Question

Bernoulli's Theroem.. I think.

Answer 1

A horizontal pipeline carries oil density of 750 kg/m^3. Section A of the pipeline has a diameter 1.5 times Section B. There is a pressure difference of 8000Pa between the two sections. What is the speed in each section?

Answer 2

@zaphodplaysitsafe

Answer 3

yes we can apply Bernoulli's theorum
which is a conservation of energy and mass in the pipeline

By conservation of mass,
mass of fluid flowing in time t through section A = mass of fluid flowing in same time through section B
hence,
\[Ma = \rho*A*Va*t\] where A is area of section A, v is velocity of section A

\[Mb = \rho * B * Vb * t\]

so Ma = Mb
hence,
\[Vb/Va = A/B = (1.5)^2 = 9/4\]



By conservation of energy,
work done due to pressure + potential energy + kinetic energy = constant
\[P + \rho*gh + 1/2*\rho*v^2 = Constant\]

hence this total mechanical energy of the fluid should be constant in both sections.
remember h(height) is constant as this is a horizontal pipeline
then

\[Pa + 1/2*\rho*Va^2 = Pb + 1/2*\rho*Vb^2\]
hence
\[1/2*\rho*(Va^2 - Vb^2) = Pb - Pa = -8000\]

put \[Vb = 9/4Va\]

\[1/2*\rho*(Va^2 - 81/16Va^2) = -8000\]

so you can solve for Va & Vb

Answer 4

you're such a wizard! ok this is exactly what I did and I got 4.21... Is that what you're getting? for Va.

Answer 5

According to the bernoulis theorem for fluid dynamic system when velocity increased causes reduced pressure

Answer 6

In fluid dynamics, Bernoulli's principle states that for an inviscid flow, an increase in the speed of the fluid occurs simultaneously with a decrease in pressure or a decrease in the fluid's potential energy.[1][2] Bernoulli's principle is named after the Swiss scientist Daniel Bernoulli who published his principle in his book Hydrodynamica in 1738.[3]
Bernoulli's principle can be applied to various types of fluid flow, resulting in what is loosely denoted as Bernoulli's equation. In fact, there are different forms of the Bernoulli equation for different types of flow. The simple form of Bernoulli's principle is valid for incompressible flows (e.g. most liquid flows) and also for compressible flows (e.g. gases) moving at low Mach numbers (usually less than 0.3). More advanced forms may in some cases be applied to compressible flows at higher Mach numbers (see the derivations of the Bernoulli equation).
Bernoulli's principle can be derived from the principle of conservation of energy. This states that, in a steady flow, the sum of all forms of mechanical energy in a fluid along a streamline is the same at all points on that streamline. This requires that the sum of kinetic energy and potential energy remain constant. Thus an increase in the speed of the fluid occurs proportionately with an increase in both its dynamic pressure and kinetic energy, and a decrease in its static pressure and potential energy. If the fluid is flowing out of a reservoir, the sum of all forms of energy is the same on all streamlines because in a reservoir the energy per unit volume (the sum of pressure and gravitational potential ρ g h) is the same everywhere.[4]
Bernoulli's principle can also be derived directly from Newton's 2nd law. If a small volume of fluid is flowing horizontally from a region of high pressure to a region of low pressure, then there is more pressure behind than in front. This gives a net force on the volume, accelerating it along the streamline.[5][6][7]
Fluid particles are subject only to pressure and their own weight. If a fluid is flowing horizontally and along a section of a streamline, where the speed increases it can only be because the fluid on that section has moved from a region of higher pressure to a region of lower pressure; and if its speed decreases, it can only be because it has moved from a region of lower pressure to a region of higher pressure. Consequently, within a fluid flowing horizontally, the highest speed occurs where the pressure is lowest, and the lowest speed occurs where the pressure is highest.