Heat transfer enhancement on saturated porous samples using electrostatic precipitator process in k-ε turbulent model

Abstract

An influence of the electrostatic precipitator process was numerically investigated for heat transfer enhancement on saturated porous samples in a k-ε turbulent model. The condition of the water entering a test section was the inlet temperature was 30 °C (303 K), and inlet velocity was tested in the range of 1 – 2.5 m/s. The electrical voltage and time varied between 0 – 30 kV and 0 – 1 s, respectively. The initial temperature of saturated porous samples was 10 °C (283 K), and both first and second samples were set in semicircle shapes. The numerical results within the water channel showed that the electric field and electric potential zone appeared and were concentrated when using the electrostatic precipitator process. The high electric voltage could increase disturbance and turbulence within the water channel. The maximum flow field zone appeared above the saturated porous sample area, and the maximum velocity field increased with the inlet velocity and electrical voltage. The maximum pressure was increased to the high voltages, but the pressure was marginally increased with high inlet velocity. The vorticity contour for an electrostatic precipitator process was more concentrated than without the electrostatic precipitator process. Therefore, the temperature contour line in case of high inlet velocity, electrical voltage, and time can be more disturbing than the other cases. In addition, heat from the water was transferred within the sample, so the temperature within the porous sample gradually increased. The fluid velocity within the front porous sample was more within the saturated porous samples than within the porous back sample. Therefore, the flow could move through and within the samples and induce temperature within both saturated porous samples. Finally, the heat transfer within samples was enhanced by fluid flow in the water channel, so the local heat transfer coefficient within samples was induced by the fluid velocity in the water channel.

1. Introduction

Sewage treatment technically removes substances that contaminate water, and after the waste treatment, water can be returned to the water cycle. For the hydrologic cycle, the water from treatment can be used in irrigation, agriculture, and various purposes [1]. Wastewater treatment plants may be distinguished by the type of wastewater to be treated. Numerous processes can be used to treat wastewater depending on the extent and type of the contamination. The treatment steps include physical, chemical, and biological treatment processes. The wastewater treatment process commonly screens for particle removal, settling, activated sludge, filtration, disinfection, and oxygen uptake. An activated sludge process is one type of wastewater treatment commonly employed using activated sludge bacteria [2,3,4,5]. This physical wastewater treatment process essentially involves the removal of large non-biodegradable and floating solids that frequently enter wastewater works. Most industrial wastewater treatment uses the physical wastewater treatment process to treat wastewater produced by industries as an undesirable by-product. The primary treatment is the evaporation of inorganic compounds. The method involves the heating of water with a solution. Process heat refers to the application of heat during industrial processes. In some processes, heat is used to manufacture many everyday products. With demanding to minimize energy consumption, alternative treatments are improved for thermal wastewater treatment. To improve sewage treatment, some researchers have paid great attention to developing thermal wastewater treatment by an electrostatic precipitator. It is a device for removing small particles from flowing water using electric force [6,7,8]. An electric heater is an electrical device that converts an electric current into heat and can enhance transport phenomena by lowering the energy supplied.

For the mechanism of the electrostatic precipitator process, a voltage of several thousand volts is applied between the wire electrode and plate ground. If the applied voltage is high enough, an electric corona discharge ionizes the air around the electrodes, which then ionizes the particles in the air stream. Due to the electrostatic force, the ionized particles are diverted toward the grounded plates. As a result, particles build upon the collection plates and are removed from the air stream. Two fluid types have been studied to improve the electrostatic precipitator performance, i.e., gas [9,10,11,12,13,14,15,16] and liquid [17,18,19,20]. The motion of dust particles in electrostatic precipitation depends on the electric field, space charge, gas flow field, and characteristics of dust particles. The induced flow causes the total reaction to lean backward in the plane of rotation; it also reduces the perpendicular component of the total reaction and reduces the total rotor thrust. The first electrostatic precipitator process research was by Yamamoto and Velkoff [21]; they used experimental and theoretical studies of the secondary flow interaction in the electrostatic precipitator. The numerical results demonstrated close agreement with the experimental. Jerzy et al. [9] measured the flow field by the PIV method. The results showed that the PIV method investigates the field in ESP models. The secondary flow can increase the turbulence flow. This electrically generated turbulence is significant due to a strong interaction between the electric charge and the fluid flow. The secondary flows could impact the precipitation of small particles. Audrey et al. [12] investigated the synthesis of gas transformed from biomass gasification to ashes. Considering this issue, the feasibility of removing the particles was studied with an electrostatic precipitator at a high temperature (500 – 1000 °C), and pressure (0.1 – 1 MPa) was studied to develop a filtration process at as high a temperature feasible. It can describe the feasibility of an electrostatic precipitator to clean syngas at temperatures above 500 °C. Beelee et al. [19] investigated the liquid phase in airborne particles’ precipitation functionality for a microfabricated electrostatic precipitator. The microfabricated electrostatic precipitator was placed into a flow control chamber with an aerosol particle from moving the gas streams. The corona current is measured during particle precipitation and can be visual observation. During a 60 s exposure of the oleic acid particles, the circuit resistivity was increased, and the electrode gap was decreased by approximately 50. In addition, the evolution of research in all electrostatic precipitator areas from the past until now is shown in

Table 1. Note: Lam = Laminar flow, Tur = Turbulent flow, N/A = Not available) Evolution of the electrostatic precipitator.

