Shape optimization of a micromixer with staggered herringbone groove

Shape optimization of a staggered herringbone groove micromixer using three-dimensional Navier-Stokes analysis has been carried out. The analysis of the degree of mixing is performed by the calculation of spatial data statistics. The calculation of the variance of the mass fraction at various nodes on a plane in the channel is used to quantify mixing. A numerical optimization technique is applied to optimize the shape of the grooves on a single wall of the channel. Two design variables, namely, the ratio of the groove depth to channel height ratio and the angle of the groove, are selected for optimization. A mixing index is used as the objective function. The results of the optimization show that the mixing is very sensitive to the shape of the groove which can be used in controlling the mixing in microdevices. The mixing is affected by the depth of the groove much more than the angle of the groove.     

Optimization analysis of the staggered herringbone micromixer based on the slip-driven method

In this paper the mixing effect of the staggered herringbone micromixer (SHM) was investigated by using the slip-driven method. This method simplified the 3D flow in the staggered herringbone micromixer into a 2D cavity flow with an axial Poiseuille flow. The solution of the 2D cavity flow was obtained by solving the biharmonic equation. An improved design with a cosine asymmetric factor P(z) was proposed, and its mixing effect was demonstrated by comparing the effect with the original design [Stroock, A.D., Dertinger, S.K.W., Ajdari, A., Mezic, I., Stone, H.A. and Whitesides, G.M., 2002, Chaotic mixer for microchannels, Science, 295: 647–651; Stroock, A.D., Dertinger, S.K.W., Whitesides, G.M. and Ajdari, A., 2002, Patterning flows using grooved surfaces, Anal Chem, 74: 5306–5312]. Four methods evaluating the mixing effect were used: (1) mixing images at different cycles; (2) Poincaré Sections; (3) segregation intensity and (4) stretching computation. Finally, an optimized value of P₀ = 1/6 was obtained, and the mixing effect of the improved design for different P₀ is discussed.     

Investigation of laminar flow in a stirred vessel at low Reynolds numbers

Many mixing applications involve viscous fluids and laminar flows where the detailed as well as overall flow structures are important. In order to understand the fluid dynamic characteristics of low Re laminar flows in mixing vessels, the flow induced by a Rushton impeller for three Re namely, 1, 10 and 28, was studied both experimentally and computationally. It was found that for the highest Re, the flow exhibited the familiar outward pumping action associated with radial impellers under turbulent flow conditions. However, as the Re decreases, the net radial flow during one impeller revolution was reduced and for the lowest Re a reciprocating motion with negligible net pumping was observed. This behaviour has not been reported in the literature in the past and represents a highly undesirable flow pattern from the standpoint of effective mixing. The CFD results successfully reproduced this behaviour. In order to elucidate the physical mechanism responsible for the observed flow pattern, the forces acting on a fluid element in the radial direction were analysed. The analysis indicated that for the lowest Re, the material derivative of radial velocity near the blade tip is small thus a balance exists between pressure and viscous forces; the defining characteristic of creeping flow. The velocity and pressure forces are in phase because the velocity is driven by the pressure field generated by the rotation of the impeller. Based on these findings, a simplified analytic model of the flow was developed that gives a good qualitative as well as quantitative representation of the flow.     

Mixing efficiency of a multilamination micromixer with consecutive recirculation zones

Adding recirculation zones to a mixer for a microplant is proposed for enhanced mixing efficiency. A multilamination interdigital micromixer has been widely used in microchemical plants for precision or small scale chemical process. The mixing efficiency of this micromixer is relatively low as the mixing of two fluids is executed by the laminar diffusion process. To assist the mixing by fluid action, a series of recirculation zones were added to the mixing chamber. The effectiveness of the recirculation zones on mixing was estimated through a numerical simulation which indicated the dependence on Reynolds number. Mixing efficiency increased at Reynolds number that is relevant to the condition that is prevalent in a microchemical plant. The proposed micromixer was fabricated by the lithography process on the photosensitive glass wafers. The mixing qualities of the fabricated micromixer were measured by two methods; the flow visualization of dilution type experiments and the reactivity measurement. The measurement of color intensity of the mixed fluid followed the predictions by the simulation. For a Reynolds number greater than 400 that was relevant in mixers for microchemical plant, a mixing efficiency higher than 90% was obtained by adding the recirculation zones.     

