First-Order Comprehensive Adjoint Sensitivity Analysis Methodology for Critical Points in Coupled Nonlinear Systems. II: Application to a Nuclear Reactor Thermal-Hydraulics Safety Benchmark

: Responses defined at critical points are particularly important for reactor safety analyses and licensing (e.g., the maximum fuel and/or clad temperature). The novel mathematical framework of the first-order comprehensive adjoint sensitivity analysis methodology for critical points (1st-CASAM-CP) is applied in this work to develop a reactor safety thermal-hydraulics benchmark model which admits exact closed-form expressions for the adjoint functions and for the first-order sensitivities of responses defined at critical points (maxima, minima, saddle points) in physical systems character-ized by imprecisely known parameters, external and internal boundaries. This benchmark model is designed for verifying the capabilities and accuracies of computational tools for modeling numerically thermal-hydraulics systems. The unique and extensive capabilities of the 1st-CASAM-CP methodology are demonstrated in this work by considering two responses of paramount importance in reactor safety, namely, (i) the maximum rod surface temperature, which occurs at the imprecisely known interface between the subsystem that models the heat conduction inside the heated rod and the subsystem modeling the heat convection process surrounding the rod; and (ii) the maximum temperature inside the heated rod, which has a critical point with two components, one located at a precisely known boundary of the subsystem that models the heat conduction inside the heated rod, while the other component depends on an imprecisely known boundary (i.e., the rod length). The exact analytical expressions developed in this work for the sensitivities of the maximum internal rod temperature and maximum rod surface temperature, as well as for the sensitivities of the locations where these respective maxima occur, provide exact benchmarks for verifying the accuracy of thermal-hydraulics computational tools. The sensitivities of such responses and of their critical points with respect to model parameters enable the quantification of uncertainties in-duced by uncertainties stemming from the system’s parameters and boundaries in the respective responses and their underlying critical points.


Introduction
The transfer of the heat generated in reactor fuel rods to the reactor's coolant and the subsequent transport of this heat by the coolant to the primary heat exchanger are processes generally modeled using three-dimensional thermal-hydraulics computational models. Quantities of particular interest in nuclear reactor safety are the peak (maximum) temperature within the "hottest" fuel rod and the peak cladding temperature (i.e., the maximum temperature on the rod's surface) of the "hottest" fuel rod within the reactor's This work is structured as follows: Section 2 presents the mathematical model of a heated rod surrounded by coolant, which simulates flow in a reactor channel or in an experimental thermal-hydraulics (TH) experimental facility. Also presented in Section 2 are the mathematical definitions of various responses of fundamental importance for reactor design and safety (e.g., maximum temperature inside the heated rod, maximum rodsurface temperature) located at critical points in the phase-space of the independent variables underlying two coupled generic nonlinear physical systems comprising imprecisely known parameters, interfaces, and boundaries. Section 3 illustrates the application of the 1st-CASAM-CP [6] mathematical framework to the obtain the expressions of the first-order sensitivities of the maximum rod surface temperature and of its critical point with respect to the physical systems' imprecisely known parameters, interfaces, and boundaries. The critical point of the maximum rod surface temperature is located at the imprecisely known interface between the subsystem that models the heat conduction inside the heated rod and the subsystem modeling the heat convection process surrounding the rod. Hence, both components of this critical point are subject to uncertainties. Section 4 presents the application of the 1st-CASAM-CP [6] to obtain the expressions of the first-order sensitivities of the maximum temperature rod surface and of its critical point with respect to the physical systems' imprecisely known parameters, interfaces, and boundaries on an interface for computing exactly and efficiently the magnitude of the response and of the phase-space location of its critical point. The critical point of the maximum rod surface temperature is located at a precisely known boundary of the subsystem that models the heat conduction inside the heated rod. Therefore, only one of the components of this critical point is subject to uncertainties. The discussion in Section 5 highlights the significance and possible future applications of the exact mathematical expressions derived for the THbenchmark presented in this work.

Mathematical Model of a Heated Rod Surrounded by Coolant
The benchmark model considered in this work simulates the steady-state heat transfer processes in an idealized reactor channel or in the test section of thermal-hydraulics experimental facilities. The heat transfer process in this benchmark model comprises two coupled "subsystems" which are defined as follows: 1. "Subsystem I" models the steady-state heat conduction in a cylindrical rod of radius a and length (height) , with a   , so that the heat conduction in the axial direction can be neglected by comparison to the heat conduction in the radial direction. The rod is heated by an internal volumetric source of the form   cos q z   , which simulates the axial power distribution in a nuclear reactor;  denotes a constant volumetric source, while z denotes the coordinate along the rod's axial (vertical) direction. The rod's conductivity,  , is considered to be a temperature-independent constant. Thus, temperature distribution within the rod,   , T r z , is governed by the following heat conduction equation: The rod's surface is cooled by forced convection to a surrounding liquid flowing along the rod's length, from the rod's lower end, taken to be located at  3. The interface (coupling) relation between the temperature distribution in the rod and the temperature distribution in the coolant is provided by the relation where the heat transfer coefficient, The system of Equation (1) The imprecisely known parameters underlying the paradigm heat transfer benchmark modeled by Equation (1)  (c) the external boundaries defined by the imprecisely known length, , of the heated rod.
The nominal values of these parameters are considered to be known and will be denoted by using the superscript "zero," i.e., Evidently, the rod surface temperature is defined on the imprecisely known interface r a  between the subsystem that models the heat conduction in the rod and the subsystem that models the heat convection in the surrounding coolant. Therefore, the maximum temperature on the rod's surface, which will be denoted as It is evident from the closed-form expressions obtained in Equation (

Maximum Rod Surface Temperature: Critical Point Located on Interface
Section 3.1 presents the derivation of the exact, closed-form expressions of the firstorder sensitivities of the maximum rod surface temperature with respect to the uncertain model parameters, interface, and boundaries. Section 3.2 presents the derivation of the exact, closed form expressions of the first-order sensitivities of the location (in the space of independent variables) of the critical point of the maximum rod surface temperature with respect to the uncertain model parameters, interface, and boundaries.

First-Order Sensitivities of the Maximum Rod Surface Temperature
The maximum temperature of the rod's surface, max s T , can be represented in the following form: where s z is implicitly defined by the relation in Equation (9). The first-order total differential, The total differential max s T  is obtained by applying the definition of the G-differential to Equation (15), which yields: The indirect-effect term,   Thus, the indirect-effect term depends on the parameter variations indirectly, through the variation in Equation (17) depends directly on parameter variations and is defined as follows: The variation   , T r z  is the solution of the "first-level forward sensitivity system" (1st-LFSS) which is obtained by G-differentiating the original system defined by Equation (1) through Equation (5). Applying the definition of the G-differential to Equation (1) through Equation (5) yields the following relations: Carrying out in Equation (21) through Equation (25) the differentiations with respect to  and setting 0   in the resulting expressions yields the following set of equations, which constitute the 1st-LFSS: The first term on the right-side of Equation (26) can be simplified by using Equation (1) to obtain the following equation: The terms containing derivatives of   , T r z in Equation (28) can also be simplified using Equations (1) and (3) to obtain the following equation: Since the equations underlying the 1st-LFSS, cf. Equations (27) and (29) through Equation (32), depend on the parameter variations, it is computationally expensive to repeatedly solve the 1st-LFSS for all possible parameter variations. The need for repeatedly solving the 1st-LFSS can be circumvented by expressing the indirect-effect term defined in Equation (18) in terms of the solution of a "first-level adjoint sensitivity system" (1st-LASS), which will be constructed next by applying the general principles of the 1st-CASAM-CP presented in [6].
The Hilbert space appropriate for the heat transport benchmark under consideration comprises the space of all square-integrable two-component vector functions of the form Using the definition provided in Equation (33), construct the inner product of a square integrable vector function  (29), respectively, to obtain the following relation: The left-side of Equation (34) is now integrated by parts (twice over the variable r and once over the variable z ) to obtain Using the boundary condition given in Equation (27) The unknown quantity , which appears in the last term on the right-side of Equation (38) is eliminated by using the boundary condition given in Equation (32); this operation transforms Equation (38) into the following form: The unknown quantity   0 , T a z  , which appears in third and fourth terms on the right-side of Equation (39) The two terms that contain the unknown function The second term on the right-side of Equation (42) Inserting the relations provided in Equations (43) Inserting the results obtained in Equations (20) and (46) The sensitivities obtained in Equation (47)

