Time scale Corrsin

Corrsin s (1957, 1964) theory considers the time scale, t, required for the decay of the concentration fluctuations. This can be expressed in terms of the intensity of segregation ... 

The mixing rate can be estimated using the Corrsin (1964) time scale (see Chapter 2), where the turbulent energy dissipation rate is calculated at the end of the jet path and the appropriate length scale is the jet s free path, Z ... 

SOLUTION The reaction time scale is taken from the first reaction in Example 13.1 Tri = 1/kRiCAo = 0.14 s. The Corrsin equation (Section 13-2.1.3) is... 

The last three steps control the small-scale or micromixing in the reactor. The characteristic time constant for the third step, that is for reduction in segregation scale (inertial-convective mixing), is (Corrsin, 1964 Baldyga, 1989)... 

Example 2-4a Blend Time. Now that we have ways to estimate s and the characteristic length scale Lg, we return to Corrsin s equations in Table 2-4. Probably the most important practical point is that the time constant of mixing scales with (Lg/e). All the rest of the terms in the equation for (Sc 1) are either constants or relatively minor effects of the Schmidt number. For mixing in a pipe, we take the radius of the feed pipe, ro, as the initial integral length scale, and the fluctuating velocity, u, as a measure of the turbulent energy. Thus we can write... 

Where T/D is held constant on scale-up, this result reduces to 1/(N N). There are many different ways to make the scaling arguments (see, e.g., Grenville et al., 1995 Grenville and Tilton, 1996, 1997 or Nienow, 1997). The point is that the end result agrees well with Corrsin s approach. The most important thing to recognize is that Lj/e, however it is estimated, must be constant on scale-up to maintain constant blend time. If the dissipation (e) is held constant on scale-up, the blend time will always increase. 

Example 13-2 Scale Effects on Mixing in Stared Vessels. Determine whether the fast reaction from Example 13-1 will be affected by mixing on scale-up if the feed point is close to the impeller. Compute the values of the Corrsin mixing time, tm, at the impeller tip and the blend time, Tb. for (a) 1 L and (b) 20 (XX) L vessels stirred by a disk turbine (Np = 6) at power per unit volume of 0.36 kW/m. Use properties of water p = 1000 kg/m v = 10 m /s Sc = 2000 for a typical solute. 

The inertial mixing (first term) is controlling since its time constant is about eight times the time constant of the Batchelor scale mixing. The Corrsin scale mixing is much faster than the reaction time constant. 

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