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

1. Introduction

Solute transport in a flowing fluid is an important topic of research in modern times and it has wide applications in diverse fields such as biology [1-3], environment fluid mechanics [4-10], and chromatography [11], etc. Among the many applications in environmental processes such as atmosphere, estuaries, and wetlands, it is important to find the dispersion phenomena for pollutant transport in river, which has important applications for environmental risk assessment [12], where actual prediction of reactive solute is of under concern.

Theoretically as well as experimentally, the longitudinal dispersion of an inert (i.e., non-adsorbing) solute in an incompressible fluid flow through a circular tube was first studied by G. I. Taylor [13]. He showed that velocity difference in the fluid layer across the tube enhanced the overall diffusivity of the solute. The effective diffusion is called as Taylor dispersion. He found the dispersion coefficient ($D_{\text{T}}$) as, $D_{\text{T}}=r^2{\langle u\rangle }^2/48D$, where $r$ is the tube radius, $\langle u\rangle$ the mean velocity of the fluid, and $D$ the molecular diffusivity. Later, Aris [14] improved the Taylor's expression by using the method of moments as $D_{\text{T}}=D+r^2{\langle u\rangle }^2/48D$.

The idea of Taylor on the solute dispersion extensively studied by many researchers. Numerical attempts include the study of Takahashi et al. [15], Stokes and Barton [16], Ekambara and Joshi [17], Lau and Ng [18], Mondal et al. [19], Dhar et al. [20], etc. Besides researchers also adopted different analytical methods to analyse the solute dispersion that includes the Aris's moment method [14, 21], mean concentration expansion method by Gill [22-26], multi-scale method of homogenization [27-31], and delay-diffusion method [32].

Effect of partitioning between phases on axial dispersion behaviour was initially analysed by Westhaver [33]. Since the pioneering work of Taylor [13], the study of solute dispersion has been extended in various ways to account for partitioning effects [34]. Shankar and Lenhoff [11] studied the gas chromatography in a coated tube within a retentive layer. Phillips and Kaye [35] studied the effect of reversible phase exchange on the solute transport in a two-phase system. Ng and Yip [36] presented a theory for the transport in open-channel flow of a chemical species under the influence of kinetic sorptive exchange between phases that are dissolved in water and sorbed onto suspended sediments. Solute transport in oscillatory Couette flow through two parallel plate with sorptive boundaries has been analysed by Ng and Bai [37]. They showed that if the phase exchange is sufficiently kinetic, the dispersion coefficient can be much higher than its inert counterpart. In his works, Ng [38, 39] has studied the effect of reversible phase exchange kinetics as well as wall absorption on dispersion coefficients in oscillatory flows. They have derived the higher-order dispersion coefficients, which arise from the combined effects of phase exchange kinetics and bed absorption. They showed that the higher-order dispersion coefficients can be comparable with their leading order counterparts in the presence of kinetic sorptive exchange with the bed.

Many literature are available on solute transport, where efforts was only made for analysing the mean concentration distribution [40-46] rather than two-dimensional transverse concentration distribution. Some of the environmental or industrial processes, the two-dimensional concentration distribution is equally important with longitudinal mean concentration distribution. Earlier, researchers believed that cross-sectional concentration variation was negligible for the Taylor dispersion stage of the transport process and mean concentration was enough to demonstrate the solute dispersion process. But, the study of Wu and Chen [47, 48] reveals that transverse variation is significant for a longer period of time. They derived the analytical expression for the two-dimensional transverse concentration distribution in steady flows using Mei's homogenization technique [49]. They also analysed the transverse uniformity of concentration cloud for laminar steady flows and proposed a time scale $\tau \sim 10$, which was an estimate of time scale for the two-dimensional concentration to reach transverse uniformity. In our study, Mei's homogenization technique is used to explore the two-dimensional concentration distribution for an open channel flow with reversible phase exchange kinetics between the channel bed (immobile phase) and fluid phase (mobile phase). Analytical expressions are derived for dispersion coefficient, mean and two-dimensional concentration distributions. This study also discussed the effects of phase exchange kinetics on the concentration variation over the channel cross-section.

The main objectives of this work include a multi-scale analysis for the concentration distribution in an open channel flow with sorptive channel bed, finding the analytical expressions for dispersion coefficient and both mean and two-dimensional real concentrations up to second-order approximation, observe the effects of phase exchange kinetics on dispersion coefficient and two-dimensional concentration distribution, discuss the effect of phase exchange kinetics on uniformity in two-dimensional concentration over the channel cross-section.

