1

Experimental and Numerical study of a Conical Helically Coiled, Single Phase Heat Exchanger

Daniel Flórez-Orregoa,1

, Walter Ariasb, Héctor Velásquez

b.

a Polytechnic School, University of São Paulo, Av. Prof. Mello Moraes, 2231, São Paulo, Brazil. b Minas Faculty, National University of Colombia, Carrera 80 #65-223, Medellín, Colombia,

wariasr@unal.edu.co, hivelasq@gmail.com

Abstract:

Complex fluid dynamic and heat transfer process present in curved tube heat exchangers confer them some advantages over the straight tubes in terms of area to volume ratio and enhancement of heat and mass transfer coefficients. In this work, an analysis of the single phase heat transfer process in a conical helically coiled heat exchanger non-previously studied was carried out. An empirical correlation for the determination of average Nusselt number along the duct, with Reynolds number ranging between 4300 and 18600 has

been developed. Also, numerical simulations were performed using ANSYS FLUENT® 12.1 software, for

simultaneously solving the equations of mass, momentum and heat transport, using realizable κ-ε two equations turbulence model. Cross section velocity contours and vector components were compared with those of conventional helical tubes configurations (non-conical). It was found there are similarities in terms of the main flow skewness, but there are slight differences on the trajectories of core fluid particles in the secondary flow. Also, pressure drop calculated using Ito and White’s correlations were compared with numerical and experimental data, concluding that experimental data lies between those results predicted by the other authors’ correlations. Finally, numerical and experimental Nusselt number has been compared with the ones obtained using the correlations proposed by Seban and McLaughlin and by Xin and Ebadian for curved tubes. It was found that those correlations overestimate the values obtained for the conical helically coiled heat exchanger, whereas experimental and a numerical results presented a good agreement.

Keywords:

Single phase, Conical helically coiled, Heat exchanger, Nusselt Correlation, Friction factor coefficient.

1. Introduction

Complex fluid dynamics occurring in curved tubes have been an invariant issue of research for last

few decades [1]. The operating principle of the curved tubes and the benefits attributed to them over

the performance of the straight tubes can be summarized as follows: (a) generation of secondary

flow in the radial direction; (b) enhanced cross-sectional mixing; (c) reduction in axial dispersion

and (d) improved heat transfer coefficient [1]. Other advantages of helical tubes are their high ratio

of heat transfer area to occupied volume, their good adaptability to cylindrical shapes and an

excellent performance in the presence of thermal expansion, behaving as a spring. Secondary flow

is a consequence of the difference in axial momentum between fluid particles in the core and the

wall regions. The core fluid experiences a centrifugal force that pushes it to the outer wall of the

coil, from where is redirected towards the inner wall in two different streams throughout the upper

and lower walls of the tube, as it is shown in Fig. 1.

1 Corresponding author. Tel.: +55-11-974982315;

E-mail address: daflorezo@usp.br (D. Flórez-Orrego).

2

Figure 1. Secondary flow in a curved tube [1].

In the literature, it is suggested that the first investigations about the flow in curved geometries can

be found in the work of Thompson [2]. Williams et al [3] observed that the location of the

maximum axial flow velocity is located in the outermost zone of the tube wall and Eustice [4]

demonstrated the existence of a secondary flow by injecting ink into water flowing in a helical tube.

Later, Grindley and Gibson [5] studied the effect of the curvature in the flow during experiments on

the viscosity of air. Dean [6, 7] was the first to develop an analytical solution for fully developed

laminar flow in a curved tube of circular cross section. Using cylindrical coordinates , ,r z , the

continuity equation, Eq. (1), and Navier Stokes equations, Eqs. (2) to (4), as well as the non-

dimensional Dean number, Eq. (5) can be expressed as:

1

0zr

uru

r r z

(1)

2

2

2

r rr

uDu upu

Dt r r r

(2)

2

2, ,

rDu u u uG r z t u

Dt r r

with 1 p

Gr

(3)

2zz

Du pu

Dt z

(4)

0.5

0

H

U D DDe

D

(5)

Jeschke [8] was the first to report the experimental results of heat transfer in two coils for a

turbulent condition, developing an empirical correlation. Merkel [9], suggested that the convective

heat transfer coefficient in curved tubes is 1 3.5 HD D times greater than in straight tubes, where

D and HD are the tube and coil diameters, respectively. Heat transfer enhancement in helical coil

systems is also reported by Prabhanjan et al. [10].

Seban and McLaughlin [11] developed two correlations for the circumferential average heat transfer

coefficient for both laminar and turbulent regimes. They introduced the friction coefficient as a

modelling parameter for the Nusselt number. Dravid et al [12] studied the effect of secondary fluid

motion on laminar flow heat transfer in helically coiled tubes. They stated that at very short

distances from the beginning of the heat transfer zone, the thermal boundary layer is too thin to be

affected by the secondary flow field, which reaches its maximum intensity at some distance away

from the tube wall. The authors considered that neglecting that zone would make the design only

slightly conservative.

Austen et al [13] carried out the study of laminar flow and heat transfer in helically coiled tubes

with substantial pitch. Significant pitch effects were noted in the friction factor and Nusselt number

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at low Reynolds numbers. These effects were attributed to free convection, and they diminished as

Reynolds number increased.

Xin and Ebadian [14] presented an experimental assessment on heat transfer in helical tubes. The

authors explored two values of curvature for Re ranging from 5000 to 1.105, and Prandtl ranging

from 0.7 to 175. They proposed the following correlation 0.92 0.40.00619Re Pr 1 3.455 HNu D D .

Cioncolini et al [15] studied the curvature effects on the laminar to turbulent flow transition from

direct inspection of experimental Nusselt number profiles. Due to the coil curvature, the process of

turbulence emergence in the tested coils was found to be much more gradual and smooth than it was

in straight tubes, enhancing the effect as the coil curvature increases. By inducing the secondary

flow, the coil curvature increases both the hydraulic resistance and the heat transfer effectiveness,

with respect to straight tube flow.

Jayakumar et al [16] carried out experimental and CFD studies in a fluid-to-fluid heat exchanger in

order to compare the transport phenomena inside a helical coil for various boundary conditions.

They also compared the effect of using constant and temperature-dependent fluid properties in the

computational modelling of heat transfer. Standard κ-ε turbulence model was implemented. A

correlation in the form 0.9112 0.40.025 PrNu De was proposed.

Conte et al [17] performed numerical simulations to understand forced heat transfer in laminar fluid

flow using rectangular coiled tubes with circular cross-section and characterized by curvature and

torsion parameters. The focus was addressed on exploring the flow pattern and temperature

distribution through the tube. Later, Conte et al [18] compared the outside convective heat transfer

coefficient for conical and conventional helical coils. The results showed a better heat transfer

performance for the conical coil case, since an effective flow arrangement triggers a more intense

turbulence. No investigations were carried out for the inner convective heat transfer coefficient.

Kumar et al [19] modelled the fluid flow and heat transfer process in a tube-in-tube helically coiled

heat exchanger for different fluid flow rates. The renormalization group (RNG) k–ε model is used

to model the turbulent flow in the heat exchanger. New empirical correlations were developed for

hydrodynamic and heat-transfer processes in the outer tube.

Di Piazza et al [20] presented a systematic attempt to assess the applicability of alternative

turbulence models for the prediction of pressure drop and heat transfer in coiled tubes. They

concluded that the SST eddy viscosity/eddy diffusivity model and the second order Reynolds

stress model give comparable results for the friction factor f correlated by Ito [21] and the

Nusselt number correlated by Pethukov’s momentum-heat transfer analogy [20].

Various correlations for the pressure drop friction factor, f, were proposed by Ito [21], White [22]

[23], Hart et al [24] and Mishra and Gupta [25]. The former is the most popular correlation.

An extensive review about the flow and heat transfer characteristics is provided by Naphon and

Wongwises [26]. The latest review was published by Vashisth et al [1].

Currently, the various types of curved tube geometries are classified as follows: (a) torus (constant

curvature and zero pitch), (b) helically coiled tube (constant curvature and pitch), (c) serpentine

tubes (periodic curved tubes with zero pitch) with bends or elbows, (d) spirals (Archimedean

spirals), and (d) twisted tubes. Other different curved chaotic configurations have been developed as

a course of technological development [27] [28]. At the author’s knowledge, only a work with a

conical helically coiled heat exchanger has been published, and the analysis was focused on the heat

transfer in the outer side of the tube [18].

At this work, both experimental and numerical investigations were performed for the purpose of

investigating the heat transfer in a conical helically coiled heat exchanger. The simulation method is

concerned with laminar and turbulent flow of an incompressible fluid (water) using commercial

computational fluid dynamics (CFD) software ANSYS® Fluent 12.1; the velocity profiles are

compared with those reported for non-conical coiled heat exchangers (Fig. 1). Experimental average

Nusselt number and pressure drop friction factors f are compared with those predicted using

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correlations proposed by other authors. Finally, a Nusselt number correlation for the conical

helically coiled heat exchanger used in this study is proposed.

2. Characteristics of the conical helically coil heat exchanger

2.1. Geometrical characteristics

Figure 2 shows the characteristics of the conical helical coil. The tube has an inner diameter of

D=7.904 mm and wall thickness of e=0.813 mm. The conical coil has major base and minor base

diameters of DMH = 150 mm and DmH = 75 mm, respectively (measured between the centers of the

tube). Meanwhile, the distance between two adjacent turns, called pitch, is the sum of inner

diameter D plus twice wall thickness. The mean coil diameter is defined as the average between

DMH and DmH, namely DH = 112.5 mm. The ratio of coil diameter to tube diameter (DH/D) is called

curvature ratio, and is equal to λ = 14.23. Coil length (L) and tube length (LP) are 400 mm and

12000 mm, respectively.

Figure 2. Coil geometry parameters (units in millimetres).

2.2. Critical Reynolds number

Similar to Reynolds number (Re) in straight tubes flow, another parameter used to characterize the

flow in a helical tube is Dean number, De, which was defined as Re HDe D D . The curvature of

the coil governs the centrifugal force while the pitch influences the torsion to which the fluid is

subjected to. The centrifugal force results in the development of a secondary flow. Many

researchers have found that transition from laminar to turbulent flow is delayed because of the

presence of secondary flow [1]. They have correlated the critical Reynolds number (or the Reynolds

number for which laminar to turbulent flow transition take place) as a function of curvature ratio.

Ito [21] derived a correlation to calculate the critical Reynolds number at which transition occurs,

given by Eq. (6): 0.6Re 2000 1 13.2c

(6)

valid in the range of 5 2000 . Srinivasan et al [29] proposed the following correlation for the

critical Reynolds number in curved tubes, Eq. (7):

0.5Re 2100 1 12c (7)

valid in the range of 7.5 100 . Cioncolini et al [30] have found that, for low values of the

curvature ratio (λ<24), an no abrupt transition from laminar to turbulent flow was observed. In

those cases, the friction factor decreased monotonically with Reynolds number, and the transition to

5

turbulence is indicated only by a change in the slope of the f - Re curve, occurring at a Reynolds

number which the authors approximated by the correlation: 0.47Re 30000c

(8)

with 7 24 .

3. Experimental set up

3.1. Experimental facility

The material used to construct the conical helically coiled heat exchanger is copper tubing. Winding

is manually performed using a special wood matrix for accomplishing the desired major and minor

diameters of the coil. Distortion of the cross section is minimized used sand filling. The conical

helically coil is provided with straight entry and exit hydrodynamic lengths. An entry length, about

21 tube diameters, is used as a hydrodynamic developing length, while the exit length is about 40

tube diameters. Both extensions are tangentially attached to the coil, which lies horizontally.

The test section of the coil is enclosed in a heat insulated AISI 430 stainless steel jacket, covered

with stone wool. The power heat supply is provided by burning liquefied petroleum gas (LPG) and

controlled by regulating the pressure at the burner inlet. Combustion gases are led through the inner

chamber of the coil and left it through a chimney. It is assumed that the flame uniformly irradiate

the surface of the coil, although condensed vapour water and sooting on its surface are also

observed. This fact can slightly alters constant wall heat flux assumption, but it is still reasonable to

assume a linear increase in the bulk temperature along the whole length, which is checked with

thermocouples attached to the outer wall of the tube (See Fig. 3).

The cold fluid enters the tube from the connection at the minor diameter of the coil and the hot fluid

leaves the tube through the connection at the major diameter of the coil, and finally it is discharged

into a water reservoir. A counter-flow heat transfer process between water and combustion product

gases is achieved (Fig. 3). A needle valve regulates the flow rate while it is measured by a

calibrated rotameter at the fluid inlet section. Pressure and temperature of the cold and hot fluids are

measured using a bourdon manometer and an immersion K-type thermocouple whose values are

available on digital displays. Before the test, the thermocouples are calibrated in situ at the water

supply temperature. Tube wall temperatures are monitored by six contact K-type thermocouples

separated each 6.7cm along the coil height. The details of the facility are given in Fig. 3.

Figure 3. Scheme of the experimental facility.

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3.2. Experimental procedure

The tests are conducted using cold water coming from the local water supply service at 295K. For

each test, cold water flow rate and heat supply are fixed and before any data is recorded, the system

is allowed to approach the steady state. Then, the cold water flow rate is increased keeping the same

heat supply and repeating the experiment. After that, heat supply is increased and new increments

of the cold water flow rate are carried out, so that different heat transfer rates, pressure drop

conditions and flow regimes are created. The temperature and pressure at the inlet and outlet fluids,

as well as the temperature at each position in the outer wall of the tube are recorded six times and

averaged in the period of time. Besides of the rotameter, the mass flow rates are also determined by

collecting the flow into a graduated vessel (600 ml) while recording the time, as an alternative way

of measurement. All the thermodynamic and transport properties of water are calculated using the

SteamTab™ software, developed by the Chemicalogic Corporation for the thermodynamic and

transport properties of water and steam [31]. In the present experimental work the heat transfer

coefficient and heat transfer rates are calculated based on the measured pressure, temperature and

flow rate. The heat transferred to the cold water flowing inside the conical helically coiled heat

exchanger is calculated from Eq. (9):

2 1 2 1Q m i m i i i i

(9)

where is calculated as an average of the outlet and inlet densities of the fluid, being a function of

pressure and temperature. The heat transfer coefficients are calculated using Eq. (10):

avg

Qh

A T

where

,2 ,2 ,1 ,1

2

w b w b

avg

T T T TT

(10)

and where w, b, 2, and 1 stands for wall, bulk, outlet and inlet, respectively. Average heat transfer

coefficients are computed in a non-dimensional form given by Nusselt number, Eq. (11):

f w f avg

hD Q DNu

k A k T

(11)

where wA and fk are the internal heat transfer area of the tube and the fluid conductivity,

respectively. The later one is calculated as an average of the inlet and outlet fluid conductivities.

The Reynolds and Prandtl numbers are defined in the conventional way as shown in Eq. (12) and

(13):

4Re

f

m

D

(12)

Prp f

f

c

k

(13)

Where f and pc are the fluid dynamic viscosity and heat capacity calculated as an average of the

inlet and outlet fluid properties.

By varying the wall heat flux and mass flow rate, a total of 25 experiments are carried out. Heat

supply ranges between 5000 and 17000 W and volumetric flow rate varies between 2.50x10-5

and

8.17x10-5

m3/s, or equivalently 1.5 and 4.9 l/min. Reynolds number ranges between 4300 and

18600, whereas Prandtl number ranges between 2 < Pr < 6.

The following simplifications are assumed: 1) Steady state flow, 2) Fully hydrodynamic and

thermal developed flow; 3) Uniform wall heat flux; 4) Axial conduction in the tube wall and fluid is

neglected and 5) Peripheral temperature is uniform [32] [12]. The later assumption is valid if it is

considered the thickness and thermal conductivity of the tube. Peripheral differences diminishes

even more with high mass flow rates [13].

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The difference between the inner and outer wall temperatures of the tube can be calculated

assuming one-dimensional (radial) heat conduction throughout the tube thickness, a uniform

volumetric rate of heat generation in the tube and an insulated boundary condition at the outer wall

[13]. Solving the appropriate conduction equation with those boundary conditions, it can be easily

shown that the temperature difference across the wall at any axial location is given by Eq. (14):

2 2 2

, , 2 ln4

ow o w i o o i

w i

rqT T r r r

k r

(14)

Where the uniform rate of heat generation can be calculated as 2 2

T o iq Q L r r .

4. Numerical Set up

A CFD methodology has been used to investigate the performance of the conical helically coiled

heat exchanger. In this research work, a commercial CFD package ANSYS® Fluent v12.1 (double

precision, 3D version) and a computer with Intel® Core™ i5-2430M Processor (3M Cache, 2.40

GHz) with 4GB RAM are used; each run has taken 12 hours.

