A Computational Study of Convective Heat Transfer to Carbon Dioxide

8/12/2019 A Computational Study of Convective Heat Transfer to Carbon Dioxide

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A computational study of convective heat transfer to carbon dioxideat a pressure just above the critical value

S. He a,*, W.S. Kim a, J.D. Jackson b

a School of Engineering, University of Aberdeen, Aberdeen AB24 3UE, UKb

Simon Building, University of Manchester, Manchester M13 9PL, UK

Received 29 January 2007; accepted 2 November 2007Available online 7 November 2007

Abstract

Computational simulations are reported of experiments on convective heat transfer to carbon dioxide at a pressure of 75.8 bar, whichis just above the thermodynamic critical value of 73.8 bar. These have been carried out using a variable property, elliptic computationalformulation incorporating low Reynolds number turbulence models ofkeand V2F types. Firstly, the simulations were compared withthe heat transfer measurements and then they were used in developing an understanding of interesting phenomena observed in the exper-iments. It has been found that the effect of buoyancy on turbulence production and heat transfer in fluids at supercritical pressure can bevery significant even under conditions of relatively low buoyancy parameter based on bulk properties. The effect of buoyancy, althoughcomplex, can be explained by relating it to the large-property-variation (LPV) region, i.e., the region within the flow field near to thelocations where the fluid temperature has the pseudo-critical value. Under certain conditions, a very non-uniform radial distributionof the buoyancy force may be present and cause some reduction of turbulence in the core but a big increase near the wall, resultingin much improved heat transfer. It is clear that new heat transfer correlations are needed to account for such effects on heat transferto supercritical pressure fluids as they come to be used more and more in new energy systems applications such as, advanced water-coolednuclear reactors, environmentally friendly air-conditioning and refrigeration systems and high pressure water oxidation plant for wasteprocessing. 2007 Elsevier Ltd. All rights reserved.

Keywords: Supercritical pressure; Mixed convection; Buoyancy influence; Computational modelling; Energy systems

1. Introduction

Considerable interest in heat transfer to fluids at pres-sures above the thermodynamic critical value was stimu-

lated during the 1950s and 1960s by the introduction offossil-fuelled power plant water steam generators operatingat supercritical pressure. A comprehensive review of suchstudies was reported by Jackson and Hall [1]. Recently,there has been renewed interest in the topic driven by activeconsideration of new applications involving fluids at super-critical pressure. These include supercritical pressure water

oxidation systems for waste processing [2], the use of car-bon dioxide at supercritical pressure in a new generationof air-conditioning systems for cars and refrigeration sys-tems [3], the development of very compact gas coolers

and internal heat exchangers[4,5], liquid hydrogenoxygenfuelled rockets [6] and the development of supercriticalpressure water-cooled nuclear reactors which can competein terms of cost, safety and reliability with other types ofpower generation systems[79].

The critical temperature and pressure of carbon dioxideare 31.1 C and 7.38 MPa, respectively. When the pressureof a fluid is below the critical value, phase change mayoccur as its temperature or pressure is varied. Above thecritical pressure a fluid does not undergo phase change.However, the physical properties of such fluids can vary

1359-4311/$ - see front matter 2007 Elsevier Ltd. All rights reserved.

doi:10.1016/j.applthermaleng.2007.11.001

* Corresponding author. Tel.: +44 1224 272799; fax: +44 1224 272497.E-mail address:[email protected](S. He).

www.elsevier.com/locate/apthermeng

Available online at www.sciencedirect.com

Applied Thermal Engineering 28 (2008) 16621675
mailto:[email protected]:[email protected]


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of the work presented here is to study the interesting

behaviour exhibited by these experimental results using

detailed information on the mean flow and thermal fields

provided by the computational simulations.

Temperature (oC)

Pressure(M

Pa) Solid

Liquid

Gas

Supercritical

Critical Point7.38

Experimental condition (P=7.58 MPa, T=0-300oC)

31.1-56.4

0.5

0

(a) Phase diagram for CO2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

10 15 20 25 30 35 40 45 50

T (oC)

[103(kg/m3)],

Cp[105(J/kgK)],

k[10-1(W/mK)],[

10-4(kg/ms)]

Conductivity Density

Viscosity

Specific heat

(b) Properties near pseudo-critical temperature at 7.58Mpa

0

20

40

60

20 25 30 35 40 45 50

Temperature (oC)

Spe

cificheat(kJ/kg-C)

8.0 MPa

9.5 MPa

7.6 MPa

(c) Pressure dependence

Fig. 1. Carbon dioxide phase diagram and fluid property variations near the pseudo-critical temperature condition.

1664 S. He et al. / Applied Thermal Engineering 28 (2008) 16621675


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2. Special features of heat transfer to fluids at supercritical

pressure

2.1. Influence of property variation

As a consequence of the strong dependence of physical

properties on temperature, the process of heat transfer ina fluid at a pressure just above the critical value is generallymore complex than for ordinary fluids. Heat transfer coef-ficients often vary significantly with axial location along aheated passage due to the non-uniformity of physical prop-erties, especially where the fluid temperature is near thepseudo-critical value. In order to illustrate this, the varia-tion of heat transfer coefficient (htc) with bulk temperatureat a flowrate of 0.041 kg/s along a tube of diameter 0.01 mis shown inFig. 2. The htc was calculated using the DittusBoelter heat transfer correlation:

Nu 0:023 Re0:8Pr0:4 1

which was reorganized to give

h 0:028 _m0:8

D1:8

k0:6l0:4

c0:4p

2

It can be seen from Fig. 2 that htc peaks at the pseudo-critical temperature (32.2 C). It increases by almost fivetimes as the bulk fluid temperature varies by only a few de-grees. It is interesting to note that the value of the htc issimilar on either side ofTpc, although the other propertiesinvolved fall significantly as fluid temperature varies from

belowTpcto above it. Clearly there is a cancelling effect. Itis clear that the peak value of htc is largely due to thesharp rise in specific heat near the pseudo-critical temper-ature. The effect of the variation of fluid properties withtemperature in a real flow system can be much more com-plex. This will be demonstrated to some extent later in thepresent paper.

2.2. Influence of buoyancy

The influence of buoyancy can cause the effectiveness ofheat transfer to be either improved or impaired, throughenhancement or suppression of the production of turbu-lence. There are two basic mechanisms[18]by which buoy-

ancy can modify turbulence, namely, the direct (structural)effect, through buoyancy-induced production of turbulencekinetic energy, and the indirect (external) effect, throughthe modification of the mean flow.

For mixed convection in a vertical channel, the indirecteffect is normally the dominant one (see later). In such a sit-uation, the semi-empirical theory of Jackson and Hall [1]can be used to explain mixed convection heat transferbehaviour and to correlate experimental data. For buoy-ancy-opposed flows (downward flow in a heated passage),onset of influence of buoyancy causes the velocity to reducenear the wall and to increase in other regions. This modifi-cation of the mean flow field causes an enhancement of tur-

bulence production and, as a result, turbulent diffusion isimproved and the effectiveness of heat transfer is improved.In the buoyancy-aided case (upward flow in a heated pas-sage), the influence of buoyancy causes a reduction in thevelocity gradient over most of the flow, except in the regionvery close to the wall. As a result, turbulence production isreduced and the turbulent diffusion of heat is impaired. Ifthe influence of buoyancy is progressively increased, byreducing the flow rate and/or increasing the heating rate,the impairment of turbulence production and the deterio-ration of heat transfer become more and more marked. Astage is reached where turbulence production in the near-

wall region virtually ceases. This is sometimes describedas laminarizationof the flow (or reversed transition). Withfurther increase of buoyancy influence, negative values ofshear stress are generated in the core region and turbulencecan then be readily produced there. Consequently, theeffectiveness of heat transfer recovers. As the buoyancyinfluence becomes increasingly stronger, heat transfer canbecome more effective than under conditions of forced con-vection in the absence of buoyancy influence. It has beenestablished (see, for instance [19]) that with normal fluidssuch as air or water at atmospheric pressure the effect ofbuoyancy influence is negligible if the following criterionis satisfied:

Bo Gr

Re3:425Pr0:8< 5:6 107 3

2.3. Influence of flow acceleration

Turbulent heat transfer in channels can also be signifi-cantly affected by bulk flow acceleration caused by thermalexpansion of the fluid due to strong heating (particularly inthe case of gases). As in the case of acceleration caused byother means (spatial or temporal) the velocity profile ismodified in such a way that turbulence production is sup-

pressed and turbulent diffusion is impaired[16].

