Instrumentation and control

Principles of Viscosity & Definitions . . . . . . . . . . . . . . . . . . . . . . . . 301

Viscosity SI Units . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 301

Dynamic, Absolute, or Simple Viscosity . . . . . . . . . . . . . . . . . . . . . 302

Kinematic Viscosity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 302

Common Viscosity Units . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 303

Other Viscosity Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 304

Measuring Viscosity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 305

Hagen-Poiseuille’s Law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 305

Stoke’s Law. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 305

Values of Viscometer Constants A and B . . . . . . . . . . . . . . . . . . . . 306

Viscosity Conversion Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 307

Poise to lb-force sec/ft2 Conversion Table . . . . . . . . . . . . . . . . . . . . 308

lb-force sec/ft2 to Pa-sec Conversion Table . . . . . . . . . . . . . . . . . . . 309

11VISCOSITY

MEASUREMENT

HB viscosity chap11.qxd 3/2/2006 11:17 AM Page 299

Principles of Viscosity & Definitions

Viscosity is a quantity describing a fluid’s resistance to flow. Fluidsresist the relative motion of immersed objects through them as well asto the motion of layers with differing velocities within them.

Formally, viscosity (represented by the symbol η) is the ratio of theshearing stress (F/A) to the velocity gradient (∆vx/∆z or dvx/dz) in a fluid.

The more usual form of this relationship is called “Newton’s equation.”It states the resulting shear of a fluid is directly proportional to the forceapplied and inversely proportional to its viscosity. Note the similarity toNewton’s second law of motion (F = ma).

Viscosity SI Units

According to NIST’s Guide for the International System of Units (SI), theproper SI units for expressing values of viscosity η (also calleddynamic viscosity) and values of kinematic viscosity ν are, respectively,the Pascal second (Pa·s) and the meter squared per second (m2/s) (andtheir decimal multiples and submultiples as appropriate).

The Pascal second [Pa·s] has no special name. And, although touted asan international system, the International System of Units (SI) has hadvery little international impact. The Pascal second is rarely used in sci-entific and technical publications today.

The most common unit of viscosity is the dyne second per squarecentimeter (dyne · s/cm2), which is given the name poise (P) after theFrench physiologist Jean Louis Poiseuille (1799-1869). Ten poise equalone Pascal second (Pa·s) making the centipoise (cP) and millipascalsecond (mPa·s) identical.

FA

vz

FA

dvdz

or

F mvt

F mdvdt

x x= =

= =

η η∆∆

∆∆

� �

η η=

÷

=

÷

FA

vz

orFA

dvdz

x x∆∆

Chapter 11/Viscosity Measurement 301


1 Pascal second = 10 poise = 1,000 millipascal second1 centipoise = 1 millipascal second

There are actually two quantities called viscosity. The quantity definedabove usually is just called viscosity. However, it sometimes is alsocalled dynamic viscosity, absolute viscosity, or simple viscosity to dis-tinguish it from the other quantity.

Dynamic, Absolute, or Simple Viscosity

whereVa = dynamic, absolute, or simple viscosityA = a viscometer constantB = a viscometer constantt = time for a volume of fluid to pass through an aperture

Kinematic Viscosity

The other quantity, called kinematic viscosity (represented by the sym-bol ν), is the ratio of the viscosity of a fluid to its density.

Kinematic viscosity is a measure of the resistive flow of a fluid underthe influence of gravity. It is frequently measured by a “capillary viscometer” — basically a graduated can with a narrow tube at the bot-tom. When two fluids of equal volume are placed in identical capillaryviscometers and allowed to flow under the influence of gravity, a viscous fluid takes longer than a less viscous fluid to flow through thetube.

v =ηρ

V AtBta = −

302 ISA Handbook of Measurement Equations and Tables

English/Metric Viscosity Units

Quantity English Metric

Viscosity Poise Pa/sec

KinematicViscosity

Stroke m2/sec


whereKv = kinematic viscosityV = viscosity of fluidD = density of fluid

The SI unit of kinematic viscosity is the square meter per second (m2/s),which also has no special name. This unit is so large it is rarely used. Amore common unit of kinematic viscosity is the square centimeter persecond (cm2/s), which has been given the name stoke [St] after the Eng-lish scientist George Stoke. Since this unit is also large, the more com-monly used unit is the square millimeter per second (mm2/s) or centis-toke (cSt).

