A macroscale mixture theory analysis of deposition and sublimation rates during heat and mass transfer in dry snow

Transcription

1 The Cryophere, 9, , doi: /tc Author) CC Attribution 3.0 Licene. A macrocale mixture theory analyi of depoition and ublimation rate during heat and ma tranfer in dry now A. C. Hanen and W. E. Folien Department of Mechanical Engineering, Univerity of Wyoming, Laramie, WY 82071, USA Correpondence to: A. C. Hanen hanen@uwyo.edu) Received: 5 February 2015 Publihed in The Cryophere Dicu.: 5 March 2015 Revied: 11 Augut 2015 Accepted: 21 Augut 2015 Publihed: 23 September 2015 Abtract. The microtructure of a dry alpine nowpack i a dynamic environment where microtructural evolution i driven by eaonal denity profile and weather condition. Notably, temperature gradient on the order of K m 1, or larger, are known to produce a faceted now microtructure exhibiting little trength. However, while trong temperature gradient are widely accepted a the primary driver for kinetic growth, they do not fully account for the range of experimental obervation. An additional factor influencing now metamorphim i believed to be the rate of ma tranfer at the macrocale. We develop a mixture theory capable of predicting macrocale depoition and/or ublimation in a now cover under temperature gradient condition. Temperature gradient and ma exchange are tracked over period ranging from 1 to 10 day. Intereting heat and ma tranfer behavior i oberved near the ground, near the urface, a well a immediately above and below dene ice crut. Information about depoition condenation) and ublimation rate may help explain now metamorphim phenomena that cannot be accounted for by temperature gradient alone. The macrocale heat and ma tranfer analyi require accurate repreentation of the effective thermal conductivity and the effective ma diffuion coefficient for now. We develop analytical model for thee parameter baed on firt principle at the microcale. The expreion derived contain no empirical adjutment, and further, provide elf conitent value for effective thermal conductivity and the effective diffuion coefficient for the limiting cae of air and olid ice. The predicted value for thee macrocale material parameter are alo in excellent agreement with numerical reult baed on microcale finite element analye of repreentative volume element generated from X-ray tomography. 1 Introduction The thermodynamically active nature of now, coupled with unuual high poroitie, poe ignificant challenge to modeling heat and ma tranfer in a now cover. A primary driver in much of the reearch on thi ubject ha been effort to explain the evolving microtructure of now that often occur in a matter of hour or day. Notably, now metamorphim, induced by trong temperature gradient in a now cover, i known to produce a highly faceted microtructure, the preence of which reult in extremely weak layer in a now cover. Weak layer have been oberved near the ground, near the urface, a well a above and below dene layer e.g., ice crut) within a now cover. While trong temperature gradient are widely accepted a the primary driver in temperature gradient metamorphim TGM), they do not fully account for the range of experimental obervation. For intance, lightly faceted crytal growth ha been oberved at low temperature gradient 3 K m 1 ) where rounded grain from intering have normally been oberved Flin and Brzoka, 2008). In contrat, Pinzer and Schneebeli 2009) note that rounded grain form have been oberved in urface layer ubjected to alternating temperature gradient of oppoite direction. An additional factor influencing now metamorphim i believed to be the rate of ma tranfer at the macrocale. The influence of ma tranfer at the macrocale i often neglected for the imple fact that depoition condenation) and ublimation rate caued by vapor diffuion and phae change are not known in a typical macrocale analyi. Vapor diffuion and the aociated phae change at the macrocale poe modeling challenge in that it force the macrocopic analyi toward a mixture theory where the ice and humid air contituent retain their identity. Mixture the- Publihed by Copernicu Publication on behalf of the European Geocience Union.

2 1858 A. C. Hanen and W. E. Folien: A macrocale mixture theory analyi ory itelf i a ubject that ha yet to fully mature and many open quetion remain. Implementing a macrocopic continuum mixture theory to elucidate the coupled heat and ma tranfer phenomena occurring in now i the central focu of thi paper. We tudy the effect of ma tranfer near the ground, near the urface including diurnal temperature effect, a well a adjacent to an ice crut within the now cover. Heat and ma tranfer rate are tracked over everal different time period ranging up to 10 day. The mixture theory analyi developed herein require an accurate aement of macrocopic propertie for effective thermal conductivity and the effective ma diffuion coefficient for now. Determining thee parameter require an analyi of heat and ma tranfer at the microcale. A major challenge in microtructural tudie of now metamorphim i the extremely complex 3-D tructure of the ice phae. Hitorically, generating an accurate geometric repreentation of the microtructure of now and further connecting it to a ubequent heat and ma tranfer analyi wa imply not poible. However, in the lat 2 decade, the ue of X-ray computed tomography ha profoundly altered experimental and theoretical reearch for now at the microtructural level. Not only can one accurately capture the true 3-D now microtructure, the evolution of the microtructure may be monitored in real time a metamorphim occur. Furthermore, finite element analyi may be coupled to experimentally produced 3-D microtructure to model heat and ma tranfer at the local cale. High-fidelity microcale numerical model, coupled with X-ray computer tomography, have been utilized by Riche and Schneebeli 2013) and Calonne et al. 2011) for prediction of macrocale effective thermal conductivity. Pinzer et al. 2012) and Flin and Brzoka 2008) ued finite element analyi with X-ray tomography to addre vapor diffuion. Evolution of the now microtructure and determining an effective diffuion coefficient for now are among their notable contribution. Finite element prediction baed on computer-generated X-ray tomography now tructure provide an excellent foundation for determining material propertie for effective thermal conductivity and the effective diffuion coefficient for now. However, intead of utilizing finite element micromechanic to generate macrocale material propertie, we rely on an intereting mathematical model developed by Folien 1994). The analytical model produce reult for effective thermal conductivity and the effective diffuion coefficient for now that are in remarkable agreement with the finite element prediction cited above. The model alo account for effective thermal conductivity and effective diffuion coefficient propertie over the entire range of denitie and temperature poible for now. The trong correlation of the analytical model material propertie compared with reult from microcale finite element analye of now lend confidence to uing material parameter baed on the analytical model in the macrocopic mixture theory analyi developed herein. 