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Performance simulation of high altitude scientific balloons
Available online at www.sciencedirect.com
Advances in Space Research 49 (2012) 1045–1052 www.elsevier.com/locate/asr
Performance simulation of high altitude scientific balloons Qiumin Dai, Xiande Fang ⇑, Xiaojian Li, Lili Tian Department of Man, Machine and Environment Engineering, College of Aerospace Engineering, Nanjing University of Aeronautics and Astronautics, 29 Yudao Street, Nanjing 210016, PR China Received 20 September 2011; received in revised form 23 December 2011; accepted 26 December 2011 Available online 5 January 2012
Abstract The design and operation of a high altitude scientific balloon requires adequate knowledge of the thermal characteristics of the balloon to make it safe and reliable. The thermal models and dynamic models of altitude scientific balloons are established in this paper. Based on the models, a simulation program is developed. The thermal performances of a super pressure balloon are simulated. The influence of film radiation property and clouds on balloon thermal behaviors is discussed in detail. The results are helpful for the design and operate of safe and reliable high altitude scientific balloons. Ó 2012 COSPAR. Published by Elsevier Ltd. All rights reserved. Keywords: High altitude balloon; Thermal model; Dynamic model; Radiation property
1. Introduction High altitude scientific balloons provide a unique and low cost way to carry out missions in a near-space environment. In recent years, the potential use of high altitude scientific balloons as observation platforms for atmosphere studies or research purposes has attracted growing interest. In order to accomplish the objectives of a high altitude scientific balloon, it is necessary to predict the thermal behaviors before it launches. The balloon mission preparation requires an accurate and reliable flight performance prediction method in order to accomplish the mission successfully. A failure prediction of the skin temperature will affect the life of the balloon. A failure to accurately predict the temperature of buoyant gas can lead to under-filling or over-filling the balloon at launch. An under-filling balloon may not reach the desired float altitude, and an over-filling balloon may lead to envelope rupture before it reaches the
⇑ Corresponding author. Tel./fax: +86 25 8489 6381.
E-mail addresses: [email protected] (Q. Dai), xd_fang@yahoo. com (X. Fang), [email protected] (X. Li), tianlili0804@163. com (L. Tian).
desired float altitude. Both of the conditions will cause potential safety risks and lead to the mission failure. In the past decades, many investigations have been carried out on the thermal characteristics of high altitude balloons. Kreith and Kreider (1974) established a simple but excellent model to predict the thermal behaviors which was served as the starting point for the subsequent research. Carlson and Horn (1983) developed a new trajectory and thermal model to analyze the flight trajectory and thermal characteristics. In their model, the lifting gas is assumed to be able to absorb solar radiation and emit infrared radiation. Cathey (1996) numerically studied the temperature distribution on the balloon. Farley (2005) developed a code to simulate the ascent and float behaviors of high-altitude balloons. The code can also be used to simulate the thermal behaviors of balloon on other planets. Xia et al. (2010) developed a transient model to predict the variation of skin and lifting gas temperatures at float conditions. The foregoing articles mainly focused on the thermal behaviors of balloons at the float condition with clear sky, especially on the film temperature distribution and average helium temperature. The effects of film radiation properties and clouds on balloon thermal performances have not been given much attention. In this paper, three-dimensional
0273-1177/$36.00 Ó 2012 COSPAR. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.asr.2011.12.026
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Q. Dai et al. / Advances in Space Research 49 (2012) 1045–1052
Nomenclature A Ap B c CC Cbv Cd Cv CR D e Pw Q QIRE QIRI Re T U V Xi,j W Z a e f h k q r /
area (m2) top projected area (m2) net buoyancy (N) specific heat (J/(kg K)) cloud cover virtual mass coefficient drag coefficient specific heat at constant volume (J/(kg K)) albedo factor drag force (N) orbital eccentricity vapor pressure (kPa) heat gain of film (W) external infrared heat gain (W) internal infrared heat gain (W) Reynolds number temperature (K) vertical velocity (m/s) volume (m3) angle factor from element i to j expansion power (W) altitude (m) solar absorptivity emissivity of the film material true anomaly angle between the element normal and the solar irradiation thermal conductivity (W/(m K)) density (kg/m3) the Stefan–Boltzmann constant (W/(m2 K4)) angle between the element normal and the gravity direction
transient thermal models and dynamic models are established to numerically study the behavior of high altitude balloons during ascent and float conditions. Based on the mathematical models, a computer program is developed. The accuracy of the models is verified by comparing the simulation data to the measured data. Then, the influence of film radiation properties and clouds on balloon thermal behaviors is simulated and discussed in detail.
g Gr h I J L m M MA P Pr Pt x u
gravitational acceleration (m/s2) Grashof number heat transfer coefficient (J/(m2 K)) solar radiation intensity (W/m2) radiosity (W/m2) characteristic length (m) air mass ratio mass (kg) mean anomaly pressure (Pa) Prandtl number atmospheric transmittance solar elevation angle view factor from element to earth
Subscripts a ambient CE external convection CI internal convection cloud cloud D direct e earth f film g ground he helium i, j element number pay payload R reflect S diffuse sky sky tot total
8 288:15 0:0065Z if Z 6 11;000 m > > > < T a ¼ 216:65 if 11;000 m < Z 6 20;000 m > > > : 216:65 þ 0:0012ðZ 20;000Þ if 20;000 m < Z 6 33;000 m 8 5:26 Z > if Z 6 11;000 m 101325 1 44330 > > > < 11000Z if 11;000 m < Z 6 20;000 m P a ¼ 22605exp 6340 > > > > 141:89þ0:003Z 11:388 : if 20;000 m < Z 6 33;000 m 2447 216:65
ð1Þ
ð2Þ
where Z is the altitude of the balloon.
2. Atmosphere and solar models
2.2. Solar models
2.1. Atmosphere models
Solar radiation is classified as direct solar radiation and diffuse solar radiation from the sky and reflected solar radiation from the earth. The value of solar radiation can be calculated using the methods introduced by Colozza (2003), Farley (2005), Ran et al. (2007) and Li et al. (2011). The direct solar radiation intensity ID can be expressed as
During the ascent phase, the temperature and pressure variation at the flight altitude is important. Based on the standard atmosphere, the temperature and pressure are modeled as functions of the altitude (Sou and He, 2004). They can be written as
Q. Dai et al. / Advances in Space Research 49 (2012) 1045–1052
2 1 þ e cos f I D ¼ 1367 pmt 1 e2
ð3Þ
where e is the orbital eccentricity, for earth e = 0.0016708, the atmospheric transmittance pt is normally in the range of 0.6–0.7. The air mass ratio m and the true anomaly f can be calculated with qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pa 2 m¼ ð4Þ 1229 þ ð614 sin xÞ 614 sin x 101325 f ¼ MA þ 0:0334 sin MA þ 3:49 104 sinð2MAÞ
ð5Þ
The sphere drag coefficient is used in this paper since the shape of balloon is close to a sphere. The relationship between the sphere drag coefficient and Reynolds number in the range of the Reynolds number up to 107 was reported by Almedeij (2008). For Reynolds values greater than 106, the drag coefficient is stable at the value of about 0.1; while for Reynolds values lower than 105, the constant drag coefficient of 0.5 can be used. Conner and Arena (2010) explored a drag coefficient model for the ascent balloon in a Reynolds number range of 105–106, which is of the form
where pa is the ambient pressure, x is the solar elevation angle (x P 0), and MA ¼ 2np=365 where n is the day number in a year, n = 1 for January 1. The diffuse solar radiation intensity is written as
C d ¼ 0:72 2:57 106 Re þ 4:71 1012 Re2 4:04
1 1 pmt I S ¼ I D sin x 2 1 1:4 ln pt
3.2. Film thermal models
ð6Þ
The reflected solar radiation intensity is given by I R ¼ C R ðI D sin x þ I S Þ
ð7Þ
where CR is the albedo factor. 3. Dynamic and thermal models 3.1. Dynamic models The net buoyancy of the balloon is B ¼ qa Vg ðM he þ M pay Þg
ð8Þ
where ambient air density qa can be determined by the ideal gas law, V is the volume of the balloon, Mhe is the mass of helium, Mpay is the mass of payload and ballonet material, and g is the gravitational acceleration. The balloon’s motion is subject to the atmosphere winds and the net lift force of the balloon. It can be assumed that the horizontal velocity of the balloon is equal to the atmospheric wind (Morani, 2009). Therefore, the horizontal motion of balloon can be assumed at the wind speed. The governing differential equation of the vertical force balance on the balloon is d 2 Z dU B þ D ¼ ¼ dt M tot dt2
1018 Re3 þ 1:31 1024 Re
M f ;i c
dT f ;i ¼ QD;i þ QS;i þ QR;i þ QIRE;i þ QIRI;i dt þ QCE;i þ QCI;i
ð9Þ
ð10Þ ð11Þ
where AP is the top projected area of the balloon, and Cd is the drag coefficient.
ð13Þ
where Mf,i is the mass of the element i, QD,i is the absorbed direct solar radiation per second, QS,i is the absorbed diffuse solar radiation per second, QR,i is the absorbed reflected solar radiation per second, QIRE,i is the absorbed external infrared radiation per second, QIRI,i is the absorbed internal infrared radiation per second, QCE,i is the
The drag force of the aerostat D can be calculated by D ¼ 0:5qa C d AP jU jU
ð12Þ
Balloon is a thermal vehicle whose thermal behaviors are relatively straight forward with the surrounding environment. Thermal environment of the high altitude balloon includes solar radiation, infrared radiation and convection, as shown in Fig. 1. The balloon film is extremely thin, so the conductive heat transfer through the film can be neglected. The film of the balloon can be divided into N triangle elements, and each element can be treated as a plane. Assuming that the film is a lumped heat capacity, the transient energy-balance equation of the ith element can be expressed as
where U is vertical velocity of the balloon, and Mtot is the total mass. It is calculated with a virtual mass coefficient Cbv. The total mass takes into account the mass of air that is necessarily dragged along with the balloon. Values of virtual mass coefficient assumed range from 0.25 to 0.5. M tot ¼ M he þ M pay þ C bv ðqa V Þ
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Fig. 1. Thermal environment of high altitude balloon.
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external convection heat gain, QCI,i is the internal convection heat gain, and c is the specific heat of the film.
