One-Phase Model

Heterogeneous Model Eqns. (3) - (9) One-phase Model Eqns. (3), (5)-(10) Specchia and co-workers (1)... 

Previous one-phase continuum heat transfer models (1), (5), (10), (11), which are all based upon "large diameter tube" heat transfer data, fail to extrapolate to narrow diameter tubes. These equations systematically underpredict the overall heat transfer coefficient by 40 - 50%, on average. When allowance is made in the one-phase model for the effect of tube diameter on the apparent solid conductivity (kr>s), Eqn. (7), the mean error is reduced to 18%. However, the best predictions by far (to within 6.8% mean error) are obtained from the heterogeneous model equations. 

The use of two-phase homogeneous continuum models in packed bed modelling has often been avoided due to the computational difficulties. Recently, Paspek and Varma (15) have found a two-phase model to be necessary to describe an adiabatic fixed-bed reactor, while Dixon and Cresswell (16) have shown that the effective parameters of the one-phase model may be interpreted in terms of the more fundamental parameters of a two-phase model, thus demonstrating more clearly their qualitative dependencies on the operating and design characteristics of the bed. When two phases and several species are involved, the computational advantages of the cubic Hermite method may be anticipated to be high. 

A semi-analytical solution to these equations was derived by Dixon and Cresswell (16), who then matched the fluid phase temperature profile to the one-phase model profile to obtain explicit relations between the parameters of the two models. 

The numerical solution to the system of equations (12) - (17) parallels that of the one-phase model almost exactly, with longer computation times due to the increased size of the collocation matrix and its bandwidth. Typical computation times to produce fluid and solid temperature profiles at each of five bed-depths were 3-4 seconds. 

Figure 1. Fitting data to one-phase (liquid) and two-phase (liquid and gas) oxygen uptake models. Key , oxygen in outlet gas (y scale) O, biomass concentration (Xt scale) , dissolved oxygen concentration dashed line, one-phase model solid line, two-phase model Xq, biomass concentration at the interface. Reprinted, with permission, from Ref. 17. Copyright 1981, John Wiley ...

A One-Phase-Model with a viscosity, which may depend from the location and may vary throughout the solution stage. 

But we omit the generalizations of equilibrium stabilities for phase transitions [1, 103-106] (for them typically criteria stability like (3.256), (3.257) are not valid), for more general materials (say sohds), and the more complicated problem of stabUity of nonequilibtium states (e.g. the vast field of dissipative structures [24, 37, 80, 107-109]) because most of these issues do not concern our (one-phase) model or are now in the stage of intensive and not completely resolved research see also Rem. 31 in Chap. 4. 

Table 5-1. Comparison of the Crystallinity am of a Po/j(ethylene) Sample Calculated from the Density p or the Specific Volume v = II p Using a Two-Phase Model with with the Percentage Number of Crystal Defects Calculated Using a One-Phase Model...

It was shown by Frith and Tuckett, that the one-phase model may explain the comparatively long melting range (compare, however, p. 53). The argument given amounts to the following. If the polymer consists of two separate phases, i.e., if a molecule belongs either to the crystalline or to the amorphous part, both the entropy and the heat content of the system will be linear functions of the fraction 0 of crystalline substance ... 

The conservation of mass equation for this situation, which is directly applied in modeling tube reactors (F. Moser, 1977) and bubble columns (Reuss, 1976), is thus identical to Equ. 3.3a. These types of one-dimensional one-phase models are not only necessary for calculating conversion They are also very useful in, for example, calculating the /cl value of a reactor with a concentration profile ... 

The types of models described thus far have already been characterized as one-dimensional one-phase models. This means that only a single coordinate (the z axis) is considered in reality, in the case of, for example, flow in a tube, processes occurring in a radial direction (coordinate R) can also be of importance. 

The formulation of a two-dimensional one-phase model for dispersion takes the following form, according to Equ. 3.3a ... 

E. Birgersson, J. Nordlund, H. Ekstrom, M. Vynnycky, and G. Lindbergh. Reduced two-dimensional one-phase model for analysis of the anode of a DMFC. J. Electrochem. Soc., 150 A1368-A1376, 2003. 

Following on this suggestion the next step would be to obtain solutions for the temperature profiles in the moving and in the stationary phases and then to evaluate effective transport parameters by interpreting these results in terms of a one-phase model of the bed. This would lead to relations between effective transport parameters and what are believed to be the major underlying and independently measurable heat transfer steps. 

Applying the same collocation technique to the one-phase model equations, assuming k = 0, gives... 

Also given are similar statistics for other models which have appeared in the recent heat transfer literature, and which are claimed to be wide-ranging. The one-phase model predictions are obtained by replacing Eqn. (28) with Eqn. (43). 

Previous one-phase continuum models [22], [23], [24,25] and [l6], which are all based on "large diameter tube heat transfer not extrapolate to narrow diameter tubes. These equations systematically underpredict the overall heat transfer coefficient by 40-50, on average. On the other hand, a one-phase model, employing the proper central core apparent solid conductivity [ 3]f and utilising wall heat transfer data measured on beds,of low tube-to-par tide diameter ratio [8], shows a mean error of only 18. ... 

U overall heat transfer coefficient (one-phase model). 

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