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Numerical modeling of the effect of number of electrodes on natural convection in an EHD fluid
ARTICLE IN PRESS
Journal of Electrostatics 65 (2007) 465–474 www.elsevier.com/locate/elstat
Numerical modeling of the effect of number of electrodes on natural convection in an EHD fluid Nat Kasayapanand School of Energy, Environment, and Materials, King Mongkut’s University of Technology Thonburi, Bangkok 10140, Thailand Received 29 August 2005; received in revised form 2 November 2006; accepted 5 November 2006 Available online 1 December 2006
Abstract Numerical modeling of the electrohydrodynamic effect on natural convection in enclosures is investigated. The interactions between electric field, flow field, and temperature field are analyzed by using the computational fluid dynamics technique. Flow pattern and temperature distribution are substantially affected by the voltage supplied at the wire electrodes, especially at low Rayleigh number. It can be concluded that the fluid velocity and heat transfer coefficient in the presence of an electric field are significantly increased with when a large number of electrodes is used. Surprisingly, a minimum value of augmented heat transfer occurs with an intermediate number of electrodes. The optimized condition between the aspect ratio of an enclosure and number of electrodes which leads to maximum heat transfer enhancement is expressed in terms of the parameters concerned. r 2006 Elsevier B.V. All rights reserved. Keywords: Electrohydrodynamic; Electric field; Wire electrode; Computational fluid dynamics; Wire electrode; Heat transfer enhancement
1. Introduction The technique of convective heat transfer utilizing electric fields or electrostatic forces from the polarization of a dielectric fluid is one of the more promising methods for enhancing heat transfer because of its many advantages. The method is easily implemented, for example, using only a transformer and electrodes, it consumes only a small amount of electric power. This technique is frequently called electrohydrodynamic (EHD) heat transfer, and it refers broadly to an interdisciplinary field dealing with the interactions between electric, flow, and temperature fields. In a typical gaseous medium, energy is transferred from free electrons to the gas molecules, and the latter move toward a grounded surface to increase the heat transfer coefficient. There exist some prior studies related to this use of EHD. Yabe et al. [1] investigated the phenomenon of a corona wind between wire and plate electrodes and found that the interaction between ionic wind and primary flow Tel.: +66 2470 8699; fax: +66 2427 9062.
E-mail address: [email protected]. 0304-3886/$ - see front matter r 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.elstat.2006.11.002
increased the heat transfer from a wall surface. Velkoff and Godfrey [2] investigated heat transfer over a horizontal flat plate using parallel wire electrodes. The ionic wind promoted mixing of the primary flow, resulting in an increase in the heat transfer coefficient. A computational method applied to an electrostatic precipitator was reported by Yamamoto and Velkoff [3], while Tada et al. [4] investigated the fundamental mechanism of heat transfer augmentation in a two-dimensional rectangular duct. Kasayapanand et al. [5] reported on numerical results investigating the effect of electrode arrangement on heat transfer enhancement in a bank of tubes. Kasayapanand and Kiatsiriroat [6] also studied corona wind-augmented heat transfer inside a wavy channel with an optimum electrode arrangement using a computational fluid dynamics technique. The EHD phenomenon in natural convection was investigated by Shu and Lai [7], Yang and Lai [8], and Yan et al. [9]. Most previous researchers in this field have concluded that EHD enhancement of heat transfer significantly increases at low Rayleigh number. However, there is no discussion regarding the effect of the number of electrodes on the flow and heat-transfer characteristics.
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Nomenclature cp De E f FE g Gr h H J k lw Lw N Nu P Pe Pr q qH r Ra Re t
specific heat, J/kg K charge diffusion coefficient, m2/V s electric field strength, V/m roughness factor, V/m EHD body force, N/m3 acceleration due to gravity, m/s2 Grashof number heat transfer coefficient, W/m2 K enclosure height, m current density, A/m2 thermal conductivity, W/m K length between wire electrodes, m length of wire electrode, m number of electrodes Nusselt number pressure, N/m2 Peclet number Prandtl number electric charge density, C/m3 heat flux, W/m2 corona wire radius, m Rayleigh number Reynolds number time, s
This numerical work investigates the electrostatic forces exerted on natural convection inside enclosures. The governing equations of the EHD phenomena are formulated, and mathematical modeling is carried out to analyze the EHD-enhanced secondary flow and heat-transfer coefficient in two dimensions via a finite-difference technique. Wire electrodes energized by high-voltage dc are simulated in the computational domain inside the enclosure. The characteristics of flow and heat transfer are discussed as a function of Rayleigh number, applied voltage, number of electrodes, and aspect ratio of the enclosure. 2. Theoretical analysis The governing equations for the EHD force per unit volume FE generated by an electric field of strength E in a fluid having electric charge density q, dielectric permittivity e, density r, and uniform temperature T can be expressed as [10] 1 1 q FE ¼ qE E2 r þ r E2 r . (1) 2 2 qr T In the symbolic notation, vectors are designated by bold-faced letters, while scalars are denoted by italic letters. The first term on the right qE is the Coulomb force exerted by the electric field upon the free charge or electrophoretic component, while the second and the
T v ue V W
temperature, K fluid velocity, m/s electric characteristic velocity, m/s voltage, V enclosure width, m
Greek symbols a b e m n y r se t o c
thermal diffusivity, m2/s volume expansion coefficient, 1/K fluid permittivity, F/m dynamic viscosity, kg/m s kinematics viscosity, m2/s dimensionless temperature density, kg/m3 electrical conductivity, 1/ohm m period vorticity, 1/s stream function, m2/s
Subscripts 0 P m w
without an electric field at the grounded plate mean value wall surface
third terms correspond to the dielectrophoretic and electrostrictive forces on and within the fluid. Eq. (1) is then included in the Navier–Stokes equation. By assuming an incompressible fluid, the conservation of momentum is given by r
dv ¼ rg þ FE rP þ mr2 v. dt
(2)
The vector rg is the gravitational force per unit volume, P the local fluid pressure and the last term on the right hand side of the equation represents the viscous terms. Introducing the vorticity x as (3)
x ¼ r v,
to get the vorticity transport equation in two-dimensional flow, the momentum equation can be rewritten in terms of the vorticity defined above, such that qx qT þ ðvrÞx ¼ n r2 x ðrV rÞq gb . qt qx
(4)
Defining the stream function c as vx ¼
qc ; qy
vy ¼
qc , qx
(5)
the vorticity transport equation can be obtained from Eqs. (4) and (5), which further gives r2 c ¼ x.
