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5.7: Untitled Page 105

  • Page ID
    18238
  • Chapter 5

    of chemical reaction and note that the species velocity, v , can be A

    replaced with the mass average velocity, v, at entrances and exits to obtain c

    dA  0

    (1)

    v n

    A

    A

    It is important to remember that the molar concentration can be expressed as

    c

    y c ,

    gas streams

    (2)

    A

    A

    where y is the mole fraction of species A and c is the total molar A

    concentration. The form given by Eq. 2 is especially useful in the analysis Figure 5.8b. Control volume

    of the air stream; however, for the wet solids stream it is convenient to work in terms of mass rather than moles and make use of

    A

    c

    ,

    wet solid streams

    (3)

    A

    MWA

    If we let species A be water and apply Eq. 1 to the control volume shown in Figure 5.8b, we obtain

    Two‐Phase Systems & Equilibrium Stages 191

      MW vn dA

    MW

    v n

    dA

    H2O

    H2O 

     H2O

    H2O 

    A 1

    A 4

    

    

    molar flow rate of water

    molar flow rate of water

    entering with the solid

    leaving with the solid

    Water:

    (4)

    y

    c v n dA

    y

    c dA

     0

    v n

    H2O

    H2O

    A 2

    A 3

    

    

    molar flow rate of water

    molar flow rate of water

    entering with the air

    leaving with the air

    Here we have used Eqs. 2 and 3 in order to arrange the fluxes in forms that are convenient, but not necessary, for this particular problem, and we can express those fluxes in terms of averaged quantities to obtain

    

    Q

    

    Q

    H2O 1 1

    H2O 4

    4

    MW

    MW

    H O

    H O

    Water:

    2

    2

    (5)

     ( y

    ) M

     ( y

    ) M

     0

    H2O 2

    2

    H2O 3

    3

    In this representation of the macroscopic mole balance for water, we have drawn upon the analysis presented in Sec. 4.5. Specifically, we have imposed the following assumptions

    Gas streams:

    c v n  constant

    (6a)

    Wet solid streams:

    v n  constant

    (6b)

    in which “constant” means constant across the area of the entrances and exits. The three phases contained in the wet solid streams are illustrated in Figure 5.8c for stream #1. The total density in these streams consists of the density of the solid, the water, and the air, and this density can be written explicitly as

      

      

       

    (7)

    solid

    H2O

    air

    The mass fraction of water in the wet solids is defined by

    

    H O

    2

    

     

    H2O

    

    (8)

    and use of this representation in Eq. 5 leads to

    index-201_1.png

    192