Table 1 The constant C and exponent n in Eq. (25). Geometry
3.2.1. Absorbed solar radiation The absorbed solar radiation of the film includes the absorption of the direct solar radiation, diffuse solar radiation from the sky and reflected solar radiation from the earth. The absorbed direct solar radiation of element i is given by QD;i ¼ aAf ;i I D cos h
ð14Þ
where a is the solar absorptivity of the film, Af,i is the area of the element, and h is the included angle between the element normal and the solar irradiation. The absorbed diffuse solar radiation from the sky is written as QS;i ¼ aAf ;i I S ð0:5 0:5 cos /Þ
ð15Þ
where / is the included angle between the plane normal and the gravity direction. The absorbed reflected solar radiation energy from Earth is expressed as QR;i ¼ aAf ;i I R ð0:5 þ 0:5 cos /Þ
ð16Þ
3.2.2. Absorbed infrared radiation The absorbed external infrared radiation includes earth and atmospheric infrared contributions. It can be calculated using the following equation: QIRE;i ¼ eAf ;i r½/ðT 4e T 4f ;i Þ þ ð1 /ÞðT 4sky T 4f ;i Þ
ð17Þ
GrPr
C
n
10 –10 109–1013 104–107 107–1011
0.59 0.1 0.54 0.15
1/4 1/3 1/4 1/3
105–1010
0.27
1/4
4
Vertical plate Upper surface of heated horizontal plate or lower surface of cooled horizontal plate Lower surface of heated horizontal plate or upper surface of cooled horizontal plate
QCE;i ¼ hCE Af ;i ðT a T f ;i Þ
where pw is the vapor pressure of the ambient air in kPa. The absorbed internal film infrared radiation of element i can be expressed as QIRI;i ¼ Af ;i ðGi J i Þ
ð19Þ
where Gi is the infrared radiation falling on element i, Ji is the infrared radiation away from element i. Ji can be expressed as the sum of radiation emitted from the internal surface and the irradiated energy reflected by it. Ji and Gi are given, respectively, by J i ¼ erT 4f ;i þ ð1 eÞ
N X
J j X i;j
ð20Þ
ð23Þ where Re is the Reynolds number, Pra is the Prandtl number of atmosphere, ka is the thermal conductivity of atmosphere, and L is the characteristic length. The internal free convection heat transfer of element i given by QCI;i ¼ hCI Af ;i ðT he T f ;i Þ
ð21Þ
ð24Þ
where the internal heat transfer coefficient hCI can be calculated with hCI ¼ CðGrPrÞ k=L
ð25Þ
where Gr is the Grashof number, Pr is the Prandtl number of helium, k is the thermal conductivity of the helium, and the constant C and exponent n are listed in Table 1. 3.3. Helium thermal model The change rate of the helium temperature during ascent can be derived from the energy equation of a close system (Dai et al., 2011). For an unsteady close system, it can be expressed as M he cv
N X dT he ¼ W QCI;i dt i¼1
ð26Þ
where cv is the gas specific heat at constant volume, W is the expansion power of the helium. The balloon will pressurize until it expands to the maximum volume. Then Eq. (26) can be reduced to
j¼1
Gi ¼ ðJ i erT 4f ;i Þ=ð1 eÞ
ð22Þ
The heat transfer coefficient hCE can be calculated with (Incropera and DeWitt, 1996) ( ð2 þ 0:47Re1=2 Pra1=3 Þka =L Re 6 5 104 hCE ¼ ð0:0262Re4=5 615ÞPra1=3 ka =L 5 104 < Re 6 108
n
where e is the emissivity of the film material, r is the Stefan–Boltzmann constant, / is the view factor from element i to earth, Te is the ground temperature in K, and Tsky is the sky equivalent temperature in K. It can be calculated with (Yan and Zhao, 1986) pffiffiffiffiffi 0:25 T sky ¼ ð0:51 þ 0:208 pw Þ T a ð18Þ
9
M he cv
N X dT he ¼ QCI;i dt i¼1
ð27Þ
where Xi,j is the angle factor from element i to element j. 3.4. Temperature on the top of cloud 3.2.3. Convection heat gain The external forced convection heat transfer of element i can be expressed as
It has been recognized that clouds affect the float thermal behaviors of high altitude balloon. The cloud below
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the balloon reflects the sunlight to the balloon and increases the solar input of the balloon. While the cloud also decreases the infrared radiation input of the balloon, because the cloud top temperature is below that of the ground. For clear sky, the constant ground albedo value of 0.2 is widely accepted in most applications. However, the cloud albedo value turned to be 0.6 for overcast sky (Menon et al., 2002). Therefore, the constant albedo can be given by C R ¼ 0:2 þ 0:4CC
ð28Þ
where CC is defined as cloud cover, for clear sky CC = 0, and for overcast sky CC = 1. The cloud temperature can be treated as ambient atmospheric temperature. The mean cloud height is about 4000 m (Gjertsen, 1997). Therefore, the earth temperature of cloudy condition can be calculated with T e ¼ T g ð1 CCÞ þ T cloud CC
ð29Þ
4. Simulation method The governing Eqs. (9), (13), and (26) can be expressed as the vector form y0 ¼ f ðt; yÞ
Fig. 2. Altitude comparison of the predicted data with the measured data.
ð30Þ
where y = (Tf,1 Tf,2 . . . Tf,N The U Z)T, f = (f1 f2 . . . fN+3)T. The standards forth order Runge–Kutta method works well for nonlinear initial value systems of ordinary differential equations and only has an error proportional to the time step to the fourth power (Dai and Qiu, 2002). Then Eq. (30) can be discretized as 8 yiþ1 ¼ yi þ Dt6 ðk1 þ 2k2 þ 2k3 þ k4 Þ > > > > > > < k1 ¼ f ðti ; yi Þ ð31Þ k2 ¼ f ðti þ Dt2 ; yi þ Dt2 k1 Þ > > Dt Dt > > k3 ¼ f ðti þ 2 ; yi þ 2 k2 Þ > > : k4 ¼ f ðti þ Dt; T b;i þ Dtk3 Þ
conditions for the experiments are implemented in the present program. It is assumed that the surrounding atmosphere environmental is standard atmosphere with clear sky, that the pumpkin balloon is flat facet, and that the partially inflated shape of the balloon is a scaled pumpkin shape. The film of the balloon is divided into 6704 elements by the software Gambit. The solar absorptivity of film is 0.06 and the infrared emissivity is 0.24. Comparing to the area of the balloon, the area of load and the ballast can be neglected, and thus they are treated as a mass point. Fig. 2 compares the predicted altitudes with the measured data. Fig. 3 compares the predicted differential pressures with the measured data. They agree with each other well. All the results in the flowing section are for the NASA super pressure balloon introduced above. At a float altitude of 31.5 km, the thermal performance of a high altitude balloon is dominated by radiation (Pankine et al., 2003). Because of the thin air and low absolute velocity, the
Based on the discrete equations expressed above, a program in FORTRAN is developed. The simulation program consists of a main program and several subroutines handling input and output and calculating solar radiation, infrared radiation, heat convection, temperature and height. 5. Results and discussions The accuracy of the models introduced in this paper is evaluated by comparison with the experimental results of a NASA super pressure balloon flying the ascent trajectory (Cathey, 2009). The 56,790 m3 balloon was launched during the day time on June 22, 2008 from 34°N, 104°W. After the balloon reaching a steady float altitude of 30.5 km, a total of 109 kg of ballast were dropped in several increments to pressurize the balloon. The balloon can reach a final float altitude of about 31.5 km. The same setup and
Fig. 3. Differential pressure comparison of the predicted data with the measured data.