References	Authors	Medium	Initial condition	Application	Methodology
[9]	Jerzy et al.	air	ui = 0.2 m/s, V0 = 24 kV	Particle precipitation process	Experiment
[10]	Lin and	air	Relam	Influence of the EHD	Simulation
	Kazimierz			flow
[11]	Farnoosh et al.	air	Retur	Influence of the EHD	Simulation
[12]	Audrey et al.	ashes particles	Ti = 200 – 400°C,	flow Biomass gasification	Experiment
		into air	Pi = 2-3 MPa
[13]	Hao et al.	wet fuel gas	V0 = 30 kV, Ti = 25°C	control the emission SO2	Experiment
[14]	Taghayi et al.	biofuel into air	ui = 1.9 m/s, V0 = 10 kV, Ti =150°C, Pi = 25 Pa	Biomass gasification	Experiment
[15]	Deylami	SO3 into air	N/A	The removal of sulfuric	Experiment
				acid mist from flue
				gases
[16]	Milad et al.	PM samples into	Qi = 75 LPM,	Toxicological studies	Experiment
[17]	Jingcai et al.	air SO3	V0 = 12 kV V0 = 15 - 35 kV	Control of sulfuric acid	Experiment
[18]	Liton and	soft nanoparticles	Qi = 50 cm3/s,	aerosol emission Industry for removal of	Experiment
	Chandra	into liquid	V0 = 0 - 10 kV	micron-sized particles
				from effluent gases.
[19]	Beelee et al.	polydispersed liquid phase oleic	V0 = 0 - 10 kV	Airborne liquid phase particles precipitation	Experiment
		acid particles
[20]	Islam et al.	aerosol is	Qi = 0.53 g/min,	Device for control of	Experiment+
		produced by	V0 = 8.1 – 16.8 kV	particle emissions	Simulation
		vaporizing
		paraffin oil

.

From a previous literature study, the main group of Electrohydrodynamics (EHD) applications was divided by Saneewong Na Ayuttaya [22]. The first group was the increased flow mechanism for EHD pumping and biomechanics application. The second group was the injection flow mechanism used in the droplet, electrospray, and microfluidics. The third group was the inducing flow mechanism used for electrostatic precipitator and EHD actuator application. The fourth group is the mixing process; it was used in boiling, heat and mass transfer, heat exchanger, and drying applications. For the third group (inducing flow mechanism), this mechanism of inducing flow requires a high Reynolds number (Re) and high electrical voltage removal of the particle into the medium. The experimental study presents most of the research because this machine is suitable for high Re or turbulent Re, so the numerical analysis is usually complex. In the different flow visualization, Re can predict flow patterns [23]. The low Re and high Re could be shown with laminar and turbulent flow, respectively. The turbulence flow appeared with differences in the fluid’s speed and direction. There have been few research studies on the electrostatic precipitator process due to the complex interactions of the turbulent flow model under the electrostatic precipitator.

The flow visualization is necessary to show the control direction of particles because the flow visualization in fluid dynamics is used to make the flow patterns visible to get qualitative or quantitative information on them [24,25]. Nowadays, Computational fluid dynamics (CFD) can solve problems that involve flow visualization and fluid mechanics field; the fluid problems are numerically solved by CFD [26,27]. In this study, the main electrostatic precipitator process involved the phenomena related to the direct conversion of electricity generation into kinetic energy, so heat transfer enhancement on saturated porous samples. This concept of the electrostatic precipitator process is a feasibility study based on the saturated porous samples improvement by heat transfer enhancement through the water channel. Furthermore, the flow visualization under the electrostatic precipitator process is conducted by numerical analysis, and all fields are systematically studied. The flow and temperature fields under the electrostatic precipitator process are numerically analyzed for heat transfer enhancement within the sample. Finally, the sample is the saturated porous medium, and the water channel is solved from the turbulent model (k − ε model).

2. Numerical analyses

To study the interaction for heat transfer enhancement of saturated porous samples in a k-ε turbulent model under the electrostatic precipitator process, the main parts of the numerical analysis were obtained as follows:

2.1. Configuration

The water channel is created from three main parts, an electric field, fluid flow, and the temperature field. The dimension of water channel is X = 15 cm (long) × Y = 2.5 cm (high) and the physical model geometry is shown in Fig. 1. The Y-axis is defined for the channel flow. In fluid mechanics and hydraulics, channel flow is an internal flow in which the confining walls change the hydrodynamic structure of the flow from an arbitrary state at the channel inlet to a certain state at the outlet [28]. From assumption, clays are removed from the water tube for wastewater treatment. Clays are assumed to be saturated porous samples composed of soil and water. The two saturated porous samples are semicircle-shaped; the radius is 1 cm, and they are stuck to the bottom wall of the water channel. The center of the first and second semicircle shapes (saturated porous sample) are fixed at (4, 0) and (8, 0), respectively. The first semicircle shape (front porous sample; T_f) and second semicircle shape (back porous sample; T_b) appear at the front and behind the water channel, respectively. The conjugate method investigates the interface zone between water and sample. Within the water channel, the diameter of the circle electrode (E) is 0.01 cm, and plate ground (G) is 1 cm (long) × 0.1 cm (high). Three-circle electrodes and the one plate ground are suspended in each sample above the saturated porous sample. From a previous study, the electric field can drive from electrode to ground and is dense between electrode and ground zone [29], so a plate ground is installed behind the circle ground when the flow direction moves from left to right. In this setup of the water entering a test section, the uniform initial temperature in the water channel (T_i) is 30 °C, and the initial temperature of the saturated porous sample (Ts) is 10 °C. The electrical voltage (V₀) is varied from 0 to 30 kV, it is incremented by 5 kV; the inlet velocity of water (u_i) is tested in the range of 1 – 2.5 m/s, it is incremented by 0.5 m/s; and the time varies in a range 0 – 1 s, it is incremented by 0.05 s. From Equation 4, the Reynolds number (Re) is composed of characteristic linear dimension (high of water channel), density and viscosity of water and inlet water flow (u_i = 1 - 2.5 m/s). The laminar flow occurs when Re_D < 2300, and turbulent flow occurs when Re_D > 2900 [29]. When u_i = 1 - 2.5 m/s, the rage of Re is 31,172 – 77,930, and Re can be considered in a turbulent flow. It can be seen that the turbulent model (k − ε model) is solved in the water channel. In the simulation, heat transfer of saturated porous samples is enhanced under an electrostatic precipitator. A domain subdivision is used for the simulation, which is made up of the inside water channel, where the electric force, fluid motion, and heat transfer enhancement equations are systematically solved in the k-ε turbulent model. The classical properties of water flow are shown in

