Yangyang Liang ,Zhengming Gao ,*,Dai'en Shi ,Haotian Li ,Yuyun Bao ,Ziqi Cai,*

1 State Key Laboratory of Chemical Resource Engineering,Beijing University of Chemical Technology,Beijing 100029,China

2 Mechanical Engineering School,Yancheng Institute of Technology,Yancheng 224051,China

Keywords:Shaft bending moment Torque Disk turbines Gas-liquid fl ow Fluid structure interaction

ABSTRACT The torque and bending moment acting on a fl exible overhung shaft in a gas-liquid stirred vessel agitated by a Rushton turbine and three different curved-blade disk turbines(half circular blades disk turbine,half elliptical blades disk turbine,and parabolic blades disk turbine)w ere experimentally measured by a customized moment sensor.The resultsshow that the amplitude distribution of torque can be fi tted by a symmetric bimodal distribution for disk turbines,and generally the distribution ismore dispersive asthe blade curvature or the gas fl ow rate increases.The amplitude distribution of shaft bending moment can be fi tted by an asymmetric Weibull distribution for disk turbines.The relative shaft bending moment manifests a “rising-falling-rising”trend over the gas fl ow number,w hich is a corporate contribution of the unstable gas-liquid fl ow around the impeller,the gas cavities behind the blades,and the direct impact of gas on the impeller.And the “falling”stage is greater and lasts wider over the gas fl ow number for Rushton turbine than for the curved-blade disk turbines.

1.Introduction

Stirred vesselsare w idely used in chemical engineering processw ith variousmaterialsand operating conditions.The motor drivestheimpeller to form aspeci fi c fl ow fi eld and it achievesthe desired effects,such as turbulence,miscible mixing,gas dispersion,and particle suspension.Thespatial structureof fl ow fi eld in stirred vesselsincludesthemain circulations in the bulk fl ow[1,2],the trailing vortices behind the impeller blades[3,4],and the turbulent eddies[5].And the temporal structure includes the macro-instabilitiesw ith low frequency[6-8],the periodicity w ith multiples of speed frequency[9,10],and the turbulent motion w ith high frequency[10].Such spatiotemporal instability results in an asymmetric and unstable fl ow fi eld and then exerts unstable reactive loads[11]on the structure such as the stirring shaft,impeller blades,baf fl es,and vessel w all.The unstable reactive loads will lead to the structure instantaneous deformation[12],w hich in turn disturbs the surrounding fl uid motion,resulting in a more non-uniform and unstable fl ow fi eld in the w hole vessel.Such interaction between the fl exible structure and the fl ow ing fl uid during the operation of stirring system is a typical example of bidirectional fl uid structure interaction(FSI)[13].

The shaft issubjected to a torque when the energy istransferred from the motor to the impeller;how ever,an undesired bending moment also actson the shaft because of the lateral de fl ection of the shaft and the lateral movement of the impeller by virtue of FSI.This lateral de fl ection and theconsequent lateral shaft bending moment,both of which arethemain performances of FSI,not only affect the fl ow fi eld and pressure fi eld,but also change the mechanical stress on the shaft and the head supporting the shaft.The torque can be derived from the power consumption of impeller;how ever,the bending moment acting on the shaft is generated from not only the unstable fl uid loads,but also the centrifugal force due to the manufacture tolerance and the impeller imbalance[13].Furthermore,in the multiphase stirred vessel,the vibration of bubbles around the impeller and the collision of particles on the impeller intensify the fl ow instability and exacerbate the fl uctuation of shaft bending moment.As a result,the shaft bending moment is an unstable nonlinear load in fl uenced by various factors,and it hardly achieves an accurate theoretical solution.On one hand,the overlarge shaft bending moment will lead to the plastic deformation and even the breakage of shaft.On the other hand,the fl uctuation characteristic of shaft bending moment may bring about the fatigue failure of the shaft.Worse more,if some periodic frequencies of structure and fl uid motion,such as speed frequency and blade passing frequency,are close to the natural frequency of stirring structure,the resonance w ill take place and the shaft bending moment will be signi fi cantly ampli fi ed[11,13],leading to a great damage on the shaft and vessel.Therefore,it is essential to determine the shaft bending moment for the design and safety check of the shaft and vessel.

