Chapter 13 - Viscosity and Specific Gravity

8/22/2019 Chapter 13 - Viscosity and Specific Gravity

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CHAPTER 13 ............................................................................................................. 13-1

13-1 Introduction. .................................................................................................................................................. 13-1

13-2 Viscosity Concepts. ........................................................................................................................................ 13-1

13-3 Rotational Viscometers. .............................................................................................................................. 13-13

13-4 Capillary Viscometers. ................................................................................................................................ 13-18

13-5 Other Viscometers. ...................................................................................................................................... 13-26

Falling Ball Viscometer .....................................................................................................................................13-26

Rising Bubble Viscometer .................................................................................................................................13-27

Comparison Rising Bubble Viscometer .............................................................................................................13-28

Float Viscometer ................................................................................................................................................13-28

13-6 Specific Gravity Concepts........................................................................................................................... 13-29

13-7 Specific Gravity Instruments ...................................................................................................................... 13-37

Picnometer and Balance .....................................................................................................................................13-37

Hydrometer ........................................................................................................................................................13-38

Specific Gravity Balance ...................................................................................................................................13-41

Pressure Devices ................................................................................................................................................13-42

VISCOSITY AND SPECIFIC GRAVITY

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CHAPTER 13


13-1 Introduction.This chapter will discuss two properties of fluids whose magnitude must be known inorder to perform meaningful measurements in the area of flow. Although theseproperties find use and application in other areas, they are, nevertheless, of primeimportance in the area of flow.

The first property, viscosity, is a relative measure of "fluid friction". It describeshow readily a fluid will flow. Fluids with a relatively low viscosity (water, for example)tend to flow readily; while a fluid like molasses which has a relatively high viscosity doesnot tend to flow easily. Viscosity is also used as a quantitative measure of how well anoil performs as a lubricant under various conditions.

The second property which is of interest is specific gravity. This is a relativemeasure of the density of a fluid, and it is important when we wish to determine thevolume of a known weight of fluid.

This chapter is not intended to be a complete treatise on viscosity and specificgravity. Such a treatment is not possible, due to the complexity of fluid behavior. Thepurpose, rather, is to introduce the basic concepts and measurement techniquesassociated with theses two properties as they apply to flow measurement and otherrelated areas.

13-2 Viscosity Concepts.

In earlier chapters we discussed various properties of solids and fluids. In the case ofsolids, terms such as stress, strain, and modules of elasticity were discussed. Young'sModules (Y) was presented to express the ratio of stress to strain in one direction bythe equation:

YF A

L L

There is, however, another modulus ofelasticity associated with solids which wasnot discussed -- the shear modulus. As thename implies, the shear modulus is used to

determine what happens to a solid when itexperiences a shearing force. A shearingforce is the force which causes parallelsections within an object to slide relative toone another. Such a force may be

Figure 13.1-Shear in a Solid


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represented as two equal and opposite forces, as illustrated in Figure 13.1. When thedeformation of the object exceeds the yield point, the object will be divided into twoparts.

Consider the action of the shearing force on the block in Figure 13.1. The solidlines indicate the configuration of the block under the action of the shear. The dotted

lines indicate the configuration of the block with no shear applied. The shear force (F)is considered to act over the area (A) of one face of the block. Thus, we can define theshear stress as:

sF

A (1)

where: F = the shear force

A = the area

and: s = the shear stress in units1

of lb/ft2

, lb/in2

, newton/m2

, etc.

The degree of deformation caused by a shear stress is measured as the ratio of theamount of "sliding" (x) to the "thickness" (L) of the block. This is shear strain and, byreferencing Figure 13.1, can be expressed in the form:

sx

L

(2)

where: x = the distance the material has moved

L = the thickness of the material

and: s (epsilon) = shear strain

The units in this expression cancel, leaving a unitless or pure number.The shear strain is numerically equal to the tangent of the angle (). If an angle

is small (


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where: = the angle in radians

The ratio of the shear stress to the shear strain is called the shear modulus (G), which

may be expressed in the form:

GF A

x L

(4)

or, using equation (3):

GF A

Notice the similarity between equation (4) and Young's Modulus, as previouslypresented. The basic difference is that the shear modulus is the ratio of the shearstress to the deformation of the solid (shear strain).

The shear modulus does not apply to fluids. This is because a fluid cannot bedeformed. When a fluid is subjected to a stress which tries to cause a deformation, thefluid reacts by flowing, not deforming. Thus, there can never be a deformation term inthe denominator of equation (4) as it applies to fluids.

A fluid, however, can be subjected to a shear stress. Thus, we can have anumerator for equation (4) as it applies to fluids, although we cannot have adenominator in terms of deformation. We must, therefore, generate a new measure to

replace the deformation term. The measureshould be related to the shear stress, if weare going to use it in a ratio form. This newratio, as will be shown later, will defineviscosity.

Since the reaction of the fluid to ashear stress is flow, then the measure weare looking for should be in terms of flow.Specifically, we will be concerned with thevelocity of the flow.

The development of a term concernedwith the velocity of flow cannot be attempteduntil a few basic concepts of flow areexplained. Viscosity and flow areinseparable. Viscosity has no meaningunless the motion or tendency for motion ina fluid is considered. Generally it isconsidered that there are two types of flow --

Figure 13.2-Laminar Flow


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laminar and turbulent. Laminar flow is a condition of flow where the fluid may beconsidered to be made up of many layers, each moving relative to the others. Thelayers of the laminar flow stream do not interact; thus, there is no mixing. Laminar flowstreams are shown in Figure 13.2. Such stream presentations may be produced byusing smoke traces.

Turbulent flow is a condition where the various layers of the stream mix and twisttogether. The layers in a lower section of the profile bend and intertwine with those in ahigher profile, and vice versa. The concept that there might be a unique velocityassociated with each layer is impossible with turbulent flow.

The definition and precise measurement of viscosity depend upon the conditionsof laminar flow. Unless otherwisestated, any reference to flow in thischapter will be considered as laminar.

In order to finalize thedevelopment of a term for thedenominator of equation (4), let us

consider a volume of fluid containedbetween two large plates. The bottomplate is stationary, and the upper plate ispulled across the fluid at some constantvelocity (v) by a force (F). The fluid incontact with the plates does not moverelative to the plate. That is, the fluid incontact with the upper plate moves withthat plate, while the fluid in contact with the lower plate is stationary. Figure 13.3 is adiagram illustrating this condition, with only a portion of the plates shown and the thinlayer of fluid indicated by small circles.

If these plates were placed very close together so that there were only threehypothetical layers of fluid, the top layer would try to drag the middle layer along with it.

However, at the same time, the bottomlayer would try to hold the middle layerstationary. Since the ability of the toplayer to influence the middle layer is thesame as that of the bottom layer, themiddle layer is not stationary, but movesat a velocity which is less than that of thetop layer. If we consider the separation(L) between the plates to be larger and to

contain more layers, we see that as weget farther from the top each layer of fluidis moving slower than the one above it,but faster than the one below it.

Figure 13.4 shows the relative

Figure 13.3-Fluid Velocities

Figure 13.4-Velocity Gradient


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velocity of several layers of a fluid between the moving and stationary plates. Noticethe similarity between this diagram and the shear diagram of Figure 13.1. The onlydifference is that in Figure 13.1 there was an increase in the amount of deformation aswe traveled from the lower surface to the upper surface, whereas in Figure 13.4 there is

an increase in the velocity of each layer as we go from the lower plate to the upperplate.The ratio of the change in velocity to the change in the separation distance

between the corresponding layers is illustrated in Figure 13.4. That is:

v

Lk (6)

The left hand side of equation (6) is called the velocity gradient. Since the ratio ofequation (6) is constant across the fluid, we can use the total separation and change invelocity to determine the velocity gradient which tells us the rate at which the velocity is

changing across the separation.

