Chemical Engineering Science 69 (2012) 382–393
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Age distribution and the degree of mixing in continuous ﬂow stirred tank reactors Minye Liu DuPont Company, 1007 Market Street, Wilmington, DE 19898, USA
a r t i c l e i n f o
a b s t r a c t
Article history: Received 18 May 2011 Received in revised form 10 October 2011 Accepted 27 October 2011 Available online 7 November 2011
New development of mean age theory is discussed for quantitative analysis of mixing and age distribution in steady continuous ﬂow stirred tank reactors. A new relationship between the moments of age and the moments of residence time are derived. With this new relationship the variance of residence time distribution can be computed much more efﬁciently and accurately. The relationships of three existing variances of age are described and a new set of variances and the degree of mixing are deﬁned. The theory is used to characterize mixing performance in a CFSTR with different layouts of an inlet and an outlet. Mean age and higher moments of age in the reactors are obtained from CFD solutions of their steady transport equations. The spatial distribution of mean age reveals details of the spatial non-uniformity in mixing. Variances of age and the degree of mixing discussed by Danckwerts and Zwietering are computed for the ﬁrst time in the literature for non-ideal stirred tank reactors. It is found that although these measures are useful, certain key features in non-uniform mixing are not reﬂected accurately. Results show that the new set of variances and the degree of mixing more accurately characterize the non-uniform mixing in the reactors. & 2011 Elsevier Ltd. All rights reserved.
Keywords: Mixing Mean age Degree of mixing Age distribution Continuous ﬂow CFD
1. Introduction In designing a continuous ﬂow stirred tank reactor (CFSTR), several factors have to be considered besides those for a batch reactor design. The locations and sizes of the inlets and the outlets are critical for efﬁcient mixing and ﬂow distribution. Many undesired effects can result, such as bypassing and dead zones, if a reactor is designed improperly. Although stirred tank reactors have been widely studied in the literature, the majority of the literature are focused on batch reactors. Research on continuous stirred tank is sparse. The spatially non-uniform mixing in CFSTRs is rarely studied. Instead, CFSTRs are often modeled as one or several ideal mixers. Widely used in practical designs of CFSTRs is a crude rule of thumb that a time ratio, the ratio of the mean residence time to the batch blend time, should be larger than 10 in order to maintain the ideal mixing assumption. One of the main methods used in studying mixing in a continuous ﬂow system is the theory of residence time distribution (RTD) (Danckwerts, 1953; Levenspiel, 1999; Nauman and Buffham, 1983). The theory is based on the probability distribution of material age at the exit of a reactor. By analyzing the shape of the distribution curve, some non-ideal mixing properties inside
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the ﬂow, such as dead zones, and bypassing paths, may be inferred. However the sizes and locations of such regions cannot be determined by the RTD curve. This is because the probability function of residence time does not contain any information about the spatial distribution of material age inside the ﬂow. Such spatial distribution of material age is critical in order to deﬁne the state of mixing. A limited number of experimental studies on the non-uniform mixing in CFSTRs exist in the literature. Cloutier and Cholette (1968) studied experimentally the effect of various parameters, impeller geometry, feed rate and feed location, on the level of mixing in CFSTRs. The level of mixing was measured by an effective volume in which mixing is complete. They found that mixing is improved when the feed location is closer to the impeller. They also found that as the feed rate is increased, the level of mixing decreases. Mavros et al. (2002a) studied ﬂow pattern in CFSTRs with an axial impeller using LDV measurements. They found that a short-circuiting is possible when the liquid is fed into the stream drawn to the impeller and the outlet is at the bottom of the reactor. A similar study with a Rushton impeller (Mavros et al., 2002b) did not ﬁnd possible circuiting due to the blockage of the impeller disk. Samaras et al. (2006) studied the effects of the continuous ﬂow stream on mixing in a CFSTR by ﬁtting the Cholette and Cloutier’s model (1959) to the measured RTD functions. From this model they estimated volume percentage of plug ﬂow, dead zone, perfectly mixed zone, and bypassing
M. Liu / Chemical Engineering Science 69 (2012) 382–393
ﬂow rate. Although their study is useful for understanding the non-uniform mixing state in the reactor, design limits and deﬁnitive recommendations could not be provided due to the limits of the methods used. Roussinova and Kresta (2008) measured the mixing time by tracking tracer concentration history experimentally at two internal points and one point in the exit of a CFSTR. They found that when the ratio of the mean residence time to the batch blend time is equal to 2, the mixing time in the tank deviated by up to 50% from the corresponding ideal mixer. Even when the time ratio is equal to 10, a 30% deviation was found. These ﬁndings seem to be different from the assumption widely used in the literature that many CFSTRs can be considered as an ideal mixer. Jones et al. (2009) measured the residence time distribution of a CFSTR with a side entering inlet and outlet. They varied the time ratio from smaller than 1 to slightly larger than 10 by changing the continuous ﬂow rate and the impeller speed. They concluded that the time ratio does not correlate the variance of the measured RTD whereas the ratio of the impeller to the jet momentum does. However, their data showed very limited improvement in data correlation except when the inlet diameter is a variable. Recent advancements in CFD and precision measurement instruments have signiﬁcantly enhanced the understanding of the ﬂow ﬁelds inside stirred tanks. Velocity vector plots can easily be obtained from 3D CFD solutions or LDV and PIV measurements. However, an effective method for quantifying spatial nonuniformity of mixing using such velocity solutions is still missing. The typical method of modeling mixing process in a CFSTR is to solve the transient tracer concentration equation. For example, Aubin et al. (2006) compared the effect of adding an additional inlet on the spatial concentration distribution by computing the spatial concentration variance. Coroneo et al. (2011) computed the time dependent history of coefﬁcient of variance of a tracer released in a batch stirred tank reactor. Good agreement with their experimental measurements was reported. Although this method can provide details of the spatial–temporal concentration distribution, it is very computing intensive. To measure the level of non-ideal mixing in a continuous ﬂow system, Danckwerts (1958) proposed the idea of spatial age distribution and the degree of segregation. Zwietering (1959) further discussed the age distribution measured by different variances and renamed Danckwerts’ degree of segregation as the degree of mixing. The degree of mixing as a concept has been widely discussed in several review articles and books (Shinnar, 1977; Rodrigues, 1981; Nauman and Buffham, 1983; Tavare, 1986). However, there have been no general methods, computational or experimental, to compute its value. There has also been no report on its application for a practical reactor design. Only very recently, Liu (2011a) developed a method to compute the degree of mixing for a general steady continuous ﬂow system using the ﬁrst two moments of age. The limited study of CFSTRs in the literature may partially be blamed for the lack of an effective method. Although the rule of thumb of the time ratio greater than 10 has been widely used, it is too crude to reﬂect any speciﬁc effects on mixing non-uniformity by different design factors. Measuring or computing ﬂow pattern is helpful for some understanding of the non-uniform mixing but no quantitative mixing measures can be obtained directly from velocity solutions. RTD theory and measuring tracer concentration at the exit, or even at selected internal points cannot provide a deﬁnitive answer on the size, locations, and intensity of the non-uniformity. More recently, a new method has been developed for mixing and ﬂow distribution studies based on mean age distribution. Using this method, Liu and Tilton (2010) demonstrated that the
moments of age can be computed at only a small fraction of the computing cost required by the transient concentration solution. Liu (2011b) computed tracer distribution history in a CFSTR equipped with a pitched blade turbine and found that the history at any point in the reactor has two stages: the initial stage that is highly position dependent and the stationary stage that has a position independent decay rate. In the stationary stage, the decay rate was found to be the inverse of the volume averaged mean age and both the spatial and the temporal variation of tracer concentration could be described by the mean age distribution. Liu (2011c) also used the mean age method to compute the blend time of a batch stirred tank reactor. He found that the probability frequency functions (PDF) of the mean age in the reactor scale with impeller speed and the ratio of the impeller diameter to the tank diameter the same way as in the experimental correlations in the literature. Using the width of the PDF as the blend time, he found that the predicted blend times agree very well with the correlations. In this article, the mean age theory will be further developed to characterize the spatial non-uniformity in a CFSTR. The effect of the continuous ﬂow on mixing due to various designs of inlet and outlet locations and ﬂow rates will be quantitatively measured. Danckwerts–Zwietering’s degree of mixing will be computed for the ﬁrst time for non-ideal stirred tank reactors. Its usefulness will be discussed as a mixing performance measure for continuous ﬂow systems. A new degree of mixing will be deﬁned to more accurately reﬂect the effect of various factors on the mixing performance. The objective of this article is to present a new and practical method for quantitatively characterizing the spatial nonuniformity of mixing in a CFSTR using CFD.
2. Flow ﬁelds in the stirred tanks The 3-D velocity ﬁelds in the CFSTRs with ﬁve different inlet/ outlet layouts are ﬁrst solved using CFD. These ﬂow solutions are the starting point for further analysis of mixing performance later in this article. In all the stirred tanks studied, a 451 pitched blade turbine (PBT) impeller is placed at a ﬁxed location as shown in Fig. 1. The diameter of the tank is T¼ 0.292 m with the ratio of the height to the diameter equal to one, H/T¼1. The diameter of the impeller is D ¼T/3, the bottom clearance is H/3, and the width of the four bafﬂes is about T/12. Both impeller blades and the bafﬂes have standard thickness. Other dimensions of the stirred tanks and locations of the inlet and the outlet are shown in Fig. 1.The ﬂuid has a density r ¼1000 kg/m3 and viscosity m ¼0.001 kg/m s.
Fig. 1. The stirred tank geometry and dimensions. T¼ H¼ 0.292 m. D ¼T/3. d¼ 25 mm. ds ¼12 mm. dt ¼ 28 mm. The box in dashed line shows the interface of rotating and ﬁxed zones in MRF method.
M. Liu / Chemical Engineering Science 69 (2012) 382–393
The impeller speed is ﬁxed at 300 rpm. The corresponding Reynolds number is about 50,000, making the ﬂow in fully turbulent regime. The batch blend time is about 8.58 s, calculated using the correlation
in Grenville and Nienow (2004). The continuous ﬂow rate varies from 0.02 kg/s to 0.5 kg/s. The corresponding mean residence time calculated from V/Q is from about 931.39 s to 37.26 s. The ratio of
Fig. 2. Velocity vector plots in the stirred tank on x ¼0 plane. The color shades and vector lengths represent velocity magnitude in m/s. Vectors are not plotted on actual computing grid. Qm ¼ 0.2 kg/s. (a) Case I. (b) Case J. (c) Case L. (d) Case T. (e) Case G. (For interpretation of the references to color in this ﬁgure legend, the reader is referred to the web version of this article.)
M. Liu / Chemical Engineering Science 69 (2012) 382–393
the mean residence time to the blend time, denoted as Z, is then in the range from 108.60 to 4.34. The solutions of the ﬂow in the stirred tanks are solved with commercial CFD package Fluent version 6.3 with a mesh of about 1.6 million cells with the majority being hexahedral type. All transport equations are solved using QUICK scheme for better accuracy and minimal effect of numerical diffusion. The turbulence is modeled with the standard k–e model. For the inlet, the ﬂow is in the laminar regime when Qm o0.1 kg/s. Therefore, the ﬂow inside the inlet tube is treated as laminar. For Qm 40.1 kg/s, the turbulent boundary condition at the inlet is set with 5% turbulent intensity. It is found that the change in the turbulent boundary condition at the inlet has very little effect on the mean age distribution for all cases. This is because the reported results in this article are all about macroscale mixing and statistical averaging. The accuracy of the inlet turbulent boundary condition could be important when local microscale or mesoscale mixing is considered. The standard wall function is used on all solid walls. The average y þ is found to be 21 on the impeller and about 26 on the tank wall. In the literature, it has often been reported that k–e model underpredicts the turbulence in a stirred tank. In this work, it was found that the ratio of the total integration of the dissipated energy in the reactor to the impeller power input calculated from impeller torque is about 0.83. The ﬂow is treated as quasi-steady using the multiple reference frame (MRF) method in Fluent. The tank is divided into two zones, one around the rotating impeller and the other near the tank wall and the bafﬂes. The box in dashed lines in Fig. 1 shows the interface of the two zones. The top interface is 0.11 m above the impeller plane, the bottom interface is 0.06 m below the impeller plane, and the radius of the interface is 0.08 m. In the outer zone all transport equations are solved in the stationary frame and in the inner zone around the impeller they are solved in a frame rotating with the impeller. With coordinate transformation, the Reynolds averaged ﬂow in both zones can be approximated as steady in its own reference frame. The absolute velocity is matched at the interface (Fluent Manual). To characterize the effect of the inlet/outlet layout on mixing efﬁciency, ﬁve different combinations are studied. For case I, the inlet is at the top around the shaft and the outlet is at the center of the tank bottom. The inlet and the outlet have the same cross sectional area. For case J, the inlet is the same but the outlet is moved to the side of the tank near the top surface. Case L also has the same inlet but the outlet is on the side at about the same elevation as the impeller plane. For case T, the outlet of case J is used as the inlet and the outlet is the same as that of case I. The last case is the reverse of case T. It is denoted as case G. At smaller ﬂow rates, the velocity distribution in the stirred tank for all cases is similar to that of batch mode. The effect of the inlet and outlet on the ﬂow can only be noticed for higher ﬂow rates. Fig. 2 shows the velocity vector plots for each case on a vertical plane between bafﬂes with a mass ﬂow rate of Qm ¼0.2 kg/s. A quick glance at all the vector plots reveals that all the ﬂows are impeller controlled with the familiar two vortices on the selected plane. This is a typical ﬂow pattern in the batch mode. Fig. 2a shows the ﬂow pattern for case I. Some local effect is noticeable in the ﬂow pattern. The inlet ﬂow creates a strong axial velocity distribution around the shaft above the impeller. Fig. 2b shows the ﬂow ﬁeld for case J. Compared with Fig. 2a, the ﬂow pattern is almost identical. The only minor difference is in the triangular zone below the impeller. Although the inlet and the outlet are close to each other in this case, there is no apparent bypassing between them. The inlet ﬂow seems to be in the same direction of the main impeller ﬂow. The ﬂow ﬁeld for case L is shown in Fig. 2c. Again, the effect of the continuous ﬂow on the main ﬂow in the stirred tank is small. For this layout, since the outlet is on the path of the main impeller stream, there seems to be some
bypassing. This bypassing should be weak since the inlet tracer passes through the impeller zone ﬁrst and has been mixed before exiting the system. Fig. 2d is the vector plot for case T. At smaller ﬂow rates, the effect of the inlet ﬂow is too weak to be noticed in the main ﬂow. However, at Qm ¼0.2 kg/s, a strong jet from the inlet is formed. This jet is expected to change the ﬂow and mixing at least in the upper portion of the tank around the jet. The ﬂow pattern for the last case, case G, is shown in Fig. 2e. A strong inlet jet is noticeable inside the triangular zone of weak ﬂow below the impeller, but its effect on the rest of the ﬂow is still difﬁcult to detect. From the above discussions, it can be seen that although the ﬂow patterns reveal some general trend of mixing in the tanks, they cannot be used to measure mixing performance quantitatively. To do so, further information is needed from the ﬂow and the material motion in the reactors.
3. Variances and the degree of mixing Quantitative characterization of mixing and age distributions can be obtained by computing several variances. In this section, the deﬁnition and the transport equation of the moments of age are ﬁrst presented. Variances and coefﬁcients of variance (CoV) for exit age (residence time) and internal age distributions are then deﬁned. A new degree of mixing is also deﬁned. 3.1. Transport equations The transport equation for mean age in a steady incompressible ﬂow was ﬁrst derived from a pulse tracer input by Spalding (1958) and Sandberg (1981). Following the same steps, Liu and Tilton (2010) derived the equations for all the moments of age. Liu (2011a) showed that the same set of equations can be derived for step changes and continuous feed of tracer as long as a proper age function is deﬁned. For a pulse input system the transport equation for the moments of age is
rUðuMn Þ ¼ rUðDrMn Þ þnM n1
where Mn is the nth moment of age deﬁned as Z 1 .Z 1 t n cðx,tÞdt cðx,tÞdt Mn ¼
The transport equation for the mean age is obtained for n ¼1,
rUðuaÞ ¼ rUðDraÞ þ1
All turbulent ﬂows are intrinsically unsteady. However, when a turbulent ﬂow is modeled with Reynolds averaging, the resulted ﬂow becomes steady with all turbulent ﬂuctuations being modeled with a large turbulent viscosity. With such a turbulence model, Eqs. (1) and (3) can also be applied to a turbulent ﬂow. The molecular diffusivity D will then be replaced with an effective turbulent diffusivity, Def f ¼ D þ nT =Sc, (Liu and Tilton, 2010). In this article, the turbulent Schmidt number Sc is ﬁxed at 0.7. The governing equations of the moments of age are also solved using MRF method. Since scalars are frame independent, they can be matched at the interface without any coordinate transformation. However, when scalar distribution is not circumferentially symmetric, large error can be resulted as will be seen later. 3.2. Variance of residence time distribution In the RTD theory, one measure for the non-ideal mixing in a continuous ﬂow reactor is the variance of the RTD (Levenspiel, 1999; Nauman and Buffham, 1983)
M. Liu / Chemical Engineering Science 69 (2012) 382–393
The RTD function E(t) is obtained from the time history of tracer concentration at the exit of a reactor from either experimental measurements or numerical computation. In order to obtain an accurate variance from the integration, a long time history has to be tracked. This could be a challenge for both experimental measurement (Jones et al., 2009) and numerical computation (Nauman and Buffham, 1983; Patwardhan, 2001; Liu and Tilton, 2010). Tracking the tracer concentration history for extremely long time computationally is often not practical. To avoid the excessive demand on computing time, researchers often truncate the upper limit of time and use large time step sizes. Both will severely affect the accuracy of the predicted results. Recently, Liu and Tilton (2010) have shown that the ﬂow averaged moments of age at the exit is equal to the moments of residence time Z Z 1 1 tn t n EðtÞdt ¼ ue M n ds Mn,e ð5Þ Q Se 0 where Q is the volumetric ﬂow rate of the system and ue is the velocity at the exit. For n ¼1, we have t ¼ M 1,e ae . Thus the mean residence time can easily be found from the ﬂow averaged mean age at the exit. Using this relation, the variance becomes
M 2,e t2
Since the 2nd moment is obtained by solving a steady transport equation, there is no error due to truncation of time derivative term or numerical integration. The variance can then be accurately calculated from Eq. (6) without the need of function E(t). The computing resources required by this method are also orders of magnitude lower than tracking the long time history of tracer concentration computationally. The relationship between the volume averaged moments in the interior of the ﬂow and the mixing cup averaged moments at the exit can also be established. Danckwerts (1953) showed that the internal age frequency function is related to the accumulative residence time distribution function by 1
where a is the molecular age in the ﬂow. Using this relation, we have Z 1 Z 1 1 n an ¼ an fðaÞda ¼ a ½1FðaÞda ð8Þ
Integration by parts, we have " # 1 Z 1 1 an þ 1 1 ð1FÞ þ an ¼ an þ 1 dF nþ1 0 t n þ1 0
It is well known that for a practical continuous ﬂow with diffusion, as time approaches inﬁnity, the E-curve approaches zero exponentially. Thus, F-curve will approach to one exponentially in the same limit. The ﬁrst term on the right hand side then becomes zero and Z 1 Z 1 1 1 tn þ 1 an ¼ an þ 1 dFðaÞ ¼ an þ 1 EðaÞda ¼ tðn þ 1Þ 0 tðn þ 1Þ 0 tðn þ1Þ ð10Þ Liu (2011a) has shown that the volume averaged moment of age is equal to the moment of internal molecular age n
M n,V ¼ a
We then have M n,V ¼
tn þ 1 ðn þ 1Þt
A similar relation was derived by Sandberg and Sjoberg (1983) for the moments of tracer concentration. Eq. (10) can also be obtained by integrating Eq. (1) over the ﬂow domain. For n¼1, from Eq. (12) we have M1,V ¼ aV ¼
t2 M 2,e ¼ 2t 2t
Substitute this relation into Eq. (6), we obtain
The advantage of this equation over Eq. (6) is that only the ﬁrst moment of age, the mean age, needs to be computed in order to compute the RTD variance. 3.3. Variances of age and the degree of mixing Variances of internal age distribution have been discussed in the literature for characterizing mixing condition. Zwietering (1959) discussed three variances for age distribution, two of which were used in the deﬁnition of the degree of mixing. Since two of the three variances require the knowledge of spatial distribution of age, and methods for ﬁnding the spatial distribution of age for a general continuous ﬂow have not been available until recently, the discussions on the variances and the degree of mixing in the literature have been limited to either the general concept level or ideal ﬂow models. Recently, a method is reported to compute these variances and the degree of mixing with the solutions of the moments of age (Liu, 2011a). In this section, the relationship of the three variances are derived and two CoVs are deﬁned with the variances for quantitative comparisons of reactor mixing performance. The variance of age in a steady continuous ﬂow system is (Zwietering, 1959; Liu, 2011a) Z 1 ðaaÞ2 ðaaÞ2 fðaÞda ¼ M 2,V a2V ð15Þ 0
Apparently this variance measures the probability distribution of age of molecules in the ﬂow system. Since both a2V and M 2,V can easily be computed from the solutions of mean age and the second moment of age, this variance can be obtained easily. It should be mentioned that Zwietering (1959) showed that !2 t3 t2 ðaaÞ2 ¼ ð16Þ 3t 2t Using Eq. (10), it can easily be seen that this equation is the same as Eq. (15). However, only second moment is needed in Eq. (15) but a higher moment is required in Eq. (16). As shown in Appendix, the variance of age can be split into three variances ðaaÞ2 ¼ ðaaÞ2 þ ðaaÞ2 þ2ðaaÞðaaÞ
with ðaaÞ2 ¼ M 2,V a2V
ðaaÞ2 ¼ a2V a2V
ðaaÞðaaÞ ¼ 0
The variance in Eq. (18) is the volume average of the variance of molecular age distribution at a point in space. Since molecular age is in time coordinate, this variance can be considered as a measure for mixing in time. The variance in Eq. (19) measures the spatial distribution of mean age in the ﬂow. Eq. (20) indicates that the spatial distribution of mean age and the time distribution of molecular age at a point in space are independent events. It is
M. Liu / Chemical Engineering Science 69 (2012) 382–393
well known that, as a measure in the traditional residence time theory, the variance of age deﬁned in Eq. (15) cannot deﬁne the state of mixing in a ﬂow (Zwietering, 1959) since no information of spatial distribution is available. Now with Eq. (17), a measure of such spatial distribution can be obtained and the mixing state can be deﬁned. Eq. (17) also deﬁnes a new measure for age mixing at a point in space. Although this measure has the potential to be valuable when mixing in time at a point is important for a ﬂow system, it will not be discussed further in this study. To measure the distributions of age, CoV can be deﬁned from these variances. To measure the probability distribution of age of all the molecules in the system, the variance of age is used to deﬁne the CoV of a qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ M 2,V a2V ðaaÞ2 sa ¼ ð21Þ a aV To measure the spatial distribution of mean age, the variance of mean age is used to deﬁne the CoV of a qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ a2V a2V ðaaÞ2 sa ¼ ð22Þ a aV For Danckwerts–Zwietering’s degree of mixing (Zwietering, 1959), Liu (2011a) has shown that it can be computed by JZ ¼
a2V a2V M2,V a2V
3.4. Variances and the degree of mixing relative to the ideal mixer It should be mentioned that in the original deﬁnition of the degree of segregation Danckwerts (1958) deﬁned both variances about the mean residence time but incorrectly stated that ‘‘the average value of t for all points in the system is t, the mean residence time,’’ where Danckwerts denoted t as the age of the ﬂuid at a point, i.e. the mean age a used in this article. In deﬁning the degree of mixing, Zwietering (1959) used a in place of t in the variances without pointing out the differences. He also derived the relation of a ¼ t 2 =2t, which is the same result as Eq. (10) for n ¼1. In fact, Danckwerts also derived this relation in his seminal paper in 1953 (Danckwerts, 1953). As we will see later in this article, Danckwerts–Zwietering’s degree of mixing fails to characterize the differences in mixing states of the stirred reactors studied in this article because it is deﬁned relative to a case dependent reference a. To compare mixing efﬁciency with an invariant reference, we deﬁne the variances about the mean residence time, or the average age for the ideal mixer. With these variances a new degree of mixing can be deﬁned. The variance of mean age ðatÞ2 is deﬁned as Z Z 1 Z 1 1 ðatÞ2 ðatÞ2 fðaÞda ¼ ðatÞ2 cðx, aÞda dx ð24Þ VI V 0 0 The variance of age ðatÞ2 is deﬁned as Z Z 1 Z 1 1 ðatÞ2 ðatÞ2 fðaÞda ¼ ðatÞ2 cðx, aÞda dx VI V 0 0
Following the same steps in Appendix, we can obtain ðatÞ2 ¼ M2,V 2taV þ t2
and ðatÞ2 ¼ a2V 2taV þ t2
A new degree of mixing can then be deﬁned as Jt ¼
a2V 2taV þ t2 M 2,V 2taV þ t2
Jt has the same value as JZ for the plug ﬂow and the ideal mixer but different values for a non-ideal ﬂow. With Eqs. (26) and (27), two new CoVs can be deﬁned qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 1 1 sat ¼ ðatÞ2 ¼ M 2,V 2taV þ t2 ð29Þ
qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 1 1 sat ¼ ðatÞ2 ¼ a2V 2taV þ t2
here, sat measures the probability distribution of molecular age and sat measures the spatial distribution of mean age about the average age of the corresponding ideal mixer.
4. Numerical results and discussions In this section, the spatial distribution of mean age will ﬁrst be presented for the ﬁve cases. Such spatial distributions readily reveal spatial non-uniformity of scalar mixing in the stirred tank reactors. To characterize the age distributions and mixing performance quantitatively, the measures discussed in the previous section will be computed. The results will be presented as a function of the ratio of the mean residence time to the batch blend time, denoted by Z. Although Jones et al. (2009) have claimed that the momentum ratio correlates the variance of the measured tracer concentration better, it can easily be shown that unless the diameter of the inlet is a variable, the momentum ratio is proportional to the square of Z. Thus, there is no beneﬁt in using the momentum ratio for the study here. 4.1. Spatial distribution of mean age in stirred tanks The spatial distributions of mean age for all the stirred tanks are obtained by solving Eq. (3) after velocity ﬁelds are available. The contour plots for each stirred tank with Qm ¼0.2 kg/s are shown in Fig. 3a–e. These contour plots are on the same plane as the velocity vector plots shown in Fig. 2. The lower bound of the mean age is all cut off at a non-zero value in order to show the spatial distribution pattern. Fig. 3a shows the distribution of mean age for case I. The distribution pattern shows that the fresh tracer material from the top inlet is drawn to the impeller along the shaft. It can be noticed that the tracer leaving the system from the bottom exit is younger than those in the main body of the tank. This indicates some level of bypassing. The oldest tracer is in the upper portion of the tank. There seems to be no dead zone in the tank. This is mainly due to the strong turbulent diffusion. The distribution pattern for case J is similar to that for Case I, as shown in Fig. 3b. Although, the outlet in this case is very close to the inlet by direct distance, it is actually the farthest away from the inlet along the tracer ﬂow path. Since the outlet is in the upper portion of the tank where mean age is the highest, the contour plot shows that the mean age in the outlet pipe is almost as old as those in the upper portion of the reactor. Fig. 3c shows the mean age distribution for case L. Since now the outlet is still in the path of the younger material but further away from the inlet than case I, some limited bypassing can still be seen by comparing the mean age in the outlet to that in the rest of the tank. Fig. 3d shows the mean age distribution for case T. Since the inlet is located at the slow mixing zone, as smaller ﬂow rate, the fresh tracer will be mixed slowly to the rest of the tank. At higher ﬂow rates, a stronger jet will be created. The jet at Qm ¼0.2 kg/s sends the fresh tracer to the center of the tank and into the
M. Liu / Chemical Engineering Science 69 (2012) 382–393
Fig. 3. Mean age contour plots on x¼ 0 plane. The color shades represent mean age in seconds. The lower limit of the mean age is cut at higher values in order to show spatial distribution patterns. (a) Case I, Qm ¼0.2 kg/s. (b) Case J, Qm ¼ 0.2 kg/s. (c) Case L, Qm ¼0.2 kg/s. (d) Case T, Qm ¼0.2 kg/s. (e) Case G, Qm ¼ 0.2 kg/s. (f) Case T, Qm ¼ 0.02 kg/s. (For interpretation of the references to color in this ﬁgure legend, the reader is referred to the web version of this article.)
impeller suction zone as shown in the ﬁgure. Fig. 3e is for case G, the reverse of case T. Since the inlet is in the weak ﬂow zone below the impeller, the fresh tracer material ﬁlls this triangular zone ﬁrst and is then slowly mixed into the main impeller loop. The main mechanism for the fresh material to enter the main loop is by turbulent diffusion. Therefore, it is expected that mixing will be poor. As the ﬂow rate is increased, the ﬂow ﬁeld below the impeller will be disturbed and mixing should be improved. When the ﬂow rate is increased beyond 0.2 kg/s, the computed results show that the inlet jet hits the bottom of the impeller and becomes non-symmetric, indicating that the jet may have become unstable.
Among the ﬁve cases discussed above, only Case T has the inlet on the side of the tank. This will create a non-symmetric distribution of mean age around the interface between the rotating and ﬁxed zone. It is found that this non-symmetric input seems to have created large errors in the scalar distribution. The effect of the interface on the mean age distribution can easily be seen in Fig. 3f at Qm ¼0.02 kg/s. In a slow feeding situation, the tracer is fed to the slow mixing zone in the upper portion of the tank. The accumulation of the fresh tracer near the entrance causes severe non-symmetry of large tracer concentration gradient. Referring to Fig. 1 for the location of the interface of the two zones for MRF method, we can see clearly the effect of the interface. This case reveals that MRF method is not
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able to handle large non-symmetric tracer distributions near the interface. This is because the circumferential symmetry of both velocity and tracer distribution has been implicitly assumed on the interface when MRF method is used to track the time dependent tracer distribution. The ﬂow in one frame zone is also assumed not affected by the ﬂow in the other frame zone when the impeller-bafﬂe orientation changes (Marshall and Bakker, 2004). It has been reported in the literature that the mean ﬂow solution using MRF method is reasonably accurate compared to the sliding mesh method (Wechsler et al., 1999; Aubin et al., 2004; Deglon and Meyer, 2006) when the only effect on the interface is the weak coupling of the impeller and the bafﬂes. Experimental measurements also revealed that tracer concentration is nearly axisymmetric far away from the impeller (Distelhoff and Marques, 2000) when the tracer is introduced into a fast mixing zone. Thus, MRF method is a good approximation in such situations. However, when there is strong non-symmetry in ﬂow or scalar distribution on the interface, such as a strong side-entering jet, MRF method can result signiﬁcant errors. The above spatial distributions of mean age have revealed many qualitative mixing features that are not present in the velocity ﬁelds. Slow mixing zones, bypassing paths can easily be identiﬁed from a contour plot of mean age. More details can be found by examining the distribution from different angles and on different planes. 4.2. Variance of residence time distribution The variance or standard deviation of a RTD has been used in the literature to detect if the ﬂow in a reactor is plug ﬂow like or has certain degree of bypassing. For an ideal mixer se ¼ 1. By comparing the standard deviation of a non-ideal ﬂow to that of the ideal ﬂow, some non-ideal behavior of the ﬂow can be revealed. When a reactor has some feature of a plug ﬂow, the tracer residence time tends to have a narrower distribution, resulting in a lower value of standard deviation than the ideal mixer. When the ﬂow has some level of bypassing, some tracer material will leave the system earlier than in an ideal ﬂow and some will be left in the system for extended time. Therefore, the E-curve will be wider with a larger standard deviation than the ideal mixer. Jones et al. (2009) have used this method to analyze their measured RTD function. The standard deviation se is computed from Eq. (14). It should be pointed out that the mean residence time computed from t ¼ M1,e ¼ ae is very accurate. For all the cases discussed in this article, the error compared to t ¼ V/Q is much smaller than 1%. Fig. 4 shows se as a function of the time ratio Z. Also shown in the ﬁgure by the thin dashed line (in gray) is the standard deviation for the ideal mixer, se ¼ 1. It can be seen clearly that all the standard deviations approach one as the time ratio increases. For cases I and L, the standard deviation approaches 1 from above and for cases J and G it approaches 1 from below the solid line. For case T, se crosses the solid line at Z E35. These curves indicate that cases J and G have some plug ﬂow feature and cases I and L have some level of bypassing. These features can also be seen by examining the spatial distribution of mean age in Fig. 3. For both case J and case G, the outlet is at the upper portion of the tank. Given the ﬂow pattern in the tank, it takes the longest time for the fresh tracer to appear at the exit. Before this ﬁrst appearance time, the ﬂuid leaving the system contains no tracer. Thus, the system behaves like a plug ﬂow. The tracer is also better mixed throughout the tank before leaving the system. On the other hand, for both case I and case L the tracer path to the outlet is on the main ﬂow stream. The worst case is case I for which the outlet is right below the exit stream of the tracer from the impeller. The mean age contour plot in Fig. 3a shows a strong bypassing. This is in agreement with the large
Fig. 4. Standard deviation of residence time distribution as a function of the time ratio Z. Solid line (black): case I; dash–double dots (tan): case J; dashed line (red): case L; dash–dot (blue): case T; dotted line (green): case G. Thin dashed line (dark gray): ideal mixer. (For interpretation of the references to color in this ﬁgure legend, the reader is referred to the web version of this article.)
values of standard deviation for the case. Case L has the smallest values of standard deviation, indicating that it is the closest to the ideal mixer. For case T, the tracer is fed into a weak ﬂow region in the tank at low ﬂow rates. Therefore, the ﬁrst appearance time of the tracer is long and the system behaves like a plug ﬂow, just like cases J and G. On the other hand, for higher ﬂow rates, the incoming tracer is injected into different locations in the tank. As shown in Fig. 3d, at Qm ¼0.2 kg/s, the tracer is sent directly into the impeller suction stream. At even higher ﬂow rates, the tracer will be sent to the opposite side of the tank. Such ﬂow behavior has been observed in lab experiments (Jones et al., 2009). Thus, the ﬂow pattern in the tank can be signiﬁcantly modiﬁed by the strong jet, causing a strong bypassing. This is reﬂected by the large values of the standard deviation as shown in Fig. 4. The standard deviation shows that the reactor has some plug ﬂow feature for Z 435 (Qm o0.07 kg/s) and some bypassing for Z o35. It should be mentioned that as discussed earlier, the severe non-symmetry of tracer concentration for case T violates the assumption of the MRF method. Relocating the interface further away from the incoming tracer stream may reduce such effect but cannot avoid such effect completely. This effect will inevitably cause some errors in the computed mean age solution. 4.3. Variances of age and the degree of mixing Results of sa for all the cases are shown in Fig. 5 as a function of Z. All the curves in the ﬁgure behave similarly to those in Fig. 4, showing that the internal age distribution follows a similar trend to the distribution at the exit, except the curve for case G. It should be pointed out that calculating sa from Eq. (21) is error prone since the values are all extremely close to 1. For Z 410, the difference from 1 for most of the values is smaller than 10 4. The most error sensitive case is case G. This is also the most different case from Fig. 4. The error seems to be related to the location of the weak ﬂow outside the main circulation loop that the fresh tracer is fed to. More accurate velocity solutions in such weak regions require much ﬁner grid. For case G, the tracer is fed to the weak secondary ﬂow below the impeller. The next most error prone case is case T, in which the tracer is fed to another weak ﬂow region near the top surface.
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Fig. 5. CoV of mean age a as a function of the time ratio Z. Solid line (black): case I; dash–double dots (tan): case J; dashed line (red): case L; dash–dot (blue): case T; dotted line (green): case G. Thin dashed line (dark gray): ideal mixer. (For interpretation of the references to color in this ﬁgure legend, the reader is referred to the web version of this article.)
condition is slightly closer to other cases. As the ﬂow rate further increases, the strong tracer jet passes the shaft and sent the fresh tracer to the opposite side of the tank. The ﬂow ﬁeld in the tank is severely altered by this tracer jet. Overall, this non-symmetric input results the worst spatial distribution of mean age. Case G is the second worst case. Since the inlet location is in a slow mixing region outside of the main ﬂow loop below the impeller, at low ﬂow rates, the fresh tracer will stay in the region for long time. As the ﬂow rate increases, the ﬂow pattern will be disturbed and fresh tracer material will be mixed into the impeller stream faster. Thus, sa values are closer to the cases with top center input. It is interesting to notice that the CoV curves for the three cases with the top center input are almost identical as shown in Fig. 6. For these three cases, the fresh tracer is introduced into the impeller suction zone and the main ﬂow loop. As has been discussed earlier, case I has a strong bypassing at higher ﬂow rates. This can be seen from both the mean age contour plot in Fig. 3b and from the CoV curves in Figs. 4 and 5. However, this effect is not reﬂected on this variance curve. The reason is that the variance is deﬁned about a case dependent volume averaged age aV . Fig. 7 shows aV scaled by t as a function of Z for all the cases. As can be seen that aV is different for each case at the same ﬂow rate. At higher ﬂow rates (small values of Z), aV is further away from the average age for the ideal mixer. As the ﬂow rate decreases, the differences become smaller. Among all the cases, case I has the largest value of aV and deviates the most from the corresponding ideal mixer. It is also interesting to notice that the curves for aV are almost identical to those for se shown in Fig. 4. This is not surprising since aV is related to se by Eq. (14). Thus, aV can in fact be used as a measure for the RTD distribution. It should be mentioned that aV is also important in measuring the transient behavior of tracer concentration decay in a CFSTR. Liu (2011b) found that it is the inverse of the exponential decay rate of the concentration in the stationary decay stage. The degree of mixing for the CFSTRs is shown in Fig. 8. For small ﬂow rates (large Z), the curves basically follow the same trend as sa since sa is close to 1 for all the cases. However, for large ﬂow rate, JZ does not reﬂect the difference in the relative level of mixing among the cases. Case L has the highest value for the largest ﬂow rate while case I has the smallest value. This is just the opposite from what the variance of residence time showed in Fig. 4. We have seen that case I has the strongest
Fig. 6. CoV of age a as a function of the time ratio Z. Solid line (black): case I; dash–double dots (tan): case J; dashed line (red): case L; dash–dot (blue): case T; dotted line (green): case G. (For interpretation of the references to color in this ﬁgure legend, the reader is referred to the web version of this article.)
The small difference of sa from 1.0 in Fig. 5 may lead to the belief that the mixing states inside these reactors are very close to that of an ideal mixer. However, the spatial distributions in Fig. 3 have shown otherwise. This reveals again that the probability distribution of molecular age alone cannot be used to measure the state of mixing. Fig. 6 shows sa for the ﬁve cases. As can be seen in the ﬁgure, sa reduces as Z increases for all ﬁve cases. This clearly indicates that mean age distribution in the tanks becomes narrower around the average value as the ﬂow rate reduces. Among the ﬁve cases, case T has the highest sa , indicating widest spatial distribution of mean age. Recall that this is a case with non-symmetric feed of fresh tracer. At a small ﬂow rate, the tracer is fed to the slow mixing zone. Thus the fresh tracer will stay near the inlet for long time, resulting in a poor spatial distribution of mean age. At higher ﬂow rate the tracer is sent closer to the impeller suction zone around the shaft as shown in Fig. 3d. Thus the mixing
Fig. 7. The ratio of volume averaged mean age to the mean residence time as a function of the time ratio Z. Solid line (black): case I; dash–double dots (tan): case J; dashed line (red): case L; dash–dot (blue): case T; dotted line (green): case G. Thin dashed line (dark gray): ideal mixer. (For interpretation of the references to color in this ﬁgure legend, the reader is referred to the web version of this article.)
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Fig. 8. Zwietering’s degree of mixing as a function of the time ratio Z. Solid line (black): case I; dash–double dots (tan): case J; dashed line (red): case L; dash–dot (blue): case T; dotted line (green): case G. (For interpretation of the references to color in this ﬁgure legend, the reader is referred to the web version of this article.)
Fig. 10. CoV of mean age a about the mean residence time as a function of the time ratio Z. Solid line (black): case I; dash–double dots (tan): case J; dashed line (red): case L; dash–dot (blue): case T; dotted line (green): case G. (For interpretation of the references to color in this ﬁgure legend, the reader is referred to the web version of this article.)
Fig. 9. CoV of age a about the mean residence time as a function of the time ratio Z. Solid line (black): case I; dash–double dots (tan): case J; dashed line (red): case L; dash–dot (blue): case T; dotted line (green): case G. Thin dashed line (dark gray): ideal mixer. (For interpretation of the references to color in this ﬁgure legend, the reader is referred to the web version of this article.)
Fig. 11. The degree of mixing about the mean residence time as a function of the time ratio Z. Solid line (black): case I; dash–double dots (tan): case J; dashed line (red): case L; dash–dot (blue): case T; dotted line (green): case G. (For interpretation of the references to color in this ﬁgure legend, the reader is referred to the web version of this article.)
bypassing at the highest ﬂow rate. Case L should be the best case since the outlet is not directly on the tracer path. These curves clearly show that Danckwerts–Zwietering’s degree of mixing fails to characterize the degree of non-uniformity in mixing.
side of the tank in case T is also shown by the large values of sat at small Z. Fig. 11 shows the new degree of mixing for all the cases. It should be mentioned that the largest value of Jt for case I at Z ¼4.34 is 0.042. In the ﬁgure, the range of Jt is cut at 0.02 in order to show details of all the curves. Comparing with Fig. 9, we can see that the new measure correctly reﬂect the difference in mixing non-uniformity. From these curves we can conclude that at small ﬂow rates, case T is the worst and case G is the second worst. In both cases the tracer is fed to a slow mixing zone. The three cases with the same feeding location have similar performance regardless of the different locations of the outlet. At higher ﬂow rates, the worst case is case I which suffers strong bypassing. The second worst case is T. All ﬂow rates considered, case L has the best performance. It should be mentioned that the general trend of the degree of mixing as a function of the time ratio shown in Figs. 8 and 11 are
4.4. The degree of mixing relative to the ideal mixer The variance of age relative to the ideal mixer,sat is shown in Fig. 9. Comparing this ﬁgure with Fig. 4, we can see that the two ﬁgures are almost identical. This is because the residence time distribution function and the age distribution function are related by Eq. (7), and both variances are about the same value t. Fig. 10 shows the variation of sat as a function of Z. Compared to Fig. 5, the curves in this ﬁgures are quite different for large ﬂow rates. The strong bypassing in case I is correctly represented by the large value of sat . The effect of the strong jet shooting to the opposite
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the same as the axial temperature difference shown by Goldstein (1973) and the deviation from ideal mixing time shown by Roussinova and Kresta (2008) in their CFSTRs, both are also as a function of the time ratio. Since these are also measures for the mixing performance in a reactor, it is not surprising that they all follow the same trend.
5. Conclusions Mixing performance in continuous ﬂow stirred tank reactors are studied with age distributions. Effect of the time ratio, the ratio of mean residence time to the batch blend time, and the layout of the inlet and the outlet are quantitatively measured with probability distribution of age and the spatial distribution of mean age inside and at the exit of the reactors. The relationship between the moments of residence time at the exit and the volume averaged moments of age in the interior of the reactor is derived. This relationship connects the residence time theory to the mean age theory, making the computation of the moments of residence time much more efﬁcient and accurate. Spatial distributions of mean age in the reactors are computed by solving the steady transport equation. The contour plots of mean age clearly show the spatial non-uniformity in mixing. Undesired ﬂow features such as bypassing and slow mixing zones can be easily identiﬁed and their sizes and locations are readily quantiﬁed. The variances for age distribution deﬁned by Zwietering are further discussed. It is found that the total variance of age for the probability distribution of molecular age in the system can be split into two uncorrelated variances, the variance of mean age and the variance about mean age. These variances can be calculated for any ﬂows with CFD solutions of mean age and the second moment of age. From these variances, Danckwerts– Zwietering’s degree of mixing can then be calculated. The variances for the age distribution and Danckwerts– Zwietering’s degree of mixing are computed for non-ideal CFSTRs for the ﬁrst time in the literature. It is found that the variance of age as a function of the time ratio behaves similarly to the variance of residence time. For the ﬁve cases studied, this variance is very close to 1, the variance for the ideal mixer. The computed variance of mean age reveals the general trend of more uniform distribution for smaller continuous ﬂow rate. However, the effects of bypassing and plug ﬂow features are not reﬂected. Since the variance of age is very close to 1, the degree of mixing is mainly determined by the variance of mean age. Therefore, Danckwerts–Zwietering’s degree of mixing fails to reveal the relative level of spatial non-uniformity in mixing among different designs of the inlet and the outlet. A new degree of mixing is deﬁned using the variances about the mean residence time which is also the average age in an ideal mixer. The newly deﬁned variance of age also behaves similarly to the variance of residence time with values very close to 1, but the new variance of mean age correctly reﬂects the effect of bypassing and plug ﬂow features in the ﬁve cases. Therefore, the newly deﬁned degree of mixing can be used to measure the relative level of spatial non-uniformity in mixing. The results discussed in this article have demonstrated that the method presented is effective and efﬁcient for quantitatively measuring mixing performance of a CFSTR. This method can also be used for other types of continuous ﬂow systems. With this method, CFD can be a practical tool for industrial reactor design and scale-up for desired mixing performance.
aV c d ds dt D D Def f E F H I JZ Jt Mn Mn,e Mn,V Qm Q Sc t tn T u V x
volume averaged mean age (s) tracer concentration (volume fraction) inlet or outlet diameter (m) shaft diameter (m) outside diameter of top inlet (m) impeller diameter (m) molecular diffusivity (m2/s) turbulent effective diffusivity (m2/s) residence time frequency function cumulative residence time distribution function tank liquid height (m) 0th moment of age, an invariant Danckwerts–Zwietering’s degree of mixing the degree of mixing about the ideal mixer nth moment of age (sn) mass averaged nth moment of age at exit (sn) volume averaged nth moment of age at exit (sn) mass ﬂow rate (kg/s) volumetric ﬂow rate (m3/s) turbulent Schmidt number time (s) nth moment of residence time (sn) tank diameter (m) velocity vector (m/s) tank volume (m3) spatial coordinate vector (m)
a a Z nT s2 se sa sa sat sat t
molecular age (s) average molecular age in the reactor (s) ratio of mean residence time to batch blend time turbulent viscosity (kg/m s) variance about mean age (s2) CoV of residence time distribution CoV of age CoV of mean age CoV of age about the mean residence time CoV of mean age about the mean residence time mean residence time (s)
Appendix The variance of age can be split into three variances. Z Z 1 1 ðaaÞ2 ¼ ðaaÞ2 cðx, aÞda dx VI v 0 Z Z 1 1 ðaa þ aaÞ2 cðx, aÞda dx ¼ VI v 0 Z Z 1 Z Z 1 1 ¼ ðaaÞ2 cðx, aÞda dx þ ðaaÞ2 cðx, aÞda dx VI v 0 v 0 Z Z 1 ðaaÞðaaÞcðx, aÞda dx ðA 1Þ þ2 v
Where I¼ 0 cðx, aÞdx is an invariant. The ﬁrst variance is the variance about mean age Z Z 1 1 ðaaÞ2 ðaaÞ2 cðx, aÞda dx ðA 2Þ VI v 0 It can easily be found that ðaaÞ2 ¼ M 2,V a2V
Nomenclature a ae
mean age (s) mass averaged mean age at exit (s)
where a2V ¼
a2 dx V
M. Liu / Chemical Engineering Science 69 (2012) 382–393
From Eq. (A-3) it can be seen that this variance is the volume average of the variance about the mean age at a point in space. The second variance in Eq. (A-1) is the variance of mean age. It is a variance about the volume averaged mean age Z Z 1 Z 1 1 ðaaÞ2 ðaaÞ2 cðx, aÞda dx ¼ ðaaV Þ2 dx ¼ a2V a2V VI v 0 V V ðA 5Þ This variance describes the spatial variation of mean age as a scalar about its volume averaged value. The third variance is a covariance of the above two variances. It can be written as Z Z 1 1 ðaaÞðaaÞ ðaaaaa2 þ aaÞcðx, aÞdadx VI V 0 Z Z 1 Z 1 Z 1 1 cðx, aÞ cðx, aÞ cðx, aÞ ¼ a a a daa daa2 V V I I I 0 0 0 Z 1 cðx, aÞ da dx þ aa ðA 6Þ I 0 R1 Recall that a ¼ 0 acðx, aÞda=I, the integrand of the above volume integral becomes a2 aaa2 aa ¼ 0. Hence the covariance is zero ðaaÞðaaÞ ¼ 0
This indicates that the variance of mean age and the variance about mean age measure two independent events. Thus, the variance of age can be split into two parts, the variance about the mean age and the variance of mean age ðaaÞ2 ¼ ðaaÞ2 þ ðaaÞ2
For this reason, we can also call the variance of age the total variance of age. Zwietering (1959) called the variance about mean age the ‘‘variance in ages within the points’’ and the variance of mean age the ‘‘variance in ages between points’’. He proved equality of Eq. (A-8). References Aubin, J., Fletcher, D.F., Xuereb, C., 2004. Modeling turbulent ﬂow in a stirred tanks with CFD: the inﬂuence of the modeling approach, turbulence model and numerical scheme. Exp. Therm. Fluid Sci. 28, 431–445. Aubin, J., Kresta, S.M., Bertrand, J., Xuereb, C., Fletcher, D.F., 2006. Alternate operating methods for improving the performance of continuous stirred tank reactors. Chem. Eng. Res. Des. 87 (A), 569–582. Cholette, A., Cloutier, L., 1959. Mixing efﬁciency determinations for continuous ﬂow systems. Can. J. Chem. Eng. 37, 105. Cloutier, L., Cholette, A., 1968. Effect of various parameters on the level of mixing in continuous ﬂow systems. Can. J. Chem. Eng. 46, 82. Coroneo, M., Montante, G., Paglianti, A., Magelli, F., 2011. CFD prediction of ﬂuid ﬂow and mixing in stirred tanks: numerical issues about the RANS simulations. Comput. Chem. Eng. 35, 1959.
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