Drop breakage model in static mixers at low and intermediate Reynolds number

Drop break-up process for the flow of liquid-liquid dispersion in a static mixer has been investigated. Two new theoretical models for the drop break-up at low and intermediate Reynolds number for variant viscosity ratio of the dispersed phase to the continuous phase have been developed assuming that the flow through the static mixer elements is analogous to the flow through porous media. This concept has recently been established by Morançais et al. (Chem. Eng. Commun. 179 (1999) 77) and Legrand et al. (Chem. Eng. Res. Des. 79 (2001) 949). The boundary-layer shear force concept has been applied to predict the drop break-up at low Reynolds number and at intermediate Reynolds number, the effect of inertia on the drop break-up has been considered. The predicted drop sizes are in reasonable agreement with experimental results.     

Dimensionless number for identification of flow patterns inside a T-micromixer

Results are presented from a numerical study examining the flow dynamics of the liquid phase inside T-type micromixers. The main aim of the study was to determine an identification number for the differentiation of the different flow regimes in the liquid phase in T-type micromixers. The critical value for the identification number at which the transition from vortex flow to engulfment flow occurs was obtained. The results were used to optimize the geometrical parameters and the operating conditions to achieve high mixing performance for the liquid phase in T-type micromixers. The model results were found to be consistent with experimental data for different T-mixers available in the literature.     

Reynolds number effects in a simple planetary mixer

Planetary mixers are widely used in a diverse range of industrial applications. This paper presents an experimental investigation of mixing in a planetary mixer, and a comparison with numerical simulations based on a simple mathematical model of the flow. The model allows an exact expression for the velocity field in the Stokes flow regime, apparently the first for a mixer with genuinely moving parts, which permits accurate numerical tracking of material interfaces. Experiments performed at low Reynolds number (Re?1) show good agreement with corresponding numerical simulations, but as the Reynolds number is increased, the agreement between experiments and Stokes-flow numerics worsens, in a manner that reflects improving experimental mixing quality. Specifically, we find that islands of poor mixing shrink as Re increases. Our results suggest that, while numerical simulations in the Stokes flow regime may be used as a ‘sieve’ to select good mixing protocols at small Re, experiments or computational fluid dynamics simulations are required properly to evaluate mixing protocols operated at finite Reynolds numbers.     

Design and characterisation of the staggered herringbone mixer

The staggered herringbone mixer was studied using computational fluid dynamics (CFD) and particle tracking methods. The positions of tracer particles as well as the stretching of a fluid element associated with each tracer particle were tracked using a fourth order Runge-Kutta integration scheme with adaptive time stepping. Striation patterns observed were in qualitative agreement with experimental work from literature. The computed stretch values were found to be log-normally distributed. The specific stretch per period for a spatially periodic flow was computed. This allows for an estimation of the required length for complete mixing by further accounting the penetration depth achieved by molecular diffusion. The microchannel lengths for complete mixing computed using the mean stretch were lower than those obtained experimentally. This was attributed mainly to the fact that the experimentally derived values were measured in the central 50% of the mixer cross-section where striation thickness reduction can be observed to be slower. Furthermore, the specific stretch per period represents the mean stretch value while in reality the stretch values are distributed log-normally. In the design of mixers, a conservative estimate of the required mixing length can be obtained by replacing the mean stretch per period with a value which represents the cut-off point for the lower 10% of the distribution. The design lengths computed using these values were slightly higher than experimental ones and found to exhibit the same trend with increasing Peclet number. The pressure drop at various Re was also investigated and was found to be slightly lower than that of an equivalent grooveless channel.     

