Analysis of Shock-plugs in Quasi-One-Dimensional Compressible Flow

Transcription

1 Proceedings of the World Congress on Mechanical, Chemical, and terial Engineering (MCM 015) Barcelona, Sain Jul 0-1, 015 Paer No. 38 Analsis of Shock-lugs in Quasi-One-Dimensional Comressible Flow Thomas Adams, Eric Chang, Benjamin Stevens, tthew Thomson Rose-Hulman Institute of Technolog 5500 Wabash Ave, Terre Haute, IN 47803, USA Abstract -With the continuing miniaturization of technolog haening a raid ace, more and more comressible flows are occurring at the microscale. At the small length scales encountered in such microflows, the standard assumtion of s having infinitesimal thickness ma no longer al. In this aer we therefore suggest the eistence of shock-lugs, standing normal shocks with finite thickness. We model shock-lugs using the same methods and assumtions as seen in standard normal analsis. The inclusion of shock thickness necessitates the inclusion of additional arameters in the analsis, however, namel differing ustream and downstream cross sectional areas, as well the ressure on the sidewalls adjacent to the shock. Predictions for changes in ch number, temerature, ressure, and entro are resented, all of which show deviation from conventional analsis. The models resented here ma rovide better estimates of shock roerties in microscale alications. Kewords: shock-lug,, comressible flow, microflows, microfluidics 1. Introduction In the quasi-one-dimensional flow of a comressible fluid in a converging/diverging nozzle, the fluid can be accelerated from subsonic flow in the converging section to suersonic flow in the diverging section if a suitabl low back ressure eists. For isentroic flow, one unique value of back ressure will roduce a constantl accelerated flow throughout the channel such that the value of back ressure eactl matches the ressure at the eit as required b isentroic conditions. For back ressures larger than this, however, isentroic flow cannot be maintained throughout, and a normal forms in the diverging section. (Fig. 1.) The ustream and downstream isentroic flows are effectivel ieced together via the. (Moran and Shairo, 008) Modelling of s is well develoed and relies on the alication of continuit, linear momentum conservation, and energ conservation across an infinitesimall thin control volume. The resence of the raidl decelerates the flow from suersonic to subsonic conditions, causing a discontinuous increase in ressure accomanied b a large generation of entro. Good treatments can be found in Anderson (1990) and NACA (1953). The simle relations that result from the standard analsis of normal s hinge on the assumtion of their infinitesimal size. Anderson (1990) notes that the actual size of shocks is on the order of several mean free aths of the fluid molecules, making the negligible thickness assumtion a more than lausible assumtion in the vast majorit of alications. Granger (1995) gives a quantitative analsis on the size of s, the basis for which is the balance between viscous shear stress fluid deceleration. The resulting thickness given b t shock, (1) C 38-1

2 0 T 0 V 0 back ressure B z subsonic flow z suersonic flow Fig. 1. A converging/diverging nozzle. For sufficientl low back ressures, suersonic conditions can eist in the diverging ortion. For a certain range of back ressures, s form, across which the flow decelerates from suersonic to subsonic seeds. where ν is kinematic viscosit, and C and are the seed of sound and ch number ustream of the shock, resectivel. For air at ambient conditions, this gives values of t shock on the order of 30 nm, well within reason for assuming zero thickness. Comressible flow in microfluidic alications, such as shock tubes within microchannels (Iancu and Müller, 005; Zeitoun et al, 009; Dodulad et al, 010) and the design of micronozzles (Louisos et al, 008), however, involve length scales in which thickness ma otentiall be of interest. Secificall, in the case a converging/diverging micronozzle, a finite thickness ma result in non-negligible differences in cross sectional flow areas ustream and downstream of the shock itself, a situation in which standard relations would no longer al. To the authors knowledge no analsis currentl eists in which shock relations take shock thickness into account. A first order analsis of such hothesized finite-sized standing shocks, hereafter referred to as shock-lugs, forms the basis of this aer.. Model Derivation Figure gives a diagram of a standard standing normal net to the hothesized shocklug. In both cases the shock would aear in the diverging section of a converging/diverging nozzle, with suersonic conditions ustream of the shock and subsonic conditions downstream. Variables describing conditions ustream are denoted with -subscrits, whereas downstream arameters utilize - subscrits. The alication of continuit, conservation of linear momentum, and conservation of energ to the in Fig. (a), along with the assumtion of a erfect ideal gas, results in the well-known normal shock relations. In these relations, the ustream suersonic ch number comletel determines the downstream subsonic ch number, as well as the corresonding increases in temerature, ressure, and densit across the shock. Ke to the shock functions deendence onl on is the s assumed thinness, which necessaril imlies the ustream and downstream flow areas be the same. The shock-lug shown in Fig. (b), however, is characterized b some thickness, t shock, resulting in the cross sectional area increasing in the flow direction. Also seen in Fig. (b) is the resence of a ressure on the sidewalls of the nozzle, side, which contributes to the flow direction linear momentum differentl than in a. For the analsis here, the ratio of areas is given the smbol r: 38-

3 r = A /A. () side T V T V T V T V A side A t shock (a) (b) Fig.. Schematic diagrams showing relevant arameters for (a) a and (b) a shock-lug b The average ressure on the sidewalls is assumed to take on some value between and as given side = + ϕ ( - ), (3) where 0 < ϕ < 1. The resence of the additional arameters r and ϕ comlicates the relations for shock-lugs. As a first order aroimation for estimating the change of arameters across a finite-sized normal shock, we emlo the same assumtions used in the derivation of standard normal shock relations, save the ecetions alread given. Namel, these include quasi-one-dimensional flow (flow arameters varing onl in the stream-wise direction), adiabatic flow, no wall friction, and the use of the erfect ideal gas model as the basis for hsical and thermal equations of state. It should also be noted that a known nozzle geometr (that is, a known variation of cross sectional area with flow direction) along with a given value of t shock comletel determine the value of r at a given location. As such, in the resent analsis we will treat r as an indeendent variable, removing the need for both detailed knowledge about channel geometr as well as estimates of shock thickness such as that given in Eq. 1. With these assumtions, the alication of continuit across the shock-lug of Fig. (b) gives ρ A V = ρ A V. For an ideal gas the densit is ρ = /RT, and the seed of sound is given b into Eq. (4), C (4) RT. Substituting A V C A V. (5) C Recognizing V/C as ch number, and emloing C RT once again, this becomes A A T T 1/ r T T 1/. (6) 38-3

4 The ressure ratio and temerature ratio in Eq. (6) are found b the alications of conservation of linear momentum and conservation of energ, resectivel. Linear momentum in the flow direction gives A + side (A -A ) + ρ A V = A + ρ A V. (7) We once again use the ideal gas relations for ρ and C along with the definition of ch number, and also incororate the relation given in Eq. (3) for side. After much maniulation this results in 1 r (1 )(1 1 (1 r) r). (8) Energ conservation across the shock-lug ields V V h h constant h0, (9) where h is enthal. Equation (9) also shows that the stagnation enthal h 0, the resulting enthal if the fluid were brought to zero velocit, remains constant across the shock-lug. Substituting c T for enthal as well as incororating relations for ρ, C and as before, Eq. (9) eventuall ields T T (10) 1 Together Eqs. (6), (8), and (10) form the counterarts of the normal shock relations for a shock-lug. The reader ma notice that the form of Eq. (10) is identical for both s and shock-lugs. This is because the form of conservation of energ resulting here deends on the shock being modelled as adiabatic, and is thus indeendent of shock thickness er se. Equations (6) and (8) reduce to normal relations as the area ratio r aroaches one. The alication of the second law of thermodnamics shows that the flow across a shock is irreversible with an entro generation er unit mass being equal to the increase in entro across it: s gen s s c ln( T / T ) Rln( / ). (11) Given that stagnation roerties reflect the values of roerties if the flow were brought to zero velocit isentroicall, and that the stagnation temerature does not increase across a shock, the entro generation can be eressed solel as a function of the decrease in stagnation ressure: s gen s s Rln( 0, /, 0). (1) As is the case with the alication of conservation of energ, the form of the relation for entro increase is identical for a and a shock-lug. 3. Results and Discussion Equations (6), (8), and (10) can be combined to eliminate ressure and temerature in order to find a relation between and. The result is 38-4

5 1/ 1/ K1 (1 K) 1 1 K1 (1 K) K1( 1)(1 K), (13) 1 K1 K 1 where r ( 1) (1 ) r (1 )(1 r) (14) and K =ϕ(1 - r). (15) The minus solution in Eq. (13) corresonds to a suersonic in which the fluid continues to accelerate through the lug, whereas the lus solution corresonds to the subsonic, shock-lug solution for. Figure 3 shows the variation of with as calculated b Eq. (13) for various values of r. In all cases ϕ is set to 0.5 so that side is the arithmetic average of and r = Fig. 3. Variation of downstream ch number for various area ratios It is evident from Fig. 3 that the larger the difference in area between shock-lug inlet and outlet (smaller r), the more the shock-lug solution deviates from the standard normal shock values. The effect is most ronounced at smaller, with becoming a function onl of r at high. Figures 4 (a) and (b) show the ressure and temerature increases across the shock-lug, resectivel. The ranges of and r are the same as in Fig. 3. From Fig. 4 (a) we see that ressure increases less across shock-lugs comared to s, whereas Fig. 4 (b) shows that temerature increases more. As is the case with, smaller area ratio results in more deviation from behaviour for both ressure and temerature. As oosed to, which shows more deviation from behaviour at lower, ressure shows more deviation at higher. The deviation in temerature comared to s is small, and does not show an clear trend with. In Fig. 5 can be seen the entro generation for the same range of arameters as in the revious comarisons. Shock-lugs clearl result larger entro generation, and therefore more irreversibilit than do s. This trend continues with larger changes in area for all ch numbers. The irreversibilit in s is often attributed to friction and conduction heat transfer within the shock itself. (Anderson, 1990) Given that the modelling equations are written assuming an adiabatic rocess with no accounting of wall friction, however, couled with the assumtion of an infinitesimall 38-5

6 s gen [kj/kg-k] thin, we ma have reason to question this elanation. Germane to this discussion is the fact that the form of Eq. (1) for entro generation is eactl the same as for an ideal gas undergoing an unrestrained, adiabatic eansion in what would otherwise be a quasistatic rocess. Seeing as how shocklugs occur over finite volumes, and that Fig. 5 shows larger irreversibilities for larger changes in area, we ma wonder if the source of entro generation in shocks is not better thought of as the result of the total ressure of a gas decreasing as it eands against nothing, more eansion leading to more irreversibilit / 0 r = 0.85 T /T 5 4 r = (a) (b) Fig. 4. (a) Pressure increase and (b) temerature increase across shock-lugs of various area ratios r = Fig. 5. Entro generation er unit mass Furthermore, the estimate for shock thickness given b Eq. 1 ma also be called into question in that it balances ressure against viscous shear stress, et it emlos the values of ressure found from standard shock relations, which, as reviousl noted, ignore wall shear stress. Modelling a shock-lug as an unrestrained eansion ma therefore also have value in roviding better estimates for shock thickness. This remains a toic for future work. The effects of varing values of the average ressure on the walls of the channel, side, via changing ϕ are not elored in this reliminar stud. Naturall questions arise as to the aroriate value of ϕ and its functional deendence. This too is left for future investigations. 4. Conclusion In this aer we have hothesized the eistence of and subsequentl modelled shock-lugs, standing normal shocks of finite thickness. In shock-lugs the assumtion of finite thickness requires analsis to include different ustream and downstream cross sectional areas, as well as an average ressure on the 38-6

7 sidewalls of the shock. The inclusion of these etra variables comlicates the analsis of roert changes across the shock, resulting in larger changes in ch number accomanied b smaller ressure increases, larger temerature increases, and larger entro increases. The analsis of shock-lugs also suggests that the entro generation across shocks ma best be visualized as a form of unrestrained eansion. The models resented here ma rovide better estimates of shock roerties in alications with ver small length scales, in which the thickness of shocks ma not necessaril be negligible. Acknowledgements The authors wish to thank ArcelorMittal and the Rose-Hulman Institute of Technolog Indeendent Project/Research Oortunities Program for their suort in this work. References Ames Research Staff (1953). Equations, Tables, and Charts for Comressible Flow, Reort 1135, National Advisor Committee for Aeronautics (NACA). Anderson, J.D. (1990). Modern Comressible Flow With Historical Persective. McGraw-Hill. Doduladab, O.I., Klossa, Y.Y., Tcheremissinebc, F.G., & Shuvalov, P.V. (010). Simulation Of Shock Wave Proagation In A Microchannel B Solving The Boltzmann Equation. tem. Mod., (6), Granger, R.A. (1995). Fluid Mechanics. Dover. Iancu, F., & Müller, N. (005). Efficienc Of Shock Wave Comression In A Microchannel. Microfluid Nanofluid, Sringer-Verlag. Louisos, W.F., Aleeenko, A.A., Hitt, D.L., &Zilic, A. (008). Design Considerations For Suersonic Micronozzles. International Journal of nufacturing Research, 3(1), Moran, M.J., & Shairo, H.N. (008). Fundamentals of Engineering Thermodnamics 6 th edition. Wile. Zeitoun, D.E., Graur, I.A., Burtschell, Y., Ivanov, M.S., Kudravstev, A.N., & Bondar, Y.E. (009). Continuum and Kinetic Simulation of Shock Wave Proagation in Long Micrcochannel. Proc. of Rarefied Gas Dnamics: 6 th Int. Smosium,