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On hot air balloons
Int. J. Mech. Sci. Vol. 22, pp. 637-649
Pergamon Press Ltd., 1980. Printed in Great Britain
ON HOT AIR BALLOONS H. M. IRVINEand P. H. MONTAUBAN Department of Civil Engineering, Massachusetts Institute of Technology, Cambridge, MA 02139, U.S.A. (Received I October 1979; in revised form l March 1980) Summary--The theory of tension shells of revolution forms the basis for an investigation of profiles suitable for use as hot air balloons. These profiles are required to obey certain constraints believed reasonable in the light of practice. A limited series of small-scale experiments was undertaken to confirm characteristics. The end results are a series of computer-produced meridional profiles, with accompanying lunes that can be used as templates for the manufacture of the balloons.
A a C h K Ks Kv L n Ap r ro rb r~ r2 r ra S S T, To T,
To V V W z z /3 Ay 0 d~ ~b0
NOTATION constant radius of sphere constant balloon height ratio of volume to surface area for given weight ratio of surface area to lrh 2 ratio of volume to ¢rh3 meridional length from crown to base exponent pressure differential radius of revolution, horizontal coordinate radius of curvature at apex (crown) radius of revolution at base or throat of balloon radius of meridional curvature distance along perpendicualr to meridian from axis of revolution to surface of balloon, radius of curvature in the normal plane perpendicular to meridian dimensionless ratio of r-h dimensionless ratio of ra-h surface area of balloon same as Ks meridional stress resultant loop stress resultant dimensionless ratio of T , - A y h 2 dimensionless ratio of Te-Ayh 2 volume of balloon same as Kv payload weight vertical coordinate with origin at crown dimensionless ratio of z - h for partially spherical balloon (see Fig. 2) difference in specific weights across balloon wall azimuthal (hoop) angle elevation angle or "co-latitude" for partially spherical balloon (see Fig. 2) !. INTRODUCTION
The principle of the hot air balloon has been known to the Chinese since ancient times when small-scale hot air balloons were used for entertainment in festivals. However, it was not until the late eighteenth century in France that balloons large enough to lift humans were constructed. The past two decades of this century have seen hot air balloon flight becoming increasingly popular. Hot air ballooning is largely a recreational activity and its popularity has grown dramatically worldwide in the last few years; there are now several hundred balloons in the U.S. whereas, a few years earlier, there were practically none. Nevertheless, hot air ballooning is an expensive sport because of the amount of high quality fabric required to make a balloon as well as ancillary equipment such as 637
638
H. M. IRVINEand P. H. MONTAUBAN
basket, burner and control system etc. For example, the lifting of a payload of 2 kN will require a balloon with a main diameter of some 12m, given an operating temperature differential across the fabric wall of 100°C (which is broadly typical and corresponds to a density difference in the air of about 25%). Constructing this balloon will require a minimum of 450 m 2 of fabric and it may well be substantially more if, as is frequently the case, relatively loose pleats occur between the meridional seams or cords that radiate out from the crown of the balloon. For larger payloads balloons with a main diameter of up to 30 m have been built. The fabrics used in balloon construction are expensive because they must be flame and heat resistant, relatively impervious and of high tear strength (to minimise the danger of accidental snagging or spiking, a fairly fine grid of seams and/or cords are deployed to arrest rips). At present high tenacity nylon and polyester are the fabrics most commonly used. Present methods of determining balloon profiles vary between trial-and-error empirical methods to semi-empirical techniques. A good example of the latter which is commonly used is the "tear d r o p " profile--based on the shape adopted by a falling droplet. The objective of the present investigation is to generate balloon profiles with the aid of the well established equilibrium theory of shells of revolution and to require these profiles to meet certain desirable objectives. We require a balloon with the characteristic light-bulb shape in which material is saved by having as low a ratio of surface area to contained volume as possible. (This is beneficial from the point of view of heat loss too). We stipulate, in addition, that the surface be free of wrinkles--that is, the stress resultants must be tensile everywhere. In practice this is likely to pose a difficulty only with the hoop stress resultants in the vicinity of the main diameter. Peak stresses, which usually occur at the crown where the radius of curvature and the differential pressure are largest, are kept in check, although material thickness is almost invariably governed by handling considerations and the need to guard against puncture. In practice the crown of a balloon is spherical so we ensure this by requiring the stress resultants in the immediate vicinity of the crown to be equal. We also require the meridians at the base of the balloon to be vertical and the radius of the opening to be a suitably small fraction of the balloon size; these aspects too may be accommodated by imposing constraints on the stress resultants. The end result is a series of profiles which, it is believed, will be useful in practice. An outline of the remainder of the paper is as follows. Attention is first focused on the equilibrium equations for a membrane of revolution under hydrostatic loading. Some time is then spent on the analysis of hot air balloons of different shapes, the only analytical solution being that for a partially spherical balloon. The balloon of constant stress is then investigated. Because of its demonstrated unsuitability a series of new profiles are then investigated by making different, but more tenable, assumptions about the constraints to be imposed on the stress resultants. Explicit expressions are derived for the stress resultants which, surprisingly, allows a simple formula to be derived for the volume of the balloon; all other data must be generated numerically. Plots of balloon profiles and associated lunes are generated and a short experimental program is reported. 2. EQUATIONS OF EQUILIBRIUM The definition diagram in Fig. 1 shows the basic geometry of a typical meridian. For our purposes the most convenient means of describing the profile is by the functional form r = r(z), so that the following relations between the "co-latitude" d, and the radii of curvature r~ and r2 are useful namely, r = r2sin d~, dr d-'~= rl cos ~b