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radiation properties, mainly the solar absorptivity and infrared emissivity, are the major factors that affect the thermal behaviors of the balloon. Figs. 4 and 5 show the effects of solar absorptivity and infrared emissivity on the helium temperature at float condition. Fig. 4 shows the effects of the solar absorptivity on the helium temperature. It can be found that the bigger the solar absorptivity is, the higher the helium temperature will be. The highest temperature of the helium increases about 9 °C when the solar absorptivity increases from 0.04 to 0.06. When the solar absorptivity increases from 0.06 to 0.08, the maximum helium temperature variation is approximately 8 °C. It also can be seen from Fig. 4 that the helium temperature changes rapidly at sunrise and sunset. At sunrise, the helium temperature increases more than 10 °C in 10 min. Fig. 5 shows the effects of the infrared emissivity on the helium temperature. It can be found that the bigger the infrared emissivity is, the lower the helium temperature difference will be. The highest temperature of the helium
decreases about 3 °C when the infrared emissivity increases from 0.12 to 0.24. But at night, the helium temperature has a 4 °C incensement when infrared emissivity changes. Besides, using the computer program, the mass of the helium filled in the balloon can be calculated. Assume the maximum pressure difference of the balloon is 300 Pa, and the volume of the balloon is known, the variation of the helium mass with radiation properties is shown in Fig. 6. The maximum variation of mass of helium is about 10 kg. The 10 kg over-filling of helium may lead the pressure difference increase to more than 400 Pa, which has critical risk to the balloon. The day-night film temperature difference has a direct influence on the safety of balloon. Fig. 7 shows the effects of radiation properties on the maximum temperature difference of the film. The maximum temperature of the film can reach as high as 120 °C when the solar absorptivity is 0.08 and infrared emissivity is 0.16. This temperature difference is rigorous to the film. The minimal day–night
Fig. 6. Effects of radiation properties on mass of helium. Fig. 4. Effects of solar absorptivity on helium temperature (e = 0.24).
Fig. 5. Effects of infrared emissivity on helium temperature (a = 0.06).
Fig. 7. Effects of radiation properties on the maximum film temperature difference.
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permit better prediction of balloon temperature which will lead to a better understanding of overall balloon behaviors. The main conclusions from the program simulation are as the following:
Fig. 8. Effects of cloud cover on helium temperature (a = 0.06, e = 0.24).
(1) Solar absorptivity and infrared emissivity are the major factors that affect the thermal behaviors of a balloon. The day time temperature of the helium increases with the increment of the solar absorptivity. The increase of infrared emissivity will reduce the helium temperature difference. (2) The max film temperature difference varies significantly with the radiation properties. For the safety consideration of the balloon, the low absorptivity and the low ratio of absorptivity to emissivity of film materials are desired. (3) The presence of clouds significantly affects balloon behaviors. The combined effects of clouds depend on the radiation properties of the balloon film. The high ratio of absorptivity to emissivity of film may cause a daytime heating, while the low ratio of absorptivity to emissivity of film may lower the gas temperature during daytime. At night, clouds always decrease the gas temperature.
Acknowledgments This work was funded by Nanjing University of Aeronautics and Astronautics, China for Outstanding Doctoral Dissertation. References Fig. 9. Effects of cloud cover on helium temperature (a = 0.06, e = 0.60).
temperature difference is shown for the material that has low absorptivity and the ratio of absorptivity to emissivity. Figs. 8 and 9 show the effects of cloud cover on the helium temperatures from which it can be seen that during night the gas temperature with clouds is lower than that without clouds. However, the combined effects of clouds during day depend on the radiation properties of the film. The film with high ratio of absorptivity to emissivity may increase the gas temperature during daytime. For the film with low ratio of absorptivity to emissivity, the decrease of infrared radiation input may overcome the increase of solar input and cause a cooling of the balloon. 6. Conclusion Trajectory and thermal models for high altitude balloons are established, based on which, a computer program is developed. The influences of film radiation properties and clouds on balloon thermal behaviors are discussed in detail. It is believed that the present models and program
Almedeij, J. Drag coefficient of flow around a sphere: matching asymptotically the wide trend. Powder Technol. 186, 218–223, 2008. Carlson, L.A., Horn, W.J. New thermal and trajectory model for highaltitude balloons. J. Aircraft. 20 (6), 500–507, 1983. Cathey, H.M. Advances in the thermal analysis of scientific balloons. AIAA 96-0695, 1996. Cathey, H.M. The NASA super pressure balloon – A path to flight. Adv. Space Res. 44, 23–38, 2009. Colozza, A. Initial feasibility assessment of a high altitude long endurance airship. NASA CR-21272, 2003. Conner, J. P., Arena, A.S. Near space balloon performance predictions. AIAA 2010-37, 2010. Dai, J.Z., Qiu, J.X. Numerical methods for partial differential equations. Southeast University Press, Nanjing, China, (In Chinese), 2002. Dai, Q.M., Fang, X.D., Li, X.J. Dynamic simulation of breakaway characteristics of tethered aerostats. Adv. Space Res. 48 (7), 1258– 1264, 2011. Farley, R.E. BalloonAscent: 3-D simulation tool for the ascent and float of high-altitude balloons. AIAA 2005-7412, 2005. Gjertsen, U. The seasonal variation of cloud parameters over Central Europe: a fuzzy approach for the analysis of NOAA-APT data. Atmos. Res. 45, 123–152, 1997. Incropera, F.P., DeWitt, D.P. Fundamentals of Heat and Mass Transfer, 4th ed John Wiley & Sons, 1996. Li, X.J., Fang, X.D., Dai, Q.M. Research on thermal characteristics of photovoltaic array of stratospheric airship. J. Aircraft. 48 (4), 1380– 1386, 2011.
