Diffusion equation, thermal

A thermal diffusion equation is expressed, as in Pick s second law of diffusion, hy... 

Similar convection-diffusion equations to the Navier-Stokes equation can be formulated for enthalpy or species concentration. In all of these formulations there is always a superposition of diffusive and convective transport of a field quantity, supplemented by source terms describing creation or destruction of the transported quantity. There are two fundamental assumptions on which the Navier-Stokes and other convection-diffusion equations are based. The first and most fundamental is the continuum hypothesis it is assumed that the fluid can be described by a scalar or vector field, such as density or velocity. In fact, the field quantities have to be regarded as local averages over a large number of particles contained in a volume element embracing the point of interest. The second hypothesis relates to the local statistical distribution of the particles in phase space the standard convection-diffusion equations rely on the assumption of local thermal equilibrium. For gas flow, this means that a Maxwell-Boltzmann distribution is assumed for the velocity of the particles in the frame-of-reference co-moving with the fluid. Especially the second assumption may break dovm when gas flow at high temperature or low pressure in micro channels is considered, as will be discussed below. 

If Dh is indeed time dependent as in eq. (5) it is not obvious that C(x, t) will follow an error function expression as in eq. (3) or that >H will be thermally activated as in eq. (4). We now show that eqs. (3) and (4) still apply with a time dependent diffusion coefficient, by making a coordinate transformation (Kakalios and Jackson, 1988). The one-dimensional diffusion equation... 

The numerical jet model [9-11] is based on the numerical solution of the time-dependent, compressible flow conservation equations for total mass, energy, momentum, and chemical species number densities, with appropriate in-flow/outfiow open-boundary conditions and an ideal gas equation of state. In the reactive simulations, multispecies temperature-dependent diffusion and thermal conduction processes [11, 12] are calculated explicitly using central difference approximations and coupled to chemical kinetics and convection using timestep-splitting techniques [13]. Global models for hydrogen [14] and propane chemistry [15] have been used in the 3D, time-dependent reactive jet simulations. Extensive comparisons with laboratory experiments have been reported for non-reactive jets [9, 16] validation of the reactive/diffusive models is discussed in [14]. 

One example would be ice melting or methane hydrate dissociation when rising in seawater. Convective melting rate may be obtained by analogy to convective dissolution rate. Heat diffusivity k would play the role of mass diffusivity. The thermal Peclet number (defined as Pet = 2aw/K) would play the role of the compositional Peclet number. The Nusselt number (defined as Nu = 2u/5t, where 8t is the thermal boundary layer thickness) would play the role of Sherwood number. The thermal boundary layer (thickness 8t) would play the role of compositional boundary layer. The melting equation may be written as... 

For sufficiently large values of the Prandtl or Schmidt number the rate of transfer is controlled by the region of thickness S near the wall. The terms of the convective-thermal diffusivity equation... 

In this section it is shown how the multicomponent equations just discussed can be applied to the discussion of binary systems. First a summary is given of the various notations used in discussing two-component systems. Then some important special results are given for the diffusion and thermal diffusion in binary systems. These equations are used as the starting point for the discussions in Sec. IV. 

The basic expressions for the mass fluxes and the equations of continuity for multi-component mixtures are given in Sec. II,B. For a -component mixture of ideal gases in a system in which there is no pressure diffusion, forced diffusion, or thermal diffusion, the fluxes are given by... 

Chapman-Enskog theory provides the basis for the multicomponent transport properties laid out by Hirschfelder, Curtiss, and Bird [178] and by Dixon-Lewis [103]. The multi-component diffusion coefficients, thermal conductivities, and thermal diffusion coefficients are computed from the solution of a system of equations defined by the L matrix [103], seen below. It is convenient to refer to the L matrix in terms of its nine block submatrices, and in this form the system is given by... 

Therefore, any result that follows from considerations of the form of Fick s second law applies to evolution of heat as well as concentration. However, the thermal and mass diffusion equations differ physically. The mass diffusion equation, dc/dt = V DVc, is a partial-differential equation for the density of an extensive quantity, and in the thermal case, dT/dt = V kVT is a partial-differential equation for an intensive quantity. The difference arises because for mass diffusion, the driving force is converted from a gradient in a potential V/u to a gradient in concentration Vc, which is easier to measure. For thermal diffusion, the time-dependent temperature arises because the enthalpy density is inferred from a temperature measurement. 

