Conjugate Heat Transfer in AirtoRefrigerant Airfoil Heat Exchangers

Yutaka ItoTokyo Institute of Technology,

4259-G3-33-402, Nagatsuta-cho,

Midori-ku, Yokohama,

Kanagawa 226-8502, Japan

e-mail: [email protected]

Naoya InokuraTokyo Institute of Technology,




Takao NagasakiTokyo Institute of Technology,




e-mail: [email protected]

Conjugate Heat Transferin Air-to-Refrigerant AirfoilHeat ExchangersA light and compact heat exchange system was realized using two air-to-refrigerant air-foil heat exchangers and a recirculated heat transport refrigerant. Its heat transfer per-formance was experimentally investigated. Carbon dioxide or water was used as arefrigerant up to a pressure of 30 MPa. Heat transfer coefficients on the outer air-contactand inner refrigerant-contact surfaces were calculated using an inverse heat transfermethod. Correlations were developed for the Nusselt numbers of carbon dioxide andwater on the inner refrigerant-contact surface. Furthermore, we proposed a method toevaluate a correction factor corresponding to the thermal resistance of the airfoil heatexchanger. [DOI: 10.1115/1.4027554]

Keywords: Nusselt number, supercritical carbon dioxide, gas turbine, airfoil heatexchanger, intercooler, recuperator

Introduction

Intercooled, recuperated aviation gas turbines (IR gas turbines)have the strong potential to reduce fuel consumption in aviation.Intercoolers and recuperators enhance the gas turbine cycle effi-ciency from a thermodynamic point of view. Wilfert et al. con-structed an IR gas turbine components demonstrator that had thepotential to achieve a 17% reduction in fuel consumption com-pared with a baseline gas turbine. However, they noted that adapt-ing recuperators, in particular, for a practical aviation gas turbineremained a future challenge [1]. Conventional IR gas turbines aretoo heavy for use as aviation gas turbines because they use a typeof tube matrix air-to-air heat exchanger [2] or a type of primarysurface air-to-air heat exchanger [3]. Although both types havehigh temperature effectiveness, they are heavy. Furthermore, hotand cold air must be collected at the heat exchanger. Therefore,long and heavy air connecting ducts are required.

To overcome these problems, a heat exchange system using aheat transport liquid or supercritical refrigerant (abbreviated inthis paper to “refrigerant”) may be used, as shown in Fig. 1. Inthis system, the refrigerant transports heat from the hot section tothe cold section. Therefore, the hot and cold sections can be in-stalled at separate sites. In the intercooling system, heat exchangerA works as an air cooler, and heat exchanger B works as a radia-tor. Conversely, in the recuperating system, heat exchanger Aworks as a heat absorber, and heat exchanger B works as an airheater. In addition, heat transport liquid and supercritical refriger-ants have higher densities and greater specific heat values than air.Therefore, the refrigerant connecting tubes require a much smallerdiameter than air connecting ducts. Although an additional recir-culation pump is needed, it must only drive the refrigerant againstthe pressure loss of the refrigerant loop. Ito proposed an IR avia-tion gas turbine that uses this heat exchange system in the form ofheat exchangers installed in already equipped components in abaseline aviation gas turbine, to reduce its weight [4]. In the inter-cooling system of this design, fixed stators and vanes in the com-pressor are used as the air cooler, and vanes in the bypass duct areused as the radiator. In the recuperating system, vanes in the com-bustor are used as the air heater, and vanes in the core nozzle areused as the heat absorber. Here, the working airflow path can

remain in the same position as a baseline aviation gas turbine.Therefore, there is no additional pressure loss in the working air-flow. Because vanes are used as heat exchangers, this type of heatexchanger is hereafter referred to as an “airfoil heat exchanger.”An airfoil heat exchanger is physically similar to the vanes of con-ventional air-cooled high-pressure turbines (HPTs) [5]. Thesewere in use prior to those of the current air-film-cooled HPTs,although the HPT vanes are not heat exchangers. In this paper, theheat transfer performance of an airfoil heat exchanger will be dis-cussed compared with that of air-cooled HPT vanes.

However, it is difficult to evaluate the heat transfer performanceof real heat exchanger components. Bejan developed the construc-tal theory of design for cooling fins [6], and Lorenzini and his col-league energetically applied this theory to optimize Y-shaped andI-shaped fins [7]. In addition, they conducted computer fluid dy-namics calculations of the heat transfer for arrays of Y-shaped, I-shaped, and T-shaped fins, and optimized the results [8]. More-over, the optimized results were compared with the constructaltheory’s results [9,10]. Similarly, modern compressor stators andguide vanes are three-dimensional airfoils optimized by computerfluid dynamics; however, a three-dimensional airfoil is too com-plex to use in experiments. Therefore, we chose a two-dimensional NACA65-(12A2I8b)10 airfoil. An NACA65 seriesairfoil is a traditional two-dimensional airfoil for a compressorstator. The estimated heat transfer rates are more suitable thanthose of a simpler plane for realizing the airfoil heat exchanger.We prepared a cascade of three NACA65-(12A2I8b)10 airfoils,each of which had five inner refrigerant channels, as shown in Fig.2, as a test model of the airfoil heat exchanger. This cascade wasinstalled in a subsonic wind tunnel at Mach 0.55–0.62. On the air-foil surfaces, as described by Nishiyama [11], a developingboundary layer changes from a laminar boundary layer to a turbu-lent boundary layer across the minimum pressure point XSmax, i.e.,the maximum point of pressure coefficient S. This is because theboundary layer is stable in regions with favorable pressure gra-dients but unstable in those with adverse pressure gradients.

To obtain the heat transfer coefficients on the outer and innersurfaces, Turner employed a heat conduction numerical analysisusing 31 discrete surface temperatures measured in experiments[12]. In contrast, we used an inverse heat transfer method. It wasconducted under conditions that included the pressure distributionaround an airfoil already reported by Dunavant et al. [13] and ourexperimentally measured temperatures at four points. This waseasier than Turner’s method because it is difficult to accurately

Contributed by the Heat Transfer Division of ASME for publication in theJOURNAL OF HEAT TRANSFER. Manuscript received October 28, 2013; final manuscriptreceived April 7, 2014; published online May 21, 2014. Assoc. Editor: GiulioLorenzini.

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measure surface temperatures without disturbing a fast airflow.Furthermore, Turner used air as the inner cooling fluid, whereaswe used a refrigerant. Lorenzini and Moretti pointed out that a liq-uid always performs better than air if the focus is exclusively onheat removal maximization [14]. Their figure seemed that therewas a greater change in the temperature distribution in the finswhen using a liquid than when using air. This difference mayaffect the heat transfer performance of an airfoil heat exchanger.

An additional contribution of our study involves the use of eithersupercritical carbon dioxide or compressed water as the inner cool-ing fluid for an airfoil heat exchanger, rather than the use of air.Liao and Zhao experimentally investigated the heat transfer coeffi-cient of supercritical carbon dioxide in the range of 7.4–12 MPa[15]. Our study extended Liao and Zhao’s pressure range up to30 MPa for carbon dioxide, and we also considered compressed(but not supercritical) water at pressures up to 30 MPa.