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Kreith, F., Kreider, J.F. Numerical prediction of the performance of high altitude balloons. NCAR Technical Note NCAR-IN/STR-65, Feb, 1974. Menon, S., Saxena, V.K., Durkee, P., et al. Role of sulfate aerosols in modifying the cloud albedo: a closure experiment. Atmos. Res. 61, 169–187, 2002. Morani, G. Method for prediction and optimization of a stratospheric balloon ascent trajectory. J. Spacecraft Rockets 46 (1), 126–133, 2009. Pankine, A.A., Heun, M.K., Schlaifer, R.S. Advanced balloon performance simulation and analysis tool. AIAA 2003-6741, 2003.
Ran, H., Thomas, R., Mavris, D. A comprehensive global model of broadband direct solar radiation for solar cell simulation. AIAA 2007– 33, 2007. Sou, R.Z., He, H.S. Aircraft air conditioning systems, Press of Beijing University of Aeronautics and Astronautics, Beijing, China, (In Chinese), 2004. Xia, X.L., Li, D.F., Sun, C., et al. Transient thermal behavior of stratospheric balloons at float conditions. Adv. Space Res. 46, 1184– 1190, 2010. Yan, Q.S., Zhao, Q.Z. Thermal processes in buildings. Press of Chinese Building Industries, Beijing, China, (In Chinese), 1986.
Advances in Space Research 49 (2012) 1045–1052 www.elsevier.com/locate/asr
Performance simulation of high altitude scientific balloons Qiumin Dai, Xiande Fang ⇑, Xiaojian Li, Lili Tian Department of Man, Machine and Environment Engineering, College of Aerospace Engineering, Nanjing University of Aeronautics and Astronautics, 29 Yudao Street, Nanjing 210016, PR China Received 20 September 2011; received in revised form 23 December 2011; accepted 26 December 2011 Available online 5 January 2012
Abstract The design and operation of a high altitude scientific balloon requires adequate knowledge of the thermal characteristics of the balloon to make it safe and reliable. The thermal models and dynamic models of altitude scientific balloons are established in this paper. Based on the models, a simulation program is developed. The thermal performances of a super pressure balloon are simulated. The influence of film radiation property and clouds on balloon thermal behaviors is discussed in detail. The results are helpful for the design and operate of safe and reliable high altitude scientific balloons. Ó 2012 COSPAR. Published by Elsevier Ltd. All rights reserved. Keywords: High altitude balloon; Thermal model; Dynamic model; Radiation property
1. Introduction High altitude scientific balloons provide a unique and low cost way to carry out missions in a near-space environment. In recent years, the potential use of high altitude scientific balloons as observation platforms for atmosphere studies or research purposes has attracted growing interest. In order to accomplish the objectives of a high altitude scientific balloon, it is necessary to predict the thermal behaviors before it launches. The balloon mission preparation requires an accurate and reliable flight performance prediction method in order to accomplish the mission successfully. A failure prediction of the skin temperature will affect the life of the balloon. A failure to accurately predict the temperature of buoyant gas can lead to under-filling or over-filling the balloon at launch. An under-filling balloon may not reach the desired float altitude, and an over-filling balloon may lead to envelope rupture before it reaches the
⇑ Corresponding author. Tel./fax: +86 25 8489 6381.
E-mail addresses: [email protected] (Q. Dai), xd_fang@yahoo. com (X. Fang), [email protected] (X. Li), tianlili0804@163. com (L. Tian).
desired float altitude. Both of the conditions will cause potential safety risks and lead to the mission failure. In the past decades, many investigations have been carried out on the thermal characteristics of high altitude balloons. Kreith and Kreider (1974) established a simple but excellent model to predict the thermal behaviors which was served as the starting point for the subsequent research. Carlson and Horn (1983) developed a new trajectory and thermal model to analyze the flight trajectory and thermal characteristics. In their model, the lifting gas is assumed to be able to absorb solar radiation and emit infrared radiation. Cathey (1996) numerically studied the temperature distribution on the balloon. Farley (2005) developed a code to simulate the ascent and float behaviors of high-altitude balloons. The code can also be used to simulate the thermal behaviors of balloon on other planets. Xia et al. (2010) developed a transient model to predict the variation of skin and lifting gas temperatures at float conditions. The foregoing articles mainly focused on the thermal behaviors of balloons at the float condition with clear sky, especially on the film temperature distribution and average helium temperature. The effects of film radiation properties and clouds on balloon thermal performances have not been given much attention. In this paper, three-dimensional
0273-1177/$36.00 Ó 2012 COSPAR. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.asr.2011.12.026
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Q. Dai et al. / Advances in Space Research 49 (2012) 1045–1052
Nomenclature A Ap B c CC Cbv Cd Cv CR D e Pw Q QIRE QIRI Re T U V Xi,j W Z a e f h k q r /
area (m2) top projected area (m2) net buoyancy (N) specific heat (J/(kg K)) cloud cover virtual mass coefficient drag coefficient specific heat at constant volume (J/(kg K)) albedo factor drag force (N) orbital eccentricity vapor pressure (kPa) heat gain of film (W) external infrared heat gain (W) internal infrared heat gain (W) Reynolds number temperature (K) vertical velocity (m/s) volume (m3) angle factor from element i to j expansion power (W) altitude (m) solar absorptivity emissivity of the film material true anomaly angle between the element normal and the solar irradiation thermal conductivity (W/(m K)) density (kg/m3) the Stefan–Boltzmann constant (W/(m2 K4)) angle between the element normal and the gravity direction
transient thermal models and dynamic models are established to numerically study the behavior of high altitude balloons during ascent and float conditions. Based on the mathematical models, a computer program is developed. The accuracy of the models is verified by comparing the simulation data to the measured data. Then, the influence of film radiation properties and clouds on balloon thermal behaviors is simulated and discussed in detail.
