A smoothed particle hydrodynamics method for evaporating. multiphase flows

Transcription

1 A smoothed prticle hydrodynmics method for evporting multiphse flows Xiufeng Yng*, nd Song-Chrng Kong** Deprtment of Mechnicl Engineering, Iow Stte University, Ames, IA 50011, USA * xyng@istte.edu ** kong@istte.edu Abstrct Smoothed prticle hydrodynmics (SPH) method hs been incresingly used for simulting fluid flows, however its bility to simulte evporting flow requires significnt improvements. This pper proposes n SPH method for evporting multiphse flows. The present SPH method cn simulte the het nd mss trnsfers cross the liquid-gs interfces. The conservtion equtions of mss, momentum nd energy were reformulted bsed on SPH, then were used to govern the fluid flow nd het trnsfer in both the liquid nd gs phses. The continuity eqution of the vpor species ws employed to simulte the vpor mss frction in the gs phse. The vpor mss frction t the interfce ws predicted by the Clusius-Clpeyron correltion. A new evportion rte ws derived to predict the mss trnsfer from the liquid phse to the gs phse t the interfce. Becuse of the mss trnsfer cross the liquid-gs interfce, the mss of n SPH prticle ws llowed to chnge. New prticle splitting nd merging techniques were developed to void lrge mss difference between SPH prticles of the sme phse. The proposed method ws tested by simulting three problems, including the Stefn problem, evportion of sttic drop, nd evportion of drop impcting on hot surfce. For the Stefn problem, the SPH results of the evportion rte t the interfce greed well with the nlyticl solution. For drop evportion, the SPH result ws compred with the result predicted by level-set method from literture. In the cse of drop impct on hot surfce, the evolution of the shpe of the drop, temperture, nd vpor mss frction were predicted. Keywords: smoothed prticle hydrodynmics, evportion, mss trnsfer, het trnsfer, multiphse 1

2 flow PACS numbers: j, C, c I. INTRODUCTION Becuse evportion is encountered in mny engineering pplictions, such s fuel droplets in engines, liquid sprys, nd mteril processing [1-5], numericl method to ccurtely predict liquid evportion is of gret importnce. Common engineering models for predicting droplet evportion ssume tht the liquid droplet is point source with homogeneous properties [1-4]. The primry concern of these models the mss trnsfer rte without considertion of the grdient in the droplet or the liquidgs interfce. While such models re useful in engineering pplictions, dvnced numericl methods re needed to revel the detils of the evportion process. The dynmics of evporting flows involves phse chnge nd energy trnsfer t the liquid-gs interfce, diffusion of vpor species in the gs phse, nd multiphse flows with shrp interfces. Becuse of the complexity of the evportion problem, it is chllenging to detiled numericl simultion. The min numericl chllenges in simulting evporting flows include the tretment of phse chnge nd the shrp discontinuity of fluid properties t the liquid-gs interfce. Phse chnge due to evportion cuses mss trnsfer from one phse to nother phse. The discontinuity t the liquid-gs interfce, of vribles such s density rtio, lso leds to numericl difficulties. Severl numericl methods to ddress the chllenges in modeling the detils of evporting flows hve been developed in recent yers. Tnguy et l. [6] presented numericl method using both the level-set method nd the ghost fluid method to cpture the interfce motion nd to hndle conditions t the interfce. Sfri et l. [7, 8] developed lttice Boltzmnn method (LBM) for simulting the phse chnge of multiphse flows with evportion. Nikolopoulos et l. [9] investigted the evportion process of n-heptne nd wter droplets impinging on hot surfce using the finite volume method coupled with the volume of fluid (VOF) method. Strotos et l. [10] studied the evportion of wter droplets depositing on heted surfce t low Weber numbers using VOF. The intent of this work is to provide numericl method, bsed on smoothed prticle hydrodynmics (SPH), to simulte multiphse flows with evportion. The SPH method is Lgrngin mesh-free

