Thermal analysis finite difference method

The analysis of polymer processing is reduced to the balance equations, mass or continuity, energy, momentum and species and to some constitutive equations such as viscosity models, thermal conductivity models, etc. Our main interest is to solve this coupled nonlinear system of equations as accurately as possible with the least amount of computational effort. In order to do this, we simplify the geometry, we apply boundary and initial conditions, we make some physical simplifications and finally we chose an appropriate constitutive equations for the problem. At the end, we will arrive at a mathematical formulation for the problem represented by a certain function, say / (x, T, p, u,...), valid for a domain V. Due to the fact that it is impossible to obtain an exact solution over the entire domain, we must introduce discretization, for example, a grid. The grid is just a domain partition, such as points for finite difference methods, or elements for finite elements. Independent of whether the domain is divided into elements or points, the solution of the problem is always reduced to a discreet solution of the problem variables at the points or nodal pointsinxxnodes. The choice of grid, i.e., type of element, number of points or nodes, directly affects the solution of the problem. 

Lineup of three diti nsional thermal hydraulics simulation is a finite difference method code AQUA SPLASH based on Arbitrary-Eulerian-Lagrangian finite element method and direct simulation code DINUS-3. The AQUA code is being applied to natural circulation and thermal stratification analysis. The SPLASH mostly simulates the free surface phenomena. The DINUS-3 code simulates thermal striping phenomena. Validations of these codes are almost completed and practical problems are currently solved. 

For numerical analysis, the governing equations were discretized using the explicit finite difference method [2]. FIGURE 3 illustrates all the governing equations and boundary conditions used with respect to their location in the one dimensional analysis. These equations were then used iteratively to simulate the thermal cycle in the injection molding process. The final thermal profile for each iteration was used as the initial condition for the next cycle. 

Experimental Methods In Differential thermal analysis (DTA) the sample and an inert reference substance, undergoing no thermal transition in the temperature range under study are heated at the same rate. The temperature difference between sample and reference is measured and plotted as a function of sample temperature. The temperature difference is finite only when heat is being evolved or absorbed because of exothermic or endothermic activity in the sample, or when the heat capacity of the sample changes abruptly. As the temperature difference is directly proportional to the heat capacity so the curves are similar to specific heat curves, but are inverted because, by convention, heat evolution is registered as an upward peak and heat absorption as a downward peak. 

Currently, numerical methods are most used to solve heat transmission problems. The method of Finite Differences is being substituted by the Finite Element Method. Most Finite Element based mechanical calculation codes include the Thermal Analysis. The temperature distribution obtained from the thermal calculation is used as a load input to the mechanical stress and deformation problem. For that, the temperatures at the nodes are transformed into initial strain by means of the equation... 

In summary, the simplified inelastic analysis rules as indicated in subsections NB 3327.6 and NB 3228.3 of the ASME Boiler and Pressure Vessel Code Section III have been critically appraised. The first rule is shown to be equivalent to a correction factor, K, to be applied to local thermal stresses, and is based on an analysis involving a modified Poisson s ratio. For a simplified situation of thermal stress in a plate with a through the thickness temperature gradient (perfect biaxiality) the solution using NB 3227.6 are comparable to the existing solutions in the literature. However, the solutions obtained using finite-element methods and a different form of Poisson s ratio than that specified in NB 3227.6 (Eq. (11.1)) typically yield higher values of K. ... 

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