First-Order Sensitivities of the Critical Point (Maximum) of the Rod Surface Temperature
As has been shown in Equation (9), the maximum value, in principle, on all of the uncertain parameters, which means that the total differential s z  will have the following expression: Since the closed-form expression provided in Equation (10) would not be available in general, the expression of s z  will be obtained by applying the general methodology presented in [6]. This application commences by writing the relations provided in Equation (9), which implicitly define the location, s z , in the following form: Taking the G-differential of Equation (66) yields the following relation: The relation in Equation (67) can be re-written in the following form: The term in Equation (68) can be expressed in terms of adjoint functions by applying the procedure outlined in [6] to obtain the following result: where the adjoint functions are the solutions of the following 1st-LASS: The subscript "zero" (which indicates "nominal values") has been omitted, for simplicity, in writing the expressions given in Equation (143) through Equation (151), but all of the quantities which appear in the respective expressions are to be evaluated/computed at the nominal values of the respective quantities.

Discussion
This work has applied the 1st-CASAM-CP methodology pioneered in [6] to develop a thermal-hydraulics benchmark model for the verification of the adjoint sensitivity functions and sensitivities of response defined at critical points for thermal-hydraulics computational tools such as the "Adjoint FLUENT Solver" [4]. It has been shown that the computation of the first-order response sensitivities of the magnitude of a response defined at a critical point of a function in the phase-space of the systems' independent variables requires a single large-scale computation of the 1st-LASS which corresponds to the magnitude of the critical point. Solving the 1st-LASS represents a "large-scale" computation, which is not more extensive, however, than solving the original coupled systems. This is because the 1st-LASS is linear in the dependent variables, whereas the original coupled systems are nonlinear in the dependent variables. Furthermore, the computation of the sensitivities of the location in phase-space of each critical point requires solving one 1st-LASS for each of the components in the phase-space of independent variables of the respective critical point. The same operators appear on the right-sides of the 1st-level adjoint sensitivity systems needed for computing the adjoint functions corresponding to the magnitude and the location (in phase-space) of a response defined at a critical point. Only the sources on the left-sides of the respective 1st-LASS differ from each other. Therefore, the same solver can be used (albeit with different sources) to compute the adjoint functions. These unique characteristics of the general 1st-CASAM-CP methodology [6] have been demonstrated in this work by considering two responses of paramount importance in reactor safety, namely, (i) the maximum rod surface temperature, which occurs at the imprecisely known interface between the subsystem that models the heat conduction inside the heated rod and the subsystem modeling the heat convection process surrounding the rod; and (ii) the maximum temperature inside the heated rod, which has two components, one located at a precisely known boundary of the subsystem that models the heat conduction inside the heated rod, while the other component depends on an imprecisely known boundary (i.e., the rod length). The exact analytical expressions developed in this work for the sensitivities of the maximum internal rod temperature and maximum rod surface temperature, as well as the sensitivities of the respective locations in phase-space where these respective maxima occur, provide accurate benchmarks for verifying the accuracy of computational tools for modeling thermal-hydraulics systems. This "solution verification" process should include the verification of the accuracy of the sensitivities of the results computed by such codes to the uncertain parameters underlying the respective codes.
Even the most advanced current thermal-hydraulics and/or CFD-computational tools, such as the Adjoint-FLUENT solver [4], are insufficiently developed and currently lack capabilities for computing sensitivities of the locations of critical points (maxima, minima, saddle points) with respect to model parameters, interfaces, and boundaries. The thermal-hydraulics benchmark presented in this work serves for the future verification/validation of the future developments/extensions of the "adjoint sensitivity analysis" capabilities of future thermal-hydraulics and/or CFD computational tools, particularly for the verification of the adjoint functions needed to compute the crucially-important (for reactor design and licensing) "maximum fuel temperature" and "maximum clad temperature" in the reactor's hot channels. Applications of the 1st-CASAM-PC [6] and use of the benchmark presented in this work for adjoint sensitivity analysis of "responses at critical points" must eventually come in the future, since such responses are crucially important for the design, optimization, and licensing of engineering systems, see e.g., Refs. [7,8] Funding: This research received no external funding.
Institutional Review Board Statement: Not applicable.