- Email: [email protected]
Hydrodynamics of the tank with a screw impeller
Chemical Engineering and Processing 44 (2005) 766–774
Hydrodynamics of the tank with a screw impeller Czeslaw Kuncewicz∗ , Krzysztof Szulc, Tomasz Kurasinski Department of Process Equipment, Lodz Technical University, ul. Wolczanska 213, 90-924 Lodz, Poland Received 21 July 2003; received in revised form 11 August 2004; accepted 11 August 2004 Available online 31 October 2004
Abstract Results of mathematical modelling of liquid flow in the tank with a screw impeller operating in the system equipped with a diffuser in a laminar regime are presented. A two-dimensional model describing liquid flow in the tank constructed in the Department of Process Equipment, LTU, was used to determine velocity fields. Results obtained were compared with experimental data given by Seichter [Coll. Czech. Chem. Commun. 46 (1981) 2032–2042]. Satisfactory agreement of the experimental and model values was obtained which confirmed that the model could be used in practical calculations. An optimisation criterion of the tank-impeller system regarding mixing time was proposed and screw impellers were optimised. © 2004 Elsevier B.V. All rights reserved. Keywords: Hydrodynamics; Screw impeller; Laminar mixing
1. Introduction Mixing is a dynamic process whose driving force is the pressure difference in various tank regions that is produced by a rotating impeller [1]. The aims of mixing can be different. They can include, for instance, the intensification of chemical reactions, production of a homogeneous mixture (emulsion or suspension) and enhancement of heat or mass transfer. The transformation of mechanical energy supplied by the impeller into kinetic energy of liquid in the tank is much more efficient in turbulent motion than in laminar one. However, in the case when liquids of viscosity exceeding 0.5 Pa s are mixed, the turbulent mixing range is practically unattainable. This refers to such technological operations as mixing of pastes, bituminous resins, molten polymers, etc. In the case of polymerisation, a precise mixing of the reactor content determines the difference of molecular masses of a polymer, hence it determines the final quality of the polymer. Polymerisation is most frequently carried out in tank reactors, in which, additionally, good heat exchange should be ensured.
∗
Corresponding author. Tel.: +48 42 636 3700; fax: +48 42 636 5663. E-mail address: [email protected] (C. Kuncewicz).
0255-2701/$ – see front matter © 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.cep.2004.08.006
So, numerous studies are dedicated to the subject of heat transfer in laminar motion [2–4]. At a significant liquid viscosity in the tank, only impellers of big diameters such as helical ribbon impellers can induce sufficient circulation in the entire tank volume. Competitive to the helical ribbon impellers are screw impellers operating in a diffuser, which despite small diameter also produce significant secondary circulation in the whole tank. This is undoubtedly a result of a remarkable pumping effect of the impeller irrespective of the liquid viscosity, because the work of this impeller can be compared to the work of a screw extruder. The screw impellers are classified to slow-speed impellers that operate at 1–4 revolutions per second and are particularly suitable for mixing of highly viscous liquids (to 105 mPa s) [1]. They can work both in unbaffled and baffled tanks. In the latter case, it is recommended that the baffles be at a distance not bigger than the baffle width from the tank walls [1]. However, despite its attractiveness, this type of impeller has not been sufficiently described in the literature. To fully estimate the impeller operation, some specific parameters such as primary and secondary circulation in the tank, time of mixing and circulation, pumping efficiency and power demand are analysed. All above-mentioned values are
C. Kuncewicz et al. / Chemical Engineering and Processing 44 (2005) 766–774
the functions of liquid velocity distributions in the tank, hence knowledge of these distributions is so important when studying the process of mixing.
2. Research approach Studies on mixing processes are usually carried out with models which differ from real machines by dimensions and technological factors. To make results of these studies applicable, e.g. in scaling-up of the process, they are generalised in the form of dimensionless equations with variables being the dimensionless similarity modules, called also dimensionless numbers. The above-mentioned analytical method is a result of a macroscopic approach to the problem. It enables a simple calculation of some utility values such as power demand, but it does not permit to wholly analyse the phenomena that take place in the tank. Another method of analysis is to compare the work of a given impeller type to the work of another rotating element, whose mathematical description is known. Worth mentioning are the studies by Chavan and Ulbrecht [5], who proposed a model of the tank where the impeller was replaced by a cylinder of a certain equivalent diameter and next the shearing rates on the model cylinder surface and inside the diffuser were analysed. One more macroscopic approach is shown in study [6], where the authors adapt the relations used in the design of screws for extruders. Recently, a quick advance of computer techniques allowed mathematical modelling to be used in studies of grinding processes. The basis for analysis is Navier–Stokes momentum transfer equations. The information that can be obtained from these solutions is incomparably richer, however, this method is very expensive from the computational point of view. To solve numerically the initial system of differential equations, some simplifying assumptions should be made and boundary conditions should be determined properly, which can present difficulties and even cause erroneous results. Many mathematical models whose complexity is very different have been and are still formulated. Owing to problems in the determination of turbulence structure in the tank, the most complicated are the three-dimensional models that describe turbulent motion. Also in the laminar motion, in spite of neglecting turbulence in the tank, the most complex models are 3D models for all three velocity components. In some cases, the complex 3D models can be simplified to a great extent. This refers particularly to the cases when the distribution of one of the velocity components can be neglected or secondary circulation in the tank can be omitted. A separate category are the professional integrated packages, such as FLUENT, FLOW 3D, FIDEAP and others, which can be used for modelling not only single-phase but also multi-phase flows of an arbitrary geometry. A disadvantage of using the integrated packages are high licence fees
767
and treating the results obtained as a consequence of the application of a “black box”, because it is absolutely impossible to learn the calculation algorithms. The authors present results of mathematical modelling of liquid flow in the tank with screw impellers operating in the diffuser. For this purpose, a two-dimensional mathematical model for pitched blade impellers [7] and helical ribbon impellers [8] working in the laminar motion was used. In the mentioned model, no tangential velocity distribution was assumed. The parameters were two constants that characterised form drags and frictional resistance of the rotating element. In the previous study [9], the applicability of the abovementioned model for screw impellers was investigated only from the point of view of power demand. Good agreement of experimental torque and the one obtained on the basis of model calculations allows us to expect that the same model can predict both axial and radial velocity distributions in the tank, i.e. it will predict correctly the value of primary and secondary circulations. Similar conclusions can be drawn on the basis of study [2], where a possibility of practical application of the hydrodynamic model in the calculation of temperature distribution and heat flux at the wall of a tank with a helical ribbon impeller was shown. An accurate description of this solution can be found in studies [2,7]. In general, solution of the model consisted in the solution of the Navier–Stokes motion equations by iterative methods in the nodes of a network with predetermined density. At the beginning, some velocity distribution satisfying boundary conditions is assumed, next for each node of the numerical network a new value was calculated on the basis of velocity in the neighbouring nodes. If the mean error of particular equations was below 0.5%, the model was assumed to be solved. The most important simplification of the model was an assumption that all tangential velocity components were stable, i.e. that for a fixed radius and height this component is constant and does not change with the angle. The parameters of the analysed model were two constants characterising the form drags and frictional resistance of the rotating element, whose values should be known when solving the model. These parameters can be determined by comparing some global values, e.g. experimental torques and axial forces acting on the impeller blades, with the values of these theoretically calculated from the model (at earlier assumed values of these constants). For instance, in study [9], numerical values of both parameters were determined on the basis of the torque acting on the screw impeller shaft that was measured in study [10].
3. Experimental set-up Experimental data needed to verify the mathematical model for screw impellers were taken from the study by Seichter [11]. Seichter gave numerical values and relevant graphical relations concerning power demand and pumping efficiency of screw impellers. He presented also many di-
768
C. Kuncewicz et al. / Chemical Engineering and Processing 44 (2005) 766–774
Fig. 1. Experimental set-up to measure axial velocities [11]: A – transducer, T – thermoanemometer.
agrams representing the dependence of axial velocities on the position of a measuring point in the space between the diffuser and the tank wall. The axial velocity field was determined using a measuring thermoanemometric system. A schematic diagram of the measuring device used in study [11] is shown in Fig. 1. The thermoanemometer was located radially in the tank in the middle of the liquid height (also in the middle of the diffuser height). Measurements were made for the tank of diameter D = 290 mm with a flat bottom. The impeller along with the diffuser was placed in the tank axis, in the middle of liquid height. The experimental liquids were the solutions of starch syrups. In all measurements, the liquid was pumped downwards. Table 1 gives dimensions of the tested screw impellers. For the sake of clarity, an original enumeration of the impellers taken from study [11] was preserved in this work. As follows from Table 1, in study [11] 10 impellers with four different diameters (the impeller diameter doubled) and seven different worm pitches were tested. For the fixed parameter d/D = 0.43, the worm pitch was changing in a broad range s/d = 0.33–1.5. In all experiments, the unchanged parameters were the ratio of inner diffusion diameter to the impeller diameter (D1 /d = 1.1) and the ratio of the impeller height to the tank diameter h/D = 2/3.
4. Comparison of model and experimental results Modelling of liquid flow in the tank with a screw impeller operating in the diffuser was started with setting up numerical values of model parameters Cn and Ct [7]. These constants were determined using experimental results related to the dimensionless number of secondary circulation given in study [11]. After the initial calculations, the values of these constants were established as Cn = 1500 and Ct = 80. They differed from similar constants determined in the previous study [8] and were a result of differences in the design of screw impellers used in both studies. In work [8] such screw impellers were tested that had the worm made of a thick sheet which constituted around 7% of the impeller diameter. This parameter is known to have a significant effect on power demand required to drive the impeller. After setting up the constant, complete calculations were made for the impellers tested by Seichter [11] for the same Reynolds number. Upon reaching a sufficient convergence of calculations (error <0.5%), the results obtained in the dimensionless form were converted (according to the equation quoted in studies [7,8]) to dimensional values used for a direct comparison with the experimental data given in study [11].
Table 1 Diameters of the tested screw impellers (d0 = 32 mm, D1 /d = 1.1) [11] Dimension
Impeller 1
d (mm) d/D s (mm) s/d h (mm)
86 0.29 86 1.0 129
2A
42 0.33 189
2B
58 0.46
2C
76 0.60
2D
94.5 0.75
2E
126 1.0
2F
2G
3
4
168 1.33
126 0.43 189 1.5
145 0.5 145 1.0 218
183 0.63 170 0.93 250
C. Kuncewicz et al. / Chemical Engineering and Processing 44 (2005) 766–774
Fig. 2. Comparison of axial velocities in the diffuser–tank wall region for impeller number 2C.
Fig. 3. Comparison of axial velocities in the diffuser–tank wall region for impeller number 2C.
Figs. 2–5 show the results of modelling in the form of a continuous line, experimental points taken from study [11] and relative percentage error of the two values. For instance, the axial velocity distributions in the external diffuser wall–inner tank wall region shown in Figs. 2–5 are practically symmetrical and have a parabolic shape.
Fig. 4. Comparison of axial velocities in the diffuser–tank wall region for impeller number 3.
769
Fig. 5. Comparison of axial velocities in the diffuser–tank wall region for impeller number 4.
The maximum axial velocities occur halfway between the diffuser–tank wall distance and these values are of the order of several cm/s. Near the tank wall and at the diffuser surface, the values of this velocity component drop to zero, which is in agreement with physical understanding of the phenomenon. It can be observed that axial velocities are proportional to the Reynolds number, which confirms that the dimensionless number of secondary circulation is constant. Good agreement of experimental and model results was obtained. The mean error for tested velocity distributions was 9.9%, which can be taken as an admissible value. In study [11], results of measurements of power demand for Re < 15 are presented in the form of product Aexp = Lm·Re. These were the mean values of parameter Aexp for each of the 10 tested impellers. In our research model, calculations were made for the same impellers for Re = 1, 2, 3, 5, 10 and 15, respectively, and on this basis the mean values of parameters Anum were determined for laminar mixing. A comparison of experimental and model results is given in Table 2. The model value of product Lm·Re obtained in the tested range of number Re for each impeller type was approximately constant (the model values differed from each other by around 1%). As follows from Table 2, the mean absolute error between experimental and model values was 10.5% (column no. 8). The value of parameter A was affected greatly by the geometric parameters of the impeller–tank system. Fig. 6 shows the dependence of parameter A on quotient s/d for the determined value d/D = 0.43. Parameter A decreases with an increase of the s/d ratio until reaching a minimum at s/d = 1.3. A further increase of s/d of the impeller causes an increase and next stabilisation of this parameter. In an extreme case at s/d → 0, parameter A reaches a maximum value Amax = 1230 and the screw impeller becomes a rotary cylinder. The pumping effect of such an impeller would be practically unnoticeable. In another extreme case, i.e. for s/d → ∞, the screw impeller becomes a single blade and then parameter A tends asymptotically to A = 495. In Fig. 7 the effect of frictional
770
C. Kuncewicz et al. / Chemical Engineering and Processing 44 (2005) 766–774
Table 2 Comparison of experimental and model values of product A = Lm·Re Impeller
D/d
s/d
i
Aexp
Anum
Acorr
Error 1 (%)
Error 2 (%)
1 2A 2B 2C 2D 2E 2F 2G 3 4
3.37 2.30 2.30 2.30 2.30 2.30 2.30 2.30 2.00 1.59
1.00 0.33 0.46 0.60 0.75 1.00 1.33 1.50 1.00 0.93
1.5 4.5 3.2 2.5 2 1.5 1.125 1 1.5 1.5
207.2 473.6 371.2 306.1 244.9 215.8 210.3 211.0 231.0 224.0
195.4 332.9 285.1 249.5 224 219 217.6 224.8 229.9 240.6
231.8 435.4 361.8 309.8 273.4 231.8 196.5 183.6 232.2 223.8
5.7 29.7 23.2 18.5 8.5 −1.5 −3.5 −6.5 0.5 −7.4
11.4 −8.1 −2.5 1.2 11.6 7.4 −6.5 −12.9 0.5 0.1
power demand, because, taking into account the definitions of power number and Reynolds number we will obtain P ∼ d3 . Only at D/d > 3 the value of A decreases and one may expect a minimum value of A at D/d → ∞. In this boundary case, the screw impeller would become a fully developed helical ribbon impeller. The data given in Table 2 can be described by a correlation equation that refers to the effect of the most important geometric parameters of the screw impeller, i.e. impeller diameter d, worm pitch s, number of worm coils i and the clearance between the worm edge and diffuser wall c defined by Eq. (1) on parameter A: Fig. 6. Dependence of constant A on s/d at constant value of invariant D/d = 2.3.
