# Cooling tower models

#### benit13

As mentioned in another thread, here's some research I've been doing on cooling towers which might come in handy when working out evaporation ratess. The method below uses the effectiveness of a counterflow heat exchanger; for a more straightforward scenario, try using the effectiveness for a flat plate heat exchanger instead.

The NTU-effectiveness method was originally developed by the Environmental Protection Agency (EPA) and is described in Reddy et al. (2016). The NTU-Effectiveness model is a good introduction to cooling tower systems and is presented for pedagogical reasons; it highlights some insights into how cooling towers behave and their calculation.

There are some common definitions and considerations which are important for all cooling tower models. Firstly, there are different kinds of cooling towers. They are usually categorised in terms of the source of the air-flow (mechanical-draft, induced-draft or natural-draft) and the direction the air flow interacts with the water flow (counterflow or crossflow). Here we focus on models describing counterflow, mechanical-draft cooling towers.

Secondly, there are different cooling tower parameters. These are aspects of the cooling tower which only change with the design of the cooling tower and do not change for a given cooling tower design. These are important because the cooling tower can be calculated by considering deviations away from design conditions. They also inform the cooling tower size (for designers). These parameters are:

1. The design entering air condition: various parameters related to the entering air condition can be calculated using psychrometric equations, so it is sufficient to specify the design entering air wet-bulb temperature.
2. The design heat rejection rate: this is set by the peak (condenser) load at design cooling load conditions. The size of the cooling tower increases linearly with load.
3. The water flow rate: this is determined by the chiller condenser specifications or the district cooling return flow rate. Generally, the water flow rate is considered constant and is not modulated at part-load conditions (for numerous reasons).
4. The range: this is the difference between the entering and leaving water temperatures. Tower size decreases with increasing range.
5. The approach: this is the difference between the water leaving temperature and the entering air wet-bulb temperature. The lower the approach, the higher the coefficient of performance (CoP).

The NTU-effectiveness method starts by equating the enthalpy increase in the air stream with the enthalpy decrease in the water stream, including an evaporative losses term. The following differential equation can be derived from this enthalpy balance

$$\displaystyle \dot{m}_a dh_a=-\dot{m}_w dh_w+\dot{m}_a dW \cdot h_{liq-vap}.$$

See Table 1 for a list of symbols and their meanings. In general, a cooling tower model can solve this differential equation for any geometry and flow conditions. However, if simplified to form a bulk model, the differential equation can be simplified to

$$\displaystyle \dot{m}_a (h_{a,o} - h_{a,i})=\dot{m}_w (h_{w,i} - h_{w,o})+\dot{m}_a (W_o - W_i) \cdot h_{liq-vap}.$$

If the cooling tower is treated as a heat exchanger, the heat exchanger effectiveness can be described as the ratio of the actual heat transfer to the maximum rate permitted by the second law of thermodynamics,

$$\displaystyle \epsilon_{tower} = \dot{Q}/(\dot{m}_a (h_{a,sat,i}-h_{a,i}) ).$$

By analogy with the counterflow heat exchanger,

$$\displaystyle \epsilon_{tower}=(1 - e^{(-NTU(1-R))})/(1 - R e^{(-NTU(1-R))} ),$$

where

$$\displaystyle R=(\dot{m}_a c_{p,a,sat})/(\dot{m}_w c_{p,w} ),$$

$$\displaystyle c_{p,a,sat}=(h_{a,sat,i}-h_{a,sat,o})/(T_{w,i}-T_{w,o} ).$$

The number of thermal transfer units (NTU) can be estimated using the empirical correlation

$$\displaystyle NTU=a\left(\dot{m}_w/\dot{m}_a \right)^n,$$

Where a has a value between 1 and 3 and n has a value between 0.2 and 0.6, depending on the cooling tower. Once effectiveness is known, the enthalpy of outlet air can be calculated using

$$\displaystyle h_{a,o}=h_{a,i}+\epsilon_{tower} (h_{a,sat,i}-h_{a,i} )$$

and, if water evaporation (mass) loss is ignored,

$$\displaystyle T_{w,o}=T_{w,i}-(\dot{m}_a (h_{a,o}-h_{a,i} ))/(\dot{m}_w c_{p,w} ).$$

The value of $$\displaystyle c_{p,a,sat}$$ is a weak function of temperature, so iteration through the above equations multiple times might be required to settle on correct values. The initial value for $$\displaystyle c_{p,a,sat}$$ can be estimated using the above equation for $$\displaystyle c_{p,a,sat}$$, but by substituting for the outside air wet-bulb temperature:

$$\displaystyle c_{p,a,sat}=(h_{a,sat,i}-h_{a,sat,o})/(T_{w,i}-T_{wb,o} ).$$

To summarise, the method proceeds as follows:
Calculate $$\displaystyle c_{p,a,sat}$$;
Calculate R and NTU;
Calculate $$\displaystyle \epsilon_{tower}$$;
Calculate $$\displaystyle h_{a,o}$$;
Calculate $$\displaystyle T_{w,o}$$;
Repeat steps 1-5 above with the new value of $$\displaystyle T_{w,o}$$ until the calculation settles on a converged result.

Table 1: List of mathematical symbols used when describing the NTU-Effectiveness method.

Symbol Description Units
a Dimensionless fitting constant (typically between 1 and 3) -
$$\displaystyle c_{p,a,sat}$$ Specific heat capacity of saturated air J/(kg.K)
$$\displaystyle c_{p,w}$$ Specific heat capacity of water J/(kg.K)
$$\displaystyle \epsilon_{tower}$$ Cooling tower heat exchange effectiveness
$$\displaystyle h_{a,i}$$ Enthalpy of inlet air J
$$\displaystyle h_{a,o}$$ Enthalpy of outlet air J
$$\displaystyle h_{liq-vap}$$ Specific enthalpy of vaporisation J/kg
$$\displaystyle h_{a,sat,i}$$ Enthalpy of (saturated) inlet air J
$$\displaystyle h_{w,i}$$ Enthalpy of inlet water J
$$\displaystyle h_{w,o}$$ Enthalpy of outlet water J
n Dimensionless fitting constant (typically between 0.2 and 0.6) -
$$\displaystyle \dot{m}_a$$ Air mass flow rate kg/s
$$\displaystyle \dot{m}_w$$ Water mass flow rate kg/s
NTU Number of Transfer Units -
$$\displaystyle \dot{Q}$$ Heat rejection rate W
R Capacity ratio -
$$\displaystyle T_{a,i}$$ Dry-bulb temperature of inlet air °C
$$\displaystyle T_{a,o}$$ Dry-bulb temperature of outlet air °C
$$\displaystyle T_{w,i}$$ Dry-bulb temperature of inlet water °C
$$\displaystyle T_{w,o}$$ Dry-bulb temperature of outlet water °C
$$\displaystyle T_{wb,i}$$ Wet-bulb temperature of inlet air °C
$$\displaystyle W_i$$ Humidity ratio of inlet air kg/kg
$$\displaystyle W_o$$ Humidity ratio of outlet air kg/kg

References:

Reddy, T. A., Kreider, J. F., Curtiss, P. S., and Rabi, A., 2016, “Heating and cooling of buildings: Principles and practice of energy efficient design”, 3rd Ed., CRC Press, ISBN-10: 1439899894, ISBN-13: 978-1439899892.

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#### donglebox

Thanks! I make sure my work is professional quality, but it helps that the references I use for the same thing are the same quality 