Heat Mass Transfer DOI 10.1007/s00231-017-2127-z

ORIGINAL

Turbulent flow regime in coiled tubes: local heat-transfer coefficient F. Bozzoli 1,2

&

L. Cattani 2 & A. Mocerino 1 & S. Rainieri 1,2

Received: 23 March 2017 / Accepted: 27 July 2017 # Springer-Verlag GmbH Germany 2017

Abstract Wall curvature represents a widely adopted technique for enhancing heat transfer: the fluid flowing inside a coiled pipe experiences the centrifugal force and this phenomenon induces local maxima in the velocity distribution that locally increase the temperature gradients at the wall by enhancing the heat transfer both in the laminar and in the turbulent flow regime. Consequently, the distribution of the velocity field over the cross-section of the tube is strongly uneven thus leading to significant variations along the circumferential angular coordinate of the convective heat-transfer coefficient at the wall internal surface: in particular, it shows higher values at the outer bend side of the coil than at the inner bend side. The aim of the present work is to estimate experimentally the local convective heat-transfer coefficient at the fluid wall interface in coiled tubes when turbulent flow regime occurs. In particular, the temperature distribution maps on the external coil wall are employed as input data of the inverse heat conduction problem in the wall and a solution approach based on the Tikhonov regularisation is implemented. The results, obtained with water as working fluid, are focused on the fully developed region in the turbulent flow regime in the Reynolds number range of 5000 to 12,000. For the sake of completeness, the overall efficiency of the coiled tubes under test is assessed under a first-law performance evaluation criterion.

* F. Bozzoli [email protected]

1

Department of Engineering and Architecture, University of Parma, Parco Area delle Scienze 181/A, I-43124 Parma, Italy

2

SITEIA.PARMA Interdepartmental Centre, University of Parma, Parco Area delle Scienze 181/A, I-43124 Parma, Italy

1 Introduction In order to save in materials and energy use, adopting techniques of heat transfer enhancement is mandatory in the design of commercial heat exchangers. Enhancement techniques can be separated into two categories: passive and active. Passive methods require no direct application of external power and they usually employ special surface geometries, which cause heat transfer enhancement. On the other hand, active schemes (e.g. electromagnetic fields and surface vibration) require external power for operation [1]. Passive techniques are commercially more attractive because no power is required to facilitate the enhancement and among them treated surfaces, rough surfaces, displaced enhancement devices, swirl-flow devices, surface-tension devices, coiled tubes, or flow additives are found [2]. Wall curvature is one of the most frequently used passive techniques; its effectiveness occurs because it gives origin to the centrifugal force in the fluid: this phenomenon induces local maxima in the velocity distribution that locally increase the temperature gradients at the wall by maximising the heat transfer [3–5]. Dean [6] solved the simplified Navier–Stokes equations for a coiled pipe of small curvature showing that the flow is governed by the Dean number De = Re·δ1/2, where Re is the Reynolds number and δ is the curvature ratio defined as the ratio of the pipe diameter to the coiling diameter. Both in the laminar and the turbulent flow regime, the distributions of the velocity field over the tube’s cross-section are asymmetrical and they lead to significant variations of the convective heat-transfer coefficient along the circumferential angular coordinate at the internal wall surface: it presents higher values at the outer bend side of the wall surface than at the inner bend side [3, 7, 8].

Heat Mass Transfer

The presence of an irregular distribution may be critical in some industrial applications, such as in those that involve a thermal process. For instance, in food pasteurisation, the irregular temperature field induced by the wall curvature could reduce the thermal killing of bacteria or could locally overheat the product. However, most of the studies available in the scientific literature did not investigate this aspect, mainly due to the practical difficulty of measuring heat flux on internal wall surface of a pipe, and they presented the results only in terms of the Nusselt number averaged along the wall circumference. Only a few authors have studied the phenomenon locally, and most of them have adopted the numerical approach. To the Authors’s knowledge, only six papers [9–14] have presented experimental results and only three of them report the real local values of the convective heat-transfer coefficient [9–11] while the others, neglecting heat conduction in the tube wall, estimate only the apparent local values [12–14]. Bai et al. [9] experimentally studied the turbulent heat transfer in helically coiled tubes using deionised water as the working fluid. The working fluid was heated by applying alternating current in the tube wall and, at each cross section eight thermocouples were placed on the external surface of the tube wall. The local heat-transfer distribution on the internal wall of the tube was estimated by solving the two-dimensional inverse heat conduction problem with the least-square method. As expected, they found that the local heat-transfer coefficient was not evenly distributed along the periphery of the cross section and that, in particular, at the outside surface of the coil, it was three or four times higher than that at the inside surface. Bozzoli et al. [10] focused their investigation on the fully developed region for the laminar flow regime in the Reynolds number range of 135 to 1050 and the Prandtl number range of 170 to 200. The temperature distribution maps on the external coil wall were employed as input data of the linear inverse heat conduction problem in the wall under a solution approach based on the Tikhonov regularisation method with the support of the fixed-point iteration technique to determine the proper regularisation parameter. The results showed that, at the outside surface of the coil, the Nusselt number is approximately five times larger than that at the inside surface and this ratio, in the conditions under test, is independent from the Dean number. Regarding local heat-transfer coefficient, some experimental data are discussed by Seban et al. [11] that investigated the laminar flow for oil and the turbulent flow of water in coiled tubes. These Authors correctly drew the attention to the difference between apparent and true local values: apparent heattransfer coefficient are obtained neglecting the circumferential heat conduction in the tube wall which means considering the average value of the convective heat flux instead of the punctual value. In terms of true heat-transfer coefficient, the ratio of

the outside to the inside coefficient found in this experimental campaign is about four, for both the laminar and the turbulent flow case. However, no details about the approach adopted to estimate the punctual convective heat flux are given in this paper. Xin and Ebadian [12], Janssen and Hoogendoorn [13] and Hadik et al. [14] conducted extensive experimental campaigns with many different fluids on a wide range of curvature ratios and Reynolds numbers. However, these Authors processed their data neglecting the circumferential heat conduction in the tube wall therefore the reported local heat-transfer coefficient has to be interpreted as the apparent one and not the real one. Extending the bibliographical research to the numerical approaches, many others Authors estimated heattransfer coefficient distributions for a wide range of conditions. In particular, the laminar fluid flow in coiled tubes is locally investigated in [15–21]. Among these studies, the results of Mitsunobu et al. [15], Futagami and Yoshiyuki [17], Yang et al. [20] are particularly relevant. Mitsunobu et al. [15] presented a finite-difference solution using a combination of line iterative method and boundary vorticity method for the hydrodynamically and thermally fully developed laminar forced convection in curved pipes. The method was applied to highlight the effect of Dean number on local Nusselt for two different fluids having Prandtl number equal to 0.7 and 100 respectively. Futagami, and Yoshiyuki [17] carried out a theoretical and experimental study on the effect of the secondary flow on heat transfer from a uniformly heated helically coiled tube to fully developed laminar flow. Both the centrifugal and buoyancy forces are taken into account in the numerical analysis. The solutions cover a range of Prandtl numbers between 1 and 100. The velocity and temperature profiles, the local friction factor and heat-transfer coefficient distribution are obtained. Yang et al. [20] presented a numerical investigation on the fully developed laminar convective heat transfer in a helicoidal pipe, with particular attention to the effects of torsion on the local heat-transfer coefficient. In particular, the authors reported the Nusselt number distribution by varying the coil pitch, and they showed that, due to the torsion, the local heattransfer coefficient, compared to the case of an ideal torus, is increased on half of the tube wall while it is decreased on the other half. Regarding the turbulent fluid flow in coiled tubes, numerical results reported in terms of local heat-transfer coefficient are present in [22–26]. The works of Jayakumar et al. [23] and of Di Piazza and Ciofalo [24] are particularly interesting. Jayakumar et al. [23] numerically analysed the turbulent heat transfer in helically coiled tubes and presented the local

Heat Mass Transfer

Nusselt number at various cross sections along the curvilinear coordinate. The results showed that, on any cross section, the highest Nusselt number is on the outer side of the coil, and the lowest one is on the inner side. Moreover, the Authors proposed a correlation for predicting the local Nusselt number as a function of the average Nusselt number and the angular location for both the constant temperature and the constant heat-flux boundary conditions. Di Piazza and Ciofalo [24] obtained computational results for turbulent flow and heat transfer in curved pipes, representative of helically coiled heat exchangers. Following a grid refinement study, grid independent predictions from alternative turbulence models (i.e., k–ɛ, SST k–ω and RSM–ω) were compared with direct numerical simulation results and experimental data in terms of local convective heat-transfer coefficient. In the present paper coiled tubes, after assessing their overall efficiency, are investigated in terms of local convective heat-transfer coefficient at the fluid-wall interface in turbulent flow regime. In particular Reynolds number values ranging from 5000 to 12,000 and Prandtl number values between 5 and 9 are considered. Following the estimation procedure presented in [10] the temperature distributions on the external wall of the coiled tube, acquired by infrared thermographic technique, are adopted as input data of the inverse heat conduction problem in the wall of the tube. The purpose of this paper is presenting results which are representative of a wide range of technical applications; these data could be employed both as a useful benchmark for CFD results as well as in the design of coiled tube heat transfer apparatuses.

Fig. 1 Sketch of the experimental setup

2 Experimental setup Two different helically coiled stainless steel type AISI 304 tubes were tested. They had smooth wall and they were characterised by eight coils following a helical profile along the axis of the tube. The tube internal diameter was 14 mm, and the wall thickness measured 1.0 mm. The helix diameter of the two tubes under test was approximately 310 mm and 425 mm, respectively while the pitch was about 200 mm for both the pipes. This geometry yields a coiled pipe length L of the two tubes of approximately 8 and 10 m respectively, and a dimensionless curvature δ of 0.045 and 0.032, respectively. To minimise the heat exchange with the environment, the heated section was thermally insulated. A small portion of the external tube wall, near the downstream region of the heated section, was made accessible to an infrared imaging camera by removing the thermally insulating layer, and it was coated by a thin film of opaque paint of uniform and known emissivity. Therefore, the test section was taken approximately sufficiently far from the inlet section, in the region of the heated section where, according to [8, 13], the turbulent boundary layers reached the asymptotic profile. This condition makes the results obtained for this particular segment representative of the thermally fully developed region. The surface temperature distribution was acquired by means of a FLIR SC7000 unit, with a 640 × 512 pixel detector array. A sketch of the experimental setup is reported in Fig. 1, and Fig. 2 shows a particular of the text section. The inlet and the outlet fluid bulk temperatures were measured with type-T thermocouples. The bulk temperature at any location in the heat transfer section was then calculated from

