empirical formula for the melting rate of snowflakes

February 1981 T. Matsuo and Y. Sasyo 1

Empirical Formula for the Melting Rate of Snowflakes

By Takayo Matsuo and Yoshio Sasyo

Meteorological Research Institute, Yatabe-cho, Tsukuba-gun, Ibaraki-ken, 305, Japan (Manuscript received 16 June 1980, in revised form 30 September 1980)

Abstract

The melting experiment has been carried out on freshly fallen snowflakes supported on a nylon net in a vertical wind tunnel at an airstream of 100 cm sec-1 in wind velocity and of 5.5* in temperature. The morphological variations of snowflakes by melting were shown in a series of photographs taken every ten seconds. Examinations of the melting

process with eyes and through a camera revealed that most of water produced at snowflake surface by melting was not accumulated on the surface but penetrated into the inside, and this is markedly different from the melting processes of ice spheres and crystals. A simple microphysical model was proposed for the melting process of snowflakes. Using the model, it was shown that the rate of decrease in snowflake radius R by melting, assuming spherical symmetry, could be expressed as -dR/dt=**(K*T+LvD**)/Lf*iR in connection with the

problem of heat transfer. * is a kind of adjustable parameter to bridge the gap between the experiment and the theory, and was evaluated experimentally as 1.75. * is a ventilation coefficient of spheres determined by Yuge (1960) which is given by *=1+0.275 Pr1/3 Re1/2.

1. Introduction

The problem of melting of snowflakes is one

of the most important ones in the studies of sleet

formation and also of melting layer formation

below freezing level in the atmosphere. The prob-

lem has been studied so far in connection with

the atmospheric conditions under which melting

takes place. In particular, the effect of air tem-

perature on the melting rate has been made to

be rather clear from statistical analyses of vertical

temperature profiles when raindrops or snow-

flakes are observed. However, there have been

few approaches to this problem from the view-

point of micophysical processes of melting. Melting experiments have been hardly carried

out for clarification of the melting processes of

snowflakes, and no determination of physical

factor affecting the melting rate has been made.

Consequently no physically meaningful formulas

expressing the melting rate are not proposed. At

present, the phenomena of melting of snow-flakes in the atmosphere are hardly elucidated

physically. The purpose of the present work is to formu-

late the melting rate of snowflakes using some

experimentally set-up values and measured fac-tors. In obtaining the formula, we adopted the as-sumptions of spherical shapes of snowflakes and a similar form of the formula to those of rain-drops and ice spheres studied by Ranz and

Marshall (1952), Macklin (1963), Goyer et al.

(1969), and Pruppacher and Rasmussen (1979). The formula of the melting rate proposed here will be useful to numerical simulations of melting of snowflakes below freezing level.

2. Experimental procedure

Falling snowflakes were collected on a wooden board covered with velvet cloth at a yard of

Meteorological Research Institute (Tokyo, Japan) from January 29 at midnight to 30 early in the morning in 1979. It was found from eye ob-servation that they consisted mainly of ice crystals of spacial dendrite and no melting took place in them.

A collected snowflake was set on a nylon net

(the thread of 100*m in diameter, mesh size of 4.2*4.2mm2) in a vertical wind tunnel in order to observe the melting of the snowflake in the

airstream. The tunnel is of the standard type and consists of a 180cm long rectangular see-

2 Journal of the Meteorological Society of Japan Vol. 59, No. 1

tion, 15*15cm2 in cross-section. Outdoor air, whose average temperature and relative humidity were 0.3* and 94% respectively during the experiment period, was drawn by a fan at the end of the tunnel into the tunnel in a laboratory

(about 19* in room temperature) through a long vynyl duct of about 10m long and 25cm in diameter. This procedure of the drawing of the almost saturated outdoor air (around the freezing point) permits water vapor density of the air-stream in the tunnel to be closely equal to equilibrium water vapor density at the freezingg

point, and can suppress latent heat accompanied by evaporation or condensation of water vapor from or on snowflake surface to be markedly small. The outdoor air became gradually warmer while passing through the duct located mostly in a warm room. Finally the air temperature was raised to 5.5* at the working section of the tunnel where melting experiment was carried out, owing to heat transfer through the wall of the duct from warmer room air.

The snowflake was subjected to the airstream of 100cm sec-1 in wind velocity, 5.5* in tem-

perature, and 4.71*10-6 gr cm-3 in water vapor density for a period of time ranging from 1-2 min, the photograph of the melting snowflakes being taken at ten second intervals from below

through a motor driven camera (35*24mm2) equipped with a Medical Nikkor lens of 200mm focal length.

Figs. 1 (a) and (b) show two examples of the melting experiment, snowflakes gradually shrink-ing to a water drop at the end by melting. After melting was completed. the mass of a water drop, namely of a snowflake, was determined by using

a filter paper method treated with water-blue reagent. Snowflake cross-sectional areas were de-termined by measuring the cross-sections enlarged on the photographic papers with a planimeter.

Seventeen snowflakes were tested in the ex-

periment. Three snowflakes of them were far from spherical symmetry and were excluded from the experimental data for obtaining the empirical

formula of the melting rate. Fig. 2 shows the relation between initial cross-sectional areas and masses of fourteen snowflakes (except these three asymmetrical ones). The average density of four-teen snowflakes is 0.036 assuming their spherical symmetry.

3. Analysis of the melting experiment

In the experimental condition employed here, heat transfer between snowflakes and the air-

stream can take place by conduction and con-vection processes, and by transfer of latent heat

Fig. 1(a) Photographs taken every ten seconds showing the shrinkage of snowflakes by melting. White horizontal and vertical lines are threads of a nylon net. Mass of the

snowflakes is indicated in the upper left-hand corner. Experimental conditions of airstream are 5.5* in temperature and 100cm sec-1 in wind velocity.


Fig. 1(b) Same as Fig. 1(a), except for snowflake mass,

F ig. 2 Relation between snowflake mass and cross-section area used. Each point is

an observation and a solid line is a line of constant average density ob-

tained assuming spherical symmetry.

of vaporization of water. Unfortunately the rate

of heat transfer to snowflakes is not known

because of the practical difficulty of the experi-

ment and the theory, although those to metal

spheres, raindrops, and ice spheres have been

precisely examined. Heat transfer rates to the surface of these spheres are generally given by

the following form,

(1)

where R' is radius of a sphere. The ventilation

coefficients * and b represent an increase in the

effective conductivity of the air for heat and

water vapor diffusion respectively because of the motion of air. The mean ventilation coefficient for heat diffusion of metal spheres * was deter-mined experimentally in the range of Reynolds number 10-1800 by Yuge (1960) in the follow-ing form,

*=1+0.275Pr1/3Re1/2 (2)

Those of raindrops and ice spheres are the same forms as metal spheres, although coefficients (i.e., 0.275 for metal spheres) between these are slightly different.

Snowflakes are usually porous and their sur-faces are rough, and these properties probably affect the heat transfer rate to snowflakes. At

present we have little knowledge on this prob-lem; therefore, the heat transfer rate to snow-flakes are obliged to be assumed. Analogically, we apply the modified heat transfer rate of spheres to spherical snowflakes as follows,

(3)where R is radius of snowflakes, and is a factor

involving the effects of various physical proper-

ties of snowflakes on the heat transfer rate. For

simplicity, * is assumed to be constant. The ven-

tilation coefficient b for water vapor diffusion is

assumed to be an analogous form by analogy

with heat diffusion, namely b=1+0.275 Sc1/3

Re1/2 Because the Prandtl and Schmidt numbers

are equal to 0.71 and 0.60 respectively for air


over a wide range of temperatures and pressures, the terms pr1/3 and Sc1/3 differ only by a few

percent. To simplify further calculations, these terms are equated to 0.87. Thus the ventilation

coefficients * and * become

*=*=1+0.24Re1/2 (4)

Assuming that the total amount of heat trans-ferred from the airstream to a melting spherical snowflake for a given time *t is consumed by melting the ice existing in the surface layer of

*R in thickness, a following equation can be obtained,

(5)

Under the experimental condition employed here, the water vapor density of airstream (4.71*10-6

gr cm-3) was adjusted so closely to equilibrium water vapor density at the freezing point (4.85* 10-6 gr cm-3) that the transfer of latent heat of evaporation does not make a substantial con-tribution to the total heat transfer rate (only a few percent). The equation (5) is then re-written, with the second term in the right hand side of the equation omitted, in the following form,

