HYDRAULIC RESISTANCE Local (Minor!) Pressure Losses

Transcription

1 HYDRULIC RESISTNCE Local (Minor!) Pressure Losses TFD-FD06 Hydraulic Resistance Local (Minor!) Losses

2 Local Pressure Losses (Minor Losses!) lthough they often account for a major portion of the total pressure loss, the additional losses due to entries and exits, fittings and ales are traditionally referred to as minor losses! These losses represent additional energy dissipation in the flow, usually caused by secondary flows induced by curature or recirculation. The minor losses are any total pressure loss present in addition to the total pressure loss for the same length of straight pipe. TFD-FD06 Hydraulic Resistance Local (Minor!) Losses

3 Modified Bernoulli Equation Bernoulli equation is alid along any streamline in any frictionless flow Howeer, this is ery restrictie. Solid walls introduce friction effects. Fittings in a piping system as well as cross sectional area changes introduce frictional pressure losses. Bernoulli s equation can be modified to include these frictional losses present in any fluid system network fl p ρ ρgh p ρ ρgh ρ ρ D TFD-FD06 Hydraulic Resistance Local (Minor!) Losses 3 h Wall Friction Local Losses is the coefficient of flow (hydraulic) resistance

4 Coefficient of Flow Resistance - Dimensionless Parameter Coefficient of flow resistance is defined as the ratio of the total energy lost oer the gien segment to the kinetic energy in the section Ratio of the total pressure lost oer the segment to the dynamic pressure in the segment Flow Resistance Total Pressure Loss Dynamic Pressure(inetic Energy) Flow resistance coefficient, for the case of uniform distribution of static pressure and density oer the segment but which are ariable along the flow ptotal ρ TFD-FD06 Hydraulic Resistance Local (Minor!) Losses 4 o

5 Base rea for Flow Resistance Flow resistance coefficients can be expressed in terms of the upstream or downstream elocity of the component Loss term can be added to upstream or downstream elocity with the same net effect pup pup ρup ρup pdown ( fl D h ρdown up ) ( ρ fl D ρdown Base area for all flow resistances () shown in this section are gien based on the smallest cross sectional area of the component hence the largest elocity. up h pdown down ) ρ down up up up down ρ up down up down down down up down up ρ p ρ down base TFD-FD06 Hydraulic Resistance Local (Minor!) Losses 5

6 TFD-FD06 Hydraulic Resistance Local (Minor!) Losses 6 Equialent Length of Fittings Local (minor) losses can be represented in a different way using the the concept of equialent length Inclusion of local losses in Bernoulli equation can make the iteratie type problems labor intensie Local losses can be significantly higher for relatiely short piping systems and may not be ignored ) ( ) ( V D fl V V D fl h h gh V p gh V p gh V p gh V p ρ ρ ρ ρ ρ ρ ρ ρ ρ ρ ρ h eq D fl

7 Equialent Length of Fittings Concept of equialent length allows us to replace the local (minor) loss term with equialent pipe length Σ fleq Dh (Σ) Leq D f h f is the friction factor that applies to the entire pipe D is the pipe hydraulic diameter (characteristic length) L eq is the equialent length: Length of pipe which can replace the fitting (local loss) to obtain the same pressure loss TFD-FD06 Hydraulic Resistance Local (Minor!) Losses 7

8 List of Configurations Sudden Expansion Sudden Expansion in One Plane Sudden Contraction Bend Vena Contracta Effect Conical Expansion Conical Contraction Uniformly Distributed Barriers Wire Screen Threaded Screen Two Plane Screen Gratings in Line With Flow Orifice In a Straight Tube Sharp Edge Rounded Beeled Thick Edge Orifice In a Tube Transition Sharp Edge Rounded Beeled Thick Edge TFD-FD06 Hydraulic Resistance Local (Minor!) Losses 8

9 Resistance to Flow Sudden Changes in Flow rea Sudden Expansion Sudden Contraction Vena Contracta Effect TFD-FD06 Hydraulic Resistance Local (Minor!) Losses 9

10 Sudden Expansion Pressure Losses n abrupt expansion of a tube cross-sectional area gies rise to so-called shock losses. The local pressure loss due to this shock depends only on the cross sectional area ratio /. exp ( ) The area ratio of / is less than.0 ssumptions: Uniform elocity distribution at cross section Reynolds number > 0 4 Vortices P P Jet P Jet P TFD-FD06 Hydraulic Resistance Local (Minor!) Losses 0 Base area is L 8- D h

11 Sudden Expansion Pressure Losses The jet formed due to sudden expansion, is separated from the remaining medium Forms a significant turbulence as shown by the ortices near the wall surfaces. It requires a pipe length (L ) of 8- hydraulic diameters (D h ) before relatiely uniform flow of elocity is established. TFD-FD06 Hydraulic Resistance Local (Minor!) Losses

12 Sudden Expansion in One Plane When an abrupt expansion of the tube cross section occurs only in one plane (as shown in the Figure) the shock losses decrease with an increase in the aspect ratio B/H. B is the width of larger cross section H is the constant height of the channel Loss coefficient is: exp k( ) Dependence of k on B/H Where k < is correction factor which depends on the aspect ratio B/H. TFD-FD06 Hydraulic Resistance Local (Minor!) Losses

13 Sudden Contraction When the cross section abruptly contracts the phenomenon is basically a similar to that obsered when shock losses occur during abrupt expansion Contraction losses occur when the jet Is further compressed after entering the section 3 - ena contracta effect Effectie flow diameter reduces to c Expands until it fills the entire section Vena Contracta c 3 of the narrow channel 3 Cont c 3 Base area is 3 TFD-FD06 Hydraulic Resistance Local (Minor!) Losses 3

14 Vena Contracta Effect Flow passage through the contraction is accompanied by distortion of the trajectories of particles with the result that they continue their motion by inertia toward the axis of the opening. This reduces the initial area of the jet cross section at C-C until the area is smaller than the area of the cross section of the opening. Plane of ena contracta. C-C Plane of Vena Contracta Starting mid-section CC, the trajectory of moing particles are straightened. Thereafter an abrupt jet expansion takes place. TFD-FD06 Hydraulic Resistance Local (Minor!) Losses 4

15 Tube Walls Following Flow Streamlines The pressure loss of a contracting cross section can be significantly reduced with a tube boundary following the flow streamlines. llowing a smooth transition from a wide section to the narrow one With the curilinear boundaries Conerging and dierging (expanding) nozzle Conerging Nozzle Pipe Wall Dierging Nozzle TFD-FD06 Hydraulic Resistance Local (Minor!) Losses 5

16 Resistance to Flow Changes of the Stream Direction Flow Losses in an Elbow Flanged Elbow Standard Threaded Elbow Miter Bend TFD-FD06 Hydraulic Resistance Local (Minor!) Losses 6

17 Change of Flow Direction In elbows the streamlines are cured and centrifugal force causes a pressure increase near the outer wall of the elbow. Starting at point pressure increases and rising to a maximum alue at B. In region B the fluid flow is opposed by an aderse pressure gradient. t the inside of the elbow, the pressure decreases to point C and then rises again in the exit section D. For this reason an aderse pressure B Higher gradient also exists from C to D at pressure region the inside wall. D These conditions may lead to a separation and turbulence and corresponding losses in flow energy. The magnitude of these losses to a large extend depend on the sharpness of the curature. C Lower pressure region TFD-FD06 Hydraulic Resistance Local (Minor!) Losses 7

18 Flow Losses In an Elbow The main portion of flow pressure losses in cured tubes is due to formation of eddies at the inner wall. Other conditions being equal, the cured tube offers the largest resistance in the case when the curature at the inner wall is a sharp corner. Rounding of the elbow corners makes the flow separation much smoother and consequently lowers the resistance. big majority of flow the flow losses can be recoered by rounding the inner corner and leaing the outer elbow sharp. Cross sectional area at the place of bending increases sharply (s shown in red x-section) Rounding the outer elbow corner and keeping the inner corner sharp does not reduce the elbow resistance. Large radius on the outer elbow will een increase flow losses since it would reduce the cross sectional area significantly (s shown in green x-sec) TFD-FD06 Hydraulic Resistance Local (Minor!) Losses 8

19 Minimizing Elbow Resistance The minimum resistance is achieed by an elbow when outer & inner radii is related as r ro 0.6 d d o o r o R o r The resistance of right angle elbows can greatly be reduced by installing a fairing on the inner corner n optimum fairing with a ratio of r o /d o 0.45 reduces the pressure losses by approx 50% n additional fairing on the outer corner with a ratio of r /d o 0.45 reduces the losses by an additional 5%. Reduction in the elbow resistance can also be attained by beeling sharp corners of the bend especially the inner corner r o d o r d o TFD-FD06 Hydraulic Resistance Local (Minor!) Losses 9

20 Minimizing Elbow Resistance Example of a Coolant Return Elbow Base Case Design Design B Computational Fluid Dynamics (CFD) was used to minimize the flow losses in a coolant return elbow. Fairing on the inner corner together with a smooth outer radius reduced the flow losses significantly. TFD-FD06 Hydraulic Resistance Local (Minor!) Losses 0

21 Minimizing Elbow Resistance Example of a Coolant Return Elbow Base Case p 6.5kPa(00%) Flow Separation Fairly small effectie flow area Design p 4.6kPa(7%) Streamlines outside corner radius Higher elocity on outside corner after 90 degree flow turn CFD results shown in terms of elocity ectors and elocity color contours Fairing Larger Inlet cross sectional area Higher elocity on inside corner before 90 degree flow turn Design B p 3.9kPa(3%) 77% reduction in pressure losses is significant! TFD-FD06 Hydraulic Resistance Local (Minor!) Losses

22 Flow Resistance of Elbows Crane 988, -9 Resistance () of 90 degree smooth, flanged elbow r o R o r d o Flanged Elbow f t is the turbulent friction factor gien in the Table aboe Resistance () of standard threaded elbow 30F t R o /d o /Ft d o (mm) Ft Standard Threaded Elbow TFD-FD06 Hydraulic Resistance Local (Minor!) Losses

23 Flow Resistance of Miter Bend Resistance of 90 degree Miter Bend 60F t 90 o Miter Bend Resistance of Non-90 degree miter bend d o Non-90 o Miter Bend Miter ngle ngle /Ft TFD-FD06 Hydraulic Resistance Local (Minor!) Losses 3

24 Resistance to Flow Smooth Change in Velocity & Flow rea Diffuser with Gradual Expansion Pressure Looses in a Diffuser Effect of Diffuser ngle on Pressure Looses Conical Expansion & Contraction Conerging Rectilinear Nozzle TFD-FD06 Hydraulic Resistance Local (Minor!) Losses 4

25 Diffuser - Gradual Expansion gradual tapered transition from one diameter to another can reduce flow losses appreciably in comparison with abrupt transitions. In a conical diffuser the pressure rises in the direction of flow Increasing diameter reduces the elocity. Higher elocity, P P Jet inetic energy at cross section is conerted to potential energy at cross section Howeer the slower fluid particles near the diffuser wall tend to stagnate and disturb the theoretical (ideal) pressure rise. This causes an energy loss. Total energy at section is less than the total energy at section α kinetic energy P < P Base area is Higher static pressure, potential energy TFD-FD06 Hydraulic Resistance Local (Minor!) Losses 5

26 Pressure Losses in a Diffuser The resistance coefficients of the diffusers as well as the flow structure in them and the separation phenomenon depend on many parameters such as Diergence angle, alpha rea ratio The shape of the elocity profile at the entrance Degree of flow turbulence at the entrance Flow regime, at the entrance Flow compressibility The effect of the Reynolds number on the resistance coefficients of the diffusers is different for different diergence angles TFD-FD06 Hydraulic Resistance Local (Minor!) Losses 6

27 Effect of Diergence ngle In Gradual Expansion - Diffuser The total resistance coefficient of a diffuser expressed in terms of the elocity at small section becomes smaller up to certain limits of diergence angle, alpha. Increase in the cross sectional area of the diffuser causes a drop in aerage elocity. Static pressure gain due to this cross sectional area change is greater than the pressure loss due to flow turbulence up to certain limits of diergence angle, alpha If the cone angle is too large, separation of the flow from the wall also occurs and causes an additional energy loss. The most adantages cone angle is 8 o degrees In flow channels of rectangular cross section, separation occurs if cone angle alpha is greater than 0 o TFD-FD06 Hydraulic Resistance Local (Minor!) Losses 7