Table 2. Classical properties.

[30,31], and the thermal properties of water flow, water (liquid; l), and clay (solid, s) are shown in Table 3 [30,31]. The saturated porous sample consists of clay with only water in the voids.

2.2. Equations

The numerical modeling based on the concept of an electrostatic precipitator process has been developed for the heat transfer enhancement within saturated porous samples in the k-ε turbulent model. Within the water channel, the fluid physical properties are constant and incompressible fluid. Concerning the velocity, a channel profile is imposed at the inlet with a turbulent intensity. The continuity equation (Eq. (1)) and Navier–Stokes equation coupled with the Coulomb force equation (Eq. (2)) are expressed as: where u is water flow velocity, t is time, P is pressure, ρ is the density of water and μ viscosity of water. Where μ_tur is the viscosity of water in a turbulent flow μ_tur/ρ = C_μk²/ε.

The vortex structures or vorticity (ω) using an electrostatic precipitator is described by Saneewong Na Ayuttaya et al. [32]. When high electrical voltage is exposed to fluid flow, the fluid flow is created by ions generated in the corona discharge near the sharp electrode drifting to the ground. As a result, the momentum of fluid flow is enhanced, and the primary flow is generated. The primary flow, ionized air, moves from electrode to ground to induce the shear flow or secondary flow, so-called vorticity. Shear flow then appeared due to the difference in fluid velocity between the charged and uncharged fluid flow.

The strength of the vortex structure subjected to an electric field is assessed through vorticity (ω), as shown in Eq. (3).

The Reynolds number (Re) is an important dimensionless quantity in fluid mechanics used to predict flow patterns in different fluid flow situations, as shown in Eq. (4) below.

The electric force (fE) relating to fluid flow can be expressed in where E is electric field and space charge density (q) is calculated by Griffiths [33]. In this idea, the electric charge is treated as a continuum of charge distributed over a region of the area. The dielectric properties are homogeneity, and the electric field line is driven from the electrode and induced to the ground. The electric field is calculated by using Maxwell’s equations (Eqs. (6), (7), (8) and (9)) shown below: where ε is dielectric permittivity, V is electrical voltage, J is current density, and b is ion mobility.

With heat transfer under the electrostatic precipitator process, the temperature in the water channel is higher than in the saturated porous sample, so the heat transfer has appeared. This idea can be used as implementation guidance for removing dry clay in the water tube under the electrostatic precipitator process. The validity of the following statements, the effect of buoyancy is negligible, and the thermal property using supercritical fluid is constant. The isotropic and homogeneous are used for the saturated porous medium, and the temperature distribution of the water is calculated from the energy equations. The governing equations describing the heat transfer enhancement within the channel flow and the saturated porous medium are calculated from Eqs. (10) and (11), respectively. where T is temperature, k thermal conductivity, and C_p the specific heat capacity. The effective thermal conductivity (k_eff) in a porous medium is calculated by k_eff = (1 − ϕ)κ_s + ϕκ_l [34] (where κ is permeability, ϕ is porosity, and subscript s and l are clay (solid, s) and water (liquid; l) phase). To investigate the local heat transfer coefficient (L_h) (Eq.(12)) within the saturated porous sample surface, the porous medium material is stuck to the bottom of the water channel, and the upper surface is exposed to hot water flow. So local convective heat transfer is defined by the thermal equilibrium equation:

2.3. Boundary condition

The boundary condition of the water channel and saturated porous sample under the electrostatic precipitator process is shown in Fig. 2. For symbol, the normal font for solving an electric field, all of the outer boundary conditions of an electric field are defined with zero charge symmetry (n ⋅D = 0). The boundary conditions of electrode and ground are V = V₀ and V = 0, respectively. In the bold font for solving the flow field, the inlet velocity boundary condition and pressure of outlet boundary condition are considered with zero viscous stress ). This boundary condition specifies vanishing viscous stress along with a Dirichlet condition on the pressure . No-slip boundary conditions are computed for the upper and lower channel flow . In the oblique font for solving the temperature field, the uniform initial temperature of the water channel is defined as T = T_i. The other surface is considered an insulated boundary condition (− n ⋅ (−k∇T) = 0). The interface area between the sample and water in the channel is exposed to water flow. The conjugate method can solve the interface between heat and fluid flow; the interface is solved by .

where u and d are the symbol of the upper and the lower interfaces.

Fig. 2. Boundary condition of the water channel and the saturated porous sample.

Fig. 2. Boundary condition of the water channel and the saturated porous sample.

2.4. Finite element method

Considering that this research investigates heat transfer within samples in water channels using an electrostatic precipitator. Calculations must be performed for the saturated porous medium, and the water channel is solved from the k − ε turbulent model. Therefore, computational fluid dynamics (CFD) using the finite element method (FEM) is conducted to investigate flow and temperature fields under the electrostatic precipitator process. The simulation and conditions are involved for all parameters, with a point of improving the device’s design to electrostatic precipitator process for removing contaminants from water tube. This research involves a complex problem that combines electrostatic and fluid mechanics and heat transfer enhancement in the water channel. This work involves a complex problem that combines electrostatics with fluid mechanics and heat transfer enhancement in the water channel. The electric field is the main parameter for the generation of corona wind. For this purpose, the commercial software (Finite Element Method Electromagnetics) [35,36] is used to assess the various parameters; as a result, the optimal design is selected. In this idea, a finite-dimensional space of some solutions, a number of points in the boundary, and the collocation points are used for the given equation. The water channel flow model is discretized using a triangular element. The Lagrange quadratic approximates each parameter’s electric field, flow field, and temperature distribution variation. The convergence test of the electrical voltage of 30 kV is carried out to identify the suitable numbers of elements required and the relationship between temperature and time from simulations at the interface between the water channel and sample for different meshes. The convergence curve resulting from the convergence test and the relationship between maximum velocity and electrical voltage from the simulation with the different number of meshes is shown in Fig. 3. The system of governing equations is solved with the unsymmetrical multi-frontal method. The number of elements independent of element density was 11,610, and a dimensionless time step of 1 × 10^-4 to ensure numerical stability and accuracy. Therefore, it can be assumed that the accuracy of the numerical results is independent of the number of elements.

3. Results and discussion

The saturated porous sample improvement through the water channel is systematically proposed. The k-ε turbulent model is combined with the electrostatic, flow field, and temperature model. This feasibility study is based on the concept of the electrostatic precipitator process. Where the electrical voltage (V₀) is tested from 0 to 30 kV, it is incremented by 5 kV; the inlet velocity of water (u_i) is tested in the range of 1 – 2.5 m/s, it is incremented by 0.5 m/s; and time vary in a range 0 – 1 s, it is incremented by 0.05 s. The uniform initial temperature of water (T_i) is 30 °C (303 K), and the initial temperature of saturated porous samples (Ts) is 10 °C (283 K). The first part shows the verification of the model for numerical flow visualization. Then, water flow transport within the k-ε turbulent model was compared to the electric field, flow field, pressure field, vorticity field, and temperature field with and without the electrostatic precipitator process. Finally, the saturated porous samples’ flow and temperature fields were compared to the first semicircle shape (front porous sample; T_f) and the second semicircle shape (back porous sample; T_b).

3.1. Verification of the mode

Fig. 4 shows the experimental and numerical setup to verify the accuracy of computational fluid dynamics (CFD) by using FEM to investigate the electric and flow fields. The numerical result is validated with the experimental result. The flow visualization from the experimental setup is generated using the oil smoke to produce a clear image of flow visualization, as shown in Fig. 4 (a). The wind tunnel is mostly made of acrylic plates. The dimension of tunnel is 1.5 m (long) × 0.3 m (high) × 0.3 m (wide). The jet airflow is supplied from a blower (u_i = 1 m/s) and is controlled by the flow meter. The acrylic inclined plate’s (the dimension of the inclined plate is fixed at 15 × 15 cm²) angle (θ) is varied from 0 to 90^o, by 30^o increments. The spotlight LED 500 W is installed at the outlet of the rectangular duct, and the light direction is averse to the flow direction. The high voltage power supply creates a high electrical voltage (V₀). The boundary condition for numerical modeling has been formulated to predict the electric field and flow field within the rectangular duct, as shown in Fig. 4 (b). The present simulated result is solved by the finite element method using the collocation method. From classical properties, ion mobility (b) is 1.8 × 10^-4 m²/Vs, dielectric permittivity (ε) is 8.85× 10^-12 F/m, density (ρ) is 1.2 kg/m³ and kinematics viscosity (η) is 1.56 × 10^-5 m²/s. The experimental result by oil smoke technique and the simulated result by cell Reynolds number are shown in the airflow motion. The cell Reynolds number characteristics or particle trajectories can be taught as the Peclet number. The cell Reynolds number characteristics or particle trajectories can describe the Peclet number. The Reynolds Number is the ratio of inertial force divided by viscous forces, while the Peclet number is the ratio of convective heat transfer divided by diffusive fluxes [37]. In all cases, the air has flowed in the cross-flow direction (left to right), avoids the inclined plate, and is the obstacle. The flow separation has appeared behind the inclined plate due to the weak adverse pressure gradients, and the momentum of upstream forces is weakening.

