**2. Numerical approach**

### **2.1 Governing equations**

In order to simulate the flow inside the rarified gas, Navier-Stokes equations are not valid and consequently, computational fluid dynamics (CFD) approaches is applicable. In fact, the continuity is not governed in low-pressure free molecular regime to near-continuum. Therefore, high order equation of Boltzmann equation should be solved to obtain the flow pattern in molecular regime. In followings, Boltzmann equation is presented.

$$\frac{\partial}{\partial t} \{\eta f\} + \mathbf{c}. \frac{\partial}{\partial \mathbf{r}} \{\eta f\} + \mathbf{F}. \frac{\partial}{\partial \mathbf{c}} \{\eta f\} = \mathbf{Q} \tag{4}$$

where *n,* **c,** and *f* are number density, molecular velocity, and velocity distribution function, respectively. In addition, *Q* = ∫−∞ +∞ ∫0 <sup>4</sup>*<sup>π</sup> n*<sup>2</sup> ( *f* ∗ *f*1 <sup>∗</sup> − *f f*1) *g* σ *d*Ω *d*<sup>3</sup> *c*1 is the collision integral which describes the change in the velocity distribution function due to intermolecular collisions.

Since solving the Boltzmann equation is hard, researchers try to find approaches that present similar results to that of Boltzmann equations. DSMC technique of Bird [48], as a particle method based on kinetic theory, is a reliable approach for simulation of rarefied gases. There are some software packages such as OpenFOAM and SPARTA in which DSMC method is developed for the simulation of the engineering problems. OpenFOAM is open-source code is proficient and flexible software for simulation of complex models [49].

#### **2.2 Numerical procedure**

In order to perform the DSMC simulations, some assumptions are made. For modeling of the collision, the variable hard sphere (VHS) collision model is used. Collision pairs are chosen based on the no time counter (NTC) method, in which the computational time is proportional to the number of simulator particles [36].

In this type of the sensor, the gap (distance between the heater and shuttle arms) is recognized as the characteristic length (L) and it is 20 μm. In this model, it is recommended to initiate 20 particles in each cell to minimize the statistical scatter.

#### **2.3 Geometry and boundary condition**

**Figure 5** illustrates the generated grid and the boundary condition applied on the model. The size of the domain is 600 × 300 μm in *x* and *y* direction. There were 150 × 65 collision cells in the *x* and *y* directions, respectively. All surfaces were assumed to be fully diffuse.

The free domain condition is applied on the top of domain while the side of the domain is symmetry. Constant temperature is applied to the hot and cold arms. The pressure of the domain varied from 0.465 to 11.2 Torr, meaning the Knudsen number varied from 4.64 to 0.19, respectively. The bottom of the domain is at constant temperature (T = 298 K). The simulations are performed for single gas of nitrogen. In this research, two types of the temperature condition (real and constant temperature) are applied on the cold and hot arm. In constant type, it is assumed that the temperature of hot and cold arm is fixed with variation of pressure and effect of four constant temperature differences (310–300, 330–300, 350–300 and 400–300 K) is investigated. In the real temperature type,

**25**

**3. Results and discussion**

**3.2 Analysis of flow structure**

**3.1 Verification**

**Figure 5.**

**Table 1.**

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

the temperature of the cold and hot arm varies with the pressure of the domain. In order to valid our results, the temperature variation of the cold and hot arm is obtained from experimental data of Strongrich et al. [21] and presented in **Table 1**.

**Pressure Kn Hot arm Cold arm (Pa) — (K) (K)** 4.48 353 303 1.8 350 303 0.72 347 303 0.29 325 302 0.18 315 300

In order to evaluate the precision and correctness of the numerical results, it is highly significant to compare simulation with experimental data. As mentioned in the previous section, the results of the SPARTA and DSMC are compared with experimental data (**Figure 6**). The comparison of results of simulations with that of experimental data of Strongrich et al. [21] for various pressure conditions shows that applied assumptions and procedures is logic and reasonable. In addition, obtained results of the SPARTA-DSMC code [21] also confirm the correctness of our results. The evaluation displays a worthy agreement of our work with other techniques.

In order to realize the main mechanism of this new gas sensor, the flow feature and temperature distribution inside the micro gas sensor are illustrated in **Figure 7**

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

*The boundary condition and grid of the present model [12].*

*Temperature of the cold and hot arm (real temperature).*

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

**Figure 5.** *The boundary condition and grid of the present model [12].*


#### **Table 1.**

*Gas Sensors*

**2. Numerical approach**

**2.1 Governing equations**

Boltzmann equation is presented.

due to intermolecular collisions.

simulation of complex models [49].

**2.3 Geometry and boundary condition**

assumed to be fully diffuse.

**2.2 Numerical procedure**

particles [36].

scatter.

\_∂ ∂*t*

tion function, respectively. In addition, *Q* =

(*nf* ) <sup>+</sup> **c.** \_<sup>∂</sup>

In order to simulate the flow inside the rarified gas, Navier-Stokes equations are not valid and consequently, computational fluid dynamics (CFD) approaches is applicable. In fact, the continuity is not governed in low-pressure free molecular regime to near-continuum. Therefore, high order equation of Boltzmann equation should be solved to obtain the flow pattern in molecular regime. In followings,

<sup>∂</sup>**r**(*nf* ) <sup>+</sup> **F.** \_<sup>∂</sup>

where *n,* **c,** and *f* are number density, molecular velocity, and velocity distribu-

collision integral which describes the change in the velocity distribution function

∫−∞ +∞ ∫0 <sup>4</sup>*<sup>π</sup> n*<sup>2</sup> ( *f* ∗ *f*1

Since solving the Boltzmann equation is hard, researchers try to find approaches that present similar results to that of Boltzmann equations. DSMC technique of Bird [48], as a particle method based on kinetic theory, is a reliable approach for simulation of rarefied gases. There are some software packages such as OpenFOAM and SPARTA in which DSMC method is developed for the simulation of the engineering problems. OpenFOAM is open-source code is proficient and flexible software for

In order to perform the DSMC simulations, some assumptions are made. For modeling of the collision, the variable hard sphere (VHS) collision model is used. Collision pairs are chosen based on the no time counter (NTC) method, in which the computational time is proportional to the number of simulator

In this type of the sensor, the gap (distance between the heater and shuttle arms) is recognized as the characteristic length (L) and it is 20 μm. In this model, it is recommended to initiate 20 particles in each cell to minimize the statistical

**Figure 5** illustrates the generated grid and the boundary condition applied on the model. The size of the domain is 600 × 300 μm in *x* and *y* direction. There were 150 × 65 collision cells in the *x* and *y* directions, respectively. All surfaces were

The free domain condition is applied on the top of domain while the side of the domain is symmetry. Constant temperature is applied to the hot and cold arms. The pressure of the domain varied from 0.465 to 11.2 Torr, meaning the Knudsen number varied from 4.64 to 0.19, respectively. The bottom of the domain is at constant temperature (T = 298 K). The simulations are performed for single gas of nitrogen. In this research, two types of the temperature condition (real and constant temperature) are applied on the cold and hot arm. In constant type, it is assumed that the temperature of hot and cold arm is fixed with variation of pressure and effect of four constant temperature differences (310–300, 330–300, 350–300 and 400–300 K) is investigated. In the real temperature type,

<sup>∂</sup>**<sup>c</sup>** (*nf* ) = *<sup>Q</sup>* (4)

<sup>∗</sup> − *f f*1) *g* σ *d*Ω *d*<sup>3</sup>

*c*1 is the

**24**

*Temperature of the cold and hot arm (real temperature).*

the temperature of the cold and hot arm varies with the pressure of the domain. In order to valid our results, the temperature variation of the cold and hot arm is obtained from experimental data of Strongrich et al. [21] and presented in **Table 1**.