0

10

20

30

40

50

60

28 29 30 31 32 33 34 35 36

T ( C)

Heattransfercoefficient,h(W/m2-C) From Dittus-Boelter equation

Mass flow rate 0.041 kg/s

Pressure 7.58 MPa

Fig. 2. Effect of fluid properties on heat transfer coefficient under

conditions of forced convection.

S. He et al. / Applied Thermal Engineering 28 (2008) 16621675 1665


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Dissipation rate (e)

oqUe

ox

1

r

orqVe

or

o

ox l

ltre

oe

ox

1

r

o

or r l

ltre

oe

or

Ce1f1 1T

Pk Ce1f1 1T

Gk

Ce2f2qe

T qE 9

where

ltqClflkT; where T k=e

Pk lt 2 oU

ox

2

oV

or

2

V

r

2( )

oU

or

oV

ox

2" #

shear production 10

Gk q0

u0

gx

bltC1t

k

e oU

or

oV

ox oT

or

gx

gravitational production 11

where gx is the acceleration due to gravity in the x-direc-tion, being, +g or gfor downward and upward flows,respectively.

V2F turbulence model[22]

Turbulent kinetic energy (k)

oqUk

ox

1

r

orqVk

or

o

ox l

ltrk

ok

ox

1r

o

or r l lt

rk

okor

Pk Gk qeqD 12

Dissipation rate (e)

oqUe

ox

1

r

orqVe

or

o

ox l

ltre

oe

ox

1

r

o

or r l

ltre

oe

or

Ce1f11

TPk Ce1f1

1

TGk

Ce2f2 qeT

qE 13

Turbulent velocity scale v2

oqU v2

ox

1

r

orqV v2

or

" #

o

ox l

ltrk

ov2

ox

" #

1

r

o

or r l

ltrk

ov2

or

" #

qkf6qv2e

k 14

and the elliptic relaxation equation for f

0 o

ox

of

ox

1

r

o

or r

of

or

1

L2f

C11

L22=3v2=k

T

C2

L21

qkPk Gk

1

L25v2=k

T 15

where Pk and Gk are defined using Eqs. (10) and (11)

respectively.

lt qClv2T; where T max

k

e; 6

ffiffiffim

e

r ;

L CL max k3=2

e ; Cg

m3

e

1=4" #

C1 1:4; C2 0:3; Cg 70; CL 0:23:

Other model constants (Cl,Ce1,Ce2,rk,re), damping func-tions (fl, f1, f2, D, E) and boundary conditions at the wallare specified inTables 13.

The complete computational domain, which covered thewhole unheated and heated lengths of the test section andranged from the centre of the tube to the inner wall, wasdiscretized into a mesh of grids, typically, 120106(axialradial). The mesh was refined in the radial directiontowards the tube wall. It was also refined in the axial direc-tion in the region where the heating commenced. The meshwas adjusted in each individual run to ensure that the near-wall flow features were properly resolved and that the y+

value at the first node of the mesh was always less than 0.5.The staggered grid arrangement was used to define the

variables. The scalar parameters were defined at the gridpoints and the velocity components were defined on the

control volume surfaces. The QUICK scheme was usedfor approximating the convection terms in the momentumequations and the SMART scheme was used for othertransport equations for reasons of numerical stability.The SIMPLE scheme was used for coupling the pressureand the velocity fields. The resultant five-point coefficientmatrix system was solved iteratively using the line-by-linealgorithm TDMA; at any time, variables at a particularline were solved simultaneously; variables at the neighbour-ing lines were assumed to be known and values from theprevious iteration were used. The TDMA was used alter-nately in the axial and radial directions following the alter-

nating-direction-implicit (ADI) approach to accelerateconvergence.The NIST Standard Reference Database 23 (REF-

PROP) Version 7 was used for calculating the temperatureand pressure dependent properties of carbon dioxide. Todo this, the relevant FORTRAN subroutines supplied withthe Database were incorporated in the CFD code SWIRL.

Table 2Constants in the turbulence models

Model Code Cl Ce1 Ce2 rk re

Abe-Kondoh-Nagano (1994)[20] AKN 0.09 1.50 1.90 1.4 1.4

V2F (1998)[21] V2F 0 .22 1.4 1.9 1.0 1.3



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They were used to generate a property table at the begin-ning of each run with temperature intervals 5104 Cand pressure intervals 1104 Pa. During the iteration,the properties were updated by interpolation using thetable. This approach was adopted because it was foundthat the calculation was unacceptably slow if the NIST sub-routines were directly used in each iteration.

