Hailstone heat and mass transfer theory and three-component nusselt numbers

ATMOSPHERIC

RESEARCH ELSEVIER

Atmospheric Research 32 (1994) 85-93

Hailstone heat and mass transfer theory and threecomponent nusselt numbers R o l a n d List Department of Physics, University of Toronto, Toronto, Ontario M5S 1A 7, Canada (Received January 12, 1993; revised and accepted July 23, 1993 )

Abstract

The search for major faults in the theory of heat and mass transfer of hailstones shows that interactions between different types of direct and indirect heat transfers cannot account for discrepancies with experiments. The residence time of cloud droplets in the boundary layer is of the order of 10-5 s, thus drastically reducing interaction possibilities with the heat transfers by conduction and convection and by the diffusion of water vapour. As a consequence, calculations from experimental data seem to be incomplete and cause Nusselt numbers to increase fourfold with increasing liquid water content and air temperature. It is proposed that the increase in surface area of up to 50% by a shape change from spheroid to wheel (at constant axis ratio) could be at the root of the problem. New experiments presently under way should clarify this point. In discussing transfers, new 2- and 3-component Nusselt numbers are defined which lead to an easier understanding of the total heat and mass transfer of hailstones.

1. Introduction

In the past the validity of the basic heat and mass transfer theory of hailstones, as developed by Schumann ( 1938 ) and improved by Ludlam (1958) and List ( 1963, 1977 ), has not been thoroughly tested. Recently, Lesins and List ( 1986 ) and Garcia-Garcia and List ( 1992 ) provided better but still incomplete experimental data. The outcome was a surprise: the deduced heat transfers differed by up to a factor of four or more from theory. Further, the Nusselt number, Nu, (assumed approximately equal to the Sherwood number, Sh) increased with increasing liquid water content, Wf, and air temperature, Ta. The first discrepancy led to the investigation of the sensitivity of the overall heat and mass transfer to individual parameters (List, 1989), which showed that only an unreasonable 0169-8095/94/$07.00 © 1994 Elsevier Science B.V. All fights reserved SSDI 0169-8095 (93)E0053-2

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R. List/Atmospheric Research 32 (1994) 85-93

roughness coefficient ( > 2 ) could balance the heat transfer. This is not satisfactory because the observed roughness of artificially grown, gyrating hailstones decreases with increasing Wf and Ta (neglecting waves and roughness at the pole). Mass transfer simulations (which also simulate heat transfer) by Schiiepp and List (1969) suggest that water skin-covered hailstones are characterized by a roughness factor of 1.0-1.2; for very rough hailstones the factor can be as large as 2.0. The spreadsheet study by List ( 1989 ) demonstrated that no adjustment of any single variable (other than unreasonable roughness) could produce agreement between theory and experiment. A combination of parameters, assuming (excessive) errors of 10-20% each in the supporting direction, may produce an artificial agreement if these errors are combined with roughness effects > 1.4. However, the errors of most measured parameters are much less than that. Multidimensional error patterns have not been explored. There are other possible explanations for the discrepancies: ( 1 ) Assumed values of parameters, which could not be measured directly, were misjudged or overlooked; (2) The heat and mass transfer theory is wrong by up to a factor of four. The two effects combined could also produce the discrepancy. The first point is valid, but will be neglected for the time being. The second point is about the adequacy of the standard heat and mass transfer equation. The basic assumption has always been that there are three components which allow the removal to the air of latent heat released on the hailstone by freezing of accreted water, Qf: (a) the heat transfer by conduction and convection, Qcc; (b) the diffusion of water molecules through air and the related heat released by evaporation/sublimation at the hailstone surface, QES. Thereby "evaporation" is from the liquid phase and "sublimation" from the solid phase and (c) the heat transfer to the hailstone by the accreted cloud droplets, caused, by the temperature difference between the droplets and the hailstone surface, QcP. Temperature equilibrium is assumed between cloud droplets and air; radiation is negligible. Thus, the sum of all heating terms is equal to zero. The discrepancy between theory and experiment is also used to review the concepts of heat and mass transfer (Section 2) and, in Section 3, to extend the concept of Nusselt number to multi-component heat and mass transfer.

2.3-Component heat transfer 2.1. General considerations The "standard" theory has assumed that the three heat transfer components Qcc, (2ESand (2cP are independent of each other and can be superimposed. Interaction terms will now be considered, since each transfer type may interact with the two others: ((2cc~QEs), (aEs~Qcp) and (Qcc~t~cp). The new heat and mass transfer equation can then be expressed by

R. List/Atmospheric Research 32 (1994) 85-93

(2cc +QEs +QcP +Interaction Terms+ Qf =0

87 (I)

The nature of the interaction could be additive as in Eq. ( 1 ) or in the form of multiplicative factors in Qcc, des and QcP, i.e.

acc,eff + aEs,eff + acP,eff + Of = 0

(2 )

where the subscript eft denotes effective heat transfer by any given component. List (1990) has stressed that transfers of any kind are gradient driven. For example, evaporation from a hailstone surface is driven by a gradient of the HzO molecule concentration through the boundary layer. If cloud droplets are moving into this boundary layer, this gradient could be affected by the presence of droplets of concentration Wr. 2.2. The heat transfer by conduction and convection, (2cc According to the standard theory, (2cc is given by (~cc - -A~kNu( Ta - Ta) D

(3)

where As (m 2) is the total surface area of the hailstone (=0.789nD 2 for an aspect ratio a = 0 . 6 7 ) , k (J m -1 s -l K -l ) the thermal conductivity of air, Nu () the bulk Nusselt number, Ta (K) the hailstone's surface temperature, Ta (K) the air temperature and D (m) the hailstone's major axis. Qcc (like QES and QcP later) carries a negative sign because it is away from the particle. The Nusselt number is given by hD NU= k

OT* Oy* ly*=o

(4)

where h (J m -2 K - l s-l ) is the surface heat transfer coefficient. Nu is a dimensionless temperature gradient at the particle surface and gives a measure of the particle's heat transfer (Incopera and De Witt, 1991 ). y* is the dimensionless coordinate y / L perpendicular to the surface, with L ( = D) a characteristic length set equal to the diameter, and the absolute temperature in non-dimensional form is given by T * = ( T - T d ) / ( T ~ - T d ) , where T~ (K) ( = Ta) is the temperature of the undisturbed air. For rough spheres and spheroids, Nu can be expressed in terms of the Reynolds number, Re, as Nu=Oz( 2.0 + O.6OPr~/3Rel /2)

(5)

where 0 is a roughness factor and Z corrects for oblateness ( = 1.4 for an oblate spheroid with a = 0 . 6 7 ) ; Pr ( =0.71 ) is the Prandtl number (-) for air. Eq. (5) was experimentally confirmed up to (subcritical) Re ~ 100,000 (Schiiepp and List, 1969). The adjustment to spheroids was also substantiated. There is some concern about using the definition of Nu of Eq. (4). During heat

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transfer from a solid body to a gas, the temperature change is discontinuous because in a layer close to the body only about half the gas molecules have a speed according to the temperature of the adjacent solid; the other half (because of a small condensation factor) comes from the opposite "gas" direction. Thus, the definition of Nu is not physically "clean". Since this situation is restricted to within a depth of the order of the mean free path length of the gas molecules, it is generally glossed over and neglected. While this does not make it negligible it is, nevertheless, automatically included in the parameterization of the transfer by Eq. (5).

2.3. The indirect heat transfer by deposition and sublimation, QEs The diffusion of water molecules away from the hailstone removes heat, QES, from the particle, is characterized by the release of latent heat due to evaporation or sublimation. This process is described by

QES -

RvD

(6)

where Lv (J kg- i ) is the latent heat of vaporization (from a liquid hailstone surface) or sublimation (vaporization of ice), Dwa (m z s -1 ) the diffusivity of water vapour in air, ed (Pa) the vapour pressure at the hailstone's surface, e the ambient vapour pressure and Rv the specific gas constant for water vapour. The Sherwood number, Sh, is defined as

Sh

ilL) Op* Dwa

(7)

Y [y*=o

where fl (m s - ~) is the mass transfer coefficient at the hailstone surface and p* is the dimensionless water vapour concentration. The Sherwood number is a dimensionless concentration gradient at the particle surface and is the driving force for mass transfer in the air boundary layer of the hailstone. For rough spheres and spheroids, Sh can be expressed as

Sh=Oz( 2.0+ O.6OScl/3Re 1/2)

(8)

where Sc ( =0.61 ) (-) is the Schmidt number for air.

