BULGARIAN ACADEMY OF SCIENCES INSTITUTE OF MECHANICS. Modeling and simulation of heat transfer and thermal phenomena in micro gas flows

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1 BULGARIAN ACADEMY OF SCIENCES INSTITUTE OF MECHANICS Modeling and simulation of heat transfer and thermal phenomena in micro gas flows Manuel Vargas Hernando Ph.D. thesis for obtaining the scientific degree DOCTOR Supervisor: Prof. Stefan Kanchev Stefanov Scientific field Mathematical modelling and application of mathematics in mechanics Sofia, 1

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3 Modeling and simulation of heat transfer and thermal phenomena in micro gas flows Preface This research work is focused on the advancement of the direct simulation Monte Carlo method (DSMC) and its efficient implementation in the solution of micro heat transfer phenomena. The aims of the dissertation may be defined as follows: Implementation and validation of more realistic boundary condition models and intermolecular potentials in the direct simulation Monte Carlo method for the study of time dependent heat transfer problems and thermal phenomena in various geometrical configurations. Extension of the DSMC method for the study of heat transfer flows through gas mixtures. Implementation of more efficient collision algorithms with improved stochastic properties. Investigation of heat and mass transfer problems by using alternative successful methodologies (Discrete Velocity Methods, DVM) and comparison with DSMC. Development of a fully 3D DSMC code for investigation of thermal transpiration effects in new designs of capacitance diaphragm gauges with helicoidal baffle system. The dissertation is structured in five chapters which may be outlined as follows: The Chapter 1 is devoted to the general concepts, theoretical background and state of the art in the field of gas microflows and the numerical simulation. The numerical methods which are employed for this investigation are also introduced. In Chapter the results on heat transfer flow in simple one-dimensional configurations are presented. Results obtained by using the more realistic Cercignani-Lampis boundary conditions, by using various intermolecular interaction models and improved collision schemes are included at this moment. The influence of these features on the heat transfer is investigated. In Chapter 3 the time dependent transient heat transfer between coaxial cylinders due to a sudden change in the temperature of the inner surface is investigated. The study covers the whole range of rarefaction and various values of the radius ratio, including the planar solution. The dynamic response of the macroscopic properties of the gas is captured and the response time estimated. The results are provided for both, simple gas and binary mixtures.

4 Manuel Vargas Hernando The thermal effects caused by non-uniform temperature at the boundaries are investigated in Chapter 4. This chapter is subdivided in two subsections. The first one is devoted to the thermal driven flow induced by non-isothermal walls in rectangular enclosures, while in the second part thermal transpiration effects in real pressure gauges with complex geometry and applications in the vacuum industry are investigated. The dissertation is concluded with the scientific contributions and possible future extensions of the present work in Chapter 5. This dissertation consists of 5 chapters written on 16 pages. The results are presented in 53 figures, 13 tables and 1 journal papers and books were cited. This investigation was accomplished for three years in the framework of the GASMEMS project: an initial training network financed by the FP7 of the European Commission with thirteen participants and six associated partners from all over the European Community. Its aim is to provide fundamental and applied research in the fields of dynamics of gas microflows and of microstructure design and manufacturing. The main part of the research was carried out at the Institute of Mechanics of the Bulgarian Academy of Sciences under the supervision of Stefan Stefanov, but this project offered the opportunity to make two secondment periods: the first one at the University of Thessaly (Volos, Greece) under the supervision of Dimitris Valougeorgis and the second one at INFICON under the supervision of Martin Wüest. The main results of this dissertation were reported in several workshops, such as the GASMEMS Workshops in 1 and 11, the 64 th International Union for Vacuum Science Technique and Applications (IUVSTA) Workshop, May 11 in Leinsweiler (Germany), or the workshop on Direct Simulation Monte Carlo 11: Theory, Methods and Applications (DSMC11), September 11 in Santa Fe (US), in international symposiums on rarefied gas dynamics (RGD7) and the 58 th American Vacuum Society (AVS) International Symposium and Exhibition, October 11 in Nashville (US), and in several international conferences on microfluidics, such as the nd European Conference on Microfluidics, December 1 in Toulouse (France), and the 1 st European Conference on Gas Micro Flows, June 1 in Skiathos (Greece). Moreover, the main results of the dissertation have been submitted or are in process of submission to international scientific journals with impact factor and published in conference proceedings of international conferences. 3

5 Modeling and simulation of heat transfer and thermal phenomena in micro gas flows Acknowledgements: First of all, I would like to express my gratitude to my supervisor Prof. S. Stefanov for guiding me during this period, sharing his wide knowledge on DSMC and for giving me support along this investigation. Also my sincere gratitude goes to Prof. D. Valougeorgis for his advice and fruitful discussions and to Dr. M. Wüest for suggesting this challenging topic and introducing me briefly in the industrial approach to the research during my secondment periods at the University of Thessaly (Greece) and INFICON (Liechtenstein). Finally, I am very grateful to my wife, Nevena, for her help, support and patience during the most difficult situations and, above all, to my little daughter, Marta. Que tu sonrisa brille siempre como hasta ahora! The finantial support for this Ph.D. has been provided by the European Community's Seventh Framework Programme (ITN - FP7/7-13) under grant agreement n

6 Manuel Vargas Hernando Table of Contents Chapter 1. Theoretical background Introduction Numerical methods Direct simulation Monte Carlo No Time Counter scheme Bernoulli trials scheme Discrete Velocity Method applied to solve kinetic models Application of kinetic models to heat transfer problems Numerical scheme Boundary conditions Intermolecular potentials Time and ensemble averaging 33 Chapter. One-dimensional steady heat transfer problem in planar and cylindrical geometry Heat transfer problem between infinite parallel plates: Application of Cercignani Lampis boundary conditions Introduction Formulation of the problem Analytical solution in the free molecular regime Application of Cercignani-Lampis boundary conditions Validation and comparison with DVM results Heat flux convergence for various sampling performances: Comparison between NTC and BT schemes 39.. Heat transfer problem between coaxial cylinders Introduction 4... Formulation of the problem Results and discussion Validation and comparison with DVM results 48 Chapter 3. Transient heat transfer flow between coaxial cylinders Introduction Problem formulation and numerical simulation Numerical results for a single gas 53 5

7 Modeling and simulation of heat transfer and thermal phenomena in micro gas flows Transient behaviour Effect of the governing parameters on the transient solution Comparison of DSMC data with Navier-Stokes-Fourier calculations Numerical results for binary mixtures Conclusions 73 Chapter 4. Flows induced by temperature field: Thermal creep flow Introduction Thermal driven flow due to non-uniform wall temperature in rectangular enclosures Introduction Problem formulation and numerical simulation Numerical results Influence of the aspect ratio Comparison with adiabatic boundary conditions Flow patterns in three dimensional enclosures Simulation of thermal driven flow using small number of particles per cell Conclusions Thermal transpiration effect in Capacitance Diaphragm Gauges with helicoidal baffle system Introduction Problem formulation and numerical simulation Steady state calculations of thermal transpiration in CDGs Flow dynamics of the system due to sudden changes in the process parameters Transient flow due to a sudden change in the temperature of the sensor Sudden change in the pressure at the inlet Conclusions 11 Chapter 5. Concluding remarks Scientific contributions Future work 115 List of publications 116 Bibliography 118 6

8 Manuel Vargas Hernando Chapter 1 Theoretical background 1.1. Introduction Rarefied (or non-equilibrium) gas flows have until recently been associated with low-density applications such as vacuum science and high-altitude applications, such as space vehicle technology. However, the advent of Micro-Electro-Mechanical Systems (MEMS) has opened up an entirely new area of research where non-equilibrium gas behaviour has become very important. The computational analysis of gaseous flows in MEMS devices operating in flow regimes at large Knudsen numbers ( Kn >.1) cannot be based on classical continuum models of fluid motion because they are not valid for non-equilibrium flow conditions. In such flows the mean free path of the gas molecules is comparable to the characteristic size of the microdevice and the kinetic effects of rarefaction and non-equilibrium must be taken into consideration. An appealing aspect of rarefied flows is that they can be analyzed in detail using the direct simulation Monte Carlo (DSMC) method and kinetic models. The DSMC technique uses model particles that move and collide in physical space to perform a direct simulation of the molecular gas dynamics. The macroscopic models treat the gas as a continuum medium and its description is in terms of the variations of the macroscopic properties of the gas (density, bulk velocity, temperature and pressure) in time and space. The Navier-Stokes equations constitute the most representative model at macroscopic level. In contrast microscopic models consider the gas as a set of discrete molecules providing their position and velocity at all times. The Boltzmann equation is the mathematical model at microscopic level. f f + ξ = ( f * f f* f ) B( g, θ) d Ω( θ)dξ* t x 3 The final aim is to calculate the distribution function (,,,, x, y, z) (1) f txyzξ ξ ξ, which defines the fraction of particles in the phase space, that means, the fraction of particles having approximately the velocity ( x, y, z) dependent problem. ξ ξ ξ near the place ( xyz,, ) and time ( ) t if it is a time 7

9 Modeling and simulation of heat transfer and thermal phenomena in micro gas flows The border which defines the region of application of the Navier-Stokes equation is given by the Knudsen number ( Kn ), which defines the degree of rarefaction of a gas. λ λ ρ Kn = = L ρ x This dimensionless parameter is defined as the ratio between the mean free path ( λ ) and the characteristic length of the system ( L ). The flow regimes can be classified in terms of the Knudsen number as follows: Kn : Inviscid limit where the transport terms vanish. The Euler equations are applicable. 3 Kn < 1 : Hydrodynamic regime. The gas may be considered as a continuum medium and the Navier-Stokes equations with no-slip boundary conditions are applicable. 3 1 < Kn <.1: Slip regime. Non-equilibrium phenomena appear in the boundary regions. The Navier-Stokes equations may be used if slip boundary conditions are applied..1 < Kn < 1 : Transition regime. A kinetic description of the gas is necessary. Intermolecular collisions are reduced and gas-surface interaction play important role. Kn > 1 : Free molecular flow. The molecules travel ballistically without interacting between them. The limits of validity of the dilute gas approximation, the continuum approach and the neglect of statistical fluctuations are shown in Figure 1. A simple gas is defined as a gas consisting of a single chemical species in which all the molecules are assumed to have the same structure. The number of molecules per unit volume, or number density, is independent of the composition of the gas, as Avogadro s law states that the volume occupied by one mole of any gas at a given temperature and pressure is the same for all gases. The average 13 volume available to a molecule is 1 n, so the mean molecular spacing is given by δ = n. As it will be explained later, a simple model of a molecule is considering a hard elastic sphere of diameter d. Then, two molecules collide if their trajectories are such that the distance between their centres decrease to d. The total cross section for such molecules is therefore () σ T = πd. The intermolecular force for such kind of molecules is zero when the distance between their centres is larger than d and is infinity when the distance is exactly d. This is a big simplification, as it is known that real molecules present attractive forces at large distances and repulsive forces at short distance. However, the results obtained by using the hard elastic molecules are accurate enough for many engineering applications. The mean free 8

10 Manuel Vargas Hernando path is defined as the average distance travelled by a molecule between two consecutive collisions with other molecules. The following relationship applies for the case of hard sphere molecules: ( n d π ) 1 λ = (3) The proportion of space occupied by the gas is actually ( d δ ) 3. A dilute gas is such gas where only a very small proportion of space is occupied by molecules ( δ >> d ), and each molecule will move outside the range of influence of other molecules. Under these conditions we can assume that only binary collisions happen within the gas. That means that the collisions in that gas can be realized between only two molecules at the same time. This is a basic assumption in the standard direct simulation Monte Carlo method. Figure 1: Effective limits of major approximations and fields of application of rarefied gas flows calculations (adapted from [1]). As it is seen in Figure 1, both MEMS applications and aerospace and vacuum applications fall in the zones where the Navier-Stokes approach breaks down. The creation of micro- and nanometer-sized devices is very important since they offer increased reliability, low cost and high efficiency [, 3] in comparison to their normal-sized counterparts. Vacuum flows are encountered in many applications, ranging from a simple pressure sensor [4] to the vacuum 9

11 Modeling and simulation of heat transfer and thermal phenomena in micro gas flows systems of fusion reactors [5]. High altitude aerodynamics needs to be investigated very carefully for the correct operation of spacecrafts [6] and satellites [7]. Thus, the accuracy and efficiency of simulations is of high importance for the design, manufacturing and optimization of these devices. Micro electro-mechanical systems (MEMS) refer to devices that have characteristic length of less than 1 mm but more than 1 µm, that combine electrical and mechanical components, and that are fabricated using integrated circuit batch-processing techniques [8]. Current manufacturing techniques for MEMS include surface silicon micromachining, bulk silicon micromachining, lithography, electrodeposition and plastic molding (or, in its original German, lithographie galvanoformung abformung, LIGA) and electrodischarge machining (EDM). MEMS have been used as sensors for pressure, temperature, mass flow, velocity and sound, as actuators for linear and angular motions, and as simple components for complex systems such as micro-heat-engines and micro-heat-pumps. In the case of MEMS with moving parts, such as microresonators [9] and comb drive sensors [1], the damping forces induced by the rarefied gas ambient can significantly alter performance and sensitivity characteristics. The flow field around micro heat flux sensors [11] plays an important role in the accuracy of the device. Flows are also rarefied in flows through porous media [1]. The read-head sliders in hard disk drives operate under conditions where the Knudsen number in the microgap is relatively large [13]. As mentioned, rarefied gases are present in the field aerospace engineering. Some indicative applications are given below. Hypersonic flows around space vehicles [6] and satellites [7] during re-entry are frequently encountered in rarefied atmospheres. The DSMC numerical method is frequently employed in such cases and large organizations, including NASA [14], develop their own code versions of this numerical algorithm. The construction of microscale propulsion devices such as mono- and bi-propellant thrusters and resistojets, has also increased the needs for the accurate simulation and measurement of rarefied flows [15]. The detailed modelling of vacuum pumps [16, 17, 18] and gas separators [19] is very important to obtain the maximum efficiency. Multi-layer insulation (MLI) blankets, extensively used in space vehicles, consist of several layers of thin sheets with vacuum conditions between them to ensure that heat is transferred only through radiation. A large factor in the performance of the insulation is its behaviour in the case of degraded vacuum []. Vacuum deposition systems are used for the fabrication of thin-film materials in the manufacture of integrated circuits, MEMS and nanocomposites [1]. 1

12 Manuel Vargas Hernando Moreover, vacuum is employed in several scientific experiments that have to be conducted in rarefied conditions. Particle accelerators [] require ultra-high-vacuum (UHV) conditions that can only be achieved by careful design and optimization of the vacuum equipment. The thermonuclear fusion reactor ITER is a promising international programme for covering future energy needs. Due to the high requirements for pumping [5] (insulation vacuum, low pressure to maintain plasma, fuel pumping), flow conditions usually correspond to the transitional or free molecular ranges. 1.. Numerical methods The numerical methods which are employed for simulating gas flows could be classified, as a first step, in molecular and continuum models [8]. The molecular models treat the gas as a collection of molecules. Among them, there are deterministic methods, such as Molecular Dynamics (MD), and methods based on statistical approaches, such as Direct Simulation Monte Carlo (DSMC) method. The evolution of a N particle system is given by the Liouville equation for the probability density function in 6N phase space. The Liouville equation is the starting point to obtain the BBGKY hierarchy of equations and finally the Boltzmann equation by truncation of the BBGKY chain and assumption of molecular chaos. Several kinetic models have been proposed (BGK, Shakhov and ellipsoidal models among others) to substitute the collision term of the Boltzmann equation in order to deal with the analytical barriers and immense computational requirements associated with its solution. The continuum models are based on the Chapman [3] and Enskog [4] theory which describes the distribution function f of molecules in terms of a deviation series from the equilibrium Maxwell distribution according to ( ) ( 1) ( ) f = f + Kn f + Kn f +... (4) By replacing this expression in the Boltzmann equation, we obtain a system of integrodifferential equations. The zeroth, first and second order terms lead to the Euler, Navier- Stokes and Burnett equations respectively [5]. Here, the detailed description of the methods which have been employed in this dissertation is given below. 11

13 Modeling and simulation of heat transfer and thermal phenomena in micro gas flows Direct Simulation Monte Carlo In general, Monte Carlo methods are methods of approximation of the solution to problems of computational mathematics, using random processes. The method can guarantee that the error of Monte Carlo approximation is smaller than a given value with a certain probability [6]. In such methods the law of large numbers ensures that the estimate converges to the correct value as the number of draws increases. The central limit theorem provides information about the likely magnitude of the error in the estimate after a finite number of draws. The direct simulation Monte Carlo (DSMC) method was proposed by Bird in late 6 s. It became popular in the 7 s in aerospace engineering and it has been improved and extended for the last decades. The main assumptions of the DSMC method are, firstly that the gas is a dilute gas the mean molecular spacing should be much larger than the molecular diameter and consequently all the collisions between particles can be considered as binary collisions, and secondly the assumption of molecular chaos the probability of finding a pair of molecules in a two particle configuration is simply the product of the probabilities of finding the individual molecules in the two corresponding one particle configurations. f (, tx, x, ξ, ξ ) = f (, tx, ξ ) f (, tx, ξ ) (5) () (1) (1) The DSMC method combines deterministic aspects for modeling particle motion with statistical approach for computing collisions between particles. It has been proved by Wagner [7] that DSMC converge to the solution of the Boltzmann equation in the limit of infinite number of particles in cell. Formally, the standard DSMC [1] makes use of the following time splitting scheme, applied to the distribution function at time a given time f ( t k ) for f t + obtaining an approximation to the solution of ( ) k 1. The real process is divided in two steps, which could be expressed as follows: 1. Collision between particles, where the molecular velocities are changed while the molecular position is kept unchanged. { } ( ) ( ) t, x ( ) ( ) f tk t, Χ tk, Ξ + tk+ t = SQ f tk, Χ tk, Ξ tk (6). Convective motion of the particles, where particle positions are changed without changing their velocities (in the simplest case without external force). { } ( ) ( ) t ( ) ( ) f tk t, Χ tk t, Ξ + + tk+ t = SD f tk, Χ tk, Ξ tk+ t (7) 1

14 Manuel Vargas Hernando Where t, x S Q and t S D are the operators approximating the action of the collision and the convective terms respectively in the kinetic equations, t is the time step and x is the cell size. Considering the global operator which evaluates the kinetic equation solution at t k + 1 from the previous state defined as t, x t, x t SQ+ D = SQ SD, it turns out according to the results obtained by Bobylev and Ohwada [8] that this splitting scheme approximates the solution of the Boltzmann equation with accuracy of first order with respect to the time step and cell size O( t+ x). The basis of the DSMC technique is the discretization of the space domain and the time, while the gas is represented by a discrete number of model particles. In summary, the DSMC algorithm could be described as follows: A. Initialization The time interval [, t f ], over which the solution is sought, is subdivided into subintervals with step t. The space domain is subdivided into cells with sides x. The gas molecules are simulated in the gap with a stochastic system of N points having positions xi ( t ) and velocities ξ ( t) i. Each point represents a certain and large number of real molecules (weighting factor) which is represented by F N. There exist thumb rules which must be fulfilled by the time step, cell size and number of particles to obtain accurate results. ( l) t τ 3 ; x λ 3 ; N > 3 (8) Where τ = λ v is the mean collision time and λ is the mean free path. Then, within each time step the following steps are repeated over a loop: B. Collision between particles: The binary collisions in each cell are calculated without moving the particles by using the chosen collision model. This stage is the most complex one and where more efforts have been made in order to improve the original Time Counter scheme [1]. The detailed explanation of the standard No Time Counter collision algorithm together with alternative collision schemes is introduced below. C. Motion of the particles: All particles in the computational domain are moved ballistically, without colliding, in accordance to the simple equation: 1 x = x + v + t g t (9) 13

15 Modeling and simulation of heat transfer and thermal phenomena in micro gas flows Where g is the acceleration due to any external force which may be applied to the molecules. During this stage the interaction of the molecules with the boundaries is performed. Molecules which hit the solid boundaries are reflected back into the domain in accordance to the chosen boundary condition model. D. Indexing: The particles are sorted out into the spatial collision cells. This stage is necessary due to fact that the intermolecular collisions are produced between the particles within the same cell. E. Sampling: The important moments are accumulated and the macroscopic properties will be derived after having a sufficiently large sample size to obtain smooth results without statistical scatter as follows: S k = 1 ρ c ( ) m N tk mn = = SV SV u = S N k= 1 i C N ξ T i ( t ) k T c (1) (11) N( t ) N( t ) 1 1 T = = xyz T= T+ T+ T α N α T k 1 i 1 N α = = T k= 1 i= 1 S k S k ( ξ, i) ξ, i, α {,, }, ( x y z) 3 (1) N( t ) N( t ) N( t ) q = + + u u v uv S k S k S k ξ, ( ξ, ξ, ξ, ) ( ξ, ) ( ξ, ξ, ) x xi xi yi zi xi xi yi NT k= 1 i= 1 NT k= 1 i= 1 NT k= 1 i= 1 N( t ) N( t ) 1 1 w uw u + + N N S k S k ( ξxi, ξzi, ) ( ξxi, ξyi, ξzi, ) (13) T k= 1 i= 1 T k= 1 i= 1 The sampling strategy is especially important in the case of the heat flux due to the fact that the heat is transported not only during the free motion of the particles, but also during the collision process, in contrast with the mass, which is transferred only during the free motion of the particles. There are several sampling performances which may lead to slightly different results, such as sampling after the motion stage, sampling after the collision stage or the so called Strang scheme or its variations, where the sampling is realized two times within each time step, one after the motion and the second after the collision stage. With the later strategy the results reach second order convergence and become more stable to the variation of the problem discretization. The convergence behaves in a similar manner by using the so called symmetric sampling, where the 14

16 Manuel Vargas Hernando sampling occurs only once every time step, but it is realized between two consecutive collision or motion stages over a time interval equal to t. As it was mentioned above, the collision stage is the most complicated one. During the last decades considerable efforts have been made in order to improve the characteristics of the initial Time Counter scheme [1].This scheme turned out to compute the binary collision frequency with a systematic error depending strongly on the inverse number of model particles occupying a grid cell in each time step. As a consequence several collision schemes with better characteristics were suggested later: Null-Collision [9], Ballot-Box [3], Modified-Nanbu [31], Majorant Frequency Scheme [3] and No Time Counter (NTC) [1]. The latter one has become the most frequently used. All these approaches are based on the principle of estimating the maximum collision frequency in cells in order to define the number of particle pairs to be checked for collision. After that the real number of collisions in cells is specified by using the acceptance-rejection procedure. These schemes differ from one another mainly in the definition of the maximum collision frequency. New approaches to reduce the variance and to deal with low speed flows were proposed [33, 34] by simulating the deviation from the local equilibrium. All these schemes require a large number of particles in cells to obtain sufficiently accurate results. More recently, an alternative collision scheme with improved stochastic properties was proposed by Stefanov [35, 36]. The so called Bernoulli Trials (BT) and its simplified version (SBT) allow the use of small number of particles in cells by avoiding the repeated collisions within the time step. The details of the standard NTC and the alternative BT and SBT are given below No Time Counter scheme It has been explained before that the probability of a collision between two particles is proportional to the product of their relative speed c r and collision cross section σ T. The nonequilibrium collision rate in a homogeneous gas, that means, the total number of collisions per unit time per unit volume of gas could be expressed as 1 Nc = ncσ r T (14) The probability of collision between two simulated molecules over the time interval t is equal to the ratio of the volume swept out by their total cross section moving at the relative speed c r to the volume of the cell. 15

17 Modeling and simulation of heat transfer and thermal phenomena in micro gas flows p F σ c t V N T r = (15) c - In the NTC scheme the number of pair selections is determined in advance. This maximum number of pairs which will be checked for collision is given by this expression: N c ( 1) N ( σ T r) max 1 N N F c t = (16) V c - The chosen pairs of particles are selected randomly within the cell and the collision is computed with probability r) max p σ c T r = (17) ( σ Tcr) max The parameter ( σ c should be updated immediately if a larger value is obtained and stored for each cell. T - If the collision is accepted in the acceptance-rejection method then velocities ( ξi, ξ j) ' ' are changed to their post-collision values ( ξi, ξ j ), otherwise they remain unchanged. The collision between particles fulfils the mass, momentum and energy conservation. The determination of the post-collision velocities in the centre of mass frame reduces to the calculation of the change in direction of the relative velocity vector. This angle is distributed randomly over the solid angle Bernoulli trials scheme An alternative collision scheme based on Bernoulli Trials (BT) and proposed by Stefanov in [35, 36] has been investigated. This method allows to work with a smaller number of particles in the grid cells ( N ( l) 1) in contrast with the NTC scheme, which requires, as mentioned above, a large number of particles in cells in order to obtain sufficiently accurate results. The reason is that the NTC algorithm allows repeated collisions between the same particle pair that lead to a reduction in the local collision frequency and to a distortion of the collision process in cells with small number of particles. BT scheme deals with two limitations of the standard NTC scheme: 1. It avoids the repeated collisions which are allowed in the NTC scheme 16

18 Manuel Vargas Hernando. It allows the collision of the potential collisions between particles of neighbouring cells. This number of collision become more important as the number of particles per cell is decreased. It has been proved that the effect of n repeated consecutive collisions of the same pair is equivalent to the realization of only one collision, which leads to a reduction in the collision rate. If we assume that ( ) ( ) ( i, j ) ξ ξ are the initial velocities of the molecules i and j, after n repeated collisions their velocities will be given by: ( ( ) ( ) ( ) ) ( ) 1 ( ) ( ) ξ = ξ + ξ + ξ ξ ω ' n n 1 n 1 n 1 n 1 n i i j i j ( ) ( ) ( ) ( ) ( ) 1 ( ) ( ) ξ = ξ + ξ ξ ξ ω ' n n 1 n 1 n 1 n 1 n j i j i j (18) (19) Taking into consideration that the sum of their velocities is conserved ( n 1) ( n 1) ( ) ( ) ξ + ξ = ξ + ξ, as well as the magnitude of their relative velocity i j i j ( n 1) ( n 1) ( ) ( ) ξ ξ = ξ ξ, then it can be seen that the effect is the same as only one collision: i j i j ( ) ( ) ( n) ( ) '( n) 1 ( ) ( ) ξi = ξi + ξ j + ξi ξ j ω ( ) ( ) ( n) ( ) '( n) 1 ( ) ( ) ξ j = ξi + ξ j ξi ξ j ω () (1) In order to allow the collisions between particles of neighbouring cells, a two-step collision algorithm applied twice on a dual grid using BT scheme is proposed. The collision scheme is divided into two steps during half of the time step. The first collision procedure is applied on the main grid, while the second is applied on a shifted grid. This shifted grid is the original grid which was moved half of the cell size in each of the directions. This strategy has two advantages: - It improves the stochastic properties of the collision algorithm, as it allows the collisions between particles in neighbouring cells. - It improves the accuracy of the splitting scheme, which becomes of second order with respect to the time step O( t ). Two alternative algorithms have been proposed to avoid this problem. Firstly, the so called BT scheme consists of the following steps. l For each cell l ( l = 1,..., L) with volume V : c ( ) 17

19 Modeling and simulation of heat transfer and thermal phenomena in micro gas flows - each pair of particles in cell l with velocities ( i, j) ξ ξ, ( l) i< j = 1,..., N is checked for collision with probability p σ c ij r, ij ij = or V ( l) c t - if the collision is accepted then velocities ( i, j) ' ' values ( ξi, ξ j ), otherwise they remain unchanged. p ij σ ijc t r, ij = () ( l) V The drawback of this algorithm is that requires a number of operations c ξ ξ are changed to their post-collision ( l) ( ) O N in cell l, as the number of collision pairs which are checked for collision every time step is ( ) ( l) ( l) N N 1. The second algorithm is the so called Simplified BT scheme realizes a number of computations proportional to the number of particles per cell pairs which are checked for collision every time step is expressed as: - the sequence of particle pairs as follows: ( l) ( N ) ( l) N ( ( l) 1) i = 1,..., 1 is chosen from, as the number of collision N. Its description may be ( l) N particles in cell l 1. The first particle i is the particle with index i in the particle list for cell l.. The second particle ( l) ( ) k = N i taking place in the list after particle i. ( l) j i+ 1, N is chosen with probability 1 from k - then the particle pair ( i, j ) is checked for collision with probability p σ c if the collision is accepted then velocities ( i, j) ' ' ( ξi, ξ j ), otherwise they remain unchanged. ij r, ij ij = k (3) V ( l) c t ξ ξ are changed to their post-collision values 1... Discrete Velocity Method applied to solve kinetic models The DVM as a way to solve different non-linear kinetic models consist of discretizing the kinetic equations in the molecular velocity and physical space and then the discretized 18

20 Manuel Vargas Hernando problem is solved in an iterative manner until the convergence criterion is fulfilled. The final aim is to calculate the distribution function (,,,, x, y, z) f txyzξ ξ ξ, which defines the fraction of particles in the phase space, that means, the fraction of particles having approximately the velocity ( x, y, z) ξ ξ ξ near the place (x,y,z) and time (t) if it is a time dependent problem. Due to the difficulty of solving the full Boltzmann equation several approaches have been investigated incorporating approximations in order to alter the form of the Boltzmann equation itself. These model equations modify the collision term on the right hand side, which poses the greatest difficulty. Two non-linear model equations for large disturbances have been studied in this work: the BGK and the Shakhov models. The collision term in the BGK equation can be written as M ( f f ) ν (4) P In this expression ν represents the collision frequency and is chosen to be ν = where P 3 µ and µ are the local pressure and gas viscosity. This expression of the collision frequency deduces the correct expression for the thermal conductivity. The local maxwellian distribution is f M 3 m m n exp π kt B kt ξ = B The Shakhov model, unlike the BGK model, provides simultaneously the correct expressions for the heat conduction and viscosity transport coefficient. The Shakhov equation express the collision term as follows: P S ( f f ) (6) µ Where (5) S M f = f 1 + q 15n kt m mξ 5 ξ y ( ) kt B B And f M represents again the local maxwellian distribution. (7) Application of kinetic models to heat transfer problems The BGK [37] and the Shakhov [38] kinetic equations where applied to the study of the 1- dimensional steady heat transfer flow between infinite parallel plates and between coaxial cylinders. For the planar geometry the distribution function obeys the following expression: 19

21 Modeling and simulation of heat transfer and thermal phenomena in micro gas flows f P ξ y = f f y µ ( ) Where f ( y, ξ ) is the unknown distribution function, f may represent (8) f M in case of employing the BGK equation or S f in case of using the Shakhov model and ξ y is the velocity component perpendicular to the plates. As mentioned before, the collision frequency for the BGK model must be modified by the factor /3 in order to predict correctly the heat conduction transport coefficient. In cylindrical geometry the governing equation is written as: f ξp sinθ f P ξp cosθ = ( f f ) (9) r r θ µ Where f ( r, ξ ) is the unknown distribution function, r is the radial spatial coordinate, ( ξr, ξθ, ξz) ( ξpcos θξ, psin θξ, z) ξ = = is the molecular velocity vector and ξ = ( ξ, ξ ) is p r θ the planar velocity vector. The macroscopic distributions can be obtained by the moments of the distribution function for planar and cylindrical configuration: ( ) ( ) ξx ξy ξz n y = fd d d (3) m = (31) T y ξ fdξxdξydξz 3n( y) kb m q( y) = ξξy fdξxdξydξz (3) ( ) p p n r = f ξ dξ dθdz (33) m T ( r) = ( ξp + ξz ) f ξpdξpdθdz 3n r k (34) ( ) B m q ( r) = ( ξp + ξz )( ξpcosθ) f ξpdξpdθdz (35) The next stage is to apply the projection procedure in order to reduce the number of independent variables. For the planar case it is possible to eliminate the x- and z- component of the molecular velocity vector, while in the cylindrical case only the z-component can be eliminated. It is convenient to introduce at this moment non-dimensional expressions: 3 fv g = (36) n

22 Manuel Vargas Hernando n ρ = (37) n T τ = (38) T δ ξ c = v PL µ v (39) = (4) Where g is the dimensionless distribution function, δ is the rarefaction coefficient, which is inversely proportional to the Knudsen number, c = ( c, c, c ) = ( cos, sin, c ) r z z θ ζ θζ θ, with ζ being the magnitude of the dimensionless molecular planar velocity vector, and ρ and τ denote dimensionless distributions of density and temperature respectively. Then, the dimensionless governing equation may be written as: For planar geometry: For cylindrical geometry: g cy = δρ τ g g y ( ) g ζ sinθ g ζ cosθ = δρ τ r r θ ( g g ) (41) (4) Where M g may represent ( ) 3 g = ρ πτ exp c τ in case of employing the BGK equation or g g qζ θ 15( ρτ ) τ S M 4 c 5 = 1 + cos in case of using the Shakhov model. At this moment the projection procedure is introduced and the reduced distribution functions are defined. As mentioned before, two velocity components can be eliminated in the planar case and only one in the cylindrical case: Planar: φ(, ) = (, c) ; ψ (, ) = ( + ) (, c ) y c g y dc dc y c c c g y dc dc y x z y x z x z Cylindrical: φ(, ζθ, ) = (, c) ; ψ(, ζθ, ) = (, c ) r g r dc r c g r dc z z z After mathematical manipulation the two coupled integro-differential equations are obtained for the unknown reduced distribution functions φ and ψ. 1

23 Modeling and simulation of heat transfer and thermal phenomena in micro gas flows Planar: Cylindrical: c c y y φ = δρ τ φ y ( φ ) ψ = δρ τ ψ y ( ψ ) φ ζ sinθ φ ζ cosθ = δρ τ ( φ φ ) r r θ ψ ζ sinθ ψ ζ cosθ = δρ τ ψ ψ r r θ ( ) (43) (44) The same projection procedure is applied to the moments to get the macroscopic quantities in terms of φ and ψ for planar and cylindrical configuration. ρ ( y) = φdc τ ( y) = ( ψ + cyφ) dc 3ρ ( ) = ( ψ + φ) q y c c dc y y y y y (45) ρ ( r) π π π τ ( r) = ( ψ + ζ φ) ζdζdθ 3ρ ( ) = ( cos )( + ) q r = φζ dζ dθ ζ θ ψ ζ φ ζdζdθ The interaction between the molecules and the walls are considered to be completely diffuse at the temperature of the wall: 3 m (46) m f + i ( ξ ) = n i exp π kt B i kt ξ (47) B i Introducing the dimensionless quantities the boundary conditions become: g + ρ i c i ( ) = c exp 3 (48) τ ( πτ ) And after applying the projection procedure:

24 Manuel Vargas Hernando φ ψ + i + i ( cy) ( cy) ρ c i y = exp 1 ( πτ i ) τ i 1 ρτ c i i y = exp 1 π τ i (49) φ ψ + i + i ( ζ ) ( ζ ) ρ i ζ = exp πτ i τ i ρ i ζ = exp π τi By applying the no penetration condition at the walls, the quantities ρ i can be obtained: π i φi cydcy τ i (5) ρ = (51) 3π π ρi = ( ζcosφφζ ) i dζdθ τ (5) i π 1... Numerical scheme The computational scheme is described in this section for solving the heat transfer problem between coaxial cylinders based on the non-linear Shakhov model by using Discrete Velocity Method (DVM) [39]. The solution is obtained by an iteration procedure. The macroscopic distributions are assumed at the right hand side of the equation and φ( r, ζθ, ) is found. Then φ( r,, ) ζθ is substituted at the right hand side of the equation and the new values of the macroscopic distributions are found and compared with the previous one. The iteration continues upon convergence. Both the molecular velocity space ( ζθ, ), with ζ [, ) and θ [, π] space r [ R, R ] 1, and the physical are discretized. The continuum spectrum of magnitudes of the molecular velocity vector is replaced by a set of discrete magnitudes ζ [, ζ ] m, m= 1,,..., M, which are taken to be the roots of the Legendre polynomial of order M accordingly mapped from [ 1,1] to [ ],ζ. max max 3

25 Modeling and simulation of heat transfer and thermal phenomena in micro gas flows Then, the spatial domain r [ R, R ] is divided in I equal intervals. Due to symmetry only 1 half of the angular space θ [, π] needs to be discretized. So a set of discrete angles θ n, θ θ n= 1,,..., N in π,π is defined. θ r R i i i + i n n 1 n + Figure : Physical and angular space discretization. I r = R = φr, ζθ, = φr, ζ, θ = φ and discretize the kinetic equation to get: Let now ( ) ( ),, imn,, i m n imn ( ) ( k+ 1 ) ( ) ( ) k+ 1 r,, ζ sinθ r,, ( k) ( k) φ ζθ φ ζθ ζ cos θ + δρ τ φ, ζ, θ = m n m n i i r ri θ imn,, imn,, By approximating: ( ) ( ) k k S i i imn,, ( ) ( k + 1 r ) imn,, = δρ τ φ (53) φ 1 = φ + φ φ φ r (54) ,,,,,,,,,, i mn i mn i mn i mn imn r φ 1 = φ + φ φ φ (55) θ θ ,,,,,,,,,, i mn i mn i mn i mn imn 1 φ = φ + φ + φ + φ imn,, i+, mn, + i+, mn, i, mn, + i, mn, 1 ri = r + r 1 1 i+ i 1 ρi = ρ + ρ 1 1 i+ i 1 τi = τ + τ It can be deduced the discrete equation for φ. 1 1 i+ i (56) (57) (58) (59) 4

26 Manuel Vargas Hernando ζmcosθn ζmsinθn δ.5 ζmcosθn ζmsinθn δ.5 φ 1 1 ρ 1τ 1 φ 1 1 ρ 1τ 1 i+, mn, ,, r r 1 θ 4 i+ i+ i+ mn r r 1 θ 4 i+ i+ i+ i+ ζmcosθn ζmsinθn δ.5 ζmcosθn ζmsinθn δ.5 φ 1 1 ρ 1τ 1 φ i, mn, r r 1 θ 4 i i 1 1 ρ 1 τ 1 i, mn, + + = r r 1 θ 4 i i i i δ.5 S S.5 S S ρ 1τ 1 φ + φ ρ 1τ 1 φ + φ (6) 4 i+ i+ i+, mn, + i+, mn, i i i, mn, + i, mn, where 4 1 ζ m ρ i ζ m φimn,, = 1 + q cos exp iζm θn 15 ρiτ i τ i πτ i τ i The corresponding equation for ψ is obtained in a similar manner. (61) 1.3. Boundary conditions As it was already mentioned, the behavior of gases in many industrial applications has been treated successful from the continuum point of view. However, the rarefied gases should be studied from a different approach due to the fact that, under such conditions and from microscopic point of view, the interaction of the molecules of the gas with the solid boundaries becomes as important as the intermolecular collisions and non-equilibrium phenomena may occur in the vicinity of the boundaries. The interaction of the molecules with the walls is a complex process where the gas molecule may be adsorbed at the wall surface, interact with the lattice and be re-emitted after a certain time interval from a different point to the incident one. Experimentally it has been seen that this process depends on many factors, such as the roughness, the kind of gas, the cleanliness and even the temperature of the wall. In the reality an adequate description of the gas flows is obtained by using simplified but reliable models which correlate the distribution function of the impinging and reflected molecules. Assuming that the adsorption time and the distance between the impinging and the reemitting points are negligible and that the solid molecules remain unaffected by collisions and are in local equilibrium at the temperature of the wall allows us to define the scattering kernel R R( ξ' ξ) = as the probability density function of the reflected state of the molecule, when it reaches the wall with velocity ξ ' and departs from it with velocity ξ. The 5

