An Analytical Study on Heat Transfer Performance Of

1451

An analytical study on heat transfer performance ofradiators with non-uniform airflow distributionE Y Ng1,2*, P W Johnson1, and S Watkins1

1Vehicle Aerodynamics Group, School of Aerospace, Mechanical and Manufacturing Engineering, RMIT University,Melbourne, Australia2Product Design, Australian Automotive Air, Australia

The manuscript was received on 19 July 2004 and was accepted after revision for publication on 24 August 2005.

DOI: 10.1243/095440705X35116

Abstract: Heat exchangers used in modern automobiles usually have a highly non-uniformair velocity distribution because of the complexity of the engine compartment and underhoodflow fields; hence ineffective use of the core area has been noted. To adequately predict theheat transfer performance in typical car radiators, a generalized analytical model accountingfor airflow maldistribution was developed using a finite element approach and applyingappropriate heat transfer equations including the e–NTU (effectiveness – number of heattransfer units) method with the Davenport correlation for the air-side heat transfer coefficient.The analytical results were verified against a set of experimental data from nine radiators testedin a wind tunnel and were found to be within +24 and −10 per cent of the experimentalresults. By applying the analytical model, several severe non-uniform velocity distributionswere also studied. It was found that the loss of radiator performance caused by airflowmaldistribution, compared with uniform airflow of the same total flowrate, was relatively minorexcept under extreme circumstances where the non-uniformity factor was larger than 0.5.

The relatively simple set of equations presented in this paper can be used independently inspreadsheets or in conjunction with computational fluid dynamics (CFD) analysis, enabling afull numerical prediction of aerodynamic as well as thermodynamic performance of radiatorsto be conducted prior to a prototype being built.

Keywords: radiators, cross-flow heat exchangers, airflow maldistribution, corrugatedlouvred fin surfaces, non-uniformity, specific dissipation

1 INTRODUCTION AND OBJECTIVES vehicle geometry strongly influences the airflow pathentering the radiator (see reference [4]).

The cooling airflow is seldom uniformly distributed There is only limited research published usingacross an automotive radiator front face, and con- analytical approaches in determining the heat transfersequently the radiator cooling performance may be performance of automotive radiators. Analyses ofimpaired [1–3]. The causes of non-uniformity of the radiator performance have often been based oncooling airflow include objects upstream and down- the inadequate assumption that the airflow has astream of the radiator, particularly the upstream uniform distribution over the radiator face. Anbumper and associated crash structure. Figure 1 important deficiency in that assumption is that theshows a normalized velocity contour measured at the air velocity distribution over the face of a real radiatorradiator front face in a typical passenger car, in in the engine compartment is always highly complexram-air condition (i.e. no fan operation), at a vehicle and non-uniform, and the airflow rate cannot easilydriving speed of 100 km/h. It is evident that the be analytically determined.

Heat exchanger performance can generally bedetermined using either one of the following* Corresponding author: 7 Ellerton Court, Donvale, Victoria, VIC

3111, Australia. email: [email protected] approaches [5]: the log mean temperature difference

D13604 © IMechE 2005 Proc. IMechE Vol. 219 Part D: J. Automobile Engineering

1452 E Y Ng, P W Johnson, and S Watkins

putation procedure) can lead to further understand-ing of heat transfer characteristics of radiators and amore accurate prediction of their effectiveness.

2 BACKGROUND TO THE ANALYSIS

Radiators used for vehicle engine cooling areusually cross-flow type heat exchangers. The airflowis generally induced by the moving vehicle (ram air)and/or the cooling fan(s) (fan air).

Radiators with extended surfaces consisting ofmultilouvred fins are commonly employed inmodern vehicles. The extended surfaces provideFig. 1 Normalized airflow velocity contour at thelarge enhancement of the heat transfer rate, notradiator face at a vehicle speed of 100 km/h.only by providing additional surface area but also(Reprinted from Ng [4])by reducing the thickness of the boundary layer byinducing a series of flat-plate leading edges, interrupt-

(LMTD) method or the effectiveness–NTU (e-NTU) ing the growth of the boundary layer along the finmethod. Selection from these two approaches is often surface [9]. Hosoda et al. [10] compared a plate finbased on what type of problem is to be solved. Use heat exchanger with one with parallel louvres andof the LMTD method is more convenient in solving found that the air-side heat transfer performance ofsizing or design problems (e.g. see reference [6]), the louvred fin structure was 60 per cent higher thanwhereas for the rating problem (performance pre- that of the plain plate fin equivalent.diction) the e-NTU method is more effective (e.g. see The direction of the coolant tubes can be eitherreferences [7] and [8]). horizontal (coolant flowing horizontally) or vertical

The work presented in this paper was aimed to (coolant flowing downward from the top tank tofulfil the following objectives: the bottom tank). According to SAE HS-40 [11], the

horizontal flow type radiator is more commonly used(a) to provide a comprehensive literature reviewin passenger vehicles. Coolant tubes are usually basedrelated to air- and coolant-side heat transferon a flat (non-circular) tube design. The advantagescoefficients applicable to modern automotiveover circular tube designs include a higher heatradiators;transfer area per unit of flow area; the wakes of the(b) to make use of existing heat exchanger theory totubes cause less reduction of heat transfer in down-solve radiator cooling problems where the air-stream regions; the small projected areas of the tubesflow distribution is far from uniform;minimize profile drag; and provision of a higher fin(c) to illustrate a finite element method, where theefficiency.radiator is mathematically treated as a finite

For the purposes of increasing the boiling pointnumber of small elemental radiators, in order toof the coolant and preventing corrosion in the cool-provide a more accurate prediction of the heating systems, coolants are generally a mixture oftransfer capability of the entire radiator withwater, antifreeze (usually ethylene glycol, or EG), andnon-uniform airflow distribution;possibly various corrosion inhibitors. It is noted that(d) to validate this method against a set of experi-the use of glycol mixture generally reduces the heatmental data; andtransfer performance compared with pure water.(e) to further the understanding of the influence

The analytical model was developed in thisof airflow maldistribution on radiator coolingstudy specifically for the most common radiatorperformance.configuration, which consists of the following features:

The method presented can be carried out with-(a) air-cooled radiator,out the need for extensive computing resources(b) cross-flow configuration,and requires only some relatively simple iteration in(c) corrugated louvred fins, anda spreadsheet. Together with computational fluid(d) flat coolant tubes with a horizontal flow structure.dynamics (CFD) simulation predicting flow pheno-

mena through radiators, use of the techniques Nevertheless, the model described here, after minoramendments, can be used to solve the radiatordescribed in this paper (including sets of governing

equations for heat transfer coefficients and a com- performance of other specified configurations.

