LIQUID DROPLET RADIATOR FOR SPACE APPLICATIONS
Following from: The radiative transfer equation; Transport approximation; Differential approximations; Two-flux approximation; P_{1} approximation of the spherical harmonics method; The simplest approximations of double spherical harmonics Radiation of the isothermal plane-parallel layer; Radiation of the nonisothermal layer of a scattering medium; Radiative equilibrium in a plane-parallel layer; The radiative boundary layer
Investigations of the liquid droplet radiator (LDR) for space applications have been conducted during last two decades because of its high power-generation efficiency, i.e., relatively small mass in comparison with other radiators of the same power. The information on the engineering problem and the main energetic parameters of droplet radiators has been presented by Mattick and Hertzberg (1981, 1982, 1985), Knap (1982), and Taussig and Mattick (1986). Droplet generators produce a flow of particles of about 50 μm diameter at a sheet divergence angle lof < 0.01 rad. The droplet sheet has usually constant thickness, but its width may decrease toward the collector. Some schemes of droplet radiators were published by White (1987). A subsequent theoretical study of the radiative cooling process in the droplet radiator has been performed by Siegel (1987a,b, 1988, 1989a,b). One should also remember the papers by Siegel (1989c, 1990, 1991), in which a similar transient problem for a rectangular region was considered. A more realistic analysis of the engineering problem of the heat rejection capability of a liquid droplet radiator through an enclosure by taking into account the effects of conducting gas among the droplets and the enclosure transmittance has been conducted by Bayazitoglu and Jones (1990). More recent estimates of the possible advantages of LDRs and liquid sheet radiators (LSRs) were presented by Tagliafico and Fossa (1997). One should also remember the paper by Konyukhov et al. (1998), in which a Monte Carlo simulation for thermal radiation from an LDR was employed and some engineering suggestions were discussed.
The energetic (power-generation) efficiency of the droplet radiator, E, is defined as a ratio of the thermal radiation power to a mass of the sprayed fluid. The value of E can be estimated by use of the following approximate formula:
(1) |
where ε is the integral hemispherical emissivity of the droplet sheet of thickness h, ρ_{p} is the droplet mass concentration, and T is the average temperature of the droplets. In the simplest case of an optically thin monodisperse droplet jet, we have
(2) |
where f_{v} is the volume fraction of the droplets, ρ_{f} is the fluid density, and a is the droplet radius. For large droplets (compared to the radiation wavelength), the absorption efficiency factor Q_{a} is equivalent to the emissivity of the fluid surface ε_{f}, and, therefore,
(3) |
According to Eq. (3), it is desirable to use a fluid of low density and high emissivity, as well as droplets of small size.
By choice of the cooling liquid, one should account for possibility of long-time operation of the radiator and the minimum mass losses of the liquid due to evaporation. Therefore, the substances with low saturated vapor pressure (<10^{-7} Torr) in the working temperature range are preferable. Some fusible metals and their eutectics, as well as silicon oil, satisfy this condition Mattick and Hertzberg (1982, 1985). Fluids having low density and applicable in a wide temperature range deserve the main attention. On this basis, eutectic Na-K (261-315 K), silicon oil Dow 705 (275-335 K), Li (453-540 K), and Al (933-975 K) were selected as preferable in Mattick and Hertzberg (1985). At higher temperatures, in the temperature range from 505 to 1000 K, one can use Sn. Unfortunately, the liquid metals have a low emissivity, on the order of 0.1-0.2, whereas Dow 705 has ε_{f} = 0.5 (Mattick and Hertzberg, 1985). Insufficiently high emissivity of the fluid results in low energetic efficiency of the droplet radiator. Mattick and Hertzberg (1981) have discussed the addition of soot as a way of increasing the emissivity of metal droplets. Because of the high surface tension of metals, the soot, having a large emissivity, would cover the particle surface.
Siegel (1987) has noted that the increase of the energy dissipation in the droplet radiator may be reached due to the thermal effect of droplet solidification. This idea may be of interest, in spite of the need for the melting of particles for the repeated usage. A computational investigation of LDR parameters in the case of solidifying droplets was reported in Siegel (1989b).
Theoretical analysis of the thermal characteristics of some droplet radiators by Siegel (1987a,b, 1988, 1989a,b) was based on the model of an isotropically scattering gray medium. The medium albedo and droplet sheet optical thickness were varied as independent parameters irrespective of the realistic droplet optical properties and size distribution. At the same time, the effect of droplet sheet narrowing was considered in Siegel (1989a). It was noted that the optical thickness increasing results in additional shielding of the internal region radiation, and diminishing of the heat transfer rate.
A problem statement discussed in this section includes a calculation of the radiative properties of single droplets. Anisotropy of scattering is taken into account. At the same time, evaporation and possible solidification of the droplets are not considered. As in Siegel (1987a,b, 1988, 1989a,b), the gray model is employed. We assume that droplets are randomly positioned in the sheet volume, and distances between neighboring droplets are large enough. In this case, the dependent scattering effects are negligible. Note that the possible regular space distribution of the droplets may lead to resonance effects in spectral emission of the radiation by the droplet sheet. These effects, considered by Averin et al. (1989a,b), cannot be explained in terms of the ordinary radiation transfer theory.
Let us consider the radiation heat transfer for the simplest geometrical model, namely, a wide droplet sheet of constant width z_{0} and thickness h = y_{0} moving along the x-axis with velocity u_{0}. The temperature of all droplets at the initial cross section is the same. By the assumption of 1D radiation energy transfer, the energy equation and initial condition are
(4a) |
(4b) |
or, in dimensionless variables,
(5a) |
(5b) |
Equation (5) is similar to the problem of a radiative boundary layer. The only difference is in the boundary conditions for radiative transfer. Two limiting cases are more obvious: small or large optical thickness of the droplet sheet. At small optical thickness, each droplet cools off due to self-emission in vacuum, as in absence of other droplets. The volume heat loss is uniform, and the temperature of monodisperse droplets is constant in any cross section of the droplet sheet. As a result, the process is described by simplified initial-value problem,
(6a) |
(6b) |
and the evident analytical solution is
(7a) |
(7b) |
The local energetic efficiency of an LDR for the known function T(τ^{*}_{a,x}) can be determined by Eq. (3). The corresponding average integral value is
(8) |
where x_{0} is the length of the droplet sheet. The other limiting case is the optically thick droplet sheet. In this case, the radiation of droplets located at some distance from the sheet surface is considerably absorbed and scattered by droplets located nearer to the surface. The radiative boundary layer is formed, in which the droplet temperature is lower than that in the isothermal central region of the droplet sheet. It was shown in the article Radiative boundary layer that the thickness of the radiative boundary layer is constant in variable τ_{tr}^{*} = τ_{tr}/√τ^{*}_{tr,x}, and one can therefore write the following relations in physical variables:
(9) |
One can see that the local energetic efficiency of the LDR, E(x), always decreases with the distance from the initial cross section of the sheet. This is explained by the decreasing in the droplet temperature. In the case of an optically thin sheet, the value of E(0) is maximal, but E(x) decreases more rapidly than at a large optical thickness (small Boltzmann number).
The droplet sheet optical thickness should be neither too small nor too large. In the first case, the total radiation power would be insufficient; in the second case, the energetic efficiency would be small. Therefore, the simple solutions for both an optically thin and thick droplet sheet are, generally speaking, inapplicable for practical problems. One can employ the diffusion approximation to get a solution for an arbitrary optical thickness of the droplet sheet. The corresponding mathematical formulation of the problem is as follows (see articles Radiation of an isothermal plane-parallel layer and The radiative boundary layer):
(10a) |
(10b) |
(10c) |
(10d) |
(10e) |
The radiation flux is
(11) |
Consider some numerical results for a silicon oil droplet radiator without taking into account the possible phase change and any temperature variation of the oil properties. The following parameters were used in the calculation: ρ_{p} = 5 kg/m^{3}, T_{0} = 335 K, y_{0} = 0.01 m, and the properties of oil were ρ_{f} = 1100 kg/m^{3}, c_{f} = 1500 J/(kg·K). It was assumed that droplets have radius a = 25 μm and Q_{a} = 0.5, Q_{s}^{tr} = 1. The resulting value of the droplet sheet optical thickness is 2τ_{tr}^{0} = 4.1 at the medium albedo ω_{tr} = 2/3 (i.e., the sheet is not optically thin or thick). The results of calculations in the form of dependences on variable τ^{*}_{a,x} are shown in Fig. 1, where both the value of the radiation flux q and the value of = q√τ^{*}_{a,x} are also presented. In contrast to the limiting case of a small optical thickness, the particle temperature at the sheet surface is appreciably less than that in the central region. This temperature difference increases initially with cooling of the sheet, but it remains almost constant at τ^{*}_{a,x} > 0.2, which corresponds to the self-similar temperature profile Siegel (1987a). The radiation flux decreases together with the average temperature of the droplets. At the same time, the value of has a maximum near to τ^{*}_{a,x} = 0.2. At larger optical thickness of the sheet, this maximum would be transformed to the long region of = const, corresponding to the self-similar solution for an optically thick radiative boundary layer.
