International Journal of Numerical Methods for Heat & Fluid FlowNumerical simulation of flow through an eccentric annulus with heat transferAmit K. Chauhan B.V.S.S.S. Prasad B.S.V. Patnaik

Article information:To cite this document:Amit K. Chauhan B.V.S.S.S. Prasad B.S.V. Patnaik , (2014),"Numerical simulation of flow through aneccentric annulus with heat transfer", International Journal of Numerical Methods for Heat & Fluid Flow, Vol.24 Iss 8 pp. 1864 - 1887Permanent link to this document:http://dx.doi.org/10.1108/HFF-07-2013-0230

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Numerical simulation of flowthrough an eccentric annulus

with heat transferAmit K. Chauhan

Department of Applied Mechanics, Indian Institute of Technology Madras,Chennai, India

B.V.S.S.S. PrasadDepartment of Mechanical Engineering, Indian Institute of Technology Madras,

Chennai, India, and

B.S.V. PatnaikDepartment of Applied Mechanics, Indian Institute of Technology Madras,

Chennai, India

Abstract

Purpose – The purpose of this paper is to investigate the effect of narrow gap on the fluid flow andheat transfer through an eccentric annular region is numerically. Flow through an eccentric annulargeometry is a model problem of practical interest.Design/methodology/approach – The approach involves standard finite volume-based SIMPLEscheme. The numerical simulations cover the practically relevant Reynolds number range of 104-106.Findings – In the narrow gap region, temperature shoot up was observed due to flow maldistributionwith an attendant reduction in the heat removal from the wall surfaces. CFD analysis is presented with theaid of, streamlines, isotherms, axial velocity contours, etc. The engineering parameters of interest such as,Nusselt number, wall shear stress, etc., is presented to study the effect of eccentricity and radius ratio.Research limitations/implications – The present investigation is a simplified model for the rodbundle heat transfer studies. However, the detailed study of sectorial mass flux distribution is a usefulprecursor to the thermal hydraulics of rod bundles.Practical implications – For nuclear reactor fuel rods, the effect of eccentricity is going to be detrimentaland might lead to the condition of critical heat flux. A thorough sub-channel analysis is very useful.Social implications – Nuclear safety standards require answers to a wide a range of what-if typehypothetical scenarios to enable preparedness. This study is a highly simplified model and a first stepin that direction.Originality/value – The narrow gap region has been systematically investigated for the first time.A detailed sectorial analysis reveals that, flow maldistribution and the attendant temperature shoot upin the narrow gap region is detrimental to the safe operation.

Keywords CFD, Annular flows, Eccentricity, Nuclear engineering, Rod bundles, Thermal hydraulics

Paper type Research paper

The current issue and full text archive of this journal is available atwww.emeraldinsight.com/0961-5539.htm

Received 26 August 2013Revised 6 March 2014Accepted 15 April 2014

International Journal of NumericalMethods for Heat & Fluid FlowVol. 24 No. 8, 2014pp. 1864-1887r Emerald Group Publishing Limited0961-5539DOI 10.1108/HFF-07-2013-0230

NomenclatureCf skin friction coefficient,

ðtoÞ=ð12 rU 2ÞDh hydraulic diameter of annulus,

D2�D1, m

E energy, J/kge eccentricity, s/r2�r1

f Darcy friction factor, 2DpDh/LrU2

G0 mass flux for e¼ 0.0, kg/m2s

The authors thank the annonymous referees for their valuable comments. The authors aregrateful to Board of Research in Nuclear Science (BRNS), India for supporting one of the authorsAmit K. Chauhan (AKC) through a Fellowship.

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1. IntroductionIn a large number of practical applications such as shell and tube heat exchangers,nuclear reactors, drilling equipment, etc., a cylindrical annular geometry forms thebasic model for analyzing the thermal and hydrodynamic characteristics of designinterest. In most of these applications, the flow in the vicinity of the inner tube can bemodeled as an annular flow. For flow through annular passages, the radius ratio (a) isdefined as inner radius (r1) to outer radius (r2). The value of a close to unity representsthe annuli with narrow gap. In the context of fuel bundles, the entire pin-bundlewill be seated on spacer pads. If this is modeled as a single inner tube, it provides amajor source of eccentricity. Furthermore, due to prolonged exposure to hightemperatures, there is a possibility of creep distortions. This can easily result in flowmaldistribution, as the attendant hydrodynamic and heat transfer characteristics aremodified. When liquids are employed as coolants, such eccentricities might even leadto undesirable phase change. It may be emphasized that, while the narrow gap may becreated by increasing either radius ratio (a) or eccentricity (e), their effect may notbe the same.

In the literature, a variety of fluid-thermal aspects have been studied either byexpermental or numerical means (Lee and Barrow, 1963; Barrow et al., 1965; Jonssonand Sparrow, 1966; Trombetta, 1971; Patel and Ingham, 1983; Ogino et al., 1987; Nouriet al., 1993; Batra and Eissa, 1994; Velusamy and Garg, 1994; Manglik and Fang, 1995;Torii and Yang, 1997; Fang et al., 1999; Yoo, 2005; Kaneda et al., 2003; Hosseini et al.,2005; Ninokata et al., 2005; Lu and Wang, 2008; El-Shaarawi et al., 2007; Nikitin et al.,2009; Sreenivasulu and Prasad, 2009; Kelessidis et al., 2011; Piot and Tavoularis, 2011).To enable a cogent view of the available literature, the details are presented in the formof a Table I. It can be observed that, a number of investigators have worked on theannular geometry, over a range of radius ratios and eccentricities. However, for largeradius ratios (close to unity) and varying eccentricities of practical interest, there arehardly any studies in this parametric space. The presence of geometric distortions in

G mass flux, kg/m2

g gap between two cylinders,r2�r1�s, m

k turbulent kinetic energy, m2/s2

Nu Nusselt number, hL/kNus Sectorial Nusselt numberq00 heat flux, W/m2

q000 heat flux, q000¼ 500 W/m2

Q Volumetric flow rate, m3s�1

r1,r2 inner and outer radii, mRe Reynolds number, rUbDh/mS1,S2 generation and source term in

turbulences distance between the two

cylinders, mT temperature, KTin inlet temperature, KUb area averaged velocity, m/sUm maximum velocity, m/s

V velocity vector (uiþ uj), m/sui,uj x and y component of velocity, m/su0,u0 x and y component of velocity

fluctuationx dimensionless distance between

gap (outer to inner cylinder)Y dissipation term in turbulenceyþ viscous grid spacing

Greek symbolsa radius ratio, r1/r2

y Angle along the circumference ofinner cylinder

F dimensionless temperature,(T�Tin)/Tin

v kinematic viscosity, m2s�1

r density, kg m�3

t wall shear stress, N/m2

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Table I.A summary of previousinvestigations oneccentric annuli

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the rod bundle of nuclear reactor may occur due to thermal effects, commonly knownas creep. This leads to a swell in the outer pressure tube. The flow maldistribution dueto geometric changes can be modeled using radius ratio changes in an annular channel.Thus, a study of combined effect of both radius ratio (a) and eccentricity (e) assumessignificance. To this end, the present study focusses on the CFD analysis of the thermalhydraulics in the narrow gap region of an eccentric annuli. In particular, the influenceof eccentricity on the mass flux distribution in the annular region and correspondingthermal-hydraulic features such as, variation of temperature, friction factor and heattransfer coefficient, etc., is studied. Inner cylinder is imposed with either a constant orsinusoidally varying heat flux, while the outer cylinder is maintained with an adiabaticboundary condition. Furthermore, the position and center of the outer cylinder areshifted to enable different radius ratios and eccentricity values.

2. The physical model and boundary conditionsThe geometric details of the physical model for fluid flow through an eccentric annulusis shown in Figure 1(a). The dimensionless eccentricity (e) is defined in terms of thedistance between the centers of the two cylinders (s) and the difference between outerand inner radius of the cylinders (Dr¼ r2-r1):

e ¼ s

Drð1Þ

It can also be expressed through the smallest gap (g), and radius ratio (a) as follows:

e ¼ 1� g

r1ð1a� 1Þð2Þ

For the purpose of analysis, the cross-sectional plane of the annulus is divided into12 different sectors, which are numbered in the anti clockwise direction, with eachsector abutting an angle of 301 window. Furthermore, angle, y is measured along the

Sector 2

Sector 12

Sector1

Mass Flow Inlet B.C.

OuterCylinder

InnerCylinder

Pressure Outlet B.C.

Inner Cylinder:No Slip,SpecifiedHeat Flux, q”

Outer Cylinder:No Slip,Adiabatic

g

x1 0

M

s

L

r

z

P2 P1

r2

r1

C1C2

(a) (b)

θ

Notes: Only sectional front view is shown. Typical side view is reflected throughFigure 1(a)

Figure 1.(a) Nomenclature for the

eccentric annulus; (b) flowdomain of interest with

boundary conditions

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circumference of the inner cylinder in anticlockwise direction, starting from theextreme right plane facing the narrow gap.

Different physical models have been considered in the present computational studyby varying the geometric parameters. Length and inner radius of these models are keptconstant, while variation in radius ratio (a) is achieved by changing only the outerradius. Length of the annular channel is taken as 2 m for each case study. Figure 1(b)presents the boundary conditions imposed on the respective boundaries of the annulargeometry. The streamwise direction of fluid flow is indicated with an arrow in theannular region z direction. Mass flow rate is specified at the inlet, whereas staticpressure is enforced at the outlet. The inlet is enforced with an ambient temperatureTin. These input conditions are estimated indirectly from the chosen Reynolds numbervalue. An adiabatic and no slip wall boundary are specified on the outer cylinder.Uniform heat flux condition is applied on the inner cylinder. However, along thecircumferential direction, two types of boundary conditions have been studied. In thefirst case, a uniform heat flux was applied, while in the second case, a cosine lawq00 ¼ qo

00ð1þ 0:5cosyÞ Reynolds (1963) was enforced.

3. Computational methodologyThe flow domain of interest is tessellated with a fine hexahedral mesh. In particular, aclustered fine mesh is generated at the walls to capture the near wall phenomenonbetter. Although transients are of interest, simulations in the present study focus onsteady operational conditions. Therefore, flow is assumed incompressible, steady andNewtonian. Fluid flow and heat transfer in the annular configuration are governed byconservation of mass, momentum and energy equations. Reynolds Averaged Navier-Stokes equations are solved with SST k-w turbulence model. In our earlier investigation(Sreenivasulu and Prasad, 2009), a wide variety of turbulence models have beenassessed and SST k�o model was found to be better suited for this class of flows. Thegoverning set of equations are as follows.

Conservation of mass:

qqxj

ðrujÞ ¼ 0 ð3Þ

Conservation of momentum:

qqxj

ðrujui � tijÞ ¼ �qP

qxi

ð4aÞ

tij ¼ 2msij �2

3mquk

qxk

dij � ru0i u0j ð4bÞ

sij ¼1

2

� qui

qxj

þ quj

qxi

�ð4cÞ

Conservation of energy:

qqxj

ðrujCpT � Fh;jÞ ¼ 0 ð5aÞ

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Fh;j ¼ KqT

qxj

� rCpu0jT

0 ð5bÞ

Turbulent kinetic energy:

qqxi

rkuið Þ ¼ qqxj

Gk

qk

qxj

� �þ Gk � Yk þ Sk ð6Þ

Specific dissipation rate:

qqxi

rouið Þ ¼ qqxj

Goqoqxj

� �þ Go � Yo þ Do þ So ð7Þ

where:

Gk ¼ mþ mt

sk

;Gw ¼ mþ mt

sk

sk;1 ¼ 1:176; so;1 ¼ 2:0; sk;2 ¼ 1:0; sk;2 ¼ 1:16

All the above equations are solved using a finite volume-based commercial solverFluent 6.3.26 (Fluent 6.3 User’s Guide, 2009) with an implicit second order accurateupwind scheme. The pressure correction strategy uses the SIMPLE algorithm(Launder and Spalding, 1974). The convergence criterion for all the variables is fixedsuch that the residual values are lower than 10�6 Water (Pr¼ 7) is chosen as theworking fluid and all thermodynamic properties are taken at inlet temperature Tin.

3.1 Grid sensitivity studiesVerifying the adequacy of grid employed is an important step in a CFD simulation.Hence, a systematic mesh senstivity study was conducted. The size of the mesh wasincreased by about 50 percent in two successive levels and yþ was maintained to beless than unity over different mesh sizes. Pressure drop per unit length (DP/L), non-dimensional velocity (U/Ub) and non-dimensional temperature (T�Tin/Tin) profilesalong plane P1 are taken as the reference parameter for ascertaining the griddependence. For a typical case, dependence of grid on the solution is shown in Table IIand Figure 2. Based on these results, a mesh size of 0.777 million cells is chosen forfurther computations.

3.2 ValidationThe normalized velocity variation in the annular region is compared against theexperimental data of Nouri et al. (1993) for Re¼ 8,900. The velocity variation in two

Sl. no. Mesh size (million) DP/DL (kPa/m)

1. 0.215 1.2242. 0.525 1.3143. 0.777 1.4204. 1.080 1.421

Table II.Pressure drop per unit

length for a¼ 0.5

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planes was chosen along narrow and wide gap regions ( P1 and P2), at the outlet sectionof the channel. This was normalized with respect to the maximum velocity in the sameplane. The results along these two planes are compared against Nouri et al. (1993)(see Figure 3) at the outlet section of the annulus. Note that the, distance is measuredfrom the outer to the inner wall on planes P1 and P2. A good match between theexperimental data and numerical simulations may be noticed. This approach is furtherextended to other simulations involving annular geometric models.

3.3 Data reductionThe engineering parameters of design interest may be expressed in the form of non-dimensional velocity (U*), skin friction coefficient (SFC) (Cf), dimensionlesstemperature (F) and Nusselt number (Nu*) which are defined as follows:

U� ¼ U

Ub

; C�f ¼Cf

ðCf Þavg

;F ¼ T � Tin

Tin

; Nu� ¼ Nu

Nuavgð8Þ

where, Ub, refers to the area averaged velocity at the outlet section. While (Cf)avg andNuavg refers to area averaged SFC and Nusselt number around the inner cylinder at theoutlet section.

U/U

b

Position (X)

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

1.2

Grid 1Grid 2Grid 3Grid 4

(a)

Wal

l She

ar S

tres

s (�

)

Axial Position (z)

0.2 0.4 0.6 0.8 1

1

1.2

1.4

1.6 Grid 1Grid 2Grid 3Grid 4

(b)

Tem

pera

ture

(φ)

Position (X)

0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Grid 1Grid 2Grid 3Grid 4

(c)Figure 2.(a) Non-dimensionalvelocity along P1 iscompared for differentgrid sizes, to establishmesh senstivity; (b) Non-dimensional wall shearstress along P1 iscompared for differentgrid sizes, to establishmesh senstivity; (c) Non-dimensionalizedtemperature (F) along P1

is plotted for differentgrid sizes

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4. Results and discussionA detailed analysis and understanding of fluid flow features and heat transfercharacteristics for an annular channel has a wider appeal in a number of engineeringapplications. Numerical simulations are performed in the Reynolds number rangeof 104-106 to study the combined effect of radius ratio and eccentricity on thefluid-thermal characteristics in each sector. The input mass flow rate is adjusted suchthat, for higher values of radius ratio, constant value of Reynolds number ismaintained. This has enabled an easy comparison between different cases. For thepurpose of ease in presentation, fluid flow features are discussed first, followed bythermal characteristics. Influence of constant as well as varying heat flux boundaryconditions are studied.

4.1 Fluid flow characteristics4.1.1 Mass flux distribution. A detailed investigation of mass flux distribution indifferent sectors of the annular channel enables a cogent view of flow redistributionand its attendant effects. To further study the flow development, sector wise massflux distribution is carefully examined. To this end, normalized mass flux (G/G0)distribution at the outlet section of the channel is plotted in Figure 4, for a¼ 0.95. Fromthe present CFD simulations, it was observed that, these valus decrease by 75 percentin the narrow gap sector (i.e. sector 1) while about 30 percent increment was noticedin the wide gap sector 7, when compared to the concentric case. While consideringa particular radius ratio, flow squeezes through the narrow gap. However, as thefluid encounters higher resistance, the fluid peripherally gets pushed from thenarrow sectors to the wider sectors. Thus it can be inferred that, for same Re, massflux will be higher in the narrow gap of higher radius ratio, while it will be much lowerthrough the narrow gap region with higher eccentricity. For lower values ofeccentricity, mass flux distribution is almost uniform in both sections, while a largescale shift in mass flux distribution from narrow regions to wide gap regions canbe noticed.

Analytical expressions for volume flow rate (and hence, mass flow rate) as a functionof radius ratio and eccentricity are available in the literature (see White Frank, 2003).

