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Article

Residence Time Distribution of Variable Viscosity Fluids in the Stirred Tank

State Key Laboratory of Chemical Resource Engineering, School of Chemical Engineering, Beijing University of Chemical Technology, Beijing 100029, China
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Authors to whom correspondence should be addressed.
Processes 2025, 13(9), 2997; https://doi.org/10.3390/pr13092997
Submission received: 31 August 2025 / Revised: 12 September 2025 / Accepted: 18 September 2025 / Published: 19 September 2025

Abstract

Stirred tanks are widely used in polymerization processes, where the residence time distribution (RTD) significantly affects monomer conversion and polymer quality. In this study, the RTD in the stirred tank with both constant and variable viscosity fluids was investigated numerically. To account for the viscosity evolution during polymerization, a model relating fluid viscosity to the mean age of the fluid was developed. After verifying mesh and time step independence, the effects of impeller speed, fluid space time, and viscosity varying on RTD were examined in both single-tank and two-tank configurations. Compared to the constant-viscosity fluids, the variable-viscosity fluid shows different flow behaviors such as dead zones and short-circuiting. Analysis based on the number of tanks in series showed that increasing impeller speed and extending space time can enhance mixing efficiency, where the improved mixing in the second stage of the two-tank configuration eliminated the concentration fluctuations caused by recirculating flow in the first tank, which may result in a more uniform RTD curves.

1. Introduction

Continuous stirred tank reactors (CSTRs) are widely used in chemical and polymerization processes due to their simple construction, operational flexibility, and excellent mixing performance [1]. In polymerization processes, the performance of reactor is strongly influenced not only by the chemical kinetics but also by the internal flow behavior and mixing efficiency [2]. The residence time distribution (RTD) plays a crucial role in evaluating reactor performance, as it reflects flow characteristics, mixing effectiveness and back mixing degree [3]. Accurate characterization of RTD is essential for predicting monomer conversion, polymer molecular weight distribution, and final product quality in polymerization processes [4,5]. This becomes particularly important in polymerization systems with high and variable viscosity [6], where non-ideal flow behaviors, such as dead zones and short-circuiting, are more likely to occur. Therefore, understanding and controlling the RTD is crucial for the optimal design and scale-up of CSTRs in viscous systems.
Currently, research on RTD in CSTRs primarily focuses on the effects of impeller type and operational parameters, such as impeller speed and feed flow rate [7]. Samaras et al. [8] compared the RTD of an axial flow impeller (mixel TT) and two radial flow impellers (Rushton turbine and NS turbine) through experiments and numerical simulations. They found that the axial flow impeller is more likely to cause feed short circuit (>10% of the flow rate) due to the low solidity ratio, while the disc structure of the radial flow impeller effectively suppresses the short circuit. Moreover, they found that when the ratio of residence time to mixing time is less than 10, the dead zone volume in the reactor increases significantly. In contrast, Jones et al. [9] analyzed the variance of RTD and pointed out that evaluating reactor performance solely based on the ratio of residence time to mixing time could result in over-design and excessive energy consumption. Instead, they proposed that the ratio of the inlet jet momentum to the impeller emission momentum is a more appropriate parameter. They suggested that enlarging the inlet pipe diameter is more energy-efficient than increasing impeller speed in reactors prone to short-circuiting. Aghbolaghy et al. [10] showed that increasing impeller speed improves mixing and approximates ideal behavior, but the benefits diminish beyond a threshold speed (approximately 375 rpm in their study), while energy consumption rises sharply. Bai et al. [11] reported that, without agitation in a three-stage mixing system, the RTD curve exhibited a broad profile with a long trailing tail. With agitation, however, the RTD showed a longer minimum residence time and a significantly shorter tail. Zhang et al. [12] found that whether in a four-stage or a two-stage agitated contactor, increasing the fluid flow rate led to a decrease in mean residence time and deterioration in mixing performance. They also investigated the RTD of sugar solutions at different concentrations, showing that increasing fluid viscosity reduced back-mixing and prolonged the residence time. Hui et al. [13] showed that when the ratio of impeller discharge flow rate to the feed rate exceeded 13.13, the reactor behavior resembled that of an ideal CSTR. When the ratio was below this value, mixing characteristics became highly dependent on the spatial configuration of the inlet and outlet. Although significant progress has been made, most RTD investigations in stirred reactors focus on turbulent flow regimes. The RTD behavior under laminar conditions has been less explored due to experimental challenges. In such systems, mixing time is often used to characterize the mixing performance [14,15], but its value can vary depending on the chosen definition, even for the same process [16].
In addition to RTD and mixing time, the fluid age distribution is another important parameter for evaluating mixing performance of the reactors. Fluid age is defined as the time elapsed since a fluid particle entered the reactor until it reaches a specific position in the reactor. When the particle eventually exits the reactor, its age equals to its residence time in the reactor [17]. Lui [18] applied steady-state numerical simulations to obtain the mean age field in a stirred tank reactor, which revealed the complex and non-uniform mixing patterns. A corresponding mean age frequency function was introduced, and it was shown that the width of this function is directly related to the mixing time. In a subsequent study [19], the impact of inlet and outlet configurations was examined, showing that short-circuiting and poorly mixed regions could be effectively identified via mean age contour maps. Under laminar flow conditions, Simcik et al. [20] validated the consistency of RTD and fluid age distributions in a 2D tubular reactor. Moreover, the concept of fluid age has been extended to other reactor types, including falling film reactors [21] and continuous polymerization reactors [22].
To understand the residence time behavior in viscous systems, this study investigates the RTD characteristics in continuous stirred tank reactors (CSTRs) equipped with a draft tube by numerical simulation. Considering that the fluid viscosity in polymerization processes varies with reaction progress, a viscosity model based on fluid mean age is proposed to account for the viscosity evolution during the reaction. After verifying grid and time step independence, the effects of impeller speed, space time, and viscosity on the RTD are analyzed under both single-stage and two-stage reactor configurations. In particular, the influence of viscosity variation on flow patterns, such as dead zones and short-circuiting, is examined. The study also quantifies mixing performance using the tank in series model. The results demonstrate that considering viscosity evolution is crucial for accurately predicting RTD and guiding the design and scale-up of stirred tank reactors.

