Multiplicative Controllability of Semilinear Parabolic Equations with Neumann Boundary Conditions

We prove the controllability and the existence of the time optimal control for semilinear parabolic equations with the homogenous Neumann boundary condition via multiplicative controls. It is worth pointing out that there is no restriction on the growth of the nonlinearity f(s) with respect to the variable s in the equation, which is a remarkable difference compared to the semilinear parabolic system with additive locally distributed controls.

     

An exact analytical solution for fractional-order thermoelasticity in a multi-stacked elliptic plate

Abstract

Thermoelastic analysis of an isotropic homogeneous multi-stacked elliptical plate has been considered in this research. For which multi-layered plate is taken into consideration on a plane-parallel elliptic geometry perpendicular to the z-direction. The governing equations are considered in the context of time-fractional derivative of the order α with temperature distribution in each s layer of the stacked plate with time-dependent sectional heat supply on the lower and upper face. The multi-stacked profile consists of s discrete plates each of a different material with perfect thermal contact at each of its s-1 interface. The general solution, which perfectly satisfies the fundamental equation of heat conduction, is obtained using an integral transformation technique. It is solved using a type of quasi-orthogonality relationship by modifying Vodicka’s method and the Laplace transformation. The analysis is based on the small-deflection theory corresponding to the fundamental solutions for the fractional-order heat conduction equation. In addition to this, the intensities of bending moments, forces, maximum normal stresses and its associated stresses are formulated involving the Mathieu functions. As a special case, a multi-stacked circular plate is also discussed in detail as a limiting case. Numerical calculations are also performed, and the results are graphically illustrated.     

Topological r-Entropy and Measure Theoretic r-Entropy for Amenable Group Actions

In this paper, we introduce the concept of measure theoretic r-entropy on a compact metric space in the context of amenable group actions. To be precise, we generalize Katok’s r-entropy formula and Brin-Katok r-entropy formula to infinite countably amenable group actions.

     

Chebyshev collocation spectral domain decomposition method for coupled conductive and radiative heat transfer in a 3D L-shaped enclosure

ABSTRACT

This paper presents a Chebyshev collocation spectral domain decomposition method (CSDDM) to study the coupled conductive and radiative heat transfer in a 3D L-shaped enclosure. The partitioned 3D L-shaped enclosure is subdivided into rectangular subdomains based on the concept of domain decomposition. The radiative transfer equation is angularly discretized by the discrete ordinate method with the SRAP_N quadrature scheme and then solved by the CSDDM using the same grid system as in solving the conduction. The effects of the conduction–radiation parameter, the optical thickness, the scattering albedo, and the aspect ratio on thermal behavior of the system are investigated. The results indicate that the 3D CSDDM has a good accuracy and can be considered as a good alternative approach for the solution of the coupled conduction and radiation problems in 3D partitioned domains.     

APPLICATIONS OF AN EXPONENTIAL FINITE-DIFFERENCE TECHNIQUE

ABSTRACT

In this paper an expontial finite-difference scheme, first presented by Bhattacharya for one-dimensional, unsteady heat conduction problems in a plane wail, is used to solve various partial differential equations. Solutions of the unsteady diffusion equation in three dimensions and of the viscous form of Burgers’ equation ere used to illustrate the method. Predicted results are compared with exact solutions or with results obtained by other numerical methods.     

Calculation of NARM‘s Equilibrium with Peng—Robinson Equation of State

The liquid molar volumes of nonazeotropic refrigerant mixtures (NARM), calculated with Peng Robinson (PR) equation, were compared with vapor -liquid equilibrium experimental data in this paper. Provided with co-reaction coefficient k _ij , the discrepancies of liquid molar volume data for R22+R114 and R22+R142b using PR equation are 7.7% and 8.1%, respectively. When HBT (Hankinson-Brobst-Thomson) equation was joined with PR equation, the deviations are reduced to less than 1.5% for both R22+R114 and R22+R142b.     