Figure 1. A general form of solution for an LDR: (a) temperature in the plane of symmetry (1) and on the surface (2) of the droplet sheet; (b) radiation flux q and the corresponding value of .
The realistic range of the droplet radiator parameters is limited by the narrow temperature interval and requirement of sufficiently high power-generation efficiency. Some results of calculations taking into account these circumstances are presented in Fig. 2. Two limiting variants for the temperature level are considered, namely, a silicon oil radiator and a radiator with molten aluminum particles. Calculations were interrupted at the solidification temperature. The parameters of the first variant were cited above; in the second variant, the following parameters were adopted at the same sheet thickness and droplet radius: ρ_{p} = 150 kg/m^{3}, T_{0} = 975 K, ρ_{f} = 2400 kg/m^{3}, c_{p} = 1080 J/(kg K), Q_{a} = 0.02, Q_{s}^{tr} = 1. One can see that the operation regime in both cases is characterized by the linear radiation flux decreasing, and is finished in the region of the radiative boundary layers joining.
Figure 2. Radiation flux from the droplet sheet for various working fluids: (a) silicon oil; (b) molten aluminum (1, pure aluminum; 2, aluminum droplets covered by soot).
The local energetic efficiency of the liquid droplet radiator is defined as
(12) |
for a high-temperature radiator with aluminum droplets, and appears to be at the same level as that for silicon oil radiator. This result is explained by the relatively high density and mainly by very low infrared emissivity of molten aluminum. Therefore, it is of interest to consider the variant of aluminum droplets covered by a thin surface layer of soot. The soot layer increases the absorption efficiency factor of the particle from 0.02 to 0.75 (Dombrovsky, 1996). As earlier, the mass concentration of aluminum particles was chosen with orientation to a not-too-large optical thickness of the droplet sheet. It was adopted that ρ_{p} = 10 kg/m^{3} in the variant presented in Fig. 2b. One can see that radiation flux increases by greater than three times in comparison with that for a pure aluminum droplet radiator. The cooling time up to solidification temperature of aluminum is less approximately by 30 times--from 1.2 to 0.04 s. The energetic efficiency of the radiator with the two-layer particles is higher on average by 50 times.
The best choice of the appropriate particle concentration is not quite clear from the above calculation. Obviously, the optimum variant should give a moderate optical thickness of the droplet sheet. To answer this practically important question more definitely, we can refer to additional calculations by the use of the method proposed. An example of the corresponding set of calculations for a silicon oil droplet radiator is shown in Fig. 3. The specific power of the unit width sheet
(13) |
where x_{0} is the length of the sheet (up to droplet solidification), increases with the surface mass concentration of the droplets ρ_{p}h = 2ρ_{p}y_{0}. At ρ_{p}h > 0.1 kg/m^{2} (τ_{tr} = 2τ_{tr}^{0} > 4), the value of W does not practically differ from the asymptotic value W_{0} for the optically thick layer. In contrast to the power, the radiator energetic efficiency E = Q/(ρ_{p}y_{0}) is maximum and approximately constant at ρ_{p}h < 0.025 kg/m^{2} (τ_{tr} < 1), but then decreases. In the region of W ≈ W_{0}, the value of E appears to be about three times less than that for an optically thin droplet sheet. One can see in Fig. 3 that the range of surface mass concentration from 0.04 to 0.1 kg/m^{2} is the most interesting. A choice of working level of ρ_{p}h inside this interval is determined by the particular practical problem.
Figure 3. Effect of surface mass concentration of the droplets ρ_{p}h on specific power W, energetic efficiency E, droplet sheet optical thickness τ = 2τ_{0}, and cooling time t for silicon oil droplet radiator (the value of W is calculated at u_{0} = 20m/s).
The main difficulty in obtaining the reliable quantitative data for the LRD energetic parameters is concerned with the uncertainty of working fluid optical constants, which may be sensible enough to admixtures. At the same time, the Mie theory as a component of the computational model enables us to estimate the effect of both the optical constants and particle size on the energetic characteristics of any liquid droplet radiator.
REFERENCES
Averin, V. V., Dmitriev, A. S., and Klimenko, A. V., Thermal Radiation of a Three-Dimensional Lattice of Spherical Particles, High Temp., vol. 27, no. 3, pp. 452-458, 1989a.
Averin, V. V., Dmitriev, A. S., and Klimenko, A. V., Thermal Radiation of Regular Spherical Particle Structures with Fluctuational Thermal Fluids Being Taken into Account, Int. Commun. Heat Mass Transfer, vol. 16, no. 3, pp. 403-414, 1989b.
Bayazitoglu, Y. and Jones, P. D., Enclosure and Conductive Effects on Thermal Performance of Liquid Droplet Radiators, J. Thermophys. Heat Transfer, vol. 4, no. 2, pp. 186-192, 1990.
Dombrovsky, L. A., Radiation Heat Transfer in Disperse Systems, Begell House, New York and Redding, CT, 1996.
Knap, K., Lightweight Moving Radiators for Heat Rejection in Space, Spacecraft Radiative Transfer and Temperature Control, Progress in Astronautics and Aeronautics, vol. 83, pp. 325-341, 1982.
Konyukhov, G. V., Koroteev, A. A., Novomlinskii, V. V., and Baushev, B. N., Modeling of Radiative Heat Transfer and Mass Transfer Processes in Drop-Flow-Based Heat Exchangers for Spacecraft, J. Eng. Phys. Thermophys., vol. 71, no. 1, pp. 87-91, 1998.
Mattick, A. T. and Hertzberg, A., Liquid Droplet Radiators for Heat Rejection in Space, J. Energy, vol. 5, no. 6, pp. 387-393, 1981.
Mattick, A. T. and Hertzberg, A., The Liquid Droplet Radiator--An Ultra Lightweight Heat Rejection System for Efficient Energy Conversion in Space, Acta Astronaut., vol. 9, no. 3, pp. 165-172, 1982.
Mattick, A. T. and Hertzberg, A., Liquid Droplet Radiator Performance Studies, Acta Astronaut., vol. 12, no. 7/8, pp. 591-598, 1985.
Siegel, R., Separation of Variables Solution for Nonlinear Radiative Cooling, Int. J. Heat Mass Transfer, vol. 30, no. 5, pp. 959-965, 1987a.
Siegel, R., Transient Radiative Cooling of a Droplet Filled Layer, ASME J. Heat Transfer, vol. 19, no. 1, pp. 159-164, 1987b.
Siegel, R., Transient Radiative Cooling of Absorbing and Scattering Cylinder--A Separable Solution, J. Thermophys. Heat Transfer, vol. 2, no. 2, pp. 110-117, 1988.
Siegel, R., Radiative Cooling Performance of a Converging Liquid Drop Radiator, J. Thermophys. Heat Transfer, vol. 3, no. 1, pp. 46-52, 1989a.
Siegel, R., Solidification by Radiation Cooling of a Cylindrical Region Filled with Drops, J. Thermophys. Heat Transfer, vol. 3, no. 3, pp. 340-344, 1989b.
Siegel, R., Some Aspects of Transient Cooling of a Radiating Rectangular Medium, Int. J. Heat Mass Transfer, vol. 32, no. 10, pp. 1955-1966, 1989c.
Siegel, R., Emittance Bounds for Transient Radiative Cooling of a Scattering Rectangular Region, J. Thermophys. Heat Transfer, vol. 4, no. 1, pp. 106-114, 1990.
Siegel, R., Transient Cooling of a Square Region of Radiating Medium, J. Thermophys. Heat Transfer, vol. 5, no. 4, pp. 495-501, 1991.
Tagliafico, L. A. and Fossa, M., Lightweight Radiator Optimization for Heat Rejection in Space, Heat Mass Transfer, vol. 32, no. 4, pp. 239-244, 1997.
Taussig, R. T. and Mattick, A. T., Droplet Radiator Systems for Spacecraft Thermal Control, J. Spacecraft Rockets, vol. 23, no. 1, pp. 10-17, 1986.