D13604 © IMechE 2005Proc. IMechE Vol. 219 Part D: J. Automobile Engineering

1453Heat transfer performance of radiators

3 METHODOLOGY OF RADIATOR ANALYSIS (d) coolant flowrate, and(e) basic radiator dimensions (refer to section 3.1).

The primary use for this model was to investigatehow airflow maldistribution affected the radiator 3.1 Definitions of the radiator geometryperformance under known conditions. The outlet

With reference to Fig. 2, the basic radiator dimensionstemperatures and the air temperature distribution atthat were required in this model are listed below:the radiator exit were the aspects to be evaluated.

For this reason, it was deemed to be more appro-Core height (B

H)

priate to utilize the e-NTU method. In this analysis,Core width (B

W)

several assumptions were made, which are stated inCore thickness (B

T)

Appendix 2. This model is based on the followingNumber of rows of tubes in the core depth

conditions being known:dimension (N

r)

Number of coolant tubes in one row (Nct

)(a) ambient temperature,(b) airflow velocity distribution across the radiator, Number of profiles (N

p)

Number of fins per metre (Nf)(c) coolant inlet temperature (top tank temperature),

Fig. 2 Definitions of the radiator dimensions



Louvre pitch (Lp

) air-side convection, fouling on the air side, wall con-Louvre length (L

l) duction, fouling on the coolant side, and coolant-side

Fin thickness (Ft) convection. In mathematical form

Fin height (Fh

)Fin pitch (F

p) 1

UA=

1

(g0hA)a

+Rf,a+Dx

(kA)w+Rf,c+

1

(g0hA)c

(10)Fin end radius (R

f)

Angle of fin (af)

It is noted that without the use of extended surfacesCoolant tube length (Yl)

(i.e. prime surface) in the coolant tubes in typicalCoolant tube cross-section length (Ycl

)radiators, the value of g

0,cbecomes unity.Coolant tube cross-section width (Y

cw)

Coolant tube thickness (Yt) For simplicity, the fouling resistances on both

Coolant tube pitch (Yp

) sides were assumed to be small and not consideredCoolant tube end radius (R

t) in this analysis. Also, coolant tubes are often made

of aluminium or copper, which have large thermal3.2 Calculation of relevant heat transfer areas conductivity, and the wall thickness is small. There-

fore, the thermal resistance of the wall conductionBased on the preceding definitions, derivations ofterm (Dx/kA)

win comparison to the other two termsvarious surface areas, which are relevant to heat

was treated as negligible. It is noted that this walltransfer, are given below:term does not include fin conduction, which is

Fin length, Fl treated separately in the next section.

With these stated restrictions and combiningFl=pRf+

Fh−2Rfcos af

(1) the nomenclature used in this study, equation (10)becomes

Radiator core frontal area, Afr,r

Afr,r=BHBW (2) 1

UAfr,r=

1

g0,ahaAa+

1

hcAc(11)

Coolant tube frontal area, Afr,t

Afr,t=YcwYlNct (3) Under typical operating conditions, the air-sideresistance is dominant (i.e. the controlling resistance).Fin frontal heat transfer area, A

fr,f Davenport [9] showed that at higher water flowrates,Afr,f=FtFlNf(per metre)YlNp (4) the coolant-side thermal resistance contributed only

Fin heat transfer area, Af

5 per cent of the total resistance.Equation (11) is the equation that governs the heatAf=2BTFlNf(per metre)YlNp (5)

transfer performance of a cross-flow type radiator.Total heat transfer area on the air side, A

a Detailed derivations of the air-side and coolant-sideheat transfer coefficients (i.e. h

aand h

c) and theirAa=Af+2NctYlNr [(Ycl−2Rt)+(2pRt)] (6)

governing equations, which involve an extensiveTotal heat transfer area on the coolant side, A

c review from the literature, are presented in section 6.Ac= [2p(Rt−Yt)+2(Ycl−2Rt)]YlNctNr (7)

Total air pass area, Ap,a 4.2 Air-side fin efficiency and total surface

Ap,a=Afr,r−Afr,f−Afr,t (8) efficiency

Total coolant pass area, Ap,c

Fins attached to the coolant tubes are primarily usedfor increasing the surface area and consequentlyAp,c= [p(Rt−Yt)2+(Ycw−2Yt)(Ycl−2Rt)]NctNr increasing the heat transfer rate. As temperature

(9) gradients along the fins extending into the air createconduction resistance, the temperature efficiency ofthe surface is reduced. Hence, the air-side heat con-

4 HEAT TRANSFER ANALYSIS ductance term (hA)a

must be multiplied by a factorto account for the temperature gradient in the fin.

4.1 Thermal circuit and overall heat transferThe fin efficiency (g

f) is defined as the ratio of the

coefficientactual heat transfer rate through the fin base (Q

f)

divided by the maximum theoretical heat transferThe overall heat transfer resistance for a radiatorcan be considered to derive from the following terms: rate through the fin base (Q

f,max), corresponding to



the entire fin surface being at the base temperature radiator on air and coolant sides are typically inthe range of a 20–35 °C rise and a 4–10 °C droprespectively. Hencegf=

QfQf,max

(12)

Cmin=Ca and Cmax=CcFor a straight fin of uniform cross-section, the finefficiency can be expressed as For a typical automotive radiator it is assumed

that the approaching air stream is directed into agf=

tanh(mL)

mL(13) large number of separate passages with no cross

mixing when the air is travelling through the radiator.where L=effective fin length and m=fin efficiency The same assumption is commonly applied toparameter. For the fins extending from wall to wall coolant flow. Hence, radiators are often considered

as a type of cross-flow arrangement heat exchangerL=

Fh2

with both fluids unmixed [13, 14]. The appropriatee-NTU relationship for this type can be found in

