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Numerical evaluation of using micropolar fluid model for EHD-induced natural convection heat transfer through a rectangular enclosure
Journal of Electrostatics 101 (2019) 103372
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Numerical evaluation of using micropolar fluid model for EHD-induced natural convection heat transfer through a rectangular enclosure
T
Ladan Samaeia, Hamed Mohaddes Deylamib,∗, Nima Amanifarda, Hesam Moayedia a b
Faculty of Mechanical Engineering, University of Guilan, Rasht, Iran Faculty of Technology and Engineering, East of Guilan, University of Guilan, Rudsar, Iran
A R T I C LE I N FO
A B S T R A C T
Keywords: Electrohydrodynamic Natural convection Enclosure Micropolar fluid model Numerical investigation
Using turbulent models for electrohydrodynamic (EHD) induced flow modeling in hydraulically laminar flow ranges has been always questionable. This study is concerned with EHD cases in natural convection heat transfer within an enclosure. Here, the micropolar model was engaged and became the main motivation of the current study as an alternative approach for the fluid dynamic behavior. The numerical investigations were performed to study the effect of the various key parameters: the applied voltage, Rayleigh number, and the gap size of the electrodes. The main challenge of using the micropolar model is the evaluation of the adequate material parameter (κω/μ); all the presented numerical investigations conducted to find the proper material parameter for various flow conditions. All cases were carried out for a two-dimensional approach with a non-uniform structured grid, which is used by a finite volume algorithm to solve the EHD natural convection governing equations. Comparing the results of the micropolar approach with those obtained from the turbulent standard k-ε model reveals that the micropolar model can be an appropriate candidate to simulate the EHD natural convection flow instead of fully turbulent models. The results depicted that by increasing Rayleigh number and the applied voltage, the discrepancy between the results of the micropolar and the standard k-ε model increases.
1. Introduction Natural convection heat transfer in enclosures plays an important role in many engineering applications, and various techniques have been provided to gain the higher performance of heat transfer. Methods of enhancing heat transfer can be classified into two main categories: active and passive methods. Using finned enclosures [1–4] is an example of the passive methods while using EHD actuator or ultrasonic waves [5–7] are some examples of active methods. As the current work focuses on EHD modeling, a brief review of the EHD actuators may be beneficial. When a high voltage is applied to a sharp electrode (emitter), the air becomes ionized and the ions are attracted toward the electrically grounded plate (collector). The transmission of the injected ions from the emitter toward the collector causes the ions to transfer their momentum to the neutral molecules by collision. As a result, a bulk flow of ionized air molecules is generated known as corona wind or the secondary flow. Recently, the EHD-induced flow became an active method for the enhancement of heat transfer among many available methods. The facility, high reliability, and short response time are among the advantages of this methodology [8].
∗
The EHD heat transfer has widely attracted the researchers to develop the technique for various cases. Most studies have been performed experimentally [8–12] or numerically by using fully turbulent flow models [13–23]. Some researchers tried to experimentally measure the velocities of the EHD flow, using Laser-Doppler Anemometers (LDA), Particle Image Velocimetry (PIV) and Hot-Wire Anemometers [9,10,12]. The researchers reported that these experimental techniques showed some uncertainties, which caused by the effect of electric field on the process of measurements. Numerical methods didn't have the same uncertainty problem, but the main challenge to modeling the EHD-induced flow numerically, is to use the fully turbulent models, even when the main flow is in the laminar regime [13–15,17–23]. Molki and Damronglerd [17] examined the effect of the EHD-induced flow on improving heat transfer in a square duct. They underlined that fully turbulent models are necessary to simulate the EHD-induced flow, even in a laminar flow regime. They compared the results of the laminar model and fully turbulent model with Large Eddy Simulation (LES) and empirical results. They showed that using laminar model failed to predict the Nusselt number and friction factor correctly, but the results of the turbulent model was in good consistency with the results of LES and experimental results. Feng et al. [13] investigated
Corresponding author. E-mail address: [email protected] (H. Mohaddes Deylami).
https://doi.org/10.1016/j.elstat.2019.103372 Received 19 May 2019; Received in revised form 26 August 2019; Accepted 4 September 2019 0304-3886/ © 2019 Elsevier B.V. All rights reserved.
Journal of Electrostatics 101 (2019) 103372
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EHD flow pattern in electrostatic preticipators, numerically. They used different turbulence models and compared the numerical results with the experimental data. Comparing the numerical results with the experimental data showed that the results obtained from the k-ε model is satisfactory. Also, Golsefid et al. [23] experimentally and numerically studied the EHD effect on enhancement of natural convection heat transfer within an enclosure with Joule heating term in the energy equation. They used the standard k-ε fully turbulent model to simulate the EHD-induced flow in the laminar regime. Their results showed that wherever Rayleigh number was small, the effect of the Joule heat parameter in the energy equation was significant. Additionally, they found an optimal configuration of the collector electrode, in which the effect of EHD-induced flow on the heat transfer enhancement reaches to its maximum. The lack of using an appropriate approach for the fluid motion in the EHD field, particularly in laminar range, and the lack of the turbulent flow concepts such as the unsteadiness and irregularity become the main reason to test the micropolar fluid model in the current work. Some studies have been carried out in micropolar fluid and its applications. The micropolar fluid theory was introduced by Eringen [24] and Lukaszewicz [25]. They introduced micropolar fluid as a fluid with microstructure which has nonsymmetrical stress tensor. Ahmadi [26] studied the boundary layer flow of micropolar fluid over a semi-infinite flat plate. The applicability of micropolar fluid theory to the motion of suspension solutions was introduced. Many cases of heat transfer of the micropolar fluid have been studied [27–32], some cases were under magnetic field [33–35] or under electric field effects [36,37]. In the natural convection cases, Aydin and Pop [29] numerically investigated the steady laminar natural convection heat transfer of the micropolar fluid in an enclosure. They discussed the effects of Rayleigh number (Ra), Prandtl number (Pr), and material parameter (K), on the micropolar fluid motion and natural convection heat transfer. They reported that by increasing Rayleigh number and decreasing the material parameter, the heat transfer increases. Muthtamilselvan et al. [31] investigated numerically natural convection of micropolar fluid in a square cavity with uniform and non-uniform heat sources. They showed that the heat transfer decreases with increasing vortex viscosity parameter. Also, the heat transfer increases with increasing heat source length for both uniform and non-uniform heat sources. In addition, with increasing Rayleigh number for all values of source nonuniformity parameter heat transfer increases. Moayedi et al. [37] used the micropolar model to simulate the forced convection heat transfer affected by EHD-induced flow in a smooth channel when the flow field was in the laminar regime. They compared results of the micropolar model with results of the standard k-ε turbulent model. They investigated the EHD-induced flow field by changing the material parameter. Their results showed that for a specific material parameter for each applied voltage and Reynolds number, the results of the micropolar model were consistent with the results of the standard k-ε model. Also, they reported that the micropolar model could be a reliable method to simulate the EHD-induced flow field in the laminar regime. Regarding the mentioned previous works, it should be underlined that the main reasons to examine a new numerical model (micropolar fluid model) as an alternative dynamic behavior model in the current work are as follows:
Table 1 Fluid properties. Property
Value
cp k0 β0 ε0 μ ρ
1006.43 J/kg. K 0.0242 W/m. K 2⨯10−4 m2/V.s 8.85⨯10−12 F/m 1.885⨯10−5 kg/m. s 1.115 kg/m3
Fig. 1. Schematic view of the computation domain and geometric parameters. Table 2 Fluid boundary conditions. Surfaces
Air flow
Temperature
Micro spin
Emitting electrode Right wall (collector) Left wall Top wall Bottom wall
ux = uy = 0 ux = uy = 0 ux = uy = 0 ux = uy = 0 ux = uy = 0
∂T/∂n=0
ωz=0
q″=cte
ωz=0
T=cte
ωz=0
∂T/∂y=0
ωz=0
∂T/∂y=0
ωz=0
Table 3 Electrical boundary conditions. Surfaces Emitting electrode Right wall (collector) Left wall Top wall Bottom wall
1. The lack of an appropriate approach for the fluid motion modeling in the presence of the EHD-induced flow, particularly in laminar range. 2. The lack of turbulent flow concepts such as the unsteadiness and irregularity.
