Steady-state diffusion equation

When electrons are injected as minority carriers into a -type semiconductor they may diffuse, drift, or disappear. That is, their electrical behavior is determined by diffusion in concentration gradients, drift in electric fields (potential gradients), or disappearance through recombination with majority carrier holes. Thus, the transport behavior of minority carriers can be described by a continuity equation. To derive the p—n junction equation, steady-state is assumed, so that = 0, and a neutral region outside the depletion region is assumed, so that the electric field is zero. Under these circumstances,... 

Equation 9.1-17 is the continuity equation for unsteady-state diffusion of A through the ash layer it is unsteady-state because cA = cA(r, a To simplify its treatment further, we assume that the (changing) concentration gradient for A through the ash layer is established rapidly relative to movement of the reaction surface (of the core). This means that for an instantaneous snapshot, as depicted in Figure 9.3, we may treat the diffusion as steady-state diffusion for a fixed value of rc i.e., cA = cA(r). The partial differential emiatm. 

Ground-state absorption studies of the probe molecules adsorbed within zeolites were performed by UV-visible diffuse reflectance spectroscopy. Cationic form of the zeolite without probe molecule was used as a comparison sample. The remission function was calculated by using the Kubelka-Munk equation. Steady-state photoluminescence studies were carried out with the Hilger spectrofluorimeter. The spectra were recorded at room temperature and 77 K, respectived. 

Transient heat conduction or mass transfer in solids with constant physical properties (diffusion coefficient, thermal diffusivity, thermal conductivity, etc.) is usually represented by a linear parabolic partial differential equation. Steady state heat or mass transfer in solids, potential distribution in electrochemical cells is usually represented by elliptic partial differential equations. In this chapter, we describe how one can arrive at the analytical solutions for linear parabolic partial differential equations and elliptic partial differential equations in finite domains using a separation of variables method. The methodology is illustrated using a transient one dimensional heat conduction in a rectangle. 

Let us consider a planar film between the position coordinates r = Tq and r = r. Mass transfer between the two edges of the film occurs purely by molecular diffusion under steady-state conditions. The thickness of the film is — Tq. The equation of continuity... 

Note that if r > 6, which occurs at. short times, ihc l/ft term predominaies, and Equation 25-22 reduces to an equation analogous to F.quaiion 25-5. If r 6, which occurs at long times, the 1/r term predominates, the elect ron-transfer process reaches a steady state, and the steady-state current then depends only on the size of the electrode. This means that if the size of the electrode is small compared to the thickness of the Ncrnsi diffusion layer, steady state is achieved very rapidly, and a constant current is produced. Because the current is proportional to the area of the electrode, it also means that microclcctrodes produce liny currents. Expressions similar in form to Equation 25-22 may be formulated for other geometries, and they all have in common the characteristic that the smaller the electrode, the more rapidly steady-state current Is achieved. 

The Gaussian expressions are not expected to be valid descriptions of turbulent diffusion close to the surface because of spatial inhomogeneities in the mean wind and the turbulence. To deal with diffusion in layers near the surface, recourse is generally made to the atmospheric diffusion equation, in which, as we have noted, the key problem is proper specification of the spatial dependence of the mean velocity and eddy diffusivities. Under steady-state conditions, turbulent diffusion in the direction of the mean wind is usually neglected (the slender plume approximation), and if the wind direction coincides with the jc axis, then =0. Thus it is necessary to specify only the lateral, Ky, and vertical, A --, coefficients. It is generally assumed that horizontal homogeneity exists so that u and A are independent of y. Hence (18.52) becomes... 

Fig. 2. Influence of the radial diffusion on the current-time response under diffusion-limited conditions. (A) at the static spherical electrode of the radius ro = 0.5mm (B) at the disc microelectrode of the radius a = 5 m. Simulated currents in dimensionless scales ipi = 1/ (nFTrrocXD ) and I/(nF7racXD ), respectively 1, currents according to the Cottrell equation 2, currents corrected for the radial diffusion 3, steady state currents. (A) = 2000, (B)...

The transient and steady-state voltammetric responses of GNEs have been analyzed by simulation, theory, and experiment (1, 2). An approximate analytical expression for the diffusion-limited steady-state current at a GNE, t,i , is given by equation (6.3.11.3) ... 

Models Considering Pore Diffusion. Pseudo steady state transport equation for SOj in a spherical porous solid reactant (CaO) can be written as follows ... 

Illustration 2.9 Laplace s Equation, Steady-State Diffusion in Three-Dimensional Space Emissions from Embedded Sources... 

The equation for the diffusion limiting steady-state current to the orifice of a silanized pipette is... 

This expression is the sum of a transient tenu and a steady-state tenu, where r is the radius of the sphere. At short times after the application of the potential step, the transient tenu dominates over the steady-state tenu, and the electrode is analogous to a plane, as the depletion layer is thin compared with the disc radius, and the current varies widi time according to the Cottrell equation. At long times, the transient cunent will decrease to a negligible value, the depletion layer is comparable to the electrode radius, spherical difhision controls the transport of reactant, and the cunent density reaches a steady-state value. At times intenuediate to the limiting conditions of Cottrell behaviour or diffusion control, both transient and steady-state tenus need to be considered and thus the fiill expression must be used. Flowever, many experiments involving microelectrodes are designed such that one of the simpler cunent expressions is valid. 

Similarly to the response at hydrodynamic electrodes, linear and cyclic potential sweeps for simple electrode reactions will yield steady-state voltammograms with forward and reverse scans retracing one another, provided the scan rate is slow enough to maintain the steady state [28, 35, 36, 37 and 38]. The limiting current will be detemiined by the slowest step in the overall process, but if the kinetics are fast, then the current will be under diffusion control and hence obey the above equation for a disc. The slope of the wave in the absence of IR drop will, once again, depend on the degree of reversibility of the electrode process. 

If tire diffusion coefficient is independent of tire concentration, equation (C2.1.22) reduces to tire usual fonn of Pick s second law. Analytical solutions to diffusion equations for several types of boundary conditions have been derived [M]- In tlie particular situation of a steady state, tire flux is constant. Using Henry s law (c = kp) to relate tire concentration on both sides of tire membrane to tire partial pressure, tire constant flux can be written as... 

Despite the very restricted circumstances In which these equations properly describe the dynamical behavior, they are the starting point for almost all the extensive literature on the stability of steady states in catalyst pellets. It is therefore Interesting to examine the case of a binary mixture at the opposite limit, where bulk diffusion controls, to see what form the dynamical equations should take in a coarsely porous pellet. 

In section 11.4 Che steady state material balance equations were cast in dimensionless form, therary itancifying a set of independent dimensionless groups which determine ice steady state behavior of the pellet. The same procedure can be applied to the dynamical equations and we will illustrate it by considering the case t f the reaction A - nB at the limit of bulk diffusion control and high permeability, as described by equations (12.29)-(12.31). 

At any point within the boundary layer, the convective flux of the macromolecule solute to the membrane surface is given by the volume flux,/ of the solution multipfled by the concentration of retained solute, c. At steady state, this convective flux within the laminar boundary layer is balanced by the diffusive flux of retained solute in the opposite direction. This balance can be expressed by equation 1 ... 

The mathematical model most widely used for steady-state behavior of a reactor is diffusion theory, a simplification of transport theory which in turn is an adaptation of Boltzmann s kinetic theory of gases. By solving a differential equation, the flux distribution in space and time is found or the conditions on materials and geometry that give a steady-state system are determined. 

Table 10 contains some selected permeabiUty data including diffusion and solubiUty coefficients for flavors in polymers used in food packaging. Generally, vinyUdene chloride copolymers and glassy polymers such as polyamides and EVOH are good barriers to flavor and aroma permeation whereas the polyolefins are poor barriers. Comparison to Table 5 shows that the large molecule diffusion coefficients are 1000 or more times lower than the small molecule coefficients. The solubiUty coefficients are as much as one million times higher. Equation 7 shows how to estimate the time to reach steady-state permeation t if the diffusion coefficient and thickness of a film are known. 

Eor t7-limonene diffusion in a 50-pm thick vinyUdene chloride copolymer film, steady-state permeation is expected after 2000 days. Eor a 50- pm thick LDPE film, steady-state permeation is expected in less than one hour. If steady-state permeation is not achieved, the effective penetration depth E for simple diffusion, after time /has elapsed, can be estimated with equation 8. 

Carbon Dioxide Transport. Measuring the permeation of carbon dioxide occurs far less often than measuring the permeation of oxygen or water. A variety of methods ate used however, the simplest method uses the Permatran-C instmment (Modem Controls, Inc.). In this method, air is circulated past a test film in a loop that includes an infrared detector. Carbon dioxide is appHed to the other side of the film. AH the carbon dioxide that permeates through the film is captured in the loop. As the experiment progresses, the carbon dioxide concentration increases. First, there is a transient period before the steady-state rate is achieved. The steady-state rate is achieved when the concentration of carbon dioxide increases at a constant rate. This rate is used to calculate the permeabiUty. Figure 18 shows how the diffusion coefficient can be deterrnined in this type of experiment. The time lag is substituted into equation 21. The solubiUty coefficient can be calculated with equation 2. 

As a reactant molecule from the fluid phase surrounding the particle enters the pore stmcture, it can either react on the surface or continue diffusing toward the center of the particle. A quantitative model of the process is developed by writing a differential equation for the conservation of mass of the reactant diffusing into the particle. At steady state, the rate of diffusion of the reactant into a shell of infinitesimal thickness minus the rate of diffusion out of the shell is equal to the rate of consumption of the reactant in the shell by chemical reaction. Solving the equation leads to a result that shows how the rate of the catalytic reaction is influenced by the interplay of the transport, which is characterized by the effective diffusion coefficient of the reactant in the pores, and the reaction, which is characterized by the first-order reaction rate constant. 

The net transport of component A in the +2 direction in the centrifuge is equal to the sum of the convective transport and the axial diffusive transport. At the steady state the net transport of component A toward the product withdrawal point must be equal to the rate at which component A is being withdrawn from the top of the centrifuge. Thus, the transport of component is given by equation 72 ... 

Usually, diffusivity and kinematic viscosity are given properties of the feed. Geometiy in an experiment is fixed, thus d and averaged I are constant. Even if values vary somewhat, their presence in the equations as factors with fractional exponents dampens their numerical change. For a continuous steady-state experiment, and even for a batch experiment over a short time, a very useful equation comes from taking the logarithm of either Eq. (22-86) or (22-89) then the partial derivative ... 

The partial pressures in the rate equations are those in the vicinity of the catalyst surface. In the presence of diffusional resistance, in the steady state the rate of diffusion through the stagnant film equals the rate of chemical reaction. For the reaction A -1- B C -1-. . . , with rate of diffusion of A limited. 

When a clean steel coupon is placed in oxygenated water, a rust layer will form quickly. Corrosion rates are initially high and decrease rapidly while the rust layer is forming. Once the oxide forms, rusting slows and the accumulated oxide retards diffusion. Thus, Reaction 5.2 slows. Eventually, nearly steady-state corrosion is achieved (Fig. 5.2). Hence, a minimum exposure period, empirically determined by the following equation, must be satisfied to obtain consistent corrosion-rate data for coupons exposed in cooling water systems (Figs. 5.2 and 5.3) ... 

A situation which is frequently encountered in tire production of microelectronic devices is when vapour deposition must be made into a re-entrant cavity in an otherwise planar surface. Clearly, the gas velocity of the major transporting gas must be reduced in the gas phase entering the cavity, and transport down tire cavity will be mainly by diffusion. If the mainstream gas velocity is high, there exists the possibility of turbulent flow at tire mouth of tire cavity, but since this is rare in vapour deposition processes, the assumption that the gas widrin dre cavity is stagnant is a good approximation. The appropriate solution of dre diffusion equation for the steady-state transport of material tlrrough the stagnant layer in dre cavity is... 

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