What is Developer Zone?

Document Type: Prentice Hall
Author: Curtis D. Johnson
Book: Process Control Instrumentation Technology
ISBN: 0-13-441305-9
NI Supported: No
Publish Date: Sep 6, 2006

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# Pressure Principles

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### Overview

Pressure is simply the force per unit area that a fluid exerts on its surroundings. If it is a gas, then the pressure of the gas is the force per unit area that the gas exerts on the walls of the container that holds it. If the fluid is a liquid, then the pressure is the force per unit area that the liquid exerts on the container in which it is contained. Obviously, the pressure of a gas will be uniform on all the walls that must enclose the gas completely. In a liquid, the pressure will vary, being greatest on the bottom of the vessel and zero on the top surface, which need not be enclosed.

## Static Pressure

The statements made in the previous paragraph are explicitly true for a fluid that is not moving in space, that is not being pumped through pipes or flowing through a channel. The pressure in cases where no motion is occurring is referred to as static pressure.

## Dynamic Pressure

If a fluid is in motion, the pressure that it exerts on its surroundings depends on the motion. Thus, if we measure the pressure of water in a hose with the nozzle closed, we may find a pressure of, say, 40 pounds per square inch (note: force per unit area). If the nozzle is opened, the pressure in the hose will drop to a different value, say, 30 pounds per square inch. For this reason, a thorough description of pressure must note the circumstances under which it is measured. Pressure can depend on flow, compressibility of the fluid, external forces, and numerous other factors.

## Units

Since pressure is force per unit area, we describe it in the SI system of units by newtons per square meter. This unit has been named the pascal (Pa), so that 1 Pa = 1 N/m2. As will be seen later, this is not a very convenient unit, and it is often used in conjunction with the SI standard prefixes, as kPa, or MPa. You will see the combination N/cm2 used, but use of this combination should be avoided in favor of Pa with the appropriate prefix. In the English system of units, the most common designation is the pound per square inch, Ib/in2. This is usually written psi. The conversion is that 1 psi is approximately 6.895 kPa. For very low pressures, such as may be found in vacuum systems, the unit Torr is often used. One Torr is approximately 133.3 Pa. Again, use of the pascal with appropriate prefix is preferred. Other units that you may encounter in the pressure description are the atmosphere (at), which is 101.325 kPa or ==14.7 psi, and the bar, which is 100 kPa. The use of inches or feet of water and millimeters of mercury will be discussed later.

## Gauge Pressure

In many cases, the absolute pressure is not the quantity of major interest in describing the pressure. The atmosphere of gas that surrounds the earth exerts a pressure, because of its weight, at the surface of the earth of approximately 14.7 psi, which defines the "atmosphere" unit. If a closed vessel at the earth's surface contained a gas at an absolute pressure of 14.7 psi, then it would exert no effective pressure on the walls of the container because the atmospheric gas exerts the same pressure from the outside. In cases like this, it is more appropriate to describe pressure in a relative sense, that is, compared to atmospheric pressure. This is called gauge pressure, and is given by

Pg = Pabs - Pat (5.30)
where pg = gauge pressure
pabs = absolute pressure
pat = atmospheric pressure

In the English system of units, the abbreviation psig is used to represent the gauge pressure.

For liquids, the expression head pressure or pressure head is often used to describe the pressure of the liquid in a tank or pipe. This refers to the static pressure produced by the weight of the liquid above the point at which the pressure is being described. This pressure depends only on the height of the liquid above that point and the liquid density (mass per unit volume). In terms of an equation, if a liquid is contained in a tank, then the pressure at the bottom of the tank is given by

P = Pgh (5.31)

where p = pressure in Pa
p = density in kg/m3
g = acceleration due to gravity (9.8 m/s2)
h = depth of liquid in m

This same equation could be used to find the pressure in the English system, but it is common practice to express the density in this system as the weight density pw in Ib/ft3, which includes the gravity term of Equation (5.31). In this case, the relation between pressure and depth becomes

p = pwh (5.32)

where p = pressure in Ib/ft2
pw = weight density in Ib/ft3
h = depth in ft

If the pressure is desired in psi, then the ft2 would be expressed as 144 in2. Because of the common occurrence of liquid tanks and the necessity to express the pressure of such systems, it has become common practice to describe the pressure directly in terms of the equivalent depth of a particular liquid. Thus, the term mm of mercury means that the pressure is equivalent to that produced by so many millimeters of mercury depth, which could be calculated from Equation (5.31) using the density of mercury. In the same sense, the expression "inches of water" or "feet of water" means the pressure that is equivalent to some particular depth of water using its weight density.

Now you can see the basis for level measurement on pressure mentioned in Section 5.2.4. Equation (5.31) shows that the level of liquid of density p is directly related to the pressure. From level measurement we pass to pressure measurement, which is usually done by some type of displacement measurement.

EXAMPLE 5.16
A tank holds water with a depth of 7.0 feet. What is the pressure at the tank bottom in psi and Pa? (density = 103 kg/m3)

Solution
We can find the pressure in Pa directly by converting the 7.0 ft into meters; thus,

(7.0 ft)(0.3048 m/ft) = 2.1 m. From Equation (5.31),

p = (103 kg/m3)(m/s2)(2.1m)
p = 21 kPa (note significant figures)

To find the pressure in psi, we can convert the pressure in Pa to psi or use Equation (5.32). Let's use the latter. The weight density is found from

pw = (103 kg/m3)(9.8 m/s2) = 9.8 X 103 N/m3
or
= (9.8 X 103 N/m3)(0.3048 m/ft3)(0.2248lb/N
= 62.4 lb/ft3

The pressure is
p = (62.4 lb/ft3)(7.0ft) = 440 lb/ft2
p = 3 psi

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