Numerical investigation of using various electrode arrangements for amplifying the EHD enhanced heat transfer in a smooth channel

Journal of Electrostatics 71 (2013) 656e665

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Numerical investigation of using various electrode arrangements for amplifying the EHD enhanced heat transfer in a smooth channel H.M. Deylami, N. Amanifard*, F. Dolati, R. Kouhikamali, K. Mostajiri Mechanical Engineering Department, Faculty of Engineering, University of Guilan, Rasht 3756, Iran

a r t i c l e i n f o

a b s t r a c t

Article history: Received 8 July 2012 Received in revised form 25 November 2012 Accepted 8 March 2013 Available online 22 March 2013

Forced convection heat transfer enhancement with electrohydrodynamic (EHD) technique of turbulent flow inside a smooth channel has been numerically investigated. A two dimensional numerical approach has been chosen to evaluate the local and average heat transfer coefficient. In addition, the swirling flow pattern in the presence of an electric field has been studied. To achieve higher enhancement while using multiple electrodes, variety of electrode arrangements have been examined for specified values of Reynolds number, applied voltage, and wire radius. The results demonstrate that different electrode arrangements cause significant improvement of the heat transfer coefficient. Ó 2013 Elsevier B.V. All rights reserved.

Keywords: EHD Numerical modeling Smooth channel Turbulent flow Heat transfer

1. Introduction Heat transfer enhancement is one of the most important research subjects of the thermal engineering field which increases the effectiveness of the heat exchangers as well as significant technical advantages and savings of costs. Enhancement techniques essentially reduce the thermal resistance in a conventional heat exchanger by promoting higher convective heat transfer coefficient. The enhancement techniques are divided into two groupings: “passive” and “active” techniques, in which the Electrohydrodynamic (EHD) phenomenon is an active method and deals with the interactions between electric field, flow field, and temperature field. When a high voltage is applied to a fine wire, air in its vicinity is ionized and the ions are drawn toward the electrically grounded plates. Traveling the injected ions from the wire (emitting electrode) toward the plate (collector electrode), causes the ions transfer their momentum to neutral molecules by collision. As a result, a bulk flow of ionized air molecules is generated known as corona wind or secondary flow. The corona wind disturbs the * Corresponding author. University of Guilan, Department of Mechanical Engineering, Campus of University of Guilan, 13th km Tehran Highway, P.O. Box 3756, Rasht, Guilan 3756, Iran. Tel.: þ98 131 6690270; fax: þ98 131 6690271. E-mail addresses: [email protected], [email protected] (N. Amanifard). 0304-3886/$ e see front matter Ó 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.elstat.2013.03.007

boundary layer on the grounded surface, creates turbulence, and thus enhances heat transfer between the grounded surface and its ambient air. This technique is called electrohydrodynamically enhanced heat transfer [1e3]. Kasayapanand and Kiatsiriroat [4] conducted a numerical study of the EHD enhanced heat transfer in a wavy channel. They demonstrated the heat transfer coefficient with the presence of electric field increases with the supplied voltage and on the other hand, it decreases by leveling up the Reynolds number or by increasing the distance between the wire electrodes and the wall surface. In addition, the number of the wire electrodes, the number of wave per length, and the wave aspect ratio can significantly change the heat transfer enhancement. Further, they numerically studied the optimized mass flux ratio of a double-flow solar air heater with EHD [5]. They found that the level of augmented heat transfer in presence of an electric field increases with the supplied voltage, but the total mass flux augmentations cause opposite effects. Besides, Lakeh and Molki [6] conducted numerical investigations to determine the effect of EHD enhancement heat transfer on natural convection with hot segments located on the perimeter of a circular tube. They revealed that the local Nusselt number in different extension angles is enhanced 2e6 times to that of the natural convection case. They also reported that increasing of applied voltage from 7500 V to 10,500 V causes significant magnification of the relative heat transfer enhancement from 4.2 times to 8.7 times, compared to natural convection at the

