U.S. De~wrttrwnt of Agrictrltlrre

Forest Seruice Research Note SE-298

CALCULATING PAYLOAD FOR A

TETHERED BALLOON SYSTEM

by Charles D. Tangren’

ABSTRACT.-A graph method to calculate payload for a tethered balloon system, with the supporting helium lift and payload equations. is described. The balloon system is designed to collect

emissions data during the convective-lift and no-convective-lift phases of a forest fire. A description

of the balloon system and a list of factors affecting balloon selection are included.

Keywords: Smoke plumes. forest fires. particulate matter. emission rate, helium lift. balloon payload. air pollution measurement.

August 1980

The Southern Forest Fire Laboratory

(SFFL) has developed a tethered balloon system

for profiling smoke plumes from forest fires (Ryan

and others 1979). The profiling method, developed by Ward and others (1974), was designed to

collect emissions of total suspended particulate

matter (TSP) from low-intensity prescribed fires

backing against the wind. Towers 12.2 m high

were used to support filters to collect particulates,

which limits this method to short periods when

winds are steady and the angle of plume rise is low

enough so that the rising smoke is below the tops

of the towers. The tethered balloon system extends this direct-measuring filter system to fires of

higher intensity and longer burning periods.

Several groups have used aircraft to sample

higher intensity and longer burning fires (Radke

and others 1978; Packham and Vines 1978; Ward

and others 1979). The behavior of these fires limits

this method to sampling only the emissions associated with a well-developed plume produced during the convective-lift (CL) phase of a fire. During

the no-convective-lift (NCL) phase of the fire,

emissions are too close to the ground to accommodate aircraft sampling. In addition to this

limitation, aircraft sampling does not always acquire fuel consumption data, it depends on indirect methods of measuring TSP (e.g., nepheITangren is Physical Scientist. Southern Forest Fire

Laboratory, Georgia Forestry Center, Macon, Georgia 3 1208.

Southeastern Forest Experiment Station - Asheuille, North Carolina

lometer, particle spectrometer), can have navigational problems, and cannot safely be used at low

altitudes in rough terrain.

The SFFL tethered balloon system surmounts the limitations of tower and aircraft

systems. It is designed to collect emissions data

from both well-defined smoke plumes up to 600 m

above ground level (AGL) during the CL phase of

a fire and from smoke drifting near the ground

during the NCL phase at sites ranging from sea

level to 2,400 m above mean sea level (MSL).

Thus, it gives us the ability to measure emissions

over the entire course of the fire. Also, being a

ground-based system, it is not subject to the navigational problems of an aircraft system, and is in a

better position to collect fuel consumption data.

Essential to a balloon system is the development of a method to determine the balloon’s static

lift and payload to determine the amount of

sampling equipment that can be suspended from

the balloon. The purpose of this paper is to describe the development of the helium lift equations and present a simplified method to calculate

the static lift of the SFFL balloon system and the

number of smoke plume sampling packages (payload) that can be lifted by this system. Field

studies with a balloon system should begin with a

review of this information.

STATIC LIFT

Helium Lift Coejjicient

The static lift of a balloon is determined from
Archimedes’ principle, but for balloon lift analysis the displaced fluid is air and the immersed

body is the helium-filled balloon. The effect of the

buoyant force of the air is to give lift to the

balloon, and the amount of this lift is the difference between the weight of the displaced air and

that of the helium and balloon. Considering first

only the effect of a unit volume of the gases air and

helium. this difference can be expressed as

(Myers 1968)

c/ = w, -jvh (1)

where

Cl = lift coefficient of helium, kg/m”

w, = weight of a unit volume of air. kg/m”

wh = weight of a unit volume of helium,

kg/m:’

The SFFL tethered balloon system was designed principally for use on prescribed fires. The

recommended range of air temperatures for these

m-es is about -4°C to 15°C. In addition, these

prescribed fires take place at elevations ranging

from sea level to 2.400 m. Two variablestemperature and pressure-are now introduced

that affect the weight per unit volume of the air

and he!ium. Humidity also affects the weight of

air, but the effect is of little consequence. especially in the range from 30 to SO percent relative

humidity preferred for prescribed burning. Two

laws of gas expansion. Charles’ or Gay-Lussac’s

and Boyle’s_ state the effects that temperature

and pressure have on the volume of a gas. Combined. they give the general law for gases

P()Vo/To = PVlT (2)

where PuVo and To are the pressure. volume. and

absolute temperature for a mass of gas at an initial

set of values and P, V, and T the same quantities

for the same mass of gas at a final set.

Gas volume has a direct and inverse relation

to gas density. Substituting density, p, for volume

in Equation (2) and solving for p gives

p = (P/PO) (To/T) PO (3)

If we define the initial set of values as those of a

standard atmosphere at MSL and the final set as

those of a standard atmosphere at a given elevation. then the variables of Equation (3) are defined as

p = air density for a standard atmosphere at

a given elevation, kg/m’

pu = air density for a standard atmosphere at

MSL, kg/m”

P = atmospheric pressure for a standard

atmosphere at a given elevation,

mm Hg

PO = atmospheric pressure for a standard

atmosphere at MSL, mm Hg

T = absolute air temperature for a standard

atmosphere at a given elevation, K

To = absolute air temperature for a standard

atmosphere at MSL. K

Values of p have been calculated and tables for

the U.S. standard atmosphere produced for a

range of elevations (Bolz and Tuve 1973). The

ratio p/,10, taken from this table, gives the fractional value of air density at a given elevation.

This fractional value is the same for any gas and,

therefore. the difference between gases. Since w,

and Wh in Equation (1) are actually densities, the

ratio p/p0 can be applied to their difference. Assuming initial values of a standard atmosphere at

MSL. this gives

GS = (P/PO) c/,, (4)

where

C!s = lift coefficient of helium for a standard

atmosphere at a given elevation,

kg/m’.

Cl, = lift coefficient of helium for a standard

atmosphere at MSL, kg/m:<.

A standard atmosphere does not frequently prevail. so we must go beyond Equation (4). Air

temperature is often much different from standard. To account for this. we go back to Charles’ or

Gay-Lussac’y law of the expansion of gases and

apply this to a volume of:rir at a given elevation

V/V,, = T/T, (5)

Substituting density for volume and solving for pa

gives

pa = (TIT,)p (6)

where

pa = air density at a given elevation, kg/m:’
T;, = absolute air temperature at a given

elevation. K.

Although Equation (6) is defined for air

density. it applies to the density of any gas and.

therefore. to the difference between gase\. Substituting the density difference between air and

helium. C/. for pa and the density difference given

in Equation (4). Cl\ for p gives

C/ = (T/T,,) C/\

= (T/T,) (pIpo) Ci,, (7)

where

C/ = lift coefficient of helium at ;I given rle\ation. kg/m”

One assumption made in the development of

Equation (7) is that the temperature and pressure

of the helium inside the balloon and that of the

ambient air are the same. This assumption introduces some error. The pressure inside the balloon

is kept at only a few inches of water higher than

ambient to prevent the balloon nose from cupping

in high winds and. therefore. contributes little to

any error. A greater error is contributed by the

increase in helium temperature over air temperature from solar radiation during the day. This

difference in temperature is called “superheat.”

The effect of superheat is explained by Equations

( I) and (5) and the construction of the balloon.

Increasing the temperature of the helium gas increases its volume. A dilation panel with bungee

restraining cord on the underside of the balloon

permits the balloon to change volume. Thi$ feature maintain\ the gas pressure inside the balloon

during changes in elevation and ambient temperature and also allows the balloon to expand from

the effect of superheat. As helium ga\ i\ not being

added to the balloon. the expanding balloon di\-

places a greater volume and weigh! of air with the

same weight of helium. A\ Equation (I) shows.

the difference w, - wh increases. thus increasing

the value of the lift coefficient. If the additional lift

provided by superheat is critical to any operation.

a method to measure the helium gas temperature

should be provided and the information used to

modify Cl, in Equation (7). Otherwise the additional lift can be considered as a bonus to ensure a

more stable operation. One must remember. however, that if the balloon is operated 14 h per day,

the superheat and accompanying additional lift

will be lost at night.

The purity of the helium has not been made a

part of E:quation (7). The high quality of the SFFL,

balloon construction and envelope material. the

nearly lo&percent purity ofthe ctelivered gas. and

the balloon operating period\ of usually le\r than

I m,k do not require a consideration for purity

decay. However. if a balloon is used that does

have a definite diffusion of air into and gas out of

the balloon, or if the balloon is to remain inflated

for several weeks. the helium purity needs to be

considered. Common practice is to assume a W

percent purity for this situation. The purity would

be entered as a direct relation to lift in Equation

(7).

Since Equation (3) gives the lift of a unit

volume of helium. it is the basic input to the development of any balloon system. Once the gross

static lift of the system has been determined. the

size of balloon required is calculated by dividing

by Cl. In determining gross static lift and selecting

a balloon. a number of factors must be considered:

l Payload.-The weight of the instrument

package. or packages. to be lifted by the balloon.

l Balloon shape.-Balloons have been built in

Cla\s C. vee. barrage. and natural shapes (Peter\

and others 1972). The Class C aerodynamic

balloon. because of its superior lift-to-drag ratio

and thus greater stability. is preferred to the vee

and barrage shapes for research experiments.

l Ralloon size.- Funding and delivery time

determine whether a balloon can be constructed

to the operator’s specification\ or whether a

standard model of closest size can be used.

0 Envelope material.-Permeability o f

balloon. along with the length of time it i\ in operation. determine\ if there i\ a need to compensate

for gas purity.

l Tethering material.--Look for good

strength-to-weight ratio with minimum elongation

propertie\.

l Tethering.-A single tether is usually adequate. lfgreater control in positioning the balloon

i\ required. a tripod arrangement with three

tethers may be needed. This will require more

lifting capability for the same payload.

l Temperature range.-The temperature’s effect on the lift coefficient must be considered

when selecting a balloon size.

3
l Launch-site elevation.-Elevation also affects the lift coefficient and, therefore, selection

of balloon size.

l Balloon’s height above ground level.--Thi\

factor must be added to launch site for elevation

effects. Also, the length of the tether affects lift

requirements.

l Flight duration.-An operation of more than

I day will have diurnal temperature change\ and

helium losses that affect lift.

The Class C aerodynamic balloon is available

as a standard model in sizes ranging from 17 to 99

m:‘. We selected the 99-m:’ balloon for the SFFL

system. which was adequate for a series of experiments scheduled at a site about 2,400 m above

MSL using a variable payload with a maximum

weight of 27.6 kg.

Payload

The next step is to develop an equation for

gross static lift obtained by combining the balloon

volume with the lift coefficient. ClO, in Equation

(7). The lift coefficient in Equation ( 1) becomes

Cl, when values of w, amd wh for a standard

atmosphere at MSL of 1.5” C air temperature and

760 mm Hg atmospheric pressure are used. For

these conditions, w, is I .225 kg/m:‘and wh is 0.169

kg/m:‘. Inserting these values into Equation (I)

gives Cl,, = 1.056 kg/m:{. Combining the balloon

volume of 99 rn:% with this C/, gives

L, = 104.5 (T/T,) (p/p01

where

(8)

L, = gross static lift of the balloon. kg.

This is the heaviest load the balloon can lift.

but not the load that should be lifted. For a stable

flight, especially during strong winds. experience

has shown that the gross static lift should exceed

the load lifted by I5 percent. Without this extra

lift, the balloon will experience large horizontal

movements in a figure 8 pattern. Also. if the tether

is short ( 100 m or less) a strong wind can force the

balloon to the ground. Modifying Equation (8) to

provide a I5percent allowance for extra lift gives

L, = 91 (T/T,) (P/PO) (9)

where

L,, = net static lift of the balloon. kg.

