ORIGINAL

Numerical investigation of heat transfer in annulus laminarflow of multi tubes-in-tube helical coil

S. A. Nada1 & H. F. Elattar1 & A. Fouda2 & H. A. Refaey3

Received: 10 May 2017 /Accepted: 7 September 2017# Springer-Verlag GmbH Germany 2017

Abstract In the present study, a CFD analysis usingANSYS-FLUENT 14.5 CFD package is used to investigatethe characteristics of heat transfer of laminar flow in annulusformed by multi tubes in tube helically coiled heat exchanger.The numerical results are validated by comparison with pre-vious experimental data and fair agreements were existed. Theinfluences of the design and operation parameters such as heatflux, Reynolds numbers and annulus geometry on the heattransfer characteristics are investigated. Different annulus ofdifferent numbers of inner tubes, specifically 1, 2, 3, 4 and 5tubes, are tested. The Results showed that for all the studiedannulus, the heat flux has no effect on the Nusselt number andcompactness parameter. The annulus formed by using fiveinner tubes showed the best heat transfer performance andcompactness parameter. Correlation of predicting Nusseltnumber in terms of Reynolds number and number of innertubes are presented.

Nomenclatureρw Water density, kg/m3

μw Water dynamic viscosity, N.s/m2

A Annulus cross sectional area, m2

Ah Rod heaters surfaces area, m2

Cp Water specific heat, J/kg.K

D Diameter of the outer tube, md Heaters rod diameter, mDc Helical coil of diameter, mDe Dean number, dimensionlessDh Hydraulic diameter, mh Average coefficient of heat transfer, W/m2.KH Pitch, mhAh Factor to measure heat exchanger compactness, W/Khx Local coefficient of heat transfer, W/m2.Kkw Thermal conductivity of water, W/m.KL Length of coil tube, mṁw Mass flow rate of water, kg/sN Number of inner rod heatersNu Average Nusselt number, dimensionlessPr Prandtl number, dimensionlessq Heat flux, W/m2

Q Rate of heat transfer, WR Coil curvature ratio, dimensionlessRe Reynolds number, dimensionlessTs Heater rods average surface temperature, oCTw Average temperature of cooling water along the

entire annulus volume, oCTwi Average temperature of inlet water, oCTwo Average temperature of exit water, oCZ Number of the coil turns

1 Introduction

Helically coils are vastly used as an efficient heat exchangerdue to their compactness and high heat transfer coefficients.The helically coiled heat exchangers are used in numerousapplications such as air conditioning systems, refrigerationsystems, thermal power plants systems, chemical processing,

* S. A. Nadasamehnadar@yahoo.com

1 Department of Mechanical Engineering, Benha Faculty ofEngineering, Benha University, Benha, Egypt

2 Department of Mechanical Power Engineering, Faculty ofEngineering, Mansoura University, El-Mansoura, Egypt

3 Department of Mechanical Engineering, Faculty of Engineering atShoubra, Benha University, Cairo, Egypt

Heat Mass TransferDOI 10.1007/s00231-017-2163-8

nuclear reactors, solar energy applications, food processingand medical equipment. Literature review revealed that thesecondary flow patterns induce in the helically-coiled heatexchangers are the key of heat transfer enhancement in thehelical annulus. Several experimental and computationalworks have been carried out to study the characteristics offlow patterns and heat transfer in helical pipe, outside helicalcoils in shell and in the annulus of tube in tube helical coils.

