Graphical solutions

The graphical method is only useful for displaying a logical sequence during the determination of volumes and inlet and outlet conditions for reactors in series under isothermal conditions. 

This expression represents a straight line that intersects the rate curve (-rAi), to give the outlet concentration of the following reactor. The slope is  

This method may be applied to PFR reactors in series as well as other different combinations of reactors. 

For a thin-walled tube with R t, the area and the second area moment of inertia are given by 

The objective function to be minimized is the weight given by F Xi,X2) = pAL = IpnRtL. 

The inequality constraints come from requiring the load to be less than the critical buckling load for the column. If 

Regardless of which failure criteria is applicable, the applied compressive load has to be less than the buckling load, 

For the problem the side constraints would normally be given by 

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An important use of the triangular equiHbrium diagram is the graphical solution of material balance problems, such as the calculation of the relative amounts of equiHbrium phases obtained from a given overall mixture composition. As an example, consider a mixture where the overall composition is represented by point M on Figure 2a. If the A-rich phase is denoted by point R (raffinate) and the B-rich phase is denoted by point E (extract), it can be shown that points R, M, and E are coUinear, and also... 

The graphics capabiUties of the CAD/CAM environment offer a number of opportunities for data manipulation, pattern recognition, and image creation. The direct appHcation of computer graphics to the automation of graphic solution techniques, such as a McCabe-Thiele binary distillation method, or to the preparation of data plots are obvious examples. Graphic simulation has been appHed to the optimisation of chemical process systems as a technique for energy analysis (84). 

Graphical solutions for concentration polarization. Uniform velocity through walls. 

FIG. 13-36 Graphical solution for a column with a partially flashed feed, a liquid side-stream and a total condenser. 

FIG. 15-15 Graphical solution to the mass-transfer-unit equations. 

The interactions discussed in this section involve rarefactions as well as shock waves. Provided that strains are small, the release path can be approximated by the Hugoniot in the P-u plane. The following graphical solutions to the interactions are approximate, but in many cases the approximation is very good. 

The predicted strain variation is shown in Fig. 2.43(b). The constant strain rates predicted in this diagram are a result of the Maxwell model used in this example to illustrate the use of the superposition principle. Of course superposition is not restricted to this simple model. It can be applied to any type of model or directly to the creep curves. The method also lends itself to a graphical solution as follows. If a stress is applied at zero time, then the creep curve will be the time dependent strain response predicted by equation (2.54). When a second stress, 0 2 is added then the new creep curve will be obtained by adding the creep due to 02 to the anticipated creep if stress a had remained... 

Rgure 8-43A-C. Graphical solution of unequal molal overflow, binary systems. 

Torres-Marchal [110] and [111] present a detailed graphical solution for multicomponent ternary systems that can be useful to establish the important parameters prior to undertaking a more rigorous solution with a computer program. This technique can be used for azeotropic mixtures, close-boiling mixtures and similar situations. 

A graphical solution is presented by Edmister [18] and handled like step-wise distillation. 

Referring to Figure 9-98 for the graphical solution Rectilying section operating line slope =... 

Note For practical purposes, a graphic solution is very advisable. 

Figure 3-14 is the graphical solution to the above equation and can be employed to estimate metals content of the E-cat, based on feed metals and catalyst addition rate. 

The balance condition between the evaporator and the compressor can be visualized in a graphic solution, superimposing the basic rating of the cooler on the compressor curves (see Figure 10.1). 

An alternative graphical solution makes use of the biphasic exponential nature of the plasma concentration function ineq. (39.16). At larger time values, when the effect of absorption has decayed, the function behaves approximately as monoexponential. Under these conditions, and after replotting the concentration data on a (decimal) logarithmic scale, one obtains a straight line for the later part of the curve (Fig. 39.8a). This line represents the P-phase of the plasma concentration and is denoted by C ... 

Figure 3 shows a graphical solution to both the °Th age equation (Eqn. 1) and the initial equation (Eqn. 2). Plotted on the ordinate and abscissa are the two measured quantities and [ °Th/ U]. Contoured with sub-vertical lines is one of the... 

ILLUSTRATION 8.7 DETERMINATION OF CSTR SIZE REQUIREMENTS FOR CASCADES OF VARIOUS SIZES— GRAPHICAL SOLUTION... 

For the case where the cascade consists of only a single reactor, only a single straight line of the form of equation 8.3.31 is involved in the graphical solution. One merely links the point on curve M corresponding to the effluent concentration of benzoquinone with the point on the abscissa corresponding to the feed concentration. The slope of this line is equal to ( — 1/t) or — io/VR). In the present instance the slope is equal to... 

Fig. A1.3. Graphical solution of Eq. (A1.76) with six equidistant non-degenerate squared frequencies a> (q) (3/V=6). The dash-dotted line connects the bending points of solid curves and defines quasilocal vibration frequencies.

The basis for a graphical solution for N = 2 is illustrated in Figure 14.11, which shows... 

Figure 14.11 Basis for graphical solution for multistage CSTR (for A +. .. - products)...

The following example illustrates both an analytical and a graphical solution to determine the outlet conversion from a three-stage CSTR. 

The Semenov criticality diagrams for fire growth are useful to understand the complex interactions of the fire growth mechanisms with the enclosure effects. These diagrams can be used qualitatively, but might also be the bases of simple quantitative graphical solutions. 

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