Prediction of heat transfer correlations for compact heat exchangersD. Taler

Abstract The following paper compares two numericalmethods for determining correlations for heat transfercoefficients in cross-flow compact heat exchangers. In thefirst method, only the correlation for the air-side heattransfer coefficient is determined. The heat transfer coef-ficient on the tube-side is calculated using the Gnielinskior Dittus-Boelter correlations. In the second method, theheat transfer correlations both on the tube and air-side aredetermined simultaneously in order to predict the air-sideheat transfer more accurately.

Bestimmung von empirischen Warmeubergangsgl-eichungen in berippten Warmeubertragern

Zusammenfassung In der vorliegenden Arbeit werdenzwei numerische Verfahren zur Bestimmung von empiri-schen Gleichungen in Kreuzstrom-Warmeubertragern mitvergroßerter Oberflache verglichen. Bei der ersten Meth-ode wird nur eine Gleichung fur den Warmeubergangs-koeffizienten auf der Gasseite gesucht, wahrend derWarmeubergangskoeffizient an der inneren Oberflache desRohres bekannt ist und nach den Gleichungen von Dittus-Boelter oder Gnielinski berechnet wird. Bei der zweitenMethode werden gleichzeitig Gleichungen fur denWarmeubergang sowohl auf der Gasseite als auch bei derRohrstromung ermittelt.

List of symbolsA area, m2

Af fin surface area, m2

Ain, Aout inside and outside cross section area of the ovaltube, m2

Amf area of the tube outer surface between fins, m2

Amin minimum free flow frontal area on the air side, m2

Atm mean heat transfer surface area of the tube,Atm ¼ Ats þ Atinð Þ=2

Atin area of the inside tube surface, m2

Ats area of the outside surface of the smooth tube, m2

c specific heat, J/kgK�c mean specific heat, J/kgK

dh hydraulic diameter of air flow passages, mdt hydraulic diameter on the liquid side, 4 Ain=Pin, mF dimensionless correction factor to account for

deviations from pure counter-flowf measured temperature, �Cf 00c;i, T00c;i measured and calculated liquid temperature at

the outlet of heat exchanger, �Ch heat transfer coefficient, W/m2Kha heat transfer coefficient on the air side, W/m2Khc inside (liquid-side) heat transfer coefficient,

W/m2Khc;d inside heat transfer coefficient in the lower

(second) pass, W/m2Khc;g inside heat transfer coefficient in the upper

(first) pass, W/m2Kho weighted heat transfer coefficient from the air

side related to outer surface area Ars of thesmooth tube, W/m2K

i specific enthalpy, J/kgja air-side Colburn j factor, Nua= ReaPr1=3

a

� �

k thermal conductivity, W/mKL length, mLch tube length in the automotive radiator, m_m mass flow rate, kg/s_mc mass flow rate of cooling liquid flowing inside

the tubes, kg/s_ma air mass flow rate, kg/s_mIc mass flow rate of cooling liquid flowing through

the first row of tubes at the first pass, kg/sNuc Nusselt number based on the liquid side,

hc dt=kc

Nua Nusselt number based on the air side, ha dh=ka

n number of data setsnt total number of tubes in a heat exchanger,

nt ¼ ng þ nd

ng , nd number of tubes in the first (upper) and thesecond (lower) pass of the heat exchanger,

P confidence interval of the estimated parameters,%

Pin, Pout inner and outer perimeter of the oval tube,respectively, m

Pr Prandtl number, l cp=k_Q total heat transfer rate of the heat exchanger, W_Qchl total heat transfer rate of the radiator, WRec liquid-side Reynolds number, wc dt=mc

Rea air-side Reynolds number, wmax dh=ma

s fin pitch, mT temperature, �CTbf fin base temperature, �C

Forschung im Ingenieurwesen 69 (2005) 137–150 � Springer-Verlag 2005DOI 10.1007/s10010-004-0148-5

137

Received: 8. October 2004

D. TalerDepartment of Power Installations, Academy of Mining andMetallurgy, Al. Mickiewicza 30, Paw. B-3, 30-059 Cracow, Poland,phone: (0-48) 12 617 30 78, fax. (0-48) 12 617 31 13e-mail: taler@imir.agh.edu.pl

T0c, T00c inlet and outlet temperature of cooling liquid,respectively, �C

Tcm outlet temperature of cooling liquid after thefirst pass, �C

T0d, T00d , T000d air temperature at the inlet and after the firstand second row of tubes at the second (lower)pass, respectively, �C

T0g , T00g , T000g air temperature at the inlet and after the firstand second row of tubes at the first(upper)pass, respectively, �C

T0g;i, T00g;i air temperature at the inlet and outlet of thecontrol volume, �C

Ta air temperature, �CT0am, T000am mean inlet and outlet temperature of air, �CT1;i, T1;iþ1 liquid temperature at the inlet and outlet of

the control volume, �CTf mean temperature of the fin, �CU overall heat transfer coefficient related to the

outer surface of the smooth tube Ats,W/m2K_Vc volume flow rate, m3/s

wd velocity of the cooling liquid in the lower(second) pass, m/s

wg velocity of the cooling liquid in the upper(first) pass, m/s

wm mean velocity of the cooling liquid in thelower and upper passes, m/s

wmax mean axial velocity of air in the minimum freeflow area, m/s

w0 air velocity in front of the radiator, m/sx, y, z Cartesian coordinates,

Greek symbolsDx, Dy control volume size in x and y direction, mdf fin thickness, mdt tube wall thickness, mgf fin efficiency,l fluid dynamic viscosity, PaÆsm kinematic viscosity, m2/sq fluid density, kg/m3

r ratio of minimum free flow area to totalfrontal area

Subscriptsa airc cooling liquidd lower (second) passg upper (first) passt tube

1IntroductionMost engineering calculations of heat transfer in heatexchangers with extended surfaces use heat transfer coef-ficients obtained from experimental data. The air-side heattransfer coefficient is usually determined from theknowledge of the overall heat transfer coefficient U basedon the air-side surface area A, the total rate of heat transfer_Qm between the hot and cold fluids and the logarithmicmean temperature difference DTlm

U ¼_Qm

A DTlm F: ð1Þ

In this paper, the overall heat transfer coefficient is basedon the outer area of the smooth tube A ¼ Ats. Since thefluid flow rates and the inlet and outlet temperatures areknown from the measurements, the total heat transfer rate_Qm is calculated as an arithmetic average of the liquid - _Qc

and air-side _Qa rate values from the following expression

_Qm ¼ _Qc þ _Qa

� �=2; ð2Þ

where

_Qc ¼ _mccp;c f 0c � f 00c� �

; and _Qa ¼ _macp;a f 00a � f 0a� �

:

Values of the air-side heat transfer coefficient ha arecalculated from the experimental data by subtracting theliquid-side and tube wall thermal resistances from theoverall thermal resistance 1=U , as given by (3)

1

ho¼ 1

U� 1

hc

Ats

Atin� dt

kt

Ats

Atm; ð3Þ

where the weighted heat transfer coefficient ho is definedby

ho ¼ haAmf

Atsþ Af

Ats� gf hað Þ

� �: ð4Þ

The foregoing procedure is repeated for a variety of testconditions. Then, following Kays and London [1], theColburn factors, ja;i,i ¼ 1; . . . ; n, are calculated and acorrelation of the form ja ¼ f Reað Þ is determined usingthe linear regression technique. The analysis describedabove aims to determine a correlation for the air-side heattransfer coefficient only. This is a typical procedure usedin data reduction in order to find empirical air-side cor-relations.

Information on the effect of the tube-side heat transferon the determined air-side heat transfer correlations andon the heat transfer performance of compact heatexchangers is limited.

The published literature and the manufacturer’s soft-ware ignore the issue of the tube-side heat transfer cor-relations and the heat transfer rate of the heat exchanger isusually over-predicted under part-load conditions.

It is assumed that the air-side resistance dominates andthe tube-side heat resistance is neglected [1, 2] orapproximately calculated from correlations published inliterature [3–13]. The water-side thermal resistance is as-sumed to be less than 10 � 20% of the overall thermalresistance [3, 4]. While this assumption is valid for highwater flow rates, the water-side heat transfer coefficientmust be determined more accurately for compact heatexchangers, which operate at low water velocities. It wasfound [5] that the water-side resistance increases rapidlyas the water velocity decreases. At a Reynolds number of2500, the water-side resistance constitutes more than 60%of the total heat transfer resistance [5]. The heat transfercoefficient on the tube-side is calculated using the Dittus-Boelter [2, 6–10], Sider-Tate [11] or Gnielinski correlations[4, 5, 12, 13]. The use of the Dittus-Boelter equation forcalculating water-side heat transfer coefficient significantly

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Forsch Ingenieurwes 69 (2005)

under-predicts the water-side heat transfer resistance atlow water flow rates [5].

When tube-side fluid velocities are low and air velocitieshigh, the heat transfer resistances are comparable on bothtube sides, since the heat transfer on the air-side is enhancedby fins. An inaccurate estimation of the inside heat transfercoefficient on the inner surface of the tube can cause largeerrors in the air-side heat transfer coefficient that is to bedetermined. This will be demonstrated by means of thefollowing example. Calculations are conducted for the au-tomotive radiator, which will be analysed in this paper.

Let us assume that the overall heat transfer coefficient Uobtained from measurements is U ¼ 600W/m2K. For thepurposes of this example, the following parameters arespecified:

kt ¼ 207 W/mK; dt ¼ 0:0004 m; Af =Ats ¼ 17:551;

Amf =Ats ¼ 0:92; Ats=Atin ¼ 1:1109:

A smooth plate fin is divided into equivalent rectangularfins. The efficiency of the fin on the oval tube is defined bythe expression: gf ¼ �Tf � Ta

� �= Tbf � Ta

� �. The mean

temperature of the fin �Tf was computed by means of theFinite Element Method (FEM) [14] for different values ofthe heat transfer coefficient ha. Then, the calculated finefficiency gf was approximated by the expression of theform

gf ¼1þ 6:45184 � 10�4 � ha

1þ 2:76144 � 10�3 � ha þ 5:16617 � 10�7 � h2a

: ð5Þ

The following procedure should be followed to calculatethe heat transfer coefficient ha on the air-side. First, ho iscalculated from (3). Then, the air-side heat transfer coef-ficient ha is determined from (4). Since the fin efficiency gf

given by (5) depends upon ha, an iterative determinationof ha from (4) is required. The results of the calculation areshown in Table 1. It can be seen that the determined air-side heat transfer coefficient ha ranges from 120.05 W/m2Kfor hc ¼ 1000W/m2K to 37.42 W/m2K for hc ¼ 10000W/m2K. The variation of the tube side heat transfer coeffi-cient hc has a large effect upon the determined air-sideheat transfer coefficient ha, especially for low values of hc.

This simple example has demonstrated that an accuratedetermination of the inside heat transfer coefficient hc isvery important.

A modified Wilson plot technique for obtaining heattransfer correlation on one side and the value of the heattransfer coefficient on the other side is presented in [15].In this method, the fluid flow rate on the unknown side,for which a correlation is to be determined, is variedsystematically. The fluid thermal resistance on the otherside, for which only the value of the heat transfer coeffi-cient is being determined, should remain approximatelyconstant. The values of the unknown constants are esti-mated using non-linear regression. The sum of squares of

measured and calculated thermal resistances is minimised.The method proposed in [15] may also be used if the fluidthermal resistance on the one side of the heat exchanger isnot dominant.

The techniques presented in this paper accept experi-mental data obtained at various flow rates on each side ofthe heat exchanger. The purpose of the paper is to presentand correlate data showing the effect of the tube-side heattransfer on the searched empirical air-side correlationsand on the total heat transfer rate of the smooth plate fin-and-tube heat exchangers. Two numerical methods forobtaining correlations for both fluid sides are presented.

2Determining heat transfer conditionson the liquid- and air- sidesThe estimation of the heat transfer coefficients of the air-and coolant-sides is the inverse heat transfer problem [16,17]. The following parameters are known from the mea-surements:

– coolant volumetric flow rate _Vc,– air velocity w0,– inlet liquid temperature f 0c ,– inlet air temperature f 0am,– outlet liquid temperature f 00c .

The construction of the heat exchanger and the materialsof which it is made are also known. The following twocases are analyzed:

1. The heat transfer coefficient hc on the cooling liquidside is known, while the heat transfer coefficient on theair side ha is to be found.

2. Both heat transfer coefficients hc and ha on the coolingliquid- and air-sides are to be determined simulta-neously.

The outlet temperatures T00c;i, i ¼ 1; . . . ; n were calculatedusing the numerical model of the heat exchanger presentedin [16].

2.1The heat transfer coefficient hc on the cooling liquid-side isknown, and the heat transfer coefficient on the air-side ha

is determinedIn the first case, the experimental value of the heat transfercoefficient he

a;i is determined from the condition that the

calculated liquid outlet temperature T00c;i hea;i

� �must be

equal to the measured temperature f 00c;i, where i denotes thedata set number.

This means that in order to determine hea;i, the following

non-linear algebraic equation has to be solved for eachdata set

f 00c;i � T00c;i hea;i

� �¼ 0; i ¼ 1; . . . ; n; ð6Þ

where n denotes the number of data sets.

