Computational Fluid Dynamics is
extensively used for the design of chemical process equipment because it
enables the understanding of flow and temperature fields in the equipment. In
addition, CFD provides a way to establish relationships between the flow fields
and the design objectives. Thus, the CFD has been useful for the optimum design
in terms of capital and operating costs. In view of the great potential of CFD,
in the area of gas-liquid bubble columns, more than 100 papers have appeared in
the published literature since 2001. The CFD solves numerically the governing equations
for mass, momentum, Reynolds stresses, turbulent kinetic energy (k), turbulent
energy dissipation rate (e) and thermal energy. All these
equations contain six terms: (1) rate of accumulation (2) rate of transport by
convection (3) diffusion (4) turbulent dispersion, (5) rate of production and
(6) rate of dissipation. The last two terms obviously are missing in the mass
conservation (equation of continuity). While deriving the conservation
equations, three turbulence models are commonly employed: k-e,
Reynolds stress modeling (RSM) and large eddy simulation (LES). These three
models are computationally intense in an increasing order. In fact, there has
been a continuous endeavour (over the past forty years) to strike a balance
between the computational power and simplicity of the model resulting into loss
of accuracy and reliability. Further, the real equipment are geometrically
complex, multiphase, heterogeneous; in terms of phase distributions and
turbulent in nature. Therefore, for large size bubble columns; > 1m, the current
computational power permits relatively simple models such as k-e and
RSM. However computationally feasible models (such as the k-e
model and RSM) make several assumptions and it is imperative to understand the
gravity of each simplifying assumption towards the loss of accuracy. This particular
feature is the subject of this paper.

            As mentioned earlier, the conservation
equations for k and e consist of convective transport,
diffusive transport, dispersive transport, production and dissipation. If each
of these terms can be quantified accurately in the exact form (i.e. without any
assumption) and in the modeled form, then the direct comparison between the two
is expected to give a clear insight into each modeling assumption. In this
direction, attempts have been made in the published literature to estimate the
exact terms from experiments. However the experiments have posed problems of
the conditions being too difficult to maintain or the occurrence of
perturbations of several physical phenomena concomitant with the experiment.
Direct simulations allow numerical experiments to be carried out that are
otherwise difficult or impossible to realize in a laboratory, and yields
detailed information concerning the flow field in individual realizations.
However, the storage of data and simulation of complex flows are still major concern
in using DNS. In order to estimate the complex correlations of the
fluctuating components, a possible choice could be to resort to the use of an
LES database and estimate the terms. It is known that the LES gives
accurate results for only the large scales of turbulence, and models the small
scales. However, for the purpose under consideration, a very fine grid
resolution in high shear regions (so that more than 95% energy is captured) is
expected to give flow information which is fairly close to DNS. In the present
work, different turbulence models have been compared against each other at
different flow conditions; characterized by the level of power consumption per
unit mass.

            In the
present work, standard k-e, RSM and LES simulation have been performed
on three different types of spargers, namely (i) single hole sparger, (ii)
sintered plate sparger and (iii) sieve plate sparger. The cylindrical column
having a computational height, H = 1000 mm with inside diameter, D
= 150 mm was employed as a bubble column. The gas inlet through the sparger was
incorporated by creating source points at the specified position to mimic the
exact spargers. The  superficial gas velocities have been varied in the range
of 20 mm/s to 100 mm/s. The grid size of 5 million has been used to ensure that
very fine scales of motion are resolved and the LES results are used to
evaluate and extract terms which are otherwise modeled in the standard k?ε
model and the RSM. In order to compare the turbulence models, the LES
results have been used to evaluate the exact terms in k and ε
transport equations in standard k?ε model and RSM. Therefore,
it was thought desirable to estimate these two terms using k- e,
RSM and LES models. From the LES simulations, the time series of velocity and
pressure were stored. These were subsequently used for the detailed estimations
of two terms. It was observed that the turbulence production and
dissipation terms are the dominant terms in the modeled transport equation. In
order to understand the error made in the modeling of turbulence production,
the production was estimated from the LES simulations which involves relatively
lesser assumptions (and that to for isotropic scales of motion). It can be
observed from Figure 1A and Figure 1B that the same order of error is made in
the estimations of production and dissipation of k in both the k-e
model and RSM approaches.Similarly, the estimations of the production and
dissipation for e in the RANS approach is shown in Figure 1C and 1D
respectively. It can be seen that the modeled production and dissipation of k
are even  an order of magnitude higher than the exact production and exact
dissipation Therefore, the conservation equations for k and e
in k-e
and RSM get satisfied. In the present work, all the transport (convective,
dissipative and diffusive) as well as production and dissipation terms in k and
e
equation in k-e and RSM have been estimated.

            The
turbulent kinetic energy has been budgeted to study the influence and impact of
the simplifying assumptions. The transport terms have been calculated i.e. the
convective term, viscous transport and turbulent transport,  the production
term and the dissipation term, and the net balance of the conservation equation
has been studied. The residuals have been plotted for each of the three
approaches (Figure 1E and Figure 1F). It can be seen that for the LES approach the
residual is zero at all the locations where as the RANS approach have a
residual depicting an imbalance in the budgeting of the transport processes.

            The
transport processes of turbulent kinetic energy and the turbulent dissipation
rate have more terms arising in their natural modeled form and that have been
modeled for closure. The additional terms arising by the correlation of the
velocity and the hold-up give rise to several terms that are non-linear in
nature whose avaeraging leads to severval more non-linear terms. The RANS
approach assumes a linear gradient similar to the molecular diffusive process for
modeling the transport of momentum due to the influence of turbulence.

            The simple
gradient transport approximation is strictly valid only when the energy
containing eddies are smaller that the distance over which the gradient of ε_Lvaries
appreciably. The absence of the other terms in the turbulent transport terms
and the approximation of the fluctuating hold up leads incorrectly estimate the
turbulent transport term. The inclusion of turbulent viscosity and the neglect
of these higher order terms grossly leads to an errorneous budgeting of the
transport terms where residuals are also left unaccounted for.