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This 5 page Class Notes was uploaded by Willa Raynor on Saturday October 3, 2015. The Class Notes belongs to ME311 at California State Polytechnic University taught by Staff in Fall. Since its upload, it has received 53 views. For similar materials see /class/218334/me311-california-state-polytechnic-university in Mechanical Engineering at California State Polytechnic University.
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Date Created: 10/03/15
Viscous Pipe Flow Part 1 Introduction Fluid ow through pipes is very important in our everyday lives Faucets HVAC systems Refrigerator coils Oil pipelines All of these systems involve pressurized uid owing through closed pipes When a pipe is closed off from the atmosphere the main driving force pushing the uid through the pipe likely is a pressure gradient although gravity can be important if the pipe is not horizontal P2 P1 1 a 1712192 Dimensionless parameters often are used to help characterize ows For pipe ow the Reynolds number Re is the most important dimensionless parameter Re p V D inertial forces m v u viscous forces Area where 0 uid density v average uid velocity D pipe diameter y dynamic viscosity also called viscosity The Reynolds number determines the type of ow Re gt 4000 turbulent 2100 lt Re lt 4000 transitional Re lt 2100 laminar Turbulent ows are characterized by unpredictable uid motion Laminar ows appear smooth and unchanging Transitional ows have turbulent outbursts ye d I g1 Re gt 4000 Turbulent S11 gt2iuUltRelt4000 Transitional Re lt 2100 Laminar 11202011 The velocity in the axial x direction as a function of time at a point is very different for laminar and turbulent ows WWTurbulent x M xlv Transitional LIA Laminar t For laminar ows the velocity is only in the axial x direction and it does not change in time For turbulent ows the velocity is largest in the axial direction but it can vary unpredictably in all three dimensions eddies can form When uid enters a pipe there is an entrance region Where the velocity profile changes in the axial direction Due to the noslip boundary condition shear forces at the wall cause momentum to be stolen from the ow producing a growing boundary layer Fully developed ow Entrance region flow fl Inviscid core Boundary layer 1 r f L x I 1 lt2 i 3 l a l 6 5 4 396 7 r5 x5 7 A Fully developed Developing f flow flow 7 After the entrance region the velocity profile no longer varies in the axial direction and the ow is fully developed The length of the entrance region 6 5 006 Re laminar 1 44Re16 turbulent Entrance region Fully developed flow flow quot33 D L lnviscid core Boundary layer l EZ 793 I i v V Q Q 6 5 4 lt r6 7 x5 x5 1 x4 Fully developed Developing flow flow 7 If a change to the piping system occurs bends contractions expansions valves etc the ow will no longer be fully developed After the change the flow will try to become fully developed again Fully developed flow Entrance region flow Inviscid core Boundary 39ayer T E1Egz quot x 1 lt2 i 3 a 6 5 4 r6 7 x5 x5 7 x4 Fully developed Developing flow flow 8 11202011 For fully developed laminar flows the velocity varies only in the radial r direction not the aXial X direction For fully developed turbulent flows the largest velocity is in the aXial X direction but unpredictable fluctuations occur in all Parabolic directions 39aminar It 11 ll 11 ll The velocity profiles for fully developed laminar and turbulent flows look very Turbulent different Al 11 It 11 ll 11 Al Uniform Pressure gradients in pipes If we neglect viscous forces in a steady incompressible flow we can use the Bernoulli equation to analyze a typical straight pipe 2 2 pV1g21P2p 2g22 l 2 13139quot P1 P2 The pressure is constant along the entire section If a pressure gradient eXisted the uid would be accelerated In real flows shear stress friction at the walls due to the noslip boundary condition steal kinetic energy from the flow This kinetic energy is dissipated as thermal energy and no longer can be used to propel the uid through the pipe Pressure forces must exactly balance the friction forces to keep the fluid flowing at a constant velocity Pinlet griCtiOn Poutlet Since pressure acts on both sides of a pipe section the inlet pressure must be greater than the outlet pressure Let s predict how pressure varies at points in a circular pipe using the Navier Stokes equations in cylindrical coordinates equation 6128 in the book What does your instinct te quotmm In the r direction 8v dvr v dvr V797 p r vx K t 8r r 80 X li dv jv iazgav 8212 Br r rar if 24 r2 r 4 0 r gsinO Oz a P pgsine 8r 11202011 VVXrX vevr 0 In the 0direction Xm ax t 8r r 80 r X 1aP r80 9 rdr r 2 r2 02 r2 0 X2 0 r gcosO 8P 0 r cosG 80 Pg Oz g f pgsine 02 3 g rpgcos9 Integrate with respect to r Integrate with respect to 0 P pgrsin0f1X9 P 0grsin0f2Xr The only way for both of these expressions to be true is if the f is a function ofjust X f1 X 0 2 f2 X r a fX P 0 g rsin 9 fX or P 10 g y fX The pressure is hydrostatically distributed y direction at a given cross section and may vary along the pipe Xdirection 1 4 7VXrX 1291r 0 In the Xdirection or W a a 0 vr vX t Br r 89 X a P liravxji2amp2amp BX pflt rdr Br r2 92 X2 9P 3 ran 8X r Br Br vx is a function of r only so the right side of the equation can be a function of r not X or 0 only What about the left side of the equation lg 7vxrX 1291r 0 For the left side of the equation aP alt pg yfltx aux d X 8X 8X 3P 2 i 8X r Br Br The left side of the equation is potentially a function of X while the right side of the equation is potentially a function of r The only way this is possible is if both terms are equal to a constant f1 X 2 f2 r constant 11202011 yVXrAlt vevr 0 In the Xdirection agzggrranjaq BX rar Br lap JCIBX PClxC2 The NayierStokes equations predict for fully developed incompressible flows P changes linearly with distance along the 1113 This theoretical result is confirmed by experiments which show that C1 is negative 17 yvxri v9vr0 L AvV39 PClxC2 The pressure also decreases linearly along tilted pipes as well The pressure gradient drops linearly with distance along a pipe for real flows in the fully developed region 39 p lt Entrance flow gt Fully developed flow 9p9x 2 constant Entrance pressure I drop A1 x1 0 x2 CF 13 x In the entrance region there is a larger pressure drop due to enhanced yiscous dissipation of useful energy We will discuss the losses that occur in entrance regions later 11202011
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