A Relationship Between Skin Thermal Conductivity and Gas Polytropic Index in an Open Atmospheric Balloon

Transcription

1 FEBRUARY 2003 DE LA TORRE ET AL. 325 A Relationship Between Skin Thermal Conductivity and Gas Polytropic Index in an Open Atmospheric Balloon A. DE LA TORRE AND P. ALEXANDER Departamento de Física, Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires, Buenos Aires, Argentina J. CORNEJO Departamento de Física, Facultad de Ingeniería, Universidad de Buenos Aires, Buenos Aires, Argentina (Manuscript received 4 March 2002, in final form 5 August 2002) ABSTRACT With the assumption of a polytropic evolution for the lifting gas, the response of an ascending open atmospheric balloon to a monochromatic gravity wave is specified among other parameters by the heat balance with the surrounding air. If one considers the bubble of gas inside the open balloon as a thermodynamic system in contact through the balloon skin with a uniform thermal source (isothermic atmosphere), a relationship between the skin thermal conductivity and the polytropic index for the lifting gas [hydrogen (H 2 ) or helium (He)] may be found. The results for both gases are extended to the case of a typical tropospheric linearly decreasing temperature profile. Constant and variable balloon skin thicknesses are studied for both background temperature profiles. The polytropic index is found to be lower for the changing skin and shows a sensitive difference between the two temperature profiles. The relationship between the thermal conductivity and polytropic index becomes abrupt only when the latter approaches the isothermal or adiabatic values. 1. Introduction The use of balloons has been increasingly attracting the interest of researchers. They have been used as a scientific platform with useful loads for experiments, space observations, and particularly for the study of the atmosphere of the earth and other planets. The advantages in the use of balloons in atmospheric soundings versus other techniques based on rocket-borne instruments or ground-based measurements are at present clearly recognized. A relevant benefit is given by the unimprovable spatial and temporal resolutions of collected data, considering the moderate speeds and the high performance of instruments that are involved. In fact, the highest vertical resolution for tropospheric and stratospheric observations stems from balloons (e.g., Shutts et al. 1994). However, many difficulties may be also encountered. For example, the trajectory of a balloon is arbitrarily affected by the local atmospheric conditions. In the last decades a considerable number of studies have been reported in relation to the behavior of closed balloons (e.g., Allen and Vincent 1995, and references Corresponding author address: Dr. Alejondro de la Torre, Departamento de Física, FCEN, Universidad de Buenos Aires, Ciudad Universitaria, Pabellón I, 1428 Buenos Aires, Argentina. delatorr@df.uba.ar therein), but bottom-open balloons have apparently received less attention. The vertical dynamics of both are essentially determined by upthrust, weight, and drag. Vertical air motions have been directly linked to the rate of ascent of radiosonde-type balloons, but open ones become in addition significantly affected by density variations, according to analyses of data obtained on board (de la Torre et al. 1996). An open balloon model needs not only to consider vertical wind oscillations but also the buoyancy effect. This force is specifically subject to unknowns related to the varying balloon size. In fact, the volume increases or decreases as the environmental conditions change. The gas usually changes temperature at its own rate because of these expansions or contractions and there is an additional reluctance to thermalize with the surrounding air, so a gradient between both exists and a heat exchange develops. This in turn leads to further volume changes, implying that the mass of displaced air or upthrust and, therefore, the vertical force balance are all modified. Moreover, the ambient radiation also influences the gas temperature in a direct or indirect way, so any appreciable variation may lead to vertical balloon motions. Convection, conduction, and radiation may all be included in a study with different degrees of significance, mainly depending on the type of balloon. Unfortunately, quantitative predictions are subject to some uncertainties that depend on several factors that are difficult to specify (Raqué 1993, 1999), 2003 American Meteorological Society

