Flux plot

In burn-out experiments, a test section is part of a loop which may be open or closed, and the question arises as to whether or not any of the loop equipment, such as condensers, heaters, pumps, or pipe fittings, have any significant effect on the burn-out flux. This issue came to prominence at the Boulder Heat Transfer Conference in 1961 with a Russian paper by Aladyev (A4) describing burn-out experiments in which a branch pipe, connecting to a small vessel, was fitted close to the test section inlet. The test section itself was a uniformly heated tube 8 mm in diameter and 16 cm long. The results are reproduced in Fig. 9, and show burn-out flux plotted against exit steam quality. Curve (A) was obtained with the branch vessel filled with cold water,... 

Fig. 7. Normalized recovered flux plotted after 600 s ( , cycle 2), 900 s ( , cycle 3), and 1200 s (O, cycle 4) of microfiltration with deposition of SMY. The SMY was deposited for tsf= 25 s, followed by forward feed filtration of 2.0 g/L of BSA for tf= 275 s and then th = 0.1,0.2,0.5,1.0,2.0, or 5.0 s of backflushing. The points are joined by straight lines for clarity. The error bars represent SD for three repeats.

A 6.0-cm-diameter pipe whose surface temperature is maintained at 2I0°C passes through the center of a concrete slab 43 cm thick. The outer surface temperatures of the slab are maintained at 15°C. Using the flux plot, estimate the heat loss from the pipe per unit length. 

A symmetrical furnace wall has the dimensions shown. Using the flux plot, obtain the shape factor for this wall. 

Repeat Example 8.6.1 (diffusion in a Stefan tube) using an explicit method of determining the fluxes. Plot the composition profiles and compare them to the results of an exact solution and to the experimental data given in Example 2.2.1. 

The heating in the bottom of cells A and B is driven by radiant energy from the flame, as demonstrated by a plot of the incident radiant flux in Figure 11.20. However, from this plot it can be deduced that peak heating near the center of the tube bank is not caused by radiation. It must, instead, be the result of convection. The net convective heat flux plot and the isosurface plot... 

Experimentally determined flux plots would have approximately the... 

Figure 28.6 Example for field eddy-correlation measurements, with the gas sampling mast located in the downwind direction of a rice field. Some time-series data of the CH4 flux plotted (lower left panel), together with a correlation of the data with the wind direction (lower right panel). Adapted from Werle and Korman, AppL Opt., 2001, 40 846, with permission of The Optical Society of America...

The total flux plot depends not only on the settling characteristics of the solids, but also on the selected underflow concentration because total flux includes the transport flux, or dead flux, contributed by movement of particles downward as underflow is withdrawn. The plot can be made independent of Cu by subtracting the transport flux contribution from G and plotting the resultant settling flux G, as done in Figure 10.23b. A line drawn tangentially... 

Results of the measurements with the 11.5- and 15.2-in.-diam spheres are given in Table I. Critical e]q>eri-ments with the 14-in.-diam sphere were previously reported . Minimum critical volumes for the plutonium nitrate solution containing 4.6 wt% Pu-240. when bare and water reflected, were determined from interpolation of the data to be about 22 and 11 t, respectively, for a Pu concentration of 175 25 g Pu/f. A concrete reflector was found to be somewhat more effective than water on the 14-in. sphere, but slightly less effective on the 11.5-in. sphere where the concentration Pu was greater. Neutron-flux measurements made in the 11.5-in. and the 15.2-in. sphere were used to determine extrapolation lengths and, hence, critical bucklings for the solutions. A typical flux plot is shown in Fig. 1. The effect of a stainless-steel shell on criticality was also evaluated in several experiments by the addition of extra shells that permitted a subsequent extrapolation to zero shell thickness. 

FIGURE 14. Water flux plotted against pressure differences AP for MBM-14 block copolymer membrane cast from CF-TFA 8 1 mixture. 

