Kathleen F. Jones April 1996

CRREL REPORT 96-2

Ice Accretion in Freezing Rain
Cover: Humpty Dympty falling off of the very high wall.

For conversion of SI units to non-SI units of measurement consult ASTM Standard

E380-93a, Standard Practice for Use of the International System of Units,

published by the American Society for Testing and Materials, 1916 Race St.,

Philadelphia, Pa. 19103.

This report is printed on paper that contains a minimum of 50% recycled

material.

Abstract

Ice accreted on structures from freezing rain causes both increased vertical

loads and increased wind loads, due to the larger projected area of the structure.

Structural failures initiated by ice loads frequently cause millions of dollars of

damage to overhead power and communication lines, towers, and other icesensitive structures. There is little information on ice loads to use in the design

of these structures, so freezing-rain models have been developed for use with

weather measurements to determine the severity of accreted ice loads from

historical data. This report describes a detailed heat-balance ice accretion

model, including the important heat fluxes in freezing rain and allowing the

accretion of runoff water in the form of icicles. It also presents a simple algorithm

for calculating the ice load on components with different diameters and cross

sections. Collision efficiency in freezing rain and the calculation of the wind-onice load are also discussed. Model results are compared with the ice load

measured during a recent freezing rain storm, and to each other using 45 years

of weather data from Des Moines, Iowa.

Cover: In this 1994 freezing-rain storm in the Southeast,

ice accretions on power lines and trees caused

over $100 million in damage. (Photo courtesy of

Entergy.)
Kathleen F. Jones April 1996

CRREL Report 96-2

U.S. Army Corps

of Engineers

Cold Regions Research &

Engineering Laboratory

Ice Accretion in Freezing Rain

Prepared for

OFFICE OF THE CHIEF OF ENGINEERS

Approved for public release; distribution is unlimited.
PREFACE

This report was prepared by Kathleen F. Jones, Research Physical Scientist, Snow

and Ice Division, Research and Engineering Directorate, U.S. Army Cold Regions

Research and Engineering Laboratory, Hanover, N.H.

Funding for this research was provided by DA Project 4A762784AT42, Design,

Construction, and Operations Technology for Cold Regions, Work Unit CS-W03, Alleviation of Structural Icing Effects.

The report was technically reviewed by Dr. Edgar Andreas and Dr. Kazuhiko

Itagaki of CRREL. The author wishes to thank Kim Mitchell of Entergy for providing

photographs taken during the 1994 freezing rain storm in the Southeast.

The contents of this report are not to be used for advertising or promotional

purposes. Citation of brand names does not constitute an official endorsement or

approval of the use of such commercial products.

ii
iii

CONTENTS

Page

Preface ..................................................................................................................... ii

Nomenclature ......................................................................................................... v

1. Introduction ........................................................................................................ 1

2. Hourly weather data ......................................................................................... 2

3. Ice accretion shape ............................................................................................. 3

4. Heat balance in freezing rain ........................................................................... 6

Cooling flux required to warm impinging precipitation ............................. 7

Convective cooling ............................................................................................ 7

Evaporative cooling........................................................................................... 9

Convective and evaporative cooling in no wind .......................................... 10

Longwave radiative cooling............................................................................. 10

Shortwave radiative heating ............................................................................ 10

Viscous heating .................................................................................................. 11

Kinetic heating ................................................................................................... 11

Resistive heating ................................................................................................ 11

Latent heat of fusion .......................................................................................... 11

Other effects........................................................................................................ 11

5. Icicles ................................................................................................................... 12

6. Simple flux model .............................................................................................. 13

7. Collision efficiency ............................................................................................ 14

8. Wind load ........................................................................................................... 14

9. Sample results .................................................................................................... 15

10. Discussion ......................................................................................................... 16

Literature cited ....................................................................................................... 18

Appendix A: Comparison of evaporative heat flux formulations ................. 21

Appendix B: Sky diffuse solar radiation flux to a horizontal cylinder .......... 23

Abstract ................................................................................................................... 25

ILLUSTRATIONS

Figure

1. Wind profile for a 10-m/s wind at 10 m above ground assuming

the 1/7 power law .................................................................................... 3

2. Freezing rain ice accretion shapes ............................................................ 4

3. Comparison of heat flux terms in typical freezing-rain conditions .... 7

4. Nusselt numbers for forced convection over rough and smooth

cylinders as a function of Reynolds number ........................................ 8

5. Equivalent uniform radial ice accretions on cylinders with circular,

rectangular, and angle cross sections for a rainfall of 1 cm ................ 13

6. Uniform radial ice thicknesses hindcast by the heat-balance model

for freezing-rain events at the Des Moines airport from 1948

to 1993 ......................................................................................................... 15

7. Comparison of uniform radial ice thicknesses at Des Moines from

the simple flux model and the heat-balance model ............................. 16
iv

CONTENTS Page

Figure (cont‘d)

8. Time series of the weather conditions and hindcast ice accretions

for the ice storm of 7–8 March 1990 at the Des Moines airport .......... 17

9. Comparison of uniform radial ice accretion thicknesses for 2.54-

and 10-cm-diameter wires using the heat-balance model .................. 18
v

NOMENCLATURE

cp specific heat of air at constant pressure =

1.006 J/g °C

cpc heat capacity of the wire (J/g °C)

cw specific heat of water = 4.22 J/g °C

CD drag coefficient

d drop diameter (μm)

D diameter

Dc wire–ice diameter (cm)

Di icicle diameter (cm)

dd diameter of protoicicle = 0.5 cm

dewT dew-point temperature (°C)

dt time interval (s)

e0 saturation vapor pressure of air over water

at T = 0°C = 6.1121 (1.0007 + 3.46 × 10–6 Pa)

mbar

eT saturation vapor pressure of air over water

at temperature T = 6.1121 (1.0007 + 3.46 ×

10–6 Pa) e[17.502T/(240.97 + T)] mbar

f fraction of impinging precipitation that

freezes to the wire

fb fraction of runoff water that forms a

protoicicle

fs fraction of runoff water that freezes to sides

of an existing icicle

g acceleration of gravity = 980 cm2/s

GrL Grashof number (based on characteristic

length L) = g(0 – T)L3/(273.15 + T)νa

2

h heat transfer coefficient (W/m2 °C)

hm vapor transfer coefficient (m/s)

I electric current (amp)

ka thermal conductivity of air =

4.186 × 10–5 (573 + 1.8 T) W/m °C

L length scale for nondimensional numbers

Lbar average icicle length = Li/45 cm

Le latent heat of vaporization for water = 2501

J/g

Li total icicle length (cm)

Lf latent heat of fusion for water = 334 J/g

M ice mass on wire/length (g/m)

Mi icicle mass/length of wire (g/m)

ma molecular weight of dry air (g/mol)

me mass of evaporated water/length of wire

(g/m)

mw molecular weight of water vapor = 18.016

g/mol

NuL Nusselt number (based on characteristic

length L) = hL/ka

P precipitation rate (mm/hr)

Pa atmospheric pressure (mbar)

Pr Prandtl number = μacp/ka

Ps solar power/length (W/m)

Qc convective heat flux (W/m2)

Qe evaporative heat flux (W/m2)

Qf latent heat flux (W/m2)

Ql longwave heat flux (W/m2)

Qk heat flux from droplet kinetic energy

(W/m2)

Qr resistive heating flux (W/m2)

Qs shortwave heat flux (W/m2)

Qv viscous heat flux (W/m2)

Qw heat flux from warming impinging precipitation (W/m2

)

R gas constant = 8.31441 J/mol K

RaL Rayleigh number (based on characteristic

length L) = GrLPr

Rc resistance/length of conductor (ohm/m)

ReL Reynolds number (based on characteristic

length L) = LV/νa

Req equivalent uniform radial ice thickness (cm)

Rh relative humidity = edewT/eT

rv overall recovery factor for viscous heating

= 0.79

S perimeter of cross section

Sc Schmidt number = νa/κm

ShL Sherwood number (based on characteristic

length L) = hmL/κm

Sr solar radiation flux (W/m2)

T air temperature (°C)

V wind speed (m/s)

Vload wind load/length =

ρaV2(Dc + Di Li/100)/2 N/m

 

r

v drop velocity (m/s)

W liquid water content (g/m3)

w flux of water (g/m2 s)

α angle from horizontal

∆t time step = 1 hour

ε ratio of the molecular weights of water

vapor and air = mw/ma

κm molecular diffusivity of water vapor in air

= 0.211 [(T + 273.15)/273.15]1.94 (1013/Pa)

cm2/s

μa dynamic viscosity of air =

1.71 × 10–4 + 5.2 × 10–7T g/cm s

νa kinematic viscosity of air = μa/ρa cm2/s

ρa density of air = 0.348 × 10–3 Pa/(T + 273.15)

g/cm3

ρc wire density (g/cm3)

ρi assumed density of glaze ice = 0.9 g/cm3

ρo density of water = 1.0 g/cm3

ρwT saturated vapor density at temperature T

(g/cm3)

σ Stefan–Boltzmann constant = 5.67 × 10–8

W/m2 K4
1. INTRODUCTION

Freezing rain occurs throughout the United

States. It may cause hazardous conditions for pedestrians and cars as the rain freezes to sidewalks

and streets. The rain may also freeze to structures.