5. Computational results and discussion

5.1. Wall temperature and Nusselt number

Fig. 3shows the axial variation of wall and bulk temper-atures in Run 1 where the Reynolds number (Re) is high

and buoyancy parameter (Bo) is low. It can be seen fromthe figure that the measured wall temperatures for theupward and downward flow cases collapsed onto the samecurve. This confirms that the influence of buoyancy wasnegligible. Also, both the AKN and V2F models giveresults which clearly reflect this fact. However, the walltemperatures predicted by both of the models are some-what higher than the measured values.

Fig. 4shows comparisons between the predictions andthe measurements of wall temperature for Run 2 withupward and downward flow.Fig. 5shows the correspond-ing variations of predicted Nusselt number. It can be seenthat the experimental wall temperature increases veryslowly along the tube in the downward flow case. However,in the upward flow case, the wall temperature distributionexhibits an abrupt increase with distance from the start ofheating. It peaks at an x/dof about 50, then reduces rap-idly and reaches a value in the downstream region (x/d> 70) which is only slightly higher than that in the down-ward flow case. The only difference between an upward anddownward flow experiment under the same conditions offlow rate and heating is in the influence of buoyancy. Con-sequently this must be responsible for the large localisedincrease of wall temperature seen in the upward flow case.It is interesting to note that the value of buoyancy param-

eter based on the bulk parameters is only 4.3 10

7, which

according to extensive experimental work with fluids suchas water or air at atmospheric pressure could be taken asindicating that the influence of buoyancy ought to be insig-nificant. However, this is clearly not the case under theexperimental conditions considered here. This will be dis-cussed later in Section5.4.

As can be seen from Fig. 4, the predictions for upwardflow using both the AKN and V2F models only manageto capture the general trend of the observed variation of

wall temperature, whereas those for the downward flowfollow the experimental measurements closely. In the caseof upward flow, the simulations using both models do pre-dict overheating, but significantly stronger than that exhib-ited in the experiment. Interestingly, some reduction of walltemperature in the downstream region is predicted by bothmodels. In this respect, the AKN model does a somewhatbetter job than the V2F model.

5.2. Buoyancy effect

As mentioned earlier, the buoyancy parameter based on

the bulk properties for Run 2 (4.3107

) is lower than thelimiting value of Bo for onset of buoyancy effects of about6107 specified on the basis of work with air at normalpressure[19]. However, the measured distributions of walltemperature with upward and downward flow in thatexperiment were distinctly different. This apparently anom-alous result can be explained with the help of the modellingresults.

Fig. 6 shows the predicted radial distributions of fluidtemperature, specific heat and density at several axial loca-tions obtained using the AKN model for upward flow inRun 2. As already seen in Fig. 1, fluid properties varyextremely sharply with temperature in the region near thepseudo-critical value. Thus, to identify the region wheresignificant variations of properties are present, a band oftemperature specified by Tpc 0.8 C has been markedon Fig. 6a, which shows the fluid temperature profiles ata number of axial locations. To facilitate the present dis-cussion, this will be referred to as the large-property-varia-tion (LPV) region. It can be seen from the figure that,upstream, this is mainly located within a narrow layer veryclose to the wall. Proceeding downstream, the LPV regionmoves away from the wall and spreads more widely. Thiscan also be inferred from Fig. 6b where specific heat isshown, knowing that the location of the maximum value

of specific heat coincides with the centre of the LPV region.

Table 3Functions in the turbulence models

Code fl f1 f2

AKN 1 5Re0:75t

exp Ret200 2 h i

1 exp y14 2

1.0 1 0:3exp Ret6:5 2 n o

1 exp y

3:1

h i2V2F v

2

k 1 0:045 ffiffiffi

k

v2q 1.0Notes:Re t

k2

me, y

y

mue.

