# Predictive Modeling of a Paradigm Mechanical Cooling Tower Model: II. Optimal Best-Estimate Results with Reduced Predicted Uncertainties

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Results

_{indirect}, is given by:

#### 2.1. Sensitivity Analysis Results and Rankings

#### 2.1.1. Relative Sensitivities of the Outlet Air Temperature, ${T}_{a}^{\left(1\right)}$

_{a,in}, T

_{db}, T

_{w,in}, T

_{dp}, a

_{0}) have relative sensitivities between ca. 10% and 50%, and are therefore the most important for the air outlet temperature response, ${T}_{a}^{\left(1\right)}$. The two largest sensitivities have values of 48%, which means that a 1% change in T

_{a,in}or T

_{db}would induce a 0.48% change in ${T}_{a}^{\left(1\right)}$. The next two parameters (i.e., a

_{1}and ω

_{in}) have relative sensitivities between 1% and 6%, and are therefore somewhat important. Parameters #8 through #16 (i.e., D

_{fan}, a

_{1f}, w

_{tsa}, Nu, A

_{surf}, μ, a

_{1,Nu}, Re

_{d}, A

_{fill}) have relative sensitivities of the order of 0.5%. The remaining 36 parameters are relatively unimportant for this response, having relative sensitivities smaller than 1% of the largest relative sensitivity (with respect to T

_{a,in}) for this response. Positive sensitivities imply that a positive change in the respective parameter would cause an increase in the response, while negative sensitivities imply that a positive change in the respective parameter would cause a decrease in the response.

#### 2.1.2. Relative Sensitivities of the Outlet Water Temperature, ${T}_{w}^{\left(50\right)}$

_{dp}, and has the value of 0.548; this means that a 1% increase in T

_{db}would induce a 0.548% increase in ${T}_{w}^{\left(50\right)}$. The sensitivities to the remaining 40 model parameters have not been listed since they are smaller than 1% of the largest sensitivity (with respect to T

_{dp}) for this response.

#### 2.1.3. Relative Sensitivities of the Outlet Water Mass Flow Rate, ${m}_{w}^{\left(50\right)}$

_{w,in}(a 1% increase in this parameter would cause a 1.01% increase in the response) and the second largest sensitivity is to the parameter T

_{w,in}(a 1% increase in this parameter would cause a 0.447% decrease in the response). The sensitivities to the remaining 42 model parameters have not been listed since they are smaller than 1% of the largest sensitivity (with respect to m

_{w,in}) for this response.

#### 2.1.4. Relative Sensitivities of the Outlet Air Relative Humidity, RH^{(1)}

_{a,in}or T

_{db}would cause a decrease in the response of 6.66% or 6.525%, respectively. On the other hand, an increase of 1% in T

_{dp}would cause an increase of 5.75% in the response. The sensitivities to the remaining 32 model parameters have not been listed since they are smaller than 1% of the largest sensitivity (with respect to T

_{a,in}) for this response.

^{(1)}, displays the largest sensitivities, so this response is the most sensitive to parameter variations. The other responses, namely the outlet air temperature, the outlet water temperature, and the outlet water mass flow rate display sensitivities of comparable magnitudes.

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

- 1.
- A model comprising N
_{α}imprecisely known system (model) parameters, α_{n}, considered as the components of a (column) vector,**α**, defined as:$$\alpha =\left\{{\alpha}_{n}|\text{\hspace{0.17em}}n=1,\text{}\dots ,\text{}{N}_{\alpha}\right\}$$

_{n}are denoted as ${\alpha}_{n}^{0}\equiv \langle {\alpha}_{n}\rangle $, and the covariances between two parameters α

_{i}and α

_{j}are denoted as cov(α

_{i},α

_{j}). The mean values ${\alpha}_{n}^{0}$ are considered to be known a priori, so that the vector ${\mathit{\alpha}}^{0}$, defined as ${\mathit{\alpha}}^{0}=\left\{{\alpha}_{n}^{0}|n=1,\dots ,{N}_{\alpha}\right\}$ is considered to be known a priori. The covariances cov(α

