Monroe L. Weber-Shirk S chool of Civil and Environmental Engineering Fluid Properties and Units CEE 331 June 15, 2015 CEE 331 June 15, 2015

Monroe L. Weber-Shirk

School of Civil and

Environmental Engineering

Fluid Properties and Units

Fluid Properties and Units

CEE 331

April 18, 2023

CEE 331

April 18, 2023

Dimensions and UnitsDimensions and Units

The dimensions have to be the same for each term in an equation

Dimensions of mechanics are length time mass force temperature

The dimensions have to be the same for each term in an equation

Dimensions of mechanics are length time mass force temperature

aF m aF m

L

T

MMLT-2

Dimensions and UnitsDimensions and Units

Quantity SymbolDimensionsVelocity V LT-1

Acceleration a LT-2

Area A L2

Volume L3

Discharge Q L3T-1

Pressure p ML-1T-2

Gravity g LT-2

Temperature T’ Mass concentration C ML-3

Quantity SymbolDimensionsVelocity V LT-1

Acceleration a LT-2

Area A L2

Volume L3

Discharge Q L3T-1

Pressure p ML-1T-2

Gravity g LT-2

Temperature T’ Mass concentration C ML-3

Show this!

Dimensions and UnitsDimensions and Units

Quantity Symbol DimensionsDensity ML-3

Specific Weight ML-2T-2

Dynamic viscosity ML-1T-1

Kinematic viscosity L2T-1

Surface tension MT-2

Bulk mod of elasticity E ML-1T-2

These are _______ properties!fluid

How many independent properties? _____4

mn

r=m

nr

=

gg r= gg r=

Definition of a FluidDefinition of a Fluid

“a fluid, such as water or air, deforms continuously when acted on by shearing stresses of any magnitude.” - Young, Munson, Okiishi

“a fluid, such as water or air, deforms continuously when acted on by shearing stresses of any magnitude.” - Young, Munson, Okiishi

Why isn’t steel a fluid?Why isn’t steel a fluid?

Fluid Deformation between Parallel Plates

Fluid Deformation between Parallel Plates

Side viewSide view

Force F causes the top plate to have velocity U.Force F causes the top plate to have velocity U.What other parameters control how much force What other parameters control how much force is required to get a desired velocity?is required to get a desired velocity?

Distance between plates (t)Distance between plates (t)

Area of plates (A)Area of plates (A)

F

t

U

Viscosity! ()Viscosity! ()

F =AUt

mAUt

m

If this parameter increases, what does F do?

Shear StressShear Stress

change in velocity with respect to distancechange in velocity with respect to distance

AFAF

2mN

2mN

tU t

U t

Ut

U

dydu dydu

tAU

F t

AUF

AUFt

AUFt

2msN

2msN

dimension of

s1

s1

Tangential force per unit area

Rate of angular deformation

rate of shear

Our general equation relating shear and viscosity

Fluid ViscosityFluid Viscosity

Examples of highly viscous fluids ______________________________

Fundamental mechanisms Gases - transfer of molecular momentum

Viscosity __________ as temperature increases. Viscosity __________ as pressure increases.

Liquids - cohesion and momentum transfer Viscosity decreases as temperature increases. Relatively independent of pressure (incompressible)

Examples of highly viscous fluids ______________________________

Fundamental mechanisms Gases - transfer of molecular momentum

Viscosity __________ as temperature increases. Viscosity __________ as pressure increases.

Liquids - cohesion and momentum transfer Viscosity decreases as temperature increases. Relatively independent of pressure (incompressible)

molasses, tar, 20w-50 oil, glycerin

increases

_______

increases

Example: Measure the viscosity of water

Example: Measure the viscosity of water

The inner cylinder is 10 cm in diameter and rotates at 10 rpm. The fluid layer is 2 mm thick and 20 cm high. The power required to turn the inner cylinder is 100x10-6 watts. What is the dynamic viscosity of the fluid?

The inner cylinder is 10 cm in diameter and rotates at 10 rpm. The fluid layer is 2 mm thick and 20 cm high. The power required to turn the inner cylinder is 100x10-6 watts. What is the dynamic viscosity of the fluid?

Outer Outer cylindercylinder

Thin layer of waterThin layer of water

Inner Inner cylindercylinder

dydu dydu

tAU

F t

AUF

Solution SchemeSolution Scheme

Restate the goal Identify the given parameters and represent the

parameters using symbols Outline your solution including the equations describing

the physical constraints and any simplifying assumptions

Solve for the unknown symbolically Substitute numerical values with units and do the

arithmetic Check your units! Check the reasonableness of your answer

Restate the goal Identify the given parameters and represent the

parameters using symbols Outline your solution including the equations describing

the physical constraints and any simplifying assumptions

Solve for the unknown symbolically Substitute numerical values with units and do the

