Cooling Tower Fans Energy Consumption

1. Total energy for each hour (Worksheet: “Step 2 Energy Calcs”). Since the measured current (Amps) is the average for an hour, Equation 10 yields the average energy per hour (kWh) assuming voltage is constant.

\begin{equation} \overline{E}_{f}(t) = \frac{i_{f}(t) * P_{f}}{I_{f}} \end{equation}

\begin{equation} \overline{E}(t) = \sum_{f=1}^{F} \overline{E}_{f}(t) \end{equation}

Where,

$\overline{E}_{f}(t) =$ Average hourly fan motor energy on for each fan, f, kWh

$i_{f}(t) =$ Average hourly current for each fan, f, Amps

$t =$ Time index

$P_{f} =$ One-time power measurement for each fan, f, kW

$I_{f} =$ One-time current measurement for each fan, f, Amps

$\overline{E} =$ Total average energy of all fan motors, kWh

$F =$ Total number of fans in the cooling tower

2. Wet-bulb temperature is calculated from the measured dry-bulb temperature and measured relative humidity with the following empirical equation (Worksheet: “Step 3 Wetbulb Calcs”)[1].

\begin{equation} T_{wb} = T_{db}tan^{-1}[0.151977(RH+8.313659)^{1/2}] \end{equation}

\begin{equation*} +tan^{-1}(T_{db}+RH) \end{equation*}

\begin{equation*} -tan^{-1}(RH-1.676331)+0.00391838(RH)^{3/2} \end{equation*}

\begin{equation*} *tan^{-1}(0.023101*RH) - 4.686035 \end{equation*}

Where,

$T_{wb} =$ Wet-bulb temperature, C

$T_{db} =$ Dry-bulb temperature, C

$RH =$ Relative Humidity, %

3. To model the total fan energy at different wet-bulb temperatures, a second-order polynomial regression model is fitted, with total fan energy as a function of wet-bulb temperature (Worksheet: “Step 4 Regression”).

\begin{equation} E(T_{wb}) = a*T_{wb}^{2} + b*T_{wb} + c \end{equation}

Where,

$E(T_{wb}) =$ Average hourly fan energy, kWh

$a,b,c =$ Regression Coefficients

4. Calculate operating hours (Worksheet: “Step 5 Daily Op Hrs”).

\begin{equation} On(t) = \begin{cases} 1 & \text{if } \overline{E}(t) > 0\\ 0 & \text{if } \overline{E}(t) = 0 \end{cases} \end{equation}

\begin{equation} \overline{On}_{d,h} = \begin{cases} 1 & \text{if } \overline{E}(n) > 0 \text{ for any } n \in N(d,h) \\ 0 & 0 \end{cases} \end{equation}

Where,

$On(t) =$ Cooling tower fans are on, binary

$\overline{On}_{d,h} =$ Cooling tower is on for a given day of the week, d, and hour of the day, h, binary

$N(d,h) =$ Total number of measured data points that fall on day of week, d, and hour of the day, h

$n \in N(d,h) =$ Index for the subset of measured data points that fall on day of week, d, and hour of the day, h

5. Calculate energy used for full year for each fan (Worksheet: “Step 6 Results”).

\begin{equation} E = \sum_{t=1}^{8760} \overline{On}_{d,h}(t) * CoolingSeason(t) * E(T_{wb}(t)) \end{equation}

Where,

$E =$ Annual energy, kWh

$CoolingSeason(t) =$ Does the given hour, t, fall within the cooling season, binary

This calculation is for a cooling tower with constant-speed fans. Measured input data is average hourly power draw (in kW) as measured for each fan in the cooling tower.

The calculator uses the following methodology to compute annual energy consumption of a constant speed cooling tower fan:

1. Total energy for each hour (Worksheet: “Step 2 Energy Calcs”). Since the measured power value (kW) is the average for an hour, it is equivalent to the energy for the hour (kWh).

\begin{equation} \bar{E}(t) = \sum_{f = 1}^F \bar{E}_{f}(t) \end{equation}

Where,

$\bar{E}_{f}(t) =$ average hourly fan motor energy on for each fan, f, kWh

$\bar{E} =$ total average energy of all fan motors, kWh

$F =$ total number of fans in the cooling tower

$t =$ time index

2. Wet-bulb temperature is calculated from the measured dry-bulb temperature and measured relative humidity with the following empirical equation (Worksheet: “Step 3 Wetbulb Calcs”)[1].

