1,721,020 research outputs found
Orthogonal Curvilinear Coordinate Systems
No full textOrthogonal curvilinear coordinates occupy a special place among general coordinate systems, due to their special properties. There exists a number of such coordinate systems where the Laplace or Helmholtz equations may be separable, thus yielding a powerful tool to solve them. Operations like gradients, divergence, Laplacian take on much simpler forms in orthogonal coordinates. In this chapter the summation convention will not be used
Introduction to Constitutive Equations
No full textWhen the conservation equations for mass, chemical species, momentum and energy were derived in the previous chapter, it became soon evident that the number of unknown functions was far larger than that of the equations. To allow the closure of the problem some quantities need to be related to others and to the properties of matter, and these are the diffusive mass fluxes, j(p), the deviatoric stress tensor, τjk, the internal energy per unity of mass, u^ (or the specific enthalpy, h^ ) and the heat flux, q. The laws that describe these quantities are known as constitutive equations, and in thermo-fluids they are inherently empirical, although they must satisfy some requirement based upon first principles, like the condition of material objectivity (material properties must be independent of observer), the symmetry properties of a material body and the law of thermodynamics (particularly, the entropy inequality)
Introduction to Tensor Analysis
No full textTensor theory provides a remarkably concise mathematical framework for the formalisation of problems in many branches of physics and engineering
Separability of PDE
No full textThe analytical modelling of heat and mass transfer phenomena relies on the analytical solutions to partial differential equations, which are used to describe the conservation of mass, chemical species, momentum and energy and their transfer mechanisms, as it will be shown in Part II of this book. Analytical methods for this kind of problems are widely used (see [1]) and among the several available techniques to solve Partial Differential Equations (PDE), separation of variable is generally the most valuable one since it may yield solutions in a form that is easily implementable for routine calculations. Separability of a PDE depends on the chosen coordinate system and this chapter is devoted to analyse conditions and methods for PDE separation
One-Dimensional Modelling of Drop Heating and Evaporation Under Steady Conditions
No full textWe have seen how modelling of drop evaporation implies the solution of a set of PDEs (momentum, energy and species conservation), which may represent a remarkable challenge, particularly when an analytical approach is chosen. The problem can be greatly simplified introducing some assumptions, like sphericity of the drop, constancy of the thermo-physical properties and steadiness
Drop Evaporation Under Unsteady Conditions
No full textIn the previous two chapters we have investigated models of drop evaporation under steady-state conditions, an assumption widely used, although clearly unphysical: a mass source inside the drop is needed to maintain the drop shape unchanged during evaporation. To relieve this assumption a time dependent problem must be set and solved, increasing the complexity of analytical approaches. In particular, even for a spherical drop shrinking by evaporation, a moving boundary problem must be solved, which is known to be a challenging task, even for the simplest geometries. In this chapter we will see how it is possible to account for unsteadiness of the heat and mass transfer processes and still approach the modelling by analytical methods
Multi-component Drop Evaporation
No full textIn many engineering applications multi-component liquids are used in spray systems, just think of fuel sprays in internal combustion engines where the fuels (Diesel, gasoline or even biofuels) are blend of a large number of different hydrocarbons [1]. In those cases the species may have quite different volatilities and diffusivities as well as other thermo-physical characteristics and this may strongly influence the rates of heating and vaporisation of the spray droplets
Conservation and Constitutive Equations in Curvilinear Coordinates
No full textThe formulation of the conservation and constitutive differential equations derived in the previous chapters was obtained under the implicit assumption that the coordinate system was a Cartesian one. In practical problems it is sometime useful to switch to more natural coordinate systems, where the actual form of the differential equations may be simplified, thanks to some symmetry properties of the problem. For example, when dealing with the heating and evaporation of a spherical drop, the natural coordinate system is the spherical one, since in such a system the governing differential equations may assume a much simpler form
Sturm–Liouville Problems
No full textIn Chap. 3 we have seen how the separability of PDEs leads to ordinary differential equations problems, usually of second order. The problem is complemented with B.C.s and the reduction of the initial PDE to second order ODEs often yield a so-called Sturm–Liouville (SL) problem (named after the French mathematicians Jacques Charles François Sturm, 1803–1855, and Joseph Liouville, 1809–1882)
Two- and Three-Dimensional Modelling of Heating and Evaporation Under Steady Conditions
No full textIn Chap. 10 we have seen that when an evaporating drop has the shape of a sphere, a spheroid or an ellipsoid, and the boundary conditions are uniform over the drop surface, the whole problem simplifies when proper coordinate systems are used and one-dimensional solutions of the conservation equations can be found. When the drop assumes different shapes, or the boundary conditions are not uniform, two- or three-dimensional solutions appear, even using proper coordinate systems. In this chapter we will explore some cases of practical interest when 2-D or even 3-D solutions can be found analytically
- …
