Damián Rivas

One-dimensional self-similar solution of the dynamics of axisymmetric slender liquid bridges

In this paper the dynamics of axisymmetric, slender, viscous liquid bridges having volume close to the cylindrical one, and subjected to a small gravitational field parallel to the axis of the liquid bridge, is considered within the context of one-dimensional theories. Although the dynamics of liquid bridges has been treated through a numerical analysis in the inviscid case, numerical methods become inappropriate to study configurations close to the static stability limit because the evolution time, and thence the computing time, increases excessively. To avoid this difficulty, the problem of the evolution of these liquid bridges has been attacked through a nonlinear analysis based on the singular perturbation method and, whenever possible, the results obtained are compared with the numerical ones.

Minimum-Fuel Cruise at Constant Altitude with Fixed Arrival Time

A N IMPORTANT problem in air traffic management (ATM) is the design of aircraft trajectories that meet certain arrival time constraints at given waypoints, for instance, at the top of descent (TOD), at the initial approach fix (IAF), or at the runway threshold. These are called 4-D trajectories, which are a key element in the trajectory-based-operations (TBO) concept proposed by NextGen and SESAR for the future ATM system. The time constraints must also be met in certain cases in which the nominal trajectories have to be modified to resolve detected conflicts (e.g., lost of separation minima) between aircraft; for example, Bilimoria and Lee [1] analyze aircraft conflict resolution with an arrival time constraint at a downstream waypoint. On the other hand, the design of fuel-optimal trajectories that lead to energy-efficient flights is another important problem which has been treated extensively in the literature, see, for instance, Burrows [2], Neuman andKreindler [3],Menon [4] and the references therein. Fuel-optimal trajectories with fixed arrival times are studied by Sorensen and Waters [5], Burrows [6] and Chakravarty [7], who analyze the 4-D fuel-optimization problem as a minimum directoperating-cost (DOC) problem with free final time, that is, the problem is to find the time cost for which the corresponding free final-time DOC-optimal trajectory arrives at the assigned time. In this work we analyze the problem of minimum-fuel cruise at constant altitudewith afixed arrival time as a singular optimal control problem, building upon the works of Pargett and Ardema [8] and Rivas and Valenzuela [9], who analyze the problem of maximumrange cruise at constant altitude also as a singular optimal control problem; the case of unsteady cruise is considered. The singular arcs and the corresponding optimal control are obtained as a function of the final time. The optimal paths are obtained as well, which define a variable-Mach cruise at constant altitude. The influence of cruise altitude on the optimal paths is analyzed, and the minimum fuel is calculated. The final-time constraint may be defined, for example, by a flight delay imposed on the nominal (preferred) cruise trajectory (which in our case is the minimum-fuel cruise trajectory with free final time); comparison with a standard constant-Mach procedure to absorb delays is made. Results are presented for a model of a Boeing 767-300ER. Problem Formulation

Minimum-Cost Cruise at Constant Altitude of Commercial Aircraft Including Wind Effects

