Simulation of the mixing process in a straight tube with sudden changed cross-section☆

Yangcheng Lü*,Shan Zhu,Kai Wang,Guangsheng Luo

State Key Laboratory of Chemical Engineering,Department of Chemical Engineering,Tsinghua University,Beijing 100084,China

1.Introduction

Microfluidics focuses on the effective manipulation of fluids at microscale,which are concerned in varieties of research fields such as DNA sequencing,synthesis,reaction and analysis,environmental monitoring,and so on[1–6].Microfluidic technologies are drawing close attentions from both industry and academia[7–9],and some microfluidic technologies have been used at industrial scale[10,11].As a well-known representative,microreactor is commonly featured with excellent heat and mass transfer as well as precise control of residence time[12–14].On the demands of chemical process intensification towards continuous production in modern chemical industry,the microreactor has potential advantages including small hold-up of chemical reagent,low equipment and energy costs[15],precise process control,high productivity,enhanced reaction selectivity,and inherent safety[16].

Since microreactor is of kernel components in integrated microfluidic systems[17],enhanced micromixing is an essential function of micromixer.Many chemical processes following reactant mixing are accompanied with relatively slow side step[18],for which the time duration before achieving good micromixing may have significant influence on the product quality.Nanoparticle preparation by precipitation is such an example.Using microreactor with high mixing performance can shorten the duration of insufficient micromixing and inhibit unexpected nanoparticle growth and aggregation[19].The mixing performance of an equipment,dependent on both geometric structure and operating conditions,is a hotspot in microchemical engineering[20].

A number of methodologies have been exploited and developed for characterizing the mixing performance of micromixer,which can be summarized into two categories:experimental method and simulation method[21–24].The “Villermaux/Dushman”reaction system is the most widely accepted experimental method,where the distribution of products from an instantaneous reaction and another rapid reaction in parallel reflects the rate of mixing on the molecular scale[25,26].Investigations on various micromixers using this method have shown that the micromixing can be distinctly enhanced after passing through some specially designed structures,like the branch,combination[27],sudden expansion[28],contraction[29],wavelike microchannels[30]and small tubular turbulent apparatuses with star-shaped diaphragms[31].Considering the expenditure in manufacture and flow distribution control,a straight single channel/tube with sudden changed(expanded/contracted)cross section is a good choice for practical use[32–35].Common experimental methods are difficult to reveal the temporal and spatial profiles of the mixing status along flow direction.Viewed from this point,the simulation methods(such as computational fluid dynamics,Lattice-Boltzmann method)providing the local three-dimensional flow information are powerful tool to understand the details of mixing process[36,37].For the commonly used T-mixer,researchers identified by simulation various flow patterns,such as vortex flow,engulfment flow and unsteady flow,and reveal their dependence on the Reynold number and geometrical parameters[38–43].However,what happens along a straight single channel/tube with sudden changed cross section is still less reported,and its exploration is quite necessary for developing easyto-access microreactor.

In this work,we investigated the mixing process along a straight tube with expanded/contracted cross section by numerical simulation.The mixing intensity of fluid across a cross section was newly defined as an index of mixing status,and the spatial/temporal profile of which,along the flow direction,was evaluated to determine the effect scope of a sudden cross section change in mixing enforcement.The relationship between the effect and the geometrical parameters was revealed as a guidance for realizing good match between mixing process and chemical process.These progresses may lead to more meaningful quantitative description of mixing process in a flow system and more precise structure design of microreactor towards some specific chemical processes.

2.Simulation Methods

2.1.Simulation conditions

Computational fluid dynamics(CFD)simulation is used to obtain the numerical solution to the equations of momentum and mass transport to describe the mixing process.Herein,an incompressible fluid was assumed,all the simulation cases corresponded that Re?2000 and Pe?1,so a direct numerical simulation was exploited and solved by COMSOL 3.4.The following are main equations.

Navier–Stokes equation and the continuity equation describing the momentum and mass transfer:

where ρ is the fluid density,u=(u,v,w)the flow-velocity field,P the fluid pressure,I the unit matrix,μ the dynamic viscosity of fluid,and F=(fx,fy,fz)the volume force affecting the fluid.In this work,we also assume that(1)the reference fluid is water with ρ of 1000 kg·m-3and μ of 0.001 Pa·s;(2)F is zero as neglecting volume forces.

The convection–diffusion equation describing mass transfer:

where c is the concentration,D the diffusion coefficient,and R the reaction rate.The default settings include D=2 × 10-9m2·s-1and R=0(no chemical reaction is involved).

