# Predictive Modeling of a Buoyancy-Operated Cooling Tower under Unsaturated Conditions: Adjoint Sensitivity Model and Optimal Best-Estimate Results with Reduced Predicted Uncertainties

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## Abstract

**:**

## 1. Introduction

## 2. Mathematical Model of a Mechanical Draft Counter-Flow Cooling Tower Operating under Unsaturated Conditions

- air and water stream temperatures are uniform at any cross section;
- the cross-sectional area of the cooling tower is assumed to be uniform;
- the heat and mass transfer only occur in the direction normal to flows;
- the heat and mass transfer through tower walls to the environment is neglected;
- the heat transfer from the cooling tower fan and motor assembly to the air is neglected;
- the air and water vapor is considered a mixture of ideal gasses;
- the flow between flat plates is unsaturated through the fill section.

- the water mass flow rates, denoted as ${m}_{w}^{(i)}\hspace{0.17em}(i=2,\mathrm{...},50)$, at the exit of each control volume, i, along the height of the fill section of the cooling tower;
- the water temperatures, denoted as ${T}_{w}^{(i)}\hspace{0.17em}(i=2,\mathrm{...},50)$, at the exit of each control volume, i, along the height of the fill section of the cooling tower;
- the air temperatures, denoted as ${T}_{a}^{(i)}\hspace{0.17em}(i=1,\mathrm{...},49)$, at the exit of each control volume, i, along the height of the fill section of the cooling tower;
- the humidity ratios, denoted as ${\omega}^{(i)}\hspace{0.17em}\hspace{0.17em}(i=1,\mathrm{...},49)$, at the exit of each control volume, i, along the height of the fill section of the cooling tower.
- the air mass flow rates, denoted as ${m}_{a}$, constant along the height of the fill section of the cooling tower.

- Liquid Continuity Equations:
- (i)
- Control Volume i = 1:$${N}_{1}^{(1)}\left({m}_{w},{T}_{w},{T}_{a},\mathsf{\omega},{m}_{a};\mathsf{\alpha}\right)\triangleq {m}_{w}^{(2)}-{m}_{w,in}+\frac{M({m}_{a},\mathsf{\alpha})}{\overline{R}}\left[\frac{{P}_{vs}^{(2)}({T}_{w}^{(2)},\mathsf{\alpha})}{{T}_{w}^{(2)}}-\frac{{\omega}^{(1)}{P}_{atm}}{{T}_{a}^{(1)}(0.622+{\omega}^{(1)})}\right]=0;$$
- (ii)
- Control Volumes i = 2,..., I − 1:$${N}_{1}^{\left(i\right)}\left({m}_{w},{T}_{w},{T}_{a},\mathsf{\omega},{m}_{a};\mathsf{\alpha}\right)\triangleq {m}_{w}^{(i+1)}-{m}_{w}^{(i)}+\frac{M({m}_{a},\mathsf{\alpha})}{\overline{R}}\left[\frac{{P}_{vs}^{(i+1)}({T}_{w}^{(i+1)},\mathsf{\alpha})}{{T}_{w}^{(i+1)}}-\frac{{\omega}^{(i)}{P}_{atm}}{{T}_{a}^{(i)}(0.622+{\omega}^{(i)})}\right]=0;$$
- (iii)
- Control Volume i = I:$${N}_{1}^{(I)}\left({m}_{w},{T}_{w},{T}_{a},\mathsf{\omega},{m}_{a};\mathsf{\alpha}\right)\triangleq {m}_{w}^{(I+1)}-{m}_{w}^{(I)}+\frac{M({m}_{a},\mathsf{\alpha})}{\overline{R}}\left[\frac{{P}_{vs}^{(I+1)}({T}_{w}^{(I+1)},\mathsf{\alpha})}{{T}_{w}^{(I+1)}}-\frac{{\omega}^{(I)}{P}_{atm}}{{T}_{a}^{(I)}(0.622+{\omega}^{(I)})}\right]=0;$$

