Comparison of Models for Heat Transfer in High-Density Fibrous Insulation.

This study evaluated different models for calculating the effective thermal conductivity of fibrous insulation by comparing predicted values with certified values of Standard Reference Material 1450c, Fibrous Glass Board. This comparison involved the coupled effects of radiation and conduction heat transfer. To support these comparisons, the fiber diameter distribution was measured using X-ray computed tomography, and this distribution was used in several heat transfer models considered in this paper. For the evaluation of the radiative heat transfer, the diffusion approximation, the Schuster-Schwarzschild approximation, and the Milne-Eddington approximation were considered. The conduction of the gas and the fibers was treated by the kinetic theory and a semi-empirical model, respectively. Two models were considered for the evaluation of the radiative properties: the large specular reflecting approach and the application of Mie theory for media composed of infinite cylinders.

Historical Review of the Metered Section Area for the Guarded-Hot-Plate Method.

Results of an extensive literature review and investigation of the metered section area for the guarded-hot-plate method, standardized as ASTM C177, Standard Test Method for Steady-State Heat Flux Measurements and Thermal Transmission Properties by Means of the Guarded-Hot-Plate Apparatus, are presented. The guarded-hot-plate apparatus is a primary linear-heat-flow method generally used to determine the thermal conductivity of insulating and building materials. The review examined technical publications from 1885 to 1990 and identified 31 papers of interest. Historical versions of ASTM C177 were also researched as well as test methods from other standard development organizations. The investigation revealed that, over the past 100 years, researchers have independently developed two main approaches for the computation of the metered section area. An assessment of the calculation techniques is presented for round plates with diameters from 250 to 1,000 mm, a guard-to-meter aspect ratio of 2, and guard gap widths of 1-4 mm. The gap effects are not negligible because large gaps (4 mm) on small plates (250 mm) can lead to errors of 10 % or more on the computation of the metered section area, ultimately affecting the uncertainty of the test results of the guarded-hot-plate method. The results of this study are applicable to other thermal conductivity test methods that employ a primary thermal guard to promote 1-D heat flow.

Transient Thermal Response of a Guarded-Hot-Plate Apparatus for Operation Over an Extended Temperature Range.

A mathematical model is presented for a new-generation guarded-hot-plate apparatus to measure the thermal conductivity of insulation materials. This apparatus will be used to provide standard reference materials for greater ranges of temperature and pressure than have been previously available. The apparatus requires precise control of 16 interacting heated components to achieve the steady temperature and one-dimensional heat-transfer conditions specified in standardized test methods. Achieving these criteria requires deriving gain settings for the 16 proportional-integral-derivative (PID) controllers, comprising potentially 48 parameters. Traditional tuning procedures based on trial-and-error operation of the actual apparatus impose unacceptably lengthy test times and expense. A primary objective of the present investigation is to describe and confirm the incremental control algorithm for this application and determine satisfactory gain settings using a mathematical model that simulates in seconds test runs that would require days to complete using the apparatus. The first of two steps to achieve precise temperature control is to create and validate a model that accounts for heating rates in the various components and interactions with their surroundings. The next step is to simulate dynamic performance and control with the model and determine settings for the PID controllers. A key criterion in deriving the model is to account for effects that significantly impact thermal conductivity measurements while maintaining a tractable model that meets the simulation time constraint. The mathematical model presented here demonstrates how an intricate apparatus can be represented by many interconnected aggregated-capacity masses to depict overall thermal response for control simulations. The major assemblies are the hot plate with four subcomponents, two cold plates with three subcomponents each, and two edge guards with three subcomponents each. Using symmetry about the hot plate, the number of components in the simulation model is reduced to 12 or 15, depending on the mode of operation for the apparatus. Configurations of the main components with embedded heating elements were carefully designed earlier using detailed finite-element analyses to give essentially isothermal surfaces and one-dimensional heat flow through test specimens. It is not tractable, or perhaps justified, to extend these prior analyses to simulate the controlled transient responses of the apparatus. The earlier design criterion does, however, support the aggregated-capacity simplification implemented in the present thermal model. The governing equations follow from dynamic energy balances on components with controlled heating elements and additional intermediate ("floating") components. Thermal bridges comprise conduction paths, with and without surface convection and radiation, between components and fixed-temperature "heat sinks." An implicit finite-difference numerical method was used to solve the resulting system of first-order differential equations. The mathematical model was initially validated using measurement data from test runs where a step change in heating rate was applied to single elements in turn, and component temperatures were recorded up to a nearly steady condition. Thermocouples and standard platinum resistance thermometers were used to measure temperatures, and thermopiles were used to measure temperature differences. Next, extensive simulations were conducted with the mathematical model to estimate suitable gain settings for the various controllers. The criteria were tight temperature control after reaching set points and acceptable times to achieve quasi-steady-state operation. Comparisons between measurements and predicted temperatures for heated components are presented. The results show that the model incorporating the above simplifying approximations is satisfactory for components comprising the hot-plate and cold-plate assemblies. For the edge guards, however, the conventional aggregated-capacity criteria are not as fully satisfied because of their configuration. Temperature variations in the edge guards, fortunately, have a lesser effect on the accuracy of the thermal conductivity measurements. Therefore, the thermal response model is deemed satisfactory for simulating PID feedback to investigate "closed-loop" control of the apparatus, thus meeting the primary objective.

Retrospective Analysis of NIST Standard Reference Material 1450, Fibrous Glass Board, for Thermal Insulation Measurements.

Thermal conductivity data acquired previously for the establishment of Standard Reference Material (SRM) 1450, Fibrous Glass Board, as well as subsequent renewals 1450a, 1450b, 1450c, and 1450d, are re-analyzed collectively and as individual data sets. Additional data sets for proto-1450 material lots are also included in the analysis. The data cover 36 years of activity by the National Institute of Standards and Technology (NIST) in developing and providing thermal insulation SRMs, specifically high-density molded fibrous-glass board, to the public. Collectively, the data sets cover two nominal thicknesses of 13 mm and 25 mm, bulk densities from 60 kg·m(-3) to 180 kg·m(-3), and mean temperatures from 100 K to 340 K. The analysis repetitively fits six models to the individual data sets. The most general form of the nested set of multilinear models used is given in the following equation: [Formula: see text]where λ(ρ,T) is the predicted thermal conductivity (W·m(-1)·K(-1)), ρ is the bulk density (kg·m(-3)), T is the mean temperature (K) and ai (for i = 1, 2, … 6) are the regression coefficients. The least squares fit results for each model across all data sets are analyzed using both graphical and analytic techniques. The prevailing generic model for the majority of data sets is the bilinear model in ρ and T. [Formula: see text] One data set supports the inclusion of a cubic temperature term and two data sets with low-temperature data support the inclusion of an exponential term in T to improve the model predictions. Physical interpretations of the model function terms are described. Recommendations for future renewals of SRM 1450 are provided. An Addendum provides historical background on the origin of this SRM and the influence of the SRM on external measurement programs.

Thermal response simulation for tuning PID controllers in a 1016 mm guarded hot plate apparatus.

A mathematical model has been developed and used to simulate the controlled thermal performance of a large guarded hot-plate apparatus. This highly specialized apparatus comprises three interdependent components whose temperatures are closely controlled in order to measure the thermal conductivity of insulation materials. The simulation model was used to investigate control strategies and derive controller gain parameters that are directly transferable to the actual instrument. The simulations take orders-of-magnitude less time to carry out when compared to traditional tuning methods based on operating the actual apparatus. The control system consists primarily of a PC-based PID control algorithm that regulates the output voltage of programmable power amplifiers. Feedback parameters in the form of controller gains are required for the three heating circuits. An objective is to determine an improved set of gains that meet temperature control criteria for testing insulation materials of interest. The analytical model is based on aggregated thermal capacity representations of the primary components and includes the same control algorithm as used in the actual hot-plate apparatus. The model, accounting for both thermal characteristics and temperature control, was validated by comparisons with test data. The tuning methodology used with the simulation model is described and results are presented. The resulting control algorithm and gain parameters have been used in the actual apparatus without modification during several years of testing materials over wide ranges of thermal conductivity, thickness, and insulation resistance values.

Assessment of Uncertainties for the NIST 1016 mm Guarded-Hot-Plate Apparatus: Extended Analysis for Low-Density Fibrous-Glass Thermal Insulation.

An assessment of uncertainties for the National Institute of Standards and Technology (NIST) 1016 mm Guarded-Hot-Plate apparatus is presented. The uncertainties are reported in a format consistent with current NIST policy on the expression of measurement uncertainty. The report describes a procedure for determination of component uncertainties for thermal conductivity and thermal resistance for the apparatus under operation in either the double-sided or single-sided mode of operation. An extensive example for computation of uncertainties for the single-sided mode of operation is provided for a low-density fibrous-glass blanket thermal insulation. For this material, the relative expanded uncertainty for thermal resistance increases from 1 % for a thickness of 25.4 mm to 3 % for a thickness of 228.6 mm. Although these uncertainties have been developed for a particular insulation material, the procedure and, to a lesser extent, the results are applicable to other insulation materials measured at a mean temperature close to 297 K (23.9 °C, 75 °F). The analysis identifies dominant components of uncertainty and, thus, potential areas for future improvement in the measurement process. For the NIST 1016 mm Guarded-Hot-Plate apparatus, considerable improvement, especially at higher values of thermal resistance, may be realized by developing better control strategies for guarding that include better measurement techniques for the guard gap thermopile voltage and the temperature sensors.