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

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.

Name thesis:degree_name

M.S.

Level thesis:degree_level

Thesis

Discipline thesis:degree_discipline

Mechanical Engineering

Grantor

University of Illinois at Urbana-Champaign

Year dc:date

2021

Author dc:creator

• Caraway IV, Willie D

Contributors dc:contributor

• Pearlstein, Arne J

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

Statement dc:rights

• Copyright 2020 Willie D. Caraway IV

Language dc:language

en

Handle dc:identifier

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

OAI identifier oai:identifier

oai:www.ideals.illinois.edu:2142/109542