Prediction of Tracer Concentration and Mixing in CFSTRs with Mean

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Prediction of Tracer Concentration and Mixing in CFSTRs with Mean Age Distribution Minye Liu* DuPont Company, 1007 Market Street, Wilmington, Delaware 19898, United States ABSTRACT: Three-dimensional time-dependent tracer concentration distributions in nonideal continuous flow stirred tank reactors with pulse input are studied computationally. Both the steady transport equation for mean age and the time-dependent transport equation for tracer concentration are solved. It is found that the time-dependent tracer concentration history can be divided into two stages: an initial stage, in which the concentration is highly location-dependent, and a stationary stage, in which the concentration decays with a location-independent, constant exponential rate. This rate is found to be the inverse of the volumeaveraged mean age. The length of the initial stage is found to be very short, on the same order of magnitude as the batch blend time. It is also found that the scaled concentration distribution in the stationary stage is almost identical to the scaled mean age distribution. The spatial and temporal concentration distribution in the stationary stage can be determined once the initial stage and the mean age solution are obtained. Thus, the CPU time required is orders of magnitude smaller than that for the full time integration of the unsteady concentration equation. With the solution of tracer concentration distribution known, mixing time and other quantitative measures for the tracer mixing can be easily calculated.

1. INTRODUCTION Stirred tank reactors are widely used the processing industry in both batch and continuous modes. The mixing process in stirred tanks in the batch mode is mainly controlled by the impeller rotation. A majority of mixing research literature has been focused on this mode. On the other hand, mixing in a continuous flow stirred tank reactor (CFSTR) also depends on the continuous flow rate and the layouts of the inlet and the outlet. Research on CFSTRs is sparse in the literature. Besides the complication of the interaction of the impeller, the continuous flow, and the inlet and outlet design, there seems to be a lack of an effective method for characterizing the nonideal mixing. As a first approximation, a CFSTR is often modeled as one or several ideal mixers. To consider the nonideality, the residence time distribution (RTD) theory1 is often used to measure the relative deviation from the ideal mixer. The RTD theory is based on the time history of the tracer concentration measured at the exit of a reactor. With this theory, an RTD function may be determined through the use of tracer addition experiments, typically by measuring exit concentration response to pulse or step-change tracer addition. By analyzing the shape of the curve, some mixing properties inside the flow may be inferred.2,3 Although the RTD theory provides useful diagnostics to mixing in a flow system, an RTD function does not have sufficient information to determine the state of mixing in the system, and within the current RTD theory, more information is difficult to obtain.4 The RTD function is also not unique to a reactor. As Danckwerts5 and Zwietering4 showed, two simple flows with a reversed order of a plug flow and an ideal mixer will result in the same RTD function but different conversion. An RTD function is a probability distribution function of molecular age at the exit of a reactor. This function does not contain spatial distribution of material age inside the flow. Such spatial distribution of material age is critical in order to define the status of r 2011 American Chemical Society

mixing. On the basis of the shape of an RTD curve relative to the exponential function of an ideal mixer, some possible dead zones and bypassing regions may be detected, but the sizes and locations of such regions cannot be determined. The RTD theory is mainly built on two ideal flows, the plug flow and the ideal mixer. A real flow system is modeled with multiple plug flows and ideal mixers with different parameters. This modeling process is just curve fitting and often does not provide much understanding of the actual process in the flow. One type of modeling method for CFSTRs using the RTD theory is the so-called compartmental model. In this model, the volume of a stirred tank reactor is divided into subvolumes of plug flow, bypassing zone, dead zone, and ideal mixing zone. Levenspiel3 discussed the effect of each type of zone on the RTD curve. Cholette and Cloutier6 developed a model for the RTD function of a nonideal CSFTR with the fractions of each subvolume as parameters to be determined from experimental measurements of tracer concentration at the exit of a reactor. Manning et al.7 divided the reactor into two zones, a perfect micromixing zone around the impeller and a perfect macromixing zone outside of the impeller. The RTD function for the macromixing zone is an exponential function in the same format as the ideal mixer but with two parameters. Although this type of method is practical, it is difficult to determine all of the parameters since in a real CFSTR there are no well-defined plug flow, bypassing, ideal mixing, and dead zones. There has also been a lack of quantitative measures for the performance of CFSTRs. One rule of thumb used in the process industry is that the ratio of the mean residence time to the batch Received: February 1, 2011 Accepted: March 29, 2011 Revised: March 19, 2011 Published: March 29, 2011 5838

dx.doi.org/10.1021/ie2002395 | Ind. Eng. Chem. Res. 2011, 50, 5838–5851

Industrial & Engineering Chemistry Research blend time should be larger than 10 in order to achieve near perfect mixing. Jones et al.8 found from their experimental measurements of the RTD function that the variance of the function cannot be scaled when the ratio is below 10. This seems to be an indication that the impeller created flow is significantly affected by the continuous flow. Roussinova and Kresta9 found from their measured concentration history both inside the reactor and at the exit that a mean residence time of 15 batch blend times or more is needed to approach the ideal condition. Knowledge of the effects of inlet/outlet positions and the continuous flow rate on mixing performance and spatial nonuniformity is still missing. For industrial applications, design guidelines and mixing performance correlations are needed. Mixing and reactive processes in a CFSTR are fundamentally controlled by the hydrodynamics of the fluid flow, and quantitative characterization of mixing performance can only be made after knowing tracer concentration distribution. Recent advancements in computational fluid dynamics (CFD) have significantly enhanced the understanding of the flow processes inside stirred tanks. However, the understanding of tracer concentration distribution is still lagging. Solving for time-dependent tracer concentration distribution using CFD is not trivial. First, small time step sizes have to be used for the time integration in order to accurately track the tracer development over time. Second, tracer history has to be followed for a very long time if accurate statistical characterization of the mixing process is desired. Nauman and Buffham10 have estimated that a 16% error in variance could result when the tracer concentration function is truncated at five mean residence times. Liu and Tilton11 found numerically that there is a 4% error in mean residence time and a 25% error in the third moment when the truncation is at four mean residence times for their 2-D test reactor. For many industrial problems, the CPU time required for accurate tracer tracking is often not practical. Several studies have been reported in the literature using CFD to model tracer concentration1217 in CFSTRs. Some insight of tracer mixing has been obtained from these studies. However, most of these studies followed tracer history for only a short period of time. None of these studies reported detailed spatial nonuniformity of tracer concentration. Only a few selected locations inside the reactor were analyzed by some of the studies. To avoid long CPU times, some of these studies used extremely large time step sizes from half up to many impeller revolutions. Numerical integrations with such large time step sizes cannot be reliable. Recently, a new method was reported that may overcome some of the weaknesses of RTD theory and is much more computationally efficient. Instead of looking for solutions of molecular age distribution, this method uses spatial distributions of mean and higher moments of molecular age. The concept of mean age in a continuous flow system was first discussed by Spalding.18 Spalding derived an equation on his way to show that, in a “closed system” with one inlet and one outlet, the mean residence time is equal to the volume divided by the volumetric flow rate, t = V/Q, a well-known fact today. On the basis of this equation, Sandberg19 derived the final transport equation for mean age and used the method in a room ventilation study. Baleo and le Cloirec20 computed mean age distribution in a 2-D axisymmetric flow through a channel containing a series of sudden expansions and contractions. They found good agreement between their computational and experimental results. Liu and Tilton11 advanced this method by deriving governing

