Chap.6 Flow in pipes Laminar flow Turbulent flow

Transcription

1 Chap.6 Flow in pipes In this chapter, howeer, a method o expressing the loss using an aerage low elocity is stated. Studies will be made on how to express losses caused by a change in the cross sectional area o a pipe, a pipe bend and a ale, in addition to the rictional loss o a pipe. Consider a case where luid runs rom a tank into a pipe whose entrance section is ully rounded. At the entrance, the elocity distribution is roughly uniorm while the pressure head is lower by V /g. As shown in below Figure,the section rom the entrance to just where the boundary layer deelops to the tube centre is called the inlet or entrance region, whose length is called the inlet or entrance length. For steady low at a known low rate, these regions exhibit the ollowing: Laminar low:a local elocity constant with time, but which aries spatially due to iscous shear and geometry. Turbulent low: A local elocity which has a constant mean alue but also has a statistically random luctuating component due to turbulence in the low. Typical plots o elocity time histories or laminar low, turbulent low, and the region o transition between the two are shown below.

2 Principal parameter used to speciy the type o low regime is the Reynolds number : V - characteristic low elocity D - characteristic low dimension μ- dynamic iscosity υ- kinematic iscosity We can now deine the critical or transition Reynolds number Re cr Re cr is the Reynolds number below which the low is laminar, aboe which the low is turbulent While transition can occur oer a range o Re, we will use the ollowing or internal pipe or duct low: Typical criteria or the length o the entrance region are gien as ollows: Le = length o the entrance region.the wall shear is constant, nd the pressure drops linearly with x,or either laminar or turbulent low.all these details are shown in the below Figure Laminar low: computation by Boussinesq experiment L = 0.065Red by Nikuradse L = O.06Red computation by Asao, Iwanami and Mori Turbulent low: L = 0.693ReI4d computation by Latzko

3 L = (5-40)d experiment by Nikuradse Deeloping pressure changes in the entrance o a duct low Velocity distribution o Laminar Flow in pipe: In the case o axial symmetry, when cylindrical coordinates are used, the momentum equation become as ollowing : () -- () --- For the case o a parallel low like this, the Naier-Stokes equation is extremely simple as ollows:. As the elocity is only u since = 0, it is suicient to use only the upper. As this low is steady, u does not change with time, so u/ t = As there is no body orce, ρx = As this low is uniorm, u does not change with position, so ul x = 0 and u/ x =0 5. Since = 0, the equation simply expresses the hydrostatic pressure ariation and has no inluence in the x direction. So, equation becomes : Integrating According to the boundary conditions, since the elocity at r = 0 must be inite c = 0 and c is determined when u = 0 at r = ro:

4 Laminar low in a circular pipe From this equation, it is clear that the elocity distribution orms a paraboloid o reolution with u max at r = 0 : The olumetric low rate passing pipe Q becomes : From this equation, the mean elocity is : The shear stress due to the iscosity is : (Since duldr < 0, T is negatie, i.e. letward.) Thus :

5 Putting the pressure drop in length L as p, the ollowing equation is obtained : ( Hagen-Poiseuille ormula ) Using this equation, the iscosity o liquid can be obtained by measuring the pressure drop p. Velocity distribution between parallel plates: Let us study the low o a iscous luid between two parallel plates as shown in below Figure, where the low has just passed the inlet length. The momentum equations in x and y directions as in the ollowing : ( ) ( ) Under the same conditions as in the preious section, the upper equation () becomes : Consider the balance o orces acting on the respectie aces o an assumed small olume dx dy (o unit width) in a luid.