28 Conical Expansion & Contraction Expansion α Conical Exp.6sin Conical Exp α > 45 α < α 45 Higher elocity, kinetic energy P P Jet Expansion P < P Higher static pressure, potential energy Contraction Conical Contraction Conical Contraction α 0.8sin 0.5 α sin α α TFD-FD06 Hydraulic Resistance Local (Minor!) Losses 8 < 45 > Base area is 45 α Rectilinear Conerging Nozzle

29 Gradual Contraction Conerging Nozzle In a conergent section flow losses are small since the pressure gradient is in the same direction as the flow. Particles of fluid near the wall are continually accelerated by this pressure gradient so that flow is maintained een in the close icinity of the wall. The resistance coefficient of a conerging nozzle with rectilinear boundaries at high Re numbers depends on: Conergence angle alpha rea ratio n o / < α Rectilinear Conerging Nozzle TFD-FD06 Hydraulic Resistance Local (Minor!) Losses 9

30 Flow Resistance of Conerging Rectilinear Nozzle t sufficiently large angles (alpha > 0 o ) and area ratios less than 0.3, the flow after passing from the contracting section of the rectilinear conerging nozzle to the straight part of the tube separates from the walls which is the main source of the flow losses. The larger the alpha and smaller the area ratio, stronger is the flow separation and greater the pressure losses Pressure losses are at maximum for the limiting case of 80 degree angle sudden contraction. Con Nozzle ( 0.05n n n 3 ( α πα 0α ) n ) where alpha is in radians degree (pi/80) radians TFD-FD06 Hydraulic Resistance Local (Minor!) Losses 30

31 Resistance to Flow at the Exit From Tubes Uniform Velocity Distribution t Tube Exit Exponential Velocity Distribution Sinusoidal Velocity Distribution symmetric Velocity Distribution TFD-FD06 Hydraulic Resistance Local (Minor!) Losses 3

32 Resistance of Exit Sections When fluid flow leaes the piping system, the kinetic energy of the discharged jet is always lost. In the case of free discharge of the flow from a straight section of the tube of constant cross section into a large olume, the total losses are reduced only to the losses of the elocity pressure at the exit. TFD-FD06 Hydraulic Resistance Local (Minor!) Losses 3 p elocity ρ The coefficient N depends on the nature of the elocity distribution at the exit. In the case of uniform exit elocity distribution, it is equal to unity In the remaining cases where elocity distribution is non-uniform, it always larger than one. Exponential, Parabolic, Sinusoidal, symmetric cases are some of the non-uniform elocity distributions. N

33 Free Discharge From a Straight Tube t Different Velocity Distributions Uniform Velocity Distribution p ρ w o Exponential Velocity Distribution p ρ 3 3 (m ) ( m ) 4 4m (m 3)( m 3) wo TFD-FD06 Hydraulic Resistance Local (Minor!) Losses 33

34 Free Discharge From a Straight Tube t Different Velocity Distributions Sinusoidal Velocity Distribution in a plane tube w w o symmetrical Velocity Distribution in a plane tube 3.67 w w sin( o b o y ) TFD-FD06 Hydraulic Resistance Local (Minor!) Losses 34

35 Resistance to Flow with Sudden Change in Velocity Orifice Plate Orifice in Straight Tube Orifice with Beeled Inlet Edge Orifice with Rounded Inlet Edge TFD-FD06 Hydraulic Resistance Local (Minor!) Losses 35

36 Orifice Plate Flow passage through the opening is accompanied by the distortion of the trajectories of particles with the result that they continue their motion by inertia toward the axis of the opening. This reduces the initial area jet cross sectional area by a factor alpha. (Vena contracta effect) Thickening, beeling and rounding the orifice edges reduces the effect of jet contraction (reduced jet elocity downstream of orifice plate) and reduce pressure losses. TFD-FD06 Hydraulic Resistance Local (Minor!) Losses 36

37 TFD-FD06 Hydraulic Resistance Local (Minor!) Losses 37 Orifice In Straight Tube Sharp, Rounded & Beeled Edges Idelchik Pg o o o Orifice Sharp 0.75 o o Orifice Rounded ζ D h o r ζ o o Orifice Beeled ζ o h o h D l D l ζ Base area is o Flow Thru Orifice