- Liquid Energy Balance Equations:
- (i)
- Control Volume i = 1:$$\begin{array}{l}{N}_{2}^{(1)}\left({m}_{w},{T}_{w},{T}_{a},\mathsf{\omega},{m}_{a};\mathsf{\alpha}\right)\triangleq {m}_{w,in}{h}_{f}({T}_{w,in},\mathsf{\alpha})-({T}_{w}^{(2)}-{T}_{a}^{(1)})H({m}_{a},\mathsf{\alpha})\\ -{m}_{w}^{(2)}{h}_{f}^{(2)}({T}_{w}^{(2)},\mathsf{\alpha})-({m}_{w,in}-{m}_{w}^{(2)}){h}_{g,w}^{(2)}({T}_{w}^{(2)},\mathsf{\alpha})=0;\end{array}$$
- (ii)
- Control Volumes i = 2,..., I − 1:$$\begin{array}{l}{N}_{2}^{\left(i\right)}\left({m}_{w},{T}_{w},{T}_{a},\mathsf{\omega},{m}_{a};\mathsf{\alpha}\right)\triangleq {m}_{w}^{(i)}{h}_{f}^{(i)}({T}_{w}^{(i)},\mathsf{\alpha})-({T}_{w}^{(i+1)}-{T}_{a}^{(i)})H({m}_{a},\mathsf{\alpha})\\ \hspace{0.17em}-{m}_{w}^{(i+1)}{h}_{f}^{(i+1)}({T}_{w}^{(i+1)},\mathsf{\alpha})-({m}_{w}^{(i)}-{m}_{w}^{(i+1)}){h}_{g,w}^{(i+1)}({T}_{w}^{(i+1)},\mathsf{\alpha})=0;\end{array}$$
- (iii)
- Control Volume i = I:$$\begin{array}{l}{N}_{2}^{(I)}\left({m}_{w},{T}_{w},{T}_{a},\mathsf{\omega},{m}_{a};\mathsf{\alpha}\right)\triangleq {m}_{w}^{(I)}{h}_{f}^{(I)}({T}_{w}^{(I)},\mathsf{\alpha})-({T}_{w}^{(I+1)}-{T}_{a}^{(I)})H({m}_{a},\mathsf{\alpha})\\ \hspace{0.17em}-{m}_{w}^{(I+1)}{h}_{f}^{(I+1)}({T}_{w}^{(I+1)},\mathsf{\alpha})-({m}_{w}^{(I)}-{m}_{w}^{(I+1)}){h}_{g,w}^{(I+1)}({T}_{w}^{(I+1)},\mathsf{\alpha})\hspace{0.17em}=0;\hspace{0.17em}\end{array}$$

- Water Vapor Continuity Equations:
- (i)
- Control Volume i = 1:$${N}_{3}^{(1)}\left({m}_{w},{T}_{w},{T}_{a},\mathsf{\omega},{m}_{a};\mathsf{\alpha}\right)\triangleq {\omega}^{(2)}-{\omega}^{(1)}+\frac{{m}_{w.in}-{m}_{w}^{(2)}}{\left|{m}_{a}\right|}=0;$$
- (ii)
- Control Volumes i = 2,..., I − 1:$${N}_{3}^{\left(i\right)}\left({m}_{w},{T}_{w},{T}_{a},\mathsf{\omega},{m}_{a};\mathsf{\alpha}\right)\triangleq {\omega}^{(i+1)}-{\omega}^{(i)}+\frac{{m}_{w}^{(i)}-{m}_{w}^{(i+1)}}{\left|{m}_{a}\right|}=0\hspace{0.17em};$$
- (iii)
- Control Volume i = I:$${N}_{3}^{(I)}\left({m}_{w},{T}_{w},{T}_{a},\mathsf{\omega},{m}_{a};\mathsf{\alpha}\right)\triangleq {\omega}_{in}-{\omega}^{(I)}+\frac{{m}_{w}^{(I)}-{m}_{w}^{(I+1)}}{\left|{m}_{a}\right|}=0\hspace{0.17em};$$