Open structures that are not designed to withstand the accreted ice load and the increased wind

load, due to the greater projected area of the structure, may fail. Recent severe freezing rain storms

in the Midwest and the South that caused structural failures initiated by ice and wind are described in Storm Data (NOAA 1959–1995: Mar.

1990; Oct., Nov. 1991; Feb., Mar. 1994). Both transmission-line design guidelines (IEC 1990, ASCE

1991) and tower standards (EIA/TIA 1991) recommend designing for accreted ice, but provide little

information on the ice load that should be used in

design.

There have been a few nationwide and regional

studies to establish the severity of freezing rain in

this country. The only systematic measurements

of accreted ice thicknesses in the United States are

described by Bennett (1959), who presents data

from nine years of measurements in the 1920s and

1930s. Tattelman and Gringorten (1973) used

Bennett’s data and information from Storm Data

and its predecessors to determine ice thicknesses

from freezing rain storms in the United States over

a 50-year period. They then calculated extreme ice

loads for return periods of up to 100 years in each

of seven regions of the country. Vilcans and

Burnham (1989) mapped the number of hours

with freezing rain in the 24 years from 1965 to 1988

at 110 airport weather stations in the eastern half of

the U.S. using archived weather data. MRI (1977)

estimated ice loads on transmission lines for 25-,

50-, and 100-year return periods in Washington,

Oregon, Idaho, and part of Montana using weather

data with their proprietary freezing rain, snow,

and rime ice accretion model. The Southeast Regional Climate Center (1993) determined the average number of freezing-rain events per year and

the average event duration in Virginia, North and

South Carolina, Georgia, Alabama, and Florida

using archived weather information.

Hourly weather information from hundreds of

weather stations across the country has been compiled at the National Climate Data Center (NCDC)

and its military counterpart, the Environmental

Technical Applications Center (ETAC). Weather

data are available in magnetic tape format, beginning in the 1940s for some stations. These hourly

measurements of the weather conditions, including precipitation rate, air temperature, dew-point

temperature, wind speed, atmospheric pressure,

and solar radiation, can be used with an ice accretion model to hindcast the amount of ice accreted

in past freezing-rain storms.

This report describes two freezing-rain models

I developed to use the historical hourly weather

data. The first, described in sections 4 and 5, is a

detailed heat-balance model that determines the

amount of the impinging rain that freezes either

directly to a structure or as icicles. The second is a

simple flux model that requires only precipitation

and wind-speed measurements to determine the

amount of accreted ice, assuming that all the impinging water freezes. This simple model, presented in section 6, gives physical insight into the

factors governing the ice load on structural elements with different cross sections. Both models

are incorporated in the Microsoft FORTRAN77

program ZRAIN.* The weather parameters used

in the two models are described in section 2. AcIce Accretion in Freezing Rain

KATHLEEN F. JONES

*Available by request from

kjones@crrel.usace.army.mil.
tual and modeled ice accretion shapes in freezing

rain are discussed in section 3. The factors affecting the collision efficiency of freezing rain drops,

in comparison with cloud droplets, are explained

in section 7. The calculation of the wind load on an

ice-covered structure using modeled ice loads is

presented in section 8. Accreted ice thicknesses

and masses predicted by the two models are compared in section 9.

2. HOURLY WEATHER DATA

Weather elements required in the detailed heatbalance model are

Present weather code

Wind speed

Precipitation rate

Air temperature

Dew-point temperature

Solar radiation

Atmospheric pressure.

Only present weather code, precipitation rate, and

wind speed are used in the simple model. Convenient, but not consistent, units are used in ZRAIN:

wind speed is in m/s, precipitation rate in mm/hr,

solar radiation in W/m2

, atmospheric pressure in

mbar, and air temperature and dew-point temperature in °C. The dew-point temperature, along with

the air temperature, is used to determine the relative humidity of the air:

Rh = edewT/eT , (1)

where eT and edewT are the saturation vapor pressures of air over water at temperatures T and dewT,

respectively. The diameter of the structural element on which the ice load is to be determined is

specified in centimeters. Appropriate factors are

incorporated in the model to convert to a consistent set of units.

Neither model predicts freezing rain. They both

rely on the weather observer having indicated by

the present weather code when freezing rain occurred. A number of present weather codes are used

to record various intensities of freezing drizzle,

freezing rain, and mixed precipitation. ZRAIN

assumes that the user has screened the weather

data, extracted the freezing-rain events, and summarized the occurrence of precipitation in each

hour as freezing rain or drizzle (Z), mixed precipitation (+), rain (R), or snow (S) by examining the

four present weather code fields in the meteorological records. The present weather code is assumed to apply for the entire hour it is reported.

Only freezing rain or drizzle, rain with the air

temperature below freezing, or mixed precipitation freeze to the structure. In particular, the models do not attempt to calculate the accretion of

snow.

ZRAIN also requires the history of the anemometer height above ground at the weather station.

The wind speed at a specified height above ground

is obtained from the measured wind speed, using

the anemometer height (ft) and assuming a 1/7

power law and exposure C (ASCE 1994). Typically, anemometer heights in the United States

have varied between 15 and 100 ft above ground,

so the correction of the wind speed to 10 m (the

worldwide default anemometer height) or 30 m

(typical height of a transmission-line wire) may be

significant (Fig. 1). Anemometer height histories

for first-order weather stations are available in

Local Climatological Data Annual Summary (NOAA,

yearly), and for military stations from ETAC in

electronic format.

Solar radiation data are used to determine the

solar heat flux at the accretion surface. Typically,

there is little solar radiation during freezing rain

because of the complete cloud cover and the relatively low solar angle in winter. During the day,

solar radiation fluxes in these conditions range up

to about 200 W/m2

, with values of about 80 W/m2

more common. If solar radiation information is

not available, then it can be assumed that there is

no incoming solar radiation. This is true at night,

but this assumption will reduce the total heat flux

from the actual amount during the day and may

slightly increase the hindcast ice load.

Atmospheric pressure is used in calculating the

density ρa and saturation vapor pressure eT of air

and the diffusivity of water vapor in air κm. The

dependence of these parameters on atmospheric

pressure is not great. For a 50-mbar pressure change

from 1000 mbar, ρa and κm change by 5% and eT by

only 0.02%. If atmospheric pressure data are not

available, then the pressure appropriate for the

elevation of the site may be assumed, with little

effect on the hindcast ice load.

Most weather elements are measured every

hour and are reported as an instantaneous value,

rather than as an average over the hour. In the

heat-balance model I assume that temperature,

dew-point temperature, atmospheric pressure, and

wind speed vary linearly between the reported

values. In each hour there are six 10-min time

2
40

30

20

10

0 5 10 15

Wind Speed (m/s)

Height Above Ground (m)

steps. In the first time step (e.g., 1400–1410) these

weather elements take on the values reported for

that hour (1400), and in the last time step (1450–

1500) they are equal to the values reported for the

next hour (1500). In each intermediate time step,

one-fifth of the difference between the hourly values is added to the value for the previous time step.

This scheme weights the reported values: each

applies for 20 minutes, and the interpolated values

apply for 10 minutes. Precipitation rate and solar

radiation are reported as a total for the hour, and

I assume them to be constant during each hour. In

the simple model, the wind speed is also assumed

to be constant for each hour.

A missing precipitation rate for an hour in

which precipitation is reported by the present

weather code is assigned 0.01 mm/hr in ZRAIN.

This gives continuity to the freezing-rain event

without arbitrarily adding significant mass to the

accretion. If wind speed, temperature, dew-point

temperature, or air pressure is missing in an hour

it is assumed to be the same as in the previous

hour.

The models contain no procedure for melting

accreted ice. I assume, in extracting freezing-rain

events from the weather data, that accreted ice

melts as soon as, and only when, the air temperature rises above freezing after the freezing rain

ends. The occurrence of T > 0°C after freezing rain

ends thus defines the end of an event. Many freezing-rain storms end with warming temperatures;

however, after a well-documented ice storm in

Hanover, N.H., the temperature remained cold,

and accreted ice remained on wires for more than

30 days (Ackley and Itagaki 1970).

3. ICE ACCRETION SHAPE

In the models, the structure on which the ice is

accreting is a horizontal circular cylinder, which is

referred to as a wire in the following. Extrapolation of the models’ results to other cylinder cross

sections is discussed in section 6.

Ice accreted on horizontal cylinders (wires and

tree branches) during freezing-rain storms in New

York, Arkansas, and Vermont is shown in Figure

2 and on the cover. The shape of the ice accretions

varies from a crescent on one side of a branch (Fig.

2d), to a round accretion with a few icicles (Fig. 2a),

to a heavily icicled barbed-wire fence (Fig. 2b) and

power line wire (cover), to an elongated accretion

shape, apparently fused icicles, on a wire. The

shape of accreted ice depends on the diameter of

Figure 1. Wind profile for a 10-m/s wind at 10 m above ground

assuming the 1/7 power law (ASCE 1994).

3
a. Tree branches, Ft. Drum, New York, March 1991 (photo, D. Fisk).

Figure 2. Freezing rain ice accretion shapes.

b. Barbed wire fence, Arkansas, February 1994 (photo, Entergy).

4
Figure 2 (cont‘d).

d) Tree branch, Ascutney, Vermont, February 1995 (photo, N. Mulherin).

c) Wires near Dermott, Arkansas, February 1994 (photo, Entergy).

5
the wire, the rate of precipitation, the wind speed,

and the rate at which the impinging water freezes.

The models, however, do not attempt to determine the ice accretion shape, but instead assume

simple shapes that, I hope, adequately represent

reality.

In the simple flux model, the ice accretes with a

uniform radial thickness around the circumference of the wire. This simple shape is consistent

with the level of detail in the model.

In the heat-balance model, the precipitation

that freezes immediately accretes with a uniform

radial thickness. The runoff water that does not

freeze immediately is allowed to freeze as icicles.

The “icicle” in this model is a single, vertically

oriented, circular cylinder of ice per meter of wire

that can be understood as the actual icicles stacked

up end to end. As runoff water freezes to the sides

of the icicle, its diameter increases from an initial

assumed diameter of 0.5 cm (Maeno et al. 1994).

Runoff water that does not freeze to the sides of the

icicle is assumed to flow down to its tip where it

may freeze, in successive 0.5 cm-diameter half

spheres, and increase the length of the icicle. Water

that does not freeze there is assumed to drip off. At

the end of each time step, the total icicle mass is

distributed evenly over the new icicle length to

form a uniform circular cylinder. For each 10-min

time step the diameter of the wire–ice accretion Dc,

and the diameter Di and length Li of the icicle are

determined as initial conditions for the next time

step.

4. HEAT BALANCE IN FREEZING RAIN

The primary process in determining the amount

of the available precipitation that freezes to a

structure is the flux of heat from the accretion

surface. Heat must be removed from the rain drops

that collide with a structure for them to freeze. In

calculating the heat fluxes in the heat-balance

model, the wire–ice accretion is assumed to be at

0°C. This is usually a good assumption in freezing

rain because, in the typical near-freezing air temperatures, relatively low winds, and high water

flux, not all the impinging precipitation freezes.

The flux of precipitation is computed using the

relationship from Best (1949) to determine the

liquid water content of the air as a function of the

precipitation rate,

W = 0.067P0.846, (2)

where W is in g/m3 and P is in mm/hr. Best

reviews relationships between precipitation rate

and liquid water content that both he and other

researchers, including Marshall and Palmer (1948),

had determined. I chose his formula rather than

the Marshall–Palmer formula (W = 0.072P0.88) for

ZRAIN because it is the average of all the W(P)

relationships, while Marshall–Palmer’s formula

gives comparatively high values of W. In ZRAIN,

the raindrops move horizontally at the wind speed,

so the flux of precipitation w is the vector sum of

the vertical and horizontal water fluxes, Pρo and

WV, respectively, converted to consistent units:

 

w

P WV o ( ) . ( )

/

g/m s2 = 



 



 +

















ρ

3 6

2

2

1 2

 . (3)

Nine heat-flux terms were considered for inclusion in the heat-balance model to determine the

fraction of the available precipitation that freezes.

They are:

• Qw (cooling flux required to warm

the incoming water to 0°C) =

–cww(0 – T)

• Qc (convective cooling flux) =

–πh(0 – T)

• Qe (evaporative cooling flux) =

–102πhmLe[e0/273.15 – RheT/(T

+ 273.15)]mw/R

• Ql (longwave radiative cooling flux)

= –πσ[273.154 – (T + 273.15)4]

• Qs (shortwave radiation heat flux) =

πSr/2 (4)

• Qf (flux of heat released by freezing

all impinging precipitation) = Lf w

• Qv (viscous heating flux) =

10–3πhrvV2/(2cp)

• Qk (heat flux from droplet kinetic

energy) =

10–3w{V2 + [Pρo/(3.6W)]2}/2

• Qr (resistive heating flux from current flowing through a conductor) =

102I

2 Rc/Dc.

The heat fluxes are in watts per unit length of the

wire per wire–ice diameter. Qc, Qe, Ql, Qs, and Qv

all include a factor of π because they are for the

outside surface of the wire–ice accretion, which

has an area per unit length of π times the diameter.

The magnitudes of these fluxes (except Qr) are

compared in Figure 3 for ranges of temperatures,

wind speeds, precipitation rates, relative humidties, and diffuse solar radiation typical for freezing-rain events. The individual heat-flux terms are

described below.

6
Figure 3. Comparison of heat flux terms in typical freezing-rain conditions. Standard conditions (black symbols) for

the parameters are V = 3 m/s, T = –3°C, P = 3 mm/hr, Rh = 0.90, Sr = 100 W/m2, Dc = 3 cm. In each plot one or two

of these parameters is varied about the standard conditions as follows: a) 0 < V < 10 m/s, b) –6 < T < –0.5°C,

c) 0.5 < P < 10 mm/hr, d) 0.5 < Dc < 10 cm, e) 0.80 < Rh < 1.00 and 0 < Sr < 200 W/m2. Minimum convective and

evaporative fluxes in a) are provided by free convection. Key: e = evaporative cooling, c = convective cooling, l =

longwave cooling, w = cooling from warming water, s = shortwave solar heating, f = heat of fusion, v = viscous heating,

k = kinetic heating.

c e lw sfv k

104

Cooling Heating

103

102

101

100

10– 1

10– 2

10– 3

a. Wind

Heat Flux (W/m )2

b. Temperature

c el wsfv k

Cooling Heating

c. Precipitation

c el wsfv k

Cooling Heating

d. Diameter

c e lw sfv k

Cooling Heating

e. Relative Humidity

and Solar Radiation

c e lw sfv k

Cooling Heating

Cooling flux required to warm impinging

precipitation

In the heat-balance model, the latent heat of

fusion given off by the fraction of precipitation

that freezes to the wire warms all the impinging

precipitation to 0°C. Assuming the raindrops are

at the air temperature T, Qw is the cooling flux

necessary to raise their temperature to 0°C.

Convective cooling

One of the primary processes that cools the

wire–ice accretion is the loss of heat from the cold

air blowing past the structure. When there is no

wind, heat is removed at a slower rate by free

convection controlled by the temperature difference between the wire–ice accretion and the air.

The Nusselt number NuL is used to define the rate

of heat transfer in both forced and free convection

in external flow:

NuL = hL/ka, (5)

where h is the heat transfer coefficient, L is the

appropriate length scale, and ka is the thermal

conductivity of the air. In forced convection, NuL

is given in terms of the Reynolds number ReL and

Prandtl number Pr of the flow. In free convection,

where there is no external velocity scale, NuL is

given in terms of the Rayleigh number RaL, which

is the product of the Grashof and Prandtl numbers,

GrL and Pr. Relationships between NuL and these

nondimensional parameters have been determined

empirically for different cylinder roughnesses and

orientations.

To decide which empirical formula to use in

forced convection, we need to know the typical

range of Reynolds numbers for freezing rain

accreting to a wire. This geometry is idealized as

flow around an infinite cylinder with the wire–ice

diameter (or icicle diameter) as the length scale.

The smallest diameter of interest is 0.5 cm for a

new icicle, and the diameter of a moderately large

wire–ice accretion is about 12 cm. Typical wind

speeds during freezing rain range from 0.5 to 10

m/s, and the kinematic viscosity of air is νa = 0.132

cm2/s at 0°C. Using these values, the Reynolds

number range of interest is 190 < ReD < 9.1 × 104,

where D is the diameter of either the icicle or the

wire–ice accretion.

Achenbach (1977) determined the total Nusselt

number for rough cylinders in subcritical flow, for

ReD > 104 with air (Pr = 0.72) as the fluid, as

7
NuD = 0.18 ReD

0.63. (6)

In these wind-tunnel experiments, the cylinder

surface was maintained at a constant temperature.

The cylinders were roughened to three different

roughness heights by knurling the surface into

regular pyramidal shapes. The data showed that,

in subcritical flow, NuD is independent of the

roughness height, but the upper value of the

Reynolds number for subcritical flow varies between 2 × 105

 and 106, decreasing with increasing

roughness. The higher Reynolds numbers for ice

accretions in freezing rain are below the upper

value of this subcritical flow range, but the lower

values of ReD extend well below 104, the lower

limit in Achenbach’s (1977) experiments. To obtain appropriate Nusselt numbers for these lower

Reynolds number conditions, I used results from

Morgan (1973). He determined NuD for bare,

stranded conductors for ReD > 102, maintaining a

constant heat flux rather than a constant surface

temperature. He compared these stranded conductor results to his Nusselt number formulation

for smooth cylinders in Morgan (1975). Morgan’s

and Achenbach’s results and a formula determined for smooth cylinders by Zhukauskas and

recommended in Incropera and DeWitt (1985) are

compared in Figure 4. This figure shows that

Zhukauskas’s smooth cylinder formula predicts

Nusselt numbers that are lower than Morgan’s

data at low Reynolds numbers and lower than

Achenbach’s at high Reynolds numbers. The scatter of the wind-tunnel data in this figure illustrates

the uncertainty in determining the Nusselt number for real freezing-rain ice accretions with irregular roughness that is different from that in

both Achenbach’s and Morgan’s controlled experiments. I incorporated Morgan’s and

Achenbach’s rough cylinder results in the heatbalance model as follows:

NuD = 0.583 ReD

0.471 35 < ReD < 1600

NuD = 0.18 ReD

0.63 1600 < ReD < 1 × 106 (7)

NuD = 0.00257 ReD

0.98 ReD > 1 × 106

,

extrapolating Achenbach’s formula to lower

Figure 4. Nusselt numbers for forced convection over rough and smooth cylinders as

a function of Reynolds number. Rough cylinder data from Achenbach (1977)

are for Reynolds numbers larger than 104

. Morgan’s (1973) rough cylinder data

are for Reynolds numbers less than 5 × 104. Zhukauskas’s curve (Incropera and DeWitt

1985) is for smooth cylinders for the entire range of Reynolds numbers shown.

104

102

Reynolds Number

Nusselt Number

3 10

2 10

1 10

0 10 3 10 4 10 5 10 6 10 7 10

Zhukauskas

Achenbach

Morgan

8
Reynolds numbers to intersect Morgan’s at ReD =

1600. In Morgan’s formula for NuD, νa and ρa are

supposed to be determined at the average of the

wire–ice surface temperature (0°C) and the air

temperature T. However, these temperatures typically differ by only a few degrees in freezing rain,

so there is very little error in using the air temperature in calculating these parameters.

Both Achenbach’s and Morgan’s experiments

were done in air, and neither specified the dependence of NuD on the Prandtl number. For air at the

temperatures and pressures appropriate for freezing rain, Pr is essentially constant. Following

Zhukauskas (Incropera and DeWitt 1985), I assume that NuD is proportional to Pr0.37. Thus eq 7

can be written

NuD = (0.583/Pr0.37) Pr0.37 ReD

0.471

35 < ReD < 1600

NuD = (0.18/Pr0.37 ) Pr0.37 ReD

0.63

1600 < ReD < 1 × 106 (8)

NuD = (0.00257/Pr0.37) Pr0.37 ReD

0.98

ReD > 1 × 106.

This formulation is used with the heat and mass

transfer analogy in determining the evaporative

cooling flux.

Evaporative cooling

Under typical freezing-rain conditions, the impinging water does not freeze immediately. There

is a vapor density gradient away from the wire

because the wire–ice surface is warmer than the air

and the saturation vapor density increases with

temperature. This gradient is enhanced if the relative humidity of the air is less than 100%. Thus, the

water film on the surface of the wire–ice accretion

is evaporating as it freezes and cooling the remaining water. The evaporative cooling flux is

 

Q L m

D dt e L h e e

c

= = – – 10 10 em w 2 6 π ρ∆ , (9)

where Le is the latent heat of evaporation, me is the

mass of evaporated water per unit length, Dc is the

wire–ice diameter, dt is the time interval, hm is the

vapor transfer coefficient, and ∆ρw is the vapor

density difference between the ice accretion surface and the air. The vapor density ρwT is related to

the saturation vapor pressure eT by the ideal gas

law

 

e m R T T w

ρwT

= + 10 273 15 4 ( .) , (10)

to give

 

∆ρ ρ ρ w w wT = − 0 Rh

T T w = +

− −

















4 10

273 15 273 15

0 e Rhe

T

m

. . R , (11)

where mw is the molecular weight of water vapor,

R is the gas constant, Rh is the relative humidity,

and eT and e0 are the saturation vapor pressures at

temperatures T and 0°C, respectively. This formulation for Qe is compared with that in other ice

accretion models in Appendix A.

The empirical formula used in the heat-balance

model for the saturation vapor pressure over water,

eT = 6.1121(1.0007 +

3.46 × 10–6Pa)e17.502T/(240.97 + T) , (12)

is from Buck (1981). The saturation vapor pressure

over water, rather than over ice, is used, both in the

air and at the accretion surface, because both the

precipitation and the accretion surface are unfrozen.

The vapor transfer coefficient hm is determined

using the mass and heat transfer analogy (Incropera

and DeWitt 1985). The nondimensional number

for mass transfer equivalent to NuL for heat transfer is the Sherwood number ShL = hmL/κm where

κm is the vapor diffusivity. The Schmidt number

Sc = νa/κm is equivalent to Pr. Like Pr, Sc is

virtually constant for air at the temperatures and

pressures we are concerned with. The heat and

mass transfer analogy states that for any function

NuL(ReL, Pr) the Sherwood number is the same

function of ReL and Sc. Thus, for mass transfer the

equivalent to eq 8 is

ShD = (0.583/Pr0.37) Sc0.37 ReD

0.471

35 < ReD < 1600

ShD = (0.18/Pr0.37) Sc0.37 ReD

0.63

1600 < ReD < 1 × 106 (13)

ShD = (0.00257/Pr0.37) Sc0.37 ReD

0.98

ReD > 1 × 106.

The vapor transfer coefficient hm = (ShDκm)/D is

then obtained from eq 13 for the appropriate

Reynolds number range. Significantly different

values for vapor diffusivity κm are specified in

different references. Incropera and DeWitt (1985)

give κm = 0.26 cm2/s at T = 25°C and Pa = 1000

mbar in their Table A8. They also say κm is proportional to (T + 273.15)3/2, which results in κm = 0.23

cm2/s at 0°C. Batchelor (1970, Appendix 1) specifies κm = 0.25 cm2

/s at 15°C. Pruppacher and Klett

9
(1980, p. 413) give

 

κm = cm s,  +

    0 211 273 15

273 15

1013 1 94

2 . .

. /

. T

Pa (14)

which results in κm = 0.21 cm2/s at Pa = 1000 mbar

and T = 0°C. They also say that conventionally

used values may be too high, and as they are a

better source for this information than either

Incropera and DeWitt (1985) or Batchelor (1970),

their formula for κm is used in the heat-balance

model.

The mass of water that evaporates in the process of freezing the impinging rain is me =

–10–2QeDcdt/Le (eq 9). In the heat-balance model,

the evaporated water is subtracted from the runoff water available for icicle formation if not

all the impinging precipitation freezes directly to

the wire.

Convective and evaporative

cooling in no wind

When there is no wind there cannot be forced

convection, so convective heat transfer occurs by

free convection only. Free convection occurs when

the temperature difference between the wire–ice

accretion and the air causes the air column to be

unstable. The wire–ice accretion is typically warmer

than the ambient air, so the warmer air over the

wire rises and is replaced continuously by colder

air. Heat transfer in free convection is governed by

the Grashof number,

 

Gr g TL

T L

a

= −

+

( )

(. )

0

273 15

3

2 ν

, (15)

and the Rayleigh number,

 

Ra Gr Pr g TL c

T k L L

a p

a a

= = −

+

( )

(. )

0

273 15

3

ρ

ν

 , (16)

where g is the acceleration of gravity, cp is the

specific heat of air, and νa is the kinematic viscosity

of air. It can be assumed that forced and free

convection do not occur simultaneously when

ReL/GrL

2

 is much different from 1 (Incropera and

DeWitt 1985). For typical wire–ice diameters, wind

speeds, and air temperatures in freezing rain, Gr

<< Re2, so this is a good assumption. I assume in

the heat-balance model that forced convection

controls the heat transfer if there is any wind, and

free convection controls only when there is no

wind (V = 0). The Nusselt number determined by

Morgan (1975) for free convection around smooth

horizontal cylinders is

NuDc = 0.85 RaDc

0.188 100 < RaDc < 104 (17)

NuDc = 0.48 RaDc

0.250 104 < RaDc < 107

.

In a separate paper, Morgan (1973) argues that

for relatively low Rayleigh numbers (100 < Ra <

4 × 105), the roughness of the surface does not have

an effect on heat transfer, so these formulas should

also apply to rough ice accretions. For most reasonable combinations of the wire–ice accretion

diameter and weather parameters in freezing rain,

the Rayleigh number is within this low range, so

eq 17 are incorporated in the heat-balance model

to determine the heat transfer coefficient for the

wire when there is no wind.

The heat and mass transfer analogy can be used

in free convection (Incropera and DeWitt 1985) to

estimate the mass transfer coefficient for evaporative cooling from eq 17:

ShDc = 0.85 RaDc

0.188 (Sc/Pr)0.188

100 < RaDc < 104

ShDc = 0.48 RaDc

0.250 (Sc/Pr)0.250 (18)

104 < RaDc < 107.