A IRCRAFT trajectory optimization is a subject of great importance in air traffic management from the point of view of defining optimal flight procedures that lead to energy-efficient flights. In practice, the airlines consider a cost index (CI) and define the direct operating cost (DOC) as the combined cost of fuel consumed and flight time weighted by the CI. Their goal is to minimize the DOC. However, in the presence of unexpected winds, the flight timemay differ considerably from the scheduled time, which leads to an arrival-error cost that can be added to the DOC to obtain the total cost (TC). Minimum-DOC trajectories have been studied by different authors [1–6]. The related problem of minimum fuel with fixed final time has been analyzed as a minimum-DOC problem with free final time in [3,5,7] (the problem is to find the time cost for which the corresponding free final time DOC-optimal trajectory arrives at the assigned time); this same problem is addressed in [8], analyzing the effects of mismodeled winds in a scenario formed by the final cruise and descent segments. The problem of minimum-cost flight, considering not only the DOC but also the arrival-error cost, is analyzed in [9,10], taking into account factors such as crew overtime cost, passenger dissatisfaction cost, and losses due to missed connections. In this Note, the problem of minimum-cost cruise at constant altitude in the presence of strong winds, including the arrival-error cost, is analyzed, considering the general unsteady problem, with variable aircraft mass, and without any restriction on cruise altitude. The main objective is to analyze the optimal trajectories that lead to minimum cost, defined as optimal speed laws (speed as a function of aircraft mass). The analysis is made using the theory of singular optimal control (see [11]), which has the great advantage of providing feedback control laws (control variables as functions of the state variables) that can be directly used to guide the aircraft along the optimal path. These optimal control laws are analyzed as well. In this work, the initial and final speeds are given, so that the optimal control is of the bang-singular-bang type, and the optimal paths are formed by a singular arc and two minimum/maximumthrust arcs joining the singular arc with the given initial and final points (see [6,12]). In previous work related to optimum cruise at constant altitude [13,14], only the singular arc was studied; hence, a more general formulation of the optimal problem is addressed now, apart from considering the arrival-error cost and including wind effects (average horizontal winds). In this analysis of the minimum-TC problem, the arrival-error cost depends on the difference between the actual and the scheduledflight times, and it is defined to be positive, so that both late and early arrivals are penalized (the objective is to achieve high arrival-time accuracy). It will be shown that, for some values of the parameters of the problem, minimum cost is obtained when the final time coincides with the scheduled time of arrival; that is, when the arrival-error cost is zero. This critical case is in fact a problem with fixed final time. Results are presented for a model of a Boeing 767-300ER.

Optimization of Multiphase Aircraft Trajectories Using Hybrid Optimal Control

An approach to optimize multiphase trajectories of commercial transport aircraft is presented. The approach is based on the theory of hybrid optimal control, and it is applied to the case of minimum-fuel trajectories. The multiphase trajectories are composed of three types of phases, climb, cruise, and descent, in a given sequence. In each phase, the optimal control is scalar and of the bang–singular–bang type, and the optimal path is formed by a singular arc and two minimum/maximum-control arcs joining the singular arc with the initial and final switching points. An indirect numerical method is developed, which takes into account the structure of the solution directly in the algorithm and exploits the singular character of the problem. In the analysis, the effects of horizontal winds are taken into account; general along-track wind and crosswind profiles are considered, dependent both on altitude and along-track position. The optimal trajectories are computed for a model of a Boeing 767-300ER performing ...

Analysis of secondary radiation (multiple reflections) in monoellipsoidal mirror furnaces

Abstract The radiation heat exchange in monoellipsoidal mirror furnaces is considered. In particular, the radiative interaction between the sample and the mirror is studied. This interaction is formulated analytically taking into account multiple reflections at the mirror. It is shown that the effect of these multiple reflections in the heating process is quite important, especially up to the third reflection, and, as a consequence, the effect of the mirror reflectance in the temperature field is quite strong. A conduction–radiation model is thus formulated to study the temperature field in slender, cylindrical samples (at this stage convection effects in the melt are not considered). This model is used to simulate the heating process in the floating-zone technique in microgravity conditions; important parameters like the Marangoni number (that drives the thermocapillary flow in the melt), and the temperature gradient at the melt-crystal interface are estimated. The model is validated comparing with experimental data, the agreement is very good both qualitative and quantitatively.

HIGH-REYNOLDS-NUMBER THERMOCAPILLARY FLOWS IN SHALLOW ENCLOSURES

Steady thermocapillary flows of low‐Prandtl‐number fluids in shallow rectangular enclosures under an imposed‐heat‐flux configuration are studied in the absence of gravitational forces. The Navier–Stokes equations are solved numerically for Reynolds numbers up to 104. The pressure correction method is used to treat the pressure‐velocity coupling, in particular, the SIMPLEC approximation is considered. In the numerical simulation, the free surface is assumed flat. This hypothesis is justified a posteriori; free surface deformations are computed by domain perturbation for small capillary numbers. The numerical results show the existence of a boundary layer along the free surface and the existence of a strong vortex close to the vertical wall. The characteristic velocity in the boundary layer responds to Ostrach’s scaling of thermocapillary boundary layers, and the wall vortex responds to Batchelor’s model for steady laminar flows with closed streamlines at large Reynolds numbers. The calculated surface defor...