2. Problem formulation

We study a laminar, unidirectional, fully developed, open channel flow, where the width between free surface and channel bed is $h$ as shown in Fig. 1. The $x^{\prime }$-axis is taken as in the direction of the flow and $y^{\prime }$-axis is vertical to the flow direction. The solute undergoes an reversible phase exchange with the channel bed and disperse throughout the channel. The flow velocity $u^{\prime }(y^{\prime })$ is taken as

where $u_0$ is the velocity of the flow at the free surface.

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

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Let us release the solute of mass $Q$ at $x^{\prime }=0$. It is assumed that the solute undergoes a reversible phase exchange between mobile and immobile phases. In the phase where solute flows with the flowing fluid is termed as mobile or fluid phase and the phase in which the solute retains at the channel bed is called immobile or stationary phase. In a reversible reaction, the solute may chemically react to the channel boundaries. The solute is assumed to be completely miscible in the fluid. During the flow, a reversible exchange between different phases of the chemical species may take place across the interface. The reversible phase exchange consists of two physical processes, adsorption and desorption. In adsorption, a part of the solute is retained at the boundary and in desorption, the retained solute come back to flow again. If the solute undergoes a phase exchange with the lower channel boundary, then the rate of change of concentration of the stationary phase is balanced by the diffusive mass concentration flux in reversible reaction and is given by

where $C^{\prime }$ is the concentration of the solute in the mobile/fluid phase, ${C^{\prime }}_{\text{S}}$ is the concentration of the solute in the immobile/stationary phase. Phase exchange will take place in either forward or backward direction, which can be expressed by a first-order linear kinetics [36] as

where $k_{\text{f}}$ and $k_{\text{b}}$ are the forward and backward rate constants for the sorption reaction, respectively. At the equilibrium state, the immobile and mobile phase concentrations will have the constant ratio

where ${\alpha }^{\prime }$ is the retention parameter or partition coefficient. From Eqs. (3) and (4) first-order kinetics reversible reaction at the channel bed can be expressed as

where $k(=k_{\text{b}})$ is the rate constant of reversible/backward reaction.

The solute transport problem can be written as

where $t^{\prime }$ is the time, $D$ is the molecular diffusivity of the solute.

The initial and boundary conditions are taken as

and

where $\delta (\cdot )$ is the Dirac delta function. Physically, the Dirac delta function is a way to represent a unit mass of highly concentrated solute in an infinitely small space [6]. In this study, the solute mass $Q$ is injected into a long, narrow channel and is distributed uniformly in the lateral direction ($y^{\prime }$-direction) of the channel and with negligible dimension in $x^{\prime }$. The initial condition $\frac{Q}{h}\delta (\frac{x^{\prime }}{h})$ (or $Q\delta (x^{\prime })$) represents a mass $Q$ which is highly concentrated into a very small space at $x^{\prime }=0$.

3. Homogenization

To apply multi-scale method at first we have to choose different scales for length and time. For length we have chosen two different scales as $h$ and $L$, which represents the channel depth and characteristic longitudinal cloud length respectively. For time we have chosen three distinct scales $T_0$, $T_1$, and $T_2$ for lateral diffusion, flow advection and longitudinal diffusion respectively. It is assumed that the reversible phase exchange will achieve equilibrium within the time scale $T_0$. The time scales can be written as

Their ratios are

where $\epsilon =h/L$ ($\ll 1$) is the dimensionless parameter used for asymptotic analysis.

Now we introduce the dimensionless parameters as

where $Pe$ is the Peclet number, $Da$ the Damkohler number, $\alpha$ the dimensionless retention parameter and the angle brackets denote section average defined as

As per the timescale we have introduced the order of $Pe$ is of $O(1)$. But the range of $Pe$ is not limited to be $O(1)$, which is given in Eq. (52).

