Chapter 6. Losses due to Fluid Friction

Transcription

1 Chapter 6 Losses due to Fluid Friction 1

2 Objectives To measure the pressure drop in the straight section of smooth, rough, and packed pipes as a function of flow rate. To correlate this in terms of the friction factor and Reynolds number. To determine the influence of pipe fittings on pressure drop

3 Calculation of Head (Energy) Losses: In General: When a fluid is flowing through a pipe, the fluid experiences some resistance due to which some of energy (head) of fluid is lost. Energy Losses (Head losses) Major Losses loss of head due to pipe friction and to viscous dissipation in flowing water Minor losses Loss due to the change of the velocity of the flowing fluid in the magnitude or in direction as it moves through fitting like Valves, Tees, Bends and Reducers. 3

4 Losses due to Friction Mechanical energy equation between locations 1 and in the absence of shaft work: P1 P V1 V ( ) ( ) ( z1 z) g g g g For flow in a horizontal pipe and no diameter change (V1=V), then : F P 1 P Hagen-Poiseuille Law-E5.10 F Q x 18 4 D o OR Or F/g= h Loss = 18μQL/(πρgD 4 ) and F/g F g hloss Because of (5.4) : F 4x D (6.13) (6.14) h L w where Thus, the shear stress at the wall is responsible for the losses due to friction F u =Friction head/unit mass P1 P g F in J/kg 4 Q m

5 Losses due to Friction Total head loss, h L (=F/g), is regarded as the sum of major losses and minor losses h L major, due to frictional effects in fully developed flow in constant area tubes, h L minor, resulting from entrance, fitting, area changes, and so on. 5

6 Losses due to Friction/The Friction Factor In order to determine an expression for the losses due to friction we must resort to experimentation. F L V D By introducing the friction factor, f: where L=length of the pipe, D=diameter of the pipe, V=velocity, F f L V D. (6.15) F P 1 P where f ( L / F D)( V / ) or f P( D / L) V / f is called Darcy friction factor 6

7 Friction Factor now later ( P gz V ) P work P The Darcy friction factor f is defined as f P( D / L) V / We know the wall shear stress f V w / and L 0 τ w P D 4 (E6-1) L, major P L The friction factor is the ratio between wall shear stress and flow inertial force. L P P L, min or 7

8 ( P gz Major Loss and Friction Factor V ) P work P L 0 P With the introduction of friction factor, we can calculate major loss by L P L, major P L, min or P L, Major f L D V (E6-) Friction factor Pipe geometry factor Dynamic pressure Therefore, our job now is find the friction factor f for various flows. 8

9 Friction Factor Case1: Laminar Pipe Flow For a pipe with a length of L, the pressure gradient is constant, the pressure drop based on Hagen-Poiseuille Law, P P P P 18Q 18( VA) 18( V 3 VL L / / D D Dividing both sides by the dynamic pressure V / and L/D L / D / 4 4), Q D 4 L / VA D 4 f P( D / L) 3VL / D D 64 V / V / L VD 64 Re We have f 64 Re 9

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11 Example 1 losses in 11

12 According to Darcy 1

13 Example 13

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15 Case : Turbulent Flow When fluids flow at higher flow rates, the streamlines are not steady, not straight and the flow is not laminar. Generally, the flow field will vary in both space and time with fluctuations that comprise "turbulence When the flow is turbulent the velocity and pressure fluctuate very rapidly. The velocity components at a point in a turbulent flow field fluctuate about a mean value. Time-averaged velocity profile can be expressed in terms of the power law equation, n =7 is a good approximation. u V max r 1 R 1/ n 15

16 Friction Factor-Turbulent Pipe Flow For a laminar flow, the friction factor can be analytically derived. It is impossible to do so for a turbulent flow so that we can only obtain the friction factor from empirical results. In addition, most pipes, except glass tubing, have rough surfaces. The pipe surface roughness is quantified by a dimensionless number, relative pipe roughness (ε / D ), 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 emerge beyond the laminar sublayer and affect the flow to a certain degree. Therefore, the friction factor f can be generally written as a function of Reynolds number and pipe relative roughness There are several theoretical models available for the 16 prediction of shear stresses in turbulent flow.

