Kevin J. Dowding

Sensitivity Analysis for Nonlinear Heat Conduction

Parameters in the heat conduction equation are frequently modeled as temperature dependent. Thermal conductivity, volumetric heat capacity, convection coefficients, emissivity, and volumetric source terms are parameters that may depend on temperature. Many applications, such as parameter estimation, optimal experimental design, optimization, and uncertainty analysis, require sensitivity to the parameters describing temperature-dependent properties. A general procedure to compute the sensitivity of the temperature field to model parameters for nonlinear heat conduction is studied. Parameters are modeled as arbitrary functions of temperature. Sensitivity equations are implemented in an unstructured grid, element-based numerical solver. The objectives of this study are to describe the methodology to derive sensitivity equations for the temperature-dependent parameters and present demonstration calculations. In addition to a verification problem, the design of an experiment to estimate temperature variable thermal properties is discussed

METHODOLOGY TO GENERATE ACCURATE SOLUTIONS FOR VERIFICATION IN TRANSIENT THREE-DIMENSIONAL HEAT CONDUCTION

This article describes the development of accurate solutions for transient three-dimensional conductive heat transfer in Cartesian coordinates for a parallelepiped which is homogeneous and has constant thermal properties. The intended use of these solutions is for verification of numerical computer programs which are used for solving transient heat conduction problems. Verification is a process to ensure that a computer code is free of errors and accurately solves the mathematical equations. The exact solutions presented in this article can have any combination of boundary conditions of specified temperature, prescribed heat flux, or imposed convection coefficient and ambient temperature on the surfaces of the parallelepiped. Additionally, spatially uniform nonzero initial condition and internal energy generation are treated. The methodology to obtain the analytical solutions and sample calculations are presented.

Development and implementation of sensitivity coefficient equations for heat conduction problems

Methods are discussed for computing the sensitivity of the temperature field to changes in material properties and initial boundary condition parameters for heat conduction problems. The most general method is to derive sensitivity equations by differentiating the energy equation with respect to the parameter of interest and solving the resulting sensitivity equations numerically. An example problem in which there are 12 parameters of interest is presented and the resulting sensitivity equations and associated boundary initial conditions are derived. The sensitivity equations are implemented in a general-purpose unstructured-grid control-volume finite-element code. Numerical results are presented for thermal conductivity and volumetric heat capacity sensitivity coefficients for heat conduction in a 2-D orthotropic body. The numerical results are compared with the analytical solution to demonstrate that the numerical sensitivity method is second-order accurate as the mesh is refined spatially.

APPLICATION OF SENSITIVITY COEFFICIENTS FOR HEAT CONDUCTION PROBLEMS

In parameter estimation considerable insight is provided by examining sensitivity coefficients. This paper focuses on the use of sensitivity coefficients in connection with estimating thermal properties in the heat conduction equation. A general methodology for computing sensitivity coefficients can be an important design tool. The use of such a tool is demonstrated in this paper. A control volume, finite element program is used, and briefly described, to implement numerical sensitivity coefficient calculations. In this approach general problems can be studied. Several example problems are presented to demonstrate the insight gained from sensitivity coefficients. The problems are selected from experimental studies to characterize the thermal properties of carbon-carbon composite. Sensitivity coefficients show that in an experiment that is not well designed, additional materials in the experimental configuration can have a larger impact on the temperature than the material of interest. Two-dimensional configurations demonstrate that there can be isolated areas of insensitivity and the difficulty of estimating multiple parameters.

Estimating Temperature-Dependent Thermal Properties

Parameter estimation techniques are applied to estimate temperature-dependent thermal properties from a series of transient experiments. Several experiments with one and two-dimensional heat flow that cover a range from room temperature to 500°C are analyzed. Temperature-dependent thermal properties are estimated by connecting the independent experiments in the series during the analysis. The techniques are applied to estimate effective properties of carbon-carbon composite. The temperature dependence of two components of thermal conductivity and volumetric heat capacity are estimated to characterize the assumed orthotropic material. The techniques can be equally applied to other homogenous materials. Combining experiments during the analysis is referred to as a sequential analysis, which uses the concepts of regularization and prior information. Regularization controls variations in the estimated parameters. Prior information carries information from a previous analysis into a subsequent analysis

Case study for model validation : assessing a model for thermal decomposition of polyurethane foam.

A case study is reported to document the details of a validation process to assess the accuracy of a mathematical model to represent experiments involving thermal decomposition of polyurethane foam. The focus of the report is to work through a validation process. The process addresses the following activities. The intended application of mathematical model is discussed to better understand the pertinent parameter space. The parameter space of the validation experiments is mapped to the application parameter space. The mathematical models, computer code to solve the models and its (code) verification are presented. Experimental data from two activities are used to validate mathematical models. The first experiment assesses the chemistry model alone and the second experiment assesses the model of coupled chemistry, conduction, and enclosure radiation. The model results of both experimental activities are summarized and uncertainty of the model to represent each experimental activity is estimated. The comparison between the experiment data and model results is quantified with various metrics. After addressing these activities, an assessment of the process for the case study is given. Weaknesses in the process are discussed and lessons learned are summarized.

