One-dimensional solutions

In Eq. (14) we took care of the fact that the differentiation in the diffusion Eq. (3) was with respect to the primed co-ordinates. Because X is an arbitrary constant, the general one-dimensional solution of the diffusion equation is ... 

EXAMPLE 2.2 Unsteady dissolution of a highly soluble pollutant (herbicides, pesticides, ammonia, alcohols, etc.) into groundwater (unsteady, one-dimensional solution with pulse boundary conditions)... 

Example 2.2.) Use the product rule, which is often applied to expand a PDE solution to three dimensions. Assume that Ci, C2, and C3 are one-dimensional solutions to the governing equation with similar boundary conditions. Then, the product rule gives a potential three-dimensional solution of... 

We will now turn our attention from one-dimensional solutions of the diffusion equation to two-dimensional solutions. 

Another one-dimensional solution that we shall employ is that for (2.2.7a) in a segment [0, A], A > 0 with boundary conditions ... 

Equation 4.40 gives the solution for one-dimensional diffusion from a point source on an infinite line, an infinite thin line source on an infinite plane, and a thin planar source in an infinite three-dimensional body (summarized in Table 5.1). Corresponding solutions for two- and three-dimensional diffusion can easily be obtained by using products of the one-dimensional solution. For example, a solution for three-dimensional diffusion from a point source is obtained in the form... 

Solution. The amount of heat added (per unit area) at x = 0 in time dt is Pdt. Using the analogy between problems of mass diffusion and heat flow (Section 4.1), each added amount of heat, P dt, spreads according to the one-dimensional solution for mass diffusion from a planar source in Table 5.1 ... 

The theoretical foundation for this kind of analysis was, as mentioned, originally laid by Taylor and Aris with their dispersion theory in circular tubes. Recent contributions in this area have transferred their approach to micro-reaction technology. Gobby et al. [94] studied, in 1999, a reaction in a catalytic wall micro-reactor, applying the eigenvalue method for a vertically averaged one-dimensional solution under isothermal and non-isothermal conditions. Dispersion in etched microchannels has been examined [95], and a comparison of electro-osmotic flow to pressure-driven flow in micro-channels given by Locascio et al. in 2001 [96]. 

Under more realistic reaction parameters, only a spatially one-dimensional solution could be obtained. Here it was assumed that the gas flow over the whole cross sectional area was reduced uniformly to 1/4 at time zero. The resulting dynamics are given in Figure 8. In this case the transient temperature is so high that the fluid concentration is completely consumed and the reaction zone moves like a front through the whole reactor. Finally a flat steady state temperature profile is established again. 

Langston, L. S. Heat Transfer from Multidimensional Objects Using One-Dimensional Solutions for Heat Loss, Int. J. Heat Mass Transfer, vol. 25, p. 149, 1982. 

Two- and Three-Dimensional Conduction The one-dimensional solutions discussed above can be used to construct solutions to multidimensional problems. The unsteady temperature of a rectangular, solid box of height, length, and width 2H, 2L, and 2 W, respectively, with governing equations in each direction as in (5-18), is... 

Mullidimensional solutions expressed as products of one-dimensional solutions for bodies that are initially at a uniform temperature F, and exposed to convection from all surfaces to a medium at... 

Note that the solution of a rwo-dimensionol problem involves the product of hvo one-dimensional solutions, whereas the solution of a three-dimensional problem involves the product of three one-dirnensional solutions,... 

Planar one-dimensional, and cylindrical two dimensional, time-dependent, solutions of equations (7.19) to (7.22) have been presented by Konig et al. [168], for various initial linear temperature gradients away from the hot spot. Their one-dimensional solutions, for po = 4 MPa, EIRTq = 20, and a hot-spot temperature, Tq, of 1000 K, with other property values the... 

Finally, we plot the wave function for the second excited state 2i(x, y) (see Fig. 4.27e). It has two maxima (positive) and two minima (negative) located at the values 0.25 and 0.75 for x and y. There are two nodal lines, along x = 0.5 and y = 0.5. They divide the x-y plane into quadrants, each of which contains a single maximum (positive) or minimum (negative) value. Make sure that you see how these characteristics trace back to the one-dimensional solutions in Figure 4.24. Figure 4.27f shows the contour plots for y). As the magnitude (absolute... 

Figure 3-19 presents equations for the transport of a conservative tracer from pulse and continuous sources in one, two, and three dimensions. Note that the one-dimensional solution for a continuous input starting at t = 0 uses a slightly different boundary condition at x = 0 than is used in the statement of Eq. [3-18] instead of letting C = C0 for all x < 0 at t = 0, a mass input at x = 0 is turned on at t = 0. The results are not distinguishable except for small times and locations very close to x = 0. (Note that unlike Eq. [3-18], porosity must be taken into account in Fig. 3-19 because the expression uses mass divided by the column area rather than concentration in the pore water.) Also note, again, that first order decay can be accomodated in these equations by multiplying by e kt. 

The general mass balance equation for one-dimensional solute flow can be written as follows ... 

R — 0.0025 m. You must expand the differential equation because the one-dimensional option does not contain cylindrical geometry. (This hmitation is removed in two- and three-dimensional problems, but here a one-dimensional solution suffices.) So, rewrite Eq. (9.6) as... 

To this point, several assumptions have been built into the theoretical development. The soil medium is being characterized as a homogeneous volume allowing the one-dimensional solute transport equation... 

Diffusion. Acid diffusion in pH modified foods was investigated with fluorescent dyes or color indicators (18). Colorimetric pH profiles compared well with electromagnetic measurements. Effective mass diffusivities for acetic, citric, gluconic and phosphoric acids in potatoes were on the order of lO10 nr/sec. Diffusion occurs mainly in an unsteady state in heterogeneous, multilayer, cellular systems (19). Diffusion can be seen mathematically predicted by one-dimensional solutions of the second Fick equation ... 

Laboratory column experiments were used to identify potential rate-controlling mechanisms that could affect transport of molybdate in a natural-gradient tracer test conducted at Cape Cod, Mass. Column-breakthrough curves for molybdate were simulated by using a one-dimensional solute-transport model modified to include four different rate mechanisms equilibrium sorption, rate-controlled sorption, and two side-pore diffusion models. The equilibrium sorption model failed to simulate the experimental data, which indicated the presence of a ratecontrolling mechanism. The rate-controlled sorption model simulated results from one column reasonably well, but could not be applied to five other columns that had different input concentrations of molybdate without changing the reaction-rate constant. One side-pore diffusion model was based on an average side-pore concentration of molybdate (mixed side-pore diffusion) the other on a concentration profile for the overall side-pore depth (profile side-pore diffusion). 

Many one-dimensional solute-transport models have been developed and used to analyze column data. For a recent review, see Grove and Stollenwerk (13). Four different models were used in the study discussed in this article to simulate the shape of the column-breakthrough curves. All four models contain a one-dimensional solute-transport equation and use the Freundlich equation to describe sorption. They differ in the rate mechanism that is assumed to control transport of Mo(VI) from flowing phase to solid surface. The essential features of each model are summarized in Table III. 

The governing equation for one-dimensional solute transport in a porous media with both mobile and immobile regions and reversible sorption in both regions is ... 

The three one-dimensional solutions presented above give important short-time results that appear in other solutions such as the external transient three-dimensional conduction from isothermal bodies of arbitrary shape into large regions. These solutions are presented in the next section. 

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