Heat Mass Transfer Fig. 2 Particular of the test section

the power supplied to the tube wall. Volumetric flow rates were obtained by measuring the time needed to fill a volumetric flask placed at the outlet of the test section. To investigate the heat transfer performance of coiled tubes in the turbulent flow regime, water was used as the working fluid. It is currently accepted that the effect of coil curvature is to suppress turbulent fluctuations arising in the flowing fluid, smoothing the emergence of turbulence and increasing the value of the Reynolds number required to attain a fully turbulent flow, with respect to a straight pipe [27]. Considering that the flow in curved pipes remains laminar up to Reynolds numbers higher at least by a factor of two than in straight pipes, in order to be sure that the flow regime was turbulent, in the present investigation the Reynolds number range 5000– 12,000 was considered [28]. In the temperature range characterising the experimental conditions, the Prandtl number of the working fluid varied in the range of 5–9. The working fluid was conveyed by a volumetric pump to a holding tank, and it entered the coiled test section equipped with stainless-steel fin electrodes, which were connected to a power supply, type HP 6671A. This setup allowed the investigation of the heat transfer performance of the tube under the prescribed condition of uniform heat flux generated by the Joule effect in the wall. The heat flux provided to the fluid was selected to make the buoyancy forces negligible compared to inertial ones for the fluid velocity values investigated here. The coiled section was inserted horizontally in a loop completed by a secondary heat exchanger, fed with city water, to keep the working fluid temperature constant at the coil inlet. A picture of the experimental equipment is shown in Fig. 3. Although the focus of the present work is the determination of the local heat transfer coefficient distribution along the cross section of coiled pipes, it is worth to know if the adopted passive technique represents an effective solution to increase the average thermal performance. Therefore, following the procedure presented in [5], the average heat transfer coefficient was estimated. In particular, the whole length of the heat

transfer section was thermally insulated and the wall temperature was measured by several thermocouples placed on the external surface of the tube at different circumferential and axial locations along the heated section. To evaluate the energy effectiveness of the technique the energy losses due to pressure drops were also measured. In the present test rig, pressure drops throughout the coiled section were measured in isothermal conditions by a BRosemount-3051S^ differential pressure transducer.

3 Estimation procedure Starting from the temperature distribution acquired on the external wall surface, it is possible to estimate the local convective heat-transfer coefficient at the fluid-internal wall interface by solving the Inverse Heat Conduction Problem (IHCP) in the wall. As it is well known, however, this procedure approach presents some complications due to the fact that the IHCP is an ill-posed problem and, consequently, it is very

Fig. 3 Experimental equipment

Heat Mass Transfer

sensitive to small perturbations in the input data. In order to bypass the ill-posedness of inverse problems, many techniques based on the processing of the experimental data have been suggested and validated in literature. Among these techniques, the function specification methods [29, 30], the iterative methods [31–33], the methods based on filtering proprieties [34–37], the regularisation techniques [38–42] and the probabilistic methods [43, 44] are found. Concerning the regularisation techniques, Tikhonov regularisation method [45] is certainly the most popular. The procedure, presented in [10], is here adopted to estimate the local convective heat-transfer coefficient in the coiled tubes under test. The temperature distribution maps on the external coil wall are employed as input data of the linear inverse heat conduction problem in the wall under a solution approach based on the Tikhonov regularisation method with the support of the fixed-point iteration technique to determine the proper regularisation parameter. This estimation procedure is based on a simplified 2-D model of the test section (sketched in Fig. 4) formulated by assuming that the temperature gradient is almost negligible along the axis of the tube. In the 2-D solid domain, the steady-state energy balance equation is expressed in the (r, α) coordinate system in the form:   k ∂ ∂T k ∂2 T r þ 2 2 þ qg ¼ 0 ð1Þ r ∂r ∂r r ∂α where qg is the heat generated by the Joule effect in the wall, k is the wall thermal conductivity and α is the angular coordinate. The following two boundary conditions completed the energy balance equation: k

∂T ðT env −T Þ ¼ ∂r Renv

ð2Þ

which is applied on surface S2 and where Renv is the overall heat transfer resistance between the tube wall and the surrounding environment with temperature Tenv; −k

∂T ¼ qðαÞ ∂r

ð3Þ

which is applied on surface S1 and where q is the local convective heat flux at the fluid-internal wall interface, assumed to be varying with the angular coordinate α. To express the problem in the discrete domain, the convective heat flux distribution can be simplified by considering that it is described by a continuous piecewise linear function. In this way, the heat flux distribution can be defined by the vector q = [q1,q2,q3,…,qn]T. Under this approach, the direct problem becomes linear with respect to the heat flux q(α) and its discrete version can be described as follows: T ¼ Xq þ Tq¼0 ;