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Figure 1. The stirred tank geometry and dimensions. T = H = 0.292 m. D = T/3. d = 25 mm. ds = 12 mm. dt = 28 mm. The “x” shows the locations of the three selected points used throughout this article. The box drawn in dashed lines shows the interface of rotating and fixed zones in the MRF method.

equations for all of the moments of age. They also made the hand waving boundary conditions by Baleo and le Cloirec20 more rigorous. The governing equations of mean age and higher moments are in the same conservative form as the NavierStokes equations. This makes the solution process simple. The same solver for the NS equations can be used within the same code. For most flows, laminar or turbulent, convergence of the solutions is very fast. It requires only a fraction of the CPU time required for the time-dependent solutions of concentration equation. Therefore, the computational efficiency makes this method of the moments of age even more attractive. In this article, a method will be discussed that characterizes mixing in a CFSTR reactor by computing full 3-D distributions of tracer mean age and concentration. With this method, the timedependent tracer history needs to be tracked for only a short period of time. Full 3-D and time-dependent tracer concentration can then be obtained from the steady solution of mean age distribution. The rest of this paper is organized in four sections. In section 2, the velocity solutions of three CFSTRs will be discussed. Obtaining the velocity solution is just the start for mixing analysis. In section 3, mean age solutions will be solved using the given velocity fields. The governing equations for the mean age and higher moments of age will be briefly presented, and the mean age solutions in the three CFSTRs will be discussed. Section 4 contains the results of time-dependent tracer concentration distributions. The relations of concentration distribution and mean age will be discussed. Section 5 concludes the article with a summary of the important findings in this study.

2. FLOW SOLUTIONS OF THE STIRRED TANK REACTOR The stirred tank reactor studied in this article is equipped with a standard 45 pitched blade turbine (PBT) impeller with four blades. The diameter of the tank is T = 0.292 m, and the ratio of the height H to the diameter of the tank is equal to 1. The diameter of the impeller is D = T/3, and the bottom clearance is H/3. The impeller speed is fixed at 300 rpm. The stirred tanks have four standard baffles with a width of about T/12. The reactors have one inlet and one outlet with two different layouts. 5839

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Figure 2. Velocity vector plots in the stirred tank on the x = 0 plane. The color shades represent velocity magnitude in m/s. (a) Case 1, Qm = 0.2 kg/s. (b) Case 2, Qm = 0.2 kg/s. (c) Case 3, Qm = 0.5 kg/s.

For case 1, the inlet is at the center of the bottom of the tank and the outlet is at the side near the top surface. For cases 2 and 3, the inlet is at the center on the top surface with a ring around the shaft. The outlet is at the center of the bottom. The ring inlet is to maintain a symmetric geometry in order to minimize numerical errors introduced by the method of multiple reference frame (MRF), which assumes rotational symmetry on the interface between the rotating zone around the impeller and the stationary zone next to the baffles and tank wall. The dimensions of the stirred tank with the inlet and the outlet are shown in Figure 1. It should be mentioned that the baffles in Figure 1 are for illustrative purpose only. The actual baffle planes are 45 from the outlet on the upper left side. The fluid is considered water with a density F = 1000 kg/m3 and a viscosity μ = 0.001 Pa s. The flow is in the turbulent regime with an impeller Reynolds number of about 47 000. To model the turbulent effect on flow and mixing, the standard kε model is used. The top surface of the reactor is fixed flat as a symmetry plane. For flow solutions, the boundary condition is set as a constant mass flow rate at the inlet and zero normal velocity gradient at the exit. The continuous mass flow rate is 0.2 kg/s for

case 1 and case 2 and 0.5 kg/s for case 3. The mean residence times calculated from V/Q are 93.14 and 37.26 s, respectively. The batch blend time is calculated at about 8.5 s using the correlation in Grenville and Nienow.21 The ratios of the mean residence time to the blend time are then 10.96 and 4.34, respectively. These two different ratios are used to study the effect of continuous flow rate on mixing later. The flow solutions are obtained with the commercial CFD package Fluent version 6.3 with full 3-D geometry and a mesh of about 1.5 million cells. QUICK scheme is used for spatial discretization of the NS equation in order to obtain a more accurate solution. The total CPU time for each steady flow solution is about 50 h on a four processor (Dell Precision 490 with Xeon chip) Linux workstation. In order to quantitatively characterize the mixing performance, a fully converged flow solution is critical. For all of the cases discussed here, the normalized residuals for all three velocity components are well below 106. The consistency of the numerical solutions should also be checked before the solutions can be trusted. For this purpose, the torque balance and energy conservation of the numerical solutions are checked 5840