2 Reflection on geometric cale: microcale v. macrocale variable The critical heat and ma tranfer mechanim for now metamorphim play out at two ditinctly different geometric and time cale. At the microcale on the order of millimeter) now exhibit an extremely complex and evolving microtructure coniting of ice grain and humid air. At the macrocale, the geometric cale of interet i aociated with the depth of the now cover typically on the order of meter. Macrocopic variable of interet include denity, temperature, temperature gradient, a well a the ma flux of water vapor and the reulting depoition and ublimation that will occur within a now cover. Thee macrocopic variable are fundamental driver for now tructure evolution occurring at the microcale, thereby coupling local phenomena driving now metamorphim with macrocale heat and ma tranfer. When developing a theory that trancend multiple geometric cale, attention mut be paid to the tranition from the microcale to the macrocale, commonly referred to a homogenization. An implicit requirement neceary for homogenization in an upcale proce i appropriate eparation of cale, both from a geometric and phyical viewpoint. Auriault et al. 2009) provide extenive dicuion of neceary condition required for eparation of cale, all of which are atified for the preent work. A notable apect of the preent homogenization proce i that a mixture theory i introduced by defining now at the macrocale to be a mixture compoed of an ice contituent and a humid air contituent. The contituent variable may, in turn, be appropriately averaged to obtain the macrocale now field variable. Allowing the contituent to retain their identity provide a vehicle to tudy ma tranfer due to condenation and ublimation at the macrocale. A a mean of formalizing an upcaling proce for now, the concept of a repreentative volume element RVE) i introduced. The RVE mut be of ufficient ize uch that volume average of the contituent variable do not change a the volume i increaed. Given an RVE, let φ α denote the volume fraction of contituent α. The mixture contituent are immicible, and the contituent volume fraction are pace filling, leading to the relation φ i + φ ha = 1, 1) where ubcript i) and ha) denote the ice and humid air contituent, repectively. The denity of now, ρ, i defined by the volume average of the local microcale) denity field, γ m x), that varie throughout the RVE, i.e., The Cryophere, 9, ,

3 A. C. Hanen and W. E. Folien: A macrocale mixture theory analyi 1859 ρ = 1 V V γ m x)dv, 2) where, for clarity, the local denity may be expreed a γ m x) = γ i χ i x) + γ ha 1 χ i x)) 3) in term of the indicator function χ i x) of the ice phae. The ubcript m) on the local denity field i ued to emphaize that the variable i defined at the microcale. In the cae of a mixture, the integral of Eq. 2) may be broken into an ice domain and a humid air domain a ρ = 1 V V i γ m x)dv + 1 V V ha γ m x)dv. 4) Moreover, the following macrocale contituent denitie are introduced a γ i = 1 V i V i γ m x)dv, 5) and γ ha = 1 V ha V ha γ m x)dv. 6) Noting Eq. 4) 6) lead to a volume average expreion for the denity of now given by ρ = φ i γ i + φ ha γ ha. 7) We emphaize that the mixture formulation i defined entirely at the macrocale. Hence, all variable in Eq. 7) repreent macrocale quantitie. Following Özdemir et al. 2008), heat tranfer propertie are introduced into the micro macro upcaling proce by defining the macrocopic heat capacity a ρc V ) = 1 γ m C V ) dv, 8) V m V where C V i the pecific heat at contant volume. Thi equation provide a definition for the pecific heat of now yielding conitent value of heat capacity at both cale. Following the ame development a for the denity of now lead to the relation ρc V ) = φ i γi Ci V ) + φha γha Cha V ), 9) where the heat capacity for contituent α i given by γα Cα V ) 1 = C V ) dv. 10) V m α V α γ m Özdemir et al. 2008) further enforce conitency of the tored heat at the microcale and macrocale through the relation ρc V ) θ = 1 V V γ m C V ) m θ mdv, 11) where θ m and θ repreent the local temperature and macrocale temperature, repectively. Again, the integral of Eq. 11) may be eparated into an ice contituent and a humid air contituent a ρc V ) = φ i 1 V i V i γ m + φ ha 1 γ m V ha V ha C V ) m θ mdv C V ) m θ mdv. 12) Contituent temperature, θ i and θ ha, are introduced through the relation γ i Ci V θ i = 1 C V ) V m θ mdv, 13) i and γ ha C V ha θ ha = 1 V i γ m γ m V ha V ha C V ) m θ mdv. 14) The heat capacity i heterogeneou at the microcale but homogeneou in the ice phae, leading to a volume average temperature for ice given by θ i = 1 V i V i θ m dv. 15) For the range of temperature of interet, the ma fraction of water vapor in dry air i on the order of Hence, the thermal propertie of the humid air may be taken to be thoe of dry air, and the heat capacity of dry air i contant for the temperature variation een at the microcale. Thi condition lead to a volume average definition for the temperature of the humid air contituent given by θ ha = 1 θ m dv. 16) V ha V ha The temperature of now may be determined from ρc V ) θ = φ i γi Ci V ) θi + φ ha γha Cha V ) θha. 17) Hence, the temperature of now doe not follow the contituent volume averaging found for the heat capacity Eq. 9) and the denity Eq. 4) but rather i baed on a volume average weighted by the contituent heat capacitie. The temperature gradient at the microcale i a critical parameter driving temperature gradient metamorphim. To thi end, volume averaged temperature gradient for the ice and humid air contituent are introduced a The Cryophere, 9, , 2015