(1)
If it is assumed that the lift force balances the payload W and we use as a datum the crown of the balloon, it is readily shown that the pressure loading along an outward normal to the profile is Ap = Ay(h - z),
0 -< z -< h,
(2)
On hot air balloons
639
td 0
rW
FIG. |. Definition diagram. where AT is the difference in specific weights between the colder ambient air and the heated contained volume, h is the height of the balloon, and z is the distance from the crown. Implicit in this is that the temperature of the hot air inside the balloon is approximately uniform. The heating process is similar to a large highly turbulent jet in which cooler air is both ingested into the flaming column and expelled down around the outer rim of the opening at the base. This facilitates heat transfer by convection and so, at the end of a heating cycle, the assumption of the absence of any significant temperature difference between points inside the balloon is probably as good as can be made in the circumstances. Our concern, it should be emphasized, is with the basic statics of the problem and not with attendant problems such as heat loss or jet noise (which, in fact, makes operation of larger balloons rather uncomfortable). The overall equilibrium of the balloon is given by W
= foL 27rrAp cos 4~ ds,
(3)
where ds is an element of the meridian and L is thus the meridional length from crown to base. However, since ds = dz]sin tb this equation may be put in the more convenient form W=
2~rrAy(h - z) cot
(4)
The left-hand side is also equal to f0h ~rr2Ay dz. In terms of the equilibrium of an element the following equations apply[l]:
d~-(~r2 sin ~bT,b) rlTo cos tb
0,
(5)
and r~
+ Te = Ay(h - z), r2
(6)
where T,~ and To are the stress resultants in the meridional and hoop directions, respectively. The symmetry of the structure and of the loading precludes the presence of shear stress resultants, and therefore the third fundamental shell equation may be eliminated. Equations (1), (5) and (6) form the basis for subsequent analyses. The effects of stretching are deemed unimportant: inflation causes an increase in diameter of perhaps 1%, which is insignificant. 3. ANALYSES OF DIFFERENT SHAPES There are numerous ways that a solution cab be approached. One may specify a geometry for the meridional profile and insert this into the equations of equilibrium to obtain expressions for the stress
640
H. M. IRVINE and P. H. MONTAUBAN
resultants. If the surface so chosen is non-developable, as it will generally be, the fact that it is not the "correct" profile for the loading applied will not matter too much in terms of the approximate applicability of the results. However, this approach is somewhat limited, as there is point to it only if a simple geometry such as a sphere is specified, for it is only then that the differential equations in the stress resultants can be integrated in closed forms. An alternative approach, and one that has been successfully employed herein, is to place constraints on the stress resultants and use numerical methods to ascertain those profiles that are considered most suitable.t (a) The partially spherical balloon Because the shape with the smallest ratio of surface area to volume is a sphere, it is logical to start the investigation with a profile which is in part spherical. The balloon shown in Fig. 2 is spherical for th < ~b0 and in this portion rl = r2 =
and
a,
/
-t
Ap = A-ya(/3 + cos ~b) "
(7)
Equations (5) and (6) reduce to i-I
)
-~-~(aT, sin d~) - aTo cos ~b = O, [.
and
T, + To=AyaZ(/3+coscb)
/
(8)
]
And these equations may be further reduced to the separated form ~-~(T, sin z ~b)
A va2(/3 + cos 4~)cos 4~ sin ~b
(9)
so that 3/3)(1 + cos tb) T, A),a22 cos 2 tk +6(1(2++ cos/3)
(10)
and from the second of equations (8) T,
"
2f4 c°sz 4' + (4 + 3/3)cos 4~ + (3/3 - 2)} 6(1 + c o s ~b) "
o =aya [
(11)
At the crown T, = To = ATa2(I +/3)/2, as expected. The meridional stress resultants are always tensile for 0-< ~b _< or, but the hoop stress resultants are tensile only if cos_l f - (4 + 313) + ~/(9/32 - 24/3 + 48)~ 6 < / 8 "f '
a il
"O
I
I
Iga
FIG. 2. Partially spherical balloon. t A s a matter of historical fact, it is interesting to note that the first such studies along these lines are due to Taylor[2], who investigated parachute profiles by considering the hoop stress resultants to be zero everywhere, while Den Hartog[3] attributes to Biezeno the practical deployment of surfaces of constant strength and constant stress. These ideas were applied to boiler design and to oil storage tanks [4].
641
On hot air balloons
so the maximum permissible value of I& (see Fig. 2) is given by this upper limit. Values of &,max for different values of p are shown in the table below.
B
d O.mar
0 1 2 5
68.5” 99.0” 120.0” 146.0” 158.0” 164.7” 171.0
:“o 50
Adequate shapes for partially spherical balloons can be obtained for 1
situation corresponds to, for example, a drop forming when the stem of the eye-dropper is full. Such a is very nearly spherical, and solutions for it may be obtained as a first-order perturbation of a circle[5] Fig. 3, with r. = 0.1, for something approaching this configuration). But the approach considered here B large is fundamentally different, since the geometry instead of the stresses is specified.
(b) Balloon of uniform stress Substituting the geometrical relations of equation (1) into equation (5) yields
so that, if T+ is set equal to TO,it follows from this equation that the stress resultants everywhere; that is, T+ = TO= T. Equation (6) becomes
I *.
\
::
0
FIG.
.
0
.lO
\ 80
RFlDIU~
.a
3. Profiles for balloons of constant stress.
Lb0
must be constant
642
H . M . IRVINE and P. H. MONTAUBAN
and if the radius at the crown of the balloon is specified as r~ we have, since, at z = 0, r~ = r2 = ra
A.3,h = 2. T
(15)
r~
However, in Cartesian coordinates 1 - d2r/dz 2 rl - {1 + (dr]dz)2} 3n'
(16)
and from equation (la)
I
1
r2 - r{ + (dr/dz)2} jl2' with the result that a suitable dimensionless form of the differential equation governing the equilibrium of this balloon of constant stress is -d2r/dz 2 {l+(dr/dz)2} m
+
!
2(l_z), ~
r{l+(dr/dz)2} m =
(17)
r = r/h, z = z/h and r = ro/h. The shape of the profile is evidently determined solely by the dimensionless parameter ra, which is simply the ratio of the radius of the crown to the height of the balloon. The scale is obviously determined by h. So far as can be determined, equation (17) does not have an analytical solutiont but a first integral exists of the form {1+ (dr~dz)2} m = r ~ r2(1- z)+
f
r2dz}.
