NCAR-TN/STR-65NCAR TECHNICAL NOTE

July 1971

Revised February 1974

Numerical Prediction of thePerformance of High Altitude Balloons

Frank KreithProfessor of Engineering, University of ColoradoandConsultant, National Center for Atmospheric Research

Jan F. KreiderResearch Associate, University of ColoradoandConsultant, Environmental Consulting Services, Inc.

ATMOSPHERIC TECHNOLOGY DIVISION

NATIONAL CENTER FOR ATMOSPHERIC RESEARCHBOULDER, COLORADO

iii

PREFACE

Originally issued in 1971, this report was prepared to assist in

predicting the vertical motion of high altitude balloon systems. In

anticipation of the increasing use of balloons as an energetically

efficient means of transporting instrumentation or commercial payloads,

it seemed desirable to enlarge the report to include a calibrated model

for predicting the vertical trajectory of zero-pressure balloons. A

number of errors in the 1971 edition have been corrected in this revi-

sion. The additions in the revised report to the dynamic and thermo-

dynamic analysis contained in the original edition provide the balloon

engineer an efficient, calibrated computer program to predict balloon

ascent rates, ceiling, and behavior at float.

We would like to thank A. L. Morris, J. H. Smalley, and K. H.

Stephan of NCAR for their help and encouragement in the preparation of

the original document.

Frank Kreith

Jan F. Kreider

November 1973

v

CONTENTS

Preface . . . . . . . . . . . . . . . . . . . . . . . . ... a iii

List of Figures .......................... vii

List of Tables . ..... . . . . . . . . .o . ix

Nomenclature . . . o . o . . o . . . . . . xi

Credits . . . . . . . . ... . o . e . . .. . .o . . xiii

I. INTRODUCTION . . . . . . . . . . . . . . . .. . . .. . . 1

II. THE EQUATIONS OF VERTICAL MOTION ............. 3

III. CONVECTION BETWEEN THE ATMOSPHERE AND THEBALLOON SYSTEM . ....... . .......... 13

IV. CONVECTION INSIDE THE BALLOON .......... o 23

V. RADIATION HEAT TRANSFER .... . ... ...... e 25

VI. EMITTED RADIATION . . .................. 33

VII. DIRECT SOLAR RADIATION ............... o 37

VIII. REFLECTED SOLAR RADIATION ................ 43

IX. INFRARED RADIATION FROM THE EARTH AND THEATMOSPHERE ............ .. ...... . 55

X. EXPERIMENTAL RESULTS ... . . . . . . . .. 69

Appendix .... . . . . . . . . . . . . . .. . . . . . . 77

References ........ . . . . . . . . . . . . . . 125

vii

FIGURES

1. Schematic illustrating force balance for verticalballoon motion ....... .......................... . 4

2. Temperature variation in the atmosphere up to 100 km .... 7

3. Schematic illustrating energy balance for lifting gas

and balloon skin ................. .. 10

4. Local free convection heat transfer over a sphere--

comparison of theory and experiments ............ 14

5. Average Nusselt number for sphere in combined free and

forced convective flow ................... 19

6. Temperature field about a heated horizontal flat plate

at a Rayleigh number of 50 ................. 21

7. Emission of radiation from a balloon skin--

the effective emittance .................. 34

8. Spectral distribution of solar radiation in space

and at sea level ................ ...... 38

9. Spectral distribution of solar radiation incident at

sea level for air masses 1.0 to 8.0 ... 39

10. Albedo as a function of latitude under various

sky conditions ......... ... 45

11. Shape factor for a small sphere and rectangular area .... 48

12. Satellite- or balloon-to-earth geometric configuration . . . 51

13. Variation of directional solar reflectance with

zenith angle (60° < C < 80°) .. ....... .... 52

14. Albedo map for the northern hemisphere during theperiod 16-28 July 1966 ................... 53

15. Variation of directional hemispherical reflectance

with zenith angle ........... . . - 54

16. Infrared radiation map for the northern hemisphere

during the period 16-28 July 1966 ...... . ... 56

17. Upward and downward radiation flux as a function

of altutude .................. *e* e 57

viii

18. Suomi-Kuhn flat plate radiometer ... ......... 61

19. Radiation environment at Green Bay, Wisconsin,in the summer . . .· . , . .... . .. . . . 63

20. Radiation environment at Green Bay, Wisconsin,in the winter ....... , 64

21. Radiation environment at a desert island, 20S . . ., 65

22. Gergen "Black Ball" radiometer ... . ...... . 66

23. Vertical trajectory of "Thermistor" Flight.October, 1964, 250,000 ft 3 polyethylene balloon,He lift gas. Solid line - model prediction; dashedline - data. ...... .... 72

24. Vertical trajectory of Stratoscope S4-2 Flight.July, 1965, 5.5 million ft 3 Mylar balloon, He liftgas. Solid line - model prediction; dashed line -data . .......................... 73

25. Vertical trajectory of Stargazer Flight. December,1962, 3.1 million ft Mylar balloon, He lift gas.Solid line - model prediction; dashed line - data. .... 74

26. Vertical trajectory of Special Hydrogen Flight.July, 1970, 1.5 million ft 3 polyethylene balloon,H2 lift gas. Solid line - model prediction; dashedline - data. .................. .... 75

ix

TABLES

1. Equivalent length dimensions of convection correlation .... 17

2. Average Nusselt numbers for a horizontal plate in freeconvection . . . . . . . . . . . . . . . . . . . . . . . . . . 21

3. Constants for the free convection equation (Eq. 27) ..... 22

4. Constants for free convection in a sphere (Eq. 29) ...... 24

5. Radiation characteristics of surfaces ... .. 30

6. Energy storage capabilities of water and batteries .. . 31

7. Relative spectral distribution of solar radiation under

various sky conditions ........... . ..... 44

8. Mean values of thickness, reflectance, and absorptancefor various cloud types ..... . .... .... . 49

9. Radiation environment for superpressure balloons ....... 59

xi

NOMENCLATURE

A -area

a -absorptance

b - thickness of fabric

CD - drag coefficient

CM - virtual displacement coefficient

c - specific heat at constant volumeV

D -diameter

E exhaust rate of balloon gas through expulsion duct and valve

e - emittance

F -force

G -radiation incident on surface

g -gravitational constant

h -heat transfer coefficient

I - intensity of radiation

LF - free lift

LG - gross lift (= p V - m )

M -molecular weight

m -mass

p -pressure

q -rate of heat transfer

R - universal gas constant

r -reflectance or radial distance

S - surface area

T - temperature

t - time

U -velocity

V -volume

v - specific volume

z -altitude or geopotential height

xii

- zenith angle

y - lapse rate of atmosphere

X- wavelength of radiation

e - polar angle or super temperature

T - superpressure

p - mass density

a - Stefan-Boltzmann constant

T - transmittance

- azimuth angle

Subscripts

a - atmospheric air

B - ballast

b - balloon system

c - convection

D - drag

d - ducting

f - balloon skin fabric

g - balloon gas

i - infrared

o - zero altitude

p - payload

s - solar

v - valve

Dimensionless Numbers

Gr - Grashof Number

Nu - Nusselt Number

Pr - Prandtl Number

Ra - Rayleigh Number

Re - Reynolds Number

Re* - Equivalent Reynolds Number

Sc - Schmidt Number

xiii

CREDITS

Below is a list of the figures used in this report which have been taken

from other publications.

Fig. 2 S. L. Valley, Ed.: Handbook of Geophysicsand Space Environments, McGraw-Hill,New York, N.Y. (Copyright 1966, usedwith permission of McGraw-Hill Book Co.)

Fig. 4 A. A. Krause and J. Schenk: Thermal freeconvection from a solid sphere. AppZ.Sci. Res. A-15. (Copyright 1965, usedwith permission of Martinus Nijhoff, N.V.)

Fig. 5 H. Borner: Heat and mass transfer of singlebodies by combined free and forced convec-tion. VDI-Forschungsheft 512. (Copyright1965, used with permission of VDI-Verlag,Dusseldorf.)

Fig. 6 F. J. Suriano and K.-T. Yang: Laminar freeconvection about vertical and horizontalplates at small and moderate Grashof num-

bers. Int. J. Heat Mass Trans. 11(3).

(Copyright 1968, used with permission of

Pergamon Press.)

Fig. 8 S. L. Valley, Ed.: Handbook of Geophysicsand Space Environments, McGraw-Hill,New York, N.Y. (Copyright 1966, usedwith permission of McGraw-Hill Book Co.)

Fig. 9 D. M. Gates: Spectral distribution of

solar radiation at the earth's surface.Science 151(3710). (Copyright 1968,

used with permission of American Asso-ciation for the Advancement of Science.)

Fig. 10 C. G. Goetzel, J. B. Rittenhouse, andJ. B. Singletary, Eds.: Space Mate-

rials Handbook, 2nd ed., TORML 64-40Air Force Materials Laboratory, Wright-

Patterson Air Force Base, Ohio, 1965.

Figs. E. Raschke: The radiation balance of

13,14 the earth-atmosphere system from radi-

15,16 ation measurements of the Nimbus IImeteorological satellite. NASA TND-4589, Washington, D.C., 1968.

xiv

Fig. 17 T. Sasamori: The radiative cooling cal-culations for application to generalcirculation experiments. J. AppZ.Meteorol. 7(5). (Copyright 1968, usedwith permission of American Meteoro-logical Society.)

Figs. Courtesy of Peter Kuhn, Atmospheric18,19 Physics and Chemistry Laboratory,20,21 Environmental Science Services

Administration, Boulder, Colo.

Fig. 22 J. L. Gergen: "Black-Ball": A devicefor measuring atmospheric infraredradiation. Rev. Sci. Instr. 27(7),1956.

1

I. INTRODUCTION

The vertical motion of balloon systems depends critically on the

heat transfer to and from the gas inside because the temperature and

the pressure of the gas determine the lift of the balloon. In the past

the thermal design of high altitude balloons has largely been based on

a combination of experience, empirical data, and approximate calcula-

tions [1,2]. Recent advances in heat transfer research make it fea-

sible, however, to calculate the temperature of the lifting gas as a

function of altitude and to predict the vertical motion of the balloon

system with the aid of high speed computers [3,4].

Any computer program designed to predict the vertical motion of a

balloon uses equations with coefficients whose values are derived from

theory or experiments. Such a program was developed several years ago

for the Office of Naval Research and has since been used by balloon

manufacturers and the National Center for Atmospheric Research. In

developing the currently available program, several of the heat transfer

and friction coefficients were adjusted to obtain agreement between the

vertical motion and the gas temperatures observed during the flights of

two balloons especially instrumented to obtain these data. Reasonably

good agreement between the computed and the measured performance was

obtained. However, in developing the program the values of several

empirical coefficients were altered simultaneously. Therefore, it is

not possible to discern which coefficients are generally applicable to

scientific ballooning and which are useful only for calculating the

motion of these two specific balloons. To improve the accuracy and

reliability of analytical predictions of balloon motion, the heat trans-

fer and friction parameters in the program should be compared with the

results of recent advances in heat transfer research and, if necessary,

revised.

There are three general problem areas in which thermal design can

improve balloon operation. The first involves rapid variations in

internal energy which occur when clouds cause sudden changes in the

radiation incident on the balloon. This type of change in the heat

2

transfer rate is usually sensed only after the altimeter registers a

change. After an altitude change is registered, altitude is maintained

by dropping ballast or exhausting gas. If one could control altitude

by reacting rapidly to a change in radiation, the loss of ballast or gas

could be reduced. The second area of thermal design is concerned with

changes in heat loss during a complete day. More accurate predictions

of heat loss and the use of appropriate materials or surface coatings

could increase the payload or the float altitude for a given balloon

system. The third area of thermal design involves long-range planning

for balloon flights, such as those currently envisioned in the equato-

rial zone, over new routes.

This report presents first the equations governing the vertical

motion of a balloon. It then reviews recent advances in heat transfer

experimentation and theory pertinent to the processes which affect the

calculation of balloon performance and the thermal design of balloon

instrument packages. Finally, the pertinent equations of fluid mechan-

ics, heat transfer, and thermodynamics are combined to predict analyti-

cally the vertical motion of balloons. The Appendix includes a complete

computer program predicting the vertical motion of balloons; this pro-

gram is based on the analysis given in the body of the report.

3

II. THE EQUATIONS OF VERTICAL MOTION

The total mass of a balloon system which must be accelerated during

ascent and descent, including a virtual mass term CMpaVg to account for

the air mass displaced by the motion of the balloons [5,6], is

Zm. = m + mf + mB + m + CMPaVg (1)i g Pg

where m is the mass of the lifting gas, mf the mass of the balloon fab-

ric, mB the ballast, and m the payload, and where m + mf + m + m = mbp g B p mb

the mass of the balloon system. The sum of the forces acting on the

balloon system (Fig. 1) is

Fi = (PaVg m - mf - mB -m p) - 1/2CDaIblUb (2)

The sum of the five terms in the parentheses times g is usually called

the "free lift," LF. The last term represents the aerodynamic drag

force, FD.

For a spherical shape in the Reynolds-number range of large bal-

loons, i.e., above 6 x 10 5 , the drag coefficient, CD, is equal to 0.1

[7], but no data for actual balloon shapes have been published. In

Ref. 8, a value of 0.3 for C was used as an average for computational

purposes.

In balloon terminology g(p Vb - m ) is called the "gross lift," LG.

Using the perfect gas law as the equation of state for the atmospheric

air and the balloon gas, LG is given by the expression

/M P T + \L /a a a_= gm M P + T (3)

where 0 = (T - T ) is the so-called gas superheat (although it is

actually a super temperature), and T = (P - Pa) is called the gassuperpressure.

superpressure.

dzVELOCITY = Ub = d

mf MASS OF FABRIC

'm--m mg - MASS OF GAS

CMP Vo v IVIRTUAL 1MASS OF BALLOONDUE TO ACCELERATION

mB + m - MASS OF PAYLOAD + BALLAST

7777r GROUNDE Fz = MASS x ACCELERATION

9 [(poV g- mm -m- m p)-CDPo UAb]

2

dt z(mg + mf+mB+mp+C V)o b o v oo oo

Fig. 1 Schematic illustrating force balance for vertical balloon motion.

5

The qualities 0 and T are usually quite small compared to T anda

P , respectively. The ratio of T/P is of the order of 0.0015 in zero-a a

pressure balloons, but may be as large as 0. 25 in superpressure balloons.

The ratio e/T depends on the heat transfer characteristics of thea

system, as discussed in detail in subsequent sections of this paper. By

expanding the fraction in Eq. (3) one can deduce the widely used approx-

imate relation

La g a 1 + a 7F (4)LO = gmg [al+ 8(-~ a)] (4)

G g -- M T-g g a a

Substituting Eqs. (1) and (2) into Newton's second law of motion

gives

F 1 d/2 Ca dt (m + mf + mB + m + C V) (5)LF - p1/2 CD a P - CMPaVpg dt2

In applying the above equations it is possible to relate the balloon

pressure to the height z through the hydrostatic balance and the equa-

tion of state of the atmosphere because there is generally only a very

slight superpressure in a balloon. In zero-pressure balloons this

superpressure is limited by the automatic exhaustion of gas through the

expulsion duct just before float altitude is reached. This gas release

keeps the balloon from bursting. Gas is also released through an

exhaust valve controlled from the ground--"valving"--to reduce the free

lift when the balloon ascends too rapidly and to initiate descent from

float altitude.

ATMOSPHERIC PROPERTY VARIATIONS

In hydrostatic equilibrium the variation of pressure in the atmo-

sphere with altitude is given by the relation

dP

a (6)dz =-gpa

6

If it is assumed that the atmospheric air obeys the perfect gas law,

the pressure as a function of altitude is expressed by

dP Ma a

P = g RT dz (7)a a

or

p ga gMa / dz

in p R / Tao a

zo

Over limited ranges in altitude the variation of the atmospheric air

temperature with altitude is often approximated by the linear relation

a = T + Y (z - z) (8)

where

ay, the lapse rate = d at z)

Figure 2 shows the temperature variation to 100 km as a function of

altitude according to various models of the atmosphere proposed during

the past 15 years [9].

If the atmospheric temperature profile is known (or taken for

purposes of calculation from Fig. 2), the pressure and density varia-

tions over an altitude change between zl and z 2 for which the lapse rate

is a constant can be written in the form

P l a T lgMa ++Ry

a a

2 \Ta 2 \aRPd p (2T) aRY

PRESSURE (mb) p

(D

21. 8l ...... ' - -.

rEOMETRICt -ATITU- (--

-v II ImIC

rtH,. m FJ- /H- -s 0o r,,

*-oiI//rt m

0

~C)~~

0r~r 0S\\/

0 2

0 3)~~~~~~DENSITY (g/m 3 )

8

BALLOON GAS EXPANSION

The variation of the gas volume in the balloon with time is

dV Rm T \ Rm dT RT dm Rm T dPg d g = g + g g _ gg g (10)

dt dt MP MP P dt M P d2 dt\gg g gg MP

gg

If the relation between atmospheric pressure and altitude from Eq. (7)

is used and superpressure is neglected, the above equation can be

written in the form

dVdV / dT dm gm T M d \m _g + T g+g d +dt P dt R dt RT (11

ga a

LIFT AND LOAD ADJUSTMENTS

As mentioned earlier, the lift of a balloon system can be reduced

by exhausting helium automatically through the gas expulsion duct when

float altitude is reached, or by valving to maintain float altitude,

to reduce the rate of ascent, or to cause the balloon to descend. If Ed

is the volumetric gas flow rate through the expulsion duct (required to

stabilize the balloon at ceiling) and E is the flow rate through the

exhaust valve, the net change in balloon mass due to loss of gas is

dm- =p E + p E (12)dt gd d d gv v

The mass of the balloon system can be reduced by dropping ballast

or part of the payload. This change in mass of the balloon system dur-

ing a time increment At = t2 - tl is

2 dmb 2 dm

- dt P

1 1

where Am represents the amount of mass dropped during the time inter-

val At.

9

THE ENERGY EQUATIONS

The conservation equation for the gas inside the skin of a balloon

can be obtained by applying the First Law of Thermodynamics (Fig. 3).

The rate of change of the internal energy of the gas equals the rate at

which heat is transferred to the gas (ql) minus both the rate at which

net work is done by the gas and the rate at which internal energy is

lost by valving. Assuming that the equation of state of the gas can be

approximated by the ideal gas law and that the temperature and pressure

of the gas inside the balloon are uniform, one obtains

RT dm dV dm

dt v g q1 M dt a dt vg dt

where the term on the left-hand side of Eq. (14) is the rate of change

in internal energy of the balloon gas, the first term on the right hand

represents the rate of heat transfer by free convection from the interior

surface of the balloon skin to the gas, the second term the flow work

done during gas release, the third term the work done on the external

pressure when the balloon volume changes because of a change in gas tem-

perature or because of valving, and the last term the loss of internal

energy by valving. If the pressure inside the balloon is uniform and

equal to the atmospheric pressure, the flow work done during valving in

the absence of heat transfer equals the work done by the atmosphere on

the balloon and the second and third terms cancel.

The assumption that the temperature of the gas inside the balloon

is uniform may lead to serious error when short-term transients occur

and the gas adjacent to the balloon skin undergoes temperature varia-

tions while the gas in the interior does not. When this situation pre-

vails, the first term in the energy equation should be replaced by

d--t Cv|c p T dV'(JJJ vggg g

g

A similar analysis can be performed on the balloon skin. The

internal energy of the skin will change only because of heat transfer

ENERGY EQUATION-SKIN ENERGY EQUATION -GAS

FLOW WORK dmg.( pv) ( Volving )dt t

I(( ^ SqSOLAR q 2 - -sKq, /I /j

/ (q CONVECTION

/; qA CONVECTION q4 X Vg C T

q RADIATION q5 q lNFRARED q 3 K COMPRESSION WORK = Pa dVdt

dt- (cpt)fdS, = q 2+q+q 4-q 5 -q,

~Sf~~ ~~ d~~~~dt t dVg. 3 S c ig e y b e for dtl (mT)9 gq- d (PV)g PbA dn.

Fig. 3 Schematic illustrating energy balance for lifting gas and balloon skin.

11

to and from its surfaces. The skin is so thin that one can neglect the

temperature drop across it at any location, but since the heat transfer

over the skin is not uniform its temperature will vary. This variation

cannot, however, be expressed analytically, and when it becomes impor-

tant only a numerical analysis can yield a satisfactory approximation of

the physical reality.

To simplify the thermodynamic analysis it will be assumed that an

averaged skin temperature can be used. A First Law analysis (Fig. 3)

then gives

d fdTfCfff qc+ d + q -q - q (15)dt cf PftfTfdSf = cfmf dt = q2 3 4 q (1 5 )

f

where c = specific heat of skin

mf = mass of skin

Tf = average skin temperature

Pf = density of the balloon skin

q = rate of absorption of solar radiation

2, direct 2, reflected)

q = rate of absorption of infrared radiation3

q = rate of heat transfer by convection from the4 atmosphere to the skin

q = rate of radiant heat transfer from the skin

tf = thickness of film

S = surface area of balloon

q = rate of heat transfer by convection from the1 skin to the lift gas

Equations (14) and (15) are the two energy equations which must be

combined with Eq. (5) to calculate the vertical position and the motion

of the balloon. The energy equations for the balloon gas and the skin

fabric can be treated analytically if one assumes that all of the gas

is at one temperature and the entire skin is at one temperature (these

two temperatures may or may not be identical). This assumption is quite

good at night but can introduce appreciable errors during the day when

solar energy heats the skin unevenly. Modifications for nonuniform

12

heating could be made in the analysis [10] for the exterior of the fab-

ric skin by numerical means, but the convection process inside a non-

uniformly heated, balloon-shaped container is very complex. At this

time the process is not well enough understood to be modeled analyti-

cally; thus an accurate calculation of the heat transfer in the interior

is not possible. Fortunately, during normal flight, balloons rotate so

all parts of their skins are exposed equally (on the average) to the sun

and the balloon gas is mixed. This makes the assumption of uniform gas

and skin temperatures valid for thermal analyses performed for times

ranging from hours to days.

