Collision heat conduction

Conduction takes place at a solid, liquid, or vapor boundary through the collisions of molecules, without mass transfer taking place. The process of heat conduction is analogous to that of electrical conduction, and similar concepts and calculation methods apply. The thermal conductivity of matter is a physical property and is its ability to conduct heat. Thermal conduction is a function of both the temperature and the properties of the material. The system is often considered as being homogeneous, and the thermal conductivity is considered constant. Thermal conductivity, A, W m, is defined using Fourier s law. 

As the pressure increases from low values, the pressure-dependent term in the denominator of Eq. (101) becomes significant, and the heat transfer is reduced from what is predicted from the free molecular flow heat transfer equation. Physically, this reduction in heat flow is a result of gas-gas collisions interfering with direct energy transfer between the gas molecules and the surfaces. If we use the heat conductivity parameters for water vapor and assume that the energy accommodation coefficient is unity, (aA0/X)dP — 150 I d cm- Thus, at a typical pressure for freeze drying of 0.1 torr, this term is unity at d 0.7 mm. Thus, gas-gas collisions reduce free molecular flow heat transfer by at least a factor of 2 for surfaces separated by less than 1 mm. Most heat transfer processes in freeze drying involve separation distances of at least a few tenths of a millimeter, so transition flow heat transfer is the most important mode of heat transfer through the gas. 

Things are quite different in a detonation wave. In this case we have a completely determined characteristic time—the duration of the chemical reaction in combination with a particular linear velocity of detonation propagation we obtain the zone width of the chemical reaction, which can no longer (as in the case of a shock wave) vary with changes in the heat conduction. The chemical reaction cannot occur in the time of a single collision many collisions of the molecules with one another will be required, and the zone width will be extended to a length many times the mean free path. 

In a collision between two spheres of different temperatures, heat conduction occurs at the interface. The contact area is usually negligibly small compared to the cross-sectional area of the spheres. Since the duration of the impact is also very short, the temperature change of the colliding particles is confined to a small region around the contact area. Therefore, the heat conduction between the two particles can be treated as that between two semiinfinite media. It is also assumed that there is no thermal resistance between the contact surfaces. Hence, the temperature and heat flux distributions are continuous across the contact area. The surfaces outside the contact area are assumed to be flat and insulated. For general information on collision mechanisms of solids, readers may refer to Chapter 2. 

For simplicity, it is assumed that the impact is a Hertzian collision. Thus, no kinetic energy loss occurs during the impact. The problem of conductive heat transfer due to the elastic collision of solid spheres was defined and solved by Sun and Chen (1988). In this problem, considering the heat conduction through the contact surface as shown in Fig. 4.1, the change of the contact area or radius of the circular area of contact with respect to time is given by Eq. (2.139) or by Fig. 2.16. In cylindrical coordinates, the heat conduction between the colliding solids can be written by... 

The boundary conditions of particle temperature may be obtained from the heat transfer due to the collision of two bodies. Sun and Chen (1988) formulated the heat transfer per impact of two elastic particles by considering the collision of two elastic particles with different temperatures and assuming the heat conduction occurs only in the normal direction. 

The kinetic equations serve as a bridge between the microscopic domain and the behavior of macroscopic irreversible processes through the description of hydrodynamics in terms of intermolecular collisions. Hydrodynamics can specify a large number of nonequilibrium states by a small number of reproducible properties such as the mass, density, velocity, and energy density of a fluid conserved during the collision of molecules. Therefore, the hydrodynamic equations can describe a wide range of relaxation processes of nonequilibrium states to equilibrium state. We call such processes decay processes represented by phenomenological equations, such as Fourier s law of heat conduction. The decay rates are determined by the transport coefficients. Reliable transport coefficients provide microscopic and macroscopic information, and validate the results of molecular dynamics. 

In the following section molecular collisions are discussed briefly in order to define the notation appearing in the exact expressions for the transport coefficients. Diffusion is treated separately from the other transport properties in Section E.2 because it has been found [7] that closer agreement with the exact theory is obtained by utilizing a different viewpoint in this case. Next, a general mean-free-path description of molecular transport is presented, which is specialized to the cases of viscosity and heat conduction in Sections E.4 and E.5. Finally, dimensionless ratios of transport coefficients, often appearing in combustion problems, are defined and discussed. The notation throughout this appendix is the same as that in Appendix D. 

To understand heat conduction, diffusion, viscosity and chemical kinetics the mechanistic view of molecule motion is of fundamental importance. The fundamental quantity is the mean-free path, i. e. the distance of a molecule between two collisions with any other molecule. The number of collisions between a molecule and a wall was shown in Chapter 4.1.1.2 to be z = CNQvdtl6. Similarly, we can calculate the number of collisions between molecules from a geometric view. We denote that all molecules have the mean speed v and their mean relative speed with respect to the colliding molecule is g. When two molecules collide, the distance between their centers is d in the case of identical molecules, d corresponds to the effective diameter of the molecule. Hence, this molecule will collide in the time dt with any molecule centre that lies in a cylinder of a diameter 2d with the area Jid and length gdt (it follows that the volume is Jtd gdt). The area where d is the molecule (particle) diameter is also called collisional cross section a. This is a measure of the area (centered on the centre of the mass of one of the particles) through which the particles cannot pass each other without colliding. Hence, the number of collisions is z = c n gdt. A more correct derivation, taking into account the motion of all other molecules with a Maxwell distribution (see below), leads to the same expression for z but with a factor of V2. We have to consider the relative speed, which is the vector difference between the velocities of two objects A and B (here for A relative to B) ... 

Equations (80a)-(80c) are called conservation equations, since their form is a direct consequence of the conservation of number of particles, momentum, and energy in the binary collisions taking place in the gas. In the phenomenological theories of fluid dynamics, equations in the form of Eqs. (80a)-(80c) are derived from the fact that mass, momentum and energy are conserved in the fluid, but in these theories one does not express the heat flow vector Jj- and the pressure tensor P in terms of a microscopic quantity, like the distribution function f(r, v, t). Instead, one relates Jr and P to n, u, and T by means of the so-called linear laws. ° For a one-component fluid with no internal structure, these linear laws are Fourier s law of heat conduction... 

Heat conduction is a type of transfer of heat in solids and liquids, interpreted as the imparting of kinetic energy resulting from collisions between disorderly moving molecules. The process occurs without any macroscopic motions in the body. The conductivity of diamond without traces of isotope C is the highest. The conductivity of a metals is also high. The lowest conductivity is that of a gas. 

The temperature instability of a two-dimensional reactive fluid of N hard disks bounded by heat conducting walls has been studied by molecular dynamics simulation. The collision of two hard disks is either elastic or inelastic (exothermic reaction), depending on whether the relative kinetic energy at impact exceeds a prescribed activation barrier. Heat removal is accomplished by using a wall boundary condition involving diffuse and specular reflection of the incident particles. Critical conditions for ignition have been obtained and the observations compared with continuum theory results. Other quantities which can be studied include temperature profiles, ignition times, and the effects of local fluctuations. 

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