One-dimensional steady-state equations

The reaction plane model with heterogeneous reactions was discussed at length for acid-base reactions in the previous section. The same modeling technique, of confining the reactions to planes, can be applied to micelle-facilitated dissolution. As with the acid-base model, one starts with a one-dimensional steady-state equation for mass transfer that includes diffusion, convection, and reaction. This equation is then applied to the individual species i, i.e., the solute, s, the micelle, m, and the drug-loaded micelle, sm, to yield... 

For the gaseous contaminants the release mechanism from the soil is generally based in the principles of diffusion across a porous medium. Basic diffusion equations are used for estimating the theoretical values of the gaseous flux from the waste material. The generic diffusion equation can be represented by a one-dimensional steady-state equation ... 

For one-dimensional steady-state flow, neglecting density changes, the above equation can be expressed as... 

Equations (5.95), (5.96) and (5.97) are suitable for constant critical melting porosity. In a one dimensional steady state melting column as a result of decompression melting, the porosity may increase from the bottom to the top of the column. If melting porosities change as a function of the spatial position, the related differential equations need to be solved numerically. More details of various melt transport models by porous flow have been given by Spiegelman and Elliott (1993), Iwamori, (1994), and Lundstrom (2000). 

Dunkle s Lecture at Picatinny Arsenal, 21 Nov 1955, p 3 (One-dimensional steady-state process and Rankine-Hugoniot equation) 78) Ibid, 13 Dec 1955, p 5 (Nature of shock waves) p 8 (Chapman-Jouguet point) pp 8-9 (Basic equations of deton) p 9... 

Reynolds number, p 46), etc 61-72 (Shock relationships and formulas) 73-98 (Shock wave interactions formulas) 99-102 (The Rayleigh and Fanno lines) Ibid (1958) 159-6l(Thermal theory of initiation) 168-69 (One-dimensional steady-state process) 169-72 (The Chapman-Jouguet condition) 172-76 (The von Neumann spike) 181-84 (Equations of state and covolume) 184-87 (Polytropic law) 188, 210 212 (Curved front theory of Eyring) 191-94 (The Rayleigh transformation in deton) 210-12 (Nozzle thepry of H. Jones) 285-88 (The deton head model) ... 

Let us consider a simplified flow, that is, a one-dimensional steady-state flow-without viscous stress or a gravitational force. The conservation equations of continuity, momentum, and energy are represented by rate of mass in - rate of mass out = 0... 

For a case that one stable steady state exists transient temperature profiles calculated agree satisfactorily with the measurements. For a case of three steady states the situation is quite complicated. The model used describes propagation of the fronts however, apparently cannot describe front multiplicity. A detailed calculation of the two-dimensional steady state equations including also the radial dispersion terms indicates that the onedimensional model is a very rough approximation for the diffusion" regime. We expect that dynamic calculations with the one-phase two-dimensional model could explain multiplicity of the fronts. 

Consider one-dimensional steady-state heat conduction in sphere that is, there is the temperature has only r dependence. The governing energy equation for a sphere with radius b, is... 

We assume that the temperature of the fin, T, is only a function of the coordinate x. In other words, the temperature is uniform at any cross section. The one-dimensional steady-state fin energy conservation equation gives... 

It is also straightforward to extend the equations to allow for only partial equihbration during transport. Iwamori (1993a) presents a one-dimensional steady-state single-porosity model for stable elements that includes diffusive re-equili-bration between melt and solid. He does not extend it to radioactive nuclides in this paper but includes this effect in his two porosity model (Iwamori, 1994) (see Section 3.14.4.3.4). The expected effects of chemical disequilibrium should be similar to those in the Qin (1992) dynamic melting model, namely he effective bulk partition coefficients of all elements will be driven towards unity. 

Obtaining analytical solutions to these differential equations requires a knowledge of the solution techniques of partial differential equations, which is beyond the scope of this introductory text. Here we limit our consideration to one-dimensional steady-state cases, since they result in ordinary differential equations... 

This is Euler s equation, ft contains, in the special case of one-dimensional steady-state flow, the relationship... 

Equation (6.1) can be readily integrated for several important cases of one-dimensional steady-state conduction, including the following ... 

The hot face temperatures were then calculated using the one-dimensional, steady-state conduction equation. Using the calculated heat flux to each ring, the hot face temperature, was calculated using ... 