In the following three sections the heat transfer phenomena in bal-

loon systems will be examined in the light of recent advances in heat

transfer research. Sections III and IV will deal with convection and

Sect. V with radiation phenomena. The final objective will be the eval-

uation of the five heat transfer terms in Eq. (15).

13

III. CONVECTION BETWEEN THE ATMOSPHERE AND THE BALLOON SYSTEM

(q in Eq. 15)4

Convective heat transfer between balloon systems and the atmosphere

occurs over wide ranges of the convectional parameters used to describe

the process. Heat is transferred between the atmosphere and the balloon

by forced convection or free convection, or both, at Reynolds numbers

from 0 to 107 and Grashof numbers from 0 to 1011. Depending on the cir-

cumstances, the flow can be laminar or turbulent. The shapes of super-

pressure balloons resemble a sphere, those of zero-pressure balloons

resemble an onion, but instrument packages come in a variety of shapes,

such as cubes, cylinders, and plates. Balloon diameters range from 10

to 400 ft, ascent velocities from near 0 to 40 ft/sec, and temperature

differences (between the skin and gas) from 0 to 50°F.

Although convective heat transfer to and from an object in air has

been studied extensively, few investigations extend into the extreme

Reynolds- and Grashof-number ranges encountered by balloons and none

have specifically treated the onion shape typical of zero-pressure bal-

loons. Therefore, approximations and extrapolations of existing data

are unavoidable.

Heat transfer from spherical shapes in forced convection has

recently been investigated experimentally by Yuge [11] and Vliet and

Leppert [12]. Local values of the heat transfer coefficient in flow

over a sphere in the neighborhood of the stagnation point have been cal-

culated by Merk [13]. The pressure distribtuion for flow over spheres

has been investigated by Fage [14], who also measured the separation

point. He found that at a Reynolds number of 1.6 x 106, separation

occurred at a polar angle ( of 70° (Fig. 4) and moved toward the rear

with increasing Reynolds number. At the highest Reynolds number of his

tests, ReD = 4.2 x 106, separation occurred at a polar angle of 100°.

Extensive correlations of experimental data indicate that in

forced convection the average Nusselt number of the entire surface of

1.0 Nu

(CorPr)V

, , , , , ,--....,..£-. *1

Ir: II/ ,

0.5 / /

°1 . . . . . . . . . .. , .. .

0 60 120 180

OAT =15.50C; "AT=19.5°C

Fig. 4 Local free convection heat transfer over a sphere--comparison of theory and experiments.

15

the sphere in air can be obtained from the relation

hDN c 0.57

Nu = = 2 + 0.30 Re m (16)m k D,m

a

for Reynolds numbers between 1.8 x 103 and 1.4 x 105 [11] or from the

relationh Dc 0.5 5

Nu = = 2 + 0.41 ReD,5 (17)m k D,m

a

for Reynolds numbers between 0.4 and 2 x 105 [12]. In both of the above

equations, the subscript m indicates that all physical properties should

be evaluated at the mean temperature between the skin and the atmospheric

temperatures.

Using Eq. (17), the rate of heat transfer by convection to or from

the surface of the balloon skin during ascent or descent can be written

in the form

q= 3.9 V 1/3 k (T - Tf) 2 + 0.472 V' 3 [a dz/dt ) (18)4 g a a g Oa p^a

Krause and Schenk [15] investigated thermal free convection from a

warmer surrounding fluid to a cooler spherical body at uniform surface

temperature in the range of Rayleigh numbers between 6 x 108 and

5 x 109. As shown in Fig. 4, the experimental results for the local

heat transfer coefficient agree reasonably well with a theoretical

analysis of Merk [16] up the the hydrodynamic separation point, which

for the narrow Grashof-number range of this investigation occurred at a

polar angle of about 145° from the vertical axis. Since Merk's theory

applies equally well for laminar free convection from a heated sphere,

it also seems reasonable to use the data in Fig. 4 to calculate heat

transfer from the upper hemispherical surface of heated balloons. No

theory or data exist at present, however, to predict the influence of

the nonuniform azimuthal temperature distribution, which is always

present when the balloon is heated by the sun.

16

Experimental evidence is contradictory regarding the influence of

the Grashof number on the separation point in laminar free convection to

or from a sphere. Garner et al. [17,18] found a shift in separation

point from 155 to 100° between mass transfer Rayleigh numbers (Gr * Sc)

from 1.5 x 108 to 5.5 x 108 for different fluids with 800 < Sc < 2,200,

whereas Schutz [19] found a much weaker dependency in the region

2 x 108 < Gr * Sc < 2 x 10l° (170 to 135°) for a fluid with Sc = 1,800.

According to conventional boundary layer theory, the separation point

should shift toward the stagnation point with increasing Grashof numbers

and the heat transfer coefficient in the region of the free convection

plume, where the flow is turbulent, will be larger than in the laminar

flow regions, as shown in Fig. 4. The results agree qualitatively with

observations of free convection about a horizontal cylinder.

The point of transition from laminar to turbulent flow in the free

convection regime is not presently known. Schlieren patterns [15] show

that purely laminar flow prevails over a sphere between the stagnation

point and the equator at Grashof numbers as high as 6 x 108 and that

some disturbances exist between 100° and separation (190 to 150°), but

real turbulence was observed only in the plume at a Grashof number of

about 109. Perhaps the stability analysis of Gebhart [20] could be

applied to natural convection over a balloon-shaped body (idealized as

a sphere to analytically predict the point of transition. Gebhart's

analysis for a flat plate predicts that turbulent instabilities could

amplify at local Grashof numbers of the same order of magnitude as have

been observed on balloons.

If the designer can use averaged values of the heat transfer

coefficient over the entire surface of the sphere, available data can

be correlated [15] by a relation of the type

DhNu = c = 2 + 0.6(GrPr)4 (19)m k m

a

for Rayleigh numbers between 105 and 2 x 10l°. The constant value of 2

applies in the limit as the Grashof number approaches zero and the heat

transfer mechanism approaches pure conduction. The theoretical

17

convergence of the Nusselt number at small Grashof numbers has recently

been elegantly verified by Fendell [21].

The influence of vibration on the heat transfer from spheres has

been investigated in free and forced convection [22], and the influence

of rotation about a vertical axis has been studied experimentally for

free convection [23,24]. The results of these investigations indicate

that under flight conditions neither vibration nor rotation will in-

fluence balloon heat transfer characteristics.

Using Eq. (19), the rate of heat transfer by free convection to or

from a balloon at float altitude can be written in the form

p2g(T - T)V \11a = .a a f -

q = 7.79 V 3k (T - Tf) 1 + 0.322 (20)

a a

For computational purposes a very convenient correlation of

averaged experimental convection data has recently been provided by

Borner [25], who reviewed 70 previous investigations of heat and mass

transfer by free convection or forced convection, or both, in flow over

single bodies, and who also conducted additional tests. One convenience

of Borner's correlation is that.data for bodies of different shapes can

be handled [26] by choosing a pertinent length dimension, 9', defined in

Table 1 for several shapes.

Table 1

EQUIVALENT LENGTH DIMENSIONS OF CONVECTION CORRELATION

Shape of Body Equivalent Length A'

Very wide plate with surface parallel to theflow in forced convection with length Z in

direction of flow. R' =

Sphere or long cylinder of diameter D with

axis perpendicular to the flow in forced

and free convection 9' = (T/2)D

Long rectangular bar of width a* andheight b* with its long axis perpendicular

to the flow in forced convection 9' - a* + b*

Vertical plate (surface parallel to gravity)

of height 9 in free convection 9' =

Long horizontal plate (surface perpendicular

to gravity) of width 9 in free convection 9' = Q/2

18

Reference 25 presents correlations of free convection data for plates

(horizontal and vertical), spheres, and cylinders (horizontal and vertical)

as plots of Nu., = h '/k versus (Gr, · Pr) = (gp23ATV'3/p 2) (c p/k)c p

and of forced convection data in the form Nu., versus Reg,. A combined

free and forced convection correlation was obtained by defining an equi-

valent Reynolds number for free convection Rei,*

x2

Reg,* = f(Pr) Gr, (21)

where the function f(Pr) for air, which is dependent upon the Prandtl

number, should be taken equal to f for "a best fit correlation," com-

pared to 0.64 predicted by analysis. With the definition of an equiva-

lent Reynolds number for free convection given in Eq. (21), free convec-

tion dominates when Re., < 2.4 Reg,* and forced convection dominates

when Re.,* < 2.4 Reg,. A transition region, where both free and forced

convection are appreciable, exists between these limits. It has been

shown that in this transition region free convection aids the forced con-

vection transfer when the motion caused by buoyancy is in the same

direction as the forced flow and retards it [27,28] when buoyancy

opposes the flow. In order to bridge the gap between forced and free

convection Borner [25] defines a third Reynolds number Re'g, as

Re' Re, + (Gr,/2) (22)

and then plots Nu., versus Re' ,. Figure 5 shows the results of this

correlation for a sphere with downward forced convection, the situation

of an ascending balloon. Under these circumstances free convection will

oppose the forced convection flow, but visualization studies for this

condition [25] have shown that the existence of a free convection field

will produce turbulence at low velocities and will also cause separation

of the boundary layer. These effects tend to offset any decrease in

the heat transfer, as predicted for purely laminar flow [27]. The

correlation function shown in Fig. 5 was also found to be applicable

for upward flow, the condition of a descending balloon. Sharma and

19

0: 2e6

O rlA = 2,7. lobiS to.YI I I i o irf= 5,0 -10 6b 6,0 10

. < Gr= = 8,0- lo Ws go -10103 2 4 6 10 2- 6 8 15 2-

IFh__38.251 Reynoldsahl Re_,; e1 r__e

Fig. 5 Average Nusselt number for sphere in

combined free and forced convective flow.

Sukhatme [29] have recently published experimental results concerning

the interaction between free and forced convection in flow over a hori-

zontal cylinder for Reynolds numbers from 10 to 5,000 and Grashof num-

103 2 610 2 6 8 1 2

bers from 3 x 103 to 7 x 10 no. These results indicate that the Reynolds

number exponent n in the parameter Gr/Re. is affected by turbulence and

separation phenomena and that a value of 3.25 is most suitable when free

convection dominates, whereas a value of 1.8 is more suitable at the

forced convection end. An average value of 2.5 was found to give a

reasonably good correlation in the transition region.

bers fBorner's correlation does not give insight into local variations

of the heat transfer coefficient, but it is convenient. By computing

Repar and Re,* phenoimultaneously and continuously from the time of launch

until the balloon has reached the float altitude, a smooth transition

between the forced and free convection regions can be made and the

influence of radiation can be superimposed directly.

20

In general, only the equilibrium temperature at float altitude is

important to the thermal design of balloon instrument packages. During

ascent forced convective heat transfer is quite effective [30] in main-

taining a small instrument package at a temperature close to that of

the ambient air, but when the balloon has reached its float altitude

only free convection can transfer heat directly between the package and

the surrounding air. The relative magnitude of the radiation to and

from the surfaces of the package dominates the thermal transfer during

the day, but convection enters prominently at night.

In certain specialized instrument packages, in addition to the

averaged heat transfer coefficient, local values at the top and bottom

surfaces are also sometimes of interest. The numerical study of Suriano

and Yang [31] provides some insight into the flow and temperature field

in the vicinity of a heated horizontal square plate at small Rayleigh

numbers. Table 2 shows averaged Nusselt numbers obtained by their cal-

culations for both the top and bottom surfaces. The temperature field

over a horizontal plate at a Rayleigh number of 50 (Fig. 6) suggests

boundary layer behavior on the lower surface but not on the upper. How-

ever, the analysis of Stewartson [32], as modified by Gill et al. [33],

indicates that a boundary layer also forms over the upper surface of a

heated horizontal strip and that the average Nusselt number for a strip

of width L in air can be predicted from the relation

h L-- c Gl5Nu = k =0.79 GrL (23)k L

a

in the laminar flow region. This result is not in complete agreement

with experimental data for the upper surface of a heated square plate

of side L in air [34]. These data give larger heat transfer coeffi-

cients that are correlated empirically by the relations

Nu = 0.50 GrL4 (105 < GrL < 2 x 107, laminar) (24)

Nu = 0.125 Grl/3 (2 x 107 < GrL < 3 x 109, turbulent) (25)L L

21

Table 2

AVERAGE NUSSELT NUMBERS FOR A HORIZONTAL PLATE IN FREE CONVECTION

NRa Nr = 0.72 Buznik andPr. .__ .. .. Bezlomtsev

x = 0- x = 0+ Average

0 1.049 1.047 1.048 1.00

0.10 1.052 1.045 1.048 1.28

5.0 1.201 0.946 1.074 1.77

10.0 1.406 0.932 1.169 1.91

50.0 2.879 1.259 2.069 2.38

100.0 4.041 1.330 2.685 2.64

200.0 6.166 1.492 3.829 2.96

250.0 7.194 1.594 4.394 3.00

300.0 7.620 1.678 4.469 3.03

1Buznik, V. M., and K. A. Bezlomtsev, 1960: A generalized equationfor the heat exchange of natural and forced convection during externalflow about bodies. Izv. Uyssh. Ucheb. Zaved. 2, 68-74; 1961: Ref. Zh.MeckZ. 6, Rev. 6V506.

\ =NRo50N p= 0.72

.6 \0.5 \0.4

0.50.7

0.8

0.9

0 .0

-0.5

0 0.50 1.0y

Fig. 6 Temperature field about a heated horizontalflat plate at a Rayleigh number of 50.

22

Heat transfer by free convection from the lower surface of finite

heated plates (or to the upper surface of cooled plates) has recently

been studied by Singh et al. [35,36] for square and circular plates and

a long strip. Local heat transfer coefficients are lowest in the center

and increase toward the edges. For a square plate of side L the local

Nusselt number at a distance x from the center is

h(x)L 0 i/58jf, R2xa2 ] , 2 4 1/4

h(x)L - 0.58 Ra1/5 1 - L ]+ 0.271 1 - + ... (26)k = ' '

The average Nusselt number for square plates, as well as for circular

plates and long strips, is given by the relation

h LNu- = k = C GrL (27)

a

where constants C and n are given in Table 3

Table 3

CONSTANTS FOR THE FREE CONVECTION EQUATION (Eq. 27)

Reference ConstantsNumbers C n

Square Platel(L is length of side)

35 (analytical) 0.89 0.234 (experimental) 1.00 0.2

Circular Platel(L is the diameter)

35 (analytical) 0.79 0.236 (experimental) 1.0 0.2

Long Stripl(L is the width)

32 and 33 (analytical) 0.80 0.2

1Heated surface facing downward, influence of side walls neglected.

Flow-visualization experiments with a number of other shapes, with

heated surface facing upward, have been reported by Husar and Sparrow

[37], but information regarding the interaction between the boundary

layer flow over vertical surfaces and the flow developing over hori-

zontal surfaces (i.e., the top and bottom of boxes or vertical cylin-

ders) is still lacking.

23

IV. CONVECTION INSIDE THE BALLOON

(q in Eqs. 14 and 15)1

The convection process inside the balloon is important because it

determines the temperature, the pressure, and the volume of the lift

gas (usually helium). Little research has been done on convection in-

side a sphere [38,39] and none has been done on convection inside

cavities resembling the shapes of high altitude balloons.

The heat transfer process is free convection. Hellums and

Churchill showed as early as 1961 [40,41] that the partial differen-

tial equations for the conservation of mass, momentum, and energy can

be solved for laminar natural convection under many different condi-

tions, but so far no solution for the heat transfer inside a balloon

has been obtained. Investigations of natural convection in annuli [42]

suggest that several types of flow patterns will exist as the tempera-

ture difference between the surface and the gas varies during a 24-hr

period. It seems likely that in the daytime, when the balloon skin is

warmer than the gas inside, there would be upward flow near the balloon

skin and downward flow in the interior. The reverse would be expected

at night. But there could also be situations during the day when gas

ascends on the side heated by the sun and descends on the cooler side.

As a first approximation, one could use existing solutions for flow in

a rectangular cavity heated on two sides as long as the flow is laminar.

But the tremendous size of a balloon makes it very likely that the flow

will be turbulent, except for very small temperature differences.

A semiempirical analysis [43] suggests that for turbulent free con-

vection over a vertical plate the Nusselt number fits the relationship

h L- = 0.021 (GrL . Pr)2 5 (28)

17 Lg

at Grashof numbers of the order of 1012, but Clark [44] suggests using

a relationship of the type

Dh gP (f g(Tf -T PrD

gg= c g · Prg (29)

24

with the constants C and n selected from Table 4, and with all physical

properties of the gas evaluated at the skin temperature.

Table 4

FREE CONVECTION IN A SPHERE(Constant C and Exponent n for Eq. 29)

Gr · Pr C n Type of Flow

104-109 0.59 1/4 Laminar

9O19lO12 0.13 1/3 Turbulent

In the absence of more concrete information, Dingwell et al. [45] used

Eq. (29) for their balloon study, with C and n equal to 0.13 and 1/3,

respectively. Equation (29) can then be used to write the rate of heat

transfer from the skin to the gas in the form

q l c(Tf - T) dAi = h T D (Tf TA i

= 4.83 h V2/3 (T - ) (30)c g f g

= 0.628 V2/3 k (T - T ) gf2 - Prg g f 2 T g

g g

Experimental data supporting the form of Eq. (29) have recently

been reported by Ulrich et al. [46] for the transient condition encoun-

tered in filling a cylindrical tank with air at Grashof numbers between

108 and 1013. The length-to-diameter ratio of their tanks varied

between 0.5 and 2.0 so their results should indicate what might occur in-

side a sphere. The experiments showed that during the initial stages of

the process the heat transfer coefficients were significantly higher

than those predicted by turbulent free convection, but agreement with

the heat transfer rate predicted by Eq. (30) was achieved after a few

seconds. Available information on the transient free convection heat

transfer characteristics of vertical surfaces in the laminar region is

extensive and has been summarized by Gebhart et al. [47]; little infor-

mation is available for the turbulent flow region [46].

25

V. RADIATION HEAT TRANSFER

The radiation heat transfer to and from balloons strongly influences

their performance and determines their short-term stability. Balloon

skins absorb direct and reflected solar radiation and radiation emitted

by the earth and the atmosphere. Over 99% of the solar radiation is in

the wavelength range between 0.2 and 4.0 p, whereas the earth and atmo-

spheric radiation is in the infrared range between 6 and 100 p, with

about 70% below 20 p. Balloon skins are usually at temperatures of

about 20°F and thus emit infrared radiation. For approximate calcula-

tions the sun and earth can be considered to be blackbodies at 9,500 and

80°F, respectively. The direct radiation from the earth and the clouds

can be as high as 150 BTU/ft2 hr [3]. However, the direct solar radia-

tion is collimated and, therefore, the effective receiving area is the

projected area, which in the case of a 1 million ft3 balloon (equivalent

to a 124 ft diam sphere) would be about 12,100 ft2. Reflected solar

radiation, on the other hand, impinges on the lower half of the total

surface area of the balloon, which is about 24,000 ft2. Earth radiation

impinges on the total area of the balloon (about 48,000 ft2) during

ascent through the atmosphere but impinges only on the lower half of the

balloon surface after it has risen to an altitude of 70,000 ft, where

less than 5% of the total air mass remains above. Thus, the total

direct solar radiation is about 5.4 x 106 BTU/hr, and the reflected ra-

diation is about 3.6 x 106 BTU/hr. Infrared radiation from the earth

and the atmosphere can amount to as much as 7.2 x 106 BTU/hr, but at

float altitude will be of the order of 2 x 106 BTU/hr. The balloon

emits radiation at a rate of 4.8 x e x 106 BTU/hr, where the effective

emissivity ef may vary between 0.2 and 0.7 for different skin materials.

In comparison, convection contributes only about 2 x 105 BTU/hr to the

heat transfer over the exterior balloon surface, but it is the only heat

transfer mechanism in the interior because helium is transparent to

radiation.

Since the actual amount of radiation absorbed depends critically on

the radiation properties of the receiving and emitting surfaces, a

26

knowledge of these properties is very important to the designer. The

two most common materials for balloon skins are polyethylene and Mylar,

which have (according to available data [48]) an effective absorptance

to ultraviolet solar radiation of about 0.12 and 0.17, respectively. The

absorptance to infrared earth radiation, which is also approximately

the emittance of the skin, is 0.21 for polyethylene and 0.63 for scrim

Mylar (0.5 mil Mylar on dacron scrim). A Mylar balloon absorbs, there-

fore, a much smaller percentage of its total radiation load from the

sun than does a polyethylene balloon, and since it also emits more

radiation by virtue of its higher emittance, a Mylar balloon will be

cooler than a polyethylene balloon. At the same time, however, because

of its high emittance a change in environmental conditions (e.g.,

setting of the sun) will reduce the gas temperature in a Mylar balloon

more quickly than in a polyethylene balloon. A Mylar balloon, there-

fore, has less altitude stability than a polyethylene balloon.

In the evaluation of the radiant contribution to the total heat

load on balloons and their instrument packages, engineers are faced with

a lack of experimental data for the pertinent surface radiation charac-

teristics of balloon materials. To calculate accurately the percentage

of direct solar incident radiation absorbed by a surface, one must know

its monochromatic absorptance for radiation of wavelengths between 0.2

and 4.0 p at various angles of incident radiation [49]. The absorptance

of thin films has a strong angular variation; radiation perpendicular

to the surface passes through more readily than does radiation at a

grazing angle. This angle variation can become particularly important

in a balloon system where the radiation incidence angle is zero at sun-

rise, rises to a maximum at noon, and then decreases again to zero at

sunset. For passive temperature control, i.e., the use of surfaces with

very different solar absorptances and infrared emittances, the influence

of the average incident angle of radiation cannot be ignored.