Radiative exchange between a cylindrical crystal and a monoellipsoidal mirror furnace

Abstract The heating of a cylindrical sample in a monoellipsoidal mirror furnace is considered. A radiation model is formulated for slender samples where the radiative exchange between the sample and the mirror is studied analytically. To analyse the behavior of the model, the temperature field in the sample is calculated using a one-dimensional conduction radiation model. The differences with respect to a model where the radiative exchange is not considered are not only quantitative but qualitative. The model predicts a strongly asymmetric heating, which may cause asymmetries in the grown crystal as observed experimentally. Comparison with experimental data is made.

An analysis of lamp irradiation in ellipsoidal mirror furnaces

Abstract The irradiation generated by halogen lamps in ellipsoidal mirror furnaces is analyzed, in configurations suited to the study of the floating-zone technique for crystal growth in microgravity conditions. A line-source model for the lamp (instead of a point source) is developed, so that the longitudinal extent of the filament is taken into account. With this model the case of defocussed lamps can be handle analytically. In the model the lamp is formed by an aggregate of point-source elements, placed along the axis of the ellipsoid. For these point sources (which, in general, are defocussed) an irradiation model is formulated, within the approximation of geometrical optics. The irradiation profiles obtained (both on the lateral surface and on the inner base of the cylindrical sample) are analyzed. They present singularities related to the caustics formed by the family of reflected rays; these caustics are also analyzed. The lamp model is combined with a conduction–radiation model to study the temperature field in the sample. The effects of defocussing the lamp (common practice in crystal growth) are studied; advantages and also some drawbacks are pointed out. Comparison with experimental results is made.

A global thermal analysis of multizone resistance furnaces with specular and diffuse samples

Abstract The heating process in multizone resistance furnaces, as applied to floating-zone crystal growth, is analyzed. A global model is formulated, where the temperature fields in the sample, the furnace and the insulation are coupled; the input thermal data is the electric power supplied to the heaters; the thermal conductivity of the insulation is modeled with the aid of experimental results. The radiation heat exchange between the sample and the furnace is formulated analytically, taking into account specular reflections on the sample; in general, for solid samples the reflectance is both diffuse and specular, and for some melts it is mostly specular. This behavior is modeled through the exchange view factors, which depend on whether the sample is solid or liquid, and, therefore, they are not known a priori. The effect of this specular behavior on the temperature field is analyzed, and compared with the case of diffuse samples; this effect is shown to be important in the analysis of the melt zone, for instance, differences of the order of 100% are obtained in parameters like the melt length or the maximum temperature difference in the melt when specular reflections are neglected. The model is used to simulate the heating process in the floating-zone technique in microgravity conditions; parameters like the Marangoni number or the temperature gradient at the melt–crystal interface are estimated. The model is validated comparing with experimental data.

Temperature field in a cylindrical crystal heated in a mono-ellipsoid mirror furnace

Abstract The temperature field in a cylindrical solid crystal heated in a mono-ellipsoid mirror furnace is studied. The radiation intensity on the surface of the sample is obtained explicity as a function of the parameters that define the problem. The resulting axisymmetric two-dimensional conduction-radiation model is solved numerically. The dependence of the maximum temperature in the crystal on several design parameters is analyzed. The importance of the radiation losses from the sample is assessed. When the slenderness of the crystal is large, the temperature field is practically unidimensional. A one-dimensional model to describe the heat transfer process is formulated; the agreement with the two-dimensional model is excellent. The simplified one-dimensional model is proposed for the analysis of the more difficult problem where the redistribution by the mirror of the radiation losses from the sample is considered.