J. of Supercritical Fluids 51 (2009) 221–229

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The Journal of Supercritical Fluids

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The significance of the buffer zone of boundary layer on convective heat transferto a vertical turbulent flow of a supercritical fluid

Majid Bazargan ∗, Mahdi MohseniDepartment of Mechanical Engineering, K.N. Toosi University of Technology, 15 Pardis St, Mollasadra Ave, P.O. Box 19395-1999, Tehran 1999 143 344, Iran

a r t i c l e i n f o

Article history:Received 22 May 2009Received in revised form 15 August 2009Accepted 17 August 2009

Keywords:Supercritical fluid flow

a b s t r a c t

A two-dimensional model is developed to simultaneously solve the momentum and energy equations inorder to predict convective heat transfer to an upward turbulent flow of supercritical carbon dioxide in around tube. It is very important to choose a proper turbulence model. An appropriate turbulence model,based on the previous studies, has been selected. The main focus of the present study is on significanceof the buffer zone in the boundary layer. The results of this study indicate that in enhanced regime ofheat transfer, the peak of heat transfer coefficient occurs when the pseudo-critical temperature, or thesituation of maximum heat capacity, lies within the buffer layer. In deteriorated regime of heat transfer,

Convective heat transferBuffer layerTN

the extent of the laminar sub-layer appears to be changed so that the buffer zone is experienced at awall.ratur

1

neltttehiea

itopawf

ta

m

0d

urbulence modelingumerical solution

farther distance from thea jump in the wall tempe

. Introduction

The variations of the fluid properties at supercritical conditionsear the critical point are large. The typical changes of the prop-rties for supercritical carbon dioxide are shown in Fig. 1. Thearge variation of the properties has a considerable effect on theurbulence and structure of the flow. In turn, these changes alterhe temperature and velocity profiles. In these circumstances heatransfer can grow considerably in comparison with constant prop-rty flows. Such heat transfer peculiarities of supercritical fluidsave resulted in an increase in the utilization of supercritical flu-

ds in engineering systems within the past several decades. Somexamples are the application of the supercritical fluids in fossil fuelnd nuclear power plants and the chemical industry [1].

Another recent application of heat transfer to supercritical fluidss in the waste management industry [1,2]. The idea is to destroyoxic aqueous waste and the method is called supercritical waterxidation (SCWO). Along with many thermodynamic and transport

roperties, the solubility properties of water dramatically changet supercritical conditions. At normal temperature and pressure,ater is a good solvent for inorganic compounds and a non-solvent

or many organics. On the contrary, supercritical water is an excel-

Abbreviations: GGDH, generalized gradient diffusion hypothesis; HTC, heatransfer coefficient; LRN, low Reynolds number; MK, turbulence model of Myoungnd Kasagi; SCF, supercritical fluid; SCWO, supercritical water oxidation.∗ Corresponding author. Tel.: +98 21 84063239; fax: +98 21 88677274.

E-mail addresses: bazargan@kntu.ac.ir (M. Bazargan),mohseni@sina.kntu.ac.ir (M. Mohseni).

896-8446/$ – see front matter © 2009 Elsevier B.V. All rights reserved.oi:10.1016/j.supflu.2009.08.004

This causes a delay in the turbulent diffusion near the wall and leading toes.

© 2009 Elsevier B.V. All rights reserved.

lent solvent for organic materials but not for inorganic salts. Thisleads to the idea of supercritical water oxidation. A simplified blockdiagram of the process is shown in Fig. 2.

Destruction efficiencies of 99.99% or higher have been reportedfor a variety of toxic and nontoxic materials treated by SCWO. Theclosed operating cycle makes it controllable and assures a completecontainment of all effluents. No harmful byproducts such as NOx areproduced and SCWO is environmentally friendly. Heat transfer datais essential to improve the design of SCWO systems. The main pur-pose of this study is to investigate heat transfer to turbulent flowof supercritical water towards optimizing the design of SCWO sys-tems. The experimental heat transfer data available in the literaturefor supercritical carbon dioxide is larger than those of supercriticalwater. Thus, supercritical CO2 has been used in this study to facili-tate the comparison of the models predictions against experiments.The results of the model developed in this study are equally validfor supercritical CO2, supercritical water and other SCFs.

Heat transfer enhancement in supercritical environment doesnot occur for the whole ranges of heat and mass fluxes. Considera heated upward turbulent flow of a supercritical fluid in a roundtube. Keeping the mass flux constant, by an increase in heat fluxthe heat transfer coefficient decreases. This trend continues untilthe enhancement of the heat transfer diminishes and the deterio-ration in heat transfer, which is manifested in large increase of thewall temperature, appears. Thus for two heated flows under dif-

ferent heat fluxes and otherwise identical, it is possible to have anenhanced heat transfer at low heat flux and degraded heat transferat high heat flux.

It is important to note that both enhancement and deteriora-tion of heat transfer to supercritical fluids may occur only when

222 M. Bazargan, M. Mohseni / J. of Supercritical Fluids 51 (2009) 221–229

Nomenclature

Cp specific heat capacity at constant pressure(J kg−1 K−1)

Cε1, Cε2 constants in the ε equationC� constant in eddy viscosity modelD source term in k equationE source term in ε equationf1, f2 functions in dissipation equationf� damping functiong gravity acceleration (m s−2)Gk buoyant production (kg m−1 s−3)h heat transfer coefficient (W m−2 K−1)H enthalpy (J kg−1)k turbulent kinetic energy (m2 s−2)p pressure (N m−2)Pk turbulent shear production (kg m−1 s−3)Prt turbulent Prandtl numberq heat flux (W m−2)Q heat transfer rate (W)Q internal heat source (W m−3)r transversal direction of flow (m)Sm source term in momentum equation (kg m−2 s−2)T temperature (K)u, v velocity components in x and r direction (m s−1)u+ dimensionless u velocityx axial direction of flow (m)y+ non-dimensional distance from wall

Greek symbolsˇ thermal expansion coefficient (K−1)ε dissipation rate of k (m2 s−3)ϕ dissipation function (s−2)� coefficient of diffusion� thermal conductivity (W m−1 K−1)� molecular viscosity (kg m−1 s−1)�t turbulent viscosity (kg m−1 s−1)� density (kg m−3)�k, �ε turbulent Prandtl number for k and ε shear stress (N m−2)

Subscriptb bulkpc pseudo-critical

tapi

porndTvfitt

s

details.In this study a two-dimensional numerical model has been

developed in order to investigate the effect of the buffer zone of theboundary layer on turbulence activities and hence heat transfer to

t turbulentw wall

he fluid bulk temperature is less than and the fluid temperaturet the wall is greater than the pseudo-critical temperature. Theseudo-critical temperature, for any given supercritical pressure,

s the temperature at which the fluid properties vary the most.In the case of heat transfer deterioration, once the bulk fluid tem-

erature exceeds the pseudo-critical temperature along the coursef flow the wall temperature, which had already been risen as theesult of heat transfer deterioration, starts decreasing. Past studiesame buoyancy and the fluid acceleration as two major causes foregradation of heat transfer under supercritical conditions [3–7].he fluid acceleration and buoyancy both are generated due to largeariation of density near the critical region. They cause velocity pro-

le to change. Consequently the local shear stress distribution andurbulence production across the flow vary and laminarization ofhe flow occurs.

There are large discrepancies between the results of varioustudies. A major source of such discrepancies, many researchers

Fig. 1. Thermo-physical properties of carbon dioxide at supercritical pressure of8.12 MPa.

believe to be the difficulty in turbulence modeling of supercriti-cal fluid flows [5,8–10]. As such, different turbulent models andexplanations about the turbulence behavior at the near wall regionhave been suggested [6,7,9–14]. Despite many interesting resultsreported in these and other researches, the detailed explanationfor deteriorated mode of heat transfer is yet incomplete. More dataand further investigations are necessary to discover the physics ofthe complicated phenomenon of the heat transfer to supercriticalfluids, especially the conditions leading to deterioration in more

Fig. 2. Block diagram for supercritical water oxidation process.

M. Bazargan, M. Mohseni / J. of Supercr

F

aBed

2

iwflieto

tlasvrntarfflt

aeidsbf

3

ttd

ig. 3. Various zones in turbulent boundary layer flows adjacent to a solid wall.

n upward flow of the supercritical carbon dioxide in a vertical pipe.oth enhanced and deteriorated modes of heat transfer are consid-red. The model is evaluated against the available experimentalata and appropriate conclusions are derived accordingly.

. Velocity distribution

The boundary layer in turbulent flows is usually divided into thenner layer adjacent to the wall and the outer layer away from the

all. It appears to be useful in many occasions to divide the innerow of the boundary layer into different layers to facilitate obtain-

ng the velocity profile. A dimensionless parameter y+ is used toxpress the normal distances from the wall. The inner layer, as illus-rated in Fig. 3, includes the laminar viscous sub-layer, the middler buffer layer, and the logarithmic law or fully turbulent layer.