27 Modeling and simulation of heat transfer and thermal phenomena in micro gas flows scattering kernel should have certain properties. The first property is related to the nonpenetration condition. If the wall is non-porous and non-absorbing, all molecules will be reemitted with some velocity. Thus the total scattering probability must be equal to unity: ξn > ( ) R ξ' ξ dξ = 1 (6) Where the subscript n denotes the normal component of the velocity. The total number of molecules impinging on a wall surface unit area per unit time interval is If we multiply Eq. 63 with R R( ξ' ξ) f ' ( ') ξ ξ dξ' n (63) = we derive the general expression for the boundary conditions: ' ( ) = ( ) ( ) ξn f + ξ ξn R ξ' ξ f ξ' dξ' (64) ' n < ξ In essence, this simply expresses the principle of mass flux equilibrium on the wall. The scattering kernel must also be non-negative for all ξ and probability density. ( ξ) ξ ' due to its definition as a R ξ' (65) Another property that the scattering kernel should satisfy is the so-called reciprocity law: ' n ( ') ( ' ) ( ) ( ') ξ nf ξ Rξ ξ = ξ f ξ R ξ ξ (66) n Where n is the unit vector normal to the surface. This means that if a gas is at equilibrium at the temperature of the wall and hence has distribution function f, the number of molecules scattered from a velocity range ( ξ', ξ' + dξ' ) to a velocity range ( ξξ, dξ) number of molecules scattered from ( ξ, ξ dξ) to ( ξ', ξ' dξ' ) + is equal to the. Thus, if the impinging distribution is the Maxwellian at the wall conditions and mass is conserved at the wall, then the distribution function of the emerging molecules is again f. Two simple models for the interaction of the molecules of a gas with a solid boundary have been suggested by Maxwell in The first one is the so called specular reflection model. Here the interaction is perfectly elastic and the molecular velocity component normal to the surface is reversed, while the components parallel to the surface remain unchanged. Mathematically can be expressed as R SPEC ( ξ' ξ) = δ ξ' ξ + n( n ξ) D (67) 6

28 Manuel Vargas Hernando Where δ D is the Dirac function. This model is not realistic in practice because it does not predict non-normal stresses on the surface. The second one is the diffuse reflection model, where the velocities of the reflected molecules are distributed in accordance to the Maxwellian distribution at the temperature T w and velocity of the wall u w. R DIF m ξ n m ξ ξ = exp ξ π ktw ( ' ) ( kt ) w ( u ) Here the velocity of the reflecting molecules is independent of their incident velocity. The kernel may be split into two tangential velocity components and one normal velocity component: w (68) R 1 m m ( ξ ' ξ ) = exp ( ξ u ) DIF, t t t t w, t π ktw ktw (69) R mξ n ( ξ ' ξ ) = exp ( ξ u ) DIF, n n n n w, n ktw ktw The diffuse reflection model has been very widely employed due to its simplicity and its small numerical effort for its implementation. In general, it provides results in agreement with real experimental ones. However, in some cases related to surfaces with specific characteristics the gas may have different scattering patterns. The sampling of these distributions in DSMC can be derived by inversion method or by acceptance-rejection method obtaining the following expressions. For the normal component of the velocity: 1 ξn = ( ln R f ) (71) β m (7) Figure 3: Specular (left) and diffuse reflection (right). As for the tangential components of the molecular velocity: 1 r = ( ln R f ) (7) β θ = πr f 1 (73) 7

29 Modeling and simulation of heat transfer and thermal phenomena in micro gas flows ξ = r cosθ ξ t,1 t, = r sinθ (74) A combination of both specular and diffuse reflection was proposed by Maxwell in order to provide a better agreement with experimental results. In this case the fraction of molecules which are reflected diffusely is defined by the accommodation coefficient α, while the fraction of particles which reflects specularly is defined by ( 1 α ). The diffuse-specular kernel can be expressed as ( ξ' ξ) α ( ξ' ξ) ( 1 α) ( ξ' ξ) RD S = RDIF + RSPEC (75) However, it is well known that momentum and energy accommodate at a different rate: momentum is changed much faster than energy and thus one coefficient is not enough to describe this process in a realistic manner. Also, as stated in [4-4], discrepancies are found in other cases between numerical and experimental data [43]. Even more, this model does not describe the lobular distribution in the direction of the re-emitted molecules observed in experiments. The Cercignani-Lampis-Lord (CLL) model [44] provides more realistic boundary conditions, satisfies the reciprocity relation and produces physically reasonable distributions of direction and energy of re-emitted molecules. In the CLL model the scattering of the normal and the tangential components are considered independent. The scattering kernel consists of three parts, one for each of the velocity components: R R ( ξ ' ξ ) CL, t t t ' 1 ξt ( 1 σt) ξ t = exp πσ ( )( ) ( ktw m) t( t) t σ t ktw m σ σ ( ξ ' ξ ) ' ' + ( 1 ) n n n 1 I n n n ( ) ( ) ξ ξ α ξ αξξ = exp kt m kt m kt m n CL, n n n αn( w ) w αn αn w Where I is the modified Bessel function of the first kind and order zero. The CLL model contains two adjustable parameters: the normal energy accommodation coefficient, α [,1], and the tangential momentum accommodation coefficient, σ [, ] n former, t (76) (77). The α n, is related to the part of kinetic energy corresponding to the normal velocity, while the latter, σ t, is related to the tangential momentum. The implementation of the model is well described by Lord [45, 46]. The value of the tangential velocity components ξ t,1 and ξ t, can be sampled as follows: 8

30 Manuel Vargas Hernando θ = πr f 1 r = σ ( σ ) ln ( R ) t t f ( ) 1 ' t,1 t t t,1 ( ) 1 ' t, t t t, ξ = σ σ ξ + r cosθ ξ = σ σ ξ + r sinθ While the value of the normal velocity component ξ n can be sampled: 1 (78) (79) θ = πr r = f 1 1 ( ) (8) αnln R f ( r r cos ) 1 ξ = + ξ + ξ θ (81) n m m ( ) 1 ' 1 ξ = α ξ (8) m n n It is important to mention that all velocities are referred to the most probable velocity at the wall temperature. v w ( kt ) 1 = (83) The CLL model reduces to the diffuse reflection model in the case that α = σ = 1 and to specular reflection in the case α = σ =. When σ > 1 more than half of impinging n t molecules experience the back scattering. The extreme case of α n = and σ t = corresponds to the reverse reflection, where an impinging molecule with velocity reemitted with velocity ξ '. w t n t ξ ' is Aside from the previously mentioned boundary condition models which are used to compute the interaction of the particles with the solid walls, there are at least two other kind of commonly used boundary conditions, namely periodic and open boundary conditions. The periodic boundary conditions are used to simulate a large system by modeling a small part that is far from its edge. It is used to simulate systems of infinite length in the direction where the periodic boundary conditions are applied. In this kind of boundary, when a molecule crosses one of these faces it is immediately reintroduced on the opposite side with the same velocity as indicated in Figure 4. 9

31 Modeling and simulation of heat transfer and thermal phenomena in micro gas flows Figure 4: Periodic boundary conditions. The open boundary conditions are used to define the conditions of the gas at the boundary. In this kind of boundary, particles which escape from the domain are removed from the simulation, but at the same time particles are produced at a given rate, which depends on the number density and temperature defined at the boundary, and with a given velocity, which is a function of the temperature and bulk velocity defined at the boundary. The inward number flux N i to a given surface is obtained by: { exp β π 1 ( β ) } n N i = u + u + erf u β 4π (84) For a stationary gas, the equation is reduced to: n N i = (85) β 4π 1.4. Intermolecular potentials A molecular model is related to the definition of the force or potential. The force between real molecules is strongly repulsive at short distances and weakly attractive at larger distances: κ κ' F = (86) r η r η' Among the repulsive-attractive models one can mention the Lennard-Jones potential, with strong repulsive forces for short distances between particles and weak attractive forces for long distances. The Generalized Hard Sphere (GHS) [47] model was suggested to deal with attractive-repulsive forces in DSMC. However, in numerical calculations the simplest molecular model which gives sufficiently accurate results is chosen. Most of these models neglect the attractive component following 3

32 Manuel Vargas Hernando an Inverse Power Law (IPL) model. In this model the purely repulsive force depends on the distance between molecules in the following way It can be proved that the total collision cross section F κ = (87) r η σ T is proportional to ( ) 4 1 c η r. The calculations of the transport coefficients, such as viscosity and conductivity, may be defined as the cross section of the Hard Sphere model which matches the coefficient. The Hard Sphere (HS) model is the limiting case of the inverse power law model with η. In this case the collision cross section is independent of the relative velocity and may be calculated by σ T = πd (88) The scattering from hard sphere molecules is isotropic in the centre of mass frame of reference. That means, all directions in the solid angle have the same probability for the postcollision relative velocity c * r. Some of the drawbacks of the HS model are that the crosssection is independent of the relative translational energy (E t ) in the collision, which is not realistic. Other that at very low temperatures, the effective cross-section of real molecules decreases as c r and E t increase. The rate of decrease is directly related to the change of the coefficient of viscosity with temperature, which for the HS model is proportional to the temperature to the power of.5. Therefore, a variable cross-section is required to match the powers of the order of.75 which are characteristic of real gases. This led to the definition of the Variable Hard Sphere (VHS) [48] model which represents a hard sphere molecule whose diameter d is function of c r. σ T σ ν cr d = = d c T, ref ref r, ref where σ T, ref, d ref and c r, ref are reference values. The deflection angle is given by the same expression as for the hard spheres: χ ( ) b d (89) 1 = cos (9) In the VHS model, the ratio of the momentum to the viscosity cross section follows the HS value. However, the difference between the viscosity and diffusion based diameters is large due to the incorrect ratio of the momentum cross section to the diffusion cross section. The 31

33 Modeling and simulation of heat transfer and thermal phenomena in micro gas flows Variable Soft Sphere (VSS) model was introduced by Koura and Matsumoto [49, 5], where the deflection angle is given by: ( ) 1 1 α χ cos b = d The VSS model matches the HS model when α is equal to 1 and υ is equal to. The following expressions for the components of the post-collision relative velocity can be expressed by ( ) 1 * r r r r (91) u = cos χu + sin χsinε v + w (9) ( ) ( ) 1 v = cos χv sin χ cwcosε + uvsinε v + w (93) * r r r r r r r r ( ) ( ) 1 w = cos χw+ sin χ cvcosε uwsin ε v + w (94) * r r r r r r r r It can be shown that the first approximation to the coefficient of viscosity µ for a monatomic gas is: µ = 5 8 ( ) ( π mk T ) 1 B B 4 7 mc r rσ µ 4kT B r m 4k T c exp dc (95) Where σ µ is the viscosity cross section defined as µ π 3 sin d σ = π σ χ χ and for a HS gas is σ T 3. Evaluating Eq. 95 we obtain the coefficient of viscosity for a HS gas: ( ) 1 mk T π 5 B µ HS = (96) 16 d The molecular diameter of a hard sphere gas that has a coefficient of viscosity µ ref at a given reference temperature T ref is d HS ( mktref π ) 1 5 = 16 µ ref For a VHS gas the coefficient of viscosity becomes: µ VHS ( π 1 ) ( 4 ) Γ( υ) σ 1 15 mk k m T = 8 4 c υ υ T, ref r, ref 1 + υ (97) (98) 3

34 Manuel Vargas Hernando This leads to a power law temperature dependence of the coefficient of viscosity for a VHS gas ( T ω ) µ, where ω = 1+ υ. The evaluation of the coefficient of viscosity for the Inverse Power Law model gives µ IPL 1 ( ) ( ) ( η 1 π mrt κ ) A ( η) Γ ( η ) 5m RT = ( ) Where A ( η ) is a numerical factor tabulated in [51]. So, for a gas that has a coefficient of viscosity proportional to T ω, where and with the value translational energy in the collision by: ( ) ( ) (99) ω = 1+ υ = 1η+ 3 η 1 (1) µ ref at temperature T ref, the molecular diameter is related to the relative d VHS 15 = 8 Γ 1 ( m ) ( kt π B ref ) ω ( 9 ω ) µ ref Et The coefficient of viscosity, when the VSS viscosity cross section is used, reads as follows: µ VSS 1 ( α + )( α + )( π ) ( ) υ 16αΓ( 4 υ) σ c 5 1 mk 4k m T = ω 1 T, ref r, ref υ 1 + υ (11) (1) The viscosity based diameter can be calculated and in a similar manner the diffusion based diameter can be obtained from D 1 ( α + ) π ( kt m ) ω r 1 = 1 16Γ 7 1 T, ref 1 r, ref ( ω1 ) ( ω ) n( σ ) c (13) 1.5. Time and ensemble averaging All macroscopic properties of a given system can be described in terms of the microscopic state of the system. In measuring any macroscopic property the associated physical measurement is never instantaneous, but it must be carried out over some finite time which may be small by refinement of the technique. The property we observe is not the instantaneous value, but a time-averaged value [5]. Another method for determining macroscopic properties from the microscopic description is the so-called method of ensembles. We consider a large number ζ of systems which macroscopically are equivalent to the system under consideration. The thermo-fluid properties of each of the ζ replicas are 33

35 Modeling and simulation of heat transfer and thermal phenomena in micro gas flows equal. The microscopic description is not specified and we can expect that this will differ a lot between replica systems, since there are large numbers of microscopic states corresponding to any given macroscopic state. This collection of systems is referred to as an ensemble. Applied to DSMC, we may use two main strategies for sampling the macroscopic quantities form the microscopic properties of the molecules in a given volume element. For steady state calculations, where the interest is in the final steady state solution of the problem which does not depend on time, the time averaging is used. This strategy consists of accumulating the important moments over one trajectory by averaging over many time steps after allowing the simulation to reach the steady state. Figure 5: Time averaging. In contrast to steady state problems, where the results from the simulations describe the final state of a system which does not change in time, the unsteady state simulations describe the behavior of a system for a discrete number of instants during the initial period before reaching the steady state (transition period) or some specific moments if the problem presents periodicity. Therefore, the macroscopic properties are obtained for each cell by ensemble averaging over a large number of process trajectories. Figure 6: Ensemble averaging. 34

36 Manuel Vargas Hernando Chapter One-dimensional steady heat transfer problem in planar and cylindrical geometry.1. Heat transfer problem between infinite parallel plates: Application of Cercignani Lampis boundary conditions.1.1. Introduction The Fourier flow, although it is one of the simplest systems, constitutes the basis of the understanding of the heat transfer problem. Its study in micro- and nano-channels, where the behaviour of the gas differs from the continuum regime one, is of great importance in several practical applications, as mentioned in Chapter 1. Many authors have intensively investigated the one-dimensional steady state heat transfer problem, most of them based on the linearization of the distribution function, which simplifies the problem. Experimental data has been obtained for small temperature difference by [53] and under non-linear temperature conditions [54]. The experiments were performed for two different gases (argon and nitrogen) and different Knudsen numbers, from the slip flow regime to the transition and free molecular regime. [55] and [56] have analysed the same problem and compared their results with the experimental ones by applying different kinetic models and have compared simulation results with the experimental ones. In most cases, the gas-surface interaction is considered to be completely diffusive or, in few cases, specular-diffuse reflection. Few works [57] employ the Cercignani-Lampis boundary conditions for solving the problem. Here, the more realistic Cercignani-Lampis boundary condition model has been employed to compute the interaction of the particles with the solid walls. The influence of the two adjustable parameters of the CL model on the heat flux is investigated. Later, the results of the heat flux subject to diffuse reflection model obtained by using DSMC are compared with those obtained by using kinetic model equations. Finally, the heat transfer flow between parallel plates is used in order to investigate the accuracy of various collision schemes for different sampling performances. 35

37 Modeling and simulation of heat transfer and thermal phenomena in micro gas flows.1.. Formulation of the problem The one-dimensional plane Fourier flow problem consists of the heat flow of a gas confined between two infinite parallel plates at rest, when they have different, but constant temperatures. For the steady state cases we consider a gas between two parallel plates at y = and y L = with temperatures T = T ( T) and T T ( T) L 1 U = + respectively as shown in Figure 7. The initial state of the gas is in equilibrium at initial pressure and temperature P and 1 T, with an initial number density n P ( RT ) = and defining an initial Knudsen number Kn = λ L, where λ is the mean free path, which is defined for hard sphere particles as ( π dn), d is the diameter of the particles and R is the gas constant. 1 To render the problem dimensionless, we define the following reference values of the variables: kb 3 yref = L ; nref = n ; Tref = TL ; vref = TL ; qref = mnvref (14) m Here k B is the Boltzmann constant and m is the molecular mass. Figure 7: Schematic diagram of the planar Fourier flow. The diffuse reflection model was used to describe the interaction of the molecules with the boundaries Analytical solution in the free molecular regime The analytical solution of the Boltzmann equation was calculated for the heat transfer problem between two infinite parallel plates with different temperatures in the limit case of free molecular regime in which the Knudsen number tends to infinity and, consequently, there is absence of intermolecular collisions. The interaction of the molecules with the solid boundaries is taken to be complete diffuse at the temperature of the walls. As mentioned before, the macroscopic quantities are given by the following integrals: 36

38 Manuel Vargas Hernando ( ) ξx ξy ξz (15) 1 Q= nq f + nq f d d d n Where n 1 and f 1 are the number density and distribution function of the particles coming from the lower wall, n and f the corresponding to the particles coming from the upper wall and Q is the quantity to be set: to obtain the mean number density Q = 1, to obtain the mean velocity Qi = mξ i, the temperature Q = mξ and the heat flux Q i = mξξ i. Before calculating the macroscopic properties of the system, it is necessary to define the distribution functions of the particles reflecting each wall. 3 m m 1 = = exp 1 π kt1 kt1 f + f v (16) 3 m m = 1 = exp π kt kt f + f v (17) n= n1+ n (18) To determine the temperature across the channel we should solve the equation ( ) 3 1 kt B = mξ ξ (19) 3 1 m m n1 n π kt 1 kt 1 ξ = ( ξx + ξy + ξz ) exp ( ξx + ξy + ξz ) + 3 m m n ( ξx + ξy + ξz ) exp ( ξx + ξy + ξz ) (11) π 1 n kt kt And from here, assuming that the bulk velocity is zero, we obtain: 1 T = [ nt 1 1+ nt ] (111) n To calculate the relationship between n and n 1, we should apply a balance of particles to the walls: n1ξy f1 = nξy f nt 1 1 = nt (11) Taking this into account the solution is obtained for the temperature: T T T + T T = = T1 + T TT 1 (113) 37

39 Modeling and simulation of heat transfer and thermal phenomena in micro gas flows.1.4. Application of Cercignani-Lampis boundary conditions The results presented in this section have been presented in [A]. The influence of the gassurface interaction parameters involved in the CLL model on the heat transfer for pure planar Fourier flow is shown in Figure 8 for a temperature difference T =.1 by using the HS model to compute the intermolecular interactions. In particular, the heat flux is plotted versus each accommodation coefficient, keeping the other constant and equal to unity. It can be seen that the influence of the accommodation coefficients is high for large Knudsen numbers, but lower when the Knudsen number decreases. This can be explained taking into account that as Knudsen is increased the collisions with the walls are dominant as the whole domain is contained within the Knudsen layers. The opposite happens when the Knudsen number is decreased and the collision between the molecules becomes more important than the collision with the boundaries. Besides, a symmetric dependence of the heat flux on σ around σ = 1 can be observed. This symmetry is understandable since in pure Fourier flow there is no macroscopic movement of the gas, where specular reflection has the same global effect as backscattering. On the contrary, the heat flux increases monotonically with α n. t t Figure 8: Heat flux dependence on the tangential momentum accommodation coefficient keeping α n = 1 (left) and on the normal energy accommodation coefficient 1 (right) for different Kn and T =.1. α n keeping σ t σ t =.1.5. Validation and comparison with DVM results Some calculations of the heat flux for the plane Fourier flow for diffuse boundary conditions were carried out in order to compare the DSMC results with those obtained by using the discrete velocity method [58]. The simulations were done for different rarefaction conditions, 38

40 Manuel Vargas Hernando from the free molecular regime until the continuum flow regime, and for the case of T = ( T T = 3 ). The results are shown in Table 1 for different Knudsen numbers. The results H C between both approaches are in very good agreement. Table 1: Heat flux in the whole range of rarefaction and T = ( T T = 3 ) obtained by DSMC and DVM. π 1 Kn DSMC DVM H C.1.6. Heat flux convergence for various sampling performances: Comparison between NTC and BT schemes As it has been explained in Chapter 1, Bernoulli trials (BT) scheme has two advantages over the standard NTC: it avoids the repeated collisions which lead to a lower collision rate and it allows collisions between particles of neighbouring cells, all this allowing the use of a small number of particles per cell. In this section, three different sampling performances were analyzed for each scheme: - For the NTC scheme collision-sample-move (CSM), collision-move-sample (CMS) and collision-sample-move-sample (CSMS). - For the BT scheme collision1-collision-move-sample (C1CMS), collision1- collision-sample-move (C1CSM) and the symmetric collision1-sample-collision- move (C1SCM). The stages collision1 and collision correspond to the collision process over a time t in the basic grid and in a grid which is shifted x in each of the directions respectively. The study case chosen for this aim was the Fourier flow defined above with purely diffuse walls and with reference pressure and temperature of the hard-sphere-like gas P and T C and with T =.1T, same conditions as in [59, 6]. The reference Knudsen is Kn =.9 and the reference heat flux for these cases is qref = mnvc, where vc = ktc m. The results presented in this section have been presented in [A, A6]. The results presented here represent the time discretization errors of the different schemes for two different values of the mean number of particles per cell. 39

41 Modeling and simulation of heat transfer and thermal phenomena in micro gas flows NTC CSM BT C1CSM -.15 BT C1SCM q y [ ] NTC CSMS -.19 NTC CMS BT C1CMS -. NTC BT t 1 [ns] C1SCM q y [ ] CSMS C1CMS CMS -.3 NTC BT t 1 [ns] Figure 9: Heat flux versus time step for the different schemes (NTC and BT) and sampling procedures. The simulations were performed dividing the domain in cells and using 5 (up) and (down) particles per cell as average. The results presented show the dependence of the heat flux on the selected scheme. When the average number of particles per cell is big enough (Figure 9 up), the behavior of both schemes (NTC and BT) are very similar. With two of the methods, NTC with double sampling (CSMS) and BT with symmetric sampling (C1SCM), the heat flux deviates slightly from the converging value when the time step is increased. The value of the heat flux is overpredicted for the BT with symmetric sampling, while for NTC with double sampling is underpredicted. For the other four DSMC models the heat flux converge to the same value as the previous methods for small time steps, but it deviates significantly when the time step increases. 4

42 Manuel Vargas Hernando However, if the average number of particles per cell is drastically reduced (Figure 9 down), it turns out that only the BT scheme with symmetric sampling can cope with a very low number of particles per cell and the increase of the time step. The NTC with double sampling gives the same value of the heat flux for small time steps, but they deviate when this time step starts increasing. The other single-sampling methods describe a similar but more pronounced tendency as with using a large number of particles per cell. Figure 1: Heat flux versus time step obtained by using NTC, BT and SBT schemes with the indicated sampling performances. The simulations were performed by dividing the domain in cells and using 3 (up) and (down) particles per cell as average. 41

43 Modeling and simulation of heat transfer and thermal phenomena in micro gas flows The convergence behaviour of the Simplified Bernoulli Trials scheme with the symmetric sampling performance (C1SCM) is shown in Figure 1 for two values of the number of particles per cell. The results are compared with the NTC for the Strang-like sampling (CSMS). Both sampling procedures show second order convergence with respect to the time step. As seen, the SBT performance is similar to the BT scheme one and better than the NTC, especially for small number of particles in cells. The reason for this deviation of the NTC results, even with second-order converging performance, for larger time steps is again the repeated collisions within the time step, which are avoided by BT and SBT schemes. For very small time steps the three schemes give similar results due to the fact that the particles remain in the same cell over many time steps and, in this way, the repeated collisions start occurring between consecutive time steps instead of within the same one... Heat transfer problem between coaxial cylinders..1. Introduction The heat transfer problem in a rarefied gas between two coaxial cylinders has plenty of technological applications, from micro heat exchangers to Pirani gauges, used for the measurement and monitoring of the pressure in vacuum systems. This problem has been investigated by many authors [61-65] using different techniques. However, the effect of the implemented intermolecular model has not been studied in detail. In the present work, the DSMC method is used to study the effect of the three parameters governing this system (temperature difference between the cylinders, curvature of the cylinders and degree of rarefaction) on the macroscopic properties of the gas in the whole range of rarefaction and from linear to strongly nonlinear conditions. The influence of different purely repulsive intermolecular potentials (HS, VHS and Maxwell molecules) on the calculation of the heat flux is analyzed. The procedure for axially symmetric flows is employed. It consists of assuming that, at the beginning of the motion stage for every time step, all the particles have the same polar z and axial x coordinates. Therefore for each particle only the radial coordinate y and the three velocity components are stored. Then, the molecules are moved according to their velocities and acquire certain coordinates x ', y ' and z ' given by x = v dt; y = y + v dt; z = v dt; y = y + z (114) ' ' ' + ' ' x y z 4

44 Manuel Vargas Hernando while y + is taken to be the new ordinate. The new velocity components, v + x, v + y and v + z, are computed by taking the components of the previous velocity in the x-direction and in the two directions parallel and orthogonal to the vector with components 11. Following this procedure it is found that x ', ( + ) ( + ) y ', as shown in Figure vy y vydt vzdt vyvzdt vz y vydt vx = vx; vy = ; v + z = (115) + y y Figure 11: Schematic representation of the transformations for axially symmetric flows.... Formulation of the problem The heat transfer through a monatomic gas confined between two concentric stationary cylinders of infinite length is studied. The cylinders are consider to have radii R 1 and R ( R1 R < ), keeping their constant temperatures T 1 and T respectively as shown in Figure 1. The gas-surface interaction is taken to be purely diffusive. Figure 1: Schematic representation of the system. 43

45 Modeling and simulation of heat transfer and thermal phenomena in micro gas flows The initial state of the gas is in equilibrium at initial pressure and temperature P and T, with an initial number density n P ( RT ) ( ) = and defining an initial Knudsen number Kn = λ R R1, where λ is the mean free path, which is defined for hard sphere particles as ( π ) 1 dn, with d being the diameter of the particles and R the gas constant. To render the problem dimensionless, we normalize the position by L ( R R ) =, the 1 temperatures by T, the velocities by the most probable molecular speed v = kt m, the number density by n and the heat flux r and m is the molecular mass. 3 q by ( nmv ). Here k is the specific gas constant Three are the parameters governing the heat transfer between cylinders: the temperature difference between the walls, the ratio of the cylindrical radii and the degree of rarefaction of the gas. These parameters are introduced in non-dimensional form as normalized temperature difference β, radius ratio γ and rarefaction coefficient δ given by: respectively. δ β T T T 1 = (116) γ = R / R (117) 1 P( R R1) π 1 = = (118) µ v Kn..3. Results and discussion The results presented in this section have been presented in [A3]. The results of the macroscopic properties of the gas are presented in terms of the parameters involved in the problem. The profile distribution of the temperature, radial heat flux and pressure are shown for several values of the normalized temperature difference β, radius ratio γ and rarefaction parameter δ. The influence of the intermolecular collision model on the macroscopic properties is analysed. The numerical results are based on the following discretization parameters: the space domain is discretized in the radial direction using 4 computational cells n C, keeping in every case the cell size much smaller than the mean free path; the time step is chosen to be approximately 5 times smaller than the cell traversal time ( ) R R nv; and the average 1 C 44

46 Manuel Vargas Hernando number of particles N per cell is kept always larger than 5, having the smallest number of particles per cell near the inner cylinder (defining in this region the accuracy of the calculations). The validation of the numerical solution was performed by comparing the results with two limiting cases: a) the solution for the planar Fourier flow when the radius ratio γ 1, and b) the solution given in [66] for shear stress driven flow between cylinders. In Figure 13 the radial heat flux, temperature and pressure profiles are represented for various values of the radius ratio for δ = 8.86 ( Kn =.1), and hard sphere molecules ( ω = 1 ). The solution of the planar Fourier flow in the same conditions is also shown. It can be seen that the solution of the planar Fourier flow represented with black solid line is in very good agreement with the solution of the cylindrical problem when γ 1. When the radius ratio increases for this value of the rarefaction parameter, the ratio between the heat flux in the inner cylinder and the heat flux in the outer cylinder is increased. This is because the product q( r) r (the heat flow) remains constant because of the energy conservation principle. The temperature profile is also modified when the radius ratio increases, showing a larger temperature jump near the hot inner cylinder and experimenting a bigger variation of the temperature near the inner than near the outer cylinder. Also the pressure profile varies, as it becomes smaller when the radius ratio increases and, as explained above, the temperature decreases. In Figure 14 the radial heat flux profiles for three different values of the normalized temperature difference, β =.1, 1 and 1, two values of the radius ratio, three values of the rarefaction coefficient and various intermolecular potentials, ω =.5 (HS), ω =.81 (VHS Argon-like gas) and ω = 1. (Maxwellian), are plotted. It is noted that when γ =.1 the rarefaction parameters areδ = 1, 1 and 1, while for γ =.5 the rarefaction parameters are δ =., and. It is seen that the variation of the heat flux along the gap between the cylinders is higher as γ is decreased (larger curvature effect). This can be explained considering that, due to the energy conservation law, the heat flow along the domain must be constant and, consequently, the heat flux (heat flow referred to surface) close to the inner cylinder will be larger than the heat flux close to the outer cylinder. The normalized heat flux in all cases increases as the rarefaction coefficient decreases and the temperature difference increases. 45

47 Modeling and simulation of heat transfer and thermal phenomena in micro gas flows 3 q r (r-r )/(R -R ) T (r-r )/(R -R ) n (r-r )/(R -R ) P (r-r 1 )/(R -R 1 ) γ = 1. γ =.91 γ =.5 γ =. γ =.1 Par.Plates Figure 13: Dependence of the radial heat flux, temperature, number density and pressure profiles on the radius ratio for δ = 8.86 ( Kn =.1), β = 1 and hard sphere molecules ( ω =.5 ) and comparison with the solution of the planar Fourier flow. As for the effect of the molecular interaction model, the deviation between the corresponding heat transfer results obtained by using the different intermolecular potentials is larger as δ increases (small degree of rarefaction where the collisions between molecules are predominant over the collisions with the boundaries) and β increases (higher temperature difference between the two cylinders and less accurate fitting of the properties of the gas which are temperature dependent). The effect of the curvature on the heat flux profiles for the two limiting intermolecular potentials can be observed better in Figure 15. For the specific 46

48 Manuel Vargas Hernando case of δ = 1 and β = 1, the deviation between the results obtained by using HS and M model is larger as γ increases (i.e., the curvature effect decreases). In all cases the HS model gives the lowest value of the heat flux, while the Maxwellian model produces the highest values of the heat flux. The results obtained by using the VHS model remains always between the two limiting intermolecular potentials q r q r (r-r 1 )/(R -R 1 ) (r-r 1 )/(R -R 1 ) q r. q r (r-r 1 )/(R -R 1 ) (r-r 1 )/(R -R 1 ) q r q r (r-r 1 )/(R -R 1 ) (r-r 1 )/(R -R 1 ) δ =1 δ =1 δ =1 δ =. δ = δ = Figure 14: Radial heat flux profiles for β =.1 (up), β = 1 (middle), β = 1 (down), with γ =.1 (left) and γ =.5 (right) and various intermolecular potentials: HS ( ω =.5 solid lines), VHS for Argon-like gas ( ω =.81 - crosses), Maxwellian ( ω = 1. - circles). 47

49 Modeling and simulation of heat transfer and thermal phenomena in micro gas flows γ =.5 q r.3 γ = 1. q r (r-r 1 )/(R -R 1 ) (r-r 1 )/(R -R 1 ) q r γ =. γ =.1 q r (r-r 1 )/(R -R 1 ) (r-r 1 )/(R -R 1 ) q r HS γ =.5 M (r-r 1 )/(R -R 1 ) Figure 15: Effect of the curvature of the cylinders on the radial heat flux profiles for δ = 1, β = 1 showing the two limiting intermolecular potentials: HS ω =.5 - circles) and M ( ω = 1. - triangles)...4. Validation and comparison with DVM results Some calculations of the heat flux for the cylindrical Fourier flow for diffuse boundary conditions were carried out in order to compare the DSMC results with those obtained by using the discrete velocity method [63]. The simulations were done for different rarefaction conditions and for various values of the temperature difference β and radius ratio γ. The results are shown in Table and, as it can be seen, the agreement is in general good. 48

50 Manuel Vargas Hernando Table : Heat flux in the whole range of rarefaction, for various values of the temperature difference β and radius ratio γ obtained by using DSMC and DVM δ = 1 δ = 1 δ = 1 δ =.1 β = 1 β = 1 β = 1 β = 1 β = 1 β = 1 β = 1 β = 1 q r at r = γ (NL) q r at r = γ (DSMC) Error (%) γ = E E- -.4 γ = E- 3.13E-.1 γ = E E+ -.3 γ = E E γ = E E γ =.5.181E-1.14E γ = E+ 4.9E+ 3.6 γ = E E+.3 γ = E E-1 1. γ = E E γ = E+ 5.78E+ 6.9 γ = E E+ 4.9 γ = E E-1 1. γ = E E-1.1 γ = E+ 5.84E+. γ =.5 7.9E E+.5 49

51 Modeling and simulation of heat transfer and thermal phenomena in micro gas flows Chapter 3 Transient heat transfer flow between coaxial cylinders 3.1. Introduction The steady-state heat transfer problem through rarefied gases confined between coaxial cylinders is a fundamental problem in fluid mechanics and has been widely investigated for a single gas by many researchers [61-65], as seen in Chapter. There are several studies in the literature dealing with the heat transfer through gas mixtures, especially in planar geometry. Numerical results were provided for the heat flux through binary gas mixtures in planar geometry based on the Boltzmann equation [67-69], on the McCormack model [7-7] and on linear extended thermodynamics [73]. The heat transfer through gaseous mixtures in cylindrical geometry was investigated [74] for small values of the normalized temperature difference. The corresponding unsteady heat transfer problem, both through a single gas or through binary mixtures, has received much less attention. Despite its simplicity, this problem has many technological applications, such as Pirani and diaphragm gauges for instrumentation, monitoring and control of vacuum processes or micro heat exchangers in microfluidics. Even more, recent experimental measurements of the heat flux between coaxial cylinders were reported [75] for large temperature difference in order to obtain the thermal accommodation coefficient. Besides, ultrafast laser heating is utilized in some manufacturing processes. In these cases the transient flow and the unsteady state behaviour of the system are of great importance. The first work which dealt with the transient evolution of a rarefied gas under a sudden change in the wall temperatures was done by Sone [76]. In this work the BGK model of the Boltzmann equation was used, assuming a small temperature change and linearizing the governing equations, to analyse the short and long time behaviour of the system. The transient heat transfer flow between plates in the free molecular limit was investigated by Perlmutter [77, 78]. The response of a diluted gas confined between parallel plates under a gradual change in the temperature of the boundaries was studied for slow [79] and fast change [8] in comparison with the acoustic scale. Their Navier-Stokes approach showed that the mechanism is dominated by a conductive-convective balance through the slab and that a bulk velocity field is developed in the gas. The problem of sudden heating and cooling of a 5

52 Manuel Vargas Hernando rarefied gas was studied numerically by using DSMC [81] from the near continuum to highly rarefied conditions and the results were compared with the Navier-Stokes solutions, showing that transient effects near the walls can not be predicted by the continuum approach. More recently, the transient heat transfer of a gas in a small-scale slab due to an instantaneous change of the wall temperature [8] as well as the response of the gas under time dependent boundary heating was investigated by using the collisionless [83] and linearized [84] Boltzmann equation. The late-time response of the system due to unsteady boundary heating was presented in [85]. However, there is a lack of studies on transient flows in rarefied gases for cylindrical configurations, where curvature effects play an important role. In the present work, the transient behaviour of a single gas and binary gaseous mixture confined at rest between two coaxial cylinders caused by a sudden change of the temperature of one of the surfaces is studied numerically by using DSMC. The study covers the response of the system from the initial moment, when the change in the temperature is introduced, until the system reaches the steady state conditions. It is found that at the early stages of the transient solution the larger momentum of the molecules reflected from the inner cylinder creates a density gradient near that boundary and a motion in the gas from the inner to the outer cylinder. The movement of the gas propagates forming waves across the channel until the system reaches the new steady state. The influence of the governing parameters (Knudsen number, temperature difference and radius ratio) on the dynamical response of the system and the time needed to achieve the steady state are also investigated, paying special attention to the transition from planar to cylindrical geometry. In the case of single gas the transient flow is calculated by using continuum compressible viscous gas model [86] described by the Navier-Stokes-Fourier equations. 3.. Problem formulation and numerical simulation The transient heat transfer through a binary mixture confined between two concentric stationary cylinders of infinite length is studied. The cylinders are consider to have radii R 1 and R ( R1 < R) and to be initially at temperaturet. At time t = a sudden change in the temperature of the inner cylinder occurs, establishing a new temperature T 1 while the temperature T of the outer cylinder remains the same ( T 1 > T ). Thus, the initial conditions at t = are 51

53 Modeling and simulation of heat transfer and thermal phenomena in micro gas flows ( ) ( ) while for t > the boundary conditions are T R = T R = T (119) 1 ( ) ; ( ) T R = T T R = T (1) 1 1 The initial state of the gas is in equilibrium at reference pressure and temperature P and T = T, with a reference number density n = P kt B. The mole fraction is defined as: X α nα = n + n α β (11) where n α ( α = 1, ) is the number density of species α. The reference Knudsen number is Kn λ = R R 1, where ( λ πd n ) 1 ( 1 ) = is the mean free path of molecules of diameter αβ dαβ = Xα dα + Xα dβ when they are in equilibrium state at rest with number density n. The gas-surface interaction is taken to be purely diffusive and the hard sphere intermolecular potential was chosen to compute the collision between particles. To render the problem dimensionless, we establish the following normalized variables: n' T' v' q' n= ; T = ; v= ; q= ; n T v nm v 3 αβ ' P' t R' P= ; t = ; R=. 1 nm v αβv λ λ where m αβ is the mean molecular mass of the mixture: and ν is defined as v kt = m αβ ( 1 ) (1) m = X m + X m (13).5 αβ α α α β and k is the Boltzmann constant. Three are the parameters governing the heat transfer through a single gas confined between cylinders: the temperature difference between the walls ( ) cylindrical radii R1/ R 1 β = T T T, the ratio of the 1 γ = < and the degree of rarefaction of the gas Kn 1 ( R R ) =. 1 The governing parameters become six for binary mixtures, as the mole fraction X α, the ratio of molecular masses m m α β and the ratio of the molecular diameters dα dβ must be taken into consideration. The effect of these governing parameters on the transient behaviour of the macroscopic quantities is a subject of the present study. The traditional Direct Simulation Monte Carlo (DSMC) method proposed by Bird [1] is used 5

