Diffusion equation localized sources

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. 

According to measurements made in the atmosphere, the Lagrangian time scale is of the order of 100 sec (Csanady, 1973). Using a characteristic particle velocity of 5 m sec", the above conditions are 100 sec and L > 500 m. Since one primary concern is to examine diffusion from point sources such as industrial stacks, which are generally characterized by small T and L, it is apparent that either one (but particularly the second one) or both of the above constraints cannot be satisfied, at least locally, in the vicinity of the point-like source. Therefore, in these situations, it is important to assess the error incurred by the use of the atmospheric diffusion equation. 

Application 2 Drug Penetration in Tissue. The diffusion equation can be used to develop a simple, quantitative method for predicting the extent of drug penetration into a tissue following the introduction of a local source. Consider the simple geometry shown in Figure 3.4d, where drug is maintained at a constant value, cq, at the interface of a semi-infinite medium. From the steady-state solution, Equation 3-58, it is possible to... 

A numte of theoretical models have been derived specifically for local chain motions [82]. These models have been usehil in stimulating thinking about the underlying mechanisms for local dynamics, and about how these mechanisms influence the shape of the correlation function. For example, it is now understood that non-exponential correlation functions can result from a variety of sources, including specific cooperative motions. Any model which resembles a one-dimensional diffusion equation will have a non-exponential decay [81]. 

In K-Ar or zircon U-Pb dating, modeling the loss of radiogenic isotopes by volume diffusion is important. If P0 is the local concentration at t = 0 of a radioactive element decaying with constant X, a source term exists in the transport equation of the radiogenic element which is the local rate of accumulation AP0e Xt. For dual decay,... 

The necessary conditions are sources of iron oxide, dissolved 804 and organic matter, and sufficiently reducing conditions for reduction of 804 coupled to intermittent or localized oxidizing conditions to produce elemental 8 or polysulfide. Potential acidity develops by the removal of alkalinity (represented by HC03 in Equation 7.2) from the sediment by diffusion and tidal action. What... 

As an illustration, consider the isothermal, isobaric diffusional mixing of two elemental crystals, A and B, by a vacancy mechanism. Initially, A and B possess different vacancy concentrations Cy(A) and Cy(B). During interdiffusion, these concentrations have to change locally towards the new equilibrium values Cy(A,B), which depend on the local (A, B) composition. Vacancy relaxation will be slow if the external surfaces of the crystal, which act as the only sinks and sources, are far away. This is true for large samples. Although linear transport theory may apply for all structure elements, the (local) vacancy equilibrium is not fully established during the interdiffusion process. Consequently, the (local) transport coefficients (DA,DB), which are proportional to the vacancy concentration, are no longer functions of state (Le., dependent on composition only) but explicitly dependent on the diffusion time and the space coordinate. Non-linear transport equations are the result. 

The mathematical model comprises a set of partial differential equations of convective diffusion and heat conduction as well as the Navier-Stokes equations written for each phase separately. For the description of reactive separation processes (e.g. reactive absorption, reactive distillation), the reaction terms are introduced either as source terms in the convective diffusion and heat conduction equations or in the boundary condition at the channel wall, depending on whether the reaction is homogeneous or heterogeneous. The solution yields local concentration and temperature fields, which are used for calculation of the concentration and temperature profiles along the column. 

The kinetic theory leads to the definitions of the temperature, pressure, internal energy, heat flow density, diffusion flows, entropy flow, and entropy source in terms of definite integrals of the distribution function with respect to the molecular velocities. The classical phenomenological expressions for the entropy flow and entropy source (the product of flows and forces) follow from the approximate solution of the Boltzmann kinetic equation. This corresponds to the linear nonequilibrium thermodynamics approach of irreversible processes, and to Onsager s symmetry relations with the assumption of local equilibrium. 

