FLUID MECHANICS Dynamics of iscous Fluid Flow in Closed Pipe: Darcy-Weisbach equation for flow in pipes. Major and minor losses in pipe lines. Dr. Mohsin Siddique Assistant Professor
Steady Flow Through Pipes Laminar Flow: flow in layers Re<000 (pipe flow) Turbulent Flow: flow layers mixing with each other Re >4000 (pipe flow)
Steady Flow Through Pipes Reynold s Number(R or Re): It is ratio of inertial forces (Fi) to viscous forces (Fv) of flowing fluid elocity olume Mass. ρ. elocity Fi Re Time Time Fv Shear Stress. Area Shear Stress. Area ρq. ρ A. ρ A. ρl L τ. A du µ. A µ. A µ υ dy L ρd D R e µ ν For laminar flow: Re<000 For transitional flow: 000<Re<4000 For Turbulent flow: Re> 4000 Where ; is avg. velocity of flow in pipe ν is kinematic viscosity L is characteristic/representative linear dimension of pipe. It is diameter of pipe (circular conduits) or hydraulic radius (non-circular conduits). alues of critical Reynolds no. 3 Note: For non-circular section, we need to use hydraulic radius (R h ) instead of diameter (D) for the linear dimension (L).
Steady Flow Through Pipes Hydraulic Radius (R h ) or Hydraulic Diameter: It is the ratio of area of flow to wetted perimeter of a channel or pipe R Area h wetted perimeter A P For Circular Pipe For Rectangular pipe D A Rh P D 4R h ( π / 4) D ) πd D 4 B A BD R h P B D R h D ν 4R ν h By replacing D with R h, Reynolds number formulae can be used for non-circular sections as well. 4 Note: hydraulic Radius gives us indication for most economical section. More the Rh more economical will be the section.
Head Loss in Pipes Total Head LossMajor Losses Minor Losses Major Loss: Due to pipe friction Minor Loss: Due to pipe fittings, bents and valves etc 5
Head Loss in Pipes due to Friction The head loss due to friction in a given length of pipe is proportional to mean velocity of flow () as long as the flow in laminar. i.e., H f But with increasing velocity, as the flow become turbulent the head loss also varies and become proportion to n H f Where n ranges from.75 to n 6 Log-log plot for flow in uniform pipe (n.0 for rough wall pipe; n.75 for smooth wall pipe
Frictional Head Loss in Conduits of Constant Cross-Section Consider stead flow in a conduit of uniform cross-section A. The pressure at section & are P & P respectively. The distance between the section is L. For equilibrium in stead flow, F ma 0 P perimeter of conduit Avg. shear stress between pipe boundary and liquid τ o Figure: Schematic diagram of conduit z L z sinα P A W sinα τ opl P A 0 7 P z z L A P A AL opl τ 0
Frictional Head Loss in Conduits of Constant cross-section 8 0 PL L z z AL P A A P o τ Dividing the equation by A ( ) 0 A PL z z P P o τ f L o h h A PL P z P z τ Therefore, head loss due to friction h f can be written as h o o f R L A PL h τ τ h L g v z P g v z P Remember!! For pipe flow For stead flow in pipe of uniform diameter v v h L z P z P This is general equation and can be applied to any shape conduit having either Laminar or turbulent flow. P A Q R h
Determining Shear Stress 9 For smooth-walled pipes/conduits, the average shear stress at the wall is Using Rayleigh's Theorem of dimensional analysis, the above relation can be written as; Rewriting above equation in terms of dimension (FLT), we get ( ),,,, R f h o ρ µ τ ( ) n c b a T L L FT L FT L K L F 4 ( ) n c b a h o R k... ρ µ τ L length R L F area force h o / τ ( ) 4 3 3 /. / / / / / FTL m N s L FT L a F L M T L µ ρ ( ) ( ) ( ) ( ) ( ) n c b a T L L FT FTL L K FL / 4