Then governing equation and boundary conditions can be written in dimensionless form as

The velocity profile is written in dimensionless form as

The homogenization technique of Mei et al. [49] is used to carry out the asymptotic analysis. The mobile and immobile phase concentrations $C$ and $C_{\text{S}}$ can be expanded as

and

Let us introduce the fast, medium and slow time variables, based on three time scales $T_0$, $T_1$, and $T_2$ as

According to the chain rule we can write,

Substituting Eqs. (16)-(18) into Eqs. (12)-(14) results in

For leading order ($O(1)$), Eqs. (19)-(21) give

The general solution of Eq. (22) becomes

The depth of the channel is considered to be narrow enough so that transverse diffusion takes place almost immediately (with in the first time scale) after the release. It is observed from the equation that the numerically value of series part decreases exponentially with fast time variable, and it become insignificant for $t_1\ge O(1)$ [50]. Considering above facts, dependency of $C_0$ on $y$ and $t_0$ in a larger time may be neglected. So, the solution can be considered as

So $C_0$ is independent of $t_0$ and $y$. Equating first and last term of Eq. (24), we have

For first order ($O(\epsilon )$), Eqs. (19)-(21) give

The time scale $t_0$ is much larger compared to $t_1$, the partial derivative terms w.r.t. $t_0$ can be neglected [7]. So, Eq. (28) becomes

Taking a section average of Eq. (31) w.r.t. $y$ and using Eqs. (29) and (30), we have

where $R(=1+\alpha )$ is called retardation factor.

Subtracting Eq. (32) from Eq. (31), we get

These equation suggest the following substitutions

where $N(y)$ and $N_{\text{S}}$ are to be obtained by matching terms associated with $\frac{\partial C_0}{\partial x}$. The function $N(y)$ to be governed by

with the following conditions

and

For second order ($O({\epsilon }^2)$), Eqs. (19)-(21) give

The time scale $t_0$ is much larger compared to $t_1$ and $t_2$, the partial derivative terms w.r.t. $t_0$ can be neglected. Equation (40) becomes

Taking a section average of Eq. (43) w.r.t. $y$ and using Eqs. (41) and (42), we get

The second, third, and fourth terms of L.H.S of Eq. (44) can be found using Eqs. (32), (34), (35), and (39) as

and

Using Eqs. (45)-(47) in Eq. (44) we get

Multiplying Eqs. (32) and (48) by $\epsilon$ and ${\epsilon }^2$ respectively and adding together results in

Rewriting Eq. (49) in original time variable as

where

The range of validity of $Pe$ can not be restricted to $O(1)$, as in the macroscopic Eq. (50), the three different time variables $t_0$, $t_1$, $t_2$, are turned into a single time variable $t$. The range of $Pe$, where Eq. (50) is applicable, can be obtained by recalling the assumptions that Taylor dispersion dominates molecular diffusion and the former dominated by the advection [51, 52]. The range of validity for $Pe$ as follows:

Without phase exchange effect the above range becomes, $7.23\ll Pe\ll \frac{52.5}{\epsilon }$, which can be large for small $\epsilon$ (for long and narrow channels).

Solving the two system of Eqs. (36)-(39), we have

and using Eqs. (15) and (53) one can find

Let us use the transformations $\tau =t$ and $\eta =\frac{x^{\prime }}{h}-Pe\frac{\langle u\rangle }{R}t$ to convert Eq. (50) into

Using the initial and boundary conditions given in Eqs. (6) and (9), the solution of Eq. (56) can be found as

Eliminating $\frac{\partial C_0}{\partial t_2}$ from Eqs. (43) and (44) one can get

After substitute the required terms of Eq. (58) one can get

and from Eq. (42) one can get

In the above equations one can neglect the first term associated with $\frac{{\partial }^2{C}_0}{\partial x^2}$ as $Pe$ is very large for practical purpose (as mentioned in Eq. (52)). In his analysis also, Taylor [13] has neglected the molecular diffusion term which is basically a good assumption for $Pe>100$ [22]. In their analysis Wu and Chen [48] mentioned that for practical purpose $Pe$ can be of order $O{(10}^4)$. Hence one can take the substitution for the equation as

where $X$ and $X_{\text{S}}$ are to be found by matching the terms associated with $\frac{{\partial }^2{C}_0}{\partial x^2}$. The problem for finding $X$ can be written as

and

On solving Eqs. (63)-(66) one can get

and

In a similar manner we can obtain the terms $C_n$ ($n=\mathrm{3,4,5,}\dots$). Solutions for these terms are omitted here as their expressions are too long and cumbersome. Without phase exchange effect ($\alpha =0$ or $R=1$), $N(y)$ and $X(y)$ becomes

which are same to those obtained in the work of Wu and Chen [48].

4. Results and discussion

In this section we have discussed about Taylor dispersivity, distributions of mean concentration and of two-dimensional transverse concentration. Taylor dispersivity and the concentrations depend on the parameters $\alpha$ and $Da$. While the parameter $\alpha$ is the normalized partition coefficient or retention parameter, which is in other word, the ratio of solute distributed between the stationary and fluid phase. The parameter $Da$, which is the ratio of the reversible reaction rate to the molecular diffusion rate, represents the significance of the kinetics of the phase exchange. Smaller and larger $Da$ implies slower and faster phase exchange kinetics respectively.

