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4.7: Untitled Page 62

  • Page ID
    18195
  • Chapter 4

    (4‐39)

    This is identical in form to the mass balance for a fixed control volume that was presented in Chapter 3; however, this result has greater physical content than our previous result for single‐component systems. In this case, the density,  , is not the density of a single component, but it is the sum of all the species densities as indicated by Eq. 4‐27. In addition, the velocity, v , is not the velocity of a single component, but it is the mass average velocity defined by Eq. 4‐38.

    4.3 Species Velocity

    In our representation of the axioms for the mass of multicomponent systems, we have introduced the concept of a species velocity indicating that individual molecular species move at their own velocities designated by v A where A  1 , 2 , ..N . In order to begin thinking about the species velocity, we consider a lump of sugar (species A) placed in the bottom of a tea cup which is very carefully filled with water (species B). If we wait long enough, the solid sugar illustrated in Figure 4‐3 will dissolve and become uniformly distributed Figure 4‐3. Dissolution of sugar

    throughout the cup. This is a clear indication that the velocity of the sugar molecules is different from the velocity of the water molecules, i.e., v

    v

    (4‐40)

    sugar

    water

    If the solution in the cup is not stirred, the velocity of the sugar molecules will be very small and the time required for the sugar to become uniformly distributed throughout the cup will be very long. We generally refer to this process as diffusio n and diffusion velocities are generally very small. If we stir the liquid in the teacup, the sugar molecules will be transported away from the sugar cube by

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    Multicomponent systems

    108

    convection as we have illustrated in Figure 4‐4. In this case, the sugar will become uniformly distributed throughout the cup in a relatively short time and we generally refer to this process as mixing. Mechanical mixing can accelerate the process by which the sugar becomes uniformly distributed throughout the teacup; however, a true mixture of sugar and water could never be achieved unless the velocities of the two species were different. The difference between species velocities is crucial. It is responsible for mixing, for separation and purification, and it is necessary for chemical reactions to occur. If all species velocities were equal, life on earth would cease immediately.

    In addition to mixing the sugar and water as indicated in Figure 4‐3 and in Figure 4‐4, we can also separate the sugar and water by allowing the water to evaporate. In that case all the water in the tea cup would appear in the surrounding air and the sugar would remain in the bottom of the cup. This separation would not be possible unless the velocity of the water were different than the velocity of the sugar. While the difference between species velocities is of crucial interest to chemical engineers, there is a class of problems for which we can ignore this difference and still obtain useful results. In the following paragraphs we want to identify this class of problems.

    Figure 4‐4. Mixing of sugar

    To provide another example of the difference between species velocities and diffusion velocities, we consider the process of absorption of SO in a falling 2

    film of water as illustrated in Figure 4‐5. The gas mixture entering the column consists of air (nitrogen and oxygen), which is essentially insoluble in water, and SO , which is soluble in water. Because of the absorption of SO in the water, 2

    2

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    109