## 103 Computational Snapshot Approach

Consider a finite volume representation of a basic conservation equation for a general variable 0:

d d0

— (VcellP0)+(area p U 0)e-(area p U fi)w+••• = (area T^—)e-----(10.1)

All the terms appearing in Eq. (10.1) are formulated following the usual finite volume method discussed in Chapter 6, except the first term containing the time derivative. Usually, the computational cell volume, Vcell, is considered constant and is written outside the time derivative. For the computational snapshot approach, it is useful to consider the above form. In the sliding mesh approach, the above equation needs to be modified to account for grid movement. In the snapshot approach, instead of considering a moving grid, the time derivative term in the above equation is approximated and a steady framework is used to solve Eq. (10.1).

Consider evaluation of Eq. (10.1) in a steady framework by assuming the cyclically repetitive flow between the impeller blades. As mentioned earlier, in a snapshot approach, blades are considered stationary at one position. For an instant, when the blades of the rotating impeller coincide with the position of the blades considered in the snapshot simulation, the following equation is solved in a steady framework:

(area pU0)e- (area pUfi)w +----= I area T^ — I-----+ S^ - — (VcellP0)

e (10.2) It is necessary to approximate the time derivative terms appearing in this equation. By separating the variables, one can write the time derivative term as (for constant density fluid)

Generally in a fixed grid simulation, the volume of any computational cell remains constant. This can be applied to all computational cells used in the snapshot approach except those directly attached to the front and rear sides of blades. As the impeller rotates, the volume of cells attached to the front side of the blade decreases. Correspondingly, the volume of computational cells attached to the rear side of the impeller blade increases (Fig. 10.4). The rate of increase or decrease can be calculated directly from the area of the interface between computational cells and impeller blade and the velocity with which the impeller is rotating. Thus, for the computational cells attached to the front and rear sides of impeller blades, the second term of the right-hand side Volume of cell increases due to impeller rotation

Sources/sinks to front/rear sides respectively

Volume of cell decreases due to impeller rotation

FIGURE 10.4 Computational snapshot approach.

Volume of cell increases due to impeller rotation

Sources/sinks to front/rear sides respectively

Volume of cell decreases due to impeller rotation

FIGURE 10.4 Computational snapshot approach.

d t where N is impeller rotational speed in revolutions per unit time. The bracketed quantity on the right-hand side is the tangential velocity of the impeller blades at radial location, r, and Abc, is the projection of the area of the interface between computational cell and impeller blade on a plane normal to the tangential velocity. Please note that for the computational cells attached to the front side, volume decreases with time and therefore, there will be a negative sign on the right-hand side of Eq. (10.4). For the mass continuity equation, that is, when <p is unity, Eq. (10.4) represents the mass source and sink at the front and rear sides respectively.

Using the assumption of cyclically repetitive flow within the impeller region, the first term on the right-hand side of Eq. (10.3) can be approximated as d d VcellP -(&) = -VcellP(2nN) — (ft>) (10.5)

This cyclically repetitive flow will occur only in certain region around the rotating impeller; baffles at the walls destroy such cyclically repetitive flow. Fortunately, for the region where cyclically repetitive flow does not exist, the magnitude of time derivative terms is quite small compared to other terms in Eq. (10.1) and therefore, the time derivative terms may be neglected. As there is no way to find in which region flow will be cyclically repetitive without solving the full time varying equations, this region has to be specified based on available empirical information. Simulations carried out with this approach, however, indicate that the predicted results are not very sensitive to choice of the assumed region of cyclically repetitive flow. Eqs (10.4) and (10.5) can be used to simulate the flow generated by a rotating impeller in a steady state framework.