Electrical potential
Space charge density
V=V0
Peek's value
V=0
∂ρc/∂x=0
∂V/∂x=0
∂ρc/∂x=0
∂V/∂y=0
∂ρc/∂y=0
∂V/∂y=0
∂ρc/∂y=0
model has few predictive power, the computational cost of the micropolar model with an adequate material parameter (κω/μ) is much lower than the computational cost of the k-ε model [36]. Therefore, the main goal of our study is twofold:
It is noteworthy that the new micropolar fluid model has advantages such as an acceptable prediction approach in hydraulically laminar flow range in which the flow is not fully turbulent. Despite the micropolar 2
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Fig. 4. Distribution of the current density on the grounded plate. (V = 8 kV, 95 μm needle 3.1 cm above the ground plate).
Fig. 2. A schematic view of computational domain grid distributions. (a) Near the wire, (b) Near the wall.
Fig. 5. Temperature distribution on the right wall of the enclosure (V = 14 kV, Ra = 106, a = 20 cm, b = 2.5 cm).
2. Governing equations 2.1. Electric field equations The EHD-induced flow is analyzed using basic electrostatic, flow and energy equations. The electric body force per unit volume can be expressed as follows [20]: Fig. 3. Grid independence test for temperature distribution (V = 14 kV, Ra = 106, a = 20 cm, b = 2.5 cm).
→ → 1 →2 →2 ⎛ ∂ε0 ⎞ ⎤ 1 Fe = ρc E − E ∇ε0 + ∇ ⎡ E ρ⎜ ⎟ ⎥ 2 2 ⎢ ⎝ ∂ρ ⎠T ⎦ ⎣
(a) Evaluation of the new numerical model (micropolar fluid model) to simulate the natural convection of the EHD-induced flow in a rectangular enclosure. (b) Presentation of a new correlation to find the adequate value of the material parameter in the micropolar model.
(1)
The first term on the right side of this equation refers to the Coulomb force. It is always employed in presence of the electric field. The second and third terms of Eq. (1) represent Electrophoretic force and Electrostrictive force, respectively. Due to the constant electric permittivity, these terms can be ignored. Whenever there is no external magnetic field and electric current is negligible, Faraday's Law can be simplified as follows [38]:
It is noted that in order to evaluate the capability and the accuracy of the micropolar fluid model, the results of the present model were compared with the results of the standard k-ε model. Finally, different cases are investigated upon various gap sizes between electrodes, different applied voltages (12 kV, 14 kV and 16 kV) and different Rayleigh numbers (105,106).
→ ∇×E =0
(2)
Therefore, the electric field is a non-rotating vector field, so that, a unique scalar function as a potential function can be defined for it:
→ E = −∇V Poisson's equation: 3
(3)
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→ → J = ρc β0 E + ρc → u − De ∇ρc
(6)
Three terms on the right side of Eq. (6) represent ion drift, convection, and diffusion of electrical charges, respectively. Since the order of magnitude of ion drift is higher in comparison to the other terms in the right side of the equation, they can be ignored. Therefore, Eq. (5) will be as follows:
∂ρc ∂t
→ + ∇ . (ρc β0 E ) = 0
(7)
Eq. (4) and Eq. (7) are the main governing equations of the EHD phenomena. Eq. (7) suffers from dispersion error [20]. So, in order to solve the charge density equation, we used artificial diffusion coeffiβ EL cient ∇ . (Γ ∇ρc ) , where Pe = 0Γ c . Hence, Eq. (7) is rewritten as → ∂ρc + ∇ . (ρc β0 E ) = ∇ . (Γ ∇ρc ) . By solving these equations at the same ∂t time, the electrical body force is obtained. Finally, the calculated body force acts as a source term in momentum equations. 2.2. Governing equations of the micropolar fluid model Governing equations of the micropolar fluid model by assuming an unsteady, two dimensional, incompressible EHD-induced flow in natural convection heat transfer can be written in the following forms [24]: 2.3. Continuity
∂ (ρui ) ∂ρ =0 + ∂x i ∂t
(8)
Momentum:
∂p ∂ (ρui ) ∂ 2u i ⎞ ∂ ⎛ ∂ωk ⎞ + (μ + κ ω) ⎜⎛ (ρui uj ) = − + ⎟ + κ ω εijk ⎜ ⎟ x x ∂x i ∂ ∂ ∂t ∂x j i j ⎝ ⎠ ⎝ ∂x j ⎠
Fig. 6. Effect of the material parameter on (a) air flow streamlines, (b) wall temperature (V = 14 kV, Ra = 106, a = 20 cm, b = 2.5 cm).
∇2 . V =
+ ρgi β (T − T0) δij + Fei
−ρc
Angular momentum:
(4)
ε0
∂ω ∂ ∂2ωi ⎞ ⎛ ∂uk ⎞ (ωi uj ) ⎤ = γ ⎛⎜ ρjω ⎡ i + ⎟ − 2κ ω ωi + κ ω εijk ⎜ ⎟ ⎢ ∂t ⎥ x ∂ ∂ j ⎣ ⎦ ⎝ x j ∂x j ⎠ ⎝ ∂x j ⎠
Conservation of space charge density equation:
∂ρc ∂t
→ + ∇. J = 0
(9)
(10)
Energy:
(5)
→ J represents electric current density which is calculated from the following equation:
∂ ∂ ∂ ⎛ ∂T ⎞ (ρe ) + [ui (ρe + p)] = ⎜k 0 ⎟ ∂t ∂x i ∂x j ⎝ ∂x j ⎠
Fig. 7. Air flow streamlines for Ra = 105, a = 20 cm, b = 2.5 cm (a) V = 12 kV, (b) V = 14 kV, (c) V = 16 kV. 4
(11)
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Fig. 8. Air flow streamlines for Ra = 106, a = 20 cm, b = 2.5 cm (a) V = 12 kV, (b) V = 14 kV, (c) V = 16 kV.
Fig. 9. The comparison of the flow velocity magnitude between the micropolar and the standard k-ε models (V = 12 kV, Ra = 105, a = 20 cm, b = 2.5 cm).
where jω represents the micro-inertia density which is equal to the square of characteristic length. Also, μ, κω and γ represent dynamic viscosity, vortex viscosity, and spin gradient viscosity, respectively.
κ K γ = ⎛μ + ω ⎞ jω = μ ⎛1 + ⎞ jω 2 ⎠ 2⎠ ⎝ ⎝
(12)
where K represents the material parameter. The properties of the fluid (air) are shown in Table 1. It should be noted that the governing equations of the standard k-ε model is presented in the Appendix.
Fig. 10. The enhancement factor on the right wall for Ra = 105, a = 20 cm, b = 2.5 cm (a) V = 12 kV, (b) V = 14 kV, (c) V = 16 kV.
4. Boundary conditions 3. Computational geometry
The boundary conditions are divided into two parts; (1) flow field, (2) electric field. The flow and electric boundary conditions are explained in sections 4.1 and 4.2, respectively.
In this study, a tall rectangular enclosure with an emitting electrode is considered, in which the emitting electrode is located in the center of the enclosure and the right wall of the enclosure is assumed to be a collecting electrode. A two-dimensional geometry of the enclosure with the dimensions of the case study is shown in Fig. 1.
4.1. Flow field boundary conditions In this study, the no-slip boundary condition is used for all walls. The right wall of the enclosure is maintained at a uniform heat flux and 5
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the emitting electrode and it is equal to zero for the right wall as a collector electrode. It is not so easy to obtain the boundary conditions for the electric space charge density. In order to determine the boundary condition of electric space charge density on the emitting electrode, the Peek's law is used. In order to find the accurate charge density, the distribution of space charge density at the wire is guessed and iterated until the electric field's strength is consistent with Peek's value [20].