38 Orifice In Straight Tube Thick Edge Orifice Idelchik Pg -4 Thick Edge 0.75 o 0.5 τ o.375 o fl D h τ (.4 l) l l 8 8 Base area is o Flow Thru Orifice TFD-FD06 Hydraulic Resistance Local (Minor!) Losses 38

39 Resistance to Flow Through Barriers Uniformly Distributed Oer the Tube Cross Section Circular Metal Wire Threaded Screens Two Plane Screens Grating In Line with Flow Gratings with an ngle of ttack TFD-FD06 Hydraulic Resistance Local (Minor!) Losses 39

40 Uniformly Distributed Barriers Barriers that are distributed uniformly oer the tube or duct cross sections create a uniform flow resistance. Types of barriers may include: Perforated sheets (grids), screens, beds, fabrics, loose material, crosswise bundles of tubes, etc. The flow resistance coefficient of a uniformly distributed barrier depends on: Shape of the barrier Free area coefficient Reynolds number Free area coefficient is defined as the percent clear area of the barrier with respect to the upstream pipe area. Idelchik pg5 open o free pipe upstream TFD-FD06 Hydraulic Resistance Local (Minor!) Losses 40

41 Circular Metal Wire Screens Idelchik Diagram 8.6 Pg 5 o 3 0 Wire Screen.3( ) ( ) 0 3 < 0 0 Wire Screen k.3( ) ( ) 0 δ Re m 0 µ 50 Re < Re < 50 Wire Screen.3( Re 0 ) ( ) 0 Base area is o o is open area on screen is pipe upstream flow area TFD-FD06 Hydraulic Resistance Local (Minor!) Losses 4

42 Two Plane & Silk Thread Screen Idelchik Pg 53 Two plane screen made from bars of circular cross section Plane Screen.8 o o Silk Thread Screen Base area is o Silk Threads. 6 Re > 500 Wire Screen TFD-FD06 Hydraulic Resistance Local (Minor!) Losses 4

43 Gratings In Line with Flow Idelchik Pg 56 Special case of l d m 5 and a S o > 0.5 k Grating 4 3 S a o β sinθ k a o is flow space between gratings S is pitch of gratings l is length of gratings d m is width of gratings Beta is grating shape constant Theta is angle of bar inclination with respect to flow Shape Beta Beta Base area is o Flow Between Gratings TFD-FD06 Hydraulic Resistance Local (Minor!) Losses 43

44 Case of Thick Grit Gratings In Line with Flow Idelchik Pg 56 Flow losses of grids made of bar gratings with different cross sectional shapes consist: Entrance loss Frictional loss oer the bars Sudden expansion (shock) losses at the exit τ arbitrary l d m and a S β sinθ Grating Thick Grid l 0.5 l 0.05 l d h o 0 open open open open.5 τ f / dh TFD-FD06 Hydraulic Resistance Local (Minor!) Losses l

45 Gratings with an ngle of ttack Reynolds number > 0 4 & a 0 /S >0.5 Grating σ σ TFD-FD06 Hydraulic Resistance Local (Minor!) Losses 45

46 HYDRULIC RESISTNCE OF NETWORS TFD-FD06 Hydraulic Resistance Local (Minor!) Losses 46

47 Hydraulic Resistance of Networks In flow networks, the portion of the total pressure which is spent in oercoming the forces of hydraulic resistance is irreersibly lost. The molecular and turbulent iscosity of the moing medium irreersibly conerts the mechanical work of the resistance forces into heat. There are two types of total pressure losses (hydraulic resistance) in flow network (pipeline system) Pressure losses resulting from friction (frictional drag) Local pressure losses TFD-FD06 Hydraulic Resistance Local (Minor!) Losses 47

48 Friction Pressure Losses Fluid friction loss is due to the iscosity (both molecular and turbulent) of fluids in motion Results from momentum transfer between the molecules and indiidual particles of adjacent fluid layers moing in different elocities molecules in laminar flow indiidual particles in turbulent flow TFD-FD06 Hydraulic Resistance Local (Minor!) Losses 48

49 Local Pressure Losses The local losses of total pressure are caused by Local disturbance of the flow Separation of flow from walls Formation of ortices and strong turbulent agitation Entrance of a fluid into the pipeline Expansion, contraction, bending and branching of flow Flow through orifices, grids, ales Friction through porous media (filtration) Discharge of fluid into atmosphere or another reseroir ll of these phenomena contribute to the exchange of momentum between the moing fluid particles Enhancing energy dissipation Increasing total pressure loss TFD-FD06 Hydraulic Resistance Local (Minor!) Losses 49

50 Total Pressure Loss Friction Local Pressure Loss The total pressure losses in any complex fluids network are inseparable. Howeer, for ease of calculation they are arbitrarily subdiided in each element of the pipeline (network) Into frictional losses Into local losses Total pressure loss for the fluid network is the summation of pressure drops in each segment p total all segments ( p friction p local The alue of friction loss (relatie to local pressure loss) is usually taken into account for Long branch pipes, diffusers with small diergence angles Long fittings TFD-FD06 Hydraulic Resistance Local (Minor!) Losses 50 )

51 Units of Pressure Loss Head is energy per unit weight of fluid. Force x Length / Weight TFD-FD06 Hydraulic Resistance Local (Minor!) Losses 5

52 References Handbook of Hydraulic Resistance 3 rd Edition I. E. Idelchik, CRC Begell House Fluid Mechanics Dr. Walther aufmann, Mc Graw Hill 958 Crane Co., Flow of Fluids Through Vales, Fittings, and Pipe, Technical Paper No. 40, Crane Co., Joliet, IL, 988. TFD-FD06 Hydraulic Resistance Local (Minor!) Losses 5

53 ppendix TFD-FD06 Hydraulic Resistance Local (Minor!) Losses 53

54 ppendix TFD-FD06 Hydraulic Resistance Local (Minor!) Losses 54

55 The total resistance coefficient of a diffuser installed in a flow network under any inlet condition can be expressed as Diffuser L L k ζ 0 > 0 d Diffuser 0 0 Total Resistance Coefficient of a Diffuser with straight section installed upstream Total Resistance Coefficient of a Diffuser L 0 /D 0 0 Exp where Grad η? TFD-FD06 Hydraulic Resistance Local (Minor!) Losses 55

56 Sudden Change in Velocity Idelchik Pg 7 Diagram 4-9 TFD-FD06 Hydraulic Resistance Local (Minor!) Losses 56

57 Velocity Distribution for a Cured Bend Increase of the pressure at the outer wall and decrease at the inner wall will create a flow elocity the lower at the altar wall and larger at the narrow Before the straight section after the turn: Flow elocity will be lower at the outer wall and larger at the inner wall. Due to higher pressure at the outer wall & lower pressure inside. Downstream of turn - Straight section: Higher flow at the outer wall and lower elocity at the inner wall. Secondary Flows: Higher pressure lower elocity Lower pressure higher elocity C B EDDY ZONES: Flow separates both at the inner and outer walls D TFD-FD06 Hydraulic Resistance Local (Minor!) Losses 57

58 TFD-FD06 Hydraulic Resistance Local (Minor!) Losses 58 Deriation of Local Loss Coefficient Sudden Expansion From the momentum law: From Modified Bernoulli Eq.: ) ( ) ( ) ( ) ( ) ( ) ( p p p p Q p p m F ext Σ ρ ρ ρ & exp ) ( ) ( p p ρ ρ exp exp exp ) ( ) ( ) ( ) ( ) ( Eliminating (p -p ) & rho exp ) ( ) ( Local Loss Coefficient, exp Ratio / is less than.0

59 Flow From Vessel With Spout Momentum Equation Between sections e & a: Forces acting on the fluid must equal to the difference in momenta between fluid leaing and entering the section per unit time ( p p ) ρ ) e o ( e pplying Bernoulli s equation at sections e & a Pressure at e is p e including the stagnant flow region ( p ) ( ) e po ρ e contractio n Combining both equations Large Vessel Contracting Jet Sharp Edged Spout Stagnant Flow Region Spout Slow Expansion Until Filling the Spout TFD-FD06 Hydraulic Resistance Local (Minor!) Losses 59

60 Orifice with Beeled Inlet Edge In Tube Transition Section Idelchik Pg 4 TFD-FD06 Hydraulic Resistance Local (Minor!) Losses 60

61 Orifice with Rounded Inlet Edge In Tube Transition Section Idelchik Pg 4 TFD-FD06 Hydraulic Resistance Local (Minor!) Losses 6