Table 1. Effect of the tube-side heat transfer coefficient on the determined air-side heat transfer coefficient

hc W/m2K 1000 1250 1500 3000 5000 7000 10 000ha W/m2K 120.05 80.65 66.13 45.58 40.53 38.69 37.42gf 0.8047 0.8580 0.8800 0.9135 0.9222 0.9254 0.9276

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D. Taler: Prediction of heat transfer correlations for compact heat exchangers

The heat transfer coefficient on the air side hea;i was

determined by searching the preset interval so that themeasured outlet temperature of the liquid f 00c;i and the

computed outlet temperature T00c;i are equal. Next, a specific

form was adopted for the correlation formula for theColburn factor ja ¼ jaðReaÞ on the air-side, containing munknown coefficients x1; x2; . . . ; xm; m � n. The coeffi-cients x1; x2; . . . ; xm are determined using the least squaresmethod from the condition

S¼Xn

i¼1

jea;i� ja;i x1;x2; . . . ;xmð Þh i2

¼min; m� n; ð7Þ

where

ja ¼ Nua=ðRea Pr1=3a Þ ð8Þ

denotes the Colburn parameter. The symbol jea;i denotes

the experimentally determined value of the factor, andja;i the calculated factor value which results from theadopted approximating function for the set value of theReynolds number Rea;i. In this paper the average heattransfer coefficient hc on the inner surface of thetube was computed using the Dittus-Boelter correlation[18]

Nuc ¼ 0:023Re0:8c Pr0:3

c 1þ dt

Lch

� �2=3" #

ð9Þ

or Gnielinski equation [19]

Nuc ¼n8 Rec � 1000ð ÞPrc

1þ 12:7 n8

� �1=2Pr2=3

c � 1� � 1þ dt

Lch

� �2=3" #

;

ð10Þwith the friction factor n given by

n ¼ 1

ð1:82 log Rec � 1:64Þ2¼ 1

ð0:79 ln Rec � 1:64Þ2:

The Colburn ja factor was approximated by the power-lawfunction

ja ¼ x1Rex2a ; ð11Þ

or by non-linear function (decay function) of the form

ja ¼x1

1þ x1 x2Rea: ð12Þ

The unknown coefficients x1 and x2 in (11) or (12) aredetermined by the Levenberg-Marquardt method [20,21]using the Table-Curve program [22]. Combining Equa-tions (8) and (11) or (12), one obtains

Nua ¼ x1Re 1þx2ð Þa Pr1=3

a ; ð13Þand

Nua ¼x1

1þ x1 x2 ReaReaPr1=3

a ; ð14Þ

where Rea ¼ wmaxdh=ma.All the air properties that appear in the dimensionless

numbers are evaluated at the average temperature taken

from the inlet temperature T0am and outlet temperatureT00am.

2.2Both heat transfer coefficients hc and ha on the coolingliquid- and air-sides are determined simultaneouslyIn the previous chapter, the heat transfer coefficient ha onthe air-side was determined under the assumption that theheat transfer coefficient hc on the cooling liquid-side isknown.

In the second method, the coefficients and exponents offormula for determining heat transfer coefficients on theair- and cooling liquid-side are now determined simulta-neously in order to make a better approximation of theexperimental data. At first, a specific form is adopted forthe correlation formulas of the heat transfer coefficient onthe air-side ha ¼ ha x1; . . . ; xp

� �and on the cooling liquid

side hc ¼ hc xpþ1; . . . ; xm

� �, where p < m.

The symbol m denotes the total number of coefficientspresent in the correlation formulas, and p the number ofunknown coefficients in the formula for the air-side co-efficient. The outlet temperature of the cooling liquid T00c isa function of all coefficients: x1; x2; . . . ; xm.

The values of the coefficients x1; x2; . . . ; xm are selectedso that the sum of squared temperature differences

S ¼Xn

i¼1

f 00c;i � T00c;i x1; . . . ; xmð Þh i 2

¼ min ð15Þ

achieves its minimum.The Levenberg-Marquardt method [20,21] is used to

find the coefficients x1; x2; . . . ; xm from (15).The heat transfer coefficient formula at the air-side is

assumed to be of the form

Nua ¼ x1Rex2a Pr1=3

a : ð16ÞThe Nusselt number Nuc for the cooling liquid is ap-proximated by the formula of the form similar to theDittus-Boelter equation

Nuc ¼ x3Rex4c Pr0:3

c 1þ dt

Lch

� �2=3" #

; ð17Þ

or to the Gnielinski equation

Nuc ¼n8 Rec � x3ð ÞPrc

1þ x4n8

� �1=2Pr2=3

c � 1� � 1þ dt

Lch

� �2=3" #

: ð18Þ

Under the assumption that x3= 1000 and x4= 12.7, theformula (18) changes over into the Gnielinski formula(10). All the physical properties of the liquid required forthe calculation of Nuc, Rec and Prc, are to be evaluated atthe mean temperature Tm ¼ 0:5 T0c þ T00c

� �.

The values of parameters x1, x2, x3 and x4 are deter-mined by minimising the sum of squares (15) using theLevenberg-Marquardt method. In order to calculate theoutlet coolant temperatures T00c;i as a function of searchedparameters xi; i ¼ 1; . . . ; 4, a mathematical model of theheat exchanger is required.

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Forsch Ingenieurwes 69 (2005)

3The numerical model of the heat exchangerThe automotive radiator for the spark-ignition combus-tion engine with a cubic capacity of 1580 cm3 is a double-row, two-pass plate-finned heat exchanger. The radiatorconsists of aluminium tubes of oval cross-section. Thecooling liquid flows in parallel through both tube rows.Figure 1 shows a diagram of the two-pass cross-flowradiator with two rows of tubes.

The heat exchanger consists of the aluminium tubes ofoval cross-section.

A discrete mathematical model [16], which defines theheat transfer was obtained using the control volumemethod. Figure 2 shows the division of the first pass

(upper) into control volumes. There are 2 ng tubes in theupper pass, with ng in the first and second row. Similarly,there are 2 nd pipes in the first and second rows in thelower pass, with nd in each of them. Mass flow rate of theliquid _mIc that passes through the first row of tubes in theupper row is equal to the half of the total _mc flow rate, e.g._mIc ¼ _mc=2. The outlets from the upper pass tubes con-

verge into one manifold. Upon mixing the cooling liquidwith the temperature T001 from the first tube row and thecooling liquid with the temperature T002 from the secondtube row, the feeding liquid temperature of the second,lower pass is Tcm. In the second, lower pass, the total massflow rate splits into two equal flow rates _mc=2. On theoutlet from the first tube row in the bottom pass the

Fig. 1. Diagram of two row cross-flowheat exchanger (automotive radiator)with two passes; 1A) front view of theradiator (a) and horizontal section of theupper pass (b); 1B) flow diagram, 1 - firsttube row in upper pass, 2 - second tuberow in upper pass, 3 - first tube row inlower pass, 4 - second tube row in lowerpass

141

D. Taler: Prediction of heat transfer correlations for compact heat exchangers

coolant temperature is T003 , and from the second row is T004 .Upon mixing the cooling liquid from the first and secondrow, the final coolant temperature that leaves the radiatoris T00c . The air stream with mass flow rate _ma flows cross-wise through both tube rows. Assuming that the inlet ve-locity w0 is in the upper and lower pass, the mass rate ofair flow through the upper pass is _mg ¼ _ma ng=nt, whereng is the number of tubes in the first row of the upper passand nt is the total number of tubes in the first row of theupper and lower pass. The air mass flow rate across thetubes in the lower pass is _md ¼ _mand=nt, where nd is thenumber of tubes in the first row of the lower pass.

A division of the upper pass of the exchanger intocontrol volumes is presented in Figure 2. In order to in-crease the accuracy of the calculations, a staggered meshwill be applied. Liquid temperatures at the control volumenodes are denoted by W1(I) and W2(I) for the first andsecond rows of tubes, respectively. P1(I) denotes air tem-perature T0g;i ¼ T0am in front of the radiator, P2(I) denotesair temperature T00g; i after the first row of tubes and P3(I)

air temperature T000g; i after the second row of tubes in thei-th control volume.

The coolant temperature flowing through the first andsecond row, in the upper and lower pass, is a function ofthe x coordinate only.

Temperatures T0am and T000am denote the mean values of theradiator’s inlet and outlet air temperatures, respectively. Inorder to determine temperature distribution of the liquidand air, a system of non-linear algebraic equations wassolved, consisting of 4N balance equations for the liquid,and 4N equations for the air. For the analysis, the liquidtemperature at the inlet to the first and second row of tubesin the upper pass was considered to be T0c and Tcm in thelower pass. Using the notation shown in Figures 1 and 2, theboundary conditions can be written in the following form

W1 1ð Þ ¼ W2 1ð Þ ¼ T0c ¼ T01 ¼ T03;P1 Ið Þ ¼ T0am ¼ T0gm I ¼ 1; . . . ;N:

ð19Þ

The temperature Tcm is a temperature of the liquid at theoutlet of the upper pass, where the liquid of temperature

W1 N þ 1ð Þ from the first row of tubes has been mixedwith the liquid of temperature W2 N þ 1ð Þ flowing out ofthe second row of tubes. In the case of the automotiveradiator, temperature T0c denotes the liquid temperature(of the engine coolant) at the inlet to the radiator whereasT0am denotes air temperature in front of the radiator.

Having determined the mean temperatures of the airT000gm and T000dm leaving the second row of tubes in the upperand lower pass, respectively, a mean temperature of the air

behind the whole radiator T000am ¼ ng T000gm þ nd T000dm

� �=nt

was calculated. If the liquid and air temperatures areknown, the heat transfer rate in the first and second rowsof tubes in the upper and lower passes can be determined.The total heat transfer rate for the radiator was calculatedusing the formula _Qchl ¼ _mc ic T0c

� �� ic T00c

� � ¼ _ma �ca

T000am � T0am

� �. The numerical model of the heat exchanger

described briefly above is used in (6) and (15) to calculatethe outlet temperature of the cooling liquid T00c . The pos-sibility of including the variable, thermophysical proper-ties as well as the variable heat transfer coefficient alongthe flow path is the advantage of the numerical model.Also, the gas flow through a part of the exchanger cross-section, caused, e.g., by fan assembly, can easily beaccounted for.

4Correlation of experimental dataCorrelations for the heat transfer coefficients in an auto-motive radiator were determined using the methods de-scribed above. The tested automotive radiator is used forcooling the spark ignition engine of a cubic capacity of1580 cm3. The cooling liquid for the engine is a 35 % watersolution of mono-ethylene glycol. The cooling liquid,warmed up by the engine is subsequently cooled down byair in the radiator. The radiator consists of 38 tubes of anoval cross-section, with 20 of them located in the upperpass with 10 tubes per row (Fig. 1). In the lower pass, thereare 18 tubes with 9 tubes per row. The radiator is 520 mmwide, 359 mm high and 34 mm thick. The outer diameters

Fig. 2. Division of the upper pass ofthe car radiator into control volumes;�- air temperature, � - cooling liquidtemperature

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Forsch Ingenieurwes 69 (2005)

of the oval tube are dmin ¼ 6:35 mm, dmax ¼ 11:82 mm.The thickness of the tube wall is dt ¼ 0:4 mm. The numberof plate fins with the thickness df ¼ 0:08 mm is 520. Theplate fins and the tubes are made of aluminium. The pathof the coolant flow is U-shaped. The two rows of tubes inthe first pass are fed simultaneously from one header. Thewater streams from the first and second row are mixed inthe intermediate header. Following that, the water is uni-formly distributed between the tubes of the first and sec-ond row in the second pass. The inlet, intermediate andoutlet headers are made of plastic. The pitches of the tubearrangement are as follows: perpendicular to the air flowdirection p1=18.5 mm and longitudinal p2=17 mm.

The hydraulic diameter of an oval tube is calculatedusing the formula dt ¼ 4 Ain=Pin. The Reynolds and Nus-selt numbers were determined on the base of the hydraulicdiameter dt. Equivalent hydraulic diameter dh on the sideof the air is specified by the following expression

dh ¼2AminL

Af þ Amf; ð20Þ

where

Af ¼ 2ðp1p2 � AoutÞ; Amf ¼ Pout s� df

� �:

The velocity wmax in the Reynolds number Rea ¼ wmaxdh=ma means the air velocity in the narrowest free flow cross-section. As the tubes in the radiator are set in line, wmax isthe air velocity in the passage between two tubes. The airvelocity increases through the radiator as its volume growsdue to warming-up from temperature T0am to T000am, whereT0am is the mean temperature in front of the radiator, andT000am the mean air temperature behind the radiator.

Assuming that the mean air temperature through theradiator is �Tam ¼ T0am þ T000am

� �=2, the maximum velocity

of air flowing between the tubes is

wmax ¼sp1

s� df

� �p1 � dminð Þ

�Tam þ 273

T0am þ 273w0 ; ð21Þ

where �Tam and T0am are in �C.The experimental results are shown in Table 2. The

measurements were conducted in an aerodynamic tunnelby a leading manufacturer of automotive radiators. The airpassed the whole front cross-section.

At the beginning, the first method described in Section2.1 was used to determine a power law correlation of theform (11). The tube side heat transfer coefficient wascalculated using the Dittus-Boelter (9) or Gnielinski (10)correlation. Using the measured data given in Table 2, theinterval was searched for each data set and the air-sideheat transfer coefficients were determined to satisfyequation (6). The physical properties of air and glycolsolution were approximated using simple functions. Theeffect of temperature-dependent properties is taken intoaccount by evaluating all the properties at the mean tem-perature of the air and glycol solution, respectively. Thecomputation yielded eighteen values of the heat transfercoefficient he

a;i, i ¼ 1; . . . ; 18 (Table 3).The discrepancy between the results of the air-side heat

transfer coefficients ha obtained for the Dittus-Boelter andGnielinski correlation does not exceed 7%. It has been

found that, relative differences tend to increase withdecreasing liquid velocity inside the tube. For the largevalues of liquid velocity both predictions are more com-patible. In the eighteenth data set a small difference in thetube-side heat transfer coefficients causes noticeable dif-ference in the determined air-side heat transfer coeffi-cients.

