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J. of Supercritical Fluids 52 (2010) 36–46

Contents lists available at ScienceDirect

The Journal of Supercritical Fluids

journa l homepage: www.e lsev ier .com/ locate /supf lu

numerical study of supercritical forced convective heat transfer of n-heptanenside a horizontal miniature tube

i-Xin Hua, Ya-Zhou Wang, Hua Meng ∗

chool of Aeronautics and Astronautics, Zhejiang University, 38 Zheda Road, Hangzhou, Zhejiang 310027, PR China

r t i c l e i n f o

rticle history:eceived 9 September 2009eceived in revised form 2 December 2009ccepted 2 December 2009

eywords:upercritical heat transferupercritical pressure

a b s t r a c t

Supercritical convective heat transfer of hydrocarbon propellants plays a key role in the regenerativecooling technology development in aerospace applications. In this paper, a numerical study of the super-critical forced convective heat transfer of a typical hydrocarbon fuel, n-heptane, has been conductedbased on a complete set of conservation equations of mass, momentum, and energy with accurate evalu-ations of the thermophysical properties. The present fundamental numerical study focuses on the effectsof many key parameters, including the inlet pressure, inlet velocity, wall heat flux, and the inlet fluidtemperature, on the supercritical heat transfer processes. Results indicate that under supercritical heat

egenerative engine coolingydrocarbon propellantumerical study

transfer processes, heat transfer deterioration could occur once the wall temperature or the fluid tem-perature in a large near-wall region reaches the pseudo-critical temperature, and increasing the fluidpressure would enhance heat transfer. The conventional empirical Gnielinski expression could only beused for supercritical heat transfer predictions of n-heptane under very limited operational conditions.It is found in the present numerical study that a supercritical heat transfer expression for CO2, H2O, andHCFC-22 applications can generally be employed for predicting the supercritical heat transfer coefficientof n-heptane when the inlet velocity is higher than 10 m/s.

. Introduction

Supercritical convective heat transfer of hydrocarbon propel-ants plays a key role in the regenerative cooling technologyevelopment for rocket and supersonic combustion ramjet (scram-

et) engines [1,2]. In these combustion engines, the chamber wallndures high heat fluxes from the high-temperature burning gases.n order to maintain engine reliability and lifetime, the combustionhamber wall has to be cooled by the engine fuel. This is called aegenerative cooling process, since the heat absorbed by the engineuel would subsequently improves the engine performance. In aooster-stage rocket engine, since the combustion chamber pres-ure is generally higher than the critical pressure of the injecteduel, i.e. RP-1, the fluid flow and heat transfer processes of theydrocarbon fuel inside the cooling channels prior to injection andurning are thus under supercritical pressures. Although the com-

ustion chamber pressure in an air-breathing scramjet engine isenerally lower than the critical pressure of the injected hydro-arbon fuel, i.e. JP-7, the operational pressure of the fluid flow andeat transfer processes inside the engine cooling channel still main-

∗ Corresponding author. Tel.: +86 571 87952990; fax: +86 571 87953167.E-mail address: menghua@zju.edu.cn (H. Meng).

896-8446/$ – see front matter © 2009 Elsevier B.V. All rights reserved.oi:10.1016/j.supflu.2009.12.003

© 2009 Elsevier B.V. All rights reserved.

tains a supercritical condition in order to avoid phase change andimprove heat transfer.

In the strictly defined term, both the temperature and pressureof the working fluid are above its critical temperature and criticalpressure at a supercritical condition. In general, however, once theworking pressure of the fluid rises above its critical pressure, thephase-changing phenomenon would disappear during the temper-ature increasing process, and the fluid would instead transits from aliquid-like to a gas-like state with abrupt property variations withina narrow temperature range. The temperature corresponding tothe maximum heat capacity is defined as the pseudo-critical tem-perature of the fluid under a specific supercritical pressure. In thispaper, the supercritical convective heat transfer refers to a processunder a working pressure above the critical pressure of the hydro-carbon fuel, while the fuel temperature could be in either sub- orsuper-critical state.

