Analytical Heat and Fluid Flow in Microchannels and Microsystems

Cotta, Renato M.; Knupp, Diego C.; Naveira-Cotta, Carolina P.

doi:10.1007/978-3-319-23312-3

Cited by 38 publications

(59 citation statements)

References 55 publications

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“…Formal Generalized Integral Transform Technique Solution: Diffusive Eigenvalue Problem. Before advancing to the proposed novel approach, the GITT [7][8][9][10][11][12][13][14], with the usual diffusive eigenvalue problem choice, is summarized, as employed in solving the general one-dimensional nonlinear convection-diffusion problem given by…”

Section: Discussionmentioning

confidence: 99%

“…where all the coefficients and the source term of the proposed equation and boundary conditions were allowed to be nonlinear, and the explicit general dependence on the space coordinate and time variable is also considered. Following the ideas in the GITT approach [7][8][9][10][11][12][13][14], the next step would be rewriting problem (1) by considering characteristic linear coefficients for the transient, diffusion, and dissipation operators, given in the form of space variable only coefficients (wðxÞ; kðxÞ, and dðxÞ), in substitution to the original general nonlinear coefficients (ŵ ðx; t; TÞ; k ðx; t; TÞ, and d ðx; t; TÞ). Then, the source term is modified accordingly, to incorporate the nonlinear convective term and the differences between the nonlinear and characteristic transient, diffusion, and dissipation terms.…”

Section: Discussionmentioning

confidence: 99%

“…The integral transformation process itself is accomplished through the operator Ð x1 x0 w i ðxÞ dx, which after manipulation of the boundary conditions from both the original partial differential equation and the eigenvalue problem, leads to the following transformed ordinary differential equations (ODE) system [7][8][9][10][11][12][13][14]:…”

Section: Discussionmentioning

confidence: 99%

“…For the last three decades, under the stimulus of constructing a hybrid numerical-analytical approach for a more flexible treatment of nontransformable problems, such as in the case of nonlinear and/or convection-diffusion formulations, the so-called generalized integral transform technique (GITT) was progressively and successfully advanced [7][8][9][10][11][12][13][14], becoming a wellestablished alternative hybrid tool in the analysis of different classes of heat and fluid flow problems. Among the various applications that have been dealt with, one may point out the analysis of heat conduction with temperature-dependent thermophysical properties, mass transfer in metal oxidation at high temperatures, heat conduction in ablative thermal protections, double-pipe heat exchanger analysis, forced convection in irregularly shaped channels, mass transfer in tracer analysis of petroleum reservoirs, convective heat transfer in nanofluids with temperature-dependent thermophysical properties, mass diffusion in fusion reactor walls, migration of radionuclides in soils from waste of uranium mining, aerodynamic heating in space vehicles during atmospheric reentry, conjugated heat transfer in microheat exchangers, anti-icing thermal systems for aeronautical Pitot probes, analysis of biodiesel synthesis in microreactors, etc., as extensively reviewed in Refs.…”

Section: Introductionmentioning

confidence: 99%

“…Among the various applications that have been dealt with, one may point out the analysis of heat conduction with temperature-dependent thermophysical properties, mass transfer in metal oxidation at high temperatures, heat conduction in ablative thermal protections, double-pipe heat exchanger analysis, forced convection in irregularly shaped channels, mass transfer in tracer analysis of petroleum reservoirs, convective heat transfer in nanofluids with temperature-dependent thermophysical properties, mass diffusion in fusion reactor walls, migration of radionuclides in soils from waste of uranium mining, aerodynamic heating in space vehicles during atmospheric reentry, conjugated heat transfer in microheat exchangers, anti-icing thermal systems for aeronautical Pitot probes, analysis of biodiesel synthesis in microreactors, etc., as extensively reviewed in Refs. [8][9][10][11][12][13][14]. In the GITT approach, practically as a first step, an eigenvalue problem is chosen that incorporates sufficient information on the originally formulated problem, yielding an appropriate basis for the eigenfunction expansions proposition.…”

Section: Introductionmentioning

confidence: 99%

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Convective Eigenvalue Problems for Convergence Enhancement of Eigenfunction Expansions in Convection–Diffusion Problems

Cotta

Naveira-Cotta

Knupp

2017

Journal of Thermal Science and Engineering Applications

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The present work considers the application of the generalized integral transform technique (GITT) in the solution of a class of linear or nonlinear convection–diffusion problems, by fully or partially incorporating the convective effects into the chosen eigenvalue problem that forms the basis of the proposed eigenfunction expansion. The aim is to improve convergence behavior of the eigenfunction expansions, especially in the case of formulations with significant convective effects, by simultaneously accounting for the relative importance of convective and diffusive effects within the eigenfunctions themselves, in comparison against the more traditional GITT solution path, which adopts a purely diffusive eigenvalue problem, and the convective effects are fully incorporated into the problem source term. After identifying a characteristic convective operator, and through a straightforward algebraic transformation of the original convection–diffusion problem, basically by redefining the coefficients associated with the transient and diffusive terms, the characteristic convective term is merged into a generalized diffusion operator with a space-variable diffusion coefficient. The generalized diffusion problem then naturally leads to the eigenvalue problem to be chosen in proposing the eigenfunction expansion for the linear situation, as well as for the appropriate linearized version in the case of a nonlinear application. The resulting eigenvalue problem with space variable coefficients is then solved through the GITT itself, yielding the corresponding algebraic eigenvalue problem, upon selection of a simple auxiliary eigenvalue problem of known analytical solution. The GITT is also employed in the solution of the generalized diffusion problem, and the resulting transformed ordinary differential equations (ODE) system is solved either analytically, for the linear case, or numerically, for the general nonlinear formulation. The developed methodology is illustrated for linear and nonlinear applications, both in one-dimensional (1D) and multidimensional formulations, as represented by test cases based on Burgers' equation.

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