It is expected that this will help to clarify a more suitablemethod for estimating the heat transfer coefficients when design-ing airfoil heat exchangers.

Experimental Setup

To experimentally estimate the heat transfer performance in ahigh-speed compressible flow, the Reynolds, Mach, and Prandtlnumbers should match those of the real gas and refrigerant flows.If scale-model experiments are conducted, all of the viscosity,thermal conductivity, specific heat, and sound speed ratios of thetested fluids should be the same as those of the real gas and refrig-erant. However, rather than using such fluids in scale-modelexperiments, it is easier to prepare a real gas, real refrigerant, andreal-size airfoil heat exchanger under real conditions.

Wind Tunnel. A closed-return wind tunnel at the Tokyo Insti-tute of Technology was employed to produce a subsonic airflow. ABE-H125 Roots blower (made by ANLET Co. Ltd.) was used asthe continuous air source. The nozzle outlet size was 60� 30 mm.Its Mach number capabilities ranged from 0 to 0.8, and Mach num-bers of 0.55–0.62 were chosen for testing the airfoil heat exchangerin an aviation gas turbine. The inlet airflow was sufficiently turbu-lent because the inlet Reynolds number Reair,nozzle, whose represen-tative length was the hydraulic diameter of the nozzle outlet,ranged from 3.92� 105 to 4.52� 105. In addition, the roots blowergenerated a turbulent airflow. Although this airflow conditioninvolved no wakes from the preceding airfoils, unlike in a practicalaxial gas turbine, it was sufficient for the time-averaged airflowcondition to be applied to the design of airfoil heat exchangers.

Airfoil Heat Exchangers. NACA 65-(12A2I8b)10 airfoils withinner refrigerant channels were employed as test models, asshown in Fig. 2. The airfoils were made of SUS304 because itsthermal conductivity is 16 W/K�m, which is almost the same asthe thermal conductivity of the materials used for practical com-pressors or vanes; for example, approximately 20 W/K�m for

titanium alloy and 11–21 W/K�m for nickel-based heat-resistantalloy. The chord length was 44 mm, and the width was 28 mm.These are average sizes for the stators or vanes in the compressorsection of middle or large class aviation gas turbines.

Four type-K thermocouples with a diameter of 0.5 mm were in-stalled in four taps with a 0.7 mm inner diameter and used to mea-sure the temperature distribution of each airfoil heat exchanger.All were located at midspan. As seen in Fig. 2, the airfoil wasdeployed at an incidence n¼ 0 (i.e., a flow direction angle fromthe airfoil camber line at its leading edge, corresponding to anangle-of-attack a¼ 9.47 deg, i.e., an inlet flow direction anglefrom the airfoil chord). x and y axes were defined in the horizontaland vertical directions. Thermocouples Ti and Tii were located onthe camber line at x¼ 3 and 41 mm, respectively. ThermocouplesTiii and Tiv were located 1.2 mm below and above the camber lineat x¼ 22 mm, respectively. Here, to enhance the temperature mea-surement accuracy and detect temperature differences smallerthan 1 K among Ti–Tiv, the potential differences between themwere measured directly. This measurement method enhances theaccuracy when detecting small temperature differences. To cali-brate all thermocouples, considering the digital voltage metererror, all of the thermocouples’ tips were placed in ice water at aconstant temperature of 273.15 K. We developed data acquisitionPC software to cancel out temperature drifts from 273.15 K.Therefore, an accuracy of 60.025 K was achieved for the temper-ature differences among all of the thermocouples.

Test Section Configurations for Cascade of Airfoil HeatExchangers. Figure 3 shows the configurations of a cascade ofthree NACA65-(12A2I8b)10 airfoil heat exchangers. Three airfoilswere deployed at the same positions as some of those tested by

Fig. 2 NACA65-(12A2I8b)10 airfoil heat exchanger as testmodel

Fig. 3 Configurations of tested cascade of NACA65-(12A2I8b)10airfoils

Fig. 1 Heat exchange system in which refrigerant transportsheat from hot section to cold section

081703-2 / Vol. 136, AUGUST 2014 Transactions of the ASME


Dunavant et al. [13]. The aerodynamic features of cascades ofNACA65-(12A2I8b)10 airfoils were previously investigated indetail by Erwin et al. [16] and Dunavant et al. [13]. In our study,the solidity r¼ LC/LG was set at 1.5, where LC and LG are the air-foil chord and tangential spacing between the airfoils’ leadingedges, respectively. In addition, the flow direction angle from per-pendicular to airfoil row b was varied from 45 deg to 70.5 deg,depending on a in the range of 0 deg–25.5 deg. To remove theboundary layer from the inlet flow, two inlet guide vanes wereused (with the distance between them being 58 mm or less), whichcould be moved to fit into the cascade position. Likewise, two slits(the distance between slits was 28 mm) mounted on the side wallswere used. The whole test section was deployed in a sufficientlylarge box with a width of 830 mm, depth of 825 mm, and height of1400 mm. This was to allow the outlet air from the cascade toleave in the appropriate direction, depending on the flow turningangle h.

Recirculation System With Carbon Dioxide or Water as Re-frigerant. The refrigerant flow loop is shown in Fig. 4. The recir-culation pump was a magnetic-coupling-driven sealed pump head,GLHH21.PFS.E-N1CH50 (made by Micropump, Inc.), connectedto an inverter-controlled ac motor. The refrigerant cooler sectionwas submersed in a TRL-N11L cooling bath (made by ThomasKagaku Co. Ltd.). This was maintained at an arbitrary temperaturebetween 253 and 353 K (�20 to 80 �C), with an accuracy of 0.1 K.A pressure gauge, KDM30-35MPaG-E (made by Krone), was in-stalled upstream of the airfoil heat exchangers. The refrigerantflowed inside the five serially connected stainless tubes in the air-foil heat exchanger with u-turn sections from the trailing to

leading edges, as shown in Fig. 4, i.e., as a multipath heatexchanger. The five stainless tubes and airfoil heat exchangerwere bonded by DM4030LD/F890 thermally conductive adhesive(made by Diemat, Inc.) with a thermal conductivity of 15 W/K�m.A structural analysis of the airfoil heat exchanger with the fivestainless tubes was conducted using ANSYS11. The maximum stresswas 32 MPa in the airfoil heat exchanger when the tubes’ innerpressure was 35 MPa. The 0.2% proof stress is 32 MPa at 1400 Kunder a strain rate of 1.0%/s for SUS304 [17]. Therefore, thestructural integrity of the airfoil heat exchanger could be main-tained. Two thermocouples Tref,C and Tref,J were installedupstream and downstream of the center airfoil heat exchanger.The potential difference between Tref,C and Tref,J was also meas-ured directly. The whole refrigerant flow loop was thermally insu-lated, and it could withstand a pressure of 34.4 MPa. Therefore,supercritical carbon dioxide or compressed liquid water could beused as the refrigerant. The refrigerant was pumped up into the re-frigerant flow loop via an 8800 series plunger pump (made by L.TEX Corporation).

Inverse Heat Transfer Method

The experimentally obtained data alone were not sufficient toenable us to determine the air and refrigerant heat transfer coeffi-cients hair and href. This was because the surface temperature dis-tributions on the outer air-contact surfaces and inner refrigerant-contact surfaces should be known to accurately evaluate hair andhref. However, these were not measured. Therefore, in order tofind the best combination of hair and href, an inverse heat transfermethod and a least square method were used to explain the experi-mentally obtained data.

Airfoil Heat Exchanger Temperature. There were only fourthermocouples in the airfoil heat exchanger, and these were notlocated on the surfaces. Therefore, the inverse heat transfer methodwas applied to estimate hair and href. Figure 5 shows the two-dimensional control volumes of the airfoil heat exchanger used toperform the inverse heat transfer method by a numerical analysis ofthe heat conduction in the airfoil heat exchanger. For each controlvolume j, the finite control volume method was employed. Namely,the steady state basic equation in an integrated form is

Qconduction;j ¼ Qref;j þ Qair;j (1)

Here, the left-hand term means the heat conduction rate of thesolid part of the airfoil heat exchanger. The right-hand terms

Fig. 4 Refrigerant recirculation loop (in case of carbon dioxideas refrigerant) and piping around airfoils

Fig. 5 Control volumes for inverse heat transfer method basedon numerical analysis of heat conduction in airfoil heatexchanger and applied boundary conditions

Journal of Heat Transfer AUGUST 2014, Vol. 136 / 081703-3


indicate the heat transfer rates through the inner refrigerant-contact surfaces (along the five bigger circular regions in Fig. 5)and outer air-contact surfaces (along the peripheral region in Fig.5). As an example, we focus on control volume j whose neighborcontrol volumes are p, q, r, and s. The discretized heat conductionrate Qconduction,j is

Qconduction;j ¼ Qconduction;j�p þ Qconduction;j�q þ Qconduction;j�r

þ Qconduction;j�s (2)

of course, the number of neighbor control volumes can be anynumber instead of four. For example, the heat conduction rateQconduction,j–p between control volume j and control volume p is

Qconduction;j�p ¼ �Aj�pksolid

dTsolid

dz� �Aj�pksolid

T pð Þ � T jð Þdj�p

(3)

where ksolid is the thermal conductivity of the solid material of theairfoil heat exchanger, z is a local coordinate along the line thatgoes through the centers of control volumes j and p, dj–p is the dis-tance between the centers of control volumes j and p, and Aj–p isthe projection area of the interfacial surface area on a plane nor-mal to the z axis. In the case of the control volume next to a refrig-erant or air boundary, the right-hand terms in Eq. (1) have values;otherwise they are 0. When control volume j contacts the refriger-ant, for example, the discretized heat transfer rate is as follows:

Qref;j ¼ Aref;jhref;jDTref;j ¼ Aref;jhref;j Tref nð Þ � TðjÞ½ � (4)

where Aref,j is the contact surface area between control volume jand the refrigerant, href,j is the local heat transfer coefficient, andTref(n) is the refrigerant temperature of the nth section in contactwith control volume j. Here, n is any location from E to I inFig. 4. Similar procedures

Qair;j ¼ Aair;jhair;jDTair;j ¼ Aair;jhair;j Tair;adiabatic;j � TðjÞ� �

(5)

should be applied for control volume j in contact with air, whereTair,adiabatic,j is the adjacent local adiabatic air temperatures in con-tact with control volume j. It depends on the air boundary layercondition, namely, whether this is laminar or turbulent.

Refrigerant Temperature. The refrigerant properties werecalculated using the procedures reported in Refs. [18,19] for car-bon dioxide and [20] for water. The properties had accuracies of62% across the critical point. In the other regions, the accuracieswere better. Figure 4 shows the refrigerant tubes’ length Lref andinner diameter Dref. The refrigerant flow is usually turbulentthrough the airfoil heat exchanger in new IR aviation gas turbinesbecause a large heat flow rate per unit flow rate is preferable to asmall pressure loss. Here, as shown in Fig. 4, each position isnamed to facilitate the discussion as follows: A, inlet pressure sen-sor; B, cross fitting; C, inlet thermocouple; D, beginning point offlow rate meter by differential pressure; E, its ending point justupstream of the center airfoil heat exchanger; F, ending point ofthe first u-bend section; G, that of the second; H, that of the third;I, that of the fourth; and J, outlet thermocouple. Section DE wasthe flow rate meter, which was made of a smooth tube. The refrig-erant pressure loss DPloss,DE was measured. The refrigerant veloc-ity uref,DE was calculated using the Darcy–Weisbach equation andBlasius’ friction coefficient in a smooth tube for turbulent condi-tions as follows:

uref;DE ¼D1:25

ref;DE

0:1582q0:75ref;Dl0:25

ref;D

DPloss;DE

Lref;DE

" # 11:75

(6)

Mass flow rate mref is found as follows:

mref ¼pD2

ref;DEqref;Duref;DE

4(7)

However, in section AB, the mass flow rate is 3mref, and

uref;AB ¼ 3qref;D

qref;A

Dref;DE

Dref;AB

� �2

uref;DE;

DPloss;AB ¼ 0:1582q0:75

ref;Al0:25ref;ALref;ABu1:75

ref;AB

D1:25ref;AB

(8)

In section BC, the mass flow rate is mref , and

uref;BC ¼qref;D

qref;B

Dref;DE

Dref;BC

� �2

uref;DE;

DPloss;BC ¼ 0:1582q0:75

ref;Bl0:25ref;BLref;BCu1:75

ref;BC

D1:25ref;BC

(9)

and similar considerations apply for sections CD, EF, FG, GH,HI, and IJ.

Therefore, the total pressure at each point from A to J is asfollows:

Ptot;ref;A ¼ Pref;A þ1

2qref;Au2

ref;AB;

Ptot;ref;B ¼ Ptot;ref;A � DPloss;AB; � � � ; and

Ptot;ref;J ¼ Ptot;ref;I � DPloss;IJ

(10)

Then, the measured Pref,A is known. Pref,B and Pref,J are found asfollows:

Pref;B ¼ Ptot;ref;B �1

2qref;Bu2

ref;AB; � � � and

Pref;J ¼ Ptot;ref;J �1

2qref;Ju

2ref;IJ

(11)

At point C, temperature Tref,C is measured and enthalpy Href,C

is defined as follows:

Tref;C ¼ TC and Href;C ¼ Href Tref;C;Pref;C

� �(12)

At each point from A to E, each refrigerant tube is adiabatic.Therefore, the total enthalpy Htot,ref at each point from A to E isthe same. Thus, the enthalpy Href at each point from A to E isfound as follows:

Href;A þu2

ref;AB

2¼ Href;B þ

u2ref;AB

2¼ Href;C þ

u2ref;BC

2

¼ Href;D þu2

ref;CD

2¼ Href;E þ

u2ref;DE

2(13)

Therefore, temperature Tref at each point from A to E can be cal-culated by using Href and Pref.

