A numerical study on the thermal hydraulic performance of finned elliptical tube with wet air flowing around

Energy conservation is a key concern for countries around the world. Enhanced heat exchange is one of the important ways to save energy. The heat transfer process of wet gas is a special heat transfer process, accompanied by a mass transfer phenomenon. Improving the thermal hydraulic performance of wet gas can greatly improve the energy utilization rate. The thermal hydraulic characteristic of finned elliptical tubes with different elliptical axis ratios (0.9 to 0.5) was numerically studied under wet air conditions. Chilton & Colburn factor was chosen as assessment parameters for the thermal performance. The mass transfer effect was characterized by a dehumidifying coefficient. The flow resistance coefficient was adopted to judge the flow characteristic and the London area goodness factor was evaluated for the overall performance. The simulations indicated the superior performance of the finned elliptical tube, j factor of the finned elliptical tube was greater and the flow resistance coefficient was less than that of the finned circular tube. Comparison between different finned elliptical tubes showed that the JF factor increased with the decrease of the elliptical axis ratio. Under the same inlet conditions, the JF factor increases by approximately 50% as the elliptical axis ratio changed from 1 to 0.5.


Introduction
Heat transfer between the inside and outside of the tube is a common phenomenon in the industrial field.Any performance increase in heat transfer on either side of the tube would be beneficial to the effective use of energy and the reduction of environmental pollution.Air is always selected as cooling media in many fields, such as dehumidifiers, coolers, and air conditions, due to its easy availability and low cost.But the air-side thermal resistance is the main thermal resistance in the heat transfer process.Thus, researchers continued to carry out research work around air-side heat transfer.
Changing the shape of the tube is a simple passive scheme to improve the performance of the heat exchange tube.Circle tube was the most common heat transfer element.From the 1960s, many types of non-circular tubes were studied.Ellipse was the earliest form of deformation, which was flexible to change between a plat and a circular depending on the axis ratio.In the 1970s, the characteristics of the flow around the elliptic tube were found to change considerably with the axis ratio [1].Then, the relevant research had been ongoing.Matos et al. [2] found that the heat transfer performance of the elliptical tube was 13% higher than that of the circular tube under the optimal tube arrangement.Khan et al. [3] found through experiments that the greater the air flow rate is, the greater the heat transfer rate of the elliptical tube is.Li et al. [4] numerically studied three kinds of elliptical tubes with different axial ratios.They found that the pressure drop and Nusselt number of elliptical tubes with an axial ratio of 0.5 were smaller than those of circular tubes.Wang et al. [5] studied a new elliptical tube, which consisted of a semi-circular tube and a semi-elliptical tube.And they found that this new elliptical tube had better performance.
Another passive method to enlarge the heat transfer rate is adding fins to the heat transfer tube.Kim et al. [6] studied the circular tube with flat fins and established the correlation between heat transfer and friction.Unger et al. [7] carried out a three-dimensional numerical analysis on circular fins and elliptical fins in heat exchangers.They analyzed the influence of axial ratio and angle of attack of elliptical fins on the performance of heat exchangers.Nemati et al [8] also conducted a detailed numerical study on elliptical annular finned tubes.Wavy fin was another type of fin.Khoshvaght et al. [9] established the mathematical model of the wavy fin-and-flat-tube heat exchanger to analyze the heat transfer and resistance performance.Of course, there were improvements around fins in recent years.The installation of vortex generators on the fins was tried to enhance heat transfer.
Numerous studies on the air side had made significant contributions to the field of heat exchange.But many enhanced heat transfer technologies mentioned above might have difficulty in processing or high cost in practical implementation.So, the finned elliptical tube has become a better choice and has been extended to apply to wet air heat transfer, due to its simpler construction, relatively easy maintenance, and lower cost.He et al. [10] investigated the influence of elliptical tubes with different shapes on the air side performance under typical operating conditions.It was found that the heat transfer coefficient of an elliptical tube was 66% higher than that of a circular tube.Zhao et al. [11] mainly studied the influence of the number of tube rows on the elliptical tube with fins.It was found that the heat transfer coefficient of finned tubes is the largest in the second row and the smallest in the first and last rows when the number of tubes exceeds two.However, few studies have paid attention to the effect of the ellipticity of finned elliptical tubes on overall performance under wet conditions.In this paper, the heat and mass transfer outside finned tubes with different elliptic axis ratios will be considered by numerical simulation.The simulation will be carried out under several conditions with different relative humidity and inlet velocity.The performance reflecting heat transfer, mass transfer, flow, and overall characteristics will be discussed to investigate the influence of the elliptical axis ratio and inlet flow conditions on the performance of heat exchangers.

Analysis of flow characteristics
A model for a staggered finned tube was investigated in this study.The total heat exchange surface of the model was composed of the fin surface and tube surface.Figure 1 showed the geometry of the model.As wet gas flowed outside the tubes, it was cooled first, then water vapor in the wet gas was condensed when the wet gas was cooled to a saturation state.Therefore, the flow process of wet gas was accompanied by heat transfer and mass transfer.Due to the periodicity of the flow domain of finned tubes and in order to save computing resources, the calculation region was set as a periodic model.The dotted lines in Figure 1 showed the symmetry boundaries of the computational domain.Moreover, taking into account the effect of the fin leading edge and avoiding numerical instabilities due to any possible backflow into the computational domain, the upstream and downstream boundaries had been extended 1 and 10 times the tube diameter from the center of the tube in the upstream and downstream respectively.The computational domain including the upstream and downstream extended regions in this study was shown in Figure 2.
Figure 2. The computational domain including upstream and downstream extended regions.

Governing equations
Water vapor and air combine to form wet air.In wet air, the mass fraction of water vapor was low.Therefore, the air was regarded as a carrier fluid and water vapor as a diluent.It was known that the thermal resistance of the condensate droplet was very small.Thus, the presence of condensate was neglected and the condensate was considered as a mass source in the present analysis.The laminar wet gas flow could be described by the equations of continuity, momentum, energy, and species transport.The governing equations [12] were: Continuity equation: where i u was velocity component, p was pressure, T was temperature, ω was water vapor mass fraction, m S was condensation amount per unit volume, h S was heat flux per unit volume, λ was thermal conductivity, μ was dynamic viscosity, p C was specific heat capacity, ρ was density, and D was mass diffusion coefficient.
In addition, for the solid zone of the fin, only an energy equation was needed: where T was the temperature of the fin.

Condensation model
When the temperature of wet gas dropped below its dew point, the water vapor would be condensed into liquid.In the process of numerical simulation, it was assumed that water vapor was only condensed on the solid surface.When the wet gas in contact with the solid surface met the condensation conditions, the condensation rate per unit area was calculated as following equations [13]: where m was condensation rate per unit area, w T was the temperature of a solid surface, sat p was saturation pressure at w T , w ω was a mass fraction of water vapor near a solid surface, and R was water vapor gas constant.

Boundary conditions
To simulate the flow of wet gas through the finned tube, boundary conditions were also required.A uniform velocity, constant temperature, and constant water vapor composition were applied at the inlet.The outlet was set as a pressure boundary condition.The pipe wall was set at a constant temperature, and the solid surface adopted a no-slip boundary condition.A fluid-solid coupling heat transfer model was established for the solid surface of finned tubes.Periodic conditions were applied to the symmetry of the domain.

Data reduction
In order to reflect the simulation results of the thermal hydraulic performance of wet gas passing through finned tubes, some non-dimensional parameters and characteristics are defined as follows.

Chilton & Colburn factor
The heat transfer characteristic could be represented by Chilton & Colburn factor j .The expressions of j factor was shown as follows [14]: where Nu was the Nusselt number, Re was the Reynolds number, Pr was the Prandtl number, s h was the convective heat transfer coefficient, λ was the thermal conductivity, and e d was the hydraulic diameter.(10) where s  was the sensible heat transfer rate, A was the total area of the heat transfer wall surface, and in T , out T , w T was the inlet temperature, outlet temperature, and average temperature of the heat transfer wall.

Dehumidifying coefficient
To analyze mass transfer characteristics, a dehumidifying coefficient was defined as Equation ( 11) [10].It represented additional heat exchange due to wet exchange under wet conditions.It could also be said that the total heat exchange under wet conditions was equal to ς times the sensible heat exchange.The greater the dehumidifying coefficient is, the greater the latent heat transfer and the greater the dehumidifying capacity is.
where  was the total heat transfer rate.

Flow resistance coefficient
To analyze flow resistance, the flow resistance coefficient was defined as f factor.If total mechanical energy loss, including pressure energy loss and the Kinetic energy loss, from the inlet to the outlet was expressed as the loss along the flow path.According to Bernoulli equation, f factor [15] was defined as: p p f (12) where 1 p , 1 u and 2 p , 2 u were the pressure and the velocity at the inlet and outlet of the fin region respectively, u was the average velocity, and L was the length of the fin region in the flow direction.

London area goodness factor
In order to evaluate the comprehensive performance of the finned tube, the characteristics of heat transfer and flow should be considered together.London area goodness factor [16] was defined as Equation ( 13).The higher the value of JF is, the better the overall performance of the finned tube is.

Implementation and Verification of numerical simulation method
Ansys Fluent was a professional software to deal with flow problems, so the realization of numerical simulation in this paper was based on this software.In order to ensure the numerical simulation method in this paper, the calculation was first made to examine the performance of one four row finned circle tube with experimental data by Jiang [17].The geometry parameters and flow condition of the finned circle tube were presented in Table 1.Other information of the finned circle tube could be found in [17].
Figure 3 shows the validation of simulated data.For the heat transfer performance, the maximum difference of Nu number was 10.4%, the minimum difference was 5.9%, and the average difference was 8.4% between the two results.For the flow characteristics, the maximum difference of f factor was 10.9%, the minimum difference was 0.25%, and the average difference was 6%.Both average differences were less than 10%.In general, there was an acceptable match between the simulations and the experiments.The comparison showed the feasibility and rationality of the numerical simulation method in the present research.

Physical model
The finned elliptical tube researched in this paper was deformed based on the finned circular tube which was experimentally researched in [17].The principle of deformations was: (1) the fin area was constant, (2) the cross-sectional area of the elliptical tube is the same as that of the circular tube.The elliptical axis ratio Ar was defined as the ratio of short axis b to long axis a. Figure 4 showed the deformed configuration with Ar of 0.5.There were five different finned elliptical tubes in present study.The values of Ar were 0.9, 0.8, 0.7, 0.6, and 0.5, respectively.

Grid generation technique
In order to improve the mesh quality, hexahedral meshes with different sizes were used in different areas.The size of the grid cell decreased as it approached the solid surface.Figure 5 showed the mesh of the finned elliptical tube with an Ar of 0.7.Grid independence was performed for each physical model.We take the numerical simulation with the Ar of 0.7, an inlet velocity of 2 m/s, an inlet temperature of 300 K, and relative humidity of 50% as an example.Three different numbers of grid elements were selected and evaluated.As seen in Table 2, when the number of grids exceeded 240 thousand, the Nu number and the f factor no longer changed.Therefore, 240 thousand grids were selected for numerical simulation.

Results and discussion
In this paper, the research objects were to study the influence of ellipticity on the performance of finned elliptical tubes.The present work carried out a numerical simulation on the wet air flow outside the finned elliptical tube and the baseline finned circle tube under different flowing conditions.Wet air flowed through the finned tube at four inlet normal velocities: 1 m/s, 2 m/s, 3 m/s, and 4 m/s.There were also four kinds of inlet relative humidity discussed.The inlet relative humidity was 50%, 60%, 70%, and 80%, respectively.We inlet temperature of wet air was 300 K.The material of the fin and tube was set as copper.The wall of the tube had a constant temperature, T w =285 K.

Numerical flow visualization
In order to describe the phenomenon of numerical simulation, the numerical results under the condition of inlet velocity of 4 m/s, and inlet relative humidity of 80% were presented as examples in the following.Figure 6 and Figure 7 showed the velocity vector and contour of wet air flowing outside the finned tube with different Ar.As could be seen that wet air struck the leading edge of the tubes first, and then was split, squeezed, and accelerated because of the profile of the tube.After bypassing the tube, a wake was formed behind the tube.That was a manifestation of the Karman vortex effect.
Due to the block of the rear rows of the tube, the Karman vortex effect was becoming increasingly evident.The wake region became bigger as the number of rows increased.Comparing the flow field of the finned tube with different Ar, it was found that the velocity distribution was different.The streamlined shape of the elliptic tube made the change of the velocity smooth, and the wake region small.The smaller the ellipticity axis ratio is, the smoother the flow is, and the smaller the wake region is. Figure 8 showed the temperature field of wet air flowing outside the finned tube with different Ar.As know, heat transfer is affected by the flow.The temperature distribution corresponded to the velocity distribution.In Figure 8, the temperature of wet air gradually decreases with the flow direction.The isotherm was dense near the upstream surface of the tube, while the isotherm was sparse in the wake region.At the same time, it could be seen the temperature in the wake region was almost equal to the surface temperature of the tube.Those indicated that the upstream surface was the main heat exchange surface.One of the differences between different finned elliptical tubes was the size of the upstream heat transfer surface.Due to the small wake region, the finned tube with a small Ar had larger upstream surfaces.As a result, the other difference between different finned elliptical tubes was the uneven degree of temperature distribution.The larger the wake region is, the more uneven the temperature field is.
For wet gas, the process of heat exchange was accompanied by condensation when the temperature was below saturation.Figure 9 showed the mass fraction distribution of water vapor in wet air flowing outside the finned tube with different Ar.Because the wet air at the inlet was not saturated, the wet air was only cooled after entering the flow region formed by finned tubes, the mass fraction remained 0.0175, the value of the mass fraction of water vapor at the inlet.After the wet gas flowed over a distance around the tube, the mass fraction began to decrease indicating the reaching of the saturation state of wet air.Then, the mass fraction of water vapor decreased continuously.Similar to temperature distribution, the mass fraction of water vapor decreased relatively quickly near the upstream surface of the tube, and almost did not change in the wake regions.

Characteristic analysis
Figure 10 showed the j factor of the finned tube with different Ar, which was calculated according to equation ( 8).As could be seen, the change in Ar had a great influence on thermal performance.j factor increased as the elliptic axis ratio decreased under any inlet conditions.In addition, it showed j factor was less sensitive to the change of inlet humidity changes (Figure 10 Dehumidifying coefficient representing the additional heat exchange of wet air was plotted as a function of the Ar in Figure 11. Figure 11 showed that as the Ar decreased, the dehumidifying coefficient was almost unchanged.That meant the change of ellipticity of elliptical tubes would not result in more mass transfer of wet air.But the Dehumidifying coefficient was slightly sensitive to inlet conditions.Under the inlet condition of high velocity and high relative humidity, the proportion of latent heat of mass transfer was relatively high.Figure 11(a) showed under the inlet condition with the velocity of 4 m/s, when the relative humidity changed from 50% to 80%, the dehumidifying coefficient increased by about 2 times.The f factor calculated from the energy loss was plotted as a function of the Ar in Figure 12.Firstly, Figure 12(a) and (b) revealed that under any inlet condition, f factor decreased as Ar became small.This implied low flow resistance was beneficial at small Ar.Besides, Figure 12(a) showed that f factor did not change with the inlet relative humidity.Figure 12(b) showed that f factor increased as the inlet velocity decreased, and the smaller the inlet velocity is, the greater the reduction amplitude of the flow resistance coefficient is. Figure 13 showed the variation of the JF factor with axial ratio.Obviously, the value JF factor increased with the decrease of the Ar value.Under the same inlet conditions, the JF factor increases by approximately 50% as the Ar changed from 1 to 0.5.In addition, the effect of inlet relative humidity on JF factor was the value of JF factor improved by about 10% when inlet relative humidity ranged from 80% to 50%.The effect of inlet velocity on JF factor was the value of JF factor improved by about 60% when inlet relative humidity ranged from 4 m/s to 1 m/s.

Conclusions
This paper investigated the outside performance of finned elliptical tubes under wet air conditions by numerical simulation.Finned elliptical tubes with five Ar were discussed and compared with finned circle tubes.The thermal hydraulic characteristics expressed by j factor, dehumidifying coefficient, flow resistance coefficient, and JF factor were analyzed.Some of the conclusions were as follows: (1) The flow field of wet air outside of finned tubes showed that there was a wake region downstream of the tube because of the effect of the Karman vortex phenomenon.Reducing the Ar made the flow field more smooth due to the reduction of the wake region.
(2) The effect of inlet relative humidity on thermal hydraulic characteristics showed that the flow resistance coefficient almost did not change, but j factor increased, and the proportion of mass transfer reduced as the relative humidity decreased.
(3) The effect of inlet velocity showed that the j factor and the flow resistance coefficient increased, while the proportion of mass transfer decreased as the inlet velocity decreased.
(4) The numerical simulation results showed that the more streamlined the elliptical tube is, the stronger the heat exchange effect is, and the smaller the flow resistance is.The finned elliptical tube with low Ar had better thermal hydraulic characteristics.Under the same inlet conditions, the JF factor increases by approximately 50% as the Ar changed from 1 to 0.5.

Figure 1 .
The geometry of the model for staggered finned tube: (a) Three-dimensional model; (b) Cross-side view.
The geometry parameters and flow condition of the finned circle tube

Figure 3 .
Verification of simulated data of the finned circle tube: (a) Nusselt number; (b) f factor.

Figure 4 .
Figure 4.The configuration of elliptical tubes deformed from circle tubes.

Figure 5 .
Figure 5.The mesh of the finned elliptical tube with an Ar of 0.7.

Figure 6 .Figure 7 .
Figure 6.The velocity vector of wet air flowing outside the finned tube with different elliptical Ar

Figure 8 .
Figure 8.The temperature distribution of wet air flowing outside the finned tube with different elliptical Ar.

Figure 9 .
Figure 9.The mass fraction of water vapor in wet air flowing outside the finned tube with different elliptical Ar.

Figure 10 .
(a)) than to the change of inlet velocity (Figure 10(b)).So the heat transfer characteristic of the finned elliptical tube would suggest that for wet gas with low parameters, especially those with low inlet velocity, better heat transfer characteristics could be obtained by using a finned elliptical tube with smaller Ar.The j factor of heat transfer as wet air flowing outside the finned elliptical tube with different Ar: (a) under the condition of inlet velocity of 4 m/s; (b) under the condition of inlet relative humidity of 80% Figure 11(b)  showed under the inlet condition with a relative humidity of 80% when the velocity changed from 1 m/s to 4 m/s, the dehumidifying coefficient increased by about 10%.So, the relative humidity of wet air was an important parameter of mass transfer.(a) (b) Figure 11.The dehumidifying coefficient of mass transfer as wet air flowing outside the finned elliptical tube with different Ar: (a) under the condition of inlet velocity of 4 m/s; (b) under the condition of inlet relative humidity of 80%.

Figure 12 .
Figure 12.The flow resistance coefficient as wet air flowing outside the finned elliptical tube with different Ar: (a) under the condition of inlet velocity of 4 m/s; (b) under the condition of inlet relative humidity of 80%.

Figure 13 .
The overall performance as wet air flowing outside the finned elliptical tube with different Ar: (a) under the condition of inlet velocity of 4 m/s; (b) under the condition of inlet relative humidity of 80%.

Table 2 .
Grid independent test