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Change in liquid temperature behind the impeller blades with impeller speed in boiling stirred tanks
chemical engineering research and design 8 8 ( 2 0 1 0 ) 1073–1077
Contents lists available at ScienceDirect
Chemical Engineering Research and Design journal homepage: www.elsevier.com/locate/cherd
Short communication
Change in liquid temperature behind the impeller blades with impeller speed in boiling stirred tanks H. Yoshikawa b , R. Fukuda a , Y. Kawase a,∗ a
Research Centre of Biochemical and Environmental Engineering, Department of Applied Chemistry, Toyo University, Kawagoe, Saitama 350-8585, Japan b Education Support Center of Experiment and Training for Students, Tokyo Institute of Technology, Meguro, Tokyo 152-8550, Japan
a b s t r a c t In our previous study (Fukuda, R., Tokumura, M., Znad, H.T. and Kawase, Y., 2009, Vapour generation from the impellers in boiling stirred tank reactors. Chem Eng Res Des, 87: 452–459), it was found that in boiling stirred tanks with multiple impeller systems vapour was generated from the heater at lower impeller speeds and with an increase in impeller speed most vapour was generated from the top impeller rather than the lower impellers and the heater. The change of nucleation sites with the impeller speed might be controlled by the local liquid temperature. Therefore we measured the liquid temperature behind the impellers blades and found the decrease in liquid temperature with increasing impeller speed. In this paper, a simple model was developed to predict the change in liquid temperature behind the impeller blades in which nucleation takes place. In the proposed model based on the results for pressure distribution on the impeller blade in the literature, the liquid temperature behind the impeller blades is estimated from the measured power consumption. The validation of the proposed model was conducted using the experimental results in our previous study and reasonable agreement was obtained. © 2010 The Institution of Chemical Engineers. Published by Elsevier B.V. All rights reserved. Keywords: Liquid temperature; Vapour generation; Stirred tank; Nucleation site; Modeling
1.
Introduction
Although boiling stirred tanks in which considerable vapour evolution occurs have been widely used to remove the large reaction heat by condensation of an evaporating reaction mixture, local phenomena of boiling have not been satisfactorily elucidated (Zhao et al., 2002; Smith, 2006). It has been found that at lower impeller speeds vapour generation mainly occurs on the heater surface but at higher impeller speeds most vapour is generated from the impeller rather than the heater (e.g., Dohi et al., 1999; Takahashi et al., 2006). The vapour generation from the impellers in boiling stirred tanks might be controlled by the local liquid temperature and vapour pressure. In our previous paper (Fukuda et al., 2009), therefore, we measured the liquid temperature behind the impeller blades and it was found to decrease with increasing impeller speed.
∗
In order to predict the local liquid temperature change with impeller speed in boiling stirred tanks, we have developed a simple model on the basis of the results for pressure distribution on the impeller blades in the literature (Tay and Tatterson, 1985; Lane et al., 2001; Vlaev et al., 2004; Mochizuki et al., 2008). We examined the validity of the proposed model using measured local liquid temperatures with dual and triple impeller systems in our previous study (Fukuda et al., 2009).
2.
