Original Russian Textc M. Barzegar Gerdroodbary, D.D. Ganji, R. Moradi, Ali Abdollahi, 2018, published in Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, 2018, No. 6, pp. 94–104.
Application of Knudsen Thermal Force for Detection of CO
2in Low-Pressure Micro Gas Sensor
M. Barzegar Gerdroodbary1*, D. D. Ganji1, R. Moradi2, and Ali Abdollahi3
1Department of Mechanical Engineering, Babol University of Technology, Babol, Iran
2Department of Chemical Engineering, School of Engineering & Applied Science, Khazar University, Baku, Azerbaijan
3Department of Mechanical Engineering, Islamic Azad University, Najafabad, Iran Received December 9, 2017
Abstract—Development of new techniques for detection of CO2gas is significant for decrease the dangers of CO2. In this research, numerical simulations are performed to evaluate the performance of a new micro gas sensor (MIKRA) for the detection of CO2 gas. This device works due to temperature difference inside a rectangular enclosure with heat and cold arms as the non-isothermal walls at low pressure condition. In this study, the pressure of CO2 is varied from 62 to 1500 Pa correspond to Knudsen number from 0.1 to 4.5 to investigate all characteristics of the thermal-driven force inside the MEMS sensor. In order to simulate a rarefied gas inside the micro gas detector, Boltzmann equations are applied to obtain high precision results. To solve these equations, Direct Simulation Monte Carlo (DSMC) approach is used as a robust method for the non-equilibrium flowfield. Ourfindings show that value of generated Knudsen force significantly different when the fraction of CO2 in N2–CO2 mixtures is varied. This indicates that this micro gas sensor could precisely detect the concentration of CO2 gas in a low-pressure environment. In addition, the obtained results demonstrate that the mechanism of force generation highly varies in the different pressure conditions.
Keywords: CO2 gas detection, Knudsen molecular force; DSMC; low-pressure gas actuators;
MEMS.
DOI:10.1134/S0015462818060149
1. INTRODUCTION
Sensors are the main device for several industrial and scientific applications. Since the first step for the separation of the mixture is detection, the development of this sensor could highly enhance the separation techniques such as membranes, cryogenic column and gas centrifuges [1–2]. In addition, several researchers have tried to develop new simple devices for the detection of the dangerous gas such as NH3, CO2, and H2S. Due to the significance of inert gas, detection of these gases is important for scientists.
Microsensors are highly developed due to their applications in the different device such as medical instruments. One of the new methods for the detection of the gas is the application of the Knudsen force which is highly sensitive to the properties of the gas. Indeed, the non-homogeneity of the temperature in the low-pressure condition produce a force known as Knudsen force. Galkin et al. [3–4] investigated some kinetic effects in continuum flows. This special characteristic motivated the researchers to use this approach for measurement of pressure. Alexandrov et al. [5–6] studied thermal stress effect and its experimental detection.
The physics of Knudsen force in rarefied conditions was investigated by various researchers. Ketsde- ver et al. [7] reviewed more than hundred papers and documents to present a comprehensive literature review on the origin of the Knudsen force and its history. In addition, the physics of the Knudsen force are explained in this paper. Among various studies, some of the recent work tried to apply this force
*E-mail:[email protected].
for diverse applications. Since Crookes radiometer [8] was thefirst device which applied Knudsen force, various researchers performed extensive studies on this device. Passian et al. [9] investigated thermal transpiration at the microscale by a Crookes cantilever. They [10] also studied Knudsen forces on micro- cantilevers and presented theoretical discussions of the magnitude of the Knudsen forces in various conditions. Moreover, several researches [11–13] investigated the Knudsen force in the rarefied gas.
Aoki et al. [14–15] performed numerical simulations to observe the main characteristics of the Knudsen force. Bosworth et al. [16] presented a study on the measurement of negative thermophoretic force.
