Skip to main content
Engineering LibreTexts

5.23: Untitled Page 120

  • Page ID
    18253
  • Chapter 5

    hydroxide leaves in the vapor stream, that none accumulates in the boiler, and that the temperature of the liquid entering and leaving the boiler is a constant.

    Assume that the solid calcium hydroxide leaving the boiler is in equilibrium with the dissolved carbon dioxide, i.e., the boiler is an equilibrium stage. The solubility is often expressed as;

    Equilibrium relation:

    solubility = S  g of Ca(OH) g of H O

    2

    2

    however, a more precise description can be constructed.

    In this problem you are asked to develop a general solution for the mass fraction of the suspended solid in the liquid stream leaving the boiler in terms of

     and S. For  = 0.50, 0.21, and 0.075, determine the mass fraction of suspended

    solid when S .

    3

    2 5  10 .

    Figure 5.29. Precipitation of calcium hydroxide in a boiler 5‐30. In problem 5‐29 an equilibrium relation between solid calcium hydroxide and dissolved calcium hydroxide was given by

    Equilibrium relation:

    solubility = S  g of Ca(OH) g of H O

    (1)

    2

    2

    In Sec. 5.3.1 we expressed the general gas/liquid equilibrium relation for species A in terms of the chemical potential and for a solid/liquid system we would express Eq. 5‐24 as

    Equilibrium relation:

    ( )

     ( )

    (2)

    A solid

    A liquid

    Two‐Phase Systems & Equilibrium Stages 221

    For the process considered in Problem 5‐29, we assume that the solid phase is pure calcium hydroxide so that Eq. 2 takes the form

    Equilibrium relation:

    (O )

     ( )

    (3)

    A solid

    A liquid

    The description of phase equilibrium phenomena in terms of the chemical potential will be the subject of a subsequent course in thermodynamics; however, at this point one can illustrate how Eqs. 1 and 3 are related.

    The development begins with a general representation for the chemical potential at some fixed temperature and pressure. This is given by ( )

    (

    F T , p, x , x , etc.)

    (3)

    A liquid

    A

    B

    where x is the mole fraction of species A (calcium hydroxide) in the liquid A

    phase. A Taylor series expansion about x  0 leads to (see Problem 5‐31) A

    2

    F

    F

    2

    ( )

    F

    ( x ) 

    ( x )  .....

    (4)

    A liquid

    x 0

    A

    2

    A

    A

    x

    A x

    x

    0

    A

    A

    x 0

    A

    The first term in this expansion is zero and when the mole fraction of species A is small compared to one, x  1 , we can make use of a linear form of Eq. 4 given A

    by

    F

    ( )

    ( x )

    A liquid

    A

    x

    (5)

    A x 0

    A

    In this problem you are asked to use Eq. 3 along with Eq. 5 and the approximation

    c

    A

    x

    ,

    when c  c

    (6)

    A

    A

    B

    cB

    to derive Eq. 1. Here c represents the molar concentration of calcium A

    hydroxide and c represents the molar concentration of water. In terms of B

    species A and species B, it will be convenient to express the solubility in the form Equilibrium relation:

    solubility = S  

    (7)

    A

    B

    and note that this can be related to the molar form by use of c MW   and A

    A

    A

    c MW   .

    B

    B

    B

    222