Experimental
All experiments were conducted in the 0.20 m i.d. mechanically stirred tank, in which a ring heater of 3.0 kW on full power was mounted close to the tank bottom. The vapour generation rate or boiling rate was varied by the change in ampere to the ring heater. Dual and triple impeller configurations, which
Corresponding author. Tel.: +81 49 239 1377; fax: +81 49 231 1031. E-mail address: [email protected] (Y. Kawase). Received 28 September 2009; Received in revised form 28 January 2010; Accepted 28 January 2010 0263-8762/$ – see front matter © 2010 The Institution of Chemical Engineers. Published by Elsevier B.V. All rights reserved. doi:10.1016/j.cherd.2010.01.039
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chemical engineering research and design 8 8 ( 2 0 1 0 ) 1073–1077
Nomenclature A a, b, c ai C1 D FD FFD g h N NIm nB P RC r T Tb Th Tm Tt Ts U Ug
projected area of blade (m2 ) parameters in Antoine’s equation area of small element i (m2 ) coefficient in Eq. (4) impeller diameter (m) drag force (N) form drag (N) gravitational acceleration (ms−2 ) vertical distance from free surface (m) impeller speed (s−1 ) impeller speed at which vapour generation from the top impeller starts (s−1 ) number of blade power consumption (W) radial distance from the impeller shaft to the centre of blade (m) radial distance from the impeller shaft (m) liquid temperature (K) liquid temperature behind the blade of bottom impeller (K) liquid temperature at heater (K) liquid temperature behind the blade of middle impeller (K) liquid temperature behind the blade of top impeller (K) liquid temperature at free-surface (K) velocity (ms−1 ) superficial vapour velocity (ms−1 )
Greek symbols ˛ dimensionless pressure on the front surface of blade ˘ f − b pressure difference between the front and back surfaces of blade = ˘ f − ˘ b (Pa) ˘b pressure on the back surface of blade (Pa) ˘f pressure on the front surface of blade (Pa) liquid density (kg m−3 ) Subscripts ave average over blade surface Exp experimental i i-th Pre predicted
consist of six-flat blade disk turbines (Rushton turbine) having diameter of 0.08 m (blade height is 0.02 m, blade width is 0.02 m, disk diameter is 0.06 m), four 45◦ pitched-blade downflow disk turbines having diameter of 0.08 m (blade height is 0.02 m, blade width is 0.015 m, disk diameter is 0.06 m) or six circular half-pipe concave-blade disk turbines having diameter of 0.07 m (blade height is 0.02 m, blade width is 0.015 m, disk diameter is 0.06 m), were used. The bottom impeller was mounted at 0.05 m above the T.L. (the transition point from the curved bottom to the vertical wall). It was very close to the top of the ring heater. For dual impeller systems, the spacing between the impellers and that between the top impeller and the clear liquid surface were 0.20 m and 0.15 m, respectively. For triple impeller configurations, they were 0.15 m and 0.05 m, respectively. All other detailed aspects of the experimental work including the configuration of the stirred tank and
the measurements of liquid temperature behind the impeller blade and power consumption are described in our previous paper (Fukuda et al., 2009).
3.
Modeling
With increasing impeller speed, trailing vortices having a low-pressure region are formed behind the impeller blades (Middleton and Smith, 2004). Lane et al. (2001) and Mochizuki et al. (2008) experimentally confirmed that the pressure on the front surface of the blade is larger than that on the back surface of the blade and found that the dimensionless pressure difference distribution or pressure coefficient on the front surface of disk turbine blades is in the range from 1.0 to 0. Vlaev et al. (2004) predicted local pressure coefficients on the blade surface for three different design impellers using the CFD analysis. The low-pressure region formed behind the impeller blade plays a very important role in nucleation. Since the boiling temperature in the low-pressure region behind the blade is lower as compared with that in the liquid bulk including the region near the heater, the liquid boiling mainly occurs behind the impeller blades and most vapour is generated from the impeller rather than the heater. In fact, rather large vapour bubbles were generated at the top impeller and only few small vapour bubbles existed at higher impeller speeds (N > NIm ). We develop a model for predicting liquid temperature behind the impeller blades. Tay and Tatterson (1985) examined the contributions of form and skin drags to power consumption for the pitchedblade turbine and found that at high Reynolds numbers the total drag force or the power consumption is dominated by form drag. Power consumption can be written as: P = FD · U
(1)
The velocity U is related to impeller tip speed, ND. Based on the results of Tay and Tatterson (1985) the drag force FD can be approximated as: FD ≈ FFD = ˘f −b · A
(2)
where ˘ f − b (=˘ f − ˘ b ) is the pressure drop occurring across the blade having a projected area A. Recently, Mochizuki et al. (2008) also found that the power consumption calculated from the pressure difference between the front and back surfaces of the blade agrees well with the power consumption measured using the torque meter. In order to evaluate the power consumption, the following summation of local pressure difference over the blade surface was used
P = 2NnB
ai ˘f −b,i ri
(3)
i
Based on this equation, we have the following approximate relationship between the power consumption and the average pressure drop across the impeller blade (˘ f − b )ave :
P = 2C1 NnB A ˘f −b
R ave C
(4)
where C1 is a coefficient introduced by considering that for simplification the average pressure drop across the impeller
chemical engineering research and design 8 8 ( 2 0 1 0 ) 1073–1077
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Fig. 1 – Change in liquid temperature behind the blade with impeller speeds for the dual Rushton turbine impeller system. blade is used instead of the local pressure difference distribution over the blade surface in Eq. (3). The values of C1 , which may be dependent of the variation in the pressure distribution on the blade surface, are discussed below. Mochizuki et al. (2008) measured the pressure distributions on the front of the blade. Their results may be approximately presented as: ˘f − gh (DN)2 /2
=˛
(5)
They obtained that the dimensionless pressure on the blade front surface ˛ varies according to the location on the blade surface but it is almost independent on the impeller speed and gas flow rate. The range of ˛ for the six blade disk turbines obtained by Mochizuki et al. (2008) is from 0 to 1.0. Lane et al. (2001) examined the pressure distribution on the surface of Rushton turbine blade. In their study, the experimental data as well as the CFD results indicate that the average value of ˛ in the centre zone of the front blade surface is around 0.5. Using the CFD analysis, Vlaev et al. (2004) found that the ranges of ˛ for the Rushton turbine and the circular half pipe blade impellers are −1.3 to 0.72 and −0.08 to 0.15 in the centre zone of front blade surface, respectively. In this study, on the basis of the results in the literature (Lane et al., 2001; Mochizuki et al., 2008) we assumed that ˛ ∼ 0.5 and furthermore by considering the results of Mochizuki et al. (2008) it is independent on the vapour generation rate as a first approximation. As well as the pressure difference in Eq. (4), we use the average value over the blade surface of ˛ instead of its local value.
