Sensitivity Analysis for Ascending Zero Pressure Balloons

Carl Olsson

Rymdteknik, master 2019

Luleå tekniska universitet Institutionen för system- och rymdteknik Abstract

This work analysed the sensitivity in ascending velocity for three different zero pressure balloons of different sizes. A program that considers the atmospheric and thermodynamic conditions was written to try and simulate the balloons ascent throughout the atmosphere. The sensitivity for ascending velocity was determined with the Monte-Carlo method and measuring the mean vertical velocity throughout the atmosphere. The performance of the simulated flight when compared to its real flight, was accurate in the troposphere and then started to deviate in the stratosphere. Whether the drag is underestimated or overestimated during the stratosphere would depend on the size of the balloon and when it is released. Changing the free lift for the balloons showed that small balloons have almost the same sensitivity in the troposphere as in the stratosphere, while a larger balloon could be more sensitive in either part, depending on the time it is launched. With a changing ground temperature, the balloons with less cloud cover would be more sensitive in the stratosphere. Balloons flying during hours with sunlight are less sensitive to a changing cloud cover than balloons that fly with little to no sunlight, but some more data should be analysed for a proper conclusion. Larger balloons are more sensitive with a changing ground albedo value compared to smaller balloons. Contents

1 Introduction 3 1.1 Previous work ...... 4 1.2 Outline of thesis work ...... 4

2 Governing equations in balloon flight 4 2.1 Dynamic model ...... 6 2.1.1 Forces ...... 6 2.1.2 The drag coefficient ...... 7 2.1.3 Equations of motion ...... 8 2.2 Geometric model ...... 8 2.3 Thermal model ...... 9 2.3.1 Heat transfer to the balloon film ...... 9 2.3.1.1 Optical properties ...... 10 2.3.1.2 Solar elevation angle ...... 10 2.3.1.3 Solar radiation ...... 11 2.3.1.4 IR radiation ...... 12 2.3.1.5 Heat convection ...... 13 2.3.2 Heat transfer to lifting gas ...... 15 2.4 Atmospheric model ...... 15

3 Sensitivity analysis 15 3.1 Balloon launch information ...... 17 3.2 Free lift analysis ...... 18 3.3 Ground temperature analysis ...... 18 3.4 Cloud factor analysis ...... 18 3.5 Ground albedo analysis ...... 18

4 Results 19 4.1 Simulation performance ...... 19 4.2 Sensitivity with an abundance/excess of buoyant gas ...... 20 4.3 Sensitivity for a changing ground temperature ...... 21 4.4 Sensitivity for a changing cloud cover ...... 22 4.5 Sensitivity for a changing ground albedo ...... 23

5 Discussion 24

6 Conclusions 25

7 References 27 Lule˚a University of Technology January 7, 2019

1 Introduction

A scientific balloon is an aerial vehicle that is used for scientific and engineering purposes. There are different kinds of scientific balloons, such as the zero-pressure balloon and the super- pressure balloon. This work will be looking at the zero-pressure balloon, where the pressure difference between the inner lifting gas and atmosphere are zero. Depending on its size, the balloon can carry payloads weighing several tons at altitudes over 30 km. The contents of the payload are usually expensive equipment such as radars and scientific data collectors that would be retrieved after the flight is done. Due to the temperature differences that occur between day and night, some operational procedures will be made during the balloon flight. During daytime, the buoyant gas will expand and the balloon will rise. If the volume of the buoyant gas exceed the designed volume of the balloon, some excessive gas will be vented out. When the sun sets over the horizon and night arrives the buoyant gas will cool down and shrink. This will cause the balloon to lose some of its lifting force and it will sink. In order to regain its altitude, the balloon will have to drop some ballasting weights. With this procedure the balloon can stay in the stratosphere for several hours or even days before it is time to terminate the balloon and drop its payload. Where it will land depends on when and where the balloon is released. To avoid diplomatic problems with neighbouring countries or having to retrieve the payload from the ocean, a trajectory prediction tool can be very useful.

Figure 1: Balloon launch in Kiruna, Sweden

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1.1 Previous work There are existing simulation tools that will predict a balloon’s trajectory, such as NASA’s Scientific Balloon Analysis Model (SINBAD). Although they give a reasonable accuracy, the prediction will hardly be the same as the real flight due to computational instabilities and existing uncertainties for the balloon flight. This work will focus on how such uncertainties affect the ascension of a balloon. The use of balloons or buoyant vehicles are not that common for space exploration, however there is a community that are exploring this kind of technology. Farley [1] wrote a manual on how to build a tool, that can describe the vertical and horizontal motions of a balloon. Van Dossalaer [2] wrote the Buoyant Aerobot Design and Simulation tool BADS, that can be used for exploration on other planets. Sensitivity studies on how uncertainties influence the balloon trajectory have been made in the past. Lee et al. [3] classifies these uncertainties into four categories: operational uncertainty, uncertainty in the prediction model, environmental uncertainty, and uncertainty by the manufacturer. They used a Monte-Carlo analysis on each uncertainty separately to identify the most dominant uncertainty parameter in the distribution of landing points. Although they did show the landing point for each uncertainty, they only tried on one type of balloon with the same mass and size. Saleh et al. [4] studied the difference in ascension speed of the balloon in the troposphere and stratosphere with different launch dates and launch sites. Their conclusion to that was that there is no significant effect on ascending velocity at troposphere, but rather in the stratosphere. Dai et al. [5] discussed the film radiation properties and clouds have on the balloon’s thermal behaviour but missed out on how the ascent of the balloon varied.

1.2 Outline of thesis work The work done for this thesis is: (1) Building a tool that simulates the balloon’s ascent and rate of climb, (2) Investigate how much of an influence different parameters have on the rate of climb. In other words, this work aims to study the sensitivities of balloons of different sizes and launch times in different parts of the atmosphere.

2 Governing equations in balloon ﬂight

To know the balloon’s trajectory during its flight, it is important to know what forces that acts on the balloon. The balloon as a vehicle works because of Archimedes principle where the buoyant force is equal to the weight of the displaced medium. The ascending velocity is determined by the buoyant force and the drag, which means that it is necessary to calculate these correctly during the balloon flight. In order to calculate the buoyant and drag forces, the surface area and volume for the balloon needs to be known. Another force that acts on the balloon is the weight from the balloon and its payload. The volume and weight of the balloon varies depending on the flight and are usually given by the manufacturers of the balloon. The forces that act on the balloon is determined by a dynamical model.A geometric model and a thermal model needs to be introduced in order to give information that is necessary for the dynamical model. Reference areas, incidence areas etc. will be given by the geometric model while the balloon film temperature, lifting gas temperature etc. will be given by the temperature model. An atmospheric model will also be needed to supply information about the surrounding air temperature, pressure and density.

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Figure 2: A block diagram describing the simulation tool

Figure 2 is a brief description on the simulation tool used for this work. Before describing it in detail, some assumptions will be made.