Let us first determine the one-dimensional steady-state absorption flux of component A (i. e., at the gas-liquid interface, z = 0) without a chemical reaction. From the equation of continuity, assuming that the bulk-motion contribution to the flux is negligible for dilute solutions we obtain... 

For one-dimensional steady-state diffusion, including a first-order homogeneous reaction, neglecting the bulk-motion contribution, the equation of continuity becomes... 

C Bolin et al. (1974) have presented a one-dimensional steady-state model based on the atmospheric diffusion equation to be able to estimate the effect of dry deposition on vertical species concentration distribution. In the model the mean concentration is governed by... 

Small-scale experiments w ere conducted with ambient temperature CO2 [ ], and an experimental run-time coefficient was determined. This coefficient is based on the one-dimensional steady-state heat conduction equation with constant material properties ... 

The integration of the Euler equations for the special case of a one-dimensional steady-state flow in z -direction leads to the Bernoulli equation ... 

One of the early mechanistic models for a PEM fuel cell was the pioneering work of Bemardi and Verbrugge [45, 46]. They developed a one-dimensional, steady state, isothermal model which described water transport, reactant species transport, as well as ohmic and activation overpotentials. Their model assumed a fully hydrated membrane at all times, and thus calculated the water input and removal requirements to maintain full hydration of the membrane. The model was based on the Stefan Maxwell equations to describe gas phase diffusion in the electrode regions, the Nemst-Planck equation to describe dissolved species fluxes in the membrane and catalyst layers, the Butler Volmer equation to describe electrode rate kinetics and Schlogl s equation for liquid water transport. 

At t = 0 we have rp t = 0). Particle transport within each pore is described by a one-dimensional steady-state diffusion/reaction equation ... 

If the fluid is considered incompressible, the density terms (p) disappear in Eqs. 5.1 and 5.2. In one-dimensional steady state flow problems (velocity components in only one direction), the mass balance equation is automatically satisfied and does not enter into the calculations. In two-dimensional flow problems (no velocity component in the third coordinate direction), the mass balance is satisfied by the introduction of a stream function [1]. 

For the specific case of one-dimensional steady-state heat transfer through a cylindrical wall, the Laplace equation reduces to... 

FIGURE 6.2 One-dimensional steady-state solution of Pennes bioheat equation for constant parameter values. 

Fourier s equation for one dimensional steady state heat... 

One-dimensional steady state flames and detonations travel with constant velocity into the unbumed gas. If the unbumed gas is stationary, such a wave actually moves in space with a constant velocity. On the other hand, if the unbumed gas moves with the proper constant velocity, the wave may be maintained at a constant position in space. In either case, it is convenient to make use of a coordinate system fixed with respect to the wave and consider the stationary state solutions of the equations. This is possible because the hydrodynamic equations apply in any system of coordinates moving with constant velocity (Galilean invariance). 

The equation for the case of one-dimensional steady-state heat conduction. Figure 6.1, is obtained as follows ... 

When reactions are affected by both deactivation and diffusion, one-dimensional, steady state pellet conservation equations are ... 

This is the one-dimensional, steady state Schrodinger equation. 

When the unsteady-state equations are being integrated forward in time either towards the Eulerian steady state or towards the Lagrangian steady propagation rate, a criterion is required for deciding when the steady situation is achieved. In both cases this is provided by the space integral reaction rates qiA dy, which for sufficiently small finite difference intervals may be replaced by the summations x y)n or an equivalent quantity, where A dy) is the volume of the nth element and A a weighted mean area within the element. For the quasi-one-dimensional steady-state flame we have from Eq. (2.12)... 

These three components must satisfy the continuity equations for one-dimensional steady-state transport across the membrane ... 

In the simplest case of one-dimensional steady flow in the x direction, there is a parallel between Eourier s law for heat flowrate and Ohm s law for charge flowrate (i.e., electrical current). Eor three-dimensional steady-state, potential and temperature distributions are both governed by Laplace s equation. The right-hand terms in Poisson s equation are (.Qy/e) = (volumetric charge density/permittivity) and (Qp // ) = (volumetric heat generation rate/thermal conductivity). The respective units of these terms are (V m ) and (K m ). Representations of isopotential and isothermal surfaces are known respectively as potential or temperature fields. Lines of constant potential gradient ( electric field lines ) normal to isopotential surfaces are similar to lines of constant temperature gradient ( lines of flow ) normal to... 

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