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Applied Thermal Engineering 62 (2014) 680e689

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Applied Thermal Engineering

journal homepage: www.elsevier .com/locate/apthermeng

On the performance of a vertical helical coil heat exchanger. Numericalmodel and experimental validation

José Fernández-Seara*, Carolina Piñeiro-Pontevedra, J. Alberto DopazoÁrea de Máquinas y Motores Térmicos, E.T.S. de Ingenieros Industriales, University of Vigo, Campus Lagoas-Marcosende 9, 36310 Vigo, Spain

h i g h l i g h t s

� A numerical model for vertical helical coils inside storage tanks was developed.� The model was validated against experimental data.� The Nusselt number improves by increasing the outer tube diameter.� The heat transfer rate depends mainly on the inner area.

a r t i c l e i n f o

Article history:Received 20 August 2012Accepted 27 September 2013Available online 25 October 2013

Keywords:Helical coilHeat exchangerExperimentalSimulation

* Corresponding author. Tel.: þ34 986 812605; faxE-mail address: jseara@uvigo.es (J. Fernández-Sea

1359-4311/$ e see front matter � 2013 Elsevier Ltd.http://dx.doi.org/10.1016/j.applthermaleng.2013.09.05

a b s t r a c t

A numerical model was developed in order to predict the heat transfer process and pressure drop in avertical helical coil heat exchanger (HCHE) located inside a fluid storage tank in which water is used asinner and outer fluid. Natural convection was considered as boundary condition for the HCHE outersurface. The model was validated with experimental data obtained from an own facility with two HCHEstested under several operating conditions. The model developed was used to evaluate the main HCHErepresentative geometrical parameter’s influence on the overall heat transfer coefficient and pressuredrop. The results show that by increasing the tube diameter causes an increase of the Nusselt numberand a larger heat transfer rate to pressure drop ratio is obtained.

� 2013 Elsevier Ltd. All rights reserved.

1. Introduction

Helical coil heat exchangers (HCHEs) are widely used in severalheat transfer applications such as steam generators, refrigerators,nuclear reactors, chemical plants and domestic hot water systems(DHW) due to their compactness in structure, ease of manufacture,maintenance and improved thermal efficiency. Moreover, it is well-known that helical pipes provide enhanced inner convection heattransfer when compared to straight pipes. Prabhanjan et al. [1]report the performance comparison between a straight tube heatexchanger and a helically coiled heat exchanger. Dravid et al. [2]analyze the laminar flow heat transfer in helically coiled tubes.

The heat transfer rate through the coil depends on the inner andouter convection processes, the conduction through the tube walland the fouling resistances on the inner and outer HCHE surfaces.For any given HCHE, and apart from the fouling effects, the innerand outer heat transfer coefficients determine the heat transferrate.

: þ34 986 811995.ra).

All rights reserved.4

General empirical correlations widely used in the HCHE innerconvection heat transfer coefficient calculation, as well as the innerfriction factor, can be readily found in basic heat transfer literature,such as Gnielinski [3] and Hewit et al. [4]. However, a literaturesurvey on the outer convection coefficient did not produce anyrelevant citations, neither general correlations to determine theconvection coefficient on the outer surface of the coil have beenfound, even when it usually controls the heat transfer process inthis kind of heat exchanger configuration. Prabhanjan et al. [5] re-ported an experimental investigation on natural convection fromvertical helical coils submerged in water and using water as innerfluid. Four different coils made from copper pipes with 15.8 mmand 13.5 mm outer diameters were analyzed. Power-law correla-tions for the outer convection coefficients using different charac-teristic lengths were obtained. Ali [6] experimentally evaluated fivecoils including different pitch-to-coil diameter ratios, tube di-ameters of 8 mm and 12 mm and different numbers of turns, andshowed the outer heat transfer coefficients as power-law correla-tions by using the coil length as characteristic length. Ali [7] alsocarried out experiments with six different vertical coil configura-tions immersed in a glycerolewater solution. Results were shownas correlations including the number of turns and coil-to-tube

Nomenclature

A area, m2

Cp specific heat capacity at constant pressure, J kg�1 K�1

D diameter, mDe Dean numberDHW domestic hot waterdi tube’s inner diameter, mdo tube’s outer diameter, mE modeled-to-experimental error valueF correction factor for the DTlmf Moody’s friction factorg gravity acceleration, m s�2

Gr Grashof numberHCHE helical coil heat exchangerh convection heat transfer coefficient, W m�2 K�1

k thermal conductivity, W m�1 K�1

N number of HCHE turnsNu Nusselt numberP pressure, kPaPe perimeter, mPr Prandtl numberp coil pitch, mQ heat transfer rate, WR coil radius, mRa Rayleigh numberRe Reynolds numberr tube radius, mT temperature, �C, KU overall heat transfer coefficient, W m�2 K�1

V volumetric flow rate, m3 s�1

z HCHE vertical axis

Greek symbolsa helical angle, �

b thermal expansion coefficient, 1 K�1

m dynamic viscosity, kg m�1 s�1

n velocity, m s�1

p Pi numberr density, kg m�3

s stress tensorq control volume angle, �

DTlm logarithmic mean temperature differenceDP pressure drop, kPa

Subscriptsc coilcri criticaldi related to the tube’s inner diameter as characteristic

lengthdo related to the tube’s outer diameter as characteristic

lengthexp related to the experimental datafi inner foulingfo outer foulingH related to the HCHE’s height as characteristic lengthho related to the HCHE outer convection heat transfer

coefficienti innerif HCHE inner fluidin inletL related to the tube’s length as characteristic lengthm meanmod related to the model resultso outer, outsideof HCHE outer fluidout outletov overalls straight tubesf surfacest storage tankt tubeti related to the inner tube wallto related to the outer tube wallw water in the tankx characteristic length

J. Fernández-Seara et al. / Applied Thermal Engineering 62 (2014) 680e689 681

diameter ratios. Ali [8], Xin and Ebadian [9] and Moawed [10] re-ported experimental researches on free convection from verticaland horizontal helical pipes in air. Fernández-Seara et al. [11] car-ried out an experimental study to analyze the thermal performanceof a vertical HCHE immersed in a DHW storage tank. The used coilwas formed from a straight stainless steel (AISI 316) tube with 8 mlength and 23/20 mm outer/inner diameters. It has 6 turns with amean diameter of 400 mm and 40 mm pitch. The average outsidecoil Nusselt number for all the experimental data was correlated asa power-law of the Rayleigh number taking the tube’s diameter, thetube’s length, and the coil’s height, as characteristic lengths.

Regarding the HCHE thermal performance numerical evalua-tion, most of the models which have been developed are based onsimplified outer boundary conditions, such as constant wall tem-perature [12] or constant wall heat flux [13]. On the other hand, inthe Prabhanjan et al. paper [5] a numerical investigation of thenatural convection heat transfer from helical coiled tubes in waterwas included. From this research a prediction model was devel-oped taking into account a natural convection heat transfer pro-cess on the outer HCHE surface which suggested that the methodwould promise to predict outlet temperatures from similarlydimensioned heat exchangers. Recently, different boundary con-ditions were proposed by Mirgolbabaei et al. [14] and Coloradoet al. [15]. In the first one, a numerical investigation was

performed on the outside mixed convection heat transfer fromvertical helical coiled tubes in a cylindrical shell by using a com-mercial fluid dynamics software. In the second, a physicaleempirical model to describe the helical coil thermal behavior in anoil and glycerol/water solution including an artificial neuralnetwork (ANN) model and outer natural convection boundarycondition was designed and validated. Rennie and Raghavan [16]numerically investigated the heat transfer performance of adouble-pipe helical heat exchanger. Ferng et al. [17] also proposeda computational fluid dynamics methodology to investigate effectsof different Dean number and pitch size on thermalehydrauliccharacteristics in a helically coil-tube heat exchanger. In 2012,Zachár [18] investigated the natural convection induced heattransfer over the outer surface of helically coiled-tube heat ex-changers. Zachár also studied the effect of the distance betweenthe wall of the tank and the outer surface of the helical pipe, andstated that distances lower than 2/3 do reverse the direction of thenatural convection induced resultant flow in the core and theouter side region of the storage tank.

In this research, the main objective was to develop a detailedsimulation model in order to predict and evaluate the performanceof two HCHE placed into the water storage tank. The numericalmodel and its implementation are detailed. Natural convectionwasconsidered as the boundary condition for the HCHE outer surface.

J. Fernández-Seara et al. / Applied Thermal Engineering 62 (2014) 680e689682

The model results are compared to the experimental data collectedfrom an own experimental facility, evaluated under several oper-ating conditions. Finally, key results such as the influence of the coiltube diameter, pitch, tube length, and coil diameter on the overallheat transfer coefficient and pressure drop are shown anddiscussed.

2. Mathematical model

The numerical model was developed in order to predict the heattransfer phenomena in a vertical HCHE located inside a fluid stor-age tank, commonly used in DHW systems. Convection heattransfer in the tube, conduction through the tube wall and con-vection heat transfer to the fluid stored in the tank were consideredas the dominant heat transfer modes. The HCHE was modeled asone helically coiled tube divided into several small control volumes.The control volumes are connected in the HCHE’s inner fluid flowdirection. Each control volume is composed from a portion of tubedefined as a model parameter, and a corresponding tank volumeportion, as shown in Fig. 1. Three systems are considered at eachcontrol volume, i.e. “inner fluid”, “tube” and “outer fluid”. In thisstudy, water was taken as the fluid in both, inner and outer fluidflows.

The model equations were formulated from the mass, energyand momentum balances applied to each system in each controlvolume. In order to formulate the model, the following assump-tions were applied: water is modeled as an incompressible fluid;the water and tube time-dependency physical properties areneglected (stationary processes); the inner water flow is unidirec-tional, i.e. the trajectory of all water particles are parallel to the tubewall; the outer water flow related to the HCHE vertical axis is alsounidirectional; the physical properties of the inner water flow andtube wall are considered uniform in the tube radial direction; theaxial heat conduction in the tubes and the variations of the waterflow’s potential and kinetic energies are considered insignificant;the pressure drop in the outer water flow is neglected; the storagetank is isolated.

Fig. 1. Schematic of the HCHE in the DHW storage tank and the established controlvolume.

The application of the mass, energy and momentum balances ineach system are given by Eqs. (1)e(4), according to the nomen-clature in Fig. 1.