Subsequently, the Colburn factors jei , i ¼ 1; . . . ; 18 were

calculated as a function of the Reynolds number using (8).A commercially available software package, Table Curve

Table 2. Thermal measurement results for the automotiveradiator

No. w0 [m/s] _Vc [l/h] f0

am [�C] f0

c [�C] f0

c � f00

c

� �

[K]

1. 1.51 5004 19.5 94.7 2.52. 2.50 5009 19.3 94.7 3.73. 4.00 5002 19.5 94.7 4.94. 5.50 5000 19.7 94.7 5.95. 7.00 5004 20.4 94.7 6.56. 8.51 5008 20.4 94.7 7.27. 1.51 3000 20.1 94.5 4.08. 2.51 3006 19.5 94.6 5.99. 4.01 3008 19.4 94.6 7.8

10. 5.50 2996 19.3 94.7 9.011. 7.00 3005 20.3 94.7 9.912. 8.51 2992 20.1 94.7 10.913. 1.50 1009 19.8 93.9 10.414. 2.51 1000 20.2 94.0 13.915. 4.00 996 20.4 94.1 16.416. 5.50 991 19.7 94.0 18.917. 7.01 1004 20.4 94.1 19.718. 8.52 1004 20.3 94.0 21.1

Table 3. Values of the air-side heat transfer coefficients ha de-termined by the first method using Dittus-Boelter or Gnielinskicorrelation for calculating tube-side heat transfer coefficient hc

No. Dittus-Boelter correlationfor hc

Gnielinski correlationfor hc

hc;g and hc;d ,W/m2K

ha,W/m2K

hc;g and hc;d,W/m2K

ha,W/m2K

1. 6740 7332 28.0 8354 9156 27.52. 6714 7304 42.1 8325 9125 41.13. 6675 7262 55.7 8279 9075 54.04. 6646 7232 68.6 8247 9040 66.05. 6636 7219 76.7 8234 9027 73.46. 6622 7204 87.0 8219 9010 82.87. 4443 4834 28.2 5271 5796 27.68. 4421 4810 42.9 5245 5768 41.69. 4390 4777 58.2 5208 5728 55.9

10. 4359 4743 67.1 5169 5685 64.011. 4354 4737 76.9 5164 5680 72.912. 4322 4703 86.9 5122 5634 81.713. 1804 1962 29.0 1701 1921 29.414. 1768 1923 41.4 1647 1864 42.415. 1746 1900 49.8 1614 1829 51.616. 1721 1872 61.6 1575 1787 64.817. 1735 1887 66.7 1594 1808 70.218. 1724 1875 76.3 1576 1789 81.3

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[22] was used to perform a non-linear regression analysis.The goodness of fit can be assessed by means of the co-efficient of determination r2 and the 95% confidenceinterval.

When the tube-side heat transfer coefficient hc wascalculated from the Dittus-Boelter correlation (9), thefollowing values of the constant x1 and exponent x2 inequation (11) were obtained: x1 ¼ 5:683 � 10�2 � 0:01528and x2 ¼ �0:37106� 0:04355. The coefficient of determi-nation is r2 ¼ 0:9529.

Substituting the estimated constants x1 and x2 into (11)and (13) gives, respectively

ja ¼ 0:05683 Re�0:3711a ; ð22Þ

200 � Rea � 1500;

Nua ¼ 0:05683Re0:6289a Pr1=3

a : ð23ÞUsing the Dittus-Boelter correlation (9) for the tube-sideheat transfer coefficient hc in conjunction with (12), thefollowing values of the constant x1 and exponent x2 with95% confidence limits were obtained: x1 ¼ 9:3982 � 10�3�6:13 � 10�4 and x2 ¼ 0:12499� 0:0137. The coefficient ofdetermination is r2 ¼ 0:9668.

Substituting the estimated constants x1 and x2 into (12)and (14) gives, respectively

ja ¼0:009398

1þ 0:00117469 Rea; ð24Þ

200 � Rea � 1500;

Nua ¼0:009398

1þ 0:00117469 ReaReaPr1=3

a : ð25Þ

The data point jea;i; i ¼ 1; . . . ; 18 and the correlations

(22) and (24) are shown in Figure 3. The difference be-tween these two predictions is noticeable for larger valuesof the Reynolds number.

When the tube-side heat transfer coefficient hc wascalculated from the Gnielinski correlation (10), the fol-lowing values of the constant x1 and exponent x2 inequation (11) with 95% confidence limits were obtained:x1 ¼ 5:80975 � 10�2 � 0:0122 and x2 ¼ �0:3765� 0:0341.The coefficient of determination is r2 ¼ 0:9715.

Substituting the estimated constants x1 and x2 intoEquations (11) and (13) gives, respectively

ja ¼ 0:05810 Re�0:3765a ; ð26Þ

200 � Rea � 1500;

Nua ¼ 0:05810Re0:6235a Pr1=3

a : ð27ÞUsing the Gnielinski correlation (10) for the tube-side heattransfer coefficient hc in conjunction with (12), the fol-lowing values of the constant x1 and exponent x2 with 95%confidence limits were obtained:x1 ¼ 9:3802 � 10�3 � 4:16�10�4 and x2 ¼ 0:1285 �9:4 � 10�3. The coefficient ofdetermination is r2 ¼ 0:9668.

Substituting the estimated constants x1 and x2 into (12)and (14) gives, respectively

ja ¼0:009380

1þ 0:00120534 Rea; ð28Þ

200 � Rea � 1500;

Nua ¼0:009380

1þ 0:00120534 ReaReaPr1=3

a : ð29Þ

The data point jea;i; i ¼ 1; . . . ; 18 and the correlations

(26) and (28) are shown in Figure 4. As in the previouscase, it has been found that the deviation between thesetwo predictions tends to increase as the values of theReynolds number increase. It can be concluded that thechoice of the function, which approximates the data is alsoof great importance.

Next, the second method described in section 2.2 wasused to estimate parameters x1, x2, x3 and x4 by minimisingthe sum of squares (21) using the Levenberg-Marquardtmethod. When the air-side Nusselt number Nua isapproximated by (16) and the tube-side Nusselt numberNuc by (17), the least squares non-linear regression analysisgives: x1 ¼ 0:068805� 0:00309, x2 ¼ 0:607209� 0:01158,x3 ¼ 0:011042� 0:00223, x4 ¼ 0:871166� 0:02453. Thevalue of the minimum sum of squares S, defined by (15), forthe determined value of the parameters equals S ¼ 0:328K2. Substituting parameters x1 and x2 in (16) and para-meters x3 and x4 in (17) gives:

Nua ¼ 0:0688Re0:6072a Pr1=3

a ; 200 � Rea � 1500; ð30Þ

Nuc ¼ 0:01104 Re0:8712c Pr0:3

c 1þ dt

Lch

� �2=3" #

;

3000 � Rec � 21000: ð31ÞWhen the air-side Nusselt number Nua is approximated by(16) and the tube-side Nusselt number Nuc by (18), theleast squares non-linear regression analysis gives:

Fig. 3. Air-side Colburn factor ja as a function of Reynoldsnumber Rea determined by using method I and the Dittus-Boelterequation for determining tube-side heat transfer coefficient;� - Colburn factor je

a;i based on the heat transfer coefficientdetermined from equation (6), 1- Equation (22), 2- Equation (24)

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x1 ¼ 0:06951� 0:0030; x2 ¼ 0:6043� 0:0065; x3 ¼442:92� 0:3431; x4 ¼ 19:25� 0:3603. The given limits ofthe 95% confidence intervals are small; this results fromgood accuracy of the developed mathematical model of theradiator and small measurement errors. The expression(16) for the Nusselt number on the air-side, as well as (18)defining the Nusselt number on the side of the coolingliquid, have the form

Nua ¼ 0:06951Re0:6043a Pr1=3

a ; 200 � Rea � 1500;

ð32Þ

Nuc ¼n8 Rec � 442:92ð ÞPrc

1þ 19:25ffiffin8

qPr2=3

c � 1� � 1þ dt

Lch

� �2=3" #

;

3000 � Rec � 21000: ð33Þ

Exponent x2 ¼ 0:6072 and x2 ¼ 0:6043 present at theReynolds number in (30) and (32), respectively, are closeto the exponent x2 ¼ 0:625 in the correlation derived bySchmidt [23] for in-line, high-fin circular tube arrays. Avery good fit of the calculated outlet temperatures T00c;i tomeasured temperatures f 00c;i,i ¼ 1; . . . ; 18 is achieved. Thevalue of the minimum sum of squares S, defined by theformula (15) for the determined value of the parameters, isextremely small and equals S ¼ 0:324 K2. This proves thatthe accuracy of the mathematical model of the radiator,described in Section 3 is good. Additionally, it confirmsthe appropriateness of the form of (26) and (27) thatapproximates Nusselt numbers both for the air- and coo-lant-sides as well as the high accuracy of the experimentaldata.

The correlation for the Colburn ja factor obtained from(30) and (32) are

ja ¼ 0:0688 Re�0:3928a ð34Þ

200 � Rea � 1500:

ja ¼ 0:06951Re�0:3957 ð35Þ

The Colburn ja factors (22) and (24) obtained from thefirst method are compared in Figure 5a with thecorresponding correlations (34) and (35) obtained fromthe second method. It can be seen that the agreementbetween the correlations obtained from the first methodis very good. Similar agreement appears betweencorrelations obtained by the second method. Thecorrelations (34) and (35) differ slightly from eachother. However, the relative differences between thecorrelations obtained from the first and second methodare larger.

The relative difference ea ¼ 100 ja � jD�Ba

� �=jD�B

a be-tween the predicted Colburn jD�B

a factors and the ColburnjD�Ba factor given by (22), which is obtained from the first

method when the tube-side heat transfer coefficient hc iscalculated using the Dittus-Boelter correlation, is shown inFigure 5b.

From the analysis of the results presented in Figure 5bwe see that the correlation (34) obtained by the secondmethod yields Nusselt number values higher by up to 9 %than (26) obtained from the first method.

Next, the Nusselt numbers Nuc on the cooling liquidside, calculated by the Dittus-Boelter (9) and the Gnielin-ski (10) correlations as well as (31) and (33) obtained fromthe second method are compared in Figure 6a. The relativedifference ec ¼ 100 Nuc � NuD�B

c

� �=NuD�B

c between thepredicted tube-side Nusselt number Nuc and the Nusseltnumber NuD�B

c given by the Dittus-Boelter equation (9) isshown in Figure 6b.

The comparison is carried out for a Prandtl number Prc

of 7. The equivalent diameter dt and length Lch of theradiator are dt= 0.00706 m and Lch= 0.52 m.

The maximum relative difference in the Nusseltnumbers predicted by the Dittus-Boelter correlation (9)and the Gnielinski correlation (10) is of the order of30% at the tube-side Reynolds numbers of 2000 and14000 (Fig. 6). It follows from Figure 6 that the Nusseltnumbers obtained from the Dittus-Boelter equation (9)are greater than those calculated by the Gnielinskiequation (10) for small Reynolds numbers and smallerfor larger Reynolds numbers. The new correlations (31)and (33) obtained from the second method are moreaccurate since they account for the oval shape of thecross-section of tubes used in the car radiator andfor the complicated flow path of the cooling liquid.It should be noted that the form of the functionapproximating the tube-side Nusselt number has onlysmall influence on the predicted Nusselt numbersNuc as the correlations (31) and (33) are in goodagreement.

In the next chapter, the power law correlations (31) and(33) for the air- and tube-side Nusselt numbers will be

Fig. 4. Air-side Colburn factor ja as a function of Reynoldsnumber Rea determined by using method I and the Gnielinskiequation for determining tube-side heat transfer coefficient;� - Colburn factor je

a;i based on the heat transfer coefficientdetermined from equation (6), 1- equation (26), 2- equation (28)

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D. Taler: Prediction of heat transfer correlations for compact heat exchangers

used in the numerical model of the heat exchanger and theobtained air temperatures will be compared with the re-sults obtained from 3D simulations using a CFD code.

5Numerical simulationIn order to determine the 3D flow and heat transfer in theair and heat conduction through the fins and tubes, theproblem will be studied numerically. In this paper, the air

Fig. 5. Comparison of correlations for Colburn factor ja obtainedfrom the method I and II; a) Colburn factor ja, b) the relativedifference ea ¼ 100 ja � jD�B

a

� �=jD�B

a between the predicted Col-burn factors ja and the Colburn factor jD�B

a given by Equation(22); 1 Method I, Equation (22), tube-side heat transfer coefficientcalculated using the Dittus-Boelter Equation (9), 2 Method I,Equation (26), tube-side heat transfer coefficient calculated usingthe Gnielinski Equation (10), 3 Method II, Equation (34), tube-side heat transfer coefficient calculated using Equation (31) of theDittus-Boelter type, 4 Method II, Equation (35), tube-side heattransfer coefficient calculated using Equation (33) of the Gni-elinski type

Fig. 6. Comparison of correlations for the tube-side Nusseltnumber Nuc used in the method I and obtained using method II;a) Nusselt number Nuc, b) the relative differenceec ¼ 100 Nuc � NuD�B

c

� �=NuD�B

c between the predicted tube-sideNusselt numbers Nuc and the Nusselt number NuD�B

c given by theDittus-Boelter Equation (9) ; 1 Dittus-Boelter Equation (9), 2Gnielinski Equation (10), 3 Equation (31) of the Dittus-Boeltertype obtained from the method II, 4 Equation (33) of the Gni-elinski type obtained from the method II

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and heat flow in the tested two-row automotive radiatorwas simulated numerically by using the CFD programFLUENT [24]. The three-dimensional (3D) flow is treatedas laminar, since the air-side Reynolds Rea number, basedon the maximum mean axial velocity in the minimum freeflow area, is less than 1550. Owing to the complicatedconstruction of the radiator, the numerical study of thewhole radiator is very difficult to carry out. Due to thesymmetry, the 3D flow through the single narrow passagebetween the fins was simulated. The temperature distri-bution in the adjacent plate fins and tube walls was alsocalculated. In this way, the effect of non-uniform heattransfer coefficient on the tube and fin surfaces is takeninto account, as well as the effect of the tube-to-tube heatconduction through the fins on the heat transferred fromthe cooling liquid to the air. The computations wereconducted for the data sets No. 5 and 17 from the Table 2.The uniform frontal velocity w0 and uniform temperatureT0am in front of the radiator were assumed. The boundarycondition of the third kind (convection surface condition)is specified at the inside surface S of the oval tubes (Fig. 7)

�kt@Tt

@nSj ¼ hc Tt S � Tcjð Þ: ð36Þ

The tube-side heat transfer coefficient hc was obtainedfrom (31). The liquid temperatures T001 and T002 at the outletof the first pass were taken as the bulk liquid temperaturesat the first and second row tubes, respectively. Thenumerical model of the radiator was used to calculatetemperatures T001 and T002 .

The 3D simulations were conducted for the followingdata:

– Data set No. 5: w0 ¼ 7:00m/s (Rea ¼ 1048),

f 0am ¼ 20:4�C, _Vc ¼ 5004 L/h,

hc ¼ 6395:7 W/m2K;T001 ¼ 90:85 �C; T002 ¼ 91:50 �C;

– Data set No. 17: w0 ¼ 7:01m/s (Rea ¼ 1061),f 0am ¼ 20:4� C, _Vc ¼ 1004 L/h,

hc ¼ 1479:7 W/m2K;T001 ¼ 82:43�C; T002 ¼ 83:63�C;

The flow passage between the fins is divided into twelvelayers of control volumes, while only two layers are onthe half of the thickness of the fin (Fig.8). The distribu-tions of the fin temperature and air temperature in thecentre plane between adjacent fins obtained from thenumerical simulation are shown in Figures 9 and 10respectively. The fin and air temperatures are signifi-cantly higher for the data set No. 5, as the tube-side fluidtemperatures and heat transfer coefficients on the innersurface of the tubes are larger, than in the case of thedata set No. 17. The simulation reveals high values of theheat flux on the fin’s leading edge due to the developingair flow. Near the fin surface, the temperature of the airincreases in the flow direction and causes a corre-sponding reduction in the heat flux. Stagnation flow onthe front of the tube in the first row produces high heattransfer near the base of the fin and at the frontal part ofthe tube circumference. Behind the tubes, the low-ve-locity wake regions exist. In the downstream regions ofthe tubes, low air velocities and very low heat fluxes canbe observed in the first and second row. Relatively lowheat fluxes are encountered on the portions of the finthat lie downstream of the minimum flow cross sections.

Fig. 7. Modelled section of the radiator(for the data set No. 5), dimensions aregiven in millimeters

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D. Taler: Prediction of heat transfer correlations for compact heat exchangers

The heat transfer rates are especially low in the re-cir-culation regions behind the tubes. In the regions withsmall air velocities, the air temperatures are large(Fig. 10). Using the correlations (30) and (31) for

calculating heat transfer coefficients the following resultsare obtained:

– Data set No.5� numerical model of the radiator (Fig.2):

hc ¼ 6395:7 W/m2K; ha ¼ 78:4 W/m2K;

U ¼ 1033:5 W/m2K; _Qchl ¼ 35462 W;

DTI;N ¼ T00g;N � T0am ¼ P2ðNÞ � P1ðNÞ ¼ 12:27 K;

DTII;N ¼ T000g;N � T00g;N ¼ P3ðNÞ � P2ðNÞ ¼ 10:24 K;

DTI ¼ DTI;N þ DTII;N ¼ 22:51K;� numerical simulation

D�TI;N ¼ 16:81K; D�TII;N ¼ 9:09 K;

D�TI ¼ D�TI;N þ D�TII;N ¼ 25:90 K:– Data set No.17� numerical model of the radiator (Fig.2):

hc ¼ 6395:7 W/m2K; ha ¼ 78:0 W/m2K; ;

DTI;N ¼ T00g;N � T0am ¼ P2ðNÞ � P1ðNÞ ¼ 6:98 K;

DTII;N ¼ T000g;N � T00g;N ¼ P3ðNÞ � P2ðNÞ ¼ 6:33 K;

DTI ¼ DTI;N þ DTII;N ¼ 13:31K;� numerical simulation

D�TI;N ¼ 8:88 K; D�TII;N ¼ 5:56 K;

D�TI ¼ D�TI;N þ D�TII;N ¼ 14:44 K:

The comparison of the results, which have been obtainedfor the data sets No.5 and No.17 using the mathematicalmodel of the radiator reveals, that for the same air velocitythe overall heat transfer coefficient U increases

Fig. 8. Mesh of finite volumes for the modelled section of theautomotive radiator

Fig. 9. Computed isotherm contours on the fin surface; a) dataset No. 5 b) data set No. 17

Fig. 10. Computed air temperature contours in the middle planebetween adjacent fins; a.) data set No. 5 b) data set No. 17

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Forsch Ingenieurwes 69 (2005)

1033:5=646:1 ¼ 1:69 times due to a higher heat transfercoefficient on the tube inner surface. It corresponds to theincrease of the total heat transfer rate of the radiator, thatis 35462=21407 ¼ 1:66. Examination of the results showsthat the tube side heat transfer coefficient has great in-fluence on the radiator performance.

It has been found from the numerical CFD simulationthat the second-row fin and tube are less effective as heattransfer surfaces than those in the first row. The 64.9% and61.5% of the total heat transfer rate is transferred in thefirst tube row, for the case No. 5 and No. 17, respectively.The heat transfer rate at the second row tube is especiallylow due to the presence of the wakes in front of and behindthe tube. The key to heat transfer enhancement on the finassociated with the first tube row is a major contributionof the developing flow region, while the portion of the finadjacent to the second tube row has no developing flowcontribution and only a small vortex contribution. Theregions behind the tubes contribute very little to the per-formance of the heat exchanger.

