# Gravity flow ## Froude Number The flow in vertical and [[Flow in partially full pipes|inclined pipes]] under gravity with respect to the transportation of air or other gases is characterised by the dimensionless superficial volumetric flux, $v^*$ (of similar form to the Froude Number, $Fr$ and sometimes known as the pipeline Froude number). A number of flow regimes are observed in gravity flow systems, but they can be reduced to three regimes of practical interest to the designer, these are: - **Self Venting Flow** $(Fr < 0.3)$. In this flow regime any air entrained in the liquid will disengage and flow upwards against - **Entraining Flow** $(0.3 < Fr < 1.0)$. In this flow regime air is likely to remain entrained in the bulk liquid flow but may disengage and accumulate at changes of direction, slope etc. causing irregular flow - **Clearing Flow** $(1.0 < Fr)$. In this flow regime any entrained air will be carried in the bulk liquid stream and any accumulated air pockets will depleted and purged through the system by the bulk liquid flow. >[!info] >The dimensionless volumetric flux boundaries between these regimes has been the subject of much research, mainly with large pipes and related to the fields of water and waste water engineering. The boundaries quoted above are selected to be conservative, i.e. self venting flow may continue at $Fr > 0.3$ and clearing flow may start at $Fr < 1.0$. The dimensionless volumetric flux in the outlet from a vessel is also the critical parameter in defining the level of liquid cover above the outlet to ensure that air is not entrained in the liquid flowing out of the vessel when designing for flooded flow or the size of outlet required to ensure non-flooded flow. Most of the work in this field has been performed on air / water systems and the justification for using the densimetric Froude number rather than the dimensionless superficial volumetric flux as a correlating parameter is unclear. Ref 17 advocates the use of the densimetric Froude number as the correlating parameter and the results generated by both parameters are sufficiently close for engineering purposes provided that the vapour density is less than ~5% of the liquid density. ### Dimensionless Superficial Volumetric Flux The Dimensionless Superficial Volumetric Flux is defined as $v^*=\frac{v}{\sqrt{gd}} \tag{1}$ Where: - $v^*$ = dimensionless superficial volumetric flux (pipeline $Fr$) - $v$ = superficial velocity in bottom outlet ($m/s$) - $g$ = acceleration due to gravity ($m/s^2$) - $d$ = pipeline internal diameter ($m$) ### Densimetric Pipeline Froude Number The Densimetric Pipeline Froude Number is defined as $Fr=\frac{v}{\sqrt{{g}'d}}$ Where: ${g}'=\frac{g\rho_{l}}{\rho_{l}-\rho_{g}}$ Where: - $\rho_{l}$ = liquid density ($kg/m^3$) - $\rho_{g}$ = vapour density ($kg/m^3$) The dimensionless superficial volumetric flux ($v^*$) is used as the correlating parameter in the remainder of this document but the densimetric pipeline Froude number can be used in any of the equations by substituting ${g}'$ for $g$. ## Ensuring self-venting flow To ensure self venting flow in vertical pipework or non-flooded flow in side outlets from vessels. Substituting $v^*=0.3$, and $Q=vA=v\frac{\pi d^2}{4}$ into equation $(1)$ we obtain: $Q<\frac{3\pi}{40}\sqrt{gd^5} \tag{2a}$ NB: $\frac{3\pi}{40}=0.2356...$ $d>\sqrt[5]{\left ( \frac{40}{3\pi} \right )^{2}}\sqrt[5]{\frac{Q^2}{g}} \tag{2b}$ NB: $\sqrt[5]{\left ( \frac{40}{3\pi} \right )^{2}}=1.7825...