## Abstract

The thermofluid characteristics of Al2O3–water nanofluid in the annulus of double-helical coiled tubes were experimentally and numerically carried out. The purpose was to investigate the effect of combined enhancement techniques of nanofluid and helicoid tube shape on the performance of a double tube heat exchanger. The effects of concentration of nanoparticles, Reynolds number, coil curvature ratio, and flow arrangement through the annulus of double-helical coiled tube were the main points of interest. Three coiled tube heat exchangers were manufactured and experimentally tested to study the design parameters on the performance of such a heat exchanger. A three-dimensional numerical computational fluid dynamic (CFD) model was developed to get additional insights on the thermal performance of double helically coiled tubes with nanofluid on a level of details not always available in experiments. It was found that the Al2O3–water nanofluid achieved an enhancement by 32% on the overall heat transfer coefficient. The heat exchanger effectiveness, heat transfer per unit pumping power, and the Nusselt number were also presented for different design parameters.

## 1 Introduction

Double-helical coil tubes’ heat exchangers are used in many kinds of industries such as refrigeration and air-conditioning, power plants, heat recovery, petrochemical, and biomedical. The double-helical coil tubes enhance the heat transfer in a small footprint of surface area compared with the double tubes heat exchanger. This increase in heat transfer rate can be attributed to the compact structure and high heat transfer coefficient which results from the flow mixing type which result from a helical flow manner. However, there is still a need for more enhancements in the effectiveness and compactness of double tube heat exchangers. A modified version of such a heat exchanger is to combine double enhancement techniques of nanofluid as working fluid and helicoid tube shape to increase the quantity of productivity per time. Pure working fluids have a low thermal conductivity for the engineering process which leads to constrains for heat exchangers compactness and effectiveness. So to improve the fluid’s thermal properties, many methods may be considered an augmentation of heat transfer. One of these methods is to add nanosolid particles to the fluid to make a mixture called nanofluids. Different types of metallic and non-metallic powder particles can be added into fluids to form slurries. The suspended particles’ thermal conductivities of the nanofluid are higher than that of pure fluids which expected to enhance heat transfer for heating or cooling processes, the thermal capacity, and the fluid heat transfer criteria improved significantly. The use of passive heat transfer enhancement technique of double-helical coiled tubes combined with enhanced working fluid by adding a nanoparticle is expected to achieve a valuable enhancement of both heat transfer coefficient and heat exchanger effectiveness. This combination of enhancement techniques to the double tube heat exchanger is affected by two main factors. One is the design of heat exchangers such as helicoid shape, coil diameter, coil pitch, number of turns, and annulus spacing. The other factor is in the enhancement of fluid thermal properties by using nanofluid such as adding nanomaterial with different nanosizes, concentration, and flow arrangement. Many researchers have investigated the heat transfer enhancements through a helical coiled tubes’ heat exchanger by using nanofluids. The thermofluid characteristics of CuO-H2O nanofluid through the helical coiled tubes heat exchanger was studied by Kannadasan et al. and Fule et al. [1,2]. The experiments carried out in the turbulent flow regimes by using CuO-H2O nanofluid with a concentration of 0.1% and 0.2%. The performance of the nanofluids into the shell and helical coil tube heat exchanger was experimentally presented by Srinivas and Vinod [3] and Kumar and Vinod [4]. Three different nanofluids types were used. The study was carried out at different nanofluid temperatures, concentrations, and flow rates. Fsadni et al. [5] presented numerically the characteristics of the turbulent flow regimes of Al2O3–water nanofluids through the helically coiled tube heat exchanger. The nanofluid concentrations from 1% to 4% and the constant wall heat flux were applied to the numerical model. Aly [6] studied numerically the thermofluid characteristics of nanofluid of Al2O3–water through a double-helical coiled tube. Realizable CFD k–ɛ turbulent model was applied to get a solution of the governing equations of the nanofluid flow through the double-helical tube heat exchanger. Enhancing the heat transfer performance in the laminar flow region using Al2O3–water nanofluid through helically coiled tubes heat exchanger was investigated numerically by Elsayed et al. [7]. A CFD model of the laminar region through helical tubes with nanofluids was developed. The thermal performance of a silver–water nanofluid in a helically coiled tube was carried out by Mirfendereski et al. [8]. Six helical coil heat exchangers with different curvature and torsion were investigated experimentally and numerically. Mahmoudi et al. [9] studied experimentally and numerically the heat transfer and fluid flow through a helically coiled tube with TiO–water nanofluid for five different curvature ratios. The experiments were carried out at Reynolds number of 3000 ≤ Re ≤ 18,000 and nanofluid concentration of 0.1% ≤ φ ≤ 0.5%. Akbarinia and Behzadmehr [10] and Akbarinia [11] investigated the thermofluid characteristics through curved tubes using an Al2O3–water nanofluid. The use of polyaniline based water nanofluid in a heat exchanger with helically coiled tubes was investigated by Bhanvase et al. [12]. The study presented the effect of nanofibers concentration and the flow velocity on the enhancement of heat transfer. El-Said et al. [13,14] investigated the heat transfer characteristics and second law analysis with the entropy generated in a new type of plate heat exchanger using a helical flow duct. El-Said and Abou Al-Sood [15,16] presented a comparative experimental study on shell and tube heat exchanger with four different segmental baffle configurations and with two air injection methods to evaluate the enhancement of heat transfer. Abdelmagied [17] derived a numerical study on thermofluid characteristics of curved tubes, and the effect of curvature ratio and Al2O3 nanofluid was presented.