(6)

Assuming that density *i maintains to be al-most constant (in this case *i=0.036) during the melting period, the differential equation (6) can be solved to give the following solution,

(7)

where a= 0.24(2 V/v)1/2 and R0 is initial snow-flake radius. Note that density *i means density of ice-skeleton structure of snowflakes incollap-sable during most of the melting period, but not real density of melting snowflakes. The solution (7) is further rewritten by using the normalized cross-sectional area of the melting snowflake S=*R2/*R02 in place of the radius R, in the following form,

(8)

where B=aR01/2, and the following experimen-tally set-up values and coefficients are already substituted in the solution (7):

In this case, temperature difference *T between the snowflake and the airstream is taken as 5.5* during the melting period on the assumption that snowflake temperature remains uniformly the freezing point during the melting period.

From the equation (8), a factor * can be evaluated experimentally by obtaining the time variation of cross-sectional area S. The factor s plays an important role for bridging the gap between the theory and the experiment. is a kind of adjustable parameter to keep a good agreement between the theory and the experi-ment. In fact, besides the influences of the

porosity and roughness, the obtained adjustable parameter possibly involves influences of all assumptions used until deriving the equation (8). Using the adjustable parameter evaluated, we can obtain the formula of the melting rate of snowflakes.

4. Experimental results

The aim of this melting experiment is to ob-tain the adjustable parameter e and to propose the formula of the melting rate of snowflakes.

Fig. 3 shows time variations of normalized cross-sectional areas caused by melting. The cross-sectional areas were determined with a planimeter from the cross-sections on the photo-graphs. The initial snowflake diameters required for the evaluation of * were calculated from the cross-sectional areas of T= 0 assuming spherical symmetry of snowflakes.

Symbols on the solid lines are experimental

points and numbers at the end of solid lines indicate the initial snowflake diameters (mm). Horizontal dashed lines indicate the normalized cross-sectional area at the end of melting (S = 0.11) calculated by using the relation S=(Rr/R0)2 = *i2/3, where Rr is radius of a water drop

formed by melting. In the figure the experi-mental points just before the end of melting are excluded because at the time the discrepancy be-tween the situation of melting and the microphys-


Fig. 3 Variation of normalized cross-sectional area caused by

melting with time for various initial snowflakes diameters

D0.

Fig. 4 Variation of adjustable parameter with time for various snowflake diameters D0. Numbers indicate the initial

snowflake diameters.


ical model proposed is great. Photographs of T=40 sec in Figs. 1 (a) and (b) show that a water drop hangs down from the thread (photo-graphs are taken from below) and transparant ice cluster floats in the upper part of a water drop. Under this condition, evidently the equa-tion (8) does not hold. For a snowflake with a initial diameter of 6.4mm, the experimental

points are missed at the last stage of melting because the melting snowflake dropped off the thread. The value of * can be calculated using the equation (8) for each experimental point ob-tained every ten seconds.

Fig. 4 shows variations of calculated values of s with time for various snowflake diameters. Symbols used are the same as in Fig. 3. * re-mains almost constant during most of the melting

period, although there is a slight tendency for the value of * of larger snowflakes (8.0-8.6mm) to decrease gradually with time. This shows a pro-

priety of the assumption made in the previous section that * is constant.

Fig. 5 shows relation between the mean value

*m and the initial snowflake diameter. The mean value *m is the average of the values of * cal-culated for a snowflake. The values are scattered in the range of 1.4-2.1 and are almost independ-

ent of the initial snowflakes diameter. The aver-age value is 1.75 as is shown by a dashed line. These results show that the assumption of * being constant is reasonable.

