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Characteristics of thermal transpiration effect and the hydrogen flow behaviors in the microchannel with semicircular obstacle
i n t e r n a t i o n a l j o u r n a l o f h y d r o g e n e n e r g y 4 4 ( 2 0 1 9 ) 2 9 7 2 4 e2 9 7 3 2
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Characteristics of thermal transpiration effect and the hydrogen flow behaviors in the microchannel with semicircular obstacle Jianjun Ye a,b,*, Junda Shao a, Zongrui Hao c, Shehab Salem a, Yuan Zhang a, Yiwen Wang a, Zhengyang Li a a
School of Power and Engineering, Huazhong University of Science and Technology, Wuhan, 430074, China The State Key Laboratory of Fluid Power and Mechatronic Systems, Zhejiang University, Hangzhou, 310027, China c Qilu University of Technology, The Institute of Oceanographic Instrumentation, Qingdao, 266001, China b
article info
abstract
Article history:
The thermal transpiration effect has great potential applications for the hydrogen energy.
Received 22 December 2018
In this paper, the thermal transpiration effect and the hydrogen flow behaviors are studied
Received in revised form
in the microchannel with the semicircular obstacles. Firstly, the slip boundary model is
14 March 2019
used in the simulation of the flow performance in the microchannel. The validity of the
Accepted 9 April 2019
model at different Kn is verified by comparing with some previous work. Then, the
Available online 7 May 2019
hydrogen flow characteristics of the thermal transpiration effect with the semicircular obstacle are investigated. The result shows that as the size of the semicircular obstacle
Keywords:
increases, the hydrogen flow path of the thermal transpiration effect becomes longer, and
Thermal transpiration effect
the temperature gradient decreases. As the characteristic length of the hydrogen flow
Hydrogen flow
decreases, there is an obviously negative influence on the thermal transpiration flow. A
Knudsen number
deeper analysis shows that the thermal driven flow and the pressure driven flow will
Slip boundary conditions
produce y-component velocity, which leads to a backflow under the effect of semicircles,
Velocity distribution
and the semicircular obstacles make the Knudsen layer spread to the channel center. © 2019 Hydrogen Energy Publications LLC. Published by Elsevier Ltd. All rights reserved.
Introduction As a clean and renewable energy, hydrogen has been used widely with the rapid development of microelectromechanical systems (MEMS) and fuel cell technology. For example, the energy consumption of the micro hydrogen sensor based on MEMS technology is only 1% compared with the conventional hydrogen sensor. Moreover, the MEMS hydrogen sensor has better sensitivity and faster test speed [1,2]. Because of the advantages of high energy density and
clean characteristic, hydrogen is also used as the energy source in the MEMS power systems, such as micro hydrogen combustors, micro hydrogen thermophotovoltaic reactor, micro hydrogen fuel cell system, etc [3e5]. Furthermore, due to the characteristic scale of the MEMS and other microdevice systems is small, the thermal transpiration effect exists in the channel. Based on this effect, the transmission and pressure increase of the gas medium can be realized only under the condition of temperature difference. There are no moving parts in the system, which has obvious advantages of long life
* Corresponding author. School of Power and Engineering, Huazhong University of Science and Technology, Wuhan, 430074, China. E-mail address: [email protected] (J. Ye). https://doi.org/10.1016/j.ijhydene.2019.04.079 0360-3199/© 2019 Hydrogen Energy Publications LLC. Published by Elsevier Ltd. All rights reserved.
i n t e r n a t i o n a l j o u r n a l o f h y d r o g e n e n e r g y 4 4 ( 2 0 1 9 ) 2 9 7 2 4 e2 9 7 3 2
and high reliability. Therefore, the thermal transpiration effect has a very active application prospect and potential for hydrogen transmission in the MEMS devices. The thermal transpiration effect, which is a gas flow along the microchannel wall in the direction of the temperature gradient, is a classical phenomenon in the microscale flow and has attracted many attentions recently [6,7]. The thermal transpiration effect leads to a pressure increase in the hot container without the pressure difference, and it can be used to manufacture a hydrogen compressor and a hydrogen power system. Actually, the understanding of the hydrogen flow behaviors is better for the design and the application of this kind of micromachines. Ye et al. adopted an original numerical method uniting the smoothed particle hydrodynamics (SPH) technique with the direct simulation Monte Carlo (DSMC) technique to simulate the hydrogen flow in multi-scale between the membranes [8]. The velocity and the mass flow rate of the hydrogen and the carbon monoxide in the multi-scale flow were discussed. The results showed that the phenomenon is different from that observed in either a macroscopic or microscopic flow and provided a reference for membrane design in hydrogen production. Karniadakis et al. constructed a theoretical model with two containers connected with a microchannel, which was filled with air [9]. In their works, the relationship between the pressure of two containers at the equilibrium state and the temperature of two containers at the initial state were derived. At equilibrium, the pressure difference of two containers was obtained by theoretical analysis and numerical simulation method. They found that it is possible to have a non-zero flow rate in a microchannel even in the case of a zero pressure gradient. The pressure difference driving some gas to move from the hot container to the cold container, which is in the reverse direction of the thermal transpiration flow, is called the Poiseuille flow. Kanki et al. used the Bhatnagar-Gross-Krook (BGK) model to discuss the Poiseuille flow and the thermal transpiration flow of a rarefied gas between two parallel plates [10]. Their results indicated that the ratio of the thermal driven flow rate to the pressure driven flow rate drops to 0 in the continuous zone, and the value is closed to 0.5 when the gas density is near zero. A notable application of the thermal transpiration effect is the Knudsen pump or compressor and its variants. Some scholars have done a lot of researches on the pressure performance, mass flow rate and application of the Knudsen compressor [11e14]. Pham-Van-Diep et al. analyzed the pressure and temperature distributions of the microchannel and the connection channel in the single-stage and multistage Knudsen compressors [15]. The pressure performance and mass flow rate of the model are improved by parallel arraying multiple microchannel connected with containers in series. As materials were developed, Vargo et al. built a physical Knudsen compressor on the foundation of the thermal transpiration effect, which can be used for practical applications for the first time. The compressor was consisted of silicon chips, aerogels, heat-resistant glass containers for improving pressure and vacuum aluminum tubes. The results showed that the ratio of the pressure is closed to 1.02 [16,17]. About the researches on the mass flow rate of the channel,
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McNamara et al. designed a miniature Knudsen compressor with a rectangular cross-section of 0.1 10 mm [18]. When the temperature difference between the two containers is 1100K, the maximum flow rate approaches to 3 105cc/min. Recently, researchers have applied the thermal transpiration effect in many more fields, such as micro/medium gas vacuum pump, high pressure gas source of MEMS, refrigeration system, gas separation equipment, gas conveyor, and so on [19e24]. Except for the experimental and applied researches, in terms of numerical methods, Alexeenko et al. numerically investigated the thermal transpiration flow in a twodimensional finite length microchannel with the length-toheight ratio of 5, the pressure ratio equal to 1 and a cylindrical tube using the DSMC method and BGK model [25]. They studied the influence of different temperature gradient on the thermal transpiration effect. Their results showed that there is a pressure difference along the channel in despite of the inlet and outlet pressure values are same. They also compared different numerical methods. Ye et al. used the DSMC method united with the SPH method to simulate the influence of rarefaction and temperature gradient on multiscale flow in the application of the Knudsen pumps [26]. These results showed that when the gas flow becomes more rarefied, the thermal transpiration effect becomes more significant. Ohwada et al. used the linearized Boltzmann equation with hard-sphere molecules to discuss the Poiseuille flow and the thermal transpiration flow of a rarefied gas among double parallel-plates [27]. They predicted the mass rates and the velocity distributions of the Poiseuille and the thermal transpiration flow. The results showed that, as the Kn number increases, the mass rate of thermal transpiration flow increases continuously. Compared with the existing numerical methods and experimental results, the proposed method has better accuracy. According to the previous studies, it is found that the researches of the thermal transpiration effect mainly involved in the experimental study and the numerical study. Most of the work focused on the characteristic of the flow, heat and mass transfer in microchannel under the thermal transpiration effect, and the microchannel studied were uniform and smooth micro-straight channels. However, the shape and characteristic length of the channel has significant influence on the thermal transpiration effect. And the gas flow in the microchannel with more complex structures often encountered in microsystems, such as the channel with obstacles, the influence of the thermal transpiration effect on the flow has not been investigated, the flow behaviors and the mechanism are also ambiguous. Therefore, in this paper, the numerical simulation of the hydrogen flow in the microchannel with semicircular obstacles is implemented by using a continuum medium model with slip boundary conditions. The semicircular obstacles are added to the inner wall of the microchannel, and the radius of the semicircular obstacle is changed to investigate the effect of the obstacle on the hydrogen flow characteristic in the microchannel. It is useful for a better understanding on the flow field in the microchannel and the design and application of microdevices based on the thermal transpiration effect in the future.
