A Numerical Study on the Effect of Pipe-Volume Reduction Using Ellipsoid Inserts on Heat Transfer Enhancement

Heat exchangers are becoming popular in this era of the fourth industrial revolution to transfer heat from one medium to another. They are being widely used in petrochemical industries, power plant industries, oil refineries, fertilizer industries, food manufacturing industries, etc. Thus, heat transfer improvement is an important aspect being focused on by researchers around the globe [1-4].

This can be achieved in two methods, active and passive. Passive methods focus on enhancing heat transfer without external power by focusing on surface extension and modification, design and orientation, shedding, geometric modification, utilizing inserts, additives, etc., while active methods focus on mechanical aid, surface vibration, fluid vibration, fluid injection and suction, fluid mixing, etc. [5, 6].

Computational Fluid Dynamics (CFD) is a reliable method for analyzing fluid flow and heat transfer in U-loop pipes. It numerically solves the Navier–Stokes equations using the FEM based on the principles of continuity, momentum, and energy [13, 14]. For fluids initially stationary, the convective heat transfer coefficient (h) and the Nusselt number (Nu) are determined as follows.

$h=\frac{Q}{T_w-T_b}$                 (1)

$N u=\frac{h D}{k}$                (2)

The efficiency ($\varepsilon$) is calculated using Eq. (3).

$\varepsilon=\frac{T_o-T_i}{T_{wav}-T_i}$                (3)

The Darcy method is applied to calculate the fluid pressure drop (ΔP) and the friction factor (f) from inlet to outlet.

$\Delta P=h \rho g$                (4)

The fluid’s head loss, h, is calculated using:

$h=\frac{f j u^2}{2 g D}$                   (5)

Eqs. (4) and (5) are used to generate Eq. (6).

$\Delta P=\frac{l}{D} \cdot \frac{f u^2 \rho}{2}$                 (6)

The thermal performance criterion ($\eta$) is calculated using Eq. (7).

$\eta=\frac{N u / N u_0}{\left(f / f_0\right)^{1 / 3}}$                  (7)

The mean average difference (M.A.D.) and the relative mean average difference (R.M.A.D.), compared to the full volume pipe with no inserts, are calculated using:

$M . A . D .=\frac{1}{n} \sum_{i=1}^n\left|f\left(x_i\right)-g\left(x_i\right)\right|$                  (8)

$R.M.A.D.=\frac{1}{n} \sum_{i=1}^n\left|\frac{f\left(x_i\right)-g\left(x_i\right)}{g\left(x_i\right)}\right|$                 (9)

The consecutive mean average difference (C.M.A.D.) and the relative consecutive mean average difference (R.C.M.A.D.), compared between two consecutive models, is calculated using:

$C.M.A.D.=\frac{1}{n} \sum_{i=1}^n\left|f\left(x_i\right)-f\left(x_{i-1}\right)\right|$                 (10)

$R.C.M.A.D.=\frac{1}{n} \sum_{i=1}^n\left|\frac{f\left(x_i\right)-f\left(x_{i-1}\right)}{f\left(x_{i-1}\right)}\right|$                  (11)

where, n is the number of observations, $f(x)$ is the insert function and $g(x)=f\left(x_0\right)$ is the no insert function.

The inlet temperature of the working fluid, water, is set as $T=T_0=293.15 \mathrm{~K}$, with a no-slip condition applied at the pipe walls. A normal stress pressure boundary condition is assumed at the outlet [25].

Numerical simulations are carried out to study the effect of 0.53–1.38% pipe-volume reduction by adding two to eight pairs of ellipsoid inserts on the heat transfer phenomenon compared to the full volume pipe with no inserts for Reynolds number 764-3056 under uniform heat flux conditions. Heat transfer characteristics such as Nusselt number, effectiveness, friction factor, thermal performance criterion, and vorticity are observed, and the results are analysed [21-24].

5.1 Nusselt number analysis

Figure 2 shows the Nusselt number distribution for the full volume, 0.53–1.38% reduced volume U-loop pipe, by adding two to eight pairs of ellipsoid inserts and an empirical relation by Bergman et al. [26] for Reynolds number 764-3056. The Nusselt number is found to increase with increasing Reynolds number. For the aforementioned Reynolds number range, the Nusselt number is greater as more inserts are added, with the highest values yielded by seven pairs of inserts for the initial Reynolds numbers and eight pairs of inserts for the latter Reynolds numbers.

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Figure 2. Nusselt number (Nu) distribution

The mean average difference (M.A.D) and the relative mean average difference (R.M.A.D) of Nusselt number compared to the full volume pipe show a fluctuating pattern and range from 10.7–27.9% and 1.76–4.56%, respectively, as seen in Figure 3(a). The consecutive mean average difference (C.M.A.D) and the relative consecutive mean average difference (R.C.M.A.D), which measure the average differences in Nusselt number between two consecutive models, can be seen in Figure 3(b). The values increase in general, indicating improvement in effectiveness compared to the previous model, with a dip at five and six pairs of inserts. This suggests that although the Nusselt number improves as more pipe volume is reduced, the addition of five and six pairs of inserts has less significance in enhancing Nusselt number compared to other consecutive additions.

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5.2 Effectiveness analysis

Figure 4 shows the effectiveness distribution for the full volume and 0.53–1.38% reduced volume U-loop pipe by adding two to eight pairs of ellipsoid inserts for Reynolds number 764-3056. The effectiveness is found to decrease with increasing Reynolds number. For the aforementioned Reynolds number range, the effectiveness is greater as more inserts are added, with the maximum effectiveness obtained for eight pairs of inserts.

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Figure 4. Effectiveness distribution

The mean average difference (M.A.D.), compared to the full volume pipe, shows a fluctuating pattern and ranges from 0.18–2.55%, as seen in Figure 5(a). The consecutive mean average difference (C.M.A.D.) and the relative consecutive mean average difference (R.C.M.A.D.), which measure the average differences in effectiveness between two consecutive models, can be seen in Figure 5(b). The values increase in general, indicating improvement in effectiveness compared to the previous model, with a dip at five pairs of inserts. This suggests that although the effectiveness improves as more pipe volume is reduced, the addition of five pairs of inserts has less impact in enhancing effectiveness compared to other consecutive additions.

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As quadratic curves are fit through the data points, the gradients of the curves for all the models lie within [−0.0003236, −0.0000944], indicating similarity in their pattern precisely for the studied Reynolds number range. This can be seen in Table 4.

Table 4. Quadratic regression models for effectiveness

5.3 Friction factor analysis

The friction factor distribution for full volume, 0.53–1.38% reduced volume U-loop pipe by adding two to eight pairs of ellipsoid inserts, and the empirical relation represented by the Hagen-Poiseuille equation [26] for Reynolds number 764-3056 can be seen in Figure 6. The friction factor is found to decrease with increasing Reynolds number. For the aforementioned Reynolds number range, the friction factor is greater as more volume of the pipe is reduced, with the maximum friction factor values obtained for eight pairs of ellipsoid inserts.

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Figure 6. Friction factor (f) distribution

The mean average difference (M.A.D.) and the relative mean average difference (R.M.A.D.), compared to the full volume pipe, increase in general and range from 1.43–6.30% and 12.3–54.3%, respectively, as seen in Figure 7(a). The consecutive mean average difference (C.M.A.D) and the relative consecutive mean average difference (R.C.M.A.D.), which measure the average differences in friction factor between two consecutive models, can be seen in Figure 7(b). The values decrease in general, indicating improvement in friction factor compared to the previous model, with a jump from five to six pairs of inserts. This suggests that although friction factor improves as more inserts are added, the addition of six pairs of inserts has less significance in enhancing friction factor compared to other consecutive additions.

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5.4 Thermal performance criterion analysis

Figure 8 shows the Thermal Performance Criterion (TPC) distribution for 0.53–1.38% reduced volume U-loop pipe by adding two to eight pairs of ellipsoid inserts for Reynolds number 764-3056. The Thermal Performance Criterion remains mostly stable with increasing Reynolds number. For the aforementioned Reynolds number range, the Thermal Performance Criterion is greater as more inserts are added, with the maximum values obtained for two pairs of inserts.

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Figure 8. Thermal performance criterion distribution

The mean average difference (M.A.D.), compared to the full volume pipe, shows an increasing pattern and ranges from 2.37–9.52%, as seen in Figure 9(a). The consecutive mean average difference (C.M.A.D.), which measures the average difference in Thermal Performance Criterion between two consecutive models, ranges from 0.41–2.37% and can be seen in Figure 9(b). The values fluctuate with a dip at four and seven pairs of inserts. This suggests that although the Thermal Performance Criterion decreases as more pipe volume is reduced, adding four and seven pairs of inserts has a greater impact in the drop in TPC compared to other consecutive additions.

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5.5 Vorticity analysis

Figure 10 shows the velocity streamlines at insert and no-insert positions. As fluid passes around the insert, its velocity increases while the pressure drops. This improves fluid mixing and induces rotational motion in the fluid, leading to increased vorticity. Figure 11(a)–(h) shows the vorticity line graph for full volume and 0.53–1.38% reduced volume U-loop pipe by adding two to eight pairs of ellipsoid inserts for Reynolds number 1910. For full volume pipe, as seen in Figure 11(a), the vorticity shows slight fluctuations ranging from 0.787–0.644 (1/s), with spikes being observed in the insert positions in Figures 11(b)–(h).

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Figures 12(a)–(h) show the vorticity slices for full volume and 0.53–1.38% reduced volume U-loop pipe by adding two to eight pairs of ellipsoid inserts for Reynolds number 1910 at maximum vorticity positions of the pipe. The maximum vorticity of 0.7417 (1/s) for full volume pipe is observed at a position of 226 mm from the inlet. As ellipsoid inserts are added, the maximum vorticities of 4.8462 (1/s), 4.7637 (1/s), 4.7978 (1/s), 4.8871 (1/s), 4.9986 (1/s), 4.867 (1/s) and 4.8978 (1/s) are obtained for two to eight pairs of inserts at positions 333 mm, 250 mm, 200 mm, 166 mm, 284 mm, 250 mm and 223 mm respectively.

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The authors gratefully recognize the Miyan Research Institute and Simulation Lab at the International University of Business Agriculture and Technology, Dhaka, and East Delta University, Chittagong, for their invaluable help and support throughout the research process.