ANALYSIS OF PRESSURE-VELOCITY COUPLING ON NONORTHOGONAL GRIDS

Extension of the SIMPLE pressure-velocity coupling algorithm to nonorthogonal grids results in a very complex pressure-correction equation (e.g., a 9-point computational molecule in a two-dimensional case, a 19-point.computational molecule in a three-dimensional case) The usual practice is therefore to further simplify this equation by neglecting the effect of nonorthogonality on the mass flux corrections, thus reducing the computational molecule to 5 or 7 points

The paper analyzes the performance of the simplified and full pressure-correction equations when the grid nonorthogonality becomes appreciable. It is demonstrated here that the efficiency of the simple coupling algorithm is not affected by the grid nonorthogonality, provided that no additional simplifications are introduced in the pressure-correction equation. However, the algorithm with the simplified equation becomes inefficient when the angle between grid lines approaches 45^° and it usually fails to converge for angles below 30^°. The problem of solving the full 9-point pressure-correction equation is best dealt with by employing the biconjugate gradient solver, which proved to be the most robust one in test calculations carried out in this study.     

ANALYSIS ON TRANSIENT THERMAL STRESSES IN AN ANNULAR FIN

Transient thermal stresses are an important consideration in production processes involving large temperature changes. Recently, thermal stresses have also become significant in design problems related to microelectronic devices through their effects on material properties and system parameters. To calculate the thermal stresses, three kinds of methods are available. The first is the analytical method, in which the elastic theory is used to find the exact solution. The second approach consists of some kind of approximate technique, such as a perturbation procedure. The third method is the use of a numerical process, such as a finite-difference or a finite-element method.

This article investigates the transient thermal stresses in an annular fin with its base subjected to a heat flux of a decayed exponential function of time. In order to obtain the solution of the governing equation, which is a partial differential equation, the following procedures of analysis are used.

1. Normalize the governing partial differential equation subject to appropriate initial and boundary conditions.

2. Take the Laplace transform of the resulting equation with respect to time.

3. Utilize the exponential-like solutions introduced by Keller and Keller to solve the transformed system.

4. Achieve the inverse Laplace transform by means of complex contour integration and the residue theorem.

5. Substitute the temperature distribution function into the governing equation of thermal stresses. Then use Simpson's rule to obtain the thermal stress distribution as a function of time and position of the fin.     

An Extension of the Flow Boiling Correlation to Transition,Laminar, and Deep Laminar Flows in Minichannels and Microchannels

The thermal performance of rectangular plate fins circumscribing elliptic tubes is presented in this paper. Based on the assumption of uniform convective heat transfer the two-dimensional conduction equation has been formulated, and the solution has been obtained through the finite element method. Performance of rectangular plate fins for both inline and staggered arrangement of tubes has been investigated for a variation of geometric and thermo-geometric parameters. The necessity of optimizing the fin geometry for a given fin surface area has also been highlighted.     

Corrosion-fatigue testing of Ni-based superalloy RR1000

Operation of gas turbines at high temperatures can lead to corrosion of turbine materials. A programme was designed to understand the corrosion-fatigue behaviour of two grain size variants of RR1000 under cyclic loading at elevated.

Tests were conducted on cylindrical specimens under a range of loading conditions. Results indicated that salted specimens tested in air-SO_x had shorter fatigue lives than unsalted specimens tested in air, and this deficit was dependent on stress and/or test duration. Examination of specimens showed that the extent of corrosion was dependent on exposure time and stress. Additional salt loading resulted in faster initial pit growth which led to lower fatigue lives at higher stresses.

This paper is part of a thematic issue on the 9th International Charles Parsons Turbine and Generator Conference. All papers have been revised and extended before publication in Materials Science and Technology.     

