Effects of bubbly flow on bending moment acting on the shaft of a gas sparged vessel stirred by a Rushton turbine☆

Dai'en Shi,Ziqi Cai,Archie Eaglesham,Zhengming Gao,

*

1School of Chemical Engineering,Beijing University of Chemical Technology,Beijing 100029,China

2Huntsman Polyurethane,Everslaan 45,B-3078 Everberg,Brussels,Belgium

Keywords:

A B S T R A C T

1.Introduction

Impeller stirred vessels play an important role in many chemical processes,enhancing chemical species transport through the input of mechanical energy into the fluid.Such vessels generally contain baffles,coils or other internals designed to enhance the mixing,heat transfer or the other desired process performances.Thus,for a typical well balanced impeller system,even though the axes of impeller and shaft are perfectly aligned with the axis of rotation,which in turn is perfectly aligned with the vessel centreline,the fluid motion produced by the impeller is not,in general,symmetric and also unsteady due to low-frequency macroinstability[1],blade passing frequency pseudo-turbulence[2]and highfrequency turbulences[3,4].This asymmetric,unsteady fluid motion exerts an imbalanced and unsteady load[5]on the impeller and makes the elastic shaft equipped with the impeller to deflect instantaneously and even vibrates severely,generating the lateral movement of the impeller.The resulting lateral oscillation of the impeller in turn induces further unstable and nonuniform flows around the impeller.In fact,this behavior in stirred vessels is a bidirectional Fluid-Structure Interaction(FSI)of a flowing fluid with a flexible structure.

However,an undesired and unsteady bending moment must also be exerted on the shaft due to the lateral deflection of the shaft and movement of the impeller produced by virtue of the complex FSI in the stirred vessel.This bending moment results in a dynamic fatigue load on the stirred vessel head supporting the shaft.In the mechanical design of mixing equipment,the underestimation of the bending moment acting on the shaft can lead to plastic deformation and fatigue failure of the shaft.However,it is difficult to theoretically determine the bending moment because of the complex nature of fluid dynamics,structure dynamics and FSI coupling dynamics in such a stirring system.

The bending moment acting on an overhung shaft equipped with a 2-blade pitched blade impeller may be simply analyzed as an example to understand the bending moment sources.

The forces acting on the pitched blade impeller are shown schematically in Fig.1.The corresponding moments balance equation is given by

where Mbis the bending moment,Mtis the shaft torque,Ffiis the fluid force acting on the ith impeller blade,Fmis the force related to mass of impeller and shaft such as the gravity and the inertia force generated by the vibration,Aiand Amare the arm lengths corresponding to Ffiand Fmrespectively,and nbis the number of impeller blades.If the instantaneous symmetries of the impeller rotation and fluid flow in the stirred vessel were perfect(i.e.such that,F1=F2,A1=A2and either Fm=0 or Am=0),the bending moment caused by the forces and the value of Fm×Amwould be zero.The formula for the moment balance is given by

Fig.1.Schematic diagram of key forces and moments acting on an overhung shaft equipped with a 2-blade pitched blade impeller.

If the impeller is operated in an empty vessel(i.e.such that,Ffi=0),the torque produced by the fluid forces should be zero.The formula for the moment balance is simpler:

When gas is sparged in the stirred vessel,the2-phase nature of flow[6]can enlarges both asymmetries and unsteadiness in the flow field beyond that of the single phase situation,which further increases the shaft bending moment by virtue of intensifying FSI in the stirred vessel.However,few researches on the study of lateral loads on gas sparged stirred vessels have been published.In this paper,an experimental study examining the impact of gas sparging on the shaft bending moment in a Rushton turbine(RT)stirred vessel is presented.The main purpose is to provide basic information of the bending moment acting on the shaft for better mechanical design of multiphase stirred vessels.

The results presented in this paper include the mean,standard deviation and the peak fluctuation of bending moment acting on an overhung shaft at different gas rates.These are respectively applied to the strength check,fatigue analysis and stiffness check for mechanical design.The combined moment of bending and torsion as a function of gas flow number is also presented.

2.Models and Method

2.1.Experimental model

The schematic diagram of the tested vessel used in the experiments is shown in Fig.2 and the key details are given in Table 1.Details of the modified RT geometry in which several holes are used to generate a practical imbalance of the impeller are shown in Fig.3.

In Table 1,the imbalance of the impeller was estimated by

Fig.2.Schematic diagram of the tested vessel.

the distance between the centers of gravity and geometry was given by

and the ratio of δ to the diameter of the impeller is con fined in 1%that meets a practical condition according to previous work[7].In Table 1,fnand frare the natural and resonant frequency of the shaft and impeller structure,respectively,which are both estimated by using vibration system with one degree of freedom[8].The latter is lower than the former due to the effect of fluid added mass.The range of the operational speeds used in the experiments was set in 0.4＜n/60fc＜0.6,thus avoiding any resonance issues.The liquid surface was sufficiently high to ensure that it had a negligible influence on the bending moment.

The gas sparging rates were chosen so that the 2-phase flow regime ranged from complete dispersion to loading at each speed.Fig.4 shows the range of the measured points(operational conditions)in the context of a 2-phase flow regime map with two dividing lines obtained by Eqs.(6)and(7)[9],which indicate two transitions between complete dispersion and loading,and between loading and flooding regimes,respectively.

2.2.Experimental method

2.2.1.Test rig

In the experiments,the bending moment and torque acting on the shaft were measured simultaneously using a high precision,high frequency response sensor.The key features of the test rig are shown in Fig.5.The drive unit consists of a motor,drive shaft and a flexible coupling.In order to reduce the influence of speed variation on the bending moment,the speed control system was set to a closed loop consisting of a frequency converter with speed feedback and a motor with shaft encoder.

Table 1 Detailed information of the stirred vessel

Fig.3.Sketch and physical map of the RT.

Fig.4.Range of the measured points in 2-phase flow regime.

The moment measurement system consists of a moment sensor(Fig.6)and a data acquisition unit customized from HBM(Germany).The specification and major parameters of the sensor are listed in Table 2.The strain gauges required to measure the three moments(the X and Y components of bending moment,and the torque)are in the sensor rotor.The signals from these are transmitted to a receiver in the stator that also contains the power supply for the system.

Fig.5.Experimental setup.

2.2.2.Data acquisition

The signals for bending moment and torque were synchronized and simultaneously sampled.The data sampling frequency and size can influence the recorded results.Therefore,it is important to select the suitable values of the two parameters appropriately.The power spectral density(PSD)of the bending moment measured by using a sampling frequency of 1200 Hz shows that approximately 99%of the spectral power(SP)is contained in frequencies below 100 Hz(Fig.7).Therefore,a sampling frequency of 600Hz was chosen to ensure sufficient resolution at low frequencies,in which most of the loading was expected,while also ensuring that no significant high-frequency loads were omitted.

Fig.8 shows the percentage deviation of the mean bending moment over the latest sampling time from that over the whole sampling period as a function of sampling time.For sampling times beyond 100 s or so,the percentage deviation is around±2%,which means the sampling time has a negligible effect on the results if the sampling time is longer than 100 s.Therefore,a 300 s sampling time was chosen in the experiments,corresponding to a sampling size 180000.

Fig.6.Moment sensor con figuration.

Table 2 Major parameters of the moment sensor

Fig.7.Power spectral density(PSD)normalized by the variance.(Stirrer speed(N)=3.6 Hz;sampling scale(SS)=217).

Fig.8.Percentage deviation of the average bending moment from the mean as a function of sampling time(stirring speed(n)=192 r·min-1;sampling frequency=600 Hz).

3.Results and Discussion

3.1.Impeller torque

Fig.9.Ratio of mean gassed to ungassed impeller torque as a function of FlG.

3.2.Shaft bending moment

Fig.10 shows the shaft bending moment normalized by its average value for the gas flow of 4.79 m3·h-1at 216 r·min-1.In this case it is clear that the bending moment is an unsteady,dynamic load with a number of different frequencies contributing to the amplitude fluctuations.

Fig.10.Instantaneous bending moment normalized by the time-averaged value.(n=216 r·min-1;QG=4.79 m3·h-1;Mb=1.86 N·m).

3.2.1.Effects of gas flow on amplitude characteristics

(1)Amplitude distribution

Fig.11 shows typical amplitude distributions measured for the bending moment and torque.The torque amplitudes were well described by a normal distribution whereas the bending moment amplitudes were much better represented using a Weibull distribution.The difference is due to the asymmetry of shaft deflection around the dynamic equilibrium position,being caused by the nature of shaft material against bending deformation,namely the elastic reaction pulling the shaft back towards the axis of rotation(static equilibrium position)after the shaft bending deformation.

The probability density function(PDF)for Weibull distribution is given by[11]

Fig.11.Typical amplitude distributions of shaft bending moment and impeller torque.(Left:bending moment;right:torque n=216 r·min-1,Ts=300s).

Fig.12.Relative scale parameters(λ)as a function of FlG.

where x is a Weibull random variable,λ (＞0)and κ (＞0)are the scaling and shape parameters respectively.The distribution is flatter for high values of λ and more symmetric for high values of κ.The mean(μ)and variance(σ2)are given by

Fig.13.Relative shape parameters(κ)as a function of FlG.

According to Eqs.(9)and(10),the scale parameter λ and the shape parameter κ are solved.The relative(gassed/ungassed)scale parameter λ and the relative shape parameter κ of the Weibull distributions for bending moment as a function of gas flow number are shown in Figs.12 and 13,respectively.Apparently,the magnitudes of the relative λ (＞1)and κ (＜1)show that the distribution tends to be flat and asymmetrical after gas is introduced,which demonstrate that the shaft is pulled away

from the axis of rotation at a low frequency due to the gas-liquid flow influences on the lateral movement of the impeller.

If any two of the four parameters,μ,σ2,λ and κ,are known,amplitudes of bending moment can be restructured by Weibull distribution,which helps obtaining a fluctuation with any one probability covering the responding amplitudes range.The purpose can be to provide basic information for a fatigue analysis.The cumulative distribution function can be obtained by integrating the Weibull probability density function:

According to Eq.(11),if the probability FCDis known,the amplitude fluctuating location x with the FCDvalue may be solved,or if the amplitude fluctuating location x is known,the probability FCDbelow the x may be solved.

(2)Amplitude mean

Fig.14 shows the relative mean bending moment/(here it is de fined as the bending moment ratio between gassed and ungassed conditions)as a function of FlG,where the ungassed bending moment had been reported by Shi et al.[12].In Fig.14,Letter A indicates the transition from compete dispersion to loading regimes.Initially the relative mean bending moment increases with the rise of FlGand reaches a peak of about 1.3 in complete dispersion regime.When the gas flow continues to increase,the mean bending moment slightly descends to a valley,going through the transition A.When the gas rate keeps increasing,the mean bending moment again starts to rise gradually in loading regime.The “S”trend of the mean bending moment over FlGfor each speed is obviously similar.

Fig.14.Relative mean bending moments as a function of FlG.

The phenomenon shown in Fig.14 demonstrates that the magnitude of the unbalanced fluid force acting on the impeller increases with the rise of the gas flow con fined in complete dispersion regime.Apparently,the increase is because the 2-phase bubbly flow enlarges the transient asymmetry of the fluid motion around the impeller,and causes the nonsynchronously sharp fluctuation of fluid pressures on the impeller blade surfaces in every moment.However,the unbalanced fluid force also decreases due to the overall decline in the fluid forces acting on all blades(the decrease in the torque also indicates the information as shown in Fig.9).Obviously,the whole decline in the fluid forces,resulting from the decrease in fluid pressure difference between the front and back of the blade,originates from the formation of gas cavities behind the blades.Therefore,the competition between those two opposite factors must lead to a peak of the bending moment when the decrease is dominant.That is to say,the transient asymmetry of the fluid motion around the impeller dominates in low-gas flow complete dispersion regime,while the cavitation behind the blades dominates in high-gas flow complete dispersion regime.Consequently,the mean bending moment shows one fluctuation in complete dispersion regime.

When the gas rate gets higher,the gas flow begins to impact the impeller unsteadily,where complete dispersion has shifted to loading,causing the stirring structure to vibrate up and down and leading to the increase in the bending moment.Being the same as the previous competition,it gives rise to a valley of the bending moment when the increase is dominant.That is to say,the cavitation behind the blades dominates in relatively low gas flow loading regime,while the gas direct impact on the impeller dominates in relatively high-gas flow loading regime.Consequently,the mean bending moment shows one fluctuation in loading regime.In order to eliminate the impeller circumferential(tangential)flu

id force influence and only examine the impeller lateral(bending)force variation with FlG,a dimensionless shaft bending moment β is de fined by Eq.(3).In the equation,Mb/L(L is the shaft overhung length)represents the lateral forces contributing to the bending moment,and Mt/D(D is the impeller diameter)represents the circumferential fluid force contributing to torque.

The relative β as a function of FlGis plotted in Fig.15,where the transition from complete dispersion to loading regimes is also marked by Letter A.Apparently,the relative β almost linearly increases to the transition A with FlG,approaching 2.2,and is independent of the five speeds.The relative β trend over FlG,further demonstrates that the increasing bubble volume fraction leads to increasing levels of transient asymmetry in the fluid forces acting on the blades.

Fig.15.Relative(gassed/ungassed)β coefficients as a function of FlG.

(3)Amplitudes fluctuation

A ratio of standard deviation to the mean represents intensity of amplitude fluctuation,while another ratio of the peak deviation(namely,the result of subtracting mean from peak)to standard deviation represents the magnitude of amplitude peak.Due to great randomness of maximum bending moment,the peak defined by the 99%probability is calculated by putting 99%into infiverse cumulative Weibull distribution function,and then a fluctuation with the99%probability covering99%minimum amplitudes is obtained by subtracting the mean of amplitudes from the previous peak.

The relative ratios both shown respectively in Figs.16 and 17,where the former approaches to 1.06 and the latter to 1.025,slightly change with FlG.Being the same as relative mean bending moment in Fig.14,the two relative ratios trend over FlGboth manifests the“S”shape.The reason for the “S”trend also results from the competition among the nonuniformity of bubbly flow around the impeller,the formation of gas cavities behind the blades,and the gas direct impact on the impeller as gas is introduced.

Fig.17.Relative ratios of the peak fluctuation to σ as a function of FlG.

3.2.2.Effects of gas flow on frequency characteristics

There is a wide range of frequencies contributing to the fluctuation of shaft bending moment.Generally,the contribution from the impeller speed frequency is caused by any imbalance of stirring structures.Contributions from low frequencies below the impeller speed typically result from macro-instabilities in the bulk flow.Contributions from frequencies higher than the impeller speed are typically more complex,resulting from periodic turbulence(e.g.blade passing frequency and its harmonics)due to the trailing vortex behind impeller blades and interactions between the impeller blades and baffles[13],and from the natural elastic vibration of stirring structures.However,these frequencies are not evident simultaneously,which depend on the specific condition which can excite the corresponding frequencies.

Fig.18 shows the PSD of the bending moment for 216r·min-1from0 to5 Hzat different gas rates,which is obtained by using the Yule-Walker autoregressive model[14].In all cases,two groups of the peaks marked by Letters A and B are evident which are both influenced by gas flow.Group A locates at rather low frequency,which should be a result from low-frequency macro-instability of bulk flow hydrodynamics in stirred tanks.And group B locates at the stirring speed frequency(3.6 Hz),which is caused by the impeller imbalance.The two frequencies give major contributions to the bending moment fluctuation when gas is introduced into stirred vessels.

A ratio of spectral power(PS)in a range of frequencies to the variance is used to represent a resultant contribution of these frequencies to the bending moment fluctuation.The relative ratio(gassed/ungassed PS)in the low-frequency range(0 to N/5)is shown in Fig.19.Initially,it increases with FlG,peaks for FlG～0.05 and then decreases with the increasing gas rate,which demonstrates that bubbly flow in complete dispersion regime has a considerable influences on bending moment fluctuation at a rather low frequency below speed.The effect is not a monotonic function but possesses a maximum approaching 9 at FlG～0.05 in complete dispersion regime.It is possible that a specific status of bubbly flow produce a maximum degree of macro-instability of bulk fluid circulation in stirred vessels.

Fig.19.Relative PS/σ 2in the range of f from 0 to N/5.

3.3.The combined moment of bending and torsion

In stirred vessels,shaft size is determined by a combination of the mean bending moment and torque.Here the mean moment combining bending with torsion is de fined as

The relative ratios of combined moment(gassed/ungassed)for the fiv e speeds are shown in Fig.20,with the transition from complete dispersion to loading regimes marked by Letter A.Apparently,it declines almost linearly to the transition A,reaching about 0.75 at FlG～0.065,and levels out for higher gas rates.This information on the combined moment could reflect more characteristics of gas-liquid flow loading on the impeller,compared with those on individual torque and bending moment.

Fig.18.Power spectral density(PSD)normalized by the variance.

Fig.20.Relative mean combined moment of bending and torsion as a function of FlG.

4.Conclusions

This paper investigated the impact of sparging gas on the shaft bending moment in an RT stirred vessel.The systematic analyses of the bending moment amplitude and power spectral density show that the influences of gas flow on the bending moment depend on the complex flow in the gas-liquid stirred vessel.The key conclusions are summarized as follows:

(1)The trend of the relative mean bending moment over FlGpresents an “S”shape,which results from the competition among the nonuniformity of bubbly flow around the impeller,the formation of gas cavities behind the blades,and the gas direct impact on the impeller.

(2)The relative dimensionless mean bending moment β almost increases linearly with FlGin complete dispersion regime,approaching to 2.2 at the transition from complete dispersion to loading regimes.

(3)The distribution of the bending moment amplitude is well described by the Weibull distribution,which reflects that the elastic reaction pulls the shaft back towards the axis of rotation after the shaft bending deformation.

(4)The relative ratio of standard deviation to the mean and that of the peak fluctuation to standard deviation approach about 1.06 and 1.025 respectively,whose changes with gas rate are similar with that of the relative mean bending moment.

(5)The rather low frequency and speed frequency in the bending moment PSD are both evident,and the low-frequency contribution to bending moment fluctuation peaks in complete dispersion regime.

(6)The relative mean combined moment almost linearly declines to the transition from complete dispersion to loading regimes with the rise of gas flow,reaching about0.75,and levels out for higher gas rates.

Nomenclature

Aiarm length of the force producing moments,mm

C clearance of impeller off bottom of vessel,mm

D diameter of impeller,mm

Dsdiameter of sparger,mm

d diameter of shaft,mm

FCDcumulative distribution function

Fffluid force,N

FlGgas flow number of impeller

Fmforce relative to mass of impeller and shaft,N

FPDprobability density function

Fr Froude number of impeller

fnfirst order laterally natural frequency,Hz

frfirst order laterally resonant frequency,Hz

H height of liquid free surface in stirred vessel,mm

L overhung length of the overhung shaft,mm

Mbbending moment acting on the overhung shaft,N·m

Mccombined moment acting on the overhung shaft,N·m

Mttorque acting on the overhung shaft,N·m

mimpmass of impeller,g

mupart unbalanced mass of impeller,g

N operational speed,Hz(rps)

n operational speed,rpm

nbnumber of blades of impeller

PSspectral power,N2·m2

PSD power spectral density,N2·m2·s

QGgas flow rate,m3·h-1

rudistance of part unbalanced mass off geometrical center,mm

S sparger height off bottom of vessel,mm

SSsampling scale(sampling number)

T diameter of vessel,mm

TSsampling time,s

Ubunbalance of impeller,g·mm

Wbwidth of baffle,mm

x Weibull random variable,N·m

β coefficient of bending moment

δ distance between centers of gravity and geometry,mm

Г Gamma function

κ shape parameter of Weibull distribution

λ scale parameter of Weibull distribution,N·m

μ mean,N·m

σ standard deviation,N·m

σ2variance,N2·m2

Subscripts

b bending

f fluid

G gas

L liquid

s structure

t torsion