1、1.Pressure drop and loss due to friction When the fluid is in steady-state laminar flow in a pipe,for a Newtonian fluid,the HagenPoiseuille equation is obtained.This can be written as 22132DLVpppf1.4-20 One of the uses of HagenPoiseuille equation is in the experimental measurement of the viscosity o
2、f a fluid by measuring the pressure drop and volumetric flow rate through a tube of known length and diameter.2.Relation between skin friction and wall shearFor horizontal pipe and constant cross section,Z2-Z1=0,and the two kinetic-energy terms can be canceled.Eq 1.3-25 becomesfhPP21The relationship
3、 between skin friction and pressure drop can be written.Then equation becomesThis is the mechanical-energy loss due to skin friction and is part of the hf term for losses in the mechanical-energy-balance equation(1.3-25).ffhp1.4-5 This term(p1-p2)f for skin-friction loss is different from the(p1-p2)
4、term,owing to velocity head or potential head changes.3.The friction factor3.The friction factor A common parameter used in fluid flow is the Fanning friction factor,f,which is defined as shear stress w at the surface divided by the product of density and velocity head.2/2Vfw1.4-7 Equation(1.4-1)can
5、 be written for entire cross section of the tube by taking=w and r=R.Equation(1.4-1)then becomes:02RLpw1.4-2 Rearranging equation(1.4-2)givesLpRw2Substituting from equation above into equation(1.4-7)gives222242/2/2/VpLDVLpRVffw Rearranging the equation and let L=L,then the equation becomes242dLfpf24
6、2dLfpf242dLfpf242dLfpf242VDLfpf1.4-9 and242VDLfpf1.4-10thus22VDLphff=4fdefining as a friction coefficientThe equation above and Eq(1.4-10)is called the Fanning equation,and the friction factor f is called the Fanning friction factor.The equation(1.4-10)is the equation usually used to calculate skin
7、friction loss in straight pipe.For laminar flow only,combining Eqs.(1.4-20)and(1.4-10).givesRe16f(1.4-221.4-22 )Re64It is not possible to predict theoretically the Fanning friction factor f for turbulent flow as was done for laminar flow.1.4.3 Turbulent Flow in 1.4.3 Turbulent Flow in Pipes and Chan
8、nelsPipes and Channels Because of the dependence of important flow parameters on the velocity distribution,theoretical and experimental study has been devoted to determining the velocity distribution in turbulent flow.Although the problem has not been completely solved,useful relationships are avail
9、able.For turbulent flow the friction factor must be determined empirically,and it not only depends upon the Reynolds number but also on surface roughness of the pipe.In laminar flow the roughness has essentially no effect.A large number of experimental data on friction factors for smooth pipe and co
10、arse pipes have been obtained and correlated.For design purposes,to predict the friction factor f and,hence,the frictional pressure drop for round pipe,the friction-factor chart can be used.It is a loglog plot of f versus Re.Completely turbulent flow zoneLaminar flow zone For the region with a Reyno
11、lds number below 2100,the line is the same as Eq.(1.4-22).For a Reynolds number above 4000 for turbulent flow,the lowest line in figure represents the friction-factor line for smooth pipes and tubes.The other lines,for higher friction factors,represent lines for different relative roughness factors,
12、/D,where D is the inside pipe diameter and is a roughness parameter.Re16fRe16fFor turbulent flow the lowest line represents the friction factor for smooth tubes.2.0Re046.0fThis applies over a range of Reynolds number from 50000 to 1106 Another equation,applicable over a range of Reynolds numbers fro
13、m 3000 to 3106,is 32.0Re125.00014.0fThe other curved lines in the turbulent range represent the friction coefficients for various types of pipe,each of which is characterized by a different value of k.1.4.4 Friction from 1.4.4 Friction from Changes in Velocity Changes in Velocity or Directionor Dire
14、ction Whenever the velocity of a fluid is changed,either in direction or magnitude,friction is generated in addition to the skin friction resulting from flow through a straight pipe.Friction Losses in Expansion,Contraction,and Pipe Fittings Skin-friction losses in flow through straight pipe are calc
15、ulated by using the Fanning friction factor.However,if the velocity of the fluid is changed in direction or magnitude,additional friction losses occur.This results from additional turbulence which develops because of vortices and other factors.1.Sudden enlargement losses If the cross section of a pi
16、pe changes suddenly,it results in additional losses due to eddies formed by the jet expanding in the enlarged section.The friction loss hfe from sudden expansion of cross section is proportional to the velocity head of the fluid in the small conduit.In this case the calculation of hfe can be madethe
17、oretically and satisfactory result obtained.The calculation utilizes the continuity equation,steady-flow momentum-balance equation,and Bernoulli equation.The equation was derived as follows The momentum equation between the station 1 and 2 gives12221VVmSpp）（12Since Z=0,mechanical energy balance equa
18、tion may be written for this situation asfehVVpp2212221 Elimination of p1-p2,since 22VSmFrom continuity equation,V2=V1(S1/S2)and equation can be written2221VVhfeso2121221VSShfe2.10-15 where Ke is the expansion loss coefficient,V1 is the upstream velocity in the smaller area.Experimentally measured v
19、alues Ke as a function of the ratio S1/S2 are shown in figure.the friction loss hfe from sudden expansion of cross section is proportional to the velocity head of the fluid in the small conduit.2211SSKfe(1.4-29)2.Friction loss from sudden contraction of cross section The friction loss from sudden co
20、ntraction is proportional to the velocity head in the smaller conduit and can be calculated by the equation2214.0221222VKVSShcc1.4-31 V is the average velocity in the smaller,or downstream section.Experimentally,for laminar flow,Kc0.1,and the contraction loss hc is negligible.For turbulent flow,Kc i
21、s given by the empirical equation3.Effect of fittings and valves Fittings and valves disturb the normal flow lines and cause friction.In short lines with many fittings,the friction loss from the fittings may be greater than that from the straight pipe.22VKhfff1.4-32 Factor Kf is found by experiment
22、and differs for each type of connection.Fitting KfGlobe valve,wide open 10.0Angle valve,wide open 5.0Gate valve,wide open 0.2 half open 5.6Return bend 2.2Tee 1.8Elbow,90 0.9 45 0.4 Sometimes,the losses in fittings is indicated as an equivalent pipe length.The data given in Table are presented as Le/
23、D,where Le is the equivalent length of straight pipe having the same frictional loss as the fitting.The K values can be converted to Le/D values.The Le values for the fittings are simply added to the length of the straight pipe to get the total length of equivalent straight pipe.4.Frictional losses
24、in mechanical-energy-balance equationForm-friction losses are incorporated in the hf term of mechanical energy balance equation.They are combined with the skin-friction losses of the straight pipe to give the total friction loss.242VKKKdlfhfcexf(1.4-34)Friction Loss in Noncircular Conduits The frict
25、ion in long straight channels of constant noncircular cross section may be estimated by use of the equation used for circular pipe if the diameter in the Re and in the definition of the friction factor is taken as an“equivalent diameter”.The friction factor for noncircular cross section can be estim
26、ated by substituting the“equivalent diameter de”for the diameter d in Reynolds number or other relevant equation.pHLSr The hydraulic radius is denoted by rH and in turn is defined as the ratio of the cross-sectional area of the channel to the wetted perimeter of the channel:Thus,for the special case
27、 of a circular tube,the hydraulic radius is 44/2dddLSrpHThe equivalent diameter is de=4rH An important special case is the annulus between two concentric pipes.Here the hydraulic radius is 44/4/22ioioioHddddddr For annular flow the Reynolds number isuddio）（ReFor laminar flow in the annulus the frict
28、ion factor is related to the Reynolds number by the equationf=C/Re Entrance Section of a Pipe If the velocity at the entrance region of a tube is the same at all positions,a certain length of tube is necessary for the velocity profile to be fully established.This length for the establishment of full
29、y developed flow is called the entry length.As the fluid progresses down the tube,the thickness of the boundary layers increases until finally they meet at the center of the pipe and the parabolic velocity profile is fully established.The approximate entry length Le of a pipe of diameter D for a ful
30、ly developed velocity profile to be formed in laminar flow isRe00575.0DLeRe0575.0DLe2.10-26 For turbulent flow,no relation is available to predict the entry length.As an approximation,the entry length is nearly independent of the Reynolds number and is fully developed after 50 diameters downstream.l
31、 Hydraulically smooth If a rough pipe is smoothed,the friction factor is reduced.When further smoothing brings about no further reduction in friction factor for given Reynolds number,the tube is said to be hydraulically smooth.Drawn copper and brass pipe may be considered hydraulically smooth.Proble
32、m 1 For laminar flow,the average velocity in a pipe is maximum velocity,for turbulent flow average velocity is maximum velocity.For laminar flow in a pipe,the flow rate is constant,when di increases,the friction coefficient will ,and friction loss will .When the pipe installation changes from horizo
33、ntal to vertical position and the velocity keeps unchanged,the frictional loss will be .A）.increased;B）.decreased;C）.the same;D）.uncertain In laminar flow,the velocity distribution with respect to radius is()at the centerline of the pipe Roughness has()on the friction factor for laminar flow unless
34、k is very large.For laminar flow,the friction coefficient is proportional to a()For turbulent flow,the friction coefficient is a function of both()For completely turbulent flow,the friction coefficient is a function of()Problem 2 1250cm3/s of water is to be pumped through a steel pipe,25mm diameter
35、and 30m long,to tank 12m higher than its reservoir.Calculate approximately the power required.What power motor(in kW)would you provide?The roughness of a steel pipe will be taken as 0.045mm;the efficiency of the pump is 50%Problem 4（1）the expressions of ,hfab,and ,hfcd,respectively.（2）the relationsh
36、ip between readings R1and R2 in the U tube.Liquid flows through the tube,shown in Fig.the length of ab equals cd，and the diameter and roughness of tube are same.Find:gpabgpcdProblem 5（1）关闭阀门，求管路中的流量。（2）阀门局部阻力系数k=30，求管路中的流量；两水池的水位差H=24m，l1=l3=l4=100m，d1=d2=d4=100 mm，d3=200mm，沿程阻力系数1=2=4=0.025，3=0.02，除阀门外，忽略其他局部阻力。The waterline difference between two reservior is 24m,
