Design Cnstraints fr iquid-prtected Divertrs S. Shin, S. I. Abdel-Khalik, M. Yda, and the ARIES Team G. W. Wdruff Schl f Mechanical Engineering, Gergia Institute f Technlgy, Atlanta, GA 3033-0405 USA [seungwn.shin@me.gatech.edu, said.abdelkhalik@me.gatech.edu, minami.yda@me.gatech.edu] Recent wrk n liquid-surface-prtected plasma facing cmpnents has resulted in the establishment f perating windws fr candidate liquids, as well as limits n the maximum allwable liquid surface temperature in rder t limit plasma impurities frm liquid evapratin. In this study, an additinal cnstraint n the maximum allwable surface temperature gradient (i.e., heat flux gradient) has been quantified. Spatial variatins in the wall and liquid surface temperatures are expected due t variatins in the incident radiatin and particle fluxes. Thermcapillary frces created by such temperature gradients can lead t film rupture and dry spt frmatin in regins f elevated lcal temperatures. Here, attentin has been fcused n nn-flwing thin liquid films similar t thse frmed n the surface f prus wettedwall cmpnents. Future analyses will include the effects f macrscpic fluid mtin, and MHD frces. A numerical mdel using the level cntur recnstructin methd was used t fllw the evlutin f the liquid free surface abve a nn-isthermal slid surface. The mdel was used t develp generalized charts fr the maximum allwable spatial temperature gradients (i.e., the critical Marangni number) as a functin f the gverning nn-dimensinal variables, viz. the Weber, Frude, and Prandtl numbers, and aspect rati. Attentin was fcused n the asympttic limit fr thin liquid films (i.e., lw aspect rati) which prvides a lwer bund fr the maximum allwable temperature gradients. Specific examples fr lithium, Flibe, lithiumlead, tin, and gallium are presented. The generalized charts develped in this investigatin will allw reactr designers t identify design windws fr successful peratin f liquid-prtected plasma facing cmpnents fr varius clants, film thicknesses, and perating cnditins. I. INTRODUCTION Wrk n liquid-surface-prtected plasma facing cmpnents, and plasma surface interactins, has been perfrmed by the Advanced Pwer Extractin (APEX) study and the Advanced imiter-divertr Plasma Facing Systems (APS) Prgram [1, ]. The attractiveness f this cncept derives frm the ability f the liquid t remve the incident heat lad, while prtecting the underlying slid wall frm radiatin damage, ersin, and thermal stress. A detailed reactr design study with liquid first wall and divertr has recently been reprted by the APEX team [3]; bth liquid metals and mlten salts have been investigated. State-f-the-art cdes have been used t mdel the plasma edge, plasma sheath at the divertr, and particle surface interactins t establish the perating windws fr candidate liquids [4]. ithium, Flibe, lithium-lead, lithium-tin, tin, and gallium were amng the liquids examined. Plasma impurity levels resulting frm liquid evapratin impse maximum surface temperature limits ranging frm 380ºC fr i t 1630ºC fr Sn [3, 4]. This study is aimed at quantifying the limits, if any, n the maximum allwable temperature gradient t assure liquid film stability n liquid-surface-prtected plasma facing cmpnents. Spatial variatins in the wall and liquid surface temperatures are expected in all magnetic fusin devices due t variatins in the incident radiatin and particle flux. Such spatial variatins in temperature induce the s-called thermcapillary effect, which derives frm the surface tensin gradient alng a twphase interface. Fr mst liquids, surface tensin decreases with temperature. Thermcapillary frces can hence lead t film rupture and dry spt frmatin in regins f elevated temperatures. While this phenmenn has been extensively examined by the fluid mechanics cmmunity, it has nt, t ur knwledge, been cnsidered by the fusin cmmunity in analyzing the feasibility f liquid prtected plasma facing cmpnents. Here, ur attentin will be fcused n nn-flwing thin liquid films similar t thse frmed n the surface f prus wetted-wall cmpnents. Future analyses will include the effects f macrscpic fluid mtin, and MHD frces. Several studies dealing with this nn-linear freebundary prblem have been reprted in the literature [5-11]. Ruckenstein and Jain used lng-wave thery t study the instability f an ultra thin liquid film n a slid surface [5]. A small perturbatin was impsed n the liquid-gas interface; the grwth f the perturbatin in time was investigated using linear thery. The Navier-Stkes equatin was used t describe the interface mtin by including a bdy frce term t accunt fr ndn/van der Waals interactin. Athertn and Hmsy derived the nnlinear partial differential (evlutin) equatins describing the mvement f a fluid-fluid interface fr lng interfacial waves in tw-phase flw, including a detailed treatment f interfacial bundary cnditins in bth Euclidean and nn-euclidean crdinates [6]. Davis examined nnlinear effects in thin film instability [7], and frmulated a strngly nnlinear evlutin equatin fr the
