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
Fundamentals of Boundary-Layer Heat Transfer With Streamwise Temperature Variations
Boundary-layer heat transfer is analyzed for the case of a sinusoidal distribution of temperature in the direction of flow. I t is shown that for both laminar and turbulent flow the spatial distribution of heat transfer is generally out of phase with the wall temperature b}^ an angle of 30° to 45°. This leads to the conclusion that in some areas the heat flow is opposite to the temperature difference as used in the definition of the heat-transfer coefficient, and points to the basic shortcomings of this concept. The physical explanation for this behavior is found to be the temperature-field distortion by the fluid motion. The distortion is measured by the Peclet number. Approximate equations representing a "conduction analogy" were used in this analysis and the validity of these equations for unsteady flow is examined with reference to limitations in frequency and wavelength. A solution of these equations is given for the case of a velocity profile which is not a straight line. The use of previously developed variational principles for the evaluation of convective heat transfer including cases of three-dimensional unsteady flow, turbulence, and nonparallel streamlines is also discussed.