Fluids Basics and Concepts

2 Fluids Basics and Concepts

TO PIC PAGE

2.1 Introduction ................................................ 22

2.2 Force, Weight , and Mass ............................ 22

2.3 Density ........................................................ 23

2.4 Specific Gravity ............................................ 23

2.5 Pressure ....................................................... 23

2.6 Temperature ................................................ 25

2.7 Viscosity ...................................................... 26

2.8 Fluid Velocity ............................................... 27

2.9 Mass and Volumetric Flow .......................... 27

2.10 Isentropic Exponent .................................... 28

2 – Fluids Basics and Concepts

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2.1 INTRODUCTIONThis chapter presents an overview of the physical properties of fluids, a review of basic concepts, and covers the fundamentals required to understand the theory of measuring flow, which is presented in Chapter 3.

A fluid is a substance that continues to deform when subjected to a shear stress. Fluids can be liquids, vapors, or gases. For most fluids, some of the fluid properties can be calculated by knowing other properties.

There are five key fluid properties that must be known to properly size and use a DP flowmeter: density or specific weight, static pressure, temperature, isentropic exponent and viscosity. These properties factor into DP Flow calculations.

2.2. FORCE, MASS AND WEIGHT

Property Symbol Units

Mass m lbm , kg

Force F lbf , N

When working with fluids in motion, it is important to review the physics behind the concept of force, mass, and weight. The DP flowmeter uses the concept of energy conversion to determine the rate of flow in a pipe by measuring a physical difference in pressures.

The US Customary unit of mass is the lbm and the SI unit is the kilogram. They are related by:

Force

Acceleration, a

Mass,m

Figure 2.2.a - Newton’s Second Law states force is equal to mass times its acceleration.

The SI unit of force is the Newton, N, and is the amount of force needed to move a 1kg mass at an acceleration of 1m/s2. So:

(2.1)

(2.2)

In SI units, gc = 1kg-m/(N-s2). Since the value of gc is 1, this factor is often left out of force calculations using the metric system.

In imperial US units, the unit of force is the pound force, lbf, and is the amount of force needed to move 1 pound mass, lbm, one foot at an acceleration of 32.174 ft/s2. Thus:

Where:

(2.3)

Newton’s Second Law of Motion describes the relationship between force, mass and acceleration as:

(2.1)

Where:

F = The force applied to or by an object

m = The mass of the object

a = The resulting acceleration of the object

To account for the force of gravity, a gravity conversion constant gc is needed. Thus, Newton’s second law becomes:


23

(2.4)

Weight of an object can be defined as:

Where:

W = The weight in force units

gl = The local acceleration of gravity

The value of earth’s gravity is slightly higher at the poles and slightly lower at the equator due to the earth’s rotation. At 45° latitude, gl = gc, and the force exerted by a Lbm is exactly a Lbf .

This concept is used to account for the weight of fluids in the energy equation (such as the elevation or “z” term in the Bernoulli flow equation), and in fluid statics when converting gravitational head to pressure, as is done with the manometer.

2.3 DENSITY


Density ρ lbm /ft2, kg/m2

The density of a fluid is its mass per unit volume. Note that for the same quantity of mass, the volume occupied by that mass will vary with temperature and pressure (see 2.9 Mass and Volumetric Flow). Fluctuations in density are typically small for liquids but is much greater for gases. Fluids whose densities change slightly with moderate temperature and pressure fluctuations are considered incompressible. If the density changes significantly with varying pressures and temperatures, it is considered a compressible fluid.

For industrial gas measurement applications it is common to use the Real Gas Law or the Ideal Gas Law. The Ideal Gas Law is applicable for moderate temperatures and low pressures. It fails to account for the interaction between gas molecules. When assumptions regarding the Ideal Gas Law do not apply the Real Gas Law is applicable. See Chapter 4 for more information on density and compressibility considerations.

2.4 SPECIFIC GRAVITYSpecific gravity (SG) is the ratio of density of one substance to the density of a second, or reference, substance. The reference substance for liquids is generally water at 68°F (20°C). The density of distilled water at 68°F is 62.316 lbm /ft3, or at 20°C, 998 kg/m3. Specific gravity thus provides a simple number that indicates whether a liquid is lighter or heavier than water.

Specific gravity of liquids is generally obtained withhydrometers, instruments whose scales read in specificgravities, degrees Baume (°B), or degrees API (American Petroleum Institute).

The reference fluid for specific gravity of gases is air. Specific gravity of a gas is defined as the ratio of the molecular weight of the gas of interest to the molecular weight of air (defined as 28.9644). This method avoids the difficulty of calculating the gas density based on temperature and pressure, as well as accounting for the non-linear behavior of gases. As long as the composition of a gas does not change, the ratio of molecular weight against that of a reference gas will remain the same regardless of temperature, pressure, or location.

2.5 PRESSURE


Pressure P psi, Pa

Pressure is the force acting on a surface in the normal direction (i.e., perpendicularly) per unit area (figure 2.2.4.a). The US unit of pressure is related by:

where the lb /in is force and not mass.

The SI unit of pressure is the Pascal and is equal to:

Where:

N = Newton and is in units of kg-m/s2


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Force Area

Figure 2.5.1.a - Pressure is force acting perpendicularly on an area.

This is a difference of 0.08% between the two reference temperatures, thus it is important to know the reference temperature.

(2.5)

The Pascal is a very small unit of pressure and is generally expressed in kPa (kiloPascals) or MPa (megaPascals). Another common SI unit of pressure is the bar which is equal to 100 kPa.

2.5.1 Absolute and Gage Pressure

The absolute pressure is the pressure relative to a perfect vacuum. The gage pressure is the pressure relative to atmospheric pressure. Thus:

The absolute pressure is used to compute the density of gases. The gage pressure, since it is relative to atmospheric pressure, is used to ensure that pressure retaining parts (ie the pipe or parts of flowmeters that retain the pressure when installed in the pipe) will remain within safe working limits.

Standard atmospheric pressure is typically defined as 1atm, or 14.69595 psi or101.325 kPa. Actual atmospheric pressure at any given location depends on that location’s elevation above sea level and day-to-day weather conditions. Typically, changes in the atmospheric pressure due to weather are not used when calculating absolute pressure. However local standard atmospheric pressure–adjusted for elevation –is used to determine the atmospheric pressure for purposes of measuring the flow rate.

The value of pressure for a flowmeter application is used to provide information for two separate but important engineering tasks:

• For the calculation of fluid parameters– especially gas or vapor density and gas expansion factor• To check the compatibility and safety margins for the mounting hardware

2.5.2 Differential Pressure

When the difference in two pressures is needed, as called for in DP Flow calculations, it is called the differential pressure or DP (figure 2.5.2.a).

The SI unit for DP is Pa or kPa.

The US unit for DP is psi or inches of water (inH2O) at a specified temperature. The inches of water unit is a carryover from the past where manometers were used to measure flow rate and indicates the pressure at the bottom of a column of water of the specified height when the water is at a specific temperature. As an example a DP of 25 inches H2O at 68°F means that this is the pressure at the bottom of a column of water that is 25 inches high when the temperature of the water is a uniform 68°F. There are two commonly used versions of this unit inches H2O at 68°F (used in the US process control industry) and inches H2O at 60°F (used in the US natural gas industry). The conversion factors in psi for each of these is:


Differential Pressure ΔP or DP inH20 @ 68°F, inHg*, kPa, mbar

*inches of mercury; can also be inches of water, alcohol, oil or other fluid

Figure 2.5.2.a - Differential pressure. P1 has higher pressure than P2.