\begin{equation} T_{wb} = T_{db}tan^{-1}[0.151977(RH+8.313659)^{1/2}] \end{equation}

\begin{equation*} +tan^{-1}(T_{db}+RH) \end{equation*}

\begin{equation*} -tan^{-1}(RH-1.676331)+0.00391838(RH)^{3/2} \end{equation*}

\begin{equation*} *tan^{-1}(0.023101*RH) - 4.686035 \end{equation*}

Where,

$T_{wb} =$ Wet-bulb temperature, C

$T_{db} =$ Dry-bulb temperature, C

$RH =$ Relative Humidity, %

3. To model the total fan energy at different wet-bulb temperatures, a second-order polynomial regression model is fitted, with total fan energy as a function of wet-bulb temperature (Worksheet: “Step 4 Regression”).

\begin{equation} E(T_{wb}) = a*T_{wb}^{2} + b*T_{wb} + c \end{equation}

Where,

$E(T_{wb}) =$ Average hourly fan energy, kWh

$a,b,c =$ Regression Coefficients

4. Calculate operating hours (Worksheet: “Step 5 Daily Op Hrs”).

\begin{equation} On(t) = \begin{cases} 1 & \text{if } \overline{E}(t) > 0\\ 0 & \text{if } \overline{E}(t) = 0 \end{cases} \end{equation}

\begin{equation} \overline{On}_{d,h} = \begin{cases} 1 & \text{if } \overline{E}(n) > 0 \text{ for any } n \in N(d,h) \\ 0 & 0 \end{cases} \end{equation}

Where,

$On(t) =$ Cooling tower fans are on, binary

$\overline{On}_{d,h} =$ Cooling tower is on for a given day of the week, d, and hour of the day, h, binary

$N(d,h) =$ Total number of measured data points that fall on day of week, d, and hour of the day, h

$n \in N(d,h) =$ Index for the subset of measured data points that fall on day of week, d, and hour of the day, h

5. Calculate energy used for full year for each fan (Worksheet: “Step 6 Results”).

\begin{equation} E = \sum_{t=1}^{8760} \overline{On}_{d,h}(t) * CoolingSeason(t) * E(T_{wb}(t)) \end{equation}

Where,

$E =$ Annual energy, kWh

$CoolingSeason(t) =$ Does the given hour, t, fall within the cooling season, binary

This calculation is for a cooling tower with constant-speed fans. Measured input data include spot measurements for true power and motor operational time per hour, as measured by motor on/off loggers on each fan in the tower.

The calculator uses the following methodology to compute annual energy consumption of a constant speed cooling tower fan:

1. Convert seconds on per hour to average energy per hour (Worksheet: “Step 2. Energy Calcs”). Repeat for all fans and then sum to get a total for the cooling tower. Note that since the data loggers cannot be perfectly synced, the time stamp for the dataset for each fan will be different. The workbook time-aligns the datasets.

\begin{equation} \overline{E}_{f}(t) = \frac{On_f(t)*P_f}{3600} \end{equation}

\begin{equation} \overline{E}(t) = \sum_{f=1}^{F} \overline{E}_{f}(t) \end{equation}

Where,

$\overline{E}_{f}(t) =$ Average hourly fan motor energy on for each fan, f, kWh

$t =$ Time index

$On_{f}(t) =$ Measured time that motor is on for each fan, f, seconds

$P_{f} =$ One-time power measurement for each fan, f, kW

$\overline{E}(t) =$ Total average energy of all fan motors, kWh

$F =$ Total number of fans

2. Wet-bulb temperature is calculated from the measured dry-bulb temperature and measured relative humidity with the following empirical equation (Worksheet: “Step 3 Wetbulb Calcs”)[1].

\begin{equation} T_{wb} = T_{db}tan^{-1}[0.151977(RH+8.313659)^{1/2}] \end{equation}

\begin{equation*} +tan^{-1}(T_{db}+RH) \end{equation*}

\begin{equation*} -tan^{-1}(RH-1.676331)+0.00391838(RH)^{3/2} \end{equation*}

\begin{equation*} *tan^{-1}(0.023101*RH) - 4.686035 \end{equation*}

Where,

$T_{wb} =$ Wet-bulb temperature, C

$T_{db} =$ Dry-bulb temperature, C

$RH =$ Relative Humidity, %

3. To model the total fan energy at different wet-bulb temperatures, a second-order polynomial regression model is fitted, with total fan energy as a function of wet-bulb temperature (Worksheet: “Step 4 Regression”).

