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Cabin heater, Caetano, EF Calcium carbonate, fouling of heat exchangers by, Calcium sulphate, fouling of heat exchangers by, CALFLO, heat transfer media, Calorically perfect gas, CANDU Reactor, fouling problems in, Carbon dioxide: Carbon disulfide: Carbon monoxide: Carbon steel: Carbon-manganese steels Carbon-molybdenum steels, Carbon tetrachloride: Carbonyl sulfide: Carboxylic acids: Carnot cycle in refrigeration, Carnot factor, Carreau fluid (non-Newtonian), Carryover of solids in fluidized beds, Cashman, B L, Cast iron, thermal and mechanical properties, Cavitation as source of damage in heat exchangers, Cell method, for heat exchanger effectiveness, Cement kilns, CEN code for pressure vessels, Centrifugal dryer, Ceramics Certification of heat exchangers, Chan, S H, Channel emissivity, Chapman-Rubescin formula for viscosity variation with temperature, Chemical exergy, Chemical formulas of commonly used fluids Chemical industry, fouling of heat exchangers in, Chemical reactions, exergy analysis of, Chemical reaction fouling, Chen correlation for forced convective boiling, Chen method, for enthalpy of vaporisation, Chenoweth, J M, Chevron troughs as corrugation design in plate heat exchangers, Chillers, construction features of, Chilton-Colburn analogy, Chisholm, D Chisholm correlations: Chlorine: Chloroacetic acid: Chlorobenzene: Chlorobutane: Chlorodifluoromethane (see Refrigerant 22) 1-Chloro-1,1-difluoroethane (Refrigerant 142b): Chloroethane (Refrigerant 160): Chloromethane (Refrigerant 40): Chloropentane: 1,2-Chloropentafluoroethane (Refrigerant 115): Chloroprene (2-Chloro-1,3-butadiene): 1-Chloropropane: 2-Chloropropane: m-Chlorotoluene: o-Chlorotoluene: Chlorotrifluoroethylene: Chlorotrifluoromethane (see Refrigerant 13) Chromium-molybdenum steels, Chudnovsky, Y, Chugging flow (gas-liquid), in shell-and-tube heat exchangers, Chung et al method, for viscosity of low pressure gases, Church and Prausnitz methods: Churchill, S W, Churchill and Chu correlations for free convective heat transfer: Churn flow, regions of occurrence of, Circles, radiative heat transfer shape factors between parallel coaxial, Circular girth flanges, design according to ASME VIII code, Circulating fluidized beds, Circulation, modes of in free convection: in enclosures heated from below, CISE correlations for void fractions, Clausius-Clapeyron relationship: Cleaning: Climbing film evaporator, Closed circuit cooling towers, Coalescence of bubbles in fluidized beds, Coatings for corrosion protection Cocurrent flow: Codes, mechanical design: Cogeneration Colburn and Drew method for binary vapor condensation, Colburn and Hougen method for condensation in presence of noncondensable gases Colburn equation for single-phase heat transfer outside tube banks, Colburn j factor: Colebrook-White equation for friction factor in rough circular pipe, Coles, law of the wake, Collier, J G, Combined free and forced convection heat transfer: Combined heat and mass transfer, Combining flow, loss coefficients in, Combustion model for furnaces, Compact heat exchangers (see Plate fin heat exchangers) Compartment dryers, Composite curves, in the pinch analysis method for heat exchanger network analysis: Compressed liquids, density of: Compressible flow: Compression, exergy analysis of Compressive stress, in heat exchanger tubes, Computer-aided design, of evaporators, Computer program for Monte Carlo calculations of radiative heat transfer, Computer simulation, of fouling, Computer software for mechanical design, Concentration, choice of evaporator type for, Concentric spheres, free convective heat transfer in, Concurrency corrections in plate heat exchangers, Condensation: Concrete, lightweight, submerged combustion system for, Condensation curves: Condenser/preheater tubes, in multistage flash evaporation, Condensers: Conduction, heat: Conductors, thermal conductivity of, Cones, under internal pressure, EN13445 guidelines for, Cones, vertical: Conical shells, mechanical design of: Conjugate radiation interactions Connors equation for fluid elastic instability, Conservation equations: Constantinon and Gani method, for estimating normal boiling point, Contact angle, Contact resistance: Continuity equation: Continuum model, for fluids, Continuum theories, for non-Newtonian fluids, Contraction, sudden, pressure drop in: Control: Control volume method, in finite difference solutions for conduction, Convection, interaction of radiation with, Convection effects, on heat transfer in kettle reboilers, Convective heat transfer, single-phase: Conversion factors: Conveyor, gravity: Cooling curves, in condensation, Cooling towers: Cooling water fouling, Cooper correlation, for nucleate boiling, Cooper, Anthony, Copper, thermal and mechanical properties, Copper alloys, Correlation, general nature of, Corresponding states principle Corrosion: Corrugation design, for plate heat exchangers Costing of heat exchangers: Countercurrent flow: Coupled thermal fields, in transient conduction, Cowie, R C, Crank-Nicolson differencing scheme, in finite difference method, Creeping flow, in combined free and forced convection around immersed bodies, m-Cresol: o-Cresol: p-Cresol: Crevice corrosion, in stainless steels, Critical constants Critical density, of commonly used fluids, Critical flow, in gas-liquid systems, Critical heat flux: Critical pressure: Critical Rayleigh number, in free convection, Critical temperature: Critical velocity, in stratification in bends and horizontal tubes, Critical volume (see also Critical density) Cross counterflow heat exchangers, Crossflow: Crude oil, fouling of heat exchangers: Cryogenic plant, entropy generation in, Crystallization Crystallization fouling, Curved ducts: Cut-and-twist factor, in enhancement of heat transfer in double pipe heat exchangers, C-value method for heat exchanger costing, Cycling, of expansion bellows, Cyclobutane: Cyclohexane: Cyclohexanol: Cyclohexene: Cyclopentane: Cyclopentene: Cyclopropane: Cylinders: Cylindrical contacts, thermal contact resistance in, Cylindrical coordinates, finite difference equations for conduction in, Cylindrical shell, analytical basis of code rules for,

Index

HEDH
A B C
Cabin heater, Caetano, EF Calcium carbonate, fouling of heat exchangers by, Calcium sulphate, fouling of heat exchangers by, CALFLO, heat transfer media, Calorically perfect gas, CANDU Reactor, fouling problems in, Carbon dioxide: Carbon disulfide: Carbon monoxide: Carbon steel: Carbon-manganese steels Carbon-molybdenum steels, Carbon tetrachloride: Carbonyl sulfide: Carboxylic acids: Carnot cycle in refrigeration, Carnot factor, Carreau fluid (non-Newtonian), Carryover of solids in fluidized beds, Cashman, B L, Cast iron, thermal and mechanical properties, Cavitation as source of damage in heat exchangers, Cell method, for heat exchanger effectiveness, Cement kilns, CEN code for pressure vessels, Centrifugal dryer, Ceramics Certification of heat exchangers, Chan, S H, Channel emissivity, Chapman-Rubescin formula for viscosity variation with temperature, Chemical exergy, Chemical formulas of commonly used fluids Chemical industry, fouling of heat exchangers in, Chemical reactions, exergy analysis of, Chemical reaction fouling, Chen correlation for forced convective boiling, Chen method, for enthalpy of vaporisation, Chenoweth, J M, Chevron troughs as corrugation design in plate heat exchangers, Chillers, construction features of, Chilton-Colburn analogy, Chisholm, D Chisholm correlations: Chlorine: Chloroacetic acid: Chlorobenzene: Chlorobutane: Chlorodifluoromethane (see Refrigerant 22) 1-Chloro-1,1-difluoroethane (Refrigerant 142b): Chloroethane (Refrigerant 160): Chloromethane (Refrigerant 40): Chloropentane: 1,2-Chloropentafluoroethane (Refrigerant 115): Chloroprene (2-Chloro-1,3-butadiene): 1-Chloropropane: 2-Chloropropane: m-Chlorotoluene: o-Chlorotoluene: Chlorotrifluoroethylene: Chlorotrifluoromethane (see Refrigerant 13) Chromium-molybdenum steels, Chudnovsky, Y, Chugging flow (gas-liquid), in shell-and-tube heat exchangers, Chung et al method, for viscosity of low pressure gases, Church and Prausnitz methods: Churchill, S W, Churchill and Chu correlations for free convective heat transfer: Churn flow, regions of occurrence of, Circles, radiative heat transfer shape factors between parallel coaxial, Circular girth flanges, design according to ASME VIII code, Circulating fluidized beds, Circulation, modes of in free convection: in enclosures heated from below, CISE correlations for void fractions, Clausius-Clapeyron relationship: Cleaning: Climbing film evaporator, Closed circuit cooling towers, Coalescence of bubbles in fluidized beds, Coatings for corrosion protection Cocurrent flow: Codes, mechanical design: Cogeneration Colburn and Drew method for binary vapor condensation, Colburn and Hougen method for condensation in presence of noncondensable gases Colburn equation for single-phase heat transfer outside tube banks, Colburn j factor: Colebrook-White equation for friction factor in rough circular pipe, Coles, law of the wake, Collier, J G, Combined free and forced convection heat transfer: Combined heat and mass transfer, Combining flow, loss coefficients in, Combustion model for furnaces, Compact heat exchangers (see Plate fin heat exchangers) Compartment dryers, Composite curves, in the pinch analysis method for heat exchanger network analysis: Compressed liquids, density of: Compressible flow: Compression, exergy analysis of Compressive stress, in heat exchanger tubes, Computer-aided design, of evaporators, Computer program for Monte Carlo calculations of radiative heat transfer, Computer simulation, of fouling, Computer software for mechanical design, Concentration, choice of evaporator type for, Concentric spheres, free convective heat transfer in, Concurrency corrections in plate heat exchangers, Condensation: Concrete, lightweight, submerged combustion system for, Condensation curves: Condenser/preheater tubes, in multistage flash evaporation, Condensers: Conduction, heat: Conductors, thermal conductivity of, Cones, under internal pressure, EN13445 guidelines for, Cones, vertical: Conical shells, mechanical design of: Conjugate radiation interactions Connors equation for fluid elastic instability, Conservation equations: Constantinon and Gani method, for estimating normal boiling point, Contact angle, Contact resistance: Continuity equation: Continuum model, for fluids, Continuum theories, for non-Newtonian fluids, Contraction, sudden, pressure drop in: Control: Control volume method, in finite difference solutions for conduction, Convection, interaction of radiation with, Convection effects, on heat transfer in kettle reboilers, Convective heat transfer, single-phase: Conversion factors: Conveyor, gravity: Cooling curves, in condensation, Cooling towers: Cooling water fouling, Cooper correlation, for nucleate boiling, Cooper, Anthony, Copper, thermal and mechanical properties, Copper alloys, Correlation, general nature of, Corresponding states principle Corrosion: Corrugation design, for plate heat exchangers Costing of heat exchangers: Countercurrent flow: Coupled thermal fields, in transient conduction, Cowie, R C, Crank-Nicolson differencing scheme, in finite difference method, Creeping flow, in combined free and forced convection around immersed bodies, m-Cresol: o-Cresol: p-Cresol: Crevice corrosion, in stainless steels, Critical constants Critical density, of commonly used fluids, Critical flow, in gas-liquid systems, Critical heat flux: Critical pressure: Critical Rayleigh number, in free convection, Critical temperature: Critical velocity, in stratification in bends and horizontal tubes, Critical volume (see also Critical density) Cross counterflow heat exchangers, Crossflow: Crude oil, fouling of heat exchangers: Cryogenic plant, entropy generation in, Crystallization Crystallization fouling, Curved ducts: Cut-and-twist factor, in enhancement of heat transfer in double pipe heat exchangers, C-value method for heat exchanger costing, Cycling, of expansion bellows, Cyclobutane: Cyclohexane: Cyclohexanol: Cyclohexene: Cyclopentane: Cyclopentene: Cyclopropane: Cylinders: Cylindrical contacts, thermal contact resistance in, Cylindrical coordinates, finite difference equations for conduction in, Cylindrical shell, analytical basis of code rules for,
D E F G H I J K L M N O P Q R S T U V W X Y Z