• The balloon is considered a 3 degree-of-freedom point mass.

• The variation of g (acceleration of gravity) is ignored.

• Lifting gas and air assumed to follow the ideal gas law.

• Eﬀects of humidity on atmospheric pressure are neglected.

• The density and pressure of the lifting gas inside the balloon is considered uniform.

• Lifting gas is transparent, so it does not absorb nor emits.

• The temperature inside of the balloon is uniform.

• The temperature along the surface of the balloon is uniform.

• The shape of the balloon is constant.

• There are no winds pushing the balloon.

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2.1 Dynamic model 2.1.1 Forces To write the diﬀerential equations that describe the balloon dynamics, it’s important to know the forces that act on the balloon. A simple analysis will give the forces of greatest importance

• net buoyancy

• weight

• aerodynamic drag

Figure 3: The forces that acts on the balloon during ﬂight

The net buoyancy, usually deﬁned as the gross inﬂation, GI is given by

GI = g · (ρair − ρgas) · V olume (1) The volume of the balloon is determined by the ideal gas law

Tgas V olume = Mgas · Rgas · (2) pgas

The temperature, Tgas, pressure, pgas and densities, ρair and ρgas, will be discussed in a later section. Rgas is the speciﬁc gas constant for the lifting gas, which in this work is helium and the mass of the helium inside of the balloon is

Mgas = 0.1602 · Mgross · (1 + F reeLift) (3) with Mgross as the gross mass and F reeLift as the desired extra lift for the balloon.

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The weight is given by W = Mgross · g (4) where Mgross is the sum of the payload mass, Mload, the ballast, Mballast, and the mass of the balloon ﬁlm, Mfilm, Mgross = Mload + Mballast + Mfilm (5) The mass of the lifting gas is considered with the net buoyancy. The aerodynamic drag for each direction is Vrelx Dragx = Drag · Vrel

Vrely Dragy = Drag · (6) Vrel

Vrelz Dragz = Drag · Vrel The magnitude of the aerodynamic drag is 1 Drag = · ρ · V 2 · C · A (7) 2 air rel d top where Atop is the top projected area of the balloon, Cd is the drag coeﬃcient and Vrel is the magnitude of the balloon’s velocity relative to the wind. For each direction the relative velocity is

Vrelx = Vx − Vwindx

Vrely = Vy − Vwindy (8)

Vrelz = Vz − Vwindz where V is the balloon’s velocity and Vwind is the wind velocity. The magnitude of the relative velocity is then q V = V 2 + V 2 + V 2 (9) rel relx rely relz

2.1.2 The drag coeﬃcient Palumbo [5] argues that for natural shape balloons, it is reasonable to consider a variable drag coeﬃcient during the ascent for the balloon. His arguments that support this are

• inconstant shape

• shape deformability

• dimensional reasoning

These arguments according to Palumbo, would explain why the drag coeﬃcient should be a function of the Reynolds number, the Froude number and another dimensionless parameter that accounts for shape variations k k1 Atop Cd = 0.2 · · + k2 · Re · (10) F r Re Atop0

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where k, k1 and k2 are constants. With an appropriate value for these constants, the simulated flight would fit with the actual flight data. The constants would need to be specified for every balloon flight however. Lee et al. [3] argues for another method to implement a varying drag coefficient. It requires to calculate the drag coefficient of the balloon during the ascent with the use of computational fluid dynamics (CFD). However, when using this method, the shape of the balloon needs to be determined and then meshed. Both the shape determination and meshing takes time and requires many computational resources. In the end, this work will be using a constant value for its drag coefficient.

2.1.3 Equations of motion As the balloon ﬂy through the air, there will be a mass of air that is dragged along. This mass will be considered in the equation of motion. The total mass is

Mtotal = Mgross + Mgas + Cadded · ρair · V olume (11) where Cadded is the virtual mass coeﬃcient and is equal to 0.37 in this work. Knowing all the forces that act on the balloon and the total mass, it is now possible to write down the equations of motion. Using local NED (North - East - Down) as a reference frame, the equations are

Mtotal · x¨ = Dragx

Mtotal · y¨ = Dragy (12)

Mtotal · z¨ = W − GI + Dragz That is it for the dynamical model. For the balloon volume, gas mass etc. it is necessary to introduce a geometric and a thermal model.

2.2 Geometric model The shape of the balloon is essential for modelling forces and heat loads. In reality the shape of the balloon will vary during its ﬂight. For the simulation tool it is assumed to have a constant shape throughout the ﬂight. The diameter viewed from the top is

1 Diameter = 1.383 · V olume 3 (13) The height from top to bottom is Height = 0.748 · Diameter (14) The gore length is 1 Lgore = 1.914 · V olume 3 (15) The surface area is 2 Asurf = 4.94 · V olume 3 (16) For convection calculations that are used in the thermal model, the eﬀective exposed surface will be

2 3 Lgore Asurf1 = 4.94 · V olumedesign · 1 − cos π (17) LgoreDesign

Aeffective = 0.65 · Asurf + 0.35 · Asurf1 (18)

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The top projected area will be used as a reference for the illuminated projected area of the balloon π A = · Diameter2 (19) top 4 The illuminated projected area depends also on the solar elevation angle, ELV , which is deﬁned as the angle between the direction of the sun and the true horizon. For a zero-pressure natural shape balloon that area will be

Aprojected = Atop · (0.9125 + 0.0875 cos(π − 2 · ELV )) (20)

The solar elevation angle among other thing will be discussed in the section about the thermal model.

2.3 Thermal model There are two thermal system to consider: the balloon ﬁlm and the lift gas inside of the balloon. The heat transfer are between the atmosphere, the balloon ﬁlm and the lift gas.

2.3.1 Heat transfer to the balloon film For the heat transfer on the balloon film, there are several factors that needs to be looked at. The differential equation that describes the temperature of the balloon film is dT Q film = film (21) dt cfilm · Mfilm where cfilm is the specific heat capacity and Qfilm is the heat of the film, which can be written as

Qfilm = QSun + QAlbedo + QIRplanet + QQIRsky + QIRfilm + QConExt − QConInt − QIRout (22)

In total there are eight factors that will be considered. They are:

• the direct solar radiation, Qsun

• the reﬂected solar radiation, QAlbedo

• the planetary IR radiation, QIRplanet

• the IR radiation from the sky, Qsky

• the absorbed self glow from the interior, QIRfilm

• the external convection, QConExt

• the internal convection, QConInt

• the emitted heat from both interior and exterior, QIRout

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2.3.1.1 Optical properties The optical properties for both solar and IR radiation are both expressed as fractions so that

α + τ + r = 1 (23) αIR + τIR + rIR = 1 with the absorptivity, α, the transmissivity, τ and the reflectivity r. Another property of the balloon film is its emissivity, ε. Kirchoff’s law of radiation transfer will state that while in thermal equilibrium, the absorptivity is equal to the emissivity at any specific wavelength. Because the balloon emits IR radiation, the emissivity is equal to the IR absorptivity.

ε = αIR (24) The film is not only affected by radiation from the outside, but also radiation that are trapped by multiple reflections. The effective reflectivity can be written as a geometric series. r r = r + r2 + r3 + ··· = eff 1 − r (25) 2 3 rIR reffIR = rIR + rIR + rIR + ··· = 1 − rIR There are measured values of the film properties for a Zero-pressure balloon made out of a type of polyethylene called SF372. They are seen in table 1.

Table 1: Zero-pressure balloon (ZPB) Material SF372

1 Layer α 0.024 ε 0.134 τ 0.916 τIR 0.176

2.3.1.2 Solar elevation angle The solar elevation angle, ELV , is addressed as

ELV = sin−1(sin(δ) · sin(ϕ) + cos(δ) · cos(ϕ) · cos(HRA)) (26) with the launch location latitude, ϕ, the hour angle, HRA and the declination angle, δ. The declination angle depends on the day of the year according to

284 + Day δ = 23.452 · sin 2 · π · (27) 365 where Day is the day number of the year, for example d = 1 for 1st January. The hour angle is expressed as 2 · π HRA = · (3600 · LST · −43200) (28) 86400

10 Lule˚a University of Technology January 7, 2019 where LST is the local solar time. During the night or twilight, when the sun is over the horizon, in other words when ELV < 0◦, the solar elevation angle will can be assumed to be 0◦.

Figure 4: Solar elevation angle

2.3.1.3 Solar radiation The amount of solar radiation that affects the balloon depends on both the direct and reflected radiation. There are several factors that affect the amount, such as the balloon’s altitude, the solar elevation angle, optical properties for the balloon, solar flux etc. The solar flux at the top of the atmosphere is " # ! 1358 1 1 + e2 Isun = 2 · 1 + · − 1 · cos(TA) (29) RAU 2 1 − e where e is the orbital eccentricity, RAU is the mean orbital radius and TA is the true anomaly. For planet Earth e = 0.016708 and RAU = 1. The true anomaly depends on the mean anomaly Day MA ≈ 2π · (30) 365 5 TA ≈ MA + 2 · e · sin(MA) + · e2 · sin(2 · MA) (31) 4 The solar flux at certain altitudes depends on the atmospheric transmittance

τatm = 0.5 · (exp(−0.65 · Airmass) + exp(−0.95 · Airmass)) (32) where Airmass (air mass ratio) is

pair p Airmass = · 1229 + (614 · sin(ELV ))2 − 614 · sin(ELV ) (33) p0 with p0 as the pressure at ground level. The resultant solar intensity is then

IsunZ = Isun · τatm (34)

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The direct solar ﬂux acting on the balloon is IsunZ · (1 − CF ), z ≤ 11000 m qsun = (35) IsunZ , z > 11000 m where CF is the cloud factor, which ranges between 0-1 and z is the balloon’s altitude. It is now possible to describe the direct solar radiation

Qsun = α · Aprojected · qsun · (1 + τ · (1 + reff )) (36) For the reﬂected solar ﬂux, it will depend on the type of ground the balloon is travelling over and the amount of cloud cover.

Albedoground · (1 − CF ) · Isun · sin(ELV ), z ≤ 11000 m qAlbedo = 2 (Albedoground · (1 − CF ) + Albedocloud · CF ) · Isun · sin(ELV ), z > 11000 m (37) The surface area that is exposed to the planet surface is

1 − cos(HalfCone ) V iewF actor = angle (38) 2 where HalfConeangle is −1 Rearth HalfConeangle = sin (39) Rearth + z

Rearth is the radius of the earth (6371000 m). The reﬂected solar radiation can now be expressed as

QAlbedo = α · Asurf · qAlbedo · V iewF actor · (1 + τ · (1 + reff )) (40)

2.3.1.4 IR radiation The IR emission from the ground is

4 qIRground = εgroundσTground (41) with the Stefan-Boltzmann constant, σ, the ground emissivity coeﬃcient εground and ground temperature, Tground. The emissivity for the ground will vary depending on the type, but the average for it is 0.95. As with visible light, the IR will be weakened by the atmospheric IR transmittance Pair Pair τatmIR = 1.716 − 0.5 · exp(−0.65 ) + exp(−0.95 ) (42) P0 P0 The IR from the ground that will reach the balloon will be

qIRgroundZ = qIRgroundZ · τatmIR (43)

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Taking the clouds in consideration, IR ﬂux from the planet will be qIRgroundZ + CF · qIRcloudZ , z ≤ 11000 m qIRplanet = (44) qIRgroundZ · (1 − CF ) + CF · qIRcloudZ , z > 11000 m where qIRcloudZ is the IR from the clouds, calculated in a similar fashion to the ground IR. For the IR ﬂux from the sky, average conditions will be used in this work

qIRsky = 265 · exp(−0.00018842 · z) (45) The absorbed IR from both the planet and the sky will be

QIRplanet = αIR · Asurf · qIRplanet · V iewF actor · (1 + τIR · (1 + reffIR)) (46)

QIRsky = αIR · Asurf · qIRsky · (1 − V iewF actor) · (1 + τIR · (1 + reffIR)) (47) The balloon itself will also emit some of its own heat from the interior and exterior.

4 QIRout = 2 · σ · ε · Asurf · Tfilm (48) It is able to absorb its own heat from the interior.

4 QIRfilm = σ · ε · αIR · Asurf · Tfilm · (1 + reffIR) (49)

2.3.1.5 Heat convection The balloon’s convective heat transfer are between the exterior and the atmosphere, and the interior and the lifting gas. The atmospheric pressure, pair and temperature, Tair, will be discussed in more detail during the section about the atmospheric model. For a zero-pressure balloon, the pressure on the inside will be the same as the pressure on the outside.

pgas = pair (50) The density of the ambient air and lifting gas depends on their temperature, pressure and speciﬁc gas constant.

pair ρair = (51) Rair · Tair

pgas ρgas = (52) Rgas · Tgas Air and gas properties Dynamic viscosity of air: −6 1.5 1.458 · 10 · Tair µair = (53) Tair + 110.4 Dynamic viscosity of helium:

T 0.647 µ = 1.895 · 10−5 · gas (54) gas 273.15

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Conductivity of air: T 0.9 k = 0.0241 · air (55) air 273.15 Conductivity of helium: T 0.7 k = 0.144 · gas (56) gas 273.15 Prandtl number for air: −4 P rair = 0.804 − 3.25 · 10 · Tair (57) Prandtl number for helium:

−4 P rgas = 0.729 − 1.6 · 10 · Tgas (58)

External convection The external convection depends on two kinds of convection, free and forced convection. The free convection in which heat transfer to the balloon skin from the warmer surrounding air. Grashof number: 2 3 ρair · g · |Tfilm − Tair| · D Grair = 2 (59) Tair · µair Nusselt number for a sphere in free convection:

0.25 Nuair = 2 + 0.45 · (Grair · P rair) (60)

External free convection heat transfer coefficient: Nu · k HC = air air (61) free Diameter When the balloon is ascending, the forced convection depends on the balloon’s relative vertical velocity. Reynolds number of the balloon: V · D · ρ Re = relz air (62) µair External forced convection heat transfer coefficient: k HC = air · (2 + 0.41 · Re0.55) (63) forced D The effective external convection coefficient is the greater value between the free and forced convection coefficient. HCfree,HCfree > HCforced HCexternal = (64) HCforced,HCforced > HCfree The heat from external convection:

QConExt = HCexternal · Aeffective · (Tair − Tfilm) (65)

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Internal convection The internal convection coeﬃcient:

1 2 ! 3 ρgas · g · |Tfilm − Tgas| · P rgas HCinternal = 0.13 · kgas · 2 (66) Tgas · µgas

The heat from internal convection:

QConInt = HCinternal · Aeffective · (Tfilm − Tgas) (67)

2.3.2 Heat transfer to lifting gas The lifting gas will lose some of its heat due to volume changes and mass loss. Because of the assumptions, the only heat load that affects the temperature of the lifting gas will be the heat convection between the balloon film and the gas. The differential equation that describes the temperature for the lifting gas is dTgas 1 g · Mgas · Rgas · Tgas · Vz = · QConInt − (68) dt cp · Mgas Rair · Tair with the specific heat capacity for constant pressure, cp.

2.4 Atmospheric model The temperature and pressure of the air depends on the altitude. It will also depend on what time of year it is. There are diﬀerent atmospheric model to choose from when designing a trajectory simulating tool such as the US standard atmosphere.   288.15 − 0.0065, 0 < z < 11000 m Tair = 216.65, 11000 m < z < 20000 m (69)  216.65 + 0.0010(z − 20000), 20000 m < z < 32000 m

 288.15−0.0065·z 5.25577  101325 · 288.15 , 0 < z < 11000 m  z−11000 pair = 22632 · exp(− 6341.62 ), 11000 m < z < 20000 m (70)  z−20000 −34.163  5474.87 · 216.65 + 0.0010 · 216.65 , 20000 m < z < 32000 m In this work, atmospheric data from two diﬀerent balloon ﬂight have been given. With interpolation of the data the pressure, temperature and density of the atmosphere at certain altitudes surrounding the balloon is retrieved during its ascent. For the third balloon, the US standard atmosphere will be used.

3 Sensitivity analysis

The program that was used for the simulations were MATLAB2018a. Even with a model that have the correct governing equations, the simulation compared to the real flight will be prone to inaccuracies. Reasons for this other than instability in the computation, is that there are uncertainties related to the balloon flight. Lee et al. [3] classified these uncertainties into four

15 Lule˚a University of Technology January 7, 2019 categories: manufactural uncertainty, operational uncertainty, environmental uncertainty and uncertainty in the prediction model. A conclusion in their work was that the manufactural uncertainty for the volume of the balloon had such a little effect on the trajectory that it could be neglected. Therefore this work will not analyse the sensitivity in small volume changes. When preparing for a launch, operational errors occurs. The most common operational uncertainty is when helium gets injected into the balloon because of existing errors when measuring the mass of the helium used for the balloon. The environmental uncertainties are unavoidable because of changing environments that affects the balloon flight. This work will analyse changes in the ground temperature, ground albedo and cloud cover. The reasoning why ground temperature are being analysed was because of the instrument that measure the temperature have the probability to show a small error. For the reason why ground albedo is studied in this work is because its value is set as constant for the simulation while the real value depends on the ground of where the balloon is flying. The reasoning why cloud coverage is being analysed is due to how it changes over time. These simulations uses a constant cloud coverage during its simulation. The uncertainty in the prediction model are involving the uncertainties with the dynamic model and the thermal model. For the dynamic model, the uncertainty is with the drag coefficient of the balloon, because of how the shape of the balloon behaves during the ascent. In reality, the drag coefficient will be changing during the flight because of the shape variation of the balloon, while in this work it is set as a constant value of 0.45. For the thermal model, the optical properties of the balloon envelope depends on the material that is used for the balloon and how thick it is. The uncertainties in prediction model will not be explored in this work because of time constraints. The balloon flights that will be simulated are based on real balloon launches. The sizes of the balloons are publicly available [7] and will be disclosed. The envelope and payload mass are not and will therefore not be disclosed. Atmospheric data have been given by the Swedish Space Corporation (SSC) for Balloon 1 and Balloon 2, so an interpolation for the atmospheric pressure and temperature could be made at the simulated altitude. No data was given for Balloon 3, so it will have to use the standard US atmosphere to calculate the surrounding airs temperature and pressure. The computation method the simulation tool used was the Euler-forward method and the method that was used to analyse the sensitivity for each parameter was the Monte-Carlo method. A normal distribution was used to get a sample of 1000 different values for each parameter.

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3.1 Balloon launch information

Table 2: Information about the ﬁrst balloon launch

Expirement Balloon 1 Maximum volume 12000 m3 Required ﬂoat altitude Above 25 km Free lift percentage 10% Date 2014-10-10 Time 09:49 UTC Ground temperature 273.75 K Cloud cover 15% Launch base Esrange, Kiruna, Sweden

Table 3: Information about the second balloon launch

Experiment Balloon 2 Maximum volume 50000 m3 Required ﬂoat altitude 28 km for ﬁrst drop Free lift percentage 12% Date 2015-06-14 Time 20:34 UTC Ground temperature 280.30 K Cloud cover 0% Launch base Esrange, Kiruna, Sweden

Table 4: Information about the third balloon launch

Experiment Balloon 3 Maximum volume More than 100000 m3 Float About 30 km Free lift percentage 10 % Date 2018-08-23 Time 06:00 UTC Ground temperature 288.15 K Cloud cover 10% Launch base Esrange, Kiruna, Sweden

Table 2-4 shows launch data that was used to analyse the ascent of balloons with the volumes of 12000 m3, 50000 m3 and a little more than 100000 m3.

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3.2 Free lift analysis Before launch, the lifting gas are injected into the balloon. The amount of gas is measured with a pressure gauge, but due to an error in the pressure gauge, the exact amount is unknown. There should be no problem with the balloon flight however, since there are ways to adjust to the amount of gas. If there is an excess amount of gas, the balloon operator could open one of the balloon’s exhaust valves and let out some of the excess gas or drop some ballasting weights if the balloon’s buoyancy is insufficient. In these simulations however, there will be no ballasting or valving until the balloon reaches its goal altitude. To determine the amount of lifting gas the balloon will be injected with, a Monte-Carlo analysis was used for the free lift. The free lift was determined with a normal distribution. The mean value is the one given as the default value for each flight and the standard deviation was 0.2%.