Inner fluid system:

�vPvq

¼ rif$g$Rc$sen aþ s$Rc$PeAti

(1)

vif$rif$Cpif$1Rc$vTifvq

þ rif$g$sen a$vif ¼ �Ui$�Tif � Tt

� 4di

(2)

Tube system:

Q ¼ Uo$do$�Tt � Tof

�¼ Ui$di$

�Tif � Tt

�(3)

Outer fluid system:

12$

R2st$p� p$d2o

2$Rc

!$vof$rof$Cpof$

vTofvz

¼ �Uo$p$do$Rc$�Tt � Tof

�(4)

The inner and outer overall heat transfer coefficients aredetermined according to Eqs. (5) and (6).

Ui ¼ 1

1hiþ 1

hfiþ

ri$ln

����rmri����

kt

(5)

Uo ¼ 1

ro$ln

���� rorm����

ktþ 1

hoþ 1

hfo

(6)

Several correlations found in technical literature were used inthe developed model to obtain the inner convection heat transfercoefficient and friction factor, as well as the outer convection heattransfer coefficients. The correlations were formulated as a functionof the dimensionless numbers commonly used in convection heattransfer processes to data reduction. The Nusselt, Reynolds, Prandtl,Rayleigh and Grashof numbers are calculated from Eqs. (7)e(11).

Nux ¼ h$xk

(7)

Rex ¼ r$n$xm

(8)

Pr ¼ Cp$mk

(9)

Rax ¼ Grx$Pr (10)

Gr ¼g$b$

�Tto � Tof

�$x3

v2(11)

2.1. Inner convection heat transfer coefficient and friction factor

Based on the Reynolds number, the inner convection heattransfer coefficient of the water flowing into the tube is determinedfrom the inner Nusselt number, which is calculated in Eq. (12) forthe laminar flow regime [19], and in Eqs. (13) and (14) for thetransitional regime [20]. For the turbulent flow condition, theNusselt number is calculated in Eq. (15) [21], as a function of the

J. Fernández-Seara et al. / Applied Thermal Engineering 62 (2014) 680e689 683

friction factor, defined in Eq. (16) [22]. The critical Reynolds numbercalculated in Eq. (17) is used to determine the transitional regimeboundaries [19].

Nudi ¼ 3:65þ 0:08$�1þ 0:8$

�diDc

�0:9�$RenPr1=3; Re < Recri

(12)

Nudi¼C$Nudi;Re¼Recriþð1�CÞ$Nudi;Re¼22;000; Recri<Re<22;000

(13)

C ¼ ð22;000� ReÞ=ð22;000� RecriÞ (14)

Nudi ¼f8$Re$Pr

1þ 12:7$ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffif8$Pr2=3 � 1

�q $

PrPrsf

!0:14

; 22;000 < Re

(15)

f ¼�0:3164Re0:25

þ 0:03$�diDc

�0:5�(16)

Recri ¼ 2300$�1þ 8:6$

�diDc

�0:45�(17)

For the laminar and transitional regimes, the tube’s inner wallfriction factor is calculated in Eqs. (18) and (19) as a function of thestraight tube correlation, Eq. (20) [23], and the Dean number, Eq.(21). The friction factor calculation for the turbulent regime iseffected according to Eq. (16).

ffs

¼ 1; De � 11:6 (18)

ffs

¼ 1þ 0:015$Re0:75$�diDc

�0:4

; 11:6 < De (19)

fs ¼ 64Re

(20)

De ¼ Re$�riRc

�0:5

(21)

2.2. Outer convection heat transfer coefficient

Regarding the HCHE outer heat transfer process, the authorshave not found general correlations to calculate the outer convec-tion coefficient. Taking into account natural convection as bound-ary condition, the water temperature around the coil is consideredas a constant. In order to calculate the outer natural convection heattransfer coefficient, the Fernandez-Seara et al. [11] correlationexpressed in Eq. (22) was established in the numerical model as thedefault option, by establishing the tube’s outside diameter as thecharacteristic length.

Nudo ¼ 0:4998$Ra0:2633do ; 4:67� 106 � Rado < 3:54� 107

(22)

In addition, several correlations found in literature have beenincluded as a model parameter, to determine and compare theouter Nusselt number estimation. These correlations are detailedbelow.

a) Fernández-Seara et al. [11] correlations, Eqs. (23) and (24),using the height and the length of the coil as characteristiclength.

0:2633 9 10

NuH ¼ 0:8181$RaH ; 5:31� 10 � RaH < 4:02� 10

(23)

0:2633

NuL ¼ 1:709$RaL ; 1:97� 1014 � RaL < 1:49� 1015

(24)

b) Churchill and Chu [24] correlations, Eqs. (25) and (26), forlaminar and turbulent flows, on horizontal straight tubesand using the tube’s outer diameter as characteristiclength.

0:25

Nudo ¼ 0:36þ 0:518$Rado�

1þ�0:559Pr

� 916�4

9

; Rado < 109 (25)

8 2 31=692

Nudo ¼><>:0:6þ0:387$64 Ra�

1þ�0:559Pr

�9=16�16=975>=>; ; 109

�Rado(26)

c) Al-Urabi and Salman [25] correlation, Eq. (27), obtained forair flow on straight tubes and using the tube’s length as thecharacteristic length.

1=3

NuL ¼ 0:158$RaL (27)

d) Ali [8] correlations obtained for helical coils immersed inwater, Eqs. (28) and (29) for 8 and 12 mm tube’s diameters,using the tube’s length as characteristic length, and Eq. (30)using the coil’s height as characteristic length.