Longwave radiative cooling

In computing the flux of longwave radiation for

the heat-balance model, I assume that the wire–ice

accretion is radiating at 0°C and the clouds and

surrounding rain-filled air are radiating at the air

temperature T. The emissivity of everything is

taken to be 1 [emissivity of ice is 0.98 and of water

is 0.96 (Incropera and DeWitt 1985)]. Then the

longwave cooling flux is given by the blackbody

radiation law:

Ql = –πσ[273.154 – (T+273.15)4] . (19)

Shortwave radiative heating

The shortwave heat flux Qs = πSr/2 is determined using the measured or modeled incoming

global diffuse radiation flux Sr. This information is

available hourly at some weather stations (NOAA

1993) and is reported as an average for each hour.

The factor of π/2 transfers the reported values for

a flat horizontal surface to a horizontal cylinder.

The derivation of this factor is presented in Appendix B. I assume in the heat-balance model that

the albedo of the wire–ice accretion is zero (the ice

is clear), that all the incoming radiation to the

accretion is absorbed in either the ice or the underlying wire, and that there is no radiation reflected

back from the ground to the wire. These assumptions are only approximately correct, but they are

somewhat offsetting in their simplicity: a) the al10
bedo of the ice is close to, but greater than zero, and

depends on how much air and/or snow is incorporated in the accretion, b) the shorter-wavelength

visible radiation may not be absorbed in the ice,

but it is probably absorbed in the underlying wire,

which is assumed to be in thermal equilibrium

with the ice accretion, and c) the ground surface

reflects some incoming radiation, depending on

the ground cover and the amount and condition of

snow and ice on the ground.

The shortwave and longwave fluxes are opposite in sign and comparable in size, but there is no

shortwave flux at night, when the air temperature

may be relatively cold and the longwave flux

relatively large. Both are small compared with the

convective and evaporative fluxes in typical freezing-rain conditions, but large compared with the

viscous and kinetic heat fluxes (Fig. 3).

Viscous heating

The formula for the viscous heat flux is from

Makkonen (1984). At wind speeds appropriate for

freezing rain, viscous heating from air moving by

the wire–ice accretion is small compared with the

other heat fluxes (Fig. 3). This term is not included

in the heat-balance model.

Kinetic heating

When precipitation drops collide with the wire–

ice accretion, some of their kinetic energy is converted to heat. At the droplet speeds associated

with freezing rain, this term is small (Fig. 3). It is

not included in the heat-balance model.

Resistive heating

If the wire conducts electricity, heat is generated from the wire resistance. The resistive heat

generated per unit length of wire I

2

Rc depends on

the current I and the wire resistance per meter Rc,

which generally is a function of the current load

and frequency. The resistive heat flux per wire

diameter is

 

Q I R

D r 2 c

c

( ) W/m = 102 2

 . (20)

The resistive heat flux in a wire varies from

zero, when there is no current, up to values larger

than any of the other heating or cooling fluxes. For

example, for a 2.63-cm (1.036-in.)-diameter ACSR

conductor with I = 400 amp and Rc = 8.14 × 10–5

ohm/m (0.131 ohms/mile), Qr = 490 W/m2. This

is comparable to the typical convective and evaporative cooling fluxes in freezing rain shown in

Figure 3. Because of the extremely large range in

the resistive heating flux and because assuming no

resistive heating is both reasonable and conservative, it is not included in the model.

Latent heat of fusion

To freeze water at 0°C, the heat of fusion must

be removed. The fraction f of the impinging precipitation that freezes is the ratio of the sum of all

the other heat fluxes to the latent heat flux Qf that

is calculated assuming that all the incoming water

freezes:

 

f QQ QQQ

Q = c e + +++ l s w

f

 . (21)

If f is 1 or greater, then all the incoming water

freezes. Properly, the temperature of the accretion

surface should be determined iteratively, and heatflux terms for cooling the ice to the calculated

surface temperature and depositing hoarfrost or

sublimating accreted ice should be included. These

are small effects in freezing-rain conditions, however, and are not incorporated in the heat-balance

model.

Other effects

Two other, possibly significant, factors in determining the heat balance at the ice accretion

surface are a) the angle of the wire to the wind

direction, and b) freezing of the impinging precipitation by the cold stored in the wire.

Wind direction is archived with the weather

data, so for a particular structure the yaw of the

wire to the wind direction during freezing rain

could be determined. The wire orientation could

then be incorporated in the calculation of the precipitation flux w across the wire by multiplying the

righthand side of eq 3 by the sine of the angle

between the wire and the direction of the precipitation flux. The Nusselt number also depends on

the yaw of the wire. Morgan (1973, 1975) discusses

the variation of the Nusselt number with yaw in

forced convection for both smooth cylinders and

bare, stranded wires.

If the air temperature is cold before freezing

rain begins and the wire is in equilibrium with the

air, impinging precipitation may freeze by conductive heat loss into the wire. The mass of water

that can be frozen by this mechanism is

 

M

D cT

L = –

10

4

2 π ρc

2

c pc

f

 , (22)

where ρc is the density and cpc is the heat capacity

of the wire. This effect is small except perhaps for

11
tained from eq 23 and 24 using the heat and mass

transfer analogy and replacing Pr with Sc both

explicitly and in Radd (Incropera and DeWitt 1985).

Of the other heat-flux terms, Qw = 0 for icicles

because the runoff water is already at 0°C. I assume that there is no solar radiation absorbed in

the icicles, Qs = 0, because a) the icicle orientation

is close to vertical and thus roughly parallel to the

incoming diffuse solar radiation flux, b) much of

the longer-wavelength visible radiation is absorbed

in the cloud cover, and c) very little of whatever

incident shorter-wave radiation that gets through

the clouds will be absorbed in the small-diameter

clear icicle.

The fraction fb of the available water that freezes

is determined as in eq 21. If fb is greater than 0.64

then all the available protoicicle water is retained,

and if fb is less than 0.64 the fraction of the protoicicle

mass that is incorporated in the icicle is fb/0.64.

After an icicle forms, runoff water may freeze to

its sides. If there is wind, the Nusselt and Sherwood

numbers for forced convection are determined

using eq 7 and 13, based on the diameter of the

icicle Di. If there is no wind, heat transfer is by free

convection. Heat transfer by free convection from

the icicle (vertical cylinder) is different from that

for the wire (horizontal cylinder). The relatively

warm air at the surface of the icicle flows upward,

drawing cooler air after it. The boundary layer

thickness increases from the bottom to the top of

the icicle, decreasing the heat transfer. The Nusselt

number for free convection for this geometry is

based on the length of the icicles rather than on

their diameter. The appropriate icicle length to use

is that for a single icicle, rather than this stack of

icicles. Following Makkonen and Fujii (1993), I

assume 45 identical icicles per meter of wire to give

an average icicle length Lbar = Li/45. Because the

icicles are closely spaced on the wire, thus inhibiting cooling, I use an empirical formula for heat

transfer from a vertical plate, rather than from a

cylinder to determine the Nusselt number

(Incropera and DeWitt 1985, eq 9.27):

 

Nu Ra

Pr

Lbar

Lbar = +

+ ( ) 



 





0 68 0 670

1

1 4

0 492 9 16 4 9 . . /. / /

for RaLbar < 109. (25)

The Sherwood number for evaporative cooling is

obtained from eq 25 by replacing Pr with Sc, both

explicitly and in RaLbar. The convective and evaporative heat fluxes are quite insensitive to the aslarge structural elements. For example, for a 3-cmdiameter aluminum wire, with cpc = 0.88 J/g° C, ρc

= 2.77 g/cm3, and air temperature T = –5°C, the

mass of water that can be frozen by heat conduction into the wire is 26 g/m, which is equivalent to

an ice thickness of 0.03 cm. The calculation of this

additional mass of ice is included as an option in

the model.

5. ICICLES

Precipitation that does not freeze to the wire

may freeze as icicles in the process of dripping off

the wire. I assume that there is no precipitation

impinging directly on the icicles. At low wind

speeds, icicles form directly under the wire and are

shielded by the wire from the falling rain. At high

wind speeds, the runoff water is blown to the lee

side of the wire so icicles form behind the wire,

shielded from the wind-driven rain. In the heatbalance model, a single icicle per meter of wire

represents the actual icicles on that meter of wire

stacked end to end. If there is no existing icicle, the

mass of water that is not frozen or evaporated

while on the wire forms a protoicicle, which is a

stack of partially frozen hemispherical droplets

with an assumed diameter dd = 0.5 cm (Maeno et

al. 1994). The protoicicle freezes from the outside

in, and when the surface water freezes the unfrozen interior water is retained and, as more heat is

removed, eventually freezes. Assuming the surface layer of ice on the hemispherical droplets is 0.1

cm thick (Fig. 2, Maeno et al. 1994), the ratio of the

volume of ice to the volume of the hemisphere is

0.64. In the heat-balance model, all the water in the

protoicicle is incorporated in the icicle if the freezing fraction fb, determined from the heat balance

for the protoicicle, is greater than 0.64.

The convective and evaporative heat fluxes for

this protoicicle are determined using Nudd for

spheres. For forced convection (Incropera and

DeWitt 1985, eq 7.55),

Nudd = 2 + Pr0.4(0.4 Redd1/4

+ 0.06 Redd2/3), (23)

and for free convection (Incropera and DeWitt

1985, eq 9.35),

 

Nu Ra

Pr

dd

dd = +

[ ] +

2 0 589

1 0 469

1 4

9 16 4 9

.

(. / )

/

/ / . (24)

The Sherwood number for vapor transfer is ob12
Figure 5. Equivalent uniform radial ice accretions on cylinders with circular, rectangular, and angle

cross sections for a rainfall of 1 cm. The width of the rectangle is 1/4 of its length. The angle has equallength legs with arbitrary thickness. The perimeter-to-“diameter” ratio S/D ranges from 3.14 for the

circle to 2.5 for the rectangle and 4 for the angle.

D

0.35 cm

S/D = π

0.44 cm

D/4

D

S/D = 2.5

0.28 cm

D

D

S/D = 4.0

sumed average icicle length as both h and hm are

proportional to Lbar–1/4. As for the protoicicle, Qw

= 0 and Qs = 0 and the freezing fraction fs is

determined using eq 21. The water that does not

evaporate or freeze to the sides of the icicle forms

a protoicicle at its tip, and the heat balance for the

protoicicle is determined as before. Finally, the

water that does not freeze at the tip is assumed to

drip off. The new total icicle mass Mi is distributed

evenly over a cylinder with length Li equal to the

original icicle length plus the length of the

protoicicle.

6. SIMPLE FLUX MODEL

The simplest case of ice accreting in freezing

rain occurs if all the precipitation impinging on the

structure freezes in a uniform radial accretion.

When that happens, the amount of ice on the

structure is directly related to the amount of rain

that falls. For example, if 1 cm of freezing rain falls

on a 2-cm-wide flat plate and freezes in place, then

a layer of ice 1(ρo/ρi) = 1.1 cm thick results. On a 4-

cm-wide plate nearby, the same thickness of ice

would form with twice the mass. A 2-cm-diameter

horizontal circular cylinder intercepts the same

depth of rain as the 2-cm-wide flat plate. If that

water depth is then spread uniformly around the

cylinder’s circumference and frozen, forming a

uniform radial accretion, then the layer of ice is

1(ρo/ρi)/π = 0.35 cm thick. The factor of π is the

ratio between the circular cylinder’s circumference, over which the ice freezes, and the diameter,

which intercepts the precipitation. This calculation can be generalized to a cylinder with any cross

section to determine the uniform ice thickness on

its perimeter:

 

R D

S

P

t

j

N

eq o

i

j = 







 ∑  =

ρ

ρ 1 10

∆ (26)

where the summation gives the total precipitation

amount in N hours and S and D are the perimeter

and horizontal dimension, respectively, of the cross

section. Equivalent uniform ice thicknesses are

shown in Figure 5 for three different cross sections

in a 1-cm rainfall.

For many shapes, but not circles, the D/S ratio

decreases as ice accretes uniformly on the perimeter. For those, Req could be determined incrementally. When this is done for the rectangle in Figure

5, using 0.2-cm increments of rain, the final uniform radial ice thickness is 0.42 cm rather than 0.44

cm. This difference is well within the uncertainties

in either this simple model or the detailed heatbalance model.

This simple precipitation model shows that, at

least to first order, the thicknesses of ice on components with the same shape cross-section but different diameters are the same. On components with

the same diameter but different cross sections, the

ice thickness ratio is equal to the inverse of the

perimeter ratio. Thus, uniform radial ice thicknesses determined by this model, or any other

model, for a circular cylinder of any diameter can

be extrapolated both to cylinders with other cross

13
sections and cylinders with other diameters.

A slightly more complex model takes the increased flux of water due to the horizontal velocity

of the rain drops in the wind into account. Using

the flux of precipitation from eq 3, the uniform

radial ice thickness is

 

R D

S

P

j

N

eq i

= ( ) j o + 

 ∑  ρ =

0 1 ρ

2

1

.

(27)

WV t + ( ) j j





 0 36

2 2 1 2

.

/

∆ .

This simple flux model with S/D = π is incorporated in ZRAIN for comparison with the heatbalance model.

7. COLLISION EFFICIENCY

In in-cloud icing, the collision efficiency of the

cloud droplets with a structure is an important

factor in determining the rate of ice accretion.

Cloud droplets are very small, typically 5 to 50 μm

in diameter, and have correspondingly small terminal velocities (0.007 m/s for 15 μm droplets,

Best 1950). Thus, in windless conditions there is

essentially no droplet flux. When there is wind, the

wind drag on the droplets carries them along and

the droplets follow the wind streamlines around

any obstacle in their path. Only the inertia of the

droplets, which tends to keep them moving in a

straight-line path, makes them diverge from the

wind streamlines. The collision efficiency of the

cloud droplets with a structure represents a balance between their drag, the tendency to follow

the streamlines, and their inertia, the tendency to

continue in a straight line. Qualitatively, the results of this balance are that a) smaller droplets,

which have less inertia, have lower collision efficiencies, b) the collision efficiency of droplets is

smaller for large obstacles than for small ones,

because streamlines diverge relatively farther in

front of large obstacles than small ones, and c) in

high winds droplets have more momentum so

their collision efficiency is higher. These observations are quantified in Langmuir and Blodgett

(1946). They numerically solved the trajectory equation, ignoring gravitational effects, to determine

droplet collision efficiency as a function of two

nondimensional numbers that depend on wind

speed, wire diameter, droplet diameter, and the

density and viscosity of air.

The Langmuir and Blodgett results cannot be

applied to precipitation droplets. The terminal

velocity of drizzle and rain drops, typically 100s of

microns to 1 or 2 mm in diameter, is between about

2 and 7 m/s (Wang and Pruppacher 1977). Thus,

even in no wind, there is a downward flux of water

that is a balance between gravity and the drag of

the still air on the drops. The drops fall through the

air and collide either with the ground or with

structures in their path. When there is wind, the

drops gain horizontal velocity in addition to their

vertical velocity. The effect of wind on drizzle- and

rain-sized drops could be determined by numerically integrating the trajectory equation (Lozowski

and Oleskiw 1983), including the gravitational

term:

 

dv

dt

C

d v Vv V g

r r r r r r = 



 



– ––  ( ) +

o

3

4

D a ρ

ρ

 , (28)

where 

r

v is the drop velocity, 

r

V is the wind velocity, CD is the drag coefficient of the drops, d is the

drop diameter, and 

r

g is the acceleration of gravity. In the absence of a solution to this equation for

rain- and drizzle-sized drops, it is reasonable to

assume that the drops fall at their terminal velocity, move horizontally at the wind speed, and

collide with all obstacles in their path.

In some freezing-rain ice accretion models, the

collision efficiency of the rain and drizzle drops is

calculated incorrectly. The resultant velocity of the

wind speed and the drops’ terminal velocity is

used in the Langmuir and Blodgett (1946) formulation to determine the collision efficiency. This

ignores the different physics of falling and windcarried droplets and leads to obviously ridiculous

results: 0.5-mm-diameter drizzle drops falling at 2

m/s in no wind are calculated to have a collision

efficiency of 0.56 with a 1-m-diameter cylinder

and 0.05 with a 10-m cylinder. In that world, if you

carried a big enough umbrella, not only would

you stay dry, so would the umbrella.

8. WIND LOAD

It is often important to know the wind load on

a structure both during a freezing-rain storm, and

for as long after the storm as ice remains on the

structure. The projected area of the structure is

larger because of the ice accretion, so at a given

wind speed the wind load is greater than it would

be on the bare structure. In ZRAIN, the wind load

per meter of wire is determined every hour using

the wire–ice diameter and the icicle diameter and

length, assuming a drag coefficient CD = 1 for both:

14
 

V VD D L

load a ci

i

100

= + 



 

 0 5  2 . ρ . (29)

The factor Li/100 converts the wind load per length

of icicle to wind load per meter of wire. The wind

load results are useful for identifying the combination of wind and ice in each event that causes the

highest horizontal load. This combination is independent of drag coefficient as long as it can be

assumed to be the same for both the wire–ice

accretion and the icicle.

9. SAMPLE RESULTS

ZRAIN was run for the freezing-rain events

that occurred at the Des Moines, Iowa, airport

between 1948 and 1993. The equivalent radial ice

thicknesses on a 2.54-cm-diameter wire 10 m above

the ground, hindcast using the heat-balance model,

are shown in Figure 6 for the 316 freezing-rain

events in those 45 years. For many events either no

ice or very little ice accreted, even though freezing

rain or drizzle was observed, typically because

there was little or no measured precipitation. The

largest accretion predicted by the model occurred

in January 1982. That event consisted of two freezing-rain storms separated by 19 days of cold

weather with temperatures as low as –30°C. In the

first 12 hours of freezing rain, 0.75 cm of ice accreted,

and in the second 19-hr period of freezing rain

there was an additional 0.75-cm uniform radial

accretion.

