Oscillatory heating in a microchannel at arbitrary oscillation frequency in the whole range of the Knudsen number

Transcription

1 Journal of Phsics: Conference Series Oscillator heating in a microchannel at arbitrar oscillation freuenc in the whole range of the Knudsen number To cite this article: O Buchina and D Valougeorgis J. Phs.: Conf. Ser Related content - Rarefied gas mixture flow between plates of arbitrar length due to small pressure difference C Tantos, S Naris and D Valougeorgis - On the role of surface shape in a microscale heat conduction problem A Dinler, I A Graur, R W Barber et al. - Design and optimization of a Holweck pump via linear kinetic theor Sterios Naris, Eirini Koutandou and Dimitris Valougeorgis View the article online for updates and enhancements. This content was downloaded from IP address on /7/8 at 6:

2 st European Conference on Gas Micro Flows (GasMems ) Journal of Phsics: Conference Series 36 () 5 doi:.88/ /36//5 Oscillator heating in a microchannel at arbitrar oscillation freuenc in the whole range of the Knudsen number O Buchina, D Valougeorgis Department of Mechanical Engineering, Universit of Thessal, Volos 38334, Greece diva@mie.uth.gr, buchina@mie.uth.gr Abstract. The oscillator heating in a microchannel is investigated on the basis of linear kinetic theor. In particular, the linearized unstead Shakhov kinetic model, subject to Maxwell boundar conditions is numericall solved in a full deterministic manner based on finite differencing scheme in the phsical space and in the discrete velocit method in the molecular velocit space. The solution of the problem is determined b two parameters: the Knudsen number and the ratio of the intermolecular collision freuenc over the oscillation freuenc. The numerical calculations are carried out for a wide range of both parameters and results are presented for the amplitude and the phase of all macroscopic uantities of phsical interest.. Introduction Recent interest in investigating unstead heating processes in rarefied gases is motivated b their applications in several fields including micro-electromechanical sstems, microelectronics and laser industr. Time dependent heat transfer configurations are common in gaseous micro devices and ma be produced b time dependent boundar cooling or/and heating. A transient flow of a gas caused b a sudden change in the boundar temperature, which is the counterpart of the periodic time-dependent problem, is a tpe of basic problem of a rarefied gas flow []. Recentl in [] the transient heat transfer in a gas confined in a small-scale slab due to the instantaneous change of a wall temperature has been investigated. In this work semi-analtical approaches have been applied in the free molecular and hdrodnamic limits, while the DSMC method has been used in the transition regime. The correct description of heat flow through a rarefied gas for arbitrar rarefaction (the ratio of a characteristic length of the flow to the molecular mean free path, i.e. it is inversel proportional to the Knudsen number) and oscillation speed (the ratio of the collision freuenc to the freuenc of the temperature on the plate) is ver important in man field including micro-electromechanical sstems, microelectronics and laser industr. Current interest in analzing unstead heating processes is motivated b the common occurrence of time-varing boundar temperatures in a wide scope of microelectromechanical and nanoelectromechanical applications, ranging from microprocessor chip heating [3] to ultrafast temperature variations encounted in the laser industr [4-5]. The oscillator heating in a microchannel has been recentl investigated b the low-variance deviational simulation Monte Carlo method motion for an arbitrar time variation of the boundar temperature [6-7]. Periodic time-dependent behaviour of a rarefied gas between two parallel planes caused b an Published under licence b Ltd

3 st European Conference on Gas Micro Flows (GasMems ) Journal of Phsics: Conference Series 36 () 5 doi:.88/ /36//5 oscillator heating of one plane has been numericall studied based on the linearized Boltzmann euation for a hard-sphere molecular gas [8]. In the present stud, an analsis of the oscillator heating of a rarefied gas confined between two parallel plates is based on linear kinetic theor. The implementation of a kinetic solution provides reliable results in the whole range of the Knudsen number with modest computational effort. In particular, the time dependent heat transfer is modelled b the linearized nonstationar kinetic euation, subject to Maxwell purel diffuse boundar conditions. The Shakhov model of the Boltzmann euation is chosen as the most appropriate one because it provides the correct Prandtl number so that both viscosit and thermal conductivit are taken into account in the solution of the kinetic euation. The solution of the problem is determined b two parameters: the Knudsen number and the ratio of the intermolecular collision freuenc over the oscillation freuenc. It is assumed that the oscillation is full established and the dependence of the solution on the time is harmonic, while its dependence on the spatial coordinate will be obtained numericall. The numerical solution of the problem is full deterministic and based on finite differencing scheme in the phsical space and in the discrete velocit method in the molecular velocit space. The aim of the present work is to calculate the heat flow induced b oscillator heating of one of the parallel plates over a wide range of both parameters. Results will be presented for the amplitude and the phase of all macroscopic uantities of phsical interest.. Statement of the problem A monoatomic gas is considered to be confined between two infinite parallel plates located at and L. The plate placed at is heated with the wall temperature oscillating harmonicall, while the temperature of the second plate is kept constant and eual to the ambient temperature T. It is assumed the oscillation to be full established through the heat flow and the solution is harmonic with respect to time. The oscillating temperature can be presented as T t T Re exp( it ) T (.) w w where is the freuenc, t is the time, Re denotes the real part of a complex expression and Tw is the amplitude of the oscillating temperature. It is assumed that Tw T. The temperature oscillation n t,, bulk on the plate causes a gas flow in the direction characterized b a number densit velocit U t,, temperature T t, and heat flux t, harmonicall as which depend on the time n( t, ) Re n( )exp( i t) n, (.) U( t, ) Re U( )exp( it ), T( t, ) Re T( )exp( it ) T, Q( t, ) Re Q( )exp( it). Here, n ( ), U( ), T( ) and Q ( ) are complex functions and n is the euilibrium number densit. The solution of the problem is determined b two parameters. The first one is the gas rarefaction defined as the ratio of a characteristic length of the flow to the molecular mean free path, i.e. it is inversel proportional to the Knudsen number: PL, (.6) Kn where P nkbt is the gas pressure, is the viscosit of the gas at T and RT is the most probable molecular velocit, with R denoting the gas constant. The second parameter characterizes the (.3) (.4) (.5)

4 st European Conference on Gas Micro Flows (GasMems ) Journal of Phsics: Conference Series 36 () 5 doi:.88/ /36//5 oscillation speed and is eual to the ratio of the intermolecular collision freuenc over the oscillation freuenc. (.7) Regarding the values of the parameters and, two limit regimes can be distinguished. Large values of both parameters, i.e., and, correspond to the hdrodnamic regime. Then, the problem can be solved on the basis of the continuum mechanics euations. When both parameters are ver small, i.e., and, the intermolecular collisions can be neglected and the kinetic euation is significantl simplified. This regime is called free molecular. For arbitrar values of the both parameters the problem must be solved on the basis of the nonstationar Boltzmann euation. Now it is convenient to introduce the dimensionless coordinate, distance and time, L L, t t. (.8) The macroscopic uantities (.)-(.5) are also written in the dimensionless form as n n T U T, u, (.9) n T T P w T T Q T,. (.) Tw P Tw Since these uantities are complex, the can be written as exp i, u u exp i u, (.) exp i, exp i, (.) where, u, and are amplitudes of the macroscopic uantities, while, u, and are their phases. We are going to calculate the functions values of the parameters and. w and i, i, u,, for various 3. Kinetic euation In order to consider arbitrar values of the both parameters the problem must be solved on the basis of the nonstationar Boltzmann euation. B considering a limited temperature difference between the plates according to Tw T, the heat flow is modelled b the unstead linearized Shakhov model euation [9-]. B assuming an oscillator behaviour of the distribution function and following a tpical procedure [] two coupled integrodifferential euations describing the problem are: 4 3 i c c u c c c, 5 (3.) 4 i c c 5. (3.) Here,, and, are the reduced complex distributions functions, with c, denoting the -component of the molecular velocit, while the macroscopic moments at the right side of euations (3.)-(3.) are given b c i e dc, (3.3) c u c e dc, (3.4) 3

