Computing sensitivity coefficients by using complex differentiation: Application to heat conduction problem

To estimate the uncertainty of the neutronic parameters it is required to calculate sensitivity coefficients to cross section changes. The sensitivity coefficients are usually calculated based on the diffusion theory. However, the accuracy of the sensitivity coefficients becomes doubtful because of the use of diffusion theory. The purpose of this paper is to derive the generalized perturbation theory using the transport theory, and to evaluate transport effect on the sensitivity coefficients. Sensitivity calculations were performed in 70 energy groups. The calculated sensitivity coefficients were compared with those obtained from direct transport calculations to check the accuracy of the present method, and those obtained from the generalized perturbation calculation based on the diffusion theory to investigate the transport effect on sensitivity coefficients. From the comparison, the difference was remarkably large for the neutronic parameters such as reaction rate distribution, reaction rate ratio, and reactivity worth. For example, the sensitivity coefficient of the 238U capture to 239Pu fission rate ratio in the lower core region of the under-moderated LWR core was large in the energy range 17.6–22.6 eV, and the sensitivity to 238U capture cross section calculated by the transport theory was 20% larger than that by the diffusion theory.

The response and operation of a heat flux gage is studied using sensitivity analysis. Sensitivity analysis is the process by which one determines the sensitivity of a model output to changes in the model parameters. This process uses sensitivity coefficients, which are defined as partial derivatives of field variables (e.g. temperature) with respect to model parameters (e.g. thermal properties and boundary conditions). Computing sensitivity coefficients, in addition to the response of a heat flux gage, aids in identifying model parameters that significantly impact the temperature response. A control volume finite element based code is used to implement numerical sensitivity coefficient calculations, allowing general problems to be studied. Sensitivity coefficients are discussed for the well known Gardon gage.

Thermal properties of compacted pharmaceutical excipients

In recent years, problems of increased energy consumption and related green-house gas emission impose new demands on building structures. Strict policies require low heat transmission of building envelope and their inbuilt materials. For the precise evaluation of the heat transmission coefficient (HTC) using numerical methods, thermal properties of building materials need to be characterized. Data on the thermal material parameters commonly used in calculations are usually obtained under laboratory conditions, and may by different from the real state of material element, especially in the case of temperature and relative humidity fluctuations of the environment. To increase the accuracy of numerical assessment of HTC, thermal properties of materials at elevated temperatures can be used as raw data for further processing. In this study, thermal properties of four types of wood-based boards (wood agglomerates) at temperatures up to 150 °C were investigated together with their basic structural characteristics. These were measured both before the samples exposure to elevated temperatures and for samples that underwent the thermal treatment. Among the structural parameters of wood-based boards, bulk density, specific density, and total open porosity were analyzed. Parameters characterizing the heat transport and storage were measured using hot-disk apparatus and the samples were placed in the heating chamber. The researched thermal characteristics were evaluated for samples heated up to 150 °C with the 25 °C steps. The examined thermophysical parameters were thermal conductivity, volumetric heat capacity, and thermal diffusivity. Up to 100 °C, increasing trend in the thermal conductivity and volume heat capacity values was identified, which is typical for this type of materials. On the other hand, the thermal diffusivity has decreased with the temperature of samples exposure. After full water release, the thermal conductivity dropped, while volumetric heat capacity still increased in a similar trend. This was partly due to the evaporation of water incorporated in intercellular spaces of wood, and mainly because of the chemical changes of wood-board adhesives. This feature was proved by porosity data, which changed about 1 % of its initial value. Based on the acquired experimental results, the thermal characteristics of wood-based boards obtained by hot-disk method may be suitable to use as raw data for the numerical evaluation of HTC of building elements or the whole building envelopes.

A method of utilizing cone penetration tests (CPTs) is presented which gives continuous profiles of both the in situ thermal conductivity and volumetric heat capacity, along with the in situ temperature, for the upper tens of meters of the ground. Correlations from standard CPT results (cone resistance, sleeve friction and pore pressure) are utilized for both thermal conductivity and volumetric heat capacity for saturated soil. These, in conjunction with point-wise thermal conductivity and in situ temperature results using a Thermal CPT (T-CPT), allow accurate continuous profiles to be derived. The CPT-based method is shown via a field investigation supported by laboratory tests to give accurate and robust results.

