Numerical modeling of the electrohydrodynamic effect to natural convection in vertical channels

International Communications in Heat and Mass Transfer 34 (2007) 162 – 175 www.elsevier.com/locate/ichmt

Numerical modeling of the electrohydrodynamic effect to natural convection in vertical channels ☆ N. Kasayapanand a,⁎, T. Kiatsiriroat b a

School of Energy, Environment and Materials, King Mongkut's University of Technology Thonburi, Bangkok 10140, Thailand b Department of Mechanical Engineering, Chiang Mai University, Chiang Mai 50200, Thailand Available online 20 December 2006

Abstract The electrohydrodynamic effect to natural convection inside the vertical channels is numerically investigated by computational fluid dynamics technique. The range of parameters considered are 104 = Ra = 107, 7.5 = V0 = 17.5 kV, and 2 = aspect ratio = 10. Flow and temperature distributions are affected with supplied voltage at the wire electrodes, and the heat transfer enhancement is significantly influenced at low Rayleigh number. The augmented volume flow rate of fluid is indicated in relation with the number of electrodes. Moreover, heat transfer enhancement also depended on the electrode arrangement while the number of electrodes is initially fixed. The relation between channel aspect ratio and number of electrodes that performs the maximum heat transfer is expressed incorporating with the optimum concerning parameters. © 2006 Elsevier Ltd. All rights reserved. Keywords: Heat transfer enhancement; Natural convection; Electrohydrodynamic; Computational fluid dynamics

1. Introduction The convective heat transfer enhancement technique utilizing electrostatic force generated from the polarization of dielectric fluid or electrohydrodynamic (EHD) can be one of the most promising methods among various active techniques because of its several advantages. This technique deals to the interdisciplinary field with subjects concerning the interactions between electric, flow, and temperature fields. There are some studies relating to the electrohydrodynamic for instances, Yabe et al. [1] investigated the heat transfer enhancement of a corona wind between wire and plate electrodes under natural convection. Velkoff and Godfrey [2] performed the heat transfer enhancement over a horizontal flat plate with parallel wire electrodes. The computational method in the electrostatic precipitator was conducted by Yamamoto and Velkoff [3]. The electrohydrodynamic phenomenon on natural convection inside the enclosures was indicated by Shu and Lai [4], Yang and Lai [5], and Yan et al. [6]. Kasayapanand and Kiatsiriroat [7,8] investigated the corona wind augmented heat transfer inside channel with the optimum electrode arrangement by computational fluid dynamics technique. Recently, Kasayapanand [9] conducted the numerical results of the electrode

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Communicated by W.J. Minkowycz. ⁎ Corresponding author. E-mail address: [email protected] (N. Kasayapanand).

0735-1933/$ - see front matter © 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.icheatmasstransfer.2006.10.005

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bank enhanced heat transfer. However, there is no previous literature concerning with the electrohydrodynamic applying to the natural convection in an open vertical channel. Natural convection in vertical channels is encountered in various engineering applications include heat exchangers, solar heating systems, cooling of electronic equipment, and chimneys. Early researchers were carried out by Bodoia and Osterle [10], and later by Levy [11] for the optimization of plate spacing. Aung [12] applied natural convection between plates with asymmetric wall conditions. The isothermal plates heated with open edges were investigated by Sparrow and Bahrami [13]. This characteristic was confirmed by [14–16]. The influence of variable property was conducted by Zamora and Hernandez [17]. Morrone et al. [18] numerically studied the optimum vertical plate separation. Lee [19] performed numerical and theoretical analysis of partially heated vertical parallel plates. This numerical study considers electrostatic forces exerting to the natural convection of air flowing inside vertical channels. Governing equations of electrohydrodynamic phenomenon are formulated and the numerical modeling has been carried out to investigate the electrohydrodynamic enhanced secondary flow and heat transfer in two-dimensional flow. The wire electrodes generated with DC high voltage are placing in the channel which installed at cross angle to the computational domain. The characteristics of flow and heat transfer are discussed in conjunction with the parameters comprising of Rayleigh number, supplied voltage, number of electrodes, electrode arrangement, and channel aspect ratio. 2. Mathematical analysis The conservation of mass for an incompressible fluid is given by Aq þ ðjd qvÞ ¼ 0 At

ð1Þ

The equations of the electrohydrodynamic force per unit volume FE generated by the electric field with strength E in a fluid of dielectric permittivity ε, density ρ, and uniform temperature T can be expressed as     1 1 Ae FE ¼ qE− E 2 je þ j E2 q : 2 2 Aq T

ð2Þ

Where q is the electric charge density in the fluid, the first term of the right qE is the Coulomb force exerted by electric field upon the free charge or electrophoretic component. The second and third terms correspond to the dielectrophoretic and electrostrictive force on and within the fluid. Eq. (2) is then

Fig. 1. Computational domain and boundary conditions.

