Experiments were conducted with the same amount of silica particles
(i.e. 570 g) at four different temperature settings: 20, 200, 400 and
600°C. For different temperature the gas volumetric flow rate was
adjusted based on the ideal gas equation to ensure that superficial gas
velocity was increased with the same interval (0.1 cm/s) from the
non-fluidized packed bed to slugging bed. In each experiment, empty bed
was first heated to the set temperature, and pressure drop across the
empty bed at different superficial gas velocity was measured. The ECT
sensor was then calibrated for empty fluidized bed as low calibration.
Silica particles were loaded into the bed and air was supplied to
fluidize the particles. After the bed temperature became stable, air
supply was gradually turned off to ensure a stable packed bed. After the
packed bed had been stable for 3 minutes, then high calibration was
conducted.
Methodology
3.1 Image reconstruction
algorithm
For an 8-electrode ECT sensor, in total 28 pairs of inter-electrode
capacitance can be obtained and used to reconstruct the relative
permittivity distribution.26 The relation between the
capacitance vector and the normalized permittivity vector can be
described by
\(\lambda=Sg\), (2)
where \(\lambda\) is the normalized capacitance vector with the
dimension of 28×1 for an 8-electrode sensor, g is the normalized
permittivity vector with the dimension of 3228×1 in a circular
region,27 and S is the normalized sensitivity
distribution matrix with the dimension of 28×3228.
The linear back-projection algorithm (LBP) and the projected Landweber
iteration algorithm with an optimal step length are commonly used to
obtain the relative permittivity distribution,26, 28
\(\hat{g_{0}}=\ S^{T}\lambda\), (3)
\(\hat{{\ g}_{k+1}}=P\left[\hat{g_{k}}-\ \alpha S^{T}\left(S\hat{g_{k}}-\lambda\right)\right]\),
(4)
\(P\left[x\right]=\left\{\par
\begin{matrix}\ \ 0\ \ \ \ \ \ \ \ \ \text{if}\text{~{}\ \ \ \ \ \ \ \ \ \ \ }x<0\\
x\text{~{}\ \ \ \ \ \ \ \ ~{}}\text{if}\ \ \ \ \ \ \ \ \ \ \ 0\leq x\leq 1\\
1\ \ \ \ \ \ \ \ \ \ \text{if}\text{~{}\ \ \ \ \ \ \ \ \ \ \ }x>1\\
\end{matrix}\ \right.\ \), (5)
where
\(e^{\left(k\right)}=\ \lambda-S\hat{g_{k}}\text{\ \ \ \ \ \ \ \ \ \ \ }\)(6)
\(\alpha=\ \frac{\left\|S^{T}e^{(k)}\right\|^{2}}{\left\|\text{SS}^{T}e^{(k)}\right\|^{2}}\).
(7)
Note that \(\hat{g_{0}}\) is derived from the LBP algorithm, which is
also regarded as the initial estimation for the projected Landweber
iteration algorithm. In Eq. (6), e is the vector of errors
between the measured and calculated capacitance vectors. The optimal
step length \(\alpha\) is computed by Eq. (7) according to Liu et
al.28. P is the function operator defined in
Eq. (5). The number of iterations is set to 200 according to our
previous work.27 The LBP algorithm is used for online
visualization and the projected Landweber iteration algorithm with an
optimal step length for results comparison considering their
characteristics.26, 28 The effect of temperature on
image reconstruction can be referred to our previous
work.23
Measurement of fluidization
characteristics
Once the normalized permittivity distribution (\(\hat{g}\)) is obtained,
the solids concentration distribution and time average solids
concentration with its standard deviation can be calculated by the
following equations.
\(p=\frac{\sum_{i=1}^{N}{\hat{g_{i}}*s_{i}}}{\sum_{i=1}^{N}s_{i}}\)(8)
\(\overset{\overline{}}{p}=\frac{1}{Q}\sum_{i=1}^{Q}p_{i}\) (9)
\(\beta=\theta\bullet p\ \) (10)
\(\overset{\overline{}}{\beta}=\frac{1}{Q}\sum_{i=1}^{Q}\beta_{i}\)(11)
\(\text{STD}=\frac{1}{Q}\sum_{i=1}^{Q}{({(\beta}_{i}-\overset{\overline{}}{\beta})}^{2}\)(12)
where p is the normalized permittivity of each frame; s is
the area of each image pixel; N is the number of pixels (3228 in
this work); \(\overset{\overline{}}{P}\) is the time average normalized
permittivity; Q is the number of frames (20000 in this work); \(p_{i}\)is the normalized permittivity of \(i_{t}\) frame; \(\beta\) is the
solids concentration of each frame, \(\overset{\overline{}}{\beta}\) is
the time average solids concentration; \(\beta_{i}\) is the solids
concentration of \(i_{t}\) frame; and STD is the standard deviation of\(\overset{\overline{}}{\beta}\).
Note that \(\theta\) is solids concentration of the packed bed, which
varies with temperature because the packed bed height can expand with
the increase in temperature. A parallel model is adopted to construct
the relation between the normalized permittivity and the solids
concentration as shown in Eq. (10).18, 29
Previous studies have shown that packed bed height increases with the
increase in inter-particle forces.14, 15, 34-36 It
should be noted that the estimation of solids concentration from ECT
images is closely related to the initial packed bed
expansion19, 31, 33, 35 because of the decrease in
coordination number of each particle (details can refer the previous
study34). A specially designed ruler was used to
measure the packed bed height. The ruler was made by a stainless-steel
rod and held by a device mounted on the outlet of the fluidized bed. It
can move freely along the axial direction of the fluidized bed. The
packed bed height was recorded according to the surface, below which
particles are clearly adhered to the rod.
Results and discussion
4.1 Visualization of fluidization transition
A series of measurements of fluidization at different temperature
(T = 20, 200, 400, and 600°C) were carried out using
high-temperature ECT. For each temperature, the superficial gas velocity
was increased from 0 to 10.0 cm/s with the stepwise increment of 0.1
cm/s. Before measurements for each temperature were made, the static
height of the packed bed without gas flow was obtained. The average
packed bed height was 22.4, 23.2, 24.4, and 25.7 cm for T = 20, 200,
400, and 600°C, respectively. The corresponding average solids
concentrations are listed in Table 3.
Table 3. Average packed bed height and solids concentration at different
temperature