Rockwell hardness testing is a widely used hardness test, and there are many scales depending on the indenter shape and test force for corresponding to various materials. The common Rockwell indenters are the diamond sphero-conical indenter and the tungsten carbide ball indenter. The influence of indenter shape error on Rockwell hardness has been acknowledged, as this shape impacts test results. To accurately estimate this uncertainty, it is essential to understand the sensitivity coefficient, which delineates the relationship between indenter shape and Rockwell hardness. Traditionally, this coefficient has been derived experimentally, but this approach faces challenges that are attributable to the nonuniformity of hardness test blocks and indenter shape measurement errors. This study introduces the estimation of sensitivity coefficients for Rockwell indenters using finite element analysis (FEA). The FEA method offers the advantage of being unaffected by the nonuniformity of test blocks and indenter shape measurement errors. We calculated the sensitivity coefficients by assessing the influence of variations in indenter angle and tip radius for diamond indenters, and indenter diameter for ball indenters, on hardness results through FEA. The sensitivity coefficients obtained via FEA for specific scales aligned with those from previous experimental studies, affirming the validity of FEA-derived coefficients for other scales. This study demonstrates that FEA is a useful tool for estimating the sensitivity coefficients of Rockwell indenters.

This paper presents an efficient inverse analysis technique based on a sensitivity coefficient algorithm to estimate the unknown boundary conditions of multidimensional steady and transient heat conduction problems. Sensitivity coefficients were used to represent the temperature response of a system under unit loading conditions. The proposed method, coupled with the sensitivity analysis in the finite element formulation, is capable of estimating both the unknown temperature and heat flux on the surface provided that temperature data are given at discrete points in the interior of a solid body. Inverse heat conduction problems are referred to as ill-posed because minor inaccuracy or error in temperature measurements cause a drastic effect on the predicted surface temperature and heat flux. To verify the accuracy and validity of the new method, two-dimensional steady and transient problems are considered. Their surface temperature and heat flux are evaluated. From a comparison with the exact solution, the effects of measurement accuracy, number and location of measuring points, a time step, and regularization terms are discussed. © 1998 Scripta Technica. Heat Trans Jpn Res, 26(6): 345–359, 1997

In this study the thermal conductivity and the volumetric heat capacity of non-Newtonian fluids are investigated. These fluids find many applications in chemical, medical and food industries. They mostly have viscoelastic behavior and generally differ from Newtonian fluids. Two non-Newtonian fluids, solutions of xanthan and carboxy methyl cellulose in water with various concentrations have been prepared and investigated. The apparatus developed has been based on two coaxial cylinder concept for measurement of thermal conductivity and volumetric heat capacity. The thermal conductivity measurement was done from steady state condition data while the volumetric heat capacity was determined from unsteady state experiments. Influence of temperature in the range of 25-80 degrees centigrade and concentration up to 20% has been studied and experimental curves are obtained for both solutions. Based on the data gathered, the empirical correlations obtained, expresses thermal conductivity and volumetric heat capacity as a function of temperature and concentration.

The aim of this work was to investigate the thermal behavior of solid wood used as building materials. Three tropical wood species namely Iroko (Milicia excelsa), Bilinga (Nauclea diderrichii), and Tali (Erythrophleum suaveolens) were chosen. The thermophysical properties in the longitudinal, radial and tangential directions were determined experimentally for different moisture contents. Asymmetrical hot plate method was used to measure the thermal conductivity and volumetric heat capacity. The influence of water content and the direction of the fibers on thermophysical properties of the species studied were investigated. The results show that thermal conductivity and volumetric heat capacity increase with moisture content, while the diffusivity decreases. In the wide range of moisture content considered, thermal conductivities varied between 0.1 and 0.6 Wm−1 K−1. The increase of moisture content by 1% induces a 0.2% increase in thermal conductivity for Bilinga, 0.4% for Tali and 0.3% for Iroko. Tali, whose density is the highest, presented high values of thermal conductivity in longitudinal direction but lowest values in radial and tangential directions compared to other species. Thermal conductivity is 2 to 3 times higher in the longitudinal direction than in the radial and tangential directions. The volumetric heat capacity was not influenced by the direction of the heat flow.

The present work summarizes the results of analysis of unique experimental data on vertical heat flow variations in different geological structures obtained from 15 scientific supper-deep and deep boreholes drilled to the depths of 1600–12262 m within Russian and ICDP programs. The new workflow was applied for the heat flow estimation which is based on (1) precise and detailed thermal conductivity measurements on more than 30000 cores with the new emerging technologies, (2) usage of more than 100 equilibrium and non-equilibrium temperature logs, and (3) determination of conductive heat flow component within 20–100 m intervals along every borehole studied. The data on conductive heat flow variations provides an estimate of vertical variations in the convective heat flow component. The latter reflects the information on variations in reservoir and formation properties and heat- and mass transfer processes in reservoirs and formations. It was established that a conductive component of the heat flow varies between 70 and 100% for the boreholes studied with essential (up to 100%) increase in heat flow within upper depth intervals of 2–4 km in some cases. Terrestrial heat flow values established from the measurements in deep and super-deep boreholes exceed the previous experimental heat flow estimates by 30…130% depending on a region of drilling. During the previous estimates the heat flow values were obtained from the measurements in shallow boreholes and heat flow was determined from averaging temperature gradient and thermal conductivity along boreholes. The established heat flow variations play an important role in the improvement of reliability of basin and petroleum system modeling and prediction of temperatures below the borehole depths. The use of calibrated heat flow distributions is shown to increase the confidence of such studies. Introduction Experimental data on heat flow density and rock thermal properties (thermal conductivity and volumetric heat capacity) are critically important for basin and petroleum system modeling. The results of the modeling depend essentially on heat flow density values and thermal property values for the sedimentary basin under studying integrated in the model. The rock thermal properties determine formation thermal regime in its natural state as borehole as at thermal methods of EOR. It is considered normally that satisfactory data on heat flow and thermal properties could be found in publications and it is a usual practice in oil/gas science and industry at basin and petroleum modeling at present.

This work deals with the identification of thermophysical properties with the line heat source probe developed by Blackwell in 1954 [1]. Such method, which is now a standard ASTM method [2],[3], relies on large time transient temperature measurements to identify the thermal conductivity of granular materials or viscous liquids. In this work we present a mathematical model that enables the estimation of the material volumetric heat capacity, in addition to the thermal conductivity, by using temperature measurements taken at short and long times. The complex step method [4] is utilized in order to verify the accuracy of the sensitivity coefficients computed with finite differences. Actual temperature measurements taken with a line heat source probe are used for estimating the properties of quartz sand. Results obtained with the developed approach are compared to those obtained with the classical method.

Changing the value of the porosity or the horizontal or vertical permeability in any grid cell in a reservoir simulator by a small amount often results in a small change in the value of a property predicted by the simulator. The ratio of change in prediction to change in reservoir property is called the sensitivity coefficient. In this paper, we describe the use of the adjoint system of equations to compute the sensitivity of wellbore pressure, water-oil ratio, and gas-oil ratio to changes in gridblock permeability and porosity. Unlike some other methods of computing sensitivity coefficients, this method is applicable for problems with large numbers of model parameters and for problems in which cross-flow and compressibility are significant. Although the adjoint system has previously been used to compute the gradient of an objective function for three-phase flow data with respect to model parameters, the sensitivity coefficients are significantly more powerful as they enable the use of Newton-like methods with quadratic convergence, i.e., estimation of the covariance of model estimates based on the inverse of the Hessian, and they provide insight into the information content of data. We use several three-phase flow examples with solution-gas drive, gas injection, and gravity segregation to illustrate these ideas.

Two-step Monte Carlo sensitivity analysis of alpha- and gamma-eigenvalues with the differential operator sampling method

Perturbation mapping method for sensitivity analysis of three-dimensional cracks near a free surface

Efficient sensitivity analysis method for chaotic dynamical systems