19 Pages

## Introduction

The cooling tower is a very common unit operation in the power generation industry, or for that matter, any process which requires a significant volume of process coolant water. For example, the UCSD CoGeneration Plant has seven large cooling towers which you should ask to visit because they're awesome.

In terms of engineering fundamentals, this unit operation is an important application of simultaneous heat and mass transfer. The goals of this experiment are for you to derive the appropriate design equations, apply the results to cooling tower operation, evaluate the tower characteristic under different operating conditions, and re-familiarize yourself with psychrometric charts.

## Background

Cooling tower theory and technology is well-established. Use your textbooks in heat and mass transfer to review transport fundamentals and review carefully the derivation below. Perry's Handbook also has an extensive discussion of evaporative cooling fundamentals .

Our cooling tower is of the mechanical draft style, specifically the forced-draft type in counterflow arrangement. Water-air contact is facilitated by an array of angled, aluminum fins which are visible through the main walls of the tower. A tray-type distributor at the top of the column distributes water evenly across the width and length of the column, and a fine metal mesh prevents droplets and spray from exiting the column.

## Theory

### Cooling Tower Balance Equations

We derive below the design equations for a countercurrent water cooling tower. True to the connotation of "design," the final result is not a rigorous model, and more so than other design equations the cooling tower takes more than the average number of approximations. In an era when we can handle a complex computational model easily, the cooling tower is a worthwhile engineering case study--it shows us how to make valid assumptions under certain circumstances and simplify a problem. The design equation provides us with a decent approximation of the problem and is a manifestation of our understanding of the underlying key physical and chemical principles.

#### Energy Balance

If we assume that the enthalpy change due to convective mass transport is negligible compared to the more potent latent heat of evaporation, we can write the differential balance for the gas phase enthalpy $$G\textrm{d}h_a$$ as the sum of the gas phase sensible heat transfer $$GC_a\textrm{d}T$$ and latent heat transfer $$G \lambda_w \textrm{d}Y$$, or

$G\textrm{d}h_a = GC_a\textrm{d}T + G \lambda_w \textrm{d}Y,$

where $$G$$ is the superficial air flow rate in units of mass of dry air per cross-sectional area per time, $$C_a$$ is the so-called "humid heat" (see below), $$h_a$$ is the enthalpy of the wet air with in units of energy per mass of dry air, $$\lambda_w$$ is the latent heat of water vaporization at some reference temperature $$T_R$$, and $$Y$$ is the absolute humidity, or

$Y = \frac{\textrm{mass of water}}{\textrm{mass of air}}.$

Absolute humidity is related to the more commonly measured relative humidity ($$RH$$) as

$\textrm{RH} = 100\frac{Y}{Y_s},$

where $$Y_s$$ is the absolute humidity at saturation. Notice that these equations are written on a mass basis; one could equally write them on a molar basis.

#### Humid Air Approximations

The humid heat $$C_a$$ is defined as

$C_a = C_{p,da} + YC_{p,w},$

where subscripts $$a, da$$ and $$w$$ denote wet air, dry air, and water, and $$C_p$$ is the mass-based heat capacity. This quantity describes the heat required to raise the temperature of one mass unit of humid air, and can be understood as the heat needed to raise the temperature of the dry air plus moisture but with units of energy per mass of dry air per degree. Similarly, the humid air enthalpy per unit mass of dry air $$h_a$$ is defined as

$h_a = h_{da} + Yh_w.$

When the pressure is near atmospheric and conditions near saturation, we can approximate $$h_a$$ as

$h_a = C_{p,da}\left(T-T_R\right) + Y\left[\lambda_w + C_{p,w}\left(T-T_R\right)\right],$

where $$T_R$$ is some reference temperature, and heat capacities of pure components are assumed constant. Combining Eq. (4) and (6), the humid air enthalpy becomes

$h_a = C_a\left(T-T_R\right) + \lambda_w Y,$

a form which will prove useful later.