For thin sheet fins (see reference [5]) graphical form in reference [5], and mathematicallyin reference [15].

m=S2hakFt

5 FINITE ELEMENT APPROACHRadiators are typically of a finned-tube con-struction. Hence, heat transfer takes place in both the

To account for the effects of airflow maldistribution,fins and the unfinned (primary) surface at the samea finite element approach was used, which dividestime. For this reason, the total surface efficiency (g

0)

the entire radiator into a number of independentis introduced to account for the weighted meansmall radiators (cells). The e-NTU method was appliedefficiency of the composite structure consisting ofto each cell, with the outlet coolant temperaturethe fins and the base structure. The total surfaceof the upstream cell being the inlet temperature ofefficiency can be calculated fromthe downstream cell, in order to determine its heatdissipation rate (see Fig. 3). As a coolant horizontal-g

0=1−

AfAa

(1−gf) (14)flow radiator was considered, the coolant flow intoeach cell was equal to the total coolant flow dividedThis equation is based on the assumption that theby the number of rows. The heat dissipation ofair-side heat transfer coefficient is unchanged bythe entire radiator was equal to the sum of the heataddition of the fins.dissipation of all cells.

This approach may be similar to techniques4.3 The e-NTU methodused in some commercial software such as KULI,

The radiator effectiveness can be expressed as a FLOWMASTER and GT-COOL, and also techniquesfunction of two dimensionless groups, NTU and C

r, developed in the European project V-THERM, except

for a given flow arrangement (such as counter-flow that this procedure can be carried out where noor cross-flow) [5] extensive computing resources or software packages

are available. This enables engineers to performe= f (NTU, Cr , flow arrangement) (15)analysis or performance predictions in house by use

whereof simple manipulations.

The number of cells, including the number of rowsNTU=

UAfr,rCmin

=1

Cmin PA U dAfr,r (m) and the number of columns (n), is given by theuser. It is assumed that the coolant flow is uniformly

C=heat capacity rate=m×cp distributed through the radiator and equally divided

between the different rows. Given the top tankCr=

CminCmax

temperature (Tci

) and the coolant mass flow, andassuming the tanks on both sides of the radiator are

Ca=ma×cp,a=AaraVa×c

p,a well insulated, it can be demonstrated that the inletcoolant temperature of each cell in the first columnCc=mc×c

p,c=AcrcVc×cp,c is equal to the top tank temperature; i.e.

Under normal driving conditions, SAE J1393 [12]states that the temperature changes across the Tci=Tci(i,1) (16)



Fig. 3 Finite element approach

where Tci(i,1)

refers to the inlet temperature of the cell Because of the complexity of airflow over louvredfins, it is difficult to determine the heat transfer(i, 1), with i=1, … , m.

Except for the cells in the first column, the coolant coefficient from conventional convection heat trans-fer equations, as the coefficient is a function of bothinlet temperature of any cell is equal to the tem-

perature at the exit of the upstream cell in the same fin geometry and flow conditions. Because the heattransfer characteristics of the louvres are closelyrowrelated to the flow structure around them, the follow-

Tco(i,j)=Tci(i,j+1) (17)ing subsection provides a brief discussion of the flowphenomena occurring in the louvre array.with i=1, … , m and j=1, … , (n−1). The value of

the bottom tank temperature (Tco

) is taken as the6.1.1 Flow structure in louvre finsaverage value of the summation of the last cells for

all rowsFollowing Beauvais [17] and Wong and Smith [18],who discovered that louvres act to realign the air-flow in a direction parallel to their own planes,

Tco=∑m

i=1Tco(i,n)

number of rows(18) Davenport [9] performed a detailed investigation on

corrugated louvred fin heat exchangers, demon-Similarly, the coolant mass flowrate of each cell strating that the flow structure within the louvredbecomes array was a function of Reynolds number.

Figure 4 illustrates a section through a louvre arraymc(i,j)=

mcnumber of rows

(19) in which two possible extreme flow directions areindicated. Davenport found that at low Reynolds

for all (i, j). numbers the flow did not pass through the louvresbut travelled axially through the fins and behaved likeduct flow (duct-directed flow). He explained that the

6 EQUATIONS FOR HEAT TRANSFER developing boundary layers on the louvres becameCOEFFICIENTS

6.1 Air side

Airflow over a louvred fin array is complex. Ratherthan acting as surface roughness or turbulentgenerators, the louvres are used to deflect airflowfrom their incident direction and consequently theflow becomes aligned with the planes of the louvres.They enhance the heat transfer by providing multiple Fig. 4 Section through louvred fin indicating possibleleading edges, associated with high heat transfer flow directions. (After Achaichia and Cowell

[19])coefficients [16].



sufficiently thick to block off the gaps between louvres. corrugated louvred-fin surfaces found in the publiclyavailable literature. The correlations are empiricalThis gradually changed to an almost complete align-

ment with the louvres as the Reynolds number was using a multiple regression technique. The corre-lation for heat transfer for corrugated louvred finincreased. At high Reynolds numbers, the flow was

directed by the louvres flowing nearly parallel to surfaces is valid when 100<Re<4000 and therecommended Colburn modulus j factor correlationthem, behaving like flat-plate flow (louvre-directed

flow). Additional studies conducted by Achaichia and was given asCowell [19, 20] and Kajino and Hiramatsu [16] agreedwith Davenport’s investigation. j=0.249Re−0.42Lp L0.33h AL lFhB1.1F0.26h (20)

6.1.2 Air-side Reynolds numbersThis correlation is valid for the ratio of the louvre

The flow path over louvres is dependent on Reynolds length to fin height ranging between 0.62 and 0.93,numbers for a given louvred fin array. However, it which is applicable to modern automotive radiators.seems that the characteristic length is rather arbitrary The Davenport correlation was reasonably accuratefor louvred fin surfaces. Davenport [21], after testing and easy to apply, and approximately 95 per cent of32 samples of multilouvred fin surfaces, suggested the data were correlated within ±6 per cent. The jthat the fin pitch and hydraulic diameter made curve (i.e. plot of the j factors versus Reynoldsno contribution to the correlation of Colburn’s numbers) had a mean gradient of −0.42 from themodulus j factor. He concluded that, although the regression analysis compared with−0.5 for the classichydraulic diameter is relevant to heat transfer in theoretical Pohlhausen equation. Also, the valuesplain fins, using the louvre-pitch-based Reynolds of the j curve were about 35 per cent below thenumber (Re

Lp) is more appropriate to describe the Pohlhausen line. Davenport explained that this was

heat transfer on louvred fin surfaces. Most of the later due to the entire heat transfer surface, not all ofresearch has been consistent with this finding and which was louvred. This study revealed that the heathas used Re

Lpas a basis – which is the approach transfer behaviour over louvred fins has a general

used here. similarity with Pohlhansen solutions for flow over aflat plate, suggesting the existence of laminar bound-

6.1.3 Air-side heat transfer coefficient ary layers on the louvres, which is consistent withthe discussions in the previous sections.Publication of the heat transfer performance of the

Since the Davenport study, questions have arisencorrugated louvred fin geometry, which is the basicas to whether his correlation is still valid, as thestructure of modern radiators, has been very limited,core structures he tested were noticeably differentperhaps for commercial reasons. Apart from thefrom the ones used nowadays in terms of materialcorrugated geometry, there are some other typesand fin geometry. A recent study conducted by Webbof heat exchanger geometry available including: aet al. [24] revealed that the correlations are still appli-flat tube and louvred plate fin [19], a corrugatedcable to current automotive radiator cores. Further-louvred fin with a rectangular channel [22], amore, a large number of recent researchers have stillcorrugated louvred fin with a splitter plate andemployed Davenport’s experimental data as a baserectangular channel [23], and a corrugated louvredsource to validate their findings, including referencesfin with a splitter plate and triangular channel [22].[24] to [27].However, these geometries are not normally used in

Another study conducted by Aoki et al. [28],automotive applications; thus discussion on them iswho measured the local heat transfer for individualoutside the scope of this paper.louvres, also suggested laminar heat transfer beingOne of the important studies on extendedpresent on the louvres. A correlation between theheat transfer surface was conducted by Kays andNussult number and the Reynolds number based onLondon [5]. However, those louvre designs are verythe louvre pitch was presented in one equation fordifferent from the multilouvred surfaces, which aredifferent louvre pitcheswidely used nowadays for automotive radiator cores,

and the data are of little relevance when applied NuLp=0.87Re1/2Lp Pr1/3 (21)to modern heat exchangers. Davenport [21], aftertesting with louvred fin cores with systematically Based on the work of Aoki et al., Webb [29]

recommended that the theoretical Pohlhausenvaried louvre geometry, presented correlations ofheat transfer and flow friction characteristics. This solution for laminar flow over a flat plate with con-

stant heat flux (see reference [30]) can be used tois the only set of experimental data published for



predict the heat transfer coefficient on louvres, i.e. is relatively insensitive to water-side heat transfer.The coolant-side heat transfer coefficient can beNuav=0.906Re1/2Lp Pr1/3 (22)evaluated by applying appropriate well-established

Based on the above findings, Sahnoun and heat transfer equations for flow inside tubes. TheWebb [25] and Dillen and Webb [26] developed coolant tubes in typical radiators are a flat-ovalan analytical model and a semi-analytical model shape. Therefore, the hydraulic diameter (D

h) is used

respectively, to predict the heat transfer coefficient to substitute for the characteristic physical diameter.of the corrugated louvred fin geometry. Both of Invalid use of the hydraulic diameter has been foundthe models were based on dividing the louvred only when calculating tubes with sections havingfin surface into four regions: the louvred areas, the very sharp corners (e.g. triangular), which givesplain leading and trailing areas, the plain middle unacceptably large errors.area, and the end region areas. In their models, the There has been a large amount of fundamentalheat transfer coefficient in the unlouvred regions work in understanding flow characteristics and heatis predicted using a fully developed laminar flow transfer in tubes; therefore only several represent-solution, while using the Pohlhansen solution for ative equations are considered here. Starting withlaminar flow over a flat plate [equation (22)] to pre- the Reynolds experiments in 1883, it has beendict louvred areas. A summation of the heat transfer demonstrated that laminar flow becomes unstable ascoefficients calculated from different regions was the velocity of flow increases in a given tube. Theused to determine the heat transfer coefficient of the transition from laminar to turbulent flow occurs at aentire louvred fin. value of Reynolds number near 2300. The transition

Instead of dividing the louvred fin surface into to turbulent flow generally takes place in the rangefour regions, an earlier work of Beard and Smith [6] of Reynolds numbers from about 2300 to 10 000, anddeveloped a simple method that approximated the a fully turbulent flow mostly occurs at a Reynoldseffects of louvres by calculating the heat transfer number above 10 000. It has also been found that thecoefficient from a mean of two coefficients obtained transition of the flow is greatly affected by the tubefrom louvred fin regions and unlouvred fin regions. inlet configuration and surface roughness.Since the proposed benefit of using louvred fins wasto give a series of leading edges to the airflow creating 6.2.1 Heat transfer in laminar flowlaminar boundaries, the heat transfer characteristics

In the laminar flow regime, the heat transfer flux isof each louvre could be similar to flow over a flat

strongly dependent on the thermal boundary con-plate. For that reason, their model was developed

dition along the whole length of the tube, while lessusing the theoretical Pohlhausen equation for flow

dependent in the turbulent flow regime for fluidsover a flat plate to predict the heat transfer coefficient

with Pr�1. The thermal boundary condition refers toon louvres.

the set of specifications describing temperature con-This simple method was shown to be valid as the

ditions and/or the heat transfer rate at the inside wallcalculated results were found to give satisfactorily

of the tube. According to Shah and London [31], theclose agreement with a series of wind-tunnel test

thermal boundary condition of automotive radiatorsresults with a maximum error of approximately 10

can be classified as the constant wall temperatureper cent.

peripherally as well as axially, since one fluid has avery much higher capacity rate than the other.6.1.4 Summary of preferred air-side heat transfer

Knudsen and Katz [32] reported an equationequationsproposed by Hansen in 1943 as representing the

Essentially, there have been only four models avail- Graetz solution for constant surface temperature,able in the existing literature to predict the heat fully developed laminar flow, and parabolic velocitytransfer coefficient for radiators (corrugated louvred distribution. The well-known Hansen equation, whichfin surfaces). Among them, it appears that the has been widely used for the mean Nusselt numberDavenport correlation [equation (20)] is relatively easy over the entire length of the tube, was used in thisto use without compromising accuracy. Therefore, this study to predict the coolant-side heat transfercorrelation was chosen in this study to determine the coefficient in the laminar flow regimeair-side heat transfer coefficient (h

a).

