Introduction

The use of heat exchangers is widespread in industrial and domestic equipments. Optimizing heat transfer enhancement techniques and in turn making the heat exchangers more efficient can be beneficial at an enormous magnitude. Therefore, increasing the thermal performance of heat exchangers has been a topic of research attracting numerous researchers for the last few decades.

In a conventional heat exchanger unit installed in the chemical industry as shown in Fig. 1, significant energy in the form of heat is under-utilized and released to the surroundings due to the lower thermal performance of the heat exchanger. To increase the thermal performance of the heat exchanger, many researchers have proposed different methods, primarily passive techniques for a broad spectrum of Reynolds number (Re), and have achieved substantial results. Improving heat transfer using physical disturbances has been a prevailing area of study and experimentation for the research community active in this field. Several insert geometries using various configurations and parameters have proved their distinct effect on the outcome of heat transfer, friction factor (f) or pressure drop, and thermal performance factor (TPF) of the heat exchanger. El-Sayed et al. [1] used longitudinal interrupted fins as turbulence promotors with inline, continuous, and staggered arrangements. According to the findings, a maximum pressure drop occurs with a staggered design, resulting in lower thermal performance. Among different insert geometries, twisted tapes are one of the most commonly used geometries reported by several researchers. Works reported by Bas et al. [2], Bhuiya et al. [3], and Bhuiya et al. [4] used twisted tapes as inserts with different combinations of twist ratio and width ratio. Their investigations reported the effect of multiple twisted tapes on heat transfer, f, and TPF and revealed that heat transfer increased with a decrease in twist ratio.

Fig. 1
figure 1

A Typical shell and tube heat exchanger used in a chemical industry

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Although incorporating twisted tapes yields significant results, there is still room for performance improvement. Moreover, twisted tapes primarily affect the core fluid region inside the tube, and the tube–wall region is widely intact; hence, various modifications have been made in twisted tape geometries. Researchers like Promvonge [6] and Promvonge et al. [7] studied the effects of twisted tape with conical insert and twisted tape with uniform coiled wire insert, respectively. They reported a notable improvement in heat transfer in respect of a simple heat exchanger having a plain tube. Eiamsa-ard et al. [5] examined the effects of a compound insert by using twisted tape with rings. This geometry influences the core and wall fluid stream, which hinders the development of the thermal boundary layer. The study revealed a significant improvement in thermal performance over a plain tube heat exchanger. Eiamsa-ard et al. [8] and Bhuiya et al. [9] incorporated serrated twisted tapes and twisted tapes with perforations, respectively, in their experimental investigation. According to the study, incorporating perforation into the geometries significantly reduces pressure drop, thereby improving thermohydraulic performance. Furthermore, the f in the perforated and serrated twisted tape was lower than the standard twisted tape. In a study, Pal and Saha [10] revealed that ribbed tubes with twisted tapes having oblique teeth yield significantly higher heat transfer improvements than individual enhancement techniques. Zhang et al. [11] employed self-rotating twisted tapes as inserts, indicating that the self-rotating inserts improved heat transfer and had anti-scaling and descaling properties. It is also noted that twisted tapes perform better in laminar than turbulent regions. Salem et al. [12] used a helical twisted tape insert in a double pipe heat exchanger and reported that increasing the height ratio and decreasing the pitch ratio (PR) of a helical twisted tape inserts increased the Nusselt number (Nu) and friction; the maximum hydro-thermal performance index was obtained at a height ratio of 0.667. The performance of peripheral u-cut twisted tapes with and without ring inserts was examined by Abolarin et al. [13]. It is reported that in the transitional flow regime, increasing the depth cut ratio and decreasing the spacing ratio of the rings considerably increased the heat transfer rate. Chu et al. [14] examined the impact of various shapes of cuts in twisted tape. The enhancements in Nu and f brought about by u-cut were identical to those brought out by isosceles V-cut configurations, according to experimental results. Fagr et al. [15] investigated the effect of a conventional twisted tape with various decreasing tapered twisted tapes. It was reported that inserting standard twisted tape and reducing tapered twisted tapes resulted in a negligible improvement in thermal performance. The thermal performance of a tube fitted with twisted tape inserts with dimples and perforations was investigated experimentally by Dagdevir and Ozceyhan [16]. The results of the experiment revealed that the novel dimpled twisted tape insert outperforms previous twisted tape geometries in terms of heat transfer performance. Chang al. [17] investigated the performance of a square duct with inclined slots and ribs paired with twist tape. The average increase in Nu was found to be 3.8–4.2 times that of a smooth tube, at the expense of an increase in frictional losses of 32.5–40.2 times that of a smooth tube. Singh et al. [18] studied the double-pipe heat exchanger with dimpled twisted tape as inserts and found the highest increase in heat transfer of 1.82 times that of a plain tube heat exchanger, with a dimple diameter to depth ratio of 0.005 m providing the best thermal performance.

