Mathematical programming model for heat exchanger design through optimization of partial objectives

Mathematical programming can be used for the optimal design of shell-and-tube heat exchangers (STHEs). This paper proposes a mixed integer non-linear programming (MINLP) model for the design of STHEs, following rigorously the standards of the Tubular Exchanger Manufacturers Association (TEMA). Bell–Delaware Method is used for the shell-side calculations. This approach produces a large and non-convex model that cannot be solved to global optimality with the current state of the art solvers. Notwithstanding, it is proposed to perform a sequential optimization approach of partial objective targets through the division of the problem into sets of related equations that are easier to solve. For each one of these problems a heuristic objective function is selected based on the physical behavior of the problem. The global optimal solution of the original problem cannot be ensured even in the case in which each of the sub-problems is solved to global optimality, but at least a very good solution is always guaranteed. Three cases extracted from the literature were studied. The results showed that in all cases the values obtained using the proposed MINLP model containing multiple objective functions improved the values presented in the literature.

A note on the real projection of the zeros of partial sums of Riemann zeta function

This paper proves that the real projection of each simple zero of any partial sum of the Riemann zeta function ζn(s):=∑nk=11ks,n>2 , is an accumulation point of the set {Res : ζ n (s) = 0}.

The Zeros of Riemann Zeta Partial Sums Yield Solutions to f(x) + f(2x) + · · · + f(nx) = 0

This paper proves that every zero of any n th , n ≥ 2, partial sum of the Riemann zeta function provides a vector space of basic solutions of the functional equation f(x)+f(2x)+⋯+f(nx)=0,x∈R . The continuity of the solutions depends on the sign of the real part of each zero.

Evolution of the last koninckinids (Athyridida, Koninckinidae), a precursor signal of the early Toarcian mass extinction event in the Western Tethys

Koninckinids are a suitable group to shed light on the biotic crisis suffered by brachiopod fauna in the Early Jurassic. Koninckinid fauna recorded in the late Pliensbachian–early Toarcian from the easternmost Subbetic basin is analyzed and identified as a precursor signal for one of the most conspicuous mass extinction events of the Phylum Brachiopoda, a multi-phased interval with episodes of changing environmental conditions, whose onset can be detected from the Elisa–Mirabile subzones up to the early Toarcian extinction boundary in the lowermost Serpentinum Zone (T-OAE). The koninckinid fauna had a previously well-established migration pattern from the intra-Tethyan to the NW-European basins but a first phase with a progressive warming episode in the Pliensbachian–Toarcian transition triggered a koninckinid fauna exodus from the eastern/central Tethys toward the westernmost Mediterranean margins. A second stage shows an adaptive response to more adverse conditions in the westernmost Tethyan margins and finally, an escape and extinction phase is detected in the Atlantic areas from the mid-Polymorphum Zone onwards up to their global extinction in the lowermost Serpentinum Zone. This migration pattern is independent of the paleogeographic bioprovinciality and is unrelated to a facies-controlled pattern. The anoxic/suboxic environmental conditions should only be considered as a minor factor of partial control since well-oxygenated habitats are noted in the intra-Tethyan basins and this factor is noticeable only in the second westward migratory stage (with dwarf taxa and oligotypical assemblages). The analysis of cold-seep proxies in the Subbetic deposits suggests a radiation that is independent of methane releases in the Subbetic basin.