Chapter Bernoulli Equation Why? For Mathematical .

Chapter 3 Bernoulli EquationWe neglectgfriction.Why? For mathematical simplicity. For quick approximation.Energy equation without frictional term.3.1 Newton’s Second LawDo you see streamlines?l?Do you see velocity?At any point, velocity is to streamline.Fig. 3.1

What is streamline?u axial component of velocity vectorV vertical component of velocity vectorSlope of streamline ?(in terms of u and v)Derivation of u/v dy/dxVelocity vector V ui vj and ds dxi dyjSince velocity vector is tangent to streamline, V and ds must be parallel.Thus,V x ds 0 orV . ds 0

3.2 Derivation of Bernoulli equationWe apply F ma along a streamline S.a acceleration along S coordinate dV/dtFigure 3.1 (p. 95).

3.2 Derivation of Bernoulli equationWe apply F ma along a streamline S.Figure 3.3 (p. 96)Free‐bodyFreebody diagram of afluid particle F sum of all the forces along the streamline

3.2 Bernoulli equationdP ρ g dz ρ VdV 0Integrating this, we have[Pressure unit, Pa][Energy per unit mass, J/kg][head unit]Show why gZ has [J/kg]Show why V2 has [J/kg]

3.4 Physical interpretation of Bernoulli equationWhich one is greater, V1 or V2?V1 3 m/sP1 ?Example:V1 3 m/sP1 75 kPaV2 ?P2 ?A1 2 cm2; A2 1 cm2Determine V2 and P2.Which one is greater, P1 or P2?

3.4 Physical interpretation of Bernoulli equationEExample:l GiGivenV1 3 m/sd1 0.1 m; d2 0.05 mP1 125 kPaDensity of water 1,000 kg/m3Objective: to determine V2 and P2.Conservation of mass:Find V2.Find P2.

Progression of atherosclerosisWhat happens at stenosis?Velocity increases or decreases?Pressure increases or decreases?What can happen to coronary artery?Scientific American Quarterly 2000 Summer Vol.11, No2

3Example: Bernoulli equationObjective: to determine the maximum height hSolution:Let us consider three locations indicated by21What is V3?Figure P3.44 (5th ed)What is P2?Whyh is theh pressure at theh tip off a nozzlel zero? Iff it is not zero, whath happens?h?

3Apply Bernoulli eq. between 1 and 2. (Answer V2 12.83 m/s)21Figure P3.44 (5th ed)Applypp y Bernoulli eq.q between 2 and 3. ((Answer h 8.4 m))

3.5 Static Pressure and Stagnation PressureFigure E3.2 (p. 101)Figure 3.5 (p. 106)Stagnation points on bodies in flowing fluids.

How to measure Static Pressure and Stagnation Pressure?What is the velocity at point 2?V2 ?Figure 3.4 (p. 105)Measurement of static and stagnationpressures.Note that we made two holes on the pipe wall to measure these pressures.Can we measure both pressures with only one hole?

How to measure Static Pressure and Stagnation PressureUsingg onlyy one hole on the ppipep wall?Figure 3.6 (p. 107)The Pitot‐static tube.Typical Pitot‐static tube designs.

How to measure a speed of an aircraft?Figure E3.6a (p. 110)Does is matter where we locate a Pitot tube?

In a similar manner, where do you position a torpedo launcher?

3.6.1 Free JetObjective: to determine the exit velocity at point 2 as a function of hSolution ((Assume A1 A2)Apply Bernoulli eq. between 1 and 2Figure 3.11 (p. 110)Vertical flow from a tank.

3.6.1 Free JetQuestion: Doe the diameter of free jet remain constant or decrease axially?Objective:jto determine the diameter of free jetj as a function of H.Figure 3.11 (p. 110)Vertical flow from a tank.

Flows from a tankExample 3.73 7 (p.113)(p 113)Objective: to determine the inlet flow rate Q to maintain water level h 2.0 mSolution: Apply Bernoulli eq. between 1 and 2Find relationship between V1 and V2 using conservation of mass.Figure E3.7 (p. 113)

Flows from a tankC we explainCanl i whath we see?

Flows from a tankExample 3.73 7 (p.113)(p 113)Objective: When we stop the inlet flow rate Q, how long will it take to drain?i.e.,, h becomes zero. Treat h h(t).()Solution: Apply Bernoulli eq. between 1 and 2Figure E3.7 (p. 113)

CavitationObjective: to estimate p2 as a fuinction of Q (or V1)Solution: Apply Bernoulli eq. between 1 and 2V1 V2P1 P2If P2 decreasesdto theh pressure,Water will boil (or evaporate).What is the pressureat the room temperature?Figure 3.16 (p. 117)Pressure variation and cavitation in a variable areapipe.

Cavitation21Figure 3.17 (p. 117)Tip cavitation from a propeller.V2 circumferential velocity R ωω angular velocity 2πn/60 where n is rpm.Given: Vapor pressure at room temperature 1770 Pa (absolute).Radius of propeller R 1 mObjective: to estimate the maximum rpm to avoid cavitation

Example 3.10Siphon and CavitationObjective: to determine the maximum value of HQuestion:QiWhyWh (How)(H ) doesdsiphoni h fail?f il? WhatWh isi theh failuref ilmoded off siphon?i h ?Do you see that P2 decreases as we raise the hose (i.e., increasing H)?Answer: At the moment of failurefailure, P2 becomes .Is P1 zero?Is P3 zero?What is P2? Pa(absolute)()Figure E3.10b (p. 118)

Example 3.10Siphon and CavitationSolutionApply Bernoulli eq. between point 1 and point 3. Find V3 35.9 ft/sApply Bernoulli eq. between point 2 and point 3. (note P2 is absolute pressure)Figure E3.10b (p. 118)

Figure 3.18 (p. 119)Typical devices for measuringflowrate in pipes.

Bernoulli principleFor unsteady flowExample 3.16 (p.129)Oscillating flowsin U‐tube:U tube:Figure E3.16 (p. 129)

Clinical disposable viscometer (Cho and Kensey)LED ArrayCCDCCCCDComputer dataacquisition systemBlood fromvacutainerCapillary tube

A new scanning capillary tube viscometer – Rheologwith disposable capillary tubeBloodenters

Pressure drop and flow ratefrom single measurement of h(t)ΔP ρ g[h1 (t ) h2 (t )]Riser columnsÆ Calculate Shear stress1Fluid fallsVelocity is a derivative of height.Fluid risesdh( t )vr ( t ) dt2BloodÆ Calculate Shear rateCapillary tubeConceptuallyμ πd 4 ΔP128 L Q

Disposable tube of RheologΔht At t 0At t

Scanning capillary tube viscometer

HW Figure P3.48 (p. 137)WindWid tunneltlUsed for measurement ofDrag of a car1V1 0V2 60 mphObjective: to calculate hP2 is vacuum? Yes No2

Chapter 3 Bernoulli Equation We neglect friction. Why? For mathematical simplicity. For quick approximation. Energy equation without frictional term. 3.1 Newton’s Second Law Do you see streaml?lines? Do you see velocity? At any point, velocity is _ to streamline. Fig. 3.1