The water permeability coefficient can be obtained via eq. V - 43 using experiments with pure water. Because the osmotic pressure difference is zero, there is a linear relationship between the hydrodynamic pressure AP and the volume (water) flux Jy (eq. V - 43), and from the slope of the corresponding flux-pressure curve the water permeability coefficient Lp can be obtained. Figure V - 3 is a schematic representation of the volume flux plotted as a function of the applied pressure for a more open membrane (high Lp) and a more dense membrane (low Lp). 

Figure VII 16. Calculated values of the permeate flux plotted as a function of the applied pressure ai varx ing bulk concentrations Cj, and the following parameters a = 100 n >...

Figure 5.10 A settling flux plot for the data in Example 5.3, showing the determination of the critical flux Gc...

Hence, on a flux plot (a plot of Ups versus concentration) ... 

Figure 33 Determination of interface and layer velocities from a batch flux plot...

Relationship Between the Height-Time Curve and the Flux Plot... 

These material balances link the total continuous flux plots for the upflow and downflow sections in the thickener. 

Figure 3.15 shows flux plots for a critically loaded thickener. The line of slope F/A represents the relationship between feed concentration and feed flux for a volumetric feed rate, F. The material balance equations [Equations (3.43) and (3.44)] determine that this line intersects the curve for the total flux in the downflow section when the total flux in the upflow section is zero. Under critical loading conditions the feed concentration is just equal to the critical value giving rise to a feed flux equal to the total continuous flux that the downflow section can deliver at that concentration. Thus the combined effect of bulk flow and settling in the downflow section provides a flux equal to that of the feed. Under these conditions, since all particles fed to the thickener can be dealt with by the downflow section, the upflow flux is zero. The material balance then dictates that the concentration in the downflow section, Cb, is equal to Cp and the underflow concentration, Cp is FCp/L. The material balance may be performed graphically and is shown in Figure 3.15. From the feed flux line, the feed flux at a feed concentration, Cp is Ups = FCf/A. At this flux the concentration in the downflow section is Cb = Cp. The downflow flux is exactly equal to the feed flux and so the flux in the upflow section is zero. In the underflow, where there is no sedimentation, the underflow flux, LCl/A, is equal to the downflow flux. At this flux the underflow concentration, Cl is determined from the underflow line.

Figure 3.16 Total flux plot for an underloaded thickener...

A common form of continuous flux plot is that exhibiting a minimum total flux shown under critical conditions in Figure 3.18. With this alternative flux plot the critical loading condition occurs when the feed concentration gives rise to a flux... 

Figure 3.19 shows this alternative flux plot in an overloaded situation. For the graphical solution in this case, the excess flux must be read from the flux axis of the downflow section plot and applied to the upflow section plot in order to determine the value of Cj and Cy. Note that in this case although there are theoretically two possible values of Cb, in practice only the higher value can stably coexist with the higher concentration region, Cj, above it. 

Figure 3.19 Alternative total flux plot shape overloaded thickener...

For the batch flux plot shown in Figure 3W3.1, the sediment has a solids concentration of... 

Using this information we can plot the concentration profile in the test vessel 50 min after the start of the test. A sketch of the profile is shown in Figure 3W3.3. The shape of the concentration profile within the variable concentration zone may be determined by the following method. Recalling that the slope of the batch flux plot (Figure 3W3.1) at a value of suspension concentration C is the velocity of a layer of suspension of that concentration, we find the slope at two or more values of concentration and then determine the positions of these layers after 50 min ... 

Lines of slope F/A, L/A and —V/A drawn on the flux plot represent the fluxes in the feed, underflow and overflow, respectively (Figure 3W4.2). The total flux plot for the section below the feed point is found by adding the batch flux plot to the underflow flux line. The total flux plot for the section above the feed point is found by adding the batch... 

In an overloaded thickener, the concentration in the bottom section of the thickener is equal to (when the total flux plot does not go through a minimum) ... 

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