- The Air/Water Vapor Energy Balance Equations:
- (i)
- Control Volume i = 1:$$\begin{array}{l}{N}_{4}^{(1)}\left({m}_{w},{T}_{w},{T}_{a},\mathsf{\omega},{m}_{a};\mathsf{\alpha}\right)\triangleq ({T}_{a}^{(2)}-{T}_{a}^{(1)}){C}_{p}^{(1)}(\frac{{T}_{a}^{(1)}+273.15}{2},\mathsf{\alpha})-{\omega}^{(1)}{h}_{g,a}^{(1)}({T}_{a}^{(1)},\mathsf{\alpha})\\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}+\frac{({T}_{w}^{(2)}-{T}_{a}^{(1)})H({m}_{a},\mathsf{\alpha})}{\left|{m}_{a}\right|}+\frac{({m}_{w,in}-{m}_{w}^{(2)}){h}_{g,w}^{(2)}({T}_{w}^{(2)},\mathsf{\alpha})}{\left|{m}_{a}\right|}+{\omega}^{(2)}{h}_{g,a}^{(2)}({T}_{a}^{(2)},\mathsf{\alpha})=0\hspace{0.17em};\end{array}$$
- (ii)
- Control Volumes i = 2,..., I − 1:$$\begin{array}{l}{N}_{4}^{(i)}\left({m}_{w},{T}_{w},{T}_{a},\mathsf{\omega},{m}_{a};\mathsf{\alpha}\right)\triangleq ({T}_{a}^{(i+1)}-{T}_{a}^{(i)}){C}_{p}^{(i)}(\frac{{T}_{a}^{(i)}+273.15}{2},\mathsf{\alpha})-{\omega}^{(i)}{h}_{g,a}^{(i)}({T}_{a}^{(i)},\mathsf{\alpha})\\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}+\frac{({T}_{w}^{(i+1)}-{T}_{a}^{(i)})H({m}_{a},\mathsf{\alpha})}{\left|{m}_{a}\right|}+\frac{({m}_{w}^{(i)}-{m}_{w}^{(i+1)}){h}_{g,w}^{(i+1)}({T}_{w}^{(i+1)},\mathsf{\alpha})}{\left|{m}_{a}\right|}+{\omega}^{(i+1)}{h}_{g,a}^{(i+1)}({T}_{a}^{(i+1)},\mathsf{\alpha})=0\hspace{0.17em};\end{array}$$
- (iii)
- Control Volume i = I:$$\begin{array}{l}{N}_{4}^{(I)}\left({m}_{w},{T}_{w},{T}_{a},\mathsf{\omega},{m}_{a};\mathsf{\alpha}\right)\triangleq ({T}_{a,in}-{T}_{a}^{(I)}){C}_{p}{}^{(I)}(\frac{{T}_{a}^{(I)}+273.15}{2},\mathsf{\alpha})-{\omega}^{(I)}{h}_{g,a}^{(I)}({T}_{a}^{(I)},\mathsf{\alpha})\\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}+\frac{({T}_{w}^{(I+1)}-{T}_{a}^{(I)})H({m}_{a},\mathsf{\alpha})}{\left|{m}_{a}\right|}+\frac{({m}_{w}^{(I)}-{m}_{w}^{(I+1)}){h}_{g,w}^{(I+1)}({T}_{w}^{(I+1)},\mathsf{\alpha})}{\left|{m}_{a}\right|}+{\omega}_{in}{h}_{g,a}({T}_{a,in},\mathsf{\alpha})=0\hspace{0.17em}.\end{array}$$

- The Mechanical Energy Balance Equation:$$\begin{array}{l}{N}_{5}\left({m}_{w},{T}_{w},\mathsf{\omega},{T}_{a},{m}_{a};\mathsf{\alpha}\right)\triangleq \\ \left[\frac{1}{2\mathsf{\rho}({T}_{db},\mathsf{\alpha})}\left(\frac{1}{{A}_{out}{(\mathsf{\alpha})}^{2}}-\frac{1}{{A}_{in}{(\mathsf{\alpha})}^{2}}+\frac{{k}_{sum}}{{A}_{fill}{}^{2}}\right)+\frac{f}{2\mathsf{\rho}({T}_{tdb},\mathsf{\alpha})}\frac{96}{{R}_{e}({m}_{a},\mathsf{\alpha})}\frac{{L}_{fill}(\mathsf{\alpha})}{{A}_{fill}{}^{2}{D}_{h}}\right]\left|{m}_{a}\right|{m}_{a}\\ -gZ(\mathsf{\alpha})\mathsf{\rho}({T}_{db},\mathsf{\alpha})-\frac{{V}_{w}{}^{2}\mathsf{\rho}({T}_{db},\mathsf{\alpha})}{2}+\Delta {z}_{rain}g\mathsf{\rho}({T}_{db},\mathsf{\alpha})+g\mathsf{\rho}({T}_{a}^{(1)},\mathsf{\alpha})\Delta {z}_{4-2}(\mathsf{\alpha})\\ +g\Delta z(\mathsf{\alpha})\frac{{P}_{atm}}{{R}_{air}}\left[\frac{1}{2{T}_{a,in}}+\frac{1}{2{T}_{a}^{(1)}}+{\displaystyle \sum _{i=2}^{I}\frac{1}{{T}_{a}^{(i)}}}\right]=0;\end{array}$$