3.3 Ground temperature analysis The amount of IR radiation that is emitted from the ground depends on the grounds temperature. A warmer ground temperature would naturally lead to more IR emission from the ground. The default ground temperature for the Balloon 1 and Balloon 2 launch was assumed to be the same as the one measured at 2 meters above ground at the time of the launch, which was given by the SSC. For the Balloon 3 launch no measurements have been given, so the default ground temperature that will be used is the same as US standard atmosphere at 0 meters above sea level. Using a normal distribution, the inﬂuence the ground temperature have on the balloon’s ascent could be determined. The mean value is the on given as the default and the standard deviation was 0.3 K.

3.4 Cloud factor analysis The amount of cloud coverage will decide the amount of IR radiation and sunlight, both direct and reflected, will reach the balloon. In reality there are different kinds of clouds that are on different parts of the atmosphere and some sunlight and IR radiation can pass through the cloud depending on its type. In these simulations none of it should pass through. These clouds will also have an albedo to them, which will be added to the reflected sunlight when the balloon is above the clouds. For these simulations there were only one type of cloud at the same altitude. The cloud type was cirrus with the albedo set to 0.23 and altitude 11 km. The cloud coverage for the launch dates for Balloon 1 and Balloon 2 were given by the SSC and for the Balloon 3 launch it was set to 10%. The effects that the cloud factor has on the balloon’s ascent was determined using a normal distribution. The mean value was the one given as a default for each flight and the standard deviation was set to 5%.

3.5 Ground albedo analysis During sunlight hours, the ground albedo will have an effect on the heat load that the balloon will receive. Depending on the ground, i.e. if there is a lake or a forest, the amount of sunlight that will reflect to the balloon will vary. For simplicity, the simulations will use an average of the area. Exact measurements with reliable results of Kiruna’s albedo are yet to be made, so Canada’s albedo average will be used due to having a similar nature. Because of different launch times, the effect the ground albedo have on the balloon flight will differ. For

18 Lule˚a University of Technology January 7, 2019 example during night or twilight, the effect the ground albedo have on the balloon flight will be negligible. The amount of cloud cover would also have an affect on the reflected sunlight received. If the cloud cover was 100%, then the uncertainty around the ground albedo would also be negligible. To determine the effect the ground albedo has on the balloon ascent, a normal distribution with mean value of 0.367 and standard deviation of 0.12 was sampled.

4 Results

4.1 Simulation performance Balloon 1 and Balloon 2 was based on real ﬂights, which makes it possible to compare the real ﬂight with the simulated one, to see how well the simulation performs.

Figure 5: The altitude over time for the real ﬂight data and simulated for Balloon 1

Figure 5 shows the real ﬂight of Balloon 1 and the simulated one with its default values. The simulation shows a decent accuracy during the troposphere, but during the stratosphere it is starting to get unstable and overshoot with its rate of climb. Table 5 shows the real mean ascending velocities that Balloon 1 had during its ﬂight.

Table 5: The mean ascending velocity in diﬀerent parts of the atmosphere for Balloon 1

Mean ascending velocity Atmosphere 3.7781 m/s Troposphere 3.6316 m/s Stratosphere 3.8957 m/s

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Figure 6: The altitude over time for the real ﬂight data and simulated for Balloon 2

Figure 6 shows the real flight of Balloon 2 and the simulated one with its default values. The simulation for Balloon 2 was less accurate than the one with Balloon 1 in the troposphere. In the stratosphere the ascent speed for the simulated balloon starts to slow down and un- dershoots compared to the real flight. Table 6 shows the real mean ascending velocities that Balloon 2 had during its flight.

Table 6: The mean ascending velocity in diﬀerent parts of the atmosphere for Balloon 2

Mean ascending velocity Atmosphere 3.8167 m/s Troposphere 4.2579 m/s Stratosphere 3.5904 m/s

4.2 Sensitivity with an abundance/excess of buoyant gas Table 7 shows the simulated mean velocities in diﬀerent parts of the atmosphere for Balloon 1 with a free lift between 9.3536-10.7157 %. The diﬀerence between the slowest and fastest rate of climb are 0.2610 m/s in the troposphere, 0.2676 m/s in the stratosphere and 0.2683 m/s from launch to 25 km. So for the simulation of Balloon 1, the simulation is just as sensitive in the troposphere and as the stratosphere when changing the free lift.

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Table 7: The simulated mean ascending velocity ranges for Balloon 1 with an uncertainty in the free lift

Mean ascending velocity Atmosphere 4.3455-4.6138 m/s Troposphere 3.7746-4.0356 m/s Stratosphere 4.9008-5.1684 m/s

Table 8 shows the simulated mean velocities in different parts of the atmosphere for Balloon 2 with a free lift between 11.3856-12.7140 %. For this flight the difference between the slowest and fastest mean vertical velocity was 0.3426 m/s in the troposphere, 0.4114 m/s in the stratosphere and 0.3960 m/s from launch to 28 km. So for the simulation of Balloon 2, the simulation was more sensitive to changes in the free lift in the stratosphere, than the troposphere.

Table 8: The simulated mean velocity ranges for Balloon 2 with an uncertainty in the free lift

Mean ascending velocity Atmosphere 3.2710-3.6670 m/s Troposphere 3.8161-4.1587 m/s Stratosphere 3.0096-3.4210 m/s

Table 9 shows the simulated mean velocities in diﬀerent parts of the atmosphere for Balloon 3 with a free lift in the range 9.3017-10.6933 %. In the simulation for Balloon 3, the diﬀerence between the slowest and fastest mean vertical velocities was 0.4270 m/s in the troposphere, 0.3012 m/s in the stratosphere and 0.3422 m/s from launch to 30 km. So for Balloon 3, the simulation will be more sensitive in changes to the free lift in the troposphere than in the stratosphere.

Table 9: The simulated mean ascending velocity ranges for Balloon 3 with an uncertainty in the free lift

Mean ascending velocity Atmosphere 3.8504-4.1926 m/s Troposphere 4.0648-4.4918 m/s Stratosphere 3.7385-4.0397 m/s

4.3 Sensitivity for a changing ground temperature Table 10 shows the simulated mean velocities in diﬀerent parts of the atmosphere for Balloon 1 with a ground temperature between 272.7804-274.8235 K. The diﬀerence between the slowest and fastest mean vertical velocities was 0.0247 m/s in the troposphere, 0.0658 m/s in the stratosphere and 0.0436 m/s from launch to 25 km. The simulation for Balloon 1 are more sensitive in the stratosphere when the ground temperature changes.

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Table 10: The simulated mean ascending velocity ranges for Balloon 1 with an uncertainty in the ground temperature

Mean ascending velocity Atmosphere 4.4542-4.4978 m/s Troposphere 3.8893-3.9140 m/s Stratosphere 4.9980-5.0638 m/s

Table 11 shows the simulated mean velocities in different parts of the atmosphere for Balloon 2 with a ground temperature between 279.3784-281.3710 K. The difference between the slowest and fastest mean vertical velocities was 0.0245 m/s in the troposphere, 0.0943 m/s in the stratosphere and 0.0753 m/s from launch to 28 km. A changing ground temperature have the greatest effect in the stratosphere for Balloon 2.

Table 11: The simulated mean ascending velocity ranges for Balloon 2 with an uncertainty in the ground temperature

Mean ascending velocity Atmosphere 3.4226-3.4979 m/s Troposphere 3.9675-3.9920 m/s Stratosphere 3.1588-3.2531 m/s

Table 12 shows the simulated mean velocities in diﬀerent parts of the atmosphere for Balloon 3 with a ground temperature between 287.1025-289.1899 K. The diﬀerence between the slowest and fastest mean vertical velocities was 0.0271 m/s in the troposphere, 0.0749 m/s in the stratosphere and 0.0595 m/s from launch to 30 km. As with the simulation for the other balloons, Balloon 3 is most sensitive to a changing ground temperature in stratosphere.

Table 12: The simulated mean ascending velocity ranges for Balloon 3 with an uncertainty in the ground temperature

Mean ascending velocity Atmosphere 3.9940-4.0538 m/s Troposphere 4.2706-4.2977 m/s Stratosphere 3.8523-3.9272 m/s

4.4 Sensitivity for a changing cloud cover Table 13 shows the simulated mean velocities in diﬀerent parts of the atmosphere with 0-32.89% cloud cover. The diﬀerence between the slowest and fastest mean vertical velocities was 0.0012 m/s in the troposphere, 0.6660 m/s in the stratosphere and 0.3040 m/s from launch to 25 km.

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Table 13: The simulated mean ascending velocity ranges for Balloon 1 with an uncertainty with the cloud cover

Mean ascending velocity Atmosphere 4.2973-4.6013 m/s Troposphere 3.9004-3.9016 m/s Stratosphere 4.6508-5.3168 m/s

Table 14 shows the simulated mean velocities in diﬀerent parts of the atmosphere with 0-17.85% cloud cover. The diﬀerence between the slowest and fastest mean vertical velocities was 0.0364 m/s in the troposphere, 0.4121 m/s in the stratosphere and 0.3035 m/s from launch to 28 km.

Table 14: The simulated mean ascending velocity ranges for Balloon 2 with an uncertainty with the cloud cover

Mean ascending velocity Atmosphere 3.1542-3.4577 m/s Troposphere 3.9791-4.0155 m/s Stratosphere 2.7904-3.2025 m/s

Table 15 shows the simulated mean velocities in diﬀerent parts of the atmosphere with 0-27.33% cloud cover. The diﬀerence between the slowest and fastest mean vertical velocities was 0.0202 m/s in the troposphere, 0.6432 m/s in the stratosphere and 0.4527 m/s from launch to 30 km.

Table 15: The simulated mean ascending velocity ranges for Balloon 3 with an uncertainty with the cloud cover

Mean ascending velocity Atmosphere 3.7284-4.1811 m/s Troposphere 4.2712-4.2914 m/s Stratosphere 3.4778-4.1210 m/s

Due to diﬀerent ranges in cloud coverage, comparing the sensitivity in the stratosphere for each balloon will be how the mean vertical velocity changes per cloud cover percentage. In the 0.0202 m/s 0.0231 m/s stratosphere, the mean vertical velocity decreases with Cloud cover % for Balloon 1, Cloud cover % 0.0235 m/s for Balloon 2 and Cloud cover % for Balloon 3.

4.5 Sensitivity for a changing ground albedo Table 16 shows the simulated mean velocities in different parts of the atmosphere for Balloon 1 when the ground albedo have values 0.2054-0.5459. The difference between the slowest and fastest mean vertical velocities was 0.0293 m/s in the troposphere, 0.0873 m/s in the stratosphere and 0.0560 m/s from launch to 25 km. Only in the troposphere can the effects of different values for the ground albedo be seen as negligible for Balloon 1.

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Table 16: The simulated mean ascending velocity ranges for Balloon 1 with an uncertainty with the ground albedo

Mean ascending velocity Atmosphere 4.4482-4.5042 m/s Troposphere 3.8870-3.9163 m/s Stratosphere 4.9878-5.0751 m/s

Table 17 shows the simulated mean velocities in different parts of the atmosphere for Balloon 2 when the ground albedo have values 0.2134-0.5455. The difference between the slowest and fastest mean vertical velocities was 0.0087 m/s in the troposphere, 0.0261 m/s in the stratosphere and 0.0214 m/s from launch to 28 km. With the difference being less than 3 cm/s in all parts of the atmosphere, the effect that the ground albedo have in this simulation can be negligible.

Table 17: The simulated mean ascending velocity ranges for Balloon 2 with an uncertainty with the ground albedo

Mean ascending velocity Atmosphere 3.4477-3.4691 m/s Troposphere 3.9750-3.9837 m/s Stratosphere 3.1904-3.2165 m/s

Table 18 shows the simulated mean velocities in diﬀerent parts of the atmosphere for Balloon 3 when the ground albedo have values 0.1924-0.5403. The diﬀerence between the slowest and fastest mean vertical velocities was 0.0867 m/s in the troposphere, 0.3397 m/s in the stratosphere and 0.2598 m/s from launch to 30 km.

Table 18: The simulated mean ascending velocity ranges for Balloon 3 with an uncertainty with the ground albedo

Mean ascending velocity Atmosphere 3.8897-4.1495 m/s Troposphere 4.2400-4.3267 m/s Stratosphere 3.7155-4.0552 m/s

5 Discussion

The performance for the simulations of Balloon 1 and 2 was decent in the troposphere and when it approached the stratosphere it started to deviate. For Balloon 1 the vertical velocity for the balloon got higher and for Balloon 2 it was getting lower than their real flights. Reasons for this deviation could be due to the shape variations for the real flight, giving it a different drag coefficient during its ascent. For the balloon that was launched during noon, the drag coefficient was underestimated and for the balloon that was launched during the night the drag coefficient was overestimated. More night launches of different balloon sizes should be analysed to see whether it is because of the balloon’s size or if it is because of its launch time.

