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Effect of gas flow on the bending moment acting on a shaft in a sparged vessel stirred by a Pitched Blade Turbine
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Effect of gas flow on the bending moment acting on a shaft in a sparged vessel stirred by a Pitched Blade Turbine Daien Shi a , Ziqi Cai a , Yangyang Liang a , Archie Eaglesham b , Zhengming Gao a,∗ a b
School of Chemical Engineering, Beijing University of Chemical Technology, Beijing 100029, China Huntsman Polyurethanes, B3078 Everberg, Belgium
a b s t r a c t The bending moment acting on the overhung shaft of a gas-sparged vessel stirred by a Pitched Blade Turbine, as one of the results of Fluid–Structure Interactions (FSI) in stirred vessels, was measured using a moment sensor equipped with digital telemetry. The amplitude and Power Spectral Density of the shaft bending moment were analyzed. It shows that the gas flow has a considerable influence on the characteristics of the bending moment, such as the amplitude mean, distribution, Standard Deviation and peak, and the low-frequency and speed frequency contributions to the fluctuation. The relative mean bending moment initially increases with gas rate till the transition from complete dispersion to loading regimes, approaching a peak, then decreases to a valley and again rises gradually, going through the transition from loading to flooding regimes. The “S” trend of the relative mean bending moment over gas flow rate, depending on the flow regime in gas–liquid stirred vessels, 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. © 2014 The Institution of Chemical Engineers. Published by Elsevier B.V. All rights reserved.
Keywords: Bending moment; Gas–liquid flow; Gas sparged stirred vessel; Fluid–structure interaction
1.
Introduction
Impeller stirred vessels play an important role in many chemical processes, enhancing chemical 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 other desired process results. Thus, for a typical well balanced impeller system where the center of gravity of the impeller and shaft is perfectly aligned with the axis of rotation and this in turn is perfectly aligned with the vessel centerline, the fluid motion produced by the impeller is not, in general, symmetric in the spatial structures including primary circulation loops, liquid swell on free liquid surface (Bruha et al., 2011) and trailing vortices behind the impeller blades (Escudie et al., 2004).
The flow in such systems is also unsteady due to lowfrequency macro-instabilities (Hasal et al., 2004; Montes et al., 1997; Roussinova and Kresta, 2004), blade passing frequency pseudo-turbulence (Vantriet et al., 1976) and high-frequency turbulent motions (Liu et al., 2008). These asymmetric, unsteady fluid motions exert an imbalanced and unsteady load (Kratena et al., 2001; Weetman and Gigas, 2002) on the impeller and lead to instantaneous deflections of the shaft and impeller. The resulting lateral movements of the impeller and shaft in turn induce further unstable and nonuniform flows around the impeller. In fact, this behavior in stirred vessels is an example of the bidirectional interactions of a flowing fluid with a flexible structure (Dowell and Hall, 2001). Various features of the flow in stirred vessels can further intensify the effects of FSI, such as instabilities in the liquid free surface
∗
Corresponding author at: Mailbox 230, School of Chemical Engineering, Beijing University of Chemical Technology, North 3rd Ring Road, Chaoyang District, Beijing No. 15, Beijing 100029, China. Tel.: +86 10 6441 8267; fax: +86 10 6444 9862. E-mail address: [email protected] (Z. Gao). http://dx.doi.org/10.1016/j.cherd.2014.03.006 0263-8762/© 2014 The Institution of Chemical Engineers. Published by Elsevier B.V. All rights reserved.
Please cite this article in press as: Shi, D., et al., Effect of gas flow on the bending moment acting on a shaft in a sparged vessel stirred by a Pitched Blade Turbine. Chem. Eng. Res. Des. (2014), http://dx.doi.org/10.1016/j.cherd.2014.03.006
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Nomenclature C d D Ds fn fr FlG Fr H L mimp mu Mb Mc Mt n N PSD QG ru S SS SP T Ub Wb x
clearance of impeller off bottom of vessel (m) diameter of shaft (m) diameter of impeller (m) diameter of sparger (m) the first order laterally natural frequency (s−1 ) the first order laterally resonant frequency (s−1 ) gas flow number of impeller Froude number of impeller height of liquid free surface in stirred vessel (m) overhung length of the overhung shaft (m) mass of impeller (kg) part unbalanced mass of impeller (kg) bending moment acting on the overhung shaft (N m) combined moment acting on the overhung shaft (N m) torque acting on the overhung shaft (N m) operational speed (rot min−1 ) operational speed frequency (rot s−1 ) Power Spectral Density (N2 m2 s) gas flow rate (m3 h−1 ) distance of part unbalanced mass off geometrical center (m) sparger height off bottom of vessel (m) sampling scale (sampling number) spectral power (N2 m2 ) diameter of vessel (m) unbalance of impeller (g mm) width of baffle (m) Weilbull random variable
Greek letters Mean (N m) Standard Deviation (N m) shaft bending moment coefficient ˇ Variance (N2 m2 ) 2 scale parameter of Weibull distribution (N m) shape parameter of Weibull distribution distance between centers of gravity and geomı etry (m) Gamma function Subscripts G gas liquid L b bending torsion t superscript — time-averaged
or the non-uniformities associated with multiphase flow, as can mechanical design features such as any imbalance of the shaft and impeller combination caused by manufacturing tolerances and the natural frequency of stirring structures in the lateral direction. In stirred vessel operation, the function of the shaft is to transmit power from the drive train to the impeller and the resulting torque must be borne by the shaft. Meanwhile, an
Fig. 1 – Sketch of the laboratory scale vessel used in the experiment.
unsteady bending moment is also exerted on the shaft due to the lateral deflection of the shaft and movement of the impeller produced by the complex FSI in the stirred vessel. This results in a dynamic load on the stirred vessel head supporting the shaft. In the mechanical design of mixing equipment, the underestimation of the shaft bending moment leads to plastic deformation and fatigue failure of shafts and vessels. However, it is difficult to theoretically determine the bending moment because of the complex nature of the fluid dynamics, the nonlinear structural dynamics and the coupling dynamics of the FSI in stirred tanks. When gas is sparged into stirred vessels, the 2-phase nature of flow can increase both asymmetries and unsteadiness in the flow field compared with the single phase situation, which further increases the shaft bending moment by virtue of intensifying the FSI. To date, little research on the lateral loads acting on gas sparged stirred vessels has been published still. In this paper, we report an experimental study examining the impact of gas flow on the shaft bending moment in a Pitched Blade Turbine stirred vessel. The main purpose is to provide quantitative data on the magnitude of the loads on gas–liquid stirred vessels for the mechanical design of these systems. The results derived from this study include the mean, Standard Deviation and peak of the shaft bending moment for different gas rates covering complete dispersion, loading and flooding regimes in sparged stirred tanks. These are respectively applied to the tensile strength check, fatigue failure analysis and yield strength check in the mechanical design. Moreover, the bending-torsion combined moment as a function of gas flow number is also presented.
2.
Model and method
2.1.
Experimental model
The laboratory scale vessel used in the experiment is shown in Fig. 1 and the key details are given in Table 1. Geometrical details of the modified PBT are shown in Fig. 2.
Please cite this article in press as: Shi, D., et al., Effect of gas flow on the bending moment acting on a shaft in a sparged vessel stirred by a Pitched Blade Turbine. Chem. Eng. Res. Des. (2014), http://dx.doi.org/10.1016/j.cherd.2014.03.006
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Table 1 – Detailed information of the stirred vessel. Stirring structure
Stirred vessel
Components
Shaft
Impeller
Fluid
Air/water mixture
Type Key dimensions (mm) Material Imbalance fn and fr Speed
Overhung d = 20;L = 1000 SS 304
Pitched Blade Turbine D = 232; C = 194 SS 304 1633 g mm
Boundaries Key dimensions (mm)
Tank with four baffles and a sparger H = 800 T = 580; Wb = 58 Ds = 155; S = 145 1–10 m3 /h
Gas rate
8.6 Hz and 7 Hz 192–240 RPM
Fig. 2 – Sketch and physical map of the modified Pitched Blade Turbine.
