ungin

PropertiesBuoyancy

icalinveede rns.. Thcto

correlation.

and hiTHEs)decadnd invids cal Energ

The term supercritical uid is used in this paper to address a

like uid with an increase in the temperature. With temperatureclose to pseudo-critical temperature (Tpc), the rate of this decreasein density intensies leading to very high thermal expansion coef-cients untypical to most single phase uids. Moreover, at a super-critical pressure, specic heat is considerably higher in the vicinityof Tpc. Pseudo-critical temperature itself depends on the pressure,

heat. A number of such correlations have been suggested in ther pipes [6,812].based on2 or wateeach othe

son, therefore, proposed a semi-empirical correlation, accto which the Nusselt number of a variable-property uidequal to that of the same uid ow with properties evaluatedat the bulk temperature using a constant property correlation corrected by two correction factors representing variations ofdensity and specic heat [9,13]:

NuVP NuCP eCpCp;b

!a1qwqb

a2; 1 Corresponding author. Tel.: +49 721 60845880.

E-mail address: p.forooghi@uq.edu.au (P. Forooghi).

International Journal of Heat and Mass Transfer 74 (2014) 448459

Contents lists availab

H

.euid with a pressure higher than its critical pressure. At any super-critical pressure, there is never two distinguishable liquid and va-por phases in equilibrium. What happens instead is a gradualtransition from high-density liquid-like uid to low-density gas-

literature for turbulent heat transfer in circulaAlthough most of the correlations were derivedments on a specic uid most commonly COcorrection factor expressions are very similar tohttp://dx.doi.org/10.1016/j.ijheatmasstransfer.2014.03.0520017-9310/ 2014 Elsevier Ltd. All rights reserved.experi-r ther. Jack-ordingow is(QGECE) has been considering PTHE as a favorite candidate forbeing used in the development of binary geothermal power cycles.An area of major focus in QGECE is the study of power cycles withsupercritical working uids to bring about higher energy conver-sion efciencies for geothermal energy resources.

ature [69]. It is partly due to the fact that temperature-dependent thermophysical properties may be considerablydifferent near the wall compared to those at bulk temperature.Obviously, new correlations were called for containing correctionsfor the unusually high wall-to-bulk ratios of density and specic1. Introduction

On account of their compactnesscients, plate-type heat exchangers (Pused in various industries in the pastments in manufacturing techniques ahigh pressure and temperature uPTHEs [3,4]. Queensland Geotherma 2014 Elsevier Ltd. All rights reserved.

gh heat transfer coef-have been increasinglyes [13]. With improve-ention of novel designs,n be pumped throughy Centre of Excellence

and approaches critical temperature as the pressure tends to criticalpressure [5].

Heat transfer of a supercritical uid with rapid changes in den-sity and specic heat, like what described above, can be differentfrom that of normal uids. It has been known since long time agothat heat transfer of supercritical uids in straight tubes does notfollow the prediction of conventional heat transfer correlationswhen uids temperature approaches the pseudo-critical temper-Plate heat exchangerConvection heat transfer for corrugation angle of 60. For corrugation angle of 30, however, buoyancy effects were found to have

some inuence, yet majority of the data points are found to be within 15% of those predicted using theExperimental analysis of heat transfer of sexchangers

Pourya Forooghi , Kamel HoomanQueensland Geothermal Energy Centre of Excellence, School of Mechanical and Mining E

a r t i c l e i n f o

Article history:Received 23 January 2014Received in revised form 7 March 2014Accepted 18 March 2014

Keywords:

a b s t r a c t

Heat transfer of a supercritature was experimentally(30 and 60) were examin3.2 to 4.2, respectively. ThtusBoelter type correlatioratio and that of buoyancytion using the correction fa

International Journal of

journal homepage: wwwpercritical uids in plate heat

eering, The University of Queensland, QLD 4072, Australia

refrigerant with highly variable properties close to pseudo-critical temper-stigated in plate heat exchangers. Two different plate corrugation angleswhile the Reynolds and the Prandtl number range from 800 to 4200 andesults are found to be different from those obtained using classical Dit-Two possible effects were investigated: effect of wall-to-bulk propertye former was found to be important and was accounted for in the correla-r proposed by Jackson and Hall. The latter was found not to be signicant

le at ScienceDirect

eat and Mass Transfer

l sevier .com/locate / i jhmt

l ofNomenclature

Latin symbolsA area [m2]C correlation constant []CP effective wall-temperature correction factor []; Eq.

(16)Cp specic heat [kJ/kg K]dh hydraulic diameter [m]G mass velocity [kg/m2 s]g gravity acceleration [m/s2]

Gr Grashof number; qbqbqwgd3hlbh enthalpy [kJ/kg]HTC convective heat transfer coefcient [kW/m2 K]i data point index []L plate length [m]m, n correlation exponents []_m mass ow rate [kg/s]N total number of data points []Nu Nusselt number; HTCdhkbPr Prandtl number; lCPkQ heat transfer power [kW]

Re Reynolds number; Gdhlb

P. Forooghi, K. Hooman / International Journawhere eCp R TbTw CpdT =Tb Tw. Exponents a1 and a2 were sug-gested to be equal to the values used in correlation of Krasnoshche-kov and Protopopov [10], which are variables themselves. In asimpler version of their correlation, Jackson and Hall [9] proposedconstant values of 0.5 and 0.3 for a1 and a2, respectively.