The supercritical convective heat transfer process exhibitsdistinct features from its subcritical counterpart and has beenextensively investigated. The existing studies in the open literaturefocused mainly on the supercritical heat transfer of carbon dioxide

and water because of their practical applications in the refrigera-tion and nuclear industries. Both numerical [3–7] and experimental[8–12] studies have been conducted with regard to these twotypes of supercritical fluid, and the past research progress has alsobeen summarized [13,14]. Based on these studies, conclusions have

Y.-X. Hua et al. / J. of Supercritical Fluids 52 (2010) 36–46 37

Nomenclature

A surface area, m2

An coefficients in BWR equationCp constant-pressure heat capacity, J kg−1 K−1

D diameter of the tube, mDω cross-diffusion term in the turbulence modelet total internal energy, J kg−1

f friction factorF� mapping function for fluid viscosityG turbulent generation termh convective heat transfer coefficient, W m−2 K−1

H enthalpy, J kg−1

k turbulent kinetic energy, J kg−1

Nu Nusselt numberp pressure, PaPr Prandtl numberq heat flux, W m−2

r radial coordinate, mRe Reynolds numberT temperature, Ku velocity, m s−1

x axial coordinate (x = 0 at the beginning of the heatedsection), m

Y turbulent dissipation termZc critical compressibility factor

Greek� coefficient in BWR equation� thermal conductivity, W m−1 K−1

� density, kg m−3

� viscous stress, N m−2

�� correction factor accounting for non-correspondence

ω specific dissipation rate, s−1, or Pitzer’s acentric fac-tor

effective diffusivity, m2 s−1

Subscriptsb averaged parameterc critical parameterk turbulent kinetic energypc pseudo-critical valuer radial directionw wallx axial directiont total value, including laminar and turbulent parts

bipmsi00

pobbt

ω specific dissipation rate0 inlet or ideal state

een drawn that a supercritical heat transfer process was stronglynfluenced by the abrupt property variations in the vicinity of theseudo-critical temperature and was further complicated by theagnitude of the enforced wall heat flux. For example, with the

upercritical water, heat transfer would be enhanced in the vicin-ty of the pseudo-critical temperature under a low heat flux, i.e..2 MW/m2, but it could be deteriorated under a high heat flux, i.e..3 MW/m2 [12].

Since the property variations of the hydrocarbon fuel at the

seudo-critical temperature region differs significantly from thosef carbon dioxide and water, and the wall heat fluxes in the com-ustion chambers of the rocket and scramjet engines, which coulde well above the MW/m2 magnitude, are generally higher thanhose in the refrigeration and nuclear applications, the supercrit-

Fig. 1. Computational configuration.

ical heat transfer phenomena of hydrocarbon fuels could exhibitmuch different characteristics from carbon dioxide and water, andhave thus attracted significant research interests in the aerospaceengineering. Hitch and Karpuk [15] conducted experimental stud-ies on the supercritical fluid flow and heat transfer of a hydrocarbonfuel, JP-7, in a vertical tube, intended for the technological devel-opment of the cooling strategies in future air-breathing propulsionsystems. When the reduced pressures (p/pc) below 1.5 and thetube wall temperatures above the corresponding pseudo-criticaltemperature, they observed significant pressure and temperatureoscillations along with declined local heat transfer coefficients.Chen and Dang [16] also performed experimental investigationson the supercritical heat transfer of JP-7 in order to determine itsviability for engine cooling applications in micro-rocket systems.They focused on the heat transfer and coking properties of super-critical JP-7 and determined that this hydrocarbon fuel performedmuch better than water in terms of the cooling capability. Zhonget al. [17] experimentally studied the heat transfer characteristicsof China no. 3 kerosene under supercritical conditions relevant tothe regenerative cooling applications in the scramjet propulsionsystems. They found drastic changes in the heat transfer charac-teristics as the hydrocarbon fuel approached its critical state, andobserved heat transfer enhancement after the fuel temperaturebecame supercritical.

Although a number of experimental studies have been con-ducted to examine the supercritical heat transfer processes ofthe hydrocarbon propellants, more fundamental investigations,including comprehensive numerical studies, are still needed to gaindeep understanding of the underlying physics and the main influ-ential factors. In this paper, a numerical study of the supercriticalforced convective heat transfer phenomena of a typical hydrocar-bon fuel, n-heptane, flowing inside a horizontal miniature tube hasbeen conducted. N-heptane is chosen as the model compound in thepresent numerical study because it is a typical pure liquid hydrocar-bon fuel with 7 carbon atoms in its molecule, sitting in the middlein terms of carbon numbers among the common pure liquid hydro-carbon materials in aerospace applications. Therefore, the resultsobtained with n-heptane might be extended to other pure liquidhydrocarbon fuels with minor modifications. The present studyfocuses on the fundamental aspects of the supercritical heat trans-fer phenomena and analyzes the effects of many key parameters,including the inlet pressure, inlet velocity, inlet fluid temperature,and the wall heat flux, on the supercritical heat transfer processes.