6.2 Coolant sideNuc=3.66+

0.0668ADh,cYl BReDh,c

Prc

1+0.04CADh,cYl BReDh,c

PrcD2/3 (23)It is emphasized that the dominant thermal resist-ance is air-side convection and that the modelling



All properties appearing in the equation are required 104<ReD

h,c

<5×106 and 0.5<Pr<2000. Thus thisequation was chosen for this study in the case ofto be evaluated at the average value of the mean

temperature. The Hansen equation is valid for the turbulent flow.Gnielinski [37] modified the Petukhov equation inconstant surface temperature condition over the

entire tube length. Considering a sufficiently long order to derive a new equation that is applicable inthe transition flow region (2300<Re<104). It wastube, it is noted that the Nusselt number approaches

a value of 3.66, which is the analytical solution for noted that in the transition region the Gnielinskiequation satisfactorily reproduced the decrease in thelaminar, fully developed conditions with a constant

surface temperature [30], i.e. Nuc=3.66. heat transfer coefficient with decreasing Reynolds

number. Thus, the following equation was employed6.2.2 Heat transfer in turbulent and transition flow in this study

Due to the complicated nature of turbulent flow,Nuc=

( f /8)(ReDh,c

−1000)Pr

1+12.7√f /8(Pr2/3−1)(27)which is transient and possesses highly irregular

fluctuations, and the fact that heat is generallytransferred by convection as well as conduction, This equation is valid for 0.5<Pr<2000 andempirical correlations of turbulent heat transfer data 2300<Re

Dh,c

<104, and was compared with approxi-for flow in tubes are often preferred for simplicity. mately 800 experimental data points, with nearlyThe correlations have been obtained based on 90 per cent of the data falling within ±20 per cent.several dimensionless groups, including the Reynoldsnumber, Nusselt number, Prandtl number, and

6.2.3 Summary of preferred coolant-side heatStanton number.transfer equationsThere were three famous correlations proposed

in the 1930s, which were the Dittus and Boelter Flow regime Equation usedequation [33], the Colburn equation [34], and the Laminar (2300>Re

Dh

) Hansen equationSieder and Tate equation [35]. Incropera and DeWitt Transition Gnielinski equation[36] reported that the use of the above equations, (10 000�Re

Dh

�2300)although they may be easily applied, may lead to Turbulent (Re

Dh

>10 000) Petukhov equationerrors as large as 25 per cent. For this reason, thesecorrelations were not chosen for this study. To examine the consistency of these three heat

Prandtl in 1944 was the first to present an equation transfer equations selected to be used in the analyticalfor heat transfer in tubes that was related to the model, the heat transfer coefficient was calculated inpressure drop. The equation was in the form of different flow regimes ranging from Reynolds number

of 400 to 13 500, as shown in Fig. 5. Discontinuitieswere found at the transitions between the regimes

Nu

Re Pr=

f /8

1+8.7√f /8(Pr−1)(24)

(i.e. at Reynolds numbers of 2300 and 10 000); never-theless, these discontinuities of the coolant-sidewhere f is the friction factor in the tube. Knowledgeheat transfer coefficient are relatively unimportant,of the friction factor in the tube is required beforesince the dominant thermal resistance is air-sideapplying this equation, and value of the frictionconvection.factor can be directly obtained from the Moody

diagram. Alternatively, the friction factor can becalculated from the following equation for isothermalflows in smooth tubes [37] 7 EXPERIMENTAL VALIDATION OF THE

ANALYTICAL MODELf= [0.79 ln(ReDh,c

)−1.64]−2 (25)

Since then, the Prandtl equation has been further A set of experimental data obtained using the RMITimproved. From the basic form, a correlation recom- University cooling test facility was used to validatemended by Petukhov [38] offered better accuracy for the analytical model.turbulent tube flow. The equation is expressed as

7.1 Test procedureNuc=( f /8)Re

Dh,c

Pr

1.07+12.7√f /8(Pr2/3−1)(26)

The detailed test procedure and equipment can befound in reference [39], and only a brief descriptionThis equation predicted experimental data with

an accuracy of 5–6 per cent over the range of is provided here. Nine sections of corrugated louvred



Fig. 5 Coolant heat transfer coefficients in different flow regimes

fin radiator cores with various fin pitch, tube spacing, To measure the heat dissipation rate of eachradiator, hot water was supplied by an externaland fin width (listed in Table 1) were tested indi-

vidually in a small open-circuit, closed-test-section heat bench and thermocouples were set at variouslocations, including the radiator inlet and outlet,wind tunnel with a cross-section of 0.3 m (width)×

0.35 m (high). Air flowed perpendicularly to, and upstream and downstream of the radiator core. Inaddition, the water flowrate was monitored anduniformly through, a section of radiator that was

located in the middle of the 0.8 m long test section. recorded via a magnetic flowmeter. The suppliedwater was set at 1 l/s and approximately 75 °C. DataAirflow velocity of 9.35 m/s was set for each of the

radiators. were taken at the equilibrium state.