thickness f a static film n a planar slid including ndn/van der Waals frces. He later extended his thery t include variable surface tensin in deriving a lng-wave evlutin equatin fr the interface shape subject t an external temperature gradient with arbitrary rientatin fr the purpse f studying thermcapillary instability [8]. Tan, et al. impsed a nn-unifrm spatially peridic temperature distributin alng the underlying plate, and examined the respnse f the liquid layer t thermcapillary effects [9]. The system is subject t gravity, which induces hydrstatic effects; bth ndn/van der Waals and surface tensin frces have been included. They develped a nnlinear lng-wave equatin fr the shape f the thin film, and fund that a cntinuus steady liquid layer culd be sustained nly if the temperature gradient at the underlying slid surface was less than a critical value. Burelbach, et al. [10] validated the theretical frmulatin f Tan, et al in a carefully designed experiment fr thin films. Ref. 11 reviews previus wrk n lng scale evlutin f thin films. Mre recently, direct numerical simulatin f the evlutin f free liquid surfaces has been perfrmed using the level cntur recnstructin methd [1, 13], which stems frm Tryggvasn s finite difference/frnt tracking methd riginally develped fr tw- and threedimensinal isthermal multi-fluid flws [14, 15]. The remainder f this paper is rganized as fllws. Sectin II includes the frmulatin f the gverning equatins fr thrmcapillary flws f a thin liquid layer n a nn-isthermal slid wall; the gverning equatin fr the shape f the liquid film fr lw aspect ratis (i.e. thin films), hereafter referred t as the asympttic slutin, is first presented. The results, including generalized nndimensinal charts fr the maximum allwable temperature gradient as a functin f the gverning nndimensinal parameters, alng with specific results fr varius liquid candidates, are presented in Sectin III. Cnclusins are given in Sectin IV. II. THEORETICA MODE II.1. Asympttic Slutin Cnsider the system shwn in Figure 1. Here, a thin liquid film with an initially unifrm thickness h rests atp a slid surface with a nn-unifrm, spatially-peridic, temperature distributin (perid ). Variatins in the z- directin (viz. tridal directin) are ignred, s that the prblem becmes tw-dimensinal. Fllwing the methd f Tan, et al. [9], fr lw aspect ratis, the lng-wave equatin fr the nn-dimensinal steady-state shape f the thin liquid film (the Asympttic Slutin) is given by: Fr 3 T + h + (M / Pr) Fr s = 0 We 3 a h h h 3 (1) y l >>h Here, Gas y x u v T = = = 0 h initial film thickness iquid Nn-unifrm wall temperature V ; ; ; Fr ; h h x ax g a = h = x = = = = 3 h h gh g ρ h c γ Th s We = ; Pr = ; M = ; ρσ h k α T Tm T = ; and Vg = T ρ h s pen B.C. at tp, Initially, quiescent liquid and gas (u=0, v=0, T=T m at t=0) are the aspect rati, nn-dimensinal film thickness, nrmalized x-crdinate, Frude, Weber, Prandtl, and Marangni numbers, nrmalized temperature, and scaling velcity. The quantity γ is the abslute value fr the rate f change f surface tensin with temperature. Fllwing the prcedure described by Tan, et al. [9], equatin (1) can be slved by apprximating the slutin fr h by an N-term truncated Furier csine series; the cefficients f the expansin f h are determined using Galerkin s methd. With a, Fr, and We as independent parameters, equatin (1) is repeatedly slved fr a given wall temperature distributin t determine the critical temperature gradient, i.e., the (am/pr) value crrespnding t zer film thickness (h =0) at the pint f maximum wall temperature (center f the wall; x= /). II.. Numerical Slutin Full direct numerical simulatin has als been perfrmed using the level cntur recnstructin methd [1, 13]. The methd has been simplified by eliminating lgical cnnectivity, and thus eliminating the assciated algrithmic burden, while retaining the accuracy and advantages f explicit agrangian surface tracking. The nn-dimensinal gverning equatins fr mass and mmentum cnservatin are given by: u v + = 0 () peridic B.C. in x directin Wall Fig. 1. Initial surface cnfiguratin and bundary cnditins used t mdel hrizntal surface subjected t temperature gradient.
