The Time After Volume Averaging Procedure

In this section we examine the multiple averaging operators with emphasis on the time after volmne averaging technique. The literature cited concerning the multiple averaging operators like the space followed by time or ensemble averaging are [12, 26,47, 50, 51, 69,78, 144, 187, 202-205, 219, 220, 244, 258]. These reports might be considered for complementary studies. 

The time after volume averaging procedure can be applied under a unified set of conditions denoting the sum of the two sub-sets of requirements formulated in Sects. 3.5.1 and 3.5.2 for the pure volume and time averaging procedures to handle the scale disparity in a proper manner. 

In this theory the covariance coefficients are affected by the size of the averaging scales in time and space [55]. However, in engineering practice these coefficients are commonly set to unity because very little data on the distribution coefficients are available [12, 244, 258], 

Let the time averaging operator, when applied to any volume averaged scalar, vector, or tensor valued function v associated with phase k, be defined by  

In accordance with the conventional Reynolds axioms (1.385), the Leibnitz s mle is given by  

It has been shown that the time-volume and volume-time averaging operators are mathematically commutative [43, 47]. Nevertheless, Sha and Slattery [189] argued that in experimental analysis the instrumentation often records space average followed by a time average data. For this reason it might be convenient to formulate a consistent model formulation for the theoretical 

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The 2D steady-state two-fluid model presented in this section is based on the early work by Torvik and Svendsen [22], Svendsen et al [21], and Jakobsen [12], The two-fluid model was derived based on the time-after volume averaging procedure, described in sect 3.4.4. 

Measurement Procedure. IGC measurements were started after the thermal and flow equilibrium in the column were stable (2 to 3 h). To facilitate rapid vaporization of the probe (0.01 yL), the injector temperature was kept 30°C above the boiling point of the probe. Measurements were made at five carrier gas flow rates. The retention volumes of six injections for each probe and twenty injections of the marker (H2) at a given flow rate were averaged. The values obtained were extrapolated to zero flow rate to eliminate the flow rate dependence of the retention data. The net retention time (tR) is defined as the time difference between the first statistical moment of the solvent peak and that of the marker gas. Thus, tR was calculated by an on-line computer statistical peak analysis rather than the retention time at the peak maximum (tp,maY). This eliminated inaccuracies arising from slight peak asymmetry, which occurs even for inert and well-coated supports. The specific retention volumes (Vg°) derived from tR and tR max differed by as much as 5% for small retention times and slightly skewed peaks (11,12). 

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