A correlation of kind Eq. (1), however, may not be adequate topredict heat transfer to or from a supercritical uid ow. In sucha uid ow, buoyancy force may affect the ow eld in a rangeof Reynolds number where buoyancy is negligible in typical uidows. Most notable occurrence of this phenomenon is reportedfor turbulent heat transfer in vertical pipes. It has been observedthat in such ows, with an increase in buoyancy forces, heat trans-fer is impaired, for upward ow direction, and is enhanced fordownward ow direction [1416]. For a laminar ow the converseis true. The reason is explained as the deformation of velocity pro-le due to the effect of buoyancy force leading to a reduction orenhancement of shear stress (depending on the ow direction) ina region of ow where turbulence production is concentrated. Sucha change in the level of turbulence production is reected by achange in the local heat transfer coefcient, which is highly sensi-tive to the amount of turbulence diffusivity. A detailed descriptionof the underlying physics can be found in [1720]. One wouldexpect this phenomenon be geometry-dependent but as allabove-mentioned references studied supercritical uid owthrough vertical pipes, it is hard to extend use of the existingexperimental results for plate heat exchangers. Recently, Forooghi

Ri Richardson number; GrReS standard deviation []T temperature [C]t students statistical factor [];TC thermal capacity [kJ/s K]; _m CPU overall heat transfer coefcient [kW/m2 K]x coordinate in ow direction [m]

Greek symbolsa1,2 correlation exponents []b corrugation angle []c channel inclination angle [] plate thickness [m]k thermal conductivity [W/m K]l viscosity [kg/m s]q density [kg/m3]

Subscriptsb bulkCP constant propertyf, f

0uid index

G Ethylene GlycolHT heat transferi data point indexin inletmean mean valueout outletplate plateR refrigerantVP variable propertyw wall

Superscripts(corr) correlation

Heat and Mass Transfer 74 (2014) 448459 449and Hooman, have reported numerical studies of this phenomenonfor inclined pipes [21], and corrugated channels [22], which wouldbe specically useful in the study of plate heat exchangers. This is-sue will be discussed in Section 3.3.

A number of experimental researches on heat transfer of super-critical uids in vertical and horizontal ducts have been publishedin the past decades. Watts and Chou [15] measured heat transfercoefcient of heated supercritical water owing in vertical pipes,and developed an empirical correlation. Their correlation accountsfor both physical effects discussed above (wall-to-bulk propertyvariation and buoyancy). More recently, a similar correlation wasdeveloped by Bae and co-workers based on extensive experimentsthey carried out on heated supercritical CO2 in vertical pipes andannuli [16,23,24]. As mentioned before, although the correlationsare developed for different uids, the suggested correction factorsare fairly similar and indicate on generality of the analysis. Jianget al. [25,26] and Kim and Kim [27] also performed experimentson supercritical heat transfer in vertical pipes under heating condi-tion to provide further evidence on the effect of property variationon heat transfer coefcient. Liao and Zhao [28] studied a similarproblem in horizontal tubes with diameters between 0.5 and2.16 mm, and reported the inuence of buoyancy in these tubes.A number of reports are available in the literature for heat transferof supercritical uids owing in horizontal tubes or tube bundlesbeing cooled by another uid owing outside tubes [12,29,30].These reports mostly concerned about nding correction factors

(exp) experimental

AbbreviationsHTC heat transfer coefcientLMTD log mean temperature differenceNWC not wall-temperature correctedPTHE plate type heat exchangerQGECE Queensland Geothermal Energy Centre of ExcellenceTHE test heat exchangerWC wall-temperature corrected

for accounting for wall-temperature properties and did not discussany buoyancy effect.

As described above, the studies on turbulent heat transfer in u-ids with highly variable properties are mainly focused on simplegeometries most commonly circular pipe and, expectedly, carehas to be taken when it comes to use the existing correlations for amuch more complicated geometry like a PTHE. A PTHE is a heat ex-changer composed of several patterned plates stacked together.The gaps between plates act as ow passages; hot and cold streamsare distributed between the passages in such a way that differentstreams are in contact with two sides of a plate acting as a (usuallythin) heat transfer surface. It is a common practice to use corru-gated plates in PTHEs. Usually the corrugation directions of adja-cent plates are in opposite angle in order to generate a complexow pattern which guarantees adequate mixing (see Fig. 1).