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Fig. 2. Grid generation (a) electric field (b) flow and temperature fields.

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Fig. 3. Stream function contours of uniform arrangement (V0 = 17.5 kV and N = 5) (a) Ra = 104, (b) Ra = 105, (c) Ra = 106, (d) Ra = 107.

included in the Navier–Stokes equation of an incompressible fluid. Thus, the conservation of momentum is defined by   Av þ ðvdjÞv ¼ qg þ FE −jP þ lj2 v; q ð3Þ At the vector ρ g is the gravitational force per unit volume, P is the local fluid pressure and the last term in the right-hand side of equation represents the viscous terms. For the energy equation of this system, without viscous dissipation effect, the energy equation can be written as   AT re E 2 V0 J þ ðvdjÞT ¼ aj2 T þ þ : ð4Þ At qcp Aw Lw qcp Maxwell equations for the electric field are as the followings jd eE ¼ q;

ð5Þ

the strength E is given by E ¼ −jV :

ð6Þ

As the current is conserved over the computational domain, the current continuity equation is expressed by jd J þ

Aq ¼ 0; At

ð7Þ

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Fig. 4. Temperature distributions of uniform arrangement (V0 = 17.5 kV and N = 5) (a) Ra = 104, (b) Ra = 105, (c) Ra = 106, (d) Ra = 107.

where the current density J is given by J ¼ qv þ re E þ ðvdjÞðeEÞ−bjq:

ð8Þ

From the combination of Eqs. (5), (6), (7) and (8), it can obtain q j2 V ¼ − ; e q2 ¼ eðjV djqÞ:

ð9Þ ð10Þ

To reduce the independent parameters investigated, all governing equations are non-dimensionalized using the dimensionless parameters described by x¯i ¼

xi vi tue T −Tc V q ; v¯i ; ¯ ; ue ¼ ðq0 V0 =qÞ1=2 ; h ¼ − ;V¯ ¼ ; ¯q ¼ : t¼ V0 q0 W ue W ðAT =AxÞw W

The dimensionless governing equations are expressed as jd ¯v ¼ 0;

ð11aÞ

Av¯ gW FE W jP j2 ¯ v þ ðv¯djÞv¯ ¼ 2 þ ; − þ A¯t ue qu2e qu2e Re

ð11bÞ

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Ah 1 1 re E2 W 1 V0 JW þ ð¯ vdjÞh ¼ ðj2 hÞ þ þ ; A¯t Pe Pe qh Pe Aw Lw qh

ð11cÞ

qq0 W 2 ¯ ; eV0

ð11dÞ

eV0 ðjV¯dj¯ qÞ; q0 W 2

ð11eÞ

j2V¯ ¼ − q2 ¼ ¯

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where Re ¼ ue dv W ; Pr ¼ av ; and Pe ¼ Red Pr are the dimensionless parameters. The Rayleigh number (based on uniform heat flux at the hot plate and characteristic length of W) is defined as Ra ¼ GrdPr ¼

gbqh W 4 dPr: kv2

ð12Þ

In an analysis of heat transfer characteristic, the local heat transfer coefficient in term of the local Nusselt number of laminar sublayer near the surface which can be applied in this work is given by Nuy ¼

ðAT =AxÞw W hW 1 ¼− ¼ : k hw ðTw −Tc Þ

ð13Þ

From which the average heat transfer coefficient along the vertical surface of channel can be integrated numerically from Eq. (13).

Fig. 5. Stream function contours of unequal number of electrodes (V0 = 17.5 kV and Ra = 106) (a) N = 5, (b) N = 9, (c) N = 17, (d) N = 33.

