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.