2. Models and Methods of Numerical Simulation

2.1. Physical Models

Numerical simulations were conducted in a flat-bottom stirred tank reactor with a diameter T0 = 476 mm and a height H = 476 mm. The reactor was equipped with a draft tube with an inner diameter T1 = 310 mm and a height of 321 mm, which promotes axial flow in the reactor. Five segmented baffles with a width w = 31 mm were evenly distributed inside the draft tube to prevent the fluid whirling in the reactor. A five-blade CBY impeller (a type of impeller with long thin blades) with a diameter D = 300 mm was used in this study, with an off-bottom clearance of 190 mm. The fluid enters the reactor through an inlet tube located on the lower part of the sidewall and exits through an outlet tube located on the upper part of the opposite sidewall. Both the inlet and outlet tube have a diameter d = 20 mm. Simulations were conducted in a single-stage stirred reactor and in two-stage stirred tank reactors in series. The key dimension and three-dimensional structure of the reactors are shown in Figure 1.

2.2. Governing Equations

The step tracer method was used to measure the RTD of the STRs. Once the flow field in the reactor stabilized, the inlet fluid was switched to a tracer with the same physical properties as the original fluid. The concentration of the tracer at the outlet is monitored over time until a stable state was reached. To describe the fluid flow in the STRs, the governing equations of fluid dynamics are employed, including the continuity equation for mass conservation and the Navier–Stokes equations for the momentum conservation.
· u = 0
ρ u t + ρ u · u = p + μ 2 u + ρ g
where u is the velocity, ρ is the density, p is the pressure, and μ is the dynamic viscosity.
The species transport equation was used to describe the mixing and mass transfer behavior of the tracer. The conservation equation of species j is given by:
Y j t + ( u Y j ) = D j Y j
where Yj is the mass fraction of species j, Dj is the molecular diffusivity.
In the polymerization process, the viscosity of the fluid strongly depends on the conversion. Since the conversion is governed primarily by the reaction time in the reactor, this study relates the fluid viscosity to its mean age in the reactor, which refers to the average time required for the fluid to travel from the inlet to a spatial position in the reactor. When the fluid enters the reactor, its age starts at zero and increases continuously until it exits. When the fluid flows out of the outlet, the age is the residence time in the reactor. When the tracer is injected into the reactor at the inlet at time 0, the concentration distribution of the tracer can be expressed as c(x, t). The frequency distribution can be defined as:
φ x , t = c x , t 0 c x , t d t
The mean age at position x can be expressed as:
a ( x ) = 0 t φ ( x , t ) d t
For incompressible fluid, the transport equation of the tracer can be expressed as:
c t + ( u c ) = ( D t c )
When Dt is the molecular diffusivity of tracer. By multiply t on both sides of the equation and then integrating, the following equation is obtained:
0 t c t d t + 0 ( u t c ) d t = 0 D t ( t c ) d t
The first term on the left can be reduced to:
0 t c t d t = t c | 0 0 c d t
When t , the tracer concentration in the reactor approaches 0. That is t c | 0 = 0 . Equation can be simplified as:
1 + u 0 t c d t 0 c d t = D t 0 t c d t 0 c d t
Thus, the transport equation of the mean age a is given as:
( u a ) = D t a + 1
The mean age is solved by user-defined scalar in Ansys Fluent. For an arbitrary scalar ϕk, the conservation equation can be expressed as:
ρ ϕ k t + x i ( ρ u i ϕ k Γ k ϕ k x i ) = S ϕ k
where Γ k is the diffusion coefficient of scalar, S ϕ k is the source term of the scalar.

