CC BY
0Russian Journal of Nonlinear Dynamics, 2022, vol. 18, no. 3, pp. 411-422. Full-texts are available at http://nd.ics.org.ru DOI: 10.20537/nd220306
NONLINEAR PHYSICS AND MECHANICS
MSC 2010: 76N15, 76T15, 76J20
Nonlinear Effects of Krypton Flow in a Micronozzle
with a Cylindrical Tube
M. A. Korepanov, M. R. Koroleva, E. A. Mitrukova, A. N. Nechay
This paper considers krypton flow in a micronozzle with a cylindrical tube. A standardized conical nozzle elongated with cylindrical portion performs gas discharge into a vacuum chamber at a pressure of 10-2 Pa. Under such conditions, a low temperature area is formed in the central part of the jet with gas condensation. The particles are entrained by the gas flow. The portion with a constant section behind the nozzle should focus the supersonic flow part and the condensed particle flow and also decrease particle dispersion behind the nozzle throat.
The paper expresses a mathematical model of homogeneous gas motion with respect to formation processes and the growth of condensation nuclei. Since the condensed particles are small, the research is carried out with a single velocity motion model. The results obtained have shown that the application of the cylindrical tube leads to nonlinear flow effects. The flow responds to: the geometrical exposure related to flow transition from the conical diverging nozzle
Received June 23, 2022 Accepted August 12, 2022
Mikhail A. Korepanov kma@udman.ru
Udmurt Federal Research Center UB RAS ul. T. Baramzinoi 34, Izhevsk, 426067 Russia, Kalashnikov Izhevsk State Technical University ul. Studencheskaya 7, Izhevsk, 426069 Russia
Maria R. Koroleva koroleva@udman.ru
Udmurt Federal Research Center UB RAS ul. T. Baramzinoi 34, Izhevsk, 426067 Russia
Ekaterina A. Mitryukova mit_e_a@mail.ru
Kalashnikov Izhevsk State Technical University ul. Studencheskaya 7 Izhevsk, 426069 Russia
Andrey N. Nechay nechay@ipmras.ru
Institute of Applied Physics RAS
ul. Ulyanova 46, Nizhny Novgorod, 603950 Russia
into the cylindrical tube, heat exposure and mass outflow due to particle formation and growth, and considerable friction force exposure due to the small sizes of the channel. The sum total ofthese factors leads to an insignificant deceleration of the supersonic flow part and highly impacts condensation.
Keywords: micronozzle, krypton, Navier-Stokes equations, condensed phase, numerical modeling
1. Introduction
For a wide range of technical applications microflows used in microchannels play a crucial role. They become more popular in microelectronics, aerospace industry, transport and energy industry. In particular, these microchannels are intensively used as cooling system elements of small-size technical equipment. The magnitude of removed heat flows in mini- and microheat exchangers generated in this way may be 1 to 2 orders greater than conventional heat exchangers [6]. Microfluid microelectromechanical systems (MF MEMS) are applied in medicine. They are intended to control parameters of the internal medium by means of biosensors and can release medical substances or hormones like insulin when the sugar blood level increases [12].
Gas microflows are also widely sought after. As opposed to microfluidics, microgas dynamics extensively uses the ability of gases to change their density, and, in this case, microchannels are replaced by micronozzles. They are the components of small-size engines for microcosmic aerial vehicles (micro-CAV) [9] and portable measuring devices [20]. Micronozzles are applied to produce high-speed collimated beams of microparticles for various technologies, namely, for a new spray method CAD-DW (Collimated Aerosol Beam Direct-Write) [8]. Nozzles for supersonic velocities are applied to form gas-jet targets of plasma gas extra ultraviolet emission sources [5, 15]. In this case, the high power of EUV emission relies heavily on the properties of the gas flow fed, as a rule, through supersonic micronozzles with a high divergence ratio into high vacuum [14]. It should be noted that such systems are usually filled with inert gases (krypton, xenon) to avoid unnecessary chemical reactions or contamination of device optical components. Gas flow in such devices has two crucial features: firstly, due to small sizes of nozzles, gas viscosity plays a decisive role in flow formation; secondly, due to discharge into vacuum gas expands greatly, causing a decrease in temperature to extremely low values and possible condensation in gas flow [10]. Experimental studies of such flows are complicated on account of small sizes of devices and extremely low temperatures in the core of the exhaust jet, while accomplishable measurements provide information far away from the nozzle outlet and report about relevant physical processes only indirectly [3, 7]. It is possible to obtain exhaustive information about the processes that take place in similar devices in theory by applying mathematical modeling methods. This approach is an essential tool to study complex physical processes and is based on the fundamental conservation laws of fluid mechanics making up the well-known Navier-Stokes equations.
The research into gas-dynamic processes in micronozzles based on mathematical modeling was carried out in [1, 5, 13, 20-22]. The studies carried out in [1, 21, 22] analyze the effect of the boundary layer being formed in micronozzles on gas dynamics and operational characteristics of micronozzle flows. It was shown that boundary layers dominate in the flow behind the nozzle throat able to decelerate the flow up to subsonic speeds. The work [20] gives an evaluation of Knudsen and Reynolds numbers at gas discharge into vacuum. The work [5] studies the processes of gas-jet targets formation at gas discharge from the nozzle. The parametric analyses
of nozzles of different geometry under various gas parameter values in front of the nozzle are made by comparing the possible relative efficiency of various nozzle configuration combinations, jet discharge modes and experiment geometry for gas-jet target calculational optimization. The work [13] studies two-phase flows in plane wedge-shaped, axially symmetric conical and 3D (rectangular section) supersonic micronozzles numerically. The second phase was introduced into the flow in the nozzle fore-chamber in the form of dust gas particles of 3.25 to 10 /m. In the works mentioned above, condensate either was not taken into account or the motion of the formed two-phase flow was studied, when condensation was finished and did not develop. Numerical studies of the condensation nuclei initiation process, their further growth and interaction with gas flow within a single velocity model of homogeneous condensation were carried out in [10, 11, 15]. The studies showed that the condensation process begins at the nozzle throat. A flow range is formed behind the nozzle filled with condensate with particle sizes up to 100 A. The Prandtl-Meyer expansion fan formed behind the nozzle causes particle projection which is undesirable in the case of making a gas-jet target. Parametric analyses showed that the location of condensate clouds is determined by the geometric features of the nozzle and the deceleration pressure affecting the initial temperature of condensation nucleation. To focus particles, it is required to transfer the condensation initiation point inside the nozzle and to decrease flow expansion at the vacuum chamber inlet. This may be gained by, for example, using a cylindrical tube for the nozzle.
The work seeks to assess the effect of the nozzle cylindrical tube on flow structure and gas condensation. The paper consists of three parts, conclusion and references. The first part states the problems and the basic mathematical relationships describing the gas-phase flow, the process homogeneous nucleation and supercritical nucleation growth. The second part briefly describes the numerical method applied. The third part presents the main results of the work.
2. Problem definition and mathematical modeling
A mathematical modeling of gas discharge into the vacuum chamber through a micronozzle with a cylindrical tube is carried out. The axially symmetrical conical supersonic nozzle is characterized by the radii of inlet Rin, throat Rcr and outlet Ra sections, and also by the lengths of convergent Lc, divergent Ld and cylindrical Lcyl parts. When modeling the flow, the gas-jet configuration formed behind the nozzle throat should be taken into account. For this reason, the computational domain includes both the nozzle and the space behind it. Sizes Rv and Lv of the volume adjoint are determined by the deceleration parameter and assessed individually for each specific case.
The basic principles and assumptions taken for calculations are:
• the working gas is krypton;
• homogeneous nucleation is studied;
• a single velocity and single temperature flow model is used;
• heat capacity is constant;
• axially symmetrical and laminar gas flow.
The problem is solved in two stages. At the first stage, gas-phase flow analysis without considering condensed particles is carried out. The modeling is done on the basis of numerical
integration of the mass conservation theorem and momentum and also energy conservation laws:
dp + dpv^ + dp% + /nv = Q (21)
dt dx dr r ' dpvx | dpvxvx | dpvxvr | pvxvr = dp | drxx | drxr dt dx dr r dx dx dr '
dpvr dpvxvr dpvrvr pvrvr = dp drxr drrr t„ - rgg dt dx dr r dr dx dr r
dpe dpvx h dpvr h pvr h dt dx dr r
d(vrTrr I VrTxr (¡r) | d((\rTrr I VrTrr (jr) | VxTrx I VrTrr (jr j,
r '
where œ and r are longitudinal and radial coordinates, respectively, t is time, p is gas density, V = (vx, vr) is the velocity vector, p is gas pressure, e = ^ + CVT is the specific energy, h = = + CpT is the total enthalpy, Cv, Cp are gas specific heat capacities under constant volume and pressure, respectively, Txx, Trx, Txr, Trr, r00 are viscous tensor components, and q = (qx, qr) is the heat flow density vector.
