4. Frictional Losses in Hydraulic Pipelines Laminar & Turbulent Flow Reynolds Number Darcy s Equation Moody Diagram Frictional Losses & Friction Factor Losses in Valves & Fittings: K Factor Equivalent-Length Technique Hydraulic Circuit Analysis 4.A Bulk Modulus 4.7.A Orifice Flow 오리피스 -1-
4.1 Energy Transfer including Energy Losses Energy Losses Frictional Fluid Flow Valves & Fittings: Bends, couplings, tees, elbows, filters, strainers Selection of the proper sizes Pipes, Valves, Fittings -2-
유체의흐름 -3-
4.2 Laminar Flow & Turbulent Flow Laminar Flow Turbulent Flow -4-
Fluid Flow in Pipe Line 유체의흐름에영향을미치는힘의종류 체적력 : 중력, 부력 관성력 내부마찰력 : 점성력 표면장력 : 전기전자력 대부분의경우, 체적력과관성력이지배적인영향을미침 Reynold Number: 점성력에대한관성력의비 관성력 ρua NR = = 점성력 μ Reynolds Number에의한유동의분류 ( 관내유동의경우 ) Laminar Flow ( 점성력에의해지배되는흐름 ): N R <2000 Transition Flow ( 천이구역 ): 2000<N R <4000 Turbulent Flow ( 관성력에의해지배되는흐름 ): 4000<N R -5-
Reynolds Experiment -6-
4.3 Reynolds Number Reynolds Experiment The nature of the flow depends on the dimensionless parameter N R = vdρ μ If N R is less than 2000, the flow is laminar. If N R is greater than 4000, the flow is turbulent. Reynolds numbers between 2000 and 4000 cover a critical zone between laminar and turbulent flow. -7-
4.4 Darcy s Equation Head Loss (H L ) Losses in pipes Losses in valves and fittings Head losses in pipes: Darcy s Equation H L 2 L v = f D 2g f: friction factor (dimensionless) L: length of pipe (m) D: pipe inside diameter (m) v: average fluid velocity (m/s) g: acceleration of gravity (m/s 2 ) -8-
4.5 Frictional Losses Laminar Flow: Hagen-Poiseuille equation f = H 64 N R 64 L v = NR D g 2 Transition Flow 2 L Turbulent Flow: The Moody diagram -9-
4.6 Effect of Pipe Roughness Pipe absolute roughness (ε) Relative roughness = ε D Typical values of absolute roughness -10-
Moody Diagram -11-
Finding Friction Factor using Moody Diagram -12-
4.7 Losses in Valves & Fittings The K Factor 2 v H L K = 2g K factors of common valves and fittings Globe Valve Gate Valve (10.0) (0.19) -13-
Bend & Elbows 45 o elbow (0.42) 90 o elbow (0.75) Tee (1.8) Return bend Ball check valve (2.2) 2) (4.0) -14-
Pressure Drop vs. Flow Rate Curves Directional Control Valve Orifice Flow : -15-
4.8 Equivalent-Length Technique The equivalent length of a valve or fitting H = H L( valve or fitting ) L( pipe) K 2 2 v L e v = f 2g D 2g L e = KD f -16-
4.9 Hydraulic Circuit Analysis Energy losses due to friction -17-
4.A Bulk Modulus 체적팽창계수 (β ) 의정의그림에서초기에 P o A 의힘이작용하는피스톤에 ΔP A 만큼의힘을추가로가하면밀폐된실린더압력의변화량을체적의내의체적변화량은체적팽창계수변화비로나눈값 β 를사용하여다음과같이나타낼 ( ΔP ΔP β = = ) 수있다. ΔV / V ΔV V 유체시스템의스프링 ( 감쇠 ) 효과를나타내는변수 유체압축율 (compressibility) 의역수를나타낸다. 즉, 유체의압축율이크면체적팽창계수가작고유체의압축율이작으면체적팽창계수가크다 ΔV = A Δl = V o ΔP β -18-
Effective Bulk Modulus 유효체적팽창계수 (β e : Effective bulk modulus) 실제체적팽창계수는유체에포함된공기의양또는용기자체의부피변화등에따라민감하게영향을받는다 이러한여러가지변수의영향을고려한체적팽창계수를유효체적팽창계수라한다. -19-
Example: Effective Bulk Modulus 예를들어가스와유체가혼합된 flexible 용기의초기체적은다음과같으며, V = V + V t l 피스톤의움직임으로압력이 ΔP 만큼증가하면, Δ V = Δ V Δ V + Δ V t 가되고전체유효체적팽창계수는다음과같이표현된다. ΔP β e = V Δ Vt 각부분의체적팽창계수 ΔP β l = Vl Δ V 식을결합하면, l l g g ΔP β g = Vg Δ V c g ΔP βc = Vc Δ V ΔVt 1 S 1 1 = + + = Vg Vg V tδp βl βg βc β S = e V V l c t -20-
Bulk Modulus in Gas & Container Gas Polytropic Process pv n dp = k V = np = β g dv isothermal compression: n=1 adiabatic compression: Container (for steel pipe) thin-walled cylinder D: inner diameter of pipe t: wall thickness E: modulus of elasticity thick-walled cylinder β = P β c g g β = C p P Cv = = γ P te D E E βc = 2(1 + +ν ) 2.5-21-
Bulk Modulus: Effect of air 유체내에는항상어느정도의공기가포함되어있으며일반적으로이것에의한체적팽창계수의변화는무시할만하다. 그러나유체내의공기가 bubble bbl 형태로존재할경우에는체적팽창계수에많은영향을주게된다. 유체내의공기에의한체적팽창계수의변화는다음식으로부터추정할수있다. n P + α β e P = o n βo α βo P n P Po 여기서 α는공기와오일체적의비 ( V a / Vo ) β o 는오일의체적팽창계수 P o는대기압 P 는유체의작동압력 n 은압축과정에따른계수 -22-
Bulk Modulus: Effect of air 그림에는유체내에함유된공기양에따른유효체적팽창계수와체적팽창계수비의변화를압력에따라나타내었다. 이때등온 (Isothermal) 압축의경우 n=1 이며, 단열 (adiabatic) 의경우에는 n=1.4 이다. 일반적으로작동유체의압력이높아질수록유체내의공기에의한체적팽창계수의변화는그영향은줄어들게된다. 공기함유량에따른체적팽창계수의변화 -23-
4.7.A Orifice Flow Orifice Flow 유동에있어갑작스런제한을가하는부분 orifice를막통과한유체의속도는연속법칙을만족시키기위하여상류의속도보다더증가하게된다. Turbulent Orifice Flow: 대부분의 orifice Flow 관성력이지배적 유체입자들의가속에의한압력강하발생 Laminar Orifice Flow 점성력이지배적 유체점성에의한내부전단력에의한압력강하발생 -24-
Turbulent Orifice Flow orifice 양단의 1과 2에서 Bernoulli s Equation 적용 2 p V gz = constant 2 u 2 2 P2 ρ + 2 + V u = P 1 P1 ρ 2 2 2 u2 u1 = ( P1 P2) A o ρ 비압축성유체에서의연속방정식 Au A 1 1= Au 2 2= Au 2 3 3 u = u A 1 2 u = P P 1 A / A ρ ( ) 2 2 1 2 Q = CAu v ( ) 2 2 2 1 1 2 1 A 1 1 2 A 2 3 CA 2 2 1 A / A ρ ρ ( ) ( ) ( ) v 2 Q= P 2 1 P2 = CdAo P1 P2 2 1-25-
Orifice Flow Equation Orifice Flow Equation 2 Q= CdAo P P ρ ( ) C d : Discharge coefficient 1 2 C v : Velocity coefficient C c : Contraction coefficient A o : Orifice area A 2: Vena-contracta area P 1 : Pressure at 1 P 2 : Pressure at 2 C C d c = = A A CC 2 c v ( ) 2 1 C A / A 2 o c o 1-26-
4.7.B Discharge Coefficient Von Mises 의해석 Steady Flow No Friction Loss (Viscous effect are negligible) ibl Irrotational Flow Two-Dimensional Flow Gravity Effect are negligible The Fluid is incompressible 일반적인 Orifice 형상 -27-
4.7.C Contraction Coefficient Contraction Coefficient and Discharge Angle for Two- Dimensional Spool Valve -28-
Contraction Coefficient Jet Angle and Contraction Coefficient at Small Clearance for Two-Dimensional Spool Valve -29-
Contraction Coefficient Contraction Coefficient Variation with Poppet Angle -30-
Contraction Coefficient Contraction Coefficient for Flapper Nozzle -31-
4.7.D Discharge Coefficient 의실험적고찰 Slot Type Orifice Short Tube Orifice -32-
Discharge Coefficient vs Reynolds No for a circular sharp edged orifice in a pipe -33-
Experimental Variation of Discharge Coefficient for segmental orifice -34-
Report Text Problems 4-22 4-28 4-36 4-38 Due date: 2 주후 -35-