Dispersion of the mean value *m around the average value of 1.75 suggests the presence of some other physical properties of snowflakes

which could affect the adjustable parameter s. For example, the effect of snowflake shapes

should be first considered. Three asymmetrical

Fig. 5 Mean value of *m of adjustable

parameters as a function of the initial snowflake diameter.

snowflakes tended to have higher melting rate than fourteen roughly spherical snowflakes available, under the same mass conditions. Half-life of decrease in snowflake cross-sectional area by melting was 14 sec for the first asymmetrical snowflake of 4.2mg in mass, while that was 18 sec for a roughly spherical snowflake available of 4.4mg in mass: 12 sec for the second of 4.3 mg, 18 sec for the same snowflake as above (4.4mg): 22 sec for the third of 10.2mg, 30 sec for 10.1mg. Besides the shapes of snowflakes, their porosity seems to be important. However, these effects could not be treated well in the

present adjustable parameter. This problem is reserved in future studies.

As a result of the melting experiment, it was found that the adjustable parameter was almost independent of time and initial snowflake diame-ter and the average value was evaluated as 1.75. Using the adjustable parameter, in the absence of evaporation or condensation of water, the rate of decrease in snowflake radius by melting, assuming spherical symmetry, became,

(9)

Using the equation (5), generally the rate can be expressed in terms of the following equation,

(10)

5. Discussions

In an analysis of the melting experiment, we used two main assumptions. We will have some discussions of them.

One may think that when melting proceeds, water produced by melting covers the snowflake surface wholly like the cases of ice spheres and crystals (coyer et al. (1969) and Knight (1979)). The experimental results, however, did not sup-

port such idea. Figs. 1(a) and (b) show that ragged ice surface of the melting snowflakes maintains during most of the melting period. The melting snowflakes look like sponges and

gauzes taking in water. From close eye-observa-tion of the melting process, it was found that water produced by melting penetrated the whole body of snowflakes through small gaps and pores among the constituent ice crystals. It is due to possible capillary action. Snowflakes consist of various ice crystals and these are mixed in it. Snowflakes are, as it were, a kind of ice-skeleton which contains a number of small pores and gaps. Under this condition, the capillary action will


prevail. Heat transferred from the airstream to snow-flake surface is first used largely by melting the ice at the surface and to a lesser extent by raising the temperature of small amount of water existing at the surface above 0*. The heat which has been consumed in order to raise the temperature of the water is then transferred from the water (above 0*) to the ice (below or equal to 0*), surrounded by the water, by thermal conductivity and is finally consumed by melting the ice at the surface. Because the heat transfer rate through water to ice is adequately large comparing with the transfer rate from airstream to substrate and because the amount of water existing at the sur-face is small in melting, it can be assumed that heat transferred to the surface is used only to melt the ice at the surface.

Figs. 1(a) and (b) also show that snowflakes maintain their original shapes during most of the melting period. This suggests that snowflakes have rather solid ice skeleton structure that is not easily collapsed by melting. This is further confirmed from three examples shown in Fig. 6. Fig. 6 shows time change in perimeter of melting snowflakes every ten seconds. Especially in this case, asymmetrical snowflakes, which are not involved in the present experimental data, are chosen because the deformation of asymmetrical ones is easier to be detected. The snowflakes can maintain their original shapes during most of the melting period and no breaking-up takes

place in spite of continuous process of melting. The ice skeleton structure, however, collapses to become a water drop by surface tension of

water when melting is nearly completed. From the facts above, we can assume that density used in the equation (6) *i maintains a constant value

(in this case, *i=0.036) during most of the melt-ing period.

Now we should refer to the availability of the empirical formula obtained. The present melting experiment was made at an airstream of 100cm sec-1 in wind velocity and for snowflakes con-sisting mainly of ice crystals of spacial dendrite. In discussing the availability, a knowledge on the dependence of the formula on wind velocity and

physical properties of snowflakes will be needed. Wind velocity may affect the formula through the ventilation coefficient of snowflakes for which that of spheres is substituted. No comparison between ventilation coefficients of snowflakes and of spheres can be made because of unknown snowflake one, and precise discussion is difficult of the velocity dependence of the formula. How-ever, it is satisfactory to consider, based on the theory of heat transfer, that the velocity de-

pendence of flow pattern in the boundary layer on the substrates may not be so different between spherical snowflakes and spheres. In view of this, the formula obtained is probably available for snowflakes with usual fall velocity from 50cm sec-1 to 200cm sec-1.

On the other hand, physical properties of snowflakes seem to exert some influence on the formula, especially on an adjustable parameter

As mentioned in the previous section, the marked discrepancy of the shape from spheres resulted in an increase in the value of *. The degree of the porosity may also affect the formula.