Table 1 Radiator dimensions, experimental data, and analytical predictions

Radiator sample numbers

1 2 3 4 5 6 7 8 9 Unit

BH

352 352 352 352 352 352 352 352 352 mmB

W302 302 302 302 302 302 302 302 302 mm

BT

38 32 32 16 40 66 38 38 32 mmN

r2 2 2 1 3 4 2 2 2

Nct

41 35 29 35 33 29 41 41 34N

p42 36 30 36 34 30 42 42 35

Nf

16 16 16 16 16 14 14 18 21 inL

p1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 mm

Ll

5.1 6.7 8.7 6.7 7.2 8.7 5.1 5.1 6.7 mmF

t0.058 0.058 0.058 0.058 0.058 0.058 0.058 0.058 0.058 mm

Fh

6.146 7.746 9.746 7.746 8.246 9.746 6.146 6.146 7.746 mmF

p1.588 1.588 1.588 1.588 1.588 1.814 1.814 1.411 1.210 mm

Rf

0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 mma

f25 25 25 25 25 25 25 25 25 degree

Yl

302 302 302 302 302 302 302 302 302 mmY

cl12.2 12.2 12.2 12.2 12.2 12.2 12.2 12.2 12.2 mm

Ycw

2 2 2 2 2 2 2 2 2 mmY

t0.127 0.127 0.127 0.127 0.127 0.127 0.127 0.127 0.127 mm

Yp

8.4 10 12 10 10.5 12 8.4 8.4 10 mmR

t1 1 1 1 1 1 1 1 1 mm

Experimental 556 434 408 282 506 555 510 579 493 W/KPrediction 509 502 506 347 555 625 489 526 541 W/KError −8.6 15.7 24.0 23.0 9.6 12.7 −4.1 −9.1 9.7 %



7.2 Test parameter transfer process in radiators is dominant and theresultant error on the coolant side becomes minor.

The specific dissipation (SD) technique was adoptedIn another study of radiator airflow maldistribution

in the measurements. This technique is commonlyin a real vehicle, this analytical model can predict

used within the automotive industry to evaluate thethe same trend as the experimental measurements

cooling performance of a vehicle in non-climaticwithin 10 per cent for various configurations (see

wind tunnels. The major advantage of using thisreference [4]). As this model is developed for use

technique is that the SD parameter, which relates toin predicting radiator performance trends before a

a radiator’s effectiveness, normalized heat transferprototype becomes available rather than predicting

with respect to the driving temperature difference,absolute values, this achieved accuracy is deemed

was proved to be insensitive to variations of air andto be acceptable. This model also accounts for the

coolant temperatures [14]. SD is defined as the heatcorrect cooling airflow distribution in a vehicle,

transfer rate of the radiator (Q) divided by the temper-giving more accurate predictions. It is emphasized

ature difference between the water and air enteringthat analytical studies are always a supplement to,

the radiator (Tci−T

ai) (note that this is the maximum

not a substitute for, performance testing in producttemperature difference available to the radiator)

development.

SD=e(macp,c)=mccp,c(Tci−Tco)

Tci−Tai(28)

8 APPLICATIONSwhere e is the radiator effectiveness

The model can be used as a general tool (with somemodifications to suit other configurations of heat7.3 Experimental versus analytical resultsexchanger applications) in early radiator design,

Table 1 shows the experimental data along with those before a prototype is built. This can allow engineersresults calculated under the same test conditions to study the effect of parameter changes, such as finby the analytical model described previously. The pitch or tube sizes. In the early design and develop-results indicate that the analytical model provides ment phase, an absolute value is not normallypredictions of the heat transfer performance of a required and the accuracy of this analytical model isrange of geometrically different radiators within+24 sufficient for the purpose of the initial conceptualand −10 per cent. It is also noted that the model study. Furthermore, this modelling can be done with-predicts the coolant temperature at the radiator out- out the need of extensive computing resources andlet well (within 1.2 °C), compared to the measured skills. In this section, two examples are presented tovalues. illustrate the use of the model.

The predicted values are generally correlated wellwith the measured values with an average deviation

8.1 Effect of airflow maldistribution on heatof 8.1 per cent and a mean deviation of 12.9 per cent,transfer performancebut in two cases errors are overpredicted by more than

23 per cent compared to the test data. Since this To understand the effect of airflow maldistribution onmodelling is developed based primarily on analytical the heat transfer performance of a radiator, severalderivation, except for the heat transfer coefficients severe non-uniform velocity distributions were con-for air and coolant flow, it is thought that such error sidered (see Figs 6 and 7). Four airflow profiles werewould arise from either or both empirically based artificially simulated, but in each case the averagecorrelations. Davenport’s air-side correlation should flow velocity through the radiator was unchangedhave given fairly accurate predictions according to and equal to 5 m/s. The coolant inlet temperature,previous publications (95 per cent of experimental ambient temperature, and coolant flowrate were alldata correlated within 6 per cent). When Reynolds fixed at 60 °C, 25 °C and 1 l/s respectively.numbers range from 2300 to 10 000, the coolant To quantify the non-uniformity of airflow acrossflow condition in the tube is transitional, which is the radiator face, a parameter is needed. Given thatthe case here. Applying the Gnielinski equation in the radiator is segmented into a finite number oftransition flow can lead to a rather large error in equal-area cells, a non-uniformly factor was definedpredicting the coolant-side heat transfer coefficient as(see section 6.2.2). On the other hand, good pre-dictions of coolant outlet temperatures can be i=

1

n∑n

k=1

|Vlocal,k−Vaverage |Vaverage

(29)explained because the air-side resistance in the heat



Fig. 6 Reduction in SD due to airflow maldistribution

Fig. 7 Increase in coolant outlet temperature due to airflow maldistribution

where flow being the same. In a case of a non-uniformityparameter of 0.4 (the typical value for passenger

n=number of cellsvehicles is generally less than 0.5), the SD value,

Vlocal= local velocity

and hence the radiator cooling performance, wouldV

average=average velocity through the entire

reduce by 8 per cent. An increase of about half aradiator

degree Celsius at the bottom tank would result as aconsequence. Except under extreme circumstances,The finite element method was utilized for a number

of non-uniformity factors and distributions, as shown it appears that the effect of airflow maldistribution onSD is relatively minor (less than 10 per cent penaltyin Figs 6 and 7. The figures clearly indicate that the

cooling capacity of a radiator is influenced by non- in SD for i=0.5). Nevertheless, it is suggested thatthe factor of non-uniformity should be realistic inuniformity of airflow, despite the total amount of air-



any radiator performance analysis, in order to pro- has been used to evaluate a radiator’s effectivenessin non-climatic type wind tunnels [14], and reflectvide accurate calculations of SD as well as radiator

performance. changes in cooling airflow [4].This analytical study reveals that:

8.2 Study of the SD parameter1. Coolant flowrates (Fig. 8). When testing at low

coolant flowrates, SD is fairly sensitive to changesThis analytical model has been also used in studyingthe effect of sensitivity of coolant flowrates, ambient in coolant flow and small fluctuations in coolant

flowrate may cause a considerable variation in SD.temperature drift, and coolant inlet temperature drifton specific dissipation (SD) [see equation (28)]. This 2. Ambient temperature drift (Fig. 9). A drift of about

4 per cent in SD would occur when the ambientparameter has been proven to be insensitive tochanges in ambient and coolant temperatures, and temperature changed from 10 to 40 °C.

