7/9/2013 TOPIC 2: FLOW IN PIPES AND CHANNELS OBJECTIVES FLOW REGIMES. laminar. turbulent

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1 7/9/03 TOPIC : FOW IN PIPES N CHNNES OBJECTIVES. Calculate te friction factor for a pipe using te Colebrook-Wite equation.. Undertake ead loss, discarge and sizing calculations for single pipelines. 3. Use ead-loss vs discarge relationsips to calculate flow in pipe networks. 4. Relate normal dept to discarge for uniform flow in open cannels. laminar FOW REGIMES turbulent V Re ν V = average (or bulk) velocity = diameter For a pipe, Re crit 300

2 7/9/03 EVEOPMENT ENGTH devel 0.06 Re /6 4.4 Re (laminar ) (turbulent) PIPE FOW: BNCE OF FORCES p z r l direction of flow mg p(πr ) ( p Δp)(πr ) mg sin θ pressure force pressure gravity friction weigt τ(πr Δ l) 0 friction m ρπr Δl Δz Δp(πr ) ρπr g Δz τ(πr Δ l) 0 sinθ Δl Δ( p ρgz)(πr ) τ(πr Δ l) 0 p* p ρgz p+p Δ( p ρgz) τ 0 Δl r Δp* τ r Δ l (ownstream) pressure Gradient: Δ p* dp* G Δl dl τ Gr MINR PIPE FOW Balance of forces: τ Gr (stress pressure gradient) Viscous stress: du τ μ dr (stress velocity gradient) du G r dr μ r R Gr u 4μ constant u 0 on r R G u ( R r ) 4μ r u u0( ) R GR u0 4μ

3 7/9/03 EXMPE, PGE 4 G u ( R r ) 4μ Pressure gradient: Δp * ρgf G Find, from te velocity distribution given above: (a) te centreline velocity, u 0 ; (b) te average velocity V; (c) (d) te flow rate, Q, in terms of ead loss and pipe diameter; te friction factor λ, defined by as a function of Reynolds number, Re. f V λ ( ) g QUESTIONS Wic forces are in balance in steady pipe flow? Pressure, gravity, friction How can one combine te effects of pressure and weigt? Via piezometric pressure p* = p + ρgz How do we convert between pressure and ead? p p = ρg ρ g How do we define (a) dynamic pressure; (b) dynamic ead? V ρv g How do we define te skin-friction coefficient? c f τ V w ρ PIPE FOW: BNCE OF FORCES p z w direction of flow p+p mg π m ρ 4 Δz sin θ p* p ρgz π Δp 4 net pressure force pressure gravity friction mg sin θ τ w π 0 weigt wall friction π π Δp ρgδz τw π 4 4 π Δ( p ρgz) τ w π 4 Δp* 4 τ w efinition of skin-friction coefficient: τ w c f ( ρv ) Δp * 4c f ( ρv ) 3

4 7/9/03 RCY-WEISBCH EQUTION Δp * λ ( ρ f V Pressure loss due to friction = λ dynamic pressure Head loss due to friction = V λ ( ) g λ ) dynamic ead λ 4c f f = frictional ead loss λ = friction factor = lengt of pipe = diameter V = average velocity ( Q) EXMPE SHEET, Q 0.75 m diameter pipe carries 0.6 cumec. t point, elevation 40 m, a Bourdon gauge fitted to te pipe records.75 bar, wile at point B, elevation 34 m and.5 km along te pipe a similar gauge reads. bar. etermine te flow direction and calculate te friction factor. CCUTING THE FRICTION FCTOR V efined by: f λ ( ) g aminar flow: 64 λ Re Turbulent flow two limits: Smoot: Roug: λ.0log 0 Re λ log0 λ k s Colebrook-Wite Equation: λ k.0log s Re λ 4

5 l Transition 7/9/03 TYPIC ROUGHNESS FOR COMMERCI PIPES Material k s (mm) Riveted steel Concrete Wood stave Cast iron 0.6 Galvanised iron 0.5 spalted cast iron 0. Commercial steel or wrougt iron rawn tubing Glass 0 (smoot) MOOY CHRT k s/ aminar l = 64/Re smoot-walled limit E0.0E03.0E04.0E05.0E06.0E07 Re = V/n OTHER OSSES oss coefficient K eadloss K ( dynamic ead) V K g Commercial pipe fittings Fitting K Globe valve 0 Gate valve wide open 0. Gate valve ½ open elbow 0.9 Side outlet of T-junction.8 Entry/exit losses Configuration K Bell-mouted entry 0 brupt entry 0.5 Protruding entry.0 Bell-mouted exit 0. brupt enlargement 0.5 5

6 7/9/03 PIPEINE CCUTIONS Q Main esign Parameters: Head loss: Quantity of flow: iameter: Q Oter Parameters: engt: Rougness: k s Kinematic viscosity: ν Minor loss coefficient: K Metod: available ead = sum of ead losses along te pipe CCUTION FORMUE. Head osses V (λ K)( ) g. oss coefficients e.g. friction factor (Colebrook-Wite): λ.0log 0 k s Re λ HES T THE ENS OF PIPES Smoot exit to a downstream reservoir: H = z H = z No residual dynamic ead at exit. z z Free jet to atmospere (or abrupt exit to a tank): H = z H = z + V /g ynamic ead must be included at exit. z z V 6

7 7/9/03 TYPIC PIPEINE CCUTIONS Type flow Know: diameter, ead Find: discarge Q Easy! Type ead Know: diameter, discarge Q Find: ead Solve Colebrook-Wite equation (iteratively) Type 3 size Know: discarge Q, ead Find: diameter Solve Colebrook-Wite and ead-loss equations simultaneously and iteratively EXMPE SHEET, Q8 Crude oil (specific gravity 0.86, kinematic viscosity m s ) is to be pumped from a barge to a large storage tank. Te pipeline is orizontal and of diameter 50 mm, lengt 400 m and rougness 0. mm. It enters te tank 8 m below te level of oil in te tank. Wen te control valve is fully open te static pressure at pump delivery is 30 5 Pa gauge. Ignore minor losses due to pipe fittings, entrance/exit losses etc. Pump Control valve 8 m Storage tank Barge Find: (a) (using ydrostatics) te gauge pressure were te pipe enters te tank; (b) (from te pressures at te two ends) te ead loss along te pipeline; (c) te volumetric flow rate in te pipeline. If te pump delivery pressure remains te same but a valve reduces te flow by alf, find: (d) te ead loss at te valve; (e) te power loss at te valve. (a) EXMPE SHEET, Q5 pipeline is to be constructed to bring water from an upland storage reservoir to a town 30 km away, at an elevation 50 m below te water level of te reservoir. In summer te pipeline must be able to convey up to 5000 cubic metres per day. If te pipe is fabricated from material of rougness 0.3 mm, find te required diameter. (b) uring te winter, water requirements fall to only 3000 cubic metres per day and te excess ead available can be used to drive a small turbine. If te turbine as an efficiency of 75% find te maximum power output. Te Colebrook-Wite equation is k.5.0log s 0 λ 3.7 Re λ were λ = friction factor, k s = rougness, = pipe diameter, V = average velocity, Re V/ν = Reynolds number. For water, take density ρ = 000 kg m 3 and kinematic viscosity ν =.00 6 m s. 7

8 energy grade line 7/9/03 EXMPE, PGE 5 reservoir is to be used to supply water to a factory 5 km away. Te water level in te reservoir is 60 m above te factory. Te pipe lining as rougness 0.5 mm. Minor losses due to valves and pipe fittings can be accommodated by a loss coefficient K = 80. Calculate te minimum diameter of pipe required to convey a discarge of 0.3 m 3 s. GRPHIC REPRESENTTION OF HE Energy Grade ine (EG) p V z ρg g Total ead Hydraulic Grade ine (HG) p z ρg Piezometric ead GRPHIC REPRESENTTION OF HE Pipe friction only reservoir ydraulic grade line pipeline V /g p/g reservoir Pipe friction wit minor losses (exaggerated), including cange in pipe diameter. entry loss pipeline EG HG exit loss EG Pumped system HG pipeline pump 8

9 7/9/03 PIPE NETWORKS: EXMPE B C Wic way does te flow go in pipe B? PIPE NETWORKS: EECTRIC NOGUE B V C 0 V 0 90 Wat are te voltages at B and? Wic way does te current go in B? PIPE NETWORKS: BSIC RUES. Continuity: at any junction, Q Q in out total flow in = total flow out. Eac point as a unique ead, H 3. Eac pipe as a ead-loss vs discarge (resistance) relation: = αq 9

10 7/9/03 EECTRIC NOGY Continuity; unique ead Kircoff s aws ead H potential V discarge Q current I Resistance law: ead loss H Q potential difference V I Wat are te ydraulic analogues of: a resistor? a capacitor? an inductor? a transistor? PIPES IN SERIES N PRE Pipes in series Q = Q = Q H = H + H Pipes in parallel same flow add ead canges α α α R R R H = H = H Q = Q + Q same ead cange add flows α α α R R R JUNCTION PROBEMS: METHO C J? B Metod: djust H J until net flow out of J = 0 (0) Establis te ead vs discarge relations for all pipes H H Q etc. α J () Guess an initial value of ead at te junction, H J. () Calculate flow rates in all pipes, Q J etc. (3) Calculate net flow out of junction, Q J (4) djust te ead at te junction, H J, until net flow out of junction = 0 Q Q J JB Q JC 0 J Q Q JB JC 0