_{i},α

_{j}) are also considered to be a priori known; these covariances are considered to be the elements of the a priori known parameter covariance matrix, denoted as ${C}_{\alpha \alpha}^{\left({N}_{\alpha}\times {N}_{\alpha}\right)}$ and defined as:

- 2.
- Also associated with the model are N
_{r}experimentally measured responses, r_{i}, considered to be components of the column vector:$$r=\left\{{r}_{i}|i=1,\dots ,{N}_{r}\right\}$$

_{i}, and the covariances, denoted as $\langle ({r}_{i}-{r}_{i}^{m})({r}_{j}-{r}_{j}^{m})\rangle $, between two measured responses, r

_{i}and r

_{j}, are also considered to be known a priori. The mean measured values ${r}_{i}^{m}$ will be considered to constitute the components of the vector

**r**

^{m}defined as:

- 3.
- In the most general case, correlations may also exist among all parameters and responses. Such correlations are quantified through a priori known parameter-response matrices, denoted as ${C}_{\alpha r}^{\left({N}_{\alpha}\times {N}_{r}\right)}$, and defined as follows:$${C}_{\alpha r}^{\left({N}_{\alpha}\times {N}_{r}\right)}\equiv \langle \left(\alpha -{\alpha}^{0}\right){\left(r-{r}^{m}\right)}^{+}\rangle ={\left[{C}_{r\alpha}^{\left({N}_{r}\times {N}_{\alpha}\right)}\right]}^{+}$$

- Optimally predicted “best-estimate” nominal values,
**α**^{pred}, for the model parameters:$${\alpha}^{pred}={\alpha}^{0}-\left({C}_{\alpha \alpha}{S}_{r\alpha}^{+}-{C}_{\alpha r}\right){\left[{D}_{rr}\right]}^{-1}\left[{r}^{c}\left({\alpha}^{0},{\beta}^{0}\right)-{r}^{m}\right],$$**D**_{rr}is defined as:$${D}_{rr}={S}_{r\alpha}{C}_{\alpha \alpha}{S}_{r\alpha}^{+}-{S}_{r\alpha}{C}_{\alpha r}-{C}_{\alpha r}^{+}{S}_{r\alpha}^{+}+{C}_{rr},$$$${S}_{r\alpha}^{{N}_{r}\times {N}_{\alpha}}\equiv \left(\begin{array}{ccc}\frac{\partial {r}_{1}}{\partial {\alpha}_{1}}& \cdots & \frac{\partial {r}_{1}}{\partial {\alpha}_{{N}_{\alpha}}}\\ \vdots & \ddots & \vdots \\ \frac{\partial {r}_{{N}_{r}}}{\partial {\alpha}_{1}}& \cdots & \frac{\partial {r}_{{N}_{r}}}{\partial {\alpha}_{{N}_{\alpha}}}\end{array}\right).$$$${C}_{rr}^{comp}={S}_{r\alpha}{C}_{\alpha \alpha}{S}_{r\alpha}^{+}.$$ - Reduced predicted uncertainties, ${C}_{\alpha \alpha}^{pred}$, for the predicted nominal parameter values, given by the expression below:$${C}_{\alpha \alpha}^{pred}={C}_{\alpha \alpha}-\left({C}_{\alpha \alpha}{S}_{r\alpha}^{+}-{C}_{\alpha r}\right){\left[{D}_{rr}\right]}^{-1}{\left({C}_{\alpha \alpha}{S}_{r\alpha}^{+}-{C}_{\alpha r}\right)}^{+};$$
- Optimally predicted “best-estimate” nominal values,
**r**^{pred}, for the model responses, given by the expression below:$${r}^{pred}={r}^{m}-\left({C}_{\alpha r}^{+}{S}_{r\alpha}^{+}-{C}_{rr}\right){\left[{D}_{rr}\right]}^{-1}\left[{r}^{c}\left({\alpha}^{0},{\beta}^{0}\right)-{r}^{m}\right];$$ - Reduced predicted uncertainties, ${C}_{rr}^{pred}$, for the predicted nominal response values, given by the expression below:$${C}_{rr}^{pred}={C}_{rr}-\left({C}_{\alpha r}^{+}{S}_{r\alpha}^{+}-{C}_{rr}\right){\left[{D}_{rr}\right]}^{-1}{\left({C}_{\alpha r}^{+}{S}_{r\alpha}^{+}-{C}_{rr}\right)}^{+};$$
- Predicted correlations, ${C}_{\alpha r}^{pred}$, between the predicted model parameters and responses, given by the expression below:$${C}_{\alpha r}^{pred}={C}_{\alpha r}-\left({C}_{\alpha \alpha}{S}_{r\alpha}^{+}-{C}_{\alpha r}\right){\left[{D}_{rr}\right]}^{-1}{\left({C}_{\alpha r}^{+}{S}_{r\alpha}^{+}-{C}_{rr}\right)}^{+}.$$

**C**

_{αα}= 0 and

**C**

_{αr}= 0), Equations (6) through (19) would yield

**α**

^{pred}=

**α**

^{0}and

**r**

^{pred}=

**r**

^{c}(

**α**

^{0},

**β**

^{0}), without any accompanying uncertainties (i.e., ${C}_{rr}^{pred}=0,\text{}{C}_{\alpha \alpha}^{pred}=0$, ${C}_{\alpha r}^{pred}=0$). In other words, for a perfect model, the PM_CMPS methodology predicts values for the responses and the parameters that would coincide with the model’s original corresponding parameter and computed responses (assumed to be perfect), and the experimental measurements would have no effect on the predictions (as would be expected, since imperfect measurements could not possibly improve a “perfect” model’s predictions). On the other hand, if the measurements were perfect, (i.e.,

**C**

_{rr}= 0 and

**C**

_{αr}= 0), but the model were imperfect, then Equations (6) through (19) would yield ${\alpha}^{pred}={\mathit{\alpha}}^{0}-{C}_{\alpha \alpha}{\mathit{S}}_{r\alpha}^{+}{\left[{S}_{r\alpha}{C}_{\alpha \alpha}{S}_{r\alpha}^{+}\right]}^{-1}{r}^{d}\left({\alpha}^{0}\right),\text{}{C}_{\alpha \alpha}^{pred}={C}_{\alpha \alpha}-{C}_{\alpha \alpha}{S}_{r\alpha}^{+}{\left[{S}_{r\alpha}{C}_{\alpha \alpha}{S}_{r\alpha}^{+}\right]}^{-1}{S}_{r\alpha}{C}_{\alpha \alpha},\text{}{r}^{pred}={r}^{m},\text{}{C}_{rr}^{pred}=0,\text{}{C}_{\alpha r}^{pred}=0$. In other words, in the case of perfect measurements, the PM_CMPS predicted values for the responses would coincide with the measured values (assumed to be perfect), while the model’s uncertain parameters would be calibrated by taking the respective measurements into account to yield improved nominal values and reduced parameters uncertainties.

**C**

_{rα}, has been already computed in [1], Equation (A5), and is reproduced below:

_{1}≡ T

_{db}, α

_{2}≡ T

_{dp}, α

_{3}≡ T

_{w,in}, and α

_{4}≡ P

_{atm}.

**C**

_{αα}, has also been already computed in [1], Equation (B1) (see the Appendix of PART I.), and is also reproduced below:

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

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

^{(1)}, have been computed using Equation (17) together with the a priori matrices given in Equations (20)–(23) and the sensitivities presented in Table 1, Table 2, Table 3 and Table 4. The resulting best-estimate predicted nominal values are summarized in Table 6. To facilitate comparison, the corresponding measured and computed nominal values are also presented in this table. Note that there are no direct measurements for the outlet water flow rate, ${m}_{w}^{\left(50\right)}$. For this response, therefore, the predicted best-estimate nominal value has been obtained by a forward re-computation using the best-estimate nominal parameter values listed in Table 5, while the predicted best estimate standard deviation for this response has been obtained by using “best-estimate” values in Equation (15), i.e.:

## 3. Discussion

^{st}-order response sensitivities to model parameters has been recently extended to computing efficiently and exactly the 2nd-order response sensitivities to parameters for linear [13] and nonlinear [14] large-scale systems. As has been shown in [15,16,17,18], the 2nd-order response sensitivities have the following major impacts on the computed moments of the response distribution: (a) they cause the “expected value of the response” to differ from the “computed nominal value of the response”; and (b) they contribute decisively to causing asymmetries in the response distribution. Indeed, neglecting the second-order sensitivities would nullify the third-order response correlations, and hence would nullify the skewness of the response. Consequently, non-Gaussian features (i.e., asymmetries, long-tails) any events occurring in a response’s long and/or short tails, which are characteristic of rare but decisive events (e.g., major accidents, catastrophes), would likely be missed. Ongoing work aims at further applications and generalization of the adjoint sensitivity analysis and the PM_CMPS methodologies, to enable the computation of 3rd- and higher-order sensitivities and response distributions. The exact and efficient computation of high-order response sensitivities for large-scale systems is expected to advance significantly the areas of uncertainty quantification, model validation, reduced-order modeling, and predictive modeling/data assimilation.

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Appendix A. Derivatives of Cooling Tower Model Equations with Respect To Model Parameters

Index i of α_{i} | Independent Scalar Parameters | C + + String | Math. Notation | Nominal Value(s) | Absolute Standard Deviation | Relative Standard Deviation (%) |

1 | Air temperature (dry bulb) (K) | tdb | T_{db} | 299.11 | 4.17 | 1.39 |

2 | Dew point temperature (K) | tdp | T_{dp} | 292.05 | 2.36 | 0.81 |

3 | Inlet water temperature (K) | twin | T_{w,in} | 298.79 | 1.70 | 0.57 |

4 | Atmospheric pressure (Pa) | patm | P_{atm} | 100586 | 401 | 0.40 |

5 | Wetted fraction of fill surface area | wtsa | w_{tsa} | 1 | 0 | 0 |

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 | μ | 1.983 × 10^{−5} | 9.676 × 10^{−7} | 4.88 |

8 | Kinematic viscosity of air at T = 300 K (m^{2}/s) | nuair | v | 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 | Δz_{dk} | 10 | 0.1 | 1 |

33 | Fan shroud height (m) | fsht | Δz_{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 | Δz_{fill} | 2.013 | 0.02013 | 1 |

36 | Rain section height (m) | rsht | Δz_{rain} | 1.633 | 0.01633 | 1 |

37 | Basin section height (m) | bsht | Δz_{bs} | 1.168 | 0.01168 | 1 |

38 | Drift eliminator thickness (m) | detk | Δz_{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} | 14221 | 3555.3 | 25 |

42 | Prandlt number of air at T = 80 C | Pr | P_{r} | 0.708 | 0.005 | 0.71 |

43 | Wind speed (m/s) | wspd | V_{w} | 1.80 | 0.92 | 51.1 |

44 | Exit air speed at the shroud (m/s) | vexit | V_{exit} | 10.0 | 1.0 | 10.0 |

Index i of α_{i} | Boundary Parameters | C + + String | Math. Notation | Nominal Value | Absolute Standard Deviation | Relative Standard Deviation (%) |

45 | Inlet water mass flow rate (kg/s) | mfwin | m_{w,in} | 44.02 | 2.201 | 5 |

46 | Inlet air temperature (K) | tain | T_{a,in} | set to ${T}_{db}$ | 4.17 | 1.39 |

47 | Inlet air mass flow rate (kg/s) | main | m_{a} | 155.07 | 15.91 | 10.26 |

48 | Inlet air humidity ratio (Dependent Scalar Parameter) | hrin | $\begin{array}{l}{\omega}_{in};\\ {\omega}_{rain}\end{array}$ | 0.0138 | 0.00206 | 14.93 |

49 | Reynold’s number | Re; Reh | Re_{d} | 4428 | 671.6 | 15.17 |

50 | Schmidt number | Sc | Sc | 0.60 | 0.074 | 12.41 |

51 | Sherwood number | Sh | Sh | 14.13 | 4.84 | 34.25 |

52 | Nusselt number | Nu | Nu | 14.94 | 5.08 | 34.00 |

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

Mass diffusivity of water vapor in air (m^{2}/s) | D_{av}(T_{a},α) | $\frac{{a}_{0,dav}{T}^{1.5}}{{a}_{1,dav}+\left({a}_{2,dav}+{a}_{3,dav}T\right)T}$ |