2.3. Numerical Details

In this study, the flow and mixing behavior in the STRs were investigated using the Ansys Fluent 2021 R2. Based on the calculated impeller Reynolds number ReD under the simulated simulations, the flow pattern was determined to be laminar flow. Therefore, the laminar model was used to describe the fluid flow. The computational domain was divided into two parts: a rotating zone and a stationary zone. The rotating zone included the impeller and its surrounding region, while the stationary zone consisted of the remaining part of the reactor. The sliding mesh approach was employed to handle the rotating impeller in transient simulations, whereas the Multiple Reference Frame approach was adopted for steady-state simulations. Date exchange between the two zones was achieved through variable interpolation at the sliding interfaces. All wall boundaries were subjected to no-slip conditions. The pressure–velocity coupling was resolved using the Coupled algorithm. To improve the calculation accuracy, the second-order upwind scheme was used to discretize the governing equations. Convergence was achieved when the residuals of all governing equations fell below 10−4.
The simulation of RTD involved two steps. First, a steady-state simulation was conducted to obtain a stable flow field. Then, the inlet fluid was switched to a tracer with the same physical properties as the original fluid, and transient simulations were performed. The cumulative residence time distribution function, F(t), was determined by monitoring the tracer concentration at the outlet.
For numerical simulation, the number of grid cells has a significant impact on the computational process and results. An insufficient number of cells can lead to significant numerical errors, while an excessive number of cells can reduce computational efficiency. Therefore, grid independence verification is essential. The single-stage STR was meshed with boundary layer grids generated near the impeller to accurately capture the shear flow. Three discretization schemes were adopted, with cell counts of 0.4 million, 0.8 million, and 2.0 million, respectively. Simulations were conducted under the conditions of N = 160 rpm, u = 0, and μ = 10 Pa∙s. Figure 2 shows the average velocity distribution along the radial direction at z/H = 0.19 under different meshing schemes. The results showed that the simulation with 0.4 million cells exhibits a large deviation, while the results with 0.8 million cells are close to those with 2.0 million cells, with a difference of only 4.41%. Therefore, the mesh with 0.8 million cells was selected for subsequent simulations to ensure both accuracy and computational efficiency. The meshing diagram of the single-state STR is shown in Figure 3. The quality of the generated mesh was verified: the maximum skewness was 0.77, the orthogonal quality values were all greater than 0.3, and the mesh expansion factor was 1.2, indicating that the mesh quality was acceptable for the present simulations. The two-stage STR was meshed using the same mesh size.
In transient simulations, the time step also affects both computational accuracy and efficiency. To verify independence from the time step size, simulations were conducted under identical conditions using four different time steps: 0.025 s, 0.05 s, 0.1 s, and 0.2 s. Figure 4 shows the average velocity distribution along the radial direction on the same plane. The results indicated that when the time step is reduced below 0.1 s, the average velocity distribution remains essentially unchanged. Further reductions in the time step do not lead to significant improvements in the results. Thus, the time step is set as 0.1 s for subsequent transient simulations.

2.4. Determination of Residence Time Distribution

Cause the step tracer method was used in this study, the cumulative residence time distribution, F(t), can be obtained by directly monitoring the mass fraction of the tracer at the outlet. Based on the discrete data obtained from the simulation, the residence time distribution, E(t), can be determined using Equation (12).
E t = F ( t ) t
where t is the time interval for monitoring, and equal to 1 s in this study. The average residence time is calculated as follows:
t ¯ = t E ( t ) t
The dimensionless average residence time ( t θ ¯ ) can be obtained by normalizing the mean residence time with the space time (τ). The space time and the dimensionless average residence time are defined as:
τ = V R Q V , i n = V R A i n v i n
t θ ¯ = t ¯ τ
where VR is the reactor volume, QV,in is the volume flow rate at the reactor inlet, Ain is the inlet area, and the vin is the inlet velocity.
The variance of the residence time distribution can be evaluated as follows:
σ t 2 = ( t t ¯ ) 2 E ( t ) t
And the dimensionless variance is defined as:
σ θ 2 = σ t 2 t ¯ 2
The inverse of the dimensionless variance is usually used to evaluate the macromixing characteristic of the reactor based on the tanks in series model, which is also called the number of tanks in series.
n = 1 σ θ 2
A higher value of n indicates that the flow pattern is closer to plug flow while a lower value suggests the flow pattern is closer to a perfectly mixed reactor.

2.5. Simulation Scheme

In this study, the effects of impeller speed (N), space time (τ), and the dynamic viscosity (μ) on RTD of single-stage and two-stage STRs were investigated. It should be noticed that, in addition to the Newtonian fluid (constant viscosity) conditions, cases considering viscosity as a function of mean age were also included, which is crucial for characterizing mixing in the polymerization reactor. During polymerization, the fluid viscosity is closely related to monomer conversion, shear stress, and the molecular weight distribution of the polymer. While viscosity models accounting for shear stress and molecular weight distribution have been reported, in this work a simplified approach by correlating the fluid viscosity with the mean age was proposed to indirectly reflect the effect of monomer conversion. Three relationships between viscosity and mean age were designed (as shown in Figure 5), and their expressions are given as follows:
μ = 0.55 a
μ = 0.005 a 2 + 0.001 0 < a < 112.5 0.55 a a 112.5
μ = 0.005 a 2 + 2 × 0.55 a + 0.001 0 < a < 112.5 0.55 a a 112.5
The simulation conditions used in this study are summarized in Table 1.

3. Results and Discussion

3.1. RTD in Single-Stage STR

3.1.1. Effect of Impeller Speed on RTD

The effect of impeller speed on RTD under both constant and variable viscosity conditions is illustrated in Figure 6. As the impeller speed increases, the peak of RTD curves appears earlier, indicating an accelerated liquid circulation in the reactor and improved mixing performance. Multiple peaks are observed in RTD curves under all impeller speeds expect at 0 rpm. This behavior is attributed to the circulation flow induced by the draft tube in the reactor. Driven by the impeller, the fluid flows downward inside the draft tube and upward outside, re-entering from the top to form a stable loop. Under laminar flow conditions, the tracer undergoes multiple circulation cycles before exiting the reactor, resulting in multiple peaks in the RTD curves. As the impeller speed increases, the circulation becomes faster, which reduces residence time differences and shortens the intervals between successive peaks. Additionally, enhanced mixing at higher impeller speeds reduces both the number and amplitude of the RTD peaks.
In Figure 7, the number of tanks in series is plotted against impeller speed for both constant and variable viscosity conditions. In both cases, the number of tanks in series decreases with increasing impeller speed, except at 0 rpm under variable viscosity. This trend reflects improved mixing performance, which gradually approaches that of an ideal reactor. However, at higher impeller speeds, the rate of improvement diminishes due to limitations in mixing enhancement under laminar flow. At 0 rpm, the number of tanks in series is less than 1 under variable viscosity conditions, indicating the presence of dead zones or short-circuiting flows.
Figure 8 shows the viscosity and the tracer mass fraction distribution at 0 rpm under variable viscosity conditions at t = 1000 s. The higher viscosity observed near the wall and the top of the reactor indicates that it takes longer for the fluid to reach these regions from the inlet, reflecting poor mixing in these areas. The flow in these regions is very slow, forming dead zones that do not actively participate in mixing. As a result, the effective reacting volume is reduced, and the number of tanks in series falls below 1. This observation is consistent with the commonly observed polymer deposition on reactor walls during polymerization processes. In these high viscosity regions, the tracer concentration is also relatively low, as shown in Figure 8b. However, the region with low tracer concentration is much smaller due to the effect of molecular diffusion.

