Cooling Tower Performance Evaluation via Merkel Integration Method
Table of Contents
The \( L/G \) ratio is simply the mass of the water (\( L \)) divided by the mass of the dry air (\( G \)) flowing through our tower.
Energy lost by water is gained by moist air:
\[ \frac{L}{G} = \frac{H_{a,out} - H_{a,in}}{C_{p,w} \ (T_{hot} - T_{cold})} \]
Where:
- \( L \): Mass flow rate of water (\( \text{kg/s} \) or \( \text{lb/hr} \))
- \( G \): Mass flow rate of dry air (\( \text{kg/s} \) or \( \text{lb/hr} \))
- \( H_{a,out} \): Enthalpy of air leaving the cooling tower (\( \text{kJ/kg} \) or \( \text{Btu/lb} \))
- \( H_{a,in} \): Enthalpy of air entering the cooling tower (\( \text{kJ/kg} \) or \( \text{Btu/lb} \))
- \( C_{p,w} \): Specific heat capacity of water (\( \text{kJ/kg}^\circ\text{C} \) or \( \text{Btu/lb}^\circ\text{F} \))
- \( T_{hot} \): Hot water temperature entering the tower (\( ^\circ\text{C} \) or \( ^\circ\text{F} \))
- \( T_{cold} \): Cold water temperature leaving the tower (\( ^\circ\text{C} \) or \( ^\circ\text{F} \))
To understand the practical implications of the \( L/G \) ratio, we have to look at it as the ultimate balancing act between CAPEX (Capital Expenditure - the physical size and cost of the tower) and OPEX (Operational Expenditure - the electricity cost to run the fans).
The Core Impact of the L/G Ratio #
- High L/G Ratio (e.g., 1.5 to 2.0): We are pushing a lot of water but using relatively little air. Because there is less air to absorb the heat, the driving force for evaporation drops. To achieve our desired cooling, we must drastically increase the NTU (Number of Transfer Units). This means we need a physically larger tower with more fill media (Higher CAPEX), but because we are running smaller fans, our electricity costs plummet (Lower OPEX).
- Low L/G Ratio (e.g., 0.5 to 0.8): We are blasting a massive amount of air through the water. The air easily absorbs the heat, so our required NTU drops. We can build a much smaller, cheaper tower (Lower CAPEX), but we need massive fan motors running at high speeds, which makes our electricity bills skyrocket (Higher OPEX).
Climate Impact #
Eastern Turkey has a harsh, continental climate: very hot, dry summers and brutally cold, snowy winters. This drastically changes how we should view our \( L/G \) ratio:
- The Summer Advantage (Hot & Dry): Because the summer air is very dry (low Relative Humidity), the Wet Bulb Temperature is significantly lower than the Dry Bulb Temperature. The air is “thirsty” for moisture. Because the air can absorb water so easily, we can actually afford to design for a slightly higher \( L/G \) ratio than we would in a humid, coastal city. We don’t need to push as much air to get the evaporative cooling we need.
- The Winter Risk (Severe Icing): In the winter, the massive drop in temperature means our cooling tower can over-cool the water, leading to severe icing on the fill and louvers. To prevent this, operators will intentionally reduce fan speeds using Variable Frequency Drives (VFDs). By slowing the fans, we artificially push our \( L/G \) ratio extremely high. The warm water prevents the fill from freezing. Therefore, our tower must be structurally capable of handling high \( L/G \) operation in the winter without suffering from poor water distribution.
Common L/G Ratios Used by Vendors #
When we solicit bids from cooling tower vendors (like SPX, BAC, or Evapco) for standard induced draft towers, they are running optimization algorithms to find the lowest total lifecycle cost (CAPEX + OPEX over 10-20 years).
- The Industry “Sweet Spot”: For induced draft mechanical towers, vendors typically design around an \( L/G \) ratio of 1.0 to 1.5. This range generally provides the best balance between tower size (fill volume) and fan motor horsepower.
- Why our Excel sheet used 0.5: The spreadsheet we analyzed used an \( L/G \) of 0.5, which is quite low. This indicates a design that relies on massive airflow (high fan power) to keep the physical size of the tower as small as possible. This is common in package towers or situations where physical space is highly constrained, and the client is willing to accept higher electricity bills.
Understanding Resistance to Cooling #
Our ultimate goal is to find the NTU (Number of Transfer Units), which tells us how “big” or capable our cooling tower needs to be. The mathematical definition of the Merkel Equation is an integral:
\[ \text{NTU} = \int_{T_{cold}}^{T_{hot}} \frac{C_p}{H_s - H_a} dT \]
Where:
- \( C_p \): Specific heat capacity of water (\( \text{kJ/kg}^\circ\text{C} \) or \( \text{Btu/lb}^\circ\text{F} \))
- \( T_{hot} \): Hot water temperature entering the tower (\( ^\circ\text{C} \) or \( ^\circ\text{F} \))
- \( T_{cold} \): Cold water temperature leaving the tower (\( ^\circ\text{C} \) or \( ^\circ\text{F} \))
- \( H_s \): Enthalpy of saturated air at water temperature (\( \text{kJ/kg} \) or \( \text{Btu/lb} \))
- \( H_a \): Enthalpy of bulk air at the same point in tower (\( \text{kJ/kg} \) or \( \text{Btu/lb} \))
- \( dT \): Differential temperature change of water (\( ^\circ\text{C} \) or \( ^\circ\text{F} \))
In calculus, the mathematical expression inside the integral sign (the part we are trying to find the area under) is called the “integrand”.
Here is why we calculate it as 1.0 / (Hs - Ha) in our loop:
- The denominator,
Hs - Ha, is our driving force. It represents how “eager” the air is to absorb heat and moisture from the water. - If the driving force is large, the cooling happens easily and quickly.
- If the driving force is small, the air is almost saturated, and it is very difficult to cool the water further.
- By taking the inverse (
1.0 / driving_force), we are calculating the “resistance” to cooling at that specific temperature.
Reference States & Visualizing the Driving Force #
The driving force for heat and mass transfer at any point in the tower is represented by the vertical distance between the saturation curve \( h_w \) and the operating line \( h_a \).
| Parameter | Operational Meaning | Impact on Driving Force |
|---|---|---|
| \( T_{\text{hot}} - T_{\text{cold}} \) | Cooling Range | Determines the span of the integration bounds. |
| \( T_{\text{cold}} - T_{\text{wb}} \) | Approach to Wet Bulb | A tighter approach significantly drops \( (h_w - h_a) \), causing NTU to spike exponentially. |
| \( L/G \) Ratio | Liquid-to-Gas Mass Ratio | Alters the slope of the operating line. A higher slope brings \( h_a \) closer to \( h_w \). |
Python Implementation: Numerical Integration #
Code is kept extremely lean without any external library dependencies. Just x1 psychometric module is loaded which is a single .py file itself. Additionally simple numerical methods like
- trapezoidal rule
- simpson’s 1/3 rule
are used to integrate & calculate NTU.
Simulation focuses on the effect of water to air flowrates on cooling tower & when you can increase or decrease it. Theoretical background is also discussed
Complete Notebook Resource #
The full calculations, along with graphical plots mapping out the saturation curve relative to the operating line across varying ranges, are included in the verification notebook: