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NATIONAL ADVISORY COMMITTEE

FOR AERONAUTICS

TECHNICAL NOTE 2738 _//'

A PROBABILITY ANALYSIS OF THE METEOROLOGICAL

FACTORS CONDUCIVE TO AIRCRAFT ICING

IN THE UNITED STATES

,By William Lewis and Norman R. Bergrun

Ames Aeronautical Laboratory

Moffett Field, Calif.

Washington

July 1952

NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS

TECHNICAL NOTE 2738

A PROBABILITY ANALYSIS OF THE METEOROLOGICAL

FACTORS CONDUCIVE TO AIRCRAFT ICING

IN THE UNITED STATES

By William Lewis I and Norman R. Bergrun 2

SUMMARY

Meteorological icing data obtained in flight in the United States

are analyzed statistically and methods are developed for the determina-

tion of: (I) the various simultaneous combinations of the three basic

icing parameters (liquid-water content, drop diameter, and temperature)

which would have equal probability of being exceeded in flight in any

random icing encounter; and (2) the probability of exceeding any speci-

fied group of values of liquid-water content associated simultaneously

with temperature and drop-diameter values lying within specified ranges.

The methods are particularly useful in the design of anti-icing equip-

ment intended to bperate through the United States, to define simulta-

neous combinations of the meteorological variables which could be

encountered, and to ascertain the effectiveness of the equipment in

withstanding the natural icing conditions to which it may be subjected.

In addition_ a mathematical basis is provided for the future statistical

analysis of meteorological icing data that might be obtained throughout

the world.

INTRODUCTION

The program of research in aircraft ice prevention which has been

conducted by the NACA during the past several years has been directed

primarily toward the development of practical methods for the design of

thermal ice-prevention equipment for various airplane components. SinCe

a rational ice-prevention design requires a knowledge of the physical

characteristics of icing conditions_ an important phase of the research

program has been an investigation of the meteorological conditions con-

ducive to icing.

The severity of an encounter with icing conditions is determined

principally by four factors; namely, the liquid-water content_ the

iLewis Flight Propulsion Laboratory, NACA

2Ames Aeronautical Laboratory, NACA

2 NACATN 2738

diameter of the drops, the temperature, and the horizontal extent of theconditions. The meteorological investigation has, therefore, been con-cerned with obtaining measurements of these quantities in a wide varietyof natural icing conditions in order to establish the range and rela-tive frequency of occurrence of various values, and combinations ofvalues, of these quantities.

During the period 1945 through 1948, a considerable amount of datawas collected by the Amesand Lewis Laboratories. These results_ anddiscussions of various phases of the investigation, have been reportedin references i through 4. All of these results_ together with similardata from observations made by other organizations_ were used as a basisfor a listing of estimated maximumicing conditions recommendedfor con-sideration in the design of anti-icing equipment (reference 5)- Theprobable maximumvalues listed in reference 5 were estimated on thebasis of what was considered a reasonable extrapolation from a limitedamount of data, and the concept of the probability of encounteringvarious simultaneous combinations of values entered only in a subjectiveand qualitative way.

In this report, the available data obtained in icing flights in theUnited States are analyzed statistically and the results are presentedgraphically in two forms. In the first form, the results are plotted toshow the various combinations of liquid-water content, drop diameter,and ambient-air temperature which have equal probability of beingexceeded. In the second form, plots are presented to determine theprobability of exceeding any specified value of liquid-water contentunder the condition that the value is associated simultaneously withvalues of temperature and drop diameter lying within specified intervals.Both plots are based on preselected values of horizontal extent; however,methods for estimating values applicable to other horizontal extents arealso presented.

Although the results of this report are directly applicable only tothe United States, an estimation of the conditions prevailing in otherlocalities of similar climatic conditions can be made based upon theresults presented herein. As meteorological icing data are accumulatedfor regions outside the United States, the methods of this report shouldprovide a framework for placing the data on a statistical basis that isnot limited in scope to the United States.

SYMBOLS

The following symbols are used throughout this report:

A,B,C designation applied to three events occurring simultaneouslybut not necessarily independently

NACA TN 2738 3

C

D

Dcr

AD

e

F

Mt

n

0

P

PA

PABC

P (A)

Pc(A,B)

Pe

PG

chord length, feet

mean-effective drop diameter, microns

the smallest drop diameter for which drops will impinge upon

a thermal element

interval of drop diameter, microns

Naperian base of logarithms

collection efficiency of an airfoil section_ percent

factor by which liquid-water-content values are multiplied

to make them applicable to other distances of cloud hori-

zontal extent, dimensionless

weight rate of water intercepted, pounds per hour per foot

of airfoil span

an integer

an icing condition considered to be defined by specifications

of the three variables, liquid-water content, temperature

depression below freezing, and drop diameter

probability of overloading any given element of a thermal

anti-icing system

probability of the occurrence of event A

probability of the simultaneous occurrence of three

events, A, B, and C

probability of the occurrence of event B, under the condi-

tion that event A will occur

probability of the occurrence of event C under the condi-tion that both events A and B will occur

probability that any random icing encounter will be char-

acterized by a combination of values of the variables

(liquid-water content, temperature depression below freez-

ing, and drop diameter), in which each variable must, res-

pectively, equal or exceed a specified value of thatvariable

probability defined by the Gumbel probability distribution

and expressed by equation (A1)

4 NACATN 2738

LkPi probability thatj for a random icing encounter, the following

three conditions will simultaneously be fulfilled:

(1) liquid-water content exceeding a specified value;

(2) temperature within a specified interval; and (3) drop

diameter within a specified interval

probability that drop diameter will have a value lying within

a specified range (AD)

probability that temperature will have a value lying within

a specified range (AT)

Pwi(AT)

Q

probability that liquid-water content will be equal to or

greater than Wi under the condition that the temperaturelies within a specified range (AT)

probability defined by the probability function Q(T') for

a fixed value of T'

Q(T') probability function, determined by the variation with tem-

perature depression below freezing for the most severe

icing conditions, of the percent of icing encounters in

which the temperature depression belOw freezing is less

than certain specified values

R probability that the maximum liquid-water content in an icing

encounter is equal to or greater than W, under the condi-

tion that the temperature depression below freezing is

equal to or greater than T'

probability function expressing R in terms of W and T'

_Y probability that the liquid-water content associated with

the maximum value of drop diameter in an icing encounter is

equal to or greater than W, under the conditions that the

temperature depression below freezing is equal to or

greater than T' and the maximum value of drop diameter

is equal to or greater than D

R' (W,D,T') probability function expressing R' in terms of W 3 D, and T'

S probability that the drop diameter is equal to or greater

than D, under the conditions that the temperature depres-

sion below freezing is equal to or greater than T' and

the liquid-water content is equal to or greater than W

S(D,W,T') probability function expressing S in terms of D 3 W, and T'

S ! probability that the maximum value of drop diameter in an

icing encounter is equal to or greater than D, under the

condition that the temperature depression below freezing

is equal to or greater than T'

NACATN 2738 5

S' (D,T')

s''(D)

t

T

T'

Tcr'

AT

AT'

U

V

W

WO .o5

Wo .6_B

AW

x

(Z

probability function expressing S' in terms of D and T'

composite distribution function consisting of parts of each

of the functions S(D,0,O') and S'(D,O)

airfoil thickness, percent of chord length

temperature existing in an icing condition# oF

temperature depression below freezing, equal to 32-T, F°

temperature depression below freezing at which the heat out-

put of the thermal element can bring the surface tempera-

ture of the element just to freezing temperature, F°

interval of temperature, OF

interval of temperature depression below freezing, F°

a constant which denotes the mode in the Gumbel probability

equation

airspeed# miles per hour

liquid-water content, grams per cubic meter

value of liquid-water content corresponding to a probability

value of 0.09 defined by the Gumbel probability equation

value of liquid-water content corresponding to a probability

value of 0.63 defined by the Gumbel probability equation

interval of liquid-water content, grams per cubic meter

independent variable in the Gumbel probability equation

a parameter which depicts the concentration of a frequencydistribution about the mode in the Gumbel distribution

equation

Subscripts

1

i

any selected icing condition

a representative value of a variable

6 NACATN 2738

ANALYSIS

The data used in this analysis consist of 1038 sets of values ofliquid-water content, drop diameter, and temperature obtained during252 encounters with icing conditions. Values of liquid-water contentand drop diameter were measured by means of the rotating multicylindermethod. All data available at the time of preparation of this reportwere used in the analysis, including (1) data obtained by the AmesandLewis Laboratories and published in references l, 2, 3, 4, and 6;(2) data contained in the monthly reports of the Air Materiel CommandAeronautical Ice Research Laboratory, the major portion of which is pre-sented in reference 7; and (3) data obtained by United Air Lines (refer-ence 8) and American Airlines (reference 9) during icing flights withDouglas DC-6 type airplanes.

For the purposes of this analysis, the basic unit of data is theicing encounter which consists of all measurements made during flightthrough a single area of continuous or intermittent icing conditions.This area was separated from other icing conditions s by relativelylarge areas in which no icing was observed. The problem consideredherein is to determine the probability that an icing encounter chosenat random will include conditions of a given severity averaged over acertain distance. Since most of the rotating-cylinder observationsrepresent averages over distances of about 3 miles in cumulus cloudsand l0 miles in layer clouds, these distances are regarded for purposesof data reduction as standard values of horizontal extent, applicableto data from the two principal cloud types. The probabilities obtainedon this basis may be applied to other distances by using the data con-cerning the relation between horizontal extent and average liquid-watercontent obtained from continuous records of the rotating-disk icing-ratemeter. (See reference 3-) It has been assumedthat the greatest valuesof liquid-water content and drop diameter measured by the rotating cyl-inders during an icing encounter represent maximumvalues averaged overthe standard distances of 3 or lO miles. This assumption is probablyapproximately true, since a particular effort was madeto obtainrotating-cylinder data during the periods of most rapid ice formation.

Since the variations of temperature during a single icing encounterare usually not very great, the most severeicing conditions measuredduring a particular icing encounter then will be either the greatestvalue of liquid-water content and the corresponding value of drop diameter_or possibly, the greatest value of drop diameter and the correspondingvalue of liquid-water content. Both of these possibilities for a severe

8The term "icing condition" as used herein denotes a state of the atmos-phere defined by a set of values of temperature, liquid-water content,drop diameter, and pressure altitude in which the temperature is belowfreezing and the liquid-water content is greater than zero.

NACA TN 2738 7

icing condition are considered in the probability analysis which, for

the sake of convenience, is developed in three phases.

The first phase of the analysis is the classification of the avail-

able data according to cloud type and geographical location; the second

phase is the determination of the particular combinations of liquid-

water content, drop diameter, and ambient-air temperature, for a given

horizontal cloud extent, that have the same probability of being simul-

taneously equaled or exceeded in a random icing encounter; and the third

phase is the determination of the probability of exceeding any specified

group of values of liquid-water content associated simultaneously with

temperature and drop-diameter values lying within specified ranges.

Classification of Data According to Cloud Type

and Geographical Location

For classification purposes in this report, clouds will be divided

into two classes, cumulus clouds and layer clouds; and the area of the

United States will be divided into three regions, the Pacific coast

region, the plateau region, and eastern United States with boundaries

as indicated in figure 1. This system of classification divides the

observations into six cases as indicated in table I, which includes

information relating to the number of icing encounters and the number

of individual measurements of liquid-water content and drop diameter for

each group. The two cases for which the greatest amount of data are

available, eastern layer clouds (case 5) and Pacific coast cumulus

clouds (case 2), are treated in considerable detail. The other cases

are treated in as much detail as the data permit, with the exception of

case 6 (cumulus clouds in eastern United States) which is omitted because

of insufficient data.

Determination of Particular Combinations of Liquid-Water

Content, Drop Diameter, and Temperature Depression

Having the Same Probability of Being Simul-

taneously Equaled or Exceeded in a

Random Icing Encounter

Exceedance probability.- The three fundamental icing variables usedto define an arbitrary icing condition are liquid-water content W,

drop diameter D, and temperature T. For purposes of probability anal-

ysis, however, it is more convenient to refer temperature to the freezing

point and to use temperature depression below freezing T' as a variable

to substitute for the temperature T. Thus, any particular icing condi-

tion can be represented by selected values of the variables WI, Dl_

and Tl'. The probability that any random icing encounter of standard

extent will provide simultaneous values of each of the icing variables

8 NACA TN 2738

equal to or greater than each of the selected values (i.e., W_WI, D_DI,

and T' _TI') will be termed the "exceedance probability" Pe correspond-

ing to the icing condition Wl, DI, and TI'.

The exceedance probability represents the probability of the simul-taneous occurrence of three events which are not necessarily independent.

In order to evaluate Pe_ use is made of the theorem of compound proba-

bility (reference 10) which states that the probability of the simulta-

neous occurrence of three events_ A, B, and C, is equal to the probability

of the occurrence of A times the conditioned probability of the occur-

rence of B, under the condition that A will occur, times the condi-

tioned probability of the occurrence of C, under the conditions thatboth A and B will occur. This theorem can be expressed symbolically

as follows:

PABC = PA PB(A) Pc(A, B) (i)

If it is considered that the probability of exceeding certain values

of T'_ W, and D is analogous to the probability of the occurrence of

events A, B, and C expressed in equation (1)3 then the exceedance

probability Pe may be represented by the product of three probability

factors Q, R_ and S_ according to the equation

Pe = QRS (2)

In applying equation (2) to the meteorological variables, the prob-

ability Q may be considered as a function of T' alone 3 R a function

of W and T', and S a function of D, T', and W. The functional rela-

tions between Q_ R_ and S3 and D_ T'_ and W are denoted, respectively,

by the expressions Q(T'), R(W_T') and S(D,T',W). The meaning of each

expression is as follows:

a relation that defines the probability Q in terms of T t

and expresses the probability for a given value of T'

of a random icing encounter having a temperature depres-

sion below freezing equal to or greater than T'

R(W,T') a relation that defines the probability R in terms of W

when T' is specified and expresses the probability for

a given value of W of a random icing encounter having

a liquid-water content equal to or greater than W_ under

the condition that the temperature depression below freez-

ing is equal to or greater than T'

S(D,W,T' ) a relation that defines the probability S_ in terms of D

when W and T' are specified and expresses the probabil-

ity for a given value of D of a random icing encounter

having a value of drop diameter equal to or greater

than D_ under the condition that the liquid-water content

is equal to or greater than W and the temperature

depression below freezing is equal to or greater than T'

2F NACA TN 2738 9

The physical significance of the term exceedance probability is

best illustrated by graphical means. Figure 2 presents a three-

dimensional plot with the variables W, D, and T' as the axes. Any

icing condition of standard duration defined by selected values of these

three variables, such as WI, DI, and Tl' , appears as a single point 01

on the plot. Corresponding to the point 01, there is an exceedance

probability Pe which expresses the probability that a random icinglencounter will providesimultaneous values of each of the icing vari-

ables equal to or greater than each of the selected values, WI, DI,

and TI'. According to equation (2), the value of Pel would be givenby the equation

Pel = QIRISI

Thus, when Pe is regarded as a constant, Pel , equation (2) defines a

surface passmng through 01, specified by all the combinations of W, T',

and D which have the same probability of being simultaneously equaled

or exceeded. This surface is shown as the equiprobability surface in

figure 2, and represents the locus of all icing conditions which have

the same probability of being equaled or exceeded. The quantitative

value of exceedance probability applying to individual points lying on

the surface is expressed by equation (2).

It should be noted that since Q, R, and S represent probabilities,

they cannot be greater than one nor less than zero. Moreover, Q is

equal to one when T' equals zero and continuously decreases as T'

increases; R is equal to one when W is zero; and for any constant

value of T', R is a continuously decreasing function of W. Finally,

S is equal to one when D is zero and, for any particular pair of

values of T' and W, S is a continuously decreasing function of D.

The point at which the equiprobability surface intersects the T' axis

is determined by the value of T' for which Q = Pe, since R and S

are both equal to one. Similarly, the surface intersects the

W and D axes at values of W and D which make R and S, respectively,

equal to Pe- A constant-temperature contour passing through the

point 01 on the equiprobability surface would be defined by the con-

dition T' = TI' , from which it follows that Q = Ql, and hence

RS = Pe/Ql" These values are depicted in figure 2.

The preceding theory concerning exceedance probability must be

modified slightly to make it applicable to the particular problem under

consideration, in which the probabilities are to apply to the most

severe icing conditions in an icing encounter. In most cases, the most

severe icing condition is represented by the maximum value of liquid-

water content and the corresponding (simultaneously observed) value of

drop diameter. For the portion of the equiprobability surface which

represents low values of liquid-water concentration and large values of

i0 NACATN 2738

drop diameter_ on the other hand_ the critical condition is representedby the maximumvalue of drop size and the corresponding value of liquid-water content. To take account of the first of these possible combina-tions_ the function R(W_T') was determined from the greatest values ofliquid-water content for each icing encounter and the function S(D_W,T')was obtained from the corresponding values of drop diameter; and to takeaccount of the second possible combinations_ another function S'(D_T')was determined from the greatest values of drop diameter and a corre-sponding function R'(W_D_T') was obtained from the corresponding valuesof liquid-water content. The functions R(W_T') and S(D,T') were thenused to determine the main portion of the equiprobability surfaces_while the functions R'(W,D_T') and S'(D,T') were used for the portionrepresenting large values of D and small values of W.

Construction of the equiprobability surfaces.- The procedure used

in the construction of equiprobability surfaces from a particular set of

data was essentially that of determining graphically the relations des-

cribing the functions Q(T'), R(W,T')_ S(D,W,T'), R'(W_D,T')_ and S'(D,T').

This procedure is illustrated in appendix A.

The mathematical functions used to represent the various distribu-

tions were chosen mainly on the basis of the empirical criterion of

goodness of fit_ although theoretical considerations also were given

considerable weight_ particularly in the selection of a function to

represent the distribution of maximum values in those cases in which

each item in the distribution is the maximum of a group of observed

values. The function chosen for representing such distributions is one

originated by Dr. E. J. Gumbel (reference ii) to describe a distribution

of extreme values. This function will be referred to hereinafter as

"Gumbel's distribution."

Because of an almost complete lack of observational data in flight

concerning icing conditions at very low temperatures_ reliance has been

placed on the results of laboratory experiments on the formation of ice

crystals to aid further in the construction of the equiprobability charts.

The experiments relied upon (summarized in reference 12) indicate that a

critical temperature exists in the neighborhood of -40 ° F_ below which

ice crystals are formed in very great numbers in atmospheric air when-

ever conditions of saturation with respect to liquid water occur. On

the basis of this fact_ it is inferred that clouds at temperatures

below -40 ° F are composed either entirely or almost entirely of ice

crystals; hence the liquid-water content is assumed to be zero at all

values of T' greater than 72 ° F.

The methods outlined in the preceding paragraphs have been used to

prepare constant-temperature contours defining the equiprobability sur-

faces for values of Pe equal to O.ij 0.01_ and 0.001 for cases i

through 5 (table I). These curves are presented in figures 3 through 7-

Details of the probability calculations for cases 2 and 5 are given as

examples in appendix A.

_ACA T_ 2738

The equiprobability surfaces defined by the contours presented in

figures 3 through 7 are constructed only for three exceedance probabil-

ities, but it may be desired to ascertain a point which lies between

the surfaces defined for the three different exceedance probabilities.

If such a point is desired, it can be identified in T'-D-W space by

logarithmic interpolation between the equiprobability surfaces, since

the probability functions decrease approximately exponentially with

increasing values of T' 3 W, and D.

The inclusion of horizontal extent as an additional variable.-

Experience hasshown that the more localized a cloud formation, the

higher will be its liquid-water content. The results of the foregoing

analysis (figs. 3 through 7), however, are directly applicable only to

conditions not differing greatly in horizontal extent from the average

distance over which the measurements were made, namely, 3 miles in

cumulus clouds and lO miles in layer clouds. The results may be made

applicablej though, to other values of horizontal extent by making use

of the data contained in reference 3 concerning the relation between

maximum average liquid-water content and distance along the flight path.

In order to modify values of liquid-water content obtained for one

horizontal extent to be applicable to another, it has been found neces-

sary to make three assumptions. These assumptions are that (i) the

variables, drop diameter_ and temperature are assumed not to be signif-

icantly related to horizontal extent; (2) the variation of average

liquid-water content with horizontal extent is assumed to be independent

of drop size and temperature; and (3) the results from reference 3 are

assumed to be representative of conditions as they would be encountered

in normal flight operations. This last assumption is uncertain because

of the limited number of flights analyzed in reference 3 (ii flights in

layer-type clouds and 26 in cumulus clouds) and because the flight paths

were frequently chosen with the aim of prolonging and maximizing the

exposure to icing conditions. On the basis of these assumptions, equi-

probability surfaces applicable to other values of horizontal extent may

be constructed by multiplying all values of liquid-water content from

the probability curves of figure 3 to 7 inclusive by the variable fac_

tor F presented in figure 8.

From figure 8, it can be seen that the horizontal extent selected

as being applicable to a particular icing encounter can have a consid-

erable effect on the factor F used to modify liquid-water-content

values. This effect is particularly pronounced in the case of layer

clouds where the horizontal extent can be very large. For example, if

an icing encounter in layer clouds is considered to extend over i00 miles

instead of lO_ the factor to use to modify liquid-water-content values

iS about 0.4 instead of io0. Consequently, it is necessary to consider

the appropriateness for design purposes of the standard values of cloud

horizontal extent (3 miles for cumulus and I0 miles for layer clouds).

For the "instantaneous" and "intermittent" classifications of reference 5

(classes I and II as defined therein)_ the horizontal extents

12 NACATN 2738

of 1/2 and 3 miles, respectively, as assigned in reference 5, are con-sidered to be reasonable values for design purposes. For the "contin-uous" classification (class III), a horizontal extent of 20 milesappears appropriate for design. Although it is realized that contin-uous icing might well extend beyond 20 miles, reduction of the liquid-water content by a factor less than 0.8 (an approximate value of Fcorresponding to 20 miles horizontal extent) does not appear justifiedfor design purposes. It should be noted that these three design valuesof horizontal cloud extent, shown in figure 8 by broken lines, are sug-gested values only and do not preclude the use of the curves for theestimation of the factor F by selecting other values of horizontalextent.

The Determination of the Probability of Exceeding Any SpecifiedGroup of Values of Liquid-Water Content Associated Simul-

taneously With Temperature and Drop-Diameter ValuesLying Within Specified Ranges

The equiprobability surfaces discussed in the preceding sectionprovide information concerning the simultaneous combinations of liquid-water content, drop diameter, and temperature having the sameprobabilityof occurrence. Another useful expedient for the designer, however, wouldbe a means for determining the probability of encountering any icingcondition not included by the values of liquid-water content, drop diam-eter, and temperature specified as design criteria for a particular com-ponent of a thermal system. The basis for such a technique lies indetermining the probability of equaling or exceeding certain values ofliquid-water content specified for a number of icing conditions.

Probability of encountering icing conditions which exceed certain

specified values.- If, for a particular element of a thermal system and

a chosen value of horizontal cloud extent, combination values of liquid-

water content_ drop diameter, and temperature are specified which the

element can just tolerate_ the values would define a surface in

T'-D-W space. Such a surface can be viewed as representing the locus

of liquid-water-content values to which the element is critical over

specified ranges of temperatures and drop diameters. Individual values

on such a surface would be obtained through knowledge of the area, rate,

and distribution of water-drop impingement for the particular element.

For purposes of explanation, figure 9 is presented to show the

locus of critical values of liquid-water content for an assumed element

of a thermal system. The locus of liquid-water-content values will be

termed a "marginal" surface to indicate that values of W lying above

the surface cannot be tolerated by the element of the thermal system

under consideration. In general_ the shape of the marginal surface in

the W-D plane for the case of a wing element having a given heat out-

put may be assumed to have a shape resembling a hyperbola. A hyperbolic

NACATN 2738 13

shape is appropriate because for drop diameters less thanthe criticaldiameter 4 for the element under consideration, an infinitely largeamount of liquid water is tolerable; and, as drop diameter progressivelyincreases, decreasing amounts of liquid water are tolerable. In anyW-T' plane, however, W will be assumed to vary from the valuesof W at T' = 0 to W = 0 at somevalue of Tl' , say Tcr' , wherethe heating capacity of the element can bring the surface temperatureof the element just up to freezing temperature (a clear-air condition).

For convenience, the base of the marginal surface (the T'-D planeat W = O) is divided into n rectangles of dimension AD and AT' suchthat variations are small in values of liquid-water content included bythe projection of each rectangle on the marginal surface. The value ofliquid-water content considered to be representative for each rectangleis designated as Wi in figure 9- The values of T'_ T, and D corre-sponding to Wi shall be called Ti' , Ti, and Di, respectively. Forevery set of Wi-Di-T i' values, there exists a probability function ZkPiwhich expresses the probability that, for a random icing encounter, theliquid-water content will exceed Wi and the drop diameter and tempera-ture depression will be within the ranges AD and AT', respectively.

The n values of APi corresponding to the n rectangles in theT'-D plane represent the probabilities of the occurrence of an event(the overloading of the anti-icing element) in n different, mutuallyexclusive ways. The probability that the event will occur is the sumof the probabilities that it will occur in any number of different waysprovided the different possibilities are mutually exclusive. (Seereference i0.) Hence, the probability of overloading the given anti-icing element is approximately equal to the sum of the partial probabil-ities for the n rectangles according to the equation

np : r.ah (3)

i

In order to evaluate 2kPi for each of the n rectangles, it

would appear that a separate determination of the relationship

between W and AP i for each rectangle would be necessary. This pro-

cedure is not practicable, however, because if the number of rectangles

is chosen large enough to make variations in W small for each rec-

tangle, then the amount of data available to define AP i as a function

of W is too small to yield statistically reliable results for the

individual rectangles. Moreover, the presentation of the results would

require separate curves or tables for each rectangle. It is possible

to overcome these di±ficulties by utilizing the fact that the

4The critical diameter may be defined as the smallest diameter for which

drops will impinge on the element. For some airfoils, the critical

size can be estimated by a technique outlined in NACA TN 2476 (refer-

ence 13). /

14 NACA TN 2738

distribution of drop diameter is approximately independent of tempera-

ture and liquid-water content. Although there is a slight tendency

toward higher frequencies of large values of drop diameter when the

liquid-water content is low_ this effect is too small to have an impor-

tant influence in the determination of the total marginal probability.

By making the assumption that the distribution of drop diameter is

independent of temperature and liquid-water content_ it is possible to

determine values of AP i for different combinations of temperature

depression below freezing_ liquid-water content_ drop diameter_ and

horizontal extent.

When_ for any given cloud horizontal extent, drop-diameter distri-

bution is assumed to be independent of temperature and liquid water, the

probability equation

aPi : PaD PaT PWi (4)

can be written from analogy with equation (i). The interpretatioo of

equation (4) is that the partial probability Z_Pi applicable to a

chosen point on the marginal surface is the product of three probabil-

ities: the probability, PAD, that the drop diameter will be within the

interval ZkD; the probability, PAT, that the temperature will lie within

the interval ZkT; and the conditioned probability, PWi(AT), that the

liquid-water content will be equal to or greater than W i under the

condition that the temperature lies within the interval AT.

To evaluate equation (4), values of PAD were obtained from the

frequency distribution of all observations of drop diameter; values

of PAT were obtained from the frequency distribution of the tempera-

ture of the most severe _ icing observations per icing encounter; and

values of PWi(AT ) were obtained from the frequency distributions of

maximum liquid-water content per encounter for the various temperature

intervals. Because of the assumption that drop size is independent of

temperature and liquid-water content, it was not possible to provide

exactly for the case in which the maximum observed drop diameter deter-

mines the most severe icing condition in an icing encounter. An approx-

imate allowance was made for this factor_ however, by substituting the

frequency distribution of the greatest value of drop diameter per

encounter over a limited range of large drop diameters.

Equation (4) was evaluated for different horizontal extents, and to

do this the frequency distribution for maximum liquid-water content per

encounter was altered according to a relation from reference 3 between

maximum average liquid-water content per encounter and the distance

along the flight path.

SFor the purposes of this report 3 the most severe icing condition is con-

sidered as the one having the largest liquid-water content.

NACA TN 2738 15

The partial probability charts.- In evaluating equation (4) for

any one value of cloud horizontal extent, numerous values of 2_Pi are

possible from the meteorological data for different combinations

of PLOD, PAT, and PWi(AT). The values of Z_Pi, however, can be conven-iently summarized in the form of partial probability charts. Such

charts are presented in figure lO for cases 1 through 5 of table I°

As is evident from figure i0, seven intervals of drop diameter and

six intervals of temperature were used in the construction of the

charts. These intervals were chosen so as to avoid large variations

of W encompassed by any particular AO-AT' rectangle projected on

the marginal surface. In all conceivable combinations, the temperature

and drop-diameter intervals provide 42 different values of LkPi which

must be summed (in accordance with equation (3)) to obtain the probabil-

ity, P, of overloading the particular element.

To obtain a specific value of Z_Pi from a chart of figure i0, the

following procedure is used: Choose a value of liquld-water content Wiand follow a vertical line until it intersects with the desired

horizontal-extent line; from this intersection, follow horizontally until

an intersection with the appropriate temperature-interval curve is

obtained; from the point of this intersection, follow a vertical line

downward to obtain an intersection with a chosen drop-diameter curve;

a horizontal line through the latter intersection yields the desired

value of 2_Pi on the ordinate scale. For ease in observing this pro-

cedure, each chart in figure l0 has a system of arrows leading from

some arbitrary value of Wi through the horizontal-extent, temperature s

and drop-diameter curves to a corresponding value of L_Pi •

It should be noted that partial probabilities presented in figure i0

can be determined, for values of horizontal extent not specifically

represented in the charts, by interpolation between the horizontal-extent

curves. It is not permissible, however, to interpolate between the

curves representing temperature and drop diameter since the partial prob-

abilities are dependent upon the size of the intervals for which the

temperature and diameter curves are constructed. The approximate median

values of temperature and drop diameter, which correspond to the intervals

of Ti and Di used in the charts of figure lO, are listed in table IIfor reference. Median values are used because the median has the advan-

tage that its value is not greatly influenced by the chance variations

of extreme items. A detailed description of the techniques used in

constructing the partial probability charts, with application to selected

cases, is presented in appendix B.

16 NACATN 2738

DISCUSSION

Reliability of the Equiprobability and thePartial Probability Charts

The reliability of the probability analysis reported herein isaffected chiefly by four factors; namely, (1) errors of measurement,(2) amount of data available, (3) representativeness of the data, and(4) limitations with respect to climate and altitude. The possibleeffect of these factors is discussed in the following paragraphs.

