First-order Differential Equations In Terms Research Articles

A VOLUME-AVERAGING APPROACH FOR ANALYZING A SPIRAL-WOUND REVERSE OSMOSIS DESALINATION MODULE

A mathematical model for spiral-wound reverse osmosis systems has been established using a porous media approach. Transport processes through membranes were investigated, fully taking account of concentration polarization associated with spiral-wound reverse osmosis desalination systems. A set of the governing equations, namely, the continuity, momentum, and concentration equations, have been derived for three distinctive phases, namely, brine, permeate, and membrane phases, following the procedure previously proposed by Nakayama and Sano for the analysis of hollow-fiber desalination modules. The first-order differential equations in terms of the average velocity, pressure, and salt concentration for the brine phase are integrated with an algebraic equation for the permeate flow rate per unit volume in order to estimate the permeate salinity, flow rate, and pressure drop in spiral-wound reverse osmosis modules. The present analytical results agree fairly well with available experiment data reported by Avlonitis et al. [Desalination, 81:191,1991; 86:273,1992; 89:227,1993; 203:218,2007; Sep. Sci. Technol., 40:2663,2005], substantiating the validity of the governing equations based on the porous media approach for spiral-wound reverse osmosis systems. The present analysis reveals that there exists an optimal brine pressure for attaining the maximum permeate flow rate for a given pumping power. The present model can be used to design spiral-wound reverse osmosis desalination systems without resorting to extensive finite difference calculations.

On numerical analysis of axisymmetric thick circular cylindrical shells based on higher order shell theories by segmentation method

Numerical analysis is carried out in this paper for thick circular cylindrical shells using higher order shell theory. In orthogonal curvilinear coordinate system, all equations are derived with the inclusion of the additional quadratic and cubic terms in the Taylor’s series expansions of the both in-plane as well as the transverse displacement components for the improvement of bending behaviour of the shell. Assuming (h/R)2 ≪ 1, a rigorous formulation involving the reduction of a three-dimensional elasticity problem to a two-dimensional one, based partly upon the Reissner’s variational principle, is presented. These equations are algebraically manipulated to be in the form of a coupled system of first-order differential equations in terms of the intrinsic dependent variables. These are then solved by a segmentation method – numerical integration technique for various combinations of material and geometric parameters. The theory is shown to result in a partial differential equation system of sixteenth order.

Mixed superposition rules and the Riccati hierarchy

Mixed superposition rules, i.e., functions describing the general solution of a system of first-order differential equations in terms of a generic family of particular solutions of first-order systems and some constants, are studied. The main achievement is a generalization of the celebrated Lie–Scheffers Theorem, characterizing systems admitting a mixed superposition rule. This somehow unexpected result says that such systems are exactly Lie systems, i.e., they admit a standard superposition rule. This provides a new and powerful tool for finding Lie systems, which is applied here to studying the Riccati hierarchy and to retrieving some known results in a more efficient and simpler way.

Symmetry Solutions of a Third-Order Ordinary Differential Equation which Arises from Prandtl Boundary Layer Equations

The similarity solution to Prandtl’s boundary layer equations for two-dimensional and radial flows with vanishing or constant mainstream velocity gives rise to a thirdorder ordinary differential equation which depends on a parameter a. For special values of a the third-order ordinary differential equation admits a three-dimensional symmetry Lie algebra L 3. For solvable L 3 the equation is integrated by quadrature. For non-solvable L 3 the equation reduces to the Chazy equation. The Chazy equation is reduced to a first-order differential equation in terms of differential invariants which is transformed to a Riccati equation. In general the third-order ordinary differential equation admits a two-dimensional symmetry Lie algebra L 2. For L 2 the differential equation can only be reduced to a first-order equation. The invariant solutions of the third-order ordinary differential equation are also derived.

Exact solutions of N -dimensional harmonic oscillator via Laplace transformation

The N-dimensional Schrodinger equation for the harmonic oscillator is reduced to a first-order differential equation in terms of the Laplace transformation and the exact bound state solutions are derived. It is shown that this method of solving Schrodinger equation may serve as a substitute for the standard functional, analytical approach also in lower dimensions.

The exact solutions of the Schr�dinger equation with the Morse potential via Laplace transforms

In this Letter, we reduce the second-order differential equation about the one-dimensional Schrödinger equation with the Morse potential reduced to the first-order differential equation in terms of Laplace transforms and then obtain the exact bound state solutions.

The exact solutions of the Schrödinger equation with the Morse potential via Laplace transforms

In this Letter, we reduce the second-order differential equation about the one-dimensional Schrödinger equation with the Morse potential reduced to the first-order differential equation in terms of Laplace transforms and then obtain the exact bound state solutions.

Statistical Thermodynamic Treatment of Multilayer Adsorption-Migration Kinetics

The transient multiple-layer (multilayer) adsorption and adsorption-migration phenomena are described with a statistical thermodynamic model. The key postulate for this model is that the transient process can be delineated by considering perturbation of the grand canonical ensemble characterizing either a multilayer adsorption or adsorption-migration system. Also, the derivative of the site partition function or adsorption site number (molecular coverage) for a given layer is affected only by the site distribution of its own layer and the two nearest-neighbor layers. With these postulates, differentiation of the grand partition function with time results in a system of first-order differential equations in terms of adsorption site numbers that describes the transient multilayer adsorption or adsorption-migration process. Its solution determines the time-varying layer-by-layer deposit buildup on a smooth surface under constant molecular impingement. In either the adsorption or adsorption-migration case, the statistical thermodynamic solution has been proven to be equivalent to that based on the classical kinetic theory, which has been verified by laboratory test data.

Buckling analysis of laminated composite conical shells

Eight first-order differential equations in terms of a particularly convenient coordinate system are obtained for linear buckling analysis of laminated conical shells, with orthotropic stretching-bending coupling, under axial compressive load. These equations are solved by using the numerical integration technique and the multisegment method. By analyzing the buckling of a series of conical shells, the effects of boundary conditions, elastic coefficients and the stretching-bending coupling, on the buckling loads and circumferential wavenumbers, are investigated.