- (a)
- the vector ${m}_{w}\triangleq {\left[{m}_{w}^{(2)},\mathrm{...},{m}_{w}^{(I+1)}\right]}^{\u2020}$ of water mass flow rates at the exit of each control volume i, $(i=1,\mathrm{...},49)$;
- (b)
- the vector ${T}_{w}\triangleq {\left[{T}_{w}^{(2)},\mathrm{...},{T}_{w}^{(I+1)}\right]}^{\u2020}$ of water temperatures at the exit of each control volume i, $(i=1,\mathrm{...},49)$;
- (c)
- the vector ${T}_{a}\triangleq {\left[{T}_{a}^{(1)},\mathrm{...},{T}_{a}^{(I)}\right]}^{\u2020}$ of air temperatures at the exit of each control volume i, $(i=1,\mathrm{...},49)$;
- (d)
- the vector $RH\triangleq {\left[R{H}^{(1)},\mathrm{...},R{H}^{(I)}\right]}^{\u2020}$, having as components the air relative humidity at the exit of each control volume i, $(i=1,\mathrm{...},49)$;
- (e)
- the scalar ${m}_{a}$, representing the air mass flow rate along the height of the cooling tower the value of the air mass flowrate.

## 3. Development of the Cooling Tower Adjoint Sensitivity Model

- considering an arbitrarily small perturbation $\delta {\alpha}_{j}$ to the model parameter ${\alpha}_{j}$;
- re-computing the perturbed response $R\left({\alpha}_{j}^{0}+\delta {\alpha}_{j}\right)$, where ${\alpha}_{j}^{0}$ denotes the unperturbed parameter value;
- using the finite difference formula$${S}_{j}^{FD}\cong \frac{R\left({\alpha}_{j}^{0}+\delta {\alpha}_{j}\right)-R\left({\alpha}_{j}^{0}\right)}{\delta {\alpha}_{j}}+O{\left(\delta {\alpha}_{j}\right)}^{2}$$
- using the approximate equality between Equations (27) and (28) to obtain independently the respective values of the adjoint function(s) being verified.

#### 3.1. Sensitivity Analysis Results and Rankings

#### 3.1.1. Sensitivity Analysis Results and Rankings for the Outlet Air Temperature, ${T}_{a}^{(1)}$

#### 3.1.2. Sensitivity Analysis Results and Rankings for the Outlet Water Temperature, ${T}_{w}^{(50)}$

#### 3.1.3. Sensitivity Analysis Results and Rankings for the Outlet Water Mass Flow Rate, ${m}_{w}^{\left(50\right)}$

#### 3.1.4. Sensitivity Analysis Results and Rankings for the Outlet Air Relative Humidity, $R{H}^{(1)}$

#### 3.1.5. Relative Sensitivities of the Air Mass Flow Rate, ${m}_{a}$

#### 3.2. Cross-Comparison of Sensitivity Results

**,**both the unsaturated case and subcase I are most sensitive to the parameter ${T}_{w,in}$, whereas subcase II is most sensitive to the parameter ${T}_{a,in}$. As a comparison, the response of water outlet temperature to the parameter ${T}_{w,in}$ ranks in 3rd place, with a value comparable to the other two cases. The next two most sensitive parameters that rank from 2nd to 3rd places of this response are also different between the operating conditions: for both the unsaturated case and subcase I, parameters ${T}_{a,in}$ and ${T}_{db}$ rank in 2nd and 3rd places, respectively; however, for subcase II, parameters that take the 2nd and 3rd places are ${T}_{db}$ and ${T}_{w,in}$, respectively. The parameters that take the 4th and 5th places are also different between the operating conditions, as shown in the table. Overall, for the response of water outlet temperature, ${T}_{w}^{(50)}$, the sensitivity behavior of subcase I is more similar to that of the unsaturated case.

#### 3.3. Experimental Data Assimilation, Model Calibration and Best-Estimate Predicted Results with Reduced Predicted Uncertainties

#### 3.3.1. Model Calibration: Predicted Best-Estimated Parameter Values with Reduced Predicted Standard Deviations

#### 3.3.2. Predicted Best-Estimated Response Values with Reduced Predicted Standard Deviations

## 4. Discussion

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Appendix A. Statistical Analysis of Experimentally Measured Responses for SRNL F-Area Cooling Towers

**Figure A2.**Histogram plot of the measured air outlet relative humidity, within the 6717 data sets collected by SRNL from F-Area cooling towers.