Errors of measurement.- As noted in reference 14_ the measurement

of liquid-water content and drop diameter by the multicylinder method

is subject to error. The following factors may be listed as possiblesources of error: (1) runoff, (2) bounce-off, (3) evaporationj

(4) variations in drop concentration in the neighborhood of the cylinders

due to disturbing influences of the airplane, (5) collection of snow

flakes as well as liquid drops in mixed clouds, (6) inaccuracies due to

the theoretical drop-size distributions differing from those assumed in

the calculations, (7) failure to fulfill, with the apparatus used, the

theoretical assumption of two-dimenslonal flow, (8) irregularities in

the surfaces of the ice-covered cylinders, (9) errors in weighing,

(lO) irregular breaking of the ice when separating cylinders, (ll) errors

in measuring the final diameter of the ice-covered cylinders or in cal-

culating the average diameter from an assumed value of ice density,

(12) errors in the measurement of airspeed and exposure time, and

(13) errors in matching the experimental data with the theoretical

collection-efficiency curves.

A detailed discussion of the possible effects of each of these

factors on the accuracy of the probability analysis presented herein is

beyond the scope of this report. The problem of runoff, though, is

Worthy of discussion since this factor imposes an upper limit to the

range of liquid-water-content values that can be measured. The results

of heat-transfer calculations reported in reference 15 indicate that the

maximum value of llquid-water content that can be measured with rotating

cylinders as they are normally used varies approximately linearly with

temperature depression below freezing, reaching about 2 grams per cubic

meter at 5° F for an airspeed of about 200 miles per hour. An examina-

tion of the data usedin the present analysis indicates that not more than

about 5 percent of the icing encounters in any of the cases analyzed con-

tained observed values of liquid-water content which were close enough to

the theoretical maximums to be regarded as likely to be influenced by

runoff. In order to estimate the effect of a small percentage of errors

of this type on the results of the probability analysis, it is necessaryto consider the manner in which the liquid-water-content distribution

functions were determined from the experimental data. In this process,

the cumulative frequency distribution of maximum liquid-water content

per icing encounter was plotted on Gumbel's distribution paper, and the

straight line of best fit was drawn to represent t1_e entire distribution;

3£ _ACA _ 2738 17

thus errors in the upper 5 percent of the distribution had only a minor

effect in the final selection of the line used to represent the distri-

bution function. Moreover, it was noted that in nearly all cases the

entire distribution followed the straight line with minor and apparently

random variations and without any consistent or marked tendency for large

negative departures at the upper end of the distribution. It may be

inferred, therefore# that the limitation on the measurement of liquid-

water content discussed in reference 15 does not have an important influ-

ence on the results of this analysis.

In view of the number and complexity of the possible sources of

error, it is not possible at this time to make a reliable estimate of the

total accuracy of the measurement of liquid-water content. In a statis-

tical analysis, however, reasonably reliable results may be obtained as

long as the errors are small compared to the real variations of the

quantity measured and are distributed approximately at random. It is

believed that these conditions are approximately fulfilled for the values

of liquid-water content measured by the multicylinder technique.

Multicylinder measurements of drop diameter, on the other hand_ while

quite accurate and reliable for small dr0ps# become increasingly inaccurateas the drop size increases. Moreover, the errors are not normally dis-

tributed, since large positive errors are more probable than large nega-

tive errors, especially at large values of drop diameter (see reference 14).

The result of these errors is an increase in the dispersion of the fre-

quency distribution and an exaggeration of the probabi_lity of occurrence of

large values of drop diameter. This effect is probably not important forobservations in the eastern United States because in this case the drop

size rarely exceeds the range of reliable measurement. In the Pacific

coast area, on the other hand, where large drops occur with much greater

frequency_ it is quite likely that the extreme values of drop diameter are

influenced considerably by this factor.

Amount of available data.- Some concept of the accuracy of the proba-

bility analyses for the various cases may be obtained by reference to the

number of icing encounters for each geographical area and each cloud-type

classification. These data are presented in table I. It is noted that

the greatest amount of data for any one case is llO encounters in layer

clouds in the eastern United States. A sample of this size is sufficient

to establish the probabilities with considerable confidence at the O.O1

probability level, and the extrapolationto a probability of O.OO1 is not

likely to introduce serious errors. In the remaining cases, the sample

size ranges from 25 to 4% encounters. In these cases, the extrapolation

to a probability of 0.O01 may introduce considerable uncertainty.

Representativeness of the data.- Another factor, which probably hasas great an influence on the reliability and applicability of the analy-

sis as either errors of measurement or sample size_ is the systematic bias

introduced by the fact that all the test data were obtained during flights

in which icing conditions were deliberately sought instead of being acci-

dentally encountered. It is difficult to estimate the effect of this

18 NACATN 2738

factor since measurements of liquid-water content and cloud-drop sizeencountered during ordinary transport operations are not yet available.It would appear that the principal difference between transport experi-ence and icing research flights would be the greater frequency of encoun-ters with icing conditions in the latter. This difference alone, however,would not affect the validity of the analysis since the icing encounter istaken as the unit of experience upon which the analysis is based. Thequestion to be considered is whether the icing enconnters which occurredduring the research flights represent an unbiased sample of the encoun-ters which would occur during normal flight operations.

The differences in severity and extent between icing conditionsexperienced during research flights and the conditions which would havebeenencountered during an ordinary flight in the samegeneral area atthe same time are of two types. The first type occurred in cases inwhich n8 icing would have been encountered on a normal flight throughthe area but, due to the existence of clouds in thin layers or localizedareas, it was possible to experience icing with the test airplane.Encounters of this type would be expected to be less Bevere and exten-sive, on the average, than typical encounters during normal operationS.Differences of the second type occurred in cases in which icing condi-tions would have been encountered on a normal flight but, due to a con-scious effort on the part of the pilot_ the conditions experienced bythe research airplane were more severe and extensive. It is evident,therefore, that the special procedures followed during research flightshave resulted in the inclusion in the test results of data from icingencounters which were both more and less severe than those which wouldhave been experienced under normal operating conditions. It is notpossible to state with certainty what is the net effect of these twotypes of differencesj but it is believed that differences of the secondtype predominate, with the result that the data are somewhat, though notgreatly_ biased by the inclusion of an abnormally high frequency ofsevere icing conditions.

Effect of climate and altitude.- Another factor having an important

bearing on the applicability of the results is the effect of climate and

flight altitude on the temperature of icing conditions. It is likely

that the distribution of temperatures of icing conditions encountered by

the test airplanes closely resembles that which would be found in normal

operations of unpressurized airplanes during winter and spring in the

areas studied. Different temperature distributions would be expected#

however, with high-altitude airplanes or with conventional airplanes in

other climates and seasons. For example, if an airplane is expected to

cruise at altitudes of from 18,000 to 28_000 feet, where the temperature

is normally in the range between 0 and -40 ° F, a much higher frequency

of icing at low temperatures is to be expected and the results of an

analysis of icing conditions occurring mostly at higher temperatures

would be applicable only to a limited extent.

NACA TN 2738 19

Comparison of Results of the EquiprobabilityAnalysis With Results of NACATN 1855

Certain values of liquid-water content, drop diameter, and temper-ature were presented in NACATN 1855 (reference 5) for consideration inthe design of anti-icing equipment. The combinations of meteorologicalvariables recommended, however, were lacking in one respect, namely_ theprobability of occurrence of these particular combinations. Fortunately,with the aid of the equiprobability curves, some indication of theseprobabilities can be obtained when corresponding meteorological condi-tions are compared.

Four classes of icing conditions are presented in NACATN 1855:I, instantaneous_ II# intermittent_ III# continuous; and IV# freezingrain. All classes_ except IV, are subdivided into two types of icingconditions, maximumand normal_ and are confined to icing conditionsassociated with definite cloud formations of characteristic horizontalextents. Hence3 classes I# II_ and III are ideal for comparison withthe results which can be obtained from the equlprobability charts(figs. 3 to 7) and the relations for horizontal extent (fig. 8).

Two groups of icing conditions are chosen as a basis for comparisonbetween the analysis of this report and NACATN 1855o The first groupconsists of icing conditions selected from the probability analysis for

the case of layer-type clouds with an exceedance probability

of Pe = O.OO1. The second group consists of icing conditions for the

case of Pacific coast cumulus, also with an exceedance probability

of 0.001. These two groups are compared_ respectively, with classes III-M

and II-M from NACA TN 1855. The comparisons are made by determining

values of liquid-water content from figure 4 and figures 3# 53 and 7 for

the various combinations of temperature and drop diameter given in

table I of NACA TN 1855 for class II-M and III-M cenditions, respectively_

The three values of liquid-water content obtained frOm figures 3, 5_

and 7 corresponding to class III-M were averaged by weighting the values

according to the horizontal extent of the region to which they apply.

The three regions (fig. I) have area ratios of about i/8_ 2/8, and 5/8

for the Pacific coast, plateau, and eastern regions, respectively; and

these area ratios were used to weight the values of liquid-water content

from the equiprobability charts so that the resulting average values

would be on a comparable basis with the values expressed in NACA TN 1855o

The weighted average values of liquid-water content were corrected by

figure8 to make them applicable to a horizontal extent of 20 miles, a

figure reasonably applicable to the layer-cloud data of NACA TN 1855o

For class II-M conditions, no correction for horizontal extent was

required because the values in NACA TN 1855 and figure 4 both apply to3 miles horizontal-cloud extent.

The results of the comparisons between NACA TN 1855 and the proba-

bility analysis are shown in tables III and IV for classes II-M and III-M,

respectively. In general_ the llquid-water-content values agree quite

20 ACA 2738

closely, except at temperatures of -S° F and below. In this low tempera-

ture range, the values of liquld-water content derived from the equiprob-

ability surfaces are considerably lower than values given in NACA TN 1855.

The reason is that the probability analysis entails some extrapolation in

this temperature range, and so may not provide exactly the correct value

of liquid-water content. On the other hand, the values listed in

NACA TN 18_ are not restricted to actual measurements, and therefore may

not be of precise magnitude. For design purposes, however, the values

given by the probability analysis should be of proper magnitude. In this

regard, it should be noted that the majority of the data utilized in the

probability analysis was taken at comparatively low altitude (13,000 feet),

whereas the temperature range between -S ° F and -40 ° F represents consider-

ably higher altitudes (18,000 to 28,000 feet). This difference in altitude

could have some bearing on the accuracy of the probability data, particu-

larly at very high altitudes.

By a procedure similar to that used in the comparisons for the two

example cases, the correspondence between classes of icing conditionsrepresented by the equiprobability charts and other classes of conditions

presented in NACA TN 18_ could be ascertained approximately. The corre-

spondence for these classes of conditions, and also for the two example

cases, are shown in table V.

An inspection of table V reveals a consistency between theclasses

of design values recommended in NACA TN 18_ and specific values of

exceedance probabilities. 0nly the instantaneous maximum icing condi-

tion (Pe = O.O001) appears to be incongruous with the maximum conditions

in classes II-M and III-M (Pe = O.OO1). This apparent incongruity is

directly attributable to the fact that the instantaneous maximum condi-

tion presented in NACA TN 18_ was calculated for tall tropical cumulus

clouds. Such extremely severe icing conditions would, of course, be

exceeded infrequently as is borne out by the probability analysis.

Another point of interest in regard to table V is the general order of

magnitude of the exceedance probabilities which are found to apply to

the various classes of icing conditions specified in NACA TN 18_. For

example, any one of the icing conditions listed under class II-M (inter-

mittent maximum) or class III-M (continuous maximum) will be exceeded once

in only about a thousand encounters with icing conditions. It is believed

that the icing conditions specified by classes II-M and III-M do not

impose too severe design requirements and, therefore, if the ice-

prevention system will cope with these conditions, the results of the

probability analysis indicate that satisfactory ice protection would be

provided for the vast majority of icing encounters.

USE OF THE EQUIPROBABILITY AND PARTIAL PROBABILITY

CHARTS AS APPLIED TO THE DESIGN OF THERMAL ICE-

PREVENTION EQUIPMENT

Heretofore, no information has been available regarding the prob-

ability of encountering icing conditions which exceed certain specified

values, so that some discussion of probability as applied to the design

NACATN 2738 21

of thermal anti-icing equipment seems worthwhile. In general, two designproblems exist: one problem is to establish consistent simultaneous com-

binations of liquid-water content, drop diameter, and temperature which

can occur in nature, and the other problem is to determine how effectively

a given thermal element of an anti-icing system will cope with the entire

array of icing conditions existing in nature. It is the purpose of the

equiprobability charts to aid in the solution of the first problem, while

the partial probability charts are intended to aid in the solution of thesecond problem.

The Equiprobability Charts

The equlprobability charts essentially area means to determine

quickly consistent sets of liquid-water content_ temperature, and drop-

diameter values which have the same probability of being exceeded during

icing encounters. Sets of such values obtained from the equiprobability

charts can be used as criteria for the design of thermal anti-icing

equipment. No need to use these charts exists if it is found that

table V will define a class of conditions from NACA TN 1855 suitable for

the needs of the particular design problem. However_ should it be

desired to design, say# a jet-englne inlet duct for an exceedance prob_

ability of O.001 (only one chance in a thousand of exceeding a specified

amount of liquid water)_ table V would not be of help because correspOnd-

ing icing conditions from NACA TN 1855 are not defined for this partic-

ular case. Hence_ reliance would have to be placed on the equiprobabilitycharts in order to establish consistent sets of values for a chosen

value of Pe-

General procedure for using the equiprObability charts.- The follow-

ing procedure for the use of the equiprobability charts is suggested:

1. Determine for what class of icing condition (I, II, or III of

NACA TN 1855) a design is to be made by considering the nature of the

component to be protected. (See table V.)

2. Establish the severity of icing condition within the class by

selecting the exceedance probability for which the component is to bedesigned. (Seetable V.)

3- With the aid of table V and steps I and 2, determine what equi-

probability chart to use in defining values of W consistent with dif-

ferent sets of T and D. The selection of charts can be made by noting

the type of cloud formation applicable to the class (i.e.# I_ II, or III)

and severity of icing condition established in steps i and 2. For cumulusclouds, it is suggested that the charts for the Pacific cOast area be

used for design purposes rather than the charts for the plateau area.

The reason is that, until more data are available for the eastern

United States, Pacific coast cumulus-cloud data should provide liquid-water-content values that are conservative for the remainder of the

United States. For layer clouds, however, it is suggested that the

charts for all three geographical areas be used for design purposes by

using a weighted average of liquid-water-content values from the three

22 NACA _ 2738

charts. The justification for using a weighted average is that layer

clouds can be regarded as being structurally continuous over exceedingly

large geographical areas; consequently, only average values of the

parameters defining an icing condition would seem to have much signif-

icance in the design of an anti-icing component for layer clouds exist-

ing throughout the United States.

4. From the appropriate equiprobability chart, choose values

of W corresponding to a variety of sets of T and D. In the case of

layer clouds, where more than one chart may be employed to determine

values of W, a suitable weighted average for the entire United States

can be obtained by weighting the liquid-water-content values for the

Pacific coast, plateau_ and east coast regions, in the ratios of 1/8,2/8

and 5/8, respectively. These weight ratios, 1/8, 2/8, and 5/8_ are pro-

portional to the three geographical areas and may be considered as

approximately accounting for the length of time that an airplane flying

transcontinentally would be in each area°

5- Multiply the values of W determined in step 4 by a factor,

chosen from figure 8_ for including the effect of horizontal extent of

cloud formation. The factor chosen should lie within the range of

horizontal extent indicated in figure 8 for the particular class icing

condition established in step l_ it should correspond also to the curve

for the exceedance probability decided upon in step 2. Recommended

values of the factor correspond to values of horizontal extent identi-

fied in figure 8 as design values. These design values are for cloud

horizontal extents of 1/2_ 3, and 20miles and correspond to icing con-

dition classes I_ II, and III_ respectively°

Example showing the use of the equiprobability charts.- For this

example_ a design of a jet-engine inlet-guide vane will be assumed in

which it is desirable to protect against those icing conditions having

a probability of being exceeded once in a thousand icing encounters

(Pe = 0o001)° The steps in determining sets of values of W_ T_ and D

are:

i. From table V_ Jet-engine intakes are recommended to be designed

for class I icing conditions.

2. The exceedance probability for which the design is to be made

is Pe = 0°001.

3- In table V, Pacific coast cumulus clouds are considered typical

for class I icing conditions. Hence, equiprobability charts for

Pacific coast cumulus (fig. 4(c)) are chosen for obtaining values of

liquid-water content corresponding to various combinations of temperature

and drop diameter.

4. Values of W selected from figure 4(c) for different arbitrar-

ily chosen sets of T and D are presented in the following table:

NACA TN 2738 23

Instantaneous Conditions Corresponding to

Pe = 0.001[Horizontal extent not considered]

W T D

2.65 32

2.39 i0

1.91 o 151.20 -I0

•55 -20

1.50 321.02 i0

•9o o 30.49 -io.18 -20

-97 32.49 i0

.32 o 45-13 -lO

•01 -20

5- From figure 8, it is determined that the factor by which to

multiply the liquid-water-content values presented in step 4 is 1.3.

This particular value corresponds to the design value of cloud horizontal

extent suggested in figure 8 for class I (instantaneous) icing conditions

with Pe taken as 0.O01. A tabulation of the final liquid-water-content

values suitable for design purposes is presented in the following table:

Instantaneous Conditions Corresponding to

Pe = O.OO1

[Horizontal extent considered]

W T D

3.44 32

3.11 io

2.48 o 151.56 -i0

•71 -2O

1.95 32

1.32 I0

1.17 0 30

•63 -lO

•23 -20

•74 32

•63 i0

.41 o 45•17 -lO

•O1 -20

24 NACATN 2738

In the mbove table, care should be taken to note that the exceedanceprobability, Pe = O.001, applies to each icing condition listed and notto the class in general.

The Partial Probability Charts

The partial probability charts essentially provide a rapid means ofdetermining the approximate probability of exceeding any specified groupof values of liquid-water content associated simultaneously with tempera-ture and drop-diameter values lying within specified ranges. Thus, ifan anti-iCing component (e.g., unit span of a Jet-inlet guide vane) isdesigned by the method of reference 16 to withstand a constant value 8of weight rate of water-drop impingement Mt, as computed from a partic-ular set of values of liquid-water content, temperature, and drop diam-eter, and if other combinations of these same variables are found whichcorrespond to the sameweight rate of drop impingement, then the prob-ability of encountering conditions under which the component will notperform satisfactorily can be determined approximately. The calculationof this probability can be performed with the aid of the partial prob-ability charts (fig. lO).

General procedure for using the partial probability charts.- To use

the partial probability charts, the following procedure is suggested:

i. Establish values of liquid-water content, Wi, for various com-

binations of drop diameter and temperature, which are regarded as crit-

ical (a marginal surface) for a given component of a thermal system.

Values of Wi can be obtained by computing values of liquid-water content

required to provide a constant weight rate of water impingement on the

component. The equation

W i = Mt (5)O.33EmVtc

eThe choice of a constant value of weight rate of water-drop impingement

rather than, for example, the use of the amount of heat to raise thesurface temperature to 32° F, provides, according to reference 16, a

reasonable basis for estimating the probability of encountering an

icing condition too severe for the iCe-protection equipment. The

amount of heat supplied for the particular weight rate of water impinge-

ment selected depends upon the type of protection required for the

component, that is, whether, for example, all the water impinging

should be evaporated instead of having the surface temperature merely

brought to 32 ° F. Since for various water-drop sizes and temperatures

the amount Of heat required for protection is not unique for a given

weight rate of water-drop impingement, the marginal surface will not

represent the same degree of protection for all combinations of drop

size and temperature. However, it will very nearly do so. As a result,

the probability value computed from use of the marginal surface is not

exactly correct.

4H NACATN 2738 25

derived in reference 16_ can be used to calculate the values of Wi asa function of drop diameter and temperature, since Em is a functionof these latter two variables. The particular value of Mt used tocalculate values of Wi is obtained by inserting into equation (5)initial values of Wi_ Em_V_ t_ and c corresponding to a selectedmeteorological design condition and airfoil section. The appropriatevalues of Em to employ in equation (5) for the chosen valuesof V, t, and c must be determined by using some type of water-drop-trajectory calculation, such as presented in reference 13. Whenusingequation (5) to obtain values of Wi, the computations should be madefor all the possible combinations of Ti and Di listed in table II.In total, there are 42 such combinations.

2. Determine the probability, APi, of exceeding each of the valuesof liquid-water content, Wi, determined in step lo To determine aspecific value of APi from a particular partial probability chartpresented in figure 10_ the procedure is as follows: Follow a verticalline representing the liquid-water content until the line intersects ahorizontal-extent line. From the horizontal-extent llne_ follow hori-zontally until an intersection with a temperature-interval contour isobtained. From this intersection, follow a vertical line downward toobtain an intersection with a diameter-interval lineo A horizontal linethrough this latter intersection yields the value of LkPi on the ordi-nate scale° When it happens that a horizontal line drawn from the

horizontal-extent curve will not intersect the desired temperature-

interval linej all that can be said about the value of LXPi for a givendiameter interval is that it is smaller than the value the horizontal

line would indicate when projected to the extreme left side of the par-

tial probability chart. Such values are usually small enough to beneglected°

3- Establish the probability, P, of exceeding all the values of

liquid-water content defined in step 1. This probability may be obtained

by summing all the individual values of LkPi, determined in step 2_ in

accordance with equation (3)- The number of APi values which must be

summed is 42, since this number represents all combinations of tempera-

ture and drop diameter possible by using the charts of figure lO.

Example showing the use of the partial probability charts.- The

example of the jet-engine inlet-guide vane will be continued to demon-

strate the use of the partial probability charts for a vane section

assumed to he heated to 100-percent chord. The starting point of this

example will be taken with established sets of values of liquid-water

content, drop diameter_ and temperature that define a marginal surface

for a unit span of the vane. These sets of values_ which are presented

in table VI, will be presumed to have been obtained by calculating

values of Wi from equation (5), using a particular design icing condi-

tion from the equiprobability charts. The particular icing condition

presumed to have been selected for the computations is one taken from

the example use of the equiprobability charts and defined by the following

26 NACATN 2738

values: W = 2.48 grams per cubic meter; T = O° F; D = 15microns; and

cloud horizontal extent, 0.Smile. With these available data, the steps

employed to use the partial probability charts are:

i. Values of Z_Pi are selected from the marginal probability

charts for Pacific coast cumulus (fig. 10(b)) to correspond with the

conditions chosen from the equiprobability charts for a horizontal

extent of 0.Smile. The results are tabulated in table VI for the 42

different possible combinations of Ti and Di.

2. The probability of encountering an icing condition lying out-

side the surface defined in W-T'-D space by the Wi-Ti-D i values

listed in table VI is obtained by adding the partial probabilities found

for the various conditions presented in the table. In the summation,

values entered in table VI as being less than a certain specified amount

are neglected because the values are exceedingly small and not exactly

determined in magnitude. These values arise from the fact that the

charts are not constructed to encompass quite small values of Z_Pi. For

the example presented in table VI_ the value of the probability, P, was

found to be 0°06_7o The value of 0.0657 can be interpreted to mean that

in approximately 93 out of 100 icing encounters the Values of liquid-

water content, for all combinations of drop diameter_ will lie below the

marginal surface and hence be encompassed by the design.

CONCLUDINGREMARKS

The equiprobability surfaces and marginal probability calculation

charts presented herein provide_ for the United States, a representation

of the data now available on the meteorological factors responsible for

aircraft icing_ expressed in terms of the probability of exceedingvarious combinations of values of these factors. The usefulness of the

results is limited somewhat by the amount of data available_ errors of

measurement, especially at large values of drop diameter; nonrepresent-

ativeness of the flight procedures during research flights; and limited

altitude and geographical extent of the research flight program_

In spite of these limitations 3 the probability analysis presented

does provide an indication_ heretofore unavailable_ of the combinations

of icing conditions having equal probability of being exceeded in the

United StateS_ and also the probability of _xceeding a special set of

icing conditions. Also_ a procedure has been established for the sta-

tistical analysis of future icing meteorological data obtained on aworld-wide ba_iso

Ames Aeronautical Laboratory

National Advisory Committee for Aeronautics

Moffett Field, California, April 18, 19_2

NACATN 2738 27

APPENDIXA

DETAILS OF DETERMININGTHEEQUIPROBABILITY

SURFACECONTOURS

The distribution functions used in the equiprobability study arethe normal and the Gumbel probability distributions. Graphical methodswere used to evaluate the distributions by using normal-probabilitygraph paper for the normal distributions and a specially constructedpaper for the G_nbel distributions. These special types of graph paperhave the property that the cumulative distribution Curves are repre-sented by straight lines.

The Gumbel distribution, taken from reference ll, is given by theequation

PG = 1-e-e-_(x-u)

In equation (A1), the constant u is the mode and _ is a quantity

which measures the concentration of the frequency distribution about the

mode. Since on the specially constructed Gumbel distribution graph paper

equation (A1) will appear as a straight line, it is desirable to define

the line by specifying two points on the curve. For the purposes of

this analysis, the points are chosen at which the probability PG has

the values of 0.63 and 0.05. The value of PG = 0.63 may be found by

placing x = u and the value of PG = 0.05 may be found byplacing _(x-u) = 3.0.

When a straight-line fit could not be obtained from the data by

using Gumbel distribution paper_ an arbitrary smooth curve on normal

probability paper was used to represent the distribution function. This

procedure conserved the essential form of the observed distribution_

smoothed out the irregularities 3 and provided a reasonable basis for

limited extrapolation.

Cases 2 and 5 (table I) are used to demonstrate how the normal and

Gumbel probability distributions were applied to the data since these

cases were the ones in which the largest number of encounters with icing

were recorded. As mentioned in the text_ the general procedure is to

evaluate the functions Q(T'), R(W,T'), S(D_W_T')_ R'(W,D_T')

and S'(D,T'). t

Case 2, Cumulus Clouds in the Pacific Coast Area

The distribution functions.- The distribution of the temperature

depression below freezing of the most severe icing condition in each

icing encounter for this case is shown in figure ll plotted on Gumbel's

25 NACA TN 2738

distribution paper. In this figure the broken line represents the

observed values and the straight line is the estimated line of best fit.

This line defines the function Q(T').

The distribution of maximum liquid-water content per encounter was

plotted for all values of T' equal to or greater than each of the

following values: O_ 8_ 12, 17, 22_ 27_ 28_ 323 and 37- The plots

for T' _0_ and T' _27 are presented in figure 12. The values of the

distribution parameters_ Wo_63 and Wo.o5 _ obtained from these distribu-

tioms and plotted as functions of T'_ are shown in figure 13. The

smoothed curves in figure 13 define the liquid-water-content distribu-

tion function, R(W_T'). These curves were drawn to the spontaneous

T' = 72 F °freezing temperature_

The distribution_ S(D_W_T') of drop size to be used with the dis-

tribution of maximum liquid-water content per encounter is to be deter-

mined for all observations with W and T' greater than certain values.

Small values of W need not be considered in this case since conditions

of small W and large D are to be expressed in terms of S T and R'o

The distribution_ S(D_W_T'), of drop diameter for all values of T' _0

and W_0.5 is shown in figure 14 plotted on normal probability paper.

S±milar plots were made for W_O°5_ T' _22_ and W_0.8_ T'_O; and

since these distributions were very similar to the one for WEO.5_

T' _0_ a statistical test, the chi-square test (reference 17)was

applied to test the significance of the differences. The results of

this test showed that the differences between these distributions were

no greater than would be expecte d as a result of random sampling_ hence

it was concluded that any dependence of S upon W and T' for values

of W>0®5 was too small to be reliably indicated by the data. Accord=

ingly_ the distribution S(D_W_T') shown in figure 14 was used for all

values of W_0.5.

In order to define the equiprobability surfaces in the region

where W is small and D is large_ the distributions S'(D_T')

and R'(W_D_T') must be determined° The distribution_ S'(D_T')_ of

the maximum value of D per icing encounter was examined for T'__O,

T' _22_ and T' _ 32_ and the application of the chi-square test to these

distributions indicated that there was no significant detectable depend-

ence of S' on temperature° The distribution_ S'(D_T')_ for T'_ 0 is

shown in figure 15o

In order to determine R'(W_D_T')_ the distribution of W for large

values of D_ this distribution was plotted for the following cases:

D_20_ T' _0_ > ' > • > ' > D >2 '> D_ ' >D_30_ T _0_ D 44, T _0_ _ 0_ T _22_ and 30_ T _22.

A study of these distributions indicated that there was apparently no

effect of T' on the variation of R' with Wo There was_ however_

a significant effect of D on the variation of R' with W°

Figure 16 shows the distributions for T v _0 and D_20, 30_ and 44.

Figure 17 shows the parameters defining the R' distributions for

T ' _0 plotted as a function of D. It is likely that the failure

of the data to show for large values of D a decrease of W with

NACA TN 2738 29

increasing T', as was the case in figure 13, is due to the fact that

insufficient data are available for large values of D to define the

function R' when T' is greater than 22, since it is only in this

range that the function R becomes strongly dependent upon T'. In

the absence of more complete information, it has been assumed that

when T' _22, the dependence of R' on T' is similar to the dependence

of R on T'. Accordingly, the values of the R' distribution param-

eters (the mode Wo_ss and the fifth percentile Wo_s) shown in fig-ure 17 are altered to obtain values of the distribution parameters which

are considered applicable when T' >22. _The factor used to alter any

particular value of W presented in figure 17 is established from fig-

ure 13 by forming the ratio of the value of W at a selected value

of T' to the value of W applicable to values of T' _22. The values

of Wo.ss and Wo.os which define the function R' were obtained inthis manner, and are presented in table VII.

Construction of the equiprobability contours.- After the distribu-tion functions have been established 3 the equiprobability contours for

various temperature depressions can be determined. The following

examples illustrate the use of the probabilities Q, R_ S, R', and S'

in determining the equiprobability surface for Pacific coast cumulus

clouds corresponding to an exceedance probability of 0.O1. For a value

of Pe = O.01, the equation

: o.oi

applies to large values of W. Similarly, the equation

QR'S' = 0.01 (A3)

applies to small values of W. In evaluating equations (A2) and (A3),

two equiprobability surface contours are selected. These contours are

for T' = 0 and T' = 32.

L__argevalues of W: Table Vlll shows the calculation of W,using equation (A2) for various arbitrarily chosen values of D.

putations are shown when T' = 0 and when T' = 32.

Com-

When T' = O, it should be noted that Q = 1.0. Thus, equa-

tion (A2) becomes

R : O.Ol/S (A4)

The values of S necessary in equation (A4) to calculate values of R

for various values of D are obtained from figure 14. Values of R

calculated from equation (A4) are used to define the desired values

of W on the curve depicting the distribution function R(W_T') for

a value of T' = O. This function is defined as a straight line on

Gumbel distribution paper by establishing the values of W for the

mode and at the fifth percentile. The values_ Wo.ssand Wo.o_ used

3o NACATN 2738

in defining the straight-line distribution of R(W,T') are obtainedfrom figure 13 corresponding to a value of T' = O. For this partic-ular case, the result would be the sameas presented in figure 12.The desired values of W chosen from the straight-line distributionfor the calculated values of R are presented in table VIII, part (a).

When T' = 32, Q is found from figure ii to have the valueof 0.176. Hence, the equation expressing values of R is

R = o.o1 (AS)0.176S

Values of S required to solve equation (AS) are obtained again from

figure 14 for various arbitrarily selected drop diameters° To find

values of W to which the calculated values of R correspond, it is

again necessary to establish R(W,T') as a straight-llne function

of W on Gumbel distribution paper. The two points required to

establish the straight-line variation are the values of W at the

mode (Wo_s) and at the fifth percentile (Wo.o_). These values, for

a value of T' = 32, are found from figure 13 to be Wo.es = 0.28

and Wo.o_ = 1.26. The values of W found from the straight-line

variation of W with R(D,T') are tabulated in table VIII, part (b),

along with the corresponding values of D, S, and R.

