- Email: [email protected]
The hydrogen flow characteristics of the multistage hydrogen Knudsen compressor based on the thermal transpiration effect
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 2 6 3 2 e2 2 6 4 2
Available online at www.sciencedirect.com
ScienceDirect journal homepage: www.elsevier.com/locate/he
The hydrogen flow characteristics of the multistage hydrogen Knudsen compressor based on the thermal transpiration effect Jianjun Ye a,b, Junda Shao a, Junlong Xie a,*, Zhenhua Zhao a, Jiangcun Yu a, Yuan Zhang a, Shehab Salem a a b
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
article info
abstract
Article history:
Multistage hydrogen Knudsen compressor based on the thermal transpiration effect has
Received 8 February 2019
very exciting prospect for the hydrogen transmission in the micro devices. Understanding
Received in revised form
of the hydrogen flow characteristic is the key issue for the designs and applications of the
28 March 2019
hydrogen energy systems. Firstly, the numerical models of the multistage hydrogen
Accepted 14 April 2019
Knudsen compressor are established. The distributions of the rarefaction, velocity and
Available online 18 May 2019
temperature at different stages of the hydrogen flow are calculated and presented. Moreover, the dimensional pressure increases of the hydrogen gas flow are analyzed, and the
Keywords:
flow behaviors in the microchannel and the connection channel are discussed. Secondly,
Thermal transpiration effect
the numerical simulation at different connection channel height is implemented, and the
Hydrogen flow
hydrogen gas flow characteristics in the connection are analyzed. Especially, the perfor-
Pressure increase
mances of the pressure drop in the connection channel under different channel heights are
Hydrogen Knudsen compressor
studied, and the hydrogen gas compression characteristics of different cases are compared
Flow vortex
and discussed. Also, the effect of the connection channel height on the hydrogen gas pressure increase in the microchannel is investigated. The studies presented in this paper could be greatly beneficial for the hydrogen detection and transmission. © 2019 Hydrogen Energy Publications LLC. Published by Elsevier Ltd. All rights reserved.
Introduction The thermal transpiration effect refers to the thermal force on a gas due to a temperature difference in microchannel, and is able to maintain a certain pressure difference and cause a gas flow [1,2]. This phenomena can be used for the design of the hydrogen Knudsen compressor, which has a great potential development for the safety of hydrogen energy devices, such as the hydrogen detectors and hydrogen sensors [3e5].
Generally, the pressure performance of the hydrogen Knudsen compressor is relative to the channel structures. Furthermore, for the multi-stage Knudsen compressor, its performance is closely related to the stage number in series. There are also many studies on the pressure increase of Knudsen compressor, Pham-Van-Diep et al. proposed a Knudsen compressor model based on the micro-electro-mechanical systems (MEMS), which can obtain a high pressure difference by connecting multiple single-stage Knudsen
* Corresponding author. E-mail address: [email protected] (J. Xie). https://doi.org/10.1016/j.ijhydene.2019.04.155 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 2 6 3 2 e2 2 6 4 2
Nomenclature Cp D H k Kn L L1 L2 mi Mj n N P P0 Patm T TH TC Tg Tw Tjump uc um uM uslip g DP l m xT r ss st sT t
thermal capacity under constant pressure (J/ (kg$K)) microchannel characteristic length (mm) connection channel height (mm) gas thermal conductivity (W/(m$K)) Knudsen number each stage length (mm) microchannel length (mm) connection channel length (mm) cross section at the microchannel cross section at the connection channel boundary normal stage number local hydrogen pressure in the channel (Pa) hydrogen pressure on the left wall of the beginning container (Pa) standard atmospheric pressure (Pa) hydrogen temperature along the channel centerline (K) high temperature condition (K) low temperature condition (K) gas temperature (K) wall temperature (K) wall jump temperature (K) hydrogen velocity along the channel centerline (m/s) hydrogen velocity in the microchannel (m/s) hydrogen velocity in the connection channel (m/s) wall slip velocity (m/s) gas specific heat ratio hydrogen pressure difference between the beginning and end container (Pa) mean molecular free path (mm) gas viscosity (Pa$s) temperature jump coefficient gas density (kg/m3) viscous slip coefficient tangential momentum accommodation coefficient temperature slip coefficient viscous stress tensor
compressors in series [6]. Miles et al. manufactured the singlestage, five-stage and ten-stage Knudsen compressors respectively and used a thermoelectric module to realize the temperature difference. It was found that the compression ratio of Knudsen compressor can be improved obviously by increasing the number of series stages [7]. This study presents theoretical and practical guidance for improving the pressure increase of the Knudsen compressor. Ye et al. applied the coupled method to study the influence of the microchannel height and length on the hydrogen flow characteristics in multi-scale channels. The results show that the length and height of the microchannel are the main factors affecting the change of
22633
hydrogen pressure and flow rate in the channel, which provides guidance for the application of the hydrogen Knudsen compressor [8]. Young et al. found that the theoretical operating pressure of a single-stage Knudsen compressor can be as low as 10 mTorr. When the pore size of microchannel material is small as few nanometers, its operating pressure can be as high as 10 atm [9]. Based on the compressible Navier-Stokes equations with velocity slip boundary conditions, Leontidis et al. constructed the research model of Knudsen compressor and investigated its working performance under different microchannel sizes [10]. Han et al. used the direct simulation Monte Carlo (DSMC) method to analyze the performance of Knudsen compressor. It was found that the gas rarefied degree and the reverse thermal transpiration effect in the connection channel of Knudsen compressor were the main factors to limit its performance, and when the reverse thermal transpiration effect in the connection channel was weak, the performance of Knudsen compressor becomes better [11]. In view of the above researches, it seems that the study of the hydrogen flow behaviors is important to the design and application of the hydrogen Knudsen compressor, and the pressure increase performance of single-stage Knudsen compressor is limited. In order to realize the practical application of the Knudsen compressor, the performance study of the multi-stage Knudsen compressor is an indispensable step. In this paper, the hydrogen flow behaviors and the pressure increases of the multi-stage hydrogen Knudsen compressor are studied, combining with the continuity equations based on velocity slip and temperature jump boundary conditions. The effect of the stages and the structure on the performance of the hydrogen Knudsen compressor is investigated, which will provide guidance for the design and application of multistage hydrogen Knudsen compressor.
Governing equations The typical characteristic length of the hydrogen Knudsen compressor is micrometer, and the typical Knudsen number of hydrogen gas in the multi-stage Knudsen compressor belongs to the slip flow regime [12]. The traditional NavierStokes equations for calculating the continuous flow is still applicable, but the phenomenon of velocity slip and temperature jump on the wall must be fully considered. It is necessary to combine the appropriate velocity slip equation and temperature jump equation to solve this problem according to the gas rarefaction degree, i.e. Knudsen number [13,14]. The Knudsen number of the hydrogen in the model is less than 0.1, which is meaning that the hydrogen gas flow belongs to the near-slip flow regime, and the gas flow is relatively rarefied. The first-order slip boundary condition is suitable [10].
Slip boundary equations Based on the micro-flow module of the Comsol Multiphysics commercial software, the numerical model of the multi-stage Knudsen compressor in this paper is constructed. And according to the rarefied characteristics of the internal flow in
22634
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 2 6 3 2 e2 2 6 4 2
the Knudsen compressor, the numerical simulation is carried out by using the first-order slip boundary conditions. Lockerby et al. consider the effect of wall temperature on the thermal transpiration effect based on previous studies, and conclude that the boundary condition of the wall velocity slip is as Eq. (1) [15]: uslip ¼ ss
l tn m
m nT tn n þ sT VTg n,VTg n rTg
(1)
where uslip is the wall slip velocity, which is the velocity difference between the gas and the wall. ss is the viscous slip coefficient, the value is equal to ss ¼ 2 at =at . Among them, at is the tangential momentum accommodation coefficient, which represents the diffusion and reflection degree of molecules from the wall. In this paper, the value is at ¼ 0:9[16]. l is the mean molecular free path, and m is the gas viscosity, n is the boundary normal. t is the viscous stress tensor, sT is the temperature slip coefficient, which is equal to sT ¼ 0:75, r is the gas density, Tg is the gas temperature, Tw is the wall temperature. Meanwhile, the wall temperature jump equation can be expressed as Eq. (2) [17]: Tjump ¼ xT ln,VTg
(2)
where Tjump is the wall jump temperature, which is equal to the difference between the gas temperature near the wall and the wall temperature. xT is the temperature jump coefficient, the g k t , where g is the gas specific value is equal to xT ¼ 2 2a at gþ1 mCp heat ratio, k is the gas thermal conductivity, and Cp is the thermal capacity under constant pressure.