6 Since there is no change o momentum between the two aces, the ollowing equation is obtained: thereore By integrating the aboe equation twice about y, the ollowing equation is obtained: (3) Using u = 0 as the boundary condition at y = 0 and h, c and c are ound as ollows: It is clear that the elocity distribution now orms a parabola. At y = h/, du/dy = 0, so u becomes u max : The olumetric low rate Q becomes : From this equation, the mean elocity is : (4) The shearing stress z due to iscosity becomes : Putting L as the length o plate in the low direction and p as the pressure dierence, and integrating in the x direction, the ollowing relation is obtained: Substituting this equation into eqn (4) gies :

7 As shown in the below Figure, in the case where the upper plate moes in the x direction at constant speed U or -U, rom the boundary conditions o u = 0 at y = 0 and u = U at y = h, c and c in eqn (3) can be determined. Thus : and Couette- Poiseuille low Velocity distribution o turbulent Flow For two-dimensional low, the elocity is expressed as ollows: where u and are the timewise mean elocities and u ' and ' are the luctuating elocities. Now, consider the low at elocity u in the x direction as the low between two lat plates as shown in the below Figure, so u = u + u ' but = '. The shearing stress τ o a turbulent low is : τ = laminar low shear stress τ t = turbulent shearing where numerous rotating eddies mix with each other. stress Now, let us examine the turbulent shearing stress only. The luid which passes in unit time in the y direction through da parallel to the x axis is ρ ' da. Since this

8 luid is at relatie elocity u ', the momentum is p' da u '. By the moement o this luid, the upper luid increases its momentum per unit area by ρ u ' ' in the positie direction o x per unit time.thereore, a shearing stress deelops on ace da. It is ound that the shearing stress due to the turbulent low is proportional to ρ u ' '. Reynolds.Thus Below Figure shows the shearing stress in turbulent low between parallel lat plates. Expressing the Reynolds stress as ollows as in the case o laminar low produces the ollowing as the shearing stress in turbulent low: This t is called the turbulent kinematic iscosity. V t is not the alue o a physical property dependent on the temperature or such, but a quantity luctuating according to the low condition. Prandtl assumed the ollowing equation in which, or rotating small parcels o luid o turbulent low (eddies) traeling aerage length, the eddies assimilate the character o other eddies by collisions with them:

9 Prandtl called this I the mixing length. According to the results o turbulence measurements or shearing low, the distributions o u' and ' are as shown in the Figure, where u ' ' has a large probability () Assuming ז to be the shearing stress acting on the wall, then so ar as this section is concerned: = ( riction o elocity) and Putting u = u δ wheneer y=y δ gies () where R δ is a Reynolds number. Next, since turbulent low dominates in the neighborhood o the wall beyond the iscous sublayer, assume ז o ז= t, and integrate eqn (): Using the relation ū = u δ when y =δ o, Using the relation in eqn (),

10 I ū/ν, is plotted against log 0 (ν,y/ν), the alue A can be obtained, A = This equation is considered applicable only in the neighbourhood o the wall rom the iewpoint o its deriation. In additional, Prandtl separately deried through experiment the ollowing equation o an exponential unction as the elocity distribution o a turbulent low in a circular pipe as shown in beow Figure : n changes according to Re, and is 7 when Re = * 0 5. Since many cases are generally or lows in this neighbourhood, the equation where n = 7 is requently used. Losses By pipe Friction Let us study the low in the region where the elocity distribution is ully deeloped ater passing through the inlet region as shown below. I a luid is lowing in the round pipe o diameter d at the aerage low elocity, let the pressures at two points distance L apart be p and p respectiely. The relationship between the elocity u and the loss head h = ( p - p ) /pg For the laminar low, the loss head h is proportional to the low elocity while or the turbulent low, it turns out to be proportional to.75-. The loss head is expressed by the ollowing equation as shown in this equation : This equation is called the Darcy-Weisbach equation', and the coeicient is called the riction coeicient o the pipe.

