Recirculating cooling water system pH together with other water quality parameters can significantly affect the scaling and corrosion potential of the water. Traditional approaches for cooling water scaling and corrosion potential evaluation rely on various saturation indices, such as the Langelier saturation index and the Ryznar stability index, that require knowledge of cooling system pH [1]. Thus, cooling water pH prediction is of great interest for scaling and corrosion control evaluation.

Several empirical equations have been proposed that express cooling water pH as different functions of cooling water alkalinity. These empirical equations include Kunz's equation [2], Puckorius' equation [3], and Caplan's equation [4]. However, because the makeup water quality and cooling system operating conditions (i.e., cycles of concentration, temperature, etc.) also affect the pH of the cooling loop, these empirical equations cannot accurately predict the pH of the cooling loop over a broad range of operating conditions.

Other traditional approaches for prediction of cooling water quality are generally based on the cooling water thermodynamic equilibria, which assume that the carbonate species in the cooling water is at equilibrium with the atmosphere. However, cooling water pH and alkalinity monitoring data collected from more than 40 cooling towers have shown that thermodynamic equilibrium is not achieved in most open-recirculating cooling systems [5].

Although comprehensive rate-based models have been developed to estimate CO_{2} and/or NH_{3} evaporations in cooling towers [e.g., [6], these models require detailed information on the properties of the water entering the cooling tower which is not usually available. To predict CO_{2} evaporation in cooling towers using readily available information such as makeup water flow rate and alkalinity, cycles of concentrations, and loop water alkalinity requires an integrated approach that combines the cooling tower with the rest of the cooling system.

Ballard and Matson [7] developed a cooling tower carbonate equilibrium model (CTCE) that considers makeup water pH and alkalinity as input parameters to predict cooling water pH based on the carbonate species mass balance in the cooling system. This model is based on the assumption that the “cooling tower is 100% efficient in achieving equilibrium between dissolved CO_{2} in the cooling water and CO_{2} in the atmosphere” [7]. They also showed that a simplified version of CTCE (i.e., SCTCE) predicts results similar to those in their study, indicating that the simplification was reasonable. The cooling loop water pH in the SCTCE model is determined from the following equation:

- (1)

Figure 1 shows the monitoring results of the cooling water pH-alkalinity relationship during a 2-month period of pilot testing, along with the results obtained by Eq. (2) (i.e., equilibrium calculations) as well as those predicted by the SCTCE model and various empirical equations including Kunz's equation, the Pukorious' equation, and Caplan's equation. The results in Figure 1 indicate that none of these models can accurately predict the pH of the cooling loop, suggesting that an improved model is needed that can take into account other important variables that significantly affect the loop pH. These include the mass transfer limitations for evaporation of CO_{2} in the cooling tower, the salt formation in the cooling system, and the extent of NH_{3} stripping when degraded waters such as municipal wastewaters are utilized for cooling.

#### Modeling Approach

Figure 2 shows the schematic diagram of a cooling tower. Air with flow rate *Q*_{y} (m^{3}/s) and total CO_{2} concentration *C*_{yb} (mol/m^{3}) enters the bottom of the tower and leaves at the top with total CO_{2} concentration *C*_{ya} (mol/m^{3}). Makeup water with flow rate *Q*_{m} (m^{3}/s) and total inorganic carbon (TIC) concentration *C*_{m} (mol/m^{3}) enters into the basin of the cooling tower. Blowdown leaves the basin with flow rate *Q*_{b} (m^{3}/s) and TIC concentration *C*_{xa} (mol/m_{3}). Cooling water leaves the basin and enters the top of the tower with flow rate *Q*_{x} (m^{3}/s) and TIC concentration *C*_{xa}. Evaporation rate in the cooling system is *Q*_{e} (m^{3}/s). Cooling water leaves the tower at the bottom and enters into the basin with flow rate of (*Q*_{x}-*Q*_{e}) and total TIC *C*_{xb} (mol/m^{3}). For this derivation, activity coefficients are considered unity for all chemical species. Usually *Q*_{e} is less than 3% of *Q*_{x} and it is reasonable to assume that *Q*_{e} is negligible when compared to *Q*_{x} (i.e., (*Q*_{x}-*Q*_{e}) is approximately *Q*_{x}). Vertical distance above the bottom of the packing is *Z* (m).

