Numerical Study on Conjugate Heat Transfer in Laminar Fully Developed Flow With Temperature...

Energy Conversion and Management 46 (2005) 633–654www.elsevier.com/locate/enconman

Numerical study on conjugate heat transfer in laminar fullydeveloped flow with temperature dependent thermal properties

through an externally heated SiC/SiC compositepipe and thermally induced stress

H€useyin Yapıcı *, Gamze Bas�t€urk, Bilge Albayrak

M€uhendislik Fak€ultesi, Erciyes €Universitesi, 38039 Kayseri, Turkey

Received 17 November 2003; received in revised form 4 February 2004; accepted 29 April 2004

Available online 19 July 2004

Abstract

This study presents numerical solutions of conjugate heat transfer in laminar, fully developed flow

through an externally heated pipe and the thermally induced stress under high temperature conditions. For

this purpose, a SiC/SiC composite and liquid metals (lithium and sodium) were considered as pipe material

and fluid, respectively. Various flow cases were considered as: the lithium and sodium flows with constantthermal properties, CTPs, and with temperature dependent thermal properties, TTPs. The calculations

were performed individually for a wide range of thermal conductivity of the solid (ks ¼ 10–100 W/m K

stepped by 10 W/m K) and various mean pipe inlet velocities (Um ¼ 0:01–0:02 m/s stepped by 0.002 m/s)

under both steady state and transient conditions. In order to keep the maximum relative temperature of the

solid within the interval, hs ¼ 190–200 K, also the temperature controlled heating case was performed.

Furthermore, a computer program, applying the SIMPSON integration method to the obtained temper-

ature distributions from the heat transfer calculations, has been developed to calculate numerically the

thermal stress distributions. The temperature difference ratios, TDRs, which are the ratios of the differenceof temperatures at the same point in both flow cases to the relative temperature at that point in the

temperature dependent thermal property cases, in the lithium flow case are lower than those in the sodium

flow case, and the averages of the TDRs in the solid, fluid and outlet regions are about 2% and 6–7% in the

lithium and sodium flow cases, respectively. Although the increase of Um substantially affects the relative

temperatures, it does not affect very much the effective thermal stress. The maximum effective thermal stress

ratios decrease exponentially with the increase of ks. As the radial distance increases, the effective thermal

stress ratios, having maximum values at the inner radius (r=D ¼ 0:4), reduce to minimum values at almost

* Corresponding author. Tel.: +90-352-437-49 01/32125; fax: +90-352-437-5784.

E-mail address: [email protected] (H. Yapıcı).

0196-8904/$ - see front matter � 2004 Elsevier Ltd. All rights reserved.

doi:10.1016/j.enconman.2004.04.009

mail to: [email protected]

634 H. Yapıcı et al. / Energy Conversion and Management 46 (2005) 633–654

r=D ¼ 0:445 and then increase. The results of this study would serve to determine the temperature distri-

butions and the thermally induced stresses in similar pipe flow applications.

� 2004 Elsevier Ltd. All rights reserved.

Keywords: Conjugate heat transfer; Pipe flow; Fully developed laminar flow; Thermal stress; Temperature controlled

heating; Computational fluid dynamics

1. Introduction

Heat transfer under laminar flow conditions in pipes is encountered in a variety of engineeringapplications. One of these applications is the heat exchangers in which conduction occurs in asolid tube and convection occurs in the fluid flowing within the outer surface. The term conjugateheat transfer is used to describe processes that involve variations of temperature within solids andfluids due to thermal interaction between the solids and fluids. Therefore, conduction and con-vection heat transfers must be simultaneously analyzed. In problems referred to as conjugated,because the thermal boundary conditions along the solid–fluid interface are not known a priori,the energy equations should be solved under the conditions of continuity in temperature and/orheat flux. In addition, in this type of heat transfer analysis, the two chief obstacles to a workabletheory are geometry and viscosity. The general theory of fluid motion is too difficult to enable theuser to attack arbitrary geometric configurations. It can be possible to apply merely numericaltechniques to arbitrary geometries. The second obstacle to theory is the action of viscosity, whichcan be neglected only in certain idealized flows. Therefore, a suitable numerical method and/orcomputational fluid dynamics (CFD) code is frequently used to solve the governing equations inthis field. The CFD code is the process by which fluid flow can be predicted through arbitrarygeometries, giving such information as flow speed, pressures, residence times, flow patterns etc.The main advantage of this approach is in its potential for reducing the extent and number ofexperiments required to describe such types of flow. In the past, many studies on conjugated heattransfer problems in pipe flow have been performed for several geometries and for differentboundary conditions. Some of them are given in Refs. [1–14].

Furthermore, non-uniform heating and/or cooling of a uniform material, or uniform heating ofnon-uniform materials cause thermal stresses that are proportional to the temperature differentialand the coefficient of thermal expansion of the material. Some of these thermal effects includethermal stress, strain and deformation. Therefore, thermally induced stresses are taken intoconsideration as important phenomena in many engineering applications. A thermal analysis isfirst conducted, providing the temperature distribution throughout the structure or product eitherat a steady state condition or as a function of time. The stress analysis is subsequently conductedusing the temperatures derived in the thermal analysis plus any additional structural ormechanical forces and constraints imposed on the structure or product. Recently, many re-searches have numerically solved the thermally induced stress analyses with heat transfer in a pipefor several pipe materials and fluids [15–18].