3 prticle method. In SPH, continuous fluid is discretized using SPH prticles, which crry physicl properties, such s mss, density, pressure, viscosity, nd velocity. Since SPH is mesh-free method, smoothing kernel is introduced to connect the neighboring prticles. The vribles nd their sptil derivtives re discretized in summtions over prticles. The SPH method ws originlly proposed by Lucy [11] nd Gingold nd Monghn [1] for stronomy problems. Since then SPH hs been pplied to wide rnge of problems [13-15]. In recent yers, the SPH method ws extended for phse chnge flows. By using the vn der Wls (vdw) eqution of stte, Nugent nd Posch [16] pplied SPH for modeling vdw fluid drop surrounded by its vpor. Their numericl results showed tht there ws more vpor round the drop t higher temperture. Using SPH with vdw eqution of stte, Siglotti et l. [17] simulted the rpid evportion nd explosive boiling of vdw liquid drop. Ry et l. [18] pplied the vdw-sph method to study the liquid-vpor equilibrium of the vdw fluid. Ds nd Ds [19] proposed model bsed on SPH to describe gs-liquid phse chnge by introducing pseudo prticles of zero mss. However, the previous phse chnge SPH methods consider the interction between the liquid nd its vpor, but do not consider the effect of the concentrtion of the vpor species in the gs phse on evportion nd the diffusion of the vpor species in the gs phse. Therefore, the bility of SPH to simulte evportion needs further improvement. In the clssicl SPH method, the mss of n SPH prticle is constnt, i.e., the mss of n SPH prticle does not chnge during simultion. In the SPH method developed for this study, the SPH prticles ner the interfce re llowed to chnge their mss to model the process of evportion t the interfce. The rte of mss chnge of SPH prticles due to evportion depends on the vpor mss frction in the gs phse nd the sturted vpor mss frction t the interfce. The sturted vpor mss frction cn be predicted by the Clusius-Clpeyron correltion. During the process of evportion, the mss of liquid SPH prticle t the interfce increses, while the mss of gs SPH prticle decreses. To constrin the mss of individul SPH prticles, prticle will split into smller prticles if its mss is lrge enough or merge into neighbor prticle if its mss is smll enough. The rest of this pper is orgnized s follows. Governing equtions re given in Sec. II, including the derivtion of evportion rte. Sec. III provide the numericl method, including the SPH formultions for liquid-gs interfce nd evportion rte, nd the prticle splitting nd merging 3

4 technique. The numericl method is tested in Sec. IV by three different numericl exmples. Then the pper ends with conclusions in Sec. V. II. GOVERNING EQUATIONS The conservtion equtions of mss, momentum nd energy re used to describe the trnsport of both the liquid phse nd gs phse. These equtions re expressed in the Lgrngin form. d u (1) dt du 1 p dt g u () dt dt 1 ( T) (3) C p Here ρ is the fluid density, u is the fluid velocity, p is the fluid pressure, µ is the dynmic viscosity, T is the temperture, C p is the specific het t constnt pressure, κ is the therml conductivity, nd g is the grvittionl ccelertion. Note tht in this pper the production of therml energy by viscous dissiption is not considered in the energy eqution becuse of its reltively smll mgnitude [6, 7, 0]. The following eqution of stte is used to clculte pressure p c ( r) pr (4) where c is numericl speed of sound, ρ r is reference density nd p r is reference pressure. At the liquid-gs interfce, the process of phse chnge due to evportion will cuse mss nd energy trnsfer. Thus, the continuity nd energy equtions, Eqs. (1) nd (3), t the liquid-gs interfce re modified s d um (5) dt dt 1 h ( T) dt C C v (6) p p where m is the mss evportion rte cross the interfce while m is the volumetric mss evportion rte, nd h v is the ltent het of vporiztion. In order to obtin the mss frction field of the vpor species in the gs phse, the continuity 4 m

5 eqution of the vpor species needs to be solved, dy dt ( DY ) (7) where Y is the vpor mss frction nd D is the mss diffusivity of the vpor. The governing equtions listed bove re not closed. An eqution to clculte the mss evportion rte is needed. A couple of equtions to describe the evportion rte hve been used in the mesh-bsed methods, however, they cnnot be directly used within the SPH method, becuse there re no mesh in SPH. Therefore, new eqution for evportion rte tht cn be used in SPH needs to be derived. Liquid Gs Liquid Gs dm m Y m + dm Y + dy Interfce Interfce FIG. 1. Schemtic of mss trnsfer from liquid element to gs element due to evportion. Figure 1 shows the mss trnsfer process t the liquid-gs interfce due to evportion. The initil totl mss nd the vpor mss frction of the gs element re m nd Y, respectively. A mss dm is trnsferred cross the liquid-gs interfce due to evportion. As result, the totl mss nd the vpor mss frction of the gs element become m dm nd Y dy, respectively. Bsed on the conservtion of vpor mss, we hve my dm ( m dm)( Y dy ). (8) The following eqution cn be obtined by neglecting the second order infinitesiml term dmdy in the bove eqution. Then the following rte eqution cn be obtined. m dm dy (9) 1 Y dm m dy (10) dt 1 Y dt 5

6 Substituting Eq. (7) in the bove eqution yields dm m( DY ). (11) dt (1 Y ) Note tht m& dm dt nd V m, Eq. (11) cn be written s V( DY) m&. (1) 1Y The volumetric mss flux cn be clculted s m& ( DY) m&. (13) V 1 Y In order to obtin the mss trnsfer rte cross the interfce, the vpor mss frction t the interfce needs to be defined. By ssuming tht equilibrium exists between the liquid nd gs phses t the interfce, the vpor mss frction t the interfce is equl to the sturted vpor mss frction. Both cn be relted to the sturted vpor molr frction s [6, 7] XM s v Ys (1 X ) M X M s g s v (14) where Y s is the sturted vpor mss frction, X s is the sturted vpor molr frction, M v is the molr mss of the vpor, nd M g is the molr mss of the dry gs (excluding the vpor species). The sturted vpor molr frction, X s, cn be relted to the sturted vpor pressure s [1] X p s s (15) pg where p s is the sturted vpor pressure, p g is the mbient gs pressure (including the vpor species). Then the sturted vpor molr frction cn be estimted by integrting the Clusius-Clpeyron eqution [6, 1] X p h M 1 1 s v v s exp pg R Ts TB (16) where R is the idel gs constnt, T s is the interfce temperture, T B is the liquid boiling temperture t the mbient gs pressure condition. 6

7 III. NUMERICAL METHODS A. Bsic formultions of the SPH method In SPH, the vlue of function f(r) t point r cn be pproximted using the following integrtion f ( r ) f( r) W( r r, h) dv (17) where W is kernel function nd dv is differentil volume element. The prmeter h is referred to s smoothing length, which determines the size of the integrl domin. In this pper, the following hyperbolic-shped kernel function in two-dimensionl spce is used [, 3] 3 s s s 6 6, W(, s h) ( ), 1 s s 3 h 0, s (18) where s = r/h. This kernel function cn void the so-clled tensile instbility [4] tht my occur in fluid simultions using SPH method [, 3]. In the SPH method, continuous fluid is discretized into set of SPH prticles. These prticles lso hve physicl properties, such s mss m, density ρ, velocity u, nd viscosity μ. Then the integrtion of Eq. (17) is discretized in prticle summtion s follows. mb f( r) f( rb) W( r r b, h) (19) b The derivtives of function cn lso be discretized into prticle summtion. For exmple, the grdient of function f cn be obtined by differentiting the kernel in Eq. (19), b b b b b b m f f W (0) where Wb denotes the grdient of W tken with respect to the coordintes of prticle. Note tht in SPH, derivtive cn be discretized into different summtion forms [5, 6]. B. SPH formultions for single phse fluid By pplying the prticle summtion, the governing equtions, Eqs (1), (), (3) nd (7), cn be replced by the following SPH prticle equtions. 7

