A Review of Particle Removal Due to Thermophoretic Deposition

*Yonggang Zhou, Mingzhou Yu and Zhandong Shi*

### **Abstract**

Thermophoretic deposition is an important technique for particle removal. The thermophoretic force of the particles under an appropriate temperature gradient can achieve a good particle removal effect. At present, there have been many studies on the deposition mechanism of ultrafine particles under the action of thermophoresis. In this chapter, the development history and current research status of the research on the thermophoretic deposition effect of ultrafine particles are summarized, and the future direction of thermophoretic deposition is proposed.

**Keywords:** ultrafine particle, deposition, thermophoresis, mathematical model, environmental gas pollution

### **1. Introduction**

Fine particles play an important role in environmental gas pollution, which are smaller than coarse particles. These particles are main driver of urban haze formation, and also the main component of tobacco aerosol and kitchen fumes. They have an aerodynamic diameter of 2.5*μm* or less (PM2.5). The fine particles which are smaller than 0.1*μm* are referred to as ultrafine particles (PM0.1). These particles can often be suspended in the air and enter the respiratory tract of the human body along with the human body's respiration. Since the surface of fine particle is usually attached to harmful substances such as bacteria and heavy metals, it is easy to cause respiratory diseases and endanger human health after entering the human body [1]. Ebenstein et al. [2] found that combined atmospheric pollution has a significant impact on life expectancy and mortality from cardiopulmonary diseases. For evey 100*μg=m*<sup>3</sup> increase in the concentration of air pollutants, the life expectancy of people under 5 years old is reduced by 1.5 years, and the life expectancy of people over 5 years old is reduced by 2.3 years. Cardiopulmonary diseases cause an increase of 79 deaths per 100,000 people. In 2016, a Polish cohort study of healthy school-age children aged 13–14 years found that for every quartile unit increase in PM1 concentration, forced vital capacity (FVC) and peak expiratory flow (PEF) decreased by 1.0 and 4.4%respectively [3].With the continuous development of manufacturing technology, in some electronic equipment and industrial products, the pollution caused by particle deposition affects both the yield and the use process. Some fine particles, after absorbing moisture in the air, will deposit on the surface of the equipment and form an electrolytic layer, which can have a corrosive effect on many metals. If the electrolyte penetrates into the protective layer of the wire

#### **Figure 1.** *Basic schematic diagram of thermophoresis.*

to form corrosion points, an arc may be generated between the wire and the conductor to burn out the components.

The thermophoretic effect plays a very important role in the deposition of fine particles, so it is necessary to study the thermophoretic deposition of fine particles. In 2012, Ström and Sasic [4] studied the improvement of particle deposition efficiency in diesel or gasoline aftertreatment systems by thermophoresis. Their results show that for a standard monolithic channel with a gas-to-wall temperature difference of 200 K, deposition efficiencies of around 15% may be possible for all particle sizes. Some scholars have also applied it to the utilization of water resources. Dhanraj and Harishb et al. [5] designed a device to extract water from the air by thermophoresis and condensation. In the microgravity environment, thermophoresis plays a more critical role, and thermophoresis may be an important way of fine particle transfer. In the microgravity environment in space, it is of great significance to consider the influence of the thermophoresis mechanism in the aerospace field.

Although thermophoresis was discovered as early as the eighteenth century, its fundamental physical process was not proposed by Maxwell until 1879. In an area with a certain temperature difference, the gas molecules in the hot and cold areas continue to hit the particles. Since the gas molecules in the hot area have a large momentum, after hitting the particles, the macroscopically shows that the particles are subjected to force from the hot area to the particle. The phenomenon of movement of the cold zone, this force is the thermophoretic force [6]. Based on the formula of thermophoretic force, scientists derived formulas of thermophoretic deposition rate. In this chapter, the studies on both thermophoretic force and deposition rate due to thermophoresis are reviewed (**Figure 1**).

### **2. Thermophoretic deposition**

In the temperature field with a certain temperature difference, the gas molecules and particles in the hot zone move to the hot zone again due to the reaction force after colliding with the particles. This phenomenon is called thermal glide. Maxwell (1879) and Reynolds (1880) analyzed the relationship between the flow field and the temperature field theoretically and experimentally [7] and determined the phenomenon of thermal slip. Maxwell proposed that the slip velocity of the gas is [8]:

*A Review of Particle Removal Due to Thermophoretic Deposition DOI: http://dx.doi.org/10.5772/intechopen.109628*

$$
\overrightarrow{\boldsymbol{v}}\_c = -\frac{3}{4} \nu \nabla l n \boldsymbol{T} \tag{1}
$$

where is the gas kinematic viscosity, *T* is the gas temperature.

#### **2.1 Models of thermophoretic force**

According to Maxwell's description and slip boundary conditions, Epstein selected spherical particles in the gas as the object, and the he solved the Navier-Stokes equation with the boundary conditions of the temperature gradient field considering the heat transfer heat balance.Finally,he first obtained the formula of thermophoretic force on the particles in the continuum region [9]:

$$F\_T = \frac{9\pi\mu\omega d\_p \overrightarrow{\nabla T}}{2T\_o} \left(\frac{k\_\text{g}}{k\_p + 2k\_\text{g}}\right) \tag{2}$$

here is fluid viscosity, is particle diameter, is fluid thermal conductivity coefficient, is particle thermal conductivity coefficient, is the average temperature of fluid near particle. Epstein's conclusion is in good agreement with the experimental results when and the thermal conductivity coefficient of the particles is low, but the theoretical value of this result is far from the experimental results when the thermal conductivity coefficient of the particles is high (L. [10]). Derjaguin and Storozhilova et al. [11]. pointed out that if high thermal conductivity particles such as sodium chloride are used, the actual thermophoretic force is two orders of magnitude higher than the theoretical [11].

In 1958, Waldmann [12] deduced the expression of the thermophoretic force of particles in the free molecular region in a single atomic gas based on the principle of molecular dynamics and the rigid body collision model between gas molecules and particles:

$$F\_{th} = -\frac{16\sqrt{\pi}}{15} \frac{R^2 k\_\mathrm{g}}{\sqrt{\frac{2k\_B T}{m\_\mathrm{g}}}} \nabla T \tag{3}$$

is Boltzmann constant, is gas molecular mass, is particle radius.

In 1962, based on Epstein's theory, Brock deduced the expressions of particle thermophoretic velocity and thermophoretic force applicable to the continuum region to the free-molecular region by applying the first-order slip-flow boundary conditions [13, 14]:

$$U\_T = -\frac{2\nu C\_s \left(\frac{k\_g}{k\_p} + 2C\_t \frac{\lambda}{d\_p}\right) \frac{(\nabla T)\_\chi}{T\_0}}{\left(1 + 4C\_m \frac{\lambda}{d\_p}\right) \left(1 + 2\frac{k\_g}{k\_p} + 4C\_t \frac{\lambda}{d\_p}\right)}\tag{4}$$

$$F\_T = -\frac{12\pi\mu\nu RC\_s \left(\frac{k\_g}{k\_p} + C\_t \frac{\lambda}{R}\right) \frac{(\nabla T)\_\chi}{T\_0}}{\left(1 + 3C\_m \frac{\lambda}{R}\right) \left(1 + 2\frac{k\_g}{k\_p} + 2C\_t \frac{\lambda}{R}\right)}\tag{5}$$

where is the mean free path of gas molecules, *Cs*ð Þ 0*:*75 , *Cm*ð Þ 1*:*14 , and are dimensionless constants, which are thermal slip coefficient, viscous slip coefficient, and temperature jump coefficient, respectively. Although the result of brock still differs from the experimental value when the particles have a high thermal conductivity coefficient, it is already much smaller than Epstein's error.

Derjaguin [15] pointed out that Brock's assumption that the gas molecules always have the same velocity distribution before hitting the particle interface is inaccurate whose velocity distributions in the Knudsen layer vary with distance from the wall in the presence of a tangential temperature gradient. Through experiments, it is pointed out that its velocity formula (Eq. 4) does not conform to the experimental results at medium and large particle sizes (0.3–0.6 *μm*).

In 1980, Talbot [10] used laser-Doppler velocimeter (LDV) to measure the thickness of the particle void region produced by the particle velocity distribution in the laminar boundary layer at the heated wall under the action of thermophoretic force. According to the experimental results and the BGK model in the particles, it is found that the reason for the large error of the Brock thermophoretic force formula (Eq. 4) is the problem of the thermal slip coefficient *Cs*, and the value of is corrected to 1.17. Talbot also proposed the expression of the thermophoresis coefficient *kth*. Although the revised thermophoretic force expression is still not perfect, it has been widely used.

