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This work continues the review and illustrative application to energy systems of the “Fourth-Order Best-Estimate Results with Reduced Uncertainties Predictive Modeling” (4th-BERRU-PM) methodology. The 4th-BERRU-PM methodology uses the Maximum Entropy (MaxEnt) principle to incorporate fourth-order experimental and computational information, including fourth (and higher) order sensitivities of computed model responses with respect to model parameters. The 4th-BERRU-PM methodology yields the fourth-order MaxEnt posterior distribution of experimentally measured and computed model responses and parameters in the combined phase space of model responses and parameters. The 4th-BERRU-PM methodology encompasses fourth-order sensitivity analysis (SA) and uncertainty quantification (UQ), which were reviewed in the accompanying work (Part 1), as well as fourth-order data assimilation (DA) and model calibration (MC) capabilities, which will be reviewed and illustrated in this work (Part 2). The applicability of the 4th-BERRU-PM methodology to energy systems is illustrated by using the Polyethylene-Reflected Plutonium (acronym: PERP) OECD/NEA reactor physics benchmark, which is modeled using the linear neutron transport Boltzmann equation, involving 21,976 imprecisely known parameters. This benchmark is representative of “large-scale computations” such as those involved in the modeling of energy systems. The result (“response”) of interest for the PERP benchmark is the leakage of neutrons through the outer surface of this spherical benchmark, which can be computed numerically and measured experimentally. The impact of the high-order sensitivities of the response with respect to the PERP model parameters is quantified for “high-precision” parameters (2% standard deviations) and “typical-precision” parameters (5% standard deviations). Analyzing the best-estimate results with reduced uncertainties for the 1st—through 4th-order moments (mean values, covariance, skewness, and kurtosis) produced by the 4th-BERRU-PM methodology for the PERP benchmark indicates that, even for systems modeled by linear equations (e.g., the PERP benchmark), retaining only first-order sensitivities is insufficient for reliable predictive modeling (including SA, UQ, DA, and MC). At least second-order sensitivities should be retained in order to obtain reliable predictions.

This work (in two parts) will review the recently developed predictive modeling methodology called “4th-BERRU-PM” and its applicability to nuclear energy systems as exemplified by an illustrative application to the Polyethylene-Reflected Plutonium (acronym: PERP) OECD/NEA reactor physics benchmark. The acronym 4th-BERRU-PM designates the “Fourth-Order Best-Estimate Results with Reduced Uncertainties Predictive Modeling” methodology, which uses the Maximum Entropy (MaxEnt) principle to incorporate fourth-order experimental and computational information, including fourth (and higher) order sensitivities of computed model responses to model parameters, while yielding best-estimate results with reduced uncertainties for the first fourth-order moments (mean values, covariance, skewness, and kurtosis) of the optimally predicted posterior distribution of model results and calibrated model parameters. The 4th-BERRU-PM methodology encompasses the scopes of high-order sensitivity analysis (SA), uncertainty quantification (UQ), data assimilation (DA) and model calibration (MC), as will be illustrated in this work by means of the above-mentioned OECD/NEA reactor physics benchmark. This benchmark is modeled using the neutron transport Boltzmann equation involving 21,976 imprecisely known parameters, the solution of which is representative of “large-scale computations”. The model result (“response”) of interest is the leakage of neutrons through the outer surface of this spherical benchmark, which can be computed numerically and measured experimentally. Part 1 of this work illustrates the impact of high-order sensitivities, in conjunction with parameter standard deviations of various magnitudes, on the determination of the expected value and variance of the computed response in terms of the first four moments of the distribution of the uncertain model parameters. Part 2 of this work will illustrate the capabilities of the 4th-BERRU-PM methodology for combining computational and experimental information, up to and including forth-order sensitivities and distributional moments, for producing best-estimate values for the predicted responses and model parameters while reducing their accompanying uncertainties.

The Second-Order Features Adjoint Sensitivity Analysis Methodology for Response-Coupled Forward/Adjoint Linear Systems (abbreviated as “2nd-FASAM-L”), presented in this work, enables the most efficient computation of exactly obtained mathematical expressions of first- and second-order sensitivities of a generic system response with respect to the functions (“features”) of model parameters. Subsequently, the first- and second-order sensitivities with respect to the model’s uncertain parameters, boundaries, and internal interfaces are obtained analytically and exactly, without needing large-scale computations. Within the 2nd-FASAM-L methodology, the number of large-scale computations is proportional to the number of model features (defined as functions of model parameters), as opposed to being proportional to the number of model parameters. This characteristic enables the 2nd-FASAM-L methodology to maximize the efficiency and accuracy of any other method for computing exact expressions of first- and second-order response sensitivities with respect to the model’s features and/or primary uncertain parameters. The application of the 2nd-FASAM-L methodology is illustrated using a simplified energy-dependent neutron transport model of fundamental significance in nuclear reactor physics.

This work introduces a new methodology, which generalizes the extant second-order adjoint sensitivity analysis methodology for computing sensitivities of model responses to primary model parameters. This new methodology enables the computation, with unparalleled efficiency, of second-order sensitivities of responses to functions of uncertain model parameters, including uncertain boundaries and internal interfaces, for linear and/or nonlinear models. Such functions of primary model parameters customarily describe characteristic “features” of the system under consideration, including correlations modeling material properties, flow regimes, etc. The number of such “feature” functions is considerably smaller than the total number of primary model parameters. By enabling the computations of exact expressions of second-order sensitivities of model responses to model “features”, the number of required large-scale adjoint computations is greatly reduced. As shown in this work, obtaining the first- and second-order sensitivities to the primary model parameters from the corresponding response sensitivities to the feature functions can be performed analytically, thereby involving just the respective function/feature of parameters rather than the entire model. By replacing large-scale computations involving the model with relatively trivial computations involving just the feature functions, this new second-order adjoint sensitivity analysis methodology reaches unsurpassed efficiency. The applicability and unparalleled efficiency of this “2nd-Order Function/Feature Adjoint Sensitivity Analysis Methodology” (2nd-FASAM) is illustrated using a paradigm particle transport model that involves feature functions of many parameters, while admitting closed-form analytic solutions. Ongoing work will generalize the mathematical framework of the 2nd-FASAM to enable the computation of arbitrarily high-order sensitivities of model responses to functions/features of model parameters.

The mathematical/computational model of a physical system comprises parameters and independent and dependent variables. Since the physical system is seldom known precisely and since the model’s parameters stem from experimental procedures that are also subject to uncertainties, the results predicted by a computational model are imperfect. Quantifying the reliability and accuracy of results produced by a model (called “model responses”) requires the availability of sensitivities (i.e., functional partial derivatives) of model responses with respect to model parameters. This work reviews the basic motivations for computing high-order sensitivities and illustrates their importance by means of an OECD/NEA reactor physics benchmark, which is representative of a “large-scale system” involving many (21,976) uncertain parameters. The computation of higher-order sensitivities by conventional methods (finite differences and/or statistical procedures) is subject to the “curse of dimensionality”. Furthermore, as will be illustrated in this work, the accuracy of high-order sensitivities computed using such conventional methods cannot be a priori guaranteed. High-order sensitivities can be computed accurately and efficiently solely by applying the high-order adjoint sensitivity analysis methodology. The principles underlying this adjoint methodology are also reviewed in preparation for introducing, in the accompanying Part II, the “High-Order Function/Feature Adjoint Sensitivity Analysis Methodology” (nth-FASAM), which aims at most efficiently computing exact expressions of high-order sensitivities of model responses to functions (“features”) of model parameters.

