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TRANSCRIPT
Tracking time-dependent production in
electrokinetic Y-microreactors
Hamed Helisaz a, Masoud Babaei b, Arman Sadeghi c,1
a Center of Excellence in Energy Conversion (CEEC), School of Mechanical Engineering, Sharif University
of Technology, Tehran 11155-9567, Iran
b School of Chemical Engineering & Analytical Science, University of Manchester, Manchester M13
9PL, United Kingdom
c Department of Mechanical Engineering, University of Kurdistan, Sanandaj 66177-15175, Iran
Abstract
We perform a theoretical study on the transient reaction-diffusion kinetics in an electrokinetic Y-shaped
microreactor. The flow is assumed to be both steady and fully developed. The governing equations are
solved in dimensionless form utilizing a 3D finite-volume based numerical algorithm, assuming a second-
order irreversible reaction between the components. Analytical solutions are also obtained for cross-stream
diffusion without reaction under a uniform velocity distribution. It is shown that the well-known butterfly-
shaped form of the production concentration profile is not immediately created and it is established only
after the system is sufficiently close to its steady-state. Furthermore, the inclination of the concentration
peak toward the component of lower diffusivity or inlet concentration is less significant at the earlier stages
of the production. Finally, it is demonstrated that the short-term influence of the parameters affecting the
advection of mass on the total efficiency, defined as the ratio of the total production to the amount of the
limiting reactant within the device, is quite the opposite of that at the steady-state. That is, whereas
increasing each of the dimensionless Debye-Hückel parameter, the pressure-driven velocity to
1 Corresponding authorE-mail addresses: [email protected] (H. Helisaz), [email protected] (M. Babaei), [email protected] (A. Sadeghi)
electroosmotic velocity ratio, and the Péclet number leads to larger short-term efficiencies, the opposite is
true at the steady-state.
Keywords: Electroosmotic flow, Lab-on-a-chip, Microreactor, Numerical modeling, Analytical solution
1. Introduction
The micro accelerometer developed in 1979 at Stanford University marked the beginning of numerous
attempts to replace the conventional large laboratories with their micro counterparts referred to later on as
lab-on-a-chip (LOC) devices [1]. As it is implied by the name, LOCs are miniaturized laboratory platforms
which are machined on a wafer of glass or silicon and are able to perform one or several laboratory
functions quickly and simultaneously. Less consumption of normally expensive samples and reagents, large
surface-to-volume ratio, and high controllability are among the main advantages of LOC devices attracting
the attention of many researchers.
LOCs are usually fabricated for a certain purpose. Therefore different devices consist of different
components. However microreactors are necessary in more or less all LOC devices due to prevalence of
mixing/reaction processes in most of laboratory functions. In their most basic design, microreactors take a
T- or Y-shaped form [2] in which fluids are entered by two arms and introduced into a chamber wherein
components are mixed together and a chemical reaction occurs among them. This form has been placed as a
base for developing other designs, such as parallel and serial lamination [3, 4], which enhance the mixing
and reaction processes. Nevertheless, considering these designs is indeed accompanied by not only an
increasing fabrication cost, but also difficulties in integrating microreactors into LOC platforms; as a result,
much research attention is still given to typical T/Y-shaped microreactors. The pioneering studies on
microscale reactors include the work of Salimi-Moosavi [5] and that of Kamholz et al. [6], both conducted
in 1990s, in which microreactors are utilized for synthesizing an azo dye and calculating a reaction rate
constant, respectively. The succeeding research works on microreactors focused on more complicated
applications, including diagnostic platforms [7-10], drug development [11-14], and nanoparticle synthesis
[15-18].
Different methods have been proposed for creating fluid flow in microstructures [19] amongst which
electroosmotic flow (EOF) is used in the growing number of microfluidic applications. This is largely
thanks to EOF’s favorable characteristics such as obviating the need for moving parts and creating pulse-
free plug flows [20]. In electroosmosis, the interaction of the electrolyte with polarized dielectric walls
form an accumulation of ions of the same electric charge adjacent to the walls. Applying an external
electric field then makes the ions move toward the electrode of opposite charge, thereby creating a net flow
upon the interaction of the ions and the solvent molecules. Since late 1990s, when the first studies on EOF-
based microsystems were performed [21, 22], electroosmosis has been considered in many microsystems
either as the only driving force [23-28] or accompanied by pressure force [29-34]. In these works, however,
more attention was paid to micromixers rather than microreactors. Few available researches on
microreactors include the experimental works of Wilson and McCreedy [35] and Watts et al. [36] whose
results proved more efficient reaction by adopting EOF-based microreactors instead of large-scale reactors.
More recently, Sharma et al. [24] and Yousefian and Saidi [37] presented theoretical
results of species transport in EOF-based catalytic microreactors.
Both mixing and reaction processes require time to complete, especially in passive microchannels
wherein flows are laminar and so mixing occurs via the typically slow mechanism of molecular diffusion
[38]. Given the transient time in current commercial micromixers/microreactors [39], it is not surprising to
see the transient time of a microreactor lasting for tens of seconds. Therefore, thorough understanding of
diffusion-reaction mechanism in microreactors is not developed unless both its transient and steady-state
behaviors are investigated. In our previous work [32], for the first time, a comprehensive parametric study
on steady-state responses of diffusion-reaction mechanism was performed by 3D finite-volume based
numerical modeling of an EOF-based Y-microreactor. Since, to our best knowledge, no time-dependent
study has yet been conducted on diffusion-reaction mechanism in T/Y-shaped microreactors, the present
study aims to fill this gap in the literature by extending our recent study to transient conditions. We
investigate the effects of various operational parameters on the performance of the microreactor before
reaching the steady-state. It is attempted to broaden the scope of research by eliminating the simplifying
assumptions made in the preceding studies. For example, the flow field, which is created by the
simultaneous action of electroosmotic and pressure forces, is evaluated without imposing any additional
approximation like Helmholtz-Smoluchowski approximation. Additionally, no predefined values are
assumed for the ratios between diffusivities and inlet concentrations of the components. We believe the
present study together with our previous work [32] pave the way for a better design of the future
microreactors.