resistance and form drags on the mixing power of screw impellers can be easily followed. For low values of parameter s/d the first resistance dominates, while for almost vertical worms (high values of s/d) form drags are dominant. The lowest values of parameter A, i.e. the lowest power demand, were revealed by the impellers with relative pitch s/d = 1.0–1.5. Fig. 7 shows the dependence of parameter A on the impeller size, i.e. D/d ratio at constant invariant s/d = 1.0. For most frequently used invariants D/d = 1.7–3, the effect of this invariant on parameter A practically does not exist. Naturally, this does not mean that the impeller diameter has no effect on
c=
D1 − d 2D1
(1)
In study [11], the effect of clearance c on power demand was not investigated (D1 /d ratio was constant and equal to 1.1) therefore, according to the results of our previous work [9], in correlation equation (2) it was assumed preliminarily that the value of exponent at variable c was equal to −0.26. After making relevant calculations, correlation equation (2) was obtained: 0.25 D − d −0.26 1 0.82 s A = 74.3i (2) d 2D1 In column 9 (Table 2), the relative correlation error for subsequent impellers is given. A mean value of this error was 6.3%. In Eq. (2) simplex (D/d) was not taken into account because, as follows from study [4], there is practically no such dependence. Eq. (2) holds for the range of parameters: i = 1–4.5, s/d = 0.33–1.5, D/d = 2–3.4. Eq. (2) is very close similar to Eq. (3) derived in one of our previous studies [9]: s 0.22 D − d −0.26 1 A = Re · Lm = 64i (3) d 2D1 The value that affects greatly the time of mixing is radial–axial circulation Vs . In study [11] this value is given as a dimensionless number of secondary circulation Ks defined by Eq. (4):
Fig. 7. Dependence of constant A on D/d at constant value of invariant s/d = 1.0.
Ks =
Vs nd 3
(4)
C. Kuncewicz et al. / Chemical Engineering and Processing 44 (2005) 766–774
771
Table 3 Comparison of experimental and model values of the secondary circulation number Impeller
D/d
s/d
Ks exp
Ks num
Error (%)
1 2A 2B 2C 2D 2E 2F 2G 3 4
3.37 2.30 2.30 2.30 2.30 2.30 2.30 2.30 2.00 1.59
1.00 0.33 0.46 0.60 0.75 1.00 1.33 1.50 1.00 0.93
0.368 0.061 0.118 0.178 0.236 0.296 0.343 0.357 0.286 0.146
0.302 0.079 0.128 0.176 0.225 0.286 0.372 0.389 0.252 0.144
18.0 −29.5 −8.5 1.1 4.7 3.4 −8.5 −9.0 12.0 1.4
For the same 10 impellers, Table 3 gives mean experimental values of Ks determined in study [11]. Next, as in the case of power demand, numerical calculations were given for the same impellers in the range of Reynolds number Re < 15. The calculated mean values of number Ks num and a comparison of model results with experimental results are also given in Table 3. The model value of number Ks num in the tested range of the Reynolds number for each impeller was approximately constant (a maximum scatter of particular values and the mean value was 4%). As follows from Table 3, the mean absolute error between the experimental and model values was 9.6%, which is satisfactory. Figs. 8 and 9 show the analysis of the effect of geometric parameters of the screw impeller–tank on secondary circulation number Ks . The value of Ks was determined from model calculations. A maximum of function Ks = 0.46 (Fig. 8) occurs at s/d = 3.0. The initial growth of the s/d ratio causes a very fast increase of Ks (the pumping effect of the impeller increases) and next a decrease with an increase of quotient s/d. In the boundary case, when the impeller becomes a rotary cylinder (s/d → 0), the number of secondary circulation reaches a minimum on the level 0.00035. Two opposite effects of the worm pitch are clearly visible here. With an increase of the pitch, the theoretical height to which liquid element will be transported during one impeller rotation grows, and
Fig. 9. Dependence of constant Ks on D/d at a constant value of invariant s/d = 1.0.
at the same time the number of effective worm coils transporting these elements decreases. Hence, at a certain pitch of the worm the flow inside the diffuser is the biggest, i.e. the value of Ks is the highest. Fig. 9 illustrates function Ks = f(D/d). Maximum of this function occurs at D/d = 3.0 and is equal to Ks = 0.3. In this case, it is difficult to analyse directly this function because the impeller diameter d is included in the definition of Ks . Satisfactory agreement of experimental and model global values, as in the case of power demand (constant A), and also the secondary circulation Ks , and local values (axial velocity) allowed us to assume that the model could be used in the practical calculations of screw impellers operating in the system with a diffuser and in the estimation of efficiency of the impeller operation.
5. Optimisation of the impeller–tank system A basis for developing an optimisation criterion was a relation that defined the energy of homogenisation E = Pτm
(5)
In the laminar region, the power number is inversely proportional to Reynolds number, hence P = An2 d 3 η
(6)
The time of circulation is equal to the ratio of mixture volume to secondary circulation: τm =
Fig. 8. Dependence of constant Ks on s/d at a constant value of invariant D/d = 2.3.
V Vs
(7)
Taking into account Eq. (3) that defines the value of Ks and assuming that the time of homogenisation is four times higher than the time of circulation [1], after some simple transformations, we obtain a formula of the modified power number E* : 3 E A H D ∗ (8) E ≡ 3 =π Ks D d nd η
772
C. Kuncewicz et al. / Chemical Engineering and Processing 44 (2005) 766–774
Table 4 Value of E* for subsequent impellers in the range Re < 15 Impeller
D/d
s/d
E*
1 2A 2B 2C 2D 2E 2F 2G 3 4
3.37 2.30 2.30 2.30 2.30 2.30 2.30 2.30 2.00 1.59
1.00 0.33 0.46 0.60 0.75 1.00 1.33 1.50 1.00 0.93
86236 161072 84092 54208 39362 29390 23140 22200 23258 21179
Both parameter A and secondary circulation number Ks are exclusively the functions of geometric simplexes of the impeller–tank system. This means that E* should also be a function of the same system. The smaller is the modified power number E* , the more efficient is the tank operation as far as the produced effect is concerned (in the form of secondary circulation) in relation to power demand. Such a quantitative description of the criterion function allows us to use the information obtained on the basis of two-dimensional model described earlier in this work. Table 4 gives the values of modified power number E* for the analysed impellers for laminar mixing. Figs. 10 and 11 show the effect of basic impeller dimensions on power number E* . From analysis of these diagrams, it follows that an optimum impeller is the one with the following parameters: s/d = 1.5 and D/d = 1.9. The shape of functions in Figs. 10 and 11 can be explained while analysing the curves shown in Figs. 6–9. It is worth mentioning that in all analysed cases the clearance between the worm edge and the diffuser was constant (D1 /d = 1.1 = const).
