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4.6: Untitled Page 61

  • Page ID
    18194
  • Chapter 4

    (4‐31)

    If one wishes to avoid the mixed‐mode nomenclature in Eqs. 4‐30 and 4‐31, one must express the mass fraction as

    (4‐32)

    while the mole fraction takes the form

    (4‐33)

    Very often x is used to represent mole fractions in liquid mixtures and A

    yA to

    represent mole fractions in vapor mixtures, thus Eq. 4‐33 represents the mole fraction in a vapor mixture while Eq. 4‐31 represents the mole fraction in a liquid mixture.

    EXAMPLE 4.1. Conversion of mole fractions to mass fractions

    Sometimes we may be given the composition of a mixture in terms of the various mole fractions and require the mass fractions of the various constituents. To convert from x

    A to

    A we proceed as follows:

    (l)

    (2)

    (3)

    (4)

    (5)

    Multicomponent systems

    106

    4.2.2 Total mass balance

    Given the total density defined by Eq. 4‐27, we are ready to recover the total mass balance for multicomponent, reacting systems. For a fixed control volume, this is developed by summing Eq. 4‐7 over all species to obtain (4‐34)

    The summation procedure can be interchanged with differentiation and integration so that this result takes the form

    (4‐35)

    On the basis of definition of the total mass density given by Eq. 4‐27 and the axiom given by Eq. 4‐11, this result simplifies to

    (4‐36)

    At this point, we identify the total mass flux according to (4‐37)

    Since  is defined by Eq. 4‐27, this result represents a definition of the velocity v that can be expressed as

    (4‐38)

    This velocity is known as the mass average velocity and it plays a key role both in our studies of macroscopic mass balances and in subsequent studies of fluid mechanics, heat transfer, and mass transfer. Use of this definition for the mass average velocity allows us to express Eq. 4‐36 as

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