Attempts for nding reliable heat transfer correlations forPTHEs with corrugated plates have started as early as 60s. In oneof the earliest, and yet most systematic, studies, Okada et al.[31], investigated different geometries by changing corrugation an-

The main part of the test loop was the test heat exchanger(THE), in which the refrigerant received heat from hot Ethylene

450 P. Forooghi, K. Hooman / International Journal ofgles and pitch-to-depth ratio of the plates. They found heat transfercorrelations of the Dittus and Boelter type to be suitable for theirexperimented heat exchangers. Kumar [32] and Thonon [33] triedto develop their own heat transfer correlations for single-phaseworking uids to come up with similar type of correlation, butslightly different in constants. Muley and Manglik [34] tried to ob-tain a generalized correlation, which can be applied to a range ofgeometries. In their correlation, the parameters are functions ofcorrugation angle and surface enlargement factor (the ratio of realto projected surface area). Apart from geometry-dependent param-eters, their correlation too has a Dittus and Boelter type. A moretheoretical approach to a generalized formula was taken by Martin[35], and consequently Dovic et al. [36], in which existing formulasfor some basic geometries were combined together to derive anequation suitable for plate heat exchangers. Flow in a straight con-duit and ow in a 2D channel between two wavy plates were thetwo basic geometries used in their approach. These geometrieswere chosen because they were believed to represent main owpatterns inside a plate heat exchanger. Such ideas are supportedby studies of ow patterns inside passages of plate heat exchangerswith corrugated plates. As an example, Focke et al. [37,38] founddifferent ow patterns in plate heat exchangers and argued thatthe corrugation angle of the plate is the determining geometricparameter. One can, as a general rule, conclude from all abovementioned studies that the corrugation angle is, by far, the mostFig. 1. Schematic gure of a passage between two adjacent cross-corrugated platesin a plate heat exchanger (narrow lines represent corrugation lines of the lowerplate).Glycol owing in the heating loop. Two commercial brazed plateheat exchangers, only different in their corrugation angle, wereused in turn as THEs, both of which similarly consist of 10 platesforming 9 passages (5 in hot side and 4 in cold side). THE was in-stalled in the loop so that the mean ow direction was vertical. Thespecications of THEs are presented in Table 1. At all four ports ofthe THE, temperatures were measured using four RTDs with Pt100elements. Pressure was also measured at both inlet and outlet ofrefrigerant side using high-accuracy pressure transducers with 03 MPa range and accuracy of 0.15%. THE and all connecting pipesbetween the measurement spots were thermally insulated usingimportant geometric parameter in the heat transfer behavior ofplate heat exchangers.

A complete review of the literature reveals that none of theexisting correlations for heat transfer in plate heat exchangers isproven to work where rates of change in thermophysical proper-ties particularly density and specic heat are signicant, likeheat transfer of supercritical uids in the vicinity of pseudo-criticaltemperature. The present experimental study is an attempt to llthis gap in the literature and provide practical guidelines to beused in applications with the possibility of occurrence of such acondition.

2. Experiments

2.1. Test facility

Fig. 2b schematically shows the test facility used for thisexperimental study. There were two loops, which formed the testfacility; the refrigerant loop or the main loop in which the work-ing uid ows and the Glycol loop or heating loop in whichEthylene Glycol ows. 98%-pure Peruoro-butane (Molecular for-mula: C4F10; CAS#: 355-25-9; critical pressure: 2.32 MPa; criticaltemperature: 113.2 C) was used as the refrigerant in this studydue to its low critical pressure compatible with the availableequipment. This uid shows expected trends in thermophysicalproperties near its critical point (see Fig. 3). The refrigerant andEthylene Glycol were circulated in their (separate) loops bymeans of a positive displacement pump and a centrifugal pump,respectively. Mass ow rate of the refrigerant was measuredusing a Coriolis owmeter. Before the refrigerant is pumped, ithad to be depressurized. This was done by means of an adjustablepressure-reduction regulator. The refrigerant cooled down in acooler, in which water at ambient temperature was used as thecoolant. A regenerator is added to the refrigerant loop to providea suitable heat balance for desired temperatures. For both coolerand regenerator, similar plate heat exchangers different fromthe test heat exchanger described below were used. In orderto reduce the uctuations of mass ow rate, an accumulator lledwith compressed N2 was installed before the inlet of the refriger-ant pump. This accumulator was also used to adjust the overallvolume of the refrigerant in different working conditions, to avoida need for recharging the loop each time. It was, in particular,necessary because of the strong sensibility of density to temper-ature for supercritical working uids leading to dramatic changein the volume once heating temperature was changed. Acostume-designed electrical oil heater was used to heat upEthylene Glycol to a controlled outlet temperature. The maximumtemperature of Ethylene Glycol was always kept below 180 C, inorder to avoid boiling.

Heat and Mass Transfer 74 (2014) 448459glass-ber wraps (with a thickness of at least 5 mm) in order toavoid any heat loss and unwanted heat transfer that might affectthe results.

l ofP. Forooghi, K. Hooman / International Journa2.2. Data collection and reduction

As mentioned before, two plate heat exchangers (with corruga-tion angles of 30 and 60) were used as THEs. For each, three

Fig. 2. Test facility (a) and its

Fig. 3. Properties of Peruoro-buthane at 2.55 MPa.Heat and Mass Transfer 74 (2014) 448459 451different cases with regard to ow arrangement and directions were studied (see Table 2 for a summary of all six cases). In eachcase, refrigerant mass-ow-rate and Ethylene Glycol temperaturevaried to create a range of working conditions. Each working con-dition leads to one data point in the experiments. Ethylene Glycolmass-ow-rate was constant for all data points within one case,and, in general, this variable was set to be considerably larger thanthat of refrigerant. It guarantees the convective thermal resistanceof the refrigerant side to be the dominant one, thus the overall heattransfer coefcient of THE would be mainly sensitive to changes inthe refrigerant side, which was the purpose of this study. It alsoleads for the hot stream temperature range to be considerablysmaller than that of the cold stream. Pressure of the mainloop was measured at both ends of THE and was controlled by

schematic diagram (b).

