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3.9: Untitled Page 35

  • Page ID
    18168
  • Chapter 3

    rate at which mass

    flows into the

      

     

    v n dA

    (3‐26)

     control volume 

    A entrance

    Use of Eqs. 3‐13, 3‐25 and 3‐26 in Eq. 3‐8 yields a precise, mathematical statement of the principle of conservation of mass for a control volume that contains only exits and entrances as they are described by Eqs. 3‐25 and 3‐26. This precise mathematical statement takes the form

    d

    dV  

    v n dA

    v n dA

    (3‐27)

    dt V

    A entrances

    A exits

    In general, any control volume will contain entrances, exits, and interfacial areas where mass transfer may or may not occur. For example, the fixed control volume illustrated in Figure 3‐11 contains an entrance, an exit, and an interfacial area which is the air‐water interface. If we assume that there is negligible mass Figure 3‐11. Entrances, exits and interfacial areas

    Single component systems

    55

    transfer at the air‐water interface, Eq. 3‐27 is applicable to the fixed control volume illustrated in Figure 3‐11 where no mass transfer occurs at the air‐water interface. This is precisely the type of simplification that was made in Example 3.1 where we neglected any mass transfer at the glass‐air interface. For the more general case where mass transfer can take place at interfacial areas, we need to express Eq. 3‐27 in terms of the fluxes at the entrances, exits and interfacial areas according to

    d

    dV  

    v n dA

    v n dA

    v n dA

    (3‐28)

    dt V

    A entrances

    A exits

    Ainterface

    In our description of the control volume, V , illustrated in Figure 3‐6, we identified the surface area of the volume as A and this leads us to express Eq. 3‐28 in the compact form given by

    d

    dV

      dA  0

    v n

    (3‐29)

    dt V

    A

    We should think of this expression as a precise statement of the principle of conservation of mass for a fixed control volume.

    3.1.2 Construction of control volumes

    The macroscopic mass balance indicated by Eq. 3‐29 represents a law of physics that is valid for all non‐relativistic processes. It is a powerful tool and its implementation requires that one pay careful attention to the construction of the control volume, V . There are four important rules that should be followed uring

    d

    the construction of control volumes, and we list these rules as: Rule I. Construct a cut (a portion of the surface area A ) where information is given.

    Rule II. Construct a cut where information is required.

    Rule III. Join these cuts with a surface located where v n is known.

    Rule IV. Be sure that the surface specified by Rule III encloses regions in which volumetric information is either given or required.

    In addition to macroscopic mass balance analysis, these rules also apply to macroscopic momentum and energy balance analysis that students will encounter in subsequent courses.

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