10.5. CONTINUOUS HUMIDIFICATION PROCESSES

10.5A. Introduction and Types of Equipment for Humidification

1. Introduction to gas–liquid contactors

When a relatively warm liquid is brought into direct contact with gas that is unsaturated, some of the liquid is vaporized. The liquid temperature will drop mainly because of the latent heat of evaporation. This direct contact of a gas with a pure liquid occurs most often in contacting air with water. This is done for the following purposes: humidifying air for control of the moisture content of air in drying or air conditioning; dehumidifying air, where cold water condenses some water vapor from warm air; and water cooling, where evaporation of water to the air cools warm water.

In Chapter 9 the fundamentals of humidity and adiabatic humidification were discussed. In this section the performance and design of continuous air–water contactors is considered. The emphasis is on cooling of water, since this is the most important type of process in the process industries. There are many cases in industry in which warm water is discharged from heat exchangers and condensers when it would be more economical to cool and reuse it than to discard it.

2. Towers for water cooling

In a typical water-cooling tower, warm water flows countercurrent to an air stream. Typically, the warm water enters the top of a packed tower and cascades down through the packing, leaving at the bottom. Air enters at the bottom of the tower and flows upward through the descending water. The tower packing often consists of slats of plastic or of a packed bed. The water is distributed by troughs and overflows to cascade over slat gratings or packing that provides large interfacial areas of contact between the water and air in the form of droplets and films of water. The flow of air upward through the tower can be induced by the buoyancy of the warm air in the tower (natural draft) or by the action of a fan. Detailed descriptions of towers are given in other texts (B1, T1).

The water cannot be cooled below the wet bulb temperature. The driving force for the evaporation of the water is approximately the vapor pressure of the water less the vapor pressure it would have at the wet bulb temperature. The water can be cooled only to the wet bulb temperature, and in practice it is cooled to about 3 K or more above this. Only a small amount of water is lost by evaporation in cooling water. Since the latent heat of vaporization of water is about 2300 kJ/kg, a typical change of about 8 K in water temperature corresponds to an evaporation loss of about 1.5%. Hence, the total flow of water is usually assumed to be constant in calculations of tower size.

In humidification and dehumidification, intimate contact between the gas phase and liquid phase is needed for large rates of mass transfer and heat transfer. The gas-phase resistance controls the rate of transfer. Spray or packed towers are used to give large interfacial areas and to promote turbulence in the gas phase.

10.5B. Theory and Calculations for Water-Cooling Towers

1. Temperature and concentration profiles at interface

In Fig. 10.5-1 the temperature profile and concentration profile in terms of humidity are shown at the water–gas interface. Water vapor diffuses from the interface to the bulk gas phase with a driving force in the gas phase of (HiHG) kg H2O/kg dry air. There is no driving force for mass transfer in the liquid phase, since water is a pure liquid. The temperature driving force is TLTi in the liquid phase and TiTG K or °C in the gas phase. Sensible heat flows from the bulk liquid to the interface in the liquid. Sensible heat also flows from the interface to the gas phase. Latent heat also leaves the interface in the water vapor, diffusing to the gas phase. The sensible heat flow from the liquid to the interface equals the sensible heat flow in the gas plus the latent heat flow in the gas.

Figure 10.5-1. Temperature and concentration profiles in upper part of cooling tower.


The conditions in Fig. 10.5-1 occur at the upper part of the cooling tower. In the lower part of the cooling tower, the temperature of the bulk water is higher than the wet bulb temperature of the air but may be below the dry bulb temperature. Then the direction of the sensible heat flow in Fig. 10.5-1 is reversed.

2. Rate equations for heat and mass transfer

We shall consider a packed water-cooling tower with air flowing upward and water countercurrently downward in the tower. The total interfacial area between the air and water phases is unknown, since the surface area of the packing is not equal to the interfacial area between the water droplets and the air. Hence, we define a quantity a, defined as m2 of interfacial area per m3 volume of packed section, or m2/m3. This is combined with the gas-phase mass-transfer coefficient kG in kg mol/s · m2 Pa or kg mol/s · m2 · atm to give a volumetric coefficient kGa in kg mol/s · m3 volumePa or kg mol/s · m3 · atm (lb mol/h · ft3 · atm).