4 1860 A. C. Hanen and W. E. Folien: A macrocale mixture theory analyi θ i ice temperature gradient θ ha humid air temperature gradient where, for example, θ i = 1 V i V i x θ m x)dv. 18) The ubcript x on the gradient operator in Eq. 18) i ued to emphaize that the gradient applie at the microcale. Given appropriate boundary condition for the RVE, the macrocale temperature gradient for now atifie the volume weighted averaging: θ = φ i θ i + φ ha θ ha. 19) Özdemir et al. 2008) develop the pecific boundary condition for the RVE that are neceary to atify Eq. 19). Thee boundary condition are preciely the one ued by Pinzer et al. 2012) and Riche and Schneebeli 2013) in their finite element analye of heat and ma tranfer at the microcale. Finally, it i extremely important to recognize difference in behavior between local microcale) temperature gradient and the volume averaged macrocale temperature gradient. For intance, Pinzer et al. 2012) provide a figure of the local temperature gradient in an RVE for an applied macrocale temperature gradient of 50 K m 1. The color bar for the microcale temperature gradient indicate that local value of the temperature gradient are a high a 300 K m 1. The high local value of the temperature gradient compared to the macrocopic temperature gradient mut be kept in mind when interpreting macrocopic reult, a it i the local temperature gradient that drive metamorphim. Hence, when macrocale temperature gradient are preented a computed by the mixture theory analyi, it i not unreaonable to aume the microcale temperature gradient may be an order of magnitude higher in ome area of the RVE. 3 A mixture theory model for macrocale heat and ma tranfer The common phae change occurring in now have motivated everal tudie uing variant of mixture theorie. Morland et al. 1990) and Bader and Weilenmann 1992) developed a four contituent mixture theory for now where one of the contituent wa water. Phenomena uch a percolation, melting, and freezing are addreed, and momentum balance play a ignificant role in the work. The preent work doe not involve momentum balance, nor doe it allow for a water contituent. Gray and Morland 1994) developed a mixture theory for dry now baed on contituent of ice and dry air. Their work i in harp contrat to the preent tudy where water vapor i a critical component of the development. Indeed, the emphai of the preent work i the prediction of depoition and/or ublimation of water vapor at the macrocale. Adam and Brown 1990) tudied heat and ma tranfer in now uing a claical form of mixture theory where water vapor wa included. Their work focued on non-equilibrium condition of the contituent, wherea the preent work i baed on equilibrium of contituent temperature and a aturated vapor denity. Equilibrium v. non-equilibrium condition amount to a focu on different time cale. Aide from the different area of emphai in the tudy of phae change phenomena in now, the mixture theorie cited are baed on a claical theory of mixture, wherea the preent work i largely baed on a volume fraction mixture theory Hanen et al., 1991). The volume fraction theory produce the ame balance equation found in the claical development of mixture theory. However, the ummed contituent balance equation are not forced to reduce to thoe of a ingle continuum except for the pecial cae of a nondiffuing mixture. A a reult of relaxing thi contraint, the phyical definition of mixture variable a well a the contraint on ma, momentum, and energy interaction term aume more appealing form. We rely on the phyical argument of Sect. 2 to define mixture quantitie of interet. Albert and McGilvary 1992) incorporated the effect of ma diffuion in a heat and ma tranfer analyi of now centered on forced convection caued by windy condition cloe to the now urface, a phenomenon known a wind pumping. The equation developed involve a velocity of the humid air and condition where the now i not aumed to be aturated with water vapor. Thee condition only occur in now under extreme circumtance. Folien 1994) performed a dimenional analyi of the condition needed for convection and howed the Rayleigh number for typical now condition wa 1 2 order of magnitude below what i needed for the onet of convection. A a conequence, convection i not conidered, and the preent paper develop a theory with no air velocity, and further, a aturated vapor denity. The work of Calonne et al. 2014a) i perhap the mot cloely related to the preent work in that they developed the governing equation for macrocopic heat and water vapor tranfer in dry now by homogenization involving a multicale expanion. We draw comparion of their work for the governing macrocale equation a well a the expreion for effective thermal conductivity and the effective diffuion coefficient in now. A unique apect of the preent approach i that analytical model, grounded in firt principle at the microcale, are developed for the effective thermal conductivity and the effective diffuion coefficient in now. By tarting at the microcale, albeit with idealized microtructure, we are afforded the advantage of uing the true thermal conductivitie of ice k i ) and humid air k ha ) a well a the known diffuion coefficient of water vapor in air D v a ). The reulting The Cryophere, 9, ,

5 A. C. Hanen and W. E. Folien: A macrocale mixture theory analyi 1861 model for the effective thermal conductivity of now and the effective diffuion coefficient for now contain no empirical adjutment and are in remarkable agreement with highfidelity numerical prediction of thee parameter baed on now microtructure obtained from X-ray tomography. The model alo generate an analytical decription of the eparation of heat tranfer due to ma diffuion and heat tranfer due to conduction. Conitent with the dicuion on homogenization, we conider now at the macrocale to be a two-contituent mixture coniting of ice and humid air. The humid air itelf i treated a a mixture of water vapor and air. A chematic of the mixture theory analyi i hown in Fig. 1. For the temperature and preure encountered in now, the humid air may be treated a a mixture of two ideal gae where each ga occupie the ame volume, i.e., φ ha = φ v = φ a, 20) where ubcript v) and a) repreent water vapor and dry air, repectively. An important conequence of repreenting the humid air a a mixture of ideal gae i that both the water vapor and the air behave a though the other ga i not preent, thereby greatly implifying the analyi and allowing one to draw on claical reult for ideal gae. The balance equation for ma, momentum, and energy for a contituent, α, are given by Hanen, 1989; Hanen et al., 1991) a follow: Ma balance ρ α + ρ α v α ) = ĉ α, 21) Momentum balance ρ α a α = T α + ρ α g + ˆp α, 22) Energy balance ρ α u α = tr T α L α ) + ρ α r α q α + ê α. 23) In the above, v α and a α repreent the velocity and acceleration of contituent α, repectively, while L α repreent the velocity gradient; u α i the internal energy, r α i the heat upply notably radiation), and g i the gravity vector. The dipered denity of contituent α i denoted by ρ α and i related to the true denity, γ α, a ρ α = φ α γ α. 24) Wherea the volume fraction, φ α, appear explicitly in the definition of the dipered denity, ρ α, the partial tre, T α, and the energy flux, q α, are implicitly caled by the volume fraction. Finally, ĉ α, ˆp α, and ê α repreent ma, momentum, and energy upply term that arie from interaction between contituent. Following Hanen et al. 1991), the mixture theory upply term atify the appealing retriction Ice Dry air Humid Air Water vapor Figure 1. Schematic howing a continuum point of now with the aociated contituent for a mixture theory analyi. ĉ α = 0, 25) α ˆp α = 0, 26) α and ê α = 0. 