arithmetic Check your units! Check the reasonableness of your answer

Viscosity Measurement: SolutionViscosity Measurement: Solution

hr

Pt322

hr

Pt322

-6-3 2

2 3

(100 10 W) (0.002 m)1.16x10 N s/m

2 (1.047/s) (0.05 m) (0.2 m)x

mp

= = ×-6

-3 22 3

(100 10 W) (0.002 m)1.16x10 N s/m

2 (1.047/s) (0.05 m) (0.2 m)x

mp

= = ×

tAU

F t

AUF U U A A

thr

F22

thr

F22

P P

thr

P322

thr

P322

Outer Outer cylindercylinder

Thin layer of waterThin layer of water

Inner Inner cylindercylinder

r = 5 cmt = 2 mmh = 20 cmP = 100 x 10-6 W10 rpm

r 2rh

Fr

Role of ViscosityRole of Viscosity

Statics Fluids at rest have no relative motion between

layers of fluid and thus du/dy = 0 Therefore the shear stress is _____ and is

independent of the fluid viscosity Dynamics

Fluid viscosity is very important when the fluid is moving

Statics Fluids at rest have no relative motion between

layers of fluid and thus du/dy = 0 Therefore the shear stress is _____ and is

independent of the fluid viscosity Dynamics

Fluid viscosity is very important when the fluid is moving

zerozero

Dynamic and Kinematic Viscosity

Dynamic and Kinematic Viscosity

Kinematic viscosity (__) is a fluid property obtained by dividing the dynamic viscosity (__) by the fluid density

Kinematic viscosity (__) is a fluid property obtained by dividing the dynamic viscosity (__) by the fluid density

3mkg

smkg

3mkg

smkg

mÞmÞ [ ]N =[ ]N =

[m2/s]

Connection to Reynolds number!

mm

nn

ReVD VDrm n

= =ReVD VDrm n

= =

nu

2mN s×é ù

ê úë û2mN s×é ù

ê úë û2s

kg m×é ùê úë û2skg m×é ù

ê úë û

Density and Specific WeightDensity and Specific Weight

Density (mass/unit volume) density of water: density of air at

atmospheric pressure and 15 C:

Specific Weight of water (weight per unit volume) __________________

Density (mass/unit volume) density of water: density of air at

atmospheric pressure and 15 C:

Specific Weight of water (weight per unit volume) __________________

950960970980990

1000

0 50 100Temperature (C)

Den

sity

(kg

/m3 )

997

998

999

1000

0 10 20

Temperature (C)

Den

sity

(kg

/m3 )

1000 kg/m3

1.22 kg/m3

= g = 9806 N/m3

Specific mass

Perfect Gas LawPerfect Gas Law

PV = nRT R is the universal gas constant T is in Kelvin

PV = nRT R is the universal gas constant T is in Kelvin

Note deviation from the text!Note deviation from the text!

R

8 314.N m

mol K

RR

Mtextgas

Mgas is molecular mass

Mgas for air is 0.029 kg/mole

Why is this Mgas for air reasonable?

N2 28 g/mol, O2 32 g/mol

Bulk Modulus of ElasticityBulk Modulus of Elasticity

Relates the change in volume to a change in pressure changes in density at

high pressure pressure waves

_________ ______ __________

Relates the change in volume to a change in pressure changes in density at

high pressure pressure waves

_________ ______ __________ 2.00

2.05

2.10

2.15

2.20

2.25

2.30

2.35

0 20 40 60 80 100

Temperature (C)

Bul

k M

odul

us o

f el

asti

city

(G

Pa)

soundsoundwater hammerwater hammer

Edp

dv /

Edp

dV Vv /

Water

v

dV dpV E

=How much does water compress?

Compression and Expansion of Gases

Compression and Expansion of Gases

Isothermal (constant temperature)

Isentropic (no heat exchanged)

Isothermal (constant temperature)

Isentropic (no heat exchanged)

p

constant

Ck

pr

= kc

cp

v

where (specific heat ratio)

pVn

constantdpd r

= Edp

dv / vE p=

RT

E kpv 1kdpCk

dr

r-= 1k

k

dp pk

dr

r r-=

dp pk

d r r=

Speed of Sound (c)Speed of Sound (c)

cEv

cdpd

Edp

dv /

E dpd

v

E kpv

For gasses, if no heat exchanged (isentropic) then we have

It can be shown that (homework) ckRTM gas

Connection to Mach number!V

Mac

=

. Solve for dpd r

and

Ev is large for fluids that are difficult to compress

Vapor PressureVapor Pressure

0

1000

2000

3000

4000

5000

6000

7000

8000

0 10 20 30 40

Temperature (C)

Vap

or p

ress

ure

(Pa)

liquid

What is vapor pressure of water at 100°C?101 kPa

Cavitation! When absolute pressure drops below vapor pressure

pR2 = 2R

Surface TensionSurface Tension

Pressure increase in a spherical droplet

Pressure increase in a spherical droplet

Rp

2R

p2

pR2

2R

Surface moleculesSurface molecules

0.0500.0550.0600.0650.0700.0750.080

0 20 40 60 80 100

Temperature (C)

Sur

face

tens

ion

(N/m

)Fp=

F=

Example: Surface TensionExample: Surface Tension

Estimate the difference in pressure (in Pa) between the inside and outside of a bubble of air in 20ºC water. The air bubble is 0.3 mm in diameter.

Estimate the difference in pressure (in Pa) between the inside and outside of a bubble of air in 20ºC water. The air bubble is 0.3 mm in diameter.

Rp

2R

p2

R = 0.15 x 10-3 mR = 0.15 x 10-3 m

= 0.073 N/m = 0.073 N/m

mx

mNp

31015.0

/073.02

mx

mNp

31015.0

/073.02

970 Pap =970 Pap =

What is the difference between pressure in a water droplet and in an air bubble?

Review: Fluid PropertiesReview: Fluid Properties

Viscosity Density and Specific Weight Elasticity Vapor Pressure Surface Tension

Viscosity Density and Specific Weight Elasticity Vapor Pressure Surface Tension

dydu dydu

Edp

dV Vv /

cEv

Rp

2R

p2