Small values of W: Sections of the equiprobability contours

corresponding to small values of W are calculated in identically the

same manner as for large values of W. In both the case of T' = 0

and of T' = 32, values of S' were obtained from figure 15 for cal-

culating values of R' from the equation

R' = P/QS'

The values of W at the mode and the fifth percentile for defining W

as a function of R'(W,D,T') were obtained from table VII. The calcu-

lations for T' = 0 are presented in table IX3 part (a)_ and the cal-

culations for T' = 32 are presented in table IX_ part (b).

Intermediate values of W: From tables Vlll and IX, equiprobabil-

ity curves can be plotted for both high and low values of W, as iS

shown in figure 18o For the intermediate values of W, however, the

final equiprobability contours must be obtained by joining the upper

and the lower portions by a smooth connecting line, shown by a broken

line in figure 18. The upper portions of the final contours represent

maximum values of liquid-water content per icing encounter and the

corresponding drop diameter; the lower portions represent maximum drop

size per encounter and the corresponding liquid-water content; and the

intermediate portions may represent either one of these combinations.

NACATN 2738 31

The equiprobability contours presented in figure 18 are anincomplete set Of those presented in figure 4(b).

Case 5, Layer Clouds in Eastern United States

The distribution functions.- The distribution of the temperature of

the most severe icing observation in each icing encounter for this case

is shown in figure 19. Since a satisfactory straight-line representa-

tion could not be found for this frequency distribution, the smooth

curve on normal probability paper given in figure 19 was used to define

the function Q(T').

The distribution of maximum liquid-water content per icing

encounter_ R(W,T'), was plotted for values of T' equal to or greater

than O, 7, 12, 17, 22, and 27. The curves for T'_0 and T' _17 are

presented in figure 20. The values of Wo.es and W o o- obtained fromeach of the six distribution curves were plotted as _ctions of T'

(fig. 21), and smooth curves were drawn to represent the distribution

parameters which define the function R(W,T').

The distribution of drop diameter for values of W_ 0.2 and T' _ 0

is shown in figure 22. As in case 2, this distribution was found to be

approximately independent of W and T' for values of W greater

than 0.2. The curve in figure 22 was, therefore, used to represent the

function S(D,WjT').

The distribution of the largest value of D_ S'(D,T) observed in

each icing encounter is shown in figure 23. This distribution curve_

which was found to be approximately independent of T', was used to

define the function S'(D_T').

In order to determine R'(W,D,T'), the distribution of W corre-

sponding to large values of D, the distribution of W was plotted for

all values of T' and for values of D equal to or greater

than 15, 17, 19, 21, 23, and 25. Values of Wo_ss and Wo_o_ for eachof these distributions were plotted against D as shown in figure 24°

As a result of the small amount of data available for large values

of D, there is a considerable scatter in these distribution parameters.

The smooth curves shown in figure 24 were drawn to represent approxi-

mately the dependence of R' upon D. To determine the dependence

of R' upon T', cases of D_17 were chosen since this was the

largest value of D for which sufficient data were available to provide

a reasonably reliable sample. Distributions of W for D_17 and T'_ 0,

12, and 17 were plotted and the resulting values of Wo_ss and Wo.o_from these distributions were used with the data from figure 24 to pro-

vide the basis for the construction of a set of curves defining the

parameters of the distribution, R'(W,D,T')_ (fig. 25). Curves repre-

senting values of Wo_s and Wo _os for D_17 were drawn through the

32 NACATN 2738

data points (fig. 25), using the shape of the curves of figure 21 (the

distribution parameters for R) as a guide_ The curves for other values

of D were then drawn from the points determined by the values

for T' = 0 (obtained from fig. 24) in such a way as to conform to the

pattern established by the curves for D_17. These curves define the

function R'(W,D,T')o

Values of the probability terms_ Q, R, S, R', and S', were obtained

with the aid of figures 19, 20, 22, 25, and 23, respectively, and were

used to construct the equiprobability surfaces for values of Pe

from O.1 to O.OO1, which are presented in figure 7o

Cases i, 3, and 4

By following procedures similar to those described in the preceding

examples, equiprobability surfaces also could be determined for the

following cases: (1) layer clouds in the Pacific coast area (fig. 3)_

(2) layer clouds in the plateau area (fig. 5); and (3) cumulus clouds in

the plateau area (fig. 6). The case of cumulus clouds in the easternUnited States was omitted because of insufficient data.

APPENDIXB

DETAILS OF THEPREPARATIONOF THE PARTIAL

PROBABILITYCHARTS

The same cases as were used in describing the construction of theequiprobability charts will be used to describe in detail th_ construc-tlon of the partial probability charts. The general procedure in con-structing the charts is first to evaluate the equation

= PWi( )

The resulting values of ZiPi for different combinations of liquid-water

content and intervals of drop diameter and temperature are then arranged

into a graphical form for convenient use.

Evaluation of Z_pi

Case 5_ layer clouds in eastern United States.- Values of PAT for

this case wereobtained from figure 19 which provides information as to

the probability that the temperature depression will be lower than any

particular value. To find a value of PAT (the probability that the

temperature will lie within the interval AT) from figure 19, it is only

necessary to take the difference between the probabilities indicated for

the two chosen limits of AT. Values of PAT determined from figure 19

are presented in table X, part (a), for selected temperature intervals.

Values of PZkD (the probability that a drop diameter will lie withinthe interval ZkD) were obtained in a m_nner similar to that used in

obtaining values of PAT. In this case, however_ a composite drop-diameter distribution curve was used in order to allow approximately for

the occurrence of icing encounters in which the conditions of maximum

severity are determined primarily bydrop size rather than liquid-watercontent. The composite distribution curve is presented in figure 26

where the probability values in this special case are called S''. As

can be seen from figure 26, the composite distribution curve follows

the distribution of all observed values of drop diameter for the lowest

90 percent of the distribution and the distribution of maximum observed

value per encounter (fig. 23) for the highest 1 percent of the cases.

The intermediate 9 percent of the cases is represented by a straight

line connecting the two distributions. Values of P_D determinedthrough use of figure 26 are presented in table X, part (b), for

selected drop-diameter intervals.

In order to determine values of PWi(AT), first the distributionsof maximum liquid-water content per encounter were plotted on Gumbel

34 NACATN 2738

distribution graph paper for each of the chosen temperature intervalsabove -9.5 ° F in a manner similar to that done in figure 20 for two

ranges of values of T'. Values of Wo.ss and W o.os thus could beobtained; and these values were plotted as a function of temperature

and extrapolated to obtain values of Wo .ss and Wo .os applicable to

temperature intervals below -9.5 ° F. The resulting values of Wo .ss

and Wo .os for all temperature intervals are presented in table X,

part (c). In addition, there are presented values of W which were

chosen arbitrarily from the straight-line distribution established by

the two points, W o .ss and Wo .os- Corresponding to the selected values

of W exist values of PWi(_T) which also can be read from the straight-

line distribution defined by the points Wo .ss and Wo .os" These values

of PWi(AT) are also presented in table X, part (c).

After the values of P2D, P21T, and PWi(_T ) are determined, valuesof _Pi can be determined simply by multiplying the constituent prob-

abilities together for various combinations of liquid-water content and

intervals of temperature and drop diameter. Such computations are sum-

marized in table XI for two temperature intervals (32.0 ° to 20.5 ° F

and 20.5 ° to 10.5 ° F), two drop-size intervals (0 to 9-5 microns

and 9-5 to 12.5 microns) and a range in liquid-water-content values.

The values of LkPi listed in the table are ones which can be obtained

through use of the partial probability charts.

Other cases.- The quantities used in calculating values of LkPi

for the remaining cases studied were obtained in the same manner as

case 5, with one exception. In case 2, the distribution of the greatest

drop diameter per encounter was used for the upper 5 percent of the

diameter distribution instead of the upper 1 percent. The reason for

this deviation was because of the greater frequency of encounters in

which the most severe icing condition was determined by the maximum

value of drop size.

Example Construction of the Partial Probability Charts

Rather than compute values of LkPi every time they are required 3

use of the data presented in tables X and XI in chart form is a great

convenience. In general, the technique of constructing a suitable chart

for evaluating values of AP i utilizes the relation

log 2_Pi = log P2D + log P2_T + log PWi(AT) (B1)

in such a manner that the chart performs the function of adding the

right-hand terms in the equation.

The various steps in the construction of a chart to solve equa-

tion (B1) (derived from equation (4)) are shown in figure 27_ Fig-

ure 27(a) establishes the liquid-water content as a function of the sum

NACATN 2738 35

of the logarithms of PATand PWi(AT)- Since the value Of PAT corre-sponding to a given temperature interval is a constant_ the liquid-water-content distribution curve is displaced horizontally from the origin bythe amount of log PAT. The values of P_T, PWi(ZkT)and Wi used inpreparing figure 27(a) were obtained from table X, parts (a) and (c),for a temperature interval of 32° to 20.5 ° F.

The next step in the construction is shown in figure 27(b ) wherethe llquid-water-content scale on the right is transformed into a corre-sponding scale at the top by use of a line drawn at an angle of 45° onthe grid system. This line may be labeled the lO-mile horizontal-extentline because the ratio of any value on the top scale to the correspond-ing value on the right-side scale is unity. For example, in the figurea value of Wi = 0.5 on the top scale leads to a value of Wi = 0.5on the vertical scale when the lO-mile horizontal-extent line is used.For other horizontal extents, however, a factor must be applied to theliquid-water content for lO-mile horizontal extent to obtain the valueappropriate to the extent under consideration. Accordingly, anotherline (derived from reference 3) has been drawn on figure 27(b) to incor-porate automatically the factor which should be applied to a 50-milehorizontal extent at different values of P_T and PWI(Z_T)° Thus, avalue of Wi = 0.5 on the top scale would be transformed into a valueof Wi = 0.82 on the right-hand scale for a horizontal extent of50 miles. The point on the liquid-water distribution curve to which thevalue of Wi = 0.82 applies is labeled point A in figure 27(b)°

The only other factor which must be included in the partial prob-ability charts is the probability factor PAD- A method of incorporat-ing the term is by using a scale-transformation reference line (similarin operation to the horlzontal-extent lines). Figure 27(c), the toppart of which is the same as figure 27(b), is presented to show thefunction of the scale-transformation reference line in determining thetotal sum of log PZ_T_log PW_(AT), and log P_D- As an example, a valueof Wi = 0._ is taken for which it is desired to find the sum of thelogs of the three probabilitiesj P_T, PWi(AT), and P2_Oapplicable toa lO-mile horizontal extent 3 a temperature interval of 320 to 20.5 ° Fand a drop-diameter interval of 0 to 9-5 microns. Accordingly, a valueof Wi = 0.5 is selected on the liquid-water-content scale and is fol-lowed downward, vertically, until an intersection with the lO-milehorizontal-extent line is obtained (point 1). From point l, a line isfollowed horizontally until an intersection with the liquid-water-contentdistribution curve for the temperature interval of 32° to 20.5 ° F(point 2). The horizontal distance that point 2 is away from the logreference line is a measure of the sum of log PZ_Tand log PWi(AT)° Ifa vertical line from point 2 is followed downward until an intersectionwith the scale transformation reference line is reached (point 3)3 theabsolute value of the sum of log PZ_Tand log PWi(AT) can be read on thescales either to the left or to the right of the point. However, if thevertical line followed from point 2 is continued downward until theintersection with the drop-diameter interval line is reached (point 4),

36 NACATN 2738

the sum of the logs of PAT_ Pwi(AT) and PAD can be read from the left-hand scale for log of APi. The reason why the total can be read isthat the drop-diameter interval line is displaced vertically downwardfrom the scale transformation reference line by the amount of log PADfor the diameter interval of 0 to 9-5 microns. The actual valueof PAD used in the construction was obtained from table X, part (b)o

The final value of log2_ i read from the partial probabilitychart in figure 27(c) is the result sought, namely, a solution of equa-tion (B1). For most rapid use of the chart, however, it is desirableto have values of _Pi obtainable directly° This added conveniencecan be incorporated in a chart by making each integer of the logarithmicscales in figure 27(c) correspond to a cycle on semilogarithmic graphpaper. The partial probability charts presented in figure lO have beenconstructed to include this feature, and hence will yield valuesof APi directly.

NACATN 2738 37

REFERENCES

lo

2_

Be

4o

o

6_

Q

.

o

lO.

ll°

12.

Lewis, William: A Flight Investigation of the Meteorological Con-

ditions Conducive to the Formation of Ice on Airplanes.

NACA TN 1393, 1947.

Lewis, William, K_line, Dwight B., and Steinmetz_ Charles P.:

A Further Investigation Of the Meteorological Conditions Conducive

to Aircraft Icing. NACA TN 1424_ 1947.

Lewis, William_ and Hoecker, Walter H., Jr.: Observations of

Icing Conditions Encountered in Flight During 1948. NACA TN 1904,1949.

Kline, Dwight B.: Investigation of Meteorological Conditions

Associated With Aircraft Icing in Layer-Type Clouds for

1947-48 Winter. NACA TN 1793, 1949.

Jones, AlunR., and Lewis, William: Recommended Values of Meteoro-

logical Factors to be Considered in the Design of Aircraft Ice-

Prevention Equipment. NACA TN 1855, 1949.

Kline, Dwight B., and Walker, Joseph A.: Meteorological Analysis

of Icing Conditions Encountered in Low-Altitude Stratiform Clouds.

NACA TN 2306, 1951.

Brock, G.W.: Liquid-Water Content and Droplet Size in Clouds of

the Atmosphere. Trans. Amer. Soc. Mech. EngT., VO1. 69, 1947,PP- 769-770°

Rasmussen_ W., Campbell, R. I., and Shaw, W. C.: Thermal Anti-Icing

Tests, DC-6o United Air Lines Rept. F-81-14, March 28, 1947.

North, David: Icing Test Flights of Curtiss No. 124171-i

Propeller De-Icing Shoes on DC-6 Airplane NX90725 .

American Airlines Rept. No. DC-6-1934XIR, April 143 1948.

Uspensky, J.V.: Introduction to Mathematical Probability.

McGraw Hill Book Co., Inc., N. Y., 1937, p. 31.

Gumbel, E. J.: On the Frequency of Extreme Values in Meteorological

Data. Bulletin of the American Meteorological Society_ vol. 23,

March 1942, pp. 95-105.

Dobson_ G. B. Mo: Ice in the Atmosphere° Quart. ,our. Roy. Meteoro-

logical Soc., vol. 75, 1949, pp. 117-130.

38 NACATN 2738

13.

14.

15.

16.

17.

Bergrun, Norman R.: An Empirical Method Permitting Rapid Deter-

mination of the Area, Rate_ and Distribution of Water-Drop

Impingement on an Airfoil of Arbitrary Section at Subsonic Speeds.

NACATN2476, 1951.

Jones, Alun R. 3 and Lewis, William: A Review of Instruments

Developed for the Measurement of the Meteorological Factors

Conducive to Aircraft Icing. NACARMA9C09, 1949.

Ludlam, F. H.: The Heat Economy of a Rimed Cylinder.

Quart. Jour. Roy. Meteorological Soc., vol. 77, no. 334,

Oct. 1951.

Hacker, P. T., and Dorsch, Robert G.: A Summary of Meteorological

Conditions Associated With Aircraft Icing and a Proposed Method

of Selecting Design Criterions for Ice-Protection Equipment.

