Predictive modeling analysis of a wet cooling tower - Adjoint sensitivity analysis, uncertainty quantification, data assimilation, model calibration, best-estimate predictions with reduced uncertainties

It is common practice, in the modern era, to base the process of understanding and eventually predicting the behavior of complex physical systems upon simulating operational situations through system codes. In order to provide a more thorough and accurate comprehension of the system dynamics, these numerical simulations are often and preferably flanked by experimental measurements. In practice, repeated measurements of the same physical quantity produce values differing from each other and from the measured quantity true value, which remains unknown; the errors leading to this variation in results can be of methodological, instrumental or personal nature. It is not feasible to obtain experimental results devoid of uncertainty, and this means that a range of values possibly representative of the true value always exists around any value stemming from experimental measurements. A quantification of this range is critical to any practical application of the measured data, whose nominal measured values are insufficient for applications unless the quantitative uncertainties associated to the experimental data are also provided. Not even numerical models can reveal the true value of the investigated quantity, for two reasons: first, any numerical model is imperfect, meaning that it constitutes an inevitable simplification of the real world system it aims to represent; in second place, a hypothetically perfect model would still have uncertain values for its model parameters - such as initial conditions, boundary conditions and material properties - and the stemming results would therefore still be differing from the true value and from the experimental measurements of the quantity. With both computational and experimental results at hand, the final aim is to obtain a probabilistic description of possible future outcomes based on all recognized errors and uncertainties. This operation falls within the scope of predictive modeling procedures, which rely on three key elements: model calibration, model extrapolation and estimation of the validation domain. The first step of the procedure involves the adjustment of the numerical model parameters accordingly to the experimental results; this aim is achieved by integrating computed and measured data, and the associated procedure is known as model calibration. In order for this operation to be properly executed, all errors and uncertainties at any level of the modeling path leading to numerical results have to be identified and characterized, including errors and uncertainties on the model parameters, numerical discretization errors and possible incomplete knowledge of the physical process being modeled. Calibration of models is performed through the mathematical framework provided by data assimilation procedures; these procedures strongly rely on sensitivity analysis, and for this reason are often cumbersome in terms of computational load. Generally speaking, sensitivity analyses can be conducted with two different techniques, respectively known as direct or forward methods and adjoint methods. The forward methods calculate the finite difference of a small perturbation in a parameter by means of differences between two independent calculations, and are advantageous only for systems in which the number of responses exceeds the number of model parameters; unfortunately this is seldom the case in real large-scale systems. In this work, this problem has been overcome by using the adjoint sensitivity analysis methodology (ASAM) by Cacuci: as opposed to forward methods, the ASAM is most efficient for systems in which the number of parameters is greater than the number of responses, such as the model investigated in this thesis and many others currently used for numerical simulations of industrial systems. This methodology has been recently extended to second-order sensitivities (2nd-ASAM) by Cacuci for linear and nonlinear systems, for computing exactly and efficiently the second-order functional derivatives of system responses to the system model parameters. Model extrapolation addresses the prediction of uncertainty in new environments or conditions of interest, including both untested parts of the parameter space and higher levels of system complexity in the validation hierarchy. Estimation of the validation domain addresses the estimation of contours of constant uncertainty in the high-dimensional space that characterizes the application of interest. The present work focuses on performing sensitivity and uncertainty analysis, data assimilation, model calibration, model validation and best-estimate predictions with reduced uncertainties on a counter-flow, wet cooling tower model developed by Savannah River National Laboratory. A cooling tower generally discharges waste heat produced by an industrial plant to the external environment. The amount of thermal energy discharged into the environment can be determined by measurements of quantities representing the external conditions, such as outlet air temperature, outlet water temperature, and outlet air relative humidity, in conjunction with computational models that simulate numerically the cooling tower behavior. Variations in the model parameters (e.g., material properties, model correlations, boundary conditions) cause variations in the model response. The functional derivatives of the model response with respect to the model parameters (called “sensitivities”) are needed to quantify such response variations changes. In this work, the comprehensive adjoint sensitivity analysis methodology for nonlinear systems is applied to compute the cooling tower response sensitivities to all of its model parameters. Moreover, the utilization of the adjoint state functions allows the simultaneous computation of the sensitivities of each model response to all of the 47 model parameters just running a single adjoint model computation; obtaining the same results making use of finite-difference forward methods would have required 47 separate computations, with the relevant disadvantage of leading to approximate values of the sensitivities, as opposed to the exact ones yielded by applying the adjoint procedure. In addition, the forward cooling tower model presents nonlinearity in their state functions; the adjoint sensitivity model possess the relevant feature of being instead linear in the adjoint state functions, whose one-to-one correspondence to the forward state functions is essential for the calculation of the adjoint sensitivities. Sensitivities are subsequently used in this work to realize many operations, such as: (i) ranking the model parameters according to the magnitude of their contribution to response uncertainties; (ii) determine the propagation of uncertainties, in form of variances and covariances, of the parameters in the model in order to quantify the uncertainties of the model responses; (iii) allow predictive modeling operations, such as experimental data assimilation and model parameters calibration, with the aim to yield best-estimate predicted nominal values both for model parameters and responses, with correspondently reduced values for the predicted uncertainties associated. The methodologies are part of two distinct mathematical frameworks: the Adjoint Sensitivity Analysis Methodology (ASAM) is used to compute the adjoint sensitivities of the model quantities of interest (called “model responses”) with respect to the model parameters; the Predictive Modeling of Coupled Multi-Physics Systems (PM_CMPS) simultaneously combines all of the available computed information and experimentally measured data to yield optimal values of the system parameters and responses, while simultaneously reducing the corresponding uncertainties in parameters and responses. In the present work, a relevantly more efficient numerical method has been applied to the cooling tower model analyzed, leading to the accurate computation of the steady-state distributions for the following quantities of interest: (i) the water mass flow rates at the exit of each control volume along the height of the fill section of the cooling tower; (ii) the water temperatures at the exit of each control volume along the height of the fill section of the cooling tower; (iii) the air temperatures at the exit of each control volume along the height of the fill section of the cooling tower; (iv) the humidity ratios at the exit of each control volume along the height of the fill section of the cooling tower; and (v) the air mass flow rates at the exit of the cooling tower. The application of the numerical method selected eliminates any convergence issue, yielding accurate results for all the control volumes of the cooling tower and for all the data set of interest. This work is organized as follows: Chapter 2 provides a description of the physical system simulated, along with presenting the mathematical model used in this work for simulating a counter-flow cooling tower operating under saturated and unsaturated conditions. The three cases analyzed in this work and their corresponding sets of governing equations are detailed in this chapter. Chapter 3 presents the development of the adjoint sensitivity model for the counter-flow cooling tower operating under saturated and unsaturated conditions using the general adjoint sensitivity analysis methodology (ASAM) for nonlinear systems. Using a single adjoint computation enables the efficient and exact computation of the sensitivities (functional derivatives) of the model responses to all of the model parameters, thus alleviating the need for repeated forward model computations in conjunction with finite difference methods. The mathematical framework of the “predictive modeling for coupled multi-physics systems” (PM_CMPS) is also detailed. Chapter 4 presents the results of applying the ASAM and PM_CMPS methodologies to all the cases listed in Chapter 2: after being calculated, sensitivities are subsequently used for ranking the contributions of the single model parameters to the model responses variations, for computing the propagated uncertainties of the model responses, and for the application of the PM_CMPS methodology, aimed at yielding best-estimate predicted nominal values and uncertainties for model parameters and responses. This methodology simultaneously combines all of the available computed information and experimentally measured data for the counter-flow cooling tower operating under saturated and unsaturated conditions. The best-estimate results predicted by the PM_CMPS methodology reveal that the predicted values of the standard deviations for all the model responses, even those for which no experimental data have been recorded, are smaller than either the computed or the measured standards deviations for the respective responses. This work concludes with Chapter 5 by discussing the significance of these predicted results and by indicating possible further generalizations of the adjoint sensitivity analysis and PM_CMPS methodologies.