2 326 JOURNAL OF APPLIED METEOROLOGY VOLUME 42 and so a simple scheme will be applied below. Specifically, the volume evolution in an open balloon flight depends on the ratio of mass and density of the lifting gas. The latter variable becomes determined by pressure (approximately uniform across the balloon and equal to the air value at the bottom interface) and temperature. The last quantity is essentially determined, if radiation effects are negligible, by the heat exchanged by the gas, which may be modeled, for example, with a polytrope [a thermodynamic evolution in which the specific heat remains constant and that is characterized by an index; adiabatic and isoprocesses are particular cases (Sears 1953)]. The transport of heat occurs mainly through the skin, so there must be a relationship between its thermal conductivity and the polytropic index appropriate for the gas description. Our aim below will be to explore how it is possible to relate both physical magnitudes. In an earlier work (Alexander et al. 1996, hereinafter ACT96) it was shown that when the gas exchanges a known amount of heat with the surroundings (characterized by a skin conductivity value between zero and infinity), the balloon response to air perturbations may be found. However, a relationship between the values of thermal conductivity and polytropic index could be easily given only in the extreme conductivity cases. An extension to the whole range requires further derivations, which will be presented here as a supplement to the previous article (but this must still be considered a preliminary simple approach to a complex problem). Consequently, whenever the thermal conductivity is known it will be possible to specify an appropriate polytropic index and predict the balloon behavior. We may then be able to improve onboard data analysis by estimating the consequences of certain balloon dynamical effects. 2. The response of an open atmospheric balloon Analytical solutions for the vertical response of an open balloon during ascent or descent under the presence of a long-period monochromatic gravity wave have already been obtained (ACT96). The simplifying assumptions that have been used are similar to those employed in other works and include a simple representation of the atmosphere with a background profile plus perturbations and the neglect of radiative effects, skin friction drag, aerodynamic lift, azimuthal rotation, or pendulous motion of the balloon. Viscous friction needed not be considered because it affects the equation of motion of the balloon only in the resistance force due to unsteady motion and a linear drag term, which must both be included at low Reynolds numbers (Tatom and King 1976), whereas the analyzed flight conditions ensure that the opposite is valid. Helium is transparent to radiation, so enveloping this gas with an ideal skin with the same property allows us to discard the corresponding effect. The heat transport through the skin depends on the specific material being employed and any example should contain the two opposite scenarios that have been considered: null and perfect thermal conductivity (i.e., when the evolution is adiabatic and when the gas mimics the air temperature). Both are particular cases of a polytropic law. The last one applies only if the atmospheric temperature is assumed to be uniform. It should be also taken into account that this is a kind of relaxed perfect conductivity: gas and air mean temperatures are equal, because air fluctuations induced by a gravity wave impose no temperature variations (as a consequence of negligible pressure fluctuations in these waves) on a polytropic gas inside an open balloon. Volume oscillations are also irrelevant (see ACT96). The lifting gas does not register the presence of atmospheric waves, but the balloon dynamics are affected by them, inducing vertical velocity fluctuations in the balloon associated with the drag and buoyancy forces. An expression for the balloon vertical velocity oscillation was found in both limiting cases as a function of the altitude and parameters of the atmosphere and the balloon. An isothermal atmosphere, a constant wave velocity amplitude with height, a nearly constant mean vertical speed of the balloon during ascent or descent, and no gas mass loss during the journey were assumed. The results showed that the wave vertical velocity variations dominate the balloon s response for perfect conductivity, but in the opposite case the air density oscillations also become significant. Although a few improvements may be included in the derivation of results by ACT96, they are not very significant and should therefore not lead to important qualitative alterations (Alexander and de la Torre 2002). In ACT96 the balloon response was derived in the adiabatic limit, with a polytropic index n equal to (ratio of specific heats at constant pressure and volume). However, under the assumption of a general polytropic gas behavior, it is possible to extend Eq. (23) of ACT96 by simply replacing with n. With parameter values from that work, we draw in Figs. 1a and 1c the balloon vertical velocity fluctuation during ascent with helium for null and perfect conductivity (n 5/3 and 1), including in Fig. 1b an intermediate case in which n 4/3. The air density and velocity profiles as a background reference are shown. The phase difference between the curves does not vary significantly either with height or with the polytropic index. In the three examples the density leads the balloon perturbation by about 20, and the latter appears approximately 70 ahead of the wind oscillation. The location of the balloon peak in the 90 interval between the maxima of both air variables may change if other parameter values for the balloon and atmosphere are used in the simulation. The most noticeable aspect among the three cases corresponds to the increase of balloon response amplitude during ascent when the balloon skin approaches adiabaticity. As expected, the intermediate case presents a vertical balloon response situated between perfect and null thermal conductivity limits. In this intermediate