In case absent electrostatic precipitator process (Fig. 5), the angle of the inclined plate (θ) is varied to show the flow visualization from the experimental (Fig. 5 (a - c)) and numerical setup Fig. 5 (d - f)). To present the simulated result, the flow visualization is shown both streamline and surface boundaries. For (θ = 15^o (Fig. 5 (a)), the airflow is moved through the inclined plate, and airflow is not turbulent, but the airflow is turbulent with the inclined plate at a 45^o angle of attack (Fig. 5 (b)). The airflow is strongly turbulent in the case of the high angle of the inclined plate, as shown in Fig. 5 (c). The flow visualization by oil smoke technique (Fig. 5 (a - c)) is a similar trend to simulation results (Fig. 5 (d - f)). The flow visualization in case of θ = 15^o (Fig. 5 (a)), θ = 45^o (Fig. 5 (b)) and θ = 75^o (Fig. 5 (c)) are similar trend in case of θ = 15^o (Fig. 5 (d)), θ = 45^o (Fig. 5 (e)) and θ = 75^o (Fig. 5 (f)), respectively.

With the electrostatic precipitator process (Fig. 6), the flow visualization of the experimental setup (Fig. 6 (a-c)) is compared with the simulated result (Fig. 6 (d - f)). The electrical voltage and the jet airflow are fixed at V₀ = 10 kV. The airflow in case of θ = 45^o (Fig. 6 (b)) is more turbulent than in case of θ = 15^o (Fig. 6 (a)). In the case of the high angle of the inclined plate (θ = 75^o (Fig. 6 (c)), the airflow is extremely turbulent. The flow visualization in case of θ = 15^o (Fig. 6 (a)), θ = 45^o (Fig. 6 (b)) and θ = 75^o (Fig. 6 (c)) are similar trend with in case of θ = 15^o (Fig. 6 (d)), θ = 45^o (Fig. 6 (e)) and θ = 75^o (Fig. 6 (f)), respectively. It can be seen that a similar trend of the airflow distribution has appeared in both cases.

3.2. Effect of electric field distribution

An electric field is a physical field composed of electrically charged particles and can affect all other charged particles in the field. The electric field within the water channel under the k-ε turbulent model is shown in Fig. 7.

In the case of the absent electrostatic precipitator process (Fig. 7 (a)), the electric field and the electric potential zone have not appeared. Instead, the electric field and potential zone appear and are concentrated in the case of using the electrostatic precipitator process. The electric field moves outward from the tip of circle electrodes (E₁, E₂, E₃, E₄, E₅, E₆) to plate grounds (G₁, G₂), and the electric field lines are turbulent at both of electrode and ground zone [32]. The electric potential zone is more expanded and more turbulent with increasing the electrical voltage, as shown in Fig. 7 (b) – 7 (d)). In addition, the semicircle shapes are considered for a feasibility study on the development of saturated porous samples improvement. The circle electrodes (E₁, E₂, E₃) and plate ground (G₁) is set up for the first semicircle shape (front porous sample; T_f), and the circle electrodes (E₃, E4, E6) and the plate ground (G₂) is set up for the second semicircle shape (back porous sample; T_b). From Fig., some electric field lines of E4 and E5 are moved to G₁ and G₂. Typically, the electric field lines are moved to the shortest distance, but the strength of electrical voltage can disturb, so the electric fields are moved to both grounds. According to the above data, the disturbance of electric field distribution can concentrate more electric potential, resulting in a turbulent area appearing. Fig. 8 shows the value of an electric field at the line of the cross-section of the X- axis through the middle of the water channel. It can be seen that the first position or edge of the ground is accumulated the high electric force, so the maximum value of the electric field appears. The electric field zone is influencedaround the electrode and ground zone above the saturated porous sample area. Therefore, the value of an electric field is increased with the electrical voltage increasing. Furthermore, the maximum value of X - electric field (E_x) and Y - electric field (E_y) are increased with the electrical voltage increasing, as shown in Fig. 9. A graph of the maximum value of X- electric field (E_x) is related to Y - electric field (E_y). The maximum value of X - electric field (E_x) and Y- electric field (E_y) show a variable straight line indicating the electrical voltage (E_max ∝ E_x ∝ E_y ∝ V₀).

Fig. 8. Value of an electric field at cross sections line of the X-axis through the middle of the water. channel (X_i, Y_i = 0, 0.0125 and X_j, Y_j = 0.15, 0.0125).

Fig. 8. Value of an electric field at cross sections line of the X-axis through the middle of the water. channel (X_i, Y_i = 0, 0.0125 and X_j, Y_j = 0.15, 0.0125).

Fig. 9. Maximum value of electric field (Emax) in various electrical voltage consider both of X- electric field (Ex) and Y- electric field (Ey).

Fig. 9. Maximum value of electric field (Emax) in various electrical voltage consider both of X- electric field (Ex) and Y- electric field (Ey).

3.3. Effect of flow field distribution

The eulerian specification specifies the flow field, and fluid motion is focused on the time the fluid move pass. the flow visualization is observed with the fluid passing the fixed location; visualized by observing the test section and watching the fluid pass the fixed location. In this study, the flow field in fluid mechanics is a space or region with a distribution of physical quantity. This water flow quantity exists at each point in this physical space. In the absent electrostatic precipitator process, the flow field within the water channel under the k-ε turbulent model is shown in Fig. 10. The flow field is moving from the left to the right and avoiding both saturated porous samples (first semicircle shape (front porous sample; T_f) and second semicircle shape (back porous sample; T_b)). The maximum flow field zone has appeared above the saturated porous sample area, and the maximum velocity field increases with inlet velocity. In the case of high inlet velocity (Fig. 10 (b)), the maximum velocity zone is higher than in cast of low inlet velocity (Fig. 10 (a)).