Industrial & Engineering Chemistry Research with a batch mode. It is found that the torque measured on the baffles and the tank wall is within 98% of that calculated on the impeller shaft. This good agreement is achieved by the deep convergence of the flow solution. It is found during the solution process that the ratio of the two torques is about 0.82 when the normalized residual of velocity is about 4 105. This ratio is about 0.95 when the residual is reduced to about 105. A ratio of 0.98 is achieved when the residual is reduced to 106 at 7000 iterations. It has been reported in the literature that the kε model significantly underpredicts turbulent kinetic energy and the energy dissipation rate in stirred tanks. It is found in this study that the volume integration of the turbulent dissipation rate is about 85% of the total energy computed from impeller torque. Although this is still not a 100% agreement, it is satisfactory given the limitations in the turbulence model and various numerical assumptions. The power number computed from impeller torque is in good agreement with those reported in the literature. In batch mode solution, the power number is calculated at 1.33. This number is in a close range of 1.27, a typical number used in the process industry.22 Velocity vector plots on x = 0, the vertical plane of inlet and outlet, are shown in Figure 2. Figure 2a is the velocity field of case 1 with the inlet at the bottom center. Figure 2b and c are for cases 2 and 3, with the inlet at the top center around the shaft. From the three vector plots, it can be seen that the effect of the inlet flow on the overall flow pattern in the stirred tank is not significant. The general structure is the familiar single loop in the tank generated by the axial down pumping action of the impeller. The circular jet leaves the impeller, impacts at the bottom of the tank at an angle, and then turns upward along the wall. The jet loses its strength before reaching the top surface. The jet is fed by the returning fluid around the impeller shaft above the impeller. This is the same typical flow pattern in a batch mode. For case 1, even though the inlet flow is in the opposite direction to the main loop in the stirred tank, its effect is the slight change in the angle of the impeller stream. There are two small but clear weak vortices between the inlet jet and the impeller stream. For case 2, since the inlet flow is in the same direction as the main loop, it enhances the strength of the impeller flow. This can be seen from the more vertical angle of the impeller stream. As the flow rate in the inlet is increased from 0.2 kg/s to 0.5 kg/s, the enhancement of the main loop is stronger. This is shown by the further change of the angle of the impeller stream. The angle change can also be seen by comparing the size of the triangular region below the impeller. Although the above-discussed flow pattern changes reveal some general trend of tracer motion in the reactor, they carry very little mixing information. For example, as the flow rate is increased from 0.2 kg/s to 0.5 kg/s, the ratio of the mean residence time to the batch blend time changes from 10.86 to 4.34. On the basis of the traditional rule of thumb, a significant change in tracer mixing should be expected. Such change is not visible from the flow field change. Given the layout of the inlet and the outlet locations for cases 2 and 3, strong bypassing of the tracer from the inlet to the outlet should also be expected. This is also not revealed clearly by the flow pattern. The feed of the tracer seems to follow the strong impeller stream and the main loop. Therefore, velocity field alone does not contain enough information to characterize mixing performance.

3. THE MEAN AGE DISTRIBUTION 3.1. Governing Equations for Mean Age. The derivation of the governing equations for the mean age and moments of age by

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Spalding,18 Sandberg,19 and Liu and Tilton11 is based on a reactor with one inlet and one outlet that formed a simple closed system. A closed system is a continuous flow system in which a fluid particle can only make one pass entering or leaving it. This means that diffusion at the inlet and the outlet is not considered.10 The flow in the reactor is assumed steady and incompressible. The nth moment of age at a spatial location x is defined as Z ¥ Z ¥ t n cðx, tÞ dt= cðx, tÞ dt ð1Þ Mn ðxÞ 0

0

where the integral in the denominator, denoted by I, Z ¥ I cðx, tÞ dt

ð2Þ

0

is an invariant and is the ratio of the total volume of the tracer in the pulse to the volumetric flow rate.18 The governing equation for the nth moment is r 3 ðuMn Þ ¼ r 3 ðDrMn Þ þ nMn 1 By its definition, mean age is the first moment of age. Z ¥ Z ¥ aðxÞ tcðx, tÞ dt= cðx, tÞ dt 0

ð3Þ

ð4Þ

0

And the governing equation for mean age is r 3 ðuaÞ ¼ r 3 ðDraÞ þ 1

ð5Þ

This method applies to both laminar and turbulent flows. Turbulent flows are intrinsically time-dependent. However, when a turbulent flow is modeled with the method of Reynolds averaging, the governing equations for the flow and tracer concentration become steady due to ensemble averaging. The eddy viscosity model introduces a turbulent viscosity and a turbulent diffusivity in the averaging process. The molecular diffusivity D in the above equations will then be replaced with an effective turbulent diffusivity which is the sum of this molecular diffusivity and the turbulent diffusivity, D eff = D þ νT/Sc. The turbulent Schmidt number Sc is chosen as 0.7 in this article. Equation 3 has simple boundary conditions. At the inlet, the mean age or moments are zero, and at the outlet, its derivative normal to the outlet is zero. On all solid walls, the normal derivatives are zero. 3.2. Mean Age Distribution in the Reactors. In order to apply the mean age theory, the flows have to be steady. This condition is satisfied by the MRF method. In the MRF method, the tank is divided into two zones, one around the rotating impeller and the other near the tank walls and baffles. The outside zone is solved in the stationary frame, and the inside zone around the impeller is solved in a frame rotating with the impeller. After the coordinate transformation, the flows in both zones become steady in their own reference frames. The flow solution is obtained first, and then the moments of age can be solved in sequence. A user-defined function is used for the turbulent diffusivity. A separate custom code is developed for further post processing of the solutions of mean age and higher moments. Contour plots of mean age for the three cases discussed in the previous section are shown in Figure 3 on the x = 0 plane. In all of the figures, the lower bound of mean age is cut above zero in order to show the spatial structure of the distribution. Figure 3a shows the mean age for case 1. Low mean age is clearly seen for 5841



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Figure 3. Mean age contour plots on the x = 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 1. (b) Case 2. (c) Case 3.

the fresh feed of the tracer near the inlet at the bottom. The triangular zone under the impeller has the lowest mean age compared to the rest of the stirred tank. On the other hand, the highest mean age is near the top surface. Compared to the mean residence time τ = 93.14 s, the maximum mean age in the tank is about 94.30 s. The location of this maximum mean age is found to be behind each baffle after a close examination of the 3-D distribution. This location is outside the main loop and inside the separation region of the flow. The ratio of the maximum mean age to the mean residence time is about 1.0125. Figure 3b shows the mean age contour plot for case 2. In this figure, the path of the fresh material can be seen clearly. The inlet and the outlet are short circuited, and a strong bypassing exists. This indicates that a portion of the fresh tracer material flows over the impeller and leaves the stirred tank without mixing with the material in the rest of stirred tank. The maximum mean age is again near the top, behind the baffle due to flow separation. The ratio of the maximum mean age to the mean residence time is 1.0958 s, slightly larger than for case 1. The spatial structure of mean age distribution for case 3 has similar pattern as for case 2, as shown in Figure 3c. However, the ratio of the maximum mean age to the mean residence time is 1.2976 s. This much larger ratio

than for case 2 indicates stronger nonuniformity of mixing due to the stronger bypassing. By comparing the spatial distributions of mean age with the velocity vector plots, it is noticed that velocity distribution can be misleading if used alone to characterize mixing. First, plots of velocity distribution, either vector or contour, do not show bypassing for cases 2 and 3, as shown in the mean age plots. Second, the strong vortex shown in the velocity field may lead to the belief that there is a dead zone at the center of the vortex. However, for a turbulent flow, turbulent diffusion contributes to mixing significantly. That is why there is no high mean age zone at the locations of the vortex center. The moments of residence time distribution can be found from mean age and higher moment solutions. Liu and Tilton11 showed that the mass averaged mean age and higher moments at the exit are the same as the moments of residence time. This relation can be used to check the accuracy of the mean age solution. For case 1, the mass averaged mean age at exit ae is found to be 93.16 s. Compared with τ = 93.14 s calculated from V/Q, the agreement is excellent. Similarly for case 2, it is found that ae = 93.09 s. For case 3, the mean residence time calculated from V/Q is τ = 37.26 s. From the mean age solution at the exit, 5842



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Figure 4. Time-dependent concentration history showing the two stages. Solid lines (black), at exit; dash-dot lines (green), point 1; dash lines (blue), point 2; dotted lines (red), point 3. (a) Case 1. (b) Case 2. (c) Case 3.

it was found that ae = 37.24 s. Therefore, the computed mean age solutions are very accurate. The second moment of age can also be checked. Nauman and Buffham10 have shown that R = th2/2τ, with R the average age of all of the tracer molecules inside the stirred tank and th2 the second moment of residence time at the exit. It can be shown that R = aV with aV the volume averaged mean age. We then have th2 = 2τaV. Since th2 is equal to the mass averaged second moment of age at the outlet, th2 = M 2,e as Liu and Tilton have shown, this relation can be used to check the accuracy of the second moment of age computed from eq 3. For case 1, it is found that aV = 91.37 s and M 2,e = 17025.03 s2. Comparing th2 = 17025.06 s2 with M 2,e, the agreement is again excellent. Similar agreements are found for cases 2 and 3. For case 2, th2 = 18491.40 s2 and M 2,e =18491.11 s2. For case 3, th2 = 3448.42 s2 and M 2,e = 3448.35 s2. The CPU time to solve for each moment is less than 20 min on the four processor Linux workstation. The traditional method of computing the moments of residence time is to track the time-dependent tracer for the full residence time distribution. This time-dependent method has to solve the unsteady concentration equation for several mean residence times at small time step sizes in order to obtain accurate integration for a moment. Compared to the CPU time of this traditional method, the CPU time of the mean age method of solving eq 3 is several orders of magnitude

smaller, making this method extremely efficient and accurate for mixing studies.

4. TRANSIENT SOLUTION OF TRACER CONCENTRATION 4.1. Solution Method. In order to study the spatial and temporal nonuniformity of tracer mixing, time-dependent tracer concentration is computed by solving the unsteady convection diffusion equation for tracer concentration.

∂c þ r 3 ðucÞ ¼ r 3 ðDef f rcÞ ∂t

ð6Þ

To obtain an accurate numerical solution, proper time step size is important. An unnecessarily small time step size will increase the computational time while a too large time step size will hurt the accuracy of the solution. An adequate time step size can be estimated by the flow CFL number, CFL = ΔtU/Δx∼1. For the given mesh, a representative mesh size is found as Δx = 2.56 103 m. An appropriate velocity scale is needed. The maximum absolute velocity is near the tip of the impeller. If this velocity is used, the maximum velocity will be about 1.5 m/s for 300 rpm. The time step size is then found to be about 0.0015 s. However around the impeller, the equation is solved on a 5843


Industrial & Engineering Chemistry Research rotating frame. Therefore, the relative velocity should be used as the velocity scale for CFL. In the fixed zone, for the same rpm, it is found that the maximum velocity is about 0.5 m/s. The time step size is then about 0.005. For the solutions discussed in the rest of this article, a time step size of 0.01 s is used in order to shorten the CPU time without sacrificing accuracy. The equivalent step size in the degree of impeller rotation for 300 rpm is 18. Compared to the CPU time needed for a batch reactor modeling, the CPU time required for a continuous flow reactor is much longer since the mean residence time is usually much longer than the batch mixing time and accurate prediction of mixing performance from the tracer concentration history requires the modeled time to be several mean residence times. For case 1, the mean residence time is 93.14 s. This corresponds to about 466 impeller revolutions. With Δt = 0.01 s, the total number of time steps will be 9314. On our four processor workstation, the CPU time for each time step is about 1.5 min. The CPU time for each mean residence time is thus about 9.7 days. Such long CPU times are often not practical for industrial demands. Therefore, a more CPU efficient method for mixing studies is extremely important for industrial applications. The order of accuracy of the time integration scheme is also important. A higher order scheme is usually more desirable. We first used the second order implicit scheme but found that it gave unrealistic undershoots with a large negative concentration at the beginning of the pulse near the inlet due to the large gradient of the concentration. This type of numerical error is typical with some higher order schemes, and the overshoots and undershoots are typically 1015% of the pulse.23 Although the negative concentration eventually disappears, its effect on the accuracy is concerned. On the other hand, we did not see such wiggles on any solution when the first order implicit scheme was used. Therefore, the first order implicit scheme is used in this study. At time zero, tracer concentration in the flow is set to zero. A pulse of c = 1 (or volume fraction = 1) is then imposed at the inlet for 20 time steps. For the initial 100 time steps or so, a smaller time step size, Δt = 0.005 s, is used in order to track the initial large concentration gradient more accurately. The spatial invariant I defined in eq 2 is then 0.1 s. This invariant will be used later to check the accuracy of numerical integrations. 4.2. Time History of Tracer Concentration. The time history of the tracer concentration is tracked at four different locations in the stirred tank reactor. Three are interior points, as shown in Figure 1, and the fourth is the mass averaged concentration at the outlet. Figure 4 shows the four time history curves for all three cases. The curves are plotted on the semilog scale. These curves are typical for nonideal CFSTRs measured at the exit. It can be observed that each curve can be divided into two stages. The first stage is the initial response of concentration at the measured location to the input pulse. This response is very much locationdependent. The time span of this stage is very short, on the order of about 10 s. Considering the batch blend time of 8.5 s, we may say that the time span of the first stage is about the same as the batch blend time. This finding is in agreement with the experimental observation reported by Voncken et al.,24 that the tracer response curve at the exit is equal to that of the perfect mixer after about five circulation times, t > 5tc. Nienow25 has suggested that homogenization can be achieved in about five circulations. In order to show more closely the location dependent properties of the curves, only the initial 15 s of the curves are replotted in Figure 5. The response curves for case 1(one) are in Figure 5a. Since point 1 is in the path of the impeller jet, tracer material first

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reaches this point about 1 s after the initial pulse at the inlet. The tracer concentration reaches the peak value of about 0.003 at 1.54 s. The tracer material reaches both point 2 and point 3 in about the same amount of time, 2 s. There is a small peak on the curve for point 2, but no peak exists on the curve for point 3. From the mean age contour plot in Figure 3a, we have seen that these two points are outside the main loop of the flow and are in the slow mixing zone. A closer check of the location of point 2 shows that it is in the separation zone behind the baffle. Therefore, the tracer reaches this point mainly by turbulent diffusion. The tracer first arrives at the outlet at about t = 1.7 s. There are two peaks on the curve at the exit: the first is at about t = 2.63 s, and the main one is at t = 3.25 s. The curves in Figure 5b for case 2 clearly show bypassing. The tracer arrives at the outlet first in about 0.8 s and reaches the large peak value at t = 1.56 s. Compared to the peak value of about 0.00125 for case 1, the peak value here is more than 0.009. Due to the early loss of tracer material, the peak at point 1 has a smaller value than for case 1, about 0.0025 compared to the value of about 0.003 for case 1. It is interesting to notice that the tracer concentration histories at point 2 and point 3 are similar to those for case 1 in both initial rising time and peak time, at t = 2.0 s and 3.3 s, respectively. When the flow rate is increased from 0.2 kg/s for case 2 to 0.5 kg/s for case 3, bypassing becomes more severe. This can be seen by comparing both the initial rising time and the magnitude of the concentration curve at the exit. The initial rising time of the curve at the exit is about 0.8 s for case 2 and about 0.45 s for case 3, as can be seen in Figure 5b and c. The peak concentration for the higher flow rate case is almost 5 times that for the lower flow rate case. The time at the peak is also early for case 3. At t = 0.84 s, it is almost half the time for case 2. Both the rising times and the peak times of the curves at the three interior points are also shorter due to the higher flow velocity. This can be seen more clearly in the inset figure in Figure 5c on smaller scales. The tracer concentrations in the second stage show exponential reduction at all locations. The exponential decay in residence time distribution has been known for a long time. Using a stagnation pocket, Nauman and Buffham10 showed that the exponential decay at a large time is due to diffusion. However, this exponential decay has never been characterized quantitatively in the literature. From Figure 4, it is noticed that the slopes of the curves in each case are the same and are independent of locations. Concentration everywhere in the reactor decays at the same exponential rate. For this reason, this stage is called the stationary stage. By contrast, the first stage is called the initial stage. In order to further characterize tracer mixing behavior, it is desirable to find an explicit function to describe the concentration history in the stationary stage. For this purpose, an exponential function similar to that for the ideal mixer is assumed, cðx, tÞ ¼ or in log form,

I t=~t e ~aðxÞ

I t ln cðx, tÞ ¼ ln ~t ~aðxÞ

ð7Þ

ð8Þ

where I is the spatial invariant, ~a(x) is a location dependent variable to be determined, and 1/~t is the slope of the curve in a semilog plot that has been found to be location-independent. Since ~t is the same for every point in the system, it must be a 5844



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Figure 5. Time-dependent concentration history showing the initial stage. Solid lines (black), exit; dash-dot lines (green), point 1; dash lines (blue), point 2; dotted lines (red), point 3. (a) Case 1. (b) Case 2. (c) Case 3.

spatial averaged quantity with a time unit. It is found that ~t = aV for every point of the three cases studied here. In fact, several more points in the three cases were checked, and the relation is found true. Figure 6 shows the slope of the concentration for the three interior points and the exit of the three cases. It can be seen that the slopes of all of the curves converge to the same value, which is 1/aV for a t greater than about 10 s. The extremely large slopes in the initial 5 s are excluded in the figures in order to show the convergence more clearly. The slope at point 2 converges to 1/aV at a slightly longer time, indicating the much slower mixing in the separated zones behind the baffles. With ~t known, ~a can then be found from any selected instant of time on a concentration curve in the stationary stage. Thus, the tracer concentration in the stationary stage can be completely determined as cðx, tÞ ¼

I t=aV e , t > ts ~a ðxÞ

evaluated at the exit with those computed from the moments of age from eq 3. Liu and Tilton11 have shown that the flow averaged moment of age at the exit of a reactor is the same as the moment of residence time, Z Z 1 1 ¥ n M n, e ¼ n uMn dx ¼ t ce ðtÞ dt ¼ ten ð10Þ Q se 3 I 0 where the spatial integration is on the surface of the exit Se, n is the out normal of the surface, and ce(t) is evaluated at the exit with flow averaging. Finding the moments by integration of concentration history takes place in two steps. From its definition, the moments are computed as Z ¥ Z ts Z ¥ n n n t ðxÞ ¼ t cðx, tÞ dt ¼ t cðx, tÞ dt þ t n cðx, tÞ dt 0

ð9Þ

where ts is the time of the beginning of the stationary stage. To check the accuracy of eq 9 and the accuracy of the numerical integration in the initial stage, comparisons of the moments of age computed from eq 3 and the explicit integration of eq 1 and eq 9 can be made. At the exit, the comparison will be between the moments of residence time computed from eq 9

0

ts

ð11Þ The first term on the right-hand side in the above equation has to be integrated numerically in the initial stage. The second term can be integrated analytically using eq 9. For n = 0, the moment is the spatial invariant I. The calculated moments at the exit using both methods are listed in Table 1 up to the third moment for all three cases. From these results, several conclusions can be made. 5845



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Figure 6. Exponential rate of tracer concentration. Solid lines (black), exit; dash-dot lines (green), point 1; dash lines (blue), point 2; dotted lines (red), point 3; long dashes, 1/aV. (a) Case 1. (b) Case 2. (c) Case 3.

Table 1. Comparisons of Moments of Age at Exit for All Three Cases I (s)

ae (s) t e (s) M 2,e (s2)

ht2e (s2)

M 3,e (s3)

Table 2. Comparison of Moments of Age for the Three Interior Points of Case 1

ht3e (s3)

I (s)

a (s)

t (s)

M2 (s2)

ht2 (s2)

M3 (s3)

ht3 (s3)

case 1 0.1000 93.16 93.71 17021.3

17129.1

4663605

4695201

point 1

0.1000

91.26

91.58

16668.7

16733.3

4567011

4586690

case 2 0.0999 93.09 93.37 18491.1

18512.9

5515268

5515431

point 2

0.1000

94.64

94.97

17302.9

17369.8

4740637

4761238

point 3

0.1000

93.45

93.76

17076.6

17138.0

4678778

4697623

case 3 0.1000 37.24 37.43

3448.35

3420.09

480910.2

475008.5

First, the computed spatial invariant for all three cases is very close to 0.1 s. This confirms that the MRF method can indeed be used to compute the moments of age using eq 3 in a stirred tank. Comparing the mean age and the moments computed from the two different methods, it is noticed that the errors are all very small; most of the errors are well below 1%. This shows the excellent accuracy of both the numerical solutions and confirms the exponential decay rate of 1/aV. Similar comparisons are made for interior points. Shown in Table 2 are the moments for case 1 at the three interior points discussed above. Again, the spatial invariant is accurately predicted. Comparing with the local mean age and higher moments from the steady governing equation, the predictions with the concentration function are again very accurate. The errors are all smaller than 1%. It should be mentioned that the same comparisons were also computed with the results using a second-order

