Mathematical modeling thermal

Mathematical Modeling. The results obtained by various dilatomet-ric techniques were compared to theoretical predictions of thermal expansion using a simple mathematical model. Thermal properties of Kevlar fabric/epoxy lamina were simulated by considering the fabric to be made of two consecutive 0° and 90° lamina of... 

In general, the desorptive behavior of contaminated soils and soHds is so variable that the requited thermal treatment conditions are difficult to specify without experimental measurements. Experiments are most easily performed in bench- and pilot-scale faciUties. Full-scale behavior can then be predicted using mathematical models of heat transfer, mass transfer, and chemical kinetics. 

For practical reasons, the blast furnace hearth is divided into two principal zones the bottom and the sidewalls. Each of these zones exhibits unique problems and wear mechanisms. The largest refractory mass is contained within the hearth bottom. The outside diameters of these bottoms can exceed 16 or 17 m and their depth is dependent on whether underhearth cooling is utilized. When cooling is not employed, this refractory depth usually is determined by mathematical models these predict a stabilization isotherm location which defines the limit of dissolution of the carbon by iron. Often, this depth exceeds 3 m of carbon. However, because the stabilization isotherm location is also a function of furnace diameter, often times thermal equiHbrium caimot be achieved without some form of underhearth cooling. 

Mathematical models that ignored kinetic forms may fit the experimental results very well but fail to predict critical performance attributes. For example, neglecting the well known exponential form of the Arrhenius fianction made one, entirely mathematical, model fail in predicting the thermal runaway. 

Horie and his coworkers [90K01] have developed a simplified mathematical model that is useful for study of the heterogeneous nature of powder mixtures. The model considers a heterogeneous mixture of voids, inert species, and reactant species in pressure equilibrium, but not in thermal equilibrium. The concept of the Horie VIR model is shown in Fig. 6.3. As shown in the figure, the temperatures in the inert and reactive species are permitted to be different and heat flow can occur from the reactive (usually hot) species to the inert species. When chemical reaction occurs the inert species acts to ther-... 

Thermal conductivity increases with temperature. The insulating medium (the air or gas within the voids) becomes more excited as its temperature is raised, and this enhances convection within or between the voids, thus increasing heat flow. This increase in thermal conductivity is generally continuous for air-filled products and can be mathematically modeled (see Figure 11.3). Those insulants that employ inert gases as their insulating medium may show sharp changes in thermal conductivity, which may occur because of gas condensation. However, this tends to take place at sub-zero temperatures. 

Farid, M.M. and Kanzawa, A., 1989, Thermal performance of a heat storage module using PCM s with different melting temperatures mathematical modeling, ASME./. Solar Energy Eng. 111 152-157. 

In the production of formic acid, a slimy of calcium formate in 50% aqueous formic acid containing urea is acidified with strong nitric acid to convert the calcium salt to free acid, and interaction of formic acid (reducant) with nitric acid (oxidant) is inhibited by the urea. When only 10% of the required amount of urea had been added (unwittingly, because of a blocked hopper), addition of the nitric acid caused a thermal runaway (redox) reaction to occur which burst the (vented) vessel. A small-scale repeat indicated that a pressure of 150-200 bar may have been attained. A mathematical model was developed which closely matched experimental data. 

Reaction (52) occurs at the gradient interface of the bolus addition until local Hb(02) concentrations have been reduced, at which point additional NO reduces the iron(III) to iron(II) which can further react with free NO to form Hb(NO). The validity of this mechanism was verified by the observation that addition of CN- ion, which binds irreversibly to metHb to form metHb(CN), significantly attenuated the formation of Hb(NO) in both cell-free Hb and RBC. Mathematical models used to simulate bolus addition of NO to cell-free Hb and RBC were compatible with the experimental results (147). In the above experiments, SNO-Hb was a minor reaction product and was formed even in the presence of 10 mM CN, suggesting that RSNO formation does not occur as a result of (hydrolyzed) NO+ formation during metHb reduction. However, formation of SNO-Hb was not detectable when NO was added as a bolus injection to RBC or through thermal decomposition of DEA/NO in cell free Hb (DEA/NO = 2-(A/ A/ diethylamino)diazenolate). SNO-Hb was observed... 

Hofelich, T.C., J.B. Powers and D.J. Frurip 1994. "Determination of Compatibility via Thermal Analysis and Mathematical Modeling." Process Safety Progress 13(4) 227. October. 

It is important to note that Vie and Kjelstrup [250] designed a method of measuring fhe fhermal conductivities of different components of a fuel cell while fhe cell was rurming (i.e., in situ tests). They added four thermocouples inside an MEA (i.e., an invasive method) one on each side of the membrane and one on each diffusion layer (on the surface facing the FF channels). The temperature values from the thermocouples near the membrane and in the DL were used to calculate the average thermal conductivity of the DL and CL using Fourier s law. Unfortunately, the thermal conductivity values presented in their work were given for both the DL and CL combined. Therefore, these values are useful for mathematical models but not to determine the exact thermal characteristics of different DLs. 

In Fig. 1, various elements involved with the development of detailed chemical kinetic mechanisms are illustrated. Generally, the objective of this effort is to predict macroscopic phenomena, e.g., species concentration profiles and heat release in a chemical reactor, from the knowledge of fundamental chemical and physical parameters, together with a mathematical model of the process. Some of the fundamental chemical parameters of interest are the thermochemistry of species, i.e., standard state heats of formation (A//f(To)), and absolute entropies (S(Tq)), and temperature-dependent specific heats (Cp(7)), and the rate parameter constants A, n, and E, for the associated elementary reactions (see Eq. (1)). As noted above, evaluated compilations exist for the determination of these parameters. Fundamental physical parameters of interest may be the Lennard-Jones parameters (e/ic, c), dipole moments (fi), polarizabilities (a), and rotational relaxation numbers (z ,) that are necessary for the calculation of transport parameters such as the viscosity (fx) and the thermal conductivity (k) of the mixture and species diffusion coefficients (Dij). These data, together with their associated uncertainties, are then used in modeling the macroscopic behavior of the chemically reacting system. The model is then subjected to sensitivity analysis to identify its elements that are most important in influencing predictions. 

Gray fit Yang (Ref 1), a mathematical model was proposed to unity the chain and thermal mechanisms of explosion. It was shown that the trajectories in the phase plane of the coupled energy and radical concentration equations of an explosive system will oive the time-dependent behavior of the system when the initial temperature and radical concentration are given. In the 2nd paper of the same investigators (Ref 2), a general equation for explosion limits (P—T relation) is derived from a unified thermal and chain theory and from chis equation, the criteria of explosion limits for either the pure chain or pure thermal theory can be deduced. For detailed discussion see Refs... 

The mathematical model developed in the preceding section consists of six coupled, three-dimensional, nonlinear partial differential equations along with nonlinear algebraic boundary conditions, which must be solved to obtain the temperature profiles in the gas, catalyst, and thermal well the concentration profiles and the velocity profile. Numerical solution of these equations is required. 

---