Passage distribution function functions

Example 7.12 Number of Passage Distribution Functions in a Batch Mixer with Recirculation We consider the batch mixer in Fig. 7.30(b). We begin by making a mass balance on the change in time of the fraction of volume that never passed through the high-shear zone at time t, g0(t) as follows... 

Z. Tadmor, Passage Distribution Functions and Their Application to Dispersive Mixing of Rubber and Plastics, AIChE J., 34, 1943 (1988). 

Z. Tadmor, Number of Passage distribution Functions, in Mixing and Compounding of Polymers - Theory and Practice, I. Manas-Zloczower and Z. Tadmor, Eds., Hanser, New York, 1994, Chapter 5. 

Next we derive a simple theoretical model to calculate the passage-distribution function (PDF) in a SSE11 (25), assuming isothermal Newtonian fluids. We examine a small axial section of length A/, as shown in Fig. 9.16. 

Fu Cumulative exit passage distribution function (Section 7.3 7.3-29)... 

Determining Fiight Clearance with Passage Distribution Function... 

According to Tadmor and Manas-Zloczower [70] the passage distribution function... 

We can use the expressions above to determine the minimum X value that will yield a Go less than 0.01, meaning that less than 1% of the fluid will not pass through the clearance at all. This is achieved when the dimensionless time X > 4.6. For certain values of L, H, W, r, cp, and w, we can then determine how large the flight clearance 8 has to be to make A, > 4.6 or Go < 0.01. The passage distribution function for A, = 4.6 is shown in Fig. 8.86(b). 

Next, we define a parallel set of NPD function in continuous flow recirculating systems. We restrict our discussion to steady flow systems. Here, as in the case of RTD, we distinguish between external and internal NPD functions. We define fk and 4 as the fraction of exiting volumetric flow rate and the fraction of material volume, respectively, that have experienced exactly k passages in the specified region of the system. The respective cumulative distribution functions, and /, the means of the distributions, the variances, and the moments of distributions, parallel the definitions given for the batch system. 

Passage times and distribution of passage times in recirculating systems were first considered by Shinnar et al. (64) in their analysis of RTD in closed-loop systems. The most important such system is that of blood circulation, but the analysis cited is also relevant to engineering systems such as fluidized-bed reactors. The main objective of this work was the analysis of tracer experiments in recirculating systems. The renewal theory discussed by Cox (65) served as the theoretical framework for their analysis. Both Shinnar et al. (64), and later Mann and Crosby (66) and Mann et al. (67) have shown that the NPD functions can be evaluated from the passage time distribution function, which in turn can be obtained from the renewal theory. 

The next steps are the following Step 1 Passage to the entropy representation and specification of the dissipative thermodynamic forces and the dissipative potential E. Step 2 Specification of the thermodynamic potential o. Step 3 Recasting of the equation governing the time evolution of the np-particle distribution function/ p into a Liouville equation corresponding to the time evolution of np particles (or p quasi-particles, Up > iip —see the point 4 below) that then represent the governing equations of direct molecular simulations. 

Once the above exponential form is established, it is further used to reconfirm the uniformity as follows. First, the average passage time is calculated in terms of the distribution function such that... 

Exercise 9.9.4. Show that the distribution function of residence times for laminar flow in a tubular reactor has the form 2z /Zp, where tp is the time of passage of any fluid annulus and the minimum time of passage. Diffusion and entrance effects may be neglected. Hence show that the fractional conversion to be expected in a second order reaction with velocity constant k is 2B[1 + j lnu5/(5 + 1)] where B = akt n and a is the initial concentration of both reactants. (C.U.)... 

When the system passes the intersection at a high velocity, that is, the above condition is not met even approximately, it will usually jump from the lower R surface (before S along the reaction coordinate) to the upper R surface (after S). That is, the system behaves in a nonadiabatic (or diabatic) fashion, and the probability per passage of electron transfer occurring is small (i.e., k< C1). The nuclear coordinates of the system change so rapidly that it cannot remain at equilibrium. At the nonadiabatic limit, the time interval for passage between the two states at point S approaches zero, that is, (tf — t,) —> 0 (infinitely rapid), and the probability density distribution functions that describe the initial and final states remain unchanged ... 

Figure 10 presents this surface for Helium molecules in a micro-structure of glassy PC at 300 K. Typically, the cavities are 5-10 A in size and contain few local maxima of the distribution function p(r) connected by small barriers (less than kT). The cavities are separated by some bottle-neck channels of 5-10 A in length and 1-2 A in width that prevent easy passage from one cavity to another. 

The cumulative distribution function for the quiescence interval is thus known from knowledge of the state of the population at time t. The random number for the interval of quiescence must be generated so that the distribution function (4.6.6) is satisfied. If the particles in the population did not grow so that the states identified at time t remained the same with the passage of time, expression (4.6.6) becomes... 

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