Position (x)

Non

Dim

ensi

onal

ized

Axi

al V

eloc

iy (

U/U

m)

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

1.2

Present (3-4)Nourie et al. (exp)(3-4)Present (1-2)Nourie et al. (exp)(1-2)

Inner wall Outer wall

1 2 3 4x x

Figure 3.Comparison of velocityvariation along narrow

and wide gap planesP1 and P2 against

experimentalmeasurements of Nouri

et al. (1993) for Re¼ 8,900

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From these closed form expressions, volume flow rate through the concentric annuli(Q0) and eccentric annuli (Q) are given as follows:

Qo ¼p8m

� q P̂

qz

0@

1A r4

2 � r41 �ðr2

2 � r21Þ

2

ln r2

r1

" #ð9Þ

Q ¼ p8m

� q P̂

qz

0@

1A r4

2 � r41 �

4c2M 2

b� a� 8c2M 2

X1n¼1

ne�nðbþaÞ

sinhðnb� naÞ

" #ð10Þ

where:

M ¼ F2 � r22

� �0:5; F ¼ r2

2 � r21 þ s2

2s; a ¼ 1

2ln

F þM

F �M; b ¼ 1

2ln

F � sþM

F � s�M

ð11Þ

Based on the Equations (9) and (10), mass flow distribution is proportioned accordingto the area ratio of each sector. However, such a proportioning assumes that pressuredrop along different axial lengths of the annuli to be uniform. The sectorial mass fluxdistribution was multiplied with a constant to enable same value of mass flux throughsector 7 (sector with highest mass flux). This way a constant normalization base wasachieved. Furthermore, it should be pointed out that, in the analytical expressions, thepressure gradient (dp/dz) is assumed to be same in both concentric and eccentricannuli, which is a restrictive assumption. This further emphasizes the fact that,sectorial mass flux distribution through CFD is different from the closed form solution.Moreover, these analytical expressions are valid for laminar flow conditions and purelybased on geometric inputs. Nevertheless, this is negated due to early flow developmentand higher pressure drops in the narrow regions than the wider zones. This furtherpromotes fluid transport between different sectors, when the eccentricity effect is more

Sector (each 30°)

Nor

mal

ized

mas

s flu

x (G

/G0)

2 4 6 8 10 120

0.2

0.4

0.6

0.8

1

1.2

1.4

0.0 (CFD)0.1 (CFD)0.1 (Eqn 9)0.3 (CFD)0.3 (Eqn 9)0.5 (CFD)0.5 (Eqn 9)0.7 (CFD)0.7 (Eqn 9)

Curve e

Note: The sectorial mass flux distribution fromclosed form analytical solution and CFDsimulations

Figure 4.Sector wise mass fluxdistribution for differenteccentricities for a¼ 0.95

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pronounced. Comparison of anticipated mass flux distribution is found to differ fromthe CFD predictions. Figure 4 depicts numerical calculations vis-�a-vis closed formsolution. Differences between the two further underscores non-trivial nature of thepresent simulations and the need for a detailed analysis. All of this translates intosector wise Reynolds number (Res) distribution along the periphery, which is shown inFigure 5. It should be mentioned that, mass flow rate and volume flow rate differ by aconstant density multiple for incompressible flows. Furthermore, this variation atteststhe fact that, flow is laminar like in the narrow zone (sector 1), whereas it is fullydeveloped and turbulent like in the wider zone (sector 7).

The sectorial Reynolds number uses a different characteristic length scale unlikethe hydraulic diameter used for the eccentric annulus. In this context, it should bepointed out that, the Reynolds number for the present simulation is 104, which isclearly in the turbulent flow regime. In Figure 5, sectorial Reynolds number ispresented. This observation helps us to realize that, it translates into peripheralvariation in thermal characteristics, which will be discussed in a subsequent section.This result is of practical significance to problems such as rod bundle heat transfer, aslow mass flux is detrimental to the reactivity of the core, as it may lead to departurefrom nucleate boiling (DNB). Furthermore, it would be interesting to study if radiusratio plays a significant role, for a given eccentricity. To this end, the variation of massflux distribution with eccentricity is plotted for different radius ratios in Figure 6. It isobserved that the radius ratio (a) does not have any significant effect on normalizedmass flux distribution. Therefore, it can be concluded that, for different values ofradius ratio, the mass flux distribution is same when eccentricity is varied.

4.1.2 Velocity distribution. The flow redistribution is inevitable either due toeccentricity or radius ratio, which in turn influences the axial velocity distributionalong the channel. The nature of flow development in different sectors of the annularchannel is revealed from the axial velocity color maps, plotted in Figures 7 and 8. Theeffect of eccentricity and radius ratio on flow development can be noticed through thesefigures. Velocity contours at the exit boundary are shown in Figure 8 for four differentradius ratios. As radius ratio is further increased, the velocity distribution increases inall the sectors of the channel. Reynolds number is kept constant, at 104. Essentially, theflow redistributes and adjusts itself in the narrow gap between the cylinders, which is

Sector (each 30°)

Re s

(ρu

bdh)

/μ

2 4 6 8 10 120

200

400

600

800

1,000

1,200

1,400

1,600

0.00.10.30.50.7

Curve e

Figure 5.Sector wise local Reynolds

number distribution fordifferent eccentricities for

a¼ 0.95

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evident from Figure 8. Stream wise development of the velocity profiles is presented inFigure 9. Due to high eccentricity, significant flow distortions and non-uniformity inflow and temperature distribution in the annular region have been observed byChoudhury and Karki (1992). These features are further revealed through the flowdevelopment depicted in Figure 9.

Eccentricity (e)

Mas

s F

lux

Dis

trib

utio

n (G

/G0)

0 0.2 0.4 0.60

0.2

0.4

0.6

0.8

1

0.500.650.800.95

�

0.33 0.36

0.65

0.66

0.67

0.68

Curve

Figure 6.Normalized mass fluxdistribution in sector 1, fordifferent radius ratios

–0.02–0.02

0.02

1

0.5

0

0 0Y X

U/U

b

–0.02–0.02

0.02

1

0.5

0

0 0Y X

U/U

b

(a) (b)

–0.02–0.02

0.02

1

0.5

0

0 0Y X

U/U

b

–0.02–0.02

0.02

1

0.5

0

0 0Y X

U/U

b

(c) (d)

Figure 7.Non-dimensional velocitycontours for radius ratio(a¼ 0.5) for eccentricity(e)¼ (a) 0.1, (b) 0.3, (c) 0.5and (d) 0.7 at z¼L