11 Pipe rictional loss Relationship between low elocity and loss head Laminar low In this case the equations and = No eect o wall roughness is seen. The reason is probably that the low turbulence caused by the wall ace coarseness is limited to a region near the wall ace because the elocity and thereore inertia are small, while iscous eects are large in such a laminar region. Turbulent low generally aries according to Reynolds number and the pipe wall roughness. Smooth circular pipe The roughness is inside the iscous sub layer i the height ε o wall ace ruggedness is In the case o a smooth pipe, the ollowing equations hae been deeloped: R e Rough circular pipe I wheneer Re > 900(ε/d), it turns out that

12 A good approximate equation or the turbulent region o the Moody chart is gien by Haaland s equation: For a new commercial pipe, can be easily obtained rom Moody diagram shown in Fig.a using ε/d in Fig.b. Fig,a Mody diagram Re Fig,b

13 Example ( Laminar low): Water, ρ=998 kg/m 3, = m /s lows through a 0.6 cm tube diameter, 30 m long, at a low rate o 0.34 L/min. I the pipe discharges to the atmosphere, determine the supply pressure i the tube is inclined 0 o aboe the horizontal in the low direction. 30 m 0 o 30*sin(0) Example An oil with ρ = 900 kg/m 3 and = m /s lows upward through an inclined pipe. Assuming steady laminar low, (a) eriy that the low is up, (b) compute h between and, and compute (c) V, (d) Q, and (e) Re. Is the low really laminar?

14 HGL < HGL hence the low is rom to as assumed. V=.7 m/s, Q= m 3 /s and Re=80 the low is laminar Example: (turbulent low) Oil, ρ = 900 kg/m 3, ν = 0-5 m /s, lows at 0. m 3 /s through a 500 m length o 00 mm diameter, cast iron pipe ε= I the pipe slopes downward 0 o in the low direction, compute h, d=00 m total head loss, pressure drop, and power required to oercome these losses. 500 m 0 o Note that or this problem, there is a negatie graity head loss ( i.e. a head increase ) and a positie rictional head loss resulting in the net head loss o 9.8 m

15 Minor losses in pipes In a pipe line, in addition to rictional loss, head loss is produced through additional turbulence arising when luid lows through such components as change o area, change o direction, branching, junction, bend and ale. The loss head or such cases is generally expressed by the ollowing equation: h s =k υ is the mean low elocity on a section loss in a suddenly expanding pipe For a suddenly expanding pipe as shown in below Figure, assume that the pipe is horizontal, disregard the rictional loss o the pipe, let h, be the expansion loss, and set up an equation o energy between sections and as : Apply the equation o momentum setting the control olume as shown in the Figure. Thus : Since Q = A = A, rom the aboe equation, Substituting into eqn( ) : ( ) This h s is called the Borda-Carnot head loss or simply the expansion loss. Flow in pipes : At the outlet o the pipe as shown in the right Figure, since = 0, the aboe equation becomes h s = k

16 Flow contraction Owing to the inertia, section (section area A ) o the luid shrinks to section (section area A c ) and then widens to section 3 ( section area A ). The loss when the low is accelerated is extremely small, ollowed by ahead loss similar to that in the case o sudden expansion. Like eqn ( ), it is expressed by : Here C c = A c / A is a contraction coeicient. Inlet o pipe line The loss o head in the case where luid enters rom a large essel is expressed by the ollowing equation: h s = k is the inlet loss actor and is the mean low elocity in the pipe. The alue o will be the alue as shown in below Figure. k= k= k= k= k= k=

17 Diergent pipe or diuser The head loss or a diergent pipe as shown in below Figure. is expressed in the same manner as or a suddenly widening pipe: h s = k Appling Bernolli equation : ( ) Putting p th or the case where there is no loss, The pressure recoery eiciency η or a diuser : Substituting this equation in equation ( ) : - k The alue o k aries according to θ. For a circular section k = 0.35 (minimum) when θ = 5 o 30'. For the rectangular section, k = 0.45 ( minimum ) when θ = 6 o, and k = ( almost constant ) wheneer θ = 50 o 60 o or more. In the case o a circular pipe, when θ becomes larger than the angle which gies the minimum alue o k, the low separates midway as in Fig.a.The loss o head suddenly increases, this phenomenon is isualized in Fig.b. Fig.a Fig.b