3.1.2. Effect of Space Time

The effect of space time on RTD was studied by varying the inlet velocity of the reactor, and the results are shown in Figure 9. With increasing space time, the peaks in the RTD curves are delayed and reduced in height, indicating a longer average residence time. This leads to a large difference in residence times along different flow paths, resulting in a broader RTD curve and more significant tailing. For example, under constant viscosity conditions, at a space time of 45 s, the RTD curve (E(t)) becomes nearly zero around 200 s, indicating nearly all fluid has been replaced by the tracer. At a space time of 360 s, the RTD curve remains non-zero even at 500 s, reflecting incomplete fluid replacement.
Furthermore, as shown in Figure 9, doubling the space time generally reduces the RTD peak height by approximately half, except for the case of variable viscosity at a space time of 45 s. In this case, a sharp early peak is observed, indicating short-circuiting flow. This is confirmed by Figure 10, where the number of tanks in series is less than 1. Due to the higher inlet velocity and non-uniform viscosity distribution, a part of fluid bypasses the main circulation and exits the reactor quickly, resulting in a narrow residence time distribution for this part of the fluid. This finding highlights the importance of selecting appropriate space time to suppress short-circuiting and ensure effective mixing in polymerization processes.
Figure 10 shows that with the increase in space time, the number of tanks in series decreases except in the case with variable viscosity at a space time of 45 s. This trend indicates improved mixing performance. For the constant viscosity fluid, the number of tanks in series decreases rapidly with increasing space time, indicating that the RTD characteristics are highly sensitive to space time. For the variable viscosity fluid, when the space time increases from 90 s to 360 s, the effective number of tanks in series decreases only from 1.29 to 1.15, a reduction of 10.85%, indicating that the space time has a relatively minor impact on the residence time distribution.

3.1.3. Effect of Viscosity

Polymerization processes are typically characterized by significant changes in viscosity in the reactor. Figure 11 illustrates the effects of both constant and variable viscosity on RTD.
For constant-viscosity fluids, increasing viscosity causes the RTD peaks to appear later and grow taller. This is because higher viscosity increases the resistance to flow, reduces the circulation induced by the impeller, and prolongs the time required for the tracer to travel from the inlet to the outlet. Under such conditions, the impeller-driven circulation becomes less effective, resulting in weaker mixing and greater concentration of the tracer around a specific residence time, which produces a sharper and narrower peak in the RTD curve. In addition, higher viscosity slows down homogenization of the tracer, so that the tracer repeatedly recirculates in the reactor and exits intermittently, leading to multiple peaks in the RTD curve. For fluids with different variable viscosities, the RTD curves show only slight differences among fluids with different viscosities. Compared to the constant viscosity cases, more fluctuations are observed in the RTD curves, indicating poorer mixing performance under variable viscosity conditions.
As shown in Figure 12, the number of tanks in series increases with viscosity under constant-viscosity conditions. Under laminar flow conditions, higher viscosity leads to a poorer mixing performance. As viscosity increases from 1 Pa·s to 500 Pa·s, the number of tanks in series rises from 1.20 to 1.36, representing an increase of only 13%, indicating that viscosity has a weaker influence on RTD compared to impeller speed or space time.
For variable-viscosity fluids described by Equations (12)–(14), the corresponding numbers of tanks in series are 1.19, 1.17, and 1.20, respectively. As shown in Figure 5, for mean ages below 112.5 s, the viscosity follows the order: Equation (21) > Equation (19) > Equation (20). Accordingly, the number of tanks in series increases in the same order, reflecting that higher viscosity reduces mixing efficiency and broadens the RTD.

3.2. RTD in Two-Stage STRs

Based on the results and discussion in Section 3.1, the variable-viscosity model, which accounts for the increase in fluid viscosity over residence time, provides a more accurate prediction of flow behavior in polymerization reactors than the constant-viscosity model. Simulations using the variable-viscosity model revealed abnormal flow behaviors. Therefore, in the analysis of residence time distribution (RTD) in two-stage STRs, only variable-viscosity conditions were considered, as simulations under constant-viscosity conditions would not provide additional meaningful insights.

