Introduction

Large solid precipitation particles, such as graupel and hailstones, grow by accretion of supercooled cloud droplets on ice crystals or frozen droplets. If the droplets freeze immediately on contact they form a coating of rime in the form of an aggregate of more-or-less deformed frozen droplets, with the interstices filled with air. This leads to low-density rimed crystals or graupel. However, if the freezing is slow, denser ice structures such as transparent hailstone shells are created, leaving the accreted droplets indistinguishable from each Other. Thus, it is no surprise that a wide variation of densities of ice particles has been reported. Natural graupel were found to range from 250 to 800 kg m⁻³ (Reference ListList, 1958; Reference Zikmunda and vali.Zikmunda and Vali, 1972; Reference HeymsfieldHeymsfield, 1978). The densities of natural hailstones collected at the ground and stored in freezers were measured at 820–920 kg m^–3 (Reference ListList, 1958; Reference Macklin, Strauch and LudlamMacklin and others, 1960; Reference List, Cantin and FerlandList and Others, 1970; Reference ProdiProdi, 1970; Reference Matson and HugginsMatson and Huggins, 1980). The actual density of hailstones in clouds can be lower than that at the ground since melting often leads to the intake of meltwater by low-density ice regions (Reference List. and Weickmann.List, 1960; Reference Macklin.Macklin, 1963; Reference Kidder and CarteKidder and Carte, 1964; Reference ProdiProdi, 1970). The density can also be higher when spongy ice, a mixture of ice and water, is grown with the density approaching the of water. For artificial hailstones produced under dry growth conditions, Reference Knight and HeymsfieldKnight and Heymslield (1983) measured densities of 310–600 kg m⁻³, values much lower than for bulk ice (ρ _i = 915 kg m ⁻³). By not considering these variations, all previous cloud and precipitation studies have implicitly assumed that the thermal properties of porous accreted) ice are equal to those of bulk ice.

Many studies of the thermal conductivity, k _i, of bulk ice have been reviewed by Reference Powell.Powell (1958). He shows that K _i varies with temperature and pressure, with a typical value of k _i = 2.18 W m ⁻¹ K⁻¹ at 0°C. Thereby, k _i, of ice is mostly dependent on temperature; it is little affected by ambient pressure. Measurements of the thermal conductivity of natural snow and low-density frost have also been reported in the literature (Reference Kondrat'evaKondrat'eva, 1954, Reference Yen.Yen, 1962; Reference Pitman. and Zuckerman.Pitman and Zuckerman, 1967; Reference Weller and Schwerdtfeger.Weller and Schwerdtfeger, 1971; Reference Dietenberger.Dietenberger, 1983; Öslin and Andersson, 1991). It is significantly lower than that of bulk ice, and mostly depends on density, not on temperature (considering a range of −40° to 0°C). Although theoretical modeling of the thermal conductivity of porous ice applicable to comet nuclei has been reported (Reference Espinasse, Klinger, Ritz and SchmittEspinasse and others, 1991; Reference Seiferlin and Kömle.Seiferlin and Kömle, 1991), little is known about the measured k _i values of porous ice with densities >600kg m⁻³ (Reference Hobbs.Hobbs, 1974; Reference Dietenberger.Dietenberger, 1983; Öslin and Andersson, 1991).

Porous ice consists of frost that is grown either over very long times by deposition of water vapor or over short times (>1h) by accretion and freezing of supercooled water droplets. The theoretical modeling has demonstrated that the thermal conductivity of porous ice is heavily dependent on the ice texture (i.e. the degree of sintering among the single grains) (Reference Espinasse, Klinger, Ritz and SchmittEspinasse and others, 1991; Reference Seiferlin and Kömle.Seiferlin and Kömle, 1991). Therefore, the present investigation will be restricted to the determination of the k _i values of accreted porous ice applicable to hailstone shells over the density range of 600–910 kg m⁻³.

