Useful concets associated with the Bernoulli equation - Static, Stagnation, and Dynamic Pressures Bernoulli eq. along a streamline + ρ v + γ z = constant (Unit of Pressure Static (Thermodynamic Dynamic Hydrostatic Piezometer tube Physical meaning of each term Velocity of stream i st term (Static ressure, : Only due to the fluid weight = 3 + γ h3 = γ h4 3 + γh3 = γh where 3 = γ h4 3 (h 4-3 : Piezometer reading ii nd term (Dynamic ressure, ρ v : Pressure increase or decrease due to fluid motion iii 3 rd term (Gravitational otential, γ z : Pressure change due to elevation change Between Point ( and Point ( on the same streamline, ρ v = + ρv + γz = constant because z = z (No elevation change v = 0 (Stagnated by a tube inserted +
Difference in between iezometer and tube inserted into a flow = γh (Stagnation ressure ρv = = γh γh (Elevation change, H h: Due to dynamic ressure Point (: Stagnation oint ( V = 0 Line through oint (: Stagnation streamline iv Total ressure T = + ρ V + γz = constant (Along a streamline Secial alication of Static and Stagnation Pressure - Determination of fluid seed (Pitot-static tube By measuring ressure at oints (3 and (4 and neglecting the elevation effect (e.g. gases At oint (3, 3 = + ρ V (Stagnation Pressure where and V: Static ressure and fluid velocity at ( At onit (4 (Small holes, 4 = = (Static ressure ~ Piezometer Then, 3 ( 4 = ρ V 3 4 V ρ = : Pitot-static tube
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How to use the Bernoulli equation (Examles Choose oints ( and ( along a streamline = + ρv + γz : 6 variables (,, V, V, z, z 5 more conditions from the roblem descrition E.g. Free Jets (A jet of liquid from large reservoir Question: Find V of jet stream from a nozzle of container Consider a situation shown Ste. Select one streamline between ( & ( Ste. Aly the Bernoulli eq. between ( & ( = + ρv + γz because = 0: Gauge ressure (The atmoshere V = 0 (Large container Negligible change of fluid level = 0 (Why? [ = 4 (Normal Bernoulli eq.] Then, γ h = ρ V (If we choose z as h and z as 0 or γh V = = gh : Velocity at the exit lane of nozzle ρ
Velocity at any oint [e.g. (5 in the figure] outside the nozzle, v = g( h + H (H: Distance from the nozzle : Conversion of P. E. to K.E. without viscosity (friction cf. Free body falling from rest without air resistance. In case of horizontal nozzle shown, v < < (Elevation difference v v3 By assuming d << h, v : Average velocity at the nozzle E.g. Confined Flows - Fluid flowing within a container connected with nozzles and ies What to know - We can t use the atmosheric as a standard - But, we have one more useful concet of conservation of mass : No change of fluid mass in fixed volume (continuity equation Mass flow rate, m& (kg/s or slug/s for a steady flow Mass of fluid entering the container across inlet er unit time = Mass of fluid leaving the container across outlet er unit time,
δm V ta m ρ δ ρ & = = (Inlet VδtA δm = = = m& (Outlet δt δt δt δt Or δm m& = δ t (kg/s = ρ Q (Q: Volume flowrate, m 3 /s For a time interval δ t, m & = = ρ Q = ρav = ρ AV = ρq m For a incomressible fluid, ρ = ρ V AV A = or Q = Q & Change in area Change in fluid velocity ( A V = AV Change in ressure ( = + ρv + γz Increase in velocity Decrease in ressure : Blow off the roof by hurricane, Cavitation damage, etc.
E.g. 3 Flowrate Measurement Case : Determine Q in ies and conduits (Closed container Consider three tyical devices (Deend on the tye of restriction Ste. At section ( : Low V, High At section ( : High V, Low Ste. Aly the Bernoulli Eq. For a horizontal flow (z = z = + ρv V ( = ρ [ ( V / V ] Ste 3. Use the continuity Eq. Q = AV = AV = A ( ρ[ ( A / A ] where A & A : Known : Measured by gages
Case : Determine Q in flumes and irrigation ditches (Oen channer Tyical devices: Sluice gate (Oen bottom & Shar-crested weir (Oen to Consider a sluice gate shown Bernoulli Eq. & Continuity Eq. [( (] = + ρv + γz g( z z V = ( V / V ( = = 0 and Q = A V = bz V = A V = bz V Finally, the flowrate, Q = bz g( z ( z z / z In case of z >> z or z z z or z / z << Q = bz gz or V gz The shar-crested weir (Oen to (If similar to horizontal free stream Then, Average velocity across the to of the weir Flow area for the weir Hb gh Q = C Hb gh = C b H 3/ g
Energy Line (EL and the Hydraulic Grade Line (HGL - Geometrical Interretation of the Bernoulli Eq. Bernoulli eq.: Conserved total energy along streamline, i.e. V + + z = constant on a streamline = H (Unit of Length γ g Heads: : Pressure head, γ H: Total head v : Velocity head, z: Elevation head g Energy line (EL: Reresentation of Energy versus Heads ( (: Elevation head Pressure head Velocity head ( ( (3 ( (3: Elevation head Pressure head Velocity head e.g. Flow from tank