流体力学第六章(Fluid Mechanics)

2nd – Year Fluid Mechanics, Faculty of Engineering and Computing, Curtin UniversityFLUID MECHANICS 230For Second-Year Chemical, Civil and Mechanical EngineeringFLUID MECHANICS LECTURE NOTESCHAPTER 6 VISCOUS FLOW IN PIPES6.1 IntroductionIn this chapter, we will consider viscous incompressible flow in pipes where the fluid is confined and bounded. Pipe systems are widely used in practice. Typical examples include drinking water distribution pipe systems, oil pipe lines etc. Figure 6-1 presents the schematic diagram of a typical pipe system.Figure 6-1 A typical pipe system [1]As shown in Figure 6-1, a pipe system may include 1. Individual straight pipes2. Pipe connectors (such as Tee-union, elbow connector etc.), for connecting pipes3. Flow rate control devices (such as valves) for adjusting the flow rate4. Inlet and outlet5. Pumps which add energy into the fluidwhere items 2, 3 and 4 are often called pipe components .6.2 Real pipe flowFor inviscid, incompressible, steady and irrotational flows, Bernoulli’s Equation appliesstreamline a along const V gz P ,22=++ρρwhich can also be written as 0)2(2=++ΔV gz P ρρ. (E6-1)Equation E6-1 indicates that for ideal fluid (0=μ), if there is no change in the sum of the fluid elevation and dynamic pressure, i.e.0)2()(2=Δ+ΔV gz ρρ, the overall pressure drop across a pipe system should be zero.However, for real fluid, energy loss hence pressure drop L P Δ across a pipe system is inevitable because of the shear force (friction) between the pipe and the fluid. This friction is a result of the viscous nature of real fluid (0≠μ). To deliver the fluid through a pipe system, pumps are often introduced into the pipe system in order to provide pressure rise work P Δ. Therefore, E6-1 is no longer applicable to real pipe flows and we need to consider both L P Δ and work P Δ, i.e.0)2(2=Δ−Δ+++ΔL work P P V gz P ρρ. (E6-2)The pressure drop L P Δ can be due to pressure drop in straight pipes (Item 1 in Figure 6-1), called major loss , and pressure drop in pipe components (Items 2, 3 and 4 in Figure 6-1), called minor loss . Therefore, we have or L major L L P P P min ,,Δ+Δ=Δ. (E6-3)Equation E6-2 shows that to deliver fluid through a pipe system, a pump needs to be properly selected to provide enough pressure rise to overcome the pressure drop due to friction in the pipe system, the changes in elevation and the dynamic pressure. It indicates that to design and operate a real pipe system, we need to have sufficient knowledge on:1. the relationship among pressure drop across a straight pipe (i.e. major loss ), the pipe properties and flow properties (This is discussed in Chapter 6)2. the relationship among pressure drop across pipe components (i.e. minor loss ), the properties of the pipe component and flow properties (This is discussed in Chapter 7)3. the energy gain by devices such as pumps (This is the topic of Chapter 10)6.3 Pioneer work on measuring pressure drop across a pipe 6.3.1 Pressure-drop test [2]Figure 6-2 illustrates is a simple experimental setup for measuring pressure drop across a pipe. Liquid flew from the tank (by elevation energy) to the pipe. There would be a long section where the flow was not uniform, before the fluid entering the test section to produce auniform flow. The test section has a length of x Δand the pressures at both ends were measured as P 1 and P 2.A flow-regulating valve was introduced to control the flow rate so that tests could be conducted to find the correlation between the pressure gradient across the test section and the fluid flow rate in pipe. Figure 6-3 presents the original experimental results of Osborne Reynolds in 1883 of one specific fluid and one specific pipe [3]. Extensive experiments showed that these observations are generic for pipe flow, regardless