Residence time distribution of high viscosity fluids falling film flow down outside of industrial-scale vertical wavy wall:Experimental investigation and CFD prediction☆

Shichang Chen,Lihao Zhang,Yongjun Wang,Xianming Zhang,Wenxing Chen

National Engineering Laboratory for Textile Fiber Materials and Processing Technology (Zhejiang),Zhejiang Sci-Tech University,Hangzhou 310018,China

ABSTRACT The flow behavior of gravity-driven falling film of non-conductive high viscosity polymer fluids on an industrial-scale vertical wavy wall was investigated in terms of film thickness and residence time distribution by numerical simulation and experiment.Falling film flow of high viscosity fluids was found to be steady on a vertical wavy wall in the presence of the large film thickness.The comparison between numerical simulation and experiment for the film thickness both in crest and trough of wavy wall showed good agreement.The simulation results of average residence time of falling film flow with different viscous fluids were also consistent with the experimental results.This work provides the initial insights of how to evaluate and optimize the falling film flow system of polymer fluid.

Keywords:Falling film flow High viscosity polymer fluid Residence time distribution Film thickness Numerical simulation

1.Introduction

Falling film flow is considered as a promising development path for chemical process industry because of its capacity in improving heat-mass transfer and production efficiency [1].Generally,the working liquid of falling film flow has quite low viscosity,such as water,aqueous solution of lithium bromide and alcohols,thus most of the falling film flows are limited into middle and high Reynolds numbers laminar flow,even into a turbulent flow.A number of researches have been reported on the falling film flow of high viscosity polymer fluids in industrial systems,such as spinning coating,latex preparation and polymer devolatilization 2-4.In combination with low processing flow rate and the large physical dimension,the polymer fluid flow is certain to be typical laminar flow with a low Reynolds number between 10-1and 10-5.Especially regarding the process of mass transfer involving chemical reaction,falling film flow in reactor was viscously dominated that the viscosity of fluid could be up to 1000 Pa·s or more [5].The application of these promising polymerization reactors is limited due to the lack of hydrodynamic data.To evaluate the performance of apparatus,a large amount of essential data of falling film flow depending on the internal structure and fluid properties are required.However,the experimental approach is usually not so feasible either,due to the high viscosity of fluid or high cost of industrial experiments.

With regard to falling film flow on vertical wall,one side of the film adheres to the wall while the other side is a free surface.The film flow is characterized with parabolic velocity profiles from the inner wall to free surface,consequently,the fluid residence time of falling filmflow on the support wall must be distributed in a certain way,i.e.residence time distribution (RTD) [6].The average residence time and the variance of RTD are the two most important parameters to characterize the performance of reaction and/or processing in equipment.The average residence time of materials can be used to evaluate the production efficiency of a reactor and the RTD variance is said to decide the property uniformity of last outlet product[7],for example,the polymer molecular weight and polydispersity index.However,the RTD measurement of falling film flow of high viscosity polymer fluids is still a great challenge.To the best of our knowledge,the RTD data of falling film flow with a fluid viscosity over 10 Pa·s is hardly available in the literature based on either experimental or numerical simulation.Therefore,it is necessary to investigate the RTD of falling film flow of high viscosity fluids on different vertical walls,in the hope of providing theoretical reference for the design and further optimization of reactors.

