A new method (homogeneous integral solution) for solving collision problems utilizing an integral equation formalism is presented. The approach is noniterative and applies equally well to purely local interactions or problems with combined local and nonlocal interactions. The method involves transforming the integral equation of scattering into a Volterra integral equation of the second kind. It follows that even for nonlocal interactions, the wavefunction may be determined noniteratively from knowledge of its value at a single point (the origin). A simple numerical procedure is proposed for solution of the Volterra equation which allows one to avoid matrix inversions completely. At the end of the calculation, the scattering T or R matrix is obtained directly from the solutions. The method is illustrated by a calculation of the singlet and triplet s-wave Hartree–Fock phase shifts for electron–H-atom scattering.

A sudden reduction in the feed temperature to a packed-bed reactor leads to a transient temperature rise, which is referred to as the wrong-way behavior. A pseudo-homogeneous plug-flow model is used to analyze the structure of this transient behavior. The key parameters which determine the magnitude of this response are the dimensionless adiabatic temperature rise, activation energy, heat transfer capacity, coolant temperature, magnitude of temperature drop and length of the reactor. A simple expression is derived for predicing the maximum transient temperature rise. SCOPEWhen the temperature of the feed to a packed-bed reactor is suddenly decreased a transient temperature rise may occur. This surprising dynamic feature is caused by the difference in the speed of propagation of the concentration and temperature disturbances and is referred to as the wrong-way behavior. This response was predicted originally by Boreskov and Slinko (1965) and Crider and Foss (1966), and was observed by many investigators (Hoiberg et al., 1971; Van Doesberg and DeJong, 1976a, 1976b; Hansen and Jorgensen, 1977;Sharma and Hughes, 1979).The wrong-way behavior may damage the catalyst and initiate undesired side reactions. The need to avoid it complicates the control policies and start-up and shut-down procedures of packed-bed reactors. At present lengthy numerical simulations are required to determine when this behavior may be encountered and its magnitude.The purpose of this work is to identify the key rate processes and parameters which cause this behavior and to develop a simple technique for a priori prediction of the highest transient temperature without solving the transient equations. This is accomplished by analyzing the dynamic response of a plug-flow pseudo-homogeneous model of a packed-bed reactor using the method of characteristics. First, we determine the structure of the solution and the conditions for which a wrong-way behavior occurs for a zeroth-order reaction in either a cooled or an adiabatic reactor. We examine then how this behavior is modified by a rate expression for which the reactants are not completely consumed and by intraparticle diffusional resistances. CONCLUSIONS AND SIGNIFICANCEThe analysis indicates that for a zeroth-order reaction the wrong-way behavior occurs only if the reactor is longer than a critical length of z,i. The highest transient temperature increases with reactor length until the reactor is of length z~,~. For 1981.downstream of zrfl.any reactor shorter than z,,, and longer than zri the highest transient temperature is encountered at the exit of the reactor. For a cooled reactor longer than zc.fl the limiting transient-peak temperature occurs at z,,,. For an adiabatic reactor the limiting transient-peak temperature is encountered at all points

The recently developed method of homogeneous integral solutions for solving integral equations of scattering is extended to include coupled channels. The formalism is developed in matrix notation and applies without change both to purely local interactions and to combined nonlocal and local interactions. A numerical quadrature method for solving the integral equations is presented. This method does not require iteration or matrix inversions. Further, during the entire calculation of the wavefunction and T (or R) matrix elements, only one or two matrix inversions need be performed (in the case of purely local or local+nonlocal interactions, respectively). It is expected that the method will provide a particularly efficient means of solving coupled equations for scattering problems.

The homogeneous integral solution formalism developed by Sams and Kouri is applied to the problems of coupled open- and closed-channel radial equations and coupled radial eigenvalue equations. The problem of coupled eigenvalue equations is considered first. Because we deal with the integral equation form of the Schrödinger equation, the proper solutions may be obtained by integrating from the origin outward, rather than both outward from and inward toward the origin. For the case of eigenvalue radial equations, quantization arises from the fact that exponentially decaying solutions are obtained only when the eigenvalue of a certain matrix equals one. Next, we consider the coupled open- and closed-channel radial equations. Unlike the case of coupled open channels, we find it convenient to consider a column vector solution rather than a matrix. Just as in the open-channel case, however, the Volterra integral equations generated in solving the coupled open- and closed-channel equations are the same for the homogeneous and inhomogeneous integral solutions. Thus, the number of equations which must be solved is not as large as first appears. The method necessitates at most two matrix inversions, one for a n0 by n0 matrix and one for a nc by nc matrix, at the very end of the calculation. Here n0 is the number of open channels and nc the number of closed channels. Finally, application of the eigenvalue procedure is made to a Lennard-Jones (12–6) potential in order to illustrate the method.

The homogeneous integral solution formalism developed by Sams and Kouri is illustrated for coupled open channel scattering calculations and coupled eigenvalue calculations. This method for scattering calculations compares favorably, both in accuracy and in computation time required, with the Lester-Bernstein close coupling solution of the differential equations for scattering and the Johnson-Secrest amplitude density method. Preliminary results for coupled eigenvalue integral equations are also encouraging. A simple technique for applying the quantization condition in searching for the eigenvalues of coupled integral equations is presented, and is employed in the calculations reported. Example results are given for 16, 23, and 30 coupled scattering channels and for 1, 2, and 3, coupled integral eigenvalue equations.