Dean [1] was the first who described and studied secondaryflow pattern in helical pipes. The existence and characteristicsof the swirling flow patterns are reported in the study. Deannumber (De) was used to investigate secondary flow profilecharacteristics. Later, Dravid et al. [2] proved that an outwarddirected flow is generated at the center of the flow in helicalpipe. Akiyama and Cheng [3] employed a boundary vorticitymethod to numerically investigate the heat transfer of laminarflow in curved pipes under constant heat flux and temperaturewall boundary conditions. Kalb and Seader [4] numericallystudied the characteristics of fully developed fluid flow andheat transfer in a helical tube under constant wall heat flux atwide ranges of Dean numbers, Prandtl numbers, and curvatureratios. Janssen and Hoogendoorn [5] studied, experimentallyand numerically, the heat transfer in the thermal entryregion of helical tubes in laminar flow under boundaryconditions of uniform heat flux and temperature. It wasreported that the boundary conditions have negligibleeffects on heat transfer characteristics. Jayakumar et al.[6] numerically studied the local Nusselt number of ahelical pipe using CFD. The effects of coil and pipe diam-eters and the coil pitch on the heat transfer rate have beenstudied. Correlations for local and average Nusselt numberhave been developed.

Several works were carried out to investigate the heat trans-fer characteristics outside helical coils in shells [7–12].Prabhanjan et al. [7] investigated the heat transfer characteris-tics of a helical coil in a shell of water bath under free convec-tion. Shokouhmand et al. [8] and Slaimpour [9] conductedexperimental works of the performance of helically coiledtube in a shell heat exchangers of different coils geometricparameters under parallel/counter flows conditions. The ther-mal performance of the shell and coil heat exchangers wasexperimentally tested by Ghorbani et al. [10, 11]. It was foundthat the temperature profiles along the axial of heat exchangerdepend on the mass flow rate ratio of the tube and shell sides.Lu et al. [12] numerically investigated the effects of the geo-metrical factors on the thermal performances of multi-streamspiral wound in a shell heat exchanger. It was found that thepressure drop and both of the shell and tube-sides Nusseltnumbers increase with increasing the center core diameterand the tube external diameter and decreasing the tube pitch.Nada et al. [13] presented experimental studies of the thermalperformance enhancement of helical coil in a shell water cool-er by attaching radial fins on the external surface of the helical

coils. Results showed enhancements in the coil thermal per-formance and compactness due the insertion of the fins.Experimental correlations of Nusselt number as a function ofRe, Gr, and the shell diameter were presented for the finnedand un-finned coils.

A numerical investigation of an incompressible-viscousflow in a helical annual was presented by Petrakis andKarahalios [14] for a Dean range 96–8000 and various coresizes. Rennie [15] and Rennie and Raghavan [16] experimen-tally studied the thermal performance of tube-in-tube helicalcoil heat exchangers for parallel and counter flow configura-tions. They concluded that the heat transfer rate for counterflow is much higher due to the higher temperature differencebetween the two fluids. The results showed that the overallheat transfer coefficient increase due to increasing the annulusDean numbers. Kunar et al. [17] conducted experimentalstudy for a tube-in-tube helical heat exchanger in countercur-rent flow configuration. The heat transfer coefficients in theinner and outer tub and for the overall heat exchanger werecalculated usingWilson plots. Louw [18] experimentally stud-ied the effect of the possibility of presence of tubes contact inthe tube-in-tube helical heat exchanger. Comparison withaligned (concentric) heat exchanger showed that annular con-tact decreases the heat transfer in the inner tube and improve itin the annular space. Seyyedvalilu and Ranjbar [19] presenteda numerical investigation on the effect of different parameterssuch as coil radius, coil pitch and diameter of the tube on thehydrodynamic and heat transfer characteristics of helical dou-ble tube heat exchangers using CFD software. The resultsindicated that a heat transfer augmentation occurs by increas-ing the inner Dean number, inner tube diameter, curvatureratio and by reducing the pitch of heat exchanger coil.Kharat et al. [20] developed a numerical correlation of heattransfer coefficient for flow between concentric helical coilsusing CFD and experimental data. They found that the devel-oped correlation provides a precise fit to the experimentalresults within an error range of 3–4%.