time integration scheme. It was found that the errors are around 3% for all three interior points and the exit using case 1. This large error is most likely caused by the undershoots of negative concentration at the beginning of the pulse. 4.3. Spatial Nonuniformity of Concentration Distribution. The spatial nonuniformity of tracer concentration can be studied by examining the contour plots at different instants. Figure 7 shows contour plots of tracer concentration for case 1 at four different times, t = 0.1 s, 1.9 s, 4.9 s, and 10.78 s. Figure 7a is at the end of the pulse, t = 0.1 s. After 1.9 s, the tracer material has been divided into two parts; one is pumped to the bottom into the triangular zone and the other into the main loop, as can be seen in Figure 7b. This may explain why there are two peaks on the concentration curve at the exit. Figure 7c shows the contour plot at t = 4.9 s. At this time, the tracer has reached every part of the tank. The slow mixing zone outside of the main 5846



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Figure 7. Contour plots of tracer concentration at different time instants for Case 1. (a) t = 0.1 s. End of the pulse. (b) t = 1.9 s. (c) t = 4.9 s. (d) t = 10.78 s.

loop has a higher concentration now, and the triangular zone below the impeller has a lower concentration due to the effect of the inlet jet. Figure 7d shows the spatial distribution of the tracer near the beginning of the stationary stage at t = 10.78 s. From this point on, the concentration will change at the same exponential rate everywhere in the reactor. Therefore, the pattern of the concentration distribution will not change. Similar development of the tracer concentration in time is also found for cases 2 and 3. The observation of the identical concentration distribution pattern seems to suggest that the concentration distribution in the stationary stage is scalable. The volume averaged tracer concentration seems to be the natural choice for the scaling, ^c ¼

cðx, tÞ cV ðtÞ

ð12Þ

If we define

cV ðtÞ ¼

I t=aV e ~ A

ð15Þ

This leads to ~ A cðx, tÞ ð16Þ ¼ ~aðxÞ cV ðtÞ Since A ~ is a constant and ~a(x) is independent of time, we therefore have shown that the scaled concentration ^c is independent of time and is a function of spatial position only. To find A~, we can use the mass balance of the tracer inside the reactor and at the inlet and the outlet. ^c ¼

V

Z Z Z 1 I 1 t=aV I 1 e dx cðx, tÞ dx ¼ dx ¼ et=aV V V V V ~aðxÞ V aðxÞ V ~

ð14Þ

we will have

The volume averaged concentration is defined as cV ðtÞ ¼

Z 1 1 1 dx ¼ ~ V V ~aðxÞ A

dcV ðtÞ ¼ Q ce ðtÞ dt

ð17Þ

where ce is the flow averaged concentration at the exit, ce ðtÞ ¼

ð13Þ 5847

I t=aV e ~ae

ð18Þ



ARTICLE

Figure 8. Tracer concentration frequency functions at different time instants. Long dash line (red), t = 3.51 s; short dash line (tan), t = 4.51 s; dash-dot (blue), t = 5.47 s; dots (green), t = 9.47 s; dash-double dot (magenta), t = 15.47 s; solid line (black), t = 21.64 s.

~ae is a constant at the exit determined in the same way as ~a(x) at x. From eq 17, we can find that ~ ¼ A

~ae τ aV

ð19Þ

~ae τ aV ~aðxÞ

ð20Þ

This leads to ^cðxÞ ¼

The scaled concentration can be better seen by its frequency functions. In Figure 8, scaled frequency curves are shown at six different time instants for case 2. From these curves, it can be seen that, after about 10 s, all of the curves collapse to a single one. 4.4. Relationship of Concentration and Mean Age. By comparing Figure 3a and Figure 7d, it can easily be recognized that there is a similarity between the spatial distribution of the mean age and the concentration in the stationary stage. Such similarity exists for all three cases. In fact, when both distributions are scaled with their own volume averaged values, the distributions become almost identical. This can be seen by comparing their frequency functions as shown in Figure 9a for case 2. Further, the near equality of the two scaled variables point by point is shown in Figure 9b along a vertical line inside the stirred tank at x = 0 and y = D/4 for case 2. From this figure, it can be seen that the differences between the two curves are negligibly small. This means that mean age can be used to characterize the spatial distribution of tracer concentration. Since mean age is obtained by the solution of a steady transport equation, only a small fraction of the CPU time is required compared to the time-dependent concentration solution. The steady solution also brings great convenience in data processing. If we assume ^c(x) = â(x) with â(x) = a(x)/aV, from eq 20 we will have IaðxÞ t=aV cðx, tÞ ¼ e τ~ae

Figure 9. Comparisons of scaled mean age distribution and scaled tracer concentration distribution for case 2. Solid lines, mean age; dashed lines, concentration. (a) Scaled mean age frequency function and scaled tracer concentration frequency function at t = 15.47 s. (b) Scaled mean age distribution and scaled tracer concentration distribution at t = 15.47 s along the line of x = 0 and y = D/4.

~ae is found, the spatial and temporal distribution of the tracer concentration in the stationary stage can then be completely determined from the mean age distribution. The conditions under which the assumption of ^c(x) = â(x) holds can be found, and the magnitude of the error of this assumption can be computed. Since Z Z Z ¥ 1 1 tcðx, tÞ dt dx aV ¼ aðxÞ dx ¼ V V V V 0 I Z Z Z 1 ¥ 1 1 ¥ ¼ t½ cðx, tÞ dx dt ¼ tcV ðtÞ dt ð22Þ I 0 V V I 0 the scaled mean age is then

âðxÞ ¼

¼

~ae can be found either from experimental data or from a numerical solution at the end of the initial stage at the exit. Once

tcðx, tÞ dt tcV ðtÞ dt

0 ts

Z

¥

tcðx, tÞ dt þ

tcðx, tÞ dt t

0

Z

ts 0

5848

¥

aðxÞ ¼ Z0 ¥ aV Z

ð21Þ

Z

Zs tcV ðtÞ dt þ

ð23Þ

¥

tcV dt ts



ARTICLE

It has been found that the initial stage is very short. The first term should then be much smaller than the second term in both the numerator and the denominator. Therefore, any approximation in the concentration distribution in this stage should only bring a small error overall. If we extend the exponential distribution of the concentration of the stationary stage into the initial stage, we will have Z ¥ Z ¥ 1 tet=aV dt tcðx, tÞ dt ~ A ~aðxÞ 0 0 Z ð24Þ ¼ âðxÞ ¼ Z ¥ 1 ¥ t=aV ~aðxÞ tcV ðtÞ dt te dt ~ 0 A 0 When this equation is compared with eq 16, we then have ^c(x) = â(x). The errors by extending the exponential function into the initial stage can be computed. For the numerator, the errors are all much smaller than 1% for all three cases for the selected three interior points and at the exit. For the denominator, the errors are also smaller than 1% for all three cases. These small errors explain why the curves for ^c and â in both Figure 9a and b are almost not distinguishable. 4.5. Mixing Time Analysis. Since we have determined the concentration distribution in a reactor, we can define some quantitative measures to evaluate and compare the mixing performance of different reactors. A basic measure widely used in the batch mode is the blend time. This blend time is often defined as the time when the concentration in the reactor has reached 95% homogeneity with a pulse tracer input at time zero.21 A similar mixing time was used for a step up input by Roussinova and Kresta9 in their study of nonideal CFSTRs. In this section, we will characterize the three cases discussed in the previous sections by comparing their mixing time with that of the corresponding ideal mixers. If we denote θc as the time when the tracer concentration in the mixer has reached within 5% of the initial mean concentration, we will have cðθc Þ ¼ 0:05c0

ð25Þ

where c0 is the initial mean concentration in the reactor after the pulse is released, c0 ¼ IQ =V ¼ I=τ

ð26Þ

For an ideal mixer with a pulse input, the nondimensional concentration is10 cðtÞ 1 ¼ et=τ f ðtÞ ¼ I τ

ð27Þ

Then it can easily be found that the mixing time for an ideal mixer is θc = 3τ. For a nonideal mixer, since the tracer concentration in the mixer varies with spatial position, the mixing time will be different and depend on the locations. Since we now know the tracer history in the exponential stage, we can easily find the mixing time at any given location. From eq 21, we can find that ~ae ð28Þ θc ðxÞ ¼ aV ln 0:05 aðxÞ From this equation, we can see that the spatial nonuniformity of the mixing time is a function of local mean age.

Table 3. Mixing Time Comparisons θc (s)

point 1

point 2

point 3

exit

ideal

case 1

275.95

279.27

278.12

277.83

279.28

case 2

291.52

294.68

293.32

284.78

279.28

case 3

128.30

131.26

130.13

118.15

111.78

To compare with the mixing time of an ideal mixer, the mixing times at the three interior points shown in Figure 1 and at the exit for all three cases discussed previously are listed in Table 3. By examining the data in the table, several interesting points can be observed: - Compared to the mixing time in the ideal mixer, all of the mixing times are smaller in case 1 but larger in case 2 and case 3. A shorter mixing time than the ideal mixer in case 1 seems counterintuitive, but a further thought explains why. As shown in Figure 5, the first appearance time at the exit for the tracer in case 1 is about 1.7 s after the tracer pulse release. Within these 1.7 s, the material leaving the system is the original unmixed one. Thus, the amount of the oldest material remaining to be mixed inside the reactor is effectively smaller. On the other hand, in case 2 and case 3, the first appearance time of the tracer at the exit is much smaller. On the contrary, strong bypassing sends unmixed tracer material out the system earlier. Therefore, a smaller amount of tracer material is left in the system for mixing. - The longest mixing time for all three cases is at the same point, point 2, which is behind the baffle. This is also the slowest mixing location when the reactor is in the batch mode. At this point, the mixing time is about the same as in the ideal mixer for case 1 and about 5.5% longer than in the ideal mixer for case 2 with bypassing. - Stronger bypassing causes more delay in mixing. This can be seen by comparing case 2 and case 3 at point 2. For case 3, the mixing time at the point is 17.43% longer than in the ideal mixer. - The mixing times at the exit are all very close to the ideal mixer, even for case 3 with strong bypassing and a small ratio of mean residence time to batch blend time of 4.34. The deviation from the ideal mixing for this worst case is only 5.7%. This seems to suggest that the mixing conditions at the exit do not reflect accurately the spatial nonuniformity in the interior of the reactor.

5. CONCLUSIONS A method is developed to study the mixing performance in nonideal CFSTRs with distributions of mean age and the time dependent tracer concentration. A steady governing equation is solved for the spatial distribution of mean age, and each higher moment of age after the flow solution is obtained. For tracer concentration, the time-dependent convectiondiffusion equation needs to be solved for only a short period of time. The exponential decay of tracer concentration can be determined as an analytical function of position and time. The CPU time required for the mean age solution is several orders of magnitude smaller than the CPU time required to solve for the full timedependent concentration solution in order to obtain accurate mixing time and moments of residence time. From the time-dependent solution of tracer concentration, it was found that the temporal variation of tracer concentration can 5849


Industrial & Engineering Chemistry Research be divided into two stages: an initial stage and a stationary stage. In the initial stage, the concentration is strongly locationdependent. In the stationary, the decay of tracer concentration has a constant exponential rate everywhere in the reactor. This exponential rate is found to be the inverse of the volume averaged mean age. With these findings, the complete history of tracer variation can be determined by the mean age solution and the time-dependent solution of tracer concentration in the initial stage only. The time span of this initial stage is found to be on the same order of magnitude as the batch blend time of the reactor. It has been shown that the scaled concentration field and the scaled mean age field are nearly identical. The small difference is due to the nonuniform variation of tracer concentration in the initial stage. The difference between the two scaled fields is smaller than 1% at the selected interior points and at the exit. Therefore, the complete spatial nonuniformity of mixing can be described by the mean age field. Strong bypassing can be revealed by smaller mean age regions connecting the inlet and the outlet. Slow mixing zones can be found with higher mean age. Quantitative measures can be calculated from the mean age distribution. From the determined concentration distribution, mixing time can be calculated to measure the mixing performance of a reactor. It was found that the defined mixing time at the selected locations can be shorter or longer than the ideal mixing time. A shorter mixing time results when the mixing has a plug flow feature, and a longer mixing time results when strong bypassing exists between the inlet and the outlet. For the three cases studied, mixing times measured at the exit are all very close to the ideal mixing time even for the case with strong bypassing. This indicates that the mixing status measured at the exit may not reflect the spatial nonuniformity inside the reactor. The method developed in this article extends the capabilities of current CFD for quantitative characterization of both spatial and temporal nonuniform mixing in a CFSTR. The method is much more efficient and accurate than the traditional method of tracking time-dependent tracer history only. This method can also be used to compute parameters of some existing compartmental models in the literature with the potential of making such models more accurate. For example, the four parameters in the model by Cholette and Cloutier6 can easily be found accurately from the concentration history at the exit using both the initial stage and the stationary stage solutions. With quantitative measures available from this method, the design and scaleup of industrial reactors using CFD becomes practical.