4.1. Computational grid

The simulation geometry is created using SOLIDWORKS® 2009 mechanical design software and

then exported to ANSYS Fluent as Parasolid geometry modelling format. The meshing is achieved

using the ANSYS Fluent Meshing module. In this model, the tube wall thickness is not considered

and only the fluid control volume is modelled. A structured grid with 5 inflation layers is used to

refine the mesh near the surface of the control volume. On the other hand, an unstructured mesh is

used in the interior volume cells. The grid for the analysis domain is shown in Fig. 4.

This grid is composed of 3.499.443 nodes and 3.312.000 elements. The maximum skewness

reported is 0.4852 and the minimum is 0.01121. This implies good-to-excellent cell quality. The

grid given in Fig. 4 is chosen because a further refinement does not result in a reduction of mass and

energy residual errors.

Fig. 4. Grid used in the analysis of the fluid flow and heat transfer process.

4.2. Boundary conditions and CFD modelling

Four representative heat transfer and flow conditions are numerically analysed (Table 1). The

steady state, pressure-based type solver with an absolute velocity formulation is used. In both

laminar and turbulent cases, energy, momentum, and turbulence model (if applicable) equations are

solved. The environmental operating conditions are defined as 295K and 0 Pascal absolute

operating pressure.

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For thermal equation, constant wall heat flux boundary condition is stated at the interface between

the wall and the fluid (See Fig. 5). For momentum equation, the wall is treated as a non-slip,

adiabatic and smooth wall. In all the cases, the cold fluid inlet temperature is kept as constant at

295K and the analyses are carried out varying the flow rate. At the fluid inlet, velocity inlet

boundary condition is specified. For the hot fluid outlet, pressure outlet boundary condition at zero

back pressure is specified. The acceleration of gravity is considered as 9.8m/s2, oriented to –X

coordinate direction (Fig. 5).

Fig. 5. Specified boundary conditions.

For the coil used in this study, the critical Reynolds number at which the transition from laminar to

turbulent flow regime occurs is calculated using Ito’s correlation [21] and it is equal to Re 7679c .

According to this, only the fourth condition shown in Table 1 represents a laminar flow regime and,

in this case, the laminar model in ANSYS Fluent simulation is considered.

For the turbulent regimes simulations, the Realizable turbulence model with Standard Wall

function is used. An immediate benefit of the realizable model is that it is likely to provide

superior performance for flows involving rotation, boundary layers under strong adverse pressure

gradients, separation, and recirculation. Initial studies have shown that the Realizable model

provides the best performance of all the model versions for several validations of separated

flows and complex secondary flow features [33]. Model constants are defined as: 1 1.44C ,

2 1.9C , 1.0 , 1.2 [33]. For fully-developed turbulent internal flow, the Intensity I and

Hydraulic Diameter D are specified. The turbulence intensity at the core of a fully-developed flow

is estimated from the Eq. (15), derived from an empirical correlation in circular ducts [34]:

18

'0.16 Re

uI

u

(15)

Turbulence intensity of 1% or less is considered as low and turbulence intensity greater than 10% is

considered as high. Turbulent Kinetic energy and turbulent dissipation rate are calculated

using the Eq. (16) and (17) [34]:

23

2avgu I

(16)

3 23 4C

(17)

Where avgu is the mean velocity inlet, is the length scale (= 0.07D) and C is an empirical

constant specified in the turbulence model, approximately 0.09. The turbulence length scale, , is a

physical quantity related to the size of the largest eddies that contain the energy in turbulent flows.

In fully-developed flows in ducts, it is restricted by the size of the duct, since the turbulent eddies

cannot be larger than the duct. The factor of 0.07 is based on the maximum value of the mixing

length in fully-developed turbulent tube flows [35]. In all the turbulent cases a turbulent length scale

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of 0.0005533m and a hydraulic diameter of 0.007904m are considered. Turbulence Intensity,

Turbulent kinetic energy and Turbulent dissipation rate are shown in Table 1.

Table 1. Boundary and Cell zone conditions.

Condition

Volumetric

flow rate

m3/s (l/min)

Mean

velocity

inlet

m/s

Expected

outlet

temp.

K

Wall

heat flux

W/m2

Reynolds

Number

(Flow

Regime)

Turbulence

intensity

%

Turbulent

Kinetic

energy

m2/s

2

Turbulent

dissipation

rate m2/s

3

1 7.50 x 10

-5

(4.5) 1.5285 339 37881

17248

(Turb.) 4.73 0.00783 0.20573

2 5.83 x 10

-5

(3.5) 1.1888 359 42639

14348

(Turb.) 4.84 0.00496 0.10371

3 4.17 x 10

-5

(2.5) 0.8491 358 29992

10220

(Turb.) 5.05 0.00275 0.04292

4 2.50 x 10

-5

(1.5) 0.5095 358 17996

6132

(Lam.) --- --- ---

At this work, thermal and transport temperature-dependent properties of water are used, as it was

proposed by [36]. For modelling temperature-dependent properties, polynomial functions, Eqs. (18)

to (21) are programmed in ANSYS Fluent User Defined Function module. The governing equations

of mass, momentum, energy and turbulence are solved with the fluid properties evaluated at the cell

temperatures.

2 3 40.33158 0.0037524 1.6028 5 3.055 8 2.1897 11T T e T e T e T

(18)

2 31227.8 3.0726 0.011778 1.5629 5T T T e T

(19)

2 31.0294 0.010879 2.261 5 1.5362 8k T T e T e T

(20)

2 34631.9 1.478 0.0031078 1.1105 5pc T T T e T

(21)

These relationships are obtained by a polynomial regression analysis using MATLAB® 9. In the

relationships above, temperature is specified in K [37].

Pressure-velocity coupling is carried out using SIMPLEC scheme with skewness correction factor

equal to 1. Momentum, energy, turbulence kinetic energy and turbulence dissipation rate equations

are discretised using second order upwind scheme. Pressure is discretised using linear scheme.

Convergence criteria used for continuity, velocity, k and ε are 1.0e−5

. Meanwhile, convergence

criterion for energy balance is 1.0e−7

. A mass flow rate surface monitor at the pressure outlet

boundary condition is used, obtaining convergence at each case studied. Under-relaxation factors

are fixed as 0.3 for pressure, 1.0 for density, 0.7 for momentum, 1.0 for body force, 0.8 for both

turbulent kinetic energy and turbulence dissipation rate, and 1.0 for energy.

5. Results and Discussion

5.1. Correlation of Experimental Data

Because there is no data in the open literature on the heat transfer process and flow characteristics

for a conical helically coiled heat exchanger, the results predicted using the present correlation are

compared with our experimental data.

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5.1.1. Experimental Nusselt number

Experiments include both laminar and turbulent flow conditions and Reynolds number varies

between 4300 and 18600, whereas Prandtl number ranges from 2 to 6. According to this, Dean

number ranges between 1000 and 6000. Figure 6 shows the variation of the experimental average

Nusselt number with the Reynolds number. As it was expected, the average Nusselt number

increases as increasing Reynolds number, since the first depends directly on the capacity of the fluid

to remove the heat conducted through the tube thickness.

Based on the nature of the correlations available in the literature, it is assumed that the Nusselt

number for the conical helically coiled heat exchanger could be calculated as a function of

Reynolds and Prandtl number in the form Re Prm nNu C , where C and m are constants to be

determined. The index of the Prandtl number, n, is chosen for being equal to 0.4. After a function

linearization and using Linear Regression analysis in MATLAB 9 software for the data shown in

Fig. 6, the following correlation to estimate the Nusselt number is proposed, which is applicable for

4300 < Re < 18600: 0.82 0.40.00797 Re PrNu

4300 Re 18600

2 Pr 6

(22)

Fig 6. Variation of experimental average Nusselt number as a function of average Reynolds and

Prandtl number.

The proposed correlation has been successfully validated using the experimental data. As it can be

seen from Fig. 7, the experimental and the average Nusselt number calculated using Eq. (22)

present a maximum deviation of ±23%.

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Fig. 7. Calculated versus Experimental Nusselt number.

5.1.2. Experimental friction factor coefficient

When the experimental data for Darcy’s friction factor at turbulent regimes is analysed and

compared it with the ones calculated using White’s [22] and Ito’s [21] correlations (See Fig.12), it

is found that for low values of Reynolds number, White correlation seems to over-predict the

friction factors obtained for the prototype used in this experiment, whereas Ito’s correlation

presents a maximum deviation of only 6%. For Reynolds numbers between 10000 and 14000, the

effect is contrary and those correlations seem under-predict the experimental friction factor, but

White’s correlation presents the lowest difference, with a maximum deviation of 9%. Finally, at

high Reynolds numbers (Re > 14000), experimental data fall between the predicted values given by

both Ito’s and White’s correlations. Since friction factor is dependent on both Reynolds number and

curvature ratio, the tapering effect for the conical helically coiled heat exchanger could be one the

causes for the observed deviations, due to those effects are not considered in the other authors’

correlations. As it can be seen from Fig. 12, scattering is high and no correlation is proposed at this

research work for friction factor values.

5.2. Numerical Results

5.2.1. Velocity contours

Cross sectional velocity magnitude contours at different angles along the first turn of the coil for a

flow rate condition of 7.50x10-5

m3/s (4.5 l/min) are shown in Fig. 8. The measurement of the angle

starts from 0 at the joint between the straight (tangential) entry section of the tube and the first

turn of the coil. As indicated for the contour 0°, the left and right sides of the circles in Figs. 8 and 9

correspond to the inner (i) and outer (o) walls of the helical coiled heat exchanger, respectively.

Figure 8 corresponds to a turbulent flow, whereas Fig. 9 shows the case of laminar flow condition

with a flow rate of 2.50x10-5

m3/s (1.5 l/min).

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i 0° o 30° 60° 90°

120° 150° 180° 210°

240° 270° 300° 330°

360°

Fig. 8. Velocity magnitude contours (m/s) at the entry section for 7.50x10-5

m3/s (4.5 l/min).

It is observed that after 300°, the velocity contour shows a fully developed flow. At 0 , cold

fluid enters the coil and, in a first instance, a typical parabolic velocity contour is observed, owed to

the fully developed flow achieved at the inlet straight section. As the fluid flows through the coil,

the flow becomes developed due to the centrifugal and torsional effects induced by the helical

nature of the coil. Unlike for fluid flow through a straight tube, it can be readily seen that the

highest velocity is skewed to nearby the outer wall of the coil. Furthermore, near the tube walls, a

velocity gradient varying from of zero to a core velocity value is obtained, since a nonslip wall

condition is assumed.

13

i 0° o 30° 60° 90°

120° 150° 180° 210°

240° 270° 300° 330°

360° 390°

Fig. 9. Velocity magnitude contours of (m/s) at the entry section for 2.50x10-5

m3/s (1.5 l/min).

Once the fully developed velocity contour is reached, skewness is maintained until the fluid reaches

the outlet cross section. After that, the flow gets the straight outlet tube and the centrifugal effect

tends to vanishes and once again a parabolic profile is achieved. These results agree with those

reported by other authors [1]. For low Dean numbers, the axial velocity profile is almost parabolic

and unaltered from the fully developed straight tube flow, but as the Dean number increases, the

maximum velocity began to be skewed toward the outer periphery and centrifugal force, secondary

flow and, consequently, mixing effect are intensified. Besides, since Dean number is a function of

the coil diameter, in the present experiment it is expected to vary along the coil length inversely

proportional to 0.5

HD . So the greater the tapering of the conical helically coiled heat exchanger, the

more pronounced the Dean number variation along the tube length. This effect is not present in

conventional coiled tubes since coil diameter is a constant. When the major diameter of the coil

tends to infinite maintaining the minor coil diameter and the coil length as constants, the conical

helically coiled heat exchanger becomes an Archimedean spiral and the torsional effect of the pitch

disappears. On the other hand, if the mean coil diameter tends to infinite (that is, the ratioHD D

tends to zero), the flow characteristics are the same of straight tubes and Dean number tends to zero.

In such case, mixing and heat transport enhancing advantages attributed to curved tubes are

negligible.

14

The combined effect of curvature, pitch and tapering in the secondary flow is shown in Fig. 10. In a

fully developed flow, the velocity vector components in a XZ-plane (See axis orientation, Fig. 5)

are compared for each simulated condition shown in Table 1. The inner (i) and outer (o) notation

follows the convention used in Figs. 8 and 9. As it can be seen from Fig. 10, the velocity field in the

secondary flow is tilted in the same direction that the generatrices of the cone do, approximately 45°

respect to the cone axis, although not with the same angle of inclination of the generatrices, which

is near 5° respect to the cone axis. Differently than conventional -non conical- helically coiled tubes

(Fig. 1), for which pitch and curvature ratio effects have been widely studied in the literature, in

conical helically tubes the tapering effect on the flow pattern and heat transfer process has not been

previously reported and must be assessed taking into accounting the variation of the instantaneous

curvature ratio.

In Fig. 10, it is also observed that near the upper and lower periphery of the tube wall, the velocity

vector components in XZ-plane are greater compared with the velocity vector components at the

core fluid. This indicates a stronger influence of the main flow (normal to XZ plane) at the core

region of the fluid, but also a weaker effect near the tube walls than that imposed by secondary

flow. On the right and left points in the periphery of the tube cross section shown in Fig. 10,

velocity vector components in XZ-plane are almost negligible (coinciding with the minimum and

maximum axial velocity, normal to XZ-plane). They increase tangentially along to the tube

periphery, starting from the left (outer) point, reach a maximum value at the top point (or bottom

point) of the periphery, and decrease to a negligible value at the right (inner) point of the tube cross

section. At this point, the fluid flows almost diametrically through the core and, with an oblique

path as described above, it reaches the outer wall of the tube. Finally, it is pointed out that a more

distorted secondary flow is presented at the condition of laminar flow, i. e., 2.50x10-5

m3/s (1.5

l/min), likely due to a weaker effect of the centrifugal force and a stronger influence of the main

flow.

o 2,5000x10

-5 m

3/s

(1.5 l/min) i

4,1667x10-5

m3/s

(2.5 l/min)

5,8333x10

-5 m

3/s

(3.5 l/min)

7,5000x10-5

m3/s

(4.5 l/min)

Fig. 10. Velocity vectors components in XZ-plane for the different flow conditions presented in

Table 1.

5.2.2. Temperature profiles

15

Temperature profiles along the coil length for the cases 1 and 4 described in Table 1 are shown in

Fig.11. The temperature increasing along the tube axis shows a linear tendency, as well the pressure

drop along the tube length. At a given cross section, temperatures around the periphery of the tube

showed a slight variation with a maximum difference of 15°C for laminar condition (condition 4 –

Table 1) and 5°C for turbulent conditions (conditions 1, 2, 3 - Table 1). Analogously to velocity

contours given in Fig. 8 and 9, cross section temperature contours are also skewed, with the

innermost part of the coil heat exchanger being the hottest one and the outermost part reporting the

lowest temperature. That behavior was expected, since those points correspond to the minimum and

maximum axial velocities, respectively, which affects the local Nusselt number and, thus the heat

transfer coefficient, so decreasing or increasing the temperature difference between surface and

fluid. Points at the top and bottom of the tube cross section have temperatures varying between both

of them.

(a) (b)

Fig. 11. Contours of static temperature (K) a) condition 4 and b) condition 1, Table 1.

5.2.3. Friction factor coefficient

Darcy’s friction factor coefficient is numerically calculated using ANSYS Fluent as four times the

area weighted-average skin friction coefficient, fC , along the tube, according to Eq. (23), with

reff and reffu being the density and velocity defined as in the inlet boundary condition:

2

4 41

2

wDarcy f

reff reff

f C

u

(23)

Figure 12 shows the comparison between the numerical and experimental friction factor coefficients

with those obtained using Ito’s and White’s correlations.

16

Fig. 12. Comparison of numerical and experimental friction factor coefficients with those obtained

using Ito’s and White’s correlation, as a function of Reynolds number.

Even though numerically calculated friction factor coefficient tends to over predict the experimental

data at low Reynolds number, the mean deviation is 14%. At low turbulent Reynolds numbers (near

10000) no well agreement has been found between numerical and experimental results.

Furthermore, it can be seen that White’s correlation attempts to predict the numerical results better

than Ito´s correlation.

5.2.4. Numerical Nusselt number

In this section, the results of the CFD simulation are used to estimate the average Nusselt number

along the tube. Figure 13 shows the comparison of the experimental and numerical average Nusselt

number. It is found that the values are well within 30%, although experimental Nusselt numbers

slightly over-predict the numerical data.

Fig. 13. Comparison of average Nusselt number obtained in the present work and using other

author’s correlations.

17

For the sake of comparison, the Nusselt numbers calculated using the Seban-Mclaughlin [11], and

Xin-Ebadian [14] correlations are also reported in the Fig. 13. Seban-Mclaughlin’s correlation is

given by Eq. (24):

1 10

0.8 0.4 1 200.023 Re Pr ReH

DNu

D

(24)

While Xin-Ebadian’s correlation is given by Eq. (25):

0.92 0.40.00619 Re Pr 1 3.455H

DNu

D

(25)

Both correlation are valid in the range of 55000 Re 10 and Pr 5 . They are also used to estimate

the variation of the “local” Nusselt number at the perimeter of the tube cross section. The data for

calculating Nusselt numbers from Eqs. (24) and (25) and plotted in the Fig. 13 is calculated using

average values for the thermal and transport properties at the inlet and outlet of the tube, attempting

to obtain an average Nusselt number along the tube length. As it can be seen, the other author’s

correlations considerably over predict the experimental and numerical Nusselt number by at least

two and a half times. Since those correlations apply only for conventional – non conical – helically

coiled tubes, Eq. (24) and (25) are not appropriate to estimate the average Nusselt number in the

heat exchanger used at this work. Seban-Mclaughlin’s correlation is widely used to calculate the

forced convective heat transfer in singles phase heat exchangers as well as the forced convective

heat transfer contribution in two phase heat exchangers [1]. However, the tapering effect, whose

influence is not considered if using Seban-Mclaughlin’s or Xin-Ebadian’s correlations, can result in

great deviations, as it is validated using the experimental data in the present work.

6. Conclusions

In this work, the study of the flow and heat transfer process in a conical helically coiled heat

exchanger was carried out. Temperature dependent properties and constant wall heat flux boundary

condition were assumed. An empirical correlation for the average Nusselt number was proposed

from experimental data and a maximum deviation of 23% was found. Numerical simulations were

validated using the experimental results since no information is found for the studied geometry in

the literature. A comparison between the Nusselt numbers calculated using Seban-Mclaughlin and

Xin-Ebadian correlations and the experimental data showed that those correlations can overpredict

at least two and a half times the experimental values. Differently from the conventional –non-

conical– helical coils, for the conical helically coiled tube the velocity vector components of the

flow at the cross section (secondary flow) presented an appreciable tilt. Nevertheless, the velocity

contours are similar to the conventional helical coils, with the particles of the fluid near the outer

wall of the conical helically coiled heat exchanger going faster due to the unbalance in centrifugal

forces. Frictions factor coefficients were found to be included in the range predicted by Ito’s and

White’s friction factor correlations, while numerical results slightly overpredicted experimental

data. Further investigations should be focused on the effect of tapering in the Dean and local

Nusselt number, as well as in the effect of the pitch, instantaneous curvature ratio and position of

the conical helically coiled heat exchanger. The fluid-to-fluid heat transfer process using

combustion gases must be also further investigated. Deviations and errors between experimental

and numerical results can be attributed to the assumption of uniform flame radiation since sooting

and condensed combustion products (due to the use of LPG) modify the conditions for a constant

wall heat flux.

18

Acknowledgments

The authors want to thank to the National University of Colombia, Medellin, for allowing us to use

their facilities and the Research Management Office, campus Medellín -DIME- for the financing of

the research work, grant No. 100502001. Also, we want to thanks to the technicians from the

Thermal Machines and Welding Laboratories of the National University of Colombia. First author

also want to the National Agency of Petroleum, Natural Gas and Biofuels (ANP), Brazil, and its

Human Resources Program (PRH/ANP) grant No. 6000.0067392.11.4.

Nomenclature

Letter symbol:

A Area, m2

cp Heat capacity, J/(kg K)

D Tube diameter, m

Re HDe D D Dean number

HD Mean coil diameter, m

mHD Minor coil diameter, m

MHD Major coil diameter, m

e Wall thickness, m

f friction factor coefficient

h Average heat transfer coefficient, W/(m2 K)

I Turbulence intensity, %

i Specific enthalpy, J/kg

k Conductivity, W/(m K)

Length scale, m

TL Tube length

L Coil Length .

m Mass flow rate, kg/s

Nu Average Nusselt number

Pr Prandtl number

Q Heat flow rate, W

r Radius, m

Re Reynolds number

T Temperature difference, °C or K

T Temperature, °C or K

u Velocity, m/s

Volumetric flow rate, m3/s

19

Greek symbols

curvature ratio

dynamic viscosity, Pa s

Turbulent kinetic energy, m2/s

2

Turbulent dissipation rate, m2/s

3

density, kg/m3

Angle, degrees

Shear stress, Pa

Subscripts and superscripts

2 outlet

1 inlet

avg average

w wall

o outer

b bulk

i inner

f fluid

reff reference

References

[1] Vashisth, S., Nigam, K., Kumar V., A review on the potential applications of curved geometries

in process industry. Ind. Eng. Chem. Res., 2008. 47: p. 3291-3337.

[2] Thompson, J., On the origin of windings of river and alluvial planes, with remarks on the flow of

water round bends in pipes Proc. R. Soc. London. Ser. A 1876: p. 5.

[3] Williams, G.S.H., C. W.; Fenkell, G. H, Experiments at Detroit, Michigan on the effect of

curvature on the flow of water pipes. Trans. Am. Soc. CiV. Eng. , 1902. 47(1).

[4] Eustice, J., Experiments of streamline motion in curved pipes. Proc. R. Soc. London, Ser. A

1911. 85(119).

[5] Grindley, J.H.G., A, On the frictional resistance of air through a pipe. Proc. R. Soc. London,

Ser. A 1908. 80(114).

[6] Dean, W.R., Note on the motion of fluid in a curved pipe. Philos. Mag. , 1927. 4(208).

