Heat Transfer Enhancement Using Twisted Tape Inserts: A Comparative Study Between Triangular and Rectangular Cross-Sections

Energy has become one of the most pressing issues facing humanity worldwide [1-3]. Heat exchangers (HEs) are extensively used in various industries and for a variety of applications such as petrochemical processing, power generation, aerospace engineering, and food engineering. There is also a lay role in energy recovery systems, particularly for the recovery of waste heat from internal combustion engines [4]. Therefore, improving the thermal performance of HEs is essential for reducing system size and operational costs. In addition, it decreased energy consumption and supports global efforts to mitigate climate change.

In recent years, various techniques have been proposed to enhance heat transfer. These techniques can be classified into three categories: active methods, passive methods, and hybrid methods. Active techniques require external energy input, such as acoustic waves or magnetic fields. Passive techniques operate without additional energy input, including the use of nanofluids, twisted tape (TT) inserts, and coiled wire inserts. Hybrid techniques, passive techniques that combine the advantages of both approaches [5-7]. Significant efforts have been devoted to the development and optimization of passive heat transfer enhancement (HTE) techniques aimed not only at improving the heat transfer rate but also at enhancing the overall thermo-hydraulic performance of HEs [8]. TT inserts are one of the most effective and practical passive HTE techniques, and it widely applied in engineering systems because it is favorable thermal performance, simple structure, ease of manufacturing and installation, and low manufacturing cost [9]. The mechanism induces swirling or secondary flow within the tube due to the insertion of TT, which enhances the mixing in the region near-wall fluid. Consequently, the thermal boundary layer is disrupted and increases in the turbulence intensity, leading to a significant enhancement in convective heat transfer performance without the need for additional external energy input [10]. Among the passive methods, TT inserts received extensive attention for their efficiency in improving the heat transfer performance of HEs. Many studies have been devoted to investigating whether and how the performance of TT inserts can be improved through geometric modifications and multi-tape configurations. For instance, Ponnada et al. [11] experimentally compared between Perforated twisted tapes with alternate axis (PATT), perforated twisted tapes (PTT) and conventional twisted tapes (CTT) under turbulent flow conditions. Their results showed that PATT provided the highest HTE, followed by PTT and TT. However, this improvement in thermal performance was accompanied by an increased friction factor $(f)$.

Previously, Bhuiya et al. [12] presented Perforated Triple Twisted Tape (PTTT) with varied porosities showing demanding rises in Nusselt number (Nu) reaching up to 320% when compared to a circular tube. However, this improvement came with a high cost and elevated pressure drop(ΔP), which suggested that there was a delicate balance between heat transfer and flow resistance.

Another study [13] also focused on the numerical investigation of single-cut and double-cut TTs. The researchers found that although double-cut configurations enhance heat transfer more effectively, higher-pressure losses were observed. Likewise, Chithra et al. [14] studied TT inserts in square ducts, and they found that even though the absolute value of heat transfer was lower with lesser friction penalties, shorter inserts turn out to be more thermally effective.

However, despite the abundance of investigations in this area, most studies have investigations have focused on conventional geometric shapes, including perforation or cuts and variations in tape length. In contrast, few attempts have been made to investigate the effect of TT cross-sectional shape. Especially non-rectangular geometries like triangular cross-sections are still not fully explored. Furthermore, previous studies did not provide systematic comparisons under controlled conditions, such as equal surface area or equal cross-sectional area, nor have they provided information on the optimum geometric parameters. Therefore, the present study systematically investigates the thermo-hydraulic performance and the effect of the geometric configuration of triangular twisted tape (TTT) inserts. The main objective is to determine the optimum b/h ratio value that achieves a maximum HTE with acceptable pressure drop. In addition, a comparison between TTT and rectangular twisted tapes (RTT) was conducted under the same conditions (equal surface area, equal cross-sectional area) to isolate the influence of geometry on flow pattern and heat transfer performance. This approach provides valuable insight into how geometric shape influences secondary flow structures, turbulence characteristics, and overall thermal performance.

