Standard Reynolds Mass Flux Model

For the convenience of derivation, the negative Reynolds mass flux u c is concerned instead of —u c. The exact u c transport equation can be derived as follows. Subtracting Eqs. (3.1) from (3.3), we have 

Multiply Eq. (3.23) by wj and multiply Eq. (1.5) by c the sum of the two equations is averaged and rearranged to yield the following Reynolds mass flux equation (in the form of fluctuating mass flux u[dy. 

The bracketed first term on the right side represents the turbulent and molecular diffusions the second term represents the influence of fluctuating pressure and concentration on the distribution of Reynolds mass flux the third term represents the production of wjc the fourth term represents the dissipation. 

Equation (3.24) should be modeled to suit computation. Applying the modeling rule, the bracketed first term on the right side of Eq. (3.24) can be considered proportional to the gradient of w-t/ and the for turbulent diffusion and molecular diffusion. The modeling form is as follows  

The modeling of second term is complicated, it can be considered to be related to the fluctuating velocity and the average velocity gradient as follows  

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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. 

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 ... 

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]...

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... 

The simulated radial averaged axial concentration distribution is compared with experimental data and the simulated result by using standard Reynolds mass flux model as shown in Fig. 4.41. These figures display no substantial difference between hybrid and standard Reynolds mass flux models. 

In this section, the standard Reynolds mass flux model (abbreviated as RMF model) is employed for simulation. 

The simulation is by using standard Reynolds mass flux model the model equation sets, the boundary conditions, and the evaluation of source terms are the same as given in Sect. 5.1.3. The simulated results are given in the following sections. 

The simulated profiles of adsorbate, methylene chloride, at different times are given in Fig. 6.11, in which the development of the concentration profiles in the column with time is seen. In comparison with Fig. 6.1, it is found that the simulation is closely similar. Yet after careful comparison, the shape of concentration distribution in the adsorption section (represented by the red brackets) is somewhat different. The parabolic shape of purge gas concentration distribution is more obvious by using standard Reynolds mass flux model due to better simulation near the column wall. 

Fig. 6.11 Simulated sequence of concentration profiles along adsorption column in mole fraction at different times by standard Reynolds mass flux model [14]...

Fig. 6.18 Simulated sequences of concentration distribution along regeneration column in mole fraction at different times by standard Reynolds mass flux model for Tjn.ads = 2.50 x 10 , = 365-365 exp (-0.352t -1.654)K, P = 1.09 atm, F = 33.5 L min [14]...

The simulated regeneration curve by using standard Reynolds mass flux model is shown in Fig. 6.19. 

Li [9] employed standard Reynolds mass flux model to simulate the water-cooled reactor as described in Sect. 7.1.3. 

Fig. 7.11 Comparison of radial temperature profiles at different packed heights between simulation obtained by the standard Reynolds mass flux model line), two-equation model dash), and experimental data circle) for Case 1 (//-distance of bed height measured from column bottom) [9]...

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