Heat Conduction

In the analysis of heat transfer of a solid by conduction, both the thermal conductivity and diffusivity need to be known. The proportionality factor, k _i, links the temperature gradient (∂T/∂n) to the conductive heat flux . Hence,

k _i is a material property which normally varies with temperature. Within the range of atmospheric temperatures for ice growth, the variation of k _i for bulk ice is within ± 10% (Reference Powell.Powell, 1958).

The thermal diffusivity, α _i is defined as the ratio of the thermal conductivity to the heat capacity per unit volume (ρ_ic_p):

where ρ_i and c _p are the density and the specific heat capacity of the solid, respectively. It represents the ability of the solid to conduct thermal energy relative to the ability to store it.

At the surface of an atmospheric ice particle, not exposed lo accretion and freezing of collected cloud droplets, the heat transfer by conduction and convection into the surroundings

, the heat released by evaporation, sublimation or deposition and by thermal radiation must balance with the heat flux from the interior by conduction :

Here it is assumed that the air flows around the test particles and that the interior heat transfer takes place only via heat conduction. The terms on the right hand side of Equation (3) are given by

where k _a is the thermal conductivity of air k_a = 0.0224 W m⁻¹ K⁻¹ at −20°C), D is the particle diameter. T is the temperature, D _v is the diffusivity of water vapor, L _s is the latent heat of sublimation. ρ_v is the water-vapor density, ϵ is the emissivity of the surface, σ is the Stefan-Boltzmann constant (σ = 5.6697 × 10⁻⁸ Wm⁻²K⁻⁴), and Nu and Sh are the Nusselt and Sherwood numbers, respectively; the subscripts “s” and “a” represent surface and air, respectively; the overbar refers to the average over the entire particle surface. Within the temperature range and conditions of the present experiments.

and are small (∼5% and <3%, respectively) compared to , and can be neglected in Equation (3). Thus, using Equation (1) for the surface, the heat-balance Equation (3) reduces to

where

is the average temperature gradient at the surface within the particle. The average Nusselt number, Nu, represents the non-dimensional, overall temperature gradient at the surface, and it provides a measure of the convective heat transfer occurring at the surface. For a given particle in a known fluid, D, k _a and T _a are known. Since Nu varies only with the physical properties of the fluid and the surface characteristics (i.e. roughness, size and geometry) and not with the internal properties (Reference Incropera and DeWittIncropera and DeWitt, 1990), the overall Nu value of bulk-ice particles can also be used for the low-density particles with any k _i. Nu has been determined by Reference Zheng and List.Zheng and List (1996):

where Re is the Reynolds number and a is the aspect ratio of the particle. The average surface temperature, T _s, in Equation (7) is continuously and remotely measured and averaged with an AGEMA infrared surface-temperature scanning system, while

can be determined by integrating over the area-weighted surface temperature gradients which are calculated by solving the heat-conduction equation

The calculation is performed with a given initial internal temperature field and the measured surface temperature held, and stepped forward in time to obtain the time variation of the internal temperature distribution (Reference Zheng and List.Zheng and list, 1996). Since the heat-conduction Equation (9) also requires knowledge of the thermal dilfusivity α _i, which is related to k _i according to Equation (2), this quantity can be calculated with Equation (7) through iterations that converge towards the exact solution for k _j. The main error sources in determining k _i are the uncertainties in T _s and

. The magnitude of the net uncertainty of (Δk_i /k₁ is <7%.

Experiment

Results and Discussion

Figure 3 gives the lime variation of the dimensionless average surface temperature,

, of the porous ice-test particles, cooling in an airflow, as established with the AGEMA system. The lower the density of the particle, the faster the cooling, and the lower the surface temperature becomes. In other words, the density changes the thermal properties, affecting the temperature field at the surface and, therefore, within the particle.

Fig. 3. The time variation of the dimensionless surface temperatures of ice particles (D = 2.1 cm, α = 0.7, T_i = (-6°C) with different densities, cooling in an airflow (T_a = −15°C, V_a = 15 ms⁻¹, Re = 2.6 × 10⁴).

The time variation of the average surface temperature was used to determine the thermal conductivity, k _i of the porous ice by Equation (7). Figure 4 is a plot of k _i of porous ice vs the particle density ρ _i at T _a = −15°C. It is apparent that porous ice exhibits lower values of k _i than bulk ice. Porous ice with ρ _i = 620 kg m ⁻³ has only 55% of k _i for bulk ice. Reference Dillard and TimmerhausDillard and Timmerhaus (1966) showed that k _i of bulk ice increased 5% and 11% as temperature decreased from 0° to −20° and −40°C, respectively. Thus, under atmospheric cloud conditions, the density of porous ice influences the thermal properties more than temperature does. The variation of k _i with temperature may be neglected when the density varies.

Fig. 4. The thermal conductivity, k_i, of porous ice as a function of density. ρ_i, at T_a = 15°C. The correlation coefficient of the linear relationship between k_i and ρ_i is 0.996. The value of k_i for bulk ice is also plotted (open circle) for comparison.

Within the experimental range of this investigation (i.e., 620 ≤ ρ _i ≤ 915 kg m⁻³), a linear-least-square fit gives

with a correlation coefficient of 0.996. [k_i] = W m⁻¹K⁻¹ and [ρi] = kg m⁻³.

The dependence of k _i on density is caused by the ice structure with its various air inclusions. The presence of air bubbles enhances the cooling (Fig. 3) because there is less ice to cool. However, the values of k _i are lower (Fig. 4) because heat is not conducted well through the air enclosures. Physically speaking. k _i of porous ice is an effective thermal conductivity which is determined by the conductivities, volume fractions and shapes of the air spaces and the ice, and the degree of sintering. As expected by the theoretical modeling (Reference Espinasse, Klinger, Ritz and SchmittEspinasse and others, 1991; Reference Seiferlin and Kömle.Seiferlin and Kömle, 1991), the thermal conductivity of porous ice depends heavily on the ice texture. Therefore, the k _i values of the present investigation are valid for accreted porous ice applicable to hailstone shells.

The thermal diffusivity. α _i, of porous ice can be determined from the indirectly measured themal conductivity, k _i, and the measured density, ρ _i, of the test particle by Equation (2), and expressed as a function of ρ _i (see Fig. 5). α _i becomes smaller as ρ _i decreases. However, the relative variation of α _i with ρ _i is small compared to k _i. since α _i is reduced by only 18% as ρ _i decreases from 915 to 620 kg m ⁻³. This compares with 55% for k _i. The ability of the porous ice to store heat is reduced with decreasing ρ _i. Thus, low-density ice with small α _i responds faster to change in its environment than bulk ice. The value of α _i can also be expressed by a linear relationship

over the range of density 620 ≤ ρ _i ≤ 915 kg m⁻³, with a correlation coefficient of 0.960, and [α _i] = m²s⁻¹.

Fig. 5. The thermal diffusivity, α_i, of porous ice as a function of density. ρ_i, at T_a = −15°C. The correlation coefficient of the linear relationship between α_i and ρ_i is 0.960.

Figure 6 summarizes the measurements of the thermal conductivity k _i of ice and snow as a function of density ρ _i. Reference Kondrat'evaKondrat'eva (1954) determined k _i of snow (with stagnant air) for ρ _i < 350 kg m⁻³, and correlated it to the square of the density (ρ _i ²). Reference Yen.Yen (1965) measured the effective thermal conductivity of unconsolidated and ventilated snow with a density of 376–472 kg m⁻³. He also found a dependence of k _i on ρ _i ². Reference Pitman. and Zuckerman.Pitman and Zuckerman (1967) and Reference Weller and Schwerdtfeger.Weller (1971) determined the thermal conductivity of snow with densities of 100–620 and 420–570 kg m⁻³, respectively, at temperatures varying from −5° to −88°C and −17° to −60°C. These authors found that the dependence of k _i on temperature became smaller as the temperature increased. Again, k _i of snow was more sensitive to changes of density than to temperature (as found for porous ice). Pitman established that the effective thermal conductivity of snow is proportional to ρ _i for 400 < ρ _i < 620 kg m⁻³ and to ρ _i ² for ρ _i < 400 kg⁻³. Reference Östin and Andersson.Östin and Andersson (1991) determined the effective thermal conductivity of frost grown in a forced air stream over a density range of 80 < ρ _i < 680 kg m⁻³. Their experimental data were described by the polynomial

