J. M. Jeremiah et al.
DOI:
10.4236/eng.2018.1010050 691 Engineering
of scalar measurements of the state of a dynamical system, then under certain
genericity assumptions,
the time delay embedding provides a one-to-one image
of the original set (
X
), provided the embedding dimension
(
m
) is large
enough”
[13]. A time-delayed variable constitute an adequate embedding, provided the
measured variable is smooth and couples to all other variables [10].
Further, it is possible to reconstruct
the phase space attractor while preserving
the topological characteristics from measurements of a single dynamical vari-
able. This renders experimental systems, like CFB reactors with attractors of
moderately low dimension, acceptable for analysis by chaos analysis tools. This
idea was introduced by Packard
et al.
in
1980 [14] and the mathematical proof of
the validity of this idea is due to Takens [12].
The reconstructed attractor consists of orbits of points corresponding to dif-
ferent state vectors,
X
(
i
), which represents one point on the orbit of the attractor
and is reconstructed from the data points of the time series. Reconstruction of
the state vectors is achieved by the method of delays where a time delay,
τ
, of the
time steps between the elements of the state vector and the number of elements,
m
, of the state vector equals to the embedding dimensions are chosen and
transposed to create a multidimensional state vector [10] [13] [15]. Suppose the
measured time series data of single variable
x
(1),
x
(2), …,
x
(
N
), is embedded
using the time delay,
τ
, and the embedding dimension,
m
, then the resulting
state vectors of the attractor is given by Equation (1)
( ) ( ) ( ) ( ) ( )
( )
T
, 2,, 1X i xi xi xi xi m
ττ
= + + +−
(1)
where the count,
i
= 1, 2, 3, …,
N.
3. Methodology
3.1. Data Collection
Data were collected from a CFB system shown in Figure 1 with a twin-riser
having 76 and 203 mm internal diameters and 10 m high operated at 50 to 550
kg/m
2
s solids flux and 4.0 to 10.0 m/s gas velocity. Fluid catalytic cracking cata-
lyst particles with 67 μm mean diameter and density of 1500 kg/m
3
were used. A
70% to 80% humid air was used for transporting the solid particles. Signals were
sampled from eight (8) axial levels (
i.e.
,
Z
= 0.98, 1.52, 2.73, 3.96, 5.13, 6.34, 8.74,
and 9.42 m) and 11 radial points (
i.e.
,
r
/
R
= 0.00, 0.16, 0.38, 0.50, 0.59, 0.67, 0.74,
0.81, 0.87, 0.92, and 0.98) at each level where
r
/
R
is the normalized radial dis-
tances from the center to the wall of the riser. To each point, 29,100 data points
of solid concentration were sampled in 30 seconds using optical fiber probe at
970 Hz.
3.2. Determination of Trajectories Distances (Euclidean Distance)
The trajectory distances of the state vectors from the reconstructed attractor
with time delay,
τ
= 6 and embedding dimension,
m
= 15 were determined using
a multi-dimensional Euclidian distances formula, used in chaos analysis. The
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