Example 1: Two plates are separated by eight inches. The top plate moves at 30ft/sec, while the bottom plate is stationary. Assuming the velocity gradientbetween the plates in constant, find its magnitude.

Using a value of 360 in/sec (30 ft/sec) forv and a value of eight inches for L andsubstituting their values into equation (6), we find:

v

L

in

360

8

in

sec

v

L

in

in 45

sec

This answer says that there is a change of velocity of 45 in/sec for each inch ofdifference between the plates. Canceling the units of the solution from Example 1, thefinal answer reduces to:

v

L 45 sec

or: = 45 sec-1

The form commonly used to express the velocity gradient is that of a number perunit time (generally reciprocal seconds). The numerical value of the gradient tells how


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fast the velocity is changing across the gap (separation of the plates) when the velocityand the gap measurements are in the same units. The velocity gradient found inExample 1 is also equal to:

45ft

ft

sec

or: 45miles

miles

sec

or: 45cm

cm

sec

or: 45km

km

sec

Now that the basic concepts have been explained, we can define VISCOSITY asthe ratio of the shear stress to the velocity gradient. Since, as we will see later, thereare other measures of viscosity, this basic definition is called absolute or dynamicviscosity and is assigned the Greek letter (eta)

2. The definition of absolute viscosity in

mathematical form is:

F A

v L (7)

Example 2: A plate with an area of 50 square cm requires a force of 100 dynes to

maintain it in motion at a constant velocity of 10 cm/sec over a 5 cmheight of fluid. Assuming that the only reactive force is that due to theviscosity, calculate its value.

Entering each known into equation (7) yields:

100

5010

5

2

dyne

cm

cm

cm sec

2-Many authors use the Greek letter (mu) to represent absolute viscosity.


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2

2

2dynes cm

sec

or:

1 2

dyne

cm

sec

The units of absolute viscosity are force "times" time over length squared.In the cgs system the unit of absolute viscosity is given the name POISE (in

honor of the French physicist, Jean L. M. Poiseuille). Thus:

1 1 2poisedyne

cm

sec

The absolute viscosity of the fluid in Example 2 is thus seen to be one poise. The units

of absolute viscosity in the MKS and FPS systems are, respectively:

MKS:

newton

meter

sec2

FPS:

lb

ft

sec2

There is no name assigned to the units of absolute viscosity in these systems.The cgs unit (poise) is, for practical measurement, quite large; thus, the absolute

viscosity of a fluid is often given in terms of the smaller unit centipoise:

1 centipoise = 0.01 poise

or: 1 centipoise = 10-2

poise

Table 13.1 lists a few approximate values for the absolute viscosity of severalcommonly encountered fluids. Notice that as the absolute viscosity increases, the fluidappears to have more resistance to flow, as judged by its pouring characteristics. Theeye is, however, an extremely poor instrument for judging viscosity, except in the range

from 10

+2

to 10

+4

centipoise where it is considered as fair.Fluids (if one considers them as such) which have absolute viscosities on theorder of 10

7centipoise or greater may appear solid if not examined closely. The

viscosity values for gases, on the other hand, are at the other end of the fluid spectrum,having values on the order of 10

-2centipoise or less. Between these two extremes (gas


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Fluid (room temperature) Viscosity (centipoise)

Hydrogen (1 atm) 0.009

Air (1 atm) 0.018

Gasoline 0.5

Water 1.0Crude Oil 10

SAE 30 (motor oil) 400

Molasses 200,000

Glass >1020

TABLE 13.1

APPROXIMATE VISCOSITIES OF SOME FLUIDS

and solids) is the range of fluids we refer to as liquids.Problems involving flow or the movement of objects through a stationary fluidoften contain a term which represents the ratio of the absolute viscosity to the density ofthe fluid. This ratio is called the kinematic viscosity and is assigned the Greek letter(nu). In mathematical form the kinematic viscosity is:

(8)

where: = absolute viscosity

= density

and: = kinematic viscosity

Example 3: What is the kinematic viscosity of a fluid whose density is 0.8 g/cm3

andwhose absolute viscosity is 4 poise?

Entering the known values of example 3 into equation (8) yields:

4

08 3poise

g cm.

5 3poise

g cm


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Replacing the poise term with its equivalent cgs units of force, time, and length, wehave:

5 2

3

dyne

cmg

cm

sec

or, replacing the unit of force with mass, length, and time, we get:

52 2

3

g cm

cmg

cm

sec

sec

5

2cm

sec

The units of kinematic viscosity are in terms of length squared per time. In the cgssystem, this unit of measure is given the name STOKE.

Thus: 1 12

stoke cm

sec

The kinematic viscosity of the fluid in Example 3 is, therefore, five (5) stokes. Again, asin the case of absolute viscosity, the stoke unit is found to be quite large, and the morepractical unit of centistoke is frequently used.

1 centistoke = 0.01 stokes

or: 1 centistoke = 10-2

stokes

The units of kinematic viscosity in the MKS and FPS systems are:

MKS

meter2

sec

FPS:

ft 2

sec


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Again, as with absolute viscosity, no name has been assigned to the FPS and MKSunits for kinematic viscosity.

Thus far in this discussion we have developed or used previously developedconcepts of shear stress, velocity gradient, absolute viscosity, density, and kinematic

viscosity. The remarks made thus far were general and, therefore, to keep the readerfrom being misled, we must impose a few restrictions on our discussion.First, let us consider the effect of the shear stress on the velocity gradient. If the

shear stress doubles, and the resultant velocity gradient doubles, we can say that theratio of equation (7) is constant. That is, the absolute viscosity of the fluid is the samefor all values of shear stress. A fluid which behaves in this manner is said to beNewtonian. A great number of commonly encountered fluids are Newtonian (water, oil,etc.); however, there are exceptions. In the case of some fluids, the ratio of equation(7) decreases with increased stress and vice versa. The problem of working with non-Newtonian fluids is extremely complex. In order to measure or specify the absoluteviscosity of a non-Newtonian fluid, one must also specify the shear stress value for

which it is quoted.This chapter will be concerned only with Newtonian fluids for two reasons. The

first is that a non-Newtonian discussion is too involved. The second is that most fluidsencountered in Physical Measurements are Newtonian. When the viscosity of a fluid ismeasured or defined, the identification as to whether it is Newtonian or not ismandatory.

The absolute viscosity of a fluid is also a function of temperature. A measure ofthe "effective drag" (viscosity) of adjacent fluid layers on each other changes withtemperature. The absolute viscosity of gases change almost directly proportional totheir temperature change, whereas liquids increase in viscosity for a decrease intemperature.

The reaction of a liquids viscosity as a function of temperature can be viewed asthinning of the liquid with increased temperature. This results in a liquid which is moreeasily poured. The reaction of a gas viscosity with temperature variations is not quiteso straightforward. The movement of the molecules in a gas are much more vigorousthan those in a liquid. Also, this molecular activity increases very rapidly with increasedtemperatures. The viscosity of a gas may be attributed to molecules of one layer

jumping to another layer. A molecule from a fast layer which jumps into a slow layertends to speed up that layer, while the opposite happens when a molecule from a slowlayer jumps into a fast layer. As the rate at which the molecules jump increases withincreased temperature, the viscosity of the gas increases.

The kinematic viscosity is affected two-fold by temperature variations. This is

because temperature causes both a change of absolute viscosity and density. Eventhough we limit ourselves to Newtonian fluids, it is important to specify the temperatureat which a viscosity is measured. Generally, a complete viscosity measurement orspecification requires a graph of the viscosity as a function of temperature. A fewtypical viscosity versus temperature graphs are shown in Figs. 13.5 and 13.6.


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Unfortunately, the basic viscosity units discussed thus far are not the only onesin use. Let us, therefore, conclude this section on viscosity with a discussion of someof the many viscosity units in use today, so that when they are mentioned in latersections the full significance of their value may be obtained.