Fig. 6. The thermal Conductivity, k_i, of bulk ice, snow, frost and porous ice as function of density, ρ_i. All measurements were made at T_a = 0° to −17°C. The dash line represents the best linear fit of the data within the range of 400 ≤ ρ_i ≤ 915 kg m⁻³, with a correlation coefficient of 0.992. This fit does not include the frost measurement of Öslin and Andersson.

Their k _i values were within a factor of 1.31 lower than those of other investigations in the density range 100 < ρ_i – 550 kg m⁻³ (Fig. 6). This is not surprising, because frost is grown by deposition of water vapor, whereas snow is produced by the combined processes of water-vapor deposition, aggregation of ice and/or other snow crystals and accretion of supercooled water droplets. Thus, frost and snow have different ice structures (texture) which, at the same density, have different thermal conductivities. As shown in Figure 6, the k _i values of porous ice are between those of bulk ice and snow or frost. The linear relationship between k _i and ρ_i for porous ice blends smoothly with that of snow with ρ_i > 400 kg m⁻³. The linear fit to the [k _i, ρ _i] curves for 400 ≤ ρ _i ≤ 915 kg m ⁻³ is

with a correlation coefficient of 0.992. This expression is close to that given by Equation (10).

Summary and Conclusions

Two sets of experiments were carried out in a Cloud Physics Wind Tunnel. First, porous ice particles with different densities were grown by accretion of supercooled Water droplets. Then the thermal conductivity and diffusivity of the accreted porous ice were measured by remotely scanning the surface temperature of cooling particles and modeling the internal heat fluxes. The conclusions are:

• (1) Under similar external cooling conditions, the surface temperature cools faster for porous ice with a low density than for bulk ice. The lower the density is, the faster the porous ice responds to the temperature of the environment because of its reduced heat capacity.

• (2) Both the thermal conductivity, k _i, and diffusivity, α _i, of accreted porous ice depend on density, ρ _i and can be described by linear relationships over the density range 400 ≤ ρ _i ≤ 915 kg m⁻³. Lowering ρ _i by 10% lowers k _i by 15% and α_i by 5%. Thus, the effect of ρ _i on k _i is more significant than on α _i.

• (3) Within the range of cloud conditions, the deusity of accreted porous ice influences the thermal properties more than temperature. Therefore, the density variation needs to be considered.

• (4) The results bridge the gap of the thermal properties of snow or low-density ice and bulk ice.

These results can be used to model the growth and heat transfer of natural and artificial hailstones and to understand better their growth and melting. They can also help in the interpretation of the growth structure of hailstones. The conclusions are restricted to artificially grown, dry hailstone shells with air enclosures. The heat conduction at a given density will vary only slightly with the ice framework, which in turn is mostly affected by the size of the accreted cloud droplets. The measurements are not directly applicable to ice densities of cometary nuclei, where ice-structure variations may be much larger and, in addition in crystalline ice, amorphous ice may be encountered. Further, the mass transfer through the gaseous phase may play a much larger role in the effective thermal conductivity because of internal convection, the long time-scales and the extremely small air pressures involved. Further investigations may consider the thermal conductivity of porous ice in the form of compacted frost, and the dependence of the thermal conductivity on ice texture and air pressure.