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From a detailed analysis of velocity profiles along the axial length of the annularregion, it can be noticed that, the nature of the flow is turbulent in the wider zone whilelaminarity prevails in the narrow sectors. Corresponding to the flow developmentalong narrow and wider planes, magnitude of velocity is plotted in Figure 10. Figure11(a) depicts the non-dimensional velocity in the narrow gap region along the line P1 atthe outlet section of the annuli. For example, for a typical eccentricity value of 0.7, thefluid velocity is found to increase with increase in a from 0.5 to 0.65 and then itdecreases with further increase in a from 0.65 to 0.95. The reason for this unexpectedbehavior is due to the fact that, with rise in radius ratio upto 0.65, the rise in localinstantaneous velocity is uniform. With further increase in radius ratio (i.e. beyond0.65), as bulk velocity increases, corresponding local velocity does not increase in thesame proportion resulting in a decrease in normalized velocity. The influence of a onthe maximum normalized velocity along P1 is plotted in Figure 11(b). It can be noticedthat higher the eccentricity, lower the maximum velocity. The plot reflects no variationin maximum velocity for different radius ratios. However, for a given radius ratio,higher the eccentricity, lower the maximum velocity. To investigate the bent nature offlow lines from the narrow gap region to wide gap zones, particles were tracked atdifferent downstream positions of the annular channel. The axial lengths were chosen

� = 0.5e = 0.7

–0.02

1

0.5

0

1.5

U/U

b

Y

X

0.02

0 0

–0.02

� = 0.8e = 0.7

–0.01

1

0.5

0

1.5

U/U

b

Y

X

0.01 0.01

00

–0.01

� = 0.65e = 0.7

–0.01

1

0.5

0

1.5

U/U

b

Y

X

0.01 0.01

0 0

–0.01

� = 0.95e = 0.7

–0.01

1

0.5

0

1.5

U/U

bY

X

0.01 0.01

00

–0.01

(a) (b)

(c) (d)

Figure 8.Non-dimensional velocitycontours for eccentricity

(e¼ 0.7) for radius ratio (a)(a) 0.5, (b) 0.65, (c) 0.8 and

(d) 0.95

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to be 0.25L, 0.5L, 0.75L, L, etc. Moreover, these particle tracks advance in the azimuthaldirection, which is naturally reflective of their axial positions. These are presented inFigures 12 and 13, as viewed from the exit end of the annular region. For the concentriccase, due to uniform fully developed conditions, particle tracks remain along a straightpath. However, at higher values of eccentricity, particles follow curved paths fromnarrow gap side toward wide gap region, indicating the fluid transport toward zonesof lower resistance. This is more pronounced for higher radius ratios at a particulare value of 0.7, as shown in Figure 13.

(a) (b)

1.3

1.3

1.4

1.3

1.3

1.21.1

1.1

0.91

0.9

0.8

0.5

0.4

0.6

1.21.2

0.9

0.8

0.3

0.5

0.8

0.8

1.4

1.4

1.3

1.3

1.2

1.1

1.1

0.7

0.3

0.4

0.6

0.5

0.7

0.4

1.4

1.4

1.3

1.2

1.1

1.2 0.8

0.7

1

0.8

0.5

0.3

0.4

0.6

0.8

U/Ub

Flow

1.4

1.3

1.2

1.1

0.7

0.4

0.7 1

.4

1.2

1.3

10.7

0.4

1.4

1.3

1.1

1.1

0.7

0.3

0.5

0.3

0.5

1.4

1.4

0.5

1.1

1

0.5

1.2

U/Ub

Flow

(d)(c)

1.4

1.4

1.30.9

0.9

0.6

0.6

0.6

1.4

1.2

1.4

0.9

0.6

0.6

0.6

0.6

1.4

1.20.6

1.4

1.2

0.6

0.6

1.4

1.2

1.4

1.2

0.6

0.61.4

1.2

1.3

0.7

1.1

1.4

0.3

0.3

0.2

0.1

0.2

U/Ub

Flow

1.4

1.3

1.2

1.1

0.8

0.4

0.3

0.4

1.4

1.3

1.2 1.0

0.6

0.3

0.4

1.0

1.4

1.4

1.4

1.3

1.2 0.8

0.8

0.6

1.4

1.3

1.0

0.6

0.4

0.61.2

1.4

0.8

0.3

0.4

0.3

U/Ub

Flow

Figure 9.Streamwise flowdevelopment as velocitycontours for (e¼ 0.7) forradius ratio (a) (a) 0.5,(b) 0.65, (c) 0.8 and (d) 0.95at z¼ 0.25L, 0.50L, 0.75L,1.00L respectively

1.8

Position (X)

U/U

b

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

z = 0.00z = 0.25z = 0.50z = 0.75z = 1.00z = 1.25z = 1.50z = 1.75z = 2.00

P1 P2

(a)P1

P2

Position (X)

U/U

b

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8z = 0.00z = 0.25z = 0.75z = 1.25z = 1.75z = 2.00

(b)

Notes: (a) e=0.1 and (b) e=0.7

Figure 10.Non-dimensional velocityalong narrow and widegap planes for a¼ 0.5

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4.1.3 Variation of friction factor. An investigation of total pressure loss is reflectedthrough the Darcy friction factor, which is found to decrease with e and a as shown inFigure 14(a). Such a decrease is inline with the experimental observations in thestandard literature such as, Lee and Barrow (1963) and Rehme (1973). Kaneda et al.(2003) have pointed out that, in the outer passages of a double pipe heat exchanger, asradius ratio approaches unity, the velocity distribution and friction factor approachesthat of flow between parallel plates. It was observed that, friction factor approaches thevalues of a round tube, when radius ratio approaches zero for both laminar andturbulent flows. Peripheral variation of SFC is found to decrease with increase in e,which are presented in Figure 14(b). However, the variation has similarity and nosignificant change is observed with increase in a when normalized with respect to theaverage value of SFC around the inner cylinder. It can be noticed that, the distributionof SFC around the wall with respect to radius ratio (a) is same. The variation of local