The Relationship Between Information, Sampling Rates, and Parameter Estimation Models

To estimate parameters from experiments requires the specification of models and each model will exhibit different degrees of sensitivity to the parameters sought. Although experiments can be optimally designed without regard to the experimental data actually realized, the precision of the estimated parameters is a function of the sensitivity and the statistical characteristics of the data. The precision is affected by any correlation in the data, either auto or cross, and by the choice of the model used to estimate the parameters. An informative way of looking at an experiment is by using the concept of Information. An analysis of an actual experiment is used to show how the information, the optimal number of sensors, the optimal sampling rates, and the model are affected by the statistical nature of the signals. We demonstrate that one must differentiate between the data needed to specify the model and the precision in the estimated parameters provided by the data

Uncertainty Quantification and Model Validation of Fire/Thermal Response Predictions

Coupled fire-environment/thermal-response models were validated using data for an object engulfed in a JP8 hydrocarbon fuel fire. Fire model predictions of heat flux were used as boundary conditions in the thermal response calculations of the object. Predictions of transient external shell temperatures as well as the surface temperatures of the embedded mass were averaged spatially and compared to data. The solution sensitivity to mesh size, time step, nonlinear iterations, and radiation rays were assessed and the uncertainties in the predictions were quantified using a Latin Hypercube Sampling (LHS) technique. The comparisons showed that the response variable was more sensitive to fire model parameters than to thermal model parameters. The observed relative difference in measurements and model predictions was also compared to the model uncertainty. The comparisons showed that the model plus uncertainty bounded the experimental data. I. Introduction Sandia National Laboratories has been engaged in testing weapon system safety in fire environments since the 1950s. Due to the high consequences involved, system safety has traditionally been demonstrated through full scale system tests, albeit with a limited number of tests. Historically developed standardized tests include the placement of a system in a fully engulfing fire for 1 hour. Systems are declared qualified and ready for production based on passage of these standardized tests and with reference to the testing and analysis during development. Beginning in the early to mid 1990’s, the DOE began a program of Science Based Stockpile Stewardship. A significant part of this program is the Advanced Simulation and Computing (ASC) program, in which modeling and simulation, through high performance computing has been applied to system development and qualification. As part of the ASC program, Sandia engaged in developing the capability to model fire environments coupled to system response in those environments. An important thrust area within the ASC program includes the advancement of the verification and validation (V&V) methodologies and uncertainty quantification techniques. Sandia National Laboratories has made strides in developing new capabilities in this area and applying them to current applications. A best estimate plus uncertainty approach has been fully adopted and incorporated into safety themes for system qualification. Providing uncertainty estimates along with deterministic results has provided value to Sandia programs and gives more insight into predictive capability. The direct contribution of this study to current and future systems is an understanding of the uncertainties in predicting internal system temperatures when an object is engulfed in a JP8 fire environment. The uncertainty in input parameters can be used with other scenarios and configurations to evaluate situations that challenge safety themes. Confidence gained in validation processes such as discussed in the current work is crucial when evaluating system qualification activities that include modeling and simulation. II. Numerical Modeling

Statistical Validation of Engineering and Scientific Models: Bounds, Calibration, and Extrapolation

Numerical models of complex phenomena often contain approximations due to our inability to fully model the underlying physics, the excessive computational resources required to fully resolve the physics, the need to calibrate constitutive models, or in some cases, our ability to only bound behavior. Here we illustrate the relationship between approximation, calibration, extrapolation, and model validation through a series of examples that use the linear transient convective/dispersion equation to represent the nonlinear behavior of Burgers equation. While the use of these models represents a simplification relative to the types of systems we normally address in engineering and science, the present examples do support the tutorial nature of this document without obscuring the basic issues presented with unnecessarily complex models.

ESTIMATING TEMPERATURE-DEPENDENT THERMAL PROPERTIES OF CARBON- CARBON COMPOSITE

Parameter estimation techniques are applied to estimate temperature dependent thermal properties for a carbon-carbon composite from a series of transient experiments. Several experiments with one and two dimensional heat flow that cover a range from room temperature to 500° C are analyzed. Temperature dependent thermal properties are estimated by connecting the independent experiments in the series during the analysis. The temperature dependence of two components of thermal conductivity and volumetric heat capacity are estimated to characterize the assumed orthotropic material. Combining experiments during the analysis is referred to as a sequential analysis. It uses the concepts of regularization and prior information. Regularization controls variations in the estimated parameters. Prior information carries information from a previous analysis into a subsequent analysis. The functional dependence of the properties estimated with a sequential analysis show excellent agreement with previous analyses that considered the experiments independently.