Stirrer power

Example 2 The Determination of the Pi Set for the Stirrer Power in the Contact Between Gas and Liquid... 

In the above relevance list, only the density and the viscosity of the liquid were introduced. The material properties of the gas are of no importance as compared to the physical properties of the liquid. It was also ascertained by measurement that the interfacial tension cr does not affect the stirrer power. Furthermore, measurements revealed that the coalescence behavior of the material system is not affected if aqueous glycerol or cane syrup mixtures are used to increase viscosity in model experiments (7). 

Under these conditions, the stirrer power must be measured and the power number Ne = P/pn d calculated. 

In Example 5.1, the working out of the power characteristics of a stirrer in a Newtonian fluid is presented in detail. It is shown how a relevance list containing five parameters stirrer power P, stirrer diameter d, density p, and viscosity p of the liquid and the stirrer speed n... 

There are well-estabhshed empirical correlations for stirrer power rcquircrnenls [6, 7]. Figure 7.8 [6] is a log-log plot of the power number for imgassed hquids versus the stirrer Reynolds number (Re). These dimensionless numbers are defined as follows ... 

Details of stirrer power requirement for non-Newtonian liquids are provided in Section 12.2. 

An aerated stirred-tank fermenter equipped with a standard Rushton turbine of the following dimensions contains a liquid with density p = 1010kgm and viscosity n = 9.8 X 10 Pa s. The tank diameter D is 0.90 m, liquid depth Hl = 0.90 m, impeller diameter d = 0.30 m. The oxygen diffusivity in the liquid Dl is 2.10 X 10 5 cm- s T Estimate the stirrer power required and the volumetric mass transfer coefficient of oxygen (use Equation 7.36b), when air is supplied from the tank bottom at a rate of 0.60 m min at a rotational stirrer speed of 120 rpm, that is 2.0 s T... 

When estimating the stirrer power requirements for non-Newtonian liquids, correlations of the power number versus the Reynolds number (Re see Figure 7.8) for Newtonian hquids are very useful. In fact, Figure 7.8 for Newtonian liquids can be used at least for the laminar range, if appropriate values of the apparent viscosity are used in calculating the Reynolds number. Experimental data for various non-Newtonian fluids with the six-blade turbine for the range of (Re) below 10 were correlated by the following empirical Equation 12.1 [1] ... 

Estimate the stirrer power requirement P for a tank fermentor, 1.8 m in diameter, containing a viscous non-Newtonian broth, of which the consistency index A = 124, flow behavior index n = 0.537, density p = 1050 kg m", stirred by a pitched-blade, turbine-type impeller of diameter d = 0.6 m, with a rotational speed AT of 1 s . ... 

Some fermentation broths are non-Newtonian due to the presence of microbial mycelia or fermentation products, such as polysaccharides. In some cases, a small amount of water-soluble polymer may be added to the broth to reduce stirrer power requirements, or to protect the microbes against excessive shear forces. These additives may develop non-Newtonian viscosity or even viscoelasticity of the broth, which in turn will affect the aeration characteristics of the fermentor. Viscoelastic liquids exhibit elasticity superimposed on viscosity. The elastic constant, an index of elasticity, is defined as the ratio of stress (Pa) to strain (—), while viscosity is shear stress divided by shear rate (Equation 2.4). The relaxation time (s) is viscosity (Pa s) divided by the elastic constant (Pa). 

Example 2 The Determination of the Pi Set for the Stirrer Power in the Contact Between Gas and Liquid. We examine the power consumption of a turbine stirrer (so-called Rushton turbine see inset in Fig. 1) installed in a baffled vessel and supplied by gas from below (see Sketch 2). We facilitate the procedure by systematically listing the target quantity and all the parameters influencing it ... 

Under these conditions the stirrer power must be measured and the power number Ne = P/ pn d ) calculated. We find Ne = 1.75. Due to the fact that Q = idem results in Ne = idem, the power consumption Pt of the industrial stirrer can be obtained ... 

Here c represents the heat capacity of the equipment that is to be identified in the experiments, together with the heat losses. The stirrer power is calculated according to Equation 2.24. The heat exchange term is... 