2.4. The heat transport by supercooled cloud droplets, Qcv, The heat transfer by accretion of supercooled cloud droplets, QcP is given by the heat required to warm the deposited water to the surface temperature of the growing hailstone and by the heat loss represented by the amount of water and ice shed from the hailstone. Lesins and List ( 1986 ) expressed it as follows

( g c p = - ~ W f V [ E n e t C w ( r d - r a ) + ( E - E n e t ) ( C w [ r s - T a ] - I s L f , s)]

(9)

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Thereby Ac (m 2) is the average cross-sectional area presented to the flow by the hailstone, with ,~c = (rt/4)KD2; x (-) is a correction factor for symmetric gyration (at precession/nutation angles of 30 ° and with a = 0.67: x = 1.0793 ). V (m s -l ) is the hailstone free fall speed, En,~ (') is the net collection efficiency, i.e. the fraction of the droplet mass in the swept-out volume that is permanently accreted, Cw (J kg- ~K - ~) is the specific heat of water averaged over the temperature range Td to Ta. E (-) is the collection efficiency or mass fraction of collected cloud droplets from the swept-out volume, Ts the temperature of the shed water drops, Is (-) the mass fraction of ice in the shed water, and Lf.s (J kg- 1) the latent heat of fusion of the shed water at the temperature of the shed water.

2.5. The freezing term, Of The latent heat released by the fraction of water permanently accreted by the hailstone, (~f, needs to be transported away from the particle. This heating rate is given by (~f =.zl c WfEne t WIfLf

(10)

Lf is the latent heat of fusion of the accreted water which becomes ice at the temperature Td and If is the mass fraction of ice produced in the deposit on the hailstone. Up to this point it has always been assumed that this released latent heat is transported away by the mechanisms outlined in Eqs. (3), (6) and (9), which are not directly interfering with each other. However, such equations do not necessarily describe the magnitude of the freezing process in nature. For the purpose of this discussion it is now postulated that the interaction between the three transport processes can explain the discrepancy between theory and experiment. 2.6. The magnitude of the interaction processes The definition of the Nusselt number in Eq. (4) can be expanded to

OT* Nu=--Oy*

\ T ~ - Td] OT D 0 (y) Oy T ~ - Td

(11)

This equation allows a calculation of the boundary layer thickness Yb, assuming that the gradient is linear. For a spherical hailstone with a diameter of 0.02 m, a temperature td=0°C [thereby the symbol T (K) has been replaced by t ( ° C ) ] , a fall speed of V=20 m s- 1in air at ta= - 2 0 ° C (resulting in Nu~ 100 for a pressure of 100 kPa) the average boundary layer thickness is Yb= 2 × 10-4 m or 0.2 mm. This implies that the residence time of droplets moving vertically through

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the boundary layer at 20 m s- 1 is 10- 5 s. The same argument can be made about the water vapour diffusion because, for Nu.~ Sh, the boundary layer thickness for the water vapour concentration field is the same as for the temperature field. It can be shown that droplets with a mean diameter of 28/tin (as in GarciaGarcia and List, 1992 ) will not react within this short time and change the original temperature and water vapour fields which control Qcc, QEs and Qcp. The same statement would also hold for 10 gm droplets. Droplet velocity components perpendicular to the gradient will not affect the general situation.

2. 7. The effect of water vapour diffusion Heat conduction requires a driving temperature gradient. In this gradient the diffusion of water vapour will increase with respect to Ta because of the warmer environment near the hailstone (at constant pressure). The general temperature/ pressure dependence of the water vapour diffusivity in air is given by

( T)1'9(~) 4

Dw"=2"l 1 × 10-5 To

(12)

The diffusivity will increase by a factor of 1.25 in the region of a gradient from t a = - 3 0 ° C to td=0°C (at constant p). Averaged over an air temperature of -20~'C and a hailstone temperature of 0°C Dwa is 8.2% higher than that at Ta. Thus, such corrections are only necessary if the calculations need to be within this range of accuracy. The reverse effect by vapour diffusion on the temperature conductivity is negligible. Other than that, and in agreement with Section 2.6, the interaction between the different processes needs not be considered.

3. 2- and 3-component Nusselt numbers The Nusselt number, Nu, is calculated on the basis of Eq. (5) as a function of Reynolds number, Re. With the additive constant "2" neglected the Sherwood number, which controls the diffusional growth (Eq. 8 ), has the same dependence on Re and nearly the same value as Nu. Thus, for smooth spheres (X0= 1 ), with no drop shedding from the growing hailstone (Enet-E) and for large Re, Nu can be calculated by substituting Eqs. ( 3 ), (6) and (9) into ( 1 ) (without interaction terms) and using the definition of Nu given in Eq. (4)

Nu =

[ rtDk(Ta-Ta)

(2f + O_.ce LvDwa ( s c ~ l / 3 ( e o ) ] 14 Rvk(Td_Ta)kPrj \Td ;,

( 13 )

When the heat components by conduction and convection, and evaporation/ sublimation, transferred at the surface of the hailstone, are compared to the heat conducted away from the hailstone by molecular heat conduction alone (represented by k), then a new 2-component Nusselt number Nu2c can be defined as

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91

Nu2c = Nu~oz~

( 14 )

where ~o2~is given by the term in the square brackets ofEq. ( 13 ). Nu2c gives the fractional contribution to (2f by conduction and convection and evaporation/ sublimation. Note that QcP ~ 0, otherwise QEs = - Qcc and Nu2c = ~o2c= O. A 3-component Nusselt number, Nu3~, can now be constructed, similar to Eq. ( 14 ), to compare all heat transfers to that by conductuion and convection alone (15)

Nu3c = Nuq93c The factor ~03cis given by

ZvDwa

-

¢o3~ = 1 4 R v k - ~ :

(Sc~l/3(ed

T a ) \er]

Td

e )-t DVEWfCw 4kNu

( 16 )

Nu3c compares the overall heat transfer at the hailstone surface, combining the three individual contributions by.Qcc,.QEs and.Qcp, to the molecular heat conduction. For growing hailstones Q c o QEs and Qce contribute to the transfer in the same sense, and Nuac> Nu because the thermal molecular conductivity k in Nu remains the same. Fig. 1 shows the ratios of Nu3c/Nu ( = ~03c) and Nu2~/Nu (=~02~) as function of air pressure for D = 0 . 0 2 m, V= 18 m s -a, Wf=0.003 kg m -3,/a = - 15°C. Varying the pressure from 100 kPa to 10 kPa changes td from - - 4 . 4 ° C to - 0 . 1 °C. A value of Nu3c,~ 3 at 40 kPa implies that the heat transfer rate at the hailstone surface, as imbedded in a corresponding h3c (a 3-component surface heat transfer coefficient), is three times the contribution of conduction and convection as expressed by h in the regular Nusselt number in Eq. (4).

Nusselt Number Ratios

7

~

"\\

6 ~, I::::

. . . . . \

~

.... Nu3d/Nu

5

~z.-. Nu2d/Nu

1 O

]0000

I - -

2oo0o

30000

~

4000o

-4-

50000

~

-

a:~oo

-

I

7o00o

8oooo

9oooo

]00000

Pressure (Pal Fig. 1. Ratios of the three-component Nusselt number Nu~ and the regular Nusselt number Nu ( = ~ )

according to Eq. ( 16 ), and the two-componentNusseit number Nu2cand Nu ( -- ~2c) accordingto Eq. (13), as functions of atmospheric pressure. These ratios give the multiplication factorsby which the heat transfer is enhanced over that by conductionand convectionalone, as represented by Nu; for the conditions outlined in the text.

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R. List/Atmospheric Research 32 (1994) 85-93

4. Summary and conclusions A discussion of the heat and mass transfer theory of hailstones has not revealed any basic factors in need of massive adjustments to force agreement with experiments. In particular, the very short residence times of accreted droplets in the boundary layer of collecting hailstones (order of 10- 5 s) makes any interaction of the transpo.rts of heat by conduction and convection, Qcc, and evaporation/ sublimation, QES, with the transport of sensitive heat by the accreted cloud droplets, QcP, very unlikely. Further, the effect of the temperature gradient in the boundary layer on the vapour diffusivity can be negl.ected unless accuracies of 510% are attempted. Thus, the interaction between Qcc and QEs does not account much when compared to a factor of up to four. Overall and for practical purposes these three transfers can be considered independent or uncoupled of each other. There is no justification to introduce either additive or multiplicative interaction terms as suggested in Eqs. ( 1 ) and (2). In this discussion, a new insight has emerged on a new concept: composite or multi-component Nusselt numbers. For hailstone growth, 3-component Nusselt numbers can be defined which comprise heat transfers by conduction and convection, vapour diffusion and droplet accretion. They are larger than the "regular" Nusselt number because the heat transfer rate is increased relative to the molecular heat conduction. A possible explanation for the apparent discrepancy between heat transfer theory and experiment could well be that hitherto unaccounted changes in shapes (at constant axis ratio ) during growth play a decisive role in reconciliating theory with experimental results. Lesins (1983) and Garcia-Garcia and List (1992) showed that the shape of hailstones remained spheroidal at low liquid water contents, Wf, and low air temperatures, T,, but that wheel shapes where produced at higher Wf and Ta. [A "wheel" is a cylinder with a diameter equal to the major spheroid axis (2a) and a length equal to the minor axis (2b). ] At equal axis ratios this represents an increase in surface area of ~ 50%. Such an increase in surface area could well explain the main increase of "measured" Nusselt numbers. The fact that it is correlated to increasing Wf and Ta supports this suggestion, particularly since the roughness is generally decreasing with higher Wf and