#### Heat and Mass Transfer Coefficients

We can use heat and mass transfer coefficients to describe the sensible and latent heat expressions as follows. The gas-phase sensible heat can be described as

$GC_a \textrm{d}T = h_c \hat{A} \left(T_i - T\right) \textrm{d}z,$

where $$h_c$$ is the convective heat transfer coefficient, $$T$$ is the air temperature, $$T_i$$ is the temperature at the air-water interface, $$\hat{A}$$ is the specific heat transfer area in units of volume per cross-section, and $$z$$ is the axial distance along the tower. Similarly, for low mass flux, the enthalpy due to latent heat transfer can be written as

$G\lambda_w \textrm{d}Y = \lambda_w k_Y \hat{A} \left(Y_i-Y\right) \textrm{d}z,$

where $$k_Y$$ is the mass transfer coefficient in units of mass of dry air per area per time (this is somewhat different from the normal definition) and $$Y_i$$ is the mass ratio at the air-water interface. Thus, the concentration driving force $$\left(Y_i-Y\right)$$ is seen to be the driving force for mass flux, and by extension, latent heat transfer. Note that we've also assumed the specific mass transfer area to be equivalent to the specific heat transfer area, which is only valid if the column packing is completely wetted.

#### Tower Characteristic

If we substitute Eqs. (8) and (9) into the general energy balance (1), we have

$G\textrm{d}h_a = h_c \hat{A} \left(T_i - T\right)\textrm{d}z + \lambda_w k_Y \hat{A}\left(Y_i-Y\right)\textrm{d}z,$

which relates $$\textrm{d}h_a$$ to $$\textrm{d}z$$, but we'll need to consolidate variables $$T$$ and $$Y$$ if we want just one design equation. The psychrometric ratio is dimensionless ratio defined as

$r=\frac{h_c}{k_YC_a} \approx 1$

where the latter (Lewis) approximation is true for the air-water system near atmospheric conditions. Substituting Eq. 11 into 10 and rearranging, we have

$G\textrm{d}h_a = k_Y \hat{A} \left[ \underbrace{\left(C_a T_i + \lambda_w Y_i\right)}_{h_a\left(T_R=0,T=T_i\right)} - \underbrace{\left(C_a T + \lambda_w Y\right)}_{h_a\left(T_R=0, T=T\right)}\right] \textrm{d}z = k_Y \hat{A} \left(h_{a,i} - h_a\right) \textrm{d}z,$

which is our central--and somewhat mysterious-- result. It is important because it forms the basis of our eventual design equation, yet mysterious because it mixes a mass transfer coefficient with an enthalpy driving force. This missing link is the invisible $$r$$, which we numerically set to approximately unity.

### The Merkel Equation

Integrating over the tower height $$Z$$ gives

$Z=\left(\frac{G}{k_Y\hat{A}}\right) \int_{h_{a,\textrm{in}}}^{h_{a,\textrm{out}}} \frac{\textrm{d}h_a}{h_{a,i}-h_a},$

where $$G / k_Y \hat{A}$$ can be interpreted as $$H_{\textrm{TG}}$$, the height of one gas-phase thermal transfer unit, and the integral as $$N_{\textrm{TG}}$$, the number of transfer units.

To simplify the enthalpy integral in Eq. (13), we equate the enthalpy change to the change in water temperature as

$LC_{p,w}\textrm{d}T_w = k_Y \hat{A} \left(h_{a,i} - h_a\right) \textrm{d}z,$

where $$L$$ is the superficial water flow rate in units of mass of water per cross-sectional area per time, and $$T_w$$ is the temperature of the water. Finally, we write the mass transfer based on the overall mass transfer coefficient $$K_y$$ such that

$k_y\left(h_{a,i} - h_a\right) = K_y \left(h_a^* - h_a\right),$

where $$h_a^*$$ is the specific enthalpy of saturated air at the bulk water temperature--importantly, a tabulated value. Substituting Eq. (15) into (12) and performing the change-of-variables implied by (14), we have a new integrated form as

$Z=\left( \frac{L C_{p,w}}{K_Y \hat{A}} \right) \int_{T_{w,\textrm{out}}}^{T_{w,\textrm{in}}} \frac{\textrm{d}T_w}{h_a^* - h_a},$

or as

$\frac{K_Y \hat{A} Z}{L} = \int_{T_{w,\textrm{out}}}^{T_{w,\textrm{in}}} \frac{C_{p,w}\textrm{d}T_w}{h_a^* - h_a},$

which is the Merkel Equation. In American Engineering units, the heat capacity of pure water is conveniently about 1, which is why this equation occasionally appears without $$C_{p,w}$$. The RHS of Eq. (17) is also referred to as the "coefficient of performance" or "tower characteristic" or "NTU" and is represented schematically as the shaded region in Figure 1.

If the LHS of Eqs. (12) and (14) are equated then integrated, we obtain the operating line which describes the change in $$h_a$$ within the column as

$h_a = h_{a,\textrm{in}} + \left(\frac{L C_{p,w}}{G}\right) \left(T_w - T_{w,\textrm{out}}\right),$

which allows numerical integration of Eq. (17). Figure 1. Specific enthalpy of air vs. water temperature plot for cooling tower design. The shaded area is related to the CoP as defined by the Merkel Equation (17).

#### Numerical Integration of the Merkel Equation

Although Perry's Handbook suggests the Tchebycheff approximation, we have considerably more capable integration methods in MATLAB (trapz or quadgk) or Mathematica (NIntegrate). To use these functions you'll need to fit $$h_a^*$$ to some function of $$T_w$$, likely by linear regression analysis. Note that $$h_a^*\left(T_w\right)$$ is best approximated through an exponential relationship, which can still be determined through linear regression analysis if appropriate linearization is performed.

To help you track down the right numbers and perform the integration correctly, Table 1 lists a few entries for $$h_a^*$$ as a function of $$T_w$$ as taken from Perry's Handbook.

Table 1. Sample values for saturated air enthalpy at various temperatures.
$$T_w$$ (°F) $$h_a^*$$ (Btu/lb)
40 15.23
80 43.69
100 71.73
120 119.54
140 205.7

As a test calculation, assume the water inlet and outlet temperatures are 105 °F and 85 °F, the ambient wet bulb temperature is 78 °F, and $$L/G = 0.97$$; for these conditions, the tower characteristic is approximately 1.62.

### Pipe Flow Measurements

Depending on flow velocity the outlet air flow profile may be laminar or profile. Recall that the deciding factor is the Reynolds number,

$\textrm{Re} = \frac{\rho \bar{u} D}{\mu},$

where $$\rho$$ is the fluid density, $$\bar{u}$$ is the average velocity, $$D$$ is the characteristic length (for pipes, usually the diameter), and $$\mu$$ is the fluid viscosity. The flow is generally considered laminar if $$\textrm{Re} < 2300$$ and turbulent if $$\textrm{Re} > 4000$$. The volumetric flow rate $$Q$$ is related to the average velocity as

$Q = \bar{u}A,$

where $$A$$ is the cross-sectional area of the pipe. In the lab we can most accurately measure the maximum velocity $$u_{max} (= u_c)$$ because we can place an anemometer at the center of the pipe. For the case of laminar flow the Navier-Stokes equations may be solved exactly to identify an important relationship between the two variables; namely,

$u_{max} = u_c = 2\bar{u},$

such that

$Q = \frac{1}{2}u_c A.$

For the case of turbulent flow the empirical power-law velocity profile

$\frac{u}{u_c} = \left(1-\frac{r}{R}\right)^{1/n},$

is often useful, where $$u_c$$ is the velocity along the centerline, $$R$$ is the pipe radius, and $$n$$ is an empirical parameter that typically varies between 6 and 10. Integrating across the mass flux over the pipe area for constant density yields

$Q = \frac{2n^2}{\left(2n+1\right)\left(n+1\right)} \pi R^2 u_c.$

For example, with a typical value of $$n$$ such as $$n = 8$$, the volumetric flow rate for turbulent flow is

$Q = \frac{128}{153} u_c A.$

Note that this information is not necessary for your Theory section but could be included in an appendix if desired.

## Standard Operating Procedure

### Safety

• If left unattended with a high set-point, the water bath can become quite hot. Use caution when hot water is present.
• Don't go poking around in the electronics or wire boxes if you like your current heart rate.
• The water reservoir is known to be somewhat leaky. There are several rags nearby to absorb excess water; if these are insufficient then use the mop and notify a TA or instructor of excessive water leakage.
• There are no hazardous chemicals used in this process.

### Tutorial Video

Check out the video below for a brief overview of this experiment, then carefully read the instructions below. Keep in mind that there may be variations in the equipment or protocol when you're in the lab compared to when this video was created. Ask a TA or instructor if you're not clear on something.

### Preparation

1. Turn on the main controller and verify that each of the six thermocouples are functional by changing the knob on the temperature display. Trace the thermocouples, which correspond to
1. Outlet water temperature,
2. Inlet water temperature,
3. Outlet air dry-bulb temperature,
4. Outlet air wet-bulb temperature,
5. Inlet air wet-bulb temperature, and
6. Inlet air dry-bulb temperature.
2. Check the inlet air wet-bulb water reservoir located inside the bottom portion of the Cooling Tower. Using the flashlight to look through the window, locate the reservoir (graduated cylinder) and refill, if empty, to keep the bulb saturated.
3. Check the outlet air wet-bulb water reservoir and re-fill, if empty, to keep the bulb saturated.
4. Check the main water reservoir to ensure the heating elements are completely submerged. Fill with deionized (DI) water to about 1 cm over the heating elements if necessary.
5. Fill the graduated cylinder to a known volume with DI water; this cylinder allows measurement of water loss.
6. The system can be leaky. Use fabric cloths or the mop to keep the area dry.

### Basic Operation

Operation Note 1: The hot water bath is a substitute for a real process unit, such as a water-cooled CSTR, which produces a hot water stream that needs to be cooled by cooling tower. A properly controlled reactor operating at steady state should produce an outlet coolant water temperature which the cooling tower must be able to handle. We use the controller on the hot water bath to approximate such as a system as closely as possible such that the hot water inlet to the tower (i.e., the water to be cooled) is nearly constant.

Operation Note 2: The heater can only be turned on (2000 W) or off (0 W) by the controller; the power can't be varied continuously between these bounds. To regulate the bath temperature, the controller instead uses pulse width modulation to vary the amount of time the heater is on as a fraction of some pre-defined period (about 500 ms); this ratio is called the duty cycle. You can see when the heater is on or off by monitoring the blinking green light on the controller: when the light is on the heater is on, and vice versa.

1. Use your engineering judgment to determine the system settings which will produce the minimum and maximum operating bath temperatures ($$T_{min}$$ and $$T_{max}$$).
1. For example, for $$T_{min}$$ should the fan be high or low? Should the water flow rate be high or low? Should the heater be on or off?
1. To turn the heater "off" (i.e., green indicator always off) , choose a very low set point (0 deg C).
2. To turn the heater "on" (i.e., steady green indicator), choose a very high set point (80 deg C).
2. Important: The variac fan controller should not be set above 115, nor the water flow rate below 0.5 gpm.
2. Set the hot water bath controller set point about half way between $$T_{min}$$ and $$T_{max}$$; this should ensure an approximately constant water bath temperature for all subsequent experiments.
3. Choose a liquid and gas flow rate, let the system reach steady state, and record all relevant data to allow calculation of the tower characteristic. Some experiments you might try (you will not have time to try them all):
1. The Pre-lab questions had you predict the effect of $$L/G$$ on the tower characteristic; determine this relationship experimentally and compare to theoretical predictions.
2. It's possible to hold $$L/G$$ constant for different $$L$$ or $$G$$. Does the Merkel equation predict any changes in the tower characteristic? Evaluate this hypothesis.
3. Arbitrarily pick one set of parameters as the "standard" set, evaluate the tower characteristic, and assume that the tower characteristic is constant at this value. Use the Merkel equation to predict the outlet water temperature for different inlet water temperatures and evaluate experimentally.
4. Check the water mass balance: how much water is leaving the system as expected and how much is leaking?

### Shutdown

1. Turn off all system components.
2. Use a towel and mop to clean up any wet areas.

## Common Report Mistakes and Suggestions

The evaluator(s) for this experiment was asked to list three or more common mistakes often made on the reports for this experiment, or to provide three or more suggestions that could improve the quality of reports for this experiment. The responses were as follows:

1. Improper formatting of units (use \SI{}{}), variables (use math environments), and cross references (use \cref{}).
2. Poor paragraph structure; use topic sentence structure to improve writing and make a cohesive story.
3. Low-res or difficult to read plots and figures (use MATLAB to make good plots as described in the wiki).
4. Regression and prediction intervals sometimes appear on plots for which there is no theoretical or empirical equation to be fitted.
5. Don't refer to "area ABCD" as somehow determining or explaining the tower characteristic.
6. The water flows through the column primarily under the influence of gravity and therefore changing the water flow rate has negligible impact on the residence time of a droplet.
7. Don't report an uncertainty on the tower characteristic unless replicate measurements were made (we don't propagate uncertainty through numerical integrations like quadgk() in this course).
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