Fig. 8 Effect of coolant flowrates on SD. Analytical predictions versus wind-tunnel test data.(Reprinted from Ng [4])

Fig. 9 Calculated heat rejection rates and SD values at ambient temperatures between 10 and45 °C in three uniform flow cases. (Reprinted from Ng [4])



Fig. 10 Calculated heat rejection rates and SD values at coolant inlet temperatures between 30and 120 °C in three uniform flow cases. (Reprinted from Ng [4])

3. Coolant inlet temperature drift (Fig. 10). A 30 °C ACKNOWLEDGEMENTSchange in coolant temperature (from 50 to 80 °C)results only in a small variation in SD and the The authors would like to acknowledge the Depart-

ment of Mechanical and Manufacturing Engineering,consequent error is less than 1 per cent.RMIT University, for financial and technical support.

9 CONCLUSIONSREFERENCES

A generalized analytical model was developed basedon applying relevant heat exchanger theory to pre- 1 Olson, M. E. Aerodynamic effects of front end

design on automobile engine cooling systems. SAEdict the heat dissipation rates and to study the effectstechnical paper 760188, 1976.of airflow maldistribution on the performance of a

2 Chiou, J. P. The effect of the flow nonuniformity onradiator. The model employed the e-NTU method inthe sizing of the engine radiator, SAE technical paper

combination with the Davenport correlation for the 800035, 1980.prediction of the air-side heat transfer coefficient. 3 Williams, J. An automotive front-end designAccording to the flow regime, the coolant-side approach for improved aerodynamics and cooling.heat transfer coefficient was calculated from the SAE technical paper 850281, 1985.

4 Ng, E. Y. Vehicle engine cooling systems: assess-Hansen equation, Gnielinski equation, or Petukhovment and improvement of wind-tunnel basedequation. The model was validated against a set ofevaluation methods. PhD Thesis, School of Aero-experimental data.space, Mechanical and Manufacturing Engineering,The loss of engine cooling performance causedRMIT University, Melbourne, 2002.

by airflow maldistribution, compared with uniform 5 Kays, W. M. and London, A. L. Compact heatairflow of the same total flowrate, was found to be exchangers, 3rd edition (Reprint edition withrelatively minor – in a typical automobile radiator corrections), 1998 (Krieger Publishing Company).

6 Beard, R. A. and Smith, G. J. A method of calculatingairflow cases are less than 10 per cent, or typicallythe heat dissipation from radiators to cool vehiclehalf a degree increase in coolant temperature at theengines. SAE technical paper 710208, 1971.radiator outlet. Nevertheless, improvement in flow

7 Emmenthal, K. D. and Hucho, W.-H. A rationaluniformity, in particular in areas influenced by theapproach to automotive radiator systems design.

bumper wake, is expected to increase the overall SAE technical paper 740088, 1974.vehicle energy efficiency since the heat transfer area 8 Eichlseder, W. and Raab, G. Calculation and designcan be used more effectively, which can lead to a of cooling systems. SAE technical paper 931088,

1993.reduction in aerodynamic drag.



9 Davenport, C. J. Heat transfer and fluid flow in 27 Chang, Y.-J. and Wang, C.-C. A generalized heattransfer correlation for louvre fin geometry. Int.louvred triangular ducts. PhD Thesis, Department

of Mechanical Engineering, Coventry (Lanchester) J. Heat and Mass Transfer, 1997, 40(3), 533–544.28 Aoki, H., Shinagawa, T., and Suga, K. An experi-Polytechnic, 1980.

10 Hosoda, T., Uzuhashi, H., and Kobayashi, N. Louvre mental study of the local heat tranfer characteristicsin automotive louvred fins. Expl Thermal and Fluidfin type heat exchangers. Heat Transfer – Jap. Res.,

1977, 6(2), 69–77. Sci., 1989, 2, 293–300.29 Webb, R. L. Principles of enhanced heat transfer,11 SAE HS-40 Principles of engine cooling systems,

components and maintenance, 1991 (Society of 1994 (Wiley Interscience, New York).30 Kays, W. M. and Crawford, M. E. Convective heatAutomotive Engineers, Warrendale, Pennsylvania).

12 SAE J1393 On-Highway Truck Cooling Test Code, June and mass transfer, 2nd edition, 1980 (McGraw-Hill,New York).1984 (Society of Automotive Engineers, Warrendale,

Pennsylvania). 31 Shah, R. K. and London, A. L. Laminar flow forcedconvection in ducts. Advances in Heat Transfer13 Yan, W.-M. and Sheen, P.-J. Heat transfer and friction

characteristics of fin-and-tube heat exchangers. Int. (Suppl. 1), 1978.32 Knudsen, J. G. and Katz, D. L. Fluid Dynamics andJ. Heat and Mass Transfer, 2000, 43, 1651–1659.

14 Lin, C. Specific dissipation as a technique for Heat Transfer, McGraw-Hill Series in ChemicalEngineering (Ed. S. D. Kirkpatrick), 1958 (McGraw-evaluating motor car radiator cooling performance.

PhD Thesis, Department of Mechanical and Manu- Hill, New York).33 Dittus, F. W. and Boelter, L. M. K. Heat transfer infacturing Engineering, RMIT University, Melbourne,

1999. automobile radiators of the tabular type. Universityof California, Berkeley Publications on Engineering,15 Holman, J. P. Heat transfer, 7th edition, 1992

(McGraw-Hill, New York). 1930, 2(13), 443–461.34 Colburn, A. P. A method of correlating forced con-16 Kajino, M. and Hiramatsu, M. Research and

development of automotive heat exchangers, in vection heat transfer data and a comparison withfluid friction. Trans. Am. Inst. Chem. Engrs, 1955,Heat transfer in high technology and power engineer-

ing (Eds W. J. Yang and Y. Mori), 1987, p. 420–432 29, 174.35 Sieder, E. N. and Tate, C. E. Heat transfer and(Hemisphere, Washington, DC).