0:295 12 14

NuL ¼ 0:685$RaL ; 3� 10 � RaL < 8� 10 (28)

0:516 11 14

NuL ¼ 0:00044$RaL ; 6� 10 � RaL < 10 (29)

0:323 8 11

NuH ¼ 0:257$RaH ; 6� 10 � RaH < 3� 10 (30)

e) Ali [7] correlations obtained for helical coils immersed inwater, Eqs. (31) and (32) for 5 and 10 turns, using the coil’slength as characteristic length and including a coil-to-tube’souter diameter relationship.

� ��1:313

NuL ¼ 0:0000253$Ra0:739L $Dc

do; 1012�RaL�1014

(31)

� ��0:702

NuL ¼ 0:00001535$Ra0:671L $Dc

do; 7� 1012 � RaL

� 8� 1014

(32)

Fig. 2. Schematic diagram of the experimental facility.

J. Fernández-Seara et al. / Applied Thermal Engineering 62 (2014) 680e689684

f) Ali [7] correlation, Eq. (33), obtained for vertical 12 mm indiameter brass helical coils immersed in a glycerolewatersolution.

0:335 12 14

NuL ¼ 0:106$RaL ; 2� 10 � RaL � 8� 10 (33)

g) Ali [7] correlation, Eq. (34), obtained for helical coils withcoil-to-tube’s outer diameter ratio of 10, and taking into ac-count the number of turns.

0:797 �1:524 13

NuL ¼ 0:00000252$RaL $N ; 2�10 �RaL

�8�1014 (34)

h) Xin and Ebadian [9] correlation, Eq. (35), obtained for helicalcoils and using the tube’s outer diameter as the characteristiclength.

0:293

Nudo ¼ 0:29$Rado ; 4� 1013 � Rado � 1015 (35)

i) Prabhanjan et al. [5] correlations, Eqs. (36) and (37), obtainedfor helical coils and using the tube’s length and the coil’sheight as the characteristic length.

0:3972

NuL ¼ 0:009759$RaL ; 5� 1014 � RaL � 3� 1015

(36)

NuH ¼ 0:0749$Ra0:3421H ; 9� 109 � RaH � 4� 1011

(37)

3. Model implementation

A finite difference approach was used to solve the modelequations. The system of discretized equations was solved in space,step by step, beginning with the control volume where the innerwater flow enters the HCHE. From the known values at the inletsection, the values of the variables at the outlet of each controlvolume are iteratively obtained, advancing in the flow direction.The procedure is repeated until reach the last control volume. Theinlet temperature, pressure and mass flow of the inner fluid, theinitial tube and outer fluid properties, and the HCHE and tank ge-ometries, are used as inputs.

A computer code was developed for the model implementationby using Visual Basic Net. From the software results, the inner fluidoutlet temperatures and the HCHE heat transfer rate are deter-mined as well as other important parameters such as the innerpressure drop, the coil temperature distribution, inner and outeroverall heat transfer coefficients, among others.

The calculation process is as follows:

1. From the initial values, determine the water flow thermody-namic properties at the HCHE inlet and the water properties inthe tank.

2. For each control volume at each turn:2.1. Calculate the heat flux as follows:

2.1.1. Guess an inner tube wall temperature, Tti.2.1.2. Calculate the inner convection coefficient hi, from

Eqs. (12)e(21).2.1.3. Calculate the heat flux between the inner water flow

and inner tube wall. Eqs. (3) and (5).2.1.4. Calculate the outer tube wall temperature, Tto.2.1.5. Calculate the external convection coefficient, ho, Eqs.

(22)e(37).2.1.6. Calculate the overall heat transfer coefficient, and

then calculate the heat transfer rate between the in-ner and outer water flows.

2.1.7. Calculate the inner tube wall temperature Tti.2.1.8. Compare the guessed and calculated inner tube wall

temperatures. If verified the convergence criterion of1.0E�2, continue, otherwise guess a new value andreturn to step 2.

2.2. Determine the inner water flow loss pressure DPi. Eq. (1).2.3. Calculate the thermodynamic properties of the inner water

flow at the control volume outlet, which will be used asinput for the following control volume. Eqs. (1), (2) and (4).

3. The process is repeated for each control volume.

4. Results and discussion

4.1. Model validation

An own experimental facility was used to obtain the experi-mental data in order to validate the model results. Two stainlesssteel HCHE of 6 turns, 35mmpitch, placed into awater storage tankand located at the top of the tank, were tested. The HCHE, labeledN� 1 is 15/12 mm tube’s outer/inner diameters, 4 m in length and150 mm in diameter while the HCHE labeled N� 2 is 20/23 mmtube’s outer/inner diameters, 9 m in length and 420 mm in diam-eter. Another stainless steel HCHE placed at the tank’s bottom andlabeled N� 3, with 15/12 mm in tube’s outer/inner diameters, 6

J. Fernández-Seara et al. / Applied Thermal Engineering 62 (2014) 680e689 685

turns, 35 mm in pitch, 9 m in length and 420 mm in diameter isused to maintain the water temperature as a constant. The tankused is a commercial hot water storage tank of 150 L, 870 mm inheight, and 480 mm in inner diameter. Fig. 2 shows a schematicdiagram of the experimental facility. A detailed description of theexperimental setup and methodology can be found in Fernández-Seara et al. [11].

