Age Distribution in the Kenics Static Micromixer ... - ACS Publications

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Age Distribution in the Kenics Static Micromixer with Convection and Diffusion Minye Liu* DuPont Company, Wilmington, Delaware 19898, United States ABSTRACT: Age distribution in a Kenics static micromixer is studied by mean age solution of a steady transport equation. The spatial distribution of mean age reveals locations of older material that could potentially cause process fouling and product defects. At the microscales of the mixer, both convection and diffusion strongly affect age distribution. A method is developed to compute an average effective diffusivity to measure false diffusion due to numerical errors. The range of Peclet number is found so that the scalar solution is not affected by false diffusion. The effect of convection and diffusion is studied for Re numbers that cover the full range of laminar flows in the mixer. Diffusion broadening is found to have a strong effect on the striation structures in the mixer. Age distribution in the mixer is quantitatively measured with a frequency distribution function and two different variances, the variance of residence time and the variance of mean age. It is found that the mean age variance decay rate increases almost monotonically as Re increases. It is shown that the distribution of mean age is always narrower than that of residence time.

1. INTRODUCTION Mixing is one of the most important applications for microfluidic devices. Many micromixers have been designed for various applications in chemical and biochemical engineering. Because of the limit of small sizes, flows in such devices are mainly in the laminar regime. One type of micromixer deals with steady continuous flows. For such a mixer, efficient mixing is usually achieved by the topological design of geometry. Some micromixers of this type are a miniature version of widely used static mixers in process industry for a long time. Over the years, a great amount of experience has been accumulated in process industries for the applications of such mixers. One of the most widely used static mixers is the Kenics static mixer due to its high efficiency and versatility of mixing for both laminar and turbulent flows. Recently, the fabrication of a Kenics static micromixer has been reported.1 A standard Kenics static mixer has a number of mixing elements inserted in a circular tube. The mixing elements are made from helical plates with 180° degree of twist. The consecutive elements are rotated in the opposite direction so that fluid is forced to rotate in the changing radial direction for better mixing. The flow and mixing in this type of mixer have been widely studied in the literature. For a Reynolds number below 10, the flow in the mixer is effectively in the creeping flow regime.4 At a higher Re, secondary flow on the crosssections starts to develop. The flow remains steady until around Re = 300.2 The change on the flow field has a strong effect on mixing. Hartung and Hiby3 studied scalar mixing in the mixer for a wide range of Re numbers experimentally. They found that there is a range of flow around Re = 50 for which mixing efficiency is the lowest. Hobbs and Muzzio4 studied mixing in the mixer using a chaotic dynamic system approach and reported that for Re < 10, the flow is a globally chaotic and mixing is independent of Re. At Re = 100, small regular islands can be observed on Poincare sections. Such regular islands are barriers for efficient mixing. As Re increases further, regular islands disappear, and the system becomes globally chaotic again. © 2012 American Chemical Society

Almost all reported computational studies of the Kenics static mixers focus on distributive mixing by means of a particle tracking method.4−6 Molecular diffusion has been completely neglected with the concern of numerical diffusion. For mixers in ordinary industrial scales, such a concern is reasonable since the diffusion effect is often very small on the mixing process. However, when such a mixer is used at microscales, diffusion becomes important, and the neglect of its effect could cause significant design errors.7 Reported studies on convective− diffusive mixing in static mixers are scarce. Fleischli et al.8 attempted to model the laminar diffusive mixing in a Kenics static mixer with a diameter of 50 mm. They found that the computed coefficient of variance of concentration is much lower than the values from experimental data due to severe numerical diffusion. They concluded that mixing cannot be studied by computing scalar distribution. One aspect of mixing in static mixers that has received little attention is the spatial age distribution. It is well-known that some polymer properties are strong functions of material age in the processing device. Many polymer materials degrade after a prolonged stay inside the processing device, and the degraded materials are often the cause of a low grade product or system fouling. Similar situations could happen to biomaterial processing since biocells are also sensitive to age. To identify the cause of such problems and to come up with methods of remedy, knowing the spatial age distribution in the system is crucial. One method in the literature for age distribution analysis is the residence time theory.9,10 Residence time distribution studies in a Kenics static mixer have also been reported in the literature.11,12 A residence time distribution is a probability distribution of material age measured at the exit of a mixing device. Received: Revised: Accepted: Published: 7081

April 5, 2011 March 19, 2012 April 17, 2012 April 17, 2012 dx.doi.org/10.1021/ie200716v | Ind. Eng. Chem. Res. 2012, 51, 7081−7094

Industrial & Engineering Chemistry Research

Article

This theory has been widely used in chemical reaction engineering for analyzing reactor performance. From a residence time function, the probability density function of material age inside the device can also be obtained, but this function is rarely used in the literature. This theory is an integral part of modern chemical reaction engineering knowledge, but it has a few weaknesses. A major drawback of this theory is that it cannot define the status of mixing in a device. Because the theory is based on probability distribution, spatial distribution of the material age is not available. Any nonideal mixing inside a mixing device can only be guessed or inferred based on the shape of a residence time function. Therefore, issues related to spatial age distributions can not be addressed with this method. The computation of a residence time distribution is also nontrivial. The complete time-dependent history of tracer particles or concentration has to be followed. This often requires a significant amount of computing resources. The predicted moments of residence time distribution often suffer poor accuracy, even the first moment, which is the mean residence time.13 Recently, a new method has emerged that can provide the spatial distribution of the mean age.14 The method solves a set of steady conservation equations for the moments of age in a steady flow. Because the equations are in the same form as the Navier−Stokes equations for velocity field, the same solver can be used for the solutions. Therefore, the time-dependent solver is avoided, the computing resources required are significantly smaller than those for the residence time distribution, and the solutions are much more accurate even for higher moments. Because the method solves for a complete spatial distribution of mean age, the exact sizes and locations of undesired mixing zones can be identified and quantified. In this article, this mean age method is used to study age distribution in a Kenics static micromixer. The effects of both convection and molecular diffusion on age distribution are considered. A method is first described to measure false diffusion in the numerical solutions and to find the conditions where false diffusion can be neglected. Then, several measures are discussed to quantify the effects of both convection and diffusion on age distribution. The rest of this article is organized as follows. In section 2, the geometry of the micromixer is first presented. Then, the theory of two main methods used in this article, the mean age method and the method for false diffusion analysis, are described. Results of age distribution in the mixer are discussed in section 3 where both the spatial distributions and the quantitative measures are discussed. Section 4 concludes the article with a summary of the main contributions and findings of this study.