In the laminar sub-layer the molecular diffusivity is much largerhan turbulent diffusivity and the velocity profile is linear. Theaminar sub-layer extends from the wall (y+ = 0) to the values of y+

round 5–10. In the log-law or fully turbulent layer the Reynoldstresses is very large compared to molecular interactions and theelocity profile follows a logarithmic law. This layer occupies theegion where 30–60 < y+ < 500. Buffer layer lies between the lami-ar sub-layer and the log-law layer as a transition region betweenhem. In the buffer layer the molecular and turbulent activitiesre both important. The outer flow is located beyond the inneregion. The measurements show that the velocity profile deviatesrom the logarithmic law in the outer region of the flow. The outerow usually occupies about 10–20% of the entire boundary layerhickness [15].

The thicknesses of various regions of the boundary layer whichre mentioned above are approximate values. The exact extent ofach layer depends on the parameters such as pressure, turbulencentensity, surface roughness and the flow configuration. That is whyifferent range of values has been reported for y+ in the literature topecify the end of the laminar sub-layer or the onset of the fully tur-ulent layer. Consequently, a variety of magnitudes may be foundor the start and end of the buffer layer [15,16].

. Mathematical Modeling

The governing equations, the boundary and flow conditions, andhe method of solution used in this study to model the convec-ion heat transfer to the turbulent flow of the supercritical carbonioxide in a round vertical tube are described below.

itical Fluids 51 (2009) 221–229 223

3.1. Governing equations

The basic governing equations including the conservations ofmass, momentum and energy, together with transport equationsmodeling the turbulence are employed to model the flow. Note thatthe flow is considered steady state and thus, the temporal termshave been eliminated. The conservation of mass states that

∇ · (�V) = 0 (1)

For turbulent flows, by using the Reynolds time averagingmethod, the momentum equation results to:

∇ · (�VV) = −∇p + ∇ · ( − t) + Sm (2)

where

= �[∇V + (∇V)T − 2

3ı∇ · V

](3)

By using the Bossinesque theorem, the turbulent shear stresscan be found from the following equation in which the Reynoldsstresses are related to the average velocity gradient.

t = �t[∇V + (∇V)T] − 23

ı(�k + �t∇ · V) (4)

This method of modeling is known as the “eddy viscosity mod-eling” in which for the eddy viscosity we have:

�t = �C�f�k2

ε(5)

The transfer equations for turbulent kinetic energy (k) and therate of its dissipation (ε) are found by the following equations:

∇ · (�Vk) = ∇ ·[(

� + �t

�k

)∇k

]+ Pk + Gk − �(ε + D) (6)

∇ · (�Vε)=∇ ·[(

�+�t

�ε

)∇ε

]+Cε1f1

ε

k(Pk+Gk)−Cε2f2�

ε2

k+�E (7)

where

Pk = −�u′iu′

j

∂ui

∂xj(8)

and

Gk = −ˇ�gi

(0.3

k

εu′

iu′

j

∂T

∂xj

)(9)

Eq. (9) is obtained with the use of GGDH modeling. More detailsmay be found in [17]. According to the results of a parallel study[18] done by the same authors of this study, the turbulence modelof Myoung and Kasagi [19] leads to the predictions of heat transfercoefficients which are in better agreement with the experimentaldata for both regimes of heat transfer. Hence, this model has beenused for turbulence modeling in this study.

The energy equation in terms of enthalpy will be in the form of:

∇ · (�VH) = ∇ ·(

�∇T + �t

Prt∇H

)+ DP

Dt+ Q + �ϕ (10)

In the above equation, as in the momentum equation, the Bossi-nesque assumption has been used for modeling turbulent heat flux.

For calculating the convective heat transfer coefficient, New-ton’s cooling law as expressed below has been used:

Q = hA(Tw − Tb) ⇒ qw = h(Tw − Tb) (11)

In which qw is the heat flux at the wall, h is the convective heattransfer coefficient, Tw and Tb are the wall surface temperature and

the average bulk temperature of the fluid in the pipe, respectively.

In other words, we will have:

h = qw

Tw − Tb= �

Tw − Tb

∂T

∂r

∣∣∣∣w

(12)

224 M. Bazargan, M. Mohseni / J. of Supercritical Fluids 51 (2009) 221–229

Table 1Test conditions for the inlet pressure of 8.12 MPa and inlet temperature of 270 K.

2 2

l

H

o

3

efldoSmts

uctptndcd

3

dt[teipn

mpTcctu

detuclau

State Description G (kg/m s) q (kW/m ) L (m)

CASE I Enhanced heat transfer 1200 50 14CASE II Deteriorated heat transfer 400 30 7

The bulk enthalpy in each cross-section is calculated by the fol-owing equation:

b = 1GA

∫ R

0

�VH(2�r dr) = 1GR2

∫ R

0

�VHr dr (13)

In order to find the bulk temperature one can also use the tablef properties for a given fluid at a given pressure and bulk enthalpy.

.2. Boundary and flow conditions

The experimental study of Song et al. [20] has been used as ref-rence in order to evaluate the results of the present study. Theow geometry under consideration is a pipe with a 9 mm internaliameter. Their data corresponding to mass flux and wall heat fluxf (1200 kg/m2 s, 50 kW/m2) and (400 kg/m2 s, 30 kW/m2) are used.uch flow conditions correspond to the enhanced and deterioratedodes of heat transfer, respectively. The supercritical pressure of

he experiments is 8.12 MPa. A summary of the flow conditions istated in Table 1.

The known mass flux and the uniform temperature profile aresed as the inlet boundary conditions. The constant heat flux isonsidered as for the thermal boundary condition at the wall. Inhe outlet, the axial gradient of properties such as temperature andressure are considered to be linear. Due to the axial symmetry ofhe flow, ∂/∂r = 0 is applicable for different properties. It should beoted that the pseudo-critical temperature and pressure of carbonioxide are T = 304.13 K and P = 7.37 MPa, respectively. For the cal-ulation of the thermodynamic properties of supercritical carbonioxide, the NIST database [21] has been used.

.3. Method of solution

The flow is axisymmetric and the equations are solved in a two-imensional cylindrical coordinate system. In order to discretizehe governing equations, the finite volume method is utilized22,23]. The SIMPLE algorithm is used alongside the staggered grido simultaneously solve the velocity and pressure equations. Tostimate the convective terms in the equations, the hybrid schemes used. The SIMPLE algorithm and the hybrid scheme used in theresent model are described extensively in [22,23]. A brief expla-ation is provided here.

The basic steps taken in the SIMPLE algorithm (semi-implicitethod for pressure-linked equations) are as follows. An initial

ressure field is used to solve the discretized momentum equation.he intermediate velocity field obtained is employed to solve theontinuity equation. Then, the values of pressure and velocity areorrected by the relevant relations. Other scalar transport equa-ions are discretized and solved next. This procedure is repeatedntil all the variables converge.

Note that an under-relaxation factor is needed to be applieduring the iterative process, otherwise the pressure correctionquation is likely to diverge. The speed of convergence and thushe cost of calculations is highly dependent on correct choice of

nder-relaxation factors. The smaller such factor is, the slower theonvergence is reached. However, a value too large causes the oscil-atory and even divergent iterative solutions. It is a matter of trialnd error for each flow configuration to find the optimum value ofnder-relaxation factor and it is a case dependent problem.

Fig. 4. Variations of wall, bulk and pseudo-critical temperatures and heat transfercoefficient for CASE I.

The hybrid scheme is based on an alternate use of central andupwind differencing methods. Depending on whether the Peclectnumber is small or large, the central differencing or the upwindschemes are employed, respectively. The non-dimensional Peclectnumber is locally defined as Pe = �ϕıx/� and reflects the relativestrength of the flow convection to diffusion magnitudes.

A larger number of grids is used close to the wall. The distancesbetween the calculating nodes grow by a factor of 1.05 as we moveaway from the wall. The optimum number of 126 for the grids in theradial direction is obtained. The greater number of the grids doesnot improve the results any further. In axial direction, dependingon the length of the tube and the regime of the flow, the number ofgrids varies between 370 and 720. In the low Reynolds k–e modelsthe grids should fully cover the flow near the wall, i.e., they areextended up to the edge of the wall covering the buffer layer as wellas the viscous sub-layer. To satisfy this requirement, the distancebetween the first calculating node and the wall is chosen to be sosmall to have y+ < 0.3. In some studies the criterion of y+ < 0.5 hasalso been used to fulfill the above condition [6,7,12–14].

4. Results

The large variation of the fluid properties in supercritical con-ditions couples strongly the equations of momentum and energy.Thus, the convergence of the solutions to the equations becomesmore difficult to reach. This phenomenon also necessitates thegreater number of meshes to be used, especially in radial direc-tion, compared to the cases at normal pressures (constant propertyflows). Thus more calculating power is required to deal withsuch highly variable property flows. The results of this study arepresented for enhanced and deteriorated modes of heat transferseparately as follows.