54 Manuel Vargas Hernando for the calculations. This method combines deterministic aspects for modelling particle motions and statistical aspects for computing collisions between particles. The collision technique that we use is the No Time Counter scheme suggested by Bird, except a slight modification in the calculation of maximum collision number in a cell, as described in [87]. The procedure for axially symmetric flows is employed as given in [1]. This procedure has been explained in Chapter. The domain was discretized in the radial direction using 4 computational cells n C and the particles are initially distributed in such way that the minimum number of particles in each cell of the domain is. As no weighting factors are used, the region, where the number of particles per cell is the smallest, corresponds to the area near the inner cylinder for geometrical reasons. The time step is chosen to be sufficiently smaller than the cell traversal L nv. The macroscopic properties are obtained by time averaging over many time time ( ) C steps when the steady state solution is sought for and by ensemble averaging over a large number of process trajectories in the case of unsteady state flow. The heat flux is volume based calculated, as well as the other macroscopic quantities, by averaging the microscopic values of the particles at a given cell q r = vc y Numerical results for a single gas The numerical results presented in this section have been presented partially in [A4] and the extended version has been submitted in [A8]. Also they were used for the validation and comparison with the results obtained by using unsteady linear BGK model in [A5]. The solution of the problem for a single gas is obtained from the general simulation of a binary gas mixture by assuming molar fraction X β =. The transient heat transfer flow through a single gas confined between two coaxial cylinders has been studied for four values of the Knudsen number ( Kn = 1, 1,.1 and.1), which cover the whole range of gas rarefaction, for three values of the temperature difference ( β =., 1 and 1), from linear to non-linear and strongly non-linear conditions, and for three values of the radius ratio ( γ =.999,.5 and.). For brevity, hereafter the ratio γ = 1 will be used instead of γ =.999. The case of γ 1 allows to compare the response of the present system to the corresponding one for the planar case, as done in [88]. For convenience, the time which is 53

55 Modeling and simulation of heat transfer and thermal phenomena in micro gas flows represented in the plots is given in acoustic time scale (the ratio of the gap width to the most probable molecular velocity) by taking ( R R1) v as reference time. In the first part of this section the transient solutions of the macroscopic properties from the initial state to their steady state values is analysed for a given typical case. In the second part, the influence of the three governing parameters on the convergence of the system is studied, investigating the effects of curvature, magnitude of the temperature change and rarefaction conditions Transient behaviour The transient behaviour of the macroscopic quantities is presented in Figure 16 by plotting the profiles of number density n, radial velocity v r, temperature T, radial heat flux q r and pressure P at certain times from the sudden change of the temperature of the inner cylinder up to the steady state conditions for a specific case of Kn =.1, β =. and γ =.. The steady state solution is represented by solid lines. It can be seen that the sudden heating of the inner wall causes a decrease in the density near the inner cylinder and consequently a wave propagates along the gap. This local difference in the density creates a macroscopic movement of the mass of gas, as indicated in the velocity profiles, from the inner to the outer cylinder, which reflects backwards after hitting the outer cylinder. The temperature profiles in Figure 16 indicate that in the early times the change affects only the region close to the inner cylinder causing locally an increase in the temperature, while the rest of the temperature profile remains unchanged. In addition, the temperature distribution is changed in a monotonic manner until the steady-state is reached. However, the heat flux evolution is different. Initially the heat flux close to the hot cylinder takes values higher than the corresponding steady-state ones, while after some distance the transient heat flux profile crosses the steady-state one taking lower values. The pressure profile evolution is shown in the last graphics of Figure 16. The pressure wave propagation toward the outer cylinder is clearly seen at the early times. After its reflection from the outer cylinder the wave dissipates quickly and its front is not well expressed. The wave continues travelling until the steady state pressure profile is reached at pressure level higher than the initial one. For the same conditions ( Kn =.1, β =. and γ =. ), the time evolution of the macroscopic properties within the transient period is represented in Figure 17 for three r R different positions in the annular gap ( r * = 1 =.1 R R 1 near the inner cylinder, r * =.5 in 54

56 Manuel Vargas Hernando the middle point and * r =.9 near the outer cylinder). As seen, the disturbance propagates along the domain and after or 3 acoustic times the wave is damped and the new equilibrium is reached. The behaviour described in Figure 16 coincides with the one represented in Figure 17, as the movement of the gas from the inner to the outer cylinder leads to forming a typical steady state density profile with maximum close to the cold wall and minimum at the hot wall. Figure 16: Distributions of number density, radial velocity, temperature, radial heat flux and pressure at the indicated times for Kn =.1, β =. and γ =.. The black solid lines represent the steady state profiles. 55

57 Modeling and simulation of heat transfer and thermal phenomena in micro gas flows Figure 17: Time evolution of the number density, radial velocity, temperature, radial heat flux and pressure at three different positions of the annular gap β =. and γ =.. * r =.1,.5,.9 for.1 Kn =, Effect of the governing parameters on the transient solution In this section, the influence of the three parameters governing the system on the dynamic response and convergence is analysed. In Figure 18 the time evolution of the radial velocity component at early times is represented for different values of Kn and γ keeping β = 1. We have chosen to represent this bulk quantity because it is more sensitive to γ at the initial instants after the temperature change. As one can observe by focusing on the results given for 56

58 Manuel Vargas Hernando the planar limit γ 1, for large values of Kn the amplitude of the travelling wave remains constant at early times. This situation changes for Kn =.1, where an increase in the amplitude of the wave is seen, and finally for small Kn the amplitude becomes flat again. It is worth noting that all the velocity profiles represented in Figure 18 correspond to times in which the outer cylinder has no influence on the travelling wave. In this situation, the Knudsen number serves only to identify the time and space scale and what is really observed in Figure 18 is the propagation and development of the same wave at different time and space scales. Formally, we separated the process of wave formation in four stages, by changing the distance between plates, cell size and time step. In conclusion, the wave formation can be split into the following stages: An initial stage in which the wave propagates with constant amplitude at very early times in kinetic scale (time normalized by λ v ) when the flow can be considered as a free molecular one. The time in acoustic scale is related to the kinetic scale multiplying the kinetic time by Kn. This behaviour is observed in Figure 18 for Kn = 1 and Kn = 1 for times in kinetic scale smaller than.3. Here the wave has a symmetric shape. A transition stage in which the amplitude of the wave is increased because of non-linear effects caused by the collision process. This stage occurs between times.3 and 1 in kinetic scale, when the effective collisions play an important role and the wave structure is evolving in time. The kinetic scale gives an idea of how many mean free paths are travelled by the wave. A final stage in which the wave propagates again with constant amplitude, but larger than in the initial stage. This stage occurs for times larger than 1 in kinetic scales (in Figure 18 for times larger than.1 for Kn =.1), when the flow has near continuum character. It can be observed that the structure of the wave is changed, as it is not anymore symmetric, but its front is steeper than its back. The effect of the radius ratio γ, as a measure of the curvature, is also seen in Figure 18 at early times. As γ is decreased, the amplitude of the velocity decreases as the wave propagates along the gap. This is a curvature effect due to the fact that the wave diverges and its surface becomes larger as it propagates from the inner to the outer cylinder. For Kn =.1, the effect of the collisions, which increases the amplitude of the wave as explained above, is dominant over the divergence of the wave for γ =.5. To confirm this effect, in Figure 19 the initial velocity profiles for Kn = 1 and β = 1 are compared with those for Kn = 1 and 57

59 Modeling and simulation of heat transfer and thermal phenomena in micro gas flows ( ) β ' = T T T = 1. This means that in this case the temperature jump occurs in the outer 1 1 cylinder. As seen for β ' = 1, the wave propagates with constant amplitude in the case of γ 1, while when γ is decreased its amplitude increases with time due to the fact that, in this case, the wave concentrates as it moves toward the inner cylinder. There is interplay between these effects what can be seen from the change of the velocity profile for different γ. Figure 18: Profiles of radial bulk velocity at early times: t =.1,.,.3 in acoustic scale for different values of Kn and γ, with β = 1. The different values of γ are represented by solid line ( γ 1), circles ( γ =.5 ) and stars ( γ =. ). The influence of the temperature difference is illustrated in Figure. In this case the time evolution of the radial velocity component and temperature is represented keeping constant the other two governing parameters ( Kn =.1 and γ =. ). Figure is complemented by Figure 17 for β =.. The main influence of β on the evolution of the system is quantitative, as the pattern of the behaviour of the properties is similar and the differences can be observed only in the magnitude of the disturbance. Only for strong non-linear conditions, the wave seems to propagate for shorter time than for moderate temperature values. In 58

60 Manuel Vargas Hernando addition, the velocity of propagation of the disturbance is larger as β is increased due to the higher kinetic energy of the molecules reflected from the inner wall. Figure 19: Profiles of radial bulk velocity at early times: t =.1,.,.3 in acoustic scale for different values of γ, with Kn = 1 and β = 1 (left) and β ' = 1 (right). The different values of γ are represented by solid line ( γ 1), circles ( γ =.5 ) and stars ( γ =. ). Figure : Time evolution of the radial velocity and temperature at three different positions of the annular gap * r =.1,.5,.9 - for.1 Kn =, γ =. and for β = 1 (left), β = 1 (right). 59

61 Modeling and simulation of heat transfer and thermal phenomena in micro gas flows Figure 1: Time evolution of the radial velocity and temperature at three different sections of the gap, near the inner hot cylinder, at the middle and near the outer cold cylinder - * r =.1,.5,.9 - for β = 1 and γ =. and for Kn = 1 (up), Kn = 1 (middle) and Kn =.1 (down). In Figure 1 the influence of the Knudsen number can be observed. In this case the time evolution of the radial velocity component and temperature is represented keeping constant the other two governing parameters ( β = 1 and γ =. ). Figure 1 is complemented by Figure (left) for Kn =.1. As it can be observed, the system shows a similar behaviour for Kn = 1 and Kn = 1, being the signal damped after few acoustic times. However, as the Kn is decreased, the time necessary to achieve the new steady state is increased, as well as the 6

62 Manuel Vargas Hernando magnitude of the perturbation of the velocity and the density. From Figure 1 (bottom left) a pseudo-damping coefficient ζ can be estimated by fitting the amplitude A of the signal to an exponential curve of the shape: A A ( ζ t ) = exp. The fitting value of the damping coefficient is around ζ =.3 when the time is given in acoustic scale. The variation of the velocity of the propagating wave can also be estimated from the same data and it is found that this velocity increases as the signal is damped and the period between two wave hills is changed from a period per 1.7 acoustic units of time initially to a period per 1 acoustic unit of time. The influence of the rarefaction conditions on the heat flux is represented in Figure. The normalized steady state heat flux is smaller as Kn is decreased and the evolution in time of the heat flux for Kn =.1 is not as uniform as for the other values of Kn. It is worth noting that the heat flow rate Q( r) rq ( r) = is represented in Figure. Because of this, the steady r state values of the heat flow rate at any point along the gap are constant due to conservation of energy. The time scale is logarithmic so that the initial transition times can be seen more clearly. At these early times the heat flux behaviour at the inner cylinder is flat for large values of Knudsen. This is because under these conditions there is absence of effective collisions and the number of particles coming from the cold wall is the same to the number of reflecting particles at the hot wall. As there is no interaction between both distribution functions, the flat behaviour of the heat flux next to the hot wall remains at least until the wave reflects from the outer cold wall. After the early times the heat flux at the inner cylinder increases until the steady state value is reached. These results are in agreement with those reported in [77, 78] for collisionless transient heat flow between parallel plates. For smaller values of Knudsen, although it is not well visible in Figure, at very early times the heat flux at the inner hot wall is also flat and takes the same value as for larger Kn, due to the fact that at these times the flow regime can be considered as free molecular. However, when the collisions start being effective the heat flux at the inner cylinder drops until the steady state value is achieved. 61

63 Modeling and simulation of heat transfer and thermal phenomena in micro gas flows Figure : Time evolution of the radial heat flow ( qr) r at three different sections of the gap, in the first cell next to the inner hot cylinder, at the middle and in the cell next to the outer cold cylinder - * r =,.5, 1 - for 1 β = and γ =. and for Kn =.1 (upper left), Kn =.1 (upper right), Kn = 1 (down left) and Kn = 1 (down right). The time necessary to reach the new steady state ( t ss ) as a function of the governing parameters is presented in Table 3. This time has been calculated considering that the steady state has been achieved when the relative difference between the macroscopic quantity ( Q ) and its value in the steady state ( Q ss, i ) is lower than the 1% of the difference between the steady state value and the initial value of that quantity ( Q,i ) for every point within the gap. i Err i Q Q ss, i i = < Q Q ss, i, i.1 (14) As it is shown, t ss is decreased as β is increased. Also it can be seen that the influence of γ on t ss changes with the rarefaction conditions. The time t ss increases as γ decreases for Kn =.1 and.1, while it decreases with γ for Kn = 1 and 1. 6

64 Manuel Vargas Hernando Table 3. Influence of β and γ on the time needed to reach the steady state t ss for different values of Kn. β = 1 β = 1. β =. Kn =.1 Kn =.1 Kn = 1. Kn = γ γ = γ = γ γ = γ = γ γ = γ =. The influence of the Knudsen number on t ss shows a minimum between Kn =.1 and Kn = 1 when the time is represented in acoustic scale ( tref = Lv ), as shown in the Table 3 for β = 1 and γ 1. This minimum is observed for all values of β, but only for γ 1, as for the other values of γ analyzed in this work t ss decreases monotonically when Kn increases. Figure 3 shows more precisely the value of the Knudsen minimum in the response time of the system. As it is shown the value of the Knudsen for which the response time is smaller for β = 1 and γ 1corresponds approximately to Kn =.3. Similar behaviour in the response time for transient fully developed Poiseuille flow through cylindrical tubes was reported in [89]. Figure 3: Time necessary to reach the 9% of the new steady state values as a function of Kn using acoustic time scale: tref = Lv for β = 1 and γ 1. 63

65 Modeling and simulation of heat transfer and thermal phenomena in micro gas flows Comparison of DSMC data with Navier-Stokes-Fourier calculations For small Knudsen numbers we study the single gas flow between two coaxial cylinders (one dimensional, axis-symmetrical problem) by using the continuum model described in [86]. The inner cylinder has radius R 1, wall temperature T 1 and peripheral velocity v 1, the outer - R, T and v respectively. The continuous model of the rarefied gas flow between two coaxial cylinders is based on the non-stationary one-dimensional Navier-Stokes (NS) equations for compressible fluid, completed with the equations of continuity and energy transport. The governing equations in cylindrical coordinate system are written as follows: ρ + div = t ( ρv) τ φφ vy vy v z P 1 ρ + vy = ( rτrr ) + t r r r r r r v v vv 1 ρ + v + = z z y z y t r r r r ( r τr φ ) (15) (16) (17) DT ρc P = div( λgradt ) PdivV + µ Φ (18) Dt P = ρrt (19) where V is the velocity vector, v y and v z are the velocity components along axis r and ϕ. A rather standard notation is used in Eqs. (15)-(16): P is the pressure, ρ is the density and T is the temperature., PTv,,, v f( rt, ) ρ =. τ i, j are the stress tensor components and Φ is y z the dissipation function [9]. For a perfect monatomic gas, the viscosity and the heat transfer coefficient read as [51]: 5 µ = µ ( T) = Cµ ρλ v T, Cµ = π (13) λ= λ( T) = Cλρλ v T, Cλ = π (131) 3 The above written equations are normalized as given in section.by using the following scales: for density, ρ = mn (m is the molecular mass, n -the average number density), for velocity v = RT - R is the gas constant, for length - the distance between the cylinders L= R R1, for time t = Lv, for temperature T - the wall temperature of both cylinders i.e. T i = T, i = 1,. The Knudsen number is Kn = λ L, where the mean free path is λ and 64

66 Manuel Vargas Hernando γ = c c P V = 5 3 ( c P and c V are the heat capacities at constant pressure and constant volume respectively). In this way in the dimensionless model the characteristic number Kn and the constants C µ and ρ, PTv,,, v are used. y z C λ take part. After the scaling, the same symbols for the dimensionless The first-order slip boundary conditions are imposed at both walls, which can be written directly in dimensionless form as follows [66, 91]: at r = Ri. In Eqs. (13)-(134) Ti Ti / T vz vz vz Aσ Kn = (13) r r v = (133) y T T ± ζ TKn = Ti r (134) = and Ri = ri L are the dimensionless temperature and radius for both cylinders (i=1, ). For diffuse scattering we have used the viscous slip and temperature jump coefficients A σ = and ζ =.194 calculated, respectively in [61, 9], from the kinetic BGK equation by using variational method (for details see [93]). A comparison of the macroscopic properties profiles obtained by using DSMC and the Navier-Stokes approach is included in Figure 4 for three different rarefaction conditions. Since we investigate the heat flux between two cylinders at rest the velocity T v z is zero everywhere and it is not shown in the figure. It is seen that both approaches show an excellent agreement for Kn =.1, even for early-time profiles, and the results differ more as the Knudsen number is increased. Even for Kn =.1 the instantaneous profiles of the macroscopic properties differ between both approaches. In Figure 5 the profiles of the macroscopic properties at different instants obtained by using DSMC and the Navier-Stokes approach with different boundary conditions are compared. In this case the boundary conditions (16)-(18) are evaluated by using the value of the local Knudsen, instead of the global Knudsen, as it was described initially. The results obtained by using global and local Knudsen show similar behaviour at the very initial time instants, while for larger values of time the results obtained by using the local Knudsen show better agreement with the DSMC data even for Kn =.1 than those obtained by using the global Knudsen in the boundary conditions. The same disagreement at early times and agreement at late times is reported in [85] in their slip-flow solution of the unsteady response of the system subject to a step-jump heating of its boundaries. 65

67 Modeling and simulation of heat transfer and thermal phenomena in micro gas flows Figure 4: Distributions of density, radial bulk velocity and temperature at the indicated times obtained by DSMC (circles) and Navier-Stokes (solid line) for Kn =.1 (up), Kn =.5 (middle) and Kn =.1 (down) for β = 1 and γ =.. 66

68 Manuel Vargas Hernando Figure 5: Distributions of density, radial bulk velocity and temperature at the indicated times obtained by DSMC (circles) and Navier-Stokes (solid lines) for Kn =.1, β = 1 and γ 1, by taking into account the average Knudsen (black line) and the local Knudsen at the boundaries (blue line) Numerical results for binary mixtures The calculations were carried out for two mixtures of noble gases: neon-argon (Ne-Ar) and helium-xenon (He-Xe). The ratio of their molecular masses and molecular diameters are m m =.55 and d d =.711 and m m =.35 and d d =.445 Ne Ar Ne Ar respectively. Simulations have been performed for four different values of the Knudsen number ( Kn = 1, 1,.1 and.1), covering the whole range of gas rarefaction, for three values of the radius ratio ( γ = 1,.5 and.). The value of the temperature difference was kept constant at β = 1. Five values of the mole fraction were considered X α =.1,.3,.5,.7 and.9, aside from the corresponding limiting values for the two simple gases when X α = and X α = 1. The transient behaviour of the macroscopic quantities in a binary mixture gas is described in Figure 6 by plotting the profiles of number density n, radial velocity v r and pressure P for both components of the mixture at certain times from the sudden change of the temperature of the inner cylinder for the specific case of a Helium-Xenon mixture with mole fraction X α =.5, γ =.5 and two values of the Knudsen number ( Kn =.1 and Kn = 1 ). He Xe He Xe 67

69 Modeling and simulation of heat transfer and thermal phenomena in micro gas flows Figure 6: Distributions of density, radial bulk velocity and pressure at the indicated times for Kn =.1 (left) and Kn = 1 (right) and γ =.5 in a mixture of Helium (in red) and Xenon (in blue) with mole fraction X α =.5. The solid lines represent the corresponding steady state profiles. The blue lines correspond to the heavier component of the mixture (Xe), while the red lines represent the profiles of the lighter component (He). The steady state solutions are represented by solid lines. It can be seen from the density profiles that, similarly to the case of a single gas, the sudden heating of the inner wall causes a decrease in the density near the 68

70 Manuel Vargas Hernando inner cylinder and consequently a wave propagates along the gap due to the larger momentum of the reflected particles from the inner wall. The perturbation propagates slightly faster for the lighter component of the mixture near continuum conditions (left). However, it is observed that for Kn = 1 (right) the signal propagates much slower for the heavier component due to the fact that in near-free molecular regime the molecules of both components move independently. As the molecules of the light component move faster, the signal is transmitted faster. A similar behaviour is observed in the bulk velocity profiles for Kn =.1, as both magnitude and phase of the perturbation are similar for the two components, while in near-free molecular regime the magnitude of the bulk velocities of both components are very different and the perturbation of the heavy component is delayed with respect to the light component. It is interesting to note the difference in the steady state profiles of the number density for the two different rarefaction conditions. The density profiles are convex in near-continuum regime, while they become concave in near-free molecular limit. Also the steady state pressure profiles show that the total pressure for small Knudsen numbers is constant across the gap, while for highly rarefied gas conditions the pressure is non-uniform across the annular gap, being larger close to the inner cylinder. The evolution of the heat flow rate Q( r) rq ( r) = at the inner (in blue) and at the outer (in r red) cylinder is given in Figure 7 for the two mixtures under study (Ne-Ar and He-Xe) for two values of Knudsen ( Kn = 1 and Kn =.1). The total heat flow rate is represented by circles, the heat flow due to the light component by stars and the heat flow due to the heavy component by squares. The steady state values are given for the total (represented by solid line) and for each of the components (represented by dashed lines). From this data it is seen that the contribution of the light component to the total heat flux is larger than the heavy component. Similarly to the behaviour of the heat flux for single gas explained in Figure, heat flux at the inner cylinder for gas mixtures at early times is constant for large values of Knudsen. This can be explained because under these conditions the motion of both components can be considered independent due to the absence of effective collisions. The heat flux at the inner cylinder increases when the wave of the corresponding component reaches the inner cylinder after reflecting back from the outer cold cylinder. This occurs at times t and t 3 for the Ne-Ar mixture and times t.35 and t for the He-Xe mixture. The latter values agree with the pattern observed in Figure 7. The jump in the heat flux due to the light component occurs always earlier than in the heat flux due to the heavy component for large values of Knudsen. At the outer cylinder the initial heat flux is equal to 69

71 Modeling and simulation of heat transfer and thermal phenomena in micro gas flows zero at early times and is increased when the wave of the corresponding component reaches the outer cylinder, which happens at times t.5 and t.8 for the Ne-Ar mixture and times t.7 and t.5 for the He-Xe mixture, in agreement with the results in Figure 7. Figure 7: Time evolution of the radial heat flow ( qr) r at the inner (in blue) and at the outer cylinder (in red) for Kn = 1 (up) and Kn =.1 (down), for Ne-Ar (left) and He-Xe (right) mixtures for β = 1 and γ =.5. The total heat flow is represented by circles, while the heat flow due to the light and the heavy component are represented by stars and squares respectively. The steady state values are indicated by black solid line (total heat flow) and dashed lines (heat flow contribution of each component of the mixture). It is worth clarifying that the slight deviation from zero at very early times for the light component of mixture He-Xe is caused by the filtering process of the statistical fluctuations. For smaller values of Knudsen, the heat fluxes of the two components at the inner cylinder experience a fast decrease at early times. It should be noted that at very early times in kinetic scale before the collisions start being effective, although it is not seen in Figure 7, the values of the heat fluxes should be constant for an initial period and should take the same values as 7

72 Manuel Vargas Hernando initially they take for Kn = 1. It is observed that the heat fluxes of the two components at the outer cylinder show a peak which starts at times t.6 for the Ne-Ar mixture and at t.7 for the He-Xe mixture. This shows that the waves of the two components of each mixture are coupled and reach the outer cylinder at the same time, which is in agreement with the results in Figure 7. The numerical results for the response time of the system are given in Table 4. These values are estimated by using the convergence criterion defined in Eq. 14. In order to provide the results in a time scale which is independent of the mole fraction, the response times shown in Table 4 are presented normalized by Lv, β (the ratio between the width of the annular gap and the most probable velocity of the heavy component of the mixture kt m β ) It is worth noting that Table 4 extends the results presented in Table 3, which provides the corresponding values of the response time for the limiting case of a simple gas when the mole fraction of the heavy component is X β = 1. From the results provided in Table 4 we can conclude that some features already described in the single gas analysis are also valid for binary mixtures: 1. The response time increases as the curvature effects increase (the radius ratio is decreased) for the smaller values of the Knudsen number.. However, the system experience an opposite behaviour for more rarefied conditions, where the response time decreases when the radius ratio is decreased. 3. There is a minimum of the response time with respect to the Knudsen number for values of the radius ratio close to unity. This minimum disappears when the radius ratio is decreased and the response time decreases monotonically as the gas rarefaction is increased. Besides, additional features related to the mole fraction of the components of the gaseous mixtures can be described: The response time of the system is monotonically increasing with the mole fraction of the heavy component for any value of the Knudsen number and any radius ratio. The variation of the response time with the mole fraction is larger as the difference of the molecular masses is increased. For this reason, the difference between the values of the response time is larger in the case of the He-Xe mixture than for the Ne-Ar mixture. 71

73 Modeling and simulation of heat transfer and thermal phenomena in micro gas flows Table 4: Response time in acoustic scale for two binary gas mixtures for various values of Kn, γ and mole concentrations. He-Xe Knudsen γ X Xe = Ne-Ar Knudsen γ X Ar = The steady state heat transfer flow between parallel plates has been investigated previously by various authors [67-73]. Here, a comparison of the steady state values of the heat flux obtained by using DSMC with those obtained on the basis of the Boltzmann equation and reported in Ref. [67] was done for various values of the Knudsen number, ratio of the molecular masses and molecular diameters for the planar configuration. The present Knudsen number and reference heat flux ( Kn and ' q ) are related to those given in [67] ( Kn and ref ' q ref ) by the following relations: d q m = + = + ' ' ref Kn Kn X1 X qref X1 X d1 m1 1 (135) 7

74 Manuel Vargas Hernando The results of the comparison are presented in Table 5. As it can be seen the results are in excellent agreement. Table 5: Comparison of the present steady state heat flux values with those obtained on the basis of the Boltzmann equation reported in [67]. m /m 1 =.5; d /d 1 = 1. m /m 1 =.5; d /d 1 =.5 X 1 Kn (Kn') q q [67] X 1 Kn (Kn') q q [67].99.1 (.1) (.1) (1) (1) (1) (1) Conclusions The transient heat transfer flow through a rarefied gas confined between two concentric cylinders due to a sudden change of the temperature of the inner cylinder surface is investigated for different curvatures extended up to near planar configuration. The instantaneous macroscopic properties profiles are obtained and the response time of the system is estimated for two different gaseous mixtures with various values of the mole fraction. A significant part of the study concerns the special case of a single gas when the mole fraction of the second component is zero. The process of the wave formation at early times and the effect of the governing parameter variations in a wide range have also been investigated. The existence of a minimum in the response time for systems close to planar geometry is an interesting finding in contrast with the monotonic behaviour shown in the response time with respect to the curvature for the different flow regimes. 73

75 Modeling and simulation of heat transfer and thermal phenomena in micro gas flows Chapter 4 Flows induced by temperature field: Thermal creep flow 4.1. Introduction Flows induced by temperature fields (for instance the flow created by a non-uniform temperature distribution at the wall) are known for a long time and they are observed especially under rarefied conditions of the gas. Nowadays these flows have several applications, such as in vacuum pumps without moving parts [94-96], due to the non-uniform pressure distribution along a channel created by the temperature gradient of the walls. The flow characterized by the temperature gradient of the boundary along it is called thermal creep flow. The thermal creep phenomenon is a rarefaction effect. For a rarefied gas it is possible to start the flow with tangential temperature gradients along the channel surface. In that case the theory says that the molecules start creeping from cold toward hot. Thermal creep effects can be important in causing variation of pressure along microchannels in presence of tangential temperature gradients. Figure 8: Thermal creep along a channel between reservoirs due to tangential temperature gradient. The physical mechanism of the thermal creep flow can be found in Sone [97]. We should consider a gas at rest with a temperature gradient along a wall and examine the momentum transmitted by the molecules impinging on a small area. The impinging molecules come from various directions directly (without intermolecular collisions) over a distance of the order of the mean free path, keeping the properties of their origins. The average speed of the 74

76 Manuel Vargas Hernando molecules coming from the hotter region is larger than that from the colder region. In the gas at rest, the pressure is uniform and the density is lower in the hotter side, so the number of molecules hitting the small area per unit time is from the hotter side is the same as that from the colder side. The contribution of the molecules reflected from the small area to the tangential component of the momentum transfer is nothing in the case of diffuse reflection. Thus, a momentum in the direction opposite to the temperature gradient is transferred to the wall from the gas and, as a reaction, the gas is subject to a force in the direction of the temperature gradient and a flow is induced in that direction. To explain the pressure variation along the microchannel, let us assume free molecular regime and define the mass flow from the hot and the cold side as: Hot: Cold: 1 M k B H = mnch = mn TH (136) m 1 M k B C = mncc = mn TC (137) m Considering that at equilibrium both mass flows should be equal and that P = ρrt, then it can be proved that in the free molecular limit the pressure ratio created by the temperature gradient along the channel is: mnhch ρ H T H P H T C P H T H = = = 1 = mnccc ρc TC PC TH PC TC (138) 4.. Thermal driven flow due to non-uniform wall temperature in rectangular enclosures 4..1 Introduction The recent development of micro electro-mechanical systems (MEMS) is leading many authors to the study of heat transfer and thermal effects in rarefied gases. Convective flows appear under rarefied conditions for some specific flow configurations. The buoyancy driven flows are an example. Here, the convection is caused by buoyancy forces which occur only in a frame which either has a gravitational field or is accelerating due to a force other than gravity. This problem in rarefied gas dynamics has been investigated by several authors [98-75

77 Modeling and simulation of heat transfer and thermal phenomena in micro gas flows 14] under the name of Rayleigh-Bénard problem. Stefanov et al. [99-11] identified the zone of convection in terms of Kn and Fr numbers and investigated the transition period from the initial state to the established flow regimes in two and three dimensional configurations. Then Manela [1] used linear stability analysis for studying the problem in the slip flow regime. The other example would be the thermal driven flow which is induced by non-uniform temperature fields at the walls. The thermal transpiration or thermal creep is a rarefaction effect which appears when there is a temperature gradient along the boundaries of a wall, which creates a flow next to the walls from the colder to the hotter side. An excellent explanation of the physical mechanism of the thermal creep flow can be found in Sone s book [97]. The flow generated by thermal creep flow through rarefied gases in rectangular enclosures was investigated by Papadopulos [15], showing vortex formation for zero-gravity conditions. Orhan [16] studied the effect of second-order slip models for such configuration and, more recently, Kosuge [17] investigated the flow pattern formation by using Cercignani-Lampis boundary conditions at the walls on the basis of the Boltzmann equation. In this study the thermal driven flow in two and three dimensional enclosures with constant temperature gradient at the vertical walls is investigated for a set of Knudsen numbers and aspect ratios in the whole range of rarefaction using completely diffusive boundary conditions. This flow configuration may have applications in microtechnology for devices which are heated only from one side or also could be applied to satellites which rotate and face a specific source of radiation. It is found that the flow pattern formation is very sensitive to the Knudsen number and the aspect ratio obtaining non-intuitive convective patterns in the opposite direction to that induced by thermal creep Problem formulation and numerical simulation The three-dimensional thermal driven flow problem for a rarefied gas treats a monatomic simple gas with average number density n studied in a rectangular computational box ( x, y, z) D L L L. The domain is confined in the vertical direction by two diffusively reflecting horizontal walls at different temperatures T H distance ( L L ) x z > T. On the four vertical walls at a mutual C =, a constant temperature gradient is applied. The reflection of the particles is taken also to be purely diffusive. The hard sphere molecular model is employed in the simulation of binary collisions. 76

78 Manuel Vargas Hernando Figure 9: Schematic representation of the system. The initial state of the gas is in equilibrium at reference pressure and temperature P and T = T, with a reference number density n P kt B H =, where λ ( π ) 1 = dn is the mean free path of molecules of diameter d when they are in equilibrium state at rest with number density n. To render the problem dimensionless, we establish the following normalized variables: n' T' v' q' P' x' n= ; T = ; v= ; q = ; P= ; x= (139) 3 n 1 TH v nmv nmv λ where m is the molecular mass, v kt H = m.5 is the most probable velocity and k is the Boltzmann constant. Three are the parameters governing this problem: the Knudsen number Kn, the temperature ratio β and the aspect ratio A given by: λ Kn = (14) L T y C β = (141) TH L L = = (14) x z A L y L y 77

79 Modeling and simulation of heat transfer and thermal phenomena in micro gas flows The aim if this study is to investigate the influence of the three parameters governing the problem, to find the region where the thermal driven flow appears and discover the different pattern formations of this flow. The traditional Direct Simulation Monte Carlo (DSMC) method proposed by Bird [1] is used for the calculations. In this method the real motion of the particles is split in two parts, the ballistic motion of all the particles proportional to their velocities and the interaction between the molecules without changing their positions. The first step is purely deterministic, while the collisions between particles are carried out in a stochastic manner. The collision technique that we use in general in this study is the No Time Counter (NTC) scheme suggested by Bird, except a slight modification in the calculation of maximum collision number in a cell, as described in [87]. In the last section of this work, a comparison between the results obtained by using the NTC scheme and those obtained by using an alternative scheme, so called Simplified Bernoulli Trails (SBT) [35, 36] which allows the use of small average number of particles in cells and avoids repeated collisions, is presented for a small set of conditions. The domain was discretized by using squared cells. The number of cells was chosen to be 1x1 for any Kn >.1 and aspect ratio A = 1, while this number was changed according to A as shown in Table 6. The number of particles per cell is fixed at 5 and the time step is chosen to be sufficiently smaller than the cell traversal time ( ) L nv. The macroscopic properties are obtained by time averaging over 1 5 time steps after the steady state regime has been achieved. The heat flux is volume based calculated, as well as the other macroscopic C quantities, by averaging the microscopic values of the particles at a given cell q r = vc y. Table 6. Number of cells employed in the calculations depending on A. A # cells A # cells. 76x x1.3 76x8 x1.5 76x15.5 5x1.8 8x1 3 3x1 1 1x1 4 4x1 1. 1x1 78

80 Manuel Vargas Hernando Numerical results As a first step, the influence of the Knudsen number on the intensity and pattern of the flow was investigated for a two-dimensional rectangle of aspect ratio A= L L = 1 (square) in the whole range of rarefaction (.1 Kn 1 ). The temperature at the two vertical walls varies linearly and the gas-surface interaction is taken to be completely diffusive. Previous works [15, 16] had shown the typical pattern under diffuse reflection boundary conditions consists on the induction of a motion in the gas in the vicinity of the walls from the cold to the hot wall due to thermal creep, while in the central region of the domain the motion is in the opposite direction due to mass conservation and two vortexes are clearly formed. In Figure 3 the velocity fields for various values of Kn are shown. As seen, the flow pattern previously described occurs for small values of Kn (the range of Kn will be determined in more detail below), while for near free molecular conditions ( Kn = 1 ) the macroscopic motion in the gas disappears. However, for values of Kn around unity the flow pattern is changed and, apart from the primary vortexes, secondary vortexes appear close to the hot plate in the regions near the vertical walls. In these secondary vortexes the flow experiences an opposite behaviour, going from hotter to colder regions in the vicinity of the vertical walls. The intensity of the flow is represented in Figure 31 by plotting the profiles of the maximum and minimum values of the vertical velocity component v y for the same values of Kn. These profiles are obtained by applying the operator max < y< ( ) and min y ( ) field v(, ) y 1 x 1 y < < to the velocity xy. The profiles of the maximum vertical velocity show that for small values of Kn the maximum vertical velocity takes place in the central region of the closure ( x =.5 ) demonstrating the existence of primary vortexes. Besides, the maximum vertical velocity is larger for Kn =.1 than for Kn =.1, which indicates the presence of maximum of the vertical velocity with respect to Kn. However, for Kn = 1 the maximum vertical velocity appears at the vertical walls (at x = and x = 1), proving the existence of the secondary vortexes. For Kn = 1 the maximum vertical velocity is close to zero in the whole domain showing absence of macroscopic motion in the gas. From the profiles of the minimum vertical velocity for this aspect ratio A = 1 it is seen that in all cases except for free molecular flow the gas moves downwards in the vicinity of the vertical walls. The magnitude of the velocity again is larger for Kn =.1 than for Kn =.1. 79

81 Modeling and simulation of heat transfer and thermal phenomena in micro gas flows Figure 3: Velocity fields for Kn =.1 (upper left), Kn =.1 (upper right), Kn = 1 (down left) and Kn = 1 (down right). Figure 31: Profiles of maximum (left) and minimum (right) vertical velocity component for Kn =.1 (circles), Kn =.1 (stars), Kn = 1 (triangles) and Kn = 1 (squares). 8

82 Manuel Vargas Hernando The dependence of the flow pattern formation and magnitude of the vertical velocity component on the Knudsen number for aspect ratio A = 1 and temperature ratio β =.1 is represented in more detail in Figure 3. As seen, the larger values of the maximum vertical velocity in the central region and the minimum vertical velocity in the vicinity of the vertical walls appear in a certain range of Knudsen (.6 < Kn <.1). The region of existence of primary vortexes can be identified between Kn =.1 and Kn =., where v y,max in the vicinity of the vertical walls and v y,min in the central region are close to zero. Then, a region of coexistence of the primary and secondary vortexes arises between Kn =.3 and Kn = 3, as the magnitude of v y,max in the vicinity of the vertical walls and y,min v in the central region increase. Finally, for Kn > 1 the motion of the gas disappears and both, v y,max and v y,min, are close to zero in the whole domain. The mass flow rate caused by the primary vortexes is estimated for various values of Kn < 1, from continuum limit up to the formation of the secondary vortexes, by integrating the mass flow rate ( nv ) which crosses the surface (line) which joins the centre of the primary vortexes x with the upper cold wall. The results, presented in Table 7, show that the mass which is affected by the primary vortexes increases in the interval from Kn =.1 until approximately Kn =.8. From Kn =.1 the mass flow rate circulating around the primary vortex decreases as the Knudsen number is increased. Figure 3: Influence of Kn on the maximum (left) and minimum (right) vertical velocity in the vicinity of the vertical walls (in blue) and in the central region (in red). 81

