**Abstract**

Development of new simple gas sensor is highly significant due to its various industrial applications. In this chapter, comprehensive studies on the new micro gas sensor of Micro In-plane Knudsen Radiometric Actuator (MIKRA) are done to evaluate the performance of its technique on the detection of gas. This new micro gas sensor works by the radiometric phenomena, which occurs due to temperature difference in rarefied gas. In order to simulate this micro gas sensor, direct simulation Mont Carlo (DSMC) is recognized as a reliable method. This study highly focused on the main mechanisms of the flow feature and force generation in low-pressure domain. In order to study this micro gas sensor, comprehensive numerical results are thoroughly investigated and the effects of primary factors and parameters are revealed. In addition, the main details of computational approach for the simulation of this micro gas sensor are fully presented. Finally, the performance of this sensor in various operating conditions is explained and main findings are discussed.

**Keywords:** micro gas sensor, Knudsen force, rarefied gas, DSMC

## **1. Introduction**

Gas sensors are widely used in various industrial applications for evaluation of the main elements inside the gas mixture. In addition, this device is significant for detection of dangerous gas such as CO2, H2 and ammonia. In addition, gas sensors are widely used for the evaluation of the main products of the combustion. Since this instrument is a main element in the various applications, considerable researches and studies have been performed to develop new techniques for the detection of the various gases. Indeed, the present gas sensor is highly expensive and spacious and these disadvantages of the current sensors have motivated the researchers to develop a simple and cheap method for the gas detection [1–6].

Various methods and techniques are applied for the gas sensors [7–10]. However, current sensors could not perform in the micro scale. Recently, scientists and researchers have investigated molecular force which is produced by the temperature difference on the solid body in high Knudsen numbers. Since this force occurs in high Knudsen numbers, it is known as Knudsen force.

Knudsen number (Eq. (1)) is mainly defined as the ratio of the mean free path of gas (λ) to specific length (*L*) as follows:

\_Assumors\_

$$Kn = \frac{\lambda}{L} = \frac{\mu}{p\_m L} \sqrt{\frac{2 \, k\_B T\_m}{m}}.\tag{1}$$

In Eq. (1), the term of mean free path of gas (λ) is proportional to the temperature and viscosity. According to this equation, high Knudsen number is obtained by decreasing either the pressure of the domain or size of our model. Previous studies [8–16] clearly showed that the Knudsen force is highly proportional to the main characteristics of the gas inside the domain.

Actually, thermal stresses are produced by the non-uniformity of the temperature within rarefied gases and create bulk fluid flows that could employ forces on solid body [17, 18]. Ketsdever et al. [17] presented broad literature reviews to reveal the origin of the Knudsen thermal force. They widely considered the technical remark of source of the Knudsen thermal force and active factor on the rate of induced forced. According to their findings, operative factors such as pressure of domain and gas component as well as the thermal gradient magnitude play significant role on the value of the exerted force. These characteristics enable researchers to apply this for the measurement of gas type.

One of the effective methods to apply the Knudsen force is to reduce the size of the model and construct micro device [19]. Micro Knudsen gauge and microscale radiometric actuator are the main conventional devices that implement this technique for industrialized applications. Numerous studies have been directed to inspect and evaluate the key features of Knudsen force in these instruments [18]. Strongrich et al. [19] performed experimental work and numerical studies to calculate Knudsen force on a non-uniformly heated beam. They highly focused on this molecular force and finally offer their new micro gas sensor as Micro In-plane Knudsen Radiometric Actuator (MIKRA) as shown in **Figure 1**. This sensor could be used either detection of gas types or measurement of gauge pressure [20, 21]. One of the significant aspects of this micro gas sensor is the micro size of this device that enables it to work in the various operating condition. Since this device is highly significant, considerable works have been performed to evaluate this micro gas sensor [11–13]. In our these papers, the main characteristics of this sensor are investigated and the precision of measured Knudsen force for different gas mixtures, for instance, hydrogen, methane/helium, methane/SO2, carbon dioxide, ammonia, and inert gas. These researches are conducted to reveal the performance and capability of this micro gas sensor in diverse operating conditions [14–18]. These works have

