Two different types of pressure ??

[bct1981]

Imagine a pipe whose diameter changes (pipe reduction or pipe expansion)
Let's define two different places in that pipe:
point 1: with a pressure P1, velocity V1 and diameter D1;
and point 2: with a pressure of P2, velocity V2 and and D2.
Knowing that the fluid flows from point 1 to point 2, the pipe is horizontal (no level change) and D2 > D1: applying a material balance we know that V2 < V1 then applying an energy balance (i.e. Benouilli's equation) we know that P2 > P1 (pressure drop in the pipe has been considered very low and irrelevant). Therefore the fluid is flowing from a low pressure place (point 1) to a high pressure place (point 2) !!! How can that be ?

 

[quark]

Fancy, ain't it? But we shouldn't worry until there is a flow from lower total pressure(the sum of the static pressure, velocity pressure and static head) to higher total pressure. Individually, it can vary anything.

That's what Bernoulli's principle is about.

If you take a pipeline vertically down to 100 meters and close it by a valve, the pressure gauge before the valve reads 10kg/sq.cm g. If the top end is open to atmosphere, the pressure there is 1kg/sq.cm (approx.). Open the valve and still you get the flow because total pressure at the high point is high. The closed valve maintains equal total pressure both sides and that is why you don't get flow.

Regards,

 

[BRIS]

Yes the water flows from a low to high pressure but you are only looking at the statics of the problem.

You need to look at the dynamic forces. The total force at 1 in the direction of flow is greater than the total force at 2 in the opposite direction and hence flow is from 1 to 2. The force is a combination of pressure and momentum and momentum is proportional to V^2.

In the same way the roller coaster rides uphill to the next crest.

 

[BRIS]

Yes the water flows from a low to high pressure but you are only looking at the statics of the problem.

You need to look at the dynamic forces. The total force at 1 in the direction of flow is greater than the total force at 2 in the opposite direction and hence flow is from 1 to 2. The force is a combination of pressure and momentum and momentum is proportional to V^2.

In the same way the roller coaster rides uphill to the next crest.

 

[25362]

To bct1981.

Fluids move from high mechanical energy points to points of lower mechanical energy. Pressure is just one component of mechanical energy.

The total mechanical energy in a point per unit mass is the sum of

p/[&rho;] the work on the fluid by displacing
it through a volumetric space 1/[&rho;] gainst the
restraint of pressure p,
V²/2 the kinetic energy per unit mass,
zg the potential energy per unit mass with
the horizontal reference level 0-0

If you multiply these summands by mass you get energy units.

Neglecting friction, as Bernoulli did for incompressible fluids, the mechanical energy balance between points 1 and 2 shows:

p₁/[&rho;] + V₁²/2 + z₁g = p₂/[&rho;] + V₂²/2 + z₂g​

Dividing them by g (acceleration of gravity) they become expressed in lengths, for example, m or ft.

Static head difference: (P₂-P₁)/[&rho;]g
Velocity head difference: [&Delta;]V²/2g
Elevation difference: [&Delta;]z

Real fluids convert some of their mechanical energy into heat due to friction. This loss of mechanical energy can be expressed as a loss in static head (or pressure), and can be expressed as [&Delta;]P_f/[&rho;]g

 

[BRIS]

Bct1981 clearly understands the Bernoulli equation - I suggest the question being asked is WHY do "Fluids move from high mechanical energy points to points of lower mechanical energy" when this means that a particle of fluid is moving from a point of low pressure to a point of high pressure when logic would suggest that flow should be from high to low pressure.

The answer is not in reinstating Bernoulli equation but in analysis of fluid dynamics and the balance of forces. (Momentum balance) .

The upstream force = the downstream force – the friction loss.

Force = pressure over area + momentum.

Momentum = V x Q. gamma/g

and since Q=VA

Momentum is proportional to V^2

The friction loss is also proportional to V^n.

What nature is able to do is to solve these iterative equations and create a balance.

 

[25362]

To BRIS, don't forget elevation (potential energy). See quark's example.

Eppur si muove...It is clear from the energy balance equation that for a fluid to move from point 1 to point 2, it needs its energy to be larger at point 1.

This can be achieved, even when P₁<P₂, only when the other energy components compensate to make energy₁ > energy₂.

 

[BRIS]

I am keeping the explanation as simple as possible andd assuming the system is horizontal and thus not including potential energy in the discusion.

The question is

Therefore the fluid is flowing from a low pressure place (point 1) to a high pressure place (point 2) !!! How can that be ?

The anser is

Because the water has greater momentum at point 1 than point 2. The force at point 1 is greater than the force at point 2 so the fluid moves from point 1 to point 2. QED.

 

[25362]

You are right, BRIS, in admitting that momentum (p=m[&times;]v) alone could not explain the draining of a vertical tank.

In fluid dynamics Bernoulli's equation expresses the conservation of energy; however one needs also conservation of mass (the continuity equation v₁A₁=v₂A₂) to explain a horizontal Venturi flowmeter. (v: velocity; A: crossflow area)

The dirt around a prairie dog's hole is mounded up in a way that forces wind to accelerate over the hole, resulting in lower pressure above the hole. Biologists speculate this design has evolved to provide natural ventilation.

One can suspend a ping-pong ball inside an inverted funnel by blowing air downward through it. Rapid divergence of the flow results in lower speed and therefore higher pressure below the ball.

All these effects and many more such as lift in flight, spinning balls curved trajectories, wind turbines, etc., can be explained by Bernoulli's formula.

 

[BRIS]

I think vertical tanks drain by the force of gravity. Sometimes in order to understand the meaning of a formula it is necessary to return to basics and understand the derivation of the formula. Bernoulli's equation is simply a statement that energy is neither created nor destroyed. It does not contain a heat component and it is not the panacea to resolving all fluid dynamic problems.

 

[25362]

BRIS, since mechanical force is the rate change of momentum (Newton's second law of motion), this force could be used to interpret motion of matter including flow in rotational fields.

Philosophically speaking about the variety of forces, you are absolutely right.

BTW, another universal law is the conservation of linear momentum. Bernoulli's law however, is quite general, and serves to explain a wide variety of effects.

There are indeed many forces in Nature besides gravity that act by affecting motion of matter in general, and liquids, in particular.

Common among them are chemical potential changes due to concentration gradients as in osmosis, or capillarity from adhesion and cohesion forces, etc.

Not to mention electromagnetic and other yet unknown forces (called colors) acting in the subatomic realm and on stars.

Would a grand unification theory (GUT) in the future merge all these forces as a manifestation of a single common interaction ? Who knows.

 

[macmet]

In quark's posting he mentioned both static pressure and static head. what's the difference?

Is static head the result of elevation differences? With static pressure being the pressure difference at the two selected points of observation.

 

[BRIS]

I should not answer for quark but the statement "there is a flow from lower total pressure (the sum of the static pressure, velocity pressure and static head) to higher total pressure". would appear to be incorrect in that it is referring to total energy not total pressure. Flow is always from higher to lower total energy but may be from lower to higher pressure. The term static head would appear to refer to potential head - i.e elevation.

"the sum of the static pressure, velocity pressure and static head" are the variables in Bernoulli’s equation

 

[quark]

Thanks Bris. I was to say total head(and ofcourse, total energy would have been much more accurate and it is always called as Steady Flow Energy Equation in thermodynamics).

Macmet,

By static head, I mean the elevation and static pressure is what we generally call as pressure.