{"id":{"repo_id":"uiuc","oai_identifier":"oai:www.ideals.illinois.edu:2142/109542"},"canonical_url":"https://search.dev.ndltd.org/etd/uiuc/oai:www.ideals.illinois.edu:2142/109542","repository":{"repo_id":"uiuc","name":"University of Illinois - Urbana-Champaign","base_url":"https://www.ideals.illinois.edu/oai-pmh"},"display":{"title":"Analysis of a partial differential equation related to pressure-driven separations of binary liquids","abstract":"Pressure-driven diffusion, in which a flux of species in a binary or multi-component fluid is driven by a pressure gradient, is known to be an important mechanism for effecting separations in a variety of chemical systems, including the separation of 238UF6 from 235UF6 in gas centrifuges, and in analytical-scale separation of proteins and other macromolecules in their aqueous solutions. In both of those cases, the pressure-gradient is either steady or varies on a time scale which is large compared to other relevant time scales. Here, we investigate the effect of a standing or traveling pressure wave on the composition distribution in a binary liquid. The governing equations involve conservation of mass, momentum, energy, and species, which we simplify to a model involving only a diffusion-like equation for conservation of species, based on neglecting the effects of mass transfer on conservation of momentum and energy and thermoacoustic effects, which allows us to set the mass-averaged velocity to zero and to neglect the Dufour contribution to the energy flux. We focus on understanding the behavior of the resulting nonlinear, time-dependent, and time-periodically forced partial differential equation, of a type that has received relatively little attention. To better understand the importance of nonlinearity to the underlying physics of pressure-driven diffusion in the context of ultrasonically-driven separation, we first linearize our governing equation, and compare the solutions of the original nonlinear equation with the solutions of the linear analogue. A spectral-element technique is employed to obtain long-time solutions of the linear and nonlinear equations as the coefficients are varied. This study analyzes these long-time solutions to better understand the behavior of the solutions and to determine whether ultrasound can affect the separation of a binary liquid mixture by means of pressure-driven diffusion.","abstract_html":"Pressure-driven diffusion, in which a flux of species in a binary or multi-component fluid is driven by a pressure gradient, is known to be an important mechanism for effecting separations in a variety of chemical systems, including the separation of 238UF6 from 235UF6 in gas centrifuges, and in analytical-scale separation of proteins and other macromolecules in their aqueous solutions. In both of those cases, the pressure-gradient is either steady or varies on a time scale which is large compared to other relevant time scales. Here, we investigate the effect of a standing or traveling pressure wave on the composition distribution in a binary liquid. The governing equations involve conservation of mass, momentum, energy, and species, which we simplify to a model involving only a diffusion-like equation for conservation of species, based on neglecting the effects of mass transfer on conservation of momentum and energy and thermoacoustic effects, which allows us to set the mass-averaged velocity to zero and to neglect the Dufour contribution to the energy flux. We focus on understanding the behavior of the resulting nonlinear, time-dependent, and time-periodically forced partial differential equation, of a type that has received relatively little attention. To better understand the importance of nonlinearity to the underlying physics of pressure-driven diffusion in the context of ultrasonically-driven separation, we first linearize our governing equation, and compare the solutions of the original nonlinear equation with the solutions of the linear analogue. A spectral-element technique is employed to obtain long-time solutions of the linear and nonlinear equations as the coefficients are varied. This study analyzes these long-time solutions to better understand the behavior of the solutions and to determine whether ultrasound can affect the separation of a binary liquid mixture by means of pressure-driven diffusion.","abstract_has_math":false,"creators":["Caraway IV, Willie D"],"institution":"University of Illinois at Urbana-Champaign","degree_name":"M.S.","degree_level":"Thesis","degree_discipline":"Mechanical Engineering","degree_department":null,"school":null,"contributors":["Pearlstein, Arne J"],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2021,"date_issued":"2021-03-05T21:42:54Z","date_published":"2021-03-05T21:42:54Z","updated_at":"2026-07-22T22:24:50Z","subjects":["pressure-driven diffusion","periodic forcing","acoustic","ultrasound","separation"],"languages":["en"],"rights":["Copyright 2020 Willie D. 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The governing equations involve conservation of mass, momentum, energy, and species, which we simplify to a model involving only a diffusion-like equation for conservation of species, based on neglecting the effects of mass transfer on conservation of momentum and energy and thermoacoustic effects, which allows us to set the mass-averaged velocity to zero and to neglect the Dufour contribution to the energy flux. We focus on understanding the behavior of the resulting nonlinear, time-dependent, and time-periodically forced partial differential equation, of a type that has received relatively little attention. To better understand the importance of nonlinearity to the underlying physics of pressure-driven diffusion in the context of ultrasonically-driven separation, we first linearize our governing equation, and compare the solutions of the original nonlinear equation with the solutions of the linear analogue. A spectral-element technique is employed to obtain long-time solutions of the linear and nonlinear equations as the coefficients are varied. This study analyzes these long-time solutions to better understand the behavior of the solutions and to determine whether ultrasound can affect the separation of a binary liquid mixture by means of pressure-driven diffusion.","Submission published under a 24 month embargo labeled 'U of I Access', the embargo will last until 2022-12-01","The student, Willie Caraway IV, accepted the attached license on 2020-12-10 at 20:56.","The student, Willie Caraway IV, submitted this Thesis for approval on 2020-12-10 at 21:13.","This Thesis was approved for publication on 2020-12-11 at 11:38.","DSpace SAF Submission Ingestion Package generated from Vireo submission #16119 on 2021-03-04 at 16:20:52","Made available in DSpace on 2021-03-05T21:42:54Z (GMT). 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In both of those cases, the pressure-gradient is either steady or varies on a time scale which is large compared to other relevant time scales. Here, we investigate the effect of a standing or traveling pressure wave on the composition distribution in a binary liquid. The governing equations involve conservation of mass, momentum, energy, and species, which we simplify to a model involving only a diffusion-like equation for conservation of species, based on neglecting the effects of mass transfer on conservation of momentum and energy and thermoacoustic effects, which allows us to set the mass-averaged velocity to zero and to neglect the Dufour contribution to the energy flux. We focus on understanding the behavior of the resulting nonlinear, time-dependent, and time-periodically forced partial differential equation, of a type that has received relatively little attention. To better understand the importance of nonlinearity to the underlying physics of pressure-driven diffusion in the context of ultrasonically-driven separation, we first linearize our governing equation, and compare the solutions of the original nonlinear equation with the solutions of the linear analogue. A spectral-element technique is employed to obtain long-time solutions of the linear and nonlinear equations as the coefficients are varied. This study analyzes these long-time solutions to better understand the behavior of the solutions and to determine whether ultrasound can affect the separation of a binary liquid mixture by means of pressure-driven diffusion.","Submission published under a 24 month embargo labeled 'U of I Access', the embargo will last until 2022-12-01","The student, Willie Caraway IV, accepted the attached license on 2020-12-10 at 20:56.","The student, Willie Caraway IV, submitted this Thesis for approval on 2020-12-10 at 21:13.","This Thesis was approved for publication on 2020-12-11 at 11:38.","DSpace SAF Submission Ingestion Package generated from Vireo submission #16119 on 2021-03-04 at 16:20:52","Made available in DSpace on 2021-03-05T21:42:54Z (GMT). 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