Liu's study discusses a possible source of error, using an effectiveness model for a counterflow cooling tower rather than a crossflow cooling tower, which is the type of tower considered in his and in the current study. Liu attempted a new effectiveness model based on a crossflow tower but the crossflow effectiveness model was less accurate in modeling the tower manufacturer's data than the counterflow effectiveness model. Liu verified that the use of Braun's counterflow model produced very accurate results.
The following assumptions were made in the modeling of the cooling tower.
and
where Tw is the water temperature, is the mass flow rate of air through the cooling tower, is the mass flow rate of water through the tower, h_{a} is the enthalpy of the moist air per pound of dry air, h_{sw} is the enthalpy of saturated air at the water conditions, dV is a differential volume, V_{T} is the total volume, and Ntu_{CT} is the number of transfer units for the cooling tower. Again, as for the condenser, Ntu is the dimensionless parameter used for heat exchanger analysis.
Braun defines a saturation specific heat, C_{s}, as the derivative of the saturated air enthalpy with respect to temperature evaluated at the water temperature.
He then rewrites Equation 4.1 in terms of air enthalpies and C_{s}.
Braun then states that Equations 4.2 and 4.3 can be solved analytically for the exit conditions if the saturation enthalpy is linear with respect to temperature, making C_{s}_{ }constant. Although air saturation enthalpy does not vary linearly with temperature, an appropriate average slope between the inlet and outlet water conditions is chosen. An effectiveness relationship can now be derived in terms of C_{s}. This air side effectiveness is defined as the ratio of the actual heat transfer to the maximum possible air side heat transfer that would occur if the exiting stream were saturated at the temperature of the incoming water.
The actual heat transfer from the tower can then be given in terms of this effectiveness as:
where is the enthalpy of saturated air at inlet water conditions,is the enthalpy of the incoming air, and e, the air side effectiveness is
where
The average value for the saturation specific heat is estimated as the average slope between outlet water states:
where is the enthalpy of saturated air at exit water conditions. From overall energy balances, the outlet air enthalpy can be determined.
The exiting water temperature is then given by:
where c and n are empirical constants specific to a particular cooling tower box. The tower manufacturer offers towers of different box sizes, each having its own dimensions and cooling capacity. To calculate the tower coefficients, c and n, and to verify the model's accuracy, Liu (1997) correlated the tower manufacturer's data for different box sizes. The manufacturer's catalog gives the approach and the tower water outlet temperature for specified water and air flow rates, entering wet bulb temperature, and tower water inlet temperature. The reader should consult Liu for the complete methodology used to calculate the tower coefficients for each box size. These coefficients were used for the current study.
where A and B are constants which correspond to a particular tower box size and cfm is the air flow rate in the tower.
In order to maintain an entering condenser water set point, the tower fan must be cycled on and off. Past studies, Joyce (1990) and Liu (1997), have found the optimum tower water set point to be in the range of 65^{ o}F to 75^{ o}F. For this study, the entering condenser water set point was taken to be 68^{o}F. It was assumed that if the water exiting the tower was greater than or equal to 68^{o}F, then the fan was on with a resulting temperature above 68^{ o}F. If the water exiting the tower was less than 68^{o}F, then the fan would cycle on and off to maintain a 68^{ o}F entering condenser water temperature. To account for the times that the fan was off, a tower duty, or ratio of time the tower fan was on to the total time for each temperature period, was calculated. This value was multiplied by the fan's power, given in Equation 4.12 above, to yield the average power consumed by the fan.
Table 4.1 lists the manufacturer's data for the towers considered in this study. The table lists a tower identification number and its corresponding air flow rate in cubic feet per minute (cfm), fan motor horsepower (hp), and dimensions (length (l), width (w), and height (h)).






















































































































































Weber developed the baseline parameters using typical values for current design practices at a condenser water flow rate of 3 gpm/ton. These typical values for the water piping system used as guidelines for Weber's model were:


















The pressure rise that must be supplied by the condenser water pump at a given flow rate can be expressed in terms of the required total dynamic head calculated in feet:
where H_{piping} is all dynamic piping and friction losses due to piping, valves, strainers, and fittings; H_{tower} is the required static head due to tower lift (11 feet); and H_{condenser} is the dynamic head loss due to water flow in the condenser (listed in Table 4.2). The head loss in the pipe and fittings is calculated using:
where s is the velocity of the water through the condenser tubes in ft/sec, g is the gravitational constant in ft/sec^{2}, d is the pipe diameter in ft, L_{eq} is the total equivalent length of straight pipe including all piping, valves, fittings, and strainers, and f is the friction factor. The friction factor can be calculated from White (1986) for traditionally turbulent and completely turbulent rough pipes as:
where e_{p}, the average roughness of black iron pipe, is assumed to be 0.00015 feet, and Re_{d} is the Reynolds number calculated using the pipe inside diameter.
The power in kW that must be applied to the pump shaft can then be calculated.
where gpm is the flow rate through the condenser, and the pump efficiency, h_{pump}, was assumed to be 0.65. Weber (1988) assumed this electricity to water pump efficiency for a constant speed pump.
The required input power to the pump motor is calculated from:
where h_{motor} , the motor efficiency, was assumed to be 0.85 by Weber.
The above cooling tower, pump, and fan relationships allow the calculations of the tower water outlet temperature, pump power, and average fan power for the chiller system under any specified chiller loading and ambient conditions.