The first term on R.H.S of the effective dispersion coefficient $D_{\text{T}}$ (Eq. (51)) represents the effects of longitudinal diffusion. If we want to confine ourselves to study the behavior of the solute dispersion due to lateral diffusion and advection, we can omit the first term $1/R$. Hence, dispersion coefficient becomes

The analytical expression for mean concentration can be found from Eqs. (16) and (57) in a new $Pe$ independent system {$\eta /Pe$,$\langle C\rangle Pe$} as

From Eq. (16), the two-dimensional transverse concentration distribution up to second order is obtained as

As we have not come across any literature that can be used directly to compare the two-dimensional concentration profile of the present problem, so results of the present study for $\alpha =0$ (i.e., without phase exchange) are compared with the results of Wu and Chen [48], and Barik and Dalal [53] without considering bed absorption effects in Fig. 2. From the figure, it can be seen that for long time evolution of the solute cloud, our analytical results gives the good agreement with the previous results though with a difference in the peak and tail ends at early times. The deviation is noticeable (at peak) with our earlier study (Barik and Dalal [53]) as in our earlier study we have derived the concentration distribution up to third order, whereas in the present study it is up to second order. As the deviation is insignificant for $t\ge 1$, so we can conclude that calculation up to second order is enough for long time evolution pattern of the solute concentration. Also following the comparison with the result of Wu and Chen [48], for early stages we observe more deviation (both peak and downstream tail), as in their analysis they have introduced some correction functions for the developed terms, that contribute to the modification of the mean as well as two-dimensional concentration distributions in early times. Sometimes, for a long channel, in order to know the solute concentration at locations far from the solute source, one does not need to know the initial transient behavior of the solute cloud. This study shows that for long time evolution, the solution obtained by perturbation technique gives excellent approximation for the concentration distribution. Our analytical study suggests that results may not be fully reliable at the initial period of time, where scalar transport is dominated by advection rather than Taylor dispersion.

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

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4.1 Dispersivity

The explicit form of the dispersion coefficient can be derived from Eqs. (54), (55), and (71) as

which is identical with the steady state dispersion coefficient obtained by Ng [39] in his study. In the inert case (when $\alpha =0$ or $R=1$), the dispersion coefficient becomes

which is same as that obtained by Wu and Chen [48].

Effects of partition coefficient ($\alpha$) and kinetics ($Da$) on dispersion coefficient are shown in Fig. 3. Figure shows that dispersion coefficient increases with the increase of $\alpha$ up to a certain value and then decreases for further increase of $\alpha$. The occurrence of this maximum value may be due to bed retention that allows solute to retain at the channel bed and after a while (depending on $Da$) it is released back into the flow at the upstream side, by that time the maximum concentration cloud moves from its original position. This creates a long tail of the solute cloud and consequently increases solute dispersion through the channel, as also discussed in Ng and Rudraiah [54]. Further increase of $\alpha$ beyond a critical value will decrease the dispersion coefficient due to the higher retardation factor, which drag down the ability of solute spreading along the concentration gradient. It is also observed that slow phase exchange kinetics ($Da\ll 1$) gives rise to larger values of dispersion coefficients than the fast kinetics ($Da>1$) as solute retains for a longer time for former case and slowly releases into the flow that enhances the solute dispersion. Such a phenomenon is reported in Ng and Yip [36]. Also it is noticed that increase in dispersion coefficient is very sharp when channel bed just turned from perfectly inert to slightly retentive. These observations can be verified mathematically.

3.  Variations of dispersion coefficient with $\alpha$ for different Da.

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From Eq. (74), the gradient of $D_{\text{T}}^{*}$ at $\alpha =0$ is

which is always positive and is large for small $Da$. This fact indicates that the curve has a very steep gradient for $Da\ll 1$, which causes sharp increase in dispersion coefficient. The dispersion coefficient can reach at maximum value at $\alpha ={\alpha }_m$, where

The values of maximum $D_{\text{T}}^{*}/P{e}^2$ for different $Da$ are listed in Table 1. The dispersion coefficients reach their maximum values when the phase exchange coefficient or retention parameter is in the range of 0.5-1.5, as limiting value of ${\alpha }_m$ as $Da$ tends to 0 is 0.5. Further increase in $\alpha$ will cause dispersion coefficient to decrease as reversible phase exchange process retard the advection speed $1/R$, which is always smaller than the initial velocity. It can be observed from Table 1 that, for $Da=0.1$, the maximum dispersion coefficient value is more than 81 times the value of the coefficient (2/105) for the inert case ($\alpha =0$). Such a huge impact of phase exchange kinetics on dispersion coefficient is extremely remarkable.

