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Article

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

Center for Nuclear Science and Energy, Department of Mechanical Engineering, University of South Carolina, Columbia, SC 29208, USA
*
Author to whom correspondence should be addressed.
Energies 2016, 9(9), 747; https://doi.org/10.3390/en9090747
Submission received: 23 June 2016 / Revised: 23 August 2016 / Accepted: 5 September 2016 / Published: 16 September 2016
(This article belongs to the Special Issue Advances in Predictive Modeling of Nuclear Energy Systems)

Abstract

:
This work uses the adjoint sensitivity model of the counter-flow cooling tower derived in the accompanying PART I to obtain the expressions and relative numerical rankings of the sensitivities, to all model parameters, of the following model responses: (i) outlet air temperature; (ii) outlet water temperature; (iii) outlet water mass flow rate; and (iv) air outlet relative humidity. These sensitivities are subsequently used within the “predictive modeling for coupled multi-physics systems” (PM_CMPS) methodology to obtain explicit formulas for the predicted optimal nominal values for the model responses and parameters, along with reduced predicted standard deviations for the predicted model parameters and responses. These explicit formulas embody the assimilation of experimental data and the “calibration” of the model’s parameters. The results presented in this work demonstrate that the PM_CMPS methodology reduces the predicted standard deviations to values that are smaller than either the computed or the experimentally measured ones, even for responses (e.g., the outlet water flow rate) for which no measurements are available. These improvements stem from the global characteristics of the PM_CMPS methodology, which combines all of the available information simultaneously in phase-space, as opposed to combining it sequentially, as in current data assimilation procedures.

1. Introduction

In the present work, the predictive modeling of the counter-flow cooling tower presented in [1] is further developed by applying the “predictive modeling for coupled multi-physics systems” (PM_CMPS) methodology recently developed in [2]. The PM_CMPS methodology constructs a prior distribution for the parameters and responses by using all of the available computational and experimental information, and by relying on the maximum entropy principle to maximize the impact of all available information and minimize the impact of ignorance. Subsequently, the PM_CMPS methodology [2] constructs formally the posterior distribution using Bayes’ theorem, and then evaluates asymptotically, to first-order sensitivities, the posterior distribution using the saddle-point method to obtain explicit formulas for the predicted optimal nominal values for the model responses and parameters, along with reduced predicted uncertainties (i.e., reduced predicted standard deviations) for the predicted model parameters and responses. The PM_CMPS methodology has been successfully applied to the analysis of large-scale experiments and the experimental validation of reactor design codes of interest to reactor physics [3,4], light water reactors [5] and sodium-cooled fast reactors [6].
The PM_CMPS methodology relies fundamentally on the sensitivities to model parameters of the measured model responses, which, in this work, are as follows: (i) the outlet air temperature; (ii) the outlet water temperature; (iii) the outlet water mass flow rate; and (iv) the air outlet relative humidity. The expressions, numerical results, and relative rankings of the sensitivities of these responses are presented in Section 2.1. These sensitivities are subsequently used in Section 2.2 for assimilating experimental data in order to “calibrate” the model parameters, and for obtaining best-estimate predicted results with reduced predicted uncertainties. Section 3 concludes this work by discussing the significance of the results presented herein in the context of ongoing work aimed at further applications and generalization of the adjoint sensitivity analysis and PM_CMPS methodologies.

2. Results

It has been shown in the accompanying PART I [1] that the total sensitivity of a model response R ( m w ,   T w ,   T a ,   ω ;   α ) to arbitrary variations in the model’s parameters δ α ( δ α 1 ,   ,   δ α N α ) and state functions δ m w ,   δ T w ,   δ T a ,   δ ω , around the nominal values ( m w 0 ,   T w 0 ,   T a 0 ,   ω 0 ;   α 0 ) of the parameters and state functions, is provided by the G-differential of the model’s response to these variations. This G-differential was denoted as D R ( m w 0 ,   T w 0 ,   T a 0 ,   ω 0 ;   α 0 ;   δ m w ,   δ T w ,   δ T a ,   δ ω ;   δ α ) , and was expressed in terms of the adjoint sensitivity functions as follows:
D R ( m w 0 ,   T w 0 ,   T a 0 ,   ω 0 ;   α 0 ;   δ m w ,   δ T w ,   δ T a ,   δ ω ;   δ α ) = i = 1 N α ( R α i δ α i ) + D R i n d i r e c t ,
where the so-called “indirect effect” term, DRindirect, is given by:
D R i n d i r e c t μ w Q 1 + τ w Q 2 + τ a Q 3 + o Q 4
and where the vector [ μ w , τ w , τ a , o ] + is the solution of the following adjoint sensitivity system:
( A 1 + A 2 + A 3 + A 4 + B 1 + B 2 + B 3 + B 4 + C 1 + C 2 + C 3 + C 4 + D 1 + D 2 + D 3 + D 4 + ) ( μ w τ w τ a o ) = ( R 1 R 2 R 3 R 4 ) .
Furthermore, the sources R ( r ( 1 ) , , r ( I ) ) ,   = 1 , 2 , 3 , 4 , for the adjoint sensitivity system represented by Equation (3) are the functional derivatives of the model responses with respect to the state functions, i.e.:
r 1 ( i ) R m w ( i + 1 ) ;   r 2 ( i ) R T w ( i + 1 ) ;   r 3 ( i ) R T a ( i ) ;   r 4 ( i ) R ω ( i ) ; i = 1 , ,   I
while the components of the vectors Q ( q ( 1 ) , , q ( I ) ) ,   = 1 , 2 , 3 , 4 , in Equation (2) are the derivatives of the model’s equations with respect to model parameters, namely:
q ( i ) j = 1 N α ( N ( i ) α j δ α j ) ;   i = 1 , , I ;   = 1 ,   2 ,   3 ,   4 .
The explicit expressions of the vectors Q ( q ( 1 ) , , q ( I ) ) ,   = 1 , 2 , 3 , 4 are provided in Appendix A. The model responses of interest in this work are the following quantities: (i) the outlet air temperature, T a ( 1 ) ; (ii) the outlet water temperature, T w ( 50 ) ; (iii) the outlet water flow rate, m w ( 50 ) ; and (iv) the outlet air relative humidity, R H ( 1 ) . Except for the water outlet flow rate m w ( 50 ) , these responses have been measured experimentally [7,8], and the first four moments of their respective statistical distributions have been quantified in [1].

2.1. Sensitivity Analysis Results and Rankings

As has been discussed in the accompanying PART I [1], there are a total of 8079 measured benchmark data sets for the cooling tower model with the “fan-on,” with a drafted air exit velocity at 10 m/s at the shroud. For this velocity (and corresponding air flow rate), the Reynolds number is around 4500, which means that the flow within the cooling tower is in the “transitional flow and heat transfer” regime. As has also been discussed in [1], 7668 benchmark data sets (out of the total of 8079 data sets) are considered to correspond to the “unsaturated conditions” which are analyzed in this work. The nominal values for boundary and atmospheric conditions used in this work were obtained, as described in [1], from the statistics of these 7668 benchmark data sets corresponding to “unsaturated conditions.” In turn, these “unsaturated” boundary and atmospheric conditions were used to obtain the sensitivity results reported, below, in this Subsection. Sub-subsections 2.1.1 through 2.1.4, below, provide the numerical values and rankings, in descending order, of the relative sensitivities computed using the adjoint sensitivity analysis methodology for the four model responses T a ( 1 ) , T w ( 50 ) , m w ( 50 ) and R H ( 1 ) . Note that the relative sensitivity, R S ( α i ) , of a response R ( α i ) to a parameter α i is defined as R S ( α i ) [ d R ( α i ) / d α i ] [ α i / R ( α i ) ] . Thus, the relative sensitivities are unit-less and are very useful in ranking the sensitivities to highlight their relative importance for the respective response. Thus, a relative sensitivity of 1.00 indicates that a change of 1% in the respective parameter will induce a 1% change in a response that is linear in the respective sensitivity. The higher the relative sensitivity, the more important the respective parameter to the respective response.