2. Problem formulation
Consideration is given toward mass transport in an electrokinetic-based Y-shaped microreactor from the
onset of reaction to steady-state. Fig. 1 provides top and front views of the microreactor under
consideration which is comprised of two arms coinciding together at the origin of the coordinate system.
Two Newtonian flows with similar flow rates and physicochemical properties are introduced by these arms
under the combined action of electroosmotic and pressure forces. The fluids, one containing reactant A at a
number concentration of c0 , A and the other containing reactant B at a concentration of c0 , B, run side-by-
side in a chamber wherein they mix together and a second-order non-reversible chemical reaction
(A+B→C) occurs among them. The solute-liquid solutions are considered dilute enough to allow
neglecting the influences of the diffusion-reaction process on the flow field which is assumed to be steady.
Thanks to the intrinsic low Reynolds number of microflows, the flow in the main channel can be
considered hydrodynamically fully developed in the presence of a uniform wall zeta potential. The last
premise is that the Debye length is much smaller than the minimum channel dimension so that there is no
EDL overlap. It should be also pointed out that, taking advantage of symmetry, we solve the flow domain
only for the channel’s first quarter and we solve concentration domain for the channel upper half to reduce
the computational costs.
2.1. Electrical potential distribution
For evaluation of the electroosmotic velocity, it is first required to predict the volumetric body force which
is dependent upon the electrostatic potential. The total electrical potential φ is related to the net ionic charge
in the channel ρe through the Poisson equation, given as
∇2 φ=−ρe
ε(1)
in which ε denotes the fluid permittivity. The electric potential in our system includes the first part
generated by the external electric field E x and the second part induced by electric charges residing on the
channel walls. Reminding that the electric field is uniform, the first part is a linear function of only x and it
can be calculated by considering the reference value of ϕ0 atx=0. The second part can be considered to be
a function of y andz, denoted here by ψ ( y , z ), since the zeta potential throughout the channel walls is
uniform and the flow is fully developed. Therefore, the following formula for the electric potential is
obtained
φ ( x , y , z )=( ϕ0−x E x )+ψ ( y , z ) (2)
Invoking the Boltzmann distribution, the net ionic charge can be related to the ionic concentration at neutral
conditionsn∞, proton chargee, Boltzmann constantk B, and absolute temperature T as
ρe=−2e Z n∞ sinh ( e Z ψk BT ) (3)
Note that this equation is obtained by assuming a symmetric electrolyte of valence Z [40]. Substituting Eqs.
(2) and (3) into Eq. (1), it may be rewritten as
∂2ψ∂ y2 + ∂2ψ
∂ z2 =2 eZ n∞
εsinh ( e Z ψ
kB T ) (4)
To generalize, Eq. (4) is made dimensionless by defining y¿= y / H ,z¿=z /H , ψ¿=e Z ψ /k B T , and
Κ=H / λD in which λD=(2 n∞ e2 Z2/ε kB T )−1 /2 indicates the Debye length. We will then have
∂2ψ¿
∂ y¿2 + ∂2 ψ¿
∂ z¿2 =Κ2 sinhψ¿ (5)
The solution to ψ¿ must satisfy the following four boundary conditions
∂ψ¿
∂ y¿|y¿=0
= ∂ ψ¿
∂ z¿ |z ¿=0
=0 ,ψ¿ ¿y ¿=1=ψ¿¿z ¿=α=ζ ¿(6)
wherein α=W /H is the channel aspect ratio and ζ ¿=e Z ζ /k B T is the dimensionless zeta potential.