of the type of liquid and kind of pipe used in such experiments.Figure 6-2 Experimental setup for pressure-drop test [2].Figure 6-3 Measured pressure gradientx PΔΔ (i.e.xP P Δ−12) of a specific pipe as a function of volumetric flow rate Q of a specific fluid [2], originally from reference [3].Depending on the flow rate Q , Figure 6-3 shows that the measured pressure gradient of the test section may fall into one of the three regions:Region I : At a low flow rate, the pressure gradient is proportional to flow rate;Region II : At an intermediate flow rate, the flow behaviour seemed to be unpredictable so that the experimental data are scattered within the regions of the two curves which are extrapolated from those in Region I and III.Region III : At a high flow rate, the pressure gradient is proportional to the flow rate to the 1.8 (for smooth pipe) or 2.0 power (for rough pipe);The questions are: why are there three regions in Figure 6-3? How should we interpret Figure 6-3 and obtain enough knowledge on pipe flows to guide the design of pipe systems?6.3.2 Laminar, transitional and turbulent flow in pipe [1]In 1883, Osborne Reynolds [3] did the pioneer work to understand Figure 6-3. Flow in the three regions were visualised using a transparent pipe with a dye streak as a tracer. The flow patterns of the three regions in Figure 6-3 are shown in Figure 6-4 while the time dependence of fluid velocity at point A is shown in Figure 6-5.1. Laminar flow: the dye streak remains a steady line as it flows through the pipe. All the flow motion is in axial direction, there is no mixing perpendicular to the axis of the pipe. This is observed at low flow rates, i.e. Region I in Figure 6-3.2. Transitional flow: the dye streak fluctuates in time and space, and intermittent bursts of irregular behaviour appear along the streak. The flow motion is not solely in axial direction anymore. Some mixing perpendicular to the axis of the pipe may occur. This is observed at higher flow rates, i.e. Region II in Figure 6-3.3. Turbulent flow: the dye streak spreads across the entire pipe in a random fashion. The flow motion is chaotic in all directions, causing rapid, crosswise mixing. This is observed at high enough flow rates, i.e., Region III in Figure 6-3.It has been found that the Reynolds number, μρVD=Re , is a key dimensionless number thatdetermines the flow in pipe as laminar, transitional or turbulent flow. Reynolds number indicates the ratio between inertial force and viscous force exert on the fluid in pipe. At low Reynolds numbers, viscous force dominates and the flow motion is well-defined. The flow is laminar flow. At high Reynolds numbers, inertial force dominates so that flow becomes turbulent and chaotic. The flow becomes turbulent flow. With the intermittent Reynolds numbers, the flow is in transition from laminar to turbulent, called transitional flow.Figure 6-4 Laminar, transitional and turbulent flows in pipe. Figure is from Reference [1], thework was originally done by Osborne Reynolds in 1883.Figure 6-5 Time dependence of fluid velocity at point A for laminar, transitional and turbulentflows in a pipe; Figure is from reference [1].It should be pointed out that besides the flow rate (corresponding to the average velocity V ), the flow character is also determined by fluid density ρ, viscosity μ and the pipe diameter D (i.e., the fluid’s Reynolds number). For flow in a round pipe, we have Re < 2100 Laminar flow 2100 < Re < 4000 Transitional flow Re > 4000 Turbulent flowTherefore, the existence of the three flow regimes leads to the experimental observations of the three regions in Figure 6-3. The first region corresponds to laminar pipe flow (Re < 2100). The second region, where pressure drop is unpredictable, is due to transitional pipe flow (2100 <Re < 4000). The third region corresponds to the turbulent pipe flow (Re > 4000). We will discuss these three regions further in Section 6.7 of this chapter.6.3.3 Entrance region flow and fully developed pipe flowWhen fluid enters a pipe, there exists a section, where the pipe flow is not fully developed, i.e., a uniform flow can only be produced after this section. The flow in this section is called entrance region flow . This is why during the pressure drop test in Figure 6-2, the test was done across the section where a uniform flow was produced. Such a uniform flow is called fully developed pipe flow .The length of the entrance flow region of a pipe flow, L e , is generally short and a function of Reynolds number.Re 06.0=D L efor laminar flow (E6-4a)6/1(Re)4.4=DL efor turbulent flow (E6-4b)The pressure drop in the entrance region is higher than the fully developed region due to extra energy loss when fluid flows into the entrance. It is generally measured by experiments and will be discussed in Chapter 7. This chapter focuses on fully developed pipe flows.6.4 Fully developed laminar pipe flowAt very low flow rates (fluid velocities), viscous effects dominate, the flow is laminar flow. In the fully develop region, the viscous forces are in equilibrium with pressure forces so that the velocity profile and pressure gradient remain constant along the pipe.6.4.1 Force balanceIt is simple to do force analysis as conditions in the fully developed laminar pipe flow are constant. Figure 6-6a illustrates the force balance system. For a fluid element at any given time, both pressure forces and shear force in action. Since there is no acceleration, these forces in the x-direction must sum to zero. At a radius r , we have Pressure force: F P = P(πr 2) – (P+δP)( πr 2)Shear force:Fs = (2πr δx)(shear stress at r)Because there is no acceleration, according to Newton’s 2nd motion law F = ma , F = 0, i.e., the two forces are balanced. We can equate the two forces and take the limit 0→x δ, yieldingShear stress at r, τ(r): dxdPr r 2)(−=τ (E6-5)It should be noted that since the force balance is generic, E6-5 is applicable to all types of fluids, both Newtonian and non-Newtonian flows. Equation 6-5 indicates that 1. The pressure gradient across a pipe is a constant.2. As shown in Figure 6-6b, the shear stress at the pipe centreline (r = 0) is zero and the shear stress reaches maximum at the pipe wall (r = R ). The shear stress is linear between the centreline and the wall;3. The wall shear stress is dxdPR w 2−=τ. The wall stress is proportional to pressure gradient across the pipe.(a) (b)Figure 6-6 (a) Force balance system in pipe flow; (b) Distribution of shear stress in pipe6.4.2 Velocity profileFor Newtonian flow, assume at a radius r , the flow velocity is V(r), Newton’s law of viscosity applies. We can write the shear stress asNewton’s law of viscosity:drr dV r )()(μτ−=. Substituting into E6-5, we have)(2)(dxdPr dr r dV −−=μ (r)wheredxdPis a constant. Therefore, we can integrate the equation and apply the boundary conditions V(R) = 0, producing the following velocity profile:⎥⎥⎦⎤⎢⎢⎣⎡⎟⎠⎞⎜⎝⎛−−=221)(4)(R r dx dp R r V μ (E6-6)Equation E6-6 indicates that1. The velocity at the pipe wall is zero, i.e., the fluid is non-slip at the wall;2. The fluid velocity V c along the pipe centreline (where r = 0), V c , reaches maximum.We have: (42max dxdp R V V c −==μ; 3. The velocity profile is parabolic, as it can be expressed in terms of V c ,⎥⎥⎦⎤⎢⎢⎣⎡⎟⎠⎞⎜⎝⎛−=21)(R r V r V c , as shown in Figure 6-7.Figure 6-7 Velocity profile of laminar pipe flow Figure 6-8 Flow through a pipe6.4.3 Flow rateThe flow is axisymmetric about the pipe centreline. As shown in Figure 6-8, through a ring at radius r when its thickness dr is thin enough at the cross-section (with an area