Fluid-Fluid Drag Corrections

Drag correction describes how concentration modifies the single-particle drag coefficient model in a multi-particle system.

Richardson Zaki Drag Correction

This is a drag correction method for spherical particles with a wide range of applicability. According to this model, the effect of increased concentration on spherical particles is that of hindering. The terminal velocity decreases faster than would be expected purely from the reduction of buoyancy force of the two-phase mixture on the particles.

This model can also cover droplets that are small enough to retain their spherical shape and whose internal viscosity is high enough for internal circulation to be negligible. This model also applies to bubbles whose surfaces are immobile due to surface contamination, as in the case of air bubbles in regular tap water.

As originally specified, the Richardson Zaki model [534] for terminal velocity correction for spherical particles is:

$$\frac{V}{V_{\infty }}={\alpha }_c^n$$

(1978)

Range	n for small tubes	n for large tubes
${\text{Re}}_{\infty }<0.2$	$4.65$	$4.65$
$0.2<{\text{Re}}_{\infty }<1$	$(4.35+17.5\frac{d}{D})R{e}_{\infty }^{-0.03}$	$4.35R{e}_{\infty }^{-0.03}$
$1<{\text{Re}}_{\infty }<200$	$(4.45+18\frac{d}{D})R{e}_{\infty }^{-0.1}$	$4.45R{e}_{\infty }^{-0.1}$
$200<{\text{Re}}_{\infty }<500$	$4.45R{e}_{\infty }^{-0.1}$	$4.45R{e}_{\infty }^{-0.1}$
${\text{Re}}_{\infty }>500$	$2.39$	$2.39$

$n$ is the Richardson Zaki exponent, $D$ is tube diameter, and $d$ is particle diameter.

The two figures below illustrate the wide range of applicability of the Richardson Zaki method. The first shows the terminal velocity correction factor that is computed in a Stokesian particle settling test ( ${\text{Re}}_{\infty }<0.2$ ) against the data of Fricke and Thompson [461]. A suspension of 44-micron resin particles in a glycerine-water mixture settles out at a rate that particle concentration affects. The rate is well-predicted either by the Richardson Zaki method, or by the Volume Fraction Exponent method using exponent $n_D$ =-8.7.

The second figure shows the terminal velocity of 3-5 mm air bubbles rising in water against the data of Simonnet [545] under conditions ${\text{Re}}_{\infty }>500$ . Even though the bubbles are ellipsoidal in this size range, agreement is good.

The original Richardson Zaki model is adapted for computation by:

• Omitting the small tube term as this would generate an unacceptable discontinuity between Reynolds number ranges
• Using min() and max() functions to make sure exact continuity between Reynolds number ranges
• Allowing you to update or adapt the original calibration constants.
• Preventing the drag correction reaching implausibly high or low levels as the continuous phase volume fraction vanishes. Set a maximum packing ${\alpha }_{d,max}$ for the sum of the dispersed phase volume fractions, so that the strength of the drag correction does not change further beyond this concentration.

  $$f_V=max({\alpha }_c,{1-\alpha }_{d,max})$$

  (1979)

  $$n=max\{\begin{array}{c}min\{\begin{array}{c}A\\ min(BR{e}_{\infty }^{n_1},CR{e}_{\infty }^{n_2})\end{array}\\ D\end{array}$$

  (1980)

  $$R{e}_{\infty }=\frac{Re}{f_V}$$

  (1981)

  $$f_D=\{\begin{array}{c}\begin{array}{c}\frac{{\alpha }_c}{f_V^2}\text{Spherical Particle Interfacial area}\end{array}\\ \begin{array}{c}\frac{1}{f_V^2}\text{Symmetric Interfacial area }\end{array}\end{array}$$

  (1982)

The single-particle Reynolds number ${\text{Re}}_{\infty }$ is applied both to the single-particle drag law as well as to the Richardson Zaki correction itself. This is essential to reproduce terminal velocity correctly in regimes where drag coefficient is a strong function of Reynolds number.

This implementation gives identical results for drag coefficient and terminal velocity regardless of which of the two interfacial area models is used, at least up to the maximum packing point.

The default calibration is applicable for spherical particles, but you can modify it:

$A$	$B$	$C$	$D$	$n_1$	$n_2$	${\alpha }_{d,max}$
4.65	4.35	4.45	2.39	-0.03	-0.1	0.66

Lockett Kirkpatrick Drag Correction

Lockett and Kirkpatrick [508] investigated the effect of bubble concentration on the terminal velocity of 5-mm air bubbles in water. They achieved stable bubbly flow of up to 66% void fraction by using a down flow. They applied the Richardson Zaki model to spherical particles to terminal velocity data for 5-mm air bubbles in water. The model assumed that both the drag coefficient and drag correction were in the Reynolds number independent range. For this range, where $\text{Re}$ > 500, the Richardson Zaki exponent, n, is 2.39. They found that a better fit to the data could be achieved by applying a further empirical factor to take bubble deformation into account.

In a figure further below, the Lockett Kirkpatrick method is compared with the Simonnet et al. [545] method for predicting Simonnet’s terminal velocity data for 7-10 mm bubbles. The agreement is good up to 15% void fraction, after which Simonnet found a swarming behavior not observed in the Lockett Kirkpatrick data from 5-mm bubbles.

The original Lockett Kirkpatrick model is:

$$\frac{V}{V_{\infty }}={\alpha }_c^{2.39}[1+2.55{\left(1-{\alpha }_c\right)}^3]$$

(1983)

This Lockett Kirkpatrick model is adapted for computation by limiting the correction for extreme values of ${\alpha }_c$ through a maximum packing criterion ${\alpha }_{d,max}$ . ${\alpha }_{d,max}$ has a default value of 0.66.

$$\widetilde{{\alpha }_c}=max({\alpha }_c,{1-\alpha }_{d,max})$$

(1984)

$$f_V={\left({\tilde{\alpha }}_c\right)}^{2.39}[1+2.55{\left(1-{\tilde{\alpha }}_c\right)}^3]$$

(1985)

$$f_D=\{\begin{array}{c}\begin{array}{c}\frac{\alpha }{f_V^2}\text{Spherical Particle Interfacial area}\end{array}\\ \begin{array}{c}\frac{1}{f_V^2}\text{Symmetric Interfacial area}\end{array}\end{array}$$

(1986)

This method gives identical results for drag coefficient and terminal velocity regardless of the interfacial area model that is used, at least up to the maximum packing point.

Simonnet Drag Correction

This is a drag correction for air / water bubbles in the 7-10 mm size range (Eotvos number 6.6-13.4) and 0-30% void fraction range. Effects are hindering in the range 0-15% and swarming in the range 15-30%.

Simonnet and others [545] measured terminal velocities, taking great care to work with uniform size distributions. Their recommended model for drag correction factor in the 7-10 mm range is:

$$\frac{C_D}{C_{D\infty }}={\alpha }_c{[{\alpha }_c^{25}+{\left(4.8\frac{1-{\alpha }_c}{{\alpha }_c}\right)}^{25}]}^{-2/25}$$

(1987)

However, they also found that for smaller bubbles (2-6 mm) only hindering behavior occurs.

Terminal velocity computed using Simonnet’s hindering/swarming method and Lockett and Kirkpatrick’s [508] hindering-only methods are compared below against Simonnet’s own data for 7-10 mm bubbles in air-water [545]. Both methods are a good fit to the data up to 15% void fraction when swarming behavior begins.

The Simonnet model is adapted for computation by making the correction go to zero smoothly and linearly as the continuous phase volume fraction vanishes, so that it can be used safely beyond the void fraction range of Simonnet’s data.

$$f_D=s{f}_{D1}+(1-s)f_{D2}$$

(1988)

$$f_{D1}={\alpha }_c{[{\alpha }_c^{25}+{\left(4.8\frac{1-{\alpha }_c}{{\alpha }_c}\right)}^{25}]}^{-2/25}$$

(1989)

$$f_{D2}={0.25\alpha }_c$$

(1990)

$$s=0.5\{1+\tanh \left(\frac{{\alpha }_c-0.7058}{0.05}\right)\}$$

(1991)

Spherical Particle Interfacial Area Density and Symmetric Interfacial Area Density models give the same results for drag coefficient and terminal velocity when the Simonnet drag correction model is used in its range of applicability of 0-30% void fraction.

