Reynolds Mass Flux Model

Besides applying the postulation similar to the Boussinesq s (or Pick s law) to solve the Reynolds mass flux — mJc in terms of isotropic turbulent mass diffusivity Dt as described in preceding Sect. 3.2 by c — Sc two-equation model, another model has been recently developed to solve the anisotropic Reynolds mass flux —M-c directly instead of using D, to close the turbulent species mass conservation equation. The Reynolds mass flux model discussed in this section could be known as a result following the turbulence closure postulations for the second-order closure turbulence model in the book of Chen and Jaw [23]. 

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Reynolds Mass Flux Model for Catalytic Reactor. 220... 

Keywords Computational mass transfer Reynolds averaging Closure of time-averaged mass transfer equation Two-equation model Turbulent mass transfer diffusivity Reynolds mass flux model... 

Reynolds mass flux models This category of models features to solve the unknown —u[d directly instead of in terms of D,. This category of models includes... 

Thus, Reynolds mass flux model involves Reynolds stress, Reynolds heat flux and Reynolds (fluctuating) mass flux equations, the unknown quantity is increased to twelve to make the total unknown quantities become twenty they are ... 

Using the Reynolds mass flux model, the directional —wjc can be calculated separately as —u c, —u c and — anisotropic turbulent mass diffusivity can be obtained. 

It should be noted that the Dy obtained from Reynolds mass flux model and the Dt obtained from two-equation model is entirely different, as Dt, is calculated from... 

Por illustrating the advantage of using anisotropic Reynolds mass flux model, the simulation of absorption of CO2 by MEA solution (see Chap. 5) using isotropic c 2 — Ec two-equation model and one-dimensional model by Tontiwachwuthikul [18] versus anisotropic Reynolds mass flux model is given in Pig. 3.6a for a packed column of 0.1 m in diameter and Pig. 3.6b for a packed column of 1.9 m in... 

In order to reduce the computer load of standard Reynolds mass flux model, the complicated Eq. (1.23a) for expressing m-m can be replaced by the simpler Eq. (1.8). Then, the model becomes the combination of Reynolds mass flux and the Boussinesq postulate (two-equation model). It is called hereafter as hybrid Reynolds mass flux model. The model equations are given below. 

The Reynolds mass flux model can also be further simplified by letting the convection terms of u c on the left side of Eq. (3.24) equal to the molecular diffusion and turbulent diffusion in terms of u d on the right side, then under steady condition, Eq. (3.25a) becomes... 

The number of unknown quantities and equations of this model is the same as the Reynolds mass flux model except using Eq. (3.31) to replace Eq. (3.25a) for calculating m c in order to reduce the load of computation. 

Reynolds mass flux model, or standard Reynolds mass flux model, in which the unknown —pw-c is calculated directly using model equation either Eq. (3.25a) or Eq. (3.25b). This model is rigorous and applicable to anisotropic case with mass and heat transfer. The model equations comprises the following equation sets ... 

Concentration distribution for the calculation of tray efficiency As stated in Chap. 3, the conventional way of using turbulent Schmidt number Sc, model for predicting the concentration distribution is not dependable for the reason that the correct Sc, is not only hard to guess but also it is varying throughout the process. Hence the recently developed c — Bc two-equation model and the Reynolds mass flux model are recommended to use as described in the subsequent sections. 

The model equations are the same as the standard Reynolds mass flux model except that the term is simplified by using Eq. (1.8) as follows ... 

The hybrid Reynolds mass flux model and algebraic Reynolds mass flux model, which only need to solve simpler Eq. (1.8) instead of complicated Eq. (1.23), may be a proper choice for multiple tray computation if their simulated results are very close to the standard Reynolds mass flux model. For comparison, the simulated column trays in Sect. 4.1.1.1 for separating n-heptane and methylcyclohexane are used. Li [17] simulated concentration profiles of all trays at different levels above the tray floor, among which the tray number 8 and tray number 6 are shown in Fig. 4.17a and b. 

Fig. 4.16 Concentration contour of x-y plan on trays by standard Reynolds mass flux model, a 20 mm above tray floor of tray number 8, b 70 mm above tray floor of tray number 8, c 20 mm above tray floor of tray number 6, and d 70 mm above tray floor of tray number 6 [16]...

The simulated result by using hybrid Reynolds mass flux model can also compared with that obtained by using two-equation model as shown in Fig. 4.17, in which the agreement between them can be seen except in the region near inlet weir and column wall where the hybrid Reynolds mass flux model gives more detailed concentration distribution than the two-equation model. 

The verification of hybrid Reynolds mass flux model can also be made by comparing with the experimental outlet concentration of each tray as shown in Figs. 4.18 and 4.19, in which the result by using two-equation model is also presented. It can be seen that the hybrid Reynolds mass flux model gives closer outlet concentration to the experimental measurement than the two-equation model, although both of them are considered to be verified by experiment. 

Fig. 4.18 Comparison between hybrid Reynolds mass flux model and two-equation model by simulated concentration contours of 20 mm above tray floor on tray number 8. a Hybrid Reynolds mass flux model (reprinted from Ref. [7], Copyright 2011, with permission from Elsevier) and b c - Be two-equation model (reprinted from Ref [4], Copyright 2005, with permission from American Chemical Society)...

Fig. 4.19 a Comparisons of simulated results by different models with experimental data [16] tray efficiency and outlet Cg concentration, b Comparisons of the outlet Ce concentration between standard and hybrid Reynolds mass flux model simulations and experimental data [16]... 

Generally speaking, the overall simulated result of a distillation tray column by using two-equation model and different Reynolds mass flux models is very close each other and checked with experimental measurements, but if detailed mass transfer and flow information on the trays are needed, the standard Reynolds mass flux model is the better choice. 

Reyuolds mass flux model at 70 mm above the tray floor, b standard Reynolds mass flux model at 20 mm above the tray floor, c hybrid Reynolds mass flux model at 70 mm above the tray floor, d hybrid Reynolds mass flux model at 20 mm above the tray floor, e algebraic Reynolds mass flux model at 70 mm above the tray floor, and f algebraic Reynolds mass flux model at 20 mm above the tray Hoot... 

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