Position (x)

U/U

b

0 0.2 0.4 0.6 0.8 1

0

0.1

0.2

0.3

0.4

0.500.650.800.95

α

(a)

Alpha (�)

(U/U

b)m

ax

0.5 0.6 0.7 0.8 0.90

0.4

0.8

1.2 e = 0.0

e = 0.1

e = 0.3

e = 0.5

e = 0.7

(b)

Curve

Figure 11.(a) Velocity variation in the

narrow gap, along P1,for e¼ 0.7 at z¼L;

(b) distribution ofmaximum velocity with

respect to radius ratio fordifferent eccentricity

values

(a) (b)

(c) (d)

Figure 12.Cross-sectional view ofparticle tracks (as seen

from the exit boundary)for a¼ 0.50 and e¼ (a) 0.0,

(b) 0.3, (c) 0.5 and (d) 0.7

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wall shear stress is lowest in the narrow sector 1, where the values decrease by 70percent, while maximum in the widest sector 7 where SFC value is higher by 65-70percent as seen from Figure 14(b).

4.1.4 Effect of Reynolds number. In this section, the effect of Reynolds number on theflow and thermal characteristics is analyzed. The cases discussed above are indeedcarried out for a particular Reynolds number of 104. However, it is desirable to extendthe present analysis over a range of Reynolds numbers. To this end, numericalsimulations are performed over the Re range of 104-106. The variation of friction factorwith Re is presented in Figure 15(a) for different radius ratios. There is a steep fall inthe friction factor value between Re¼ 104 to 5.5 � 105 and then it further graduallydecreases with higher Re. This observation is in agreement with previously publishedliterature (see Kaneda et al., 2003; Kelessidis et al., 2011; Rehme, 1973). Figure 15(b)presents variation of SFC around the inner cylinder which is normalized with respect to

(a) (b)

(c) (d)

Figure 13.Cross-sectional view ofparticle tracks (as seenfrom the exit boundary)for e¼ 0.7 and a¼ (a) 0.50;(b) 0.65; (c) 0.80and (d) 0.95

Eccentricity (e)

Dar

cy F

rictio

n Fa

ctor

0 0.2 0.4 0.6 0.8 10.02

0.025

0.03

0.035

0.04

0.500.650.800.95

α

(a)

Angle (�)

Ski

n Fr

ictio

n C

oeffi

cien

t, C

f/(C

f) avg

0 100 200 3000

0.4

0.8

1.2

1.6

2

0.00.10.30.50.7

(b)

Curve

Curve e

Figure 14.(a) Darcy-friction factoragainst eccentricity (e) fordifferent radius ratios (a);(b) variation of skinfriction coefficient aroundthe inner cylinderfor a¼ 0.5

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the average value. It is found that SFC is almost independent of Reynolds number inthe narrow gap region of the annular channel, while it decreases with Re in the widegap region.

4.2 Thermal characteristicsIn the present simulations, temperature is treated as a passive scalar. Hence, energyequation is decoupled from the momentum equations. Toward thermal characterizationof the present study, two types of practically usefull boundary conditions, namely,uniform heat flux and circumferentially varying heat flux are considered.

4.2.1 Effect of uniform heat flux on the inner cylinder. 4.2.1.1 Temperaturedistribution. In the annular region of the channel, with change in the eccentricity andradius ratio, heat transfer from the inner cylinder to the coolant are indeed functions ofhydrodynamic characteristics. Figure 16 depicts dimensionless isotherms for thecoolant region at the outlet section of the annuli for a typical eccentricity of 0.7. Areduction in flow area, causes an attendant decrease in mass flow rate. Hence, due tolack of sufficient coolant through that sector, heat transfer decreases, which leads to anincrease in temperature of both coolant and the surface temperature of the heated innercylinder. These features are indeed reflected in Figure 16. The above discussion can beeasily understood from the color map of temperature distribution for differenteccentricities and radius ratios at outlet section, which is shown in Figures 16 and 17.On the contrary, in the wider regions of the annulus, the temperature of both fluid andsurface is lower. It can be clearly observed from these plots that at a particular radiusratio, the temperature is maximum on the cylinder surface in the narrow sector 1, whileminimum in the wide gap sector 7. But, for higher radius ratios, flow rate through thenarrow gap increases as the Reynolds number is maintained constant. Although theannular gap is reduced, higher amount of heat is extracted compared to lower radiusratio cases (for the same e). In the present study, the radius ratio range that was studiedis from 0.50 to 0.95. In this range, non-dimensionalized temperature, was observed tobe maximum in case of a¼ 0.50, i.e, 4.5 and minimum for 0.95, i.e. 2.4, in narrowsector 1 (see Figure 17(a) and (b)). Inner cylinder surface is the zone of interest for theanalysis of temperature distribution and its associated gradients, for practicalapplications, such as, nuclear rod bundles, heat exchangers, etc. Figure 18 depicts thedimensionless temperature profile along the circumference of the inner (heated)

Log10 (Re)

Log 1

0(f)

104 105 106

0.01

0.02

0.030.500.650.800.95

Curve �

(a)

Angle (�)

Ski

n Fr

ictio

n C

oeffi

cien

t, C

f/(C

f)avg

0 100 200 3000

0.4

0.8

1.2

1.6

2

1.0X104

5.5X104

1.0X105

5.5X105

1.0X106

(b)

Curve Re Figure 15.(a) Variation of friction

factor for differentReynolds number on

logarithmic scale; (b) skinfriction coefficient along

the inner cylinder, fora¼ 0.8 and e¼ 0.7

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cylinder with respect to angle (y). It is evident from Figure 18 that, the cylinderencounters maximum temperature where the gap is minimum. It is also observed thatwith increase in eccentricity to 0.7 (e¼ 0.7), for lower radius ratio (a¼ 0.5) temperature(F) rises by 400 percent with reference to the concentric case (see Figure 18a). Such adrastic shoot-up in temperature may lead to change in the physical and mechanicalproperties of both the coolant and cylinder surface. PHWR designs are such that forsafe reactor operations, single phase flow need to be maintained. However, thissituation might change and may lead to undesirable DNB and CHF condition, where inthe coolant enters two phase. It is also observed that, the temperature peak value on theinner cylinder surface in the narrow gap increases uniformly with eccentricity uptoe¼ 0.5. However, this increase is drastic with further increase in e value. For a typicalcase of a¼ 0.5, it is found that, there is an increase in temperature on the cylindersurface by 100 percent upto e¼ 0.5, but it shoots up by 400 percent for e¼ 0.7. Whenthe gap reduces further, beyond e¼ 0.5, the fluid gets squeezed through the narrow gapand tries to escape into wider sectors of the annulus. This results in a reduction in theamount of coolant to extract heat from the cylinder surface. The coolant present in thenarrow gap region gets heated up quickly and thus looses the ability to extract heatany further. This leads to an imminent shoot up in cylinder surface temperature. At thesame time, it is evident from Figure 18 that, with increase in radius ratio, peaktemperature decreases because the coolant flow improves the flow redistribution

(a)� = 0.5e = 0.1

Temperature (φ)

4

2

0

0 0Y X

–0.02 –0.02

(c)� = 0.5e = 0.5

Temperature (φ)

4

2

0

00

Y X–0.02

–0.02

(d)� = 0.5e = 0.7

Temperature (φ)

4

2

0

00

Y X–0.02

–0.02

(b)� = 0.5e = 0.3

Temperature (φ)

4

2

0

0 0Y X

–0.02 –0.02

Figure 16.Non-dimensionalisotherms for radiusratio (a¼ 0.5) fordifferent eccentricityvalues(e)¼ (a) 0.1;(b) 0.3; (c) 0.5and (d) 0.7

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(see Figure 8). Thus it can be concluded that, the effect of eccentricity on the narrowgap is more pronounced than radius ratio, as far as heat transfer is concerned.

4.2.1.2 Nusselt number variation. Non-dimensional form of the temperature gradientis exemplified through the local variation of Nusselt number around the periphery ofthe inner cylinder. To start with for a concentric annuli of a¼ 0.5, the local Nusseltnumber variation was obtained. This value was integrated around the periphery of theinner cylinder to obtain an average value of Nusselt number. These values of Nuaverage for different values of Reynolds number is plotted in Figure 19(a) againstthe experimental data of Lu and Wang (2008). It is evident from Figure 19(a) that thepresent CFD data agrees well with the experimental studies. For a detailed comparisonof thermal characterization in rod bundle heat transfer, refer to Chauhan et al.(2013). This approach is further extended to other simulations involving annulargeometric models.

The variation of Nusselt number (Nu) for different radius ratios is shown inFigure 19(b) and (c). It is observed that, the maximum decrease or increase in the valueof Nu around the inner cylinder is for higher radius ratio and higher eccentricity, i.e.a¼ 0.95 and e¼ 0.7, by 85 percent in the narrow and 90 percent in wide gap region ofthe annular channel, respectively. From Figure 19(b) and (c), it can be noticed that,although the flow rate increases in the gap region with increase in radius ratio (Reconstant), local Nu decreases. From earlier discussion it was observed that,temperature is maximum for the lower radius ratio rather than higher radius ratioannular channel for the same eccentricity. It further reflects in heat transfer

(a)� = 0.5e = 0.7

Temperature (φ)

4

2

0

00

Y X–0.02

–0.02

(c)� = 0.8e = 0.7

Temperature (φ)

4

2

0

0 0Y X

–0.01

0.01

–0.01

0.01 0.01

(d)� = 0.95e = 0.7

Temperature (φ)

4

2

0

0 0Y X

–0.01 –0.01

0.01

(b)� = 0.65e = 0.7

Temperature (φ)

4

2

0

0 0

Y X–0.01

0.01

–0.01

0.01

Figure 17.3-D non-dimensional

isotherms for eccentricity(e¼ 0.7) for radius ratio, a(a) 0.5; (b) 0.65; (c) 0.8 and

(d) 0.95

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enhancement for higher radius ratio, which contradicts Nu behavior presented inFigure 19(b) and (c). For higher eccentricity values, the fluid escapes from narrow gapsectors toward the wide gap sectors. The local and sectorially average heat transfercoefficients are calculated as follows:

Nuy ¼hl

k; where; Nus ¼

1p6

Zp60

NuðyÞdy; h ¼ q00

Tw � Tf

� � ð12Þ

where:

Tw ¼1

As

ZTdA; Tf ¼

RTr V:dAj jRr V:dAj j ð13Þ

4.2.2 Effect of reynolds number. Thermal characterization is influenced byhydrodynamics, which is analyzed over the Re range of 104-106. A significantvariation in non-dimensional temperature is observed around the inner cylinder

Angle (�)

Tem

pera

ture

(φ)

0 100 200 3000

1

2

3

4

50.00.10.30.50.7

(b)Curve e

Angle (�)

Tem

pera

ture

(φ)

0 100 200 3000

1

2

3

4

50.00.10.30.50.7

(a)Curve e

Angle (�)

Tem

pera

ture

(φ)

0 100 200 3000

1

2

3

4

5

0.00.10.30.50.7

(c)

Curve e

Angle (�)

Tem

pera

ture

(φ)

0 100 200 3000

1

2

3

4

5

0.00.10.30.50.7

(d)

Curve e

Figure 18.Variation of dimensionlesstemperature around theinner cylinder wall fora¼ (a) 0.5; (b) 0.65; (c) 0.8and (d) 0.95

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surface which is shown in Figure 20(a). It is observed that for lower Re, the flow rate isless and temperature gradient between the cylinder and coolant is less. This results inhigher wall temperature for the narrow gap region. Corresponding Nu variationaround the inner cylinder for a typical case of a¼ 0.8 and e¼ 0.7, is shown in Figure20(b). For higher values of Re, due to better cross-mixing, the Nusselt number variationis more uniform, compared to lower Re values.

