FUNDAMENTALS OF FLUID MECHANICS Chapter 8 Pipe Flow 1 MAIN TOPICS GlChiifPiFlGeneral Characteristics of Pipe Flow Fully Developed Laminar Flow Fully Developed Turbulent Flow Dimensional Analysis of Pipe Flow Pipe Flow Examples Pipe Flowrate Measurement 2 Introduction Flows completely bounded by solid surfaces are called INTERNAL FLOWS which include flows through pipes (Round cross section), ducts (NOT Round cross section ), nozzles, diffusers, sudden contractions and expansions, valves, and fittings. The basic principles involved are independent of the cross -sectional shape, although the details of the flow may be dependent on it. The flow regg(ime (laminar or turbulent )of internal flows is ppyrimarily a function of the Reynolds number ((-->inertial>inertial force/viscous force). Laminar flow: Can be solved analyyytically. Turbulent flow: Rely heavily on semi-semi-empiricalempirical theories and experimental data. 3 General Characteristics of Pipe Fl ow Laminar vs. Turbulent Entrance Region vs. Fully Developed Flow 4 Pipe System A pipe system include the pipes themselves (perhap s of more than one diameter), the various fittings, the flowrate control devices valves) , and the pumps or turbines. 5 Pipe Flow vs . Open Channel Flow Pipe flow: Flows completely filling the pipe. (a) The ppgressure gradient alonggpp the pipe is main driving force. Open channel flow: Flows without completely filling the pipe.(b). (b) The gravity alone is the driving force. 6 Laminar or Turbulent Flow 1/2 The flow of a fluid in a pipe may be Laminar ? Or Turbulent ? Osborne Reynolds, a British scientist and mathematician, was the first to distinguish the difference between these classification of flow by using a simple apparatus as shown. 7 Laminar or Turbulent Flow 2/2 For “small enough flowrate” the dye streak will remain as a wellwell--defineddefined line as it flows along, with only slight blurring due to mol ecul ar diff usi on of th e d ye i nt o th e surroundi ng wat er. For a somewhat larger “intermediate flowrate” the dye fluctuates i n ti me and space, and i ntermi ttent b ursts of i rregul ar behavior appear along the streak. F“For “lhfltlarge enough flowrate”h” the d ye streak al most immediately become blurred and spreads across the entire pipe in a random fashion. 8 Time Dependence of Fluid Velocity at a Point 9 Indication of Laminar or Turbulent Flow The term flowrat e shldbhould be repldbRldlaced by Reynolds number, R e VL / ,where V is the average velocity in the pipe, and L is the characteristic dimension of a flow. L is usually D (diameter) in a pipe flow. -->> a measure of inertial force to the viscous force. IiIt is not onlhflidlily the fluid velocity thdhat determ ines th e ch aracter of fh the flow – its density, viscosity, and the pipe size are of equal importance. For general engineering purpose, the flow in a round pipe Laminar R e 2100 Transitional Turbulent R e＞4000 10 Entrance Region and Fully Developed Flow 1/5 Any fluid flowing in a pipe had to enter the pipe at some location. The region of flow near where the fluid enters the pipe is termed the entrance (entryyg) region or developpgfing flow region.. 11 Entrance Region and Fully Developed Flow 2/5 The fluid typically enters the pipe with a nearly uniform velocity profile at section (1). As the fluid moves through the pipe, viscous effects cause it to stick to the pppipe wall (the no slippy boundary condition).. 12 Entrance Region and Fully Developed Flow 3/5 A boundary layer in which viscous effects are important is produced along the pipe wall such that the initial velocity profile changes with distance along the pipe, x , until the fluid reaches the end of the entrance length, section (2), beyond which the velocity profile does not vary with x.. The boundary layer has grown in thickness to completely fill the pipe. 13 Entrance Region and Fully Developed Flow 4/5 Viscous effects are of considerable importance within the boundary layer. Outside the boundary layer, the viscous effects are negligible. The shappype of the velocity profile in the pppipe dep ends on whether the flow is laminar or turbulent, as does the length of the entrance region, .. For laminar flow For turbulent flow 1/ 6 0.06R 4.4R D e D e Dimensionless entrance length 14 Entrance Region and Fully Developed Flow 5/5 Once the fluid reaches the end of the entrance region, section (2), the flow is simpler to describe because the velocity is a function of only the distance from the pipe centerline, r, and independent of x.. The flow between (2) and (3) is termed fully developed.. 15 Pressure Distribution along Pipe In the entrance region of a pipe , the fluid accelerates or decelerates as it flows. There is a balance between pressure, viscous, and inertia (acceleration) force.. p p constant 0 x u 0 x There is a balance between pressure and The magn itud e of th e viscous force. pressure gradient is larger The magnitude of the than that in the fully ppgressure gradient is constant. developed region. 16 Fully Developed Laminar Flow There are numerous ways to derive important results pertaining to fully developed laminar flow: From F=ma applied directly to a fluid element. From the NavierNavier--StokesStokes equations of motion From dimensional analysis methods 17 From F=ma 1/8 Considering a fully developed axisymmetric laminar flow in a long, straight, constant diameter section of a pipe. The fluid element is a circular cylinder of fluid of length l and radius r centered on the axis of a horizontal pppipe of diameter D. 18 From F=ma 2/8 Because the velocity is not uniform across the pipe, the initially flat end of the cylinder of fluid at time t become distorted at time t+t when the fluid element has moved to its new location along the pipe. If the flow is fully developed and steady, the distortion on each end of the fluid element is the same, and no part of the fluid experiences any acceleration as it flows. V u Steady 0 Fully developed V V u i 0 t x 19 From F=ma 3/8 Apply the Newton’s second Law to the cylinder of fluid Fx ma x The force (pressure & friction) balance 2 2 p 2 rx p1r p1 pr rx2r 0 p p1 p2 r Basic balance in forces needed to drive each fluid particle alhiihlong the pipe with constant vel liocity Not function of r Independent of r 2 r p 2 ? Cr w rx rx rx r D BCB.C. r=0 rx=0 Not function of r r=D/2 rx= w < 0 20 From F=ma 4/8 The pressure d rop and wall sh ear stress are rel ated b y 2 r p 2 w rx 4 w rx p D r D Valid for both laminar and turbulent flow.. Laminar du rx dr 21 From F=ma 5/8 du du p p 2 rx Since r dr dr 2 r p p LiLaminar du rdr u r 2 C 1 2 4 With the boundary conditions: u=0 at r=D/2 pD 2 C 1 2 2 2 16 pD 2r 2r u(r) 1 VC 1 16 D D Velocity distribution 2 D r 4 u(r) w 1 w p 4 R D 22 From F=ma 6/8 The shear stress distribution du rp rx 2 2 dr 2 pD2 2r 2r u(r) 1 V 1 Volume flowrate C 16 D D 2 R R V Q u dA u(r)2rdr ..... C 0 A 2 D 4 p Q Poiseuille’s Law 128 Valid for Laminar flow only 23 From F=ma 7/8 Average velocity 2 Q Q pD D4p Vaverage 2 Q A R 32 128 Point of maximum velocity 2 2 pD2 2r 2r du u(r) 1 VC 1 0 at r=0 16 D D dr R 2 p u u U 2V max 4 average 24 From F=ma 8/8 Making adjustment to account for nonhorizontal pipes >0 if the flow is uphill p p sin <0 if the fl ow i s d ownhill 2 2 2 p gsin 2 rx pr p p r rx 2r gr sin 0 r 4 2 p sin D p sin D Specific weight Q Vaverage , g 128 32 25 Example 8 .2 Laminar Pipe Flow An oil with a viscosity of μ= 0.40 NN··s/ms/m2 and density ρ= 900 kg/m3 flows in a pipe of diameter D= 0.20m . -5 (a) What pressure drop , p 1-p2, is needed to produce a flowrate of Q =2 .0 ×10 3 m /s if the pipe is horizontal with x1=0 and x2=10 m? (b) How steep a hill, θ,must the pipe be on if the oil is to flow through the pipe at the same rate as in part (a), but with p1=p2? (c) For the conditions of part (b), if p1=200 kPa, what is the pressure at section, x3=5 m, where x is measured along the pipe? 26 Example 8. 2 Solution1/2 R e VD / 2.87 2100 Q V 0.0637 m / s A The flow is laminar flow 128 Q p p p ... 20.4kPa 1 2 D 4 If the pipe is on the hill of angle θ with Δp=0 128 Q sin ... 13.34 gD4 27 Example 8. 2 Solution2/2 With p1=p2 thlhe length hfh of the pi pe, , dihfldoes not appear in the flowrate equation Δp=0 for all p1 p2 p3 200 kPa 28 From the Navier-Stokes Equations 1/3 General motion of an incompressible Newtonian fluid is governed by the continuity equation and the momentum equation Mass conservation Navier-Stokes Equation in a cylindrical coordinate Acceleration 29 From the Navier-Stokes Equations 2/3 Simplify the NavierNavier--StokesStokes equation axial component: z The fl ow i s governed b y a b al ance of pressure, wei ght , and vi scous forces in the flow direction.