4.1. Enhancement of heat transfer

The peak of heat transfer coefficient occurs once Tb < Tpc < Tw, i.e.,the pseudo-critical temperature lies between the fluid bulk tem-perature and the temperature of the fluid at the wall. This can be

seen in the results of the numerical solution, shown in Fig. 4, wherethe variations of the wall temperature, bulk temperature, pseudo-critical temperature, and the heat transfer coefficient are presented.A simple explanation for this phenomenon is that the peak of spe-cific heat of the fluid occurs at the pseudo-critical temperature, as

M. Bazargan, M. Mohseni / J. of Supercritical Fluids 51 (2009) 221–229 225

F[

wrtIidet

pF

tlTocTici

Ffbo

ig. 5. Comparison of numerical results with the experimental data of Song et al.20] for the CASE I.

as shown earlier in Fig. 1. Thus the condition Tb < Tpc < Tw war-ants that a portion of the fluid, somewhere between the wall andhe flow centerline, experiences the pseudo-critical temperature.t is true that by transition of the fluid from liquid like to vapor liken the course of a heated flow, the thermal conductivity of the fluidecreases. The effect of large increase of the specific heat is, how-ver, dominant over the effect of thermal conductivity and heatransfer enhancement will be observed.

To evaluate the present numerical model, the results are com-ared with experimental data of Song et al. [20] and are shown inig. 5.

Some researchers, e.g. [24], have stated that the peak of heatransfer coefficients occur at the flow cross-sections where theargest amount of fluid exists at the pseudo-critical temperature.his is almost equivalent to say that while holding the conditionf Tb < Tpc < Tw, the closer the bulk fluid temperature to the pseudo-ritical temperature, the larger the heat transfer coefficients will be.

his idea is quite worthwhile to be further investigated, becauset can help to better understand the physics behind the compli-ated phenomenon of the heat transfer to supercritical fluids. It ismportant to discover the profiles of various flow parameters, espe-

ig. 6. Radial variation of the flow region with peak values of Cp along the test sectionor CASE I. Shaded area represents the portion of flow experiencing temperaturesetween 0.999Tpc and 1.001Tpc . Heat transfer coefficient is also shown for the sakef comparison.

Fig. 7. Temperature profiles normalized by pseudo-critical temperature for CASE I.Different profiles correspond to various locations along the test section.

cially temperature profile, coinciding with occurrence of the localmaximum heat transfer coefficient.

The variations of the heat transfer coefficient along the tube areshown in Fig. 6. In addition, the flow region where the fluid tem-perature is very close to pseudo-critical temperature is specifiedin Fig. 6 by dashed lines. Note that the 2nd vertical access “y/R”in Fig. 6 refers to the dashed lines and has nothing to do with theheat transfer coefficients. The criterion of “a temperature close tothe pseudo-critical temperature” has been arbitrarily defined as0.999 < T/Tpc < 1.001. It can be seen that for the flow cross-sectionwhere Hb < 280 kJ/kg, the fluid temperature at the wall has notreached the pseudo-critical temperature yet, i.e., Tb < Tw < Tpc. Fardownstream of the flow where, Hb > 360 kJ/kg the entire flow isheated enough so that the fluid bulk temperature is larger thanthe pseudo-critical temperature, i.e., Tpc < Tb < Tw. For the fluid bulkenthalpies from 280 to 360 kJ/kg, the pseudo-critical region movesfrom the region adjacent to the wall towards the flow centerline.It can be seen from Fig. 6 that the maximum heat transfer coef-ficient occurs where 315 < Hb < 330 kJ/kg. It is interesting to notethat in this region the pseudo-critical temperature is experiencedwithin the buffer layer. The direct correlation of the maximum heattransfer coefficient with the flow conditions inside the buffer layerbecomes more evident when we see that for the flow cross-sectionswith 335 < Hb < 355 kJ/kg the larger portion of the flow is in thepseudo-critical region, but the heat transfer coefficient are not thelargest. That is simply because the buffer layer, though occupy-ing a very narrow layer of the flow, is not at the pseudo-criticaltemperature.

To better appreciate the above fact, a number of temperatureprofiles at different cross-sections along the flow are plotted andshown in Fig. 7. The temperature profiles are normalized by makinguse of the pseudo-critical temperature to facilitate their compar-ison. Note that the temperature profiles are shown only for theregion close to the wall 0 < y+ < 50 where the temperature variationis significant. Away from the wall, in the core region, the tem-perature variations is small anyway and there will be no point tocompare them at different flow cross-sections along the tube.

