Integral transport method

C. L. BENNET, GRANIT A Code for Calculating Position Dependent Thermal Neutron Spectra in Doubly Heterogeneous Systems by the Integral Transport Method, BNWL-1634 (Dec. 1971). 

Yan, Q. (2003) Bioinformatics and data integration in membrane transporter studies, in Membrane transporters methods and protocols (Q. Yan, ed.). Methods in Molecular Biology, Humana, Totowa, NJ, pp. 37-60. 

Selected entries from Methods in Enzymology [vol, page(s)j Boundary analysis [baseline correction, 240, 479, 485-486, 492, 501 second moment, 240, 482-483 time derivative, 240, 479, 485-486, 492, 501 transport method, 240, 483-486] computation of sedimentation coefficient distribution functions, 240, 492-497 diffusion effects, correction [differential distribution functions, 240, 500-501 integral distribution functions, 240, 501] weight average sedimentation coefficient estimation, 240, 497, 499-500. 

It is well known that a flow-equilibrium must be treated by the methods of irreversible thermodynamics. In the case of the PDC-column, principally three flows have to be considered within the transport zone (1) the mass flow of the transported P-mer from the sol into the gel (2) the mass flow of this P-mer from the gel into the sol and (3) the flow of free energy from the column liquid into the gel layer required for the maintenance of the flow-equilibrium. If these flows and the corresponding potentials could be expressed analytically by means of molecular parameters, the flow-equilibrium 18) could be calculated by the usual methods 19). However, such a direct way would doubtless be very cumbersome because the system is very complicated (cf. above). These difficulties can be avoided in a purely phenomenological theory, based on perturbation calculus applied to the integrated transport Eq. (3 b) of the PDC-column in a reversible-thermodynamic equilibrium. 

Liu, C., J.E. Szecsody, J.M, Zachara, and W.P, Ball, Use of the generalized integral transform method for solving equations of solute transport in porous media. Advances in Water Resourses, 2000, 23 pp, 483 92... 

Exact calculation of reactivity and of the effects of perturbation on other integral parameters has to take into account the effect of the perturbation on the flux (or adjoint) distribution. Several methods allowing for the flux perturbation have been recently developed, or are already routinely in use. Three of these methods are reviewed in this work the direet calculation of the flux perturbation (Section III) integral transport theory methods (Section IV), and generalized perturbation theory (Section V, B). 

The formulations presented in this section are for the calculation of perturbed fluxes. Analogous formulations can be derived and applied for the calculation of perturbed adjoints, kernels for the integral transport equations (see Section IV), generalized functions (see Section V), and other distribution-function perturbations. The presentation is restricted to exposition of the general formulation of the methods, without considering the technical details of the solution. 

Many methods and computer codes have been developed for the solution of Eq. (47) for the purpose of reactivity calculations. The majority of them are based on integral transport theory formulations. They include the methods of Karam et al. (20), Collins and Palmer (27), Kier and Salvatores (22,23), McGrath and Foell (24), Fischer (25), Oosterkamp (26), and McGrath and Fischer (27). 

Interest in integral transport theory methods for reactivity calculations has recently increased, mainly because of two features peculiar to integral formulations. 

The collision probability is one of several possible formulations of integral transport theory. Three other formulations are the integral equations for the neutron flux, neutron birth-rate density, and fission neutron density. Oosterkamp (26) derived perturbation expressions for reactivity in the birth rate density formulation. The fission density formulation provides the basis for Monte Carlo methods for perturbation calculations (52, 55). 

E. A. Fischer, A Method to Calculate Reactivity Worth by Integral Transport Theory, KFK-995. Kemforchungszentrum Karlsruhe (1969). 

The approximate methods used frequently for criticality calculations are classified into several broad categories Monte Carlo, discrete ordinates, integral transport, and diffusion theory. Within each category there exist several computer programs, each with a different treatment of spatial detail and energy group structure. Several of these methods are topics for later papers in this session. In the remainder of this paper we will describe some general aspects of each method and indicate the types of situations where each is used. 

The probability table method of treating unresolved resonances, developed for Monte Carlo calculations, has been extended to and tested for use in a one-dimensional integral transport theory slowing down calculation using a modified version of the HABBLE code. 

I. An attempt was made to ensure consistency of the cross sections used in the three calculations as far as was possible with the differing code formats. The resolved resonance cross sections were taken from ENDF Version I data. Other cross sections were similar but not identical to ENDF/B. The Standard Argonne Method used integral transport rather than Sn methods. Leakage corrections were not included in any of the calculations. 

Again, the traditional calculational tool for this step has been integral transport (collision probability) methods, but discrete ordinates and Monte Carlo are seeing increased use. 

It is of special interest to note that the diffusion-theory methods, even in the Pa approximation, yield significantly different solutions from those of the integral (transport) theory model. This is to be expected in the present application since the system in question is relatively narrow... 

An alternative method is to move the reactor core in one piece, which is also described in the DOE environmental impact statement (EIS) (1993) and which is the preferred transport method for the eight deactivated reactors at the Hanford Site. However, because the N Reactor sits on a pedestal and the thermal and radiation shielding are not integrated into the core structure, the preferred option may not be feasible. 

There are two approaches to the nodal transport method, the first, almost conventional, being based on the transversal integration procedure [4.45], the second being the variational nodal transport method [4.60,4.61]. 

The use of underground mining methods requires integration of transportation, ventilation, ground control, and mining methods to form a system that provides the highest possible degree of safety, the lowest cost per ton of product, the most suitable quality of final product, the maximum possible recovery of coal, and the minimum disturbance of... 

---