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3. Methodology The computational domain and boundary conditions required for the governing equations are illustrated in Fig. 1. The left plate is maintained at uniform heat flux and the right plate is insulated, both upper and lower of the channel are opening. All plates are electrically grounded and the channel size is 5 × 30 cm2. This numerical modeling conducts grid generation method to convert the physical plane in Cartesian coordinates (x, y) into the computational plane in curvilinear coordinates (ξ, η) by Poisson's equation. The illustration of grid generation is shown in Fig. 2. In electric field (Fig. 2(a)), the first radial discrete value is computed using geometrical progression in such a way to allow high nodal density near the wire and the remaining region being subdivided into equispaced nodes. The starting conic coincides with the wire, for greater distance from the wire, the circular symmetry changes gradually fitting into Cartesian reference at the grounded plates and the symmetrical axes. The obtained charge and potential distributions are converted into polynomial function of quadrilaterals. This function must pass through the values of change and potential distributions at the rectangle nodes and the result can be expressed by interpolation functions. Finally, these values are mapping into computational fluid dynamics grid generation for calculate velocity and temperature in Fig. 2(b). All equations can be non-dimensionalized and the numerical calculations can be carried out with the computational grid side of 751 × 141 which is achieved within 1.4% of error at Re = 107 by independent grid refinement test. A non-dimensional time step of 5 × 10− 5 is used to guarantee numerical stability and accuracy. Several models have been proposed for the calculation of electric field and charge density distribution in a wire-plate electrostatic precipitation system using finite difference method. Firstly, we must assume q0 at the wire electrode according to the semi-empirical formula by Peek [20]. The space charged density near the wire can be given by q0 ¼

lw JP kbrf ð30d þ 0:9ðd=rÞ1=2 Þ

 10−5 ;

ð14Þ

where δ is T0P/TP0, T0 is 293 K, P0 is 1.01 × 105 N/m2, f is 1, T is the operating temperature at 300 K, and P is the operating pressure at 1.01 × 105 N/m2. The JP is an initial current density at the grounded plate. The electric field and the charge density

Fig. 6. Temperature distributions of unequal number of electrodes (V0 = 17.5 kV and Ra = 106) (a) N = 5, (b) N = 9, (c) N = 17, (d) N = 33.

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distribution are obtained responding to Eqs. (11d) and (11e) by successive under relaxation method and upwind difference scheme to avoid divergence in the iterative solution. The obtained electric field and charge distribution are used to calculate electric current density at the grounded plate and compared with the initial value. When the calculated value is differed, the above calculation is repeated by changing the value of space charge density at the wire electrode until convergence. Embarking an upwind difference scheme on velocity field at all nodes are calculated. The velocity at each time step is hence calculated from Eqs. (11a) and (11b) in accordance with the electric field and electric charge density by SIMPLEC (SIMPLE-Consistent) algorithm. The temperature field is finally acquired from Eq. (11c). The above mentioned procedures are repeated until the convergence of velocity and temperature is reached. A convergence criterion is selected when the results are in steady state, periodic state, or non-periodic state. 4. Results and discussion The results of non-electrohydrodynamic effect are validated against with a bench mark solution of the natural convection in vertical channel by Aung and Worku [21] at aspect ratio = 4 and Gr/Re = 250. The calculated values agree very well within the maximum error of 2.7%. Fig. 3 investigates the stream function contours of electrohydrodynamic applying while the Rayleigh number is varied between 104–107 in which an effect of Joule heating at the wire electrode is neglected (V0 = 17.5 kV and N = 5). The flow patterns are oscillated due to the interaction between thermal buoyancy force and electrical body force. Moreover, two categories of oscillatory are observed in these figures; periodic state (Fig. 3(a)–(b)) and non-periodic state (Fig. 3(c)–(d)). There is an effect of secondary flow induced by ionic wind at the wire electrode which causes four of rotating cellular motions occurring at each electrode in Fig. 3(a)–(b). It can be observed that flow pattern in the channel is activated by an electric field at low Rayleigh number, but dominated by convective regime instead of an electric field as Rayleigh number increases further to 106–107 in Fig. 3(c)–(d). A convective regime is established at high Rayleigh number, the cold air is embarked through the bottom of opening, rises along the hot plate, and discharges at the top of opening. At low Rayleigh number, the temperature fields have been affected due to an electric field as represented in Fig. 4. A thermal boundary layer corresponding to the

Fig. 7. Stream function contours of various electrode arrangements (V0 = 17.5 kV, Ra = 106, and N = 17) (a) bottom denseness, (b) top denseness, (c) ends denseness, (d) middle denseness.