(18)
This may be obtained formally or, more simply, from the overall equilibrium of a slice of the balloon. Equation (17) was solved numerically using the subroutine DVOGER of the IBM 370 IMSL library for three different values of r~.:~ None of the resulting profiles proved to be adequate (see Fig. 3). For small values of r, the curves never open out enough and are in any event pinched well before the base. It is clear that small values of ro give rise to the formation of a pendant drop discussed earlier. For larger values of r~ the curves never narrow (this too is evident from Fig. 3). Overall, therefore, the use of a constraint of constant stress resultants is inappropriate because the resulting profiles are quite unsatisfactory. (c) A new shape In an attempt to overcome the difficulties presented by the other approaches, a lengthy investigation was undertaken into a suitable form for the stress resultants. Two things had to be accomplished; equality of hoop and meridional stress resultants at the crown and a pinched meridional profile at the base of the balloon. This suggested a relation between the stress resultants of the form To = T ~ ( I - f ( r ) ) ,
(19)
where 0 -< [(r) < 1; the lower limit applying at the crown (r = 0) and the upper limit being necessary to guard against a possible, and unacceptable, change in sign of the hoop stress resultant. The functional form f(r) was chosen as this allows equation (5) to be integrated directly to yield
T6 = C e x p { -
f ' [ ( r ) dr}
Jo r
J' (20)
To = C(l - f(r))exp { - f" f(r) dr} , Jo r where C is a constant that is found by substituting these results in equation (6), the governing differential equation. Noting that at the crown r~ = r2, while f(r) = 0, we have
C = AvhrJ2.
(21)
The differential equation assumes the dimensionless form
{1 + (dr/dz)2}3n * r{i + (dr/dz)2} m
(22)
It is evident that an unlimited series of possibilities present themselves for f(r). But again, after trying ?Except when the rhs is zero (this occurs when the pressure loading is zero, i.e. Ay = 0), in which case a solution of the form r = A cosh (z/A + B ) - - t h e c a t e n o i d - - m a y be found. This is the classical problem of the profile taken up by a soap film when stretched between two rings. We thus know the nature of the solution at the crown of the balloon, and its base. ~:See Appendix for more details on the numerical solution.
On hot air balloons
643
several functions it was found that the following simple algebraic form best suited our purposes, namely f(r) = Ar",
(23)
and equation (22) becomes - d2r/dz 2 (1 - Ar") {1 + (dr/dz)2}3/2 + r{l + (dr/dz)2} I/2 =
1 - z) exp (Ar~/n).
(24)
This equation was solved numerically (see Appendix) for different combinations of A, ra and n. It was found that for given ra and n there was only one value of A that met the requirement of vertical meridians at the base. Of the values explored for n (namely, n = 1/4, 1/2 and l) n = 1/2 generated the most satisfactory profiles. The profiles generated with n = 1/4 required very large values of r= if the profiles were to be properly pulled in at the throat of the balloon. Consequently, the stress resultants are unnecessarily high at the crown. With n = l the opposite was essentially true and only small values of r~ could be used before the hoop stress resultants became compressive at the widest p o i n t - - t h e main diameter. The profiles so obtained are not wide enough in the main body of the balloon. Fig. 4 show balloon profiles obtained using n = 1/2 and different values of r= and A. Next to each profile is a lune which is a development of one-eighth of the surface area of the balloon. As these are intended as an aid for manufacture, it is worth noting that lunes representing one-twelfth and one-twentieth of the surface area (these are common divisions too) are easily obtained from the "one-eighth" plots. Before comparing the profiles it is as well to list what is known analytically of them. From equations (20) we have T~, = ~- exp ( - 2Ar'/2), (25) To = 2 (1 - A r '/2) exp (-2Art/2), where T 6 = T J A T h 2, To = To/A3,h 2 and the value of A that is associated with the chosen value of ra may be found from Table 1. The surface area is S = 2¢rh z fo I r{l + (dr/dz)2}"2dz, or
(26)
S=Ks, where S = S/rrh 2 and Ks must be found from numerical integration (see Table 1). Similarly the volume V is given by V = gv,
(27)
-r
t"i"-
Z ._1
"1-
~
==
N
.
. RROIU$
BRLLOON
R
.do==,"_
(RR=I.5OoR=I,OqoN=0.50I
FIG. 4(a). MS Vol. 22, No. 10---D
.do HIOTH
.10
644
H. M. IRVINE and P. H. MONTAUBAN
g
"I-
k---r
•
Z L~
//
~_~
"r
.
. RflDIU5
.Jo~-
~0
. dO WIDTH
.~o
BRLLOON B {RR=2.OO,R=I.20,N=O.SO) F]G. 4(b). where V = V/zrh3 and Kv( = fd r2dz) may be found either by numerical integration or, more simply, from overall equilibrium, namely K~ = r.rb exp ( - 2Arb 112),
(28)
where rb is the radius of the throat of the balloon (see Table I). The final result of interest is the ratio of contained volume to surface area for a given payload W. Using the previous equations, it is readily shown that
V
S
=
K(W/TrAy) I/3,
(29)
where K Ko2/3/Ks. Thus for given W and A-y the constant K provides a basis for comparing the balloons and a suitable basis for that comparison is clearly the sphere in which Ks = 1, Ko = 1/2, so that K = 0.303 (see Table 1). The information supplied in Table 1 is thought to cover all points of interest regarding the Balloons A through E. The sixth column contains entries for a sphere for comparative purposes. Balloon E has the =
I-"r
"1-
,o
Jo 4o flRDIU5
.do:-
o WIDTH
BRLLOON C ( R R I 2 . 5 O , R , , I . 3 2 , N = , O . 5 0 ) FIG. 4(C).
645
On hot air balloons
o I-"r MJ
-r
(.~
°
°
LLJ --J (J
.
.~o
.40
°~tl
t'-
I0
.dO
WIDTH
RADIUS
BALLOON D IRA--S.OO,A=I.U,IoN=O.5Ol FIG. 4(d).
largest crown radius ro possible that still meets the requirement of tensile stress resultants everywhere. Peak stress resultants always occur at the crown and are very easily calculated (i.e. T,,a~= Ayhra/2). It is noteworthy that the peak radius rmaxvaries little between the different profiles and seems always to occur at approximately a one-third point of the balloon height. The radius at the throat rb shows a significant variation, and it is this that may in many instances determine which of the profiles to use, for rb is obviously related to the size of basket that can be attached. Finally we note the essential constancy of the ratio of contained volume to surface area K over all the profiles considered. The shapes are not as "efficient" as the sphere, but it is somewhat surprising to see how well they compare. The above discussion indicates that all of Balloons A through E are suitable, although the last three, C, D and E are probably the most suitable. In view of this the experimental program was confined to simple tests on models of balloons C and E.
