Mikhail D. Mikhailov

Integral transform solution of Luikov's equations for heat and mass transfer in capillary porous media

Abstract The Luikov system of equations for coupled heat and mass transfer within capillary porous bodies is analytically handled through application of the generalized integral transform technique. The problem of temperature and moisture distribution during contact drying of a moist porous sheet is considered to illustrate the development of the present approach. The classical coupled auxiliary problem with the related complex eigenvalues is completely avoided and, instead, two decoupled eigenvalue problems for temperature and moisture are chosen, which are of the conventional Sturm-Liouville type. A set of benchmark results is generated and critically compared with previously reported approximate solutions.

Contaminant transport in finite fractured porous medium: integral transforms and lumped-differential formulations

Abstract Mass transfer within saturated porous media with discrete finite fractures is examined, by simultaneously solving the convection–diffusion equation for the contaminant transport along the fracture and the two-dimensional diffusion equation for contamination within the porous matrix. A lumped-differential formulation based on Hermite integration is proposed for the porous matrix thus eliminating the dependence on the transverse direction. The resulting coupled partial differential equations are then handled through the generalized integral transform technique (GITT), which yields analytical expressions for the space dependence and numerical estimates for the concentration fields as a function of time. Different analytical filtering strategies are proposed and analyzed in terms of convergence rates. An illustrative example is considered for both the constant and time-variable contamination physical situations.

Periodic laminar forced convection : solution via symbolic computation and integral transforms

Abstract Transient laminar forced convection within the thermal entrance region of parallel-plate chennels is analytically solved for by making use of the Generalized Integral Transform Technique (GITT) and mixed symbolic-numerical computation ( Mathematica 3.0 system). The physical situation involves a periodic variation in time of the fluid inlet temperature, together with a fifth kind boundary condition at the channel walls that includes external convection and wall thermal capacitance effects. A mixed symbolic-numeric algorithm is constructed, which provides fully automatic derivation of all the analytical steps and offers a straightforward visualization of the numerical results, in both tabular and graphical forms. Amplitudes and phase lags of the system thermal response are then presented and interpreted, while critically compared with previously reported results.

Transient Flow and Thermal Analysis in Microfluidics

The present lecture summarizes some of the most recent joint research results from the cooperation between the Federal University of Rio de Janeiro, Brasil, and the University of Miami, USA, on the transient analysis of both fluid flow and heat transfer within microchannels. This collaborative link is a natural extension of a long term cooperation between the two groups, in the context of fundamental work on transient forced convection, aimed at the development of hybrid numerical-analytical techniques and the experimental validation of proposed models and methodologies [19]. The motivation of this new phase of the cooperation was thus to extend the previously developed hybrid tools to handle both transient flow and transient convection problems in microchannels within the slip flow regime. The analysis of internal flows in the slip-flow regime recently gained an important role in association with the fluid mechanics of various microelectromechanical systems (MEMS) applications, as well as in the thermal control of microelectronics, as reviewed in different sources [10-16]. For steady-state incompressible fully developed flow situations and laminar regime within simple geometries such as circular microtubes and parallel-plate microchannels, explicit expressions for the velocity field in terms of the Knudsen number are readily obtainable, and have been widely employed in the heat transfer analysis of microsystems, such as in [17-23]. Only quite recently, attention has been directed to the analysis of transient flow in microchannels [24-33]. Unsteady one-dimensional models have been extended from classical works, and analytical solutions have been sought for fully developed flows in simple geometries. These recent works are also concerned with situations in which a simple and well-defined functional form for the pressure gradient time variation is prescribed or for the time dependence of the wall imposed velocity, in the case of a Couette flow application. Research findings are yet to be further pursued in the analytical and robust solution of more generalized models, which will accommodate more general conditions and parameter specifications, and thus offer a wider validation range for the automatic general purpose numerical codes. Mikhailov and Ozisik [34] presented a unified solution for transient one-dimensional laminar flow models, with the usual no-slip boundary condition, based on the classical integral transform method. Their solution was then specialized to two situations: step change and periodically varying pressure gradient. The knowledge in regular size channels is therefore fairly well consolidated for models that use simple functional forms for the pressure gradient variation such as for the two cases cited above. One of the objectives of this paper is to illustrate the solution of a onedimensional mathematical model for transient laminar incompressible flow in microchannels such as circular tubes and parallel-plate channels, that accounts for a source term time variation in any functional form, including electrokinetic effects for liquid flows, by making use of the Generalized Integral Transform Technique (GITT) [35-40], and thus yielding analytical expressions for the time and space dependence of the velocity fields in the fully developed region. We then demonstrate this hybrid numerical-analytical solution for transient internal slip flow, obtained employing mixed symbolic-numerical computations with the Mathematica platform [41]. The goal here is to improve and complement existing analytical solution implementations to study laminar fully developed flows in micro-ducts subjected to arbitrary source term disturbances in space and time. On the other hand, the heat transfer literature of the last decade has demonstrated a vivid and growing interest in thermal analysis of flows in micro-channels, both through experimental and analytical approaches, in connection with cooling techniques of micro-electronics and with the development of micro-electromechanical sensors and actuators (MEMS), as also pointed out in recent reviews [12-16]. Since the available analytical information on heat transfer in ducts could not be directly extended to flows within microchannels with wall slip, a number of contributions have been recently directed towards the analysis of internal forced convection in the micro-scale. In the paper by Barron et al.