Validation of the numerical method In order to verify the applicability of the continuity equation with the slip boundary conditions in the slip flow region, the pressure variation of pressure driven flow in the microchannel is compared with that calculated by Nance et al. using DSMC method [18]. Nance et al. established the microchannel model, the length of the microchannel is 31.8 mm, the characteristic length of the microchannel is 0.53 mm. The pressure boundary condition is adopted for the inlet and outlet of the microchannel. The inflow pressure is P ¼ 2.5 105 Pa, the outflow pressure is P ¼ 1 105 Pa, the inlet temperature is 300 K, and the working gas is oxygen gas. It is found that the Knudsen number of the gas at the microchannel centerline ranges from about 0.04 at the inlet to about 0.07 at the outlet. It can be considered that the gas in the microchannel is in the slip-flow regime. The pressure distribution along the microchannel centerline is compared in Fig. 1, it can be found that the pressure distribution calculated by the two method is almost identical, especially at the beginning and end of the microchannel. In the middle of the microchannel, there is a gap in the pressure value. The maximum difference of the oxygen pressure between the slip boundary method and the Nance's method is less than 2%. This indicates that the continuity equation with the first-order slip boundary is adopted in calculating the gas flow in the slip-flow regime.
Fig. 1 e Comparisons of the pressure distribution along the microchannel centerline.
Numerical model Multistage hydrogen Knudsen compressor model The model of multi-stage hydrogen Knudsen compressor presented in this paper is shown in Fig. 2. The model of single-stage Knudsen compressor established in this paper is consistent with the thermal transpiration effect research in Karniadakis's works [13]. It consists of the beginning container, the end container, several micro-channels and connection channels connected in series. Among them, the microchannel characteristic length is D and the length is L1. The connection channel height is H and the length is L2. The beginning container and the end container have a height H and a length L2/2, that is to say, the height is same as the connection channel height, and the length is half of the connection channel length. As shown in Fig. 2, stage N (N ¼ 1, 2, …, N), i.e. one stage is defined as a container (or half of the connection channel along the length direction), a microchannel and half of the connection channel along the length direction. The length of each stage is defined as L, so L ¼ L1þL2. Define the cross section at the microchannel vertical centerline of each stage as mi (i ¼ 1, 2, …, N). The cross section at the connection channel vertical centerline is defined as Mj (j ¼ 1, 2, …, N-1).
Boundary conditions of the multistage hydrogen Knudsen compressor The multi-stage Knudsen compressor studied in this paper is closed at the beginning container and the end container, the inside and outside of the model are isolated, so the wall boundary conditions only need to set the temperature boundary conditions. As can be seen from Fig. 2, the high temperature condition is set on the connection channel left wall and the end container wall, which is defined as TH, and TH ¼ 400 K. The low temperature condition is set on the connection channel right wall and the beginning container wall, which is defined as TC, and TC ¼ 300 K. The ambient
22635
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 2 6 3 2 e2 2 6 4 2
Fig. 2 e Model of the multi-stage hydrogen Knudsen compressor.
temperature is the room temperature at standard atmospheric pressure, i.e. 293.15 K. In the initial conditions, the hydrogen gas pressure in the Knudsen compressor is the standard atmospheric pressure, which is defined as Patm.
Numerical cases of the multistage hydrogen Knudsen compressor In this paper, the effects of stages and connection channel height on the hydrogen pressure increase and the hydrogen flow characteristics of the Knudsen compressor are studied. In Table 1, the structure of the microchannel and the connection channel is the same, and the stage number of the hydrogen Knudsen compressor is changed from 1 to 5, to investigate the pressure performance of the hydrogen Knudsen compressor under different stages. As shown in Table 2, the stage of the hydrogen Knudsen compressor is 3, and the connection channel height is changed from 10 mm to 30 mm, to study the flow characteristics of the hydrogen Knudsen compressor.
Table 2 e Numerical details of the connection channel with different heights. Quantity Microchannel length L1/mm Microchannel height D/mm Connection channel length L2/mm Connection channel height H/mm Stages of hydrogen Knudsen compressor N
Case 6
Case 7
Case 8
25 2.5 25 10
25 2.5 25 20 3
25 2.5 25 30
Grid independence verification In this paper, the multistage hydrogen Knudsen compressor applied the unstructured grids, and the number and quality of grids are achieved. Fig. 3 is the grid independence of the twostage Knudsen compressor, which is verified according to the pressure difference of the hydrogen gas between the beginning container and the end container. The results show that when the grid number is 1.7 105 to 1.9 105, the difference of the DP between the two models is less than 0.1%. Other models are also validated for the grid independence. Considering the results of grid independence verification and computing resources, the numbers of grids used for the Knudsen compressor model in this paper are as follows. When the stage of the Knudsen compressor is 1, 2, 3, 4 and 5, the
Table 1 e Numerical details of the Knudsen compressors with different N. Quantity Microchannel length L1/mm Microchannel height D/mm Connection channel length L2/mm Connection channel height H/mm Stages of hydrogen Knudsen compressor N
Case Case Case Case Case 1 2 3 4 5
1
2
25 2.5 25 20 3
4
5
Fig. 3 e Grid independence verification of the hydrogen Knudsen compressor.
grids numbers are 0.5 105, 1.7 105, 2.6 105, 3.4 105 and 4.3 105, respectively. The grid quality histograms show that the grid quality of the multistage Knudsen compressor model and the three-stage Knudsen compressor model with variable connection channel height are all above 0.56.
Effect of the stage number on the performance of the hydrogen Knudsen compressor Kn number distribution of the hydrogen flow in the hydrogen Knudsen compressor Fig. 4 shows the Knudsen number distribution of the hydrogen flow along the centerline of the hydrogen Knudsen compressor. It can be seen that the Knudsen number of the hydrogen gas distributes between 0.001 and 0.1, which
22636
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 2 6 3 2 e2 2 6 4 2
Fig. 4 e Knudsen number of the hydrogen gas distribution along the centerline.
Fig. 5 e Dimensionless pressure distribution along the centerline.
indicates the hydrogen gas flow in the Knudsen compressor belongs to the slip flow regime. In the beginning and end containers, the Knudsen number hardly changes. This is duo to the pressure and temperature of the hydrogen gas in the container are almost unchanged in both ends, which makes the molecular mean free path and Knudsen number of the hydrogen gas invariant. Whereas, it is obvious that the Knudsen number of hydrogen gas in the end container is higher than that in the beginning container. This is because that the temperature of the end container is higher than that of the beginning container, and the molecular number density of the end container is lower, therefore the molecular mean free path and Knudsen number of the hydrogen gas in the end container is higher than that in the beginning container. In the microchannel and connection channel, the Knudsen number of the hydrogen gas presents a gradually change characteristic. In the microchannel, Knudsen number increases from 0.050 to 0.068. This is because the temperature of hydrogen gas in microchannel increase along the direction of temperature rise, which will lead to the gradual increase of the hydrogen gas mean free path in the microchannel. In the connection channel, Knudsen number decreases from 0.0084 to 0.0063. This is due to the hydrogen gas temperature in the channel decreases gradually along the x direction, while the hydrogen gas pressure decreases slightly or invariants nearly, therefore the molecular number density increases and Knudsen number decreases.
With the stage number increases, the pressure rise of the hydrogen Knudsen compressor increases proportionally. In the microchannel, the hydrogen gas pressure rises along the x direction because of the temperature difference. The temperature gradient along the wall drives the hydrogen gas molecular creeping from the cold-side to the hot-side, resulting in the pressure difference between the two sides of the microchannel. When the pressure difference achieves a certain value, it will drive the hydrogen gas moving from the hot-end to the cold-end, which is the Poiseuille flow. Finally the thermal creeping flow and the Poiseuille flow achieve an equilibrium state. Moreover, there is a slight pressure drop occurs in the connection channel. This is because that there is a reverse temperature difference along the wall in the connection channel, which leads to a pressure drop. Otherwise the height of the connection channel is higher than that of the microchannel, the thermal transpiration effect in the connection channel is weaker compared with the microchannel, therefore the pressure drop in the connection channel is much slighter than the pressure increase in the microchannel. Fig. 6 is the temperature distribution of the hydrogen flow along the centerline of Knudsen compressor. It shows that the temperature variation in the channel is periodic between 300 K and 400 K along the x-direction, which is related to the wall temperature condition.
Dimensionless pressure increase of the hydrogen flow The dimensionless pressure increase along the channel centerline of the Knudsen compressor is shown in Fig. 5, where P represents the local hydrogen pressure in the channel, P0 represents the hydrogen pressure on the left wall of the beginning container. It is shown that the pressure increase of one stage and multistage is almost coincide, which indicates that the pressure increase of each stage is not sensitive to the number of the stage. Also, the result presents that the hydrogen Knudsen compressor has a replication characteristic.
Velocity distribution of the hydrogen flow Fig. 7 is the velocity distribution of the hydrogen flow along the centerline of the Knudsen compressor. The positive value of the velocity denotes that the flow direction agrees the x direction, and the negative velocity value denotes the flow direction is reversed. In the microchannel, the Poiseuille flow direction is the -x direction. The maximum hydrogen flow velocity is 0.44 m/s, which occurs near the junction between the microchannel and the hot-end of the connection channel. The flow velocity decreases gradually along the flow direction. This is due to the hydrogen pressure increases gradually along the x direction, and the pressure achieves a maximum value at the junction between the microchannel and the hot-end of
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 2 6 3 2 e2 2 6 4 2
Fig. 6 e Temperature distribution of the hydrogen flow at the centerline.
22637
microchannel increases gradually. Fig. 8(b) shows the velocity distribution of each microchannel cross section of the fivestage Knudsen compressor. Along the x direction, the velocity of Poiseuille flow decreases gradually. Because the temperature difference between the two ends of the microchannel is constant, the velocity of the thermal transpiration flow near the wall along the microchannel cross section changes little. Fig. 9 is the velocity distribution profile of the multistage hydrogen Knudsen compressor along the connection channel cross section Mj. The negative value of uM represents the thermal transpiration flow velocity in the connection channel, while the positive value represents the Poiseuille flow velocity in the connection channel. Fig. 9(a) is the velocity profile distribution of the multi-stage Knudsen compressor at the cross section M1 in the connection channel, while Fig. 9(b) is the velocity profile of the fivestage Knudsen compressor at the cross section Mj of each connection channel. It is obvious that the velocity distributions of the thermal transpiration flow and the Poisson flow are not affected by the stage number.
Pressure fields of hydrogen flow in the connection channel
Fig. 7 e Velocity distribution of the hydrogen flow along the channel centerline.
the connection channel. Therefore, the pressure increase decreases along the flow direction, and the hydrogen velocity at the center of the microchannel also decreases gradually. In the connection channel, the velocity of the hydrogen flow along the channel center increases firstly and then decreases gradually, and the maximum flow velocity is 0.50 m/s. This is because the size of hydrogen flow path expands suddenly and then shrinks abruptly, and the hydrogen flow direction in the microchannel is opposite to that in the connection channel. The flow at the channel center generates a pair of vortexes with the thermal transpiration flow near the wall, so the velocity of hydrogen gas at the center channel increases and then decreases gradually, and achieves the maximum at the vertical centerline of the connection channel. The hydrogen gas flow velocity distribution curve of the microchannel cross section mi is shown in Fig. 8. The positive value of um represents the velocity of the thermal transpiration flow in the microchannel, and the negative value represents the velocity of the Poiseuille flow in the microchannel. As shown in Fig. 8(a), with the stage number increases, the velocity of the Poiseuille flow at the centerline of the
Fig. 10 is the hydrogen pressure fields and velocity streamlines of hydrogen flow in the intermediate connection channel, and the 10a,c is the two-stage Knudsen compressor and the 10b,d is the four-stage compressor. It can be found that the hydrogen gas pressure changes little in the connection channel. There is the thermal transpiration flow creeps from the cold-end to the hot-end near the channel wall, and the Poiseuille flow from the hot-end to the cold-end at the channel center. The two flows generate a symmetrically distributed vortexes, and there is a low-speed zone in the center of the vortexes, which makes the velocity of the two flow increase firstly and then decrease.
Effect of connection channel height on the performance of the three-stage hydrogen Knudsen compressor In view of the above research, it can be concluded that the compression performance of multistage hydrogen Knudsen compressor is linearly related to the stage number, which indicates that the multistage Knudsen compressor established has a single-stage replication characteristic. The results present that there is a hydrogen gas pressure drop in the connection area of the Knudsen compressor. Therefore, many stages may lead to a significant pressure drop for the hydrogen Knudsen compressor, which is obviously bad to the design and application of the hydrogen Knudsen compressor. Considering the effect of the microchannel height on the compression characteristic is investigated in the previous researches [19,20], therefore the understanding of the connection structure influence on the compressor performance is necessary. In this paper, the effect of the connection channel height on the three-stage Knudsen compressor performance is studied.
22638
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 2 6 3 2 e2 2 6 4 2
Fig. 8 e x-component velocity distribution of hydrogen flow at the cross section of mi.
Fig. 9 e x-component velocity distribution of hydrogen flow at the cross section of Mj.
Kn number distribution in the three-stage Knudsen compressor The Knudsen number of the hydrogen gas distribution along the channel centerline with different connection channel heights are presented in Fig. 11. When the connection channel height is H ¼ 10 mm, the maximum value of the Knudsen number in the first connection channel is Kn ¼ 0.0169. When H ¼ 20 mm and H ¼ 30 mm respectively, the maximum value of the Knudsen number is Kn ¼ 0.0084 and Kn ¼ 0.0056 respectively. The result shows that as the connection channel height increasing, the Knudsen number of hydrogen gas in the connection channel decreases gradually, whereas the Knudsen number of different cases almost coincide in the micro channel. It indicates that the change of the connection channel height has little effect on the gas rarefaction in the micro channel.
heights. When the connection channel height H is 10 mm, the hydrogen pressure increase of the three-stage Knudsen compressor is 1084.1 Pa. When the height of the cases are H ¼ 20 mm and H ¼ 30 mm, the pressure increase are 1128.7 Pa and 1141.0 Pa, respectively. It shows that as the connection channel height increases, the pressure increase performance of Knudsen compressor is gradually improved. The main reason for this phenomenon is the thermal transpiration effect. When the connection channel height is lower, the hydrogen flow is more rarefied and the thermal transpiration effect is stronger, and the negative temperature difference will lead to a significant pressure drop. As the connection channel height increases, the flow in this area is closer to continues regime, and the thermal transpiration effect is weaker, also the pressure drop is less significant. Therefore, the compression of the hydrogen Knudsen compressor is higher if the connection channel height is higher.
Dimensionless pressure increase in the three-stage Knudsen compressor
Velocity distribution of hydrogen flow in the three-stage Knudsen compressor
Fig. 12 is the dimensionless pressure distribution of the Knudsen compressor with different connection channel
The velocity distribution of the hydrogen flow along the channel centerline of hydrogen Knudsen compressor is
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 2 6 3 2 e2 2 6 4 2
22639
Fig. 10 e Pressure fields and velocity streamlines for the connection channel.
Fig. 11 e Kn number distribution at the centerline.
Fig. 12 e Pressure distribution at the centerline.
depicted in Fig. 13. When the connection channel height is H ¼ 10 mm, the maximum velocity of hydrogen flow in the connection channel occurs near the high temperature wall, which is 0.52 m/s. The velocity in the connection channel decreases slowly at first, and then suddenly decreases to 0 near the low temperature wall at last. This can be interpreted as that the height of the connection channel is lower,
and the vortexes in the y direction is narrower, which is benefit to the fully developing of the hydrogen flow along the x direction. Whereas the pressure drop exists in the connection channel, so the velocity of the hydrogen gas flow decreases gradually. When the connection channel height is H ¼ 20 mm and H ¼ 30 mm, the maximum velocity in the connection channel is 0.49 m/s and 0.35 m/s, respectively. As the
22640
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 2 6 3 2 e2 2 6 4 2
Fig. 13 e Velocity distribution of the hydrogen flow at the centerline.
