Analysis of a partial differential equation related to pressure-driven separations of binary liquids

Caraway IV, Willie D

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https://hdl.handle.net/2142/109542 Copy

Description

Title

Analysis of a partial differential equation related to pressure-driven separations of binary liquids

Author(s)

Caraway IV, Willie D

Issue Date

2020-12-11

Director of Research (if dissertation) or Advisor (if thesis)

Pearlstein, Arne J

Department of Study

Mechanical Sci & Engineering

Discipline

Mechanical Engineering

Degree Granting Institution

University of Illinois at Urbana-Champaign

Degree Name

M.S.

Degree Level

Thesis

Keyword(s)

• pressure-driven diffusion
• periodic forcing
• acoustic
• ultrasound
• separation

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.

Graduation Semester

2020-12

Type of Resource

Thesis

Permalink

http://hdl.handle.net/2142/109542

Copyright and License Information

Copyright 2020 Willie D. Caraway IV

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