Several works performed to apply a Direct Simulation Monte Carlo (DSMC) for the simulation of the rarefied gas. Balaj et al. [17, 18] focused on the effects of shear work on non-equilibrium heat transfer characteristics of rarefied gasflows through micro/nanochannels. Poozesh and Mirzaei [19, 20]
applied the lattice Boltzmann method for flow simulation around cambered airfoil by using conformal mapping and intermediate domain. Eskandari and Nourazar [21] focused on the time relaxed Monte Carlo computations for the lid-driven microcavityflow. Strongrich et al. [22] performed experimental measurements and numerical simulations of the Knudsen force on a non-uniformly heated beam.
In 2016, Strongrich et al. [23, 24] introduced a new device (Fig. 1a) for sensing the pressure by Knudsen force. They constructed in-plane Knudsen Radiometric Actuator (MIKRA) sensor which operates by the temperature difference between the two arms in low-pressure condition. In this sensor, the hot arm isfixed while the cold arm known as shuttle arm could move and the capacitor is attached to shuttle arm. Since the gap of these two arms is too small, the Knudsen force exerts force on the cold side and this could be measured by the capacitor. Numerical simulations showed that there are two other types of the mechanism which induce force on the cold arm. Fig. 1b schematically presents main mechanisms inside the MIKRA. The description of each type offlows will be comprehensively presented in the next chapters.
Although numerous scholars investigated the radiometric force, most of works focused on the study of vane radiometer in which hot and cold sides are on the two sides of the vane. Indeed, the characteristics of Knudsen thermal force are not well studied when hot and cold elements are existed in front of each other. In our previous works [25], the effect of Knudsen thermal force on the performance of the low-pressure micro gas sensor is completely investigated. However, the performance of MIKRA was neither experimentally nor numerically investigated for gas mixtures with approximately similar chemical properties. In fact, the effects of mass concentration of the each component are not revealed. Therefore, the study of theflow feature and main mechanisms of force generation inside the MIKRA in different conditions is essential for the development of the device.
In this study, low pressure micro gas sensor (MIKRA) is comprehensively investigated to detect CO2gas in various conditions. This study applied a DSMC approach to reveal theflow feature and main mechanism of force generation in low pressure domain. In addition, influences of geometrical parameters on thermal force generation are studied to improve the application of this device. Firstly, the details of the MIKRA is explained and the characteristic of the domain is briefly presented. Then, boundary conditions and working situation of the MIKRA are determined. Next, the governing equations and numerical approach are explained. Comprehensive parametric studies on roles of pressure condition, length and gap of the arms are presented and the significant effects of each parameter on theflow feature inside the device are explained. Also, the mechanism of force generation inside the microactuator due to temperature difference is explained. Finally, the extensive comparisons along these parameters are done and details of the all aspects of force generation are compared.
2. NUMERICAL APPROACH 2.1. Governing Equations and Solver
In order to simulate rarefied gas, molecular approach of the kinetic gas theory is as a reliable and robust method for obtainingflow feature. In this method, the Boltzmann equation (Eq. 2.1) are solved.
∂
∂t(nf) +c.∂
∂r(nf) +F. ∂
∂c(nf) =Q, (2.1)
wheren,candf are number density, molecular velocity and velocity distribution function, respectively.
In addition,Q=+∞
−∞
4π
0 n2(f∗f1∗−f f1)gσdΩd3c1is the collision integral which describes the change in the velocity distribution function due to intermolecular collisions.
Shuttle
Shuttle Arm
Actuation Capacitor
Sense Capacitor
Suspension 4 mm
4 mm Heater Arm
Main Circulation (thermal creeping)
Molecular force (thermal stress)
Thermal Edge flow Shear Force
Gold Hot
(a)
(b)
Fig. 1.(a) MIKRA device [17], (b) Schematic offlow inside the MEMS sensor.
In order to solve the Boltzmann equation, some approaches are available. Several studies [16–18]
showed that the DSMC method of Bird [27], as a particle method based on kinetic theory for simulation of rarefied gases, is a reliable approach for such molecular regime. In the present study, open-source dsmcfoam code is applied to obtain from a proficient and flexible implementation of complex models [28]. dsmcfoam has been established within the framework of the open source software of OpenFOAM version 1.7. Numerical technique is widely used for solving engineering problems [29–34].
2.2. Numerical Procedure
In this work, the variable hard sphere (VHS) collision model is applied to simulate the collision.
Moreover, 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 [23].
In the present model, the size of the gap (distance between the heater and shuttle arms) is known as the characteristic length (L) and it is approximately 20μm. The total number of simulated particles was approximately2.48×10+5. We calculated the time step and it is1×10−8 (s), smaller than the mean collision time. The physical time of these simulations was 0.015 s to reach steady state condition. For a typical simulation, twenty particles are initially set in each cell to minimize the statistical scatter. Table 1
Table 1.Numerical specification
Parameter Unit Value
Cell size μm 4
Number of particles in each cell – 20
Time step s 1×10−8
The number of time step – 3×10+6
Total number of grid – 9910
Table 2.Temperature of the cold and hot arm (real temperature)
Pressure Kn Hot Heater Cold Heater
Hot Arm Cold Arm Hot Arm Cold Arm
(pa) – (K) (K) (K) (K)
62 4.48 370 306 326 303
155 1.8 363 306 329 303
387 0.72 356 306 327 303
966 0.29 331 304 315 302
1500 0.18 321 303 306 300
presents the detail of problem such as cell size, number of particles in each cell, time step, number of time steps, and number of samples.
2.3. Geometry and Boundary Condition
Figure 2 illustrates the generated grid and the boundary condition applied on the model. The size of the domain is600×300μm in xandy direction. There were 150 by 65 collision cells in thexandy 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 = 298K). The simulations are performed for single gas of nitrogen. In this research, the real temperature condition of experimental examinations (Strongrich et al. [17]) are applied on the cold and hot arm and presented in the Table 2.
3. RESULTS AND DISCUSSION 3.1. Verification
In order to evaluate the precision of the obtained data, it is necessary to compare the simulated results with experimental data. Hence, the Knudsen forces on the shuttle arm are compared with experimental data of Strongrich et al. [17] for nitrogen gas in various pressure conditions (Fig. 3). In this study, the net force on the cold arm is calculated by following equation:
net F orce=Fr−Fl, (3.1)
where Fr andFl represent the force exerted on the right and left side of the cold arm. Figure 3 also compares the results of the dsmcfoam for hot and cold heater with experimental results of Strongrich et al. [17] to evaluate the precision of the data. The comparison shows a good agreement between obtained data (dsmcfoam) with experimental and numerical results. The numerical results show some
Wall : T = 296 K
Substrate (wall : T = 298 K)
Side Wall Side Wall
600 m
300 m
50 50 m 50 m 5050 50 mm
ColdArm Hot
Arm Y
Z X
Fig. 2.The boundary condition and grid of the present model.
3.5
Experimental
Dsmcfoam-COLD Heater Dsmcfoam-HOT Heater Forse, N
3.0 2.5 2.0 1.5
0.5
101 102 103 104
Pressure, Pa 0
1.0
Fig. 3.Comparison of the obtained results (dsmcfoam) with experimental and numerical of Strongrich et al. [17] for N2gas.
deviation in the force value in the vicinity of the maximum force. This is likely caused for several different reasons. The main reason for this difference is related to the gap distance. In the numerical simulation, it is assumed that the heater-shuttle arm pair gap is 20μm, but wasn’t necessarily true for all the cases in the experiments. Second, coupling and 3D effects likely caused some differences. In addition, we assumed that the temperature isfixed while the temperature of the real model is not constant and varied in the third dimension.
3.2. Flow Pattern Inside Microsensor
Figure 4 shows the temperature contour along with streamline patterns for various operating pressures when the average temperature of hot and cold heater (Table 2) is applied on the hot and cold arm. The contour clearly shows that flow feature is highly influenced by the operating pressure. In addition, temperature distribution inside the domain is highly different as the pressure of the domain increases.
