Formula for fluid pressure in physics. Formula for pressure of air, steam, liquid or solid. How to find pressure (formula)

Let us take a cylindrical vessel with a horizontal bottom and vertical walls, filled with liquid to a height (Fig. 248).

Rice. 248. In a vessel with vertical walls, the pressure force on the bottom is equal to the weight of the entire poured liquid

Rice. 249. In all the vessels depicted, the pressure on the bottom is the same. In the first two vessels it is more than the weight of the liquid poured, in the other two it is less

The hydrostatic pressure at each point on the bottom of the vessel will be the same:

If the bottom of the vessel has an area, then the force of pressure of the liquid on the bottom of the vessel, i.e., is equal to the weight of the liquid poured into the vessel.

Let us now consider vessels that differ in shape, but with the same bottom area (Fig. 249). If the liquid in each of them is poured to the same height, then the pressure is on the bottom. it is the same in all vessels. Therefore, the pressure force on the bottom is equal to

is also the same in all vessels. It is equal to the weight of a column of liquid with a base equal to the area of ​​the bottom of the vessel and a height equal to the height of the liquid poured. In Fig. 249 this pillar is shown next to each vessel with dashed lines. Please note that the force of pressure on the bottom does not depend on the shape of the vessel and can be either greater or less than the weight of the liquid poured.

Rice. 250. Pascal's device with a set of vessels. The cross sections are the same for all vessels

Rice. 251. Experiment with Pascal's barrel

This conclusion can be verified experimentally using the device proposed by Pascal (Fig. 250). The stand can hold vessels of various shapes that do not have a bottom. Instead of a bottom, a plate suspended from the balance beam is tightly pressed against the vessel from below. If there is liquid in the vessel, a pressure force acts on the plate, which tears the plate off when the pressure force begins to exceed the weight of the weight standing on the other pan of the scale.

In a vessel with vertical walls (cylindrical vessel), the bottom opens when the weight of the poured liquid reaches the weight of the weight. In vessels of other shapes, the bottom opens at the same height of the liquid column, although the weight of the poured water may be greater (a vessel expanding upward) or less (a vessel narrowing) than the weight of the weight.

This experience leads to the idea that with the proper shape of the vessel, it is possible to obtain enormous pressure forces on the bottom using a small amount of water. Pascal attached a long thin vertical tube to a tightly caulked barrel filled with water (Fig. 251). When the tube is filled with water, the force of hydrostatic pressure on the bottom becomes equal to the weight of a column of water, the base area of ​​which is equal to the area of ​​the bottom of the barrel, and the height is equal to the height of the tube. Accordingly, the pressure forces on the walls and upper bottom of the barrel increase. When Pascal filled the tube to a height of several meters, which required only a few cups of water, the resulting pressure forces ruptured the barrel.

How can we explain that the force of pressure on the bottom of a vessel can be, depending on the shape of the vessel, greater or less than the weight of the liquid contained in the vessel? After all, the force acting on the liquid from the vessel must balance the weight of the liquid. The fact is that the liquid in the vessel is affected not only by the bottom, but also by the walls of the vessel. In a container expanding upward, the forces with which the walls act on the liquid have components directed upward: thus, part of the weight of the liquid is balanced by the pressure forces of the walls and only part must be balanced by the pressure forces from the bottom. On the contrary, in a vessel that tapers upward, the bottom acts upward on the liquid, and the walls act downward; therefore, the force of pressure on the bottom is greater than the weight of the liquid. The sum of the forces acting on the liquid from the bottom of the vessel and its walls is always equal to the weight of the liquid. Rice. 252 clearly shows the distribution of forces acting from the walls on liquid in vessels of various shapes.

Rice. 252. Forces acting on liquid from the walls of vessels of various shapes

Rice. 253. When water is poured into the funnel, the cylinder rises up.

In a vessel that tapers upward, a force directed upward acts on the walls from the liquid side. If the walls of such a vessel are made movable, the liquid will lift them. Such an experiment can be carried out using the following device: the piston is fixedly fixed, and a cylinder is put on it, turning into a vertical tube (Fig. 253). When the space above the piston is filled with water, pressure forces on the areas and walls of the cylinder lift the cylinder upward.

Liquids and gases transmit in all directions not only the external pressure exerted on them, but also the pressure that exists inside them due to the weight of their own parts. The upper layers of liquid press on the middle ones, those on the lower ones, and the latter ones on the bottom.

The pressure exerted by a fluid at rest is called hydrostatic.

Let us obtain a formula for calculating the hydrostatic pressure of a liquid at an arbitrary depth h (in the vicinity of point A in Figure 98). The pressure force acting in this place from the overlying narrow vertical column of liquid can be expressed in two ways:
firstly, as the product of the pressure at the base of this column and its cross-sectional area:

F = pS ;

secondly, as the weight of the same column of liquid, i.e. the product of the mass of the liquid (which can be found by the formula m = ρV, where volume V = Sh) and the acceleration of gravity g:

F = mg = ρShg.

Let us equate both expressions for the pressure force:

pS = ρShg.

Dividing both sides of this equality by area S, we find the fluid pressure at depth h:

p = ρgh. (37.1)

We got hydrostatic pressure formula. Hydrostatic pressure at any depth inside a liquid does not depend on the shape of the container in which the liquid is located and is equal to the product of the density of the liquid, the acceleration of gravity and the depth at which the pressure is considered.

The same amount of water, being in different vessels, can exert different pressure on the bottom. Since this pressure depends on the height of the liquid column, it will be greater in narrow vessels than in wide ones. Thanks to this, even a small amount of water can create very high pressure. In 1648, this was very convincingly demonstrated by B. Pascal. He inserted a narrow tube into a closed barrel filled with water and, going up to the balcony of the second floor of the house, poured a mug of water into this tube. Due to the small thickness of the tube, the water in it rose to a great height, and the pressure in the barrel increased so much that the fastenings of the barrel could not withstand it, and it cracked (Fig. 99).
The results we obtained are valid not only for liquids, but also for gases. Their layers also press on each other, and therefore hydrostatic pressure also exists in them.

1. What pressure is called hydrostatic? 2. What values ​​does this pressure depend on? 3. Derive the formula for hydrostatic pressure at an arbitrary depth. 4. How can you create a lot of pressure with a small amount of water? Tell us about Pascal's experience.
Experimental task. Take a tall vessel and make three small holes in its wall at different heights. Cover the holes with plasticine and fill the vessel with water. Open the holes and watch the streams of water flowing out (Fig. 100). Why does water leak out of the holes? What does it mean that water pressure increases with depth?

Plumbing, it would seem, does not provide much reason to delve into the jungle of technologies, mechanisms, or engage in scrupulous calculations to build complex schemes. But such a vision is a superficial look at plumbing. The real plumbing industry is in no way inferior in complexity to the processes and, like many other industries, requires a professional approach. In turn, professionalism is a solid store of knowledge on which plumbing is based. Let’s dive (albeit not too deeply) into the plumbing training stream in order to get one step closer to the professional status of a plumber.

The fundamental basis of modern hydraulics was formed when Blaise Pascal discovered that the action of fluid pressure is constant in any direction. The action of liquid pressure is directed at right angles to the surface area.

If a measuring device (pressure gauge) is placed under a layer of liquid at a certain depth and its sensitive element is directed in different directions, the pressure readings will remain unchanged in any position of the pressure gauge.

That is, the fluid pressure does not depend in any way on the change in direction. But the fluid pressure at each level depends on the depth parameter. If the pressure meter is moved closer to the surface of the liquid, the reading will decrease.

Accordingly, when diving, the measured readings will increase. Moreover, under conditions of doubling the depth, the pressure parameter will also double.

Pascal's law clearly demonstrates the effect of water pressure in the most familiar conditions for modern life.

Therefore, whenever the speed of movement of a fluid is set, part of its initial static pressure is used to organize this speed, which subsequently exists as a pressure speed.

Volume and flow rate

The volume of fluid passing through a particular point at a given time is considered as flow volume or flow rate. Flow volume is usually expressed in liters per minute (L/min) and is related to the relative pressure of the fluid. For example, 10 liters per minute at 2.7 atm.

