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Abstract

This study looked at optimizing the geometrical shape of a simple T-mixer using Bernstein polynomials-based shape optimization technique to improve the mixing of the T-mixer. Passive micromixers of planar geometry are preferred in a wide range of applications such as lab-on-chips and chemical processing applications, due to their ease of fabrication and low processing costs. Studies conducted on T-mixers have revealed that the performance of T-mixers at low Re (<30) is dismal. At low Reynolds number flows, the mixing is completely dominated by diffusion because of laminar flow conditions. In the present work, an attempt to improve the mixing performance of the T-mixer was made and a nearly three-fold improvement in performance was reported. The adjoint-based shape optimization technique was employed to optimize the wall profile without losing the advantage of the ease of fabrication. The T-mixer boundaries were represented parametrically using Bernstein polynomials that could take any shape within a constrained plane. Different shapes can be generated for different polynomial orders. A limit on the minimum channel thickness (60 microns) was imposed, while the inlet and outlet boundary lengths were fixed. For this particular geometry, the 12^th-order polynomial exhibits an optimized shape for maximum mixing performance. The optimized shape of the T-mixer also shows significant improvement in mixing compared to a conventional T-mixer with a reduced channel thickness of 60 microns.

Keywords

Micro mixing shape optimization T-mixer Bernstein polynomials microfluidics

Introduction

Micromixers are an important component in microfluidic devices and microfluidic systems. They have significant applications in biomedical and chemical analysis.^1–5 Mixing has prime importance in microfluidics. At the same time, mixing is very challenging in microchannels. At the macroscale, we exploit the velocity of the fluid so that the flow can easily turn into turbulent mode by increasing the flow velocity so that turbulence in the flow helps in the mixing of the fluids at macroscale. At microscale, however, the turbulence effects are not present, and flows are inherently laminar because of the small length scale of the system. Micromixers can be primarily classified into two types: (1) Passive micromixers and (2) Active micromixers. Active micromixers require external force to induce mixing such as electrodynamic, magnetohydrodynamic, acoustic, and ultrasonic.^6–8 The pressure-driven passive micromixers require no external force to induce mixing. In passive micromixers, diffusion and chaotic advection are dominant physical phenomena. Passive micromixers can increase mixing by stretching, folding, splitting, and recombining mechanisms.^9–11 Different passive micromixers are designed by considering the mechanisms mentioned above to enhance mixing efficiency.

Due to ease of design and fabrication, many passive micromixers have been developed to improve mixing efficiency. Chen et al.¹² carried out a numerical study to investigate the mixing behavior of stacking and folding type E-shaped micromixers. It was found that mixing enhanced significantly due to chaotic advection. Above Re of 5, chaotic advection played a major role in improving the mixing in stacking and folding type E-shaped micromixers. Strook et al.¹³ carried out an experimental study to investigate mixing performance in a staggered herringbone mixer. Ridges were placed on the bottom wall of the mixer. A transverse flow was created in the channel due to the ridges. Moreover, transverse flow helped in improving the mixing of the channel. Hsiao et al.¹⁴ carried out both experimental and numerical studies in a T-mixer with rectangular winglet pairs (RWPs) placed on the bottom wall of the mixer. At higher Re, strong vortices were created behind RWPs, and thus, mixing efficiency improved significantly. Ottino¹⁵ studied chaotic mixing in microchannels. It was observed an increase in interfacial contact area by creating repeated stretching and folding of fluids in a microchannel. Ekta et al.¹⁶ carried out a numerical study to investigate the mixing characteristics and pressure drop of a spiral micromixer by varying width and depth of the micromixers. They found that for Re > 50, the mixing efficiency does not improve significantly. Dundi et al.¹⁷ developed passive-type T-T mixer with cylindrical elements. And they numerically proved that the developed mixer showed significant improvement in micro mixing compared to the basic T-mixer. Muhammad et al.¹⁸ developed non-aligned input M-type micromixers with differently shaped obstacles. They found that the proposed four novel micromixers showed improved mixing performance. Masoud et al.¹⁹ studied the confluence angle, flow rate, and flow rate ratio effect on passive mixing. The effectiveness increased with an increase in flow rate ratio and decrease in angle. Mranal et al.²⁰ studied the effect of heterogeneous charge patterns at the microchannel bottom on electrokinetic mixing performance. They identified optimal heterogeneous charge pattern by formulating a binary numerical optimization problem. Furthermore, other studies were able to create chaotic mixing by different geometry modifications in microchannel shapes such as serpentine channels, zig-zag channels, and curved microchannels.^21–23 Jain et al.²⁴ used a shape optimization technique to optimize groove shape in a microchannel. In this study, the groove shape has been represented parametrically using Bezier curves. The control points of the Bezier curve were chosen as optimization parameters to identify the optimal groove shape, which maximizes mixing for a given set of operating conditions.

It was observed that different passive micromixers provide improved mixing efficiency.^12–14 Other studies^25,26 showed that simple planar designs such as square, wave, and serpentine micromixers that showed better mixing were easy to manufacture and had minimum pressure drops during the flow. All micromixers mentioned previously were developed by trial-and-error method. And there was a need to develop a systematic approach to address this problem. This was achieved by employing the features of Bernstein polynomials. Sergei Natanovich Bernstein (1880–1968) proposed the Bernstein basis in order to offer a useful demonstration of Weierstras’ theorem. Due to Bernstein polynomials’ sluggish convergence as function approximants, they were not commonly used until the development of digital computers. The Bernstein polynomials were eventually used widely once it was discovered that the coefficients of these polynomials could be easily changed to alter the shape of the curves they described. In the 1960s, two French car engineers, Paul de Casteljau and Pierre Etienne Bezier, were interested in putting this idea into action. Clay model sculpting turned out to be a time-consuming and expensive procedure for creating intricate designs for car bodies. De Casteljau and Bezier set out to create mathematical tools that would enable designers to make and modify complex objects naturally. Bernstein polynomials, also called Bezier curves, brought Bezier's name to people's attention. This is because de Casteljau published most of his research when at work. Bernstein polynomials are based on current research and modern technologies, and they have a number of useful properties that can be used in many fields. Inspired by the successful application of the Bernstein polynomial-based optimization approach for numerous aerodynamic applications, this approach has been employed for the outer wall profile of the T-mixer also.²⁷ In this approach, the shape variation is governed by equation (8) such that the need for grid generation at every iteration is eliminated. On the flipside, a minor drawback of restriction of the available shape owing to the choice of deformation vectors exists. However, this could be solved by varying the order of parameterization as well as the control points’ displacement. The present article attempts to perform optimization by choosing the former method.

In the present work, the basic T-mixer and the effect of channel shape on the mixing efficiency of the T-mixer were studied. Shape optimization technique to enhance mixing efficiency by altering the shape of T-mixer channel was used. The optimized T-mixer showed significant improvement in mixing compared with the basic T-mixer. Numerical analysis was conducted in 2D to reduce computational time and the complexity of setting up the shape optimization problem. At low Re (Re < 30), the extent of mixing evaluated using 3D and 2D models was nearly the same. This was due to the absence of secondary flows and flow vortices at low Re.²⁸

Methodology

Mathematical modeling

A two-dimensional, laminar-incompressible flow of Newtonian fluid through a planar T-micromixer is considered. The schematic diagram of the T-micromixer is shown in Figure 1. The governing equations to solve the fluid flow and species transfer in the planar type T-micromixer are given by the following equations:

\nabla \cdot V = 0

(1)

V \cdot \nabla V = - \frac{1}{ρ} \nabla P + v \nabla^{2} V

(2)

(V \cdot \nabla) c = D \nabla^{2} c

(3)

Figure 1.

Schematic diagram of the T-micromixer and its mesh (a) Schematic diagram of the computational domain and (b) free triangular mesh of T-micromixer.

where $V, ρ, P$ , and v are velocity vector, density, pressure, and kinematic viscosity of the fluid, respectively. The species transport in T-micromixer is governed by diffusion–convection (equation (3)), where c is the concentration and D is the diffusion coefficient of species.

In the present study, a parameter employing variance to measure the extent of mixing is chosen due to two reasons mentioned in the literature.²⁹ The first reason is that the variance measures the mean of the fluctuations between the current state and the perfectly mixed state. Here, variance = 0 indicates a perfectly mixed state, while variance = 1 indicates no mixing. The second reason is that, for an incompressible flow, the governing advection–diffusion equation also tends to zero in the absence of any sources. This logical similarity between the variance and the advection–diffusion equations tending to zero in the absence of any external sources has led to wide acceptance of the mixing quality for Low-Reynold number flows without any stirrers. However, the variance-based mixing quality fails to accurately predict the extent of mixing in cases where stirrers are used for mixing.

A parameter called mixing quality is defined using the variances of the species concentrations to measure the extent of mixing. The mixing quality “α” is mathematically defined as

α = 1 - \sqrt{\frac{σ_{M}^{2}}{σ_{m a x}^{2}}}

(4)

where “

σ_{M}^{2}

” is variance of the mixture at the outlet and “

σ_{m a x}^{2}

” is the maximum variance of the mixture (0.5). Here, variance of the mixture “

σ_{M}^{2}

” is calculated using equation (5):

σ_{M}^{2} = \frac{1}{n} \sum_{i = 1}^{n} (c_{i} - c_{M})^{2}

(5)

^{'} c_{M}^{'}

represents the mean value of concentration taken over n elements of the grid at the cross-section considered. The value of mixing quality “

α

” is “zero” for no mixing and “one” for complete mixing.

Numerical modeling

The 2D planar type T-micromixer geometry was created using COMSOL Multiphysics 5.5 software, shown in Figure 1(a). The length of the microchannel (L_c) is 3000 µm, width (W_c) is 200 µm, height (h_c) is 800 µm, and inlet channel width (W_i) is 100 µm.

In this problem, two liquid species A and B were chosen for mixing in planar-type T-micromixer channel. Both species had the same properties as liquid water at 20°C, mentioned in Table 1. Concentration gradient was created by providing the boundary conditions at the inlet, so that mass fraction was set to unity for species “A” and zero for species “B” at inlet one and vice versa at inlet two. Normal inflow velocity boundary condition (Re = 2–30) was given at inlet one and inlet two along with mass fraction boundary conditions. At the outlet, the pressure boundary condition was provided. It was set to atmospheric pressure. Except for inlet and outlet, all other boundaries were applied with no-slip boundary conditions.

Table 1.

Properties of the species materials.

Properties	Species A & Species B
Type of material	Liquid Water
Density, ρ (Kg/m³)	998.2
Dynamic viscosity, μ (Pa·s)	0.001
Diffusion coefficient, D [m²/s]	2 × 10⁻⁹

Fluid flow equations were solved using Laminar flow physics interface model, while mass fraction equations were solved using the Transport of concentrated species model. Reacting flow Multiphysics coupling feature was used to simulate mass transport where the fluid flow depends on mixture composition. For solving the fluid flow problem, P2 + P1 discretization scheme was used, and for the concentration field, a quadratic discretization scheme was used.

A free triangular mesh was used for the whole computational domain, as shown in Figure 1(b). Because this mesh is like a “workhorse” for 2D Computational fluid dynamics problems, it was easy to create this kind of mesh with high element quality.³⁰ However, this ease comes with a cost. This free-unstructured triangular mesh gives a greater numerical diffusion. The triangular meshes must be extra fine to overcome the difficulty. In this study, the extremely fine mesh was used with a maximum element size of 5.36 µm, and the number of elements was 63864. The details of mesh independence are mentioned in the following section. The different meshes that were considered for the mesh independence test are given in Table 2.

Table 2.

Grid independence test parameters.

Meshes	Maximum element size	Minimum element size	Maximum element growth rate	Number of elements	Mixing quality, α
Mesh 1	176	5.6	1.4	18704	0.2497
Mesh 2	69.6	3.2	1.25	23966	0.1826
Mesh 3	36	1.6	1.15	30208	0.1726
Mesh 4	22.4	0.32	1.1	37052	0.1684
Mesh 5	10.4	0.12	1.08	41974	0.1674
Mesh 6	5.36	0.016	1.05	63864	0.1669
Mesh 7	5.00	0.016	1.05	73280	0.1668

Grid independence study

To determine the optimum mesh size for planar-type T-mixer analysis, a grid independence study was performed. This study considers axial velocity and mixing quality distribution at the outlet of planar type T-mixer for Re = 5. Figure 2(a) depicts the axial velocity distribution at the outlet, while Figure 2(b) depicts the mixing quality distribution at the outlet. Between meshes 6 and 7, the axial velocity has the smallest variation. In addition to axial velocity, the mixing quality difference between mesh 6 and mesh 7 is also minimal. Hence, mesh 6 was considered for the current study.

Figure 2.

Grid independence study and validation (a) Average axial velocity profiles at outlet of T-micromixer for different meshes, (b) Mixing quality at outlet of T-micromixer for different meshes and (c) Comparison of Mixing quality with various Reynolds number for T-mixer of the present study and Dundi et al.¹⁷

Figure 3.

Geometry bounding box realization on the microchannel.

Validation

The governing equations of the flow field and species transport of the T-mixer have been solved in this study using conditions comparable to those used by Dundi et al.¹⁷ To validate the accuracy of the current numerical scheme, the results obtained were compared to the results of Dundi et al.¹⁷ Equations (4) and (5) were used to assess mixing quality at the T-mixer outlet. Figure 2(c) shows that the current study's findings are in close accord with those of Dundi et al.¹⁷

Optimization approach

The effect of the outer wall's shape of the T-micromixer on the mixing performance was examined. Bernstein polynomial was used to deform the geometry. After deformation, mixing quality was evaluated for different Reynolds numbers from 2 to 30. The following equations represent the Bernstein polynomial of n^th degree with n + 1 control point:

P (t) = \sum_{i = 0}^{n} B_{i} J_{n, i} (t) 0 \leq t \leq 1

(6)

In the above equation, B_i represents the control points of the Bezier curve, and

J_{n, i}

(t) is the Bernstein basis polynomial mathematically defined as follows:

J_{n, i} (t) = \frac{n!}{i! (n - i)!} t^{i} (1 - t)^{n - i}

(7)

In this study, the COMSOL Multiphysics 5.5 optimization module's features were used to deform the geometry. The two external walls of T-micromixer were represented by n^th-order Bernstein polynomial, which is available through the polynomial boundary feature. Inspired by the successful application of the Bernstein polynomial-based optimization approach for numerous aerodynamic applications, the approach was employed for the outer wall profile of the T-mixer also.²⁷ In this approach, the shape variation is governed by equation (8) such that the need for grid generation at every iteration is eliminated. This representation helps in deforming geometry. It controls the deformation of interior nodes by a smoothing equation, while the deformation of walls is given by:

d = \sum_{i}^{n} B_{i}^{n} (s) c, - d_{m a x} \leq c_{i} \leq d_{m a x,}

(8)

B_i represents the control points of Bezier curve. The initial value of B_i could be anything depending on the COMSOL parameter(s), which ranges from

0 \leq s \leq 1

. For example, in a linear curve, when S = 0.25, B_i(S) is one-quarter of the way from starting point to ending point. “

d_{m a x}

” represents the maximum displacement of Bernstein polynomial of n^th order.

On the flipside, a minor drawback of restriction of the available shape owing to the choice of deformation vectors exists while using Bernstein polynomials. However, this could be overcome by varying the order of parameterization (n) and/or control points’ displacement (d_max). The present study attempts to perform optimization by choosing the former method.

Maximum displacement was limited to 5% of the total length of the curve which was to be altered. In this problem, the upper and lower walls of microchannel were considered to alter the channel shape. Both walls were of equal length of 2900 µm, and the maximum displacement of each wall was 145 µm, which was 5% of total length of inlet/outlet boundary. The geometry bounding box is a square box with a side length of 2dmax as shown in Figure 3. It is to be noted that the mass of the species held in T-mixer at any point of time is not constrained and it varies with the wall profile. Initially, the polynomial order was fixed to second order. Bernstein polynomials satisfy the bounds of coefficients across the whole line. It means that each point on the line is confined to move in a square box. The higher-order and higher maximum displacement lead to a clashing of boundaries and reduce the tendency for inverted elements. So, it is necessary to find a balance between them for the convergence of the solution. The optimization technique uses the adjoint sensitivity method^31,32 to perform sensitivity analysis. Initially, shape optimization was performed at Reynolds number (Re = 5). Here, the flow rate was fixed, and the maximization of mixing quality (α) was considered an objective function to perform shape optimization.

The COMSOL numerical solver initially solves the fluid flow and species conservation equations for the rectangular domain and transfers the data to the optimization module. In the optimization module, the objective function, that is, mixing quality for different data points in control space is searched. If the stopping criteria are reached, that is, the optimum value is obtained, the corresponding outer wall profile is given as the output and the search procedure is terminated. However, if the optimum profile is not found in the initial run I₀, further iterations are performed in the optimization module to search for the optimum geometric configuration, which exhibits maximum mixing quality. The flowchart depicted in Figure 4 shows the implementation of the optimization algorithm in the COMSOL-based module.

Figure 4.

Flowchart showing steps for optimization problem implementation in COMSOL Multiphysics.

Results and discussions

The profile of the walls of the T-micromixer was modified using the adjoint-based shape optimization technique. Figure 5(a) shows the geometric model of T-mixer with the walls being highlighted. Initially, the length of both walls was 2900 µm each. Numerical simulations were carried out by varying the inlet velocity such that the Reynolds number varied from 2 to 30. As the Reynolds numbers pertain to the laminar flow regime throughout the channel as shown in Figure 5(b), the mixing was observed to be diffusion dominant. Therefore, the residence times played a pivotal role in the effective mixing of the constituent species. The effect of Reynolds number on the mixing quality depicted in Figure 2(c) reinforces the dependence of mixing quality on residence time. The increase in Reynolds number represents an increase in inlet velocity. As the inlet velocity increases, the residence time available for the constituent species inside the mixer decreases. This adversely affects diffusion and thus results in ineffective mixing.

Figure 5.

Streamlines and polynomial boundary representation (a) polynomial boundary representation of boundary walls of T-micromixer, and (b) Streamlines based on Spec B Mass fraction at re = 5.

Optimized T-micromixer

The optimized order of the Bernstein polynomial for improved mixing significantly depends on the geometric bounding box size and its geometry. The side of the square bounding box selected in the present study was 5% of the total length of the straight channel T-micromixer (2900 microns), that is, 145 microns. Considering the intersection of the walls at higher orders, the order of the polynomial was limited to “12” in this study. Figure 6 shows different T-micromixer configurations obtained by varying the order of the polynomial. Each wall profile results in different mixing quality as well as pressure drop.

Figure 6.

Optimized shapes for different polynomial order (a) 2^nd, (b) 4^th, (c) 6^th, (d) 8^th, (e) 10^th, (f) 12^th orders, and (g) Overlapping images of all orders.

Table 3 shows the effect of the order of the polynomial on the mixing quality as well as pressure drop at Re = 5. It is observed that with increasing order of the polynomial, the mixing quality monotonously increases accompanied by an increase in pressure drop. The channel profile following the 12^th-order Bernstein polynomial was observed to exhibit the highest mixing quality. It is observed from the table that a significant increase in the mixing quality, that is, 139.9% was also accompanied by an increase in pressure drop. However, the relative increase in the mixing quality as well as the pressure drop with increasing polynomial order decreased at higher orders of the polynomial (i.e., n = 4–12). It is to be noted that the aforementioned maximum mixing quality in the case of 12^th-order polynomial was obtained by only a 1% increase in the length of the curved boundary, that is, 2932.4 µm. A similar kind of optimization procedure was also performed at higher Reynolds numbers. However, the optimized wall profile obtained by performing the aforementioned optimization procedure does not depend on the Reynolds number of the flow.

Table 3.

Mixing quality and pressure drop variation for different polynomial order at Re = 5.

Polynomial order	Mixing quality, α	Pressure drop (Pa)
Linear	0.16692	18.113
2^nd	0.29598	195.63
4^th	0.34675	324.92
6^th	0.35914	320.4
8^th	0.38013	366.32
10^th	0.39265	397.65
12^th	0.4005	419.82

Figure 7 shows the effect of Reynolds number on the mixing quality and pressure drop at various Reynolds number for the optimized 12^th order T-micromixer. From Figure 7(a), it is observed that the mixing quality decreases with an increase in Re. It is understood that with an increase in Re, the sample's residence time is reduced which resulted in reduced mixing quality as discussed earlier. On the contrary, it is observed that with an increase in Re, the inlet pressure requirement increases monotonously as expected. It is also observed that the major portion of the pressure loss across the channel occurs in the throat region owing to the gradually reduced cross section. This indicates that the minimum throat area and the channel profile significantly impact pressure losses across the channel.

Figure 7.

Variation of mixing quality and pressure with Re of Optimized T-mixer (a) Variation of Mixing quality with Re of Optimized T-mixer and (b) Pressure variation of optimized T-micromixer along with the axial length of the channel for different Re.

Comparison between basic T-micromixer, Optimized T-micromixer, and basic T-mixer with width equals to the throat width of the optimized channel

In this section, the mixing performance of the optimized T-mixer was compared with the basic T-mixer and thin T-mixer. The thin T-mixer refers to the configuration where the thickness of the straight channel is equal to throat thickness of the optimized channel. The geometric model of the aforementioned thin T-mixer is shown in Figure 8(a). It is observed that the mixing quality monotonously decreases with Re for all geometries. However, the rate at which the mixing quality decreases significantly depends on the profile of the mixer walls. Although all the mixer configurations exhibit similar mixing performance at very low Re, the effects are amplified with increasing Re. The optimized T-mixer exhibits maximum mixing quality followed by thin T-mixer and the basic T-mixer.

Figure 8.

Schematic of T-mixer (60 μm), the mixing quality and Pressure drop variation with Re of T-mixer, Optimized T-mixer, and T-mixer with width equals to the throat width (60 μm) of the Optimized channel.

From Figure 8(b) and Table 4, an improvement of 100%–300% in the mixing quality of the optimized T-mixer is observed compared to the basic T-mixer. There is an increase in mixing quality with decreasing channel thickness. At the same values of Re at the inlet, that is, same flow rates of the species, the decrease in the channel gap results in a decrease in the transverse distance between the fluid streams, thereby reducing diffusion length. This reduction in diffusion length at smaller channel gaps significantly improves the rate of diffusion. However, the improvement of the optimized T-mixer over the thin T-mixer is due to increased residence times owing to the diverging section at the outlet. The residence time is calculated for both mixers at Re = 10. A streamline along with the centerline of the channel is considered and the time taken for a particle to traverse this length is numerically calculated by considering the velocities on the discrete nodes of the aforementioned streamline. The residence time calculated for the optimized T-mixer is 0.136 s and for the thin T-mixer, it is 0.039 s. The residence time of the Optimized T-mixer is nearly 2.5 times that of the thin T-mixer.

