Bessel function rate constants

In the absence of simplifying assumptions a non-closed form analytical solution of these equations giving the concentration of each species as a function of time has been accomplished through the use of Bessel functions U1 ). Simplifying assumptions, such as a large hydroxyl excess, are acceptable theoretically but are not helpful in determining the rate constants of coatings materials with standard compositions. 

Here, e is the maximum electron density in the reactor, r is the radial position in the reactor, rt is the radius of the reactor cylinder, z is the axial coordinate, L is the height of the reactor (distance between the two electrodes), and J0 is the zero-order Bessel function of the first kind. Clearly, the electron density is a maximum at the center of the reactor (r = 0, z = L/2). The rate constants are ... 

Unfortunately, the explicit solutions to these systems usually involve fairly complicated functions such as Bessel, Haenkel, theta, and incomplete gamma functions in which the arguments depend on the rate constants. The result is that these solutions are of use only if the ratios of the rate constants are known. Otherwise an inordinate amount of arithmetical labor is involved. 

At higher temperatures, when the Bessel function argument becomes large, it is possible to apply its asymptotic expansion. Furthermore, for temperatures at which intermolecular vibrations are classical (T > fe 2/4Kj), the rate constant takes the form... 

Henry constant for absorption of gas in liquid Free energy change Heat of reaction Initiator for polymerization, modified Bessel functions, electric current Electric current density Adsorption constant Chemical equilibrium constant Specific rate constant of reaction, mass-transfer coefficient Length of path in reactor Lack of fit sum of squares Average molecular weight in polymers, dead polymer species, monomer Number of moles in electrochemical reaction Molar flow rate, molar flux Number chain length distribution Number molecular weight distribution... 

Lag coefficient for particle in a pore Bessel function of the first kind of order 0 Mass flux of component A (mg/cm s) in a binary mixture with respect to a coordinate system that is moving with the mass average velocity of the mixture Enhanced friction for particle within a pore Equilibrium partition coefficient First-order elimination rate constant (s )... 

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