CHAPTER 3 The Bernoulli Equation

Bernoulli Equation (B.E) ? 4- Unknowns : 4- equations : 1- Continuity Equation & 3 - Momentum Equations (E.E for inviscid flow, N-S for viscous flow) B.E gives simple form btw Very convenient to be used in many practical flow problems B.E is another form of momentum equation (should not be used with momentum equation simultaneously)

3.1 A Simple Form of the B.E - 2 d steady flow, constant fluid density Equations of motions (gravity in z-dir): A B Stream line dx dy : x – force per unit mass : y – force per unit mass

Integrating equation of motions from A to B, Adding the two equations (work done from A to B), Rewriting,

3.2 Conservative Body Forces (ex. Gravity force) Conservative ?, Potential ? Gravity in (-y) dir. (same process)

considering then Energy equation per unit mass Heads (in Civil engineering) energy equation per unit volume

2.3 Unsteady Flow Considering only unsteady terms, Adding,

Thus,

(ex) A1A1 V1V1 ,g h A2A2 V2V2 A 1 >>A 2, V 1 <<V 2, P 1 =P 0 =0 V 2 = ? (ex) x x P1P1 P2P2 U D h P 1, P 2, D are given h & U = ?

3.4 Barotropic Fluids Fluid density is function of pressure only - Assume gravity in z-dir. Steady flow in 3.1, then let

Then, (Examples)

BAC - CAB rule : (considering  is differential operator) from E.E. 3.5 The B.E in Irrotational Flow

thus, another form of E.E. is : integrating this E.E. along any line connecting point 1 & 2 : 3 possibilities for this term to be 0 (1)(2) (3)

for irrotational flow(most general case) : B.E. for unsteady, irrotational, incompressible flow B.E. for steady, irrotational, incompressible flow

inviscid flow viscous flow C.E. Eq. Of motion E.E. N-S eq. B.E. ?

(ex) steady, uniform flow around circular cylinder U, P 0 V r =U(1-R 2 /r 2 )cos  is given R (r,  ) x y g is in z dir. (1) v , (2) P(r,  ) (3) P-P 0 drawing on the surface

(ex) A1A1 A2A2 1.2m Steady flow A 1 =0.0013m 2, A 2 = m 2 V 1, Q = ?

Work - Energy equation (1) B.E. (steady state) : no energy influx or efflux btw 1 & 2 (2) Energy supply btw 1 & 2 Where, E : energy supply per unit flow volume thus power supply = E · Q (Q : flow rate) [energy/m 3 x m 3 /sec = energy/sec] 3.6 Extended Bernoulli Equation

(3) Energy extract btw 1 & 2 (ex)

3.8 Engineering Applications of B.E. (ex) Flow in Pipe Circuits y1y pump y4y4 y5y5 Head losses : h f1-2 =h f3-4 =5m Discharge rate : Q p =0.5 m 3 /sec Cross sectional area : A 4 =0.04m 2 Head across the pump=? Pump power=? Elevations : y 1 =20m, y 5 =40m Pump efficiency=75%

(sol) B.E btw 1 & 5 h p h f1-2 h f3-4 h f4-5 Where, E p : input energy by pump per unit volume per unit time = input power by pump per unit volume unknown

One more equation : B.E btw 4 & 5 If the fluid surrounding the jet, 4, is assumed stationary, hydrostatic relation can be applied to 4~5

(ex) 1 2 y1y1 y2y2 Pipe friction coeff.(Darcy coeff.) f=0.02 l/d=1000 Discharge velocity=? (with head loss, w/o head loss)

Type of Orifices A jet AoAo A o : orifice area A jet : jet area Where, C c : contraction coeff. (vena contracta : minimum section) V Where, C d : discharge coeff.

(ex)