3 FEBRUARY 2003 DE LA TORRE ET AL. 327 FIG. 1. Helium-filled ( 5/3) balloon vertical velocity fluctuation during ascent for (a) null, (b) finite, and (c) perfect skin conductivity. Air oscillations are also included for reference. case there is a heat transfer between the lifting gas and the surrounding air through a skin with finite thermal conductivity. A relationship between this material property and the gas evolution is modeled below, taking into account some thermodynamic considerations. As explained above, atmospheric waves do not affect the gas thermodynamics calculations below. 3. Skin thermal conductivity and gas polytropic evolution It is assumed that the open balloon basically resembles a sphere regardless of the volume of the gas bubble inside it. The thermodynamic air and gas variables may not be completely uniform across the whole balloon, but it can be considered that unique representative values may be taken. In addition, at the bottom interface between air and gas we assume that both pressures are equal. We will consider molecular hydrogen and helium as lifting gases. The latter is often preferred because of certain advantages (e.g., nonflammable). We consider an open atmospheric balloon as a thermodynamic system in which the lifting gas is separated from the surrounding atmosphere (thermal source) by a skin of thermal conductivity K. The lifting gas may be considered as ideal. We assume a polytropic expansion. The first principle of thermodynamics, du Q W, (1) relates the internal energy variation of the gas during the ascent of the balloon du, the heat absorbed Q, and the work done by the gas during its expansion W. In a polytropic evolution, (1) may be written as NC (Tf T i) NC(Tf T i) [ ] V i 1n PV i i Vf 1, (2) n 1 where N, C, C, T, P, V, and n are, respectively, the number of gas moles, the molar specific heat, the molar specific heat at constant volume, the gas temperature, the atmospheric (and gas) pressure, the balloon (and gas) volume, and the polytropic index. The subindices i and f refer to the initial and final states of the gas evolution. We shall consider an arbitrary polytropic evolution with n between 1 and (perfect and null thermal conductivity). In this range, the value of C is less than or equal to zero. Note from the following relation for a polytropic process of an ideal gas (e.g., Yavorsky and Detlaf 1972) W 1 1 (3) Q n that W Q. Moreover, on the right-hand side of (2) we see that W 0, Q 0, which assures us that du 0. (4) This is consistent with the sign of the internal energy variation on the left-hand side of (2), knowing that the value of T f is less than T i and C is greater than zero. Now we shall consider that all Q absorbed by the gas during the ascent penetrates uniformly through the balloon skin. From the Fourier law of heat conduction F KT, the heat flux may be integrated in time t over the balloon surface S to obtain f dt Q K ds dt, (5) i S dr where r is a radial coordinate directed outwards from the center of the spherical balloon and ds is a differential skin surface element. The factor dt/dr may be written as (T )/r, where is the atmosphere temperature and r is the skin thickness. It will be assumed in what follows that K is constant; T and are taken just inside

4 328 JOURNAL OF APPLIED METEOROLOGY VOLUME 42 and outside the balloon skin. We neglect any influence in the heat balance arising from the opening at the bottom of the balloon. We write (5) as (T ) r f 2 Q 4K R dt, (6) i where R is the balloon radius. In (6) we must take into account the variation of T and R with time during the ascent. From the polytropic relations n/1n n PT const, PV const. (7) Considering the log pressure reference system, where the vertical coordinate z H ln(p/p 0 ) (e.g., Holton 1979), we obtain [ ] dzb d P wb H ln, (8) dt dt P 0 where w b, z b, P 0, and H are the mean balloon vertical velocity (assumed to be constant), the mean vertical position of the balloon, the initial level pressure, and the atmospheric scale height. We obtain, replacing (8) for the pressure in (7), [ ] w b (1 n) wb T Ti exp t, V Vi exp t. (9) H n Hn Also, 1/3 3 R V. (10) 4 To solve (6) and obtain a relation between K and n, we shall consider two different cases: a uniform atmospheric temperature and a linearly decreasing with height profile. In both cases, it will be considered that the balloon skin density is constant and we shall evaluate the skin thickness evolution r. Let us consider the balloon skin limited by internal and external radii R and R r. By equating the total volume occupied by the skin at the begining of the flight and at a time t, we obtain /3 r [(Ri r i) Ri R ] R. (11) To make (6) integrable, note that the expression in (11) may be reduced if we use the fact that the value of r is much less than R. After replacing (9) and (10) in (11) and applying a Taylor series expansion around r 0, (11) is simplified: r ri exp(t), (12) where 2w b /3nH. Two illustrative different cases will be considered below: the ascent of an open balloon in an isothermal atmosphere and then the ascent of an open balloon in a tropospheric typical linearly decreasing temperature profile. FIG. 2. Balloon skin thermal conductivity (J m 2 s 1 K 1 ) per initial thickness vs gas polytropic index for an isothermic atmosphere. a. Uniform atmospheric temperature In this case i and (6) may be written as t K Q {exp[( 2)t] exp(2t)} dt, (13) r i 0 where 4 i (3V i /4) 2/3 H/w b and (w b /H)[(1 n)/n]. This integral yields K 1 Q [exp(2t) 1] r 2 i 1 {exp[( 2)t] 1}. (14) 2 The relationship between K and n is obtained through the definitions of and, replacing (14) in the first right-hand term of (2), with T and V given by (9). Choosing a representative time t H/w b, 1 (1n)/n PV[1 i i e ] C n 1. (15) 3 1 4/3n (7/3n)/n n (e 1) [e 1] 4 7 n 3 Here we define a thermal conductivity per initial thickness, K/r i. Note the dependence with n and the initial launching and atmospheric conditions. In Fig. 2 the result may be appreciated on a logarithmic scale. We have chosen H 8000 m, w b 5ms 1, V i 1200 m 3, and P i hpa for H 2 and He, with C equal to 1.50R g and 1.45R g respectively, where R g is the universal constant of gases. As expected, the conductivity tends to infinite for n approaching 1 (perfect thermal conductivity) and 0 as it reaches the adiabatic value (5/3 and 7/5 for He and H 2, respectively). We have also plotted the curves corresponding to constant skin thickness.