2. Theoretical formulation

The present numerical study is based on a complete set ofconservation equations of mass, momentum, and energy, andincorporates accurate evaluations of the thermodynamic andtransport properties under various supercritical conditions. Thecomputational configuration is illustrated in Fig. 1. In the presentstudy, since the tube diameter is very small, i.e. 2 mm, while the

forced flow velocity is sufficiently high, i.e. >7.5 m/s, the buoy-ancy effect is small and is temporarily neglected, focusing solelyon the forced-flow heat transfer phenomena. Therefore, the com-putational domain can be simplified to an axisymmetric plane. The

3 rcritical Fluids 52 (2010) 36–46

c

M

E

�

Tt

omtaatT

Siae

2

namfltio

p

u

Odb

Table 1Thermodynamic properties of n-heptane [28].

8 Y.-X. Hua et al. / J. of Supe

onservation equations are presented in the following forms:Mass conservation:

∂

∂x(�ux) + 1

r

∂

∂r(�rur) = 0 (1)

omentum conservation:

∂

∂x(�uxux) + 1

r

∂

∂r(r�uxur) = −∂p

∂x+ ∂�xx

∂x+ 1

r

∂

∂r(r�xr) (2)

∂

∂x(�urux) + 1

r

∂

∂r(r�urur) = −∂p

∂r+ ∂�rx

∂x+ 1

r

∂

∂r(r�rr) (3)

nergy conservation:

Cpux∂T

∂x+ �Cpur

∂T

∂r= ∂

∂x

(�t

∂T

∂x

)+ 1

r

∂

∂r

(r�t

∂T

∂r

)

− T

�

(∂�

∂T

)p

�u · ∇p (4)

he relevant variables in these equations are defined in Nomencla-ure.

In the present numerical study of the supercritical heat transferf n-heptane, given the high wall heat fluxes and the strong ther-ophysical property variations in the near-wall region, we chose

he SST k–ω turbulent model [18] for the internal turbulent flownd heat transfer calculations, as this turbulent model is directlypplicable in the near-wall region, including the viscous sub-layer,hus avoiding the implementation of an uncertain wall function.he turbulent equations are presented as follows.

∂

∂x(�kux) + 1

r

∂

∂r(r�kur) = ∂

∂x

[k

∂k

∂x

]+ 1

r

∂

∂r

[rk

∂k

∂r

]+ Gk − Yk

(5)

∂

∂x(�ωux) + 1

r

∂

∂r(r�ωur) = ∂

∂x

[ω

∂ω

∂x

]+ 1

r

∂

∂r

[rω

∂ω

∂r

]+ Gω − Yω + Dω (6)

tandard terms in the SST k–ω turbulent model are implementedn the present numerical study. The relevant variables in Eqs. (5)nd (6) are also defined in Nomenclature, and more details can beasily found in the Ref. [18].

.1. Boundary conditions

The computational configuration is shown in Fig. 1. The begin-ing unheated section with a length of 150 mm is used to obtainfully-developed flow-field prior to the heat transfer process, theiddle section of 300 mm long is heated with a constant wall heat

ux, while the ending unheated section of 150 mm long is includedo avoid the effects of out-flow boundary conditions on the numer-cal results. The following boundary conditions are defined basedn the internal flow physics.

The inlet boundary conditions are used to set the incoming flowroperties:

x = u0, ur = 0, T = T0, p = p0 (7)

ut-flow boundary conditions are used to describe a fully-

eveloped flow at the mini tube outlet (an unheated section haseen included to meet the physical requirement):

∂ux

∂x= ∂ur

∂x= ∂T

∂x= ∂p

∂x= 0 (8)

pc/MPa Tc/K ω Zc

2.74 540.3 0.349 0.263

Axisymmetric boundary conditions are defined at the tube centeraxis:

∂ux

∂r= ∂T

∂r= ∂p

∂r= 0, ur = 0 (9)

Wall boundary conditions for the heated section describe a viscousflow under a constant wall heat flux:

ux = ur = 0, q = qw,∂p

∂r= 0 (10)

Wall boundary conditions for the two unheated sections describeviscous flows with thermally isolated tube walls:

ux = ur = 0, q = 0,∂p

∂r= 0 (11)

2.2. Property evaluations

Since property variations significantly influence the heattransfer processes, accurate evaluations of the thermophysicalproperties become the key for the supercritical heat transfer cal-culations. To obtain the thermodynamic properties of n-heptaneunder various supercritical conditions, including its heat capacityand internal energy, fundamental thermodynamic theories couldbe applied to derive the relevant expressions [19,20]. For example,the following expression for the heat capacity can be derived [19]:

Cp(T, �) = Cv0(T) −∫ �

�0

[T

�2

(∂2p

∂T2

)�

]T

d� + T

�2

(∂p/∂T)2�

(∂p/∂�)T(12)

where all the variables are defined in Nomenclature. Inorder to obtain the numerical values of the heat capacity, aSoave–Redlich–Kwong (SRK) equation of state is used along withEq. (12), as this equation of state is relatively simple and accurate[19,20].