’ AUTHOR INFORMATION Corresponding Author

*E-mail: [email protected].

’ NOMENCLATURE A = area (m2) ~ = a volume averaged age constant (s) A a = mean age (s) ~a = an age constant (s) â = scaled mean age (s) ae = mass averaged mean age at exit (s) ~ae = an age constant at the exit (s) aV = volume averaged mean age (s) c = tracer concentration (volume fraction) ^c = scaled tracer concentration cV = volume averaged concentration

ARTICLE

d = inlet or outlet diameter (m) ds = shaft diameter (m) dt = outside diameter of top inlet (m) D = impeller diameter (m) D = molecular diffusivity (m2/s) D eff = turbulent effective diffusivity (m2/s) H = tank liquid height (m) I = spatial invariant (s) Mn = nth moment of age (sn) M n,e = mass averaged nth moment of age at exit (sn) n = unit normal (m) N = impeller rotating speed (rev/s) Qm = mass flow rate (kg/s) Q = volumetric flow rate (m3/s) r = radius coordinate (m) Sc = Turbulent Schmidt number t = time (s) ts = time when the stationary stage starts (s) ~t = a time constant (s) t = first moment of residence time (s) tn = nth moment of residence time (sn) T = tank diameter (m) u = velocity vector (m/s) U = mean velocity (m/s) V = tank volume (m3) x = spatial coordinate (m) y = spatial coordinate (m) z = spatial coordinate (m) Greek Letters

r = tracer molecular age (s) R = average molecular age in the flow system (s) Δt = time step size (s) Δx = mesh size (m) μ = viscosity (kg/m 3 s) νT = turbulent viscosity (kg/m 3 s) θb = blend time at 95% homogeneity in batch stirred tanks (s) θc = mixing time at 95% homogeneity in continuous flow stirred tanks (s) F = density (kg/m3) τ = mean residence time (s)

’ REFERENCES (1) Danckwerts, P. V. Continuous Flow Systems. Distribution of Residence Times. Chem. Eng. Sci. 1953, 2, 1–13. (2) Wen, C. Y.; Fan., L. T. Models for Flow Systems and Chemical Reactors; Marcel Dekker: New York, 1975. (3) Levenspiel, O. Chemical Reaction Engineering; 3rd ed.; Wiley: New York, 1999. (4) Zwietering, T. N. The Degree of Mixing in Continuous Flow Systems. Chem. Eng. Sci. 1959, 11, 1. (5) Danckwerts, P. V. The Effect of Incomplete Mixing on Homogeneous Reactions. Chem. Eng. Sci. 1958, 8, 93. (6) Cholette, A.; Cloutier, L. Mixing Efficiency Determinations for Continuous Flow Systems. Can. J. Chem. Eng. Sci. 1959, 37, 105. (7) Manning, F. S.; Wolf, D.; Keairns, D. L. Model Simulation of Stirred Tank Reactors. AIChE J. 1965, 11, 723. (8) Jones, P. N.; Ozcan-Taskin, N. G.; Yianneskis, M. The Use of Momentum Ratio to Evaluate the Performance of CSTRs. Chem. Eng. Res. Des. 2009, 87, 485. (9) Roussinova, V.; Kresta, S. M. Comparison of Continuous Blend Time and Residence Time Distribution Models for a Stirred Tank. Ind. Eng. Chem. Res. 2008, 47, 3532–3539. 5850



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(10) Nauman, E. B.; Buffham, B. A. Mixing in Continuous Flow Systems; Wiley: New York, 1983. (11) Liu, M.; Tilton, J. N. Spatial Distributions of Mean Age and Higher Moments in Steady Continuous Flows. AIChE J. 2010, 56, 2561. (12) Patwardhan, A. W. Prediction of Residence Time Distribution of Stirred Reactors. Ind. Eng. Chem. Res. 2001, 40, 5686–5695. (13) Choi, B. S.; Wan, B.; Philyaw, S.; Dhanasekharan, K.; Ring, T. A. Residence Time Distributions in a Stirred Tank: Comparison of CFD Predictions with Experiment. Ind. Eng. Chem. Res. 2004, 43, 6548–6556. (14) Khopkar, A. R.; Mavros, P.; Ranade, V. V.; Bertrand, J. Simulation of Flow Generated by an Axial-Flow Impeller: Batch and Continuous Operation. Chem. Eng. Res. Des. 2004, 82 (A6), 737. (15) Samaras, K.; Mavros, P.; Zamboulis, D. Effect of Continuous Feed Stream and Agitator Type on CFSTR Mixing State. Ind. Eng. Chem. Res. 2006, 45, 4805–4815. (16) Singh, K. K.; Mahajani, S. M.; Shenoy, K. T.; Ghosh, S. K. Computational Fluid Dynamics Modeling of a Bench-scale PumpMixer: Head, Power and Residence Time Distribution. Ind. Eng. Chem. Res. 2007, 46, 2180–2190. (17) Zhang, L.; Pan, Q.; Rempel, G. L. Residence Time Distribution in a Multistage Agitated Contactor with Newtonian Fluids: CFD Prediction and Experimental Validation. Ind. Eng. Chem. Res. 2007, 46, 3538–3546. (18) Spalding, D. B. A Note on Mean Residence-times in Steady Flows of Arbitrary Complexity. Chem. Eng. Sci. 1958, 9, 74. (19) Sandberg, M. What is Ventilation Efficiency?. Build. Environ. 1981, 16, 123. (20) Baleo, J.-N.; Le Cloirec, P. Validating a Prediction Method of Mean Residence Time Spatial Distributions. AIChE J. 2000, 46, 675. (21) Grenville, R. K.; Nienow, A. W. Blending of Miscible Liquid. in Handbook of Industrial Mixing; Paul, E. L., Atiemo-Obeng, V. A., Kresta, S. M., Eds.; Wiley: Hoboken, NJ, 2004. (22) Oldshue, J. Y. Fluid Mixing Technology; McGraw-Hill: New York, 1983. (23) Oran, E. S.; Boris, J. P. Numerical Simulation of Reactive Flow, 2nd ed.; Cambridge University Press: Cambridge, U.K., 2001. (24) Voncken, R. M.; Rotte, J. W.; Houten, A. T. Circulation Model for Continuous-Flow Turbine-Stirred, Baffled Tanks. Proc. AIChEIChemE Symp. Ser. 10, 1965. (25) Nienow, A. W. On Impeller Circulation and Mixing Effectiveness in the Turbulent Flow Regime. Chem. Eng. Sci. 1997, 52, 2557.

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Prediction of Tracer Concentration and Mixing in CFSTRs with Mean

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