Figure 1. Geometry of the Kenics static micromixer with six elements.

Table 1. Geometry of the Kenics Static Micromixer mixer element length diameter thickness twist angle

1.8 mm 1.2 mm 96.μm 180.° mixer

entrance length exit length overall length

0.9 mm 1.8 mm 13.5 mm

than about 300, the flow becomes unsteady.2 The Reynolds number is based on the open pipe diameter and the mean velocity in the open pipe, Re = ρUd/μ. The mixer is discretized into a total of about 3.6 million hexahedral cells. The average size of the cells is about 16 μm. The flow solution in the mixer is solved using commercial package Ansys Fluent 12.0. The third order QUICK scheme is used for better solution accuracy. The inlet velocity boundary condition is set with the analytical parabolic solution in an open pipe. The outflow boundary condition in the package is used at the exit. It is found that the zero gradient is not enforced at this boundary since for an upwinding scheme, the boundary condition at the exit is not needed.15 The velocity solution is solved first. All flow solutions are converged to a normalized residual of smaller than 10−9 to ensure accurate solutions for quantitative scalar analysis. The mean age is solved after the flow solution is obtained. 2.2. The Theory of the Moments of Age. The transport equation for mean age in a steady flow was first derived from a pulse tracer input by Spalding16 and Sandberg.17 Recently, Liu18 has shown that the same equation applies to other types of inlet concentration changes. For a pulse input flow system, the steady flow has zero tracer concentration at the inlet. At time zero, a pulse of the tracer material that has the same molecular properties as the fluid in the flow is injected uniformly at the inlet. If we measure the time history of the tracer concentration at a spatial location x inside the flow, we would obtain a distribution function c(x, t). An age frequency function can be defined with c(x, t),

2. GEOMETRY AND METHODS 2.1. Mixer Geometry and CFD Model. A six-element Kenics micromixer is studied numerically in this article. The diameter of the mixer is d = 1.2 mm. The length to diameter ratio of the elements is 1.5, and the thickness of the elements is 0.08d. A short section of open pipe is added before and after the elements as the entrance and the exit of the mixer. The entrance is half the element length, and the exit is equal to one element length. The geometry of the mixer is shown in Figure 1, and the dimensions of the mixer are listed in Table 1. A density of 1000 kg/m3 and viscosity of 0.001 kg/m-s are used as reference values in calculating the Reynolds number. Only steady flows in the mixer are studied with the range of Reynolds numbers from 0.12 to 240 since for Reynolds numbers greater

ϕ(x , t ) =

c(x , t ) ∞

∫0 c(x , t ) dt

(1)

It has been shown that for a steady continuous flow with one inlet and one outlet, the denominator is an invariant of the flow,16 I= 7082

∫0

∞

c(x , t ) dt = const.

(2)

dx.doi.org/10.1021/ie200716v | Ind. Eng. Chem. Res. 2012, 51, 7081−7094


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With this age function, the mean age of all of the tracer materials that pass through x is defined as 1 a(x ) = I

∫0

where ae2 is the flow rate weighted average values at the exit Se, ae2 =

∞

tc(x , t ) dt

(3)

∫S n·ua2 dS

(10)

e

Because the mean age is zero at the inlet and the gradient of the mean age is zero at the exit and on solid walls, the first term on the right-hand side is zero. The volume integration in the second term on the right-hand side can be calculated by volume averaging,

From the time-dependent transport equation for concentration, the governing equation for the mean age can be derived as14 ∇·(ua) = ∇·(+∇a) + 1

1 Q

(4)

From its definition, the mean age is the first moment of age. Following the same steps, the governing equation for all of the higher moments of age can be derived,14

(∇a)V2 =

1 V

∫V (∇a)·(∇a) dV ,

aV̅ =

1 V

∫V a dV (11)

∇(uMi) = ∇·(+∇Mi) + iMi − 1

where V is the volume of the flow domain between the inlet and the exit. Equation 8 can then be written as

(5)

where Mi is the i-th moment of age defined as 1 M i (x ) = I

∫0

∞

Qae2 = 2V [aV̅ − + (∇a)V2 ]

i

t c(x , t ) dt

This equation can be rearranged as

(6)

2.3. Numerical Diffusion Analysis. Numerical diffusion is the main obstacle for computational analysis of scalar mixing in a laminar convection−diffusion flow. Obtaining the solution of the velocity field is trivial, but the scalar solution is often plagued by the large numerical diffusion. For a steady incompressible flow, the leading source of numerical diffusion is the discretization of the convection term. The resulting numerical diffusion from this term is often called false diffusion. Although higher order schemes help to reduce false diffusivity, the resulting false diffusivity can still be orders of magnitude larger than the molecular diffusivity for many industrial flows. Analyzing numerical diffusion is extremely difficult for general 3D flows. For the simple first order upwind scheme in the finite volume method, it has been found that false diffusion is a function of grid size, velocity magnitude, and the angle between the grid line and the local velocity.15 For flows in microchannel mixers with complicated geometries, even with an uniform grid, both the magnitude and the direction of velocity change spatially. Therefore, the magnitude of false diffusion also varies spatially. To the author's knowledge, there has been no method reported to quantitatively measure the spatial variation of numerical diffusivity in a given scalar solution. Recently, Liu7 developed a simple method to compute an average effective diffusivity from a given numerical solution of species concentration. Here, a similar method is extended to compute an averaged effective diffusivity in a mean age solution. Consider the mean age transport equation (eq 4). Multiplying this equation by a, we will obtain an equation for a2, 2

2

2

∇·(ua ) = ∇·(+∇a ) − 2+(∇a) + 2a

+=

2τaV̅ − ae2 2τ (∇a)V2

(13)

where τ is the mean residence time of the mixer, τ = V/Q. From this equation, a volume-averaged effective diffusivity can be computed for the mixer. It will be denoted as +eff in the rest of this article. This method can also be used to compute a volume-averaged diffusivity for a selected region in the mixer. For example, for a selected element, the term on the left-hand side in eq 8 will be

∫s n·ua2 dS = Q ( ae2 − ai2)

(14)

where the subscripts i and e represent the inlet and the exit of the element. The first term on the right-hand side in eq 8 also needs to be considered.