Design of microfluidic slug mixing based on the correlation between a dimensionless mixing rate and a modified Peclet number

A method to design mixing in microfluidic slugs using a modified Peclet number, , has been reported by the authors, but it was limited to mixing at constant diffusivity D. This paper reports an improved method to quantitatively determine the effect of D on a relation between Pe^* and mixing rates. Computational fluid dynamics (CFD) simulations were used for the investigation. We introduce D into the mixing rate term in the relation between Pe^* and mixing rates, and found that (mixing ) becomes a function of only Pe^*. Thus, slug mixing can be designed using the new dimensionless number, (mixing ), and Pe^*. This allows us to use mixing rate data at any value of D to estimate mixing rates at another value of D. Though Pe^* includes effects of D, l, d_s, and U_s, effects of initial arrangements of reactants inside a slug and slug cross-sectional shapes are not considered. Thus, the relations between (mixing ) and Pe^* (referred as Pe^* correlation) are quantitatively determined to cover the effects of these parameters. Furthermore, we used the Pe^* correlation to show theoretically that channel contraction is an effective microfluidic operation to enhance mixing in slugs.     

Evaluation of the mixing performance of three passive micromixers

This work presents a numerical investigation on mixing and flow structures in microchannels with different geometries: zig-zag; square-wave; and curved. To conduct the investigation, geometric parameters, such as the cross-section of the channel, channel height, axial length of the channel, and number of pitches, are kept constant for all three cases. Analyses of mixing and flow fields have been carried out for a wide range – 0.267–267 – of the Reynolds number. Mixing in the channels has been analyzed by using Navier–Stokes equations with two working fluids, water and ethanol. The results show that the square-wave microchannel yields the best mixing performance, and the curved and the zig-zag microchannels show nearly the same performance for most Reynolds number. For all three cases, the pressure drop has been calculated for channels with equal streamwise lengths. The curved channel exhibits the smallest pressure drop among the microchannels, while the pressure drops in the square-wave and zig-zag channels are approximately the same.     

Low Reynolds number flow in a channel with oscillating wavy-walls: An analytical study

An analytical study using perturbation methods has been investigated on the flow profiles and power requirements in an oscillating flow field with a wavy-walled boundary. The velocity field is found for a fluid in between two wavy-walled plates that oscillate in phase. The wavy-walled boundary causes the flow to produce regions of recirculations within the recesses of the boundary. Once the velocity profiles are identified, the power required to drive the oscillatory fluid motion is then calculated and compared to a flat plate geometry without recesses. As expected, the analytical results show that more power is needed to drive a wavy-walled system than a flat plate configuration.     

三角槽道低Reynolds数脉动流的流动特性分析

以水为工质对三角槽道低Reynolds数脉动流的流动特性进行了实验研究与理论分析。通过流动实验,分析了Reynolds数（Re）、Womersley数（W）、脉动振幅（A）等参数对流动阻力的影响。同时,建立了脉动流压降的数学模型,从理论层面解析了各脉动参数对流动阻力产生影响的作用机理,揭示了瞬时流速曲线与瞬时压降曲线之间相位差产生的原因。研究发现,相较于稳态流,脉动流会增加流动阻力,这是因为脉动流压降方程中存在三角函数项Δp_r,其周期平均值P始终大于0;就脉动参数而言,W对总压降的形阻压降部分影响较为显著,而A与总压降呈正相关;相位差的产生是由于总压降方程中存在加速度压降,而相位差的大小取决于沿程阻力、形阻压降、加速度压降三者对总压降波动的影响程度。     

Design of mixing in microfluidic liquid slugs based on a new dimensionless number for precise reaction and mixing operations

