Understanding the Origin of Pressure Oscillations in Fluidized Beds

The origin of pressure oscillations in fluidized beds has stood for a long time as a major question in the understanding of two-phase granular flow. Pressure oscillations are important to understand because they can easily be measured on fluidized beds of all sizes and their relation to the passage of void regions has led to their use in defining different flow regimes of fluidization [1]. While it has long been understood that pressure oscillations correlate well with bubble passage through fluidized beds, there has been contention as to whether pressure oscillations are tied to bubble formation [2], coalescence [3], or eruption [4]. This paper compares detailed results from a recently developed computational model [5] with experimental results [4] to shed light on the origin of pressure oscillations in bubbling fluidized beds.

A bubbling fluidized bed with a diameter of D_bed = 50 mm, filled with d_p = 1.2 mm diameter particles to various tapped bed heights, H₀, was modelled computationally for direct comparison to combined magnetic resonance (MR) imaging and pressure sensor experiments conducted by Müller et al. [4] on the same bed. A 3D cylindrical discrete element model with computational fluid dynamics (DEM-CFD) described and validated for simulating bubbling fluidized beds elsewhere [4] was used to model the fluidized bed. The bed was fluidized at excess gas velocity (U-U_mf) = 0.263 m/s for direct comparison of model and experiment.

Figure 1 shows simulation results of the bed with initial height H₀ = 40 mm over 275 ms of steady bubbling. The first two rows show voidage (ε) and differential pressure drop in the vertical direction (-dp/dz) maps of the bed every 25 ms. The third row gives the pressure drop across the bed over the time period and the fourth row gives the 1D voidage traces at the bottom and top of the bed, capturing bubble formation and eruption, respectively, with spikes in voidage. The results show that bubbles form and erupt at the same frequency, since bubbles form, rise and erupt without interacting with bubbles above or below them in this short bed. Pressure drop occurs most rapidly in regions dense with particles due to the drag force on these particles. Almost no pressure drop occurs in the vertical region spanning from the bottom to the top of a bubble, since the lack of particles in the bubble provides a free path through which fluid can pass. The pressure drop across the entire bed is at a minimum when a single, fully-formed bubble is rising through the bed because the bubble provides a long path of low resistance for fluid. The pressure drop reaches a maximum just as a bubble erupts because the free path for fluid disappears and the fluid now has to pass through a tall region dense with particles.

Figure 2 gives the same sequence of simulation results of the same bed, except with initial height H₀ = 50 mm, and voidage and differential pressure drop maps every 50 ms over a period of 550 ms. Approximately every 260 ms, a large bubble forms and begins to rise, spanning almost the entire diameter of the bed. Soon after, a smaller bubble forms in its wake and rises more quickly behind the first large bubble. The first bubble erupts, followed by the second in quick succession. The pressure drop across the entire bed is at a minimum the entire time the first bubble rises, but begins to increase dramatically as the first bubble erupts since the free path for fluid created by this large bubble suddenly disappears. The pressure drop reaches a maximum just as the second bubble erupts, since the next large bubble has not formed yet and the bed is left with a large height of dense particles, through which fluid must pass.

Table 1 gives the frequency of oscillations in pressure drop, bubble formation and bubble eruption for various bed heights, determined by taking the Fourier transform of the respective traces shown in the third and fourth rows of Figures 1 and 2 over 2.1 seconds of steady bubbling. In the 40 mm tall bed, bubble formation, eruption and pressure oscillations all occur at the same frequency, since the bubbles do not interact, and thus the same particle arrangements which create easy and difficult paths for fluid passage occur for every bubble formed. In the 50 mm tall bed, the frequency of bubble formation is twice that of bubble eruption and pressure oscillations. Even though the bubbles do not fully coalesce, they strongly interact, causing the arrangement of particles around the first bubble to be vastly different from the second. Since, due to drag, arrangements sparse in particles allow for little pressure drop over a vertical distance and arrangements dense in particles create large pressure drops, maxima in pressure drop occur only once for every two bubbles formed. While the bubbles do not erupt at exactly the same time, the second bubble erupts in the wake of the first eruption, causing the frequency of the oscillations of the voidage trace at the top of the bed to be half of that at the bottom.

