Search

In this paper, a novel formula expressing explicitly the fractional-order derivatives, in the sense of Riesz-Feller operator, of Jacobi polynomials is presented. Jacobi spectral collocation method together with trapezoidal rule are used to reduce the fractional L\'{e}vy-Feller advection-dispersion equation (LFADE) to a system of algebraic equations which greatly simplifies solving like this fractional differential equation. Numerical simulations with some comparisons are introduced to confirm the effectiveness and reliability of the proposed technique for the L\'{e}vy-Feller fractional partial differential equations., Comment: 23 pages, 4 figures

The basic features of nucleus-acoustic (NA) envelope bright and dark solitons, which exist in degenerate quantum plasmas, have been theoretically investigated by deriving the nonlinear Schr\"odinger (NLS) equation. The reductive perturbation method, which is valid for a small but finite amplitude limit, is employed. It is found that the bright envelope solitons are modulationally unstable, whereas the dark ones are stable. It is also observed that the fundamental properties (viz. Modulational instability (MI) growth rate, width and energy concentration of NA waves, etc.) of NA unstable bright envelope solitons are significantly modified by constituent particles number density. The implications of our results obtained from our present investigation in astrophysical compact objects like white dwarfs and neutron stars are briefly discussed., Comment: Submitted to Advances in Space Research

The nonlinear propagation of heavy-ion-acoustic (HIA) waves (HIAWs) in a four component multi-ion plasma (containing inertial heavy negative ions and light positive ions, as well as inertialess nonextensive electrons and positrons) has been theoretically investigated. The nonlinear Schr\"{o}dinger (NLS) equation is derived by employing the reductive perturbation method. It is found that the NLS equation leads to the modulational instability (MI) of HIAWs, and to the formation of HIA rogue waves (HIARWs), which are due to the effects of nonlinearity and dispersion in the propagation of HIAWs. The conditions for MI of HIAWs, and the basic properties of the generated HIARWs are identified. It is observed that the striking features (viz. instability criteria, growth rate of MI, amplitude and width of HIARWs, etc.) of the HIAWs are significantly modified by effects of nonextensivity of electrons and positrons, ratio of light positive ion mass to heavy negative ion mass, ratio of electron number density to light positive ion number density, and ratio of electron temperature to positron temperature, etc. The relevancy of our present investigation to the observations in the space (viz. cometary comae and earth's ionosphere) and laboratory (laser plasma interaction experimental devices) plasmas is pointed out., Comment: Submitted to Chaos

Modulational instability (MI) of ion-acoustic waves (IAWs) has been theoretically investigated in a plasma system which is composed of inertial warm adiabatic ions, isothermal positrons, and two temperature superthermal electrons. A nonlinear Schr\"odinger (NLS) equation is derived by using reductive perturbation method that governs the MI of the IAWs. The numerical analysis of the solution of NLS equation shows the existence of both stable (dark envelope solitons exist) and unstable (bright envelope solitons exist) regimes of IAWs. It is observed that the basic features (viz. stability of the wave profile and MI growth rate) of the IAWs are significantly modified by the superthermal parameter ($\kappa$) and related plasma parameters. The results of our present investigation should be useful for understanding different nonlinear phenomena in both space and laboratory plasmas., Comment: Submitted to Physics of plasmas

We used a permanent-magnet MRI system to estimate the integral and spatially- and/or temporally-resolved void-fraction distributions and flow patterns in gas-liquid two-phase flows. Air was introduced at the bottom of the stagnant liquid column using an accurate and programmable syringe pump. Air flow rates were varied between 1 and 200 ml/min. The cylindrical non-conducting test tube in which two-phase flow was measured was placed in a 2.67 kGauss MRI with MRT spectrometer/imager. Roughly linear relationship has been obtained for the integral void-fraction, obtained by volume-averaging of the spatially-resolved signals, and the air flow rate in upward direction. The time-averaged spatially-resolved void fraction has also been obtained for the quasi-steady flow of air in a stagnant liquid column. No great accuracy is claimed as this was an exploratory proof-of-concept type of experiment. Preliminary results show that MRI a non-invasive and non-intrusive experimental technique can indeed provide a wealth of different qualitative and quantitative data and is especially well suited for averaged transport processes in adiabatic and diabatic multi-phase and/or multi-component flows.

The overall objective of this work is to study nucleate boiling heat transfer under microgravity conditions in such a way that while providing basic knowledge of the phenomena, it also leads to development of simulation models and correlations that can be used as design tools for a wide range of gravity levels. In the study a building block type of approach is used and both pool and low velocity flow boiling are investigated. Starting with experiments using a single bubble, the complexity of the experiments is increased to two or three inline bubbles, to five bubbles placed on a two-dimensional grid. Finally, experiments are conducted where a large number of prescribed cavities nucleate on the heater and when a commercial surface is used. So far experiments have been conducted at earth normal gravity and in the reduced gravity environment of the KC-135 aircraft whereas experiments on the space station are planned. Modeling/complete numerical simulation of the boiling process is an integral part of the total effort. Experiments conducted with single bubbles formed on a nucleation site microfabricated on a polished silicon wafer show that for gravity levels (g) varying from 1.5g(sub e) to 0.01g(sub e), the bubble diameter at departure varies approximately as (g(sub e)/g)(exp 1/2) and the growth period as (g(sub e)/g). When bubbles merge either inline or in a plane, the bubble diameter at departure is found to be smaller than that obtained for a single bubble and shows a weaker dependence on the level of gravity. The possible reason is that as the bubbles merge they create fluid circulation around the bubbles, which in turn induces a lift force that is responsible for the earlier departure of the bubbles. The verification of this proposition is being sought through numerical simulations. There is a merger of two inline, three inline, and several bubbles in a plane in the low gravity environment of the KC-135 aircraft. After merger and before departure, a mushroom type of bubble with several stems attached to the heater surface is clearly evident. Local heat fluxes during growth and departure of a single bubble were also measured. It was found that during most of the growth period of the bubble, generally the wall heat flux decreased with time because of the increased dry area under the bubble. However, the heat flux increased rapidly just prior to departure of the bubble because of the transient conduction into the cold liquid rushing to fill the space vacated by the bubble as the bubble base shrinks. The measured heat fluxes at various radial locations are found to be in qualitative agreement with the numerical predictions. Single bubble studies at earth normal gravity have also been performed on surfaces oriented at different angles to the gravitational acceleration with flow parallel to the surface. It is found that in all cases the bubbles slide along the surface before lift-off from the surface. The lift force generated as a result of the relative motion between the sliding bubbles and the imposed flow is found to play an important role when the normal force due to buoyancy is reduced. An experimental apparatus for the study of the bubble behavior with imposed flow under reduced gravity conditions has been developed and will soon be employed for experiments in the KC-135 aircraft.

Experimental studies of growth and detachment processes of a single bubble and multiple bubbles formed on a heated surface have been conducted in the parabola flights of KC-135 aircraft. Distilled water and PF5060 were used as the test liquids. A micro-fabricated test surface was designed and built. Artificial cavities of diameters 10 microns, 7 microns and 4 microns were made on a thin polished Silicon wafer that was electrically heated by a number of small heating elements on the back side in order to control the surface superheat. Bubble growth period, bubble size and shape from nucleation to departure were measured under subcooled and saturation conditions. Significantly larger bubble departure diameters and bubble growth periods than those at earth normal gravity were observed. Bubble departure diameters as large as 20 mm for water and 6 mm for PF5060 were observed as opposed to about 3 mm for water and less than 1 mm for PF5060 at earth normal gravity respectively. It is found that the bubble departure diameter can be approximately related to the gravity level through the relation D(sub d) proportional 1/g(exp 1/2). For water,the effect of wall superheat and liquid subcooling on bubble departure diameter is found to be small.The growth periods are found to be very sensitive to liquid subcooling at a given wall superheat. However,the preliminary results of single bubble dynamics using PF5060 showed that the departure diameter increases when wall superheat is elevated at the same gravity and subcooling. Growth period of single bubbles in water has been found to vary as t(sub g) proportional g(exp -.93). For water, when the magnitude of horizontal gravitational components was comparable to that of gravity normal to the surface, single bubbles slid along the heater surface and departed with smaller diameter at the same gravity level in the direction normal to the surface. For PF5060, even a very small horizontal gravitational component caused the sliding of bubble along the surface. The numerical simulation has been carried out by solving under the condition of axisymmetry, the mass, momentum, and energy equations for the vapor and the liquid phases. In the model the contribution of micro-layer has been included and instantaneous shape of the evolving vapor-liquid interface is determined from the analysis. Consistent with the experimental results, it is found that effect of reduced gravity is to stretch the growth period and bubble diameter It is found that effect of reduced gravity is to stretch the growth period and bubble diameter at departure. The numerical simulations are in good agreement with the experimental data for both the departure diameters and the growth periods. In the study on dynamics of multiple bubbles, horizontal merger of 2,3 4,and 5 bubbles was observed. It is found that after merger of 2 and 3 bubbles the equivalent diameter of the detached bubble is smaller than that of a single bubble departing at the same gravity level. During and after bubble merger, liquid still fills the space between the vapor stems so as to form mushroom type bubbles. The experimental and numerical studies conducted so far have brought us a step closer to prediction of nucleate boiling heat fluxes under low gravity conditions. Preparations for a space flight are continuing.

The present work is aimed at the experimental studies and numerical modeling of the bubble growth mechanisms of a single bubble attached to a heating surface and of a bubble sliding along an inclined heated plate. Single artificial cavity of 10 microns in diameter was made on the polished Silicon wafer which was electrically heated at the back side in order to control the surface nucleation superheat. Experiments with a sliding bubble were conducted at different inclination angles of the downward facing heated surface for the purpose of studying the effect of magnitude of components of gravity acting parallel to and normal to the heat transfer surface. Information on the bubble shape and size, the bubble induced liquid velocities as well as the surface temperature were obtained using the high speed imaging and hydrogen bubble techniques. Analytical/numerical models were developed to describe the heat transfer through the micro-macro layer underneath and around a bubble formed at a nucleation site. In the micro layer model the capillary and disjoining pressures were included. Evolution of the bubble-liquid interface along with induced liquid motion was modeled. As a follow-up to the studies at normal gravity, experiments are being conducted in the KC-135 aircraft to understand the bubble growth/detachment under low gravity conditions. Experiments have been defined to be performed under long duration of microgravity conditions in the space shuttle. The experiment in the space shuttle will provide bubble growth and detachment data at microgravity and will lead to validation of the nucleate boiling heat transfer model developed from the preceding studies conducted at normal and low gravity (KC-135) conditions.

Experiments have been conducted to investigate the effect of fluid mixing on the depressurization of a large liquid hydrogen storage tank. The test tank is approximately ellipsoidal, having a volume of 4.89 m(exp 3) and an average wall heat flux of 4.2 W/m(exp 2) due to external heat input. A mixer unit was installed near the bottom of the tank to generate an upward directed axial jet flow normal to the liquid-vapor interface. Mixing tests were initiated after achieving thermally stratified conditions in the tank either by the introduction of hydrogen gas into the tank or by self-pressurization due to ambient heat leak through the tank wall. The subcooled liquid jet directed towards the liquid-vapor interface by the mixer induced vapor condensation and caused a reduction in tank pressure. Tests were conducted at two jet submergence depths for jet Reynolds numbers from 80,000 to 495,000 and Richardson numbers from 0.014 to 0.52. Results show that the rate of tank pressure change is controlled by the competing effects of subcooled jet flow and the free convection boundary layer flow due to external tank wall heating. It is shown that existing correlations for mixing time and vapor condensation rate based on small scale tanks may not be applicable to large scale liquid hydrogen systems.

The paper presents experimental observations of explosive boiling from a large (relative to bubble sizes) flat heating surface at very low heat fluxes in microgravity. The explosive boiling is characterized as either a rapid growth of vapor mass over the entire heating surface due to the flashing of superheated liquid or a violent boiling spread following the appearance of single bubbles on the heating surface. Pool boiling data with saturated Freon 113 was obtained in the microgravity environment of the space shuttle. The unique features of the experimental results are the sustainability of high liquid superheat for long periods and the occurrence of explosive boiling at low heat fluxes (0.2 to 1.2 kW/sq m). For a heat flux of 1.0 kW/sq m a wall superheat of 17.9 degrees C was attained in ten minutes of heating. This was followed by an explosive boiling accompanied with a pressure spike and a violent bulk liquid motion. However, at this heat flux the vapor blanketing the heating surface could not be sustained. Stable nucleate boiling continued following the explosive boiling.

The Tank Pressure Control Experiment (TPCE) made its first space flight on STS-43 in 1991. Its objective was to test the effectiveness of low-energy axial jet mixing at controlling pressures in low gravity. The experiment used refrigerant 113 at near-saturation conditions, at an 83 percent fill level, to simulate the fluid dynamics and thermodynamics of cryogenic fluids in future space applications. Results from this flight were reported previously. TPCE was again flown in space on STS-52 in 1992, this time primarily to study boiling and related thermal phenomena which will be reported elsewhere. However additional mixing and pressure control data were obtained from the reflight that supplement the data from the first flight.

Experiments were conducted to investigate the effect of axial jet-induced mixing on the pressure reduction of a thermally stratified liquid hydrogen tank. The tank was nearly cylindrical, having a volume of about 0.144 cu m with 0.559 m in diameter and 0.711 m long. A mixer/pump unit, which had a jet nozzle outlet of 0.0221 m in diameter was located 0.178 m from the tank bottom and was installed inside the tank to generate the axial jet mixing and tank fluid circulation. The liquid fill and jet flow rate ranged from 42 to 85 percent (by volume) and 0.409 to 2.43 cu m/hr, respectively. Mixing tests began with the tank pressure ranging from 187.5 to 238.5 kPa at which the thermal stratification results in 4.9 to 6.2 K liquid sub cooling. The mixing time and transient vapor condensation rate at the liquid-vapor interface are determined. Two mixing time correlations, based on the thermal equilibrium and pressure equilibrium, are developed. Both mixing time correlations are expressed as functions of system and buoyancy parameters and compared well with other experimental data. The steady state condensation rate correlation of Sonin et al. based on steam-water data is modified and expressed as a function of jet subcooling. The limited liquid hydrogen data of the present study shows that the modified steady state condensation rate correlation may be used to predict the transient condensation rate in a mixing process if the instantaneous values of jet sub cooling and turbulence intensity at the interface are employed.

The results are presented of an experimental study of nucleate pool boiling performed in the low gravity environment of the space shuttle. Photographic observations of pool boiling in Freon 113 were obtained during the 'Tank Pressure Control Experiment,' flown on the Space Transportation System, STS-43 in August 1991. Nucleate boiling data from large (relative to bubble size) flat heating surfaces (0.1046 by 0.0742 m) was obtained at very low heat fluxes (0.22 to 1.19 kW/sq m). The system pressure and the bulk liquid subcooling varied in the range of 40 to 60 kPa and 3 to 5 C respectively. Thirty-eight boiling tests, each of 10-min duration for a given heat flux, were conducted. Measurements included the heater power, heater surface temperature, the liquid temperature and the system pressure as functions of heating time. Video data of the first 2 min of heating was recorded for each test. In some tests the video clearly shows the inception of boiling and the growth and departure of bubbles from the surface during the first 2 min of heating. In the absence of video data, the heater temperature variation during heating shows the inception of boiling and stable nucleate boiling. During the stable nucleate boiling, the wall superheat varied between 2.8 to 3.8 C for heat fluxes in the range of 0.95 to 1.19 kW/sq m. The wall superheat at the inception of boiling varied between 2 to 13 C.

Thermal stratification and self-pressurization of partially filled liquid hydrogen (LH2) storage tanks under microgravity condition is studied theoretically. A spherical tank is subjected to a uniform and constant wall heat flux. It is assumed that a vapor bubble is located in the tank center such that the liquid-vapor interface and tank wall form two concentric spheres. This vapor bubble represents an idealized configuration of a wetting fluid in microgravity conditions. Dimensionless mass and energy conservation equations for both vapor and liquid regions are numerically solved. Coordinate transformation is used to capture the interface location which changes due to liquid thermal expansion, vapor compression, and mass transfer at liquid-vapor interface. The effects of tank size, liquid fill level, and wall heat flux on the pressure rise and thermal stratification are studied. Liquid thermal expansion tends to cause vapor condensation and wall heat flux tends to cause liquid evaporation at the interface. The combined effects determine the direction of mass transfer at the interface. Liquid superheat increases with increasing wall heat flux and liquid fill level and approaches an asymptotic value.

The 'Tank Pressure Control Experiment/Thermal Phenomena (TPCE/TP)' is a reflight of the tank pressure control experiment (TPCE), flown on STS-43 in a standard Get-Away Special (GAS) container in August 1991. The TPCE obtained extensive video and digital data of the jet induced mixing process in a partially filled tank in low gravity environments. It also provided limited data on the thermal processes involved. The primary objective of the reflight of TPCE is to investigate experimentally the phenomena of liquid superheating and pool nucleate boiling at very low heat fluxes in a long duration low gravity environment. The findings of this experiment will be of direct relevance to space based subcritical cryogenic fluid system design and operation. Experiment hardware and results from the first TPCE are described in outline and graphic form.

Experimental results are presented for the self pressurization and thermal stratification of a 4.89 cu m liquid hydrogen storage tank subjected to low heat flux (2.0 and 3.5 W/sq m) in normal gravity. The test tank was representative of future spacecraft tankage, having a low mass to volume ratio and high performance multilayer thermal insulation. Tests were performed at fill levels of 29 and 49 pcts. (by volume) and complement previous tests at 83 pct. fill. As the heat flux increases, the pressure rise rate at each fill level exceeds the homogeneous rate by an increasing ratio. Herein, this ratio did not exceed a value of 2. The slowest pressure rise rate was observed for the 49 pct. fill level at both heat fluxes. This result is attributed to the oblate spheroidal tank geometry which introduces the variables of wetted wall area, liquid-vapor interfacial area, and ratio of side wall to bottom heating as a function of fill level or liquid depth. Initial tank thermal conditions were found to affect the initial pressure rise rate. Quasi steady pressure rise rates are independent of starting conditions.

The effect of liquid surface motion on the vapor condensation in a tank mixed by an axial turbulent jet is numerically investigated. The average value (over the interface area) of the root-mean-squared (rms) turbulent velocity at the interface is shown to be linearly increasing with decreasing liquid height and increasing jet diameter for a given tank size. The average rms turbulent velocity is incorporated in Brown et al. (1990) condensation correlation to predict the condensation of vapor on a liquid surface. The results are in good agreement with available condensation data.

Self-pressurization of cryogenic storage tanks due to heat leak through the thermal protection system is examined along with the performance of various pressure control technologies for application in microgravity environments. Methods of pressure control such as fluid mixing, passive thermodynamic venting, and active thermodynamic venting are analyzed using the homogeneous thermodynamic model. Simplified equations suggested may be used to characterize the performance of various pressure control systems and to design space experiments.

The results of a numerical study of the direct contact condensation on a slowly moving horizontal liquid surface are presented. The geometrical configuration and the input conditions used to obtain numerical solutions are representative to those of experiments of Celata et al. The effects of Prandtl number (Pr), inflow Reynolds number, and Richardson number on the condensation rate are investigated. Numerical predictions of condensation rate for laminar flow are in good agreement with experimental data. The effect of buoyancy on the condensation rate is characterized by Richardson number. A correlation based on the numerical solutions is developed to predict the average condensation Nusselt number in terms of Richardson number, Peclet number, and inflow Reynolds number.

Results are presented for an experimental investigation of self-pressurization and thermal stratification of a 4.89 cu m liquid hydrogen (LH2) storage tank subjected to low heat flux (0.35, 2.0, and 3.5 W/sq m) under normal gravity conditions. Tests were performed at fill levels of 83 to 84 percent (by volume). The LH2 tank was representative of future spacecraft tankage, having a low mass-to-volume ratio and high performance multilayer thermal insulation. Results show that the pressure rise rate and thermal stratification increase with increasing heat flux. At the lowest heat flux, the pressure rise rate is comparable to the homogenous rate, while at the highest heat flux, the rate is more than three times the homogeneous rate. It was found that initial conditions have a significant impact on the initial pressure rise rate. The quasi-steady pressure rise rates are nearly independent of the initial condition after an initial transient period has passed.

Experimental results are presented for the self pressurization and thermal stratification of a 4.89 cu m liquid hydrogen storage tank subjected to low heat flux (2.0 and 3.5 W/sq m) in normal gravity. The test tank was representative of future spacecraft tankage, having a low mass to volume ratio and high performance multilayer thermal insulation. Tests were performed at fill levels of 29 and 49 pcts. (by volume) and complement previous tests at 83 pct. fill. As the heat flux increases, the pressure rise rate at each fill level exceeds the homogeneous rate by an increasing ratio. Herein, this ratio did not exceed a value of 2. The slowest pressure rise rate was observed for the 49 pct. fill level at both heat fluxes. This result is attributed to the oblate spheroidal tank geometry which introduces the variables of wetted wall area, liquid-vapor interfacial area, and ratio of side wall to bottom heating as a function of fill level or liquid depth. Initial tank thermal conditions were found to affect the initial pressure rise rate. Quasi steady pressure rise rates are independent of starting conditions.

The steady-state motion exhibited by a large gas slug that is contained in a liquid-filled tube and subjected to a linear temperature variation is analyzed, taking the thermally-induced gradient of the gas-liquid surface tension into account. An expression which characterizes the terminal velocity of the gas slug has been derived.