This paper reports mixing characteristics inside a microfluidic liquid slug using the computational fluid dynamics (CFD) simulations. Each slug is modeled as a single-phase flow domain. Slug-based microfluidics offers rapid mixing by internal circulation and transport with narrow residence time distribution, making it suitable for precise reaction and mixing operations. Miniaturizing the slug size to microscale allows high interactions between the slug internal fluid and the channel wall, leading to a highly effective internal circulation. However, quantitative understanding of mixing characteristics and the influences of operating parameters on mixing rate is crucial for the design of a liquid slug that ensures desired mixing rates. The simulation results provide insights into the influences of operating parameters on slug-based mixing rates. Based on the simulation results, the modified Peclet number, , is proposed for designing mixing in liquid slugs. A novel method using Pe^* to estimate mixing rates and design liquid slugs to obtain desired mixing rates is discussed. Using this method, both short (ms) and long (min) mixing timescales can be accessed in the same microfluidic device by simply varying the slug velocity.     

Thin film flow over structured packings at moderate Reynolds numbers

A model describing the dynamic evolution of waves on laminar falling wavy films at low to moderate Reynolds numbers (Re<30) over corrugated surfaces is presented. This model is based on the integral balance method, which is used to simplify the analysis by assuming a parabolic velocity profile within the bulk of the film. The predictions of this model are compared to those obtained from a computational fluid dynamics (CFD) code for which no assumptions regarding the film velocity profile are made. The film evolution profiles, obtained via numerical solution of the model equations, are used to calculate the ratio of film interfacial area to that of the substrate. The model predicts suppression of wave growth on a corrugated surface. The model is also used to predict the effect of packing geometry on the flow characteristics, which may improve packing design. Solutions obtained for Re∼200 using the CFD code demonstrate the degree of fluid accumulation within the ‘valleys’ of the structured substrate wherein re-circulation occurs.     

Investigation of impeller modification and eccentricity for non-Newtonian fluid mixing in stirred vessels

A shear thinning fluid (1% carboxymethyl cellulose) was used to investigate mixing under laminar flow conditions in an unbaffled vessel. The effects of impeller modification in addition to eccentricity were studied. Quantitative measurements such as percentage of uncovered area and coefficient of variance (CoV) of a tracer solution distributed inside the vessel were obtained using planar laser-induced fluorescence (PLIF) method. Increased eccentricity was found to be more effective than increasing rpm alone in reducing isolated mixing regions size (determined by the percentage of uncovered area). The dual-flow pitched blade turbine (DF-PBT), which was the modified version of a standard pitched blade turbine (PBT), was designed to provide both upward and downward flow at the same time to induce more chaotic flow. Though numerical analysis showed this type of flow generated, DF-PBT did not return lower values for the percentage of uncovered area and CoV than PBT did. Power consumption data were also compared between the two impeller types and eccentric locations. Further analyses focusing on the interactions between the impeller blades and fluid rheology is needed to improve laminar mixing in stirred vessels by impeller modification.     

Drag force on a circular cylinder midway between two parallel plates at very low Reynolds numbers—Part 1: Poiseuille flow (numerical)

At very low Reynolds numbers, we calculate the drag force exerted on a circular cylinder in cross flow fixed midway between two parallel plane walls which are fixed while the fluid experiences a Poiseuille profile at upstream and downstream. The drag wall correction factor is numerically investigated from a very weak interaction to the lubrication regime. The Navier-Stokes and continuity equations are expressed in the stream function and vorticity formulation and are rewritten in an orthogonal system of curvilinear co-ordinates. These equations are solved with using a finite differences method. The generation of the grid was carried out by the singularities method. We calculated the separate contributions of the pressure and viscous forces numerically. At very weak interactions, our numerical results are in good agreement with those obtained analytically by Harrison (Trans. Camb. Phil. Soc. 23 (1924) 71) and Faxèn (Proc. Roy. Swed. Acad. Eng. Sci. 187 (1946) 1). In the lubrication regime these numerical calculations are in very good agreement with those we carried out by asymptotic expansion. So that, the accuracy of the numerical code is tested. This analysis allowed us to show how that the pressure term prevails over the viscosity term in the lubrication regime. At very weak interaction, these forces have the same value.     