This work presents a perspective on deterministic predictive modeling methodologies, which aim at extracting best-estimate values for model responses and parameters along with reduced predicted uncertainties for these best-estimate values. The two oldest such methodologies are the data-adjustment method, which stems from the nuclear energy field, and the data-assimilation method, which is implemented in the geophysical sciences. Both of these methodologies attempt to minimize, in the least-square sense, a user-defined functional that represents the discrepancies between computed and measured model responses. These two methodologies were briefly reviewed and shown to be inconsistent even to first-order in the sensitivities of the response to the model parameters. In contrast to these methodologies, it was shown that the “maximum entropy”-based predictive modeling methodology (called BERRU-PM) that was developed by the author not only dispenses with the subjective “user-chosen functional to be minimized” but is also inherently amenable to high-order formulations. This inherent potential was illustrated by presenting a novel, higher-order, MaxEnt-based predictive modeling methodology, labelled BERRU-PM-2+, which is complete and exact to second-order sensitivities and moments of both the a priori and posterior distributions of responses and parameters, while explicitly including third- and fourth-order sensitivities and correlations, thus indicating the mechanism for incorporating information of orders higher than second in predictive modeling. The presentation of this new predictive modeling methodology also aims at motivating a widespread application of predictive modeling principles and methodologies in the energy sciences for obtaining best-estimate results with reduced uncertainties.

This work reviews from a unified viewpoint the concepts underlying the “nth-Order Comprehensive Adjoint Sensitivity Analysis Methodology for Response-Coupled Forward/Adjoint Linear Systems” (nth-CASAM-L) and the “nth-Order Comprehensive Adjoint Sensitivity Analysis Methodology for Nonlinear Systems” (nth-CASAM-N) methodologies. The practical application of the nth-CASAM-L methodology is illustrated for an OECD/NEA reactor physics benchmark, while the practical application of the nth-CASAM-N methodology is illustrated for a nonlinear model of reactor dynamics that exhibits periodic and chaotic oscillations. As illustrated both by the general theory and by the examples reviewed in this work, both the nth-CASAM-L and nth-CASAM-N methodologies overcome the curse of dimensionality in sensitivity analysis. The availability of efficiently and exactly computed sensitivities of arbitrarily high order can lead to major advances in all areas that need such high-order sensitivities, including data assimilation, model calibration, uncertainty reduction, and predictive modeling.

The Special Issue “Sensitivity Analysis, Uncertainty Quantification and Predictive Modeling of Nuclear Energy Systems” comprises nine articles that present important applications of concepts for performing sensitivity analyses and uncertainty quantifications of models of nuclear energy systems [...]

The computational models of physical systems comprise parameters, independent and dependent variables [...]

This work presents the mathematical framework of the nth-Order Comprehensive Adjoint Sensitivity Analysis Methodology for Response-Coupled Forward/Adjoint Linear Systems (abbreviated as “nth-CASAM-L”), which is conceived for obtaining the exact expressions of arbitrarily-high-order (nth-order) sensitivities of a generic system response with respect to all of the parameters (including boundary and initial conditions) underlying the respective forward/adjoint systems. Since many of the most important responses for linear systems involve the solutions of both the forward and the adjoint linear models that correspond to the respective physical system, the sensitivity analysis of such responses makes it necessary to treat linear systems in their own right, rather than treating them as particular cases of nonlinear systems. This is in contradistinction to responses for nonlinear systems, which can depend only on the forward functions, since nonlinear operators do not admit bona-fide adjoint operators (only a linearized form of a nonlinear operator admits an adjoint operator). The nth-CASAM-L determines the exact expression of arbitrarily-high order sensitivities of responses to the parameters underlying both the forward and adjoint models of a nonlinear system, thus enable the most efficient and accurate computation of such sensitivities. The mathematical framework underlying the nth-CASAM is developed in linearly increasing higher-dimensional Hilbert spaces, as opposed to the exponentially increasing “parameter-dimensional” spaces in which response sensitivities are computed by other methods, thus providing the basis for overcoming the “curse of dimensionality” in sensitivity analysis and all other fields (uncertainty quantification, predictive modeling, etc.) which need such sensitivities. In particular, for a scalar-valued valued response associated with a nonlinear model comprising TP parameters, the 1st-−CASAM-L requires 1 additional large-scale adjoint computation (as opposed to TP large-scale computations, as required by other methods) for computing exactly all of the 1st-−order response sensitivities. All of the (mixed) 2nd-order sensitivities are computed exactly by the 2nd-CASAM-L in at most TP computations, as opposed to TP(TP + 1)/2 computations required by all other methods, and so on. For every lower-order sensitivity of interest, the nth-CASAM-L computes the “TP next-higher-order” sensitivities in one adjoint computation performed in a linearly increasing higher-dimensional Hilbert space. Very importantly, the nth-CASAM-L computes the higher-level adjoint functions using the same forward and adjoint solvers (i.e., computer codes) as used for solving the original forward and adjoint systems, thus requiring relatively minor additional software development for computing the various-order sensitivities.

