Droplet Vaporization in a Supercritical Environment. M. F. Trujillo Applied Research Laboratory Pennsylvania State University State College, PA 16803

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1 ILASS Americas 19th Annual Conference on Liquid Atomization and Spray Systems, Toronto, Canada, May 2006 Droplet Vaporization in a Supercritical Environment M. F. Trujillo Applied Research Laboratory Pennsylvania State University State College, PA D.J. Torres Theoretical Division Los Alamos National Laboratory Los Alamos, NM Abstract A theoretical framework for the description of the physical processes involved in droplet vaporization leading towards the thermodynamic critical condition is proposed here. As opposed to previous work, an explicit expression for the di cult-to-determine thermal di usion coe cient is presented. This coe cient is responsible for the cross-e ect in heat and di usion fluxes and it is valid for non-ideal mixtures, which are typically encountered in the conditions of interest. Preliminary calculations of droplet vaporization at ambient pressures are shown to agree very well with micro-gravity data. At higher pressures a discrepancy in the heat-up period is noted, although the vaporization rate constants in the D 2 law portion of the droplet lifetime agree fairly well. Corresponding Author mft10@psu.edu

2 Introduction The description and modeling of droplet vaporization in supercritical environments has been the subject of a sizable number of investigations as documented in reviews written by Givler and Abraham [1], Yang [2], and Bellan [3]. While the behavior of vaporization prior to the attainment of a thermodynamic critical condition is well established, the phenomena occurring at the onset and after the interface has reached this point is much less understood. In particular, the physics governing the critical surface regression have received seemingly contradictory interpretations in the literature [4 6]. Our long term objectives are to study the onset critical point attainment taking into account as precisely as possible the anomalies encountered in transport coe cients and to consider the issue of thermodynamics instability for the entire droplet system. Prior to this, however, a complete theoretical framework needs to be put in place to incorporate all the dominant physics, in particular the cross-terms occurring in the heat and di usion fluxes, which have been shown to be important near the critical regime [7]. Among the various investigators that have contributed to the field, only Bellan and coworkers have advanced such a generalization of the physics. The general expressions for both the heat and mass fluxes that resulted from their work have already been known for quite some time in the nonequilibrium thermodynamics field. The new aspect was the application of this theory to the problem of droplet vaporization where it is expected that both the liquid and gas mixtures are non-ideal. The current work distinguishes itself from Bellan s group principally in the manner in which the essential thermal di usion coe cients are determined. These coe cients are responsible for the cross-e ects given by the Dufour and Soret terms. Borrowing from work aimed at hydrocarbon reservoirs [8, 9], an explicit expression for the thermal di usion coe - cient is presented that is valid for non-ideal mixtures. In what follows, the governing equations are described along with the interfacial conditions and the thermophysical properties. The expressions for heat and di usion flux are derived and the expression for the thermal di usion coe cient is developed. Preliminary results consisting of surface temperature and droplet diameter time series are presented for various conditions of temperature and pressure. Governing Equations The system under consideration is multicomponent with species =1,...,N, and with transport coe cients that are functions of pressure, temperature, and chemical composition. The equations are: for Y for species mass for momentum; + r ( u) + r ( Y u)= r j, + r ( v v) = rp + r, DT C p Dt ) ) P,Y Dt + : rv + X r q r j H, for energy. The di usion flux, j, and heat flux, q, originate from formulations found in nonequilibrium thermodynamics and are described below. Additionally the Equation-of-State (EoS) employed in this work is the Peng-Robinson EoS (PR- EoS) given by P = R ut b m a m 2 +2b m b 2, (1) m where the parameters a m and b m are calculated from mixing rules [10] and discussed in more detail in [11]. In order to isolate the core physics from geometrical complexities produced by buoyancy and/or convective forces, we consider vaporization in a micro-gravity environment. This leads to significant simplifications of the governing equations by a reduction to a spherical symmetry. As a consequence every vector and scalar field is only a function of the radial coordinate, i.e. u =(U, 0, 0), q =(q, 0, 0), and j =(j, 0, 0). This reduces the respective gov-

3 erning + r (r2 ( Y )+ r (r2 Y U)= r (r2 ( U)+ r (r2 U + @r r C ( T )+ r (r2 TU) ) ) P,Y Dt r (r2 q)+ X r (r2 j )H " 2 U + µ 2 r 3 r + 2U #. r Interfacial relations (2) The first of interfacial relation is the phaseequilibrium condition. In general, phase equilibrium is expressed by equating the chemical potential of species at the interface. This can be expressed as an equation for fugacities f g i = f l i, (3) where for the PR EoS, fugacities are given by, ln f i = b i (Z 1) ln(z B ) Px i b m P A N 2 p 2 j=1 x! j(1 k ji )(a j a i ) 1/2 b i 2B a m b m ln Z + (1 + p! 2)B Z ( p. (4) 2 1)B Additionally, we solved the following set of jump conditions for mass; g sy g,s(u g s g s( r s u g s) l s( r s u l s)=0, (5) Ṙ) l sy,s(u l l s r s )= j g + j l n, (6) for species; X L [ l sy (Ṙ l ul s) j ]+k l g rts g k l rts l +(P l P g )Ṙ = R utj A k T W avg Y A Y B g s R u Tj A k T W avg Y A Y B for energy. The interfacial constraints introduce the following (6+2N) unknown quantities; they are ( g s, l s, Ṙ, ug s,u l s,t s, Y, g Y )., l assuming that pressure is given, i.e. that the pressure change across the interface is of secondary importance (this assumption becomes even more accurate as the crit- µu r ical point is approached and the surface tension be- l s (7) gins to vanish). The interfacial system reduces to (6+2(N-1)) variables due to P Y g = P Y l = 1. For a binary system the result is 8 unknowns. The number of equations include two phase-equilibrium relations (3), jump condition for mass (5), for species (6), and energy (7), as well as the EoS (for both liquid and gas) giving 7 equations. Due to the excess of unknowns a variable will remain undetermined. This variable is chosen to be the liquid phase velocity which is solved iteratively until both governing equations and interfacial conditions are satisfied. Numerical Solution The numerical solution of the above equations follows an explicit version of an ALE method presented by Torres et al. [12]. We employ a staggered grid where all of the variables are defined at the cell center with the exception of the fluid velocity which is defined at the cell edges. Figure 1 shows this arrangement. A hyperbolic tangent is used to cluster the nodes close to the interface to allow the e cient capture of the thermal and composition boundary layers. Heat and Di usion Fluxes Most of the literature in droplet vaporization expresses the heat and di usion flux in terms of the Fourier heat conduction and Fick s Law of mass diffusion, respectively. Near the critical regime, however, it has been reported by Bellan and co-workers [13], that other modes of heat and mass di usion become important. To account for them consistently, we employ here the Maxwell-Stefan Form of these fluxes derived from irreversible thermodynamics [14, 15]. For a n-component mixture, the heat flux q is given by q = krt + NX =1 NX =1;6= NX =1 j h W + R u TX X D T j C j, (8) where j and j are the di usion fluxes for species and, respectively (to be described shortly). A constraint for these di usion fluxes is that they must satisfy P n =1 j = 0. Moreover, the multicomponent