83 Modeling and simulation of heat transfer and thermal phenomena in micro gas flows Table 7: Mass flow rate of the primary vortexes for various Kn, A = 1 and β =.1. Kn Flow rate 1.5 (-3) 4.7 (-3) 6.1 (-3) 6.5 (-3) 6.4 (-3) 5. (-3).6 (-3) 1.1 (-3) Influence of the aspect ratio The results presented in the previous section correspond to a rectangular enclosure with aspect ratio fixed at A = 1. In this section the influence of the aspect ratio on the flow pattern is investigated at a given Kn = 1, which corresponds to the range of Kn where the secondary vortexes appear. Simulations were carried out for an interval of A [.,4], by varying the distance in the x -direction L x and keeping L y. In Figure 33 the streamlines and the vertical component of the velocity v y are represented for three different values of the aspect ratio A =., 1 and 4 for a given Kn = 1. It is clearly seen in the case of A = 1 the formation of the primary vortexes in the upper part centred approximately at the points [.,.8] and [.8,.8], as well as the secondary vortexes turning in the opposite direction centred at [.1,.] and [.9,.]. It is possible the existence of another secondary vortex situated at [.35,.1], as seen in Figure 33, although the noise prevent the visualization of the antisymmetric vortex which should be at [.65,.1]. It is seen that the vertical velocity is positive in the central region, but the largest magnitude of the upward vertical velocity occurs at the vicinity of the vertical walls. This effect is a combination of the fact that the probability of the particles to cross the domain without collide is large but at the same time the small number of collisions seems to play a very important role, since, as shown in the previous section, in free molecular regime this effect disappears. As seen in Figure 33, when the aspect ratio is decreased by shortening the x -dimension until A =., the flow pattern changes drastically as the flow is induced in the vicinity of the walls from the hot to the cold wall, while in the central region of the domain the motion is in the opposite direction due to mass conservation Two long vortexes which are extended along the whole domain are clearly seen. This is a completely opposite behaviour to the typical motion induced by thermal creep (from the cold to the hot plate in the vicinity of the walls). It is seen from the vertical velocity representation that the magnitude of the velocity is larger for both, upward and downward vertical velocities, in the lower half of the domain close to the hot wall. Besides, the streamlines show small recirculations within the two main vortexes. In contrast, when the aspect ratio is increased until A = 4, the typical flow induced by thermal creep is developed and two clear vortexes are observed. The motion of the gas in 8

84 Manuel Vargas Hernando central region of the domain is upwards (from the hot to the cold plate) while close to the vertical walls the gas moves from the cold to the hot plate and two big vortexes are clearly seen. It is worth noting that even for A = 4 an upward vertical motion of the gas appears in the most adjacent layers of the gas close to the vertical walls in the lower part of the simulation domain. 1 x x 1-3 y x 5-5 y x y x Figure 33: Streamlines and vertical component of the velocity (in colors) for A =. (upper left), A = 1 (upper right) and A = 4 (down) for Kn = 1 and T T =.1. C H x

85 Modeling and simulation of heat transfer and thermal phenomena in micro gas flows The dependence of the different patterns and magnitude of the vertical velocity component on the aspect ratio for Kn = 1 and temperature ratio β =.1 is represented in more detail in Figure 34. As seen, the values of the maximum vertical velocity both in the central region and near the vertical walls show a maximum with respect to the aspect ratio. For small values of the aspect ratio the maximum vertical velocity v y,max in the central region is practically zero, which corresponds to the flow pattern showed in Figure 33 for A =.. This maximum value of the vertical velocity increases until A = because of the fact that what we called primary vortexes start developing and growing as the aspect ratio is increased. Then, for A > the value of v y,max at the central region decreases as the area through which the gas rises becomes larger. The values of v y,max in the vicinity of the vertical walls increase from small values of A until A 1 where it reaches its maximum. In the interval A [.,1] the flow develops from a configuration with two vortexes with upward flow next to the vertical walls and downward flow in the central region to the initialization and growth of the primary vortexes in the opposite direction. For A > 1.5 the effect of upward vertical velocity adjacent to the vertical walls start vanishing. In a similar manner, v y,min in the central region is larger for small values of A which confirms the presence of the two vortexes in direction opposite to the classical induced flows due to thermal creep. For A 1 the minimum vertical velocity in the central part of the domain tends to zero because the primary vortexes become dominant. In the vicinity of the walls v y,min decreases monotonically as A increases. Figure 34: Influence of A on the maximum (left) and minimum (right) vertical velocity in the vicinity of the vertical walls (in blue) and in the central region (in red). 84

86 Manuel Vargas Hernando In Figure 35 we show the spatial coordinates of the centre of the primary vortex situated at the left half space in terms of Kn and A. As the flow is antisymmetric with respect to x= A the abscissa of the right primary vortex will be ( A x). It is seen that for a fixed A = 1 the centre of the primary vortex is shifted slightly towards the plane of antisymmetry in x -direction and towards the upper cold wall in the y -direction as Kn is increased. The effect of A when the Kn is fixed at Kn = 1 on the position of the centre of the left primary vortex is the following: the centre of the primary vortex clearly travels from near the upper cold wall towards the middle plane at y =.5 as the aspect ratio is increased. As for its horizontal coordinate, the distance between the centre of the primary vortex and the vertical wall increases monotonically with A. However, if the distance x is normalized by the corresponding aspect ratio the dependence is monotonically decreasing as A increases. Figure 35: Influence of Kn for A = 1 (left) and of A for Kn = 1 (right) on the position of the centres of the left primary vortexes Comparison with adiabatic boundary conditions The thermal driven flow in a two-dimensional rectangular enclosure of A = 1 is now investigated by using adiabatic boundary conditions at the vertical walls. In this case the temperature along the vertical walls is not defined, but is an output from the simulation and may not coincide with the constant gradient established in the formulation of the problem. The results are compared with those obtained by using completely diffusive boundary conditions. The implementation in DSMC of adiabatic boundary conditions can be done similarly as for diffuse reflection, by reflecting the particles according to the Maxwellian distribution at a given temperature, but keeping the energy of the particle equal before and 85

87 Modeling and simulation of heat transfer and thermal phenomena in micro gas flows after the reflection. The heat flux through the vertical walls is then equal to zero when adiabatic boundary conditions are employed. The comparison between both boundary condition models is done by presenting the macroscopic properties at the perimeter of the enclosure as indicated in Figure 36 starting and finishing at point A. The number density and temperature distributions along the indicated points are shown in Figure 37. It can be seen that the profiles obtained by using the two different boundary conditions show a good agreement for Kn =.1, but the deviation between them becomes larger as Kn is increased. As seen, the variation in the temperature of the gas along the vertical walls for Kn = 1 and Kn =.1 is larger for diffuse than for adiabatic boundary conditions. This leads to a large discrepancy between the results at the hot plate. Besides, the variation of the macroscopic properties along the walls which are kept at constant temperature is larger when the diffuse boundary conditions are employed. Especially at the corners of the rectangular enclosure the temperature of the gas becomes closer to its respective value when the diffuse reflection is used. Similar peaks at the corners are observed for the number density, which are absent for adiabatic walls. However, in spite of these differences in the macroscopic properties along the walls, the flow for isothermal and adiabatic walls follow the same patterns: the gas moves upward in the central region of the domain and downwards next to the walls showing two primary vortexes for Kn =.1 and Kn =.1, while secondary vortexes rotating in the opposite direction coexist with the primary vortexes for Kn = 1. Figure 36: Schematic representation of the points in which the macroscopic properties are shown. 86

88 Manuel Vargas Hernando Figure 37: Number density and temperature distributions obtained by using completely diffusive boundary conditions (solid line) and adiabatic boundary conditions (circles). The different colors indicate the profiles for different Kn : Kn =.1 in black, Kn =.1 in red and Kn = 1 in blue Flow patterns in three dimensional enclosures In this section the three-dimensional thermal driven flow problem due to non-uniform wall temperatures is investigated in a rectangular computational box ( x, y, z) D L L L. As it was already formulated, the domain is confined in the vertical direction by two diffusively reflecting horizontal walls at different temperatures T H > T. On the four vertical walls at a C mutual distance ( L L ) x =, a constant temperature gradient is applied. The gas-surface z interaction is taken also to be purely diffusive and the hard sphere molecular model is employed for the computation of intermolecular binary collisions. The computational domain was divided in 1x1x1 cells, the number of particles per cell is fixed at 15 and the time step is chosen to be sufficiently smaller than the cell traversal time. The macroscopic properties are obtained by time averaging over time steps after the steady state regime has been achieved. The calculations were carried out for A = 1, β =.1 and for two values of the Knudsen number: Kn =.1 and Kn = 1. In Figure 38 the vertical component of the velocity v y is plotted at four different sections of the box: at z =.5, y =., y =.5 and y =.8. 87

89 Modeling and simulation of heat transfer and thermal phenomena in micro gas flows Figure 38: Vertical component of the velocity v y for Kn =.1 (up) and Kn = 1 (down) at four different sections of the domain: z =.5, y =., y =.5 and y =.8. It is clearly seen that, for Kn =.1, in the central region of the domain the vertical velocity v is positive and the gas moves from the hot plate (situated at y = ) to the cold plate (at y y = 1). On the contrary, the gas flows downwards (as indicated by the negative vertical 88

90 Manuel Vargas Hernando velocity) in the regions adjacent to the vertical walls, especially at the four corners. This flow consisting of the gas moving from the cold to the hot plate in the vicinity of the vertical walls and from hot to cold in the centre of the domain, represents the typical flow induced by thermal creep, which in two dimensional geometry corresponds with the formation of the two primary vortexes. However, for Kn = 1 the vertical velocity v y is positive in two areas: the central region especially for y >.4 and the layers of gas adjacent to the four vertical walls (situated at x and y = and 1) especially for y <.6. The velocity v y is negative in the regions near the vertical walls beyond the influence of the positive vertical velocity and especially at the four corners. This flow pattern would correspond with the coexisting primary and secondary vortexes described in two-dimensional geometry. This proves that the effect of inversion of the convective patterns do not occur exclusively in two-dimensional enclosures, but also in three-dimensional systems Simulation of thermal driven flow using small number of particles per cell In this section the results obtained by using the standard NTC scheme are compared with an alternative collision scheme (Simplified Bernoulli Trials SBT) which allows the use of a small average number of particles in cells as it avoids repeated collisions within the time step [35, 36]. Besides SBT scheme makes use of a dual grid which allows the potential collisions with particle pairs in neighbouring cells. The collision stage is subdivided in two parts over a time t : the first part is carried out in the basic grid, while the second collision step is carried out in a grid which is shifted x in every direction with respect to the basic one. The SBT collision algorithm consists of the following steps: 1) The sequence of ( l) ( N ) i = 1,..., 1 particle pairs is chosen from the ( l) N particles in cell l as follows: the first particle i is the particle with index i in the particle list for cell l. ) The second particle ( l ) 1, is chosen with probability 1 k from ( l) j i+ N taking place in the list after particle i. 3) Then the particle pair ( i, j ) is checked for collision with probability p ij ijcr, ij t c ( ) k = N i σ = k (143) ( l) V 89

91 Modeling and simulation of heat transfer and thermal phenomena in micro gas flows In Figure 39 the number density, temperature, pressure and vertical heat flux profiles at the position y =.5 obtained by using various collision schemes with different average number of particles in cells for the thermal driven flow in a two-dimensional enclosure for Kn =. and A = 1 are presented. Results are given for the standard NTC, the SBT and, additionally, the Majorant Frequency scheme [3]. The domain was discretized in x cells and the time step was taken to be at t =.3λ v. The reference solution is assumed to be that obtained with 1 particles in cells by NTC. As seen, when the number of particles is drastically decreased both NTC and MFS lose accuracy and deviate significantly from the reference solution. However, the SBT scheme maintains its accuracy even when one particle per cell in average is used. The main reason is that the probability of having repeated collisions increases rapidly when the number of particles per cell decreases, as demonstrated in [35]. The effect of the repeated collisions can be clearly seen in the heat flux profiles. The NTC and MF schemes overestimate the value of the heat flux when small number of particles in cells is used. This is because the collision frequency is underestimated due to the fact that n repeated collisions have the same effect as only one collision. The computational cost is included in Table 8 for the NTC and SBT schemes for 5, and 1 particle per cell, keeping the same sample size for all the cases. The times are normalized by that for NTC with 5 particles per cell. As seen the computational cost of the NTC is considerably smaller than the SBT when large number of particles are used. However, the computational cost of SBT is comparable to NTC if a smaller number of particles in cells are employed. In Table 9 an estimation of the time consumed by each subroutine is presented for the NTC and SBT schemes. As seen the majority of the time is consumed by the collision process. The time of the collision process is reduced by the SBT with small number of particles in cells without loss of accuracy. The higher times of the indexing process for the SBT is caused by the fact that two indexing process are carried out every time step due to the dual grid. Improvements in order to reduce this effect must be investigated. 9

92 Manuel Vargas Hernando n.76 T x x P x q y.1.1 Ref. NTC 1pc SBT 1pc MFS 1pc NTC 1pc x Figure 39: Number density, temperature, pressure and vertical heat flux profiles at y =.5 for Kn =. and A = 1. The results were obtained by using NTC (green line), SBT (red dashed line) and MFS (pink dashed-dotted line) with 1 particle per cell and NTC scheme with 1 particles per cell (black solid line). Table 8: Computational cost of the different collision schemes. NTC 5 pc NTC pc NTC 1 pc SBT 5 pc SBT pc SBT 1pc Total time Table 9: Percentage of time consumed by each of the subroutines. NTC 5pc NTC pc NTC 1pc SBT 5pc SBT pc SBT 1pc Total Collisions Motion Index Others

93 Modeling and simulation of heat transfer and thermal phenomena in micro gas flows Conclusions The steady behaviour of a rarefied gas in two-dimensional rectangular enclosures with nonuniform wall temperature was investigated by using direct simulation Monte Carlo (DSMC) method for a wide range of Knudsen number and aspect ratios. The temperature in the vertical completely accommodated walls varies linearly. It is found that the flow pattern has a strong dependence on the Knudsen number. For small values of Kn the thermal creep induces a flow from colder to hotter regions in the vicinity of the vertical walls and in the opposite direction in the central region and two primary vortexes are developed. However, in the transition regime, the primary vortexes coexist with secondary vortexes turning in the opposite direction: from hotter to colder wall in the regions adjacent to the vertical walls. In the free molecular regime there is absence of global motion due to the lack of intermolecular collisions. The results obtained by using diffuse reflection at the vertical walls are compared with those obtained by using adiabatic boundary conditions. The flow patterns developed by using both boundary condition models are similar, although some differences in the temperature and density distributions are observed. The aspect ratio also plays an important role, since for Kn = 1 and small aspect ratios the typical flow pattern induced by thermal creep is completely inversed in the whole domain going from hotter to colder regions in the vicinity of the vertical walls. Results for three-dimensional enclosures where obtained for various Kn proving that the secondary vortexes and inversion of the typical thermal creep flow pattern occurs also in three-dimensional systems. Some calculations were carried out by using an alternative collision scheme (SBT) which allows the use of small number of particles per cell by avoiding repeated collisions within the time step. The comparison with the results obtained by NTC show that the SBT scheme is more accurate when small number of particles in cells are needed, such as in three dimensional configurations, and that the computational cost is comparable to the standard NTC Thermal transpiration effect in Capacitance Diaphragm Gauges with helicoidal baffle system Introduction Many techniques have been developed for the measurement of pressure and vacuum. Instruments used to measure pressure are called pressure gauges or vacuum gauges. A manometer could also be referring to a pressure measuring instrument, usually limited to 9

94 Manuel Vargas Hernando measuring pressures near to atmospheric. A vacuum gauge is used to measure the pressure in a vacuum which is further divided into two subcategories, high and low vacuum (and sometimes ultra-high vacuum). Among the different kind of electronic pressure sensors, the main kinds of vacuum gauges are the following: - Pirani gauge consists of a metal wire open to the pressure being measured. The wire is heated by a current flowing through it and cooled by the gas surrounding it. If the gas pressure is reduced, the cooling effect will decrease; hence the equilibrium temperature of the wire will increase. The resistance of the wire is a function of its temperature: by measuring the voltage across the wire and the current flowing through it, the resistance (and so the gas pressure) can be determined. Figure 4: Scheme of a Pirani gauge. - Ionization gauges are the most sensitive gauges for very low pressures (also referred to as hard or high vacuum). They sense pressure indirectly by measuring the electrical ions produced when the gas is bombarded with electrons. Fewer ions will be produced by lower density gases. Thermionic emission generates electrons, which collide with gas atoms and generate positive ions. The ions are attracted to a suitably biased electrode known as the collector. The current in the collector is proportional to the rate of ionization, which is a function of the pressure in the system. Hence, measuring the collector current gives the gas pressure. Most ion gauges come in two types: hot cathode and cold cathode. In the hot cathode version, an electrically heated filament produces an electron beam. The electrons travel through the gauge and ionize gas molecules around them. The resulting ions are collected at a negative electrode. The current depends on the number of ions, which depends on the pressure in the gauge. The principle behind cold cathode version is the same, except that electrons are produced in the discharge of a high voltage. 93

95 Modeling and simulation of heat transfer and thermal phenomena in micro gas flows Figure 41: Example of ionization gauge. - The Capacitance Diaphragm Gauge (CDG) is one of the most widely used vacuum gauges in low and middle vacuum ranges. This device consists basically of a very thin ceramic or metal diaphragm which forms one of the electrodes of a capacitor. The pressure is determined by measuring the variation in the capacitance due to the deflection of the diaphragm caused by the pressure difference established across the membrane. In order to minimize zero drift, some CDGs are operated keeping the sensor at a higher temperature. This difference in the temperature between the sensor and the vacuum chamber makes the behavior of the gauge be non-linear due to thermal transpiration effects [18]. This effect becomes more significant when we move from the transitional flow to the free molecular regime. Figure 4: Capacitance diaphragm gauge as built by INFICON. 94

96 Manuel Vargas Hernando Besides, CDGs may incorporate different baffle systems to avoid the condensation on the membrane or its contamination. Figure 43: Contamination of CDGs by deposition of Ti (left) and TiN (right). The thermal transpiration or thermal creep is a rarefaction effect which appears when there is a temperature gradient along the boundaries of a channel, which creates a flow next to the walls from the colder to the hotter side. The tube of the gauge connecting vacuum chamber and membrane is such a channel filled with gas under rarefied gas conditions. In the free molecular limit the pressure ratio created by the temperature gradient along the channel is:.5 H T H = C TC where T H and T C are the temperatures of both ends of the channel ( T H P P > T ). C (144) An excellent explanation of the physical mechanism of the thermal creep flow can be found in Sone s book [97]. A critical review of the numerical data and analytical results available was made by Sharipov and Seleznev [19]. The thermal creep flow and the thermomolecular pressure difference are described and experimental and analytical solutions are given for different configurations, rarefaction conditions and pressure and temperature drops. Several authors have performed experiments in order to measure the thermal transpiration either in capacitance manometers or in microchannels. Hobson [11] carried out the experiments under very low pressures matching the ideal limiting value in the free molecular regime when using a leached pyrex tube joining the cold and hot regions. However, the results did not match the theoretical value when the tube was smooth due to the fact that the reflection of the molecules in this case in not purely diffusive. Storvick et al. [111] measured the absolute thermal transpiration differential pressure in a capillary tube for different gases and compared their measurements with their numerical results obtained by solving the BGK kinetic model 95