Fig. 10. Dependence of E* on s/d at a constant value of invariant D/d = 2.3.
Fig. 11. Dependence of E* on D/d at a constant value of invariant s/d = 1.0.
Fig. 12. Liquid circulation in the tank with impeller number 2C at Re = 1.0.
C. Kuncewicz et al. / Chemical Engineering and Processing 44 (2005) 766–774
773
Fig. 13. Liquid circulation in the tank with impeller number 3 at Re = 20.
6. Liquid circulation in the tank Figs. 12 and 13 show tangential and axial velocity distributions and the way of liquid circulation in the tank for selected screw impellers operating in the diffuser. This enables better understanding of the phenomena that take place in the tank. The direction in which liquid circulates in the tank shown in Figs. 12 and 13 is a result of a pumping character of the screw impeller operation. Irrespective of geometric parameters, for low values of the Reynolds number of the order of unity, there is one big vortex of secondary circulation in the tank. It covers the whole tank region, and the direction of flow is in agreement with the direction of pumping of the impeller. At smaller values of the Reynolds number, a very weak additional vortex of secondary circulation appears in the bottom part of the tank (below the worm) that rotates in the direction opposite to the main vortex. The power of this vortex increases with a further growth of the Reynolds number. As follows from the right-hand sides of the two figures, in the region between the diffuser and the tank wall there is practically no liquid circulation in the tangential direction. A reason is the presence of the diffuser that efficiently suppresses liquid rotation in the tangential direction. Liquid in the diffuser is pumped downwards first of all by the end of the band. The axial velocity grows nearly linearly with the radius almost to the worm edge, just to drop to zero due to a small distance to the inner diffuser surface. This means that most of the liquid in the diffuser is pumped by external worm edges. With an increase of the impeller diameter and a decrease of liquid viscosity, axial velocity in the diffuser still grows with a radius but at a slightly smaller rate. Results obtained confirm the conclusions presented in study [9] where also the effect
of the impeller–tank system geometry on liquid circulation in the tank with a screw impeller was analysed.
7. Conclusions • Results of model axial velocity distributions were compared with experimental data given in study [11]. The error of comparison equal to 9.0% provides an evidence that the analysed mathematical model can be applied in practical calculations and in the investigation of phenomena that occur in the tank with a screw impeller. • An optimisation criterion in the form of a modified power number was proposed. On the basis of values calculated from the model, an optimum choice of geometry of the impeller–tank system can be made. • Universal information obtained as a result of mathematical modelling enables a complete analysis of phenomena that occur in the tank within the laminar motion.
Acknowledgment This study was carried out within the project W10/1/2004/ DS. Appendix A. Nomenclature A product of power and Reynolds numbers (–) c clearance between worm edge and diffuser wall (–) Cn , Ct parameters of model (–) d external diameter of the worm band (m)
774
d0 D D1 E E* h H H1
C. Kuncewicz et al. / Chemical Engineering and Processing 44 (2005) 766–774
i Ks Lm n n* P r r* Re s u ur uz uϕ v V Vs w z z*
inner diameter of the worm band (hub diameter) (m) inner tank diameter (m) inner diffuser diameter (m) energy of homogenisation (J) modified power number, E* = E/nd3 η (–) screw impeller height (m) liquid height in the tank (m) distance from liquid surface to the diffuser or from the diffuser to the tank bottom (m) number of worm coils (–) secondary circulation number Ks = Vs /nd3 (–) power number, Lm = P/n3 d5 ρ (–) tangential velocity of the impeller (s−1 ) relative angular velocity (–) power (W) tank radius (m) dimensionless tank radius, r* = 2r/D (–) Reynolds number, Re = nd2 ρ/η (–) height of one coil of the impeller ribbon (m) dimensionless tangential velocity, u = uϕ /πDn (–) radial velocity (m/s) axial velocity (m/s) tangential velocity (m/s) dimensionless axial velocity, v = uz /πDn (–) liquid volume in the tank (m3 ) volumetric rate of radial–axial circulation (m3 /s) dimensionless radial velocity, w = ur /πDn (–) liquid height in the tank (m) dimensionless height, z* = z/H (–)
Greeks η ρ τc τm Φ
dynamic viscosity (Pa s) density (kg/m3 ) time of circulation (s) time of mixing (s) current function (–)
Subscripts corr calculated with correlation values exp experimental values num numerical values
References [1] F. Str˛ek, Mixing and Stirring Apparatuses (Mixer), WNT, Warsaw, 1981. [2] C. Kuncewicz, M. Pietrzykowski, A. Niedzielska, Modelling of heart transfer in the tank with ribbon impellers, In˙z. Chem. Proc. 23 (2002) 21–38. [3] M. Kuriyama, K. Arai, S. Saito, Mechanism of heat transfer to pseudoplastic fluids in an agitated tank with helical ribbon impeller, J. Chem. Eng. Jpn. 16 (6) (1983) 489–494. [4] M. Kaminoyama, M. Watanabe, K. Nishi, M. Kamiwano, Numerical simulation of local heat transfer coefficient in stirred vessel with impeller for highly viscous fluid, J. Chem. Eng. Jpn. 32 (1) (1999) 23–30. [5] V.V. Chavan, J. Ulbrecht, Power correlations for close-clearance screw impellers in non-Newtonian liquids, Ind. End. Chem. Process Des. Dev. 12 (1973) 472–476. [6] P. Seichter, J. Dohnal, F. Rieger, Process characteristics of screw impellers with a draught tube for Newtonian liquids: the power input, Coll. Czech. Chem. Commun. 46 (1981) 2007–2020. [7] C. Kuncewicz, M. Pietrzykowski, Hydrodynamic model of a mixing vessel with pitched-blade turbines, Chem. Eng. Sci. 56 (2001) 4659–4672. [8] C. Kuncewicz, M. Pietrzykowski, K. Szulc, Modelling the hydrodynamics of agitator for ribbon impellers working in laminar flow, In˙z. Chem. Proc. 22 (2001) 461–481. [9] C. Kuncewicz, K. Szulc, P. Budzy´nski, Modelling of highly viscous liquid flow in a tank with screw impellers, In˙z. Chem. Proc. 22 (3C) (2001) 837–842. [10] P. Budzy´nski, Reometr rotacyjny typu “on-line” do pomiaru własno´sci reologicznych zawiesin fermentacyjnych, Ph.D. thesis, Faculty of Process and Environmental Engineering, Technical University of Lodz, 1992. [11] P. Seichter, Process characteristics of screw impellers with a draught tube for Newtonian liquids: pumping capacity of the impeller, Coll. Czech. Chem. Commun. 46 (1981) 2032–2042.
