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Title: [Kinetic theory and boundary conditions for highly inelastic spheres]. Quarterly progress report, April 1, 1993--June 30, 1993

Abstract

In this quarter, a kinetic theory was employed to set up the boundary value problem for steady, fully developed, gravity-driven flows of identical, smooth, highly inelastic spheres down bumpy inclines. The solid fraction, mean velocity, and components of the full second moment of fluctuation velocity were treated as mean fields. In addition to the balance equations for mass and momentum, the balance of the full second moment of fluctuation velocity was treated as an equation that must be satisfied by the mean fields. However, in order to simplify the resulting boundary value problem, fluxes of second moments in its isotropic piece only were retained. The constitutive relations for the stresses and collisional source of second moment depend explicitly on the second moment of fluctuation velocity, and the constitutive relation for the energy flux depends on gradients of granular temperature, solid fraction, and components of the second moment. The boundary conditions require that the flows are free of stress and energy flux at their tops, and that momentum and energy are balanced at the bumpy base. The details of the boundary value problem are provided. In the next quarter, a solution procedure will be developed, and it will be employed tomore » obtain sample numerical solutions to the boundary value problem described here.« less

Authors:
Publication Date:
Research Org.:
Worcester Polytechnic Inst., MA (United States)
Sponsoring Org.:
USDOE, Washington, DC (United States)
OSTI Identifier:
10117962
Report Number(s):
DOE/PC/90185-T10
ON: DE94005772; TRN: 94:001582
DOE Contract Number:
AC22-91PC90185
Resource Type:
Technical Report
Resource Relation:
Other Information: PBD: [1993]
Country of Publication:
United States
Language:
English
Subject:
99 GENERAL AND MISCELLANEOUS//MATHEMATICS, COMPUTING, AND INFORMATION SCIENCE; 42 ENGINEERING; 01 COAL, LIGNITE, AND PEAT; SPHERES; TRANSPORT; GRAVITATIONAL INTERACTIONS; SOLIDS FLOW; BOUNDARY-VALUE PROBLEMS; KINETIC EQUATIONS; MATHEMATICAL MODELS; MASS TRANSFER; COAL; PROGRESS REPORT; 990200; 420400; 013000; MATHEMATICS AND COMPUTERS; HEAT TRANSFER AND FLUID FLOW; TRANSPORT, HANDLING, AND STORAGE

Citation Formats

Richman, M. [Kinetic theory and boundary conditions for highly inelastic spheres]. Quarterly progress report, April 1, 1993--June 30, 1993. United States: N. p., 1993. Web. doi:10.2172/10117962.
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Richman, M. [Kinetic theory and boundary conditions for highly inelastic spheres]. Quarterly progress report, April 1, 1993--June 30, 1993. United States. doi:10.2172/10117962.
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Richman, M. Fri . "[Kinetic theory and boundary conditions for highly inelastic spheres]. Quarterly progress report, April 1, 1993--June 30, 1993". United States. doi:10.2172/10117962. https://www.osti.gov/servlets/purl/10117962.
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@article{osti_10117962,
title = {[Kinetic theory and boundary conditions for highly inelastic spheres]. Quarterly progress report, April 1, 1993--June 30, 1993},
author = {Richman, M.},
abstractNote = {In this quarter, a kinetic theory was employed to set up the boundary value problem for steady, fully developed, gravity-driven flows of identical, smooth, highly inelastic spheres down bumpy inclines. The solid fraction, mean velocity, and components of the full second moment of fluctuation velocity were treated as mean fields. In addition to the balance equations for mass and momentum, the balance of the full second moment of fluctuation velocity was treated as an equation that must be satisfied by the mean fields. However, in order to simplify the resulting boundary value problem, fluxes of second moments in its isotropic piece only were retained. The constitutive relations for the stresses and collisional source of second moment depend explicitly on the second moment of fluctuation velocity, and the constitutive relation for the energy flux depends on gradients of granular temperature, solid fraction, and components of the second moment. The boundary conditions require that the flows are free of stress and energy flux at their tops, and that momentum and energy are balanced at the bumpy base. The details of the boundary value problem are provided. In the next quarter, a solution procedure will be developed, and it will be employed to obtain sample numerical solutions to the boundary value problem described here.},
doi = {10.2172/10117962},
journal = {},
number = ,
volume = ,
place = {United States},
year = {Fri Dec 31 00:00:00 EST 1993},
month = {Fri Dec 31 00:00:00 EST 1993}
}
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Technical Report:
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DOI:  10.2172/10117962
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  • In the last quarter, we focused on steady, fully developed, gravity-driven flows of identical, smooth spheres down bumpy inclines, with flow depths greater than one particle diameter, solid fraction profiles everywhere less than .65, and dimensionless granular temperature T(y={beta}) at the top between 0 and 20. In this quarter, we continue the same parameter study by choosing other sets of values of r,{delta}, e, e{sub w}, and {phi}for which we know that at least one solution of the type described above may be maintained, and determine the complete range of T(y={beta}) (within O

  • In this quarter, we extended our study of the effects of isotropic boundary vibrations to steady, gravity driven, inclined granular flows. These flows are more complex than those considered last quarter because of the presence of slip and mean velocity gradients at the boundary. Consequently, it was first necessary to modify the boundary conditions derived by Richman (1992) to account for corrections to the flow particle velocity distribution function from velocity gradients. In what follows we only summarize the results obtained.

  • In this quarter a solution procedure is developed and numerical solutions are obtained for steady, fully developed, gravity-driven flows of identical, smooth, highly inelastic spheres down bumpy inclines. The boundary value problem for these flows was described in detail in the previous quarterly progress report (4/1/93--6/30/93). Solutions to the problem had not been obtained. The solution procedure is described in detail and numerical solutions are obtained that are most easily parameterized in terms of the granular temperature at the tops of the flows. To each such temperature there corresponds a value of mass hold-up and mass flow rate. For fixedmore » coefficients of restitution, boundary bumpiness, and angle of inclination, the variation of mass flow rate with mass hold-up is calculated, and for a prescribed value of mass hold-up the profiles of solid fraction, mean velocity, normal components of the second moment, and normal stresses are calculated. For completeness the author begins by summarizing the boundary value problem, which was developed in greater detail in the previous quarterly report (4/1/93--6/30/93).« less

  • In this quarter, we conducted a study to determine a range of parameters over whichsteady, fully developed, gravity driven granular flows of identical, smooth, inelastic spheres down bumpy inclines could be maintained. The appropriate boundary value problem has been described in detail in our Quarterly Progress Report from April 1, 1993 to June 30, 1993. The numerical solution procedure that we employed to solve the boundary value problem has been described in detail in our Quarterly Progress Report from July 1, 1993 to September 30, 1993. In what follows, we describe the parameters relevant to the inclined flows investigated, themore » range of these parameters that we included in our study, and the resulting ranges over which steady, fully developed flows could be maintained.« less

  • In the last quarter, we focused on steady, fully developed, gravity-driven flows of identical, smooth spheres down bumpy inclines, with flow depths greater than one particle diameter, solid fraction profiles everywhere less than .65, and dimensionless granular temperature T(y={beta}) at the top between 0 and 10. For prescribed boundary bumpiness (r=l and {delta}=.414), restitution coefficients of the boundary (e{sub w}=.5) and the flow particles (e=.5), and angles of inclination ({phi}0=25.5{degrees}, 27.505{degrees}, 28{degrees} and 34{degrees}), we determined what appeared to be complete range of T(y={beta}) (within 0