On the other hand, in each section from EF to IJ, heat inflowsfrom the airfoil heat exchangers. The steady state basic equationin an integrated form for section EF, for example, is as follows:

Qconvection;EF ¼ Qref;EF (14)

Based on the heat balance in section EF, the heat convectionrate Qconvection,EF is

Qconvection;EF ¼ �mref Htot;ref Tref Fð Þ;Pref;F; uref;EF

� ��Htot;ref Tref;E;Pref;E; uref;DE

� ��(15)



and the refrigerant heat gain rate Qref,EF in section EF from theairfoil heat exchanger is

Qref;EF ¼ REF

jQref;j

� ¼ R

EF

jAref;jhref;j Tref;E � TðjÞ

� �� (16)

where Qref,j was described in Eq. (4), and the summation of Qref,j

at all of the control volumes j in contact with the refrigerant insection EF is used. Similar procedures can be applied for sectionsFG, GH, HI, and IJ.

Adiabatic Air Temperature. Although the airfoil heatexchanger surfaces are rigorously nonadiabatic, the air tempera-ture in a boundary layer of a high-speed compressible airflow on asolid surface is close to the adiabatic air temperature. Recently,Pinilla et al. investigated the effects of the adiabatic temperatureon the heat load of the blades of a gas turbine [21]. They men-tioned that the adiabatic temperature plays a role in determiningthe heat flux through the air-contact surfaces.

On the outer air-contact surfaces, the adjacent local static airpressure can be calculated using the adjacent local pressure coeffi-cient Sj as follows:

Pair;j ¼ Ptot;air;in � Sj1

2qair;inu2

air;in (17)

where Sj is obtained from Ref. [13] and interpolated. Note that adecrease in Sj implies an adverse pressure gradient, and vice versa.Thus, the adjacent local air Mach number and the adjacent localstatic air temperature are found as follows:

Mair;j ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2

c� 1

Ptot;air;in

Pair;j

� �c�1c

�1

( )vuut ;Tair;j ¼Ttot;air;in

1þ c�12

Mair;j

(18)

The adjacent local adiabatic air temperatures Tair,adiabatic,j in theair’s laminar or turbulent boundary layer is formulated [22] as

Tair;adiabatic;j ¼ Tair;j þ Ttot;air;in � Tair;j

� �rj (19)

where rj is the adjacent local recovery coefficient. Each rj repre-sents a laminar or turbulent boundary layer as follows:

rj ¼ Pr1=2 for air laminar boundary layer

Pr1=3 for air turbulent boundary layer

�(20)

If a pure air laminar flow enters the system, a transition to a tur-bulent boundary layer occurs across the maximum S pointXSmaxLC, i.e., Reair,transition�XSmaxReair, in a cascade of airfoils.In the present test setup, there may be many main flow instabil-ities. Thus, the transition may occur slightly upstream of XSmaxLC.However, it probably occurs not far from XSmaxLC, as shown inFig. 6.

Here, Tair,adiabatic,j was determined using only the airflow condi-tions, i.e., air Reynolds, Prandtl, and Mach numbers, and can besubstituted into Eq. (5).

Calculation Procedure of Inverse Heat Transfer Method.The following procedure was conducted for the assumed hair,j andhref,j to calculate the distribution of solid temperatures T(j) in theairfoil heat exchanger, as well as to calculate the refrigerant tem-peratures Tref(F)–Tref(J).

First, we calculated the already determined values according tothe experimental conditions before using an inverse heat transfermethod. The already determined values were the distribution ofthe adiabatic air temperatures Tair,adiabatic,j around the airfoil heatexchanger, and the refrigerant temperatures Tref,A–Tref,E. The dis-tribution of Tair,adiabatic,j around the airfoil heat exchanger wasgiven in the Adiabatic Air Temperature subsection when the airinlet conditions and cascade configuration were determined. Tre-

f,A–Tref,E were given in the Refrigerant Temperature subsectionwhen the refrigerant inlet conditions were determined.

Second, for the assumed values of hair,j and href,j, the solid tem-peratures T(j) and the refrigerant temperatures Tref(F)–Tref(J) werefound numerically by solving Eqs. (1) and (14). Here, Eqs. (2),(4), and (5) were substituted into Eq. (1) at all of the solid controlvolumes in the airfoil heat exchanger, and Eqs. (15) and (16) weresubstituted into Eq. (14) at all of the refrigerant sections. Thus, wecould generate simultaneous temperature equations for all the con-trol volumes of the solid and refrigerant sections. To solve these,all of the T coefficients were arranged for control volume j asfollows:

cj;jTðjÞ þ cp;jTðpÞ þ cq;jTðqÞ þ cr;jTðrÞ þ cs;jTðsÞ¼ cj;ETref;E þ cj;nTref nð Þ þ cj;airTair;adiabatic;j (21)

where only Tref,E and Tair,adiabatic,j are known, and Tref(n), i.e.,Tref(F)–Tref(J), are unknown. Therefore, a large coefficient matrixfor all the airfoil temperatures for all the control volumes of thesolid and refrigerant sections was constructed. Then, this coeffi-cient matrix was diagonalized. Finally, the distribution of the solidtemperatures T(j) in the airfoil heat exchanger and the refrigerantsections’ temperatures Tref(F)–Tref(J) were obtained for theassumed href,j and hair,j.

Third, the heat removal rate from the hot air Qair,whole and theinput rate into the cold refrigerant Qref,whole were found asfollows:

Qair;whole ¼Xwhole

j

Qair;j

� �(22)

Qref;whole ¼ mref;DE Htot;ref Tref;J;Pref;J; uref;IJ

� ��Htot;ref Tref;E;Pref;E; uref;DE

� �� (23)

Finding Heat Transfer Coefficients by Least SquareMethod. In the Calculation Procedure of Inverse Heat TransferMethod subsection, a procedure was described for using theassumed hair,j and href,j to calculate the solid temperatures T(j), re-frigerant sections’ temperatures Tref(F)–Tref (J), and heat removalrate from the hot air Qair,whole. In the present subsection, we dis-cuss how to find the best combination of hair,j and href,j. Here, thecalculation results for T(i), T(ii), T(iii), and T(iv) are expressed interms of the experimentally measured Ti, Tii, Tiii, and Tiv values tofacilitate the discussion. We divided the airfoil heat exchangerinto three parts: the front part forward of tube IJ, the rear partbehind tube EF, and the part between them, as shown in Fig. 6.Additionally, as shown in Fig. 6, the space-averaged heat transfercoefficients on the front part surface hair,front, rear part surface hair,-

rear, middle upper surface hair,middle,up, and middle lower surfacehair,middle,low were defined. href was locally determined based on

Fig. 6 Schematic view of air boundary layers around airfoilheat exchangers



the local adjacent refrigerant Reynolds number. However, the dif-ference in the local refrigerant Reynolds numbers between theinlet and outlet was negligible. Thus, the space-averaged href overall of the refrigerant-contact surfaces in the airfoil heat exchangerwas considered.