\begin{equation} E(T_{wb}) = a*T_{wb}^{2} + b*T_{wb} + c \end{equation}

Where,

$E(T_{wb}) =$ Average hourly fan energy, kWh

$a,b,c =$ Regression Coefficients

4. Calculate operating hours (Worksheet: “Step 5 Daily Op Hrs”).

\begin{equation} On(t) = \begin{cases} 1 & \text{if } \overline{E}(t) > 0\\ 0 & \text{if } \overline{E}(t) = 0 \end{cases} \end{equation}

\begin{equation} \overline{On}_{d,h} = \begin{cases} 1 & \text{if } \overline{E}(n) > 0 \text{ for any } n \in N(d,h) \\ 0 & 0 \end{cases} \end{equation}

Where,

$On(t) =$ Cooling tower fans are on, binary

$\overline{On}_{d,h} =$ Cooling tower is on for a given day of the week, d, and hour of the day, h, binary

$N(d,h) =$ Total number of measured data points that fall on day of week, d, and hour of the day, h

$n \in N(d,h) =$ Index for the subset of measured data points that fall on day of week, d, and hour of the day, h

5. Calculate energy used for full year for each fan (Worksheet: “Step 6 Results”).

\begin{equation} E = \sum_{t=1}^{8760} \overline{On}_{d,h}(t) * CoolingSeason(t) * E(T_{wb}(t)) \end{equation}

Where,

$E =$ Annual energy, kWh

$CoolingSeason(t) =$ Does the given hour, t, fall within the cooling season, binary

This calculation is for a cooling tower with variable-speed fans. Measured input data is average hourly power draw (in kW) as measured by a power logger for one fan in the cooling tower. It is assumed that all the remaining fans are run at the same speed.

The calculator uses the following methodology to compute annual energy consumption of a variable speed cooling tower fan:

1. Total energy for each hour (Worksheet: “Step 2 Energy Calcs”). Since the measured power value (kW) is the average for an hour, it is equivalent to the energy for the hour (kWh).

\begin{equation} \overline{E}(t) = \overline{E}_{1}(t) * F \end{equation}

Where,

$\overline{E}_{1}(t) =$ Average hourly fan motor energy for the measured fan, kWh

$\overline{E} =$ Total average energy of all fan motors, kWh

$F =$ Total number of fans in the cooling tower

$t =$ Time index

2. Wet-bulb temperature is calculated from the measured dry-bulb temperature and measured relative humidity with the following empirical equation (Worksheet: “Step 3 Wetbulb Calcs”)[1].

\begin{equation} T_{wb} = T_{db}tan^{-1}[0.151977(RH+8.313659)^{1/2}] \end{equation}

\begin{equation*} +tan^{-1}(T_{db}+RH) \end{equation*}

\begin{equation*} -tan^{-1}(RH-1.676331)+0.00391838(RH)^{3/2} \end{equation*}

\begin{equation*} *tan^{-1}(0.023101*RH) - 4.686035 \end{equation*}

Where,

$T_{wb} =$ Wet-bulb temperature, C

$T_{db} =$ Dry-bulb temperature, C

$RH =$ Relative Humidity, %

3. To model the total fan energy at different wet-bulb temperatures, a second-order polynomial regression model is fitted, with total fan energy as a function of wet-bulb temperature (Worksheet: “Step 4 Regression”).

\begin{equation} E(T_{wb}) = a*T_{wb}^{2} + b*T_{wb} + c \end{equation}

Where,

$E(T_{wb}) =$ Average hourly fan energy, kWh

$a,b,c =$ Regression Coefficients

4. Calculate operating hours (Worksheet: “Step 5 Daily Op Hrs”).

\begin{equation} On(t) = \begin{cases} 1 & \text{if } \overline{E}(t) > 0\\ 0 & \text{if } \overline{E}(t) = 0 \end{cases} \end{equation}

\begin{equation} \overline{On}_{d,h} = \begin{cases} 1 & \text{if } \overline{E}(n) > 0 \text{ for any } n \in N(d,h) \\ 0 & 0 \end{cases} \end{equation}

Where,

$On(t) =$ Cooling tower fans are on, binary

$\overline{On}_{d,h} =$ Cooling tower is on for a given day of the week, d, and hour of the day, h, binary

$N(d,h) =$ Total number of measured data points that fall on day of week, d, and hour of the day, h

$n \in N(d,h) =$ Index for the subset of measured data points that fall on day of week, d, and hour of the day, h

5. Calculate energy used for full year for each fan (Worksheet: “Step 6 Results”).

\begin{equation} E = \sum_{t=1}^{8760} \overline{On}_{d,h}(t) * CoolingSeason(t) * E(T_{wb}(t)) \end{equation}

Where,

$E =$ Annual energy, kWh

$CoolingSeason(t) =$ Does the given hour, t, fall within the cooling season, binary