In Table 1, the imbalance of the impeller was estimated by Eq. (1), and the distance between the centers of gravity and geometry was given by Eq. (2), the ratio of which to the diameter of the impeller is confined in 1%. Ub = ı=
(mu ru )
(1)
Ub mimp
(2)
In Table 1, fn and fr are the resonant and natural frequency of the shaft and impeller structure, respectively, which are both estimated by using vibration system with one degree of freedom (Berger et al., 2003; Harnby et al., 1992). 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/nc < 0.6, thus avoiding any resonance issues. The liquid surface was sufficiently high to ensure that it had a negligible influence on the lateral FSI. The gas sparging rates were chosen so that the 2-phase flow regime ranged from complete dispersion through loading to flooding at each speed. Fig. 3 shows the range of the measured points (operational conditions) in the context of a 2-phase flow regime map with two dividing lines marked by the letters A and C obtained by experimental observation. The lines indicate two transitions between complete dispersion 0.45 Complete dispersion
A
C Flooding
Loading
Fr (N2 D/g)
0.4 0.35 0.3 0.25 0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
Fl (Q /(N3D )) G
G
Fig. 3 – Range of the measured points in 2-phase flow regime.
0.08
and loading, and between loading and flooding regimes, respectively. Fig. 4 shows the images of gas distribution at 216 RPM for complete dispersion, loading and the transition state between loading and flooding regimes. Referencing those defined by Harnby et al. (1992), complete dispersion and the transition state between loading and flooding regimes are defined as completely dispersing bubbles at the tank bottom and horizontally sparging gas along the sparger, respectively, while loading regime is considered as a tapered distribution between above two states as showed in Fig. 4c.
2.2.
Experimental method
2.2.1.
Experimental setup
In these 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 experimental setup are shown in Fig. 5. The drive unit consists of a motor, drive shaft and a flexible coupling. The supporting structure employs two sets of ball bearings to support the drive shaft, which makes the drive shaft end a clamped point for the shaft and impeller. 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. The sparging system included an air pressure stabilizer for the air source, a regulator valve to control the gas rate, a flowmeter to measure the gas rate, a pressure gauge to measure gas pressure of the air inlet and a cut-off valve. The moment measurement system consists of a moment sensor (Fig. 6) and a data acquisition unit. The specification and main 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. Communications between the rotor fixed at overhung point of the shaft and the stator employ digital telemetry technology.
Please cite this article in press as: Shi, D., et al., Effect of gas flow on the bending moment acting on a shaft in a sparged vessel stirred by a Pitched Blade Turbine. Chem. Eng. Res. Des. (2014), http://dx.doi.org/10.1016/j.cherd.2014.03.006
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Fig. 4 – Gas distributions in the stirred tank (n = 216RPM).
Table 2 – Specification and main parameters of the moment sensor. Moment sensor (HBM, Germany) Type
Customized
Communication between rotor and stator
Digital telemetry
Parameters
Unit
Index
Rated bending moment X Rated bending moment Y Rated torque Linearity deviation Sampling frequency Torsional stiffness Bending stiffness
Nm Nm Nm % kHz kN m/rad kN m/rad
20 20 10 <1 <6.7 62.21 77.91
2.2.2.
Fig. 5 – Experimental setup.
Fig. 6 – Moment sensor configuration.
Data acquisition
The signals for bending moment and torque were synchronized and simultaneously sampled. The data sampling frequency and size can influence the experimental results. Therefore, it is important to select the appropriate sampling frequency and size. The Power Spectral Density (PSD) of the bending moment measured by using a sampling frequency of 1200 Hz showed that approximately 99% of the Spectral Power (SP) is contained in frequencies below 100 Hz (Fig. 7). Therefore, a sampling frequency of 600 Hz was chosen to ensure sufficient resolution at low frequencies, where most of the loading was expected, whilst also ensuring that no significant high-frequency loads were omitted. Fig. 8 shows the percentage deviation of the mean bending moment as a function of time. For sampling times beyond 150 s or so, the percentage deviation is around ±3% (i.e. the sampling time has a negligible effect on the results after 150 s). Therefore, a 300 s sampling time was chosen in these
Fig. 7 – Power Spectral Density (PSD) normalized by the variance. (Speed frequency (N) = 3.6 Hz; sampling scale (SS ) = 217 ).
Please cite this article in press as: Shi, D., et al., Effect of gas flow on the bending moment acting on a shaft in a sparged vessel stirred by a Pitched Blade Turbine. Chem. Eng. Res. Des. (2014), http://dx.doi.org/10.1016/j.cherd.2014.03.006
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3 Mb / Mb
2 1 0
0
0.2
0.4
0.6
0.8
1
1.2
1.4
time(s)
Fig. 10 – Instantaneous Bending Moment normalized by the time-averaged value. (n = 216 RPM; QG = 5.42 m3 /h).
Fig. 8 – Percentage deviation of the average bending moment from the mean as a function of sampling time (speed (n) = 216 RPM; sampling frequency = 600 Hz). experiments, corresponding to a sampling size 180,000 (300 s times 600 Hz).
3.
Results and discussion
3.1.
Impeller torque
The impeller mean torque (Mt ) is an important factor in estimating the bending moment acting on the shaft in stirred tanks, which reflects the magnitude of circumferential fluid forces on the impeller. Fig. 9 shows a ratio of gassed to ungassed mean torque as a function of flow number at five impeller speeds, being in accordance with the two transitions are marked by the letters A and C shown in Fig. 3. For each speed the ratio firstly decreases slightly to the line A, then descends sharply to the line marked by the letter B, and finally increases slightly through the line C with the rise of gas flow number. The ratio trend over FlG is consistent with that by Chapman et al. (1983), where the sharp decrease in the ratio is due to the formation of large cavities behind the impeller blades. In such decrease, the loading regime between A and B is considerable unstable by means of rather low-frequency, which was also observed by Chapman et al. (1983). The reason could be that the competition between two different ways of loading gas causes a macro-instability of the bulk flow. According to the works by Warmoeskerken et al. (1984), the ways of PBT loading gas are divided into indirect and direct loadings. In the former way, the gas below the downward PBT cannot enter upward the impeller but is discharged down when the discharging flow capacity of the impeller is beyond the gas buoyancy, while in the latter the gas firstly enters the impeller and is discharged down since the discharging flow capacity of the impeller is below the gas buoyancy. Therefore, in complete dispersion the way of PBT loading gas should be considered as direct loading while in the loading regime between B and C as indirect loading because of high gas flow. In the loading regime between A and B, the loading way can be the rather slow shift of two loading way, and causes a macro-stability of the bulk flow.
Shaft bending moment
Fig. 10 shows the shaft bending moment normalized by its average value for the gas flow of 5.42 m3 /h at 216 RPM. 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.