Minimum | Maximum | Range | Mean | Std. Dev. | Variance | Skewness | Kurtosis |
---|---|---|---|---|---|---|---|

38.2 | 104.1 | 65.9 | 89.61 | 13.63 | 185.72 | −1.01 | 3.22 |

**Figure A3.**Histogram plot of the air outlet temperature measured using “Tidbit” sensors, within the 6717 data sets collected by SRNL from F-Area cooling towers.

**Figure A4.**Histogram plot of the air outlet temperature measured using “Hobo” sensors, within the 6717 data sets collected by SRNL from F-Area cooling towers.

**Figure A5.**Histogram plot of water outlet temperature measurements, within the 7688 data sets collected by SRNL from F-Area cooling towers.

**Table A2.**Statistics of the air outlet temperature distribution [K], measured using “Tidbit” sensors.

Minimum | Maximum | Range | Mean | Std. Dev. | Variance | Skewness | Kurtosis |
---|---|---|---|---|---|---|---|

292.94 | 309.52 | 16.58 | 299.21 | 2.92 | 8.55 | 0.59 | 2.71 |

Minimum | Maximum | Range | Mean | Std. Dev. | Variance | Skewness | Kurtosis |
---|---|---|---|---|---|---|---|

292.93 | 308.90 | 15.97 | 299.00 | 2.77 | 7.68 | 0.58 | 2.75 |

Minimum | Maximum | Range | Mean | Std. Dev. | Variance | Skewness | Kurtosis |
---|---|---|---|---|---|---|---|

293.08 | 301.70 | 8.62 | 298.10 | 1.39 | 1.94 | −0.51 | 3.31 |

Minimum | Maximum | Range | Mean | Std. Dev. | Variance | Skewness | Kurtosis |
---|---|---|---|---|---|---|---|

292.93 | 309.10 | 16.17 | 299.11 | 2.84 | 8.09 | 0.58 | 2.71 |

## Appendix B. Model Parameters for the SRNL F-Area Cooling Towers

Index $\mathit{i}$ of ${\mathit{\alpha}}_{\mathit{i}}$ | Independent Scalar Parameters | C++ String | Math. Notation | Nominal Value(s) | Absolute Std. Dev. | Rel. Std. Dev. (%) |

1 | Air temperature (dry bulb) (K) | tdb | ${T}_{db}$ | 298.882 | 4.034 | 1.35 |

2 | Dew point temperature (K) | tdp | ${T}_{dp}$ | 292.077 | 2.287 | 0.78 |

3 | Inlet water temperature (K) | twin | ${T}_{w,in}$ | 298.893 | 1.687 | 0.56 |

4 | Atmospheric pressure (Pa) | patm | ${P}_{atm}$ | 100,588 | 408.26 | 0.41 |

5 | Wind speed (m/s) | wspd | ${V}_{w}$ | 1.859 | 0.941 | 50.7 |

6 | Sum of loss coefficients above fill | ksum | ${k}_{sum}$ | 10 | 5 | 50 |

7 | Dynamic viscosity of air at T = 300 K (kg/m·s) | muair | $\mu $ | 1.983 × 10^{−5} | 9.676 × 10^{−7} | 4.88 |

8 | Kinematic viscosity of air at T = 300 K (m^{2}/s) | nuair | $\mathsf{\nu}$ | 1.568 × 10^{−5} | 1.895 × 10^{−6} | 12.09 |

9 | Thermal conductivity of air at T = 300 K (W/m·K) | tcair | ${k}_{air}$ | 0.02624 | 1.584 × 10^{−3} | 6.04 |