∫s

⎡ ⎛ 2⎞ ⎛ ∂a 2 ⎞ ⎤ ∂a ⎟⎟ − Ai ⎜⎜ ⎟⎟ ⎥ +n·∇a 2 dS = +⎢Ae ⎜⎜ ⎢ ⎝ ∂n ⎠ ∂n ⎠ ⎥⎦ ⎝ ⎣ e i

(15)

where Ai and Ae are the cross-sectional areas of the inlet and the exit, and n is the local unit norm. Equation 8 then becomes ⎡ ⎛ 2⎞ ⎛ ∂a 2 ⎞ ⎤ ∂a ⎟⎟ − Ai ⎜⎜ ⎟⎟ ⎥ − 2Ve + (∇a)V2 Q ( ae2 − ai2) = +⎢Ae ⎜⎜ ⎢ ⎝ ∂n ⎠ ∂ n ⎝ ⎠i ⎥⎦ ⎣ e + 2VeaV̅

(16)

where Ve is the mixer volume between the selected inlet and the exit. Rearranging this equation, we obtain

(7)

Integrating this equation over the flow domain and using the Stokes theorem to convert both the convection term and the diffusion term to integrations over the boundary surfaces of the flow, we obtain

+=

2τeaV̅ − ( ae2 − ai2) ⎡1 2 1 2τe (∇a)V2 − ⎢ U ∂a − U ∂ n i ⎣ e e

( )

∫s n·ua2 dS = ∫s +n·∇a2 dS − 2 ∫V [+(∇a)2 − a] dV

⎤

( ) ⎥⎦ ∂a 2 ∂n

i

(17)

where Ui and Ue are the area-weighted mean velocities at the selected inlet and exit, respectively, and τe is the mean residence time between the inlet and the exit, τe = Ve/Q. For most channel mixers with periodic mixing elements, the gradient of a scalar in the flow direction can be several orders of magnitude smaller than in the lateral direction. This is because such a mixer is designed to achieve efficient mixing in the radial/lateral direction. The computed results for the Kenics micromixer

(8)

with n the unit normal of a surface. Because velocity u is zero on solid walls and the mean age is zero at the inlet, the term on the left-hand side becomes

∫s n·ua2 dS = Qae2

(12)

(9) 7083



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confirmed that the second term in the denominator is 3−4 orders of magnitude smaller than the first term for the flows studied in this article. Therefore, the second term in the denominator in the above equation can be neglected. Equation 17 can then be simplified as +=

2τeaV̅ + ( ai2 − ae2) 2τe (∇a)V2

(18)

Thus, eq 13 is a special case when the standard deviation of the scalar at the inlet is zero. Using eq 13 or 17, an effective diffusivity can be computed from a numerical scalar solution of mean age. When the diffusion from numerical error is much smaller than molecular diffusion, the computed effective diffusivity should be the same as the molecular diffusivity. If the numerical diffusion is comparable or larger than molecular diffusion, the computed effective diffusivity will be larger than the molecular diffusivity. For industrial scale mixers, the resulted false diffusivity can be orders of magnitude larger than molecular diffusivity. This large numerical diffusivity is the main obstacle in modeling convective−diffusive mixing in industrial laminar flow mixers.

3. RESULTS AND DISCUSSION 3.1. Velocity Solutions. The laminar flows in the Kenics static mixer in industrial scales have been widely studied in the literature.19 Good agreement between the predicted friction factor and the experimental measurements has been reported in the literature with a much coarser grid than the one used in this article. In this study, velocity solutions in the Kenics mixer are solved for Reynolds numbers from 0.12 to 240. Figure 2a shows the velocity vector plot for Re = 0.12 at the middle cross-section of element 4. As can be seen, even at such a small Re, there is a strong cross-sectional flow due to the helical structure of the element. By design, the rotating direction will reverse between consecutive elements. At Re = 60, a strong recirculation on the cross-section is found in the first 1/6 of each element length but disappears quickly as the flow develops further down the element. As Re is increased further, the cross-sectional recirculation sustains for longer distance. Beyond Re = 180, the recirculation flow exists in the full length of each element. Figure 2b shows the velocity vector plot on the cross-section in the middle cross-section of element 4 for Re = 240. A secondary flow with recirculation can clearly be seen. These findings are in agreement with the study by Van Wageningen et al.,2 who computed the flow solutions in a Kenics static mixer for Re = 10−1000. The strength of the cross-sectional flow also increases with Re. The ratio of the maximum velocity of the cross-sectional flow to the maximum velocity magnitude in the mixer is about 0.5 for Re = 0.12. At Re = 60, this ratio increases to about 0.57. This ratio further increases to about 0.7 for Re greater than 180. These findings are in agreement with those by Regner at al.,20 who computed helicity of the cross-sectional flow along the axial direction of an element as a function of Re. They reported a monotonic increase of helicity as Re increases, indicating stronger cross-sectional flow at higher Re. Figure 3 shows the velocity component parallel to the element along the radius normal to the element and in the middle cross-section of an element for several Re numbers. The velocity is scaled by the average velocity at the entrance of the mixer. As can be seen from these curves, even in the creeping flow regime, the magnitude of the cross-sectional flow is larger than the mean average velocity in an open pipe. This is still true

Figure 2. Velocity vector plots on the middle cross-section of element 4. The color shades and vector lengths show velocity magnitude in m/s. (a) For Re = 0.12 and (b) for Re = 240.

Figure 3. Scaled x-velocity distribution along the y-axis in the direction normal to element on the middle cross-section of an element. Solid line (black), Re = 0.12; short dashes (red), Re = 12; dots (green), Re = 60; dash-dots (blue), Re = 120; and long dashes (tan), Re = 240. 7084



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even considering the difference in cross-sectional areas between the open pipe and the mixer with the element, which is about 11%. The increase of the strength of the cross-sectional flow with Re can also be seen in the figure. At small Re, the velocity distribution is nearly a parabola. The distribution deviates further and further as Re increases. Because of these strong differences in velocity distribution, we can expect that mixing and age distribution will be strongly affected. 3.2. Numerical Diffusion. Molecular diffusion is often important in a laminar mixing process, and CFD-predicted laminar mixing is often dominated by numerical diffusion. Without knowing the exact quantitative levels of the numerical diffusion, it is difficult to determine whether an obtained scalar distribution is accurate or not. Figure 4 shows the mean age distributions at

Figure 5. Effective diffusivity vs molecular diffusivity. (a) Re = 0.12 and (b) Re = 120. The straight line is for +eff = + . Figure 4. Mean age distribution along radius at 135° with element wall on the middle cross-section of element 4. Solid line (black), + = 0; thin solid line (magenta), + = 1 × 10−13 m2/s; dash-dots (tan), + = 1 × 10−12 m2/s; sdots (green) + = 1 × 10−11 m2/s; short dashes (blue), + = 1 × 10−10 m2/s; and long dashes (red), + = 1 × 10−9 m2/s.

smaller than the molecular diffusion for Pe < 1200; thus, the molecular diffusion effect can be accurately computed on average. As the specified molecular diffusivity further reduces, the computed effective diffusivity starts to deviate from the linear zone shown by the straight line. This is in agreement with the results shown in Figure 4 for mean age distribution. For + = 1.0 × 10−11 m2/s, the computed effective diffusivity is about 2.6 × 10−11 m2/s. The computed effective diffusivity converges to about 2.5 × 10−12 m2/s when molecular diffusivity is set at 0. This corresponds to an effective Pe number of Pef = Ud/+eff = 4.8 × 104. Figure 5b shows the comparison of diffusivity for Re = 120. The curve is similar to that for Re = 0.12, but the linear range is at much higher diffusivity. The linear range ends at about 1.0 × 10−7 m2/s, and the effective diffusivity starts to deviate from the specified molecular diffusivity as it is further reduced. This is because false diffusion increases with the magnitude of velocity.21 However, the corresponding Peclet number is the same at 1200 as the previous case. The actual effect on the diffusion broadening of age distribution is thus the same when considering the diffusion penetration distance since this distance is a function of Pe, δ ∼ (+ τ)1/2 = (dL/Pe)1/2. Although at higher Re the false diffusion is higher, the mean residence time, which is the time the diffusion takes effect, is shorter. When the specified molecular diffusivity is set at zero, the resulted numerical diffusivity is 5.3 × 10−9 m2/s, and the corresponding Pef is 2.3 × 104. This is about half of the corresponding Pef for Re = 0.12, indicating a slightly higher nondimensional numerical diffusion. Because false diffusion is also a function of the angle between the local grid line and the velocity direction,21 stronger cross-sectional flow and recirculation for Re = 120 will naturally

different diffusivity along a radius about 135° with the element wall at z = 7.2 mm for Re = 0.12. This radius is about normal to the striations of the mean age mixing as shown in the insert. The range of specified molecular diffusivity is from 0 to 1.0 × 10−9 m2/s. For + = 0, all diffusion should be due to numerical error. At the upper end, the diffusivity is a typical value for water. The results in Figure 4 show that for water, the diffusivity is large enough to destroy the manifold of older material due to a wall effect. The striations have also been smeared out almost completely after 4.5 periods of mixing. When diffusivity is reduced by an order of magnitude to 1.0 × 10−10 m2/s, the distribution of mean age is pronouncedly different, showing the strong effect of diffusion. For another order of magnitude decrease in diffusivity, the strong effect can still be seen from the distribution curve. However, a further decrease in diffusivity has little effect on the distribution. This is a clear indication that numerical diffusion has masked the molecular diffusion. To better understand the effect of numerical diffusion on the mean age solution, eq 13 is used to compute the effective diffusivity. Figure 5 compares the computed effective diffusivity with molecular diffusivity specified for each of the numerical solutions. At Re = 0.12, Figure 5a shows that the computed effective diffusivity is the same as the specified molecular diffusivity for + > 1.0 × 10−10 m2/s. The corresponding Peclet number is 1200. This means that numerical diffusion is much 7085



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striations are smeared by diffusion. As Re is increased to 60, the striations are no longer straight anymore, as shown in Figure 6b. This is due to the cross-sectional recirculation near the entrance of each element as discussed in section 3.1. Figure 6c shows the mean age distribution for Re = 120. The crosssectional flow at this Re number is stronger, and the crosssectional recirculation exists almost in the full length of each element. It is interesting to notice some fine structures in the distribution, for example, the three-finger structure inside the odd-numbered elements. Figure 6d shows the mean age distribution for Re = 240. The effect of strong recirculating flow can clearly be seen from the spatial structure of the mean age distribution. The material with older mean age shed off from the previous element is stretched and folded by the recirculating flow. Several other interesting observations can also be made from Figure 6. First, the spatial structure of mean age distribution for all Re numbers are periodically self-similar. The similar structures are repeated, and finer structures are added periodically, a typical chaotic flow property of self-similarity. Second, the positions of older material repeat themselves periodically, indicating some well-defined manifolds. Such manifolds can be further examined by the mean age distribution on the crosssections of the mixer. Figure 7 shows the mean age contour plots at the middle of element 4 for different Re numbers. The lower bounds in the plots are the minimum value of mean age on the cross-section, and the upper bounds are twice of the mean residence time at the location. From these contour plots, the effect of Reynolds number can be seen readily. At Re < 12, the striations are similar as shown in Figure 7a,b. The flow is in the creeping flow regime, and mixing is purely by flow redistribution due to the helical elements. The creeping flow is affected by the inertial effect of the flow when Re is increased to 24. This is reflected by the shapes of the mean age striations in Figure 7c. At Re = 60 and 90, as shown in Figure 7d,e, the simple lamellar structures disappear completely, and the typical chaotic folding structures emerge. Such spatial distributions suggest that there is a mixing mechanism change, and the periodic distributive mixing becomes chaotic stretching and folding due to the secondary flow on cross-sections. A similar mixing mechanism change of species concentration has been reported by Liu7 in another microchannel mixer. At Re = 120, the distribution structures change further, and the three-fingers structure shown in Figure 6c can be identified on the cross-section. Figure 7g,h show the mean age distributions for Re = 180 and 240. The large structures of distributions are similar in these two plots, with some differences in finer details. From all of the mean age contour plots above, it can be noticed that the area of mean age older than the upper bounds in each plot reduces as Re increases. This seems to suggest that the stronger secondary flows on the cross-sections enhance mixing of age. The evolvement of age distribution along the mixer can be studied by inspecting the cross-sectional variation along the mixer. Figure 8 shows the mean age distributions at the middle of each element at Re = 0.12. Again, the lower bounds of the plots are the minimum value, and the upper bounds are twice of the mean residence time at each cross-section. In the first element, a layer of larger mean age material is clearly shown in Figure 8a. The younger material is at the center of each half of the cross-section. In the second element, the same wall layer of older material remains, and a new layer of older material is added that divided the younger material region into two in each

