Net Flow Patterns

The columns that have been dealt with thus far all have a commonality in that there is only one viable flow pattern to consider in the design. By flow pattern, we mean the general direction in which material flows up or down the column. For instance, in a simple one-feed-two-product column, the only possible flow pattern is that of material flowing upward in the CS above the feed, that is, the rectifying CS characterized by A 0, and material flowing downward in the CS below the feed, that is, the stripping CS characterized by A 0. Other columns can be 

FIGURE 7.9 Material balance across the Petlyuk feed stage. 

FIGURE 7.13 Five different global FPs for the Petlyuk column, labeled 1 5. 

It is interesting to note that for all FPs other than FP3, the net flows in the CSs above and below the side-draw stage (CS3 and CS5) are in the same direction. This means that both these CSs will have the same sign for the net flow as well as reflux. This does not seem to be a problem on flkce value, but one needs to consider how the refluxes in these CSs come about and what their roles are. It can be shown that 

Now that we have identified the live different FPs that permit feasible Petlyuk operation and qualitatively understood each of their effects, it is now of interest to obtain a more tangible understanding. First consider then Table 7.1, which shows the net flow direction in each CS for each of the live FPs. 

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While the net flows patterns for each structure are now clearer, it still remains difficult to comprehend the effects of changing the reflux in one CS in the rest of the column. The reflux ratio in a specific CS is an important parameter in finding feasible structures and therefore it is necessary to fully understand these effects. As shown in previous chapters, the reflux in a specific CS is a limitless parameter valid anywhere from negative to positive infinity. Notice howev that in the side-stripper configuration, for example, that the liquid stream is split into two parts. 

Since the reflux ratios are, by definition, dependent on the net flow of a particular CS, the same arguments put forward in Sections 7.3.2 wiU hold for reflux ratios with regard to net FPs. For instance, choosing a FP that will lead to CS2 having a negative net flow will cause the value of R 2 to also be negative in this CS. However, because is defined as the ratio of liquid flowrate to the net flowrate in a particular CS, the actual value of in a particular CS depends on the quality of the feed, and whether the side-draw product is liquid or vapor. Just as we have conveniently represented net flow patterns in >-space, we can also then construct lines of constant reflux in the >-space, which will ultimately allow us to hone in on feasible designs, as will be illustrated in Section 7.4. In order to do this, it is necessary to write all our reflux equations in terms of v and as shown in the equations below... 

Figure 13. Schematic diagram showing overall net flow patterns for solids in a circulating fluidised bed. The diagram is simplified in only showing a unidirectional flux of solids from core to wall at the top rather than a net flux in this direction.

Net Present Va.Iue, Each of the net annual cash flows can be discounted to the present time using a discount factor for the number of years involved. The discounted flows are then all at the same time point and can be combined. The sum of these discounted net flows is called the net present value (NPV), a popular profit criterion. Because the discounted positive flows first offset the negative investment flows in the NPV summation, the investment capital is recovered if the NPV is greater than zero. This early recovery of the investment does not correspond to typical capital recovery patterns, but gives a conservative and systematic assumption for investment recovery. 

This gives two choices ia interpreting calculated NRR values, ie, a direct comparison of NRR values for different options or a comparison of the NRR value of each option with a previously defined NRR cutoff level for acceptabiUty. The NPV, DTC, and NRR can be iaterpreted as discounted measures of the return, iavestment, and return rate, analogous to the parameters of the earher example. These three parameters characterize a venture over its entire life. Additional parameters can be developed to characterize the cash flow pattern duting the early venture years. Eor example, the net payout time (NPT) is the number of operating years for the cumulative discounted cash flow to sum to zero. This characterizes the early cash flow pattern it can be viewed as a discounted measure of the expected operating time that the investment is at risk. 

In horizontal flow, the flow pattern will inevitably be more complex because the gravitational force will act perpendicular to the pipe axis, the direction of flow, and will cause the denser component to flow preferentially nearer the bottom of the pipe. Energy transfer between the phases will again occur as a result of the difference in velocity, but the net force will be horizontal and the suspension mechanism of the particles, or the dispersion of the fluid will be a more complex process. In this case, the flow will not be symmetrical about the pipe axis. 

Figure 2.2 shows the cash flow pattern for a typical project. The cash flow is a cumulative cash flow. Consider Curve 1 in Figure 2.2. From the start of the project at Point A, cash is spent without any immediate return. The early stages of the project consist of development, design and other preliminary work, which causes the cumulative curve to dip to Point B. This is followed by the main phase of capital investment in buildings, plant and equipment, and the curve drops more steeply to Point C. Working capital is spent to commission the plant between Points C and D. Production starts at D, where revenue from sales begins. Initially, the rate of production is likely to be below design conditions until full production is achieved at E. At F, the cumulative cash flow is again zero. This is the project breakeven point. Toward the end of the projects life at G, the net rate of cash flow may decrease owing to, for example, increasing maintenance costs, a fall in the market price for the product, and so on.

In this chapter, we extend the discussion of the previous chapter to nonspherical shapes. Only solid particles are considered and the discussion is limited to low Reynolds number flows. The flow pattern and heat and mass transfer for a nonspherical particle depend on its orientation. This introduces complications not present for spherical particles. For example, the net drag force is parallel to the direction of motion only if the particle has special shape properties or is aligned in specific orientations. 

