Universal correlation for the rise velocity of long gas bubbles in round pipes

Abstract

We collected all of the published data we could find on the rise velocity of long gas bubbles in stagnant fluids contained in circular tubes. Data from 255 experiments from the literature and seven new experiments at PDVSA Intevep for fluids with viscosities ranging from 1 mPa s up to 3900 mPa s were assembled on spread sheets and processed in log–log plots of the normalized rise velocity, $\hbox{\it Fr} \,{=}\,U/(gD)^{1/2}$ Froude velocity vs. buoyancy Reynolds number, $R\,{=}\,(D^{3}g (\rho_{l}-\rho_{g}) \rho_{l})^{1/2}/\mu $ for fixed ranges of the Eötvös number, $\hbox{\it Eo}\,{=}\,g\rho_{l}D^{2}/\sigma $ where $D$ is the pipe diameter, $\rho_{l}$, $\rho_{g}$ and $\sigma$ are densities and surface tension. The plots give rise to power laws in $Eo$; the composition of these separate power laws emerge as bi-power laws for two separate flow regions for large and small buoyancy Reynolds. For large $R$ ($>200$) we find \[\hbox{\it Fr} = {0.34}/(1+3805/\hbox{\it Eo}^{3.06})^{0.58}.\] For small $R$ ($<10$) we find \[ \hbox{\it Fr} = \frac{9.494\times 10^{-3}}{({1+{6197}/\hbox{\it Eo}^{2.561}})^{0.5793}}R^{1.026}.\] The flat region for high buoyancy Reynolds number and sloped region for low buoyancy Reynolds number is separated by a transition region ($10\,{<}\,R\,{<}\, 200$) which we describe by fitting the data to a logistic dose curve. Repeated application of logistic dose curves leads to a composition of rational fractions of rational fractions of power laws. This leads to the following universal correlation: \[ \hbox{\it Fr} = L[{R;A,B,C,G}] \equiv \frac{A}{({1+({{R}/{B}})^C})^G} \] where \[ A = L[\hbox{\it Eo};a,b,c,d],\quad B = L[\hbox{\it Eo};e,f,g,h],\quad C = L[\hbox{\it Eo};i,j,k,l],\quad G = m/C \] and the parameters ($a, b,\ldots,l$) are \begin{eqnarray*} &&\hspace*{-5pt}a \hspace*{-0.8pt}\,{=}\,\hspace*{-0.8pt} 0.34;\quad b\hspace*{-0.8pt} \,{=}\,\hspace*{-0.8pt} 14.793;\quad c\hspace*{-0.8pt} \,{=}\,\hspace*{-0.6pt}{-}3.06;\quad d\hspace*{-0.6pt} \,{=}\, \hspace*{-0.6pt}0.58;\quad e\hspace*{-0.6pt} \,{=}\,\hspace*{-0.6pt} 31.08;\quad f\hspace*{-0.6pt} \,{=}\, \hspace*{-0.6pt}29.868;\quad g\hspace*{-0.6pt}\,{ =}\,\hspace*{-0.6pt}{ -}1.96;\\ &&\hspace*{-5pt}h = -0.49;\quad i = -1.45;\quad j = 24.867;\quad k = -9.93;\quad l = -0.094;\quad m = -1.0295.\end{eqnarray*} The literature on this subject is reviewed together with a summary of previous methods of prediction. New data and photographs collected at PDVSA-Intevep on the rise of Taylor bubbles is presented.

The thickness of the falling film of liquid around a Taylor bubble

“…Much previous work has focussed on the ascent velocity v b of a Taylor bubble (see Viana et al 2003 for a recent review of experimental data, and Funada et al 2005 for a theoretical perspective). The ascent velocity has been shown to depend on the dynamic viscosity of the liquid m, its density r, the liquid-gas interfacial tension s, the internal diameter of the pipe D and the gravitational acceleration g. These quantities can be combined to form various dimensionless groups, including (White & Beardmore 1962;Wallis 1969;Seyfried & Freundt 2000) These are, respectively, the Froude number-which is a dimensionless velocity, representing the ratio of inertial and gravitational forces; the Morton numberwhich represents the ratio of viscous and interfacial tension forces; and the Eötvös number-which represents the ratio of buoyancy and interfacial tension forces.…”

“…The ascent velocity has been shown to depend on the dynamic viscosity of the liquid m, its density r, the liquid-gas interfacial tension s, the internal diameter of the pipe D and the gravitational acceleration g. These quantities can be combined to form various dimensionless groups, including (White & Beardmore 1962;Wallis 1969;Seyfried & Freundt 2000) These are, respectively, the Froude number-which is a dimensionless velocity, representing the ratio of inertial and gravitational forces; the Morton numberwhich represents the ratio of viscous and interfacial tension forces; and the Eötvös number-which represents the ratio of buoyancy and interfacial tension forces. Interfacial tension plays a negligible role in determining the behaviour of a Taylor bubble when Eo > 40 (Viana et al 2003), in which case, the morphology and ascent velocity of the bubble are controlled by inertial and viscous forces. These conditions are met for all volcanic slugs (Seyfried & Freundt 2000), as well as for many situations of industrial and engineering importance.…”

“…However, Viana et al note that Fr is largely independent of Eo for Eo > 40, which holds for the current study, allowing us to simplify the correlation considerably to give hence, for the situation of interest (negligible interfacial tension), Froude number is a function of inverse viscosity only. The data synthesized by Viana et al (2003) for Eo > 40, cover the inverse viscosity range 10 −1 < N f < 10 5 , so equation (2.5) has a very broad validity; we recommend this equation for calculating the ascent velocity of a Taylor bubble when interfacial tension can be ignored. A further dimensionless quantity that is employed by some workers to describe Taylor bubbles is the bubble (or slug) Reynolds number (e.g.…”

The existence and behaviour of large diameter Taylor bubbles

“…Theoretical predictions for F r exist in the range 0.328 − 0.369 [3,2,15,22,25] with Dumitrescu's 0.351 being regarded as the most accurate [4]. The stability of large diameter bubbles is less well understood [5,1,12,14].…”

Friction factor correlations for laminar, transition and turbulent flow in smooth pipes

“…Two power law correlations for sediment transport have been studied by Wang et al (2003) and for friction factors in turbulent gas-liquid flows by Garcia et al (2003Garcia et al ( , 2005. Correlations of families of bi-power laws depending on a third parameter have been constructed by Viana et al (2003) who correlated data for the rise velocity of Taylor bubbles in round vertical pipes as a family of bi-power laws of the Froude number vs. Reynolds number indexed by the Eotvos number. A file of papers correlating large data sets from real and numerical experiments can be found at (http://www.aem.umn.edu/people/faculty/joseph/PL-correlations).…”