The uppermost curve in Fig. 7, containing the highest tempera-

tures, corresponds to the flow downstream where both the wall andbulk temperatures have already exceeded the pseudo-critical point.The most below curve of Fig. 7 include the lowest temperaturesassociated with the flow upstream where neither of the bulk andwall temperatures has reached the pseudo-critical point. Variations

226 M. Bazargan, M. Mohseni / J. of Supercritical Fluids 51 (2009) 221–229

ottctaom

tttsfiflahlt

raaTstiti

toicoohscv(pr

Table 2functions, coefficients and boundary conditions in MK and Chien k–e models.

Quantity MK model Chien model

f1 1 1

f2

[1 − 0.22 exp

(−Re2

t36

)][1−exp−y+

5

]21 − 0.22 exp

(−Re2

t36

)f�

[1 − exp

(−y+70

)](1+3.45Re0.5

t

)1 − exp(−0.0115y+)

D 0 2� ky2

E 0 −2� εy2 exp(−0.5y+)

kw 0 0

εw � ∂2k∂y2 0

C� 0.09 0.09Cε1 1.4 1.35

The Van Driest eddy viscosity model is as follows:

�t = �l2m

∣∣∣∣∂u+

∂y+

∣∣∣∣ (14)

Fig. 8. Cp profiles corresponding to the temperature profiles shown in Fig. 6.

f temperature in these two curves are the largest, because the heatransfer coefficients are low in both situations. It is apparent fromhe cooling law of Newton, q = h(Tw − Tb), that once the heat flux isonstant, the low heat transfer coefficient is synonyms to the largeemperature variation. On contrary, high heat transfer coefficientccompanies low temperature variations. For instance, the peakf heat transfer coefficient occurs at the flow cross-section withinimum difference between the wall and bulk fluid temperatures.Between these two regions, where the entire flow is either at

emperatures less than or greater than the pseudo-critical point,here exists the flow region where the pseudo-critical tempera-ure occurs somewhere between the wall and the centerline. Astated earlier, for a heated flow the pseudo-critical temperaturerst appears at the wall and moves away from the wall towards theow centerline along the course of flow. The temperature profilet the flow cross-section where the maximum heat transfer occursas been specified in Fig. 7. It is clear that out of the laminar sub-

ayer the fluid temperature remains almost constant and equal tohe pseudo-critical temperature.

Wherever the fluid reaches its pseudo-critical temperature (theegion of maximum heat capacity), the temperature profile flattenst that point. Consequently, the overall difference between the wallnd bulk temperature reduces that is heat transfer is enhanced.he variations of the Cp corresponding to the temperature profileshown in Fig. 7 are illustrated in Fig. 8. It is interesting to see thathe largest portion of the fluid experiences the region of the max-mum heat capacity, if the pseudo-critical temperature lies withinhe buffer layer. Note that the variations of the fluid properties,ncluding Cp, for y+ > 50 are very small.

Here is an explanation that why it is the temperature condi-ion at the buffer layer that matters with respect to occurrencef the heat transfer enhancement. Since the temperature profiles already flat at the core region, the coincidence of the pseudo-ritical temperature in this region may further flatten the profilenly slightly and cannot significantly contribute to enhancementf heat transfer. Adjacent to the wall, in the laminar sub-layer, theeat transfer is poor in general and the temperature profile is veryteep. The existence of the pseudo-critical temperature in this layerannot leave a noticeable effect as it may be sensed only within a

ery narrow portion of the flow cross-section. In the buffer layer10 < y+ < 50), however, the occurrence of the pseudo-critical tem-erature may make a substantial change. Compared to the coreegion the temperature profile is not flat and there is a plenty of

Cε2 1.8 1.8�k 1.4 1�ε 1.3 1.3

room for its further flattening. Meanwhile the temperature pro-file is not as steep as that of the laminar sub-layer. In addition thethickness of the buffer layer is much larger than that of the lam-inar sub-layer so that the temperature profile gets the chance toeffectively become flattened if the fluid reaches its pseudo-criticaltemperature there.

4.2. Effect of turbulence modeling

It is important to know whether the significance of the bufferlayer as explained above is restricted to the specific type of tur-bulence model used or not. To investigate this matter a variety ofturbulence models were examined. Again in all cases, the peak ofheat transfer coefficients was observed where the pseudo-criticalregion occurred in the buffer layer. The results obtained by makinguse of the algebraic eddy viscosity model of Van Driest [25] and theLRN k–e turbulence model of Chien [26] are presented here as twoexamples.