5 FEBRUARY 2003 DE LA TORRE ET AL. 329 b. Atmospheric temperature linearly decreasing with height A more realistic calculation may be performed by taking a linear atmospheric vertical temperature profile of the form (z) mzb i mwbt i. (16) This fits very well a tropospheric mean temperature profile under stable and calm conditions. A new expression must be added to Q, b t m Q w t exp(2t) dt. (17) i 0 This contributes with an additive term in the denominator of (15), and now: 1 (1n)/n PV[1 i i e ] C n 1. (18) 3 1 (mh) /3n (7/3n)/n 4/3n 4/3n n (e 1) [e 1] e ne n 4 7 i n 3 FIG. 3. Same as Fig. 2, but for a linear background atmospheric temperature profile. In Fig. 3 the results for H 2 and He and identical parameters as in Fig. 2 show similar behavior to that figure but a less abrupt tendency to infinity and zero is observed at n 1 and, respectively. For known skin thermal conductivity and tropospheric temperature profile, the expected polytropic expansion may be estimated from the graph. 4. Discussion In a simple preliminary approach to a complex problem, we model the relationship between the skin thermal conductivity and polytropic index of the lifting gas of an open balloon ascending in the atmosphere under the presence of a monochromatic gravity wave. Because the existence of the latter does not affect the polytropic evolution of the lifting gas, mean values both for the atmospheric and for the thermodynamic lifting gas variables must only be considered for our calculations. Two cases, constant and variable balloon skin thickness, are studied. If the balloon skin thermal conductivity is known and its thickness decreases during the ascent, n is found to be lower than in the constant case. Although a real tropospheric temperature profile departs less than 15% from an isothermic atmosphere, n shows a sensitive variation from one case to the other. The relationship between and n is abrupt only close to 1 and. As expected, in these limits the conductivity tends to infinity and zero, respectively, either for constant or variable skin thickness, both for uniform or linear background temperature profile. The above description is applicable to H 2 and He, taking into account the corresponding values. Acknowledgments. Alejandro de la Torre and Peter Aexander are members of Conicet. This work has been supported by grants UBA X058 and Conicet PID 4554/ 96. REFERENCES Alexander, A., and A. de la Torre, 2002: Q program for the simulation and analysis of open atmospheric balloon soundings. Comput. Phys. Commun., in press., J. Cornejo, and A. de la Torre, 1996: The response of an open stratospheric balloon in the presence of inertio-gravity waves. J. Appl. Meteor., 35, Allen, S. J., and R. A. Vincent, 1995: Gravity wave activity in the lower atmosphere: Seasonal and latitudinal variations. J. Geophys. Res., 100, de la Torre, A., H. Teitelbaum, and F. Vial, 1996: Stratospheric and tropospheric gravity wave measurements near the Andes Mountains. J. Atmos. Terr. Phys., 58, Holton, J., 1979: An Introduction to Dynamic Meteorology. Academic Press, 319 pp. Raqué, S. M., 1993: Sinbad 3.0: NASA s scientific balloon analysis model: User s manual. NASA Goddard Space Flight Center, Wallops Flight Facility, Wallops Island, VA, 120 pp.

6 330 JOURNAL OF APPLIED METEOROLOGY VOLUME 42, 1999: A spreadsheet tool for terrestrial and planetary balloon design. Proc. Int. Balloon Technology Conf., Norfolk, VA, AIAA, AIAA , [Available from AIAA, 1801 Alexander Bell Drive, Suite 500, Reston, VA ] Sears, F. W., 1953: An Introduction to Thermodynamics: The Kinetic Theory of Gases, and Statistical Mechanics. Addison-Wesley, 374 pp. Shutts, G. J., P. Healey, and S. D. Mobbs, 1994: A multiple sounding technique for the study of gravity waves. Quart. J. Roy. Meteor. Soc., 120, Tatom, F. B., and R. L. King, 1976: Determination of constant-volume balloon capabilities for aeronautical research. Summary Rep., NASA Space Sciences Laboratory, MSFC, Huntsville, AL, 154 pp. Yavorsky, B., and A. Detlaf, 1972: Handbook of Physics. Mir, 964 pp.