Reynolds stirring

Re Reynolds (stirred tank) Q TS =S-si II n d Stirrer frequency Diameter of agitator... 

Impeller Reynolds Number The presence or absence of turbulence in an impeller-stirred vessel can be correlated with an impeller Reynolds number defined... 

FIG. 20-38 Newton number as a Function of Reynolds number for a horizontal stirred bead mill, with fluid alone and with various filling fractious of 1-mm glass beads [Weit and Schwedes, Chemical Engineering and Technology, 10(6), 398 04 (1987)]. (N = power input, W d = stirrer disk diameter, m n = stirring speed, 1/s i = liquid viscosity, Pa-s Qj = feed rate, mVs.)... 

The above analysis is restricted to high Reynolds numbers, although the definition of high is different in a stirred tank than in a circular pipe. The Reynolds number for a conventionally agitated vessel is defined as... 

In the case of stirred vessels the values A/riL can be calculated by the following equation using the geometry parameter d/D, H/D, the Newton number Ne, the Reynolds number Re = nd /v, the energy dissipation ratio e/e and the related macro scale A/d. For standard turbines e.g. Mockel [24] found the value A/d = 0.08 close to the impeller. Corresponding to this the maximum of the dissipation ratio ,/ has to be used which can be estimated by Eq. (20). 

Re Reynolds number STR stirred tank reactor t time... 

Another complication, caused by stirring the two phases at the same speed, occurs when the two solutions have different viscosities, which is common for immiscible liquids. The key fluid flow parameter is the Reynolds number. Re, which is the ratio of inertial to viscous forces in the solution, as indicated by... 

The stirring rate is given by N in the Reynolds number expression. A characteristic length of the system is given by d in the Reynolds number expression and was defined as the length of the agitator. 

The rate of agitation, stirring, or flow of solvent, if the dissolution is transport-controlled, but not when the dissolution is reaction-con-trolled. Increasing the agitation rate corresponds to an increased hydrodynamic flow rate and to an increased Reynolds number [104, 117] and results in a reduction in the thickness of the diffusion layer in Eqs. (43), (45), (46), (49), and (50) for transport control. Therefore, an increased agitation rate will increase the dissolution rate, if the dissolution is transport-controlled (Eqs. (41 16,49,51,52), but will have no effect if the dissolution is reaction-controlled. Turbulent flow (which occurs at Reynolds numbers exceeding 1000 to 2000 and which is a chaotic phenomenon) may cause irreproducible and/or unpredictable dissolution rates [104,117] and should therefore be avoided. 

Even nowadays, a DNS of the turbulent flow in, e.g., a lab-scale stirred vessel at a low Reynolds number (Re = 8,000) still takes approximately 3 months on 8 processors and more than 17 GB of memory (Sommerfeld and Decker, 2004). Hence, the turbulent flows in such applications are usually simulated with the help of the Reynolds Averaged Navier- Stokes (RANS) equations (see, e.g., Tennekes and Lumley, 1972) which deliver an averaged representation of the flow only. This may lead, however, to poor results as to small-scale phenomena, since many of the latter are nonlinearly dependent on the flow field (Rielly and Marquis, 2001). 

The question whether or not stirred tank flow is locally isotropic, may be investigated with the help of a LES which resolves a great deal of the Reynolds stresses. To this end, the Reynolds stress data are best presented in terms of the so-called anisotropy tensor and its invariants Ah A2< and A3. 

Derksen (2006a) continued along this line of approach and—by means of a clever strategy—mimicked the long-time behavior of solids suspension in an unbaffled tall stirred tank equipped with four hydrofoil impellers (Lightnin A310). The time span covered by his LES amounted to some 20,000 impeller revolutions (some 20 min). Running a LES for a Reynolds number of 1.6 x 10 over the entire time span is not an option, and for that reason a particular flow... 