There is a strong demand to make high performance ceramic matrix composites (CMCs) foradvanced energy systems such as nuclear fusion reactors and advanced gas turbine engines.Among them, the silicon carbide composite (SiC/SiC), which is also known as a very low

Nomenclature

CP specific heatCTC constant thermal conductivityCTP constant thermal propertyD diameter of pipeE modulus of elasticityk thermal conductivityL length of pipeP pressureq0 value of heat flux per unit areaq00 heat flux per unit arear radial coordinateri inner radius of pipero outer radius of pipet timeT temperatureTDC temperature dependent thermal conductivityTDR temperature difference ratioTCH temperature controlled heatingTTP temperature dependent thermal propertyUDF User Defined Functionu velocity component in axial directionv velocity component in radial directionx axial coordinate

Greek symbolsb thermal expansion coefficient/ any arbitrary variablel dynamic viscositym Poisson’s ratioh relative temperature (T � Tm)q densityr thermal stressr� effective thermal stress ratio

Subscripts

avg averageeff effectivef fluidm meanmax maximum

H. Yapıcı et al. / Energy Conversion and Management 46 (2005) 633–654 635

r radials solidx longitudinalh tangential


activation material, is the primary composite material being evaluated and developed world-widefor fusion structural applications because of their advantages [19–24]: (1) high specific strength,(2) high temperature strength, (3) fracture toughness compared with monolithic ceramics, (4)insulating property (prevent energy loss by conduction), (5) controllability to improve conduc-tivity, (6) low induced radioactivity under nuclear environments and (7) chemical compatibilitywith liquid metal coolants (such as pure Li, liquid sodium, eutectic lead-lithium, Flibe etc.). Thesecharacteristics are beneficial to achieve high plant efficiency in nuclear fusion systems with higherreliability on safety, which makes fusion an attractive energy option for the future. However, theirbehaviors and performances at high temperatures and under irradiation are still not well knownand need to be better characterized. Although these composites are already widely used in theaerospace industry, they are relatively new materials for fusion applications, and a number of keyissues still remain to be solved. Because of its brittle nature, SiC is often considered in a fiberreinforced ceramic matrix composite form. This enables its use as a structural and/or pressurevessel material subjected to high thermal and pressure stresses. SiC/SiC pipes and ducts can bemanufactured at reasonable cost. The thermophysical properties of SiC/SiC composites aredependent on the fabrication procedure.

Liquid metal coolants have several attractive thermophysical properties that would make thempotentially interesting candidates for use in the liquid metal blankets of fusion reactors. Amongthem, pure Li (melting point 454 K and boiling point 1620 K) is used for the dual purposes ofcoolant and tritium breeder. Liquid sodium metal (melting point 371 K and boiling point 1156 K)has properties that make it suitable as a coolant in some nuclear reactors. Furthermore, becausesodium has a very high boiling point, reactor coolant loops can be operated at high temperatureand at essentially atmospheric pressure without boiling, and no heavy pressure vessel is required.The high coolant temperature also leads to high temperature, high pressure steam and, thus, highplant efficiency. Finally, sodium, unlike water, is not corrosive to many structural materials.

2. Geometry and mathematical modeling

2.1. Geometry of the pipe

In this study, numerical solutions of conjugate heat transfer in laminar, fully developed flowthrough an externally heated pipe under high temperature conditions were analyzed with a CFDcomputer code. For this purpose, a SiC/SiC composite that is resistant to high temperature con-ditions and liquid metals (lithium and sodium) that have very high boiling points were considered aspipe material and fluid, respectively. The two dimensional axisymmetric model of this consideredpipe is shown in Fig. 1. As is known, temperature differences occurring in a solid subjected to athermal process may cause thermal stress because of non-uniformities of heating, cooling and/or

Liquid

x

r ro

ri

L q”

Um

Tm

CL

Solid

Fig. 1. Coordinate system and two dimensional axisymmetric model of the pipe (the dimensions are not in scale).


thermal properties of the solid material. This study also considers this thermal stress, which isproportional to the temperature differential and the coefficient of thermal expansion of the material.

As is apparent from the above explanations, in this analysis, three phenomena are considered asfollows: (i) conduction inside the pipe materials, (ii) convection from the pipe surface to thesurrounding fluid and (iii) thermally induced stress in the pipe. The analysis is based on twodimensional continuity, momentum and energy equations.

The following important assumptions are made in the present analysis:

• The flow is two dimensional, laminar and incompressible.• The thermal conductivity of the pipe material does not vary with temperature (constant thermal

conductivity, CTC, case).• The slip condition is assumed at the pipe walls.• The modulus of elasticity (E), thermal expansion coefficient (b) and Poisson’s ratio (m) do not

vary with temperature.