97 Modeling and simulation of heat transfer and thermal phenomena in micro gas flows and recently experimental measurements of the mass flow rate caused by thermal transpiration due to a temperature gradient along a microchannel has been reported in [11]. The calibration of CDGs with heated heads has been analyzed by Poulter et al. [113], studying the thermal transpiration effect in such devices and comparing their results with some empirical equations proposed by Liang [114] for different pressures and gases. Additional empirical equations relating the pressure measured at the sensor to the actual pressure for various gases can be found in [115]. These results have been widely used for calibrating CDGs for different gases to those employed in the calibration procedure. Complete descriptions on the operation of CDGs and analysis of their deviations and calibration errors have been done by Jousten [116]. Šetina suggested [117] later a universal expression for the thermal transpiration correction in CDGs as a function of normalized pressure using viscosity and molecular mass of the gas to simplify the expression. More recently Nishizawa and Hirata [118] applied the DSMC method to compute numerically the pressure distribution along the connecting tube of the gauge in the whole range of rarefaction and Iwata and Kusumoto [119] analysed by using DSMC the limits of the Takaishi-Sensui formula and suggested a rescaling factor for short pipes. In the present work, the thermal transpiration effect on the behavior of a rarefied gas and on the measurements in a CDG with a helicoidal baffle system is investigated by using the Direct Simulation Monte Carlo method (DSMC). The study covers the behavior of the system under the whole range of rarefaction, from the continuum up to the free molecular limit for two different geometries: a) Short cylindrical pipe connecting the vacuum chamber and the diaphragm and b) Channel confined in the double helix which acts as a baffle system to protect the diaphragm of the instrument. The results obtained are compared with the empirical Takaishi-Sensui formula and the effect of different temperature distributions, channel lengths and accommodation coefficients are studied. Moreover, the dynamic response of the system is investigated by performing some non-steady state calculations under two different situations: 1) the temperature of the sensor is changed suddenly and ) sudden change in the pressure of the vacuum chamber. In this way, the evolution of the macroscopic properties of the gas is studied from the initial moments until the steady state is achieved and the time response is calculated for various rarefaction conditions and different values of the magnitude of the perturbation. From the code for 1D axially symmetric flows for cylindrical configuration, the first step was to add a second position variable to the set of data which is saved during the simulation. This second position variable was chosen to be the polar angle. In this way, the three velocity 96

98 Manuel Vargas Hernando components and two position components (radius and polar angle) are saved. Among other modifications, it was necessary to add two new boundaries at θ = θ1 and θ = θ (the two extreme values of the polar angle which limit the domain) and to find expressions to calculate the new variable of the polar angle after a given time and the time needed by a particle to hit the boundary. Once the two dimensional code for cylindrical configuration was implemented, a brief validation was made. The case chosen for the validation was the heat transfer problem between two coaxial cylinders ( R1 R =.5 ) at rest with surfaces at different temperatures ( T 1 T = ) at a given Knudsen number ( Kn =.1). This case was run by the 1D code (assuming diffuse reflection at the surfaces of the inner and outer cylinders) and the D code (assuming diffuse reflection at the surfaces of the cylinders and specular reflection at the new boundaries). The results obtained with both codes were in very good agreement. The second step was to add the third position variable to the set of data which is saved during the simulation. This position variable is the axial coordinate ( z ). In this way, the three velocity components and the three position components (radial, polar and axial coordinates) are saved. Among other modifications, it was necessary to add two new boundaries at z = z1 and z = z (the two limits of the domain in the axial direction) and to compute the new value of the axial coordinate, as well as the time needed by a particle to hit the boundaries. The validation was done in a similar way as with the D code, assuming specular reflection at the new boundaries. The results obtained with the 1D, D and 3D approaches are in very good agreement as it can be seen in Figure D D 3D n T 1.4 P (r-r 1 )/(R -R 1 ) (r-r 1 )/(R -R 1 ) (r-r 1 )/(R -R 1 ) Figure 44: Comparison between the results obtained by using 1D, D and 3D codes. 97

99 Modeling and simulation of heat transfer and thermal phenomena in micro gas flows Besides, open boundaries at θ = θ1 and θ = θ for both codes (D and 3D) were implemented. In this way, pressure driven flows or transient flows due to changes in the pressure at the boundaries could eventually be simulated. In these so called open boundaries, particles which hit such boundaries leave the domain. At the same time new particles are produced at the boundaries at a rate and with velocities following a maxwellian distribution at a given pressure and temperature. This 3D code served as basis for building the code with which the helicoidal geometry of the CDGs baffle system is studied. This system consists of a double helix between two coaxial vertical cylinders. Due to its symmetry, our approach is to simulate only one of the two channels in which the system is divided. The assumption was to consider one of the two unwrapped rectangular channels. The section of this channel would be ( R R ) x( z) the length n( π r) out in and, where n is the number of completed turns of the helix. The domain was divided into cells in the following way: The radial component was divided in n r cells of size ( R R ) The polar angle was divided in n θ cells defining an angle of out n r in ( θ θ ) 1 n θ (145) (146) The vertical component (axial considering the cylinders between which the helix is situated, but vertical considering the unwrapped channel) was divided in n z cells of size zn z With this configuration there will be six different boundaries: the boundaries with the inner and the outer cylinder ( r = Rin and r = Rout, diffuse reflection), the boundaries with the upper and lower ramp of the helix ( z = z w,1 and z = z w,, diffuse reflection), the boundary between the baffle and the chamber ( θ = θ1,considered as open boundary at the pressure and temperature of the reservoir) and the boundary at the end of the helicoidal system ( θ = θ, the reflection at this boundary is considered to be diffuse when the thermal transpiration is studied, but it can be turned to open boundary when the flow rate due to a pressure gradient is to be calculated). 98

100 Manuel Vargas Hernando The z-coordinate of the upper and lower ramp of the helix will vary with the θ-coordinate of the particle, so the collision of the particles with the lower boundary is computed if z + ε z < z1 + θ and the collisions with the upper boundary will be computed when π z + ε z > z + θ. For calculating the time necessary for the particle to hit the upper and π lower walls, the bisection method is used. Furthermore, the slope of these two boundaries depends on the radial coordinate of the particle at the moment of hitting the wall. Due to this, the slope must be calculated for every collision of the particles with these boundaries and the velocity components of the reflected molecules must be transformed according to the local slope Problem formulation and numerical simulation The traditional Direct Simulation Monte Carlo (DSMC) method proposed by Bird [1] is used for simulation of the gas flow in gauge system. This method combines deterministic aspects for modeling particle motions and statistical aspects for computing collisions between particles. The collision-sample technique that we use is the No Time Counter scheme suggested by Bird, except for a slight modification in the calculation of maximum collision number in a cell, as described by Stefanov et al. [87]. The simulation is carried out in threedimensional computational domain defined by the geometry of the baffle system. Figure 45: Scheme of the investigated baffle system. 99

101 Modeling and simulation of heat transfer and thermal phenomena in micro gas flows The geometry of the studied baffle system is a double helix as it is shown in Figure 45. The inner radius of the helix is R = 1.7 mm, its outer radius R in out = 4.8 mm, the vertical distance between the ramps is z = 4. mm and the thickness ε = 1. mm. The helix contains two channels with rectangular cross section (width 3.1 mm and height 4. mm) and, as each channel corresponds to 1.5 turns, the total length of the channel is 34.1 mm at its middle point. Only one of the two channels confined within the helix is taken into consideration in the numerical calculations under the assumption that the behavior of the gas in both channels is the same because of the system symmetry. The calculations are realized in cylindrical coordinates, by using the procedure for axially symmetric flows, and the 3D-space domain is then divided into uniform cells in the radial direction r, in the axial direction z and in the polar angular direction θ. The cell size is always kept smaller than the mean free path, the time step smaller than the cell traversal time and the average number of particles in cells was never smaller than 15. The most demanding calculation which has been carried out employed 5x3x5 basic cells with approximately simulators. The two ramps which form z + ε the lower and upper boundaries of the channel are situated at z 1 = θ and π z + ε z = z+ θ. The boundary conditions are chosen to be purely diffusive at the π temperature of the wall and the hard sphere model is used to compute the interaction between molecules. The simulations are carried out in a dimensionless manner and the results expressed in non-dimensional form can be applied for any gas which is simulated as a hard sphere gas. The reference quantities are taken to be n, T and P = nkt B, which define the number density, temperature and pressure at the inlet and are kept constant during the calculations. Moreover, we define the reference Knudsen number as: Kn = R out λ R in (147) For the dimensional results, the gas used for the calculations was nitrogen, with molecular 1 6 diameter and molecular mass d = m and m = kg respectively. The slope of the upper and lower boundaries must be taken into account when computing the reflection of the particles, due to the fact that the slope of the ramp varies depending on the radial coordinate, being larger next to the inner cylinder and smaller near the outer cylinder. 1

102 Manuel Vargas Hernando z + ε This slope ( m ) can be determined by m = arctan as a function of the radial π r coordinate for each molecules hitting any of those two boundaries. Besides, it is assumed that any change in the vacuum chamber has an immediate effect at the inlet of the baffle system, as the inlet is defined as an open boundary where particles crossing it are automatically removed, while, at the same time, particles are produced at a given rate and with a certain molecular velocity depending on the pressure and temperature defined at that boundary. For this reason, all mentions to inlet properties in this work are equivalent to the properties at the vacuum chamber in the reality. Part of the following results have been presented in [A7] Steady state calculations of thermal transpiration in CDGs In this section the effect of the thermal transpiration on the pressure distribution along the channels at equilibrium is investigated in the whole range of rarefaction. The vacuum chamber is maintained at T = 3ºC, while the diaphragm is kept at a higher temperature of T diaph = 8ºC. This temperature difference between the inlet and the diaphragm is large enough to reduce the statistical fluctuations and hence the sample size of the calculations. The temperature is considered to vary linearly along the z-axis in a certain region of the helix, as defined in Figure 46. Figure 46: Temperature distribution within the helix. The calculations have been performed for different values of the pressure at the inlet of the helix. In Figure 47 the pressure distribution inside the helix is shown for a pressure in the 11

103 Modeling and simulation of heat transfer and thermal phenomena in micro gas flows vacuum chamber of P 1 = Pa. Under these conditions the Knudsen number takes a value of Kn, which corresponds practically to free molecular regime. As it can be seen, a pressure variation along the helix appears in the steady state as a consequence of the temperature difference between the diaphragm and the vacuum chamber. Figure 47: Pressure distribution in the helicoidal baffle system for T diaph = 8ºC, T = 3ºC and P 1 = Pa. In order to obtain a better quantitative representation of the results, the distributions of the Rout + Rin macroscopic properties along the central line of the channel, at r = and z z+ ε z = + θ, are plotted in Figure 48 for different values of the reference pressure at the π inlet. 1

104 Manuel Vargas Hernando Figure 48: Density, temperature and pressure distribution in the central line of the channel defined by the helicoidal baffle system for T diaph = 8ºC, T = 3ºC and different reference pressures at the inlet. The effect of the thermal transpiration can be clearly seen. The ratio between the pressure measured at the diaphragm at equilibrium and the pressure at the vacuum chamber is increased as the reference pressure in the vacuum chamber is decreased (the Kn is increased). It can be seen that for Kn = 8, which defines practically free molecular conditions, the numerical results match the theoretical value of this pressure ratio due to thermal transpiration effect in the free molecular limit. P diaph P = = (148) This value tends to unity when the conditions are close to continuum regime. As for the density distributions, its value at the hotter region is smaller than at the inlet region in the continuum limit, while for rarefied conditions this difference between the density at the diaphragm and at the inlet decreases significantly and the density at the hotter region gets closer to the reference inlet pressure. In Figure 49 the ratio between the pressure measured at the diaphragm at equilibrium and the pressure at the vacuum chamber is represented as a function of the reference pressure. The results obtained for the pressure in the helicoidal baffle system are compared with those obtained when the baffle system is absent. In this case, the geometrical configuration is the cylindrical tube in which the helix is inserted. Then, its radius is R out = 4.8 mm and its length z + ε is L= z+ θ, which for this case takes value of L = 19 mm. The temperature is π considered again to vary linearly along the z-axis in a certain region of the helix, as shown in Figure 46. As it could be expected for the same reference pressure the thermal transpiration 13

105 Modeling and simulation of heat transfer and thermal phenomena in micro gas flows effect through the helicoidal baffle system is larger than through the tube without baffle system. This is because the characteristic length of the system without helix (the diameter of the tube) is larger than the characteristic length of the helix (width or height of the channel confined by the helix). Under the same pressure conditions, the Knudsen number is larger for the system with helicoidal baffle than without it, and then the rarefaction is larger, as well as the thermal transpiration effect. Additional reasons for the observed difference between the results obtained for both systems might be sought for in the different geometry, especially those reasons concerning the curvature effects and the difference in the length of the two channels. Figure 49: Variation of the ratio between the pressure at the diaphragm and at the vacuum chamber as a function of the reference pressure obtained with helicoidal baffle system (blue circles) and without baffle system (red triangles) for T diaph = 8ºC, T = 3ºC. The results obtained by using DSMC are compared with those obtained by using the Takaishi-Sensui (TS) formula [115] for both geometries investigated in this work. ( Pdiaph P ) 1 = 1 T T AX + BX+ C X+ diaph * * * 1 1 (149) 14

106 Manuel Vargas Hernando where X = PdiaphD T, D is the diameter of the tube or channel and T is the mean temperature along the channel. For the cylindrical geometry without baffle system the diameter of the channel (9.6 mm) was used for the comparison, while for the helicoidal geometry the hydraulic diameter of the channel (3.49 mm) was chosen. The empirical * * * constants ( A, B and C ) for nitrogen are taken from Table 1 of [115]. The two curves are shown in Figure 49. Besides, as the cylindrical tube under investigation is short (aspect ratio LD= ), the TS curve with the modification suggested in [119] is included. For this aspect ratio, the corresponding correction factor is α =.851. It can be seen from Figure 49 that the agreement between the results is better when this correction factor is included, especially in the transition regime. Although near the free molecular regime the DSMC results deviate slightly from the TS curves, in general the agreement between the results is good. In the following part of this section, the effect of the temperature distribution at the walls is investigated. Three different temperature configurations have been taken into account, as shown in Figure 5. Firstly, linear variation of the temperature in the whole length of the baffle system, then, linear variation of the temperature at a certain region of the helix and, finally, a stepwise variation in the centre of the system. Figure 5: Three different temperature distributions under investigation, linear in the whole length, linear at certain region and stepwise. 15

107 Modeling and simulation of heat transfer and thermal phenomena in micro gas flows The calculations are performed for conditions near free molecular regime, as under these conditions the effect of the thermal transpiration is more evident. In Table 1 the pressure at the diaphragm normalized by the pressure at the vacuum chamber is indicated for the three different temperature profiles shown in Figure 5 and for four different values of the channel length. These values were obtained by averaging the last 9 data obtained by time averaging over 1 5 time steps, after a time sufficiently larger than the transient time, in order to avoid the statistical fluctuations. As it can be seen from the results, the pressure at the diaphragm has a small dependence on temperature distribution. The larger difference is observed between the linear and the stepwise configuration for shorter channels. This difference becomes smaller as the length of channel is increased. These results are in agreement with those showed in [1] for a long tube and in [19] for a short channel and small temperature gradient. The observed disagreements with those results is more evident for short tubes and could be explained by the fact that the conditions for applying linear theory for short tube are not fulfilled. Table 1: Ratio between the pressure measured at the diaphragm and the pressure at the vacuum chamber for various temperature distributions for initial conditions of the gas P 1 = Pa and T = 3ºC. Number of turns Linear 1% Linear 4% Stepwise In the following part of this section, the effect of the accommodation coefficient (α ) on the pressure difference created by thermal transpiration effects is investigated. The speculardiffuse reflection model is used for this purpose, where α defines the fraction of particles reflected diffusively and ( 1 α ) is the fraction of particles which reflect specularly. The results for various values of the accommodation coefficient and different rarefaction conditions are shown in Table 11. It can be deduced from the results that, in the free molecular regime, the value of the accommodation coefficient in the range analyzed here has no effect on the induced pressure gradient. However, in the transition and slip flow regimes, 16

108 Manuel Vargas Hernando the induced pressure gradient depends slightly, but clearly, on the values of the accommodation coefficient included in this work. As shown in Table 11, the pressure gradient for the same rarefaction conditions becomes larger as the accommodation coefficient is increased. It is expected that in the continuum limit the accommodation coefficient has again no effect on the pressure, because the thermal transpiration effect is negligible under those conditions Table 11: Pressure measured at the diaphragm for various accommodation coefficients and rarefaction conditions. α P =.1 Pa P =.1 Pa P = 1. Pa P = 1 Pa (Kn =8) (Kn =.8) (Kn =.8) (Kn =.8) Flow dynamics of the system due to sudden changes in the process parameters In this section, the dynamic response of the flow in the helicoidal baffle system is considered and compared to the response of the flow in the cylindrical tube without baffle system, in the case of a sudden change in one of the following process parameters: The temperature of the sensor from initial isothermal conditions. The pressure at the inlet keeping isothermal conditions Transient flow due to a sudden change in the temperature of the sensor The aim of this analysis is to investigate the dynamic response observed in the profiles of the macroscopic properties along the helix (see Figure 45) caused by a sudden change in the sensor temperature from initial isothermal conditions, to evaluate the response time of the system until the steady state is achieved and to compare the results with those obtained for the cylindrical tube without helicoidal baffle system. The results presented in this section correspond to the conditions of some of the cases already studied in the previous section. It is assumed that the gas initially is in equilibrium at a given temperature ( T = 3ºC) and pressure ( P = 1 Pa). At time t = a sudden change in the temperature of the sensor (initially at T diaph = 3 ºC) occurs, establishing a new temperature T diaph = 8 ºC at t >. 17

109 Modeling and simulation of heat transfer and thermal phenomena in micro gas flows Figure 51: (Left) Pressure distributions in the central line of the channel defined by the helix at the indicated time instants due to a sudden change of T diaph from 3ºC to 8ºC for initial conditions of the gas P 1 = Pa and T = 3ºC. The black solid line indicates the steady state solution. (Right) Evolution of the pressure in the diaphragm for the system without helicoidal baffle system. The black dotted line indicates the final value of the pressure at the diaphragm in the steady state. In Figure 51 (left) the evolution of the pressure in the central line of the channel defined by the helix at the indicated time instants is represented. As can be seen, the system shows a fast overshoot at initial times (from to.3 ms). Then the pressure decreases slowly until it reaches the steady state value at times larger than 1.5 ms. In Figure 51 (right) the evolution of the pressure in the diaphragm is shown for the alternative geometry under study, cylindrical without baffle system. Here the magnitude of the overshoot is smaller and the response is faster for the system without helicoidal baffle system. It is interesting to note that the temperature of the gas (not shown here) recovers its steady value after only 3 µs, the time when the maximum overshoot in the pressure is reached Sudden change in the pressure at the inlet The aim of this section is to investigate the evolution of the macroscopic properties within the helix for different values of the pressure jump in the whole range of rarefaction, as well as to evaluate the response time of the system to reach the steady state. 18

110 Manuel Vargas Hernando It is assumed that the system initially is in equilibrium at a given temperature (3ºC) and pressure (which depends on the case). At time t =, a jump in the pressure P inlet P = at the P inlet is produced. The geometrical configuration is the same as that described in Figure 45. Similarly to the steady state calculations the cell size and the time step here are always kept smaller than the mean free path and the cell traversal time respectively. However, this is not the case for the number of particles in cells, which varies throughout the simulations. The maximum number of particles in cells occurs in the final state of the calculations, where the final pressure in the whole domain is reached. This maximum number of particles in cells coincides in all the simulations and was chosen to be 1. Hence the initial number of particles in cell depends on the pressure ratio which is simulated. It turned out that even for free molecular flow initial conditions for the cases with large pressure ratio, the insufficient initial number of particles per cell has no effect on the results. To come to this conclusion test calculations with different number of simulators have been carried out and no differences in the behavior of the system were observed. Figure 5: Pressure distribution at the middle plane in the vertical direction at three different time instants: after 7.7 µs (left), 4.6 µs (centre) and 154 µs (right) for initial conditions P 1 1 = Pa (.8 Kn = ), T = 3ºC and a pressure jump P = 1. 19