Net flow rate

The surface integral represents the net flow rate of mass out of the control volume. This is easily seen from Figure 3.5, where v — w is the relative velocity at the fluid giving a flow... 

Figure 6.6 uses a particular problem to illustrate some of the salient differences between the semi-infinite and finite-gap configurations. In both cases the net flow rates are the same, and the streamlines in both plots have the same values. The streamlines for the finite-gap problem show pure axial flow at the inlet, whereas the semi-infinite case shows radial spreading everywhere. The radial-velocity profiles are quite different, with the finite-gap profile showing no slip at both boundaries. 

Finally, it can easily be shown (see Problem 2.12) from Eq. E2.5.9 that, for a given net flow rate q there is an optimum H = 3q/Vo for a maximum pressure rise of... 

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. 

This equation shows that, when the pressure to drag flow ratio equals —1/3, the shear rate at the stationary plate is zero, when it equals +1/3, the shear rate at the moving plate is zero, and when it equals zero, the shear is constant and equals Vq/H. In this range the velocity profile exhibits no extremum. In terms of the net flow rate, the condition of no extremum in velocity is ... 

Optimum Gap Size in Parallel Plate Flow Show that for the flow situation in Example 2.5, for a given net flow rate the optimum gap size for maximum pressure rise is... 

Equations 6.3-19 is the well-known isothermal Newtonian extrusion theory. Since it was obtained by the solution of a linear differential equation, it is composed of two independent terms, the first representing the contribution of drag flow Q,j, and the second, the pressure flow, Qp. The net flow rate is the linear superposition of the two. 

So far we have neglected the effect of the flight clearance. As small as the clearance is, polymer melt is being dragged across the clearance by the barrel surface and the pressure drop may pump melt across the flight width. This creates a continuous leakage flow from downstream locations to (one turn back) upstream locations, reducing net flow rate. 

In Eq. 6.6-16 the first term on the right-hand side is the drag flow and the second term is the pressure flow. The net flow rate is their linear superposition, as in the case of the Newtonian model in single screw extrusion. The reason that in this case this is valid for non-Newtonian flow as well is because the drag flow is simply plug flow. 

We now extend the model to the positive net flow situation, and assume that the differential volume moves axially. Although the axial flow is not plug flow, this is not an unreasonable approximation because as we recall the RTD is rather narrow. In this case, the elapsed time t becomes the mean residence time in the extmder given by the ratio of screw channel volume and net flow rate... 

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. 

For generalized fluidization, the diffusion flux Jm2 of Eq. (5.6) and the segregation flux Js2 of Eq, (5.7) will not be equal, their difference being the net flow rate of particles 2 ... 

CA, the conversion per cycle CAc approaches zero if the recycle ratio Rr is increased more and more. Thus, at sufficiently high recycle flow rates, the concentration and temperature gradients along the catalyst bed can be kept small in the same way as in the differential reactor. The overall conversion in the reactor can be set at any desirable, easily measurable level by regulating the net flow rate < . A further benefit of high recycle flow rates lies in the fact that, due to the high fluid velocities past the catalyst particles, any possible interface transport effects can be eliminated. The recycle reactors, therefore, are called gradient-free reactors. 

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). 

Direct integration is not possible as the three equations depend on each other. Fish and Carr (1989) suggested eliminating the reaction term by addition of Eq. 8.31 and Eq. 8.32 or 8.33. After integration and introducing a net flow rate (Wyn) for components B and C, Eqs. 8.32 and 8.33 are transformed into Eq. 8.35 ... 

The net flow rate is constant for each component and does not depend on the position within the section. Based on this equation the concentration of components B and C can be calculated as a function of the reactant concentration and the operating parameters ... 

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 ... 

Now the concentration at the outlet of section III can be determined by introducing the net flow rates (Eqs. 8.39 and 8.40) into Eqs. 8.38. Based on this concentration the conversion rate of the TMBR can be calculated. Notably, the outlet concentration has to be evaluated numerically, as Bln depends on CDLjAjIIi- For a given limit of the conversion rate the flow rate in section III is determined so as to guarantee that the conversion calculated by Eq. 8.29 corresponds to the required conversion. 

The concentration of component C at the outlet of section II, CDLjCjII(0), can be calculated using the net flow rate in this section, Wc,n- If this concentration is known the inlet concentration of section III can be determined by the mass balance of the feed node ... 

Here Vsoiid is the apparent solid flow rate, HA and ffB describe the slopes of the adsorption isotherm, which are calculated in the nonlinear case by linearization of the adsorption isotherm for the feed concentration Cfeed, . The transformation reflects the fact that, in a counter-current process, it is not the net flow rates that are important but rather their values relative to the apparent solid movement. For this reason, Morbidelli et al. introduced the m factors in their graphical design (known as the triangle theory) (Biressi et al., 2000 and Mazzotti et al., 1997c). 

Only specific combinations of these values are compatible with a complete separation. The conditions that the flow rate ratios must satisfy in order to achieve such a separation were derived from the equilibrium theory [28,38]. These constraints imply that in sections II and III the net flow rate of components B and A must be negative and positive, respectively and that in sections I and IV the net flow rates of these two components must be positive and negative, respectively. 

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. 

NET FLOW RATES. Quantity Z> is the difference between the flow rates of the streams entering and leaving the top of the column. A material balance around the condenser and accumulator in Fig. 18.8 gives... 

Thus quantity D is the net flow rate of material upward in the upper section of the column. Regardless of changes in V and L, their difference is constant and equal to D. 

In the lower section of the column the net flow rates are also constant but are in a downward direction. The net flow rate of total material equals that of... 

A decrease in hydrostatic pressure along the fiber due to resistance to substrate solution flow occurs so that at a definite distance from the inlet, say Lc, transmembrane pressure is nil. Fiber-to-shell solution flux from that point on is negative and becomes a shell-to-fiber flux. Neglecting the shell pressure drop, the overall fiber-to-shell ultrafiltration net flow rate can then be obtained upon integration of the flux equation over the length of the fiber from the inlet to Lc, that is ... 

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