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

Mass transfer coefficient (m/s) | k_{m}(α) | $\frac{{f}_{mt}{S}_{h}{D}_{av}\left({T}_{db},\mathit{\alpha}\right)}{{D}_{h}}$ |

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

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

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

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

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 | $\u2206{z}_{dk}+\u2206{z}_{fan}-\u2206{z}_{bs}$ |

Height above fill section (m) | Δz_{4–2} | $Z-\u2206{z}_{fill}-\u2206{z}_{rain}$ |

Fill section control volume height (m) | Δz | $\frac{\u2206{z}_{fill}}{I}$ |

Fill section length, including drift eliminator (m) | L_{fill} | $\u2206{z}_{fill}+\u2206{z}_{de}$ |

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

Fan shroud flow area (m^{2}) | A_{out} | $\pi {{r}_{fan}}^{2}$ |

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},α) | ${a}_{0f}+{a}_{1f}{T}_{w}$ |

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

H_{g}(T_{a}) = saturated vapor enthalpy (J/kg) | h_{g,a}(T_{a},α) | ${a}_{0g}+{a}_{1g}{T}_{a}$ |

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

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

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

_{1}through α

_{4}(i.e., the dry bulb air temperature, dew point temperature, inlet water temperature, and atmospheric pressure) were measured at the SRNL site at which the F-area cooling towers are located. Among the 8079 measured benchmark data sets [8], 7688 data sets are considered to represent “unsaturated conditions”, which have been used to derive the statistical properties (means, variance and covariance, skewness and kurtosis) for these model parameters, as shown in Figures B1 through B4 and Tables B4 through B7 in Appendix B of PART I [1].

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

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

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

- The air/water vapor energy balance equations:
- (i)
- Control Volume i = 1:$$\begin{array}{l}{N}_{4}^{(1)}\left({m}_{w},{T}_{w},{T}_{a},\omega ;\alpha \right)\triangleq ({T}_{a}^{(2)}-{T}_{a}^{(1)}){C}_{p}^{(1)}(\frac{{T}_{a}^{(1)}+273.15}{2},\alpha )-{\omega}^{(1)}{h}_{g,a}^{(1)}({T}_{a}^{(1)},\alpha )\\ \text{\hspace{1em}}+\frac{({T}_{w}^{(2)}-{T}_{a}^{(1)})H({m}_{a},\alpha )}{\left|{m}_{a}\right|}+\frac{({m}_{w,in}-{m}_{w}^{(2)}){h}_{g,w}^{(2)}({T}_{w}^{(2)},\alpha )}{\left|{m}_{a}\right|}+{\omega}^{(2)}{h}_{g,a}^{(2)}({T}_{a}^{(2)},\alpha )=0;\end{array}$$
- (ii)
- Control Volumes i = 2,..., I − 1:$$\begin{array}{l}{N}_{4}^{(i)}\left({m}_{w},{T}_{w},{T}_{a},\omega ;\alpha \right)\triangleq ({T}_{a}^{(i+1)}-{T}_{a}^{(i)}){C}_{p}^{(i)}(\frac{{T}_{a}^{(i)}+273.15}{2},\alpha )-{\omega}^{(i)}{h}_{g,a}^{(i)}({T}_{a}^{(i)},\alpha )\\ \text{\hspace{1em}}+\frac{({T}_{w}^{(i+1)}-{T}_{a}^{(i)})H({m}_{a},\alpha )}{\left|{m}_{a}\right|}+\frac{({m}_{w}^{(i)}-{m}_{w}^{(i+1)}){h}_{g,w}^{(i+1)}({T}_{w}^{(i+1)},\alpha )}{\left|{m}_{a}\right|}+{\omega}^{(i+1)}{h}_{g,a}^{(i+1)}({T}_{a}^{(i+1)},\alpha )=0;\end{array}$$
- (iii)
- Control Volume i = I:$$\begin{array}{l}{N}_{4}^{(I)}\left({m}_{w},{T}_{w},{T}_{a},\omega ;\alpha \right)\triangleq ({T}_{a,in}-{T}_{a}^{(I)}){{C}_{p}}^{(I)}(\frac{{T}_{a}^{(I)}+273.15}{2},\alpha )-{\omega}^{(I)}{h}_{g,a}^{(I)}({T}_{a}^{(I)},\alpha )\\ \text{\hspace{1em}}+\frac{({T}_{w}^{(I+1)}-{T}_{a}^{(I)})H({m}_{a},\alpha )}{\left|{m}_{a}\right|}+\frac{({m}_{w}^{(I)}-{m}_{w}^{(I+1)}){h}_{g,w}^{(I+1)}({T}_{w}^{(I+1)},\alpha )}{\left|{m}_{a}\right|}+{\omega}_{in}{h}_{g,a}({T}_{a,in},\alpha )=0.\end{array}$$

