Mixing, time

Now let us consider the mixing time, t. This will be estimated by an order of magnitude estimate for diffusion to occur across the boundary layer thickness, 5Bl- If we have turbulent natural conditions, it is common to represent the heat transfer in terms of the Nusselt number for a vertical plate of height, , as 

we see that this mixing time is also very small. However, this analysis implicity assumed that the pilot is located where the fuel vapor evolved. A pilot downstream would introduce an additional transport time in the process. 

The variation of measured mixing time with gas fiow rate for three different vessels are presented in Figs. 5.38,5.39 and 5.40. The straight line in each figure represents the best fit of the measured data for the same aspect ratio. The mixing time decreases with an increase in the gas flow rate, but increases with an increase in the bath diameter. The following empirical relation has been proposed for the mixing time in a bottom blown bath in the absence of swirl motion [52], 

In the above equations, g is the acceleration due to gravity and Re is the Reynolds number based on the superficial velocity of gas, E p. The conditions describing the onset of swirl motion have been given Sect. 5.2.3.1 [29,30]. 

Inspection of Fig. 5.41 suggests that the coefficient ak is a function of the vessel diameter D. 

Referring to Stokes s first problem [56], we introduce the following representative length to non-dimensionalize the vessel diameter as follows 

For convenience, k is assumed to be unity and (5.30) is substituted into (5.29) to give 

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A proposal based on Onsager s theory was made by Landau and Lifshitz [27] for the fluctuations that should be added to the Navier-Stokes hydrodynamic equations. Fluctuating stress tensor and heat flux temis were postulated in analogy with the Onsager theory. Flowever, since this is a case where the variables are of mixed time reversal character, tlie derivation was not fiilly rigorous. This situation was remedied by tlie derivation by Fox and Ulilenbeck [13, H, 18] based on general stationary Gaussian-Markov processes [12]. The precise fomi of the Landau proposal is confimied by this approach [14]. 

The interval between the second and third pulse is called the mixing time, during which the spins evolve according to the multiple-spin version of equation B 1.13.2 and equation B 1.13.3 and the NOE builds up. The final pulse converts the longitudinal magnetizations, present at the end of the mixing time, into detectable transverse components. The detection of the FID is followed by a recycle delay, during which the equilibrium... 

Figure Bl.13.7. Simulated NOESY peak intensities in a homoniielear two-spin system as a fiinetion of the mixing time for two different motional regimes. (Reprodiieed by pennission of Wiley from Neiihaiis D 1996 Encyclopedia of Nuclear Magnetic Resonance ed D M Grant and R K Harris (Chiehester Wiley) pp 3290-301.)...

The time resolution of these methods is detennined by the time it takes to mitiate the reaction, for example the mixing time in flow tubes or the laser pulse width in flash photolysis, and by the time resolution of the detection. Relatively... 

Single-base propellants are mixed in a similar fashion by adding the ingredients to the nitrocellulose in the mixer together with the required amounts of ether and alcohol. The mixing time is about one-half hour, and the temperature is kept below 25°C. The pardy colloidal mixture looks like moist cmde sugar. A maceration step may be included to increase homogeneity. 

Bakery Products. Sorbates are used in and/or on yeast-raised and chemically leavened bakery products. The internal use of sorbates in yeast-raised products at one-fourth the amount of calcium—sodium propionate that is normally added provides a shelf life equal to that of propionate without adversely affecting the yeast fermentation. Sorbates added at one-tenth the propionate level reduce the mix time by 30% (126). This internal treatment combined with an external spray of potassium sorbate can provide the same or an increased shelf life of pan breads, hamburger and hot-dog buns, English muffins, brown-and-serve roUs, and tortillas. The total sorbate useful in or on these baked goods ranges from 0.03 wt % for pan breads to 0.5 wt % for tortillas 0.2—0.3 wt % sorbic acid protects chemically leavened yellow and chocolate cakes (127). Emit-pie fillings and icings can be protected with 0.03—0.1 wt % sorbates. 

SBR mbber compounds are prepared in two stages the nonproductive and the productive phases. In the nonproductive phase the compound ingredients are mixed, usually in internal mixers (Banbury). The mixing time is usually short and the compound temperature is in the 160—195°C range. 

Not only is the type of flow related to the impeller Reynolds number, but also such process performance characteristics as mixing time, impeller pumping rate, impeller power consumption, and heat- and mass-transfer coefficients can be correlated with this dimensionless group. 

Likewise, large-diameter impellers (D > Df/2) are useful for (1) avoiding stagnant regions in slurries, (2) short mixing times to obtain uniformity throughout a vessel, (3) promotion of heat transfer, and (4) laminar continuous averaging of slurries. 

Propeller size, pitch, and rotational speed may be selected by model tests, by experience with similar operations, or, in a few cases, by published correlations of performance data such as mixing time or heat transfer. The propeller diameter and motor power should be the minimum which meet process requirements. 

A double helix shortens mixing time but requires more power. The disadvantages of the higher torque requirement are freqiientlv offset bv the better mixing and heat transfer,... 

Care should be taken to avoid mixing too long, as in some cases this will result in a poorer blend. A graph of degree of mixing versus time should be made to selecl the proper mixing time quantitatively. 

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