Due to the importance of RTD for the film-forming flow in device,several conventional techniques have been proposed in the open literature to evaluate the mixing performance of some typical equipment,such as spinning disc reactor(SDR),micromixers,multistage agitated contactor,and even for the rotary calciner under the harsh conditions 8-13.In most cases,the RTD tracer is a kind of dyed solution which has a similar feature as the fluid of falling liquid film.For SDR,liquid film is approximate to plug flow behavior and the film surface is normally covered with numerous ripples which can change the film hydrodynamics and thereby affect the RTD [8].Based on RTD measurements of a single phase rotor-stator SDR,Visscher et al.[9]described a reactor model consisting of a plug flow model and a tanks-in-series model.The proposed reactor model was based on turbulence effect and the formation of boundary layer effect.As a critical complementary tool to experimental RTD studies,CFD simulation was adopted to evaluate the performance of mixed laminar flow systems with a T-junction microchannel [11].The RTD investigation of annular centrifugal extractors indicated a single backmixing stage in spite of the presence of multiple vortices in the annulus,and the CFD predictions were well in agreement with the experimental measurement [12].Yet in another approach,Zhang et al.[14,15]calculated the local RTD of polymer processing in two-screw extruders by deconvolution based on a statistical theory.The results were confirmed theoretically and experimentally by using a new in-line RTD measurement instrument.Mighri et al.[16]investigated the flow behavior of polymer systems in extruder from the perspective of RTD by using an online fluorescence monitoring device.Interestingly,Laske et al.[17]determined the RTD of polypropylene in an extruder with near infrared spectroscopy and using an UV-absorber as tracer.

Before the investigation of RTD,it is an urgent need but still a significant challenge to understand the basic hydrodynamic behavior of falling film of high viscosity polymer fluids flowing on vertical wall.With a low viscous fluid of water,falling liquid film will show surface wave instability,which varies from the flat-film flow to a two-dimensional periodic wave train,then evolves into solitary waves and further to three-dimensional solitary waves and complex flows[18].Because of the rapid progress of the computational performance,extensive numerical studies of thin film flow have been done based on either the improving numerical methods or the combination with additional model.There have been attempts to clarify the instabilities of thin films with or without forcing perturbations,which have made a great contribution to the hydrodynamic behaviors of thin film flow on multifarious wall[19-26].Tryggvason et al.[19]presented the direct numerical simulations (DNS) of multiphase flows by using a front-tracking method.Subsequently,Gao et al.[21]proposed a continuum surface force model (VOF-CSF) to track free surfaces and to account for dynamic boundary conditions at free surfaces in investigation of the surface wave dynamics of vertical falling films.Thanks to the use of higher order,hybrid and adaptive techniques,the level set method has become an incredibly powerful and accurate tool used in many application areas[22].Recently,the statistical learning technique of Model Averaging Neural Network for the DNS of bubbly multiphase flows was developed to generate closure relationships for a simplified two-fluid model [24-26].The resulting model worked well for predicting the evolution of the various initial conditions and it was in reasonably good agreement with DNS results,that it could provide a new sight for the gas-liquid two phase flow filed.Apparently,the variation of free surface and film thickness also affect the hydrodynamic behavior.When falling film of high viscosity polymer fluids flows on a vertical wavy wall,it is difficult to characterize the flowing film,which is not only due to the complexity of the falling film flow itself,but also the working fluids and the supporting structure of flow.The measurement technique for falling film of non-conductive high viscosity polymer fluids flowing on a complex vertical wall has not yet been realized.

Existing measurement methods of film thickness can be divided into three categories:(1) intrusive measurement method by using various probes,such as the conventional needle-contact device[27];(2)non-intrusive methods by measuring the signal originated from the variation of volume,light intensity and conductance/capacitance [28-30];(3) digital image processing,such as shadow graphic imaging and fluorescent imaging [18,31].The latter two methods have been widespread application owing to the high accuracy and precision without causing disturbance in film flowing.Particularly,the capacitance technique is popular in experimental research because of its simple operation and directly readable data.The output voltage changes along with the capacitance between two parallel electrical conductors and the voltage signal can be converted into the value of film thickness based on the feedback principle of high input impedance amplifier.Hence,the linear relation between the displaying value of film and the actual film thickness can be acquired by calibrating the capacitance detector.It is noteworthy that this linear relation is generally applicable to electric fluids,such as aqueous solution.

In this work,an attempt has been made to investigate the basic hydrodynamic behavior of falling film of high viscosity polymer fluids flowing outside of industrial-scale vertical wavy wall by CFD-VOF method and experimental validation with capacitance technique,providing essential understanding of RTD.By performing the so-called tracer with one-step input,the RTD data of falling film flow were obtained experimentally as well as numerically,then the effect of flow rate,fluid viscosity and the structure of falling film tube on the RTD were investigated.