8 d dt mbu u b Wb (1) b d u p p b mb( b)( b) Wb gmb bwb r r ( u ) u b () dt b b b b( rb ) dt 1 m ( )( r r ) W ( T T b) dt C r b b b b p b b( b ) dy m ( D D )( r r ) W ( Y Y b) dt b b b b b b b( rb ) (3) (4) Here the term 0.01h is dded to prevent the singulrity when two prticles re too close to ech other [5]. Note tht Eq. (4) is vlid for the gs phse SPH prticles. A gs SPH prticle hs property of vpor mss frction Y, thus there re no prticles tht only represent vpor species. In Eq. (), b is the rtificil viscosity proposed by Monghn [5] ( c cb) b b, ( u ub) ( r rb) 0 b ( b) 0, ( u ub) ( r rb) 0 (5) where b h( u u ) ( r r ). (6) b b rb The prmeters α nd β re used to control the strength of the rtificil viscosity. α is relted to the sher viscosity, nd β is relted to the bulk viscosity. For SPH simultion, the density nd pressure fields my undergo lrge fluctutions numericlly. In order to reduce the fluctution, the Shephrd filtering [7] is pplied to reinitilize the density field. b b b % VW b b b (7) In this pper, the summtion is only executed for the prticles from the sme phse. The density reinitiliztion is conducted every 50 time steps for the liquid phse nd every 500 time for the gs phse. To prevent prticle penetrtion, the XSPH correction introduced by Monghn [5] is used to move 8 mw

9 prticles dr dt ) mb u u ( u ) W u b b b b (8) Following Colgrossi nd Lndrini [8], the XSPH correction is lso used in the mss eqution. C. SPH formultions for interfce For multiphse flow, especilly for liquid-gs flow, there exists discontinuity t the interfce for certin fluid properties, such s density, viscosity nd therml conductivity. The discontinuity my led to numericl difficulties. Therefore, the SPH equtions for single phse fluid need to be modified for the liquid-gs interfce. Following Clery nd Monghn [9], when two prticles from different phses interct with ech other, the following therml conductivity is used. b b b (9) Similrly, the viscosity between the gs nd liquid prticles is b b. (30) b For the pressure term, the inter-prticle pressure proposed by Hu nd Adms [30] is used to replce the prticle pressure in Eq. () t the liquid-gs interfce p b p p b b b. (31) The contributions of prticle b to the momentum eqution nd the energy eqution of prticle re s follows. du dt b p b mb b+ RbWb b m ( r r ) W ( u u ) b b b b b( rb ) b (3) dt m ( r r ) W ( T T b) dt C r b b b b b p b( b ) (33) Here R b on the right hnd side of Eq. (3) is n rtificil repulsive force with the following form 9

10 R p R b b (34) b The prmeter ε R is in the rnge of 0 nd 0.1. This repulsive force is similr to tht used by Monghn [31] nd Grenier et l. [3]. When gs prticle intercts with liquid prticle, Eq. (1) tends to overestimte the contribution of the liquid prticle to the density of the gs prticle, becuse the mss of liquid prticle is much lrger thn the mss of gs prticle. In order to void the overestimtion, the contribution of the liquid prticle to the rte of chnge of the density of the gs prticle is clculted by d dt gl g l g l g gl V u u W (35) The subscripts g nd l denote the gs prticle nd the liquid prticle, respectively. The contribution of the gs prticle to the rte of chnge of the density of the liquid prticle is d lg mg l g lwlg dt u u. (36) For the eqution of the vpor mss frction t the interfce, liquid prticle is treted s gs prticle, nd its vpor mss frction is defined by the sturted vpor mss frction, Eq. (14). The contribution of liquid prticle to the rte of chnge of the vpor mss frction of gs prticle is dy m D ( r r) W ( Y g Y l). (37) dt gl l g g l g gl l( rgl ) It should be noted tht ll the formultions in this section (i.e., Section III.C) re only used for the interfce. Tht is, the interctions between two prticles from different phses re clculted using the formultions in this section, while the interctions between prticles from the sme phse re clculted using the formultions in Section III.B. D. SPH formultions for evportion rte The rte of mss trnsfer from liquid prticle to gs prticle due to evportion, Eq. (1), is discretized s mmd( r r) W m ( Y Y). (38) g l g g l g gl gl g l l( rgl )(1 Yg) 10