18 Loss wheneer the low direction changes Bend In a bend, in addition to the head loss due to pipe riction, a loss due to the change in low direction is also produced. The total head loss h b is expressed by the ollowing equation: h b =( + k ) Here, k is the loss actor due to the bend eect. In a bend, secondary low is produced as shown in the igure owing to the introduction o the centriugal orce, and the loss increases. I guide blades are ixed in the bend section, the head loss can be ery small. Below table shows alues o k or the bends. Table, loss actor k or bends (smooth wall Re=5000, coarse wall ace Re=46000 ) Elbow The section where the pipe cures sharply is called an elbow. The head loss h b is gien in the same orm as aboe equation o the bend. Since the low separates rom the wall in the curing part, the loss is larger than in the case o a bend. Below table shows alues o k or elbows. k Table, Loss actor k or elbows k

19 Pipe branch and pipe iunction Pipe branch As shown in below Figure, a pipe diiding into separate pipes is called a pipe branch. Putting h s as the head loss produced when the low runs rom pipe to pipe 3, and h s as the head loss produced when the low runs rom pipe to pipe, these are respectiely expressed as ollows: h s = k h s = k Since the loss actors k, k ary according to the branch angle θ, diameter ratio d /d or d / d 3 and the discharge ratio Q /Q or Q /Q 3. Pipe junction Two pipe branches conerging into one are called a pipe junction. Putting h s as the head loss when the low runs rom pipe to pipe 3, and h s as the head loss when the low runs rom pipe to pipe 3, these are expressed as ollows: h s = k h s = k Vale and cock Head loss on ales is brought about by changes in their section areas, and is expressed by this equation proided that indicates the mean low elocity at the point not aected by the ale. h s =k k Gate ale Global ale k cock k

20 The alues o k or the arious ales such as relie ale, needle ale,pool ale, disc ale ball ale..etc are also depend on the ratio o the ale area to pipe area. Total loss along a pipe line h t = h + h s h t or h t These equations would be appropriate or a single pipe size ( with aerage elocity V ). For multiple pipe/duct sizes, this term must be repeated or each pipe size. Hydraulic grade line and energy line As shown in the Figure, wheneer water lows rom tank to tank, the energy equations or sections, and 3 with losses are as ollowing:

21 h and h3 are the losses o head between section and either o the respectie sections. Example Water, ρ=000 kg/m 3 and =.0 0-6, is pumped between two reseroirs at m 3 /sthrough m o 5.08 cm diameter pipe and seeral minor losses,as shown. The roughness ratio is ε/d = Compute the pump power required. Take the ollowing minor losses. K i Loss element Sharp entrance 0.5 Open globe ale 6.9 bend, R/D = 0.5 Threaded, 90Þ, reg., 0.95 elbow Gate ale, / closed.7 Submerged exit Z =36 m Z =6 m m o pipe, d=5.08 c m Write the steady-low energy equation between sections and, the two reseroir suraces: h s where h p is the head increase across the pump. A=π /4

22 V=.8 m/s = the low is turbulent and Haaland s equation can be used to determine the riction actor: = 0.04 But since p = p = 0 and V =V = 0, sole the aboe energy equation or the pump head : Z = 36 m, Z = 6 m, L = m h P = m The power required to be deliered to the luid is gie by : = 39 W I the pump has an eiciency o 80 %, the power requirements would be speciied P in = P / η = 39 /0.8 P in = W Example : Sketch the energy grad line or below Figure. Take H=0 m, K A =, K B =(-(A /A )), K C (ale) =3.5, K D = and =0.05 (or all pipes ) ()

23 (.667) (.667) (.667) (6.667) ) ( (6.667) (6.667) 0 0 /.667..,.. / ()..... tan ) ( ) ( ) ( ) 9.8 ( ) ( () ) ( ) ( surace D C C B B A Surace L m m L L L H m H m H m H m H m H m H m H s m s m Eq in ce subs h h g K h h A A gd L gd L h h h h Z g g p h Z g g p L l Multiple-Pipe Systems Series Pipe System: The indicated pipe system has a steady low rate Q through three pipes with diameters D, D, & D3. Two important rules apply to this problem.