The experimental facility was equipped with a data acquisitionsystem based on a 12-bit data acquisition card (USB-5008, NationalInstruments) and a PC. Water was used as both fluids, cold as innerfluid and hot as outer fluid. The water temperature distributioninside the tank, the HCHEs inner water flow inlet and outlet tem-peratures were measured by using A-Pt100 sensors with an accu-racy of �(0.15 þ 0.002T) (�C), inserted in 6 mm diameter stainlesssteel pockets and 190 mm long inside the tank, at 100 mm from thetank wall and 50 mm long inside the HCHE. The inner HCHE N� 1and N� 2 water flow rate was measured by using a volumetric flowmeter model SITRANS F M MAG 3100 and the transmitter modelSITRANS F M MAG 5000 with an accuracy of �0.25% of themeasured value. The pressure drop in the HCHE N� 1 and N� 2 innerwater flow was measured by using a differential pressure trans-mitter model SITRANS P DSIII with an accuracy of �0.075% of themeasured value. The location of all sensors at the experimentalsetup is also shown in Fig. 2. The estimated overall error of the dataacquisition system and sensors is �0.2 �C for temperature and�0.5% for flow rate measurements.

48 sets of experimental data were gathered from the HCHE N� 1and N� 2 by combining several operating conditions. Hot water waspumped through the HCHE N� 3 to heat up the water stored in thetank while cold water was introduced into the HCHE located at thetop of the tank. The operating conditions included 8 water volu-metric flow rates inside the tested HCHE from 250 to 2000 L h�1 atflow rates intervals of 250 L h�1 and 3 tank water (outer fluid)average temperatures at 50, 60 and 70 �C. The tested HCHE waterinlet temperature was fixed at 30 �C. The tank water averagetemperatures were kept constant by using the HCHE N� 3 withwater flow inlet temperatures at 65, 80 and 95 �C, and ranging thewater volumetric flow rate inside the HCHE N� 3 from 330 to515 L h�1.

In addition, 19 sets of experimental data were carried out inorder to study the pressure drop in the HCHE N� 1. In this cases, thewater HCHE N� 1 inlet and tank temperatures were keeping con-stant at 30 �C, and the water volumetric flow rate inside the HCHEN� 1 was varied from 500 L h�1 to 1400 L h�1 at intervals of 50 L h�1.

All the experimental data was scanned and recorded understable working conditions, i.e. when the variations in all

Fig. 3. Outer water temperature distribution experimental measurements from sensorTw1 to Tw10 (see Fig. 2).

temperature and flow ratemeasurements werewithin a�1% range,in a time frame of 15 min. The water properties in the datareduction process were obtained from Refprop Database [26].

The outer natural convection coefficients obtained from theexperimental data were calculated as follows. The heat flowtransferred through the coil was determined from an energy bal-ance on the inner fluid, according to Eq. (38).

Q ¼ Vif$rif$Cpif$�Tif ;out � Tif ;in

�(38)

The overall heat transfer coefficient from the outer tube surfacewas calculated from Eq. (39). The logarithmic mean temperaturedifference was obtained from Eq. (40), where Tof refers to the watertemperature in the tank around de coil, i.e. the temperaturemeasured by sensors Tw6 to Tw10, which are nearly the same.

Uov ¼ QF$Ao$DTlm

(39)

DTlm ¼�Tof � Tif ;in

���Tof � Tif ;out

�

ln

Tof�Tif ;inTof�Tif ;out

! (40)

Since neither the wall temperature, nor the film temperatureused to evaluate the Reynolds and Prandtl numbers, are known, aniterative approach is used to determine the wall temperature andthe inner Nusselt number. The correlations to calculate the innerNusselt number were the same as used in the model. The insideheat transfer coefficient was determined from Eq. (41) and the in-ner wall temperature was determined from Eq. (42).

hi ¼Nudi$k

di(41)

Tti ¼ Tif þQ

Ati$hi(42)

In Eq. (42) Tif is the average bulk temperature based on themeaninlet and outlet temperatures. The film temperature was obtainedas the mean value of wall and fluid temperatures. The outside heattransfer coefficient was calculated from the thermal resistance Eq.(43). The water properties were evaluated at the mean bulk tem-perature. The outer tubewall temperaturewas calculated accordingto Eq. (44).

ho ¼ 1

1Uo

� dodi$hi

� do$ln dodi

2$k

(43)

Tto ¼ Tof �Q

Ato$ho(44)

The experimental performance of the HCHE N� 1 and N� 2 testedwas numerically simulated by using the developed model. Themaximum running time observed was 3 min, approximately, on anIntel Core2 Duo-processor PC with 3 GB of RAM. Prior to theanalysis, a grid dependency of the solution was studied and aproper control volumes number for the tube’s geometry waschosen.

Experimental results of water temperature profiles along theheight of the tank for test set of 50 �C water tank temperature areshown in Fig. 3. Results from both HCHEs are represented. It can beobserved that the water temperature around the HCHE N� 1 ispractically constant for each flow rate. Therefore the factor F in Eq.(39) can be considered approximately 1 for this HCHE.