Using Eq. (4) we can evaluate (˘ f − b )ave from the measured power consumption, P. The data for P in the boiling stirred tank with the dual impeller systems are presented in Fig. 2 of our previous paper (Fukuda et al., 2009). Similar results were obtained for the triple impeller systems. From the calculated (˘ f − b )ave and Eq. (5), we can evaluate ˘ b and then estimate liquid temperature corresponding to ˘ b , T, using Antoine’s equation. The liquid temperature behind the impeller blade, T, in Kelvin can be calculated by Antoine’s equation using the pressure ˘ b in Pa. By rearranging Antoine’s equation for vapour pressure we have the following equation for the liquid temperature: T = −c +
b a − ln ˘b
(6)
where a = 23.1964, b = 3816.44 and c = −46.13 for water (Read et al., 1977). By the procedure described above the liquid temperature behind the impeller blades can be estimated from the measured power consumption.
4.
Results and discussion
The model predictions are compared with the liquid temperatures measured using the thermocouple attached to the back of the impeller blade in our previous study (Fukuda et al., 2009). The values of C1 in Eq. (4) used in this study for dual Rushton turbine, pitched-blade downflow disk turbine and concaveblade disk turbine systems are 1.5, 1.8 and 2.0, respectively. They were obtained by fitting the present experimental data. Vlaev et al. (2004) found that the pressure distribution for the
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flat blade is more uniform as compared with that for the circular half-pipe blade. By considering that as can be seen from Eq. (4) C1 is inversely proportional to the pressure difference (˘ f − b )ave at constant power consumption, the values of C1 for different blade designs correspond to the uniformity of pressure difference on different blade designs. This implies that the difference between the average and local pressures over the blade surface is somewhat affected by the impeller designs. For the triple impeller systems the values of C1 are 1.2 times of those for the dual impeller systems. This may be attributed to an interaction among the impellers in the multiple impeller systems. Fig. 1 shows the changes in liquid temperature behind the blade with impeller speed for the dual Rushton turbine impeller system at Ug = 0.014 and 0.027 ms−1 . In our previous study (Fukuda et al., 2009), the impeller speed at which the vapour generation from the top impeller starts, NIm , was defined and measured to characterize the vapourization in the boiling stirred tanks. When the impeller speed was less than NIm , the liquid temperatures behind the blades of top and bottom impellers were nearly constant, respectively and the nucleation was localized on the heater surface. It can be seen in Fig. 1 that the continuous and steep decrease in liquid temperature with increasing impeller speed starts at NIm . For N > NIm , since the lower pressure region was formed behind the impeller blade and then the liquid temperature and the vapour pressure decreased, the nucleation was taken place in the low-pressure region formed behind the blades rather than the heater. The liquid temperatures behind the blade of the top and bottom impellers, Tt and Tb , decreased as the impeller speed increased and could be satisfactorily predicted in the whole range of N by the proposed model. As expected from static pressure gradient, Tt was lower than Tb . The changes in Tt and Tb with impeller speed were similar. For reference the measured liquid temperature at the free surface, Ts , is also presented in the figure. While Ts slightly decreased due to the liquid mixing by the impeller at lower impeller speeds, it was nearly independent of the impeller speed at higher impeller speeds. The vapour generation rate was found to have a rather minor effect on the liquid temperature behind the impeller blade. For reference, the calculated results for pressure on the back surface of blades of the bottom and top impellers are presented in Fig. 1. It can seen from the figure that the pressure on the back surface of blades decreased around 1.75 and 1.5 kPa from the static pressure or pressure on the blade surfaces with N = 0 s−1 with impeller speed at Ug = 0.014 and 0.027 ms−1 , respectively. These decreases in the pressure are caused by the rotation of the impeller. The changes in liquid temperature behind the blade for three dual impeller systems (Rushton turbines, pitched-blade downflow disk turbines and concave-blade disk turbines) are depicted in Fig. 2. The liquid temperatures behind the blades of the top and bottom impellers, Tt and Tb , decreased with impeller speed for N > NIm . On the whole, the model could predict the experimental data reasonably well. The decrease in liquid temperature behind the blade for the concave-blade disk turbines was somewhat steeper as compared with those for Rushton turbines and pitched-blade downflow disk turbines. For reference, the boiling temperatures at 101 kPa and the heater evaluated using Eq. (6) are presented in the figure. At higher impeller speeds the liquid temperature behind the blade of the top impeller was lower than the boiling temperature at 101 kPa. It can be seen in Fig. 2 that the change of nucleation site from the heater to the top impeller is due
Fig. 2 – Change in liquid temperature behind the blade with impeller speeds for the dual impeller systems at Ug = 0.012 ms−1 . to the formation of low-pressure regions behind the blades and the corresponding decrease in boiling temperature with increasing impeller speed. In Fig. 3, the model predictions are compared with the experimental results obtained at Ug = 0.012 ms−1 for triple impeller systems. Again the continuous decreases in liquid temperature behind the blades of top, middle and bottom impellers, Tt , Tm and Tb , started at N = NIm . The liquid temperatures behind the blades of the impellers, Tt and Tb , somewhat sharply decreased with impeller speed for N > NIm . At higher impeller speeds the liquid temperature behind the blade of the top impeller was reduced to be lower than the boiling temperature at 101 kPa as well as the dual impeller systems. The comparison between the model and the experimental data shows reasonably good agreement.
chemical engineering research and design 8 8 ( 2 0 1 0 ) 1073–1077
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the literature (Mochizuki et al., 2008). In the model the liquid temperature behind the impeller blades in which nucleation takes place is readily estimated from the measured power consumption. The model could predict the decreases in pressure and liquid temperature behind the impeller blades with impeller speeds. Predictions of the proposed model for liquid temperature behind the blades were in reasonable agreement with the experimental results carried out in a laboratory-scale boiling stirred tank. Therefore, the model may be used to provide a reasonable prediction of liquid temperature behind the impeller blade in boiling stirred tanks. It should be noted that our experiments were performed in rather small stirred tank reactor and the effect of hydrostatic pressure on the change in the vapour generation site may be rather small. This subject is one of our future studies. Further understanding of the local temperature homogeneities and vapour–liquid two-phase mixing around the heater and impellers is also desirable.
References
Fig. 3 – Change in liquid temperature behind the blade with impeller speeds for the triple impeller systems at Ug = 0.012 ms−1 .
5.