In addition, since the impeller blades are modeled as stationary walls, it is also necessary to include additional source terms to computational cells attached to the edges of impeller blades, representing the shear caused by the rotating impeller blades (Fig. 10.4). For all the computational cells attached to the four thin edges of the impeller blade, a momentum source needs to be added when solving for the tangential velocity. Though the standard wall function formulation is not strictly applicable to very thin walls, it may be used in the absence of better information to define the momentum source:

where WBLD is the tangential velocity of the blade averaged over AB, which is the contact area between the computational cell and impeller blade. y+ is the dimensionless distance from the wall, defined by:

If the calculated value of y+ is less than 11.225, the near wall cell center lies in the viscous sub-layer and therefore the factor in the square bracket of Eq. (10.6) is set to unity.

Ranade and Tayalia (2000) validated the snapshot approach by considering a two-dimensional case of rotating flows. Application of this approach to simulating complex, three-dimensional flows in stirred tank reactors is discussed below. The next section will discuss application of this approach to cases relevant to reactor engineering.

10.3.1. Simulation of Flow Generated by a Disc (Rushton) Turbine

Recently Ranade et al. (2001a, 2001b) used a computational snapshot approach to simulate the flow generated by a disc (Rushton) turbine in a fully baffled vessel. The predictions were compared with the comprehensive data available in the open literature. Computations were carried out for the stirred vessel configuration used in the experimental studies by Schafer et al. (1997) and Perrard et al. (2000). Standard wall functions were used to specify boundary conditions at all the stationary walls (Launder and Spalding, 1974). The top surface of the liquid was also modeled as a wall, since Schafer et al. (1997) used a closed vessel in their experiments. For the impeller shaft, disc and hub, an angular velocity corresponding to the impeller rotational speed was specified as boundary condition. Half of the vessel was considered as the solution domain. Cyclic boundary conditions were imposed at the open surfaces of the solution domain. Based on available experimental information, a region surrounding impeller blades was selected in which time derivative terms were included in the governing equations (using the approximation given by Eq. 10.5). Beyond this region, surrounding the impeller blades, time derivative terms were assumed to be negligible. The solution domains and boundary conditions used in their work are shown in Fig. 10.5.

Turbulence was modeled using the standard k-e model. All the governing equations were discretized using a QUICK discretization scheme with SUPERBEE limiter function (Fluent User Guide, 1997). The SIMPLE algorithm (Patankar, 1980) was

Cyclic planes

Cyclic planes used to iteratively solve discretized equations. Preliminary numerical experiments were carried out to examine the influence of grid size on predicted results. The findings of these preliminary simulations were in line with those reported by Ng et al. (1998). These studies indicated that in order to capture trailing vortices, it is necessary to use more than 100 cells to cover the impeller blades. Results of Ranade et al. (2001a) obtained with the 630 800 grids (80 x 83 x 95 : : r x 0 x z) are discussed below. A blade of a Rushton turbine was discretized with 29 x 3 x 25 cells (r x 0 x z). Typical grids used are shown in Fig. 10.6. Iterations were continued until the sum of normalized residues fall below 1 x 10-3 and none of the local as well as volume integrated variables of interest showed any appreciable change with further iterations.

Numerical simulations carried out with the computational snapshot approach show the well-known flow patterns generated by the Rushton turbine. Predicted flow fields for typical r-z planes are shown in Fig. 10.7 (vector plot and contours of turbulent kinetic energy). Simulations indicate the upward inclination of the radial jet issuing from the impeller, which is in agreement with published experimental evidence (for example, Ranade and Joshi, 1990). The predicted turbulent kinetic energy contours show the high turbulence region in the impeller stream. Apart from the qualitative agreement, the predicted results in the bulk region also show satisfactory agreement with the experimental data of Ranade and Joshi (1990). The predicted pumping number for the Rushton turbine (calculated at r/T = 0.18) is 0.6, which is lower than the generally accepted value. It is, however, within the range reported in the published literature (0.75 ± 0.15). Recent angle-resolved velocity measurements also indicate a pumping number value of about 0.61 (Perrard et al., 2000). The snapshot approach may underpredict the radial velocities in the immediate vicinity of the blades, since the blades are modeled as stationary walls. This leads to lower 0 0