Table 4Damping terms D and Eand wall boundary conditions for kand e

Code D E Wall BC

AKN 0 0 kw 0; ew 2m ky2V2F 0 0 kw v2w fw 0; ew 2m

ky2



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It is interesting to note that although fluid temperature var-ies sharply near the wall, the variation of specific heat nearthe wall at downstream locations is much smaller than thatin the vicinity of the pseudo-critical temperature where thetemperature gradient is relatively small. This is generallytrue for the other fluid properties as well. Thus, the majorchanges of fluid properties occur in the vicinity of the loca-tion of the pseudo-critical temperature, which can be at alocation away from the wall. This is an important featureof heat transfer with a supercritical pressure fluids. Witha normal fluid, the maximum variation of fluid propertiesalways occurs near the wall where the maximum tempera-ture gradient occurs.

The influence of buoyancy can now be studied by exam-ining the distribution of the density (Fig. 6c). At x/d= 5,the density of the fluid varies extremely sharply very closethe wall (around y+ = 3). Such a large change of densitymight be taken as implying a strong local buoyancy influ-

ence leading to a significantly modified mean velocity pro-

file. However, since the influence is mainly limited to theviscous sub-layer, turbulence should not be significantlymodified and therefore no significant effect on the wall tem-perature distribution should be expected. Indeed this doesnot occur (seeFig. 4).

As the flow proceeds downstream, the LPV regionmoves away from the wall, reaching about y+ = 20 at thelocation x/d= 40. Further out, the density increases shar-ply, leading to a situation where lighter fluid occupies thewhole near-wall region of the shear flow with much heavierfluid in the core flow region. Consequently a large buoy-ancy force is experienced by the fluid in the viscous sub-layer and buffer layer regions and a significant influenceon turbulence production is expected and therefore a sig-nificant reduction in the effectiveness of heat transfer. Thisis supported by the wall temperature results shown inFig. 4, where the wall temperature peaks at this axial loca-tion. Further downstream (e.g. x/d= 80 & 120), the LPV

region moves further away from the wall and the spread

0

20

40

60

80

0 20 40 60 80 100 120

0 20 40 60 80 100 120

x/D

Temperatu

re(C)

wall temperature (Exp.)

upflow and downflow

wall temperature (AKN)

upflow and downflow

bulk temperature (Exp.)

bulk temperature (AKN)

(a) Predictions using the AKN model

0

20

40

60

80

x/D

Temperature(C)

wall temperature (Exp.)

upflow and downflow

wall temperature (V2F)upflow and downflow

bulk temperature (Exp.)

bulk temperature (V2F)

(b) Predictions using the V2F model

Fig. 3. Variations of wall and bulk fluid temperature for Run 1 from the present computational simulations and from the experiments of Fewster[11].



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0

50

100

150

200

250

0 20 40 60 80 100 120

0 20 40 60 80 100 120

x/D

Walltempera

ture(C)

upflow (AKN)

upflow (Exp.)

downflow (Exp.) downflow (AKN)

(a) Predictions using the AKN model

0

50

100

150

200

250

x/D

Walltemperature(C)

upflow (V2F)

upflow (Exp.)

downflow (Exp.) downflow (V2F)

(b) Predictions using the V2F model

Fig. 4. Variations of wall and bulk fluid temperature for Run 2 from the present computational simulations and from the experiments of Fewster[11].

0

50

100

150

200

250

300

0 20 40 60 80 100 120 x/D

Nu

AKN(downflow)

V2F(downflow)

AKN(upflow)

V2F(upflow)

Fig. 5. Predictions of Nusselt number with upward and downward flow for Run 2.



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is much wider. Here, the density first increases sharply nearto the wall as a result of the steep gradient of fluid temper-ature, then a slower but still very significant variation ofdensity occurs over the LPV region in the vicinity of thepseudo-critical temperature. Thus, the buoyancy force issignificant over a wide region although the distribution isvery different from that which would occur in a normalfluid. This has an interesting impact on turbulence whichwill be discussed in Section5.4.