Fig. 9. Temperature profiles normalized by pseudo-critical temperature for CASE Iby using of Van Driest model.

M. Bazargan, M. Mohseni / J. of Supercritical Fluids 51 (2009) 221–229 227

FI

w

l

Tm

cfophwl

nocFfl

Fl

of heat transfer, are shown in Fig. 13. The degraded heat transfercoefficients occur at the flow cross-section where the fluid bulkenthalpy varies from 235.9 to 300.5 kJ/kg.

Since the heat transfer coefficients in CASE II are smaller thanthose of CASE I, the difference of the fluid bulk and wall tempera-

ig. 10. Temperature profiles normalized by pseudo-critical temperature for CASEby using of Chien model.

here

m = 0.41y+[

1 − exp

(− y+

26

)]

he functions, coefficients and boundary conditions in Chien k–eodel together with those of MK model are listed in Table 2.Fig. 9 shows the temperature profiles normalized by pseudo-

ritical temperature for CASE I with the Van Driest model used asor turbulence model. Fig. 10 represents the corresponding resultsnce Chien Model is implemented. The most flatness of the tem-erature profiles which is associated with the enhancement of theeat transfer coefficients are clearly observed for flow conditionshere the pseudo-critical region occurs in buffer layer (the solid

ine carrying no symbols) in both Figs. 9 and 10.It should be noted that the use of algebraic models usually does

ot lead to acceptable predictions of heat transfer in the deteri-ration regime. Nevertheless, they give satisfactory results in thease of enhancement regime of heat transfer. This can be seen in

ig. 11 where the heat transfer coefficients corresponding to theow conditions of Figs. 9 and 10 are shown.

ig. 11. Comparison of numerical results by using of Van Driest and Chien turbu-ence models with the experimental data of Song et al. [20] for the CASE I.

Fig. 12. Comparison of numerical results with experimental data of Song et al. [20]in the case of deterioration (CASE II).

4.3. Deterioration of heat transfer

The results of the present study for the heat transfer coefficientversus the fluid bulk enthalpy for the deteriorated regime of heattransfer, CASE II, are shown in Fig. 12. The experimental data ofSong et al. [20] are also shown for the sake of comparison. As canbe seen the degraded heat transfer coefficients are followed by asmall enhancement returning the heat transfer coefficients to theirvalues before deterioration. Note that the values of heat transfercoefficients in CASE I are much greater than those of CASE II, i.e., thescale of the vertical axes in Figs. 5 and 12 are different. Consideringthe fact that the deteriorated mode of heat transfer to supercriticalfluids is a complicated phenomenon, the qualitative agreement ofthe results of the present numerical solution with the experiments,shown in Fig. 12, seems to be promising. The temperature profilesat a number of the flow cross-sections, before and after degradation

Fig. 13. Temperature profiles normalized by pseudo-critical temperature at variousflow cross-sections for CASE II.

228 M. Bazargan, M. Mohseni / J. of Supercr

Fflt

ttwciietso

ctfiaotbmrtectw

tbNpltipiBpcsnbflv

ig. 14. Variations of the turbulent viscosity and turbulent kinetic energy at theow cross-sections before (x = 3.315 m) and after (x = 3.365) deterioration in heatransfer for CASE II.

ures gets larger in CASE II. It can be seen from the well behavedemperature profiles that the most of the difference between theall and bulk temperatures occurs across the laminar sub-layer. It is

lear that the temperature difference across the laminar sub-layern CASE II, as shown in Fig. 13, is much larger than the correspond-ng values for CASE I, shown in Fig. 7. Therefore, it is important toxamine the laminar sub-layer more carefully. It is deduced fromhe results of the present study that the thickness of the laminarub-layer grows in the flows where deterioration in heat transferccurs. The explanation follows.

The increase of the heat flux in a supercritical environmentauses the heat transfer coefficient to decrease. The mass flux, onhe other hand, has a direct correlation with the heat transfer coef-cient. Thus, the ratio of the heat flux to the mass flux, q/G, mayct as an important deciding parameter with respect to the extentf the enhancement of heat transfer. That is mainly due to the facthat q/G can be considered as a measure of the significance of theuoyancy effect. Under certain circumstances the buoyancy effectay prohibit turbulence activities near the wall and cause the local

e-laminarization of the flow and hence heat transfer deteriora-ion. Fig. 14 shows the turbulent viscosity and the turbulent kineticnergy at two different flow cross-sections, one before (x = 3.315 m,orresponding to the flow cross-section where Hb = 299.9 kJ/kg) andhe other after (x = 3.365 m, corresponding to the flow cross-sectionhere Hb = 301.64 kJ/kg) the occurrence of deterioration.