**α**, which appears in Equations (A1)–(A12), comprise the model parameters, i.e.:

_{α}denotes the total number of model parameters. These model parameters are described in Table A1.

#### A1. Derivatives of the Liquid Continuity Equations with Respect to the Parameters

^{(1)}≡ T

_{db}are as follows:

^{(2)}≡ T

_{dp}are as follows:

^{(3)}≡ T

_{w,in}are as follows:

_{2,}

_{ρ}= –0.26847207; a

_{3,}

_{ρ}= –1.8113691 × 10

^{–3}; a

_{4,}

_{ρ}= –1.7041217 × 10

^{–6}.

^{(4)}≡ P

_{atm}are as follows:

^{(5)}≡ w

_{tsa}are as follows:

^{(6)}≡ k

_{sum}are as follows:

^{(7)}≡ μ are as follows:

^{(8)}≡ υ are as follows:

^{(9)}≡ k

_{air}are as follows:

^{(10)}≡ f

_{ht}are as follows:

^{(11)}≡ f

_{mt}are as follows:

^{(12)}≡ f are as follows:

^{(13)}≡ a

_{0}are as follows:

^{(14)}≡ a

_{1}are as follows:

^{(15)}≡ a

_{0,cpa}are as follows:

^{(16)}≡ a

_{1,cpa}are as follows:

^{(17)}≡ a

_{2,cpa}are as follows:

**α**≡

^{(18)}**a**are as follows:

_{0,dav}^{(19)}≡ a

_{1,dav}are as follows:

^{(20)}≡ a

_{2,dav}are as follows:

^{(21)}≡ a

_{3,dav}are as follows:

^{(22)}≡ a

_{0f}are as follows:

^{(23)}≡ a

_{1f}are as follows:

^{(24)}≡ a

_{0g}are as follows:

^{(25)}≡ a

_{1g}are as follows:

^{(26)}≡ a

_{0,Nu}are as follows:

^{(27)}≡ a

_{1,Nu}are as follows:

^{(28)}≡ a

_{2,Nu}are as follows:

^{(29)}≡ a

_{3,Nu}are as follows:

^{(30)}≡ W

_{dkx}are as follows:

^{(31)}≡ W

_{dky}are as follows:

^{(32)}≡ Δz

_{dk}are as follows:

^{(33)}≡ Δz

_{fan}are as follows:

^{(34)}≡ D

_{fan}are as follows:

^{(35)}≡ Δz

_{fill}are as follows:

^{(36)}≡ Δz

_{rain}are as follows:

^{(37)}≡ Δz

_{bs}are as follows:

^{(38)}≡ Δz

_{de}are as follows:

^{(39)}≡ D

_{h}are as follows:

^{(40)}≡ A

_{fill}are as follows:

^{(41)}≡ A

_{surf}are as follows:

^{(42)}≡ Pr are as follows:

^{(43)}≡ V

_{w}are as follows:

^{(44)}≡ V

_{exit}are as follows:

^{(45)}≡ m

_{w,in}are as follows:

^{(46)}≡ T

_{a,in}are as follows:

^{(47)}≡ m

_{a}are as follows:

^{(48)}≡ ω

_{in}are as follows:

^{(49)}≡ Re

_{d}are as follows: