Effects on the mixing process of a coiled tube after a T-junction:Simulation and correlation☆

Shan Zhu,Kai Wang,Yangcheng Lü*

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

Keywords:Mixing T-junction Simulation Coiled tube Correlation Flow

A B S T R A C T Simulations were performed to examine the effects of a coiled tube after a T-junction on the mixing and flow characteristics.Acoiled tube was found to have two effects:inducing a radial flow and flattening the axial velocity distribution,which enhances and weakens the mixing,respectively.In the straight tube section connecting the T-junction and coiled tube,the latter may dominate and cause the mixing to deteriorate.An experiment was performed with the Villermaux/Dushman method to verify the simulation results.Based on a mixing performance simulation with various fluid and geometric structure parameters,a dimensionless correlation was obtained that can be used to determine the mixing intensity along the coiled tube with a deviation of less than 1.5%.These results provide guidance for designing a coiled tube or optimizing the operating conditions to meet the mixing requirements of specific chemical processes.

1.Introduction

The coiled tube is an essential component of a variety of equipment,such as mixers[1],heat exchangers[2],and reactors[3].It presents outstanding performance in terms of mixing,heat transfer,and mass transfer[4].It is applied in almost all industrial processes[5],including chemical industries,electronics,environmental engineering,waste heat recovery,manufacturing industries,and refrigeration.Much attention has been paid by academia to fundamental research on the flow and mixing performance to develop guidelines for choosing and designing coiled tubes[6,7].

Methodologies for characterizing the flow and mixing performances can be summarized into two categories:experimental and simulation methods[8–11].The Villermaux/Dushman method is the most commonly used experimental method and adopts paralleled instantaneous and rapid reactions to scale the rate of mixing at the molecular scale based on the measured reaction selectivity[12,13].However,experimentally determining the detailed temporal and spatial profiles of the mixing states is difficult.Alternatively,simulation methods such as computational fluid dynamics(CFD)and the lattice Boltzmann method can provide enriched local three-dimensional flow information and are a powerful tool for understanding the details of the flow and mixing processes[14].

So far,many research articles have been published on experimental and simulation methods for determining the flow and mixing in coiled tubes[18–26].They have revealed possible secondary and swirling flows in curved tubes and the global influence of various dimensionless numbers on mixing.To our knowledge,however,there have been few reports on the temporal and spatial profiles of the flow and mixing state throughout an integrated system comprising a mixer and delay tube,which is what actually determines the sensitivity of chemical processes to mixing.

The T-type micromixer is widely investigated and used,and the flow and mixing considering different structures have been studied in the literature[15–17].In this work,we established a micro flow system comprising a T-junction mixer and a coiled tube as the delay loop.The mixing process coupled with the micro flow was systematically investigated by simulation and correlation.The main contributions are as follows:(1)determining the spatial profile of the mixing state(characterized by the flow mixing intensity[27])along the coiled tube;(2)revealing the dependence of the mixing performance in the T-junction mixer on the geometric structure of the coiled tube;(3)exploring the relationship between the flow and mixing;and(4)establishing a dimensionless correlation equation for describing and predicting the mixing process along the coiled tube.Our goal was to achieve a quantitative understanding of the mixing process in detail as support for designing the coiled tube or optimizing the operating conditions to intensify specific chemical processes.

2.Simulation and Experimental Methods

2.1.Physical model and numerical formulation

Fig.1 shows the typical integrated system used for the CFD simulation in this work.The system comprised a T-junction micromixer and coiled tube.The T-junction micromixer had three straight tubes of the same size.Two were inlet branches,and the other was the mixing tube.The straight mixing tube before the coiled tube was 20 mm long.The coiled tube was constructed according to the parametric curve expressed by x=R cos θ,y=R sin θ,and z=-0.3θ,where R is the coil radius.For all simulation cases,the length of the coiled tube was 188.4 mm,and the radius of the tube(r)was 0.45 mm.

The flow and mixing behavior in the system were analyzed by using the commercial CFD code COMSOL Multiphysics 4.0,which uses the finite element method to solve the three-dimensional(3-D)equations of momentum and mass transport.The incompressible laminar flow model was adopted as the physical model in the simulation.

The momentum transfer can be described by the Navier–Stokes and continuity equations:

where ρ is the fluid density,u is the flow velocity,P is the fluid pressure,I is the unit diagonal matrix,μis the dynamic viscosity of the fluid,and F is the volume force affecting the fluid.In this work,we set(i)the reference fluid to water with ρ =1000 kg·m-3and μ =0.001 Pa·s and(ii)F to zero because the volume forces were neglected.

The mass transfer can be described by the convection–diffusion equation:

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

The boundary conditions were set as follows:(i)tube walls,no slip,and no flux;(ii)two inlets,uniform velocity through the cross-section;and(iii)outlet pressure of 0 Pa(G).We also set the concentrations of the two feeds as 20 and 2 mol·L-1,respectively.

Fig.1.Geometric construction of the integrated system comprising a T-shaped mixer and coiled tube.

The mixing intensity(IM)can be used to quantify the mixing efficiency[21]and is expressed by

where δ2is the variance of the tracer concentration in the mixing andis the theoretical maximum of δ2,which is the value(81 in this work)corresponding to the case where the two flows are totally separated from each other.In Eq.(5),Ciis the concentration at the sampling point i,Cmis the average concentration of the mixing,and ui,fis the velocity along the flow direction.The concentration and velocity of each sampling point were read from the computational grids of the cross-section to calculate IMat any specific position along the flowing direction.IMconsiders the influences of the flow and mass transfer on mixing simultaneously;it reaches a value of 0 for a completely segregated system and 1 for a perfect mixing system.

In this work,the physics-controlled mesh was adopted,and the element size was set to normal.The convergence test was performed to ensure accuracy.In detail,we selected the meshes(from coarser to coarse,normal,and fine)to calculate the profile of the mixing intensity(IM).The construction with R=5 mm is used as an example in Fig.2 to show the calculation results of IMat various cross-sections,where s is the axial distance from the T-junction to the cross-section,i.e.,mixing length.The results with the normal mesh(70 triangles in the crosssection,cell size in the flow direction:0.076–0.25 mm,205,145 cells)almost coincided with the results using a fine mesh(104 triangles,0.038–0.20 mm,451,429 cells),and clearly deviated from those using the coarse mesh(40 triangles,0.11–0.38 mm,101,366 cells)or coarser mesh(40 triangles,0.15–0.49 mm,36,838 cells).This indicates that mesh-independent convergence was almost achieved with the normal mesh and that the normal mesh should be selected to balance the calculation accuracy and cost.

2.2.Experimental characterization of micromixing

Fig.2.Test of the mesh dependency for the mixing intensity simulation.The flow velocity at each inlet was 0.3 m·s-1(r=0.45 mm,R=5 mm,D=2 × 10-9 m2·s-1,μ =0.001 Pa·s,ρ =1000 kg·m-3).

Fig.3.Profiles of the mixing intensity with s for various constructions.The flow velocity at each inlet was 0.3 m·s-1(r=0.45 mm,D=2 × 10-9 m2·s-1,μ =0.001 Pa·s,ρ =1000 kg·m-3).

The Villermaux/Dushman parallel competitive reactions were adopted in an experiment to help analyze the simulation results.One feed contained 0.03 mol·L-1KI,0.006 mol·L-1KIO3,and 0.09 mol·L-1H3BO3;the other feed contained 0.02 mol·L-1H2SO4.A stainless steel tee was used to start the mixing.Two feed tubes and one mixing tube connected the tee.The mixing tube before an online ultra-violet detector(HD-9707,Jingke,China)was fixed at a length of2.5 m,which included a 1.0 m long straight tube and 1.5 m long coiled tube.As reported in other studies[12,13],the Villermaux/Dushman method involves an instantaneous neutralization reaction and fast I2generation reaction with hydrogen ions as the shared reactant.XS,which is commonly used as an indicator of the micromixing efficiency,reflects the degree that the slower reaction occurs before hydrogen ions are completely consumed by the faster reaction.If the faster reaction is regarded as the main reaction and the slower reaction is a side reaction,a lower XScorresponds to higher selectivity.XSis defined by

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where Y is the ratio of acid consumed by the I2generation reaction divided by the total acid added and YSTis the value of Y in the total segregation case.The value of XScan be from 0(perfect mixing)to 1(total segregation).According to spectrum measurements,the error of XSwas less than 5%.