+ u u u a ρ + u + v t p + u + u = + a + + v σ + a + n + t ˆi σ κ δds (3) y x + v v v a 4 ρ + u + v t + p = + ρ 4 + v + u + a + a Fr + v n + t ˆ + σ + a a j σ κ δds (4) y y Here p is the pressure and κ is twice the mean interface curvature. We assume that the surface tensin at the liquid-gas interface σ varies linearly with the temperature T frm the reference value at T m ; the nn-dimensinal surface tensin cefficient, σ, can then be expressed by: σ = 1/ We ( M/Pr) T (5) Neglecting evapratin at the liquid-gas interface, the energy equatin becmes: + + + + ct ct ct a ρ + u + v t (6) a + T 1 + T = k + k Pr x Pr One set f transprt equatins (), (3), (4), and (6), valid fr bth fluids, is slved. This lcal single-field frmulatin incrprates the effect f the interface in the equatins as delta-functin surce terms which act nly at the interface and autmatically satisfy the interface bundary cnditins. The delta-functin surce term in equatins (3) and (4) represents the surface tensin frce exerted n a differential element at the interface (Fig. ): where σ A ( σt) δ Fe = σκ n+ t ds = ds B s = ( σ t σ t ) A A B B (7) σ t σ ( σt) σκ n + t = σ + t = (8) The first and secnd term in equatin (8) represent the nrmal surface tensin frce and the thermcapillary frce, respectively. A mre detailed descriptin f the evel Cntur Recnstructin techniques and prcedures can be fund in Refs. 1 and 13. σ B t B -σ C t C III. RESUTS III.1. Asympttic Slutin The results f the asympttic slutin can be presented as generalized charts fr the maximum nndimensinal temperature gradient (i.e., critical value fr am/pr) as a functin f the Weber and Frude numbers (Fig. 3) with a given aspect rati (a=0.0) and a prescribed sinusidal wall temperature distributin: [ x ] T ( x ) = cs π( 0.5) (9) s B Fig.. Schematic diagram f the surface tensin frce frmulatin. (n : unit vectr in nrmal directin. t : unit vectr in tangential directin) The maximum nn-dimensinal temperature gradient (am/pr) is defined by: ( Ts / ) am/pr = (10) ( / ρ γ h ) Referring t Figure 3, fr small values f (1/Fr), the maximum temperature gradient is prprtinal t (1/We), while at high (1/Fr) values, it is independent f the Weber number. Plts similar t thse shwn in Figure 3 have been btained fr ther aspect ratis. It can be shwn that in the limit f zer aspect rati, (am/pr) cr appraches a value f (π a/1fr), independent f the Weber number. III.. Numerical Slutin Fr high aspect ratis, the level cntur recnstructin methd described in Sectin II. abve has been used t slve the full set f cnservatin equatins. Generalized charts fr the maximum nn-dimensinal temperature gradients as a functin f the Weber, Frude, and Prandtl numbers, and aspect rati have been btained. The simulatin gemetry is the same as that used t develp the asympttic slutin (Fig. 1). The liquid interface is assumed t be initially flat, while bth the liquid and gas are assumed t be initially quiescent (u,v=0 and T=T m at t=0). A nn-unifrm temperature distributin is impsed n the bttm wall (Eq. 9). The evlutin f e -σ B t B n σ A t A s A