2.Experimental

2.1.Working fluid and experimental apparatus

The selected four polymer fluids (Polydimethylsiloxane) in this work are different in viscosity,while other physical properties are similar.All fluids are Newtonian type at low shear rate and turn to weak non-Newtonian at high shear rate(Supplementary Fig.S1).In consideration of the low flow rate,the fluids can be always treated as Newtonian-type during falling film flowing.The zero shear viscosity of four fluids(named fluid a to fluid c)at 25°C are 0.92 Pa·s,5 Pa·s,130 Pa·s and 623 Pa·s,respectively.The corresponding densities are little affected by viscosity with a range from 970 kg·m-3to 978 kg·m-3.

Fig.1.Schematic diagram of the experimental setup.

Table 1 The experimental parameters of falling film flow

The experimental geometry for falling film flow and RTD tracer is schematically shown in Fig.1.The falling film tube is 3 m in length and the effective height for falling film flow is about 2.5 m,which consists of straight tube (0.495 m) and wavy tube(1.98 m).The repeat unit of wavy wall is 0.033 m in length.The falling film tube is installed vertically and an annular distributing plate is concentric with falling film tube,with three teeth distributed at an interval of 120°.The fluid flows outside of straight tube and then flows along wavy wall.The flow rate can be controlled by the gap between annular film distributing plate and falling film tube and by the height of top level tank.When the flow rate is steady,the film thickness value is recorded synchronously,and then the RTD experiment can be performed.The operating conditions(flow rate and fluid viscosity)and the structure parameter of falling film tube (the height of wavy wall) are listed in Table 1.

The height of wavy wall is defined as wave amplitude by the equation:

2.2.Measurement correction of film thickness

In the experiments,film thickness tester (JDC-2008,Tianjin University) based on capacitance technique was used to measure the film thickness of falling film of high viscosity polymer fluids on vertical wavy wall.Since the film in crest and trough of wavy wall was the main concern,two detectors were allocated to focus on these two particular position,respectively.As measuring on wavy wall,the detector probe was designed to be suitable for the measurement of hoop face and an average film thickness value in detection area was received by detector.Theoretically,the measuring error is inevitable in most salient points,and the error will increase with the decrease of curvature in a modest range of falling film thickness for sphere or cylinder.The maximum error should be no more than 3%.

It was noted that the experiment fluids were totally nonconducting,so the detectors were needed to be corrected to find out the relationship between the displaying value in detector and the actual film thickness on wavy wall.The correction graphs of film thickness for four fluids in both crest and in tough of wavy wall were acquired in a static state(Supplementary Fig.S2)Therefore,the corresponding fitting equations for the relationship between the actual value and the displaying value of detector can be given,respectively (Supplementary Table S1 and Eq.(1)).

2.3.RTD tracer detection

High viscosity polymer fluids show poor fluidity,which leads to difficulty in measurement and analysis.It is an urgent need to find out a solvent that can disperse the tracer in fluids and improve the fluidity in the concentration detection.According to previous dissolving experiments,it can be known that the selected fluids were water-insoluble polymer;both the low and high viscosity polymer fluids were insoluble in some common organic solvents,such as alcohol,dimethyl sulfoxide and dimethylformamide,but soluble in benzenes,halogenated hydrocarbons and ethers,it was noticeable that toluene was a better solvent than petroleum,ether and chloroform in favor of diluting the RTD samples.With respect to the RTD tracer,disperse blue 291 (AR) could dye four fluids uniformly under the ultrasonic and heating(of course,more time will be costed for high viscosity fluid),thus the dyed fluids could be used as RTD tracer.The UV-Vis absorption spectrum (UV-2550,Shimada) of tracer is shown in Fig.2.The maximal absorbance peak was close to 604 nm,and the absorption of RTD tracer was well linearly related to tracer concentration (Supplementary Fig.S3).

