Specific energy dissipation

The model is able to predict the influence of mixing on particle properties and kinetic rates on different scales for a continuously operated reactor and a semibatch reactor with different types of impellers and under a wide range of operational conditions. From laboratory-scale experiments, the precipitation kinetics for nucleation, growth, agglomeration and disruption have to be determined (Zauner and Jones, 2000a). The fluid dynamic parameters, i.e. the local specific energy dissipation around the feed point, can be obtained either from CFD or from FDA measurements. In the compartmental SFM, the population balance is solved and the particle properties of the final product are predicted. As the model contains only physical and no phenomenological parameters, it can be used for scale-up. 

P = specific energy dissipation rate per unit mass of liquid... 

The specific energy dissipation rate for mixing was equated to the input energy rate so that Baird and Rice (1.) used... 

G)giobai effective global specific energy dissipation rate m s )... 

In the right-hand side there are probabilities of the average specific energy dissipation being smaller than the critical value at the moment t, and larger than the critical value at the moment t + At. In the denominator, there is the probability that at the initial moment t, the value of specific energy dissipation is smaller than the critical value. 

Here t = (ve/so) is the characteristic time scale in a turbulent flow with specific energy dissipation so, W — AnfiRlrio/ i is the volume concentration of drops of type 2 in the flow, C is the parameter whose value depends on the chosen model of drop coagulation in a turbulent flow. 

In order to work safely with multi-injection reactor, an accumulation of the limiting reactant and of heat in the main channel has to be thoroughly prevented. As the considered reactions are mostly Umited by mixing, the time required to mix can be estimated by using the correlation between specific energy dissipation and mixing time. Sufficient residence time should be provided between two injection points to minimize the hot spot. 

Many workers tried to correlate k a in function of specific energy dissipated in two-phase when neglecting the gas superficial velocity (6, 15, 27, 31, 39). There are also many empirical correlation cited (12, 14, 15, 37, 41). One may remark that the influence of the liquid viscosity has not yet been investigated. 

Mean total specific energy dissipation rate (W/kg) Pressure drop (N/m )... 

The most important impact on the power input necessary for suspension comes from the impeller type. Axially working impellers are the most economic tools for suspending. A comparison of a Viscoprop impeller with a Rushton turbine shows that the Viscoprop impeller is able to reach the same result with less than 1/3 of the energy consumption. It has to be considered that for shear-sensitive particles such as needle- or plate-shaped crystals the power input is of less importance than particle exposure to the local specific energy dissipation rate. 

The first term on each side of Eq. (3.4.1) represents the kinetic energy (J kg = the second term the contribution of the static pressure, and the third term the specific potential energy with h as height above a reference height. The term is the specific energy dissipation (Jkg ), that is, the ratio of the friction power,... 

In turbulent flow, micro-mixing takes place at the level of the smallest eddies. According to turbulence theory, the size of the smallest eddies is related with the specific energy dissipation e. 

The diffusion time can be calculated easily from the specific energy dissipation rate with eq. (4.7) ... 

In this example it was silently assumed that the mean specific energy dissipation is a characteristic for the whole stirred tank. In reality local values of e differ considerably in the vicinity of the impeller blades the ratio T = e/e may be about 30 to 50, and in the quiet areas away from the impeller, indeed in most of the... 

Note that on scale-up situation a may change into bl We shall first consider situation a, which is usually applicable to well stirred continuous reactors with a mean residence time that is not very short. The following approximation may be used One can relate the turbulent diffusivity with the local specific energy dissipation (Tj e) and the integral length scale X (Jeurissen et al. 1994)... 

It was shown above that local values of the specific energy dissipation e (= T e) vary considerable across the reactor. The ratio r is highest in the impeller region, particularly in the flow leaving the impeller. Obviously, it is best to introduce the feed there. Meso-mixing times could be at least 3 times shorter than when the feed is introduced into the bulk of the liquid. 

When this ratio is on the order of 10, the parameter (p will be < 1 when T n > 10. For bench scale reactors, this criterion is usually fulfilled. If one wants to be certain that (p is the same for large scale reactors, the following scale-up criterion would apply for constant specific energy dissipation (see eq. (4.3)) ... 

The average specific energy dissipation (e) should be the same that means that for larger vessels the stirrer speed has to be lower, and proportional to the impeller (or tank) diameter to the power -2/3. 

The reciprqcal value of k can be considered as a micro-mixing time. The average shear rate Y can be replaced by c n, and the impeller speed n may be related to the specific energy dissipation with eq. (4.15). The initial striation thickness is put equal to the inlet tube diameter. We find then for the micro-mixing time... 

On scale-up, the ratio of the specific energy dissipation and the square of the scale of initial segregation (e.g., inlet tube diameter) should be kept constant. Since this may require the same stirrer speed on the larger scale, Ae required torques on the propeller shaft may become prohibitive. This will limit the maximum applicable size of stirred reactors for very viscous liquids. 

Note that these relations are similar, jince e n. The local specific energy dissipation is found from e = q e see sections 4.2.2.1 and 4,2,2,3, Tike coefficients a and P are dependent on geometrical factors. Some authors found that P s 0.5, p s 0.33 and q s 0.5. WiA eq. (4.3) this would mean that, e.g. for standard turbine impellers (with q s 0.1), a s 7.7. There is however quite some controversy concerning the exponents p and q. Some authors found that the exponent p approached 1 for very smsJl particles (Molerus and Latzel, 1987), and for large particles p approached 0, which would seem contrary to expectation (see review by Voit and Mersmann, 1986). Others found p s 0.33 for intermediate sizes (see review by Geisler, et al, 1991). Some authors found little influence of the solids fraction, particularly for low values, others found that q = 0.5 in the range of solids fractions between 0.01 and 0.3 (see review by Geisler, et al., 1991). Obviously, the exponential relation cannot be extrapolated to e 0. 

It was found that the minimum specific energy dissipation for standard turbines and downward pumping propellers, in baffled vessels, is approximately the same. 

When the particles are larger, the turbulent eddies may disrupt the boundary layers around them and the mass transfer is increased. In that case, a modified Reynolds number may be used, that is based on the specific energy dissipation e ... 

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