The First-Order Comprehensive Sensitivity Analysis Methodology (1 st -CASAM) for Scalar-Valued Responses: II. Illustrative Application to a Heat Transport Benchmark Model

This work illustrates the application of the 1 st -CASAM to a paradigm heat transport model which admits exact closed-form solutions. The closed-form expressions obtained in this work for the sensitivities of the temperature distributions within the model to the model’s parameters, internal interfaces and external boundaries can be used to benchmark commercial and production software packages for simulating heat transport. The 1 st -CASAM highlights the novel finding that response sensitivities to the imprecisely known domain boundaries and interfaces can arise both from the definition of the system’s response as well as from the equations, interfaces and boundary conditions that characterize the model and its imprecisely known domain. By enabling, in premiere, the exact computations of sensitivities to interface and boundary parameters and conditions, the 1 st -CASAM enables the quantification of the effects of manufacturing tolerances on the responses of physical and engineering systems. The First-Order Comprehensive Sensitivity Analysis Methodology (1 st -CASAM) for Sca-lar-Valued Responses: II. Illustrative Application to a Heat Transport Benchmark Model.


Introduction
An accompanying work [1] has presented the mathematical framework of the first-order comprehensive adjoint sensitivity analysis methodology (1 st -CASAM) for computing efficiently, exactly and exhaustively, the first-order response sensitivities to imprecisely known parameters that describe the system, the imprecisely known physical interfaces between systems, and the systems' imprecisely known external boundaries for coupled nonlinear physical systems. This work presents an illustrative application of the 1 st -CASAM to a benchmark model [2] [3] [4] that models coupled heat conduction and convection in a physical system comprising an electrically heated rod surrounded by a coolant which simulates the geometry of an advanced ("Generation-IV") nuclear reactor [5]. This benchmark model [2] [3] [4] admits exact closed-form solutions for the sensitivities of the temperature distribution in the coupled rod/coolant system which can be used to benchmark thermal-hydraulics production codes. Notably, this model [2] [3] [4] was used to verify the numerical results produced by the FLUENT Adjoint Solver [6], showing, in particular, that the current version of the FLUENT Adjoint Solver cannot compute sensitivities for the temperature distribution within the solid rod.
This work is structured as follows. Section 2 presents the mathematical modeling of the heat conduction process in the electrically heated rod coupled to the convective heat transport in the coolant surrounding the heated rod. This mathematical model admits exact closed-form solutions for the temperature distributions, which can be used to benchmark thermal-hydraulics production codes. Section 3 presents the application of the 1 st -CASAM to the heat conduction/convection model to obtain the exact expressions of the sensitivities of the temperature distribution in the coupled rod/coolant system to the imprecisely known model, internal interface and external boundary parameters. The exact closed-form expressions obtained in this work for the respective sensitivities can also be used to benchmark thermal-hydraulics production codes. Section 4 offers concluding remarks. Ongoing research will generalize the methodology presented in this work, aiming at computing exactly and efficiently higher-order response sensitivities for coupled systems involving imprecisely known interfaces, parameters, and boundaries. As is well known [7], the availability of response sensitivities to imprecisely known parameters, interfaces and boundaries is essential for a variety of subsequent uses, including uncertainty quantification, optimization, data assimilation, model calibration and validation, and reduction of uncertainties in predicted model results.
 , is considered to be a temperature-independent constant. 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 2 z = −  , towards the rod's upper end, located at 2 z =  . The heat transfer coefficient,  , from the rod's surface to the coolant is considered to be constant. For this benchmark, the rod's length is typically two orders of magnitude larger than its diameter, so the heat conduction process in the rod's axial direction can be neglected by comparison to the heat conduction in the rod's radial direction. Under these conditions, the steady-state temperature distributions, ( ) T r z k z r q r a z r r r The imprecisely known parameters underlying the paradigm heat transfer benchmark modeled by Equations (1)- (5) are: , , , , , , , p inlet q k h W c T a  . A list of these parameters is provided in the Nomenclature Section at the end of this work. The known nominal values of these parameters will be denoted by using the superscript "zero," i.e.,

Application of the 1 st -CASAM to the Coupled Heat Conduction and Convection Benchmark Model
The arbitrary variations in the imprecisely known model parameters, around the respective nominal values, will be denoted as follows: The variations in the rod and coolant temperatures, respectively, caused by the imprecisely known parameters will be denoted as follows:

First-Order Sensitivities of the Coolant's Temperature
The sensitivities to model and boundary parameters of several typical responses, including the value of the coolant temperature at a point, the average coolant temperature, and the coolant temperature itself, will be determined in this Section by applying the 1 st -CASAM presented in Reference 1. The 1 st -LFSS corresponding to Equations (4) and (5) is obtained by determining the G-differentials of these equations to obtain: Carrying out in Equations (10) and (11) the differentiations with respect to ε and setting 0 ε = in the resulting expressions yields the following set of equations: It is evident that the expression obtained in Equation (14) is the total differential with respect to the model and boundary parameters of the expression of ( ) fl T z given in Equation (7).

First-Order Sensitivities of the Coolant's Temperature at a Point in Phase-Space
The imprecisely known model and boundary parameters that characterize the heat transport benchmark modeled by Equations (1) through (5) and including the imprecisely known location p z , the response defined in Equation (15) are: , , , , , , , , The total sensitivity of the response defined in Equation (15) is given by its total G-differential, which is while the indirect-effect term is defined as follows: The direct-effect term Using the definition provided in Equation (19), construct the inner product of to obtain the following relation: Integrating by parts the term on the left-side of Equation (20) Identify the first term on the right-side of Equation (21) with the indirect-effect term defined in Equation (18) to obtain the following relations: The boundary condition given in Equation (13) where the adjoint function It also follows from Equation (25) that Solving the 1 st -LASS comprising Equations (22) and (24), which is notably independent of any parameter variation, yields the following expression for the adjoint function Replacing the expression obtained in Equation (32) (17) provides the additional sensitivity of the response with respect to its imprecisely known location, which is computed directly from Equation (7), namely: It is evident from the above illustrative example that the 1 st -CASAM is the most efficient way to compute exactly the 1 st -order response sensitivities to model and boundary parameters, since it requires a single large-scale computation to solve the 1 st -LASS, namely Equations (22) and (24) for determining the 1 st -level adjoint function needed in the subsequent quadrature formulas to compute all of the response sensitivities using Equations (26)-(31).

First-Order Sensitivities of the Average Coolant Temperature
The average coolant temperature, denoted as ave fl T , is given by the expression The total sensitivity of ave fl T is given by its total G-differential, which is obtained, by definition, by evaluating the following expression: where the direct-effect term is defined as follows: while the indirect-effect term is defined as follows: The indirect-effect term { } where the adjoint function

First-Order Sensitivities of the Rod's Temperature
The value of the heated rod temperature, ( ) , T r z , at some location ( ) This 1 st -LFSS is obtained by determining the G-differentials of Equations (1)- (5), which are as follows: The first term on the right-side of Equation (69) can be simplified by using Equation (1) to obtain the following equation: The terms containing derivatives of ( ) , T r z in Equation (71) can also be simplified using Equations (1) and (3) Using the boundary condition given in Equations (70) and imposing the boundary condition The two terms that contain the unknown function ( ) fl T z δ are grouped together, transforming Equation (84) into the following form: