Nusselt numbers from numerical investigations of turbulent flow in highly eccentric horizontal annuli

International Communications in Heat and Mass Transfer 109 (2019) 104344

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Nusselt numbers from numerical investigations of turbulent flow in highly eccentric horizontal annuli☆

T

⁎

Conrad Zimmermann , Felix J. Lemcke, Stephan Kabelac Institut für Thermodynamik, Leibniz Universität Hannover, Callinstrasse 36, D-30167 Hannover, Germany

A R T I C LE I N FO Keywords: Heat transfer Nusselt number Annulus Eccentric Forced convection Cooled cable

Abstract: Fluid flow and heat transfer in annuli appear in various technical applications. One of these are actively cooled underground cables, which are a solution for high-voltage transmission grids that face overloads due to a growing capacity of renewable sources. The cooling of these systems depends on precise prediction of heat transfer of a highly eccentric annulus in a way that has not been subject to any known literature or former research. This article presents Nusselt numbers obtained from CFD calculations for turbulent flow (20,000 < Re < 150,000) in highly eccentric annuli (dimensionless eccentricity e up to 0.95). The mesh and turbulence models are validated with well-known correlations from literature, which show very good accordance. Nusselt numbers show a high dependency on the eccentricity. They strongly decrease for e > 0.5 and drop to a third and less for e = 0.95 in comparison to concentric annuli. This has significant impact on the assurance of safe operation in actively cooled cable systems. Results also indicate a change of flow characteristics from laminar to turbulent depending on the eccentricity that should be considered in order to improve heat transfer.

1. Introduction Annuli have a wide range of boundary conditions as they consist of two different surfaces with independent geometrical and thermal characteristics. These two independent surfaces, however, are still coupled concerning the heat transfer. Heat transfer correlations for standard cases of concentric annuli are given by Gnielinski [1]. Still, the large amount of heat transfer configurations leads to several cases that are yet not entirely investigated in literature, although annuli appear in many forms of heat exchangers, in nuclear power engineering, oil drilling, superconducting cables and solar thermal energy systems. Furthermore, there is great potential for actively cooled cables in transmission grids to enable tight laying of cables and higher loads both of which could significantly reduce the cost for transmission grids. Since a new installation system for actively cooled underground cables was recently awarded and certified [2], there is a need for precise predictions of heat transfer in highly eccentric annuli. The cooled cable system consists of a heat generating cable lying on the bottom of a horizontal underground pipe. The annulus, which is formed between cable and pipe, has a dimensionless eccentricity of e = 1.0 (cf. Eq. (9) and Fig. 1 for definition) and is flooded with moving water. Our objective is to evaluate the cooling of the cable with calculation of the

☆ ⁎

turbulent convective heat transfer in an eccentric annulus. To our knowledge, there is no validated Nusselt correlation from experimental data and only very little literature about this case. In general, the evaluation of heat transfer in an eccentric annular duct is based on approaches of concentric annular ducts. The characteristics of concentric annuli are well-described in literature and are very much concentrated on circular tubes. Gnielinski [1] summed up several works and modifications on Nusselt numbers for concentric annuli that are based on a semi-empirical correlation suggested by Prandtl [3]. The Nusselt number Nu, is defined as

Nu =

hLchar λ

with the heat transfer coefficient h, a characteristic length Lchar and the thermal conductivity λ. The characteristic length Lchar,ann of an annulus is the hydraulic diameter Dh, which is the difference of the outer diameter Do and the inner diameter Di (cf. Fig. 1).

Lchar,ann = D h = Do − Di

0735-1933/ © 2019 Elsevier Ltd. All rights reserved.

(2)

For typical situations, the Nusselt numbers of forced convective single phase heat transfer can be calculated from empirical correlations that are functions of the Reynolds number Re and Prandtl number Pr as

The manuscript was written through contributions of all authors. All authors have given approval to the final version of the manuscript. Corresponding author. E-mail addresses: [email protected] (C. Zimmermann), [email protected] (F.J. Lemcke), [email protected] (S. Kabelac).

https://doi.org/10.1016/j.icheatmasstransfer.2019.104344

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International Communications in Heat and Mass Transfer 109 (2019) 104344

C. Zimmermann, et al.

S1

Trombetta [7] and several others [8–11] studied some of these for fully developed laminar flow in annuli with different eccentricities. While Trombetta investigated forced convection with laminar flow numerically, Kuehn and Goldstein [12] analyzed natural convection and temperature fields in eccentric horizontal annuli experimentally with two isothermal cylinders and an optical measuring method. In turn, Mokheimer and El-Shaarawi [13] numerically developed correlations for laminar natural convection in vertical eccentric annuli. A review article from Dawood et al. [14] published in 2015 gives an overview and classification of a few dozen older and younger works on all kind of convective heat transfer in annuli. It reveals that only Lu and Wang [15] are declared to work on the case of a horizontal eccentric annulus with turbulent forced convection. But a look into the article reveals that their investigations were actually carried out in a concentric test section. Riyi et al. [16] have recently provided Nusselt numbers from experimental and numerical studies on a vertical annulus with different radius ratios and eccentricities, but their Reynolds numbers indicate laminar flow with a maximum dimensionless eccentricity of e = 0.8. Moreover, the boundary condition of the inner surface is supposed to be at constant uniform temperature while there is no distinct boundary condition mentioned at the outer surface. In the light of the above, there is a necessity for further investigations of turbulent flow in a highly eccentric annulus, which represents our given case of a cooled cable lying on the bottom of a horizontal pipe. Therefore, we will conduct CFD simulations in this article for numerical results that serve as a contribution to evaluate heat transfer in applications with high eccentricities.

Ro

r1

E Ri

S2

r2

Fig. 1. Schematic geometric model of eccentric annulus.

in Eq. (3).

Nu = f (Re, Pr)

(3)

The correlation given in Eq. (4) as derived by Gnielinski [1] on the basis of the analogy between heat and momentum transport uses Petukhov and Kirillov [4] for the friction factor fann and fits a large number of experimental data on turbulent flow in concentric annular ducts.

Nu =

2/3

(fann /8)RePr k1 + 12.7 fann /8

(Pr 2/3

⎡1 + ⎛ D h ⎞ − 1) ⎢ ⎝ L ⎠ ⎣

⎤F ⎥ ann ⎦

(4)

The annular friction factor fann is defined here as in

fann = (1.8log10Re∗ − 1.5)−2

2. Setup and modeling

(5)

with the modified Reynolds number Re∗

Re∗ = Re

2.1. Geometry and boundary conditions

(1 + a2) ln a + (1 − a2) (1 − a2) ln a

We approach the given case from reality with a simplified theoretical model, in which an outer tube with constant wall temperature represents the pipe and the heat generating cable is represented by an eccentric inner tube with constant uniform heat flux. Furthermore, our CFD calculations are limited to a dimensionless eccentricity of e = 0.95, because otherwise the calculation becomes unstable due to a missing fluid continuum. The basic geometry is shown in Fig. 1. For all calculated cases presented in paragraph 3, we choose an inner radius Ri of 72 mm and set the radius ratio to a = 0.5, which leads to an outer radius Ro of 144 mm. The eccentricity E is the distance between the centers of both cylinders. The dimensionless eccentricity e is then defined by

(6)

which also considers the radius ratio a of inner radius Ri and outer radius Ro (cf. Fig. 1).

a=

Ri Ro

(7)

The term k1 is approximately equal to 1. The additional factor Fann is dependent on the boundary condition and the radius ratio a. As an example, Eq. (8) taken from [1] shows the factor Fann for “heat transfer at the inner surface and outer surface insulated”:

Fann = 0.75a−0.17

(8)

e=

Eq. (4) to Eq. (8) represent the basic concept which is used to calculate heat transfer on the surface of the inner tube for fully developed turbulent flow in concentric annular ducts. There are also correlations for laminar flow or the transition from laminar to turbulent flow [5], as well as correlations that take entrance effects into account [6]. Moreover, all kinds of boundary conditions of inner or outer surface with uniform heat flux or constant surface temperature form a conceptual matrix of many cases that obviously appear at eccentric annuli as well.

E R o − Ri

(9)

The fluid (water) is assumed to be incompressible with constant fluid properties and an inlet temperature Tin = 288.15 K for all cases. Therefore, the default values from ANSYS Fluent are used for water, which are density ρ = 998.2 kg/m3, specific heat capacity cp = 4182 J/ (kg K), thermal conductivity λ = 0.6 W/(m K) and dynamic viscosity μ = 0.001003 kg/(m s). Concerning heat transfer, we simulate three

Table 1 Overview of the investigated cases with characteristics and boundary conditions. Reynolds number

Thermal boundary conditions Inner surface

Outer surface Constant temperature To = 288.15 K Insulated q̇o = 0 W/m2

0 | 0.25 | 0.5 | 0.75 | 0.90 | 0.95

Constant temperature To = 288.15 K

0 | 0.5 | 0.75 | 0.90 | 0.95

Case A – Trombetta [7]

1000

Uniform heat flux q̇i = 66.3 W/m2

Case B – Gnielinski [1]

20,000 | 30,000 | 50,000 | 80,0000

Constant temperature Ti = 298.15 K Uniform heat flux q̇i = 2500 W/m2

Case C – Own CFD research

20,000 | 30,000 | 50,000 | 80,0000 | 110,000 | 150,000

Dimensionless eccentricity e

2

0

International Communications in Heat and Mass Transfer 109 (2019) 104344

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difficulty of an eccentric annulus lies in forming a structured volume mesh with hexahedral cells that meets high quality criteria concerning cell skewness and orthogonal quality. As an example, a dimensionless eccentricity of e = 0.9 leads to an average skewness of 0.25, the maximum is 0.50, whereas the average orthogonal quality is 0.86 and the minimum is 0.27. In order to save computation time, a symmetric geometry model is used and a swept mesh technique is applied. Boundary layers are applied at the inner and outer wall surfaces. The grid has 60 circumferential subdivisions on the inner and outer perimeter of the semicircle related to the symmetric geometry shown in Fig. 2 and 30 radial subdivisions between inner and outer cylinder for each angle. The resulting cross-section grid for the case of e = 0.9 is exemplarily shown in Fig. 2. It supports that the high amount of radial subdivisions leads to a very high resolution in the smallest gap, which certainly is a critical spot. The numerical simulations are carried out using the solver FLUENT 19.1. Pressure-velocity coupling is done by the SIMPLE algorithm and for higher accuracy Second-Order-UPWIND algorithm is used. For the non-laminar cases the Reynolds-averaged Navier-Stokes equations (RANS) and k-ω SST turbulence model are used. Here, the default values for the turbulence models' constants are applied. The k-ω SST turbulence model is well known for its ability to produce accurate results in the near wall region as well as in the free-stream. Neto et al. [17] stated that this turbulence model shows the best agreement for the velocity profile in eccentric annulus in comparison to other turbulence models. In all cases, a mass-flow-inlet is used and for the turbulent case we estimate the turbulent intensity and the turbulent length scale beforehand with recommended equations from ANSYS Fluent User's Guide. The residuals for all cases show monotonically decreasing behavior and fell well below 10−5. Moreover, all monitor points like inlet pressure, outlet temperature, temperature/heat flux on the walls, become constant.

3. Results and discussion Fig. 2. Exemplary cross-section grid of the CFD setup for e = 0.9.

3.1. Validation and comparison with literature

different cases with varying boundary conditions, two for validation purposes (cases A and B) and one for advanced research (case C). They are summarized in Table 1 and will be explicitly presented and discussed later in paragraph 3. Another validation case only for the velocity profile is not listed in Table 1, where we do not consider any heat transfer (cf. paragraph 3.1).

We validate our procedure of meshing with the case from Neto et al. [17] with Re = 26,600 and compare the velocity field to both, experimental results from Nouri and Whitelaw [18] and numerical results from Neto et al. [17]. The boundary conditions are taken from [17]. Fig. 3 shows the velocity profiles for the concentric case and eccentric case (e = 0.5). The latter one is evaluated in the smallest and widest gap. Therefore, a dimensionless velocity is built by the ratio of local velocity u and bulk velocity Ub. The position in the gap is given by r1/S1 at the widest gap and r2/S2 at the smallest gap, just as defined in Fig. 1. There is good agreement between our numerical simulation and the

2.2. Meshing and numerical model The eccentric annulus is meshed using ANSYS Meshing. The

Concentric Nouri & Whitelaw Neto et al. Own CFD

0.5

1

0.5

0 0.5

r/D h (-)

1

1

0.5

0 0

Nouri & Whitelaw Neto et al. Own CFD

1.5

u/Ub (-)

1

Eccentric - smallest gap

Nouri & Whitelaw Neto et al. Own CFD

1.5

u/Ub (-)

1.5

u/Ub (-)

Eccentric - widest gap

0 0

0.5

r1 /S1 (-)

1

0

0.5

1

r2 /S2 (-)

Fig. 3. Velocity profiles from experimental (Nouri and Whitelaw) and numerical investigation (Neto et. al) in concentric and eccentric annuli in comparison to own CFD results. 3

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data of Nouri and Whitelaw [18] and Neto et al. [17] for the concentric case. In case of eccentricity there is a good agreement for the wide gap, but the numerical simulations, both Neto et al. [17] and ours, differ from the experimental results of Nouri and Whitelaw [18]. Neto et al. [17] do not state a clear reason for this deviation. However, as Nouri and Whitelaw [18] used a mixture of tetralin and turpentine and only gave a constant value for viscosity and density, it seems arguable that the real fluid properties and transport behavior of this special mixture could be a reason for the under prediction by our calculation. Anyway, the behavior of the velocity field matches well in large parts of the system, which should be sufficient for further investigations. The heat transfer in this article is generally evaluated from circumferential average Nusselt numbers Nu at the inner surface obtained from Eq. (1). For the laminar case A we insert a heat transfer coefficient h of a single cross-section calculated by

h=

qi̇ (TW − Tcup)

600 500

Nu m (-)

400

100 0 20,000

hm =

ΔTln =

Nu (-)

(TW − Tin ) − (TW − Tcup,out ) ln T

TW − Tin

(13)

with the wall temperature TW (here expressed by the boundary condition at the inner surface), Tin the inlet temperature and Tcup, out the outlet mixing cup temperature. All other boundary conditions are given in Table 1. The axial length for the calculated case is L = 350Do, which is sufficiently long to neglect entrance effects. Fig. 5 shows the mean turbulent Nusselt numbers Num from our CFD simulations and those calculated with the correlation from Gnielinski [1] as in Eq. (4) to (8) for the same Reynolds numbers and boundary conditions. Our results deviate from the correlation because the correlation does not take entrance effects into account. For this reason, our results underestimate Gnielinski [1] for low Reynolds numbers, which usually cause larger sections of entrance effects with low Nusselt numbers. Yet, our CFD results agree well with the literature. 3.2. Results for turbulent flow in highly eccentric annuli Our simulation is validated for laminar heat transfer in eccentric annuli in case A and it is validated for turbulent heat transfer in concentric annuli in case B. These sophisticated preparations with different validation cases are time-consuming. They have not been reproduced with CFD that way before and in combination, they build a necessary basis to obtain reliable results in our research case C for turbulent heat transfer in highly eccentric annuli. In our CFD simulation, the heat emission of the cable is applied as a uniform heat flux on the inner surface of the annulus and constant temperature at the outer surface. Thus, we reassemble the validated thermal boundary conditions from case A (cf. Table 1), but with turbulent flow. Therefore, we conduct first investigations of the technically favorable turbulent heat transfer in a highly eccentric annulus with the aim to predict active cable cooling. The heat transfer coefficient h is calculated just the same way as in Eq. (10) from case A. Nusselt numbers are again circumferentially averaged at a specific cross-section. Fig. 6 shows the circumferential average Nusselt number at a local dimensionless position z/Do with z representing the axial distance to the inlet in the direction of flow.

