**3.2 Analysis of flow structure**

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**

**Figure 6.**

*Comparison of the obtained results (dsmcfoam) with experimental and numerical of Strongrich et al. [21].*

#### **Figure 7.**

*Flow pattern and temperature distribution inside the MIKRA for different pressure conditions with real arm temperature [11].*

when the real temperature is applied on the arms. As shown in the figures, the main characteristics of the flow feature significantly varies with change of the temperature. Since the main difference of flow structure inside the model is related to the temperature distribution, this study also considers the temperature distributions as well as flow pattern.

In low pressure (P = 62 Pa), one big circulation as well as a few small ones are noticed. As the pressure of the domain increases, three main circulations are observed in which two of them is on the top of the hot arm. The main circulation

**27**

**Figure 8.**

*Tcold = 300) [10].*

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

fully covered the whole domain. It is significant to note that the temperature diffusion strengthen as the pressure of the domain is raised. In high pressure (P = 966 Pa), the temperature of the hot arm is not high enough and the number of the particles considerably increases. Due to these reasons, the diffusion of the temperature inside the domain highly declines. Therefore, the temperature gradient

As mentioned in the previous section, the effect of the temperature is significant in the performance of this type of micro gas sensor. In order to recognize the main effect of the temperature, constant temperature is applied for all pressure to investigate the effect of pressure (or number of particles) in the performance of the system. **Figure 4** compares the temperature contour along with streamline patterns for various operating pressures when the temperature of the hot and cold arm is fixed 350 and 300 K for all pressure domains, respectively. In this figure, hot arms are colored according to the temperature of particles in the vicinity of arms, while the temperature of the hot solid arm is 350 K. This coloring method improves the perceptibility of the temperature difference in various

As shown in **Figure 8**, the temperature diffusion to particles that exist in the vicinity of the hot arm increases by raising the pressure of the domain. Indeed, the number of the particles increases when the pressure is raised. Therefore, the particles interaction to hot surface increases in high pressure. The evaluation of the flow feature inside the micro gas sensor will reveal significant results. The main circulation inside domain occurs due to thermal creeping. As the pressure increases inside the model, the main circulation moves to the right side on the top of the gap. Contours clearly show that the strength of the circulation intensifies by growing the pressure till 387 Pa. Then, the circulation weakens inside the

The temperature gradient alters meaningfully from the high pressure (P = 966 Pa) situation to rigorously rarefied (P = 62 Pa) case where noticeable kinks in the contour lines are perceived. These kinks are originated at the sharp angles on the top of the

*Flow pattern and temperature distribution inside the MIKRA for different pressure conditions (Thot = 350,* 

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

as the main source of the circulation reduces.

**3.3 Effect of the temperature difference**

pressures.

domain.

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

fully covered the whole domain. It is significant to note that the temperature diffusion strengthen as the pressure of the domain is raised. In high pressure (P = 966 Pa), the temperature of the hot arm is not high enough and the number of the particles considerably increases. Due to these reasons, the diffusion of the temperature inside the domain highly declines. Therefore, the temperature gradient as the main source of the circulation reduces.

### **3.3 Effect of the temperature difference**

*Gas Sensors*

**Figure 6.**

**26**

**Figure 7.**

*temperature [11].*

well as flow pattern.

when the real temperature is applied on the arms. As shown in the figures, the main characteristics of the flow feature significantly varies with change of the temperature. Since the main difference of flow structure inside the model is related to the temperature distribution, this study also considers the temperature distributions as

*Flow pattern and temperature distribution inside the MIKRA for different pressure conditions with real arm* 

*Comparison of the obtained results (dsmcfoam) with experimental and numerical of Strongrich et al. [21].*

In low pressure (P = 62 Pa), one big circulation as well as a few small ones are noticed. As the pressure of the domain increases, three main circulations are observed in which two of them is on the top of the hot arm. The main circulation

As mentioned in the previous section, the effect of the temperature is significant in the performance of this type of micro gas sensor. In order to recognize the main effect of the temperature, constant temperature is applied for all pressure to investigate the effect of pressure (or number of particles) in the performance of the system. **Figure 4** compares the temperature contour along with streamline patterns for various operating pressures when the temperature of the hot and cold arm is fixed 350 and 300 K for all pressure domains, respectively. In this figure, hot arms are colored according to the temperature of particles in the vicinity of arms, while the temperature of the hot solid arm is 350 K. This coloring method improves the perceptibility of the temperature difference in various pressures.