There are viscosity units which have been adopted by and are particular toalmost every industry. This is especially true for the petroleum industry. The singularfact that there are many viscosity units in existence is not in itself bad; however, the factthat there are no exact relationships between these units and the basic defined units iswhat causes dissatisfaction. The reason that this condition exists today may have beenthat an urgent need for viscosity measurements did not allow time for the designers ofvarious viscometers to wait for the publication and acceptance of a detailed viscositytheory. Even today, many viscosity relationships are not fully understood because ofthe complexity of the theory. It appears that originally each expert set out to design aviscometer to suit his needs.

In Chapter 5 (Temperature) a passing remark was made regarding how anyonecould set up a "Jones" temperature scale. Table 13.2 shows a list of viscosity unitswhich are based upon certain types of viscometers. Although these units are accepted

(a) Absolute Viscosity (b) Kinematic ViscosityFigure 13.5-Typical Viscosity vs.Temperature Charts for Liquids


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and used in various industries, there is no direct traceable relationship between themand absolute or kinematic viscosity.

(1) Saybolt Universal Seconds (SUS)

(2) Saybolt Furol Seconds (SFS)

(3) Redwood Seconds

(4) Engler Degrees

(5) Ford No. 4 Seconds

(6) Zahn No. 3 Seconds

(7) Zahn No. 5 Seconds

(8) Demmier No. 10 Seconds

TABLE 13.2

VISCOSITY UNITS

(a) Absolute Viscosity (b) Kinematic VelocityFigure 13.6-Typical Viscosity vs.

Temperature Chart for Gases


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The eight units listed are associated with viscometers which operate on theprinciple of the amount of time required for a known volume of fluid to pass through anorifice. All units are somewhat related to kinematic viscosity, as all efflux times are a

function of both the absolute viscosity and the density of the fluid.The first four units are used in the oil industry for measuring the viscosity oflubricating and fuel oils. Saybolt units are used in the United States; Redwood (No. 1and No. 2) units are used in the United Kingdom, while Germany uses Engler degrees.The Ford, Zahn, and Demmier units are used almost exclusively by the paint industry.

The problem of relating these units to the stoke or poise is not due to theinstrumentation or control in use, but rather because the flow during measurement isturbulent. The relationships which do exist between the units in Table 13.2 andabsolute or kinematic viscosity have been empirically determined. The values listed inTable 13.3 should provide the reader with an idea of the relative size of the variousunits, as compared to the stoke. It must be noted that the values in this table are only

approximate and vary with temperature. One other unit, the Gardner Second, is notlisted, but its value is approximately the same as the stoke. It is also traceable, beingbased on the time required for a bubble to rise through a fluid.

STOKE 1 5 10 15

SUS 400 2200 4500 6700SFS 50 240 460 680Redwood 400 2100 4200 6200Engler 17 65 130 200

Ford #4 30 135 270 405Zahn #3 18 58 -- -Zahn #5 -- 27 50 72Demmier #10 -- 15 31 -

TABLE 13.3

RELATIVE VALUES OF COMMON VISCOSITY UNITS

13-3 Rotational Viscometers.

The basic instrument for measuring absolute viscosity is the Rotational Viscometer.

This instrument derives its name from the manner in which the shear stress isgenerated; namely, through the rotation of a cylinder. There are two major advantagesof the rotational viscometer over all the other types. The first is that the relationshipsinvolved are those employed in the basic definition -- shear stress and velocity gradient.The second advantage is that most rotational viscometers can be operated continually


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for extended periods of time. This allows time for the measurement conditions tostabilize and also provides for time studies of viscosity.

The basic rotational viscometer consists of two cylinders, one stationary and onewhich rotates. The cylinders are mounted coaxially (common rotational axis), with onecylinder having a radius which is slightly larger than the other.

Referencing Figure 13.7, let us consider that the outer cylinder is rotating. Thefluid to be measured is trapped between the walls of the two cylinders. The viscosity ofthe fluid transfers the motion of the outer cylinder to the inner one and tries to set it intorotation. This action is countered by the action of a spring and lever arm. Thedeflection of the pointer is a measure of the shearing force imposed on the stationary

cylinder. Since the system is in equilibrium, and the two drums have a common axis,the torques about the axis must be equal and opposite, or:

FsR1 = FR3 (9)

where: Fs = the shearing force acting on the inner drum

R1 = the radius of the inner drum

R3 = the displacement from the axis to the spring

and: F = the restoring force of the spring

Equation (9) can be solved for the shearing force as:

Figure 13.7-Rotational Viscometer


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F FR

Rs 3

1

(10)

If the scale in Figure 13.7 reads directly in pounds, and if R 1 is the same as R3,

then the shear force Fs is read directly from the scale.Since we know the shear force, if we compute the surface area (A) of the inside

cylinder we can find the shear stress (s) using the equation:

ssF

A

But, the surface area of the inside cylinder is:

A = 2R1h

where: h = the height of the fluid (width of the cylinder)

Thus, the expression for shear stress becomes:

s

sF

R h

2 1(11)

The other expression needed to calculate the absolute viscosity is derived fromthe difference in radii of the cylinders (fluid width) and the rotational rate. In Figure 13.7the velocity of the fluid at R1 is zero. The velocity at R2 is R2. The velocity gradient is

thus found to be:

v

L

R

R R

2

2 1

(12)

where: = the rotational velocity in RPS, degrees/sec. etc.

Combining equations (11) and (12) to find the absolute viscosity we get:

=

F

2 R hR

R - R

s

1

2

2 1

(13)


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Example 4: A rotational viscometer similar to the one in Figure 13.7 has the followingvalues associated with it:

R1 = 5.00 cm

R2 = 5.01 cm

h = 3 cm

Assuming the outer cylinder (R2) rotates, what is the absolute viscosity ofthe fluid?

Since the radius of the indicator (R3) and the inside drum (R1) are equal, then the forcegiven may be considered as the shear force (Fs). Let us start our solution by convertingRPM to RPS.

Thus: 30 160

05rev revmin

minsec

.sec

Substituting the information given in Example 4 into equation (13) we get:

=

10 dynes

2 5 3(cm )

.5 5.01(cm/ )

(5. -5)cm

4

2

sec

01

Simplifying and inverting we get:

10 001

15 501

4

2

.

.

secdyne

cm

100

2361.poise

thus: = .42 poise

or: = 420 centipoise

An actual rotational viscometer is not as simple as presumed by equation (11),nor is it as easy to use as presented in Example 4. There are some problems whichchange the use versus the theoretical equation of the rotational viscometer.


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The first problem is that the shear stress is not the same across the gap. Sincethe shear stress is not constant across the gap, the velocity gradient is not constant.The finite height of the fluid causes effects at the edges. There is also the self-heatingof the rotating fluid. The effect of all these deviations on equation (13) can be

computed mathematically and an exact equation generated, but it would be veryinvolved. We can avoid the use of a complicated corrected expression by computingan instrument constant. This will reduce equation (13) in its simplified corrected formto:

k

Fs(14)

where: k = instrument constant (determined by the manufacturer)

Fs = measure of the shear force

and: = angular velocity

The units of Fs and in equation (14) depend upon the choice of k. Usually Fs isdimensionless; that is, it is the numerical value indicated by the pointer and scale of theinstrument. The most popular choice for is in units of RPM.

A practical viscometer may use aconstant speed motor, making constantfor all measurements. In such cases theviscosity of the fluid under test is readdirectly from the indication of the pointer

and scale, which is a function of therestoring torque required for equilibrium.