${contentEle.labelText}.  Table 1 Values of maximum dispersion coefficients at maximum $\alpha$ for different Da

$Da$	Max{$D_{\text{T}}^{*}/P{e}^2$}	${\alpha }_m$	$\frac{\text{Max}\{D_{\text{T}}^{*}/P{e}^2\}}{\{D_{\text{T}}^{*}/P{e}^2{\left\}\right|}_{\alpha =0}}$
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
$\cdots$	$\cdots$	$\cdots$	$\cdots$
$\infty$	0.0876	1.4557	4.5990

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4.2 Mean concentration distributions

From Eq. (72), the explicit form of the mean concentration distribution can be written as

The present study is not intended to focus on the longitudinal skewness of concentration cloud at initial times. The expression for mean concentration becomes exactly similar to that of Taylor dispersion model. The expression of mean (Eq. (78)) can give an excellent result after some initial time when the longitudinal skewness completely dies out.

Longitudinal distributions of mean concentration in the channel are shown in Fig. 4. Figure 4a shows the distribution curves of mean for different values of$\alpha$. Figure shows that peak value of the mean concentration distribution decreases with the increase of $\alpha$ up to a maximum value ${\alpha }_m\approx 0.6588$ (graph not shown) as dispersion coefficient increases in that range. Further increase of $\alpha$ causes the peak value to increase. As large $\alpha$ delays the solute transport by slowing down the transport velocity. The result is reported in Table 2. Variations of longitudinal distribution of mean concentration with$Da$ can be observed from Fig. 4b. Figure shows that for a fixed value of $\alpha$, peak value of the mean concentration increases as $Da$ increases. For $Da\gg 1$, the reversible reaction rate is much faster than the diffusion rate. Solute moves back quickly to the fluid from the immobile phase (i.e., the channel bed) and increases the solute concentration in the mobile phase. The result is reverse in the case of $Da\ll 1$, as molecular diffusion rate dominates the reversible reaction rate and makes the mean concentration distribution more blunt and flatter.

4.  Longitudinal distribution of mean concentration at$\tau =1$, a $Da=1$, b $\alpha =1$.

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${contentEle.labelText}.  Table 2 Peak values of mean concentration distributions for different values of$\alpha$ and $Da$ at $\tau =1$

$Da$	${\alpha }_m$	$\alpha =0.001$	$\alpha =0.01$	$\alpha =0.1$	$\alpha =0.3$	$\alpha =0.5$	$\alpha =1$	$\alpha =3$	$\alpha =5$	$\alpha =10$
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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4.3 Two-dimensional concentration distribution

The two-dimensional concentration distributions for mobile and immobile phases can be written explicitly up to second order as

Concentration contour plots are given in Figs. 5 and 6, which demonstrate the effects of reversible phase exchange. Initially, the solute is introduced at $\eta =0$ and then spread throughout the channel. Here the coordinate system is moving with velocity $Pe/R$ along the $x$-axis to fix the centroid of the solute concentration at the origin. From Fig. 5 it can be seen that how reversible phase exchange affects the solute dispersion compared to its inert counterpart. When the channel bed just changes from inert to slightly retentive (say, $\alpha =0.01$) and $Da=0.1$, concentration distribution becomes more dispersed due to slow kinetics, which increase the solute dispersion in the flow. Further increase of $Da$ increase the solute concentration in the channel as the faster rate of exchange between the channel bed and fluid phase increases solute concentration in the fluid phase. It is possible that for a slightly retentive channel bed ($\alpha \ll 1$), the dispersion process is more akin to the inert bed case subject to a fast phase exchange ($Da\gg 1$) between mobile and immobile phases. The maximum concentration of the solute after $\tau =1$ for $\alpha =0$ is 2.1367 and that for $\alpha =0.01$ is 0.8865, 1.7237, and 2.0307 for $Da=$ 0.1, 1, and 10, respectively. From the figure it is also observed that for large $\alpha$ (say, $\alpha =25$), concentration is higher than that of the flow with inert channel bed owing to higher retention capacity of the channel bed. As bed retention increases, retardation factor reduces the ability of the solute to move along the concentration gradient, which also reported in Ramon et al. [55]. The maximum concentration of the solute after $\tau =1$ for $\alpha =25$ is 3.0753, 5.1054, and 5.4903 for $Da=$ 0.1, 1, and 10, respectively. Plots in Fig. 6 show the two-dimensional concentration distributions for different combinations of $\tau$ and $\alpha$. It is observed that for small bed retention, higher concentration zone appears in the downstream free surface due to flow advection and with the increase in retention parameter, the solute concentration decreases as the channel bed holds some fraction of solute temporally which allows to enhances the solute dispersion. Further increase of $\alpha$ will retard the flow advection, consequently causes the higher concentration zone in both down-stream and upstream sides. It is also noticeable that concentration is higher in the upstream channel bed than downstream free surface with large value of $\alpha$. The maximum solute concentration of upstream and downstream sides at $\tau =0.5$ are 3.8724 and 3.0550 respectively for $\alpha =10$.