Determining Shear Stress 0 According to dimensional homogeneity, the dimension must be equal on either side of the equation, i.e., ( ) ( ) ( ) ( ) ( ) n c b a T L L FT FTL L K FL / 4 ) ( 0 : ) ( 4 : ) ( : iii n c b T ii n c b a L i c b F Solving three equations, we get ; ; n c n b n a Substituting values back in above equation ( ) ( )...... R k R k R k n h n n n n h n c b a h o ρ µ ρ ρ µ ρ µ τ ( ) R k n e τ o ρ Setting we get C f o ρ τ Where, C f is coefficient of friction ( ) n R e k / C f
Determining Shear Stress Now substituting the equation of avg. shear stress in equation of head loss, h f C f ρ R h For circular pipe flows, R h D/4 h f g4d L C f gr 4C f L 4C f f h L L D g L D g τ h o f C f ρ τ ol R h / Where, f is a friction factor. i.e., f 4C f f ( Re) The above equation is known as pipe friction equation and as Darcy- Weisbach equation. It is used for calculation of pipe-friction head loss for circular pipes flowing full (laminar or turbulent)
Friction Factor for Laminar and Turbulent Flows in Circular Pipes Smooth and Rough Pipe Mathematically; Smooth Pipe Rough Pipe e < δ v e >4δ v Transitional mode e δ v Turbulent flow near boundary Roughness height Smooth pipe Rough pipe e < δ v e > δ v Thickness of viscous sub-layer δ v e 4δ v δ v 4.4ν f 4.4D Re f
Friction Factor for Laminar and Turbulent Flows in Circular Pipes For laminar flow Re < 000 f 64 Re For turbulent flow From Nikuradse experiments Re log f f.5 Re.8log f 6.9 Blacius Eq. for smooth pipe flow Re > 4000 for smooth pipe flow Colebrook Eq. for smooth pipe flow on-karman Eq. for fully rough flow f 3.7 log f e / D Colebrook Eq. for turbulent flow in all pipes e / D.5 log 3.7 Re f f 0.36 0.5 Re 3000 Re 0 5 u u max y r o / 7 Seventh-root law Halaand Eq. For turbulent flow in all pipes f e / D.8log 3.7. 6.9 Re 3
Friction Factor for Laminar and Turbulent Flows in Circular Pipes The Moody chart or Moody diagram is a graph in nondimensional form that relates the Darcy-Weisbach friction factor, Reynolds number and relative roughness for fully developed flow in a circular pipe. The Moody chart is universally valid for all steady, fully developed, incompressible pipe flows. 4
Friction Factor for Laminar and Turbulent Flows in Circular Pipes For laminar flow For non-laminar flow f 64 e / D.5 log f 3.7 Re f R e Colebrook eq. 5
Friction Factor for Laminar and Turbulent Flows in Circular Pipes The friction factor can be determined by its Reynolds number (Re) and the Relative roughness (e/d) of the Pipe.( where: e absolute roughness and D diameter of pipe) 6
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Problem Types Type : Determine f and hf, Type : Determine Q Type 3: Determine D 8
Problem Find friction factor for the following pipe e0.00 ft Dft Kinematic iscosity, ν4.x0-6 ft /s elocity of flow, 0.4ft/s Solution: e/d0.00/0.00 RD/ν x0.4/(4.x0-6 )0000 From Moody s Diagram; f0.034 f f / log e D 3.7.5 Re f 9
Problem-Type Pipe dia 3 inch & L00m Re50,000 ʋ.059x0-5 ft /s (a): Laminar flow: f64/re64/50,0000.008 Re D ν 50000 (3/).059 0 5. ft / s fl 0.008(00)(. ) H Lf 0. 0357 gd (3.)(3/) ft 0
Problem-Type Pipe dia 3 inch & L00m Re50,000 ʋ.059x0-5 ft /s (b): Turbulent flow in smooth pipe: i.e.: e0 f f / log e D 3.7 0.009.5 Re f log 0 3.7.5 50000 f fl 0.009(00)(. ) H Lf 0. 58 ft gd (3.)(3/)
Problem-Type Pipe dia 3 inch & L00m Re50,000 ʋ.059x0-5 ft /s (c): Turbulent flow in rough pipe: i.e.: e/d0.05 f f / log e D 3.7 0.070.5 Re f log 0.05 3.7.5 50000 f fl 0.070(00)(. ) H Lf. 0 ft gd (3.)(3/)
Problem-Type h L? L 000 m; D 0.5m; e 0. 0005m 3 6 Q 0.05m / s; ν.306 0 m / s R D / ν 3 6 (.039 0.5) /(.306 0 ) e / D 0.0005/ 0.5 0.00 From Moody's Diagram f 0.045 0 fl h L 5. 39m gd 5 Q Q / A.039m / s
Problem-Type h L fl / gd 4
Problem-Type For laminar flow For non-laminar flow f 64 e / D.5 log f 3.7 Re f R e Colebrook eq. 5