Another type of rotating drumviscometer holds the force constant, andthe rotational rate is used as a directindication of the viscosity of the fluid.Such a device is shown in Figure 13.8.The torque provided by weight (W) isconstant during the test. After the initialacceleration, the angular velocitybecomes constant and is measured by

the tachometer. The biggestdisadvantage to this system is that it canonly operate for a fixed period of time(while the weight is falling), and it ispossible that constant velocity may not beFigure 13.8-Constant Force Viscometer


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reached. The time is controlled by the length of the string and the viscosity of the fluid.The easiest and most straightforward way of finding the instrument constant

mentioned is to measure it. This can be done by the use of "standard" Newtonianfluids. Standard fluids which have known viscosities are available from the NationalInstitute of Standards and Technology (NIST) and other testing laboratories. The

standard fluid is placed in the viscometer in the same manner as a test fluid and theviscometer operated normally. The values for, Fs, and are entered into equation(14), from which the instrument constant is calculated. This constant includes all smallcorrections which apply to equation (13).

The instrument constant is for Newtonian fluids only. Theconstant should be calculated at all normal test

temperatures which will be used, since a change ofdimensions with temperature will affect the instrument

constant.

There are many variations to the basic rotational viscometer of Figure 13.7.Some use dynamic (strain gage) torque sensors on the shaft of the rotating member.Either cylinder may be rotated. The cylinder of some rotational viscometers is replacedby a disk which is completely submerged in the test fluid.

Although there are many models and variations of the basic design, the betterrotational viscometers have uncertainties of +0.5% to +1.0% of FS. Rotationalviscometers theoretically can be used over a range from 0.1 centipoise to 10

7

centipoise, but most units have a range of 0.1 to 105

centipoise.

13-4 Capillary Viscometers.

The most commonly encountered viscometers in the field belong to a group called

Capillary Viscometers. Most capillary viscometers are designed to measure kinematicviscosity on the basis of the time required for a known volume of fluid to flow through acapillary. Capillary viscometers are usually divided into three classes which are:

(1) piston-cylinder

(2) glass capillary

(3) orifice

The principle of operation of the piston cylinder class is that a fluid is forced through a

capillary by the action of a piston. The actual theory of operation is of littleconsequence, since this class of viscometer does not find wide use. Suffice to say thatthis type of viscometer measures in terms of absolute viscosity.

Both of the other classes of capillary viscometers use the hydrostatic pressuregenerated by the fluid itself to drive it through a capillary. Because the hydrostatic


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pressure is a function of the density ofthe fluid, glass capillary and orificeviscometers measure viscosity in termsof kinematic viscosity. Glass capillary

viscometers generally measure directly interms of centistokes; whereas orificeviscometers measure kinematic viscosityin one of the various other viscosity units,as previously discussed. The range ofthis type of viscometer is somewhatrestricted, since the driving force isprovided solely by head pressure. Theirrange is normally from 0.4 to 20,000centistokes.

The basic theory of a capillary viscometer is related to the characteristics of fluid

flow in a tube. Consider the circular section of tubing shown in Figure 13.9. As wasmentioned earlier and is true for all cases, the finite fluid layer in contact with a surfacedoes not move. If the surface is moving, the fluid layer will move at the same velocity.If the surface is stationary, the fluid layer will be stationary. A fluid flowing through astationary tube is considered to be made up of concentric segments which "telescope"with respect to one another. If the viscosity of the fluid increases, the effect is to makethe telescoping action more difficult. There is a basic similarity between this type offluid flow and that described by Figure 13.3; however, the shear stress and velocitygradient in the tube flow are not as easy to describe as they were in the case of parallelplates. The discussions of the glass capillary and orifice viscometers, therefore, will notattempt or include any mathematical development, analysis, or examples.

These instruments are all basically the same as the original instrument developedby Wilhelm Ostwald. The design consists of two (or more) reservoir bulbs connected bymeans of a U-tube, as shown in Figure 13.10. The dark section of the viscometerindicates the capillary, while the dotted lines show where the volume measurement aremade. All three viscometers illustrated in Figure 13.10 were named after theirdevelopers.

The Ostwald viscometer has fallen into disuse, while the Ubbelohde viscometeris used primarily by persons interested in investigating kinematic viscosity with respectto temperature. The Cannon-Fenske viscometer is representative of the mostcommonly used glass capillary viscometer.

There are two versions of the Cannon-Fenske (and Ubbelohde) viscometer.

One is considered as routine, while the other is a master. The basic differencebetween the routine and master viscometer lies in the length of the capillary. Themaster viscometer has a longer capillary than does the routine viscometer. This longercapillary holds the head pressure more constant during a measurement. For this andother reasons, the master viscometer is more precise than the routine viscometer.

Figure 13.9-Fluid Elements


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The operation ofany glass capillaryviscometer is basedupon the efflux time of aknown volume of fluid

through a known sizecapillary, underconditions of laminarflow. The laminar flowis assured by asufficiently long capillaryand smooth, gradualtransition sectionsbetween where thecapillary joins the bulbs.If these transition

sections are notdesigned properly, notonly is there anunpredictable departurefrom theory, but therepeatability ofsuccessivemeasurements is lost.

The critical factor which determines accuracy is the dimensions of theviscometer, with the volume and capillary being the most important. The designdimensions have tolerances which can cause slightly different viscometer response;

therefore, each viscometer must be calibrated. An instrument constant is thenassigned to the viscometer for each temperature of use so that the kinematic viscosityof the test fluid may by computed by use of the equation:

= kt (15)

where: = viscosity at temperature

K = instrument constant at temperature

and: t = efflux time

The units in equation (15) are usually in terms of centistokes for , centistokesper second for k, and seconds for t. However, MKS or FPS units could also beemployed.

Temperature plays a two-fold role in glass capillary viscometry. The kinematic

Figure 13.10-Various Styles of Glass CapillaryViscometers


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viscosity of the fluid changes with temperature, and the instrument constant alsochanges with temperature

3. The net result is that excellent temperature control must be

maintained at all times during a measurement. For this reason, all glass capillaryviscometers are operated in a temperature bath.

The constant in equation (15) may be found by one of two calibration methods.The fundamental calibration is performed by using distilled water as the viscositystandard ( for water = 1.0036 centistokes at 60F). This method is used to calibratemaster viscometers with relatively small diameter capillaries. In order to calibrate themaster viscometers with larger diameter capillaries, a "step-up" procedure is used.

The viscosity of a fluid slightly more viscous than water is measured with a smalldiameter master viscometer, and then this fluid is used to determine the calibrationconstant for a larger diameter master viscometer. The reason the "step-up" technique

Absolute Viscosity Kinematic Viscosity

OIL CODE 100F 210F 100F 210FD 1.4 0.6 1.9 0.8I 6.6 1.7 1.8 2.2L 37 4.9 43 6.0M 100 9.9 110 12N 400 25 460 30

TABLE 13.4

NIST STANDARD OILS (PARTIAL LIST)

is needed is that the efflux time for water in the larger capillary viscometers would bevery short thus creating problems in timing, flow stability, and other related areas.

The second calibration method for capillary viscometers is by means of standardoils. This technique is used to calibrate routine viscometers. As was mentionedpreviously, such standard oils are available from the National Institute of Standards andTechnology (NIST), the American Petroleum Institute (API), and the American Societyfor Testing Materials (ASTM). Table 13.4 lists the code letters and viscosities for someof the standard oils available from NIST.

Since the viscosity of a standard oil may change with age, a master viscometer is

required to test it periodically. The major use of master viscometers is to determine the

3-The instrument constant does not change with temperature in Ubbelohde

viscometers.


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viscosity of oils which will then be used to calibrate routine viscometers.Temperature has already been pointed out as one possible source of error in

glass capillary viscometry. Other sources of error include efflux time determination,vertical alignment of the viscometer, initial head pressure, and the degree ofcleanliness. Other less important variables which may cause small errors are

accounted for by the instrument constant.The error in the measurement of the efflux time is kept small, if the total effluxtime is large. A 0.1 second timing error in a total efflux time of 30 seconds representsabout 0.3 percent. In 200 seconds, the same timing error reduces to about 0.05percent. Normally, the minimum efflux time is about 200 seconds. This can be done bychoosing the proper standard oil for the temperature of interest.