The results show that in deteriorated region of heat transferhe turbulent viscosity is zero near the wall up to y+ ≈ 25. That isecause the turbulent kinetic energy is negligible near the wall.ote that the turbulent viscosity is proportional to the secondower of the turbulent kinetic energy. In other words, the turbu-

ence has been dissipated and the flow is almost laminar. It seemshat in this situation the laminar sub-layer is thickened. A carefulnspection and comparison of Figs. 7 and 13 shows that the tem-erature profiles get almost flat at y+ > 10 in CASE I whereas that

s the case for y+ > 20 in the most well behaved profiles in CASE II.y “well behaved temperature profiles” the temperature profiles atost deteriorated region, as shown in Fig. 13, are meant. It can beoncluded that in CASE II the buffer layer, and hence turbulent diffu-

ivity, starts at a farther distance from the wall. Note that the thick-esses of the various flow layers near the wall are defined on theasis of the experimental data obtained from the constant propertyows. It seems to be necessary to revisit those definitions for highlyariable property flows of supercritical fluids in future studies.

itical Fluids 51 (2009) 221–229

On the same basis the evolution of the temperature profiles inCASE II, shown in Fig. 13, may be explained as follows. The flowis heated until the pseudo-critical temperature is felt at the wall.Similar to CASE I, the pseudo-critical point is expected to cross thelaminar sub-layer and enter the buffer layer along the course ofthe flow. Unlike CASE I, however, it takes longer in CASE II for thepseudo-critical temperature to march the width of the laminar sub-layer. That is because the laminar sub-layer, as mentioned earlier,is thicker in CASE II and is extended even beyond y+ > 20. Conse-quently the temperature difference between the wall and bulk getsthe opportunity to noticeably grow. Note that the growth of thedifference between the bulk and wall temperatures is equivalentto decrease in heat transfer coefficient. Once the pseudo-criticaltemperature leaves the laminar sub-layer and gets into the bufferlayer, the heat transfer enhances locally due to turbulent diffu-sivity in the buffer layer and the temperature profile graduallyflattens. This will stop further degradation of heat transfer and theflow becomes stable. The temperature difference created across thelaminar sub-layer, however, is already large enough to have heattransfer coefficient much smaller than those of CASE I.

5. Conclusion

A two-dimensional numerical code is developed to simultane-ously solve the momentum and energy equations. The model ofMyoung and Kasagi [19] has been selected as the main turbulencemodel. Both enhanced and deteriorated modes of heat transfer areconsidered. The most important conclusions derived in this studyare as follows.

Neither of the laminar sub-layer and log-law layer can take theresponsibility of large heat transfer enhancement in the supercriti-cal fluids. The peak of heat transfer coefficient is reached when thepseudo-critical region is located in the buffer zone. In other words, itcan be said that at the flow cross-section where the pseudo-criticalpoint does not lie within the buffer layer, the peak of heat transferwill not occur even if the largest fraction of the fluid experiencesthe pseudo-critical temperature. In fact, the pseudo critical regioncovers almost the entire width of the buffer layer causing the peakof heat transfer coefficient.

The significance of the buffer layer is pronounced in all solutionsregardless of the turbulence model used in the numerical model.In all cases, no matter how close the predicted heat transfer coeffi-cients are to the experiments, the peak values of HTC coincide withthe conditions where the pseudo-critical region exists in the bufferlayer.

In the case of deteriorated regime, because of turbulence reduc-tion and flow laminarization, the laminar sub-layer is thickened. Itmeans that the buffer layer locates at a farther distance from thewall. This causes a delay in the turbulent diffusion near the wall andleading to a jump in the wall temperatures. It means that unlikethe enhanced regime of heat transfer, which the temperature pro-file starts effective flattening at y+ ≥ 10, for the deteriorated of heattransfer, this happens for y+ ≥ 20. As the pseudo-critical tempera-ture is swept away from the extended laminar sub-layer and entersthe buffer layer, turbulent activities grow and slight improvementof heat transfer occurs. It causes the local rise of the wall tempera-ture to stop and the normal mode of heat transfer to resume.

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