of rdr dA π2=) of a pipe, the velocity can be considered as constant. Therefore, the flow rate Q through the pipe can be calculated asrdrR r V rdr R r dx dp R dAr V Q Rc Rpipe ∫∫∫⎥⎥⎦⎤⎢⎢⎣⎡⎟⎠⎞⎜⎝⎛−=⎥⎥⎦⎤⎢⎢⎣⎡⎟⎠⎞⎜⎝⎛−−==020221221)(4)(ππμi.e. (D is the diameter of the pipe, D = 2R )⎟⎠⎞⎜⎝⎛−=⎟⎠⎞⎜⎝⎛−==dx dp D dx dp R V R Q c μπμππ12882442The average velocity V can be calculated as⎟⎠⎞⎜⎝⎛−=⎟⎠⎞⎜⎝⎛−=⎟⎠⎞⎜⎝⎛−===dx dp D dx dp R Rdx dp R V A Q V c μμπμπ328822224If the pressure drop across a pipe with a length of L is ΔP , i.e., LPdx dp Δ=−, we have LPD Q Δ=μπ1284 (E6-7)LPD V Δ=μ322 (E6-8) Note, E6-7 and E6-8 are ONLY suitable for laminar flow!6.5 Fully developed turbulent pipe flow 6.5.1 Randomness of turbulent pipe flowIn turbulent flow (Re > 4000), the fluid experiences random, chaotic motion, including strong eddy transport on a macro scale, compared with the molecular motion in laminar flow. As a result, there is much more dissipation of energy as fluid molecules experience velocity changes in all directions. We can consider the two velocity components, i.e. j i v u V +=. Figure 6-9 shows an example of the axial component, u(t),measured at a given location.Figure 6-9 Axial component of velocity at a given location in turbulent pipe flowAlthough the flow is chaotic, the velocity can be described in terms of a mean value (denoted with an overbar) on which the fluctuations (denoted with a prime) are superimposed.'u u u += (E6-9a) where∫+⋅=Tt t dt u T u 0010'100'=⋅=∫+T t t dt u T u .Similarly, in the radial direction of the pipe flow, we have 'v v v += (E6-9b) where0=v 0'100'=⋅=∫+Tt t dt v T v .6.5.2 Turbulent shear stressIn turbulent flow, in addition to the motion of fluid particles discussed in laminar flow, there is also motion across the flow direction. The parcels of fluid experience relatively larger shear stress, resulting in momentum transfer. Figure 6-10 illustrates the momentum transport in a control volume in turbulent pipe flow.Figure 6-10 Momentum transport in turbulent pipe flowIn the x-direction we have:0int =⎟⎟⎠⎞⎜⎜⎝⎛+⎟⎟⎠⎞⎜⎜⎝⎛+CV the o transfered momentum of rate Net CV the in increase momentum of Rate forces applidofSum1. We need to consider: Pressure force: F P = P(πr 2) – (P+δP)( πr 2) Viscous shear force: Fs = – ( 2πr δx τ) Sum of applied forces: – τ 2πr δx – πr 2δP2. Rate of momentum increase in the CV: 03. Net rate of momentum transferred into the CV: x r u u v δπρ2)(''+Therefore, we get02)(2''2=++−−x r u u v P r x r δπρδπδπτ Collecting the terms, rearranging then taking the limit 0→x δ,2'''rdx dP v u u v ⎟⎠⎞⎜⎝⎛−=−−ρρτApplying time-averaging gives2'''rdx P d v u u v ⎟⎟⎠⎞⎜⎜⎝⎛−=−−ρρτ As 0'=v , it reduces to2)(''rdx P d v u ⎟⎟⎠⎞⎜⎜⎝⎛−=−+ρτ (E6-10)where drud μτ=. From E6-5, if we consider a total shear stress at r , total τ, which include fluid viscous shear stress and the shear stress due to eddy transport, we have2rdx P d total ⎟⎟⎠⎞⎜⎜⎝⎛−=τ . (E6-11) A comparison between E6-10 and E6-11 leads to()turb lam total u v drud ττρμτ+=′′−+=. (E6-12)Therefore, in turbulent flow, the turbulent shear stress consists of two components:• drud lam μττ==, as a result of fluid viscous effect;• )(''v u turb ρτ−=, termed as Reynolds stress, as result of eddy transport.Figure 6-11 Distribution of shear stress in turbulent pipe flowFor turbulent pipe flow, the distribution of the two components of turbulent shear stress is shown in Figure 6-11. Governed by E6-11, the total shear stress is linear between the centreline and the wall. However, the contribution of viscous shear stress and Reynolds stress are significantly different from the pipe wall to the pipe centreline. As shown in Figure 6-11, near the pipe wall, viscous shear stress dominates while at the pipe centreline, Reynolds stress dominates and viscous shear stress is negligible. Therefore, the structure of turbulent pipe flow may be divided into three parts, as shown in Figure 6-12. 