Finite-Element Methods for Conduction

DOI 10.1615/hedhme.a.000166

2.4.8 Finite-element methods for conduction

A. Introduction

(a) Preamble

The development of finite-element methods (FEMs) for the solution of conduction heat transfer problems is relatively recent in comparison with their finite-difference method counterparts. The FEM development has followed a philosophically different approach from that followed for finite-difference methods, although both are based on a volumetric discretization of the solution domain. One of the prime motivating factors leading to the development of finite-element methods has been the desire to enhance the compatibility of thermal models for use in thermal stress analysis. In stress analysis, there exists a variational extremum principle such that the minimization of the strain energy within a continuum domain leads naturally to the formulation of a discrete model to determine an approximate displacement field from which the approximate stress distribution can be calculated. An additional motivation for the development of finite-element methods for conduction heat transfer has been the desire to achieve geometric independence of the resulting procedure. Thus, whereas a finite-difference code can be constructed rapidly for the analysis of a Cartesian domain of rectangular configuration, the extension of such a code to arbitrary configurations is painful and complex. In contrast, the construction of a finite-element code entails separate consideration of many different aspects of the discrete problem formulation. Once constructed, however, extension of the code to consider geometries of relatively arbitrary configuration is straightforward.

In conduction heat transfer, as opposed to stress analysis, a natural integral formulation having a clear physical interpretation docs not exist. The search for a variational formulation of the heat conduction problem, apparently in the interest of preserving the underlying principles leading to the development of the method, has, however, led to several variational formulations of the problem. The approach adopted by Visser (1965), Zienkiewicz and Parekh (1970), and Zienkiewicz (1971) considers a particular instant in time such that time derivatives of temperature and of physical parameters can be treated as prescribed functions of the spatial coordinate. In this way, a quasi-variational statement can be formulated and converted to a restricted variational statement as provided by Finlayson and Scriven (1967). Biot (1955 and 1956) has also provided quasi-variational statements for the heat conduction problem. These statements for the conduction problem are in contrast to the true variational statement given by Gurtin (1964) through the use of convolution integrals in time. The true variational approach has been used by Wilson and Nickell (1966) for analysis in a Cartesian coordinate frame.

In contrast to the variational approach, the method of weighted residuals (MWR) Finlayson (1972) is being used with increasing frequency to formulate finite-element models for conduction heat transfer problems (Finlayson and Scriven, 1967; Warzec, 1970; Yalamanchili and Chu, 1973). In particular, the Galerkin method of weighted residuals has received the greatest amount of attention. Finlayson and Scriven (1967) state that while both the quasi-variational and restricted variational approaches retain some variational formalism, and they refer to these methods as ad hoc variational principles, the variational method of approximation to which they lead is indistinguishable from the Galerkin method applied to the original equations.

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