Problem-Type 3 6
Problem h f Lf e / D log 3.7 fl gd.5 Re f 7
Problem 8
Problem 9
MINOR LOSSES Each type of loss can be quantified using a loss coefficient (K). Losses are proportional to velocity of flow and geometry of device. H m K g Where, H m is minor loss and K is minor loss coefficient. The value of K is typically provided for various types/devices NOTE: If L > 000D minor losses become significantly less than that of major losses and hence can be neglected. 30
Minor Losses These can be categorized as. Head loss due to contraction in pipe. Sudden Contraction. Gradual Contraction. Entrance loss 3. Head loss due to enlargement of pipe 3. Sudden Enlargement 3. Gradual Enlargement 4. Exit loss 5. Head loss due to pipe fittings 6. Head loss due to bends and elbows 3
Minor Losses Head loss due to contraction of pipe (Sudden contraction) A sudden contraction (Figure) in pipe usually causes a marked drop in pressure in the pipe because of both the increase in velocity and the loss of energy of turbulence. Head loss due to sudden contraction is H m K c g Where, k c is sudden contraction coefficient and it value depends up ratio of D/D and velocity ( ) in smaller pipe 3
Minor Losses Head loss due to enlargement of pipe (Gradual Contraction) Head loss from pipe contraction may be greatly reduced by introducing a gradual pipe transition known as a confusor as shown Figure. Head loss due to gradual contraction is H m K c ' g Where, k c is gradual contraction coefficient and it value depends up ratio of D /D and velocity ( ) in smaller pipe 33
Minor Losses Entrance loss The general equation for an entrance head loss is also expressed in terms of velocity head of the pipe: H m Ke g The approximate values for the entrance loss coefficient (Ke) for different entrance conditions are given below 34
Minor Losses head loss due to enlargement of pipe (Sudden Enlargement) The behavior of the energy grade line and the hydraulic grade line in the vicinity of a sudden pipe expansion is shown in Figure The magnitude of the head loss may be expressed as ( ) H m g 35
Minor Losses head loss due to enlargement of pipe (Gradual Enlargement) The head loss resulting from pipe expansions may be greatly reduced by introducing a gradual pipe transition known as a diffusor The magnitude of the head loss may be expressed as H m K e ( ) g The values of Ke vary with the diffuser angle (α). 36
Minor Losses Exit Loss A submerged pipe discharging into a large reservoir (Figure ) is a special case of head loss from expansion. 37 Exit (discharge) head loss is expressed as H m K d ( ) g where the exit (discharge) loss coefficient K d.0.
Minor Losses Head loss due to fittings valves Fittings are installed in pipelines to control flow. As with other losses in pipes, the head loss through fittings may also be expressed in terms of velocity head in the pipe: H m K f g 38
Minor Losses Head loss due to bends The head loss produced at a bend was found to be dependent of the ratio the radius of curvature of the bend (R) to the diameter of the pipe (D). The loss of head due to a bend may be expressed in terms of the velocity head as H m K b g For smooth pipe bend of 90 0, the values of Kb for various values of R/D are listed in following table. 39
Minor Losses 40
4 Numerical Problems
4 Numerical Problems
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Fluid Mechanics Fluid Dynamics: (ii) Hydrodynamics: Different forms of energy in a flowing liquid, head, Bernoulli's equation and its application, Energy line and Hydraulic Gradient Line, and Energy Equation Dr. Mohsin Siddique Assistant Professor
Forms of Energy (). Kinetic Energy: Energy due to motion of body. A body of mass, m, when moving with velocity,, posses kinetic energy, KE m (). Potential Energy: Energy due to elevation of body above an arbitrary datum PE mgz (3). Pressure Energy: Energy due to pressure above datum, most usually its pressure above atmospheric PrE h!!! m and are mass and velocity of body Z is elevation of body from arbitrary datum m is the mass of body
Forms of Energy (4). Internal Energy: It is the energy that is associated with the molecular, or internal state of matter; it may be stored in many forms, including thermal, nuclear, chemical and electrostatic. 3
HEAD Head: Energy per unit weight is called head Kinetic head: Kinetic energy per unit weight KE Kinetic head m / mg QWeight mg Weight g Potential head: Potential energy per unit weigh PE Potential head / Weight ( mgz ) mg Z Pressure head: Pressure energy per unit weight Pressure head PrE Weight P 4
TOTAL HEAD TOTAL HEAD Kinetic Head Potential Head Pressure Head g Z P Total Head H Z P g 5
Bernoulli s Equation It states that the sum of kinetic, potential and pressure heads of a fluid particle is constant along a streamline during steady flow when compressibility and frictional effects are negligible. i.e., For an ideal fluid, Total head of fluid particle remains constant during a steady-incompressible flow. Or total head along a streamline is constant during steady flow when compressibility and frictional effects are negligible. Total Head Z P g constt Z H P g H Z P g Pipe 6
Derivation of Bernoulli s Equation Consider motion of flow fluid particle in steady flow field as shown in fig. Applying Newton s nd Law in s- direction on a particle moving along a streamline give Assumption: Fluid is ideal and incompressible Flow is steady Flow is along streamline elocity is uniform across the section and is equal to mean velocity Only gravity and pressure forces are acting F s ma s Eq() Where F is resultant force in s- direction, m is the mass and a s is the acceleration along s-direction. d dsd dsd a s dt dsdt dtds d ds Eq() 7 Fig. Forces acting on particle along streamline
Derivation of Bernoulli s Equation F s PdA ( P dp) da W sinθ Substituting values from Eq() and Eq(3) to Eq() ( P dp) PdA da W sinθ dz dpda ρ gdads ρdads ds m d ds Eq(3) d ds sinθ dz ds Cancelling da and simplifying dp ρ gdz ρd 8 Note that d d dp ρ gdz ρ d Eq(4) Eq(5) Fig. Forces acting on particle along streamline Wweight of fluid W mg ( ρdads)g Wsin( θ) component acting along s-direction da Area of flow dslength between sections along pipe
Derivation of Bernoulli s Equation Dividing eq (5) by ρ dp ρ Integrating gdz d dp gdz ρ d Assuming incompressible and steady flow P gz ρ Dividing each equation by g contt 0 contt Eq (6) Eq (7) Eq (8) Hence Eq (9) for steadincompressible fluid assuming no frictional losses can be written as Z P Z g ( Total Head) ( Total Head) P g Above Eq(0) is general form of Bernoulli s Equation Eq (0) 9 P ρg z g contt Eq (9)
Energy Line and Hydraulic Grade line Static Pressure : P Dynamic pressure : P Hydrostatic Pressure: ρgz z g H Pressure head Elevation head elocity head P ρgz ρ contt ρ / Total Head Multiplying with unit weight,, Stagnation Pressure: Static pressure dynamic Pressure P ρ P stag 0
Energy Line and Hydraulic Grade line Measurement of Heads Piezometer: It measures pressure head ( P / ). Pitot tube: It measures sum of pressure and velocity heads i.e., P g What about measurement of elevation head!!
Energy Line and Hydraulic Grade line Energy line: It is line joining the total heads along a pipe line. HGL: It is line joining pressure head along a pipe line.
Energy Line and Hydraulic Grade line 3
Energy Equation for steady flow of any fluid Let s consider the energy of system (Es) and energy of control volume(ecv) defined within a stream tube as shown in figure. Therefore, E s E C E out C E in C Eq() Because the flow is steady, conditions within the control volume does not change so E C Hence E 0 s E out C E in C Eq() Figure: Forces/energies in fluid flowing in streamt ube 4
Energy Equation for steady flow of any fluid Now, let s apply the first law of thermodynamics to the fluid system which states For steady flow, the external work done on any system plus the thermal energy transferred into or out of the system is equal to the change of energy of system External work done heat transferred change of energy ( flowwork shaftwork ) heat transferred Es out in ( flowwork shaftwork ) heat transferred E E C C Eq(3) Eq(4) Flow work: When the pressure forces acting on the boundaries move, in present case when p A and p A at the end sections move through s and s, external work is done. It is referred to as flow work. 5 Flow work p A s p A s p p Flow work g m p A s Q ρ A s p A s ρ A s m Eq(5) Steady flow
Energy Equation for steady flow of any fluid Shaft work: Work done by machine, if any, between section and Shaft work Shaft work weight time energy weight time A ( A s ) h m ( g m) h m Where, h m is the energy added to the flow by the machine per unit weight of flowing fluid. Note: if the machine is pump, which adds energy to the fluid, h m is positive and if the machine is turbine, which remove energy from fluid, h m is -ve HeatTransferred: The heat transferred from an external source into the fluid system over time interval t is Heat transferred A Heat transferred ds dt Q ds dt h ( A s ) Q H ( g m) Q H H t m t Eq(6) Eq(7) 6 Where, Q H is the amount of energy put into the flow by the external heat source per unit weight of flowing fluid. If the heat flow is out of the fluid, the value Q H is ve and vice versa
Energy Equation for steady flow of any fluid 7 Change in Energy: For steady flow during time interval t, the weight of fluid entering the control volume at section and leaving at section are both equal to g m. Thus the energy (PotentialKineticInternal) carried by g m is; ( ) ( ) I g z m g E I g z A ds t I g z dt ds A E I g z m g E I g z A ds t I g z dt ds A E out C out C in C in C α α α α α α α is kinetic energy correction factor and ~ Eq(8) Eq(9)
Energy Equation for steady flow of any fluid 8 Substituting all values from Eqs. (5),(6), (7), (8), & (9) in Eq(4) ( ) in C out E C E shaftwork lowwork ferred heat trans f ( ) ( ) I g z m g I g z m g m Q g m h g p p m g H m α α I g z I g z Q h p p H m α α I g z p Q h I g z p H m α α This is general form of energy equation, which applies to liquids, gases, vapors and to ideal fluids as well as real fluids with friction, both incompressible and compressible. The only restriction is that its for steady flow. Eq(0)
Energy Equation for steady flow of incompressible fluid 9 For incompressible fluids Substituting in Eq(0), we get ( ) I I g z p Q h g z p H m ( ) H m Q I I g z p h g z p m h L g z p h g z p Eq() ( ) H L Q I I h Q Where h L (I -I )-Q H head loss. It equal to is gain in internal energy minus any heat added by external source. H m is head removed/added by machines. It can also be referred to head loss due to pipe fitting, contraction, expansion and bends etc in pipes.
Energy Equation for steady flow of incompressible fluid 0 In the absence of machine, pipe fitting etc, Eq() can be written as When the head loss is caused only by wall or pipe friction, h L becomes h f, where h f is head loss due to friction h L g z p g z p Eq()
Power Rate of work done is termed as power PowerEnergy/time Power(Energy/weight)(weight/time) If H is total headtotal energy/weight and Q is the weight flow rate then above equation can be written as In BG: In SI: Power(H)(Q)QH Power in (horsepower)(h)(q)/550 Power in (Kilowatts)(H)(Q)/000 horsepower550ft.lb/s
Reading Assignment Kinetic energy correction factor Limitation of Bernoulli s Equation Application of hydraulic grade line and energy line
NUMERICALS 5.. 3
5..3 4
5.3. 5
5.3.4 6
5.3.6 7
5.9.6 8
Momentum and Forces in Fluid Flow We have all seen moving fluids exerting forces. The lift force on an aircraft is exerted by the air moving over the wing. A jet of water from a hose exerts a force on whatever it hits. In fluid mechanics the analysis of motion is performed in the same way as in solid mechanics - by use of Newton s laws of motion. i.e., F ma which is used in the analysis of solid mechanics to relate applied force to acceleration. In fluid mechanics it is not clear what mass of moving fluid we should use so we use a different form of the equation. 9 F ma d ( m) dt s
Momentum and Forces in Fluid Flow Newton s nd Law can be written: The Rate of change of momentum of a body is equal to the resultant force acting on the body, and takes place in the direction of the force. F d ( m) dt s F m Sum of all external forces on a body of fluid or system s Momentum of fluid body in direction s The symbols F and represent vectors and so the change in momentum must be in the same direction as force. F dt d ( m) s 30 It is also termed as impulse momentum principle
Impact of a Jet on a Plane 3
Impact of a Jet on a Plane 3
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