The difficulty of determining radiation properties for balloon

skins is exacerbated by the transparency of these skins, and the trans-

mittance has to be considered [50]. As will be shown,- to calculate

27

accurately the radiation heat load on a balloon, one needs to know the

total hemispherical emittance of the skin at its temperature, the direc-

tional and the angular hemispherical absorptance in the solar spectrum

(since surfaces are irradiated directly and indirectly by the sun), the

directional reflectances of clouds and terrestrial surfaces in the

solar spectrum, and the hemispherical absorptance in the infrared (for

the surfaces exposed to the earth or clouds).

In 1969, Edwards [51] summarized our knowledge of the radiative

transfer characteristics of materials and surveyed techniques available

to measure surface properties [52]. It does not seem feasible to mea-

sure spectral-angular surface properties for all potentially useful

materials. The amount of data would be unmanageable, and the cost with

present equipment would be unreasonably high. It would, therefore, be

desirable to classify materials according to their physical surface

properties and to develop working relationships to estimate "effective"

angular properties from measurements of a few select properties. One

should also know the influence on the radiation surface properties of

launch procedures, aging, solar radiation, and atmospheric phenomena so

thermal predictions can be made not only for idealized laboratory

samples but also for actual operational systems.

The flight lifetime of a balloon depends critically on the relation-

ship between the amount of radiant energy absorbed during the day and

the amount of radiant energy lost at night. In the morning the balloon

is usually at the lowest float altitude because the gas is at its lowest

temperature. After sunrise the gas is warmed by transient free convec-

tion from the skin after the skin has been warmed by the absorption of

solar radiation. The part of the skin exposed to the sun transfers heat

by transmission and internal emission to the rest of the skin. As the

temperature of the gas increases, the buoyancy of the balloon also in-

creases and the system begins to rise. This rise will continue until

shortly after sunset, whereupon the net loss of energy exceeds the net

input of energy, and the balloon begins to sink. At present the total

balloon heat balance over a 24-hr period shows a small loss in internal

energy. One of the long-range objectives of balloon designers is a

28

system which will passively maintain its average float altitude, i.e.,

the altitude for which the net change in lift-gas energy over a 24-hr

period is zero.

Various schemes to achieve a zero net change have been tried without

success. But even if a "permanent balloon" is not possible, increased

balloon lifetime and altitude stability would materially contribute to

programs aimed at permitting long-term weather prediction and eventual

weather control.

At altitudes below 70,000 ft the radiation incident on a balloon is

subject to considerable variation, and quantitative estimates are uncer-

tain. The amounts of infrared radiation from below and above will differ

and will both depend on the weather and cloud cover. The incident solar

radiation will also depend on the altitude and the clouds. At altitudes

over 70,000 ft a balloon is above the weather and receives nearly all of

its infrared radiation from below. Under these conditions the infrared

radiation heat load can be estimated with considerably more confidence.

Fortunately, large balloons are generally launched in good weather so

that changes in cloud cover during ascent are minimized.

Although measurements of upward and downward radiation in the atmo-

sphere have been made for many years, accurate evaluation of rapid

changes in radiation flux are unreliable because it is not possible to

predict local weather changes in advance. One could, however, modify

the radiation heat flux calculations to include observed weather data,

such as the types of clouds and their altitudes. Calculations based on

recent observations made at ESSA by Kuhn [53] clearly correlate changes

in the radiation flux with cloud cover. Application of available

knowledge of atmospheric radiation to balloon performance will, however,

require the close cooperation of meteorologists, cloud physicists, and

ballooning engineers.

For small packages used on superpressure balloons at intermediate

altitudes, an engineering analysis of the thermal control problem and a

summary of the experience gained in several flights by NCAR have been

presented by Lichfield and Carlson [30]. The basic problems of

29

temperature control at float altitude are quite similar to those en-

countered in spacecraft. In space, where only radiation can transfer

heat, the equilibrium temperature T developed by an opaque body sub-

jected only to direct solar radiation at the rate G over a projecteds

surface area S normal to the sun, having an averaged directional ab-s

sorptance per unit projected area, a , and a surface area, S, with

an average hemispherical emittance in the infrared eH is

a GS S

T / s, avg s s(31)

\ H,i

Equation (31) shows that in this case the ratio of the absorptance in

the solar spectrum to the emittance in the infrared controls the equi-

librium temperature.

In the balloon packages the heat transfer problem is more compli-

cated because of the addition of reflected solar radiation, convection,

and the influence of the atmospheric radiation, all of which depend

strongly on the cloud cover and weather. In daytime flights, the tem-

perature can be controlled by proper treatment of the surface of the

package. The properties of a number of materials and surface coatings

for such use are presented in Refs. 54-57; Table 5 presents a typical

selection. Silver sulfide, which has an average infrared emittance of

only 0.03 and an average solar absorptance of 0.60, has been used suc-

cessfully in balloon packages for which a temperature between 44 and

60°F was desirable. A further increase in temperature during the day

was achieved by covering the package with a thin film of material such

as Mylar which is transparent to radiation only in the solar spectrum

between 0.2 and 3.0 D, but has a large reflectance for infrared radia-

tion. This method of trapping the radiation, using this so-called

greenhouse effect, should be used cautiously to avoid an excessive tem-

perature rise during the middle of the day.

For nighttime balloon flights, energy must be stored to maintain

the internal temperatue of the flight package. A simple reservoir is

water. When a pound of water freezes, 42 W-hr of energy are released.

In addition, 0.3 W-hr of energy are released for each pound of water

30

Table 5

RADIATION CHARACTERISTICS OF SURFACES

Material IR Solar RatioEmissivity Absorption

Silver (polished) 0.02 0.07 0.28

Platinum 0.05 0.10 0.50

Aluminum 0.08 0.15 0.53

Nickel 0.12 0.15 0.80

Stellite 0.18 0.30 0.60

Aluminum paint 0.55 0.55 1.00

White lead paint 0.95 0.25 3.80

Zinc oxide paint 0.95 0.30 3.20

Gray paint 0.95 0.75 1.26

Black paint 0.95 0.95 1.00

Lamp black 0.95 0.97 0.98

Silver sulfide 1 0.03 0.60 0.05

Nickel blackl' 2 0.10 0.90 0.11

Cupric oxide1' 2 0.15 0.90 0.16

1These are special surfaces where a metal is covered with a very thinlayer of absorbing material. The layer is so thin that it is a frac-tion of a wavelength thick in the infrared and is, therefore, almosttransparent to IR. The result is that the IR emissivity is nearlythat of the underlying metal. However, the thickness is large com-pared to the wavelength of the maximum solar spectrum so the absorp-tivity is large for solar radiation.

2H. Tabor 1961: Solar Energy.

31

cooled 1F°. Table 6 compares the energy storage capacity of water with

that of zinc-air and silver-zinc batteries. The batteries have greater

energy storage capacity than water, but zinc-air batteries require a

supply of oxygen at balloon float altitudes, and water is cheaper and

easier to handle.

Table 6

ENERGY STORAGE CAPABILITIES OF WATER AND BATTERIES

W-hr/lb W-hr/kg W-hr/in.3

Water (heat of fusion) 42 93 1.5

Water (per °C) 0.52 1.16 0.02

Zinc-air 80 176 5

Lead-acid 11 24.2 1.2

Nickel-cadmium 8 17.6 0.4

Silver-cadmium 35 77.0 3.5

Silver-zinc 55 121 4.5

For flights lasting several days, the package must be able to

absorb as much heat during the day as it loses during the night. With

suitable black paint daytime surface temperatures of 6 to 100F can be

attained, but since a typical nighttime surface temperature is -67°F,

it is necessary to achieve a daytime surface temperature of about 1300F

to maintain temperatue equilibrium in the water. This requires the use

of special coatings, such as silver sulfide, or the use of one or more

greenhouse covers. One can also use solar cells to generate energy

within the package during the day, but this adds weight and complicates

the system.

33

VI. EMITTED RADIATION

(q in Eq. 15)

The balloon fabric, polyethylene or Mylar, transfers heat to the

atmosphere by infrared radiation. A typical Mylar balloon fabric

(0.35 mil thick Mylar with dacron scrim, 4 x 6 strands per inch) has

at its operating temperature an average transmittance of about 0.55, an

average reflectance of about 0.20, and an average absorptance of 0.25.

A typical polyethylene film has an average transmittance of 0.75, an

average reflectance of only 0.05, and an average absorptance of 0.20.

However, the monochromatic properties of fabric materials vary consid-

erably. Mylar, for example, has radiation windows with transmittances

as high as 0.80 for wavelengths between 3 and 6 p and opaque ranges with

transmittances as low as 0.10 for wavelengths between 13 to 15 p [58].

To calculate the emitted radiation it is necessary to know the

spectrally averaged hemispherical emittance in the infrared region.

The evaluation of an average hemispherical emittance for a given wave-

length range or a given temperature offers no difficulties for an opaque

surface [51,55,59]. As shown in Fig. 7, however, a surface element

of a balloon skin dS radiates not only directly into space, but also

into the interior where radiation can pass through the fabric into

space, can be reflected from the interior surface of the balloon fabric,

or can be absorbed by the fabric. To calculate accurately the "effec-

tive emittance" of a balloon from data on the surface radiation proper-

ties of its fabric skin, bidirectional values of the monochromatic

emittance, absorptivity, and reflectivity of the interior surface would

have to be known for the infrared wavelengths between 6 and 100 p [51].

Such measurements are difficult and are generally too expensive. For

balloon design it would actually be much more desirable to measure the

actual emittance of a spherical sample of the fabric material filled

with helium. However, no such data have as yet been taken, and calcula-

tions have been based on a model proposed by Germeles [8]. This model

assumes that the inner fabric surface obeys Lambert's law, i.e., it

emits and reflects diffusely. It also assumes that average values can

T-A^:-ei---r"7ei - 4T 4 xdS

<iei~zot 4 adSTi er 2 o- T 4 dS. -S '

ri e; T 4 dS ri ei oc T 4 d S

Ef=f f| e i (X) - T 4 dSdX ei rD2 Tf 4so

Q total e i w Dg 0 T 4 [I+-(+ +rI2.]Fig. 7 Emission of radiation from a balloon skin-the effective emittance.

Fig. 7 Emission of radiation from a balloon skin--the effective emittance.

35

replace the spectrum of values for the emittance, absorptivity, and

reflectivity of the inner surface over the wavelength range (between

6 and 20 p) for which data are available [53]. The net rate of emis-

sion from the entire balloon is then equal to the radiation directly

emitted from the outer surface, ei ~' D2 a T, plus that portion of the

radiation emitted by the interior surface which eventually passes

through the fabric, ei Ti T D2 a T4 (1 + ri. + .) + ) By summing

the series, one obtains the effective emissivity of the fabric

e =ff ei 1 + i (1 + ri + r . )eff i r1 i (lri .i ''

(32)

= -1 - l--i

where

100 /100

a = a(X) I (X) d / I (X) dX (33a)

p100 / .100

= r(X) I (X) dX I (X) dX (33b)

6

T = - r. - a. (33c)i 1 1

ei (Tf) = ai (Tf) (33d)

Use of the effective emissivity gives the rate of heat transfer from

the balloon fabric in the form

q = 4.83 eeff V213 oT4 (34)

where T4 is the average of the fourth power of the absolute temperature

of the balloon fabric.

36

The infrared hemispherical emittance of the surface of an opaque

body can be measured easily, so the determination of radiation emitted

by the surface of a balloon instrument package generally offers no

problem.

37

VII. DIRECT SOLAR RADIATION

(q in Eq. 15)2,d

The solar radiation spectrum has been investigated in great detail,

and summaries of the current state of knowledge are presented in Refs. 9

and 54. Figure 8 shows the solar spectrum at the outer fringes of the

atmosphere and at the surface of the earth after attenuation and absorp-

tion by the atmosphere. Over 99% of the solar energy is contained

within a narrow wavelength band between 0.2 and 4 p, and for most engi-

neering heat transfer calculations the sun's spectrum can be approxi-

mated by that of a blackbody at 9,500°F. The solar constant, i.e., the

radiation received by a surface placed perpendicular to the rays of the

sun outside the earth's atmosphere, is 2.0 cal/cm2 min, or 442 BTU/ft2 hr.

The solar radiation per unit area on a horizontal surface outside the

earth's atmosphere depends only on the zenith angle, i.e., the angle be-

tween a line normal to the surface and the rays of the sun. This angle

can be determined from the relationship

cos C = sin X sin a + cos X cos a cos h (35)S S

As shown in detail in Refs. 59 and 60, the radiation per unit area

on a surface tilted at an angle T to the horizontal is

G = G cos C cos ( (36)s

where

cos < = cos (A - a) cos (90 - %) sin T + sin (90 - C) cos T

T = the angle between the surface and the horizontal

a= the westward declination (measured from the southmeridian) of the projection on the earth's surfaceof the normal to the surface

X = latitude

a = solar declinations

h = local hour angle

A = azimuth of the sun, measured westward from the southmeridian

G = average incident solar radiations

0Q25aj , | I, g l | I | l | l | I | I' I I I ' I I I i I J I

0.20

'I t\ Solar Radiation Curve Outside Atmosphereo. ,s I ' o Solar Radiation Curve at Sea Level

0< . / \.-Curve for Blackbody at 59000 K"E

\ H20I o2 , H20

1010 IQ

I H,2'H20

H20

- / *n ©H0.05 - HO

::::· / / f j o ^,_ co°I H209 CoII IliiH201 C0

/3 I ;· HO CO2

0 02 0.4 Q6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6 2.8 3.0 3.2

WAVELENGTH (^)

Fig. 8 Spectral distribution of solar radiation in space and at sea level.

39

Once the angle between the sun and the surface, i.e., (% - T), is

known the solar radiation absorbed by a surface outside the atmosphere

is simply a (% - T)G , where a (% - T) is the effective directional

absorptance for that angle.

In passing through the atmosphere, the intensity and spectrum of

the solar energy are altered by absorption and scattering [61]. There-

fore, the radiation on an object is strongly dependent on the atmo-

spheric path length of the solar rays, usually expressed in terms of

the "optical air mass." Exact calculations of the attenuated spectrum

are quite cumbersome [61,62] and for engineering purposes such

calculations are only useful when the directional absorptances of the

receiving surfaces are known [63,64]. Figure 9 shows the distribution

of direct solar radiation incident at sea level on a horizontal surface,

as a function of wavelength, for several short paths corresponding to

optical air masses between 1.0 and 8.0. The optical air mass is unity

WAVELENGTH (L)0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 2.0 o

7^~~~0 ~~ _ E

.5 60 aEE | TOTAL ENERGY ' :

Air Mass W/cm2

cal/min cmr2

/

0 50 -- 1.0 0.0865 1.24 /> 1.5 0.0750 .080 2.0 0.0640 0.917

4.0 0.0372 0533 c1 40_ 6.0 0.0240 0.344 /

E 8.0 0.0152 0.218 ~~~~~~~~~~~~~~~~~~~~~-O 0.5 E

0.o

dettealeelfoar20masses1.0 0

Z WW 1.0 Z

06.0

35,000 30,000 25,000 20,000 15,000 10,000 5000 0nO

WAVE NUMBER (cm - ')

Fig. 9 Spectral distribution of solar radiation inci-dent at sea level for air masses 1.0 to 8.0.

40

when the sun is directly overhead and the body receiving radiation is

on the earth's surface. Under these conditions only 70.2% of the solar

constant is received. For other values of optical air mass (m), the

fraction of solar radiation received is expressed to within 3% accuracy

[8] by the empirical relation

sZ 0.5 (e-065m + e-0095m (37)

s,o

where optical air mass depends on the altitude and the sun's zenith

angle C. The variation of the optical air mass at sea level with zenith

angle, m(o,C), is given in Table 16-18 of Ref. 9 and can be closely

approximated [8] by the relationship

m(o,C) = [1229 + (614 cos %)2]½ - 614 cos C (38)

for 0 < C < 90°. Since the attenuation is proportional to the number

of air molecules in the path, for a given zenith angle the optical air

mass at altitude z can be related approximately to the air mass at sea

level by the relationship

m(z) = m(o) [p(z)z/p(o)] (39)

Using Eqs. (35) - (39) one can determine radiation on a surface at

any altitude, geographical location, and time between sunset and sunrise.

There are two short periods

90° < C < 90° + cos- kDearth + 2~ = \ Dea h+ 2z

before sea-level sunrise and, after sea-level sunset during which a bal-

loon also receives solar radiation. A method for determining the air

mass during these periods is presented in Ref. 8, but for most purposes

it is sufficiently accurate to double the value of the air mass calcu-

lated at sea-level sunrise or sunset.

41

The portion of the solar radiation absorbed at the surface of an

opaque body, such as an instrument package, can be calculated once the

orientation of the surface relative to the sun is known. The amount of

incident radiation absorbed per unit area is

r 4qabsorbed = a( - T, X) G (X,z)dX = a ( - T) G(z) (40)

0.2

However, as mentioned previously, the problem is more complicated for

the balloon because the skin is transparent, and a portion of the inci-

dent solar radiation is absorbed, a portion reflected, and the rest

transmitted into the interior where some of the radiation transmitted

on the first pass will be absorbed as a result of internal reflection

and absorption.

Except for occasional measurements in connection with basic inves-

tigation of radiation [51,63], only Dingwell [48] has attempted to

make systematic measurements of the radiation properties of balloon fab-

rics. Since balloon skins are transparent, absorptances are difficult

to measure directly. Therefore, Dingwell measured the monochromatic

transmittance of solar radiation using a Beckman D.K. spectrophotometer.

The reflectance was then estimated by observing interference fringes

caused by internal reflections and applying Fresnel's formula for nor-

mal incidence. Finally the monochromatic absorptance for incident ra-

diation, presumably normal to the surface, was calculated from the

relationship

a(X) = 1 - r(X) - T(X) (41)

and the integrated normal absorptance of the fabric over the solar spec-

trum was determined by numerical integration of the relationship

r4.0 / 4.0

a (normal) = | a(X) I (X) dX I (X) dX (42)

0.2 0.2

42

No measurements have yet been made of the effect on the radiation prop-

erties of balloon films of the solar radiation angle of incidence.

Edwards [64] has shown, however, that predicted equilibrium temperatures

can be as much as 40°F in error if the angular dependence of radiation

characteristics is neglected, and it would be desirable to measure the

directional absorptance, a(90,,X,T), for some typical materials.

To estimate from available information the percentage of the solar

radiation actually absorbed by the skin of a balloon, it will be assumed

that the absorptance of the skin is independent of the angle of inci-

dence and that radiation transmitted through the skin emerges diffusely

from the interior surface. Furthermore, the shape of the balloon will

be idealized as a sphere so that the projected area is D/D24. The

fraction of the incident solar energy absorbed on the first pass is

a G(z)D 2/4, and the fraction transmitted is T G(z)D2/4. Of the frac-s g_ _ s _tion transmitted, a will be absorbed by the fabric and r will again

5 s

be reflected. The total amount of the incident solar radiation even-

tually absorbed, obtained by summing this series of interreflections,

will be

qd = G(z) D2/4 a + s (43)\ r

or in terms of the gas volume in the balloon

q = G(z) 1.21 V23 a 1 + s (44)2,d g ls \ 1 - r

s

The zenith angle r can be defined in a time coordinate system re-

lated to the flight of the balloon. If the Greenwich hour angle at the

time of launch is GHA, then at any subsequent time t the hour angle is

6h = GHA - LONG + (t/240) (45)

where GHA and the longitude LONG are in degrees and t is in seconds,

43

VIII. REFLECTED SOLAR RADIATION

(q in Eq. 15)2 ,r

Until recently, the heat load of greatest uncertainty in Eq. (15)

was the reflected solar radiation. It was known from an overall heat

balance that the portion of the solar radiation reflected by the earth

and its atmosphere (the albedo), when averaged over time and space, was

about 34% [65]. It was also known from balloon and aircraft observa-

tions that the albedo can vary widely, but no accurate long-term mea-

surements could be made without an observation station outside the

atmosphere. The early measurements of Explorer VII and several Tiros

satellites resulted in data for only part of the globe because these

satellites were not in polar orbits. In 1967 and 1968, however, the

meteorological satellite Nimbus II measured the incoming and outgoing

radiation over the entire globe for several weeks, and Raschke of the

Goddard Space Flight Center correlated the results and presented them

in convenient graphs [65].1 These results are very useful for estimates

of the reflected solar radiation above the atmosphere at the geographic

locations scanned by Nimbus II, but they cover only a limited period and

do not indicate variations with weather or direction.

The works of Houghton [66] and Fritz [67] are, therefore, still

very useful for engineering design. Houghton established a convenient

graph for the approximate albedo as a function of latitude for clear,

partially overcast, and completely overcast skies (Fig. 10). Although

these curves were designed for conditions above the atmosphere, they

also will give approximate results for balloons in the atmosphere as

long as the sky above is not completely overcast. The accuracy of the

method, which is very good for altitudes above 50,000 ft, depends upon

the altitude and the amount of cloud cover. From the work of Fritz [67]

and Coulson [68] an estimate of the spectrum of the reflected solar radi-

ation can be made. Table 7 presents the results [54] as the ratio of the

solar radiation received from below at a wavelength X to the radiation

at the wavelength of maximum intensity for three sky conditions. The

1According to a private communication all the curves are shifted by 2.5°

due to a drafting error.

