Useful concepts associated with the Bernoulli equation. Dynamic


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1 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 43 : 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 +
2 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 (Pitotstatic 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 = : Pitotstatic tube
3 Here Here Photodetector Wheel How to detect the wheel seed (e.g. Car and Comuter mouse etc.
4 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
5 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,
6 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.
7 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
8 Case : Determine Q in flumes and irrigation ditches (Oen channer Tyical devices: Sluice gate (Oen bottom & Sharcrested 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 sharcrested 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
9 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
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