Standard orifice plate

Rosemount ConditioningOrifice Plate

Nozzle

Venturi

P1

P2


25

Differential pressure can be calculated with a simple relation:

The simplest of instruments used for the measurement of a small difference in pressures is the manometer A manometer uses a U-shaped tube or two vertical tubes connected at the bottom, with a liquid filling the tubes part-way (figure 2.5.2.b). When the two pressures are applied at the top of each tube, the liquid in the two tubes change height, and a scale fixed to the manometer is used to measure the height or elevation difference, h. The differential pressure is then calculated using the relation:

(2.6)

(2.7)

Where:

ΔP = Differential pressure, in appropriate units

gl = The local acceleration of gravity

gc = Units conversion constant from mass to

force

pm = The density of the manometer fluid

pf = The density of the fluid conveying the

pressure

h = Elevation difference or height of the fluid

at the points of measure

The standard manometer fluid for gas flow DP measurement is water, with the height indicated in inches, and the DP expressed as “Inches of Water Column.” Of course, if water or other liquids are being measured, a heavier manometer fluid is needed. Typically, mercury (S.G. = 13.5), or bromide-based fluids are used (S.G. = 2.5 to 3.0). Any fluid that is heavier than water could be used, but it must be “immiscible,” or unable to mix with the fluid in contact with the manometer.

P

P0Unknown Pressure

Ambient Fluid(Atmosphericpressure inmost cases)

Reference FluidDensity p (Liquid,e.g., water or mercury)

hFluid of Interest(Gas in most cases)

Figure 2.5.2.b - Example of how a manometer works.

Note that manometers and mechanical pressure gages are old technology. Current best practices exploit electronic differential pressure and static pressure transmitters (figure 2.5.2.c), which provide extremely accurate readings over very large ranges of pressure or DP, and can operate over a wide range of ambient temperatures without external correction. The electronic signal output is easily fed into microprocessors for calculating the flow rate or logging data.

Figure 2.5.2.c - Modern differential pressure transmitter.

2.6 TEMPERATURE


Temperature T °C, °F, °K, °R

1 More in-depth information can be found in The Engineer’s Guide to Industrial Temperature Measurement, Rose- mount literature reference number 00805-0100-1036 or go to Rosemount.com/temperature.

Outside of the engineering world, measurements are generally done on the relative Fahrenheit or Celsius scales, which were originally devised to measure the earth’s temperate range. However, flow engineering problems require a different temperature scale, one that represents an absolute temperature. Absolute temperature has units of Kelvin in SI units and Rankine in US customary units. The relationships between absolute temperature and the conventionally used units of °F and °C are:

The absolute temperature is required in the calculation of the fluid properties (i.e., density, viscosity and insentropic exponent). The calculation of thermal expansion effects involves temperature differences so it is common to use °F or °C.

Absolute zero Rankine and absolute zero Kelvin are equivalent. The Rankine scale increments by degrees Fahrenheit, while the Kelvin scale increments by degrees Celsius.


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2.7 VISCOSITY

(2.10)

Every real fluid has viscosity, which changes primarily with temperature. For this reason, fluid viscosity is usually plotted against temperature, and equations are developed that allow the calculation of viscosity once the temperature is known. For a liquid, the viscosity decreases with temperature; for a gas, the viscosity increases with temperature.

Kinematic viscosity is the absolute viscosity divided by the density of the fluid at the same temperature, or:

Fluids are classified by the relationship between fluid stress (the force needed to overcome viscosity) and the strain (the fluid velocity). Figure 2.7.b shows the plot for different types of fluids based on the behavior of μ. DP flowmeters are restricted to “Newtonian” type fluids, or those where the slope of the fluid stress/strain curve (μ) is constant. Shear-thinning fluid viscosities decrease with increasing shear stress; examples include ketchup, lava, or polymer solutions and molten polymers. Shear thickening (with viscosity increasing as shear stress increases) include suspensions—corn starch in water for example. Bingham plastics do not flow until a critical stress yield is exceeded; examples of this type of fluid include toothpaste.

(2.11)

boundary plate (2D, stationary)

boundary plate(2D, moving)

shear stress,τ

velocity,ν

dimensionγ

δυδγ

gradient,�uid

Figure 2.7.a - Viscosity defines the resistance to movement of a fluid.

Absolute viscosity defines the resistance to movement of a fluid (figure 2.7.a). It is a measure of the resistance of fluid molecules to velocity change due to shear stress. Stated differently, viscosity tends to resist one particle from moving faster than an adjacent particle. Absolute viscosity is defined as:

459.67

(2.8)

(2.9)


Absolute Viscosity μ centipoise (μcp), lbm/ft-s, kg/m-s

Dynamic Viscosity ν centistokes

Where:

τ = The shearing stress in the fluid, or the

force required to move the fluid against a

surface per unit area

= The change in velocity or “strain” between

the wall or surface and the free-stream

velocity

Industrial methods of measuring temperature are based on substances that change electrical resistance with temperature, such as RTDs (resistance temperature detectors), or thermocouples that generates a voltage at the junction of dis-similar metals that is based on temperature.


Shea

ring

stre

ss,τ

δυ

μ

δγRate of shearing strain,

Shear thickening

Shear thinning

Bingham plasticNewtonian

1

Figure 2.7.b - Fluid classification based on the behavior of viscosity, m.

Velocity is not a fluid property, but can be used to predict the behavior of fluids in motion and will frame the application of DP flowmeters. In general, velocity is the rate of change of an object’s position relative to a reference, and is equivalent to the specification of speed and direction of an object. As applied to fluid dynamics, velocity defines the speed of a particle of fluid with respect to a stationary reference such as a pipe. When a fluid flows around an object or through a pipe, the viscosity of the fluid creates a velocity profile. If there were no viscosity, the velocity of a flowing fluid in a pipe would be uniform across a pipe section. With the slightest of viscosity, however, shearing between adjacent fluid particles produces a non-uniform velocity profile in the pipe, with a velocity of zero at the pipe wall and maximum at the pipe centerline for developed flow.

Fluid flow in a pipe also defines a velocity field. The flow of fluids through pipes has been studied extensively and velocity fields can be predicted when the rate of flow and fluid properties are known.

2.8 FLUID VELOCITY


Velocity v ft/s, m/s

2.9 MASS AND VOLUMETRIC FLOW

Volumetric flow is measured in terms of volume, as the name implies, yielding how much volume is passing through a given area, as follows:


Volumetric Flow Rate Qv

1/hr, m3/hr

Mass Flow Rate Qm

lb/hr, kg/hr

(2.14)

(2.13)

(2.12)

Mass flow is dependent on density and the volumetric flow rate, as follows:

Volumetric flow and mass flow can be related by the following:

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In other words, when the mass flow rate (units of mass/unit time) is divided by the density at reference conditions, the flow rate is equivalent to the volume the fluid would occupy if its pressure and temperature were adjusted from the flowing conditions to reference conditions. So if the flowing pressure is 146.96 psia, the temperature is 68 °F, and the reference conditions are 14.696 psia and 68°F when the mass flow rate is converted to standard volume flow rate (by dividing by the density of the fluid at reference conditions) the numerical value will increase by a factor of 10. The mass will remain the same, but in order for the pressure to be at 14.696 psia it must occupy 10 times the volume as it did under flowing conditions.


For a given quantity of liquid, mass does not change, but volume can with changes in pressure and temperature (figure 2.9.a).

The measurement of mass flow is preferred for most gases and liquids, while volumetric flow can be acceptable for stable liquids. Flowmeters suitable for mass flow include multivariable DP flowmeters or Coriolis meters. Volumetric flowmeters include DP flowmeters, turbine, vortex, magmeter, or variable area meters.

2.10 ISENTROPIC EXPONENTAs gases flow through a restriction in a pipe the density changes due to pressure changes. The expansion of the gas is assumed to be an isentropic process and the effect of the density changes on the flow rate can be determined theoretically for some obstructions or empirically for others. The relevant fluid property is the isentropic exponent of the gas, designated by k (sometimes designated by g) and primarily is a function of temperature. It is common practice to determine k at a nominal temperature and use this value for all flow rates. Typical values of k can range from 1.0 to 1.4.

40.9 gal

342 lb 342 lb

20̊ F 60̊ F

42.0 gal2.7%

0.0%

Figure 2.9.a - Mass does not change with fluctuations in pressure and temperature, but volume can. This illustration depicts an example of how much volume can change with temperature.

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Literature reference number: 00805-0100-1041 Rev AA March 2015

Fluids Basics and Concepts

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