This work illustrates the application of the nth-order comprehensive adjoint sensitivity analysis methodology for response-coupled forward/adjoint linear systems (abbreviated as “nth-CASAM-L”) to a paradigm model that describes the transmission of particles (neutrons and/or photons) through homogenized materials, as encountered in radiation protection and shielding. The first-, second-, and third-order sensitivities of responses that depend on both the forward and adjoint particle fluxes are obtained exactly, in closed-form, underscoring the principles and methodology underlying the nth-CASAM-L. The results presented in this work underscore the fundamentally important role of the nth-CASAM-L in the quest to overcome the “curse of dimensionality” in sensitivity analysis, uncertainty quantification and predictive modeling.

This work aims at underscoring the need for the accurate quantification of the sensitivities (i.e., functional derivatives) of the results (a.k.a. “responses”) produced by large-scale computational models with respect to the models’ parameters, which are seldom known perfectly in practice. The large impact that can arise from sensitivities of order higher than first has been highlighted by the results of a third-order sensitivity and uncertainty analysis of an OECD/NEA reactor physics benchmark, which will be briefly reviewed in this work to underscore that neglecting the higher-order sensitivities causes substantial errors in predicting the expectation and variance of model responses. The importance of accurately computing the higher-order sensitivities is further highlighted in this work by presenting a text-book analytical example from the field of neutron transport, which impresses the need for the accurate quantification of higher-order response sensitivities by demonstrating that their neglect would lead to substantial errors in predicting the moments (expectation, variance, skewness, kurtosis) of the model response’s distribution in the phase space of model parameters. The incorporation of response sensitivities in methodologies for uncertainty quantification, data adjustment and predictive modeling currently available for nuclear engineering systems is also reviewed. The fundamental conclusion highlighted by this work is that confidence intervals and tolerance limits on results predicted by models that only employ first-order sensitivities are likely to provide a false sense of confidence, unless such models also demonstrate quantitatively that the second- and higher-order sensitivities provide negligibly small contributions to the respective tolerance limits and confidence intervals. The high-order response sensitivities to parameters underlying large-scale models can be computed most accurately and most efficiently by employing the high-order comprehensive adjoint sensitivity analysis methodology, which overcomes the curse of dimensionality that hampers other methods when applied to large-scale models involving many parameters.