g Gr h I J L m M MA P Pr Pt x u
gravitational acceleration (m/s2) Grashof number heat transfer coefficient (J/(m2 K)) solar radiation intensity (W/m2) radiosity (W/m2) characteristic length (m) air mass ratio mass (kg) mean anomaly pressure (Pa) Prandtl number atmospheric transmittance solar elevation angle view factor from element to earth
Subscripts a ambient CE external convection CI internal convection cloud cloud D direct e earth f film g ground he helium i, j element number pay payload R reflect S diffuse sky sky tot total
8 288:15 0:0065Z if Z 6 11;000 m > > > < T a ¼ 216:65 if 11;000 m < Z 6 20;000 m > > > : 216:65 þ 0:0012ðZ 20;000Þ if 20;000 m < Z 6 33;000 m 8 5:26 Z > if Z 6 11;000 m 101325 1 44330 > > > < 11000Z if 11;000 m < Z 6 20;000 m P a ¼ 22605exp 6340 > > > > 141:89þ0:003Z 11:388 : if 20;000 m < Z 6 33;000 m 2447 216:65
ð1Þ
ð2Þ
where Z is the altitude of the balloon.
2. Atmosphere and solar models
2.2. Solar models
2.1. Atmosphere models
Solar radiation is classified as direct solar radiation and diffuse solar radiation from the sky and reflected solar radiation from the earth. The value of solar radiation can be calculated using the methods introduced by Colozza (2003), Farley (2005), Ran et al. (2007) and Li et al. (2011). The direct solar radiation intensity ID can be expressed as
During the ascent phase, the temperature and pressure variation at the flight altitude is important. Based on the standard atmosphere, the temperature and pressure are modeled as functions of the altitude (Sou and He, 2004). They can be written as
Q. Dai et al. / Advances in Space Research 49 (2012) 1045–1052
2 1 þ e cos f I D ¼ 1367 pmt 1 e2
ð3Þ
where e is the orbital eccentricity, for earth e = 0.0016708, the atmospheric transmittance pt is normally in the range of 0.6–0.7. The air mass ratio m and the true anomaly f can be calculated with qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pa 2 m¼ ð4Þ 1229 þ ð614 sin xÞ 614 sin x 101325 f ¼ MA þ 0:0334 sin MA þ 3:49 104 sinð2MAÞ
ð5Þ
The sphere drag coefficient is used in this paper since the shape of balloon is close to a sphere. The relationship between the sphere drag coefficient and Reynolds number in the range of the Reynolds number up to 107 was reported by Almedeij (2008). For Reynolds values greater than 106, the drag coefficient is stable at the value of about 0.1; while for Reynolds values lower than 105, the constant drag coefficient of 0.5 can be used. Conner and Arena (2010) explored a drag coefficient model for the ascent balloon in a Reynolds number range of 105–106, which is of the form
where pa is the ambient pressure, x is the solar elevation angle (x P 0), and MA ¼ 2np=365 where n is the day number in a year, n = 1 for January 1. The diffuse solar radiation intensity is written as
C d ¼ 0:72 2:57 106 Re þ 4:71 1012 Re2 4:04
1 1 pmt I S ¼ I D sin x 2 1 1:4 ln pt
3.2. Film thermal models
ð6Þ
The reflected solar radiation intensity is given by I R ¼ C R ðI D sin x þ I S Þ
ð7Þ
where CR is the albedo factor. 3. Dynamic and thermal models 3.1. Dynamic models The net buoyancy of the balloon is B ¼ qa Vg ðM he þ M pay Þg
ð8Þ
where ambient air density qa can be determined by the ideal gas law, V is the volume of the balloon, Mhe is the mass of helium, Mpay is the mass of payload and ballonet material, and g is the gravitational acceleration. The balloon’s motion is subject to the atmosphere winds and the net lift force of the balloon. It can be assumed that the horizontal velocity of the balloon is equal to the atmospheric wind (Morani, 2009). Therefore, the horizontal motion of balloon can be assumed at the wind speed. The governing differential equation of the vertical force balance on the balloon is d 2 Z dU B þ D ¼ ¼ dt M tot dt2
1018 Re3 þ 1:31 1024 Re
M f ;i c
dT f ;i ¼ QD;i þ QS;i þ QR;i þ QIRE;i þ QIRI;i dt þ QCE;i þ QCI;i
ð9Þ
ð10Þ ð11Þ
where AP is the top projected area of the balloon, and Cd is the drag coefficient.
ð13Þ
where Mf,i is the mass of the element i, QD,i is the absorbed direct solar radiation per second, QS,i is the absorbed diffuse solar radiation per second, QR,i is the absorbed reflected solar radiation per second, QIRE,i is the absorbed external infrared radiation per second, QIRI,i is the absorbed internal infrared radiation per second, QCE,i is the
The drag force of the aerostat D can be calculated by D ¼ 0:5qa C d AP jU jU
ð12Þ
Balloon is a thermal vehicle whose thermal behaviors are relatively straight forward with the surrounding environment. Thermal environment of the high altitude balloon includes solar radiation, infrared radiation and convection, as shown in Fig. 1. The balloon film is extremely thin, so the conductive heat transfer through the film can be neglected. The film of the balloon can be divided into N triangle elements, and each element can be treated as a plane. Assuming that the film is a lumped heat capacity, the transient energy-balance equation of the ith element can be expressed as
where U is vertical velocity of the balloon, and Mtot is the total mass. It is calculated with a virtual mass coefficient Cbv. The total mass takes into account the mass of air that is necessarily dragged along with the balloon. Values of virtual mass coefficient assumed range from 0.25 to 0.5. M tot ¼ M he þ M pay þ C bv ðqa V Þ
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Fig. 1. Thermal environment of high altitude balloon.