The process is carried out adiabatically; the various streams and conditions are shown in Fig. 10.5-2, where

Figure 10.5-2. Continuous countercurrent adiabatic water cooling.



The enthalpy Hy as given in Eq. (9.3-8) is

Equation 9.3-8


The base temperature selected is 0°C or 273 K (32°F). Note that (TT0)°C = (TT0) K.

Making a total heat balance for the dashed-line box shown in Fig. 10.5-2, an operating line is obtained:

Equation 10.5-1


This assumes that L is essentially constant, since only a small amount is evaporated. The heat capacity cL of the liquid is assumed constant at 4.187 × 103 J/kg · K (1.00 btu/lbm · °F). When plotted on a chart of Hy versus TL, Eq. (10.5-1) is a straight line with a slope of LcL/G. Making an overall heat balance over both ends of the tower,

Equation 10.5-2


Again, making a heat balance for the dz column height and neglecting sensible-heat terms compared to the latent heat,

Equation 10.5-3


The total sensible heat transfer from the bulk liquid to the interface is (refer to Fig. 10.5-1)

Equation 10.5-4


where hLa is the liquid-phase volumetric heat-transfer coefficient in W/m3 · K (btu/h · ft3 · °F) and Ti is the interface temperature.

For adiabatic mass transfer the rate of heat transfer due to the latent heat in the water vapor being transferred can be obtained from Eq. (9.3-16) by rearranging and using a volumetric basis:

Equation 10.5-5


where qλ/A is in W/m2 (btu/h · ft2), MB = molecular weight of air, kGa is a volumetric mass-transfer coefficient in the gas in kg mol/s · m3 · Pa, P = atm pressure in Pa, λ0 is the latent heat of water in J/kg water, Hi is the humidity of the gas at the interface in kg water/kg dry air, and HG is the humidity of the gas in the bulk gas phase in kg water/kg dry air. The rate of sensible heat transfer in the gas is

Equation 10.5-6


where qS/A is in W/m2 and hG a is a volumetric heat-transfer coefficient in the gas in W/m3 · K.

Now from Fig. 10.5-1, Eq. (10.5-4) must equal the sum of Eqs. (10.5-5) and (10.5-6):

Equation 10.5-7


Equation (9.3-18) states that

Equation 10.5-8


Substituting PkGa for kya,

Equation 10.5-9


Substituting Eq. (10.5-9) into Eq. (10.5-7) and rearranging,

Equation 10.5-10


Adding and subtracting cST0 inside the brackets,

Equation 10.5-11


The terms inside the braces are (HyiHy), and Eq. (10.5-11) becomes

Equation 10.5-12


Integrating, the final equation to use for calculating the tower height is

Equation 10.5-13


If Eq. (10.5-4) is equated to Eq. (10.5-12) and the result rearranged,

Equation 10.5-14


10.5C. Design of Water-Cooling Tower Using Film Mass-Transfer Coefficients

The tower design is done using the following steps:

1.
The enthalpy of saturated air Hyi is plotted versus Ti on an H-versus-T plot as shown in Fig. 10.5-3. This enthalpy is calculated by means of Eq. (9.3-8) using the saturation humidity from the humidity chart for a given temperature, with 0°C (273 K) as a base temperature. Calculated values are tabulated in Table 10.5-1.

Table 10.5-1. Enthalpies of Saturated Air–Water Vapor Mixtures (0°C Base Temperature)
TLHyTLHy
btuJbtuJ
°F°Clbm dry airkg dry air°F°Clbm dry airkg dry air
6015.618.7843.68 × 10310037.863.7148.2 × 103
8026.736.184.0 × 10310540.674.0172.1 × 103
8529.441.897.2 × 10311043.384.8197.2 × 103
9032.248.2112.1 × 10311546.196.5224.5 × 103
9535.055.4128.9 × 10314060.0198.4461.5 × 103

Figure 10.5-3. Temperature enthalpy diagram and operating line for water-cooling tower.


2.
Knowing the entering air conditions TG1 and H1, the enthalpy of this air Hy1 is calculated from Eq. (9.3-8). The point Hy1 and TL1 (desired leaving water temperature) is plotted in Fig. 10.5-3 as one point on the operating line. The operating line is plotted with a slope LcL/G and ends at point TL2, which is the entering water temperature. This gives Hy2. Alternatively, Hy2 can be calculated from Eq. (10.5-2).

3.
Knowing hLa and kGa, lines with a slope of −hLa/kGaMBP are plotted as shown in Fig. 10.5-3. From Eq. (10.5-14), point P represents Hy and TL on the operating line, and point M represents Hyi and Ti, the interface conditions. Hence, line MS or HyiHy represents the driving force in Eq. (10.5-13).

4.
The driving force HyiHy is computed for various values of TL between TL1 and TL2. Then the function 1/(HyiHy) is integrated from Hy1 to Hy2 by numerical or graphical integration to obtain the value of the integral in Eq. (10.5-13). Finally, the height z is calculated from Eq. (10.5-13).

10.5D. Design of Water-Cooling Tower Using Overall Mass-Transfer Coefficients

Often, only an overall mass-transfer coefficient KGa in kg mol/s · m3 · Pa or kg mol/s · m3 · atm is available, and Eq. (10.5-13) becomes

Equation 10.5-15


The value of is determined by going vertically from the value of Hy at point P up to the equilibrium line to give at point R, as shown in Fig. 10.5-3. In many cases the experimental film coefficients kGa and hLa are not available. The few experimental data available indicate that hLa is quite large; the slope of the lines −hLa/(kGaMBP) in Eq. (10.5-14) would be very large and the value of Hyi would approach that of in Fig. 10.5-3.

The tower design using the overall mass-transfer coefficient is done using the following steps:

1.
The enthalpy–temperature data from Table 10.5-1 are plotted as shown in Fig. 10.5-3.

2.
The operating line is calculated as in steps 1 and 2 for the film coefficients and plotted in Fig. 10.5-3.

3.
In Fig. 10.5-3 point P represents Hy and TL on the operating line and point R represents on the equilibrium line. Hence, the vertical line RP or Hy represents the driving force in Eq. (10.5-15).

4.
The driving force Hy is computed for various values of TL between TL1 and TL2. Then the function 1/( Hy) is integrated from Hy1 to Hy2 by numerical or graphical methods to obtain the value of the integral in Eq. (10.5-15). Finally, the height z is obtained from Eq. (10.5-15).

If experimental cooling data from an actual run in a cooling tower with known height z are available, then, using Eq. (10.5-15), the experimental value of KGa can be obtained.

EXAMPLE 10.5-1. Design of Water-Cooling Tower Using Film Coefficients

A packed countercurrent water-cooling tower using a gas flow rate of G = 1.356 kg dry air/s · m2 and a water flow rate of L = 1.356 kg water/s · m2 is to cool the water from TL2 = 43.3°C (110°F) to TL1 = 29.4°C (85°F). The entering air at 29.4°C has a wet bulb temperature of 23.9°C. The mass-transfer coefficient kGa is estimated as 1.207 × 107 kg mol/s · m3 · Pa and hLa/kGaMBP as 4.187 × 104 J/kg · K (10.0 btu/lbm · °F). Calculate the height of packed tower z. The tower operates at a pressure of 1.013 × 105 Pa.

Solution: Following the steps outlined, the enthalpies from the saturated air–water vapor mixtures from Table 10.5-1 are plotted in Fig. 10.5-4. The inlet air at TG1 = 29.4°C has a wet bulb temperature of 23.9°C. The humidity from the humidity chart is H1 = 0.0165 kg H2O/kg dry air. Substituting into Eq. (9.3-8), noting that (29.4 − 0)°C = (29.4 − 0) K,


Figure 10.5-4. Graphical solution of Example 10.5-1.