Hydrodynamics of the tank with a screw impeller Czeslaw Kuncewicz∗ , Krzysztof Szulc, Tomasz Kurasinski Department of Process Equipment, Lodz Technical University, ul. Wolczanska 213, 90-924 Lodz, Poland Received 21 July 2003; received in revised form 11 August 2004; accepted 11 August 2004 Available online 31 October 2004
Abstract Results of mathematical modelling of liquid flow in the tank with a screw impeller operating in the system equipped with a diffuser in a laminar regime are presented. A two-dimensional model describing liquid flow in the tank constructed in the Department of Process Equipment, LTU, was used to determine velocity fields. Results obtained were compared with experimental data given by Seichter [Coll. Czech. Chem. Commun. 46 (1981) 2032–2042]. Satisfactory agreement of the experimental and model values was obtained which confirmed that the model could be used in practical calculations. An optimisation criterion of the tank-impeller system regarding mixing time was proposed and screw impellers were optimised. © 2004 Elsevier B.V. All rights reserved. Keywords: Hydrodynamics; Screw impeller; Laminar mixing
1. Introduction Mixing is a dynamic process whose driving force is the pressure difference in various tank regions that is produced by a rotating impeller [1]. The aims of mixing can be different. They can include, for instance, the intensification of chemical reactions, production of a homogeneous mixture (emulsion or suspension) and enhancement of heat or mass transfer. The transformation of mechanical energy supplied by the impeller into kinetic energy of liquid in the tank is much more efficient in turbulent motion than in laminar one. However, in the case when liquids of viscosity exceeding 0.5 Pa s are mixed, the turbulent mixing range is practically unattainable. This refers to such technological operations as mixing of pastes, bituminous resins, molten polymers, etc. In the case of polymerisation, a precise mixing of the reactor content determines the difference of molecular masses of a polymer, hence it determines the final quality of the polymer. Polymerisation is most frequently carried out in tank reactors, in which, additionally, good heat exchange should be ensured.
∗
Corresponding author. Tel.: +48 42 636 3700; fax: +48 42 636 5663. E-mail address: [email protected] (C. Kuncewicz).
0255-2701/$ – see front matter © 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.cep.2004.08.006
So, numerous studies are dedicated to the subject of heat transfer in laminar motion [2–4]. At a significant liquid viscosity in the tank, only impellers of big diameters such as helical ribbon impellers can induce sufficient circulation in the entire tank volume. Competitive to the helical ribbon impellers are screw impellers operating in a diffuser, which despite small diameter also produce significant secondary circulation in the whole tank. This is undoubtedly a result of a remarkable pumping effect of the impeller irrespective of the liquid viscosity, because the work of this impeller can be compared to the work of a screw extruder. The screw impellers are classified to slow-speed impellers that operate at 1–4 revolutions per second and are particularly suitable for mixing of highly viscous liquids (to 105 mPa s) [1]. They can work both in unbaffled and baffled tanks. In the latter case, it is recommended that the baffles be at a distance not bigger than the baffle width from the tank walls [1]. However, despite its attractiveness, this type of impeller has not been sufficiently described in the literature. To fully estimate the impeller operation, some specific parameters such as primary and secondary circulation in the tank, time of mixing and circulation, pumping efficiency and power demand are analysed. All above-mentioned values are
C. Kuncewicz et al. / Chemical Engineering and Processing 44 (2005) 766–774
the functions of liquid velocity distributions in the tank, hence knowledge of these distributions is so important when studying the process of mixing.
2. Research approach Studies on mixing processes are usually carried out with models which differ from real machines by dimensions and technological factors. To make results of these studies applicable, e.g. in scaling-up of the process, they are generalised in the form of dimensionless equations with variables being the dimensionless similarity modules, called also dimensionless numbers. The above-mentioned analytical method is a result of a macroscopic approach to the problem. It enables a simple calculation of some utility values such as power demand, but it does not permit to wholly analyse the phenomena that take place in the tank. Another method of analysis is to compare the work of a given impeller type to the work of another rotating element, whose mathematical description is known. Worth mentioning are the studies by Chavan and Ulbrecht [5], who proposed a model of the tank where the impeller was replaced by a cylinder of a certain equivalent diameter and next the shearing rates on the model cylinder surface and inside the diffuser were analysed. One more macroscopic approach is shown in study [6], where the authors adapt the relations used in the design of screws for extruders. Recently, a quick advance of computer techniques allowed mathematical modelling to be used in studies of grinding processes. The basis for analysis is Navier–Stokes momentum transfer equations. The information that can be obtained from these solutions is incomparably richer, however, this method is very expensive from the computational point of view. To solve numerically the initial system of differential equations, some simplifying assumptions should be made and boundary conditions should be determined properly, which can present difficulties and even cause erroneous results. Many mathematical models whose complexity is very different have been and are still formulated. Owing to problems in the determination of turbulence structure in the tank, the most complicated are the three-dimensional models that describe turbulent motion. Also in the laminar motion, in spite of neglecting turbulence in the tank, the most complex models are 3D models for all three velocity components. In some cases, the complex 3D models can be simplified to a great extent. This refers particularly to the cases when the distribution of one of the velocity components can be neglected or secondary circulation in the tank can be omitted. A separate category are the professional integrated packages, such as FLUENT, FLOW 3D, FIDEAP and others, which can be used for modelling not only single-phase but also multi-phase flows of an arbitrary geometry. A disadvantage of using the integrated packages are high licence fees
767
and treating the results obtained as a consequence of the application of a “black box”, because it is absolutely impossible to learn the calculation algorithms. The authors present results of mathematical modelling of liquid flow in the tank with screw impellers operating in the diffuser. For this purpose, a two-dimensional mathematical model for pitched blade impellers [7] and helical ribbon impellers [8] working in the laminar motion was used. In the mentioned model, no tangential velocity distribution was assumed. The parameters were two constants that characterised form drags and frictional resistance of the rotating element. In the previous study [9], the applicability of the abovementioned model for screw impellers was investigated only from the point of view of power demand. Good agreement of experimental torque and the one obtained on the basis of model calculations allows us to expect that the same model can predict both axial and radial velocity distributions in the tank, i.e. it will predict correctly the value of primary and secondary circulations. Similar conclusions can be drawn on the basis of study [2], where a possibility of practical application of the hydrodynamic model in the calculation of temperature distribution and heat flux at the wall of a tank with a helical ribbon impeller was shown. An accurate description of this solution can be found in studies [2,7]. In general, solution of the model consisted in the solution of the Navier–Stokes motion equations by iterative methods in the nodes of a network with predetermined density. At the beginning, some velocity distribution satisfying boundary conditions is assumed, next for each node of the numerical network a new value was calculated on the basis of velocity in the neighbouring nodes. If the mean error of particular equations was below 0.5%, the model was assumed to be solved. The most important simplification of the model was an assumption that all tangential velocity components were stable, i.e. that for a fixed radius and height this component is constant and does not change with the angle. The parameters of the analysed model were two constants characterising the form drags and frictional resistance of the rotating element, whose values should be known when solving the model. These parameters can be determined by comparing some global values, e.g. experimental torques and axial forces acting on the impeller blades, with the values of these theoretically calculated from the model (at earlier assumed values of these constants). For instance, in study [9], numerical values of both parameters were determined on the basis of the torque acting on the screw impeller shaft that was measured in study [10].
