Bernoulli's equation?

Question

This is with regard to bernoulli's equation.

In one of the books (fundamentals of physics.. by resnick, halliday and walker) he gives the reason for the increase in velocity of fluid as it flows from a portion of higher cross section area in a horizontal tube to a smaller cross section as being...and i quote " the link between a change in speed and a change in pressure makes sanse if you consider a fluid particle . when the particle nears a narrow region , the higher pressure behind it accelerates it so that it then has a greater speed in the narrow region.. And when it nears a wide region , the higher pressure ahead of it decelerates it so that it then has a lesser speed in the wide region"

What i am unable to comprehend is .. why is the pressure in wide region higher compared to that in narrow region..???

Answer 1

In this case, Bernoulli's Principle deals with a constant Volumetric Flow through the pipeline. Without a Diagram, this is a bit difficult to explain but, here goes.

A Metering device like an o*****e Plate or Venturi Tube gives a decrease in the cross-sectional area of the pipe.

In order to maintain the flowrate, the fluid particles have to increase Velocity for them to pass through the restriction in the same numbers as in the larger section of pipe.

This velocity increase causes a Pressure Decrease across the restriction as the particles upstream of the restriction 'Crowd together' as they push against it trying to get through. This maintains the 'Upstream Pressure' of the system.

On passing through, due the particles' velocity, they can't immediately move apart to take up the pipe's volume downstream. This therefore causes the lower pressure just downstream of the restriction.

As the fluid decreases velocity and takes up the pipe's volume again a little further downstream, the Pressure builds back up but not to the same level as before the restriction.

(Fluid flow requires a ΔP).

The Pressure Drop (ΔP) is measured and, together with other parameters of upstream and downstream pressure and temperature, the cross-sectional area of the restriction and the density of the fluid, the 'Mass Flow Rate' is computed by the Instrumentation (calibrated by Instrumentation Engineers), to whatever Flow Units are required.

Answer 2

Let me venture a detailed protocol:

Remeber, PV = nRT, where P= pressure, V = volume, n = number of moles, R is a constant, T= temperature. Without any external forces, at steady state (temperature constant throughout) PV (at narrow end) = PV (at wide end).

Now this is where you have concerns:

Because the volume at the narrow end is smaller, the pressure there is greater! However this greater pressure is the resultant of the volume of the large section forcing particles to the small section to maintain steady state. The volume produces that "push" so the particles speed up and makes the narrow end more presurized NOT that the larger end has more pressure. It pushes more by virtue of its larger volume, exerting greater force on indivual particles, hence the particle picks up speed getting into the narrow end... and vice versa.

Answer 3

Bernoulli's Equation is basically a statement of the conservation of energy per unit volume along the pipe.

Static Pressure + Dynamic Pressure = Total pressure

The equation states that the static pressure ps in the flow plus the dynamic pressure, one half of the density r times the velocity V squared, is equal to a constant throughout the flow. We call this constant the total pressure pt of the flow.

This is the reason  for the statement in the book.  Also it is based on certain fundamental assumption about the fluid.