3.Results and Discussion

3.1.Comparison of the spatial evolution of the mixing intensity with various constructions

Fig.3 compares the calculated profiles of the mixing intensity with the mixing length for various constructions.For the construction without a coiled tube(the delay loop is a completely straight tube,i.e.,reference construction),IMsmoothly increased with s,and the slope became very small after IMreached 0.60 because of a continuous decrease.When the coiled tube was used,the IMvs.s curve showed a distinct change with two quite different segments.Because their boundary corresponds to the transition from the straight tube to the coiled tube,we denote these as the straight tube segment and coiled tube segment.In the straight tube segment,the tendency of IMwith s was similar to that in the reference construction.However,the value of IMwas relatively low at only 70%–80%of that in the reference construction.This indicates that introducing a coiled tube can weaken the mixing in the upstream straight tube.If mixing intensification in the straight tube segment is important,introducing a coiled structure may lead to a negative effect.In the coiled tube segment,however,IMincreased very steeply with increasing s,which allowed IMto quickly exceed the value that could be obtained in the reference construction.In general,these results indicate that the coiled tube has a remarkable but delayed mixing intensification effect in the system.We found that decreasing the coil radius can shorten the delay time or length for the mixing intensification effect to emerge.For example,the delay length,which was captured by IMin the reference construction,was 43 mm when R=50 mm and 25 mm when R=5 mm.For the cross-section at s=60 mm,IMexceeded 0.96 when R was 25 mm or less,which corresponds to high uniformity.Fig.4 illustrates this more intuitively.For the scale covering an average concentration of 4%,the mixing status in the cross-section could be perfectly determined for the constructions with R=5 and 25 mm because the ratios of the maximal concentration difference to the average concentration were 0.5%and 4%,respectively.For the reference construction,however,the two fluids were almost completely segregated.

To verify the effects of the configuration of the downstream tube on the mixing in the upstream straight tube,we changed only the coil radius and calculated the mixing intensity at various positions within the straight tube segment for comparison.As shown in Fig.5a,increasing the coil radius caused the mixing intensity at the end of the straight tube segment to first decrease(from R=5 mm to R=125 mm)and then increase(from R=125 mm to R=40000 mm).The worst mixing in the straight tube segment was obtained at R=125 mm.The reference construction can be regarded as R=zero or infinity,which provides the best mixing performance.As shown in Fig.5b,the curves of IMvs.s do not intersect with each other in the straight tube segment.This confirms that the downstream coiled tube weakens the mixing throughout the straight tube segment.Fig.6 presents the concentration distribution along the cross-section at the end of the straight tube segment.The profile of the concentration at R=5 mm is similar to that at R=40000 mm.Both are much more uniform than that at R=125 mm.

Fig.4.Concentration distributions on the cross section at s=59.45 mm:(a)R=5 mm,(b)R=25 mm,and(c)straight tube.The flow velocity at each inlet was 0.3 m·s-1(r=0.45 mm,D=2 × 10-9 m2·s-1,μ =0.001 Pa·s,ρ =1000 kg·m-3).The scale bar is in units of moles per liter(mol·L-1).

Fig.5.Effect of the coil radius on the mixing process in the straight tube segment.The velocity in each branch of the T-junction was 0.30 m·s-1.(a)Profile of the mixing intensity with s in the straight tube.(b)Mixing intensity at the end of the straight tube segment(s=20 mm)(r=0.45 mm,D=2 × 10-9 m2·s-1,μ =0.001 Pa·s,ρ =1000 kg·m-3).