Identification of Heat Flux Imposed by an Oxyacetylene Torch

This paper deals with the use of inverse analysis t echniques for the identification of the heat flux imposed by an oxyacetylene torch, in an experimental setup for the identification of thermophysical properties of ablating materials. Parameter and function estimation techniques are used for the identifica tion of the boundary heat flux on the surface of a specimen with known thermophysical properties. Results are presented by using temperature measurements taken within the specimen. Nomenclature C(T) = volumetric heat capacity I = number of transient measurements per sensor k(T) = thermal conductivity M = number of sensors q(t) = boundary heat flux T(x,t) = estimated temperature Y(t) = measured temperature ΔT(x,t) = sensitivity function λ(x,t) = Lagrange multiplier I. Introduction he Brazilian Space Agency is dedicating efforts to build a satellite that will reenter the atmosphere after staying a couple of days in orbit. The Laboratory of Heat Transmiss ion and Technology of COPPE is involved in the design of the heat shield for such satellite, which shall be constituted of ablating and non -ablating materials. An important requirement for the thermal design of heat shields of vehicles re -entering the atmosphere, which are subjected to extremely high heat loads, is to have prior accurate information regarding the thermal properties of the materials utilized. In Ref. 1 we presented the solution of the inverse problem of parameter estimation use d for the identification of the thermal properties of ablating materials. The D -optimum approach was used, together with the analysis of the sensitivity coefficients, for the design of the experiment for the estimation of thermal conductivity, volumetric heat capacity and heat of ablation of materials with negligible thermal decomposition. A combination of the Levenberg-Marquardt method 2,3

Covalidation of Integral Transforms and Method of Lines in Nonlinear Convection-Diffusion with Mathematica

The Mathematica system (version 4.0) is employed in the solution of nonlinear difusion and convection-difusion problems, formulated as transient one-dimensional partial diferential equations with potential dependent equation coefficients. The Generalized Integral Transform Technique (GITT) is first implemented for the hybrid numerical-analytical solution of such classes of problems, through the symbolic integral transformation and elimination of the space variable, followed by the utilization of the built-in Mathematica function NDSolve for handling the resulting transformed ODE system. This approach ofers an error-controlled final numerical solution, through the simultaneous control of local errors in this reliable ODEs solver and of the proposed eigenfunction expansion truncation order. For covalidation purposes, the same built-in function NDSolve is employed in the direct solution of these partial diferential equations, as made possible by the algorithms implemented in Mathematica (versions 3.0 and up), based on application of the method of lines. Various numerical experiments are performed and relative merits of each approach are critically pointed out.

Integrals of radial basis functions for boundary element method

The technique for transferring the domain integrals into equivalent boundary integrals, published in this journal by Wen, Aliabadi and Rooke, requires analytical expressions for five integrals of radial basis functions. Exact solutions for these integrals are obtained by using Mathematica. The solutions are equivalent to the published one, but more convenient for computation. One of the published solutions is wrong.