Rushton turbine(RT)is a classic impeller since 1950s;how ever,the draw back of the considerable pow er input drop in the aerated system is serious.A great improvement is the modi fi cation of blade shape from the fl at to the curved w ith different blade curvature and are now commercially available[14].As for these curved-blade disk turbines,w ork related to the trailing vortex and turbulent structures in pure liquid system[15,16]and the pow er consumption[17-19],turbulent dispersion[20],gas holdup[20-22],and mass-transfer[20,23,24]in gas-liquid system have been performed.Compared w ith RT,the blade curvature of curved-blade disk turbines brings about smaller vortices[15]and has advantages in mass transfer and gas dispersion,handling more gas even w ith a low rotation speed.

The curved-blade disk turbines,such as half circular blades disk turbine(CD),half elliptical blades disk turbine(HEDT),and parabolic blades disk turbine(PDT),are now w idely used in gas-liquid stirred vessel.How ever,the effect of the gas activity on the shaft bending moment w ith these curved-blade disk turbines is still not clear now.Furthermore,the gas sparger is mostly right under the impeller,and the effect of the ejected gas fl ow w ith high speed on the shaft bending moment should be revealed.As for the shaft bending moment in the gas-liquid stirred vessel,experimental studies on RT and pitched blade turbine have been carried out in Shi's work[25,26];however,no researches on these curved-blade disk turbines have been published up to now.Therefore,this paper investigates the characteristics of the shaft bending moment and torque in a gas-liquid stirred vessel agitated by three different curved-blade disk turbines(CD,HEDT,and PDT)and compares with RTon the mean values and amplitude distributions of the torque,shaft bending moment,and combined moment.This w ork,asafollow-up research of Shi'sw ork[25,26],can provide valuable information for the mechanical design of the agitator in gas-liquid stirred vessels under the speci fi c operating condition(rotation speed and gas fl ow rate et al.),such as the safety evaluation of the shaft.

2.Model and Method

2.1.Experimental model

As show n in Fig.1,the experiments w ere carried out in a cylindrical stirred vessel with a fl at bottom and four equispaced baf fl es,and a sparger w as mounted at the bottom of the vessel.The geometrical details of the vessel are listed in Table 1.The fl exible shaft w as designed to be slim(length/diameter=55.6)to enhance the fl exibility.The liquid level w as high enough to eliminate the surface effect on FSI.

The four disk turbines used in the experiments were RT,CD,HEDT,and PDTw ith the same diameter(both disk and impeller),blade length and thickness.The schematic draw ings of the four disk turbines are show n in Fig.2,and the material and key dimensions are listed in Table 2.The curvature factor w as de fi ned as c=Ebla/Hbla,w hich indicates the degree of the blade curvature.Therefore,the values of c were 0,0.5,1,and 1.44 for RT,CD,HEDT,and PDT,respectively.

The natural frequency and operating conditions of the four disk turbines in the experiments are listed in Table 3,w here Froude number and gas fl ow number are de fi ned as

where N istheimpeller rotation speed,s-1;D istheimpeller diameter,m;g is the gravity acceleration,m·s-2;and QGis the gas fl ow rate,m3·s-1.The rotation speed N w as set as 0.4＜N/fn＜0.6 to avoid resonance as much as possible,and the fl uid fl ow is completely turbulent since the Reynolds number exceeds 120000,which is de fi ned as

where fnis the natural frequency of the shaft-impeller,Hz;andνis the kinematic viscosity of liquid,m2·s-1.

Fig.1.Schematic diagram of the stirred vessel.

Table 1 Detailed information of the stirred vessel

Fig.2.Schematic drawings of the four disk turbines.

2.2.Experimental method

2.2.1.Experimental setup

As show n in Fig.3,the experimental setup consists of the stirred vessel,impeller,stirring shaft,moment sensor,sparging system,andthe driver system.A sparger is connected to the external sparging system,w hich includes a pressure stabilizer to keep stable gas rate.The torque and bending moment acting on the shaft w ere simultaneously measured by a three-component moment sensor w ith high frequency and high precision,as show n in Fig.4.The detailed principle,structure,and parameters of the moment sensor can be referred to Shi et al.[25,26].

Table 2 Materials and dimensions of the four disk turbines

Table 3 Natural frequency and operating conditions of the four disk turbines

Fig.3.Experimental setup.