Table 1Specication of test heat exchangers.

Corrugation angle 30, 60Number of plates (including end plates) 10Total heat transfer area 0.226 m2

Cross section area (cold/hot) 8.64/10.8 104 m2Plate spacing 2 mmPlate material SS316Passage hydraulic diameter 3.2 mmMaximum working pressure 4.5 MPaMaximum /minimum working temperature 100/150 C

Table 2Flow arrangement and directions in different cases.

b Case Flow arrangement

30 I Parallel-owII Parallel-owIII Counter-ow

60 IV Parallel-owV Parallel-owVI Counter-ow

452 P. Forooghi, K. Hooman / International Journal ofthe adjustable valve located downstream of the outlet port so thatits value remains constant during measurements of one data point.The pressure variations during the whole experiment were alsoslight; the working pressure was 2.55 0.1 MPa for all points.The pressure drop through heat exchanger found to be negligible.Totally, 90 data points were recorded in all six cases. For every datapoint, data logging started after the system reached a steady statecondition, i.e. no time-dependency in the measured quantities wasobserved except for random uctuations. The total duration of datacollection for each data point was a few times more than the timeit takes for the system to reach steady state. During the data collec-tion time of every single data point, numerous measurements (atleast 100) were made to reduce the uncertainty due to uctuationsas will be further discussed in the error analysis section. Asummary of working conditions during the whole experiments ispresented in Table 3.

For each data point, total heat transfer rate of THE is determinedusing the rst law of thermodynamics from the measured refriger-ant mass-ow-rate as well as inlet and outlet enthalpies (asfunctions of temperatures):

Q exp _mRhout hinR: 2Average overall heat transfer coefcient of the heat exchanger canbe found from Q(exp) and the measured temperatures at four portsof the heat exchanger:

Uexp Qexp

AHT DTmean : 3

Overhead line sign j indicates an area-averaging over thewhole heat transfer surface area;

U ZAHT

UxdA=AHT ; 4

where U(x) indicates local overall heat transfer coefcient which isinverse-linearly proportional to the overall thermal resistance (de-ned as the of summation of convective thermal resistances of bothrefrigerant and Ethylene Glycol sides plus conductive thermal resis-tance of the plate), i.e.;

Ux 11=HTCRx 1=HTCGx 2 =kplate : 5Table 3Working conditions of test loop.

Main loopHigh pressure (THEa working pressure) 2.452.65 MPaLow pressure

P. Forooghi, K. Hooman / International Journal ofof uncertainties due to all sources of error.1 The contribution of ran-dom uncertainties, however, can be reduced to a high extent byusing the mean value out of an increased number of independentmeasurements. As mentioned before, for each data point, more than100 measurements have been made to reduce this contribution asfar as possible. According to Moffat [40], uncertainty due to randomerrors is equal to t S=

N

pwhere S is standard deviation of all

measurements and N is the number of measurements. t is Studentsmultiplier, which is approximately 2 for N > 60.

As explained before, there are two types of measured quantitiesin the present study: mass-ow-rate and temperature. Alluncertainties introduced to the experiments necessarily stem fromerrors in the measurements these two quantities.Fig. 4. Error in calculation of mean temperature reference using LMTD as a functionof refrigerant outlet temperature.2.3.1. Uncertainty of temperatureThe RTDs used for measuring temperatures have been already

calibrated with tolerance of 0.35 C. This can be regarded as thexed component of the instrument error for all measurementsthrough the course of experiments. Fixed errors due to systemmay be caused by numerous factors. Attempts has been made toeliminate all of them as much as possible; the test heat exchangersand all pipes in between temperature sensors were insulated toprevent any unwanted heat transfer in thermocouples or any heatloss. Besides, RTDs are placed as close to the heat exchangers portsas possible. Another source of xed error could be temperaturevariation at different point of a cross-section where the measure-ment takes place. This error was ruled out by doing an auxiliaryexperiment in which the intrusion of RTD is changed a few times;no systematic error was observed in this auxiliary experiment.

Random errors in temperature measurement can be simplydetermined for every data point based on the observed scatter inthe measurements. This scatter varied from one data point to an-other, but the standard deviation was always smaller than 0.1 C.Combining this with the instrument error mentioned above withan assumption that the xed errors due to system are negligible the overall uncertainty in temperature measurements would beless than 0.4 C.

1 Unless specied otherwise, uncertainty, in this paper, is based on 95%probability (20 to 1 odds in favor of the real value being within the specied interval).2.3.2. Uncertainty in mass-ow-rateBased on manufacturers specication, the instrument error for

the Coriolis owmeter is 0.1% of the absolute measured value.However at the beginning, some severe uctuations wereobserved in the measurement of mass-ow-rate. The uctuationswere reduced to some degree by minimizing the vibrations in thesystem. In the main experiments, the uctuations varied frompoint to point, with standard deviation being always less than7%. The remaining uctuations possibly originate in the functionof the reciprocal positive displacement pump. Fixed error com-bined with the uctuations was used into determine the totaluncertainty in mass-ow-rate for each data point. The biggestobtained value of uncertainty for a data point was 1.5% of theabsolute value.