The point Hy1 = 71.7 × 103 and TL1 = 29.4°C is plotted. Then substituting into Eq. (10.5-2) and solving,


Hy2 = 129.9 × 103 J/kg dry air (55.8 btu/lbm). The point Hy2 = 129.9 × 103 and TL2 = 43.3°C is also plotted, giving the operating line. Lines with slope −hLa/kGaMBP = −41.87 × 103 J/kg · K are plotted, giving Hyi and Hy values, which are tabulated in Table 10.5-2 along with derived values as shown. Values of the function 1/(HyiHy) are used with numerical integration between the values Hy1 = 71.7 × 103 to Hy2 = 129.9 × 103 to obtain the integral

Table 10.5-2. Enthalpy Values for Solution to Example 10.5-1 (enthalpy in J/kg dry air)
HyiHyHyi − Hy1/(Hyi − Hy)
94.4 × 10371.7 × 10322.7 × 1034.41 × 105
108.4 × 10383.5 × 10324.9 × 1034.02 × 105
124.4 × 10394.9 × 10329.5 × 1033.39 × 105
141.8 × 103106.5 × 10335.3 × 1032.83 × 105
162.1 × 103118.4 × 10343.7 × 1032.29 × 105
184.7 × 103129.9 × 10354.8 × 1031.82 × 105


Substituting into Eq. (10.5-13),



10.5E. Minimum Value of Air Flow

Often the air flow G is not fixed but must be set for the design of the cooling tower. As shown in Fig. 10.5-5, for a minimum value of G, the operating line MN is drawn through the point Hy1 and TL1 with a slope that touches the equilibrium line at TL2, point N. If the equilibrium line is quite curved, line MN could become tangent to the equilibrium line at a point farther down the equilibrium line than point N. For the actual tower, a value of G greater than Gmin must be used. Often, a value of G equal to 1.3 to 1.5 times Gmin is used.

Figure 10.5-5. Operating-line construction for minimum gas flow.


10.5F. Design of Water-Cooling Tower Using Height of a Transfer Unit

Often another form of the film mass-transfer coefficient is used in Eq. (10.5-13):

Equation 10.5-16


Equation 10.5-17


where HG is the height of a gas enthalpy transfer unit in m, and the integral term is called the number of transfer units. The term HG is often used since it is less dependent upon flow rates than kGa.

In many cases another form of the overall mass-transfer coefficient KGa in kg mol/s · m3· Pa or kg mol/s · m3 · atm is used, and Eq. (10.5-15) becomes

Equation 10.5-18


where HOG is the height of an overall gas enthalpy transfer unit in m. The value of is determined by going vertically from the value of Hy up to the equilibrium line, as shown in Fig. 10.5-3. This method should be used only when the equilibrium line is almost straight over the range used. However, the HOG is often used even if the equilibrium line is somewhat curved because of the lack of film mass-transfer-coefficient data.

10.5G. Temperature and Humidity of Air Stream in Tower

The procedures outlined above do not yield any information on the changes in temperature and humidity of the air–water vapor stream through the tower. If this information is of interest, a graphical method by Mickley (M2) is available. The equation used for the graphical method is derived by first setting Eq. (10.5-6) equal to GcSdTG and then combining it with Eqs. (10.5-12) and (10.5-9) to yield Eq. (10.5-19):

Equation 10.5-19


10.5H. Dehumidification Tower

For the cooling or humidification tower discussed above, the operating line lies below the equilibrium line, and water is cooled and air humidified. In a dehumidification tower, cool water is used to reduce the humidity and temperature of the air that enters. In this case the operating line is above the equilibrium line. Similar calculation methods are used (T1).

..................Content has been hidden....................

You can't read the all page of ebook, please click here login for view all page.
Reset
3.234.253.152