According to NIST’s Guide for the International System of Units (SI), theCGS units commonly used to express values of these quantities, thepoise (P) and the stokes (St), respectively [and their decimal submulti-ples the centipoise (cP) and the centistoke (cSt)], are not to be used.However, since CGS units are, in fact, the most widely used terms, theyare included in this ISA Handbook.

Common Viscosity Units

1 m2/s = 10,000 cm2/s (stoke) = 1,000,000 mm2/s (centistokes)1 cm2/s = 1 stoke1 mm2/s = 1 centistoke

1 Poise = 1 dyne sec/cm2

1 Poise = 0.1 Pa sec

1 Centipoise = 0.001 Pa/sec

1 Centipoise = 1 cm2/sec

1 cP = viscosity of water at 68°C

1 lb-force sec/ft2 = 1 slug/ft sec

KvVD

=



Other Viscosity Equations

where V = viscosity of a fluidSs = shear stress, force per areaSr = shear rate, velocity per layer thickness

Ratio of Shear Stress to Shear Rate, Hagen-Poiseuille Law

whereV = viscosityPd = pressure differential of liquidR = inside radius of tubeQ = rate of liquid flowL = length of tube

Apparent Viscosity (Consistency)

whereC = consistency, percentAd = dry-weight of solidWs = weight of solid plus liquid

CA

Wd

s= x 100

VP RQLd=

π 4

8

VSS

s

r=




Measuring Viscosity

Hagen-Poiseuille’s Law

French physician and physiologist Jean Poiseuille, while developing animproved method for measuring blood pressure, formulated a mathe-matical expression for the flow rate for the laminar (nonturbulent) flowof fluids in circular tubes. Discovered independently by Gotthilf Hagen,a German hydraulic engineer, this relation is also known as the Hagen-Poiseuille equation, or Hagen-Poiseuille Law.

For laminar, non-pulsatile fluid flow through a uniform straight pipe, theflow rate (volume per unit time) is:

• directly proportional to the pressure difference between theends of the tube,

• inversely proportional to the length of the tube, • inversely proportional to the viscosity of the fluid, and • proportional to the fourth power of the radius of the tube.

Stoke’s Law

George Gabriel Stokes, an Irish-born mathematician who spent much ofhis life working with fluid properties, is most famous for his workdescribing the motion of a sphere through viscous fluids. This led to thedevelopment of Stokes’s Law – an equation that shows the forceneeded to move a small sphere through a continuous, quiescent fluid ata certain velocity. It is based primarily on the radius of the sphere and the viscosity of the fluid. He found what has become known asStokes’ Law:

The drag force on a sphere of radius (R) moving through a fluid of vis-cosity η at speed Vc is given by:

WhereR = the radius of the sphereη = the viscosityVc = the velocity through a continuous fluid

The faster a sphere falls through a fluid, the lower the viscosity. Themeasurement involves dropping a sphere through a measured distance of fluid and measuring how long it takes to traverse the distance.