(6)
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Without any viscous dissipation effect, the energy equation can be written as dT se E2 ¼ a r2 T þ . dt r cp
(7)
Gauss’s law for the electric field is as follows: r E ¼ q,
(8) (9)
Because electric current is conserved over the computational domain, the current continuity equation holds, expressed as qq ¼ 0, qt where the current density J in this case is given by
rJþ
J ¼ qv þ se E þ ðv rÞðEÞ De rq. Combining Eqs. (8)–(11), one can obtain q r2 V ¼ , q2 ¼ ðrV rqÞ.
(10)
(11)
(12) (13)
The boundary conditions required for solving Eqs. (12) and (13) are V ¼ V 0 ; q ¼ q0 at the wire electrodes; and
(14a)
qq qq or ¼ 0 along all boundaries. qx qy
(14b)
V ¼ 0;
To reduce the independent parameters investigated, all governing equations are non-dimensionalized using the following dimensionless parameters: V q x y T TC ; y¯ ¼ ; y¼ V¯ ¼ , ; q¯ ¼ ; x¯ ¼ V0 q0 W W qH W =k vx vy xW ¯ ¼ c ; x v¯ x ¼ ; v¯ y ¼ ; c ¯ ¼ , ue W ue ue ue tu ¯t ¼ e ; ue ¼ q0 V 0 =r 1=2 . W Finally, the dimensionless governing equations take the form q¯ q0 W 2 , V 0
(15a)
V 0 ðrV¯ rq¯ Þ, q0 W 2
(15b)
r2 V¯ ¼ q¯ 2 ¼
where Re ¼ ueW/n, Pr ¼ n/a, and Pe ¼ Re Pr are dimensionless parameters. The Rayleigh number, based on a uniform wall heat flux having characteristic length W, and defined as gbqH W 4 Pr, (16) kn2 has been used to correlate the heat transfer in natural convection. In an analysis of heat transfer characteristics [11–13], the local heat transfer coefficient in terms of the local Nusselt number of the laminar sublayer near the surface, which can be applied in this study, is found from the relation Ra ¼ Gr Pr ¼
where the field E is given by E ¼ rV .
467
qx ¯ 1 Ra Pr qy þ ð¯v rÞx ¯ ¼ ð r2 xÞ , ¯ ðrV¯ rÞ¯q q¯t Re Pe2 qx¯ (15c) r2 c ¼ x, ¯
(15d)
qy 1 þ ð¯vrÞy ¼ ðr2 yÞ, q¯t Pe
(15e)
Nuy ¼
hW ðqT=qxÞW 1 ¼ ¼ . k ðT w T C Þ yw
(17)
From this equation, the local Nusselt number is obtained by using a second degree Lagrangian polynomial, while an average Nusselt number along the vertical surface of the enclosure is calculated from Eq. (18) by numerical integration using trapezoidal rule for nonuniform step sizes: Z W H=W 1 Nu ¼ d y¯ . (18) H 0 yw For periodic flow, the average Nusselt number is computed by integrating Eq. (18) over a period as follows: Z 1 tp Num ¼ Nu d¯t, (19) tp 0 where tp is a period of the oscillation. For nonperiodic flow, the average Nusselt number is also determined by integrating over an entire time span ts by Z 1 ts Num ¼ Nu d¯t. (20) ts 0 3. Methodology The computational domain and related boundary conditions are summarized in Fig. 1(a). The right thermally conducting plate is maintained at a uniform heat flux, and the left thermally conducting plate is maintained at a uniform temperature of 300 K. Both upper and lower plates are thermal insulators. All plates are electrically grounded, and the enclosure has a size of 5 40 cm2. Our numerical simulation uses a grid generation method to convert the physical plane (x, y) into the computational plane in curvilinear coordinates (x, Z) via Poisson’s equation. The illustrations of grid generation are shown in Figs. 1(b) and (c). The radial discrete value of the electric field in Fig. 1(b) is computed using geometrical progression in such a way as to allow high nodal density near the wire. The remaining region is subdivided into equi-spaced nodes. The starting conic coincides with the wire, but at greater distances from the wire, the circular symmetry changes, gradually changing into a Cartesian reference at the
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a
b
c
Table 1 Grid independence test for N ¼ 7 (V0 ¼ 17.5 kV, Ra ¼ 108) Mn
Nu
% change in Nu
71 376 106 566 141 751 176 941 211 1126
2.592 2.332 2.260 2.250 2.247
— 10.03 3.09 0.44 0.13
Table 2 Grid independence test for N ¼ 51 (V0 ¼ 17.5 kV, Ra ¼ 108)
Fig. 1. Boundary conditions and generated grid.
grounded plates and the symmetrical axes. The computed 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’s nodes. The result can be expressed using interpolation functions. Finally, these values are mapped into a computational fluid-dynamic mesh arrangement for calculating the velocity and temperature in Fig. 1(c). All equations are nondimensionalized, and the numerical calculations are carried out within computational grid using the grid independence test shown in Tables 1 and 2. A grid sensitivity analysis is conducted for N ¼ 7, V0 ¼ 17.5 kV, and Ra ¼ 108 in Table 1. As seen in the data, a grid size of 141 751 is selected for this study. As the number of electrodes becomes large (for example, at N ¼ 51), the grid independence test is done by fixing m with varying n in Table 2. The difference in Nusselt number between grid sizes of 141 1501 and 141 1751 is sufficiently small. From this observation, the grid pattern used for the present number of electrodes is 141 1501. Based on this methodology, the grid sizes obtained in this work are 140 500 for N ¼ 3, 141 751 for N ¼ 7 and 13, 141 1001 for N ¼ 26, and 141 1501 for N ¼ 51.