$ Where: - $Q$ = volumetric flowrate ($m^3/s$) - $g$ = acceleration due to gravity ($m/s^2$) - $d$ = pipeline internal diameter ($m$) >[!info] >If transitioning from self venting to flooded flow in vertical pipework then the self venting section should be the greater of 5 diameters or 0.5m in length before any reduction in diameter. >>[!warning] >>Citation needed on this point. This is a heuristic I have seen used multiple times, but have never seen this referenced to any technical publication. Use with caution! Or, more generally, for a target Dimensionless Superficial Volumetric Flux, $v^*$: $d=\sqrt[5]{\frac{16}{\pi^2}}\sqrt[5]{\frac{Q^2}{g{v^*}^2}} \tag{2c}$ NB: $\sqrt[5]{\frac{16}{\pi^2}}=1.1014...$ ## Ensuring flooded flow ### Bottom outlet flooded flow #### Bottom outlet non-rotational flooded flow To ensure flooded flow from the bottom outlet of a vessel with non-rotational flow. i.e. with a vortex breaker or similar device, the criterion below must be fulfilled. $v^*<1.6{\left ( \frac{h}{d} \right )}^2 \tag{4}$ Where: - $v^*$ = dimensionless superficial volumetric flux (pipeline $Fr$) - $d$ = bottom outlet internal diameter ($m$) - $h$ = liquid level above bottom outlet ($m$) Re-arranging we find: $h>\frac{\sqrt{20}}{\sqrt{8\pi}}\sqrt[4]{\frac{Q^2}{gd}} \tag{4a}$ NB: $\frac{\sqrt{20}}{\sqrt{8\pi}}=0.89206...$ Where: - $Q$ = volumetric flowrate ($m^3/s$) - $g$ = acceleration due to gravity ($m/s^2$) - $d$ = pipeline internal diameter ($m$) - $h$ = liquid level above bottom outlet ($m$) #### Bottom outlet rotational flooded flow To ensure flooded flow from the vertical downwards or upwards outlet of a vessel with rotational flow. i.e. without a vortex breaker or similar device, the criterion below must be fulfilled. $1+2.3v^*<\frac{h}{d} \tag{5}$ Where: - $v^*$ = dimensionless superficial volumetric flux (pipeline $Fr$) - $d$ = bottom outlet internal diameter ($m$) - $h$ = liquid level above top of bottom outlet ($m$) Re-arranging we find: $h>d+\frac{9.2}{\pi}\sqrt{\frac{Q^2}{gd^3}}\tag{5a}$ NB: $\frac{9.2}{\pi}=2.92845...$ Where: - $Q$ = volumetric flowrate ($m^3/s$) - $g$ = acceleration due to gravity ($m/s^2$) - $d$ = bottom outlet internal diameter ($m$) - $h$ = liquid level above top of bottom outlet, or bottom of dip-pipe for dipped outlets ($m$) ### Side outlet flooded flow To ensure flooded flow from the side outlet of a vessel, the criterion below must be fulfilled. Refs. 7, 8, 9, 13. $v^*<\sqrt{\frac{2h}{d}} \tag{6}$ Where: - $v^*$ = dimensionless superficial volumetric flux (pipeline $Fr$) - $d$ = bottom outlet internal diameter ($m$) - $h$ = level above the **top of the side outlet** ($m$) Re-arranging we find: $h>\frac{8}{\pi^2}\frac{Q^2}{gd^4}\tag{6a}$ NB: $\frac{8}{\pi^2}=0.81057...$ ## References 1. Open Channel Flow Resistance, Yen B C, Journal of Hydraulic Engineering, Jan 2002 2. Computer Applications in Hydraulic Engineering, Halsted Methods Inc., 2002 3. Water & Wastewater Enginering Hydraulics, Casey T J, Aquaverra Research Ltd, 2004 4. A Review of Explicit Approximations of Colebrook's Equation, Genic et al, FME Transactions (2011) 39, 67-71 5. Perrys Chemical Engineers Handbook, 7th Edition 6. ASME BPE 2012 7. Designing Piping for Gravity Flow, P D Hills, Chemical Engineering, September 1983 8. HTFS Handbook, FM8, Design of Gravity Flow Systems 9. HTFS Handbook, FP3, Gravity Flow 10. Purging & Air Removal, Richard A Beier, Mechanical Engineering Technology Department, Oklahoma State University, 2009 11. Air In Pipelines, A Literature Review, HR Wallingford, Report SR 649, April 2005 12. Co-current air-water flow in downward sloping pipes, Pothof, Technical University of Delft, 2011 13. Simpson L.L. 1968 "Sizing Piping for Process Plants" Chem Eng 75 No. 13 (June 1968) p192 14. Not used. 15. Air Problems in Pipelines, A Design Manual, HR Wallingford, 2005. ISBN-10 1-898485-11-9 16. ANSI / HI 9.8 - 1998, American National Standard for Pump Intake Design