Forced convection heat transfer enhancement using a coaxial wire-tube corona system

Journal of Electrostatics 103 (2020) 103415

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Forced convection heat transfer enhancement using a coaxial wire-tube corona system Navid Zehtabiyan-Rezaie a, Majid Saffar-Avval a, *, Kazimierz Adamiak b a b

Mechanical Engineering Department, Amirkabir University of Technology, Tehran, Iran Department of Electrical and Computer Engineering, University of Western Ontario, London, Ontario, Canada

A R T I C L E I N F O

A B S T R A C T

Keywords: Heat transfer enhancement Electrohydrodynamics Corona discharge Ionic wind Numerical simulation

Laminar forced convection heat transfer enhancement in a coaxial wire-tube corona system is numerically examined. Governing equations were solved using the finite volume method implemented in a developed solver using the OpenFOAM platform. The proposed corona system could change the flow pattern pushing the central high-velocity flow towards the hot surface. For a specific wire length, an enhancement of 100% in heat transfer could be achieved at a Reynolds number of 600 and the applied voltage of 8 kV. Finally, considering both heat transfer enhancement and electrical power consumption, the most effective case was recommended among scenarios considered in this study.

1. Introduction Electrohydrodynamics (EHD) is an interesting research field with many engineering applications, including heat transfer enhancement [1], mass transfer enhancement [2], pumping [3], and electrostatic precipitation [4], etc. Low energy consumption and lack of moving parts are the most important advantages of the EHD systems. Some previous EHD studies were based on the conduction phenomenon in dielectric liquids, which is associated with the non-equilibrium process of disso­ ciation of neutral species and recombination of generated ions in a finite thickness in the vicinity of electrodes. In this layer, an exceeding rate of dissociation creates a unipolar layer of space charge near the electrodes [5,6]. Another form of the EHD flow is generated by the electric corona discharge in gas between two electrodes with substantially different radii of curvature. Ions are generated in a thin layer near the discharge electrode. As they drift towards the collecting electrode, a drag force on the fluid particles results in a net gas flow, which is called the ionic wind and could be used for gas pumping or heat transfer enhancement [7]. In this paper, some of the studies relevant to the heat transfer enhancement by using the ionic wind are reviewed. Kasayapanand and Kiatsiriroat [8] numerically examined the laminar EHD flow in a wavy channel. The results showed that the heat transfer enhancement de­ grades with increasing the Reynolds number. Esmaeilzadeh et al. [9] studied laminar heat transfer augmentation by using a high voltage wire placed perpendicularly to the air flow in a 2D channel. The obtained

results revealed that the heat transfer enhancement is higher for a flow with a Reynolds number smaller than 1000. Kasayapanand and Kiat­ siriroat [10] studied the EHD effect on laminar natural convection in an open vertical channel. The results showed that the heat transfer enhancement is significantly larger at low Rayleigh numbers. Ahmedou and Havet [11] analyzed the effects of corona discharge on heat transfer in a flat duct. They observed that by increasing the Reynolds number, the heat transfer enhancement deteriorates. Kasayapanand [12] con­ ducted a computational study on the EHD effects on natural convection in a square cavity with multiple fins. The results revealed that the heat transfer is significantly improved by EHD at higher number of fins and longer fin lengths. Deylami et al. [13] examined the effects of several charged wires placed in a 2D channel on the forced convection heat transfer enhancement. The results showed that different arrangements of wires could improve the heat transfer coefficient. Gallandat and Mayor [14] numerically examined the laminar heat transfer enhance­ ment in a vertical open channel. The ionic pump section of the system was simulated first, and then the calculated velocity profile was chosen as the boundary condition for studying the heat transfer. Peng et al. [15] numerically studied heat transfer enhancement in a rectangular chan­ nel. The main goal of the study was to find an optimum longitudinal position for single electrodes, longitudinal arrangement for multiple electrodes, and electrode number. Taghavi Fadaki et al. [16] investi­ gated the effects of collecting plate geometry on the heat transfer enhancement in a 2D channel. The results indicated that the

* Corresponding author. Mechanical Engineering Department, Amirkabir University of Technology, 424 Hafez Ave, P.O. Box 15875-4413, Tehran, Iran. E-mail address: [email protected] (M. Saffar-Avval). https://doi.org/10.1016/j.elstat.2019.103415 Received 24 August 2019; Received in revised form 29 October 2019; Accepted 11 December 2019 Available online 18 December 2019 0304-3886/Crown Copyright © 2019 Published by Elsevier B.V. All rights reserved.

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Journal of Electrostatics 103 (2020) 103415

multi-collector arrangement significantly modifies the flow pattern and heat transfer. Wang et al. [17] numerically investigated the EHD-enhanced heat transfer in a rectangular channel. Two wires were placed in the channel acting as the emitting electrodes, while multiple scenarios were assumed for the collecting electrodes. Zhang and Lai [18] numerically studied the heat transfer enhancement using the corona wind in a square channel. Three configurations of the emitting elec­ trodes with a different number of pins were considered. Limited number of studies investigated the flow and heat transfer in a coaxial configuration consisting of an emitting wire and a tube/duct. Molki and Bhamidipati [19] experimentally examined turbulent heat transfer in the developing region of a circular tube with the emitting wire located on the centerline of the tube. The maximum enhancements of the local and average heat transfer coefficients were equal to 14–23% and 6–8%, respectively. Molki and Damronglerd [20] numerically and experimentally studied the heat transfer enhancement in a duct with the wire placed on the centerline. The results showed that the corona discharge could be an effective method to increase the forced convection heat transfer. In the present study, forced convection heat transfer enhancement in a novel coaxial wire-tube corona system is numerically investigated. According to our best knowledge, there has been no numerical study published yet, which would solve mass continuity, momentum, electric potential distribution, conservation of space charge, and energy equa­ tions for a coaxial wire-tube corona system with a finite length charged wire placed on the centerline in the hot section of the tube. The effect of charged wire length on the heat transfer enhancement is seemingly nonexistent in the published literature. The governing equations are solved using a developed solver in the open-source computational software OpenFOAM version 4.1. The effects of the Reynolds number, the charged wire length and the applied voltage on velocity profile, tem­ perature field, and heat transfer enhancement are examined.

tube [21,22]. The main idea lies in increasing the velocity of fluid near the hot surface. These methods are effective from the heat transfer point of view, but the pressure drop increases, too. It seems that a coaxial wire-tube EHD system could alter the flow pattern of air in a similar way to the above-mentioned methods, while the power consumption would be relatively low. The proposed configuration has the potential to improve the air-side convective heat transfer coefficient, especially for mini-scale devices. The schematic view of the problem is given in Fig. 1. A wire with a diameter of d and length of Lw is placed on the centerline of the middle section of a pipe with a diameter of D. The length of the charged wire has an effect on the power consumption; therefore, this is an important parameter of the present study. The system has three sections with lengths of L1, L2, and L3. At the tube inlet, air is flowing into the system 0 0 with a uniform velocity of win and temperature of Tin . The wire is sup­ plied with a high DC voltage equal to ϕw . The hot surface is assumed to 0 be maintained at a temperature equal to TH . Preliminary simulations revealed that the heat transfer rate in a simple tube (without wire) is nearly equal to that of a coaxial wire-tube system with an applied voltage of 0 kV, because the wire radius to tube radius ratio is very small. A coaxial wire-tube system with an applied voltage of 0 kV shows a heat transfer enhancement less than 1% compared to a tube without wire. Therefore, to implement the proposed geometry for a real-world application, an energized wire which is electrically insulated, except of the middle section having the length of Lw, could be put on the centerline of the tube and fixed at both ends. 0

3. Mathematical modeling and governing equations The non-dimensional form of the governing equations for a laminar incompressible EHD flow can be derived using the following dimen­ sionless parameters:

2. Problem statement

0

r¼

Temperature control of electronic devices is an important issue, which is traditionally performed using air as the cooling fluid but there is one problem; air is a poor media with some limitations such as low density and low specific heat, resulting in an unfavorable heat carrying capacity. Moreover, it has a low thermal conductivity; therefore, the airside convective heat transfer coefficient is usually small and heat transfer enhancement is mostly achieved by increasing the heat transfer area using external fins. Smaller heat sink volumes, elimination of audible noise, the ability to move rejected heat away from the user area and applicability in vacuum environments could not be easily achieved with the traditional method. However, due to its availability, low cost, ease of maintenance, and low pumping power characteristics, aircooling is still the preferred choice. Using multiple air tubes/channels with enhanced heat transfer coefficient embedded inside the heat pro­ ducing device could be a solution. In some studies published in the field of heat transfer enhancement without EHD, it has been suggested to replace the tube with an annular tube or place a porous media inside the

0

0

p¼

0

r z w ​t ; z ¼ ; t ¼ in D D D 0

0

p

pin 02

ρw in

; U¼

0

0

0

0

0

U q ϕ E T ; T¼ 0 0 ; q ¼ 0 ; ϕ ¼ 0 ; E ¼ 0 win qw EPeek TH ϕw

0

Tin 0 Tin

(1)

0

Re ¼

win D ν ; Pr ¼

ν

α

where t is the dimensionless time, p is the dimensionless pressure, ρ is the air density, U is the dimensionless velocity, q is the dimensionless 0 space charge density, qw is the space charge density on the surface of the emitting electrode, ϕ is the dimensionless electrical potential, E is the 0 dimensionless electric field, EPeek is the Peek’s value for the critical electric field, T is the dimensionless temperature, Re is the Reynolds number, ν is the kinematic viscosity of air, Pr is the Prandtl number, and α is the thermal diffusivity of air. By using the dimensionless parameters given in Eq. (1) and consid­ ering the recommendations made by the IEEE-DEIS-EHD Technical Committee [23], the non-dimensional numbers EHD, C and ReE appear

Fig. 1. Schematic view of the problem. 2

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Journal of Electrostatics 103 (2020) 103415

in the dimensionless governing equations. Theses dimensionless pa­ rameters are defined as: 0

EHD ¼

4. Numerical method The governing equations, comprising of continuity, momentum, energy and electrical equations, have been numerically solved by using a solver in the OpenFOAM version 4.1, an open source software based on the Finite Volume Method (FVM). The solver is not an original solver of OpenFOAM and it is developed by the authors. The developed solver is able to handle the coupled equations. The Euler and Gauss linear discretization schemes were implemented for the time derivatives and gradient terms, respectively. The Gauss linear corrected scheme was used for the Laplacian terms. For the dis­ cretization of the convective and diffusive terms of the momentum equation, the bounded Gauss upwind and Gauss linear schemes were used. The divergence term of the space charge density conservation equation was discretized by using the Gauss upwind scheme. The pressure-velocity coupling was done through the PISO-semi-Implicit Method for Pressure-Linked Equations (PIMPLE) algorithm [28]. The solver used an adjustable time step controlling method to keep the Courant number below a specified value. The algorithmic structure of the developed solver is shown in Fig. 3. The numerical grid was based on three-dimensional hexahedral blocks comprised of hexagonal cells.

0

0

D3 I q D2 ϕ b ; C ¼ w 0 ; ReE ¼ 0w ρν2 bA win D εϕw

(2)

where A is the surface area of the electrode, b is the ion mobility in air, ε 0 is the electrical permittivity of air, and I is the discharge current. The dimensionless forms of the continuity and momentum equations are given as [3,24,25]: (3)

r⋅U ¼ 0

∂U þ ðU ⋅ rÞU ¼ ∂t

rp þ

1 EHD qE r ⋅ ðrUÞ þ Re Re2

(4)

The electric variables are governed by Eqs. (5) and (6) to calculate the electrical potential and the space charge density distributions [2,3]: r2 ϕ ¼

(5)

Cq

∂q þ ReE r ⋅ ð q rϕÞ ¼ 0 ∂t

(6)

The governing equation for temperature could be given as [6]:

∂T 1 r2 T þ r ⋅ ðUTÞ ¼ Re Pr ∂t

5. Results and discussion

(7)

In order to validate the solver, geometries and operating conditions of some experimental and analytical studies have been entered to the created solver and the output results are compared with the results re­ ported in the published references. The results of the grid independency tests are also given. Then, the results for the velocity profile, the streamlines, the temperature distribution, the heat transfer enhance­ ment, and electrical power consumption are presented. Important pa­ rameters of the study are given in Table 1.

Natural convection effects are not considered in this study since Ri ¼ Gr/Re2 <<1 [26]. In Eq. (7), the Joule heating effect is also neglected since it is not significant in problems with forced convection [18,27]. To evaluate the thermal performance of the proposed system, the Nusselt number and heat transfer enhancement are calculated using the following equations [6]: h¼

Q_ 0

0

As TH

Tin

�; Nu ¼

hD k

5.1. Solver validation

Z 0

Q_ ¼

(8)

kðn ⋅ rT ÞdA

(9)

To validate the solver, three tests have been carried out. First, the forced convection of air in a channel without EHD was modeled and compared with analytical results. Then, the performance of the solver in predicting the ionic wind in an isothermal wire-plate corona discharge configuration was studied. Finally, local heat transfer coefficient in a wire-plate corona system is compared with experimental data. To analyze the performance of the solver in simulation of heat transfer without EHD, the analytical results presented by ZehtabiyanRezaie et al. [29] are used. They derived analytical relationships for the fully developed flow and heat transfer of two immiscible fluids in a parallel plate channel. After assuming the same thermo-physical prop­ erties for the two phases (both air), the fully developed velocity and temperature profiles for the present solver and the analytical solution are shown in Fig. 4, showing maximum relative errors of 0.26% and 0.85% for velocity and temperature profiles, respectively. The bulk temperature is calculated from Eq. (11) [29].

Hot wall

� eh ¼

Nu Nu0

� (10) Re

3.1. Boundary conditions The governing equations are solved in transient form using the boundary conditions given in Fig. 2 to achieve the steady-state solution. For the pressure field, the zero gradient boundary condition has been used at the solid surfaces. For the thermal boundary condition, the constant temperature boundary condition is considered because most of the electronic devices should be kept at a constant temperature. The Kaptzov hypothesis has been used to calculate the value of the space charge density over the emitting electrode; an initial value is guessed and iterated until the electrical field is sufficiently close to Peek’s value [7].

Fig. 2. Boundary conditions. 3

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flux of 187 W/m2 is compared with the experimental results of Owsenek and Seyed-Yagoobi [31]. Three different voltages are applied to the wire placed at a distance of 2 cm above the heated plate. Results obtained from the developed solver show a very good agreement with the experimental data. Small differences can result from the experimental measuring uncertainties. The maximum and minimum uncertainties for the local heat transfer coefficient in the study of Owsenek and Seyed-Yagoobi [31] are equal to 16.2% and 3.2%. The tests reveal that the solver is capable of modeling EHD-enhanced heat transfer with good accuracy. 5.2. Grid independency test The grid independency test was performed for the proposed geom­ etry and for the most critical condition, in which the applied voltage and the Reynolds number were at their maximum values. As it is seen in Table 3, grid No. 3 is good enough and there is no need for a further mesh refining. 5.3. Velocity distribution and flow pattern In Fig. 6, the local distribution of the dimensionless axial velocity component is given for two values of applied voltage and four values of the wire length at Re ¼ 600. The hot wall is marked with the red line. By

focusing on the results for ϕw ¼ 6 kV, it is seen that the maximum value of the dimensionless axial velocity component is greater than that of a flow without EHD. According to the analytical solution of Hagen–Poiseuille [32], the maximum value of dimensionless axial ve­ locity component in a pipe is equal to 2.0. A region with negative axial velocity component is seen beneath the hot surface which means there is a vortex in that region. The vortex pushes the main flow stream towards the tube centerline. By increasing Lw/L2, the negative velocity area is enlarged covering larger portion of the hot surface. At the same time, the high-velocity region occupies larger part of the region near the centerline which means the highvelocity region is far from the hot surface. This may cause the heat transfer at a lower Lw/L2 to be greater than that of a larger Lw/L2. In the right column of Fig. 6, the results for a higher value of the applied voltage are given. The local dimensionless axial velocity component is significantly affected when compared to the case with a lower applied voltage. A high-velocity zone is observed near the hot surface. In a tube without EHD, as a result of the wall-effects, the maximum axial velocity would be at the tube centerline. While here, for all values of Lw/L2 ratio, there exists a high velocity zone near the hot surface; increasing Lw/L2, pushes the zone towards the end of the hot surface. Looking at the centerline, we can see two zones with positive and negative axial ve­ locities showing that there is a vortex near the wire. The vortex diverts the flow from the centerline towards the hot surface which could have positive effects on the heat transfer. By increasing Lw/L2, the vortex occurs further from the inlet and the influence on the high velocity flow near the centerline weakens. This could have an adverse effect on the heat transfer. In Fig. 7, it can be seen that the streamlines are different from the flow without EHD, which has streamlines parallel to the tube centerline. A vortex at the leading edge of the hot section pushes the flow towards the centerline and that is the reason why we observed a higher axial velocity than that in the case without EHD in Fig. 6. A very weak vortex occurs near the ending part of the wire, which is not able to significantly affect the flow. The vortices are created because of the electric force is directed from the wire to the ground electrode. The streamlines are slightly diverted towards the hot surface after passing the first vortex. By increasing Lw/L2, deviation of the streamlines towards the hot surface is delayed which could decrease the heat transfer. By increasing the voltage, the streamlines are significantly affected. Two vortices are generated, one at the leading edge of the hot surface and the other near 0

Fig. 3. Solver algorithmic structure. Table 1 Important parameters of the numerical model. Parameter

Unit

Value

d D L1, L2, L3

mm mm mm F/m m2/V⋅s kg/m3 m2/s J/kg⋅K W/m⋅K K

0.08 20.08 100 8.85419 � 10 2.3 � 10 4 1.225 1.0 � 10 5 1000 0.02 300

K

320

ε

b

ρ ν

cp k 0 Tin TH 0

Z

1

,Z

ρcp wTdy

Tbulk ¼ 0

1

ρcp wdy 0

12

(11)

In Table 2, the maximum vertical component of velocity beneath the wire in an isothermal wire-plate corona discharge is compared with the experimental results of Fylladitakis et al. [30]. The small relative error proves that the solver works properly in the EHD flow modeling. In Fig. 5, the local heat transfer coefficient over a plate with a heat 4

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Journal of Electrostatics 103 (2020) 103415

Fig. 4. Profiles of fully developed velocity and temperature compared with the analytical results of Zehtabiyan-Rezaie et al. [29] for Re ¼ 600 and wall heat flux of 75 W/m2 (without EHD). Table 2 Maximum vertical component of velocity [m/s] beneath the wire in an isothermal wire-plate corona system. ϕw [kV]

qw [C/ m3]

Fylladitakis et al. [30]

Developed solver

Relative error

8 10

5.254e-04 8.219e-04

0.7731 1.052

0.7787 1.0799

0.72% 2.65%

0

0

Table 3

Results of grid independency test for Re ¼ 900, Lw/L2 ¼ 0.25 and ϕw ¼ 8 kV. 0

No.

Number of elements

eh ¼ Nu/Nu0

½ðRefined

1 2 3 4

1800 4500 7200 13500

1.8948 1.8556 1.8306 1.8301

– 2.11% 1.37% 0.03%

CoarseÞ =Refined� � 100

the ending point of the wire. These vortices are stronger than those generated with a voltage of 6 kV. The first vortex pushes the flow to­ wards the centerline. As it reduces the area in front of the flow, the velocity increases, then the second vortex diverts the high-velocity flow towards the hot surface. At higher values of Lw/L2, the high-velocity zone is created by the first vortex, but the second vortex occurs further from the inlet, therefore, the flow pushed towards the hot surface could sweep smaller area of the hot surface. To understand the local variation of the dimensionless axial velocity, local profiles of the dimensionless axial velocity component at different z-positions are given in Fig. 8. For Lw/L2 ¼ 0.25, the z-positions of 5.00 and 6.25 are inside the first and second vortices. The first vortex em­ braces z-position of 5.00 since we see a negative axial velocity near the hot surface and the second vortex covers z-position of 6.25 showing a negative axial velocity near the wire and positive axial velocity near the hot surface. At z-positions of 7.50, 8.75 and 10.0 the axial velocity profile does not change considerably because the flow has passed the second vortex and it is starting to get fully developed. Without EHD, we would expect to have hydrodynamically fully developed flow beneath the hot surface, but the EHD disturbs the velocity profile making a nonfully developed region under the hot surface, which is very desirable for the heat transfer enhancement applications. For Lw/L2 ¼ 0.5, the first vortex covers the z-positions of 5.00 and 6.25 showing that larger parts of the hot surface is covered with the first vortex. The second vortex had happened further from the inlet when compared to Lw/L2 ¼ 0.25 at a z-position of 7.50. Z-positions of 8.75 and 10.0 are beyond the second vortex and the flow is starting to get fully developed. For Lw/L2 ¼ 0.75 and 1.0, four and three z-positions are inside the first vortex, respectively. It is expected to see a smaller heat transfer enhancement in these two cases when compared to Lw/L2 ¼ 0.25 and 0.5 since the hot surface is exposed to the high-velocity flow over a smaller

Fig. 5. Local heat transfer coefficient with a single wire electrode placed 2 cm above the plate electrode for a heat flux of 187 W/m2; a comparison between experimental results of Owsenek and Seyed-Yagoobi [31] and results of the present solver.

5

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Journal of Electrostatics 103 (2020) 103415

Fig. 6. Contour of dimensionless axial velocity component as a function of the applied voltage and wire length at Re ¼ 600.

Fig. 7. Flow streamlines as a function of the applied voltage and wire length at Re ¼ 600.

6

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Journal of Electrostatics 103 (2020) 103415

Fig. 8. Dimensionless axial velocity profile for ϕw ¼ 8 kV and Re ¼ 600. 0

area.

surface has absorbed the heat and consequently the outlet temperature has been increased. One can see that by increasing the wire length, the outlet temperature is decreasing, which means that the thermal per­ formance of the system is reduced compared to a system with a shorter charged wire. In Fig. 10(a), the Nusselt number of a flow without EHD is given. The Nusselt number is enhancing with an increasing Reynolds number, which is mainly because of the effect of the Reynolds number on the entrance length of flow. In the laminar flow, the entrance length is proportional to the Reynolds number. Since the developing region has a significant contribution to heat transfer, a higher Reynolds number re­ sults in a higher heat transfer rate in a flow without EHD. Fig. 10(a) also shows some results of the heat transfer enhancement of the proposed EHD system as a function of the Reynolds number and the Lw/L2 ratio for

5.4. Temperature distribution and heat transfer enhancement The effect of the proposed EHD system on the local distribution of the dimensionless temperature is shown in Fig. 9. Considering the results for the applied voltage of 6 kV, we can see two hot zones beneath the hot surface. The left zone is placed inside the first vortex, which does not have a considerable contribution in heat transfer. The second hot zone is the area of flow diverted towards the hot surface, which advects the heat from the surface. By increasing Lw/L2, the entire lower surface of the hot wall is occupied by the first vortex and less heat is being advected to the main flow, which was predicted from the velocity distribution contours. For the applied voltage of 8 kV, the high-velocity flow near the hot 7

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Journal of Electrostatics 103 (2020) 103415

Fig. 9. Contour of the dimensionless temperature as a function of the applied voltage and wire length at Re ¼ 600.

Fig. 10. Heat transfer enhancement as a function of the wire length for four different Reynolds numbers of 600, 700, 800 and 900 (a) ϕw ¼ 7kV (b).ϕw ¼ 8kV 0

8

0

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Journal of Electrostatics 103 (2020) 103415

the applied voltage of 7 kV. The gray rectangular region shows the cases which are not recommended for the heat transfer enhancement appli­ cations. For each Reynolds number, increasing the Lw/L2 ratio results in a reduction in the heat transfer enhancement. As the Reynolds number increases, the heat transfer enhancement decreases because the influ­ ence of the second vortex on a flow with higher axial velocity is lower than that for a flow with lower axial velocity. For the voltage of 7 kV, at all Reynolds numbers considered in this study, the results for Lw/L2 ¼ 1.0 are located in the gray rectangular zone, which shows a reduction in the heat transfer compared to the flow without EHD. The reason lies behind the first vortex, which separates the hot surface from high ve­ locity flow. In the base case without EHD, the main axial flow has a better contact with the hot surface than the case with a Lw/L2 ¼ 1.0. In Fig. 10(b), heat transfer enhancement for the applied voltage of 8 kV is shown. The enhancement for each Reynolds number and all Lw/L2 ratios is higher than that for the voltage equal to 6 kV. Assuming the applied voltage of 8 kV, Lw/L2 ¼ 0.25, 0.5 and 0.75 can show heat transfer enhancement, but the case with Lw/L2 ¼ 1.0 is not recommended. As mentioned earlier, a coaxial wire-tube system with an applied voltage of 0 kV shows a heat transfer enhancement less than 1% compared to a tube without wire. Therefore, heat transfer enhancement only results from two vortices generated because of the corona discharge and not from the hydrodynamic effects of putting a thin solid wire in the centerline of a tube. Fig. 10 showed that the maximum heat transfer enhancement is observed at Re ¼ 600. In Fig. 11(a), the effect of charged wire length and applied voltage are examined on the heat transfer enhancement for a Re ¼ 600. It is seen that for all applied voltages, Lw/L2 ¼ 0.25 gives the maximum heat transfer enhancement among the cases studied in this paper. The electrical power consumption of the proposed EHD system is shown in Fig. 11(b). An ascending behavior is observed when the applied voltage increases from 6 kV to 8 kV. A higher voltage results in a higher electrical current, which consequently increases the power con­ sumption. Another output of this figure is the effect of Lw/L2 on the power consumption. Longer charged wires would have higher electrical current for the same applied voltage and electrode gap, which affects the

power consumption. The previous results revealed that the case Lw/L2 ¼ 1.0 does not show considerable heat transfer enhancement at the voltage of 8 kV. At the voltage of 7 kV, all the cases of Lw/L2 ¼ 1.0 showed heat transfer reduction. By looking at the power consumption graph, it can be seen that Lw/L2 ¼ 1.0 is not a good choice both from the heat transfer and power consumption points of view, while the cases with Lw/L2 ¼ 0.25 enhance the heat transfer and have the lowest power consumption. Therefore, a coaxial wire-tube system with Lw/L2 ¼ 0.25 is the best configuration among the cases studied in this paper. 6. Conclusions In this paper, a novel coaxial wire-tube corona system is proposed to enhance forced convection in a tube. The computational model including continuity, momentum, energy, and electrical equations is simulated using a developed solver in the OpenFOAM software. Com­ parison between experimental data and analytical solutions reveals that the developed solver is an effective tool in EHD-enhanced heat transfer modeling. The effect of the proposed system on heat transfer enhance­ ment is numerically examined for the first time. The effects of the applied voltage, the Reynolds number, and the wire length on the ve­ locity and temperature distribution, as well as the heat transfer enhancement, are studied. A parametric optimization for the investi­ gated system is proposed as the future work on this subject. The following conclusions can be highlighted: � The results revealed the coaxial corona system generates two vortices in the middle section of the tube; one at the leading edge of the hot surface and the other at the ending section of the wire. � The first vortex pushes the main flow towards the tube centerline and the second one diverts the flow to the hot surface. � By increasing the wire length, the first vortex covers the hot surface and the second vortex occurs at the end of the middle section of the tube; therefore, the high-velocity flow loses the opportunity to sweep the hot surface.

Fig. 11. (a) Heat transfer enhancement as a function of the wire length and applied voltage at Re ¼ 600 (b) Electrical power consumption as a function of wire length and applied voltage. 9

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Journal of Electrostatics 103 (2020) 103415

� By increasing the Reynolds number at a constant applied voltage and wire length, heat transfer enhancement showed a descending behavior. � For Lw/L2 ¼ 1, the proposed system did not show considerable heat transfer enhancement. Assuming the applied voltage of 7 kV, a reduction of the heat transfer was observed for this configuration. � The highest heat transfer for all voltage levels and Reynolds numbers was observed for Lw/L2 ¼ 0.25. The lowest power consumption was

also seen for this wire length. Therefore, Lw/L2 ¼ 0.25 was the best scenario among the cases examined in this study. Declaration of competing interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Nomenclature A b C cp d D e E 0 EPeek EHD g Gr I0 k L Nu p P_ Pr q 0 qw Q_

r Re Ri t T U w y z

surface area, m2 ion mobility of air, m2/V⋅s dimensionless injection parameter specific heat at constant pressure, J/kg⋅K wire diameter, m tube diameter, m enhancement dimensionless electric field Peek’s value for the critical electric field, V/m dimensionless electrohydrodynamic number gravitational acceleration, m/s2 0 0 Grashof number ¼ gβðTH Tin ÞD3 =ν2 current, A thermal conductivity, W/m⋅K length, m Nusselt number ¼ hD=k dimensionless pressure power, W Prandtl number ¼ ν=α dimensionless space charge density space charge density on the surface of the emitting electrode, C/m3

heat transfer rate, W dimensionless radial distance from the tube centerline Reynolds number Richardson number ¼ Gr/Re2 dimensionless time dimensionless temperature dimensionless velocity vector dimensionless axial velocity dimensionless vertical distance from lower wall of the channel used in Zehtabiyan-Rezaie et al. study [29] dimensionless axial distance from the tube inlet

Greek letters thermal diffusivity, m2/s β coefficient of thermal expansion, 1/K ε electric permittivity of air, F/m ν kinematic viscosity of air, m2/s ρ density of air, kg/m3 ϕ dimensionless electrical potential

α

0

ϕw

voltage of the emitting electrode, V

Superscripts ‘ dimensional variables Subscripts E F.D. h H in s w

electric fully developed heat transfer hot wall inlet surface wire 10

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Journal of Electrostatics 103 (2020) 103415

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