In this study, the Levenberg–Marquardt algorithm [23] wasused. It is one of the least square methods. In this algorithm, fivefitting functions [f(1)¼ {T(i)� Ti}/Ti, f(2)¼ {T(ii)� Tii}/Tii,f(3)¼ {T(iii)�Tiii}/Tiii, f(4)¼ {T(iv)� Tiv}/Tiv, f(5)¼ {Qair,whole

�Qref,whole}/Qref,whole] were set. The best combination of five in-dependent variables [hair,front, hair,middle,up, hair,middle,low, hair,rear,href] was found to realize the minimum of [f(1)2þ f(2)2þ f(3)2

þ f(4)2þ f(5)2]. In other words, we numerically found the calcula-tion results that were the closest to the experimental results. Thisnumerical analysis was performed by a VisualBasic2010 code thatwe developed using the Levenberg–Marquardt algorithm packageprovided by the ALGLIB Project [24].

Nusselt Numbers and Modified Stanton Numbers. Based onthe calculation results, an average refrigerant Nusselt numberNuref and an average refrigerant modified Stanton number Stref areobtained as follows:

Nuref ¼hrefDref

kref

; Stref ¼hrefpDref5Wfringe

qrefp4D2

refurefCPref

¼ Nuref

RerefPrref

20Wfringe

Dref

(24)

where pDref5Wfringe is the area of the refrigerant-contact surfaces.Here, Stref is a dimensionless number that measures the ratio ofthe heat transferred to the refrigerant to the total heat capacity ofthe refrigerant passing through the airfoil heat exchanger.

The average hair over all the surfaces is calculated using theoverall energy balance as follows:

hair ¼Qair;wholePair�contact

j Aair;jDTair;j

� � (25)

where DTair;j ¼ Tair;adiabatic;j � T jð Þ for control volume j on theouter air-contact surfaces. Then, an average air Nusselt numberNuair and an average air modified Stanton number Stair are calcu-lated as follows:

Nuair ¼hairLC

kair

; Stair ¼hairLCW

qairLG sin bð ÞWuairCPair

¼ Nuair

Reair;channelPrair

(26)

where LGsin(b)W is the cross section of the air stream handled bya unit of the airfoil heat exchanger. Here, the representative length

of Reair,channel is the height LGsin(b) of the air stream handled by aunit of the airfoil heat exchanger. Stair is a dimensionless numberthat measures the ratio of the heat transferred to the air stream tothe total heat capacity of the air stream handled by a unit of theairfoil heat exchanger.

Results and Discussion

Experimental Conditions. The carbon dioxide critical pointswere TC,CO2¼ 304.2 K and PC,CO2¼ 7.38 MPa, whereas the watercritical points were TC,H2O¼ 647.3 K and PC,H2O¼ 22.12 MPa.The refrigerant temperature Tref,E and refrigerant pressure Pref,E

were set to 315 K and 10, 20, and 30 MPa, respectively. Theywere chosen to verify the effects of the supercritical carbon diox-ide and compressed water. The condition at point E in the refriger-ant tube corresponded to the inlet condition immediately upstreamof the airfoil heat exchanger inlet, as shown in Fig. 4. RefrigerantReynolds number Reref,E increased in proportion to the increasingrefrigerant mass flow rate mref,E for each refrigerant for each pres-sure because Reref,E depended only on the inlet velocity. HigherReref values of 8.3� 104 at 10 MPa, 4.8� 104 at 20 MPa, and3.8� 104 at 30 MPa for carbon dioxide were achieved at the samemref,E of 5 g/s compared with an Reref of 0.4� 104 at all pressuresfor water. It is said that carbon dioxide has a better heat transferperformance in comparison with water because the Nusselt num-ber is generally larger in a case with a larger Reynolds number. At10 MPa, the specific heat CPref,E of 6696 J/kg�K for carbon dioxidewas much larger than that of water (4156 J/kg�K). In contrast, thedensity qref,E of 586.0 kg/m3 for carbon dioxide was much smallerthan the value of 995.8 kg/m3 found for the water. The heattransport performance is proportional to qref,ECPref,E, i.e.,qref,ECPref,E¼ 3.92� 106 for carbon dioxide and qref,ECPref,E

¼ 4.14� 106 for water. This produced a better heat transfer per-formance (through a heat transfer surface) for carbon dioxidecompared with water, and almost the same heat transport perform-ances (between different sites) at 10 MPa. Conversely, at 30 MPa,the CPref,E of 1961 J/kg�K for carbon dioxide was smaller than theCPref,E of 4113 J/kg�K for water. Moreover, the density qref,E

of 902.7 kg/m3 for carbon dioxide approached the qref,E of1004 kg/m3 for water. The heat transport performance wasqref,ECPref,E ¼ 1.77� 106 for carbon dioxide and qref,ECPref,E

¼ 4.13� 106 for water. Thus, carbon dioxide has a better heattransfer but less heat transport compared with water at 30 MPa.Given these results, we can say that carbon dioxide is superior towater as a heat transfer medium at all pressures, but the optimumheat transport medium should be determined on a case-by-casebasis.

Figure 7 shows the air Prandtl number Prair,in, Mach numberMair,in, and Reynolds number Reair,in for different values of awithin a range of 0 deg–25.5 deg. Here, the values of Prair,in wereunrelated to the air temperature and pressure. Thus, the inlet air

Fig. 7 Experimental air Mach, Prandtl, and Reynolds numbers versus airflow direc-tion angle from airfoil chord



temperature could be maintained at a convenient value. Addition-ally, to conduct intercooling experiments, the adiabatic air tempera-ture Tair,adiabatic had to be higher than the refrigerant temperatureTref,E. Therefore, the target inlet air total temperature Ttot,air,in wasset to approximately 360 K. Mair,in became smaller in regions farfrom a¼ 10.5 deg. The airfoil incidence n was 0 at a¼ 9.47 deg.Thus, the air pressure loss in a closed wind tunnel increased inthose regions far from n¼ 0. The same thing was true for Reair,in.

All of the experimental data were obtained after reachingdynamic steady states and thermal equilibrium. The following“experimental results” represent the results of inverse heat transferanalyses, in which the corresponding experimental results wereused as boundary conditions and fitting parameters.