-r
Z W
LeJ ~Q
•
.
.~0~
DO
RADIUS BALLOON
E
WIDTH
(RR=3.EO,R=I.qI~,N=0.50]
FIG. 4(e). FIG. 4. New Profiles and accompanying lunes.
.~10
646
H. M. IRVINE and P. H. MONTAUBAN TABLE 1. Balloon
A
B
[a
1.500
2.000
C 2.500
D 3.000
E 3.200
A
1.040
1.200
1.320
1.410
1.440
n
0.500
0.500
0.500
0.500
0.500
1.000
1.250
1.500
1.600
~@,max' ~e,max 0.750
Sphere 0.500
~¢,min
0.192
0.201
0.208
0.215
0.217
~e,min
0.062
0.040
0.022
0.006
0.0003
[max
0.428
0.446
0.461
0.476
0.481
[b
0.180
0.146
0.120
0.I04
O.lO0
rb/rmax
0.420
0.327
0.261
0.220
0.207
Ks
0.775
0.808
0.825
0.853
0.867
l.O00
Kv
0.112
O.ll7
0.120
0.126
0.128
0.167
K
0.299
0.296
0.296
0.294
0.294
0.303
0.500
4. EXPERIMENTS A short series of experiments was conducted to provide a visual check on the theory. The balloons experimented on were models of Balloon C (ra = 2.5) and Balloon E (ra = 3-2). This latter balloon is on the boundary above which tensile hoop stress resultants cannot be assured and is therefore of interest on that account (in fact, the small but finite effects of fabric elasticity, in the form of lateral strains due to Poisson's ratio, means that slightly larger values of ra may be possible). The balloons were made by gluing together eight lunes of silicon-coated nylon fabric. They were 40 cm high and operating conditions were simulated by filling them with water and hanging them upside down from a small frame. A circumferential ring, placed inside and attached to the balloon throat, ensured that the load was evenly transferred to the balloons. In both cases the stress resultants were tensile everywhere. The theory predicts that the hoop stress resultants are low in the vicinity of the major diameter and this was confirmed qualitatively, for meridional wrinkles appeared in both models during filling and stayed there, in the case of Balloon E, almost until the balloon was full. From this we infer that in Balloon E, r, has reached its practical upper limit, if the stress resultants are to remain tensile everywhere. Fig. 5 shows views of Balloon C full and in the filling process when wrinkles were still apparent. An interesting phenomenon which was observed with both balloons was the slight negative (or concave) curvature around the major circumference (and confined to that vicinity). This is believed due to the lay of the fabric, the stiffness of the glued seams and the sizable portion of the circumference that the seams occupy in the model. It is also noticeable on other pneumatic modes[6]. On full-scale prototypes its presence is likely to be marginal only, especially if twelve or twenty lunes are used, and if each lune is "shaped" by constructing it from many smaller fabric pieces. Measurements of rm~x taken from Fig. 5(a) give a value of 0-41 as against the theoretical value of 0-46 from Table 1. This is believed to be a direct result of the observed negative curvature. The profiles considered here compare favorably with those used by balloon manufacturers. Our discussions with manufacturers indicate that the profiles presently used have been developed with little recourse to analytical methods. 5. C O N C L U S I O N S
A new set of profiles for hot air balloons is presented. The profiles are obtained by an analytical-numerical solution of the governing equilibrium equations for shells of revolution. They are believed to be of use for practical purposes, and it is for this reason that both meridional profiles and lunes are presented for each of the five shapes (see Fig. 3). An efficient use of material is assured, since the ratio of surface area to contained volume is low. The contained volume may be calculated from a simple formula, so it is a straightforward matter to determine balloon dimensions for a given payload and operating density differential. Finally, we note that the techniques developed here may be extended to investigate and produce other shapes. For example, profiles in which the meridians are inclined to the vertical at the base of the balloon are an obvious adaptation.
On hot air balloons
FIG. 5. Experimental profile for Balloon C. (a) Balloon full. (b) Balloon partially full.
647
On hot air balloons
649
Acknowledgement--The authors gratefully acknowledge the support of the A. D. Little Research and Innovation Fund of M.I.T. in conducting this research.
REFERENCES 1. W. FLOGGE, Stresses in Shells, Chap. 2. Springer Verlag, New York (1973). 2. G. I. TAYLOR, The Scientific Papers of G. I. Taylor, (Edited by G. K. Batchelor) Vol. 3, p. 26. Cambridge University Press, Cambridge (1963). 3. J. P. DEN HARTOG, Advanced Strength of Materials, Chap. 3. McGraw-Hill, New York 0952). 4. C. L. DAY, Engng News Record, 103, 416-419 (1929). 5. A. K. CHESTERS, J. Fluid Mech. 81(4), 609--624 (1977). 6. M. KAWAGUCHI,Bull. IASSS 18(63), 3-11 (1977). A P P E N D I X - - N U M E R I C A L SOLUTIONS If the subroutine DVOGER is to be used, the second-order differential equations must be transformed into a system of first-order equations namely, the pair dr ~zz =y,
y2 dZY= 1 + - 9"(I + y2)3/2(idz r ra
z),
for equation (17) and the pair dr ~=y,
d y = (1 - y2)(1 - Ar") dz r
2 (1 + y2)3/2(1 -- Z) exp (Ar"/n). ra
for equation (24). To obtain numerical solutions of either system, initial conditions must be specified. In both cases at the crown of the balloon r = 0, y = oo and z = 0. However, the infinite slope cannot be used, so an alternate set of starting values must be sought. It is noted that for very small values of z both systems reduce to dr dz=Y,
dYdz= (1 +y2)r b l
+ y)3/2,
which may be integrated to give the equation of a circle. Therefore, by swinging a short arc of radius ra, we obtain r = ra sin &, y = cot (h and z = ra(I --COS ~b). A value of 4~ = I ° was considered sufficiently small to accurately specify the initial conditions from which a step-by-step integration could proceed. The error in each step was specified to be less than 10-6. No problems of numerical instabilities were encountered.