2.2.2.Data acquisition

The accuracy of the data acquisition is associated w ith the sampling frequency and duration.Fig.5 show s the power spectral density(PSD)of the shaft bending moment of HEDT collected by using a sampling frequency of 1200 Hz.The result show s that a majority,namely beyond 97%,of the spectral pow er is contained in frequencies below 100 Hz.Consequently,600 Hz w as chosen as the sampling frequency to ensure enough resolution at low frequencies and to avoid the loss of high frequencies.

Fig.6 shows the deviation of the mean shaft bending moment of HEDT as a function of sampling time under different gas fl ow rates.The deviation reduces to be w ithin±2%w hen the sampling time goes beyond 150 s,indicating that the in fl uence of the sampling time is acceptable after 150 s.Therefore,300 s w as chosen as the sampling time,so the corresponding sampling size is 180000.

Fig.4.Moment sensor con fi guration.

3.Results and Discussion

3.1.Impeller torque

3.1.1.Mean amplitude

The torque re fl ects the magnitude of tangential fl uid force on the impeller and the gas status/fl ow fi eld in a gas-liquid stirred vessel.The power numbers(/（ρN2D5）)for RT,CD,HEDT,and PDTin the completely turbulent regime under ungassed status in the experiments w ere 6.14,3.45,2.43,and 1.73,respectively.Fig.7 shows the relative pow er demand RPD(the ratio of the pow er consumption under gassed to the ungassed,and is equal to the ratio of the mean amplitude of torque under gassed to the ungassed under the same rotation speed,for the four disk turbines as a function of FlG.Generally,for RT,HEDT,and PDT,the RPD keeps decreasing w ith the increase of FlG.How ever,for CD,the RPD fi rstly rises to slightly greater than 1 and then decreases,w hich is consistent w ith the published results[17,20].The black solid lineistheintegrative trend over FlGof all rotation speeds and the relative error of each speed is w ithin±2%,and thus it can be seen that the RPD is rarely in fl uenced by the rotation speed or Froude number.The overall dow nw ard trend indicates that the tangential fl uid force on the impeller is reduced after the introduction of gas,and it is associated w ith a close relation betw een the stirring pow er and thegascavitiesbehind the impeller blades.The formation of gascavities reduces the pressure difference betw een the front and back sides of the blades,as a result,the mean torque and pow er decrease.

The fl at bladesof RTproduce large low-pressure regionsbehind the blades and there generate large gas cavities once the gas is introduced in.However,the formation and size of the low-pressure regionsbehind the streamlined blades are reduced,leading to smaller gas cavities behind thecurved blades,especially for the highly curved[20].Therefore,the pow er drop due to the gas introduction is largely reduced for the curved-blade disk turbines as the increase of the blade curvature.

3.1.2.Amplitude distribution

Fig.8 show s the amplitude distribution of torque normalized by the mean valueμMtunder gassed status for the four disk turbines,and the PDFrepresents for the probability density function.The amplitude distribution can be fi tted by a symmetric bimodal distribution for the four disk turbines.For RT,the bimodal distribution is concentrated and close to a normal distribution;how ever,for the curved-blade disk turbines,the distribution is dispersive and not concentrated around the mean value anymore,and the distance betw een the tw o peaks increases as the blade curvature increases.

Fig.5.PSD normalized by the variance(Impeller type:HEDT,Q G=7.29 m3·h-1,n=228 r·min-1).

Fig.6.Deviation of the mean shaft bending moment as a function of sampling time(Impeller type:HEDT,n=228 r·min-1).

Fig.7.Relative pow er demand RPD as a function of Fl G.

Fig.8.The amplitude distribution of torque(Q G=8.56 m3·h-1,n=228 r·min-1).

Since the amplitude distribution of torque can be fi tted by a symmetric bimodal distribution for the four disk turbines,then the left peak μ1and the right peak μ2are symmetric about Mt/μMt=1.That is to say,μ1+ μ2=2.The left peak μ1of the amplitude distribution of torque for the four disk turbines as a function of FlGis show n in Fig.9.It can be seen thatμ1generally decreases w ith the increase of FlGfor the four disk turbines.As the gas fl ow rate increases,the gas dispersion becomes worse,and the gas cavities behind the blades are more unstable and the size varies frequently.Therefore,the torque fl uctuation is intensi fi ed and the amplitude distribution is more dispersive at high gas fl ow rates.Note that for CD,HEDT,and PDT,theμ1is low er than 1 under ungassed status as w ell,w hich means the bimodal distribution of torque is an inherent characteristic of the curved-blade disk turbines,and the gas fl ow just intensify the distribution dispersity.Furthermore,theμ1decreases as thebladecurvatureincreasesunder thesame FlG,w hich meanstheamplitude distribution of torque is more dispersive as the blade curvature increases.