The hydrodynamic parameters that are required for stirred tank design and analysis include phase holdups (gas, liquid, and solid) volumetric gas-liquid mass-transfer coefficient liquid-solid mass-transfer coefficient liquid, gas, and solid mixing and heat-transfer coefficients. The hydrodynamics are driven primarily by the stirrer power input and the stirrer geometry/type, and not by the gas flow. Hence, additional parameters include the power input of the stirrer and the pumping flow rate of the stirrer. 

Due to the approximate uniformity of the intensively mixed gas/liquid system and, therefore, the intensity character of the target quantity kLa, the influencing process quantities (stirrer power P, air throughput q) have to be formulated in an intensive manner as well. Now, the question arises whether, in addition to the volume-related stirrer power, P/V, the gas throughput also has be formulated as a volume-related one (q/V), or if its inclusion as the so-called superficial velocity vG = q/S (as accurate... 

The influence of the baffles is, understandably, nil in the laminar flow region. However, it is extremely strong at Re > 5 x 104. The installation of baffles under otherwise unchanged operating conditions increases the stirrer power by a factor of 20 here ... 

In operating a hollow stirrer, both target quantities - gas throughput q and the stirrer power P -adapt themselves simultaneously. Both target quantities depend on the following parameters ... 

In the optimization of stirrers for an optimal heat transfer, it may not be forgotten that the removable heat flow, Q [kW], increases according to the heat transfer characteristic with Re2/3 n2/3, whereas the thereby associated stirrer power (stirring heat)... 

The conditions used were those from which Fig. 32 is based. With Pr = 5 x 104 and the abscissa value I It Pr (H/D)-1 = 2.82 x 108, the optimum conditions Reopt Pr1/2 = 4.8 x 103 and the ordinate value (n2)opt= 8 x 101 follow from the work-sheet, producing nopt = 20 min-1 and Rmax = (R/V)opt V = 28.5 kW (see the optimum operating point in Fig. 32). At this stirrer speed the stirrer power amounts to ca. 6 kW, which is ca. 20% with respect to the maximum removal of reaction heat. From the auxiliary diagram in inset (b), it can be inferred that the rotation speed interval, in which at least 90% of the maximum achievable value (R90% = 25.6 kW) could be removed, lies between 8 and 32 min-1. 

In contrast, Fig. 72 shows the results of mass transfer in the system aqueous 1-n sodium sulphite solution/air. These measurements were carried out under steady-state conditions in vessels with hollow stirrers on the scale p = 1 5 [58/1, 92]. In this material system, the high salt concentration (70 g/1) fully suppresses bubble coalescence. In the case of the self-aspirating hollow stirrer (see Fig. 28), the stirrer power and gas throughput were coupled via the stirrer speed and were therefore dependent on each other. Consequently, v does not occur explicitly in the representation in Fig. 72, because it is a function of (P/V)". 

The optimum operating conditions under which the desired gas flow rate is attained at the minimum stirrer power (per unit of gas flow) can be found by combining the dimensionless number NA, Ne, and Fr. The appropriate dimensionless group in this case is Ne Fr/NA = P/(qtpgdt), and is likewise a function of Fr H /d,. The optimum operating condition for the tube stirrer is given by Fr d,/H = 1.80 here H — HL — Hx (i.e., liquid height above the stirrer). 

Experiments (Weinspach, 1967) have shown that the flow behavior of suspensions with solid volume fractions ips up to 0.25-0.30 approximates that of a Newtonian fluid. In this range of ij/s values, the stirrer power required for suspension can be calculated from the corresponding power characteristics Ne = /(Re) for homogeneous fluid (see Section II), only the physical properties p and p must be replaced with the effective values ps and ps for the suspension ... 

Conventional stirred-tank polymeric reactors normally use turbine, propeller, blade, or anchor stirrers. Power consumption for a power-law fluid in such reactors can be expressed in a dimensionless form, Ne = Reynolds number based on the consistency coefficient for the power-law fluid. Various forms for the function f(m) in terms of the power-law index have been proposed. Unlike that for Newtonian fluid, the shear rate in the case of power-law fluid depends on the ratio dT/dt and the stirrer speed N. For anchor stirrers, the functionality g developed by Beckner and Smith (1962) is recommended. For aerated non-Newtonian fluids, the study of Hocker and Langer (1962) for turbine stirrers is recommended. For viscoelastic fluids, the works of Reher (1969) and Schummer (1970) should be useful. The mixing time for power-law fluids can also be correlated by the dimensionless relation NO = /(Reeff = Ndfpjpti ) (Tebel et aL 1986). 

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