0.308 ⎞ EPeek = 3.1 × 106 ⎜⎛1 + ⎟ re ⎠ ⎝
(13)
where re (cm) represents the radius of the emitting electrode. 5. Numerical solution 5.1. Numerical procedure In this study, OpenFOAM 4.0 is used to simulate the EHD phenomena. OpenFOAM is an open source computational fluid dynamics (CFD) package which is written in C++. Its open source property has some advantages such as the implementation of any addition or modification for solvers. A pressure based solver by using PISO-SIMPLE (PIMPLE) algorithm was used to solve the continuity, linear and angular momentum, and energy equations. To solve the linear momentum, angular momentum, energy, turbulence kinetic energy, and turbulence dissipation energy, PBICG solver is employed. PBICG is a preconditioned biconjugate gradient solver for anti-symmetric matrices using a run-time selectable and good parallel scaling pre-conditioner. In addition, to solve the electric potential PCG solver is employed. The PCG solver is a pre-conditioned conjugate gradient solver for symmetric matrices using a runtime selectable and good parallel scaling pre-conditioner. For the solution of space charge density “smooth” solver is employed. This smoother is a Gauss Siedel smoother which is used for symmetric and anti-symmetric matrices, forward and reverse sweeps [39]. The computations were continued until the residuals decreased to below 10−11 for the electric field equations and 10−6 for the flow field equations.
Fig. 11. The enhancement factor on the right wall for Ra = 106, a = 20 cm, b = 2.5 cm (a) V = 12 kV, (b) V = 14 kV, (c) V = 16 kV. Table 4 The adequate material parameters for various sets of Rayleigh number and applied voltage (a = 20 cm, b = 2.5 cm). Ra
K=κω/μ
105 106
5.2. Mesh generation
V = 12 kV
V = 14 kV
V = 16 kV
0.2 0.5
0.3 0.6
0.5 0.8
In this study, a two-dimensional non-uniform structured mesh with tetrahedral elements was generated. The adjacent cells of the wire and the walls of the enclosure were smaller (Fig. 2). In order to ensure that the numerical results are independent of the computational grid, a grid dependency test was conducted for the temperature distribution of the right wall of the enclosure. The temperature distributions of the right wall of the enclosure for four different grid densities were calculated in Fig. 3. The difference in solution predicted by the mesh densities of 37435 cells and 27690 cells was negligible. Hence, the final computations were performed using the mesh density of 27690 cells.
Table 5 Various positions of the emitting electrode. NO.
a (cm)
b (cm)
(1) (2) (3) (4) (5)
30 20 10 20 20
2.5 2.5 2.5 4 1
5.3. Data reduction In order to investigate the EHD-induced flow effect on heat transfer enhancement, the local heat transfer coefficient hx along the right wall is calculated as follows [23]:
the left wall is maintained at a constant temperature (293 K). Also, the bottom and top walls of the enclosure are insulated. The details of flow boundary conditions are shown in Table 2.
hx = 4.2. Electric field boundary conditions
q" Tw − Tref
(14)
where Tw is the local temperature of the right wall, and Tref is the temperature of the left wall. Also, the Rayleigh number is obtained as follows [23].
The boundary conditions for electric potential are straightforward (see Table 3). So, the electric potential is equal to a specific value, V0 for 6
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Fig. 12. Air flow streamlines for V = 12 kV, Ra = 106 and b = 2.5 cm (a) a = 10 cm, (b) a = 20 cm, (c) a = 30 cm.
Fig. 13. Air flow streamlines for V = 14 kV, Ra = 106 and b = 2.5 cm (a) a = 10 cm, (b) a = 20 cm, (c) a = 30 cm.
Ra =
gβq"H 4 ναk 0
6.2. Flow field validation (15) In order to validate the flow field in numerical results of the micropolar model, the temperature distribution on the right wall of the enclosure was compared with the experiments test and numerical results of the standard k-ε model [23]. The comparison shows that the results are in good agreement with each other (Fig. 5).
6. Code validation To verify the accuracy of the results, the electric field and the micropolar flow results are presented, in section 6.1 and 6.2, respectively.
7. Results and discussion 6.1. Electric field validation
In order to gain an adequate value of the material parameter, the results of the micropolar model are compared with the results of the standard k-ε turbulent model and the results of ordinary Navier-Stokes equations, in all cases. The computations are conducted for two Rayleigh numbers, 105 and 106 at different applied voltages. To obtain the appropriate results for the micropolar model, different material parameters (K) are examined. In order to obtain the appropriate material parameter, the ratio hx-EHD/hx-noEHD (enhancement factor) was investigated for each set of Rayleigh numbers and applied voltages.
The computed results of the relative current density distribution on the grounded plate are compared with the Warburg's law and the experimental data obtained by Adamiak and Atten [40] and shown in Fig. 4. It should be noted that Pe = 100 was used, in order to solve the charge density equation. It is shown that the numerical results are consistent with the Warburg's law and the experimental results. It should be noted that Warburg's law is explained by the following equation:
J (θ) = J (0)cos4.82 (θ) θ ≤ 60∘
(16) 7
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Fig. 16. The enhancement factor for Ra = 106, V = 12 kV and a = 20 cm.
streamlines and wall temperature distribution are shown in Fig. 6. As well, to achieve the adequate material parameter, the results were compared with results of the standard k-ε turbulent model. It is found that the size and location of vortices in the enclosure are remarkably influenced by changing the material parameter. According to Fig. 6a, it is evident that the streamlines obtained from the micropolar model at K = 0.6 are more similar to the standard k-ε model results. Additionally, it is shown that the temperature distribution is affected by the material parameter. As shown in Fig. 6b, in terms of wall temperature, the micropolar results at K = 0.6 are in good agreement with the experimental data and k-ε model (V = 14 kV, Ra = 106). To more study, the streamlines of micropolar and k-ε models are compared to each other for a = 20 cm and b = 2.5 cm in different applied voltages and Rayleigh numbers (Figs. 7 and 8). It can be seen that the patterns for both models are similar to each other. It should be mentioned that by increasing the applied voltage, the discrepancy between the flow patterns increases. Moreover, the deviation among the stream patterns increases near the bottom wall of the enclosure. Further, the result of the velocity magnitude of the micropolar model was compared with the results of the k-ε model (V = 12 kV and Ra = 105, a = 20 cm and b = 2.5 cm). As shown in Fig. 9, the results of the velocity profiles obtained for both models are in good agreement. Furthermore, the enhancement factor of the right wall of the enclosure in various sets of the Rayleigh numbers and the applied voltages were computed upon both the micropolar and k-ε models. In order to
Fig. 14. The enhancement factor for Ra = 106 and b = 2.5 cm (a) V = 12 kV, (b) V = 14 kV.
7.1. Effect of the material parameter on the flow field As mentioned before, the material parameter represents the ratio of vortex viscosity to the dynamic viscosity of the fluid (κω/μ). This parameter has the key role in simulations of EHD-induced flow by using the micropolar model. It is noted that for each set of Rayleigh numbers and applied voltages, there is a specific material parameter so that the results of the micropolar model are in good agreement with the results of the k-ε model and experimental data. It is noted, the results of ordinary Navier-Stokes equations failed to predict the EHD-induced flow behavior, correctly [17]. In order to show the effect of the material parameter on flow characteristic and temperature fields, the
Fig. 15. Air flow streamlines for V = 12 kV, Ra = 106 and a = 20 cm (a) b = 1 cm, (b) b = 2.5 cm, (c) b = 4 cm. 8
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doi.org/10.1016/j.elstat.2019.103372.
evaluate the quality and accuracy of the micropolar model, the results are compared in Figs. 10 and 11. It is shown that the enhancement factors obtained by the micropolar model are consistent with the results of the standard k-ε model. The difference between the two models increases by increasing the applied voltage. The adequate material parameters obtained from the computations for various sets of Rayleigh numbers and applied voltages are classified in Table 4.
Nomenclature a b
vertical position of emitting electrode, cm distance between the emitting electrode and grounded plate, cm cp specific heat coefficient, J/kg. k d diameter, cm E electric field strength, V/m e total energy, J Fe electrohydrodynamic force, N/m3 H height of the enclosure, m h convective heat transfer coefficient, W/m2.K i index, i = 1, 2, 3 for x, y and z direction J current density, A/m2 jω micro-inertia density, m2 K material parameter k0 thermal conductivity, W/m.K k turbulence kinetic energy, J Lc characteristic length Pe Peclet number Prt turbulent Prandtl number r radius, cm r Ra radius, cm Rayleigh number T temperature, K t time, s ui velocity component, m/s V electric potential, V W width of the enclosure, m Greek symbols
7.2. Effect of emitting electrode position on the material parameter of the micropolar model To study the effect of emitting electrode's position on the heat transfer coefficient and the adequate material parameter, various positions of the emitting electrode were selected (Table 5). 7.2.1. Moving the emitting electrode in the vertical direction The vertical position of the emitting electrode is changed at different Rayleigh numbers and applied voltages (Ra = 106, V = 12 kV and 14 kV), and the streamlines of the micropolar model and k-ε model are depicted in Figs. 12 and 13, respectively. In addition, the enhancement factor of the two models was compared in Fig. 14 According to Figs. 12–14, the material parameter is not sensitive to movement the emitting electrode in the vertical direction due to almost constant electric body force. 7.2.2. Moving the emitting electrode in the horizontal direction The effect of the emitting electrode location in the horizontal direction was investigated at Ra = 106 and V = 12 kV. According to Fig. 15, the streamlines obtained from the micropolar model were compared with the results of the standard k-ε model. Also, the enhancement factors gained from the two models were compared in Fig. 16. It should be noted that the material parameter is dependent on the flow field and the EHD force [37]. According to Figs. 15 and 16, by decreasing the gap size of the electrodes (b), the adequate material parameter of micropolar model increases due to rising the EHD force. The strength of the vortices is another remarkable pattern in Fig. 15 which is increased by locating the wire closer to the grounded plate (right wall). Moreover, as shown in Fig. 16, it is apparent that by decreasing the gap size between the wire and the grounded electrode, the deviation of the two models increases. This may be due to the fact that the stronger vortices cause more perturbation in the main flow. 8. Conclusion In this study, natural convection heat transfer is numerically investigated in a tall rectangular enclosure in the presence of EHD actuator using micropolar fluid model. The results of the micropolar model have been compared with the results of the standard k-ε model which has been commonly used for EHD simulations. Therefore, the following achievements can be highlighted:
• The results highlight that the material parameter highly depends on the Rayleigh number and the applied voltage. • The effect of the position of the wire was studied and it was shown that the major effect relates to the gap size of the electrodes. • The results reveal that using the micropolar approach with an •
α β β0 ε ε0 εijk Γ γ κω μ μt ρ ρc σk σε ωi Subscripts
thermal diffusivity, m2/s thermal expansion, 1/K ion mobility, m2/V.s turbulent dissipation rate, W dielectric permittivity, F/m permutation symbol (Levi-Civita symbol) diffusion coefficient, m2/s spin gradient viscosity, kg.m/s vortex Viscosity, kg/m.s dynamic Viscosity, kg/m.s turbulence dynamic viscosity, kg/m.s fluid density, kg/m3 space charge density, C/m3 Prandtl number for turbulent kinetic energy Prandtl number for turbulent dissipation rate angular velocity component, 1/s
e i t x w 0
emitting electrode value ith component turbulence value right wall local value at axial location wall emitting electrode value, initial value
References
adequate material parameter can model the EHD-induced natural convection problems, when the flow cannot be strongly assumed to have a fully turbulent regime. As the final statement, the current study showed that the micropolar approach can be a reliable model for the semi-turbulent EHD cases.
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Appendix A. Supplementary data Supplementary data to this article can be found online at https:// 9
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[4] A. Elatar, M.A. Teamah, M.A. Hassab, Numerical study of laminar natural convection inside square enclosure with single horizontal fin, Int. J. Therm. Sci. 99 (2016 Jan 1) 41–51. [5] J. Cai, X. Huai, R. Yan, Y. Cheng, Numerical simulation on enhancement of natural convection heat transfer by acoustic cavitation in a square enclosure, Appl. Therm. Eng. 29 (10) (2009 Jul 1) 1973–1982. [6] B. Tajik, A. Abbassi, M. Saffar-Avval, A. Abdullah, H. Mohammad-Abadi, Heat transfer enhancement by acoustic streaming in a closed cylindrical enclosure filled with water, Int. J. Heat Mass Transf. 60 (2013 May 1) 230–235. [7] M. Talebi, M. Setareh, M. Saffar-Avval, R.H. Abardeh, Numerical investigation of natural convection heat transfer in a cylindrical enclosure due to ultrasonic vibrations, Ultrasonics 76 (2017 Apr 1) 52–62. [8] S.N. Vatan, E.S. Nia, A. Merdasi, Empirical correlation for performance evaluation of electric/corona wind on natural convection, J. Electrost. 72 (1) (2014 Feb 1) 82–90. [9] G.L. Leonard, M. Mitchner, S.A. Self, An experimental study of the electrohydrodynamic flow in electrostatic precipitators, J. Fluid Mech. 127 (1983 Feb) 123–140. [10] P. Béquin, K. Castor, J. Scholten, Electric wind characterisation in negative point-toplane corona discharges in air, Eur. Phys. J. Appl. Phys. 22 (1) (2003 Apr) 41–49. [11] S.R. Mahmoudi, K. Adamiak, P. Castle, M. Ashjaee, The effect of corona discharge on free convection heat transfer from a horizontal cylinder, Exp. Therm. Fluid Sci. 34 (5) (2010 Jul 1) 528–537. [12] L. Zhao, K. Adamiak, EHD flow in air produced by electric corona discharge in pin–plate configuration, J. Electrost. 63 (3–4) (2005 Mar 1) 337–350. [13] Z. Feng, Z. Long, K. Adamiak, Numerical simulation of electrohydrodynamic flow and vortex analysis in electrostatic precipitators, IEEE Trans. Dielectr. Electr. Insul. 25 (2) (2018 Apr 19) 404–412. [14] Z. Feng, Z. Long, S. Cao, K. Adamiak, Characterization of electrohydrodynamic (EHD) flow in electrostatic precitators (ESP) by numerical simulation and quantitative vortex analysis, J. Electrost. 91 (2018 Feb 1) 70–80. [15] Y. Liao, Z. Feng, X. Zhou, Predicting the pumping effects of electrohydrodynamic (EHD) gas pumps by numerical simulations and quantitative pressure drop vs. flow rate curves, J. Electrost. 96 (2018 Dec 1) 160–168. [16] S. Mantach, K. Adamiak, A double-vortex EHD flow pattern generated by negative corona discharge in point-plane geometry, J. Electrost. 93 (2018 Jun 1) 118–124. [17] M. Molki, P. Damronglerd, Electrohydrodynamic enhancement of heat transfer for developing air flow in square ducts, Heat Transf. Eng. 27 (1) (2006 Jan 1) 35–45. [18] F. Dolati, N. Amanifard, H. Mohaddes Deylami, Numerical investigation of mass transfer enhancement through a porous body affected by electric field, Dry. Technol. 36 (13) (2018 Oct 3) 1563–1577. [19] F. Dolati, N. Amanifard, H.M. Deylami, Numerical investigation of moisture removal and energy consumption of porous body affected by EHD, Energy 154 (2018 Jul 1) 352–364. [20] H.M. Deylami, N. Amanifard, F. Dolati, R. Kouhikamali, K. Mostajiri, Numerical investigation of using various electrode arrangements for amplifying the EHD enhanced heat transfer in a smooth channel, J. Electrost. 71 (4) (2013 Aug 1) 656–665. [21] H.M. Deylami, N. Amanifard, F. Dolati, R. Kouhikamali, K. Mostajiri, Numerical investigation of electrohydrodynamic effects on the airflow through corrugated
