Simplified Stationary Balancing

Since the fiber phase is not stationary, the surface integral cannot be set to zero without further considerations. As shown earlier, dBr/dt = 1/V js Ur hids (see Eq. 5.10). Because der/dt = 0 in the IP process, the contribution of the surface integral to the overall mass balance is negligible. Based on this observation Equation 5.50 can be simplified mid the appropriate equation for a conservation of mass in this process can be obtained (i.e., V Ur) = 0). Using this, Equation 5.18 can be simplified and the appropriate species balance equation for the IP process can be obtained. This equation is similar to the equation obtained for the RTM process. 

This quick test does not, however, tell us that there will be only one stable limit cycle, or give any information about how the oscillatory solutions are born and grow, nor whether there can be oscillations under conditions where the stationary state is stable. We must also be careful in applying this theorem. If we consider the simplified version of our model, with no uncatalysed step, then we know that there is a unique unstable stationary state for all reactant concentrations such that /i < 1. However, if we integrate the mass-balance equations with /i = 0.9, say, we do not find limit cycle behaviour. Instead the concentration of B tends to zero and that for A become infinitely large (growing linearly with time). In fact for all values of fi less than 0.90032, the concentration of A becomes unbounded and so the Poincare-Bendixson theorem does not apply. 

The population balance simulator has been developed for three-dimensional porous media. It is based on the integrated experimental and theoretical studies of the Shell group (38,39,41,74,75). As described above, experiments have shown that dispersion mobility is dominated by droplet size and that droplet sizes in turn are sensitive to flow through porous media. Hence, the Shell model seeks to incorporate all mechanisms of formation, division, destruction, and transport of lamellae to obtain the steady-state distribution of droplet sizes for the dispersed phase when the various "forward and backward mechanisms become balanced. For incorporation in a reservoir simulator, the resulting equations are coupled to the flow equations found in a conventional simulator by means of the mobility in Darcy s Law. A simplified one-dimensional transient solution to the bubble population balance equations for capillary snap-off was presented and experimentally verified earlier. Patzek s chapter (Chapter 16) generalizes and extends this method to obtain the population balance averaged over the volume of mobile and stationary dispersions. The resulting equations are reduced by a series expansion to a simplified form for direct incorporation into reservoir simulators. 

The stationary phase concentration is given by the isotherm equation (Eq. 2.4). The mobile phase concentration is denoted simply by Q. Since the axial dispersion coefficient is nil, the mass balance equation for component i (Eq. 2.2) simplifies to... 

It should be obvious that this term is volumetric, which means that the accumulation rate process applies to the entire system contained within the control volume. The stipulation that the control volume be stationary simplifies the mathematics to some extent, but the final form of the force balance does not depend on details pertaining to the movement of the control volume. Possibilities for this motion and the appropriate time derivatives are summarized in Table 8-1. The substantial derivative operator... 

Sensors, Calibration of. Fig. 3 Simplified principle behind the force balanced accelerometer. The displacement transducer normally uses a capacitor C, whose capacitance varies with the displacement of the mass. A current, proportional to the displacement transducer output, will force the mass to remain stationary relative to the frame (Figure from Havskov and Alguacil 2010)... 

To analyze the mass balance of DO, a useful control volume is a stationary segment of river Ax units thick in the direction of flow, as shown in Fig. 2.30. A simplified steady-state mass conservation expression for oxygen in this slice is... 

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