Table 4.

Mixing quality of basic T-mixer and Optimized T-mixer.

Reynolds number (Re)	Mixing quality of basic T-mixer	Mixing quality of Optimized T-mixer	Increase percentage (%) of mixing quality
2	0.326	0.72	120
4	0.193	0.477	147.15
6	0.148	0.375	153.37
8	0.125	0.321	156.8
12	0.099	0.275	177.77
16	0.085	0.273	221.17
20	0.076	0.285	275
25	0.068	0.276	305.88
30	0.063	0.265	320.63

Figure 8(c) shows the pressure drop variation for different configurations of the T-mixer vis-à-vis Reynolds number. The pressure drop across a T-mixer is calculated by evaluating the average pressures at the inlet and outlet and their corresponding difference. It is observed that though the pressure drop across the channel monotonously increases with Re, the rate of increase depends on the mixer configuration. The thin and optimized T-mixers exhibit significantly higher pressure drop compared to the basic T-mixer. It is due to the deformed boundaries of the T-mixer. Figure 8(d) shows a comparison of the longitudinal pressure distribution of the basic T-mixer, Optimized T-mixer, and thin T-mixer at Re = 5. Basic T-mixer has a very low pressure drop when compared to Optimized T-mixer and thin T-mixer. The pressure drop of an Optimized T-mixer is roughly identical to that of T-mixer with a width equal to Optimized channel throat width. The difference in inlet pressure requirement between thin T-mixer and Optimized T-mixer is relatively insignificant. The longitudinal distribution of pressure along with the centerline, however, depends on the channel profile. It is observed that in locations upstream, pressure in the Optimized T-mixer is slightly higher than its value in the thin T-mixer. On the contrary, in the location downstream, the thin T-mixer has higher pressure compared to Optimized T-mixer. This is due to the converging and diverging profiles at the inlet and outlet. Due to the converging and diverging sections, the pressure gradient is relatively low at the inlet and outlet, as shown in the figure. These effects result in nearly the same values of pressure drops.

Conclusions

An adjoint-based shape optimization scheme was employed, and the channel walls were modified to form Bernstein polynomial profiles. The shape optimization scheme was run by varying the order of Bernstein polynomial and Reynolds number. An upper limit on the maximum order of the polynomial was imposed as “12” in order to prevent the intersection of the channel walls. The optimized shape of T-mixer wall is independent of Reynolds number at which the algorithm is run. The mixing quality monotonously decreases with Re owing to reduced residence times. Also, the mixing quality monotonously increases with the increasing order of the Bernstein polynomial. This is attributed to reduced throat thickness. The pressure drop across the T-mixer increases with an increasing order of Bernstein polynomial owing to reduced throat thickness. A thin T-mixer with a channel width equal to the throat width of the optimized T-mixer was also designed, and the performance was evaluated. The mixing quality in the case of the thin T-mixer was lower than the optimized T-mixer. This is attributed to the converging–diverging profile in the case of the latter. However, the effect of the wall profile on the pressure drop is relatively insignificant.

Therefore, it may be concluded that the developed shape optimization procedure is beneficial in providing a geometric configuration of the T-mixer with enhanced mixing performance without losing the advantage of the ease of fabrication. Various studies could be performed by dividing the upper and lower walls into discrete line segments. It is to be noted that as the shape of the mixer changes, there will be a change in the amount of mass that the mixer can hold. There might be a possibility of change in the optimization result also. Therefore, future studies could be performed by using the mass held in the mixer also as one of the geometrical constraint while designing the shape of T-mixer. However, the application of this adjoint-based shape optimization procedure is limited. If the solution space contains multiple minima or maxima, it may fail to find global minima or maxima. Hence, it is difficult to apply for non-differential loss functions and non-convex problems.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

ORCID iD

V. R. K. Raju

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Mixing enhancement of passive type T-mixer through shape optimization