Expérience de pompage par air-lift
a
F. Raveleta Arts et Metiers ParisTech, DynFluid,
151 boulevard de l'Hôpital, 75013 Paris, France. contact:
[email protected] 5 novembre 2013
1 Introduction • Le principe du pompage par air-lift est d'injecter de l'air sous forme de bulles en bas d'une colonne d'eau (voir Fig. 1).
Figure 1 Schéma de principe du pompage par air-lift • Le montage sur lequel s'eectue le TP (voir photo en Fig. 2) se compose principalement : d'une cuve au niveau du sol ; d'une cuve séparatrice située à environ 5 m de hauteur ; d'un tuyau souple de diamètre interne 40 mm, pour la montée ; d'un tuyau rigide de diamètre interne 100 mm, équipé d'un débimètre électromagnétique, pour la descente ; d'un troisième tuyau rigide de diamètre interne 100 mm, muni d'une vanne, servant à injecter des solides (non utilisé dans ce TP). 1
Figure 2 Photographie de l'expérience • On fournit en annexe un document présentant un modèle pour déterminer en fonction du débit volumique d'air injecté le débit d'eau pompée. • Ce modèle a été implémenté dans un logiciel • La gure 3 présente une carte des régimes établie selon la méthode de Taitel & Duckler, pour un tuyau de diamètre 40 mm.
2 Travail à eectuer 2.1
Modélisation
• Etudier et comprendre les équations du modèle, identi er les hypothèses et les limites de l'application du modèle à l'expérience présente. • Faire tourner le modèle en variant quelques paramètres et eectuer une analyse de l'in uence de ces divers paramètres. 2.2
Dispositif expérimental
• Le dispositif est plus complexe que le schéma de principe ci-dessus : identi er les parties du dispositif, relever les cotes signi catives. • Quels sont les eets non pris en compte dans le modèle ? Peut-on les négliger ? Pourrait-on les mesurer ? Comment ? Proposer des pistes d'amélioration de l'expérience (instrumentation ou autre). 2.3
Expériences
• Remplir l'expérience à divers niveaux. 2
Figure 3 Carte établie pour air / eau D = 40 mm, zobs = 3 m • Varier le débit d'air, relever le débit d'eau et prendre des photos. • Placer les points obtenus dans la carte des régimes. 2.4
Confrontation au modèle et analyse critique
• A vous de jouer.
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Modeling of air-lift in a vertical pipe.
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F. Ravelet1 , S. Khelladi1 DynFluid, Arts et Metiers ParisTech,
151 boulevard de l'Hôpital, 75013 Paris, France. contact:
[email protected] December 6, 2010 1
Introduction
We have made a software that solves a model of air-lift in a vertical pipe [1]. This model has also been validated by other authors on a dierent experiment [2]. This is a three-phase model. Here we have restricted it to 2-phase as a starting point. Extending it to three-phase is straightforward once the present model would have been validated and if it is thought of interest with respect to our project.
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Gas-Liquid Model
2.1
Parameters
The physical parameters of the model are:
• g the acceleration of gravity; • µl the dynamic viscosity of the liquid; • ρl its density; • µg the dynamic viscosity of the gas; • ρg its density. The geometrical parameters of the model are (see Fig. 1):
• D the pipe diameter; • L1 the length of the pipe above the gas injection; 1
Figure 1: Scheme of the air-lift installation and model parameters [1].
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• L2 the length of the pipe below the gas injection point; • α the submergence ratio: this is the ratio between the height of the freesurface outside of the pipe1 and the height of the pipe exit: α = L3 /L1 . Two other free parameters of the model are:
• ξi the singular head loss at point I (gas injection); • ξe the singular head loss at point E (pipe entrance). 2.2
Control and Order parameters
The only control parameter is the volumetric ux of gas, presented as a discharge velocity2 :
• Jg the volumetric ux of gas. The order parameters are:
• Jl the volumetric ux of liquid; • g the in-situ concentration of gas in the upper part of the pipe; • l the in-situ concentration of liquid in the upper part of the pipe; • ul,O the real velocity of liquid at the pipe exit; • ug,O the real velocity of air at the pipe exit;
1 see
the de nition of L3 in Fig. 1. This corresponds to the static head that exists at the bottom of the pipe. 2 if Q is the volumetric ux and A the cross-sectional area of the pipe, J = Q/A.