The vertical alignment affects the fluid head pressure. This source of error isreduced by checking the alignment against some known vertical reference. Thepercent error caused by a misalignment may be computed by the equation:

e% = 100 (1 - cos ) (16)

where: = the angle between the true vertical and the axis of theviscometer

If the fluid head (pressure) in the viscometer is not the same at the start of twosimilar measurements, there will not be compatibility in the efflux time measurementsfor a single fluid. To overcome this, the viscometer must always be charged (filled)initially with a given volume of fluid. The charge volume is normally between 7 to 10milliliters. The error caused by a variation in charging is difficult to evaluate. Themethod of minimizing such error is to fill the viscometer exactly as directed by themanufacturer.

Glass capillary viscometers must be kept extremely clean, if accurate results areto be obtained. The cleaning needs to be performed only when different oils areemployed. That is, a viscometer used for multiple runs with the same oil over a shorttime period would not require cleaning between runs. When cleaning is required, it isdone with a solvent chosen to provide the necessary amount of cleaning.

Routine cleaning (between runs) may be done with approved cleaning solvents,and drying may be accomplished with filtered dry air. More thorough cleaning isperformed with approved cleaning solvents in overnight baths. Although the cleaningprocess is time consuming and costly, it is a must with the glass capillary viscometerbecause of the small bore size. Any slight amount of dirt can cause large errors in aviscosity determination.

The capillary viscometer is used as a comparison standard, not an absolutestandard; therefore, its quality is defined by its precision. The master viscometer iscalibrated in terms of water. The instrument constant is as good as the precision ofrepeated measurements, since the viscosity of water is accurately known. The same istrue of the standard oils and routine viscometers. The uncertainty of master


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viscometers is about 0.1%, while the uncertainty for routine viscometers is about0.3% of the value calculated, using equation (15).

One final word about glass capillary viscometers. Since they are normallycalibrated by a standard oil, the instrument constant assigned is good only for the

location of the calibration. If the viscometer is moved to a new location where thegravitational constant is different from that of the gravitational constant at the calibrationlocation, the instrument constant must be revised. This may be done by using theequation:

k kg

g2 12

1

(17)

where: k2 = new instrument constant

k1 = calibrated constant

g2 = gravitational constant at new location

g1 = gravitational constant at calibration location

The orifice viscometer differs from other types of capillary viscometers in thedesign of its capillary. The orifice is a short capillary, that is, the ratio of capillary lengthto diameter is less than 10:1. Also, the entrance and exit sections of the orifice are notdesigned to assure laminar flow during the entire efflux time. Finally, because of theshort capillary section, the variation in fluid head pressure is large during the efflux time.

These factors contribute to an instrument which is not a predictable or accurateas the glass capillary viscometer. As was stated before, the relationship of efflux timeto the basic kinematic viscosity units cannot be mathematically derived for an orificeviscometer. Whereas the glass capillary type viscometer was used and calibrated inunits of centistokes, the orifice device is almost always used and calibrated in the otherkinematic viscosity units, such as Saybolt, Redwood or Ford seconds, etc., as listed inTable 13.2. However, within reasonable limitations, the orifice viscometer providesadequate results. Because of their simplicity and ease of operation, orifice viscometershave found their way into considerable general usage.

The orifice viscometer provides a measure of kinematic viscosity in terms of thefollowing equation:

= t (18)

where: = an arbitrary kinematic measure at temperature


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and: t = efflux time in seconds

The unit of kinematic viscosity in equation (18) is given a name which reflectswhich model or style of orifice viscometer is used for a particular measurement; that is, is in units of Saybolt seconds for the Saybolt viscometer, Redwood seconds for the

Redwood viscometer, etc.There are two restrictions which must be imposed upon an orifice viscometer forreliable results. The first is that the dimensions of the viscometer must be well specifiedand controlled. The second is that the test fluids must not be too different from the"standard fluids" which were used to calibrate the viscometer.

The first restriction says, in effect, that each orifice viscometer of a given classmust be a copy of all others in that class. The second restriction states that, since theaction of these viscometers cannot be predicted, test fluids must be close to a pre-calibrated operating point to have any significant meaning.

The saybolt viscometer is the most widely used type of orifice viscometer in theUnited States. It is used chiefly to classify motor oils. There are two styles of Saybolt

viscometers-- the Saybolt Universal Viscometer and the Saybolt Furol

4

Viscometer.The Universal viscometer is used for the lighter oils, while the Furol viscometer is usedfor heavier oils. The basic difference in these viscometers is the size of the capillary

(orifice). If the efflux time for a given oil is greaterthan 200 seconds in the Universal viscometer,then the Furol instrument is employed.

Figure 13.11 is an illustration of a SayboltUniversal viscometer. The dimensions of thisinstrument are specified in an ASTM publication.In addition, the publication also specifies the typeof bath used, receiver, test fluid and bath

temperature, measuring instruments, bath fluidlevel, type of timer, and the test method. Thedetail of the measurement specification isprovided to assure that all viscosity determinationsare made as similar as possible to achievecompatible results. The viscosity of motor oils inthe United States is classified by an SAE

5weight

number. An SAE weight number is assigned to anoil if it falls within a certain viscosity range in termsof Saybolt Universal Seconds. The oils are firsttested at two temperatures, usually 100F and

210F. They are then assigned a weight number, depending upon the viscosity rangewithin which they fall.

4-Furol is a contraction of the words fuel and road oils.

5-SAE is the Society of Automotive Engineers.

Figure 13.11-Saybolt Viscometer


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Table 13.5 indicates that the viscosity of lighter oils must be known at 0F beforethey can be classified. Usually one does not perform viscosity measurements at 0F.In order to determine a viscosity at 0F, an extrapolation process is used. The oil ismeasured at two points at least 60F apart, and these points are plotted on semi-log

graph paper

6

with temperature plotted along the linear axis. Then a straight line isdrawn through these points and extended to 0F. This temperature vs viscosity graphfor a given oil thus dictates the assigned SAE weight number. Even under excellenttest conditions, this method cannot be considered as being precise. The extrapolatedviscosities may have errors as high as 20%, even when the measured pointsthemselves are not in error by more than 1%.

SAEViscosity No.

Saybolt Universal Seconds0

oF 210

oF

5 W 4 x 103

(max)

10 W 6 - 12 (x 103) 40

20 W 12 - 48 (x 103) 45

20 45 TO 58

30 58 TO 70

40 70 TO 85

50 85 TO 110

TABLE 13.5

SAE VISCOSITY WEIGHT NUMBER CLASSIFICATION

In addition to the viscosity classification, motor oils are also classified by theSociety of Automotive Engineers according to the type of maximum service under whichan oil should be expected to perform properly. In 1947 there were three grades -Regular, Premium, and Heavy Duty (detergent). It was found, however, that a motor oilgraded in this manner did not perform equally well in gasoline and diesel engines, andthe names, as applied, did not fully reflect the type of service under which each gradecould be used. Therefore, in 1960 the existing three grades were replaced by six newgrades, three for gasoline and three for diesel engines. The names were dropped anda letter code assigned. The code and description of the current motor oil grades are

given Table 13.6.

6-Semi-log graph paper has linear scale in one direction and a logarithmic scale in the

other direction.


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SAE Grade

Gasoline Diesel Recommended Usage

MS DS Most severe operating conditions

Continuous Start and Stop operation

MM DM Moderate to severe operating conditions

ML DG Light and favorable operating conditions

TABLE 13.6SAE MOTOR OIL GRADE CLASSIFICATION

Another type of orifice viscometer which has received some acceptance in thepaint industry is the cup type. Basically, this viscometer is a cup with a known volumewith a hole of precise dimension in the bottom. The viscosity is measured according to

equation (18), with the units being in terms of which style cup is used.The operation of the cup type viscometer is simple. The fluid to be tested and

the cup are heated (or cooled) to the desired measurement temperature. The cup isthen dipped into the fluid, filled, and lifted above the fluid surface. As the cup is liftedabove the fluid, a stop watch is started, and the time required for the cup to emptydetermines the viscosity.

One disadvantage of this type of viscometer is that it must be suspended by asling system in such a manner that it will be vertical during runoff, to insure that thehead pressure and volume for each measurement are constant. The uncertaintyobtained with this type of device is from +2 to +5%.

13-5 Other Viscometers.As we stated previously, there is almost an endless variety of viscometers. Any deviceconnected with the flow of fluids, by proper design, can be made to function as aviscometer. Let us, however, restrict this section to a discussion of three or the morecommonly encountered viscometers. They are the falling or rolling ball viscometer, therising bubble viscometer, and the float viscometer.

Falling Ball Viscometer

The falling ball viscometer is designed on the basis of Stoke's Law which saysthat the terminal velocity of a sphere (or any other object) falling freely through a fluid iscontrolled by the density of the sphere and the absolute viscosity of the fluid. The

equation normally associated with a falling ball viscometer is:

= k (f- ) t (19)

where: = absolute viscosity


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k = instrument constant

f= density of the fluid

= density of the sphere

and: t = the time required for the ball to fall through a givendistance

It is mandatory that the ball reach its terminal velocity prior to entering themeasured distance over which it will be timed. This is usually assured by providing along section of tubing (filled with the test fluid) above the measurement zone. It is alsoimportant that the ball release device should not "throw" the ball downward, as theterminal velocity must be determined solely by the fluid and not be influenced by other

factors. The falling ball viscometer tube must be vertical and aligned, so the ball fallsdown the center without contacting the walls of the tube.This last requirement of noninterference is very difficult to achieve; therefore, the

tube on some viscometers is purposely tipped at a known angle, so the ball is forced toroll down the tube. This modification complicates the mathematics needed to describethe device, but it provides precise results. The equation for the rolling ball viscometer isbasically the same as presented in equation (19).

Regardless of its basic configuration, the viscometer is calibrated with standardfluids at known temperature points to determine its constant. The constant may bederived in such a manner, that when it is applied, the viscosity will be in terms ofabsolute units. The constant may also be computed so that when it is applied, the

viscometer yields a special viscosity unit directly, such as a Saybolt second. As theviscosity of a fluid increases, the fall time of the ball increases. To provide aninstrument which covers a wide range of viscosities, several balls of various densitiesare usually furnished with each tube.

Rising Bubble Viscometer

The rising bubble viscometer is much like the falling ball viscometer, with theexception that the density of the "ball" (bubble) is less than that of the fluid. The netforce acting on the bubble is generated by the buoyant force of the liquid. The timerequired for a bubble of known size to rise a measured distance reflects the viscosity ofthe fluid. One measurement unit which is applied to this type of device is the Gardner

second.


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The rising bubble viscometer is a sort oftest tube graduated at given linear intervals.Two lines scribed on the tube are used toassist in forming a bubble of known size, asillustrated in Figure 13.12. The other lines are

used for timing the rise of the bubble. Theinstrument is operated by placing a test fluid inthe tube to a "fill" line, inserting a stopper to theuppermost line, and then inverting the tube.The time required for the bubble to rise andpass between a pair of lines is measured andreflects the viscosity of the fluid.

Comparison Rising Bubble Viscometer

A modification of the rising bubbleviscometer is a "comparison rising bubble viscometer". This device has one tube for a

test fluid and a number of pre-filled tubes which contain standard fluids. All the tubesare mounted in a single rack, so that they can act as a single unit and can be invertedat the same time. It is not necessary to time the rise of the test liquid bubble, as it isassigned a viscosity value which is the same as the standard fluid whose bubble hasapproximately the same rise time. This device eliminates many errors, such as theneed for the tubes to be vertical; however, at best, it only provides a general grouping

for the viscosity value of the test fluid.

Float Viscometer

The float viscometer, illustrated inFigure 13.13, is very similar to a falling ball

viscometer. In this case, however, the floatis stationary, and the fluid is in motion. Thistype of device is also used to measure flow.When it is used as a viscometer, however,the flow rate must be know. Assuming thatthe velocity is always adjusted to someconstant value for all fluids, the height towhich the float rises is a relative measure ofthe viscosity of the fluid.

Most viscometers are relativelyimprecise devices with uncertainties of +1%

or greater. Precision viscositymeasurements are made with glass

capillary viscometers or rotational viscometers. All other types of viscometers must beconsidered as secondary instruments. This does not imply they should not be used,since most are more than adequate for their purpose. However if the precise

Figure 13.12-Rising Bubble Viscometer

Figure 13.13-Float Viscometer


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measurement of viscosity and traceability to defined standards is the object, the properinstrument must be used.

13-6 Specific Gravity Concepts.

The density of a substance has previously been described in two forms -- the massdensity () and the weight density (D). There is another method of expressing thedensity of a substance. This expression is dimensionless and is the ratio of the densityof a given substance to the density of water. This ratio has been assigned the nameSPECIFIC GRAVITY (spgr), although it actually represents the relative density of asubstance to the density water. The use of the word "gravity" is also very bad, becausethe effect of gravity is of no consequence on the specific gravity of a substance. Thisis, however, the name of the measure, and its use is so widespread that a change isn'tvery probable. Specific gravity is mathematically defined as:

spgrx

w

(20)

or: spgrD

D

x

w

(21)

where: the subscripts x and w represent the unknown substanceand water, respectively

The density of water, as with all substances, varies as a function of temperature.Since water attains its maximum density (1 gm/cm

3) at +4C, this will be considered as

its reference temperature for all values of specific gravity, unless otherwise specified.

Example 5: What is the specific gravity of a substance which has a mass density of 8.4g/cm

3?

Using equation (20) and substituting our known values we get:

spgrg cm

g cm

84

1

3

3

.

spgr = 8.4

Notice that the value of the specific gravity for any substance is equal to the numericalvalue of its mass density in the cgs system.

Since the volume of a substance changes with temperature, but the massremains fixed, it follows that the density of a substance also changes with temperature.


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A given value of specific gravity for a particular substance, therefore, applies at onlyone temperature. One further point -- if a substance is readily compressible (such as agas), the density will change radically as a function of pressure. The term specificgravity, therefore, must include both temperature and pressure specifications whenapplied to a gas.

Equations (20) and (21) are only applicable for determining specific gravity if oneknows the density of an unknown substance at the temperature of interest. Suchinformation may be obtained in one of two ways. If the makeup of the substance isknown, its density at some temperature can be found in a reference table, similar toTable 13.7. If the exact makeup of the substance is not known, or if there is no density-temperature table available, the density may be measured.

The empirical determination of density requires the measurement of both massand volume. The measured value of the mass must be precise enough for theaccuracy requirement of the specific gravity. The mass measurement would, mostprobably, be performed using a balance and one of the weighing techniques discussedin Chapter 10.

Volume measurements for liquids are readily and accurately made by use ofexact volume standards such as graduated cylinders, standard size beakers, etc. Thevolume of a solid can be determined by noting the change in volume indication of abeaker of liquid (water) before and after the solid is submerged. This method is seldom

Substance (g/cm3) D(lbs/ft

3) Conditions

Alchohol,ethyl .791 49.4 20oC

Asbestos slate 1.8 112.0 -

Ebonite 1.15 72.0 0oC

Ether .736 45.9 15oC

Ice .917 57.2 15oC

Oil, linseed .942 58.5 -

Oil, olive .918 57.3 -

Turpentine .87 54.3 -

Wood:

Balsa .12 8.0 -

Elm .57 35.5 -

TABLE 13.7

DENSITIES OF VARIOUS SUBSTANCES


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satisfactory, however, because of the lack of resolution of the scale on the beaker.Another method of volume determination for solids utilizes the concepts of

buoyancy. The difference in apparent mass of an object in air and in water is a function

of the buoyant force, which is dependent upon the volume of water displaced. Sincethe volume of water displaced is the same as the volume of the object which displacesit, we may use this relationship to find the volume of an unknown. The development ofthis relationship might be as follows, using the basic expression:

Wa = mg (22)

where: Wa = the weight of the object in air

m = the apparent mass

and: g = gravitational acceleration

The weight of the same object in water may be expressed as:

Ww = Wa - Fb (23)

where: Fb = the buoyant force of the water

and: Ww = the weight of the object in water

The buoyant force of the water can be expressed as:

Fb = mwg (24)

where: mw = the mass of the water displaced

and: g = the gravitational acceleration

The mass of the water displaced can be expressed as:

mw = wV (25)

where: w = the density of the water

and: V = the volume of the object

Combining equations (23 through 25) we get:


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Ww = Wa - V wg (26)

Solving for the volume we get:

V W Wg

a w

W

(27)

But: Dw = wg

where: Dw = the weight density of water

So, equation (27) can be rewritten as:

VW W

D

a w

w

(28)

And, since the only difference between mass and weight is the action of gravity, we canalso write equation (28) as:

Vm ma w

w

(29)

where: ma = apparent mass in air

mw = apparent mass in water

and: w = density of water

Example 6: An object is weighed in air and then in water. The values found are 855grams and 645 grams, respectively. What is the specific gravity of theobject?

Using equation (29), we first find the volume of the object. Thus:

V

g

g cm

855 645

1 3

V = 210 cm3

Using the apparent mass in air of the object as its mass value, and its volume as


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determined, we get:

x 855

210 3g

cm

x = 4.07 g/cm3

Since the units of density are in the cgs system, the numerical value is also the specificgravity; that is:

spgrg cm

g cm

4 07

1

3

3

.

or: sp.gr. = 4.07

Example 7: An object weighs 11 pounds in air and 8.5 pounds in water. What is itsspecific gravity?

At first this problem may seem difficult to handle in terms of units. Let us, therefore,consider equation (21) which said:

spgrD

D

x

w

Substituting for Dw from equation (28) we get:

spgrD

W W

V

x

a w

or, taking Dx in terms of its units we get:

spgr

W

VW W

V

a

a w

Since the volume is the same, it cancels, leaving:


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spgrW

W W

a

a w

(30)

Or, equation (30) might be rewritten in the form of apparent mass as:

spgrm

m m

a

a w

(31)

Applying equation (30) to Example 7 yields:

spgr

11

11 85

lbs

lbs lbs.

11

2 5..

spgr = 4.4

To indicate the validity of equation (31) by means of an example, let us rework Example7 as follows:

spgr

855

855 645

g

g g

855

210

spgr = 4.07

Let us now consider the case of how the specific gravity of an unknown liquidmay be determined. An object of convenient size is first weighed in the unknown liquid.This yields:

Wx = Wa - DxV (32)

where: Wx = weight in the fluid

Wa = weight in air

Dx = weight density of the liquid


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and: V = volume of the object

The object is then weighed in water which gives us:

Ww = Wa - DwV (33)

Solving equations (32) and (33) in terms of the difference in weight (in and out of thetwo relative liquids) yields:

Wa - Wx = DxV (34)

Wa - Ww = DwV (35)

Solving these two equations for Dx and Dw respectively, canceling the volume term, andplacing the equations in the ratio of specific gravity, we get:

DW W

Vxa x

and: DW W

Vwa w

Since: spgrD

D

x

w

then: spgr W WW W

a x

a w

(36)

Again, using apparent mass, we may also say:

spgrm m

m m

a x

a w

(37)

Example 8: An object weighed in a liquid of unknown density has a value of 76 grams.When weighed in water it has a value of 74 grams. If the weight of the

object is 84 grams in air, what is the specific gravity (or cgs density) of theliquid?

Entering the values of the example into equation (37) we obtain:


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spgr

g

g

84 76

84 74

8

10

or: spgr = 0.8

Although we have presented most equations in the form of both weight andmass, the reader should note that the terms are interchangeable, if all units are thesame, since specific gravity is a ratio; thus, the gravitational term always cancels.

There are more than a dozen different common measures for specific gravity.Some of them, as in the case of viscosity, refer to the type of instrument used for themeasurement, while other are variations in the basic definition. All specific gravitymeasurements, however, are firmly related to one another.

The terminology used by the scientific world and industry to describe specificgravity differs. This is a possible source of error when referencing various documents.The basic accepted definitions of specific gravity are as follows:

A. Scientific Usage

1. Specific gravity - the ratio of the true mass density of a substanceat a given temperature to the true mass density of water at +4C.

2. Apparent specific gravity - the ratio of the apparent mass density ofa substance at a given temperature to the apparent mass densityof water at +4C.

B. Industrial Usage

1. Specific gravity - the ratio of the weight in air of a given substanceat a known temperature to the weight in air of an equal volume ofwater at the same or some other temperature. This measure isdesignated as:

spgr tx/tw7

2. Absolute specific gravity - the ratio of the weight in a vacuum of agiven volume of the substance at a known temperature to theweight in a vacuum of an equal volume of water at the same orsome other temperature. This measure is designated as:

7-The reference temperatures of the unknown (tx) and of water (tw) are always given.


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abs. spgr tx/tw

There are other terms which are also used in industry; however, they are mostly

concerned with the difference between porous and nonporous solids.As stated before, unless otherwise specified, the specific gravity values apply tothe apparent mass of the substance at the measurement temperature to the apparentmass of water at +4C. One commonly encountered term of reference used in industryis "specific gravity 60/60". This is the ratio of the apparent mass of a substance at 60Fto the apparent mass of water at 60F. This measure is widely used in the petroleumindustry.

Another commonly used measure of specific gravity in the petroleum industry is"Degrees API". This measure is taken from a hydrometer

8with a scale designed to

meet the following relationship:

Degrees API

141560 60

1315. .spgr

The range of the Degrees API used to indicatespecific gravity is from 0 to 100. If we solved equation(38) for specific gravity and substituted for variousDegrees API, we would find that the equivalent specificgravity 60/60 ranges from 1.076 to .6112. Note that asthe Degree API increases, the specific gravitydecreases.

13-7 Specific Gravity InstrumentsThe specific gravity of a solid is generally determinedby use of an equal arm balance. Such measurementsare, in reality, density measurements. These methodswere outlined in section 13.6. There are, however,

various other types of instruments available for determining the specific gravity ofliquids.

Picnometer and Balance

This basic instrument consists of a standard vessel (cup, beaker, etc.) and a balance.The weight of the empty picnometer cup, the picnometer cup completely filled withwater, and the picnometer cup completely filled with the liquid under test are measuredand entered into equation (39) to determine the specific gravity. That is:

8-A hydrometer is an instrument which directly measures specific gravity.

Figure 13.14-Hydrometer


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13-38

spgrW W

W W

a p

b p

(39)

where: Wp - weight of the empty picnometer

Wa - weight of the picnometer vessel and test liquid

and: Wb - weight of the picnometer vessel and water

Since the volume of both fluids is the same, the ratio of the weight of theunknown fluid to water must be the specific gravity of the unknown. The notationassigned to the specific gravity thus measured depends on whether or not the weighttaken is in air or in vacuum and on the temperatures at the time of measurement.Usually, all measurements are referenced to 60F, and weight of the liquids in air isused. This gives the measure in terms of specific gravity 60/60.