The extended corresponding-state principle [21,22] could beused for evaluating the density and the transport properties ofn-heptane, including its viscosity and thermal conductivity, at dif-ferent temperatures and pressures. In this approach, the transportproperties of n-heptane are evaluated via conformal mappings oftemperature and density to those of a reference material, usuallymethane. For example, the viscosity of n-heptane can be calculatedas:

�(T, �) = �0(T0, �0)F��� (13)

where the parameter � refers to the viscosity of n-heptane at anytemperature and density, �0 is the viscosity of the reference mate-rial at the mapped temperature, T0, and density, �0, the parameterF� represents the mapping function, and �� is a correction fac-tor accounting for the effect of non-correspondence. The relevantthermodynamic properties of n-heptane are listed in Table 1.

In order to use this expression, the density of n-heptane has to becalculated at a specified temperature and pressure. This is accom-plished using a modified Benedict–Webb–Rubin (BWR) equationof state first to calculate the density of the reference material,i.e. methane, and the density of n-heptane can be subsequentlyobtained via conformal mapping. The BWR equation of state for

methane can be expressed in the following form:

p =9∑

n=1

An(T)�n +15∑

n=10

An(T)�2n−17e−��2(14)

Y.-X. Hua et al. / J. of Supercritical Fluids 52 (2010) 36–46 39

TTdldb

tpcFhcwaeFian

bitemontuisfven

Fig. 2. Thermophysical property variations, (a) density; (b) heat capacity.

he empirical parameters in this equation can be found in [21].his complex equation of state is needed to obtain the methaneensity very accurately, because the precision of the density calcu-

ation would drastically affect that of the transport properties. Moreetails concerning the extended corresponding-state method cane found in the Refs. [21,22].

The property-evaluation methods briefly described in this sec-ion have been used to calculate the thermodynamic and transportroperties of n-heptane, including its density, heat capacity, vis-osity, and thermal conductivity, as presented in Fig. 2 andig. S1 in the online supplementary material. The calculated resultsave been compared with the NIST data [23]. For density cal-ulation, the largest difference between the two sets of data isithin 3%, for heat capacity, it is within 10%, for viscosity, it is

lso within 10%, and for thermal conductivity, the largest differ-nce slightly increases to around 15%. Details can be found inig. S2 in the online supplementary material. These comparisonsndicate that the property-evaluation methods are fairly accuratend can be used for predicting the thermophysical properties of-heptane.

The numerical model and property-evaluation methods haveeen implemented into a commercial CFD package, Fluent, through

ts user coding capability. An incompressible flow solver based onhe finite-volume scheme and the SIMPLEC algorithm has beenmployed for the present numerical studies. In order to conductodel validation, numerical calculations have first been carried

ut to study the supercritical heat transfer of carbon dioxide. Theumerical results have been compared with available experimen-al data [6], as shown in Fig. 3. These calculated cases are allnder supercritical pressures with the inlet temperatures rang-

ng from the pseudo-critical to the supercritical region. Therefore,

trong property variations occur during the supercritical heat trans-er processes, rendering them appropriate for the present modelalidation. The relative error between the two sets of results is gen-rally less than 10%, further verifying the reliability of the presentumerical method.

Fig. 3. Model validation against experimental data in Table 1 in Ref. [6], (a) P01:9.59 MPa, P02: 9.54 MPa, P03: 9.5 MPa, P04: 9.43 MPa; (b) P51: 8.46 MPa, P52:8.46 MPa, P53: 8.47 MPa, P54: 8.48 MPa.

3. Results and discussion

The physical problem of concern involves a hydrocarbon fuel,n-heptane, flowing inside a horizontal mini tube with a constantwall heat flux enforced in the middle section, as shown in Fig. 1.Prior to detailed numerical studies, a grid-independence study hasbeen conducted to ensure the accuracy of the present numericalinvestigations. A grid system with 8000 and 60 meshes in the axialand radial directions, respectively, has been proved to be sufficient.The meshes in the radial direction are clustered towards the tubewall with a stretch ratio of 1.1. Increasing the meshes from 60 to90 in the radial direction renders less than 3% relative numericalerror in terms of the maximum wall temperature, while increasingthe axial meshes from 8000 to 14,000 causes less than 1% relativenumerical error.

After this grid-independence study, the present numericalmethod described in the early section has been employed toinvestigate the supercritical convective heat transfer of a typicalhydrocarbon fuel, n-heptane, focusing on the fundamental effectsof the key influential parameters, including the inlet pressure, inletfluid velocity, inlet temperature, and the wall heat flux, on the fluidflow and heat transfer processes.

3.1. Pressure effects

The effects of the inlet pressure on the supercritical heat transferprocesses of n-heptane are studied in this section. The inlet fluidvelocity is 20 m/s, the inlet temperature at 400 K, while the constant

2

wall heat flux at 5 MW/m . Variations of the wall and the averagedfluid temperature under four different inlet pressures, ranging from4 to 7 MPa, are illustrated in Fig. 4. Under a supercritical pressure of4 MPa, the inlet Reynolds and Prandtl numbers are 1.48 × 105 and4.34, respectively, clearly in the turbulent heat transfer regime. The

40 Y.-X. Hua et al. / J. of Supercritical Fluids 52 (2010) 36–46

Fa

a

T

ewhat

tbttt5rAotrnttu

waaavva

Fig. 5. Thermophysical property variations along the tube wall, (a) density; (b) heatcapacity.

ig. 4. Temperature variations under four different inlet pressures, (a) wall temper-ture; (b) the averaged fluid temperature.

veraged fluid temperature is defined in the present study as

b =∫

A��uCpTdA∫

A��uCpdA

(15)

As shown in Fig. 4b, the inlet pressure produces essentially noffect on the averaged fluid temperature, but the wall temperatureould decrease as the inlet pressure increases, indicating improvedeat transfer, as depicted in Fig. 4a. Once the inlet pressure rises tobove 6 MPa or p/pc just above 2, however, the pressure effect onhe heat transfer enhancement becomes very weak.

As shown in Fig. 4a, at a supercritical pressure of 4 MPa, the wallemperature suffers a sharp increase from x/D = 45 (x/D = 0 at theeginning of the heated section), indicating heat transfer deteriora-ion. From Fig. 2b and based on the definition of the pseudo-criticalemperature of a supercritical fluid, we can easily determine thathe pseudo-critical temperature of n-heptane at 4 MPa is around70 K. As clearly depicted in Fig. 4a, the tube wall temperatureeaches the pseudo-critical temperature of n-heptane at x/D = 45.fter this location, the thermodynamic and transport propertiesf n-heptane experience abrupt variations. As illustrated in Fig. 5,he density and heat capacity of n-heptane suffer abrupt decrease,esulting in the decreased total heat capacity of the supercritical-heptane and consequently, the sharply increased wall tempera-ure. The same phenomenon has also been observed in a study ofhe convective heat transfer of the cryogenic-propellant methanender supercritical pressures [24].

The abrupt variations of the thermophysical properties at 4 MPaould also cause slight flow-field unsteadiness. As shown in Fig. 6

nd Fig. S3 in the online supplementary material, variations of the

xial and radial velocity under two supercritical pressures of 4nd 7 MPa have been displayed and compared. Although the axialelocity under the two pressures shows no difference, the radialelocity at 4 MPa experiences much stronger variation than thatt 7 MPa, especially at x/D = 50 immediately following the location

Fig. 6. Radial velocity variations under two different inlet pressures, (a) at 4 MPa;(b) at 7 MPa.

rcritic

rsfltTi

ittttG

N

w

f

Tt

0

Tn

N

tFas

Fcm

Y.-X. Hua et al. / J. of Supe

eaching the pseudo-critical temperature. Since the radial velocityignificantly influences the convective heat transfer process, thisow-field unsteadiness would lead further to the slight oscilla-ion of the tube wall temperature, as clearly illustrated in Fig. 4a.his unsteady heat transfer phenomenon has also been observedn experiments [15].

Heat transfer coefficients calculated from the present numer-cal study have first been compared with those obtained fromhe conventional heat transfer expressions with consideration ofhe variable properties in order to obtain a full understanding ofhe supercritical heat transfer phenomena, and in the mean time,o verify the applicability of these empirical expressions, i.e. thenielinski formula [25] in the following form:

uD = (f/8)(Reb − 1000)Prb

1 + 12.7(f/8)1/2(Pr2/3b

− 1)(16)

here the friction factor, f, in a smooth pipe is calculated as

= (0.790 ln Reb − 1.64)−2 (17)

he Gnielinski formula can be applied for the heat transfer predic-ion with mild property variations under the following conditions:

.5 ≤ Pr ≤ 2000, 3000 ≤ Reb ≤ 5 × 106 (18)

he Nusselt number for the present supercritical heat transfer of-heptane is defined as

uD = hD

�b(19)