Conclusions
In order to predict the change in liquid temperature behind the impeller blades with impeller speeds in boiling stirred tanks a simple model has been developed. The model is based on the results for pressure distribution on the impeller blade in
Dohi, N., Matsuda, Y., Itano, N., Shimizu, K., Minekawa, K. and Kawase, Y., 1999, Mixing in multiple-impeller stirred tank reactors with boiling liquids. Can J Chem Eng, 77: 786–792. Fukuda, R., Tokumura, M., Znad, H.T. and Kawase, Y., 2009, Vapour generation from the impellers in boiling stirred tank reactors. Chem Eng Res Des, 87: 452–459. Lane, G.L., Rigby, G.D. and Evans, G.M., 2001, Pressure distribution on the surface of Rushton turbine blades—Experimental measurement and prediction by CFD. J Chem Eng Jpn, 34: 613–620. Middleton, J.C. and Smith, J.M., 2004, Gas–liquid mixing in turbulent systems, in Handbook of Industrial Mixing: Science and Practice, Paul, E.L., Kresta, S.M., & Atiemo-Obeng, V. (eds) (John Wiley & Sons, Inc, New Jersey, USA), pp. 585–638. (John Wiley & Sons, Inc, New Jersey, USA). Mochizuki, M., Sato, H., Doida, Y., Saita, Y., Amanuma, T. and Takahashi, T., 2008, Pressure distribution on the blade surface and form drag of impeller in a gas–liquid stirred vessel with disc turbine. Kagaku Kougaku Ronbunshu, 34: 557–561. Read, R.C., Prausnitz, J.M. and Sherwood, T.K., (1977). The Properties of Gases and Liquids (3rd ed.). (McGraw-Hill, New York, USA), pp. 184–185 Smith, J.M., 2006, Large multiphase reactors. Some open questions. Chem Eng Res Des, 84: 265–271. Takahashi, T., Tagawa, A., Atsumi, N., Dohi, N. and Kawase, Y., 2006, Liquid-phase mixing time in boiling stirred tank reactors with large cross-section impellers. Chem Eng Process, 45: 303–311. Tay, M. and Tatterson, G.B., 1985, Form and skin drag contributions to power consumption for the pitched-blade turbine. AIChE J, 31: 1915–1918. Vlaev, S.D., Staykov, P. and Popov, R., 2004, Pressure distribution at impeller blades of some radial flow impellers in saccharose and xanthan gum solutions. A CFD visualization approach. Trans IChemE, Part C, 82: 13–20. Zhao, D., Muller-Steinhagen, H. and Smith, J.M., 2002, Micromixing in boiling and hot sparged systems. Development of a new reaction pair. Trans IChemE, 80: 880–886.
Contents lists available at ScienceDirect
Chemical Engineering Research and Design journal homepage: www.elsevier.com/locate/cherd
Short communication
Change in liquid temperature behind the impeller blades with impeller speed in boiling stirred tanks H. Yoshikawa b , R. Fukuda a , Y. Kawase a,∗ a
Research Centre of Biochemical and Environmental Engineering, Department of Applied Chemistry, Toyo University, Kawagoe, Saitama 350-8585, Japan b Education Support Center of Experiment and Training for Students, Tokyo Institute of Technology, Meguro, Tokyo 152-8550, Japan
a b s t r a c t In our previous study (Fukuda, R., Tokumura, M., Znad, H.T. and Kawase, Y., 2009, Vapour generation from the impellers in boiling stirred tank reactors. Chem Eng Res Des, 87: 452–459), it was found that in boiling stirred tanks with multiple impeller systems vapour was generated from the heater at lower impeller speeds and with an increase in impeller speed most vapour was generated from the top impeller rather than the lower impellers and the heater. The change of nucleation sites with the impeller speed might be controlled by the local liquid temperature. Therefore we measured the liquid temperature behind the impellers blades and found the decrease in liquid temperature with increasing impeller speed. In this paper, a simple model was developed to predict the change in liquid temperature behind the impeller blades in which nucleation takes place. In the proposed model based on the results for pressure distribution on the impeller blade in the literature, the liquid temperature behind the impeller blades is estimated from the measured power consumption. The validation of the proposed model was conducted using the experimental results in our previous study and reasonable agreement was obtained. © 2010 The Institution of Chemical Engineers. Published by Elsevier B.V. All rights reserved. Keywords: Liquid temperature; Vapour generation; Stirred tank; Nucleation site; Modeling
1.
Introduction
Although boiling stirred tanks in which considerable vapour evolution occurs have been widely used to remove the large reaction heat by condensation of an evaporating reaction mixture, local phenomena of boiling have not been satisfactorily elucidated (Zhao et al., 2002; Smith, 2006). It has been found that at lower impeller speeds vapour generation mainly occurs on the heater surface but at higher impeller speeds most vapour is generated from the impeller rather than the heater (e.g., Dohi et al., 1999; Takahashi et al., 2006). The vapour generation from the impellers in boiling stirred tanks might be controlled by the local liquid temperature and vapour pressure. In our previous paper (Fukuda et al., 2009), therefore, we measured the liquid temperature behind the impeller blades and it was found to decrease with increasing impeller speed.
∗
In order to predict the local liquid temperature change with impeller speed in boiling stirred tanks, we have developed a simple model on the basis of the results for pressure distribution on the impeller blades in the literature (Tay and Tatterson, 1985; Lane et al., 2001; Vlaev et al., 2004; Mochizuki et al., 2008). We examined the validity of the proposed model using measured local liquid temperatures with dual and triple impeller systems in our previous study (Fukuda et al., 2009).
2.
