Dispersion Coefficients

The dispersion coefficients for the gas, liquid and solid phases were estimated from the available literature, although possible experimental methods for measuring the dispersion coefficients in the pilot-scale SBCR were considered as presented in Appendix A.

Liquid-Phase Dispersion

The axial liquid phase dispersion coefficient (DL) was predicted using Equation (6-34)

introduced by Baird and Rice [310] who used a large number of experimental data reported by various authors to obtain such an equation.

DL= 0.35dR1.33(gUG)0.33 (6-34)

It should be noted that Equation (6-35) proposed by Deckwer et al. [108, 316] _{can also be used} instead of Equation (6-34) and predicts similar values for the dispersion coefficient.

D𝐿 = 0.768𝑑𝑅1.34𝑈𝐺0.32 (6-35)

The dependency of the liquid phase dispersion coefficient (DL) on the reactor diameter was

accounted for by the scale-up index (n), introduced by Yang et al. [329] _{and defined as:}

DL~dRn (6-36)

These authors found that the operating pressure has an effect on the scale-up index since the axial dispersion coefficient was found to decrease with increasing pressure. They proposed the following equation to take into account the effect of pressure:

n

n0 = 1 − 0.11 ln � ρG

ρG,0� (6-37)

In the above equation, n0 and ρG,0 represent the scale-up index and the gas density at atmospheric

pressure, respectively.

It should be noted that the decrease of the liquid phase axial dispersion coefficient with increasing gas density could be related to the increase of the gas holdup or more precisely to the increase and decrease of the populations of small gas bubbles and large gas bubbles, respectively [120]_{. The decrease of the large gas bubbles population results in less back-mixing in the SBCR} i.e., lower values of the liquid-phase axial dispersion coefficient (DL).

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Gas-Phase Dispersion

The axial gas phase dispersion coefficient (DG) can be predicted using several correlations

available in the literature, such as those by Mangartz and Pilhofer [330]_{, Towell and Ackerman} [331] _{or Field and Davidson} [332]_{. These correlations, however, cannot be applied to the 2-class} ADM because they were developed considering the gas bubbles as only a single-phase. They also overpredict the axial dispersion coefficient values for large reactor diameters. While the small-bubbles axial dispersion coefficient can adequately be assumed to be identical to that of the liquid phase since the small bubbles are entrained by the liquid recirculation, the axial dispersion of the large gas bubbles, which rise fast in the reactor in a plug flow, is more difficult to estimate. It should be mentioned that to our knowledge, no one has measured the axial dispersion of the small or large gas bubbles. de Swart and Krishna [272] assumed a constant value of 100 for the Peclet Number of the large gas bubbles (Pe)Large in order to account for this

phenomena. Their assumption, however, neglected the impact of the other operating variables on the size and population of the large gas bubbles and thus on their axial dispersion. In this study, the axial dispersion coefficient of the gas-phase was assumed to be a function of the diameters of the gas bubbles. For small gas bubbles, the axial dispersion coefficient should be equal or similar to that of the liquid-phase, whereas for large gas bubbles the axial dispersion coefficient should be small to obtain plug flow like conditions. The following relationship is therefore proposed to estimate the dispersion of each class of gas bubbles:

D𝐺

D𝐿= 1 −

𝑑𝐵𝑎

𝑑𝐵,0𝑎+ 𝑑𝐵𝑎 (6-38)

This relationship shows that for small gas bubbles diameter (dB ≈ 0) the gas phase dispersion is

equal to the liquid-phase dispersion. For large gas bubbles (dB is large) the gas phase dispersion

approaches 0. The constant dB,0 in Equation (6-38) represents the diameter of the gas bubbles at

the point of designation as large gas bubbles in the 2-class model and was set at 0.01 m. The value of the exponent in Equation (6-38) was chosen so that Equation (6-38) predicts the large gas bubbles Peclet number assumption of de Swart and Krishna [272] _{under the conditions theses} authors used.

Solid-Phase Dispersion Coefficient and Particles Settling Velocity

The direct measurements of the solid concentration performed along the pilot SBCR have shown that the dispersion-sedimentation model was able to fit well the data obtained (see Section 6.1.8).

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Behkish [127] _{performed similar measurements of the solid concentration using the same reactor;} and also found that the dispersion-sedimentation model fitted well the data obtained with average absolute relative error and standard deviation of 1.5 and 2.5 %, respectively. Unfortunately the number of experimental data points obtained by Behkish [127] _{and in this present study was not} sufficient to allow a precise estimation of the particles settling velocity and solid dispersion coefficient. These two parameters were therefore estimated using the data of several other authors [173, 186, 270, 271]. Using the data/correlations developed by the authors listed in Table 31, the following two correlations were developed:

PeS = 8.5FrG0.76ReG−0.052+ 0.025ReP1.07FrG−0.067 (6-39)

UP = 1.37UG0.17UTS0.78(1 − cV)2.43 (6-40)

Table 31: Models used for Predicting the Axial Solid Dispersion Coefficient and Particle Settling Velocity

Authors Solid System Correlation

Kato et al. [270] Glass beads ρP = 2520 kg/m3 75.5<dP<163 µm CS: 48-202 kg/m3 UGdR DS = 13FrG 1 + 0.009RePFrG−0.8 1 + 8FrG0.85 UP= 1.33Ut,∞�_UUG t,∞� 0.25 (1 − cV)2.5

Kojima et al. [173] 105 <dGlass beads P<125 µm CS: 3.1-62 kg/m3 UGdR DS = 10FrG 0.76 O’Dowd et al. [186] Glass beads ρP = 2420 kg/m3 88<dP<105 µm CS: up to 420 kg/m3 UGdR DS = 7.7 � FrG6 ReG� 0.098 + 0.019ReP1.1 UP= 1.69UG0.23UT,∞0.8(1 − cV)1.28

Smith and Reuther [271]

Glass beads ρP = 2420, 3990 kg/m3 48.5<dP<164 µm CS: up to 420 kg/m3 UGdR DS = 9.6 � FrG6 ReG� 0.1114 + 0.019ReP1.1 UP= 1.1UG0.026UT,∞0.8(1 − cV)3.5

The particles terminal settling velocity was estimated as follows [333]_:

UTS =(ρS− ρ_18µLL)gdP2 (6-41)