Cha, McCoy [16] and Wood [17] deduced a thermophoretic calculation model with good applicability in a wide range of Knudsen numbers. Although the theoretical calculation results and the experimental data reported by other scholars at that time have small numerical errors, they are in good agreement with the overall trend. After correcting the correlation coefficient, the calculated result is in good agreement with the result of Talbot's formula when *kn* <3. The corrected expression is:

$$F\_{th} = 1.15 \frac{Kn}{4\sqrt{2}a\left(1 + \frac{\pi\_1}{2}Kn\right)} \cdot \left[1 - \exp\left(-\frac{a}{Kn}\right)\right] \cdot \left(\frac{4}{3\pi}\phi\pi\_1Kn\right)^{0.5} \frac{k\_B}{d\_m^2} \nabla T d\_p^2 \tag{6}$$

$$a = 0.22 \left[ \frac{\frac{\pi}{6} \phi}{\left(1 + \frac{\pi\_1}{2} K n\right)} \right]^{\frac{1}{2}} \tag{7}$$

$$\pi\_1 = \frac{0.18\left(\frac{36}{\pi}\right)}{\left[\frac{(2-S\_n+S\_t)4}{\pi}+S\_n\right]}\tag{8}$$

where is the equivalent diameter of the channel, *Cv* is the constant volume-specific heat capacity, is the conventional momentum adjustment coefficient, and is the tangential momentum adjustment coefficient.

In 2003, Li and Wang [18, 19] deduced the theoretical calculation formula of the thermophoretic force on nanoparticles in the free molecular region based on the nonrigid collision model:

$$\mathbf{F}\_{T, \text{Li}} = -\frac{8}{3} \sqrt{\frac{2\pi m\_{\text{r}}}{k\_{\text{B}}T}} \kappa R^2 \nabla T \left(\frac{6}{5} \mathbf{\Omega}^{(1,2)^\*} - \mathbf{\Omega}^{(1,1)^\*}\right) \tag{9}$$

In the formula, is the mass of particles, is thermal conductivity of gases, *mr* ¼ *mgmp= mg* þ *mp* � �, is the reduced mass of gas molecules and particles; Ωð Þ 1,1 <sup>∗</sup> and Ωð Þ 1,2 <sup>∗</sup> are the dimensionless collision integral, for rigid body collision,

*A Review of Particle Removal Due to Thermophoretic Deposition DOI: http://dx.doi.org/10.5772/intechopen.109628*

#### **Figure 2.**

*The potential energy Ep, kinetic energy Ek and density functiong on the surface of nanoparticles Cui et al. [20].*

Ωð Þ 1,1 <sup>∗</sup> <sup>¼</sup> <sup>Ω</sup>ð Þ 1,2 <sup>∗</sup> ¼ 1. Eq. (3) is a special case of Eq. (9). But Li and Wang's theoretical model has not been experimentally demonstrated. Due to the insufficiency of nanoscale particle measurement technology, it is often difficult to study the corresponding thermophoresis phenomenon in experiments. Cui et al. [20] used molecular dynamics simulation to verify the theoretical calculation model of Waldmann [12] and Li [18, 19] found that when the particle size was reduced to the nanometer scale, Waldmann's calculation model is no longer applicable due to the enhancement of the non-rigid collision effect, and the theoretical model established by Li and Wang after considering the non-rigid collision effect is closer to the molecular dynamics simulation results. At high gas-solid bonding strength, gas molecules are easily adsorbed on nanoparticles, which changes the true particle size of the particles (**Figure 2**), resulting in a certain error in the correction formulas of Li and Wang [18, 19]. Cui et al. [20] corrected the particle size error and the results were basically consistent with the simulated values.

In terms of thermophoresis measurement technology, it is mainly divided into particle group measurement and single particle measurement. In the measurement using the particle group, the gas stream with particles enters a vacuum chamber with a temperature gradient in the vertical direction. Calculations are often inaccurate. Particle size can be determined when individual particles are measured, so it tends to be more precise. Li and Davis [21] used electrodynamic balance (EDB) to measure individual particles and compared with previous theories and obtained better data result (**Figure 3**).

#### **2.2 Models of thermophoretic deposition efficiency**

In the calculation of the deposition efficiency of thermophoresis, many scholars have established their expressions. Since the theoretical calculation models of various scholars under laminar flow conditions can be well in line with the experimental or simulated values, the calculation model of thermophoretic deposition efficiency under turbulent flow conditions is mainly introduced here.

In 1969 Byers and Calver [22] measured the influence of flow parameters and particle collector size on deposition efficiency and established the thermophoretic deposition efficiency expression:

$$\eta\_L = 1 - \exp\left(-\frac{\rho C\_\text{p} f \operatorname{Re}\_\text{D}}{4Dh} \frac{K\_\text{th} \nu (T\_\text{e} - T\_\text{w})}{\overline{T}} \left(1 - \exp\left(\frac{-4hL}{\mu\_\text{m} \rho C\_\text{p} D}\right)\right)\right) \tag{10}$$

**Figure 3.**

*A cross-section of the electrodynamic balance and vacuum chamber used by Li and Davis for thermophoretic force measurements [21].*

where is the specific heat capacity of the gas at constant pressure, *Te* and are the fluid inlet and tube wall temperatures respectively, *K*th is thermophoretic coefficient, is the average temperature, *h*is the convective heat transfer coefficient, is the tube length, is the tube diameter, is the gas density, *f* is the coefficient of friction, is the average radial velocity.

In 1974, Nishio experimentally determined the deposition of aerosols on the temperature gradient along the length of the heat exchanger tube wall,

$$\eta\_{L} = 1 - \exp\left(-\frac{\rho C\_{\rm p} K\_{\rm th} \nu (T\_{\rm e} - T\_{\rm w})}{k\_{\rm g} \overline{T}} \left(1 - \exp\left(\frac{-4hL}{\mu\_{\rm m} \rho C\_{\rm p} D}\right)\right)\right) \tag{11}$$

Batchelor and Shen [23] used the similarity method to analyze the deposition rate as the pipe length by thermophoretic effects in flow over plates, cylinders, and rotating bodies,

$$\eta\_{\rm L} = \text{PrK}\_{\rm th} \left( \frac{T\_{\rm e} - T\_{\rm w}}{T\_{\rm e}} \right) \left( \mathbf{1} + (\mathbf{1} - \text{PrK}\_{\rm th}) \left( \frac{T\_{\rm e} - T\_{\rm w}}{T\_{\rm e}} \right) \right) \tag{12}$$

Where is Prandtl number of air.

#### *A Review of Particle Removal Due to Thermophoretic Deposition DOI: http://dx.doi.org/10.5772/intechopen.109628*

In 1998, Romay et al. [24] conducted experiments in turbulent pipelines, respectively measured the influence of inlet flow velocity, flow rate, particle size, and inlet fluid temperature on the deposition efficiency of the pipeline, and compared them with the existing theoretical model Eqs. (10) and (11) of deposition efficiency at that time. For comparison, an expression for the deposition efficiency of thermophoresis in a turbulent tube is derived:

$$\eta\_L = \mathbf{1} - \left[\frac{T\_w + (T\_\epsilon - T\_w) \exp\left(-\frac{\pi DhL}{\rho Q C\_p}\right)}{T\_\epsilon}\right]^{P\_c K\_{th}} \tag{13}$$

*Q* is the volume flow (**Figure 4**).

The above one-dimensional expressions are all derived under specific assumptions, and cannot be well applied to general engineering specifications. In 2005, Housiadas and Drossinos [25] developed a two-dimensional model including radial section effects after establishing the one-dimensional long-tube deposition efficiency expression suitable for laminar and turbulent flow and carried out extensive verification. Its one-dimensional expression is:

$$\eta\_L = \mathbf{1} - \left(\frac{\boldsymbol{\theta}^\*}{\mathbf{1} + \boldsymbol{\theta}^\*}\right)^{\text{Pr}K\_{th}} = \mathbf{1} - \left(\frac{T\_w}{T\_\varepsilon}\right)^{\text{Pr}K\_{th}} \tag{14}$$

There are many studies on the calculation of key parameters under the thermophoretic effect. Because the preconditions selected by each scholar are different, the establishment of the final expression is also different. Early studies on thermophoretic deposition effects mostly focused on the mechanism. In recent years,

*Schematic diagram of the experimental apparatus by Romay et al. [24].*

with the maturity of numerical simulation and particle detection technologies, more scholars have begun to consider the performance of thermophoretic deposition in different scenarios. The development direction is gradually diversified.