27) α In what follow, the mixture theory balance equation are further pecialized to tudy the macrocale coupled heat and ma tranfer problem for now. 3.1 Ice contituent ma balance The balance of ma for the ice phae i given by ρ i + ρ i v i ) = ĉ i. 28) Auming the ma upply i poitive during condenation, we can write ĉ = ĉ i = ĉ ha. 29) Neglecting any ettling velocity lead to a ma balance for the ice contituent given by γ i φ i = ĉ, 30) where the ma denity of ice i taken a contant at 917 kg m Water vapor ma balance The development of the humid air ma balance differ from that of the ice contituent in that we begin at the microcale. Furthermore, only the ma balance of the water vapor i conidered becaue the air act only a a medium through which the water vapor diffue. The Cryophere, 9, , 2015

6 1862 A. C. Hanen and W. E. Folien: A macrocale mixture theory analyi Ma tranfer of the water vapor may be expreed a Bird and Lightfoot, 1960) γ v v v = γ v γ ha γ a v a + γ v v v ) + j v. 31) Equation 31) ay that the ma flux of the water vapor i due to the bulk fluid motion the barycentric velocity) plu a relative velocity due to diffuion. In the abence of a preure gradient, the barycentric velocity i zero, i.e., γ ha v ha = γ a v a + γ v v v ) = 0. 32) Ma balance due to diffuion may be expreed in the form of Fick law Bird and Lightfoot, 1960) a ) γv j v = γ ha D v a x, 33) γ ha where D v a i the binary diffuion coefficient for water vapor in air and x denote the gradient operator at the microcale. The diffuive flux can be expanded to give j v = D v a x γ v + γ v γ ha D v a x γ ha, 34) but the econd term on the right i negligibly mall becaue the ma fraction of aturated water vapor in air at 273 K i about Hence, ma tranfer of water vapor at the microcale may be decribed by γ v v v = D v a x γ v. 35) In the tranition to the macrocale, the ame phyical principle apply but one mut now ue an effective diffuion coefficient for water vapor. The need to introduce an effective diffuion coefficient for water vapor i attributed to the preence of the ice microtructure in now. Specifically, the preence of the ice contituent introduce vapor tranfer mechanim that both enhance and retard ma tranfer of water vapor when compared to a medium of humid air only. Thee ma tranfer mechanim are briefly dicued in Sect Defining D eff a the effective diffuion coefficient for the humid air contituent at the macrocale follow φ v γ v v v = ρ v v v = γ v, 36) where v v and γ v now repreent appropriately volume averaged macrocale variable. Note that the ma flux of water vapor i baed on the dipered denity, ρ v, in order to account for the reduced volume occupied by the humid air in the mixture. Finally, ince only the humid air contituent i aociated with diffuion in a mixture of ice and humid air, D eff alo repreent the effective diffuion coefficient for now. Again, noting air i imply the medium for ma tranfer of water vapor, the balance of ma for the vapor phae may be written a ρ v + ρ v v v ) = ĉ v. 37) Subtitution of the diffuive flux into Eq. 37) and noting ĉ v = ĉ ha = ĉ lead to ρ v ) γ v = ĉ. 38) Expanding the time derivative of the dipered denity of the water vapor give ρ v but φ v = γ v φ v = φ ha + φ v γ v, 39) = φ i. 40) The above reult, along with the ma balance for the ice contituent Eq. 30), can be ued to write Eq. 38) a γ ) ) v φ v γv γ v = ĉ 1, 41) γ i but the quantity γ v γ i 1. Neglecting thi term and noting φ v = φ ha, the ma balance equation for the water vapor become φ ha γ v ) γ v = ĉ. 42) Equation 42) tate that change in the water vapor denity at the macrocale are due to the divergence of the water vapor flux and ublimation or condenation a defined through the ma upply. 3.3 Momentum balance The momentum balance for the ice phae can be ued to find the tre and train in the ice phae. However, the effect that the ice tre ha on the vapor denity of the water i neglected, o the ice phae momentum balance i not conidered further. The momentum balance for the humid air phae become important when bulk fluid motion occur a in the cae of convection. Folien 1994) ha hown the Rayleigh number for a typical now cover i more than an order of magnitude below the critical value for the onet of convection, o convection i unlikely to occur except in extreme circumtance. Therefore, the momentum balance of the humid air phae i not conidered further. 3.4 Ice contituent energy balance The energy balance for the ice contituent may be expreed at the macrocale a ρ i u i = tr T i L i ) + ρ i r i q i + ê i. 43) The Cryophere, 9, ,

7 A. C. Hanen and W. E. Folien: A macrocale mixture theory analyi 1863 In the above, any velocity gradient in the ice, L i, i attributed to ettling and may be neglected. Moreover, heat generation from olar radiation i alo neglected but could eaily be included a Colbeck 1989) and McComb et al. 1992) have done. Thee aumption reduce the energy balance for ice to ρ i u i = q i + ê i. 44) The internal energy of the non-deforming ice i aumed to be a function of temperature only and i given by u i = C V i θ i θ ref ), 45) where Ci V i the pecific heat of ice at contant volume and θ ref i the reference temperature. The heat flux at the macrocale i expreed a Fourier law of heat conduction a q i = φ i k eff i θ i, 46) where ki eff i the effective thermal conductivity for the ice phae in now. Thi parameter hould not be confued with the thermal conductivity of pure ice k i ) a difference arie due to the complex microtructural network of the ice phae in now. The tortuoity of the ice phae, for example, play a role in ki eff. The only microtructure where k i and ki eff would be equal for 1-D heat tranfer would be the pore microtructure dicued in the preent paper. In a 3-D analyi of now, the two parameter are fundamentally different. Combining Eq. 44) 46), the energy balance for the ice phae i given by φ i γ i C V i i ) = φ i ki eff θ i + ê i. 47) 3.5 Humid