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impinging point of the corona jet. Moreover, a numerical study of EHD-enhanced forced convection using two electrodes in a horizontal channel has been reported by Mathew and Lai [7]. The variable voltage and Reynolds numbers were set in their study. In their work, with a fixed electric field, flow and temperature fields were reported oscillatory at small Reynolds numbers, which tends to stabilize when the Reynolds number was increased. They also concluded the heat transfer was increased about 375% in comparison with the cases without electric field in presence of the oscillatory secondary flows. Ahmedou and Havet [8]carried out a numerical simulation based on finite element method to study the EHD-enhanced heat transfer for a wire-plate. They used COMSOL commercial software and their results demonstrate a three-fold increase of local heat transfer coefficient which was gained at the Reynolds number of 3846 with a single-wire plate configuration and no significant heat transfer enhancement was produced at higher Reynolds numbers. In addition, the average heat transfer coefficient decreased with the number of wires increased. This implies that multiple wire-plate cases, the spacing between wires plays a critical role in the heat transfer enhancement. Zhao and Adamiak [9] also conducted experiments and numerical simulations in order to demonstrate the effect of corona wind on the air velocity. According to their results, the axial velocity reaches its maximum at the point on the axis not far from the pin tip and then, it decreases to zero at the ground plane. In addition, the airflow velocity profiles and the pressure distribution are greatly affected by the applied voltage as well as by the corona device configuration. Molki and Bhamidipati [10] presented an experimental study on enhancement of convective heat transfer in the developing region of the circular tubes using corona wind under a variable set of Reynolds number range from 2500 to 13,000. They reported the maximum percent of the local and average heat transfer coefficients enhancement is in the range of 14e23% and 6e8% respectively, with a same applied voltage of 10.5 kV. Alamgholilou and Esmaeilzadeh [11] experimentally studied the hydrodynamics and heat transfer of fluid flow into channel for cooling of the rectangular ribs by passive and EHD active enhancement methods. They showed the corona wind in low Reynolds numbers affects on main flow more significant than high Reynolds numbers, with an average heat transfer enhancement of 4.02 (402%). Shakouri Pour and Esmaeilzadeh [12] performed experimental studies on convective heat transfer enhancement from a 3D-shape heat source in presence of EHD technique in a duct. Their reported results indicated the higher effect of electrode arrangement with wires over the ribs than between the ribs. Regarding the above mentioned reports and based on the fact that, the electrodes arrangement is important in the production of an efficient and stable corona discharge, the main purpose of this study is to illustrate the effect of electrodes arrangement on the heat transfer enhancement in a smooth channel. The current numerical study is provided under the effect of different parameters such as Reynolds number, applied voltage, wire radius, number of electrodes, and electrodes arrangement on heat transfer enhancement. 2. Governing equations 2.1. Electric field equations The electric body force per unit volume is the main driving force of corona-induced flow mixing. It is expressed as:

rc

V2 V ¼ 

(1)

(2)

ε

The electric field strength is given by:

! E ¼ VV

(3)

Conservation of space charge density is defined by:

! vrc þ V$ j ¼ 0 vt

(4)

where j is the electric current density:

! ! ! j ¼ rc b E þ rc u  De Vrc

(5)

in which b is the mobility of ions, De is the ions diffusion coefficient ! and u is the gas velocity vector. The three terms on the right hand side of the Eq. (5) are drift, convection and diffusion currents, respectively. Ignoring the diffusion and convective terms of the electric current density and substituting Eqs. (3) and (5) into Eq. (4), the charge conservation equation becomes [6]:

vrc þ V$ðrc bVVÞ ¼ 0 vt

(6)

Therefore, the electrohydrodynamic governing equations must be simultaneously solved are Eqs. (2) and (6) to determine the electric potential and the space charge density distributions. 2.2. Flow and energy equations The Reynolds number range studied in this article varies from 4000 to 12,000, which means that the flows can be turbulent at the different flow velocities. For this case, among the turbulence models, the standard k  ε model seems to be the most commonly used and the most efficient to study the EHD flow [13e15]. The governing equations, describing the fluid flow and heat transfer by assuming an incompressible fluid, two-dimensional and steadystate can be written in the following form [16]: Continuity:

vðrui Þ ¼ 0 vxi

(7)

Momentum:

" !#  v  vp v vui vuj 2 vuk rui uj ¼  m þ þ  d vxj vxi vxj vxj vxi 3 ij vxk   v  ru0i u0j þ rgi þ Fei þ vxj Energy:


  ! ! 1!2 1 !2 vε F e ¼ rc E  E Vε þ V E r 2 2 vr

657

! In this equation E , rc, r, and ε are the electric field strength, space charge density, flow density, and air permittivity, respectively. Many EHD applications use air as the working fluid. For corona discharge in air under atmospheric pressure and at room temperature, electric permittivity can be assumed to be constant. The second and third terms on the right-hand side of the Eq. (1) become negligible. In such ! cases, the electric body force is simply the Coulomb force ðrc E Þ. This is the force applied by the electric field on the electrons and ions that may exist in the fluid. The EHD governing equations for EHDinduced flows can be summarized as follows. Poisson’s equation which is defined as:

v v ½u ðrE þ pÞ ¼ vxi i vxj

 Kþ

cp mt Prt



vT vxj

(8)

! (9)

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The last terms in the momentum equations are the electric body forces, which can be calculated once the electric potential and the charge density distribution are known. In standard k  ε turbulent model the turbulent viscosity, mt, is computed by combining k and ε as follows:

mt ¼ rCm

k2

(10)

ε

The model constants have the following default values [16]:

C1ε ¼ 1:44

C2ε ¼ 1:92

Cm ¼ 0:09

Table 1 Numerical values of the dimensions for the different configurations. Configuration number

Number of electrodes

Wire 1 Distance (h1-m)

Wire 2 Distance (h2-m)

Wire 3 Distance (h3-m)

1 2 3 4 5 6 7

1 2 3 3 3 3 3

e 0.05 0.05 0.03 0.05 0.02 0.05

0.05 0.05 0.05 0.05 0.03 0.03 0.03

e e 0.05 0.03 0.05 0.05 0.02

sk ¼ 1 sε ¼ 1:3 Peek’s formula specifies the onset electric field on the corona electrode which is the following forms [18]:

3. Computational geometry For the current numerical analysis, a 2-D computational domain has been assumed. A schematic view of the complete domain used for the simulation is shown in Fig. 1. The dimensions comprise L ¼ 0.6 m, L1 ¼ L2 ¼ 0.1 m, H ¼ 0.1 m. In order to find the best arrangement of the electrodes to improve heat transfer, various electrode configurations were examined. The different configurations used in this study are summarized in Table 1. 4. Boundary conditions 4.1. Flow and thermal boundary conditions Boundary conditions specify the flow and thermal variables on the boundaries of the physical model. Therefore, they play a key role in numerical simulations. Boundary conditions applied to the numerical model are as follows. As shown in Fig. 2 air at a uniform velocity (ux ¼ UN) and temperature (T ¼ TN) is introduced at the inlet. The pressure outlet with zero gauge pressure was set at the outlet of domain. All walls assumed no slip condition (ux ¼ uy ¼ 0). Upper wall is adiabatic (q00 ¼ 0) and lower wall under constant heat flux (q00 ¼ cte). Also, wire surface act as stationary wall and was considered thermally insulated. 4.2. Electric field boundary conditions The boundary conditions for the potential are rather straightforward to describe: a given positive DC potential V0at the corona electrode and zero at the ground plane. But, the appropriate boundary conditions to determine the electric charge density are difficult. For this purpose, the Kaptzov hypothesis is used to provide proper boundary conditions for the space charge density, which suggests the electric field increases proportionally to the corona electrode voltage below the corona onset, but it will preserve its value after the corona is initiated [17]. According to this theory the distribution of space charge density at the wire is guessed and iterated until the electric field is sufficiently close to Peek’s value.

  0:308 E0 ¼ 3:1  106 1 þ pffiffiffi r

(11)

where r is the wire radius in centimeter. In addition, the space charge density gradient and potential gradient is zero on the other boundaries. 4.3. Turbulence boundary conditions To modeling the turbulent flow at the inlet boundary, turbulence properties must be specified. So, turbulence kinetic energy (k) and turbulence dissipation rate (ε) were calculated as [16]:

k ¼

3 ðIUN Þ2 2

(12)

3 3 k2 ε ¼ c4 m l

(13)

where I is the turbulence intensity and l is the turbulence length scale which is expressed as:

I ¼ 0:16ðReÞ8

(14)

l ¼ 0:07Dh

(15)

1

in which Reynolds number based on the hydraulic diameter of the channel, Dh, is given by:

Re ¼

UN Dh

n

(16)

in which n is the air kinematic viscosity and Dh ¼ 2H. 5. Numerical solution FLUENT 6.3 as a commercial CFD software was used to perform all numerical simulations. The electric field and the

Fig. 1. A schematic view of the main computational domain and geometric parameters.

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Fig. 2. Boundary conditions for 2D model domain.

space charge density are predicted by simultaneously solving Poisson’s and charge conservation equations to determine the Coulomb force. They are used to model the corona discharge in gases. The solution of the charge density equation suffers from dispersion errors. The reason is that charge equation does not have a diffusion term and only carries a convective term. The PDEs without diffusion term may lead to distorted values of charge density, which may distort the electric body forces and eventually affect the secondary flows generated by these forces. These unstable results could be fixed by adding artificial viscosity. Therefore, the charge density equation is rewritten in the following form [19]:

vrc þ V$ðrc bVVÞ ¼ V$ðGVrc Þ vt

(17)

The diffusion coefficient G is used to define the Peclet number, Pe. The definition of Pe ¼ (bEDh)/G is used here. In this study, the solution for Pe ¼ 100 showed the best result for charge density. A generalized User-Defined-Function (UDF) was written in the C programming language and interpreted to the FLUENT to generate the exact value of body force at each cell. After adding the source terms for each cell, the governing equations of continuity, momentum, energy, turbulence kinetic energy, and the turbulence dissipation rate were simulated using finite volume method in the steady state regime. A pressure-based solver was used along with implicit formulation of the discredited functions. Also, the SIMPLE pressureevelocity coupling algorithm was used. Moreover, the convective and diffusive terms in the equations is solved by a second order upwind implicit approximation. The physical properties of the air as working fluid assumed to be constant with density (r) of 1.225 kg/m3, permittivity (ε) of 8.85  1012 F/m, ion mobility (b) of 2  104 m2/V s, thermal conductivity (K) of 0.0242 W/m K,

specific heat coefficient (cp) of 1006.43 J/kg K, and the dynamic viscosity (m) of 1.789  105 kg/m s. For the aforementioned equations, the computations were continued until to reduce the residuals as a convergence criteria less than1010. This criterion was utilized for all simulations. 6. Data reduction The local heat transfer coefficient is used to investigate the heat transfer, which is defined as follows:

hx ¼

q00 ðTwx  TN Þ

(18)

where Twx is the local wall temperature. The average heat transfer coefficient for the channel surface was obtained by integrating the local heat transfer coefficient over the channel wall and given by the following relation:

h ¼

1 L

ZL hx dx

(19)

0

In order to investigate the effect of electric field on the heat transfer, the strength of the EHD induced secondary flow can be scaled by the ratio of the EHD number to the Reynolds number square:

NEHD ¼

Ehd Re2

(20)

in which the EHD number is the ratio of the electrical body force to the inertial body force and defined as follows [20]:

Fig. 3. A schematic view of 2D model domain grid distribution for configuration No.1., a) Near the wire b) near the wall.

