c2 and c3 denote practically 0 stress

fluctuations while 3D stress fluctuations demonstrate to be up to 40% of the maximum stress

value for the extreme c3 condition.

Fig. 21. Thermal response of 3D pipe wall for cases c1 and c3 at specific time when a

fluid hot spot passes the studied D4 RIGHT point. The results show distributions of pipe wall temperature fluctuations.

Linear correlation factors between hoop and axial stress fluctuations at the surface r(θ-

z) are represented in right vertical axis of Fig. 20 after being subtracted from the unity. It can

be clearly observed that also the stress correlations, which denote equi-biaxiality of stress

fluctuations, increase with wall thermal-thickness. The differences between 3D and 1D results

can be observed in Fig. 20 indicated by bars and dots, respectively. 1D correlations are in

general higher than 3D ones except for the three thermally-thin wall cases that presented clear

34

differences between 3D and 1D profiles of temperature fluctuations, i.e., c6, c2 and c3 in Fig.

15. These 1D correlations, as it is shown next, differ from 3D ones due to higher axial

stresses. This effect arises by the fixed boundary condition (BC) in axial direction which

proves to be stiff in these thermally-thin wall cases. However 3D correlations are, in general,

very high with the exception of c2 and c3 which present moderated values (r(θ-z)~0.8). Also

note that the 1D correlation for c4 is 1 which means perfectly equal axial and hoop stresses.

Nevertheless, it has been observed that these results are also representative of correlations

between temperature and stress fluctuations, i.e., r(F-z) and r(F-θ) in Fig. 8.

1D wall axial thermal stresses, defined in Eqs. (7), assume free (,}=1) or fixed (,}=,)

axial BC. Fig. 22 shows the implications of using one or the other pipe end condition by

comparing them to the 3D results. The normalized profiles of axial stress fluctuations,

:<126��, are shown for 4 different cases. From results in Fig. 22 one can see that, as the wall

becomes thermally-thinner, 1D axial stress fluctuations are over predicted (too stiff BC) or

under predicted using fixed or free boundary conditions respectively, when compared to 3D

axial stresses. For a very thermally-thick wall (c4) both 1D profiles and the 3D profile are

almost equivalent. In this case, surface stress correlations are also shown to be very high in

Fig. 20. For a very thermally-thin wall (c3), on the other hand, 1D and 3D profiles present

same minima levels with free end BC, Fig. 22. But this BC would give same axial and hoop

surface stresses while it has been shown that 3D correlation is 0.8 in Fig. 20.

Fig. 22. Radial profiles of axial stress fluctuations, :<126��, normalized with the inner

surface value, :<126���. Results shown for 3D and 1D analyses using fixed and free end pipe boundary conditions of selected cases defined in Fig. 14.

35

5. Conclusions

Numerical study has been performed on the investigation of thermal stresses

developing in a pipe wall under turbulent fluid mixing conditions downstream of a T-junction.

Experimental input data of a selected case study from the literature has been used to generate

thermal loads using a recently developed numerical approach. The computed loads have been

used as a thermal boundary condition in 3D analyses of piping, including heat transfer and

mechanical analyses.

Results of the first part of the study agree with previous observations in the literature.

Namely, it has been demonstrated that in the selected reference case study (i) the surface

stress fluctuations are almost linearly correlated with the surface temperature fluctuations, and

that (ii) the surface axial and hoop stress fluctuations, calculated on pipe cross sections away

from system boundaries, are almost equi-biaxial. In addition, new insights have been provided

for the inner surface stress state of a pipe when applying more realistic mechanical boundary

conditions. It has been shown that (iii) when further piping installation is modeled around the

pipe, a more realistic deformation of the pipe is obtained with regions of nonzero mean stress

values. At those regions, such a stress state is no longer equi-biaxial and may favor crack

initiation and propagation.

In the second part of the study sensitivity analyses have been performed to evaluate

the scope of previous findings by analyzing the results using different heat transfer related

parameters, namely the Fourier and Biot numbers. Also, a comparison between 1D and 3D

thermo-mechanical approaches has been performed and analyzed in terms of given results. It

has been demonstrated that (iv) both earlier findings, (i) and (ii), also hold for a wide range of

Fourier and Biot numbers around the reference case, and that (v) within this range the 1D

approach, although with a slight over prediction of the results, yields very similar results as

the 3D one. Eventually, it has also been shown that (vi) the wall thermal-thickness influences

the surface stress state: thicker the wall, more equi-biaxial the stress state.

Immediate future work will be dedicated to reproduce surface thermal loads from

other fluid flow patterns (wall jet and impinging jet). Also, further efforts will be devoted to

analyze the influence of different fluid power spectral density profiles on surface stresses and

to assess the thermal fatigue lifetime of a pipe wall under turbulent fluid mixing conditions.

36

Acknowledgments

The authors gratefully acknowledge the financial contribution of the Slovenian

Research Agency through the research program P2-0026.

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