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channels, Proc. Inst. Mech. Eng. C J. Mech. Eng. Sci. 227 (12) (2013 Dec) 2730–2741. M. Peng, T.H. Wang, X.D. Wang, Effect of longitudinal electrode arrangement on EHD-induced heat transfer enhancement in a rectangular channel, Int. J. Heat Mass Transf. 93 (2016 Feb 1) 1072–1081. S.S. Golsefid, N. Amanifard, H.M. Deylami, F. Dolati, Numerical and experimental study on EHD heat transfer enhancement with Joule heating effect through a rectangular enclosure, Appl. Therm. Eng. 123 (2017 Aug 1) 689–698. A.C. Eringen, Microcontinuum Field Theories: II, Fluent Media, Springer Science & Business Media, 2001 Mar 30. G. Lukaszewicz, Micropolar Fluids: Theory and Applications, Springer Science & Business Media, 1999 Feb 1. G. Ahmadi, Self-similar solution of imcompressible micropolar boundary layer flow over a semi-infinite plate, Int. J. Eng. Sci. 14 (7) (1976 Jan 1) 639–646. T.H. Hsu, Natural convection of micropolar fluids in a rectangular enclosure, Int. J. Eng. Sci. 34 (4) (1996 Mar 1) 407–415. A. Mrabti, K. Gueraoui, A. Hihi, O. Terhmina, Effects of microstructure on natural convection flow of micropolar fluids in a vertical cylinder heated from below, Int. J. Eng. Sci. 38 (7) (2000 May 1) 823–831. O. Aydin, I. Pop, Natural convection from a discrete heater in enclosures filled with a micropolar fluid, Int. J. Eng. Sci. 43 (19–20) (2005 Dec 1) 1409–1418. M. Saleem, S. Asghar, M.A. Hossain, Natural convection flow of micropolar fluid in a rectangular cavity heated from below with cold sidewalls, Math. Comput. Model. 54 (1–2) (2011 Jul 1) 508–518. M. Muthtamilselvan, K. Periyadurai, D.H. Doh, Effect of uniform and nonuniform heat source on natural convection flow of micropolar fluid, Int. J. Heat Mass Transf. 115 (2017 Dec 1) 19–34. I.V. Miroshnichenko, M.A. Sheremet, I. Pop, Natural convection in a trapezoidal cavity filled with a micropolar fluid under the effect of a local heat source, Int. J. Mech. Sci. 120 (2017 Jan 1) 182–189. G.M. Abdel-Rahman, Studying effect of MHD on thin films of a micropolar fluid, Phys. B Condens. Matter 404 (21) (2009 Nov 15) 3859–3866. M. Ezzat, A.A. El-Bary, S. Ezzat, Combined heat and mass transfer for unsteady MHD flow of perfect conducting micropolar fluid with thermal relaxation, Energy Convers. Manag. 52 (2) (2011 Feb 1) 934–945. A. Borrelli, G. Giantesio, M.C. Patria, An exact solution for the 3D MHD stagnationpoint flow of a micropolar fluid, Commun. Nonlinear Sci. Numer. Simul. 20 (1) (2015 Jan 1) 121–135. H. Moayedi, N. Amanifard, H.M. Deylami, Evaluation of using micropolar fluid approach for the EHD-enhanced forced convection through a rectangular channel using multiple electrode arrangements, Appl. Therm. Eng. (2019 May 29) 113857. H. Moayedi, N. Amanifard, H.M. Deylami, F. Dolati, Numerical investigation of using micropolar fluid model for EHD flow through a smooth channel, J. Electrost. 87 (2017 Jun 1) 51–63. R.B. Lakeh, M. Molki, Enhancement of convective heat transfer by electrically-induced swirling effect in laminar and fully-developed internal flows, J. Electrost. 71 (6) (2013 Dec 1) 1086–1099. Open CF, OpenFOAM User Guide vol 2, OpenFOAM Foundation, 2011 1. K. Adamiak, P. Atten, Simulation of corona discharge in point–plane configuration, J. Electrost. 61 (2) (2004 Jun 1) 85–98.
Contents lists available at ScienceDirect
Journal of Electrostatics journal homepage: www.elsevier.com/locate/elstat
Numerical evaluation of using micropolar fluid model for EHD-induced natural convection heat transfer through a rectangular enclosure
T
Ladan Samaeia, Hamed Mohaddes Deylamib,∗, Nima Amanifarda, Hesam Moayedia a b
Faculty of Mechanical Engineering, University of Guilan, Rasht, Iran Faculty of Technology and Engineering, East of Guilan, University of Guilan, Rudsar, Iran
A R T I C LE I N FO
A B S T R A C T
Keywords: Electrohydrodynamic Natural convection Enclosure Micropolar fluid model Numerical investigation
Using turbulent models for electrohydrodynamic (EHD) induced flow modeling in hydraulically laminar flow ranges has been always questionable. This study is concerned with EHD cases in natural convection heat transfer within an enclosure. Here, the micropolar model was engaged and became the main motivation of the current study as an alternative approach for the fluid dynamic behavior. The numerical investigations were performed to study the effect of the various key parameters: the applied voltage, Rayleigh number, and the gap size of the electrodes. The main challenge of using the micropolar model is the evaluation of the adequate material parameter (κω/μ); all the presented numerical investigations conducted to find the proper material parameter for various flow conditions. All cases were carried out for a two-dimensional approach with a non-uniform structured grid, which is used by a finite volume algorithm to solve the EHD natural convection governing equations. Comparing the results of the micropolar approach with those obtained from the turbulent standard k-ε model reveals that the micropolar model can be an appropriate candidate to simulate the EHD natural convection flow instead of fully turbulent models. The results depicted that by increasing Rayleigh number and the applied voltage, the discrepancy between the results of the micropolar and the standard k-ε model increases.
1. Introduction Natural convection heat transfer in enclosures plays an important role in many engineering applications, and various techniques have been provided to gain the higher performance of heat transfer. Methods of enhancing heat transfer can be classified into two main categories: active and passive methods. Using finned enclosures [1–4] is an example of the passive methods while using EHD actuator or ultrasonic waves [5–7] are some examples of active methods. As the current work focuses on EHD modeling, a brief review of the EHD actuators may be beneficial. When a high voltage is applied to a sharp electrode (emitter), the air becomes ionized and the ions are attracted toward the electrically grounded plate (collector). The transmission of the injected ions from the emitter toward the collector causes the ions to transfer their momentum to the neutral molecules by collision. As a result, a bulk flow of ionized air molecules is generated known as corona wind or the secondary flow. Recently, the EHD-induced flow became an active method for the enhancement of heat transfer among many available methods. The facility, high reliability, and short response time are among the advantages of this methodology [8].
∗
The EHD heat transfer has widely attracted the researchers to develop the technique for various cases. Most studies have been performed experimentally [8–12] or numerically by using fully turbulent flow models [13–23]. Some researchers tried to experimentally measure the velocities of the EHD flow, using Laser-Doppler Anemometers (LDA), Particle Image Velocimetry (PIV) and Hot-Wire Anemometers [9,10,12]. The researchers reported that these experimental techniques showed some uncertainties, which caused by the effect of electric field on the process of measurements. Numerical methods didn't have the same uncertainty problem, but the main challenge to modeling the EHD-induced flow numerically, is to use the fully turbulent models, even when the main flow is in the laminar regime [13–15,17–23]. Molki and Damronglerd [17] examined the effect of the EHD-induced flow on improving heat transfer in a square duct. They underlined that fully turbulent models are necessary to simulate the EHD-induced flow, even in a laminar flow regime. They compared the results of the laminar model and fully turbulent model with Large Eddy Simulation (LES) and empirical results. They showed that using laminar model failed to predict the Nusselt number and friction factor correctly, but the results of the turbulent model was in good consistency with the results of LES and experimental results. Feng et al. [13] investigated
Corresponding author. E-mail address: [email protected] (H. Mohaddes Deylami).
https://doi.org/10.1016/j.elstat.2019.103372 Received 19 May 2019; Received in revised form 26 August 2019; Accepted 4 September 2019 0304-3886/ © 2019 Elsevier B.V. All rights reserved.
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EHD flow pattern in electrostatic preticipators, numerically. They used different turbulence models and compared the numerical results with the experimental data. Comparing the numerical results with the experimental data showed that the results obtained from the k-ε model is satisfactory. Also, Golsefid et al. [23] experimentally and numerically studied the EHD effect on enhancement of natural convection heat transfer within an enclosure with Joule heating term in the energy equation. They used the standard k-ε fully turbulent model to simulate the EHD-induced flow in the laminar regime. Their results showed that wherever Rayleigh number was small, the effect of the Joule heat parameter in the energy equation was significant. Additionally, they found an optimal configuration of the collector electrode, in which the effect of EHD-induced flow on the heat transfer enhancement reaches to its maximum. The lack of using an appropriate approach for the fluid motion in the EHD field, particularly in laminar range, and the lack of the turbulent flow concepts such as the unsteadiness and irregularity become the main reason to test the micropolar fluid model in the current work. Some studies have been carried out in micropolar fluid and its applications. The micropolar fluid theory was introduced by Eringen [24] and Lukaszewicz [25]. They introduced micropolar fluid as a fluid with microstructure which has nonsymmetrical stress tensor. Ahmadi [26] studied the boundary layer flow of micropolar fluid over a semi-infinite flat plate. The applicability of micropolar fluid theory to the motion of suspension solutions was introduced. Many cases of heat transfer of the micropolar fluid have been studied [27–32], some cases were under magnetic field [33–35] or under electric field effects [36,37]. In the natural convection cases, Aydin and Pop [29] numerically investigated the steady laminar natural convection heat transfer of the micropolar fluid in an enclosure. They discussed the effects of Rayleigh number (Ra), Prandtl number (Pr), and material parameter (K), on the micropolar fluid motion and natural convection heat transfer. They reported that by increasing Rayleigh number and decreasing the material parameter, the heat transfer increases. Muthtamilselvan et al. [31] investigated numerically natural convection of micropolar fluid in a square cavity with uniform and non-uniform heat sources. They showed that the heat transfer decreases with increasing vortex viscosity parameter. Also, the heat transfer increases with increasing heat source length for both uniform and non-uniform heat sources. In addition, with increasing Rayleigh number for all values of source nonuniformity parameter heat transfer increases. Moayedi et al. [37] used the micropolar model to simulate the forced convection heat transfer affected by EHD-induced flow in a smooth channel when the flow field was in the laminar regime. They compared results of the micropolar model with results of the standard k-ε turbulent model. They investigated the EHD-induced flow field by changing the material parameter. Their results showed that for a specific material parameter for each applied voltage and Reynolds number, the results of the micropolar model were consistent with the results of the standard k-ε model. Also, they reported that the micropolar model could be a reliable method to simulate the EHD-induced flow field in the laminar regime. Regarding the mentioned previous works, it should be underlined that the main reasons to examine a new numerical model (micropolar fluid model) as an alternative dynamic behavior model in the current work are as follows:
Table 1 Fluid properties. Property
Value
cp k0 β0 ε0 μ ρ
1006.43 J/kg. K 0.0242 W/m. K 2⨯10−4 m2/V.s 8.85⨯10−12 F/m 1.885⨯10−5 kg/m. s 1.115 kg/m3
Fig. 1. Schematic view of the computation domain and geometric parameters. Table 2 Fluid boundary conditions. Surfaces
Air flow
Temperature
Micro spin
Emitting electrode Right wall (collector) Left wall Top wall Bottom wall
ux = uy = 0 ux = uy = 0 ux = uy = 0 ux = uy = 0 ux = uy = 0
∂T/∂n=0
ωz=0
q″=cte
ωz=0
T=cte
ωz=0
∂T/∂y=0
ωz=0
∂T/∂y=0
ωz=0
Table 3 Electrical boundary conditions. Surfaces Emitting electrode Right wall (collector) Left wall Top wall Bottom wall
1. The lack of an appropriate approach for the fluid motion modeling in the presence of the EHD-induced flow, particularly in laminar range. 2. The lack of turbulent flow concepts such as the unsteadiness and irregularity.
Electrical potential
Space charge density
V=V0
Peek's value
V=0
∂ρc/∂x=0
∂V/∂x=0
∂ρc/∂x=0
∂V/∂y=0
∂ρc/∂y=0
∂V/∂y=0
∂ρc/∂y=0
model has few predictive power, the computational cost of the micropolar model with an adequate material parameter (κω/μ) is much lower than the computational cost of the k-ε model [36]. Therefore, the main goal of our study is twofold:
It is noteworthy that the new micropolar fluid model has advantages such as an acceptable prediction approach in hydraulically laminar flow range in which the flow is not fully turbulent. Despite the micropolar 2
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Fig. 4. Distribution of the current density on the grounded plate. (V = 8 kV, 95 μm needle 3.1 cm above the ground plate).
Fig. 2. A schematic view of computational domain grid distributions. (a) Near the wire, (b) Near the wall.
Fig. 5. Temperature distribution on the right wall of the enclosure (V = 14 kV, Ra = 106, a = 20 cm, b = 2.5 cm).
2. Governing equations 2.1. Electric field equations The EHD-induced flow is analyzed using basic electrostatic, flow and energy equations. The electric body force per unit volume can be expressed as follows [20]: Fig. 3. Grid independence test for temperature distribution (V = 14 kV, Ra = 106, a = 20 cm, b = 2.5 cm).
→ → 1 →2 →2 ⎛ ∂ε0 ⎞ ⎤ 1 Fe = ρc E − E ∇ε0 + ∇ ⎡ E ρ⎜ ⎟ ⎥ 2 2 ⎢ ⎝ ∂ρ ⎠T ⎦ ⎣
(a) Evaluation of the new numerical model (micropolar fluid model) to simulate the natural convection of the EHD-induced flow in a rectangular enclosure. (b) Presentation of a new correlation to find the adequate value of the material parameter in the micropolar model.
(1)
The first term on the right side of this equation refers to the Coulomb force. It is always employed in presence of the electric field. The second and third terms of Eq. (1) represent Electrophoretic force and Electrostrictive force, respectively. Due to the constant electric permittivity, these terms can be ignored. Whenever there is no external magnetic field and electric current is negligible, Faraday's Law can be simplified as follows [38]:
It is noted that in order to evaluate the capability and the accuracy of the micropolar fluid model, the results of the present model were compared with the results of the standard k-ε model. Finally, different cases are investigated upon various gap sizes between electrodes, different applied voltages (12 kV, 14 kV and 16 kV) and different Rayleigh numbers (105,106).
→ ∇×E =0
(2)
Therefore, the electric field is a non-rotating vector field, so that, a unique scalar function as a potential function can be defined for it:
→ E = −∇V Poisson's equation: 3
(3)
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→ → J = ρc β0 E + ρc → u − De ∇ρc
(6)
Three terms on the right side of Eq. (6) represent ion drift, convection, and diffusion of electrical charges, respectively. Since the order of magnitude of ion drift is higher in comparison to the other terms in the right side of the equation, they can be ignored. Therefore, Eq. (5) will be as follows:
∂ρc ∂t
→ + ∇ . (ρc β0 E ) = 0
(7)
Eq. (4) and Eq. (7) are the main governing equations of the EHD phenomena. Eq. (7) suffers from dispersion error [20]. So, in order to solve the charge density equation, we used artificial diffusion coeffiβ EL cient ∇ . (Γ ∇ρc ) , where Pe = 0Γ c . Hence, Eq. (7) is rewritten as → ∂ρc + ∇ . (ρc β0 E ) = ∇ . (Γ ∇ρc ) . By solving these equations at the same ∂t time, the electrical body force is obtained. Finally, the calculated body force acts as a source term in momentum equations. 2.2. Governing equations of the micropolar fluid model Governing equations of the micropolar fluid model by assuming an unsteady, two dimensional, incompressible EHD-induced flow in natural convection heat transfer can be written in the following forms [24]: 2.3. Continuity
∂ (ρui ) ∂ρ =0 + ∂x i ∂t
(8)
Momentum:
∂p ∂ (ρui ) ∂ 2u i ⎞ ∂ ⎛ ∂ωk ⎞ + (μ + κ ω) ⎜⎛ (ρui uj ) = − + ⎟ + κ ω εijk ⎜ ⎟ x x ∂x i ∂ ∂ ∂t ∂x j i j ⎝ ⎠ ⎝ ∂x j ⎠
Fig. 6. Effect of the material parameter on (a) air flow streamlines, (b) wall temperature (V = 14 kV, Ra = 106, a = 20 cm, b = 2.5 cm).
∇2 . V =
+ ρgi β (T − T0) δij + Fei
−ρc
Angular momentum:
(4)
ε0
∂ω ∂ ∂2ωi ⎞ ⎛ ∂uk ⎞ (ωi uj ) ⎤ = γ ⎛⎜ ρjω ⎡ i + ⎟ − 2κ ω ωi + κ ω εijk ⎜ ⎟ ⎢ ∂t ⎥ x ∂ ∂ j ⎣ ⎦ ⎝ x j ∂x j ⎠ ⎝ ∂x j ⎠
Conservation of space charge density equation:
∂ρc ∂t
→ + ∇. J = 0
(9)
(10)
Energy:
(5)
→ J represents electric current density which is calculated from the following equation:
∂ ∂ ∂ ⎛ ∂T ⎞ (ρe ) + [ui (ρe + p)] = ⎜k 0 ⎟ ∂t ∂x i ∂x j ⎝ ∂x j ⎠
Fig. 7. Air flow streamlines for Ra = 105, a = 20 cm, b = 2.5 cm (a) V = 12 kV, (b) V = 14 kV, (c) V = 16 kV. 4
(11)
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Fig. 8. Air flow streamlines for Ra = 106, a = 20 cm, b = 2.5 cm (a) V = 12 kV, (b) V = 14 kV, (c) V = 16 kV.
Fig. 9. The comparison of the flow velocity magnitude between the micropolar and the standard k-ε models (V = 12 kV, Ra = 105, a = 20 cm, b = 2.5 cm).
where jω represents the micro-inertia density which is equal to the square of characteristic length. Also, μ, κω and γ represent dynamic viscosity, vortex viscosity, and spin gradient viscosity, respectively.
κ K γ = ⎛μ + ω ⎞ jω = μ ⎛1 + ⎞ jω 2 ⎠ 2⎠ ⎝ ⎝
(12)
where K represents the material parameter. The properties of the fluid (air) are shown in Table 1. It should be noted that the governing equations of the standard k-ε model is presented in the Appendix.
Fig. 10. The enhancement factor on the right wall for Ra = 105, a = 20 cm, b = 2.5 cm (a) V = 12 kV, (b) V = 14 kV, (c) V = 16 kV.
4. Boundary conditions 3. Computational geometry
The boundary conditions are divided into two parts; (1) flow field, (2) electric field. The flow and electric boundary conditions are explained in sections 4.1 and 4.2, respectively.
In this study, a tall rectangular enclosure with an emitting electrode is considered, in which the emitting electrode is located in the center of the enclosure and the right wall of the enclosure is assumed to be a collecting electrode. A two-dimensional geometry of the enclosure with the dimensions of the case study is shown in Fig. 1.
4.1. Flow field boundary conditions In this study, the no-slip boundary condition is used for all walls. The right wall of the enclosure is maintained at a uniform heat flux and 5
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the emitting electrode and it is equal to zero for the right wall as a collector electrode. It is not so easy to obtain the boundary conditions for the electric space charge density. In order to determine the boundary condition of electric space charge density on the emitting electrode, the Peek's law is used. In order to find the accurate charge density, the distribution of space charge density at the wire is guessed and iterated until the electric field's strength is consistent with Peek's value [20].
0.308 ⎞ EPeek = 3.1 × 106 ⎜⎛1 + ⎟ re ⎠ ⎝
(13)
where re (cm) represents the radius of the emitting electrode. 5. Numerical solution 5.1. Numerical procedure In this study, OpenFOAM 4.0 is used to simulate the EHD phenomena. OpenFOAM is an open source computational fluid dynamics (CFD) package which is written in C++. Its open source property has some advantages such as the implementation of any addition or modification for solvers. A pressure based solver by using PISO-SIMPLE (PIMPLE) algorithm was used to solve the continuity, linear and angular momentum, and energy equations. To solve the linear momentum, angular momentum, energy, turbulence kinetic energy, and turbulence dissipation energy, PBICG solver is employed. PBICG is a preconditioned biconjugate gradient solver for anti-symmetric matrices using a run-time selectable and good parallel scaling pre-conditioner. In addition, to solve the electric potential PCG solver is employed. The PCG solver is a pre-conditioned conjugate gradient solver for symmetric matrices using a runtime selectable and good parallel scaling pre-conditioner. For the solution of space charge density “smooth” solver is employed. This smoother is a Gauss Siedel smoother which is used for symmetric and anti-symmetric matrices, forward and reverse sweeps [39]. The computations were continued until the residuals decreased to below 10−11 for the electric field equations and 10−6 for the flow field equations.
Fig. 11. The enhancement factor on the right wall for Ra = 106, a = 20 cm, b = 2.5 cm (a) V = 12 kV, (b) V = 14 kV, (c) V = 16 kV. Table 4 The adequate material parameters for various sets of Rayleigh number and applied voltage (a = 20 cm, b = 2.5 cm). Ra
K=κω/μ
105 106
5.2. Mesh generation
V = 12 kV
V = 14 kV
V = 16 kV
0.2 0.5
0.3 0.6
0.5 0.8
In this study, a two-dimensional non-uniform structured mesh with tetrahedral elements was generated. The adjacent cells of the wire and the walls of the enclosure were smaller (Fig. 2). In order to ensure that the numerical results are independent of the computational grid, a grid dependency test was conducted for the temperature distribution of the right wall of the enclosure. The temperature distributions of the right wall of the enclosure for four different grid densities were calculated in Fig. 3. The difference in solution predicted by the mesh densities of 37435 cells and 27690 cells was negligible. Hence, the final computations were performed using the mesh density of 27690 cells.
Table 5 Various positions of the emitting electrode. NO.
a (cm)
b (cm)
(1) (2) (3) (4) (5)
30 20 10 20 20
2.5 2.5 2.5 4 1
5.3. Data reduction In order to investigate the EHD-induced flow effect on heat transfer enhancement, the local heat transfer coefficient hx along the right wall is calculated as follows [23]:
the left wall is maintained at a constant temperature (293 K). Also, the bottom and top walls of the enclosure are insulated. The details of flow boundary conditions are shown in Table 2.
hx = 4.2. Electric field boundary conditions
q" Tw − Tref
(14)
where Tw is the local temperature of the right wall, and Tref is the temperature of the left wall. Also, the Rayleigh number is obtained as follows [23].
The boundary conditions for electric potential are straightforward (see Table 3). So, the electric potential is equal to a specific value, V0 for 6
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Fig. 12. Air flow streamlines for V = 12 kV, Ra = 106 and b = 2.5 cm (a) a = 10 cm, (b) a = 20 cm, (c) a = 30 cm.
Fig. 13. Air flow streamlines for V = 14 kV, Ra = 106 and b = 2.5 cm (a) a = 10 cm, (b) a = 20 cm, (c) a = 30 cm.
Ra =
gβq"H 4 ναk 0
6.2. Flow field validation (15) In order to validate the flow field in numerical results of the micropolar model, the temperature distribution on the right wall of the enclosure was compared with the experiments test and numerical results of the standard k-ε model [23]. The comparison shows that the results are in good agreement with each other (Fig. 5).
6. Code validation To verify the accuracy of the results, the electric field and the micropolar flow results are presented, in section 6.1 and 6.2, respectively.
7. Results and discussion 6.1. Electric field validation
In order to gain an adequate value of the material parameter, the results of the micropolar model are compared with the results of the standard k-ε turbulent model and the results of ordinary Navier-Stokes equations, in all cases. The computations are conducted for two Rayleigh numbers, 105 and 106 at different applied voltages. To obtain the appropriate results for the micropolar model, different material parameters (K) are examined. In order to obtain the appropriate material parameter, the ratio hx-EHD/hx-noEHD (enhancement factor) was investigated for each set of Rayleigh numbers and applied voltages.
The computed results of the relative current density distribution on the grounded plate are compared with the Warburg's law and the experimental data obtained by Adamiak and Atten [40] and shown in Fig. 4. It should be noted that Pe = 100 was used, in order to solve the charge density equation. It is shown that the numerical results are consistent with the Warburg's law and the experimental results. It should be noted that Warburg's law is explained by the following equation:
J (θ) = J (0)cos4.82 (θ) θ ≤ 60∘
(16) 7
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Fig. 16. The enhancement factor for Ra = 106, V = 12 kV and a = 20 cm.
streamlines and wall temperature distribution are shown in Fig. 6. As well, to achieve the adequate material parameter, the results were compared with results of the standard k-ε turbulent model. It is found that the size and location of vortices in the enclosure are remarkably influenced by changing the material parameter. According to Fig. 6a, it is evident that the streamlines obtained from the micropolar model at K = 0.6 are more similar to the standard k-ε model results. Additionally, it is shown that the temperature distribution is affected by the material parameter. As shown in Fig. 6b, in terms of wall temperature, the micropolar results at K = 0.6 are in good agreement with the experimental data and k-ε model (V = 14 kV, Ra = 106). To more study, the streamlines of micropolar and k-ε models are compared to each other for a = 20 cm and b = 2.5 cm in different applied voltages and Rayleigh numbers (Figs. 7 and 8). It can be seen that the patterns for both models are similar to each other. It should be mentioned that by increasing the applied voltage, the discrepancy between the flow patterns increases. Moreover, the deviation among the stream patterns increases near the bottom wall of the enclosure. Further, the result of the velocity magnitude of the micropolar model was compared with the results of the k-ε model (V = 12 kV and Ra = 105, a = 20 cm and b = 2.5 cm). As shown in Fig. 9, the results of the velocity profiles obtained for both models are in good agreement. Furthermore, the enhancement factor of the right wall of the enclosure in various sets of the Rayleigh numbers and the applied voltages were computed upon both the micropolar and k-ε models. In order to
Fig. 14. The enhancement factor for Ra = 106 and b = 2.5 cm (a) V = 12 kV, (b) V = 14 kV.
7.1. Effect of the material parameter on the flow field As mentioned before, the material parameter represents the ratio of vortex viscosity to the dynamic viscosity of the fluid (κω/μ). This parameter has the key role in simulations of EHD-induced flow by using the micropolar model. It is noted that for each set of Rayleigh numbers and applied voltages, there is a specific material parameter so that the results of the micropolar model are in good agreement with the results of the k-ε model and experimental data. It is noted, the results of ordinary Navier-Stokes equations failed to predict the EHD-induced flow behavior, correctly [17]. In order to show the effect of the material parameter on flow characteristic and temperature fields, the
Fig. 15. Air flow streamlines for V = 12 kV, Ra = 106 and a = 20 cm (a) b = 1 cm, (b) b = 2.5 cm, (c) b = 4 cm. 8
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doi.org/10.1016/j.elstat.2019.103372.
evaluate the quality and accuracy of the micropolar model, the results are compared in Figs. 10 and 11. It is shown that the enhancement factors obtained by the micropolar model are consistent with the results of the standard k-ε model. The difference between the two models increases by increasing the applied voltage. The adequate material parameters obtained from the computations for various sets of Rayleigh numbers and applied voltages are classified in Table 4.
Nomenclature a b
vertical position of emitting electrode, cm distance between the emitting electrode and grounded plate, cm cp specific heat coefficient, J/kg. k d diameter, cm E electric field strength, V/m e total energy, J Fe electrohydrodynamic force, N/m3 H height of the enclosure, m h convective heat transfer coefficient, W/m2.K i index, i = 1, 2, 3 for x, y and z direction J current density, A/m2 jω micro-inertia density, m2 K material parameter k0 thermal conductivity, W/m.K k turbulence kinetic energy, J Lc characteristic length Pe Peclet number Prt turbulent Prandtl number r radius, cm r Ra radius, cm Rayleigh number T temperature, K t time, s ui velocity component, m/s V electric potential, V W width of the enclosure, m Greek symbols
7.2. Effect of emitting electrode position on the material parameter of the micropolar model To study the effect of emitting electrode's position on the heat transfer coefficient and the adequate material parameter, various positions of the emitting electrode were selected (Table 5). 7.2.1. Moving the emitting electrode in the vertical direction The vertical position of the emitting electrode is changed at different Rayleigh numbers and applied voltages (Ra = 106, V = 12 kV and 14 kV), and the streamlines of the micropolar model and k-ε model are depicted in Figs. 12 and 13, respectively. In addition, the enhancement factor of the two models was compared in Fig. 14 According to Figs. 12–14, the material parameter is not sensitive to movement the emitting electrode in the vertical direction due to almost constant electric body force. 7.2.2. Moving the emitting electrode in the horizontal direction The effect of the emitting electrode location in the horizontal direction was investigated at Ra = 106 and V = 12 kV. According to Fig. 15, the streamlines obtained from the micropolar model were compared with the results of the standard k-ε model. Also, the enhancement factors gained from the two models were compared in Fig. 16. It should be noted that the material parameter is dependent on the flow field and the EHD force [37]. According to Figs. 15 and 16, by decreasing the gap size of the electrodes (b), the adequate material parameter of micropolar model increases due to rising the EHD force. The strength of the vortices is another remarkable pattern in Fig. 15 which is increased by locating the wire closer to the grounded plate (right wall). Moreover, as shown in Fig. 16, it is apparent that by decreasing the gap size between the wire and the grounded electrode, the deviation of the two models increases. This may be due to the fact that the stronger vortices cause more perturbation in the main flow. 8. Conclusion In this study, natural convection heat transfer is numerically investigated in a tall rectangular enclosure in the presence of EHD actuator using micropolar fluid model. The results of the micropolar model have been compared with the results of the standard k-ε model which has been commonly used for EHD simulations. Therefore, the following achievements can be highlighted:
• The results highlight that the material parameter highly depends on the Rayleigh number and the applied voltage. • The effect of the position of the wire was studied and it was shown that the major effect relates to the gap size of the electrodes. • The results reveal that using the micropolar approach with an •
α β β0 ε ε0 εijk Γ γ κω μ μt ρ ρc σk σε ωi Subscripts
thermal diffusivity, m2/s thermal expansion, 1/K ion mobility, m2/V.s turbulent dissipation rate, W dielectric permittivity, F/m permutation symbol (Levi-Civita symbol) diffusion coefficient, m2/s spin gradient viscosity, kg.m/s vortex Viscosity, kg/m.s dynamic Viscosity, kg/m.s turbulence dynamic viscosity, kg/m.s fluid density, kg/m3 space charge density, C/m3 Prandtl number for turbulent kinetic energy Prandtl number for turbulent dissipation rate angular velocity component, 1/s
e i t x w 0
emitting electrode value ith component turbulence value right wall local value at axial location wall emitting electrode value, initial value
References
adequate material parameter can model the EHD-induced natural convection problems, when the flow cannot be strongly assumed to have a fully turbulent regime. As the final statement, the current study showed that the micropolar approach can be a reliable model for the semi-turbulent EHD cases.
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