2.2. Velocity distribution
For evaluation of the velocity field, the mathematical representation of the momentum conservation law
should be solved. For a laminar incompressible flow of Newtonian fluids with constant physical properties,
the general form of the momentum equation reads
ρ D uDt
=−∇ p+μ∇2u+ f (7)
wherein t , ρ, p, μ, and u represent the time, density, pressure, dynamic viscosity, and velocity vector,
respectively. Moreover, f is the body force vector reflecting the interaction of the electric field with the free
ions within the solution. Reminding that the flow is assumed steady and fully developed, the left hand side
of Eq. (7) vanishes. The momentum conservation equation, therefore, can be solved for velocity since both
the applied pressure gradient ∇ p and the body force f are known; particularly in the streamwise direction,
∂ p/∂ x=d p /d x which is known a priori and f x= ρe Ex which can be calculated with the aid of the
electric potential solution. Accordingly, the momentum conservation equation in the axial direction reduces
to
0=−d pd x
+μ( ∂2 ux
∂ y2 +∂2 ux
∂ z2 )+ ρe Ex (8)
wherein ux is the axial velocity. This equation is made dimensionless by scaling the velocity asu¿=ux /uHS
with uHS=−εζ Ex /μ being the Helmholtz-Smoluchowski velocity. Since there is a pressure-driven
contribution to the fluid velocity, the parameter Γ=uPD /uHS is required to quantify the ratio of the
pressure-driven velocity scale uPD=−H2 (d p /d x ) /2 μ to the electroosmotic velocity scale uHS. Based on
the newly defined quantities, Eq. (8) modifies into
∂2u¿
∂ y¿2 +∂2 u¿
∂ z¿2 =−2 Γ−Κ2
ζ ¿ sinhψ¿ (9)
The four necessary boundary conditions for Eq. (9), which are provided by symmetry and no-slip
conditions, take the forms
∂u¿
∂ y¿|y ¿=0
=∂ u¿
∂ z¿|z¿=0
=0 , u¿¿ y¿=1=u¿¿z ¿=α=0 (10)
2.3. Concentration distribution
Generally, advection, diffusion, and reaction are mechanisms responsible for species transport. Here, we
intend to study the effects of these mechanisms over time and so we need to consider the transient form of
reaction-advection-diffusion equation for each species, that is
∂ c A
∂t+u
∂ c A
∂ x=DA ∇2 c A−k cA cB (11)
∂ c A
∂t+u
∂ cB
∂ x=DB ∇2 cB−k c A cB (12)
∂ c A
∂t+u
∂ cC
∂ x=DC ∇2 cC+k c A cB (13)
In the above equations, c andD represent, respectively, the number concentration and diffusion coefficient
of the species whose type is shown as subscript. The terms including the chemical rate constant k are
reaction terms written based on the law of mass action [41]. Eqs. (11) to (13) can be scaled by introducing
new dimensionless parameters as x¿= x /H , β=DA / DB, γ=DA / DC, c¿=c /c0 , A, and t ¿=t DA/ H2.
Substituting the already defined parameters into Eqs. (11) to (13), there appear two well-known
dimensionless quantities namely the Péclet numberPe=uHS H / DA and Damköhler number
Da=Hk c0 , A/uHS. The dimensionless forms of Eqs. (11) to (13), hence, become
∂ c A¿
∂t ¿ +Pe u¿ ∂ c A¿
∂ x¿ =( ∂2 cA¿
∂ x¿2 +∂2 c A
¿
∂ y¿2 +∂2 c A
¿
∂ z¿2 )−PeDac A¿ cB
¿ (14)
∂ cB¿
∂ t¿+ βPeu¿ ∂c B
¿
∂ x¿ =( ∂2 cB¿
∂ x¿2 +∂2c B
¿
∂ y¿2 +∂2 cB
¿
∂ z¿2 )−βPeDac A¿ cB
¿ (15)
∂ cC¿
∂ t ¿ +γPeu¿ ∂ cC¿
∂ x¿ =( ∂2 cC¿
∂ x¿2 +∂2 cC
¿
∂ y¿2 +∂2cC
¿
∂ z¿2 )+γPeDa cA¿ cB
¿ (16)
Eqs. (14) to (16) require the following boundary and initial conditions to be solved
∂2c A ,B ,C¿
∂ x¿2 |x¿=l
=∂c A ,B ,C
¿
∂ y¿ |y¿=0,1
=∂ c A ,B ,C
¿
∂ z¿ |z¿=± α
=0
(17)c A
¿|x¿=0={ 1 z¿≥ 0¿0 z¿<0
, cB¿|x ¿=0={ 0 z¿>0
¿1/ω z¿≤0, cC
¿|x¿=0=0
c A, B , C¿ |t¿=0=0
wherein l=L/ H and ω=c0 , A/c0 , B.
2.4. Numerical method
The governing equations in dimensionless form are solved by adopting the finite volume method in which
the first step is generating the grid structure. Considering the problem physics, we expect higher electric
potential and velocity gradients at the vicinity of the walls; the opposite is true for solute concentration as
chemical reaction produces intense concentration gradients at the fluid-fluid interface. Therefore, the grid
structure for solving Eqs. (5) and (9) should be different from that of Eqs. (14) to (16), not to mention that
for the latter the grids should also be clustered near the channel entrance so as to capture high gradients
therein. To avoid the computational difficulties of dealing with non-uniform grid structures, we transform
the current coordinatesx¿, y¿, andz¿ into new ones named x, y, and z, along which grid nodes are uniformly
distributed, by applying the same transformations as those given in our previous work [32]. Adopting this
new coordinate system, Eqs. (5), (9), and (14) to (16) are converted to
Q12 ( y ) ∂2ψ¿
∂ y2 +Q2 ( y ) ∂ ψ¿
∂ y+Q3
2 ( z ) ∂2ψ¿
∂ z2 +Q4 ( z ) ∂ ψ¿
∂ z=Κ 2sinh ψ¿
(18)
Q12 ( y ) ∂2u¿
∂ y2 +Q2 ( y ) ∂u¿
∂ y+Q3
2 ( z ) ∂2 u¿
∂ z2 +Q4 ( z ) ∂ u¿
∂ z=−2 Γ− Κ2
ζ ¿ sinh ψ¿(19)
∂ c A¿
∂t ¿ +Pe u¿Q5 ( x )∂ c A
¿
∂ x=[Q 5
2 ( x )∂2 c A
¿
∂ x2 +Q6 ( x )∂ cA
¿
∂ x+
∂2 c A¿
∂ y2 +Q72 ( z )
∂2c A¿
∂ z2 +Q8 ( z)∂ cA
¿
∂ z ]−PeDac A¿ c B
¿ (20)
∂ cB¿
∂ t¿+ βPeu¿Q5 ( x )
∂ c B¿
∂ x=[Q5
2 ( x )∂2c B
¿
∂ x2 +Q6 ( x )∂ cB
¿
∂ x+
∂2c B¿
∂ y2 +Q72 ( z )
∂2 cB¿
∂ z2 +Q8 ( z )∂ cB
¿
∂ z ]−βPeDa cA¿ cB
¿ (21)
∂ cC¿
∂ t ¿ +γPeu¿Q5 ( x )∂ cC
¿
∂ x=[Q5
2 ( x )∂2 cC
¿
∂ x2 +Q6 ( x )∂ cC
¿
∂ x+
∂2 cC¿
∂ y2 +Q72 ( z )
∂2 cC¿
∂ z2 +Q8 ( z )∂ cC
¿
∂ z ]+γPeDa c A¿ cB
¿ (22)
in which the functions Q1 ,.. , 8 are given in the Appendix. Now, it is time to integrate the equations over each
volume segment and then place the variables at cell centers by practicing the power-law scheme; the
transient terms are also handled by integrating Eqs. (20) to (22) implicitly over the time. This process
introduces a set of algebraic equations each of the governing equations. The algebraic equations
corresponding to Eqs. (18) and (19) have already been introduced in our previous work [32] and those
pertinent to Eqs. (20) to (22) are as follows
aP , A c A,i , j , k¿ −aW , A c A ,i−1 , j , k
¿ −aE , A c A , i+1 , j ,k¿ −aB , A c A ,i , j−1, k
¿ −aT , A c A ,i , j+1, k¿ −aS , A c A, i, j , k−1
¿ −aN , A c A,i , j , k+1¿ =SA(23)
aP , B cB, i , j , k¿ −aW , B cB,i−1 , j ,k
¿ −aE , B cB, i+1 , j ,k¿ −aB ,B cB ,i , j−1 ,k
¿ −aT , B cB , i , j+1 ,k¿ −aS ,B c B,i , j ,k−1
¿ −aN , B cB ,i , j , k+1¿ =SB(24)
aP , CcC , i, j , k¿ −aW ,C cC ,i−1 , j ,k
¿ −aE ,C cC ,i+1 , j , k¿ −aB, C cC , i , j−1 , k
¿ −aT ,C cC ,i , j+1 , k¿ −aS , C cC , i, j , k−1
¿ −aN , C cC , i , j , k+1¿ =SC
(25)
Here, i, j, and k denote the location of volume blocks in x, y, and z directions, respectively. As seen, Eqs.
(23) to (25) relate the variables at each central cell P to those at the south, north, west, east, bottom, and top
neighbor cells denoted respectively by S, N , W , E, B, and T . Note that, because of considering implicit
time integration, the values of variables are all calculated at time levelt+∆ t and the results of the previous
time step are reflected only in the source terms S which are listed in the Appendix along with the other
coefficients appeared in the equations. The algebraic equations (23) to (25) are solved by applying the Tri-
Diagonal Matrix Algorithm (TDMA): we start from the electric potential equation whose solution is
required for solving velocity field. The velocity equation is then recalled and its results are extracted for
solving the mass transport equations (concentration fields). It is noteworthy that, because of using different
grid systems for the velocity and concentration fields, the velocity values at the concentration grid points is
obtained utilizing the cubic spline interpolation, prior to their use in the concentration fields. Ultimately,
Eqs. (23) to (25) are solved to get the concentrations of components A, B, and C at desired locations and
times.
For validation of the numerical method developed, the results for t → ∞ were compared with those of
our previous study [32] in which the steady-state of the present problem was investigated, whereby a very
good agreement was observed. This validation, however, does not necessarily confirm the transient
responses of our numerical algorithm; these responses are verified by using an analytical solution
developed for a special case in the next section.
2.5. Analytical solution for u¿=1 and Da=0
In this section, we intend to present an analytical solution for the problem by considering two more
assumptions including
EDLs are so thin that the velocity over the majority of the channel cross-sectional area equals uHS (
u¿ (x , y , z )=1).
No chemical reaction occurs among the components, that is Da=0, implying that the inlet components
only mix together.
The first assumption obviates the need for solving the electric potential and velocity equations; it also
reduces our problem to a two-dimensional one dependent only on x and zcoordinates [42]. Moreover, the
reaction terms in mass transport equations vanish by virtue of the second assumption; hence Eqs. (14) and
(15) are no longer and can be handled separately. Here, we develop our analytical model only for c A¿ as
exactly the same procedure can be followed for cB¿ . Based on the new premises, Eq. (14) simplifies to
∂ c A¿
∂t ¿ +Pe∂ c A
¿
∂ x¿ =∂2 cA
¿
∂ x¿2 +∂2 c A
¿
∂ z¿2 (26)
We express c A¿ as the summation of a steady-state component c ( x¿ , z¿ ) and a transient component
c ( x¿ , z¿ , t ¿), governed respectively by the following equations
Pe ∂ c∂ x¿=
∂2 c∂ x¿2 +
∂2 c∂ z¿2
(27)
∂ c∂ t¿
+Pe ∂ c∂ x¿=
∂2 c∂ x¿2 +
∂2 c∂ z¿2 (28)
whose solutions must conform to the following boundary conditions
∂2 c∂ x¿2|
x¿=l
= ∂2 c∂ x¿2|
x ¿=l
= ∂ c∂ z¿|
z ¿=± α= ∂ c
∂ z¿|z¿=± α
=0
(29)
c|x¿=0={ 1 z¿≥0¿0 z¿<0
, c|x ¿=0=0 , c|t ¿=0=−c
Eqs. (27) and (28) are solved invoking the separation of variables method; the analytical solution of the
former satisfying the boundary conditions (29) can be readily derived as
c ( x¿ , z¿ )=12−∑
n=1
∞ 2nπ
sin(nπ2 )cos [nπ
2 ( z¿
α+1)]exp¿¿¿ (30)
wherein
δ−¿
+¿=Pe2
±√ Pe2+n2 π2
α2
2¿¿
(31)
Solving Eq. (28), however, poses more difficulties due to the presence of three independent variables. The
general solution of c satisfying the pertinent boundary conditions may be written as
c ( x¿ , z¿ , t ¿)=−2i∑m=1
A0 m exp( Pe2
x¿)sin( mπl
x¿)exp [−( Pe2
4+ m2 π2
l2 ) t ¿]−2i∑n=1
∑m=1
Anm exp( Pe2
x¿)sin(mπl
x¿)cos [ nπ2 ( z¿
α+1)]exp [−( Pe2
4+ m2 π2
l2 + n2 π2
4 α 2 ) t¿ ](32)
Applying the initial condition and following the orthogonality conditions, the coefficients A0m and Anm
appeared in Eq. (32) are calculated as
A0 m=−i
πm2 [1−cos ( πm) exp(−Pe
2l)]
Pe2 l2
4+m2 π2
(33)
Anm=−i 2 mn
sin( nπ2 )¿¿ (34)
Therefore, considering c A¿ =c+c, the final expression for c A
¿ is obtained as
c A¿ (t ¿ , x¿ , z¿)=1
2−∑
n=1
∞ 2nπ
sin( nπ2 )cos [nπ
2 ( z¿
α+1)]exp¿¿¿ (35)
Now that we have presented an analytical solution for mixing in electrokinetic microreactors at thin EDL
limit, it is convenient to quantify the degree of mixing. Different parameters are used in the literature to
quantify the degree of mixing in T/Y-shaped microchannels among which is the mixing index. The
mathematical representation of this parameter appropriate to the present analysis is given as [43]
I mix(x¿ , t¿)=2∫
−W
0
c A ( x , z , t ) d z
W c0 , A=
2∫−α
0
cA¿ ( x¿ , z¿ , t¿ ) d z¿
α c0 , A¿
(36)
Substituting Eq. (35) into Eq. (36) provides
I mix(x¿ , t¿)=1−∑n=1
∞ 8n2 π 2 sin2( nπ
2 )exp¿¿¿ (37)
The predictions of Eq. (37) are compared with the results of the numerical simulations based on a uniform
fluid velocity in Fig. 2 at different x¿ and Pe. A good agreement is observed between the results, revealing
the correctness of the numerical procedure conducted; the small deviation between the results stems from
applying the no-slip boundary condition in the numerical code. That is why the error is smaller for a lower
x¿ at which there is more dominance of the diffusion over advection due to larger concentration gradients.
3. Results and discussion
In this section, we intend to investigate the reaction-diffusion behavior of components over time by
studying the effect of each governing parameter on the system transient response. The results are all
presented assumingα=2andl=20. To start with, the axial progress of the concentration zones over the
time is depicted in Fig. 3 where red and blue contours correspond respectively to components A and B. For
drawing this figure, the inlet concentrations of the reactants are assumed to be the same but the diffusion
coefficient of component A is twice that of B. Therefore, while A puts its effort into moving both in x and
z directions, B is more focused on moving along the streamwise direction and so, as observed in the figure,
it takes less time for B to pass the channel. Bearing in mind that the components are introduced with the
same flow rate, this implies an important conclusion: providing the same concentrations of the reactants in
the channel, which is crucial for efficient production, requires supplying more amount of the reactant of
lower diffusivity. It should be noted that the contours in Fig. 3 are primarily depicted for the mid-plane (
y¿=0¿ wherein the components are faster than those traveling adjacent to the wall; therefore, less progress
of the concentration zones is expected close to the walls as it is demonstrated by the dashed lines.
Fig .3 can be redrawn for the product of reaction to achieve Fig. 4 wherein the time development of C
concentration profile is shown at both x¿=7 and y¿=0. It is interesting that the expected butterfly-shaped
profile, which has been explained in our previous work [32], does not appear immediately and C profile
firstly emerges in a spindle-like form. That is because in a short duration, slower axial movement of the
reactants traveling near the horizontal walls gives rise to less accumulation of A and B therein as compared
with the mid-plane; so less product is generated near the walls forming the spindle-like profile observed in
Fig. 4a. Giving more time, A and B find more opportunity to cover the entire channel and now the
difference between the residence times of the components shapes the C concentration profile: at the
centerline, where the velocity is maximum and the components have the least time to react with each other,
the production is minimum whereas in the near-wall regions the opposite is true and the production reaches
its maximum (Figs.4b and 4c).
Like Fig. 3, DA is assumed to be double of DB in Fig. 4 causing more penetration of component A into
B stream, thereby forcing the product concentration peak toward there. This inclination can be seen even in
Fig. 4a where reaction is in its beginning stages. Another point worth noting is that the upstream sections
obtain their steady-state in advance of the downstream locations. This can be readily recognized by
comparing Figs. 4b and 4c in which although the y-z profiles do not change, x-z profiles experience
significant alteration in the downstream region. In order to consider this matter in more depth, we plot the
transverse distributions of component C at the mid-plane for differentx¿andt ¿ in Fig. 5. As observed, the
concentration profile of C at x¿=5 does not change over time meaning that it has obtained its steady-state
before other axial positions, especially x¿=20in which the production has not even started at t ¿=0.1. The
larger transient times of the downstream sections let components located therein to participate more in
diffusion process, thereby increasing the near-wall production of C by moving toward the channel end. It is
exactly for the same reason that the inclination of the concentration peak is magnified at larger values of x¿
(Fig. 5c).