Recently, multi tubes in tube helically coiled heat ex-changers are proposed to improve the compactness of the heatexchangers and the coil thermal performance due to increasingthe encourage secondary flow. In spite of multi tubes in tubehelical coil heat exchanger are currently available in the mar-ket for engineering applications; only two recent types of re-search [21, 22] are available in the literature for heat transfer/pressure drop characteristics in the annulus of multi tubes intube heat exchangers. An experimental investigation on multitubes in tube helically coiled heat exchanger was presented inthese works. The effects of the operating and designparameters such as Re, heat flux, annulus hydraulic di-ameter, and number of inner tubes on the heat exchang-er thermal performance and compactness were investi-gated. Correlations of the average Nusselt number werededuced from the experimental data.

Heat Mass Transfer

The previous literature survey reveals that although a hugeamount of researches have been conducting for investigatingflow and heat transfer characteristics in curved and helicallycoiled pipes, the multi tubes in tube helical heat exchangersstill needs to be investigated. The experimental investigationof these studies is not an easy task and time consuming.Therefore, the present study aims to present numerical modelto investigate the effects of the different operating and geo-metric parameters on the thermal performance of multi tubesin tube helical heat exchanger. A validated numerical study inthis filed will open the work for comprehensive parametricstudy for design development of such devices.

2 Physical model

A geometry of the physical model under consideration isshown in Fig. 1a. It describes a flow in the helical annulus.The annulus is created by a solid heating rod or multi-heatingrods of diameter d inserted inside an outer tube of innerdiameter D. The outer tube and rods are turned in theform a helically coil of diameter Dc. The distance be-tween two adjacent turns, called a pitch, is H. Fiveannuluses using 1, 2, 3, 4, and 5inner rods are investi-gated. The number of the coil turns and the coil length isdenoted by Z and L, respectively. The rod(s) is (are) concen-tric inside the tube as shown in Fig. 1b.

Table 1 gives the geometrical and physical parameters ofthe five annuluses under investigation in the present study.These conditions were taken identical to the ones used in theprevious experimental work [21, 22] for the sake of validationof the present numerical work.

3 Mathematical formulation, analysis and governingequations

The velocity and temperature distributions and the heat trans-fer characteristics in a helical annulus is a complex problem.Proper selection of the computational domain, grid genera-tion, and numerical solution techniques lead to accurate con-vergent simulation process. ANSYS-FLUENT 14.5 CFDcode is used in the present study, it provides complex meshcapability, fast solution algorithm, and post-processing facili-ties. A personal computer hardware system consisting of Intelcore ™ i5 four core 3.1 GHz PC with 16 GB memory RAMwas used for the numerical runs and analysis. The data wasanalyzed to obtain the velocity distributions, temperature con-tours and heat transfer coefficients for the fluid flow in thehelical annulus of multi tubes in tube heat exchanger for dif-ferent numbers of inner tubes and at different operating con-ditions. Steady, incompressible and Newtonian flow are as-sumed in the analysis. Heat transfer by radiation and the grav-itational effects are neglected during the analysis. The

Coil 1

Coil 2

Coil 5

Coil 4

Coil 3

Coil 1

Coil 2

Coil 5

Coil 4

Coil 3

(b)(a)

Fig. 1 Physical model: (a)Geometry of multi-tubes in-tubehelical coils, (b) Section views ofthe studied coils

Heat Mass Transfer

following governing equations (continuity, momentum andenergy equations, respectively) for flow field, temperaturedistribution and heat transfer in the annulus of multi tubes intube helical heat exchanger were applied and solved in theCartesian coordinate system.

∂∂xi

ρUið Þ ¼ 0 ð1Þ

∂∂xi

ρU jUi� � ¼ ∂

∂xiμ∂U j

∂xi

� �−∂P∂xi

ð2Þ

∂∂xi

ρCpU jT� � ¼ ∂

∂xik∂T∂xi

� �ð3Þ

where, Ui and T are the time-averaged velocity and tempera-ture in i direction. ρ, Cp, k, μ and P are the density, specificheat, thermal conductivity, viscosity and static pressure, re-spectively. For the sake of accurate results, fluid propertieswere not assumed to be constant but they are analyzed andprogrammed as temperature dependent properties as given bythe following polynomials [23]

ρ Tð Þ ¼ 999:98þ 4:69� 10−2T−7:54� 10−3T2 þ 4:36

� 10−5T 3−1:46� 10−7T 4 ð4Þ

k Tð Þ ¼ 0:547þ 2:05� 10−3T−4:71� 10−6T2−8:89

� 10−8T 3 þ 4:81� 10−10T4 ð5Þ

μ Tð Þ ¼ 1:77� 10−3−5:49� 10−5T þ 9:93

� 10−7T2−9:44� 10−9T3 þ 3:55� 10−11T4 ð6Þ

Cp Tð Þ ¼ 4:23−7:15� 10−3T þ 4:42� 10−4T 2−1:32

�10−5T 3 þ 2:03� 10−7T4−1:53

�10−9T5 þ 4:53� 10−12T6

ð7Þ

Where T is the temperature in °C.The annulus cross sectional area (A), the heat transfer area

(Ah) and the annulus hydraulic diameter (Dh) are calculated by:

A ¼ π D2−N d2� �

=4 ð8Þ

Ah ¼ π d L N ð9Þ

Dh ¼ 4Aπ Dþ N dð Þ ð10Þ

Table 1 Geometrical and physicalparameters of the studied coils Parameter Coil-1 Coil-2 Coil-3 Coil-4 Coil-5

Outer diameter, D (mm) 23 23 23 23 23

Rod diameter, d (mm) 6 6 6 6 6

Number of rods, N 1 2 3 4 5

Tube/rod length, L (m) 2.675 2.675 2.675 2.675 2.675

Number of turns, Z 5 5 5 5 5

Pitch, H (mm) 25 25 25 25 25

Annulus area, A (mm2) 387 359 331 303 274

Hydraulic diameter, Dh (mm) 17 13 10.3 008.2 006.6

Coil Diameter, Dc (cm) 17 17 17 17 17

Curvature ratio, Dh /Dc 0.1 0.076 0.061 0.048 0.039

Heat transfer area, Ah (m2) 0.05 0.1 0.15 0.15 0.25

Recr 10,069 9047 8304 7621 7077

q (kW/m2) 10.53–52.65 5.55–26.2 3.91–17.42 3.91–13.06 3.92–13.07

Fig. 2 Mesh generation

Heat Mass Transfer

Reynolds number in the annulus flow is determined basedon Dh as given by:

Re ¼ m˙ Dh

A μwð11Þ

The dean number for flow in helical annulus is defined by

De ¼ Re

ffiffiffiffiffiffiDh

Dc

rð12Þ

The present study is under laminar flow conditions; there-fore, the critical Reynolds number is determined usingSrinivasan et al. [24] correlation as follows:

Recr ¼ 2100 1þ 12

ffiffiffiffiffiffiDh

Dc

r� �ð13Þ

The outer surface of the annulus is taken to be adi-abatic and the inner surface is assumed to be heated at

constant heat flux q. Cooling water enters and exits theannulus at average temperatures of Twi and Two, re-spectively. The heat transfer rate Q from the inner rodsto the annulus flow is calculated from the enrgy balanceof the heater rods as follows:

Q ¼ q Ah ¼ m˙ wCp Two−Twið Þ ð14Þ

Where mw is cooling water flow rate and Cp is specific heat ofcooling water and calculated at mean cooling water temperature.

The local heat transfer coefficient at the inner surface at anydistance x along the length of the helical tube is calculatedfrom:

h xð Þ ¼ q″

Ts xð Þ−Tw xð Þð Þ ð15Þ

Where Ts (x) and Tw (x) are the average surface temperature ofthe heating rods and the cooling water temperature in the pipeat distance x from the start of the coil tube.

Table 2 Mesh independence study

Coil 1 Q = 527 W, N = 1,Re = 2222, Twi = 297 K

No. of cells 453,299 629,034 790,326 889,823 1,030,807 1,226,533 1,409,681

Two (K) 299.7 299.4 299.3 299.6 299.7 299.8 299.7

Coil 2 Q = 560 W, N = 2,Re = 1734, Twi = 297 K

No. of cells 435,656 576,543 813,245 1,135,467 1,343,435 1,546,543 1,787,678

Two (K) 298.8 298.7 298.7 298.6 298.6 298.5 298.6

Coil 3 Q = 593 W, N = 3,Re = 1895, Twi = 297 K

No. of cells 421,781 775,883 926,923 1,421,781 1,558,298 1,996,598 2,024,245

Two (K) 299.6 299.6 299.5 299.5 299.4 299.4 299.4

Coil 4 Q = 791 W, N = 4,Re = 1987, Twi = 297 K

No. of cells 539,281 756,030 891,123 1,083,439 1,136,616 1,264,576 1,490,534

Two (K) 306.6 305.7 305.2 305.2 305.1 305.3 305.1

Coil 5 Q = 989 W, N = 5,Re = 2813, Twi = 297 k

No. of cells 515,753 694,637 888,803 1,018,340 1,061,453 1,193,187 1,224,623

Two (K) 300.1 300 300 299.9 299.9 299.8 299.9

100 1,000 10,000

Re

1

10

100

Nu

Coil 3

Exp., Nada et al. [21-22]

Numerical

Fig. 3 Model validation

Heat Mass Transfer

The average heat transfer coefficient is either calculatedfrom Eqs. 16 or 17

h ¼ ∫L

0h xð Þ dx ð16Þ

h ¼ QπNLd Ts−Twð Þ ð17Þ

Where Ts and Tw are the average heating rods surfaces tem-peratures and the cooling water temperature along the entireannulus volume. For model validation and comparison withexperimental works, the average heat transfer coefficient iscalculated using Eq. 16.

The average Nusselt number is calculated from:

Nu ¼ hDh

kwð18Þ

(a) Effect of heat flux (Re=1220, N=3) (b) Effect of Re (q=10.44 kW/m2, N=3)

q= 3.91 kW/m2

q= 10.44 kW/m2

q= 17.42 kW/m2

Re= 1889

Re= 3394

Re= 4684

Fig. 4 Effect of Re and heat fluxon Temperature contour

Heat Mass Transfer

4 Numerical solution and validations

4.1 Numerical solution and grid independence study

An element-based finite volume method was utilized byANSYS-FLUENT, which firstly discretize the spatial domainusing a mesh. Finite volumes are constructed using this mesh.

The governing equations are discretized by using the finitevolume method. The near-wall region was solved by usingenhanced wall treatment. The pressure-velocity couplingwas achieved by SIMPLE algorithm. Mass, momentum andenergy conservation are applied for each finite volume. In thepreset work, the grid was generated with tetrahedral elementsand wedges elements as shown in Fig. 2. Distributed axial

100 1,000 10,000

Re

100

1000

10000

h (

W/m

2K

)

N= 1

q= 10.53 kW/m2

q= 13.15 kW/m2

q= 26.32 kW/m2

q= 39.49 kW/m2

q= 52.65 kW/m2

100 1,000 10,000

Re

100

1000

10000

h (

W/m

2K

)

Coil 2

q= 5.555 kW/m2

q= 7.207 kW/m2

q= 14.44 kW/m2

q= 19.66 kW/m2

q= 26.20 kW/m2

100 1,000 10,000

Re

100

1000

10000

h (

W/m

2K

)

N= 3

q= 3.910 kW/m2

q= 5.218 kW/m2

q= 10.44 kW/m2

q= 13.06 kW/m2

q= 17.42 kW/m2

100 1,000 10,000

Re

100

1000

10000

h (

W/m

2K

)

N= 4

q= 3.913 kW/m2

q= 5.225 kW/m2

q= 7.837 kW/m2

q= 10.95 kW/m2

q= 13.06 kW/m2

100 1,000 10,000

Re

100

1000

10000

h (

W/m

2K

)