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coefficient. The generalized Maxwell's model gave the above three coefficients by:
Numerical equations Continuity equations For the numerical simulation of macro-scale fluids, three equations of mass conservation, momentum conservation and energy conservation in traditional fluid mechanics [28], namely Navier-Stokes equations (NeS equations), are often used, as shown in equations (2.1)e(2.3). The NeS equations ignore the molecular properties of gases and liquids, and regard fluids as continuous media. It is called continuum model by describing the changes in density, velocity, pressure, temperature and other macroscopic flow with time and space. vu v þ ðruk Þ ¼ 0 vt vxk
(2.1)
P vui vui v ki þ uk r þ rgi ¼ vt vxk vxk
(2.2)
ve ve vqk X vui þ uk r þ ki ¼ vt vxk vxk vxk
(2.3)
where r is its density, uk is the instantaneous velocity P component (u, v, w), ki is the second-order stress tensor (surface force per unit area), gi is the volume force per unit mass, e is the internal energy, qk is the sum of the heat flux vectors of conduction and radiation.
Boundary conditions For the calculation of the model in this paper, the internal gas of the microchannel is a rarefied gas in the micro scale. The Knudsen number describing the gas rarefaction is more than 0.01. With the characteristic length of the microchannel decreases, the interaction between the gas and the wall is far more than that between the gas molecules, so the numerical results based on the traditional NeS equations are far from the experimental results. Therefore, more and more researchers added reasonable equations of boundary conditions to make the NeS equations more suitable for calculating the micro-scale flow. Based on previous studies, Lockerby and Kennard proposed new boundary conditions, which take into account the effect of wall temperature on the thermal transpiration effect on the basis of ideal gas equation [29,30], as shown in equations (2.4)e(2.7): uslip ¼ u uw ¼
lss sT m tn;t þ Vt Tg rTg m
(2.5)
Vt Tg ¼ VTg VTg $n n
(2.6)
nT tn n
2 at at
(2.8)
sT ¼
3 4
(2.9)
xT ¼
2 at 2g k at g þ 1 mCp
Numerical modeling A 2D numerical model under the thermal transpiration effect is shown in Fig. 1, made of silicon material with the thickness of d ¼ 1mm. The model has a length L and a height H. The cavity of the model filled with hydrogen gas, which is divided into the cold container, the hot container and the microchannel connected them. The length of the microchannel is l and the height is h. At the inner wall of the microchannel along the vertical centerline, two semicircular obstacles with a radius of R are distributed symmetrically. The temperature of the cold container and the hot container is indicated by Tc and Th respectively. In order to prevent the external temperature field from interfering with the temperature gradient inside the microchannel, assuming the exterior wall of the microchannel is adiabatic. Before the calculation, the gas pressure of the model is P0. When the flow achieves the equilibrium state, the pressure of the hydrogen gas in the cold container and the hot container is expressed by Pc and Ph respectively, which is the average relative pressure compared with P0. DP represents the pressure difference between the cold container and the hot container.
(2.7)
Where uslip is the slip velocity, l is the mean free path of gas, n is the boundary normal, t is the viscous stress tensor, m is the viscosity of the gas, r is its density, and Tg is its temperature. Tw is the wall temperature, ss is the viscous slip coefficient, sT is the thermal slip coefficient, and xT is the temperature jump
(2.10)
Where at is the tangential momentum accommodation coefficient, the value of at means the fraction of molecules which are reflected diffusely from the wall, g is the specific heat ratio of the gas, k is the thermal conductivity of the gas and Cp is the heat capacity at constant pressure. In this paper, the NeS equations with the above boundary conditions is adopted for the numerical calculation of the microchannel model. According to Sharipov's work, the present work chooses st ¼ 0.9 as the tangential momentum accommodation coefficient [31].
(2.4)
Tw ¼ Tg xT ln$VTg
tn;t ¼ tn
ss ¼
Fig. 1 e Computational model under the thermal transpiration effect.
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In order to study the effect of the different obstacles on the thermal transpiration effect, this paper added two semicircular obstacles with varying radius to the inner wall of the microchannel. The radius of semicircular obstacles is R ¼ 0 mm, R ¼ 0.1 mm, R ¼ 0.2 mm, R ¼ 0.3 mm and R ¼ 0.4 mm respectively. The numerical details of the studied cases are shown in Table 1. The computational model is meshed with unstructured grids, and the local encryption is performed at the half arcs. Grid detection histogram shows that the quality of the grid is all above 0.54. It is verified the mesh number by the maximum pressure difference between the cold container and the hot container. Finally, when the radius of the semicircular obstacles is R ¼ 0mm, R ¼ 0.1mm, R ¼ 0.2mm, R ¼ 0.3mm and R ¼ 0.4mm, the mesh number of the model is set at 5.7 104, 6.9 104, 6.3 104, 6.0 104 and 5.9 104 respectively.
Results and discussions Verification of the numerical method used in the model To verify the validity of the numerical model in this article, the simulation result for the thermal transpiration effect is compared with the analytic solution from Karniadakis’s work [9]. In their work, the air gas flow in a Knudsen compressor with a straight microchannel is studied. The average value of Knudsen number in the microchannel is 0.052. Both in the present work and the previous work, the temperature of two containers are set at 300 K and 400 K, respectively. Table 2 indicates the comparison results and the deviation of maximum pressure difference. The normalized pressure variation along the central channel under the standard atmospheric pressure in present work are compared with the analytical solution in the literature, which is shown in Fig. 2. It can be seen from Table 2, the maximum pressure difference of the analytical solution in Ref. [9] and the numerical result in the present work are 342 Pa and 341 Pa, respectively. The deviation of the maximum pressure difference between the results in this paper and the analytical solution is 0.29%. It shows that the present results are in good agreement with the analytical solution, which proves the calculation method has good applicability in simulating the thermal transpiration effect in the microchannel.
Kn number distribution in the model The Kn number distribution along the microchannel centerline is shown in Fig. 3, which represents the rarefaction of the gaseous medium.