Consequences of an asymmetrical confinement in the transfer phenomena for a cylinder at low Reynolds numbers

In most of the systems where heat or mass transfer occur geometrical symmetries are often present. In this paper, we wish to know if the global flux transferred to cylindrical particles in motion (sedimentation, etc.) could be influenced when introducing some geometrical disturbance breaking the general symmetry of the system. To answer this question, we numerically investigate the simple configuration of a single cylindrical particle moving at constant speed between two parallel walls when the particle is off the symmetry plane and when the thermal boundary conditions are of the Dirichlet or Neumann type. When the geometrical disturbance of the system increases, the results show that the backflow that appears in such confined situations yields a non-intuitive evolution of the transfer in convective regimes. In this case a minimum of the flux appears off the symmetry plane. However, in purely diffusive regimes, we find a monotonical evolution of the transfer.     

Electroviscous effects in low Reynolds number liquid flow through a slit-like microfluidic contraction

A finite volume numerical method is used to predict electroviscous effects in steady state, pressure-driven liquid flow in a slit-like microfluidic contraction at low Reynolds number. A uniform charge density is assumed on the channel walls and the liquid is taken to be a symmetric 1:1 electrolyte. It is shown that predicted profiles of electrical potential, charge, and velocity across the channel at locations half way along the contraction and outlet sections are almost coincident with their uniform slit counterparts. A simple theory is developed that calculates the pressure drop along the channel by adding the pressure losses in the inlet, contraction and outlet sections (based on the classical fully developed electrokinetic flow solution in a uniform slit) to the entry and exit losses due to the contraction, approximated using the low Reynolds number analytical solution for a slit orifice without electrokinetic effects. For the parameter range investigated, the simple theory overestimates the apparent viscosity by up to 5-10%, compared with that determined by the numerical solution, but the differences are smaller when the surface charge density or EDL thickness is small, or the overall pressure drop is large (as occurs for long contractions).     

Mixing performance of a planar micromixer with circular obstructions in a curved microchannel

A numerical investigation of the mixing and fluid flow in a new design of passive micromixer employing several cylindrical obstructions within a curved microchannel is presented in this work. Mixing in the channels is analyzed using Navier–Stokes equations and the diffusion equation between two working fluids (water and ethanol) for Reynolds numbers from 0.1 to 60. The proposed micromixer shows far better mixing performance than a T-micromixer with circular obstructions and a simple curved micromixer. The effects of cross-sectional shape, height, and placement of the obstructions on mixing performance and the pressure drop of the proposed micromixer are evaluated.     

三角槽道低 Reynolds 数脉动流与柔性壁耦合特性研究

以水为工质对三角槽道低 Reynolds 数脉动流与柔性壁耦合特性进行了实验研究。通过传热与流动实验,分析了脉动频率(W)、脉动振幅(A)、柔性壁特性对脉动流传热及流动的影响。同时,通过可视化实验,研究了柔性壁与脉动流之间的响应特性,解析了柔性壁形变与振频对脉动流传热及流动的作用机制及分离贡献。研究结果表明,柔性流道脉动流可以实现强化传热与流动减阻双重效果,但强化传热效果相对较弱(传热效率提升0~50%),适用于以减阻为主要目的的换热场合;柔性壁减阻与削弱强化传热效率,源于柔性壁形变造成时均流通截面积增大(流体流速下降)、W与A的增大减弱脉动能量;W的增加将使得柔性壁振动对脉动流强化传热效率的削减逐步趋于主导地位,而A的增加将使得柔性壁变形对脉动流强化传热效率的削减逐步趋于主导地位;脉动流阻力的削减主要来自于柔性壁的变形(D₁>70%),而柔性壁振频对于脉动流能量耗散的抑制作用较为次要。