T.. In the past the Toronto Cloud Physics Group has always tried to reduce the number of experiment variables by coupling temperature to atmospheric pressure (as observed in Colorado and Alberta hailstorms) and relative speeds between icing tunnel air speed and model hailstone to calculated free fall speeds at that temperature/pressure combination. It is obvious that uncoupled experiment series are now required in which only one initial single parameter is changed at any given time. Further, it is also necessary to remeasure the basic heat and mass transfer rates of hailstone models without accretion of cloud, first of solid, inert spheroids with varying surface roughness, followed by ice spheroids where the water molecule diffusion is a contributing factor. Such studies have been made possible by the availability of a scanning infrared microscope, a computerized

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AGEMA system. Initial studies have already been reported (Cober and List, 1992, 1993; Greenan and List, 1992; Zheng and List, 1992), and extensive articles are in the works in an attempt to fully clarify the discrepancies. Acknowledgments This research has been sponsored by the Natural Sciences and Engineering Research Council of Canada and the Canadian Atmospheric Environment Service. Particular thanks go to my present and former students for the stimulating discussions and the reviewers for helpful suggestions. References Cober, S.G. and List, R., 1992. Measurements of the bulk collision efficiency and Nussdt number for growing graupel. In: Proc. 11th Int. Conf. Clouds and Precipitation, Montreal, 17-21 Aug., 1992, Vol. 1, pp. 48-51. Cober, G.S. and List, R., 1993. Measurements of the heat and mass transfer parameters characterizing graupel growth. J. Atmos. Sci., 50:1591-1609. Greenan, B.J.W. and List, R., 1992. Hailstone heat and mass transfer measurements using a thermal imaging system in an icing wind tunnel. In: Proc. 1lth Int. Conf. Clouds and Precipitation, Montreal, 17-21 Aug., 1992, Vol. 1, pp. 31-34. Garcia-Garcia, F., and List, R., 1992. Laboratory measurements and parameterizations of supercooled water skin temperatures and bulk properties of gyrating hailstones. J. Atmos. Sci., 49: 20582073. Incopera, F.P. and De Witt, D.P., 1991. Fundamentals of Heat and Mass Transfer, 3rd Edition. Wiley, N.Y., 919 pp. Lesins, G.B., 1983. Hailstone studies in an icing tunnel. PhD Thesis, Univ. Toronto, 118 pp. Lesins, G.B. and List, R., 1986. Sponginess and drop shedding of gyrating hailstones in a pressurecontrolled icing wind tunnel. J. Atmos. Sci., 43: 2813-2825. List, R., 1963. General heat and mass exchange of spherical hailstones. J. Atmos. Sci., 20:189-197. List, R., 1977. Ice accretions on structures. J. Glaciol., 19:451-465. List, R., 1989. Analysis of sensitivities and error propagation in heat and mass transfer of spheroidal hailstones using spreadsheets. J. Appl. Meteorol., 28:1118-1127. List, R., 1990. Physics of supercooling of thin water skins covering gyrating hailstones. J. Atmos. Sci., 47: 1919-1925. Ludlam, F.H., 1958. The hail problem. Nubila, l: 12-96. Schiiepp, P.H. and List, R., 1969. Mass transfer of rough hailstone models in flows of various turbulence levels. J. Appl. Meteorol., 8: 254-263. Schumann, T.E.W., 1938. The theory of hailstone formation. Q. J. R. Meteorol. Soc., 64:3-21. Zheng Guoguang and List, R., 1992. Measurement of convective heat transfer of hailstone Models. In: Proc. 1lth Int. Conf. Clouds and Precipitation, Montreal, 17-21 Aug., 1992, Vol. l, pp. 4447.