5 st European Conference on Gas Micro Flows (GasMems ) Journal of Phsics: Conference Series 36 () 5 doi:.88/ /36//5 c e dc 3 c, (3.5) 3 c c ce dc. (3.6) The interaction between the particles and the walls is modelled according to Maxwell purel diffuse boundar conditions, which are written for the outgoing distributions at the walls as c v at and c, c, (3.7) vl at L and c, at and c, c, (3.8) at L and c, where the constants v and v L are 4. Numerical scheme c, (3.9) v c, c e dc c L, v c L c e dc. (3.) To reduce the number of the velocit nodes the complex perturbation functions c, c, are split into two parts as follows, c, c, c, c, c, c where the functions c, and c, satisfies the following euations and, (4.), (4.) i c, (4.3) i c, (4.4) with the following boundar conditions c v at and c, c, v L at L and c, (4.5) at and c, c, at L and c. (4.6) Euations (4.3)-(4.4) have the following analtical solutions c v exp i c, c, c, v L exp i L c, c, (4.7) exp i c, c, c,, c. (4.8) The constants v and v L are obtained from euations (3.9)-(3.) and (4.5)-(4.6) as 4

6 st European Conference on Gas Micro Flows (GasMems ) Journal of Phsics: Conference Series 36 () 5 doi:.88/ /36//5 where the special functions 3 4I i L 4I i L I i L I i L v 4I i L, (4.9) In I3 i L I i L v L x [] are defined as The kinetic euations for the functions c, and c,, (4.) n x In x c exp c dc c. (4.) are 4 3 i c c u c c c, 5 (4.) 4 i c c 5, (4.3) with the boundar conditions c c, c e dc at and c, c, c c, at and, L c e dc L c (4.4) at and c, c, at L and c. (4.5) The macroscopic moments (3.3)-(3.6) are also decomposed into two parts as, (4.6) where the terms u u u, u, and, (4.7), (4.8), (4.9) are obtained b substituting the solutions (4.7)- (4.8) into (3.3)-(3.6) to ield v LI il I i v I i (4.) u v LI il I3 i v I i (4.) 4 v L I i L I i L v I i I i, (4.) 3 I i I i I i 5

7 st European Conference on Gas Micro Flows (GasMems ) Journal of Phsics: Conference Series 36 () 5 doi:.88/ /36//5 3 v L I il I i L 3 v I i I i. (4.3) 3 I4 i I i I i Finall, the uantities u,, and in euations (4.6)-(4.9) are calculated b euations (3.3)-(3.6) using and instead of and, respectivel. The governing euations (4.) and (4.3) subject to the boundar conditions ( ) are solved numericall in the whole range of the rarefaction δ and oscillation speed θ parameters. The computational scheme is full deterministic and all spaces are accordingl discretized. In particular, the discretization in the molecular velocit space is performed b using the discrete velocit method. The continuum spectrum is substituted b a discrete set of velocities m, m,,... M, which are taken to be the roots of the Legendre polnomial, with M denoting the degree of the polnomial, accordingl mapped into the interval of interest. The phsical space is divided into I segments and it is consisting of i,,... I nodes, while the discretization is performed b a second order central difference scheme. The discretized set of euations (4.-4.3) is solved for each molecular velocit marching through the phsical space, while the macroscopic uantities defined b euations ( ) are estimated b a Gauss-Legendre uadrature. Based on the above the results presented in the next section have been 5 obtained with M and I. 5. Results The amplitudes and phases of the normal heat flux at the surfaces, i.e., at = and L = δ/θ, are given in Tables,. From these data it is seen that at δ = the value of remains as expected constant. For δ > at an fixed value of the rarefaction parameter the heat flux amplitude at the oscillator heated plate decreases b increasing the oscillation speed parameter θ, while L increases. Also, this behaviour can be observed in Figure, where the graph of the heat flux amplitude vs. is shown. Table. Amplitudes of the heat flux vs. δ and θ at = and = L. α δ θ =. θ = θ = θ = = = L = = L = = L kin.e

