# 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*