If a chemical process starts with mixing and involves competitive reactions,the mixing process determines the selectivity.The Villermaux/Dushman reaction system is just an example.A faster mixing process during hydrogen ion consumption corresponds to higher selectivity of the instantaneous reaction product and a lower XS.Introducing a coiled tube may not have a single effect on the reaction selectivity;the previous simulation indicated that the mixing process can be weakened in the upstream straight tube but intensified thereafter.Comparing the reaction selectivity of different fluid mixing systems may provide evidence for multiple effects of introducing a coiled tube to the mixing process.We established an experimental setup comprising a T-junction mixer and replaceable delay tube to carry out the Villermaux/Dushman reaction and determine XS.Two kinds of delay tubes were used:a sole straight tube and straight tube plus a coiled tube.Fig.7 shows the experimental results.XSalways decreased with an increasing flow capacity,which indicates improved mixing.However,XSwas not always largest in the sole straight tube,which agrees with the simulation results showing that the mixing intensity in the sole straight tube is not always inferior to that in the straight tube plus coiled tube.Specifically,when the flow capacity is relatively high,a short length for the straight tube can allow the mixing intensity to be high enough to deplete hydrogen ions.Meanwhile,the mixing performance in the straight tube would be weakened by the connected coiled tube.So the sole straight tube with better mixing performance in this section can have higher selectivity.When the flow capacity is relatively low,a longer mixing length is required to achieve a mixing intensity high enough to deplete hydrogen ions,so the mixing intensification effect in the coiled tube may effectively increase the selectivity.In general,the requirements of a specific reaction system should be considered when optimizing the structure of a micromixer.

3.2.Comparison of flows with various constructions

The homogeneous mixing process is determined by the coupling of flow and diffusion.Thus,recognizing flow phenomena is important for understanding the mixing performance.With regard to the flow,Fig.8 presents the distribution of the velocity component in the flow direction(uf)at the cross-section for the simulation cases shown in Fig.8.For the reference construction,the profile of ufwas centrosymmetric,and the largest ufwas obtained at the center of the cross-section.The other constructions with a coiled tube segment showed eccentricity in the ufprofile.With a decreasing coil radius,the position of the maximal ufmoved away from the coil center,while the variance of ufalong the x axis became smooth.Considering that the contact plane of the two fluids was perpendicular to the x axis and that more fluid flowing near the contacting plane is preferred for mixing,the homogenization of ufalong the x axis is not good for mixing.We also calculated the variances of the velocity component in the radial direction of the coil(ur)along the symmetry axis of the cross-section,as shown in Fig.9.For the reference construction,urwas almost 0.However,remarkable radial flow with urat the scale of centimeters per second existed in the constructions with a coiled tube segment,especially with a smaller coil radius.In details,urwhen R=5 mm was almost twice that when R=25 mm.Compared with a tube having a diameter of less than 1 mm,urrelated with the secondary flow[28–30]in the coiled tube is large enough to strengthen mixing and offset the negative effect on mixing of homogenization of ufalong the x axis.

Similarly,the coil radius was set to each of 5,125,and 40000 mm to calculate and analyze the flow in the straight tube segment.As shown in Fig.10(a),the flow in the coiled tube segment also produced a radial velocity with the scale of millimeters per second at the end of the straight tube segment when R=5 mm.However,the value was much smaller than that observed in the coiled tube segment.When R was 125 mm or even larger,urbecame so small that its effect on mixing could be neglected.On the other hand,Fig.10(b)shows the variance of ufalong the symmetry axis of the cross-section in the radial direction of the coil.The curve for R=125 mm was less than the curves for R=5 and 40000 mm,which reflects the decrease in ufat the radius of 125 mm.Based on the mass balance of the flow system,a rational inference is that the variance of ufalong another direction,i.e.,the x axis,becomes smooth for R=125 mm.This is not good for mixing.In general,using the coiled tube causes the radial flow to have little influence on the mixing in the straight tube,and the homogenization of the velocity component in the flow direction along the x axis increases the distance of diffusion,which has an adverse effect on mixing.

Fig.6.Concentration distributions on the cross-section at the end of the straight tube:(a)R=5 mm,(b)R=125 mm,and(c)R=40000 mm.The flow velocity at each inlet was 0.3 m·s-1(r=0.45 mm,D=2 × 10-9 m2·s-1,μ =0.001 Pa·s-1,ρ =1000 kg·m-3).The scale bar is in units of moles per liter(mol·L-1).

Fig.7.Comparison of X S after the flow through various delay loops.The inner diameter of the tube was 0.90 mm.The total length of the delay tube was 2.5 m.The length of the straight tube section in the straight tube+coiled tube system was 1.0 m.