3.1 Governing equations

The investigated fluid flow is three-dimensional, incompressible, turbulent flow, and the steady-state behavior. The heat transfer processes are governed by the fundamental conservation equations, namely the continuity, Navier–Stokes, and energy equations as expressed below [16-18].

-Continuity equation

$\frac{\partial u}{\partial x}+\frac{\partial u}{\partial y}+\frac{\partial u}{\partial z}=0$   (1)

-Momentum equation

In x-direction: 

$\rho\left(u \frac{\partial u}{\partial x}+v \frac{\partial u}{\partial y}+w \frac{\partial u}{\partial z}\right)=-\frac{\partial p}{\partial x}+\mu\left(\frac{\partial^2 u}{\partial x^2}+\frac{\partial^2 u}{\partial y^2}+\frac{\partial^2 u}{\partial z^2}\right)$    (2)

In y-direction:

$\rho\left(u \frac{\partial v}{\partial x}+v \frac{\partial v}{\partial y}+w \frac{\partial v}{\partial z}\right)=-\frac{\partial p}{\partial y}+\mu\left(\frac{\partial^2 v}{\partial x^2}+\frac{\partial^2 v}{\partial y^2}+\frac{\partial^2 v}{\partial z^2}\right)$    (3)

In z-direction: 

$\rho\left(u \frac{\partial w}{\partial x}+v \frac{\partial w}{\partial y}+w \frac{\partial w}{\partial z}\right)=-\frac{\partial p}{\partial z}+\mu\left(\frac{\partial^2 w}{\partial x^2}+\frac{\partial^2 w}{\partial y^2}+\frac{\partial^2 w}{\partial z^2}\right)$    (4)

- Energy equation

$\mathrm{u} \frac{\partial \mathrm{T}}{\partial \mathrm{x}}+\mathrm{v} \frac{\partial \mathrm{T}}{\partial \mathrm{y}}+\mathrm{w} \frac{\partial \mathrm{T}}{\partial \mathrm{z}}=\alpha\left(\frac{\partial^2 \mathrm{~T}}{\partial \mathrm{x}^2}+\frac{\partial^2 \mathrm{~T}}{\partial \mathrm{y}^2}+\frac{\partial^2 \mathrm{~T}}{\partial \mathrm{z}^2}\right)$   (5)

The (SST) k-ω turbulence model governing equations are as follows [19]:

The turbulence kinetic energy k:

$\frac{\partial\left(k u_j\right)}{\partial x_j}=\frac{1}{\rho} \cdot \frac{\partial}{\partial x_j}\left\{\left[\frac{\partial k}{\partial x_j}\left(\mu_l+\frac{\mu_t}{\sigma_k}\right)\right]\right\}+\frac{G_k}{\rho} \varepsilon$   (6)

where,

$\mu_t=\rho C_\mu \frac{k^2}{\varepsilon}, G_k=2 \mu_t E_{i j} . E_{i j}, E_{i j}=\frac{1}{2}\left\{\left[\frac{\partial u_i}{\partial x_i}+\frac{\partial u_i}{\partial x_j}\right]\right\}$

The constant for the current turbulent model is set as below [20]:

$C_\mu=0.09 ; C_{1 \varepsilon}=1.44 ; C_{2 \varepsilon}=1.92 ; \sigma_k=1.0 ; \sigma_{\varepsilon}=1.3$

3.2 Data reduction

For the analyzed turbulent flow, the dimensionless parameters, namely the f, Re, heat transfer coefficient (h) and Nu were evaluated using the following mathematical expressions [21]:

$\mathrm{h}_{(\mathrm{x})}=\frac{q}{T_w-T_b}$   (7)

where, T_w and T_b represented the wall temperature of heated tube and bulk temperature for fluid, respectively. As a result, enhancement of heat transfer depending on local Nusselt number (Nu_(x)) can be defined on [22]:

$\mathrm{Nu}_{(\mathrm{x})}=\frac{h(x) D}{k_f}$   (8)

where, k_f thermal conductivity for air.

To calculate the average Nusselt number (Nu_average) [23]:

$\mathrm{Nu}_{\text {average}}=\frac{1}{L} \int_0^L N u_{(x)} d x$   (9)

The Re is defined as [24]:

$R e=\frac{\rho u D}{\mu}$   (10)

where, $\rho$ density, u velocity of air, $D$ diameter of tube and $\mu$ dynamic viscositv.

The $\Delta \mathrm{p}$ can be calculated by the equation [25]:

$\Delta \mathrm{p}=\frac{f L \rho \mathrm{u}_{\mathrm{m}}^2}{2 \mathrm{D}}$   (11)

where, $f$: friction factor, u_m: the mean fluid velocity in the tube, L: the length of tube.

The HTE was quantified by the thermal performance factor (η) which depended on the comparison between a tube with and without TT inserts. It can be calculated using the following expression [26]:

$\eta=\left(\frac{N u_c}{N u_p}\right)\left(\frac{f_p}{f_c}\right)^{\frac{1}{3}}$      (12)

where, Nu_c, Nu_p, and f_c, f_p are Nu and f with and without TT in tube, respectively.

The Nu and f for the tube without a twist can be calculated from [27]:

Dittus–Boelter correlation for turbulent flow:

$\mathrm{Nu}=0.023 \mathrm{Re}^{0.8} \operatorname{Pr}^{1 / 3}$   (13)

Correlation of Gnielinski:

$\mathrm{Nu}=\frac{(f / 8)(R e-1000) P r}{1+12.7\left(\frac{f}{8}\right)^{1 / 2}\left(\operatorname{Pr}^{1 / 3}-1\right)}$   (14)

Blasius equation for turbulent flow:

$f=0.079 \mathrm{Re}^{-0.25}$    (15)

Correlation of Petukhov:

$f=(0.79 \ln R e-1.64)^{-2}$     (16)

3.3 Boundary condition

In the present work, it focused on air as a working fluid and the thermophysical properties of air are constant in all the Computational Fluid Dynamics (CFD) simulations. The inlet velocity was determined to obtain a Re of 5,000–25,000 as required for the study and the inlet temperature (T₁) was fixed at 300 K supplemented with a pressure outlet condition on the tube exit. The tube wall is uniformly heated by a heat flux of about 8000 W/m² to introduce the same thermal load on it. A no-slip boundary condition is employed at all solid walls, which include the inner wall of the tube and TT surfaces. The air properties used in the analysis are given as under: Density (1.225 kg/m³), Specific heat at constant pressure (Cp = 1006.43 J/kg·K), the thermal conductivity (K = 0.0242 W/m·K) and the dynamic viscosity ($\mu$ = 0.0000178 kg/m·s) [28].

3.4 Numerical method

The CFD simulations were performed by using Ansys Fluent 2020 to solve the governing equations and associated boundary conditions in numerical algorithms within a suitable computational domain. The governing equation is discretized and solved using the finite volume method (FVM) with the given boundary conditions. The SST k-ω turbulence model is employed to simulate turbulence effects because of its capability to accurately predict near-wall flows, and provide accurate predictions of flow separation and swirling motion. This model is especially applicable to internal flows with increased turbulence intensity induced by TT inserts. Furthermore, the SST k-ω model is widely used in previous numerical investigations of TT geometries, and it demonstrated good agreement with experimental and numerical results [29-31]. Medium mesh resolution and second-order upwind discretization schemes are applied for momentum and energy equations. Convergence is assumed when the residuals for continuity, velocity, and energy fall below 10⁻⁶.

3.5 Grid independence

Mesh sensitivity analyses were conducted to ensure that the numerical results were independent of the mesh density. The computational grid was progressively refined until reaching no noticeable variation in the computed quantities. Due to the geometric complexity of the domain, an unstructured tetrahedral mesh was employed. In addition, 20 inflation layers were generated near the tube wall to accurately resolve the velocity and temperature gradients within the boundary layer. Figure 2 illustrates the overall computational mesh refined regions and the surface mesh employed in the three-dimensional simulations with non-uniform mesh distribution. A grid independence test at Re = 5000 was run by using the Nu and f as shown in Table 1. Both Nu and f are seen to increase with mesh refinement yet converge towards a stable value at larger element counts. At some grid size the difference between consecutive meshes becomes negligible, which means that the result is mesh-independent. As a result, the selected mesh has given valid and accurate predictions.