Maximum Nn-dimensinal Temperature Gradient maximum y lcatin T s /=0 [K/cm] T s /=40 [K/cm] T s /=60 [K/cm] minimum y lcatin Fig. 3. Generalized nn-dimensinal charts fr the maximum nndimensinal temperature gradient frm asympttic slutin the free surface is then fllwed until steady state cnditins are reached. Typical results fr a lithium film with three different temperature gradients (0, 40, and 60 K/cm) are shwn in Figure 4. These tw-dimensinal simulatins were perfrmed using a 50 50 mesh reslutin with a 1[cm] 0.1[cm] bx size, and an initial film thickness f 0. mm. Fr each case, the transient evlutins f the maximum and minimum film thickness alng the surface are depicted; as the temperature gradient increases, the minimum thickness reaches zer indicating film rupture. A typical generalized nn-dimensinal chart fr the maximum nn-dimensinal temperature gradient as a functin f the Weber and Prandtl numbers is shwn in Figure 5. These results crrespnd t a (1/Fr) value f 10 3 and an aspect rati f 0.0. The crrespnding values predicted by the asympttic slutin are als shwn. Similar charts fr ther values f (1/Fr) and aspect ratis have been btained. These results indicate that the maximum allwable nn-dimensinal temperature gradient increases with Pr and that the asympttic slutin represents the case f zer Prandtl number (i.e. the cnductin limit), and can, therefre, be viewed as a cnservative lwer bund. Fig. 4. Transient prfile f the maximum and minimum interface pint by D simulatin f full equatin using the evel Cntur Recnstructin Methd III.3. Applicatin t Candidate iquids Bth the asympttic slutin and the full numerical simulatin have been used t determine the maximum allwable temperature gradients fr varius candidate clants, namely, lithium, lithium-lead, Flibe, tin, and gallium. Table I lists values f the nn-dimensinal parameters fr the five fluids at different temperatures with a nminal film thickness f 1.0 mm. The crrespnding maximum temperature gradients predicted by bth the asympttic slutin and the numerical simulatin are shwn in Table II. These results shw that the allwable temperature gradients differ significantly amng the candidate fluids, and that, in sme cases, such cnstraint may be highly restrictive. IV. CONCUDING REMARKS In this study, limits n the maximum allwable wall surface temperature gradients (i.e., spatial heat flux gradients) in liquid-surface-prtected plasma facing cmpnents have been quantified. Operatin beynd these limits may lead t film rupture and dry spt frmatin in regins f elevated temperatures due t Table I. Values f the nndimensinal variables fr different Clants with nminal thickness f 1 mm Parameter ithium ithium-ead Flibe Tin Gallium 573K 773K 573K 773K 573K 773K 1073K 1473K 873K 173K Pr 0.04 0.06 0.031 0.031 0.013.4 0.0047 0.0035 0.0058 0.009 1/Fr 1. 10 4. 10 5 1.9 10 5 6.3 10 5 1. 10 3 3.8 10 4 5.3 10 5 6.7 10 5 5.7 10 5 8. 10 5 1/We 7.8 10 5 1.3 10 6 9.4 10 5 3.0 10 6 1.3 10 4 4.0 10 5 4. 10 6 4.8 10 6 6.9 10 6 1.0 10 7 /(ρ γ h )[K/m].8 1.5 4.4 1.3 140 4.3 0.7 0.55 1.6 1.1