8 st European Conference on Gas Micro Flows (GasMems ) Journal of Phsics: Conference Series 36 () 5 doi:.88/ /36//5 For a fixed value of oscillation parameter θ the amplitude has a nonmonotonical dependence on the rarefaction parameter δ and it is seems that there is a minimum at the transition regime in the interval.θ < δ < θ. Then, in ever case of tends to and the amplitude finall reaches a constant value. Next, it is interesting to note that in the low speed oscillation regime, i.e. at large values of θ, and in highl rarefied atmosphere (,. in Table ) due to the small number of collisions there is a ver small variation of the heat flux between the plates. For comparison purposes, in the last column of Table the values of the heat flux are presented for the case of stationar heat flow between two plates (θ = ). Table. Phase of the heat flux vs. δ and θ at = and = L. φ δ θ =. θ = θ = = = L = = L = = L In Table at an value of δ the phase on the oscillator heated plate is alwas negative and tends to zero in the free-molecular regime ( ). Also, the phase of the heat flux similar to its amplitude has a nonmonotonical behaviour with respect to δ and there is a maximum approximatel in the interval. L increases b increasing the rarefaction parameter δ. The phase when the oscillation speed parameter θ is fixed. Some of the results presented in Table have been compared with the tabulated results of Doi [8] and ver good agreement has been found for the amplitude of the heat flux for the specific cases of (δ,θ) = (.,.), (,) and (,). As explained in Section in Figures 4 the amplitudes i and phases i, i,, u, of the heat flux, temperature, bulk velocit and number densit are shown. Three values of the rarefaction parameters δ =.,, and and three values of the oscillation parameters θ =.,, and and phases are considered. One can see that for δ =. the dependence of the amplitudes i on the dimensionless coordinate is ualitativel the same for all values of θ. The var just i uantitativel b increasing the oscillation parameter θ. However, for δ = and the behaviours of change both ualitativel and uantitativel b increasing the oscillation parameter and i i θ. Also, as it was shown above at δ =. and θ = the heat flow reaches nearl the nonstationar heat flow with a zero bulk velocit and constant heat flux. Figure shows the heat flux amplitudes. Due to the oscillator heating and phases of the plate at =, an oscillating heat flow propagates towards the second plate. For ever δ the number of collisions results in the decrease of the amplitudes and increase of phases with respect to the distance towards the nonheated plate located at = L. Since L = δ/θ then, for instance, at δ =. (Figure a) and θ = the distance L =. is the shortest, therefore at the relativel high rarefaction the oscillations of the heat flux have almost no dela in propagating, hence the constant amplitude and 7

9 st European Conference on Gas Micro Flows (GasMems ) Journal of Phsics: Conference Series 36 () 5 doi:.88/ /36//5 phase are observed. In general the low speed oscillation regime, i.e., provides less increase of the phase and decrease of the amplitude at smaller rarefaction parameter δ, and the heat flux as well as the other macroscopic parameters tends to ones corresponding to the stationar heat flow. So, at δ = (Figure b) and θ = the heat flux is also constant. The phases of the heat flux var significantl for the small value of the parameter θ, i.e. when the temperature oscillation freuenc ω is large. This variation of the phases becomes larger as the rarefaction δ is increased (Figure c: θ = ), providing ver fast deca of the amplitude. It is noted that due to the dependenc of the macroscopic parameters on the distance L = δ/θ the profiles of the amplitudes and phases of all macroscopic parameters are uantitativel the same, when values of δ and θ give the same value of L (e.g. Figure b: θ =. and Figure c: θ = ). The temperature amplitudes and phases are given in Figure as functions of the dimensionless coordinate. The behaviour of the temperature is similar to that of the heat flux, but as opposed to the negative values of the heat flux phases at the plate =, the is positive. The negative values of indicate that the plate. In Figure 3 the bulk velocit amplitudes is slower relativel to the phase of the temperature of and phases u are shown. As it was expected for the large value of θ, where the heat flow tends to the stationar behaviour, the phases of the bulk velocit var weakl. At θ = the are nearl constant, while the amplitude is eual to zero, what is practicall corresponds to the stationar heat flow between plates. Compared to the phases of the heat fluxes and temperature at θ =, the phase of the bulk velocit is much more far from zero. The behaviour of the densit distribution (Figure 4) is uite different from those for the bulk velocit, temperature and the heat flux. The densit waves propagate and reflect off the other plate at = L, and forward and backward waves are superimposed. It also causes and defines the bulk velocit distribution profiles. It is noted that the distribution of the number densit at an moment obes the mass conservation law, which implies that the integral of the perturbed densit between the plates is zero. 6. Conclusion The oscillator heat flow of a rarefied gas contained between two parallel plates has been investigated on the basis of the linearized unstead Shakhov kinetic model, subject to Maxwell diffuse boundar conditions. The kinetic euation has been solved numericall in a full deterministic manner based on finite differencing scheme in the phsical space and on the discrete velocit method in the molecular velocit space. Numerical calculations have been carried out for a wide range of the rarefaction parameter and for tpical small, moderate and large values of the oscillation speed. Results are presented for the amplitude and the phase of all macroscopic uantities of phsical interest in graphical and tabulated forms. Both the amplitude and the phase of all macroscopic uantities strongl depend on the two main parameters (δ, θ) and their dependenc is further increased as the gas rarefaction is decreased and as the oscillation speed is increased. When the oscillation speed parameter tends to infinit the flow field correspond to that for the stead flow. Tabulated results for the amplitude of the heat flux has been compared with the results available in the literature and ver good agreement has been found. Acknowledgments The research leading to these results has received funding from the European Communit's Seventh Framework Programme (ITN - FP7/7-3) under grant agreement n 554. u 8