4 thermal di usion coe cients obey the following relationship P n =1 DT = 0 [14], and the Maxwell-Stefan di usivities C are symmetric [15] (i.e.c = C ). For a binary mixture consisting of species A and B, the heat flux becomes q = krt + h A j A + h B j B W A W B + R utx A X B A + R utx B X A B DA T ja C AB A DB T jb C BA B j B B j A A. (9) Since the mixture is binary, we have j A = j B, DA T = DT B, and C AB = C BA which give ha hb q = krt + j A W A W B D T A 1 + R u TX A X B j A + 1. (10) C AB A B A B where = A + B. Simplifying this further we obtain a working version of the heat flux ha hb q = krt + j A W A W B + R utd T A C AB j A W avg A B W A W B. (11) This expression clearly shows the contributions from the traditional Fourier conduction term, the enthalpy di usion, and the Dufour term. With respect to the di usion flux, j, we begin with the definition of the barycentric or mass averaged velocity v, v = P N =1 v. The di usion flux is then defined as j = (v v). (12) This expression can be written in many cases in terms of the gradient of species concentration, resulting in the well known Fick s di usion form. In general, however, the di usion flux is influenced by other gradients besides mass fraction. A useful general expression is given by Bird et al. [15] relating the di erence in molecular velocities to these gradients. For zero external forces, we have NX =1;6= X X C (v v )= X (r ln a ) T,P 1 R u T (! )rp NX X X D T C =1;6= D T! r ln(t ) (13) For a binary mixture of A and B, this reduces to X A X B (v A v B )= X A (r ln a A ) T,P C AB 1 Y A )rp R u T ( A X A X B D T A C AB A From (12) and j A = v A v B = D T B B j B we have ja A r ln(t ) (14) j B B = j A A B. (15) Additionally, since DA T = DT B, (14) becomes j A = A B C h AB X A r ln a A X A X B + 1 R u T ( A Y A )rp + X AX B C AB D T A r ln T A B i. (16) Typically, we would like to deal with something more tangible than the gradient of the activity a A in the di usion flux equation. First we can rearrange this term to ln aa X A r ln a A = rx ln X A The activity is defined as the ratio a A = f A (T,P,X)/fA o (T,P), where f A(T,P,X) and fa o (T,P) are the fugacities at some arbitrary state (T,P,X) and a reference state, respectively. With this definition of activity and noting that fa o (T,P) is not a function of composition, the above expression ln fa (T,P,X) X A r ln a A = rx ln X A Substituting this into (16) and ignoring the expected weak contribution from the pressure gradient, yields j A = A B C AB X A X B + X AX B D T A C AB ln(fa ) rx ln X A A B r ln T i. (17) This expression can be further simplified by noting that rx A = WavgrY 2 A /(W A W B ) and performing some manipulations to give ln(f A ) j A = C AB ry A + D T rt A. ln X A T The binary di usion coe cient, D AB = C AB (@ ln(f A )/@ ln X A ), which is typically the

5 one provided in measurements, is equal to the Maxwell-Stevan di usivity in the case of ideal mixtures. The second term in this equation represents the Soret e ect. The major di culty in predicting this e ect is that for non-ideal mixtures, which are the type encountered in this work, the determination of the thermal di usion coe cient, DA T, is considerably di cult. Actually even in ideal mixtures the determination of DA T is far from easy [16]. Firoozabadi and co-workers [7 9] have provided a means of predicting the thermal di usion factor that can be related to the thermal di usion coe cient. This is examined below. Thermal Di usion Coe cient Under the same conditions examined above, i.e. no external forces (micro-gravity) and negligible pressure gradient, Ghorayeb and Firoozabadi [7] derive a similar expression to (18) for the di usion flux of a multicomponent mixture. Casting this relationship for a binary mixture yields ln f A j A = H AB X A W avg rx A W avg where and + H AB W avg K T A A i, (19) K T A = M AX A M B X B L 0 Aq W avg R u TL AA (20) H AB = L 0 Aq = L AA Q A W A W avgrl AA M A M B X A X B. (21) KA T is the thermal di usion ratio. The phenomenological coe cients are given respectively by L 0 Aq and L AA (their introduction comes from the theory of non-equilibrium thermodynamics where they appear in the linear relationship between the fluxes, j and q, and the thermodynamic forces rt, rx, rp [17].) An expression exists [8] between L 0 q and L,, which for a binary mixture reduces to. (22) Here Q = Q where [8] Q B W B H is the net heat of transport, q = nx =1 Q W j, (23) i.e. Q is the heat of transport of component. Substituting (22) into the equation for KA T gives KA T = M AX A M B X B Q A Q B. (24) W avg R u T W A W B This clearly shows as documented in [8] that for a binary mixture the thermal di usion ratio is independent of the Onsager phenomenological coe cients. Firoozabadi et al. [8] extends this statement to conclude that the thermal di usion coe cients themselves (D T ) are independent of the molecular di usion coe cients (D AB ) in the binary mixture case. However, as we will see shortly the expression for D T still retains L AA through H AB. Unless L AA can be expressed in terms of something other than a molecular di usion parameter, D T cannot be made independent from it. By comparing (18) with (20), we have D T A = W avg H AB K T A. (25) If we were to use (24) in this expression, L AA remains as previously mentioned. If instead we substitute (20), then L AA cancels, but L 0 Aq remains. To overcome this problem, another comparison between (18) and (20) yields A B C ln(f A ) = ln f A 1 H AB X A W avg. X A X ln X A W ln X A X A (26) This can be solved for H AB giving H AB = W AW B Wavg 2 C AB. (27) This equation along with (24) can be substituted into (25) resulting in a final expression for the thermal di usion coe cient DA T = W A W B W avg Wavg 2 C AB Q A M A X A M B X B W avg R u T W A Q B. (28) W B Relations for the net heat of transport, Q, have been developed by Firoozabadi and co-workers [8, 9] improving on the earlier work of Dougherty and Drickamer [18]. Namely, the net heat of transport for a binary mixture is given as apple XA Ī A A + X B ĪB B Q A = Ī A A + VA X A VA + X B VB, (29) where I is the partial molar internal energy departure and is the ratio of latent heat to the energy of viscous flow [19], for species respectively.