Flow velocity (fluid speed) is defined as the average speed at which a fluid moves past a given point. Typically expressed in meters per second (m/s) or meters per minute (m/min). Flow rate is an important factor when sizing hydraulic lines.

Volume and flow rate are often considered simultaneously. All other things being equal (assuming the input volume remains constant), the flow rate increases as the cross-section or size of the pipe decreases, and the flow rate decreases as the cross-section increases.

Thus, a slowdown in flow speed is observed in wide parts of pipelines, and in narrow places, on the contrary, the speed increases. At the same time, the volume of water passing through each of these control points remains unchanged.

Bernoulli's principle

The well-known Bernoulli principle is built on the logic that a rise (fall) in the pressure of a fluid fluid is always accompanied by a decrease (increase) in speed. Conversely, an increase (decrease) in fluid velocity leads to a decrease (increase) in pressure.

This principle underlies a number of common plumbing phenomena. As a trivial example, Bernoulli's principle is responsible for causing the shower curtain to "retract inward" when the user turns on the water.

The pressure difference between the outside and inside causes a force on the shower curtain. With this forceful effort, the curtain is pulled inward.

Another clear example is a perfume bottle with a spray, where a low pressure area is created due to high air speed. And the air carries the liquid with it.

Bernoulli's principle also shows why windows in a house tend to break spontaneously during hurricanes. In such cases, the extremely high speed of air outside the window leads to the fact that the pressure outside becomes much less than the pressure inside, where the air remains practically motionless.

A significant difference in force simply pushes the windows outward, causing the glass to break. So when a major hurricane approaches, you essentially want to open the windows as wide as possible to equalize the pressure inside and outside the building.

And a couple more examples when Bernoulli’s principle operates: the rise of an airplane with subsequent flight due to the wings and the movement of “curve balls” in baseball.

In both cases, a difference in the speed of air passing past the object from above and below is created. For airplane wings, the difference in speed is created by the movement of the flaps; in baseball, it is the presence of a wavy edge.

Home Plumber Practice

Hydrostatics is the branch of hydraulics that studies the laws of equilibrium of fluids and considers the practical application of these laws. In order to understand hydrostatics, it is necessary to define some concepts and definitions.

Pascal's law for hydrostatics.

In 1653, the French scientist B. Pascal discovered a law that is commonly called the fundamental law of hydrostatics.

It sounds like this:

The pressure on the surface of a liquid produced by external forces is transmitted into the liquid equally in all directions.

Pascal's law is easily understood if you look at the molecular structure of matter. In liquids and gases, molecules have relative freedom; they are able to move relative to each other, unlike solids. In solids, molecules are assembled into crystal lattices.

The relative freedom that the molecules of liquids and gases have allows the pressure exerted on the liquid or gas to be transferred not only in the direction of the force, but also in all other directions.

Pascal's law for hydrostatics is widely used in industry. The work of hydraulic automation, which controls CNC machines, cars and airplanes, and many other hydraulic machines, is based on this law.

Definition and formula of hydrostatic pressure

From Pascal’s law described above it follows that:

Hydrostatic pressure is the pressure exerted on a fluid by gravity.

The magnitude of hydrostatic pressure does not depend on the shape of the vessel in which the liquid is located and is determined by the product

P = ρgh, where

ρ – liquid density

g – free fall acceleration

h – depth at which pressure is determined.

To illustrate this formula, let's look at 3 vessels of different shapes.

In all three cases, the pressure of the liquid at the bottom of the vessel is the same.

The total pressure of the liquid in the vessel is equal to

P = P0 + ρgh, where

P0 – pressure on the surface of the liquid. In most cases it is assumed to be equal to atmospheric pressure.

Hydrostatic pressure force

Let us select a certain volume in a liquid in equilibrium, then cut it into two parts by an arbitrary plane AB and mentally discard one of these parts, for example the upper one. In this case, we must apply forces to the plane AB, the action of which will be equivalent to the action of the discarded upper part of the volume on the remaining lower part of it.

Let us consider in the section plane AB a closed contour of area ΔF, which includes some arbitrary point a. Let a force ΔP act on this area.

Then the hydrostatic pressure whose formula looks like

Рср = ΔP / ΔF

represents the force acting per unit area, will be called the average hydrostatic pressure or the average hydrostatic pressure stress over the area ΔF.