Fig.9.The left peak of amplitude distribution of torque(n=228 r·min-1).

Fig.10.Relative shaft bending moment RMBas a function of Fl G.

3.2.Shaft bending moment

3.2.1.Mean amplitude

The relative shaft bending moment RMBthe ratio of the mean shaft bending moment under gassed to the ungassed)for the four disk turbines as a function of FlGis show n in Fig.10.For RT,HEDT,and PDTat all rotation speeds and CDat 240 and 252 r·min-1,the trends of RMBover FlGare similar and manifest three-staged.The RMB fi rstly rises to a peak and then declines as soon as FlGup to a certain value and fi nally keeps increasing w ith FlG.Such a trend results from the complex activity of gas and liquid since the shaft bending moment is in fl uenced by both the intensity and fl uctuation of the gas-liquid motion and loads.Initially the introduction of gas results in a more unstable 2-phase fl uid motion around the impeller w hich exerts more unstable and asymmetric fl uid force on the impeller than the single-phase liquid motion does,leading to the increase of RMB.Meanwhile,the gas cavities formed behind the blades reduce the pressure difference between the front and back sides of the blades,leading to the overall decline of the fl uid force on the impeller(the decrease in impeller pow er or torque show n in Fig.7 also demonstrated)and it results in the decrease of RMB.When the gas fl ow rate increases to a certain value,the decrease is dominant and then the RMBpresents a dow nw ard trend.Finally,the gas fl ow rate keeps increasing so high that the gas masses begin to impact directly on the impeller and shaft.Therefore,the inhomogeneity and fl uctuation of the fl uid force on the impeller is intensi fi ed and the axial vibration of the shaft is exacerbated.Then the RMBpresents an upw ard trend over FlGagain.Totally,such three-staged(rising-falling-rising)trend is the corporate contribution of the three factors,and theunstable fl uid motion,thegascavitiesbehind theblades,and thegasdirect impact on theimpeller are dominant in the stage of“rising”,“falling”,and “rising again”,respectively.

The w eights of above three factors for different disk turbines lead to a modi fi cation of the three-staged trend between disk turbinesin some extent.For example,because the gas cavities behind the blades are largest for RT,the second factor(the pressure difference between the front and back sides of the blades because of the gas cavities)isdominant for RT.Then the decrease stage is greater and lasts w ider over FlGfor RTthan for the curved-blade disk turbines.The transitions between the “rising-falling”and “falling-rising”are located at FlG～0.062,0.040,0.020,0.025 and 0.120,0.055,0.023,0.04 for RT,CD,HEDT,and PDT,respectively and are slightly varied with the rotation speed.

Unlike the three-staged trend described previously,for CDat 216 and 228 r·min-1,the trend of RMBis different and decreases sharply over FlGat fi rst.This is because an intense vibration of the shaft-impeller took place under ungassed status in the experiments and is attributed to the system resonance,leading to a much high shaft bending moment.However,when the gas in introduced,the resonance excitation was disturbed and thevibration of shaft-impeller isweakened,leading to thedecreaseof RMBover FlGat fi rst.With the continuous increase of gas fl ow rate,the effect of resonance is relatively small to be considered and then the RMBmanifests the similar three-staged trend again.

In order to only investigate the lateral force on the impeller w ithout taking the effect of tangential force in,the shaft bending moment is normalized by the torque and is de fi ned as

w here the numerator and denominator represent for the lateral force and tangential force on the impeller,respectively.The relativeβ(the ratio ofβunder gassed to the ungassed)for the four disk turbines as a function of FlGis show n in Fig.11.It also presents a three-staged(rising-falling-rising)trend,w hich is similar to that of RMBshow n in Fig.10 and may also be the corporate contribution of the unstable fl uid motion,the gas cavities behind the blades,and the gas direct impact on the impeller.

Fig.11.Relativeβas a function of Fl G.