If the temperature difference 0C between the heat bath and the inflow is greater than zero, we can have the opposite effect to Newtonian cooling, with a net flow of heat into the reactor through the walls. With his possibility, two more stationary-state patterns can be observed, giving a total of seven different forms—the same seven seen before in cubic autocatalysis with the additional uncatalysed step (the two new patterns then required negative values for the rate constant) or with reverse reactions included and c0 > ja0 ( 6.6). 

Estimation of Entrance Pressure-Pressure Losses from the Entrance Flow Field17 Consider the entrance flow pattern observed with polymer melts and solutions in Fig. 12.16(a). The flow can be modeled, for small values of a, as follows for 0 < a/2 the fluid is flowing in simple extensional flow and for a/2 < 0 < rc/2 the flow is that between two coaxial cylinders of which the inner is moving with axial velocity V. The flow in the outer region is a combined drag-pressure flow and, since it is circulatory, the net flow rate is equal to 0. The velocity V can be calculated at any upstream location knowing a and the capillary flow rate. Use this model for the entrance flow field to get an estimate for the entrance pressure drop. 

Dynamic similarity is especially important in fluid flow systems. A fluid s flow pattern is determined by the forces acting on the fluid elements. The net force acting on a fluid element gives the acceleration of that element over the next time interval and, hence, determines its motion. If the net forces on corresponding fluid elements at corresponding times are similar, then their motion will be similar. 

Flow is then instituted along the axis of the chamber and a differential flow pattern develops. This differential flow carries each component along in some uneven pattern. Different components distributed over the cross section with different concentration profiles will be carried downstream unequally because of the nonuniformity of flow. The net result is that the different components will be displaced at different average velocities in the tube and separation will be realized. 

Figure 11.7 Flow patterns associated with a rotating disk a) a three-dimensional representation of flow trajectories that lead to a net flow toward the disk and in the radial direction. The axial scale was expanded greatly in order to allow visualization of the flow trajectories, b) a projection of the flow trajectories onto a plane at a fixed axial position.

The flow pattern for solids is generally downward near the wall and upward in the central core. This particle movement also affects the gas flow in the emulsion phase. In this model the net rise velocity of solids is neglected, hence... 

In an exact treatment, values of Ci and C5 in the stripping section would differ sli tly from the enriching section because of the slightly different flow profile. In the present approximate treatment, the constants are to be evaluated for the total reflux case in which the flow patterns in both sections are the same. If the net flow rate is a small fraction of the circulation rate, studies by Parker [PI] and others have shown that the effect on Ci and C5 of the changed flow pattern with net flow is small. 

If the surface is first saturated with a monolayer of protein exposed to steady-state concentration cQ, and then is exposed to a second treatment at concentration 2c0, a second front emerges. The second profile represents the situation where no net protein is adsorbed and thus, in principle, is representative of the diffusion-shifted flow pattern of the nonadsorbed protein. Figure 7 shows both the initial (cQ) and second (2c0) fronts and the subtraction curve which is very close to the ideal step function. If the data are interpreted as solution-borne molecules passing over an inert surface, then (a) adsorption must be essentially instantaneous and (b) the surface must become covered by exhausting the concentration of solute at the front as it moves down the column. The slope of the difference profile should represent the rate of uptake of material on the column, and that is essentially infinite on the time scale of the experiment. The point of inflection of the subtracted front indicates the slowing of the sorption process due to filling of sites on the surface. 

A measure of the strength of the ebullience to be expected is the ratio of the vapour velocity based on the net cross sectional area Upip to the velocity of the rising bubble Ububbie for the two individual flow pattern. The vapour velocity based on the net cross sectional area is obtained finm the volumetric vapour generation rate referenced to the cross-sectional area of the reactor. If this velocity and the viscosity of the reaction mixture are known for the relief conditions, then the liquid level occupied due to the boil-up may be estimated wifli the help of one of the flow models (c.f. Figure 7-3). 

Therefore, the net flow direction in each CS in the side-stripper arrangement shown in Figure 6.30 is the only one that is physically realizable. Furthermore, it can be shown fairly easily that in the side stripper the feed has to lie below the thermally coupled junction. Analogous mass balance arguments can be made for the side rectifier, and its only viable flow pattern is shown in Figure 6.30b. 

Notice that there are six possibilities of positive and negative net flow combinations, each producing unique pinch point curves, but their pinch point curve behavior suggests that these six different flow possibilities can be divided into three pairs because of the continuation of these curves. A pair essentially consists of two opposite flow patterns. These three pairs are imp ative for the design of this simplified system, since a mass balance around the thermally coupled junction in Figure 7.2 gives... 

Thus, CSb and CSc operate with opposite difference vectors, that is, one pair of the three conceivable flow patterns pairs. Placing either difference vector automatically specifies the other. Physically, this means that components that are in rectifying mode in CSb are in stripping mode in the other CSc- In other words, a component will have a net movement upwards in one CS, and downwards in the other this is expected from the coupled CSs configuration in Figure 7.2. A CPM for the simplified DPE in Equation 7.2 is shown in Figure 7.4. 

Figure 7.10 illustrates that there are four possible flow patterns across the feed stage with different combinations of positive and negative net flows. However, the flow directions shown in Figure 7.10d are not considered feasible because this choice of flow directions violates the material balance formulated in Figure 7.9. If A2 is a negative value and A4 is a positive value, the result is that material has to be removed across the feed stage (and not added) because the quantity F will be negative. Depending on the phase equihbrium behavior and the objectives of the column, some of these flow patterns may hold advantages over other modes of operation, as will be highlighted in subsequent discussions.

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