3.2.1. Effect of gas flow on amplitude characteristics 3.2.1.1. Amplitude mean. Fig. 11 shows the relative mean bending moment (Mb )G /(Mt )L (here it is defined as the bending moment ratio between gassed and ungassed conditions) as a function of FlG . Being the same as those in Fig. 9, in Fig. 11 the lines A and C indicate the two transitions, and the line B is the critical location for stable and unstable loadings in loading regime. Initially the relative mean bending moment increases with the rise of FlG until the flow regime varies from complete dispersion to loading, and then goes through a peak. When the gas flow keeps increasing, the mean bending moment drops to a valley and again starts to rise gradually, going through the transition from loading to flooding regimes. The “S” trend of the mean bending moment over FlG for each speed is obviously similar. The phenomenon shown in Fig. 11 demonstrates that the magnitude of the unbalanced fluid force acting on the impeller increases with the rise of the gas flow confined 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 non-synchronously sharp fluctuation of fluid pressures on the impeller blades surfaces in every moment. However, the unbalanced fluid force also decreases when the gas distribution in the vessel move on to the loading regime. This is 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
1.5
A
B
C
n=192 (RPM); Fr=0.242 n=204 (RPM); Fr=0.273
0.9 0.8
n=216 (RPM); Fr=0.306 n=228(RPM); Fr=0.341
A
B
C
0.7
n=240 (RPM); Fr=0.378
(Mb)G/(Mb)L
1.4
1
( Mt )G/( Mt )L
3.2.
1.3 1.2 n=192(RPM) n=204(RPM)
1.1
0.6 0.5
n=216(RPM) n=228(RPM)
1
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
Fl (Q /(N3D )) G
G
Fig. 9 – Ratio of gassed to ungassed mean torque as a function of FlG .
0.08
n=240(RPM)
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
Fl (Q /(N3D)) G
G
Fig. 11 – Relative (gassed/ungassed) mean bending moments as a function of FlG .
Please cite this article in press as: Shi, D., et al., Effect of gas flow on the bending moment acting on a shaft in a sparged vessel stirred by a Pitched Blade Turbine. Chem. Eng. Res. Des. (2014), http://dx.doi.org/10.1016/j.cherd.2014.03.006
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=1.9042; =2.0464;
C
=0.512
2.2
G
/ L
A
B
1.6
n=192(RPM) n=204(RPM) n=216(RPM)
1.3 1
n=228(RPM) n=240(RPM)
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
10000 5000
0.08
0
Fl (Q /(N3D)) G
exp. weibull
15000
sampling number
1.9
G
0
1
Fig. 12 – Relative (gassed/ungassed) ˇ as a function of FlG .
2 M b / Mb
3
Fig. 13 – Typical measured amplitude distributions for shaft bending moment. (QG = 7.31 m3 /h; n = 216 RPM; SS = 180,000). in the fluid forces, resulting from the decrease in pressure difference between the front and back of blades, 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 complete dispersion regime, while the cavitation behind the blades dominates in loading regime. Consequently, the mean bending moment shows one fluctuation as the gas distribution in the vessel goes from complete dispersion to loading regime. When the gas rate gets higher, the gas flow begins to impact the impeller unsteadily, where loading gas way has shifted to indirect 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. As the gas rate continues to increase and the gas–liquid flow transits to the flooding regime, the gas flow begins to completely dominate the fluid motion around the impeller and directly impacts the impeller, which leads to the bending moment increasing monotonously in flooding regime. In order to eliminate the impeller circumferential (tangential) fluid force influence and only examine the impeller lateral (bending) forces variation with FlG , a dimensionless shaft bending moment ˇ is defined by Eq. (3). In the equation, Mb /L (L is the shaft overhung length) represents the lateral forces contributing to the bending moment, and Mb /D (D is the impeller diameter) represents the circumferential fluid force contributing to torque. ˇ=
Mb /L Mt /D
(3)
Fig. 12 shows the relative ˇ plotted as a function of FlG , where the two transitions A and C and the critical location B are also marked by the dashed lines. Apparently, the relative ˇ increases slightly with FlG in complete dispersion regime, and then sharply rises in loading regime. Such trend of the relative ˇ further demonstrates that the asymmetry of the fluid motion around the impeller, causing the unequal fluid force on each blade, slightly increases because of the increasingly nonuniform bubbles around blades in complete dispersion regime, and sharply rises due to the increasingly asymmetrical cavities behind the blades in loading regime. When the gas rate continues to increase, the rising of the relative ˇ becomes to slow down from the beginning of the line B in loading regime and keeps in flooding regime. Such slowing down of the relative ˇ reflects that
the unsteady impact of gas flow on the impeller, resulting in an up-and-down vibration of the impeller and shaft due to the shaft material elasticity and the gas beating up the impeller, causes the moderate increase in the bending moment.
3.2.1.2. Amplitude distribution. The shaft bending moment is a considerably unsteady load, whose mean is mainly used to explain some phenomena, such as fluid flow instabilities and structure vibrations, or served as a basic reference for fluctuating load. However, for mechanical design, the fluctuating characteristics of the bending moment, such as Standard Deviation and peak value, are significant for the material fatigue analysis and yield strength check. Apparently, amplitude of bending moment is characterized by polydispersity. Therefore, amplitude distribution is firstly checked by means of a statistical method, whose purposes are to quantify the polydispersity and extract the fluctuating characteristics with any probability. Fig. 13 shows the typical shaft bending moment amplitude distribution which is well fitted by a Weibull distribution. Apparently, the distribution is not symmetry about the amplitude mean. It is due to the shaft deflections’ asymmetry around the dynamic equilibrium position, being caused by the nature of shaft against bending deformation. This nature is the elastic reaction pulling the shaft back toward the axis of rotation (static equilibrium position) after the shaft bending deformation. The Probability Density Function (PDF) for Weibull distribution (Papoulis and Pillai, 2002) is given by:
PDF(x; , ) =
⎧ ⎪ ⎨ x −1 ⎪ ⎩
x
−
e
x≥0
(4)
x≤0
0
where x is a Weilbull random variable, (>0) and (>0) are the scaling and shape parameters respectively. The mean () and variance ( 2 ) are given by Eqs. (5) and (6), respectively.
=
2 = 2
1+
1
1+
(5)
2
− 2
(6)
Please cite this article in press as: Shi, D., et al., Effect of gas flow on the bending moment acting on a shaft in a sparged vessel stirred by a Pitched Blade Turbine. Chem. Eng. Res. Des. (2014), http://dx.doi.org/10.1016/j.cherd.2014.03.006
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/ Mb)G / ( / Mb)L
or if the amplitude fluctuating location x is known, the probability CDF below the x may be solved.
n=192(RPM)
1.06
n=204(RPM) n=216(RPM) n=228(RPM)
1.04
3.2.1.3. Amplitude Standard Deviation. A ratio of Standard Deviation to mean represents intensity of amplitude fluctuation, which may be served as the basic information for material fatigue analysis. The relative ratio (gassed/ungassed) shown in Fig. 14, whose maximum approaches to 1.065, slightly changes with gas flow number. Its trend over FlG is similar with that of relative bending moment shown in Fig. 11. According to Eq. (7), when x = Mb + , the corresponding probability below Mb + approaches 84% as shown in Fig. 15, which means that only 16% amplitudes are greater than Mb + .
n=240(RPM)
1.02 1 0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
Fl (Q /(N3D)) G
G
Probability (%)
Fig. 14 – Relative (gassed/ungassed) /Mb as a function of FlG . 83.95
n=192(RPM)
83.9
n=216(RPM) n=228(RPM)
83.85
n=240(RPM)
n=204(RPM)
3.2.1.4. Amplitude peak. A ratio of peak fluctuation to Standard Deviation ((peak − Mb )/) represents the dimensionless peak value, by means of which can easily solve the peak served as the basic information for material yielding check. Because the peak sampled is too unstable, the peak is defined by a 99% probability, which means the CDF is 0.99 according to Eq. (7). Thus the corresponding amplitude peak can be calculated. Fig. 16 shows the relative ratio (gassed/ungassed) for each speed, whose trend over FlG is the same as the previous Figs. 11 and 14 essentially. The ratio maximally approaches to 1.025, but slightly changes with gas flow number.