Fig. 15 e Velocity of the hydrogen flow at the cross section M1.
connection channel height increases, the centerline velocity of hydrogen flow in the connection channel decreases. The velocity of Poiseuille in the connection channel increases first and then decreases, this is because the channel of hydrogen Knudsen compressor expands suddenly and then shrinks abruptly at the connection channel, which causes the vortexes occur. Fig. 14 is the hydrogen gas flow velocity profiles of threestage Knudsen compressor at the cross section m2 in the microchannel. It is clearly that the velocity in the microchannel is not affected by the connection channel height. The velocity profiles of the compressor at the cross section M1 in the connection channel is presented in Fig. 15. When the connection channel height are H ¼ 10 mm, H ¼ 20 mm and H ¼ 30 mm, the velocity of the thermal transpiration flow near the connection channel wall are 1.03 m/s, 1.05 m/s and 1.09 m/s, respectively. The velocity of the Posieuille flow at the connection channel centerline are 0.51 m/s, 0.49 m/s and 0.35 m/s, respectively. The result shows that when the connection channel is higher, the thermal transpiration flow near the connection channel wall increases gradually, and
the Posieuille flow at the connection channel centerline decreases gradually. Fig. 16 shows the mass flow rate distribution of the thermal transpiration flow and the Poiseuille flow in the cross-section m2 of the Knudsen compressor. From the figure, it can be seen that the thermal transpiration flow and the Poiseuille flow in the microchannel have the same flow rate but the different directions, which also verifies that the calculation model in this paper has achieved a steady state. It shows that the change of the connection channel height has little influence on the flowrate of thermal transpiration flow in the microchannel.
Fig. 14 e Velocity of the hydrogen flow at the cross section m2.
Pressure fields of hydrogen flow in the three-stage Knudsen compressor Fig. 17 is the pressure fields and velocity vectors of the first connection channel of the three-stage Knudsen compressor. It can be conclude that with the connection channel height increases, the hydrogen gas pressure in the connection channel decreases gradually. Also, because of the width of the
Fig. 16 e Distribution of hydrogen flowrate at m2 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 2 6 3 2 e2 2 6 4 2
22641
Fig. 17 e Pressure fields and velocity vectors for the connection channel with different heights.
Poiseuille flow path increases, the velocity of the Poiseuille flow in the connection channel decreases gradually.
Conclusions In this paper, the hydrogen flow performance of the multistage hydrogen Knudsen compressor is studied, and the effect of the stage and the connection channel height on the hydrogen flow and pressure characteristic are investigated. The conclusions are as follows: (1) With the increase of the stage number, the pressure increase performance of the hydrogen flow increases linearly and the velocity changes little, which indicates that the multi-stage Knudsen compressor has a singlestage replication characteristic. (2) The change of connection channel height has an effect on the performance of Knudsen compressor. With the increase of the connection channel height, the pressure drop at the center line of the connection channel decreases, which is more conducive to the improvement of the overall pressure increase performance of the hydrogen Knudsen compressor. Whereas the change of the connection channel height has little effect on the pressure change in the microchannel.
(3) For the hydrogen flow fields, there are symmetrical vortexes in the connection channel, and there is also a low-speed zone in the center of the vortexes. When the connection channel height becomes higher, the thermal transpiration flow velocity near the connection channel wall increases slightly, and the Posieuille flow velocity at the connection channel centerline decreases gradually.
Acknowledgements This project was supported by the National Natural Science Foundation of China, Grants No. 51106137, the Major Technological Innovation Projects in Hubei Province, Grants No. 2017AAA035, and the Open Foundation of the State Key Laboratory of Fluid Power and Mechatronic Systems, Grants No. GZKF-201811.
references
[1] Kugimoto K, Hirota Y, Yamauchi T, Yamaguchi H, Niimi T. A novel heat pump system using a multi-stage Knudsen compressor. Int J Heat Mass Transf 2018;127:84e91. https:// doi.org/10.1016/j.ijheatmasstransfer.2018.06.072. [2] Ye JJ, Yang J, Zheng JY, Ding XT, Wong I, Li WZ, et al. Thermal transpiration effect on the mass transfer and flow behaviors
22642
[3]
[4]
[5]
[6]
[7]
[8]
[9]
[10]
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 2 6 3 2 e2 2 6 4 2
of the pressure-driven hydrogen gas flow. Int J Hydrogen Energy 2012;37(17):12474e80. https://doi.org/10.1016/j. ijhydene.2012.05.164. Das S, Sarkar CK, Roy S. Development of integrated microsystem for hydrogen gas detection. IET Circuits, Devices Syst 2018;12(4):453e9. https://doi.org/10.1049/iet-cds. 2017.0243. El Matbouly H, Domingue F, Palmisano V, Boon-Brett L, Post MB, Rivkin C, et al. Assessment of commercial micromachined hydrogen sensors performance metrics for safety sensing applications. Int J Hydrogen Energy 2014;39(9):4664e73. https://doi.org/10.1016/j.ijhydene.2014. 01.037. Buttner WJ, Post MB, Burgess R, Rivkin C. An overview of hydrogen safety sensors and requirements. Int J Hydrogen Energy 2011;36(3):2462e70. https://doi.org/10.1016/j.ijhydene. 2010.04.176. 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. Miles SD, McNamara S, Pharas K. Ten stage Knudsen gas pump. ASME. International conference on nanochannels, microchannels, and minichannels, ASME 2012 10th international conference on nanochannels, Microchannels, and Minichannels. 2012. p. 287e93. https://doi.org/10.1115/ ICNMM2012-73209. 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. Young M, Muntz EP, Shiflett G. The Knudsen compressor as an energy efficient micro-scale vacuum pump. In: Presented at the 2nd NASA/JPL Miniature Pumps Workshop, Pasadena, CA; 2002. Leontidis V, Chen J, Baldas L, Colin S. Numerical design of a Knudsen pump with curved channels operating in the slip
[11]
[12]
[13] [14]
[15]
[16]
[17]
[18]
[19]
[20]
flow regime. Int J Heat Mass Transf 2014;50(8):1065e80. https://doi.org/10.1007/s00231-014-1314-4. Han YL, Muntz EP. Implications of imposing working gas temperature change limits on thermal creep driven flows. In: AIP Conference Proceedings, 1084(1); 2008. p. 305e10. https:// doi.org/10.1063/1.3076491. Colin S. Rarefaction and compressibility effects on steady or transient gas flows in microchannels. In: ASME 2nd International Conference on Microchannels and Minichannels; 2004. p. 1e12. https://doi.org/10.1115/ ICMM2004-2316. Karniadakis GE, Beskok A. Micro flows: fundamentals and simulation. New York: Springer Verlag; 2002. Shu JJ, Teo JBM, Chan WK. Fluid velocity slip and temperature jump at a solid surface. Appl Mech Rev 2017;69(2). 020801(1-13), https://doi.org/10.1115/1.4036191. Lockerby DA, Reese JM, Emerson DR, Barber RW. Velocity boundary condition at solid walls in rarefied gas calculations. Phys Rev 2004;70(1). 017303(1-4), https://doi.org/10.1103/ PhysRevE.70.017303. Sharipov F. Data on the velocity slip and temperature jump on a gas-solid interface. J Phys Chem Ref Data 2011;40(2). 023101(1-28), https://doi.org/10.1063/1.3580290. Colin S. Gas microflows in the slip flow regime: a critical review on convective heat transfer. J Heat Transf 2012;134(2):020908. https://doi.org/10.1115/1.4005063. Nance RP, Hash DB, Hassan HA. Role of boundary conditions in Monte Carlo simulation of Microelectromechanical Systems. J Thermophys Heat Transf 1998;12(3):447e9. https://doi.org/10.2514/2.6358. Ye JJ, Yang J, Zheng JY, Ding XT, Wong I, Li WZ, et al. Rarefaction and temperature gradient effect on the performance of the Knudsen pump. Chin J Mech Eng 2012;25(4):745e52. https://doi.org/10.3901/CJME.2012.04.745. Alexeenko AA, Gimelshein SF, Muntz EP, Ketsdever AD. Kinetic modeling of temperature driven flows in short microchannels. Int J Therm Sci 2006;45(11):1045e51. https:// doi.org/10.1016/j.ijthermalsci.2006.01.014.
Available online at www.sciencedirect.com
ScienceDirect journal homepage: www.elsevier.com/locate/he
The hydrogen flow characteristics of the multistage hydrogen Knudsen compressor based on the thermal transpiration effect Jianjun Ye a,b, Junda Shao a, Junlong Xie a,*, Zhenhua Zhao a, Jiangcun Yu a, Yuan Zhang a, Shehab Salem a a b
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
article info
abstract
Article history:
Multistage hydrogen Knudsen compressor based on the thermal transpiration effect has
Received 8 February 2019
very exciting prospect for the hydrogen transmission in the micro devices. Understanding
Received in revised form
of the hydrogen flow characteristic is the key issue for the designs and applications of the
28 March 2019
hydrogen energy systems. Firstly, the numerical models of the multistage hydrogen
Accepted 14 April 2019
Knudsen compressor are established. The distributions of the rarefaction, velocity and
Available online 18 May 2019
temperature at different stages of the hydrogen flow are calculated and presented. Moreover, the dimensional pressure increases of the hydrogen gas flow are analyzed, and the
Keywords:
flow behaviors in the microchannel and the connection channel are discussed. Secondly,
Thermal transpiration effect
the numerical simulation at different connection channel height is implemented, and the
Hydrogen flow
hydrogen gas flow characteristics in the connection are analyzed. Especially, the perfor-
Pressure increase
mances of the pressure drop in the connection channel under different channel heights are
Hydrogen Knudsen compressor
studied, and the hydrogen gas compression characteristics of different cases are compared
Flow vortex
and discussed. Also, the effect of the connection channel height on the hydrogen gas pressure increase in the microchannel is investigated. The studies presented in this paper could be greatly beneficial for the hydrogen detection and transmission. © 2019 Hydrogen Energy Publications LLC. Published by Elsevier Ltd. All rights reserved.
Introduction The thermal transpiration effect refers to the thermal force on a gas due to a temperature difference in microchannel, and is able to maintain a certain pressure difference and cause a gas flow [1,2]. This phenomena can be used for the design of the hydrogen Knudsen compressor, which has a great potential development for the safety of hydrogen energy devices, such as the hydrogen detectors and hydrogen sensors [3e5].
Generally, the pressure performance of the hydrogen Knudsen compressor is relative to the channel structures. Furthermore, for the multi-stage Knudsen compressor, its performance is closely related to the stage number in series. There are also many studies on the pressure increase of Knudsen compressor, Pham-Van-Diep et al. proposed a Knudsen compressor model based on the micro-electro-mechanical systems (MEMS), which can obtain a high pressure difference by connecting multiple single-stage Knudsen
* Corresponding author. E-mail address: [email protected] (J. Xie). https://doi.org/10.1016/j.ijhydene.2019.04.155 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 2 6 3 2 e2 2 6 4 2
Nomenclature Cp D H k Kn L L1 L2 mi Mj n N P P0 Patm T TH TC Tg Tw Tjump uc um uM uslip g DP l m xT r ss st sT t
thermal capacity under constant pressure (J/ (kg$K)) microchannel characteristic length (mm) connection channel height (mm) gas thermal conductivity (W/(m$K)) Knudsen number each stage length (mm) microchannel length (mm) connection channel length (mm) cross section at the microchannel cross section at the connection channel boundary normal stage number local hydrogen pressure in the channel (Pa) hydrogen pressure on the left wall of the beginning container (Pa) standard atmospheric pressure (Pa) hydrogen temperature along the channel centerline (K) high temperature condition (K) low temperature condition (K) gas temperature (K) wall temperature (K) wall jump temperature (K) hydrogen velocity along the channel centerline (m/s) hydrogen velocity in the microchannel (m/s) hydrogen velocity in the connection channel (m/s) wall slip velocity (m/s) gas specific heat ratio hydrogen pressure difference between the beginning and end container (Pa) mean molecular free path (mm) gas viscosity (Pa$s) temperature jump coefficient gas density (kg/m3) viscous slip coefficient tangential momentum accommodation coefficient temperature slip coefficient viscous stress tensor
compressors in series [6]. Miles et al. manufactured the singlestage, five-stage and ten-stage Knudsen compressors respectively and used a thermoelectric module to realize the temperature difference. It was found that the compression ratio of Knudsen compressor can be improved obviously by increasing the number of series stages [7]. This study presents theoretical and practical guidance for improving the pressure increase of the Knudsen compressor. Ye et al. applied the coupled method to study the influence of the microchannel height and length on the hydrogen flow characteristics in multi-scale channels. The results show that the length and height of the microchannel are the main factors affecting the change of
22633
hydrogen pressure and flow rate in the channel, which provides guidance for the application of the hydrogen Knudsen compressor [8]. Young et al. found that the theoretical operating pressure of a single-stage Knudsen compressor can be as low as 10 mTorr. When the pore size of microchannel material is small as few nanometers, its operating pressure can be as high as 10 atm [9]. Based on the compressible Navier-Stokes equations with velocity slip boundary conditions, Leontidis et al. constructed the research model of Knudsen compressor and investigated its working performance under different microchannel sizes [10]. Han et al. used the direct simulation Monte Carlo (DSMC) method to analyze the performance of Knudsen compressor. It was found that the gas rarefied degree and the reverse thermal transpiration effect in the connection channel of Knudsen compressor were the main factors to limit its performance, and when the reverse thermal transpiration effect in the connection channel was weak, the performance of Knudsen compressor becomes better [11]. In view of the above researches, it seems that the study of the hydrogen flow behaviors is important to the design and application of the hydrogen Knudsen compressor, and the pressure increase performance of single-stage Knudsen compressor is limited. In order to realize the practical application of the Knudsen compressor, the performance study of the multi-stage Knudsen compressor is an indispensable step. In this paper, the hydrogen flow behaviors and the pressure increases of the multi-stage hydrogen Knudsen compressor are studied, combining with the continuity equations based on velocity slip and temperature jump boundary conditions. The effect of the stages and the structure on the performance of the hydrogen Knudsen compressor is investigated, which will provide guidance for the design and application of multistage hydrogen Knudsen compressor.
Governing equations The typical characteristic length of the hydrogen Knudsen compressor is micrometer, and the typical Knudsen number of hydrogen gas in the multi-stage Knudsen compressor belongs to the slip flow regime [12]. The traditional NavierStokes equations for calculating the continuous flow is still applicable, but the phenomenon of velocity slip and temperature jump on the wall must be fully considered. It is necessary to combine the appropriate velocity slip equation and temperature jump equation to solve this problem according to the gas rarefaction degree, i.e. Knudsen number [13,14]. The Knudsen number of the hydrogen in the model is less than 0.1, which is meaning that the hydrogen gas flow belongs to the near-slip flow regime, and the gas flow is relatively rarefied. The first-order slip boundary condition is suitable [10].
Slip boundary equations Based on the micro-flow module of the Comsol Multiphysics commercial software, the numerical model of the multi-stage Knudsen compressor in this paper is constructed. And according to the rarefied characteristics of the internal flow in
22634
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 2 6 3 2 e2 2 6 4 2
the Knudsen compressor, the numerical simulation is carried out by using the first-order slip boundary conditions. Lockerby et al. consider the effect of wall temperature on the thermal transpiration effect based on previous studies, and conclude that the boundary condition of the wall velocity slip is as Eq. (1) [15]: uslip ¼ ss
l tn m
m nT tn n þ sT VTg n,VTg n rTg
(1)
where uslip is the wall slip velocity, which is the velocity difference between the gas and the wall. ss is the viscous slip coefficient, the value is equal to ss ¼ 2 at =at . Among them, at is the tangential momentum accommodation coefficient, which represents the diffusion and reflection degree of molecules from the wall. In this paper, the value is at ¼ 0:9[16]. l is the mean molecular free path, and m is the gas viscosity, n is the boundary normal. t is the viscous stress tensor, sT is the temperature slip coefficient, which is equal to sT ¼ 0:75, r is the gas density, Tg is the gas temperature, Tw is the wall temperature. Meanwhile, the wall temperature jump equation can be expressed as Eq. (2) [17]: Tjump ¼ xT ln,VTg
(2)
where Tjump is the wall jump temperature, which is equal to the difference between the gas temperature near the wall and the wall temperature. xT is the temperature jump coefficient, the g k t , where g is the gas specific value is equal to xT ¼ 2 2a at gþ1 mCp heat ratio, k is the gas thermal conductivity, and Cp is the thermal capacity under constant pressure.
Validation of the numerical method In order to verify the applicability of the continuity equation with the slip boundary conditions in the slip flow region, the pressure variation of pressure driven flow in the microchannel is compared with that calculated by Nance et al. using DSMC method [18]. Nance et al. established the microchannel model, the length of the microchannel is 31.8 mm, the characteristic length of the microchannel is 0.53 mm. The pressure boundary condition is adopted for the inlet and outlet of the microchannel. The inflow pressure is P ¼ 2.5 105 Pa, the outflow pressure is P ¼ 1 105 Pa, the inlet temperature is 300 K, and the working gas is oxygen gas. It is found that the Knudsen number of the gas at the microchannel centerline ranges from about 0.04 at the inlet to about 0.07 at the outlet. It can be considered that the gas in the microchannel is in the slip-flow regime. The pressure distribution along the microchannel centerline is compared in Fig. 1, it can be found that the pressure distribution calculated by the two method is almost identical, especially at the beginning and end of the microchannel. In the middle of the microchannel, there is a gap in the pressure value. The maximum difference of the oxygen pressure between the slip boundary method and the Nance's method is less than 2%. This indicates that the continuity equation with the first-order slip boundary is adopted in calculating the gas flow in the slip-flow regime.
Fig. 1 e Comparisons of the pressure distribution along the microchannel centerline.
Numerical model Multistage hydrogen Knudsen compressor model The model of multi-stage hydrogen Knudsen compressor presented in this paper is shown in Fig. 2. The model of single-stage Knudsen compressor established in this paper is consistent with the thermal transpiration effect research in Karniadakis's works [13]. It consists of the beginning container, the end container, several micro-channels and connection channels connected in series. Among them, the microchannel characteristic length is D and the length is L1. The connection channel height is H and the length is L2. The beginning container and the end container have a height H and a length L2/2, that is to say, the height is same as the connection channel height, and the length is half of the connection channel length. As shown in Fig. 2, stage N (N ¼ 1, 2, …, N), i.e. one stage is defined as a container (or half of the connection channel along the length direction), a microchannel and half of the connection channel along the length direction. The length of each stage is defined as L, so L ¼ L1þL2. Define the cross section at the microchannel vertical centerline of each stage as mi (i ¼ 1, 2, …, N). The cross section at the connection channel vertical centerline is defined as Mj (j ¼ 1, 2, …, N-1).
Boundary conditions of the multistage hydrogen Knudsen compressor The multi-stage Knudsen compressor studied in this paper is closed at the beginning container and the end container, the inside and outside of the model are isolated, so the wall boundary conditions only need to set the temperature boundary conditions. As can be seen from Fig. 2, the high temperature condition is set on the connection channel left wall and the end container wall, which is defined as TH, and TH ¼ 400 K. The low temperature condition is set on the connection channel right wall and the beginning container wall, which is defined as TC, and TC ¼ 300 K. The ambient
22635
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 2 6 3 2 e2 2 6 4 2
Fig. 2 e Model of the multi-stage hydrogen Knudsen compressor.
temperature is the room temperature at standard atmospheric pressure, i.e. 293.15 K. In the initial conditions, the hydrogen gas pressure in the Knudsen compressor is the standard atmospheric pressure, which is defined as Patm.
Numerical cases of the multistage hydrogen Knudsen compressor In this paper, the effects of stages and connection channel height on the hydrogen pressure increase and the hydrogen flow characteristics of the Knudsen compressor are studied. In Table 1, the structure of the microchannel and the connection channel is the same, and the stage number of the hydrogen Knudsen compressor is changed from 1 to 5, to investigate the pressure performance of the hydrogen Knudsen compressor under different stages. As shown in Table 2, the stage of the hydrogen Knudsen compressor is 3, and the connection channel height is changed from 10 mm to 30 mm, to study the flow characteristics of the hydrogen Knudsen compressor.
Table 2 e Numerical details of the connection channel with different heights. Quantity Microchannel length L1/mm Microchannel height D/mm Connection channel length L2/mm Connection channel height H/mm Stages of hydrogen Knudsen compressor N
Case 6
Case 7
Case 8
25 2.5 25 10
25 2.5 25 20 3
25 2.5 25 30
Grid independence verification In this paper, the multistage hydrogen Knudsen compressor applied the unstructured grids, and the number and quality of grids are achieved. Fig. 3 is the grid independence of the twostage Knudsen compressor, which is verified according to the pressure difference of the hydrogen gas between the beginning container and the end container. The results show that when the grid number is 1.7 105 to 1.9 105, the difference of the DP between the two models is less than 0.1%. Other models are also validated for the grid independence. Considering the results of grid independence verification and computing resources, the numbers of grids used for the Knudsen compressor model in this paper are as follows. When the stage of the Knudsen compressor is 1, 2, 3, 4 and 5, the
Table 1 e Numerical details of the Knudsen compressors with different N. Quantity Microchannel length L1/mm Microchannel height D/mm Connection channel length L2/mm Connection channel height H/mm Stages of hydrogen Knudsen compressor N
Case Case Case Case Case 1 2 3 4 5
1
2
25 2.5 25 20 3
4
5
Fig. 3 e Grid independence verification of the hydrogen Knudsen compressor.
grids numbers are 0.5 105, 1.7 105, 2.6 105, 3.4 105 and 4.3 105, respectively. The grid quality histograms show that the grid quality of the multistage Knudsen compressor model and the three-stage Knudsen compressor model with variable connection channel height are all above 0.56.
Effect of the stage number on the performance of the hydrogen Knudsen compressor Kn number distribution of the hydrogen flow in the hydrogen Knudsen compressor Fig. 4 shows the Knudsen number distribution of the hydrogen flow along the centerline of the hydrogen Knudsen compressor. It can be seen that the Knudsen number of the hydrogen gas distributes between 0.001 and 0.1, which
22636
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 2 6 3 2 e2 2 6 4 2
Fig. 4 e Knudsen number of the hydrogen gas distribution along the centerline.
Fig. 5 e Dimensionless pressure distribution along the centerline.
indicates the hydrogen gas flow in the Knudsen compressor belongs to the slip flow regime. In the beginning and end containers, the Knudsen number hardly changes. This is duo to the pressure and temperature of the hydrogen gas in the container are almost unchanged in both ends, which makes the molecular mean free path and Knudsen number of the hydrogen gas invariant. Whereas, it is obvious that the Knudsen number of hydrogen gas in the end container is higher than that in the beginning container. This is because that the temperature of the end container is higher than that of the beginning container, and the molecular number density of the end container is lower, therefore the molecular mean free path and Knudsen number of the hydrogen gas in the end container is higher than that in the beginning container. In the microchannel and connection channel, the Knudsen number of the hydrogen gas presents a gradually change characteristic. In the microchannel, Knudsen number increases from 0.050 to 0.068. This is because the temperature of hydrogen gas in microchannel increase along the direction of temperature rise, which will lead to the gradual increase of the hydrogen gas mean free path in the microchannel. In the connection channel, Knudsen number decreases from 0.0084 to 0.0063. This is due to the hydrogen gas temperature in the channel decreases gradually along the x direction, while the hydrogen gas pressure decreases slightly or invariants nearly, therefore the molecular number density increases and Knudsen number decreases.
With the stage number increases, the pressure rise of the hydrogen Knudsen compressor increases proportionally. In the microchannel, the hydrogen gas pressure rises along the x direction because of the temperature difference. The temperature gradient along the wall drives the hydrogen gas molecular creeping from the cold-side to the hot-side, resulting in the pressure difference between the two sides of the microchannel. When the pressure difference achieves a certain value, it will drive the hydrogen gas moving from the hot-end to the cold-end, which is the Poiseuille flow. Finally the thermal creeping flow and the Poiseuille flow achieve an equilibrium state. Moreover, there is a slight pressure drop occurs in the connection channel. This is because that there is a reverse temperature difference along the wall in the connection channel, which leads to a pressure drop. Otherwise the height of the connection channel is higher than that of the microchannel, the thermal transpiration effect in the connection channel is weaker compared with the microchannel, therefore the pressure drop in the connection channel is much slighter than the pressure increase in the microchannel. Fig. 6 is the temperature distribution of the hydrogen flow along the centerline of Knudsen compressor. It shows that the temperature variation in the channel is periodic between 300 K and 400 K along the x-direction, which is related to the wall temperature condition.
Dimensionless pressure increase of the hydrogen flow The dimensionless pressure increase along the channel centerline of the Knudsen compressor is shown in Fig. 5, where P represents the local hydrogen pressure in the channel, P0 represents the hydrogen pressure on the left wall of the beginning container. It is shown that the pressure increase of one stage and multistage is almost coincide, which indicates that the pressure increase of each stage is not sensitive to the number of the stage. Also, the result presents that the hydrogen Knudsen compressor has a replication characteristic.
Velocity distribution of the hydrogen flow Fig. 7 is the velocity distribution of the hydrogen flow along the centerline of the Knudsen compressor. The positive value of the velocity denotes that the flow direction agrees the x direction, and the negative velocity value denotes the flow direction is reversed. In the microchannel, the Poiseuille flow direction is the -x direction. The maximum hydrogen flow velocity is 0.44 m/s, which occurs near the junction between the microchannel and the hot-end of the connection channel. The flow velocity decreases gradually along the flow direction. This is due to the hydrogen pressure increases gradually along the x direction, and the pressure achieves a maximum value at the junction between the microchannel and the hot-end of
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 2 6 3 2 e2 2 6 4 2
Fig. 6 e Temperature distribution of the hydrogen flow at the centerline.
22637
microchannel increases gradually. Fig. 8(b) shows the velocity distribution of each microchannel cross section of the fivestage Knudsen compressor. Along the x direction, the velocity of Poiseuille flow decreases gradually. Because the temperature difference between the two ends of the microchannel is constant, the velocity of the thermal transpiration flow near the wall along the microchannel cross section changes little. Fig. 9 is the velocity distribution profile of the multistage hydrogen Knudsen compressor along the connection channel cross section Mj. The negative value of uM represents the thermal transpiration flow velocity in the connection channel, while the positive value represents the Poiseuille flow velocity in the connection channel. Fig. 9(a) is the velocity profile distribution of the multi-stage Knudsen compressor at the cross section M1 in the connection channel, while Fig. 9(b) is the velocity profile of the fivestage Knudsen compressor at the cross section Mj of each connection channel. It is obvious that the velocity distributions of the thermal transpiration flow and the Poisson flow are not affected by the stage number.
Pressure fields of hydrogen flow in the connection channel
Fig. 7 e Velocity distribution of the hydrogen flow along the channel centerline.
the connection channel. Therefore, the pressure increase decreases along the flow direction, and the hydrogen velocity at the center of the microchannel also decreases gradually. In the connection channel, the velocity of the hydrogen flow along the channel center increases firstly and then decreases gradually, and the maximum flow velocity is 0.50 m/s. This is because the size of hydrogen flow path expands suddenly and then shrinks abruptly, and the hydrogen flow direction in the microchannel is opposite to that in the connection channel. The flow at the channel center generates a pair of vortexes with the thermal transpiration flow near the wall, so the velocity of hydrogen gas at the center channel increases and then decreases gradually, and achieves the maximum at the vertical centerline of the connection channel. The hydrogen gas flow velocity distribution curve of the microchannel cross section mi is shown in Fig. 8. The positive value of um represents the velocity of the thermal transpiration flow in the microchannel, and the negative value represents the velocity of the Poiseuille flow in the microchannel. As shown in Fig. 8(a), with the stage number increases, the velocity of the Poiseuille flow at the centerline of the
Fig. 10 is the hydrogen pressure fields and velocity streamlines of hydrogen flow in the intermediate connection channel, and the 10a,c is the two-stage Knudsen compressor and the 10b,d is the four-stage compressor. It can be found that the hydrogen gas pressure changes little in the connection channel. There is the thermal transpiration flow creeps from the cold-end to the hot-end near the channel wall, and the Poiseuille flow from the hot-end to the cold-end at the channel center. The two flows generate a symmetrically distributed vortexes, and there is a low-speed zone in the center of the vortexes, which makes the velocity of the two flow increase firstly and then decrease.
Effect of connection channel height on the performance of the three-stage hydrogen Knudsen compressor In view of the above research, it can be concluded that the compression performance of multistage hydrogen Knudsen compressor is linearly related to the stage number, which indicates that the multistage Knudsen compressor established has a single-stage replication characteristic. The results present that there is a hydrogen gas pressure drop in the connection area of the Knudsen compressor. Therefore, many stages may lead to a significant pressure drop for the hydrogen Knudsen compressor, which is obviously bad to the design and application of the hydrogen Knudsen compressor. Considering the effect of the microchannel height on the compression characteristic is investigated in the previous researches [19,20], therefore the understanding of the connection structure influence on the compressor performance is necessary. In this paper, the effect of the connection channel height on the three-stage Knudsen compressor performance is studied.
22638
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 2 6 3 2 e2 2 6 4 2
Fig. 8 e x-component velocity distribution of hydrogen flow at the cross section of mi.
Fig. 9 e x-component velocity distribution of hydrogen flow at the cross section of Mj.
Kn number distribution in the three-stage Knudsen compressor The Knudsen number of the hydrogen gas distribution along the channel centerline with different connection channel heights are presented in Fig. 11. When the connection channel height is H ¼ 10 mm, the maximum value of the Knudsen number in the first connection channel is Kn ¼ 0.0169. When H ¼ 20 mm and H ¼ 30 mm respectively, the maximum value of the Knudsen number is Kn ¼ 0.0084 and Kn ¼ 0.0056 respectively. The result shows that as the connection channel height increasing, the Knudsen number of hydrogen gas in the connection channel decreases gradually, whereas the Knudsen number of different cases almost coincide in the micro channel. It indicates that the change of the connection channel height has little effect on the gas rarefaction in the micro channel.
heights. When the connection channel height H is 10 mm, the hydrogen pressure increase of the three-stage Knudsen compressor is 1084.1 Pa. When the height of the cases are H ¼ 20 mm and H ¼ 30 mm, the pressure increase are 1128.7 Pa and 1141.0 Pa, respectively. It shows that as the connection channel height increases, the pressure increase performance of Knudsen compressor is gradually improved. The main reason for this phenomenon is the thermal transpiration effect. When the connection channel height is lower, the hydrogen flow is more rarefied and the thermal transpiration effect is stronger, and the negative temperature difference will lead to a significant pressure drop. As the connection channel height increases, the flow in this area is closer to continues regime, and the thermal transpiration effect is weaker, also the pressure drop is less significant. Therefore, the compression of the hydrogen Knudsen compressor is higher if the connection channel height is higher.
Dimensionless pressure increase in the three-stage Knudsen compressor
Velocity distribution of hydrogen flow in the three-stage Knudsen compressor
Fig. 12 is the dimensionless pressure distribution of the Knudsen compressor with different connection channel
The velocity distribution of the hydrogen flow along the channel centerline of hydrogen Knudsen compressor is
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 2 6 3 2 e2 2 6 4 2
22639
Fig. 10 e Pressure fields and velocity streamlines for the connection channel.
Fig. 11 e Kn number distribution at the centerline.
Fig. 12 e Pressure distribution at the centerline.
depicted in Fig. 13. When the connection channel height is H ¼ 10 mm, the maximum velocity of hydrogen flow in the connection channel occurs near the high temperature wall, which is 0.52 m/s. The velocity in the connection channel decreases slowly at first, and then suddenly decreases to 0 near the low temperature wall at last. This can be interpreted as that the height of the connection channel is lower,
and the vortexes in the y direction is narrower, which is benefit to the fully developing of the hydrogen flow along the x direction. Whereas the pressure drop exists in the connection channel, so the velocity of the hydrogen gas flow decreases gradually. When the connection channel height is H ¼ 20 mm and H ¼ 30 mm, the maximum velocity in the connection channel is 0.49 m/s and 0.35 m/s, respectively. As the
22640
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 2 6 3 2 e2 2 6 4 2
Fig. 13 e Velocity distribution of the hydrogen flow at the centerline.
Fig. 15 e Velocity of the hydrogen flow at the cross section M1.
connection channel height increases, the centerline velocity of hydrogen flow in the connection channel decreases. The velocity of Poiseuille in the connection channel increases first and then decreases, this is because the channel of hydrogen Knudsen compressor expands suddenly and then shrinks abruptly at the connection channel, which causes the vortexes occur. Fig. 14 is the hydrogen gas flow velocity profiles of threestage Knudsen compressor at the cross section m2 in the microchannel. It is clearly that the velocity in the microchannel is not affected by the connection channel height. The velocity profiles of the compressor at the cross section M1 in the connection channel is presented in Fig. 15. When the connection channel height are H ¼ 10 mm, H ¼ 20 mm and H ¼ 30 mm, the velocity of the thermal transpiration flow near the connection channel wall are 1.03 m/s, 1.05 m/s and 1.09 m/s, respectively. The velocity of the Posieuille flow at the connection channel centerline are 0.51 m/s, 0.49 m/s and 0.35 m/s, respectively. The result shows that when the connection channel is higher, the thermal transpiration flow near the connection channel wall increases gradually, and
the Posieuille flow at the connection channel centerline decreases gradually. Fig. 16 shows the mass flow rate distribution of the thermal transpiration flow and the Poiseuille flow in the cross-section m2 of the Knudsen compressor. From the figure, it can be seen that the thermal transpiration flow and the Poiseuille flow in the microchannel have the same flow rate but the different directions, which also verifies that the calculation model in this paper has achieved a steady state. It shows that the change of the connection channel height has little influence on the flowrate of thermal transpiration flow in the microchannel.
Fig. 14 e Velocity of the hydrogen flow at the cross section m2.
Pressure fields of hydrogen flow in the three-stage Knudsen compressor Fig. 17 is the pressure fields and velocity vectors of the first connection channel of the three-stage Knudsen compressor. It can be conclude that with the connection channel height increases, the hydrogen gas pressure in the connection channel decreases gradually. Also, because of the width of the
Fig. 16 e Distribution of hydrogen flowrate at m2 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 2 6 3 2 e2 2 6 4 2
22641
Fig. 17 e Pressure fields and velocity vectors for the connection channel with different heights.
Poiseuille flow path increases, the velocity of the Poiseuille flow in the connection channel decreases gradually.
Conclusions In this paper, the hydrogen flow performance of the multistage hydrogen Knudsen compressor is studied, and the effect of the stage and the connection channel height on the hydrogen flow and pressure characteristic are investigated. The conclusions are as follows: (1) With the increase of the stage number, the pressure increase performance of the hydrogen flow increases linearly and the velocity changes little, which indicates that the multi-stage Knudsen compressor has a singlestage replication characteristic. (2) The change of connection channel height has an effect on the performance of Knudsen compressor. With the increase of the connection channel height, the pressure drop at the center line of the connection channel decreases, which is more conducive to the improvement of the overall pressure increase performance of the hydrogen Knudsen compressor. Whereas the change of the connection channel height has little effect on the pressure change in the microchannel.
(3) For the hydrogen flow fields, there are symmetrical vortexes in the connection channel, and there is also a low-speed zone in the center of the vortexes. When the connection channel height becomes higher, the thermal transpiration flow velocity near the connection channel wall increases slightly, and the Posieuille flow velocity at the connection channel centerline decreases gradually.
Acknowledgements This project was supported by the National Natural Science Foundation of China, Grants No. 51106137, the Major Technological Innovation Projects in Hubei Province, Grants No. 2017AAA035, and the Open Foundation of the State Key Laboratory of Fluid Power and Mechatronic Systems, Grants No. GZKF-201811.
references
[1] Kugimoto K, Hirota Y, Yamauchi T, Yamaguchi H, Niimi T. A novel heat pump system using a multi-stage Knudsen compressor. Int J Heat Mass Transf 2018;127:84e91. https:// doi.org/10.1016/j.ijheatmasstransfer.2018.06.072. [2] Ye JJ, Yang J, Zheng JY, Ding XT, Wong I, Li WZ, et al. Thermal transpiration effect on the mass transfer and flow behaviors
22642
[3]
[4]
[5]
[6]
[7]
[8]
[9]
[10]
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 2 6 3 2 e2 2 6 4 2
of the pressure-driven hydrogen gas flow. Int J Hydrogen Energy 2012;37(17):12474e80. https://doi.org/10.1016/j. ijhydene.2012.05.164. Das S, Sarkar CK, Roy S. Development of integrated microsystem for hydrogen gas detection. IET Circuits, Devices Syst 2018;12(4):453e9. https://doi.org/10.1049/iet-cds. 2017.0243. El Matbouly H, Domingue F, Palmisano V, Boon-Brett L, Post MB, Rivkin C, et al. Assessment of commercial micromachined hydrogen sensors performance metrics for safety sensing applications. Int J Hydrogen Energy 2014;39(9):4664e73. https://doi.org/10.1016/j.ijhydene.2014. 01.037. Buttner WJ, Post MB, Burgess R, Rivkin C. An overview of hydrogen safety sensors and requirements. Int J Hydrogen Energy 2011;36(3):2462e70. https://doi.org/10.1016/j.ijhydene. 2010.04.176. 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. Miles SD, McNamara S, Pharas K. Ten stage Knudsen gas pump. ASME. International conference on nanochannels, microchannels, and minichannels, ASME 2012 10th international conference on nanochannels, Microchannels, and Minichannels. 2012. p. 287e93. https://doi.org/10.1115/ ICNMM2012-73209. 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. Young M, Muntz EP, Shiflett G. The Knudsen compressor as an energy efficient micro-scale vacuum pump. In: Presented at the 2nd NASA/JPL Miniature Pumps Workshop, Pasadena, CA; 2002. Leontidis V, Chen J, Baldas L, Colin S. Numerical design of a Knudsen pump with curved channels operating in the slip
[11]
[12]
[13] [14]
[15]
[16]
[17]
[18]
[19]
[20]
flow regime. Int J Heat Mass Transf 2014;50(8):1065e80. https://doi.org/10.1007/s00231-014-1314-4. Han YL, Muntz EP. Implications of imposing working gas temperature change limits on thermal creep driven flows. In: AIP Conference Proceedings, 1084(1); 2008. p. 305e10. https:// doi.org/10.1063/1.3076491. Colin S. Rarefaction and compressibility effects on steady or transient gas flows in microchannels. In: ASME 2nd International Conference on Microchannels and Minichannels; 2004. p. 1e12. https://doi.org/10.1115/ ICMM2004-2316. Karniadakis GE, Beskok A. Micro flows: fundamentals and simulation. New York: Springer Verlag; 2002. Shu JJ, Teo JBM, Chan WK. Fluid velocity slip and temperature jump at a solid surface. Appl Mech Rev 2017;69(2). 020801(1-13), https://doi.org/10.1115/1.4036191. Lockerby DA, Reese JM, Emerson DR, Barber RW. Velocity boundary condition at solid walls in rarefied gas calculations. Phys Rev 2004;70(1). 017303(1-4), https://doi.org/10.1103/ PhysRevE.70.017303. Sharipov F. Data on the velocity slip and temperature jump on a gas-solid interface. J Phys Chem Ref Data 2011;40(2). 023101(1-28), https://doi.org/10.1063/1.3580290. Colin S. Gas microflows in the slip flow regime: a critical review on convective heat transfer. J Heat Transf 2012;134(2):020908. https://doi.org/10.1115/1.4005063. Nance RP, Hash DB, Hassan HA. Role of boundary conditions in Monte Carlo simulation of Microelectromechanical Systems. J Thermophys Heat Transf 1998;12(3):447e9. https://doi.org/10.2514/2.6358. Ye JJ, Yang J, Zheng JY, Ding XT, Wong I, Li WZ, et al. Rarefaction and temperature gradient effect on the performance of the Knudsen pump. Chin J Mech Eng 2012;25(4):745e52. https://doi.org/10.3901/CJME.2012.04.745. Alexeenko AA, Gimelshein SF, Muntz EP, Ketsdever AD. Kinetic modeling of temperature driven flows in short microchannels. Int J Therm Sci 2006;45(11):1045e51. https:// doi.org/10.1016/j.ijthermalsci.2006.01.014.