The SFFL tethered balloon system (fig. 1) is

designed to fly the balloon 100 m above the smoke

plume from a prescribed forest fire with a vertical

array of a maximum of I2 particulate matter and

gas sampling packages suspended below the

balloon and through the plume. The balloon is

controlled by a single tether from a variable-speed

winch and a rapid deflation line attached to a wire

inside the balloon and secured to the ground. A

failure of the tether will put tension on the rapid

deflation line and rip open the balloon. Data and

sampling-control signals to and from the sampling

packages and a ground station are transmitted

through an instrumentation line that also supports

the packages by attachment to the balloon (fig. 2).

Connectors for sampling packages are located at

3-m intervals along the length of the line. The

spacing between sampling packages is determined by the height of the plume, and can be

arranged in any pattern on the instrumentation

line within the limits of a maximum plume height

of 600 m AGL and minimum spacing of 3 m.

The load for the SFFL tethered balloon system consists of the tether line. rapid deflation line.

instrumentation line. balloon. and sampling

packages. The maximum for this load can be expressed in terms of the net static lift

L,=w,+wb+wP

where

(10)

WI = weight of the tether. rapid deflation.

and instrumentation lines. kg

Wb = weight of the balloon. kg

W, = weight of the payload. kg.

This introduces another variable to the calculation of balloon lift. WI. Because the balloon is

flown above the smoke plume. its height AGL

depends on the plume’s height. The plume’s

height-determined by the fire’s intensity and

meteorological conditions-at the balloon’s proximity to the fire will vary from a few meters to

several hundred meters AGL. As the balloon’s

height varies. so do the lengths ofthe tether. rapid

deflation. and instrumentation lines. and their

weights. Their combined weight per unit length is

0.0485 kg/m.

The payload. W,. is a set of sampling

packages (fig. 3). and the number of packages

used with the system at any time depends on the

net static lift minus the weights of the lines and

balloon. Therefore. before the balloon system can

4
DEFLATION INSTRUMENTATION

LINE TETHER LINE

LINE

Figure I .-The SFFL tethered balloon system. \u\pending sampling packages in the plume from a fire. i\ controlled by a Tmgle tether

and rapid deflation line. The upper drawing \how\ the rapid deflation line tied off at the point where the tether and instrumentation

lines are connected to the confluence of the load line\.

5
6
he
BALLOON ELEVATION. E(m)

Figure 4.-Graph used to determine net static lift and payload.

BALLOON ELEVATION. E (ml

-NET STATIC LIFT, L&kg)

Figure 5.-Net static lift is determined by drawmg a nortzonrar une tram rue dn rrmper~~ure varue ,r\, a~ LIIC IFLL-llallU ,,“e or

the graph to its intersect with the diagonal at the balloon’s elevation (AI). then drawing a vertical line from that point to its

intersect with the bottom line of the graph (A2). Payload is determined at the intersect of this vertical line with the diagonal at

the balloon’s height (A3) by drawing a horizontal line to the right-hand side of the graph (A4).

8
Net static lift is determined by the intersect of

air temperature at the balloon’s elevation. To

determine net static lift. begin with the air temperature (left-hand side of the graph) at the

balloon’s elevation. Draw a horizontal line from

the temperature point to its intersect with the

diagonal that indicates the balloon’s elevation

above MSL (400 m). Draw a vertical line from this

point to the bottom of the graph that gives the net

static lift at that intersect-in this case. 87. I kg.

The intersect of this same vertical line with the

diagonal at the balloon’s height AGL (300 m) determines the payload. Draw a horizontal line from

this point to the right-hand side of the graph that

gives the payload at that intersect. which is 29.5

kg for this example.

In the second example. B represents a

balloon with height set at 200 m AGL at a site that

is 1.900 m above MSL. making the balloon’s elevation 2.100 m above MSL at an air temperature

of 1°C. Again. determine the net static lift by

starting with the air temperature at the balloon’s

elevation. Draw a horizontal line from this point

to its intersect with the diagonal that indicates the

balloon’s elevation above MSL (9.100 m). From

this point. draw a vertical line to the bottom of the

graph that gives the net static lift at the intersect.

which is 74.2 kg for example B. Extend this vertical line upward to the point on the diagonal line

that indicates the balloon’s height AGL (200 m) to

determine payload. Draw a horizontal line from

this point to the right-hand side of the graph which

gives payload at that intersect. which is 2 I .5 kgfor

this example.

The purpose of Equation (I I) and the graph is

to determine the number of sampling packages

that can be attached to the balloon’s instrumentation line for any set of conditions. i\t present. the

system has I1 sampling packages. each weighing

2.3 kg. In the examples, the payload for balloon A

of 29.5 kg can accommodate 12 packages. with lift

to spare. The payload for balloon B of 21.5 kg

limits the number of packages to nine.

Gmph AccLrrac>

In addition to offering speed and simplicity.

the graph (fig. 4) must have enough accuracy built

into it to give us confidence that the number of

sampling packages selected is correct. For the

first entry on the graph, temperature data can be

entered with an accuracy of? O.OYC. Proceeding

through the graph, a 100-m separation between

elevation lines gives an interpolation accuracy

within ? IO m. a 50-m separation between height

lines gives an interpolation accuracy within 5 5

m, and the intersect with the payload scale can be

interpreted within 2 0.1 kg. A plot through the

graph of the limits of accuracy produces an error

of less than 2 0.5 kg of payload. Since this is only

about 4 percent of the excess lift needed for a

stable balloon fight. an error caused by the graph

will not overload the system.

Any error in the graph should be considered

in view of the approximation of Equation (I I). A

day’s operation will see a continuous change in

temperature and the predicted temperature may

be in error by several degrees. This has a direct

effect on Equation (I I). The amount of superheat

from solar radiation will have an effect on the

helium lift coefficient, Cl,. that is not accounted

for. In addition. the method of inflating the

balloon allows the volume to be estimated only at

the start of an operation. The accumulation of all

of these factors overshadows any error in the

graph.

In addition to determining net static lift and

payload for any set of conditions, figure 5 describes the general relationship between the

balloon system and its environment. A quick look

at the graph shows how lift decreases with either

increasing elevation or temperature. Since lift decreases as conditions move to the left and top of

the graph. this is where the limits ofthe system are

to be found. If the balloon is required to be at 600

m AGL. the highest elevation possible for the

balloon with just one package ranges from 1,510 m

at WC temperature to 2,260 m at -SC. However, the system can lift nine packages to 3,000 m

at -5°C if the balloon’s height AGL is only 50 m.

This illustrates the options facing a~?alloon operator. At the beginning of the project. the operator

can choose the balloon size and payload, but after

that, only the payload can be changed. The

weather can be chosen only by waiting for the day

and time when operating conditions are optimum.

The elevation may also be selected if the higher

elevation projects can be set aside for days of

optimum conditions. However. experience will

show that schedules do not always allow for time

to wait for the best conditions. which leaves the

choice restricted to the size of the payload.

SUMMARY

Two laws of gas expansion. Charles’ or GayLussac’s and Boyle’s, applied to helium and air

give an equation of the helium lift coefficient. C,.

9
as it is affected by changes in air temperature and

air density. The right balloon for the tethered

balloon system is selected by considering a number of factors. Combining balloon size with the

helium lift coefficient gives the gross static lift

from which equations for net static lift and payload are developed. The equation5 are replaced

by a graph that eliminates the need for an engineering handbook and calculator when determining payload for sampling packages.

LITERATURE CITED

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U.S. Department of Agriculture or Forest Service

of any product or service to the exclusion of

others which may be suitable.

r

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use management of the Nation’s

forest resources for sustained

yields of wood, water, forage,

wildlife, and recreation. Through

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