Logarithmic coordinates are adopted for a better understanding. At standard atmospheric pressure, the Kn number in the cold container and the hot container is 0.0061 and 0.0086 nearly, which indicates the hydrogen gas in containers is in the continuum regime. At the entrance of the microchannel, due to the reduction of the microchannel characteristic length, the Kn number increased to 0.050. For the changing of the Kn number along with the microchannel, when the radius is R ¼ 0 mm, it means the channel is straight, the value of Kn increased from 0.050 to 0.068 linearly. When there are the semicircular obstacles existing in the microchannel, the Knudsen number distribution changes obviously. From the beginning of the semicircular obstacles, the characteristic length of the channel decreases gradually, and the Knudsen number increases. At the top of the semicircular obstacles, the Knudsen number reaches the maximum value, and then decreases gradually. When the radius of semicircular obstacles is R ¼ 0.1mm, R ¼ 0.2 mm, R ¼ 0.3 mm and R ¼ 0.4 mm, the maximum of the Kn number along the channel is 0.064, 0.070, 0.078 and 0.087 respectively. It proves that the hydrogen gas of the microchannel is in the slip regime. In the whole process, the flow state of the hydrogen gas skips from the continuum regime to the slip regime, finally it changes to the continuum regime again. The change of the flow regime occurs in the model. As discussed in the previous work, when the Kn number is in the range of (103, 101), the continuum medium model with slip boundary conditions is satisfied with the calculation in the model.
Semicircular obstacles influence on the distribution of pressure and velocity in the microchannel As it is shown in Fig. 4, the normalized hydrogen pressure varies along the centerline of the channel with a different radius of semicircular obstacles. Through the analysis of the fixed R, the hydrogen gas pressure in the container is nearly constant. Nevertheless, there is an obvious pressure rise in the microchannel. According to the gas equation of state P ¼ nkT, where P is pressure changes of the monitoring points, n is the molecular number density, k is Boltzmann's constant and T is temperature changes of the monitoring points. The temperature gradient along the microchannel wall can drive the thermal transpiration flow to creep from the cold container towards the hot container along walls, which made the hydrogen gas pressure increased in the channel by degrees. In the containers, since the distribution of the temperature and the molecular number density is equal in equilibrium, the pressure barely changed. Finally, because the hydrogen pressure in the hot container is higher than that in the cold
Table 1 e Numerical details of the cases. Quantity Model length L/mm Model height H/mm Microchannel length l/mm Microchannel height h/mm Type of gaseous medium
30 20 15 2.5 Hydrogen
Quantity Temperature in cold container Tc/K Temperature in hot container Th/K Pressure in cold container Pc/Pa Pressure in hot container Ph/Pa Operate pressure P0/Pa
300 400 0 0 1 atm
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Table 2 e Compared results in R ¼ 0.mm Quantity Average of Kn in microchannel Kn Maximum pressure difference DPmax /Pa
Fig. 2 e Normalized pressure variation along the central channel at 1 atm
container, the hydrogen gas pressure difference produced between the two containers. As the radius of semicircular obstacles increases, the hydrogen gas pressure in the hot container first increased from 202.18 Pa at R ¼ 0 mm to 204.67 Pa at R ¼ 0.1 mm, and then dropped little by little. When the radius of semicircular obstacles is R ¼ 0.2 mm, R ¼ 0.3 mm and R ¼ 0.4 mm, the hydrogen pressure in the hot container is 196.17Pa, 196.46Pa and 195.32Pa, respectively. The average pressure of the hydrogen gas in the cold container first increased from 172.40Pa to 155.19Pa and then decreased to 165.87Pa gradually. On the one hand, compared with the straight channel, semicircular
Fig. 3 e Kn distribution along the centerline of the microchannel with different R.
Analytic solution
Our Results
0.052 342
0.054 341
Fig. 4 e Normalized pressure profile of hydrogen flow along the centerline with different R.
obstacles weakened the temperature gradient, which influenced the thermal driven flow and produced a diversion of the flow in the y-direction. On the other hand, the obstacles hindered the Poiseuille flow to move from the hot container towards the cold container in the central channel. When the radius of the semicircular obstacles is 0.1 mm, the hydrogen gas pressure in the cold container increases 17.20Pa, and the pressure in the hot container only increases 2.50Pa. The results show that the hydrogen gas pressure at both containers increases, whereas the pressure increase in the cold container is higher than that in the hot container, therefor the pressure difference of the hydrogen Knudsen compressor decreases. As the radius of the semicircular obstacles enlarge, the characteristic length of the microchannel deceased deeply, the Kn number increased, and then the value of QT/QP also increased, where QT is the thermal driven flow coefficient, QP is the pressure driven return flow coefficient. The influence of semicircular obstacles on the thermal transpiration flow is higher than the Poiseuille flow. With the increase of the thermal transpiration flow path, the temperature gradient along the microchannel is further reduced, which weakened the thermal transpiration effect and made the pressure values of the hot container are lower and that of the cold container are also lower. Fig. 5 shows the x-component of velocity along the vertical centerline with different radius of the semicircular obstacles. As the radius of the semicircular obstacles increases from R ¼ 0 mm to R ¼ 0.4 mm, the velocity of the Poiseuille flow through the central channel is getting lower and the decreasing amplitude is getting lower. That is because, with the enlarging of the semicircular obstacles radius, the characteristic length of the microchannel along the vertical
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Fig. 5 e X-component of velocity along the vertical centerline with different R.
centerline decreased. For the thermal driven flow along the wall, the maximum of velocity dropped by 0.63 m/s from R ¼ 0 mm to R ¼ 0.1 mm. That is because, when semicircular obstacles exist, the velocity of the thermal driven flow produced a diversion of the hydrogen flow in the y-direction. Besides, semicircular obstacles extended the flow path, which made the temperature gradient decreased and the velocity dropped. The velocity of the hydrogen flow had a maximum cut. When the radius of the semicircular obstacles is R ¼ 0.2 mm, the microchannel characteristic length decreased, the Kn number increased, and then the value of QT/QP also increased, which made the influence of the thermal transpiration effect is more intense and the hydrogen flow velocity is higher than before. As the radius continues to rise, the increase of the ratio of QT/QP is lower than the increase of the thermal driven flow path. The drop of the temperature gradient lead to a decrease of the thermal driven flow velocity. In summary, semicircular obstacles set at the wall of the microchannel caused the backflow to occur in the pressure driven flow in the central channel and the thermal driven flow along the wall. The backflow made the Knudsen layer spread to the central channel. The region of the Poiseuille flow decreased and the velocity of the Poiseuille flow also dropped. The hydrogen mass flowrate of the microchannel is shown in Fig. 6. The mass flowrate is calculate based on the accumulation. The positive flow direction represents the thermal transpiration flow and negative direction represents the Poiseuille flow. It can be found that the mass flowrate of the thermal transpiration flow is as the same value of the Poiseuille flow, just with different flow direction. Which indicates that the flow achieves the equilibrium state in the microchannel with the thermal transpiration flow and the Poiseuille flow. When the obstacle radius increases from R ¼ 0 mm to R ¼ 0.4 mm, the mass flowrate of the thermal transpiration flow decreases from 5.0 108 kg/s to 1.7 108 kg/s. It shows that the semicircular obstacles affect the thermal transpiration flow as well as the Poiseuille flow. As it is indicated in Fig. 7 that the x-component of velocity along the centerline of the model with a different radius of semicircular obstacles, is actually the x-component of the
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Fig. 6 e Hydrogen mass flowrate of the microchannel with different semicircular obstacles. velocity distribution of the Poiseuille flow. When R ¼ 0 mm, the velocity of the microchannel centerline increased linearly. The maximum velocity of the hydrogen flow nearly occurred in the connection between the microchannel and the hot container is 0.75 m/s. The x-component of velocity along the vertical centerline with R ¼ 0 mm is 0.75 m/s. When semicircular obstacles exist, the velocity along the vertical centerline suddenly dropped. The value of velocity reduced by 0.50 m/s at R ¼ 0.1 mm, that declined 33.5% compared with the velocity at R ¼ 0 mm. When the radius of semicircular obstacles is R ¼ 0.2 mm, R ¼ 0.3 mm and R ¼ 0.4 mm, the value of the velocity reduced by 0.43 m/s, 0.39 m/s and 0.37 m/s respectively, and the velocity of hydrogen flow declined 41.8%, 47.3% and 50.9% compared with the velocity at R ¼ 0 mm. The results show clearly that the velocity of the Poiseuille flow is getting lower and the decline range is getting lower. That is because, on the one hand, the existence of the semicircular obstacles lessened the characteristic length of the microchannel, which makes the Knudsen layer spread to the central channel and hinders the Poiseuille flow. On the other hand, due to the existence of the semicircular obstacles, there is a backflow
Fig. 7 e X-component of velocity along the centerline of the model with different R.
