Net-flow point

In accordance with (11-3), the net flow point P in Fig. 11.4 is located at the intersection of the material balance lines, which are extensions of the lines drawn from Lw to Vw and from L +, to V 2. 

It is now possible to step off stages for the enriching section, stopping when the feed line FP is crossed. At this point, the stripping section net flow point P" must be located. Since P" = B - S = F - P, this is easily done. 

To handle the section of the column between F] and it is necessary to find a new difference point. The construction is shown in Fig. 10.17. For the section of the column between the top and a stage between Fi and F2, the net flow point... 

An analogous minimum-solvent, maximum-stage situation exists for the V> L enriching section situation. Here, the net flow point P would be lowered until an operating line-equilibrium line pinch is encountered that is no further change in composition takes place as we add stages. 

Since there are both stripping and enriching sections, there are two net flow points, P and P", above and below the feed stage, respectively. 

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. 

Equation 10-5 is the unsteady, three-dimensional mass eonservation or eontinuity equation at a point in a eompressible fluid. The first term on the left side is the rate of ehange in time of the density (mass per unit volume). The seeond term deseribes the net flow of mass leaving the element aeross its boundaries and is ealled the eonveetive term. 

Now the liner has been rotated counterclockwise. which opens the seal point at the top This allows part of the liquid to be recirculated, reducirtg the net flow by a proportionate amount... 

When the Imer is rotated past the no flow" point, the volume above the rotor exceeds that below, and net flow reverses direction even though pump speed and rotation have not changed (Limited to approx 30 cof lull forward How on all models)... 

In the present discussion only the problem of steady flow will be considered in which the time average velocity in the main stream direction X is constant and equal to ux. in laminar flow, the instantaneous velocity at any point then has a steady value of ux and does not fluctuate. In turbulent flow the instantaneous velocity at a point will vary about the mean value of ux. It is convenient to consider the components of the eddy velocities in two directions—one along the main stream direction X and the other at right angles to the stream flow Y. Since the net flow in the X-direction is steady, the instantaneous velocity w, may be imagined as being made up of a steady velocity ux and a fluctuating velocity ut, . so that ... 

During the migration of cations and anions towards their respective electrodes, each ion tends to carry solvated water along with it. As cations are usually more solvated than anions, a net flow of water towards the cathode occurs during the separation process. This effect, known as electro-osmosis, results in a movement of neutral species which would normally be expected to remain at the point of application of the sample. If required, a correction can be applied to the distances migrated by ionic species by measuring them... 

The basic point here is that ECT causes a net flow of water from the anode to the cathode, causing electroosmotic dewatering (EOD) of the tissue the experimentally observed flow of water from anodic site (dry) to the cathodic site (oedema) of the tumor tissue can be explained only by an electroosmotic mechanism. 

Considering a countercurrent system consisting of n thickeners, as shown in Figure 10.19, the net flow to the right must necessarily be constant throughout the system, if no material enters or leaves at intermediate points. 

The point representing the stream F will be a difference point, since it will represent the composition of the stream which must be added to w h to give W h r In general, in a countercurrent flow system of this sort, the net flow of all the constituents will not be in the same direction. Thus one or more of the fractional compositions in this difference stream will be negative, and the difference point will lie outside the triangle. A balance on the whole system, as shown in Figure 10.19, gives ... 

The total net flow of material to the right at some intermediate point... 

Eyring et al. (226) examine the entanglement problem from a somewhat different point of view, the activated complex theory of liquid viscosity. In a monomeric liquid the molecules move by random jumps from one equilibrium position to another. The jump frequency is controlled by an activation barrier between neighboring sites. According to activated complex theory, a shear stress lowers the barrier in the direction of the stress and raises it in the opposite direction, producing a bias in jump frequency and a net flow of molecules in the stress direction. For low stresses, the expression for viscosity in a monomeric system is ... 

Let us determine the change in p with time at a given position (p, q) in phase space. We surround the point with a small volume element Q, sufficiently small to make the value of p(p, q, t) the same at all points in the volume element, and sufficiently large to contain enough system points so that p is well defined and not dominated by large fluctuations. Then the change in the number of system points per second in 0, is equal to the net flow of system points across the bounding surface S(Q.) of Q, that is,... 

Equation E2.5-9 further indicates that, in the absence of a pressure drop, the net flow rate equals the drag flow rate. Note that qp is positive if Pq > PL and pressure flow is in the positive z direction and negative when Pp > Po- The net flow rate is the sum or linear superposition of the flow induced by the drag exerted by the moving plate and that caused by the pressure gradient. This is the direct result of the linear Newtonian nature of the fluid, which yields a linear ordinary differential equation. For a non- Newtonian fluid, as we will see in Chapter 3, this will not be the case, because viscosity depends on shear rate and varies from point to point in the flow field. 

We can now employ our usual methods to calculate, at any point in the tube, the net flows of species 1 and 2 across a unit area. If we take a plane perpendicular to the direction of the concentration gradients (i.e., the X axis) then there will be a net flow of species 1 across this plane owing to the fact that there is a difference in the number of molecules of species 1 that strike it from opposite sides (Fig. VIII.5). By making the usual assumption of local equilibrium distributions we can use our formula [Eq. (VII.6.6)] for the number of collisions per unit area per second. We then have for the number of collisions of species 1 with a unit area per second, made on the high-concentration side,... 

High ratios of gas (or vapor) to catalyst volumes, and velocities of 15 to 40 ft./second, are used in carrier lines in order to maintain the catalyst in dilute suspension and to prevent accumulation of stagnant catalyst at any point (53). The net flow rates of catalyst and vapors in the carrier lines typically correspond to an aerated bulk density of the order of 5 Ib./cu. ft. (68). However, the actual density is about twice the calculated value because of slip factor (105). 

When zero voltage is reached in the cell in Figure 18-2b, the concentrations of Cu(II) and Ag(I) ions will have values that satisfy the equilibrium-constant expression shown in Equation 18-4. At this point, no further net flow of electrons will occur. It is important to recognize that the overall reaction and its position of equilibrium are totally independent of the way the reaction is carried out, whether it is by direct reaction in a solution or by indirect reaction in an electrochemical cell. 

To evaluate Eqs. 8.38, the constant net flow rate (WiJ) has to be specified for the different separation regions. If the operating point is within region T3 and the separation condition inside section III is fulfilled, the stronger adsorbed component B cannot reach the raffinate node, i.e. CBi m(h < < ) is equal to zero. In this case Eq. 8.35 can be reduced and the net flow rate of component B is ... 

The net flow rate of component C can be derived by considering the inlet of section III. As long as the operating point is within the separation region, component C propagates towards the raffinate node. Therefore, its concentration at the inlet of section III is zero and Wc.iii becomes ... 

Dynamic osmometers reach equilibrium pressures in 10 to 30 minutes and indicate osmotic pressure automatically. Several types are available. Some commonly used models employ sensors to measure solvent flow through the membrane and adjust a counteracting pressure to maintain zero net flow. A commercially available automatic osmometer operates on the null-point principle. In this high-speed membrane osmometer schematically represented in Fig. 4.4, the movement of an air bubble inside the capillary immediately below the solvent cell indicates the solvent flow to the solution cell. Such movement is immediately detected by a photocell, which in turn is coupled to a servomechanism. If any movement of the air bubble is detected by a photocell, the servomechanism is stimulated to move the solvent reservoir upward or downward in order to adjust the hydrostatic pressure such that the solvent flow is completely arrested. The pressure head of the reservoir gives the osmotic head. Some osmometers also use strain gauges on flexible diaphragms to measure the osmotic pressure directly. 

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