Genera Zized Variationa I Prhcz $ Zes for Convective Heat Transfer and Irreversz ’ b k Thermodynamics
Variational principles for convective heat transfer are extended to nonhomogeneous fluids and to fluids with thermal expansion. The results apply to nonlinear problems with temperature dependent properties. The context is restricted to heat transfer but the derivation and a discussion of the physics brings out the general validity of the variational principles for a large category of irreversible processes.
Further Developments of New Methods in Heat-Flow Analysis+
SUMMARY Lagrangian methods in heat-flow problems and transport phenomena were introduced by the writer in some previous work. The present paper develops further one particular aspect of the method,-i.e., the elimination of “ignorable coordinates.” This is accomplished by a special choice of generalized coordinates, each of which is constituted by an arbitrary temperature distribution and an “associated flow field.” The latter is a vector field which is derived from the corresponding scalar field by a variational method. The procedure is valid for a certain class of nonlinear problems, provided we replace the temperature by the heat content as the unknown. It is shown that for normal coordinates derivation of the associated flow field is immediate. The use of normal coordinates and their associated flow fields is illustrated by an example. Introduction of Dirac functions and associated flow fields yields a procedure which constitutes a generalization of the classical formulation by Green’s functions and integral equations. This is illustrated by application to onedimensional problems of heating of a homogeneous or composite slab and directly verified by classical methods in the Appendix. N A PREVIOUS PUBLICATION~~~ haveintroducednew methods in heat-flow analysis. These methods have a two-fold basis; first, a new formulation of the thermodynamics of irreversible processes, and, second, the application of Lagrangian techniques to the mathematical analysis itself. The earlier developments were carried out in references 2, 3, and 4. We have recently reviewed in more detail in reference 5 the basic thermodynamic concepts and principles. The formulation of dissipative phenomena into a variational language can be achieved in many ways.$ The particular method which we have chosen is different from the classical variational approaches. It appears to be the most general and fits into a unified thermodynamic theory embracing a large category of physical phenomena, which leads to equations of the same type as in Lagrangian mechanics. A particularly useful concept which has been introduced is the generalized thermodynamic force by a @inciple of virtual work. In the case of thermal problems this results in a representation of the physical system by means of a vectorial flow field and to the use of a generalized thermal force to represent the externally applied tem
A Short-Term and High-Resolution Distribution System Load Forecasting Approach Using Support Vector Regression With Hybrid Parameters Optimization
This paper proposes an approach for distribution system load forecasting, which aims to provide highly accurate short-term load forecasting with high resolution utilizing a support vector regression (SVR) based forecaster and a two-step hybrid parameters optimization method. Specifically, because the load profiles in distribution systems contain abrupt deviations, a data normalization is designed as the pretreatment for the collected historical load data. Then an SVR model is trained by the load data to forecast the future load. For better performance of SVR, a two-step hybrid optimization algorithm is proposed to determine the best parameters. In the first step of the hybrid optimization algorithm, a designed grid traverse algorithm (GTA) is used to narrow the parameters searching area from a global to local space. In the second step, based on the result of the GTA, particle swarm optimization is used to determine the best parameters in the local parameter space. After the best parameters are determined, the SVR model is used to forecast the short-term load deviation in the distribution system. The performance of the proposed approach is compared to some classic methods in later sections of this paper.