1. Viscous sublayer , where turb lam ττ>>.In the viscous sublayer, the viscous effects are dominant, and the flow is laminar. The mean velocity, u , is zero at the wall and increases rapidly with r . 2. Buffer layer or Overlap layer , where turb lam ττ≈Since both turbulent and laminar stresses are acting, this layer demonstrates properties of both laminar and viscous flow.3. Turbulent core or Outer layer , where turb lam ττ<<Viscous stresses are negligible, resulting in almost constant velocity and little shear. This zone occupies 80-90% of the cross-sectional area of the pipe.Figure 6-12 Structure of turbulent pipe flow6.5.3 Velocity profile of turbulent pipe flowFor laminar flow, the velocity profile can be analytically derived as E6-6. However, it is impossible to do so for a turbulent flow so that we can only obtain semi-empirical results. As the importance of viscous shear stress and Reynolds stress varies in different layers (see Figures 6-11 and 6-12), it is expected that the velocity profile varies in various layers.Viscous sublayerWe define a friction velocity, ρτw u =*. Please note that *u is not a real physical quantitybut *u has the same unit of velocity. In the viscous sublayer, experimental data show that**u y uu ν= (E6-13) where ρμν/= and r R y −=. Equation E6-13 is only valid within the viscous sublayer, which is taken as5*≤νyu . (E6-14)Therefore, Equation E6-14 is often used to estimate the thickness of viscous sublayer.In the buffer layer and turbulent coreExperimental data in these two layers can be described by the following empirical equation.4.5ln5.2**+⎟⎠⎞⎜⎝⎛=u y u u ν (E6-15) For the buffer layer: 505*≤<νyu (E6-16)For the turbulent core:50*>νyu (E6-17)The turbulent flow profile is shown in Figure 6-13. Experimental data are plotted along with the predictions using E6-13 and E6-15.Figure 6-13 Turbulent flow profile, modified from [1]6.5.4 Velocity profile of turbulent pipe flow – the power lawFor turbulent pipe flow, the viscous sublayer is generally thin and the flow structure is dominated by the turbulent core. In 1932, Nikuradse proposed the following empirical power law velocity profile :nR r u u1max 1⎟⎠⎞⎜⎝⎛−=,(E6-18) where n is a function of Reynolds number, and u max is the velocity on the pipe centreline. For many practical fluid, n = 7 is a reasonable approximation. Certainly, E6-18 CANNOT be used near the wall (i.e., viscous sublayer).Figure 6-14 Correlation between n and Reynolds number [1]*u uFigure 6-15 Correlation between n and Reynolds number, modified from [1]Figure 6-14 shows that the value of n increases with Reynolds number, indicating the stronger the turbulence, the higher the n value. Figure 6-15 shows the velocity profiles at various n values. It can be seen that at strong turbulence (with a high n value), the pipe flow has nearly uniform velocity profile, for example the profile at n = 10.6.6 Friction factor 6.6.1 DefinitionLet’s introduce a dimensionless number, friction factor f . For a pipe with a length of L and a diameter of D , friction factor (Darcy friction factor) is defined as2/)/(2V L D P f ρΔ= (E6-19)To understand the meaning of the friction factor in E6-19, we recall that E6-5 applies for all flows therefore the wall stress can be written as)/(412L D P L P R w Δ=Δ=τ (E6-20)Substituting into E6-19, we have the definition of friction factor f as2/42V f wρτ= (E6-21) Fundamentally, the friction factor f is the ratio of wall shear stress and inertial force of the flow. E6-19 and E6-21 is generic for laminar or turbulent flows. With the introduction of friction factor f , rewriting E6-19, the pressure drop can be calculated as22V D L f P ρ=Δ (E6-22)or in head formgV D L f h major L 22,= (E6-22a)E6-22 (or E6-22a) is