It is interesting to compare the modeled ice

thicknesses with the Storm Data descriptions of the

severe ice storms in 1990 and 1991. For the 7–8

March 1990 storm at the Des Moines airport, the

heat-balance model hindcast a 0.5-cm-thick uniform radial accretion with 26-cm-long icicles (assuming 45 icicles/m), which is equivalent to a 1.3-

cm uniform radial ice accretion. According to Storm

Data this ice storm affected most of Iowa and

resulted in 1.3 to 7.6 cm (0.5 to 3 in.) of ice on

exposed surfaces and 7.6- to 12.7-cm (3- to 5-in.)-

thick ice on some power lines. Central Iowa from

near Des Moines and up to the north and west,

particularly Carroll and Crawford counties (about

80 miles from Des Moines’ Polk County), was

hardest hit.

According to the model, the 31 October 1991 ice

storm was less severe than the 1990 storm in the

Des Moines area. The hindcast uniform radial ice

accretion is only 0.2 cm thick with 3-cm-long icicles.

Storm Data describes this ice storm as affecting

about a third of Iowa, including the forecast zone

where Des Moines is located. From 2.5 to 5.1 cm (1

to 2 in.) of ice accumulation was reported “in many

areas,” and much of the damage was attributed to

the high winds that accompanied the storm. However, Des Moines’ Polk County was not included

in the Presidential Disaster Declaration for 43 Iowa

counties following that storm (FEMA 1991), and

the measured wind speeds at Des Moines airport

were lower during this storm than in the 1990

storm.

The simple flux model results are compared

with the heat-balance model results in Figure 7.

The models agree well for the vast majority of the

freezing-rain events at Des Moines. In the cases for

which the results from the simple model are significantly higher than those from the heat-balance

model, the air temperature was at or above 0°C

with relative humidity near 100%, and moderate

Figure 6. Uniform radial ice thicknesses hindcast by the heat-balance model for freezing-rain events at the Des Moines

airport from 1948 to 1993.

1.5

1.0

0.5

0

19501960197019801990

Heat-Balance Model R (cm) eq

15
or heavy precipitation. A time series of one of these

events, the 7–8 March 1990 ice storm, is shown in

Figure 8. This storm was characterized by high

precipitation rates, up to 1 cm/hr, increasing temperature and relative humidity, and high diffuse

solar radiation during the day. As the air temperature increased and freezing rain continued near

the end of Julian day 66, little additional ice accreted

directly on the wire, but much of the runoff water

froze as icicles. It is during this time period that the

heat-balance model results diverge from the simple

flux model, which accretes all the impinging precipitation.

The heat-balance model was also run for a 10-

cm-diameter wire to determine whether the lack

of dependence of the ice thickness on wire diameter predicted by the simple model also holds

when the heat fluxes, which are diameter-dependent (Fig. 3), are incorporated in the ice accretion

calculation. The equivalent radial thicknesses for

these two diameters are compared in Figure 9 and

show almost no difference. This is undoubtedly

due to the accretion of runoff water as icicles,

which freeze independent of the wire diameter.

One small but well-documented freezing-rain

event at CRREL on 28 February 1995 serves as a

test for the two models against reality. This event

was characterized by cold temperatures, very light

winds, and light freezing drizzle that froze quickly

rather than as icicles. The ice thickness determined

from the measured mass of ice on a 2.65-cmdiameter horizontal rod (Itagaki, in prep.) at

CRREL’s freezing-rain weather station was 0.19

cm. Both models predict equivalent radial ice accretions 0.18-cm thick. While this good agreement

is encouraging, further testing of the models in

more extreme freezing-rain conditions is desirable.

10. DISCUSSION

This report describes two models to calculate

freezing-rain ice loads on horizontal cylinders from

weather data. One model is very simple and requires only the information that freezing rain is

occurring along with the concurrent precipitation

rate and wind speed. The more detailed model

requires as additional information the air temperature, dew-point temperature, and solar radiation flux, as well as the diameter of the wire on

which ice is accreting. A comparison of the two

models using the historical weather data at Des

Moines indicates that the simple flux model is

conservative, as expected, but generally agrees

well with the detailed heat-balance model.

As freezing-rain events occur at CRREL, measurements of accreted ice on horizontal rods will

be compared with the models’ predictions. This

Figure 7. Comparison of uniform radial ice thicknesses at Des

Moines from the simple flux model and the heat-balance model.

2.5

2.0

1.5

1.0

0.5

0 0.5 1.0 1.5 2.0 2.5

Heat-Balance Model R (cm) eq

Simple Model R (cm) eq

16
10

5

Precipitation (mm/hr)

0

10

5

0

Wind

Speed

Temperature

Temperature (°C)

Wind Speed (m/s)

1200

800

400

0

Ice

on Wire

Icicles

Ice Mass (g/m)

2

1

0

66 67

Julian Day (GMT)

Simple

Model

Heat-Balance

Model

R (cm) eq

200

150

100

50

0

Solar

Radiation

Relative

Humidity

Relative Humidity (%)

Solar Radiation (W/m )2 Figure 8. Time series of the weather conditions and hindcast ice accretions

for the ice storm of 7–8 March 1990 at the Des Moines airport. The

formation of icicles in the heat-balance model occurs in relatively warm,

humid conditions. In those conditions the results from the heat-balance

model and the simple flux model diverge.

field data can be used to fine-tune the heat-balance

model. Parameters that may have a significant

effect on the calculated ice load and may need to be

adjusted, include a) the 0.64 threshold for incorporating all of the runoff water in the protoicicle, and

b) the projected area of the ice accretion that is

assumed to intercept the precipitation flux.

The good agreement between the heat-balance

model and the simple flux model is interesting. It

indicates that the modeled ice thicknesses can be

easily extrapolated so that consistent ice thicknesses can be used in the design of structural

elements with different cross sections. It also shows

that weather data can be used to determine conservative, but apparently realistic, ice loads using

back-of-the-envelope calculations. It will be interesting to see if this good agreement holds as the

heat-balance model is fine-tuned.

17
LITERATURE CITED

Ackley, S.F. and K. Itagaki (1970) Distribution of

icing in the Northeast’s ice storm of 26–27 December 1969. Weatherwise, 23: 274–279.

Achenbach, E. (1977) The effect of surface roughness on the heat transfer from a circular cylinder to

the crossflow of air. International Journal of Heat and

Mass Transfer, 20: 359–369.

ASCE (1991) Guidelines for Electrical Transmission

Line Structural Loading. ASCE Manuals and Reports on Engineering Practice No. 74, American

Society of Civil Engineers, New York.

ASCE (1994) Minimum Design Loads for Buildings

and Other Structures. ASCE Standard 7-93, American Society of Civil Engineers, New York.

Batchelor, G.K. (1970) An Introduction to Fluid

Dynamics. New York: Cambridge University Press.

Bennett, I. (1959) Glaze: Its Meteorology and Climatology, Geographical Distribution and Economic Effects. Quartermaster Research and Engineering

Center, Environmental Protection Research Division, Technical Report EP-105.

Best, A.C. (1949) The size distribution of raindrops. Quarterly Journal of the Royal Meteorological

Society, p. 16–36.

Best, A.C. (1950) The size of cloud droplets in

layer-type clouds. Quarterly Journal of the Royal

Meteorological Society, p. 241–248.

Buck, A. (1981) New equations for computing

vapor pressure and enhancement factor. Journal of

Applied Meteorology, 20: 1527–1532.

EIA/TIA (1991) Structural Standards for Steel Towers and Antenna Supporting Structures. EIA/TIA222-E, Electronics Industries Association, Washington, D.C.

FEMA (1991) Hazard Mitigation Opportunities in

Iowa. FEMA-928-DR-Iowa, Kansas City, Missouri.

IEC (1990) Loading and Strength of Overhead Transmission Lines. International Standard 826, 2nd edition. International Electrotechnical Commission,

Technical Committee 11, Geneva, Switzerland.

Incropera, F.P. and D.P. DeWitt (1985) Fundamentals of Heat and Mass Transfer. New York: John

Wiley and Sons.

Iqbal, M. (1983) An Introduction to Solar Radiation.

Don Mills, Ontario: Academic Press Canada.

Itagaki, K. (in prep.) Mass measurement of wet

snow and freezing rain accumulation.

Langmuir, I. and K. Blodgett (1946) Mathematical

investigation of water droplet trajectories. In The

Collected Works of Irving Langmuir (C.G. Suits, Ed.)

(1960), p. 335–393. Elmsford, New York: Pergamon

Press.

2.0

1.5

1.0

0.5

0 0.5 1.0 1.5 2.0

R (cm) for Wire Diameter = 10 cm eq

R (cm) for Wire Diameter = 2.54 cm eq

Figure 9. Comparison of uniform radial ice accretion thicknesses

for 2.54- and 10-cm-diameter wires using the heat-balance model.

18
Lozowski, E.P. and M.M. Oleskiw (1983) Computer modeling of time dependent rime icing in the

atmosphere. CRREL Report 83-2, Cold Regions Research and Engineering Laboratory, Hanover, New

Hampshire.

Lozowski, E.P., E.M. Gates and L. Makkonen

(1987) Recent progress in the incorporation of

convective heat transfer into cylindrical ice accretion models. In 1987 International Symposium on

Cold Regions Heat Transfer, American Society of

Mechanical Engineers.

Maeno, N., L. Makkonen, K. Nishimura, K.

Kosugi and T. Takahashi (1994) Growth rates of

icicles. Journal of Glaciology, 40(135): 319–326.

Makkonen, L. (1984) Modeling of ice accretion on

wires. Journal of Climate and Applied Meteorology,

23: 929–939.

Makkonen, L. and Y. Fujii (1993) Spacing of icicles.

Cold Regions Science and Technology, 21: 317–322.

Marshall, J.S. and W. Palmer (1948) Relation of

drop size to intensity. Journal of Meteorology, 5: 165–

166.

Morgan, V.T. (1973) The heat transfer from bare

stranded conductors by natural and forced convection in air. International Journal of Heat and Mass

Transfer, 16: 2023–2034.

Morgan, V.T. (1975) The overall convective heat

transfer from smooth circular cylinders. In Advances in Heat Transfer (T.F. Irvine and J.P. Hartnett,

Eds.), vol. 11, p. 199–264. New York: Academic

Press.

MRI (1977) Pacific Northwest Icing Study, MRI 77

FR-1515, prepared for Bonneville Power Administration, Portland, Oregon.

NOAA (1959–1995) Storm Data and Other Unusual

Weather Phenomena. National Climate Data Center,

Asheville, North Carolina.

NOAA (1993) Solar and Meteorological Surface

Observation Network 1961–1990 Version 1.0. September. National Climate Data Center, Asheville,

North Carolina.

NOAA (yearly) Local Climatological Data Annual

Summary. National Climate Data Center, Asheville,

North Carolina.

Pruppacher, H.R. and J.D. Klett (1980) Microphysics of Clouds and Precipitation. Boston: D. Reidel

Publishing Co.

Southeast Regional Climate Center (1993) Freezing Rain and Sleet Climatology of the Southeastern

United States. NOAA, Columbia, South Carolina.

Tattelman, P. and I.I. Gringorten (1973) Estimated

Glaze Ice and Wind Loads at the Earth’s Surface

for the Contiguous United States. Air Force Cambridge Research Laboratories Report AFCRL-TR73-0646.

Vilcans, J. and D. Burnham (1989) Climatological

study to determine the impact of icing on the Low Level

Windshear Alert System. DOT-TSC-FAA-89-2, U.S.

Department of Transportation.

Wang, P.K. and H.R. Pruppacher (1977) Acceleration to terminal velocity of cloud and rain

drops. Journal of Applied Meteorology, 16: 275–280.

19
APPENDIX A: COMPARISON OF EVAPORATIVE

HEAT FLUX FORMULATIONS

The formulation of the evaporative heat flux term in the heat-balance model is

different from that of Lozowski et al. (1987) and Makkonen (1984). The different Qes

are compared in this appendix. Assuming consistent units and considering the heat

flux across an arbitrary surface, Qe is (eq 4)

 

Q hL e Rhe

T

m

R e me T w = + −















 0

273 15 273 15 . . , (A1)

Using the definition of the Sherwood number ShD and the relationship between ShD

and the Nusselt number NuD specified by the heat and mass transfer analogy gives

NuD = hD/ka = CPr0.37 ReD

b (A2)

ShD = hmD/κm = CSc0.37 ReD

b

where C and b depend on ReD. Equation A2 can be used to write hm in terms of NuD:

 

h

Nu

D

Sc

Pr m

m D =



  κ

0 37 .

 . (A3)

Using this relationship and Sc/Pr = ka/(ρacpκm), eq A1 can be written finally as

 

Q Nu

D

k

c

m L

R

e Rhe

T e D a

a p m e T = 









 +





 





ρ

κ

0 37

0 63 0

273 15 273 15

.

. w

. – . . (A4)

The evaporative heat transfer term in Lozowski et al. (1987) is given as

 

Q h Pr

Sc

L e Rhe

c Pa e e T

p

= 



 















 − 0 63

0 .

ε , (A5)

where ε = mw/ma is the ratio of the molecular weights of water vapor and air. Using

the ideal gas law,

 

Pa R T

m = ρa +

a

( .) 273 15 , (A6)

and the definition of NuD in eq A2, eq A5 becomes

 

Q Nu

D

k

c

m L e Rhe

R T e D a

a p m e T = 









 ( )

ρ +

κ

0 37

0 63 0

273 15

.

. w –

( .) . (A7)

Thus, Lozowski and Gates’s formulation (eq A7) is different from this model’s

formulation (eq A4) only in the temperature used to calculate the saturation vapor

density at the accretion surface.

In Makkonen (1984) the evaporative cooling flux is given as

 

Q hL e e

c Pa e e T

p

= ε ( ) 0 − . (A8)

20
Again using the ideal gas law and eq A2, this can be rewritten as

 

Q Nu

D

k

c

mLe e

R T e D a

a p

e T = 









 ( )

ρ +

w –

( .)

0

273 15 . (A9)

Makkonen’s formulation differs from this model’s by the temperature used to

calculate the saturation vapor density at the freezing surface, by the dependence on

ka/(ρacp) rather than on [ka/(ρacp)]0.37κm0.63, and by using the saturation vapor

density of the air rather than the vapor density at the ambient humidity.

The effect of these differences is to make Lozowski and Gates’s Qe 4% larger and

Makkonen’s Qe 32% smaller than this model’s at typical freezing-rain conditions

T = –3°C, Rh = 90% and Pa = 1000 mbar. The differences between the models increase

somewhat as air temperature decreases and increase substantially as humidity

decreases.

This comparison addresses the differences in the physical models of evaporative

cooling and in the assumed dependence of the Nusselt number on the Prandtl and

Schmidt numbers. There are further differences between the models in the empirical

formulas used to relate the Nusselt number to the Reynolds number and Rayleigh

number.

21
APPENDIX B: SKY DIFFUSE SOLAR RADIATION FLUX TO A

HORIZONTAL CYLINDER

Sky diffuse radiation under an overcast that would be typical during freezing rain

is approximately isotropic over the sky dome. The flux of diffuse solar radiation to

a surface inclined at an angle α to the horizontal (Fig. B1) is

Qst = Sr(1 + cosα)/2 (B1)

for the top of the surface, and

Qsb = Sr(1 – cosα)/2 (B2)

for the bottom of the inclined surface (Iqbal 1983). The circumference of a cylinder

is made up of elements with area dC = Ddα/2 per unit length, inclined at angles to

the horizontal varying from –π/2 to π/2 (Fig. B2). The contributions of the sky diffuse

solar radiation to each of these surface elements are summed, taking advantage of

the symmetry of the cylinder about the vertical, to determine the solar power per

unit length of the cylinder:

 

P Sr D d D s = d 

   +  +     ∫ ∫ 



 



2  2

1

2

2

2

1

2 2

0

0

2 cos – cos

– /

/ α

α

α

α

π

π

 . (B3)

Evaluating the integrals results in Ps = πDSr/2, so the flux of sky diffuse solar

radiation to the cylinder per diameter is

 

Q Sr s = π

2 , (B4)

which was given in eq 4.

α

dα

α

α = π/2

D/2

α = 0

Ddα/2

α = – π/2

Figure B1. Sky diffuse solar radiation incident on surface inclined at

an angle α to the horizontal.

Figure B2. Cylinder surface elements.

22
April 1996

Ice Accretion in Freezing Rain

Kathleen F. Jones

U.S. Army Cold Regions Research and Engineering Laboratory

72 Lyme Road

Hanover, N.H. 03755-1290

CRREL Report 96-2

Approved for public release; distribution is unlimited.

Available from NTIS, Springfield, Virginia 22161

31

UNCLASSIFIED UNCLASSIFIED UNCLASSIFIED UL

Collision efficiency Ice load Wind-on-ice load

Freezing rain Icicles

Ice Rain

Ice accreted on structures from freezing rain causes both increased vertical loads and increased wind loads, due to

the larger projected area of the structure. Structural failures initiated by ice loads frequently cause millions of dollars

of damage to overhead power and communication lines, towers, and other ice-sensitive structures. There is little

information on ice loads to use in the design of these structures, so freezing-rain models have been developed for

use with weather measurements to determine the severity of accreted ice loads from historical data. This report

describes a detailed heat-balance ice accretion model, including the important heat fluxes in freezing rain and

allowing the accretion of runoff water in the form of icicles. It also presents a simple algorithm for calculating the ice

load on components with different diameters and cross sections. Collision efficiency in freezing rain and the

calculation of the wind-on-ice load are also discussed. Model results are compared with the ice load measured during

a recent freezing rain storm, and to each other, using 45 years of weather data from Des Moines, Iowa.

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