Investigators also looked into a few novel geometries to further increase the thermal performance of the heat exchanger. Promvonge & Eiamsa-ard [19] used conical turbulators and found that closely spaced turbulator produces better outcomes concerning heat transfer than widely spaced turbulators. Kongkaitpaiboon et al. [20] employed a perforated conical ring as an insert geometry and reported that it performed better than a smooth tube heat exchanger. Gunes et al. [21] used triangular cross-sectioned coiled wire in their study with a scope confined to surface alteration resulting in less significant findings. The circular rings were employed as insert geometry by Kongkaitpaiboon et al. [22], who varied the geometrical parameters PR and diameter ratio. It was reported that the lowest value of PR and diameter ratio pertains to maximum heat transfer. In their investigation, Eiamsa-ard and Promvonge [23] used double-sided delta wing tape and found that employing the double-sided delta wings increased the Nu and f by 165% percent and 14.8 times, respectively, over the plain tube, with a maximum TPF of 1.19. Kumar et al. [24] used a solid and perforated circular disk as insert geometry and found that using perforation in the circular disk geometry reduces f, resulting in improved thermal performance. Singh et al. [25] examined perforated hollow circular cylinder inserts and found that the heat transfer rate is 150–230% higher than that of plain tube values; the TPF is in the range of 1.21–1.47, maximum at diameter ratio of 0.65, and a perforation index (PI) of 8%. Straight tape with solid and perforated curved winglets was utilized as inserts by Skullong et al. [26]. They reported that perforated curved winglet tapes had a 9% higher thermal enhancement factor than solid curved winglet tapes. Wijayanta et al. [27] used double-sided delta-wing tape as inserts and found that at a higher wing width ratio, the double-sided tape inserts had higher convective heat transfer enhancements and frictional losses than the longitudinal strip insert and plane tube; the maximum thermohydraulic performance factor was 1.15 at wing width ratio of 0.63. Chamoli et al. [28] used anchor-shaped inserts in their numerical study, which resulted in a significant increase in heat transfer and f, with a maximum thermal enhancement factor of 1.72. Gururatana and Skullong [29] employed airfoil-shaped inserts, claiming that heat transfer increase by utilizing a 45° inclination angle insert is three times better than a conventional tube heat exchanger. Nakhchi et al. [30] used perforated louvered strips as inserts and found that Nu for double perforated louvered strip inserts is higher than for single perforated louvered strip inserts for the same slant angles; the double perforated louvered strip insert has the highest thermal enhancement factor of 1.84. Liang et al. [31] used a center tapered wavy tape insert and reported that a maximum performance enhancement criterion of 2.62 is achieved in respect of a plain tube heat exchanger. Hong et al. [32] examined the transverse corrugated tube (TCT) using twin and triple wire coils in an experimental setting. The Nu enhancement ratio for TCT and TCT combined with multiple wire coil inserts ranged from 1.35 to 1.49 and 1.74 to 2.61, respectively, while the maximum performance evaluation criterion was found to be 1.09 when utilizing twin wire coils. Paneliya et al. [33] employed X-shaped inserts and found that compared to a simple twisted tape, X-shaped inserts demonstrated a 1.27-fold increase in heat transfer. Multiple helical tape inserts were employed by Bahuguna et al. [34] and an increase in thermal performance of 1.23 times than the smooth tube heat exchanger was reported for twist ratio 2 and number of tapes 1. Singh et al. [35] employed conical ring turbulators with protrusion and dimple roughness and found that for protrusion roughness of 0.006 m diameter and pitch space of 0.240 m, the best thermohydraulic factor of 2.44 was reached. Zarei et al. [36] investigated the heat transfer increase produced by a Koflo inline mixture as a swirl creator placed in a circular tube with a crossing angle of 90° and 120° between the blades. It was reported that raising the crossing angle reduces heat transfer enhancement. Ebrahimi and Naranjani [37] numerically studied the flat-plate channel configured with pyramidal protrusions in laminar range and reported that the overall efficiency of the channels with pyramidal protrusions is improved by 12.0–169.4% compared to the plain channel for the conditions studied here. Ebrahimi and Roohi [38] investigated numerically the flow patterns inside mini twisted oval tubes (TOTs) heated by constant-temperature walls and recorded that the improvement in synergy between velocity and temperature gradient and lower irreversibility cause heat transfer enhancement for TOTs. Manoram et al. [39] numerically investigated the influence of dimples and its various parameters on the thermal efficiency of the solar collector and reported a 2.5 times improvement in Nu and 11.1% increase in f concerning smooth tube. Bhattacharyya et al. [40] studied the effect of broken twisted tape on thermal performance of triangular axial corrugated tubes and reported that for all examined cases the performance factors are larger than one.

From the vast discussion of the literature review, it is noticed that the vortex generators are capable of providing disturbances in the core as well as in the wall region of the fluid flow inside the tube and have performed better. The circular rings, flapped wings, and wiglets significantly improve the performance of different insert geometries embedded heat exchangers over the smooth tube. In light of these findings, the vortex generator inserts appear to be a promising alternative to other inserts. Hence, it is critical to develop a new vortex-inducing insert geometry that is capable of increasing heat transfer by disrupting the thermal boundary layer in the heated wall region, allowing the heat exchanger's thermal performance to improve. To propose a novel vortex-inducing insert geometry, a modification in the solid disk is provided by making it into three equal blades and keeping these blades at a certain angle for providing better fluid flow streamlining over them. Furthermore, the perforation in the blades is done for reducing the pressure drop across the tube, thereby increasing the TPF of the heat exchanger. This novel insert geometry used in the present research work is named as “triple-blade vortex generator” (TBVG). This paper investigates and presents the effect of various geometrical and flow parameters of TBVG inserts on heat transfer, fluid flow, and thermal performance of tubular heat exchangers.

Experimental facility

Insert geometry

The TBVG insert is used as a turbulence promoter inside the heat exchanger tube in this work. Different geometrical and flow parameters of TBVG inserts embedded heat exchanger tube used in the experiments are stated in Table 1. Pictorial views of the TBVG insert are shown in Fig. 2, and its dimensional details are presented in Fig. 3.

Table 1 Experimental parameters used in the present study
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Fig. 2
figure 2

a. Pictorial view of TBVG insert. b. Pictorial view of TBVG insert with different PR

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Fig. 3
figure 3

Dimensional details of TBVG insert

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Experimental setup

Experimental apparatus primarily composed of an inlet section, also known as a calming section, a test section with TBVG inserts, a heating arrangement for the test section, and a blower for supplying air. The length of the inlet section is 2.5 m, which ensures that a fully developed fluid flow enters the test section. The test section which is 1.5 m long comes after the inlet section. A heating mechanism is used in the test section, after which insulation is applied to the tube's outer surface. A total of sixteen T-type thermocouples are used to measure temperatures using a data logger which is linked to a computer having LabVIEW software. Twelve thermocouples are fixed near the wall of the tube to measure the tube wall temperature: one at the inlet and three at the exit of flow to determine the inlet and outlet temperature of the air. Thermocouples on the tube's wall are fixed at the fluid–wall interface. The thermocouple is fixed in a hole bored in the tube wall up to the closest point of interaction between the fluid and the tube wall. The pressure drop across the test section is measured with a digital micromanometer connected to knobs at the inlet and exit of the test segment. The flow system follows the test segment, which includes a 3-HP blower with a 3-phase power supply. Before the blower, a flow measuring device is connected, which comprises an orifice plate and a u-tube manometer using distilled water as the manometric fluid. A flow control valve is installed near the blower to regulate the rate of fluid flow during experimentation. Figure 4 shows a schematic diagram of the experimental test setup.

Fig. 4
figure 4

Schematic view of the experimental setup

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The approach proposed by Kline and McClintock [41] is used to calculate the uncertainty in the measurements of different parameters during experiments. The calculated uncertainty for different parameters, i.e., Re, Nu, and f, is ± 7.42%, ± 2.87%, and ± 7.45%, respectively. The uncertainty analysis is presented in "Appendix".

