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.

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

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 expansion^{19, 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.

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