3.1. Effect of velocity on production
This section is dedicated to investigating the effect of the velocity profile on the system transient response.
The velocity here is assumed to be created by the combined action of the electroosmotic and pressure forces
and, therefore, Κ ad Γ are the governing parameters specifying how fast the solutes are moving by the
flow. Based on the definition, Κ is inversely proportional to the Debye length and so the larger Κ is the
thinner EDL becomes. Accordingly, the electroosmotic body force is concentrated more near the wall for a
higher K, leading to a faster movement of the species, especially near the wall. The time development of
the transverse C concentration profile at the mid-plane for pure electroosmotic flow at thick and thin EDLs
are respectively studied in Figs. 6a and 6b. The two remaining parts of Fig. 6 are assigned to examine the
time development of the concentration profile at positive and negative values of Γ . Recalling its definition,
Γ measures the share of the pressure gradient force in the driving components and its sign indicates
whether the pressure gradient is favorable (Γ>0) or unfavorable (Γ<0) to electroosmotic force; therefore,
increasingΓ is equivalent to rising the fluid velocity.
The first point drawing attention in Fig. 6 is that the inclination of the concentration peak increases
with time. Moreover, by comparing the left and right parts of the figure, one can conclude that producing
the same amount of C at a fixed axial position takes much less time when the flow rate increases either by
shrinking EDL or applying a favorable pressure gradient. For example, whereas the profile of t ¿=0.25 in
Fig. 6b passes through c¿=0.18 at its peak, its counterpart in the LHS figure does not even exceed
c¿=0.05. This is firstly because charging the channel with reacting components takes more time when the
particles are slower, causing a delay in the start of production. Moreover, slower flow rates allow the
reacting components to spread throughout a larger area and so the aggregation of C at a given point takes
longer. In return for the decline in the production rate, introducing the components with a lower velocity
gives them a longer time to participate in the reaction-diffusion process and so leads to higher production at
the steady-state, particularly in the lateral regions.
3.2. Efficiency
One of the most important attributes of a microreactor, especially when it comes to application, is its total
production. This parameter, however, is dependent not only on the microreactor performance, but also on
the inlet concentration of the reactants; hence, it cannot be considered as the only factor for examining the
microreactor performance. Therefore, in order to omit the effect of the inlet concentration of the reactants
on the total production, we measure the total amount of C with respect to the limiting reactant, that is to say
that the microreactor efficiency is calculated. Recalling that in this studyω≥ 1 for all the cases, component
B can always be treated as the limiting reactant and so the total efficiency of system in percent can be
written as
η (t )=∫−W
W
∫0
H
∫0
L
cC ( x , y , z , t ) d xd y d z
∫−W
W
∫0
H
∫0
L
cB (x , y , z ,t ) d x d yd z×100=
∫−α
α
∫0
1
∫0
l
cC¿ ( x¿ , y¿ , z¿ , t¿ ) d x¿d y¿ d z¿
∫−α
α
∫0
1
∫0
l
cB¿ ( x¿ , y¿ , z¿ ,t ¿) d x¿d y¿ d z¿
×100 (38)
Due to its importance, the efficiency is studied in Fig. 7 under the effect of all the governing
parameters, namelyΓ ,Κ ,β ,ω,Pe, and Da. As mentioned in the previous section, increasing the flow rate
diminishes the production in long-term and so it is not surprising that the efficiency grows by decreasing
either ΓorΚ . The supplementary point added by this figure is the extent to which each of these parameters
is effective: according to Figs. 7a and 7b, the efficiency seems to be more influenced by the pressure
gradient than EDL thickness. Consequently, applying an unfavorable pressure gradient should be
considered as the first hydrodynamic factor in designing efficient microreactors. This conclusion, however,
is reversed in short-term intervals: considering an unfavorable pressure gradient in Fig. 7a leads to the
lowest efficiency beforet ¿≅ 0.17. This is the immediate result of a higher production rate in the presence of
higher velocities which was pointed out to in the previous section. The opposite behavior of the system in
short-term duration is important especially when it is not allowed to obtain its steady-state; as an example,
for cases in which there is a time-dependent injection of the reacting components instead of a constant
injection rate.
The second row of Fig. 7 is dedicated to specifying the effect of component B on the system
performance for which the efficiency is plotted versus the diffusivity and inlet concentration of this reactant
in parts (c) and (d). It should be noted that although βandωare dependent on both A and B, changing them
at fixed Pe and Da only affects the properties of B component. Enhancing DB (decreasingβ) increases the
region over which B can penetrate into A stream and so raises the production of C and efficiency. The RHS
figure, however, tells a more interesting story: increasingc0 , Braises the initial concentration of the limiting
reactant, which of course increases the production; nonetheless, this should not be interpreted as efficiency
enhancement since the production growth occurs solely at the expense of injecting more reactants. In fact,
the ratio of the production to the limiting reactant, that is the efficiency, is reduced by increasing c0 , B
because a lower number of A particles surround each B particle.