2.1.1. Relative Sensitivities of the Outlet Air Temperature, T a ( 1 )

The sensitivities of the air outlet temperature with respect to all of the model’s parameters have been computed using Equations (1) and (2). The numerical results and ranking of the relative sensitivities, in descending order of their magnitudes, are provided in Table 1 below, along with their respective relative standard deviations.
As the results in Table 1 indicate, the first five parameters (i.e., Ta,in, Tdb, Tw,in, Tdp, a0) have relative sensitivities between ca. 10% and 50%, and are therefore the most important for the air outlet temperature response, T a ( 1 ) . The two largest sensitivities have values of 48%, which means that a 1% change in Ta,in or Tdb would induce a 0.48% change in T a ( 1 ) . The next two parameters (i.e., a1 and ωin) have relative sensitivities between 1% and 6%, and are therefore somewhat important. Parameters #8 through #16 (i.e., Dfan, a1f, wtsa, Nu, Asurf, μ, a1,Nu, Red, Afill) 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 Ta,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 ( 50 )

The results and ranking of the relative sensitivities of the outlet water temperature with respect to the most important 12 parameters for this response are listed in Table 2.
The largest sensitivity of T w ( 50 ) is to the parameter Tdp, and has the value of 0.548; this means that a 1% increase in Tdb would induce a 0.548% increase in T w ( 50 ) . 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 Tdp) for this response.

2.1.3. Relative Sensitivities of the Outlet Water Mass Flow Rate, m w ( 50 )

The results and ranking of the relative sensitivities of the outlet water mass flow rate with respect to the most important 10 parameters for this response are listed in Table 3. This response is most sensitive to mw,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 Tw,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 mw,in) for this response.

2.1.4. Relative Sensitivities of the Outlet Air Relative Humidity, RH(1)

The results and ranking of the relative sensitivities of the outlet air relative humidity with respect to the most important 20 parameters for this response are listed in Table 4. The first three sensitivities of this response are quite large (relative sensitivities larger than unity are customarily considered to be very significant). In particular, an increase of 1% in Ta,in or Tdb would cause a decrease in the response of 6.66% or 6.525%, respectively. On the other hand, an increase of 1% in Tdp 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 Ta,in) for this response.
Overall, the outlet air relative humidity, RH(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

This subsection presents the results of applying the Predictive Modeling of Coupled Multi-Physics Systems (PM_CMPS) methodology [2] to the counter-flow cooling tower model. The PM_CMPS methodology [2] encompasses into a unified conceptual and mathematical framework, the concepts of both “forward” and “inverse” modeling, including data assimilation, model calibration and prediction of best-estimate values for model parameters and responses, with reduced predicted uncertainties. For the simplest case of a single computational model, such as the counter-flow cooling tower model analyzed in this work, the PM_CMPS methodology considers the following a priori information:
1.
A model comprising Nα imprecisely known system (model) parameters, αn, considered as the components of a (column) vector, α, defined as:
α = { α n | n = 1 ,   ,   N α }
The mean values of the model parameters αn are denoted as α n 0 α n , and the covariances between two parameters αi and αj are denoted as cov(αi,αj). The mean values α n 0 are considered to be known a priori, so that the vector α 0 , defined as α 0 = { α n 0 | n = 1 , , N α } is considered to be known a priori. The covariances cov(αi,αj) 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 α α ( N α × N α ) and defined as:
C α α ( N α × N α ) [ cov ( α i , α j ) ] N α × N α ( α i α i 0 ) ( α j α j 0 ) N α × N α ;   i , j = 1 ,   ,   N α
2.
Also associated with the model are Nr experimentally measured responses, ri, considered to be components of the column vector:
r = { r i | i = 1 , , N r }
The mean values, denoted as r i m , of the measured responses, ri, and the covariances, denoted as ( r i r i m ) ( r j r j m ) , between two measured responses, ri and rj, 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 rm defined as:
r m = { r i m | i = 1 , , N r } , r i m r i ,   i = 1 , , N r ,
and the covariances ( r i r i m ) ( r j r j m ) of the measured responses are considered to be components of the a priori known measured covariance matrix, denoted as C r r ( N r × N r ) , and defined as:
C r r ( N r × N r ) ( r i r i m ) ( r j r j m ) N r × N r ,   i , j = 1 , , N r .
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 α r ( N α × N r ) , and defined as follows:
C α r ( N α × N r ) ( α α 0 ) ( r r m ) + = [ C r α ( N r × N α ) ] +
To keep the notation simple, the dimensions of the various vectors and matrices will not be shown in subsequent formulas. For a single multi-physics system, as is the case of the cooling tower model under consideration in this work, the quantities predicted by the PM_CMPS methodology [2] are as follows:
  • Optimally predicted “best-estimate” nominal values, αpred, for the model parameters:
    α p r e d = α 0 ( C α α S r α + C α r ) [ D r r ] 1 [ r c ( α 0 , β 0 ) r m ] ,
    where the matrix Drr is defined as:
    D r r = S r α C α α S r α + S r α C α r C α r + S r α + + C r r ,
    and the components of the matrix S r α ( N r × N α ) are the first-order sensitivities (i.e., functional derivatives) of all responses with respect to all model parameters, defined as follows:
    S r α N r × N α ( r 1 α 1 r 1 α N α r N r α 1 r N r α N α ) .
    It is important to note that the first term on the right side of Equation (13) is the covariance matrix of the computed responses, C r r c o m p , when only the first-order sensitivities are taken into account, i.e.:
    C r r c o m p = S r α C α α S r α + .
  • Reduced predicted uncertainties, C α α p r e d , for the predicted nominal parameter values, given by the expression below:
    C α α p r e d = C α α ( C α α S r α + C α r ) [ D r r ] 1 ( C α α S r α + C α r ) + ;
  • Optimally predicted “best-estimate” nominal values, r pred, for the model responses, given by the expression below:
    r p r e d = r m ( C α r + S r α + C r r ) [ D r r ] 1 [ r c ( α 0 , β 0 ) r m ] ;
  • Reduced predicted uncertainties, C r r p r e d , for the predicted nominal response values, given by the expression below:
    C r r p r e d = C r r ( C α r + S r α + C r r ) [ D r r ] 1 ( C α r + S r α + C r r ) + ;
  • Predicted correlations, C α r p r e d , between the predicted model parameters and responses, given by the expression below:
    C α r p r e d = C α r ( C α α S r α + C α r ) [ D r r ] 1 ( C α r + S r α + C r r ) + .
The expressions given in Equations (6) through (19) can also be obtained from the results presented originally in [9] for the particular case of a time-independent single multi-physics system. Note that if the model is perfect (which means that Cαα = 0 and Cαr = 0), Equations (6) through (19) would yield αpred = α0 and rpred = rc(α0,β0), without any accompanying uncertainties (i.e., C r r p r e d = 0 ,   C α α p r e d = 0 , C α r p r e d = 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., Crr = 0 and Cαr = 0), but the model were imperfect, then Equations (6) through (19) would yield α p r e d = α 0 C α α S r α + [ S r α C α α S r α + ] 1 r d ( α 0 ) ,   C α α p r e d = C α α C α α S r α + [ S r α C α α S r α + ] 1 S r α C α α ,   r p r e d = r m ,   C r r p r e d = 0 ,   C α r p r e d = 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.
The a priori response-parameter covariance matrix, C, has been already computed in [1], Equation (A5), and is reproduced below:
C o v ( T a ,   o u t m e a s ,   T w ,   o u t m e a s ,   R H m e a s ,   α 1 ,   ,   α 52 ) C r α = ( 12.96 3.51 2.33 447.09 0 0 3.35 3.05 1.89 93.58 0 0 54.16 1.73 2.27 1831.03 0 0 ) .
where the measured correlated parameters are: α1 ≡ Tdb, α2 ≡ Tdp, α3 ≡ Tw,in, and α4Patm.
The a priori parameter covariance matrix, Cαα, has also been already computed in [1], Equation (B1) (see the Appendix of PART I.), and is also reproduced below:
C α α ( V a r ( α 1 ) C o v ( α 1 , α 2 ) C o v ( α 1 , α 52 ) C o v ( α 2 , α 1 ) V a r ( α 2 ) C o v ( α 2 , α 52 ) C o v ( α 52 , α 1 ) V a r ( α 52 ) ) = ( 17.37 2.83 1.81 529.26 0 0 2.83 5.56 2.31 87.16 0 0 1.81 2.31 2.90 47.22 0 0 529.26 87.16 47.22 160597.01 0 0 0 0 0 0 0 0 0 0 0 0 0 25.81 )
The a priori covariance matrix of the computed responses, C r r c o m p , is obtained by using Equations (15) and (21) together with the sensitivity results presented in Table 1, Table 2, Table 3 and Table 4; the final result is given below:
C r r c o m p C o v ( T a ( 1 ) , T w ( 50 ) , R H ( 1 ) ) = S r α C α α S r α + = ( T a ( 1 ) α 1 , , T a ( 1 ) α N α T w ( 50 ) α 1 , , T w ( 50 ) α N α R H ( 1 ) α 1 , , R H ( 1 ) α N α ) ( V a r ( α 1 ) C o v ( α 1 , α 2 ) C o v ( α 1 , α 52 ) C o v ( α 2 , α 1 ) V a r ( α 2 ) C o v ( α 2 , α 52 ) C o v ( α 52 , α 1 ) V a r ( α 52 ) ) ( T a ( 1 ) α 1 , , T a ( 1 ) α N α T w ( 50 ) α 1 , , T w ( 50 ) α N α R H ( 1 ) α 1 , , R H ( 1 ) α N α ) + = ( 10.87 7.19 34.81 7.19 7.72 13.97 34.81 13.97 221.88 ) .
The a priori covariance matrix, C o v ( T a , o u t m e a s , T w , o u t m e a s , R H o u t m e a s ) C r r , of the measured responses (namely: the outlet air temperature, T a , o u t m e a s [ T a ( 1 ) ] m e a s u r e d ; the outlet water temperature, T w , o u t m e a s [ T w ( 50 ) ] m e a s u r e d , and the outlet air relative humidity, R H o u t m e a s [ R H ( 1 ) ] m e a s u r e d was also computed in [1], Equation (A4), and is reproduced below:
C o v ( T a , o u t m e a s ,   T w , o u t m e a s ,   R H o u t m e a s ) C r r = ( 11.29 3.55 43.85 3.55 2.53 5.31 43.85 5.31 252.49 ) .

