1、1CHAPTER 5Incompressible Flow in Pipes and Channels2content 5.1 SHEAR STRESS AND SKIN FRICTION IN PIPES 5.2 LAMINAR FLOW IN PIPES AND CHANNELS 5.3 TURBULENT FLOW IN PIPES AND CHANNELS 5.4 FRICTION FROM CHANGES IN VELOCITY OR DIRECTION 5.5 DESIGN OF PIPE SYSTEM* 35.1 SHEAR STRESS AND SKIN FRICTION IN
2、 PIPES 1. Shear-stress distribution 2. Relation between skin friction and wall shear 3. The friction factor 4. Relations between skin friction parameters 5. Flow in noncircular channels41. Shear-stress distributionConsider the steady flow of fluid of constant density in fully developed flow through
3、a horizontal pipe. 5 Apply momentum equation (4.42) between two faces of the disk.)(aabbVVmF(4.42)0)2()(22rdLdpprprF02rdLdp(5.1)rearranging6Subtracting Eq. (5.1) from Eq. (5.2) gives 02wwrdLdp(5.2)rrww(5.3)also02rdLdp(5.1)7A straight line with slop rww82. Relation between skin friction and wall shea
4、r Writing Bernoulli equation over a definite length L of the complete stream. L9fsbahppEq. (5.2)(5.2)02wwrLpbecome(5.5)LDLrhwwwfS42fshp02wwrdLdp1022422VDLVhwfsLDLrhwwwfS42(5.5)f113. The friction factor -Fanning friction factory 2222/VVfww(5.6) f4-Blasius or Darcy friction factor124. Relations betwee
5、n skin friction parameters The relation of common quantities used to measure skin friction in pipes. fshspwfPressure drop caused by friction loss阻力降, Pafsshp2422VDLfpLrhSwwfS(5.7)f4135. Flow in noncircular channelsequivalent diameter eqDhydraulic radius HeqrD4pHLSr Hr14Flow between parallel plates,
6、when the distance between them b is much smaller than the width of the plates bDeq2A annulus between two concentric pipes 44/4/22iooiioHDDDDDDr(5.15)A square duct with a width of side b bbbDeq4/42155.2 LAMINAR FLOW IN PIPES AND CHANNELS 1. Laminar flow of newtonian fluids 2. Hagen-Poiseuille equatio
7、n 3. Laminar flow of non-newtonian liquids 4. Laminar flow in an annulus 161. Laminar flow of newtonian fluids Newtonian fluid is flowing in a circular channel in laminar flow.17Velocity distribution Newtons law drdu/(5.13)Eliminating byrrwwTherefore rrdrduww(5.14)18Integration of Eq. (5.14) with th
8、e boundary condition u = 0, r=rw gives drrrdurrwwuw0222rrruwww(5.15)When r=02maxwwru(5.16)19The ratio of the local velocity to the maximum velocity 2max1wrruu(5.17)In laminar flow the velocity distribution with respect to the radius is a parabola. 2021Average velocity SudSSV 12wrSrdrdS2222rrruwww402
9、23wwrwwwrrdrrrrVw(5.18)225 . 0maxuV(5.19)The average velocity is precisely one-half the maximum velocity. 2maxwwru4wwrV 23Kinetic energy correction factor Momentum correction factor 0 . 234SVdSuS33 dSVuSS21242. Hagen-Poiseuille equationto eliminate w(5.7)4wwrV Using In Eq. 5.7swwfspLrh225(5.20)232DV
10、Lps- Hagen-Poiseuille equation Compare with 22422VDLVDLfphsfsRe1616VDfRe64(5.22)Therefore In laminar flow, friction factor is only influenced by Re.26 3. Laminar flow of non-newtonian liquids27 4. Laminar flow in an annulus Velocity distribution for the laminar flow of a newtonian fluid through an a
11、nnular space where = radius of outer wall of annulus = ratio = radius of inner wall of annulus022020ln)/1ln(114rrrrLrpuS(5.28) orkiroirr /28 For annular flow the Reynolds number isVDDio)(Re(5.29)afRe16(5.30)29305.3 TURBULENT FLOW IN PIPES AND CHANNELS 1. Velocity distribution for turbulent flow 2. U
12、niversal velocity distribution equations 3. Limitations of universal velocity distribution laws 4. Flow quantities for turbulent flow in smooth round pipes31 5. Relations between maximum velocity and average velocity 6. Effect of roughness 7. The friction factor chart 8. Reynolds numbers and frictio
13、n factor for non-newtonian fluids 9. Drag reduction in turbulent flow 10. Nonisothermal flow 11. Turbulent flow in noncircular channels 321. Velocity distribution for turbulent flow Flow in turbulent through a closed channel 33Viscous stresses 粘性应力 Viscous stress + Turbulent stress Turbulent stress
14、or Reynold stress湍流应力or 雷诺应力 viscous sublayer:buffer layer : turbulent core:34viscous sublayer:buffer layer : turbulent core:Velocity gradient large middle small3536 It is customary to express the velocity distribution in turbulent flow in terms of dimensionless parameters wfVu2*(5.31)Friction veloc
15、ity摩擦速度，m/swwuy/*Friction distance,摩擦距离，m.37*uuu (5.32)velocity quotient, dimensionless 无量纲速度*yyy wyyuy*(5.33) dimensionless distance, 无量纲距离 y = distance from wall of tubeyrrw(5.34)382. Universal velocity distribution equations通用速度分布方程通用速度分布方程39viscous sublayer wdydu(5.35)1dydu yu(5.36)5yfor the vis
16、cous sublayer: 40buffer layer : an empirical equation 05. 3ln00. 5yu(5.37)for the buffer zone:305y41turbulent core : 5 . 5ln5 . 2yu(5.38)for the turbulent core: y3042433. Limitations of universal velocity distribution laws 444. Flow quantities for turbulent flow in smooth round pipes45Average veloci
17、ty wrwurdrrV02275. 32/1*cufuV(5.46)2/75. 311maxfuV(5.47)46The Reynolds number-friction factor law for smooth tubes von Karman equation 75. 18Reln5 . 22/1ff(5.50)47The kinetic energy and momentum correction factors )9 .1515(78. 01fff91. 31(5.51)(5.52)48For a Reynolds number of 104, the friction facto
18、r for a smooth tube is 0.0079, is 1.084, and is 1.031. For example: For Re = 106 the values are f= 0.0029, =1.032, and = 1.011. For turbulent flow the error is usually very small ifandare assumed to be unity 495. Relations between maximum velocity and average velocity Experimentally measured values
19、of as a function of the Reynolds number are shown in Fig. 5.8, max/uV50516. Effect of roughness52k and is called the roughness parameter粗糙度 .k/D is defined as relative roughness相对粗糙度.53 Smooth pipe光滑管 Rough pipe粗糙管hydraulically smooth pipe: hydraulically rough pipe:Drawn copper and brass pipeOld, fo
20、ul, and corroded pipe54Fig. 5.10 give the roughness parameter of several material of new pipe. (p112)55The effect of roughness on the friction factorRoughness has no appreciable effect on the friction factor for laminar flow unless k is too large.56From dimensional analysis. f is a function of both
21、Re and the relative roughness k/D 577. The friction factor chartFrom dimensional analysisf is a function of both Re and the relative roughness k/D Friction factor chart is a log-log plot of f versus Re. 5859二、管内湍流的摩擦系数606162Laminar flowBuffer TurbulenceComplete turbulence （完全湍流区,阻力平方区）63Laminar flow
22、Re1616VDf64Turbulent flow: for hydraulically smooth pipeCoburn equation 2 . 0Re046. 0fBlasius equation 25. 0Re079. 0f32. 0Re125. 00014. 0fThis applies over Re from about 50,000 to 1 x 106. Applicable over Re from 3,000 to 3 x 106 65Complete turbulent flow:Turbulent flow: for rough pipeDkfRe,Dkf66 8. Reynolds numbers and friction factor for non-newtonian fluids for pseudoplastic fluids and laminar flow 21132nnnnnVDKf(5.56)
