and 0o as follows. Assume that initially r = r 0 , r = 0 and 0 = 0. Then, by Equation [46] 1 GM , . ^ 7? -= -77 + 5 o sin 0o r 0 h 2 r 0[50]and, from the first derivative of Equation [46] 0 = s 0 (l -r) cos 0o [51] It then follows that the initial position is at pericenter with 0o = TT/2 [52] So = -rr 1 53 h 2 r 0 r 0 hence, using Equation [48] so ~ v/ro [54] which in conjunction with Equation [48] implies that rj is approximately equal to the eccentricity. To get the maximum excursion of the radial distance r, it is only necessary to take the difference between r max and r m i n as obtained from Equation [46] and write n = 2r 0 s 0 = 2 77 [55]in agreement with Dobrowolski.In the case of circumferential thrust, the maximum radial distance in each cycle does not remain stationary. Its rate of growth is given by Equation [26] for the nearly circular orbit, i.e., Ar a /r a ~ 4 7rrj. This result agrees directly with that of Perkins (5) as can be seen by substituting A0 = 2 w in Equation [29] of Perkins 7 paper. ConclusionBriefly, application of the nonlinear techniques of Kryloff and Bogoliuboff has been shown to provide concise solutions to the problems of small radial thrust and small circumferential thrust previously discussed in the literature, and to the problem of intermittent thrust, which has not had previously published solutions. It is hoped that this work will inspire more extensive application of the method. Powered trajectory calculations have been carried out for missions from Earth to Venus, Mars,Jupiter and return. The procedure used was based upon an optimization which minimizes total energy requirements and maximizes payload. Data are presented which relate payload fraction, total mission time, and minimum specific power requirements. It is shown that propulsion plant weight-to-exhaust power ratios of 122 lb/kw and 22 lb/kw are adequate for missions ranging in difficulty from one-way trips to Mars with 10 per cent payload fraction to round trips to Mars with 30 per cent payload, respectively. In both cases the transfer is assumed to be between low altitude satellite orbits about the planets in question. D URING the last year a study of electrical propulsion has been carried out at the Lawrence Radiation Laboratory. In order to lay a proper foundation for such a study it was necessary to determine the relationship among mission requirements (characteristic velocity), propulsion system specific power, mission time, and payload fraction. Consequently, some theoretical studies of a simple nature were made, supplemented by some powered trajectory calculations on the IBM-704. The results, while certainly not optimal, do provide the desired results in an approximate form with minimum expenditure of effort and computer time. 28 ARS JOURNAL Downloaded by UNIVERSITY OF CALIFORNIA -DAVIS on February 5, 2015 | http://arc.aiaa.org |

This report describes in detail the technical findings of the DOE Award entitled "Development, Verification, and Validation of Multiphase Models for Polydisperse Flows." The focus was on high-velocity, gas-solid flows with a range of particle sizes. A complete mathematical model was developed based on first principles and incorporated into MFIX. The solid-phase description took two forms: the Kinetic Theory of Granular Flows (KTGF) and Discrete Quadrature Method of Moments (DQMOM). The gas-solid drag law for polydisperse flows was developed over a range of flow conditions using Discrete Numerical Simulations (DNS). These models were verified via examination of a range of limiting cases and comparison with Discrete Element Method (DEM) data. Validation took the form of comparison with both DEM and experimental data. Experiments were conducted in three separate circulating fluidized beds (CFB's), with emphasis on the riser section. Measurements included bulk quantities like pressure drop and elutriation, as well as axial and radial measurements of bubble characteristics, cluster characteristics, solids flux, and differential pressure drops (axial only). Monodisperse systems were compared to their binary and continuous particle size distribution (PSD) counterparts. The continuous distributions examined included Gaussian, lognormal, and NETLprovided data for a coal gasifier.FINAL TECHNICAL 4 DE-FC26-07NT43098 5 EXECUTIVE SUMMARYThis report is the final technical report for the award DE-FC26-07NT43098 entitled "Development, Verification, and Validation of Multiphase Models for Polydisperse Flows." Below is a summary of work completed under the grant for each of the major goals. Goal I: Continuum Theory for Solid PhaseTwo separate and complementary continuum theories were derived to model polydisperse solids: the kinetic theory of granular flow (KTGF) and the discrete quadrature method of moments (DQMOM). Both theories are based on first principles with no adjustable parameters. The polydisperse KTGF and DQMOM were then encoded in MFIX, and underwent a wide range of verification testing to ensure, to the best possible extent, that no coding errors were present. These verification tests included both granular and gas-solid flows, including cases where an analytical solution and/or DEM (discrete element method) data and/or simple test cases. For the case of KTGF, another set of constitutive equations were derived which rigorously incorporated the gas phase for a simplified case of monodisperse systems at low Reynolds numbers. This derivation used the acceleration model developed in Task 2 from direct numerical simulations (DNS) in the starting kinetic equation, and indicated the effect of the fluid phase on the resulting constitutive relations. Goal II: Improved Gas-Particle Drag Laws -effect of particle size distributionIn this portion of the effort, two types of direct numerical simulations (DNS) were used to extract the drag force experienced by particles in a polydisperse suspension. These two methods, na...

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This document contains Secret-Restricted Data relating to civilian applications of atomic energy. LEGAL NOTICE This report was prepared as an account of Government sponsored work. Neither the United States, nor the Commission, nor any person acting on behalf of the Conamission: A. Makes any warranty or representation, express or implied, with respect to the accuracy, completeness, or usefulness of the information contained in this report, or that the use of any information, apparatus, method, or process disclosed in this report may not infringe privately owned rights; or B. Assumes any liabilities with respect to the use of, or for damages resulting from the use of any information, apparatus, method, or process disclosed in this report. As used in the above, "person acting on behdlf of the Commission" includes any employee or contractor of the Commission to the extent that such employee or contractor prepares, handles or distributes, or provides access to, any information pursuant to his employment or contract with the Commission.

An approximate solution is derived for the pressure ratio in high-speed subsonic flow through constant-cross-section circular tubes with friction and heating. The results are compared with those obtained from an accurate numerical calculation. The gas used is air. A wide range of variables such as temperature ratio, exit Mach number (up to 0.90), tube length-to-diameter ratio, and shape of the power distribution function is examined. The derived expression is found to be quite accurate in all cases.