Fig.2.The UV-Vis spectroscopy of RTD tracer.

With a steady flow,50 ml of RTD tracer was injected into the beginning of falling film flow as fast as possible and within less than one unit sampling time (time-step size),which could be approximately considered as a step input.In the end of falling film tube,the fluid was collected with interval time until the fluid in outlet was completely colorless.The collected fluid sample was well-mixed under ultrasonic and heating in several hours,then 2 ml of sample fluid was taken out and added into 8 ml of solvent for ultrasonic and heating treatment again in relatively less time.The sampling time under different flow conditions and the dilution ratio of four viscous fluid samples are shown in Supplementary Table S2 and Table S3,respectively.On the basis of the absorbance of RTD tracer,the average residence time and the variance of residence time distribution can be figured out successfully.

2.4.Data analysis

Mathematically,at any moment the RTD of falling film is given by the function E(t),and the fraction of fluid unit in a system having the residence time between t and t+dt is equal to E(t)dt [32].For a step input,the E(t) is related to the time-dependent output tracer concentration C(t) that can be defined by the expression given by:

The time during falling film flows on wall as plug flow is the average residence time (tav),which can be given by the integral of E(t):

The variance of RTD is evaluated withand the falling film flow is closer to plug flow with a smaller

With the different fluids and wall structure in the falling film flow,the non-dimensional residence time θ and the varianceare compared more conveniently:

The concept of skewness S based on the symmetry axis of θ=1 was proposed to evaluate the deviation from a symmetrical distribution[33].A left-skew distribution of RTD curve exists when S ＜0 and a right-skew distribution exists when S ＞0:

3.CFD Simulations

In our previous work [34],the numerical simulations of high viscosity polymer fluid falling film flow down wavy wall were performed as the number of wave node of wall was set as 10 and the height of falling film was 0.5 m.Herein,the numerical strategy mainly referred to the previous work,such as the width of computational domain(W)for different viscous fluid,the grid system,the initial conditions and the basic numerical scheme.

3.1.Computational domain

Considering the large-scale experimental setup,the numerical simulation would be a huge cost.Alternatively,it was implemented by convolution calculation of the small scale computational domain.A small scale two-dimensional falling film flow with four fluids on a vertical wall with equal interval wave node was numerically simulated in this paper.The computational domain is shown in Fig.3.The wavy wall was consisted of five repeat units of wave node and the vertical height of wall was set to 165 mm.The wavy wall and flat wall can be combined into a new wall via multiple superposition,which is the same scale with the effective falling film height in experiment.The width of computational domain (W) changed with fluid viscosity,as described in reference [34].

Fig.3.The computational domain of falling film flow,(a) wavy wall,(b) flat wall.

3.2.Governing equations

The falling film flow is considered to be a laminar,incompressible and isothermal gravity-driven two-dimensional steady-state flow (or unsteady-state for RTD analysis),so the governing equations of conservation of mass,momentum and species mass transport for Newtonian flow take the following forms:

The subscript q denotes properties related to the liquid phase(denoted by subscript l) and gas phase (denoted by subscript g).For the local value of αq,the appropriate properties and variables were determined by the presence of the component phases in each control volume.

3.3.CFD simulation strategy

The modeling and grid were created and meshed using CFD preprocessor.The numerical simulations were performed using a commercial CFD code installed on a 64-bit master workstation(HP-z640,Dual 2.50 GHz Intel Xenon CPU E5-2680 v3 with 128G of RAM).As previously mentioned,most of the numerical strategies in this work followed the our previous work[34].The computation of flow field based on different grid mode derived from different mesh sizes and mesh generation were implemented and improved until it resulted in a steady gas-liquid interface,then the mesh sizes and grid numbers could be determined to validate the grid independence.On the other hand,the flow pattern of falling film flow was compared with that under similar flow conditions as reported by Zhao and Cerro’s work [36].The CFD simulations was satisfied to predicted the flow dynamic behavior of falling film flow could be well predicted by CFD simulations,and the results were in agreement with the public work.The three-dimensional overall computational region was considered that the width of physical domain was going to be at least triple,the meshes number increased dozens of times in the conditions of guaranteeing the fundamental accuracy.A Trial calculation for mesh numbers of 3.83 million was implemented with a time cost of 5 days,while the RTD calculation cost was up to 1 month.Therefore,the computational cost is a reasonable concern.Besides,the film thickness of 3D simulation is a little bit less than 2D simulation owing to the less accuracy,but the deviation is considered insufficient.

In the fluid inlet,uniform velocity(U0) and fixed film thickness(δ0) were specified according to the corresponding flow rate and were also used as the initial conditions of the calculation.U0and δ0conform to the following relationship:

A segregated implicit solver method was used for the momentum equations.A solver technique based on finite volume method was implemented to convert the governing equation into algebraic equation.Second-order upwind scheme was selected as the discretization scheme for the momentum equation and volume of fraction.PRESTO! was employed for the pressure discretization scheme.The PISO algorithm was used for pressure-velocity coupling,which is recommended for steady simulations.Once the steady state flow field was attained,the solver is needed to be transformed into unsteady state.Then the above-mentioned species transport equation was added,and the concentration of the tracer at the outlet surface was monitored,so that the RTD curve was obtained from the monitored data.The time-step size was set to 0.001 s.

It should be mentioned that the flow analysis was dependent on the gas-liquid interface.It was determined by designating the phase contour level in postprocessing software.Herein,the width of gas-liquid transition region was less than 0.2 mm.Once the gas-liquid interface was determined,the film thickness could be calculated from subtracting from the coordinates of interface and the wavy wall.

3.4.RTD convolution

For a system consisting of two statistically independent subsystems in series,the RTD function can be calculated with a statistical theory in RTD,as described by Chen and Hu[37].That theory stipulates a closed system composed of two idealized elements,i.e.E1(t) and E2(t).The overall RTD density function E(t) is related to the two elements,E1(t) and E2(t),as shown by the following equation:

This equation implies that knowing any two of the three RTD density functions makes it possible to calculate the remaining one either by convolution or devolution.Zhang et al.[15]calculated the local RTD in a co-rotating two-screw extruder by deconvolution based on the statistical theory.

To compare simulated RTD results with experimental data at the same height of vertical wall of falling film flow,the RTD density functions of the flat part (Ez15(t)) and the wavy part (Eb60(t)) were respectively calculated by convolution of the RTD results of repeated units (Ez5(t) and Eb5(t)).After that,the two results were combined by convolution to constitute the total RTD density function E75(t).The whole convolution process is schematically illustrated in Supplementary Fig.S4.

The numerical results at different heights of wall were obtained by repetitive convolution of RTD data of Ez5(t) and Eb5(t) in MATLAB program,as shown in Fig.4.The subscript letters “b”and “z”represent wavy wall and straight wall respectively,and the Arabic numerals of subscripts mean the number of repeating units(a unit height of 33 mm).The RTD results for different viscous fluids obtained by convolution are listed in Supplementary Table S2.Compared to the convolution unit of flat wall,the tavof wavy wall was longer,and thetended to be reduced except the lowest viscosity fluid.It was obvious that a doubled height of wall would lead to a doubled tavof falling film flow as well as an approximately halvedAs a matter of fact,the tavof any two identical subsystems in series by convolution was exactly twice of that of one subsystem for the steady-state falling film flow,which indicates that both the tavandvaried linearly with the height of falling film wall and were not affected by the convolution calculation.This result is consistent with the variation of RTD multistage ideal mixed model.

4.Results and Discussion

4.1.Film thickness of falling film flow of high viscosity polymer fluids

Fig.4.The RTD convolution of falling film flow of high viscosity fluids with flow rate Q=15 kg·h-1,λ=4 mm,(a)μ=0.92 Pa·s,(b)μ=623 Pa·s.

