Mixing parameter

Parameters that are used to characterize stirred tank flows and mixing processes in general can be computed by correlations that can be found in the literature. In many cases, these parameters can also be computed from the CFD results. Examples of how to compute some of these parameters are given below. 

1 Power Number. The power number is a dimensionless parameter that provides a measure of the power requirements for the operation of an impeller. It is defined as 

(5-47) P is the power applied to the impeller of diameter D, p the density, and N the impeller rotation speed in Hertz. Correlations are available that provide the dependence of Np on the Reynolds number. Thus, if CITD is not available, the power requirements can generally be obtained from one of these correlations. The eorrelations can break down, however, if they do not address the D/T or C/T ratios of single impellers or the presence and spacing of multiple impellers. In sueh eases, CFD results can be used to compute Np, or simply, the power requirements. 

The power delivered to the fluid is the product of the impeller speed, 2irN, in rad/s, and torque, x, which is obtained by integration of the pressure on the impeller blade  

Reports are usually available for the torque delivered to the fluid by the impeller. In some cases, reports of power or even power number can be obtained from the 

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Fig. 11. Changes ia mixing parameters on scale-up to 125 times the volume of the pilot plant.

Between 1 s and 1 min specific contact time, conduction heat-transfer performance decreases theoretically as the 0.29 power of contact time. This is consistent with empirical data from several forms of indirect-heat dryers which show performance variation as the 0.4 power of rotational speed (21). In agitator-stirred and rotating indirect-heat dryers, specific contact time can be related to rotational speed provided that speed does not affect the physical properties of the material. To describe the mixing efficiency of various devices, the concept of a mixing parameter is employed. An ideal mixer has a parameter of 1. 

In what follows, both macromixing and micromixing models will be introduced and a compartmental mixing model, the segregated feed model (SFM), will be discussed in detail. It will be used in Chapter 8 to model the influence of the hydrodynamics on a meso- and microscale on continuous and semibatch precipitation where using CFD, diffusive and convective mixing parameters in the reactor are determined. 

The coalescence-redispersion (CRD) model was originally proposed by Curl (1963). It is based on imagining a chemical reactor as a number population of droplets that behave as individual batch reactors. These droplets coalesce (mix) in pairs at random, homogenize their concentration and redisperse. The mixing parameter in this model is the average number of collisions that a droplet undergoes. 

In the three and four environment (3B and 4B) models (Ritchie and Togby, 1979 Mehta and Tarbell, 1983), the reactor is divided into two segregated entering environments and one or two fully mixed leaving environments. The mixing parameter is the transfer coefficient between the environments. 

This is an occupied-virtual off-diagonal element of the Fock matrix in the MO basis, and is identical to the gradient of the energy with respect to an occupied-virtual mixing parameter (except for a factor of 4), see eq. (3.67). If the determinants are constructed from optimized canonical HF MOs, the gradient is zero, and the matrix element is zero. This may also be realized by noting that the MOs are eigenfunctions of the Fock operator, eq. (3.41). 

Grossing and Zeilinger have studied this periodicity invariance quantitatively, as a function of 5 [gross88c]. Generalizing the parameter 6 (which they call a mixing parameter) to have possibly non-equal real and imaginary components, we write S... 

The Voigt function is a convolution product ( ) between L and G. As the convolution is expensive from a computational point of view, the pseudo-Voigt form is more often used. The pseudo-Voigt is characterized by a mixing parameter r], representing the fraction of Lorentzian contribution, i.e. r] = 1(0) means pure Lorentzian (Gaussian) profile shape. Gaussian and Lorentzian breadths can be treated as independent parameters in some expressions. 

Set Klu to a suitable value and experiment with the influence of the mixing parameters FLl and FL2 in GASLIQl and FGRl and FGR2 in GASLIQ2,... 

In those cases where we are dealing with nuclear transitions which are a mixture of multipolarity Ml and E2 with a mixing parameter 5 defined by = ((/i E2 h))l ((/i I M111/2)) (positive or negative), one obtains the extended relation... 

The magnetic moment and the E2IM1 mixing parameter of the 69.6 keV level (5/2 ) of Os have been determined by several authors [254, 255, 257, 259]. 

There are 26 experimental osmotic coefficient data and they are given in Table 15.2 (Park and Englezos, 1999 Park, 1999). Two sets of the binary parameters for the NaOH and Na2Si03 systems and two mixing parameters. 

Park has also obtained osmotic coefficient data for the aqueous solutions of NaOH-NaCl- NaAl(OH)4 at 25°C employing the isopiestic method (Park and Englezos, 1999 Park, 1999). The solutions were prepared by dissolving AlCl r6H20 in aqueous NaOH solutions. The osmotic coefficient data were then used to evaluate the unknown Pitzer s binary and mixing parameters for the NaOH-NaCI-NaAl(OH)4-H20 system. The binary Pitzer s parameters, [3(0), P0). and C9, for NaAI(OH)4 were found to be -0.0083, 0.0710, and 0.00184 respectively. These binary parameters were obtained from the data on the ternary system because it was not possible to prepare a single (NaAl(OH)4) solution. 

Fig. 6 Simulational cooling curves of disorder parameters (solid lines) and mixing parameters (dashed lines) for 32-mers with different sets of energy parameters in a 64-sized cubic box (the concentration is fixed at 0.150). The mixing parameter is defined as the mean fraction of neighboring sites occupied by the solvent for each chain unit [84]... 

Lowering the pH value of the suspension from pH 4.7 to pH 3.2 or pH 2.3 by adding an acid PEG-solution a totally different stability behaviour is observed. Ere (24h) and T (1 ) are zero independent of the mixing parameters at both polymer concentrations. 

The variation of the mixing procedure at pH 2 3 implies that the critical concentration of the polymer necessary for flocculation is independent of the mixing parameters. This behaviour differentiates the flocculation behaviour of precipitated silica from that of pyrogenic silica at pH 7. However, it should be remembered that at low pH values the flocculation of pyrogenic silica is also independent of mixing conditions. 

Occasionally, various methods for evaluating tracer data and for estimating the mixing parameter in the TIS model lead to different estimates for t and N In these cases, the accuracy of t and N must be verified by comparing the concentration-versus-time profiles predicted from the model with the experimental data. In general, the predicted profile can be determined by numerically integrating N simultaneous ordinary differential equations of the form ... 

The mixing parameter Q must be chosen to yield the correct mixture-fraction-variance dissipation rate. However, inertial-range scaling arguments suggest that its value should be near unity.165... 

The simplest inverse approach consists, given the end-member coordinates, in finding the mixing parameters of any individual mixture, i.e., finding the parameter q defined by equation (1.3.1). We will treat this case with an example. 

Mattioli G. S. and Bishop F. (1984). Experimental determination of chromium-aluminum mixing parameter in garnet. Geochim. Cosmochim. Acta, 48 1367-1371. 

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