doi:10.1007/s10409-021-09037-y

Analytical solution for concentration distribution in an open channel flow with phase exchange kinetics

doi: 10.1007/s10409-021-09037-y

Barik Swarup^{1, 2,*
, ,},
Dalal D. C.¹

1.
Department of Mathematics, Indian Institute of Technology Guwahati, Guwahati, 781039, India;
2.
Department of Mathematics, SRM Institute of Science and Technology, Chennai, 603203, India;

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Corresponding author: Barik Swarup, E-mail address: swarupb@srmist.edu.in (Swarup Barik)

摘要

摘要: 压痕试验是一种局部试验技术, 因此材料尺寸效应和局部不均匀性的作用非常重要. 尺寸无关材料的非均匀性影响已有研究. 本文采用尺寸相关应变梯度理论详细研究了材料尺寸效应和非均匀性(压头尖端附近的夹杂物)对压痕硬度的影响. 研究发现, 与基于常规塑性理论的尺寸无关材料相比, 考虑材料尺寸效应时, 非均质材料中的浅硬夹杂物更显著地提高了材料的压痕硬度. 这种硬化效应被认为与载荷的升高和大变形的局部约束有关. 特别是当涉及尺寸效应且夹杂物模量影响的过渡范围非常窄时, 材料的不均匀性影响主要来自于结构的不均匀性, 而非夹杂物模量本身. 初始夹杂物深度大于其直径后, 不均匀性的影响变得可以忽略. 压头与夹杂物的水平偏移对非均匀压痕的影响也非常敏感. 本文重点研究了微压痕和纳米压痕中的标度关系, 并对微观材料中的非均匀性的影响进行了研究和补充.
Abstract: This study is about an analytical attempt that explores the two-dimensional concentration distribution of a solute in an open channel flow. The solute undergoes reversible sorption at the channel bed. The method of multiple scales is used to find the two-dimensional concentration distribution, which is important for modern day application in industry, environmental risk assessment, etc. Study deduces an analytic expression of two-dimensional concentration distribution for an open channel flow with sorptive channel bed. Effects of retention parameter, Damkohler number on the solute dispersion are also discussed in this paper. Results reveal that slow or strong kinetics (small value of Damkohler number) increases solute dispersion. It is also observed that for slow phase exchange kinetics between bulk flow and small retentive channel bed, solute concentration distribution will uniform faster than their inert counterpart.
- Multi-scale method /
- Dispersion coefficient /
- Two-dimensional concentration distribution /
- Phase exchange kinetics

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1. Schematic diagram for the open channel flow with phase exchange kinetics.

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2. Comparison between present and previous results of two-dimensional concentration distributions at different times, a $τ = 0.5$ , b $τ = 0.75$ , c $τ = 1$ , d $τ = 2$ (where $y = 0$ and $α = 0$ ).

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3. Variations of dispersion coefficient with $α$ for different Da.

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4. Longitudinal distribution of mean concentration at $τ = 1$ , a $D a = 1$ , b $α = 1$ .

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5. Mobile phase concentration contours at $τ = 1$ (horizontal coordinate: $η / P e$ , vertical coordinate: $y$ ).

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6. Mobile phase concentration contours for $D a = 1$ (horizontal coordinate: $η / P e$ , vertical coordinate: $y$ ).

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7. Mean concentration with immobile phase concentration for different $α$ at $τ = 6$ , where $D a = 1$ , a small values of $α$ , b large values of $α$ .

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8. Axial distribution of the two-dimensional concentration variation rate, a $α = 0$ , b $α = 0.1, D a = 1$ , c $α = 1, D a = 1$ , d $α = 10, D a = 1$ , e $α = 1, τ = 1$ .

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. Table 1 Values of maximum dispersion coefficients at maximum $α$ for different Da

$D a$	Max{ $D_{T}^{*} / P e^{2}$ }	$α_{m}$	$\frac{Max {D_{T}^{} / P e^{2}}}{{D_{T}^{} / P e^{2} {} \|}_{α = 0}}$
0.1	1.5490	0.5154	81.3225
1	0.2194	0.6588	11.5185
10	0.0981	1.2386	5.1502
100	0.0886	1.4317	4.6515
$\dots$	$\dots$	$\dots$	$\dots$
$\infty$	0.0876	1.4557	4.5990

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. Table 2 Peak values of mean concentration distributions for different values of $α$ and $D a$ at $τ = 1$

$D a$	$α_{m}$	$α = 0.001$	$α = 0.01$	$α = 0.1$	$α = 0.3$	$α = 0.5$	$α = 1$	$α = 3$	$α = 5$	$α = 10$
0.1	0.5154	1.6528	0.8238	0.3190	0.2369	0.2267	0.2442	0.3820	0.5223	0.8395
1	0.6588	1.9868	1.6313	0.8666	0.6497	0.6081	0.6163	0.8027	0.9767	1.3253
10	1.2386	2.0326	1.9389	1.4404	1.1011	0.9861	0.9056	0.9892	1.1276	1.4368

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. Table 3 Peak values of $R$ with increasing $α$ , where $τ = 1$

$D a$	$α = 0$	$α = 0.001$	$α = 0.01$	$α = 0.1$	$α = 0.3$	$α = 0.5$	$α = 1$	$α = 3$	$α = 5$	$α = 10$
0.1	40.32	35.94	30.97	30.42	30.69	31.05	32.22	39.82	49.62	72.45
1	40.32	39.74	36.55	33.64	35.33	37.47	43.12	62.78	76.52	98.33
10	40.32	40.31	40.22	41.26	44.94	48.18	54.88	72.71	84.41	103.34
15	40.32	40.33	40.41	42.08	45.94	49.16	55.72	73.20	84.77	103.55

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Analytical solution for concentration distribution in an open channel flow with phase exchange kinetics

doi: 10.1007/s10409-021-09037-y

Corresponding author: Barik Swarup, E-mail address: swarupb@srmist.edu.in (Swarup Barik)

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Analytical solution for concentration distribution in an open channel flow with phase exchange kinetics

doi: 10.1007/s10409-021-09037-y

Corresponding author: Barik Swarup, E-mail address: swarupb@srmist.edu.in (Swarup Barik)

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