Fig. 7shows the predicted radial profiles of the temper-

ature, specific heat, and density at several axial locations

for Run 2 for downward flow. Again there are regionswhere fluid properties (including density) vary dramaticallyover a narrow radial band. This might be expected to causesome enhancement of turbulence production. But, for theupstream stations (x/d = 5, 20 & 40), the LPV regionsare mostly restricted to a region very close to the wall,and therefore the effect on turbulence is actually quitesmall. This is consistent with the wall temperature varia-tions shown earlier where no significant influence of buoy-ancy was observed at those axial locations. For the last twostations (x/d= 80 & 120), the buoyant band moves into the

buffer layer region and a significant effect on turbulence

[ x/D=5 (o), x/D=20 (+), x/D=40 (*), x/D=80 () , x/D=120 ( ) ]

0 50 100 150 20020

25

30

35

40

45

50

y+

T(oC)

(a) Fluid temperature

0 50 100 150 2000

2

4

6

8

10

12

14x 10

4

y+

cp(J.kg

-1.K

-1)

(b) Specific heat

0 50 100 150 2000

100

200

300

400

500

600

700

800

900

y+

(kg.m

-3)

(c) Density

Fig. 6. Near-wall variations of temperature, specific heat and density for

Run 2 with upward flow predicted using the AKN model.

0 20 40 60 80 10020

25

30

35

40

45

50

y+

T(oC)

(a) Fluid temperature

0 20 40 60 80 1000

2

4

6

8

10

12

14x 10

4

y+

cp(J.kg

-1.K

-1)

(b) Specific heat

0 20 40 60 80 1000

100

200

300

400

500

600

700

800

900

y+

(kg.m

-3)

(c) Density

[ x/D=5 (o), x/D=20 (+), x/D=40 (*), x/D=80 () , x/D=120 ( ) ]Fig. 7. Near-wall variations of temperature, specific heat and density forRun 2 with downward flow predicted using the AKN model.



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associated with the particular distribution of buoyancyforce resulting from the condition where the LPV regionextends right into the core of the pipe. It benefits heattransfer since turbulence produced near the wall is of primeimportance in determining the effectiveness of heat trans-fer. As a result the wall temperature falls to a low level.

This is seen in both the experiment and the simulations.

5.4. Effect of buoyancy on turbulence production

As discussed in the Introduction, there are two mecha-nisms by which buoyancy can affect turbulence, namely,the indirect (external) effect, through production by shear,on which attention has been focussed so far in this paperand the direct (structural) effect, through production bybuoyancy [18]. The latter is associated with the fact thatdensity fluctuation can contribute directly to turbulence

production/destruction by correlating with the turbulentfluctuation of axial velocity component and creating a tur-bulent mass flux and hence a transfer of potential energy.This is represented in the transport equations for the turbu-lent kinetic energy and its dissipation by means of source(sink) terms. These terms can be modelled using the simple

gradient diffusion hypothesis (SGDH) or the general gradi-ent diffusion hypothesis (GGDH) in the eddy viscositymodel framework[23]. GGDH is generally regarded as amore soundly-based approach and has, therefore, beenused in the current study.

Fig. 11 shows the predicted turbulence production byshear and that by buoyancy, in Run 2 for upward anddownward flow at several axial locations using the AKNmodel. It can be seen fromFig. 11df that with downwardflow buoyancy production is always positive, acting toincrease turbulence, but the values are two orders of mag-

0

500

1000

1500

2000

2500

3000

0 0.2 0.4 0.6 0.8 1y/R

Turbulentshearproduction(kg.m

-2)

akn-downward

akn-upward

x/D=20

-30

-20

-10

0

10

20

30

0 0.2 0.4 0.6 0.8 1y/R

Buoyancyproduction(kg.m

-2)

akn-downward

akn-upward

x/D=20

(a) Turbulent shear production (x/D=20) (d) Buoyancy production (x/D=20)

0

500

1000

1500

2000

2500

3000

0 0.2 0.4 0.6 0.8 1y/R


-2)

akn-downward

akn-upward

x/D=40

-30

-20

-10

0

10

20

30

0 0.2 0.4 0.6 0.8 1y/R


-2)

akn-downward

akn-upward

x/D=40

(b) Turbulent shear production (x/D=40) (e) Buoyancy production (x/D=40)

0

1000

2000

3000

4000

5000

6000

7000

0 0.2 0.4 0.6 0.8 1y/R


-2)

akn-downward

akn-upward

x/D=120

-30

-20

-10

0

10

20

30

0 0.2 0.4 0.6 0.8 1y/R


-2)

akn-downward

akn-upward

x/D=120

(c) Turbulent shear production (x/D=120) (f) Buoyancy production (x/D=120)

Fig. 11. Turbulent shear production and buoyancy production predicted for Run 2 using the AKN model.



