Heat Transfer Through a Gas Between Parallel Plates

1977 ◽  
Vol 32 (9) ◽  
pp. 914-926 ◽  
Author(s):  
L. Waldmann
Keyword(s):  
Heat Transfer ◽  
Particle Distribution ◽  
Temperature Jump ◽  
Exact Formula ◽  
Parallel Plates ◽  
Total Heat Transfer ◽  
Linearized Boltzmann Equation ◽  
Scattering Kernel ◽  
Interfacial Scattering ◽  
Transport Relaxation

Within the framework of boundary conditions recently developed for the linearized Boltzmann equation 1 the problem of heat transfer between parallel plates can be solved in terms of "transport- relaxation eigenfunctions". The particle distribution function and the total heat transfer in the Knudsen case are exactly expressed by integrals over the interfacial scattering kernel occurring in the new scheme. A detailed discussion of the general case gives an exact formula and sign statement for the temperature jump at parallel plates. An approximation, which encompasses v. Smoluchowski's approach, lies at hand. This approximation is also readily confirmed by a moment method.

scholarly journals Differential Equations and Boundary Conditions for Higher Gas Kinetic Moments. Heat Transfer Between Parallel Plates

1977 ◽  
Vol 32 (7) ◽  
pp. 667-677 ◽  
Author(s):  
H. Vestner ◽  
L. Waldmann
Keyword(s):  
Heat Transfer ◽  
Heat Flux ◽  
Differential Equations ◽  
Boundary Conditions ◽  
Parallel Plates ◽  
Formula One ◽  
Linearized Boltzmann Equation ◽  
Transport Relaxation ◽  
Monatomic Gas

Abstract Transport-relaxation equations for seventeen moments are derived from the linearized Boltzmann equation for a monatomic gas. Besides the well-known thirteen moments (i. e. density n, tempera­ture T, velocity ν, heat flux q, friction pressure tensor ), one additional scalar A and one ad­ditional vector A are taken into account. In steady state, differential equations for T, v. A and constitutive laws for q, , A follow from the transport-relaxation equations. Boundary conditions for T, v, A are obtained by the thermodynamical method from the interfacial entropy production. The role of the higher moments A and A for heat transfer in a gas between parallel plates is dis­cussed. The heat flux has the correct low pressure limit. Due to the presence of A and A, expo­nential terms occur in the temperature profile near the boundary.

Theory of Boundary Conditions for the Boltzmann Equation

1977 ◽  
Vol 32 (6) ◽  
pp. 521-531 ◽  
Author(s):  
L. Waldmann
Keyword(s):  
Boltzmann Equation ◽  
Boundary Conditions ◽  
Temperature Jump ◽  
Functional Relation ◽  
Linearized Boltzmann Equation ◽  
Kinetic Boundary ◽  
The Difference ◽  
Molecular Distribution Function ◽  
Transport Relaxation ◽  
Slip Coefficients

Abstract In preceding papers, Refs. 1,2, boundary conditions were developed for transport-relaxation equations by aid of a general reciprocity postulate for the interface. The same method is now used for the linearized Boltzmann equation. A new scheme emerges: the kinetic boundary conditions consist in a linear functional relation between interfacial "forces and fluxes" - in the sense of non-equilibrium thermodynamics - which are, broadly speaking, given by the sum and the difference of the molecular distribution function and its time-reversed, at the wall. The general properties of the kernels occurring in this atomistic boundary law are studied. The phenomenological surface coefficients of (generalized) linear thermo-hydrodynamics, as e. g. temperature jump, slip coefficients etc., can in a simple way be expressed by the kernel of the atomistic boundary law. This kernel is explicitly worked out for completely thermalizing wall collisions.

Energy Loss and Entropy Generation Analysis in Fundamental Convective Heat Transfer Problems

2005 ◽  
Author(s):  
S. R. Javadinejhad
Keyword(s):  
Heat Transfer ◽  
Energy Loss ◽  
Entropy Generation ◽  
Circular Cross Section ◽  
Dimensionless Number ◽  
Parallel Plates ◽  
Heat Transfer Characteristics ◽  
Total Heat Transfer ◽  
Transfer Problems ◽  
Steady Laminar

Energy loss characteristics of heat transfer and fluid flow due to forced convection of steady laminar flow of incompressible fluid inside channel with circular cross section and channel made of two parallel plates is analyzed. Energy loss profiles and heat transfer characteristics for different problems have been discussed. In each case energy loss due to heat transfer effect and fluid friction have been drived analytically. For energy loss calculations a new dimensionless number have been introduced that is the energy loss to total heat transfer rate ratio. The energy loss dimensionless number behavior have been compared with entropy generation dimensionless number behavior for various problems.

Upper and Lower Bounds for the Heat Flux in a Gas Between Parallel Plates

1981 ◽  
Vol 36 (6) ◽  
pp. 568-578
Author(s):  
H. Vestner
Keyword(s):  
Heat Transfer ◽  
Boundary Condition ◽  
Heat Flux ◽  
Lower Bounds ◽  
Upper And Lower Bounds ◽  
Parallel Plates ◽  
Starting Point ◽  
Kinetic Boundary ◽  
Transport Relaxation ◽  
Monatomic Gas

Abstract Waldmann's result for heat transfer through a monatomic gas between parallel plates is the starting point for the derivation of upper and lower bounds for the heat flux as a function of Knudsen number. In order to obtain numerical results, one transport-relaxation eigenfunction and its eigenvalue are determined approximately, and a simple model for the interfacial kernel of the kinetic boundary condition for the distribution function is used

Thermally Optimum Spacing of Vertical, Natural Convection Cooled, Parallel Plates

1984 ◽  
Vol 106 (1) ◽  
pp. 116-123 ◽  
Author(s):  
A. Bar-Cohen ◽  
W. M. Rohsenow
Keyword(s):  
Heat Transfer ◽  
Unit Volume ◽  
Parallel Plates ◽  
Total Heat Transfer ◽  
Various Boundary Conditions ◽  
Operating Modes ◽  
The Core ◽  
Optimum Spacing ◽  
The Heat Transfer Coefficient ◽  
System Operating

While component dissipation patterns and system operating modes vary widely, many electronic packaging configurations can be modeled by symmetrically or asymmetrically isothermal or isoflux plates. The idealized configurations are amenable to analytic optimization based on maximizing total heat transfer per unit volume or unit primary area. To achieve this anlaytic optimization, however, it is necessary to develop composite relations for the variation of the heat transfer coefficient along the plate surfaces. The mathematical development and verification of such composite relations as well as the formulation and solution of the optimizing equations for the various boundary conditions of interest constitute the core of this presentation.

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