air contituent energy balance A with the ice phae, the work term and the energy ource term of the humid air contituent are neglected, thereby reducing the energy equation to ρ ha u ha = q ha + ê ha. 48) The internal energy for the humid air mixture of ideal gae i given by γ ha u ha = γ a C V a θ ha θ ref ) + γ v C V v θ ha θ ref ) + u g ), 49) where u g i the latent heat of ublimation of ice. The above aume the reference value of the internal energy of ice wa et to zero a wa the cae. The definition for the energy flux vector for a mixture may be written a Bird and Lightfoot, 1960) q = q c + q d, 50) where q c i the conductive flux and q d repreent a contribution from the interdiffuion of variou pecie preent. In the cae of a mixture of water vapor and air, the energy flux i given by q ha = k ha x θ ha + u g γ v v v, 51) where γ v v v, i the ma flux of water vapor diffuing through air. Now conider now at the macrocale compoed of a mixture of humid air and ice. At thi cale, Eq. 51) mut be modified a q ha = φ ha k eff ha θ ha + φ ha u g γ v v v. 52) The interpretation of the volume fraction in each term on the right-hand ide of the above equation i clear when one view the energy flux acro a urface of a macrocale control volume of now. Specifically, the true energy flux of humid air mut be caled by the area fraction of the humid air at the control urface. From quantitative tereology, the area fraction i equal to the volume fraction, reulting in Eq. 52). Noting Eq. 36), ma tranfer of the humid air may be expreed a a diffuive flux, leading to q ha = φ ha k eff ha θ ha u g γ v, 53) where D eff repreent an effective diffuion coefficient for now. A in the cae of the ice phae, one mut recognize that kha eff repreent an effective thermal conductivity of the humid air in now, and thi parameter i different from the true thermal conductivity of humid air a a pure ubtance. The difference in the two parameter i again attributed to the complex microtructure of the humid air phae in now. In brief, jut a the effective thermal conductivity of now, k eff i influenced by microtructure, o are ki eff and kha eff a all three parameter are macrocale quantitie. A uch, they depend on a hot of microtructural variable other than temperature. Subtituting Eq. 49) and 53) into Eq. 48) lead to φ ha γa Ca V + γ vcv V ) ha γ ) ) v + u g φ ha γ v ) = φ ha kha eff θ ha + ê ha, 54) but ĉ = γ v ) φ ha γ v, 55) from the ma balance of the water vapor given by Eq. 42). Therefore, Eq. 54), governing the energy balance of humid air, aume the form φ ha γa Ca V + γ vcv V ) ) ha = φ ha kha eff θ ha + ê ha + u g ĉ. 56) Hence, the change in internal energy for the humid air i attributed to the divergence of the heat flux, energy exchange with the ice contituent through the energy upply, and energy exchange through phae change accounted for by the ma upply. The Cryophere, 9, , 2015

8 1864 A. C. Hanen and W. E. Folien: A macrocale mixture theory analyi 4 Separation of cale: macrocale obervation In thi ection, we dicu ome obervation that lead to important implification in the macrocale heat and ma tranfer olution. Moreover, we demontrate eparation of the time cale for local and global heat and ma tranfer, a condition required for homogenization. 4.1 Macrocale temperature An important implification in the analyi of heat and ma tranfer at the macrocale i to aume the contituent temperature are equal and write θ = θ i = θ ha, where θ i the macrocale temperature of now. Jutification for auming the ice and humid air temperature are equal tart by writing a 1-D heat conduction equation at the microcale given by α ) kα 2 θ α = γ α Cα V x 2. 57) Equation 57) i non-dimenionalized by introducing the following dimenionle variable: t = t/t o, x = x/l c, and θ = θ θ init θ f θ init. The reulting non-dimenional equation i ) = to k α 2 θ L 2 c γ αcα V. 58) x 2 The time cale, to micro, for heat conduction on the microcale i introduced a t micro o = γ αc V α L2 c k α. 59) The time cale, to macro, for heat conduction in a now cover i imilarly defined a to macro φi γ i Ci V + φ ha γ ha C V ) ha H 2 =, 60) k eff where H i the height of the nowpack and k eff repreent the effective thermal conductivity for now. Riche and Schneebeli 2013) provide an expreion for the effective thermal conductivity of now a a function of now denity. Auming a now denity of 200 kg m 3, a depth of 1 m, and a microcale characteritic length of 1 mm, the ratio of the time cale for heat conduction on the macrocale of the nowpack to the time cale for heat conduction on the microcale i on the order of 10 6, which ugget that macrocale thermal equilibrium between the ice and humid air contituent i a good aumption. Moreover, the large The Cryophere, 9, , 2015 eparation of cale in the time domain i conitent with the dicuion of Auriault et al. 2009) regarding eparation of time cale neceary for homogenization. The aumption of uniform contituent temperature at the macrocale hould not be confued with the local microcale) temperature. Under a macrocale temperature gradient, local contituent temperature in the interior of the RVE differ due to different thermal conductivitie of the ice and humid air. Further, temperature gradient within individual contituent are alo preent at the microcale. A warmer ice grain i eparated from a colder ice grain by pore pace, for example. Thee temperature differential drive the ma tranfer proce at the microcale. Again, an excellent inight into microcale thermal behavior i provided in Fig. 4 of Pinzer et al. 2012). Thermal equilibrium of the ice and humid air contituent at the macrocale allow the contituent energy equation, Eq. 47 and 56), to be added together to yield an energy equation for now with a ingle temperature a φha γ ha C V ha + φ iγ i C V i ) = k eff θ ) + ĉu g, 61) where θ i the temperature of the now. Notably, the contituent energy upply term um to zero in the energy equation for now and the volume averaged contituent effective thermal conductivitie have been aborbed into an effective thermal conductivity for now, k eff, a k eff = φ i ki eff + φ ha kha eff. 62) While the effective thermal conductivitie, ki eff and kha eff, are never computed, it would be important to do o if one wanted to tudy non-equilibrium contituent temperature on a hort time cale with a mixture theory. One can make a direct connection of ki eff and kha eff with the work of Calonne et al. 2014a). Specifically, the tenorial form of the effective thermal conductivity for now i defined in Eq. 25) of Calonne et al. 2014a) a k eff = 1 V V a k a t a + I)dV + V i k i t i + I)dV, 63) where t α characterize the temperature fluctuation in contituent α and I i the identity tenor. The above equation may be rearranged a k eff 1 1 = φ a k a t a + I)dV + φ i k i t i + I)dV. 64) V a V i V a Comparing Eq. 62) and 64) provide a clear mathematical interpretation of ki eff and kha eff a k eff ha = 1 k a t a + I)dV, 65) V a V a V i

