Mathematical modeling on gas turbine blades/vanes under variable convective and radiative heat flux with tentative different laws of cooling

In the last twenty years the modeling of heat transfer on gas turbine cascades has been based on computational fluid dynamic and turbulence modeling at sonic transition. The method is called Conjugate Flow and Heat Transfer (CHT). The quest for higher Turbine Inlet Temperature (TIT) to increase electrical efficiency makes radiative transfer the more and more effective in the leading edge and suction/ pressure sides. Calculation of its amount and transfer towards surface are therefore needed. In this paper we decouple convection and radiation load, the first assumed from convective heat transfer data and the second by means of emissivity charts and analytical fits of heteropolar species as CO2 and H2O. Then we propose to solve the temperature profile in the blade through a quasi-two-dimensional power balance in the form of a second order partial differential equation which includes radiation and convection. Real cascades are cooled internally trough cool compressed air, so that we include in the power balance the effect of a heat sink or law of cooling that is up to the designer to test in order to reduce the thermal gradients and material temperature. The problem is numerically solved by means of the Finite Element Method (FEM) and, subsequently, some numerical simulations are also presented

Boundedness for a Fully Parabolic Keller–Segel Model with Sublinear Segregation and Superlinear Aggregation

This work deals with a fully parabolic chemotaxis model with nonlinear production and chemoattractant. The problem is formulated on a bounded domain and, depending on a specific interplay between the coefficients associated to such production and chemoattractant, we establish that the related initial-boundary value problem has a unique classical solution which is uniformly bounded in time. To be precise, we study this zero-flux problem {ut=Δu−∇⋅(f(u)∇v) in Ω×(0,Tmax),vt=Δv−v+g(u) in Ω×(0,Tmax), where Ω is a bounded and smooth domain of Rn, for n≥ 2 , and f(u) and g(u) are reasonably regular functions generalizing, respectively, the prototypes f(u) = uα and g(u) = ul, with proper α, l> 0. After having shown that any sufficiently smooth u(x, 0) = u(x) ≥ 0 and v(x, 0) = v(x) ≥ 0 produce a unique classical and nonnegative solution (u, v) to problem (◊), which is defined on Ω × (0 , Tmax) with Tmax denoting the maximum time of existence, we establish that for any l∈(0,2n) and 2n≤α<1+1n−l2, Tmax= ∞ and u and v are actually uniformly bounded in time. The paper is in line with the contribution by Horstmann and Winkler (J. Differ. Equ. 215(1):52–107, 2005) and, moreover, extends the result by Liu and Tao (Appl. Math. J. Chin. Univ. Ser. B 31(4):379–388, 2016). Indeed, in the first work it is proved that for g(u) = u the value α=2n represents the critical blow-up exponent to the model, whereas in the second, for f(u) = u, corresponding to α= 1 , boundedness of solutions is shown under the assumption 0<2n

Improvements and generalizations of results concerning attraction-repulsion chemotaxis models

We enter the details of two recent articles concerning as many chemotaxis models, one nonlinear and the other linear, and both with produced chemoattractant and saturated chemorepellent. More precisely, we are referring respectively to the papers “Boundedness in a nonlinear attraction-repulsion Keller–Segel system with production and consumption,” by S. Frassu, C. van der Mee and G. Viglialoro [J. Math. Anal. Appl. 504(2):125428, 2021] and “Boundedness in a chemotaxis system with consumed chemoattractant and produced chemorepellent,” by S. Frassu and G. Viglialoro [Nonlinear Anal. 213:112505, 2021]. These works, when properly analyzed, leave open room for some improvement of their results. We generalize the outcomes of the mentioned articles, establish other statements, and put all the claims together; in particular, we select the sharpest ones and schematize them. Moreover, we complement our research also when logistic sources are considered in the overall study

A nonlinear attraction-repulsion Keller–Segel model with double sublinear absorptions: criteria toward boundedness

This paper deals with the zero-flux attraction-repulsion chemo-taxis model {u(t) = del center dot ((u + 1)(m1-1)del u-chi u(u + 1)(m2-1)del v in Omega x (0, T-max), +xi u(u + 1)(m3-1)del w) + h(u) (lozenge) v(t) = Delta v - f (u)v in Omega x (0, T-max), w(t) = Delta w - g(u)w in Omega x (0, T-max), in the unknown (u, v, w)= (u(x, t), v(x, t), w(x, t)). Here, x is an element of Omega, a bounded and smooth domain of R-n(n >= 1), t, chi, xi > 0, m(1), m(2), m(3) is an element of R, and f (u), g(u) and h(u) sufficiently regular functions generalizing the prototypes f(u) = K(1)u(alpha), g(u) = K2u(gamma) and h(u) = ku - mu u(beta), with K-1, K-2, mu > 0, k is an element of R, beta > 1 and suitable alpha, gamma > 0. Besides, further regular initial data u(x, 0) = u(0)(x), v(x, 0) = v(0)(x), w(x, 0) = w(0)(x) >= 0 are given, whereas T-max is an element of (0, infinity] stands for the maximal instant of time up to which solutions to the system exist. We will derive relations between the parameters involved in (>) capable to warrant that u, v, w are global and uniformly bounded in time. The article generalizes and extends to the case of nonlinear effects and logistic perturbations some results recently developed in [3] where, for the linear counterpart and in the absence of logistics, criteria towards boundedness are established

Decay in chemotaxis systems with a logistic term

This paper is concerned with a general fully parabolic Keller-Segel system, defined in a convex bounded and smooth domain Ω of RN , for N ∈ {2, 3}, with coefficients depending on the chemical concentration, perturbed by a logistic source and endowed with homogeneous Neumann boundary conditions. For each space dimension, once a suitable energy function in terms of the solution is defined, we impose proper assumptions on the data and an exponential decay of such energies is established

Comparación entre el análisis 2-D y el Método de la Densidad de Fuerzas (discreto) para el equilibrio en estructuras de membrana

This paper deals with the equilibrium problem of a membrane, presenting a comparison between the well-known 1-D discrete Force Density Method, using spatial cable networks as membrane surface approximations, and the 2-D continuous analysis of such a surface. Although Force Density is a practical and powerful method in structural membrane design, it will be checked that the 2-D continuous analysis is not only more accurate and general but necessary for new structural membrane applications such as footbridges. In this way, once summarized the discrete Density Force Method, the continuous approach is presented. Then, a comparison process between both methods is proposed, being developed for specific membrane examples. Finally, some conclusions are pointed out.Este trabajo analiza el problema del equilibrio de una membrana y propone una comparación entre el conocido Método de la Densidad de Fuerzas, discreto y unidimensional (1-D), que aproxima la superficie de la membrana mediante una red espacial de cables, y el método continuo y bidimensional (2-D) sobre la propia superficie. Aunque el Método de la Densidad de Fuerzas representa una estrategia práctica y útil en el diseño de estructuras de membrana, se comprobará que el análisis continuo bidimensional no solo es más preciso y general sino que es más fiable, especialmente en aquellos casos en los que la membrana es el mismo tablero de una estructura portante (por ejemplo, una pasarela). En particular, una vez resumido el Método de la Densidad de Fuerzas, se planteará el problema continuo del equilibrio de membrana. A continuación se definirá un proceso de comparación entre ambos métodos, analizándolo por medio de ejemplos concretos. Finalmente, se señalarán algunas conclusiones

Industrial steel heat treating: Numerical simulation of induction heating and aquaquenching cooling with mechanical effects

This paper summarizes a mathematical model for the industrial heating and cooling processes of a steel workpiece corresponding to the steering rack of an automobile. The general purpose of the heat treatment process is to create the necessary hardness on critical parts of the workpiece. Hardening consists of heating the workpiece up to a threshold temperature followed by a rapid cooling such as aquaquenching. The high hardness is due to the steel phase transformation accompanying the rapid cooling resulting in non-equilibrium phases, one of which is the hard microconstituent of steel, namely martensite. The mathematical model describes both processes, heating and cooling. During the first one, heat is produced by Joule’s effect from a very high alternating current passing through the rack. This situation is governed by a set of coupled PDEs/ODEs involving the electric potential, the magnetic vector potential, the temperature, the austenite transformation, the stresses and the displacement field. Once the workpiece has reached the desired temperature, the current is switched off an the cooling stage starts by aquaquenching. In this case, the governing equations involve the temperature, the austenite and martensite phase fractions, the stresses and the displacement field. This mathematical model has been solved by the FEM and 2D numerical simulations are discussed along the paper