The true pressure at different points of this area may be different: at some points it may be greater, at others it may be less than the average hydrostatic pressure. It is obvious that in the general case, the average pressure Рср will differ less from the true pressure at point a, the smaller the area ΔF, and in the limit the average pressure will coincide with the true pressure at point a.

For fluids in equilibrium, the hydrostatic pressure of the fluid is similar to the compressive stress in solids.

The SI unit of pressure is newton per square meter (N/m2) - it is called pascal (Pa). Since the value of the pascal is very small, enlarged units are often used:

kilonewton per square meter – 1 kN/m 2 = 1*10 3 N/m 2

meganewton per square meter – 1MN/m2 = 1*10 6 N/m2

A pressure equal to 1*10 5 N/m 2 is called a bar (bar).

In the physical system, the unit of pressure intention is dyne per square centimeter (dyne/m2), in the technical system it is kilogram-force per square meter (kgf/m2). In practice, liquid pressure is usually measured in kgf/cm2, and a pressure equal to 1 kgf/cm2 is called technical atmosphere (at).

Between all these units there is the following relationship:

1at = 1 kgf/cm2 = 0.98 bar = 0.98 * 10 5 Pa = 0.98 * 10 6 dyne = 10 4 kgf/m2

It should be remembered that there is a difference between the technical atmosphere (at) and the physical atmosphere (At). 1 At = 1.033 kgf/cm 2 and represents normal pressure at sea level. Atmospheric pressure depends on the altitude of a place above sea level.

Hydrostatic pressure measurement

In practice, various methods are used to take into account the magnitude of hydrostatic pressure. If, when determining hydrostatic pressure, the atmospheric pressure acting on the free surface of the liquid is also taken into account, it is called total or absolute. In this case, the pressure value is usually measured in technical atmospheres, called absolute (ata).

Often, when taking pressure into account, atmospheric pressure on the free surface is not taken into account, determining the so-called excess hydrostatic pressure, or gauge pressure, i.e. pressure above atmospheric.

Gauge pressure is defined as the difference between the absolute pressure in a liquid and atmospheric pressure.

Rman = Rabs – Ratm

and are also measured in technical atmospheres, called in this case excess.

It happens that the hydrostatic pressure in a liquid is less than atmospheric. In this case, the liquid is said to have a vacuum. The magnitude of the vacuum is equal to the difference between atmospheric and absolute pressure in the liquid

Rvak = Ratm – Rabs

and is measured from zero to the atmosphere.

Hydrostatic water pressure has two main properties:
It is directed along the internal normal to the area on which it acts;
The amount of pressure at a given point does not depend on the direction (i.e., on the orientation in space of the site on which the point is located).

The first property is a simple consequence of the fact that in a fluid at rest there are no tangential and tensile forces.

Let us assume that the hydrostatic pressure is not directed along the normal, i.e. not perpendicular, but at some angle to the site. Then it can be decomposed into two components - normal and tangent. The presence of a tangential component, due to the absence of forces of resistance to shearing forces in a fluid at rest, would inevitably lead to the movement of the fluid along the platform, i.e. would upset her balance.

Therefore, the only possible direction of hydrostatic pressure is its direction normal to the site.

If we assume that the hydrostatic pressure is directed not along the internal, but along the external normal, i.e. not inside the object under consideration, but outside from it, then due to the fact that the liquid does not resist tensile forces, the particles of the liquid would begin to move and its equilibrium would be disrupted.

Consequently, the hydrostatic pressure of water is always directed along the internal normal and represents compressive pressure.

From this same rule it follows that if the pressure changes at some point, then the pressure at any other point in this liquid changes by the same amount. This is Pascal's law, which is formulated as follows: The pressure exerted on a liquid is transmitted inside the liquid in all directions with equal force.

The operation of machines operating under hydrostatic pressure is based on the application of this law.

Another factor influencing the pressure value is the viscosity of the liquid, which until recently was usually neglected. With the advent of units operating at high pressure, viscosity also had to be taken into account. It turned out that when the pressure changes, the viscosity of some liquids, such as oils, can change several times. And this already determines the possibility of using such liquids as a working medium.