10 st European Conference on Gas Micro Flows (GasMems ) Journal of Phsics: Conference Series 36 () 5 doi:.88/ /36//5 (a) =. = =.6 α =. = =.8.55 φ (b) =. = =.5 =. = = 4.4 α. φ.3.. (c) =. = =.6. =. = = α φ Figure. Heat flux amplitudes and phases at (a) δ =., (b), (c) and θ =.,,. 9

11 st European Conference on Gas Micro Flows (GasMems ) Journal of Phsics: Conference Series 36 () 5 doi:.88/ /36//5 (a).4 =. = =.6 =. = =. ατ φτ (b) =. = = =. = = ατ φτ (c) =. = =.6..4 =. = = 3.4 ατ φτ Figure. Temperature amplitudes and phases at (a) δ =., (b), (c) and θ =.,,.

12 st European Conference on Gas Micro Flows (GasMems ) Journal of Phsics: Conference Series 36 () 5 doi:.88/ /36//5 (a).8 αu =. = =.6 φu.4 (b) =. = = =. = = =. = = 6. 4 αu φu.5 (c) =. = =.. =. = = 3.8 αu. φu Figure 3. Velocit amplitudes and phases at (a) δ =., (b), (c) and θ =.,,.

13 st European Conference on Gas Micro Flows (GasMems ) Journal of Phsics: Conference Series 36 () 5 doi:.88/ /36//5 (a).3 =. = =.5 αd =. = = 3 φd (b)..4.6 =. = =.3 αd.8. 8 φd =. = = (c) =. = = =. = = 3.4 αd.3 φd Figure 4. Number densit amplitudes and phases at (a) δ =., (b), (c) and θ =.,,.

14 st European Conference on Gas Micro Flows (GasMems ) Journal of Phsics: Conference Series 36 () 5 doi:.88/ /36//5 References [] Sone Y 965 Effect of sudden change of wall temperature in a rarefied gas J. Phs. Soc. Jpn. [] Manela A Hadjiconstantinou N G 7 On the motion induced in a small-scale gap due to instantaneous boundar heating J. Fluid Mech [3] Ho C M Tai Y C 998 Micro-electro-mechanical sstems and fluid flows Annu. Rev. Fluid Mech [4] Tzou D Y 997 Macro-to-Microscale Heat Transfer (Talor and Francis Washington DC) [5] Tzou D Y Beraun J E Chen J K Ultrafast deformation in femtosecond laser heating ASME J. Heat Transfer 4 84 [6] Manela A Hadjiconstantinou N G 8 Gas motion induced b unstead boundar heating in a small-scale slab, Phs. Fluids 74 [7] Manela A Hadjiconstantinou N G Gas flow animation b unstead heating in a microchannel, Phs. Fluids 6 [8] Doi T Numerical analsis of the time-dependent energ and momentum transfers in a rarefied gas between two parallel planes based on the linearized Boltzmann euation J. Heat Transfer [9] Shakhov E M 968 Approximate kinetic euations in rarefied gas theor Fluid Dn [] Shakhov E M 968 Generalization of the Krook kinetic relaxation euation Fluid Dn [] Sharipov F Kalempa D 9 Sound propagation through a gas confined between source and receptor at arbitrar Knudsen number and sound freuenc Phs. Fluids [] Abramowitz M Stegun I 965 Handbook of Mathematical Functions (Dover New York) p [3] Buchina O Vargas M Stefanov S Valougeorgis D Transient micro heat transfer in a gas confined between parallel plates due to a sudden increase of the wall temperature 3rd Micro and Nano Flows Conference Thessaloniki Greece August -4 3