The Mathematics of Fluid Flow Through Porous Media - Myron B. Allen, III

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the second category of multiconstituent continua, segregation of constituents is observable only at molecular length scales, so continuum‐scale interfaces between the constituents do not exist. Saltwater is an example: The particles of Na Superscript plus, Cl Superscript minus, and normal upper H 2O are segregated at length scales of roughly 1 0 Superscript negative 10 m, far smaller than the continuum scale. For such multispecies or miscible multiconstituent continua, the concept of a continuum‐scale volume fraction does not apply.

      With this framework in place, we define several functions associated with the continuum. The mixture density is

rho equals sigma-summation Underscript alpha equals 1 Overscript upper N Endscripts rho Subscript alpha Baseline comma

      which we can write for multiphase continua as follows:

rho equals sigma-summation Underscript alpha equals 1 Overscript upper N Endscripts phi Subscript alpha Baseline gamma Subscript alpha Baseline period

      The mass‐weighted or barycentric velocity is

bold v equals StartFraction 1 Over rho EndFraction sigma-summation Underscript alpha equals 1 Overscript upper N Endscripts rho Subscript alpha Baseline bold v Subscript alpha Baseline period

      Sometimes it is useful to refer to the barycentric derivative, which for a differentiable function f left-parenthesis bold x comma t right-parenthesis has the form

      (2.26)StartFraction upper D f Over upper D t EndFraction left-parenthesis bold x comma t right-parenthesis equals StartFraction partial-differential f Over partial-differential t EndFraction left-parenthesis bold x comma t right-parenthesis plus bold v left-parenthesis bold x comma t right-parenthesis dot nabla f left-parenthesis bold x comma t right-parenthesis period

      Finally, the diffusion velocity of constituent alpha is

      (2.27)bold-italic nu Subscript alpha Baseline equals bold v Subscript alpha Baseline minus bold v period

      Exercise 2.14 Show that

sigma-summation Underscript alpha equals 1 Overscript upper N Endscripts rho Subscript alpha Baseline bold-italic nu Subscript alpha Baseline equals bold 0 period

      2.5.3 Multiconstituent Mass Balance

      For the differential mass balance, the extension has the following form:

      To see how this equation allows for exchanges of mass among constituents, rewrite it as follows:

      where

      Mathematically, this new form amounts to a trivial reformulation. Physically, it captures the exchange of mass into each constituent script upper B Subscript alpha from other constituents, at a rate given by the mass exchange rate r Subscript alpha, having dimension ML Superscript negative 3 Baseline normal upper T Superscript negative 1. Mass exchange can occur via several mechanisms:

       Phase changes, such as melting, freezing, evaporation, and condensation;

       Interphase mass transfer, such as dissolution or adsorption;

       Chemical reactions, which transform molecular species into different molecular species.

 StartFraction upper D Superscript alpha Baseline Over upper D t EndFraction left-parenthesis phi Subscript alpha Baseline gamma Subscript alpha Baseline right-parenthesis plus phi Subscript alpha Baseline gamma Subscript alpha Baseline nabla dot bold v Subscript alpha Baseline equals r Subscript alpha Baseline comma alpha equals 1 comma 2 comma ellipsis comma upper N

      It is common to write the multiconstituent mass balance in terms of constituent mass fractions, defined as omega Subscript alpha Baseline equals rho Subscript alpha Baseline slash rho and having dimension (mass of alpha)slash(total mass). Doing so yields the following equivalent forms for the mass balance equation for each constituent alpha, all subject to the constraint (2.30):

StartLayout 1st Row 1st Column StartFraction upper D Superscript alpha Baseline Over upper D t EndFraction left-parenthesis rho omega Subscript alpha Baseline right-parenthesis plus rho omega Subscript alpha Baseline nabla dot bold v Subscript alpha 2nd Column equals r Subscript alpha Baseline comma 3rd Column alpha 4th Column equals 1 comma 2 comma ellipsis comma upper N semicolon 2nd Row 1st Column StartFraction partial-differential Over partial-differential t EndFraction left-parenthesis rho omega Subscript alpha Baseline right-parenthesis plus <p style= Скачать книгу