As shown in **Figure 8**, the temperature diffusion to particles that exist in the vicinity of the hot arm increases by raising the pressure of the domain. Indeed, the number of the particles increases when the pressure is raised. Therefore, the particles interaction to hot surface increases in high pressure. The evaluation of the flow feature inside the micro gas sensor will reveal significant results. The main circulation inside domain occurs due to thermal creeping. As the pressure increases inside the model, the main circulation moves to the right side on the top of the gap. Contours clearly show that the strength of the circulation intensifies by growing the pressure till 387 Pa. Then, the circulation weakens inside the domain.

The temperature gradient alters meaningfully from the high pressure (P = 966 Pa) situation to rigorously rarefied (P = 62 Pa) case where noticeable kinks in the contour lines are perceived. These kinks are originated at the sharp angles on the top of the

#### **Figure 8.**

*Flow pattern and temperature distribution inside the MIKRA for different pressure conditions (Thot = 350, Tcold = 300) [10].*

arms. Dissimilar to the high-pressure conditions wherein intermolecular collisions promptly smooth out those kinks in the gap of the arms, the absence of adequate intermolecular collisions in the rarefied situations lets these kinks to diffuse much additional away from the hot arms as displayed in **Figure 7**. Therefore, the temperature of hot arm simply enters inside the domain and the noticeable temperature gradient observed in the vicinity of hot arm. In the next sections, it will be clarified how this temperature gradient influences on the induced flow field.

In order to recognize the main impact of the temperature in our problem, **Figure 9** illustrates the flow structure and temperature distribution inside the micro gas sensor in various temperature differences of 10, 30, 50, and 100 K at pressure of 387 Pa. Our findings reveal that the strength of the main circulation intensifies as the temperature difference of the hot and cold arm increases. It was predicted that this would occurs as the temperature gradient inside the model increases. One of important findings of this contour is the temperature penetration. In fact, temperature difference plays significant role on the particles direction. **Figure 10** shows the temperature distribution in the vicinity of the arms. The figure displays that the temperature gradient is intensive on the edges of the hot arm. In order to distinguish the induced flow pattern nearby of the edge, it is supposed that molecules within a mean free path away from

#### **Figure 9.**

*Flow pattern and temperature distribution inside the MIKRA for different temperature differences (P = 387 Pa) [11].*

**29**

top of the cold arm.

**Figure 11.**

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

this area arrive at the surface without experiencing any intermolecular collision. As is shown in **Figure 6**, the temperature molecules coming from points B and C is high, while those from point A have low temperatures. Since the diffuse condition is applied as a function of the wall, the tangential velocity of the molecules after collision with the wall is related to the wall temperature. Hence, the tangential velocity of the cold molecules (A) highly increases while hot molecules (B and C) do not experience any change in their velocity. Therefore, the direction of cold molecules after collision is more dominant and they induce a vortex (blue lines) in the edge of the hot arm. Since the temperature of the cold arm is not varied, this flow is not observed on

**Figure 11** plots the variation of the net force on the cold arm for various temperature differences of 10, 30, 50, and 100 K. Obtained results clearly demonstrate

In order to evaluate the primary factors on this micro gas sensor, the effect of force on the both sides of the cold arm is investigated. Since the exerted force should be normalized, Eq. (3) is applied to compare the change of the force as the ratio to

**Figure 12** illustrates the variation of the FR for various pressures of domain when the temperature of the hot arm is 30, 50, and 100 K. Comparison of the Knudsen force on both sides of the cold arm clearly reveals that FR declines on right side as the pressure of the domain is increased. This shows that the effect of molecular thermal force within gap is limited due to high interactions of molecules. On the other side, the Knudsen force on the left side of the increases with rising of the pressure of domain. This confirms that the influence of the thermal creeping on the

\_*F FT*=10 )*p*

. (5)

that main inflation occurs in the maximum Knudsen force.

