Second averaging

Looking back at (A3.2.7). we see that a second average over n(0) can be perfonned using PF Uq). This second type of averaging is denoted by (...). Using (A3.2.9) we obtain... 

Such spatial variations in, e.g., mixing rate, bubble size, drop size, or crystal size usually are the direct or indirect result of spatial variations in the turbulence parameters across the flow domain. Stirred vessels are notorious indeed, due to the wide spread in turbulence intensity as a result of the action of the revolving impeller. Scale-up is still an important issue in the field of mixing, for at least two good reasons first, usually it is not just a single nondimensional number that should be kept constant, and, secondly, average values for specific parameters such as the specific power input do not reflect the wide spread in turbulent conditions within the vessel and the nonlinear interactions between flow and process. Colenbrander (2000) reported experimental data on the steady drop size distributions of liquid-liquid dispersions in stirred vessels of different sizes and on the response of the drop size distribution to a sudden change in stirred speed. 

The second average is viscosity average molecular weight, M . This expression is obtained by using the exponent from the limiting viscosity number-molar mass relationship, a, as a power for the molecular weight of each molecule in the distribution. The formula for this average is... 

The SLAB dispersion model was used for this situation. Figures 7.13 and 7.14 show isopleths for the two sets of meteorological conditions (D/5 and F/2) for a 3600-second averaging time. As in the carbon disulfide and n-pentane example, the more stable meteorological conditions (F/2) result in the larger hazard zone. 

The second set of brackets isn t written on the right here because the second averaging is explicitly written out. This is understood by recognizing that the is the normalized thermal distribution of configurations of the... 

The second average is not easy. Kirkwood and Riseman introduced the following approximation... 

In the following, some aspects of multi-pulse excitation are reviewed which are relevant for space encoding. The effects of time-invariant and pulsed gradients are addressed as well as methods to increase the line-narrowing efficiency through second averaging and CRAMPS (cf. Section 3.3.4). 

Fig. 8.7.2 The 48-pulse sequence used for imaging with phase and frequency encoding. For second averaging the phases of the rf pulses are shifted in small increments (phase shifter). The signal is observed stroboscopically during (2- Adapted from [Corl3] with permission from Wiley-VCH.

Absorption ratio and pyrometer temperature measurements. Time histories (L) (Lighter Traces 30 second average. Darker Traces 30 minute moving average) (Reprinted from Teichert, H., Fernholz, T, and Ebert, V., "Simultaneous In Situ Measurement of CO, H2O, and Gas Temperatures in a Full-Sized Coal-Fired Power Plant by Near-Infrared Diode Lasers," Applied Optics 42,2043-51,2003. With permission.) and temperature versus absorption rate (R). 

Second, average standard deviation of the test signals relative to their mean was... 

Fig. 2. One hour swimming depth record for salmon F. The continuous line shows successive 10 second averages, the dotted portions indicate periods of low signal-to-noise ratio and data drop-out (from Dj ving et al., 1985).

As the duration of the flehmen response is between 1-3 seconds, differences were possibly undetected. Compared to flehmen duration reported in other species such as black-tailed deer, 20 seconds (average) (Muller-Schwarze, 1979), and giraffe, as along as 2 minutes observed (Dagg and Taub, 1970), the flehmen response in the bull elephant is of very short duration. The response to estrous urine by the male elephant is unique in its high frequency. 

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