Numerical thermal performance study in a heat exchanger tube with inclined elliptical rings

The paper deals with a numerical study on the effect of inclined elliptical ring (IER) on heat transfer augmentation in a uniform heat-fluxed heat exchanger tube. In the present work, the 60° IER was mounted repeatedly in the tube with six different eccentricity ratios (ER = b/a = 1, 0.9, 0.8, 0.7, 0.6 and 0.5) at a single ring-pitch ratio PR = 1.0. Air as the test fluid flows into the tube for Reynolds number ranging from 4000 to 20,000. To find the optimum thermal performance, the effect of ER values on the heat transfer and pressure loss is investigated. The study indicates that the use of IER can induce higher turbulent intensity imparted to the flow leading to higher heat transfer in range of about 237 to 461% above the smooth tube.


Introduction
Heat exchanger is a device that facilitates the exchange of heat between two different temperature fluids while keeping them from mixing with each other. Heat exchanger is widely applied in production process such as drying furnace, ventilation, air conditioning, refrigeration, electronic device, engine of machine and equipment in process industry, etc. The high performance heat exchanger is required in industry field, no matter small or large industrials take heat exchanger application in their process. At present, due to high competition and advanced development of those processes, improving their thermal performance is needed to increase efficiency. Principles used to design the device or thermal system for transfer or exchange of heat must be selected with aware of the features such as work according, laminar or turbulent flow [1], production cost, heat transfer efficacy, structure, compact size, maintenance & repair, reliability and safety. To achieve all features required, the basic key is the great allocation of resources, fuel and materials, so for allocation of resources we need to develop the capacity and potential of the heat transfer system to the best efficient, to minimize the power consumption. The design of compact size can reduce material and installation cost. Those are important that will lead to sustainable development in the future. Turbulators (turbulence promotors) are devices that can be described as static mixers. It can generate the secondary flows. Typically located inside the tubes of heat exchange equipment, turbulators can help increase the tube-side heat transfer efficiency of these units. Custom-made to fit the inside diameter of the tube, turbulators are designed to create and maintain turbulent swirl flow. There are many types of turbulators such as wire coils [2 three relative ring width ratios and four relative ring pitch ratios. Air was employed as the test fluid for the Reynolds number from 5000 to 26,000. The IVR yields the highest TEF of about 1.4 at BR=0.1 and PR=0.5. According to the literature review above, several experimental works on heat transfer enhancement in tubes by using turbulators have been extensively reported. However, only a few numerical works on that issue have been done. Also, the application of inclined elliptical rings in a circular tube has never found in the literature. Therefore, numerical investigations for six different eccentricity ratios of 60 inclined elliptical rings (IER) are conducted to study the flow structure and thermal behaviours and to understand the mechanism of heat transfer enhancement in the tube with IER insert. In the present work, air as the test fluid enters the tube inserted with IER elements for Reynolds number ranging from 4000 to 20,000.

Physical flow model
A schematic of the IER tube in this work is shown in Fig. 1. The tube has inner diameter, D = 50 mm while the IER having outer diameter of 48 mm has a ring pitch, P = D. For the elliptical ring, it has a major radius, a = 23 mm which is kept constant and a minor radius, b to be varied (see Fig.1). The inner ring shape is changed to be elliptic with eccentricity ratio, E R = b/a (= 0.5, 0.6, 0.7, 0.8, 0.9 and 1) where E R = 1 means that the ring shape is circular. The heat is transferred to air through the tube wall due to different temperatures between air and tube wall. The heat transfer coefficient and pressure drop are calculated by numerical computations. The tube flow model of this work is expected to attain a periodically fully developed flow where the velocity field repeats itself from one module to another [9]. Therefore, only one module is used for the current simulation. Due to tube symmetry, only left or right half of the tube flow model is employed as computational domain as shown in Fig. 1.

Governing equations
The numerical tube flow model in the tube is developed with assumptions as follows: 1) Steady threedimensional, turbulent and incompressible flow. 2) Constant fluid properties and 3) omitting body forces, viscous dissipation and radiation heat transfer. Based on assumptions above, the tube flow model is governed by the Reynolds averaged Navier-Stokes (RANS) equations and the energy equation. In the Cartesian tensor system these equations can be expressed as: Continuity equation: where is fluid density, is a mean component of velocity in direction , p is the pressure, is dynamic viscosity, and is a fluctuating component of velocity. Repeated indices indicate summation from one to three for 3-dimensional problems.
Energy equation: Where Γ and Γ t are molecular thermal diffusivity and turbulent thermal diffusivity, respectively and are given by The Reynolds-averaged approach to turbulence model requires that the Reynolds stresses, in (2) needs to be modeled. The Boussinesq hypothesis relates the Reynolds stresses to the mean velocity gradients as seen in (5) below:   (5) Where k is the turbulent kinetic energy, as defined by , and is the Kronecker delta. An advantage of the Boussinesq approach is the relatively low computational cost associated with the computation of the turbulent viscosity,  t given as . The realizable k- turbulence model is an example of the two-equation models that use the Boussinesq hypothesis. The steady state transport equations are expressed as: In the above equations,  k and   are the inverse effective Prandtl numbers for k and , respectively. C 1 , C 2 , C 3 , and C 1 are constants. All the governing equations are discretized by the QUICK numerical scheme, coupling pressure-velocity with the SIMPLE algorithm and solved using a finite volume approach. For closure of the equations, the realizable k- model is used in the present study. The solutions are converged when the normalized residual values are less than 10 −6 for all variables but less than 10 −9 only for the energy equation. More details on boundary conditions are similar to the case in [8]. There are four parameters of interest in the present work, namely, the Reynolds number, friction factor, Nusselt number and thermal enhancement factor. The Reynolds number is defined as The friction factor, f is computed by pressure drop, P  across the length of the tube, L as where k a is thermal conductivity of air. The area-average Nusselt number can be obtained by The thermal enhancement factor (TEF) is defined as the ratio of the heat transfer coefficient of an inserted tube, Nu to that of a smooth tube, Nu 0 , at an equal pumping power and given by

Grid independence
Grid independence method is tested to make sure that numerical solutions are free from grids by using four levels of grid systems performed at Re = 12,000 as shown in Fig 2. Comparing the solution of Nu obtained from four different grid levels, 90,749, 187,527, 261,275 and 385,239. It is seen that the Nu difference of the two grids between 90,749 and 187,527 is 1.021% while those between 187,527 and 261,275 and between 385,239 and 261,275 were less than 1.00%. In terms of convergence time and solution precision, the grid size of 187,527 is adopted in the current work. Figure 3. Validation Nu and f with measurements of inclined ring insert.

Model validation
It is necessary to verify the tube flow model and the method adopted in the numerical simulation. The validation of simulated Nu and f with Re of the tube with measurements for using  30 inclined ring insert [3] is depicted in Fig. 3. In the figure, it is found that the present numerical results and experimental data are in good agreement. The deviation of both numerical and measured data is within  8% for Nu and  18% for f.

Flow structure and heat transfer
The flow structure and vortex coherent in the flow model fitted with 60 IER at E R = 0.5, Re = 12,000 can be displayed in the form of the streamlines in transvers planes and the local Nu x contours as shown in Fig. 4(a) and 4(b), respectively. In Fig. 4(a), the use of IER can induce two main vortices flow appearing on the left and right sides of the tube. Closer examination reveals that both are counterrotating vortices with common-flow-down to the bottom wall. The local Nusselt number contours on the tube surface for E R = 0.5 and Re = 12,000 are presented in Fig. 4(b). It is apparent that high Nusselt number values appear on large areas over the tube wall, especially for the area around the IER and on the bottom wall. The peaks can be observed at the impingement areas on surface close to the IER. Also, larger regions of higher heat transfer values can be found on the bottom wall area of the tube due to common-flow-down vortices as can be seen in Fig. 4(b). This indicates that the IER enhances the heat transfer rate effectively.

Heat transfer result
Effect of using IER turbulators with different six ER values on the heat transfer enhancement in the form of Nu/Nu 0 is depicted in Fig. 5. In the figure, Nu/Nu 0 decreases steeply with increasing Re and ER especially for Re = 4000 to 8000 but shows a gradual decrease for further Re. The maximum and minimum Nu/Nu 0 is about 6.7 and 3.6 times for E R = 0.5, Re = 4000 and E R = 1.0, Re = 20,000, respectively. This is because the presence of IER turbulator can interrupt the development of thermal boundary layer and increases the degree of turbulence level apart from inducing two counter-rotating vortices. The smaller E R yields the considerable increase in the heat transfer because of higher flow blockage leading to higher vortex strength. The E R = 0.5 provides the Nusselt number in the range of 4.7 to 6.7 times higher than the smooth tube.

Friction factor result
The employ of IER insert comes with higher pressure drop in terms of friction factor. Fig. 6  .9 is about 38 to 57 times that is nearly the same as that for circular ring (E R =1.0). The friction losses mainly come from the dissipation of the dynamical pressure of the air due to high viscous losses near the wall, to the extra forces exerted by reversing flow and to higher friction of increasing surface area. Figure 6. Friction factor against Reynolds number. Fig. 7 depicts the variation of TEF with Re for using the IER. In the figure, it is observed that TEF shows the decreasing trend with increasing Re and E R values. The IER with E R = 0.5, 0.6, 0.7, 0.8, 0.9 and 1 at Re=4000 provides the maximum TEF of about 1.54, 1.54, 1.54, 1.51, 1.49 and 1.46, respectively. This indicates the merit of using the IER for enhancing the heat transfer in the heat exchanger tube.

Conclusions
Heat transfer and flow friction characteristics in a round tube with 60 inclined elliptical rings with various E R values has been numerically investigated for the turbulent regime, Re = 4000 -20,000 under a uniform heat-flux condition. The findings in the present study can be drawn as follows: The insertion of 60 inclined elliptical rings yields a considerable heat transfer enhancement at about 3.6 to 6.7 times above the smooth tube while the friction factor is 36 to 130 times.