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external convection heat gain, QCI,i is the internal convection heat gain, and c is the specific heat of the film.
Table 1 The constant C and exponent n in Eq. (25). Geometry
3.2.1. Absorbed solar radiation The absorbed solar radiation of the film includes the absorption of the direct solar radiation, diffuse solar radiation from the sky and reflected solar radiation from the earth. The absorbed direct solar radiation of element i is given by QD;i ¼ aAf ;i I D cos h
ð14Þ
where a is the solar absorptivity of the film, Af,i is the area of the element, and h is the included angle between the element normal and the solar irradiation. The absorbed diffuse solar radiation from the sky is written as QS;i ¼ aAf ;i I S ð0:5 0:5 cos /Þ
ð15Þ
where / is the included angle between the plane normal and the gravity direction. The absorbed reflected solar radiation energy from Earth is expressed as QR;i ¼ aAf ;i I R ð0:5 þ 0:5 cos /Þ
ð16Þ
3.2.2. Absorbed infrared radiation The absorbed external infrared radiation includes earth and atmospheric infrared contributions. It can be calculated using the following equation: QIRE;i ¼ eAf ;i r½/ðT 4e T 4f ;i Þ þ ð1 /ÞðT 4sky T 4f ;i Þ
ð17Þ
GrPr
C
n
10 –10 109–1013 104–107 107–1011
0.59 0.1 0.54 0.15
1/4 1/3 1/4 1/3
105–1010
0.27
1/4
4
Vertical plate Upper surface of heated horizontal plate or lower surface of cooled horizontal plate Lower surface of heated horizontal plate or upper surface of cooled horizontal plate
QCE;i ¼ hCE Af ;i ðT a T f ;i Þ
where pw is the vapor pressure of the ambient air in kPa. The absorbed internal film infrared radiation of element i can be expressed as QIRI;i ¼ Af ;i ðGi J i Þ
ð19Þ
where Gi is the infrared radiation falling on element i, Ji is the infrared radiation away from element i. Ji can be expressed as the sum of radiation emitted from the internal surface and the irradiated energy reflected by it. Ji and Gi are given, respectively, by J i ¼ erT 4f ;i þ ð1 eÞ
N X
J j X i;j
ð20Þ
ð23Þ where Re is the Reynolds number, Pra is the Prandtl number of atmosphere, ka is the thermal conductivity of atmosphere, and L is the characteristic length. The internal free convection heat transfer of element i given by QCI;i ¼ hCI Af ;i ðT he T f ;i Þ
ð21Þ
ð24Þ
where the internal heat transfer coefficient hCI can be calculated with hCI ¼ CðGrPrÞ k=L
ð25Þ
where Gr is the Grashof number, Pr is the Prandtl number of helium, k is the thermal conductivity of the helium, and the constant C and exponent n are listed in Table 1. 3.3. Helium thermal model The change rate of the helium temperature during ascent can be derived from the energy equation of a close system (Dai et al., 2011). For an unsteady close system, it can be expressed as M he cv
N X dT he ¼ W QCI;i dt i¼1
ð26Þ
where cv is the gas specific heat at constant volume, W is the expansion power of the helium. The balloon will pressurize until it expands to the maximum volume. Then Eq. (26) can be reduced to
j¼1
Gi ¼ ðJ i erT 4f ;i Þ=ð1 eÞ
ð22Þ
The heat transfer coefficient hCE can be calculated with (Incropera and DeWitt, 1996) ( ð2 þ 0:47Re1=2 Pra1=3 Þka =L Re 6 5 104 hCE ¼ ð0:0262Re4=5 615ÞPra1=3 ka =L 5 104 < Re 6 108
n
where e is the emissivity of the film material, r is the Stefan–Boltzmann constant, / is the view factor from element i to earth, Te is the ground temperature in K, and Tsky is the sky equivalent temperature in K. It can be calculated with (Yan and Zhao, 1986) pffiffiffiffiffi 0:25 T sky ¼ ð0:51 þ 0:208 pw Þ T a ð18Þ
9
M he cv
N X dT he ¼ QCI;i dt i¼1
ð27Þ
where Xi,j is the angle factor from element i to element j. 3.4. Temperature on the top of cloud 3.2.3. Convection heat gain The external forced convection heat transfer of element i can be expressed as
It has been recognized that clouds affect the float thermal behaviors of high altitude balloon. The cloud below
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the balloon reflects the sunlight to the balloon and increases the solar input of the balloon. While the cloud also decreases the infrared radiation input of the balloon, because the cloud top temperature is below that of the ground. For clear sky, the constant ground albedo value of 0.2 is widely accepted in most applications. However, the cloud albedo value turned to be 0.6 for overcast sky (Menon et al., 2002). Therefore, the constant albedo can be given by C R ¼ 0:2 þ 0:4CC
ð28Þ
where CC is defined as cloud cover, for clear sky CC = 0, and for overcast sky CC = 1. The cloud temperature can be treated as ambient atmospheric temperature. The mean cloud height is about 4000 m (Gjertsen, 1997). Therefore, the earth temperature of cloudy condition can be calculated with T e ¼ T g ð1 CCÞ þ T cloud CC
ð29Þ
4. Simulation method The governing Eqs. (9), (13), and (26) can be expressed as the vector form y0 ¼ f ðt; yÞ
Fig. 2. Altitude comparison of the predicted data with the measured data.