From these time-scales, it may be assumed in most circumstances that the free electrons have a Maxwellian distribution and that the dominant populations of impurities in the plasma are those of the ground and metastable states of the various ions. The dominant populations evolve on time-scales of the order of plasma diffusion time-scales and so should be modeled dynamically, that is in the particle number continuity equations, along with the momentum and energy equations of plasma transport theory. The excited populations of impurities on the other hand may be assumed relaxed with respect to the instantaneous dominant populations, that is they are in a quasi-equilibrium. The quasi-equilibrium is determined by local conditions of electron temperature and electron density. So, the atomic modeling may be partially de-coupled from the impurity transport problem into local calculations which provide quasi-equilibrium excited ion populations and effective emission coefficients (PEC coefficients) and then effective source coefficients (GCR coefficients) for dominant populations which must be entered into the transport equations. The solution of the transport equations establishes the spatial and temporal behaviour of the dominant populations which may then be re-associated with the local emissivity calculations, for matching to and analysis of observations. 

Finally, the last two terms on the r.h.s. are needed If the potential and the pressure gradients In the normal direction are considered, when no fluid flow occurs In that direction. This is often the case because the surface cannot act as a source of liquid (o = 0 at x = 0). Formation of a diffuse layer leads to local excesses of ions exerting a certain osmotic pressure, which Is just equal and opposite to Vp, because otherwise the liquid would start to flow. Equation 14.6.6) then reduces to... 

The influence of hydrogen on fracture depends on hydrogen concentration, C, in the sites where localized material damage might occur. The accumulation of hydrogen in these zones proceeds by diffusion from external or internal sources, i.e., local fracture event takes place when and where hydrogen concentration reaches some critical value, Cor, which is conditioned by the stress-strain state in material [1], This is expressed by the following equation... 

A generic local instantaneous equation for a general conserved quantity ip, where J represents the diffusive fluxes and cp the sources of ip, can be expressed as ... 

The conservation equations do not by themselves constitute a closed set of relaxation equations. In order to close Eq. (10.3.8) we must specify a constitutive relation relating the flux Jfi(r, t) to the density A(r, t). As an example we consider the simple example of diffusion in a binary mixture, in which there are no chemical reactions. Then Eq. (10.3.8) applies with no source term. According to Fick s second law the local average current of the solute is... 

The following discussion represents a detailed description of the mass balance for any species in a reactive mixture. In general, there are four mass transfer rate processes that must be considered accumulation, convection, diffusion, and sources or sinks due to chemical reactions. The units of each term in the integral form of the mass transfer equation are moles of component i per time. In differential form, the units of each term are moles of component i per volnme per time. This is achieved when the mass balance is divided by the finite control volume, which shrinks to a point within the region of interest in the limit when aU dimensions of the control volume become infinitesimally small. In this development, the size of the control volume V (t) is time dependent because, at each point on the surface of this volume element, the control volnme moves with velocity surface, which could be different from the local fluid velocity of component i, V,. Since there are several choices for this control volume within the region of interest, it is appropriate to consider an arbitrary volume element with the characteristics described above. For specific problems, it is advantageous to use a control volume that matches the symmetry of the macroscopic boundaries. This is illustrated in subsequent chapters for catalysts with rectangular, cylindrical, and spherical symmetry. 

It is not obvious, but other studies (Ryan etal., 1981) have shown that the reaction source in equations 1.124 and 1.125 makes a negligible contribution to c y. In addition, one can demonstrate (Whitaker, 1999) that the heterogeneous reaction, k CAy, can be neglected for all practical problems of diffusion and reaction in porous catalysts. Furthermore, the non-local diffusion term is negligible for traditional systems, and under these circumstances the boundary value problem for the spatial deviation concentration takes the form... 

Vc) = 0 (D diffusion coefficient, c H-atoms per unit volume ), is limited to some rare cases (e.g. to steels with saturated traps resulting from low density of defects or high H-content). In fact, a source term must be added to the right side of the equation, which was firstly developed by McNabb and Foster, (McNabb Foster, 1963), and later by Oriani assuming local equilibrium, (Oriani, 1970). Thus, the following equations hold for diffusion of hydrogen under the presence of local stress fields (c << 1), (Serebrinsky et al., 2004) ... 

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