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Equilibrium and Transfer in Porous Media 3 Applications, Isothermal Transport, Coupled Transfers von Da?an, Jean-Fran?ois (eBook)

  • Erscheinungsdatum: 16.04.2014
  • Verlag: Wiley-ISTE
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Equilibrium and Transfer in Porous Media 3

A porous medium is composed of a solid matrix and its geometrical complement: the pore space . This pore space can be occupied by one or more fluids. The understanding of transport phenomena in porous media is a challenging intellectual task. This book provides a detailed analysis of the aspects required for the understanding of many experimental techniques in the field of porous media transport phenomena. It is aimed at students or engineers who may not be looking specifically to become theoreticians in porous media, but wish to integrate knowledge of porous media with their previous scientific culture, or who may have encountered them when dealing with a technological problem. While avoiding the details of the more mathematical and abstract developments of the theories of acroscopization, the author gives as accurate and rigorous an idea as possible of the methods used to establish the major laws of macroscopic behavior in porous media. He also illustrates the constitutive laws and equations by demonstrating some of their classical applications. Priority is to put forward the constitutive laws in concrete circumstances without going into technical detail. This third volume in the three-volume series focuses on the applications of isothermal transport and coupled transfers in porous media.

Produktinformationen

    Format: PDF
    Kopierschutz: AdobeDRM
    Seitenzahl: 332
    Erscheinungsdatum: 16.04.2014
    Sprache: Englisch
    ISBN: 9781118931301
    Verlag: Wiley-ISTE
    Größe: 16444 kBytes
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Equilibrium and Transfer in Porous Media 3

1.4. Appendices and exercises

1.4.1. Diffusion and diffusion - convection equations

We have already discussed the diffusion equation, particularly in relation to thermal conduction and isothermal fluid transport outside gravity within a porous medium. This partial differential equation a state variable or "contingent" variable (e.g. temperature, saturation or concentration), which plays the role of potential , function of the time and space f ( t , ), and involves a diffusivity α , expressed in m2 s-1. If the diffusivity is constant, the equation is linear. We will also contemplate the case where the diffusivity depends on the unknown potential f . For problems with a single-space variable x , we need to handle the two following versions of the diffusion equation:

[1.41]

For problems that bring diffusion and advection into play, we also need to solve the following linear equation, which in addition, involves the diffusivity α , Darcy's velocity V and the capacity γ ; all of these are assumed to be constant:

[1.42]
1.4.1.1. Linear diffusion equation: several solutions

To write the equation in simple form, we introduce the dimensionless variables of space and time, which take diffusivity out of the equation: 32

[1.43]

If we are looking for a specific solution for the form , function of the only dimensionless Boltzmann variable :

the linear equation may be written as:

which yields:

Figure 1.17. The Gaussian functions erf and erfc in the domain ξ 0

The "error function" erf is the primitive 33 of the Gaussian function ,

[1.44]

In the domain ξ 0 we also use the "complement" erfc ξ = 1 - erf ξ .
Properties linked to the linearity of equations

Other solutions of practical interest may be obtained by using the following properties common to all linear differential equations:
- The linear combination with constant coefficients of several solutions is also a solution. - The temporal derivative of a solution is also a solution. This is because the variables x and t are independent. We thus obtain, for example, the following solution for the diffusion equation:
[1.45]
- We also obtain a new solution by temporal integration of a solution. - This is also the case for the spatial derivative. 34 Thus, for example, a new solution may be obtained by integration of [1.45] in relation to :
[1.46]
- Regarding the diffusion equation, the opposite of the potential gradient represents, in dimensionless form, the flux density. As a result, the flux density obeys the same differential equation as the potential. Any solution of the linear diffusion equation can thus be indifferently interpreted either as a spatiotemporal distribution of the potential ( T , c or ρ for example) or of the flux density ( q or J ). The spatial integral of a flow response is, except for one sign, the potential response to the same stress. 1.4.1.2. Stress and response

Generally, when a semi-infinite or limited monodimensional domain which is initially in a state of equilibrium is exposed to a stress at one of its boundaries by a disturbance imposed at the potential f (e.g. temperature, pressure or concentration) or to the flux, the spatiotemporal evolution of the potential or of the flux which follows on from this, , is the response of this d

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