Experimental
All experiments were conducted in the 0.20 m i.d. mechanically stirred tank, in which a ring heater of 3.0 kW on full power was mounted close to the tank bottom. The vapour generation rate or boiling rate was varied by the change in ampere to the ring heater. Dual and triple impeller configurations, which
Corresponding author. Tel.: +81 49 239 1377; fax: +81 49 231 1031. E-mail address: [email protected] (Y. Kawase). Received 28 September 2009; Received in revised form 28 January 2010; Accepted 28 January 2010 0263-8762/$ – see front matter © 2010 The Institution of Chemical Engineers. Published by Elsevier B.V. All rights reserved. doi:10.1016/j.cherd.2010.01.039
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Nomenclature A a, b, c ai C1 D FD FFD g h N NIm nB P RC r T Tb Th Tm Tt Ts U Ug
projected area of blade (m2 ) parameters in Antoine’s equation area of small element i (m2 ) coefficient in Eq. (4) impeller diameter (m) drag force (N) form drag (N) gravitational acceleration (ms−2 ) vertical distance from free surface (m) impeller speed (s−1 ) impeller speed at which vapour generation from the top impeller starts (s−1 ) number of blade power consumption (W) radial distance from the impeller shaft to the centre of blade (m) radial distance from the impeller shaft (m) liquid temperature (K) liquid temperature behind the blade of bottom impeller (K) liquid temperature at heater (K) liquid temperature behind the blade of middle impeller (K) liquid temperature behind the blade of top impeller (K) liquid temperature at free-surface (K) velocity (ms−1 ) superficial vapour velocity (ms−1 )
Greek symbols ˛ dimensionless pressure on the front surface of blade ˘ f − b pressure difference between the front and back surfaces of blade = ˘ f − ˘ b (Pa) ˘b pressure on the back surface of blade (Pa) ˘f pressure on the front surface of blade (Pa) liquid density (kg m−3 ) Subscripts ave average over blade surface Exp experimental i i-th Pre predicted
consist of six-flat blade disk turbines (Rushton turbine) having diameter of 0.08 m (blade height is 0.02 m, blade width is 0.02 m, disk diameter is 0.06 m), four 45◦ pitched-blade downflow disk turbines having diameter of 0.08 m (blade height is 0.02 m, blade width is 0.015 m, disk diameter is 0.06 m) or six circular half-pipe concave-blade disk turbines having diameter of 0.07 m (blade height is 0.02 m, blade width is 0.015 m, disk diameter is 0.06 m), were used. The bottom impeller was mounted at 0.05 m above the T.L. (the transition point from the curved bottom to the vertical wall). It was very close to the top of the ring heater. For dual impeller systems, the spacing between the impellers and that between the top impeller and the clear liquid surface were 0.20 m and 0.15 m, respectively. For triple impeller configurations, they were 0.15 m and 0.05 m, respectively. All other detailed aspects of the experimental work including the configuration of the stirred tank and
the measurements of liquid temperature behind the impeller blade and power consumption are described in our previous paper (Fukuda et al., 2009).
3.
Modeling
With increasing impeller speed, trailing vortices having a low-pressure region are formed behind the impeller blades (Middleton and Smith, 2004). Lane et al. (2001) and Mochizuki et al. (2008) experimentally confirmed that the pressure on the front surface of the blade is larger than that on the back surface of the blade and found that the dimensionless pressure difference distribution or pressure coefficient on the front surface of disk turbine blades is in the range from 1.0 to 0. Vlaev et al. (2004) predicted local pressure coefficients on the blade surface for three different design impellers using the CFD analysis. The low-pressure region formed behind the impeller blade plays a very important role in nucleation. Since the boiling temperature in the low-pressure region behind the blade is lower as compared with that in the liquid bulk including the region near the heater, the liquid boiling mainly occurs behind the impeller blades and most vapour is generated from the impeller rather than the heater. In fact, rather large vapour bubbles were generated at the top impeller and only few small vapour bubbles existed at higher impeller speeds (N > NIm ). We develop a model for predicting liquid temperature behind the impeller blades. Tay and Tatterson (1985) examined the contributions of form and skin drags to power consumption for the pitchedblade turbine and found that at high Reynolds numbers the total drag force or the power consumption is dominated by form drag. Power consumption can be written as: P = FD · U
(1)
The velocity U is related to impeller tip speed, ND. Based on the results of Tay and Tatterson (1985) the drag force FD can be approximated as: FD ≈ FFD = ˘f −b · A
(2)
where ˘ f − b (=˘ f − ˘ b ) is the pressure drop occurring across the blade having a projected area A. Recently, Mochizuki et al. (2008) also found that the power consumption calculated from the pressure difference between the front and back surfaces of the blade agrees well with the power consumption measured using the torque meter. In order to evaluate the power consumption, the following summation of local pressure difference over the blade surface was used
P = 2NnB
ai ˘f −b,i ri
(3)
i
Based on this equation, we have the following approximate relationship between the power consumption and the average pressure drop across the impeller blade (˘ f − b )ave :
P = 2C1 NnB A ˘f −b
R ave C
(4)
where C1 is a coefficient introduced by considering that for simplification the average pressure drop across the impeller
chemical engineering research and design 8 8 ( 2 0 1 0 ) 1073–1077
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Fig. 1 – Change in liquid temperature behind the blade with impeller speeds for the dual Rushton turbine impeller system. blade is used instead of the local pressure difference distribution over the blade surface in Eq. (3). The values of C1 , which may be dependent of the variation in the pressure distribution on the blade surface, are discussed below. Mochizuki et al. (2008) measured the pressure distributions on the front of the blade. Their results may be approximately presented as: ˘f − gh (DN)2 /2
=˛
(5)
They obtained that the dimensionless pressure on the blade front surface ˛ varies according to the location on the blade surface but it is almost independent on the impeller speed and gas flow rate. The range of ˛ for the six blade disk turbines obtained by Mochizuki et al. (2008) is from 0 to 1.0. Lane et al. (2001) examined the pressure distribution on the surface of Rushton turbine blade. In their study, the experimental data as well as the CFD results indicate that the average value of ˛ in the centre zone of the front blade surface is around 0.5. Using the CFD analysis, Vlaev et al. (2004) found that the ranges of ˛ for the Rushton turbine and the circular half pipe blade impellers are −1.3 to 0.72 and −0.08 to 0.15 in the centre zone of front blade surface, respectively. In this study, on the basis of the results in the literature (Lane et al., 2001; Mochizuki et al., 2008) we assumed that ˛ ∼ 0.5 and furthermore by considering the results of Mochizuki et al. (2008) it is independent on the vapour generation rate as a first approximation. As well as the pressure difference in Eq. (4), we use the average value over the blade surface of ˛ instead of its local value.
Using Eq. (4) we can evaluate (˘ f − b )ave from the measured power consumption, P. The data for P in the boiling stirred tank with the dual impeller systems are presented in Fig. 2 of our previous paper (Fukuda et al., 2009). Similar results were obtained for the triple impeller systems. From the calculated (˘ f − b )ave and Eq. (5), we can evaluate ˘ b and then estimate liquid temperature corresponding to ˘ b , T, using Antoine’s equation. The liquid temperature behind the impeller blade, T, in Kelvin can be calculated by Antoine’s equation using the pressure ˘ b in Pa. By rearranging Antoine’s equation for vapour pressure we have the following equation for the liquid temperature: T = −c +
b a − ln ˘b
(6)
where a = 23.1964, b = 3816.44 and c = −46.13 for water (Read et al., 1977). By the procedure described above the liquid temperature behind the impeller blades can be estimated from the measured power consumption.
4.