11 7/9/03 EXMPE SHEET, Q7 In a water-storage sceme tree reservoirs, B and C are connected by a single junction J as sown. Te water levels in, B and C are 300 m, 00 m and 40 m respectively. Te pipeline properties are given below. Friction factors may be assumed constant and minor losses may be neglected. 300 m Pipeline J JB JC engt (m) iameter (m) Friction factor λ m B Calculate te total flow in eac pipe and te direction of flow in pipe JB if: J C 40 m (a) tere is a valve-regulated flow of 50 s to reservoir C but water flows freely under gravity in te oter pipes; (b) water flows freely under gravity in all pipes. FOW IN PIPES N OPEN CHNNES PIPE FOW OPEN-CHNNE FOW Fluid: IQUIS or GSES IQUIS (free surface) riven by: PRESSURE, GRVITY or BOTH GRVITY (down slope) Size: IMETER HYRUIC RIUS Volume: FIS pipe epends on EPTH Equations: RCY-WEISBCH (ead loss) MNNING S FORMU COEBROOK-WHITE (friction factor) NORM FOW Q Normal flow = steady, uniform flow (constant-dept flow under gravity) t best an approximation for rivers / natural cannels For any given Q tere is a particular normal dept

12 7/9/03 NORM FOW V /g EG HG (free surface): p = 0 f In normal flow: Equal ydrostatic pressure forces at any cross section ownslope component of weigt balances bed friction Cannel bed, free surface (= HG) and EG are parallel; i.e. loss of fluid ead equals drop in eigt Usual to assume small slopes PRT : BNCE OF FORCES = area of fluid cross-section P = wetted perimeter P b mg downslope component of weigt = friction on sides mgsinθ τb wetted surface area ρg sinθ τbp ρg sinθ τb P Hydraulic radius (*** depends on dept ***): R cross sectional area wetted perimeter P Normal-flow relationsip: τ b ρgr S PRT : EXPRESSION FOR FRICTION τ b ρgr S R is te ydraulic radius c f ( ρv ) ρgrs definition of te skin-friction coefficient V g RS c f Cézy s Formula: V C R S Robert Manning (compilation of experimental data): C R function of rougness /6 R n /6 Manning s Formula: V R n / 3 S /

13 7/9/03 MNNING S ROUGHNESS COEFFICIENT Cannel type Surface n (m /3 s) Glass 0.0 Brass 0.0 Steel, smoot 0.0 painted 0.04 riveted 0.05 Cast iron 0.03 rtificial lined cannels Concrete, finised 0.0 unfinised 0.04 Planed wood 0.0 Clay tile 0.04 Brickwork 0.05 spalt 0.06 Corrugated metal 0.0 Rubble masonry 0.05 Clean 0.0 Excavated eart cannels Gravelly 0.05 Weedy 0.03 Stony, cobbles Clean and straigt 0.03 Natural cannels Sluggis, deep pools 0.04 Major rivers Pasture, farmland Floodplains igt brus 0.05 Heavy brus Trees 0.5 CCUTION FORMUE (SUMMRY) Manning s Formula: V R n / 3 S / Metod For a given cannel: V = average velocity n = Manning s rougness parameter S = slope (gradient) R = ydraulic radius P cross- sectionalarea wettedperimeter. Write area and perimeter P as functions of a parameter (often dept, ). Calculate ydraulic radius 3. Calculate average velocity 4. Calculate quantity of flow Two Main Types of Problem Given find Q Given Q find EXMPE SHEET, Q0 V-saped cannel wit sides sloping at 30º to te orizontal as a gradient of m in 00 m and an estimated Manning s n of 0.0 m /3 s. Calculate: (a) te discarge for a dept of 0.5 m; (b) te dept wen te discarge is m 3 s. 3

14 7/9/03 EXMPE SHEET, Q concrete pipe 750 mm in diameter is laid to a gradient of in 00. Te estimated value of Manning s n is 0.0 m /3 s. Calculate te discarge wen: (a) te pipe is full; (b) te dept is 90% of maximum. Explain wy te answer in (b) exceeds tat in (a). EXMPE SHEET, Q5 culvert used to divert run-off as a rectangular cross section wit base widt 0.4 m and side eigts of 0.3 m. Manning s coefficient may be taken as n = 0.0 m /3 s. (a) Find te minimum slope S necessary to carry a discarge Q = 0.3 m 3 s. (b) If te slope from part (a) is doubled for te same discarge, calculate dept of flow. CONVEYNCE (a) (b) Manning s formula: iscarge: V R n Q V / 3 S / R P / Q n P 3 / S / Q KS K n P 5/3 /3 conveyance For compound cannels (e.g. river plus flood plain) simply add te conveyances: 3 flood plain river flood plain K eff K K K3 4

15 7/9/03 COMMON SHPES OF CHNNE rectangle trapezoid circle R b b area b b tanα R ( θ sin θ) wetted perimeter P b b sin α Rθ 5