8 6 4 2 0 0.8

(12)

W − Tcup,out

Trombetta Own CFD

0.6

qṁ ΔTln

The mean wall heat flux q̇m of the inner surface for the total axial length can be taken from CFD-Post and ΔTln is the logarithmic mean temperature, which is defined as

10

0.4

80,000

again inserted into Eq. (1) and it is calculated from our CFD results as follows:

(11)

0.2

60,000

Fig. 5. Comparison of mean Nusselt numbers from own CFD simulation with calculated mean Nusselt numbers from [1] for turbulent flow in concentric annulus depending on Reynolds numbers.

In the post-processing tool CFD-Post from ANSYS Fluent the mixing cup temperature Tcup is represented by the mass-flow-average temperature in the particular cross-section. Taking the configurations from the numerical investigation of Trombetta [7], we validate our CFD-model for laminar flow, high eccentricities and with the same thermal boundary conditions that are present in our advanced research case C of an actively cooled cable. Further boundary conditions are listed in Table 1. The mass flow inlet is applied with a laminar viscous model on the semicircle cross-section of Fig. 2. Just as Trombetta [7], we consider a fully developed laminar flow verified by the temperature gradient dTcup/dz < 10−3 K/m. The resulting Nusselt numbers of our simulation are shown in Fig. 4. In comparison to Trombetta [7], there is a deviation in the Nusselt numbers smaller than 5% for all investigated eccentricities. It should be mentioned that the results from Trombetta [7] can only be read off from printed graphs, which leads to an inaccuracy that may explain the deviation in part. Nevertheless, there is a good confirmation of our CFD results for laminar convective heat transfer in highly eccentric annuli, which serves as basic validation for our research case C. For turbulent flow, we validate our simulation with the well-known concentric case B from Gnielinski [1] that delivers mean turbulent Nusselt numbers Num. Here, the mean heat transfer coefficient hm is

0

40,000

Re (-)

(10)

∫A ρ·u·cp·T dA ∫A ρ·u·cp dA

300 200

TW is the average wall temperature of the inner surface at the crosssection and Tcup is the mixing cup temperature at the same cross-section, generally defined by the following equation:

Tcup =

Gnielinski Own CFD

1

e (-) Fig. 4. Comparison of Nusselt numbers from own CFD simulation with results from [7] for different eccentricities. 4

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Re = 20,000 1000

e=0 e = 0.5 e = 0.75 e = 0.9 e = 0.95

800

e=0 e = 0.5 e = 0.75 e = 0.9 e = 0.95

800

600

Nu (-)

Nu (-)

Re = 50,000 1000

600

400

400

200

200

0

0 0

100

200

300

0

100

z/Do (-) Re = 80,000 1000

e=0 e = 0.5 e = 0.75 e = 0.9 e = 0.95

800

600

Nu (-)

Nu (-)

300

Re = 150,000 1000

e=0 e = 0.5 e = 0.75 e = 0.9 e = 0.95

800

200

z/Do (-)

600

400

400

200

200

0

0 0

100

200

300

0

100

z/Do (-)

200

300

z/Do (-)

Fig. 6. Local Nusselt number depending on the distance z to the inlet in the direction of flow for differerent eccentricities and Reynolds numbers.

1000

800

Re = 20,000 Re = 30,000 Re = 50,000 Re = 80,000 Re = 110,000 Re = 150,000

800

600

Nu (-)

Nu (-)

1000

e=0 e = 0.5 e = 0.75 e = 0.9 e = 0.95

600

400

400

200

200

0

0 0

100,000

200,000

0

Re (-)

0.2

0.4

0.6

0.8

1

e (-)

Fig. 7. Nusselt numbers for eccentric annuli depending on Reynolds number and eccentricity at z/Do = 175.

There are four of six investigated Reynolds numbers and five different eccentricities. In total, we have investigated 32 computing-intensive operation points. First of all, it is obvious to see that all Nusselt numbers increase as the Reynolds number increases, which is consistent with the general understanding of heat transfer. In addition to that, the Nusselt numbers of all eccentricities decrease with longer distances and they seem to reach asymptotically a lower limit, which is also justifiable for this kind of thermal boundary condition. But in contrast to Fig. 4 (comparison to Trombetta [7]), where the Nusselt numbers increase with higher eccentricities, we can clearly see in Fig. 6 that the eccentricity has a negative impact on the heat transfer at least for turbulent Reynolds numbers, as Nusselt numbers are smaller with increasing eccentricity.