Fig.5.Falling film of high viscosity polymer fluids flows outside of the vertical wavy tube (λ=4 mm,Q=15 kg·h-1).

Fig.6.The comparison of the film thickness in crest (a) and in trough (b) between CFD simulation and experiment at different flow rates,λ=4 mm.

The experiments of falling film flow with a wide range of high viscosity polymer fluids were performed on a vertical wavy tube.The film thickness was still large even if the Reynolds numbers was as low as 10-4under low flow rate,as shown in Fig.5.As the fluid viscosity increased,the flow behavior of falling film can be characterized as a creeping movement.No wave pattern was observed throughout the process of falling film flow in present work.Fig.5 indicates that the film thickness is very sensitive to the fluid viscosity.It has been found that the fluid properties can influence the flow structure in many reported works [38-40].The film thickness of falling flow of the lowest viscous fluid was relatively small.With the increase of fluid viscosity,the viscous shear stress of film as well as the flow resistance both increased,and thus therefore the film thickness enlarged remarkably.When the falling film of low viscous fluid flows along the wavy wall,the flow pathline of free surface is almost the same as that on wavy wall,while the film surface tends to become flat for the high viscosity fluid c and fluid d.This is consistent with the falling film flow on a miniature rod surface reported by Zhao and Cerro [36].A similar but less significant result could be achieved by increasing the flow rate.

It is necessary to verify the experimental results,although there is insufficient effective data of falling film flow for comparison owing to the lack of reports on various high viscosity polymer fluids.However,CFD tools can be used as an alternative approach to provide an indirect validation.With the aim of obtaining basic hydrodynamic description of falling film flow field,the film thickness of falling film flow on vertical wavy wall was determined based on the experimental data and CFD simulations.Fig.6 shows the film thickness in crest (δc) and in trough (δtr) of wavy wall obtained from experiments and the data obtained from CFD simulation.Both of them increase gradually with the flow rate.It is quite noticeable that the results of film thickness obtained via the CFD simulation are in very good agreement with the experimental results for the wide range of fluid viscosity.It also can be seen that the difference of film thickness between in crest and in trough tends to be slightly reduced as the flow rate increases.For falling film flow of high viscosity fluids,it is more inclined to accumulate fluids in the trough of wall and at higher flow rate [41],thus the film thickness in trough increases rapidly.In a case of infinite flow rate,the film surface is rather flat like the case of sky-high fluid viscosity,it can be expected that the difference of film thickness between in crest and in trough will tend to be a constant,which is expectedly equal to the wave amplitude of wall.

Fig.7.The comparison of the film thickness in crest (a) and in trough (b) between CFD simulation and experiment at different wave amplitudes of wall,μ=130 Pa·s.

Fig.8.The RTD studies of falling film flow,Q=15 kg·h-1,λ=4 mm.(a)the injection of RTD tracer,(b)the diluting samples,μ=5 Pa·s,(c)the diluting samples,μ=130 Pa·s.

Fig.9.The plot of E(θ) vs.θ curves from CFD simulations and experiment at different flow rate,μ=130 Pa·s,λ=4 mm.

Fig.7 shows the comparison of the film thickness between CFD simulation and experiment at different wave amplitudes of wavy wall.It should be noted that the δcdecreases gradually with the wave amplitude of wall,while the δtrdisplays an opposite trend.The difference between the δcand δtrenlarges slightly as the flow rate increases.This can be attributed to the accelerated accumulation of viscous fluids in the presence of higher wave amplitude of wave wall.In most cases,both δcand δtrin experiment are very close to the numerical results and the error between them is less than 5% through further analysis.In conclusion,the comparison results show good agreement regardless of the change of flow rate,fluid viscosity and wave amplitude of wall.

4.2.Residence time distribution

Considering the qualitative feature of high viscosity,the offline testing method was adopted to characterize the RTD tracer and samples were collected with the uninterrupted equal timeinterval at the bottom of falling film tube.Before the detection of tracer concentration,the ultrasonic accompanied with heating is necessary both before and after the sample diluting,which is very important for two higher viscosity fluids.Certainly,the collection and preparation of testable sample probably account for the largest workload in the RTD experiments.Fig.8a shows the tracer falling film flows on wavy wall.In most cases,the number of sample collected at the outlet was over 30,and the points where the concentration of samples approximated to zero were generally not plotted in the RTD curves.The testable samples for fluid of 5 Pa·s and fluid of 130 Pa·s containing tracer are shown in Fig.8b and c,respectively.For relative low viscosity fluid,the collected samples had less volume due to the less tavand the good solubility with fewer solvent.Consequently,the sample concentration of fluid of 5 Pa·s for absorbance detection was higher than the sample concentration of fluid of 130 Pa·s.

Fig.10.The plot of E(θ) vs.θ curves from CFD simulations and experiment at different fluid viscosity,Q=15 kg·h-1,λ=4 mm.

As before-mentioned,the sampling time and diluting ratio may differ according to flow rate or fluid viscosity.For comparison of the mixing characteristics of falling film flow at different flow conditions,the variance of RTD was nondimensionalized (Figs.9 to 11).The corresponding original RTD curves obtained from experiment and numerical simulation were demonstrated respectively in Supplementary Information (Fig.S5 to Fig.S7).

The curves of four flow rates obtained for the normalized RTD function E(θ) as a function of dimensionless time θ with fluid viscosity of 130 Pa·s and wave amplitude of 4 mm were shown experimentally and numerically in Fig.9.The curves in this Figure displays the expected deviations that the peak of the RTD function for CFD simulation is far higher than that of experiment,the curves are well symmetrical that the tavapproach to the main residence time.The results of the RTD analysis have been listed as Table 2.

It seems that the difference of tavbetween experiment and CFD prediction is already within the acceptable range,with just a few exceptions that the deviation is over 10%.In general,the tavbased on experiment is less than the value of CFD prediction.However,the difference ofbetween them is so obvious that theof experiment is several times more than theof simulation,even as much as dozens of times.In consideration of multistep process for off-line measurement technique in this work,there is a constraint on the time-step size and the associated data points obtainable with the RTD experiments.Besides,the skewness S which represents the deviation from the symmetry axis of θ=1 for experiment is significantly less than that of simulation.Hence,the results derived from CFD simulation accords with the theoretical situation that seems to be more advisable.This is owing to the distinct semi-parabolic velocity distribution of falling film from inner wall to free surface which determines the RTD in CFD simulation,yet the RTD tracer of experiment is probably taken away by the surface film as a whole for falling flow that the residence time gets shortened and the peak of residence time appears early,therefore,the remarkable center-left RTD curves with trailing phenomena can be seen in experimental.

Fig.11.The plot of E(θ) vs.θ curves from CFD simulations and experiment at different wave amplitude of wall,Q=15 kg·h-1,μ=130 Pa·s.

Table 2 The results of the RTD analysis at different flow rates for CFD simulations and experiment

Table 3 The results of the RTD analysis with different high viscosity fluids

Fig.10 shows the comparison of normalized RTD obtained via CFD simulation and experiment with different viscous fluids.Zhang et al.[12].described the effect of fluid viscosity with range of 0.001-0.0125 Pa·s on the RTD curves,finding that the RTD curves showed similar trends as in Fig.10.The peak height of curves in this figure had the expected deviations considering the wide range of fluid viscosity and the characteristic of high viscosity,and the deviation was more significant for the unnormalized curves (Supplementary Fig.S6).The expected deviation considering the possible combinations of tracer injection and measurement techniques [11].With the application of one-step input method,the on-flow-line tracer injection technique took only a few seconds to perform smoothly.More importantly,the time-step size was rather longer compared to the time-step size in CFD simulation,thus the sampling quantity was limited to several dozens.Because of the poor fluidity of high viscosity polymer fluids,the initial sample in outlet would inevitably experience the process of dissolution and dilution before the measurement of tracer absorbance.In the process of metastasis,the error would be increased inadvertently.It should be noted that the difference of tavbetween CFD prediction and experiment was satisfactorily less than 10%for a wide range of fluid viscosity,as listed in Table 3,which validates the dependability of RTD results.It was also worth noting that the S obtained via CFD simulation increased with the fluid viscosity,which is quite the contrary for the value obtained via experiment.The tracer injection technique and sample aftertreatment for higher viscous fluid could result in more error for the RTD testing.

The desirable structure design of device is supposed to be appropriate for the RTD of materials.The normalized RTD curves obtained by CFD simulation and experiment at different wave amplitudes of wall are shown in Fig.11.As the wave amplitude increased,either the RTD curves of simulation or that of experiment shifted fractionally.This is most probably because the increase of wave amplitude was too small to tell the difference.The tavonly increased a few seconds for each 1 mm when the waveamplitude was smaller than 3 mm (see Table 4).Predictably,the growth of tavwould be accelerated with the increase of wave amplitude.It can be seen that the wavy wall has a limited effect on the behavior of falling film flow when wave amplitude is small.The falling film flow on wave node generates horizontal stretching deformation and the structure of wave node enhances the mixing of film,which increases the residence time and decreases the variance.However,in the case of overhigh wave amplitude,more accumulation of film in the trough of wavy wall will produce dead zone thus increasing the trailing of RTD [36],which leads to expectedly unfavorable effects on the material production especially invovling the condition of high temperature reaction.On the basis of variance of RTD,simulation results show that neither too high nor too low wave amplitude of wall is suitable,and an optimal wave amplitude should exist to lengthen the average residence time and reduce the discreteness of residence time.

Table 4 The results of the RTD analysis with different wave amplitude of wall

From the data in Table 4,it can be seen that the results of CFD simulation are basically consistent with the experimental results,which verifies the reliability of present investigation.On an industrial-scale vertical wavy wall,it was difficult to achieve a high degree of verticality and uniformed film distribution,although the setup of falling film flow was proved to have a limited impact on experiments.On the other hand,the gas phase was not loaded an initial velocity in the numerical simulation of gas-liquid two phase flow that did not take into account of the influence of gas flow.It was also noticed that the falling film flow of high viscosity polymer fluids was mainly dominated by viscous effect,the mixing was probably generated because of the stretching in radial direction in the presence of wavy wall.Nevertheless,the CFD predicted value of tavwas in good agreement with experiment result of tav.The RTD investigation of falling film flow of high viscosity polymer fluids outside of industrial-scale vertical wavy wall can is expected to provide a new insight into the design,evaluation and optimization of falling film flow system.

5.Conclusions

The flow behavior and RTD of falling film of high viscosity polymer fluids on an industrial-scale vertical wavy wall were investigated by using numerical simulation and experimental approach,respectively.Results showed the falling film flow of high viscosity fluid was a steady flow owe to its large film thickness.The result of the film thickness both in crest and in trough of wavy wall by CFDVOF prediction was in good agreement with the experimental value obtained by capacitance detector.The RTD experiments under different flow conditions were carried out with a pulse injection of a tracer and off-line measurement,and the results were validated numerically based on convolution calculation of RTD data.The RTD curves obtained by experiment implied a phenomenon of trailing,but the CFD simulation result indicated the RTD curve was well symmetric and the variance was quite small.The CFD prediction value of tavwas in good agreement with the experimental value of tav,however,it was not comparable to each other for the variance and skewness of RTD.Both the tavand variance of residence time increased with the increase of fluid viscosity or with the decrease of flow rate.As the wave amplitude increased,the growth of tavwas accelerated,while the variance of residence time firstly decreased and then increased.The deviation could be attributed to the imperfect numerical strategy and experiment system in the presence of high viscosity fluid.

Supplementary Material

Supplementary data to this article can be found online at https://doi.org/10.1016/j.cjche.2018.12.022.