Adiabatic reaction path

Fehrensen B, Luckhaus D and Quack M 1999 Inversion tunneling in aniline from high resolution infrared spectroscopy and an adiabatic reaction path Hamiltonian approach Z. Phys. Chem., NF 209 1-19... 

We have established an important principle in electron transfer theory that is not present in conventional one-dimensional models. The reaction coordinate is always localizing and corresponds to coordinate Aj. The coordinate X2 corresponds to the direction in which the matrix element between ground and excited states is switched on. If this coordinate has zero length then the branching space becomes one dimensional and an adiabatic reaction path does not exist. We now consider two examples. 

Because the adiabatic reaction path is linear, a graphical solution, also... 

Because the adiabatic reaction path is linear a graphical solution, also applicable to multi-bed reactors, is particularly apposite. (See Example 3.7 as an illustration.) If the design data are available in the form of rate data for various temperatures and conversions they may be displayed as contours of equal reaction rate in the (T, Y) plane. Figure 3.14 shows such contours upon which is superimposed an adiabatic reaction path of slope cp/(- AH) and intercept T0 on the abscissa. The reactor size may be evaluated by computing ... 

Thus the feed stock is represented by the origin in the c,t plane and Eq. (6.1.2) for an adiabatic reaction path becomes... 

Fig. 3.5 Adiabatic reaction paths for exothermic and endothermic reversible reactions.

Sample adiabatic reaction paths determined by this equation are shown in Figure 5. Kinetic parameters obtained in this work are summarized in Table I. 

Two adiabatic reaction paths for epoxy + AEP, at T = 41.0 C and = 51.9 C. Solid curves are predictions based on complex kinetics (Equation 7). 

Equation 11.5.d-5 means that the reaction has to be stopped when the rate has reached a value of v. Beyond that point the increase in cost outweighs the increase in profit resulting from the conversion. It is clear that this point is situated on that part of the adiabatic reaction path that is beyond and That part of the rate contour that has a value v and that is to the right of F is represented by F,. 

The second condition in Eq. (11.5.d-6) is satisfied only when 5r/5T the partial derivative of the rate with respect to the temperature is partly positive and partly negative. Substituting into this partial derivative the relation between x and T along an adiabatic reaction path starting from F, (condition Eq. 11.5.d-5) turns 5p/5T, = /i(x, r,) into a function dp/dTi = /2(T,). The root of this equation is easily found by a one-dimensional search procedure and is the optimum inlet temperature leading to the exit conditions represented by the point chosen on Fi. This procedure is repeated for a certain number of points on F, to obtain the locus of optimum inlet conditions for bed 1, represented by F, in Fig. 11.5.d-l. It follows from Eq. 11.5.d-6 that F, and F, intersect on F , not on F. ... 

In this diagram of possible outlet conditions for various inlet conditions, the adiabatic reaction path corresponding to given x(0) and 7(0) is represented by the straight line (b) having a slope i and ending in a point of (a). Curve (a) and the line (b) have only one point in common. 

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