10 | Heat transfer coefficient multiplier | mlthtc | ${f}_{ht}$ | 1 | 0.5 | 50 |

11 | Mass transfer coefficient multiplier | mltmtc | ${f}_{mt}$ | 1 | 0.5 | 50 |

12 | Fill section frictional loss multiplier | mltfil | $f$ | 4 | 2 | 50 |

13 | P_{vs}(T) parameters | a0 | ${a}_{0}$ | 25.5943 | 0.01 | 0.04 |

14 | a1 | ${a}_{1}$ | −5229.89 | 4.4 | 0.08 | |

15 | C_{pa}(T) parameters | A(1) | ${a}_{0,cpa}$ | 1030.5 | 0.2940 | 0.03 |

16 | A(2) | ${a}_{1,cpa}$ | −0.19975 | 0.0020 | 1.00 | |

17 | A(3) | ${a}_{2,cpa}$ | 3.9734 × 10^{−4} | 3.345 × 10^{−6} | 0.84 | |

18 | D_{av}(T) parameters | A(1) | ${a}_{0,dav}$ | 7.06085 × 10^{−9} | 0 | 0 |

19 | A(2) | ${a}_{1,dav}$ | 2.65322 | 0.003 | 0.11 | |

20 | A(3) | ${a}_{2,dav}$ | −6.1681 × 10^{−3} | 2.3 × 10^{−5} | 0.37 | |

21 | A(4) | ${a}_{3,dav}$ | 6.55266 × 10^{−6} | 3.8 × 10^{−8} | 0.58 | |

22 | h_{f}(T) parameters | a0f | ${a}_{0f}$ | −1,143,423.78 | 543 | 0.05 |

23 | a1f | ${a}_{1f}$ | 4186.50768 | 1.8 | 0.04 | |

24 | h_{g}(T) parameters | a0g | ${a}_{0g}$ | 2,005,743.99 | 1046 | 0.05 |

25 | a1g | ${a}_{1g}$ | 1815.437 | 3.5 | 0.19 | |

26 | Nu parameters | - | ${a}_{0,Nu}$ | 8.235 | 2.059 | 25 |

27 | - | ${a}_{1,Nu}$ | 0.00314987 | 0.001 | 31.75 | |

28 | - | ${a}_{2,Nu}$ | 0.9902987 | 0.327 | 33.02 | |

29 | - | ${a}_{3,Nu}$ | 0.023 | 0.0088 | 38.26 | |

30 | Cooling tower deck width in x-dir. (m) | dkxw | ${W}_{dkx}$ | 8.5 | 0.085 | 1 |

31 | Cooling tower deck width in y-dir. (m) | dkyw | ${W}_{dky}$ | 8.5 | 0.085 | 1 |

32 | Cooling tower deck height above ground (m) | dkht | $\Delta {z}_{dk}$ | 10 | 0.1 | 1 |

33 | Fan shroud height (m) | fsht | $\Delta {z}_{fan}$ | 3.0 | 0.03 | 1 |

34 | Fan shroud inner diameter (m) | fsid | ${D}_{fan}$ | 4.1 | 0.041 | 1 |

35 | Fill section height (m) | flht | $\Delta {z}_{fill}$ | 2.013 | 0.02013 | 1 |

36 | Rain section height (m) | rsht | $\Delta {z}_{rain}$ | 1.633 | 0.01633 | 1 |

37 | Basin section height (m) | bsht | $\Delta {z}_{bs}$ | 1.168 | 0.01168 | 1 |

38 | Drift eliminator thickness (m) | detk | $\Delta {z}_{de}$ | 0.1524 | 0.001524 | 1 |

39 | Fill section equivalent diameter (m) | deqv | ${D}_{h}$ | 0.0381 | 0.000381 | 1 |

40 | Fill section flow area (m^{2}) | flfa | ${A}_{fill}$ | 67.29 | 6.729 | 10 |

41 | Fill section surface area (m^{2}) | flsa | ${A}_{surf}$ | 14,221 | 3555.3 | 25 |

42 | Prandlt number of air at T = 80 °C | Pr | ${\mathrm{P}}_{r}$ | 0.708 | 0.005 | 0.71 |

43 | Wetted fraction of fill surface area | wtsa | ${w}_{tsa}$ | 1 | 0 | 0 |

Index $\mathit{i}$ of ${\alpha}_{i}$ | Boundary Parameters | C++ String | Math. Notation | Nominal Value | Absolute Std. Dev. | Rel. Std. Dev. (%) |

44 | Inlet water mass flowrate (kg/s) | mfwin | ${m}_{w,in}$ | 44.0193 | 2.201 | 5 |

45 | Inlet air temperature (K) | tain | ${T}_{a,in}$ | set to ${T}_{db}$ | 4.034 | 1.35 |

46 | Inlet air humidity ratio (Dependent Scalar Parameter) | hrin | $\begin{array}{l}{\omega}_{in};\\ {\omega}_{rain}\end{array}$ | 0.01379 | 0.00192 | 13.80 |

Index $\mathit{i}$ of ${\alpha}_{i}$ | Special Dependent Parameters | C++ String | Math. Notation | Nominal Value | Absolute Std. Dev. | Rel. Std. Dev. (%) |

47 | Schmidt number | Sc | $Sc$ | 0.5999 | 0.0159 | 2.66 |

Dependent Scalar Parameters | Math. Notation | Defining Equation or Correlation |
---|---|---|

Mass diffusivity of water vapor in air (m^{2}/s) | ${D}_{av}({T}_{a},\mathsf{\alpha})$ | $\frac{{a}_{0,dav}{T}^{1.5}}{{a}_{1,dav}+({a}_{2,dav}+{a}_{3,dav}T)T}$ |

Heat transfer coefficient (W/m^{2}·K) | $h(\mathsf{\alpha})$ | $\frac{{f}_{ht}{N}_{u}{k}_{air}}{{D}_{h}}$ |

Mass transfer coefficient (m/s) | ${k}_{m}(\mathsf{\alpha})$ | $\frac{{f}_{mt}{S}_{h}{D}_{av}({T}_{db},\mathsf{\alpha})}{{D}_{h}}$ |

Heat transfer term (W/K) | $H({m}_{a},\mathsf{\alpha})$ | $h\left(\mathsf{\alpha}\right){w}_{tsa}{A}_{ff}$ |

Mass transfer term (m^{3}/s) | $M({m}_{a},\mathsf{\alpha})$ | ${M}_{{H}_{2}O}{k}_{m}\left(\mathsf{\alpha}\right){w}_{tsa}{A}_{ff}$ |

Density of dry air (kg/m^{3}) | $\mathsf{\rho}(\mathsf{\alpha})$ | $\frac{{P}_{atm}}{{R}_{air}{T}_{db}}$ |

Air velocity in the fill section (m/s) | ${v}_{a}({m}_{a},\mathsf{\alpha})$ | $\frac{\left|{m}_{a}\right|}{\mathsf{\rho}(\mathsf{\alpha}){A}_{fill}}$ |

Fill falling-film surface area per vertical section (m^{2}) | ${A}_{ff}$ | $\frac{{A}_{surf}}{I}$ |

Rain section inlet flow area (m^{2}) | ${A}_{in}$ | ${W}_{dkx}{W}_{dky}$ |

Height for natural convection (m) | $Z$ | $\Delta {z}_{dk}+\Delta {z}_{fan}-\Delta {z}_{bs}$ |

Height above fill section (m) | $\Delta {z}_{4-2}$ | $Z-\Delta {z}_{fill}-\Delta {z}_{rain}$ |

Fill section control volume height (m) | $\Delta z$ | $\frac{\Delta {z}_{fill}}{I}$ |

Fill section length, including drift eliminator (m) | ${L}_{fill}$ | $\Delta {z}_{fill}+\Delta {z}_{de}$ |

Fan shroud inner radius (m) | ${r}_{fan}$ | $0.5{D}_{fan}$ |

Fan shroud flow area (m^{2}) | ${A}_{out}$ | $\mathsf{\pi}\hspace{0.17em}{r}_{fan}{}^{2}$ |

Thermal Properties (Functions of State Variables) | Math. Notation | Defining Equation or Correlation |
---|---|---|

h_{f}(T_{w}) = saturated liquid enthalpy (J/kg) | ${h}_{f}({T}_{w},\mathsf{\alpha})$ | ${a}_{0f}+{a}_{1f}{T}_{w}$ |

H_{g}(T_{w}) = saturated vapor enthalpy (J/kg) | ${h}_{g,w}({T}_{w},\mathsf{\alpha})$ | ${a}_{0g}+{a}_{1g}{T}_{w}$ |

H_{g}(T_{a}) = saturated vapor enthalpy (J/kg) | ${h}_{g,a}({T}_{a},\mathsf{\alpha})$ | ${a}_{0g}+{a}_{1g}{T}_{a}$ |

C_{p}(T) = specific heat of dry air (J/kg K) | ${C}_{p}(T,\mathsf{\alpha})$ | ${a}_{0,cpa}+({a}_{1,cpa}+{a}_{2,cpa}T)T$ |