A general brief summary of the previous studies is presented in Table 1. Although a significant amount of researches have been performed on the flow patterns of nanofluid inside or outside the helically coiled tube which was in a shell, there is still much need to investigate the heat transfer enhancement through the annulus of the double-helical coiled tube with nanofluid as a combined enhancement technique. Hence, the present study aims to investigate experimentally and numerically the thermofluid characteristics of nanofluid in the annulus of a double-helical coiled tube using Al2O3–water mixture. The effect of geometric parameters of the heat exchanger, such as annulus curvature ratio, nanoparticles concentration, Reynolds number, as well as the flow arrangements on the heat transfer criteria of the double-helical coiled tube heat exchanger. The main concern of the study focused on the enhancement of the effectiveness and on the overall heat transfer coefficient of such heat exchanger to minimize the process cost of the products.

## 2 Experimental Apparatus

The experimental test rig consists of three main circuits which are the chilled water closed-loop circuit, the closed loop of hot water circuit, and the test specimen circuit which contains the nanofluid in the annulus of the double-helical tube as illustrated in Fig. 1. The hot water circuit which comprises an insulated tank of 0.25 m3 capacity, three heaters 2 kW each, 1 HP centrifugal pump, and a rotameter (0–18 l/min) to produce the hot water flow path. The temperature inside the hot water tank was maintained constant at 60 ± 0.5 °C. On the other side, the cold water produced from a chilled water system consists of a chilled water refrigeration circuit, an insulated tank of 0.15 m3 capacity, 1 HP centrifugal pump, a ball valve, and a rotameter. The test specimens of the double-helical coiled tubes consist of two concentric copper tubes in a helical shape. The outer diameter of the external tube was 15.87 mm (5/8 in.) and the outer diameter of the internal tube 6.35 mm. The details of the geometrical parameters of the helically coiled tubes are presented in Fig. 2. A three double-helical coiled tube heat exchangers with a helical coil pitch of 31.75 mm and different curvature ratios of 0.026, 0.032, and 0.043 were used. Details of the dimensions of the physical coils are given in Table 2. The outer tube of the helical coiled was thermally insulated by rubber foam pipe insulation (k = 0.031 W/m K). The heat loss from the coil outer surface was estimated to be less than 4%. The accuracy and uncertainty of measuring devices are given in Table 3.

## 3 Uncertainty Analysis

The implication of the experimental error specifies the error of the measuring quantities. For the different calculated parameters, the uncertainty analysis was established according to Holman [18]. For independent variables (X1, X2, X3, … Xn), given Y1, Y2, Y3, …, Yn uncertainties and WR was the uncertainty in the result for the same order of magnitude which can be given as
$YR=[(∂R∂X1Y1)2+(∂R∂X2Y2)2+(∂R∂X3Y3)2+⋯⋯+(∂R∂XnYn)2]1/2$
(1)
The uncertainty values of calculated Nusselt number, Dean number, and friction factor were ±5.3%, ±3.5%, and ± 4.25%, respectively.

## 4 Nanofluid Preparation

Alumina (Al2O3) nanoparticle of size varied from 20 to 70 nm was used to form the nanofluid. A photo of alumina nanoparticles which was taken by a technique of transmission electron microscope (TEM) is shown in Fig. 3. The nanofluid was prepared by mixing commercial spherical-shape Al2O3 powders (Sigma-Cairo, Egypt) with Al2O3 content >99.98%. The average diameter of 35 nm was used with polyacrylic acid, in ammonia–water base solution. The Al2O3 nanoparticles were prepared by weighing the mass of Al2O3 powder using an electronic balance and then put into the weighed distilled in base solution gradually with the agitation of the mixture. The solution was heated up to 150 °C for 2 h using a heater with continuous magnetic stirrers. A supersonic homogenizer (ultrasonic 250 model, China) was used in the preparation of different concentrations of the nanofluids [19]. The suspension stability of the nanofluids was then compared. The Alumina (Al2O3) nanoparticle of concentration 0.5%, 1%, and 2% was added to the cold water tank of 0.15 m3 and tested for each test specimen. The thermophysical properties of the water have been changed due to the addition of Alumina (Al2O3) nanoparticle. The suspended Al2O3 nanoparticles increase the surface area, thermal conductivity, and the heat capacity of the nanofluid, and consequently, the heat transfer characteristics will be affected positively. The changes in the viscosity and in the thermal conductivity do not obey the relation of the mixture rules. There were many correlations for predicting the thermophysical properties of the nanofluid mixture [1-2-6-9-20], which depend on many parameters. Aly [6] and Heyhat et al. [20] proposed correlations for calculating the thermophysical properties of nanofluid such as thermal conductivity, density, viscosity, and specific heat. The same correlations (Eqs. (2)(5)) were applied in this study to calculate the nanofluid thermal properties.

Thermal conductivity
$knf=kbf(1+8.733ϕ)$
(2)
Density
$ρnf=ϕρn+(1−ϕ)ρbf$
(3)
Specific heat
$Cnf=ϕ(ρC)n+(1−ϕ)(ρC)bfϕρn+(1−ϕ)ρbf$
(4)
Dynamic viscosity
$μnf=μbfexp(5.989ϕ0.287−ϕ)$
(5)

The thermophysical properties of water, Alumina, and nanofluid are presented in Table 4.

## 5 Numerical Model

A three-dimension CFD model was created to study the thermofluid characteristics of the nanofluid through a double helically coiled tube using the finite volume method software of ANSYS-15. The SIMPLEC-based solution algorithm with the second-order upwind scheme was employed to solve the momentum and energy equations. Conjugate heat transfer was applied to the numerical domain with the coupling of the conduction heat transfer with convection before and after the solid of the inner tube wall. The numerical solution algorithm was adjusted to couple the conduction equation which was in the tube wall with the two convective heat transfer equations (the hot water stream in the inner tube and the cold nanofluid stream in the annulus) to solve these three equations simultaneously.

