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.
ASYMPTOTICS OF THE FILTRATION PROBLEM WITH ALMOST CONSTANT COEFFICIENTS