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Ehd ¼

IL3c

(21)

ry2 bA

where Lc ¼ 2H is the characteristic length, A is the discharge area providing that the entire current density crosses it, and I is the total electrical current flowing between electrodes, which can be calculated by:

Z I ¼ A

! j dA ¼ 

Z

rc b A

vV dA vn

(22)

where n is the independent space variable perpendicular to the equipotential lines, such as the grounded electrode, and it is positive in the direction from the wire toward the grounded electrode. The potential gradient is negative in the direction from the wire electrode toward the grounded electrode. Therefore the value of corona current is positive [19]. 7. Mesh generation and grid study The pre-processing commercial software, Gambit 2.4, is used to create the geometry and discrete the computational domain into a number of structured quadrilateral grids with a non-uniform density. It was important to maintain a fine mesh resolution in the boundary layer region near the wall of the internal flow and near the wires, where it is expected the more representative gradients will take place. Fig. 3 illustrates the view of grid distribution for the configuration No.1. Grid convergence is the term used to describe the improvement of results by using successively smaller cell sizes for the calculations. Therefore, different structured grid distributions are tested on different simulations to ensure that the numerical results are grid independent. The local heat transfer coefficient distribution at the lower wall for four different grid densities is depicted in Fig. 4. The computed results show that there was no change in local heat transfer coefficient distribution when the cells number exceed of 20,160. Thus the number of mesh is used for numerical analysis is 20,160. Also, the grid sensitivity for other configurations has been studied. 8. Code validation To validate the predicted results, the current density and the electric field distribution at the grounded plate are compared with

Fig. 4. Grid independence test for local heat transfer coefficient (V ¼ 18 kV, Re ¼ 4000).

Fig. 5. Comparison for current density at the grounded plate (V ¼ 18 kV, Re ¼ 0).

semi-empirical Warburg’s law and experimental data. From Figs. 5 and 6, it is clear that the numerical simulation is in good agreement with the experimental data and Warburg’s law which is described by the following equations [21,22]:

Eq ¼ E0 cos1:5 q

(23)

Jq ¼ J0 cos4 q

(24)

in which, q is the angle of the wire respect to normal axis. 9. Result and discussion In order to evaluate the flow and heat transfer characteristics in presence of EHD-induced flow in a smooth channel, numerical studies were conducted for turbulent regime in which the Reynolds

Fig. 6. Comparison for electric field at the grounded plate (V ¼ 18 kV, Re ¼ 0).

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661

number is evaluated on the basis of hydraulic diameter of the channel. In the first step of the computations, the effect of Reynolds number and applied voltage are studied to gain an optimal set up. In the next step, the effect of wire radius and number of electrodes on enhance heat transfer are investigated. Finally, as the main objective of this study, the electrode arrangement effects are evaluated carefully. 9.1. Effect of Reynolds number By increasing the Reynolds number as one of the most important parameters, the ionic wind seems to have an insufficient effect to divert the flow toward the wall. As depicted in Fig. 7, the corona wind cannot dominate the inertia of the bulk flow and the effect of the electric body force is a weak recirculating cell under the wire. The heat transfer enhancement is defined in current set of studies, as the ratio of the local heat transfer resulting from the electric field to that without the electric field. Concerning to the depicted effect of the Reynolds number, Fig. 8 illustrates the trend of local heat transfer coefficient ratio hxEHD/hxnoEHD, as an enhancement factor along the lower wall of the channel for different Reynolds number. According to this figure, the heat transfer enhancement ratio seems to be more sensitive at the lower values of Re than for higher Reynolds number cases. As it is expected, the corona wind is more effective in low Reynolds number to enhance heat transfer. 9.2. Effect of applied voltage As shown in Fig. 9, implementing higher voltages significantly increases the current passes through grounded plate which makes the effect of corona wind to be increased in comparison with lower voltages. Regarding the previous studies [7,20], in order to investigate the effect of Reynolds number and applied voltage simultaneously, an important factor in EHD study as the strength of EHD is defined. If NEHD / 0, the flow inertia is dominant and the airflow is not affected by the corona wind. Thus, the electric body force effects may be neglected. Conversely, when NEHD / N, the flow field is dominated by the electric body force and forced convection effects

Fig. 7. Stream function contours in the smooth channel for configuration No.1 (V ¼ 18 kV), (a) Re ¼ 4000 (b) Re ¼ 6000 (c) Re ¼ 8000.

