Page 118 - Chemical process engineering design and economics

102 Chapters

is no chemical reaction, and the tower operates at steady state. Thus, Equation 3.2
reduces to the flow rate into the tower equal the flow rate out of the tower. Table
3.2.1 lists the appropriate relations (the subscript three means Chapter 3, two, ex-
ample two, and one, table one). As established before, the first subscript in the
composition variable, y, indicates the stream number, shown in Figure 3.2.1, and
the second subscript the component, one for water and two for air. The primed
variables indicate specified variables. Thus, in Table 3.2.1, Equation 3.2.1 is the
water mole balance and Equation 3.2.2 the air mole balance (three means Chapter
3, two, Example two, and two Equation 1). Nitrogen and oxygen are only slightly
soluble in water and, therefore, we treat air as a single, unabsorbed component.
The water and air mole balances together with the mole fraction summations,
given by Equations 3.2.3 and 3.2.4, are all the mole balance relations that we can
write. If it is more convenient to use the total mole balance instead of a compo-
nent balance, then drop one of the equations in the set from Equations 3.2.1 to
3.2.4.
Because cooling water is not an isothermal process, we must use the energy
equation. The general energy balance, Equation 3.10, is modified to fit the cooling
tower. We define the system by a boundary that cuts across all the streams and
encloses the tower, but not the fan, which is located in the upper part of the tower.
The kinetic and potential energy changes of the streams across this boundary are
small compared to the enthalpy change. Although the fan does work on the air
stream to overcome the resistance to air flow in the tower, no work crosses the
boundary selected. At a later stage in the design, we will need a mechanical en-
ergy balance to calculate the fan power. Finally, because no heat flows across the
boundary, the heat-transfer term will be zero. Therefore, enthalpy is conserved,
and the cooling-tower energy equation reduces to Equation 3.2.5 in Table 3.2.1.
Equation 3.2.6 gives the concentration of water vapor in the inlet air as
function of \v, yiw> and Ahw> where the subscript, w, means wet bulb. The
t
t V
equations are in functional notation to indicate that these data may be available in
tables, graphs or equations. The wet-bulb temperature, tiw, will be discussed later.
Equation 3.2.7 expresses the mole fraction of water vapor in the exit air in terms of
the vapor pressure at saturation. The air leaving the tower is assumed to be 90%
saturated, a value recommended by Walas [12].
Before solving the equations, we need system property data, which, in this
case, are thermodynamic properties. Equations 3.2.9 and 3.2.11 states that we may
obtain vapor pressures for water from steam tables, such as those compiled by
Chaar et al. [13]. Equation 3.2.10 also states that we can find the enthalpy of va-
porization in the steam tables. We assume that the air-water mixture is ideal to
calculate the enthalpy of air, so we can use the mole-fraction average of the pure-
component enthalpies. Equations 3.2.12 and 3.2.13 in Table 3.2.1 give the mole
fraction average of the inlet and outlet enthalpy. Table 3.2.1 also lists pure com-
ponent enthalpies for water vapor (Equations 3.2.14 and 3.2.16) and for air (Equa-

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