9 A. C. Hanen and W. E. Folien: A macrocale mixture theory analyi 1865 and k eff i = 1 k i t i + I)dV. 66) V i V i Finally, recent reearch work ha hown the effective thermal conductivity of now to be aniotropic, ee for example Schertzer and Adam 2011) and Riche and Schneebeli 2013). We avoid thi complexity at preent a it become a non-iue for the 1-D heat and ma tranfer theory developed ubequently. To ummarize, the governing equation for heat and water vapor tranfer in now are given by Eq. 42) and 61). Thee equation are identical to macrocale equation developed by Calonne et al. 2014a) through a decription at the pore cale uing the homogenization of multiple cale expanion. The equality i bet hown by multiplying the right-hand ide of Eq. 20) in Calonne by ρ i /ρ i ) and relabeling L g /ρ i ) a u g, reulting in Eq. 61) of the preent paper. Equation 42) i already identical in form to Eq. 21) of Calonne et al. 2014a). While the equation of Folien 1994) and Calonne et al. 2014a) governing the macrocale repone of heat and ma tranfer in now are identical, the emphai of Calonne work i on upcaling, wherea the preent paper focue on olution of the macrocale behavior. We alo addre imilaritie and difference in the calculation of effective thermal conductivity and the effective diffuion coefficient for now, critical parameter affecting macrocale ublimation and depoition rate in a now cover. 4.2 Saturated vapor denity at the macrocale A phyical interpretation of the ma upply term, ĉ, i the ma rate at which water vapor i condening to form ice per unit volume of now. Hobb 1974) provide an expreion for the condenation of water vapor to ice driven by a difference in the vapor preure and the aturated vapor preure over ice, p p at ), a α c m mol p p at ) 2πm mol θ) 1/2 kg m 2 1, where m mol i the ma per molecule of water, i Boltzman contant, and α c i the condenation coefficient. Multiplying the above expreion by the pecific urface area of now, ξ, and utilizing the ideal ga law for water vapor provide an explicit expreion for the ma upply driven by a difference in vapor denity given by ) ĉ = ξrθα cm mol γv γv at 2πm mol θ) 1/2. 67) In the abence of diffuion, Eq. 67) can be combined with the ma balance equation Eq. 42) for the water vapor a γ v φ v = ξrθα cm mol γv γ at ) v 2πm mol θ) 1/2. 68) If the aturated vapor denity over the ice i held contant, the time for the vapor denity difference between the pore denity and the aturated vapor denity to become 0.1 % of the initial denity difference can be computed. Delaney et al. 1964) meaured the condenation coefficient, α c, of ice to be for temperature between 271 and 260 K. For a now denity of 200 kg m 3 and a pecific urface area of 1400 m 1, the time for the vapor denity in the pore to reach equilibrium i approximately Hence, the vapor denity in a pore can be aumed to be the aturated vapor denity throughout the proce of heat and ma tranfer occurring at the macrocale where the time cale of interet i on the order of hour or day. The knowledge that the vapor denity may be aumed aturated in a macrocale analyi afford a critical implification in the mixture theory analyi in that a contitutive law for the ma upply i no longer needed. Intead, the ma upply i computed from Eq. 42) by noting the water vapor i alway aturated at the now temperature, leading to ĉ = γv at ) γv at φ ha. 69) We emphaize that Eq. 67) i not utilized in the nowpack modeling of water vapor depoition and ublimation found in Sect. 6 a it i replaced by Eq. 69). 4.3 Formulation ummary At thi point, we retrict the development to a 1-D model and write the energy equation, Eq. 61), a φha γ ha C V ha + φ iγ i C V i ) = x k eff x ) + ĉu g. 70) The ma upply equation, Eq. 69), repreenting phae change due to condenation or ublimation aume the 1-D form ĉ = x γ at v x ) γv at φ ha. 71) The aturated vapor denity may be expreed a purely a function of temperature Dorey, 1968) leading to γ at v x dγ at = v dθ x and γ v at = dγ at v dθ. Noting the above, the ma upply equation, Eq. 71), i expreed a ĉ = x dγ at v dθ x ) φ ha dγ at v dθ. 72) Finally, ubtituting Eq. 72) into Eq. 70) lead to a ingle partial differential equation governing the energy balance for The Cryophere, 9, , 2015

10 1866 A. C. Hanen and W. E. Folien: A macrocale mixture theory analyi now given by φ ha γ ha C Vha + φ iγ i C Vi dγv at + u g φ ha dθ = x where k con+d k con+d = k eff x ) ), 73) + u g D eff dγv at dθ. 74) The thermal conductivity k con+d i the apparent effective thermal conductivity of now that account for heat conduction, k eff, a well a energy tranfer due to water vapor diffuion. Rather than combining Eq. 70) and 72) and olving Eq. 73), it i more inightful to olve Eq. 70) and 72) eparately. Retaining a eparate equation for the ma upply allow one to quantify macrocale depoition and ublimation rate, a fundamental objective of the theory developed herein. 5 Evaluation of the effective thermal conductivity and the effective diffuion coefficient for now Solution of the energy equation Eq. 70) and the ma balance equation Eq. 72) require knowledge of macrocale parameter for effective thermal conductivity a well a the effective diffuion coefficient for now. Calonne et al. 2011) and Riche and Schneebeli 2013) have performed extenive numerical tudie uing finite element analyi coupled with X-ray tomography to quantify the effective thermal conductivity for now a a function of denity at a fixed temperature. Calonne et al. 2011) alo provide effective thermal conductivity prediction at two eparate temperature. Pinzer et al. 2012) and Chriton et al. 1994) performed numerical tudie aimed at determining the effective diffuion coefficient for now. Calonne et al. 2014a) alo ued finite element micromechanic to predict an effective diffuion coefficient for now although the pecific numerical approach to evaluate thi parameter followed a fundamentally approach. Regardle of the parameter being tudied, a drawback of microcale finite element analyi micromechanic) i that the reult provide heat and ma tranfer propertie at a ingle temperature and denity. Hence, a complete characterization of thee parameter a a function of denity and temperature require a ignificant number of micromechanic olution at multiple denitie and temperature followed by a curve-fitting exercie. Rather than relying