Volume Fraction Exponent Drag Correction

This drag correction model assumes that the drag correction factor is of the form:

$$f_D={\alpha }_c^{n_D}$$

(1992)

where $n_D$ is some constant power.

This method complements the other drag correction methods by covering several regimes of practical interest, such as

• Small spherical particles settling under Stokesian drag condition.
• Large spherical particles under turbulent wake drag conditions.
• Spherical cap bubbles swarming under a churn-turbulent flow regime.

Regime	Reference	n	$n_D$ (Spherical particle Interfacial Area)	$n_D$ (Symmetric Interfacial Area)
small spherical particles ${\text{Re}}_{\infty }<0.2$	[534]	+4.65	-8.3	-9.3
large spherical particles ${\text{Re}}_{\infty }>500$	[534]	+2.39	-3.78	-4.78
spherical cap bubbles, churn-turbulent conditions	[481]	-0.5	+2	+1

The first of these three regimes has already been shown in the figure in the Richardson Zaki section on the dilute particle settling under Stokesian conditions. The settling velocity is a strong function of concentration. Using a drag correction exponent of $n_D=-8.7$ is equivalent to using the velocity correction exponent recommended by Fricke and Thompson, $n=4.85$ , which provides a good fit to their own data [461].

The drag correction method in the churn-turbulent regime is taken from Ishii and Zuber [481] and is relevant for large spherical cap bubbles in bubble columns. In such columns, the bubble size is controlled by coalescence and breakup, and is unrelated to the initial size of the bubbles that are introduced at the inlet.

Ishii and Zuber suggested that bubbles in this regime are characterized by a Weber number of 8 and a drag coefficient of 8/3. Both of these numbers are defined in terms of their drift flux model velocity. This leads to a model predicting pure swarming behavior. Bubble equivalent diameter, terminal velocity, and drag coefficient vary as continuous phase volume fraction raised to the power of -1/2, -3/4 and +2 respectively.

Ishii and Zuber found that this model under-predicts Akita and Yoshida data for superficial velocity for air bubbles in dilute sodium sulfite solution that is presented in Figure 16 of Ishii and Zuber [481]. However it does show the correct trend of superficial velocity against bubble concentration and is a reasonable approximation for the data from the bubble column with the largest diameter of 60 cm.

In the verification calculation reproduced below, the equivalent bubble diameter varies from 11 mm to 13 mm as the void fraction increases from 1% to 30%. The terminal velocity increases from 23 cm/sec to 30 cm/sec, as expected from the relevant exponents.

The constant exponent method is adapted for computation by limiting the correction for extreme values of ${\alpha }_c$ . This is done through a maximum packing criterion ${\alpha }_{d,max}$ , which has a default value of 0.66:

$${\tilde{\alpha }}_c=max({\alpha }_c,{1-\alpha }_{d,max})$$

(1993)

$$f_D={\left({\tilde{\alpha }}_c\right)}^{n_D}$$

(1994)

$$f_V=\{\begin{array}{c}\begin{array}{c}\sqrt{\frac{\alpha }{f_D}}\text{Spherical Particle Interfacial Area}\end{array}\\ \begin{array}{c}\sqrt{\frac{1}{f_D}}\text{Symmetric Interfacial Area }\end{array}\end{array}$$

(1995)

This velocity correction factor $f_V$ is used to compute the single-particle terminal velocity correctly where the drag coefficient depends on Reynolds number. However, this Reynolds number adjustment is not relevant in cases where the single particle drag coefficient is not a function of Reynolds number. For example, in the Ishii Zuber model for large bubbles in the churn turbulent regime.