Pergamon Press Ltd., 1980. Printed in Great Britain
ON HOT AIR BALLOONS H. M. IRVINEand P. H. MONTAUBAN Department of Civil Engineering, Massachusetts Institute of Technology, Cambridge, MA 02139, U.S.A. (Received I October 1979; in revised form l March 1980) Summary--The theory of tension shells of revolution forms the basis for an investigation of profiles suitable for use as hot air balloons. These profiles are required to obey certain constraints believed reasonable in the light of practice. A limited series of small-scale experiments was undertaken to confirm characteristics. The end results are a series of computer-produced meridional profiles, with accompanying lunes that can be used as templates for the manufacture of the balloons.
A a C h K Ks Kv L n Ap r ro rb r~ r2 r ra S S T, To T,
To V V W z z /3 Ay 0 d~ ~b0
NOTATION constant radius of sphere constant balloon height ratio of volume to surface area for given weight ratio of surface area to lrh 2 ratio of volume to ¢rh3 meridional length from crown to base exponent pressure differential radius of revolution, horizontal coordinate radius of curvature at apex (crown) radius of revolution at base or throat of balloon radius of meridional curvature distance along perpendicualr to meridian from axis of revolution to surface of balloon, radius of curvature in the normal plane perpendicular to meridian dimensionless ratio of r-h dimensionless ratio of ra-h surface area of balloon same as Ks meridional stress resultant loop stress resultant dimensionless ratio of T , - A y h 2 dimensionless ratio of Te-Ayh 2 volume of balloon same as Kv payload weight vertical coordinate with origin at crown dimensionless ratio of z - h for partially spherical balloon (see Fig. 2) difference in specific weights across balloon wall azimuthal (hoop) angle elevation angle or "co-latitude" for partially spherical balloon (see Fig. 2) !. INTRODUCTION
The principle of the hot air balloon has been known to the Chinese since ancient times when small-scale hot air balloons were used for entertainment in festivals. However, it was not until the late eighteenth century in France that balloons large enough to lift humans were constructed. The past two decades of this century have seen hot air balloon flight becoming increasingly popular. Hot air ballooning is largely a recreational activity and its popularity has grown dramatically worldwide in the last few years; there are now several hundred balloons in the U.S. whereas, a few years earlier, there were practically none. Nevertheless, hot air ballooning is an expensive sport because of the amount of high quality fabric required to make a balloon as well as ancillary equipment such as 637
638
H. M. IRVINEand P. H. MONTAUBAN
basket, burner and control system etc. For example, the lifting of a payload of 2 kN will require a balloon with a main diameter of some 12m, given an operating temperature differential across the fabric wall of 100°C (which is broadly typical and corresponds to a density difference in the air of about 25%). Constructing this balloon will require a minimum of 450 m 2 of fabric and it may well be substantially more if, as is frequently the case, relatively loose pleats occur between the meridional seams or cords that radiate out from the crown of the balloon. For larger payloads balloons with a main diameter of up to 30 m have been built. The fabrics used in balloon construction are expensive because they must be flame and heat resistant, relatively impervious and of high tear strength (to minimise the danger of accidental snagging or spiking, a fairly fine grid of seams and/or cords are deployed to arrest rips). At present high tenacity nylon and polyester are the fabrics most commonly used. Present methods of determining balloon profiles vary between trial-and-error empirical methods to semi-empirical techniques. A good example of the latter which is commonly used is the "tear d r o p " profile--based on the shape adopted by a falling droplet. The objective of the present investigation is to generate balloon profiles with the aid of the well established equilibrium theory of shells of revolution and to require these profiles to meet certain desirable objectives. We require a balloon with the characteristic light-bulb shape in which material is saved by having as low a ratio of surface area to contained volume as possible. (This is beneficial from the point of view of heat loss too). We stipulate, in addition, that the surface be free of wrinkles--that is, the stress resultants must be tensile everywhere. In practice this is likely to pose a difficulty only with the hoop stress resultants in the vicinity of the main diameter. Peak stresses, which usually occur at the crown where the radius of curvature and the differential pressure are largest, are kept in check, although material thickness is almost invariably governed by handling considerations and the need to guard against puncture. In practice the crown of a balloon is spherical so we ensure this by requiring the stress resultants in the immediate vicinity of the crown to be equal. We also require the meridians at the base of the balloon to be vertical and the radius of the opening to be a suitably small fraction of the balloon size; these aspects too may be accommodated by imposing constraints on the stress resultants. The end result is a series of profiles which, it is believed, will be useful in practice. An outline of the remainder of the paper is as follows. Attention is first focused on the equilibrium equations for a membrane of revolution under hydrostatic loading. Some time is then spent on the analysis of hot air balloons of different shapes, the only analytical solution being that for a partially spherical balloon. The balloon of constant stress is then investigated. Because of its demonstrated unsuitability a series of new profiles are then investigated by making different, but more tenable, assumptions about the constraints to be imposed on the stress resultants. Explicit expressions are derived for the stress resultants which, surprisingly, allows a simple formula to be derived for the volume of the balloon; all other data must be generated numerically. Plots of balloon profiles and associated lunes are generated and a short experimental program is reported. 2. EQUATIONS OF EQUILIBRIUM The definition diagram in Fig. 1 shows the basic geometry of a typical meridian. For our purposes the most convenient means of describing the profile is by the functional form r = r(z), so that the following relations between the "co-latitude" d, and the radii of curvature r~ and r2 are useful namely, r = r2sin d~, dr d-'~= rl cos ~b
