General approach to transient flows of rarefied gases through long capillaries

Vacuum 100 (2014) 22e25

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Rapid communication

General approach to transient flows of rarefied gases through long capillaries Felix Sharipov a, *, Irina Graur b a b

Departamento de Física, Universidade Federal do Paraná, Curitiba, Brazil IUSTI, UMR CNRS 7343, Polytech Marseille, Aix Marseille University, Marseille, France

a r t i c l e i n f o

a b s t r a c t

Article history: Received 13 May 2013 Received in revised form 10 July 2013 Accepted 12 July 2013

An approach to model a transient flow of rarefied gas through a long capillary based on numerical data for flow rate previously obtained from the linearized and stationary kinetic equation was proposed. As an example, non-steady flow through a long circular tube connecting two reservoirs is considered. Pressures in the reservoirs and flow rates in the inlet and outlet are obtained as a function of time. It is shown that the behaviors of these quantities vary by changing the gas rarefaction from the free-molecular regime to hydrodynamic one. The typical time to reach the equilibrium state is calculated. Ó 2013 Elsevier Ltd. All rights reserved.

Keywords: Transient flows Rarefied gas Flow rate

Nowadays, steady flows of rarefied gases through tubes are well studied, see e.g. the review [1], and some recent works [2e6]. However, transient flows of rarefied gases are quite usual in vacuum systems, but the information about this topic in open literature is very limited. On our knowledge, only the papers [7e10] report some results on such problems. Calculations of transient flows of rarefied gas are based on a numerical solution of the nonstationary kinetic equation or on the direct simulation Monte Carlo method. Both approaches require significant computational efforts, but in many practical situations, a transient flow can be modeled applying numerical results obtained from the linearized stationary kinetic equation. Some applications of such results to gas flows trough long capillaries for arbitrary pressure and temperature drops were shown in Refs. [11e14]. The aim of this communication is to show that the same idea can be used to model a transient flow of rarefied gas through a long capillary so that the computational effort is not so significant in comparison to a direct solution of the full time-dependent kinetic equation. Consider a long capillary connecting two reservoirs of volumes VA and VB containing the same gas as is shown in Fig. 1. Here, we will consider an arbitrary shape of the capillary cross section denoting as a its characteristic transversal size, e.g. a is a radius of a cylindrical tube or a height in case of rectangular channel. Like the

* Corresponding author. E-mail addresses: sharipov@fisica.ufpr.br polytech.univ-mrs.fr (I. Graur).

(F.

Sharipov),

irina.graour@

0042-207X/$ e see front matter Ó 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.vacuum.2013.07.017

previous works [11e14], the main assumption here is that the capillary length L is much larger that its transversal size, i.e. L [ a, in order to neglect by the end effects. According to Refs. [15,16], the end influence in the hydrodynamic regime has the order of a/L, while in the free-molecular regime [17] its order is (a/L)ln(L/a). These estimations are confirmed by the numerical results for a tube flow [6]. The state of the gas in this system can be disturbed by several non-steady forces, e.g. the pressures pA and pB, volumes VA and VB and temperatures TA and TB of the containers can be time variables. The temperature of the capillary T can also be variable. Since the capillary is long, the gas in its each cross section quickly reaches an equilibrium with the capillary surface so that we may assume that the temperature distribution of the gas along the capillary is the same as that of the capillary wall and given as a function of the time t and position x, i.e. T ¼ T(t,x). If we assume that the containers do not have any other gas inflow or outflow besides that through the capillary, the pressures pA(t) and pB(t) in the containers are determined by their volumes VA and VB, temperatures TA and TB and mass _ through the capillary ends. Thus, the dependence of the flow rate M pressures pA(t) and pB(t) on time is unknown a priori. We are going to obtain a closed system of equations describing the evolution of the pressures pA(t) and pB(t) in the reservoirs, the pressure distribution in the capillary p(t,x) and the mass flow rate _ Mðt; xÞ through each cross section as a function of time. As was shown in Ref. [8], the characteristic time to establish a steady flow over a cross section of capillary has the order of a/vm. According to the numerical results for short tubes [10], the time to

F. Sharipov, I. Graur / Vacuum 100 (2014) 22e25

23

MA ¼ mpA VA =ðkB TA Þ;

MB ¼ mpB VB =ðkB TB Þ;

(7)

where MA and MB are the total masses of gas in the containers, and considering that

 dMA _  ; ¼ M dt x¼0 Fig. 1. Scheme of flow and coordinates.



_ ðt; xÞ ¼ apS G vln p þ G vln T ; M P T vm vx vx

(1)

where S is the cross section area, vm is the most probable molecular speed give as

vm ¼

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2kB T=m;

(2)

kB being the Boltzmann constant and m being the molecular mass. The Poiseuille GP and thermal creep GT coefficients are functions of the local rarefaction parameter d, i.e.

GP ¼ GP ðdÞ;

GT ¼ GT ðdÞ;

(3)

(4)

and determined by the local pressure p(t,x), most probable speed vm ¼ vm(T) and viscosity m ¼ m(T) both depending on the local temperature T(t,x). The numerical data on the coefficients GP and GT for cylindrical tube and planar channel can be found in Ref. [1]. Some results on these coefficients for rectangular channel are reported in Refs. [18,19]. A gas flow through a long elliptical tube was calculated in Refs. [20,21]. Thus, the information about the Functions (3) is widely presented in many previously published papers. The mass balance in a cross section can be expressed in term of the local mass density r(t,x) as

_ Sðvr=vtÞ ¼ vM=vx:

(5)

Then with the help of Eq. (1) and the state equation r ¼ pm=ðkTÞ ¼ 2p=v2m , the mass balance takes the form






 dln pA aSvmA vln p vln T  d VA  GT ln ¼ ; GP  x¼0 vx vx dt dt 2VA TA

 dlnpB aSvmB vln p vln T  d VB GT ln ¼ ; GP  x¼L dt vx vx dt 2VB TB



vp av2 v p vln p vln T ¼ m  GT ; GP vt vx vx 2 vx vm

(6)

which describes the evolution of the pressure p(t,x) inside the capillary. To solve Eq. (6), an initial condition, i.e. p(0,x), and two boundary conditions, i.e. p(t,0) ¼ pA(t) and p(t,L) ¼ pB(t) are needed. The initial pressure distribution p(0,x) is usually known, while the pressures pA(t) and pB(t) are obtained from the mass balance in the containers. Starting from the state equation in each container

(9)

(10)

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi where vmA ¼ 2kB TA =m and vmB ¼ 2kB TB =m. Thus, Eq. (6) with the boundary Conditions (9) and (10) completely determines the solution of the problem on nonsteady flow of rarefied gas through a long capillary. Once the pressure distribution p(t,x) is known, the mass flow rate through any cross section is given by Eq. (1). These equations take a more concise and elegant form in terms of dimensionless quantities. Let p0 and T0 be reference pressure and temperature, respectively. Then, the rarefaction parameter determining the solution can be defined for these reference quantities, i.e.

d0 ¼ ap0 =ðm0 vm0 Þ;

(11)

where m0 and vm0 correspond to the reference temperature T0. The dimensionless quantities can be defined as

p0 ¼ p=p0 ;

which is defined as

dðt; xÞ ¼ ap=ðmvm Þ;

(8)

the pressure variations are obtained as

establish a steady flow over the whole capillary increases by increasing its length. Thus, under the assumption L [ a, the time to establish a steady flow in a cross section is significantly smaller than that to establish a steady flow in the whole capillary. It allows us to apply a steady solution for each cross section, i.e. the mass _ flow rate Mðt; xÞ at the time moment t through a cross section fixed at x can be written as [1]




 dMB _  ; ¼ M dt x¼L

T 0 ¼ T=T0 ;

t 0 ¼ t=s;

x0 ¼ x=L;

(12)

where s is the characteristic time to establish a steady flow in the capillary given as

s ¼ 2L2 =ðavm0 Þ;