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

The best-estimate nominal parameter values have been computed using Equation (12) in conjunction 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 nominal values are listed in Table 5, below. The corresponding best-estimate absolute standard deviations for these parameters are also presented in this table. These values are the square-roots of the diagonal elements of the matrix C α α p r e d , which is computed using Equation (16) in conjunction with the a priori matrices given in Equations (20)–(23) and the sensitivities presented in Table 1, Table 2, Table 3 and Table 4. For comparison, the original nominal parameter values and original absolute standard deviations are also listed. As the results in Table 5 indicate, the predicted best-estimate standard deviations are all smaller or at most equal to (i.e., left unaffected) the original standard deviations. The parameters are affected proportionally to the magnitudes of their corresponding sensitivities: the parameters experiencing the largest reductions in their predicted standard deviations are those having the largest sensitivities.

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

Using the a priori matrices given in Equations (20)–(23) together with the sensitivities presented in Table 1, Table 2, Table 3 and Table 4 in Equation (18) yields the following predicted response covariance matrix, C r r p r e d :
C r r p r e d C o v ( [ T a ( 1 ) ] b e , [ T w ( 50 ) ] b e , [ R H ( 1 ) ] b e ) = ( 6.71 2.73 22.80 2.73 2.37 1.79 22.80 1.79 145.19 ) .
The best-estimate response-parameter correlation matrix, C α r p r e d , is obtained using Equation (19) 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 non-zero elements with the largest magnitudes are as follows:
r e l . c o r . ( R 1 , α 4 ) = 0.278 ; r e l . c o r . ( R 1 , α 41 ) = 0.070 ; r e l . c o r . ( R 1 , α 49 ) = 0.039 ; r e l . c o r . ( R 2 , α 4 ) = 0.108 ; r e l . c o r . ( R 2 , α 41 ) = 0.019 ; r e l . c o r . ( R 3 , α 4 ) = 0.232 ; r e l . c o r . ( R 3 , α 41 ) = 0.127 ; r e l . c o r . ( R 3 , α 49 ) = 0.072 .
The notation used in Equation (25) is as follows: R 1 T a ( 1 ) ,   R 2 T w ( 50 ) ,   R 3 R H ( 1 ) ,   α 4 P a t m , α 41 A s u r f and α 49 Re d .
The best-estimate nominal values of the (model responses) outlet air temperature, T a ( 1 ) ; outlet water temperature T w ( 50 ) ; and outlet air relative humidity, RH(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 ( 50 ) . 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.:
[ C r r c o m p ] b e = [ S r α ] b e [ C α α ] b e [ S r α + ] b e .
The results presented in Table 6 indicate that the predicted standard deviations are smaller than either the computed or the experimentally measured ones. This is indeed the consequence of using the PM_CMPS methodology in conjunction with consistent (as opposed to discrepant) computational and experimental information. Often, however, the information is inconsistent, usually due to the presence of unrecognized errors. Solutions for addressing such situations have been proposed in [10]. It is also important to note that the PM_CMPS methodology has improved (i.e., reduced, albeit not by a significant amount) the predicted standard deviation for the outlet water flow rate response, for which no measurements were available. This improvement stems from the global characteristics of the PM_CMPS methodology, which combines all of the available simultaneously on phase-space, as opposed to combining it sequentially, as is the case with the current state-of-the-art data assimilation procedures [11,12].

3. Discussion

In the present work, the adjoint sensitivity model of the counter-flow cooling tower derived in the accompanying PART I [1] was used to obtain the expressions and relative numerical rankings of the sensitivities, to all model parameters, of the following responses (quantities of interest): (i) the outlet air temperature; (ii) the outlet water temperature; (iii) the outlet water mass flow rate; and (iv) the air outlet relative humidity. These sensitivities were subsequently used within the “predictive modeling for coupled multi-physics systems” (PM_CMPS) methodology [2] to obtain explicit formulas for the predicted optimal nominal values for the model responses and parameters, along with reduced predicted standard deviations for the predicted model parameters and responses. These explicit formulas embody the assimilation of experimental data and the “calibration” of the model’s parameters.
The results presented in this work indicate that the predicted standard deviations are smaller than either the computed or the experimentally measured ones. It is also important to note that the PM_CMPS methodology has improved (i.e., reduced, albeit not by a significant amount) the predicted standard deviation for the outlet water flow rate response, for which no measurements were available. This improvement stems from the global characteristics of the PM_CMPS methodology, which combines all of the available information simultaneously in phase-space, as opposed to combining it sequentially, as is the case with the current state-of-the-art data assimilation procedures [11,12]. This is indeed the consequence of using the PM_CMPS methodology in conjunction with consistent (as opposed to discrepant) computational and experimental information. Often, however, the information is inconsistent, usually due to the presence of unrecognized errors. Solutions for addressing such situations have been proposed in [10].
The adjoint sensitivity analysis methodology used in PART I [1] for computing exactly and efficiently the 1st-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

This work has been partially sponsored by the US Department of Energy (James J. Peltz, Program manager) with the University of South Carolina.

Author Contributions

Ruixian Fang performed all of the numerical calculations in this paper. Dan Cacuci conceived and directed the research reported herein, and wrote the paper. Madalina Badea contributed the programming of the equations underlying the “predictive modeling” formalism.

Conflicts of Interest

The authors declare no conflict of interest.