3.3.Correlation of mixing intensity profiles in the coiled tube

As noted above,the mixing performance along the tube can be determined by simulation,but the modeling and calculation are complicated and time-consuming processes.Fitting the simulation results to formulas may be a more convenient alternative.We carried out a series of simulation cases covering various fluids and constructions in an attempt to establish a correlation equation to predict the variance of the mixing intensity in a coiled tube.Considering that the mixing in a coiled tube is influenced by the flow rate(u),coil radius(R),tube radius(r),fluid viscosity(μ),and mixing length(Δs),four dimensionless parameters were selected for the correlation equation:the Reynolds numbers(Re,2ruρ/μ,ratio of the inertia force to the viscous force)characteristic time(T,(Δs/u)/(r2/D),ratio of the residence time to the diffusion time),Dean numberratio of the centrifugal force to the viscous force on circumferential motion)[31,32],and Schmidt number(Sc,μ/(ρD),ratio of the kinematic viscous force to the diffusion coefficient).Considering that IMranges from 0 to 1,we used the following equations:

Fig.9.Variance of the velocity component in the radial direction(u r)along the symmetry axis of the cross-section in the radial direction at s=59.45 mm(r=0.45 mm,D=2 × 10-9 m2·s-1,μ =0.001 Pa·s,ρ =1000 kg·m-3).

Fig.11 plots the calibratedΔM(output)from Eq.(9)against the simulated ΔM(target).They agree with each other well.Therefore,the mixing intensity at any position in the coiled tube can be conveniently calculated with the fitting formulas.We established a new construction to test the adaptability of the fitting formulas.In detail,the coil radius was 30 mm,and the length between the coil and T-junction was 25 mm.First,we used a construction with a short coiled tube rotating π/6 to obtain IM0in the simulation.Then,ifa coiled tube with a sufficient length is considered,IMafter a rotation of π/4 can be calculated with the fitting formulas(Re=90,T=3.8765×10-4,De=11.023,Sc=1000).The calculated IMwas 0.8033.Because the simulated IMwas 0.7912,the deviation between the calculated and simulated IMvalues was 1.5%,which indicates good agreement.Thus,we could quantitatively analyze and design the mixing process in the coiled tube with the correlation model based on a simplified simulation.

Fig.8.Distributions of the velocity component in the flowing direction(u f)on the cross-section,where s=59.45 mm:(a)R=5 mm,(b)R=25 mm,and(c)straight tube(r=0.45 mm,D=2 × 10-9 m2·s-1,μ =0.001 Pa·s,ρ =1000 kg·m-3).The scale bar is in units of meters per second(m·s-1).

Fig.10.Variances of the velocity components in the(a)radial and(b) flowing directions along the symmetry axis of the cross-section in the radial direction at the end of the straight tube(s=20 mm).

Fig.11.Comparison of the calculated and simulation results for ΔM in the coiled tubes.

4.Conclusions

The coiled tube is a widely used structural component in industry and laboratories for mixing and heat exchange.Specific combinations of the spatial position,residence time,and mixing status exist for a steady flow in a coiled tube.Thus,the mixing process can be described by the dependence of the mixing status with the position to determine the outcome of a chemical process controlled by reactants mixing.In this work,we established various geometric constructions composed of a T-junction mixer and coiled tube and investigated the effect on the mixing intensity of the mixing process along the tube by CFD simulation.The conclusions are as follows:

1)Compared with a sole straight tube,introducing a coiled tube after the straight tube can decrease the mixing intensity in the upstream straight tube segment by 20%–30%while distinctly intensifying the mixing in the coiled tube.

2)The mixing performance in the straight tube segment is poorest when the coil radius of the following coiled tube is at a medium value(around 125 mm in this work).

3)In the coiled tube,the radial flow dominates the mixing intensification,and a smaller coil radius can enhance the radial flow;in the straight tube before the coiled tube,a more uniform velocity component along the main mass transfer direction results in weakened mixing.

4)A dimensionless correlation of the mixing intensity with the Reynolds number,Dean number,characteristic time,and Schmidt number was successfully obtained to predict the details of the mixing process in a coiled tube accurately and conveniently.

Overall,this work demonstrates that combining CFD simulation with engineering-oriented correlation can achieve quantitative understanding of the mixing process details.This can help in designing the coiled tube and optimizing the operating conditions according to the mixing requirements of specific chemical processes.