INTRODUCTION Bernoulli’s theorem which is known as Bernoulli’s principle, states that an increase in the speed of moving air or a flowing fluid is accompanied by a decrease in the air or fluid’s pressure. Swiss scientist, Daniel Bernoulli (1700- 1782), demonstrated that, in most cases the pressure in a liquid or gas decreases as the liquid or gas move faster. This is an important principle involving the movement of a fluid through the pressure difference. Suppose a fluid is moving in a horizontal direction and encounters a pressure difference. This pressure difference will result in a net force, which is by Newton’s Second Law will cause an acceleration of the fluid. Bernoulli’s theorem states that the total energy (pressure energy, potential energy and kinetic energy) of an incompressible and non-viscous fluid in steady flow through a pipe remains constant throughout the flow, provided there is no source or sink of the fluid along the length of the pipe. This statement is due to the assumption that there is no loss energy due to friction. This theorem deals with the facts that when there is slow flow in a fluid, there will be increase in pressure and when there is increased flow in a fluid, there will be decrease in pressure. If the elevation remains constant, velocity and pressure, energy to or from the system can be calculated by this equation- Static pressure + dynamic pressure = total pressure = constant
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Transcript
INTRODUCTION
Bernoulli’s theorem which is known as Bernoulli’s principle, states that an increase in the
speed of moving air or a flowing fluid is accompanied by a decrease in the air or fluid’s
pressure. Swiss scientist, Daniel Bernoulli (1700-1782), demonstrated that, in most cases the
pressure in a liquid or gas decreases as the liquid or gas move faster. This is an important
principle involving the movement of a fluid through the pressure difference. Suppose a fluid
is moving in a horizontal direction and encounters a pressure difference. This pressure
difference will result in a net force, which is by Newton’s Second Law will cause an
acceleration of the fluid.
Bernoulli’s theorem states that the total energy (pressure energy, potential energy and
kinetic energy) of an incompressible and non-viscous fluid in steady flow through a pipe
remains constant throughout the flow, provided there is no source or sink of the fluid along
the length of the pipe. This statement is due to the assumption that there is no loss energy
due to friction.
This theorem deals with the facts that when there is slow flow in a fluid, there will be
increase in pressure and when there is increased flow in a fluid, there will be decrease in
pressure. If the elevation remains constant, velocity and pressure, energy to or from the
system can be calculated by this equation-
Static pressure + dynamic pressure = total pressure = constant
Static pressure + 1/2 x density x velocity2 = total pressure = constant
Fig 1: Bernoulli Equation
So the main objective of this experiment is to justify the validity of Bernoulli’s theorem for
water flow through a circular conduit.
The converging-diverging nozzle apparatus is used to show the validity of Bernoulli’s
Equation. It is also used to show the validity of the continuity equation where the fluid flows
is relatively incompressible. The data taken will show the presence of fluid energy losses,
often attributed to friction and the turbulence and eddy currents associated with a
separation of flow from the conduit walls.
Theory
Clearly stated that the assumptions made in deriving the Bernoulli’s equation is:
The liquid is incompressible.
The liquid is non-viscous.
The flow is steady and the velocity of the liquid is less than the critical velocity for
the liquid.
There is no loss of energy due to friction.
Derivation of Bernoulli equation from Newton’s second law:
The Bernoulli equation for incompressible fluids can be derived by integrating the Euler
equations, or applying the law of conservation of energy in two sections along a streamline,
ignoring viscosity, compressibility, and thermal effects.
The simplest derivation is to first ignore gravity and consider constrictions and expansions in
pipes that are otherwise straight, as seen in Venturi effect. Let the x axis be directed down
the axis of the pipe.
Define a parcel of fluid moving through a pipe with cross-sectional area "A", the length of
the parcel is "dx", and the volume of the parcel A dx. If mass density is ρ, the mass of the
parcel is density multiplied by its volume m = ρ A dx. The change in pressure over distance
dx is "dp" and flow velocity v = dx / dt.
Apply Newton's Second Law of Motion Force F =mass . acceleration and recognizing that the
effective force on the parcel of fluid is -A dp. If the pressure decreases along the length of
the pipe, dp is negative but the force resulting in flow is positive along the x axis.
In steady flow the velocity is constant with respect to time, v = v(x) = v(x(t)), so v itself is not
directly a function of time t. It is only when the parcel moves through x that the cross
sectional area changes: v depends on t only through the cross-sectional position x(t).
With density ρ constant, the equation of motion can be written as
by integrating with respect to x
where C is a constant, sometimes referred to as the Bernoulli constant. It is not a universal
constant, but rather a constant of a particular fluid system. The deduction is: where the
speed is large, pressure is low and vice versa.
In the above derivation, no external work-energy principle is invoked. Rather, Bernoulli's
principle was inherently derived by a simple manipulation of the momentum equation.