Fig. 8. Variation of the heat transfer enhancement factor for different Reynolds number (V ¼ 18 kV).

may be neglected. Variation of the EHD number to Reynolds number square ratio as a function of Reynolds number for different voltages is shown in Fig. 10. The significant effect of the corona wind occurs at Re ¼ 4000 and V0 ¼ 18 kV, which is cleared in Fig. 10. Thus, in order to analyze other effective parameters in presence of the EHD technique, we were set Re ¼ 4000 and V0 ¼ 18 kV as the optimal condition. 9.3. Effect of wire radius In this section, the effect of wire radius on the average convection heat transfer coefficient is investigated. Three different radiuses of wire were chosen for this purpose. As mentioned above, the Reynolds number was 4000 and the applied high voltage was 18 kV. From computations can be resulted by increasing the radius of wire, the average convection heat transfer coefficient is decreased. According to Table 2, it is clear that the electrode with a

Fig. 9. The corona current as a function of applied voltage.

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Fig. 10. Variation of the EHD number to Reynolds number square ratio as a function of Reynolds number for different applied voltages.

radius of 0.2 mm has more effects on heat transfer enhancement using EHD-induced flow in comparison with another radius. In addition, for a higher radius of wire, obtaining a specific corona current requires to higher applied high voltage. Therefore, it must be noted that the lower radius of the wire generally has more electrohydrodynamically effect on heat transfer enhancement. Fig. 11. Stream function contours in the smooth channel (V ¼ 18 kV, Re ¼ 4000)., (a) No EHD (b) configuration No.1 (c) configuration No.2 (d) configuration No.3.

9.4. Effect of number of electrodes As shown in Fig. 11, in presence of EHD-induced flow, the flow patterns have significantly more induced vortices and complexities than those without electric field. In this regard, the effect of the number of electrodes plays the main role of the mentioned complexity. In this figure, the recirculating cells appear directly under the wires location. This causes thermal boundary layer perturbations on the heated wall and consequently, an improvement in heat transfer. In order to analyze the effect of number of recirculating cells on enhance heat transfer, Fig. 12 shows the distribution of local heat transfer coefficient along the lower wall. It can be seen clearly, increasing the number of electrodes causes different distribution of local heat transfer coefficient. To evaluate the performance of various wires number, Table 3 summarizes the average heat transfer coefficient obtained from simulations. One can observe from Table 3, when the number of electrodes is increased, the heat transfer has a tendency to decrease. Although production of vortices causes to divert the flow toward the wall and increase the heat transfer, but on the other hand, reduction of heat transfer is due to a kind of barrier effect appears when the air arrives on the first wire [14]. This phenomenon can be explained upon the fact that the nearby vortices make the main flow to be deflected upward and consequently, to be

prevented from the fresh air exposure to the heated wall. This effect vanishes after the last wire and the flow is then directed toward the plate. Of course, this phenomenon may be disappeared with increasing the distance between the wire electrodes while fresh air can be contacted with the heated surface after each electrode.

Table 2 Average heat transfer coefficient and corona current for different wire radiuses (V ¼ 18 kV, Re ¼ 4000). Wire radius

hðW=m2 *KÞ

I (mA)

r¼0.2 mm r¼0.3 mm r¼0.4 mm

6.75 5.08 4.06

160 102 60

Fig. 12. Distribution of local heat transfer coefficient for three configurations (V ¼ 18 kV, Re ¼ 4000).

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Table 3 Average heat transfer coefficient (V ¼ 18 kV, Re ¼ 4000). Configuration number

hðW=m2 *KÞ

No EHD 1 2 3

4.03 6.75 6.49 6.34

9.5. Effect of electrodes arrangement As mentioned before, breaking and destabilizing in the thermal boundary layer are promoted in presence of EHD-induced flow. Therefore, EHD technique has a significant effect on the heat transfer enhancement as well as more than 1.5 times to that in absence of EHD. But contrary to what was expected, by increasing the number of electrodes the heat transfer is decreased. It must be pointed out that the increase of number of electrodes is necessary but not a sufficient condition to achieve higher enhancement.