4.2.3 Influence of varying heat flux on the inner cylinder. All the simulationsdiscussed so far, involve only constant heat flux boundary condition on the innercylinder. However, in practice heat flux does not remain uniform circumferentially dueto the non-uniform neutron flux distribution along the circumference of the fuel pins.Thus the analysis with circumferentially varying heat flux conditions assumessignificance. Nevertheless, constant heat flux is imposed along the axial direction. Thethermal boundary condition, on the inner cylinder, is therefore specified in the form ofcircumferentially varying heat flux according to cosine law, q00 ¼ qo

00 (1þ 0.5 cosy)(Reynolds, 1963), for which a user defined function was implemented into the boundarycondition. Since the heat flux imposed on the wall is a function of cosy, it has amaximum value of wall heat flux at y¼ 0 (narrow gap zone) while minimum is at 1801(wide gap zone). For the purpose of analysis, this indeed refers to severe condition.

Reynolds number (Re)

Nus

selt

Num

ber

(Nu)

0 100 200 300 4000

2

4

6

8

10Present studyLu and Wang (2008)

(a)

Angle (�)

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eCurve

eCurve

(b) (c)

Figure 19.(a) Comparison of Nusselt

number againstexperimental

measurements of Lu andWang (2008) for Re¼ 10 to

400; (b-c) variation ofNusselt number around

the inner cylinder wall for(b) a¼ 0.5 and (c) a¼ 0.95

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4.2.3.1 Temperature distribution. The variation of temperature with eccentricity (e)around the inner cylinder is shown in Figure 21. Temperature of the heated wall inthe narrow gap shoots up with increase in eccentricity (e), by 428 percent, when e¼ 0.7for a radius ratio of 0.5. It is higher than the maximum value of temperature in caseof constant heat flux condition by 28 percent. The most interesting part to be notedis that, as the radius ratio increases, the distribution of temperature is observedto be uniform in the wider zone, while for the lower radius ratio the profile ismore pronounced due to the cosine nature of the boundary condition imposed on theinner cylinder. It can be concluded from Figure 21 that, when the pressure tubeof rod bundle is subjected to any thermal or mechanical distortions, it could effect therate of heat transfer depending on the intensity of distortion. Thus temperatureshoot up in the narrow gap zone could be detrimental, as it may lead to unexpectedphase change.

4.2.3.2 Nusselt number variation. The variation of local Nu around the wall of theinner cylinder is presented in Figure 22. It is observed that Nu variation is similar to theprevious case of constant heat flux boundary condition. Nusselt number changes byapproximately 90 percent in the narrow gap and wide gap of the channel. With

Angle (�)

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1.0×104

5.5×104

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Angle (�)

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Curve Re

Curve Re

(a) (b)

Figure 20.(a) Variation oftemperature along theinner cylinder for a¼ 0.8and e¼ 0.7; (b) variation ofNu along the innercylinder for a¼ 0.8 ande¼ 0.7

Angle (�)

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eCurve eCurve

(a) (b)

Figure 21.Variation of dimensionlesswall temperature aroundthe inner cylinder wall fora¼ (a) 0.5 and (b) 0.95,with varying heat flux atthe wall

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increase in a, Nu decreases for a particular eccentricity in the narrow gap, while it isfound to increase in the wide gap sectors. It is also found that, for constant heat fluxcondition, Nu is almost uniform around the inner cylinder, but for the present case, ithas a cosine nature, i.e. for e¼ 0, where Nu is maximum in the zone where the gap isnarrow. Nusselt number decreases in the narrow gap with increase in eccentricity (e).

5. Summary and conclusionsIn the present CFD study, attention was focussed on the fluid flow and heat transfercharacteristics in the narrow gap region of an eccentric annuli. For the parametricstudy, radius ratio of 0.5-0.95 and eccentricity range of 0.0-0.7 was chosen. The studyemploys either a uniform heat flux on the inner cylinder wall or a sinusoidally varyingheat flux boundary condition on the periphery of the inner wall. Based on the presentstudy, specific conclusions are as follows:

(1) In comparison with a concentric case, mass flux was found to decrease byabout 75 percent in the narrow gap sector, and increase by about 30 percent inthe wide gap sector for a typical case of a¼ 0.95 and e¼ 0.7.

(2) Increase in eccentricity was found to reduce the velocity maxima through thenarrow gap region, while the radius ratio had little role to play.

(3) Friction factor decreases with eccentricity as well as with radius ratio. Fora¼ 0.5 with either of these changes, it was found to decrease and follow auniform trend.

(4) The SFC was almost independent of radius ratio. However, with increase ineccentricity it was found to decrease in the narrow gap. Even with increase inRe, it was found that it remains constant.

(5) The dimensionless wall temperature of inner cylinder rises drastically forhigher values of eccentricity at lower values of radius ratio, though either casechanges the size of the narrow gap in sector 1. For example, for e¼ 0.7 anda¼ 0.5, the wall temperature increases by about 400 percent for constant heatflux case and by about 428 percent for varying heat flux case, for the samegeometric and flow parameters. Thus, the effect of eccentricity is moredominant, compared to radius ratio effect.

Angle (�)

Nus

selt

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Angle (�)N

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lt nu

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Figure 22.Variation of Nusselt

number around the innercylinder wall for a¼ (a) 0.5and (d) 0.95, with varying

heat flux at the wall

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(6) Local Nusselt number increases with increase in Re in the narrow gap regionwhile it decreases in the wide gap region.

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Corresponding authorDr B.S.V. Patnaik can be contacted at: bsvp@iitm.ac.in

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