P_{vs}(T_{w}) = saturation pressure (Pa) | ${P}_{vs}({T}_{w},\mathsf{\alpha})$ | ${P}_{c}\cdot {e}^{{a}_{0}+\frac{{a}_{1}}{{T}_{w}}},\hspace{0.17em}in\hspace{0.17em}which\hspace{0.17em}{P}_{c}=1.0\hspace{0.17em}Pa$ |

P_{vs}(T_{a}) = saturation pressure (Pa) | ${P}_{vs}({T}_{a},\mathsf{\alpha})$ | ${P}_{c}\cdot {e}^{{a}_{0}+\frac{{a}_{1}}{{T}_{a}}},\hspace{0.17em}in\hspace{0.17em}which\hspace{0.17em}{P}_{c}=1.0\hspace{0.17em}Pa$ |

**Figure B1.**Histogram plot of dry-bulb air temperature data collected by SRNL from F-Area cooling towers.

**Figure B2.**Histogram plot of dew-point air temperature data collected by SRNL from F-Area cooling towers.

**Figure B3.**Histogram plot of inlet water temperature data collected by SRNL from F-Area cooling towers.

**Figure B4.**Histogram plot of atmospheric pressure data collected by SRNL from F-Area cooling towers.

Minimum | Maximum | Range | Mean | Std. Dev. | Variance | Skewness | Kurtosis |
---|---|---|---|---|---|---|---|

289.50 | 309.91 | 20.41 | 298.88 | 4.03 | 16.27 | 0.36 | 2.38 |

Minimum | Maximum | Range | Mean | Std. Dev. | Variance | Skewness | Kurtosis |
---|---|---|---|---|---|---|---|

282.58 | 298.06 | 15.48 | 292.08 | 2.29 | 5.23 | −0.66 | 3.11 |

Minimum | Maximum | Range | Mean | Std. Dev. | Variance | Skewness | Kurtosis |
---|---|---|---|---|---|---|---|

293.93 | 303.39 | 9.46 | 298.89 | 1.69 | 2.85 | −0.16 | 2.91 |

Minimum | Maximum | Range | Mean | Std. Dev. | Variance | Skewness | Kurtosis |
---|---|---|---|---|---|---|---|

99617 | 101,677 | 2060 | 100588 | 408.6 | 166,678 | 0.079 | 2.57 |

Minimum | Maximum | Range | Mean | Std. Dev. | Variance | Skewness | Kurtosis |
---|---|---|---|---|---|---|---|

0.00 | 6.60 | 6.60 | 1.859 | 0.94 | 0.89 | 0.71 | 3.42 |

## Appendix C. Derivative Matrix (Jacobian) of the Model Equations with Respect to the State Functions

- (1)
- For ${\mathrm{Re}}_{d}<2300$$$\frac{\partial {N}_{1}^{\left(i\right)}}{\partial {m}_{a}}\equiv {e}_{1}^{i}=0;\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}i=1,\mathrm{...},I;\hspace{0.17em}\hspace{0.17em}$$
- (2)
- For $2300\le {\mathrm{Re}}_{d}\le 10,000$$$\frac{\partial {N}_{1}^{\left(i\right)}}{\partial {m}_{a}}\equiv {e}_{1}^{i}=\left[\frac{{P}_{vs}({T}_{w}^{(i+1)},\mathsf{\alpha})}{\overline{R}\cdot {T}_{w}^{(i+1)}}-\frac{{\omega}^{(i)}{P}_{atm}}{\overline{R}\cdot {T}_{a}^{(i)}(0.622+{\omega}^{(i)})}\right]\cdot \frac{\partial {M}_{2}({m}_{a},\mathsf{\alpha})}{\partial {m}_{a}};\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}i=1,\mathrm{...},I;$$
- (3)
- For ${\mathrm{Re}}_{d}>10,000$$$\frac{\partial {N}_{1}^{\left(i\right)}}{\partial {m}_{a}}\equiv {e}_{1}^{i}=\left[\frac{{P}_{vs}({T}_{w}^{(i+1)},\mathsf{\alpha})}{\overline{R}\cdot {T}_{w}^{(i+1)}}-\frac{{\omega}^{(i)}{P}_{atm}}{\overline{R}\cdot {T}_{a}^{(i)}(0.622+{\omega}^{(i)})}\right]\cdot \frac{\partial {M}_{3}({m}_{a},\mathsf{\alpha})}{\partial {m}_{a}};\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}i=1,\mathrm{...},I;$$