produce more numerical diffusion. Similar results were reported by Liu7 who found that at higher Re, a slightly larger diffusion penetration distance resulted for species concentration. The results shown in both Figure 5a,b indicate that for the mean age solutions computed with the given mesh, the effect of numerical diffusion can be neglected when the Peclet number is smaller than about 1200. For a larger Peclet number, the effective diffusivity in the solution is larger than the specified molecular diffusivity. In such a situation, the solution can be considered as an approximate one with the effective diffusivity as the actual molecular diffusivity. Thus, the range of the numerical solution can be expanded. For example, for Re = 120, by specifying a zero molecular diffusivity in the problem setup, the solution can be considered approximately as one with a molecular diffusivity of the order of 5.0 × 10−9 m2/s. The range is then expanded from Pe < 1200 to Pef < 2.3 × 104. 3.3. Mean Age Spatial Distributions. In this section on, the effect of convection and diffusion on mean age distribution in the mixer is studied at different Re numbers but the same effective Peclet number of 12000. From Pef = Re(ν/+eff ) with the reference viscosity of 1.0 × 10−6 m2/s, a target +eff can be calculated for each Re. With a few trials for each Re, the desired molecular diffusivity + can be found such that +eff of the mean age solution is near the target value. For Re = 0.12, the specified molecular diffusivity is 4.0 × 10−12 m2/s, and the computed effective diffusivity is 1.00 × 10−11 m2/s. For Re = 240, the specified molecular diffusivity is found to be 8.0 × 10−10 m2/s, and the resulting effective diffusivity is 2.01 × 10−8 m2/s. The contour plots in Figure 6 show the spatial distributions of mean age on x = 0 plane across the length of the mixer at

Figure 6. Contour plots of mean age on x = 0 plane across the mixer. (a) Re = 0.12, (b) Re = 60, (c) Re = 120, and (d) Re = 240. The color shades show the mean age scaled by the mean residence time of the mixer.

different Re numbers. The upper limit of the mean age is cut at the mean residence time to show the spatial structures. The color shades represent the local mean age scaled by the mean residence time of the mixer. The qualitative effect of Re on mean age distribution in the mixer can be seen clearly from these contour plots. At low Re when the flow is in the creeping flow regime, the striations of different mean age are mainly straight, as can be seen in Figure 6a. These striations are generated by the material coming off the element walls from the previous periods that have a higher mean age than those in the interior of the flow. The number of striations increases as the material flows down the mixer. The boundaries between 7086



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Figure 7. Contour plots of mean age on the middle cross-section of element 4. The lower bounds of the color shades are the minimum mean age on the cross-section, and the upper bounds are two mean residence time at the cross-section. (a). Re = 0.12, (b) Re = 12, (c) Re = 24, (d) Re = 60, (e) Re = 90, (f) Re = 120, (g) Re = 180, and (h) Re = 240.

half of the section. The mechanism of mixing in a Kenics static mixer in the Stokes regime is similar to the baker's transformation. The mixed material is repeatedly divided into two parts after each period. Without diffusion, the number of striations after m period will be 2m.22,23 This doubling process can also be seen in Figure 8c,d after 3 and 4 periods. The younger materials are repeatedly divided by the older materials shedding off from the previous element walls. However, this striation doubling process ceases in Figure 8e,f. On the contrary, these two contour plots are almost identical and have fewer striations

than in the fourth period in Figure 8d. Apparently, a diffusion broadening effect7 is the cause of the result. From all of the contour plots after the first period, the oldest striation is at the same relative position on each of the cross section. The mean age contour plots on the middle cross-section of each element for Re = 240 are shown in Figure 9. The effect of cross-sectional recirculation on mean age distribution can clearly be seen in Figure 9a in the first element. As compared to the case of Re = 0.12, the element walls are only partially covered by the older material. The strong recirculation washes 7087



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Figure 8. Contour plots of mean age distribution on the middle cross-sections of elements for Re = 0.12. The color shades are mean age in s. The lower bounds are the minimum mean age on the cross-section. The upper bounds are two mean residence times at the cross-section. (a) Element 1, (b) element 2, (c) element 3, (d) element 4, (e) element 5, and (f) element 6.

studied by mean age frequency function.24 For a 3D CFD solution of a scalar field, the frequency function can be defined as

away the older material from part of the walls as shown in Figure 9a. The striation doubling mechanism has completely disappeared at this high Re number. Instead, the mixing mechanism is the typical chaotic stretching and folding. As the material flows down the mixer, less and less older material can be found near walls as shown from Figure 9b−f, indicating a more efficient mixing of age than for the lower Re cases. Selfsimilar distribution structures can also be found in these figures. The structures are strongly influenced by diffusion. 3.4. Mean Age Frequency Distributions. The convection and diffusion effects on age distribution can be further

g (a) =

1 d v (a) V da

(19)

where dv is the differential volume with mean age between a and a + da. Two ideal flows can be used to compare mixing efficiency. One is the plug flow, and the other is the laminar flow in an open pipe with zero diffusion. The residence time 7088



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distribution function is F(t) = 0 for t < τ and F(t) = 1 for t > τ for a plug flow, and F(t) = 0 for t < τ/2 and F(t) = 1 − τ2/4t2 for t > τ/2 for the flow in an open pipe.25 Danckwerts9 showed that the internal age frequency function φ(α) for all of the molecules in a flow system is related to the residence time function by φ(α) = [1 − F(α)]/τ. Since for a flow with zero diffusion the mean age at any point in space is the same as the molecular age,14 the internal age frequency function is then the same as the mean age frequency function. Thus, for a plug flow, g (a) =

1 , a ≤ τ; τ

g (a) = 0, a > τ

range when the creeping low is perturbed by inertia. From Re = 48 to Re = 150, the relative first appearance mean age increases almost linearly with Re. This is the flow regime with stronger secondary flow on cross-sections. However, the cross-sectional recirclation is still not in the full length of each element. Thus, the mixing mechanism in this regime is the combination of the secondary flow and the baker's transformation in the creeping flow regime. Beyond Re = 180, the increase of the relative first appearance mean age is still almost linear with Re but at a smaller slope. This is the range where the cross-sectional recirculation exists in the full length of each element. The first appearance mean age can be used to analyze crosssectional mixing by comparing to a plug flow. The curve in Figure 11 shows in general that as Re increases, the mixing approached that in a plug flow. However, a minimum value appears at about Re = 50. As discussed in section 3.2, there is a mixing mechanism change due to the secondary flow. For Re < 50, the weak secondary flow hinders the cross-sectional mixing; thus, some materials move faster than others and reach the exit much earlier. For Re > 50, the secondary flow starts to enhance the mixing. This finding is in agreement with the experimental finding for species mixing by Sir and Lecjaks, 2 6 who found that for a given degree of homogeneity, the largest number of elements required is around Re = 50. The Peclect number effects on mean age frequency function are shown in Figure 12 for Re = 0.12. The specified values of molecular diffusivity are + = 0, 1.0 × 10−12, 1.0 × 10−11, 1.0 × 10−10, and 1.0 × 10−9 m2/s, and the corresponding Pe numbers are from infinity to 120. The computed effective diffusivity are +eff = 2.49 × 10−12, 5.54 × 10−12, 1.68 × 10−11, 1.02 × 10−10, and 0.99 × 10−9 m2/s. The diffusion penetration distances in the mixer are in the range of 45.6−352 μm based on the effective diffusivity. From the figure, it can be seen that diffusion has a strong effect on the distribution curve. At small values of diffusivity, the curves are closer to those for the open pipe and have a longer tail of large mean age values. As diffusivity increases, the curves approach that for plug flow. The tail is also shorter for larger diffusivity. The overall trend is similar to the effect of Re numbers. This clearly shows that both convection and diffusion are important to the mixing process in the mixer. 3.5. Variances of Age. 3.5.1. Decay of Mean Age Distribution. Age distribution in the mixer can be quantitatively measured with variances at cross-sections down the mixer. The variance of mean age on a cross-section can be defined as

(20)

and for the flow in an open pipe, g (a) =

τ 1 ,a≤ ; τ 2

g (a ) =

τ τ ,a> 2 4a 2

(21)

The frequency functions of the mean age solutions discussed in the previous section for different Re numbers in the mixer are plotted in Figure 10. Also plotted in the figure are the nondimensional functions of eqs 20 and 21. From the figure, it can be seen that all of the curves for the mixer deviate from the ideal flows. The two ideal flows are two limits of the mixer. At smaller Re numbers, the curves have longer tails, and the tails start early as compared to the curves at higher Re numbers. There is a clear trend that when Re increases, the curves deviate further from that for the open pipe and approach that for plug flow. The difference between the curves for Re = 0.12 and 12 is small, indicating that the mixing state in the mixer is similar. This is in agreement with the previous discussions. At Re = 60, the curve shows a noticeable difference. The peaks and valleys are at different mean age locations. The tail starts at higher mean age but is shorter, indicating a slightly more uniform distribution. At Re = 120, the curve can be further differentiated, but the tail after the mean residence time is similar to that for Re = 60. As Re is further increased to 180 and 240, the peaks and valleys are more pronounced, and the tails are much shorter. Overall, the curves move closer to that for plug flow at higher Re. For ideal flows, the distribution of mean age is constant before the first appearance age. For the mixer, the curves have peaks and valleys, reflecting the complex mixing structure due to the circulations in the flow field and the spiraling motion of fluid. These results show that mean age frequency function is an effective way of measuring relative mixing efficiency to the ideal flows. The first appearance age at the exit can clearly be identified on the curves for the ideal flows. For the plug flow, all material arrives at the exit at the same time. Thus, the first appearance age scaled by the mean residence time is 1. For the empty pipe, the nondimensional first appearance age is 0.5 and is where the nonlinear tail starts. For a nonideal mixer, a similar first appearance mean age can be defined as the minimum mean age at the exit. It should be pointed out that this mean age is different from the first appearance time defined for residence time. It is generally higher than the first appearance residence time due to averaging with older material, and it is the same only when diffusion is not present. Figure 11 shows the ratio of the first appearance mean age to the mean residence time for different Re numbers. There seem to be four different regions on the curve. For Re < 24, the relative first appearance mean age increases as Re increases. This is the range with small inertial effect of the flow. From Re = 24 to Re = 48, the relative first appearance mean age decreases as Re increases. This is the

σa =

(a − a )2 a̅

=

a2 − a ̅ 2 a̅

(22)

where the over bars indicate flow averaging on a crosssection. This definition is the same as the variance of concentration in literature27 since mean age is also a scalar. Figure 13 shows the variance along the mixer for the cases discussed in section 3.3. The first data point is at z = 0.9 mm at the entrance of the first element, and the subsequent data points are at the end of each element. Small differences at the beginning of the first element can already be seen. This is due to the effect of different degree of inertia in the flow. At higher Re numbers, the mean age distribution is dominated by the incoming fully developed pipe flow. At lower Re numbers, the leading edge of the first element affects both the flow and the mean age distribution. Thus, lower 7089



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Figure 9. Contour plots of mean age distribution on the middle cross-sections of elements for Re = 240. The color shades are mean age in s. The lower bounds are the minimum mean age on the cross-section. The upper bounds are two mean residence times at the cross-section. (a) Element 1, (b) element 2, (c) element 3, (d) element 4, (e) element 5, and (f) element 6.

variances result at lower Re numbers on this cross-section. Down the mixer, the decay rate is generally higher at higher Re numbers, with an exception of Re = 90. At this Re number, the decay rate is higher than that of Re = 60 before element 3 but becomes lower afterward. For Re < 60, scalar mixing is

mainly due to the division of material of different ages by the helical elements, a mechanism similar to the baker's transformation. For higher Re numbers, strong crosssectional recirculation develops, and the mixing mechanism is a combination of distributive mixing and dynamic mixing 7090



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Figure 10. Mean age frequency function for different Re values. Solid line (black), Re = 0.12; short dashes (tan), Re = 12; dash-double dots (magenta), Re = 60; dash-dots (blue), Re = 120; dots (green), Re = 180; and long dashes (red), Re = 240. Thin dashes, plug flow; thin solid line, open pipe flow.

Figure 12. Mean age frequency function for different diffusivity. Solid line (black), + = 0; short dashes (tan), + = 1 × 10−12 m2/s; dash-dots (blue), + = 1 × 10−11 m2/s; dots (green), + = 1 × 10−10 m2/s; long dashes (red), + = 1 × 10−9 m2/s; thin dashes, plug flow; and thin solid line, open pipe flow.

Figure 11. First appearance mean age, or minimum mean age at the exit, scaled by the mean residence time as a function of Re. Figure 13. Mean age decay rate along mixer for different Re. Solid line with open squares (black), Re = 0.12; long dashes with open squares (magenta), Re = 12; short dashes with open squares (red), Re = 60; dots with solid circles (blue), Re = 90; dash-dots with solid circles (green), Re = 120; long dashes with solid circles (tan), Re = 180; and solid line with solid circles (gray), Re = 240.

of stretching and folding by the flow. Hartung and Hiby have found that around Re = 50, the most elements are needed to achieve the same mixing quality for species concentration. The same results were confirmed later by Sir and Lecjaks.26 Hobbs and Muzzio4 have found that with zero diffusivity, material particle mixing has a higher decay rate at lower Re. Liu7 have found that in a topologically periodic micromixer, the species concentration decay is faster at low Re when the mixing mechanism is baker's transformation than at higher Re when the mixing mechanism is a combination of kinematic redistribution and dynamic stretching and folding. A maximum variance is also found around Re = 60. These findings are in agreement with the theoretical prediction of Wiggins and Ottino28 that baker's transformation is the optimal mixing mechanism. On the other hand, the results in Figure 13 show that age distribution does not necessarily follow the same trend as the species concentration distribution. This is because in the study of concentration mixing, molecules with different age are not distinguished. Therefore, the mixing of age reveals a new dimension in material 3

mixing process. At higher Re, some efficiency may be lost in mixing materials, but a more uniform distribution in material age is gained. For some applications, such a trade-off may be desirable. 3.5.2. Variance of Residence Time Distribution. Scalar mixing in a continuous flow is often characterized with the variance of residence time distribution. σe2 =

1 τ2

∫0

∞

(t − τ )2 E(t ) dt =

te2 − τ 2 τ2

(23)

where E(t) is the residence time frequency function. To compute the variance of residence time from eq 23, the full distribution function E(t) has to be known. However, Liu and Tilton14 have shown that the moments of residence time are 7091



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equal to the flow-averaged moments of age defined in eq 5 at the exit. Using this relation, eq 23 becomes

σe2 =

It can be shown that mean age variance is only part of the variance of residence time. Zwietering30 showed that the total variance of molecular age inside a steady continuous flow system is equal to the sum of the variance of mean age and the variance about the mean age. Following the same method, starting from eq 23, it can be shown that

M̅ 2, e − τ 2 τ2

(24)

The mean residence time τ is equal to the flow-averaged mean age at the exit, τ = ae̅ , and M̅ 2,e is the flow-averaged second moment of age at the exit. More recently, Liu29 has found that the variance can simply be computed by

σe2 =

2aV̅ − τ τ

σe2 = σ12 + σa2

(26)

where σ1 is the variance about mean age. From its definition, the residence time frequency function E(t) is

(25)

E (t ) =

Thus, the variance of residence time can be computed from the mean age solution without the need of knowing the full function of residence time distribution. The computing cost for the steady transport equation of mean age is usually several orders of magnitude smaller than that for the full time-dependent function of E(t).14 The solid line in Figure 14 shows the variance of residence time at the end of the mixer for different Re values. In general,

1 QI

∫S u(x)c(x , t ) dx = Q1 ∫S u(x)ϕ(x , t ) dx e

e

(27)

where Q is the volumetric flow rate of the mixer. Equation 23 can then be written as 1 τ2 1 = Qτ 2

∫0

σe2 =

∞

(t − τ )2 E(t ) dt

∞

∫0 ∫S u(x)(t − τ)2 ϕ(x , t ) dx dt e

(28)

In the above integral, (t − τ) can be expanded as (t − a + a − τ)2 = (t − a)2 + (a − τ)2 + 2(t − a)(a − τ). Then, the above equation becomes 2

σe2 =

∞ 1 u(x)(t − a)2 ϕ(x , t ) dx dt 2 Se Qτ 0 ∞ 1 u(x)(a − τ )2 ϕ(x , t ) dx dt + 2 Se Qτ 0 ∞ 2 + u(x)(t − a)(a − τ )ϕ(x , t ) dx dt 2 Se Qτ 0

∫ ∫

∫ ∫ ∫ ∫

(29)

The first term is a variance of molecular age about mean age. 1 Qτ 2 1 = Qτ 2

σ12 = Figure 14. Age variances at the end of the mixer for different Re. Solid line, variance of residence time σe; dashes, variance of mean age σa.

=

the variance decreases as Re increases except arount Re = 90 where the variance reaches a local maximum. This seems to be in agreement with the findings of Hobbs and Muzzio4 that regular islands appear around Re = 100. The regular islands are separated from chaotic mixing region by KAM surfaces. Inside the regular islands, the stretching rate is linear, while in the chaotic region, it is exponential. 3.5.3. The Relationship of Variances. It is important to understand the differences and the relationship between the two variances discussed above. A residence time function is a probability distribution function for molecular age at the exit of a mixing device. The mean age distribution at the exit is about the spatial distribution of mean age at all points on the surface. The distribution of mean age is always narrower than the residence time distribution due to diffusion.14 At the limit of zero diffusion, the two distributions are the same. The dotted line in Figure 14 shows the variance of mean age at the end of the mixer at different Re values. As compared to the variance of residence time, the values are all much smaller, confirming its narrower distribution than residence time.

∞

∫0 ∫S u(x)(t − a)2 ϕ(x , t ) dx dt e

∞

∫0 ∫S u(x)(t 2 − a2)ϕ(x , t ) dx dt e

ae2

M̅ 2, e − τ2

(30)

In the above derivation, the definitions of mean age and moments of age of eqs 3 and 6 have been used. The second term in eq 29 is a variance of mean age about the mean residence time, 1 Qτ 2 1 = Qτ 2

σ22 =

=

ae2

∞

∫0 ∫S u(x)(a − τ)2 ϕ(x , t ) dx dt e

∞

∫0 ∫S u(x)(a2 − τ 2)ϕ(x , t ) dx dt e

−τ τ2

2

(31)

Because the flow-averaged mean age at exit is the same as the mean residence time on the cross-section, a̅ = τ, this term is then the same as the variance defined in eq 22 for exit, σ2 = σa. It can be shown that the third term in eq 29 is zero. Therefore, the variance of residence time can be split into two variances, a variance of mean age and a variance about mean age. 7092



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of the variance of mean age and the variance of residence time is derived from their definitions. It is found that the variance of residence time consists of two variances, the variance of mean age and the variance about mean age. Therefore, the variance of residence time is always greater than the variance of mean age. The distribution of mean age on the exit of the mixer will always be narrower than residence time distribution.

The distance between the two variance curves in Figure 14 is thus the variance about mean age defined in eq 30.

4. CONCLUSIONS Undesirable age distribution in a mixing device is a potential source of process fouling and product degradation for polymers and biomaterials. In this article, detailed spatial age distributions in a practical micromixing device are analyzed quantitatively by considering both convection and diffusion. A method is developed to measure numerical diffusion in a numerical solution of mean age. Numerical diffusion has been the main obstacle of computational analysis of scalar distribution in laminar flows. In such a flow solution, numerical diffusion is often much higher than molecular diffusion and thus masks the true effect of molecular diffusion on the scalar distribution. Without knowing the quantitative level of numerical diffusion, it is not possible to study a scalar distribution in a convection−diffusion flow quantitatively. That is why it is rare to find such analysis in the literature. This method is applied to the mean age distribution in the Kenics static micromixer. It is found that the computed effective diffusivity is in agreement with the computed diffusion effect on mean age distributions. A range of Peclet numbers can be determined within which numerical diffusion is smaller than molecular diffusion. This range is Reynolds number-dependent and can include most of the aqueous materials in chemical and biochemical process industries. The computed spatial mean age distributions reveal chaotic and self-similar structures in the mixing elements. As Re increases, the striations in the distribution change from simple lamellar structures to complicated stretching and folding structures. Besides the nonuniform flow distribution, the walls of both the mixer tube and the mixing elements are also main sources for nonuniform age distribution. Because of the periodic placement of the mixing elements, some older materials are created on the element walls and return back to the interior of the mixer at the end of the elements. Both contour plots and frequency functions reveal that higher Re flows improve mean age distribution with stronger secondary flows in cross-sectional directions. The wall layers of older materials are thinner due to the stronger radial convection. The mean age frequency functions show that as Re increases, the distribution curves approach to that for plug flow. This is in contrast to species concentration distribution that has a maximum in variance or number of elements around Re = 50 as reported in literature. Diffusion has a similar effect on the frequency function. As diffusivity increases, the function approaches that for plug flow with shorter tail of older materials. Variances are often used in the literature to measure scalar mixing and distribution. Because mean age is a scalar, its decay along the mixer can be measured with the variance on crosssections. It is found that the decay rate is higher for higher Re values except around Re = 90. This is consistent with the mean age distribution shown on contour plots. The faster decay is due to the more uniform distribution by the secondary flow at higher Re. Age distribution is also measured with variance of residence time distribution. With the current method in the literature, the full function of residence time distribution has to be known to compute this variance. However, with the solution of mean age computed from a steady transport equation, this variance can be computed with a much lower computing cost. The relationship

■

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

■

NOMENCLATURE A area (m2) a mean age (s) ae̅ mass averaged mean age at exit (s) ai̅ mass averaged mean age at inlet (s) aV̅ volume averaged mean age (s) c tracer concentration d mixer pipe diameter (m) + molecular diffusivity (m2/s) +eff effective diffusivity (m2/s) E residence time frequency function (s) F accumulative residence time function (s) g mean age frequency function I spatial invariant (s) L mixer length (m) Mn n-th moment of age (sn) M̅ n,e mass averaged n-th moment of age at exit (sn) n unit normal (m) Pef effective Peclet number Pef = Ud/+eff Q volumetric flow rate (m3/s) r radius coordinate (m) S surface area (m2) t time (s) u velocity vector (m/s) U mean velocity (m/s) V mixer volume (m3) Ve volume of one element of mixer (m3) x spatial coordinate (m) y spatial coordinate (m) z spatial coordinate (m) Greek Letters

δ μ ρ σ σa σe τ τe

■

diffusion penetration distance (m) viscosity (kg/m-s) density (kg/m3) variance variance of mean age variance of residence time mean residence time (s) mean residence time in one element (s)

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