ð4Þ

where T is the vector of the discrete temperature data at the external coil surface, q is the heat flux vector at the fluidinternal wall interface, Tq=0 is a constant term and X is the sensitivity matrix. The sensitivity matrix X was calculated using the two-point finite difference approach: X i; j ¼

    T i q1 ; q2 ; …; q j þ Δq; …; qn −T i q1 ; q2 ; …; q j ; …; qn Δq

ð5Þ

where Ti is the temperature value at the i sensor position obtained by solving Eqs. (1–3) with an internal heat flux distribution as defined in Eq. (5). In the same way, the constant term Tq=0 was obtained by imposing a null internal heat flux. This set of equations was easily solved by the finite element method within Comsol Multiphysics® environment. The direct formulation of the problem is concerned with the determination of the temperature distribution on the tube external wall when the convective heat flux vector q is known. In the inverse formulation considered here, q is instead regarded as being unknown, whereas the surface temperature Tmeas is measured. As the inverse problem is ill-posed, in order to cope with the presence of noise in the measured temperature some type of regularisation is required. The Tikhonov regularisation method [39] makes it possible to reformulate the original problem as a well-posed problem that consists of minimising the following objective function:

2 J ðqÞ ¼ T meas −Xq−Tq¼0 2 þ λ2 kLqk22 ; λ > 0;

Fig. 4 Geometrical domain with a coordinate system

;

ð6Þ

where k⋅k22 stands for the square of the 2-norm, λ is the regularisation parameter, L is a discrete derivative operator and T is the distribution of the external surface temperature

Heat Mass Transfer

derived from a direct numerical solution of the problem obtained by imposing a given convective heat flux distribution on the internal wall side q. Often, L is the zero, first or second derivative operator: in this work the second-order derivative formulation was chosen to preserve the local variation in the heat-flux distribution. An appropriate choice of λ is a crucial point to find a reliable approximation of the wanted solution and, in this paper, this choice was made by the fixed-point method [45]. Once the heat-flux distribution at the fluid-wall interface compatible with the experimental temperature data has been determined through the strategy described above, the local convective heat-transfer coefficient can be easily determined, as follows: hint ðαÞ ¼

j qλ ð α Þ j T ðα; r ¼ rint Þ−T b

ð7Þ

where qλ(α) is the heat flux distribution estimated under the solution approach based on the Tikhonov regularisation method, Tb is the bulk-fluid temperature on the test section, calculated from the energy balance on the heated section as described in [5, 8] and T(α, r = rint) is the temperature distribution on the tube internal wall efficiently estimated by numerically solving the direct problem by imposing a convective heat flux equal to qλ(α). The convective heat transfer coefficient can be suitably expressed in a dimensionless form by means of the local Nusselt number, as follows: NuðαÞ ¼

hint ðαÞ⋅Dint kf

ð8Þ

where kf is the fluid thermal conductivity, evaluated at the bulk temperature.

The impact of the curvature first on the overall thermo-fluid dynamics performance of the pipes under test and later on the local convective heat transfer characteristics of the coiled tubes was investigated. 4.1 Average thermal and fluid dynamics performance The overall average performance of the studied coiled pipe was evaluated by the well-established enhancement efficiency coefficient [46, 47] based on a first-law analysis defined as follows: Nue =Nu0 ð f e =f 0 Þ1=3

2π

Nu ¼

ð9Þ

1 ∫α¼0 qðαÞ⋅rdα Dint ⋅ kf 2πrint T w −T b

ð10Þ

where 2π

∫ T ðα; r ¼ rint Þ⋅rdα T w ¼ α¼0 2πrint

ð11Þ

The bulk temperature at any location in the heat transfer section was calculated from the energy balance on the heated pipe as follows, by considering that the circumferential averaged heat flux does not depend on the axial coordinate z: Tb ¼

Q z ⋅ þ T inlet ˙ ⋅c m p L

ð12Þ

where Q is the heat power provided to the pipe, m˙ and cp are the fluid mass flowrate and specific heat, respectively, Tinlet is the bulk temperature of the fluid at the inlet section of the coil, L is the coiled pipe length and z is the distance from the inlet section of the coil, taken along the curvilinear coordinate. From the distributions of the circumferentially averaged Nusselt number, estimated at different values of axial coordinate, the asymptotic Nusselt number was evaluated. In fact, for the experimental configurations here studied, the fully developed conditions were always reached in the downstream region of the heated section. The Darcy friction factor was computed as follows: f ¼

4 Results

η¼

where Nu is the average Nusselt number and f is the Darcy friction factor evaluated on the test section. The subscripts e and 0 refer to the enhanced and reference geometry respectively. The straight smooth tube was chosen as the reference geometry. The circumferentially averaged Nusselt number was evaluated, for different axial locations, as follows:

Δp D 2 ⋅ ⋅ ρ L w2

ð13Þ

where w is the mean fluid axial velocity and Δp is the pressure drop along the coiled section having length L. In Fig. 5 it is reported the asymptotic Nusselt number distribution obtained for the studied coiled pipes compared with smooth wall straight tube behavior in turbulent flow regime as described by the Dittus-Boelter correlation [48]: Nuasym ¼ 0:023⋅Re0:8 ⋅Pr0:4

ð14Þ

In the same figure, it is also reported the estimated friction factor compared to the solution for turbulent regime in a straight pipe [49] with the same relative roughness ε:     ε 1:11 6:9 −2 f ¼ −1:8⋅log þ 3:7 Re

ð15Þ

Heat Mass Transfer

4.2 Local convective heat transfer coefficient

Fig. 5 Average Nusselt number, Darcy Moody friction factor and enhacement efficiency of the studied pipes and comparison with the straight smooth wall tube behaviour

Eventually, it is reported in the same figure also the enhancement efficiency as defined in Eq. (9): it clearly confirms the goodness of the investigated coiled pipes as passive heat transfer enhancement technique in the range of Reynolds number values analysed.

Figure 6 reports a representative temperature map of a portion of the coil tube. The figure clearly reveals that the tube wall is colder at the outer bend side of the coil than at the inner bend side while the temperature gradients are almost negligible along the axis of the tube. This observation confirms that adopting a 2-D numerical model for this type of problem is appropriate for the flow conditions under test. The temperature values on the test section wall along the whole circumference are reported in Fig. 7, where the angular coordinate origin is taken at the inner side of the coil. This distribution was obtained processing multiples thermal images of the test section, taken from different point of view around the coil. To implement the direct problem in the discrete domain, the convective heat-flux distribution was approximated by a continuous, piecewise linear function composed of 36 sections; this approach represents a good compromise between model precision and the computational cost. To calculate the Tq = 0 and X terms of Eq. (4), the numerical solution of Eqs. (1–3) was calculated by the finite element method implemented in Comsol Multiphysics® environment with a mesh of approximately 2600 triangular elements. A mesh refinement study, as reported in [10], was performed to verify the appropriateness of the numerical model. The overall heat transfer resistance between the tube wall and the surrounding environment Renv, which was assumed to be known in the inverse problem considered here, was taken equal to 0.2 m2K/W, which is a representative value for natural convection in air compounded with radiative heat transfer with the environment. The wall thermal conductivity k was certified by the manufacturer equal to 15 W/m K; the heat generated by the Joule effect in the wall qg was assumed to be uniformly distributed and was calculated by the ratio of the power supplied and the volume of the tube wall. The distribution of the convective heat-transfer coefficient restored by the minimisation procedure presented above is reported in Fig. 8 for the same case of Figs. 6 and 7. These data, as expected, highlight that the convective heattransfer coefficient is minimal close to the inner bend side of the coil, while it reaches its maximum at the outer bend side. Moreover, Fig. 8 shows the effect of torsion induced by the coil pitch: it creates a rotation force that affects the flow pattern. Consequentially, the location of the minimum Nusselt number shifts slightly from zero to higher angular coordinate values. In Fig. 8 it is reported also the Bapparent^ convective heattransfer coefficient, estimated following the procedures suggested by many Authors [12–14] that neglected the heat conduction in the tube wall. Comparing real and apparent heattransfer coefficient values it is clear that neglecting heat conduction in the tube wall misleads the estimated convective

Heat Mass Transfer

Fig. 6 Representative infrared image of the coil wall (Re = 7443, Pr = 8)

heat transfer distribution, increasing the minimum local values and decreasing the maximum ones. The 95% confidence interval associated with the estimated values was determined by parametric bootstrap [50], assuming the uncertainties in the input data reported in Table 1. To identify the main contributions to the uncertainty of the estimated heat flux distribution a sensitivity analysis and the calculation of the influence coefficient values [51] were performed:  2 ∂Z Z ωξ Jξ ¼ ð16Þ ∂ξ where Z is the estimated quantity and ξ is the considered input parameter with an uncertainty equal to ωξ. For the problem here investigated, the analytical determination of the sensitivity ∂Z/∂ξ was not feasible, and finite difference approach was adopted. The sensitivity analysis pointed out that the main contributions to the uncertainty were the k, qg and Tb measurements 301

Tmeas (K)

300

299

298

297

296

-3

-2

-1

0

α (rad)

1

2

3

Fig. 7 Temperature distribution on the coil external wall (Re = 7443, Pr = 8)

Fig. 8 Restored real convective heat-transfer coefficient distribution with 95% confidence interval and its corresponding apparent distribution (Re = 7443, Pr = 8)

while the uncertainties on Renv and Tenv are not very significant. From this observation it is possible to state that the heat exchanged between the tube wall and the environment is negligible in comparison to the heat exchanged between the tube wall and the working fluid and that particular attention to the accuracy of k, qg and Tb should be given. The whole estimation procedure was repeated for various Reynolds number values and representative results for the turbulent regime are plotted in Fig. 9. To locally compare the Nusselt distributions estimated for the various Re values, the shifting effect of the torsion was compensated by introducing a relative angle α* whose origin was taken where the Nusselt number reaches its minimum. Fig. 10 reports the Nu/Numax ratio for various Reynolds numbers: by accounting for the experimental uncertainty, it can be stated that this ratio is almost independent of the Reynolds number, analogously to the laminar fluid flow in coils [10]. The best fit of these experimental Nu/Numax distributions is plotted in Fig. 11 and compared to the distribution found by Bozzoli et al. [10] for the laminar regime. Some differences between the behaviour in the two different flow regimes can be observed: in the turbulent regime, at the outside surface of the coil, the Nusselt number is about ten times larger than that at the inside surface while in the laminar regime it is only five time larger. Moreover, in turbulent regime the Nu/Numax pattern shows a typical BV-shape^ while in the laminar regime the pattern is more flat near the outside surface of the coil. In Fig. 12 the best fit of the experimental Nu/Numax distributions is also compared with the distribution obtained experimentally by Bai et al. [9] and with the result of the direct numerical simulation performed by Di Piazza et al. [24] for a similar value of Reynolds and Prandtl number.

Heat Mass Transfer The 95% confidence interval of the main physical quantities involved in the estimation procedure

Table 1 Y (K)

α (°)

k (W/m·K)

Tenv (K)

Renv (m2·K/W)

qg (W/m3)

Tb (K)

± 0.1 K

± 4°

± 5%

± 0.1 K

± 50%

± 4%

± 0.1 K

The distributions obtained in the present investigation show a good correspondence with the numerical data obtained by Di Piazza et al. [24]. Only in correspondence of the inner side of the coil the numerical results [24] are more flat than the ones obtained in the present work. This difference could be due to some problems in mesh refinement in [24] because the inner side of the coil is the region with the higher values of the heat flux gradient. Acceptable agreement is found also with the data obtained by Bai et al. [9] even though the ratio between the maximum and the minimum Nusselt number is significantly different. A possible reason for the difference between the two distributions could be that Bai et al. [9] adopted a limited number of temperature sensors to measure the temperature and it prevented the accurate detection of the minimum and the maximum temperature values.

5 Conclusions In this paper, it was experimentally investigated first the overall thermo-fluid dynamics performance and later the local convective heat-transfer coefficient in coiled tubes when turbulent flow regime is present. The research was focused on the fully developed region in the Reynolds number range of 5000 to 12,000 and the Prandtl number range of 5 to 9.

For assessing the local thermal behaviour, the temperature distribution maps on the external coil wall were employed as input data of the linear inverse heat conduction problem in the wall under a solution approach based on the Tikhonov regularisation method with the support of the fixed-point iteration technique to determine the proper regularisation parameter. The results showed that the variation in the convective heat-transfer coefficient along the boundary of the duct section is very significant: at the outside surface of the coil, the Nusselt number is approximately ten times larger than that at the inside surface and this ratio is almost independent of the Reynolds number. In addition, in turbulent regime the Nu/Numax pattern shows a typical BV-shape^ different from the one found by Bozzoli et al. [10] for the laminar regime where the pattern is more flat near the outside surface of the coil. Moreover, the distribution obtained in the present investigation are in good agreement with the numerical one obtained by Di Piazza et al. [24]. It is important to underline that the purpose of this paper is not only presenting results which are representative of a wide range of technical applications but also providing a useful benchmark for CFD results. cp, Specific heat at constant pressure (J/kg·K); f, Darcy friction factor; h, Convective heat-transfer coefficient (W/m 2 ·K); k, Thermal conductivity (W/mK); m˙ , Mass

800 Re=5384 Re=5601 Re=6153 Re=6527 Re=6546 Re=7164 Re=7443 Re=7551 Re=9218 Re=10028 Re=12218

600

Nu

500 400 300

Re=5384 Re=5601 Re=6153 Re=6527 Re=6546 Re=7164 Re=7443 Re=7551 Re=9218 Re=10028 Re=12218

1

0.8

Nu/Numax

700

0.6

0.4

200 0.2

100 0

-3

-2

-1

0

α* (rad)

1

2

3

Fig. 9 Restored convective heat-transfer coefficient distribution for different Reynolds number values

0

-3

-2

-1

0

α* (rad)

1

2

3

Fig. 10 Normalised local Nusselt number for different Reynolds numbers

Heat Mass Transfer Acknowledgements This work was partially supported by the EmiliaRomagna Region (POR-FESR 2014-2020). MBS S.r.l. (Parma, Italy) is gratefully acknowledged for the set-up of the experimental apparatus.

1

Nu/Numax

0.8

Compliance with ethical standards Conflict of interest On behalf of all authors, the corresponding author states that there is no conflict of interest.

0.6

0.4

References 0.2 turbulent laminar [10] 0

-3

-2

-1

0

α* (rad)

1

2

1. 3

Fig. 11 Normalised local Nusselt number and comparison with the data for laminar regime [10]

2.

3. 4.

5.

flowrate (kg/s); p, Pressure (Pa); q, Convective heat flux per unit area (W/m2); qg, Internal heat generation per unit volume (W/m3); r, Radial coordinate (m); w, Mean axial velocity (m/s); z, Axial coordinate (m); D, Tube diameter (m); De, Dean number; L, Pipe length (m); Nu, Nusselt number; Q, Heat power (W); Re, Reynolds number; T, Temperature (K); α, Angular coordinate (rad); δ, Curvature ratio; η, Enhancement efficiency; λ, Regularisation parameter; ρ, Density (Kg/m3); asym, Asymptotic; b, Bulk; e, Enhanced geometry; env, Environment; ext, External; int, Internal; meas, Measured; 0, Reference problem.

6. 7.

8.

9.

10.

11. 12.

13.

14.

15.

16.

Fig. 12 Normalised local Nusselt number and comparison with the data by Bai et al. [9] and Di Piazza et al. [24]

17.

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