A results validation between three turbulence models including standard k-ɛ model, re-normalisation group (RNG) k-ɛ model, and realizable k-ɛ model with experimental and previously published data was performed and presented in Fig. 4 in order to validate and select the appropriate turbulent model for the helical flow through the coiled tube. According to the experimental validation of the numerical model and according to Aly [6], the realizable k-ɛ model with enhanced wall treatment was the most appropriate for modeling the fluid flow in curved tubes which was considered here. The governing equations that describe the nanofluid flow pattern through the annulus of double-helical tube are a group of non-linear partial differential equations. Generally, the governing equations can be written as [21]:

Mass
$∂ρ∂t+div(ρu)=0$
(6)
Momentum
$∂(ρu)∂t+div(ρuu)=−∂p∂x+div(μgradu)+[−∂(ρu′2¯)∂x−∂(ρu′ν′¯)∂y−∂(ρu′w′¯)∂z]$
(7)

$∂(ρν)∂t+div(ρνu)=−∂p∂y+div(μgradν)+[−∂(ρu′ν′¯)∂x−∂(ρν′2¯)∂y−∂(ρν′w′¯)∂z]$
(8)

$∂(ρw)∂t+div(ρwu)=−∂p∂z+div(μgradw)+[−∂(ρu′w′¯)∂x−∂(ρν′w′¯)∂y−∂(ρw′2¯)∂z]$
(9)
Energy
$∂(ρE)∂t+div(ρEu)=div(kgradT)+[−∂(ρu′E′¯)∂x−∂(ρv′E′¯)∂y−∂(ρw′E′¯)∂z]$
(10)
The transport equations of the k-ɛ model can be given as
$∂∂t(ρk)+∂∂xi(ρkui)=∂∂xj(αkμeff∂k∂xj)+Gk+Gb−ρε+Sk$
(11)

$∂∂t(ρε)+∂∂xi(ρεui)=∂∂xj(αεμeff∂ε∂xj)+C1εεk(Gk+C3εGb)−C2ερε2k−Rε+Sk$
(12)
where $Gk=−ρui′uj′¯∂uj∂xi$, $Rε=Cμρξ3(1−ξ/ξo)1+θξ3ε2k$, $Gb=giμtρPrt∂ρ∂xi$, $C3ε=tanh|vu|$, $Sk=ξε,$$C1ε=1.42,C2ε=1.68,$$αk=αε≈1.393$; $ξ=4.34$ and $θ=0.012$ [6].

The above equations are discretized by integration on a control-volume to yield Eq. (13):

An algebraic equation can be obtained by applying Eq. (13) to each cell in the computational domain
$∫CV∂(ρϕ)∂t+∫CVdiv(ρϕu)=∫CVdiv(Γgradϕ)+∫CVSϕ$
(13)

$∑fNfacesufφfAf=∑fNfacesΓφ(divφ)nAf+SφV$
(14)

### 5.1 Grid-dependence Check.

The grid density and its distribution have significant effects on the solution accuracy. It is useful to perform a grid-dependence check to obtain an appropriate grid size. A multi-block structured grid technique was specified in the computational domain. After some trails of mesh refinements, the accuracy of the Nusselt number and friction factor of course mesh, medium mesh, fine mesh, and the very fine mesh was established as illustrated in Table 5. The fine grid system of 2.94 × 106 cells was chosen as a compromise between the computational accuracy and computational time cost.

### 5.2 Boundary Condition.

The boundary condition was applied in which the hot water flowrate (velocity inlet) and temperature were specified in the upstream of the inner tube, while a zero pressure gauge was specified on the downstream as illustrated in Fig. 5. The same boundary condition was applied on the annulus of the nanofluid side with a counter flow arrangement. On the outer tube surface, no-slip condition and zero heat flux were specified. The convergence criterion of the computational run was specified at 1 × 10−5 for continuity, momentum, k, and ɛ. while the convergence criterion for the energy equation was 1 × 10−8.

## 6 Data Reduction

To minimize the effect of both heat loss from the outer tube surface and measurement uncertainties, the average heat transfer between the hot fluid which was in the inner tube and the nanofluid which was in the annulus was chosen in calculations [22,23] as
$Q˙avg=Q˙h+Q˙c2$
(15)
where the heat released and the heat absorbed was calculated as in Eqs. (16) and (17), respectively
$Q˙h=m˙hCh(Th,i−Th,o)$
(16)

$Q˙c=m˙cCc(Tc,o−Tc,i)$
(17)
The overall heat transfer coefficient, Uo, was calculated according to White [24] as
$Uo=Q˙avgAoΔTLMTD$
18)
The annulus heat transfer coefficient, ho, of double-helical coil was determined from the overall heat transfer coefficient relationship:
$1Uo=AoAihi+Aoln(do/di)2πkL+1ho$
(19)
The heat transfer coefficient of the inner tube, hi, was calculated from Gnielinski [25] correlation as
$hi=kdi⋅(fi/8)RePr1+12.7fi/8(Pr2/3−1)$
(20)
The Nusselt number can be calculated as
$Nu=hdhyk$
(21)
In which the inner tube friction factor, fi, was given by
$fi=[0.3164Re0.25+0.03(dhy2R)0.5]$
(22)
The effectiveness of the double-helical coiled tube is the ratio of the actual heat transfer to the maximum possible heat transfer and it can be calculated as
$ε=Q˙avQ˙max=Q˙av(Cm˙)min(Th,i−Tc,i)$
(23)
The annulus friction factor was calculated as
$f=2ΔpdhyLρv2$
(24)
In most cases of passive enhancement techniques, the heat transfer enhancement comes on the expense of pressure drop. Therefore, it was necessary to estimate the amount of heat transfer enhancement at the expense of loss in pressure which was defined as the heat transfer per unit pumping power through the double-helical tubes. The heat transfer per unit pumping power for the annulus side can be calculated as
$Q˙/Pp=cpΔT[ΔP/ρ]$
(25)

## 7 Results and Discussion

The thermofluid performance criteria of the heat exchanger with double-helical coiled tube and nanofluid were presented in the form: Nusselt number, overall heat transfer coefficient, friction factor, heat exchanger effectiveness, and heat transfer per unit pumping power. This criterion was demonstrated at different Al2O3 nanofluid concentrations, coil curvature ratio, and flow configurations.

### 7.1 Effect of Nanoparticles Concentration.

The thermal performance of the double-helical coiled tube due to different Al2O3 concentrations was considered with a particular reference of pure water. The results were presented for test specimen A as a sample of the total results. Figure 6 illustrated the Nusselt number though the annulus of the double-helical coiled tube versus Reynolds number at h = 0.05 kg/s and different nanoparticles concentration. It evident that the presence of Al2O3 nanoparticles in working fluid (water) augment the heat transfer Nusselt number which can be referred to the enhanced thermal properties of the nanofluid. At a certain value of Re = 10,000, the Nusselt number of the 2% nanoparticles concentration and curvature ratio = 0.026 was 44% higher than that of the base fluid (zero concentration). The friction factor of the nanofluid in the annulus of the double-helical coiled tube was illustrated in Fig. 7. Although the pressure drop of nanofluid flow through the annulus increased when compared with the base fluid flow of pure water, the friction factor for all nanoparticles concentration (0%, 0.5%, 1%, and 2%) was close together. This can be attributed to an increase in nanofluid viscosity and density when compared with the pure water which led to a compensation of the increase in pressure drop according to Eq. (23). The effect of nanoparticles with a concentration of 0% ≤ φ ≤ 2% on the annulus friction factor can be considered negligible [26]. Figure 8 demonstrates the effect of nanofluid concentration on the effectiveness of a double helically coiled tube compared with pure water. Both the experimental and numerical results were applied for test specimen A with curvature ratio of 0.026 and h = 0.05 kg/s. There was a significant effect of adding Alumina nanoparticles on the effectiveness of double helical comparing with pure water. At a certain value of annulus Re = 10,000, the effectiveness at a nanoparticles concentration of 2% was increased by 20.52% compared with pure water. In most heat exchanger applications, the overall heat transfer coefficient which comprises the heat transfer coefficients of inside and outside the annulus of double-helical tube besides the heat conduction through the solid wall is the most common factor on how to measure the enhancement will be achieved. Figure 9 illustrated the overall heat transfer coefficient versus annulus Reynolds number of a double helically coiled tube for test specimen A with curvature ratio of 0.026 and h = 0.05 kg/s. Clearly, the overall heat transfer coefficient increases with the increase of Reynolds number. For a certain value of the Reynolds number of Re = 10,000, there was a 32% increase of the overall coefficient at a concentration of Al2O3 nanofluid = 2%. This can be explained to the annulus flow mixing due to the presence of nanoparticles and increase in nanofluid thermal conductivity which in turn increases in heat transfer coefficient.

The heat transfer rate per unit pumping which expresses the enhancement in heat transfer at the expense of the drop in pressure was illustrated in Fig. 10. The heat transfer per unit pumping power decreases with the increase of Reynolds number which referred to the increase in pumping power which was achieved more rapidly than the increase in heat transfer according to Eq. (24). Higher values of the heat transfer per unit pumping power can be obtained at lower values of nanoparticle concentration. The numerical solution of the double-helical coiled tube model which produced flow contours is illustrated in Fig. 11. The velocity and temperature contours of nanofluid through the helical annulus of the three test specimens A, B, and C were indicated that a secondary flow induced due to a helicoid shape. As the curvature ratio decreased, the secondary flow exceeds which in turn enhances the flow mixing and hence the enhancement of heat transfer occurred.

### 7.2 Effect of Coil Curvature Ratio.

In order to investigate the coil curvature ratios on the performance characteristics of the double-helical heat exchanger, three tested specimens with different curvature ratio of the double-helical coiled tube were examined. The experimental and numerical data were presented for test specimens A, B, and C. Figure 12 presented the Nusselt number versus Dean number for counterflow through a double helically coiled tube. For h = 0.041 kg/s and Dean number of 1500, the Nusselt number of the test specimen with a curvature ratio of 0.026 was increased by 21% compared with the test specimen with a curvature ratio of 0.043. This referred to the increase in the secondary flow which was generated by the effect of helicoid shape and hence the enhancement of heat transfer was evidenced.

The heat exchanger effectiveness versus annulus Dean number with different curvature ratios was illustrated in Fig. 13. At a certain value of h = 0.041 kg/s, the double-helical coil effectiveness with curvature ratio of 0.026 was higher than that of the helical coil with a curvature ratio of 0.043 by 25%. This can be attributed to an increase in heat transfer surface area which in turn reflects on the rate of heat transfer and then on the heat exchanger effectiveness. Figure 14 illustrates the overall heat transfer coefficient versus annulus Dean number for a range of 800 ≤ De ≤ 3575 at h = 0.041 kg/s. Clearly, the overall heat transfer coefficient increases with the increase of Dean number. For the annulus Dean number of 1500, the overall heat transfer coefficient with a curvature ratio of 0.026 was higher than that of the helical coil with a curvature ratio of 0.043 by 20%. The total tube length of the helical coil decrease as the curvature ratio increases for the same number of turns which led to a reduction in the heat transfer surface area and then in heat transfer rate. Figure 15 illustrates the heat transfer rate per unit pumping power versus annulus Dean number of the counter flow at different curvature ratios for a range of 800 ≤ De ≤ 3575 at h = 0.041 kg/s. The higher values of heat transfer per unit pumping power were obtained at lower values of the annulus Dean number. At lower values of Dean number, a reduction in pump power consumption due to a decrease in volumetric flow rate occurred. In this case, a higher temperature difference through the heat exchanger was obtained which results in an increase the heat transfer rate and hence increase in the heat transfer rate per unit pumping power. An increase of 28% of heat transfer per unit pumping power was acquired of the test specimen with a curvature ratio = 0.043 compared with the test specimen with a curvature ratio = 0.026.

### 7.3 Effect of Flow Configuration.

The effect of the parallel and counters flow arrangement on the overall heat transfer coefficient and heat transfer per unit pumping power are illustrated in Figs. 16 and 17. These results were shown for test specimen “B” as a sample of test specimens in which the data of test specimens “A, B, and C” were similar in this case. Figure 16 illustrates the overall heat transfer coefficient versus the annulus Dean number for a range of 1000 ≤ De ≤ 3160 at hot water mass flowrate of 0.066 kg/s and curvature ratio of 0.032. For the annulus Dean number of 1500, the overall heat transfer coefficient for counter flow was higher than that of parallel flow by 44.7% [23]. This can be attributed to the increase in heat transfer rate in counter flow which corresponds to the increase in logarithmic means temperature difference. The heat transfer per unit pumping power versus annulus Dean number for a range of 1000 ≤ De ≤ 3160 at hot water mass flowrate of 0.066 kg/s and curvature ratio of 0.032 is illustrated in Fig. 17. It can be noticed that the heat transfer per unit pumping power increases to the maximum value at Dean number = 1600 and decreases gradually. For the annulus Dean number of 1500, the heat transfer per unit pumping power for counter flow was higher than that of the parallel flow by 18%.

## 8 Conclusion

The combination of two enhancement techniques of the helicoid shape of double tube heat exchanger incorporated with the addition of Alumina nanoparticles to water as working fluid was investigated experimentally and numerically. The effects of Al2O3 nanofluid concentration, coil curvature ratio, Reynolds number, and flow configuration were discussed. This study focused on the thermofluid characteristics of the flow inside the annulus of the double-helical coiled tube. The effectiveness, overall heat transfer coefficient, and the heat transfer per unit pumping power of double-helical coil were presented for all investigated parameters. The main conclusions were epitomized as

• A considerable heat transfer enhancement can be achieved with double enhancement techniques of nanoparticles and a helicoid shape of the double tube heat exchanger.

• At the same Reynolds number and curvature ratio, the increase of Al2O3 nanoparticles concentration of 2% produces an increase in the Nusselt number up to 44% and an increase in the overall heat transfer coefficient by up to 32%.

• Higher values of the heat transfer per unit pumping power can be obtained at lower values of both nanofluid concentration and curvature ratio.

• As the annulus curvature ratio decreased from 0.043 to 0.026, the Nusselt number was augmented by 21% at the expense of 13.5% in the friction factor.

• The Nusselt number of counterflow arrangement with nanofluid was higher than that of the parallel flow arrangement by 10.4% at the same operating condition.

## Acknowledgment

The principal investigator would like to express his sincere gratitude to the College of Technological Studies–Public Authority for Applied Education, Kuwait, Research project No. TS-17-12 for supporting and funding this research work. He would also like to thank the RHVAC department, Faculty of Industrial Education, Helwan University, Egypt, that facilitated this research work.

## Nomenclature

• b =

pitch, m

•
• d =

diameter of inner tube, m

•
• f =

friction factor

•
• h =

heat transfer coefficient, W/m2 K

•
• k =

thermal conductivity, W/m °C

•
• v =

velocity, m/s

•
• A =

surface area, m2

•
• C =

specific heat, kJ/kg K

•
• D =

diameter of outer tube, m

•
• L =

length, m

•
• R =

•
• T =

temperature, K

•
• U =

overall heat transfer coefficient, W/m2 °C

•
• $m˙$ =

mass flow rate, kg/s

•
• $Q˙$ =

heat transfer rate, W

•
• dhy =

hydraulic diameter (Di-do), m

•
• De =

Dean number (Re.δ0.5)

•
• LMTD =

log mean temperature difference, °C

•
• Nu =

Nusselt number (h.dhy/k)

•
• Pp =

pumping power

•
• Pr =

Prandtl number, µ C/k

•
• Re =

Reynolds number (ρvdhy/µ)

•
• Δp =

pressure drop, Pa

### Greek Symbols

• ɛ =

effectiveness

•
• λ =

annulus curvature ratio, dhy/2R

•
• µ =

dynamic viscosity, kg/m s

•
• ρ =

density, kg/m3

•
• φ =

concentration, %

• avg =

average

•
• b =

base

•
• c =

cold

•
• f =

fluid

•
• h =

hot

•
• i =

inner/inlet

•
• max =

maximum

•
• n =

nano

•
• p =

particles

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