*Variation of the thermal Knudsen force in various temperature differences [11].*

exerted force when temperature difference is 10 K.

*FR* = 100 × (

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

**Figure 10.** *Schematic illustration of the flow feature in the vicinity of the arms [11].*

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

**Figure 11.** *Variation of the thermal Knudsen force in various temperature differences [11].*

this area arrive at the surface without experiencing any intermolecular collision. As is shown in **Figure 6**, the temperature molecules coming from points B and C is high, while those from point A have low temperatures. Since the diffuse condition is applied as a function of the wall, the tangential velocity of the molecules after collision with the wall is related to the wall temperature. Hence, the tangential velocity of the cold molecules (A) highly increases while hot molecules (B and C) do not experience any change in their velocity. Therefore, the direction of cold molecules after collision is more dominant and they induce a vortex (blue lines) in the edge of the hot arm. Since the temperature of the cold arm is not varied, this flow is not observed on top of the cold arm.

**Figure 11** plots the variation of the net force on the cold arm for various temperature differences of 10, 30, 50, and 100 K. Obtained results clearly demonstrate that main inflation occurs in the maximum Knudsen force.

In order to evaluate the primary factors on this micro gas sensor, the effect of force on the both sides of the cold arm is investigated. Since the exerted force should be normalized, Eq. (3) is applied to compare the change of the force as the ratio to exerted force when temperature difference is 10 K.

 $\rightarrow$ - $-$ - $-$ - $-$ - $-$ - $-$ - $-$ - $-$ - $-$ - $-$ - $-$ - $-$ - $-$ - $-$ - $-$ - $-$ - $-$ - $-$ - $-$ - $-$ - $-$ - $-$ - $-$ - $-$ - $-$ - $-$ - $-$ - $-$ - $-$ - $-$ - $-$ - $-$ - $-$ - $-$ - $-$ - $-$ - $-$ - $-$ - $-$ - $-$ - $-$ - $-$ - $-$ - $-$ - $-$ - $-$ - $-$ - $-$ - $-$ - $-$ - $-$ - $-$ - $-$ - $-$ - $-$ -- $}-$ - $-$ - $-$ - $-$ - $-$ - $-$ - $-$ - $-$ - $--$ - $-$ -- $-$ -- $-$ -- $--$ -- $--$ -- $--$ -- $--$ -- $---$ -- $---$ --- $---$ ---- $-----$ -----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------

**Figure 12** illustrates the variation of the FR for various pressures of domain when the temperature of the hot arm is 30, 50, and 100 K. Comparison of the Knudsen force on both sides of the cold arm clearly reveals that FR declines on right side as the pressure of the domain is increased. This shows that the effect of molecular thermal force within gap is limited due to high interactions of molecules. On the other side, the Knudsen force on the left side of the increases with rising of the pressure of domain. This confirms that the influence of the thermal creeping on the

*Gas Sensors*

arms. Dissimilar to the high-pressure conditions wherein intermolecular collisions promptly smooth out those kinks in the gap of the arms, the absence of adequate intermolecular collisions in the rarefied situations lets these kinks to diffuse much additional away from the hot arms as displayed in **Figure 7**. Therefore, the temperature of hot arm simply enters inside the domain and the noticeable temperature gradient observed in the vicinity of hot arm. In the next sections, it will be clarified

In order to recognize the main impact of the temperature in our problem, **Figure 9** illustrates the flow structure and temperature distribution inside the micro gas sensor in various temperature differences of 10, 30, 50, and 100 K at pressure of 387 Pa. Our findings reveal that the strength of the main circulation intensifies as the temperature difference of the hot and cold arm increases. It was predicted that this would occurs as the temperature gradient inside the model increases. One of important findings of this contour is the temperature penetration. In fact, temperature difference plays significant role on the particles direction. **Figure 10** shows the temperature distribution in the vicinity of the arms. The figure displays that the temperature gradient is intensive on the edges of the hot arm. In order to distinguish the induced flow pattern nearby of the edge, it is supposed that molecules within a mean free path away from

how this temperature gradient influences on the induced flow field.

**28**

**Figure 10.**

**Figure 9.**

*(P = 387 Pa) [11].*

*Schematic illustration of the flow feature in the vicinity of the arms [11].*

*Flow pattern and temperature distribution inside the MIKRA for different temperature differences* 

**Figure 12.** *Variation of the exerted force on hot and cold side [11].*

**Figure 13.** *Comparison of (a) normalized pressure (b) flow pattern and temperature distribution in various gap sizes [12].*

**31**

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

left side is strengthened. Obtained results also indicate that the rate of FR augments

In order to determine the main characteristics of the each term, the pressure of

*Pave*

**Figure 14** illustrates the variation of the Knudsen thermal force on the cold arm.

Since the gap size is crucial in the main characteristics of our problem, the impact of gap size on the normalized pressure and flow structure are depicted in **Figure 13a** and **b**, respectively. As the gap size increases in our model, the thermal creeping effect declines due to high gap of the hot and cold arm. Meanwhile, the

Our findings show that increasing the gap size declines the value of the exerted Knudsen force on the model. The variation of the Knudsen force on cold arm presents significant note about the value of the Knudsen number. Since the gap size is known as the specific length (l) in our model, change of this size significantly influence on the value and pressure of the maximum Knudsen force. In low gap size (10 μm), the maximum Knudsen force occurs at 600 Pa while it declines as the gap size is increased to 50 μm. The main impact of gap size on the Knudsen force could be noticed in the pressure distribution. As shown in **Figure 13a**, the pressure gradient hardly reach to the cold arm. This confirms that the pressure gradient is

. (6)

with rising of the temperature difference of the hot and cold arm.

*Comparison of applied Knudsen force on the cold arm for various gap sizes [12].*

the domain is normalized by the average pressure of domain as follows:

*<sup>I</sup>* = \_*<sup>P</sup>*

considerably significant on the exerted force.

number of small circulations increases inside the model.

**3.4 Effect of gap size**

**Figure 14.**

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

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

**Figure 14.** *Comparison of applied Knudsen force on the cold arm for various gap sizes [12].*

left side is strengthened. Obtained results also indicate that the rate of FR augments with rising of the temperature difference of the hot and cold arm.

#### **3.4 Effect of gap size**

*Gas Sensors*

**Figure 12.**

*Variation of the exerted force on hot and cold side [11].*

**30**

**Figure 13.**

*Comparison of (a) normalized pressure (b) flow pattern and temperature distribution in various gap sizes [12].*

In order to determine the main characteristics of the each term, the pressure of the domain is normalized by the average pressure of domain as follows: *<sup>I</sup>* = \_*<sup>P</sup>*

$$I = \frac{P}{P\_{\text{ave}}}.\tag{6}$$

Since the gap size is crucial in the main characteristics of our problem, the impact of gap size on the normalized pressure and flow structure are depicted in **Figure 13a** and **b**, respectively. As the gap size increases in our model, the thermal creeping effect declines due to high gap of the hot and cold arm. Meanwhile, the number of small circulations increases inside the model.

**Figure 14** illustrates the variation of the Knudsen thermal force on the cold arm. Our findings show that increasing the gap size declines the value of the exerted Knudsen force on the model. The variation of the Knudsen force on cold arm presents significant note about the value of the Knudsen number. Since the gap size is known as the specific length (l) in our model, change of this size significantly influence on the value and pressure of the maximum Knudsen force. In low gap size (10 μm), the maximum Knudsen force occurs at 600 Pa while it declines as the gap size is increased to 50 μm. The main impact of gap size on the Knudsen force could be noticed in the pressure distribution. As shown in **Figure 13a**, the pressure gradient hardly reach to the cold arm. This confirms that the pressure gradient is considerably significant on the exerted force.