Oily Products from Mosses and Algae via Pyrolysis

In this study, the fuel properties of mosses and algae, and the effect of pyrolysis temperature on the yield of bio-oil from moss and alga samples, were investigated. The yield of bio-oil from pyrolysis of the samples increased with temperature. The yields were increased up to 750 K in order to reach the plateau values at 775 K. The maximum yields were 39.1, 34.3, 33.6, 37.0, 35.4, 48.2 and 55.3% of the sample for Polytrichum commune, Dicranum scoparium, Thuidium tamarascinum, Sphagnum palustre, Drepanocladus revolvens, Cladophora fracta and Chlorella protothecoides, respectively. The bio-oil yield for Chlorella protothecoides (a microalga sample) rose from 5.7 to 55.3% as the temperature rose from 525 to 775 K, and then gradually decreased to 51.8% and was obtained at 875 K with a heating rate of 10 K/s. Formulas can be developed to calculate higher heating value (HHV) of different moss and alga samples. The calculated HHV using these new correlations showed mean differences ranging from ?2.3% to +0.06%. The equation developed in this study showed good agreement with experimental results on moss and algae samples. The HHVs for bio-oils from mosses 21.5–24.8 MJ/kg and the HHVs for bio-oils from algae and microalga 32.5 and 39.7 MJ/kg, respectively, were obtained at temperature ranging from 775 to 825 K. In general, algae bio-oils are of higher quality than bio-oils from mosses. In general, microalgae bio-oils are higher quality than bio-oil from wood.     

The Maximum Principle for Optimal Control of BSDEs with Locally Lipschitz Coefficients

The present paper studies a stochastic control problem for a locally Lipschitz backward stochastic differential equation. Assuming that the control domain is not necessarily convex, we establish a necessary and sufficient condition for optimality satisfied by all optimal controls. These conditions are described by a linear locally Lipschitz SDE and a maximum condition on the Hamiltonian. We first prove, under some convenient conditions, the existence of a unique solution to the resulting adjoint equation. Then, with the help of an approximation argument on the coefficients, we define a family of control problems with globally Lipschitz coefficients whereby we derive a stochastic maximum principle for near optimality to such approximated systems. Thereafter, we turn back to the original control problem by passing to the limits. As far as the authors are aware, this is the first version of the stochastic maximum principle covering the locally Lipschitz case.

     

To recover heat source G(x) + H(t) by using the homogenized function and solving rectangular differencing equations

ABSTRACT

For recovering an unknown heat source F(x, t) = G(x) + H(t) in the heat conduction equation, we develop a homogenized function method and the expansion methods by polynomials or eigenfunctions, which can solve the inverse heat source recovery problem by using collocation technique. Because the initial condition/boundary conditions/supplementary condition are satisfied automatically and a rectangular differencing technique is developed, a middle-scale linear system is sufficient to determine the expansion coefficients. After deriving a multiscale postconditioning matrix, the present methods converge very quickly, and are accurate and stable against large noise, as verified by numerical tests.     

Resonant Confluence of Singular Points and Stokes Phenomena

We consider a linear ordinary differential equation with resonant irregular singularity of generic type. For its generic deformation that splits the irregular singularity of the unperturbed equation into Fuchsian singularities of the perturbed one, the nonformal analytic classification invariants (Stokes operators) of the unperturbed equation are expressed via limit transition operators that compare appropriate monodromy eigenbases of the perturbed equation. We do this for all values of the Poincaré rank and the dimension (denoted by k and n respectively), except for the case where k = 1 and n 3. We show (Theorems 2.1 and 2.2 in Sec. 2) that appropriate branches of the monodromy eigenfunctions of the perturbed equation converge to appropriate canonical solutions of the unperturbed equation. In the case where k = n = 2, this statement implies that the Stokes operators are limits of transition operators between appropriate monodromy eigenbases of the perturbed equation (Corollary 2.1). We give a generalization of the last statement for the case of higher Poincaré rank and dimension (Corollary 2.2).     

Transient stability analysis of a power system with one generator connected to an infinite bus system

This paper presents a novel method to identify the fault that would affect the transient stability of one machine to an infinite bus. Here, the swing equation of each machine is expressed in the form of a closed-loop transfer function in Laplace ss-domain. In the transfer function, the duration of the fault and the changes in maximum electrical power output of the machine during and after the fault are considered. Then, the real value of the dominant root of each machine's characteristic equation is identified on the real axis of the s-plane. In this method, the generator has been considered unstable based on the position of the real value of the dominant root in the left half or very close to the imaginary axis or in the right half of the s-plane. The results obtained by the proposed method are compared with those of the conventional step-by-step method on the basis of different fault clearing times, and good agreements have been demonstrated.     