J. Fernández-Seara et al. / Applied Thermal Engineering 62 (2014) 680e689686

Fig. 4 shows a comparison between experimental and numericalresults of the heat transfer rates (a), inner water flow outlet tem-peratures (b) and inner pressure drops (c), for the HCHE N� 1. Ingeneral, reasonable agreement can be appreciated between

Fig. 4. Comparison of the experimental and modeled results of the HCHE heat transferrate (a), inner water flow outlet temperature (b) and inner water flow pressure drops(c).

experimental and model results of the three aforementioned vari-ables. The results show that almost all the compared data remainwithin an error band of �4% for the heat transfer rate, below �1%for the inner water flow outlet temperature and �8% for thepressure drops. Results of the uncertainty analysis are also includedin Fig. 4. These results revealed that the uncertainties in thedetermination of the heat transfer rate increasewith the increase ofthe water flow ranging from �0.036 to �0.285 kW. The tempera-ture and the pressure drop uncertainties values are so small thatcan hardly be appreciated. The obtained temperature uncertaintyvalue is 0.2 �C and the obtained pressure drop uncertainties valuesrange from 0.004 to 0.025 kPa.

Moreover, a comparison between the heat transfer rate pre-dictions based on Eqs. (22)e(37) for the outer convection coeffi-cient calculation was performed by using the numerical model.Fig. 5 shows the model-to-experimental error values on the heattransfer rate and on the outer convection coefficient, calculatedfrom the HCHE N� 1 and HCHE N� 2 for the three water tanktemperatures tested. In general, it can be observed that for each testthe lower the error on the outer convection coefficient prediction,the lower the error on the heat transfer rate estimation. A goodprediction on the outer convection heat transfer coefficient resultsin a good HCHE heat transfer estimation. From HCHE N� 1 test N�

19, an error value of 16.56% on the outer convection coefficient andof 13.10% on the heat transfer rate predictions were observed byusing the Ali correlation (Eq. (28)), whereas in test N� 9 lower errorvalues on both the outer convection coefficient and heat transferrate were 0.31% and 0.04%, when the Fernández-Seara correlation(Eq. (22)) was used. A summary of the maximum and minimumerror values on the HCHE heat transfer rate and outer convectioncoefficient calculations is shown in Table 1.

Taking into account the results showed above, the model wasconsidered validated.

4.2. Parametric study

Once themodel validationwas carried out, the developedmodelwas used to evaluate the main HCHE representative geometricalparameters influence on the overall heat transfer coefficient. Twotypes of parametric analysis have been carried out. Firstly, one ofthe geometric parameters was changed (tube’s outer diameter, coildiameter, pitch and length) whilst keeping the remaining param-eters constant. This methodology was used in several researchesfound in literature [6,7,9,11]. However, the change of one of theparameters implies a change of the coil’s heat exchange area. Hencea second type of analysis has been performed inwhich the variationof the parameters was effected whilst the inner heat exchange areaof the coil remained unchanged. In both types of analysis the tubewall thickness was kept constant.

4.2.1. Parametric analysis with change of heat exchange areaThe values of the geometric parameter analyzed are summa-

rized in Table 2. One of themwas varied while the rest of themweremaintained constant, so 72 different geometries were studied. Inorder to simplify the figures, 36 HCHE geometries were selected asthe most representative. The operating conditions included watervolumetric flow rate inside the HCHE at 1000 L h�1, tank watertemperature at 60 �C and water inlet temperature at 30 �C.

Mean values of the Nusselt numbers from the HCHE outer fluid,Nuo,do, calculated with the tube’s diameter as the characteristiclength, are plotted in Fig. 6 as a function of L/Dc ratio. From thefigure it can be appreciated that, for a tube’s diameter value andfixing an L/Dc ratio, similar Nusselt numbers were obtained, inde-pendently of the pitch and the coil’s diameter values. Therefore thesame Nuo,do are obtained with the same do, Dc, L and different pitch.

Fig. 5. Errors on the prediction of the heat transfer rate and outer convection coefficient of the a) HCHE N� 1 and b) HCHE N� 2 by using several outer convection correlations.

J. Fernández-Seara et al. / Applied Thermal Engineering 62 (2014) 680e689 687

This is due to the fact that both have the same L/Dc ratio for thesame tube’s diameter. It is also observed that the highest Nuo,dovalues are obtained with the larger do and the smaller Dc and L. Onthe other hand Nuo,do values are almost constant with the increaseof the L/Dc relation for a fixed L and do, i.e. with the increase of thecoil’s turns. HoweverNuo,do values decreasewith the increase of the

Table 1Errors on the estimation of the HCHE N� 1 and HCHE N� 2 heat transfer rate and outer c

Correlation for outer convection heat transfer coefficient

(22) (23) (24) (25) (27) (28) (2

HCHE N� 1 EQ Max 3.86 10.50 10.51 21.05 8.45 13.14 2(%) Min 0.04 5.94 2.73 13.72 3.55 5.53Eho Max 4.97 13.96 13.27 27.23 11.14 16.56 2(%) Min 0.28 6.84 4.39 18.75 4.17 10.49

HCHE N� 2 EQ Max 2.38 2.94 3.31 21.96 3.54 7.64 2(%) Min 0.03 0.07 0.26 8.92 0.15 2.46Eho Max 3.47 4.25 4.77 28.90 7.54 10.73 4(%) Min 0.15 0.94 1.47 22.51 0.30 5.49

coil’s turns for the same Dc, p and do, i.e. with the increase of heatexchange area. This effect is higher for the larger do due to the factthat the changes in the geometric parameters give a higher heatexchange area with the largest do.