Fig. 14. Distribution of local heat transfer coefficient for different configurations (V ¼ 18 kV, Re ¼ 4000).

Therefore, in order to diminish the barrier effects of vortices and find the best arrangement of three wires, results for four configurations were presented and compared with configuration No.3. Fig. 13 represents the stream function contours for the different electrodes configuration. It is clear that the variation of electrodes distance away from the grounded plate causes different formation of vortices. It seems that with reducing the distance between electrodes and grounded plate in configurations No.4, 5, 6 and 7, the number of effective vortices and their strength for the deviation of flow toward the hot plate is increased. Of course, to prove it, the local and average heat transfer coefficient has been checked. The local heat transfer coefficient over the lower wall for five configurations is illustrated in Fig. 14.The results show that the distribution of local heat transfer is different in comparison with the configuration No.3. This effect can be explained on the basis of flow streamlining and influences of different electrodes configuration on the recirculating cells behavior. In addition, to compare the heat transfer performances of the different configurations, the average heat transfer coefficient are given in Table 4. From Table 4, it is clear that changing electrodes arrangement has improved the heat transfer from heated wall. This observation can also be explained on the basis of the shape, number, and the strength of vortices. The main reason for this enhancement is the same as before, that is, decreased wire distance from grounded plate fortifies the effective vortices. In addition, the comparison of streamlines between configuration No.6 with other configurations demonstrate that the first wire generates an effective recirculating cell and when the main flow tend to deflect upward, second and third wires divert fresh air to the heated wall and cause

Table 4 Average heat transfer coefficient (V ¼ 18 kV, Re ¼ 4000).

Fig. 13. Stream function contours in the smooth channel for different configurations (V ¼ 18 kV, Re ¼ 4000)., (a) Configuration No.3 (b) configuration No.4 (c) configuration No.5 (d) configuration No.6 (e) configuration No.7.

Configuration number

hðW=m2 *KÞ

3 4 5 6 7

6.34 11.04 9.96 15.05 13.25

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10. Conclusions Heat transfer enhancement and the swirling flow pattern due to the EHD-enhanced heat transfer inside a smooth channel under a constant heat flux and turbulent flow regime were studied numerically, and the following main conclusions can be highlighted:

Fig. 15. Stream function contours in the smooth channel for different wireewire distances (V ¼ 18 kV, Re ¼ 4000)., (a) L1 ¼ L2 ¼ 5 cm (b) L1 ¼ L2 ¼ 10 cm (c) L1 ¼ L2 ¼ 15 cm (d) L1 ¼ L2 ¼ 20 cm

significantly enhancements. Therefore, configuration No.6 is the best arrangement for heat transfer enhancement, in this study. 9.6. Effect of wireewire distance In order to evaluate the effect of wireewire distance on the heat transfer enhancement and the swirling flow pattern, three arrangements were considered and compared with configuration No.6. Fig.15 shows the stream function contours at the different wireewire distances. It can be seen that a significant difference in the number of induced vortices, their influence on each other, and the size of the recirculating zones. It is important, therefore, to determine the wiree wire distance in order to optimize the arrangement of the electrodes. To demonstrate the effect of wireewire distance on the heat transfer, average heat transfer coefficient is reported in Table 5 for an applied voltage of 18 kV at the Reynolds number of 4000 for all four electrodes distances. As reported, average heat transfer coefficient is reduced with decreasing and increasing the wireewire distance. However, heat transfer enhancement can be optimized by the proper selection of the wireewire distance. Table 5 Average heat transfer coefficient for different wireewire distances (V ¼ 18 kV, Re ¼ 4000). Wireewire distance L1 L1 L1 L1