Although it might have been presumable to see this behavior of decreasing Nusselt numbers for higher eccentricities, it is an important finding because results from the laminar investigations of Trombetta [7] could have misled to the assumption that eccentricity might have positive effects on heat transfer. Moreover, the results show that the impact on the Nusselt numbers for a change of eccentricity from e = 0 to e = 0.5 is only about 20% and therefore mostly in the range of uncertainty when designing heat exchangers. However, if it comes to eccentricities of 75% and more, the negative impact is quite strong, reducing the heat transfer to a third and even less in comparison to the concentric case. This unfavorable behavior must be taken into account for the design of cooling systems of eccentric cables in a tube. Furthermore, in Fig. 6 it seems that for low Reynolds numbers of 5

International Communications in Heat and Mass Transfer 109 (2019) 104344

C. Zimmermann, et al.

Re ≤ 50,000 there is almost no difference between the Nusselt number for high eccentricities of e = 0.9 and e = 0.95. This relation is similar to laminar flow behavior, where heat transfer is also only slightly depending on Reynold numbers. As the Reynolds number increases, the graph for e = 0.9 does result in higher Nusselt numbers than the graph for e = 0.95. This leads to the assumption that there is a change of the heat transfer characteristics from laminar to turbulent. Consequently, Fig. 7 shows Nusselt numbers as a function of Reynolds numbers calculated at z/Do = 175. For high eccentricities we can see that the curves of e = 0.9 and e = 0.95 almost match for low Reynolds numbers, but clearly drift away from each other for higher Reynolds number. We can also identify a change of the gradient for e = 0.9 at about Re = 50,000 and for e = 0.95 a slight change between Re = 150,000 and Re = 200,000. This behavior supports our initial assumption from Fig. 6 that the eccentricity has an impact on the Nusselt number and promotes the change from laminar to turbulent flow and heat transfer. As for the cooling of a cable with an eccentricity of e = 1.0 we must consider that the increase of Nusselt numbers due to increasing Reynolds numbers is small. Besides, the critical Reynolds number that indicates a change from laminar to turbulent heat transfer is probably > 200,000. It seems reasonable to consider a certain improved surface structure that may provoke a switch from laminar to turbulent heat transfer characteristics at smaller Reynolds numbers. Additionally, Fig. 7 emphasizes again the large impact of the eccentricity on the Nusselt number. A change of eccentricity from e = 0 to e = 0.5 decreases the Nusselt numbers about 20%, whereas a change from e = 0.5 to e = 0.95 leads to a far bigger decrease of the Nusselt numbers. This impact amplifies especially for higher Reynolds numbers, and we see almost hyperbolic behavior in this function. Hence, if there is no enhancement of the laminar-turbulent change, it does not seem energetically rational to reach higher Reynolds numbers while the increase of the heat transfer is relatively small. Equally, if possible, high eccentricities should be prevented by all means for other heat transfer applications.

the surface design either of the cable coating or of the outer tube, which may trigger turbulent flow at smaller Reynolds numbers to improve the overall efficiency. This way, cooled cables could cover overloads in high-voltage transmission grids and reduce losses in transmission, as well as reduce cost for land use because of narrow cable paths. In general, small constructional supplements like spacers can avoid narrow gaps, contact of heat emitting elements, or eccentricities, which reduce the performance of heat exchangers. One parameter that has not been discussed in this article due to massive necessary computing time is the radius ratio a. A sophisticated study on that topic as well as experimental data will be part of future work. Long-term objective of this research could be a modification of Eq. (4) that consists of an additional correcting factor, which takes eccentricity into account for design purposes of heat transfer applications. Declaration of Competing Interest None. References [1] V. Gnielinski, Heat transfer coefficients for turbulent flow in concentric annular ducts, Heat. Tran. Eng. 30 (6) (2009) 431–436. [2] R. Hamann, W. Spiegel, Aktiv gekühlte Stromübertragung in Schmaltrassen, Netzpraxis 54 (1–2) (2015) 48–54. [3] L. Prandtl, Führer durch die Strömungslehre, Vieweg Verlag, Braunschweig, Germany, 1944. [4] B. Petukhov, V. Kirillov, On heat exchange at turbulent flow of liquids in pipes, Teploenergetika 4 (1958) 63–68. [5] V. Gnielinski, Ein neues Berechnungsverfahren für die Wärmeübertragung im Übergangsbereich zwischen laminarer und turbulenter Rohrströmung, Forsch. Ingenieurwes. 61 (9) (1995) 240–248. [6] V. Gnielinski, G2 Heat transfer in concentric annular and parallel plate ducts, VDI Heat Atlas, Springer-Verlag, Berlin Heidelberg, 2010, pp. 701–708. [7] M.L. Trombetta, Laminar forced convection in eccentric annuli, Int. J. Heat Mass Transf. 14 (1971) 1161–1173. [8] K. Cheng, G.J. Hwang, Laminar forced convection in eccentric annuli, AICHE J. 14 (3) (1968) 510–512. [9] R.K. Shah, A.L. London, Laminar Flow Forced Convection Heat Transfer, Academic Press, New York, 1978, pp. 322–340. [10] W.T. Snyder, G.A. Goldstein, An analysis of fully developed laminar flow in an eccentric annulus, AICHE J. 11 (3) (1965) 462–467. [11] R.M. Manglik, P.P. Fang, Effect of eccentricity and thermal boundary conditions on laminar fully developed flow in annular ducts, Int. J. Heat Fluid Flow 16 (4) (1995) 298–306. [12] T. Kuehn, R. Goldstein, An experimental study of natural convection heat transfer in concentric and eccentric horizontal cylindrical annuli, J. Heat Transf. 100 (4) (1978) 635–640. [13] E.M. Mokheimer, M.A. El-Shaarawi, Correlations for maximum possible induced flow rates and heat transfer parameters in open-ended vertical eccentric annuli, Int. Commun. Heat. Mass Tran. 34 (2007) 357–368. [14] H.K. Dawood, H. Mohammed, N. Azwadi Che Sidik, K. Munisamy, M. Wahid, Forced, natural, mixed-convection heat transfer and fluid flow in annulus: a review, Int. Commun. Heat. Mass Tran. 62 (2015) 45–57. [15] G. Lu, J. Wang, Experimental investigation on heat transfer characteristics of water flow in a narrwo annulus, Appl. Therm. Eng. 28 (1) (2008) 8–13. [16] L. Riyi, W. Xiaoqian, X. Weidong, J. Xinfeng, J. Zhiying, Experimental and numerical study on forced convection heat transport in eccentric annular channels, Int. J. Thermal Sci. 136 (2019) 60–69. [17] J. Neto, A.L. Martins, A.S. Neto, C. Ataide, M.A.S. Barozo, CFD applied to turbulent flows in concentric and eccentric annuli with inner shaft rotation, Can. J. Chem. Eng. 89 (4) (2011) 636–646. [18] J. Nouri, J. Whitelaw, Flow of Newtonian and non-Newtonian fluids in an eccentric annulus with rotation of the inner cylinder, Int. J. Heat Fluid Flow 18 (2) (1997) 236–246.

4. Conclusions and prospect This article gives an overview of Nusselt numbers for eccentric annuli with turbulent flow, which does not exist to this extent and validated with well-known correlations. The results show that heat transfer in annuli strongly depends on the eccentricity. Especially high eccentricities of e ≥ 0.5 drastically reduce Nusselt numbers for turbulent Reynolds numbers. It reveals that results from Trombetta [7] for laminar flow are in contrast with those for turbulent flow, which is an important finding of this article. This behavior needs further investigation and is probably caused in the smallest gap between the two surfaces, as there are the lowest local velocities. Consequently, local Nusselt numbers on the perimeter are another interesting field of research, which co-occurs with a detailed consideration of the practically non-uniform distribution of heat emission on the surface of the cable. The change of the heat transfer characteristics for high eccentricities depending on the Reynolds number is another aspect that needs to be analyzed in the future. Simple modifications on the surfaces of the pipes may lead to an enhancement of the heat transfer for high eccentricities. In case of an actively cooled cable, solutions are required that reshape

6