Results and discussion
The model predictions are compared with the liquid temperatures measured using the thermocouple attached to the back of the impeller blade in our previous study (Fukuda et al., 2009). The values of C1 in Eq. (4) used in this study for dual Rushton turbine, pitched-blade downflow disk turbine and concaveblade disk turbine systems are 1.5, 1.8 and 2.0, respectively. They were obtained by fitting the present experimental data. Vlaev et al. (2004) found that the pressure distribution for the
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flat blade is more uniform as compared with that for the circular half-pipe blade. By considering that as can be seen from Eq. (4) C1 is inversely proportional to the pressure difference (˘ f − b )ave at constant power consumption, the values of C1 for different blade designs correspond to the uniformity of pressure difference on different blade designs. This implies that the difference between the average and local pressures over the blade surface is somewhat affected by the impeller designs. For the triple impeller systems the values of C1 are 1.2 times of those for the dual impeller systems. This may be attributed to an interaction among the impellers in the multiple impeller systems. Fig. 1 shows the changes in liquid temperature behind the blade with impeller speed for the dual Rushton turbine impeller system at Ug = 0.014 and 0.027 ms−1 . In our previous study (Fukuda et al., 2009), the impeller speed at which the vapour generation from the top impeller starts, NIm , was defined and measured to characterize the vapourization in the boiling stirred tanks. When the impeller speed was less than NIm , the liquid temperatures behind the blades of top and bottom impellers were nearly constant, respectively and the nucleation was localized on the heater surface. It can be seen in Fig. 1 that the continuous and steep decrease in liquid temperature with increasing impeller speed starts at NIm . For N > NIm , since the lower pressure region was formed behind the impeller blade and then the liquid temperature and the vapour pressure decreased, the nucleation was taken place in the low-pressure region formed behind the blades rather than the heater. The liquid temperatures behind the blade of the top and bottom impellers, Tt and Tb , decreased as the impeller speed increased and could be satisfactorily predicted in the whole range of N by the proposed model. As expected from static pressure gradient, Tt was lower than Tb . The changes in Tt and Tb with impeller speed were similar. For reference the measured liquid temperature at the free surface, Ts , is also presented in the figure. While Ts slightly decreased due to the liquid mixing by the impeller at lower impeller speeds, it was nearly independent of the impeller speed at higher impeller speeds. The vapour generation rate was found to have a rather minor effect on the liquid temperature behind the impeller blade. For reference, the calculated results for pressure on the back surface of blades of the bottom and top impellers are presented in Fig. 1. It can seen from the figure that the pressure on the back surface of blades decreased around 1.75 and 1.5 kPa from the static pressure or pressure on the blade surfaces with N = 0 s−1 with impeller speed at Ug = 0.014 and 0.027 ms−1 , respectively. These decreases in the pressure are caused by the rotation of the impeller. The changes in liquid temperature behind the blade for three dual impeller systems (Rushton turbines, pitched-blade downflow disk turbines and concave-blade disk turbines) are depicted in Fig. 2. The liquid temperatures behind the blades of the top and bottom impellers, Tt and Tb , decreased with impeller speed for N > NIm . On the whole, the model could predict the experimental data reasonably well. The decrease in liquid temperature behind the blade for the concave-blade disk turbines was somewhat steeper as compared with those for Rushton turbines and pitched-blade downflow disk turbines. For reference, the boiling temperatures at 101 kPa and the heater evaluated using Eq. (6) are presented in the figure. At higher impeller speeds the liquid temperature behind the blade of the top impeller was lower than the boiling temperature at 101 kPa. It can be seen in Fig. 2 that the change of nucleation site from the heater to the top impeller is due
Fig. 2 – Change in liquid temperature behind the blade with impeller speeds for the dual impeller systems at Ug = 0.012 ms−1 . to the formation of low-pressure regions behind the blades and the corresponding decrease in boiling temperature with increasing impeller speed. In Fig. 3, the model predictions are compared with the experimental results obtained at Ug = 0.012 ms−1 for triple impeller systems. Again the continuous decreases in liquid temperature behind the blades of top, middle and bottom impellers, Tt , Tm and Tb , started at N = NIm . The liquid temperatures behind the blades of the impellers, Tt and Tb , somewhat sharply decreased with impeller speed for N > NIm . At higher impeller speeds the liquid temperature behind the blade of the top impeller was reduced to be lower than the boiling temperature at 101 kPa as well as the dual impeller systems. The comparison between the model and the experimental data shows reasonably good agreement.
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the literature (Mochizuki et al., 2008). In the model the liquid temperature behind the impeller blades in which nucleation takes place is readily estimated from the measured power consumption. The model could predict the decreases in pressure and liquid temperature behind the impeller blades with impeller speeds. Predictions of the proposed model for liquid temperature behind the blades were in reasonable agreement with the experimental results carried out in a laboratory-scale boiling stirred tank. Therefore, the model may be used to provide a reasonable prediction of liquid temperature behind the impeller blade in boiling stirred tanks. It should be noted that our experiments were performed in rather small stirred tank reactor and the effect of hydrostatic pressure on the change in the vapour generation site may be rather small. This subject is one of our future studies. Further understanding of the local temperature homogeneities and vapour–liquid two-phase mixing around the heater and impellers is also desirable.
References
Fig. 3 – Change in liquid temperature behind the blade with impeller speeds for the triple impeller systems at Ug = 0.012 ms−1 .
5.
Conclusions
In order to predict the change in liquid temperature behind the impeller blades with impeller speeds in boiling stirred tanks a simple model has been developed. The model is based on the results for pressure distribution on the impeller blade in
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