(1)
If it is assumed that the lift force balances the payload W and we use as a datum the crown of the balloon, it is readily shown that the pressure loading along an outward normal to the profile is Ap = Ay(h - z),
0 -< z -< h,
(2)
On hot air balloons
639
td 0
rW
FIG. |. Definition diagram. where AT is the difference in specific weights between the colder ambient air and the heated contained volume, h is the height of the balloon, and z is the distance from the crown. Implicit in this is that the temperature of the hot air inside the balloon is approximately uniform. The heating process is similar to a large highly turbulent jet in which cooler air is both ingested into the flaming column and expelled down around the outer rim of the opening at the base. This facilitates heat transfer by convection and so, at the end of a heating cycle, the assumption of the absence of any significant temperature difference between points inside the balloon is probably as good as can be made in the circumstances. Our concern, it should be emphasized, is with the basic statics of the problem and not with attendant problems such as heat loss or jet noise (which, in fact, makes operation of larger balloons rather uncomfortable). The overall equilibrium of the balloon is given by W
= foL 27rrAp cos 4~ ds,
(3)
where ds is an element of the meridian and L is thus the meridional length from crown to base. However, since ds = dz]sin tb this equation may be put in the more convenient form W=
2~rrAy(h - z) cot
(4)
The left-hand side is also equal to f0h ~rr2Ay dz. In terms of the equilibrium of an element the following equations apply[l]:
d~-(~r2 sin ~bT,b) rlTo cos tb
0,
(5)
and r~
+ Te = Ay(h - z), r2
(6)
where T,~ and To are the stress resultants in the meridional and hoop directions, respectively. The symmetry of the structure and of the loading precludes the presence of shear stress resultants, and therefore the third fundamental shell equation may be eliminated. Equations (1), (5) and (6) form the basis for subsequent analyses. The effects of stretching are deemed unimportant: inflation causes an increase in diameter of perhaps 1%, which is insignificant. 3. ANALYSES OF DIFFERENT SHAPES There are numerous ways that a solution cab be approached. One may specify a geometry for the meridional profile and insert this into the equations of equilibrium to obtain expressions for the stress
640
H. M. IRVINE and P. H. MONTAUBAN
resultants. If the surface so chosen is non-developable, as it will generally be, the fact that it is not the "correct" profile for the loading applied will not matter too much in terms of the approximate applicability of the results. However, this approach is somewhat limited, as there is point to it only if a simple geometry such as a sphere is specified, for it is only then that the differential equations in the stress resultants can be integrated in closed forms. An alternative approach, and one that has been successfully employed herein, is to place constraints on the stress resultants and use numerical methods to ascertain those profiles that are considered most suitable.t (a) The partially spherical balloon Because the shape with the smallest ratio of surface area to volume is a sphere, it is logical to start the investigation with a profile which is in part spherical. The balloon shown in Fig. 2 is spherical for th < ~b0 and in this portion rl = r2 =
and
a,
/
-t
Ap = A-ya(/3 + cos ~b) "
(7)
Equations (5) and (6) reduce to i-I
)
-~-~(aT, sin d~) - aTo cos ~b = O, [.
and
T, + To=AyaZ(/3+coscb)
/
(8)
]
And these equations may be further reduced to the separated form ~-~(T, sin z ~b)
A va2(/3 + cos 4~)cos 4~ sin ~b
(9)
so that 3/3)(1 + cos tb) T, A),a22 cos 2 tk +6(1(2++ cos/3)
(10)
and from the second of equations (8) T,
"
2f4 c°sz 4' + (4 + 3/3)cos 4~ + (3/3 - 2)} 6(1 + c o s ~b) "
o =aya [
(11)
At the crown T, = To = ATa2(I +/3)/2, as expected. The meridional stress resultants are always tensile for 0-< ~b _< or, but the hoop stress resultants are tensile only if cos_l f - (4 + 313) + ~/(9/32 - 24/3 + 48)~ 6 < / 8 "f '
a il
"O
I
I
Iga
FIG. 2. Partially spherical balloon. t A s a matter of historical fact, it is interesting to note that the first such studies along these lines are due to Taylor[2], who investigated parachute profiles by considering the hoop stress resultants to be zero everywhere, while Den Hartog[3] attributes to Biezeno the practical deployment of surfaces of constant strength and constant stress. These ideas were applied to boiler design and to oil storage tanks [4].
641
On hot air balloons
so the maximum permissible value of I& (see Fig. 2) is given by this upper limit. Values of &,max for different values of p are shown in the table below.
B
d O.mar
0 1 2 5
68.5” 99.0” 120.0” 146.0” 158.0” 164.7” 171.0
:“o 50
Adequate shapes for partially spherical balloons can be obtained for 1
situation corresponds to, for example, a drop forming when the stem of the eye-dropper is full. Such a is very nearly spherical, and solutions for it may be obtained as a first-order perturbation of a circle[5] Fig. 3, with r. = 0.1, for something approaching this configuration). But the approach considered here B large is fundamentally different, since the geometry instead of the stresses is specified.
(b) Balloon of uniform stress Substituting the geometrical relations of equation (1) into equation (5) yields
so that, if T+ is set equal to TO,it follows from this equation that the stress resultants everywhere; that is, T+ = TO= T. Equation (6) becomes
I *.
\
::
0
FIG.
.
0
.lO
\ 80
RFlDIU~
.a
3. Profiles for balloons of constant stress.
Lb0
must be constant
642
H . M . IRVINE and P. H. MONTAUBAN
and if the radius at the crown of the balloon is specified as r~ we have, since, at z = 0, r~ = r2 = ra
A.3,h = 2. T
(15)
r~
However, in Cartesian coordinates 1 - d2r/dz 2 rl - {1 + (dr]dz)2} 3n'
(16)
and from equation (la)
I
1
r2 - r{ + (dr/dz)2} jl2' with the result that a suitable dimensionless form of the differential equation governing the equilibrium of this balloon of constant stress is -d2r/dz 2 {l+(dr/dz)2} m
+
!
2(l_z), ~
r{l+(dr/dz)2} m =
(17)
r = r/h, z = z/h and r = ro/h. The shape of the profile is evidently determined solely by the dimensionless parameter ra, which is simply the ratio of the radius of the crown to the height of the balloon. The scale is obviously determined by h. So far as can be determined, equation (17) does not have an analytical solutiont but a first integral exists of the form {1+ (dr~dz)2} m = r ~ r2(1- z)+
f
r2dz}.