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Fig. 8. Temperature distributions of various electrode arrangements (V0 = 17.5 kV, Ra = 106, and N = 17) (a) bottom denseness, (b) top denseness, (c) ends denseness, (d) middle denseness.

secondary flow along the channel enhances convection and heat transfer along the hot plate. It can be seen that temperature gradient (line density) at the left plate becomes higher with the increasing of Rayleigh number that causes high heat transfer coefficient to develop along the hot plate. However, flow patterns and temperature distributions remain almost resembling to the non-electrohydrodynamic phenomenon at high Rayleigh number which can be indicated that there is no significant change in the heat transfer enhancement. The oscillatory stream function and isotherm line contours in categories of various numbers of electrodes from 5, 9, 17, and 33 are expressed in Figs. 5 and 6, respectively (V0 = 17.5 kV and Ra = 106). The effect of number of electrodes plays much important role

Fig. 9. Comparison of heat transfer enhancement (V0 = 7.5–17.5 kV and N = 5).

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Fig. 10. Effect of number of electrodes to the volume flow rate enhancement (V0 = 17.5 kV).

on the air flow pattern in channel. It can be observed that the number of vortices increases when the number of electrodes is increased from 5 to 9 in Fig. 5(a)–(b), but the small vortices are combined and the large vortices occur at the upper and lower of the channel when the number of electrodes has sufficient excess at 17 to 33 in Fig. 5(c)–(d). The isotherm lines show formation of a boundary layer heat transfer along the hot plate in Fig. 6. As seen that temperature gradient at the left plate of Fig. 6(d) is highest compared with the other that obtains greatest heat transfer coefficient. Therefore, one can conclude that the effect of the number of electrodes has more significance at a high value. However, it should be compared to the extra received convective heat transfer per power of electrical energy input for an optimized design. Fig. 7 conducts the effect of electrode arrangements in bottom, top, ends, and middle denseness by exponential function while the number of electrodes is initially kept as the same at 17 (V0 = 17.5 kV and Ra = 106). It is found that there are many vortices around the electrode strip especially at the high density zone and the flow patterns are oscillated due to the high intensity of an electric field. The temperature fields in channel are represented in Fig. 8. It can be seen that the large heat trap occurs in Fig. 8(b) due to the high flow resistance at the top of the channel, while the other cases have uniform temperature gradient along the hot plate especially in Fig. 8(c) (ends denseness). This can be perceived that the temperature gradient, also convective heat transfer, should be highest in this case. The average heat transfer enhancement over the period of periodic state or over the entire time span of non-periodic state (evaluated using the ratio of an average Nusselt number of the presence of an electric field to that without electric field) along the channel at Rayleigh numbers between 104–106 are compared with those of experimental results in Fig. 9 (V0 = 7.5–17.5 kV and N = 5). The simulation results have a good agreement to the experimental results within value around 10%. It is indicated that electrohydrodynamic enhancement of heat transfer has high effectivity at low Rayleigh number and it diminishes at high Rayleigh number. Thus, the convective heat transfer is consequently dependent with Rayleigh number and supplied voltage. An augmented ratio of volume flow rate (per unit depth) inside channel is indicated Figs. 10 and 11 (V0 = 17.5 kV). The number of electrodes has high effect to the volume flow rate of fluid through the channel in Fig. 10. The enhanced volume flow rate ratio is starting from 1 at N = 0 and increases to the maximum point at N = 5 due to the optimum electrode distance ratio (which will be discussed later) and decreases at high number of electrodes due to the raising of pressure drop. Fig. 11 shows a comparison of electrode arrangement

Fig. 11. Effect of electrode arrangement to the volume flow rate enhancement (V0 = 17.5 kV and N = 17).

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Fig. 12. Effect of number of electrodes to the heat transfer enhancement (V0 = 17.5 kV).

effect to the augmented volume flow rate which was found that bottom denseness performs highest performance. The reason of this result can be described by the fluid that is rapidly conveyed by this arrangement along the hot plate from Fig. 7(a) that is an advantage for convective phenomenon. Fig. 12 demonstrates the relation between heat transfer enhancement and number of electrodes (V0 = 17.5 kV). Augmented Nusselt number significantly varies with respect to the number of electrodes, the trend of curve reduces at the intermediate number (the combination between oscillatory cells occurs here) and becomes raise again by the effect of high intensity of an electric field. This result can be achieved by considering the isotherm line density in Fig. 6 at N = 5 which was found to be higher than at N = 9 due to lower number of vortices and heat trap that cause higher heat transfer coefficient. Since the number of electrodes is rather high as N = 33, many vortices are formulated along the channel and combined when number of electrodes is too high even though there is a good turbulent mixing, but the flows are recirculating and oscillating around the top and bottom zones of a channel. Nevertheless, the heat transfer coefficient is higher due to more ionic wind effect. Thus, the number of electrodes installed in a channel should be compromised between all concerning parameters. This is clearly seen from Fig. 13, number of electrode being the same, for which an optimum arrangement (bottom denseness) that ascertained from the predicted value of the flow arrangement in Fig. 11 has dramatically contrasted with this figure. The ends denseness (Fig. 8(c)) has more uniform distribution of temperature gradient in Fig. 8, which causes the maximum heat transfer enhancement (from the thinness thermal boundary layer) occurring approximately around 25% higher than in the case of non-EHD in Fig. 13. The ratios between the augmented heat transfer of air and the number of electrodes are compared in Fig. 14 (Ra = 104). It is interesting that high number of electrodes cannot yield proportionally performance because high power consumption is required for this case (in relation with the electrical power consumption (V0JP × area of the grounded plate)). However, the power consumption is generally neglected due to very lower compared with the extra received heat transfer especially on natural convection. Fig. 15 represents the relation between the heat transfer enhancement and channel aspect ratio (V0 = 17.5 kV and W = 5 cm). In general, the augmented ratio decreases at high gap between wire and plate electrodes due to low intensity of an electric field. When a gap is initially fixed, the enhanced ratio reaches the first maximum point at a rather low number of electrodes and reduces slowly when the number is augmented. However, the enhanced ratio becomes increasingly again when the number of electrodes is sufficiently

Fig. 13. Effect of electrode arrangement to the heat transfer enhancement (V0 = 17.5 kV and N = 17).

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Fig. 14. Augmented heat transfer per number of electrodes (Ra = 104).

exceeded due to the high intensity of an electric field. The reason can be described by considering the value of ratio between distance between wire electrodes and distance between vertical plates, the maximum point occurs when this value immediately approaches 1 (or at N = 1 for H = 10 cm, N = 5 for H = 30 cm, and N = 9 for H = 50 cm). The electric field from each wire electrode emanates from a point charge symmetrically in all directions. Field lines begin on positive (wire electrode) and end on negative (grounded plate), they cannot simply terminate in computational domain. At this electrode distance ratio, it may be an advantage for polarization of ionic wind to the grounded plate. Thus, it can be noticed that the ratio of heat transfer enhancement increases in relation with the small gap and the suitable electrode distance ratio.

5. Conclusion Numerical simulations are carried out to analyze the mechanism of natural convection inside open channels incorporating with electrohydrodynamic effect. The conclusions are obtained as follows. Flow pattern of fluid is affected with the supplied voltage. The thermal boundary layer along the surface is strongly perturbed by an electric field effect and decreases at high supplied voltage. The enhancement of flow and heat transfer with the presence of an electric field increases in relation with the supplied voltage but decreases with the Rayleigh number. The volume flow rate enhancement of fluid inside the channel decreases in relation with the number of electrodes due to pressure drop. The heat transfer enhancement reaches the maximum point at the intermediate number of electrodes.

Fig. 15. Distribution of augmented ratio in relation with channel aspect ratio (V0 = 17.5 kV and W = 5 cm).

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The effect of electrode arrangement in case of bottom denseness performs highest volume flow rate, while ends denseness arrangement yields a maximum heat transfer enhancement due to the highest temperature gradient along hot plate. Moreover, the channel aspect ratio which obtains the optimization between augmented heat transfer and power consumption when number of electrodes is fixed has the recommended value between channel width and distance between wire electrodes as 1. Thus, for high efficiency and economy design, it should be compromised among all concerning parameters. Nomenclature Aw Cross section area of wire electrode, m2 b Ion mobility, m2/V·s cp Specific heat, J/kg·K E Electric field strength, V/m f Roughness factor, V/m FE Electrohydrodynamic body force, N/m3 g Acceleration due to gravity, m/s2 Gr Grashof number h Heat transfer coefficient, W/m2·K H Channel height, m J Current density, A/m2 k Thermal conductivity, W/m·K lw Length between wire electrodes, m Lw Length of wire electrode, m N Number of electrodes Nu Nusselt number P Pressure, N/m2 Pe Peclet number Pr Prandtl number q Electric charge density, C/m3 qh Heat flux, W/m2 Q Volume flow rate per unit depth, m2/s r Corona wire radius, m Ra Rayleigh number Re Reynolds number t Time, s T Temperature, K v Fluid velocity, m/s V Voltage, V W Channel width, m Greek symbols α Thermal diffusivity, m2/s β Volume expansion coefficient, 1/K ε Fluid permittivity, F/m μ Dynamic viscosity, kg/m·s ν Kinematics viscosity, m2/s ρ Density, kg/m3 σe Electrical conductivity, 1/ohm·m Subscripts 0 Without electric field P At the grounded plate w Wall surface