The flow field using the electrostatic precipitator process (V₀ = 20 kV) is shown in Fig. 11. The three cells of circulating fluids have appeared within the water channel under the k-ε turbulent model. The big cell circulated counterclockwise for the first cell, appearing above the first semicircle shape (front porous sample) zone. For the second cell, the small cell circulated clockwise, and it appears between the saturated porous samples (first semicircle shape (front porous sample) and second semicircle shape (back porous sample)). The big cell circulated in the clockwise direction for the third cell, appearing above the second semicircle shape (back porous sample) zone. In addition, the main effect of the first and the third cells is the shear flow effect. The shear flow is influenced by differences in velocities for the fluid caused by fluid charging effects [36]. The charged fluid flow from primary flow leads to secondary flow. The electric force is influenced by the first and the third cells, but the initial force from the pore between the first and second semicircle shape is influenced by the second cell. The three cells in case of high inlet velocity (u_i = 2.0 m/s) (Fig. 11 (b)) are stronger and more expanded than the three cells in case of low inlet velocity (u_i = 1.0 m/s) (Fig. 11 (a)).

As the electrostatic precipitator process increases (electrical voltage increases), inlet velocity is fixed at u_i = 1.5 m/s, as shown in Fig. 12. The maximum flow has appeared around the electrode and ground zone. Fig. 12 (a)) shows the low electrical voltage (V₀ = 10 kV); the two cells have appeared within the water channel under the k-ε turbulent model. The two cells are circulated clockwise; the first and the second cells appear between the first and the second semicircle shapes and behind the second semicircle shape, respectively. Fig. 12 (b)) shows the medium electrical voltage (V₀ = 20 kV); the three cells have appeared within the water channel under the k-ε turbulent model. The first cell circulated counterclockwise and appeared above the first semicircle shape zone. The second and the third cells are circulated clockwise, and they have appeared behind both of the second semicircle shapes. In the case of high electrical voltage (V₀ = 30 kV), the four cells are circulated, as shown in Fig. 12 (c). The first big cell is circulated counterclockwise, appearing above the first semicircle shape zone. The second cell is circulated in the clockwise direction, and it has appeared between the first and the second semicircle shapes. The third biggest cell is circulated in the clockwise direction, appearing behind the back semicircle shape zone. From the above data, the main effect of the first and third cells in the shear flow and the main effect of the second cell is the initial force. As a result, the small fourth cell is circulated on the top of the water channel under the k-ε turbulence model, which is influenced by the initial force of the first cell. It can be seen that the increasing electrical voltage influences the strength of the electric force. The high electric voltage can increase disturbance and turbulence within the water channel. The above result showed that electric force is induced by the electric field (Fig. 7), resulting in the circulating flow that can induce fluid flow (Fig. 11 and Fig. 12) within the water channel under the k-ε turbulence model. Increasing electric field (Fig. 12 (c)) causes the strength of flow field distribution (Fig. 7 (d)).

Fig. 13 and Fig. 14 show the relation between inlet velocity and electrical voltage. The inlet velocity of water (u_i) is tested in the range of 1 – 2.5 m/s, it is incremented by 0.5 m/s, and the electrical voltage (V₀) is varied from 0 to 30 kV, is incremented by 5 kV. The maximum velocity (umax) in various inlet velocities and electrical voltage is considered in Fig. 13 (a). The maximum velocity is increased with inlet velocity and electrical voltage increasing. The maximum velocity ratio ((u_R = umax/ u_i) is shown in Fig. 13 (b); it can be seen that in the case of inlet velocity is high, the maximum velocity ratio is higher than that case of inlet velocity is low. The maximum velocity ratio in the case of high electrical voltage is lower than in the case of low electrical voltage. The maximum velocity (u_max) and maximum velocity ratio (u_R) are related to inlet velocity. The maximum velocity (u_max) graphs show a variable straight line indicating the inlet velocity increasing. In other words, the maximum velocity ratio (u_R) is inversely proportional to inlet velocity increasing. The correlation of data is u_max ∝ u_i ∝ 1/u_R).

Fig. 14 (a) shows that the maximum velocity (umax) is increased with electrical voltage and inlet velocity increasing. The maximum velocity ratio (uR) increases with the electrical voltage but decreases with inlet velocity increases, as shown in Fig. 14 (b). The graphs of maximum velocity (umax) and the maximum velocity ratio (uR) are variable exponential lines for the electrical voltage (V₀), umax ∝ uR ∝ V ². The maximum velocity ratio in case of V₀ = 0 kV and u_i = 2.5 m/s (Fig. 13 (b)) and V₀ = 30 kV and u_i = 1.0 m/s (Fig. 14 (b)) are about 1.90 and 2.96 times, respectively. The maximum velocity ratio in case of V₀ = 30 kV and u_i = 1.0 m/s is higher than in case of V₀ = 0 kV and u_i = 2.5 m/s. As can be seen that electric force is superior to inertial force.