The two remaining figures in Fig. 7 are devoted to studying the influences of the Péclet and
Damköhler numbers on η. Considering the fact that only one governing parameter is permitted to vary in
each figure, the increase of the former means a decrease in DA while the latter is altered only through
changing the chemical rate constant. Accordingly, an increase in Pe will be accompanied by a lower cross-
stream diffusion of A particles, which enhances their overall streamwise movement; this leads to more
accumulation of the reacting components downstream and, ultimately, larger short-term efficiencies. At
long-term, however, higher cross-stream diffusion of component A provides a wider engagement of the
reacting components, this is to say that the efficiency is magnified by decreasing Pe (Fig. 7e). The last
figure we discuss here is Fig. 7f in which the impact of the Damköhler numberon the system efficiency is
demonstrated. Due to its definition,Da is directly proportional to the chemical rate constant and, therefore,
the observed monotone-increasing relation between Da and η is not strange.
4. Conclusions
A theoretical study was conducted to investigate the transient response of a Y-shaped microreactor. It was
assumed that the fluid flow, created under a combined influence of the pressure and electroosmotic forces,
is steady and fully developed. A second-order irreversible chemical reaction was assumed to take place
between the components. In general, the problem was treated numerically utilizing a 3D finite-volume-
based numerical code; however, analytical solutions were also presented assuming that the velocity is
uniform and there is no chemical reaction between the components. The analytical and numerical results
were shown to be in a good agreement. A complete parametric study revealed that the well-known
butterfly-shaped profiles of the product concentration are not immediately established. In fact, spindle-like
profiles are created at the beginning and are evolved into butterfly-shaped profiles only near the steady-
state. The inclination of the concentration peak toward the component with either less diffusivity or less
inlet concentration, which has already been reported for a steady-state, was shown to be less significant at
the earlier stages of the production. Moreover, the inspection of the total efficiency, defined as the ratio of
the whole production to the amount of the limiting reactant within the device, demonstrated that the long-
term influence of the factors affecting the advection of mass is quite the opposite of their impact on the
transient response. More precisely, whereas increasing (a) the dimensionless Debye-Hückel parameter, (b)
the velocity scale ratio, and (c) the Péclet number give rise to higher efficiencies at short time scales; quite
the opposite is true at the steady-state. That is, increasing (a), (b) and (c) at steady state decreases the
efficiencies. The influence of other parameters on the efficiency is not time-dependent. For example,
increasing either the inlet concentration ratio or the Damköhler number is accompanied by better
efficiencies for non-steady and steady states. In contrast, increasing the diffusivity ratio always decreases
the efficiency.
Appendix
The functions Qi=1 ,… ,8appeared in Eqs. (18) to (22) are as follows
Q1 ( y )= eΩ y+e−Ω y+22 τ Ω
, Q2 ( y )= e2 Ω y+2eΩ y−2e−Ω y−e−2 Ω y
4 τ2 Ω(A1)
Q3 ( z )= eΩ z+e−Ω z+22 ατ Ω
,Q 4 ( z )= e2 Ω z+2eΩ z−2 e−Ω z−e−2Ωz
4 α2 τ2 Ω
Q5 ( x )= eΩ' ( x−1)+e−Ω' ( x−1 )+22l τ x
' Ω' , Q6 ( x )= e2Ω' (x−1 )+2 eΩ' (x−1 )−2 e−Ω' ( x−1)−e−2 Ω' ( x−1)
4 l2 τ x' 2 Ω'
Q7 ( z )=sinh(0.5 τ z
' )
τ z' α cosh [ τ z
' ( z−0.5 ) ], Q8 ( z )=
−sinh2(0.5 τ z' )sinh [τ z
' ( z−0.5)]τ z
' α2 cosh3[τ z' ( z−0.5)]
where Ω=ln ( τ+1τ−1 ), Ω'=ln ( τ x
' +1τ x
' −1 ), τ=1.01, τ x' =1.05, andτ z
' =10. The coefficients of algebraic
equations (23) to (25) are also given as
aP , A=aW , A +aE, A +aS , A+aN , A +aB, A+aT , A+PeDa cB, P¿ g Δ3+Δ3/δ
(A2)
aP , B=aW , B+aE , B+aS , B+aN ,B+aB ,B+aT , B+βPeDa cA , P¿ g Δ3+Δ3 /δ
aP , C=aW ,C+aE , C+aS ,C+aN , C+aB,C+aT ,C+ Δ3/δ
aW , A=Δ2 [ Pe u¿Q5 ( x )−Q6 ( x ) ] (1+G1 ) , aW , B=Δ2 [ βPe u¿Q5 ( x )−Q6 ( x ) ] (1+G2 )
aW ,C=Δ2 [γPeu¿Q5 ( x )−Q6 ( x ) ] ( 1+G3 ) ,aE , A=Δ2 [ Pe u¿Q5 ( x )−Q6 ( x ) ] G1
aE , B=Δ2 [ βPe u¿Q5 ( x )−Q6 ( x ) ] G2 , aE, C=Δ2 [ γPeu¿Q5 ( x )−Q6 ( x ) ]G3
aS , A=aS , B=aS ,C=−Δ2 Q8 ( z ) (G4+1 ) , aN , A=aN , B=aN ,C=−Δ2 Q8 ( z ) G4
aB , A=aB , B=aB ,C=Δ, aT , A=aT ,B=aT ,C=Δ
SA=c A , P¿h Δ3/δ , SB=cB, P
¿h Δ3/δ ,SC=γPeDac A ,P¿ c B , P
¿ Δ3+cC , P¿h Δ3/δ
in which Δ is the spatial interval. A 100 ×100 ×100 mesh structure is considered in this study as it has
been found to provide sufficiently mesh independent results. Moreover, δ is the time interval which is
multiplied by 1.05 in each time step and its value for the first time step is δ 0=5 ×10−4; this value is
specified because further time refinement does not affect the results. In addition, the superscripts gandh
mark the results corresponding to previous iteration and previous time step, respectively. Finally, the
parameters Gi=1 ,⋯ ,4 are defines as
G1= {e[ Pe u¿Q5 ( x )−Q6 ( x )]/Q52 ( x ) ∆−1}−1
,G 2={e [ βPe u¿ Q5 ( x )−Q6 ( x )] /Q52( x ) ∆−1}−1
(A3)
G3= {e [γPeu¿ Q5 ( x )−Q6 ( x ) ]/Q52 ( x ) ∆−1}−1
,G4=[e−Q8 ( z )/Q72 ( z )∆−1 ]−1
Acknowledgment
The authors sincerely thank the Iranian National Science Foundation (INSF) for their financial support
during the course of this work.