Fig. 6 Three examples of schematic drawing of time change in

the perimeter of melting asymmetrical snowflakes

observed at ten seconds intervals. Symbols plus indicate

crossings of the threads.


In an extreme case, for example, for snow aggregates of a few plate-like ice crystals capillary action will not prevail and the present formula may not hold. Based on the above discussions, it is considered that the formula may be available for snow aggregates (not far from spherical) con-sisting of a number of ice crystals.

In any way, in future the empirical evaluation of * should be made of various types of snow-flakes over a wide range of wind velocity. If

precise meaning of * can be elucidated through the micro-physical processes of melting in these experiments, the study of snowflake melting will make a great progress.

6. Conclusions

In summary, the conclusions to be drawn from this melting experiment of snowflakes are:

(1) Most of water produced by melting is not accumulated on the surface but penetrates into the inside. It is due to possible capillary action.

(2) Snowflakes can be regarded to have rigid ice-skeleton structure which does not collapse easily during most of the melting period.

(3) By using a simple micro-physical model of the melting process proposed on the basis of the results above, the rate of decrease in snow-flake radius by melting, in the absence of vapori-zation or condensation of water vapor, can be expressed in the following form,

and generally the rate is as follows,

where * is an adjustable parameter and was evaluated empirically as 1.75.

Acknowledgements

The authors would like to thank Mr. Jiro Kubo (Head) and the Staff members of Physical Meteorology Division, Meteorological Research Institute of Japan for their valuable comments and discussions.

Appendix: List of Symbols

*

: ventilation coefficient of snowflakes for heat diffusion (that of spheres is sub-

stituted for)

b : ventilation coefficient of snowflakes for water vapor diffusion (that of spheres is

analogously used for) D : coefficient of molecular diffusion of

water vapor in air K : thermal conductivity of air

L f : latent heat of fusion of water Lv : latent heat of vaporization of water Pr : Prandtl number Re : Reynolds number

R : radius of melting snowflake Sc : Schmidt number

*T : temperature difference between snowflake and airstream

t : time V : airstream wind velocity

*i : density of ice-skeleton structure of snowflakes

** : difference between water vapor density

of airstream and equlibrium water vapor density at snowflake surface

v : coefficient of kinematic viscosity of air * : adjustable parameter of snowflake

References

Goyer, G. G., S. S. Lin, S. N. Gitlin and M. N. Plooster, 1969: On the heat transfer to ice spheres

and the freezing of spongy hail. J. Atmos. Sci., 26, 319-326.

Knight, C. A., 1979: Observations of the mor-

phology of melting snow. J. Atmos. Sci., 36, 1123-1130.

Macklin, W. C., 1963: Heat transfer from hail- stones. Quart. J. R. Met. Soc., 89, 360-369.

Pruppacher, H. R, and R. Rasmussen, 1979: A wind tunnel investigation of the rate of evaporation

of large water drops falling at terminal velocity in air. J. Atmos. Sci., 36, 1255-1260.

Ranz, W. E. and W. R. Marshall, 1952: Evapora- tion from drops. Chem. Eng. Prog., 48, 141-

146. Yuge, T., 1960: Experiments on heat transfer from

spheres including combined natural and forced convection. J. Heat Transfer, 82, 214-220.


雪片の融解速度式

松尾敬世・佐粧純男

気象研究所

垂直風洞内に取り付けたナイロンネット上で,自然雪片の融解実験を行なった。実験中の気流温度,風速は,

それぞれ5.5*,100cm sec-1とした。融解に伴う形体変化を10秒間隔で写真にとった。その結果,雪片の融

解過程は氷球,氷晶のそれとは異なり,融解により生成した水は表面に蓄積されないで内部にしみ込むことがわ

かった。結果に基づき融解に関する微物理モデルを作りこれを使用すると,雪片の融解速度(雪片半径の減少速

度)は球形を仮定して次式で表現出来た。

ここで εは理論と実験を合わせるパラメータで,1.75の実験値を得た。なお,*は熱輸送に関する球のventila-

tion係数で*=1+0.275Pr1/3Re1/2である。

empirical formula for the melting rate of snowflakes

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