17 Surface Roughness Additional dimensionless group /D need to be characterized Thus more than one curve on friction factor- Reynolds number plot Fanning diagram or Moody diagram Depending on the laminar region. If, at the lowest Reynolds numbers, the laminar portion corresponds to f =16/Re Fanning Chart (or f = 64/Re Moody chart) 17

18 Pipe Surface Roughness 18

19 Friction Factor of Turbulent Flow If the surface protrusions are within the viscous layer, the pipe is hydraulically smooth; f Re 1/ 4 If the surface protrusions extend into the buffer layer, f is a function of both Re and /D; f (Re, D) For large protrusions into the turbulent core, f is only a function of /D. f ( D) 19

20 Friction Factor for Smooth, Transition, f P L Smooth pipe, Re>3000 and Rough Turbulent flow D U Rough pipe, [ (D/ε)/(Re ƒ) <0.01] Transition function for both smooth and rough pipe Or 1 f 1 f * logre* f 0.4 f 0.079Re 0.5 f 4.0 * log D * log D D/ * log 4.67 Re f 1 0

21 Fanning Diagram 1 f 4.0 * log D D/ * log 4.67 Re f 1 1 f 4.0 * log D.8 f =16/Re 1

22 Friction Factor The Moody Chart

23 Friction Factor The Moody Chart 3

24 Example: Comparison of Laminar or Turbulent pressure Drop Air under standard conditions flows through a 4.0-mmdiameter drawn tubing with an average velocity of V = 50 m/s. For such conditions the flow would normally be turbulent. However, if precautions are taken to eliminate disturbances to the flow (the entrance to the tube is very smooth, the air is dust free, the tube does not vibrate, etc.), it may be possible to maintain laminar flow. (a) Determine the pressure drop in a 0.1-m section of the tube if the flow is laminar. (b) Repeat the calculations if the flow is turbulent. Straight and horizontal pipe and same diameters give same velocity: Z 1 =Z =0 V 1 =V and thus p / g h Loss 4

25 Solution 1/ Under standard temperature and pressure conditions =1.3kg/m 3, μ= Ns/m The Reynolds number R e VD /... 13,700 Turbulent flow If the flow were laminar and using Darcy friction f=64/re=` = p f 1 V kPa D If the flow were laminar and using Fanning friction 1 p 4 f V kPa D f=16/re=` =

26 Solution / If the flow were turbulent From Moody chart f=φ(re, smooth pipe) =0.08 p f 1 V kPa D From Fanning chart f=φ(re, smooth pipe) = p 4 f V kPa D 6

27 Example Straight and horizontal pipe and same diameters give same velocity: Z 1 =Z =0 V 1 =V and thus p gh Loss pump pressure 7

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30 Example: Determine Head Loss Crude oil at 140 F with γ=53.7 lb/ft 3 and μ= lb s/ft (about four times the viscosity of water) is pumped across Alaska through the Alaska pipeline, a 799-mile-along, 4-ft-diameter steel pipe, at a maximum rate of Q =.4 million barrel/day = 117ft 3 /s, or V=Q/A=9.31 ft/s. Determine the horsepower needed for the pumps that drive this large system. 30

31 Solution 1/ The energy equation between points (1) and () p 1 V1 p V z1 hp z hl g g h P is the head provided to the oil by the pump. Assume that z 1 =z, p 1 =p =V 1 =V =0 (large, open tank) Minor losses are negligible because of the large lengthto-diameter ratio of the relatively straight, uninterrupted pipe. V hl hp f ft D g f=0.014 from chart ε/d=( ft)/(4ft), Re=.. 31

32 Solution / The actual power supplied to the fluid power=q P =Qρgh unit of power (SI): Watt =N.m/s=J/s Power 1hp gqh P hp 550 ft lb / s 3