Gradient profiles mathematical model

Reaction rates of nonconservative chemicals in marine sediments can be estimated from porewater concentration profiles using a mathematical model similar to the onedimensional advection-diffusion model for the water column presented in Section 4.3.4. As with the water column, horizontal concentration gradients are assumed to be negligible as compared to the vertical gradients. In contrast to the water column, solute transport in the pore waters is controlled by molecular diffusion and advection, with the effects of turbulent mixing being negligible. 

Preliminary residence time distribution studies should be conducted on the reactor to test this assumption. Although in many cases it may be desirable to increase the radial aspect ratio (possibly by crushing the catalyst), this may be difficult with highly exothermic solid-catalyzed reactions that can lead to excessive temperature excursions near the center of the bed. Carberry (1976) recommends reducing the radial aspect ratio to minimize these temperature gradients. If the velocity profile in the reactor is significantly nonuniform, the mathematical model developed here allows predictive equations such as those by Fahien and Stankovic (1979) to be easily incorporated. 

Fig. 6. (a) Interstitial pressure gradients in the mammary adenocarcinoma R3230AC as a function of radial position. The circles ( ) represent data points (Boucher et al., 1990), and the solid line represents the theoretical profile based on our previously developed mathematical model (Jain and Baxter, 1988 Baxter and Jain, 1989). Note that the pressure is nearly uniform in most of the tumor, but drops precipitously to normal tissue values in the periphery. Elevated pressure in the central region retards the extravasation of fluid and macromolecules. In addition, the pressure drop from the center to the periphery leads to an experimentally verifiable, radially outward fluid flow. (Reproduced from Boucher et al., 1990, with permission.) (b) Microvascular pressure (MVP) in the peripheral vessels of the mammary adenocarcinoma R3230AC is comparable to the central interstitial fluid pressure (IFP) (adapted from Boucher and Jain, 1992). These results suggest that osmotic pressure difference across vessel walls is small in this tumor. 

Milton et al. (1997) proposed the manometric temperature measurement (MTM) the transient pressure response is mathematically modeled under the assumption that four mechanisms contribute to the pressure rise, namely the direct sublimation of ice through the dried product layer at a constant temperature, the increase in the ice temperature due to continuous heating of the frozen matrix during the measurement, the increase in the temperature at the sublimation interface when a stationary temperature profile is obtained in the frozen layer and, finally, the leaks in the chamber. The four contributions are considered purely additive the values of the thickness and of the thermal gradient are needed but they are not known exactly. The values of the vapor pressure over ice, of the product resistance and the heat transfer coefficient at the vial bottom are determined with regression analysis. 

In practice, the fluid velocity profile is rarely flat, and spatial gradients of concentration and temperature do exist, especially in large-diameter reactors. Hence, the plug-flow reactor model (Fig. 7.1) does not describe exactly the conditions in industrial reactors. However, it provides a convenient mathematical means to estimate the performance of some reactors. As will be discussed below, it also provides a measure of the most efficient flow reactor—one where no mixing takes place in the reactor. The plug-flow model adequately describes the reactor operation when one of the following two conditions is satisfied ... 

By and large we can describe the results of the analysis of distributed parameter systems (i.e., flow reactors other than CSTRs) in terms of the gradients or profiles of concentration and temperature they generate. To a large extent, the analysis we shall pursue for the rest of this chapter is based on the one-dimensional axial dispersion model as used to describe both concentration and temperature fields within the nonideal reactor. The mass and energy conservation equations are coupled to each other through their mutual concern about the rate of reaction and, in fact, we can use this to simplify the mathematical formulation somewhat. Consider the adiabatic axial dispersion model in the steady state. 

The applicable mathematical form of the flux equation is model dependent. For example, the first type model consists of differential equations (DEs). They are developed to yield concentration profiles in the sediment layers as well as the flux. These DEs typically use Equation 4.1 as a boundary condition. The solutions to these DEs require one or more of the following boundary condition categories the Dirichlet condition, the Neuman condition, or a third condition. The first two types are the most common these require mathematical functions containing gradients of the dependent variable (i.e., Cw) as well as functions of the dependent variable itself. For these diffusive-type fluxes, the transport parameter is a diffusion coefficient such as Dg. Several other transport parameters are commonly used and represent diffusion in air and the biodiffusion or bioturbation of soil/sediment particles. 

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