Staring from a T-junction for initiating mixing,three typical geometries are introduced in simulation and illustrated in Fig.1,corresponding to sudden expansion,sudden contraction,and their combination.The three branches of the tee are the same in diameter.The length of the mixing tube before the cross section changing is 20 mm,guaranteeing a nearly fully developed flow after the collision of two flows at T-junction.The other geometric sizes were set according to the objective of simulation and illustrated case by case.

The boundary conditions were set as follows:(1)the no slip and no flux boundary conditions were applied on the tube walls;(2)the velocity through the cross section of each inlet was uniform;(3)the gauge pressure at the outlet was 0(P=0 Pa);(4)the two feeds had the same physical properties except for solute concentration;and(5)the concentration of two feeds was 20 mol·L-1and 2 mol·L-1,respectively.

The physics-controlled mesh was adopted and the elementsize is normal.The mesh was tetrahedral.The convergence test was conducted to ensure the accuracy.In detail,we selected several meshes,from coarser,coarse,normal, fine to finer,to calculate the profile of mixing intensity(IM,described later).Taking a 20 mm length straight tube(i.d.0.45 mm)as an example,Fig.2 shows calculation results(IMof fluid at the end cross section of the tube)using various types of mesh.As seen,the results corresponding to the normal mesh(70 triangles in the cross section,cell size in flow direction 0.076–0.25 mm,95381 cells)and fine mesh(104 triangles,0.038–0.20 mm,219073 cells)almost coincide with each other,deviating apparently from those corresponding to the coarse mesh(40 triangles,0.11–0.38 mm,50927 cells)and the coarser mesh(40 triangles,0.15–0.49 mm,25997 cells).From this point of view,the requirement of grid(mesh)independence could be met for using normal mesh to characterize IM.On the other hand,using normal mesh could evade the difficulty in convergence at high velocity or critical cross section change as using mesh with higher precision.

Fig.2.The comparison of simulation results using various meshes.The flow velocity at each inlet is 0.1 m·s-1.

2.2.Index for characterizing mixing status

Based on Danckwerts'intensity of segregation[44],the mixing intensity(IM)is commonly used to quantify the mixing efficiency,and deifned as

Fig.1.Straight tube with(a)suddenly expanded cross-section,(b)suddenly contracted cross-section,or(c)their combination.

where δ2is the variance of the tracer concentration in the mixture,the theoretical maximum of δ2,is the value(81 in this work)corresponding to the case that the two flows are totally separated with each other.N the number of the sampling points,cithe concentration at the sampling point i,cmthe average concentration of the mixture,determined by the concentrations of two entering streams(of equal flow rate in this work).Since IMis normalized,it reaches a value of 0 for a complete segregated system and a value of 1 for a perfect mixing system.

In this work,the concentration and velocity of each sampling point was read from a uniformly distributed two-dimensional grid in the cross section(not the three-dimensional grids used for COMSOL simulation)to calculate IMat any specific position along the flowing direction(negative y axis).A plot of IMvs.tm(nominated mixing time)can be obtained.Herein,we defined the nominated mixing time as the volume between the T-junction and the cross section divided by the flow rate of mixing fluid.In details,before the position of cross section change,tm=l/u,where l was the distance away from the T-junction,u the flow velocity;after the position of cross section change,tm=l0/u+(x/u)(r?2/r2),where l0was the length of the straight tube,x the distance away from the position of cross section change,r the radius of the straight tube,and r?the radius of expanded/contracted tube.Fig.3(a)shows two typical results.For the straight tube without expansion,IMcomes to around 0.4 within 10 ms,and then increases gradually and slowly with tm.For the straight tube with an extremely large expansion(radius from 0.45 mm to 6.0 mm),a sudden increase of IM(from 0.5 to 0.95)could be observed.An oddity,however,is that we cannot find the continuity of the plot reflecting an actual mixing process between fluids.It implies that using the original definition of IMin a direct way may not fit for characterizing the mixing status of fluids and its time profile in a flow system.

It is well understood that a chemical process is controlled by the fluid behavior,while a snapshot of cross section cannot reflect the fluid behavior with the deficiency in velocity distribution information.Fig.4 gives typical calculation results of concentration distribution and streamlines along a tube with suddenly expanded cross section.Fig.4(d)shows clearly that the flow becomes non-uniform abruptly after the sudden expansion,and most of the streamlines concentrate towards the extended region of the original mixing tube.It implies that the fluid close to the wall of the expanded tube has much more time to achieve good mixing,although the mixing at some other location might be still poor,as shown in Fig.4(a)–(c).Unfortunately,most of what collected in the downstream is just the poor mixed fluid.

