AN ANALYTIC SOLUTION FOR DENSITY DISTRIBUTION IN A PLANETARY EXOSPHERE
Derivation of an analytical expression for the ballistic density distribution in a planetary exosphere
EXOSPHERE↗
SEARCH · Engineering Papers
Results for “analytical solution”
Search indexed NASA NTRS and DOE OSTI research on propulsion, heat transfer, battery materials and energy systems. Follow report and document links to the original sources.
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Analytical solutions to the problem of transient heat transfer in living tissue.
An analytical model of transient heat transfer in living biological tissue is considered. The model includes storage, generation, conduction, and convective transport of heat in the tissue. Solutions for rectangular and cylindrical coordinates are presented and discussed. Transient times for reaching the ?locally fully developed' temperature profile were found to be of the order of 5 to 25 min. These transients are dominated by a geometrical parameters and, to a lesser extent, by a parameter representing the ratio of heat supplied by blood flow to heat conducted in the tissue.
Shitzer, A.↗
On the use of approximate analytical solutions in solving optimum trajectory problems
Method for numerical solutions of nonlinear optimum trajectory problems using approximate analytic method
APPROXIMATION METHOD↗
Analytic Solutions Of Equations For Q-Switched Lasers
Approximate closed-form, analytic equations for characteristics of Q-switched, continuous-wave-pumped laser express relationships among characteristics of output, pumping rate above threshold pumping rate, and rate of repetition of switch. Equations describe performance in terms of peak output power, energy in each pulse, average output power, and duration of each pulse as functions of pulse-repetition rate, actual pump power, threshold pump power, volume occupied by laser electromagnetic mode, and reflectivity of output coupler. Useful for analyzing performances of diode-pumped solid-state lasers.
Wilson, Keith E.↗
A complete analytical solution for the inverse instantaneous kinematics of a spherical-revolute-spherical (7R) redundant manipulator
Using a method based upon resolving joint velocities using reciprocal screw quantities, compact analytical expressions are generated for the inverse solution of the joint rates of a seven revolute (spherical-revolute-spherical) manipulator. The method uses a sequential decomposition of screw coordinates to identify reciprocal screw quantities used in the resolution of a particular joint rate solution, and also to identify a Jacobian null-space basis used for the direct solution of optimal joint rates. The results of the screw decomposition are used to study special configurations of the manipulator, generating expressions for the inverse velocity solution for all non-singular configurations of the manipulator, and identifying singular configurations and their characteristics. Two functions are therefore served: a new general method for the solution of the inverse velocity problem is presented; and complete analytical expressions are derived for the resolution of the joint rates of a seven degree of freedom manipulator useful for telerobotic and industrial robotic application.
Podhorodeski, R. P.↗
Analytical solution for boundary heat fluxes from a radiating rectangular medium
Reference is made to the work of Shah (1979) which demonstrated the possibility of partially integrating the radiative equations analytically to obtain an 'exact' solution. Shah's solution was given as a double integration of the modified Bessel function of order zero. Here, it is shown that the 'exact' solution for a rectangular region radiating to cold black walls can be conveniently derived, and expressed in simple form, by using an integral function, Sn, analogous to the exponential integral function appearing in plane-layer solutions.
Siegel, R.↗
Analytical solution for partial heating on the exterior of the pipe with application to measuring fluid flow rate
An analytical model using Green’s functions for partial external heating of a pipe is developed, which results in an exact mathematical solution for the radial and axial temperature distribution in the pipe wall. Partial heating consists of a constant heat flux function imposed over a small section of the exterior of a pipe and for a limited time duration. The solution comprises steady-state and transient parts, and an algebraic identity is used to decrease the number of summation terms in the slowly-converging steady- state part. Intrinsic verification principles are used to verify the solution. As an example application, this transient solution is applied toward the development of a simple, noninvasive method for in-field measurement of the flow rate in pipes. Furthermore, to simulate this application, a pulse of energy is imposed to the wall of the pipe, and the developed mathematical solution is used to find the flow rate of the fluid inside the pipe. An optimal experiment is designed to find the best measurement location and time.
32 ENERGY CONSERVATION, CONSUMPTION, AND UTILIZATI↗
Additional considerations in analytical solution for time-dependent heat conduction in a three-dimensional multilayer sphere
This work presents an analytical method to solve the heat conduction equation in three dimensions for problems consisting of multilayer concentric spheres. The method can be used to treat time-varying heat conduction problems where the heat source that drives the transient is time-invariant. Equally applicable to all Poisson-type problems with concentric spherical geometry, the method consists of representing the solution as a summation of weighted eigenfunctions. The weights for each eigenfunction are computed algebraically. Previous work has already established the core constituents of the methodology. The current work augments the existing methods by including consideration of nonzero interface resistance between layers and explicit discussion on the boundary condition homogenization required to treat inhomogeneous problems. Also, two demonstration problems are presented. One demonstration problem is based on the method of manufactured solutions and therefore allows for comparison with exact expressions for the solution temperature distribution. The second, more complex, demonstration problem relies on the finite element method for comparisons. The expected convergence behavior is observed for both demonstration problems.
97 - MATHEMATICS AND COMPUTING↗
Analytic solutions of the radial pulsation equation for rotating and magnetic star models
Eddington's (1926) form of wave equation for small-amplitude, radial, adiabatic stellar pulsations of spherically symmetric, gaseous stars is generalized to include the effects of axial rotation and tangled magnetic fields. Equilibrium quantities possessing dimensions are affected, and the relative importance of rotation and magnetism in affecting pulsation characteristics of the models depends on the choices of gamma and the type of model. Solutions are obtained in closed form for adiabatic pulsation periods of fine analytic stellar models, and nonadiabatic stability criteria are determined by means of the one-zone stellar model. Results are discussed for a range of physical parameters such as rotational angular momenta, central condensations, and magnetic energies; and applications are made to the case of classical Cepheids and other variable giant stars.
Stothers, R.↗
Continental Scale Hydrostratigraphy: Comparing Geologically Informed Data Products to Analytical Solutions
Abstract This study synthesizes two different methods for estimating hydraulic conductivity (K) at large scales. We derive analytical approaches that estimate K and apply them to the contiguous United States. We then compare these analytical approaches to three‐dimensional, national gridded K data products and three transmissivity (T) data products developed from publicly available sources. We evaluate these data products using multiple approaches: comparing their statistics qualitatively and quantitatively and with hydrologic model simulations. Some of these datasets were used as inputs for an integrated hydrologic model of the Upper Colorado River Basin and the comparison of the results with observations was used to further evaluate the K data products. Simulated average daily streamflow was compared to daily flow data from 10 USGS stream gages in the domain, and annually averaged simulated groundwater depths are compared to observations from nearly 2000 monitoring wells. We find streamflow predictions from analytically informed simulations to be similar in relative bias and Spearman's rho to the geologically informed simulations. R ‐squared values for groundwater depth predictions are close between the best performing analytically and geologically informed simulations at 0.68 and 0.70 respectively, with RMSE values under 10 m. We also show that the analytical approach derived by this study produces estimates of K that are similar in spatial distribution, standard deviation, mean value, and modeling performance to geologically‐informed estimates. The results of this work are used to inform a follow‐on study that tests additional data‐driven approaches in multiple basins within the contiguous United States.
54 ENVIRONMENTAL SCIENCES↗
An approximate analytic solution of the steady- state thermal design problem of space radiators.
Space radiator steady state thermal design using approximate analytic method
Chapman, A. J.↗
Analytical solution for the wind-driven circulation in a lake containing an island
An analysis was carried out to determine analytically the effect of an island on the wind driven currents in a shallow lake (or sea). A general analysis is developed that can be applied to a large class of lake and island geometries and bottom topographies. Detailed numerical results are obtained for a circular island located eccentrically or concentrically in a circular lake with a logarithmic bottom topography. It is shown that an island can produce volume flow (vertically integrated velocities) gyres that are completely different from those produced by a normal basin without an island. These gyres in the neighborhood of the island will produce different velocity patterns, which include the acceleration of flow near the island shore.
Goldstein, M. E.↗
Construction of approximate analytical solutions to a new class of non-linear oscillator equation
The principle of harmonic balance is invoked in the development of an approximate analytic model for a class of nonlinear oscillators typified by a mass attached to a stretched wire. By assuming that harmonic balance will hold, solutions are devised for a steady state limit cycle and/or limit point motion. A method of slowly varying amplitudes then allows derivation of approximate solutions by determining the form of the exact solutions and substituting into them the lowest order terms of their respective Fourier expansions. The latter technique is actually a generalization of the method proposed by Kryloff and Bogoliuboff (1943).
