Transcendental equation

This gives a transcendental equation in q, the root of which occurs when q = q ... 

Due to the inherent uncertainty of the Langmuir model and difficulties in solving the transcendental equation (41), probably the most accurate treatment in the near-equilibrium cases is a numerical or graphical integration of the expression... 

In estimating the value of Ed by means of the transcendental equations (28), the circumstance utilized is that the variation of em for a given change in Tm is much less than the variation of exp(em) (31). Until now, only particular solutions have been available for the hyperbolic and linear heating schedules and for the first-order and second-order desorptions. They can be found for example in the fundamental papers by Redhead (31) and Carter (32) or in the review by Contour and Proud homme (106), and therefore will not be repeated here. Recently, a universal procedure for the... 

The other global method in common use is generally called the method of Qraejfe in western Europe and in the United States, and the method of Lobachevskii in the USSR. The first to state the principle was Dandelin, but Graeffe devised the algorithm, simple enough in itself, that is normally used. The method is widely known, and is described in many places. Hence, it will not be described here. But it is not widely known that the method is applicable, with rather trivial modifications, to solving transcendental equations. 

Differentiation and setting dbout/dT = 0 gives a transcendental equation in Toptimai that cannot be solved in closed form. The optimal temperature must be found numerically. 

Asymptotically for large n the solution of this transcendental equation is... 

The prefactor of the exponential in (C.7) is less easily obtained. To get it one has to solve the transcendental equation de(h)/h = 0 for / and insert this into Numerically one obtains that this factor is close to 1/2. 

Between the two roots of Eq. (5-140), one must take into consideration the higher value in order to assure the positivity of (z — L0). Substituting this value of qM into Eq. (5-138) yields a transcendental equation its solution gives the value of z corresponding to the burnout point. 

This transcendental equation must be solved for r using the indicated numerical values and recognizing that 0 < r < 1 for the product distribution cited. 

From equation 12.7.20 it is evident that W must be at least as large as hC9 and this fact provides a useful starting point for solving these transcendental equations. We begin by assuming that W = 35 BTU/(hr-ft2-°R). From equation 12.7.23,... 

This is a transcendental equation, which is not easily solved by ordinary methods. Nowadays, however, computers make the solution of such equations by successive approximations easy. In this case, again using EXCEL , we find that the value of T that makes the left-hand side of equation (42-50) become zero, which thus gives the value corresponding to the transmittance corresponding to minimum relative error, is 0.32994, rather than the previously accepted value of 0.368... 

Equation 48-119 is a much simpler equation than most of the ones we have had to deal with before, including equation 42-50 (which is the corresponding equation for the constant-detector-noise case [2]) nevertheless, it is still a transcendental equation and is best solved by successive approximations. 

These two transcendental equations define a pair of closely spaced energy levels, respectively associated with symmetric and antisymmetric wave functions as defined by the arbitrary choice of D = C. 

We notice that the positive blackbody contributions for E and P dominate in the high-temperature limit, while the energy and the pressure are negative for low T. From Eq. (32), we can determine the critical curve (/3C = XoL) for the transition from negative to positive values of P, by searching for the value of the ratio x = j3/L for which the pressure vanishes this value, xo, is the solution of the transcendental equation... 

As before, for any given value of L, the pressure changes from negative to positive when the temperature is raised. The critical curve for this transition is f3c = xoL, where Xo is the value of ratio % = f3/L for which the pressure vanishes, that is the solution of the transcendental equation... 

Because solving the transcendental equation (1.5.23) requires iterative methods of root finding, the numerical solution to this equation is postponed to Section 3.1. o... 

The parameter , is therefore obtained as a solution of the transcendental equation... 

Numerics has been used intensively in the field of integrated optics (10) since its early days, simply due to the fact that even the basic example of the slab waveguide requires the solution of a transcendental equation in order to calculate the propagation constants of the slab- guided modes. Of course, the focus was directed to analytical methods, primarily, as long as the power of a desktop computer did allow for a few coupled equations and special functions only e.g. to describe the nonlinear directional coupler in a coupled mode theory (CMT) picture. During the years, lots of analytic and semi-analytic approaches to solve the wave equation have been developed in order... 

These equations are significantly more complicated to solve than those for constant density. If we specify the reactor volume and must calculate the conversion, for second-order kinetics we have to solve a cubic polynomial for the CSTR and a transcendental equation for the PFTR In principle, the problems are similar to the same problems with constant density, but the algebra is more comphcated. Because we want to illustrate the principles of chemical reactors in this book without becoming lost in the calculations, we win usually assume constant density in most of our development and in problems. 

We shall defer detailed discussion of the ripple structure, which is considerably more complicated both mathematically and physically than the interference structure, until Chapter 11. Suffice it to say for the moment that the ripple structure has its origins in the roots of the transcendental equations (4.54) and (4.55), the conditions under which the denominators of the scattering coefficients vanish. 

Upon insertion of Eq. (82) into Eq. (81) one has an implicit transcendental equation for z that must in general be solved numerically for specified values of x, T, and the interaction coefficients. Then yt and y3 can be obtained with Eq. (57) and various thermodynamic properties calculated. Analytical expressions can be obtained for the various properties in terms of x, , and z. However, these are somewhat cumbersome and in general we shall only write them for... 

Derive an expression for the temperature Tp at which the desorption rate is a maximum (this may be a transcendental equation). Let the surface coverage at T = Tp be defined as 0 = Gp. 

Transcendental equations, such as for A in Eq. 20.10, appear frequently in moving interface problems and can be solved using numerical methods. 

Therefore, the interface again moves parabolically, and A is given by a transcendental equation. 

Consider again the problem posed in Exercise 20.6, whose solution had the form of a transcendental equation. A simple and useful approximate solution can be found by using the linear approximation to the diffusion profile shown in Fig. 20.14. Find the solution based on this approximation. 

This transcendental equation cannot be solved in elementary functions. However, we may easily construct the desired curves of the dependence of on r for given values of a and b by introducing the parameter 6, which has the simple physical meaning of the temperature ... 

At the limit the derivative du/da —> oo and the curve is tangent to the vertical. The dashed continuation of the curve corresponds to a spurious root of the transcendental equation, an unstable, physically unrealizable regime. 

Here and /3, are the first roots of a certain transcendental equation [see Hlavacek and Kubicek (11)]. Evidently after inspection of Eq. (5) and Tables II and III the value of (yj3) is essentially reduced so that Eq. (5) can be satisfied for a number of catalytic exothermic systems. 

Polynomial equations are not rare in chemical/biological engineering problems, they are typically met in most local stability problems. However, nonalgebraic or transcendental equations are more common in our subject. A function in one variable x is called transcendental, if it is not merely a polynomial in x, nor a ratio of polynomials (called rational function), but contains nonalgebraic transcendental expressions in x, such as... 

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