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

For convenience, the model parameters are reproduced in Table A1 below from Appendix B of PART I [1]. The independent model parameters are used for computing various dependent model parameters and thermal material properties, as shown in Table A2 and Table A3, below.
Table A1. Parameters for SRNL f-area cooling towers.
Table A1. Parameters for SRNL f-area cooling towers.
Index i of αiIndependent Scalar ParametersC + + StringMath. NotationNominal Value(s)Absolute Standard DeviationRelative Standard Deviation (%)
1Air temperature (dry bulb) (K)tdbTdb299.114.171.39
2Dew point temperature (K)tdpTdp292.052.360.81
3Inlet water temperature (K)twinTw,in298.791.700.57
4Atmospheric pressure (Pa)patmPatm1005864010.40
5Wetted fraction of fill surface areawtsawtsa100
6Sum of loss coefficients above fillksumksum10550
7Dynamic viscosity of air at T = 300 K (kg/m·s)muairμ1.983 × 10−59.676 × 10−74.88
8Kinematic viscosity of air at T = 300 K (m2/s)nuairv1.568 × 10−51.895 × 10−612.09
9Thermal conductivity of air at T = 300 K (W/m·K)tcairkair0.026241.584 × 10−36.04
10Heat transfer coefficient multipliermlthtcfht10.550
11Mass transfer coefficient multipliermltmtcfmt10.550
12Fill section frictional loss multipliermltfilf4250
13Pvs(T) parametersa0a025.59430.010.04
14a1a1−5229.894.40.08
15Cpa(T) parametersA(1)a0,cpa1030.50.29400.03
16A(2)a1,cpa−0.199750.00201.00
17A(3)a2,cpa3.9734 × 10−43.345 × 10−60.84
18Dav(T) parametersA(1)a0,dav7.06085 × 10−900
19A(2)a1,dav2.653220.0030.11
20A(3)a2,dav−6.1681 × 10−32.3 × 10−50.37
21A(4)a3,dav6.55266 × 10−63.8 × 10−80.58
22hf(T) parametersa0fa0f−1,143,423.78543.0.05
23a1fa1f4186.507681.80.04
24hg(T) parametersa0ga0g2,005,743.9910460.05
25a1ga1g1815.4373.50.19
26Nu parameters-a0,Nu8.2352.05925
27-a1,Nu0.003149870.00131.75
28-a2,Nu0.99029870.32733.02
29-a3,Nu0.0230.008838.26
30Cooling tower deck width in x-dir. (m)dkxwWdkx8.50.0851
31Cooling tower deck width in y-dir. (m)dkywWdky8.50.0851
32Cooling tower deck height above ground (m)dkhtΔzdk100.11
33Fan shroud height (m)fshtΔzfan3.00.031
34Fan shroud inner diameter (m)fsidDfan4.10.0411
35Fill section height (m)flhtΔzfill2.0130.020131
36Rain section height (m)rshtΔzrain1.6330.016331
37Basin section height (m)bshtΔzbs1.1680.011681
38Drift eliminator thickness (m)detkΔzde0.15240.0015241
39Fill section equivalent diameter (m)deqvDh0.03810.0003811
40Fill section flow area (m2)flfaAfill67.296.72910
41Fill section surface area (m2)flsaAsurf142213555.325
42Prandlt number of air at T = 80 CPrPr0.7080.0050.71
43Wind speed (m/s)wspdVw1.800.9251.1
44Exit air speed at the shroud (m/s)vexitVexit10.01.010.0
Index i of αiBoundary ParametersC + + StringMath. NotationNominal ValueAbsolute Standard DeviationRelative Standard Deviation (%)
45Inlet water mass flow rate (kg/s)mfwinmw,in44.022.2015
46Inlet air temperature (K)tainTa,inset to T d b 4.171.39
47Inlet air mass flow rate (kg/s)mainma155.0715.9110.26
48Inlet air humidity ratio (Dependent Scalar Parameter)hrin ω i n ; ω r a i n 0.01380.0020614.93
49Reynold’s numberRe; RehRed4428671.615.17
50Schmidt numberScSc0.600.07412.41
51Sherwood numberShSh14.134.8434.25
52Nusselt numberNuNu14.945.0834.00
Table A2. Dependent scalar model parameters.
Table A2. Dependent scalar model parameters.
Dependent Scalar ParametersMath. NotationDefining Equation or Correlation
Mass diffusivity of water vapor in air (m2/s)Dav(Ta,α) a 0 , d a v T 1.5 a 1 , d a v + ( a 2 , d a v + a 3 , d a v T ) T
Heat transfer coefficient (W/m2·K)h(α) f h t N u k a i r D h
Mass transfer coefficient (m/s)km(α) f m t S h D a v ( T d b , α ) D h
Heat transfer term (W/K)H(ma,α) h ( α ) w t s a A f f
Mass transfer term (m3/s)M(ma,α) M H 2 O k m ( α ) w t s a A f f
Density of dry air (kg/m3)ρ(α) P a t m R a i r T d b
Air velocity in the fill section (m/s)va(ma,α) | m a | ρ ( α ) A f i l l
Fill falling-film surface area per vertical section (m2)Aff A s u r f I
Rain section inlet flow area (m2)Ain W d k x W d k y
Height for natural convection (m)Z z d k + z f a n z b s
Height above fill section (m)Δz4–2 Z z f i l l z r a i n
Fill section control volume height (m)Δz z f i l l I
Fill section length, including drift eliminator (m)Lfill z f i l l + z d e
Fan shroud inner radius (m)rfan 0.5 D f a n
Fan shroud flow area (m2)Aout π r f a n 2
Table A3. Thermal properties (dependent scalar model parameters).
Table A3. Thermal properties (dependent scalar model parameters).
Thermal Properties (Functions of State Variables)Math. NotationDefining Equation or Correlation
hf(Tw) = saturated liquid enthalpy (J/kg)hf(Tw,α) a 0 f + a 1 f T w
Hg(Tw) = saturated vapor enthalpy (J/kg)hg,w(Tw,α) a 0 g + a 1 g T w
Hg(Ta) = saturated vapor enthalpy (J/kg)hg,a(Ta,α) a 0 g + a 1 g T a
Cp(T) = specific heat of dry air (J/kg·K)Cp(T,α) a 0 , c p a + ( a 1 , c p a + a 2 , c p a T ) T
Pvs(Tw) = saturation pressure (Pa)Pvs(Tw,α) P c e a 0 + a 1 T w , in which Pc = 1.0 Pa
Pvs(Ta) = saturation pressure (Pa)Pvs(Ta,α) P c e a 0 + a 1 T a , in which Pc = 1.0 Pa
Note: The parameters α1 through α4 (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].
Recall that the cooling tower model comprises conservation balances representing mathematically the following physical phenomena: A. liquid continuity; B. liquid energy balance; C. water vapor continuity; D. air and water vapor energy balance. For easy reference, these conservation equations are reproduced below from Section 2 of PART I [1]:
  • Liquid continuity equations:
    (i) 
    Control Volume i = 1:
    N 1 ( 1 ) ( m w , T w , T a , ω ; α ) m w ( 2 ) m w , i n + M ( m a , α ) R ¯ [ P v s ( 2 ) ( T w ( 2 ) , α ) T w ( 2 ) ω ( 1 ) P a t m T a ( 1 ) ( 0.622 + ω ( 1 ) ) ] = 0 ;
    (ii) 
    Control Volumes i = 2,..., I − 1:
    N 1 ( i ) ( m w , T w , T a , ω ; α ) m w ( i + 1 ) m w ( i ) + M ( m a , α ) R ¯ [ P v s ( i + 1 ) ( T w ( i + 1 ) , α ) T w ( i + 1 ) ω ( i ) P a t m T a ( i ) ( 0.622 + ω ( i ) ) ] = 0 ;
    (iii) 
    Control Volume i = I:
    N 1 ( I ) ( m w , T w , T a , ω ; α ) m w ( I + 1 ) m w ( I ) + M ( m a , α ) R ¯ [ P v s ( I + 1 ) ( T w ( I + 1 ) , α ) T w ( I + 1 ) ω ( I ) P a t m T a ( I ) ( 0.622 + ω ( I ) ) ] = 0 ;
  • Liquid energy balance equations:
    (i) 
    Control Volume i = 1:
    N 2 ( 1 ) ( m w , T w , T a , ω ; α ) m w , i n h f ( T w , i n , α ) ( T w ( 2 ) T a ( 1 ) ) H ( m a , α )           m w ( 2 ) h f ( 2 ) ( T w ( 2 ) , α ) ( m w , i n m w ( 2 ) ) h g , w ( 2 ) ( T w ( 2 ) , α ) = 0 ;
    (ii) 
    Control Volumes i = 2,..., I − 1:
    N 2 ( i ) ( m w , T w , T a , ω ; α ) m w ( i ) h f ( i ) ( T w ( i ) , α ) ( T w ( i + 1 ) T a ( i ) ) H ( m a , α )      m w ( i + 1 ) h f ( i + 1 ) ( T w ( i + 1 ) , α ) ( m w ( i ) m w ( i + 1 ) ) h g , w ( i + 1 ) ( T w ( i + 1 ) , α ) = 0 ;
    (iii) 
    Control Volume i = I:
    N 2 ( I ) ( m w , T w , T a , ω ; α ) m w ( I ) h f ( I ) ( T w ( I ) , α ) ( T w ( I + 1 ) T a ( I ) ) H ( m a , α )      m w ( I + 1 ) h f ( I + 1 ) ( T w ( I + 1 ) , α ) ( m w ( I ) m w ( I + 1 ) ) h g , w ( I + 1 ) ( T w ( I + 1 ) , α ) = 0 ;
  • Water vapor continuity equations:
    (i) 
    Control Volume i = 1:
    N 3 ( 1 ) ( m w , T w , T a , ω ; α ) ω ( 2 ) ω ( 1 ) + m w . i n m w ( 2 ) | m a | = 0 ;
    (ii) 
    Control Volumes i = 2,..., I − 1:
    N 3 ( i ) ( m w , T w , T a , ω ; α ) ω ( i + 1 ) ω ( i ) + m w ( i ) m w ( i + 1 ) | m a | = 0 ;
    (iii) 
    Control Volume i = I:
    N 3 ( I ) ( m w , T w , T a , ω ; α ) ω i n ω ( I ) + m w ( I ) m w ( I + 1 ) | m a | = 0 ;
  • The air/water vapor energy balance equations:
    (i) 
    Control Volume i = 1:
    N 4 ( 1 ) ( m w , T w , T a , ω ; α ) ( T a ( 2 ) T a ( 1 ) ) C p ( 1 ) ( T a ( 1 ) + 273.15 2 , α ) ω ( 1 ) h g , a ( 1 ) ( T a ( 1 ) , α ) + ( T w ( 2 ) T a ( 1 ) ) H ( m a , α ) | m a | + ( m w , i n m w ( 2 ) ) h g , w ( 2 ) ( T w ( 2 ) , α ) | m a | + ω ( 2 ) h g , a ( 2 ) ( T a ( 2 ) , α ) = 0 ;
    (ii) 
    Control Volumes i = 2,..., I − 1:
    N 4 ( i ) ( m w , T w , T a , ω ; α ) ( T a ( i + 1 ) T a ( i ) ) C p ( i ) ( T a ( i ) + 273.15 2 , α ) ω ( i ) h g , a ( i ) ( T a ( i ) , α ) + ( T w ( i + 1 ) T a ( i ) ) H ( m a , α ) | m a | + ( m w ( i ) m w ( i + 1 ) ) h g , w ( i + 1 ) ( T w ( i + 1 ) , α ) | m a | + ω ( i + 1 ) h g , a ( i + 1 ) ( T a ( i + 1 ) , α ) = 0 ;
    (iii) 
    Control Volume i = I:
    N 4 ( I ) ( m w , T w , T a , ω ; α ) ( T a , i n T a ( I ) ) C p ( I ) ( T a ( I ) + 273.15 2 , α ) ω ( I ) h g , a ( I ) ( T a ( I ) , α ) + ( T w ( I + 1 ) T a ( I ) ) H ( m a , α ) | m a | + ( m w ( I ) m w ( I + 1 ) ) h g , w ( I + 1 ) ( T w ( I + 1 ) , α ) | m a | + ω i n h g , a ( T a , i n , α ) = 0 .
The components of the vector α, which appears in Equations (A1)–(A12), comprise the model parameters, i.e.:
α ( α 1 , , α N α )
where Nα denotes the total number of model parameters. These model parameters are described in Table A1.
The following notation will be used for the derivatives of the above equations with respect to the parameters:
a i , j N ( i ) α ( j ) ;   = 1 ,   2 ,   3 ,   4 ;   i = 1 , , I ;   j = 1 , , N α .

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

The derivatives of the “liquid continuity equations” [cf. Equations (A1)–(A3)] with respect to the parameter α(1)Tdb are as follows:
N 1 ( i ) α ( 1 ) = N 1 ( i ) T d b a 1 i , 1 = 1 R ¯ [ P v s ( i + 1 ) ( T w ( i + 1 ) , α ) T w ( i + 1 ) ω ( i ) P a t m ( 0.622 + ω ( i ) ) T a ( i ) ] M ( m a , α ) D a v ( T d b , α ) D a v ( T d b , α ) T d b ; = 1 ;   i = 1 , , I ;   j = 1 ,
where:
M ( m a , α ) D a v ( T d b , α ) = 2 3 M ( m a , α ) D a v ( T d b , α )
D a v ( T d b , α ) T d b = 1.5 a 0 d a v T d b 0.5 D a v ( T d b , α ) ( a 2 d a v + 2 a 3 d a v T d b ) a 1 d a v + a 2 d a v T d b + a 3 d a v T d b 2
The derivatives of the “liquid continuity equations” [cf. Equations (A1)–(A3)] with respect to the parameter α(2)Tdp are as follows:
N 1 ( i ) α ( 2 ) = N 1 ( i ) T d p a 1 i , 2 = 0 ; = 1 ;   i = 1 , , I ;   j = 2 .
The derivatives of the “liquid continuity equations” [cf. Equations (A1)–(A3)] with respect to the parameter α(3)Tw,in are as follows:
N 1 ( 1 ) α ( 3 ) = N 1 ( 1 ) T w , i n a 1 1 , 3 = m w , i n T w , i n ; = 1 ;   i = 1 ;   j = 3 ,
where:
m w , i n T w , i n = T w , i n [ ρ ( T w , i n ) 700.0 15850.32 ] = [ a 2 , ρ + 2 a 3 , ρ ( T w , i n 273.15 ) + 3 a 4 , ρ ( T w , i n 273.15 ) 2 ] 700.0 15850.32 ;
and where a2,ρ = –0.26847207; a3,ρ = –1.8113691 × 10–3; a4,ρ = –1.7041217 × 10–6.
N 1 ( i ) α ( 3 ) = N 1 ( i ) T w , i n a 1 i , 3 = 0 ; = 1 ;   i = 2 , , I ;   j = 3 .
The derivatives of the “liquid continuity equations” [cf. Equations (A1)–(A3)] with respect to the parameter α(4)Patm are as follows:
N 1 ( i ) α ( 4 ) = N 1 ( i ) P a t m a 1 i , 4 = M ( m a , α ) R ¯ ω ( i ) T a ( i ) ( 0.622 + ω ( i ) ) + 1 R ¯ [ P v s ( i + 1 ) ( T w ( i + 1 ) , α ) T w ( i + 1 ) ω ( i ) P a t m ( 0.622 + ω ( i ) ) T a ( i ) ] M ( m a , α ) N u ( Re , α ) N u ( Re , α ) m a m a P a t m ;     = 1 ;   i = 1 , , I ;   j = 4 ,
where:
M ( Re , α ) N u ( Re , α ) = M ( m a , α ) N u ( Re , α ) ,
N u ( Re , α ) m a = { 0 Re d < 2300 a 1 , N u Re ( m a , α ) / m a 2300 Re d 10000 0.8 N u ( Re , α ) / m a Re d > 10000 ,
m a P a t m = 1 R a i r T a , i n V e x i t π D f a n 2 4 ;
Note: The term on the right hand side of Equation (A25) stems from the following relation:
m a = ρ ( T a ) V e x i t π D f a n 2 4 = P a t m R a i r T a V e x i t π D f a n 2 4 .
The derivatives of the “liquid continuity equations” [cf. Equations (A1)–(A3)] with respect to the parameter α(5)wtsa are as follows:
N 1 ( i ) α ( 5 ) = N 1 ( i ) w t s a a 1 i , 5 = 1 R ¯ [ P v s ( i + 1 ) ( T w ( i + 1 ) , α ) T w ( i + 1 ) ω ( i ) P a t m ( 0.622 + ω ( i ) ) T a ( i ) ] M ( m a , α ) w t s a ; = 1 ;   i = 1 , , I ;   j = 5 ,
where:
M ( m a , α ) w t s a = M H 2 O f m t N u ( Re , α ) ( ν Pr ) 1 3 [ D a v ( T d b , α ) ] 2 3 A s u r f D h I
The derivatives of the “liquid continuity equations” [cf. Equations (A1)–(A3)] with respect to the parameter α(6)ksum are as follows:
N 1 ( i ) α ( 6 ) = N 1 ( i ) k s u m a 1 i , 6 = 0 ; = 1 ;   i = 1 , , I ;   j = 6 .
The derivatives of the “liquid continuity equations” [cf. Equations (A1)–(A3)] with respect to the parameter α(7)μ are as follows:
N 1 ( i ) α ( 7 ) = N 1 ( i ) μ a 1 i , 7 = 1 R ¯ [ P v s ( i + 1 ) ( T w ( i + 1 ) , α ) T w ( i + 1 ) ω ( i ) P a t m ( 0.622 + ω ( i ) ) T a ( i ) ] M ( m a , α ) μ ; = 1 ;   i = 1 , , I ;   j = 7 ,
where:
M ( m a , α ) μ = { 0 Re d < 2300 a 1 , N u M ( m a , α ) Re ( m a , α ) N u ( Re , α ) μ 2300 Re d 10000 0.8 M ( m a , α ) μ Re d > 10000 .
The derivatives of the “liquid continuity equations” [cf. Equations (A1)–(A3)] with respect to the parameter α(8)υ are as follows:
N 1 ( i ) α ( 8 ) = N 1 ( i ) υ a 1 i , 8 = 1 R ¯ [ P v s ( i + 1 ) ( T w ( i + 1 ) , α ) T w ( i + 1 ) ω ( i ) P a t m ( 0.622 + ω ( i ) ) T a ( i ) ] M ( m a , α ) υ ; = 1 ;   i = 1 , , I ;   j = 8 ,
where:
M ( m a , α ) υ = 1 3 M ( m a , α ) υ .
The derivatives of the “liquid continuity equations” [cf. Equations (A1)–(A3)] with respect to the parameter α(9)kair are as follows:
N 1 ( i ) α ( 9 ) = N 1 ( i ) k a i r a 1 i , 9 = 0 ; = 1 ;   i = 1 , , I ;   j = 9 .
The derivatives of the “liquid continuity equations” [cf. Equations (A1)–(A3)] with respect to the parameter α(10)fht are as follows:
N 1 ( i ) α ( 10 ) = N 1 ( i ) f h t a 1 i , 10 = 0 ; = 1 ;   i = 1 , , I ;   j = 10 .
The derivatives of the “liquid continuity equations” [cf. Equations (A1)–(A3)] with respect to the parameter α(11)fmt are as follows:
N 1 ( i ) α ( 11 ) = N 1 ( i ) f m t a 1 i , 11 = 1 R ¯ [ P v s ( i + 1 ) ( T w ( i + 1 ) , α ) T w ( i + 1 ) ω ( i ) P a t m ( 0.622 + ω ( i ) ) T a ( i ) ] M ( m a , α ) f m t ; = 1 ;   i = 1 , , I ;   j = 11 ,
where:
M ( m a , α ) f m t = M H 2 O N u ( Re , α ) ( ν Pr ) 1 3 [ D a v ( T d b , α ) ] 2 3 w t s a A s u r f D h I .
The derivatives of the “liquid continuity equations” [cf. Equations (A1)–(A3)] with respect to the parameter α(12)f are as follows:
N 1 ( i ) α ( 12 ) = N 1 ( i ) f a 1 i , 12 = 0 ; = 1 ;   i = 1 , , I ;   j = 12 .
The derivatives of the “liquid continuity equations” [cf. Equations (A1)–(A3)] with respect to the parameter α(13)a0 are as follows:
N 1 ( i ) α ( 13 ) = N 1 ( i ) a 0 a 1 i , 13 = M ( m a , α ) R ¯ 1 T w ( i + 1 ) P v s ( i + 1 ) ( T w ( i + 1 ) , α ) a 0 ; = 1 ;   i = 1 , , I ;   j = 13 ,
where:
P v s ( i + 1 ) ( T w ( i + 1 ) , α ) a 0 = P v s ( i + 1 ) ( T w ( i + 1 ) , α ) .
The derivatives of the “liquid continuity equations” [cf. Equations (A1)–(A3)] with respect to the parameter α(14)a1 are as follows:
N 1 ( i ) α ( 14 ) = N 1 ( i ) a 1 a 1 i , 14 = M ( m a , α ) R ¯ 1 T w ( i + 1 ) P v s ( i + 1 ) ( T w ( i + 1 ) , α ) a 1 ; = 1 ;   i = 1 , , I ;   j = 14 ,
where:
P v s ( i + 1 ) ( T w ( i + 1 ) , α ) a 1 = P v s ( i + 1 ) ( T w ( i + 1 ) , α ) T w ( i + 1 ) .
The derivatives of the “liquid continuity equations” [cf. Equations (A1)–(A3)] with respect to the parameter α(15)a0,cpa are as follows:
N 1 ( i ) α ( 15 ) = N 1 ( i ) a 0 , c p a a 1 i , 15 = 0 ; = 1 ;   i = 1 , , I ;   j = 15 .
The derivatives of the “liquid continuity equations” [cf. Equations (A1)–(A3)] with respect to the parameter α(16)a1,cpa are as follows:
N 1 ( i ) α ( 16 ) = N 1 ( i ) a 1 , c p a a 1 i , 16 = 0 ; = 1 ;   i = 1 , , I ;   j = 16 .
The derivatives of the “liquid continuity equations” [cf. Equations (A1)–(A3)] with respect to the parameter α(17)a2,cpa are as follows:
N 1 ( i ) α ( 17 ) = N 1 ( i ) a 2 , c p a a 1 i , 17 = 0 ; = 1 ;   i = 1 , , I ;   j = 17 .
The derivatives of the “liquid continuity equations” [cf. Equations (A1)–(A3)] with respect to the parameter α(18)a0,dav are as follows:
N 1 ( i ) α ( 18 ) = N 1 ( i ) a 0 , d a v a 1 i , 18 = 1 R ¯ [ P v s ( i + 1 ) ( T w ( i + 1 ) , α ) T w ( i + 1 ) ω ( i ) P a t m ( 0.622 + ω ( i ) ) T a ( i ) ] M ( m a , α ) D a v ( T d b , α ) D a v ( T d b , α ) a 0 , d a v ; = 1 ;   i = 1 , , I ;   j = 18 ,
where M ( m a , α ) D a v ( T d b , α ) was defined previously in Equation (A16), and:
D a v ( T d b , α ) a 0 , d a v = T d b 1.5 a 1 d a v + a 2 d a v T d b + a 3 d a v T d b 2 .
The derivatives of the “liquid continuity equations” [cf. Equations (A1)–(A3)] with respect to the parameter α(19)a1,dav are as follows:
N 1 ( i ) α ( 19 ) = N 1 ( i ) a 1 , d a v a 1 i , 19 = 1 R ¯ [ P v s ( i + 1 ) ( T w ( i + 1 ) , α ) T w ( i + 1 ) ω ( i ) P a t m ( 0.622 + ω ( i ) ) T a ( i ) ] M ( m a , α ) D a v ( T d b , α ) D a v ( T d b , α ) a 1 , d a v ; = 1 ;   i = 1 , , I ;   j = 19 ,
where M ( m a , α ) D a v ( T d b , α ) was defined previously in Equation (A16), and:
D a v ( T d b , α ) a 1 , d a v = a 0 d a v T d b 1.5 ( a 1 d a v + a 2 d a v T d b + a 3 d a v T d b 2 ) 2 .
The derivatives of the “liquid continuity equations” [cf. Equations (A1)–(A3)] with respect to the parameter α(20)a2,dav are as follows:
N 1 ( i ) α ( 20 ) = N 1 ( i ) a 2 , d a v a 1 i , 20 = 1 R ¯ [ P v s ( i + 1 ) ( T w ( i + 1 ) , α ) T w ( i + 1 ) ω ( i ) P a t m ( 0.622 + ω ( i ) ) T a ( i ) ] M ( m a , α ) D a v ( T d b , α ) D a v ( T d b , α ) a 2 , d a v ; = 1 ;   i = 1 , , I ;   j = 20 ,
where M ( m a , α ) D a v ( T d b , α ) was defined previously in Equation (A16), and
D a v ( T d b , α ) a 2 , d a v = a 0 d a v T d b 2.5 ( a 1 d a v + a 2 d a v T d b + a 3 d a v T d b 2 ) 2 .
The derivatives of the “liquid continuity equations” [cf. Equations (A1)–(A3)] with respect to the parameter α(21)a3,dav are as follows:
N 1 ( i ) α ( 21 ) = N 1 ( i ) a 3 , d a v a 1 i , 21 = 1 R ¯ [ P v s ( i + 1 ) ( T w ( i + 1 ) , α ) T w ( i + 1 ) ω ( i ) P a t m ( 0.622 + ω ( i ) ) T a ( i ) ] M ( m a , α ) D a v ( T d b , α ) D a v ( T d b , α ) a 3 , d a v ; = 1 ;   i = 1 , , I ;   j = 21 ,
where M ( m a , α ) D a v ( T d b , α ) was defined previously in Equation (A16), and
D a v ( T d b , α ) a 3 , d a v = a 0 d a v T d b 3.5 ( a 1 d a v + a 2 d a v T d b + a 3 d a v T d b 2 ) 2 .
The derivatives of the “liquid continuity equations” [cf. Equations (A1)–(A3)] with respect to the parameter α(22)a0f are as follows:
N 1 ( i ) α ( 22 ) = N 1 ( i ) a 0 f a 1 i , 22 = 0 ; = 1 ;   i = 1 , , I ;   j = 22 .
The derivatives of the “liquid continuity equations” [cf. Equations (A1)–(A3)] with respect to the parameter α(23)a1f are as follows:
N 1 ( i ) α ( 23 ) = N 1 ( i ) a 1 f a 1 i , 23 = 0 ; = 1 ;   i = 1 , , I ;   j = 23 .
The derivatives of the “liquid continuity equations” [cf. Equations (A1)–(A3)] with respect to the parameter α(24)a0g are as follows:
N 1 ( i ) α ( 24 ) = N 1 ( i ) a 0 g a 1 i , 24 = 0 ; = 1 ;   i = 1 , , I ;   j = 24 .
The derivatives of the “liquid continuity equations” [cf. Equations (A1)–(A3)] with respect to the parameter α(25)a1g are as follows:
N 1 ( i ) α ( 25 ) = N 1 ( i ) a 1 g a 1 i , 25 = 0 ; = 1 ;   i = 1 , , I ;   j = 25 .
The derivatives of the “liquid continuity equations” [cf. Equations (A1)–(A3)] with respect to the parameter α(26)a0,Nu are as follows:
N 1 ( i ) α ( 26 ) = N 1 ( i ) a 0 , N u a 1 i , 26 = 1 R ¯ [ P v s ( i + 1 ) ( T w ( i + 1 ) , α ) T w ( i + 1 ) ω ( i ) P a t m ( 0.622 + ω ( i ) ) T a ( i ) ] M ( m a , α ) N u ( Re , α ) N u ( Re , α ) a 0 , N u ; = 1 ;   i = 1 , , I ;   j = 26 ,
where M ( m a , α ) N u ( Re , α ) was defined previously in Equation (A23), and
N u ( Re , α ) a 0 , N u = { 1 Re d < 2300 0 2300 Re d 10000 0 Re d > 10000 .
The derivatives of the “liquid continuity equations” [cf. Equations (A1)–(A3)] with respect to the parameter α(27)a1,Nu are as follows:
N 1 ( i ) α ( 27 ) = N 1 ( i ) a 1 , N u a 1 i , 27 = 1 R ¯ [ P v s ( i + 1 ) ( T w ( i + 1 ) , α ) T w ( i + 1 ) ω ( i ) P a t m ( 0.622 + ω ( i ) ) T a ( i ) ] M ( m a , α ) N u ( Re , α ) N u ( Re , α ) a 1 , N u ; = 1 ;   i = 1 , , I ;   j = 27 ,
where M ( m a , α ) N u ( Re , α ) was defined previously in Equation (A23), and:
N u ( Re , α ) a 1 , N u = { 0 Re d < 2300 Re ( m a , α ) 2300 Re d 10000 0 Re d > 10000 .
The derivatives of the “liquid continuity equations” [cf. Equations (A1)–(A3)] with respect to the parameter α(28)a2,Nu are as follows:
N 1 ( i ) α ( 28 ) = N 1 ( i ) a 2 , N u a 1 i , 28 = 1 R ¯ [ P v s ( i + 1 ) ( T w ( i + 1 ) , α ) T w ( i + 1 ) ω ( i ) P a t m ( 0.622 + ω ( i ) ) T a ( i ) ] M ( m a , α ) N u ( Re , α ) N u ( Re , α ) a 2 , N u ; = 1 ;   i = 1 , , I ;   j = 28 ,
where M ( m a , α ) N u ( Re , α ) was defined previously in Equation (A23), and:
N u ( Re , α ) a 2 , N u = { 0 Re d < 2300 1 2300 Re d 10000 0 Re d > 10000
The derivatives of the “liquid continuity equations” [cf. Equations (A1)–(A3)] with respect to the parameter α(29)a3,Nu are as follows:
N 1 ( i ) α ( 29 ) = N 1 ( i ) a 3 ,   N u a 1 i , 29 = 1 R ¯ [ P v s ( i + 1 ) ( T w ( i + 1 ) , α ) T w ( i + 1 ) ω ( i ) P a t m ( 0.622 + ω ( i ) ) T a ( i ) ] M ( m a , α ) N u ( Re , α ) N u ( Re , α ) a 3 , N u ; = 1 ;   i = 1 , , I ;   j = 29 ,
where M ( m a , α ) N u ( Re , α ) was defined previously in Equation (A23), and:
N u ( Re , α ) a 3 , N u = { 0 Re d < 2300 0 2300 Re d 10000 [ Re ( m a , α ) ] 0.8 Pr 1 3 Re d > 10000 .
The derivatives of the “liquid continuity equations” [cf. Equations (A1)–(A3)] with respect to the parameter α(30)Wdkx are as follows:
N 1 ( i ) α ( 30 ) = N 1 ( i ) W d k x a 1 i , 30 = 0 ; = 1 ;   i = 1 , , I ;   j = 30 .
The derivatives of the “liquid continuity equations” [cf. Equations (A1)–(A3)] with respect to the parameter α(31)Wdky are as follows:
N 1 ( i ) α ( 31 ) = N 1 ( i ) W d k y a 1 i , 31 = 0 ; = 1 ;   i = 1 , , I ;   j = 31 .
The derivatives of the “liquid continuity equations” [cf. Equations (A1)–(A3)] with respect to the parameter α(32) ≡ Δzdk are as follows:
N 1 ( i ) α ( 32 ) = N 1 ( i ) Δ z d k a 1 i , 32 = 0 ; = 1 ;   i = 1 , , I ;   j = 32 .
The derivatives of the “liquid continuity equations” [cf. Equations (A1)–(A3)] with respect to the parameter α(33) ≡ Δzfan are as follows:
N 1 ( i ) α ( 33 ) = N 1 ( i ) Δ z f a n a 1 i , 33 = 0 ; = 1 ;   i = 1 , , I ;   j = 33 .
The derivatives of the “liquid continuity equations” [cf. Equations (A1)–(A3)] with respect to the parameter α(34)Dfan are as follows:
N 1 ( i ) α ( 34 ) = N 1 ( i ) D f a n a 1 i , 34 = 1 R ¯ [ P v s ( i + 1 ) ( T w ( i + 1 ) , α ) T w ( i + 1 ) ω ( i ) P a t m ( 0.622 + ω ( i ) ) T a ( i ) ] M ( m a , α ) N u ( Re , α ) N u ( Re , α ) m a m a D f a n ; = 1 ;   i = 1 , , I ;   j = 34 ,
where M ( m a , α ) N u ( Re , α ) and N u ( Re , α ) m a were defined previously in Equations (A23) and (A24), respectively, and:
m a D f a n = 2 m a D f a n .
The derivatives of the “liquid continuity equations” [cf. Equations (A1)–(A3)] with respect to the parameter α(35) ≡ Δzfill are as follows:
N 1 ( i ) α ( 35 ) = N 1 ( i ) Δ z f i l l a 1 i , 35 = 0 ; = 1 ;   i = 1 , , I ;   j = 35 .
The derivatives of the “liquid continuity equations” [cf. Equations (A1)–(A3)] with respect to the parameter α(36) ≡ Δzrain are as follows:
N 1 ( i ) α ( 36 ) = N 1 ( i ) Δ z r a i n a 1 i , 36 = 0 ; = 1 ;   i = 1 , , I ;   j = 36 .
The derivatives of the “liquid continuity equations” [cf. Equations (A1)–(A3)] with respect to the parameter α(37) ≡ Δzbs are as follows:
N 1 ( i ) α ( 37 ) = N 1 ( i ) Δ z b s a 1 i , 37 = 0 ; = 1 ;   i = 1 , , I ;   j = 37 .
The derivatives of the “liquid continuity equations” [cf. Equations (A1)–(A3)] with respect to the parameter α(38) ≡ Δzde are as follows:
N 1 ( i ) α ( 38 ) = N 1 ( i ) Δ z d e a 1 i , 38 = 0 ; = 1 ;   i = 1 , , I ;   j = 38 .
The derivatives of the “liquid continuity equations” [cf. Equations (A1)–(A3)] with respect to the parameter α(39)Dh are as follows:
N 1 ( i ) α ( 39 ) = N 1 ( i ) D h a 1 i , 39 = 1 R ¯ [ P v s ( i + 1 ) ( T w ( i + 1 ) , α ) T w ( i + 1 ) ω ( i ) P a t m ( 0.622 + ω ( i ) ) T a ( i ) ] M ( m a , α ) D h ; = 1 ;   i = 1 , , I ;   j = 39 ,
where:
M ( m a , α ) D h = { M ( m a , α ) / D h Re d < 2300 a 2 , N u M ( m a , α ) N u ( Re , α ) D h 2300 Re d 10000 0.2 M ( m a , α ) / D h Re d > 10000 .
The derivatives of the “liquid continuity equations” [cf. Equations (A1)–(A3)] with respect to the parameter α(40)Afill are as follows:
N 1 ( i ) α ( 40 ) = N 1 ( i ) A f i l l a 1 i , 40 = 1 R ¯ [ P v s ( i + 1 ) ( T w ( i + 1 ) , α ) T w ( i + 1 ) ω ( i ) P a t m ( 0.622 + ω ( i ) ) T a ( i ) ] M ( m a , α ) A f i l l ; = 1 ;   i = 1 , , I ;   j = 40 ,
where:
M ( m a , α ) A f i l l = { 0 Re d < 2300 a 1 , N u M ( m a , α ) Re ( m a , α ) N u ( Re , α ) A f i l l 2300 Re d 10000 0.8 M ( m a , α ) / A f i l l Re d > 10000 .
The derivatives of the “liquid continuity equations” [cf. Equations (A1)–(A3)] with respect to the parameter α(41)Asurf are as follows:
N 1 ( i ) α ( 41 ) = N 1 ( i ) A s u r f a 1 i , 41 = 1 R ¯ [ P v s ( i + 1 ) ( T w ( i + 1 ) , α ) T w ( i + 1 ) ω ( i ) P a t m ( 0.622 + ω ( i ) ) T a ( i ) ] M ( m a , α ) A s u r f ; = 1 ;   i = 1 , , I ;   j = 41 ,
where:
M ( m a , α ) A s u r f = M ( m a , α ) A s u r f
The derivatives of the “liquid continuity equations” [cf. Equations (A1)–(A3)] with respect to the parameter α(42) ≡ Pr are as follows:
N 1 ( i ) α ( 42 ) = N 1 ( i ) Pr a 1 i , 42 = 1 R ¯ [ P v s ( i + 1 ) ( T w ( i + 1 ) , α ) T w ( i + 1 ) ω ( i ) P a t m ( 0.622 + ω ( i ) ) T a ( i ) ] M ( m a , α ) Pr ; = 1 ;   i = 1 , , I ;   j = 42 ,
where:
M ( m a , α ) Pr = { M ( m a , α ) / ( 3 Pr ) Re d 10000 0 Re d > 10000 .
The derivatives of the “liquid continuity equations” [cf. Equations (A1)–(A3)] with respect to the parameter α(43)Vw are as follows:
N 1 ( i ) α ( 43 ) = N 1 ( i ) V w a 1 i , 43 = 0 ; = 1 ;   i = 1 , , I ;   j = 43 .
The derivatives of the “liquid continuity equations” [cf. Equations (A1)–(A3)] with respect to the parameter α(44)Vexit are as follows:
N 1 ( i ) α ( 44 ) = N 1 ( i ) V e x i t a 1 i , 44 = 1 R ¯ [ P v s ( i + 1 ) ( T w ( i + 1 ) , α ) T w ( i + 1 ) ω ( i ) P a t m ( 0.622 + ω ( i ) ) T a ( i ) ] M ( m a , α ) N u ( Re , α ) N u ( Re , α ) m a m a V e x i t ; = 1 ;   i = 1 , , I ;   j = 44 ,
where M ( m a , α ) N u ( Re , α ) and N u ( Re , α ) m a were defined previously in Equations (A23) and (A24), respectively, and
m a V e x i t = P a t m R a i r T a , i n π D f a n 2 4
The derivatives of the “liquid continuity equations” [cf. Equations (A1)–(A3)] with respect to the parameter α(45)mw,in are as follows:
N 1 ( 1 ) α ( 45 ) = N 1 ( 1 ) m w , i n a 1 1 , 45 = 1 ; = 1 ;   i = 1 ;   j = 45 ,
N 1 ( i ) α ( 45 ) = N 1 ( i ) m w , i n a 1 i , 45 = 0 ; = 1 ;   i = 2 , , I ;   j = 45 .
The derivatives of the “liquid continuity equations” [cf. Equations (A1)–(A3)] with respect to the parameter α(46)Ta,in are as follows:
N 1 ( i ) α ( 46 ) = N 1 ( i ) T a , i n a 1 i , 46 = 0 ; = 1 ;   i = 1 , , I ;   j = 46 .
The derivatives of the “liquid continuity equations” [cf. Equations (A1)–(A3)] with respect to the parameter α(47)ma are as follows:
N 1 ( i ) α ( 47 ) = N 1 ( i ) m a a 1 i , 47 = 1 R ¯ [ P v s ( i + 1 ) ( T w ( i + 1 ) , α ) T w ( i + 1 ) ω ( i ) P a t m ( 0.622 + ω ( i ) ) T a ( i ) ] M ( m a , α ) N u ( Re , α ) N u ( Re , α ) m a ; = 1 ;   i = 1 , , I ;   j = 47 ,
where M ( m a , α ) N u ( Re , α ) and N u ( Re , α ) m a were defined previously in Equations (A23) and (A24), respectively.
The derivatives of the “liquid continuity equations” [cf. Equations (A1)–(A3)] with respect to the parameter α(48)ωin are as follows:
N 1 ( i ) α ( 48 ) = N 1 ( i ) ω i n a 1 i , 48 = 0 ; = 1 ;   i = 1 , , I ;   j = 48 .
The derivatives of the “liquid continuity equations” [cf. Equations (A1)–(A3)] with respect to the parameter α(49) ≡ Red are as follows:
N 1 ( i ) α ( 49 ) = N 1 ( i ) Re d a 1 i , 49 = 1 R ¯ [ P v s ( i + 1 ) ( T w ( i + 1 ) , α ) T w ( i + 1 ) ω ( i ) P a t m ( 0.622 + ω