THEORETICAL ANALYSIS OF USING AIRFLOW TO PURGE RESIDUAL WATER IN AN INCLINED PIPE

Applied Mathemalics and MechanicsPublished by Shanghai Liniverity .( English Edition, Vol 23, No 6. Jun 2002)Shanghai, ChinaArticle ID: 0253 4827<2002)06-0694-09THEORETICAL ANALYSIS OF USING AIRFLOW TO PURGERESIDUAL WATER IN AN INCLINED PIPE”SHEN Fang (沈芳)， YAN Zong-yi (严宗毅)'，ZHAO Yao-hua (赵耀华)2，Kiyoshi Hori'1.. Departmcnt of Mechanics and Engineering Science, PekingUniversity, Bejng 100871. P R China;2. Institute of Engineering Thernophysics, Chinese Academy ofSciences, Bejing 100080. P R China;3. Shirayuri College. Tokyo, Japan)(Communicaled by WU Wang-yi)Abstract: A refined theoretical analysis for using the spiral airflow and carial airflow' topurge residual water in an inclined pipe was presented. The computations reveal that, inmost cases. the spiral flow can purge the re sidual water in the inclined pipe indeed while theaxrial flowr may induce back flow of the water , just as predicted in the experiments presentedby Horii and Zhao et ul. In addition. the effects of warious initial conditions on water purg.ing were studied in derail for both the spiral and axial flow cases .Key words: spiral flow; axial flow; watcr purging; two-phase flow; pipe flowCLC number: 035° .1Document code: AIntroductionPurging residual liquid in a U-shaped pipeline is challenge for chemical and transportationindustry'! . Many techniques have been employed to purge out residual water'l ，but none haveheen completely sutisfactory. One possible imethod for doing this has been to blow a greatquantity of compressed air through lhe pipeline. When a shor line is plugged with watercolumns. it is possible to purge out the residual waler . However. when the length of pipeline islong and U-shaped sections are included ，it is inadequate to remove it completely.In the early 1990s， Hori put forward the conception of spiral flow'3.4. Different from theaxial flow (airflow in a pipeline which is parallel to the axis). the airflow in spiral flow movesnot only along the axis, but also around the circumference of the pipeline . So when spiral airflowas employed. the surface of residual water was sheared off and blown toward the inclincdsection of the pipe to form a water film, and was purged out completely in spiral way . Recently .Received date: 2000-11 !5; Revised date: 2001-11-28Foundation item: the Nationa! High Capability Calculation Foundation of China (98 1006)Biography: SHEN Fang (1977- ), Doctor (E- mail: shcnfang @ sina . com)中国煤化工MYHCNMHGUsing Airlow lo Purge lesidual Water in Pipe695Hori. Zhao， Tomita and Shimo have made experiments to show that the spiral airflow couldremove the residual waler completely in an inc lined pipeline :Fig. 1 shows their experimentmodel . The pipeline consisted of 79mm-diameter pipes over a distance of 129m including the U-shaped pipe section of 2.6m. Experiments were performed to test removal of 0.07m' -water. Thpressure of both device (ubes arose to 200 kPa-gage and developed an average pipe flow velocityof 20m/s. In order to make comparison，airflow was generated by either the spiral flow device orthe axial flow device and experiments were conducted varying the inclination angle of the pipe .Their experimcntal results showed that the spiral flow could efficiently purge the residual water outof the inclined pipe while in the axial turbulent flow the water often flows back after it was drivenforward for a distance .0.18 m2.6mFig. 1 The experimental apparalus of Rer.[5]Referring 10 the experimental result of Ref.[5], we present a simplified theoretical modelherein. The principle of mass and momentum conservation is applied to the thin water layer fromLagrangian viewpoint and, thus ，goverming equations similar to the momentum integral equationin the boundary layer theory are derived. Variable thickness of the layer and a third orderpolynomial approximation for the velocity profile are considered.1 The Varying Thickness Model for Axial Flow1.1 Simplification of geometryFig.2 shows a circular pipe of radius ro， which is inclined at angle Q with respect to thehorizontal direction. The residual water inside it corresponds to angle θ with respect to the centerof the circle. We usual consider the thickness of the residual water to be far smaller than theradius of pipeline. and we may treal it as a problem of water flowing u over an inclined plane ofwidth rqθ. The equivalent thickness of the waterx,layer ho can be derived . From the equality of thecross-section areas of the water in the twoAgeometries, we have ro0ho = rH(0 - sin0)/2,and thuswater -AA sectionho =号(1- sing).(1)For a small segment of air with cross- sectionFig.2 Geonetry of waler purging (ul o[area πrand thickness dx in the circular pipe, thean inclined pipemoment conservation in tbe x direction requires出dxπ活= τro0dx + rτ。(2π- 0)rodx.(2)wherep is the pressure. τ。is air' s shear stress on the pipe wall and τi is air's shear stress on its中国煤化工MYHCNMHG596SHEN Fang，YAN Zong-yi, ZHAO Yao-hua et alinterface with water. The gravity force is neglected here for air. Then the pressure gradient isgiven by .(3)=-r品-。(2-号).1.2 Calculation of the shear stress on the interfaceRef.[6] gave empirical formulae for the stress on the interface on experiments. τg is givenbyτp = λxρ。号, λg = 0.485 Re:0:28, Re, = PpU,(2ro)/μg，(4)whereρp，μp, U。are the density, dynamic viscosity, and mean velocity of air, respectively .Air' s shear stress τ; on the interface with water is computed byt=之λiPq吸.where the friction coefficient 入, is dependent on the flow pattern. The flow is Ripply when thewater velocity l; on the interface is greater than the wave speed C , otherwise the flow is Pebbly .λ. =入g[1 +6.1 x 10~*(X0.8 Re:)°6Re[7] for Ripple flow (U, > G)，(6)lx[! + 4xl2]for Pebble flow (U; < Gx)，where Rer = pw/μ, ρ and μ are the density and dynamic viscosity of water, w is the volumetricflow rate of water per unit width. X is called Martinelli parameterf6l, defined byA:pU}X2=λxρxζ '(7)where U; = w/h is the mean velocity of water in the thin layer, λi is the friction coefficient ofwater flowing over the plane, given byj96Rei'for laminar flow (Rer < 300),.λ::l0.485 Re70.28for turbulentr flow ( Re; > 300).(8)The wave speed Cx is given byC, =一U:+7Up/12+gh+ ok*h/p,whereg = 9.81 m/s2 is the gravity acceleration, σ = 0.073 N/m is the surface tension cofficientof the air-water interface, k = 2π/λ is the wave number(λ = 0.09m is the wave length'S! ).Substituting Eqs.(6), (7) and (8) into Eq. (5) gives[Tg + cqh-1:2w0.6for Pebble flow (U; < Cx),for Ripple flow (U; > Cx),where c: and (2 are constants independent ofh, U;, r{7.1.3 Mass and momentum conservationConsider a water element of thickness h and small length I from Lagrangian viewpoint(Fig.3). The volumetric flow rate of water per unit width is defined as w =}。udy, massconservation of water requires that中国煤化工MYHCNMHGUising Arflow 1o Purge Residual Water in P'ipe697du = dhu,+[" dud, =0.(10)di=duJo dThe momentum conservation for the water element requiresuldy)= (r.-t.)1- phlsina - h出(11)where τ is water' s shear stress on the wall. It should be noted that both length l and thuckness hof the water element change with the axial position x during is movement (for a water particle.its traveled distancex is dependent on the time I, of course). It iswaterevident that dl/di = l0u/ax. under the assumption of steady statethat the quantities al a certain axial position x are independent of l.Afier some laborious reduction including suslitution of Eq.(3)，we以havetu=-(1+0+0- 5与(2-员)+ ghsina.πFo1p~pro(12) Fig.3 The waler element1.4 The nonlinear velocity profileWe assume that the velocity profile may be represented by a third order polynomial. and it iscxpressed as.口= ao+a1η+ (427“+ a3η'The coefficients ao. a. a2 and ag can be solved by applying the boundary conditions below(They are not listed here for the sake of saving space. The details of them are listed in Ref.[7]).)u.7 = l:u=ti,μg"=T;yη=0:u=0，0”2=,+pgsina=- t,+ pg sina,scu = U,(aη+ a2y + an').(13)Subtiuting Eq .(13) into the definition of w gives=udr = u,h+军小(2- !)- ghsine]n,. (14a)48rπμrol 48μrp48μgulja!. 3μU. t;τ。= μ2h( 2i。-1)+台[r.(2-品- pg sina](14b)1.5 Governing equations and initial conditionsdx/dl = l; will give the distance x traveled by the water particle on the interface.Substituting Eqs.(13) and (14) into Eqs.(10) and (1I)，and differentiating Eq. (9) withrespect I, we can completc a closed set of four ordinary differential equations aboutx, h, τ, and中国煤化工MYHCNMHG698SHEN Fang, YAN Zong-yi, ZHAO Yao-hua el alas follows:[dx|dhτr;I出=16dτ，=- 0.52c1h-1.52w091(for Ripple flow),(15)}dld(for Pebble flow)di|。-品[(uitbry,h2 +b2.h+0.则+ dr(boh2 +bosh2}!5hldl 8In the above bi , b... b, are constants indcpendent of h, U;,r!"At time1 = 0. we getxo = 0, and from Eq.(1), ho = 0.5ro(1 - sin0/0): we selectlio =7.5m/s to coincide with Ref. [5]. The initial shear stress τ;o on the interface isdetermined by(16)(10o= ((rioho+号明)+ (0。n + b.h)r.o .In our calculations，the bisection method is used to numerically solve Eq.(16) for T;n.2 The Varying Thickness Model for Spiral FlowAll the derivations are in parallel to those for axial flow. The only difference lies in thegeometry. For spiral flow the water is spread over the whole circumference rather than constrictedto the angle θ. To render the results comparable between the axial flow and the spiral flow, weassume that the water layers in the two flows have the same initial cros8-section area. Thus theinitial thickness ho of the axial flow should satisfty the following equation[0.5r子(0 - sin0)] =2πhoro. and then we have h。= (ro/4π)(0 - sin0), which is much smaller than the initialthickness ho of the axial flow. For the spiral flow, the volumetric flow rate of water per unitwidth should be defined as u =| u(1 - y/ro)dy and the momentim should be| ul(ro -r)dyv. where y is the coordinate perpendicular t0 the wall and starting from the wall. Keepingthese differences in mind , one may finally obtain the following governing equations for the spiralflow:dx; = li;，(17a)dhτ;(3ro - 2h)h(3ro - 2h)1(17b) .(ro- 1)20+d:' U,(ro- h)+d3 h(ro-石'1- 0.52c2h-152w0411 dh(U > G),(17c).d中国煤化工MYHCNMHGLsing Airflow to Purge Residual Water in Pipe699Hi=- 1.2c1h22u0.6 dh(1i< (a),(17d)d[dh[a。 s.h2(3ro -2h)+ dov, + dyr;h(ro- h)+dgh2 +di二一 (dgh-0.4h2)ldi[u4- (r。- h)dsh' -0.81,b1ls -;↓d,h}(3ro -2h)]},(17e)where d. d2. ”，dg are constants independent of h, U，τ,!"The determination of the initial conditions of the spiral flow also can be given just like theaxial flow.3 Results and DiscussionIn our examples，the following parameters are taken: ρ = 1 000kg/m', μ = 0.001 Pars,ro = 39.5mm, P: = 1.2kg/m', μg= 1.8x 10- Pa's， U。= 20 m/s, the swirl velocity forthe spiral flow V. = 0.25 m/s'sThe ordinary differential Eqs. (15) and (17) with the initial conditions can be solvednumerically through the fourth-order Runge-Kutta method. To discuss the effects of theinclination angle of the pipe and the CrOSs section area of the residual water on the water purging，we leta = 0，a =30°, a =60° and 0 = 70°，0 =50°，0 = 118.115° separately, and getvarious initial conditions .The results for the variation ofx, U; and h with i are shown in Fig.5a and Fig .5b, and thedevelopment of the water velocity profile with time t are shown in Fig .4a, Fig .4b and Fig 6a LoFig .9b.From these figures, we can draw the following conclusions :1) When the inclined angle of the pipe is zero, both kinds of airflow can purge the residualwater out, but the spiral flow spends much shorter time than the axial flow to reach the steadystate. In the steady stale , the velocity of the water film all became positive, so the residual watercan flow out of the pipe at last ( Please compare Fig . 4a with Fig .4b) .waler vekoclly prolile:waler velocily prolile(asial 10m1 Uo=7.5 m/s(spiral 1ow)Uo=7.5 m/s1515厂- 0 -t=010-←1=1.2只o-t=1.6-0-t=0.15-←t=1.8- +t=0.2).2亡0.610.20.6 1.0y/hFig.4a Axial flow(0 = 70 °。Fig .4b Spiral flow(θ =70。.u =0°)a =00)中国煤化工MYHCNMHG700SHEN F'ang, YAN 7ong-yi, ZHAO Yao-hua et al2) The spiral aitow can purge the residual water in inclined pipe out completely when theamount of water is not to0o much. The water layer thickness h reduces as l increases (i.e. xincreases). and, a1 last, the water velocity profile can become all positive and then the water filmflow out at this stcady velocity profile (Fig .b to Fig . 8b) . But the axial airflow can remove thewater only for cases with small inclined angle (e.g.，a =30°) and thin water film (e.g. θ =50*) (Fig.7a). In most cases , back flow appears on the water bottom for the axial flow, and thewaler layer thickness h increases as l increases (i,e, x increases) (Fig.5a to Fig.7a). This iscoincident with the experimental result of Ref.[5] which showed a back flow afer the residualwater was driven u for a distance by lhe axial airlow. It should be noted that, as x increases ，the water thickness h in the axial flow may increases 10 such degree that the back flow of thebotom water may lead lo the surface insablity. Hence the water surface may be broken intodrops and then they flow down, so the residual water can' I flow out. In this case, the movementof water can'↑be described by our continuous model. This prevented from our calcultion of thelong time behavior for the axial flow cases .axial lom日= 70spiral low:0 = 70-a- U,a=30°,Uo=7.5 m/s .a=30, Ua=7.5 m/s0F厂0.0090.006S0.003000..5Fig.5u Axial flow(θ = 70。，Fig.5b Spiral flow(0 = 70。，a =300)a =30°)watcer velocrty profile:waler velocity profile:axidl flov} Uo=7.5 m/s .(spiral f1ow)Ug=7.5 m/s20F20-o- -t=0---1=005办-1=0.05-o -t=1.2+-t=0.1-0-1=1.6o -1=0.15- +-i=0.2- 100.20.6 1.00.6 T.0y/hFig.6a Axial 1low(0 = 70 °，Fig.6b Spiral f1ow(θ =70。.中国煤化工MYHCNMHGLIsing Airlow to Purge Residual Water in Pipe7013) When the amount of water is small and the inclined angle is small, both kinds of fnowcan remove the residual water (compare Fig.8a with Fig 8b), but the time that the axial flowspends to purge the water out completely is 10 limes as the spiral flow spends. This suggests thatusing the spiral flow is much more eficient than using the axial flow.4) In the case of the residual water with very thick film, if the given spiral flow can'tremove the water, largetr velocity can often be used to purge the water out. For example. al theinitial arflow velocity ofli,o = 7.5 m/sa spiral nlow could not remove the residual water( 0 =18.115, Fig .9a), bul at U,o = 12m/s, the water was sucssully removed (Fig 9b). While .for the axial flow, our computation reveals that one can't remove lhe residual water by increasingthe velocity of airflow (Figures were not given for the sake of space) .water velocity profile:waler velocity profile(axlal 1low) Uo=7.5 m/s(spiral f0ow)Uj=7.5 m/s3020- + -1=0.6---i=0; 1010-0-1=0.05-。-t=1.6-0-1=0.1-←1=1.8-0-1=0.15~ 10- 10+t=0.2~ 20L- 20L0.6T.00.20.6 T.0y/hFig.7a Axial flow(θ = 70。,Fig.7b Spiral flow(0 =70。.a =600)a =60°)waler vclocity protilc.nater velocity profile:(avial 1ow) Uo=7.5 m/s(spiral fIow)Uj=7.5 m/s15-0-1=0- 。t=0.6- -!=0.02-t=0,8-力1=0.04-0-1=1.0-o_t=0.06士-t=1.2-一t=0.120.6 T.oFig.8a Axial flow(θ =50。,Fig.8b Spiral tlow(θ = 50 °.a =300)a =30°)5) From the figures. we can easily find oul hat, it is more difcult to remove the residualwater when (he inclined angle a of the pipe and the thickness of the water gets larger. Thesefindings are the same as the experimental results in Ref.[5].In summary . our theoretial computation reveal that the spiral flow can purgc the residual in中国煤化工MYHCNMHG702SHFN Fang，YAN Zong-yi, ZHAO Yao-hua et alwiler \clocily prolilew ater velocity prolile:(spiral 1om1 Uo=7.5 m/s(sprad 1ow)Uo= 12 m/g1510-0 -1=01+ -8=0.4-o -t=0.8- +1=4-←1=1.8- 100.6 1.00.20.6T.0y/hFig .9a Spiral flow(θ = 118.115°，Fig.9b Spiral flow(θ = 118.115。a =300)a =30°)the inclined pipe eficiently, while the axial flow can't in most cases, just as predicted in theexperiment5l. These results are of significance in guiding the design of water purging frompipeline.As we stated above . the back flow on the surface of the axial flow may induce great changesin the water surface shape and thus the steady state assumption as we adopted herein will no .longer be valid. In order to calculate such complicate cases ，unsteady flowing model should beconsidered in the future research.References:[ 1 ] Schweinstein A M. Change of flow condition in U- Shaped conduits[J]. Proc Congr Int AssOCHydraul Res，1987 ,22(3):280 - 281.! 2. Underwood M P, Kendal C. Vacum technology for pipeline and system drying[A.. In: TheProc Int Pipeline Technol Exhib Conf[C].1984.12 :209 - 226 .”3] Hori K. Using spiral flow for optical cord passing[ J]. Mechanical Engineering , 1990, 112(8):68- 69.4° Hori K. Matsumae Y，Ohsumi K, el al. Novel optical fiber intallation by use of spiral airlow[小Journal of Fluids Engineerings , 1992 ,114(3):375 - 378.{ 5] Hori K, Zhao Y H. Tomita Y, et al. High performance spiral air- flow apparatus for purging resid-ual waler in a pipeline[A]. In: D P Telionis Ed. ASME Fluids Engineering Division SummerMeeting[C]. Vancouver; ASME, FEDSM 97-3035,1997,1-6.[6 ] Fukano T. Liquid films flowing concurrently with air in horizontal duct[J]. Trans .JSME Series B.1985 .51(462) :494 - S02.l 7 ] SHEN Fang. Theoretica! analysis of using airlow to purge residual water in pipe[ D]. Thesis forBachelor Degrce . 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