The tensor components and the heat flow density vectors in Eqs. (2.2)-(2.4) with regard to the axially symmetrical flow are determined as follows:
T -o n^ + + (2V
T™-2r]dx -^{dx + dr + r)' (2-5)
j + ^ + (2.6)
rr dr 3 V dx dr r J '
2 ( dvx dvr v,
(dv dv \
dT dT
= "A^, ?r = "A^:, (2.9)
where n is the dynamic viscosity coefficient, A is the gas heat-conductivity factor, and T is the gas temperature.
The modeling takes into account variation of these coefficients with regard to temperature in accordance with recommendations for pure gases under low pressure values [18] within the framework of the Chapman-Enskog theory for the intermolecular interaction model based on the Lennard-Jones potential. Viscosity in relation to temperature is calculated by the formula
r¡{T)= 26.69-^|-, (2.10)
a2 U(1)
where p. is the gas molar mass, a is the Lennard-Jones potential parameter characterizing the distance of zero intermolecular interaction, and Q is the collision integral depending on the interaction law and can be found by the formula [18]
1.16 0.52 2.16
^ = =n-nr +-= +-=, 2.11
e0.77 T e2.44T' v '
where T is the nondimensional temperature which can be defined by the energy parameter of the Lennard-Jones potential e and the Boltzmann constant k = 1.38 • 10_23 J/K:
- T
T = t-
k
The heat conductivity factor is determined by the viscosity coefficient as follows:
A(T) = A ; p, (2.12)
where Pr is the Prandtl number.
The gas considered without due regard to the condensation effect is described by the ther-modynamically perfect gas equation of state with the specific heat ratio 7:
/ v2 + v2\
p = (2.13)
The basic physical values of krypton used in formulae (2.1)-(2.13) will be found as follows: Cp = 246 J/(kgK), Cv = 147 J/(kgK), R = 99 J/(kgK), 7 = 1.67, f = 83.79 g/mol, a = = 3.542 A, f = 178.9 K, Pr = 0.7.
As mentioned above, at the first stage the gas flow is analyzed regardless of condensation effects. When a stationary solution is found by the equality of flow rates in the inlet and outlet sections, equations and new terms specifying nucleation and the following growth of supercritical clusters are introduced into the analysis. This requires the initial model of Eqs. (2.1)-(2.13) to be supplemented by the equation of forming and growth of supercritical clusters within the flow [10, 11]:
dpG + dpGvx + dpGi y + pGi y = ~ 9t dx dr r '
where G is the condensate mass fraction and G is the condensation mass flowrate calculated by the formula
a = P -PS n (2.15)
Equation (2.15) uses the following notation: rcl is the cluster radius, pS is the pressure of the saturated gas, R0 = 8.31441 J/(molK) is the molar gas constant, and N is the particle concentration per unit volume.
Due to the short range of liquid krypton (116-119 K) and the negligibly small probability of condensation at the stated temperature range, the condensation of solid state krypton was considered.
The cluster size rcl is determined as the ratio of the cluster mass m = ^ and the mass of a single krypton molecule m0 = (NA = 6.022 • 1023 mol-1 is Avogadro's number), and
is found as rcl = r0where r0 is the radius of the krypton atom, here the corresponding
parameter of the Lennard-Jones potential a (2.10) [18].
An extra term taking into account the latent heat of phase transition is added to the right-hand side of the energy equation (2.4) to consider the condensation effect [10]:
dpe dpvxh dpvr h pvr h
dt dx dr r
_ ^^xTxx ^ Qx dvxTrx VrTrr Qr VxTrx VrTrr Qr (2 16)
dx dx r si •
where Ahs is the sublimation heat [4] determined by the formula
s t \ / t \ 0-8755"
R0. (2.17)
AHS = 83.8058
T \ / T \ 0-8755
11.1269(1 --1+ 11.1884'
83.8058 83.8058
The relation of krypton-saturated gas and gas temperature is set by the formula [18]
S 0 _ S 0 AW
lnps = ^^ - (2.18)
R0 R0T
where S0 is the gas-phase entropy [16] and S° is the solid-phase entropy [4]. The equation of state will be changed as follows because of the flow condensed phase:
p = p(7 - 1) (e - (1 -G). (2.19)
The condensation model (2.14)-(2.19) was implemented after getting stationary distribution fields of the basic flow parameters: pressure, velocity, density, and temperature. The condition to form supercritical clusters is gas cooling below 95 K. Until the required number of supercritical clusters is formed (N = 1015 m_3), their growth is neglected, while after their formation nucleation is taken to be cut off and only the growth of supercritical clusters is considered.
3. Numerical algorithm
The axially symmetrical conical nozzle with the cylindrical tube under study has the following dimensions:
Rin = 500 fm, Rcr = 250 fm, Ra = 600 fm, Rv = 3 cm, Lc = Ld = 5 mm, Lcyl = 2 mm, Lv = 1 cm.
Fig. 1. Computational domain
The computational domain is sketched in Figure 1. The initial space problem is reduced to the axially symmetrical one due to geometry. The problem is solved by the control-volume
method [2, 19] based on the author's code. For the computational domain shown in Figure 1 a block structured finite-volume mesh with quadrilateral elements taking the form of squares in a cylindrical tube and the outlet volume was plotted. The mesh becomes more concentrated at the nozzle walls. The total number of mesh elements is 450 000 cells. The computational mesh of the nozzle contains 45 000 elements (450 cells horizontally and 100 cells vertically). Mesh convergence was studied by considering the motion problem of viscous heat-conducting gas without due regard to condensation based on comparison of temperature, velocity and pressure distribution diagrams on the nozzle axis with corresponding distributions specific for nonviscous and non-heat-conducting gas flow.
Control volumes correspond to mesh elements in the applied approach. The main variables are determined at central points of cells. Convection and diffusion flows on the control cell border are found by means of linear approximation at each cell. The value of convection flow through the cell border is determined by means of the upwind difference scheme. Determination of flow parameter values at the cell centers allows one to avoid the divide-by-zero problem of a numerical algorithm for Eqs. (2.1)-(2.19), since the radius does not vanish in this case. To integrate the equations over time, the three-stage Runge-Kutta method was applied. The time step used for gas phase analysis is 10_10 s, and that used to analyze the gas flow with condensation is 10_12 s.
The border of the computational domain is divided into portions that correspond to the solid wall "walls", inlet "inlet", outlet "outlet" and the axis of symmetry "axis" (Fig. 1). There is a chamber in front of the nozzle where decelerated parameters are sustained: p0 = 25 • 105 Pa and T0 = 300 K. The nozzle walls and vertical walls of the vacuum chamber behind it are impermeable adiabatic walls that meet the condition of flow adhesion. In the case of subsonic discharge constant pressure pa = 10_2 Pa is sustained in outlet sections, and the deflection of the flow parameters is implemented in the case of supersonic discharge.
4. Calculation results
The stationary distribution of single-phase parameters in the nozzle obtained by the method of relaxation are shown in Fig. 2. This figure represents the outlet flow from the nozzle into the vacuum chamber at a larger scale. The Reynolds number calculated by the mean flow velocity in the nozzle throat is Re = 1890, and the Knudsen number is Kn = 8 • 10_4. These numbers confirm the satisfiability of the laminar flow model and the legitimacy of application of the Navier-Stokes equations for flow description in the micronozzle under study.
0.002
öS
0.001
0.002 0.004 0.006 0.008
x
(a)
0.002
0.001
■
p: 5.0 7.7 10.3 13.0
0.002 0.004 0.006 0.008
x
(b)
Fig. 2. Distribution: (a) temperature and (b) density at the nozzle without condensed phase
Due to adiabatic gas expansion in the supersonic nozzle, a considerable part of initial enthalpy turns into kinetic energy. The peripheral flow on the walls of the supersonic part of the nozzle due to viscous deceleration is characterized by high temperature and low density. On the
contrary, a cold and dense core is formed in the central part. Such a flow structure is kept in the case of the classical Laval nozzle [11]. The application of the cylinder tube results in the flow being restructured. The supersonic flow decelerates when it is transferred into the cylindrical part, causing local pressure and temperature increase in the core flow, while the gas density decreases. As a result, the flow core is divided into two parts that are divided by the portion of high temperature between each other. The obtained temperature field in its turn affects the formation and growth of clusters.
The obtained stationary distribution of basic flow parameters is the initial condition for condensed phase analysis. Figure 3 shows the pressure, temperature, density, and mass fraction distribution of condensate in percents, and the longitudinal velocity and the Mach number of the flow with respect to the two-phase flow.
0.002
Q5
0.001
P: 108000 124000 140000
0.002 0.004 0.006 0.008
x
(a)
0.002
o;
0.001
0.002 0.004 0.006 0.008
x
(c)
0.002
Q5
0.001
0.002 0.004 0.006 0.008
x
(e)
0.002
0.001
0.002 0.004 0.006 0.008
x
(b)
0.002
0.001
0.002 0.004 0.006 0.008
x
(d)
0.002
0.001
0.002 0.004 0.006 0.008
x
(f)
Fig. 3. Distribution of the pressure (a), temperature (b), density (c), mass fraction of condensate (%) (d), longitudinal velocity (e) and Mach number (f) in the nozzle with cylindrical tube
Gas excessive cooling leads to formation of condensed phase particles (Fig. 3d). Formation and growth of condensation nuclei goes with heat release and the following gas heating. This correlation can be observed in Figs. 3b, 3d. These processes take place in the dynamically changed flow that respond to the change of channel geometry. The growth of friction forces at the near-wall part of the cylindrical tube leads to temperature and pressure increase and decrease in flow velocity. These processes take place in the subsonic flow part. The radius of the supersonic jet is reduced by approximately 10 % in comparison with the expanding nozzle portion, while the maximum velocity is decreased (Figs. 3e, 3f). The subsonic flow part is shown
in white in Fig. 3f. Flow deceleration leads to the appearance of local portions of pressure and density increase within the point x = 6 pm (Figs. 3a, 3c).
-0.005
0 0.005
x
(c)
0.01 -0.005
0.005
0.01
0.01
(d)
Fig. 4. Distribution of the pressure (a), temperature (b), density (c) and longitudinal velocity (d) on the nozzle axis
A change in the energy flow in the cylindrical tube affects the flow in front of it. The arising nonlinear effects are clearly evident in the pressure field (Fig. 3a). There is no visible pressure change in the radius of the nozzle without a tip. The inclination of equal pressure lines is just noticeable at the transition to the critical section and when approaching the nozzle throat. In Figure 3a alternating areas of low and high pressure formed within the upper wall and along the nozzle axis are observed. However, the flow dynamics tend to changes when approaching the cylindrical portion at the point x = 4 ¡m. More specifically, this can be observed in the diagrams showing the parameter distribution along the nozzle axis in Fig. 4. The vertical dashed lines in all diagrams indicate the beginning of the cylindrical nozzle. Local maximum and minimum points are clearly seen in the temperature diagram. The first minimum point x = 4 ¡m corresponds to the onset of the condensation nuclei. The origin of the condensed phase with heat release leads to the following temperature rise and condensation stops. The next temperature increase
at x = 6 ¡m is related to flow transformation inside the tip. The gas temperature goes down to the level when condensation nuclei are formed near the nozzle throat.
Geometrical, kinematic and heat effects that take place in the nozzle can be well described in this case by means of the Vulis influence reduction equation:
(-,r2 \ dV dF dm dlt .dq , „ .
where M is the Mach number, V is the gas velocity, F is the channel cross-section area, m is the mass rate of flow, lt is the work done by friction force, a is the sound velocity, and q is the heat supply.
According to Eq. (4.1), the velocity increase at the supersonic flow area may take place when the flow area and the mass outflow increase, while heat supply and internal friction lead to velocity decrease. In this case the velocity changes due to geometrical, mass, heat exposure and friction. Balanced exposure results in insignificant deceleration of the supersonic flow part and fragmentation of the condensed particle formation area.
5. Conclusion
In this work we have studied krypton flow in the Laval micronozzle with a cylindrical tube. The Navier-Stokes equations describing the viscous heat conducting gas has been the basis for analysis of the two-phase krypton flow initiated as a result of condensation due to medium excessive cooling. To describe the process of nucleation and condensation nuclei growth, an extra equation to keep the condensation mass fraction was introduced, the energy conservation equation was supplemented by a new term, taking into account the heat of phase transition, while the perfect gas equation of state was modified with respect to condensation in the flow.
The analysis was performed at krypton discharge into the vacuum chamber with a static pressure of 10_2 Pa, at deceleration pressure and temperature values 25 • 105 Pa and 300 K, respectively. The effect of low temperatures on the viscosity and pressure of krypton-saturated vapor was taken into account when describing the flow. The obtained results showed that the cylindrical tube greatly affects the flow. Small nozzle transverse sizes lead to considerable increase of flow viscous friction forces, which leads to essential energy flow redistribution in addition to the portion of the constant section behind the supersonic part.
Nonlinear effects observed in the flow are caused by a combination of internal and external actions. The flow is intensively affected by viscous friction forces, channel geometry, changes in the flow rate and temperature effects. These effects result in nonlinear pressure, density and temperature distribution and in the withdrawal of the condensed phase formation area. Thus, the introduction of a cylindrical tube with a length of 2 mm resulted in a considerable change of the flow structure. Flow deceleration and temperature increase due to friction and condensation heat release prevented the flow formation with sufficient concentration of condensed particles. The increase in the conical nozzle part and the reduction of the tube's length are considered to hold much promise in this respect.
Conflict of interest
The authors declare that they have no conflicts of interest.
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