In the above equations the coefficient “*α*” accounts for the influence of CO_{2}/HCO_{3}^{−} speciation and co-diffusion through the water film. *R*_{g} is the resistance in the air film (s/m) and is defined as 1/(*k*_{g}*K*_{H}), *R*_{w} is the resistance in water film (s/m) and is defined as 1/(*k*_{w}*α*), and *R*_{g}/*R*_{w} represents the ratio of the gas to liquid phase resistances. Equation (6) shows that the gas to liquid phase resistance increases as the pH of the aqueous phase increases (increase in *α*).

The CO_{2} concentration in the water (*C*_{x}) and the total inorganic carbon concentration (*C*_{TX}) are related through the following equation:

- (7)

In the pH range of interest (i.e., pH of 6–9), alkalinity is typically dominated by bicarbonate, that is

- (8)

and therefore, TIC concentration is related to CO_{2} through the alkalinity:

- (9)

Since the evaporation rate is significantly lower than the recirculating water flow rate (*Q*_{e} << *Q*_{x}), the TIC mass balance in the packing section of the vertical distance from 0 to *Z* can be expressed as:

- (10)

By substituting *C*_{Tx} from Eq. (9) and *C*_{y} from Eq. (10) into Eq. (3), and assuming that alkalinity is conserved in the system (i.e., ALK_{xa}=ALK_{xb}=ALK_{x} and Q_{m}ALK_{m}=Q_{b}ALK_{x}), Eq. (3) can be integrated over the ranges of *C*_{x} from *C*_{xb} to *C*_{xa} and *Z* from 0 to *Z*_{T} [packing total height, (m)], to obtain the following equation:

- (11)

In Eq. (11), *k* and *K*_{H} can be assumed to be constant over the height of the tower (due to near isothermal tower operation), *A* is the effective total packing surface area (*A*_{b}*aZ*_{T}), and *E* is calculated from the following equation:

- (12)

Since *Q*_{y}*K*_{H}>>*Q*_{x}, *E* becomes:

- (13)

and TIC mass balance in the basin can be expressed as:

- (14)

Applying Eq. (9) and alkalinity conservation (i.e., ALK_{xa}=ALK_{xb}=ALK_{x} and Q_{m}ALK_{m}=Q_{b}ALK_{x}), Eq. (14) becomes:

- (15)

By substituting *C*_{xa} from Eq. (9) into Eq. (13) and after rearrangement, the CO_{2} concentration in the blowdown as a function of the flow rates and cooling tower properties can be obtained by Eq. (16), while the pH of the cooling water in the basin is given by Eq. (17):

- (16)

- (17)

In Eq. (17), *C*_{xa} is [H_{2}CO_{3}*]_{xa}, and *K*_{a1} is the equilibrium constant for the dissociation reaction of [H_{2}CO_{3}*]:

- (18)

Since *k* and therefore *E* are functions of pH, pH_{xa} can be calculated by solving Eqs. (16) and (17) using an iterative scheme.

As indicated above, Eqs. (16) and (17) are obtained based on the assumption that the alkalinity is conserved in the system. However, if alkalinity is not conserved in the cooling system due to salt formation, acid addition, or ammonia stripping, Eqs. (16) and (17) cannot be applied. If carbonate salts are precipitating in the basin at a rate of S (mol/s) and alkalinity is also consumed by acid addition at a rate of *R* (equivalent/s^{−1}), Eqs. (16) and (17) become:

- (19)

- (20)

where *S* and *R* are the sink terms, representing alkalinity consumption in the cooling system due to the carbonate salts precipitation and acid addition, respectively. Equations (19) and (20) indicate that the cooling water pH can be estimated using the makeup water properties, the makeup and the blowdown flow rates (Qm and Qb), and the overall mass transfer constant (*k*).