[7] Dean, W.R., The streamline motion of fluid in a curved pipe. Philos. Mag., 1928. 7(673).

[8] Jeschke, D., Heat transfer and pressure loss in coiled pipes. Ergaenzungsheft Z. Ver. Disch. Ing.

, 1925. 68: p. 24-28.

[9] Merkel, E., Die grundlagen Der Warmeubertragung. 1927: p. 51.

[10] Prabhanjan D, R.G., Rennie T., Comparison of heat transfer rates between a straight tube

heat exchanger and a helically coiled heat exchanger. Int Commun Heat Mass Transfer, 2002.

29: p. 185-191.

[11] R.A. Seban, E.F.M., Heat transfer in tube coils with laminar and turbulent flow. Int. J. Heat

Mass Transf, 1963. 6: p. 387.

[12] Dravid A, S.K., Merrill E, Brian P, Effect of secondary fluid motion on laminar flow heat

transfer in helically coiled tubes. AiChE Journal, 1971. 27(5): p. 1114-1122.

[13] Austen, D., Soliman, H., Laminar Flow and Heat Transfer in Helically Coiled Tubes with

Substantial Pitch. Experimental Thermal and Fluid Science 1988. 1: p. 183-194.

20

[14] R.C. Xin, M.A.E., The effects of Prandtl numbers on local and average convective heat

transfer characteristics in helical pipes. J. Heat Transf. , 1997. 119: p. 467–473.

[15] Cioncolini, A., Santini, L,, On the laminar to turbulent flow transition in diabatic helically

coiled pipe flow. Experimental Thermal and Fluid Science, 2006. 30: p. 653–661.

[16] Jayakumar J, M.S., Mandal P, Experimental and CFD estimations of heat transfer in helically

coiled heat exchangers. Chemical Engineering Research And Design, 2008. 86.

[17] Conté, I., Peng, X, , Numerical and experimental investigations of heat transfer performance

of rectangular coil heat exchangers. Applied Thermal Engineering, 2009. 29: p. 1799–1808.

[18] Conté, I., Peng, X, Wang, B, , Numerical Investigation of Forced Fluid Flow and Heat

Transfer from Conically Coiled Pipes. Numerical Heat Transfer, Part A, 2008. 53: p. 945–965.

[19] Kumar, V., Faizee B, Mridha, M, Nigam, K, , Numerical studies of a tube-in-tube helically

coiled heat exchanger,. Chemical Engineering and Processing, 2008. 47: p. 2287–2295.

[20] Di Piazza, I., Ciofalo, M,, Numerical prediction of turbulent flow and heat transfer in

helically coiled pipes. International Journal of Thermal Sciences, 2010. 49: p. 653–663.

[21] Ito, H., Friction factors for turbulent flow in curved pipes. J. Basic Eng., 1959. 81: p. 123–

134.

[22] White, C.M., Fluid friction and its relation to heat transfer. Trans. Inst. Chem. Eng. , 1932.

10: p. 66-86.

[23] White, C.M., Streamline Flow through Curved Pipes. Proc. R. Soc. Lond. , 1929. 123: p. 645-

663.

[24] Hart, J.E., J.; Hamersma, P. J, Single and two-phase flow through helically coiled tubes.

Chem. Eng. Sci. , 1988. 45(4): p. 775.

[25] P. Mishra, S.N.G., Momentum transfer in curved pipes: Newtonian fluids. Ind. Eng. Chem.

Des. Dev. , 1979. 18: p. 130–137.

[26] Naphon, P., Wongwises,S., A review of heat transfer and flow characteristics in curved tubes.

Renewbable and sutainable energy reviews, 2006. 10: p. 463-490.

[27] Slominski, C., Stuart W. Churchill,, Helical and Lemniscate Tubular Reactors. Ind. Eng.

Chem. Res. , 2011. 50: p. 8842–8850.

[28] Vashisth, S., Nigam, K, Prediction of flow profiles and interfacial fenomena for two phase

flow in coiled tubes. Chemical Engineering and Process, 2008. 48(29): p. 452-463.

[29] S. Srinivasan, S.N., F.A. Holland, Friction factors for coils. Trans. Inst. Chem. Eng. , 1970.

48: p. T156–T161.

[30] A. Cioncolini, L.S., An experimental investigation regarding the laminar to turbulent flow

transition in helically coiled pipes. Exp. Thermal Fluid Sci, 2006. 30: p. 367–380.

[31] Chemicalogic Corporation. 30/11/2011]; Available from: http://www.chemicalogic.com

[32] Futagami, K., Aoyama, Y, , Laminar heat transfer in helically coiled tube. Int. J. Heat Mass

Transfer, 1988. 31(2): p. 387-396.

[33] Shih, H., Liou, W Shabbir, A Yang, Z, Zhu, Y, A, A New k-e Eddy-Viscosity Model for High

Reynolds Number Turbulent Flows - Model Development and Validation. Computers Fluids,

1995. 24(3): p. 227-238.

[34] Naphon P, S.J., Effect of curvature ratios on the heat transfer and flow developments in the

horizontal spirally coiled tubes. Journal of Heat and Mass Transfer 2007. 50: p. 444–451.

[35] ANSYS FLUENT 12.0/12.1 Documentation. 11/11/11]; Available from:

https://www.sharcnet.ca/Software/Fluent12/index.htm

[36] Kumar V, G.P., and Nigam K, Fluid Flow and Heat Transfer in Curved Tubes with

Temperature-Dependent Properties. Ind. Eng. Chem. Res., 2007. 46: p. 3226-3236.

[37] Jayakumar J, M.S., Mandala J, Iyer K, Vijayanb V, CFD analysis of single-phase flows inside

helically coiled tubes. Computers and Chemical Engineering, 2010. 34: p. 430–446