1 Introduction

The need to achieve high thermo-hydraulic performance and compact heat exchangers have pushed many industries to find for new methods to promote heat transfer. Applying dimpled surface is a unique passive technique for heat transfer enhancement (HTE). In this regard, the dimpled surface has emerged as one way to increase thermo-hydraulic performance and minimize the thermal boundary layer thickness of the heat exchangers’ surfaces. The enhanced surfaces creates rotating and/or secondary flows that also increases the effectiveness of heat transfer surface area. It interrupts thermal and velocity boundary layer development with increasing degrees of turbulence near the rough tube wall. This, in turn, increases the convective heat transfer coefficient (h) with a consequent increase in the pressure drop along the tube. The studies of heat transfer enhancement in dimpled tubes were early reported by Kalinin et al. [1] and Giovannini et al. [2]. Several investigators have reported that dimpled tubes are effective devices for augmenting the rate of heat transfer in heat exchangers, especially for shell-tube and concentric-tube heat exchangers [3,4,5,6,7,8,9,10]. Suresh et al. [10] studied the thermo-hydraulic behavior in a dimpled tube with CuO-water nanofluids having various concentrations of nanoparticles (0.1, 0.2 and 0.3% by volume). It was concluded the use of nanofluids together with the dimpled tube resulted in better heat transfer as compared to the use of water as a testing fluid. Chen et al. [11] investigated the thermo-hydraulic behavior in a dimpled tube and found that the heat transfer rate with dimpled tubes increased by 25 to 137% over a plain tube. Wang et al. [12] studied the thermo-hydraulic behaviors in the dimpled tubes for both staggered and aligned arrangements. They reported that the dimpled tubes in aligned arrangement enhanced heat transfer rate and friction factor up to 22.7 and 25.9% as compared to those of a plain tube. Wang et al. [13] also studied the thermo-hydraulic behavior in an ellipsoidal dimpled tube. Their results indicated that ellipsoidal dimpled tubes provided 175% higher heat transfer rate than a smooth straight tube with the nearly 75% higher pressure loss. García et al. [14] investigated the effect of dimpled tubes on the heat transfer rates in in laminar and turbulent flow regimes. It was also reported that corrugated and dimpled tubes provided superior thermo-hydraulic performance to coiled wire which was attributed the lower pressure drop for similar convective heat transfer coefficients (h) encountered at Reynolds numbers higher than 2000. Kim [15] studied the influences of the roughness geometries on thermo-hydraulic behavior to optimize the dimpled tube configuration and found that dimpled tubes gave better thermo-hydraulic performance than a tube with diamond-shaped roughness. Li et al. [16] examined the thermo-hydraulic performance of a dimpled tube in steady state single phase (liquid-to-liquid) fluid flow for Reynolds numbers ranging from 500 to 8000 and for a water/glycol solution with Re ranging from 150 to 2000. It was reported that dimples disturbed the boundary layers, generating secondary flows which increased turbulence. Li et al. [17] investigated the heat transfer rate and pressure drop in helically dimpled tubes. Their results showed that the protrusion of dimples changed the fluid flow pattern, which enhancing turbulence through flow interaction and originated secondary flow. The maximum and average heat transfer coefficients of the helically dimpled tube were 2.48 and 1.74 times of the plain tube, respectively. Sarmadian et al. [18] investigated condensation heat transfer in a helically dimpled tube using refrigerant R-600a as a working fluid. Their results showed that the heat transfer rate was increased up to two times greater than that of a smooth straight tube. Kumar et al. [19] carried out analyses of the influence of a dimpled rib on the thermo-hydraulic performance of a heat exchanger tube. The maximum thermo-hydraulic performance and heat transfer rate were 2.87 times 3.18 times that of a plain tube, respectively. Ganjbakhsh et al. [20] reported the effects of surface roughness of semicircular curved horizontal dimple tubes on heat transfer, friction coefficient and thermal performance. Their results showed that the dimpled tube had better thermal performance than that a smooth tube especially at the low Reynolds number. Xie et al. [21] studied the heat transfer and thermal performance of enhanced tube with cross ellipsoidal dimples. They observed that the cross ellipsoidal dimples induced the transverse and longitudinal dimples which caused the reattachment and periodic impingement flows that helped in improving thermal performance. Again, Xie et al. [22, 23] examined the heat transfer and thermal enhancement performance of enhanced tube with dimples and protrusions [22] and enhanced tube with teardrop dimples [23]. They reported that the enhanced tube with dimples and protrusions yielded thermal enhancement performance up to 1.65 while the enhanced tube with teardrop dimples yielded performance up to 2.06. Aroonrat and Wongwises [24] studied the effect of dimpled depth on heat transfer enhancement, pressure loss and overall performance of dimpled tubes. Their results showed that the dimpled tube with the largest dimpled depth gave the highest heat transfer enhancement and pressure loss up to 83 and 892% over those of the smooth tube.

Recently, nanofluids were applied in dimpled tubes for further heat transfer enhancement. Kumar et al. [25] investigated the thermo-hydraulic behaviors in a square passage with rib shape protrusions using Al2O3-water, CuO-water, and ZnO-water nanofluids as working fluids in turbulent flow regime (4000 ≤ Re ≤ 18,000). The effects of nanofluid concentration, stream wise distance, and protrusion height to cross-section diameter ratio on thermo-hydrodynamic behaviors were reported. It was observed that the thermal performance increased with the concentration and decreased with nanoparticle diameter. Akyürek et al. [26] studied thermo-hydraulic behaviors in a concentric tube heat exchanger with Al2O3-water nanofluids as the test fluids and used wire coil inserts as a turbulence promoter. The effects of nanofluid concentration and the pitch ratio of the wire coils on the thermo-hydraulic behavior were also studied. It was reported that pressure loss and heat transfer rate increased with increasing nanofluid concentration and decreasing pitch ratio of the wire coils. Suresh et al. [27] investigated the convective heat transfer characteristics of helically dimpled tubes using CuO-water nanofluid as the testing fluid. It was found that the use of nanofluids in a dimpled tube resulted in obvious enhancement of heat transfer coefficient with an insignificant increase of pressure loss. Recently, the combined effects of the nanofluids and helical/tape inserts were examined [28,29,30,31] and their results showed that the compound technique could improve the heat transfer rate and reduce exergy loss.

The above review shows that of the uses dimpled tubes, twisted tapes and nanofluids are the promising methods for thermal enhancement. The results of the dimpled tube, nanofluid, twisted tape and combined heat transfer enhancement techniques reported in literature motivates the current study of heat transfer augmentation using dimpled tubes with a nanofluid and twisted tape as a compound technique. Dimpled surfaces were used to boost the blending of the fluids via formation of secondary flow nearby the trough of the dimpled surfaces, nanofluids were applied to improve the thermo-fluid properties and twisted tapes were employed to promote the swirling flow along the flow pass. Current study was performed to evaluate the thermo-hydraulic performance in a heat exchange with compound technique. The effects of (1) dimpled tubes with four dimple angles of 0, 15, 30 and 45°, (2) twisted tapes with twist ratios of 3.0, 4.0 and 5.0, and (3) TiO2-water nanofluids with volume concentrations of 0.05, 0.1 and 0.15 vol.%, on thermo-hydraulic performance were investigated. The study encompassed Reynolds numbers (Re) ranging from 5000 to 15,000. The heat transfer rate and friction factor as well as empirical correlations for the Nusselt number, friction factor and thermo-hydraulic performance were also developed and reported.

2 Dimpled tube and twisted tape

In the present work, concentric double pipe heat exchanger was employed. Inner tubes (dimpled tubes) were made of copper with an inner diameter of 18 mm while the annulus (smooth tube) was made of stainless steel with an inner diameter of 52 mm as displayed in Fig. 1. The dimpled tubes were fabricated with various dimple angles (θ) of 0, 15, 30 and 45°. The concentric tube heat exchanger with a length of 1100 mm was well insulated to minimize heat losses to environment. An image of the dimpled tube alone and dimpled tube with twisted tape inserted are shown in Figs. 1 and 2.

Fig. 1
figure 1

Images of a dimpled tube and twisted tape inserts

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Fig. 2
figure 2

Images of plain tube and dimple tubes at various dimpled attach angles (θ)

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The dimpled surfaces were formed in a staggered manner, with a constant dimple pitch (p) = 12 mm or pitch ratio of p/D = 0.67. The diameter of an ellipsoidal dimple and its depth (e) were constant at 6 × 3 mm2 and 1.5 mm, respectively. Twisted tapes were inserted into dimpled tubes to generate swirls in the flow through the whole test section. Each twisted tape was formed from a stainless sheet with a thickness of 0.8 mm, length (L) of 1100 mm and width (W) of 15 mm. The twisted tapes used in the present study were fabricated with three twist lengths (y) of 45, 60, and 75 mm, corresponding to twist ratios (y/W) of 3.0, 4.0 and 5.0, respectively.

3 Experimental apparatus and methodology

A schematic diagram and photographs of the experimental setup for this heat transfer test are given in Figs. 3 and 4, respectively. The experimental apparatus consisted of a concentric tube heat exchanger, a set of resistance temperature detectors (RTDs) and T-type thermocouples, cooling and heating water batches, a differential pressure transducer, a mixer, centrifugal water pump, heater controller, a data logger, a personal computer and two rotameters to measure the volumetric flow rates of water as well as the nanofluids. The cold water flow was fed through a 2.0 m calming section to form a fully developed flow prior to being directed through the test section. A set of resistance temperature detectors (RTD) and T-type thermocouple sensors were utilized to measure bulk fluid and wall temperatures. Four RTDs were mounted to measure the inlet and outlet water temperatures of the working fluids. During the experiments, a differential pressure transducer was used to monitor pressure loss (ΔP) of the fluids in the test section. The pressure loss across the test section was monitored under isothermal flow conditions (constant fluid temperature). Titanium dioxide (TiO2) nanoparticles with an average diameter of 50 nm were purchased from Nanostructured and Amorphous Material, Inc., USA. Nanofluids were prepared with the required volume concentrations, 0.05, 0.1 and 0.15%, by dispersing the TiO2 nanoparticles in de-ionized water. Then, the mixture was sonicated for 5 h prior to use as the working fluid.

Fig. 3
figure 3

Details of the experimental facility: (1) outer tube with PVC,(2) inner tube with plain copper tube or dimpled tube, (3–4) twisted tapes, (5) hot water tank, (6) electrical heater, (7) heater controller, (8) hot water pump, (9) rotameter for hot water, (10) resistance temperature detectors (RTDs), (11) U-tube manometer or digital pressure gage, (12) data logger, (13) personal computer, (14) water chiller for suppling cold water, (15) cold water pump and (16) rotameter for cold water

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Fig. 4
figure 4

Procedure of TiO2-water nanofluid preparation

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During experiments, hot fluids (water/nanofluids) flowed through the dimpled tube while cold water as a cooling medium flowed through the annulus of the counter-current flow heat exchanger. The temperature of inlet cooling water was kept constant at 25 °C and the flow rate of cold water was maintained at 500 l/h. The hot fluids were fed with the inlet temperature of 80 °C and Reynolds numbers ranging from 5000 to 15,000. All experimental data were recorded under steady state conditions.

The uncertainties of non-dimensional/dimensional parameters [32, 33] of the present experimental results are described in Table 1. The experimental results are within the reported uncertainty range.

Table 1 Uncertainties of experimental parameters
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4 Data reduction

4.1 Thermophysical properties of nanofluids

Thermophysical properties of nanofluids were evaluated using the widely used correlations which are mentioned below. The density of TiO2-water nanofluid (ρnf) was determined using the standard formula for the mixture:

$$ {\rho}_{nf}=\left(1-\phi \right){\rho}_w+\phi {\rho}_{np} $$
(1)

The specific heat (cp,nf) of the TiO2-water nanofluid was determined by:

$$ {c}_{p, nf}=\frac{\phi {\rho}_{np}{c}_{p, np}+\left(1-\phi \right){\rho}_w{c}_{p,w}}{\rho_{nf}} $$
(2)

The above equations were found to be appropriate for TiO2-water nanofluid through equation below [34, 35]:

The thermal conductivity TiO2-water nanofluid (knf) was determined using Eq. (3).

$$ \frac{k_{nf}}{k_w}=\frac{k_{np}+2{k}_w+2\phi \left({k}_{np}-{k}_w\right)}{k_{np}+2{k}_w-\phi \left({k}_{np}-{k}_w\right)} $$
(3)

The viscosity of TiO2-water nanofluid (μnf) was evaluated using Einstein’s formula expressed as:

$$ {\mu}_{nf}={\mu}_w\left(1+\eta \phi \right) $$
(4)

4.2 Heat transfer, friction loss and thermo-hydraulic performance

In the experiments, the cold water absorbed heat from hot fluids flowing in the test section. The heat absorbed by the cold water and supplied by hot fluids were calculated from

$$ {\overset{.}{Q}}_c={\overset{.}{m}}_c{c}_{pc}\left({T}_{c, out}-{T}_{c, in}\right) $$
(5)
$$ {\overset{.}{Q}}_h={\overset{.}{m}}_h{c}_{ph}\left({T}_{h, out}-{T}_{h, in}\right) $$
(6)

where \( {\overset{.}{m}}_c={\rho}_c{\overset{.}{V}}_c \) and \( {\overset{.}{m}}_h={\rho}_h{\overset{.}{V}}_h \).

The cold and hot fluid flow rates, temperatures and the pressure losses in a dimpled tube were measured on both sides of the concentric tube heat exchanger using calibrated rotameters, resistance temperature detectors (RTD) and U-tube manometer or digital pressure gage. All the data were recorded and displayed on a personal computer (PC).

The mean heat transfer rates were given by:

$$ {Q}_{ave}=\left({Q}_c+{Q}_h\right)/2 $$
(7)

For fluid flows in a double pipe heat exchanger, the overall heat transfer coefficient (U) was determined from

$$ {Q}_{ave}=U{A}_i\varDelta {T}_{LMTD} $$
(8)

where

$$ {A}_i={\pi d}_iL $$
(9)

In the present work, the Wilson plot method was employed to develop the Nusselt number correlation for the flow of TiO2-water nanofluid through the dimpled tubes mounted with twisted tape inserts. The correlation development was based on the separation of the overall thermal resistance into the internal convective thermal resistance and the other thermal resistances involving the heat transfer process. In the experiments, the tube-wall temperature cannot be measured directly. Therefore, the tube-side convective heat transfer coefficient (hi) was determined from the overall heat transfer coefficient (U) as found from Eqs. (8) and (9). The tube-side convective heat transfer coefficient (hi) was evaluated as:

$$ 1/U= 1/{h}_i+{A}_i\ \mathit{\ln}\left({d}_o/{d}_i\right)/ 2\pi kL+{A}_i/{A}_o{h}_o+{R}_f $$
(10)

Note that Rf is a fouling resistance [11]. When the last three terms (Ailn(do/di)/2πkL, Ai/Aohoand Rf) on the right-hand side of Eq. (10) were kept constant and set to B, then Eq. (10) can be re-written as

$$ 1/U= 1/{h}_i+B $$
(11)

For the turbulent flow, the tube-side convective heat transfer coefficient (hi) is proportional to the Reynolds number (Re) powered m that can be write in a function shown below [11]:

$$ {h}_i={CRe}^m $$
(12)

where C and m stand for constant and power index values.

Substituting Eq. (12) into Eq. (11) yields:

$$ 1/U=1/{CRe}^m+B=A\ {\mathit{\operatorname{Re}}}^{-m}+B $$
(13)

Equation (13) implies that the plot between 1/U and Re-m is a straight line with its slope of A and intercept at B in Y-axis (1/U) [11]. Rearranging Eq. (13) yields

$$ {h}_i= 1/\left( 1/U-B\right) $$
(14)

The mean heat transfer rate (Nu) is then determined from

$$ Nu={h}_i{d}_H/k $$
(15)

The local thermal conductivity (k) of the fluid is determined from the fluid properties at the local mean bulk fluid temperature.

The Reynolds number (Re) is based on the flow rate at the inlet of the test section. The Reynolds number at the entry of the dimpled tube is

$$ \mathit{\operatorname{Re}}={\rho Vd}_H/\mu $$
(16)

where ρ and μ are the density and dynamic-viscosity of the test fluid and dH is mean dimpled tube diameter.

The friction factor (f) in the dimpled tube can be calculated from

$$ f= 2\Delta {Pd}_H/\left({\rho LV}^2\right) $$
(17)

Finally, the thermo-hydraulic performance can be written as [36, 37]:

$$ \eta =\left(\frac{N{u}_E}{N{u}_0}\right)/{\left(\frac{f_E}{f_0}\right)}^{1/3} $$
(18)

To evaluation the thermal performance the dimpled tube with enhance devices (twisted tape and nanofluid), the performance of the dimpled tube with enhance devices is evaluated relatively to the plain/smooth tube at an identical pumping power in the form of thermo-hydraulic performance (η).

5 Experimental results

The results of thermo-hydraulic behaviors showing the effects of (1) dimpled tubes with four dimple angles of 0, 15, 30 and 45°, (2) twisted tapes with twist ratios of 3.0, 4.0 and 5.0, and (3) TiO2-water nanofluids with concentrations of 0.05, 0.1 and 0.15 vol.% are reported.

5.1 Validation test of a dimpled tube

Calibrated experimental test set-up was tested by using a conventional plain/smooth tube, for validation which is essential test to ensure the accuracy of the experimental data. The average Nusselt number and friction factor values were compared with the values calculated from the correlations developed by Dittus-Boelter and Gnielinski for Nusselt numbers (Nu) and Blasius and Petukhov correlations [38] for friction factors (f). The comparisons are presented in Fig. 5a–b. The Nusselt numbers obtained from the Dittus-Boelter and Gnielinski correlation were within ±6.7 to ±11.8% of experimental values whereas the friction factors calculated from Blasius and Petukhov correlation were between ±6.1 and ± 9.2% of experimental data. The comparisons showed that the experimental data were within an acceptable range. Moreover, the experimental Nusselt numbers and friction factors data of the dimpled tube were also validated using published data [5, 13] as presented in Fig. 6a–b. The results of Thianpong et al. [5] and Wang et al. [13] results were comparable to the present experimental data and deviations within ±3.6 and ± 10.5% for Nusselt number and ± 50.6 and ± 48.5% for friction factor.

Fig. 5
figure 5

Validation of the plain tube

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Fig. 6
figure 6

Validation of the dimpled tube

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5.2 Effect of dimpled angle (θ)

The influence of the dimple angles (θ = 0, 15, 30 and 45°) on the heat transfer rate (Nu) of the dimpled tube inserted with a twisted tape insert is demonstrated in Fig. 7a. Evidently, heat transfer (Nu) increased significantly with increasing dimple angle since the rougher dimples cause stronger turbulence than the smoother ones. At a given Reynolds number, the highest Nusselt number (Nu) was obtained by using the dimpled tube with the dimple angle of 45°. The heat transfer (Nu) of the dimpled tube with a 45° dimple angle was greater than that of tubes with dimple angles, 0, 15 and 30°, by 2.3, 1.6 and 0.7%, respectively. Figure 7b shows the Nusselt number enhancement ratio for dimple angles ranging from 0 to 45°. Nusselt number enhancement ratios ranged from 1.33 to 1.41 over the range of Reynolds numbers.

Fig. 7
figure 7

Effect of dimple attach angle (θ) on the heat transfer rate

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Figure 8a–b shows the effect of dimple angle on friction losses. It was found that friction loss and friction factor enhancement ratio increased with increasing dimple angle. At dimple angle of 45°, friction factor enhancement ratios ranged from 2.26 to 2.34, over the entire range of Re.

Fig. 8
figure 8

Effect of dimple angle (θ) on the friction factor

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Figure 9 demonstrates the variation in the thermo-hydraulic performance ratio with Reynolds number at various dimple angles (0, 15 and 30°). Obviously, thermo-hydraulic performance ratio increased with increasing dimple angle, resulting from the better tradeoff between improved heat transfer and increased friction loss penalty. At dimple angle of 45°, thermo-hydraulic performance ratios ranged from 1.035 to 1.063, over the entire range of Re.

Fig. 9
figure 9

Effect of dimple angle (θ) on the thermo-hydraulic performance

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5.3 Effect of twist ratio

The variation in the heat transfer rate (Nu) with Re for the dimpled tubes mounted with a twisted tape swirl generator (TT) is demonstrated in Fig. 10a–b. The heat transfer (Nu) of the plain tube is also reported for comparison. At comparable operating conditions, the heat transfer rate (Nu) of the dimpled tubes with TTs were consistently higher than those of the plain or the dimpled tubes alone, attributed to the combined effects of the turbulence/re-circulation flow induced by the dimpled rough surfaces and the swirling flow induced by TTs.

Fig. 10
figure 10

Influence of twist ratio on heat transfer enhancement

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Figure 10a–b also reveals that Nusselt number and heat transfer enhancement ratio (NuE/Nu0) increased as increasing dimpled angle of dimpled tubes and decreasing twist ratio of twist tapes. The mean Nusselt number (Nu) of the dimpled tube with twisted tape swirl generators having y/W of 3.0, 4.0 and 5.0 were higher than those of the dimpled tubes alone by 46.5, 39.2 and 32.7%, respectively, and higher than that of the plain tube alone by 106.9, 96.6 and 87.4%, respectively. In other words, the heat transfer enhanced by TT insert having y/W of 3.0 was higher than that promoted by TTs with y/W of 4.0 and 5.0 by 5.3 and 10.4%, respectively. It can be explained by the fact that the twisted tape with smaller y/W generates stronger swirl intensity which is more efficient in promoting fluid mixing, breaking the thermal boundary layer and thereby enhancing heat transfer.

The effect of the dimpled tubes combined with a TT insert on the friction factor (f) is displayed in Fig. 11a–b. It can be observed that friction factor (f) gradually increased with decreasing Reynolds numbers (Re). Under similar conditions, the friction factor (f) of the dimpled tubes mounted with TT insert caused higher friction factor (f) than the dimpled tubes alone and the plain tube due to (1) the larger surface area the dimple tube together with that of the twisted tapes and 2) the additional dissipation caused by the compound device. The friction factors of the dimpled tubes with twisted tape swirl generators were found to be1.73–2.03 times over those of the dimpled tube alone and 3.96–4.69 times over those of the plain tube alone.

Fig. 11
figure 11

Influence of twist ratio on friction factor

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In Fig. 11a, shows that the maximum friction factor was caused by the compound device consisting of dimpled tube having the dimple angle of 45° and twisted tape having y/W of 3.0. The maximum friction factor was found to be higher than those caused in the dimpled tube with TTs having twist ratios of 4.0 and 5.0 by 7.8 and 16.0%, respectively. It was also found that the effects of dimple angle and twist ratio on friction factor enhancement ratio are in similar manner with those on friction factor (Fig. 11b).

Figure 12 shows the influence of the twist ratio (y/W) on thermo-hydraulic performance. It was observed that the thermo-hydraulic performance increased with decreasing y/W. At Reynolds number of 5000, the highest thermo-hydraulic performance provided by the dimpled tubes combined with TT inserts having twist ratios of 3.0, 4.0 and 5.0 were 1.24, 1.20 and 1.18, respectively. In other words, the dimpled tube with the TT insert possessing twist ratio of 3.0 gave higher thermo-hydraulic performance than the identical tube with the TT inserts having twist ratios (y/W) of 4.0 and 5.0 by around 2.7 and 5.1%, respectively.

Fig. 12
figure 12

Influence of twist ratio on the thermal performance

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5.4 Effect of nanofluid concentration (ϕ)

Figures 13a–b, 14a–b and 15 show the effect of TiO2-water nanofluid concentration on heat transfer (Nu), friction loss and thermo-hydraulic performance of the dimpled tube with the dimple angle of 45°. It was observed that heat transfer (Nu), friction loss and thermo-hydraulic performance increased as the TiO2-water nanofluid concentration increased. Over the investigated range, the highest thermo-hydraulic performance of 1.258 was achieved by using the nanofluid with ϕ = 0.15 vol.% in the dimple angled having the dimple angle of 45°, inserted with twisted tape possessing the twist ratio of 3.0.

Fig. 13
figure 13

Influence of nanofluid concentration (ϕ) on the heat transfer enhancement

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Fig. 14
figure 14

Influence of nanofluid concentration (ϕ) on friction factor

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Fig. 15
figure 15

Influence of nanofluid concentration (ϕ) on the thermal performance

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5.5 Empirical correlation

Experimentally obtained data were used to develop correlations of Nu, f and thermo-hydraulic performance as function of flow Re, twist ratio and dimple angle using regression analysis. The Nu, f and thermo-hydraulic performance correlations for dimpled tube, dimpled tube with TT inserts and dimpled tubes containing TT inserts using a TiO2-water nanofluid are shown as follows:

  • Dimpled tube:

$$ N{u}_{DT}=0.024{\operatorname{Re}}^{0.836}{\Pr}^{0.3}{\left(1+\tan \theta \right)}^{0.034} $$
(19)
$$ {f}_{DT}=1.317{\operatorname{Re}}^{-0.31}{\left(1+\tan \theta \right)}^{0.044} $$
(20)
$$ {\eta}_{DT}=1.281{\operatorname{Re}}^{-0.023}{\left(1+\tan \theta \right)}^{0.018} $$
(21)

For 5000 ≤ Re ≤ 15,000 and 0° ≤ θ ≤ 45°.

  • Dimpled tube inserted with twisted tape:

$$ N{u}_{DT- TT}=0.057{\operatorname{Re}}^{0.805}{\Pr}^{0.3}{\left(1+\tan \theta \right)}^{0.039}{\left(y/W\right)}^{-0.189} $$
(22)
$$ {f}_{DT- TT}=3.269{\operatorname{Re}}^{-0.3}{\left(1+\tan \theta \right)}^{0.054}{\left(y/W\right)}^{-0.28} $$
(23)
$$ {\eta}_{DT- TT}=2.224{\operatorname{Re}}^{-0.059}{\left(1+\tan \theta \right)}^{0.02}{\left(y/W\right)}^{-0.095} $$
(24)

For 5000 ≤ Re ≤ 15,000, 0° ≤ θ ≤ 45°, and 3.0 ≤ y/W ≤ 5.0.

  • Dimpled tube inserted with twisted tape and nanofluid:

$$ N{u}_{DT- TT- NF}=0.062{\operatorname{Re}}^{0.8}{\Pr}^{0.3}{\left(y/W\right)}^{-0.197}{\left(1+\phi \right)}^{0.135} $$
(25)
$$ {f}_{DT- TT- NF}=3.463{\operatorname{Re}}^{-0.301}{\left(y/W\right)}^{-0.29}{\left(1+\phi \right)}^{0.133} $$
(26)
$$ {\eta}_{DT- TT- NF}=2.36{\operatorname{Re}}^{-0.063}{\left(y/W\right)}^{-0.1}{\left(1+\phi \right)}^{0.09} $$
(27)

For 5000 ≤ Re ≤ 15,000, 0° ≤ θ ≤ 45°, 3.0 ≤ y/W < ≤ 5.0, and 0.05 vol.% ≤ ϕ ≤ 0.15 vol.%.

Experimental data are plotted against predicted data as demonstrated in Fig. 16a–c. It was found that the predicted Nusselt number, friction factor and thermo-hydraulic performance data agreed well with the experimental data. A summary of experimental results for Nu, f and thermo-hydraulic performance are given in Tables 2, 3 and 4.

Fig. 16
figure 16

Experimental data versus predicted data

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Table 2 Summary of experimental results for dimpled tubes
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Table 3 Summary of experimental results for dimpled tube with twisted tape inserts
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Table 4 Summary of experimental results for dimpled tube with twisted tape inserts at various nanofluid concentrations
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6 Conclusions

The experimental results of heat transfer enhancement, friction factor and thermo-hydraulic performance using TiO2-water nanofluids under turbulent flow (4000 ≤ Re ≤ 18,000). in dimpled tubes with a TT inserts are presented. The effect of following parameters are reported: (1) the dimple angles of the dimpled tubes, θ = 0, 15, 30 and 45°, (2) twist ratio of twisted tapes, y/W = 3.0, 4.0 and 5.0, and (3) TiO2-water nanofluid concentrations, ϕ = 0.05, 0.1 and 0.15 vol%.

  • The dimpled tube with TT inserts consistently yielded higher heat transfer enhancement and caused higher friction factor than those of the dimpled tube or the plain tube alone.

  • The dimpled tubes with the largest dimple angle, 45°, provided higher thermo-hydraulic performance than those at 0, 15 and 30° by 1.5, 0.9 and 0.4%, respectively. The heat transfer rate at θ = 45° was found to be higher than for dimple angles of 0, 15 and 30° by 2.6, 1.6 and 0.7%, since the rougher dimples cause stronger turbulence than the smoother ones.

  • For dimpled tubes with TT inserts, the heat transfer rate as well as pressure loss increased as y/W decreased. Over the entire range of parameters investigated, the heat transfer rates of dimpled tube with TT inserts at y/W = 3.0, 4.0 and 5.0 were higher than those of the dimpled tube alone by 46.5, 39.2 and 32.7%, corresponding to higher thermo-hydraulic performances of 16.3, 13.1 and 10.6%, respectively.

  • Over the investigated range, the highest thermo-hydraulic performance of 1.258 was achieved by using the nanofluid with ϕ = 0.15 vol.% in the dimple angled having the dimple angle of 45°, inserted with twisted tape possessing the twist ratio of 3.0.