NACA TN2569, 1951.

Hoel, Paul G.: Introduction to Mathematical Statistics.

John Wiley and Sons, N. Y., 1947, pp. 36 and 186.

NACA TN 2738 39

TABLE I.- CLASSIFICATION OF ICING CONDITIONS

AND AMOUNT OF DATA AVAILABLE FOR EACH CASE

Case

r'

i

2

3

4

5

6

Geographical Cloud

area type

Pacific coast

Pacific coast

Plateau

Plateau

Eastern U.S.

Eastern U.S.

Layer

Cumulus

Layer

Cumulus

Layer

Cumulus

Number of icingencounters

39

44

30

25

Ii0

4

Number of observations

of liquid-water content

and drop diameter

171

227

i19

91

4o4

26

40 NACA TN 2738

TABLE II.- INTERVALS OF TEMPERATURE AND DROP DIAMETER

USED IN THE DETERMINATION OF MARGINAL PROBABILITIES,AND MEDIANVALUES IN EACH INTERVAL TO BE USED IN

CALCULATING THE MARGINAL LIQUID-WATER CONTENT

Intervals of temperature_ _T Intervals of drop diameter# AD(microns)

Lower

limit

20.5

10,5

.5

-9.5

_9.5

-_.o

(°F)

Upperlimit

32.0

2O.5

10.5

.5

-9.5

-19.5

Median of

interval_ Ti

Lower

limit

24.0

15.5

6.0

-4.0

-13.0

-25.o

Median of

interval_ D i

Upperlimit

o 9.5

9.5 12.5

z2.5 15.5

15.5 19.5

z9.5 29.5

29.5 49.5

49.5

8

ll

14

17

24

37

6o

6 NACA TN 2738 41

TABLE III.- COMPARISON OF VALUES OF LIQUID-WATER CONTENT

FOR PACIFIC COAST CUMULUS CLOUDS, Pe = 0-0011 WITHCORRESPONDING VALUES FOR INTERMITTENT MAXIMUM

CONDITIONS LISTED IN NACA TN 1855

Temperature _ T(°F)

32

_2

14

14

-k

-4

-22

-22

-22

-4o

-40

-4o

Drop diameter_ D

(microns)

2O

30

5o

20

3o

5o

20

30

5o

20

3o

5o

20

3o

5o

Liquid water content_ W(g/m s)

TN 1855 Probability analysis

2.5

1.3

.4

2.2

1.0

.3

1.7

.8

.2

1.0

.5

.1

2.46

1.50

.42

2.3O

1.35

•37

1.45

.70

.15

.37

.15

0

.2

.i

<.i

0

0

0

42 NACA TN 2738

/

TABLE IV.- COMPARISON OF WEIGHTED-AVERAGE VALUES OF

LIQUID WATER CONTENT FOR LAYER CLOUDS, Pe = 0.001,

WITH CORRESPONDING VALUES FOR CONTINUOUS MAXIMUM

CONDITIONS LISTED IN NACA TN 1855

Temperature

(OF)

32

32

32

14

14

14

-4

-4

-4

-22

-22

-22

-4o

-40

-40

TDrop diameter_ D

(microns)

15

25

4O

15

25

4o

15

25

4o

15

25

4o

15

25

4o

Liquid--water content_ W

(g/m 3)'TN 1855 iProbability analysis

0.80

.5o

.15

•60

•30

.l0

•30

.15

.06

.20

.10

.04

.05

.03

.O1

0.77

.49

.20

•50

.28

.08

.17

.09

.02

.02

.01

.oo3

0

0

0

NACATN 2738 43

TABLEV.- CORRESPONDENCEBETWEENTHEEQUIPROBABILITYANALYSISANDTHEANALYSISIN NACATN 1855

Type of cloudformation

Tropical ctm_ulus

Pacific coastcumulus

DoDo ....

Pacific coastcumulus

Do

Do

Layer clouds

occurring anywhere

in the U.S.(Pacific

coast, plateau andeast coast regions)

Exceed-

ance

proba-

bility_

Pe

a0.o001

.001

.O1

.1

.001

.01

.1

.001

Approximate corre-

sponding class icingcondition as definedby TN 1855

Instantaneous maxlm_m

(I-M)

None

None

Instantaneous normal

(I-N)

Intermittent maximum

(II-M)

None

Intermittent normal

(II -N)

Continuous maximum

(III-M)

None

Continuous normal

(III-N)

Equipment to

which condition

is applicable

Any part of

airplane, such as

guide vanes_ inlet

ducts_ where a

sudden large mass

of supercooled

water would be

critical, even forshort duration

Any critical com-

ponent of the

}airplane_ where ice

iaccretions, even

though slight and

!of short duration_could not be

itolerated

All components of

the airplane; that

is, every part of

the airplane shouldbe examined wlth

the question in

mind_"Wil_ this partbe affected seri-

ouslyby accretions

during continuous

flight in icingconditions ?"

aExtrapolated from Pacific coast cumulus data.

TABLE VI°- EXAMPLE PARTIAL PROBABILITY CALCULATIONS FOR PACIFIC COAST CUMULUS CLOUDS

um temperature, Ti(°F)

Medium

drop

diameter,

Di

(micro )

8

ll

i4

1724

376o

Temperature

Diam._terval, AT 32.0

intLrval,_

(microns) _ wi

o to 9.5

9.5 to 12.5

12.5 to 15.5

15.5 to 19.5

19.5 to 29.5

29.5 to 49.5

49.5 to

TOTALS

24.0

to 20.5

aPi

5.00 0.00001"

4.10 .00001"

3.63 .00002"

3.25 .00004*

2.62 .00007"

1.75 .00004

•85 .0012o

.... .0o124

15.5

20.5 to lO.5

Wi _Pi

4.50 0.00001"

3.70 .00001"

3.26 .00003

2.92 .000142.35 .00140

1.56 .OO32O

•75 .01600

.... .02077

6.0

ii0.5 to 0.5

Wi APi

3.93 0.00001"

3.231 .00009

2.86 .00045

2.55 .001302.05 .OO8OO

1.37 .00700.66 .01400

.... .03084

-4.0

0.5 to -9.5

Wi APi

3.35 0.00001

2.75 .00007

2.42 .000302.16 •00080

1.75 .00380

1.16 .00230

•55 .00430

.... .01158

2.80

2.30

2.031.82

i.46

.97

.47

-13.0

-9.5 to -19.5

J

Wi I _Pi

0.00001"

.00001"

.00002*

.00004*

.00007*

.OOO15

.00100

.... .O0ll5

n

P =f AP i = 0.0012 + 0.0208 + 0.0308 + 0.0116 + 0.0012 + 0.0001 = 0.0657

1

-25.0

-19.5 to -40

ii

Wi APi

ii2.08!0.00001"

1.761 .00001"

1.51 I .00002*1.36 _ .00004*

1.08 •00007*

.73 I .00001"

•35 .00013

.... ! .00013E

NOTE: Values marked with an asterisk, *, denote that the value is actually smaller than the value stated.

Since such values arise because of termination of the partial probability charts_ they are usually

so small that the values can be neglected in the summation for total probability.

c_

_o

ooOO

(%) 32 to i0

TABLE VII.- PARAMETERS DEFINING THE FUNCTION_ R'_

FOR PACIFIC COAST CUMULUS CLOUDS (CASE 2)

T', (FO)

D

(microns) Wo.o 5

3o

35

4o

45

5o

55

6o

65

7o

75

0 to 22

0

32

-lO

_2

-2O

52

-3o62

_o

72

.e_ %.os %.6s

0

##

F3-..4

03

%,-I

46 NACA TN 2738

TABLE VIII.- EXAMPLE CALCULATIONS OF EQUIPROBABILITY

CONTOURS FOR SMALL VALUES OF DROP DIAMETER FOR

PACIFIC COAST CUMULUS CLOUDS (CASE 2)

(a) Pe = 0.01; T' = O; Q = 1.00

I ,

Drop di_eter_ D

(microns)

0

i0

152o

253o

Probability value s

S

1.00

.96

.78

.415

.i16

.o15

R

0.0100

.OlO_

.oi28

.o24

.o86

.67

Liquid-_ter content_ W

(g/m 3)

1.96

1.941.88

1.67

.4o

(b) Pe = O.O1; T' = 32; Q = 0.176

Drop diameter, D

(microns)

0

l0

152O

25

Probability values

S R

1.00 0.057

•96 •059

•78 .073

.415 .137

.i16 .490

Liquid-water content 2 W

(g/m s )

1.22

1.20

1.14

.91

.40

NACA TN 2738 47

TABLE IX.- EXAMPLE CALCULATIONS OF EQUIPROBABILITY

CONTOURS FOR LARGE VALUES OF DROP DIAMETER FOR

PACIFIC COAST CUMULUS CLOUDS (CASE 2)

(a) Pe = 0.O1; T' = 0; Q = 1.00

Drop diameter, D

(microns)

3O

354o455o5562

Probability values Liquid--water content, W

S t R T

0.33 0.030

.2O .O5O

.12 .083

•07 .143

.o4 .25o

.023 .43

.010 1.00

Wo .05 Wo .68

0.7O 0.2O

.60 .16

.51 .12

.42 .o9•351 .o6

.28[ .o45

W

0.79

.60

.43•30.18

.08

(b) Pe = 0.01; T' = 32; Q = 0.176

Drop diameter 3 D

(microns)

3O

35_o47

Probability values Liquid-water content, W

O. 12

.i0

.o75

S _ R'

0.33 0.172

.20 .284

.12 .47

.o57 l.O0

Wo.o5 Wo.68

0.62.53.45

W

0.40

.26

.13

0

48 NACATN 2738

TABLEX.- VALUES OF THE FACTORSUSED IN THE DETERMINATIONOF PARTIALPROBA_ILICX_SAPPLICABLETO EASTERN

LAYERCLOUDS (CASF5)

(a) Temperature

Temperature

interval, AT(OF)

32 to 20.5

20.5 to i0.5

lO.5 to o.5

o.5 to -9.5

-9.5 to -19.5

-19.5 to -40.0

Upper limit of

temperature interval

ValuelProbability, Q

of T' of exceeding

(F° )

0

11.5

21.5

31.5

i41.5

51.5

T !

i. 000

•520

.175

.063

.016

.oo3

Lower limit of

temperature interval

Value Probability, Q

of T' of exceeding

(F 0) T'

I ll.5

21.5

31.5

41.5

51•5

72.0

Probability,

IPAT, that

temperature

lies within

the interval

o.520

.175

.063

•016

.oo3

0

AT

0.480

•345

•112

•047

.013

.oo3

Drop -diameter

interval,

AD,

(microns)

o to 9.5

9.5 to 12.5

12.5 to 15.5

15.5 to 19.5

19.5 to 29.5

29.5 to 49.5

49.5 to

(b) Drop diameter

Lower limit of

diameter interval

Value Probability_of D S'', of

micron@ exceeding D

0 1.000

9.5

12.5

15.5

19.5

29.5

49.5

Upper limit of

diameter interval

Value

of D,micron_

9.5

Probability,

S'', of

exceeding D

0.780

Probability, PZkD I

that drop diameter

lies within the

interval ZkD i

.780 12.5

•I_0 15.5

.230 19.5

.090 29.5

.022 49.5

.0O06

.48o

.230

.o9o

.022

.ooo6

0

0.220

.3oo

.250

.140

.o68

.0214

.ooo6

C_

CO

b-OJ

c_

L_

0[J

I

4_

0o

+_

v

o

I

n _ _ 0

©

-Or-I

._,_-.._o'_ oq O,J 0

kO _-I 0 0• • • •

0

eq o Lr_ 00,1 _"_ D- 0

0

I I i

0 i t o

O_er_ co ._.d- 0

kO_O0

.-_" 0 Lf_ 0OJ I._ L'_- 0

• • • o

O_cf'_O Lr'xO

_O_O0

.-.-1"L_ _- 0,--t OJ ('_ Lr_

•..-d- 0,1

LO r'-I 00

0000

eel0

I I I

! I !

b-

oqOJO_000

b-O000_ ('_

! I

.-..d"Or'_ LrX 0

_000

.-.-d" 0 _"_Or-t_

0 I I

! !

(Y_ I I I

0 u n

I I I0

0 I I IOJ

I I I

0000

OJ

I I I

! ! !

_111

dill

III

0000

d

_111

0111

III

0000

d

0 I I I

! ! I

_111

_111

i

ill

0000

0

b'-- I I

I I

la_ i i

I I I

000

Lr_

!

0 1 I

! I I

0 0 0

t.fN

,---t!

50 NACA TN 2738

TABLE XI.- PARTIAL PROBABILITY VALUES FOR EASTERN LAYER

CLOUDS (CASE 5) FROM DATA PRESENTED IN TABLE X

Temperature

interval_ AT

(°F)

32.0 to 20.5

20.5 to i0.5

32.0 to 20.5

20.5 to 10.5

Diameter

interva_AD

(microns)

0 to 9.5

0 to 9.5

Liquid -water

content_ Wi3for AT and AD

(g/m- 3)

0.23

•50

-75

i .00

i. 30

.24

.50

.75i .00

9.5 to 12.5

9.5 to 12.5

.23

•50

.75

1.00

.24

•50

.751.00

PAT PAD PWi (AT)

0.48 0.22!

.345 .22

.48 .3o

.345 •3o

APi

0.63 J0.0665

.13 .o137

.o24 .oo25

.oo4 .ooo4

.ooo5 .oooo5

.63 .0478

.18 •0136

.o4 .oo3o

.oo9 .ooo7

.63 .0906

.13 .0187

.024 .0034

.004 .0006

.63 .0653

.18 .0186

.o4 .oo41

.oo9 .ooo9

I

/

Pocific Coost

PloteouEostern United Stores

Figure L-Mop of the United Stoles

oreos used in the geogrophicol

showing opproximote

clossificotion

boundories of

of icing doto.

92NACA _N 2738

I

of

Figure 2.--Grophicol presentotion of the equiprobobility surface which

presents the locus of oll combinotions of liquid-woter content, drop

dlbmefer ond temperoture depression below freezing hoving the

some probobility, Pe,, of being exceeded in ony single icingencounter.

NACA TN 2738 53

.5

\

(o) Pe "" 0.1

Figure 3.- Consfont-femperoture contours defining theequiproboMlify surfoce for Cose I, (Loyer clouds tPocific coost).

54 NACA TN 2738

.8

\

o_

.7

.6

.5

.4

.3

.2

./

%

\\

\,\

\

\

.I0

32" F,,,20°F

IOOF

OOF

. iOOF

- 20OF

/Jr

/\

\\J / _

!

../ \"_'-.._ \

"-__. _. \_0 ,.:70 40

Drop diomefer t D,

(T'=O)

(T'-12)(T%22)

(T" 3=o)(T'42)

(T'=52)

\\

\

5O

microns

\

(a; Pe. o.o/Figure 3.-- Gonfinued.

6O

NACA TN 2738 55

LO

.t

I0

\

\\

\\\,

"\ \"\\.\\,.

\ \\ \,, \ \

50 60microns

\

\\\ \'\\\\70 80

(c) Pe = 0,00/

Figure 3.- Concluded,

56 NACA TN 2738

\

"o_

!

.J

1.2

, , , !

1.0

.8

.6

.4

.2

0 o

\

I0

Drop

,- 52" F (T'-- O)/r20* F (T'= I2)

///-I0 ° F (T'-22)

_ 0" F (T'=32)

20 30 40 50

d/ometer, D, microns

(o) Pe - 0.1

Figure 4.- Constont-temperofure

probobilify surfoce for Cose 2.

COOS t).

contours defining(Oumulus c/ouds,

the equi-Pocific

81 _CA _ 2738 _7

2.0

L8

L6

L4

L2 _

.8

-.J .6

.4

.2

O0 I0

\ '2/ \\\

2O

Drop

--32 ° F (T'=O)

-- 2o° F (Ti.12)I0° F (T =22)0 ° F (T'=32)

--I0 ° F (T'=42)

--20 ° F (T'-52).,

30 40 50 60 70

diameter S D, microns

(b) Pe=O.O/

Figure 4. - Continued.

58 NACA TN 2738

\

2.8

2.6

2.4

2.2

2.0

/.8

_ 1.4

_ /.2I

g"" /.0.M

.8

.6 _ _

.4

.2

0 o

\\

\\

\

\\

\\

\\

I 32" F---- 20" f

\II/_---- 0 ° f\I/f -_ooF

\' I r--2oo F

,ItY

_1\\\

II_,

(r_=O)(r =/2 •

/0 ° F -('T:=ZJ)

(r =32)(T'--42)(T'=52)

.-30 ° F (T'=62)

• \_

80/0 20 30 40 50 60 70Drop diameter, D, microns

Pe = 0.001

Figure 4.- Concluded.

NACA_ 2738

59

.5

4.

\

3°

!

_J

00 8

Drop

coo ;r- (T _7 2)

16 _4 3E

diameter, D t microns

(o) Pe = 0.1

Figure 5.- Consfant_temperafure contours defining theequiprobabilify surface for Case 3. (Layer clouds tPlateau area).

60 NACA TN 2738

.8

.7 \

!

%

\ 32°F (l"- O), 2O oF (r'-/2)

lO° f (7"'-'22]

O° f (r'-32)

__ --IO°F (r'-42)

"_. \ \8 16 24 J_

Drop diometer, D, microns40

_b)& - O.Ol

Figure 5.- Gonfinued.

NACA TN 2738 61

\

,LI

,LO

.9

.8

.7

.6

.5

.4

.3

.2

.I

\

\

8

\\

\\

\\

16

Drop

--32 °

20 °

-ig:

_///-_o.\

\ ,)\

\\

\

.. ".,,24 32

diometer s D,

F (T'=F (T'=F CT;--F (T =F (T?F (T'--

\\

\4O

m/c tons

(c) Pe - o.ool

F7gure 5.- Concluded.

O)12)22)32)42)52)

48 56

62 NACA TN 2738

/.O

\

.8

0O !0 20

Drop diomefers Ds

\

3Omicrons

4O

(o) P e = 0.1

Figure 6.- Gonsfont-temperoture contoursequiprob ability surfoce for 0 ose 4.

clouds t Plofeou ores),

de fining(Oumulus

the

NACA TN 2738 63

1.4

\

I

t_

o_

_J

1.2

1.0

.8

.6

.4

.2

00

\

\

I0

\

\//

- 3B°to20 (T:= OtolB)I0 ° ; (T = 2B)

0 ° F (T_32)

/--- -- I0 ° ( T_ 42)

/--_oo_ _.-_

\

\\%

20

Drop diometer t

30 40 50Dt microns

(b) Pe = 0.01

Figure 6.- Oont/nued.

64 NACA TN 2738

/.8

1.4 -_\ \ _-32°to20 ° F (T'=Otol2)I0 ° F _T',,22)

T-_-- (T'=42)

_" \/ er --eej

-- i"4 .6

-'-'. \\

._ \ \__

I0 _0 30 40 50

\

6O

Drop diometer, D, microns

(c) Pe - O.OOl

Figure 6. - Concluded.

91 NACA TN 2738 65

.6

,f

q_

bI

.5

.4

.3

.2

,/

00

n

- s2. (r_--o)i _--25o F (r --?)

\ \_ :--to° (r:-/e)_ is* _"iT'-iT)

),?\

16 24

dlomorer, Da

8

Drop microns

fa) Pe - oJ

Figure 7.- Gonstont--femperoture contours defining the

equiprobobility surface for Gose 5. (Layer clouds aEastern United States).

66 NACA TN 2738

.9

.8

.7

\--...__'.:o,

25 ° (T - 7)20" F (T'-- 12)

I0 ° F (T'(T'-22)

5 ° F (T'27)

F- - 5 ° r ( 7"-30

I/_/ ; -/0° F (T=42)° \o_ \___. .4

°_

.2

00

.I

-\ ,,_,\\'\\\\8 16 24 32

Drop diameter, D, microns

(b) Pe - 0.01

4O

Figure Z - Continued.

NACA _N 2738 67

L2

/.o "X

\.9

_ .8 _

_ .5

!

.3

.2

,I

0

, 32" F (r'- O)' \_\ _5"::r-'T:'/-- /5 ° F (T'-- /7) --

\ \\\///,-,o. ,/: 5: ::r--2_:F (T'= 27)

-_ \\-,,,\\,\\\'. ,,, ,",,I I8 16 24 32 40 48 56

Drop diameter, D, microns

¢c) P,-o.oo/

Figure Z- Concluded.

O_

NAGA TN 1855

distonce

Stondord

Design I Design Design

vo/ue _

.2.I I /0 I00 I000

Cloud horizontol extent, miles

Figure 8.-Voriotion, with horizontol extent, of the foctor by which volues of Iiquid-wotercontent from the equiprobobility chorts should be multiplied to include the effect

of horizonfol cloud extent for different exceedonce probobilities.

0

CO

J

_ Dcr , below which an inh'nite amount of liquid water is tolerable

because the element collection-efficiency is zero

Pi is the probability applicable to all points lying in a

\ prism above .the marginal surface

surface as determined by design

of the thermal-system element

P =E_ Pi is the probability applicable to all points

falling anywhere above this surface

As D approaches oo, o

progressively smaller amount

of liquid water is tolerable

FO

_oCo

Figure 9,- Marginal surface for

some temperature depression T_r,.\

_o Iiquid water is tolerable because heat

in the thermal element will bring the

surface temperature just up to 32°F.

in clear air

a particular element of on assumed thermal system.

o__o

70 NACA TN 2738

O

Liquid-water content W1, grams per cubic meter

•2 .4 .6 .8 /. 0 I. 2 1.4

I0 -/

li

(a) Case I-Layer clouds, Pacific Coast.

Figure I0.-Partial probability charts.

_CA TN 2738 71

i

/0-4

0L/qu/d-wofer content Wi, groins per cubic meter

.5 /.0 /.5 2.0 2.5 3.0

(b) Cose 2-Cumulus c/ouds, Pocific Goost.

3.5

Figure I0. -- Gont/nued.

72 NACA TN 2738

O

Liquid-water conteM Wi, grams per cubic meter

•2 .4 .6 B 1.0 I. 2 1.4

i0__

over 49.5

(c) Case 3-Laver clouds, Plateau Region.

Figure IO.--Oontinued.

NACA TN 2738 73

OLiquid-wafer content W1, grams per cubic meter

.4 .8 1.2 1.6 2.0 2.4 2.8

IO -!

Horizontal exfent_ miles

(d) Case 4- Cumulus clouds, Plateau Region.

Figure I0,-- Continued.

74 NACA TN 2738

O

Liquid-wafer content

.2 .4 .6Wt, grams per cubic meter

.8 1.0 L2 1.4

I0-/

microns

(e) Case 5 - Layer

Figure

clouds, Eastern United

I0.- Concluded.

States.

Lf_b--

CO0"3b-OJ

•spnolo snlnwno 4soo3 oWODd Joj lJawnooua 6u/o! dad uo!t!puoo 6u./o! a_aAas

isow aq/ jo 6u./zaa,_j Mo/aq uo/ssaddap adn/odadu./a_ aq/ jo uoNnq.u_s!p %ouanbadj- 7/ a,/n6!__

Du/zaaq

B'66666IIIIIIIIII II IIIIIIHMI............ II......

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IIIIHI

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t[ I IIIIl(pl I

.........,,,, !!!!!!!IIIIrll

i',il:lii',l::I!!',I',',",,!!r::,,IIIrllll:llll rrr,,,,,,,IIIllllrllllt 1111111111

__i_;ill'l'l'l'

!l!!'rHILII [111111]11Illl,1_, :1ill

!!!!"_'_'II',"'!!'..11111I(1111 III IN....... , ....... INI

ii111111111!!!!III/IIIIH

',I',,,,,,_,.......ii lililllllllllIIIIIiiiiiiii[iiiiHiiii!!!11111111!!:,,I,,............... 111111111

IIIIIIHillll j

III I Jsrl [1111111

J I I_'l I I I III

111111111 III IIIIIIIII

:,,i,,,,,,,::!!!,,,,,,,,,,I I LIIII I IIIIIII.................... II

i00" _00" .qO0"

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Molaq uo!ssa.ldap adnlo./adu./al aq4 qo!qM u! s./a/unooua do luao./ad

_'66 66 06 #6 OG OZ Og 0£ OI S I #" I" I0"0

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" "......." II'* ,,," .....i,ii,,;i;;i,"_'ii', I'" ........:.... i '''''_''' ..... ,,,, ............. 'r"iiiiiiiiii[iI=i" !!!rIII[II IIII( lllilll

I I I I I llllllll;llIllll III IIIlllIII IIII I I I IJPFII I IIII

I I II IIII Illllllllllll:lllllllllll I .11I I lllllllll I ..... I I I I I I III II, II IIllllllIIIIIIIIIIiIIIIIIIII II

!!! lllllll ........ 111iii II IlIl'illl .......................... Iiil..... iiII_rIl l ,IIlH l,Illl: I l I l II lIl'[Jrl lIII l II lll_i l I I ..... : l,l I I I Ilr II llllllll;I ,Ill ,, II, I l l(llll __

......."...... ! ;1111 III IHII •II1' ...... iii!........!!!'_'_''_'...... ',I'"'" ....'""'_"_':'_"

.... Oil,t1111111 Iltlllllillllll r IIIlllIll

! ,,,,..................... III...."....................' '"'""iiiiii......iii1111111,4'111111 III I II'III ........ Li'_4lll I I I I I lllll Illll I I lllllllllllillrlll I lllll,,,_,,,,,,................IIIIII

i11!'"'"............. !!! !!lllltllrl!i rl

:'"' ........ rHII!! ii 1!111!1!! I" 11!!!IIill !!: '............ 1111111 _I IIIIIII I I llllllllll[illlllllll I IIlllll

"1'_'1" IIiitl II_ll I l,li,iiil, Ii = IiIlllllll ......=III II '1ii.......Illillllllll IllllII l' tl " IlllIii II= II i '1II i IIIIi II= : III' i i J =I I ' i FI IIIll IIlll !,, ! IIIlll=l' Ill illllllilillll'lllII=llIIIIIII_II'IIIIP'.................lililtlllilllllll ii!!1111I......IIIII Illilllllql itPl_ _°

Illllll II ............ II IIIIIIIIII.... Ill'Illl ll_l_11_,llllllllll fill,,,,,,,,,,,,,.................NIII,,, ,,'I

II"'"' ..... EIII'"'"" ................. 11!1! !1_''*''''' ...................... Iiiii1'"'"'111,,,,,,, ,,,,,,,,,.........iirll',','",,, ,,,, Irll'"'"'_'".'_,,,,,,,,,,,,,,,,,,,,,,,",_,,,,,I ..,.iII III II O_

l_lltlll Ill IIIII IIII'Ilil I I I IIIII I I

IIIlllllllfl IIIIIII111

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lO" CO" _0" Ol" 0_" 0_" OZ" 06" 66" 09

SSalUO!Sua_/_ "O "_Pl!qaqo_d

76 NACA TN 2738

Probobility, R, d/rnens_nless

.99 .90 70 .50 .30 JO .05 .02 .01 .005

.............. " kiLIIL" "2.0 IIIII',',I11[Irllllllllllilllllllll.......... llflllll 111li liI 11I I I [II Ill II 1Ill I'IllIII]lII1lll,ll l{[lltllllllllllll Illl ll}

I;'I'"'1"iIi i'' fILl!L!II II --11, III lili',""lll I 1I III ,III[III111 II!III1111I L,,k_1111

I .J'11II IllllllIllllll III lIIIHI{IIlIIIIIII I I I IIlllillllllllllllll I I ' II 'L [ II II III I,l . ......... ..... I

tllllllll!llllllll III I I I I I I II III IIII III I_1HIII I I 1.1,,_11 I,,,,,,,I ,, ,,, '"' i'" 'I ,, IIIIIIIIIHIIIIIIIII', ,

I I I II'[I

,,,,,I,"H1,,,,,_,,,,,,,I,,,,,,,I,,,,I i'", '_' '"_ J [11','''''''''_'''' L,15 I11111111;1111111111111111 I II Ill Ilil I

, ,_rHk,,,K, I IIIIj1111'111111111111 I I I I I I I 1 IILJ"_II I I I I Illl 1- ,, ,,I,1,,,,,,I,,,,_,1,,,,,,,I,, ,, , , I I ?-'.._. I_,_1,1,,,,I 1,1III II I I ..... I I I I £t"F I IIIIIIIlil I I

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• _ " Illlj[llll II I __

11111111 _ IIIIIIilil II II I --_, _;. IIIIIIIIIII I _

i II J,¢- II I I IIIII II_.I II I,,;.,_,,t.i, rT+ ,,I,L,II I_,.,,,, ,,1_ I._ /.0 , _ ,,,,,,,lu,lj,,lll,,,,,_,, _, ,_,,,"III I I I I I I I ill II I I L.L,@ Wf_ f'_" I I i i II I

--. I' lill I 11111 IIL I,J,l_,#,_t lIlt II

I IIIIltlllllll _ F IIII II I I1,11L_rll 11,111111 .11 U.qiL II III I I _ I I t ]1 i i

I IIIII _ IIIIIIIIIIIIl IIII II I

U,_ ll_tllilllllllllllllilllll II I I I I ,_r IIIlllllll ] I_ll II

J_FI jILl IIIIII1_ II1_1 1111111111111 I __L,_ I-IT] IIII1111111 IIII II I [-

_ IIIIIIII I IIIIIIIIIIIIA_F 11111111111 I • "1 I IIIIIIlill nun nL._ I. i Illllllllll i Illi II I _1

__ IIIU4_II I I Iltlilkl'l 1111171 t l 11j1111'111111111 ?-_--II I I IJl,t"lll I I I I IIIIIIIIIiIIIIIIIIItl I il I _ I I I I I IIIII I I I I I IIIII I I I • _. :

_:"_o0 ' i'l Ill111',':_,I',I_ iiIII_1111 ' ' IIiiiiii !L_T"!!/I II

.... iiilllli III I I',111 I_ " : I IIItllll . ,

p, /5

.01 J _ I 5 I0 30 50 70 90 95 98 99 9_.5

Percent of/bin_ encounters with moximum liquid woter less thon W

R'gure 12,-Frequency disfribuf/bns of moximum Iiquid-woter content per

icing encounter for two volues of moximum femperoture forPocific Coosf cumulus clouds.

NACA TN 2738 77

L6

1.4 _

"_ L2

_" .O

0 0u C) _

percentile, WOO5

= \._ n _ Mode, W063 \"_ _ _ C_,,_' \

-_ .z "n..

0 0 I0 20 30 40 50 60 70

Temperoture depression below freezing, 7"; E °

8O

Figure 13.-Distribution parameters which define the probolx'iity

function R(Wj T') for Pocific Coosf cumulus clouds.

.99940 _

l

Probob/_'ty, S, dimensionless•99 .98 .95 .90 .80 .70 .60.50 .40.30 J_O .10 .05 .OZ .01.005 .001 .0001

V

20

t° I0

0.1 .5 I 5 I0 30 50 70 90 95 98 99 99.9 99.99

Percent of observations wifh drop diomefer less fhon D

Figure 14.-Frequency distribution of drop o_'omefer for observofionsin which liquid-woter content is of leost 0.5 groins per cubic meter

for Pocific Coosf cumulus clouds; T'_ O.

c__>

ro

L.oOo

12T- NACA TN 2738 81

.9

.8

_.5

/

5 th percentile, WO.05

\

' "_'_ Mode, W0.63

.I _

020 30 40 50 60 70 80

Drop diometer, Do microns

Figure 17.--Porometers defining the distribution of liquidwofer content for volues of drop diometer greofer thonD for Pocific Coost cumulus clouds; T' __. O.

82 NACA _ 2738

LO

From table 3ZZZT

From fable 2_

- - -Faired

I

t= 0

I0 20 30 40 50 60 70

Drop diameter, Ds microns

Figure 18.- Equiprobability contours constructed

from the data of tables 1771T and _

Pe = O.OL

0

.9OlillHIIIIIIflIIIllllilFIil I

!!!!!!!

i]iiiiIIIII Illl!l I

!!!!!!

Liii_

!:i!!!!!i!!

!i!!!!i;iiii_[i!: ::: :

i i,:

ii!J :::

i!_!:!!!!,Ii1',Iiiii

iii_!!i .... i,::;:]1],1

rll!l ili_,.,.-_,_/"[ I I I

_!!!! i

llllli

Probobilify , S', dimensionless

.70 .50 .30 .10 .05!!!!!!iiiiii_lltll

iiiiii!!!!!!

!!!!!!

_i_LL111T_

!!ii',i

!!!!!!iiiiiiI I_UllIll]IF1

IIlilllIt111!1I III I ] ,G,

.qff! ! ! !

iiiiiiiiiiii

.02 .O1 .005

iiil;i

!]!i!!iiiiii

frequency of maximum diometer

,,!,,, I greofer thon 50 microns IL Ililll;iiiiii (tOO Iorge for relioble meosurement w#h mul#cy/inders) i

r!!!!! Illllllllll;lllllllll]ll]l I I I I I I I llll, r ; r i i iiiiiiiii i iiiiiiFiiii[ii i iliillii i-]

;;;;;i ' _ r I' _ I..... r ll!!llZl!l;lll]llllllllll i I I I ] I r II',lll ! I i I IIIII III I I Illll!llll] i r i i iiiilll i ] i l

.01 .I .51 5 I0 30 50 70 90 95 98 99 9_.5

Cumulotive frequency_ percent

Figure 15- Frequency distribution of moximum volue of meon-effec#ve

drop diometer observed in eoch icing encounter for Pocific Coosfcumulus clouds, T' __ O.

C3

DO

UJCO

-q_0

80 NACA TN 2738

Probobility, R; dimensionless

.99 .90 .70 .50 .30 .10 .05 .02 .01 .005 .002 .0011.6

-i t.4; .... ' ' II',_'F' ,_,, i ,,I,I,I

_ : Ill,,,,,,_H-H,,,,,,0 5 I0 30 50 70 90 95 98 99 99.5 99.8 9:_.9

Cumulative frequency, percent

Figure 16.--Frequency distribution of liquid-woter content observed withvolues of drop diometer greoter fhon 20, 30 ond 44 microns;

T'___ O; Pocific Coost cumulus clouds.

.9999 .999/I_ ¸ I IIIIIII

I IIIINIt I Illllli I Ikllll

III .....I]11(I t IlllllI I 111111

iii .....I Ilill

IliliIII1!

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I IIIII

_2 ......I I t11111

II ......IIIIII

II ......I IIIII

II ......e_/_ IIIIII

o_ I I IlllllI I IIIIII

I IIIIHIIIIIIIi I IIII1!

i i IIIIIII I Illltl

l',.,",,,iIf" ,,rill

i J iPlllJI IIIIII

i I III111I Illlll

22 ........I I I!IIIII lillll

• i I I IIINIIJIIHI1111111

12 i! IIII!IIi_ i iiiiir

I IltlllI IIII!1

I IIIIIII

I JliJill

=']i!"'i I1|I[ illll III

III .....IllliI _ IIIlll

0.01 Od

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III ,,....,, II TM,,,

lllll IIII

II .........iiitl Iiiiii

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Iltll IIII

II ..... I ....IIIII I111

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II ..... I ....I!111 IIIII I IIIII I IIII

II ..... I ....IIIII Itll

II ..... I ....IIIII IIII

II ..... I ....IIIII II11I Illlllillil

II ..... I""Illll PillI IIitll lilll

I ...... 1....I I IIII llll

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I]I,,,,.........,,,,,III ...... ll--Illl III I I I U_I._ ii

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0.5 /

Probobility, O, dimensionless.95.90 .70 .50 .30 dO .05

',"III ................................................. IIII',I II I IIIIIIIIII I I I I I IIIIIIIIIIIIIIIIIII I I i i rllllllllll

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I IIIII .......... II ................. I"I .................. III I IIIIIllllll IIIIIIIIIIIIIIIII II IIIII IIIIIIIllllllI I IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIlllllllllllllllllllilllll I I I

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":""'":'",,'"'"!"i'"".............,,""=............."""'' = ,,,....""'IIIIIIIII f III IIIIIIIIIIIIIIIII,., I I ,, ,_.,,,,,+.,,,,,'"IIIIII ,,,111i1III

IIII ; I I I . ,,t1111............ III I I I I I _NI I I I.................. I I I I i I I I I I ', Illlll.......... Ill, II I ' 'IiI! Ill,.... " ,,,,lillI ' II I I I t I I I IFIII!II I I I I I.L.ll_"ll I I III I I I I I I I I t I I I I I Illltllll I t I I i I IIIlillil!l

I I t I I I i_L./...f'l']llll

-..--_,,_rl] ......... III ............... I"IIIII ........... llll'll I "' III I r I IIIIIIIIIIIi IllIIIIIIIIIIII II IIIII IIIIII IIIII IIIIII

III,,,,,,,.+,,,.III,,,+,,.,,,,,,,,,.I"IIIIII,, .,,,,.,,,,IIIIIII ...........

III,,.,,,,.,,,,,,II'"I,., ,,.,,,,.,,,,.,,,ll',lll,,,,.,,,,ll]l'rl 1

I I I IIl[rlllllllllllillllllll!llllllllll I I I i I IIIIIII1(I III I !: I i

5 I0 30 50 70 90

Percen# of encounters in which

below freezing is

.00/ .00_II I"'lll ]IIIII

II ..... IIIlilllI I IIIIIIi iI i IIIIii I i /

IIIIII I

II ........illll i_il- i

I I IIIII/I I

II" Jlllll i

Lrl._ IIIIII iIIIIII I i

iHilll

i .......i. I IIJIIIII I ....... ,i IIIIII i

II ....... IIIIIII II I lililli I

f. ,,,,,., ]r[lll i _rl

II ,,.,I1[III ....... IIIIIII I

II """' IIIIIII II I IIIIIII i

I ....... !, I ...... I|11111 I

I ] ...... I Iillll II I IIIIIII II I lillill I

II INI"',i Hi!!il,,I i IIIll I

...... [ I

I lillll I i

II I111111III ...... I]IIIIII

II ..... II!illliI I lillll I

i mllji iIIII

I i,,,,I, iI[ ..... IIIIIHIJ I IIIII I I I

"'II ................II! ltllll I I IIIIIII I!! IIII I I IIIIII I.,,,, ........

II ....|111II III11 .... -v-

iiii_;iil[ I I I ,.,,IH--F-

99 99.9 99.99

the #emperoture depression

less thon T'

Figure _.-Frequency distribution of the #emperoture depression below

freezing of the most severe icing condi#ion per icing

encounfer for Eos#ern U. S. layer clouds.

C3

_0

tJOO0

O0OD

8_ NACA TN 2738

L,2 L4

LO L2 " '

.6 .8

__ .4.6

I .,2 .4

.4 0.2

Probab/litx, /7, dimensionless

.99 .90 .70 .50 .50 .10 .05 .02 .01 .005 .002

IlllIIIlll ' "'"'................. ii!!! .............'..................=,,;,, .,,,,,,,i..........iIIIIIII................ ,t1,,., 111'"11.,,..... 111111.......................... I,II II.GNIiil1!! ill ...........I NtlIIIIIIIII I I IIIIIIII 11111111111...................................I;I;1111IIII ,,,,_,,,=,,,,,,,I,.,,,,.,,'"'............ Ill '11 ' .... 111111 , .,,,,,,,,,,,,,, ._,,,,, '11 11111.... 1

LI ........... iI_11111 [I I 1111111111 IIIIIIIIItlIII IIII11111111.......................... I111' I, 1 .........

.............................. 11111111,,,,i...............,,,i,,,, 11111111.... ,,,,,,,,,,,'.......... I"I,_ .,,_,,,H"_'IIIIIIII I I It IIIIIILIIIIIIIII III I I I I I I I

NIl...... "F', 4-I_+t-tddd_5½tfi_ H-t-Ft/tHt41H+t+Hd4444I+i+I-V ......_ I I I IIIIIIII I I IIIIIIl111111 I I IIIIIIII I I I 111111bi_FIIIIL.l,t_ll I II

* //" II IIIIII I 1111111111111 _ I IIIIt111 I I IIII1_1111111111111 II I III/11 III I 111111111111111 IIIIIIII............. /111 N .................. ill _..,,,_,.......

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IIIII I I I I IIIIIIII I I II IIIIIIIII I_111111111 I II IIIIIIIII

ii

IIIIIIII I I I I_"si IIII I I I I I I I_ I i

LIILIIIII .................. III ................... ,111. I .,,,,,_,.........• III I I 111111111 I IIIIIIIII_,'FII I I 1111111111 I

IIIIIilljl

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It I I I I I IIIL.I.'_GII I I III I I t1111 I I I I I I i ........... IIIIIIIIII.iiiili_ ..... IIIIIIII ..... _, .......... ,, _,,,,_

111111111111111 IIII II I 1111 I I lli/'_'l I I I I I IIIIIIIII I I I I I I II _ I_ IIIItlllll.,,,,e _. #,,i III ...... _IIIIII , I_111/11',I I. ' "' .......i _d,_T IIII I 111111111111 III I I -I'_T

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............I, _ _ Itll, I , t11111 I III

I IIIIIIIII

III I I i111111 I I I IIIIIIIIIIII I I I IIIIII I L I I I...................................... IiiI'111111IIIII I 1 IIIIIIII1_11 I I I 1111111111 I,,11i................................. ,,,111111,!

......................... 11NIIIIIIIII ...................................... _ ,,,,,,,,,IIL,i,_.Dt , i i iiiii li i i i IIIIIliilil I I i i illllll I I I I

. i i i IIIIIII I I I IIIIIIII I I IIIIIIIIIIIII I I IIIit111 I I I I 111111111

IIIIIIIII IIIIIIIIIII I I IIIIili111111 I / Illllllll IIIIII IIIII I I IIIIIIIIIII I I I I I_11111 I I I I IIIIIIIII

i IIII I Iit11111ti IIIIIIIIII I IIIItll i..........Illi11111 !!i ii i i ! Iiiiiiiiiiii I I I IIIIIIII I I I

i i I I IIIIIIII ] ' IIIIIIII IIII I I IIIIIIII I I I

Illllllll illltll1111 I I I I IIIIit I I I I I

IIIIIII IIIil11111 I IIIIIIII r II............................ IIII ....... I

iiiiiiiii III111lll 11111111111 I I I I IIIIII I1111111................................ ,, ,,, ,_,,, , f111

/ lO 30 50 70 90 95 98 99 9_.5 998

Percent of encounters with maximum //quid water contentless than W

Figure 20.--Frequency distribution of the maximum liquid-water

content observed during each icing encounter with T" equalto or greater than the indicated value.

NACA TN 2738 85

da/au./ o!qno .#ad su./o./# _ "/ua,_uoo./a,_M-p./nb!7

COo'3b--

"0 _,l :sa4o/s pal!u/? uJalso3 u! spnolo _a,_Ol

f_alaw o!qno _ad sword _'0 uoq/ _a/oa_6 _o o/ Ionba lua/uoo _aloM-p!nb!l 4o sanloAol Du!puodsaJ_oo _a_a_uo!p do_p aA#oajja-uoaw aq_ jo uo!_nq.u_s!p ,¢ouanba_,J-'_ ax_!.,-/

g6"66 9"66O unql ssal ,la4awa!p q/./M SUO!4DA,lasqo lu_o_ad

g'66 66 86 g6 06 OZ Og

gO0" I0" _0" gO" 01" 0£" Og"

SSalUO!SUau#p"S "_/!l!qoqo_d

gO00" I00" RO0"

0£ Ol /0"g

OZ 06"

I I"

m O/

0£

_£

I_ACA TN 2738 87

4O.99 .80

Probobility, ,9,' dimensionless

.7_ .50 .30 .10 D5 .O2 .01 .005

35

15

.0/./ / I0 3o 5o 7o 9o 95 98 99 99.5Percent of icing encounters with maximum observed value of drop

diameter less than D

Figure 23.-Frequency distribution of the. Iorgest value of drop diometer

observed during each icing encounter i layer clouds, Eostern UnitedStores.

88 NACA TN 2738

.5O

.40

.3O()

/

(

015 17 19 21 23 25

/-W0.63

I

I27 29 31 33 35

Drop diomefer, D, microns

Figure 24.-Oisfribufion poromefers defining fhe funcfion R'(W,D)

for the cose of T'___ O; Ioyer clouds s EosfernUnited Sfotes.

00

• ,,_ I I I I

Drop diometer, micronsI I

17k------

0

rO--4L_oOO

00 I0 20 30 40 50 60 70 80

Temperoture depression below freezing t Tt' Fohrenhe/f degrees

F/gure.25.--Disfribufion porometers def/hing the function R'(W,D,T') ; Ioyer clouds, EosfernUnited Stores.

Oo_)

5O

Probobflits, S," dimensionless

.90 .70 .50 .30 .10 .05 .02 .01 .005

Distribution of greotest observed volue of i

drop diometer in eoch icing encounter

510 30 50 70 90 95 98 99 995

Percent of coses hoving drop diomefer less thon D

.002 .001 .0005

99.8 99.9 99.95

Figure 26.--Composite frequency distribution of drop diometer used in colculofing port�o� probob#ities forlover clouds in Eostern United Stores.

C3_>

PO

_0CO

NACA TN 2738 91

-4,0

Log PAT + Log Pwi(AT)

-3.0 -2.0 -I.0

Liquid- water content distributioncurve for the temperatureinterval 32 to 20.5 °F \

/

//

//

.

/Log P_T

00

.4

.8C3

zo!

1,2 "_

1,4

(a} Step I

Figure 2F.- Example construction of o partial

probability chart.

92 NACA TN 2738

Liquid- water

0 .2

(b) Step 2

content, Wi, grams per cubic

.4 .6 .8

extent

Liquid- wafer content distributioncurve for the temperature

Log P_T + Log

meter

1.00

.2

.4

.6

.8

1.4

_b

Ca

C_

I

-4

Figure 27.- Continued.

14 _ NAOATN 2738 93

Liquid- wofer

0

0

-I.0

-2.O

-3.O

-4.O

content t

.25

_ Horizontal extent lines_\ I I I I /\ "_o _,,,, I/_ N Log. reference

\ _. _o,.,_

Wi, grams per cubic meter

.50 .75 LO0

/#oe

\

I _ \ "I _T+

content distribution / PwI(_TJ

7,temperature \

interval 32 /tO 20.5 °F--/i

11

_Log_o " I //

-2.0

// _l_- Scole transformation-3.0/ /_ reference line

//_ /_:%oo'- I I ,Drop- diameter interval

line for 0 to 95 microns' 9

(c) Step 3

Figure 27. -- Concluded.

',,,I

NACA-Langley - 7-1"/-52 - 1000