Dimensional heterogeneity

Fischer CH, Quarini GL. Three-dimensional heterogeneous modeling of distillation tray hydraulics. Proceedings of AIChE Annual Meeting, Miami Beach, 1998. 

An extension of this one-dimensional heterogeneous model is to consider intraparticle diffusion and temperature gradients, for which the lumped equations for the solid are replaced by second-order diffu-sion/conduction differential equations. Effectiveness factors can be used as applicable and discussed in previous parts of this section and in Sec. 7 of this Handbook (see also Froment and Bischoff, Chemical Reactor Analysis and Design, Wiley, 1990). 

Dynamic Model A two-dimensional heterogeneous dynamic model was developed, which describes the mass and energy balances in both phases. In dimensionless form, for the nfc component and the temperature in the gas phase,... 

The reformer tube operation was simulated on the basis of a set of continuity-, energy- and momentum equations using one and two dimensional heterogeneous models. Intraparticle gradients in the rings were accounted for by the use of the generalized modulus concept. 

The one dimensional heterogeneous model equations are as follows Continuity equations ... 

Liu J, Wilson JL (1995) Modeling travel time and source location probabilities in two-dimensional heterogeneous aquifer. Proc 5th WERC Technology Development Conference, Las Cruces, New Mexico, pp 59-76... 

The transport of a sorbing contaminant in two-dimensional, heterogeneous, saturated porous media, resulting from the dissolution of a single component NAPL pool, as illustrated in Fig. 5, is described by the following partial differential equation [44] ... 

Burr DT, Sudicky EA, Naff RL (1984) Nonreactive and reactive solute transport in three-dimensional heterogeneous porous media mean displacement, plume spreading, and uncertainty. Water Resour Res 30 791-815... 

Takebe et al. [28] studied the effect of temperature and molecular weight on the crystallization rate of SPS by DSC. When SPS was melted at 300 °C, then rapidly cooled to the crystallization temperature, T, the evolution of crystallization showed a sigmoidal curve with reference to crystallization time (Figure 18.6). The crystallization rate becomes slower as T approaches close to melting point. When the crystallization rate of SPS is analyzed based on Avrami theory, the Avrami index, n, is equal to 3, which suggests that crystallization of SPS proceeds via three-dimensional heterogeneous growth [28,29]. 

W. D. Rannie, Perturbation Analysis of One-Dimensional Heterogeneous Flow in Rocket Nozzles, in Detonation and Two-Phase Flow, vol. 6 of Progress in Astronautics and Rocketry, S. S. Penner and F. A. Williams, eds.. New York Academic Press, 1962, 117-144. 

Equation 7.119 seems to be consistent with the flow in real life. The explanation may be extended to the difference between Eq. 7.103 and Eq. 7.118 at an arbitrary flow angle. Equation 7.118 appears to model the two-dimensional flow in a homogeneous and isotropic porous media. The trapping numbers for two-dimensional and three-dimensional heterogeneous, anisotropic porous media were also derived in Jin (1995). 

A two-dimensional heterogeneous model is developed for the catalyst tubes, without catalyst deactivation, taking into consideration the steam reforming... 

Ri, Rj, R3 are ratesof reaction for reactions(1), (2), (3) respectively. The two-dimensional heterogeneous model equations takes the form,... 

Johnson, D.N., Pedit, J.A. and Miller, C.T. (2004) Efficient, near-complete removal of DNAPL from three-dimensional, heterogeneous porous media using a novel combination of treatment technologies. Environ. Sci. Technol, 38(19), 5149-5156. 

Gelhar, L. W. Axness, C. L. Stochastic Analysis of Macrodispersion in Three-Dimensionally Heterogeneous Aquifers" Hydrology Res. Center, New Mexico Inst it. Mining Technol. ... 

FIGURE 3.11 Proliie of rate of methane disappearance dependence upon steam feed partial pressure and the corresponding exit methane conversion predicted by a one-dimensional heterogeneous model, operating conditions are given as follows tube length/O.D./LD. 11.95/ 0.102/0.0795 m feed composition for point a (mol%) 12.98,... 

Attempts have been made to develop two-dimensional heterogeneous models (McGreavy and Cresswell, 1968, 1969, Deasch and Froment, 1971). McGreavy and Cresswell proceded by adding to the one-dimensional heterogeneous model the terms accounting for radial heat and mass transfer in the bed. 

Figure 11.8.a-1 One-dimensional heterogeneous model with interfacial gradients. 

Figure I I.8.a-2 One-dimensional heterogeneous model with interfacial gradients. Nonmique steady-state case Pq = 0.15 atm. To = 393°C. Initial T, A 393°C, B = 560f°C (after Liu and Amundson 91], from Froment [9]).

Figure II.8.b-I One-dimensional heterogeneous model with interfacial gradients. Start up of reactor, transient temperature profiles. AT = temperature increase of gas phase above feed value AT = increase of solid temperature above initial value.

The mathematical model used by Capelli et al. to simulate the reactor may be classified as a one-dimensional heterogeneous model considering external and internal gradients, but not axial difiusion and conduction. The steady-state model equations are, in terms of the partial pressures... 

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