83.8 83.75 0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
Fl (Q /(N3D)) G
G
((peak-Mb)/ )G/((peak-Mb)/ )L
Fig. 15 – Probability of the amplitudes below Mb + as a function of FlG . 1.025
n=192(RPM) n=204(RPM)
1.02
3.2.2.
n=216(RPM) n=228(RPM)
1.015
n=240(RPM)
1.01 1.005 1 0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
FlG(QG/(N3D))
Fig. 16 – Relative (gassed/ungassed) (peak − Mb )/ as a function of FlG . The Cumulative Distribution Function (CDF) can be obtained by integrating the Weibull probability density function:
CDF(x; , ) =
⎧ ⎪ ⎨ ⎪ ⎩
x
−
1−e
x≥0
(7)
x≤0
0
Effect 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 (Hasal et al., 2004; Montes et al., 1997) 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 (Vantriet et al., 1976) and interactions between the impeller blades and between the blades and baffles (Kratena et al., 2001), and from the natural elastic vibration of stirring structures. However, the particular frequencies are not all evident simultaneously, depending on the specific condition which can excite the corresponding frequency. Fig. 17 shows the PSD of the bending moment for 216 RPM from 0 to 5 Hz at different gas rates, which is obtained by using the Yule-Walker autoregressive model. In all cases, two groups of the peaks marked by the letters D and E are evident which are influenced by gas flow. Group D locates at rather low
According to Eq. (7), if the probability CDF is known, the amplitude fluctuating location x with the CDF may be solved,
PSD/ 2 [ 1/Hz ]
2
D
1.5
E
1
0.5
0 10.5 9.8 9.2 8.6
7.97.3 6.7 6.1
5.44.8 4.2 3.5 2.9 2.3 1.7
gas flow [m3/h]
1
0
0
N/5
1
2
3
N-0.5
4
N+0.5
frequency [ Hz ]
Fig. 17 – PSD normalized by the variance (N = 3.6 Hz (216 RPM); SS = 217 ). Please cite this article in press as: Shi, D., et al., Effect of gas flow on the bending moment acting on a shaft in a sparged vessel stirred by a Pitched Blade Turbine. Chem. Eng. Res. Des. (2014), http://dx.doi.org/10.1016/j.cherd.2014.03.006
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n=192(RPM)
3
n=216(RPM) n=228(RPM)
2.5 2
n=240(RPM)
1.5 1 0.5
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
C
1 n=192(RPM)
0.9
n=204(RPM) n=216(RPM) n=228(RPM)
0.8 0.7
Fl (Q /(N3D)) G
A
1.1
n=204(RPM)
( Mc)G/( Mc )L
(SP/ 2)g/(SP/ 2)l
3.5
B
n=240(RPM)
0
0.01
0.02
G
0.03
0.04 G
Fig. 18 – Relative (gassed/ungassed) SP/ 2 from 0 to N/5. frequency, which should result from low-frequency macroinstability of bulk flow in stirred tanks. And group E locates at the speed frequency (3.6 Hz), which is caused by the impeller imbalance. A ratio of Spectral Power in a range of frequencies to the variance is used to represents a resultant contribution of these frequencies to the fluctuation of bending moment. The relative ratio (gassed/ungassed) in the low-frequency range (0-N/5) is shown in Fig. 18. Initially, it increases with FlG , peaks at FlG ∼ 0.02 and then decreases with the increasing gas rate. It is noted that the location of the maximum ratio just occurs in the considerable unstable regime between A and B shown in Fig. 9. This demonstrates that the instability observed firstly by Chapman et al. (1983) is reflected in shaft bending moment fluctuation and the low frequency should be the shifting frequency between the two loading ways: indirect and direct loading presented by Warmoeskerken et al. (1984). The relative (gassed/ungassed) ratio in the speed frequency range (N ± 0.5) is shown in Fig. 19. Initially, it slightly increases in the range of low gas rates and then decreases with FlG , which demonstrates that the contribution to the total fluctuation of shaft bending moment declines as gas flow increases because of the dominant gas-fluid flow.
3.3.
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 defined as:
Mc =
Mb2 + Mt2
(8)
The relative combined moment (gassed/ungassed) for different speeds is shown in Fig. 20. At first, it increases slightly to one peak, approaching the transition between complete dispersion and loading regimes, then decreases rapidly, and finally increases slightly with the rise of FlG . This information on the combined moment could reflect more overall the
n=192(RPM)
1.2 (SP/ 2)g/(SP/ 2)l
0.05
0.06
0.07
0.08
Fl (Q /(N3D))
n=204(RPM) n=216(RPM)
1
n=228(RPM) n=240(RPM)
0.8 0.6
G
Fig. 20 – Relative (gassed/ungassed) combined moment as a function of FlG . characteristics of gas–liquid flow loading on the impeller, compared with those on individual torque and bending moment.
4.
Conclusion
This paper investigated the impact of gas sparged on the shaft bending moment in a Pitched Blade Turbine 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 were summarized as follows: (1) The trend of the relative mean bending moment over FlG presents 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 slightly increases with FlG in complete dispersion regime, approaching to 1.4 at the transition between complete dispersion and loading regimes, then ascends sharply in loading regime and almost levels out for higher gas rates, reaching about 2.2 in flooding regime. (3) The distribution of the bending moment amplitude is well described by Weibull distribution, which reflects that the elastic reaction pulls the shaft back toward 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 peak to about 1.065 and 1.025 respectively, whose changes with gas rate are similar with that of the relative mean bending moment. (5) The low-frequency contribution to bending moment fluctuation peaks in an unstable loading regime, which is caused by the low-frequency shift of gas indirect and direct loading ways. The speed frequency contribution gradually becomes weak as gas flow increases because of the dominant gas-fluid flow. (6) The relative mean combined moment firstly increases slightly, then decreases rapidly, and finally increases slightly with FlG , which overall reflects the characteristics of gas–liquid flow loading on the impeller.
Acknowledgments 0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
FlG(QG/(N3D))
Fig. 19 – Relative (gassed/ungassed) SP/ 2 from N − 0.5 to N + 0.5.
The financial support from the “A Computational and Experimental Study of Fluid Structure Interactions in Stirred Vessels” PhD Studentship provided by Huntsman Europe (Belgium) is gratefully acknowledged.
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References Berger, T., Fischer, M., Strohmeier, K., 2003. Fluid–structure interaction of stirrers in mixing vessels. Trans. ASME 125, 440–445. Bruha, T., Smolka, P., Jahoda, M., Fort, I., 2011. Dynamics of flow macro-formation and its interference with liquid surface in mixing vessel with pitched blade impeller. Chem. Eng. Res. Des. 89 (11A), 2279–2290. Chapman, C.M., Nienow, A.W., Cooke, M., Middleton, J.C., 1983. Particle gas–liquid mixing in stirred vessels. 2. Gas–liquid mixing. Chem. Eng. Res. Des. 61, 82–95. Dowell, E.H., Hall, K.C., 2001. Modeling of fluid–structure interaction. Annu. Rev. Fluid Mech. 33, 445–490. Escudie, R., Bouyer, D., Line, A., 2004. Characterization of trailing vortices generated by a Rushton turbine. AIChE J. 50, 75–86. Harnby, N., Edwards, M.F., Nienow, A.W., 1992. Mixing in the Process Industries, 260/261., 2nd ed. Butterworth-Heinemann, Oxford, pp. 337. Hasal, P., Fort, I., Kratena, J., 2004. Force effects of the macro-instability of flow pattern on radial baffles in a stirred vessel with pitched-blade and Rushton turbine impellers. Chem. Eng. Res. Des. 82 (A9), 1268–1281.