Figure 2. The structure mesh

Table 1. Grid independent study

3.6 Validation

In order to validate the present numerical model, a comparison between Nu and f with the numerical results reported by Cabello [32] for TT at Re from 5000 to 25000, as illustrated in Figures 3 and 4. The present numerical predictions showed good agreement with the reference data across of Re for both Nu and f. The deviation between the present results and the reference data is within 5%. These minor differences may be attributed to variations in boundary conditions, geometric simplifications and the mesh resolution used for discretizing the governing equations.

Figure 3. Validation of Nu with result [32]

Figure 4. Validation f with result [32]

The variation of the Nu and f with Re is presented in Figures 5 and 6. Among the considered correlations, the Gnielinski equation predicted the highest Nu values, followed by the Dittus–Boelter correlation. In contrast, the present numerical simulations on a plain tube produced slightly lower values. This deviation is expected because the Gnielinski correlation considers the f and generally provides high accuracy with turbulent flow conditions. However, for all the cases considered, Re increased and f decreased with increasing Re in the smooth tube under turbulence flow. In addition, Blasius correlation slightly underestimated the f in comparison with the updated numerical results, while Petukhov equation showed closer agreement with the present study. It should be noted that the present numerical predictions showed good agreement with a set of empirical correlations. The deviations between present results and the correlations remain within an acceptable limit at a maximum error not cross about 5% for Nu and about 3% for f. These discrepancies are due to the use of different turbulence models, boundary conditions, simplifying geometries and numerical discretization methods.

Figure 5. Confirmation of Nu for plain tube

Figure 6. Confirmation of f for plain tube

TT inserts are commonly used as passive HTE devices due to their ability to generate swirl flow, boundary layer separation, and significantly improve the performance of heat transfer without requiring external power. As tape geometry affects HTE and pressure drop, the understanding of their effects is significant in the optimization of thermal systems such as HEs and other energy-intensive processes. Therefore, this study investigates characterize and compares the thermo-hydraulic performance of TTT with different geometric parameters within tubular HEs relative to their counterpart RTT under both constant surface area and cross-sectional area boundary conditions. The proposed TTT design is low-cost, easily manufactured, and fabricated for application as a passive internal component in conventional HE systems. The following are the main results of this study:

- The highest thermo-hydraulic performance of TTT is obtained at b/h = 0.35-0.40, where swirl intensity and turbulence are more pronounced. This condition enhances wall interaction and fluid mixing that contribute to uniform temperature distribution.

- Nu decreases from a ratio of 0.35 to around 0.6 and then remains almost constant with an increment in the b/h ratio. Both turbulence intensity and f decrease slowly with increasing in Re as the cross section of the tape expands and reduces disturbance flow.

- At constant cross section area, the TTT configuration exhibits a higher η of about 1.97 and 1.54 for TTT when compared with RTT configuration, about 1.64 and 1.33, respectively, and it is achieved from approximately 20% to 12% for Re 5000 and 25000, respectively. The thermos-hydraulic performance is mainly enhanced due to the sharper edges in the triangular geometry, which generate more strong swirl flow and an increase in fluid mixing, leading to a higher heat-transfer enhancement with a little increment of hydraulic resistance.

- At constant surface area, the TTT consistently outperforms the RTT across all Re. It achieves a maximum thermal performance factor of 1.92 at Re = 5000, about 17 % higher than the RTT and maintains superior Nu with enhancements ranging from 19% to 10% across Re = 5000 – 25,000. Although the TTT results in moderately higher f about 18–12% compared to the RTT, the substantial improvement in heat transfer ensures a superior overall thermo-hydraulic performance.