Feature of Airfoil Heat Exchanger. Figure 8 shows the tem-perature distributions for large and small values of Stair under thesame airflow conditions: a¼ 10.5 deg and Mair,in¼ 0.60. Theleft-hand figure in Fig. 9 shows the local air pressure coefficient Sdistribution plotted from data taken from Ref. [13]. The others inFig. 9 show the corresponding local air Mach number and adiabaticair temperature distributions on the air-contact surfaces plotted byusing Eqs. (17)–(20) under the same conditions as those shown inFig. 8. In this case, an air turbulent boundary layer formed overalmost the entire lower surface. However, on the upper surface, theair boundary layer transitioned from laminar to turbulent flow atX¼ 0.6, causing the adiabatic air temperature to increase discontin-uously. This adiabatic air temperature was used to conduct theinverse heat transfer analysis shown in Fig. 8. In Fig. 8, the blackindicates a high temperature, whereas the white indicates a lowtemperature. The highest temperatures were found in those regionsnear the leading and trailing edges. This was because the heatentered with the hot air and flowed out into tubes IJ and EF. In thepart between them, however, the heat flowed in from the air-contact surfaces and flowed out into the nearest refrigerant tubes.

Figure 10 shows the air Nusselt number Nuair for each part ver-sus the air modified Stanton number Stair. Each part has a front,upper middle, lower middle, or rear corresponding to each hair in

Fig. 6. In each experiment, the largest was Nuair,middle,low, fol-lowed, in order, by Nuair,middle,up, Nuair,front, and Nuair,rear. On thelower surface, the minimum pressure coefficient Smax was foundat the leading edge for a¼ 0, and gradually shifted to X¼ 0.2 fora¼ 25.5 deg. Thus, on the lower middle surface, there was anadverse pressure gradient, and the air boundary layer switched toa turbulent one near the leading edge. Therefore, Nuair,middle,low

was large. On the upper surface, Smax was located at X¼ 0.5–0.6for a� 13.5 deg, and swiftly shifted at X¼ 0–0.05 fora 16.5 deg. Thus, on the upper middle surface at a� 13.5 deg,there was a favorable pressure gradient, the airflow accelerated,and the air boundary layer became thin, although it remained lam-inar. On the upper middle surface at a 16.5 deg, a turbulent tran-sition occurred. Therefore, Nuair,middle,up was almost equal to orwas slightly smaller than Nuair,middle,low. On the rear surfaces, theair boundary layers were turbulent. However, the boundary layerthickness became much larger, and Nuair,rear was the smallest airNusselt number.

Without cooling the airfoil heat exchanger, the temperature onthe middle air-contact surfaces would become the air adiabatictemperature. Their temperature profiles would be smooth in theairflow direction. Nuair would be determined by the airflow condi-tions. If the refrigerant cooled the airfoil heat exchanger onlyslightly, in right-hand figure in Fig. 8, the temperature gradient(which is indicated by the distance from the gray dotted line to thegray solid and dashed lines) became small. The heat flow rate(Stair) between the middle surfaces and the refrigerant tubes alsobecame small. Thus, the temperature profiles (which are the graysolid and dashed lines for Stair¼ 0.00198) remain smooth. Nuair

would be nearly the same as those when there is no refrigerantcooling. If, on the other hand, the refrigerant cooled the airfoilheat exchanger considerably, the temperature gradient (which isindicated by the distance from the black dotted line to the blacksolid and dashed lines in right-hand figure in Fig. 8) became large.The heat flow rate (Stair) between the middle surfaces and the re-frigerant tubes also became large. Then, the airfoil heat exchangertemperature varied widely depending on the distance from eachrefrigerant tube along each heat flow path. Therefore, there were

Fig. 9 Distributions of air pressure coefficient, air local Mach number, and adiabatic air temperature corresponding to experi-mental results shown in Fig. 8 (air pressure coefficient data were taken from Ref. [13])

Fig. 8 Comparison of temperature distributions of airfoil heat exchangers for large and smallair modified Stanton numbers



many temperature fluctuations on the middle surfaces in the air-flow direction, such as the black solid and dashed lines forStair¼ 0.00348. Thus, strong thermal entrance effects may occurin the air boundary layers on the middle surfaces, making Nuair

larger, and the temperature gradient larger. Namely, there may bea positive feedback effect until the temperature on the middlesurfaces approaches the air adiabatic temperature. This is whyNuair,middle,low and Nuair,middle,up increase with an increase in Stair,as shown in Fig. 10. This conclusion will be confirmed if transientexperiments are performed in the future. However, the front andrear parts of the airfoil heat exchanger resemble forced-convection fins. Thus, a uniform temperature gradient forms, sothat there are no temperature fluctuations along the airflow direc-tion. Therefore, Nuair,front and Nuair,rear are determined dependingon the airflow condition but independently of Stair.

Figure 11 shows the average air Nusselt number versus airReynolds number at the outlet. The solid lines are from Ref. [12],summarizing the correlations of the air-cooled airfoil dataobtained by Ainley [25], Fray and Barnes [26], Hodge [27], Wil-son and Pope [28], Bammert and Hahnemann, and Andrews andBradley [29], as well as data obtained by Turner himself. In addi-tion, the dashed line is the correlation of the liquid-cooled airfoildata obtained by Freche and Diaguila [30]. Experiments were con-ducted in which the average airfoil temperature was held constantat 220 F (378 K). The symbols in the figures indicate the results ofour study, indicating a wide range of air Nusselt numbers even forthe same air Reynolds number. In the air-cooled gas turbine, thecooling airflow rate was almost proportional to the main airflowrate. This was because the cooling air was bled from the compres-sor, such that the ratio of the airflow rates was almost constant.Thus, the temperature effectiveness and airfoil temperature didnot vary significantly. However, in the airfoil heat exchangers, thecooling refrigerant flow rate could be set independent of the main

airflow rate. Furthermore, the refrigerant showed better a heattransfer performance than air. Lorenzini and Moretti conductedcomputer fluid dynamics calculations for fins in a liquid or airflow [14]. Their results seemed that the temperature distributionof the structure in a liquid varied over a wider range than that inair. The temperature distribution of the refrigerant-cooled airfoilheat exchanger varied over a wider range. Therefore, the air Nus-selt numbers are affected not only by the air Reynolds and Prandtlnumbers but also by the temperature fluctuations of the airfoilsurfaces, as mentioned above. Freche and Diaguila also conductedother experiments while allowing the average airfoil temperatureto vary widely by altering the refrigerant flow rate [30]. Theyfound that the air Nusselt numbers were scattered even at thesame air Reynolds number, although they suggested that experi-mental error caused this phenomenon. The reason for this scatter-ing of the air Nusselt numbers found by Freche and Diaguila islikely to be the same as that found in our study. Basically, the airNusselt number of a refrigerant-cooled airfoil heat exchanger can-not be determined using only the air Reynolds and Prandtlnumbers.

Moreover, the right-hand figure in Fig. 11 shows the distribu-tion of the average of all the Nuair values for each angle-of-attacka in the left-hand figure. The average in ranges far froma¼ 9.47 deg corresponding to n¼ 0 may be higher than those inranges close to a¼ 9.47 deg. It was found that a larger absolute nwas preferred, unless a large separation occurred, from a heattransfer performance point of view.