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Kratena, J., Fort, I., Bruha, O., Pavel, J., 2001. Distribution of dynamic pressure along a radial baffle in an agitated system with standard Rushton turbine impeller. Trans IChemE 79, 819–823. Liu, X.H., Bao, Y.H., Li, Z.P., Gao, Z.M., Smith, J.M., 2008. Particle image velocimetry study of turbulence characteristics in a vessel agitated by a dual Rushton impeller. Chin. J. Chem. Eng. 16, 700–708. Montes, J.L., Boisson, H.C., Fort, I., Jahoda, M., 1997. Velocity field macro-instabilities in an axially agitated mixing vessel. Chem. Eng. J. 67 (2), 139–145. Papoulis, A., Pillai, S.U., 2002. Probability, Random Variables, and Stochastic Processes, 4th ed. McGraw-Hill, Europe, pp. 89–90. Roussinova, V.T., Kresta, S.M., 2004. Resonant geometries for circulation pattern macro-instabilities in a stirred tank. AIChE J. 50, 2986–3005. Vantriet, K., Bruijn, W., Smith, J.M., 1976. Real and pseudo-turbulence in the discharge stream from a Rushton turbine. Chem. Eng. Sci. 31, 407-112. Warmoeskerken, M.M.C.G., Speur, J., Smith, J.M., 1984. Gas–liquid dispersion with pitched blade turbines. Chem. Eng. Commun. 25, 11–29. Weetman, R.J., Gigas, B., 2002. Mixer mechanical design – fluid forces. In: Proceedings of the 19th International Pump Users Symposium, pp. 203–213.
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Contents lists available at ScienceDirect
Chemical Engineering Research and Design journal homepage: www.elsevier.com/locate/cherd
Effect of gas flow on the bending moment acting on a shaft in a sparged vessel stirred by a Pitched Blade Turbine Daien Shi a , Ziqi Cai a , Yangyang Liang a , Archie Eaglesham b , Zhengming Gao a,∗ a b
School of Chemical Engineering, Beijing University of Chemical Technology, Beijing 100029, China Huntsman Polyurethanes, B3078 Everberg, Belgium
a b s t r a c t The bending moment acting on the overhung shaft of a gas-sparged vessel stirred by a Pitched Blade Turbine, as one of the results of Fluid–Structure Interactions (FSI) in stirred vessels, was measured using a moment sensor equipped with digital telemetry. The amplitude and Power Spectral Density of the shaft bending moment were analyzed. It shows that the gas flow has a considerable influence on the characteristics of the bending moment, such as the amplitude mean, distribution, Standard Deviation and peak, and the low-frequency and speed frequency contributions to the fluctuation. The relative mean bending moment initially increases with gas rate till the transition from complete dispersion to loading regimes, approaching a peak, then decreases to a valley and again rises gradually, going through the transition from loading to flooding regimes. The “S” trend of the relative mean bending moment over gas flow rate, depending on the flow regime in gas–liquid stirred vessels, 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. © 2014 The Institution of Chemical Engineers. Published by Elsevier B.V. All rights reserved.
Keywords: Bending moment; Gas–liquid flow; Gas sparged stirred vessel; Fluid–structure interaction
1.
Introduction
Impeller stirred vessels play an important role in many chemical processes, enhancing chemical 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 other desired process results. Thus, for a typical well balanced impeller system where the center of gravity of the impeller and shaft is perfectly aligned with the axis of rotation and this in turn is perfectly aligned with the vessel centerline, the fluid motion produced by the impeller is not, in general, symmetric in the spatial structures including primary circulation loops, liquid swell on free liquid surface (Bruha et al., 2011) and trailing vortices behind the impeller blades (Escudie et al., 2004).
The flow in such systems is also unsteady due to lowfrequency macro-instabilities (Hasal et al., 2004; Montes et al., 1997; Roussinova and Kresta, 2004), blade passing frequency pseudo-turbulence (Vantriet et al., 1976) and high-frequency turbulent motions (Liu et al., 2008). These asymmetric, unsteady fluid motions exert an imbalanced and unsteady load (Kratena et al., 2001; Weetman and Gigas, 2002) on the impeller and lead to instantaneous deflections of the shaft and impeller. The resulting lateral movements of the impeller and shaft in turn induce further unstable and nonuniform flows around the impeller. In fact, this behavior in stirred vessels is an example of the bidirectional interactions of a flowing fluid with a flexible structure (Dowell and Hall, 2001). Various features of the flow in stirred vessels can further intensify the effects of FSI, such as instabilities in the liquid free surface
∗
Corresponding author at: Mailbox 230, School of Chemical Engineering, Beijing University of Chemical Technology, North 3rd Ring Road, Chaoyang District, Beijing No. 15, Beijing 100029, China. Tel.: +86 10 6441 8267; fax: +86 10 6444 9862. E-mail address: [email protected] (Z. Gao). http://dx.doi.org/10.1016/j.cherd.2014.03.006 0263-8762/© 2014 The Institution of Chemical Engineers. Published by Elsevier B.V. All rights reserved.
Please cite this article in press as: Shi, D., et al., Effect of gas flow on the bending moment acting on a shaft in a sparged vessel stirred by a Pitched Blade Turbine. Chem. Eng. Res. Des. (2014), http://dx.doi.org/10.1016/j.cherd.2014.03.006
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Nomenclature C d D Ds fn fr FlG Fr H L mimp mu Mb Mc Mt n N PSD QG ru S SS SP T Ub Wb x
clearance of impeller off bottom of vessel (m) diameter of shaft (m) diameter of impeller (m) diameter of sparger (m) the first order laterally natural frequency (s−1 ) the first order laterally resonant frequency (s−1 ) gas flow number of impeller Froude number of impeller height of liquid free surface in stirred vessel (m) overhung length of the overhung shaft (m) mass of impeller (kg) part unbalanced mass of impeller (kg) bending moment acting on the overhung shaft (N m) combined moment acting on the overhung shaft (N m) torque acting on the overhung shaft (N m) operational speed (rot min−1 ) operational speed frequency (rot s−1 ) Power Spectral Density (N2 m2 s) gas flow rate (m3 h−1 ) distance of part unbalanced mass off geometrical center (m) sparger height off bottom of vessel (m) sampling scale (sampling number) spectral power (N2 m2 ) diameter of vessel (m) unbalance of impeller (g mm) width of baffle (m) Weilbull random variable
Greek letters Mean (N m) Standard Deviation (N m) shaft bending moment coefficient ˇ Variance (N2 m2 ) 2 scale parameter of Weibull distribution (N m) shape parameter of Weibull distribution distance between centers of gravity and geomı etry (m) Gamma function Subscripts G gas liquid L b bending torsion t superscript — time-averaged
or the non-uniformities associated with multiphase flow, as can mechanical design features such as any imbalance of the shaft and impeller combination caused by manufacturing tolerances and the natural frequency of stirring structures in the lateral direction. In stirred vessel operation, the function of the shaft is to transmit power from the drive train to the impeller and the resulting torque must be borne by the shaft. Meanwhile, an
Fig. 1 – Sketch of the laboratory scale vessel used in the experiment.
unsteady bending moment is also exerted on the shaft due to the lateral deflection of the shaft and movement of the impeller produced by the complex FSI in the stirred vessel. This results in a dynamic load on the stirred vessel head supporting the shaft. In the mechanical design of mixing equipment, the underestimation of the shaft bending moment leads to plastic deformation and fatigue failure of shafts and vessels. However, it is difficult to theoretically determine the bending moment because of the complex nature of the fluid dynamics, the nonlinear structural dynamics and the coupling dynamics of the FSI in stirred tanks. When gas is sparged into stirred vessels, the 2-phase nature of flow can increase both asymmetries and unsteadiness in the flow field compared with the single phase situation, which further increases the shaft bending moment by virtue of intensifying the FSI. To date, little research on the lateral loads acting on gas sparged stirred vessels has been published still. In this paper, we report an experimental study examining the impact of gas flow on the shaft bending moment in a Pitched Blade Turbine stirred vessel. The main purpose is to provide quantitative data on the magnitude of the loads on gas–liquid stirred vessels for the mechanical design of these systems. The results derived from this study include the mean, Standard Deviation and peak of the shaft bending moment for different gas rates covering complete dispersion, loading and flooding regimes in sparged stirred tanks. These are respectively applied to the tensile strength check, fatigue failure analysis and yield strength check in the mechanical design. Moreover, the bending-torsion combined moment as a function of gas flow number is also presented.
2.
Model and method
2.1.
Experimental model
The laboratory scale vessel used in the experiment is shown in Fig. 1 and the key details are given in Table 1. Geometrical details of the modified PBT are shown in Fig. 2.
Please cite this article in press as: Shi, D., et al., Effect of gas flow on the bending moment acting on a shaft in a sparged vessel stirred by a Pitched Blade Turbine. Chem. Eng. Res. Des. (2014), http://dx.doi.org/10.1016/j.cherd.2014.03.006
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Table 1 – Detailed information of the stirred vessel. Stirring structure
Stirred vessel
Components
Shaft
Impeller
Fluid
Air/water mixture
Type Key dimensions (mm) Material Imbalance fn and fr Speed
Overhung d = 20;L = 1000 SS 304
Pitched Blade Turbine D = 232; C = 194 SS 304 1633 g mm
Boundaries Key dimensions (mm)
Tank with four baffles and a sparger H = 800 T = 580; Wb = 58 Ds = 155; S = 145 1–10 m3 /h
Gas rate
8.6 Hz and 7 Hz 192–240 RPM
Fig. 2 – Sketch and physical map of the modified Pitched Blade Turbine.
In Table 1, the imbalance of the impeller was estimated by Eq. (1), and the distance between the centers of gravity and geometry was given by Eq. (2), the ratio of which to the diameter of the impeller is confined in 1%. Ub = ı=
(mu ru )
(1)
Ub mimp
(2)
In Table 1, fn and fr are the resonant and natural frequency of the shaft and impeller structure, respectively, which are both estimated by using vibration system with one degree of freedom (Berger et al., 2003; Harnby et al., 1992). 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/nc < 0.6, thus avoiding any resonance issues. The liquid surface was sufficiently high to ensure that it had a negligible influence on the lateral FSI. The gas sparging rates were chosen so that the 2-phase flow regime ranged from complete dispersion through loading to flooding at each speed. Fig. 3 shows the range of the measured points (operational conditions) in the context of a 2-phase flow regime map with two dividing lines marked by the letters A and C obtained by experimental observation. The lines indicate two transitions between complete dispersion 0.45 Complete dispersion
A
C Flooding
Loading
Fr (N2 D/g)
0.4 0.35 0.3 0.25 0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
Fl (Q /(N3D )) G
G
Fig. 3 – Range of the measured points in 2-phase flow regime.
0.08
and loading, and between loading and flooding regimes, respectively. Fig. 4 shows the images of gas distribution at 216 RPM for complete dispersion, loading and the transition state between loading and flooding regimes. Referencing those defined by Harnby et al. (1992), complete dispersion and the transition state between loading and flooding regimes are defined as completely dispersing bubbles at the tank bottom and horizontally sparging gas along the sparger, respectively, while loading regime is considered as a tapered distribution between above two states as showed in Fig. 4c.
2.2.
Experimental method
2.2.1.
Experimental setup
In these 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 experimental setup are shown in Fig. 5. The drive unit consists of a motor, drive shaft and a flexible coupling. The supporting structure employs two sets of ball bearings to support the drive shaft, which makes the drive shaft end a clamped point for the shaft and impeller. 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. The sparging system included an air pressure stabilizer for the air source, a regulator valve to control the gas rate, a flowmeter to measure the gas rate, a pressure gauge to measure gas pressure of the air inlet and a cut-off valve. The moment measurement system consists of a moment sensor (Fig. 6) and a data acquisition unit. The specification and main 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. Communications between the rotor fixed at overhung point of the shaft and the stator employ digital telemetry technology.
Please cite this article in press as: Shi, D., et al., Effect of gas flow on the bending moment acting on a shaft in a sparged vessel stirred by a Pitched Blade Turbine. Chem. Eng. Res. Des. (2014), http://dx.doi.org/10.1016/j.cherd.2014.03.006
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Fig. 4 – Gas distributions in the stirred tank (n = 216RPM).
Table 2 – Specification and main parameters of the moment sensor. Moment sensor (HBM, Germany) Type
Customized
Communication between rotor and stator
Digital telemetry
Parameters
Unit
Index
Rated bending moment X Rated bending moment Y Rated torque Linearity deviation Sampling frequency Torsional stiffness Bending stiffness
Nm Nm Nm % kHz kN m/rad kN m/rad
20 20 10 <1 <6.7 62.21 77.91
2.2.2.
Fig. 5 – Experimental setup.
Fig. 6 – Moment sensor configuration.
Data acquisition
The signals for bending moment and torque were synchronized and simultaneously sampled. The data sampling frequency and size can influence the experimental results. Therefore, it is important to select the appropriate sampling frequency and size. The Power Spectral Density (PSD) of the bending moment measured by using a sampling frequency of 1200 Hz showed that approximately 99% of the Spectral Power (SP) is contained in frequencies below 100 Hz (Fig. 7). Therefore, a sampling frequency of 600 Hz was chosen to ensure sufficient resolution at low frequencies, where most of the loading was expected, whilst also ensuring that no significant high-frequency loads were omitted. Fig. 8 shows the percentage deviation of the mean bending moment as a function of time. For sampling times beyond 150 s or so, the percentage deviation is around ±3% (i.e. the sampling time has a negligible effect on the results after 150 s). Therefore, a 300 s sampling time was chosen in these
Fig. 7 – Power Spectral Density (PSD) normalized by the variance. (Speed frequency (N) = 3.6 Hz; sampling scale (SS ) = 217 ).
Please cite this article in press as: Shi, D., et al., Effect of gas flow on the bending moment acting on a shaft in a sparged vessel stirred by a Pitched Blade Turbine. Chem. Eng. Res. Des. (2014), http://dx.doi.org/10.1016/j.cherd.2014.03.006
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3 Mb / Mb
2 1 0
0
0.2
0.4
0.6
0.8
1
1.2
1.4
time(s)
Fig. 10 – Instantaneous Bending Moment normalized by the time-averaged value. (n = 216 RPM; QG = 5.42 m3 /h).
Fig. 8 – Percentage deviation of the average bending moment from the mean as a function of sampling time (speed (n) = 216 RPM; sampling frequency = 600 Hz). experiments, corresponding to a sampling size 180,000 (300 s times 600 Hz).
3.
Results and discussion
3.1.
Impeller torque
The impeller mean torque (Mt ) is an important factor in estimating the bending moment acting on the shaft in stirred tanks, which reflects the magnitude of circumferential fluid forces on the impeller. Fig. 9 shows a ratio of gassed to ungassed mean torque as a function of flow number at five impeller speeds, being in accordance with the two transitions are marked by the letters A and C shown in Fig. 3. For each speed the ratio firstly decreases slightly to the line A, then descends sharply to the line marked by the letter B, and finally increases slightly through the line C with the rise of gas flow number. The ratio trend over FlG is consistent with that by Chapman et al. (1983), where the sharp decrease in the ratio is due to the formation of large cavities behind the impeller blades. In such decrease, the loading regime between A and B is considerable unstable by means of rather low-frequency, which was also observed by Chapman et al. (1983). The reason could be that the competition between two different ways of loading gas causes a macro-instability of the bulk flow. According to the works by Warmoeskerken et al. (1984), the ways of PBT loading gas are divided into indirect and direct loadings. In the former way, the gas below the downward PBT cannot enter upward the impeller but is discharged down when the discharging flow capacity of the impeller is beyond the gas buoyancy, while in the latter the gas firstly enters the impeller and is discharged down since the discharging flow capacity of the impeller is below the gas buoyancy. Therefore, in complete dispersion the way of PBT loading gas should be considered as direct loading while in the loading regime between B and C as indirect loading because of high gas flow. In the loading regime between A and B, the loading way can be the rather slow shift of two loading way, and causes a macro-stability of the bulk flow.
Shaft bending moment
Fig. 10 shows the shaft bending moment normalized by its average value for the gas flow of 5.42 m3 /h at 216 RPM. 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.
3.2.1. Effect of gas flow on amplitude characteristics 3.2.1.1. Amplitude mean. Fig. 11 shows the relative mean bending moment (Mb )G /(Mt )L (here it is defined as the bending moment ratio between gassed and ungassed conditions) as a function of FlG . Being the same as those in Fig. 9, in Fig. 11 the lines A and C indicate the two transitions, and the line B is the critical location for stable and unstable loadings in loading regime. Initially the relative mean bending moment increases with the rise of FlG until the flow regime varies from complete dispersion to loading, and then goes through a peak. When the gas flow keeps increasing, the mean bending moment drops to a valley and again starts to rise gradually, going through the transition from loading to flooding regimes. The “S” trend of the mean bending moment over FlG for each speed is obviously similar. The phenomenon shown in Fig. 11 demonstrates that the magnitude of the unbalanced fluid force acting on the impeller increases with the rise of the gas flow confined 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 non-synchronously sharp fluctuation of fluid pressures on the impeller blades surfaces in every moment. However, the unbalanced fluid force also decreases when the gas distribution in the vessel move on to the loading regime. This is 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
1.5
A
B
C
n=192 (RPM); Fr=0.242 n=204 (RPM); Fr=0.273
0.9 0.8
n=216 (RPM); Fr=0.306 n=228(RPM); Fr=0.341
A
B
C
0.7
n=240 (RPM); Fr=0.378
(Mb)G/(Mb)L
1.4
1
( Mt )G/( Mt )L
3.2.
1.3 1.2 n=192(RPM) n=204(RPM)
1.1
0.6 0.5
n=216(RPM) n=228(RPM)
1
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
Fl (Q /(N3D )) G
G
Fig. 9 – Ratio of gassed to ungassed mean torque as a function of FlG .
0.08
n=240(RPM)
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
Fl (Q /(N3D)) G
G
Fig. 11 – Relative (gassed/ungassed) mean bending moments as a function of FlG .
Please cite this article in press as: Shi, D., et al., Effect of gas flow on the bending moment acting on a shaft in a sparged vessel stirred by a Pitched Blade Turbine. Chem. Eng. Res. Des. (2014), http://dx.doi.org/10.1016/j.cherd.2014.03.006
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=1.9042; =2.0464;
C
=0.512
2.2
G
/ L
A
B
1.6
n=192(RPM) n=204(RPM) n=216(RPM)
1.3 1
n=228(RPM) n=240(RPM)
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
10000 5000
0.08
0
Fl (Q /(N3D)) G
exp. weibull
15000
sampling number
1.9
G
0
1
Fig. 12 – Relative (gassed/ungassed) ˇ as a function of FlG .
2 M b / Mb
3
Fig. 13 – Typical measured amplitude distributions for shaft bending moment. (QG = 7.31 m3 /h; n = 216 RPM; SS = 180,000). in the fluid forces, resulting from the decrease in pressure difference between the front and back of blades, 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 complete dispersion regime, while the cavitation behind the blades dominates in loading regime. Consequently, the mean bending moment shows one fluctuation as the gas distribution in the vessel goes from complete dispersion to loading regime. When the gas rate gets higher, the gas flow begins to impact the impeller unsteadily, where loading gas way has shifted to indirect 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. As the gas rate continues to increase and the gas–liquid flow transits to the flooding regime, the gas flow begins to completely dominate the fluid motion around the impeller and directly impacts the impeller, which leads to the bending moment increasing monotonously in flooding regime. In order to eliminate the impeller circumferential (tangential) fluid force influence and only examine the impeller lateral (bending) forces variation with FlG , a dimensionless shaft bending moment ˇ is defined by Eq. (3). In the equation, Mb /L (L is the shaft overhung length) represents the lateral forces contributing to the bending moment, and Mb /D (D is the impeller diameter) represents the circumferential fluid force contributing to torque. ˇ=
Mb /L Mt /D
(3)
Fig. 12 shows the relative ˇ plotted as a function of FlG , where the two transitions A and C and the critical location B are also marked by the dashed lines. Apparently, the relative ˇ increases slightly with FlG in complete dispersion regime, and then sharply rises in loading regime. Such trend of the relative ˇ further demonstrates that the asymmetry of the fluid motion around the impeller, causing the unequal fluid force on each blade, slightly increases because of the increasingly nonuniform bubbles around blades in complete dispersion regime, and sharply rises due to the increasingly asymmetrical cavities behind the blades in loading regime. When the gas rate continues to increase, the rising of the relative ˇ becomes to slow down from the beginning of the line B in loading regime and keeps in flooding regime. Such slowing down of the relative ˇ reflects that
the unsteady impact of gas flow on the impeller, resulting in an up-and-down vibration of the impeller and shaft due to the shaft material elasticity and the gas beating up the impeller, causes the moderate increase in the bending moment.
3.2.1.2. Amplitude distribution. The shaft bending moment is a considerably unsteady load, whose mean is mainly used to explain some phenomena, such as fluid flow instabilities and structure vibrations, or served as a basic reference for fluctuating load. However, for mechanical design, the fluctuating characteristics of the bending moment, such as Standard Deviation and peak value, are significant for the material fatigue analysis and yield strength check. Apparently, amplitude of bending moment is characterized by polydispersity. Therefore, amplitude distribution is firstly checked by means of a statistical method, whose purposes are to quantify the polydispersity and extract the fluctuating characteristics with any probability. Fig. 13 shows the typical shaft bending moment amplitude distribution which is well fitted by a Weibull distribution. Apparently, the distribution is not symmetry about the amplitude mean. It is due to the shaft deflections’ asymmetry around the dynamic equilibrium position, being caused by the nature of shaft against bending deformation. This nature is the elastic reaction pulling the shaft back toward the axis of rotation (static equilibrium position) after the shaft bending deformation. The Probability Density Function (PDF) for Weibull distribution (Papoulis and Pillai, 2002) is given by:
PDF(x; , ) =
⎧ ⎪ ⎨ x −1 ⎪ ⎩
x
−
e
x≥0
(4)
x≤0
0
where x is a Weilbull random variable, (>0) and (>0) are the scaling and shape parameters respectively. The mean () and variance ( 2 ) are given by Eqs. (5) and (6), respectively.
=
2 = 2
1+
1
1+
(5)
2
− 2
(6)
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/ Mb)G / ( / Mb)L
or if the amplitude fluctuating location x is known, the probability CDF below the x may be solved.
n=192(RPM)
1.06
n=204(RPM) n=216(RPM) n=228(RPM)
1.04
3.2.1.3. Amplitude Standard Deviation. A ratio of Standard Deviation to mean represents intensity of amplitude fluctuation, which may be served as the basic information for material fatigue analysis. The relative ratio (gassed/ungassed) shown in Fig. 14, whose maximum approaches to 1.065, slightly changes with gas flow number. Its trend over FlG is similar with that of relative bending moment shown in Fig. 11. According to Eq. (7), when x = Mb + , the corresponding probability below Mb + approaches 84% as shown in Fig. 15, which means that only 16% amplitudes are greater than Mb + .
n=240(RPM)
1.02 1 0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
Fl (Q /(N3D)) G
G
Probability (%)
Fig. 14 – Relative (gassed/ungassed) /Mb as a function of FlG . 83.95
n=192(RPM)
83.9
n=216(RPM) n=228(RPM)
83.85
n=240(RPM)
n=204(RPM)
3.2.1.4. Amplitude peak. A ratio of peak fluctuation to Standard Deviation ((peak − Mb )/) represents the dimensionless peak value, by means of which can easily solve the peak served as the basic information for material yielding check. Because the peak sampled is too unstable, the peak is defined by a 99% probability, which means the CDF is 0.99 according to Eq. (7). Thus the corresponding amplitude peak can be calculated. Fig. 16 shows the relative ratio (gassed/ungassed) for each speed, whose trend over FlG is the same as the previous Figs. 11 and 14 essentially. The ratio maximally approaches to 1.025, but slightly changes with gas flow number.
83.8 83.75 0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
Fl (Q /(N3D)) G
G
((peak-Mb)/ )G/((peak-Mb)/ )L
Fig. 15 – Probability of the amplitudes below Mb + as a function of FlG . 1.025
n=192(RPM) n=204(RPM)
1.02
3.2.2.
n=216(RPM) n=228(RPM)
1.015
n=240(RPM)
1.01 1.005 1 0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
FlG(QG/(N3D))
Fig. 16 – Relative (gassed/ungassed) (peak − Mb )/ as a function of FlG . The Cumulative Distribution Function (CDF) can be obtained by integrating the Weibull probability density function:
CDF(x; , ) =
⎧ ⎪ ⎨ ⎪ ⎩
x
−
1−e
x≥0
(7)
x≤0
0
Effect 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 (Hasal et al., 2004; Montes et al., 1997) 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 (Vantriet et al., 1976) and interactions between the impeller blades and between the blades and baffles (Kratena et al., 2001), and from the natural elastic vibration of stirring structures. However, the particular frequencies are not all evident simultaneously, depending on the specific condition which can excite the corresponding frequency. Fig. 17 shows the PSD of the bending moment for 216 RPM from 0 to 5 Hz at different gas rates, which is obtained by using the Yule-Walker autoregressive model. In all cases, two groups of the peaks marked by the letters D and E are evident which are influenced by gas flow. Group D locates at rather low
According to Eq. (7), if the probability CDF is known, the amplitude fluctuating location x with the CDF may be solved,
PSD/ 2 [ 1/Hz ]
2
D
1.5
E
1
0.5
0 10.5 9.8 9.2 8.6
7.97.3 6.7 6.1
5.44.8 4.2 3.5 2.9 2.3 1.7
gas flow [m3/h]
1
0
0
N/5
1
2
3
N-0.5
4
N+0.5
frequency [ Hz ]
Fig. 17 – PSD normalized by the variance (N = 3.6 Hz (216 RPM); SS = 217 ). Please cite this article in press as: Shi, D., et al., Effect of gas flow on the bending moment acting on a shaft in a sparged vessel stirred by a Pitched Blade Turbine. Chem. Eng. Res. Des. (2014), http://dx.doi.org/10.1016/j.cherd.2014.03.006
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n=192(RPM)
3
n=216(RPM) n=228(RPM)
2.5 2
n=240(RPM)
1.5 1 0.5
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
C
1 n=192(RPM)
0.9
n=204(RPM) n=216(RPM) n=228(RPM)
0.8 0.7
Fl (Q /(N3D)) G
A
1.1
n=204(RPM)
( Mc)G/( Mc )L
(SP/ 2)g/(SP/ 2)l
3.5
B
n=240(RPM)
0
0.01
0.02
G
0.03
0.04 G
Fig. 18 – Relative (gassed/ungassed) SP/ 2 from 0 to N/5. frequency, which should result from low-frequency macroinstability of bulk flow in stirred tanks. And group E locates at the speed frequency (3.6 Hz), which is caused by the impeller imbalance. A ratio of Spectral Power in a range of frequencies to the variance is used to represents a resultant contribution of these frequencies to the fluctuation of bending moment. The relative ratio (gassed/ungassed) in the low-frequency range (0-N/5) is shown in Fig. 18. Initially, it increases with FlG , peaks at FlG ∼ 0.02 and then decreases with the increasing gas rate. It is noted that the location of the maximum ratio just occurs in the considerable unstable regime between A and B shown in Fig. 9. This demonstrates that the instability observed firstly by Chapman et al. (1983) is reflected in shaft bending moment fluctuation and the low frequency should be the shifting frequency between the two loading ways: indirect and direct loading presented by Warmoeskerken et al. (1984). The relative (gassed/ungassed) ratio in the speed frequency range (N ± 0.5) is shown in Fig. 19. Initially, it slightly increases in the range of low gas rates and then decreases with FlG , which demonstrates that the contribution to the total fluctuation of shaft bending moment declines as gas flow increases because of the dominant gas-fluid flow.
3.3.
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 defined as:
Mc =
Mb2 + Mt2
(8)
The relative combined moment (gassed/ungassed) for different speeds is shown in Fig. 20. At first, it increases slightly to one peak, approaching the transition between complete dispersion and loading regimes, then decreases rapidly, and finally increases slightly with the rise of FlG . This information on the combined moment could reflect more overall the
n=192(RPM)
1.2 (SP/ 2)g/(SP/ 2)l
0.05
0.06
0.07
0.08
Fl (Q /(N3D))
n=204(RPM) n=216(RPM)
1
n=228(RPM) n=240(RPM)
0.8 0.6
G
Fig. 20 – Relative (gassed/ungassed) combined moment as a function of FlG . characteristics of gas–liquid flow loading on the impeller, compared with those on individual torque and bending moment.
4.
Conclusion
This paper investigated the impact of gas sparged on the shaft bending moment in a Pitched Blade Turbine 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 were summarized as follows: (1) The trend of the relative mean bending moment over FlG presents 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 slightly increases with FlG in complete dispersion regime, approaching to 1.4 at the transition between complete dispersion and loading regimes, then ascends sharply in loading regime and almost levels out for higher gas rates, reaching about 2.2 in flooding regime. (3) The distribution of the bending moment amplitude is well described by Weibull distribution, which reflects that the elastic reaction pulls the shaft back toward 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 peak to about 1.065 and 1.025 respectively, whose changes with gas rate are similar with that of the relative mean bending moment. (5) The low-frequency contribution to bending moment fluctuation peaks in an unstable loading regime, which is caused by the low-frequency shift of gas indirect and direct loading ways. The speed frequency contribution gradually becomes weak as gas flow increases because of the dominant gas-fluid flow. (6) The relative mean combined moment firstly increases slightly, then decreases rapidly, and finally increases slightly with FlG , which overall reflects the characteristics of gas–liquid flow loading on the impeller.
Acknowledgments 0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
FlG(QG/(N3D))
Fig. 19 – Relative (gassed/ungassed) SP/ 2 from N − 0.5 to N + 0.5.
The financial support from the “A Computational and Experimental Study of Fluid Structure Interactions in Stirred Vessels” PhD Studentship provided by Huntsman Europe (Belgium) is gratefully acknowledged.
Please cite this article in press as: Shi, D., et al., Effect of gas flow on the bending moment acting on a shaft in a sparged vessel stirred by a Pitched Blade Turbine. Chem. Eng. Res. Des. (2014), http://dx.doi.org/10.1016/j.cherd.2014.03.006
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Please cite this article in press as: Shi, D., et al., Effect of gas flow on the bending moment acting on a shaft in a sparged vessel stirred by a Pitched Blade Turbine. Chem. Eng. Res. Des. (2014), http://dx.doi.org/10.1016/j.cherd.2014.03.006