Question 13.3: Water is to be cooled from 100 to 85 F in a counterflow cool......

Figure 13-8 is the cooling diagram for the given conditions. As the water is cooled from t_{l1}  to  t_{l2}, the enthalpy of the saturated air i_{l} follows the saturation curve from A to B. The air entering at wet bulb temperature t_{wb1}  has  enthalpy  i_{l}. (This assumes that the air enthalpy is only a function of wet bulb temperature.) The leaving water temperature t_{l2}  and  the  enthalpy  i_{l} define point C, and the initial driving potential is represented by the distance BC. The enthalpy increase of the air is a straight-line function with respect to the water temperature as defined by Eq. 13-28a. The slope of the air operating line CD is therefore c_{l} \dot{m}_{l}/\dot{m}_{a}.

G_{a}  d_{i}  =  G_{l}  c_{l}  dt_{l}  =  U_{i} a_{m}  (i_{i}  –  i)dL                  (13-28a)

Point C represents the air conditions at the inlet, and point D represents the air conditions leaving the tower. Note that the driving potential gradually increases from the bottom to the top of the tower. Counterflow integration calculations start at the bottom of the tower where the air conditions are known. Evaluation of the integral of Eq.13-29 may be carried out in a manner similar to that described in Example 13-1 by plotting t_{l}  versus  1/(i_{l}  –  i); however, another method will be used here (6, 7). The stepby-step procedure is shown in Table 13-1. Water temperatures are listed in column 1 in increments of one or two degrees. Smaller increments will give greater accuracy. The film enthalpies shown in column 2 are the enthalpy of saturated air at the water temperatures. Column 3 shows the air enthalpy, which is determined from Eq. 13-28c:

Δi  =  (i_{l}  –  i)  =  \frac{c_{l}  \dot{m}_{l} }{\dot{m}_{a}}  Δt_{l}                    (13-28c)

where the initial air enthalpy i_{l}  is  38.5  Btu/lbma,  c_{l}  =  1.0   Btu/(lbmw-F), \dot{m}_{l}/\dot{m}_{a}  =  1.0, and ∆t is read in column 1 of Table 13-1. The data of columns 4 and 5 are obtained from columns 2 and 3. Column 6 is the average of two steps from column 5 multiplied by the water temperature increment (column 1) for the same step. The number of transfer units is then given in column 7 as the summation of column 6. Column 8 gives the temperature range over which the water has been cooled. The last entry in column 7 is the number of transfer units required for this problem.

It is evident from Table 13-1 that either an increase in the cooling range or a
decrease in the leaving water temperature will increase the number of transfer units. As mentioned earlier, these two factors are quite important in cooling-tower design. The heat exchangers with which the cooling tower is connected should be designed with the cooling tower in mind. It may be more economical to enlarge the heat exchangers and/or increase the flow rate of the water than to increase the size of the cooling tower.

To continue the problem of tower design, we need information on the overall
mass-transfer coefficient per unit volume, U_{i}  a_{m}. There is little theory to predict this coefficient; therefore, we must rely on experiments. After many tests have been made on towers of a similar type, it is possible to predict U_{i}  a_{m} with reasonable accuracy. Then the volume of the tower required for a given set of conditions is given by

V  =  \frac{N \dot{m}_{l} c_{l}}{U_{i}  a_{m}}                          (13-30)

where N is the number of transfer units given by Eq. 13-29. The cross-sectional area of the tower is defined by

A_{c}  =  \frac{\dot{m}_{a}}{G_{a}}  =  \frac{\dot{m}_{l}}{G_{l}}                              (13-31)

and the height of the tower is given by

L  =  \frac{V}{A_{c}}              (13-32)