Design Method for Airfoil Heat Exchanger (RefrigerantNusselt Number). Figure 12 shows the refrigerant Nusselt num-ber for a range of carbon dioxide refrigerant turbulent flows. Itcompares the refrigerant Nusselt numbers estimated according toDittus–Boelter, Liao–Zhao, and our proposed correlations. TheLiao–Zhao correlation [15] is

Nuref ¼ 0:128Re0:8ref;wallPr0:3

ref;wall

Grref

Re2ref;bulk

!0:205qref;bulk

qref;wall

!0:437

� CPref;bulk

CPref;wall

� 0:411

for CO2

(27)

where the subscripts bulk and wall indicate values evaluated atbulk and wall temperatures, respectively. It covers carbon dioxideflow up to 12 MPa. In Fig. 12, the values calculated by theDittus–Boelter correlation were underestimated. The thermal en-trance regions through the airfoil heat exchanger in the refrigerantboundary layers and secondary Dean flows due to u-turn sectionsmay have affected the enhancement of the refrigerant turbulentNusselt number. On the other hand, the values calculated by theLiao–Zhao method were overestimated. Their method evaluatesthe buoyancy effect in a horizontal tube; however, our refrigeranttubes were set vertically. Therefore, our study’s buoyancy effectin the radial direction may be smaller than the Liao–Zhao estima-tion. Our correlation results for the carbon dioxide refrigerantNusselt number were obtained as follows:

Nuref;turbulent ¼ 0:0231Re0:823ref Pr0:300

ref for CO2 (28)

by using a least square method for experimental carbon dioxideNuref,turbulent. This correlation converged to within �25% andþ50%, relative to the experimental results. The average values ofthe current results were located between the Dittus–Boelter andLiao–Zhao correlations. The Dittus–Boelter correlation was ac-ceptable, although there were slight differences in the constantscompared with our correlation.

Our correlation results for the water refrigerant Nusselt numberwere obtained as follows:

Fig. 11 Average air Nusselt number versus air Reynolds num-ber at outlet at various angles-of-attack a

Fig. 10 Experimental air Nusselt number for each part versusair modified Stanton number



Nuref;turbulent ¼ 0:0230Re0:808ref Pr0:300

ref for H2O (29)

by the same method as that used for carbon dioxide. Our correla-tion is within �10% and þ50% compared with the experimentalresults. The values estimated by the Dittus–Boelter correlationwere also underestimated for the previously given reason. How-ever, the Dittus–Boelter correlation was acceptable, althoughthere were also slight differences in the constants compared withour correlation.

Design Method for Airfoil Heat Exchanger (ThermalResistance). The air Nusselt number Nuair for the refrigerant-cooled airfoil heat exchanger had a wide range, so it was difficultto predict Nuair by a correlation using only Reair and Prair. Instead,the relationships between the heat transfer rate Qwhole, a logarith-mic mean temperature difference DTlm,whole, and the refrigerantNusselt number Nuref were used to estimate Nuair.

Temperature changes in the air and refrigerant through the air-foil heat exchanger were evaluated by the temperature effective-ness /air and /ref as follows:

/air ¼Tair;adiabatic;in � Tair;adiabatic;out

Tair;adiabatic;in � Tref;E;/ref ¼

Tref;J � Tref;E

Tair;adiabatic;in � Tref;E

(30)

Based on the heat balance of the airfoil heat exchanger, the rela-tionships of /air and /ref are found as follows by using the ratio ofthe refrigerant heat capacity rate to air heat capacity rate eRA:

/air ¼ eRA/ref ; eRA ¼mrefCPref

mairCPair

(31)

The heat input into the refrigerant Qwhole is explained by usingEq. (30) as follows:

Qwhole � mrefCPref Tref;J � Tref;E

� �¼ mrefCPref/ref Tair;adiabatic;in � Tref;E

� �(32)

The logarithmic mean temperature difference DTlm,whole for acounter-flow heat exchanger was derived by its definition asfollows:

DTlm;whole ¼ U Tair;adiabatic;in � Tref;E

� �(33)

where the variable U is found as follows:

U ¼ 1 for eRA ¼ 1; U ¼ /ref � /air

ln1�/air

1�/ref

h i for eRA 6¼ 1

Here, based on its definition, the ideal overall heat transfercoefficient g is

g ¼ 11

hrefþ 1

hair

Aref

Aair

(34)

if the heat exchanger is an ideal counter-flow heat exchanger with-out thermal resistance in the airfoil heat exchanger’s solid mate-rial. Furthermore, the correction factor w is the ratio of thepractical heat transfer rate Qwhole to the ideal heat transfer rate,which includes thermal resistance of the airfoil heat exchanger’ssolid material, as follows:

Qwhole ¼ wgArefDTlm;whole (35)

Generally, the overall heat transfer coefficient includes thermal re-sistance by using the term d/k in its denominator, where d is thethickness of the solid material, and k is the thermal conductivity.However, it is difficult to define the value of d because there is noconstant thickness in the airfoil heat exchanger. Therefore, thecorrection factor w is introduced instead of d/k. Equations (32)and (33) were substituted into Eq. (35). Thus, the practical overallheat transfer coefficient wg is as follows:

wg ¼ mrefCPref

Aref

/ref

U(36)

Therefore, from Eqs. (34) and (36), the air heat transfer coeffi-cient is obtained by using the refrigerant heat transfer coefficienthref derived from Eqs. (24), (28), and (29) as follows:

hair ¼Aref

Aair

1Arefw

mref CPref

U/ref� 1

href

(37)

and air Nusselt number Nuair is calculated using Eq. (26).The values of correction factor w are obtained using a least

square method for all experimental w as follows:

w ¼ 0:1236 0:02093 nj j þ 1½ �/ref � exp �0:5eRAð Þ½ � þ 1 for 4:3� 105 � Reair � 5

� 105;Prair � 0:73 (38)

Fig. 12 Comparison of experimental refrigerant Nusselt number with Dittus–Boelter, Liao–Zhao, and our proposed correla-tions for carbon dioxide turbulent flows



where n is the incidence in degrees, which is the flow directionangle from the airfoil camber line at its leading edge. Under con-stant /ref and eRA, obviously, the correction factor w increaseswith an increase in absolute incidence n in Eq. (38). Figure 13shows a profile example of w. Under constant n and eRA, wincreases and asymptotically approaches a certain large value as/ref decreases; conversely, w drastically approaches zero as /ref

increases. Please note that the value of w estimated by Eq. (38)would be inappropriate at a value greater than /ref that results inw¼ 0. This indicates the existence of a maximum permissible /ref

for each eRA. On the other hand, under constant n and /ref, wincreases as eRA decreases and vice versa. In Eq. (38), w implicitlycontains the effects of Reair and Prair. Basically, w should explic-itly include the Reair and Prair terms. However, the effects cannotbe expressed because experiments in small ranges of Reair andPrair were conducted in this study.