2.3.3. Uncertainty in Q(exp) and Uexp

As explained in the previous section, Q(exp) and Uexp are calcu-lated based on the measured values of temperature and mass-ow-rate. In order to evaluate the uncertainty in each calculatedvariable, it is necessary to determine its sensitivity to all mea-sured variables rst. The values of sensitivity are equal to partialderivative of the calculated variable with respect to the measuredvariable. They are straightforward to obtain from the equationsalready presented except for DTmean, for which computerizedanalysis introduced in [40] was employed. Once the sensitivityis known, the uncertainties in Q(exp) and Uexp can be determinedbased on uncertainties in the measured values of temperatureand mass-ow-rate, which were already discussed. For eachpoint, the analysis is based on all measurements for that datapoint.

It must be mentioned here that the errors which are xed dur-ing measurements of a single data point may have a variable partwhen different data points are taken into account since for record-ing every new data point, changes had to be made to the test facil-ity. Fixed errors of the instrument may vary when the value of theactual measurable quantity changes. On the other hand, it is possi-ble that system errors, which are xed for one data point, varyfrom one data point to another. Note that six different cases havebeen studied, for each of which parts of pipe work and insulationshould have been be redone. Moreover, variables such as pump fre-quency, accumulator pressure and heater outlet temperature weredifferent in different data points. These variable parts of xed er-rors, if exist, cannot be smoothed out by averaging among themeasurements of a single data point. Therefore, if a variable hasto be calculated based on various data points, some data scattermay arise. This issue will be further discussed in the next section.

3. Results and discussion

3.1. Test results

Fig. 5 presents the values of heat transfer rate (Q(exp)) plottedagainst refrigerant mass-ow-rate for all data points, distinguishedby the case numbers. For all cases, Q(exp) increases monotonicallywith mass-ow-rate. It must be mentioned that inlet temperaturesof refrigerant and Ethylene Glycol are not identical for differentpoints. According to Fig. 5, no big difference is observed betweencases with parallel- and counter-ow arrangements; it is reason-able since, as mentioned before, the hot stream temperature is al-most constant, compared to that of the cold stream.

Fig. 6 presents the obtained values of overall heat transfer coef-cient Uexp for all data points, plotted against average Reynoldsnumber dened as

Heat and Mass Transfer 74 (2014) 448459 453Re G dhlb

; 9

both

l ofFig. 5. Heat transfer power plotted against refrigerant mass-ow-rate for

454 P. Forooghi, K. Hooman / International Journawhere G is mass velocity (mass ow rate divided by total passagearea) and dh is the hydraulic diameter of the passage. The averageReynolds number is obtained by averaging among local values ofReynolds number obtained from a 1D analysis which will beexplained later.

Although, as expected, Uexp shows a generally increasing trendwith Reynolds number, a big deal of irregularity is also observedfor both heat exchangers. It is, as already discussed, believed tobe due to the strong dependence of physical properties on temper-ature. As such, a detailed investigation is called for.

As discussed in the introduction, heat transfer correlations ofDittus and Boelter type are widely used for PTHEs. Similarly, in thisstudy a correlation of the same type is used, i.e.

Nu C RenPrm: 10However, it was observed, as will be shown later, that the abovecorrelation cannot satisfactorily reect the physics of the problem;then the idea of using a correction factor of Jackson and Hall typeemerges, i.e. correcting Nusselt number from Eq. (10) using walltemperature correction factor from Eq. (1). The result would be

Nu C RenPrmeCpCp;b

!a1qwqb

a2: 11

To avoid unnecessary complication of the problem, the simpliedvalues of 0.5 and 0.3 are used for a1 and a2 in the present study.

If onewishes to use a correlation of above types, it must be notedthat Reynolds and Prandtl numbers as well as the property ratiosvary considerably along the heat exchanger, at least for the refriger-

Fig. 6. Overall heat transfer coefcient plotted against refrigerant average Reynolds numtest heat exchangers. Error bars indicate on 20 to 1 uncertainty interval.

Heat and Mass Transfer 74 (2014) 448459ant side, which is the dominant side in terms of heat transfer resis-tance. Therefore, either a 0D approach must be used in which someaveraging practice is required or a 1D one, which accounts for thevariation of temperature, and thereby variations of other parame-ters. The former approach would lead to some extra uncertaintyin the results, in spite of its simplicity. Therefore, the latter ap-proach was chosen for this study, for which the following heat bal-ance equation was numerically solved for both refrigerant andEthylene Glycol streams in the entire heat transfer surface area:

_mfCP;f dTfdx Ux Tf x Tf 0 x;f ; f 0 R;G; ff 0; Tf x0;f Tf ;in; 12

where positive and negative signs refer to the ow directions.Refrigerant ow direction is always assumed positive, so the signfor Ethylene Glycol is positive and negative for parallel- and coun-ter-ow, respectively. Here, x0,f denotes the inlet position, whichis always zero for f = R; for f = G it is zero or L for parallel- and coun-ter-ow arrangements, respectively. The equations of the twostreams are coupled by heat transfer term in the right hand side. Lo-cal heat transfer coefcient, U(x), can be found from Eq. (5), inwhich:

HTCf x Nuf x kb;fdh : 13

Nuf(x) can be determined based on local parameters using eitherEq. (10) or Eq. (11) depending on which type of correlation is beingexamined. Since both sides of the heat exchanger are identical,

ber for both test heat exchangers. Error bars indicate on 20 to 1 uncertainty interval.

same correlation can be used for both sides (f = R, G). It is worthmentioning that, the convective thermal resistance of the refriger-ant side, has always been found to be at least an order of magnitudelarger than the other two thermal resistances, and, in general, Ucould be suitably approximated by HTCR. Obviously, all thermo-physical properties appearing in the solution are functions of localtemperatures.

In order to solve Eq. (12), heat transfer surface area have beenone-dimensionally discretized using 500 grid points, which guar-antees mesh independence with less than 0.01% error when com-pared to results obtained on a 1000 grid points. First orderforward difference scheme was used for the temperature deriva-tive and a time-marching iterative method was adopted by adding

constant Prandtl number equal to 3.6, which is the average refrig-erant Prandtl number for the present study, was used in these

[31]. This is quite satisfactory noting the difference between avail-able experimental results in the literature obtained for differentheat exchangers. Furthermore, the present correlation, althoughobtained for a different working uid from those of other works,matches well with other correlations, which is an approval of thegeneric nature of our analysis.

In order to investigate the wall temperature correction, it is nec-essary to compare the scattering intervals of both correlations withregards to experimental data. To do that, for every data point, thevalue of constant C, which exactly leads to the experimental heattransfer coefcient Uexp for that point, is obtained. Obviously,the value of C reported in Table 4, for either WC or NWC case,should be the mean of all single-data-point values of C for that

P. Forooghi, K. Hooman / International Journal of Heat and Mass Transfer 74 (2014) 448459 455graphs. Both WC and NWC cases were shown in Fig. 7; since theother works do not account for any wall-to-bulk ratio of densityand/or specic heat, the points for WC correlation in these graphsare calculated by neglecting the correction factors in Eq. (11), i.e.

equatingeCpCp;b

a1qwqb

a2with unity. The prediction of WC case lies

perfectly within those of other correlations, while for NWC corre-lation, some underestimation is observed. Considering WC to bethe correct correlation (this issue will be discussed in depth later),the present results are in best agreement with those of Okada et al.

Table 4Correlation parameters; comparison of the present results with other reports.

b Reference

30 Kumar [84]Muley and Manglik [99]Okada et al. [72]Thonon [95]Present WC

NWC

60 Kumar [84]Muley and Manglik [99]Okada et al. [72]Thonon [95]an articial transient term to the left hand side of the equation. Thevalidity of the solution was checked in comparison with the analyt-ical expression for mean temperature difference (Eq. (6)). The nalresults are also in a reasonable agreement with other correlationsavailable in the literature as will be discussed later.

The parameters n, m and C were found by minimizing a targetfunction total deviation of the correlation from experimentaldata, which is straightforward apart from the fact that a differentialequation had to be solved rather than calculating an algebraicexpression for heat transfer coefcient.

3.2. Effect of wall-to-bulk property ratios

Optimum values for (n,m) were found to be almost the same forboth not wall temperature corrected (NWC) and wall tempera-ture corrected (WC) correlations, i.e. Eqs. (10) and (11), respec-tively. These values are (0.74,0.35) for b = 30 and (0.71,0.35) forb = 60. The value of constant C is however different for the twocorrelations being 0.09 (WC) and 0.076 (NWC) for b = 30 and0.187 (WC) and 0.165 (NWC) for b = 60. A summary of the param-eters is presented along with the values reported by other experi-mentalists in Table 4. To better compare the present correlationwith other correlations their predicted variation of Nusselt numberwith Reynolds number are plotted all in the same graph (Fig. 7). APresent WCNWCcase. To distinguish between mean and single-data-point C, respec-tively, subscripts mean and i are used:

Cmean PN

i CiN

: 14

It can be shown that the ratio of Ci to Cmean is the ratio of heattransfer coefcient obtained in experiments to that predicted bythe correlation, or

CiCmean

Uexp

Ucorr

i

: 15

In Fig. 8, all values of Ci obtained using both WC and NWC correla-tions are presented. Ideally, there must be no scatter in these val-ues. It would be the case if an ideal correlation was used, i.e. acorrelation that captures the physics of the problem in full details.In this sense, the scatter in Fig. 8 could be considered a result ofconceptual error. There is yet another possible source of data scat-ter in Fig. 8 which could be the result of what addressed in the erroranalysis section as the variable part of xed errors. Although carehas been taken to block any source of xed error, except for theinevitable instrument error, it is always possible that unknown er-ror sources emerge during the experiment. Such an error, if exists,however, must be almost the same for both heat exchangers sincethe whole experimental procedure is exactly similar for all casespresented in Table 2. Therefore, any difference in the amount ofscatter for the two heat exchangers can only be due to the concep-tual error not experimental error.