very important for the engineering design and operation of pipe systems. It shows that the pressure drop of a fully-developed pipe flow is a function of three parameters, i.e., the friction factor f , pipe geometry (L/D) and dynamic pressure 2/2V ρ (or velocity head g V 2/2). Therefore, pipe flow problems are converted to find f for the flows under different conditions.6.6.2 Friction factor for laminar pipe flowFor laminar pipe flow, according to E6-7, we know 2/32D VL P μ=Δ This is for laminar flow only! If we divided both sides by the dynamic pressure (22V ρ) andDL, we haveRe 64642//322/)/(222=⎟⎟⎠⎞⎜⎜⎝⎛==ΔVD L DV D VL V L D P ρμρμρ. Therefore, for laminar flow, the friction factor is proportional to Re1, i.e. Re64=f . (E6-23)6.6.3 Friction factor for turbulent flowEquations E6-19, E6-21 and E6-22 are applicable to turbulent pipe flow. However, for turbulent flow, it is impossible to analytically derive the friction factor f , which can ONLY be obtained from experimental data. In addition, most pipes, except glass tubing, have rough surfaces. The pipe surface roughness is quantified by a dimensionless number, relative pipe roughness (D /ε), where ε is pipe roughness and D is pipe diameter. For laminar pipe flow, the flow is dominated by viscous effects hence surface roughness is not a consideration.However, for turbulent flow, the surface roughness may protrude beyond the laminar sub-layer and affect the flow to a certain degree. Therefore, the friction factor f can be generally written as a function of Reynolds number Re and pipe relative roughness Dε:)(Re,Df εΦ= (E6-24)For smooth pipesFor turbulent flow in smooth pipes with Re < 105, Blasius [4] calculated the friction factor for a large variety of pipe-flow experiments and found that the friction factor f is indeed only a function of Reynolds number,4/1Re316.0=f (E6-25)For rough pipesDepending on the depth of surface protrusions, one would expect that it is possible to determine the following three regimes,1. If the surface protrusions are within the viscous sublayer, the pipe can be treated as hydraulically smooth so that the flow is dependent only on the Reynolds number, i.e. E6-24 can be simplified to (Re)Φ=f , i.e., E6-25.2. If the surface protrusions extend into just the buffer layer, the flow is dependent on both the Reynolds number and the relative roughness. Details of E6-24 needs to be obtained from experimental investigations.3. For larger protrusions into the turbulent core, the effect of surface roughness is sostrong that the flow is dependent only on pipe roughness, i.e. )(Df εΦ=.6.6.4 Moody chartAs discussed in the previous two subsections, the friction factor for turbulent flow is a function of Reynolds number and pipe roughness, described in the generic equation E6-24. In order to find the detailed equation of E6-24, large quantities of experiments have been carried out using pipes of various relative surface roughness under different flow conditions.Figure 6-16 shows the functional dependence of f on Re and D ε and is called the Moody chart [5]. The Moody Chart covers all three flow regions: laminar, transitional and completely turbulent. Table 6-1 lists the typical roughness from various new, clean pipe surfaces. To use the Moody chart, we need to1. Calculate the relative roughness (D ε) and the Reynolds number (μρVD =Re ) for the flow under consideration.2. Follow the line for the particular roughness value until we locate the Reynolds number, and read the friction factor from the left axis.3. Take extra care when reading the Reynolds number axis, which goes up to 108. In particular, note the pattern of values along the axis, e.g., 105 2 4 6 8 106, etc. If a value 8.5 falls into the region, the exponent needs to be read TO THE LEFT (i.e. 105), not to the right. This