Fig. 7 shows the relationship between thermal powerexchanged and pressure drop, Q/DP, as a function of the L/Dc, where

onvection coefficient.

calculation

9) (30) (31) (32) (33) (34) (35) (36) (37)

1.87 14.94 107.69 31.01 30.92 57.50 8.14 53.73 41.755.25 6.34 29.48 5.19 20.84 0.98 4.92 38.25 29.358.37 20.10 187.57 38.71 36.10 85.23 10.45 58.67 46.946.28 11.37 61.47 6.21 31.53 2.12 5.64 53.09 42.94

8.33 16.79 92.42 25.26 25.59 128.71 5.75 41.17 36.681.84 5.15 16.21 0.13 9.77 22.80 1.75 17.41 15.646.63 24.60 227.72 40.95 31.35 412.83 8.67 48.12 43.454.13 14.42 64.62 0.04 26.95 131.88 5.93 41.25 38.86

Table 2Geometrical parameters of HCHE analyzed in parametric analysis.

do (m) Dc (m) p (m) L (m)

0.015 0.1 0.025 20.025 0.25 0.05 5

0.5 0.075 100.1

Fig. 7. Relation between thermal power exchanged and drop pressure, Q/DP, against L/Dc from the 36 HCHE geometries.

Fig. 8. Mean values of Nusselt numbers from HCHE outer fluid plotted against L/Dc

relation obtained from 18 HCHE geometries with equal inner exchanger area.

J. Fernández-Seara et al. / Applied Thermal Engineering 62 (2014) 680e689688

each data series are the same as those shown in Fig. 6. This figureshows the large influence of the increasing diameter on thereduction of the pressure drop. As the pitch increases, the relationQ/DP decreases caused by the increase of the pressure drop. Thelargest Q/DP value is obtained from a larger Dc, a smaller p and ahigher do.

4.2.2. Parametric analysis without change of exchange areaIn order to keep constant the inner heat exchanger area a second

parametric analysis was carried out. In this case, the variation of theparameters was effected whilst the inner exchange area of the coilremained unchanged, so length was calculated with the rest ofparameters in this analysis. The operating conditions includedwater volumetric flow ratio inside the HCHE at 1000 L h�1, tankwater temperature at 60 �C and water inlet temperature at 30 �C.The geometries studied were: 2 HCHE outer pipe diameters at0.015, 0.025 m; 3 HCHE coil’s diameters at 0.1, 0.25 and 0.5 m and 4HCHE pitches at 0.025, 0.05, 0.075, 0.1 m. Thereby 24 geometrieswere studied. In order to simplify the figures, 18 HCHE geometrieswere selected as the most representative.

Mean values of the Nusselt numbers from the HCHE outer fluid,Nuo,do, are plotted in Fig. 8 as a function of L/Dc ratio, where eachdata series represents HCHE with same do, L and pitch and differentDc with values of 0.1, 0.25 and 0.5 m. This figure shows the effect ofthe correlation used for calculating Nusselt, based on tube’sdiameter as the characteristic length, so the same tube’s diameterleads to almost same Nusselt numbers independently of the othergeometric parameters, contrary to the observed in Fig. 6. On theother hand the highest Nuo,do is also obtained for the higher do inthis analysis.

Fig. 9 shows the relation between thermal power exchanged andpressure drop,Q/DP, as a function of L/Dc, where each data series arethe same as shown in Fig. 8. From Figs. 7 and 9 it is observed thatthe low Q/DP values obtained with smallest do are independent ofthe geometric parameters. This is attributed to the high pressure

Fig. 6. Mean values of Nusselt numbers from HCHE outer fluid plotted against L/Dc

relation obtained from 36 HCHE geometries.

drop produced by the decrease of do, despite the fact that Q isgreater. This figure also shows the large influence of the increasingdiameter and the decreasing pitch on the reduction of the pressuredrop.

Fig. 9. Relation between thermal power exchanged and drop pressure, Q/DP, against L/Dc from the 18 HCHE geometries with equal inner exchanger area.

J. Fernández-Seara et al. / Applied Thermal Engineering 62 (2014) 680e689 689

5. Conclusions

A numerical model was developed in order to predict the heattransfer phenomena and pressure drop in a vertical HCHE locatedinside a fluid storage tank. Natural convection was considered asboundary condition for the HCHE outer surface. The model equa-tions were formulated from the mass, energy and momentumbalances. Several correlations found in technical literature wereused in the model, which was developed to obtain the inner con-vection heat transfer coefficient and friction factor, as well as theouter convection heat transfer coefficients. The model was vali-dated with experimental data obtained from an own facility withtwo HCHE under several operating conditions. The results showthat from amongst the 15 correlations used that of Fernández-Searaet al. [11] (Eq. (22)) would appear best fitted to the experimentalresults.

The model developed was used to evaluate the main HCHErepresentative geometrical parameter’s influence on the overallheat transfer coefficient. The results show that the Nusselt number,calculatedwith the outer tube diameter as the characteristic length,improves by increasing the outer tube diameter. Moreover the heattransfer is independent of the other geometric parameters for agiven value of inner heat exchanger area.

References

[1] D.G. Prabhanjan, G.S.V. Raghavan, T.J. Rennie, Comparison of heat transferrates between a straight tube heat exchanger and a helically coiled heatexchanger, Int. Commun. Heat Mass Transfer 29 (2) (2002) 185e191.