The results for the four bed heights in Table 1 demonstrate H₀ = 50 mm to be a critical height at which two consecutive bubbles begin to interact, and thus the frequency of bubble eruption and pressure oscillations suddenly half, deviating significantly from the frequency of bubble formation. This result matches well with experiments conducted on the same bed [4], which showed a sudden decrease in bubble eruption and pressure oscillation frequency at H₀ = 50 mm, as shown in Table 1. Finding the same frequency of bubble eruption and pressure oscillation beds with H₀ ranging from 20-161 mm, but vastly different frequencies of bubble formation in beds 50 mm or taller, Müller et al. [4] postulate that pressure oscillations in fluidized beds are derived from bubble eruption. The detailed results provided by the DEM-CFD simulations show that maxima in pressure drop across narrow fluidized beds occur just after bubble eruption due to the large height of dense particles left behind by the bubbles. Müller et al. [4] also found that the transition point in pressure fluctuations occurred at a bed height at which the theoretical value for effective bubble diameter, D_e, became large enough that the bed entered the slugging regime, defined as D_e/D_bed > 0.6.  This hypothesis is confirmed by DEM-CFD results, shown in Table 1, which saw a drastic increase in average D_e, with the bed entering the slugging regime between H₀ = 40 mm and H₀ = 50 mm.

This paper will further investigate the origin of pressure oscillations in fluidized beds by using the DEM-CFD model to look at pressure oscillations in beds ranging from 30-156 mm in tapped bed height and further comparing to the results of Müller et al. [4]. Investigating this larger range of heights will provide understanding on how bubble coalescence and a variety of types of slugging affect pressure oscillations.

References:

[1] Bi, H.T. (2007). Chem. Eng. Sci., 62, 3473-3493.

[2] Fan, L.T., Ho, T., Hiraoka, S. and Walawender, W.P. (1981). AIChE J., 27, 388-396.

[3] Littman, H. and Homolka, G. (1973). Chem. Eng. Sci., 28, 2231-2243.

[4] Müller, C.R., Davidson, J.F., Dennis, J.S., Fennell, P.F., Gladden, L.F., Hayhurst, A.N., Mantle, M.D., Rees, A.C., Sederman, A.J. (2007). Powder Technol., 177, 87-98.

[5] Boyce, C.M., Holland, D.J., Scott, S.A., Dennis, J.S. (submitted 2013). Phys. Rev. E.

Figures:

Figure 1. Analysis of pressure oscillations in a fluidized bed with H₀ = 40 mm and D_bed = 50 mm. Over 275 ms this figure gives: voidage maps (first row) and differential pressure drop in the vertical direction maps (second row) every 25 ms, as well as traces of pressure drop across the entire bed (third row) and 1-D voidage at the top and bottom of the bed (fourth row).

Figure 2. Analysis of pressure oscillations in a fluidized bed with H₀ = 50 mm and D_bed = 50 mm. Over 550 ms this figure gives: voidage maps (first row) and differential pressure drop in the vertical direction maps (second row) every 50 ms, as well as traces of pressure drop across the entire bed (third row) and 1-D voidage at the top and bottom of the bed (fourth row).

Tables:

Table 1. Frequencies of bubble formation (f_form), bubble eruption (f_erupt) and oscillations in pressure drop (f_ΔP), as well as the ratio of effective bubble diameter to bed diameter (D_e/D_bed) in a 50 mm diameter bubbling fluidized bed with varying tapped bed heights (H₀). The results are given for a DEM-CFD model as compared to experiments [4] using a combination of magnetic resonance (MR) imaging and a pressure sensor.

	DEM-CFD Results				Experimental Results
H0 (mm)	fform (Hz)	ferupt (Hz)	fΔP (Hz)	De/Dbed	ferupt (Hz)	fΔP (Hz)
30	6.96	6.96	6.96	0.41	6.4	6.0
40	7.12	7.14	7.15	0.44	5.6	5.5
50	7.42	3.73	3.72	0.66	2.9	3.0
60	7.57	3.75	3.79	0.65	2.9	3.1