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Another reason for the deviation could be due to computational instabilities. Since small errors will add over time, a solution to this would be to use another computational method that is more stable, such as the 4th order Runge-Kutta method. It could also be because of the tool that measured the temperature and pressure in the atmosphere having error in it since the simulated flight for Balloon 1 doesn’t reach the real flight float altitude. But it did reach its goal altitude of 25 km. The difference in float altitude could be because of faulty measurements for the temperature and pressure in the atmosphere leading to a different air density at the same altitude. The possibility of erroneous measurement of the atmosphere makes it worth to see how the simulation would work if the US standard atmosphere was used instead. When there was an abundance or excess amount of lifting gas, the sensitivity of the simulations varied. There could be several reasons to this, such as size difference in the balloons, different launch times and even different atmospheric models. A smaller balloon could be just as sensitive to an excess or abundance to lifting gas in different parts of the atmosphere compared to a bigger balloon. And a balloon launched at night could be more sensitive in the stratosphere than the troposphere with an excess or abundance to lifting gas compared to a balloon launched at day. Or the simulation could be more sensitive in the troposphere because it is using the US standard atmosphere to calculate the surrounding temperature and pressure. For a changing ground temperature, the sensitivity does seem to increase as the balloon size increases in the troposphere, but still seem to be negligible due to the differences being less than 3 cm/s. In the stratosphere it is hard to tell, since all of the simulations uses a different cloud cover. The size of the balloon would seem to have an effect overall, but it does seem to diminish as the cloud cover increases. When the cloud cover changed, the mean vertical velocity in the troposphere for Balloon 1 and Balloon 2, increased when the cloud cover increased, while it decreased for Balloon 3. Either way, the effects in the troposphere were so small that it can be negligible. In the stratosphere however, the mean vertical velocity decreased with a bigger cloud coverage and it had a noticeable effect in all of the simulations. The sensitivity with changing cloud cover seems to increase for bigger balloons and for balloons that are released during hours with little sunlight. More flights that are launched during the night should be analysed for a proper conclusion to this. When testing the sensitivity for a changing ground albedo value, Balloon 3 was the most sensitive to the different values for the ground albedo and Balloon 2 was the least sensitive. So balloons released during hours with little to no sunlight does not have to think about the ground albedo, because there is no sunlight to reflect. During the day however there is sunlight to reflect and a bigger balloon would absorb more. The balloon would also be more inflated in the stratosphere, making it more sensitive there when compared to the troposphere.

6 Conclusions

This work presented a prediction model for a high altitude zero pressure balloon. The mean ascending velocities in the troposphere and stratosphere was measured to evaluate its performance. This work also tried to see what changing a parameter would do to a balloon’s ascent. Seeing which parameter the balloons are most sensitive to would be an unfair comparison, since the parameters are of different units and the spread of each test are different. Instead the work compares the same parameter for balloons of different sizes and different launch times. Two of the balloons used atmospheric data from the flights its based of to test the simulation

25 Lule˚a University of Technology January 7, 2019 with a realistic atmospheric model and one used the US standard atmosphere. Conclusions that can be made from these simulations are:

1. The simulations performance are accurate in the troposphere, while in the stratosphere the simulated ﬂight starts to deviate from the real ﬂight. The drag is underestimated for the small sized balloon released at noon and overestimated for the medium sized balloon released at night.

2. With an abundance or excessive amount of lifting gas injected at launch, the size of the balloon and the time it is released do have an importance:

• Small sized balloons released at noon have almost the same sensitivity in the troposphere as in the stratosphere. • Medium sized balloons released at night are more sensitive in the stratosphere. • Large sized balloons released at morning are more sensitive in the troposphere.

3. With a changing ground temperature during the balloon’s ascent, the effects are negligible in the troposphere. In the stratosphere, a larger balloon would be more sensitive than a smaller balloon with the same cloud cover. A balloon that is flying with cloud cover would be less sensitive to a changing ground temperature, than a balloon of same size but flying with no cloud cover.

4. With a changing cloud cover during the ascent, the sensitivity depends on the size of the balloon, the launch time of the balloon and where in the atmosphere the balloon is.

• In the troposphere, the effects can be considered as negligible. However, for the smaller sized balloon launched at noon and the medium sized balloon released at night, the mean ascent speed increased with more cloud cover, while for the larger sized balloon released at morning it decreased with increased cloud cover. • In the stratosphere, larger balloons are more sensitive to changes to the cloud cover and the balloons that flied during the night was more sensitive than the balloon that flied during the day.

5. The eﬀects of a changing ground albedo depends on the balloon’s size, on the launch time of the balloon, where in the atmosphere the balloon is and how much cloud cover there is during the balloon’s ascent.

• Larger balloons are more sensitive to a changing ground albedo compared to a smaller balloon. • The eﬀects of a changing ground albedo are negligible for balloons launched at hours with little to no sunlight. • The balloons are more sensitive to a changing ground albedo in the stratosphere compared to when it is in the troposphere. • A larger cloud cover should make the balloons less sensitive to changes in the ground albedo, since the clouds would block the sunlight that will be reﬂected to the balloon.

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Future works should be analysing data from balloons that are launched at night and compare simulations that use the US standard atmosphere and the atmospheric data for the same balloon launch. Another way to improve the simulation could be to give the clouds some opacity, making it closer to how it works in reality.

7 References

[1]: Farley, R. E., ”BalloonAscent: 3-D Simulation Tool for the Ascent and Float of High- Altitude Balloons”, AIAA 5th Aviation, Technology, Integration, and Operations Conference (2005). [2]: Van Dossalaer, I. ”Buoyant Aerobot Design and Simulation Study BADS”, Ph.D. Disser- tation, Delft University of Technology, Delft, Netherlands, 2014. [3]: Lee, Y. and Yee, K., ”Numerical Prediction of Scientiﬁc Balloon Trajectories While Considering Various Uncertainties”, Journal of Aircraft, Vol. 54, No. 2, March-April 2017. DOI: 10.2514/1.C033998 [4]: Saleh, S. and He, W., ”Ascending Performance Analysis for High Altitude Zero Pressure Balloon”, Advances in Space Research, Volume 59, Issue 8, 2017-04-15, Pages 2158-2172, DOI: 10.1016/j.asr.2017.01.040. [5]: Dai, Q., Fang, X., Li, X. and Tian, L., ”Performance Simulation of High Altitude Scientiﬁc Balloons”, Advances in Space Research, Volume 49, Issue 6, 2012-03-15, Pages 1045-1052, DOI: 10.1016/j.asr.2011.12.026 [6]: Palumbo, R., ”A Simulation Model for Trajectory Forecast, Performance Analysis and Aerospace Mission Planning with High Altitude Zero Pressure Balloons”, Ph.D Dissertation, University of Naples Federico II, Italy, 2008. [7]: https://stratocat.com.ar/globos/indexe.html, 2018-08-09

Appendix A: Analysis of the free lifts inﬂuence on the ascent

Figure 7-9 shows the altitude over time for all of the simulations for the different the free lifts for the balloons. Figures 10-12 shows how the mean velocities of the simulations for Balloon 1 changes with the free lift over different parts of the atmosphere. Figures 13-15 shows how the mean velocities of the simulations for Balloon 2 changes with the free lift. Figures 16-18 shows how the mean ascending velocities of the simulation for Balloon 3 changes with the free lift. Every figure shows how the mean vertical velocity increases with an increased free lift. This makes sense in that with more lifting gas, the balloon’s volume would increase and then give the balloon more buoyancy, which will accelerate the balloon even more.