White, K. A., Liquid Droplet Radiator Development Status, AIAA Paper No. 1537, 1987.
References
- Averin, V. V., Dmitriev, A. S., and Klimenko, A. V., Thermal Radiation of a Three-Dimensional Lattice of Spherical Particles, High Temp., vol. 27, no. 3, pp. 452-458, 1989a.
- Averin, V. V., Dmitriev, A. S., and Klimenko, A. V., Thermal Radiation of Regular Spherical Particle Structures with Fluctuational Thermal Fluids Being Taken into Account, Int. Commun. Heat Mass Transfer, vol. 16, no. 3, pp. 403-414, 1989b.
- Bayazitoglu, Y. and Jones, P. D., Enclosure and Conductive Effects on Thermal Performance of Liquid Droplet Radiators, J. Thermophys. Heat Transfer, vol. 4, no. 2, pp. 186-192, 1990.
- Dombrovsky, L. A., Radiation Heat Transfer in Disperse Systems, Begell House, New York and Redding, CT, 1996.
- Knap, K., Lightweight Moving Radiators for Heat Rejection in Space, Spacecraft Radiative Transfer and Temperature Control, Progress in Astronautics and Aeronautics, vol. 83, pp. 325-341, 1982.
- Konyukhov, G. V., Koroteev, A. A., Novomlinskii, V. V., and Baushev, B. N., Modeling of Radiative Heat Transfer and Mass Transfer Processes in Drop-Flow-Based Heat Exchangers for Spacecraft, J. Eng. Phys. Thermophys., vol. 71, no. 1, pp. 87-91, 1998.
- Mattick, A. T. and Hertzberg, A., Liquid Droplet Radiators for Heat Rejection in Space, J. Energy, vol. 5, no. 6, pp. 387-393, 1981.
- Mattick, A. T. and Hertzberg, A., The Liquid Droplet Radiator--An Ultra Lightweight Heat Rejection System for Efficient Energy Conversion in Space, Acta Astronaut., vol. 9, no. 3, pp. 165-172, 1982.
- Mattick, A. T. and Hertzberg, A., Liquid Droplet Radiator Performance Studies, Acta Astronaut., vol. 12, no. 7/8, pp. 591-598, 1985.
- Siegel, R., Separation of Variables Solution for Nonlinear Radiative Cooling, Int. J. Heat Mass Transfer, vol. 30, no. 5, pp. 959-965, 1987a.
- Siegel, R., Transient Radiative Cooling of a Droplet Filled Layer, ASME J. Heat Transfer, vol. 19, no. 1, pp. 159-164, 1987b.
- Siegel, R., Transient Radiative Cooling of Absorbing and Scattering Cylinder--A Separable Solution, J. Thermophys. Heat Transfer, vol. 2, no. 2, pp. 110-117, 1988.
- Siegel, R., Radiative Cooling Performance of a Converging Liquid Drop Radiator, J. Thermophys. Heat Transfer, vol. 3, no. 1, pp. 46-52, 1989a.
- Siegel, R., Solidification by Radiation Cooling of a Cylindrical Region Filled with Drops, J. Thermophys. Heat Transfer, vol. 3, no. 3, pp. 340-344, 1989b.
- Siegel, R., Some Aspects of Transient Cooling of a Radiating Rectangular Medium, Int. J. Heat Mass Transfer, vol. 32, no. 10, pp. 1955-1966, 1989c.
- Siegel, R., Emittance Bounds for Transient Radiative Cooling of a Scattering Rectangular Region, J. Thermophys. Heat Transfer, vol. 4, no. 1, pp. 106-114, 1990.
- Siegel, R., Transient Cooling of a Square Region of Radiating Medium, J. Thermophys. Heat Transfer, vol. 5, no. 4, pp. 495-501, 1991.
- Tagliafico, L. A. and Fossa, M., Lightweight Radiator Optimization for Heat Rejection in Space, Heat Mass Transfer, vol. 32, no. 4, pp. 239-244, 1997.
- Taussig, R. T. and Mattick, A. T., Droplet Radiator Systems for Spacecraft Thermal Control, J. Spacecraft Rockets, vol. 23, no. 1, pp. 10-17, 1986.
- White, K. A., Liquid Droplet Radiator Development Status, AIAA Paper No. 1537, 1987.