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4.9: Untitled Page 64

  • Page ID
    18197
  • Chapter 4

    th

    e mass average velocity given by Eq. 4‐38 and express the z‐component of v according to

    (4‐47)

    in which the mass fractions are constrained by

    (4‐48)

    n

    O the basis of the approximation given by Eq. 4‐

    41 we conclude that

    (4‐49)

    However, the radial component of the species velocities is an entirely different matter. Once again we can use Eq. 4‐38 to obtain

    (4‐50)

    but on the basis of Eq. 4‐42 this reduces to

    (4‐51)

    nder

    U

    these circumstances we see that

    (4‐52)

    nd

    a

    the type of approximation indicated by Eq. 4‐46 is not valid. In this text, we will study a series of macroscopic mass balance problems for which Eq. 4‐46

    represents a reasonable approximation; however, one must always remember that neglect of the diffusion velocity is a very delicate matter, and it is considered further in Problem 4‐10. As we have mentioned before, the difference between species velocities is responsible for separation and purification, and it is absolutely necessary in order for chemical reactions to occur.

    4.4 Measures of Velocity

    In the previous section we defined the mass average velocity according to (4‐53)

    and we noted that the total mass flux vector was given by

    Multicomponent systems

    112

    (4‐54)

    By analogy we define the molar average velocity by

    (4‐55)

    and it follows that the total molar flux vector is given by

    (4‐56)

    In the previous section we used a decomposition of the species velocity of the form

    (4‐57)

    o

    s that the species mass flux vector could be expressed as (4‐58)

    In dealing with the molar flux vector one finds it convenient to express the species velocity as

    (4‐59)

    o

    s that the molar flux takes the form

    (4‐60)

    When convective transport dominates, the species velocity, the mass average velocity and the molar average velocity are all essentially equal, i.e., (4‐61)

    This is the situation that we encounter most often in our study of material balances, and we will make use of this result repeatedly to determine the flux of species A at entrances and exits. While Eq. 4‐61 is widely used to describe

    113