The lower values of the air temperature differenceacross the radiator obtained from the numerical model ofthe heat exchanger, which used the empirical heat transfercorrelations based on the experimental data, may resultfrom the contact resistance between the fins and the tubes.The numerical CFD simulation does not account for thethermal contact conductance, because the homogenoustube and fin material is assumed. On the other hand, in thedeveloped data reduction method, contact resistance be-tween the tube and fin has implicitly been included in theair-side heat transfer coefficient, which was used in thenumerical model of the radiator. The air temperaturedifferences across the first and second row of tubes ob-tained from the mathematical model of the radiator arenot so large as in the case of the numerical simulation. Thesmall differences between the temperature drop over thefirst and second row are caused by the uniform air-sideheat transfer coefficient assumed in the mathematicalmodel of the radiator.

6ConclusionsTwo different methods were used to determine correla-tions for the heat transfer coefficients.

In the first method, only the correlation for the air-sideheat transfer coefficient was determined from the condi-tion that the calculated and measured coolant outlettemperatures are equal.

In the second method the problem of determining heattransfer correlations both on the air- and tube-side isformulated as a parameter estimation problem by selectingthe functional form for the Nusselt number Nu ¼ f ðRe,PrÞ.There are m parameters to be determined in such a waythat computed outlet tube side liquid temperatures agreein the least-squares sense with the experimentally obtainedtemperatures. A numerical model of the heat exchangerwas used to calculate the outlet liquid temperature as thefunction of searched parameters.

The results of the experimental investigation of theplate-fin-tube automotive radiator are presented. Thetested two-pass radiator consists of two inline rows of oval

tubes with smooth plate fins. The new correlations for theheat transfer coefficients on the air and coolant sides weredeveloped using presented techniques.

The temperature differences across the first and secondtube rows obtained from the mathematical model of theradiator were compared with the results obtained from thenumerical simulation of the 3D fluid and heat flow, per-formed by means of the commercially available CFD code.The empirical correlations obtained from the method IIwere used in the mathematical model for calculating heattransfer coefficients on the air- and tube-side. The ob-tained results are in good agreement

The heat transfer correlations should be predicted si-multaneously both on the tube and air-side in order toproperly predict the heat transfer correlations on the air-side. The first method, in which only the air-side heattransfer correlation is determined, can be less accuratewhen the fluid velocities inside the tubes are low or whenthe inner tube surface is fouled.

References1. Kays WM, London AL (1984) Compact Heat Exchangers.

3rd Ed, McGraw-Hill, New York2. McQuiston FC, Tree DR (1971) Heat transfer and flow -

friction data for two fin – tube surfaces. Transactions of theASME, Journal of Heat Transfer, Vol. 93: 249-250

3. Rich DG (1975) The effect of the number of tube rows on heattransfer performance of smooth plate fin-and-tube heatexchangers. ASHRAE Transactions, Vol. 81, Part 1, No. 2345:307–319

4. Wang CC, Hsieh YC, Lin YT (1997) Performance of platefinned tube heat exchangers under dehumidifying conditions.Transactions of the ASME, Journal of Heat Transfer, Vol. 119:109–117

5. Mirth DR, Ramadhyani S, Hittle DC (1993) Thermal per-formance of chilled - water cooling coils operating at lowwater velocities. ASHRAE Transactions, Vol. 99, Part 1: 43–53

6. Senshu T, Hatada T, Ishibane K (1981) Heat and masstransfer performance of air coolers under wet conditions.ASHRAE Transactions,Vol. 87, No. 2641: 109–115

7. Beecher DT, Fagan TJ (1987) Effects of fin pattern on the air –side heat transfer coefficient in plate finned - tube heat ex-changers. ASHRAE Transactions, Vol. 93, Part 2: 1961–1986

8. McQuiston FC (1978) Correlation of heat, mass and mo-mentum transport coefficients for plate-fin-tube heat transfersurfaces with staggered tubes. ASHRAE Transactions, Vol. 84,Part 1: 294–309

9. Kayansayan N (1993) Heat transfer characterization of flatplain fins and round tube heat exchangers. ExperimentalThermal and Fluid Science, Vol. 6: 263–272

10. Yun JY, Lee KS (1999) Investigation of heat transfer char-acteristics on various kinds of fin-and-tube heat exchangerswith interrupted surfaces. International Journal of Heat andMass Transfer, Vol. 42: 2375–2385

11. Wanniarachchi S, Marto P J, Rose JW (1986) Film conden-sation of steam on horizontal finned tubes: effect of finspacing. Transactions of the ASME, Journal of Heat Transfer,Vol. 108: 960–966

12. Mirth DR, Ramadhyani S (1994) Correlations for predictingthe air side Nusselt numbers and friction factors in chilled -water cooling coils. Experimental Heat Transfer, Vol. 7, pp.143–162

13. Wang CC, Lee CJ, Chang CT, Lin SP (1999) Heat transfer andfriction correlation for compact louvered fin-and-tube heatexchangers. International Journal of Heat and Mass Transfer,Vol. 42: 1945–1956

149

D. Taler: Prediction of heat transfer correlations for compact heat exchangers

14. ANSYS (2003) Release 5.7. ANSYS, Inc., Southpointe, 275Technology Drive, Canonsburg, PA, USA

15. Roetzel W, Na Ranong C (2003) On the application of theWilson Plot Technique. Heat and Technology, Vol. 21, No. 2:pp. 125–130

16. Taler D (2004) Experimental determination of heat transferand friction correlations for plate fin-and-tube heat ex-changers, Journal of Enhanced Heat Transfer, Vol. 11, No. 3,pp. 183–204

17. Taler D (2004) Determination of heat transfer correlations forplate-fin-and-tube heat exchangers, Heat and Mass Transfer,Vol. 40, pp. 809–822

18. Dittus FW, Boelter LMK (1930) Heat transfer in automobileradiators of the tubular type, University of California,Berkeley, Publications on Engineering, Vol. 2, No. 13: 443–461, Oct. 17; reprinted in (1985) Int. Comm. Heat MassTransfer, Vol. 12: pp. 3–22

19. Gnielinski V (1975) Neue Gleichungen fur den Warme- undden Stoffubergang in turbulent durchstromten Rohren undKanalen. Forschung im Ingenieurwesen, Bd. 41, Nr. 1: 8–16

20. Marquardt DW (1963) An algorithm for least squares esti-mation of non-linear parameters, SIAM Journal 11: 431–441

21. Seber GAF, Wild CJ (1989) Nonlinear Regression, John WileySons, New York

22. Table Curve (2002) Automated curve fitting software, JandelScientific, San Rafael, CA94901, USA

23. Schmidt Th E (1963) Der Warmeubergang an Rippenrohrenund die Berechnung von Rohrbundel-Warmeaustauschern,Kaltetechnik 15 No. 4, 98; (1963) 15 No. 12, 370

24. FLUENT 6.0. (2003) Fluent Inc., 10 Cavendish Court, Leb-anon, NH03766, USA

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