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Fig. 8 e Velocity vector of hydrogen flow in the model with R ¼ 0.
appeared in the Poiseuille flow, which deceases the flow velocity.
Semicircular obstacles influence on the state of hydrogen flow in the microchannel Fig. 8 shows the distribution of the velocity vector of hydrogen flow in the model with the straight channel. The flow
directions are indicated by the logarithmic arrows. There are the temperature gradient drove the thermal transpiration flow to creep from the cold container to the hot one along the channel walls. The pressure difference between the two containers drove the Poiseuille flow to move from the hot container towards the cold container through the central channel, which agrees with the theoretical analysis [32]. The return pressure-driven flow through the centerline occupies a significant portion of the channel cross-section whereas the thermal driven flow in the direction of the temperature gradient is confined to a small region along the wall. According to the results of the previous studies, when the Kn number is 0.059, the ratio of the thermal driven flow coefficient to the pressure driven flow coefficient is QT/QP z 0.01, which marks the quantity of the thermal driven flow is much less than the pressure driven flow. The velocity vector showed that, in the above of containers, there are two clockwise vortices, and in the below of containers, there are two anticlockwise vortices, which caused by the separation of the thermal driven flow due to the adverse pressure difference. It can be seen in Fig. 9, the influence of semicircular obstacles on hydrogen flow state in the microchannel mainly embodied in the y-component of velocity and the backflow with different R.
Fig. 9 e Velocity vector of hydrogen flow at semicircle in the microchannel.
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In Fig. 9aee, when semicircular obstacles exist, the velocity of hydrogen flow suddenly dropped at the semicircular arcs. As the enlarging of the semicircular radius, the decreasing amplitude in velocity of the thermal driven flow enhanced, and the interface between the thermal transpiration flow and the Poiseuille flow gradually diffuses to the microchannel center, which agree with x-component of velocity along the vertical centerline in Fig. 5 of the previous work. In Fig. 9bee, because of the existence of semicircle obstacles, the thermal driven flow along the wall of the channel produced y-component velocity and even caused the backflow, which enhanced the Poiseuille flow in the central channel. The phenomenon is different from that at R ¼ 0 mm. Affected by the semicircular obstacles, the Poiseuille flow produced y-component velocity and even caused the backflow, which further strengthens the thermal driven flow to creep at the right side of the semicircle. Comparing with Fig. 9bee, it can be found that with the radius of semicircular obstacles increase, the reflux trend of the thermal driven flow and the pressure driven flow become more and more intense.
Conclusions This paper added the semicircular obstacles to the inner wall of the microchannel, and the radius of the semicircular obstacle is changed to study the influence of obstacles on the thermal transpiration effect and the hydrogen flow state in the microchannel. The conclusions are as follows: (1) Semicircular obstacles change the pressure increase of the hydrogen Knudsen compressor. When the flow state in the channel belongs to the slip region, with the semicircular radius increasing, the lift-pressure capability in the channel is lower than the straight microchannel. The velocity of the thermal driven flow and the pressure driven flow is more and more slow. (2) Semicircle obstacles have an influence on the state of the hydrogen flow through the channel. The larger semicircular radius extended the hydrogen flow path, which weakens the temperature gradient and the thermal transpiration effect. The decline of the microchannel characteristic length along the vertical centerline decreases the velocity of the Poiseuille flow in the central channel. The thermal driven flow and the pressure driven flow produce y-component velocity and even lead to the backflow under the effect of semicircles. With the increase of semicircle obstacles radius, the backflow in the microchannel becomes more remarkable. (3) Semicircular obstacles of the microchannel have an influence on the Knudsen layer, which is made of the thermal transpiration flows. Semicircular obstacles make the Knudsen layer spread to the central channel. The junction of the thermal transpiration flow and the Poiseuille flow at semicircular obstacles move towards the channel center.
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Acknowledgements This project was supported by the National Natural Science Foundation of China, Grants No. 51106137, and the Open Foundation of the State Key Laboratory of Fluid Power and Mechatronic Systems, Grants No. GZKF-201811.
references
[1] Gerdroodbary MB, Anazadehsayed A, Hassanvand A, Moradi R. Calibration of low-pressure MEMS gas sensor for detection of hydrogen gas. Int J Hydrogen Energy 2018;43(11):5770e82. https://doi.org/10.1016/j.ijhydene.2017. 11.087. zquez RM, Gra cia I, Cane C, [2] Mozalev A, Calavia R, Va Correig X, et al. MEMS-microhotplate-based hydrogen gas sensor utilizing the nanostructured porous-anodic-aluminasupported WO3 active layer. Int J Hydrogen Energy 2013;38(19):8011e21. https://doi.org/10.1016/j.ijhydene.2013. 04.063. [3] Wang Q, Zhao Y, Wu F, Bai J. Study on the combustion characteristics and ignition limits of the methane homogeneous charge compression ignition with hydrogen addition in micro-power devices. Fuel 2019;236(15):354e64. https://doi.org/10.1016/j.fuel.2018.09.010. [4] Tang AK, Cai T, Huang QH, Deng J, Pan JF. Numerical study on energy conversion performance of microthermophotovoltaic system adopting a heat recirculation micro-combustor. Fuel Process Technol 2018;180:23e31. https://doi.org/10.1016/j.fuproc.2018.07.034. [5] Lee J, Kim T. Micro space power system using MEMS fuel cell for nano-satellites. Acta Astronaut 2014;101:165e9. https:// doi.org/10.1016/j.actaastro.2014.04.010. [6] Sharipov F. Rarefied gas flow through a long rectangular channel. J Vac Sci Technol A 1999;17(5):3062e6. https://doi. org/10.1116/1.582006. [7] Cardenas MR, Graur I, Perrier P, Meolans JG. Thermal transpiration flow: a circular cross-section microtube submitted to a temperature gradient. Phys Fluids 2011;23(3):031702. https://doi.org/10.1063/1.3561744. [8] Ye JJ, Yang J, Zheng JY, Ding XT, Wong I, Li WZ, et al. A multiscale flow analysis in hydrogen separation membranes using a coupled DSMC-SPH method. Int J Hydrogen Energy 2012;37(1):894e902. https://doi.org/10.1016/j.ijhydene.2011. 03.163. [9] Karniadakis GE, Beskok A. Micro flows: fundamentals and simulation. New York: Springer Verlag; 2002. [10] Kanki T, Luchi S. Poiseuille flow and thermal creep of a rarefied gas between parallel plates. Phys Fluid 1973;16(5):594e9. https://doi.org/10.1063/1.1694393. [11] Aoki K, Sone Y, Takata S, Takahashi K, Bird GA. One-way flow of a rarefied gas induced in a circular pipe with a periodic temperature distribution. AIP Conf Proc 2001;585(1):940e7. https://doi.org/10.1063/1.1407660. [12] Sone Y, Sugimoto H. Vacuum pump without a moving part and its performance. AIP Conf Proc 2003;663(1):1041e8. https://doi.org/10.1063/1.1581654. [13] Han YL, Young M, Muntz EP, Shiflett G. Knudsen compressor performance at low pressures. AIP Conf Proc 2005;762(1):162e7. https://doi.org/10.1063/1.1941530. [14] Young M, Han YL, Muntz EP, Shiflett G. Characterization and optimization of a radiantly driven multi-stage Knudsen compressor. AIP Conf Proc 2005;762(1):174e9. https://doi.org/ 10.1063/1.1941532.