44

Table 7

RELATIVE SPECTRAL DISTRIBUTION OF SOLAR RADIATIONUNDER VARIOUS SKY CONDITIONS

R (1) R ( 1)__________ R R()

X Clear Mean Overcast X Clear Mean Overcast(p) skies skies (p) skies skies

0.29 0 0 0 0.53 0.378 0.668 0.807

0.30 0.882 0.721 0.477 0.54 0.252 0.663 0.812

0.31 0.892 0.761 0.534 0.55 0.333 0.637 0.795

0.32 0.848 0.749 0.553 0.56 0.312 0.615 0.770

0.33 1.000 0.920 0.705 0.57 0.288 0.590 0.752

0.34 0.858 0.813 0.648 0.58 0.276 0.581 0.748

0.35 0.815 0.802 0.661 0.59 0.262 0.563 0.734

0.36 0.744 0.746 0.634 0.60 0.241 0.541 0.717

0.37 0.768 0.796 0.700 0.70 0.134 0.396 0.554

0.38 0.671 0.714 0.632 0.80 0.084 0.293 0.425

0.39 0.575 0.624 0.570 0.90 0.057 0.224 0.331

0.40 0.758 0.840 0.772 1.00 0.042 0.179 0.266

0.41 0.781 1.000 0.950 1.10 0.028 0.149 0.219

0.42 0.790 0.937 0.925 1.20 0.019 0.118 0.181

0.43 0.675 0.835 0.835 1.30 0.017 0.095 0.145

0.44 0.708 0.908 0.934 1.40 0.015 0.076 0.115

0.45 0.734 0.958 1.000 1.50 0.013 0.063 0.093

0.46 0.649 0.892 0.960 1.60 0.010 0.051 0.078

0.47 0.713 0.864 0.954 1.70 0.008 0.042 0.063

0.48 0.563 0.834 0.935 1.80 0.006 0.034 0.052

0.49 0.487 0.749 0.853 1.90 0.005 0.029 0.045

0.50 0.524 0.727 0.840 2.00 0.005 0.025 0.037

0.51 0.422 0.697 0.825 4.00 0.000 0.002 0.028

0.52 0.378 0.650 0.780

R = the ratio of the solar radiation at wavelength X to thesolar radiation at the wavelength of maximum intensity.

45

1001 0 0 -\-.-I . I I .I I .- I _\I -- I -'

80

o 60 II OVERCAST SKIES

0

o 40 III MEAN2 0 ~ I CLEAR SKIES

0 10 20 30 40 50 60 70 80 90

LATITUDE (°)

Fig. 10 Albedo as a function of latitudeunder various sky conditions.

reflected solar radiation incident on a satellite or a balloon from a

surface element of the earth-atmosphere dS ise

dG = (r D/4) Glar rs(0) cos 0 cos C dS/12b (46)

where

dGs = reflected solar radiation incident on dSe

r (0,%) = the bidirectional reflectance of dSe for a zenith

angle C in the direction of the balloon, e

1 = the distance between the balloon and dSs-b e

For a known geographical distribution of r (09,), the reflected radia-

tion can be calculated numerically. If one assumes that the earth-

atmosphere system reflects uniformly and diffusely, Eq. (46) can be

approximately integrated [69] over the portion of the earth visible

46

from the balloon, and the radiation heat load resulting from albedo

reflection q becomes2 r

q (7 D/4) as solar [2 ra (- z/ cos s] (47)2,r g s solar s,a s

where

Go = solar constant (442 BTU/ft2 hr)solar

a = average effective absorptance of the skin in the solar

spectrum

rS = average hemispherical albedo of the earth-atmosphere fors,aa given zenith angle

z = altitude of the balloon

D = earth diameter (7,920 mi)

s = angle between a vertical line through the balloon and

the earth's center and a line through the earth's center

and the sun

When the atmosphere below a balloon is partially covered by clouds,

the reflection from below will not be uniform. In such a situation one

can improve the accuracy of the calculations by dividing the visible

earth-atmosphere system into areas of uniform, but not equal, reflec-

tion. Assuming that each area reflects diffusely, one can approach

this problem by means of shape-factor algebra [59,69,70,71] just as in

calculations of radiation between two diffuse surfaces.

Although the atmospheric layers are curved, one can approximate

the atmosphere below the balloon by a flat surface without introducing

an appreciable error. The shape factor between a small sphere dA and1

47

a plane rectangle A when the sphere is located at one corner of a2

second rectangle having a common side with A (Fig. 11) is2

F = -tan1 x y (48)dA -A 4r l+x2 +y/

1 2 VI + x' + y"

where x = b/c and y = a/c.

The reflectance of various types of clouds can be estimated from

Table 8 in conjunction with the relationship [77]

/ -b t\r= r - e ) (49)

where

r = reflectance of cloud of thickness t

r = reflectance of cloud of infinite depth00

t = cloud thickness in meters

b = constant whose value depends on liquid water1

content or cloud type

An approximate method, sufficiently accurate for most purposes, is

to assume that: (a) the atmosphere ends at the intersection of a

straight line between the balloon and the horizon, and (b) the atmo-

spheric surface below the balloon is flat.

As x and y approach infinity, the shape factor for radiation be-

tween a sphere and an infinitely large plane approaches TrD /2, which is

larger by a factor of 1 - 4 z/Darth J than the correct value between

a small and a large sphere. At an altitude of 20 mi, the error would

be about 10%. A calculation taking the curvature into account is pre-

sented by Cunningham [72].

d A x x=b/c-TS- .(+.... y=a/c

C bb

C

A a 0.

I/ 9 0 0 o

I , /an' x yI -IdA1 -A2 4 tan T

g 1 S 2 + y 2

Fig. 11 Shape factor for a small sphere and rectangular area.

49

Table 8()

MEAN VALUES OF THICKNESS, REFLECTANCE, AND ABSORPTANCE

AS WELL AS CONSTANTS b AND b FOR VARIOUS CLOUD TYPES1 2

Mean Mean Mean

Reflectance Absorptance Thicknesst

b i b2

r() (2) a ( 3) (10 - 4/m) (x 10- 4/m)

Low cloud 0.60 0.07 450 21.9 4.5

Middle cloud 0.48 0.04 600 10.5 1.8

High cloud 0.21 0.01 1,700 1.2 0.15

Nimbostratus 0.70 0.10 4,000 4.2 0.78

Cumulonimbus 0.70 0.10 6,000 2.8 0.52

Stratus 0.69 0.06 100 155 16.8

(1)Adopted from Ref. 77

2 ) = r, (1 - e ;t) value of r corresponds to mean thicknesst co t

(3)a = a (1 - e ); value of a corresponds to mean thickness

Data from which the geographic distribution of reflected solar

radiation above the atmosphere can be calculated were gathered by the

Nimbus II meteorological satellite, which was launched 16 May 1966 and

remained in a polar, synchronous, circular orbit at a mean altitude of

707 mi until 28 July 1966. Its orbital period was 108.6 min, and during

its 13 daily orbits the entire globe could be observed day and night.

Reflected solar radiation in the wavelength range between 0.2 and 4 i

and the earth-emitted long-wave radiation between 5 and 30 p were mea-

sured with a radiometer. Details of the data reduction and instrumen-

tation are presented by Raschke [65]. In the following, only the

application of the averaged data to the evaluation of the reflected

solar radiation will be discussed.

50

Using the parameters defined in Fig. 12, the solar radiation

reflected from a surface element dS depends on the location of d e one Sthe globe, the time of year, the weather conditions (primarily the cloud

cover), and the zenith angle of the sun. The total hemispherical radi-

ation reflected from dS will bee

27 7/2

qreflected = I (I ,e) sin 0 cos 0 dedi (50)

O 0

where I is the intensity of the reflected solar radiation and 0 and i

are the zenith and azimuth angles of measurement, respectively. Since

the intensity of radiation reflected from the earth-atmosphere system

depends on 0, p, and C1 (the instantaneous zenith angle of the sun),

pl, the instantaneous bidirectional reflectance of the area element

dS at longitude X and latitude (, can be defined ase

i, (X,()co P (51)cos l I+

s

where I+ is the instantaneous direct intensity of the direct solar radi-s

ation incident on dSe

The hemispherical directional reflectance r(%) is the ratio of

the total reflected solar radiation (as would be received by a black

hemisphere placed over dS ) to the incoming solar radiation. It de-

pends only on the zenith angle C and the time of year. From airplane

and balloon measurements Raschke prepared diagrams relating the ratio

between the directional hemispherical reflectance (r) and the bi-

directional reflectance (p1 = p/7) at various azimuth angles within

relatively narrow ranges of the zenith angle. Figure 13 shows one of

these diagrams in which isolines of x = r/p are drawn as functions of e

and i for a range of solar zenith angles between 60 and 80°. With the

aid of these intermediate approximations, Raschke calculated the hemi-

spherical directional reflectance from the Nimbus II data and presented

his results as maps of the average albedo of the earth-atmosphere sys-

tem during a two-week period [65]. Figure 14 shows such a map for the

northern hemisphere during the period 16-28 July 1966. The albedo

ZENITH

\ I SATELLITE

~~~~~z----

OBSERVED AREA (X,4{)

Fig. 12 'Satellite- or balloon-to-earth geometric configuration.

52

I= 180°

= 135

O = 60 ° 0

x ( , ,'= CONST. )' ( ~' = CONST. ) j=o" = 9

p ( i9, ~, ~'= CONST. ) e 'O"~V P~ = 04

0.4-- = 45°

0.3

0= 300V3 = 690=0°

Fig. 13 Variation of directional solar reflectancewith zenith angle (60° < C < 80°)

53

.--t Nio t m,,t . ,* e*tW" , ,. S S" "I it - t"l· ~~~~ ~ ~~~~~-- i /ALBEDO 1%1 c.· 7-w:4N NORTHERN HEMISPHER ENIMBUS A 16-28 ALy 1966 . . .i / '

mm~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ .'./..'~ '.'\(,,

r ;~~~~~~~~~~~~, :'~/ 7' .' ._/7z ~ - ~. -~

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ,,

Fig. 14 Albedo map for the northern hemisphereuigt

.r · ~ ~ ~ ~ ~ ~ ~ ~ ·c~~

.. 1 :'~~~~4jl· p L/5-5

g ·-,~~~~~~~~~~~ I~~~

or\40~ ~~~~t40I

$Q$O r ··~4

r~~~~~~~~~~~~~~~~~~~~~~~~3

oar ~~~~~~~~~~~~~~, t: " ev " W W " "1 .

54

varies from 20 to 30% over coastal parts of the North American continent

to 60% over Greenland and the Sahara Desert.

In addition to the albedo maps, Raschke also presented a correla-

tion of available data showing the variation of the directional hemi-

spherical reflectance with zenith angle (Fig. 15) and the ratio between

the directional hemispherical reflectance and the bidirectional reflec-

tance at various azimuth angles for snow, stratocumulus clouds, and

cloudless ocean areas. With the aid of these diagrams one can make pre-

dictions of the reflection at altitudes above 50,000 ft. Such predic-

tions are quite accurate except when there are changes in cloud cover.

For engineering design it is usually satisfactory, however, to use

Eq. (47) in combination with the graph in Fig. 15 relating the average

hemispherical reflectance to the zenith angle, which can be calculated

(using Eq. 45) in a time-coordinate system related to a balloon flight.

3.00

2.8

2.6 0-0 FUNCTION USED IN THIS WORK

*-2.4 -* *TIROS E (0.2p-5.0p): AFTER ARKING, A., PRIVATE COMMUNICATION, 1967

A---A TIROS II (0.55p-0.75#): AFTER LEVINE, J. S., PRIVATE COMMUNICATION, 1967 I

2.2 [2.2 -- O TIROS I (0.55M-0.75p): REFERENCE 34 /

2.0 -

1.8 -.,--O .,,

1.4 s2--- -,--..

^-^ y \ I

1.4 - '-O;O•• c -

.8 I I I I i I I0 10 20 30 40 50 60 70 80 90

SOLAR ZENITH ANGLE C (°)

Fig. 15 Variation of directional hemisphericalreflectance with zenith angle.

55

IX. INFRARED RADIATION FROM THE EARTH AND THE ATMOSPHERE

(q in Eq. 15)3

The contribution to the total heat load on a balloon by the radia-

tion from the earth and the atmosphere is an important variable in the

energy balance formulated by Eq. (15). Unfortunately, the quantitative

prediction of this portion of the total heat load is subject to some

uncertainty because it depends on several factors difficult to specify.

The infrared radiation from the earth and the atmosphere varies as

the balloon ascends. Immediately after its launch, the balloon receives

radiation from the atmosphere over its entire surface, but as it ascends

the amount of air above the balloon continuously decreases. Eventually,

only the lower part of the balloon receives radiation from the atmo-

sphere. This infrared radiation is dependent on the altitude and cloud

cover, and since the cloud cover often changes rapidly, a balloon can

experience unexpected and unpredictable fluctuations in the radiation

from the earth-atmosphere. The situation improves considerably, how-

ever, as the balloon ascends; once it has risen above the clouds and the

weather to an altitude of about 60,000 ft, the contribution of radiation

from the earth and the atmosphere can be predicted with a considerable

degree of accuracy.

Radiation from the earth and the atmosphere is of considerable

interest to meteorologists. Comprehensive treatises on the radiation

characteristics of the atmosphere have been published during the past

decade by outstanding meteorologists such as Goody [73], Feigelson [62],

Budyko [74], and Kondratyev [75]. The radiation characteristics have

been investigated and reduced to convenient graphs and charts by

Simpson [76], Yamamoto [77], and Elsasser and Culbertson [78]. Using

Nimbus II data Raschke also prepared infrared radiation maps. Figure 16

shows the infrared radiation emitted by the earth and the atmosphere

into space over the northern hemisphere during the period 16-28 July 1966.

In connection with efforts to construct a general circulation

model of the atmosphere, several researchers including Houghton [66],

London [79], Manabe and Moller [80], and Davis [81] have studied

56

OUTGOING LONGWAVE RADIATION (cal cm mmin / ..

r~~~~76.. ~ ~ .' ', ,,IBM~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~.

NORTHERN HEMISPHERENIMBUSX116-28 JULY,1966 o~J4 j

/ Q 0.36 04

;' 0 310.482 .

/. ·. ~. ~ ~ 033 / / 4 042

03 33'1

_ ~~~~~,1~~~~~~~~~~~~~~~~. ;.M?~r0.3155-A'.: ..

0.33~~~~~~~~~~~~~~~~3

-' ' 033'~~~~~~~~' 0367 ~~~~~~ K033-. .4 I " ·: ...

sphere·during the period 16-28 July 1

-,'0.30,

i~~~~~~~~a -0.42~~~~~~~~~~~~~~~~~~~~~4

0.33 4rv

·,~~~~~~~~~~~~~~~~~:,

.36~~~~~~ ,'k

,0.42:0 0, it ': O36 x

I ~ ~ ~ ~ '''~~~~ ~ ~~~.I. '0.!2 ~.iIlrr~~~~~~~~~~~~~~~~~~~~~~'' ·

7-._ i. :~ ,~ . ,~0.33 ·/,,Y ,, -f)~~~~~~~~3

i .I~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

cm f 1( -L-- \· 2~~~~~~a o" VV T

Fig. 16 Infraredradainmpfrtenrhreisphere dr i n the' p erio 162 ul96

57

theoretical aspects of atmospheric radiation, but most of them calcu-

lated the meridional distribution and seasonal variation of the radia-

tion balance and not its global distribution. From a practical point of

view, Simpson's simple model [76,82] is still very useful for estimating

radiation heat loads, although Budyko [74] presents more accurate heat

balance calculations at the earth's surface. Neither of them, however,

considers the radiation within the atmosphere. Recently, Katayama [83]

has made valuable calculations of the radiation budget of the troposphere

over the northern hemisphere, and Sasamori [84] has developed a method

for calculating the upward and downward radiation flux in a cloudless

atmosphere. Sasamori's calculations are based on empirical formulas for

the spectral variations of the absorptances of water vapor, carbon diox-

ide, and ozone (the constituents of the atmosphere with absorption bands

in the infrared range) and on the assumption that the surface of the

earth is perfectly black. A summary of his results is shown for various

latitudes in Fig. 17, in which the upward and downward radiation fluxes

are plotted as a function of altitude. The spectral characteristics of

the radiation are not shown but can be calculated from the empirical

relations in Ref. 84.

-25 LAYERS ( YMAMOTO CHART).o 6 LAYERS(EMPIRICAL FORMULA)

20 a, , 1 ,T1 - '0- 20

0-10 N 30-40'N 60-N70'N

15 *sb 9· \Pi 0 1

E E

to .4 o .1 . » . *! - 150o Io

Ft-F' FFF F

1;) 6 -FX \F W

F'-F4 5

200 400 600 00 200 400 600 20 400 600

FLUX (cal/cm 2 day)

Fig. 17 Upward and downward radiation flux as a function of altitude.

58

It should be noted that the atmosphere is almost transparent to

wavelengths between 8 and 12 p. Consequently, a balloon will receive

such radiation directly from the earth. At longer and shorter wave-

lengths the atmospheric water vapor and carbon dioxide have strong

absorption bands, and in those parts of the spectrum the infrared radia-

tion received by a balloon is emitted by the atmosphere, usually at a

temperature lower than that of the earth. Some balloon materials ex-

hibit considerable variations in the monochromatic absorptance in the

infrared, and to reliably estimate the infrared radiation absorbed by

the skin, integrations over the spectrum between 6 and 100 p, as shown

by Eq. (33), are necessary [59,85]. For these integrations, simplified

models of the variations in the monochromatic radiation of the atmo-

sphere, such as those given in Refs. 9 or 59, are quite satisfactory.

In view of these complicated conditions it is not surprising that

several approaches have been taken to calculate the radiation from the

earth and its atmosphere which is absorbed by a balloon. Lally [3],

dealing only with superpressure balloons, divided the incoming infrared

radiation into four parts: from below, from above, and from two sides,

each illuminating an area of ( D2/4). The radiation environment is\ 6 / /

then estimated using the following simplified assumptions:

1. Downward radiation.

a. If clouds above: downward flux equals blackbody radiationfrom a source at the temperature of the cloud base.

b. If clear above: balloon above 9 mi--3.2 BTU/ft 2hr;balloon between 6-9 mi--6.3 BTU/ft 2hr; balloon below6 mi--12.7 BTU/ft2hr.

2. Upward radiation.

a. If clouds below: upward flux equals blackbody radiationfrom a source at the temperature of the cloud tops.

b. If clear below: upward radiation will vary from 48 to144 BTU/ft2hr depending on altitude and air mass. Table 9provides estimates of upward flux for several altitudesand air masses.

Table 9

RADIATION ENVIRONMENT FOR SUPERPRESSURE BALLOONS1

Average nighttime clear sky Mylar balloon (a = 0.05)

balloon super temperatureAltitude Season Air

Temperature Mylar balloon Metallized top Temperature in- Maximum added Maximum daytime

(°C) (°C) balloon crease per W/m2 solar flux temperature in-

(°C) increment (W/m2 ) crease

(*C) (°C)

Temperate,winter -10 0 5 0.24 35 8

3 km Temperature,(700 mb) summer 5 - 3 2 0.21 35 8

Tropic 10 - 5 0 0.20 35 8

Temperate,winter -30 0 8 0.30 35 10

6 km Temperate,(500 mb) summer -15 - 5 5 0.27 35 9

Tropic - 5 -10 2 0.25 35 9

Temperate,winter -50 5 15 0.36 40 14

9 km Temperate,(300 mb) summer -35 - 5 7 0.34 40 13

Tropic -30 -10 2 0.34 40 13

Temperate,winter -55 10 20 0.36 45 16

12 km Temperate,(200 mb) summer -55 10 20 0.36 45 16

Tropic -50 5 15 0.36 45 17

Temperate,winter -60 5 0.42 45 19

16 km Temperate,(100 mb) summer -65 10 0.42 45 21

Tropic -80 15 0.47 45 21

Temperate,winter -55 - 5 0.45 45 20

24 km Temperate,

(30 mb) summer -55 - 5 0.45 45 20

Tropic -55 - 5 0.45 45 20

1Adopted from Ref. 3.

60

3. The radiation from the sides can be estimated as equal toblackbody radiation from a source at the temperature of the airat balloon altitude.

London [85] considered only the upward and downward fluxes. The

advantage of this approach is that theoretical analyses and measurements

of atmospheric radiation usually provide the upward and downward, but

not horizontal, radiation fluxes. An instrument which has been widely

used at ESSA to obtain measurements of the upward and downward atmo-

spheric radiation was developed by Suomi and Kuhn [86]. This instrument

(Fig. 18) is basically a double-faced, hemispherical bolometer with

broad-response, blackened sensing surfaces shielded from convection cur-

rents by thin membranes of polyethylene. The upward and downward radi-

ation flux can be calculated from the temperatures, measured with tiny

thermistors, of the sensor surfaces and the air. A flight test of this

instrument, also called an "economical net-radiometer," is reported in

Ref. 87. Data taken during several balloon flights have been published

[e.g., 88-91], but data from other flights are still being processed at

ESSA [53]. Whenever the spectra of the upward and downward infrared

radiation are known, the rate at which infrared radiation is transferred

to the balloon skin can be calculated from the relationship

100

q = (7 D2 2) a (X) Gu (X) + a (X) i, (X) d3 g eff i,up eff i,down

(52)

where aeff (X) is the effective monochromatic absorptance of the balloon

film, which can be approximated from Eq. (43) by

(a ) [1 +1 -a(X) - r(X) ]a (X) [1 + 1- r(X)

and where G. (X) is the upward monochromatic radiation as measured byi,up

a hemispherical flat surface radiometer, and G do(X) is the downwardimonochromatic radown

monochromatic radiation, measured likewise.

61

Thermal Isolation

0.00025in. Aluminum Stretched Black Coated 7 7/ _a .--_i

____ __ __ -i41CBI Bead Thermistor

4.75 9a -. 75 -- 7

0 0.0005 in, Polyethylene Film

Fig. 18 Suomi-Kuhn flat plate radiometer.

62

Unfortunately, only the total average radiation over all wavelengths

is usually measured. Typical data from meteorological measurements are

illustrated in the graphs of Figs. 19, 20, and 21. In all three figures

the air temperature and the total radiation on a horizontal surface from

above and below are plotted as a function of altitude. Also shown are

the net radiation flux (i.e., the difference between the radiation from

above and from below) and the atmospheric cooling rate--quantities which

are of meteorological interest only.

Figures 19 and 20 show the average radiation environment at Green

Bay, Wisconsin, during summer and winter, and Fig. 21 shows the radia-

tion environment of a desert island at 2°S. The downward radiation in-

creases markedly with decreasing altitude, whereas the upward radiation

decreases slowly with increase in altitude. In winter, when the atmo-

spheric temperature variation with altitude is less than in summer, the

changes in radiation are also smaller. Near the equator where variation

in the atmospheric temperature with altitude is even larger than at

Green Bay during the summer, the changes in the infrared radiation en-

vironment are also more pronounced. It should be noted that the infra-

red radiation graphs in Figs. 19, 20, and 21 represent averages of the

total hemispherical radiation over all wavelengths. When the spectrum

is not known, an average absorptance must be estimated to calculate the

rate of absorption of infrared radiation from Eq. (52). There will, of

course, always be variations with changes in weather conditions, and the

amount of radiation absorbed by a balloon skin or an instrument package

will depend not only on the spectrum of the incoming radiation but also

on the variation with wavelength, as yet not known, of the directional

hemispherical absorptance of the receiving surface.

A third method of estimating the infrared heat load has been pro-

posed by Germeles [8]. It uses the measurements obtained by a "black-

ball" radiometer (Fig. 22), an instrument widely used by meteorologists

to measure the radiation in the atmosphere. The instrument, developed

by Gergen [92], consists essentially of a small balsa dodecahedron

painted black and surrounded by a convection shield. A thermistor im-

bedded in the center is used to measure T , the "equilibrium radiation

TEMPERATURE (°C)-90 -80 -70 -60 -50 -40 -30 -20 -10 0 10 20 30

100,|^~ ,Xq ~~ 4 |LEGEND

~200 0t \~ v 7 ~ o °DOWNWARD FLUX

300 03 UPWARD FLUXX AIR TEMPERATURE

.5 400E

500\ \U)

u) 600iti

700

800

900

1000 I-0.I 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

RADIATION (ly/min)

Fig. 19 Radiation environment at Green Bay, Wisconsin, in the summer.

TEMPERATURE (°C)-90 -80 -70 -60 -50 -40 -30 -20 -10 0 10 20 30

00

100

~ .8 {¥~ [|~ ~LEGEND

~200~ ~~ t k i~ °0o DOWNWARD FLUX300 \ 0 UPWARD FLUX

^~_ i R N~~ Q |~ X AIR TEMPERATURExQ 400E

V)

500

39 600-

70 0 \ \

800 -

900- : -

1000-0o. 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

RADIATION (.y/min)

Fig. 20 Radiation environment at Green Bay, Wisconsin, in the winter.

TEMPERATURE (°C)-90 -80 -70 -60 -50 -40 -30 -20 -10 0 10 20 30

100

X1 00 "o ~ > ~ \q |~ :LEGEND200-

~ 2 0 0 0t~ ~:~ ^Y "~0 ° DOWNWARD FLUX

300- UPWARD FLUX] K -Ad : x AIR -TEMPERATURE

-- 400

500

O 600

(t 700 \

800

900

1000-0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

RADIATION (.y/min)

Fig. 21 Radiation environment at a desert island, 2°S.

66

Fig. 22 Gergen "T Black Ballo" radiometer.

67

temperature" of the device. From this measurement, the radiation is

then calculated from the relationship [92]

G. = a T4 (53)1 r

Measurements with this device are carried out during the night so

the detector is not affected by the sun. It is claimed that the temper-

ature recorded by the black ball is representative of the total radia-

tion field to which it is exposed. Black-ball equilibrium temperatures

have been recorded as a function of altitude at different geographic

locations, and it has been observed that large changes in the black-ball

temperature profile occur within days at the same location. The results

of these measurements support the following approximate, simple, general

rule. At ground level, the radiation temperature is usually less than

the air temperature, but the deviation is about 10°F. The equivalent

radiation temperature decreases almost linearly with altitude up to the

tropopause, where it becomes about 75% of its ground value. From there

to higher altitudes the radiation temperature remains approximately con-

stant, indicating that most of the radiation is received from below.

Germeles [8] used data obtained by means of a black-ball radiometer

to calculate the infrared radiation on a balloon. He assumed that the

black-ball temperature profile is known. Since the black ball has a

shape similar to that of a balloon, the radiation absorbed by the bal-

loon can be obtained from the relationship

= 4.83 a a V213 T4 (54)q3 ' aeff g r

It is apparent that the estimates of radiation emitted by the atmosphere

and the earth require considerable improvement and elaboration. It may

well be that such improvement has not been made because the properties

of balloon fabrics are not precisely known, so that even if the heat

load could be calculated with considerably higher accuracy, the uncer-

tainty in the absorption load would remain.

68

At this juncture I would recommend: (a) using black-ball radiometer

data, when available, or (b) calculating the heat load from above and be-

low, making use of measurements obtained with an instrument such as the

Suomi-Kuhn radiometer, and calculating the actual amount of radiation

with a simple model of the atmosphere such as that proposed by Simpson

[76], which considers only water and carbon dioxide absorption. This

approach, described in Ref. 58 for use in spacecraft design, is also

directly applicable to calculations on balloon and instrument packages.

For calculations of heat loads in areas where the Suomi-Kuhn instrument

has not been flown, I suggest use of the Nimbus II data, which are sim-

ilar to albedo data in that they include the total radiation flux emit-

ted by the earth's atmosphere. This approach, of course, will only be

useful for altitudes above 60,000 ft since satellite data are taken far

outside the atmosphere. An aid to further improvement of the calcula-

tion of the heat load would be to fly several balloons instrumented to

measure gas temperature, the radiation heat load, and air temperature

simultaneously. Although such measurements have not been made, they are

feasible with existing instrumentation.

69

X. BALLOON TRAJECTORY COMPUTER MODEL

DESCRIPTION AND VALIDATION

A FORTRAN code has been created which utilizes the analyses of

balloon dynamics and thermodynamics contained in the previous chapters

to predict the vertical motion of zero-pressure balloons. Such a model

has many uses, It can predict ascent rates, time to reach ceiling, and

ceiling altitude in advance of an actual flight. An optimal ballasting

or valving schedule can also readily be determined by preliminary model

runs. The program and its use are described in detail in the appendix

which follows.

With a model as complex as that required to predict balloon motion,

many parameters enter which could be specified in such a way that the

model predictions would represent any single flight to good accuracy.

If, however, such a model is to be valid for a variety of balloon types

and flight conditions, a more fundamental approach than a simple curve-

fitting exercise is needed. In order to calibrate the present computer

simulation, the following approach has been used. Four recent, accurate-

ly monitored flights, which include a twenty-fold range in balloon volume,

daytime and nighttime launches, both hydrogen and helium lift gas, and

Mylar and polyethylene fabric materials, have been selected for model

validation. Of the thirty-three parameters which must be specified to

permit a unique solution to the equations of motion and energy, all but

two were determined by independent means. The parameter list is detailed

in the appendix and includes:

balloon gas transport properties, molecular weight andspecific heat

balloon fabric radiation properties (IR and UV)

balloon system weight and balloon volume

launch-time parameters - time of launch, initial freelift, fabric and gas temperatures, locationand altitude of site

atmospheric properties - temperature profile andblack ball profile

70

In principle, all but eight of the thirty-three parameters can be

specified readily to good accuracy from handbooks, The American Ephemeris

and Nautical Almanac, sonde data, and measurements normally made at a

balloon launch site. The eight parameters which are more difficult to

evaluate are:

1) internal convection coefficient

2) external convection coefficient

3) & 4) fabric UV and IR absorptances

5) initial free lift

6) atmospheric "black ball" profile

7) & 8) initial fabric and gas temperature

The latter two parameters, 7) and 8), were evaluated by assuming that

initially the fabric and lift gas are at launch site ambient temperature.

To evaluate the black ball profile, the simple rule described on p. 67

was used. Such an approach was justified by a sensitivity analysis which

showed that infrared profiles are not one of the sensitive factors during

the ascent phase of a flight. At float altitudes, the black ball tem-

perature has always been measured as approximately constant, implying

that all the IR radiation sources lie below the balloon. Consequently,

the only important component of a black ball profile is its value at

tropopause and above, where the temperature is nearly uniform.

The multiple pass fabric radiation properties 3) and 4) were

evaluated by the method described in [48] where the authors measured

absorptances in the IR and UV bands for more than 50 samples of balloon

fabrics. Computation of effective balloon values from such samples

(vide pp. 33-35 above) does not require the spherical balloon assumption

and should be quite accurate. However, the lubricating powder normally

used on plastic balloons has an unknown effect on the IR and UV absorp-

tance values.

The initial free lift of a balloon is not measured directly, but is

determined by semi-empirical means at most balloon launch facilities. A

sensitivity analysis has shown that this is one of three parameters

which must be specified quite accurately if a computer model is to

71

replicate an actual flight, Although the accuracy of the method

presently used to determine initial free lift could easily be improved,

the free lift specified in the balloon data sheets was usually assumed

to be a non-adjustable parameter for this study within its normal limit

of accuracy (+ 1%).

The only two parameters which were considered to be adjustable

during the model calibration were the inner and outer convective heat

transfer coefficients. As noted earlier, no heat transfer measurements

have been made on balloon shaped bodies and the expressions proposed for

the convective heat transfer coefficients (Eqs. 18, 20, 28) were at the

outset recognized to be only approximate. As a result of the present

model calibration, it is suggested that for zero pressure balloons the

external coefficient be increased by 50% over that predicted by Eqs, 18

and 20 and the internal free convection coefficient be increased by a

factor of three over the value computed from Eq. 28. In a survey of

convection in enclosures such as spheres and rectangular boxes, Ostrach

[98] has also shown that free convection Nusselt numbers for these

geometries contain variations of the same order, of 300%.

The balloon drag coefficient was calculated throughout all phases

of the flight by assuming the balloon to be a sphere. The data in [97]

were curve-fitted in several ranges of Reynolds number and their use

gave good replication of measured flight trajectories.

The measured time-altitude history of the four flights used for

model validation are plotted in Figs. 23-26 along with the model pre-

dictions. The agreement is seen to be good for all of the flights

during the ascent as well as the float regimes. From flight to flight

no adjustments were required in any parametric representations of the

balloon dynamics and thermodynamics to achieve this agreement.

In addition to altitude correlations, it is possible to attempt to

correlate model predictions of balloon gas and fabric temperatures with

measured flight values. In order to determine flight values of Tf and

Tg, upon which the model computations are based, an unmanageably large

72

90000 -

8000070000 ~- /

70000 //

60000 /

50000

40000

F 30000H:2 200000

F --.J 10000 /

o I I -- I I , 1 - , I l . -

0 2000 4000 6000 8000 10000 12000

TIf E-SEC THERMISTOR FLIGHT

Fig. 23 Vertical trajectory of "Thermiscor" Flight.

October, 1964, 250,000 ft 3 polyethylene balloon,He lift gas. Solid line - model prediction;dashed line- data.

73

80000

60000

50000 -

40000

-'1'

30000 -

Cl)

LL.20000 -

- / / -JI

- 10000-

0 2000 4000 6000 8000 10000 12000

TIME-SEC STRATOSCOPE S4-2

Fig. 24 Vertical trajectory of Stratoscope S4-2 Flight.

July, 1965, 5.5 million ft 3 Mylar balloon, Helift gas. Solid line - model prediction; dashedline - data,

74

90000

80000 -,//

70000

60000

50000

40000

30000

y 20000

10000

00 2000 4000 6000 8000 10000 12000

TIME-SEC STARGAZER FLIGHT

Fig. 25 Vertical trajectory of Stargazer Flight.

December, 1962, 3.1 million ft 3 Mylar balloon,He lift gas. Solid line - model prediction;dashed line - data.

75

140000 -

120000 - /

100000

8 0 0 0 0 -/ /80000 -

60000 - /

I-

_ 20000

. _ /

0 1000 2000 3000 4000 5000 6000 7000 8000

T I ME-SEC HYDROGEN FL I TE 571 PT

Fig. 26 Vertical trajectory of Speclal Hydrogen Flight,

July, 1970, 4 .5 million ft 3 polyethylene balloon,H2 lift gas. Solid line -model prediction; dashedline - data.

76

grid of thermocouple o thermistors would be needed. In the tThermistor'

Flight two film nd two balloon gas temperatures were -measured, These

measurements show the same trends as the model, but numerical equivalence

was not achieved simply because such a small sampling of gas and fabric

temperatures does not permit an accurate computation of space-averaged

values of T and T.f g

Caution should be exercised in accepting the claims of some previous

investigators who have asserted that they were able to predict balloon

motion from a mathematical model. Germeles [8], for example, has made

such a claim, but close inspection of his model showed a 20-fold error

in balloon gas transport properties, an improper use of black ball tem-

perature data, and an assumption that the heat transfer mechanism at the

balloon outer surface was by mixed forced and free convection mode at

Eckert numbers (Gr/Re 2 ) where such mixed mode convection has never been

observed in nature or the laboratory. As noted above, the proper selec-

tion of the large number of parameters present in this model can give

the illusion of a successful model when, in fact, only a good curve fit

has been achieved without any fundamental representation of reality.

77

APPENDIX

A COMPUTER PROGRAM FOR PREDICTING THE VERTICAL MOTION OF

ZERO-PRESSURE BALLOONS

JAN KREIDER

I

79

INTRODUCTION

The analysis presented in the body of this report can be used to

predict the vertical motion of balloons by means of a computer. This

appendix gives a complete outline of the steps in the computer program

developed to solve the equations describing the balloon motion and to

determine parameters of interest in the design and performance analysis

of high altitude balloons. The program is similar in concept and exe-

cution to that presented in Ref. 8, but it has been updated to account

for recent developments in heat transfer and has been coded in FOR-

TRAN IV.

The fluid mechanical and thermodynamic model study in the body of

this report requires six nonlinear, ordinary differential equations

(Eqs. 5, 11, 12, 14, 15, and z = U) to describe the vertical motion of

zero-pressure high-altitude balloons. The six dependent variables

listed below are all functions of the independent variable, time t:

z(t) - altitude of balloon above sea level

T (t) - balloon gas temperatureg

Tf(t) - balloon skin fabric temperature

V (t) - volume of balloon gas

U(t) - balloon vertical velocity

m (t) - mass of balloon gasg

The following parameters must be independently specified to obtain

a unique solution of the six model equations:

1. Atmospheric properties: density (p ), pressure (P ), temper-a *a

ature (Ta), thermal conductivity (ka), viscosity (p ), Prandtl number

(Pr ), and molecular weight (M ), as functions of altitude z.a

2. Balloon gas properties: thermal conductivity (k ) and viscos-

ity (p ) as functions of balloon gas temperature (T ); also Prandtl

number (Pr ), molecular weight (Mg), and specific heat at constantgvolume (

volume (c ).v~g

80

3. Infrared radiation field: specified by "black ball" radiation

equilibrium temperature (T ) as a function of z according to Eq. (53).

Alternatively, the rigorous approach outlined in Sect. IX concerning the

evaluation of q can be used.

4. Balloon fabric properties: mass (mf), specific heat (Cf),

effective infrared emittance and absorptance (e ff i and aeff i ) , and

effective solar absorptance (aeff,s).

5. Payload: ballast mass (mB), fabric mass (mf), payload mass

(m ), and a ballasting schedule if any.

6. Valving schedule: pounds of lift lost as a function of time

because of valving.

7. Other parameters: drag coefficient (CD), apparent additional

mass coefficient (CM), maximum balloon volume (VBM), and correction fac-

tors for convective heat transfer equations (C4 and C6).

8. Launch information: latitude (X) and longitude ((), local hour

angle (h), and solar declination (a ).

9. Initial conditions: the initial values of the six dependent

variables specified at t = 0.

The program integrates the six differential equations numerically using

a fourth-order Runge-Kutta technique once the listed parameters are

specified. The symbols used in the program are shown in Table Al. The

Fortran listing of the program is also provided.

INPUT DATA

The input data contain all the parameters and initial values re-

quired for a unique solution of the equations along with certain other

information concerning output, the integration, and duration of flight.

The data are read in from cards of two types:

1. Control cards: for identifying the type of data, these have an

integer number from 0 to 14 in 12 format in the first two columns.

81

Table Al

LIST OF IMPORTANT FORTRAN SYMBOLS

NCARFORTRAN Report Description Units

ABIR a Effective infrared absorptance ofballoon fabric for multiple absorp-tion of infrared radiation (InputData)

ABUV a Effective ultraviolet absorptance ofeffs balloon fabric for multiple absorp-

tion of solar radiation (Input Data)

AIRM m(z) Optical air mass used to completeatmospheric transmittance to solarradiation

ALPHA -Balloon gas viscosity parameter lb /ft sec(Input Data)

B (dm /dt)g/g Instantaneous ballasting rate lb/sec( BB(I) + BO )

BO -Fixed ballasting rate (Input Data) lb/sec

BB(I) - Ballasting schedule, corresponding lb/secto TB(I) (Input Data)

BETA -Balloon gas viscosity parameter(Input Data)

Cl cf Specific heat of balloon fabric ft lb/lb °R(Input Data)

C4 -Correction constant for convectionfrom air to balloon fabric (Eqs. 18and 20) (Input Data)

C6 -Correction constant for free convec-tion from balloon gas to fabric(Eqs. 28, 29) (Input Data)

CB CM Apparent additional mass coefficientof balloon (Input Data)

CAIR k Thermal conductivity of air ft lb/ft sec°Ra

CCSI(I) - Array of solar radiation factors -Type 7 data (Input Data)

82

Table Al (cont.)

NCARFORTRAN Report Description Units

CD CD Drag coefficient of balloon

CG k Thermal conductivity of balloon gas ft lb/ft sec°Rg

CIR(I) - Array of infrared radiation factors -Type 13 data (Input Data)

CLATD cos X * Cosine of latitude x cosine ofcos a solar declination

s

CP Specific heat of balloon gas at ft lb/lb °Rconstant pressure (c + R/M) m(Input Data)

CV c Specific heat of balloon gas at ft lb/lb °R9 g ~ constant volume (Input Data)

DDRAG - 1.21p ACDIUU2V /3/2g ft lb/sec

DEC a Solar declination during flight - degassumed constant (Input Data)

DEH2 q3 Absorption of IR radiation by ft lb/secballoon fabric

DEH3 q5 Emission of IR radiation by balloon ft lb/secfabric

DEH4 q Heating of balloon fabric by air ft lb/secthrough convection

DEH6 q Heating of balloon gas by fabric ft lb/secthrough free convection

DEH7 q Absorption of solar radiation by ft lb/secballoon fabric

DEVE - Rate of energy loss when gas is ft lb/secvalved or exhausted

(c g + R/Mg) (T )v,g g gg

DEWA - Rate of work done by balloon gas in ft lb/secexpanding against the atmosphere

(Pavg)

83

Table Al (cont.)

NCARFORTRAN Report Description Units

DKE -Rate of change of kinetic energy of ft lb/secballoon system

(dz(mg + mf + mB + mp + CPaVg) dt /c

DPE -Rate of change of potential energy ft lb/secof balloon

(P V m - - mr) dzag g f -mp dt g/gc

DPLT - Plotting routine time interval - must secbe an integer multiple of H (InputData)

DPR -Printing time interval (Input Data) sec

E E Rate of volumetric gas flow due to ft /secexhausting

EAIRT(I) T Array of observed atmospheric air °Ftemperatures corresponding to timesETIME(I) (Input Data)

EALT(I) z Array of observed balloon altitudes ftcorresponding to times ETIME(I)(Input Data)

ECHK -ENET - EP Energy check ratioENET (Eq. A10)

EFILMT(I) Tf Array of observed balloon film tem- °Fperatures corresponding to timesETIME(I) (Input Data)

EGAST(I) T Array of observed balloon gas ten- °Fg peratures corresponding to times

ETIME(I) (Input Data)

EMIR eeff,i Effective infrared emissivity ofef1 balloon fabric for multiple emission

(Input Data)

84

Table Al (cont.)

NCARFORTRAN Report Description Units

ENET -Difference between internal energy ft lbof gas and fabric at time t and at

beginning of flight, t ER dt (Eq. Al)

0

EP -Same as ENET but computed by ft lbsummation

EP = (Z ERi ) * Hi i

ER - Rate of change of mechanical and ft lb/secthermal energy of balloon (Eq. Al)

(- DKE + DPE - DDRAG - DEWA + DEVE+ q2 + q3 + q4 - q5)

ETIME(I) -Array of time of observed flight secdata points (Input Data)

ET0T -Internal energy of balloon gas and ft lbfabric (c m T + cfmfTf)

v,g g g f f f

EVEL(I) -Array of observed balloon velocities ft/seccorresponding to times ETIME(I)(Input Data)

F(1) dz/dt Vertical velocity of balloon ft/sec

F(2) d2 z/dt2 Vertical acceleration of balloon ft/sec2

F(3) Tf Rate of change of fabric temperature °F/secf

F(4) T Rate of change of gas temperature °F/secg

F(5) m Rate of change of gas mass lb /secg m

F(6) V Rate of change of balloon volume ft3/secg

FL - Free lift (pV - m - m lb

- mp)g/g

FLO - Initial free lift (Input Data) lb

85

Table Al (cont.)

NCARFORTRAN Report Description Units

4 2FLUXIR aT Atmospheric radiation flux at ft lb/sec ft

r balloon surface - computed fromblack ball profile RTIR(I)

2FS0L G Solar constant outside the ft lb/ft2sec

~s atmosphere ( = 96)

G g Acceleration due to gravity ft/sec2

( = 32.17)

GAMMA -Balloon gas conductivity parameter ft lb/lb °R(Input Data)

GHA GHA Greenwich hour angle of the sun at degstart of flight - positive westfrom the prime meridian (Input Data)

GMTS -Greenwich Mean Time at start of secflight (Input Data)

H - Time interval for computer integra- section (Input Data)

NBI -Number of Type 9 TB/BB data cards

NCIR -Number of Type 13 data cards

NEX -Number of Type 12 data cards

NIR - Number of Type 11 data cards

NSI -Number of Type 7 data cards

NT -Number of Type 10 ZZ/TZ data cards

NVI -Number of Type 8 TV/VV data cards

NYC -Number of model points to be plottedby YPLOT

P P Atmospheric pressure lb/fta

PERB -Cumulative percent of ballastdropped

86

Table Al (cont.)

NCARFORTRAN Report Description Units

PERE -Cumulative percent of gas exhausted

PERV - Cumulative percent of gas valved

PHI V/3 One third power of balloon volume ftg D = 1.24), Ab = 1.21)2, S = 4.834)2

PRG Pr Prandtl number of balloon gasg (Input Data)

R R Molar universal gas constant ft lb/lb mole°R( = 1545)

RA Gas constant for air (R/XMA) ft lb/lb °Rm

REN Re Balloon Reynolds number basedon ascent (descent) rate andequivalent diameter of balloon gas(1.24 x PHI)

RG Gas constant for balloon gas (R/XMG) ft lb/lb °Rm

RHO Pa Local density of atmosphere lb /ft 3

RH00 Density of air at ZZ(1) (q.v.) lb /ft 3

level (Input Data)

RH0G p Density of balloon gas lb /ft 3

RH~G Pg m

RTIR(I) T Array of black ball equilibrium °Rradiation temperatures corres-ponding to ZIR(I) (Input Data)

RTZ(I) (pT) Array of products of air density and lb °R/ft 3

temperature computed from atmos-pheric data corresponding to ZZ(I)

SB0LZ Stefan-Boltzmann constant ft lb/ft2 ecR 4

( = 3.6995 x 10' 1 0

SLATD sin A * Product of sines of latitude andsin a solar declination

s

TA T Atmospheric air temperature °Ra

87

Table Al (cont.)

NCARFORTRAN Report Description Units

TB(I) Array of times for scheduled secballasting - measured from startof flight (Input Data)

TCSI(I) -Array of times for solar radiation secfactor CCSI(I) measured from startof flight (Input Data)

TEMP T Air temperature computed from RHOT °Ra

TFO - Initial temperature of balloon °Rfabric (Input Data)

TGO -Initial temperature of balloon °Rgas (Input Data)

TIR(I) - Array of times for infrared secirradiation factor CIR(I) measuredfrom start of flight (Input Data)

TITLE - Descriptive title of model run

TMASS (mf + m Total mass of balloon system lbf g m

+ mB + mp)

TRANS G /G Transmittance of atmosphere tos Z s, solar radiation

TV(I) - Array of specified time for scheduled secvalving VV(I) measured from start offlight (Input Data)

TZ(I) -Array of atmospheric temperatures °Rcorresponding to ZZ(I) measuredupward from mean sea level (InputData)

TZA(I) -Atmospheric temperature profile °R

TZB(I) interpolation variables

V - Instantaneous valving rate (lb lift lb/seclost per sec) (VO + W(I))

VO - Constant valving rate (Input Data) lb/sec

88

Table Al (cont.)

NCARFORTRAN Report Description Units

VALD -Cumulative mass of gas valved lbm

VBM -Design maximum balloon volume ft(Input Data)

VISA a Viscosity of air lb /ft sec-a m

VISG pg Viscosity of balloon gas lbm/ft sec

W(I) -Scheduled rate of gas valving lb/sec(lb lift lost per sec) (InputData)

WF mf Mass of balloon fabric (does not lbinclude fittings) (Input Data)

WGO -Initial mass of balloon gas lbm

WGR0SS (m + m Total mass of payload (includes lbP B fittings and tapes), fabric, and

+ mf) ballast

WPB -Total initial mass of payload and lbballast (Input Data)

X t Time measured from start of balloon secflight to be modeled - may differfrom time measured from start ofmodel run XO

XO Time during flight to be modeled secat which model run is initiated(Input Data)

XLAT X Latitude of launch site - positive degnorth of equator (Input Data)

XL0NG LONG Longitude of launch site - measured degwestward from prime meridian (InputData)

XLAH 6 Local hour angle of the sun deg(XLAH = GHA - XLONG + X/240,)

89

Table Al (cont.)

NCARFORTRAN Report Description Units

XMA M Molecular weight of aira

XMG M Molecular weight of balloon gasg (Input Data)

XT -Specified final flight termination sectime measured from start of flight,not from start of model run XO(Input Data)

Y(1) z Altitude of balloon above mean ftsea level

Y(2) U Vertical velocity of balloon ft/secsystem

Y(3) Tf Mean temperature of balloon fabric °R

Y(4) T Mean temperature of balloon gas °Rg

Y(5) m Mass of balloon gas lbg m

Y(6) V Volume of balloon ft3

Y20 - Initial balloon velocity at XO ft/sec(Input Data)

ZO - Initial balloon altitude above ftmean sea level at XO (Input Data)

ZIR(I) - Array of altitudes above mean ftsea level for black balltemperatures RTIR(I) (Input Data)

ZMIN - Lowest permissible altitude - ftusually corresponding to groundlevel above mean sea level

(ZMIN > ZZ(1) > 0)

(Input Data)

90

Table Al (cont.)

NCARFORTRAN Report Description Units

ZZ(I) - Array of atmospheric temperature ftprofile altitudes above meansea level corresponding to TZ(I)

ZZ(l) < ZMINZZ(1) .< ZO

(Input Data)

91

2. Data cards: these have the first two columns blank and then

input data in F10.4 or E10.0 format.

There are 12 data types (coded Types 3-14), each of which must be

preceded by a control card identifying it. A tabular representation of

a data deck is shown in Table A2. The order of the control cards is to

be noted. The control card with the number 1 initiates computation; it

is, therefore, placed after all the data. The control card with the num-

ber 2 initiates plotting and after plotting is completed, terminates the

program. It is, therefore, the last card in the deck as shown.

TYPE 3 DATA

These data include:

C1 - balloon fabric specific heat (cf) (ft lb/lbm°R)

C4 - convection heat transfer factorsC6 J

ABIR - effective infrared absorptance (a ff )eff,i

EMIR - effective infrared emittance (e ff i)

ABUV - effective solar absorptance (aeff,s)

TYPE 4 DATA

These data include:

WPB - payload mass at start of model run (mB + mp) (lb m )

WF - balloon fabric mass (mf) (lb m )

VBM - maximum balloon volume as specified by manufacturer (ft )

FLO - free lift at start of model run (lb)

CB - apparent additional mass coefficient for a sphere (CM)

ZMIN - lowest permissible altitude (ft) (ZMIN 2 ZZ(1) > 0)

TYPE 5 DATA

These data include:

TGO - gas temperature at start of model run (°R)

TFO - fabric temperature at start of model run (°R)

ZO - balloon altitude above MSL at start of model run (ft)

(ZO ' ZZ(1) > 0)

Y20 - balloon vertical speed at start of model run (ft/sec)

DEC - solar declination on launch day - positive N of equator

(Source: The Air Almanac) (deg)_.......... . -.. ..

92

XLAT - latitude of launch site - positive N of equator (deg)

XLONG - longitude of launch site - positive W of prime meridian(deg)

TYPE 6 DATA

These data include:

H - integration time interval (sec)

DPR - output printing interval (sec)

XO - model run start time measured from flight start time (sec)

XT - model run termination time measured from flight start

time (sec)

GMTS - Greenwich Mean Time at model run start time (sec)

GHA - Greenwich Hour Angle of the sun at flight start timemeasured positive W from the prime meridian (Source: TheAir Almanac) (deg)

DPLT - output plotting interval (sec). If zero, plots are notgenerated.

TYPE 7 DATA

These data include two arrays of time TCSI in seconds from the start

of flight and the "solar radiation factor" CCSI which accounts for

cloud cover, etc. They are in order of increasing TCSI and are

limited to 100 cards. If no cloud effects are considered, the

correct data are the two cards:

TCSI(1) = XO CCSI(1) = 1.0

TCSI(2) = XT CCSI(2) = 1.0

CCSI data are extrapolated if the balloon flies longer than

TCSI(NSI). NSI is the number of Type 7 data cards.

TYPE 8 DATA

These data include:

VO - The constant valving rate on the first card (lb lift lost

per sec). If no gas is valved uniformly, this card must

still lead Type 8 cards with a value 0.0.

TV and VV - Two arrays of time measured from start of flight (sec)

and valving rate (lb lift lost per sec) schedule. They are

in order of increasing TV and are limited to 100 cards.

TV(1) must be equal to XO, TV(NVI) must be equal to or

93

greater than XT. NVI is the number of Type 8 data

cards containing TV/VV data.

Example: XO = 0. , XT = 36,000.

VO = 1.0

TV(1) = 0. VW() = 0.

TV(2) = 18,000. VV(2) = 0.3

TV(3) = 18,100. VV(3) = 0.

TVC4) = 36,000. W(4) = 0.

These data mean that the constant valving rate is 1.0 lb/sec

and that scheduled valving takes place only from 18,000 to

18,100 sec at a rate of 0.3 lb/sec.

TYPE 9 DATA

These data include:

BO - The constant ballasting rate (lb/sec) on the first card.

If there is no uniform ballasting, this card must still be

present with a value of 0.0.

TB and BB - Two arrays of time (sec) measured from start of

flight and ballasting rate (lb/sec) schedule.

The rules for these data are similar to those for Type 8.

TYPE 10 DATA

These data include:

RH00 - Air density (lbm/ft 3 ) at ZZ(1) level on the first card

ZZ and TZ - Two arrays of altitude above mean sea level (ft)

and the corresponding air temperature (°R). They are in

order of increasing ZZ and are limited to 100 cards.

Properties of the first layer are extrapolated for properties below ZZ(1)

if necessary (as required for optical air mass computation, for example).

Note that ZZ(1) may be below ZMIN but may not be greater than ZMIN. ZZ(1)

may also differ from ZO but must not exceed it. Thus, ZZ(1) may be

chosen at a level more convenient for density specification than the

downstream ground level ZMIN or the initial altitude ZO. ZZ(NT) should

be equal to or greater than the expected maximum altitude of the balloon.

Atmospheric data for the model are extrapolated from the uppermost layer

whenever the balloon goes above ZZ(NT); NT is the number of Type 10 data

94

cards containing ZZ/TZ data.

TYPE 11 DATA

These data include two arrays of altitude above mean sea level

ZIR (ft) and the corresponding values of black ball radiation equilibrium

temperature TIR (°R). The rules for these data are the same as for Type

10 ZZ/TZ data. Whenever the balloon goes above ZIR(NIR), infrared data

from the uppermost data layer are extrapolated; NIR is the number of

Type 11 data cards.

TYPE 12 DATA

These data include six arrays of observed altitude EALT, gas tem-

perature EGAST, film temperature EFILMT, air temperature EAIRT, and

ascent rate EVEL corresponding to flight times ETIME from an actual

flight which the program is to model. These data may be used for flight

correlation checks. The data must be in order of increasing ETIME and

are limited to 400 cards.

TYPE 13 DATA

These data consist of two arrays of time TIR from the start of

flight and corresponding values of the "infrared radiation factor" CIR.

This factor accounts for possible transient cold cloud cover infrared

effects. The rules for these data are similar to those for Type 7 data.

TYPE 14 DATA

These data include the following information about the balloon gas:

PRG - balloon gas Prandtl number

ALPHA)

BETA 3- empirical constants for viscosity computation from an

expression of the form:

BETA= ALPHA * T BETA

g g

ALPHA has units (lbm/ft sec)m

GAMMA - constant for computing conductivity from viscosity

assuming constant Pr ;g

kg = GAMMA * ,g

GAMMA has units (ft lbf/lbm - OR)i m

95

XMG - balloon gas molecular weight

CP - specific heat of balloon gas at constant pressure

(ft - lbf/lbm 0R)

CV - specific heat of balloon gas at constant volume

(ft - lbf/lbm°R)

96

Table A2

CONTENT AND ORDER OF INPUT CARDS FOR ONE FLIGHT

Field Format

2 12 22 32 42 52 62 72

TITLE 8A10

3 I2

C1 C4 C6 ABIR EMIR ABUV F10.4

4 I2

WPB WF VBM FLO CB ZMIN F10.4

5 I2

TGO TFO ZO Y20 DEC XLAT XL0NG F10.4

6 I2

H DPR XO XT GMTS GHA DPLT F10.4

7 12

TCSI(1) CCSI(1) F10,4

*: F10.4

TCSI(NSI) CCSI NSI) F10.4

8 I2

VO F10.4

TV(1) VV(1) F10.4

*':~~~~ FlO.4;C, ~~F10.4

TV(NVI) VV(NVI) F10.4

~~~~~~~~~~~9 ~~~~1~~I2

BO F10.4

TB(1) BB(1) F10.4

' ' F10.4

TB(NBI) BB(NBI) F10.4

97

Table A2 (cont.)

Field Format

2 12 22 32 42 52 62 72

10 12

RHO F10.4

ZZ(1) TZ(1) F10.4

: F10.4

ZZ(NT) TZ(NT) F10.4

11 12

ZIR(1) RTIR(1) F10.4

F10.4

ZIR(NIR) RTIR(NIR) F10.4

12 12

ETIME(1) EALT(1) EAIRT(1) EFILMT(1) EGAST(1) EVEL(1) F10.4

......: :i~ :~ :~ :~ ~ F10.4

ETIME(NEX) EALT(NEX) EAIRT(NEX) EFILMT(NEX) EGAST(NEX) EVEL(NEX) F10.4

13 I2

TIR(1) CIR(1) F10.4

: ' F10.4

TIR(NCIR) CIR(NCIR) F10.4

14 I2

PRG ALPHA BETA GAMMA XMG CP CV 10.0

1 I2

2 I2

98

THE PROGRAM

The program is composed of a MAIN routine and six subroutines (RNGKTA,

YPRIME, RHOT, DRAGC, YPLOT, and TIME) which perform special tasks.

The MAIN Routine

This routine reads the input data from the input tape. When a run

is made with more than one flight, the computer will read the input data

and compute one flight at a time. The cards containing the input data

as well as the control cards must be punched correctly and must be in

the proper order as described in the INPUT DATA section. Otherwise,

the first time that the MAIN finds an error, it will terminate the

entire run after writing a comment in the output tape which will help

locate the error in the data.

As a further check on the input data, before proceeding to the com-

putation of the flight, the MAIN will write on the output tape all the

data used for the flight. The input data printed on the output should

be examined carefully to insure that they are identical to the input

data.

A flight will be terminated automatically when:

1. The entire payload, including the gondola, has been

dropped. This could be caused by excessive ballasting

(see below). A comment will be written on the output

tape indicating that this has happened.

2. The balloon hits the ground.

3. The flight time exceeds the flight time specified in the

input data.

The program will then proceed to the next flight.

The MAIN does a considerable amount of computation involved in

interpolating the input data and reducing it to a suitable form. It

also computes some of the variables in the desired output form and

stores them on the output tape.

99

The MAIN also exhausts gas artificially. The integration is

carried out in time intervals equal to H. At the end of each interval

the instantaneous volume Y(6) of the gas is compared with the inflated

volume of the balloon VBM as specified by its manufacturer. If Y(6) is

equal to or smaller than VBM, no gas is exhausted and the computer pro-

ceeds to the next step. If Y(6) is greater than VBM, the exhausting

rate is set equal to (VBM - Y(6))/H and the same step is repeated. If

Y(6) again comes out greater than VBM, the exhausting rate is increased

by (VBM - Y(6))/H, etc. Twenty such iterations per integrating step are

included in the basic program.

Finally, the MAIN does an energy check, which will be described

later.

Subroutine RNGKTA

The actual integration of the six differential equations is carried

out by this subroutine with Gill's fourth-order Runge-Kutta method,

which is a stable, self-starting, accurate numerical integration tech-

nique; RNGKTA requires another subroutine YPRIME in which the differen-

tial equations are stated and derivatives computed.

Subroutine YPRIME

This subroutine contains the six first-order differential equations

and evaluates the first derivatives of the dependent variables. It also

evaluates the various heat transfer rates. Thus, YPRIME is called fre-

quently by MAIN, as well as by RNGKTA, whenever information is needed

concerning the derivatives of the dependent variables or the heat trans-

fer rates.

To determine the rate of absorption of solar radiation, YPRIME

computes the intensity of solar radiation and, when sunrise or sunset

occurs at the balloon, it computes the Greenwich Mean Time (GMT) and

writes directly on the output tape the appropriate comment (for

instance, GMT - 2.20.13 SUNSET AT BALLOON),

100

When the balloon goes above the infrared radiation field specified

in the input data (Type 11), YPRIME will print on-line an appropriate

comment. The flight will be continued using the infrared data of the

highest altitude for as long as the balloon remains above the specified

infrared field.

In computing the convective heat transfer rates to the

atmosphere, YPRIME gets information about the atmosphere from subroutine

RHOT.

Subroutine RHOT

This subroutine computes the atmospheric temperature and the

product of atmospheric temperature and density for given altitude.

When the balloon goes above the atmospheric temperature profile speci-

fied in the input data (Type 10), RHOT will print on-line an appropri-

ate comment. The flight will be continued using the atmospheric data

of the highest altitude for as long as the balloon remains above the

specified atmosphere.

Subroutine RHOT is called frequently by MAIN and YPRIME to give

information about the atmosphere.

Subroutine DRAGC

This subroutine computes the drag coefficient for the balloon, the

shape of which is assumed to be spherical. For Reynolds numbers less

than 1.0, Stokes Law is used. For Reynolds numbers between 1.0 and

300,000, the data presented in Ref. 97 are used. Three equations are

required to fit the data, and the form of these equations depends on

which subinterval of the Reynolds number interval between 1.0 and

300,000 is of interest. For Reynolds numbers larger than 300,000 a

value of CD = 0.5 is used. Subroutine DRAGC is called by YPRIME each

time YPRIME is called.

101

Subroutine TIME

This subroutine is called by MAIN and YPRIME when the GMT is

required during the model run.

Subroutine YPLOT (written by J.H. Smalley)

This subroutine is called by MAIN after all calculations have been

completed for a given flight. YPLOT plots the following variables ver-

sus local time in hours:

Variable Symbol on Plot

Air temperature (°F) A

Balloon Fabric Temperature (°F) F

Balloon gas temperature (°F) G

Balloon altitude (ft) H

Balloon vertical velocity (ft/min) V

Percent free lift L

If two or more variables have the same ordinate value at any time, only

one symbol is printed. A maximum of 1,000 values of any of the six

variables listed above may be plotted. If DPLT = 0, no plot is generated.

ACCURACY OF INTEGRATION

With a program as massive as this it is desirable to have a scheme

by which the accuracy of integration can be estimated for a given

integrating-time interval. With such a scheme the integrating-time

interval can be maximized for a given desired accuracy and, thus, the

total computer time per flight can be minimized.

To this end, an energy check is most effective. If Eq. (5) is

multiplied by U and Eqs. (14) and (15) are added, after some rearrange-

ment, the following energy equation is obtained:

d (ETOT) = - DKE + DPE - DDRAG - DEWA + DEVE + q + q - q + q (Al)dt 3 4 5 .2

102

where

ETOT = cv m T + c mf T (A2)vg g g f f f

DKE = (m + mf + mB +m + CM p V ) U (A3)g f B p Ma g g

DPE = (p V -m - m -m - m) U (A4)ag g f p B

DDRAG = (1.21/2g ) p CD U | 2 V2/3 (A5)g

DEWA = p V (A6)a g

DEVE = c T m (A7)P g g

and where c is the specific heat at constant pressure of the gas.P

Note that Eq. (Al) relates the rates at which the various forms of

energy of the gas-fabric system are exchanged. The interpretations of

the various terms are: ETOT is the internal energy; DKE and DPE are

the rates of change of kinetic and potential energy, respectively;

DDRAG is the rate at which the energy is expended to overcome drag;

DEWA is the rate at which work is done (on the atmosphere) when the bal-

loon expands; and DEVE is the rate of energy lost when gas is valved or

exhausted. Finally, the sum of the q terms is the rate of net heat

transfer to the system.

Let the right hand side of Eq. (Al) be denoted by ER. Integrating

this equation from time zero to t, one obtains:

ENET = ETOT - ETOTI = ER dt (A8)Let

nEP = HZ ER. (A9)

i=l 1

where ER. is the value of ER at the end of each time step H.1

103

At the end of each integration by RNGKTA, the MAIN routine eval-

uates ETOT and, therefore, ENET from the integrated variables. Simi-

larly, it evaluates the increment in EP, which is H(ERi), resulting

from each step and keeps track of the total EP. Clearly, ENET and EP

thus evaluated are not equal for two reasons (if all integrations were

exact, ENET and EP would be identical). First, there is an error in-

volved in the integrations done by RNGKTA. Second, there is another

error involved in the above approximate first order integration of ER.

The second error is certainly larger than the first one. Nevertheless,

the ratio

ECHK ENET-EP (A10)ENET

which is evaluated by MAIN and stored on the output tape each time

output is stored there, has some bearing on the accuracy of RNGKTA.

If ECHK is a small number, this means, at least, that RNGKTA is very

accurate.

OUTPUTS

As stated above, the first part of output is a printout of the

input data used for the flight. The printout of input data is followed

by the computed output. A unit of output contains one line; the con-

tents of the line are the values of the variables appearing at the head

of each page of output. These variables are:

GMT - Greenwich Mean Time (hr, min, sec)

TIME - Time from beginning of flight (min)

ALTITUDE - Altitude of balloon from sea level (ft)

VEL - Velocity of balloon (ft/min)

PRESS - Atmospheric pressure (mb)

TA - Atmospheric temperature (°R)

TF - Balloon fabric temperature (°R)

104

TG - Balloon gas temperature (OR)

VOLUME - Balloon volume (ft3)

TMASS - Mass of payload ballast, gas and balloon fabric (lb)m

GMASS - Mass of balloon gas (lb)m

FR LIFT - Free lift (lb)

PERB - Cumulative ballast dropped (percent of initial load--

payload plus ballast)

PERV - Cumulative gas valved (percent of initial gas weight)

PERE - Cumulative gas exhausted (percent of initial gas weight)

IRS - Number of iterations required to exhaust an adequate

amount of gas during the most recent integrating step

ECHK - Energy check, as explained above

When sunset or sunrise occurs at the balloon, the GMT of the occur-

rence, together with the appropriate comment, are printed on the output.

When a flight is aborted because of errors in the data, the output

contains comments which will help locate the errors.

105

OTHER RULES FOR USAGE

In MAIN, a card reading "KIP = 0" is noted near the start of the

listing. The function of KIP is as follows:

KIP = 0: input data are printed

KIP = 1: input data are not printed.

To achieve greater accuracy during the initial stages of the flight,

a one second integration step is used instead of that specified by the

input value H.

Valving and ballasting events must span at least one integration

interval H. If not, the event will not be sensed by the model.

A model run may be begun at any point in an actual flight if the

investigator wishes to examine a specific portion of the flight in

detail. This is done by specifying the appropriate values of FLO, TGO,

TFO, ZO, Y20, XO, etc. in the input data which correspond to the time

at which the model run is to be initiated.

More than one flight may be computed in one computer run. This

is done by simply placing the decks of input cards in consecutive order

replacing those intermediate control data cards with a 2 in Column 2

by blank cards. If certain data is the same from one flight to the

next, it is carried over automatically from the first flight to subse-

quent flights. Each new data deck must be preceded by a TITLE card

(format 8A10).

A microfilm plotting routine has been developed for use with this

program on the NCAR computers; it is used to plot model prediction and

actual flight data for model validation purposes only and has not been

included in this report. A listing of subroutine MFP is available upon

request from the authors or from the NCAR Palestine Launch Site, Box 1175,

Palestine, Texas 75801.

106

In the data description section above a distinction is made between

"start of flight" and "start of model run" times. "Start of flight"

refers to the time at which the balloon launch took place. "Start of

model run" refers to the point at which the computer begins its compu-

tations; it may be at the start of the flight or at a later time at the

user s option.

Selection of the optimum integration interval H is facilitated by

the printout of ECHK. It has been found that a value of H resulting in

ECHK < .010 is satisfactory. A smaller integration interval will not

materially change the output values but will consume more computer time

unnecessarily.

107

PROGRAM MAINTDMENSION Y(16),F (6) ZZ(lOO) ,TZ( 100,RTZ(100) ,TZA( 100) TZO(100)

1TCSI (100) CCSI (100)TV (100) ,VV (100) ,TB (100) BB (10) ,XX (7) T I TLE(122) ,RT IR( 100) ,Z R( 1 0 ),DIRrZ ( 1 0) ,S IDT( 100) ET IME(400)FALT(400),TTR( InO)CTR(l00),r CIRnT(l00),YPH(OOO,1 ),XPT ( 1O),4YPT(1000,3),YPL(1000,1),YPV(1000 1) ,Q(6)'DIMENSION FAIRTT(0),EFILMT(400)FGAST(400),FVEL(400)COMMON/AA/ N IRZIRDIRDZ,TRARTIRAR IRMIRSBOLZ,RADGHA XLONGRHOG,1'SIATnCLATD,P ,PZFRFSOLA ,AUV LP ,GAMMA,RG ,GC4 C6 ,PRG ,WGPOSS,CR,2ClTWFCP,VE,ALPHARFTA,Fr)FH2,OFH3,'FH4,)FH7,CSI CCIRCOMMON/RB/ZZ,NT,TZA ,TZBRTZ,TZCOMMON/CC/GMTSCOMMON/DD/TARHO,PHI,CDCOMMON/EE/QCOMMON/SHARE/RAXXO,F,Y,AHAM AS RFN,RT

KIP=OTHIPDn=. /3.FSOL=96.SBOLZ=3.6995F-10XMA=28.9644R=1545.32PA = R/XMAG=32.1740RAD = (4.0*ATAN(1.O))/180.0

1001 READ 995,(TITLE(I),I=1,8)RFAD 20,ITTF(TT) 998,9QR,1000

998 WRITE(6,994) (TITLE(I),I=1!8)WRITF (6950)CALL FXIT

1000 ,GOTc( lO0,?On ,40, ,OO O ,7nn ^ , oO, 00,^O o, , ^ /^On) T

C**DATA PRINTOUT SECTION****************************************

100 IF (KIP.FQ.1) GO TO 2013WRITF (6,994) (TITLF(I),T=i,8)WRITF (6,2000)1=3WRITF (692001) IWRITF (6,3000)WRITF (6,200?) Cl C4 ,r6,ARITRFMIRARUV1=4WRITF (69 2001) IWRITE (694000)WRITF (6,2002) WPB,WF ,VRMFL-n, CRZMIN1=5

WRITE (6, 2001) IWRITF (6,5000)WRITE (6, 2002) TGO,TFOZO, Y20,DEC,XLAT,XLONGt=6WRITE (6, 2001) IWRITF ( 6, 6000)WRITF (6, 200?) HDPR,Xf,XT,GMTS,GHADPLTT=7WRITF (6, 2001) IWRITE (6,7000)

108

rO 2004 T=1,NST2004 WRITE (6,2005) TCSI(I),CCSI(I)

T=8WRITE (6, 2001) IWRITE(6,8000) VODO 2007 T=1,NVI

2007 WRITE (6,2005)TV(I),VV(I)I=9WRTTF (69 2001) IWRITF (6,9000)R1n 200R T=1,NRT

200R WRITF (6,2005) TR(I),BR(T)T=10WRITE (6, 2001) IWRITE (6,1010) RHOO00 2009 1=1 ,NT

2009 WRITF (6,2005) ZZ(I),TZ(I)I=11WRITF(6,2001) IWRITF(6,1100).o 2010 T=1 NTR

2010 WRITE (6,2005) 7IR(I),RTTR(I)T=12WRYTF (6, 2001) IWRITF(6,1111)DO 2011 T=1,NFX

2011 WRITF (6,2005)FTIMF() ,FALT( I)EAIRT(I),EFILMT(I),EGAST(I) FVEL(I)1=13WRTTF (6, 2001) TWRITF (6 1300)) 72012 =l ,NCTP

201? WRITF (6.,2?Oo) TIR(I),CIR(T)T = 14WRITF (692001) TWRITF (6,1405)WRITE(6,1500) PRG, ALPHA, FFTA, GAMMA,XMGCP,CVc, Tn 2013

200 PRINT 994CALL FXIT

C**DATA REAnIN AND INTERPOLATION SECTION********************************

300 RFAD 2nITT(XX(I),I=1,7)IP=3IF(IT) 997,302,970

o0? (l = XX(1)

C4=XX(2)C6 = XX(3)ARIR = XX(4)FMIR = XX(5)ARUV = XX(6)PFAD 20,ITIF(IT) 997,971 1 100

400 RFAD 20, IT, ( XX(I), =1,7 )IP=4IF(IT) 997,402,970

109

402 WPR=XX(1)WF =XX (?)VRM=XX (3)FLO=XX (4)CR = XX(5)ZM IN=XX (6)READ 20,ITIF(TT) 997,971,1000

50, RFAD 20,TT,(XX(I),I=1,7)

IP=5IF(TT) 997 502,970

50? TGO=XX(1)TFO=XX(2)ZO=XX (3)Y20=XX(4)DFC=XX (5XLAT=XX(6)XLONC=XX(7)RFAn ?0,TT

TF ( TT) 097,971 ,1 OnC

95, READ 20,IT,(XX(I),I=1,7)TP=6IF(IT) 997,952 970

95? H = XX( )DPR=XX(2)XO=XX(3)XT=XX ()GMTS=XX (5)CHA=XX (6)DPLT = XX(7)HSTOR=HPFAtn 7n,TTTF(TT) 997,971 1000

70, TP=7

nO 701 T=1,100READ 20 , IT, TCSI(I ) CCST(I)IF (T)9977701 ,704

701 CONTINUFRFAD 20, ITIF(TT) 997,071 706

7nh T= T+704 NSI=T-1

nr 705 J=2,NST

DnsIDT(J-1)=(CCST (J)-CS (J-1))/(TCSIJ! -TSI(J-1))705 (ONTTINUF

GOTO 1000

800 READ 20, IT, XX(1)TP=RIF (T)997,802s970

80? VO = XX(1)nO 801 T= 1,100RFAD 20, TT. TV(I), VV(T)I (IT) 99-7801,804

801 CONTTNUJE

110

RfAfn ?0, TTIF(IT) 997 071t 05

805 I=1+1804 NVI=I-1

GO TO 1000

900 READ 20, IT, XX(1 )IP=oIF(IT) 997,902,970

90? CONT INUFPR = XX ( )nnO 903 T=1,100)READ 20, IT, TR(I ), BR(I)IF (T) 997,90 ,905

903 CONTINU JFREAD 20, ITIF(IT) 997,971,904

004 1=1+1900 NRI=T-1

GO TO 00o0

6n PFAD ?0, IT, XX(I )TP=1TF( T) 907 602? 70

60? RHOO= XX(1)rn 6n03 T=1 100RFAD 20, IT, Z7(I), TZ(T)IF (IT) 997, 603, 608

603 CONTINUFREAD 20, ITIF(IT) 997,971,609

609 I = 1+16hn NIT = T-1

PTZ(1) = RHnO * T7(1)rn 604 =?, NITIF ( T7(I) -TZ(I-l)) 605, 6n6, 605

605 TZA(I-1) = ( TZ(I - TZ(T-1))/(ZZ(TI -ZZ(I-l))TZR( -1 ) =(ZZ( ) *TZ(I-1) - ZZ(T-1) * TZ(T)) / (ZZ(T) - ZZ(T-l))RTZ(T) = RTZ(I-1) *(TZ(1-1)/T Z(I))**(1./(RA*TZA(T-1)) )GO TO 604

606 TZA( 1-1) = 0.0TZB( I-1) = TZ(I)RTZ(I) = RTZ(I-1) *EXP ((ZZ(I-l)-ZZ(I))/(RA*TZ(I-1)) )

604 CONTINUEr)0 TO Innn

96^ TP= 11no 961 T=1,100RFAn 20,TTZTR( T) ,RTTR(I)

C** FOUATION 51 - INFRARED HEAT FLUX TO FABRIC FROM BLACK PALL TFMP PROFILE ****IF( IT )997,961 962

961 CONTINUERFAD 20, ITIF ( IT) 997,971 964

964 T=1+1962 NIR=I-1

n 9P63 J=2?NIPDIRDZ(J-I)=(RTIR(J)-RTIR(J-1))/(ZIR(J)-ZIR(J-1))

11

963 CONTINUEGOTO 1000

qsn IP=1?DO 984 MZY=1 400ETIME(MZY)=0.FALT(MZY)=0.FAIRT(MZY)=O.EFILMT(MZY)=0.FGAST(MZY)=0.FVEL(MZY)=0.

984 CONTINUEDO 981 I=1,400READ 20,ITET IME(I),EALT(I),EAIRT(I)EFILMT(I) EGAST(I),EVEL(I)IF(TT)997,981 982

981 CONTTNUFREAD 20, ITIF(IT) 997,971,983

983 - I+l982 NFX= I-

GO TO 1000

1400 TP=13nO 1401 T=1t,00READ 20, IT,TTR(fI) CIR(I)IF ( IT)997,1401 1402

1401 CONTINUEREAD 20, ITTF(IT) 997,97191403

1403 1=1+1140? NCIR=I-1

D01404 I=2?NCTRDC IRDT( I-1)=(CIR(I)-CIR(T-1l/(TIR(I)-TIR(I-1 )

1404 CONTINUEGO TO 1000

3020 READ 21,IT,(XX(I),I=1,7)IP = 14IF(TT) 997,3022, 970

3022 CONTINUEPRG = XX()1ALPHA = XX(2)BETA = XX(3)GAMMA = XX(4)XMG = XX(5)CP = XX(6)CV = XX(7)RG = R/XMGRFAD 20, ITIF(IT) 997,97111000

997 WRITF(6,994) (TITLF(I),I=19l )WRITF (6,80) IPCALL FXIT

970 WRITE(6,994) (TITLE(I),I=1,8)WRITE(6,60) IPCALL FXIT

971 WRITE(6,994) (TITLE(I)tI=1,8)WRITE (6,70) IP

112

CALL EXIT

C**INTT ALTZTNG VARTARLFS AND PARAMFTFRS*******************************

2013 CLATD=COS (XLAT*RAD)*COS (DEC*RAD)SLATD=SIN (XLAT*RAD)*SIN (DrC*RAD)

C FOR THE FIRST 60 SECONDS, ONE SECOND INTERVALS ARE USED.H = 1.0CALL RHOT(0.,TEMP)PZER=RT*RAY(1 = ZO

Y(?)=Y;'0Y(3) = TFOY(4)=TGOX = XO

IF (X -TV(1)) 102,1021 9021102 WRITE (6,40) X, TV(1)

WRITE (6,45)GO TO 996

1021 CONTINUFIF(X-TR(l)) } 111 112,t11?

ll WRITF (6,56) XTP(1)WRITF (6,45)GO TO 996

111? CONT NUFIT(X-TCSI (1) )1?211211 ,121

121 WRITF(6,65) X, TCSI(1)WRITF (6, 45)GO TO 996

1211 CONTINUEIF(X-TTR(1) )126,1261,1261

126 WRITE (6,81) XTIR(1)WRITE (6, 45)GOTO 996

1261 CONTINUFXP= XOVTF=XOVA.LD=O.LP=5SIY=ONYC=0LAST=0tH=0C1TWF=C1*WFCALL RHOT(Y(1),TFMP)RHO = RT / TFMPP=RT*RATA = TFMPWGROSS=WPR+WFFL=ELOY(5)=(FL+WGROSS)/((XMA*Y(4) /(XMG*TA)-1,WGO=Y( 5)Y(6)=RG*Y(5 )*Y(4)/PDO 99 I=1,6Q( I )=0.

99 CONTINUECSI=CCSI (1)CCIR=CTR( 1)

113

PHI=Y(6)**THYRD

C**INITIALIZING TOTAL ENERGY, E1****************************************

CALL YPRTMFDKE=(WGROSS+Y(5)+CB*RHO*Y(6))*Y(2)*F(2)/GDPE= (RHO*Y(6)-WGROSS-Y(5))*Y(2)DDRAG=CD*.60375*PHI**2.*RHO*Y(2)*ABS (Y(2))*Y(2)/GDEWA=P*F(6)DEVE=Y(4)*CP*F(5)ETOT=CV*Y(5)*Y (4)+C1*WF*Y(3)FTOT1=FTOTENET=.OFP=0.OECHK=0.

C**BEGIN NEXT ITERATIVE STEP********************************************

101 LAMBDA=0DO 104 I=2,100IF (TV(I)- X) 104,104,105

104 CONTINUFWRITF(6,108)GO TO 996

105 V= VV(I-t)+VOCONTINUE

Dn 11t3 = 2, 100IF ( X - TR(I)) l16, 11 3 113

11 CONTINUFWRITF (6,125)GO TO 996

116 B = PR (-I )+ROCONT INUF

WGROSS=WGROSS-R*HIF (WGROSS-WF) 109,109,9120

109 WRITF(6,15)GO TO 996

120 00 12? I=2,NSI

IF(X-TCSI(I )+.5*H)124,123,123123 CONT NUF124 CSI=DSIDT(T-1)*(X-TCSI(I-1))+CCSI(I-1)

DO 128 I=2,NCTRIF(X-TIR(I)+.5*H)129*128,128

17R CONTINUF12q CCIR=CIR(I-1)+OCIRDT(I-1)*(X-TIR(I-1))

IF (IY)130,153,1301.5 3 TY=1

GOT0140

130 F = 0.XS = X

DO 131 I=16131 Y(I+10) = Y(I)

114

CALL RNGKTA (Ho 6 1 1 )IF(Y(6)-VRM)160,160,133

16^ CONTINUEPHI = Y(6)**THIRD

C**INCREMENTING TOTAL FNERGY********* ************ **-******************

CALL YPRIMEDKE=(WGROSS+Y(5)+CB*RHO*Y(6))*Y(2)*F(2)/GDPE= (RHO*Y(6)-WGROSS-Y(5))*Y(2)DDRAG=CD*.60375*PHI*PHI*RHO*Y(2)*ABS (Y(2))*Y(2)/GDEVE=Y(4)*CP*F(5)DFWA=P*F(6)FL=Y(5)*((XMA*Y(4))/(XMG*TA)-l1)-WGROSSEP=EP+(-DKE+DPE-DDRAG-DEWA+DFVE+DEH2-DEH3+DEH4+DEH7)*HVALn=VALr)+H*V/( ((XMA*Y(4)))/(XMG*TA )-.1 )ETOT=CV*Y(5 )*Y(4)+C1TWF*Y(3)FNET=FTOT-~TOTECHK=(ENET-EP)/ENETGO TO 140

133 CONTINUEDO 135 LAMRDA=19 20F = F + ( VRM - Y(6)) / HX = XSDr 1t6 T=I1Y(I) = Y(+l10)

136 CONTINUECALL RNGKTA ( H, 6, 1, 1)IF(Y(6)-VRM)160,160,135

13, CONTINUFPRINT 25GO TO 133

140 IF( (Y(1)+(0.5*F(2)*H+Y(2))*H) .GT. ZMIN )GO TO 142IF (LAST.EQ.1)GO TO 141HSTOR=HLAST=InZMTN=Y(I )-ZMTN

148 HST=HH=H-((0.5*F(2)*H+Y(2))*H+nZMIN)/(F(2)*H+Y(2))IF(ABS(HST-H).GT0.1 )148,142

141 IX = 1GO TO 150

142 IF ( X -XT) 1449 1419 141144 IX=2150 IF(X-(X0+60.))156,154,155154 H=HSTOR-AMOD(60.,HSTOR) $ GO TO 156155 IF(IH.EQ.1)GO TO 156 $ IH=1 $ H=HSTOR156 CONTINUE

C**CONVERT PROGRAM VALUES TO PRINTFR AND PLOTTER OUTPUTS****************

IF( (DPLT.EQ.O.) .OR. (X.LT.VTF) )GO TO 171CALL TIMEVTM = AH + (AM*60.0 + AS)/3600.0NYC = NYC + 1TC = 459.67

115

YPT(NYC1 ) = TA - TCYPT(NYC,2) = Y(3) - TCYPT(NYC.3) = Y(4) - TCYPH(NYC91) = Y(l)YPV(NYC,1) = 60,0*Y(2)YPL(NYC,1) = FL/ WGROSS * 100.0XPT(NYC) = VTMVTF=X+DPL T

171 CONTINUEC**PRINT OUTPUT*********************************************************

IF(TX.FQ.1),O TO 143IF(X-XP) 157,143*143

143 CONTINUEXP = XP + rPRIF(LP-55)146,146,147

147 CONTINUEWRITE (69994) ( T+TLF(JS)9 JS 3 IT 8DWRITE (60976)LP = 0

146 CONTINUEPERB=100.*(WF+WPR-WGROSS)/(WPB)PERV=100.*VALD/WGOPERE=100.*(WGO-Y(5))/(WGO)-PERVVEL=60.*Y(2)PM = 0.478802516 * PXMIN=X/60.CALL TIMEWGP=WGROSS+Y(5)WRITF (69 977) AHAMASXMIN,Y(1),VFL,PMTA,Y(3)IY(4),Y(6 ,W

1GPY(5) ,FLPFRR PERV PERF LAMPBAFCHKLP= LP+1

157 CONTINUEC**CALL PLOT ROUTINE****************************************************

IF(IX.EQ.2)GO TO 101IF(DPLT.EQ.O.)GO TO 996CALL YPLOT(NYC,3,XPTYPT,1)CALL YPLOT(NYC91,XPT,YPH,2)CALL YPLOT(NYC,1,XPTYPV,3)CALL YPLOT(NYC,1,XPT,YPL-4)

996 READ 209 ITIF(IT.EQ.2) GO TO 200IF(IT.EQ.O) GO TO 1001WRITE (69993)CALL EXIT

C**INPUT AND OUTPUT FORMATS*********************************************15 FORMAT (77H1 YOU HAVE NOW THROWN AWAY ALL OF THE BALLAST - - INCLU

1DING THE GONDOLA./1HO/1HO)20 FORMAT ( TI2 7F10.4)21 FORMAT( 12* 7F100 )25 FORMAT(* ITFRATION DOES NOT CONVFRGF*//)40 FORMAT ( 7H1 TIME=F10.598H TV(1)=F10.5)45 FORMAT(15H TNPUT IS WRONG/IHO/1HO)50 FORMAT (31H CARD AFTER TTTLE CARD IS WRONG/1HO)55 FORMAT(7H1 TIME=F10.58H TB(1)=F10.5)60 FORMAT (9H NO TYPE I295H DATA/1HO)

116

65 FORMAT ( 7H1 TIME=F10O.59H TCSI(1)=F10.5)70 FORMAT (15H TOO MANY TYPE I295H DATA/1HO)80 FORMAT (20H IT AFTER DATA TYPE I2,12H IS NEGATIVE/1HO)81 FORMAT(7H1 TIME=F10.58H TIR(1)=F10.5)

108 FORMAT (28H1 X IS GREATER THAN TV(100)/1HO/1HO)125 FORMAT (28H1 X IS GREATER THAN TB(100)/1HO/1HO)976 FORMAT(1HO03X,3HGiMT 6X 4HT MF 2X 8HALT ITUDF,4X3HVFL,3X,5HPRESS 4X