ð30Þ
where y = (Tf,1 Tf,2 . . . Tf,N The U Z)T, f = (f1 f2 . . . fN+3)T. The standards forth order Runge–Kutta method works well for nonlinear initial value systems of ordinary differential equations and only has an error proportional to the time step to the fourth power (Dai and Qiu, 2002). Then Eq. (30) can be discretized as 8 yiþ1 ¼ yi þ Dt6 ðk1 þ 2k2 þ 2k3 þ k4 Þ > > > > > > < k1 ¼ f ðti ; yi Þ ð31Þ k2 ¼ f ðti þ Dt2 ; yi þ Dt2 k1 Þ > > Dt Dt > > k3 ¼ f ðti þ 2 ; yi þ 2 k2 Þ > > : k4 ¼ f ðti þ Dt; T b;i þ Dtk3 Þ
conditions for the experiments are implemented in the present program. It is assumed that the surrounding atmosphere environmental is standard atmosphere with clear sky, that the pumpkin balloon is flat facet, and that the partially inflated shape of the balloon is a scaled pumpkin shape. The film of the balloon is divided into 6704 elements by the software Gambit. The solar absorptivity of film is 0.06 and the infrared emissivity is 0.24. Comparing to the area of the balloon, the area of load and the ballast can be neglected, and thus they are treated as a mass point. Fig. 2 compares the predicted altitudes with the measured data. Fig. 3 compares the predicted differential pressures with the measured data. They agree with each other well. All the results in the flowing section are for the NASA super pressure balloon introduced above. At a float altitude of 31.5 km, the thermal performance of a high altitude balloon is dominated by radiation (Pankine et al., 2003). Because of the thin air and low absolute velocity, the
Based on the discrete equations expressed above, a program in FORTRAN is developed. The simulation program consists of a main program and several subroutines handling input and output and calculating solar radiation, infrared radiation, heat convection, temperature and height. 5. Results and discussions The accuracy of the models introduced in this paper is evaluated by comparison with the experimental results of a NASA super pressure balloon flying the ascent trajectory (Cathey, 2009). The 56,790 m3 balloon was launched during the day time on June 22, 2008 from 34°N, 104°W. After the balloon reaching a steady float altitude of 30.5 km, a total of 109 kg of ballast were dropped in several increments to pressurize the balloon. The balloon can reach a final float altitude of about 31.5 km. The same setup and
Fig. 3. Differential pressure comparison of the predicted data with the measured data.
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radiation properties, mainly the solar absorptivity and infrared emissivity, are the major factors that affect the thermal behaviors of the balloon. Figs. 4 and 5 show the effects of solar absorptivity and infrared emissivity on the helium temperature at float condition. Fig. 4 shows the effects of the solar absorptivity on the helium temperature. It can be found that the bigger the solar absorptivity is, the higher the helium temperature will be. The highest temperature of the helium increases about 9 °C when the solar absorptivity increases from 0.04 to 0.06. When the solar absorptivity increases from 0.06 to 0.08, the maximum helium temperature variation is approximately 8 °C. It also can be seen from Fig. 4 that the helium temperature changes rapidly at sunrise and sunset. At sunrise, the helium temperature increases more than 10 °C in 10 min. Fig. 5 shows the effects of the infrared emissivity on the helium temperature. It can be found that the bigger the infrared emissivity is, the lower the helium temperature difference will be. The highest temperature of the helium
decreases about 3 °C when the infrared emissivity increases from 0.12 to 0.24. But at night, the helium temperature has a 4 °C incensement when infrared emissivity changes. Besides, using the computer program, the mass of the helium filled in the balloon can be calculated. Assume the maximum pressure difference of the balloon is 300 Pa, and the volume of the balloon is known, the variation of the helium mass with radiation properties is shown in Fig. 6. The maximum variation of mass of helium is about 10 kg. The 10 kg over-filling of helium may lead the pressure difference increase to more than 400 Pa, which has critical risk to the balloon. The day-night film temperature difference has a direct influence on the safety of balloon. Fig. 7 shows the effects of radiation properties on the maximum temperature difference of the film. The maximum temperature of the film can reach as high as 120 °C when the solar absorptivity is 0.08 and infrared emissivity is 0.16. This temperature difference is rigorous to the film. The minimal day–night
Fig. 6. Effects of radiation properties on mass of helium. Fig. 4. Effects of solar absorptivity on helium temperature (e = 0.24).
Fig. 5. Effects of infrared emissivity on helium temperature (a = 0.06).
Fig. 7. Effects of radiation properties on the maximum film temperature difference.
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permit better prediction of balloon temperature which will lead to a better understanding of overall balloon behaviors. The main conclusions from the program simulation are as the following:
Fig. 8. Effects of cloud cover on helium temperature (a = 0.06, e = 0.24).
(1) Solar absorptivity and infrared emissivity are the major factors that affect the thermal behaviors of a balloon. The day time temperature of the helium increases with the increment of the solar absorptivity. The increase of infrared emissivity will reduce the helium temperature difference. (2) The max film temperature difference varies significantly with the radiation properties. For the safety consideration of the balloon, the low absorptivity and the low ratio of absorptivity to emissivity of film materials are desired. (3) The presence of clouds significantly affects balloon behaviors. The combined effects of clouds depend on the radiation properties of the balloon film. The high ratio of absorptivity to emissivity of film may cause a daytime heating, while the low ratio of absorptivity to emissivity of film may lower the gas temperature during daytime. At night, clouds always decrease the gas temperature.
Acknowledgments This work was funded by Nanjing University of Aeronautics and Astronautics, China for Outstanding Doctoral Dissertation. References Fig. 9. Effects of cloud cover on helium temperature (a = 0.06, e = 0.60).
temperature difference is shown for the material that has low absorptivity and the ratio of absorptivity to emissivity. Figs. 8 and 9 show the effects of cloud cover on the helium temperatures from which it can be seen that during night the gas temperature with clouds is lower than that without clouds. However, the combined effects of clouds during day depend on the radiation properties of the film. The film with high ratio of absorptivity to emissivity may increase the gas temperature during daytime. For the film with low ratio of absorptivity to emissivity, the decrease of infrared radiation input may overcome the increase of solar input and cause a cooling of the balloon. 6. Conclusion Trajectory and thermal models for high altitude balloons are established, based on which, a computer program is developed. The influences of film radiation properties and clouds on balloon thermal behaviors are discussed in detail. It is believed that the present models and program
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