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method are described. In the following Section 4, the mixing pat
terns are discussed and the mixing times are compared with th
PLIF result. Finally, the main conclusions are summarized in
Section 5.
2. Stirred tank configuration
Configurations of the investigated stirred tank and the impelle
is given in Fig. 1. The system is an unbaffled, flat-bottomed, cylin
drical tank (of diameter T = 0.15 m) agitated by a down-pumping
pitched-blade turbine with four blades (named as PBT-4), each
angled at 30 to the horizontal and attached to a hub that i
mounted on the impeller shaft with a diameter d = 0. 008 m. Th
thickness and width of the impeller blade are tb = 0. 001 m and
wb = 0. 01 m, respectively. The tankwas filledwithwater to a heigh
of H = T. In the numerical simulations, the fluid is assumed to b
incompressible with a density of q =1 103
kg m3
and a dynami
viscosity of l =1 103
Pa s. As for the concentric configuration
the shaft of the impeller was concentric with the axis of the tank
For the eccentric agitation, the impeller was positioned at thre
off-axis locations, i.e. at e =2E/T = 0.2, 0.3 and 0.5 fromthe tank axis
For all cases, the diameter of the impeller is D = T/3 and the impelle
off-bottom clearance is C = T/3. The impeller rotates clockwise (a
viewed from the above) with a speed of N =5s
1
, which corre
sponds to a Reynolds number of Re ¼ qND2
l ¼ 1:25 104
.
3. Methodology
3.1. Mathematical formulation of DES model
The turbulent fluid flow in the stirred tank is predicted by th
DES model, which is formulated by replacing the distance function
d in the one-equation Spalart–Allmaras (S–A) model [30] with
modified distance function:
~ d ¼ minfd; CDESDgð1
where CDES = 0.65 is the model empirical constant and D is the larg
est dimension of the grid cell in question. This modification of th
S–A model changes the interpretation of the model substantially
In regions close to the wall, where d < CDESD, it behaves as a RAN
model. Away from the wall, where d > CDESD, it behaves in a Smago
rinsky-like manner and is changed to the LES model. The governin
equation of DES model can be given as follows:DES model is proposed based on the one-equation S–A model.
Since then, some variants, such as the DES model based on the
SST k–x and Realizable k–e model, were proposed by [31,32],
respectively. No matter what kind of RANS model was used, the
principle was the same. In this paper, DES model proposed by
Spalart et al. [33], which is referred to as the standard edition,
was adopted. The closure coefficients in the governing equation
of DES model are given as follows: r = 2/3, c = 0.41, Cb1 = 0.1335,
Cb2 = 0.622, Cw1 = Cb1/k2
+(1 + Cb2)/r, Cw2 = 0.3, Cw3 =2, Cv1 = 7.1,
Ct1 = 1.1 and Ct2 =2.
3.2. Simulation of the mixing time
Mixing time was predicted by using a virtual scalar tracer and
monitoring the scalar concentration variations with the time. In
the present work, the origin of coordinate system coincide with
the projective point of the impeller shaft on the bottom plane of
the stirred tank (see Fig. 1). As shown in Fig. 2, the scalar tracer
was feeded from the top surface at a point, which is located in
the plane y = 5 mm, with a distance of 3 mm to the stirred tank
wall. The concentration of the tracer was initialized as 1 in the
feeding region, and in the rest region as 0. A total of 15 monitoring
points in the same plane were selected in regions of different agi-
tation intensity, and the axial and radial positions were (z
=z/
T = 0.2,0.5,0.9) and (x
=2x0
/T = ±0.9, ±0.5,0, where x0
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