3.4. Effect of pressure field distribution

The pressure field within the water channel with and without the electrostatic precipitator process is compared in Fig. 15. The pressure field is a general field component, represented in the Lagrangian and Hamiltonian of an arbitrary physical system, including the term with the energy of particles in the pressure field and the term with field energy. The upstream pressure is generally greater than the downstream pressure. The pressure field within the water channel under the k-ε turbulent model is not concentrated in case of an absent electrostatic precipitator process (Fig. 15 (a - b)). The contour lines are swirl around the porous samples and the pressure field in case of low inlet velocity (u_i = 1.5 m/s) (Fig. 15 (a)) and high inlet velocity (u_i = 2.5 m/s) (Fig. 15 (b)) are gradually constant in both cases. In case of electrostatic precipitator process (V₀ = 15 kV (Fig. 15 (c - d)), the contour lines are move above porous samples and the upstream pressure field in case of low inlet velocity (u_i = 1.5 m/s) (Fig. 15 (c)) and high inlet velocity (u_i = 2.5 m/s) (Fig. 15 (d)) are greater than the downstream pressure. The pressure field with the electrostatic precipitator process is more concentrated than without. It can be seen that the electrostatic precipitator process is more influenced than the inlet velocity within the water channel under the k-ε turbulent model. The maximum pressure (P_max) in various inlet velocities (u_i) and electrical voltage (V₀) is shown in Fig. 16. For Fig. 16 (a), the maximum pressure is increased with the electrical voltage increasing, but it is little increased with inlet velocity increasing. On the other hand, the maximum pressure increased as the voltage increased, and it is a little different with various inlet velocities, as shown in Fig. 16 (b). However, higher inlet velocity and high electrical voltage can induce more pressure than in other cases. The graph of maximum pressure (P_max) is the variable exponential line for the electrical voltage (V₀), P_max ∝ V ².

3.5. Effect of vorticity field distribution

The vorticity field within the water channel with and without the electrostatic precipitator process is compared in Fig. 17. The vorticity is a pseudovector field that can describe the trend and direction of particle rotation. Conceptually, vorticity could be determined by marking parts of a continuum in a small neighborhood of the point in question and watching their relative displacements as they move along the flow. In all cases, the contour of vorticity is swirling and turbulent around the porous samples. The vorticity contour with the electrostatic precipitator process (Fig. 17 (b - c)) is more concentrated than without (Fig. 17 (a)). In addition, high inlet velocity and electrical voltage can more disturb the contour of vorticity than low inlet velocity and absent electrical voltage. The vorticity would be twice the mean angular velocity vector of those particles relative to their center of mass, oriented according to the right-hand rule. The value of vorticity in various inlet velocities and electrical voltage is shown in Fig. 18. The maximum vorticity (Ꞷ_max) is increased with inlet velocity (Fig. 18 (a)) and electrical voltage increasing (Fig. 18 (b)). The graph of maximum vorticity (Ꞷ_max) is a variable straight line for inlet velocity increasing (u_i), and it is the variable exponential line for the electrical voltage (V₀), Ꞷ_max ∝ u_i ∝ V ². But the minimum vorticity (Ꞷ_max) is decreased with electrical voltage and inlet velocity increasing (Fig. 18 (c)).

3.6. Effect of temperature field distribution

The graph of the isothermal surface is explained in terms of the temperature field; all points of the same temperature are connected in the field. The temperature field within the water channel with and without the electrostatic precipitator process is compared in Fig. 19. The heat transfer from water transport through the sample, so the temperature within the porous sample gradually increases. Temperature field due to time-dependent heat sources in a large rectangular grid. With absent electrostatic precipitator process, u_i = 1.0 m/s and first stage (t = 0 s) (Fig. 19 (a)), uniform initial temperature of water (T) is 30 °C (303 K) and initial temperature of saturated porous samples (Ts) are 10 °C (283 K). As time progresses (t = 0.5 s) (Fig. 19 (b)), heat from water transfers into the sample, so the temperature within the porous sample gradually increases. Within water channel under the k-ε turbulent model, the contour line of temperature in case of high inlet velocity, electrical voltage and time (u_i = 2.5 m/s, V₀ = 30 kV and t = 0.6 s) (Fig. 19 (d)) can more disturb than in case of low inlet velocity, electrical voltage and time (u_i = 1.0 m/s, V₀ = 5 kV and t = 0.2 s) (Fig. 19 (c)). Fig. 20 shows the relationship between the interface zone (center of surface temperature) of saturated porous samples and time with various electrical voltage and u_i = 1.0 m/s. The first semicircle shape (front porous sample; T_f) and second semicircle shape (back porous sample; T_b) are shown in Fig. 20 (a) and Fig. 20 (b), respectively. The graph of a parabolic function is shown at the interface zone of temperature. As time progresses, the heat is transferred from water to the sample, so the water flow temperature is decreased. In the final process, the temperature of the sample is closest to the airflow temperature. As a result, it is almost the thermal equilibrium. Therefore, the surface temperature of high electrical voltage (V₀ = 30 kV) is higher than in the case of absent electrical voltage (V₀ = 0 kV), as shown in Fig. 20 (c). In addition, increasing the electrical voltage and inlet velocity speed up the sample’s surface temperature to close to the water flow temperature.