¼ ¼ ¼ ¼

L2 L2 L2 L2

¼ ¼ ¼ ¼

5 cm 10 cm 15 cm 20 cm

1. To achieve to a robust basis about the Reynolds number effect and its exact effective range for the main study, first of all, the strength of the corona wind were examined while the Reynolds number was forced to increase. The results demonstrated that, at the lower Reynolds numbers; Re ¼ 4000, the heat transfer enhancement factor exceeds to 2.5 times of its value in higher Reynolds number; Re ¼ 12000, in which the corona winds were clearly vanished. This reality not only validates the expectation of EHD dominant in low Reynolds numbers, but also gave the required assurance to choose the lower Reynolds number for the further studies of the current work; Re ¼ 4000. In the same way, it has been realized that applying higher voltages caused the same effect, which means that the EHD influence was significantly augmented by increasing the applied voltages, and consequently the 18 kV was also chosen as the adequate applying voltage for further steps of current study. 2. According to this research, it is observed that the lower radius of the wire in a constant applied voltage and Reynolds number has a better enhancement in heat transfer. Also, it is understood that using the electrode with a lower radius has more efficiency due to more increase of the corona current. 3. The results indicate that, using EHD-induced flow causes considerable improvements on heat transfer, but not in a regular and linear way. It means the heat transfer slightly decreases as the number of electrodes increases one by one, with the same constant distances from the grounded surface. This reality became the main motivation of providing numerical observation of variable electrodes configuration effects to give a recovery technique when increasing the number of electrodes cause lower performances. 4. Furthermore, in consequence of the previous steps, the main emphasis of this study is placed on identifying the effect of electrodes arrangement on the flow pattern and heat transfer coefficient. It can be clearly seen the heat transfer significantly depends on the electrodes arrangement; variable vertical distance from the grounded surface. This main conclusion has been highlighted in Table 4 in which the heat transfer coefficient exceeds twice than that of electrode configuration No. 3, and which also means consequently, diminishing of the barrier effect of vortices. 5. In this study, the heat transfer enhancement mechanism is obtained through vortex generation by utilizing the EHDinduced flow. According to stream function contours, it is clear that the induced vortices by corona discharge have a significant effect on the change in flow structure. In addition, the strength of these vortices depends on the electrodes arrangement. 6. Finally, It is found that at the same EHD conditions, the heat transfer enhancement is dependent on the wireewire distance. Moreover, Considerable improvement in heat transfer enhancement is obtained by employing the optimum distance between the wires.

hðW=m2 *KÞ 12.65 15.05 14.66 14.26

Regarding all the mentioned results and conclusions, it is obvious that the increasing of number of electrodes can highly increase the EHD performance, but of course in an optimum manner regarding the other conditions.

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Nomenclature A surface area of the grounded plate, m2 cp specific heat coefficient, J/kg K Cm, C1ε, C2ε the turbulence model constant De charge diffusion coefficient, m2/s Dh hydraulic diameter, m E electric field strength, V/m E total energy, J Ehd EHD number Fe electrohydrodynamic force, N/m3 g acceleration due to gravity, m/s2 H channel height, m h distance between wire and ground plate, m h convective heat transfer coefficient, W/m2 K I corona current, A I turbulence intensity j current density, A/m2 K thermal conductivity, W/m K k turbulence kinetic energy, J L channel length, m Lc characteristic length, m l turbulence length scale, m NEHD the EHD number to Reynolds number square ratio n normal space variable t turbulent P pressure, Pa Pe Peclet number Pr Prandtl number q00 wall heat flux, W/m2 Re Reynolds number r wire radius, m T temperature,  C U mean velocity, m/s u velocity, m/s V voltage, V

Greek symbols b ion mobility, m2/V s dij Kronecker delta G diffusion coefficient, m2/s ε permittivity, F/m ε turbulence dissipation rate, W m dynamic viscosity, kg/m s n kinematic viscosity, m2/s q angle with respect to normal axis,  r density, kg/m3 rc space charge density, C/m3 sk Prandtl number for turbulent kinetic energy sε Prandtl number for turbulent dissipation rate

Subscripts wire value x local value in x direction



i, j w N

665

1,2 denote xey space coordinate wall value inlet

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