24 . The low rate is the same through each pipe section.. The total rictional head loss is the sum o the head losses through the arious sections. Example: Gien a pipe system as shown in the preious igure. The total pressure drop is Pa Pb = 50 kpa and the eleation change is Zb Za = 5 m. Gien the ollowing data, determine the low rate o water through the section. The luid is water, ρ = 000 kg/m 3 and = m /s. Calculate the low rate Q in m 3 /h through the system..() Begin by estimating,, and 3 rom the Moody-chart ully rough regime Substitute in Eq. () to ind : V =0.58 m/s, V =.03 m/s, V 3 =.3 m/s

25 Hence, rom the Moody chart, e/d with R e Substitute in Eq. () : Parallel Pipe System: Example : Assume that the same three pipes in aboe Example are now in parallel. The total pressure drop is Pa Pb = 50 kpa and the eleation change is Zb Za = 5 m. Gien the ollowing data. Compute the total low rate Q, neglecting minor losses. The luid is water, ρ = 000 kg/m 3 and = m /s. Calculate the low rate Q in m 3 /h through the system Guess ully rough low in pipe : = 0.06, V = 3.49 m/s; hence Re = 73,000. From the Moody chart Re with e/d =0.067; recomputed V =3.46 m/s, Q = 6.5 m 3 /h.

26 Next guess or pipe : =0.034, V =.6 m/s ; then Re =53,000, From the Moody chart Re with e/d = 0.046, V =.55 m/s, Q = 5.9 m 3 /h. Finally guess or pipe 3: 3 = , V 3 =.56 m/s ; then Re 3 = 00,000 From the Moody chart Re with e/d 3 =0.033, V 3 =.5 m/s, Q3 =.4 m 3 /h. This is satisactory conergence. The total low rate is These three pipes carry 0 times more low in parallel than they do in series. Branched pipes Consider the third example o a three-reseroir pipe junction as shown in the igure. I all lows are considered positie toward the junction, then.() which obiously implies that one or two o the lows must be away rom the junction. The pressure must change through each pipe so as to gie the same static pressure p J at the junction. In other words, let the HGL at the junction hae the eleation where p J is in gage pressure or simplicity. Then the head loss through each, assuming P = P = P 3 = 0 (gage) at each reseroir surace, must be such that

27 We guess the position h J and sole the aboe Equations or V, V, and V 3 and hence Q, Q, and Q 3, iterating until the low rates balance at the junction according to Eq.(). I we guess h J too high, the sum Q + Q + Q 3 will be negatie and the remedy is to reduce h J, and ice ersa. Example : Take the same three pipes as in the preious example, and assume that they connect three reseroirs at these surace eleations Find the resulting low rates in each pipe, neglecting minor losses. As a irst guess, take h J equal to the middle reseroir height, Z 3 = h J = 40 m. This saes one calculation (Q 3 = 0) and enables us to get the lay o the land : Since the sum o the low rates toward the junction is negatie, we guessed h J too high. Reduce h J to 30 m and repeat : This is positie Q, and so we can linearly interpolate to get an accurate guess: h J = 34.3 m. Make one inal list :

28 Hence we calculate that the low rate is 5.4 m3/h toward reseroir, balanced by 47. m3/h away rom reseroir and 6.0 m3/h away rom reseroir 3. One urther iteration with this problem would gie h J = m, resulting in Q = 5.8, Q = 47.0, and Q 3 =5.8 m3/h, so that Q = 0 to three-place accuracy.