**21**

*π* σ<sup>2</sup>

*Application of Knudsen Force for Development of Modern Micro Gas Sensors*

tried to disclose the influence of temperature difference of cold and hot arm, the gap size, and pressure of domain on the value of the exerted force on the cold side. The physics of the Knudsen force in the Knudsen gage are widely investigated by the researchers and scholars. Passian et al. [22–24] as pioneer research group initiated to reveal the main characteristic of the Knudsen force at the microscale. They mainly studied on a micro cantilever which includes two surfaces with dissimilar temperatures separated by a gap in rarefied domain. Theoretical and experimental studies have been conducted via a U-shaped silicon microcantilever to disclose the main parameters. The impact of thermal difference on the Knudsen forces in the transitional regime is examined by Lereu et al. [25]. The measurement of these forces at ordinary environment on test configurations made by surface micromachining of polysilicon are done by Sista and Bhattacharya [26]. Kaajakari and Lal [27] studied Knudsen forces produced within molecular flow regime to examine surface micromachined hinged structure assembly. Furthermore, negative thermophoretic force is studied by different scholars [28] and the influence of valuable factors on radiometric force is disclosed [29, 30]. In order to simulate the model, DSMC approach is a reliable technique for evaluation of the exerted Knudsen force in the rarefied domain. This method is highly popular and conventional for the simulation of the problems with low-pressure condition. Hence, numerous scholars and scientists [31–40] applied this for the

Recognition of the force value in the low-pressure domain is the primary challenge in this field. Indeed, scholars have performed various studies to obtain the reliable and comprehensive correlation which offers the main value of Knudsen force in various operating and geometrical conditions [41–46]. Following the above description and containing the historical perspective, the broadly established modern appreciative is such that the major force related to vane rotation is the force generated close to the edges of the vane, in a zone with the dimensions of a mean free path according to Einstein. At very low pressures, the mean free path is great and the entire area of the vane is involved in force generation. As the pressure rises and flow enters the transition regime, the mean free path shrinks and the effective force-producing area of the vane is reduced. At some pressure (where the free path is on the order of the vane thickness according to Einstein), a maximum is gotten and force generation thereafter initiates to weaken as thermal creep and then convective currents initiate to lead the flow. A brief visual summary of the expected force output of a Nichols radiometer vane in free-space is shown in **Figure 2**, where comparisons are made for several of the dominant theories of the previous century. Here, FM denotes free molecular, B&L denotes Brüche and Littwin experimental

For a Crookes type radiometer, Scandurra et al. [46] have offered a first expression for radiometric force that includes both pressure and shear components. For the normal force per unit area (pressure difference) on a thin vane, they offered

> 32 √ \_ 2 *π* σ<sup>2</sup>

where α*E* is the energy accommodation coefficient, *k* is the Boltzmann constant,

is the total collision cross section of the gas molecule, and *l* is the vane perim-

\_15 *k*

 \_ *ΔT*

64 √ \_ 2 *π* σ<sup>2</sup>

where *τ* is the vane thickness. One of the key assumptions of that work is

\_15 *k*

Δ*Tl*, (2)

<sup>λ</sup> (τ*l*), (3)

*Fn* = (2 − α*E*)

*F*<sup>τ</sup> = α*<sup>E</sup>*

constant pressure in the gas surrounding the heated vane.

*DOI: http://dx.doi.org/10.5772/intechopen.86807*

simulation of scientific and engineering problems.

measurements and "Einstein" denotes his correlation.

eter. For the shear stress, the expression is

**Figure 1.** *Micro In-plane Knudsen Radiometric Actuator (MIKRA) [21].*

#### *Application of Knudsen Force for Development of Modern Micro Gas Sensors DOI: http://dx.doi.org/10.5772/intechopen.86807*

tried to disclose the influence of temperature difference of cold and hot arm, the gap size, and pressure of domain on the value of the exerted force on the cold side.

The physics of the Knudsen force in the Knudsen gage are widely investigated by the researchers and scholars. Passian et al. [22–24] as pioneer research group initiated to reveal the main characteristic of the Knudsen force at the microscale. They mainly studied on a micro cantilever which includes two surfaces with dissimilar temperatures separated by a gap in rarefied domain. Theoretical and experimental studies have been conducted via a U-shaped silicon microcantilever to disclose the main parameters. The impact of thermal difference on the Knudsen forces in the transitional regime is examined by Lereu et al. [25]. The measurement of these forces at ordinary environment on test configurations made by surface micromachining of polysilicon are done by Sista and Bhattacharya [26]. Kaajakari and Lal [27] studied Knudsen forces produced within molecular flow regime to examine surface micromachined hinged structure assembly. Furthermore, negative thermophoretic force is studied by different scholars [28] and the influence of valuable factors on radiometric force is disclosed [29, 30].

In order to simulate the model, DSMC approach is a reliable technique for evaluation of the exerted Knudsen force in the rarefied domain. This method is highly popular and conventional for the simulation of the problems with low-pressure condition. Hence, numerous scholars and scientists [31–40] applied this for the simulation of scientific and engineering problems.

Recognition of the force value in the low-pressure domain is the primary challenge in this field. Indeed, scholars have performed various studies to obtain the reliable and comprehensive correlation which offers the main value of Knudsen force in various operating and geometrical conditions [41–46]. Following the above description and containing the historical perspective, the broadly established modern appreciative is such that the major force related to vane rotation is the force generated close to the edges of the vane, in a zone with the dimensions of a mean free path according to Einstein. At very low pressures, the mean free path is great and the entire area of the vane is involved in force generation. As the pressure rises and flow enters the transition regime, the mean free path shrinks and the effective force-producing area of the vane is reduced. At some pressure (where the free path is on the order of the vane thickness according to Einstein), a maximum is gotten and force generation thereafter initiates to weaken as thermal creep and then convective currents initiate to lead the flow. A brief visual summary of the expected force output of a Nichols radiometer vane in free-space is shown in **Figure 2**, where comparisons are made for several of the dominant theories of the previous century. Here, FM denotes free molecular, B&L denotes Brüche and Littwin experimental measurements and "Einstein" denotes his correlation.

For a Crookes type radiometer, Scandurra et al. [46] have offered a first expression for radiometric force that includes both pressure and shear components. For the normal force per unit area (pressure difference) on a thin vane, they offered \_15 *k*

$$F\_n = \left(\mathbb{Z} - \alpha\_E\right) \frac{15 \, k}{32 \, \sqrt{2} \, \pi \sigma^2} \, \Delta T l,\tag{2}$$

where α*E* is the energy accommodation coefficient, *k* is the Boltzmann constant, *π* σ<sup>2</sup> is the total collision cross section of the gas molecule, and *l* is the vane perimeter. For the shear stress, the expression is \_15 *k*

$$F\_{\pi} = \text{ or} \begin{aligned} \text{or} \quad & \text{or} \begin{aligned} \text{tr}\_{E} & \frac{15}{\lambda} \frac{k}{\pi} (\text{\textdegree t}), \end{aligned} \tag{3}$$

where *τ* is the vane thickness. One of the key assumptions of that work is constant pressure in the gas surrounding the heated vane.

*Gas Sensors*

*Kn* = \_

characteristics of the gas inside the domain.

to apply this for the measurement of gas type.

λ *L* = \_ *μ <sup>p</sup>*<sup>∞</sup> *<sup>L</sup>* √

In Eq. (1), the term of mean free path of gas (λ) is proportional to the temperature and viscosity. According to this equation, high Knudsen number is obtained by decreasing either the pressure of the domain or size of our model. Previous studies [8–16] clearly showed that the Knudsen force is highly proportional to the main

Actually, thermal stresses are produced by the non-uniformity of the temperature within rarefied gases and create bulk fluid flows that could employ forces on solid body [17, 18]. Ketsdever et al. [17] presented broad literature reviews to reveal the origin of the Knudsen thermal force. They widely considered the technical remark of source of the Knudsen thermal force and active factor on the rate of induced forced. According to their findings, operative factors such as pressure of domain and gas component as well as the thermal gradient magnitude play significant role on the value of the exerted force. These characteristics enable researchers

One of the effective methods to apply the Knudsen force is to reduce the size of the model and construct micro device [19]. Micro Knudsen gauge and microscale radiometric actuator are the main conventional devices that implement this technique for industrialized applications. Numerous studies have been directed to inspect and evaluate the key features of Knudsen force in these instruments [18]. Strongrich et al. [19] performed experimental work and numerical studies to calculate Knudsen force on a non-uniformly heated beam. They highly focused on this molecular force and finally offer their new micro gas sensor as Micro In-plane Knudsen Radiometric Actuator (MIKRA) as shown in **Figure 1**. This sensor could be used either detection of gas types or measurement of gauge pressure [20, 21]. One of the significant aspects of this micro gas sensor is the micro size of this device that enables it to work in the various operating condition. Since this device is highly significant, considerable works have been performed to evaluate this micro gas sensor [11–13]. In our these papers, the main characteristics of this sensor are investigated and the precision of measured Knudsen force for different gas mixtures, for instance, hydrogen, methane/helium, methane/SO2, carbon dioxide, ammonia, and inert gas. These researches are conducted to reveal the performance and capability of this micro gas sensor in diverse operating conditions [14–18]. These works have

\_ \_ 2 *kB T*<sup>∞</sup>

*<sup>m</sup>* . (1)

**20**

**Figure 1.**

*Micro In-plane Knudsen Radiometric Actuator (MIKRA) [21].*

**Figure 2.** *Nichols radiometer: force prediction.*

For a Crookes radiometer, practically, all earlier analytical estimates of the radiometric force, with the exception of the Brüche and Littwin bell-shaped correlation, were implicitly or explicitly assuming a collision-dominated flow, where the radiometer vane area is much larger than the gas mean free path. This is essentially a slip flow regime, where the impacts of the free molecular, area-related forces are relatively small. This explains that the proposed expressions depend on the perimeter of the radiometer vane, and not on its area. While this is a reasonable approach for many cases, where the velocity distribution function is close

**23**

explained.

*Application of Knudsen Force for Development of Modern Micro Gas Sensors*

to equilibrium and the pressures at the centers of two sides of the radiometer are equilibrated, it is not obvious that such an approach is applicable to the regime where the flow is far from equilibrium, and both the area and the edge contribute

The recognition of the main effective term on this type of sensor is highly significant for the evaluation and performance analysis of this device. According to previous works, three main flow patterns are recognized in this model. As shown in **Figure 4**, the Molecular force Known as thermal stress is the main effective factor that produce the Knudsen force within the gap of two arms. The direction of this force is from hot to cold side. The second dominant term in this model is known as thermal creeping. This flow pattern is produce shear force on the top of cold arm and the direction of this flow is from cold to hot side. The least important flow pattern is thermal edge flow which occurs in the vicinity of the sharp edge with high temperature. In the following, comprehensive details of each factor will be

In the following, the governing equations and the main technical approach for the simulation of this micro gas sensor is presented. In addition, the boundary condition of this model according to the real working condition is defined. Then, the results of various codes are compared to evaluate the performance of each methods. In the next step, the main flow feature inside the model is studied to define the impact of main parameters. Moreover, the effect of the pressure and temperature difference of the hot and cold arm is determined. Finally, the performance of this

The authors of Ref. [47] used an assumption similar to Einstein's, and calculated force with n = 1. They found that the radiometric force computed with this simple empirical expression gives surprisingly close agreement with experimental results, as shown in **Figure 3**. The assumption of n = 1 works very well, even though, the pressure imbalance occurs over a region of 10 mean free paths. The agreement is fairly good in the free molecular and nearly free molecular flows (pressures below or about 0.1 Pa, or Knudsen numbers above 0.5 that are characterized by a nearly linear increase in the radiometric force, and the area-related radiometric forces are dominant). Then, even though the empirical expression stems from the free molecular formula, the agreement is also quite reasonable in the transitional flow where the collisions start to reduce the radiometric force, and both area- and edgerelated radiometric forces are important (Knudsen numbers between 0.5 and 0.05,

*DOI: http://dx.doi.org/10.5772/intechopen.86807*

*Flow feature and main stream inside the MIKRA [10].*

to the radiometric forces.

**Figure 4.**

where the maximum radiometric force is observed).

sensor in detection of gas mixture will be explained.

**Figure 3.** *Crookes radiometer: force prediction and comparison with experimental data.*

*Application of Knudsen Force for Development of Modern Micro Gas Sensors DOI: http://dx.doi.org/10.5772/intechopen.86807*

**Figure 4.**

*Gas Sensors*

**Figure 2.**

*Nichols radiometer: force prediction.*

**22**

**Figure 3.**

*Crookes radiometer: force prediction and comparison with experimental data.*

For a Crookes radiometer, practically, all earlier analytical estimates of the radiometric force, with the exception of the Brüche and Littwin bell-shaped correlation, were implicitly or explicitly assuming a collision-dominated flow, where the radiometer vane area is much larger than the gas mean free path. This is essentially a slip flow regime, where the impacts of the free molecular, area-related forces are relatively small. This explains that the proposed expressions depend on the perimeter of the radiometer vane, and not on its area. While this is a reasonable approach for many cases, where the velocity distribution function is close

*Flow feature and main stream inside the MIKRA [10].*

to equilibrium and the pressures at the centers of two sides of the radiometer are equilibrated, it is not obvious that such an approach is applicable to the regime where the flow is far from equilibrium, and both the area and the edge contribute to the radiometric forces.

The authors of Ref. [47] used an assumption similar to Einstein's, and calculated force with n = 1. They found that the radiometric force computed with this simple empirical expression gives surprisingly close agreement with experimental results, as shown in **Figure 3**. The assumption of n = 1 works very well, even though, the pressure imbalance occurs over a region of 10 mean free paths. The agreement is fairly good in the free molecular and nearly free molecular flows (pressures below or about 0.1 Pa, or Knudsen numbers above 0.5 that are characterized by a nearly linear increase in the radiometric force, and the area-related radiometric forces are dominant). Then, even though the empirical expression stems from the free molecular formula, the agreement is also quite reasonable in the transitional flow where the collisions start to reduce the radiometric force, and both area- and edgerelated radiometric forces are important (Knudsen numbers between 0.5 and 0.05, where the maximum radiometric force is observed).

The recognition of the main effective term on this type of sensor is highly significant for the evaluation and performance analysis of this device. According to previous works, three main flow patterns are recognized in this model. As shown in **Figure 4**, the Molecular force Known as thermal stress is the main effective factor that produce the Knudsen force within the gap of two arms. The direction of this force is from hot to cold side. The second dominant term in this model is known as thermal creeping. This flow pattern is produce shear force on the top of cold arm and the direction of this flow is from cold to hot side. The least important flow pattern is thermal edge flow which occurs in the vicinity of the sharp edge with high temperature. In the following, comprehensive details of each factor will be explained.

In the following, the governing equations and the main technical approach for the simulation of this micro gas sensor is presented. In addition, the boundary condition of this model according to the real working condition is defined. Then, the results of various codes are compared to evaluate the performance of each methods. In the next step, the main flow feature inside the model is studied to define the impact of main parameters. Moreover, the effect of the pressure and temperature difference of the hot and cold arm is determined. Finally, the performance of this sensor in detection of gas mixture will be explained.
