Dynamic model

The ascent phase usually lasts 2-4 hours in contrast to the 55 days that LDB complete flights can last up. Since the air density is very thin at high altitudes, it is very common to drop a ballast [90] at a certain altitude to keep a positive ascent rate. Besides, the use of that extra mass allows the ascent rate to be controlled, usually between 2-8 m/s. This value is a consequence of the acceleration resulting from the buoyancy and drag forces. The atmospheric winds provoke balloons to rotate slowly due to friction forces [91]. A rotatory control mode can be used during the ascent phase with the aim of providing solar illumination and heat to some instruments [10], although this mode may be limited by the battery power. Some profiles of the altitude, latitude and longitude during the ascent phase, which are obtained from data available [92] of four flights launched from Esrange [10, 58, 93, 94, 95], are shown in Figure 4.39. The variation in the latitude and longitude is no greater than ±1° and ±1.5°, respectively, in all flights.

As has been depicted, the obtention of the ascent rate and the relative velocity of the balloon-borne system with the wind speed profile with the altitude are required for the ascent phase thermal analysis. With that purpose, in this section, the dynamic model implemented to obtain these profiles is presented.

The forces acting on the balloon-borne system are the balloon weight-borne, the drag force and the buoyancy force as it is shown in Figure 4.40. The acceleration ¨r in North-East-Down reference frame [80] is given by the following equation:

d²r

dt² = mgross(h)g(h) +g(h) (ρ_air(h)−ρgas(h))V(h) +D(h)

m_gross(h) +m_gas+m_add , (4.33)

Figure 4.39: Altitude, latitude and longitude profiles during the ascent phase of PMC- Turbo, HIWIND, SUNRISE I and SUNRISE II.

where g is the gravity acceleration, ρ_air is the air density, ρ_gas is the lifting gas density, V is the balloon volume and D is the drag force.

The total mass of the system is computed as the sum of three terms, the gross mass, m_gross, the lifting gas mass, m_gas and the additional mass, m_add.

• m_grossis the sum of the payload, balloon film and ballast mass (meaning the non-gaseous mass).

• mgas is the lifting gas mass.

• m_add , is the additional mass that takes into account the added mass effect that appears when a body is accelerated immersed in a fluid (air). It is computed as,

m_add=C_addρ_air(h)V(h) (4.34)

4. Planetary thermal environment for stratospheric balloon thermal design 97

Figure 4.40: Forces acting on the balloon-borne system during the ascent phase.

The values of the coefficient C_add are assumed to be in the range 0.25 <

C_add < 0.5 [66].

The lifting gas mass can be estimated using Equation (4.35), taking into account the excess of gas with which the zero-pressure balloons are always launched,

m_gas = m_gross100^FL + 1

Mair

Mgas −1 , (4.35)

where FL is the percentage of the extra gas (typical values are around 10-20% [80]).

M_gas and M_air are the molecular mass of the lifting gas and the air, respectively.

The drag force is computed according to Equation (4.36).

D(h) = 1

2ρ_air(h)v²(h)A(h)C_d(h)v

v, (4.36)

wherevis the balloon relative velocity to the wind speed, computed as the difference between the balloon velocity, ˙r and the wind speed, v_w, which can be obtained from experimental data such as [49], A is approximated by the balloon top view cross section area, and C_d is the drag coefficient.

During the initial phases, the drag coefficient drives the result of the dynamic problem due to the high air density and relative velocity expected. The literature provides different values for the drag coefficient by different approaches:

1. Musso et al. [96] use a constant value drag during the ascent phase,C_d= 0.45, based on comparison with the data flights.

2. Comparing the balloon to a sphere, Farley et al. [66] give values ranging from 0.1-0.47 for the drag coefficient as a function of Reynolds number.

3. Conrad and Robbins [97] provides another approach adding a Froude number dependent term, to the sphere drag.

C_d(h) = 4 3

1 Fr

(

1− R_airT_air R_gasT_gas

"

1 + m_gross

m_gas +m_gas+m_add m_gasg

dv_z dt

#)

+C_d,Re(Re), (4.37) where R_air is the specific air constant and R_gas is the specific lifting gas constant.

The characteristic length to compute both non dimensional numbers is the balloon diameter. Due to their better results, this is the selected approach used here.

The volume of the balloon is computed assuming that the lifting gas follows the ideal gas law, under the condition of zero-pressure, meaning the air and the lifting gas pressure are the same. Besides, it has been assumed that the volume of the lifting gas is the same that the volume of the displaced air,

V = m_gas ρgas

=m_gasR_gasT_gas pgas

, (4.38)

where ρ_gas, T_gas and p_gas are the density, the temperature and the pressure of the lifting gas respectively.

As has been shown in the previous equation, the dynamic model of the balloon is coupled with the thermal model as the temperature of the lifting gas changes the volume of the balloon and therefore the buoyancy force.

The thermal environment during the ascent phase must be previously defined (with either an analytical model [66] or from experimental data [31]) in order to compute the heat loads affecting the balloon-borne system during the ascent phase.

The input variables profile required are summarized in Table 4.1 . Besides, the source where these variables must be defined is shown together with information from SUNRISE I mission.

In the following figures, some of the parameters computed in the model are compared to flight data. The horizontal dash lines stand for the drop of ballast mass. At each marked altitude a ballast of 27 kg is dropped.

The altitude profile, which is shown in Figure 4.41, presents two slopes clearly distinguished as it happens in the flight data.

Besides, the horizontal trajectory obtained in the model follows the same behaviour than the experimental data. The maximum error is lower than 0.1°

in both latitude and longitude as it shown in Figure 4.42a and Figure 4.42b.

Finally, the relative velocity obtained, comparing the total magnitude with the horizontal component, is shown in Figure 4.43.

4. Planetary thermal environment for stratospheric balloon thermal design 99

Table 4.1: Input variables required to compute relative velocity.

Input variables SUNRISE I Air density, [kg/m³]

ECMWF Air temperature, [K]

Air pressure, [Pa]

Wind speed, [m/s]

Gross mass, [kg] 6098

Drag coefficient, [-] Conrad and Robbins model.

Balloon film thermo-optical properties

α= 0.024 ε= 0.134 τ = 0.916 τ_IR = 0.176

Payload mass, [kg] 1920

Ascent ballast mass, [kg] 81 Balloon film mass, [kg] 2330

Auxiliary mass, [kg] 1767

Volume at float altitude, [MCF]

34

Launch site and date 08/06/2009. Esrange.

Percentage Free Lift 10%

Figure 4.41: SUNRISE I GPS ascent profile compared to simulation outputs.