Bernoulli’s Equation Terms Term Meaning Relationship SI Units

a Acceleration - ��

���

�2sec

m

F Force onacceleratimass × ��

���

� ⋅2secmkg

,

Newton (N)

g Gravitational Acceleration

- 2secm

9.81

W Weight gmass × ��

���

� ⋅2secmkg

,

Newton (N) A area 2length [ ]3m

V Volume 3length [ ]3m

� Density volumemass

��

���

�3m

kg

γ Specific Weight

gvolume

gmassvolumeweight ρ=⋅= ��

���

�3m

N

P Pressure areaforce

��

���

�2m

N,

Pascal (Pa)

w Work distanceforce × [ ]mN ⋅ , Joule (J)

E Energy Ability to do work [ ]mN ⋅ ,

Joule (J) z Elevation - [ ]m

� Velocity time

distance ��

���

�

secm

H Hydraulic

Head ggP

zgmass

Energy2

2µρ

++=⋅

[ ]m

Derivation of Bernoulli’s Equation Bernoulli’s Equation is the equation for the total energy in water normalized by the product of the unit mass and gravitational acceleration.

The three types of energy in water are:

1. Potential Energy (EP) due to gravity

2. Kinetic Energy (Ek) due to motion

3. Non flow energy (En) associated with work on system by pressure and volume (hydraulics)

��

�

��

−=

=

=

)(21

01

2

PPVE

mE

mgzE

n

k

p

µ Equations for different types of energy (Equation 1)

+→++= mgzEEEE nkP 2

21 µm + ( )01 PPV − (Equation 2)

Remember:

ρ

ρ mV

Vm =→= (Equation 3)

Substituting equation 3 for the volume (V) in equation 2:

( ) ( )012

012

21

21

PPm

mmgzPPVmmgzE −++→−++=ρ

µµ (Equation 4)

In natural systems we assume the P0 is atmospheric pressure (Pa), which one can assume is a constant over the entire system. P1 is pore pressure (Pp) including atmospheric pressure. So, we can express the difference between P1 and P0 as just the pore pressure (Equation 5).

papa

a

pa

PPPPPP

PP

PPP

=−+=−=

+=

01

0

1

(Equation 5)

Equation 4 can be rewritten as:

Pm

mmgzEρ

µ ++= 2

21

(Equation 6)

Finally, if one normalizes the energy (E) by the force of a unit mass of water (mg), equation 6 can be rewritten as:

ρµ

ρµ

ρµ

ρµ

gP

gzH

mgE

gP

gz

gmPm

gmm

gmzgm

mgmP

mgm

mgmgz

mgE

++==

++=//+

//+

////=++=

2

2222

222

(Equation 7)

Equation 7 defines hydraulic head (H) as the sum of three terms. Hydraulic head has units of length. The three terms in the equation for hydraulic head are elevation head, velocity head and pressure head (Equation 8).

headpressureg

P

headvelocityg

headelevationz

2

2

≡

≡

≡

ρ

µ (Equation 8)