N= 5

q= 3.919 kW/m2

q= 5.229 kW/m2

q= 7.843 kW/m2

q= 10.46 kW/m2

q= 13.07 kW/m2

Fig. 5 Effect of Re and heat flux on Nu for different numbers of inner tubes

Heat Mass Transfer

velocity and temperature were imposed at the inlet ofthe annulus and conditions of zero pressure was consid-ered at the outlet. Conditions of no slip were assumedat the inner and outer surfaces of the annulus. The so-lution is considered converged when the normalized re-sidual of continuity, momentum, and energy are less than10−4, 10−4 and 10−6, respectively for determining accurateresults.

Grid refinement study has been conducted by using differ-ent number of cells and study its effect on the accuracy of thenumerical solution and solutions convergence. The cells werehighly concentrated nears the walls of the annulus, re-gion of high flow and temperature gradients, to capturethe important features near the walls. Temperatures andvelocity distributions and heat transfer inside the annu-lus were obtained for the different numbers of cells. Theeffect of the number of cells and grid size on the wateroutlet temperature from the heat exchanger is shown inTable 2 for a typical test runs having different numberof cells for each coil. It can be observed that the wateroutlet temperature is converged at 1030807, 1,135,467,926,923, 891,123 and 888,803 number of cells for coil1, 2, 3, 4 and 5, respectively. Thus these numbers ofcells are sufficient to produce accurate results and canbe considered as grid independence. These number ofcells were utilized in all the numerical simulation of the pres-ent work.

4.2 Model validations

To validate the present numerical model, comparisons withprevious experimental works have been performed. This hasbeen conducted by adjustments of the present model to matchthe geometric, operating, boundary and thermal conditions ofthe previous related works. According to the author knowl-edge, the only available work for annulus flow of multi-tubesin tube helical coils is the experimental work of Nada et al.[21, 22] which was conducted for multi heater rods insidehelical tube. The number of turns of the helical coilswas 5 turns, a straight sections of lengths 35 cm wereexisting at the start and end of the helical coil. Thenumber of heater rods, heater rod(s) diameter, the annu-lus outer diameter, pitch between turns and the totallength of the helical coil were 3 rods (coil 3), 6 mm,23 mm, 25 mm and 3.5 m, respectively. The inner sur-face of the annulus (heater rods outer surface) wasmaintained at constant heat flux and the outer surfaceof the annulus was adiabatically thermally insulated. Forthe sake of comparison, the model was run under the sameoperating and geometric conditions of this experimental work.Comparison between the presented predicted Nusselt numberand Nada et al. [21, 22] measurements gives a fair agreementas shown in Fig. 3.

5 Results and discussions

Numerical runs have been conducted using water flowin the annuls for different flow, geometric and bound-ary conditions as given in Table 1. The aim of theruns was to study the effects of Reynolds number,heat flux and inner tubes numbers on the thermal per-formance of the coil, namely on heat transfer coeffi-cient, Nusselt number, and compactness factor, hAh

(the factor that measure performance and compactnessof heat exchangers).

N= 1

N= 5

N= 3

Fig. 6 Effect of number of inner tubes on the temperature contour(Re = 2920, q = 13.06 kW/m2)

Heat Mass Transfer

5.1 Effects of Reynolds number and heat flux

Figure 4 shows the temperature contour of the flow inside thehelical annulus at different heat flux and Reynolds number forN = 3 as an example. The figure shows the increase of the flowtemperature as it passes inside the annulus due to the contin-uous heating of the flow during its path in the annulus. Thefigure also shows that temperature level of the fluid contourincreases with increasing the heat flux and decreasing the fluidReynolds number. This can be attributed to the increase of theheat input to the fluid with increasing the heat flux and thedecrease of the mixing level leading with the decrease ofReynolds number and both lead to the increase of the heattransfer surface and fluid temperatures.

Figure 5 shows the variation of the average heat transfercoefficients versus Reynolds number with the input heat fluxas a parameter for coils 1–5, respectively. As shown in thefigure, the heat transfer coefficient increases with increasingReynolds number and is not affected by the heat flux. Thetrend is the same for all the tested coils; coils 1–5. The increaseof the heat transfer coefficient with the increase of theReynolds number is attributed to the increase of the flow ve-locity which cause an increase of the mixing levels inside theflow filed as well as the increase of the secondary flow insidethe annulus leading to the dropping in the temperature differ-ence between the flowing fluid and the rod heaters surfaces(see Fig. 4) and consequently improvement of the heat transfercoefficient. The trend is matched with the scientific physics

0 0.5 1 1.5 2 2.5 3

X (m)

1200

1400

1600

1800

2000

2200

hx

(W

/m2K

)

N= 1

N= 5

N= 3

q 10.5 kW/m2

Re 1870

hx= 1406 W/m2K

1408

1410

1412

1414

1431

1470

1512

1465

1428

14191503

1450

1563

1484

1525

1426

1453

1500

1549 1580

(a)

(a)

(b)

Fig. 7 Local heat transfercoefficient: along the coil atq = 10.5 kW/m2 K, Re = 1870 fordifferent number of inner tubes

Heat Mass Transfer

where the heat transfers coefficient increases in power orderwith the increases of Reynolds number. Fig. 5 also shows thatthe heat flux has negligible effect on the heat transfer coeffi-cient at constant Reynolds number. This may be attributed tothe increase of the temperature difference between the rodsheater surface and the flowing fluid with increasing the heatflux (see Fig. 4) with the same rate of increasing the heat fluxwhich leads to a constant heat transfer coefficient. Referring tothe studied range of Reynolds number, one can notice that atthis range in a flow inside plain tube and straight annulus, theflow regime lies in the mixed convection range. The indepen-dence of the heat transfer coefficient on the heat flux for thepresent helical annulus for this range of Re can attributed tothe high mixing level that exists at low Reynolds number incase of flow in helical annulus. As Nusselt number and thecompactness factor (hAh) are linearly depends on heat transfercoefficient for a certain coil (see Eq. 16), the results showthat Nusselt number and the compactness factor, havethe same trend of variation of the heat transfer coeffi-cient with Reynolds number and heat flux for the differentcoils.

5.2 Effects of inner tubes number (N)

The effect of inner tubes numbers on temperature contour andthe heat transfer coefficients are shown in Figs. 6 and 7 re-spectively. As shown in Fig. 6, the lowest temperature level ofthe contour occurs at N = 3 which leads to lower heat transfersurface temperature. The figure also shows that, at the firstturns of the coil at flow inlet (bottom of coil), the temperaturelevel of the contour increase with the increase of the numberof the inner tube (compare the cases for N = 5 and N = 1) andthis is attributed to the increase of the surface are of the heattransfer.

Figure 7 gives the variation of the local heat transfer coef-ficient along the length of the tube for different numbers of theinner tubes. As shown in the figure, the local heat transfercoefficient increases along the length of the tube. /this canbe attributed to the decrease of the temperature differencebetween the fluid and the tube surface with the increase ofthe length of the tube.

Figure 8 shows the increase of the average heat transfercoefficient with the increase of the number of the inner tubes.This may be attributed to the decay in temperature differencebetween the flow and heat transfer surface in the annuluscaused by the increase of the flow velocity, mixing level withincreasing the number of the inner tubes for the same heat fluxandmass flow rate. Figure 8 also shows that the increase of theaverage heat transfer coefficient with increasing number ofinner tubes converges with increasing Reynolds number untilit finishes at Re =10,000. This can be attributed to that athigher Reynolds number the effect of the number of the innertubes on the temperature difference decreases because the

effect of increasing Re overcome on the decrease of the tem-perature difference with increasing the number of inner tubes.

The compactness factor, hAh (the product of the heat trans-fer coefficient times the heat transfer surface area) is utilized topredict the thermal performance and the size compactness ofthe heat transfer devices. For the same volume occupied by theheat exchangers, increasing hAh means the increase of its ca-pability of heat transfer. In the present study the volume of theheat exchanger is fixed for all coils and equals to the volumeoccupied by the outer helical shell. Therefore, the coil whichgives higher hAh will be recommended. Figure 9 shows theeffect of the inner tubes number on hAh at the studied range ofReynolds number. As shown in the figure hAh increases with

100 1,000 10,000

Re

100

1000

10000

h (

W/m

2K

)

N= 1

N= 2

N= 3

N= 4

N= 5

Fig. 8 Effect of number of inner tubes on the heat transfer coefficient

100 1,000 10,000

Re

10

100

1000

hA

h(W

/K)

N= 1

N= 2

N= 3

N= 4

N= 5

Fig. 9 Effect of inner tubes numbers on hAh

Heat Mass Transfer

increasing the inner tubes number. This may be returned to theincrease of both of h (see Fig. 8) and Ah (see Eq. 9) with theincrease tubes numbers. This means that coil 5 is better thancoil 3 from point of heat transfer view but may be not theoptimal one form constructing and operating cost point ofview where more pumping power is needed. Therefore, opti-mization study including pumping power cost must beaddressed.

Effect of the inner tubes numbers on Nusselt number isshown in Fig. 10. As shown in the figure, Nusselt numberdecreases with increasing the tubes number. This may be

attributed based on Eqs. 8, 10 and 16 and Fig. 8. Increasingthe number of tubes increases heat transfer coefficient (seeFig. 8) and dramatically decrease the hydraulic diameterof the annulus Dh (see Eq. 8, 10). The decrease in Dh

overcomes on the increase of h to reduce Nusselt numberas per Eq. 18.

Figure 10 shows that Nusselt number varies with Re in theform of power functions for any number of tubes. All the dataof Fig. 10 are regressed using least square method to give acorrelation of Nusselt number in terms of Re and n. The bestfitting correlation is obtained on the following form as shownin Fig. 11

Nu ¼ 8:185Re0:121Pr0:227N−0:395 ð19Þ

Figure 12 shows the comparison between the prediction ofthis equation and the present results. As shown in the figure,the equation can predict 95% of the present results within anerror of 6%.

6 Summary and conclusions

CFD study of thermal performance of laminar flow inhelical annulus heat exchanger formed by multi-tubes intube helically coiled has been presented to explore theeffects of the geometric and fluid flow parameters onthe heat exchanger thermal performance. Different coilshaving different numbers of inner tubes, 1–5 tubes weresimulated. A correlation for predicting Nusselt numberas function of Reynolds number and the inner tubesnumber is predicted from the numerical results. The

0 10 20 30 40

Nu (Cal. results)

0

10

20

30

40

Nu

(Cor

r. re

sults

)

+6%

-6%

Fig. 12 Comparison between the present data and the prediction of Eq. 17

1 2 3 4 5

Re0.121

Pr0.272

N-0.396

0

10

20

30

40

Nu

Numerical correlation

Numerical results

8.185 Re0.121

Pr0.272

N-0.396

Fig. 11 Correlation of the present data

100 1,000 10,000

Re

10

100N

u

N= 1

N= 2

N= 3

N= 4

N= 5

Fig. 10 Effect of inner tubes numbers on Nusselt number

Heat Mass Transfer

conclusions of the present work are summarized asfollows:

& For all tested coils, the heat flux has approximately noeffect on h, hAh and Nuat the studied range of Re.

& Annulus formed by five inner tubes showed higher ther-mal performance (h, hAh and Nu) and bitter heat transfercharacteristics comparing with other coils other coils.

& Optimization study including constructing and op-erating costs is needed as a further study to detect theoptimal coil.

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Heat Mass Transfer