Mickens, R. E.↗
A simplified analytical solution for thermal response of a one-dimensional, steady state transpiration cooling system in radiative and convective environment
A simplified analytical method for calculation of thermal response within a transpiration-cooled porous heat shield material in an intense radiative-convective heating environment is presented. The essential assumptions of the radiative and convective transfer processes in the heat shield matrix are the two-temperature approximation and the specified radiative-convective heatings of the front surface. Sample calculations for porous silica with CO2 injection are presented for some typical parameters of mass injection rate, porosity, and material thickness. The effect of these parameters on the cooling system is discussed.
Kubota, H.↗
A canopy reflectance model based on an analytical solution to the multiple scattering equation
An approximation to the radiative transfer equation for solar radiation in relatively full, homogeneous plant canopies is presented and solved analytically for solar zenith angles less than 60 deg. The model predicts reflectance at any depth in the canopy and in any direction and may be inverted with bidirectional reflectance measurements. The model was fit to data at two sun angles and two wavebands (visible and NIR) to within the assumed errors on the reflectance data. The calculated albedos are insensitive to achievable measurement errors. Some of the parameter values themselves found by the inversion agree reasonably well with independent measurements, but the uncertainties introduced by the data noise are rather large. However, the agreement is good enough to demonstrate that the model is physically realistic.
Camillo, Peter↗
Analytical solutions with Generalized Impedance Boundary Conditions (GIBC)
The diffraction by a material discontinuity in a thick dielectric/ferrite layer is considered by modeling the layer as a distributed current sheet obeying generalized sheet transition conditions (GSTC's). The sheet currents are then formulated and solved via the standard dual integral equation approach. This yields the diffracted field in terms of unknown constants which underscore the non-uniqueness of the GSTC current sheet representation. The constants are dependent on the geometry and properties of the discontinuity and are determined by enforcing field continuity across the material junction. This requires the field internal to the slab which are determined from the external ones via analytic continuity. Results are given which validate the solution and demonstrate the importance of the constants.
Ricoy, Mark A.↗
A class of analytic solutions for the thermally balanced magnetostatic prominence sheet
A theoretical study is presented for the nonlinear interplay between magnetostatic equilibrium and energy balance in a Kippenhahn-Schlueter type solar prominence sheet. A class of theoretical models is presented, expressed in closed analytic forms, thus facilitating the direct illustration of the nonlinear physical properties. The model couples the equilibrium between magnetic field, plasma pressure, and weight on the one hand, with the balance between a rho-squared T radiative loss, a rho wave heating (where rho equals plasma density, and T equals plasma temperature), and thermal conduction channeled along magnetic field lines on the other. The steady solutions are divided into three classes, and are characterized by the total wave heating in the prominence sheet which is greater than, equal to, or less than the total radiative loss. The compaction of the plasma along the field lines, under its own weight, and the energy transport effects determine which of the three basic behaviors obtains in a particular situation. A discussion is presented of the implications of the steady solutions for the formation of prominences.
Low, B. C.↗
Numerical and Analytical Solutions of Hypersonic Interactions Involving Surface Property Discontinuities
The local viscous-inviscid interaction field generated by a wall temperature jump on a flat plate in supersonic flow and on the windside of a Reusable Launch Vehicle in hypersonic flow is studied in detail by both a Navier-Stokes numerical code and an analytical triple-deck model. Treatment of the rapid heat transfer changes both upstream and downstream of the jump is included. Closed form relationships derived from the triple-deck theory are presented. The analytically predicted pressure and heating variations including upstream influence are found to be in generally good agreement with the Computational Fluid Dynamic (CFD) predictions. These analyses not only clarify the interactive physics involved but also are useful in preliminary design of thermal protection systems and as an insertable module to improve CFD code efficiency when applied to such small-scale interaction problems. The analyses only require conditions at the wall and boundary-layer edge which are easily extracted from a baseline, constant wall temperature, CFD solution.
Gnoffo, Peter A.↗