6 Thermophysical Properties For gas di usion, we employ the Chapman- Enskog dilute gas expression [10] modified for high pressure by using the Takahashi correction factor [20]. For liquid di usivity, the Hayduk and Minhas correlation is employed [21]. The method of Chung et al. [22] is used to calculate gas conductivity. Liquid conductivity is obtained from the fuel libraries of KIVA-4; it has been tested against data found in [10] for C 7 H 16. Latent heat of vaporization is calculated directly from the PR-EoS (details in [11]). Liquid specific heats are calculated based on a corresponding-of-states principle [10] and validated against data found in Vargaftik [23]. Gas specific heat are first obtained from ideal predictions [10] and corrected for high pressure using the internal energy departure calculated from the PR- EoS. Results The numerical scheme has been tested to ensure that the results presented here are numerically convergent with respect to time step and grid size. Due to the explicit nature of the numerical scheme, stability imposed a much higher restriction on t than accuracy. This meant that all of the calculations which successfully ran were already converged with respect to t. The calculations presented here are preliminary; the incorporation of the Soret and Dufour e ects previously described are still being incorporated into the code. Furthermore, the viscous dissipation term in the energy equation and the spatial variation of pressure are not yet accounted for. Nevertheless, for the conditions investigated in these initial runs the incorporation of these terms should not cause significant impacts. To validate our predictions, comparisons have been performed against the micro-gravity experimental data of Nomura et al. [24] for C 7 H 16 droplets vaporizing in a pure Nitrogen environment. In their experiments, a droplet in the diameter range of 0.6 to 0.8 mm and having an initial temperature of 300 o K, was generated at the tip of a silica fiber and introduced into an electric furnace. The elapsed time for the introduction was 0.16 seconds. In Figure 2 the surface temperature histories for cases ran at various ambient conditions are shown. As expected the rate of the temperature rise significantly increases with both ambient pressure and temperature. Comparisons of (d/d 2 o) as a function of time for the low pressure case are shown respectively in Figures 3 and 4. The agreement to experiments is particularly good at this pressure. As reported by Harstad and Bellan [25], one of the most unreliable transport coe cient predictions corresponds to the pressure e ect on mass di usivity. They proposed an alternate method for correcting for high pressure; in this work we have opted for the traditional Takahashi correction factor [20]. Comparisons can be made between both of these methods for the system being investigated here. In any case at ambient pressures, the predictions of the thermophysical properties are the most accurate; hence, the agreement shown in Figures 3 and 4 is expected. It should be noted that during the 0.16 seconds of elapsed time between the temperature measurement instant and the subsequent introduction into the chamber, the droplet was exposed to a convective environment caused by this translation which significantly enhances mass and heat transfer. The degree to which the temperature field was a ected can be estimated from the surface temperature history profiles shown in Figure 2; although because of the convective environment this plot actually shows a much milder rise. The key finding is that even though the elapse time is small, 5%, compared to the droplet lifetime, it occurs in a period of time when the rates of temperature increase are the largest. Hence, it is expected that for cases at higher pressures and temperatures, the initial field may deviate substantially from 300 K o. Additionally, other discrepancies are incurred by heat transfer through the supporting fiber [26] or from buoyancy e ects which become important at high pressure, even under micro-gravity conditions [3]. Motivated by this potential significant change in initial conditions, the next set of plots shown in Figures 5 (P=1.0 MPa, Ta=466K o ), 6 (P=2.0 MPa, Ta=452K o ), and 7 (P=2.0 MPa, Ta=656K o ) have an additional calculation performed with an initial droplet temperature of 340 K o. This additional calculation is not performed with the objective of replicating the exact details of the initial time period, but to show the impact on the results. In all of these latter plots, the lower temperature calculations show a considerably greater amount of initial droplet swelling. Once the droplets reaches the D 2 law region, the calculated vaporization rate constants are practically the same in both sets of predictions and agree reasonably well with experiments. Comparing the predictions between the fully resolved physics in both the gas and the liquid to those of only the liquid phase agree with the results of Zhu et al. [27]. It is observed that when fully resolving the gas phase and its unsteadiness, the calculations show a more pronounced vaporization behavior.

7 Summary The governing equations for the physics governing droplet phase change in supercritical environments have been advanced based on the generalization of the di usion and heat fluxes from irreversible thermodynamics [14, 15]. An explicit expression has also been obtained for the thermal diffusion coe cient which is responsible for the crosse ect between fluxes. Calculations for this are currently in progress. Computations for droplet diameter as a function of time have been performed at various pressures and temperatures. Good agreement is found at low pressures. At higher pressures a consistent deviation exists between the predictions and the data. This will be studied more thoroughly in the coming weeks. A potential source of experimental error, however, is the unaccounted convective transport occurring during a short, but significant initial elapsed time. This is almost negligible at lower pressures but becomes important at higher pressures.

8 Nomenclature a m,b m Peng-Robinson parameters C Maxwell-Stefan multicomponent di usivity C p Constant pressure specific heat C v Constant volume specific heat D Mass di usion coe cient D T Thermal di usion coe cient f Fugacity k Thermal conductivity H, h Enthalpy, Enthalpy per unit mass I Internal energy per unit mass j Di usion flux L Latent heat of vaporization M Mass P Pressure Pvap o Pure species vapor pressure q Heat flux r s Droplet radius r s Time rate of change of droplet radius R u Universal gas constant R Gas constant R u /W avg S Entropy per unit mass t Time T Temperature u, u Velocity V Volume v Droplet velocity Specific volume W Molecular weight X, X Mole fraction, mole fraction vector Y Mass fraction Z Compressibility factor Greek Specific heat ratio C p /C v µ Viscosity Density Viscous stress tensor! Acentric factor Subscripts, Corresponding to species, 1 Cell quantity c Critical r Reduced quantity, e.g. P r = P/P c o Reference state (ideal-gas) Superscripts X Specifies a molar quantity for X s Surface value g Gas phase l Liquid phase References [1] S.D. Givler and J. Abraham. Progress in Energy and Combustion Science, 22:1 28, [2] V. Yang. Proceedings of the Combustion Institute, 28: , [3] J. Bellan. Progress in Energy and Combustion Science, 26: , [4] P. Haldenwang, C. Nicoli, and J. Daou. Int. J. Heat Mass Transfer, 39: , [5] V. Yang, K.C. Hsieh, and J.S. Shuen. AIAA , 31st Aerospace Sciences Meeting and Exhibit, [6] W.A. Sirignano. Fluid Dynamics and Transport of Droplets and Sprays. Cambridge University Press, New York, [7] K. Ghorayeb and A. Firoozabadi. AIChE Journal, 46: , [8] A. Firoozabadi, K. Ghorayeb, and K. Shukla. AIChE Journal, 46: , [9] K. Shukla and A. Firoozabadi. Ind. Eng. Chem. Res., 37: , [10] B.E. Poling, J.M. Prausnitz, and J.P. O Connell. The Properties of Gases and Liquids, 5th Edition. Mc Graw Hill, New York, [11] M.F. Trujillo, D.J. Torres, and P.J. O Rourke. Int. J. Engine Res., 5: , [12] D.J. Torres, P.J. O Rourke, and A.A. Amsden. Combustion Theory Modelling, 7:67 86, [13] K. Harstad and J. Bellan. International Journal of Multiphase Flow, 26: , [14] C.F. Curtiss and R.B. Bird. Ind. Eng. Chem. Res., 38: , [15] R.B. Bird, W.E. Stewart, and E.N. Lightfoot. Transport Phenomena. John Wiley & Sons, Inc., 2 edition, [16] J.H. Ferziger and H.G. Kaper. Mathematical Theory of Transport Processes in Gases. North-Holland Publishing Company, Amsterdam, [17] S.R. de Groot and P. Mazur. Non-Equilibrium Thermodynamics. North-Holland Publishing Company, Amsterdam, [18] E.J. Dougherty and H.G. Drickamer. J. Chemical Physics, 23: , 1955.

9 [19] S. Glasstone, K.J. Laidler, and H. Eyring. The Theory of Rate Processes. McGraw-Hill, New York, [20] Takahashi S. J. Chem. Engng. Japan, 6: , [21] W. Hayduk and B.S. Minhas. Canadian Journal of Chemical Engineering, 60, [22] T-H. Chung, M. Ajlan, L.L. Lee, and K. Starling. Ind. Eng. Chem. Res., 27: , [23] N.B. Vargaftik. Tables on the Thermophysical properties of Liquids and Gases. Hemisphere Publishing Corporation, Washington, D.C., [24] H. Nomura, Y. Ujhe, H.J. Rath, J. Sato, and M. Kono. Twenty-Sixth Symposium on Combustion, pp , [25] K. Harstad and J. Bellan. Ind. Eng. Chem. Res., 43: , [26] C. Morin, C. Chauveau, P. Dagaut, I. Gokalp, and M. Cathonnet. Combust. Sci. and Tech., 176: , [27] G.S. Zhu, R.D. Reitz, and S.K. Aggarwal. International Journal of Heat and Mass Transfer, 44: , 2001.

10 Drop boundary Nodes nl 2 nl 1 nl nl+1 nl+2 nl+ng 1 nl+ng Cells nl 1 nl nl+1 nl+2 nl+ng 1 nl+ng Cell quantities: T, Y, Vol, and ρ Figure 1. Staggered grid arrangement used for numerical solution of drop vaporization. 420 Initial elapse period Droplet Surface Temperature (K) P= 0.1 MPa Ta=471 P=1.0 MPa Ta=466 K P=2.0 MPa Ta=452 K P=2.0 MPa, Ta= 656 K t/d 2 o (s/mm 2 ) Figure 2. Droplet surface temperature time series under various conditions.

11 P=0.1MPa, Ta=471 K Calculations Experiments (d/d o ) t/d 2 o (s/mm 2 ) Figure 3. Droplet size as a function of time, P=0.1 MPa, ambient temperature = 471 o K P=0.1 MPa Ta=741 K Calculations Experiments (d/d o ) t/d 2 o (s/mm 2 ) Figure 4. Droplet size as a function of time, P=0.1 MPa, ambient temperature = 741 o K.

12 1 0.8 (d/d o ) P=1 MPa Ta= 466 K 0.4 Calculations T init =340 K Calculations T init =300 K Experiment t/d 2 o (s/mm 2 ) Figure 5. Droplet size as a function of time, P=1.0 MPa, ambient temperature = 466 o K (d/d o ) Pa = 2.0 MPa and Ta=452 K 0.4 T_init=340K T_init=300 K Experiment t/d 2 o (s/mm 2 ) Figure 6. Droplet size as a function of time, P=2.0 MPa, ambient temperature = 452 o K.

13 1 0.8 (d/d o ) Pa= 2 MPa Ta=656 K Calculations T init =340 K Calculations T init =300 K Experiments t/d 2 o (s/mm 2 ) Figure 7. Droplet size as a function of time, P=2.0 MPa, ambient temperature = 656 o K.