2.2. Mathematical modeling

The governing equations for the flow and heat transfer in the flow region (0 < x6 L and06 r6 ri) can be written as follows:

continuity :ouox

þ 1

roðr � vÞor

¼ 0; ð1Þ

x-momentum : qouot

�þ 1

ro

oxðruuÞ þ 1

ro

orðruvÞ

�

¼ � oPox

þ 1

ro

oxrl 2

ouox

�� 2

3

ouox

�þ ov

orþ v

r

��þ 1

ro

orrl

ouor

��þ ovox

��ð2aÞ

r-momentum : qovot

�þ 1

ro

oxðruvÞ þ 1

ro

orðrvvÞ

�¼ � oP

orþ 1

ro

oxrl

ovox

��þ ou

or

��

þ 1

ro

orrl 2

ovor

�� 2

3

ouox

�þ ov

orþ v

r

�� 2l

vr2þ 2

3

lr

ouox

�þ ov

orþ v

r

�

ð2bÞ


energy :o

otððqCP ÞfhfÞ þ

o

oxððqCP ÞfuhfÞ þ

1

ro

orððqCP ÞfvhfÞ

¼ o

oxkf

ohf

ox

� �þ 1

ro

orkfr

ohf

or

� �ð3Þ

where hf is the relative temperature, described as hfðx; r; tÞ ¼ Tfðx; r; tÞ � Tm.Boundary conditions: At the pipe inlet (x ¼ 0),

uð0; r; tÞ ¼ 2 � Um � 1

�� r2

r2i

�; vð0; r; tÞ ¼ 0 and hfð0; r; tÞ ¼ 0 �C: ð4a–cÞ

Since the profiles of velocity and temperature are both symmetric with respect to the axis of thepipe, the relevant boundary conditions at the pipe axis ðr ¼ 0Þ are

o/ðx; 0; tÞor

¼ 0; ð4dÞ

where / is any arbitrary variable.At the inner surface of the pipe (r ¼ ri),

vðx; ri; tÞ ¼ 0; ð4eÞ

and the no slip condition is assumed as follows:

uðx; ri; tÞ ¼ 0: ð4fÞ
The governing conduction equation in the solid region (06 x6 L and ri 6 r6 ro) can be written
as follows:

ohs

ot¼ 1

ðq � CP Þs

� rðks � rhsÞ; ð5Þ

where hsðx; r; tÞ ¼ Tsðx; r; tÞ � Tm.Boundary conditions: At the pipe inlet and exit planes,

ohsð0; r; tÞox

¼ 0 andohsðL; r; tÞ

ox¼ 0: ð6a;bÞ

At the outer surface of the pipe (r ¼ ro), a uniform heat flux is assumed as follows:

q00 ¼ ks

ohsðx; ro; tÞor

; ð6cÞ

where q00 ¼ q0, and in the temperature controlled heating, TCH, case, the value of q0 is regularlydecreased to one tenth its first value and increased again to its first value to keep the maximumtemperature of the solid within the assumed interval.

At the inner surface of the pipe (r ¼ ri), it is assumed that the temperature and heat flux at thesolid–fluid interface are continuous, and so, the relevant boundary conditions can be written asfollows:

hsðx; ri; tÞ ¼ hfðx; ri; tÞ and ks

ohsðx; ri; tÞor

¼ kf

ohfðx; ri; tÞor

: ð6d; eÞ


Initial conditions:

hfðx; r; 0Þ ¼ hsðx; r; 0Þ ¼ 0 �C or Tfðx; r; 0Þð ¼ Tsðx; r; 0Þ ¼ TmÞ: ð7a;bÞ

The governing thermal stress equations for hollow cylinders: To obtain the thermal stress dis-tribution, it is assumed that the pipe reaches thermal equilibrium at the end of each time interval.Therefore, three thermal stress components in the tangential (h), radial (r) and longitudinal (x)directions are calculated at each time interval. The general equations for hollow cylinders aregiven as follows [25]:

rhðrÞ ¼E � b

ð1 � mÞ � r2� r2 þ r2

i

r2o � r2

i

�Z ro

ri

T ðrÞ � r � dr�

þZ r

ri

T ðrÞ � r � dr � T ðrÞ � r2

�; ð8aÞ

rrðrÞ ¼E � b

ð1 � mÞ � r2� r2 � r2

i

r2o � r2

i

�Z ro

ri


�Z r

ri

T ðrÞ � r � dr�; ð8bÞ

rxðrÞ ¼E � b1 � m

� 2

r2o � r2

i

�Z ro

ri


� T ðrÞ�: ð8cÞ

The effective stress according to the Von-Mises theory [25] is calculated by:

reffðrÞ ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi½rhðrÞ�2 þ ½rrðrÞ�2 þ ½rxðrÞ�2 � ½rhðrÞ � rrðrÞ þ rhðrÞ � rxðrÞ þ rrðrÞ � rxðrÞ�

q: ð8dÞ

As is apparent from these equations, the thermally induced stresses in a solid are proportionalto the temperature differences in the solid. Their values will, therefore, vary (increase or decrease)depending on the applied heat flux q00. In order to obtain the effective thermal stress distributionsindependently of the physical properties (E, b and m) of the material, the effective thermal stressratio, r�ðrÞ, was expressed as follows:

r�ðrÞ ¼ reffðrÞE�b1�m

: ð8eÞ

2.3. Thermal properties and simulation values

Temperature dependent thermal properties, TTPs, of the considered fluids:For liquid Li at T ¼ 455–1500 K [26],

qðT Þ ¼ 278:5 � 0:04657 � T þ 274:6 � 1

�� T

3500

�0:467

; ð9aÞ

CP ðT Þ ¼ 4754 � 0:925 � T þ 2:91 � 10�4 � T 2; ð9bÞ

kðT Þ ¼ 22:28 þ 0:05 � T � 1:243 � 10�5 � T 2; ð9cÞ


lðT Þ ¼ exp

�� 4:164 � 0:6374 � lnðT Þ þ 292:1

T

�; ð9dÞ

at T ¼ 650 K, constant thermal properties, CTPs: q ¼ 497:708 kg/m3, CP ¼ 4276 J/kg K,k ¼ 49:528 W/m K, and l ¼ 3:925 � 10�4 kg/m s.

For liquid Na at T ¼ 371–1154 K,

qðT Þ ¼ 1016 � 0:235 � T ½27�; ð10aÞ

CP ðT Þ ¼ 1622 � 0:784 � T þ 4:147 � 10�4 � T 2 ½27�; ð10bÞ

kðT Þ ¼ 124:67 � 0:11381 � T þ 5:5226 � 10�5 � T 2 � 1:1842 � 10�8 � T 3 ½28�; ð10cÞ

lðT Þ ¼ exp

�� 6:4406 � 0:3958 � lnðT Þ þ 556:835

T

�½29�; ð10dÞ

at T ¼ 700 K, the CTPs: q ¼ 851:5 kg/m3, CP ¼ 1276 J/kg K, k ¼ 68:002 W/m K, and l ¼ 2:644 �10�4 kg/m s.

For SiC/SiC composite, q ¼ 3000 kg/m3, CP ¼ 700 J/kg K, and k ¼ 10; 20; . . . ; 100 W/m K.Simulation values: D ¼ 0:05 m, ri ¼ 0:4 � D, ro ¼ 0:5 � D, L ¼ 10 � D, q0 ¼ 105 W/m2, Tm ¼ 500

K, Um ¼ 0:01; 0:012; . . . ; 0:02 m/s, and t ¼ 0; 0:05; . . . ; 250 s.

3. Numerical results

3.1. Calculational tools and procedure

A suitable computer code can be used to solve numerically the governing equations (1)–(3) and(5) along with the boundary and initial condition equations (4), (6) and (7) by a finite differencemethod. For this work, the FLUENT 5.3 program [30] was chosen as the computer code becauseof the ease with which the analysis model can be created and because the software allows users tomodify the code for special analysis conditions through the use of user subroutines. The FLU-ENT computer code uses a finite volume procedure to solve the Navier–Stokes equations of fluidflow in primitive variables such as u-velocity, v-velocity and pressure.

In this study, various flow cases were considered as follows:

(a) The lithium flows with the CTPs (at 650 K) and with the TTPs given by Eqs. (9).(b) The sodium flows with the CTPs (at 700 K) and with the TTPs given by Eqs. (10).

The calculations were performed individually for wide ranges of thermal conductivity of thepipe material (ks ¼ 10–100 W/m K stepped by 10 W/m K) and mean pipe inlet velocities(Um ¼ 0:01–0:02 m/s stepped by 0.002 m/s) under both steady state and transient conditions. Theresults of the steady state calculations are valid also for any solid material having a thermalconductivity in this range. In addition, in order to keep the maximum temperature of the solidwithin the desired interval, an interpreted UDF (user defined function) file, which supplies reg-


ularly increasing and decreasing heat flux (see Appendix A), was introduced to the preparedFLUENT case file as the heat flux input condition (TCH case). For this case, the desired relativetemperature interval is assumed as hs ¼ 190–200 K. The results of the conduction heat transfercalculations in the solid were checked by using the HEATING7 [31] computer code with the innersurface temperature boundary conditions obtained from the conjugate heat transfer calculations.Furthermore, a computer program, applying the SIMPSON integration method to the obtainedtemperature distributions from the heat transfer calculations, has been developed to calculatenumerically the thermal stress distributions defined by Eqs. (8a–e).