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nitude smaller than the shear production, thereby makingit a negligible contribution in real terms. The buoyancyproduction with the upward flow can be positive or nega-tive, i.e., causing either turbulence energy production ordestruction, but again the values are much smaller thanthe shear production. Therefore the direct effect of buoy-

ancy on turbulence is negligible in real terms. This conclu-sion is consistent with the observations made in the studyof buoyancy-influenced flow with normal fluids, see Cottonand Jackson[24].

Thus, the effect of buoyancy on flow and heat transferidentified earlier in this paper has almost entirely resultedfrom the indirect effect, which can be studied by examiningthe shear production shown in Fig. 11ac. For downwardflow, the shear production essentially remains unchangedat x/d locations of 20 and 40. As already seen in Fig. 7,although the radial variation of density is very large atthose locations, it is restricted to a region very close tothe wall (y+ < 5), thereby having little influence on turbu-

lence production. At location x/d= 120, however, thelighter fluid occupies the region up to y+ 10 and there-fore, the shear production is greatly increased as can beseen fromFig. 11c.

For upward flow, the turbulence shear production is sig-nificantly reduced at x/d= 20. It is further reduced (nearlyto zero) at x/dof 40. This is consistent with the buoyancyforce being large at those locations, as seen earlier in Fig. 6.The shear production at x/d= 120 is interesting: it is lar-gely damped out in the centre but greatly increased nearthe wall. It can be seen from Fig. 6c that the buoyancyforce is very big at this axial location which implies that

with a normal fluid the flow would have reached the recov-ery regime. Under such conditions, with a normal fluid,turbulence would be produced in the core but wouldremain low near the wall, which is exactly opposite to whatexhibited in this study of flow and heat transfer with a fluidat supercritical pressure. Clearly, this is due to the particu-lar distribution of the buoyancy force which can be presentin a supercritical pressure fluid which apparently can pro-mote very effective renewed turbulence production in thenear-wall region. This buoyancy distribution is associatedwith the condition where the large-property-variation(LPV) region is situated in a region away from the wall,as was illustrated earlier (seeFig. 6a).

6. Conclusions

Both the low Reynolds number k e model and theV2F model were able to capture the general trends ofthe interesting wall temperature behaviour observedwith upward flow in some experiments with a fluid ata pressure just above the critical value. However thedetailed variation of the wall temperature predictedusing each of the models was rather different from thatin the experiments. The prediction of the ke model

was the better of the two for the conditions considered.

With downward flow, the measured distribution of walltemperature did not exhibit the localised non-uniformityseen with upward flow and the predicted wall tempera-ture distributions were closely consistent with observedbehaviour.

It has been found that the effect of buoyancy on heat

transfer to fluids at supercritical pressure cannot be reli-ably assessed just using a buoyancy parameter based onbulk properties. Strong effects of buoyancy may occureven when the buoyancy parameter is lower than thevalue for which such effects have been found to be smallin the case of normal fluids.

The effect of buoyancy on turbulence is largely depen-dent on the extent of the large-property-variation(LPV) region, i.e., the region in the vicinity of the pointwhere the fluid temperature takes the pseudo-criticalvalue.

When LPV region is located very close to the wall, buoy-ancy has little influence on turbulence production. When

it extends into the buffer layer, it causes a large reduc-tion in turbulence production with upward flow which,in turn, causes a significant reduction in the effectivenessof heat transfer.

Under some conditions, the very non-uniform distribu-tion of buoyancy force may cause a reduction of turbu-lence in the core but a significant enhancement near thewall leading to enhanced heat transfer. This interestingphenomenon could prove to be an important featureof heat transfer to fluids at supercritical pressure whichmight possibly be exploited in practical systems.

Under the conditions examined in this study, the direct

effect of buoyancy on turbulence energy production wasfound to be negligible in comparison with the indirecteffect. Thus, the effect on heat transfer seen was almostentirely due to the shear production effect caused bythe distortion of the mean flow as a result of the influ-ence of buoyancy.

Acknowledgements

The authors gratefully acknowledge the funding for thisproject provided by UK Engineering and Physical SciencesResearch Council (EPSRC) through Grant GR/S19424/02and the use of the valuable experimental data produced byJonathan Fewster in his PhD work.

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A Computational Study of Convective Heat Transfer to Carbon Dioxide

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