1,2HTA,5X,2HTXHTF,5X2HTG5X,6HVOLUME3X,5HTMASS92X6H GMASS,2X97HFR LI.2FT,2X,4HPERB,2X,4HPERV 2X,4HPERE,2X 3HIRS,3X,4HECHK/2X*H M S*35X*MIN*5X*FT*7X*FPM*5X*MR*6X*R*6X*R*6X*R*6X*CU FT*4X*LBM*5X*LBM*47X*LR* / )

977 FORMA ( 3F3.0,1XF7.0,2XF7.0,2XF6.0,2F61XF6512XF51 2XF5.lt2XF15.12XF8 0XF*Ot2 2XF2XF8.02XF6 2XF6 2XF7 2XF412XF4.12XF4.1,2X I22XF63)

993 FORMAT(1H1952H CARD AFTER CARD TYPE 1 OF PREVIOUS PROBLEM IS WRONG1/1HO)

994 FORMAT(1H1,8A]0)995 FORMAT(8A10)1010 FORMAT(1H+8X*RHOO =*F10.5/11X*ALT*12X*TEMP*)1100 FORMAT(1H+1OX*ALT*11X*RTEMP*)1111 FORMAT(1H+8X,*FTIME*912X,*EALT*11X*EAIRT*O1X*EFILMT*11X,

l*FGAST*12X*EVFL*)1300 FORMAT(1H+8X*TIME ETHIR*8X*CIR*)1405 FORMAT(1H+9X*PR.NO.* 12X*ALPHA* 13X*BETA* 12X*GAMMA* 10X*MOL.WT.*

115X,*CP*, 15X,*CV*)1500 FORMAT(4X9 7(E15.6,2X))2000 FORMAT(1HO,11H INPUT DATA)2001 FORMAT(1HO,4H IT=I2)2002 FORMAT(7F16.5)

2005 FORMAT(6F16.3)3000 FORMAT(1H+11X*C1*14X*C4* 14X*C6*12X*ABIR*12X*EMIR*12X*ABUV*)4000 FORMAT(1H+7X*PAY+BLST*6X*FABRIC WT*12X*VOL*12X*F.L.*14X*CB*12X*ZMI

1N*)5000 FORMAT

* (1H+10X*TGO*13X*TFO*14X*ZO*12X*ZVEL*13X*DEC*13X*LAT*12X*LONG*)6000 FORMAT(1H+8X*DTIME*10X*DPRINT*11X*ZTIME*11X*ENDTIME*12X*GMTS*

* 13X*GHA*12X*DPLT*)7000 FORMAT(1H+8X*TIMF SUNIR*7X*CCSI*)8000 FORMAT(1H+8X*MAN VALVE =*F1O.5/9X*VTIME*12X*VDOT*)9000 FORMAT(1H+8X*MAN BLST =*Fl0.5/9X*BTIME*12X*BDOT*)

FND00562

117

SUBROUTINE YPRIMETDMFNSION Y(16),ZIR(100),F(6),DIRDZ(100),RTIR(100)

COMMON/AA/ NIR,ZIRDIRDZRTIR,ABIREMIRSBOLZRAC GHAXLONGRHOG,1SLATDCLATDP,PZERFSOL,ABUV LP,GAMMARGG C4 C6,PRGWGROSS CB2C1TWF,CP,V,E,ALPHABETA,DEH2,DEH3,DEH4,DEH7,CSI CCIRCOMMON/DD/TARHOPHICOMMON/SHARE/RAX,XO,F,YAHAMASRENRTDATA THIRD, TRZ, RETH/ .33333333333339 459.67, 20903520. /SBOLZ=3.6995E-10CALL DRAGC(CD)CALL RHOT(Y(1),TA)P=RT*RARHO = PT / TAPHI=Y(6)**THIRDPHISQ=PHT*PHIDO 181 I=2,NIRTF(Y(1)-ZIR(I ))182,181,18l

181 CONTINUEGO TO (184,183),IXY

184 PRINT 20,XY(1)IXY=2GO TO 183

182 IXY=1183 TIR=DIRDZ(I-1)*((Y(l)-ZIR(I-1))) +RTIR(I-1)

FLUX R=SROLZ*T R**4C** EQUATION 50 - ABSORPTION OF INFRARED RADIATION BY BALLOON FABRIC *****N*****

DEH2=CCIR*ABIR*4.83*PHISO*FLUXIRC** EQUATION 32 - EMISSION OF INFRARFD RADIATION BY BALLOON FABRIC *************

DEH3=EMIR*4.83/PHISQ*SBOLZ*(Y(3)**4)C** FQUATION 43 *************-*************************************************

XLAH=RA)*(GHA-X ONG+X/240.)C-** EQUATION 33 ****-********************************************************

CAM= SLATD + CLATD* COS (XLAH)IF (CAM) 100,101,101

C** EQUATION 36 - AIR MASS CALCULATION ****************************************101 AIRM= (P/PZER)*(SQRT(1228.6 +376750.44*CAM*CAM)-613.8*CAM)

C** FQUATION 35 - ATMOSPHFRIC TRANSMITTANCE TO SOLAR RADIATION **********.*****TRANS=.5*(FXP (-.65*AIRM)+EXP (-.095*AIRM))FLUXUV= FSOL*TRANS *CSIGOTO 110

100 CAL=RETH/(RETH+Y(1))CAMM=-SQRT(1.-CAL*CAL)IF (CAMM-CAM)102,103,103

102 ZZ1=(RETH+Y(1))*SQRT(1.-CAM*CAM)-RETHCALL RHOT(ZZ1,TTT)PZ1 = RT*RAAIRM1=35.1*PZI/PZERAIRM2=(P/PZER)*(SQRT(1228.6+376750.44*CAM*CAM)+613.8*CAM)TRAN1=.5*(EXP (-.65*AIRM1)+EXP (-.095*AIRM1))TRAN2=.5*(FXP (-.65*AIRM2)+EXP (-.095*AIRM2))FLUXUV=FSOL*CSI*TRAN1*TRAN1/TRAN2GOTO 110

103 FLUXUV=0.110 CONTINUE

DEHH7=DEH7C** EQUATION 42 - RATE OF ABSORPTION OF SOLAR RADIATION BY BALLOON FABRIC*L*****

DEH7= ABUV*1.208*PHISQ*FLUXUV

118

IF(DFHH7) 111 111 115111 IF(DFH7) 119119,114114 IF(X-XO)119,119,120120 INM=1

GO TO 121117 WRITF (6,113) AH, AM, AS

118 LP=LP+3 $ GO TO 119115 IF(DFH7.GT.O.)GO TO 119

INM=2121 CALL T IM

GO TO(112,116),INM116 WRITE (69117) AH, AM, AS $ GO TO 118119 VISA = (7.30248F-07 * TA *SQRT(TA))/(TA + 198.72)

VISG = ALPHA * Y(4)**RFTACG = GAMMA * VISGCAIR = 277.28 * VTSARHOG = P/(RG*Y(4))GR = ABS(RHO*RHO*G*(TA-Y(3))*Y(6))/(TA*VISA*VISA)*1.91EC = 0.0IF(RFN.EQ.0.0) FC = 2.0IF(EC.GT.1.O) GO TO 200EC = GR/(REN*REN)

200 CONTINUEC** EQUATION 20 - CONVECTION TO OR FROM FABRIC AT FLOAT ***********************

IF(EC.GT.1.0)DEH4=7.79*PHI*CAIR*(TA-Y(3) )*(1+.322*((RHO*RHO*G** ABS(TA-Y(3))*Y(6))/(TA*VTSA*VISA))**0.25)*C4

C** EQUATION 18 - CONVECTION TO OR FROM FABRIC DURING ASCFNT OR DESCENT ********

IF(EC.LE.1.O) DEH4 = 3.9*PHI*CAIR*(TA-Y(3))*(2.0+O.472*PHI**0.55*(1(RHO*ABS(Y(2)))/VISA)**0.55)*C4

C** EQUATION 28 AND 29 - CONVECTION HEATING OF BALLOON GAS BY FABRIC ***********XNUSG=0.1612*PHI*((RHOG*RHOG*G*PRG*ABS(Y(4)-Y(3)))/(VISG*VISG*Y(4)*))**THIRDDEH6 = 3.895 * C6 * CG * PHI * XNUSG * (Y(3)-Y(4))F(1) = Y(2)

C** EQUATION 5 - BALLOON ACCELERATION FROM BALLOON FORCE BALANCE ***************F(2)=((RHO*Y(6)-WGROSS-Y(5))*G-.6038*CD*PHISQ*RHO*Y(2)*ABS (Y(2)))

1/(WGROSS+Y(5)+CR*RHO*Y(6))C** EQUATION 15 - BALLOON FABRIC ENERGY BALANCE ********************************

F(3)=(DEH2-DFH3+DEH4-DEH6+DEH7)/(C1TWF)C** EQUATION 14 - BALLOON GAS ENERGY BALANCE ***********************************

F(4)=( DEH6-RHO*Y(6)*Y(2))/(Y(5)*CP)C** EQUATION 12 - BALLOON GAS MASS BALANCE ***********************************

F(5)=RHOG-RHOGRHOE-RHOG*V/(RHO-RHOG)C** EQUATION 11 - RATE OF CHANGE OF BALLOON VOLUME *****************************

F(6)=((F(4)/Y(4))+(F(5)/Y(5))+(Y(2)/(RA*TA)))*Y(6)RETURN

20 FORMAT(1HO9H AT TIME=F7.0,75H SEC THE BALLOON WENT ABOVE THE INFR1ARED FIELD SPECIFIED IN THE INPUT DATA./18H BALLOON ALTITUDE=F7.0O26H FEET./123H START TO CONTINUE FLIGHT USING INFRARED DATA OF UPPE3R LAYER OF SPECIFIED FIELD FOR AS LONG AS BALLOON REMAINS ABOVE FI4FLD.)

113 FORMAT (1HO,25X,4HGMT=3F3.0,5X,18HSUNRISE AT BALLOON / )117 FORMAT (1HO925X,4HGMT=3F3.0,5X17HSUJNSET AT BALLOON / )

FND00112

119

SUBROUTINE RHOT(ZT)DIMFNSION ZZ(100),TZ(100),RTZ(100) ,TZA(100) ,TZ( 100) Y( 16) F(6)

COMMON/BB/ZZ NT,TZATZB,RTZ TZCOMMON/SHARE/RA,X,XO,F ,YAH,AM,ASRENRTIF ( Z- ZZ(1)) 100,200, 200

100 J=lGO Tf 203

20,, NM1 = NT-1DO 201 J=1, NM1IF ( Z - ZZ( J+1)) 300, 300, 201

201 CONTINUEJ=NM1GO TO (202,203),IYX

202 PRINT 209XY(1)IYX=2GO TO 203

300 IYX=1203 IF ( TZA(J)) 301, 302, 301301 T = TZA( J) * 7 + TZB (J)

RT=RTZ(J)*(TZ(J)/T)**(I./(RA*TZA(J)))RETURN

302 T = TZB (J)RT=RTZ(J)*FXP ((ZZ(J)-Z)/(RA*T))RETURN

20 FORMAT(1H099H AT TIME=F7.0,63H SEC THE BALLOON WENT ABOVE ATMOSPHE

1RE SPECIFIED IN INPUT DATA./18H BALLOON ALTITUDE=F7.0,6H FEET./1282H START TO CONTINUE FLIGHT USING ATMOSPHERIC DATA OF UPPER LAYER 0

3F SPECIFIED ATMOSPHFRE FOR AS LONG AS BALLOON REMAINS AROVE IT.)

END00030

120

SUBROUTINE TIMEDIMENSION Y(16),F(6)COMMON/CC/GMTSCOMMON/SHARE/RAX XO F ,Y ,AH AM AS RFN RTGT TMF=X-XO+GMTSAH = 0.0HH = GTIME

10 HH = HH - 3600.AH = AH + 1.0

IF(HH.EQ.0.) GO TO 20IF(HH.GT.0.) GO TO 10AH = AH - 1*0

20 HH = (GTIME) - 3600.0 * AHAM = 0.0

30 HH = HH - 60AM = AM + 1.0IF(HH.FQO00) GO TO 40IF(HH.GT.0.0) GO TO 30AM = AM - 1.0

40 AS=(GTIME) - 3600.0 * AH - 60.0 * AM

RFTURNEND

00023

121

SUBROUTINE DRAGC(XCD)C**WRITTEN BY J.H. SMALLEY

DIMENSION Y(16) F(6)COMMON/DD/TARHOPHI , CDCOMMON/SHARE/RA ,XXO ,F,YAH AM AS REN RT

C SUTHERLANDS EQUATION FOR VISCOSITY OF AIRVISA=7.30248E-7*TA*SQRT(TA)/(TA+198.72)REN=1.24*RHO*ABS(Y(2))*PHI/VISAIF( REN.GT. 0.01 ) GO TO 10CD=2400.GO TO 100

10 IF( REN.GT. 1.0 ) GO TO 20C STOKES LAW FOR DRAG COEFFICIENT

CD=24./RFNGO TO 100

2, IF( REN.GT.3E+3 ) GO TO 30XZ=ALOG0 (RFN)Z=(( 0.0185*XZ+ 0.0095 )*XZ-0.748 )*XZ+ 1.385

GO TO 9030 IF( REN.GT.1E+5 ) GO TO 40

CD=0.47GO TO 100

40 IF( REN.GT.3E+5 ) GO TO 50XZ=ALOG10(REN)Z= -1 406*XZ+ 6.702GO TO 90

50 CO=0.5GO TO 100

9, CD=FXP( Z/0.4342944819 )

100 CONTINUEXCD = CDRETURNEND

00034

122

SUBROUTINE RNGKTA(HI N1 ,N2.N3)DIMENSION Q(6),Y(16),F(6)COMMON/FEE/QCOMMON/SHARE/RAX ,XOF ,Y ,AHAM ,ASRFNRTTF(N3-1)2,1,2

1 H=H1HH=.5*HN=!N1M=N?

? Dn 4 J=1 MCALL YPRTMFDO 5 I=1. NS=F(I)*HT=.5*(S-2.*Q(I))Y(I)=Y(I)+T

5 Q( I )=Q( I )+3.*T-.5*SX=X+HHCALL YPRIMFDO 6 I=1,NS=F (I )*HT=.2928932?*(S-0(I))Y(I )=Y(I )+T

6( I)=Q(IT+3.*T -. 29289322*SCALL YPRTMFDO 7 T=19NS=F(I) *HT=1.7071067*(S-( I))Y( I )=Y( I )+T

7 Q(I)=Q(I)+3.*T-1.707106*SX=X+HHCALL YPRTMF

n0 ( TR l=1NS=F ( )*HT=(S-2.*O( T )/6.Y(I)=Y(I)+T

8 Q(I)=Q(I)+3.*T-.5*S4 CONTINUF

RFTURNFN0

00040

123

SUBROUTINE YPLOT(N,MXYNG)C**WRITTFN BY J.H. SMALLFY

DIMENSION Y(10003) ,IA(120),PTS(11) IHP(10),X(1)DIMENSION IS(12), IL(24), IY(8), CC(10), IFMT(2)DATA IF,TF /1HE1HF/

C N=NO. OF DATA POINTS, M=NO. OF CURVES PER GRAPH, NG=NO. OF GRAPHSC CC=ARRAY OF ADMISSIBLE RANGES OF THE Y-AXIS SCALEC IS=SYMBOL ARRAY, DIMENSION=NSYM*NGC IL=LEGEND ARRAY, DIMENSION=NSYM*NG*2C IY=Y-AXIS LABLE ARRAY, DIMENSION=2*NGC NSYM=MAXIMUM NO. OF SYMBOLS (CURVES) PER GRAPH

DATA NSYM/ 3 /DATA CC/ 1.,2.,3.,4.,5.,6.,8.,10.,15.,20. /DATA IS/ 1HA,1HF,1HG9 1HH,1H ,1H * 1HV,1H ,1H , 1HL,1H ,1HDATA IY/ 20HY AXIS = DEGREES (F), 20HY AXIS = FEET* 20HY AXIS = FEET/MINUTE, 20HY AXIS = PERCENTDATA TL/ 20HA = AMBIFNT AIR TFMP, 20HF = FILM TFMP

* 2CHG = BALLOON GAS TFMP,* 20HH = BALLOON HEIGHT , H ,1H ,1H ,1H* 20HV= VERTICAL VELOCITY, 1H ,1H ,1H ,1H 9* 20HL = FREE LIFT , 1H ,1H ,1H ,1H

CNGM1=NG-1 $ NSNG1=NSYM*NGM1NCONI=2*NGM1+1 $ NCON2=2*NGMI+2DO 100 I=1 ,M

100 IHP(I) = IS(NSNG1+I)DO 2 J=1,170

2 IA(J)=1H

C FIND Y-SCALE FXTREMES AND CHARACTFRISTICSYMAX=Y(5,1) $ YMIN=Y(1,1)DO 1 I=1,M $ DO 1 J=1l,YMAX=MAX1F(Y(J,I),YMAX)

l YMIN=MIN1F(Y(J,I),YMIN)YDIF=YMAX-YMIN $ YM=AMAX1(ABS(YMAX),ABS(YMIN))PT=INTF(ALOGlO(YDIF)) $ IF(YDIF.LT.1.)PT=PT-1.PP=ATNT(ALOGl0(YM)) $ IF(YM.LT.1.)PP=PP-1,IF(YMIN*YMAX.LTO.)PT=PPFM=10.**(PT-1.)SU=AINT(YMAX/FM) $ SL=AINT(YMIN/EM)TF((YMAX.GT.O.) .AND. (YMAX.GT.SU*EM))SU=SU+1.IF((YMIN.LT.O.) *AND. (YMINLT.SL*FM))SL=SL-1.

C MODIFY Y-SCALF TO A MORE USEFUL SCALE10 SS=SU-SL $ IF(SS.GT.?0.)SS=SS/10.

DO 1? =1,l10TF(SS-CC(I)) 13, 17, 12

13 CCM=CC(I)*EM $ IF(CCM.LT.YDIF)CCM=10.*CCM $ 1=10SD=AINT(CCM/EM+0.5) $ SC=SD/10.

12 CONTINUEIF(SU.FQ.0,)GO TO 14IF(SL.FQ.O.)GO TO 15IF(SU*SL.GT.O.)GO TO 18IF(AMOD(SLSC) )14915

14 SL=SL-1. $ GO TO 1015 SU=SU+l. $ GO TO 1018 IF(ABS(SU).GT.ABS(SL))GO TO 19

124

SUA=SU-AMOD(SU,10() $ IF(SUA-SD.GT.SL)SUASU-AMOD(SU,5 )IF(SUA-SD.GT.SL)GO TO 15SU=SUA $ SL=SU-SD $ GO TO 17

19 SLA=SL-AMOD(SL910.) $ IF(SLA+SD.LT.SU)SLA=SL-AMOD(SL,5*)TF(SLA+Sn.LT.SU)GO TO 14SL=SLA $ SLU=SL+SD $ GO TO 17

C

C START THE GRAPH17 PRINT 1616 FORMAT( 1H1 )

Dn 32 I=1.M $ K=2*(NSNGI+I)-132 PRINT 111, IL(K), IL(K+1

111 FORMAT( 1X9 2A10 )PRINT 110, IY(NCON1), IY(NCON2)

1l1 FORMAT( 41X, *X AXIS = GMT (HOURS) * // 41X, 2A10 / )PTS(1)=SL*FM $ DTS=FM*(SU-SLL)/1 .D n 'O j =I , 1n

30 PTS(J+1) = (PTS(J) + DTS)C nFTERMINF THF FORMAT FOR THE Y-ScAl.F

IP=INT(ALOG10(DTS)) $ TF(rOTS.LT..) IP=IP-lIF((SD.FQ.15.).OR.(SD.EQ.150.))IP=IP-1Tr)=0 $ TF (IP.LT.O) in=lD AS( IP )

IF((ID.NE.O).AND.(ID+PP.lGT.8.))II=INT(8.-PPIENCODF( 20,31,IFMT ) IFID, IF,Tn

31 FORMAT( *(1H * A1, *10.* I19 *,10* Al *11.* I1, * /* )II=l $ IF((PP.LT.O.).AND.(ID.GT.91 )ENCODE(2031,IFlMT)IFEIIIETIPRINT IFMT, (PTS(J) J=1,11)SC=SL*FM $ DTS=SU*EM-SCNZ= 0.+INTF ( -SC/FTS* 1 10.+.5 )IF(NZ.LT.10)NZ=1O $ IF(NZ.GT. 120)NZ=120rXP=(X (N)-X ( 1 ) / ( - ) I=n

C PLOT POINTS, AXIS AND X-SCALFnn 40 J=1 ,XDO=X(I)+(J-1)*nXP

42 DO 29 I= ,M46 II=II+l $ IF(II.LE.N)GO TO 447

TI=N-1 $ rO TO 48447 IF(XDO-X( I ))47,46,46

47 TT=T1-1 $ TF(II.G..1)GO TO 448t=1 $ cO TO 48

448 IF(XnO-X( T))47,48,4848 YP=Y(IIT I)+(Y(T +1,I )-Y(IIT T /(X(1+ 1)-X( II )*(XDO-X(I I

NW=10.+INTF((YP-SC)/DTS*110.+.5)

TF(TA(NZ).F0.1H )TA(NZ)=1HT29 IA(NW)=IHP(I)

WRITE (6,33 (IA(K) ,K= ,120)33 FORMAT( 20?A1

IF(XMODF(J-1,10).EQ.O) PRINT 44, XDO44 FORMAT( 1H+, F6.2, 2X1H+, 1(lOX1H+ ))

nO 34 K=l,120

' 4 IA(K)=1H40 CONTINUE

RFTURN'ND

00113

125

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