This work reviews the state-of-the-art methodologies for the deterministic sensitivity analysis of nonlinear systems and deterministic quantification of uncertainties induced in model responses by uncertainties in the model parameters. The need for computing high-order sensitivities is underscored by presenting an analytically solvable model of neutron scattering in a hydrogenous medium, for which all of the response’s relative sensitivities have the same absolute value of unity. It is shown that the wider the distribution of model parameters, the higher the order of sensitivities needed to achieve a desired level of accuracy in representing the response and in computing the response’s expectation, variance, skewness and kurtosis. This work also presents new mathematical expressions that extend to the sixth-order of the current state-of-the-art fourth-order formulas for computing fourth-order correlations among computed model response and model parameters. Another novelty presented in this work is the mathematical framework of the 3rd-Order Comprehensive Adjoint Sensitivity Analysis Methodology for Nonlinear Systems (3rd-CASAM-N), which enables the most efficient computation of the exact expressions of the 1st-, 2nd- and 3rd-order functional derivatives (“sensitivities”) of a model’s response to the underlying model parameters, including imprecisely known initial, boundary and/or interface conditions. The 2nd- and 3rd-level adjoint functions are computed using the same forward and adjoint computer solvers as used for solving the original forward and adjoint systems. Comparisons between the CPU times are also presented for an OECD/NEA reactor physics benchmark, highlighting the fact that finite-difference schemes would not only provide approximate values for the respective sensitivities (in contradistinction to the 3rd-CASAM-N, which provides exact expressions for the sensitivities) but would simply be unfeasible for computing sensitivities of an order higher than first-order. Ongoing work will generalize the 3rd-CASAM-N to a higher order while aiming to overcome the curse of dimensionality.

The most general quantities of interest (called “responses”) produced by the computational model of a linear physical system can depend on both the forward and adjoint state functions that describe the respective system. This work presents the Fourth-Order Comprehensive Adjoint Sensitivity Analysis Methodology (4th-CASAM) for linear systems, which enables the efficient computation of the exact expressions of the 1st-, 2nd-, 3rd- and 4th-order sensitivities of a generic system response, which can depend on both the forward and adjoint state functions, with respect to all of the parameters underlying the respective forward/adjoint systems. Among the best known such system responses are various Lagrangians, including the Schwinger and Roussopoulos functionals, for analyzing ratios of reaction rates, the Rayleigh quotient for analyzing eigenvalues and/or separation constants, etc., which require the simultaneous consideration of both the forward and adjoint systems when computing them and/or their sensitivities (i.e., functional derivatives) with respect to the model parameters. Evidently, such responses encompass, as particular cases, responses that may depend just on the forward or just on the adjoint state functions pertaining to the linear system under consideration. This work also compares the CPU-times needed by the 4th-CASAM versus other deterministic methods (e.g., finite-difference schemes) for computing response sensitivities These comparisons underscore the fact that the 4th-CASAM is the only practically implementable methodology for obtaining and subsequently computing the exact expressions (i.e., free of methodologically-introduced approximations) of the 1st-, 2nd, 3rd- and 4th-order sensitivities (i.e., functional derivatives) of responses to system parameters, for coupled forward/adjoint linear systems. By enabling the practical computation of any and all of the 1st-, 2nd, 3rd- and 4th-order response sensitivities to model parameters, the 4th-CASAM makes it possible to compare the relative values of the sensitivities of various order, in order to assess which sensitivities are important and which may actually be neglected, thus enabling future investigations of the convergence of the (multivariate) Taylor series expansion of the response in terms of parameter variations, as well as investigating the range of validity of other important quantities (e.g., response variances/covariance, skewness, kurtosis, etc.) that are derived from Taylor-expansion of the response as a function of the model’s parameters. The 4th-CASAM presented in this work provides the basis for significant future advances towards overcoming the “curse of dimensionality” in sensitivity analysis, uncertainty quantification and predictive modeling.

The European Union’s environmental goal by 2050 is to become the first climate-neutral continent in the world. This means specific efforts for diversifying the energy mix and investing in low-carbon energy. Our study investigates the nexus among carbon emissions, energy consumption and mix, and economic growth in a modified framework that includes the contribution of inward foreign direct investments and international trade to lowering air pollution. We have used a two-step approach to explore in more detail the links between these variables in 24 EU countries over the period 1995–2018, followed by a panel VECM analysis. Our results indicate that there is a unidirectional link between economic growth and CO2 emissions, which should imply a decoupling of environmental improvement measures from the pace of economic growth. We also find bidirectional causal relationships between low-carbon energy shares in consumption and CO2 emissions, as well as between low-carbon energy share in consumption and GDP per capita, which confirms both pollution haven and the halo effect hypotheses for FDI on gas emissions. However, in the long term, FDI, exports, and imports have positively impacted the reduction in CO2 emissions; therefore, stronger EU investment and trade integration should be promoted to improve the quality of the environment.

By applying the Second-Order Adjoint Sensitivity Analysis Methodology (2nd-ASAM) to the polyethylene-reflected plutonium (PERP) benchmark, this work presents results for the first- and second-order sensitivities of this benchmark’s leakage response with respect to the spontaneous fission source parameters. The numerical results obtained for these sensitivities indicate that the 1st-order relative sensitivity of the leakage response to the source parameters for the two fissionable isotopes in the benchmark are all positive, signifying that an increase in the source parameters will cause an increase in the total neutron leakage from the PERP sphere. The 1st- and 2nd-order relative sensitivities with respect to the source parameters for 239Pu are very small (10−4 or less). In contradistinction, the 1st-order and several 2nd-order relative sensitivities of the leakage response with respect to the source parameters of 240Pu are large. Large values (e.g., greater than 1.0) are also displayed by several mixed 2nd-order relative sensitivities of the leakage response with respect to parameters involving the source and: (i) the total cross sections; (ii) the average neutrons per fission; and (iii) the isotopic number densities. On the other hand, the values of the mixed 2nd-order relative sensitivities with respect to parameters involving the source and: (iv) the scattering cross sections; and (v) and the fission cross sections are smaller than 1.0. It is also shown that the effects of the 1st- and 2nd-order sensitivities of the PERP benchmark’s leakage response with respect to the benchmark’s source parameters on the moments (expected value, variance and skewness) of the PERP benchmark’s leakage response distribution are negligibly smaller than the corresponding effects (on the response distribution) stemming from uncertainties in the total, fission and scattering cross sections.

This work presents the Third-Order Adjoint Sensitivity Analysis Methodology (3rd-ASAM) for response-coupled forward and adjoint linear systems. The 3rd-ASAM enables the efficient computation of the exact expressions of the 3rd-order functional derivatives (“sensitivities”) of a general system response, which depends on both the forward and adjoint state functions, with respect to all of the parameters underlying the respective forward and adjoint systems. Such responses are often encountered when representing mathematically detector responses and reaction rates in reactor physics problems. The 3rd-ASAM extends the 2nd-ASAM in the quest to overcome the “curse of dimensionality” in sensitivity analysis, uncertainty quantification and predictive modeling. This work also presents new formulas that incorporate the contributions of the 3rd-order sensitivities into the expressions of the first four cumulants of the response distribution in the phase-space of model parameters. Using these newly developed formulas, this work also presents a new mathematical formalism, called the 2nd/3rd-BERRU-PM “Second/Third-Order Best-Estimated Results with Reduced Uncertainties Predictive Modeling”) formalism, which combines experimental and computational information in the joint phase-space of responses and model parameters, including not only the 1st-order response sensitivities, but also the complete hessian matrix of 2nd-order second-sensitivities and also the 3rd-order sensitivities, all computed using the 3rd-ASAM. The 2nd/3rd-BERRU-PM uses the maximum entropy principle to eliminate the need for introducing and “minimizing” a user-chosen “cost functional quantifying the discrepancies between measurements and computations,” thus yielding results that are free of subjective user-interferences while generalizing and significantly extending the 4D-VAR data assimilation procedures. Incorporating correlations, including those between the imprecisely known model parameters and computed model responses, the 2nd/3rd-BERRU-PM also provides a quantitative metric, constructed from sensitivity and covariance matrices, for determining the degree of agreement among the various computational and experimental data while eliminating discrepant information. The mathematical framework of the 2nd/3rd-BERRU-PM formalism requires the inversion of a single matrix of size Nr Nr, where Nr denotes the number of considered responses. In the overwhelming majority of practical situations, the number of responses is much less than the number of model parameters. Thus, the 2nd-BERRU-PM methodology overcomes the curse of dimensionality which affects the inversion of hessian matrices in the parameter space.