mn
Nu
% change in Nu
141 751 141 1001 141 1251 141 1501 141 1751
4.495 3.810 3.551 3.393 3.372
— 15.24 7.85 3.36 0.62
To prove that the oscillatory flows are not due to instability of the numerical results, the simulation has been recalculated using reduced nondimensional time steps of 1 104, 5 105, 2.5 105, and 1 105. The results obtained are the same as those with a time step of 5 105, thus defining a suitable time step for use in this work. This study assumes that the radius of the wire electrode is small enough to treat the wire as a single nodal point. Several models have been proposed for calculating the electric field and charge density distribution in a wire–plate precipitation system using the finite-difference method [14–18]. First, q0 must be assumed at the wire electrode according to the semi-empirical formula by Peek [19]. The space charge density near the wire is thus given by q0 ¼
l w JP pDe rf ð30d þ 0:9ðd=rÞ1=2 Þ
105 ,
(21)
where d is T0P/TP0, T0 is 293 K, P0 is 1.01 105 N/m2, f is 1, and T and P are the operating temperature and pressure at 300 K and 1.01 105 N/m2 , respectively. The vector JP is an initial current density at the grounded plate. Only positive corona is considered in the present study. The electric field and the charge density distribution are obtained from Eqs. (15a) and (15b) by an upwind difference scheme and successive under relaxation method to avoid divergence of the iterative solution. The electric field and charge distribution obtained are used to calculate electric current density at the grounded plate from the relation Z H qV JP ¼ De Lw dy, q (22) qx 0 The result is compared to the initial value in Eq. (21). When the calculated value differs, the above calculation is repeated by changing the corona ion value generated from the wire electrode until convergence is obtained. Using an
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¯ ¼ 5:0 105 , (b) Fig. 2. Stream-function contours inside enclosure for various Rayleigh numbers (V0 ¼ 17.5 kV, N ¼ 7): (a) Ra ¼ 104, t¯ ¼ 15; 000, Dc 5 6 7 4 4 4 ¯ ¯ ¯ Ra ¼ 10 , ¯t ¼ 15; 000, Dc ¼ 1:0 10 , (c) Ra ¼ 10 , ¯t ¼ 15; 000, Dc ¼ 2:0 10 , (d) Ra ¼ 10 , ¯t ¼ 15; 000, Dc ¼ 4:0 10 , and (e) Ra ¼ 108, ¯ ¼ 8:0 104 . ¯t ¼ 15; 000, Dc
upwind difference scheme, the velocity field at all nodes is also calculated. The vorticity at each time step is thus obtained from Eq. (15c) in accordance with the stream function determined from Eq. (15d), and the temperature field is finally acquired from Eq. (15e). The above procedures are repeated until convergences of the stream function and temperature are reached. Convergence is
assumed when the results are in steady state or in an oscillatory state. 4. Results and discussion When the numerical results of a nonEHD system are validated against the bench mark numerical solutions of
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natural convection in an enclosure [20], the calculated Nusselt numbers agree well within a maximum error of 3.2%. Fig. 2 shows the stream-function contours obtained by applying the EHD effect while the Rayleigh number is varied between 104 and 108. In the latter cases, the effect of Joule heating at the wire electrode is negligible (V0 ¼ 17.5 kV, N ¼ 7). There is an effect of the secondary
flow induced by the ionic wind at the wire electrode, which causes four rotating cellular motions at each electrode in Fig. 2(a). The Rayleigh number is next increased further, until the fluid inside the enclosure is dominated by the effect of Rayleigh number instead of the electric field. The flow patterns are oscillatory due to the interaction between the thermal buoyancy force and electrical body force. Two
Fig. 3. Temperature distributions inside enclosure for various Rayleigh numbers (V0 ¼ 17.5 kV, N ¼ 7): (a) Ra ¼ 104, ¯t ¼ 15; 000, y ¼ 1.0 102, (b) Ra ¼ 105, ¯t ¼ 15; 000, y ¼ 4.0 103, (c) Ra ¼ 106, ¯t ¼ 15; 000, y ¼ 2.6 103, (d) Ra ¼ 107, ¯t ¼ 15; 000, y ¼ 1.7 103, and (e) Ra ¼ 108, ¯t ¼ 15; 000, y ¼ 9.1 104.
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categories of oscillation—the periodic state of Figs. 2(a) and (b), and the nonperiodic state of Figs. 2(c)–(e)—are observed in these figures. Fig. 3 represents the temperature fields (isotherm lines) inside the enclosures defined in Fig. 2. The thermal boundary layer is perturbed by the electric field when it extends over the recirculation region. It can be seen that the temperature gradient, represented by the line density, at the right plate becomes larger with an increasing Rayleigh number, resulting in a high heat transfer coefficient. It can be concluded that for low
a
b
c
471
Rayleigh number, the flow and temperature fields have been substantially affected by an electric field. However, the effect of EHD is diminished at high Rayleigh number which is an indication of no significant change in heat transfer enhancement. The oscillatory stream function and isotherm line contours for 3, 7, 13, 26, and 51 electrodes are shown in Figs. 4 and 5 (V0 ¼ 17.5 kV, Ra ¼ 106). Altering the number of electrodes causes different flow patterns to occur in the enclosure. The number of vortices is seen to increase when the number of electrodes is augmented from
d
e
Fig. 4. Stream-function contours inside enclosure for various numbers of electrodes (V0 ¼ 17.5 kV, Ra ¼ 106): (a) N ¼ 3, ¯t ¼ 12; 500, Dc¯ ¼ 1:7 104 , (b) ¯ ¼ 2:0 104 , (c) N ¼ 13, ¯t ¼ 15; 000, Dc ¯ ¼ 2:2 104 , (d) N ¼ 26, ¯t ¼ 17; 500, Dc ¯ ¼ 2:8 104 , and (e) N ¼ 51, ¯t ¼ 22; 500, N ¼ 7, ¯t ¼ 15; 000, Dc Dc¯ ¼ 3:1 104 .
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3 to 13, but remains relatively constant as the number of electrodes increases to 26 and 51. Large vortices appear at the upper and lower zones of the enclosures of Figs. 4(d) and (e), especially around the extremes of the electrode strips, due to the low-pressure region. The temperature gradient along the right plate of Fig. 5(e) is highest in comparison to other configurations that cause the max-
imum heat transfer coefficient to be a maximum. One can thus conclude that the effect of the number of electrodes is more significant at larger number values. This effect should be considered together with the extra heat transfer per unit input power of electrical energy for optimization. A comparison of the average heat transfer enhancement over a cycle of the periodic state, or over the entire time
Fig. 5. Temperature distributions inside enclosure for various numbers of electrodes (V0 ¼ 17.5 kV, Ra ¼ 106): (a) N ¼ 3, ¯t ¼ 12; 500, y ¼ 2.6 103, (b) N ¼ 7, ¯t ¼ 15; 000, y ¼ 2.6 103, (c) N ¼ 13, ¯t ¼ 15; 000, y ¼ 2.6 103, (d) N ¼ 26, ¯t ¼ 17500, y ¼ 2.6 103, and (e) N ¼ 51, ¯t ¼ 22; 500, y ¼ 2.6 103.
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span of the nonperiodic state, along an enclosure is shown for various Rayleigh numbers in Fig. 6 (N ¼ 7). The latter is evaluated using the ratio of the average Nusselt number in the presence of an electric field to that without an electric field. The results show that EHD enhancement of flow and heat transfer plays an important role in regions of low Rayleigh number, while convective heat transfer is consequently dependent on the relation between Rayleigh number and supply voltage. Fig. 7 demonstrates the relationship between heat transfer enhancement and number of electrodes for V0 ¼ 17.5 kV. The enhancement ratio reaches a minimum at an intermediate number of electrodes (the combination between oscillatory cells occurs here) and increases again due to the high intensity of the electric field at higher numbers of electrodes. This phenomenon can be described by considering the isotherm line density at the right plate of Fig. 5(a). For the case N ¼ 3, the density is found to be higher than for N ¼ 7 due to the lower number of vortices and also the smaller heat trap that yields a higher heat transfer coefficient. Since the number of electrodes is rather high at N ¼ 13, 26, and 51, many vortices are formulated along the domain of
8.00 17.5 kV 15.0 kV 12.5 kV 10.0 kV 7.5 kV
7.00
Nu/Nu0
6.00 5.00 4.00 3.00 2.00 1.00 104
105
106 Rayleigh Number
107
108
Fig. 6. Heat transfer enhancement in relation to supply voltage (N ¼ 7).
Ra = 104
Ra = 106
473
enclosure, and they combine when the number of electrodes is large, even though there is good turbulent mixing. The flow and temperature fields are recirculating and oscillating, especially around the top and bottom zones of the enclosure. Therefore, the heat transfer coefficient is augmented due to the larger ionic wind effect. Thus, the optimum number of electrodes installed in an enclosure is significantly affected. The enhanced ratio between average fluid velocity ðv ¼ ðv2x þ v2y Þ1=2 Þ inside the enclosure and various numbers of electrodes, averaged along the computational domain and over a period or an entire time span, is investigated in Fig. 8 for V0 ¼ 17.5 kV. The velocity enhancement increases monotonically with the number of electrodes. A comparison between Figs. 7 and 8 shows that the velocity enhancement does not display a similar tendency and the heat transfer enhancement, instead increases further with number of electrodes, while the augmentation to heat transfer shows a minimum value at an intermediate number of electrodes. Fig. 9 compares the distribution of the heat transfer enhancement with enclosure dimensions for V0 ¼ 17.5 kV, Ra ¼ 104 and N ¼ 7 by calculating for the cases W ¼ 5, 7.5, 10, 12.5, and 15 cm and H ¼ 20, 40, 60, 80, and 100 cm. In total, 25 computations were performed and contours generated by using a Lagrange interpolating polynomial. In general, the augmented ratio decreases at a larger gap W between the vertical plates due to the low intensity of the electric field. However, for a fixed gap, the enhancement ratio reaches a maximum at an intermediate height H; the maximum point shifts slightly towards a larger height value as the gap is increased. The reason for this phenomenon can be described by considering the ratio of the distance between wire electrodes to the distance between vertical plates. The maximum point occurs as this value approaches unity. The electric field from each wire electrode emanates from a point symmetrically in all directions. Field lines begin on the positive wire electrode and terminate on the negative grounded plate; They cannot simply terminate within the computa-
Ra = 108
Ra = 104
11.00
Ra = 106
Ra = 108
30.00
10.00
25.00
9.00 7.00
v/v0
Nu/Nu0
8.00 6.00
20.00 15.00
5.00 4.00
10.00
3.00
5.00
2.00 N=3
N=7
N = 13
N = 26
N = 51
Fig. 7. Effect of number of electrodes to the heat transfer augmentation (V0 ¼ 17.5 kV).
N=3
N=7
N = 13
N = 26
N = 51
Fig. 8. Average velocity enhancement for various numbers of electrodes (V0 ¼ 17.5 kV).
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Acknowledgment The author gratefully acknowledges the support provided by the Thailand Research Fund and Commission on Higher Education for carrying out this study.
7
Nu:Nu 0
6
References
5 100
4
80
3 60
5 7.5 10 W, m
H, m
40 12.5 15
20
Fig. 9. Effect of aspect ratio of enclosure on heat transfer enhancement (V0 ¼ 17.5 kV, Ra ¼ 104, N ¼ 7).
tional domain. Thus, at this ratio, it may be advantageous for the polarization of ionic wind at the grounded plate when the number of electrodes is fixed. It can be concluded that the ratio of heat transfer enhancement increases in relation with small gap and a suitable height for the enclosure. From these results, we can conclude that using electrodes in the optimum number produces the best performance with respect to both efficiency and economy.
5. Conclusion Numerical simulations are performed to analyze the mechanism of natural convection inside enclosures incorporating electric field effect by using a computational fluiddynamics technique. The follows conclusions are reached:
5.1 The flow pattern of the fluid is affected by the supply voltage. The thermal boundary layer along the surface is perturbed by the electric field effect and also decreases at higher voltages. The enhancement of flow and heat transfer due to the EHD effect increases in relation with the supply voltage but diminishes with Rayleigh number. 5.2 Heat transfer enhancement is significantly increased at a higher number of electrodes, but it yields a minimum value at an intermediate number of electrodes. Moreover, the aspect ratio of the enclosure that results in the best augmented heat transfer for a fixed number of electrodes yields a recommended ratio of unity for the distance between wire electrodes to the distance between the vertical plates.
[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] Y. Tada, A. Takimoto, Y. Hayashi, Heat transfer enhancement in a convective field by applying ionic wind, J. Enhanced Heat Transfer 4 (2002) 71–86. [5] N. Kasayapanand, J. Tiansuwan, W. Asvapoositkul, N. Vorayos, T. Kiatsiriroat, Effect of the electrode arrangements in tube bank on the characteristic of electrohydrodynamic heat transfer enhancement: low Reynolds number, J. Enhanced Heat Transfer 9 (2002) 229–242. [6] N. Kasayapanand, T. Kiatsiriroat, EHD enhanced heat transfer in wavy channel, Int. J. Comm. Heat Mass Transfer 32 (2005) 809–821. [7] 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. [8] 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. [9] Y.Y. Yan, H.B. Zhang, J.B. Hull, Numerical modeling of electrohydrodynamic (EHD) effect on natural convection in an enclosure, Num. Heat Transfer Part A 46 (2004) 453–471. [10] L.D. Landau, E.M. Lifshitz, Electrohydrodynamics of Continuous Media, Pergamon, New York, 1963. [11] G.E. Karniadakis, Numerical simulation of forced convection heat transfer from a cylinder in cross flow, Int. J. Heat Mass Transfer 31 (1988) 407–418. [12] W. Chun, R.F. Boehm, Calculation of forced flow and heat transfer around a cylinder in cross-flow, Num. Heat Transfer 15 (1989) 101–122. [13] W.S. Fu, B.H. Tong, Numerical investigation of heat transfer from a heated oscillating cylinder in a cross-flow, Int. J. Heat Mass Transfer 45 (2002) 3033–3043. [14] J.R. McDonald, W.B. Smith, H.W. Spencer III, L.E. Sparks, A mathematical model for calculating electrical conditions in wire duct electrostatic precipitation devices, J. Appl. Phys. 48 (1977) 2231–2243. [15] P.A. Lawless, L.E. Sparks, A mathematical model for calculating effects of back corona in wire-duct electrostatic precipitators, J. Electrostat. 51 (1980) 242–256. [16] E. Lami, F. Mattachini, R. Sala, H. Vigl, A mathematical model of electrostatic field in wires-plate electrostatic precipitators, J. Electrostat. 39 (1997) 1–21. [17] M.R. Talaie, M. Taheri, J. Fathikaljahi, A new method to evaluate the voltage–current characteristics applicable for a single-stage electrostatic precipitator, J. Electrostat. 53 (2001) 221–233. [18] J. Anagnostopoulos, G. Bergeles, Corona discharge simulation in wire-duct electrostatic precipitator, J. Electrostat. 54 (2002) 129–147. [19] F.W. Peek, Dielectric Phenomena in High-Voltage Engineering, McGraw-Hill, New York, 1929. [20] G. De Vahl Davis, Natural convection of air in a square cavity: a bench mark numerical solution, Int. J. Num. Methods Fluid 3 (1983) 249–264.
Journal of Electrostatics 65 (2007) 465–474 www.elsevier.com/locate/elstat
Numerical modeling of the effect of number of electrodes on natural convection in an EHD fluid Nat Kasayapanand School of Energy, Environment, and Materials, King Mongkut’s University of Technology Thonburi, Bangkok 10140, Thailand Received 29 August 2005; received in revised form 2 November 2006; accepted 5 November 2006 Available online 1 December 2006
Abstract Numerical modeling of the electrohydrodynamic effect on natural convection in enclosures is investigated. The interactions between electric field, flow field, and temperature field are analyzed by using the computational fluid dynamics technique. Flow pattern and temperature distribution are substantially affected by the voltage supplied at the wire electrodes, especially at low Rayleigh number. It can be concluded that the fluid velocity and heat transfer coefficient in the presence of an electric field are significantly increased with when a large number of electrodes is used. Surprisingly, a minimum value of augmented heat transfer occurs with an intermediate number of electrodes. The optimized condition between the aspect ratio of an enclosure and number of electrodes which leads to maximum heat transfer enhancement is expressed in terms of the parameters concerned. r 2006 Elsevier B.V. All rights reserved. Keywords: Electrohydrodynamic; Electric field; Wire electrode; Computational fluid dynamics; Wire electrode; Heat transfer enhancement
1. Introduction The technique of convective heat transfer utilizing electric fields or electrostatic forces from the polarization of a dielectric fluid is one of the more promising methods for enhancing heat transfer because of its many advantages. The method is easily implemented, for example, using only a transformer and electrodes, it consumes only a small amount of electric power. This technique is frequently called electrohydrodynamic (EHD) heat transfer, and it refers broadly to an interdisciplinary field dealing with the interactions between electric, flow, and temperature fields. In a typical gaseous medium, energy is transferred from free electrons to the gas molecules, and the latter move toward a grounded surface to increase the heat transfer coefficient. There exist some prior studies related to this use of EHD. Yabe et al. [1] investigated the phenomenon of a corona wind between wire and plate electrodes and found that the interaction between ionic wind and primary flow Tel.: +66 2470 8699; fax: +66 2427 9062.
E-mail address: [email protected]. 0304-3886/$ - see front matter r 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.elstat.2006.11.002
increased the heat transfer from a wall surface. Velkoff and Godfrey [2] investigated heat transfer over a horizontal flat plate using parallel wire electrodes. The ionic wind promoted mixing of the primary flow, resulting in an increase in the heat transfer coefficient. A computational method applied to an electrostatic precipitator was reported by Yamamoto and Velkoff [3], while Tada et al. [4] investigated the fundamental mechanism of heat transfer augmentation in a two-dimensional rectangular duct. Kasayapanand et al. [5] reported on numerical results investigating the effect of electrode arrangement on heat transfer enhancement in a bank of tubes. Kasayapanand and Kiatsiriroat [6] also studied corona wind-augmented heat transfer inside a wavy channel with an optimum electrode arrangement using a computational fluid dynamics technique. The EHD phenomenon in natural convection was investigated by Shu and Lai [7], Yang and Lai [8], and Yan et al. [9]. Most previous researchers in this field have concluded that EHD enhancement of heat transfer significantly increases at low Rayleigh number. However, there is no discussion regarding the effect of the number of electrodes on the flow and heat-transfer characteristics.
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Nomenclature cp De E f FE g Gr h H J k lw Lw N Nu P Pe Pr q qH r Ra Re t
specific heat, J/kg K charge diffusion coefficient, m2/V s electric field strength, V/m roughness factor, V/m EHD body force, N/m3 acceleration due to gravity, m/s2 Grashof number heat transfer coefficient, W/m2 K enclosure height, m current density, A/m2 thermal conductivity, W/m K length between wire electrodes, m length of wire electrode, m number of electrodes Nusselt number pressure, N/m2 Peclet number Prandtl number electric charge density, C/m3 heat flux, W/m2 corona wire radius, m Rayleigh number Reynolds number time, s
This numerical work investigates the electrostatic forces exerted on natural convection inside enclosures. The governing equations of the EHD phenomena are formulated, and mathematical modeling is carried out to analyze the EHD-enhanced secondary flow and heat-transfer coefficient in two dimensions via a finite-difference technique. Wire electrodes energized by high-voltage dc are simulated in the computational domain inside the enclosure. The characteristics of flow and heat transfer are discussed as a function of Rayleigh number, applied voltage, number of electrodes, and aspect ratio of the enclosure. 2. Theoretical analysis The governing equations for the EHD force per unit volume FE generated by an electric field of strength E in a fluid having electric charge density q, dielectric permittivity e, density r, and uniform temperature T can be expressed as [10] 1 1 q FE ¼ qE E2 r þ r E2 r . (1) 2 2 qr T In the symbolic notation, vectors are designated by bold-faced letters, while scalars are denoted by italic letters. The first term on the right qE is the Coulomb force exerted by the electric field upon the free charge or electrophoretic component, while the second and the
T v ue V W
temperature, K fluid velocity, m/s electric characteristic velocity, m/s voltage, V enclosure width, m
Greek symbols a b e m n y r se t o c
thermal diffusivity, m2/s volume expansion coefficient, 1/K fluid permittivity, F/m dynamic viscosity, kg/m s kinematics viscosity, m2/s dimensionless temperature density, kg/m3 electrical conductivity, 1/ohm m period vorticity, 1/s stream function, m2/s
Subscripts 0 P m w
without an electric field at the grounded plate mean value wall surface
third terms correspond to the dielectrophoretic and electrostrictive forces on and within the fluid. Eq. (1) is then included in the Navier–Stokes equation. By assuming an incompressible fluid, the conservation of momentum is given by r
dv ¼ rg þ FE rP þ mr2 v. dt
(2)
The vector rg is the gravitational force per unit volume, P the local fluid pressure and the last term on the right hand side of the equation represents the viscous terms. Introducing the vorticity x as (3)
x ¼ r v,
to get the vorticity transport equation in two-dimensional flow, the momentum equation can be rewritten in terms of the vorticity defined above, such that qx qT þ ðvrÞx ¼ n r2 x ðrV rÞq gb . qt qx
(4)
Defining the stream function c as vx ¼
qc ; qy
vy ¼
qc , qx
(5)
the vorticity transport equation can be obtained from Eqs. (4) and (5), which further gives r2 c ¼ x.
(6)
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Without any viscous dissipation effect, the energy equation can be written as dT se E2 ¼ a r2 T þ . dt r cp
(7)
Gauss’s law for the electric field is as follows: r E ¼ q,
(8) (9)
Because electric current is conserved over the computational domain, the current continuity equation holds, expressed as qq ¼ 0, qt where the current density J in this case is given by
rJþ
J ¼ qv þ se E þ ðv rÞðEÞ De rq. Combining Eqs. (8)–(11), one can obtain q r2 V ¼ , q2 ¼ ðrV rqÞ.
(10)
(11)
(12) (13)
The boundary conditions required for solving Eqs. (12) and (13) are V ¼ V 0 ; q ¼ q0 at the wire electrodes; and
(14a)
qq qq or ¼ 0 along all boundaries. qx qy
(14b)
V ¼ 0;
To reduce the independent parameters investigated, all governing equations are non-dimensionalized using the following dimensionless parameters: V q x y T TC ; y¯ ¼ ; y¼ V¯ ¼ , ; q¯ ¼ ; x¯ ¼ V0 q0 W W qH W =k vx vy xW ¯ ¼ c ; x v¯ x ¼ ; v¯ y ¼ ; c ¯ ¼ , ue W ue ue ue tu ¯t ¼ e ; ue ¼ q0 V 0 =r 1=2 . W Finally, the dimensionless governing equations take the form q¯ q0 W 2 , V 0
(15a)
V 0 ðrV¯ rq¯ Þ, q0 W 2
(15b)
r2 V¯ ¼ q¯ 2 ¼
where Re ¼ ueW/n, Pr ¼ n/a, and Pe ¼ Re Pr are dimensionless parameters. The Rayleigh number, based on a uniform wall heat flux having characteristic length W, and defined as gbqH W 4 Pr, (16) kn2 has been used to correlate the heat transfer in natural convection. In an analysis of heat transfer characteristics [11–13], the local heat transfer coefficient in terms of the local Nusselt number of the laminar sublayer near the surface, which can be applied in this study, is found from the relation Ra ¼ Gr Pr ¼
where the field E is given by E ¼ rV .
467
qx ¯ 1 Ra Pr qy þ ð¯v rÞx ¯ ¼ ð r2 xÞ , ¯ ðrV¯ rÞ¯q q¯t Re Pe2 qx¯ (15c) r2 c ¼ x, ¯
(15d)
qy 1 þ ð¯vrÞy ¼ ðr2 yÞ, q¯t Pe
(15e)
Nuy ¼
hW ðqT=qxÞW 1 ¼ ¼ . k ðT w T C Þ yw
(17)
From this equation, the local Nusselt number is obtained by using a second degree Lagrangian polynomial, while an average Nusselt number along the vertical surface of the enclosure is calculated from Eq. (18) by numerical integration using trapezoidal rule for nonuniform step sizes: Z W H=W 1 Nu ¼ d y¯ . (18) H 0 yw For periodic flow, the average Nusselt number is computed by integrating Eq. (18) over a period as follows: Z 1 tp Num ¼ Nu d¯t, (19) tp 0 where tp is a period of the oscillation. For nonperiodic flow, the average Nusselt number is also determined by integrating over an entire time span ts by Z 1 ts Num ¼ Nu d¯t. (20) ts 0 3. Methodology The computational domain and related boundary conditions are summarized in Fig. 1(a). The right thermally conducting plate is maintained at a uniform heat flux, and the left thermally conducting plate is maintained at a uniform temperature of 300 K. Both upper and lower plates are thermal insulators. All plates are electrically grounded, and the enclosure has a size of 5 40 cm2. Our numerical simulation uses a grid generation method to convert the physical plane (x, y) into the computational plane in curvilinear coordinates (x, Z) via Poisson’s equation. The illustrations of grid generation are shown in Figs. 1(b) and (c). The radial discrete value of the electric field in Fig. 1(b) is computed using geometrical progression in such a way as to allow high nodal density near the wire. The remaining region is subdivided into equi-spaced nodes. The starting conic coincides with the wire, but at greater distances from the wire, the circular symmetry changes, gradually changing into a Cartesian reference at the
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a
b
c
Table 1 Grid independence test for N ¼ 7 (V0 ¼ 17.5 kV, Ra ¼ 108) Mn
Nu
% change in Nu
71 376 106 566 141 751 176 941 211 1126
2.592 2.332 2.260 2.250 2.247
— 10.03 3.09 0.44 0.13
Table 2 Grid independence test for N ¼ 51 (V0 ¼ 17.5 kV, Ra ¼ 108)
Fig. 1. Boundary conditions and generated grid.
grounded plates and the symmetrical axes. The computed 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’s nodes. The result can be expressed using interpolation functions. Finally, these values are mapped into a computational fluid-dynamic mesh arrangement for calculating the velocity and temperature in Fig. 1(c). All equations are nondimensionalized, and the numerical calculations are carried out within computational grid using the grid independence test shown in Tables 1 and 2. A grid sensitivity analysis is conducted for N ¼ 7, V0 ¼ 17.5 kV, and Ra ¼ 108 in Table 1. As seen in the data, a grid size of 141 751 is selected for this study. As the number of electrodes becomes large (for example, at N ¼ 51), the grid independence test is done by fixing m with varying n in Table 2. The difference in Nusselt number between grid sizes of 141 1501 and 141 1751 is sufficiently small. From this observation, the grid pattern used for the present number of electrodes is 141 1501. Based on this methodology, the grid sizes obtained in this work are 140 500 for N ¼ 3, 141 751 for N ¼ 7 and 13, 141 1001 for N ¼ 26, and 141 1501 for N ¼ 51.
mn
Nu
% change in Nu
141 751 141 1001 141 1251 141 1501 141 1751
4.495 3.810 3.551 3.393 3.372
— 15.24 7.85 3.36 0.62
To prove that the oscillatory flows are not due to instability of the numerical results, the simulation has been recalculated using reduced nondimensional time steps of 1 104, 5 105, 2.5 105, and 1 105. The results obtained are the same as those with a time step of 5 105, thus defining a suitable time step for use in this work. This study assumes that the radius of the wire electrode is small enough to treat the wire as a single nodal point. Several models have been proposed for calculating the electric field and charge density distribution in a wire–plate precipitation system using the finite-difference method [14–18]. First, q0 must be assumed at the wire electrode according to the semi-empirical formula by Peek [19]. The space charge density near the wire is thus given by q0 ¼
l w JP pDe rf ð30d þ 0:9ðd=rÞ1=2 Þ
105 ,
(21)
where d is T0P/TP0, T0 is 293 K, P0 is 1.01 105 N/m2, f is 1, and T and P are the operating temperature and pressure at 300 K and 1.01 105 N/m2 , respectively. The vector JP is an initial current density at the grounded plate. Only positive corona is considered in the present study. The electric field and the charge density distribution are obtained from Eqs. (15a) and (15b) by an upwind difference scheme and successive under relaxation method to avoid divergence of the iterative solution. The electric field and charge distribution obtained are used to calculate electric current density at the grounded plate from the relation Z H qV JP ¼ De Lw dy, q (22) qx 0 The result is compared to the initial value in Eq. (21). When the calculated value differs, the above calculation is repeated by changing the corona ion value generated from the wire electrode until convergence is obtained. Using an
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¯ ¼ 5:0 105 , (b) Fig. 2. Stream-function contours inside enclosure for various Rayleigh numbers (V0 ¼ 17.5 kV, N ¼ 7): (a) Ra ¼ 104, t¯ ¼ 15; 000, Dc 5 6 7 4 4 4 ¯ ¯ ¯ Ra ¼ 10 , ¯t ¼ 15; 000, Dc ¼ 1:0 10 , (c) Ra ¼ 10 , ¯t ¼ 15; 000, Dc ¼ 2:0 10 , (d) Ra ¼ 10 , ¯t ¼ 15; 000, Dc ¼ 4:0 10 , and (e) Ra ¼ 108, ¯ ¼ 8:0 104 . ¯t ¼ 15; 000, Dc
upwind difference scheme, the velocity field at all nodes is also calculated. The vorticity at each time step is thus obtained from Eq. (15c) in accordance with the stream function determined from Eq. (15d), and the temperature field is finally acquired from Eq. (15e). The above procedures are repeated until convergences of the stream function and temperature are reached. Convergence is
assumed when the results are in steady state or in an oscillatory state. 4. Results and discussion When the numerical results of a nonEHD system are validated against the bench mark numerical solutions of
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natural convection in an enclosure [20], the calculated Nusselt numbers agree well within a maximum error of 3.2%. Fig. 2 shows the stream-function contours obtained by applying the EHD effect while the Rayleigh number is varied between 104 and 108. In the latter cases, the effect of Joule heating at the wire electrode is negligible (V0 ¼ 17.5 kV, N ¼ 7). There is an effect of the secondary
flow induced by the ionic wind at the wire electrode, which causes four rotating cellular motions at each electrode in Fig. 2(a). The Rayleigh number is next increased further, until the fluid inside the enclosure is dominated by the effect of Rayleigh number instead of the electric field. The flow patterns are oscillatory due to the interaction between the thermal buoyancy force and electrical body force. Two
Fig. 3. Temperature distributions inside enclosure for various Rayleigh numbers (V0 ¼ 17.5 kV, N ¼ 7): (a) Ra ¼ 104, ¯t ¼ 15; 000, y ¼ 1.0 102, (b) Ra ¼ 105, ¯t ¼ 15; 000, y ¼ 4.0 103, (c) Ra ¼ 106, ¯t ¼ 15; 000, y ¼ 2.6 103, (d) Ra ¼ 107, ¯t ¼ 15; 000, y ¼ 1.7 103, and (e) Ra ¼ 108, ¯t ¼ 15; 000, y ¼ 9.1 104.
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categories of oscillation—the periodic state of Figs. 2(a) and (b), and the nonperiodic state of Figs. 2(c)–(e)—are observed in these figures. Fig. 3 represents the temperature fields (isotherm lines) inside the enclosures defined in Fig. 2. The thermal boundary layer is perturbed by the electric field when it extends over the recirculation region. It can be seen that the temperature gradient, represented by the line density, at the right plate becomes larger with an increasing Rayleigh number, resulting in a high heat transfer coefficient. It can be concluded that for low
a
b
c
471
Rayleigh number, the flow and temperature fields have been substantially affected by an electric field. However, the effect of EHD is diminished at high Rayleigh number which is an indication of no significant change in heat transfer enhancement. The oscillatory stream function and isotherm line contours for 3, 7, 13, 26, and 51 electrodes are shown in Figs. 4 and 5 (V0 ¼ 17.5 kV, Ra ¼ 106). Altering the number of electrodes causes different flow patterns to occur in the enclosure. The number of vortices is seen to increase when the number of electrodes is augmented from
d
e
Fig. 4. Stream-function contours inside enclosure for various numbers of electrodes (V0 ¼ 17.5 kV, Ra ¼ 106): (a) N ¼ 3, ¯t ¼ 12; 500, Dc¯ ¼ 1:7 104 , (b) ¯ ¼ 2:0 104 , (c) N ¼ 13, ¯t ¼ 15; 000, Dc ¯ ¼ 2:2 104 , (d) N ¼ 26, ¯t ¼ 17; 500, Dc ¯ ¼ 2:8 104 , and (e) N ¼ 51, ¯t ¼ 22; 500, N ¼ 7, ¯t ¼ 15; 000, Dc Dc¯ ¼ 3:1 104 .