So as to evaluate the change of mixing status of the fluid passing through the cross section during an interval of Δt,the net number of fluid elements passing through each grid along the flow direction,supposed to be niat the grid numbered as i,should be considered in the calculation of the variance of the tracer concentration.Therefore,Eq.(5)should be rewritten as

Eq.(8)could be regarded as weigh factors being introduced in Eq.(5).Aubry et al.[45]and Mauri et al.[46]also introduced weigh factors to modify δ2for calculating the mixing intensity in T-mixer,but Aubry's work at very low Reynold number(～0.3)did not consider the deviation between the local velocity direction and the axial direction,and Mauri's work using bulk flow variance instead of concentration variance did lose the physical significance of the mixing intensity to some extent.In this work,the Reynolds number is up to 270.

Using Eq.(8)for IMcalculation can derive a quite different time profile of mixing status in a tube with expansion,as shown in Fig.3(b).The gradual change of IMis reasonable in physics.Besides,we can find that an expansion does not always bring favorable effect on mixing.In the context,we use the modified calculation method to explore the flow mixing process in a tube with changed cross section.

3.Results and Discussion

3.1.Mixing performance in straight tube with suddenly expanded cross-section

3.1.1.Effects of the expansion size

Fig.3.Comparison of the mixing intensity of Case a and a straighttube calculated by(a)the originalEq.(4)and(b)the modified equation.The branches ofT-junction are 0.45 mm in radius and the expanded tube is 6.0 mm in radius.The flow velocity at each inlet is 0.1 m·s-1.Those dashed lines indicate the abrupt expanded section.

Fig.4.Calculations of a tube with sudden expanded cross section.(a)the concentration distribution and(d)streamline.(b)and(c)correspond magnifications in x–z section and x–y section,respectively.The tube of 0.45 mm in radius is expanded to 6.0 mm in radius.The flow velocity at tee inlets was 0.1 m·s-1.

Fig.5.The influence of the expansion size on the profile of mixing intensity with(a)mixing length and(b)time.The tube is 0.45 mm in radius.The flow velocity at each inlet was 0.1 m·s-1.

We composed a series of straight-tube systems with various cross section radius(r?)after the sudden change position and investigated the effects of this characteristic structural size on the mixing performance.The profiles of mixing intensity with mixing length and time were simulated underthe same operating condition,and the results were shown in Fig.5.As seen in Fig.5(a),the mixing intensity at the same position always increases with the increasing r?.But,the enlargement of cross-section will lead to the increase of residence time in the tube with the same length which may give main contribution in the increasing of mixing intensity.Herein,Fig.5(b)gives a totally different scenario.Compared with the straight tube as the reference,the introduction of sudden expanded shows clear mixing intensification effect within a limited period,not always,with the exception of the case of r?=6.0 mm.Moreover,the duration time of the mixing intensification period is highly dependent on r?.It is 190 ms for r?=0.8 mm,740 ms for r?=1.5 mm,1190 ms for r?=2.0 mm,respectively.Beyond the period,the straight tube shows better mixing performance.Since the radial mass transfer distance is determined by the cross section diameter,the expanded cross section corresponds to poor mixing performance only if some extra mixing intensification mechanism exists.A reasonable mechanism is that the sudden expansion of the cross-section improves the vortices,which may be supported by the simulated streamlines along the tube with sudden expansion.Fig.6 gives some simulation results.The sparse streamlines at the expanded location imply that the flow velocity is small due to cross section expansion as well as some dead zone existence.Obvious vortices could be found in a segment of tube following the sudden expansion position,which were resulted from the separation of boundary layer due to frictional drag and positive pressure gradient.With the enlargement of the expanded cross-section,corresponding to the sub-graphs from left to right in Fig.6,the length of the vortex segment increases first and then decreases,in accordance with the changing tendency of the mixing intensification period.In general,the sudden expansion of cross section has two influences on mixing.Firstly,the mass transfer distance increases with the increasing of the tube diameter,which is adverse for mixing.Secondly,the vortex may be induced,and enhanced with flow capacity increasing,which is favorable for mixing.Therefore,the sudden expansion of cross section could be a choice to provide mixing intensification effect in the time scale of sub-second,but the working window should be explored in which the vortex shows overwhelmed effect,and the optimization of structure parameter based on vortex analysis is necessary towards specific chemical system and process.