3.2. Temperature distributions

In the calculations, consideration of the SiC/SiC composite with temperature dependentthermal conductivity, TDC, was also desired. However, suitable TDC expressions for the thermalconductivities in the range considered in this study are not found in the literature pertaining to theSiC/SiC composites because of the fabrication dependent thermal conductivity of SiC/SiC com-posites. Nevertheless, in order to compare the results of at least one thermal conductivity value inboth thermal conductivity cases, it is assumed that the thermal conductivity decreases linearlywith temperature. Thus, as an example, it was assumed as follows:

Table

Temp

Flo

Um

ks [W

Reg

Soli

Flu

Out

ksðT Þ ¼280

3� 1

15� T ðT in KelvinÞ: ð11Þ

The average of these TDCs between T ¼ 500 and 800 K corresponds almost to ks ¼ 50 W/m Kin the CTC case. The comparison of the results in both cases (ksðT Þ and ks ¼ 50 W/m K) bringsout that the relative temperatures in the TDC case are higher by only about 1% relative to those inthe CTC case. This value is a negligible quantity. Therefore, the calculations were performed forthe CTC case.

In order to determine the effects of the variations of the thermal properties of the fluid withtemperature on the heat transfer, the conjugate heat transfer calculations were also performed forthe flow with the CTP. The temperature differences between the average temperatures in someregions of the pipe in both flow cases are given in Table 1 for the lower and upper limit values ofUm and ks as temperature difference ratio, TDR, which is the ratio of the difference of tempera-tures at the same point in both flow cases to the relative temperature at that point in the TTP case.This ratio can also be defined mathematically as follows:

1

erature difference ratios, TDRs [%], calculated by using the average temperatures of relevant regions

w Lithium Sodium

[m/s] 0.01 0.02 0.01 0.02

/m K] 10 100 10 100 10 100 10 100

ion

d 2.2 2.6 1.2 1.7 5.7 6.2 6.6 7.7

id 2.3 2.3 2.2 2.2 6.7 6.7 8.1 8.1

let 1.6 1.6 2.2 2.2 5.3 5.3 7.0 7.0

Fig. 2

lithium


TDR ¼ T jin CTP � T jin TTP

T jin TTP � Tm

� 100: ð12Þ

As is apparent from this table, the TDRs in the lithium flow case are lower than in the sodiumflow case, and the averages of the TDRs in the regions are about 2% and 6–7% in the lithium andsodium flow cases, respectively. If it is assumed that these values are a negligible quantity, thecalculations can be performed with the assumption of CTP. In this section, only the resultsbelonging to the TTP cases are considered and discussed. The results in the CTP case can be easilyobtained by using the values in the TTP cases and Table 1.

Fig. 2 shows the increases of the maximum relative temperatures in the solid for the variousUms, depending on time. The increases of the average relative temperatures at the outlet are also

0 50 100 150Time [s]

0

50

100

150

200

250

300

350

[K]

0.0100.0120.0140.0160.0180.020

(a)

θ

Um [m/s]

ks=10 W/m-K

0 50 100 150Time [s]

0

50

100

150

200

250

300

350

[K]

θ

ks=100 W/m-K

0 50 100 150 200 250Time [s]

0

50

100

150

200

250

300

350

400

450

500

550

600

[K]

(b)

θ

ks=10 W/m-K

0 50 100 150 200 250Time [s]

0

50

100

150

200

250

300

350

400

450

500

550

600

[K]

θ

ks=100 W/m-K

. Variations of the maximum relative temperatures in the solid for the various Ums, depending on time for (a) the

and (b) sodium flow cases.

0 50 100 150Time [s]

0

50

100

150

200

250

300

350

[K]

0.0100.0120.0140.0160.0180.020

θ

(a)

Um [m/s]

0 50 100 150 200 250Time [s]

0

50

100

150

200

250

300

350

400

450

500

550

600

[K]

θ

(b)

0.010 0.012 0.014 0.016 0.018 0.020Um [m/s]

125

150

175

200

225

250

275

300

[K]

θ

0.010 0.012 0.014 0.016 0.018 0.020Um [m/s]

250

300

350

400

450

500

550

[K]

θFig. 3. Variations of the average relative temperatures at the outlet for the various Ums depending on time (ks ¼ 10 W/

m K) and the sub-graphics show the variations of these temperatures in the steady state, depending on Um, for (a) the

lithium and (b) sodium flow cases.


plotted in Fig. 3. The times to converge to steady state (the converging times) for the Ums aredetermined approximately from these figures. These converging times decrease from about 125 to100 s and from about 200 to 125 s with the increase of Um in the lithium and sodium flow cases,respectively. As apparent from Fig. 3 (especially the sub-figures), increasing Um substantiallyaffects the average relative temperatures at the outlet. It can be easily seen in the sub-figures thatthese relative temperatures decrease exponentially from 288 to 147 K and from 539 to 269 K withthe increase of Um in the lithium and sodium flow cases, respectively.

Figs. 4 and 5 exhibit the variations of the maximum relative temperatures in the solid for thelithium and sodium flow cases, respectively, depending on Um and ks. The maximum effectivethermal stress ratio distributions presented in these figures will be interpreted in Section 3.3. Onecan see in these figures that the increase in Um has much more effect on the relative temperaturethat in ks and that the relative temperatures in the lithium flow case are lower than those in thesodium flow case. It is also seen that in the lithium flow case, the maximum relative temperaturesvary in the range of 166–354 K, whereas in the sodium flow case, this range is between 283 and607 K.

The relative temperature profiles in both the solid and fluid regions are plotted in Figs. 6 and 7for the lower and upper limit values of Um and ks in the lithium and sodium flow cases as twodimensional (axial and radial) contour lines, respectively. As the distance from the inlet along thepipe length increases, the relative temperatures increase due to the uniform heat supplied from theouter surface of the pipe. These contours reach considerably high values near the outer surfaceand extend further inside the fluid region, and no radial uniformity is observed. The rapid change

0

100

200

300

400

500

600

700

1020

3040

5060

7080

90100

k s[W

/m-K]0.010

0.012

0.014

0.016

0.018

0.020

Um [m/s]

θ[K

]

0

4

8

12

16

20

24

28

32

36

40

1020

3040

5060

7080

90100

k s[W

/m-K]0.010

0.012

0.014

0.016

0.0

0.018

20

Um [m/s]

The

rmal

Stre

ssR

atio

[K]

Fig. 4. Variations of the maximum relative temperature and the maximum effective thermal stress ratio distributions in

the solid for the lithium flow case depending on Um and ks.


in temperature gradient occurs in the solid close to the inner surface. Furthermore, the effects ofUm and ks on the heat transfer and the relative temperature profiles in both regions are moreobviously seen in these figures. As expected, the relative temperatures in the solid regions decreasewith the increases of ks and Um, and as mentioned above, the increase of Um reduces the relativetemperatures more than the increase of ks.

In the temperature controlled heating case: In some pipe flow applications, it is frequently re-quired that a pipe is heated by keeping the temperature of solid and/or fluid within a certain

0

100

200

300

400

500

600

700

1020

3040

5060

7080

90100

k s[W

/m-K]0.010

0.012

0.014

0.016

0.018

0.020

Um [m/s]

θ [Κ

]

0

4

8

12

16

20

24

28

32

36

40

1020

3040

5060

7080

90100

k s[W

/m-K]0.010

0.012

0.014

0.016

0.018

0.020

Um [m/s]

The

rmal

Stre

ssR

atio

[K]

Fig. 5. Variations of the maximum relative temperature and the maximum effective thermal stress ratio distributions in

the solid for the sodium flow case depending on Um and ks.


desired interval along a certain time. Hence, this type of application involves controlled heating ofthe pipe. In this study, the TCH case with the assumed relative temperature interval hs ¼ 190–200K is also considered, and in order to investigate this case, the externally applied heat flux on thepipe wall is regularly increased and decreased by means of the prepared interpreted UDF file,which performs the heat flux regulating process (see Section 2.2) by controlling the maximumrelative temperature of the solid.

0 1 2 3 4 5 6 7 8 9 10x/D

0.0

0.1

0.2

0.3

0.4

0.5

r/D

5050

5050

50

100100

100

100100

150

200200

200200

250

250

k s=10

0W

/m-K

0 1 2 3 4 5 6 7 8 9 10x/D

0.0

0.1

0.2

0.3

0.4

0.5

r/D

50

50

50

50 100

100100

100100100

100 150 150150150150

150

k s=10

0W

/m-K

0 1 2 3 4 5 6 7 8 9 10x/D

0.0

0.1

0.2

0.3

0.4

0.5

r/D

5050

5050

5050

100100100

100100

100 100100100100100100 100 100100 100

150150150 150150150150150150 150150150150

20

k s=10

W/m

-K

Um= 0.02 m/s

0 1 2 3 4 5 6 7 8 9 10x/D

0.0

0.1

0.2

0.3

0.4

0.5

r/D

50

5050

5050 505050

100

100

100100 100100

150

150

150150

200200200

200200 200200200 250

250250

250 300300 300

350

k s=10

W/m

-K

Um= 0.01 m/s

Fig. 6. The relative temperature contours for the lithium flow case.


Fig. 8 exhibits how the relative temperatures in the solid and at the outlet vary in the TCH casefor the lithium flow cases with Um s of 0.01 and 0.02 m/s depending on time. As is apparent from

0 1 2 3 4 5 6 7 8 9 10x/D

0.0

0.1

0.2

0.3

0.4

0.5

r/D

5050

50

50

100100100

100100100100100100100100100

150150

150150

200

200

250

250300 300300

350350

350

400

400400

450450450450

500

500500550550550

k s=10

W/m

-K

Um= 0.01 m/s

0 1 2 3 4 5 9 10x/D

0.0

0.1

0.2

0.3

0.4

0.5

r/D

5050

505050 100

100

100

100 150150

150150

150150150150150 150150150

200200

200200200

200200 250

250250

250250250250250250

300300

k s=10

W/m

-K

Um= 0.02 m/s

0 1 2 3 4 5 6 7 8 9 10x/D

0.0

0.1

0.2

0.3

0.4

0.5

r/D

505050

505

505050

100100

200200 250

250

250

300

350350350

400

450450450 500

500500

500

k s=10

0W

/m-K

0 1 2 3 4 5 6 7 8 9 10x/D

0.0

0.1

0.2

0.3

0.4

0.5

r/D

5050

100

150150

150

150

200200

200

250

k s=10

0W

/m-K

6 7 8

Fig. 7. The relative temperature contours for the sodium flow case.


this figure, the reaching time to the upper limit of the relative temperature of the solid increaseswith the increase of Um, and the relative temperature profiles continue as periodic curves after the

0 5 10 15 20 25 30 35 40 45 50 55 60 65 70Time [s]

0

20

40

60

80

100

120

140

160

180

200

[K]

max in the solid

max at the outlet

avg at the outlet

θ

θθ

θ

Um=0.01 m/s

0 5 10 15 20 25 30 35 40 45 50 55 60 65 70Time [s]

0

20

40

60

80

100

120

140

160

180

200

[K]

θ

Um=0.02 m/s

Fig. 8. Variations of the relative temperatures in the temperature controlled heating case for the lithium flow case

depending on time.


relative temperature reaches this upper limit. The periods of these profiles grow longer with time(in the case of Um ¼ 0:02 m/s after 70 s). It can be easily read from this figure that these reachingtimes are 42 and 52 s in the cases of Um ¼ 0:01 and 0.02 m/s, respectively. It also appears in thisfigure that the maximum and average relative temperatures at the outlet reach about 180 and 177K in the case of Um ¼ 0:01 m/s and about 151 and 137 K in the case of Um ¼ 0:02 m/s, respec-tively.

3.3. Thermal stress distributions

The thermally induced stresses, numerically calculated with Eqs. (8a–d) under the thermalequilibrium assumption mentioned in Section 2.2, are plotted as the effective thermal stress ratio,r�ðrÞ, defined with Eq. (8e). Figs. 4 and 5 also present the variations of the maximum effective


thermal stress ratios in the solid for the lithium and sodium flow cases, respectively, dependingon Um and ks. Some specific maximum effective thermal stress ratio values taken from thesefigures for the lower and upper limit values of Um and ks are as follows: while they decreaseexponentially from �36 to �9 K and from �35 to �8 K in the cases of Um ¼ 0:01 and 0.02 m/sfor the lithium flow case with the increase of the thermal conductivity, respectively, for thesodium flow case, these ranges are from �35 to �8.5 K and from �34 to �7 K. It is apparentfrom these figures and values that although the maximum effective thermal stress ratios in bothflow cases (the lithium and sodium flows) decrease exponentially rapidly with the increase of ks,they are not very much affected by the increase of Um and the fluid type because the effects of Um

and fluid type on the relative temperature differences in the radial direction are quite low in theconsidered cases.

The calculations bring out that the effective thermal stresses at the inner radius, ri, reach thehighest values for all axial planes (especially the inlet plane) in the solid. In order to exhibitdthe variations of the effective thermal stress ratio distributions at the inlet plane depending on theradial distance, they are plotted in Fig. 9 for Ums ¼ 0:01 and 0.02 m/s in the lithium flow casedepending on the dimensionless radial distance and the thermal conductivity of the solid. One canalso see in this figure that the effect of the increase of Um on the effective thermal stress ratio isquite low. Because of the fact that the tendencies of the effective thermal stress ratio profiles inboth flow cases are similar to each other, the variations in the sodium flow case are not plotted. Inall investigated cases, as the radial distance increases, the effective thermal stress ratios having,maximum values at the inner radius (r=D ¼ 0:4), reduce to minimum values at almost r=D ¼ 0:445and then increase. The reason for this is the temperature gradient variation in the solid. Theseeffective thermal stress ratios reduce exponentially with the increase of ks while almost notchanging their shapes. Furthermore, the variations of the effective thermal stress ratio distribu-tions inside the solid are depicted in Fig. 10 for ks ¼ 10 W/m K and Um ¼ 0:01 m/s in the lithiumflow case, depending on the dimensionless axial and radial distances. As is apparent from thisfigure, the effective thermal stress ratios decrease quasi-linearly also with the increase of axialdistance.

The results of the effective thermal stress ratios would serve to determine the thermallyinduced stresses occurring in the pipe during the heating process for a wide spectrum of SiC/SiC composites having the thermal conductivities considered in this study. In addition, if theflow with the CTP case is considered, the effective thermal stress ratios will be per q0. For thephysical properties (E, b and m) of any SiC/SiC composite having a thermal conductivity in theconsidered range, the effective thermal stress, reff , can be easily calculated with Eq. (8e). Theeffective thermal stress values calculated for the maximum effective thermal stress ratios inFigs. 4 and 5 may exceed the yield stress of selected SiC/SiC composites. As is known, whenthe effective thermal stresses exceed the yield stress at the corresponding temperature, plasticdeformation occurs. Moreover, the thermally induced stresses occurring in the solid pipe(especially at the points where they reach the maximum values) may lead to micro-cracks, adecrease in strength and fatigue life and possibly catastrophic failure. They may also remain inthe structure or product as residual stresses and algebraically add to stresses developed underservice and/or extreme loading conditions. Therefore, care must be taken whether the ther-mally induced stress arising during the heating process exceeds the yield stress of the con-sidered solid material or not.

0

10

20

30

40

1020

3040

5060

7080

90100

k s[W

/m-K]0.40

0.42

0.44

0.46

0.48

0.50

r/D

The

rmal

Stre

ssR

atio

[K]

Um=0.02 m/s

0

10

20

30

40

1020

3040

5060

7080

90100

k s[W

/m-K]0.40

0.42

0.44

0.46

0.48

0.50

r/D

The

rmal

Str

ess

Rat

io[K

]

Um=0.01 m/s

Fig. 9. Variations of the effective thermal stress ratio distributions at the inlet plane in the solid for the lithium flow case

depending on dimensionless radial distance and ks.


3.4. Comparison with similar previous studies

The thermally induced stresses in an externally uniformly heated pipe subjected to fullydeveloped laminar flow under low temperature conditions were analyzed numerically for variousfluids and pipe materials in previous studies [15,16,18]. The numerical thermal stress analysis wasalso performed for pulsating flow in Ref. [17] and also for a non-uniform heat flux case in Ref.[15]. In these studies, water and steel were used as fluid and solid materials with constant thermal

0

10

20

30

40

01

23

45

67

89

10

x/ D0.4

0.420.44

0.460.48

0.5

r/D

The

rmal

Stre

ssR

atio

[K]

Fig. 10. The effective thermal stress ratio distributions inside the solid for the lithium flow case versus the dimensionless

axial and radial distances (ks ¼ 10 W/m K and Um ¼ 0:01 m/s).


properties, respectively. In Ref. [18], also coolanol-25 and mercury as fluid and copper as pipematerial are considered.

The main differences of the present study from Refs. [15–18] are as follows:

• The SiC/SiC composite is used as the pipe material.• The lithium and sodium with temperature dependent thermal properties are selected as the

fluid.• The flows are under high temperature conditions.• The calculations are performed for a wide thermal conductivity range belonging to the SiC/SiC

composites.

In spite of these differences, the tendencies of the temperature and thermal stress profiles in thisstudy are similar to those in Refs. [15,16,18] because the flow types are the same. However, thethermal stress values in this study evidently cannot be compared with the present references due tothe fact that they were calculated by taking into account independent pipe material properties.Furthermore, the fluid and solid properties in the present and reference studies are quite differentfrom each other.

4. Conclusions

The main conclusions derived from this study may be listed briefly as follows:

• The relative temperatures in the TDC case are higher by only about 1% than those in the CTCcase.


• The averages of the TDRs in the regions are about 2% and 6–7% in the lithium and sodiumflow cases, respectively.

• The increase of Um

(1) reduces the converging times from about 125 to 100 s and from about 200 to 125 s in thelithium and sodium flow cases, respectively,

(2) substantially affects the average relative temperatures at the outlet and exponentially lowersthem from 288 to 147 K and from 539 to 269 K in the lithium and sodium flow cases,respectively,

(3) reduces the relative temperatures in the solid more than the increase of ks,(4) increases the reaching time from 42 to 52 s in the TCH case and(5) does not, however, affect the effective thermal stress very much.

• The maximum effective thermal stress ratios decrease exponentially from �36 to �9 K andfrom �35 to �8 K in the cases of Um ¼ 0:01 and 0.02 m/s, for the lithium flow case with theincrease of ks, respectively. For the sodium flow case, these ranges are from �35 to �8.5 Kand from �34 to �7 K.

• The effective thermal stress is also not very much affected from the considered flow cases.• As the radial distance increases, the effective thermal stress ratios, having maximum values at

the inner radius, reduce to minimum values at almost r=D ¼ 0:445 and then increase. However,they decrease quasi-linearly with the increase of axial distance.

Consequently, the results of this study clearly demonstrate the numerical simulations ofconjugate heat transfer in laminar, fully developed flow through the externally heated pipe andthe thermally induced stress for the various cases. They would be useful for similar pipe flowapplications. The results of steady state calculations are valid also for any solid material havinga thermal conductivity in the range of 10–100 W/m K. However, care must be taken whetherthe thermally induced stresses arising during the heating exceed the yield stress of the solid ornot.

Appendix A. User defined function file for the temperature controlled heating case

#include ‘‘udf.h’’DEFINE_PROFILE (heatflux_constant, thread, position){real tmax¼ 0.; real tmin¼ 0.; real temp; real tupper¼ 200.; real tlower¼ 190.;real qh¼ 1.0e5; real ql¼ 1.0e4;face_t c; face_t f;begin_c_loop (c, thread){temp ¼ C_T (c, thread); temp ¼ temp - 500;if (temp < tmin jj tmin ¼ ¼ 0.) tmin¼ temp;if (temp > tmax jj tmax ¼ ¼ 0.) tmax¼ temp;}end_c_loop (c, thread)


begin_f_loop (f, thread){if (tmax < tlower jj tmax ¼ ¼ tlower) F_PROFILE (f, thread, position) ¼ qh;if (tmax > tupper) F_PROFILE (f, thread, position) ¼ ql;}end_f_loop (f, thread)}

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