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[15] Pham-Van-Diep G, Keeley P, Muntz EP, Weaver DP. A micromechanical Knudsen compressor. In: Harvey J, Lord G, editors. Rarefied gas dynamics. vol. 1. Oxford: Oxford University Press; 1995. p. 715e21. [16] Vargo SE, Muntz EP, Shiflett GR. Knudsen compressor as a micro-and macroscale vacuum pump without moving parts or fluids. J Vac Sci Technol A 1999;17(4):2308e13. https://doi. org/10.1116/1.581765. [17] Vargo SE, Muntz EP. Initial results from the first MEMS fabricated thermal transpiration-driven vacuum pump. AIP Conf Proc 2001;585(1):502e9. https://doi.org/10.1063/1. 1407602. [18] McNamara S, Gianchandani YB. On-chip vacuum generated by a micromachined knudsen pump. J Microelectromech Syst 2005;14(4):741e6. https://doi.org/10.1109/JMEMS.2005. 850718. [19] Gupta NK, An S, Gianchandani YB. A si-micromachined 48stage Knudsen pump for on-chip vacuum. J Micromech Microeng 2012;22(10):105026. https://doi.org/10.1088/09601317/22/10/105026. [20] Wang BT, Lu W, Xu K, Wang X. Effect of thermal transpiration on vacuum pumping characteristics: a theoretical study. Chin J Vacuum Sci and Technol 2017;37(9):866e72. https://doi.org/10.13922/j.cnki.cjovst.2017. 09.05. [21] Yang L, Lu W, Cao C, Liu JY, Chen H, Wang Y. Design and performance analyses of refrigeration systems with Knudsen compressors. Cryogenics 2015;1:37e44. € rneklett A, Nilsson P, [22] Muthuvijayan I, Antelius M, Bjo Thorslund R. Design and development of a novel knudsen compressor as a part of a joule-thomson cryocooler. J Phys Conf Ser 2017;922(1e10):012020. https://doi.org/10.1088/17426596/922/1/012002. [23] Takata S, Sugimoto H, Kosuge S. Gas separation by means of the Knudsen compressor. Eur J Mech B Fluid
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Characteristics of thermal transpiration effect and the hydrogen flow behaviors in the microchannel with semicircular obstacle Jianjun Ye a,b,*, Junda Shao a, Zongrui Hao c, Shehab Salem a, Yuan Zhang a, Yiwen Wang a, Zhengyang Li a a
School of Power and Engineering, Huazhong University of Science and Technology, Wuhan, 430074, China The State Key Laboratory of Fluid Power and Mechatronic Systems, Zhejiang University, Hangzhou, 310027, China c Qilu University of Technology, The Institute of Oceanographic Instrumentation, Qingdao, 266001, China b
article info
abstract
Article history:
The thermal transpiration effect has great potential applications for the hydrogen energy.
Received 22 December 2018
In this paper, the thermal transpiration effect and the hydrogen flow behaviors are studied
Received in revised form
in the microchannel with the semicircular obstacles. Firstly, the slip boundary model is
14 March 2019
used in the simulation of the flow performance in the microchannel. The validity of the
Accepted 9 April 2019
model at different Kn is verified by comparing with some previous work. Then, the
Available online 7 May 2019
hydrogen flow characteristics of the thermal transpiration effect with the semicircular obstacle are investigated. The result shows that as the size of the semicircular obstacle
Keywords:
increases, the hydrogen flow path of the thermal transpiration effect becomes longer, and
Thermal transpiration effect
the temperature gradient decreases. As the characteristic length of the hydrogen flow
Hydrogen flow
decreases, there is an obviously negative influence on the thermal transpiration flow. A
Knudsen number
deeper analysis shows that the thermal driven flow and the pressure driven flow will
Slip boundary conditions
produce y-component velocity, which leads to a backflow under the effect of semicircles,
Velocity distribution
and the semicircular obstacles make the Knudsen layer spread to the channel center. © 2019 Hydrogen Energy Publications LLC. Published by Elsevier Ltd. All rights reserved.
Introduction As a clean and renewable energy, hydrogen has been used widely with the rapid development of microelectromechanical systems (MEMS) and fuel cell technology. For example, the energy consumption of the micro hydrogen sensor based on MEMS technology is only 1% compared with the conventional hydrogen sensor. Moreover, the MEMS hydrogen sensor has better sensitivity and faster test speed [1,2]. Because of the advantages of high energy density and
clean characteristic, hydrogen is also used as the energy source in the MEMS power systems, such as micro hydrogen combustors, micro hydrogen thermophotovoltaic reactor, micro hydrogen fuel cell system, etc [3e5]. Furthermore, due to the characteristic scale of the MEMS and other microdevice systems is small, the thermal transpiration effect exists in the channel. Based on this effect, the transmission and pressure increase of the gas medium can be realized only under the condition of temperature difference. There are no moving parts in the system, which has obvious advantages of long life
* Corresponding author. School of Power and Engineering, Huazhong University of Science and Technology, Wuhan, 430074, China. E-mail address: [email protected] (J. Ye). https://doi.org/10.1016/j.ijhydene.2019.04.079 0360-3199/© 2019 Hydrogen Energy Publications LLC. Published by Elsevier Ltd. All rights reserved.