3. Experimental set-up Experimental data needed to verify the mathematical model for screw impellers were taken from the study by Seichter [11]. Seichter gave numerical values and relevant graphical relations concerning power demand and pumping efficiency of screw impellers. He presented also many di-
768
C. Kuncewicz et al. / Chemical Engineering and Processing 44 (2005) 766–774
Fig. 1. Experimental set-up to measure axial velocities [11]: A – transducer, T – thermoanemometer.
agrams representing the dependence of axial velocities on the position of a measuring point in the space between the diffuser and the tank wall. The axial velocity field was determined using a measuring thermoanemometric system. A schematic diagram of the measuring device used in study [11] is shown in Fig. 1. The thermoanemometer was located radially in the tank in the middle of the liquid height (also in the middle of the diffuser height). Measurements were made for the tank of diameter D = 290 mm with a flat bottom. The impeller along with the diffuser was placed in the tank axis, in the middle of liquid height. The experimental liquids were the solutions of starch syrups. In all measurements, the liquid was pumped downwards. Table 1 gives dimensions of the tested screw impellers. For the sake of clarity, an original enumeration of the impellers taken from study [11] was preserved in this work. As follows from Table 1, in study [11] 10 impellers with four different diameters (the impeller diameter doubled) and seven different worm pitches were tested. For the fixed parameter d/D = 0.43, the worm pitch was changing in a broad range s/d = 0.33–1.5. In all experiments, the unchanged parameters were the ratio of inner diffusion diameter to the impeller diameter (D1 /d = 1.1) and the ratio of the impeller height to the tank diameter h/D = 2/3.
4. Comparison of model and experimental results Modelling of liquid flow in the tank with a screw impeller operating in the diffuser was started with setting up numerical values of model parameters Cn and Ct [7]. These constants were determined using experimental results related to the dimensionless number of secondary circulation given in study [11]. After the initial calculations, the values of these constants were established as Cn = 1500 and Ct = 80. They differed from similar constants determined in the previous study [8] and were a result of differences in the design of screw impellers used in both studies. In work [8] such screw impellers were tested that had the worm made of a thick sheet which constituted around 7% of the impeller diameter. This parameter is known to have a significant effect on power demand required to drive the impeller. After setting up the constant, complete calculations were made for the impellers tested by Seichter [11] for the same Reynolds number. Upon reaching a sufficient convergence of calculations (error <0.5%), the results obtained in the dimensionless form were converted (according to the equation quoted in studies [7,8]) to dimensional values used for a direct comparison with the experimental data given in study [11].
Table 1 Diameters of the tested screw impellers (d0 = 32 mm, D1 /d = 1.1) [11] Dimension
Impeller 1
d (mm) d/D s (mm) s/d h (mm)
86 0.29 86 1.0 129
2A
42 0.33 189
2B
58 0.46
2C
76 0.60
2D
94.5 0.75
2E
126 1.0
2F
2G
3
4
168 1.33
126 0.43 189 1.5
145 0.5 145 1.0 218
183 0.63 170 0.93 250
C. Kuncewicz et al. / Chemical Engineering and Processing 44 (2005) 766–774
Fig. 2. Comparison of axial velocities in the diffuser–tank wall region for impeller number 2C.
Fig. 3. Comparison of axial velocities in the diffuser–tank wall region for impeller number 2C.
Figs. 2–5 show the results of modelling in the form of a continuous line, experimental points taken from study [11] and relative percentage error of the two values. For instance, the axial velocity distributions in the external diffuser wall–inner tank wall region shown in Figs. 2–5 are practically symmetrical and have a parabolic shape.
Fig. 4. Comparison of axial velocities in the diffuser–tank wall region for impeller number 3.
769
Fig. 5. Comparison of axial velocities in the diffuser–tank wall region for impeller number 4.
The maximum axial velocities occur halfway between the diffuser–tank wall distance and these values are of the order of several cm/s. Near the tank wall and at the diffuser surface, the values of this velocity component drop to zero, which is in agreement with physical understanding of the phenomenon. It can be observed that axial velocities are proportional to the Reynolds number, which confirms that the dimensionless number of secondary circulation is constant. Good agreement of experimental and model results was obtained. The mean error for tested velocity distributions was 9.9%, which can be taken as an admissible value. In study [11], results of measurements of power demand for Re < 15 are presented in the form of product Aexp = Lm·Re. These were the mean values of parameter Aexp for each of the 10 tested impellers. In our research model, calculations were made for the same impellers for Re = 1, 2, 3, 5, 10 and 15, respectively, and on this basis the mean values of parameters Anum were determined for laminar mixing. A comparison of experimental and model results is given in Table 2. The model value of product Lm·Re obtained in the tested range of number Re for each impeller type was approximately constant (the model values differed from each other by around 1%). As follows from Table 2, the mean absolute error between experimental and model values was 10.5% (column no. 8). The value of parameter A was affected greatly by the geometric parameters of the impeller–tank system. Fig. 6 shows the dependence of parameter A on quotient s/d for the determined value d/D = 0.43. Parameter A decreases with an increase of the s/d ratio until reaching a minimum at s/d = 1.3. A further increase of s/d of the impeller causes an increase and next stabilisation of this parameter. In an extreme case at s/d → 0, parameter A reaches a maximum value Amax = 1230 and the screw impeller becomes a rotary cylinder. The pumping effect of such an impeller would be practically unnoticeable. In another extreme case, i.e. for s/d → ∞, the screw impeller becomes a single blade and then parameter A tends asymptotically to A = 495. In Fig. 7 the effect of frictional
770
C. Kuncewicz et al. / Chemical Engineering and Processing 44 (2005) 766–774
Table 2 Comparison of experimental and model values of product A = Lm·Re Impeller
D/d
s/d
i
Aexp
Anum
Acorr
Error 1 (%)
Error 2 (%)
1 2A 2B 2C 2D 2E 2F 2G 3 4
3.37 2.30 2.30 2.30 2.30 2.30 2.30 2.30 2.00 1.59
1.00 0.33 0.46 0.60 0.75 1.00 1.33 1.50 1.00 0.93
1.5 4.5 3.2 2.5 2 1.5 1.125 1 1.5 1.5
207.2 473.6 371.2 306.1 244.9 215.8 210.3 211.0 231.0 224.0
195.4 332.9 285.1 249.5 224 219 217.6 224.8 229.9 240.6
231.8 435.4 361.8 309.8 273.4 231.8 196.5 183.6 232.2 223.8
5.7 29.7 23.2 18.5 8.5 −1.5 −3.5 −6.5 0.5 −7.4
11.4 −8.1 −2.5 1.2 11.6 7.4 −6.5 −12.9 0.5 0.1
power demand, because, taking into account the definitions of power number and Reynolds number we will obtain P ∼ d3 . Only at D/d > 3 the value of A decreases and one may expect a minimum value of A at D/d → ∞. In this boundary case, the screw impeller would become a fully developed helical ribbon impeller. The data given in Table 2 can be described by a correlation equation that refers to the effect of the most important geometric parameters of the screw impeller, i.e. impeller diameter d, worm pitch s, number of worm coils i and the clearance between the worm edge and diffuser wall c defined by Eq. (1) on parameter A: Fig. 6. Dependence of constant A on s/d at constant value of invariant D/d = 2.3.