is a significant source of error when determining friction factors, and hence leading to possible faulty pipe designs.Table 6-1 Equivalent roughness for new pipes [5]Equivalent roughness, εPipeFeet MillimetresGlass, plastic Smooth Smooth Drawn tubing (e.g. copper) 0.000005 0.0015 Commercial steel or wrought iron 0.00015 0.045 Galvanised iron 0.0005 0.15 Cast iron 0.00085 0.26 Concrete 0.001 – 0.01 0.3 – 3 Riveted steel 0.003 – 0.03 0.9 – 9.0It should be noted that the Moody chart is to help us understand the fundamentals and practice hand solutions. Working with the Moody chart directly can sometimes be tedious and error-prone. Most modern engineers prefer to use computer programs to solve flow problems. In this case, besides the analytical solutions from laminar flow, empirical correlations are developed to simulate partially regions of the Moody chart with good accuracy, includingFor laminar flow: Re 64=f (E6-26a) For turbulent flow in smooth pipes, Re < 105 4/1Re 316.0=f (E6-26b)For the entire turbulent region:⎟⎟⎠⎞⎜⎜⎝⎛+−=f D f Re 51.27.3/log 0.21ε (E6-26c)6.7 Revisit of Figure 6-3 Region I: Laminar pipe flowThe test section length is Δx . For laminar pipe flow, according to E6-22, we have the pressure gradient as212VD f x P ρ⎟⎠⎞⎜⎝⎛=ΔΔ (E6-27) The fluid velocity V can be calculated from flow rate Q and pipe cross-section area A (which is determined by pipe diameter D ) ,24DQA Q V π== (E6-28) According to E6-23, for laminar pipe flow (with the substitution of V using E6-28),Q D VD f ρπμρμ1664Re 64=== (E6-29) Substituting E6-28 and E6-29 into E6-27, yieldingQ D D Q D Q D VD f x P 2222128)4(2111621πμπρρπμρ==⋅⎟⎠⎞⎜⎝⎛=ΔΔ In experiments leading to Figure 6-3, 2128Dπμis a constant (assumed as c 1), therefore, we have Q c xP1=ΔΔ (E6-30)In Region I, E6-30 therefore holds. The pressured gradient should be proportional to the flow rate. This was indeed validated by the experimental data in Figure 6-3.Figure 6-16 The Moody Chart, adapted from [1] which the original data were from [5]Prepared by Dr Hongwei Wu, since 2006 Chapter 6 – Page 17 of 19Region III: Turbulent flowE6-27 and E6-28 are still applicable. We need to use the friction factor in turbulent flow.1. For rough pipesFrom the Moody chart, one can see that when Reynolds number is high enough, the flow is wholly turbulent flow. The friction factor f is only a function of relative pipe roughness, independent of Reynolds number. The same pipe was used in experiment therefore the relative pipe roughness should be a constant. Consequently, the friction factor should be constant (denoted as c ),c Df =Φ=)(ε(E6-31)Substituting E2-28 and E2-31 into E6-27, we have2522228)4(21121Q D f D Q D f VD f x P πρπρρ==⋅⎟⎠⎞⎜⎝⎛=ΔΔ. In the experiments leading to Figure 6-3, 528Df πρis a constant (denoted as c 2), we have22Q c xP=ΔΔ (E6-30) Therefore the pressure gradient is proportional to the square of the flow rate.2. For smooth pipesBlasius equation E6-25 (also E6-26b) can be used to calculate the friction factor f forturbulent flow in smooth pipes. Substituting μρVD=Re and E6-28 into E6-25, we have()()()25.04/14/14/14/14/14316.0316.0Re 316.0−⎟⎟⎠⎞⎜⎜⎝⎛===Q D V D f ρμπρμ (E6-32)Substituting E2-28 and E2-32 into E6-27, we have75.1224/12)4(214316.021Q DD D VD f x P πρρμπρ⎟⎟⎠⎞⎜⎜⎝⎛=⋅⎟⎠⎞⎜⎝⎛=ΔΔ. In the experiments leading to Figure 6-3, 224/1)4(214316.0D D D πρρμπ⎟⎟⎠⎞⎜⎜⎝⎛ is a constant (denotedas c 3), we have75.13Q c xP=ΔΔ (E6-33)Hence the pressure gradient is proportional to the flow rate to ~1.8 power.6.8 Noncircular ConduitsMany practical engineering situations use conduits of non-circular cross section. Certainly, the details of the flows in noncircular conduits depend on the exact cross-sectional shape. However, the discussion on flow in round pipe can be applied in noncircular conduits with slight modifications.。