17 Beauvais, F. N. An aerodynamic look at automotive pressure drop of liquids in tubes. Ind. Engng Chem.,1936, 28, 1429.radiators. SAE technical paper 650470, 1965.

18 Wong, L. T. and Smith, M. C. Airflow phenomena 36 Incropera, F. P. and DeWitt, D. P. Fundamentals ofheat and mass transfer, 4th edition, 1996 (Johnin the louvred-fin heat exchanger. SAE technical

paper 730237, 1973. Wiley, New York).37 Gnielinski, V. New equations for heat and mass19 Achaichia, A. and Cowell, T. A. Heat transfer

and pressure drop characteristics of flat tube and transfer in turbulent pipe and channel flow. Int.Chem. Engng, 1976, 16(2), 359–367.louvred plate fin surfaces. Expl Thermal and Fluid

Sci., 1988, 1, 147–157. 38 Petukhov, B. S. Heat transfer and friction in20 Achaichia, A. and Cowell, T. A. A finite difference turbulent pipe flow with variable physical proper-

analysis of fully developed periodic laminar flow in ties. In Advances in heat transfer (Eds J. P. Hartnettinclined louvre arrays. In Proceedings of the 2nd UK and T. F. J. Irvine), 1970, 503–564 (Academic Press,National Heat Transfer Conference, Glasgow, 1988. New York).

21 Davenport, C. J. Correlations for heat transfer and 39 Blatti, A. Drag and heat dissipation from passengerflow friction characteristic of louvred fin. AIChE vehicle radiators. BEng Thesis, Department ofSymp. Ser., 1983, 79(225), 19–27. Mechanical and Manufacturing Engineering, RMIT

22 Webb, R. L. and Jung, S.-H. Air-side performance University, Melbourne, 2002.of enhanced brazed aluminium heat exchangers.ASHRAE Trans., 1992, 98(2), 391–401.

23 Rugh, J. P., Pearson, J. T., and Ramadhyani, S. Astudy of a very compact heat exchanger used forpassenger compartment heating in automobiles. APPENDIX 1ASME Symp. Ser. HTD, 1992, 201, 15–24.

24 Webb, R. L., Chang, Y.-J., and Wang, C.-C. Heat Notationtransfer and friction correlations for the louvre fingeometry. Proc. Veh. Thermal Managmt System, A surface area for heat transfer1995, 2, 533–541. A

atotal heat transfer area on the air side

25 Sahnoun, A. and Webb, R. L. Prediction of heat Ac

total heat transfer area on the coolanttransfer and friction for the louvre fin geometry.

sideJ. Heat Transfer, 1992, 114, 893–900.A

ffin heat transfer area26 Dillen, E. R. and Webb, R. L. Rationally based heat

Afr,f

fin frontal heat transfer areatransfer and friction correlations for the louvre fingeometry, SAE technical paper 940504, 1994. A

fr,rradiator core frontal area



Ycw

coolant tube cross-section widthAfr,t

coolant tube frontal areaA

p,atotal air pass area Y

lcoolant tube length

Yp

coolant tube pitchAp,c

total coolant pass areaB

Hcore height Y

tcoolant tube thickness

BT

core thicknessa

fangle of finB

Wcore width

e effectivenesscp

specific heat capacityg

ffin efficiencyC heat capacity rate

g0

total surface efficiency of an extended finDh

hydraulic diametersurfacef core friction factor

r densityFh

fin heightF

lfin length

SubscriptsFp

fin pitchF

tfin thickness

a airh heat transfer coefficient

av averagei non-uniformity parameter

c coolantj Colburn factor

i inletk thermal conductivity

Lp louvre pitchL effective fin length

max maximumL

hlouvre height

min minimumL

llouvre length

o outletL

plouvre pitch

r ratiom fin efficiency parameter

w wallm number of rowsm mass flowraten number of columnsn number of cellsN

ctnumber of coolant tubes in one row

APPENDIX 2N

fnumber of fins

Np

number of profilesList of assumptions

Nr

number of rows of tubes in the coredepth dimension 1. The cooling system operates under steady state

conditions, i.e. a constant coolant flowrate andNTU number of heat transfer unitsNu Nusselt number fluid temperatures at both inlets.

2. Heat carried by the coolant only transfers toPr Prandtl numberQ overall heat transfer rate of the the airflow that travels through the radiator. Any

other heat losses to, or heat gains from, theradiatorQ

factual heat transfer rate through the fin surroundings are negligible.

3. There are no phase changes in the fluid streamsbaseQ

f,maxmaximum theoretical heat transfer rate flowing through the radiator.

4. Longitudinal heat conduction in the fluid and inthrough the fin baseQ

maxthermodynamically permitted maximum the wall is negligible.

5. The coolant flowrate in coolant tubes is uniformlypossible heat transfer rateRe Reynolds number distributed through the radiator, with no flow

maldistribution, flow stratification, flow bypass-Rf

fouling resistanceR

ffin end radius ing, or flow leakage occurring.

6. Coolant flow is in a fully developed condition inRt

coolant tube end radiusSD specific dissipation each tube.

7. Both fluids are considered incompressible flow,St Stanton numberT temperature and unmixed at any cross-section between passes.

The term ‘unmixed’ is defined as each fluidU overall heat transfer coefficientV velocity through the radiator that behaves as if it was

divided into a large number of separate passagesDx wall thicknessY

clcoolant tube cross-section length with no lateral mixing.



8. All dimensions are uniform throughout the 11. The thermal conductivity of the radiator materialis constant.radiator and the heat transfer surface area is

consistent and distributed uniformly. 12. The thermal resistance (fouling) induced by fluidimpurities, rust formation, or other reactions9. There are no heat sources and sinks in the

radiator walls or fluids. between the fluids and the material is assumedto be small.10. Pure water is used as the coolant.


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