Many texts, such as Crank s treatise on diffusion [2], contain solutions in terms of simple functions for a variety of conditions—indeed, the number of worked problems is enormous. As demonstrated in Section 4.1, the differential equation for the diffusion of heat by thermal conduction has the same form as the mass diffusion equation, with the concentration replaced by the temperature and the mass diffusivity replaced by the thermal diffusivity, k. Solutions to many heat-flow... 

Solution. Using Eq. 4.61 and the analogy between mass diffusion and thermal diffusion, the basic differential equation for the temperature distribution in graphite can be written... 

As seen in the above, many approaches are now appearing aiming at providing a new driving source like muscles. Responses of thermosensitive gels are mainly governed by thermal diffusivity, and the size of gel as seen from a thermal diffusion equation. In general, the thermal diffusivity of a solvent such as water... 

In the absence of catalysis on the surface, similarity of the concentration and temperature fields is achieved precisely at the ignition limit if the coefficients of diffusion and thermal diffusivity are equal, since in this case both the diffusion gradient and the temperature gradient at the igniting surface are equal to zero, and the equations of diffusion and thermal conductivity with the chemical reaction may be reduced to the form of an identity (see our work on flame propagation [3]). 

This similarity was established in [2] by consideration of the second-order differential equations of diffusion and heat conduction. Under the assumptions made about the coefficient of diffusion and thermal diffusivity, similarity of the fields, and therefore constant enthalpy, in the case of gas combustion occur throughout the space this is the case not only in the steady problem, but in any non-steady problem as well. It is only necessary that there not be any heat loss by radiation or heat transfer to the vessel walls and that there be no additional (other than the chemical reaction) sources of energy. These conditions relate to the combustion of powders and EM as well, and were tacitly accounted for by us when we wrote the equations where the corresponding terms were absent. 

The balance over the ith species (equation IV. 5) consists of contributions from diffusion, convection, and loss or production of the species in ng gas-phase reactions. The diffusion flux combines ordinary (concentration) and thermal diffusions according to the multicomponent diffusion equation (IV. 6) for an isobaric, ideal gas. Variations in the pressure induced by fluid mechanical forces are negligible in most CVD reactors therefore, pressure diffusion effects need not be considered. Forced diffusion of ions in an electrical field is important in plasma-enhanced CVD, as discussed by Hess and Graves (Chapter 8). 

Concentration grating Due to the Ludwig-Soret effect, the temperature grating is the driving force for a secondary concentration grating, which starts to build up and is superimposed upon the thermal one. Its temporal and spatial evolution is obtained from the one-dimensional form of the extended diffusion equation... 

Traditionally, experimental values of Zeff have been derived from measurements of the lifetime spectra of positrons that are diffusing, and eventually annihilating, in a gas. The lifetime of each positron is measured separately, and these individual pieces of data are accumulated to form the lifetime spectrum. (The positron-trap technique, to be described in subsection 6.2.2, uses a different approach.) An alternative but equivalent procedure, which is adopted in electron diffusion studies and also in the theoretical treatment of positron diffusion, is to consider the injection of a swarm of positrons into the gas at a given time and then to investigate the time dependence of the speed distribution, as the positrons thermalize and annihilate, by solving the appropriate diffusion equation. The experimentally measured Zeg, termed Z ), is the average over the speed distribution of the positrons, y(v,t), where y(v,t) dv is the number density of positrons with speeds in the interval v to v + dv at time t after the swarm is injected into the gas. The time-dependent speed-averaged Zef[ is therefore... 

A more detailed picture of the 3D temperature and concentration distribution can be obtained by an appropriate numerical model. Besides the diffusion equations for heat and mass, convection caused by both thermal and solutal expansion needs to be taken into account. 

The high thermal conductivity of the sapphire windows ensures a fixed temperature T = To at the boundaries z = Ls/2 and x = LX/2. The boundary condition for the diffusion equation is a vanishing flux at the walls (normal vector en) ... 

Furthermore, asB in the previously developed theory (7), because the thermal diffusivity is so much greater than the mass diffusivities Equation (2) simplifies to... 

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