Hydrometer

The most widely used instrument for determining the specific gravity of a liquid is thehydrometer. This instrument operates on the buoyancy principle. This principle,paraphrased, states that when the weight of the volume of fluid displace is the same asthe weight of the object which displace it, the object will neither sink nor rise, but willfloat in equilibrium. The depth to which the hydrometer sinks in a given liquid is afunction of the specific gravity of that liquid. Figure 13.14 illustrates the basichydrometer design.

The volume of the stem, buoyancy chamber, and weight chamber is known (or atleast fixed). Sufficient weight in the form of shot, etc. is added to the hydrometer tocause it to float with a fixed portion of the stem above water. The scale is marked1,000 at the point of immersion. Other points on the scale may be found and markedby calculation or by actual calibration of the hydrometer in liquids of known specificgravity.

If calculations are to be used for marking the scale, they are made as follows:Assume the total weight of the hydrometer is W and that its volume is Vb + Vs, where Vbis the volume of everything except the stem, and Vs is the volume of the stem. Assumealso that the stem is a cylinder of diameter d. For equilibrium, the weight of the waterdisplaced must equal the weight of the hydrometer.

thus: W = DwV

where: Dw = the weight density of water

V = the volume of the hydrometer submerged


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and: W = the weight of the hydrometer

This expression can be rewritten in terms of the volume of the hydrometer bulb and

stem as:

W D Vd

hw b

2

4(40)

where: h = the height of the stem submerged in the water

and:d

h

2

4

= the stem volume submerged in water

Note that the air buoyant force on the stem portion above the water is not accounted forby this equation; thus, there is a small error introduced. Equation (40) is used to markthe 1.000 point on the scale by solving it for h (h is measured from the top of thebuoyancy chamber up the stem).

thus:

hW D V

D d

w b

w

4

2(41)

Once the location of h is found, the other points are found by noting how the change inheight of the hydrometer is related to specific gravity. At equilibrium in a test fluid we

have:

W = Dx (Vw + V) (42)

where: Vw = total volume of the hydrometer which is submergedwhen it is floating in water

and: V = the change in volume of the hydrometer which issubmerged when it is floating in a test liquid

Also, when the hydrometer was in water it was found that:

W = DwVw (43)

But the weight of the hydrometer is constant in either case, so, by combining equations(42) and (43) we get:


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Dx(Vw + V) = DwVw (44)

Solving for Dx/Dw we get:

DD

VV V

x

w

w

w

or: spgrv

Vw

1

1

(45)

Equation (45) states that to measure a liquid whose specific gravity is 0.5, the change insubmerged volume of the hydrometer must equal the volume of the hydrometer whichwas submerged in water. Inspection of fig. 13.14 illustrates the impracticality of

achieving a value of this magnitude with respect to a reference specific gravity of water.Hydrometers, therefore, are designed to measure only a small range about a givenvalue. If a hydrometer is well designed, the ratio ofV/Vw in equation (45) is kept small.Since this is the case, equation (45) may be rewritten, as a good approximation, in theform:

spgrV

Vw 1

or: spgrd

V

h

w

1

4

2

Notice that the diameter and volume terms in the parenthesis are constant for allmeasurements. The equation may, therefore, be rewritten utilizing an instrumentconstant (k) in the form:

spgr = 1 - kh (46)

As the hydrometer sinks deeper in a test fluid, h is positive; thus resulting in anindication on the hydrometer of a lower specific gravity. If the hydrometer floats higherin the test fluid than it does in water, h is negative; thus, the specific gravity indicated is

higher than unity.A general form of equation (46) is:

spgr = R - krh (47)


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where: R = Specific gravity of reference9

for the hydrometer

kr= instrument constant with respect to the reference

and: h = change in height from the reference mark in units whichare compatible with the units of k r

Specific Gravity Balance

A specific gravity balance is a device whichsimplifies the specific gravity determination ofa liquid. The balance has a known plummetsuspended from one end of its beam and acounter weight on the other end. Thus, thebalance is, initially, in equilibrium.

When the plummet is submerged in aliquid, the balance reads the weight of theliquid displaced. If the plummet is designedproperly, it is already known to displace acertain weight of water; thus, with oneweighing the specific gravity of a test liquidcan be determined, using the equation:

spgrW

W

x

w

where: Wx = the weight of the displace test liquid

and: Ww = known weight of water displaced by the plummet

A modified type of specific gravity balance is shown in Figure 13.15. This deviceprovides a direct comparison to water and is often used to monitor specific gravity ofliquids in production areas. The flow through the test fluid chamber is adjusted toeliminate viscous drag which might cause errors. The weights are matched in that theyare made of the same materials and have the same volume.

The instrument indicates directly the difference in the buoyant forces acting onthe weights between the test fluid and water. The buoyant forces (F) acting on each

weight are:

9-Hydrometer references are selected to be compatible with the desired range of

operation.

Figure 13.15-Comparison Balance


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F1 = DwV

F2 = DxV

where: V = the volume of either weight

Since the volume of both weights is the same, we can set up a proportion such that:

D

D

F

F

x

w

2

1

(48)

We can express F2 as (F1 + F). Since the ratio of Dx to Dw is the specific gravity, wecan also say:

spgr

F F

F

1

1

or: spgrF

F 1

1

(49)

Since the difference in buoyant force (F)is a function of the difference in specificgravity of the two fluids, the scale of thebalance indicates F, but is markeddirectly in units of specific gravity.

Pressure Devices

Another method of measuring the specificgravity of a test fluid directly is by use of aninstrument similar to the one shown inFigure 13.16. This is referred to as aninverted Y-tube. One leg of the Y is inwater and the other leg in a test liquid.

The center of the tube is connected to a vacuum system with a controllable pressure.The Y is evacuated until the test liquid reaches some predetermined height, say 10inches. At that time the head pressures (P) of the two liquids are given as:

Pw = Dwhw

and: Px = Dxhx

Figure 13.16-Head Pressure Instrument


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But, since atmospheric pressure is the reference for both liquids, their head pressuresare equal, and thus:

Dwhw = Dxhx

or:D

D

h

h

x

w

w

x

(50)

If the height of the unknown liquid (hx) is always set to 10 inches, then the heightof the water column divided by ten is the specific gravity of the test liquid, and theinstrument can be marked directly in units of specific gravity, as shown. This instrumenthas basically the same configuration as two cistern manometers and as thus, issubjected to the same errors. Primarily changes in temperature and the ability toaccurately measure heights of the liquid columns.

The inverted Y-tube does not work well when the vapor pressure of the test liquid

is high, since as the tube is evacuated, the test liquid tends to boil. To overcome thisproblem the basic configuration may be modified, as shown in Figure 13.17. Thedevice illustrated uses positive pressure, rather than a vacuum. The pressure acting atthe bottom of the tube in the test fluid is Dxhx. The pressure acting at the bottom of thetube in the water is Dwhw. If the pneumatic pressure applied on the right side of themanometer and water is adjusted until bubbles just appear, then this pressure isequivalent to:

Pw = Dwhw

If the pneumatic pressure on the test fluid and left side of the manometer is adjusteduntil bubbles just appear, then the pressure is:

Px = Dxhx

If the depths of the two tubes are made the same, then:

P

P

D

D

x

w

x

w

(51)

We can write Px in the form:

Px = Pw + P

Substituting for Px in equation (51) we get:


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D

D

P P

P

x

w

w

w

(52)

or: spgrP

Pw 1

Since Pw

can be fixed, the change ofP is a direct measurement of the specific gravity.To facilitate a permanent scale on the U-tube, a plunger arrangement is incorporated in

the base of the tube, so that the mercury can always be set to a reference mark on thewater side. The value of the specific gravity is then read from the top of the mercury onthe test fluid side.

Figure 13.17-U-Tube Instrument


Documents

Chapter 13 - Viscosity and Specific Gravity