Fig.6.The streamlines in several systems.The branches of T-junction are 0.45 mm in radius.The flow velocity at each inlet was 0.1 m·s-1.From left to right,the radius of expanded cross section was 0.8 mm,1.5 mm,3.0 mm and 6.0 mm,respectively.

3.1.2.Effects of the flow rate of mixing fluid

It is well known that the generation of vortices in a straight tube needs enough flow capacity,but it becomes easy intuitively even if the sudden expansion of cross section is introduced.As for understanding the effects of flow capacity,we fixed the geometric structure parameters and changed the flow rate of mixing fluid to simulate the profiles of mixing intensity with mixing length and time.The results were shown in Fig.7.As seen in Fig.7(a),for each position in the system,we can find that(1)the mixing intensity increases monotonously with the increase of the flow rates;and(2)a limitation of mixing intensity seems to exist corresponding to a certain geometric structure.As seen in Fig.7(b),for all the simulation cases,the increase of mixing intensity with mixing time always presents an obvious acceleration as the mixed fluid flows through the sudden expansion position.It indicates that the mixing performance limitation of the straight tube could be broken through the introduction of sudden expansion as the local vortex is intensified by the increase of flow rate.Besides,the mixing time for achieving the same mixing intensity is almost in reverse proportion to the flow rate of mixing fluid,as it being large enough.

We further carried out some Villermaux/Dushman experiments to help understand the simulation results and the relationship between what happens in a competitive reaction system and apparent mixing kinetics(described as the time profile of mixing intensity).As for the working system,one feed contained 0.03 mol·L-1KI,0.006 mol·L-1KIO3,and 0.09 mol·L-1H3BO3;the other feed contained 0.005 mol·L-1H2SO4.In experiments,a tee was used for starting mixing.Two feed tubes(i.d.1.0 mm)and one mixing tube(i.d.1.0 or 2.0 mm)connected the tee.150 mm away from the tee,a joint was used to connect an extended mixing tube,which was 350 mm in length until passing through an online ultra-violet detector(HD-9707,Jingke,China).Fig.8 shows the changing of the segregation index(Xs)with the flow capacity.As widely reported elsewhere[18],the Villermaux/Dushman method involves two competitive parallel reactions(an instantaneous neutralization reaction and a fast I2generation reaction)with hydrogen ions as the shared reactant.Xs,commonly used as an indicator of micromixing efficiency,reflects the degree that the slower reaction conducts before hydrogen ions are completely consumed by the faster reaction.If regarding the faster reaction as a main reaction and the slower reaction a side reaction,lower Xscorresponds higher selectivity.The definition of Xs,is

where Y is the ratio of acid mole number consumed by I2generation reaction divided by the total acid mole number added,Ystthe value of Y in total segregation case.The error of Xsdetermination,mainly coming from the spectrum measurements,is less than 5%.

Fig.8.The effect of flow rates on X s.

Fig.7.The influence of the flow rates on the profile of mixing intensity with(a)mixing time and(b)length.The branches of T-junction are 0.45 mm in radius,and the expanded tube is 1.5 mm in radius.

As seen in Fig.8,comparing the straight tube with the expansion tube(the radius changing from 1 mm to 2 mm),we can conclude that when both Q1(the flow rate of feed 1)and Q2(the flow rate of feed 2)are 2.0 ml·min-1or lower,the introduction of sudden expansion leads to lower selectivity.But,at any mixing length,the mixing intensity of the straight tube is always not higher than that of the expanded tube[as shown in Fig.5(a)].If the selectivity of competitive reactions has nothing to do with time profile of mixing intensity,the results of the experiment in Fig.8 should be that the line of the straight tube is always above the line of the expansion tube.Considering that the experiments gave contradictory results,we infer that the selectivity of competitive reactions are just dependent on the time profile of mixing intensity,and the vortices may be too weak at low flow capacity to improve the time profile of mixing intensity.On the other hand,when both Q1and Q2are 2.5 ml·min-1or higher,the introduction of sudden expansion leads to higher selectivity,corresponding to the largely intensified local vortices at higher flow capacity and mixing intensification effect within a certain period.These experimental results verified the importance to explore the time profile of mixing intensity and carry out the vortex analysis by simulation for enhancing the selectivity of a complicated reaction system.

3.2.Mixing performance with suddenly contracted cross-section

In this section,setting the original inner radius of mixing tube at 2.0 mm and the flow velocity at each inlet wat0.01 m·s-1,we investigated the effects of the radius of the contracted cross-section on the profiles of mixing intensity with length and time by simulation.Fig.9 shows the results,indicating that the cross section contraction could intensify the mixing but the increase of mixing intensity seems to be relatively smaller than the cross section expansion does.For achieving the same mixing intensity within the limited increase scope,the decrease of crosssection diameter could decrease the mixing length and time remarkably.Particularly,the introduction of sudden contraction may be very effective for an instantaneous and weak mixing intensification,the expenses of which include the increased pressure drop and much more serious clogging problem.The mixing intensification effect of the cross section contraction comes from two aspects.One is the decrease of the radial mass transfer distance.The other is the local vortices due to boundary layer separation.Fig.10 shows the comparison of the contracted tube and the expanded tube on velocity vector profiles.In both cases,obvious vortices could be found at the downstream from the cross section change position.But,the sudden contraction may restrain the development of local vortices to some extent,weakening the mixing intensification effect somehow.

3.3.Microchannel with combined expansion and contraction

Considering that the sudden expansion may provide remarkable mixing intensification effect within a certain length and the sudden contraction may provide instantaneous and weak mixing intensification effect,we may combine them in sequence,i.e.,a local expansion,to realize sufficient mixing at high efficiency and low cost.Herein,the length of local expansion needs optimization to match with the time limitation of the mixing intensification effect.Setting the inner radius of the initial and expanded mixing tube at 0.45 mm and 1.5 mm,respectively,Fig.11 shows the effects of the length of expanded tube(L)on the time pro file of the mixing intensity at various flow capacities.Selected from the simulation cases,the optimal L was 7 mm at u=0.1 m·s-1,9 mm at u=0.2 m·s-1,10 mm at u=0.3 m·s-1,and 12 mm at u=0.4 m·s-1,respectively,increasing with the increase of flow capacity.Besides,the mixing intensification effect of the local expansion structure with an optimal length is obvious and well guaranteed.Fig.12 shows the streamline in a tube with suddenly expanded cross section at various flow capacity.The vortex intensification region could be observed easily,and its length just increases with the increase of the flow capacity.Therefore,it is possible to find a mixing intensification strategy based on local expansion by vortex analysis towards a sudden expanded structure,and describe the mixing kinetics in details by simulation on the profile of mixing intensity.

4.Conclusions

In this work,the expression of mixing intensity was revised to describe the mixing output through a cross section in a flow system based on the consideration of heterogeneity of flow velocity in quantity and direction,using which the continuity of the mixing process in a steady-state flow system could be well reflected.According to simulation results of the profile of mixing intensity with mixing time and length and the streamlines of flow velocity,it concludes that:

(1)A sudden expansion of cross section has remarkable mixing intensification effect within a limited time(on the sub-second scale)or tube-length(on the millimeter scale),corresponding to the generation of a considerable local vortices.The intensity and scope of the local vortices is determined by both the flow capacity and the ratio of cross section change.High flow capacity and medium ratio of cross section change favor for mixing intensification.

Fig.9.The influence of the contracted section's size on the profile of mixing intensity with(a)mixing time and(b)length.The branches of T-junction are 2.0 mmin radius.The flow velocity at each inlet was 0.01 m·s-1.

Fig.10.The comparison of(a)contracted tube and(b)expanded tube on velocity vector profiles.The smaller cross section is 0.45 mmin radius,the larger cross section is 1.5 mm in radius.For(a),the flow velocity at each inlet was 0.01 m·s-1;for(b),the flow velocity at each inlet was 0.1 m·s-1.

Fig.11.The effect of the length of expanded tube on the time profile of mixing intensity.The branches of T-junction are 0.45 mm in radius.The expanded cross section is 1.5 mm in radius.From(a)to(d),the flow velocity at each inlet was 0.1 m·s-1,0.2 m·s-1,0.3 m·s-1,and 0.4 m·s-1,respectively.

(2)A sudden contraction of cross section has instantaneous but weak mixing intensification effect,in which the local vortices resulted from the boundary-layer separation,as an important factor,may be restrained by the contracted tube.

(3)Through introducing a local expansion structure with proper length,as the combination of sudden expansion and sudden contraction,the mixing intensification effects could be superposed to decrease the mixing time for achieving sufficient mixing(the mixing intensity over 90%)distinctly.

The present progresses may lead to more meaningful quantitative description of mixing process in a flow system and more precise structure design of microreactor towards some specific chemical processes.

Fig.12.The velocity streamline around the suddenly expanded cross section at various flow rates.The branches of T-junction are 0.45 mm in radius.The expanded cross section is 1.5 mm in radius.From left to right,the flow velocity at each inlet was 0.1 m·s-1,0.2 m·s-1,0.3 m·s-1,and 0.4 m·s-1,respectively.

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