# Joule Heating Effects in Electrokinetic Remediation: Role of Non-Uniform Soil Environments: Temperature Profile Behavior and Hydrodynamics

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## Abstract

**:**

## 1. Introduction

## 2. Problem Formulation of the Heat Transfer Model

**R**indicates the electrical resistance of the physical domain. Q specifies the generation of heat due to Joule heating [22]. As mentioned before, the function Q is assumed uniform and constant within the bulk of the domain of the capillary. The thermal conductivity of the capillary domain is given by k, the temperature is represented by T (x,y) which is a function of x and y within the domain of the capillary. The direction “z” (see Figure 1) is not relevant because we assume “symmetry” in that direction. Also for the domain, geometry is assumed that L >> H and W >> H. From the geometric point of view, the domain can be thought as of a bounded domain with two parallel surfaces, one located at y = 0 and the other one, at y = H.

_{H}and h

_{0}have been used to indicate heat transfer coefficients at the position y = H and y = 0, respectively. Now, by following dimensionless variables defined in Equation (4), Equation (7) becomes:

## 3. Formulation of the Hydrodynamic Model

- (1)
- Identifying Geometry: The geometry used in this study is of rectangular shape with dimensions L, H, W (Figure 1).
- (2)
- Selection of the Coordinate System: The coordinate system is “anchored” in one of the vertices of the capillary domain to simplify the calculation of the flow rate (or flow) (Figure 1).
- (3)
- Kinematics of Flow: Components of the velocity profile are described. Therefore, for the case under analysis:
- a.
- ${V}_{x}\ne 0$ (the flow of the fluid is in the axial direction only since ends effects are neglected)
- b.
- ${V}_{y}=0$ (there is no net flow perpendicular to the axial flow in the domain)
- c.
- ${V}_{z}=0$ (there is no net flow in the direction perpendicular to the xy plane of the domain)Based on the description above, we can postulate that the hydrodynamics is given by$\underset{V}{\to}$ (v
_{z}, 0, 0) as the unidimensional vector.

- (4)
- Boussinesq Approximation: This approximation uses the incompressibility condition everywhere except in the buoyancy term [24]. By using this approximation, all the effects of temperature on the system and due to Joule heating are considered only in the density variation with the temperature. Other properties such as viscosity and the heat capacity are considered constant in the model equations. Basically, the fluid behaves as incompressible, i.e., as it is assumed that there is no density variation in the case of conservation of total mass and, therefore, the continuity equation with the divergence of the vector field equated to zero is valid. This condition leads to the following conclusion:$${V}_{x}=f(y)$$
- (5)
- Application of the Conservation of Momentum: Since the buffer is assumed to exhibit Newtonian fluid behavior, the Navier-Stokes equation is valid [21]. In the axial flow direction and with the assumption indicate above, the Navier-Stokes equation for the x-component is:$$-\mu \frac{{d}^{2}{V}_{x}}{d{y}^{2}}=\frac{\partial p}{\partial x}+\rho (T){g}_{x}$$

## 4. Solution and Obtaining Both the Heat Transfer and Velocity Profile Equations

#### 4.1. Solution to Differential Model: The Distinct Heat Transfer Regimes

_{1}and C

_{2}are +integration constants. These constants can be determined using the boundary conditions (8) to find that:

_{1}and C

_{2}can be rewritten as:

#### 4.2. Solution and Obtaining the Velocity Profile Equation

_{1}and C

_{2}, of which are functions of the R and Nu numbers. C

_{1}and C

_{2}are given by Equations (23) and (24). This transformation gives the following modified Navier-Stokes equation as a function of the Grashof number, Gr, the Joule heating number, ${\Phi}^{2}$, and the medium temperature $\overline{\Theta}$ that will be determined later in Section 5 (see below):

_{1}and D

_{2}are two integration constants to be determined using the non-slip boundary conditions,

_{1}, I

_{2}, y I

_{3}such as:

- Case 1: Symmetrical case, (R = 1) i.e., both Nu values at the different walls are the same
- Case 2: Adiabatic case, (R = 0) at the wall positioned at $\eta =1$. No heat flux is present at this location.
- Case 3: Well-mixed case, (R → ∞) at the wall positioned at $\eta =1$. This case has an excellent heat transfer at the position.

## 5. Different Heat Transfer and Hydrodynamic Regimes: Analysis, Numerical Illustration, and Discussion

#### 5.1. Case 1: Symmetrical Case

#### 5.1.1. Case 1: Applied to the Heat Transfer Regime

#### 5.1.2. Case 1: Applied to the Hydrodynamic Regime

#### 5.2. Case 2: Adiabatic Case for the Capillary Wall Located at $\eta =1$

#### 5.2.1. Case 2: Applied to the Heat Transfer Regime

#### 5.2.2. Case 2: Applied to the Hydrodynamic Regime

#### 5.3. Case 3: Well Mixed Case at the Wall Located at $\eta =1$

#### 5.3.1. Case 3: Applied to the Heat Transfer Regime

#### 5.3.2. Case 3: Applied to the Hydrodynamic Regime

#### 5.4. Case 4: Favorable Heat Transfer at the Wall Located at $\eta =1$ When R > 1

#### 5.4.1. Case 4: Applied to the Heat Transfer Regime

#### 5.4.2. Case 4: Applied to the Hydrodynamic Regime

#### 5.5. Case 5: Unfavorable Heat Transfer at the Wall Located at $\eta =1$, (R < 1)

#### 5.5.1. Case 5: Applied to the Heat Transfer Regime

#### 5.5.2. Case 5: Applied to the Hydrodynamic Regime

## 6. Average (Mean) Temperature Analysis of the Capillary

- (a)
- Symmetrical Case: R = 1, F(R = 1, Nu) = 1, gives:$${\overline{\Theta}}_{r}^{S}=\frac{3}{10}+(\frac{Nu-2}{2Nu})$$
- (b)
- Adiabatic Case for the Wall located at $\eta =1$. For this case $R=0$,$F(R=0,Nu)=2$ and it leads to:$${\overline{\Theta}}_{r}^{A}=\frac{3}{10}+[\frac{Nu-2}{Nu}]$$
- (c)
- Well-Mixed Case for the Wall Located at $\eta =1$. For this case ($R\to \infty $), we have, $F(R\to \infty ,Nu)=\frac{Nu}{1+Nu}$, and it leads to:$${\overline{\Theta}}_{r}^{WM}=\frac{3}{10}+\frac{(Nu-2)}{2(1+Nu)}$$Regarding the general case, R < 1 and R > 1, these can be obtained directly from Equation (68).

## 7. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Nomenclature

A,B,C | Parameters |

C_{1},C_{2} | Integration constants |

D_{1},D_{2} | Integration constants |

E_{x} | Electric field in the axial direction (Vm^{−1}) |

F | Function of the ratio between the Nusselt numbers at both walls of the capillary and Nusselt = $\left[\right(RNu+2)/(1+R+RNu\left)\right]$ |

G | Gravity (m s^{−2}) |

Gr | Grashof number $Gr=\frac{g\beta (T-{T}_{\infty}){L}^{3}}{{\upsilon}^{2}}$ |

H | Heat transfer coefficient (Wm^{−2} K^{−1}) |

H | Capillary height (m) |

I | Current (A) |

$\hat{{\mathrm{I}}_{1}},\hat{{\mathrm{I}}_{1}},\hat{{\mathrm{I}}_{1}}$ | Integrals of currents |

K | Thermal conductivity (Wm^{−1}K^{−1}) |

L | Capillary length (m) |

Nu | Nusselt number = [hL/k] |

Q | Electric charge (C) |

Q | Heat generation due to Joule heating |

R | Electrical resistance (Ω) |

R | Ratio between the Nusselt numbers at both walls of the capillary = [Nu^{1}/Nu^{0}] |

T | Time (s) |

T | Temperature (K) |

T_{∞} | Temperature outside of the capillary domain (K) |

V_{x} | Dimensionless velocity |

W | Capillary depth (m) |

$\widehat{x}$ | Dimensionless capillary length |

x,y,z | Coordinates (m) |

## Greek Symbols

α | Inclination angle of capillary with respect to the orientation of gravity |

η | Dimensionless capillary height |

ρ | Fluid density (kg m^{−3}) |

$\overline{\rho}$ | Average density |

β | Volumetric thermal expansion coefficient |

$\Theta $ | Dimensionless temperature = [(T − T_{∞})/T_{∞}] |

${\Theta}_{}^{R}$ | Dimensionless reduced temperature |

$\overline{\Theta}$ | Average dimensionless temperature |

Φ | Dimensionless Joule heating generation = [QH^{2}/k T_{∞}] |

## Sub-Indexes

0 | Indicates any dimensional and/or non-dimensional variable located at the capillary wall y = 0 |

H | Indicates any variable located at the capillary wall y = H |

1 | Indicates any dimensionless variable located at the capillary wall |

S | Symmetrical |

A | Adiabatic |

WM | Well-mixed |

r | Reduced |

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**Figure 1.**A sketch of the capillary domain in rectangular coordinates showing key dimensions and coordinate system, showing the length (L), width (H), depth (W) and an inclination of an angle α (0–90°) with respect to the orientation of gravity “g”.

**Figure 2.**Behavior of the reduced non-dimensional temperature profiles vs. the Nusselt number: symmetric case (R = 1).

**Figure 3.**Reduced Non-dimensional Velocity Profile for Symmetrical Case (R = 1). (Velocity values have been scaled ×10

^{3}).

**Figure 4.**Behavior of the non-dimensional reduced temperature profiles vs. the Nusselt number: Adiabatic Case (R = 0).

**Figure 5.**Reduced non-dimensional velocity profile for adiabatic Case ($R=0$). (Velocity values have been scaled ×100).

**Figure 6.**Behavior of the reduced non-dimensional temperature profiles vs. the Nusselt number: Well-Mixed Case (R → ∞).

**Figure 7.**Reduced Non-dimensional Velocity Profile for Well-Mixed Case (R → ∞) and with different values for the Nu numbers (Velocity values have been scaled ×10

^{3}).

**Figure 8.**Behavior of the non-dimensional reduced temperature profiles vs. the Nusselt Number: Heat Transfer Favorable Case at $\eta =1$, (R > 1), for different values for R (3/2, 2, 5, and 10). (

**a**–

**f**), correspond to the different values for Nu of 2, 4, 6, 8, 10 and 50, respectively.

**Figure 9.**Reduce non-dimensional Velocity Profile: Heat Transfer Favorable Case for R > 1 and Nu = 5, 15, and 50 (Velocity values have been scaled ×10

^{3}). (

**a**–

**d**), correspond to the different values for R of 1.5, 5, 10 and 100, respectively.

**Figure 10.**Behavior of the non-dimensional reduced temperature profiles vs. the Nusselt number at values of R (0.001, 0.1, 0.05, and 0.5): Heat Transfer Unfavorable Case at $\eta =1$, (R < 1). (

**a**–

**f**), correspond to the different values for Nu of 2, 4, 6, 8, 10 and 500, respectively.

**Figure 11.**Non-dimensional Reduced Velocity Profile: Heat Transfer Unfavorable Case for R < 1 and Nu = 5, 15, 50 (Velocity values have been scaled ×10

^{3}). (

**a**–

**d**), correspond to the different values for R of 0.001, 0.01, 0.1 and 0.8, respectively.

a. Non dimensional Navier Stokes equation and boundary conditions [21]: | |

$\frac{{d}^{2}{V}_{x}}{d{\eta}^{2}}=-Gr\mathrm{sin}(\alpha )[\Theta (\eta )-\overline{\Theta}]$ | (16) |

${V}_{x}(\eta =0)={V}_{x}(\eta =1)=0$ | (17) |

b. Non dimensional function of density and temperature derived from the Taylor approximation [21]: | |

$\frac{\rho (\Theta )}{\overline{\rho}}=1-\beta (\Theta -\overline{\Theta})$ | (18) |

c. Heat transfer model, where the function is defined by: | |

$F(R,Nu)\equiv \frac{(RNu+2)}{(1+R+RNu)}$ | (19) |

d. Non dimensional total mass conservation [21]: | |

$\underset{0}{\overset{1}{\int}}\rho (\overline{\Theta}){V}_{x}(\eta )d\eta =0$ | (20) |

e. Mean value theorem for integration: | |

$\underset{0}{\overset{1}{\int}}{V}_{x}(\eta )d\eta =0$ | (21) |

**Table 2.**General and Limiting Cases of Heat Transfer and Hydrodynamic Regimes Based on the Parameter R Values.

Case | R-Value | F-Value | Description | Comments (Heat Transfer Regime) | Comments (Hydrodynamic Regime) |
---|---|---|---|---|---|

1 | 1 | F(R, Nu) = 1 | Symmetrical (limiting) case with same Nu values at both boundaries | The profile will have a “symmetrical” shape with the maximum temperature located at the center of the capillary | The velocity profile will have a “symmetrical” shape with the maximum velocity (upward) located at the center of the capillary |

2 | 0 | F(R, Nu) = 2 | Adiabatic (limiting) case at the wall located at $\eta =1$ | The temperature located will be located at the boundary located at $\eta =1$ | The velocity profile will show two regions: one upward positioned near the boundary located at $\eta =1$ and the other one downward, located near the wall. at $\eta =0$ |

3 | ∞ | F(R, Nu) = Nu/(1 + Nu) | Well-mixed (limiting) case at the wall located at $\eta =1$ | The temperature at the wall located at $\eta =1$ will reach the environmental temperature | The velocity profile will show two regions: One upward located near at the wall at $\eta =0$ and the other downward located near the wall at $\eta =1$ |

4 | R > 1 | $F(R,Nu)\equiv \frac{(RNu+2)}{(1+R+RNu)}$ | General case where the boundary at $\eta =1$ has a “favorable” heat transfer rate than the one at $\eta =0$ | These cases will show temperature profiles similar to Case 3 but the temperature will not reach the environmental one | These cases will show hydrodynamic velocity profiles with behaviors between Case 1 and Case 3 |

5 | R < 1 | $F(R,Nu)\equiv \frac{(RNu+2)}{(1+R+RNu)}$ | General case where the boundary at $\eta =1$ has an “unfavorable” heat transfer rate than the one at $\eta =0$ | The general shape of the temperature profiles will follow Case 2 showing a “slower” heat transfer rate at $\eta =1$ | These cases will show hydrodynamic velocity profiles with behaviors between Case 2 and Case 1 |

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**MDPI and ACS Style**

Torres, C.M.; Arce, P.E.; Justel, F.J.; Romero, L.; Ghorbani, Y.
Joule Heating Effects in Electrokinetic Remediation: Role of Non-Uniform Soil Environments: Temperature Profile Behavior and Hydrodynamics. *Environments* **2018**, *5*, 92.
https://doi.org/10.3390/environments5080092

**AMA Style**

Torres CM, Arce PE, Justel FJ, Romero L, Ghorbani Y.
Joule Heating Effects in Electrokinetic Remediation: Role of Non-Uniform Soil Environments: Temperature Profile Behavior and Hydrodynamics. *Environments*. 2018; 5(8):92.
https://doi.org/10.3390/environments5080092

**Chicago/Turabian Style**

Torres, Cynthia M., Pedro E. Arce, Francisca J. Justel, Leonardo Romero, and Yousef Ghorbani.
2018. "Joule Heating Effects in Electrokinetic Remediation: Role of Non-Uniform Soil Environments: Temperature Profile Behavior and Hydrodynamics" *Environments* 5, no. 8: 92.
https://doi.org/10.3390/environments5080092