The Nusselt number from both the present numerical calcula-ions and the empirical Gnielinski equation has been compared inig. 7a. The Nusselt number from the Gnielinski equation decreasess the supercritical pressure increases, but the changes are quitemall. However, the Nusselt number from the present numerical

ig. 7. Variations of the Nusselt number under four different inlet pressures, (a)omparison with the Gnielinski formula; (b) comparison with the Bae and Kimodified expression.

al Fluids 52 (2010) 36–46 41

calculations shows an opposite trend, increasing significantly asthe inlet pressure increases, particularly when the pressure is lessthan 6 MPa. As the inlet supercritical pressure becomes higher than5 MPa, the relative errors between the two sets of data are less than20%, verifying that the Gnielinski equation is only applicable underthese limited operational conditions.

The conventional Gnielinski formula works poorly for the super-critical heat transfer of n-heptane with the inlet pressure less than5 MPa, because it cannot accommodate strong property variations.In order to find a proper heat transfer formula for supercriticaln-heptane, we have tested many other empirical expressions, par-ticularly those for supercritical CO2 and water [26,27]. We find thatthe following heat transfer formula originally proposed by Jacksonand Hall [26] and further modified by Bae and Kim [27] works verywell for the supercritical n-heptane:

NuD = 0.021Re0.82b Pr0.5

b

(�w

�b

)0.3(

Cp

Cpb

)n

(20)

where the parameters, Cp and n, are defined as

Cp = Hw − Hb

Tw − Tb(21)

n =

⎧⎪⎪⎨⎪⎪⎩

0.4 for Tb < Tw≤Tpc or 1.2Tpc≤Tb < Tw,

0.4+0.2

(Tw

Tpc−1

)for Tb≤Tpc < Tw,

0.4+0.2

(Tw

Tpc−1

)[1−5

(Tb

Tpc−1

)]for Tpc < Tb ≤ 1.2Tpc and Tb < Tw.

(22)

Because it could account for strong property variations, this formulahas been proved to work well for the supercritical heat transfer ofCO2, H2O, and HCFC-22. Its application range in terms of Reynoldsand Prandtl numbers is, however, presently not determined. Moredetails concerning this formula can be found in the Refs. [26,27].

As illustrated in Fig. 7b, the empirical formula in Eq. (20) showsexcellent agreement with our present numerical results. Heattransfer deterioration and its transitional locations at the pseudo-critical temperatures under two inlet pressures of 4 and 5 MPa havebeen successfully predicted by this empirical expression. The max-imum relative error between the two sets of results is well within10%, indicating that Eq. (20) can be used for predicting the super-critical heat transfer coefficients of n-heptane under the specifiedconditions in Fig. 7.

3.2. Effects of inlet velocity

The effects of the inlet fluid velocity on the supercritical heattransfer of n-heptane are next studied. The inlet fluid temperatureis at 400 K, the inlet pressure at 5 MPa, the constant wall heat flux at5 MW/m2, while the inlet velocity ranges from 7.5 to 20 m/s. Fig. 8presents variations of the wall and the averaged fluid temperature.As the inlet velocity increases, both types of temperature woulddecrease because of the enhanced convective heat transfer.

As shown in Fig. 8a, at a low inlet velocity of 7.5 m/s, thewall temperature experiences a relatively strong increase atx/D = 105, it reaches a maximum around 1100 K at x/D = 125, andthen gradually decreases. The temperature distributions at thefour different inlet velocities are further illustrated in Fig. 9 andFig. S4 in the online supplementary material.

As shown in Fig. 8a, at an inlet velocity of 7.5 m/s, the walltemperature drastically increases immediately after entering theheated section and rises above the corresponding pseudo-criticaltemperature of n-heptane at 5 MPa, which is around 580 K fromFig. 2b. Therefore, there is a narrow region close to the tube wall, in

42 Y.-X. Hua et al. / J. of Supercritica

F(

wtturioai

ig. 8. Temperature variations at four different inlet velocities, (a) wall temperature;b) the averaged fluid temperature.

hich the fluid temperature is in the vicinity of the pseudo-criticalemperature. As the fluid flows along the axial direction, the wallemperature further increases, and this narrow region moves grad-ally towards the center of the tube. Starting from x/D = 105, this

egion grows larger and visibly separates from the tube wall, aslluminated in Fig. 9. This process results in the abrupt decreasef the fluid density and heat capacity in a large near-wall region,s illustrated in Fig. 10, which leads to stronger wall temperaturencrease.