i n t e r n a t i o n a l j o u r n a l o f h y d r o g e n e n e r g y 4 4 ( 2 0 1 9 ) 2 9 7 2 4 e2 9 7 3 2
and high reliability. Therefore, the thermal transpiration effect has a very active application prospect and potential for hydrogen transmission in the MEMS devices. The thermal transpiration effect, which is a gas flow along the microchannel wall in the direction of the temperature gradient, is a classical phenomenon in the microscale flow and has attracted many attentions recently [6,7]. The thermal transpiration effect leads to a pressure increase in the hot container without the pressure difference, and it can be used to manufacture a hydrogen compressor and a hydrogen power system. Actually, the understanding of the hydrogen flow behaviors is better for the design and the application of this kind of micromachines. Ye et al. adopted an original numerical method uniting the smoothed particle hydrodynamics (SPH) technique with the direct simulation Monte Carlo (DSMC) technique to simulate the hydrogen flow in multi-scale between the membranes [8]. The velocity and the mass flow rate of the hydrogen and the carbon monoxide in the multi-scale flow were discussed. The results showed that the phenomenon is different from that observed in either a macroscopic or microscopic flow and provided a reference for membrane design in hydrogen production. Karniadakis et al. constructed a theoretical model with two containers connected with a microchannel, which was filled with air [9]. In their works, the relationship between the pressure of two containers at the equilibrium state and the temperature of two containers at the initial state were derived. At equilibrium, the pressure difference of two containers was obtained by theoretical analysis and numerical simulation method. They found that it is possible to have a non-zero flow rate in a microchannel even in the case of a zero pressure gradient. The pressure difference driving some gas to move from the hot container to the cold container, which is in the reverse direction of the thermal transpiration flow, is called the Poiseuille flow. Kanki et al. used the Bhatnagar-Gross-Krook (BGK) model to discuss the Poiseuille flow and the thermal transpiration flow of a rarefied gas between two parallel plates [10]. Their results indicated that the ratio of the thermal driven flow rate to the pressure driven flow rate drops to 0 in the continuous zone, and the value is closed to 0.5 when the gas density is near zero. A notable application of the thermal transpiration effect is the Knudsen pump or compressor and its variants. Some scholars have done a lot of researches on the pressure performance, mass flow rate and application of the Knudsen compressor [11e14]. Pham-Van-Diep et al. analyzed the pressure and temperature distributions of the microchannel and the connection channel in the single-stage and multistage Knudsen compressors [15]. The pressure performance and mass flow rate of the model are improved by parallel arraying multiple microchannel connected with containers in series. As materials were developed, Vargo et al. built a physical Knudsen compressor on the foundation of the thermal transpiration effect, which can be used for practical applications for the first time. The compressor was consisted of silicon chips, aerogels, heat-resistant glass containers for improving pressure and vacuum aluminum tubes. The results showed that the ratio of the pressure is closed to 1.02 [16,17]. About the researches on the mass flow rate of the channel,
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McNamara et al. designed a miniature Knudsen compressor with a rectangular cross-section of 0.1 10 mm [18]. When the temperature difference between the two containers is 1100K, the maximum flow rate approaches to 3 105cc/min. Recently, researchers have applied the thermal transpiration effect in many more fields, such as micro/medium gas vacuum pump, high pressure gas source of MEMS, refrigeration system, gas separation equipment, gas conveyor, and so on [19e24]. Except for the experimental and applied researches, in terms of numerical methods, Alexeenko et al. numerically investigated the thermal transpiration flow in a twodimensional finite length microchannel with the length-toheight ratio of 5, the pressure ratio equal to 1 and a cylindrical tube using the DSMC method and BGK model [25]. They studied the influence of different temperature gradient on the thermal transpiration effect. Their results showed that there is a pressure difference along the channel in despite of the inlet and outlet pressure values are same. They also compared different numerical methods. Ye et al. used the DSMC method united with the SPH method to simulate the influence of rarefaction and temperature gradient on multiscale flow in the application of the Knudsen pumps [26]. These results showed that when the gas flow becomes more rarefied, the thermal transpiration effect becomes more significant. Ohwada et al. used the linearized Boltzmann equation with hard-sphere molecules to discuss the Poiseuille flow and the thermal transpiration flow of a rarefied gas among double parallel-plates [27]. They predicted the mass rates and the velocity distributions of the Poiseuille and the thermal transpiration flow. The results showed that, as the Kn number increases, the mass rate of thermal transpiration flow increases continuously. Compared with the existing numerical methods and experimental results, the proposed method has better accuracy. According to the previous studies, it is found that the researches of the thermal transpiration effect mainly involved in the experimental study and the numerical study. Most of the work focused on the characteristic of the flow, heat and mass transfer in microchannel under the thermal transpiration effect, and the microchannel studied were uniform and smooth micro-straight channels. However, the shape and characteristic length of the channel has significant influence on the thermal transpiration effect. And the gas flow in the microchannel with more complex structures often encountered in microsystems, such as the channel with obstacles, the influence of the thermal transpiration effect on the flow has not been investigated, the flow behaviors and the mechanism are also ambiguous. Therefore, in this paper, the numerical simulation of the hydrogen flow in the microchannel with semicircular obstacles is implemented by using a continuum medium model with slip boundary conditions. The semicircular obstacles are added to the inner wall of the microchannel, and the radius of the semicircular obstacle is changed to investigate the effect of the obstacle on the hydrogen flow characteristic in the microchannel. It is useful for a better understanding on the flow field in the microchannel and the design and application of microdevices based on the thermal transpiration effect in the future.
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coefficient. The generalized Maxwell's model gave the above three coefficients by:
Numerical equations Continuity equations For the numerical simulation of macro-scale fluids, three equations of mass conservation, momentum conservation and energy conservation in traditional fluid mechanics [28], namely Navier-Stokes equations (NeS equations), are often used, as shown in equations (2.1)e(2.3). The NeS equations ignore the molecular properties of gases and liquids, and regard fluids as continuous media. It is called continuum model by describing the changes in density, velocity, pressure, temperature and other macroscopic flow with time and space. vu v þ ðruk Þ ¼ 0 vt vxk
(2.1)
P vui vui v ki þ uk r þ rgi ¼ vt vxk vxk
(2.2)
ve ve vqk X vui þ uk r þ ki ¼ vt vxk vxk vxk
(2.3)
where r is its density, uk is the instantaneous velocity P component (u, v, w), ki is the second-order stress tensor (surface force per unit area), gi is the volume force per unit mass, e is the internal energy, qk is the sum of the heat flux vectors of conduction and radiation.
Boundary conditions For the calculation of the model in this paper, the internal gas of the microchannel is a rarefied gas in the micro scale. The Knudsen number describing the gas rarefaction is more than 0.01. With the characteristic length of the microchannel decreases, the interaction between the gas and the wall is far more than that between the gas molecules, so the numerical results based on the traditional NeS equations are far from the experimental results. Therefore, more and more researchers added reasonable equations of boundary conditions to make the NeS equations more suitable for calculating the micro-scale flow. Based on previous studies, Lockerby and Kennard proposed new boundary conditions, which take into account the effect of wall temperature on the thermal transpiration effect on the basis of ideal gas equation [29,30], as shown in equations (2.4)e(2.7): uslip ¼ u uw ¼
lss sT m tn;t þ Vt Tg rTg m
(2.5)
Vt Tg ¼ VTg VTg $n n
(2.6)
nT tn n
2 at at
(2.8)
sT ¼
3 4
(2.9)
xT ¼
2 at 2g k at g þ 1 mCp
Numerical modeling A 2D numerical model under the thermal transpiration effect is shown in Fig. 1, made of silicon material with the thickness of d ¼ 1mm. The model has a length L and a height H. The cavity of the model filled with hydrogen gas, which is divided into the cold container, the hot container and the microchannel connected them. The length of the microchannel is l and the height is h. At the inner wall of the microchannel along the vertical centerline, two semicircular obstacles with a radius of R are distributed symmetrically. The temperature of the cold container and the hot container is indicated by Tc and Th respectively. In order to prevent the external temperature field from interfering with the temperature gradient inside the microchannel, assuming the exterior wall of the microchannel is adiabatic. Before the calculation, the gas pressure of the model is P0. When the flow achieves the equilibrium state, the pressure of the hydrogen gas in the cold container and the hot container is expressed by Pc and Ph respectively, which is the average relative pressure compared with P0. DP represents the pressure difference between the cold container and the hot container.
(2.7)
Where uslip is the slip velocity, l is the mean free path of gas, n is the boundary normal, t is the viscous stress tensor, m is the viscosity of the gas, r is its density, and Tg is its temperature. Tw is the wall temperature, ss is the viscous slip coefficient, sT is the thermal slip coefficient, and xT is the temperature jump
(2.10)
Where at is the tangential momentum accommodation coefficient, the value of at means the fraction of molecules which are reflected diffusely from the wall, g is the specific heat ratio of the gas, k is the thermal conductivity of the gas and Cp is the heat capacity at constant pressure. In this paper, the NeS equations with the above boundary conditions is adopted for the numerical calculation of the microchannel model. According to Sharipov's work, the present work chooses st ¼ 0.9 as the tangential momentum accommodation coefficient [31].
(2.4)
Tw ¼ Tg xT ln$VTg
tn;t ¼ tn
ss ¼
Fig. 1 e Computational model under the thermal transpiration effect.
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In order to study the effect of the different obstacles on the thermal transpiration effect, this paper added two semicircular obstacles with varying radius to the inner wall of the microchannel. The radius of semicircular obstacles is R ¼ 0 mm, R ¼ 0.1 mm, R ¼ 0.2 mm, R ¼ 0.3 mm and R ¼ 0.4 mm respectively. The numerical details of the studied cases are shown in Table 1. The computational model is meshed with unstructured grids, and the local encryption is performed at the half arcs. Grid detection histogram shows that the quality of the grid is all above 0.54. It is verified the mesh number by the maximum pressure difference between the cold container and the hot container. Finally, when the radius of the semicircular obstacles is R ¼ 0mm, R ¼ 0.1mm, R ¼ 0.2mm, R ¼ 0.3mm and R ¼ 0.4mm, the mesh number of the model is set at 5.7 104, 6.9 104, 6.3 104, 6.0 104 and 5.9 104 respectively.
Results and discussions Verification of the numerical method used in the model To verify the validity of the numerical model in this article, the simulation result for the thermal transpiration effect is compared with the analytic solution from Karniadakis’s work [9]. In their work, the air gas flow in a Knudsen compressor with a straight microchannel is studied. The average value of Knudsen number in the microchannel is 0.052. Both in the present work and the previous work, the temperature of two containers are set at 300 K and 400 K, respectively. Table 2 indicates the comparison results and the deviation of maximum pressure difference. The normalized pressure variation along the central channel under the standard atmospheric pressure in present work are compared with the analytical solution in the literature, which is shown in Fig. 2. It can be seen from Table 2, the maximum pressure difference of the analytical solution in Ref. [9] and the numerical result in the present work are 342 Pa and 341 Pa, respectively. The deviation of the maximum pressure difference between the results in this paper and the analytical solution is 0.29%. It shows that the present results are in good agreement with the analytical solution, which proves the calculation method has good applicability in simulating the thermal transpiration effect in the microchannel.
Kn number distribution in the model The Kn number distribution along the microchannel centerline is shown in Fig. 3, which represents the rarefaction of the gaseous medium.
Logarithmic coordinates are adopted for a better understanding. At standard atmospheric pressure, the Kn number in the cold container and the hot container is 0.0061 and 0.0086 nearly, which indicates the hydrogen gas in containers is in the continuum regime. At the entrance of the microchannel, due to the reduction of the microchannel characteristic length, the Kn number increased to 0.050. For the changing of the Kn number along with the microchannel, when the radius is R ¼ 0 mm, it means the channel is straight, the value of Kn increased from 0.050 to 0.068 linearly. When there are the semicircular obstacles existing in the microchannel, the Knudsen number distribution changes obviously. From the beginning of the semicircular obstacles, the characteristic length of the channel decreases gradually, and the Knudsen number increases. At the top of the semicircular obstacles, the Knudsen number reaches the maximum value, and then decreases gradually. When the radius of semicircular obstacles is R ¼ 0.1mm, R ¼ 0.2 mm, R ¼ 0.3 mm and R ¼ 0.4 mm, the maximum of the Kn number along the channel is 0.064, 0.070, 0.078 and 0.087 respectively. It proves that the hydrogen gas of the microchannel is in the slip regime. In the whole process, the flow state of the hydrogen gas skips from the continuum regime to the slip regime, finally it changes to the continuum regime again. The change of the flow regime occurs in the model. As discussed in the previous work, when the Kn number is in the range of (103, 101), the continuum medium model with slip boundary conditions is satisfied with the calculation in the model.
Semicircular obstacles influence on the distribution of pressure and velocity in the microchannel As it is shown in Fig. 4, the normalized hydrogen pressure varies along the centerline of the channel with a different radius of semicircular obstacles. Through the analysis of the fixed R, the hydrogen gas pressure in the container is nearly constant. Nevertheless, there is an obvious pressure rise in the microchannel. According to the gas equation of state P ¼ nkT, where P is pressure changes of the monitoring points, n is the molecular number density, k is Boltzmann's constant and T is temperature changes of the monitoring points. The temperature gradient along the microchannel wall can drive the thermal transpiration flow to creep from the cold container towards the hot container along walls, which made the hydrogen gas pressure increased in the channel by degrees. In the containers, since the distribution of the temperature and the molecular number density is equal in equilibrium, the pressure barely changed. Finally, because the hydrogen pressure in the hot container is higher than that in the cold
Table 1 e Numerical details of the cases. Quantity Model length L/mm Model height H/mm Microchannel length l/mm Microchannel height h/mm Type of gaseous medium
30 20 15 2.5 Hydrogen
Quantity Temperature in cold container Tc/K Temperature in hot container Th/K Pressure in cold container Pc/Pa Pressure in hot container Ph/Pa Operate pressure P0/Pa
300 400 0 0 1 atm
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Table 2 e Compared results in R ¼ 0.mm Quantity Average of Kn in microchannel Kn Maximum pressure difference DPmax /Pa
Fig. 2 e Normalized pressure variation along the central channel at 1 atm
container, the hydrogen gas pressure difference produced between the two containers. As the radius of semicircular obstacles increases, the hydrogen gas pressure in the hot container first increased from 202.18 Pa at R ¼ 0 mm to 204.67 Pa at R ¼ 0.1 mm, and then dropped little by little. When the radius of semicircular obstacles is R ¼ 0.2 mm, R ¼ 0.3 mm and R ¼ 0.4 mm, the hydrogen pressure in the hot container is 196.17Pa, 196.46Pa and 195.32Pa, respectively. The average pressure of the hydrogen gas in the cold container first increased from 172.40Pa to 155.19Pa and then decreased to 165.87Pa gradually. On the one hand, compared with the straight channel, semicircular
Fig. 3 e Kn distribution along the centerline of the microchannel with different R.
Analytic solution
Our Results
0.052 342
0.054 341
Fig. 4 e Normalized pressure profile of hydrogen flow along the centerline with different R.
obstacles weakened the temperature gradient, which influenced the thermal driven flow and produced a diversion of the flow in the y-direction. On the other hand, the obstacles hindered the Poiseuille flow to move from the hot container towards the cold container in the central channel. When the radius of the semicircular obstacles is 0.1 mm, the hydrogen gas pressure in the cold container increases 17.20Pa, and the pressure in the hot container only increases 2.50Pa. The results show that the hydrogen gas pressure at both containers increases, whereas the pressure increase in the cold container is higher than that in the hot container, therefor the pressure difference of the hydrogen Knudsen compressor decreases. As the radius of the semicircular obstacles enlarge, the characteristic length of the microchannel deceased deeply, the Kn number increased, and then the value of QT/QP also increased, where QT is the thermal driven flow coefficient, QP is the pressure driven return flow coefficient. The influence of semicircular obstacles on the thermal transpiration flow is higher than the Poiseuille flow. With the increase of the thermal transpiration flow path, the temperature gradient along the microchannel is further reduced, which weakened the thermal transpiration effect and made the pressure values of the hot container are lower and that of the cold container are also lower. Fig. 5 shows the x-component of velocity along the vertical centerline with different radius of the semicircular obstacles. As the radius of the semicircular obstacles increases from R ¼ 0 mm to R ¼ 0.4 mm, the velocity of the Poiseuille flow through the central channel is getting lower and the decreasing amplitude is getting lower. That is because, with the enlarging of the semicircular obstacles radius, the characteristic length of the microchannel along the vertical
i n t e r n a t i o n a l j o u r n a l o f h y d r o g e n e n e r g y 4 4 ( 2 0 1 9 ) 2 9 7 2 4 e2 9 7 3 2
Fig. 5 e X-component of velocity along the vertical centerline with different R.