Studying the effect of the force convection boundary condition and radius magnetic field on fluid flow and heat transfer between coaxial pipes

An investigation of the heat transfer of Newtonian fluid flow through coaxial two pipes with variable radius ratio has been conducted with the boundary conditions of forced convection on the inner pipe walls and a radius magnetic field. This paper presents an exact analytical solution to the momentum equation and a novel semi-analytic collocation method for solving the full-term energy equation that takes Joule heating into account as well as viscous dissipation. Based on the results of the numerical fourth-order Runge–Kutta method, it was found that increasing the magnetic parameter decreased the amount of friction on the surface of the pipe walls and the rate of heat transfer. As the radius ratio of the two pipes increases, so does the skin friction and heat transfer rate on the internal pipe walls. As Eckert (Ec) and Prandtl (Pr) numbers increase, the mean temperature as well as the dimensionless temperature between the two pipes increases. The increase in Biot number (Bi) has the opposite impact on the mean temperature. As Ec, Pr, and Bi increase, so does the rate of heat transfer on the inner wall of the pipe.     

Analysis of Transient Heat Transfer in Annular Fins of Various Shapes with a Base Subjected to a Decayed Exponential Function of Time in Heat Flux

Abstract

This theoretical study of the heat flow within finned heat exchangers is of considerable practical importance because of the extensive utilization of finned surfaces for heat transfer enhancement in applications varying from air-cooled heat exchanges in the process industries to heat-rejection equipment in space vehicles.

This article investigates the transient heat transfer in two-dimensional annular fins of various shapes with a base subjected to a decayed exponential function of time in heat flux. In order to obtain the solutions of the governing partial differential equation, the following procedures of analysis are performed:

1. Normalize the governing partial differential equation subject to the appropriate initial and boundary conditions.

2. Take the Laplace transform with respect to time.

3. Utilize the integral method to solve the transformed system.

4. Achieve the inverse Laplace transform by the Fourier series technique, and the transient temperature distribution of the annular fins of various shapes are obtained as a base subjected to a decayed exponential function of time in heat flux.

5. Typical results in the form of graphs are also presented.     

EFFECT OF REYNOLDS AND PRANDTL NUMBERS ON TURBULENT CONVECTIVE HEAT TRANSFER IN A THREE-DIMENSIONAL SQUARE DUCT

Abstract

This paper presents the numerical prediction of thermal developing processes in a square cross-section duct for turbulent flow with isothermal walls. The algebraic stress model has been employed to predict the fully developed turbulent flow. This fully developed turbulent flow was numerically solved using the Teach Program. The three-dimensional energy equation was discretized and solved by the method of lines. According to this method, the energy equation is reformulated by a system of first-order differential equations control-ting the temperature along each line. A nonuniform grid of 8 × 8 nodes was used for calculating the temperature profile at each cross section. The effects of Prandtl and Reynolds numbers on the thermal behavior in the entrance region are investigated. The computed results for the fully developed region are shown to be in good agreement with the measured data publisked in the open literature     

The Symplectic Semigroup and Riccati Differential Equations

In this paper, we study close connections that exist between the Riccati operator (differential) equation that arises in linear control systems and the symplectic group and its subsemigroup of symplectic Hamiltonian operators. A canonical triple factorization is derived for the symplectic Hamiltonian operators, and their closure under multiplication is deduced from this property. This semigroup of Hamiltonian operators, which we call the symplectic semigroup, is studied from the viewpoint of Lie semigroup theory, and resulting consequences for the theory of the Riccati equation are delineated. Among other things, these developments provide an elementary proof for the existence of a solution of the Riccati equation for all t ≥ 0 under rather general hypotheses. 2000 Mathematics Subject Classification. 49N10, 93B27, 93B03, 22E15.     

Surface waves in nonlocal thermoelastic medium with state space approach

Abstract

In this paper, a new model based on Eringen’s nonlocal thermoelasticity is constructed. The propagation of Rayleigh surface waves in homogeneous isotropic medium is considered under the purview of this new nonlocal thermoelasticity theory in the context of energy dissipation theory. The normal mode analysis is employed to the considered equations to obtain vector matrix differential equation which is then solved by state space approach. The frequency equations for different cases are derived. The phase velocity, attenuation coefficients, specific loss and penetration depth of Rayleigh surface waves are computed numerically and presented graphically with respect to frequency and the effects of non-locality on the considered parameters are presented in the figures.