Fig. 9. Temperature distribution in the entire

Fig. 10. Thermophysical property distributions at an inle

l Fluids 52 (2010) 36–46

Along the flow direction, density decreases as the fluid isheated. In order to maintain a constant mass flux, the axialvelocity increases accordingly, as shown in Fig. 11. Starting fromx/D = 125, the strong temperature increase and the consequent den-sity decrease cause the significant velocity increase in the near-wallregion. The increased velocity would enhance the convective heattransfer and leads to gradual wall temperature decrease after thislocation, as shown in Fig. 8a.

Variations of the Nusselt number from both the present numer-ical calculations and the Gnielinski expression are depicted inFig. 12a. In general, as the inlet velocity increases, both sets of Nus-selt number increase. At low inlet velocities, i.e. 7.5 and 10 m/s, therelative errors between the two sets of data could reach more than100%, indicating that the Gnielinski equation is invalid under theseconditions. However, as the inlet velocity increases to 15 m/s, therelative error between the two sets of data decreases to within 25%,and consequently, the Gnielinski equation can be used for super-critical heat transfer predictions of n-heptane with an acceptableaccuracy only at high inlet velocities, i.e. ≥15 m/s.

Fig. 12b compares the present numerical results with thosefrom Eq. (20). In general, Eq. (20) performs much better than theGnielinski formula for the supercritical heat transfer of n-heptane.However, at a low inlet velocity of 7.5 m/s, the maximum differencebetween the two sets of results is still close to 100%. Increasing theinlet velocity to 10 m/s, the difference could quickly drop within20%, except at the very end of the heated section where a maximumdifference can reach around 30%. Therefore, Eq. (20) can be used forpredicting the supercritical heat transfer coefficient of n-heptanewith an acceptable accuracy once the inlet velocity becomes higherthan 10 m/s.

3.3. Effects of wall heat flux

The effects of the wall heat flux on the supercritical heat transferof n-heptane are illuminated in Fig. 13 with variations of the wall

and the averaged fluid temperature under four different wall heatfluxes, ranging from 1 to 7 MW/m2. The inlet velocity is 20 m/s,the inlet fluid temperature at 400 K, the inlet pressure at 5 MPa. InFig. 13, as the wall heat flux increases, the wall and the averagedfluid temperature both increase, as expected.

flow-field at an inlet velocity of 7.5 m/s.

t velocity of 7.5 m/s, (a) density; (b) heat capacity.

Y.-X. Hua et al. / J. of Supercritical Fluids 52 (2010) 36–46 43

7.5 m

ptec1flb

Fpe

Fig. 11. Velocity distributions at an inlet velocity of

Fig. 14a presents variations of the Nusselt number from both theresent numerical calculations and the Gnielinski expression. Ashe wall heat flux increases, the Nusselt number from the Gnielinskiquation increases. The Nusselt number from the present numeri-

al calculations also increases as the wall heat flux increases fromto 5 MW/m2, but it decreases significantly when the wall heat

ux further increases to 7 MW/m2. Furthermore, the relative erroretween the two sets of Nusselt number is within 20% when the

ig. 12. Variations of the Nusselt number at four different inlet velocities, (a) com-arison with the Gnielinski formula; (b) comparison with the Bae and Kim modifiedxpression.

/s, (a) velocity contour; (b) axial velocity variations.

wall heat flux is less than 5 MW/m2, indicating that the Gnielin-ski equation is only applicable at these specified conditions. As thewall heat flux increases to 7 MW/m2, the difference between thetwo sets of data grows significantly, and as a result, the Gnielinskiequation is invalid under a high wall heat flux.

Comparisons between the present numerical results and thosefrom Eq. (20) are provided in Fig. 14b. The two sets of results showvery good agreement, and the maximum relative error betweenthem is within 15% under the four different wall heat fluxes. TheNusselt number calculated from Eq. (20) increases as the wall heat

Fig. 13. Temperature variations under four different wall heat fluxes, (a) wall tem-perature; (b) the averaged fluid temperature.

44 Y.-X. Hua et al. / J. of Supercritical Fluids 52 (2010) 36–46

F(m

flflTnaEcp

iActFacu

3

hT

ig. 14. Variations of the Nusselt number under four different wall heat fluxes,a) comparison with the Gnielinski formula; (b) comparison with the Bae and Kim

odified expression.

ux increases from 1 to 3 MW/m2; further increasing the wall heatux to 5 MW/m2, however, the Nusselt number starts to decrease.his trend is consistent with the numerical results, except that theumerically calculated Nusselt number only starts to decrease athigher wall heat flux of 7 MW/m2. Based on these comparisons,q. (20) can be used for predicting the supercritical heat transferoefficient of n-heptane under the four different wall heat fluxesresently studied.

Fig. 15 and Fig. S5 illustrate variations of the thermal conductiv-ty in the entire flow-field under the four different wall heat fluxes.s the wall heat flux increases, the fluid temperature rises signifi-antly from the tube inlet to the outlet in the heated section, leadingo drastic variations of the thermophysical properties, as shown inig. 15. Because the conventional Gnielinski equation could onlyccount for mild property variations during the heat transfer pro-ess, it is not applicable in cases with significant property variationsnder a high wall heat flux at supercritical pressures.

.4. Effects of inlet temperature

The effects of the inlet fluid temperature on the supercriticaleat transfer of n-heptane are finally investigated in this section.he inlet fluid velocity is 20 m/s, the inlet pressure at 5 MPa, the

Fig. 15. Distribution of the thermal conductivity in the en

Fig. 16. Temperature variations at four different inlet fluid temperatures, (a) walltemperature; (b) the averaged fluid temperature.

wall heat flux at 5 MW/m2, while the inlet fluid temperature rangesfrom 300 to 450 K. As shown in Fig. 16, the wall and the aver-aged fluid temperature both increase as the inlet fluid temperatureincreases.

Variations of the Nusselt number from both the present numer-ical calculations and the Gnielinski expression are presented inFig. 17a. The Nusselt number from the Gnielinski equation steadilyincreases as the inlet fluid temperature increases. For the numericalresults, the Nusselt number also increases when the inlet temper-ature varies from 300 to 400 K, but it changes little when the inlettemperature further rises to 450 K and even decreases at the end ofthe heated section. With the inlet fluid temperature less than 400 K,the Gnielinski equation is applicable for the supercritical heat trans-fer predictions of n-heptane with an uncertainty less than 20%. Oncethe inlet temperature rises beyond 400 K, however, the differencebetween the two sets of Nusselt number grows drastically becauseof strong thermophysical property variations, a phenomenon dis-cussed in the early section.

Fig. 17b compares the numerical results with those from Eq. (20)at the four different inlet fluid temperatures. The two sets of results

show excellent agreement with a maximum relative error within10%. As the inlet fluid temperature increases from 400 to 450 K, theNusselt number obtained from Eq. (20) also slightly decreases, atrend consistent with the numerical results.

tire flow-field under a wall heat flux of 7 MW/m2.

Y.-X. Hua et al. / J. of Supercritic

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(

(

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ig. 17. Variations of the Nusselt number at four different inlet fluid temperatures,a) comparison with the Gnielinski formula; (b) comparison with the Bae and Kim

odified expression.

. Conclusions

In this paper, a thorough numerical study of the supercriti-al forced convective heat transfer of a typical hydrocarbon fuel,-heptane, has been conducted based on a complete set of conser-ation equations of mass, momentum, and energy with accuratevaluations of the thermophysical properties. The fundamen-al investigation is intended to gain deep understanding of thenderlying physics of the regenerative cooling technology in theerospace applications. The present numerical study focuses on theffects of the key influential parameters, including the inlet pres-ure, inlet velocity, wall heat flux, and the inlet fluid temperature,n the supercritical fluid flows and heat transfer processes. Theollowing conclusions could be reached:

1) Under supercritical pressures, decreasing the inlet pressure ofn-heptane below 5 MPa could cause the tube wall temperatureor the fluid temperature in a large near-wall region to reachthe corresponding pseudo-critical temperature and lead to heattransfer deterioration during the heating processes;

2) In order to improve heat transfer and avoid heat transfer dete-rioration under supercritical pressures, the cooling channelpressure should be raised to satisfy p/pc larger than 2;

3) The conventional empirical Gnielinski expression could onlybe used for supercritical heat transfer predictions of n-heptaneunder the following limited conditions: p ≥ 5 MPa, u0 ≥ 15 m/s,qw ≤ 5 MW/m2, and T0 ≤ 400 K, because it cannot account forstrong property variations during the fluid flow and heat trans-

fer processes. We expect this conclusion holds for other liquidhydrocarbon fuels, but this certainly requires further investiga-tions;

4) The supercritical heat transfer expression originally proposedby Jackson and Hall and later slightly modified by Bae and

[

al Fluids 52 (2010) 36–46 45

Kim for CO2, H2O, and HCFC-22 applications can generally beemployed for predicting supercritical heat transfer coefficientsof n-heptane when the inlet fluid velocity is higher than 10 m/s.The applicability of this form of heat transfer expression forother liquid hydrocarbon fuels also requires further investiga-tions.

Acknowledgement

This research work is financially supported by the National Nat-ural Science Foundation of China (No. 10972197).

Appendix A. Supplementary data

Supplementary data associated with this article can be found, inthe online version, at doi:10.1016/j.supflu.2009.12.003.

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