F R Vc(drag) = 6π η

φπ

η=

∆Pr4

8 �



Since you know distance and time, you also know velocity, which is dis-tance/time. A formula for determining the viscosity in this manner is:

Where∆p = difference in density between the sphere and the liquid g = acceleration of gravity a = radius of sphere v = velocity = d/t = (Distance sphere falls/time it takes to fall)

viscosity2( )ga2

= η =∆p

v9

Values of Viscometer Constants A and B

Viscometer Constant A Constant B Time of Efflux

Saybolt Universal 0.2260.220

195135

32-100over 100

Saybolt Furol 2.24 184 25-40

Redwood #1 0.2600.247

17950

34-100over 100

Redwood #2 2.462.45

100-

32-90over 90



Viscosity Conversion Table

To Convert from To Multiply by:

Centipoise Pascal/sec 0.001

Centistroke m2/sec 0.000001

cm3/sec ft3/min 0.00211888

cm3/sec liter/hr 3.6

ft3/hr cm3/sec 7.865791

ft3/hr liter/min 0.4719474

ft3/min cm3/sec 471.9474

ft3/sec cm3/hr 101.9406

ft3/sec liter/min 1699.011

in3/min cm3/sec 0.2731177

Dyne-sec/cm2 Poise 1.0

Geepound Slug 1.0

Gram-force Dyne 980.665

kilogram-force Dyne 0.0000980665

liter/sec ft3/hr 127.1328

liter/sec ft3/min 2.11888

liter/sec gallon/hr 951.0194

part per million mg/kg 1.0

part per million ml/cm3 1.0

Poise Dyne-sec/cm2 1.0

Poise gram/cm-sec 1.0

Poise Pascal-sec 0.1

lb-force-sec/ft2 Pascal-sec 47.8803

lb-force-sec/in2 Pascal-sec 6894.76

Slug kg 14.5939



Conversion Table, Poise to lb-force sec/ft2

Poise lb-force sec/ft2 Poise lb-force sec/ft2

1 478.80 800 383,040

2 957.60 900 430,920

3 1436.40 1000 478,800

4 1915.20 2000 957,600

5 2394.00 3000 1,436,400

6 2872.80 4000 1,915,200

7 3351.60 5000 2,394,000

8 3830.40 6000 2,872,800

9 4309.20 7000 3,351,600

10 4788.00 8000 3,830,400

20 9576.00 9000 4,309,200

30 14,364.00 10,000 4,788,000

40 19,152.00 20,000 9,576,000

50 23,940.00 30,000 14,364,000

60 28,728.00 40,000 19,152,000

70 33,516.00 50,000 23,940,000

80 38,304.00 60,000 28,728,000

90 43,092.00 70,000 33,516,000

100 47,880.00 80,000 38,304,000

200 95,760.00 90,000 43,092,000

300 143,640.00 100,000 47,880,000

400 191,520.00 110,000 52,668,000

500 239,400.00 120,000 57,456,000

600 287,280.00 130,000 62,244,000

700 335,160.00 140,000 67,032,000



Conversion Table, lb-force sec/ft2 to Pa-sec

lb-force sec/ft2 Pa/sec lb-force sec/ft2 Pa/sec

100 4788.03 600,000 28,728,180

200 9576.06 700,000 33,516,210

300 14,364.09 800,000 38,304,240

400 19,152.12 900,000 43,092,270

500 23,940.15 1,000,000 47,880,300

600 28,728.18 2,000,000 95,760,600

700 33,516.21 3,000,000 143,640,900

800 38,304.24 4,000,000 191,521,200

900 43,092.27 5,000,000 239,401,500

1000 47,880.30 6,000,000 287,281,800

2000 95,760.60 7,000,000 335,162,100

3000 143,640.90 8,000,000 383,042,400

4000 191,521.20 9,000,000 430,922,700

5000 239,401.50 10,000,000 478,803,000

6000 287,281.80 20,000,000 957,606,000

7000 335,162.10 30,000,000 1,436,409,000

8000 383,042.40 40,000,000 1,915,212,000

9000 430,922.70 50,000,000 2,394,015,000

10,000 4,788,030.00 60,000,000 2,872,818,000

20,000 9,576,060.00 70,000,000 3,351,621,000

30,000 14,364,090.00 80,000,000 3,830,424,000

40,000 19,152,120.00 90,000,000 4,309,227,000

50,000 23,940,150.00 100,000,000 4,788,030,000


Engineering

Instrumentation and control