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2.3
Equations
The model is based on the momentum equation:
ρl Jl2 − (ρl Jl ul,O + ρg Jg ug,O ) dPl − ( L2 + ∆PE ) dz dPl,g − ( L1 + ∆PI ) dz − (ρl l + ρg g )gL1 + ρl g(αL1 ) = 0
(1) (2) (3) (4) (5) (6)
The term (1) is the momentum balance. The term (2) corresponds to the head losses (regular and singular) in the bottom part. The term (3) corresponds to the head losses (regular and singular) in the upper part. The term (4) is the static weight inside the pipe. And the term (5) is the pressure force coming from the surrounding water and acting on E 3 . There are also the immediate equations:
g + l = 1 ul,O = Jl l ug,O = Jg g The regular head losses are modeled with the Blasius formula:
dPl 1 = 0.316Re−0.25 ρl J 2 l dz 2D l For the bottom part of the pipe, where Rel = Jl Dρl /µl is a Reynolds number based on the volumetric ux of liquid. For the upper part of the pipe, the Chisholm & Laird [3] equation is used:
dPl,g dPl 21 1 = (1 + + 2) dz dz χ χ 3 There
are extra terms in the bottom part L2 that cancell in (4) and (5) for the twophase case.
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with
v u dP u l u dz χ=u t dPg dz
and
dPg 1 = 0.316Re−0.25 ρg Jg2 g dz 2D
The only remaining equation is a correlation that gives the in-situ concentration of gas in the upper part of the pipe. Yoshinaga e t al. [1] propose to use the correlation of Smith 1969 [4]. This equation has also been used by Fujimoto e t al. [5].
"
ρg 1 ρg 1 g = 1 + 0.4 ( − 1) + 0.6 ( − 1) ρl x ρl x with
x=
3
ρl + 0.4( 1 − 1) 0.5 #−1 ρg x 1 + 0.4( x1 − 1)
ρg Jg ρg Jg + ρl Jl
Implementation
Figure 2: Graphical User Interface for version 2.1 of the code.
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This system of ve nonlinear equations has been implemented in Matlab for the sake of simplicity. Once it has been validated, the code has been written in C++, with the Newton iterative method from the Numerical Recipes as the nonlinear system solver. We have then made a Graphical User Interface (gui) with QtCreator. An illustration of the gui is given in Fig. 3. The default values are given in the gure. Since version 1 of the code, a loop on the values of Jg has been added. The results are given in a text le "res.txt" the structure of which is the following:
• A header
Jg,Jl,ulo,ugo,el,eg • The values of Jg , Jl , ul,O , ug,O , l and g separated by tabulations
0.75 1.11814 1.66135 2.2938 0.673032 0.326968
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Validation
As a test case we have taken the parameters of Fig. 7a in the article of Yoshinaga e t al. [1]. The results are compared in Fig. ??.
Figure 3: Test case for validation: Fig. 7a of [1] (left) and results of our code (right).
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conclusion
The code seems to work pretty well and may be easily extended to a threephase ow model. S. Balducci has done a parametric study using that code in order to see what are the in uence of the dierent geometrical parameters and to help us in the design the air-lift test loop:
• D = 40mm • L1 = 4.3m • L2 = 0.3m • α = 0.93m • ξE = 1.56 • xiI = 1 give a maximum value of Jl = 1.7m.s−1 for Jg = 1.75m.s−1 and an in-situ concentration of air of 40%. With a security coe cient of 2 on the singular losses, the maximum is Jl = 1.5m.s−1 for Jg = 1.2m.s−1 .
References [1] T. Yoshinaga and Y. Sato. Performance of an air-lift pump for conveying coarse particles. Int. J. Multiphase FLow, 22:223238, 1996. [2] S. Z. Kassab, H. A. Kandil, H. A. Warda, and W. H. Ahmed. Experimental and analytical investigations of airlift pumps operating in three-phase ow. Chem. Eng. J, 131:273281, 2007. [3] D. Chisholm and A. D. K. Laird. Two-phase ow in rough tubes. ASME, 80:276286, 1958.
Trans.
[4] S. L. Smith. Void fractions in two-phase ow: a correlation based upon an equal velocity head model. Proc. Inst. Mech. Engs, 184:647664, 1969. [5] H. Fujimoto, S. Ogawa, H. Takuda, and N. Hatta. Operation of a small air-lift pump for conveying solid particles. J. Energy Resources Technology, 125:1725, 2003. 7