resistance and form drags on the mixing power of screw impellers can be easily followed. For low values of parameter s/d the first resistance dominates, while for almost vertical worms (high values of s/d) form drags are dominant. The lowest values of parameter A, i.e. the lowest power demand, were revealed by the impellers with relative pitch s/d = 1.0–1.5. Fig. 7 shows the dependence of parameter A on the impeller size, i.e. D/d ratio at constant invariant s/d = 1.0. For most frequently used invariants D/d = 1.7–3, the effect of this invariant on parameter A practically does not exist. Naturally, this does not mean that the impeller diameter has no effect on
c=
D1 − d 2D1
(1)
In study [11], the effect of clearance c on power demand was not investigated (D1 /d ratio was constant and equal to 1.1) therefore, according to the results of our previous work [9], in correlation equation (2) it was assumed preliminarily that the value of exponent at variable c was equal to −0.26. After making relevant calculations, correlation equation (2) was obtained: 0.25 D − d −0.26 1 0.82 s A = 74.3i (2) d 2D1 In column 9 (Table 2), the relative correlation error for subsequent impellers is given. A mean value of this error was 6.3%. In Eq. (2) simplex (D/d) was not taken into account because, as follows from study [4], there is practically no such dependence. Eq. (2) holds for the range of parameters: i = 1–4.5, s/d = 0.33–1.5, D/d = 2–3.4. Eq. (2) is very close similar to Eq. (3) derived in one of our previous studies [9]: s 0.22 D − d −0.26 1 A = Re · Lm = 64i (3) d 2D1 The value that affects greatly the time of mixing is radial–axial circulation Vs . In study [11] this value is given as a dimensionless number of secondary circulation Ks defined by Eq. (4):
Fig. 7. Dependence of constant A on D/d at constant value of invariant s/d = 1.0.
Ks =
Vs nd 3
(4)
C. Kuncewicz et al. / Chemical Engineering and Processing 44 (2005) 766–774
771
Table 3 Comparison of experimental and model values of the secondary circulation number Impeller
D/d
s/d
Ks exp
Ks num
Error (%)
1 2A 2B 2C 2D 2E 2F 2G 3 4
3.37 2.30 2.30 2.30 2.30 2.30 2.30 2.30 2.00 1.59
1.00 0.33 0.46 0.60 0.75 1.00 1.33 1.50 1.00 0.93
0.368 0.061 0.118 0.178 0.236 0.296 0.343 0.357 0.286 0.146
0.302 0.079 0.128 0.176 0.225 0.286 0.372 0.389 0.252 0.144
18.0 −29.5 −8.5 1.1 4.7 3.4 −8.5 −9.0 12.0 1.4
For the same 10 impellers, Table 3 gives mean experimental values of Ks determined in study [11]. Next, as in the case of power demand, numerical calculations were given for the same impellers in the range of Reynolds number Re < 15. The calculated mean values of number Ks num and a comparison of model results with experimental results are also given in Table 3. The model value of number Ks num in the tested range of the Reynolds number for each impeller was approximately constant (a maximum scatter of particular values and the mean value was 4%). As follows from Table 3, the mean absolute error between the experimental and model values was 9.6%, which is satisfactory. Figs. 8 and 9 show the analysis of the effect of geometric parameters of the screw impeller–tank on secondary circulation number Ks . The value of Ks was determined from model calculations. A maximum of function Ks = 0.46 (Fig. 8) occurs at s/d = 3.0. The initial growth of the s/d ratio causes a very fast increase of Ks (the pumping effect of the impeller increases) and next a decrease with an increase of quotient s/d. In the boundary case, when the impeller becomes a rotary cylinder (s/d → 0), the number of secondary circulation reaches a minimum on the level 0.00035. Two opposite effects of the worm pitch are clearly visible here. With an increase of the pitch, the theoretical height to which liquid element will be transported during one impeller rotation grows, and
Fig. 9. Dependence of constant Ks on D/d at a constant value of invariant s/d = 1.0.
at the same time the number of effective worm coils transporting these elements decreases. Hence, at a certain pitch of the worm the flow inside the diffuser is the biggest, i.e. the value of Ks is the highest. Fig. 9 illustrates function Ks = f(D/d). Maximum of this function occurs at D/d = 3.0 and is equal to Ks = 0.3. In this case, it is difficult to analyse directly this function because the impeller diameter d is included in the definition of Ks . Satisfactory agreement of experimental and model global values, as in the case of power demand (constant A), and also the secondary circulation Ks , and local values (axial velocity) allowed us to assume that the model could be used in the practical calculations of screw impellers operating in the system with a diffuser and in the estimation of efficiency of the impeller operation.
5. Optimisation of the impeller–tank system A basis for developing an optimisation criterion was a relation that defined the energy of homogenisation E = Pτm
(5)
In the laminar region, the power number is inversely proportional to Reynolds number, hence P = An2 d 3 η
(6)
The time of circulation is equal to the ratio of mixture volume to secondary circulation: τm =
Fig. 8. Dependence of constant Ks on s/d at a constant value of invariant D/d = 2.3.
V Vs
(7)
Taking into account Eq. (3) that defines the value of Ks and assuming that the time of homogenisation is four times higher than the time of circulation [1], after some simple transformations, we obtain a formula of the modified power number E* : 3 E A H D ∗ (8) E ≡ 3 =π Ks D d nd η
772
C. Kuncewicz et al. / Chemical Engineering and Processing 44 (2005) 766–774
Table 4 Value of E* for subsequent impellers in the range Re < 15 Impeller
D/d
s/d
E*
1 2A 2B 2C 2D 2E 2F 2G 3 4
3.37 2.30 2.30 2.30 2.30 2.30 2.30 2.30 2.00 1.59
1.00 0.33 0.46 0.60 0.75 1.00 1.33 1.50 1.00 0.93
86236 161072 84092 54208 39362 29390 23140 22200 23258 21179
Both parameter A and secondary circulation number Ks are exclusively the functions of geometric simplexes of the impeller–tank system. This means that E* should also be a function of the same system. The smaller is the modified power number E* , the more efficient is the tank operation as far as the produced effect is concerned (in the form of secondary circulation) in relation to power demand. Such a quantitative description of the criterion function allows us to use the information obtained on the basis of two-dimensional model described earlier in this work. Table 4 gives the values of modified power number E* for the analysed impellers for laminar mixing. Figs. 10 and 11 show the effect of basic impeller dimensions on power number E* . From analysis of these diagrams, it follows that an optimum impeller is the one with the following parameters: s/d = 1.5 and D/d = 1.9. The shape of functions in Figs. 10 and 11 can be explained while analysing the curves shown in Figs. 6–9. It is worth mentioning that in all analysed cases the clearance between the worm edge and the diffuser was constant (D1 /d = 1.1 = const).