11 The totl mss chnge rte of gs prticle is dm m m D ( r r) W m ( Y Y). (39) dt r Y g g l g g l g gl gl g l l l l( gl )(1 g) The totl mss chnge rte of gs prticle is dm mmd( r r) W l m ( Y Y). (40) dt r Y g l g g l g gl gl g l g g l( gl )(1 g) Eqs. (39) nd (40) indicte tht the totl mss of the liquid nd gs prticles does not chnge. Thus, the mss conservtion is stisfied in the process of evportion. The volumetric mss flux, Eq. (13), is m md ( r r) W m ( Y Y). (41) g g l g g l g gl g g l Vg l l( rgl )(1 Yg) E. Prticle splitting nd merging The phse chnge due to evportion will increse the mss of gs prticle nd decrese the mss of liquid prticle t the interfce. The mss chnge rte of gs prticle nd liquid prticle re given by Eqs. (39) nd (40), respectively. In order to ensure tht the mss of prticle is not excessively lrge or smll, prticle splitting nd merging techniques re developed here. Both the splitting nd merging process stisfy the conservtion of mss, momentum nd energy. b b FIG.. Schemtic of simulting prticle splitting. If the mss of prticle is lrger thn given vlue, the prticle will split into two smller prticles, s shown in Fig.. The process of prticle splitting is s follows. 11

12 1) A reference mss is set to m r ds r d, where ρ r is the reference density, ds is the initil prticle distnce, nd the superscript d is the number of sptil dimension. In this study, twodimensionl cse is considered, thus m r rds. ) If the following condition is stisfied, prticle will be split into two smller prticles. m m (4) r mx Here m is the mss of prticle. mx is prmeter to control the mximum limit of prticle mss, whose rnge is 1.5. Both the two smller prticles hve the mss tht is hlf mx of the mss of the originl prticle, nd the sme density nd velocity of the originl prticle. 3) The next step is to find the nerest prticle b of prticle. The two new prticles re on the perpendiculr line of the line connecting prticles nd b. The distnce between the two new prticles is m. The reson to find the nerest prticle is to void tht the new smller r prticles re too close to the neighboring prticles. If the mss of prticle is less thn given vlue, it will merge to its nerest prticle, s shown in Fig. 3. The process of prticle splitting is s follows. 1) A reference mss is set to m r rds. ) If the following condition is stisfied, the prticle will merge with its nerest prticle. m m (43) r min Here min is prmeter to control the minimum limit of prticle mss, whose rnge is 0.5. The reson to merge into the nerest prticle is to void tht the new min mx prticle is too close to the neighboring prticles nd to reduce to influence re of the merging process. 3) The next step is to find the nerest prticle b of prticle. The new prticle is locted t the center of mss of prticles nd b. 1

13 b FIG. 3. Schemtic of simulting prticle merging. IV. NUMERICAL EXAMPLES The evportion model bsed on the SPH method will be vlidted in this section, by simulting three different cses. Tble 1 shows the physicl properties of the liquid nd the gs used in the following numericl exmples in this pper. Note tht the density in the tble is the initil density. The liquid density will chnge slightly during the simultion becuse the numericl method used in the pper is the so-clled wekly compressible SPH method, which llows the density for up to one percent vrition from the initil density. On the other hnd, the gs density does vry becuse of evportion. Tble 1. Physicl properties of the liquid nd gs phses [33]. ρ (kg/m 3 ) μ (kg/m/s) κ (W/m/K) C p (J/kg/K) M (kg/mol) h v (J/kg) D v (m /s) T B (K) Gs Liquid A. The Stefn problem To vlidte the new evportion rte, Eq. (1), which ws derived in this pper, nd its SPH form, Eq. (38), the Stefn problem ws simulted. As illustrted in Fig. 4, n open continer ws prtilly filled with liquid, nd the reminder with gs. The liquid, then evportes from the liquid-gs interfce, nd the vpor diffuses from the interfce to the open end of the continer. The vpor mss frction t the interfce is ssumed to be constnt (i.e., sturted vpor condition, Y v,s ), nd the vpor mss frction t the open end is lso constnt (Y v,h ). In other words, the system is t stedy stte, nd the nlyticl 13