Pave = 62 Pa overa11T Pave = 155 Pa
321 318 315 312 309306 303 300
overa11T 321318 315 312 309 306 303 300
overa11T 321 318 315 312 309306 303 300 overa11T
321318 315 312 309 306 303 300
Pave = 387 Pa Pave = 966 Pa
Fig. 4.Flow pattern and temperature distribution inside the MIKRA for CO2gas in different pressure conditions with average arm temperature.
The tracking of theflow pattern inside the domain for various pressures reveals the main parameters which are significant in the performance of the device. In low pressure (P = 62Pa), the temperature of hot arm easily penetrates inside the domain and the temperature gradient between the hot region and cold side wall induces two distinct circulation inside the domain. As the pressure of the domain is increased, the number of the molecules rises and the circulation is formed due to thermal creepingflow from cold to hot arm. In addition, the temperature gradient as the main source of the circulation declines due to the presence of high number of particles in high pressure domain. In low pressure (P = 966Pa), the temperature of hot arm could not penetrate into the main domain and the molecular interactions significantly block the enthalpy of the reflected particles from the hot surface. Indeed, the temperature of the hot arm is low and this leads the formation of several circulation in the vicinity of the arms.
3.3. Effect of Arm Length (L)
Figure 5 compares the flow pattern and temperature distribution when the length of the arm is increased from 1×L to 3×L. The thickness (50 μm) and pressure (P = 387 Pa) of the model remains constant. The contour clearly illustrates that increasing the length of the arms divides the main circulations into three distinct circulations. Moreover, the temperature gradient inside the domain significantly intensifies by increasing the length of the arms.
Figure 6 plots the effect of length arm on variation of the net force generation in various domain pressures. The comparison of the net force for different lengths shows that the value of the force intensifies when the length increases. If we precisely inspect the value of the net force in high pressure condition (P = 1500Pa), it is observed that the increase of generated force on cold arm is more than three times when the length of the arm is elevated form1×Land3×L. The contour of the streamline in Figure 5 also shows that the strength of the circulation is considerable in high length. These results confirm that the net force is directly proportional with length of arms.
3.4. Effect of Gap Distance (G)
One of the main characteristics of the present sensor is the distance of the two arm that are known as gap. In the rarefied gas, the Knudsen number (Eq. 3.2) was introduced to determine the main characteristics of the molecular interaction. Knudsen number is defined by the ratio of the mean free path of gas (λ) to specific length (L) as follows:
Kn= λ
L. (3.2)
overa11T 324 320 316 312 308 304 300 296 292
overa11T 324 320 316 312 308 304 300 296 292
overa11T 324 320 316 312 308 304 300 296 292 3 L
2 L 1 L
Fig. 5.Effect of the arm length (L) onflow pattern and temperature distribution inside the gas sensor (P= 387Pa).
Since the specific length in this study is the distance of the two arms, the variation of the gap could significantly influence on theflow feature and the force generation on cold arm.
Figure 7 depicts the variation of the Knudsen thermal force on the cold arm in various pressure conditions. As displayed in thefigure, decreasing the gap distance from 50 to 10μm significantly reduces Knudsen force on the right side of the cold arm and the absolute net force on the cold arm increases. As the gap size increases, the difference of the two sides is low and the net force is close to zero. The difference of the net force intensifies by increasing the operating pressure.
In order tofind the main reason for different models, the streamline and temperature contour of various gap sizes in P = 966Pa are compared in the Fig. 8. It is clearly observed that the formation of the circulation intensifies inside the gap and hot molecules hardly reach to cold arm. This confirms that the formation of the circulation decreases the exerted force on the right side of the cold arm.
Forse, μN 2
−2
−4
−6
−8
−10
−12
−14
−16
−18 0
CO2, L = 50 μm
101 102 103
Pressure, Pa 3 ×L
2 ×L 1 ×L
L
Fig. 6.Effect of the arm length (L) on the force generation.
Forse, N CO2 gas
101 2
2 4 6 8 10 12 0
Gap
Gap = 10 m Gap = 20 m Gap = 50 m
102 103
Pressure, Pa
Fig. 7.Effect of the arm length (L) onflow pattern and temperature distribution inside the gas sensor (P = 387Pa).
3.5. Effect of Mass Concentration
Since primary goal of this study is to develop microsensor (MIKRA) for detection of the CO2gas, it is crucial to examine the effect of CO2 mass concentration on the generated force. Figure 9 plots the variation of the generated force for various CO2-nitrogen mixture for MIKRA device when the average temperature is applied on arms. Thefigure shows that the presence of the nitrogen as the second gas significantly influences on the generated force. The comparison of generated force on the cold arm for the pure nitrogen and CO2 gases indicate a significant difference in the produced force. It is found that the net force on the pure nitrogen is reached to maximum about P = 387Pa while the net force in CO2gases steeply decrease by increasing the pressure of the domain. Since the chemical properties (molecular mass, viscosity and degree of freedom) of these two gases are highly different, this difference is expectable. The results show that the device is able to detect a different mass concentration of CO2
gas.
overa11T Gap = 10 m
Сlose up
Сlose up
Сlose up Gap = 20 m
Gap = 50 m 313
309 305 301 297 293 289
overa11T 313 309 305 301 297 293 289
overa11T 313 309 305 301 297 293 289
Fig. 8.Streamline and temperature contour for various gap distances inside the gas sensor (P = 966Pa) with average temperature.
Forse, μN 2
−2
−3
−1
−4
−5
−6 0
100% CO2
50% CO2 + 50% N2 100% N2
1
101 102 103
Pressure, Pa
Fig. 9.Effect of the CO2mass concentration on Knudsen thermal force.
100% CO2
50% CO2 + 50% N2
100% N2
overa 11T 325 320 315 310 305 300
overa 11T
overa 11T 325 320 315 310 305 300
325 320 315 310 305 300
Fig. 10.Effect of CO2 mass concentration onflow pattern and temperature distribution inside the gas sensor (P = 387Pa).
In order tofind the main effect of CO2mass concentration, Figure 10 compares theflow pattern and temperature variation for three cases with different CO2–N2 mass concentrations inside the domain at P = 387Pa. The contour shows that one large circulation forms up to gap and one small circulation is also found in the right side of the hot arm when pure CO2 fills whole domain. In the mixture of CO2–N2, the temperature penetration declines and thermal edgeflow becomes dominant. This leads to form several small circulations in the vicinity of the top of hot arms. As the domain fills with pure N2, the temperature gradient become penetrates inside the domain and thermal creeping induce a large circulation region. According to theflow feature, it is found that the mixture of the CO2–N2 does not allow the temperature to the domain. Indeed, non-homogeneity of the particles damp the molecular
transmission in the domain. This phenomenon is also known as thermal transpiration that has been applied as the main driving mechanism for force generation inside Knudsen pumps [35–36].
4. CONCLUSIONS
In this study, a DSMC method is applied to simulate rarefied CO2 gas inside the low-pressure gas sensor (MIKRA). This work has tried to investigate and calibrate the performance of the MIKRA as a detector of CO2 gas in different operating conditions. In addition, the influences of ambient pressure, length, gap distance and CO2 mass concentration in theflow feature and force generation mechanism are investigated. The governing equations of the current problem are Boltzmann equations and DSMC approach is then applied to simulate rarefied gas. Meanwhile, comprehensive physical explanations on the mechanism of force generation andflow feature inside the gas sensor are presented. Obtained results show that micro gas sensor (MIKRA) is highly sensitive to geometrical parameters such as arm length and gap distance. Furthermore, our obtained results show that the CO2mass concentration and significantly influence on the MIKRA, and this enables users to apply this MEMS device for detection of the CO2in various applications.
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