3.2.1. Effect of Impeller Speed

Figure 13 shows the residence time distribution of fluid viscosity described by Equation (19) under different impeller speeds in two-stage STRs. The results indicate that the effect of impeller speed on RTD is relatively limited except at 0 rpm. At 0 rpm, the peak in the RTD curve appears later and is significantly higher, which is consistent with the behavior observed in the single-stage reactor. This behavior indicates the formation of substantial dead zones in the absence of agitation. Compared to the single-stage reactor, the RTD curves of the two-stage reactor are smoother and show few fluctuations. This behavior may be attributed to the addition mixing provided by the second reactor, which enhances the dispersion of the tracer from the first reactor and reduces local concentration peaks.
As the impeller speed increases, the corresponding numbers of tanks in series are 1.82, 2.31, 2.21, and 2.20, respectively. Except for the 0 rpm case, the decrease in the number of tanks in series with increasing speed reflects enhanced mixing performance. At 0 rpm, the number of tanks in series is lower than the actual number of tanks, further confirming the presence of dead zones. Figure 14 shows the distributions of viscosity and tracer concentration in the two-stage STRs at t =1000 s. Dead zones are observed in both reactors. However, their extent is significantly reduced compared to the single-stage configuration. This result demonstrates that multistage reactor configurations can improve mixing performance and reduce stagnant regions.

3.2.2. Effect of Space Time

The influence of space time on RTD in two-stage STRs is shown in Figure 15. As space time increases, the RTD peaks are delayed and reduced in magnitude, while the overall curve becomes broader. This indicates an increase in average residence time and a more uniform residence time distribution, which corresponds to improved mixing and flow behavior approaching that of an ideal stirred reactor. The increasing tailing of the RTD curves with space time further supports this observation. The decrease in the number of tanks in series also reflects enhanced mixing with longer space times. These trends are consistent with previously reported RTD characteristics in multistage agitated contactors [12].

3.2.3. Effect of Viscosity

As shown in Figure 16, viscosity has a negligible effect on the RTD in two-stage STRs. The RTD curves under different viscosity conditions are nearly identical, and the corresponding number of tanks in series show minimal variation. This phenomenon can be attributed to three main factors. First, the differences among the three viscosity models used in this study are relatively small. At the same mean age, the maximum viscosity difference is around 30 Pa∙s, and the viscosities become identical when the mean age exceeds 112.5 s. Second, as discussed in Section 3.1.3, the RTD characteristics of STR are not significantly affected by viscosity. Third, the presence of the second stirred tank enhances mixing and further mitigates the effect of viscosity differences. Consequently, the influence of viscosity on RTD in two-stage STRs can be considered negligible.

4. Conclusions

In this paper, a simplified viscosity model based on the mean age of the fluid was proposed to describe viscosity evolution during polymerization. The residence time distribution (RTD) of the Newtonian fluids and the variable-viscosity fluids in the stirred tank reactors equipped with the draft tubes was investigated numerically. The results demonstrated that considering viscosity evolution is essential to capture abnormal flow behaviors such as dead zones and short-circuiting, underscoring the importance of incorporating viscosity variation for accurate RTD prediction in polymerization reactors.
The study also showed that impeller speed and space time are the dominant factors controlling the RTD in laminar stirred tanks, whereas viscosity has a relatively minor effect. The comparison between single- and two-stage configurations revealed that the second stage significantly improves mixing uniformity, smooths RTD curves, and mitigates the influence of viscosity differences.
Beyond reproducing flow characteristics, these findings emphasize the significance of the proposed simplified viscosity model. By linking viscosity to the mean age of the fluid, the model provides an efficient way to capture the effect of viscosity evolution on RTD without introducing excessive complexity. This approach provides practical guidance for reactor design, scale-up, and operation in polymerization processes. Since the effect of shear stress on viscosity was not considered, the proposed model is applicable mainly to Boger-type fluids, which are insensitive to shear stress. In future work, we will extend the model to incorporate the effect of shear stress on viscosity, in order to more accurately capture viscosity variations in polymerization reactors.

Author Contributions

G.W.: Writing—original draft, methodology, software, conceptualization. L.L.: Methodology, Software. Z.L.: Review and editing. J.W.: Investigation, supervision. Z.G.: Resources, supervision. All authors have read and agreed to the published version of the manuscript.