In some way, introducing an increased particle drag by means of Eq. (17) resembles the earlier proposal raised by Bakker and Van den Akker (1994b) to increase viscosity in the particle Reynolds number due to turbulence (in agreement with the very old conclusion due to Boussinesq, see Frisch, 1995) with the view of increasing the particle drag coefficient and eventually the bubble holdup in the vessel. Lane et al. (2000) compared the two approaches for an aerated stirred vessel and found neither proposal to yield a correct spatial gas distribution. 

Heat exchange in stirred reactors is described in [207]. By using dimensional analysis of heat flow and energy balance equations, the Nusselt number, containing hT, can be expressed as a function of the Reynolds number and the Prandtl number ... 

As mentioned earlier, Reynolds numbers determined for the bulk flow have to be discerned from Reynolds numbers characterizing a particle-liquid dissolution system. The latter were calculated for drug particles of different sizes using the Reynolds term according to the combination model. The kinematic viscosity of the dissolution medium at 37°C is about 7 x 10-03 cm2/sec. The fluid velocities (Ua) employing the paddle method at stirring rates of 50-150 rpm can be taken from the literature and may arbitrarily be used as the slip velocities at the particle surfaces. 

Correlations are available for mixing times in stirred-tank reactors with several types of stirrers. One of these, for the standard Rushton turbine with baffles [13], is shown in Figure 7.9, in which the product of the stirrer speed N (s ) and the mixing time t (s) is plotted against the Reynolds number on log-log coordinates. For (Re) above approximately 5000, the product N t (-) approaches a constant value of about 30. 

Deriving the conservation equations that describe the behavior of a perfectly stirred reactor begins with the fundamental concepts of the system and the control volume as discussed in Section 23. Here, however, since the system is zero-dimensional, the derivation proceeds most easily in integral form using the Reynolds transport theorem directly to relate system and control volume (Eq. 2.27). 

Maintenance of proper temperature is a major aspect of reactor operation. The illustrations of several reactors in this chapter depict a number of provisions for heat transfer. The magnitude of required heat transfer is determined by heat and material balances as described in Section 17.3. The data needed are thermal conductivities and coefficients of heat transfer. Some of the factors influencing these quantities are associated in the usual groups for heat transfer namely, the Nusselt, Stanton, Prandtl, and Reynolds dimensionless groups. Other characteristics of particular kinds of reactors also are brought into correlations. A selection of practical results from the abundant literature will be assembled here. Some modes of heat transfer to stirred and fixed bed reactors are represented in Figures 17.33 and 17.18, and temperature profiles in... 

Here the Reynolds number is expressed for a stirred tank where the flow rate corresponds to the tip speed (n-ds) of the agitator. 

When a liquid warms up, its density decreases, which results in buoyancy and an ascendant flow is induced. Thus, a reactive liquid will flow upwards in the center of a container and flow downwards at the walls, where it cools this flow is called natural convection. Thus, at the wall, heat exchange may occur to a certain degree. This situation may correspond to a stirred tank reactor after loss of agitation. The exact mathematical description requires the simultaneous solution of heat and impulse transfer equations. Nevertheless, it is possible to use a simplified approach based on physical similitude. The mode of heat transfer within a fluid can be characterized by a dimensionless criterion, the Rayleigh number (Ra). As the Reynolds number does for forced convection, the Rayleigh number characterizes the flow regime in natural convection ... 

Figure 4.6 Dimensionless mixing time inside a stirring tank with and without baffles as a function of Reynolds number.

In microfluidics, segmentation by stirring or creation of turbulent flow can not be expected because Reynolds numbers do not exceed 2000 (the limit for turbulent flow). Some methods for mixing in microfluidics have been developed. 

It is decided to model a full-scale prototype, unbaffled, stirred vessel with a one-tenth scale model. The liquid in the prototype has a kinematic viscosity, v. of 10 7 m2 s As we have seen above, power number is a function of both Reynolds number and Froude number for unbaffled vessels. To ensure power number similarity, we need to ensure both Reynolds number and Froude number are similar from prototype to model. 

In stirring, is represented by n 1, whereby n [T-1] stands for the stirrer speed. With d as the stirrer diameter, the Reynolds number then becomes ... 

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