on finite element micromechanic olution, we preent an analytical approach developed by Folien 1994) to predict value for the effective thermal conductivity and the effective diffuion coefficient of now. Folien model ha everal attractive feature including the following: b) Lamellae Microtructure Ice layer Humid Air layer a) Pore Microtructure Figure 2. Idealized microtructure ued to model heat and ma tranfer in parallel a) and erie b). there i excellent correlation with cited finite element reult for effective thermal conductivity and effective diffuion coefficient for now; denity effect are explicitly introduced in the analytical model through volume fraction, while temperature effect appear implicitly through thermal conductivity propertie for ice and air; the effect of ma diffuion on the energy flux are explicit and the relative influence on the energy flux i readily determined; the model provide elf-conitent reult for effective thermal conductivity and effective diffuion coefficient for now for the limiting cae of air and ice; the model i developed from firt principle of heat and ma tranfer applied to imple microtructure and contain no empirical coefficient of adjutment. Folien development begin by formulating microcale heat and ma tranfer model for claic microtructure coniting of ice and humid air acting in parallel and erie, repectively. Heat and ma tranfer propertie for now are then propoed uing argument from quantitative tereology. Figure 2a how an ice matrix with humid air pore in parallel to an applied temperature gradient. In thi pore microtructure, energy i tranferred in parallel through the nowpack. The energy fluxe for the ice q i ) and humid air q ha ) contituent are imply added together to obtain the total energy flux through the nowpack. Becaue the thermal conductivity of ice i roughly 100 time larger than for the humid air, the ice phae play a dominant role in heat tranfer for thi microtructure. The Cryophere, 9, ,

11 A. C. Hanen and W. E. Folien: A macrocale mixture theory analyi 1867 The econd microtructure tudied, referred to a a lamellae microtructure, conited of ice and humid air layer oriented perpendicular to the energy flux Fig. 2b). In thi cae, energy flow in erie through the repective layer. Hence, the energy flux in the humid air contituent mut equal the energy flux through the ice contituent. An intereting feature of ma tranfer in the lamellae microtructure i that diffuion via the hand to hand model decribed by Yoida 1955) i naturally preent and accounted for in the development. Specifically, diffuion i enhanced a the total path length for diffuion i reduced by the ice layer which act a both a ource and ink for water vapor. The two microtructure tudied by Folien 1994) were firt conidered by de Quervain 1963) and produce two very different heat and ma tranfer reult that are believed to repreent the extreme poible for ice and humid air mixture. 5.1 Pore microtructure Folien heat and ma tranfer analyi of the pore microtructure begin by writing energy flux expreion for the ice and humid air contituent at the macrocale. The energy flux of the ice i attributed to heat conduction, leading to q i = k i x. 75) The energy flux of the humid air i attributed to conduction of the humid air and the ma flux of water vapor. Following Bird and Lightfoot 1960) we can write q ha = k ha x u dγ v at gd v a dθ x. 76) The energy flux of the pore microtructure i introduced a q pore = k pore x. 77) Energy tranfer in the pore microtructure occur in parallel and the energy flux i imply the volume average of the energy fluxe of the ice and humid air leading to k pore = φ i k i + φ ha k ha + φ ha u g D v a dγ v at dθ. 78) 5.2 Lamellae microtructure The dicontinuou nature of the lamellae microtructure in the direction of interet introduce a complexity in the patial gradient, a the contituent gradient mut be defined with repect to a differential length, dx α. Hence the contituent energy fluxe aume the form q i = k i x i, 79) and dγ v at q ha = k ha u g D v a. 80) x ha dθ x ha 1-D Heat Tranfer Figure 3. Tet line through a now urface ection howing 1-D heat and ma tranfer at the microcale. The average temperature gradient expreed in term of the macrocale coordinate x i given by x = φ i + φ ha. 81) x i x ha The energy flux through the lamellae microtructure i introduced a q lam = k lam x. 82) Equation 79) 82) may be combined to arrive at ) dγv k i k ha + u g D at v a dθ k lam = ). 83) dγv φ i k a + u g D at v a + φ ha k i 5.3 Snow propertie dθ The energy flux for now account for heat conduction a well a energy tranfer due to water vapor diffuion. From Eq. 73) and 74), the energy flux may be identified a q = k eff + u g dγv at ) dθ x. 84) Folien 1994) propoed an energy flux for now that include energy tranfer due to heat conduction and ma diffuion a q = φ i q pore + φ ha q lam. 85) Jutification for Eq. 85) i provided by conidering a now urface ection a hown in Fig. 3. When a tet line i arbitrarily drawn through the urface ection, a fraction of the total The Cryophere, 9, , 2015

12 1868 A. C. Hanen and W. E. Folien: A macrocale mixture theory analyi length will pa through the ice contituent, and the remainder will pa through the humid air contituent. If one imagine a 1-D heat tranfer occurring along the tet line, heat tranfer through the ice phae i dominated by the pore microtructure where the thermal conductivity of ice i nearly 100 time that of air. In contrat, anytime the tet line pae through the humid air contituent, heat tranfer would be dominated by the lamellae microtructure. Uing the lineal fraction a the weighted behavior of the thermal conductivity and recognizing the lineal fraction i identical to the volume fraction under condition of iotropy Underwood, 1970) lead directly to Eq. 85). Combining Eq. 77) 78) and Eq. 82) 83) with Eq. 85) lead to an expreion for the energy flux of now given by q = φ i φ ha k ha + φ i k i ) + φ ha k i k ha ) dγ φ i k ha + u g D v at v a dθ + φ ha k i x φ i φ ha D v a ) + φ ha k i D v a ) dγ φ i k ha + u g D v at v a dθ + φ ha k i dγv at u g dθ x. 86) Motivated by the functional form of Eq. 84) and 86), we define the effective thermal conductivity and effective diffuion coefficient a k eff and =φ i φ ha k ha + φ i k i ) + φ ha k i k ha ), 87) dγv φ i k ha + u g D at v a + φ ha k i dθ =φ i φ ha D v a ) + φ ha k i D v a ). 