[2] A.N. Dravid, K.A. Smith, E.W. Merrill, P.L.T. Brian, Effect of secondary fluidmotion on laminar flow heat transfer in helically coiled tubes, AIChE J. 17(1971) 1114e1122.

[3] V. Gnielinski, Helically coiled tubes of circular cross sections, in: G.F. Hewitt(Ed.), Heat Exchanger Design Handbook, vol. 2, Begell House, Inc., New York,USA, 2002 (Section 2.2.14).

[4] G.F. Hewit, G.L. Shires, Y.V. Polezhaev, International Encyclopedia of Heat andMass Transfer, CRC Press, Florida, USA, 1997, pp. 167e170.

[5] D.G. Prabhanjan, T.J. Rennie, G.S.V. Raghavan, Natural convection heat transferfrom helical coil tubes, Int. J. Therm. Sci. 43 (2004) 359e365.

[6] M.E. Ali, Experimental investigation of natural convection from vertical helicalcoiled tubes, Int. J. Heat Mass Transfer 37 (4) (1994) 665e671.

[7] M.E. Ali, Free convection heat transfer from the outer surface of verticallyoriented helical coils in glycerolewater solution, Int. J. Therm. Sci. 43 (2004)615e620.

[8] M.E. Ali, Laminar natural convection from constant heat flux helical coiledtubes, Int. J. Heat Mass Transfer 41 (1998) 2175e2182.

[9] R.C. Xin, M.A. Ebadian, Natural convection heat transfer from helical pipes,J. Thermophys. Heat Transfer 12 (2) (1996) 297e302.

[10] M. Moawed, Experimental investigation of natural convection from verticaland horizontal helical pipes in HVAC applications, Energy Convers. Manage.46 (2005) 2996e3013.

[11] J. Fernández-Seara, R. Diz, F.J. Uhía, J. Sieres, J.A. Dopazo, Thermal analysis of ahelically coiled tube in a domestic hot water storage tank, in: On Proceedingof the 5th International Conference on Heat Transfer, Fluid Mechanics andThermodynamics, 2007. Paper number: SJ5, Sun City, South Africa.

[12] I. Di Piazza, M. Ciofalo, Numerical prediction of turbulent flow and heattransfer in helically coiled pipes, Int. J. Therm. Sci. 49 (2010) 653e663.

[13] A.K. Satapathy, Thermodynamic optimization of a coiled tube heat exchangerunder constant wall heat flux condition, Energy 34 (9) (2009) 1122e1126.

[14] H. Mirgolbabaei, H. Taherian, G. Domairry, N. Ghorbani, Numerical estimationof mixed convection heat transfer in vertical helically coiled tube heat ex-changers, Int. J. Numer. Meth. Fluids (2010), http://dx.doi.org/10.1002/fld.2284.

[15] D. Colorado, M.E. Ali, O. García-Valladares, J.A. Hernández, Heat transfer usinga correlation by neural network for natural convection from vertical helicalcoil in oil and glycerol/water solution, Energy 36 (2011) 854e863.

[16] T.J. Rennie, V.G.S. Raghavan, Numerical studies of a double-pipe helical heatexchanger, Appl. Therm. Eng. 26 (2006) 1266e1273.

[17] Y.M. Ferng, W.C. Lin, C.C. Chieng, Numerically investigated effects of differentDean number and pitch size on flow and heat transfer characteristics in ahelically coil-tube heat exchanger, Appl. Therm. Eng. 36 (2012) 378e385.

[18] A. Zachár, Investigation of natural convection induced outer side heat transferrate of coiled-tube heat exchangers, Int. J. Heat Mass Transfer (2012), http://dx.doi.org/10.1016/j.ijheatmasstransfer.2012.08.014.

[19] E.F. Schmidt, Wärmeübergang und Druckverlust in Rohrschlangen, Chem. Ing.Tech. 39 (1967) 781e789.

[20] V. Gnielinski, Heat transfer and pressure drop in helically coiled tubes, in:Proc. 8th Int. Heat Transfer Conf., San Francisco, Hemisphere, Washington,D.C., vol. 6, 1986, pp. 2847e2854.

[21] B.S. Petukhov, Heat transfer and friction in turbulent pipe flow with variablephysical properties, in: J.P. Hartnett, T.F. Irvine (Eds.), Advances in HeatTransfer, vol. 6, Academic Press, New York, USA, 1970.

[22] P. Mishra, S.N. Gupta, Momentum transfer in curved pipes. 1. NewtonianFluids, Ind. Eng. Chem. Process. Des. Dev. 1 (1979) 130e137.

[23] H. Ju, Z. Huang, Y. Xu, B. Duan, Y. Yu, Hydraulic performance of small bendingradius helical coil-pipe, J. Nucl. Sci. Technol. 18 (2001) 826e831.

[24] S.W. Churchill, H.H.S. Chu, Correlating equations for laminar and turbulentfree convection from a horizontal cylinder, Int. J. Heat Mass Transfer 18 (1975)1049e1053.

[25] M. Al-Urabi, Y.K. Salman, Laminar natural convection heat transfer from aninclined cylinder, Int. J. Heat Mass Transfer 23 (1980) 45e51.

[26] E.W. Lemmon, M.O. McLinden, M.L. Huber, NIST Reference Fluid Thermody-namic and Transport Properties Database (REFPROP), Version 8.0, NationalInstitute of Standards and Technology (NIST), Colorado, USA, 2008.