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Fig. 1. 2D views of the microreactor under consideration including the coordinate system, dimensions,
and arrangement of components.
t*
I mix
0 0.2 0.4 0.6 0.8 10
0.1
0.2
0.3 NumericalAnalytical
(b)Pe = 5
x*= 2
x*= 1
t*
I mix
0 0.2 0.4 0.6 0.8 10
0.05
0.1
0.15 NumericalAnalytical
(a)Pe = 1
x*= 1
x*= 2
Fig. 2. Comparison among the analytical results of I mix over time and the predictions of the numerical
code considering a uniform velocity field at two different axial positions for a) Pe=1 and b) Pe=5.
x*
z*
0 5 10 15 20
(a)
x*
z*
0 5 10 15 20
(b)
x*
z*
0 5 10 15 20
(d)
x*
z*
0 5 10 15 20
(c)
Fig. 3. Axial movement of the concentration front over the time for components A (red counters) and B
(blue contours); the corresponding times for each part include a) t ¿=0.01, b)t ¿=0.04, c)t ¿=0.07, and d)
t ¿=0.1. The contours are depicted for the mid-plane ( y¿=0) while the dashed lines delineate the
concentration zones at the upper wall ( y¿=1). The other parameters include Κ=10, Γ=0, Pe=100,
β=2, ω=1, γ=1, and Da=1.
0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2
(a)
(b)
(c)
Fig. 4. Product contours at the mid-plane and at x¿=7 for a)t ¿=0.1, b)t ¿=0.2, and c)t ¿=0.3. The
evolution of the concentration profile from spindle-like form into butterfly-shaped form is observed in the
cross-sectional contours; the asymmetry of the production zone is also evident in these figures. The
parameter set considered include Κ=10, Γ=0, Pe=100, β=2, ω=2, γ=1, and Da=1.
z*
c*
-2 -1 0 1 20
0.05
0.1
0.15
0.2
x* = 5x* = 10x* = 15
(a)
z*
c*
-2 -1 0 1 20
0.05
0.1
0.15
0.2
x* = 5x* = 10x* = 15x* = 20
(b)
z*
c*
-2 -1 0 1 20
0.05
0.1
0.15
0.2
x* = 5x* = 10x* = 15x* = 20
(c)
Fig. 5. Profiles of c¿ vs. z¿ at the mid-plane and at different axial positions for a) t ¿=0.1, b)t ¿=0.2, and
c)t ¿=0.3. The profile of x¿=20 is not shown in the first figure because the production has not already
started therein. The parameter set considered include Κ=10, Γ=0, Pe=100, β=2, ω=2, γ=1, and
Da=1.
z*
c*
-2 -1 0 1 20
0.05
0.1
0.15
0.2
SSt* = 0.25t* = 0.2t* = 0.15
(b)
z*
c*
-2 -1 0 1 20
0.05
0.1
0.15
0.2
SSt* = 0.45t* = 0.35t* = 0.25
(a)
z*
c*
-2 -1 0 1 20
0.05
0.1
0.15
0.2
SSt* = 0.90t* = 0.75t* = 0.60
(c)
z*
c*
-2 -1 0 1 20
0.05
0.1
0.15
SSt* = 0.15t* = 0.125t* = 0.1
(d)
Fig. 6. Profiles of c¿ vs. z¿ at different times while keeping x¿=20 and y¿=0 for a)Κ=2, b)Κ=50, c)
Γ=−1, and d)Γ=1. Besides the transient profiles, the results of the steady-state, denoted by SS, are also
shown in the figures. The default parameters include Κ=10, Γ=0, Pe=100, β=2, ω=2, γ=1, and
Da=1.
t*
t*
(%
)
(%
)
0 0.2 0.4 0.6 0.8 10
10
20
30
40
= -1 = 0 = 1
(a)
t*
(%
)
0 0.2 0.4 0.6 0.8 10
10
20
30
K = 2K = 10K = 50
(b)
t*
(%
)
0 0.1 0.2 0.3 0.40
5
10
15
20
= 1 = 2 = 4
(c)
t*
t*
(%
)
(%
)
0 0.2 0.4 0.6 0.8 10
5
10
15
20
25
30
Pe = 50Pe = 100Pe = 500
(e)
t*
(%
)
0 0.1 0.2 0.3 0.40
5
10
15
20
25
Da = 4Da = 2Da = 1
(f)t*
(%
)
0 0.1 0.2 0.3 0.40
5
10
15
20
25
= 4 = 2 = 1
(d)
Fig. 7. Efficiency enhancement over the time at different values of a)Γ , b)Κ , c)β, d)ω, e)Pe, and d)Da.
The default parameters include Κ=10, Γ=0, Pe=100, β=2, ω=2, γ=1, and Da=1.