centerline decreased. For the thermal driven flow along the wall, the maximum of velocity dropped by 0.63 m/s from R ¼ 0 mm to R ¼ 0.1 mm. That is because, when semicircular obstacles exist, the velocity of the thermal driven flow produced a diversion of the hydrogen flow in the y-direction. Besides, semicircular obstacles extended the flow path, which made the temperature gradient decreased and the velocity dropped. The velocity of the hydrogen flow had a maximum cut. When the radius of the semicircular obstacles is R ¼ 0.2 mm, the microchannel characteristic length decreased, the Kn number increased, and then the value of QT/QP also increased, which made the influence of the thermal transpiration effect is more intense and the hydrogen flow velocity is higher than before. As the radius continues to rise, the increase of the ratio of QT/QP is lower than the increase of the thermal driven flow path. The drop of the temperature gradient lead to a decrease of the thermal driven flow velocity. In summary, semicircular obstacles set at the wall of the microchannel caused the backflow to occur in the pressure driven flow in the central channel and the thermal driven flow along the wall. The backflow made the Knudsen layer spread to the central channel. The region of the Poiseuille flow decreased and the velocity of the Poiseuille flow also dropped. The hydrogen mass flowrate of the microchannel is shown in Fig. 6. The mass flowrate is calculate based on the accumulation. The positive flow direction represents the thermal transpiration flow and negative direction represents the Poiseuille flow. It can be found that the mass flowrate of the thermal transpiration flow is as the same value of the Poiseuille flow, just with different flow direction. Which indicates that the flow achieves the equilibrium state in the microchannel with the thermal transpiration flow and the Poiseuille flow. When the obstacle radius increases from R ¼ 0 mm to R ¼ 0.4 mm, the mass flowrate of the thermal transpiration flow decreases from 5.0 108 kg/s to 1.7 108 kg/s. It shows that the semicircular obstacles affect the thermal transpiration flow as well as the Poiseuille flow. As it is indicated in Fig. 7 that the x-component of velocity along the centerline of the model with a different radius of semicircular obstacles, is actually the x-component of the
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Fig. 6 e Hydrogen mass flowrate of the microchannel with different semicircular obstacles. velocity distribution of the Poiseuille flow. When R ¼ 0 mm, the velocity of the microchannel centerline increased linearly. The maximum velocity of the hydrogen flow nearly occurred in the connection between the microchannel and the hot container is 0.75 m/s. The x-component of velocity along the vertical centerline with R ¼ 0 mm is 0.75 m/s. When semicircular obstacles exist, the velocity along the vertical centerline suddenly dropped. The value of velocity reduced by 0.50 m/s at R ¼ 0.1 mm, that declined 33.5% compared with the velocity at R ¼ 0 mm. When the radius of semicircular obstacles is R ¼ 0.2 mm, R ¼ 0.3 mm and R ¼ 0.4 mm, the value of the velocity reduced by 0.43 m/s, 0.39 m/s and 0.37 m/s respectively, and the velocity of hydrogen flow declined 41.8%, 47.3% and 50.9% compared with the velocity at R ¼ 0 mm. The results show clearly that the velocity of the Poiseuille flow is getting lower and the decline range is getting lower. That is because, on the one hand, the existence of the semicircular obstacles lessened the characteristic length of the microchannel, which makes the Knudsen layer spread to the central channel and hinders the Poiseuille flow. On the other hand, due to the existence of the semicircular obstacles, there is a backflow
Fig. 7 e X-component of velocity along the centerline of the model with different R.
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Fig. 8 e Velocity vector of hydrogen flow in the model with R ¼ 0.
appeared in the Poiseuille flow, which deceases the flow velocity.
Semicircular obstacles influence on the state of hydrogen flow in the microchannel Fig. 8 shows the distribution of the velocity vector of hydrogen flow in the model with the straight channel. The flow
directions are indicated by the logarithmic arrows. There are the temperature gradient drove the thermal transpiration flow to creep from the cold container to the hot one along the channel walls. The pressure difference between the two containers drove the Poiseuille flow to move from the hot container towards the cold container through the central channel, which agrees with the theoretical analysis [32]. The return pressure-driven flow through the centerline occupies a significant portion of the channel cross-section whereas the thermal driven flow in the direction of the temperature gradient is confined to a small region along the wall. According to the results of the previous studies, when the Kn number is 0.059, the ratio of the thermal driven flow coefficient to the pressure driven flow coefficient is QT/QP z 0.01, which marks the quantity of the thermal driven flow is much less than the pressure driven flow. The velocity vector showed that, in the above of containers, there are two clockwise vortices, and in the below of containers, there are two anticlockwise vortices, which caused by the separation of the thermal driven flow due to the adverse pressure difference. It can be seen in Fig. 9, the influence of semicircular obstacles on hydrogen flow state in the microchannel mainly embodied in the y-component of velocity and the backflow with different R.
Fig. 9 e Velocity vector of hydrogen flow at semicircle in the microchannel.
i n t e r n a t i o n a l j o u r n a l o f h y d r o g e n e n e r g y 4 4 ( 2 0 1 9 ) 2 9 7 2 4 e2 9 7 3 2
In Fig. 9aee, when semicircular obstacles exist, the velocity of hydrogen flow suddenly dropped at the semicircular arcs. As the enlarging of the semicircular radius, the decreasing amplitude in velocity of the thermal driven flow enhanced, and the interface between the thermal transpiration flow and the Poiseuille flow gradually diffuses to the microchannel center, which agree with x-component of velocity along the vertical centerline in Fig. 5 of the previous work. In Fig. 9bee, because of the existence of semicircle obstacles, the thermal driven flow along the wall of the channel produced y-component velocity and even caused the backflow, which enhanced the Poiseuille flow in the central channel. The phenomenon is different from that at R ¼ 0 mm. Affected by the semicircular obstacles, the Poiseuille flow produced y-component velocity and even caused the backflow, which further strengthens the thermal driven flow to creep at the right side of the semicircle. Comparing with Fig. 9bee, it can be found that with the radius of semicircular obstacles increase, the reflux trend of the thermal driven flow and the pressure driven flow become more and more intense.
Conclusions This paper added the semicircular obstacles to the inner wall of the microchannel, and the radius of the semicircular obstacle is changed to study the influence of obstacles on the thermal transpiration effect and the hydrogen flow state in the microchannel. The conclusions are as follows: (1) Semicircular obstacles change the pressure increase of the hydrogen Knudsen compressor. When the flow state in the channel belongs to the slip region, with the semicircular radius increasing, the lift-pressure capability in the channel is lower than the straight microchannel. The velocity of the thermal driven flow and the pressure driven flow is more and more slow. (2) Semicircle obstacles have an influence on the state of the hydrogen flow through the channel. The larger semicircular radius extended the hydrogen flow path, which weakens the temperature gradient and the thermal transpiration effect. The decline of the microchannel characteristic length along the vertical centerline decreases the velocity of the Poiseuille flow in the central channel. The thermal driven flow and the pressure driven flow produce y-component velocity and even lead to the backflow under the effect of semicircles. With the increase of semicircle obstacles radius, the backflow in the microchannel becomes more remarkable. (3) Semicircular obstacles of the microchannel have an influence on the Knudsen layer, which is made of the thermal transpiration flows. Semicircular obstacles make the Knudsen layer spread to the central channel. The junction of the thermal transpiration flow and the Poiseuille flow at semicircular obstacles move towards the channel center.
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Acknowledgements This project was supported by the National Natural Science Foundation of China, Grants No. 51106137, and the Open Foundation of the State Key Laboratory of Fluid Power and Mechatronic Systems, Grants No. GZKF-201811.
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