(13)

vm0 is given by Eq. (2) with T0. Below, only dimensionless pressure p0 , temperature T0, time t0 and coordinate x0 will be used so that the prime will be omitted. In terms of the dimensionless quantities, Eqs. (6), (9) and (10) take the form


 vp v p vln p vln T ¼ T pffiffiffi GP  GT ; vt vx T vx vx

(14)



 dln pA Vc vln p vln T  d VA  GT ln ¼ ; TA GP  x¼0 vx vx dt dt VA TA

(15)



 dln pB Vc vln p vln T  d VB  GT ln ¼  TB GP ;  x¼1 vx vx dt dt VB TB

(16)

respectively. Here, Vc ¼ SL is the capillary volume. Thus, the characteristic times to establish a constant pressure in the containers are equal to sVA/Vc and sVB/Vc. If the dimensionless flow rate is defined as

_ Gðt; xÞ ¼ Mðt; xÞLvm0 =ðap0 SÞ;

(17)

it can be calculated via the pressure and temperature gradient by

24

F. Sharipov, I. Graur / Vacuum 100 (2014) 22e25






p vln p vln T þ GT : Gðt; xÞ ¼ pffiffiffi GP vx vx T

(18)

As an example of application of the approach, consider a nonsteady isothermal flow of rarefied gas through a circular tube of radius a. The temperatures in both containers and that of the tube are maintained constant, i.e. TA ¼ TB ¼ T ¼ 1. Let us use the initial pressures in the container A as the reference pressure p0 so that the dimensionless pressure pA ¼ 1 at t ¼ 0. The pressure in the tube and left container is initially so small that can be assumed to be zero, i.e. p(0,x) ¼ 0 and pB(0) ¼ 0. We are going to calculate the evolution of both pressures pA and pB for several values of the reference rarefaction parameter defined by Eq. (11) assuming VA ¼ VB ¼ Vc. Under the isothermal conditions Eqs. (14)e(18) are simplified to

dpB vp  ¼ GP x¼1 ; vx dt

(20)

(21)

In case of tube, the Poiseuille coefficient GP(d) obtained in several previous publications, see e.g. Refs. [1,12,21], can be presented by the following formula

1:505 þ 0:0524d

0:75

0:78

1 þ 0:738d

ln d


þ

d 4

 þ 1:018

d 1:073 þ d

; (22)

which interpolates the numerical data within the uncertainty of 0.2%. In order to apply a finite difference scheme, the dimensionless time t and coordinate x are discretized as

xiþ1 ¼ xi þ Dx;

x0 ¼ 0;

Dx ¼ 1=N;

(23)

where N is integer. The time is advances by the step Dt, i.e.

tkþ1 ¼ tk þ Dt;

0.6 0.4 0.2 pB 0 0

0.5

1

1.5

t

2

(19)

Gðt; xÞ ¼ GP ðdÞð vp=vxÞ:

GP ðdÞ ¼

δ0=0.01 10 100

Fig. 2. Dimensionless pressures pA and pB vs dimensionless time t, see definitions Eq. (12).

vp vGP vp v2 p ¼ þ GP 2 ; vt vx vx vx dpA vp  ¼ GP x¼0 ; vx dt

1 pA 0.8

t0 ¼ 0:

(24)

Then, Eq. (19) is approximated as

n       pkþ1;i ¼ pk;i þ GP dk;iþ1  GP dk;i1 pk;iþ1  pk;i1 =4   o þ GP dk;i pk;iþ1  2pk;i þ pk;i1 Dt=ðDxÞ2 ;