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Acknowledgement The author gratefully acknowledges the financial support provided by the Thailand Research Fund and Commission on Higher Education for carrying out this research. References [1] A. Yabe, Y. Mori, K. Hijikata, EHD Study of the corona wind between wire and plate electrodes, AIAA J. 16 (1978) 340–345. [2] H.R. Velkoff, R. Godfrey, Low velocity heat transfer to a flat plate in the presence of a corona discharge in air, J. Heat Transfer 101 (1979) 157–163. [3] T. Yamamoto, H.R. Velkoff, Electrohydrodynamics in an electrostatic precipitator, J. Fluid Mech. 108 (1981) 1–18. [4] H.S. Shu, F.C. Lai, Effect of electrical field on buoyancy-induced flows in an enclosure, Conf. Rec. IAS Annu. Meet. 2 (1995) 1465–1471. [5] H. Yang, F.C. Lai, Effects of heat generation on EHD-enhanced natural convection in an enclosure, Conf. Rec. IAS Annu. Meet. 3 (1997) 1851–1859. [6] Y.Y. Yan, H.B. Zhang, J.B. Hull, Numerical modeling of electrohydrodynamic (EHD) effect on natural convection in an enclosure, Numer. Heat Transfer, Part A 46 (2004) 453–471. [7] N. Kasayapanand, T. Kiatsiriroat, EHD enhanced heat transfer in wavy channel, Int. J. Commun. Heat Mass Transfer 32 (2005) 809–821. [8] N. Kasayapanand, T. Kiatsiriroat, Optimized electrode arrangement in solar air heater, Renew. Energy 31 (2006) 439–455. [9] N. Kasayapanand, Numerical study of electrode bank enhanced heat transfer, Appl. Therm. Eng. 26 (2006) 1471–1480. [10] J.R. Bodoia, J.F. Osterle, The development of free convection between heated vertical plates, ASME J. Heat Transfer 84 (1962) 40–44. [11] E.K. Levy, Optimum plate spacings for laminar natural convection heat transfer from parallel vertical isothermal flat plates, ASME J. Heat Transfer 93 (1971) 463–465. [12] W. Aung, Fully developed laminar free convection between vertical plates heated asymmetrically, Int. J. Heat Mass Transfer 15 (1972) 1577–1580. [13] E.M. Sparrow, P.A. Bahrami, Experiments on natural convection from vertical parallel plates with either open or closed edges, ASME J. Heat Transfer 102 (1980) 221–227. [14] R.A. Wirtz, R.J. Stutzman, Experiments on free convection between vertical plates with symmetric heating, ASME J. Heat Transfer 104 (1982) 501–507. [15] E.M. Sparrow, L.F. Azevedo, Vertical channel natural convection spanning between the fully-developed limit and the single-plate boundarylayer limit, Int. J. Heat Mass Transfer 28 (1985) 1847–1857. [16] B.W. Webb, D.P. Hill, High Rayleigh number laminar natural convection in an asymmetrically heated vertical channel, ASME J. Heat Transfer 111 (1986) 649–656. [17] B. Zamora, J. Hernandez, Influence of variable property effects on natural convection flows in asymmetrically-heated vertical channels, Int. Commun. Heat Mass Transfer 24 (1997) 1153–1162. [18] B. Morrone, A. Campo, O. Manca, Optimum plate separation in vertical–parallel plate channels for natural convective flows incorporation of large spaces at channel extremes, Int. J. Heat Mass Transfer 40 (1997) 993–1000. [19] K.T. Lee, Natural convection heat and mass transfer in partially heated vertical parallel plates, Int. J. Heat Mass Transfer 42 (1999) 4417–4425. [20] F.W. Peek, Dielectric Phenomena in High-Voltage Engineering, McGraw-Hill, New York, 1929. [21] W. Aung, G. Worku, Developing flow and flow reversal in a vertical channel with asymmetric wall temperatures, ASME J. Heat Transfer 108 (1986) 299–304.