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4.8: Untitled Page 63

  • Page ID
    18196
  • Chapter 4

    the exit gas is less detrimental to the local environment. It should be intuitively appealing that the species velocities in the axial direction are constrained by (4‐41)

    in which k is the unit vector pointing in the z‐direction. The situation for the Figure 4‐5. Absorption of sulfur dioxide

    radial components of v

    and v

    is quite different because the radial

    SO2

    air

    components are normal to the gas‐liquid interface. Since both nitrogen and oxygen are only slightly soluble in water, we neglect the transport of air at the gas‐liquid interface and write

    (4‐42)

    On the other hand, the sulfur dioxide is very soluble in water and the rate at which it leaves the gas stream and enters the liquid stream is significant.

    Multicomponent systems

    110

    Because of this the radial component of vSO must be positive and we express 2

    this idea as

    (4‐43)

    It should be clear that

    is a velocity associated with a diffusion process

    while

    is a velocity

    iated

    assoc

    with a convection process. The latter process

    is generally uch,

    m

    much larger than the former, i.e.,

    (4‐44)

    The motion of a chemical species can result from a force applied to the fluid, i.e., a fan might be used to move the gas mixture through the tube illustrated in Figure 4‐5. The motion of a chemical species can also result from a concentration gradient such as the gradient that causes the sugar to diffuse throughout the teacup illustrated in Figure 4‐3. Because the motion of chemical species can be caused by both applied forces and concentration gradients, it is reasonable to decompose the species velocity into two parts: the mass average velocity and the mass diffusion velocity. We represent this decomposition as (4‐45)

    At entrances and exits, such as those illustrated in Figure 4‐5, the diffusion velocity in the z‐direction is usually small compared to the mass average velocity in the z‐direction and Eq. 4‐45 can be approximated by (4‐46)

    In this text we will repeatedly make use of this simplification in order to olve s

    a

    variety of problems without the need to predict the diffusion velocity. However, in subsequent courses the subject of mass transfer across fluid‐fluid interfaces will be studied in detail, and those studies will require a complete understanding of the role of the diffusion velocity.

    In order to reinforce our thoughts about the species velocity, the mass average velocity, and the mass

    sion

    diffu

    velocity, we return to the definition of

    111