Fig. 10. Dependence of E* on s/d at a constant value of invariant D/d = 2.3.
Fig. 11. Dependence of E* on D/d at a constant value of invariant s/d = 1.0.
Fig. 12. Liquid circulation in the tank with impeller number 2C at Re = 1.0.
C. Kuncewicz et al. / Chemical Engineering and Processing 44 (2005) 766–774
773
Fig. 13. Liquid circulation in the tank with impeller number 3 at Re = 20.
6. Liquid circulation in the tank Figs. 12 and 13 show tangential and axial velocity distributions and the way of liquid circulation in the tank for selected screw impellers operating in the diffuser. This enables better understanding of the phenomena that take place in the tank. The direction in which liquid circulates in the tank shown in Figs. 12 and 13 is a result of a pumping character of the screw impeller operation. Irrespective of geometric parameters, for low values of the Reynolds number of the order of unity, there is one big vortex of secondary circulation in the tank. It covers the whole tank region, and the direction of flow is in agreement with the direction of pumping of the impeller. At smaller values of the Reynolds number, a very weak additional vortex of secondary circulation appears in the bottom part of the tank (below the worm) that rotates in the direction opposite to the main vortex. The power of this vortex increases with a further growth of the Reynolds number. As follows from the right-hand sides of the two figures, in the region between the diffuser and the tank wall there is practically no liquid circulation in the tangential direction. A reason is the presence of the diffuser that efficiently suppresses liquid rotation in the tangential direction. Liquid in the diffuser is pumped downwards first of all by the end of the band. The axial velocity grows nearly linearly with the radius almost to the worm edge, just to drop to zero due to a small distance to the inner diffuser surface. This means that most of the liquid in the diffuser is pumped by external worm edges. With an increase of the impeller diameter and a decrease of liquid viscosity, axial velocity in the diffuser still grows with a radius but at a slightly smaller rate. Results obtained confirm the conclusions presented in study [9] where also the effect
of the impeller–tank system geometry on liquid circulation in the tank with a screw impeller was analysed.
7. Conclusions • Results of model axial velocity distributions were compared with experimental data given in study [11]. The error of comparison equal to 9.0% provides an evidence that the analysed mathematical model can be applied in practical calculations and in the investigation of phenomena that occur in the tank with a screw impeller. • An optimisation criterion in the form of a modified power number was proposed. On the basis of values calculated from the model, an optimum choice of geometry of the impeller–tank system can be made. • Universal information obtained as a result of mathematical modelling enables a complete analysis of phenomena that occur in the tank within the laminar motion.
Acknowledgment This study was carried out within the project W10/1/2004/ DS. Appendix A. Nomenclature A product of power and Reynolds numbers (–) c clearance between worm edge and diffuser wall (–) Cn , Ct parameters of model (–) d external diameter of the worm band (m)
774
d0 D D1 E E* h H H1
C. Kuncewicz et al. / Chemical Engineering and Processing 44 (2005) 766–774
i Ks Lm n n* P r r* Re s u ur uz uϕ v V Vs w z z*
inner diameter of the worm band (hub diameter) (m) inner tank diameter (m) inner diffuser diameter (m) energy of homogenisation (J) modified power number, E* = E/nd3 η (–) screw impeller height (m) liquid height in the tank (m) distance from liquid surface to the diffuser or from the diffuser to the tank bottom (m) number of worm coils (–) secondary circulation number Ks = Vs /nd3 (–) power number, Lm = P/n3 d5 ρ (–) tangential velocity of the impeller (s−1 ) relative angular velocity (–) power (W) tank radius (m) dimensionless tank radius, r* = 2r/D (–) Reynolds number, Re = nd2 ρ/η (–) height of one coil of the impeller ribbon (m) dimensionless tangential velocity, u = uϕ /πDn (–) radial velocity (m/s) axial velocity (m/s) tangential velocity (m/s) dimensionless axial velocity, v = uz /πDn (–) liquid volume in the tank (m3 ) volumetric rate of radial–axial circulation (m3 /s) dimensionless radial velocity, w = ur /πDn (–) liquid height in the tank (m) dimensionless height, z* = z/H (–)
Greeks η ρ τc τm Φ
dynamic viscosity (Pa s) density (kg/m3 ) time of circulation (s) time of mixing (s) current function (–)
Subscripts corr calculated with correlation values exp experimental values num numerical values
References [1] F. Str˛ek, Mixing and Stirring Apparatuses (Mixer), WNT, Warsaw, 1981. [2] C. Kuncewicz, M. Pietrzykowski, A. Niedzielska, Modelling of heart transfer in the tank with ribbon impellers, In˙z. Chem. Proc. 23 (2002) 21–38. [3] M. Kuriyama, K. Arai, S. Saito, Mechanism of heat transfer to pseudoplastic fluids in an agitated tank with helical ribbon impeller, J. Chem. Eng. Jpn. 16 (6) (1983) 489–494. [4] M. Kaminoyama, M. Watanabe, K. Nishi, M. Kamiwano, Numerical simulation of local heat transfer coefficient in stirred vessel with impeller for highly viscous fluid, J. Chem. Eng. Jpn. 32 (1) (1999) 23–30. [5] V.V. Chavan, J. Ulbrecht, Power correlations for close-clearance screw impellers in non-Newtonian liquids, Ind. End. Chem. Process Des. Dev. 12 (1973) 472–476. [6] P. Seichter, J. Dohnal, F. Rieger, Process characteristics of screw impellers with a draught tube for Newtonian liquids: the power input, Coll. Czech. Chem. Commun. 46 (1981) 2007–2020. [7] C. Kuncewicz, M. Pietrzykowski, Hydrodynamic model of a mixing vessel with pitched-blade turbines, Chem. Eng. Sci. 56 (2001) 4659–4672. [8] C. Kuncewicz, M. Pietrzykowski, K. Szulc, Modelling the hydrodynamics of agitator for ribbon impellers working in laminar flow, In˙z. Chem. Proc. 22 (2001) 461–481. [9] C. Kuncewicz, K. Szulc, P. Budzy´nski, Modelling of highly viscous liquid flow in a tank with screw impellers, In˙z. Chem. Proc. 22 (3C) (2001) 837–842. [10] P. Budzy´nski, Reometr rotacyjny typu “on-line” do pomiaru własno´sci reologicznych zawiesin fermentacyjnych, Ph.D. thesis, Faculty of Process and Environmental Engineering, Technical University of Lodz, 1992. [11] P. Seichter, Process characteristics of screw impellers with a draught tube for Newtonian liquids: pumping capacity of the impeller, Coll. Czech. Chem. Commun. 46 (1981) 2032–2042.