5.  Mobile phase concentration contours at $\tau =1$ (horizontal coordinate: $\eta /Pe$, vertical coordinate: $y$).

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6.  Mobile phase concentration contours for $Da=1$ (horizontal coordinate: $\eta /Pe$, vertical coordinate: $y$).

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Variation of immobile phase concentration distribution is depicted in Fig. 7. As a comparison, mean concentration in the mobile phase is shown in the figure. Figure 7 reveals that increase in retention parameter always increases the immobile flow concentration as retention capacity of the channel bed increases with the increase in $\alpha$. It is also observed from the figure that for large value of $\alpha$ ($\alpha \ge 1$), immobile phase concentration can be higher compared to the mean concentration in the fluid phase.

7.  Mean concentration with immobile phase concentration for different$\alpha$ at $\tau =6$, where $Da=1$, a small values of $\alpha$, b large values of $\alpha$.

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In this study an indicator $\mathcal{R}$ is adopted [47, 56] to discuss the uniformity of solute concentration as

which measures the two-dimensional concentration variation rate over the channel cross-section against the center of solute concentration.

The axial distribution of the two-dimensional concentration variation rate is presented in Fig. 8. From Fig. 8a it can be seen that the variation rates are exactly similar around the vertical axis for the inert case and with the increase of time the maximum variation rates moves away from the solute centroid. Effects of retention parameter on concentration variation rate are plotted in Fig. 8b-d. Figures show that for $\alpha =0.1$, initially variation rate higher at some downstream locations. As higher concentration appears due to advection near the free surface and bed retention takes some solute from the flow at the channel bed. This leads to increase in concentration variation rate in the downstream and consequently the non-uniformity increases. In the upstream, due to slow advection initially concentration is high at channel bed and it reduces when phase exchange takes place. Hence the concentration variation rates decrease. When bed retention increases variation rate increases in both upstream and downstream side. But maximum variation rate appears in upstream bed due to slow advection and strong retention effect. Influence of $Da$ on two-dimensional concentration variation rate are displayed in Fig. 8e. It is shown from the figure that as $Da$ increases, maximum variation rates on both upstream and downstream increases. When $Da\ll 1$, molecular diffusion rate dominates reversible reaction rate and consequently reduce the maximum variation rate. But result is reverse for the case of $Da\gg 1$, where the reversible reaction rate dominates over diffusion and increases the solute concentration in the flow and also the maximum variation rate. With the increase of $Da$, maximum variation rate in upstream increases faster than that of in the downstream and for fast kinetics ($Da=10$), the maximum variation rates in both sides are almost equal to each other (variation rates for $Da=10$ in upstream and downstream sides are 54.86$\%$ and 54.88 $\%$, respectively). Maximum variation rates for different combinations of $\alpha$ and $Da$ are reported in Table 3. It can be noted that for slow kinetics ($Da=0.1$), subject to small retention capacity of the channel bed, variation rates are smaller than that in the inert case ($\alpha =0$). So, it can be concluded from the table that for small values of $\alpha$ and $Da$, i.e., for slow kinetics with small retentive channel bed, solute concentration distribution will become uniform faster.

8.  Axial distribution of the two-dimensional concentration variation rate, a$\alpha =0$, b $\alpha =\mathrm{0.1,}Da=1$, c $\alpha =\mathrm{1,}Da=1$, d $\alpha =\mathrm{10,}Da=1$, e $\alpha =\mathrm{1,}\tau =1$.

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