It is observed in Fig. 8 that, for b = 60, the standard deviation ofdata scatter can be signicantly reduced from 12.6% to 4.6% byincluding wall temperature correction. For b = 30, however, thereduction is from 11.4% to 9.7%, which can hardly be called a mean-ingful improvement. In view of the above, one may argue thatusing a heat transfer correlation, which accounts for wall-to-bulkproperty variations, can bring about better results for b = 60 andnot for b = 30. This deduction will be further investigated in thefollowing paragraphs.

C n m

0.108 0.703 0.330.109 0.703 0.330.157 0.66 0.40.2267 0.631 0.330.090 0.74 0.350.076

0.348 0.663 0.330.098 0.782 0.330.327 0.65 0.40.2946 0.700 0.33

0.187 0.71 0.350.165

l of456 P. Forooghi, K. Hooman / International JournaIn Fig. 8, data were plotted against both the average Reynoldsand Prandtl numbers to observe no signicant dependency. Asignicant trend, however, can be found when data points areplotted against a third variable, which is the value of wall

Fig. 8. Scattering of experimental C constant obtained from no wall corrected (up) anrefrigerant average Reynolds number and Prandtl number.

Fig. 7. Comparison of the present correlation for Nusselt number with other wHeat and Mass Transfer 74 (2014) 448459temperature correction itself as indicated by Fig. 9. The variableappearing on abscissa in this gure is the effective value of walltemperature correction applied in each data point, which wascalculated as

d wall corrected (down) correlations. For each case, data are plotted against both

orks for both corrugation angles in the studied range of Reynolds number.

rect

l ofCF RAHT

RenPrmeCpCp;b

a1qwqb

a2RAHT

RenPrm: 16

In fact, CF measures how much, for each data point, heat transfercoefcient is corrected because of wall-to-bulk property ratios. Itis clearly observed that, in NWC case, the value of constant C is pro-portional to CF. For both corrugation angles b = 30, 60, the numer-ical value of the slope is of the order of magnitude of one althoughthe trend line is somewhat steeper for b = 60. Comparing WC andNWC results, it can be argued that, when the employed heat trans-fer correlation does not include the required correction due to wall-to-bulk property ratio (NWC), the required correction emerges inthe value of Ci. In other word, there is a physical variable on whichthe heat transfer coefcient depends but is not accounted for by theheat transfer coefcient. For WC case, i.e. with a correlation of typeEq. (11), this dependence is already accounted for, so C shows nodependence on CF. It is observed that the trend lines in the righthand side picture are also not perfectly horizontal. This is mostprobably because of the use of a simplied correction factor withconstant exponents. As mentioned in the introduction, there aremore elaborated correction equations available in the literature,among which the right choice can be made depending on theexpected accuracy. The important nding of Fig. 9 is existence ofa dependence on wall-to-bulk property ratio, which can be satisfac-torily eliminated by use of a right correction equation.

Fig. 9. Experimental C constant obtained from no wall corrected (left) and wall cor

P. Forooghi, K. Hooman / International Journa3.3. Effect of buoyancy

It was shown in the previous section that, for both plate geom-etries, use of a heat transfer correlation, which contains a walltemperature correction expression can remove, to a satisfactoryextent, the dependency of error on the property ratios, whichwould appear if a Dittus and Boelter type correlation was used.However, as both Figs. 8 and 9 suggest, the amount of data scatteris only reduced by this correction for b = 60 and not for b = 30.When WC correlation is used, the experimental data scatters with-in around 20% of the mean value for b = 30which is more than 10%bigger than that of b = 60 (considering root sum square combina-tion of uncertainties, the extra uncertainty is approximately 15%).As explained in the previous section, this difference in the amountof scatter should be a result of some conceptual error. It suggeststhe existence of a physical effect for b = 30 which is absent inb = 60. It was mentioned in the introduction that buoyancy canbe able to affect the heat transfer coefcient in supercritical uidswith similar density variations to that of this problem, and itseffect is also highly geometry-dependent.Forooghi and Hooman [21,22], numerically investigated the pos-sible effect of buoyancy on the ow patterns and heat transfer thatcan arise in a plate heat exchanger. They estimated that, in an in-clined conduit, the effect of buoyancy on turbulent heat transferis most severe around 50% in the worst scenario for c anglessmaller than 10, where c is the angle of the conduit with the ver-tical direction. This effect decays as c increases and is negligiblefor c larger than 50. On the other hand, studies of ow pattern in-side plate heat exchangers suggest that, for both b = 30 and 60, thedominant regime of ow inside a plate heat exchanger is ow alongfurrows, which can be regarded analogous to ow in conduits thatmake an angle equal to b with the vertical direction [35,38]. Basedon this fact, the effect of buoyancy in the heat exchangers witheither b angle can be roughly estimated by what Forooghi and Hoo-man [21] reported for the same c angle. The result would be a neg-ligible effect for b = 60 and one with around 15 to 20% inuence forb = 30. The gures approximately match the value of extra datascatter observed for b = 30 in the present study. It must be re-minded that the buoyancy effect could be positive or negativedepending on the ow direction being downward or upward.

To better investigate the effect of buoyancy, values of Ci for alldata points, distinguished by the direction of ow, are plottedagainst Richardson number in Fig. 10. For the test heat exchangerwith b = 60, all points are within approximately 10% of the meanvalue. For the one with b = 30, one out of four to ve points lie outof this accuracy interval, all of which belong to Richardson num-

ed (right) correlations plotted against effective wall temperature correction factor.

Heat and Mass Transfer 74 (2014) 448459 457bers higher than a certain threshold. For the latter heat exchanger,data scattering generally increases with Richardson number.

Richardson number is widely used as a measure of buoyancyforces relative to viscous forces. In this work it is dened as:

Ri GrRe

; Gr qbqb qw g d3h

lb: 17

Although Fig. 10 provides some general information about the effectof buoyancy, it is hard to identify a clear trend. It can be due to atleast two factors; rst, the range of Reynolds number chosen for thisstudy is very close to the laminarturbulence transition region. It isgenerally hard, if possible, to dene a clear transition threshold inthe plate heat exchangers, mainly due to the complicated and mul-ti-pattern nature of ow. Experiments of Focke et al., for example,suggest ranges of 150 < Re < 600 for b = 60and 1000 < Re < 3000for b = 30, none of which being clear-cut [38]. Hence, for the prob-lem under consideration, probably some data points are in the tran-sition regime, where the behavior of turbulent heat transfer isconsiderably complicated and, as a matter of fact, all previousworks on supercritical uid ow considered fully turbulence ows.

d fr

458 l ofThus, it is hard to expect the theory to be perfectly reproduced bythe present experiments. In addition to that, even in the experi-ments dealing with straight pipes [14,16,41,42], similar scatter isobserved within the experimental points. In other words, the effectof buoyancy on turbulent supercritical uid ows is in general onewith some inherent uncertainty.

It is worth mentioning that b = 60, is presently, by far, the mostwidely used corrugation angles for plate heat exchangers, forwhich one can comfortably ignore the effect of buoyancy; this ismost probably the case for all corrugation angles larger than 50.For b = 30, the present data are enough to raise the concern of pos-sible deviations from what one would expect using the availablecorrelations in the literature.

For corrugation angles smaller than 30 (including the case ofcombined corrugation angle with one angle smaller than 30)experimental proof is required; for this case numerical observa-tions [21,22] suggest possibility of more severe deviations.

4. Conclusion

Experiments have been done on two plate-type heat exchangerswith corrugation angles of b = 30, 60 to investigate the perfor-mance of this type of heat exchangers in the working conditionswhere density and specic heat both strongly depend on tempera-ture, like in supercritical uids. The ranges of Reynolds and Prandtlnumber are 8004200 and 3.24.2, respectively. It was found that:

In a PTHE, Whenever a working uid exchanges heat in condi-tions where the rates of variations of its thermophysical proper-ties (specic heat and density) with temperature are high, it is

Fig. 10. Experimental C constant obtaineexpeffetusspesurbultem

Totramaicabe(seP. Forooghi, K. Hooman / International Journaected that heat transfer correlations not accounting for thect of wall-to-bulk property ratios (for example those of Dit-and Boelter type), fail to make satisfactory predictions. Acic example of such uids is a supercritical uid in a pres-e slightly larger than its critical pressure when either/both ofk or/and wall temperatures is/are close to the pseudo-criticalperature.tackle the above mentioned problem, a two-component heatnsfer correlation is suggested: the rst component is the nor-l heat transfer correlation that would be used if thermophys-l properties were constant. This correlation should, however,multiplied by a correction factor of Jackson and Hall typee Eq. (1)). The correlations found in this research are:

Nu 0:187 Re0:71Pr0:35eCpCp;b

!0:5qwqb

0:3; b 60;

Nu 0:09 Re0:74Pr0:35eCpCp;b

!0:5qwqb

0:3; b 30;but any other correlation obtained through the above two-com-ponent approach can bring about satisfactory results. This ideacan be, in particular, useful because a specic correlation devel-oped for a specic design may work the best for that design.

For a plate heat exchanger with corrugation angle of 30 theremight be up to 20% deviation in results due to the effect ofbuoyancy. For the corrugation angle of 60 the buoyancy effectsare negligible.

Conict of interest

None declared.

Acknowledgments

This research paper was made possible through substantialtechnical support from Mr Jason Czapla. Authors would also liketo appreciate help from staff of the School of Mechanical andMining Engineering Workshops in University of Queensland, inparticular, safety ofcer, Mr Hugh Russell.

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P. Forooghi, K. Hooman / International Journal of Heat and Mass Transfer 74 (2014) 448459 459

Experimental analysis of heat transfer of supercritical fluids in plate heat exchangers1 Introduction2 Experiments2.1 Test facility2.2 Data collection and reduction2.3 Error analysis2.3.1 Uncertainty of temperature2.3.2 Uncertainty in mass-flow-rate2.3.3 Uncertainty in Q(exp) and ?

3 Results and discussion3.1 Test results3.2 Effect of wall-to-bulk property ratios3.3 Effect of buoyancy

4 ConclusionConflict of interestAcknowledgmentsReferences