Graphical Solution of Equations

Instead of approximating an equation and then solving the approximate equation algebraically, we can apply the graphical method to obtain a munerical approximation to the correct root. This method is sometimes very useful because you can see what you are doing and you can usually be sure that you do not obtain a different root than the one you want to find. The equation to be solved is written in the form 

Drawing a graph by hand is tedious, and almost no one actually makes graphs by hand, since it is possible to make graphs more easily with a spreadsheet program 

When one first opens Excel a window is displayed on the screen with a number of rectangular areas called cells arranged in rows and columns. This window is called a worksheet. The rows are labeled by numbers and the columns are labeled by letters. Any cell can be specified by giving its column and its row (its address). For example, the address of the cell in the third row of the second column is B3. A list of menu headings appears across the top of the screen and a double strip of small icons called a toolbar appears under the menu headings. 

Any cell can be selected by using the arrow buttons on the keyboard or by moving the mouse until its cursor is in the desired cell and clicking the left mouse button. One can then type one of three kinds of information into the cell a number, some text, or a formula. For example, one might want to use the top cell in each column for a label for that column. One would first select the cell and then t3 e the label. As the label is typed, it appears in a line above the cells. It is then entered into the cell by moving the cursor away from that cell or by pressing the Return key. A number is entered into another cell in the same way. To enter a munber but treat it as text, precede the number with a single quotation mark ( ). 

EXAMPLE 3.6 Enter a formula into cell Cl to compute the sum ofthe number in cell A1 and the number in cell B2, divide by 2, and take the common logarithm 

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A graphic solution of Equation 1 in coordinates JCobs — (S) produces a straight line with the slope equal to Ks. Figure 2 shows the results of the treatment according to Table I. In all three cases the value of the decay constant Km is approximately the same. While the experiments are carried out with different systems but under reproduced conditions, Km is stable in above mentioned limits ( 20% ) and the value of Ks is steady too. 

Figure 2. Graphic solution of Equation 1 from data of Table I. 0—H202 A—HCIO D—Cu(CIOa)2...

Fig. 5. Graphical solution of equation (64). Curve A right-hand side cmve B left-hand side of the equation for a = 0, /3 = -1, t= -0.2 and A/ = 8. The root is the abscissa of intersection point 0o, while 0 , is the limiting value of the root for A/— 00.

An alternative way for the graphical solution of equation 3 was proposed by Foster and Fyfe [49]. According to this technique, equation (4) is rewritten in the following form ... 

Similarly, the graphical solution of Equation 5.28 is identical to the previous one. 

Fig. 8. Scheme of graphic solution of equation 12. Full and broken lines relate to blends with and without a compatibilizer. Y-axis shows F R) and (4/w)yi Pc(P) in arbitrary units. 

Figure 2.45. Graphic solution of equation (2.112) assuming that the ratio klq)/(l+R/P) increases with temperature. The curve L corresponds to the left part of equation (2.112), curves R andiij correspond to its right part at T, and T, (T <7 j).

The solution of equations (A ) and (B ) gives the stationary-state values of fA and I The solutions for the three cases (a), (b), and (c) are shown graphically in Figure 14.5. The sigmoid-shaped curve is constructed from (A ), and the three straight lines from (B ), each with slope 0.0201 K-1, for the three values of Tr Note that for adiabatic operation, T = T0 at /A = 0. 

The solution of equation 14.4-1 to obtain V(given /A) or fA (given V) can be carried out either analytically or graphically. The latter method can be used conveniently to obtain fA for a specified set of vessels, or when ( —rA) is not known in analytical form. 

Sometimes it is desirable to have a solution of Equation (b) in even approximate analytical form rather than in the tabular or graphical form that a numerical solution provides. Suitable methods are described in such books as Bender Orszag (Advanced Mathematical Methods for Scientists and Engineers, 1978)... 

As depicted in Figure 4, all the solutions for can be found graphically at each intersection of / ss (which is a straight line of slope — DM/ro) with the curve for /u = /u i + /u,2 (which is the sum of two hyperbolae with their corresponding vertical asymptotes at = Km,i and at = Km,2)- Due to the positive character of all the physical constants, one concludes that there is only one positive (physically meaningful) solution of equation (23). 

Fig. 15. Graphical solution of the design equation for a cascade of continuous stirred tank reactors.

The volume of tabular information necessary to record in detail the thermodynamic data for the paraffin hydrocarbons and their mixtures, as was done for steam, is excessive. It appears hopeful that graphical generalizations typified by the work of Edmister (19) will prove adequate for the less rigorous requirements of design, whereas the Benedict equation of state (4) may be employed where precision is necessary. However, the effective application of this equation of state to compounds containing more than four carbon atoms per molecule still awaits the evaluation of the constants. After the composition and specific volume have been established for a particular state, the solution of equations of state to establish enthalpy and entropy is a straightforward process. 

Extraction calculations involving more than three components cannot be done graphically but must be done by numerical solution of equations representing the phase equilibria and material balances over all the stages. Since extraction processes usually are adiabatic and nearly isothermal, enthalpy balances need not be made. The solution of the resulting set of equations and of the prior determination of the parameters of activity coefficient correlations requires computer implementation. Once such programs have been developed, they also may be advantageous for ternary extractions,... 

Fig, 9. Graphic solution of the balance equation for the reaction aa + -methyl-styrene aa its living dimer and aax its living trimer. 1// is the reciprocal of the resident time in the stirred-fiow reactor. Reproduced, with permission, from Lee, Smid, and Szwarc J. Am. Chem. Soc. 85,912... 

Stable states can be found, for example, by graphical solution of the equation 1 /x(4> — 4>o) = 7(potential minima [42,65], and it can be shown immediately that OB arises only if the system is biased by a sufficiently strong external field, that is, when it is far away from thermal equilibrium. If the noise intensity is weak, the system, when placed initially in an arbitrary state, will, with an overwhelming probability, approach the nearest potential minimum and will fluctuate near this minimum. Both the fluctuations and relaxation... 

Fig. 25. Real and imaginary parts of the 4 p self-energy (E) and straight lines E-Ej(zlSCF), i = 4 pi/2.3/2. giving graphical solutions of the Dyson equation (Eq. (15))...

Table V via graphical solution of the series of simultaneous equations.

A graphical solution for equation 2 is possible but somewhat complicated as some assumptions must be made (8). Graphical evaluation of more complex equilibria becomes very difficult if not impossible. As will be shown in this work, appropriate nonlinear least squares (NLLSQ) programs can be used to quickly evaluate and solve a variety of simple and complex equilibria problems. In addition to solving a number of the more difficult cyclodextrin complexation problems, data in the literature will be re-evaluated. 

In Figure 3-28A it can be seen that all the observed counts cluster around a central value, the mean. This distribution of data closely approximates the graphic solution of the Poisson distribution equation. 

Figure 8-7 Graphical solution of equilibrium and energy balance equations to obtain adiabatic temperature and equilibrium conversion.

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