BERNOULLI’S PRINCIPLE l EQUATION





Bernoulli equation is arrived from the following assumptions:

1. Steady flow - common assumptions applicable to many flows.

2. Incompressible flow - acceptable if the flow Mach number is less than 0.3.

3. Frictionless flow - very restrictive; solid walls introduce friction effects.

4. No shaft work - no pump or turbines on the streamline.

5. No transfer of heat - either added or removed.



Range of validity of the Bernoulli Equation:

Bernoulli   equation   is   valid   along   any   streamline   in   any   steady,   inviscid, incompressible
flow. There are no restrictions on the shape of the streamline or on the geometry of
the overall flow. The equation is valid for flow in one, two or three dimensions.


Modifications on Bernoulli equation:

Bernoulli equation can be corrected and used in the following form for real cases.


































Derivation of Bernoulli equation from Euler's Equation of Motion:

Mass in per unit time = ȡ Av = m
For steady flow, mass out per unit time = m
Rate of momentum in = m v
Rate of momentum out = m (v + dv)
Rate of increase of momentum from AB to CD = m (v + dv) - m v = ȡ Avdv
Force due to p in the direction of motion = pA
Force due to p + dp opposing the direction of motion = (p + dp)(A + dA)
Force due to pside producing a component in the direction of motion = p side dA
Force   due   to  mg   producing   a   component   opposing   the   direction  of   motion=
mgcos(ș)
Resultant force in the direction of motion
= pA - (p + dp)(A + dA) + p side dA - mgcos(ș)
The value of pside will vary from p at AB to p + dp at CD, and can be taken as p +
kdp
where k is fraction.
Mass of fluid element ABCD = m =  ȡ g(A + 1/2 dA) ds
And ds = dz/cos(ș); since cos(d z/ds) Substituting in equn.2,
Resultant force in the direction of motion
pA - (p + dp)(A + dA) + p + kdp - ȡ g(A + 1/2 dA) dz
= -Adp - dpdA + k ȡ pdA - ȡ gAdz - 1/2 dAdz
Neglecting products of small quantities,
Resultant force in the direction of motion = -Adp - ȡ gAdz Applying Newton's second law, (i.e., equating equns.1 & 3) dAv dv = -Adp - ȡ gAdz
Dividing by dAds,



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