Thermoacoustic waves in pulsed Pirani sensor MEMS. Numerical Investigation

The transient heat transfer process is studied in rarefied gas confined between two stationary concentric cylinders. The inner cylinder (filament) is subjected to a periodically heating-cooling cycle. The energy transfer is modeled with continuous model based on Navier-Sockes Fourier equations of motion and energy transfer and with a statistical DSMC model. Numerical results for the temperature, thermodynamic pressure and pressure difference between thermodynamic pressure and radial stress tensor component are obtained for different circular frequencies of heating cooling cycle of filament and for different filament radii. The pressure variation at the end of any local heating stage of heating-cooling cycle is close to the value of equilibrium thermodynamic pressure. The results are applicable in designing the pulsed Pirani sensors.


Introduction
Pirani vacuum gauges usually consist of an electrically heated filament suspended in a cylindrical tube.The filament is kept at a constant temperature.A new gauge is described for measurement of pressures in the range 10 −4 mbars to 5 bars.The gauge uses a conventional Pirani sensor, but pulses the filament temperature and measures filament heating and cooling rates [4].The physical foundations of this measurement method are considered in [8].A novel Pirani transmitter which is operated in pulsed mode reveals significant advantages.Its efficiency can be compared to the classical Pirani system.The filament is no longer operated steadily at a constant temperature, but cyclically heated up to a certain temperature threshold by an increasing voltage ramp [10].Numerical calculations of heat flux in the Pirani sensor were carried out for a wide range of the gas rarefaction based on the kinetic Boltzmann equation.To take into account a non-diffusible gas-surface interaction, the Cercignani-Lampis scattering kernel was used in the boundary condition [11].Sound waves, propagating through a rarefied gas confined between two coaxial cylinders and induced by an unsteady heating of the inner boundary, are modeled on the basis of a kinetic model to the linearized and non-stationary Boltzmann equation [16].
We study a rarefied monatomic gas between two coaxial, stationary, unconfined cylinders (one dimensional, axis-symmetrical problem).The temperature of the outer cylinder is constant and equal to the gas equilibrium temperature, while the temperature of the inner cylinder varies in time.The inner cylinder (filament) is subjected on periodically heating-cooling cycle.

Continuous Model (NSF) and Numerical Simulation
The continuous model is based on the Navier-Stokes-Fourier (NSF) equations for compressible fluid, completed with the equations of continuity and energy transport.For details see [15,17,19].The tangential velocity component in the gas flow is The coordinates in radial and tangential direction are    , r .
The following notations are used to describe the rarefied gas flow: u is the velocity component in r directions,  is the density and T is the temperature, P is the thermodynamic pressure.The nonzero stress tensor components taking part in the equation of motion [7] can be expressed as.
The dissipation function Φ [7] in the equation of energy transport is For a perfect monatomic gas, the viscosity and the heat transfer coefficients read as [1,2,3,6]: The NSF model is completed with the equation of state for ideal gas:

RT P  
The governing equations are normalized by using the following scales: for density 0 0 mn   (m is the molecular mass, n0 -the average number density), for velocity -R is the gas constant, for length -the distance between the cylinders , where the mean free path is o l and For the problem formulated first-order slip boundary conditions are imposed at both walls which can be written directly in dimensionless form as follows [19]: at r=Ri ,: i=1, 2. In Eqs. ( 7)-( 8) is the dimensional wall velocity).For diffuse scattering we have used the temperature jump coefficients ζT = 2.1904 calculated, respectively, from the kinetic Bhatnagar-Gross-Krook (BGK) equation [2,14], The dimensionless wall pulsating temperature on the filament 1 T depends on the dimensionless time t harmonically: , and 0.1 A  is the oscillations amplitude.The condition ( 8)-( 9) describes the filament heating stage from the heating-cooling cycle.
For 8) is applied the adiabatic condition (10), corresponding to the cooling stages of the heating-cooling cycle.
T  is the outer cylinder wall constant temperature.
The boundary conditions are modeled by using the local Knudsen number Knlocal.[20]   (11) In (11) with l is denoted the local mean free path,  -molecular diameter and  is the dimensionless density.
Following [9,12,15] we introduce a Stokes number for temperature oscillations , where  is kinematic viscosity and ω -temperature circular frequency.The relation between dimensionless temperature frequency  and Stokes number  can be written, using eq.Eq.( 4).
The dimensionless value of thermodynamic pressure 0 P corresponding to initial equilibrium state can be calculated as: In Eq.( 13) 0  and T0 are the equilibrium dimensionless density and temperature.
The heating-cooling cycle is the reason why the thermoacoustic waves arise in the gas media.In fact, the measured pressure is the pressure difference defined as [16]:

Direct Simulation Monte Carlo (DSMC) Method
The gas considered is simulated as a stochastic system of N particles [5].All quantities used are non-dimensional, so that the mean free path at equilibrium is equal to 1.The basic steps of simulation are as follows: A. The time interval   G. Boundary conditions are: diffusive at the outer cylinder wall; diffusive or adiabatic at the inner cylinder wall depending on the time t; periodic along axis Oz.
We denote the particle velocity before collision with wall with   , ,  .This gas-surface interaction model can be described by keeping the particle velocity magnitude invariant and the velocity direction of the reflected particle set based on isotropic scattering boundary conditions in the half unite sphere.We discuss this problem first in the spherical coordinate system   [13,18].The three velocity components on the inner cylinder for adiabatic case reflected by: .cos .sin.cos.sin.sin and for diffusion case reflected by: The difference between diffusion and adiabatic reflection is only the coefficient.At the diffusion reflection the coefficient depends on the wall temperature 1 T and the particle mass m , while at the adiabatic reflection the coefficient c is the particle velocity magnitude before collision on the particle with the wall.The temperature on the filament 1 T depends on the dimensionless time t harmonically for   . The boundary condition is modeled by (16).The adiabatic condition ( 16) is applies for 15), corresponding to the cooling stage from the heating-cooling cycle.

Numerical results and discussion
The numerical solution of the NSF and DSMC models in detail is described in [15,17,19].
We study three main cases with different frequency - Numerical results for filament temperature (Fig. 1, 2 ,3 and 5), pressure P (Fig. 4) and pressure difference Π (Fig. 1, 2, 3, 4) are presented.The values of the macrocharacteristics for a fixed dimensionless time value are obtained as a mean value in the radial direction.The filament temperature variation 300 0   t is shown on the Fig. 1, 2 and 3.At the lowest frequency, the dimensionless time to reach sustained oscillations for filament temperature and macrocharacteristics is less than the half period -Fig. 1, 2, 3, while at higher frequencies this time is in the order of 30 (Fig. 5 left) and 50 (Fig. 5

right).
The filament temperature pulsation generates thermoacoustic waves in the direction of the radius that are reflected between the two cylinder walls.This causes the deviation of the gas pressure from its equilibrium value.The pressure difference Π is the actual measured value with the gauge [16].The stress tensor variation limits are negligible and the difference between thermodynamic pressure and pressure difference is negligible (Fig. 4).The pressure difference local extreme coincides at the time with the local extreme temperature after reaching the sustained oscillations.No phase difference was observed for the studied cases.It can be concluded that, while maintaining the amplitude of filament temperature, the pressure variation limits are narrowed at increasing the pulses' forced frequency.
The filament radius decreasing decreases the amplitudes of filament temperature and pressure differential oscillations.
The local maxima of sustained oscillations of pressure difference are close to the value of the equilibrium thermodynamic pressure when decreasing the filament radius.(Fig. 6)

Conclusion
The differences in numerical results between the two models appear visually large but as values are below 0.5%.This means that there were no significant differences in numerical results obtained by two methods -NSF and DSMC.
cV. are the heat capacities at constant pressure and constant volume respectively).In this way, the characteristic number Kn and the constants  C and  C take part in the dimensionless model.
, respectively the velocity magnitude with

Fig. 5 .
Fig. 5.The local maxima of sustained oscillations of pressure difference.
Nm particles are located in the m-th cell at any given time.This number varies during the computer simulation by the following two stages:Stage 1. Binary collisions in each cell are calculated, whereas particles do not move.Collision modeling is realized using Bird's scheme "no time counter".Stage 2. Particles move with new initial velocities, acquired after collisions and no external forces act on particles.No collisions are accounted for at this stage.