d0 ¼ 1 showed that the functions pA(t) and pB(t) are practically the same as those in the free-molecular regime d0 ¼ 0.01. From Fig. 2 it can be seen that in all cases the pressures pA and pB tend to the value 1/3 because initially the gas fills only one container, while in the equilibrium state (t / N) the same gas fills both container and capillary having the same volumes. The time to establish the equilibrium pressures decreases by increasing the rarefaction parameter d0. The dependence of the dimensionless flow rates G on the dimensionless time t in the inlet and outlet are plotted in Fig. 3 for the limit values of the rarefaction parameter d0 ¼ 0.01 and d0 ¼ 100. It shows that the inlet flow rate G(t,0) has a high value in the beginning because of the high pressure gradient at x ¼ 0. The outlet flow rates G(t,1) increase from zero then vanishes in the equilibrium state (t / N). The maximum values and moment to reach this value depends on the rarefaction parameter d0. In the hydrodynamic regime (d0 ¼ 100), the maximum of G is equal to 3.9 and reached at t ¼ 0.034, while in the free-molecular regime (d ¼ 0.01) the maximum is quite smaller, i.e. Gmax ¼ 0.63, and reached quite later, i.e. at t ¼ 0.16. It is interesting to know the time teq needed to reach the equilibrium when the pressures pA and pB are the same and the flow rate G is zero. Since the flow rate for the problem in question is never vanishes but always approaches to zero, the time to reach the equilibrium could be defined as that when both flow rates in the inlet and outlet decreasing reach the value 0.01. The values of teq are given in Table 1. It is curious that the time teq reaches its maximum in the transitional regime, i.e. at d0 ¼ 1. This time tends to a constant value in the free-molecular regime (d ¼ 0.01) and decreases in the hydrodynamic regime, i.e. at d / N. An approach to model a transient flow of rarefied gas through a long capillary is proposed. Since the approach is based on

(25) where pk,i ¼ p(tk,xi) and dk,i ¼ d0pk,i. Eqs. (20) are approximated as

   pkþ1;0 ¼ pk;0 þ GP dk;0 pk;1  pk;0 Dt=Dx;

(26)

  

pkþ1; N ¼ pk; N  GP dk; N pk; N  pk; N1 Dt=Dx:

(27)

The numerical calculations were carried out for the scheme parameters N ¼ 100 and Dt ¼ 106, which provide the numerical error of the flow rate G and pressures pA and pB less than 1%. The calculations take a few minutes using an ordinary workstation. The evolution of the dimensionless pressure pA and pB for three values of the rarefaction parameter d0 ¼ 0.01, 10 and 100 are plotted on Fig. 2. An analysis of the data corresponding to d0 ¼ 0.1 and

4 G 3 2 1

δ0=0.01 δ0=100

0 0

0.1

0.2

0.3

0.4

t

0.5

Fig. 3. Dimensionless flow rate G, Eq. (17), vs dimensionless time t, Eq. (12): solid line e inlet x ¼ 0, dashed line e outlet x ¼ 1.

F. Sharipov, I. Graur / Vacuum 100 (2014) 22e25 Table 1 Dimensionless time teq needed to establish the equilibrium vs d.

d0 teq

0.01 1.84

0.1 1.88

1 1.94

10 1.46

100 0.41

numerical data for flow rate obtained previously, it requires modest computational effort. As an example, non-steady flow through a long circular tube connecting two reservoirs is considered. Pressures in the reservoirs and flow rate are obtained as a function of the time. It is shown that the behavior of these quantities varies by changing the gas rarefaction from the free-molecular regime to the hydrodynamic one. The typical time to reach the equilibrium state is calculated. Following Ref. [14], the proposed method can be applied to a capillary with a variable cross section. Acknowledgments The authors acknowledge Programme Action en Région de Coopération Universitaire et Scientifique (ARCUS, France) for the support of their mutual visits. One of the authors (F.Sh.) also thanks the Brazilian Agency CNPq for the support of his research. References [1] Sharipov F, Seleznev V. Data on internal rarefied gas flows. J Phys Chem Ref Data 1998;27(3):657e706. [2] Varoutis S, Valougeorgis D, Sazhin O, Sharipov F. Rarefied gas flow through short tubes into vacuum. J Vac Sci Technol A 2008;26(2):228e38. [3] Sharipov F. Benchmark problems in rarefied gas dynamics. Vacuum 2012;86(11):1697e700.

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