88) dγv φ i k ha + u g D at v a + φ ha k i dθ Depite the preence of the binary diffuion coefficient of water vapor in air in the expreion for k eff, it hould be emphaized that the reult given in Eq. 87) repreent the effective thermal conductivity for now a predicted by the analytical model. Similarly, contituent thermal conductivity parameter appear in the equation for the effective diffuion coefficient of now, D eff. Thee reult are a conequence of a direct application of heat and ma tranfer principle for the lamellae microtructure the parameter of thermal conductivity and diffuion imply do not eparate at the macrocale for thi microtructure. A good deal of clarity in the phyical interpretation of k eff and D eff may be achieved through an order of magnitude The Cryophere, 9, , 2015 analyi of the variou term in Eq. 87) and 88). To begin, for the range of temperature of interet, one may how dγv k ha and u g D at v a dθ ) are of the ame order of magnitude. Now rearrange Eq. 87) and 88) by dividing numerator and denominator of the lat term in each by k i, leading to k eff and = φ i φ ha k ha + φ i k i ) + φ ha [ φ i = φ i φ ha D v a ) + φ ha [ φ i k ha k ha +u g D dγ v at v a dθ k i D v a ], 89) + φ ha k ha +u g D dγ at ] v v a dθ k i + φ ha. 90) The value of the thermal conductivity of ice i on the order dγv of 100 time that of the term k ha + u g D at v a dθ ). Therefore, neglecting the term in quare bracket in the above expreion for k eff and D eff lead to k eff = φ i φ ha k ha + φ i k i ) + k ha, 91) and = φ i φ ha D v a + D v a. 92) Equation 91) and 92) reveal a deirable conitency in term. Specifically, the effective thermal conductivity of now depend only on the thermal conductivitie of ice and humid air, repectively, while the effective diffuion coefficient for now depend only on the binary coefficient of water vapor in air. Hence, the thermal conductivity and diffuion expreion decouple from one another. Owing to the clean nature of the implified form for k eff and D eff, one might be tempted to ue them at all time. That approach i, indeed, valid for the effective thermal conductivity a the implified effective thermal conductivity curve i nearly identical to the original propoed by Folien. However, important difference arie in the diffuion prediction. Figure 4 how the effective diffuion curve predicted by Eq. 88) and 92), repectively. The two curve are identical over a wide range of denitie from approximately 0 to 400 kg m 3. A the curve deviate at higher denitie, the original form propoed by Folien i neceary to drive D eff to the known limiting value of zero for olid ice. The conitency of Folien model i impreive in thi regard. There i yet another phyically pleaing apect of Folien model for the effective diffuion coefficient for now. Uing the implified form of Eq. 92), one can write the effective diffuion coefficient a = φ i φ ha D v a ) + φ ha Dv a φ ha ). 93)

13 A. C. Hanen and W. E. Folien: A macrocale mixture theory analyi Predicted Diffuion Coefficient Enhancement for Snow 2.5 Folien 1994): 253K Riche et. al. 2013): 253K Calonne et. al. 2011): 271K Snow Thermal Conductivity k ice 1 2 D / D v a k W m 1 K 1 ) Folien 1994) Folien modified by order of magnitude analyi Snow Denity kg m 3 ) Snow Denity kg m 3 ). Figure 4. Predicted diffuion coefficient enhancement for now uing the original model of Folien and a implified model baed on an order of magnitude analyi. Figure 5. Thermal conductivity analytical prediction of Folien 1994) v. finite element prediction of Calonne et al. 2011) and Riche and Schneebeli 2013). The leading volume fraction in each of the term in the above equation i attributed to the volume fraction weighting of the now model propoed by Folien, allowing u to identify effective diffuion coefficient for the pore and lamellae microtructure a ) Dv a D pore = φ ha D v a ) and D lam =. φ ha In the cae of the effective diffuion coefficient for the pore microtructure, the humid air volume fraction lead D v a. The interpretation of φ ha i quite clear, a the ice phae act a a blockage and limit the amount of area for humid air ma tranport to occur. The influence of the ice phae on the effective diffuion of water vapor i fundamentally different for the lamellae microtructure compared to the pore microtructure. Firt, the ice doe not act a a blockage of diffuion path in the lamellae microtructure a it doe in the pore microtructure. Secondly, the ice phae actually enhance water vapor diffuion in the lamellae microtructure by hortening the pathway needed to travel via the hand to hand mechanim decribed by Yoida 1955). For example, given an ice volume fraction of 0.5, one would expect the diffuion coefficient of the lamellae microtructure to be double that found in humid air a water vapor would only have to travel half the ditance compared to the ditance traveled in humid air alone. Taken collectively, thee factor ugget the influence of φ ha on the diffuion coefficient D lam hould cale a 1/φ ha ), preciely a Folien model predict. While the idealized microtructure utilized by Folien are not repreentative of the complex microtructure of now, the ma tranfer mechanim decribed above that are aociated with each microtructure are clearly preent in now. Im- portantly, the propoed diffuion model capture thee mechanim Effective thermal conductivity Calonne et al. 2011) and Riche and Schneebeli 2013) provide curve fit of now effective thermal conductivity a a function of denity baed on their finite element micromechanic analye. Calonne data included analyi of crytal tructure of all type, while Riche data were limited to depth hoar and faceted crytal which produce higher thermal conductivitie in the direction of interet normal to the ground). Figure 5 provide the prediction of Eq. 87) for a temperature of 253 K againt the curve fit of Calonne et al. 2011) and Riche and Schneebeli 2013). The correlation of the analytical model i excellent a the model virtually track the numerical reult of Riche and Schneebeli 2013) whoe data were alo generated at 253 K. Folien predicted curve at 271 K hift downward toward the curve generated by Calonne et al. 2011), alo generated at 271 K, but remain well within the bound of both curve generated through finite element analyi of real microtructure. Furthermore, the mot ignificant deviation of the analytical model occur at a denity for olid ice where Folien model predict the elf-conitent correct reult of thermal conductivity for ice. Change in effective thermal conductivity a a function of temperature were oberved by Calonne et al. 2011) for temperature of 271 and 203 K, repectively. Figure 6 how the effective thermal conductivity line predicted by Folien along with the numerical micromechanic prediction of Calonne et al. 2011). Excellent correlation of the analytical model and the finite element analye i again oberved. The Cryophere, 9, , 2015