Funding

This research was funded by the China Petrochemical Corporation (Sinopec Group, project number: 222129, 224285, and 224108).

Data Availability Statement

The original contributions presented in this study are included in the article. Further inquiries can be directed to the corresponding authors.

Conflicts of Interest

The China Petrochemical Corporation (Sinopec Group) had no role in the design of the study; in the collection, analyses, or interpretation of data; in the writing of the manuscript, or in the decision to publish the results.

Abbreviations

RTDResidence Time Distribution
CSTRContinuous Stirred Tank Reactor
STRStirred Tank Reactor

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Figure 1. (a) Key dimension of the reactor. (b) Three-dimensional structure of the single-stage stirred reactor. (c) Three-dimensional structure of the two-stage stirred reactor.
Figure 1. (a) Key dimension of the reactor. (b) Three-dimensional structure of the single-stage stirred reactor. (c) Three-dimensional structure of the two-stage stirred reactor.
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Figure 2. Distribution of average velocity magnitude at z/H = 0.19 with 0.4 million, 0.8 million, and 2.0 million cells, respectively (N = 160 rpm, u = 0, μ = 10 Pa∙s).
Figure 2. Distribution of average velocity magnitude at z/H = 0.19 with 0.4 million, 0.8 million, and 2.0 million cells, respectively (N = 160 rpm, u = 0, μ = 10 Pa∙s).
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Figure 3. Meshing diagram of the single-stage STR: (a) axial section view; (b) boundary layer near impeller.
Figure 3. Meshing diagram of the single-stage STR: (a) axial section view; (b) boundary layer near impeller.
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Figure 4. Distribution of average velocity magnitude at z/H = 0.19 with time steps of 0.025 s, 0.05 s, 0.1 s, and 0.2 s, respectively (N = 160 rpm, u = 0, μ = 10 Pa∙s).
Figure 4. Distribution of average velocity magnitude at z/H = 0.19 with time steps of 0.025 s, 0.05 s, 0.1 s, and 0.2 s, respectively (N = 160 rpm, u = 0, μ = 10 Pa∙s).
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Figure 5. Relationships between viscosity and mean age.
Figure 5. Relationships between viscosity and mean age.
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Figure 6. Effect of impeller speed on RTD in single-stage STR: (a) constant viscosity of 10 Pa∙s; (b) variable viscosity described by Equation (19) (τ = 180 s).
Figure 6. Effect of impeller speed on RTD in single-stage STR: (a) constant viscosity of 10 Pa∙s; (b) variable viscosity described by Equation (19) (τ = 180 s).
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Figure 7. Effect of impeller speed on the number of tanks in series in single-stage STR (τ = 180 s).
Figure 7. Effect of impeller speed on the number of tanks in series in single-stage STR (τ = 180 s).
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Figure 8. The viscosity distribution (a) and mass fraction of tracer distribution (b) in the single-stage STR at t = 1000 s (N = 0 rpm, τ = 180 s, viscosity described by Equation (19)).
Figure 8. The viscosity distribution (a) and mass fraction of tracer distribution (b) in the single-stage STR at t = 1000 s (N = 0 rpm, τ = 180 s, viscosity described by Equation (19)).
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Figure 9. Effect of space time on RTD in single-stage STR: (a) constant viscosity of 10 Pa∙s; (b) variable viscosity described by Equation (19) (N = 80 rpm).
Figure 9. Effect of space time on RTD in single-stage STR: (a) constant viscosity of 10 Pa∙s; (b) variable viscosity described by Equation (19) (N = 80 rpm).
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Figure 10. Effect of space time on the number of tanks in series in single-stage STR (N = 80 rpm).
Figure 10. Effect of space time on the number of tanks in series in single-stage STR (N = 80 rpm).
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Figure 11. Effect of viscosity on RTD in single-stage STR: (a) constant viscosity; (b) variable viscosity (N = 80 rpm, τ = 180 s).
Figure 11. Effect of viscosity on RTD in single-stage STR: (a) constant viscosity; (b) variable viscosity (N = 80 rpm, τ = 180 s).
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Figure 12. Effect of viscosity on the number of tanks in series in single-stage STR (N = 80 rpm, τ = 180 s).
Figure 12. Effect of viscosity on the number of tanks in series in single-stage STR (N = 80 rpm, τ = 180 s).
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Figure 13. Effect of impeller speed on RTD in two-stage STRs with variable viscosity described by Equation (19) (τ = 180 s).
Figure 13. Effect of impeller speed on RTD in two-stage STRs with variable viscosity described by Equation (19) (τ = 180 s).
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Figure 14. The viscosity distribution (a) and mass fraction of tracer distribution (b) in the two-stage STRs at t = 1000 s (N = 0 rpm, τ = 180 s, viscosity described by Equation (19)).
Figure 14. The viscosity distribution (a) and mass fraction of tracer distribution (b) in the two-stage STRs at t = 1000 s (N = 0 rpm, τ = 180 s, viscosity described by Equation (19)).
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Figure 15. Effect of space time on RTD in two-stage STRs with variable viscosity described by Equation (19) (N = 80 rpm).
Figure 15. Effect of space time on RTD in two-stage STRs with variable viscosity described by Equation (19) (N = 80 rpm).
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Figure 16. Effect of viscosity on RTD in two-stage STRs (N = 80 rpm, τ = 180 s).
Figure 16. Effect of viscosity on RTD in two-stage STRs (N = 80 rpm, τ = 180 s).
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Table 1. Simulation schemes of RTD in STRs under various conditions.
Table 1. Simulation schemes of RTD in STRs under various conditions.
Caseμ/Pa∙sN/rpmτ/sSTRReD t ¯ /s σ t 2 / s 2 n
1100180single-stage0177.419270.313.40
21040180single-stage8.28179.2919,778.111.63
31080180single-stage16.56177.7724,093.701.31
410120180single-stage24.84177.4125,140.811.25
510160180single-stage33.12176.1626,448.091.17
6Equation (19)0180single-stage 170.1339,042.690.74
7Equation (19)40180single-stage 176.3124,530.081.27
8Equation (19)80180single-stage 176.8626,359.141.19
9Equation (19)120180single-stage 176.3526,692.591.17
10Equation (19)160180single-stage 176.1026,913.411.15
11108045single-stage16.5644.98793.462.55
12108090single-stage16.5689.884793.251.69
131080360single-stage16.56353.15105,419.951.18
14Equation (19)8045single-stage 42.002150.330.82
15Equation (19)8090single-stage 88.436053.831.29
16Equation (19)80360single-stage 353.07108,041.541.15
17180180single-stage165.6176.7026,101.461.20
1810080180single-stage1.66177.6523,444.821.35
1950080180single-stage0.33178.1123,325.301.36
20Equation (20)80180single-stage 176.3726,509.601.17
21Equation (21)80180single-stage 176.4225,847.251.20
22Equation (19)0180two-stage 168.7915,612.041.82
23Equation (19)40180two-stage 176.6713,529.082.31
24Equation (19)80180two-stage 176.5314,099.552.21
25Equation (19)120180two-stage 176.7014,183.522.20
26Equation (19)8090two-stage 87.373260.042.34
27Equation (19)80360two-stage 353.4157,664.462.17
28Equation (20)80180two-stage 176.8613,863.832.26
29Equation (21)80180two-stage 176.8113,891.572.25
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Wu, G.; Li, L.; Li, Z.; Wang, J.; Gao, Z. Residence Time Distribution of Variable Viscosity Fluids in the Stirred Tank. Processes 2025, 13, 2997. https://doi.org/10.3390/pr13092997

AMA Style

Wu G, Li L, Li Z, Wang J, Gao Z. Residence Time Distribution of Variable Viscosity Fluids in the Stirred Tank. Processes. 2025; 13(9):2997. https://doi.org/10.3390/pr13092997

Chicago/Turabian Style

Wu, Guangshuo, Linxi Li, Zhipeng Li, Junhao Wang, and Zhengming Gao. 2025. "Residence Time Distribution of Variable Viscosity Fluids in the Stirred Tank" Processes 13, no. 9: 2997. https://doi.org/10.3390/pr13092997

APA Style

Wu, G., Li, L., Li, Z., Wang, J., & Gao, Z. (2025). Residence Time Distribution of Variable Viscosity Fluids in the Stirred Tank. Processes, 13(9), 2997. https://doi.org/10.3390/pr13092997

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