Part VIII – August 1969 – Papers - Mathematical Models of a Transient Thermal System

Mathematical models of the transient thermal behavior of a high-temperature solution calorimeter1-3 have been developed. The thermal behavior of the calorimeter is appoxirrzated by linear lumped-parameter models, and hence is described by sets of linear ordinary differential equations with constant coefficients The response of the models to various inputs is shown to agree with the response of the real system. Application of the modeling to experimental design and analysis of data illustrates the usefulness of simple models of complex systems. The early eperiments1,2 with the high-temperature solution calorimeter indicated that the change in the temperature of the bath resulting from the addition of a solute sample to the bath involved not only the direct effect due to the solution process but also possibly a secondary effect arising from the change in coupling between the bath and the induction heating coil. Consequently, an extensive analysis of the calorimeter was carried out, and models of the transient thermal processes of the instrument were developed to aid in improving the design and interpreting the behavior of the system. This paper describes the dynamic modeling; the use of it in treating experimental results has been reported earlier.3 The high-temperature solution calorimeter was constructed to measure directly the partial molar heats of solution of solute elements in a variety of liquid metal solvents.1-3 The calorimeter consists of an induction-heated liquid metal bath into which small samples of a solute element can be dropped. The bath temperature is recorded continuously, and the change in the measured bath temperature with time, dTm = f(t), resulting from the solute addition are the raw data from which the enthalpy change caused by the addition is determined. To extract the rmodynamic results from the data, the temperature change must be compared with that resulting from calibration additions of known enthalpy change. Accordingly, it is necessary to understand the transient thermal processes arising as a result of the addition to the bath. Neither modeling nor experimentation alone could provide the required insight into the working of the calorimeter. The alternate use of both methods in conjunction greatly assisted the design of the equipment and experiments, and the interpretation of the data. THE PHYSICAL CHARACTER OF THE SYSTEM The essential parts of the calorimeter, Fig. 1, for model studies are the thermocouple, the liquid metal bath and the surrounding refractories. The system is the solvent metal bath and those refractories around it which undergo a temperature change as a result of an addition to the bath, and which determine the way the temperature of the bath responds to an input. The inputs are the combined transient thermal effects arising when an addition is made to the bath. They include the thermal effects of the addition itself and the results of changed coupling between the bath and the induction coil. The response is the variation in the measured bath temperature, dTm(t) = Tm(t) - Tm(O), from an initial steady state resulting from the inputs. It was assumed in this study that the physical properties of the various elements of the system are independent of the inputs and time, although these properties may vary as the result of changes in the composition and size of the bath during a series of additions. This separation of inputs and the system is equivalent to assuming that the system is linear, i.e., that its behavior can be described by linear differential equations with constant coefficients. Linear behavior can be expected whenever the departure of each portion of the system from its steady-state condition is small enough to cause negligible changes in the thermal properties of the materials and in the various heat-transfer coefficients. Radiative heat transfer is important in this system, so the assumption of linearity should be valid only for small temperature deviations. Several conclusions were drawn from operation of the calorimeter in earlier experimental studies: 1) Radiative heat transport from the top of the bath is a significant portion of the total heat lost from the bath. However, for small changes in the bath temperature the change in transport by this path could be assumed to be proportional to the change in the bath temperature. 2) A very small portion of the heat input is lost through the thermocouple to its water-cooled holder. The thermal resistance and thermal capacity of the thermocouple protection tube are small, so the temperature of the thermocouple should follow closely that of the bath. 3) The remainder of the total heat lost from the bath will pass by conduction through the crucible to, and through, the other refractories, eventually being absorbed by the water-cooled induction coil or by the water-cooled sides and bottom of the enclosure. 4) The thermal resistance between the bath and crucible is very small. Thus the thermal capacity of the crucible will affect the temperature of the bath very soon after an addition of heat to the bath. 5) The thermal resistance between the crucible and the silica sleeve is large, especially if a radiation shield is placed in the gap. The effect of the thermal capacity of the sleeve thus will be significant only at longer times. The thermal resistance through the packing below the crucible also is large, so the packing and the silica sleeve will have similar effects on the behavior of the system. 6) A large temperature drop exists across the gap containing the water-cooled induction coil. Thus for relatively small changes in the thermal input to the bath, the refractories beyond the sleeve