This work continues the presentation commenced in Part I of the second-order sensitivity analysis of nuclear data of a polyethylene-reflected plutonium (PERP) benchmark using the Second-Order Adjoint Sensitivity Analysis Methodology (2nd-ASAM). This work reports the results of the computations of the first- and second-order sensitivities of this benchmark’s computed leakage response with respect to the benchmark’s 21,600 parameters underlying the computed group-averaged isotopic scattering cross sections. The numerical results obtained for the 21,600 first-order relative sensitivities indicate that the majority of these were small, the largest having relative values of O (10−2). Furthermore, the vast majority of the (21600)2 second-order sensitivities with respect to the scattering cross sections were much smaller than the corresponding first-order ones. Consequently, this work shows that the effects of variances in the scattering cross sections on the expected value, variance, and skewness of the response distribution were negligible in comparison to the corresponding effects stemming from uncertainties in the total cross sections, which were presented in Part I. On the other hand, it was found that 52 of the 21600 × 180 mixed second-order sensitivities of the leakage response with respect to the scattering and total microscopic cross sections had values that were significantly larger than the unmixed second-order sensitivities of the leakage response with respect to the group-averaged scattering microscopic cross sections. The first- and second-order mixed sensitivities of the PERP benchmark’s leakage response with respect to the scattering cross sections and the other benchmark parameters (fission cross sections, average number of neutrons per fission, fission spectrum, isotopic atomic number densities, and source parameters) have also been computed and will be reported in subsequent works.

Nuclear and other large-scale energy-producing plants must include systems that guarantee the safe discharge of residual heat from the industrial process into the atmosphere. This function is usually performed by one or several cooling towers. The amount of heat released by a cooling tower into the external environment can be quantified by using a numerical simulation model of the physical processes occurring in the respective tower, augmented by experimentally measured data that accounts for external conditions such as outlet air temperature, outlet water temperature, and outlet air relative humidity. The model’s responses of interest depend on many model parameters including correlations, boundary conditions, and material properties. Changes in these model parameters induce changes in the computed quantities of interest (called “model responses”), which are quantified by the sensitivities (i.e., functional derivatives) of the model responses with respect to the model parameters. These sensitivities are computed in this work by applying the general adjoint sensitivity analysis methodology (ASAM) for nonlinear systems. These sensitivities are subsequently used for: (i) Ranking the parameters in their importance to contributing to response uncertainties; (ii) Propagating the uncertainties (covariances) in these model parameters to quantify the uncertainties (covariances) in the model responses; (iii) Performing model validation and predictive modeling. The comprehensive predictive modeling methodology used in this work, which includes assimilation of experimental measurements and calibration of model parameters, is applied to the cooling tower model under unsaturated conditions. The predicted response uncertainties (standard deviations) thus obtained are smaller than both the computed and the measured standards deviations for the respective responses, even for responses where no experimental data were available.

Cooling towers discharge waste heat from an industrial process into the atmosphere, and are essential for the functioning of large energy-producing plants, including nuclear reactors. Using a numerical simulation model of the cooling tower together with measurements of outlet air relative humidity, outlet air and water temperatures enables the quantification of the rate of thermal energy dissipation removed from the respective process. The computed quantities depend on many model parameters including correlations, boundary conditions, material properties, etc. Changes in these model parameters will induce changes in the computed quantities of interest (called “model responses”). These changes are quantified by the functional derivatives (called “sensitivities”) of the model responses with respect to the model parameters. These sensitivities are computed in this work by applying the general Adjoint Sensitivity Analysis Methodology (ASAM) for nonlinear systems. These sensitivities are needed for: (i) ranking the parameters in their importance to contributing to response uncertainties; (ii) propagating the uncertainties (covariances) in these model parameters to quantify the uncertainties (covariances) in the model responses; (iii) performing predictive modeling, including assimilation of experimental measurements and calibration of model parameters to produce optimal predicted quantities (both model parameters and responses) with reduced predicted uncertainties.

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.