3.2.2.Amplitude distribution

Fig.12(a)and(b)show sthe amplitude distribution of shaft bending moment normalized by the mean valueμMbfor CDunder gassed status at 228 r·min-1.The amplitude distribution can be well fi tted by an asymmetric Weibull distribution,and the shape of the distribution curve is in fl uenced by the gas fl ow rate.The similar distributions can be also found in the other three disk turbines.The asymmetric performance is attributed to the asymmetric shaft lateral de fl ection:w hen thelateral de fl ection takesplacebecauseof FSI,the shaft elastic reaction pulls the shaft back to the rotation axis,and thusthe shaft ismore likely to close to the rotation axis rather than away from it.Therefore,the distribution of lateral de fl ection is asymmetric and inclined to be small.The shaft bending moment is largely determined by the lateral de fl ection,and generally a small lateral de fl ection meansa low bending moment.As a result,an asymmetric distribution of lateral de fl ection result in an asymmetric distribution of shaft bending moment and it can be fi tted by a Weibull distribution,and a larger proportion of bending moment is located at the left side of the distribution curve.

In addition,Fig.12(c)show s the amplitude distribution of shaft bending moment under ungassed status for CD at 216 r·min-1w ith system resonance.The distribution is more symmetric and cannot be well fi tted by a Weibull distribution anymore.

The PDFof a Weibull distribution is expressed as following

Tw o parameters that control Weibull distribution are the scale parameterλ and shape parameterκ.A high value ofλ means a fl atter distribution,and ahigh value ofκmeansamore symmetric distribution.The mean(μ)and variance(σ2)are determined byλ and κand are given by

According to Eqs.(6)and(7),λandκcan also be recalculated byμ and σ2.Therefore,once any tw o of the four parameters,λ,κ,μ,and σ2,are known,the amplitude of shaft bending moment can be restructured by the Weibull distribution.Besides of the mean shaft bending momentμ(show n as the relative value under gassed to the ungassed)that already show n in Fig.10,the relative shape parameter κ(theratio ofκunder gassed to the ungassed)isshow n in Fig.13.For RT and CD,the relativeκdeclines w hen FlGincreases to a certain value and is low er than 1,w hich means the Weibull distribution is more asymmetric due to the more unstable fl uid motion and fl uid force.How ever,for HEDTand PDT,the relativeκvaries slightly around 1.

Fig.12.The amplitude distribution of shaft bending moment(Impeller type:CD).

Fig.13.Relative shape parametersκof Weibull distribution as a function of FlG.

As for CD at 216 r·min-1,the relative κ is much less than 1.This is because an obvious system resonance took place under ungassed status and is much beyond the shaft elastic resilience,and the shaft iseasily to be kept away from the rotation axis.Therefore,the amplitude distribution of shaft bending moment becomes wider and relatively symmetric under ungassed status as show n in Fig.12(c).When the gas is introduced,the resonance is weakened and the distribution becomes asymmetric again,leading to the decline ofκ.

3.2.3.Frequency characteristics

There are mainly three groups of frequency contributing to the fl uctuation of shaft bending moment,w hich are the impeller speed frequency caused by therotation of stirring structures,the low frequencies caused by macro-instabilitiesin thebulk fl ow,and the high frequencies,such asblade passing frequency due to the pseudo-turbulence originatingfrom thetrailingvorticesbehind theimpeller bladesand theinteraction betw een the blades and baf fl es.Furthermore,the multiple natural frequenciesof thestirringstructuresmight bestimulated.Thebehaviors of these frequencies depend on the operating conditions and the properties of fl uid and structure.

The PSD of shaft bending moment for the four disk turbines under the same ratio of speed frequency to natural frequency is obtained by the Yule-Walker autoregressive model and is show n in Fig.14.In the PSD of the four disk turbines,there are tw o groups of obvious peaks,and the one is located at frequencies below speed frequency(marked as A),w hile the other one is located at speed frequency(marked as B).The peaks A and Bindicate the contribution of macro-instabilities in the bulk fl ow and the rotation of stirring structures to the fl uctuation of shaft bending moment,respectively.In the PSD of the higher curved blade disk turbines(HEDT and PDT),there is another group of peak located at 6-6.2 Hz(marked as C),w hich may result from the linear combination of the natural frequency,macro-instabilities frequency,speed frequency,and blade passing frequency.How ever,this peak cannot be found in the PSD of less curved blade disk turbines(RTand CD).

3.3.Combined moment

The impeller torque and shaft bending moment are mainly determined by the tangential fl uid force and the lateral force on the impeller,respectively.In order to effectively indicate the overall loads,the combined moment is de fi ned as

And the relative combined moment RMCthe ratio of the mean combined moment under gassed to the ungassed)for the four disk turbines as a function of FlGis show n in Fig.15.For RT,the RMCdecreases signi fi cantly as FlGincreases in general.For CD at 240 and 252 r·min-1,the RMCincreases until FlG～0.04 because of the both increase trends of the RMB(see Fig.10)and RPD(see Fig.7)and then keeps decreasing as FlGincreases.How ever,for CD at 216 r·min-1,the RMC decreases sharply at fi rst due to the system resonance under ungassed status.For HEDT,the RMCvariesslightly over FlGand is slightly less than 1.For PDT,the RMCvaries slightly over FlGand is around 1.Therefore,generally,the RMCincreases w ith the increase of the blade curvature except for CDunder resonance.

Fig.14.PSDof shaft bending moment normalized by the variance as a function of Fl G.

4.Conclusions

The torque and bending moment acting on a fl exible overhung shaft agitated by four disk turbines(RT,CD,HEDT,and PDT)in a gas-liquid stirred vessel w ere experimentally studied,and this w ork is an extension of our previous research that focused on RT[25]and pitched blade turbine[26].

The relative pow er demand keeps decreasing as the gas fl ow number increasesfor RT,HEDT,and PDT,but for CDit risesto slightly greater than 1 and then decreases.And the declinedegreeof power decreaseswith the increase of the blade curvature.The amplitude distribution of torque can be well fi tted by a symmetric bimodal distribution for the four disk turbines.And the distribution is concentrated and closes to a normal distribution for RTbut is more dispersive for the curved-blade disk turbines as the blade curvature increases.Generally,the distribution is more dispersive as gas fl ow number increases for the four disk turbines.

Generally,therelativeshaft bending moment for thefour disk turbines is a corporate contribution of the unstable fl uid motion,the gas cavities behind the blades,and the gas direct impact on the impeller,manifesting a “rising-falling-rising”trend over the gas fl ow number.And the falling stage is greater and lasts wider over the gas fl ow number for RTthan for the curved-blade disk turbines.The amplitude distribution of shaft bending moment can be w ell fi tted by an asymmetric Weibull distribution for the four disk turbines.Particularly,for CDat 216 and 228 r·min-1under ungassed status,theshaft bending moment ishigh and theamplitudedistribution is relatively symmetric because of the system resonance.The resonanceisweakened after theintroduction of gas,and thereforetherelative shaft bending moment decreases sharply over the gas fl ow number at fi rst and the amplitude distribution becomes asymmetric again.

The macro-instabilities w ith low frequency and stirring structure rotation w ith the speed frequency are the tw o key factors contributing for the fl uctuation of shaft bending moment.Another dominant frequency located at 6-6.2 Hz also exists for the higher curved blade disk turbines(HEDTand PDT)but disappears for the less curved blade disk turbines(RTand CD).

Therelativecombined moment,asacombination of torqueand shaft bending moment,increases w ith the increase of the blade curvature in general except for CDunder resonance.

Nomenclatures

C clearance of impeller off bottom of vessel,mm

c curvature factor of blade

D diameter of impeller,mm

D s diameter of sparger,mm

d diameter of shaft,mm

FlGgas fl ow number of impeller

Fr Froude number

f frequency,Hz

fnthe natural frequency of shaft-impeller,Hz

g gravity acceleration,m·s-2

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

N rotation speed frequency,s-1

n rotation speed,r·min-1

PDF probability distribution function

PSD pow er spectral density,N2·m2·s

QGgas fl ow rate,m3·h-1

Re Reynolds number

RMB relative shaft bending moment

RMC relative combined moment

RPD relative power demand

S sparger height off bottom of vessel,mm

T diameter of vessel,mm

Wbwidth of baf fl e,mm

x Weibull random variable

Γ gamma function

β dimensionless coef fi cient of shaft bending moment to torque

Fig.15.Relative combined moment RMCas a function of Fl G.

κ shape parameter of Weibull distribution

λ scale parameter of Weibull distribution,N·m

μ mean value,N·m

ν kinematic viscosity of liquid,m2·s-1

ρ density of liquid,kg·m-3

σ2variance,N2·m2

Subscripts

b bending

G gas

L liquid

t torsion

Superscript

— time-averaged