Data reduction and validation

The air is used as a working fluid in the experiments, which is forced to flow in the heated test segment. The data points are recorded when the system reaches a quasi-steady state. When there is no change in any of the measurement quantities recorded within 5 min, the steady state is assumed to have been reached. The heat losses from the tube to the surroundings are neglected; thus, in a steady state the convective heat transfer is equal to the heat carried away by the flowing air and is expressed as:

$${Q}_{\mathrm{air}}={Q}_{\mathrm{conv}}$$
(1)

where

$${Q}_{\mathrm{air}}=\dot{m}{C}_{\mathrm{p}}\left({T}_{\mathrm{o}}-{T}_{\mathrm{i}}\right)$$
(2)

where Ti and To are the temperatures of air at the inlet and outlet of the test segment. To is calculated by area-weighted averaging of the temperatures gauged by the thermocouples, three in number, positioned in the airstream at the outlet of the test section, and is given as:

$${T}_{\mathrm{o}}=\frac{{A}_{\mathrm{c}1}{T}_{\mathrm{o}1}+{A}_{\mathrm{c}2}{T}_{\mathrm{o}2}+{A}_{\mathrm{c}3}{T}_{\mathrm{o}3}}{{A}_{\mathrm{c}}}$$
(3)

where To1, To2, and To3 are the temperatures gauged at the outlet of transverse section area realm Ac1, Ac2, and Ac3, respectively, and Ac is the transverse section area of the tube.

The convection heat transfer rate from the test segment or heat exchanger tube wall is given by:

$${Q}_{\mathrm{conv}}=h{A}_{\mathrm{s}}\Delta {T}_{\mathrm{m}}$$
(4)

where ΔTm is log mean temperature difference of heat exchanger tube and is given as:

$$\Delta T_{{\text{m}}} = \tfrac{{\Delta T_{{\text{i}}} - \Delta T_{{\text{o}}} }}{{\left( {\ln \left( {\frac{{\Delta T_{i} }}{{\Delta T_{o} }}} \right)} \right)}}$$
(5)

where ΔTi and ΔTo are the differences in temperatures of the mean wall temperature of the tube (Twm) with Ti and To, respectively.Twm is calculated by averaging the temperature readings of all the thermocouples (total 12) placed on the wall surface of the test segment and is given as:

$$T_{{{\text{wm}}}} = \frac{{\sum {T_{{\text{w}}} } }}{{12}}$$
(6)

The average heat transfer coefficient (h) is calculated as:

$$h=\frac{\dot{m}{C}_{\mathrm{p}}\left({T}_{\mathrm{o}}-{T}_{\mathrm{i}}\right)}{{A}_{\mathrm{s}}\Delta {T}_{\mathrm{m}}}$$
(7)

where \(\dot{m}\), the mass flow rate of air, is calculated as:

$$\dot{m}={C}_{\mathrm{d}}{A}_{\mathrm{o}}{\left[\frac{2\rho \Delta {P}_{\mathrm{o}}}{1-{\beta }^{4}}\right]}^{0.5}$$
(8)

Mean fluid velocity (V) inside the tube is determined as:

$$V=\frac{\dot{m}}{\rho {A}_{\mathrm{c}}}$$
(9)

The values of Re at different fluid flow rate are obtained by the following standard equation:

$$\mathrm{Re}=\frac{\rho \mathrm{VD}}{\mu }$$
(10)

The average Nu is further calculated as:

$$\mathrm{Nu}=\frac{h\mathrm{D}}{k}$$
(11)

The values of f is obtained as:

$$f=\frac{\Delta P}{\left[\left(\frac{\mathrm{L}}{\mathrm{D}}\right)\left(\frac{\rho {\mathrm{V}}^{2}}{2}\right)\right]}$$
(12)

TPF of the tubular heat exchanger is then calculated as:

$$\mathrm{TPF}=\frac{\mathrm{Nu}/{\mathrm{Nu}}_{s}}{{(f/{f}_{\mathrm{s}})}^{1/3}}$$
(13)

All thermophysical properties of air are taken at fluid bulk mean temperature. Smooth tube validation is performed before collecting heat transfer and f data of TBVG inserts embedded tube. The validation of the smooth tube assures the correctness of the methodology and the instruments used in the experimental work. The average values of Nu and f are determined for smooth tube and are compared with the predicted values from the smooth tube correlations available in the literature and depicted in Eqs. 14 and 15 for f and Nu, respectively.

Blasius equation

$${f}_{\mathrm{s}}=0.316{\mathrm{Re}}^{-0.25}$$
(14)

Dittus–Boelter equation

$${\mathrm{Nu}}_{\mathrm{s}}=0.023{\mathrm{Re}}^{0.8}{\mathrm{Pr}}^{0.4}$$
(15)

The variation of ± 5% and ± 9% from the predicted values is obtained for the Nu and f, respectively, and is shown in Fig. 5. This confirms the accuracy of data measurement and validates the test facility.

Fig. 5
figure 5

Validation of smooth tube for Nu and f

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Numerical simulation

A numerical simulation is done to better understand the fluid flow behavior and heat transfer mechanism inside the heat exchanger tube with TBVG inserts. For numerical simulation, the geometric parameters of the TBVG insert are kept the same as in experimental work. At constant Re = 6000, TBVG inserts with PR = 1 and PI = 25% are used to achieve the fluid flow pattern. During the numerical simulation, the following assumptions are taken into account.

  • The flow is 3D, steady and incompressible.

  • No variation in pressure along the y-direction.

  • No shear force along the y-direction.

  • The body's gravitational force has been ignored.

  • It is a fully developed flow at the test section's inlet.

  • The fluid's axial heat conduction is negligible.

  • At atmospheric pressure and temperature, the thermophysical properties of air do not change.

Turbulent flows are characterized by fluctuating velocity fields. There can be small-scale and high-frequency changes in the transported quantities. Variations in velocity fields cause momentum and energy to be mixed; hence, these quantities vary as well. To remove small scales, the governing equations can be modified that are computationally little expensive to resolve. The k-ε models include independent transport equations for the turbulence length scale, or some comparable parameter, as well as the turbulent kinetic energy, in which turbulent production and dissipation balance. Therefore, a typical (RNG) k-ε model is chosen for simulations because several studies [8, 19, 20, 22, 30] have used the (RNG) k-ε model to produce nearly accurate results in their numerical simulations. The RANS (Reynolds averaged Navier–Stokes) equations and energy equations are applied for simulation. These governing equations along with k and ε are solved using Ansys Fluent, which is a CFD software based on the finite volume method. In the Cartesian tensor system, these governing equations can be written as:

Continuity equation:

$$\frac{\partial }{\partial {x}_{\text{i}}}(\rho {u}_{\text{i}})=0$$

Momentum equation:

$$\frac{\partial }{\partial {x}_{\text{i}}}\left(\rho {u}_{\text{j}}{u}_{\text{j}}\right)=-\frac{\partial p}{\partial {x}_{\text{i}}}+\frac{\partial }{\partial {x}_{\text{j}}}\left[\mu \left(\frac{\partial {u}_{\text{i}}}{\partial {x}_{\text{j}}}+\frac{\partial {u}_{\text{j}}}{\partial {x}_{\text{i}}}\right)\right]+\frac{\partial }{\partial {x}_{\text{j}}}\left(-\rho \overline{{{u}_{\text{i}}}^{^{\prime}}{{u}_{\text{j}}}^{^{\prime}}}\right)$$

Energy equation:

$$\frac{\partial }{\partial {x}_{\text{i}}}\left(\rho {u}_{\text{j}}T\right)=\frac{\partial }{\partial {x}_{\text{j}}}\left(\left(\Gamma +{\Gamma }_{\text{t}}\right)\frac{\partial T}{\partial {x}_{\text{j}}}\right)$$

where \(\Gamma +{\Gamma }_{\text{t}}\) are the molecular thermal diffusivity and turbulent thermal diffusivity respectively and are given by,

$$\Gamma = \mu /Pr\quad \Gamma_{\text{t}} = \mu_{\text{t}} /Pr_{\text{t}}$$

Transport equations used for Renormalization group (RNG) k-ε model, to determine turbulence kinetic energy, k and rate of dissipation, ε for computational domain analysis, are:

$$\frac{\partial }{\partial {x}_{\text{i}}}\left(\rho k{u}_{\text{i}}\right)=\frac{\partial }{\partial {x}_{\text{j}}}\left({\alpha }_{\text{k}}{\mu }_{\text{eff}}\frac{\partial k}{\partial {x}_{\text{j}}}\right)+{G}_{\text{k}}-\rho \varepsilon$$

and

$$\frac{\partial }{\partial {x}_{\text{i}}}\left(\rho \varepsilon {u}_{\text{j}}\right)=\frac{\partial }{\partial {x}_{\text{j}}}\left({\alpha }_{\text{k}}{\mu }_{\text{eff}}\frac{\partial \varepsilon }{\partial {x}_{\text{j}}}\right)+{C}_{1\varepsilon }\frac{\varepsilon }{k}({G}_{\text{k}})-{C}_{2\varepsilon }\rho \frac{{\varepsilon }^{2}}{k}-{R}_{\varepsilon }$$

where Gk represents the generation of turbulence kinetic energy due to mean velocity gradients and is expressed as

$$G_{{\text{k}}} = -\rho \overline{{\stackrel{^{\prime}}{u}}_{\text{i}}{\stackrel{^{\prime}}{u}}_{\text{j}}}\frac{\partial {u}_{\text{j}}}{\partial {x}_{\text{i}}}$$

µeff represents the effective turbulent viscosity and is given as:

$${\mu }_{\mathrm{eff}}=\mu +{\mu }_{\text{t}}$$

The turbulent viscosity, μt, is computed by combining k and ε as follows:

$${\mu }_{\text{t}}=\rho {C}_{\mu }\frac{{k}^{2}}{\varepsilon }$$

where Cμ = 0.09 is constant. The quantities \({\alpha }_{k}\) and \({\alpha }_{\varepsilon }\) are the inverse effective turbulent Prandtl numbers for k and ε, respectively. The model constants have the following values, C1ε = 1.44, C2ε = 1.92, αk = 1 and αε = 1.30. An elaborate explanation of these standard governing equations is provided in [42].

A SIMPLE algorithm is used to link the pressure and velocity. QUICK, which is a higher-order differencing scheme, is used to discretize governing equations. For all variables, the convergence criteria for normalized residual values are set to less than 10–7. The grid independence test is conducted using the solution refinement method. As shown in Fig. 6, the variance in Nu is negligible after 4.16 × 106 cells; hence, mesh with 4.16 × 106 cell numbers will be analyzed for simulation.

Fig. 6
figure 6

Results of grid independence test in graphical form

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The part section of meshed modeled geometry of TBVG inserts is shown in Fig. 7. The boundary conditions applied to a numerical model for simulation are given in Table 2.

Fig. 7
figure 7

Mesh generation of TBVG inserts geometry

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Table 2 Boundary conditions for a geometric model
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Numerical results and discussion

The results of the numerical simulation of TBVG inserts integrated heat exchanger tube are presented in Fig. 8.

Fig. 8
figure 8

a. 3D streamlines, b. Vortex core regions, c. Temperature contours along with velocity vectors, for TBVG insert integrated tube with PR = 1 and PI = 25%, at Re = 6000

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The modified fluid flow structure due to the presence of TBVG insert with PR = 1, PI = 25%, at Re = 6000, is shown in the form of 3D streamlines inside the tube in Fig. 8a. The presence of the TBVG insert causes a swirl flow inside the tube, resulting in better mixing of fluid flowing near the wall and in the tube's center. This new fluid flow pattern aids in the generation of high turbulence intensity, resulting in an improved rate of heat transfer. Furthermore, the presence of perforation causes jet impingement, which aids in the formation of a thinned thermal boundary layer and reduces flow hindrance to the fluid flow, lowering frictional losses. Because of the jet impingement through the perforations, the fluid velocity is also increased.

Figure 8b depicts the vortices created by the recirculation of fluid caused by TBVG inserts. The lambda 2 criterion is used to show the vortex core region and its direction. The vortex core in a tube is represented by the isosurface of a specific vortex strength. The vortices are produced downstream of each blade of the TBVG inserts resulting in three vortex core regions between two consecutive inserts, as shown in Fig. 8b. These vortices aid in the mixing up of fluid inside the tube and increase the fluid's residence time inside the tube.

Temperature contours with velocity vectors are shown at different transverse planes depicting the temperature transformation along the tube in Fig. 8c to demonstrate the temperature distribution through the tube due to the presence of TBVG inserts with PR = 1 and PI = 25% at Re = 6000. The presence of TBVG inserts increases the temperature of the air, and this increase in temperature is the result of a better mix-up of fluid flowing through the tube's center and wall regions.

Experimental results and discussion

The experimental results are plotted in graphical form, and the effects of each parameter on heat transfer and f are examined. Every insert with a different set of geometrical parameters for a specific flow rate performed differently and had a different impact on Nu, f, and TPF. In the following subsections, the effects of different geometrical and flow parameters on Nu, f, and TPF are discussed. The insert geometry and flow parameter values that are most desirable are determined by the heat exchanger's intended use.

Effect on the heat transfer

Figure 9a and b shows the effect of geometrical parameters such as PR and PI on Nu and enhancement in heat transfer over a smooth tube (Nu/Nus) with varying Re. With ascending Re, Nu shows an uptrend. This is because higher velocities of air in the tube produce more turbulence increasing heat transfer.

Fig. 9
figure 9

Influence of different geometrical and flow parameters with error bars on: a. Nu, b. Nu/Nus, c. f, d. f/fs, e. TPF. f. 3D surface plot of TPF at Re = 6000

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Since PR is proportional to the insert spacing, the number of TBVGs in the test section plays a significant role in generating turbulence in the fluid streams. As PR increases, heat transfer decreases significantly, and vice versa. This is primarily due to higher turbulence at lower PR values, which leads to a higher number of vortex and eddy generation in the fluid stream, ultimately resulting in a higher heat transfer value. Chamoli et al. [28], Nanan et al. [43], and Li et al. [44] reported similar results in their numerical investigations of inserts for various PR values. For PR = 1, the highest heat transfer is achieved. For PI = 0%, a decrease in Nu is in the range of 10.7–12%, 12.2–14.8%, 19.2–20.5% for changes in PR from 1–2, 2–3, and 3–4, respectively.

It has been observed that as PI rises, the rate of heat transfer decreases slightly. Perforation in the insert causes jet impingement, which increases the heat transfer rate; however, perforation in the insert also reduces flow blockage, reducing the fluid's resident time, which decreases the heat transfer rate. These two aspects of perforation are of opposing nature. The aspect of flow blockage appears to dominate the jet impingement aspect of perforation in the current geometry. Convective heat transfer is observed to be maximum for 0% PI, which is due to the fact that the solid vortex generator has a large flow hindrance and generates strong vortex flow, which helps to generate a thinned thermal boundary layer over a tube wall and a high convective heat transfer rate. Skullong et al. [26] and Nakhchi et al. [30] reported similar results in their numerical investigation of the influence of perforation on Nu. When PI is increased from 0 to 25%, Nu decreases in the range of 4.6% to 9.7%. At Re = 24,000, the highest Nu of 252.5 is seen for PI = 0% and PR = 1. The range of heat transfer augmentation over smooth tube is 2.4–4.35 as shown in Fig. 9b.

Effect on the friction factor

The f value has a significant impact on such systems' thermal performance. Because thermal performance is inversely proportional to f, it is highly desirable to obtain the optimal configurations that result in the least amount of f penalty. Figure 9c and d shows the effects of PR and PI on f and f/fs, respectively. Because f has an inverse relationship with the square of the fluid's velocity, the values of f show a downward trend as Re increases. The values of f decrease as PR increases and vice versa. This is due to the significant decrease in dynamical pressure head caused by low flow obstruction. For PI = 0%, a decrease in f is in the range of 18.4%-18.8%, 30.7%-31.2%, 38.8%-40% for changes in PR from 1–2, 2–3, and 3–4, respectively.

The values of f decrease as PI increases, which is due to the fact that high perforation values impart very little flow blockage, resulting in a lower pressure drop. As PI increases from 0 to 25%, there is a significant decrease in f, with its values declining ranging from 38 to 54% for all parameters. At Re = 6000, the highest value of f, about 3.15, is found for the lowest PR of 1 and PI = 0%; this results in a f enhancement of the order of about 88 times over a smooth tube. Furthermore, at Re = 24,000, where f/fs is 15, the minimum value of f is 0.43 for PR = 4 and PI = 25%.

Effect on the thermal performance factor

The improved heat transfer always accommodates the increased pumping power requirement, as shown by the above results for heat transfer and f. As a result, a new parameter called the thermal performance factor (TPF) is considered to accumulate both parameters at the same time. The TPF value for equal pumping power in a smooth tube is unity, and any value greater than unity indicated that the system is superior to a smooth tube. Figure 9e shows the TPF values for various values of Re, PR, and PI. The result shows that when the value of PR is reduced, the value of TPF increases. Heat transfer is high for lower PR values, resulting in a higher TPF.

The effect of PI on TPF is also quite distinct. The values of TPF show an upward trend as PI rises. This is due to a significant decrease in f as PI increases from 0 to 25%, with values ranging from 38 to 54%. TPF shows a downward trend as Re rises, which is due to a decrease in Nu/Nus as Re rises as opposed to f/fs. The highest TPF value achieved is 1.16; this highest value was obtained for PI = 25%, PR = 1, at Re = 6000.

TPF is in the range of (0.96–1.16), (0.92–1.09), (0.91–1.08), and (0.88–1.006) for values of PR of 1, 2, 3, and 4, respectively. In Fig. 9f, a 3D surface plot of the best parameters of the vortex generator insert shows that the maximum TPF of 1.16 is achieved at PR = 1 and PI = 25%. This vortex generator is deemed to be the most efficient in terms of energy consumption.

Experimental correlations

Correlations for Nu, f, and TPF are developed based on the experimental results and include various experimentally tested parameters, as shown in Table 1. Table 3 shows the statistical correlations that have been developed. Figure 10 shows the deviations of predicted values from experimental values for Nu, f, and TPF, demonstrating that the developed correlations are in good agreement with experimental results, with maximum deviations of 8% for Nu, 15% for f, and 5% for TPF.

Table 3 Statistical correlations for Nu, f, and TPF
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Fig. 10
figure 10

Comparison of predicted values with experimental values for a. Nu for solid TBVG, b. Nu for perforated TBVG, c. f for solid TBVG, d. f for perforated TBVG, e. TPF for solid TBVG, f. TPF for perforated TBVG

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Comparison with previous inserts

The maximum TPF of present TBVG inserts is compared with some of the inserts investigated before by other researchers and is presented in Fig. 11. It could be seen from the plot that the maximum TPF of TBVG inserts is higher than the other inserts. The maximum TPF of TBVG insert is 0.87%, 11.54%, 2.65%, 10.48% higher than the maximum TPF of double-sided delta-wing tape (Wijayanta et al. [27]), drainage inserts (Li et al. [44]), delta-winglet twisted tape (Eiamsa-ard et al. [45]), tapered twisted tape (Piriyarungrod et al. [46]) respectively. Hence, one can infer that the TBVG inserts generally perform better than the other inserts that are compared.

Fig. 11
figure 11

Comparison of maximum TPF achieved for TBVG inserts with previous inserts

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Conclusions

A novel insert geometry is developed in this study to improve the thermal performance of a tube heat exchanger. The following conclusions are presented as a result of the current study.

  • By generating recirculation and vortices in the fluid flow, TBVG inserts in a heat exchanger tube can achieve a better mix-up of fluid between core and at the wall region of fluid flow which helps in disrupting the thermal boundary layer at the wall region of a tube resulting in a better convective heat transfer.

  • The convective heat transfer increases with an increase in Re. Values of Nu ascend as PR drops from 4 to 1. The maximum enhancement in Nu value for vortex generator embedded tube over smooth tube is about 4.35.

  • Heat transfer decreases as PI increases due to a reduction in flow hindrance. For Re = 6000 and PR = 2, the highest reduction in heat transfer of 9.7% is observed as PI increases from 0 to 25%.

  • PI and PR significantly affected the values of f due to their direct effect on flow blockages. With an increase in Re, there is a decrease in f, whereas, with a reduction in PI and PR, the trend is reversed. The highest enhancement in f of 88 times over smooth tube is achieved by using TBVG inserts.

  • TBVG inserts with PI = 25% give better TPF values than at PI = 0% for all values of PR. A downtrend is seen in the values of TPF with an increase in the values of Re. The best TPF for TBVG inserts achieved is 1.16.

  • Modifications of the inserts, such as changing the size of the TBVG insert or the angle of the blades of the insert, could be considered for further research that could lead to the improved thermal performance of the heat exchanger tube.