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Figure 7: Altitude over time for all of the simulations of Balloon 1 when the free lift changed

Figure 8: Altitude over time for all of the simulations of Balloon 2 when the free lift changed

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Figure 9: Altitude over time for all of the simulations of Balloon 3 when the free lift changed

Figure 10: The simulated mean ascending velocity vs the free lift in the entire atmosphere for Balloon 1

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Figure 11: The simulated mean ascending velocity vs the free lift in the troposphere for Balloon 1

Figure 12: The simulated mean ascending velocity vs the free lift in the stratosphere for Balloon 1

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Figure 13: The simulated mean ascending velocity vs the free lift in the atmosphere for Balloon 2

Figure 14: The simulated mean ascending velocity vs the free lift in the troposphere for Balloon 2

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Figure 15: The simulated mean ascending velocity vs the free lift in the stratosphere for Balloon 2

Figure 16: The simulated mean ascending velocity vs the free lift in the atmosphere for Balloon 3

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Figure 17: The simulated mean ascending velocity vs the free lift in the troposphere for Balloon 3

Figure 18: The simulated mean ascending velocity vs the free lift in the stratosphere for Balloon 3

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Appendix B: The mean ascending velocity analysis of a changing ground temperature

Figure 19-21 shows the altitude over time for all of the simulations for the balloons when there is a diﬀerent ground temperature. Figure 22-24 shows how the mean ascending velocities of the simulation for Balloon 1 changes with the ground temperature. Figure 25-27 shows how the mean ascending velocities of the simulation for Balloon 2 changes with the ground temperature. Figure 28-30 shows how the mean ascending velocities of the simulation for Balloon 3 changes with the ground temperature. In all simulations, the mean vertical velocity increased with the ground temperature. With an increased ground temperature, more IR- radiation will be emitted from the ground, warming the balloon and expanding it, giving it more buoyancy.

Figure 19: Altitude over time for all of the simulations of Balloon 1 with diﬀerent ground temperature

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Figure 20: Altitude over time for all of the simulations of Balloon 2 with diﬀerent ground temperature

Figure 21: Altitude over time for all of the simulations of Balloon 3 with diﬀerent ground temperature

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Figure 22: The simulated mean ascending velocity vs the ground temperature in the atmosphere for Balloon 1

Figure 23: The simulated mean ascending velocity vs the ground temperature in the troposphere for Balloon 1

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Figure 24: The simulated mean ascending velocity vs the ground temperature in the stratosphere for Balloon 1

Figure 25: The simulated mean ascending velocity vs the ground temperature in the atmosphere for Balloon 2

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Figure 26: The simulated mean ascending velocity vs the ground temperature in the troposphere for Balloon 2

Figure 27: The simulated mean ascending velocity vs the ground temperature in the stratosphere for Balloon 2

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Figure 28: The simulated mean ascending velocity vs the ground temperature in the atmosphere for Balloon 3

Figure 29: The simulated mean ascending velocity vs the ground temperature in the troposphere for Balloon 3

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Figure 30: The simulated mean ascending velocity vs the ground temperature in the stratosphere for Balloon 3

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Appendix C: The mean ascending velocity analysis of a changing cloud cover

Figure 31-33 shows the altitude over time for all of the simulations for the balloons when there is a different cloud cover. Figure 34-36 shows how the mean ascending velocities of the simulation for Balloon 1 changes with the cloud cover. Figure 37-39 shows how the mean ascending velocities of the simulation for Balloon 2 changes with the cloud cover. Figure 40-42 shows how the mean ascending velocities of the simulation for Balloon 3 changes with the cloud cover. Most of the figures shows that an increased cloud cover gives a lower mean vertical velocity in the atmosphere, but in figures 35 and 38 will show that the mean vertical velocity for Balloon 1 and Balloon 2 increases in the troposphere when the cloud cover increases. It can be because the IR-radiation from the clouds are greater than the sunlight the balloon would have absorbed.

Figure 31: Altitude over time for all of the simulations of Balloon 1 with diﬀerent cloud covers

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Figure 32: Altitude over time for all of the simulations of Balloon 2 with diﬀerent cloud covers

Figure 33: Altitude over time for all of the simulations of Balloon 3 with diﬀerent cloud covers

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Figure 34: The simulated mean ascending velocity vs the cloud factor in the atmosphere for Balloon 1

Figure 35: The simulated mean ascending velocity vs the cloud factor in the troposphere for Balloon 1

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Figure 36: The simulated mean ascending velocity vs the cloud factor in the stratosphere for Balloon 1

Figure 37: The simulated mean ascending velocity vs the cloud factor in the atmosphere for Balloon 2

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Figure 38: The simulated mean ascending velocity vs the cloud factor in the troposphere for Balloon 2

Figure 39: The simulated mean ascending velocity vs the cloud factor in the atmosphere for Balloon 2

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Figure 40: The simulated mean ascending velocity vs the cloud factor in the atmosphere for Balloon 3

Figure 41: The simulated mean ascending velocity vs the cloud factor in the troposphere for Balloon 3

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Figure 42: The simulated mean ascending velocity vs the cloud factor in the stratosphere for Balloon 3

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Appendix D: The mean ascending velocity analysis of a changing ground albedo

Figure 43-45 shows the altitude over time for all of the simulations when there is a different value for the ground albedo. Figure 46-48 shows how the mean ascending velocities of the simulation for Balloon 1 changes with the ground albedo. Figure 49-51 shows how the mean ascending velocities of the simulation for Balloon 2 changes with the ground albedo. Figure 52-54 shows how the mean ascending velocities of the simulation for Balloon 3 changes with the ground albedo. Every figure shows that the mean vertical velocity for the balloons increases with the ground albedo. With a higher albedo value, more sunlight will get reflected, giving the balloon more sunlight to absorb, making it expand more and get more buoyancy.

Figure 43: Altitude over time for all of the simulations of Balloon 1 with diﬀerent ground albedo

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Figure 44: Altitude over time for all of the simulations of Balloon 2 with diﬀerent ground albedo

Figure 45: Altitude over time for all of the simulations of Balloon 3 with diﬀerent ground albedo

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Figure 46: The simulated mean ascending velocity vs the ground albedo in the atmosphere for Balloon 1

Figure 47: The simulated mean ascending velocity vs the ground albedo in the troposphere for Balloon 1

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Figure 48: The simulated mean ascending velocity vs the ground albedo in the stratosphere for Balloon 1

Figure 49: The simulated mean ascending velocity vs the ground albedo in the atmosphere for Balloon 2

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Figure 50: The simulated mean ascending velocity vs the ground albedo in the troposphere for Balloon 2

Figure 51: The simulated mean ascending velocity vs the ground albedo in the stratosphere for Balloon 2

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Figure 52: The simulated mean ascending velocity vs the ground albedo in the atmosphere for Balloon 3

Figure 53: The simulated mean ascending velocity vs the ground albedo in the troposphere for Balloon 3

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Figure 54: The simulated mean ascending velocity vs the ground albedo in the stratosphere for Balloon 3