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3 to 13, but remains relatively constant as the number of electrodes increases to 26 and 51. Large vortices appear at the upper and lower zones of the enclosures of Figs. 4(d) and (e), especially around the extremes of the electrode strips, due to the low-pressure region. The temperature gradient along the right plate of Fig. 5(e) is highest in comparison to other configurations that cause the max-
imum heat transfer coefficient to be a maximum. One can thus conclude that the effect of the number of electrodes is more significant at larger number values. This effect should be considered together with the extra heat transfer per unit input power of electrical energy for optimization. A comparison of the average heat transfer enhancement over a cycle of the periodic state, or over the entire time
Fig. 5. Temperature distributions inside enclosure for various numbers of electrodes (V0 ¼ 17.5 kV, Ra ¼ 106): (a) N ¼ 3, ¯t ¼ 12; 500, y ¼ 2.6 103, (b) N ¼ 7, ¯t ¼ 15; 000, y ¼ 2.6 103, (c) N ¼ 13, ¯t ¼ 15; 000, y ¼ 2.6 103, (d) N ¼ 26, ¯t ¼ 17500, y ¼ 2.6 103, and (e) N ¼ 51, ¯t ¼ 22; 500, y ¼ 2.6 103.
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span of the nonperiodic state, along an enclosure is shown for various Rayleigh numbers in Fig. 6 (N ¼ 7). The latter is evaluated using the ratio of the average Nusselt number in the presence of an electric field to that without an electric field. The results show that EHD enhancement of flow and heat transfer plays an important role in regions of low Rayleigh number, while convective heat transfer is consequently dependent on the relation between Rayleigh number and supply voltage. Fig. 7 demonstrates the relationship between heat transfer enhancement and number of electrodes for V0 ¼ 17.5 kV. The enhancement ratio reaches a minimum at an intermediate number of electrodes (the combination between oscillatory cells occurs here) and increases again due to the high intensity of the electric field at higher numbers of electrodes. This phenomenon can be described by considering the isotherm line density at the right plate of Fig. 5(a). For the case N ¼ 3, the density is found to be higher than for N ¼ 7 due to the lower number of vortices and also the smaller heat trap that yields a higher heat transfer coefficient. Since the number of electrodes is rather high at N ¼ 13, 26, and 51, many vortices are formulated along the domain of
8.00 17.5 kV 15.0 kV 12.5 kV 10.0 kV 7.5 kV
7.00
Nu/Nu0
6.00 5.00 4.00 3.00 2.00 1.00 104
105
106 Rayleigh Number
107
108
Fig. 6. Heat transfer enhancement in relation to supply voltage (N ¼ 7).
Ra = 104
Ra = 106
473
enclosure, and they combine when the number of electrodes is large, even though there is good turbulent mixing. The flow and temperature fields are recirculating and oscillating, especially around the top and bottom zones of the enclosure. Therefore, the heat transfer coefficient is augmented due to the larger ionic wind effect. Thus, the optimum number of electrodes installed in an enclosure is significantly affected. The enhanced ratio between average fluid velocity ðv ¼ ðv2x þ v2y Þ1=2 Þ inside the enclosure and various numbers of electrodes, averaged along the computational domain and over a period or an entire time span, is investigated in Fig. 8 for V0 ¼ 17.5 kV. The velocity enhancement increases monotonically with the number of electrodes. A comparison between Figs. 7 and 8 shows that the velocity enhancement does not display a similar tendency and the heat transfer enhancement, instead increases further with number of electrodes, while the augmentation to heat transfer shows a minimum value at an intermediate number of electrodes. Fig. 9 compares the distribution of the heat transfer enhancement with enclosure dimensions for V0 ¼ 17.5 kV, Ra ¼ 104 and N ¼ 7 by calculating for the cases W ¼ 5, 7.5, 10, 12.5, and 15 cm and H ¼ 20, 40, 60, 80, and 100 cm. In total, 25 computations were performed and contours generated by using a Lagrange interpolating polynomial. In general, the augmented ratio decreases at a larger gap W between the vertical plates due to the low intensity of the electric field. However, for a fixed gap, the enhancement ratio reaches a maximum at an intermediate height H; the maximum point shifts slightly towards a larger height value as the gap is increased. The reason for this phenomenon can be described by considering the ratio of the distance between wire electrodes to the distance between vertical plates. The maximum point occurs as this value approaches unity. The electric field from each wire electrode emanates from a point symmetrically in all directions. Field lines begin on the positive wire electrode and terminate on the negative grounded plate; They cannot simply terminate within the computa-
Ra = 108
Ra = 104
11.00
Ra = 106
Ra = 108
30.00
10.00
25.00
9.00 7.00
v/v0
Nu/Nu0
8.00 6.00
20.00 15.00
5.00 4.00
10.00
3.00
5.00
2.00 N=3
N=7
N = 13
N = 26
N = 51
Fig. 7. Effect of number of electrodes to the heat transfer augmentation (V0 ¼ 17.5 kV).
N=3
N=7
N = 13
N = 26
N = 51
Fig. 8. Average velocity enhancement for various numbers of electrodes (V0 ¼ 17.5 kV).
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Acknowledgment The author gratefully acknowledges the support provided by the Thailand Research Fund and Commission on Higher Education for carrying out this study.
7
Nu:Nu 0
6
References
5 100
4
80
3 60
5 7.5 10 W, m
H, m
40 12.5 15
20
Fig. 9. Effect of aspect ratio of enclosure on heat transfer enhancement (V0 ¼ 17.5 kV, Ra ¼ 104, N ¼ 7).
tional domain. Thus, at this ratio, it may be advantageous for the polarization of ionic wind at the grounded plate when the number of electrodes is fixed. It can be concluded that the ratio of heat transfer enhancement increases in relation with small gap and a suitable height for the enclosure. From these results, we can conclude that using electrodes in the optimum number produces the best performance with respect to both efficiency and economy.
5. Conclusion Numerical simulations are performed to analyze the mechanism of natural convection inside enclosures incorporating electric field effect by using a computational fluiddynamics technique. The follows conclusions are reached:
5.1 The flow pattern of the fluid is affected by the supply voltage. The thermal boundary layer along the surface is perturbed by the electric field effect and also decreases at higher voltages. The enhancement of flow and heat transfer due to the EHD effect increases in relation with the supply voltage but diminishes with Rayleigh number. 5.2 Heat transfer enhancement is significantly increased at a higher number of electrodes, but it yields a minimum value at an intermediate number of electrodes. Moreover, the aspect ratio of the enclosure that results in the best augmented heat transfer for a fixed number of electrodes yields a recommended ratio of unity for the distance between wire electrodes to the distance between the vertical plates.
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