14 solution of the vporiztion mss flux is [1] D 1 Y v v, H m& v ln. (44) H 1 Y v,s Y v (H) = Y v,h y = H Gs Y v (0) = Y v,s Liquid y = 0 FIG. 4. Schemtic of the Stefn problem. Since the vpor mss frction t the interfce is ssumed to be constnt, nd the interfce is ssumed to be sttionry, the numericl simultion is only conducted in the gs phse. The bottom boundry of the computtionl domin is the liquid-gs interfce, t which the vpor mss frction is set from 0.1 to 0.9. The top boundry is gs boundry, t which the vpor mss frction is set t 0. The periodic boundry condition is used for the left nd right boundries. The height nd width of computtionl domin re.0 mm nd 0.5 mm, respectively. The initil SPH prticle spcing is 0.05 mm. Fig. 5 shows the SPH results of evporting mss flux compred with the nlyticl solution. The SPH prediction grees well with the nlyticl solution. 14

15 FIG. 5. SPH prediction nd nlyticl solution of evporting mss flux s function of vpor mss frction t the interfce. Another numericl test ws conducted by solving only the eqution for vpor mss frction, Eq. (7), using SPH Eq. (4), without solving ny other governing equtions. The results re shown in Fig. 6. The numericl solution closely greed with the nlyticl solution when the vpor mss frction is less thn 0.5. However, s the vpor mss frction incresed beyond 0.5, the numericl solution devited from the nlyticl solution. According to Sfri et l. [7], the divergence of the velocity t the liquidgs interfce is nonzero becuse of evportion, which leds to the over-prediction of the evporting mss flux. Therefore, the eqution for vpor mss frction, Eq. (7), lone does not ccurtely simulte the evportion process. Therefore, for simultion of evportion, ll the governing equtions listed in Section II need to be solved. 15

16 FIG. 6. Anlyticl solution nd numericl prediction by considering only Eq. (4). B. Evportion of sttic drop The evportion of sttic drop ws simulted using the proposed SPH method. Figure 7 shows the initil SPH prticle distribution for simulting the evportion of sttic drop. The initil rdius of the drop is R 0 = 0.15 mm. The initil temperture of the drop is 353 K. The drop ws locted t the center of squre computtionl domin, which ws filled with gs. The length of the squre ws 1. mm. The initil temperture of the gs ws 373 K. The temperture of the boundry ws lso 373 K, nd did not chnge during the simultion. These tempertures were chosen in order to be consistent with nd to llow comprisons with the conditions in the literture [6]. The initil vpor mss frction in the gs phse ws zero. The vpor mss frction of the boundry remined zero. The initil prticle spcing ws 0.0 mm. 16

17 FIG. 7. Initil SPH prticle distribution of sttic drop, nd the computtionl domin. The interction between the SPH prticles long the interfce ws not bsolutely symmetric. Thus, the shpe of the interfce ws not perfect circle, nd the drop moved slightly. Although the movement of the drop ws very slow, the drop hd noticeble displcement when time llowed. To void the movement, the drop ws fixed t the center of the computtionl domin by use of the following equtions. r rr, uuu (45) c Here r c nd u c re the displcement nd velocity of the center of mss of the drop, respectively. Figure 8 shows the snpshots of the evporting drop t different times. The shpe of the interfce chnged slightly with time, but it is very close to circle. Figure 8 lso shows tht the size of the drop decresed slightly. The decrese in the drop size, s compred with the result from D xisymmetric level set method [6], is shown in Fig. 9. It should be noted tht the D circle used in this study corresponded to the cross section of 3D cylinder of infinite length, while the D xisymmetric circle used in Ref. [6] corresponded to 3D sphere. Therefore, the comprison in Fig. 9 qulittively demonstrtes the ccurcy of the proposed SPH method. Since the rtio of surfce re to volume of D drop (this study) is less thn tht of D xisymmetric drop (Ref. [6]), the decrese in the size of the D drop is less thn tht of the D xisymmetric drop, s shown in Fig. 9. Nonetheless, the trends re similr. At the initil stge, the size of the drop decresed quickly, becuse initilly there ws no vpor in the gs phse, nd becuse the evportion rte ws fst. As the vpor concentrtion in the gs c phse incresed, the evportion rte decresed. 17

18 As cn be seen in Fig. 8, the SPH prticle distribution ws not uniform. The reson for this is tht the sizes of the prticles were not the sme. As discussed in Section E, the rtio of the prticle mss to the corresponding reference mss my vry from min to mx. Initilly, the distribution of the prticles ws uniform, s shown in Fig. 7. Then the mss of the gs prticles ner the interfce incresed becuse of the mss trnsfer from the liquid prticles to the gs prticles due to evportion. When the mss rtio of gs prticle ws lrger thn mx, the prticles were split into two smller prticles. At the sme time, the mss of the liquid prticles ner the interfce decresed. When the mss rtio of liquid prticle ws less thn min, the liquid prticle merged into its nerest liquid prticle. The mss of the gs prticles ner the boundry lso decresed becuse of the mss trnsfer from the gs prticles to the boundry prticles. When the mss rtio of gs prticle ws less thn min, the gs prticle merged into its nerest gs prticle. t = 0.1 s t = 0. s t = 0.5 s t = 1.0 s FIG. 8. Snpshots of the evporting drop t different times. 18

19 FIG. 9. Normlized squre of rdius versus time. t = 0.1 s t = 0. s t = 0.5 s t = 1.0 s FIG. 10. Evolution of vpor mss frction. 19

20 t = 0.1 s t = 0. s t = 0.5 s t = 1.0 s FIG. 11. Evolution of temperture. Figure 10 shows the evolution of the vpor mss frction surrounding the drop. As time incresed from 0.1 s to 1.0 s, the corresponding sturted vpor mss frction t the interfce decresed from 0.3 to The reson for this is tht evportion consumed energy, nd thus decresed the drop temperture, nd consequently decresed the vpor concentrtion t the interfce. The evolution of the temperture field is shown in Fig. 11 to clerly show tht the temperture of the drop decresed due to evportion. Figure 11 lso shows tht the temperture of the drop ws lower thn its initil temperture, nd tht it decresed until reching n equilibrium temperture. At certin times, the temperture t the interfce ws lower thn the temperture t the drop center. Eventully, the temperture difference between the interfce nd the drop center decresed until reching n equilibrium temperture. If the detils of mss nd energy trnsfer t the interfce hd not been considered, the temperture of the drop would hve been higher thn its initil temperture, nd the temperture t the interfce would hve been higher thn the temperture t the drop center, becuse the surrounding gs would hve heted the liquid drop, s is commonly seen in trditionl evportion models. 0

21 C. Evportion of drop impcting on hot surfce The proposed method ws lso used to simulte the evportion of drop impcting on hot surfce, s shown in Fig. 1. The initil rdius of the drop ws R = 0.5 mm. The initil velocity of the drop ws U = m/s. The height nd length of the computtionl domin were 1.5 mm nd 5.0 mm, respectively. The drop ws locted t the center of the domin nd surrounded by gs. The initil temperture of the drop ws 353 K. The initil temperture of the gs ws 373 K. The temperture of the boundries ws lso 373 K, nd did not chnge during the simultion. The initil vpor mss frction in the gs phse ws zero. The vpor mss frction of the boundry remined zero. The initil prticle spcing ws 0.0 mm. R U FIG. 1. Schemtic of drop impct on surfce. Figure 13 shows the evolution of drop impct on hot surfce. After the drop touched the surfce, it spred nd formed film on the surfce. At pproximtely 1.0 ms, tiny crown-like structure ws formed round the rim. Lter, the crown merged with the film, nd the film receded. Finlly, the film reched n equilibrium size. The evolution of the temperture field, nd vpor mss frction, re shown in Figs. 14 nd 15, respectively. Since the initil temperture of the drop ws lower thn the gs temperture, the het trnsfer from the surrounding gs to the drop led to the decrese in the locl gs temperture. However, the drop temperture lso decresed slightly becuse evportion consumed energy, s discussed erlier. As cn be seen in Fig. 14 (t = 1.0 nd.0 ms), the rim hd the lowest temperture, becuse the evportion rte in the re is lrge. When the drop spreds on the hot surfce nd forms film, het trnsfer from the hot surfce to the film incresed the temperture of the film. 1

22 t = 0. ms t = 0.3 ms t = 0.5 ms t = 1.0 ms t =.0 ms t = 5.0 ms FIG. 13. Evolution of drop impct on hot surfce.

23 t = 0. ms t = 0.3 ms t = 0.5 ms t = 1.0 ms t =.0 ms t = 5.0 ms FIG. 14. Evolution of temperture field of drop impct on hot surfce. 3

24 t = 0. ms t = 0.3 ms t = 0.5 ms t = 1.0 ms t =.0 ms t = 5.0 ms FIG. 15. Evolution of the vpor mss frction of drop impct on hot surfce. V. CONCLUSION The intent of this pper ws to present n SPH method to simulte evporting multiphse flows. This method ccurtely models the process of evportion t the liquid-gs interfce nd the diffusion of the vpor species in the gs phse. An evporting mss rte ws derived to clculte the mss trnsfer t the interfce. To model the process of phse chnge from the liquid phse to the gs phse, 4

25 mss ws llowed to trnsfer from liquid SPH prticle to gs SPH prticle. Thus this proposed method, unlike the trditionl SPH method, llows chnge in the mss of n SPH prticle. Additionlly, prticle splitting nd merging techniques were developed to void the lrge difference in the SPH prticle mss. Three numericl exmples were tested nd compred with nlyticl solutions nd results from level-set method. In generl, the results show tht the method proposed in this pper successfully replicted the physicl process of evporting flows, such s het nd mss trnsfers nd the diffusion of the vpor species. The first exmple were the Stefn problem, in which the mss evportion rtes t different conditions were predicted; the numericl results showed tht the evportion rte incresed quickly s the vpor mss frction t the interfce incresed, nd tht the results gree well with the nlyticl solution. The second exmple ws to simulte the evportion of sttic drop becuse of evportion, the present SPH method predicts the decreses of both the temperture of the interfce nd the size of the drop. The lst exmple ws to simulte the evportion of drop impcting hot surfce. The temperture of the liquid-gs interfce decresed t first becuse of evportion, especilly t the rim of the film. Then the temperture incresed becuse of the het trnsfer from the hot surfce to the liquid. In summry, the results of this study indicte tht the numericl method proposed in this pper cn be successfully used to produce n evporting flow simultion. References [1] W. A. Sirignno, Fluid dynmics nd trnsport of droplets nd sprys (Cmbridge University Press, 014), Third edn. [] K. Hrstd nd J. Belln, Combustion nd flme 137, 163 (004). [3] L. Zhng nd S.-C. Kong, Combustion nd Flme 158, 1705 (011). [4] L. Zhng nd S.-C. Kong, Combustion nd Flme 157, 165 (010). [5] S. S. Szhin, Progress in energy nd combustion science 3, 16 (006). [6] S. Tnguy, T. Ménrd, nd A. Berlemont, J. Comput. Phys. 1, 837 (007). [7] H. Sfri, M. H. Rhimin, nd M. Krfczyk, Phys. Rev. E 90, (014). [8] H. Sfri, M. H. Rhimin, nd M. Krfczyk, Phys. Rev. E 88, (013). [9] N. Nikolopoulos, A. Theodorkkos, nd G. Bergeles, Int. J. Het Mss Trn. 50, 303 (007). [10] G. Strotos, M. Gvises, A. Theodorkkos, nd G. Bergeles, Int. J. Het Mss Trn. 51, 1516 (008). [11] L. B. Lucy, Astron. J. 8, 1013 (1977). [1] R. A. Gingold nd J. J. Monghn, Mon. Not. R. Astron. Soc. 181, 375 (1977). [13] J. J. Monghn, Eur. J. Mech. B/Fluid 30, 360 (011). [14] M. B. Liu nd G. R. Liu, Arch. Comput. Methods Eng. 17, 5 (010). 5

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