4. Heat transfer enhancement on saturated porous samples using electrostatic precipitator process

This section investigates the flow and temperature fields with the improvement of thermal efficiency within saturated porous samples. This feasibility study is based on the concept of the electrostatic precipitator process; the electrical voltage (V₀) and time are varied from 0 to 30 kV and 0 – 1 s. The inlet velocity of water (u_i) is compared with 1 and 2.5 m/s. From the initial time of the k-ε turbulent model (t = 0 s), the uniform initial temperature of water (T) is 30 °C (303 K), and the initial temperature of saturated porous samples (T) is 10 °C (283 K). The inlet velocity is fixed at u_i = 2.0 m/s, as shown in Fig. 21 and Fig. 22. Fig. 21 shows the flow field within saturated porous samples in various electrical voltages. The flow can move from the left to right through both saturated porous samples and is concentrated with high electrical voltage. In all cases, swirling flow is circulated clockwise between the front and back porous samples due to the space between samples. In cases of low electrical voltage (V₀ = 0 – 5 kV) (Fig. 21 (a - b)), fluid velocity within the front porous sample is similar to fluid velocity within the back porous samples. But fluid velocity within the front porous sample is higher than (V₀ = 10 – 15 kV) (Fig. 21 (c - d)). In cases of high electrical voltage (V₀ = 20 – 30 kV) (Fig. 21 (e - g)), fluid velocity within the front porous sample is fluid velocity within the back porous sample, as shown in cases of medium electrical voltage higher than fluid velocity within the back porous sample. The fluid velocity of the left side is more than the right side of the front porous medium, and the fluid velocity of the left and right sides of the back porous medium is rather steady.

From the above date, the flow can move from the left to right direction within the samples, and it can induce temperature within both saturated porous samples, as shown in Fig. 22. The temperature field in case of V₀ = 0 – 30 kV (Fig. 22 (a - g)) is increased by flow field in Fig. 21 (a - g), respectively. In addition, the inlet velocity of the water channel under the k-ε turbulent model (u_i) is compared with 1 and 2.5 m/s, as shown in Fig. 23. In t = 0.05 s, the temperature within the saturated porous sample surface between the first semicircle shape (front porous sample; T_f) and the second semicircle shape (back porous sample; T_b) is compared in various electrical voltage and initial velocity. In all cases, the value of the temperature of the interface zone (center of surface temperature) in the case of high inlet velocity (u_i = 2.5 m/s) is higher than in the case of low inlet velocity (u_i = 1.0 m/s). The temperature value in the first semicircle shape (front porous sample; T_f) is higher than in the case of the second semicircle shape (back porous sample; T_b). In addition, the graph of the value of temperature (T) is a variable straight line for electrical voltage increasing (V₀), T∝ V₀.

Fig. 24 compares the local heat transfer coefficient ratio (h_R) with t = 0.1 s between the electrical voltage in various inlet velocities within the saturated porous sample. In mechanics and thermodynamics, the heat transfer coefficient is the proportionality constant between the heat flux and the thermodynamic driving force for the flow of heat. The local heat transfer coefficient (hL) at a particular point on the heat transfer surface is equal to the local heat flux, at the same point, per the local temperature drop. The first semicircle shape (front porous sample; T_f) and second semicircle shape (back porous sample; T_b) are investigated in Fig. 24 (a) and 24 (b), respectively. The local heat transfer coefficient ratio (T) is referred to as the local heat transfer coefficient when using the electrostatic precipitator process (EP) per local heat transfer coefficient in the absent electrostatic precipitator process (no EP), i.e., h_L,EP/h_L,noEP. The heat transfer within the samples is enhanced by the fluid flow in the water channel, so the local heat transfer coefficient within the samples is induced by fluid velocity in the water channel.

The graph of the local heat transfer coefficient ratio (h_R) (Fig. 24) shows a similar trend to the fluid velocity ratio (Fig. 14 (b)). The local heat transfer coefficient ratio decreases with inlet velocity increasing and increases with the electrical voltage increases. The local heat transfer coefficient ratio graphs are variable exponential lines for the electrical voltage (V₀), h_L ∝ V₀²; this is because the fluid velocity within the front porous sample is higher than the back porous sample. Consequently, the first semicircle shape’s local heat transfer coefficient ratio (front porous sample; T_f) is higher than the second semicircle shape (back porous sample; T_b).

5. Conclusions

The heat transfer enhancement on saturated porous samples is numerically investigated using the electrostatic precipitator process. The water channel is solved in this study using the k-ε turbulent model. In the first part, the simulated result of flow visualization is validated against the experimental result. It can be seen that a similar trend of the airflow distribution has appeared in both cases.

For the second part, the disturbance of electric field distribution can create a more concentrated electrical potential, resulting in the turbulent area appearing. The flow field moves from the left to the right direction, avoiding both saturated porous samples. The electric force is induced from the electric field resulting in the circulating flow that can induce the fluid flow within the water channel under the k-ε turbulence model. The pressure field in the case of the electrostatic precipitator process is more concentrated than in the absent case. In addition, higher inlet velocity and high electrical voltage can induce greater pressure than in other cases. The contour of vorticity is swirling and turbulent around the porous samples. The contour line of temperature in case of high inlet velocity, electrical voltage, and time can be more disturbing than in the case of low inlet velocity, electrical voltage, and time.

The flow and temperature fields were investigated with heat transfer enhancement within saturated porous samples for the last part. The flow can move from the left to right direction within both saturated porous samples and induce temperature within both. The value of the temperature in the case of the first semicircle shape is higher than in the case of the second semicircle shape. The heat transfer within samples is enhanced from fluid flow in the water channel, so the local heat transfer coefficient within samples is induced from fluid velocity in the water channel.