(18)
This may be obtained formally or, more simply, from the overall equilibrium of a slice of the balloon. Equation (17) was solved numerically using the subroutine DVOGER of the IBM 370 IMSL library for three different values of r~.:~ None of the resulting profiles proved to be adequate (see Fig. 3). For small values of r, the curves never open out enough and are in any event pinched well before the base. It is clear that small values of ro give rise to the formation of a pendant drop discussed earlier. For larger values of r~ the curves never narrow (this too is evident from Fig. 3). Overall, therefore, the use of a constraint of constant stress resultants is inappropriate because the resulting profiles are quite unsatisfactory. (c) A new shape In an attempt to overcome the difficulties presented by the other approaches, a lengthy investigation was undertaken into a suitable form for the stress resultants. Two things had to be accomplished; equality of hoop and meridional stress resultants at the crown and a pinched meridional profile at the base of the balloon. This suggested a relation between the stress resultants of the form To = T ~ ( I - f ( r ) ) ,
(19)
where 0 -< [(r) < 1; the lower limit applying at the crown (r = 0) and the upper limit being necessary to guard against a possible, and unacceptable, change in sign of the hoop stress resultant. The functional form f(r) was chosen as this allows equation (5) to be integrated directly to yield
T6 = C e x p { -
f ' [ ( r ) dr}
Jo r
J' (20)
To = C(l - f(r))exp { - f" f(r) dr} , Jo r where C is a constant that is found by substituting these results in equation (6), the governing differential equation. Noting that at the crown r~ = r2, while f(r) = 0, we have
C = AvhrJ2.
(21)
The differential equation assumes the dimensionless form
{1 + (dr/dz)2}3n * r{i + (dr/dz)2} m
(22)
It is evident that an unlimited series of possibilities present themselves for f(r). But again, after trying ?Except when the rhs is zero (this occurs when the pressure loading is zero, i.e. Ay = 0), in which case a solution of the form r = A cosh (z/A + B ) - - t h e c a t e n o i d - - m a y be found. This is the classical problem of the profile taken up by a soap film when stretched between two rings. We thus know the nature of the solution at the crown of the balloon, and its base. ~:See Appendix for more details on the numerical solution.
On hot air balloons
643
several functions it was found that the following simple algebraic form best suited our purposes, namely f(r) = Ar",
(23)
and equation (22) becomes - d2r/dz 2 (1 - Ar") {1 + (dr/dz)2}3/2 + r{l + (dr/dz)2} I/2 =
1 - z) exp (Ar~/n).
(24)
This equation was solved numerically (see Appendix) for different combinations of A, ra and n. It was found that for given ra and n there was only one value of A that met the requirement of vertical meridians at the base. Of the values explored for n (namely, n = 1/4, 1/2 and l) n = 1/2 generated the most satisfactory profiles. The profiles generated with n = 1/4 required very large values of r= if the profiles were to be properly pulled in at the throat of the balloon. Consequently, the stress resultants are unnecessarily high at the crown. With n = l the opposite was essentially true and only small values of r~ could be used before the hoop stress resultants became compressive at the widest p o i n t - - t h e main diameter. The profiles so obtained are not wide enough in the main body of the balloon. Fig. 4 show balloon profiles obtained using n = 1/2 and different values of r= and A. Next to each profile is a lune which is a development of one-eighth of the surface area of the balloon. As these are intended as an aid for manufacture, it is worth noting that lunes representing one-twelfth and one-twentieth of the surface area (these are common divisions too) are easily obtained from the "one-eighth" plots. Before comparing the profiles it is as well to list what is known analytically of them. From equations (20) we have T~, = ~- exp ( - 2Ar'/2), (25) To = 2 (1 - A r '/2) exp (-2Art/2), where T 6 = T J A T h 2, To = To/A3,h 2 and the value of A that is associated with the chosen value of ra may be found from Table 1. The surface area is S = 2¢rh z fo I r{l + (dr/dz)2}"2dz, or
(26)
S=Ks, where S = S/rrh 2 and Ks must be found from numerical integration (see Table 1). Similarly the volume V is given by V = gv,
(27)
-r
t"i"-
Z ._1
"1-
~
==
N
.
. RROIU$
BRLLOON
R
.do==,"_
(RR=I.5OoR=I,OqoN=0.50I
FIG. 4(a). MS Vol. 22, No. 10---D
.do HIOTH
.10
644
H. M. IRVINE and P. H. MONTAUBAN
g
"I-
k---r
•
Z L~
//
~_~
"r
.
. RflDIU5
.Jo~-
~0
. dO WIDTH
.~o
BRLLOON B {RR=2.OO,R=I.20,N=O.SO) F]G. 4(b). where V = V/zrh3 and Kv( = fd r2dz) may be found either by numerical integration or, more simply, from overall equilibrium, namely K~ = r.rb exp ( - 2Arb 112),
(28)
where rb is the radius of the throat of the balloon (see Table I). The final result of interest is the ratio of contained volume to surface area for a given payload W. Using the previous equations, it is readily shown that
V
S
=
K(W/TrAy) I/3,
(29)
where K Ko2/3/Ks. Thus for given W and A-y the constant K provides a basis for comparing the balloons and a suitable basis for that comparison is clearly the sphere in which Ks = 1, Ko = 1/2, so that K = 0.303 (see Table 1). The information supplied in Table 1 is thought to cover all points of interest regarding the Balloons A through E. The sixth column contains entries for a sphere for comparative purposes. Balloon E has the =
I-"r
"1-
,o
Jo 4o flRDIU5
.do:-
o WIDTH
BRLLOON C ( R R I 2 . 5 O , R , , I . 3 2 , N = , O . 5 0 ) FIG. 4(C).
645
On hot air balloons
o I-"r MJ
-r
(.~
°
°
LLJ --J (J
.
.~o
.40
°~tl
t'-
I0
.dO
WIDTH
RADIUS
BALLOON D IRA--S.OO,A=I.U,IoN=O.5Ol FIG. 4(d).
largest crown radius ro possible that still meets the requirement of tensile stress resultants everywhere. Peak stress resultants always occur at the crown and are very easily calculated (i.e. T,,a~= Ayhra/2). It is noteworthy that the peak radius rmaxvaries little between the different profiles and seems always to occur at approximately a one-third point of the balloon height. The radius at the throat rb shows a significant variation, and it is this that may in many instances determine which of the profiles to use, for rb is obviously related to the size of basket that can be attached. Finally we note the essential constancy of the ratio of contained volume to surface area K over all the profiles considered. The shapes are not as "efficient" as the sphere, but it is somewhat surprising to see how well they compare. The above discussion indicates that all of Balloons A through E are suitable, although the last three, C, D and E are probably the most suitable. In view of this the experimental program was confined to simple tests on models of balloons C and E.
-r
Z W
LeJ ~Q
•
.
.~0~
DO
RADIUS BALLOON
E
WIDTH
(RR=3.EO,R=I.qI~,N=0.50]
FIG. 4(e). FIG. 4. New Profiles and accompanying lunes.
.~10
646
H. M. IRVINE and P. H. MONTAUBAN TABLE 1. Balloon
A
B
[a
1.500
2.000
C 2.500
D 3.000
E 3.200
A
1.040
1.200
1.320
1.410
1.440
n
0.500
0.500
0.500
0.500
0.500
1.000
1.250
1.500
1.600
~@,max' ~e,max 0.750
Sphere 0.500
~¢,min
0.192
0.201
0.208
0.215
0.217
~e,min
0.062
0.040
0.022
0.006
0.0003
[max
0.428
0.446
0.461
0.476
0.481
[b
0.180
0.146
0.120
0.I04
O.lO0
rb/rmax
0.420
0.327
0.261
0.220
0.207
Ks
0.775
0.808
0.825
0.853
0.867
l.O00
Kv
0.112
O.ll7
0.120
0.126
0.128
0.167
K
0.299
0.296
0.296
0.294
0.294
0.303
0.500
4. EXPERIMENTS A short series of experiments was conducted to provide a visual check on the theory. The balloons experimented on were models of Balloon C (ra = 2.5) and Balloon E (ra = 3-2). This latter balloon is on the boundary above which tensile hoop stress resultants cannot be assured and is therefore of interest on that account (in fact, the small but finite effects of fabric elasticity, in the form of lateral strains due to Poisson's ratio, means that slightly larger values of ra may be possible). The balloons were made by gluing together eight lunes of silicon-coated nylon fabric. They were 40 cm high and operating conditions were simulated by filling them with water and hanging them upside down from a small frame. A circumferential ring, placed inside and attached to the balloon throat, ensured that the load was evenly transferred to the balloons. In both cases the stress resultants were tensile everywhere. The theory predicts that the hoop stress resultants are low in the vicinity of the major diameter and this was confirmed qualitatively, for meridional wrinkles appeared in both models during filling and stayed there, in the case of Balloon E, almost until the balloon was full. From this we infer that in Balloon E, r, has reached its practical upper limit, if the stress resultants are to remain tensile everywhere. Fig. 5 shows views of Balloon C full and in the filling process when wrinkles were still apparent. An interesting phenomenon which was observed with both balloons was the slight negative (or concave) curvature around the major circumference (and confined to that vicinity). This is believed due to the lay of the fabric, the stiffness of the glued seams and the sizable portion of the circumference that the seams occupy in the model. It is also noticeable on other pneumatic modes[6]. On full-scale prototypes its presence is likely to be marginal only, especially if twelve or twenty lunes are used, and if each lune is "shaped" by constructing it from many smaller fabric pieces. Measurements of rm~x taken from Fig. 5(a) give a value of 0-41 as against the theoretical value of 0-46 from Table 1. This is believed to be a direct result of the observed negative curvature. The profiles considered here compare favorably with those used by balloon manufacturers. Our discussions with manufacturers indicate that the profiles presently used have been developed with little recourse to analytical methods. 5. C O N C L U S I O N S
A new set of profiles for hot air balloons is presented. The profiles are obtained by an analytical-numerical solution of the governing equilibrium equations for shells of revolution. They are believed to be of use for practical purposes, and it is for this reason that both meridional profiles and lunes are presented for each of the five shapes (see Fig. 3). An efficient use of material is assured, since the ratio of surface area to contained volume is low. The contained volume may be calculated from a simple formula, so it is a straightforward matter to determine balloon dimensions for a given payload and operating density differential. Finally, we note that the techniques developed here may be extended to investigate and produce other shapes. For example, profiles in which the meridians are inclined to the vertical at the base of the balloon are an obvious adaptation.
On hot air balloons
FIG. 5. Experimental profile for Balloon C. (a) Balloon full. (b) Balloon partially full.
647
On hot air balloons
649
Acknowledgement--The authors gratefully acknowledge the support of the A. D. Little Research and Innovation Fund of M.I.T. in conducting this research.
REFERENCES 1. W. FLOGGE, Stresses in Shells, Chap. 2. Springer Verlag, New York (1973). 2. G. I. TAYLOR, The Scientific Papers of G. I. Taylor, (Edited by G. K. Batchelor) Vol. 3, p. 26. Cambridge University Press, Cambridge (1963). 3. J. P. DEN HARTOG, Advanced Strength of Materials, Chap. 3. McGraw-Hill, New York 0952). 4. C. L. DAY, Engng News Record, 103, 416-419 (1929). 5. A. K. CHESTERS, J. Fluid Mech. 81(4), 609--624 (1977). 6. M. KAWAGUCHI,Bull. IASSS 18(63), 3-11 (1977). A P P E N D I X - - N U M E R I C A L SOLUTIONS If the subroutine DVOGER is to be used, the second-order differential equations must be transformed into a system of first-order equations namely, the pair dr ~zz =y,
y2 dZY= 1 + - 9"(I + y2)3/2(idz r ra
z),
for equation (17) and the pair dr ~=y,
d y = (1 - y2)(1 - Ar") dz r
2 (1 + y2)3/2(1 -- Z) exp (Ar"/n). ra
for equation (24). To obtain numerical solutions of either system, initial conditions must be specified. In both cases at the crown of the balloon r = 0, y = oo and z = 0. However, the infinite slope cannot be used, so an alternate set of starting values must be sought. It is noted that for very small values of z both systems reduce to dr dz=Y,
dYdz= (1 +y2)r b l
+ y)3/2,
which may be integrated to give the equation of a circle. Therefore, by swinging a short arc of radius ra, we obtain r = ra sin &, y = cot (h and z = ra(I --COS ~b). A value of 4~ = I ° was considered sufficiently small to accurately specify the initial conditions from which a step-by-step integration could proceed. The error in each step was specified to be less than 10-6. No problems of numerical instabilities were encountered.