Clant Mean Temperature [K] ( T s /) [K/cm] Numerical Slutin Asympttic Slutin ithium 573 30 13 ithium -ead 673 570 173 Flibe 673 76 38 Tin 173 113 80 Gallium 1073 600 11 Table II. The maximum temperature gradient limit fr different clants thermcapillary frces. Generalized charts have been develped t allw designers f fusin systems t identify design windws fr successful peratin f liquidprtected plasma facing cmpnents. An asympttic slutin which prvides a cnservative lwer bund n the allwable temperature gradient has been develped. Detailed numerical simulatins have als been used t btain mre accurate limits fr cases with high aspect ratis. The results have been applied t five different candidate fluids, namely, lithium, lithium-lead, flibe, tin, and gallium. These results shw that the maximum allwable temperature gradients differ by mre than an rder f magnitude, with lithium having the lwest allwable temperature gradient and gallium the highest. ACKNOWEDGMENTS This wrk has been perfrmed as a part f the ARIES-CS Study. We gratefully acknwledge the financial supprt prvided by the U.S. Department f Energy Office f Fusin Energy Sciences thrugh cntract number DE-FG0-01ER54656. REFERENCES [1] M. A. ABDOU, and APEX TEAM, Explring Nvel High Pwer Cmpnents fr Attractive Fusin Systems, Fusin Eng. Design 45, 145 (1999). [http://www.fusin.ucla.edu/apex/] [] R. MATTAS, and APS TEAM, APS - Advanced imiter-divertr Plasma Facing Systems, Fusin Eng. Design, 49-50, 17 (000). [http://fusin.anl.gv/aps_inf_center/calls.html] [3] R. E. NYGREN, ET A., A Fusin Reactr Design with a iquid First Wall and Divertr, t appear in Fusin Eng. Design, (004). [4] T. D. ROGNIEN and M. E. RENSINK, Interactin between iquid-wall Vapr and Edge Plasmas, J. Nucl. Materials, 90-93, 31 (001). Fig. 5. Generalized nn-dimensinal charts fr the maximum nndimensinal temperature gradient frm full numerical simulatin with 1/Fr=10 3 [5] E. RUCKENSTEIN and R. K. JAIN, Spntaneus rupture f Thin iquid Film, J. Chem. Sc. Farad. T., 70, 13 (1974). [6] R. W. ATHERTON and G. M. HOMSY, On the Derivatin f Evlutin Equatin fr Interfacial Waves, Chem. Eng. Cmmun.,, 57 (1976). [7] S. H. DAVIS, Rupture f Thin iquid Films, Waves n Fluid Interfaces, 91, R. E. Meyer, Ed., Academic Press, New Yrk (1983). [8] S. H. DAVIS, Thermcapillary Instability, Annu. Rev. Fluid Mech., 19, 403 (1987). [9] M. J. TAN, S. G. BANKOFF, and S. H. DAVIS, Steady Thermcapillary Flws f Thin iquid ayers: I. Thery, Phys. Fluids A,, 313 (1990). [10] J. P. BUREBACH, S. G. BANKOFF, and S. H. DAVIS, Steady Thermcapillary Flws f Thin iquid ayers: II. Experiment, Phys. Fluids A,, 3 (1990). [11] A. ORON, S. H. DAVIS, and S. G. BANKOFF, ng-scale Evlutin f Thin iquid Films, Reviews f Mdern Physics, 69, 931 (1997). [1] S. SHIN, A evel Cntur Recnstructin Methd fr Three-Dimensinal Multiphase Flws and Its Applicatin, Ph.D. Thesis, Gergia Institute f Technlgy (00). [13] S. SHIN and D. JURIC, Mdeling Three- Dimensinal Multiphase Flw Using a evel Cntur Recnstructin Methd fr Frnt Tracking Withut Cnnectivity, J. Cmput. Phys., 180, 47 (003). [14] S. O. UNVERDI and G. TRYGGVASON, A Frnt- Tracking Methd fr Viscus Incmpressible Multi- Fluid Flws, J. Cmput. Phys., 100, 5 (199). [15] S. O. UNVERDI and G. TRYGGVASON, Cmputatins f Multi-Fluid Flws, Physica D, 60, 70 (199).