To verify the adequacy of w, we can estimate Nuair from w,Nuref, and Qwhole by using these relations. Figure 14 compares theexperimental values of Nuair with values estimated using this pro-cedure. These estimations are sufficiently accurate within 610%.When practically designing airfoil heat exchanger, conversely,Qwhole can be estimated by using Nuref, w, and Nuair if Nuair willbe obtained by an appropriate method.

Conclusions

The heat transfer performance of two air-to-refrigerant airfoilheat exchangers and a recirculated heat transport refrigerant was

experimentally investigated. This constituted a light and compactheat exchange system. The Reynolds and Mach numbers of the air-flow were in the ranges of 4.3� 105–5� 105 and 0.55–0.62, respec-tively. Carbon dioxide or water was used as a refrigerant up to apressure of 30 MPa. Thus, the carbon dioxide was supercritical, andthe water was a compressed liquid over its critical pressure.

The heat transfer coefficients on the outer air-contact and innerrefrigerant-contact surfaces were calculated using an inverse heattransfer method. The inverse heat transfer calculations were con-ducted under conditions that included our experimentally measuredtemperatures at four points and another researcher’s alreadyreported pressure distribution around an airfoil. The correlation ofthe carbon dioxide Nusselt number on the inner refrigerant-contactsurface was Nuref,turbulent¼ 0.0231 � Re0:823

ref � Pr0:300ref , and that

of the water Nusselt number was Nuref,turbulent¼ 0.0230 � Re0:808ref

� Pr0:300ref . Their correlations were very close to the Dittus–Boelter

correlation. Furthermore, we proposed a method to evaluate a cor-rection factor corresponding to the thermal resistance of the air-to-refrigerant airfoil heat exchanger. The correction factor increasedwith an increase in the absolute incidence, which was the angle ofthe flow direction from the airfoil camber line at its leading edge.The correction factor increased and asymptotically approached acertain value close to unity with a decrease in the refrigerant tem-perature effectiveness. Conversely, the correction factor drasticallyapproached zero with an increase in the refrigerant temperatureeffectiveness. This indicates the existence of a maximum permissi-ble temperature effectiveness for each ratio of the refrigerant heatcapacity rate to that of air. On the other hand, the correction factorincreased with a decrease in the ratio of the refrigerant heatcapacity rate to that of air, and vice versa.

These correlations and correction factor can be used for design-ing airfoil heat exchangers.

Acknowledgment

The authors gratefully thank Professor Emeritus Toshio Naga-shima, Professor Takehiro Himeno, and Professor Koji Okamotoat the University of Tokyo, who gave us useful advice. This workwas supported by JSPS KAKENHI Grant No. 22760622.

Nomenclature

A ¼ areac ¼ constant number

CP ¼ isobaric specific heatd ¼ distance between control volume centersD ¼ refrigerant tube inner diameter

Gr ¼ Grashof numberh ¼ heat transfer coefficientH ¼ enthalpyk ¼ thermal conductivityL ¼ refrigerant tube length

LC, LG ¼ airfoil chord length, tangential spacing between air-foils’ leading edges

m ¼ mass flow rateM ¼ Mach number

Nu ¼ Nusselt numberP ¼ pressure

Pr ¼ Prandtl numberQ ¼ heat flow rater ¼ recovery coefficient

Reair ¼ air Reynolds number whose representative length isairfoil chord length

Reair,channel ¼ air Reynolds number whose representative length isair inlet height handled by the unit airfoil

Reair,nozzle ¼ air Reynolds number whose representative length isnozzle hydraulic diameter

Reref ¼ refrigerant Reynolds number whose representativelength is circular tube diameter

Fig. 13 Example of correction factor for heat transfer with re-frigerant temperature effectiveness /ref and the ratio of refriger-ant heat capacity rate to that of air eRA at a 5 10.5 deg, i.e.,n 5 1.03 deg

Fig. 14 Comparison of experimental air Nusselt number withestimated values calculated by our proposed method



S ¼ pressure coefficient, S ¼ Ptot;air;in � Pair;local

� �=

qair;inu2air;in=2

h iSt ¼ modified Stanton numberT ¼ temperatureu ¼ velocity

W, Wfringe ¼ airfoil width, airfoil width with both side fringesx, y, z ¼ coordinates

X ¼ position to airfoil chord length, leading edge is atX¼ 0, trailing edge is at X¼ 1

Parentheses

(j) ¼ variable to be calculated at jth control volume ofsolid

(n) ¼ variable to be calculated at nth refrigerant sectionnext to jth control volume (n is either E or I)

(T, P, u) ¼ function of T, P, and u

Greek Symbols

a ¼ angle-of-attack in degrees, flow direction anglefrom airfoil chord

b ¼ inlet angle in degrees, flow direction angle fromperpendicular to airfoil row

c ¼ specific heat ratioDPloss ¼ pressure loss

DT, DTlm ¼ temperature difference, logarithmic mean tempera-ture difference

eRA ¼ ratio of refrigerant heat capacity flow rate to airheat capacity flow rate

g ¼ ideal overall heat transfer coefficienth ¼ turning angle in degreel ¼ viscosityn ¼ incidence in degrees, flow direction angle from air-

foil camber line at its leading edge

q ¼ densityr ¼ solidity LC/LG

R ¼ summation/air ¼ air temperature effectiveness,

/air ¼ Tair;adiabatic;in � Tair;adiabatic;out

� �=

Tair;adiabatic;in � Tref;E

� �/ref ¼ refrigerant temperature effectiveness,

/ref ¼ Tref;J � Tref;E

� �= Tair;adiabatic;in � Tref;E

� �U ¼ variable, U ¼ 1 for eRA ¼ 1, U ¼ /ref � /air½ �=

ln 1� /airf g= 1� /reff g½ � for eRA 6¼ 1w ¼ correction factor of practical overall heat transfer

coefficient compared with g

Subscripts

A to J ¼ points in refrigerant flow tube in Fig. 4adiabatic ¼ adiabatic temperature on air-contact surface in

high-speed airflowair ¼ air or outer air-contact surfaceC ¼ critical point

CO2 ¼ carbon dioxidefront ¼ front part of airfoil surfaceH2O ¼ water

in ¼ inleti, ii, iii, iv ¼ thermocouple number

j ¼ jth control volume of solidlow ¼ lower (concave) surfacemax ¼ maximum

middle ¼ middle part of airfoil surfacen ¼ nth refrigerant section next to jth control volume (n

is either E or I)p, q, r, s ¼ neighbor control-volumes index

rear ¼ rear part of airfoil surfaceref ¼ refrigerant or refrigerant-contact surface

Smax ¼ maximum point of pressure coefficient Ssolid ¼ airfoil material

tot ¼ totaltransition ¼ transition point from laminar to turbulent flows

up ¼ upper (convex) surfacewhole ¼ whole airfoil

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Conjugate Heat Transfer in AirtoRefrigerant Airfoil Heat Exchangers

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