A transition in the spatially integrated reaction rate of bimolecular reaction‐diffusion systems. Issue 9 (20th September 2015)

• Record Type: Journal Article
• Authors: Arshadi, Masoud; Rajaram, Harihar
• Is Part Of: Water resources research. Volume 51:Issue 9(2015:Sep.)
• Page Start: 7798
• Page End: 7810
• Publication Date: 2015-09-20
• Subjects: Hydrology -- Periodicals; 333.91
• DOI: 10.1002/2015WR017674
• Languages: English
• ISSNs: 0043-1397

Abstract

<abstract abstract-type="main"> <title>Abstract</title> <p>Numerical simulations of diffusion with bimolecular reaction demonstrate a transition in the spatially integrated reaction rate—increasing with time initially, and transitioning to a decrease with time. In previous work, this reaction‐diffusion problem has been analyzed as a Stefan problem involving a distinct moving boundary (reaction front), leading to predictions that front motion scales as <inline-formula><alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgkkzc6mrq" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley:00431397:media:wrcr21688:wrcr21688-math-0001" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msqrt><mml:mi>t</mml:mi></mml:msqrt></mml:mrow></mml:math></alternatives></inline-formula>, and correspondingly the spatially integrated reaction rate decreases as the square root of time <inline-formula><alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgkkzc6mq6" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley:00431397:media:wrcr21688:wrcr21688-math-0002" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mrow><mml:mn>1</mml:mn><mml:mo>/</mml:mo><mml:mrow><mml:msqrt><mml:mi>t</mml:mi></mml:msqrt></mml:mrow></mml:mrow></mml:mrow></mml:math></alternatives></inline-formula>. We present a general nondimensionalization of the problem and a perturbation analysis to show that there is an early time regime where the spatially integrated reaction rate scales as <inline-formula><alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgkkzc6mpp" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley:00431397:media:wrcr21688:wrcr21688-math-0003" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msqrt><mml:mi>t</mml:mi></mml:msqrt></mml:mrow></mml:math></alternatives></inline-formula> rather than <inline-formula><alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgkkzc6mn5" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley:00431397:media:wrcr21688:wrcr21688-math-0004" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mrow><mml:mn>1</mml:mn><mml:mo>/</mml:mo><mml:mrow><mml:msqrt><mml:mi>t</mml:mi></mml:msqrt></mml:mrow></mml:mrow></mml:mrow></mml:math></alternatives></inline-formula>. The duration of this early time regime (where the spatially integrated reaction rate is kinetically rather than diffusion controlled) is shown to depend on the kinetic rate parameters, diffusion coefficients, and initial concentrations of the two species. Numerical simulation results confirm the theoretical estimates of the transition time. We present illustrative calculations in the context of in situ chemical oxidation for remediation of fractured rock systems where contaminants are largely dissolved in the rock matrix. We consider different contaminants of concern (COCs), including TCE, PCE, MTBE, and RDX. While the early time regime is very short lived for TCE, it can persist over months to years for MTBE and RDX, due to slow oxidation kinetics.</p> </abstract>