Shmuel Olek

Convergence and accuracy of Adomian’s decomposition method for the solution of Lorenz equations

Abstract The convergence and accuracy of Adomian’s decomposition method of solution is analysed in the context of its application to the solution of Lorenz equations which govern at lower order the convection in a porous layer (or respectively in a pure fluid layer) heated from below. Adomian’s decomposition method provides an analytical solution in terms of an infinite power series and is applicable to a much wider range of heat transfer problems. The practical need to evaluate the solution and obtain numerical values from the infinite power series, the consequent series truncation, and the practical procedure to accomplish this task, transform the analytical results into a computational solution evaluated up to a finite accuracy. The analysis indicates that the series converges within a sufficiently small time domain, a result that proves to be significant in the derivation of the practical procedure to compute the infinite power series. Comparison of the results obtained by using Adomian’s decomposition method with corresponding results obtained by using a numerical Runge–Kutta–Verner method show that both solutions agree up to 12–13 significant digits at subcritical conditions, and up to 8–9 significant digits at certain supercritical conditions, the critical conditions being associated with the loss of linear stability of the steady convection solution. The difference between the two solutions is presented as projections of trajectories in the state space, producing similar shapes that preserve under scale reduction or magnification, and are presumed to be of a fractal form.

Transitions and chaos for free convection in a rotating porous layer

Abstract The non-linearity which is inherently present in centrifugally driven free convection in porous media raises the problem of multiple solutions existent in this particular type of system. The solution to the non-linear problem is obtained by using a truncated Galerkin method to obtain a set of ordinary differential equation for the time evolution of the Galerkin amplitudes. It is demonstrated that Darcys model when extended to include the time derivative term yields, subject to appropriate scaling, the familiar Lorenz equations although with different coefficients, at a similar level of Galerkin truncation. The system of ordinary differential equations was solved by using Adomians decomposition method. Below a certain critical value of the centrifugally related Rayleigh number the obvious unique motionless conduction solution is obtained. At slightly super-critical values of the centrifugal Rayleigh number a pitchfork bifurcation occurs, leading to two different steady solutions. For highly supercritical Rayleigh numbers transition to chaotic solutions occurs via a Hopf bifurcation. The effect of the time derivative term in Darcys equation is shown to be crucial in this truncated model as the value of Rayleigh number when transition to the non-periodic regime occurs goes to infinity at the same rate as the time derivative term goes to zero. Examples of different convection solutions and the resulting rate of heat transfer are provided.

Unsteady conjugated heat transfer in laminar pipe flow

Abstract This study analyses the transient conjugated heat transfer in laminar pipe flow, where the flow is both hydrodynamically and thermally fully developed. Two cases are considered: a prescribed constant wall temperature and a constant heat flux at the wall. A non-standard method of separation of variables is applied, which treats the fluid and the solid as one region with certain discontinuities. The resulting eigenfunctions, which are not orthogonal to each other with respect to the usual weight function according to the Sturm-Liouville theorem, are made orthogonal to each other with respect to a special weight function. It is concluded that the degree of conjugation and viscous dissipation may have a great impact on the temperature distribution in the fluid.

Computational recovery of the homoclinic orbit in porous media convection

Abstract A simple procedure for computational recovery of the homoclinic orbit in porous media convection is presented.

A two-fluid model for critical flashing flows in pipes

Abstract A one-dimensional two-fluid model is described for the analysis of critical flows in pipes of nondiverging cross-sectional area. The model accounts for thermal nonequilibrium between the liquid and vapor bubbles and for interphase relative motion. Closure of the set of governing equations is performed with constitutive relationships which determine the pressure drop along the flow channel as a function of the flow regime, and the number and rate of growth of vapor bubbles in a variable temperature field in terms of the problems primary state variables and geometrical configuration. An empirical correlation is derived which fits the number density of bubble nuclei in the flow as a function of the flow channel length-to-diameter ratio. Model predictions compare favorably with experimental data in small- and large-scale systems over a wide range of pressures and pipe diameters and lengths.

Quenching of hot oxidizing surfaces

Abstract A one-dimensional time-dependent rewetting model is developed and assessed to describe the interrelated processes of conduction, convective cooling and exothermic steam–metal reactions at the vapor zirconium-cladding interface during quenching of degraded fuel rods. Upstream of the quench front a constant heat transfer coefficient is assumed whereas a heat flux profile of general spatial shape may be prescribed downstream. The quenching velocity history and the temperature profile are computed analytically via a Greens function approach. Numerical results of the present model compare favorably with published experimental data.

Analytical solution of two-dimensional diffusion in a composite medium with application to cooling of reactor fuel elements

Abstract A special eigenfunction expansion is applied to the solution of two-dimensional, steady state, heat transfer from a composite medium to a well-stirred fluid. The composite medium may consist of an arbitrary number of parallel slabs or cylinders, possibly with contact resistance between them, and heat sources or sinks of an arbitrary spatial distribution. The composite medium is treated in a unified way as a single region with discontinuities. The advantages of the present solution method are demonstrated through its application to a well-known problem of heat transfer in reactor fuel elements. Results of the current solution compare favorably with a previous analytical solution by Fourier series expansions for a slab geometry.

Rewetting of a solid cylinder with precursory cooling

A model for the rewetting of a solid cylinder is solved by the Wiener-Hopf technique. A constant heat transfer coefficient is assumed in the wetted part of the cylindrical rod, whereas an exponentially decaying heat flux is assumed in that part of the solid which is cooled by a mixture of vapor and liquid droplets (the precursory cooling). Accurate predictions of the rewetting velocity are obtained for a wide range of model parameters. The results of the present solution are found to compare favorably with a separation of variables solution obtained by Olek (1987). Values of the rewetting velocity where precursory cooling is neglected are recovered as a particular case. The decomposition in the form of infinite products presented here is shown to yield results which agree well with those derived from a decomposition by Evans (1984), employing the Cauchy theorem.