Principles and Practice of Engineering PE Chemical Reference Handbook by National Council of Examiners for Engineering and Surveying (NCEES) (z-lib.org)

Chapter 6: Fluid Mechanics

6.2.7 Pressure Drop for Turbulent Flow

6.2.7.1 Head Loss in Pipe or Conduit

The Darcy-Weisbach equation is

=hL f=DL 2ug2 K u2
2g

where

f = friction factor

D = inside diameter of the pipe or hydraulic diameter (DH) of conduit

L = length over which the pressure drop occurs

f L = K = the loss coefficient
D

The total loss coefficient for a system is

K = / Ki

where Ki = the loss coefficient for individual fittings, valves, and other components

Changes in K for different pipe internal diameter are

Ka = Kb e Da 4
Db
o

An alternative formulation is

hL = 2fFanning Lu2
Dg

where the Fanning friction factor is

fFanning = f
4

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Chapter 6: Fluid Mechanics

Loss Coefficients and Equivalent Lengths for Fittings and Valves

Loss Coefficient

Fitting Equivalent K = K1 + K3d1 + 1 n
Length* Re IDinches

L K1 K3
D

Standardc r = 1m, threaded 35 800 0.40
d

90o Standardc r = 1m, flanged or welded 20 800 0.25
d

Long radiusc r = 1.5 m 16 800 0.20
d

Mitered 100 1000 1.15

Standardc r = 1m, threaded 16 500 0.20
d

Elbows 45o Long radius c r = 1.5 m 13 500 0.15
Tees d 20 500 0.25

Mitered, 1 weld (45°)

Standardc r = 1m, threaded 60 1000 0.70
d

180o Standardc r = 1m, flanged or welded 30 1000 0.35
d

Long radius c r = 1.5 m 25 1000 0.30
d

Standard, threaded 60 500 0.70

Used as Long radius, threaded 35 800 0.40
elbows
Standard, flanged or welded 65 800 0.80
Run through
Stub-in branch 85 1000 1.00

Threaded 10 200 0.10

Flanged or welded 40 150 0.50

Stub-in branch 5 100 0.05

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Chapter 6: Fluid Mechanics

Loss Coefficients and Equivalent Lengths for Fittings and Valves (cont'd)

Loss Coefficient

Equivalent K1 1
Re IDinches
Fitting Length* K = + K3d1 + n

L K1 K3
D

Full line sizef Dopening = 1.0 p 10 300 0.10
Dpipe

Gate, ball, or Reduced trimf Dopening = 0.9 p 12 500 0.15
plug Dpipe

Reduced trimf Dopening = 0.8 p 20 1000 0.25
Dpipe

Valves Globe Standard 330 1500 4.00
Diaphragm
Butterfly Angle or Y type 165 1000 2.00

Check Fully open 165 1000 2.00

Full open 20 800 0.25
Lift
Swing 830 2000 10.00
Tilting disk
125 1500 1.50

40 1000 0.50

* Approximated from the loss coefficient equation using friction factors for fully turbulent flow
for pipe sizes 1" through 24"

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Chapter 6: Fluid Mechanics

6.2.7.2 Loss Coefficients for Contraction and Expansion
Notes:

1. Reynolds Number (Re) and friction factor (f) are based on inlet velocity.

2. b = d
D
CONTRACTION
Contraction:

FLOW Dθ d

When θ < 45° and

Re < 2500, then K = 1.6c1.2 + 160 m e 1 − 1 o sinc i m
Re b4 2

Re > 2500, then K = 1.6`0.6 + 1.92f jf1 − b2 p sin i
b4 2

When θ > 45° and 1

Re < 2500, then K = 1.6c1.2 + 160 m e 1 − 1 o<sinc i 2
Re b4 2
mF

jf1 − b2 i 1
b4 2 mF2
Re > 2500, then K = `0.6 + 0.48f p <sin c

Expansion: EXPANSION

FLOW d θ D

When θ < 45° and

Re < 4000, then K = 5.2`1 − b 4j sin c i m
2

Re >4000, then K = 2.6`1 + 3.2f j`1 − b4 j sin c i m
2

When θ > 45° and
Re < 4000, then K = 2`1 − b4j

Re > 4000, then K = `1 + 3.2f j`1 − b4j2

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Chapter 6: Fluid Mechanics

6.2.7.3 Loss Coefficients for Pipe Entrance and Exit
Loss Coefficients

Loss Coefficient

Fitting Type Configuration K = K1 + K3d1 + 1 n
Re IDinches

K1 K∞

Inward projecting FLOW 160 1.0
or reentrant

Sharp-edged FLOW 160 0.5
Entrance

r/d K∞
0.02 0.28

0.04 0.24

Rounded FLOW d 160 0.06 0.15
r
0.10 0.09

0.15 & up 0.04

Exit All geometries 0.0 1.0

6.2.7.4 Valve Flow Coefficient (Cv)

Valve flow coefficient (Cv ) is a value of the relationship between the pressure drop across a valve and the corresponding flow rate:

Cv = Vo SG
DP

Also:

Cv = ad2
K

where m3
0.0352 m2 s Pa
a = constant, 29.9 gpm or
in2 psi

d = effective diameter of the valve, in inches or meters
K = loss coefficient
Note: Values of Cv are not interchangeable between unit systems.

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Chapter 6: Fluid Mechanics

The estimated flow rate with a known K value is

Vogpm = ad2 DP
K SG

where ΔP = pressure drop (psi or Pa)

6.2.8 Flow Through an Orifice
6.2.8.1 Submerged Orifice

Submerged Orifice Operating Under Steady-Flow Conditions
V

h1 – h2
h1 h2

D D2

Vo = A2 u2 = CA 2g _h1 − h2i

where
u2 = velocity of fluid exiting the orifice
A = cross-sectional area at diameter D
A2 = vena contracta cross-sectional area at diameter D2
C = coefficient of discharge
6.2.8.2 Orifice Discharging Freely into Atmosphere

Orifice Discharging Into Atmosphere

Atm

h

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Chapter 6: Fluid Mechanics

Torricelli's equation is
u = 2gh
Vo = CA 2gh

where
h = distance from the liquid surface to the centerline of the orifice opening
A = cross-sectional area at diameter D
C = coefficient of discharge

6.2.9 Particle Flow

The force exerted by a fluid that opposes the weight of an immersed object (buoyant force) can be expressed in terms of
differential densities:

FG = `t p − tf jg Vp
gc

where

FG = buoyant force

rp = particle density

rf = fluid density

Vp = volume of particle

The force exerted by a fluid flowing past a solid body (drag force) can be expressed in terms of a drag coefficient (CD):

FD = CD tf u32 AP
2gc

where

FD = drag force
u3 = approach velocity

AP = the projected area of object with axes perpendicular to the flow

6.2.9.1 Stokes Law or Stokes Flow

For a sphere moving through a fluid at Re << 1:

CD = 24
Re

where

Re = Dp u3 t
n

Dp = the particle diameter

In Stokes flow, viscosity can be determined using:

n = D p2 g`tp − tf j
18ut

where ut = terminal (or settling) velocity of particle

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Chapter 6: Fluid Mechanics

Drag Coefficients

For spheres in a flowing fluid with Reynolds numbers (1 < Re < 2×105), the Dallavalle equation applies:

CD = e0.632 + 4.8 2
Re
o

For cylinders in a flowing fluid with Reynolds numbers (1 < Re < 2×105) and with the axis normal to the flow, this equation

applies:

CD = e1.05 + 1.9 2
Re
o

DRADGrCaOgECFoFeICffIiEcNieTnStFsOfoRrSSPpHhEeRrEeSsAaNnDdFFLlAaTt DDIiSsKksS

10 2
8
6

4V d

2

10 8

6
4

CD 2 CIRCULAR DISK

STOKES LAW:
18
CD = 24/Re
6

4

2 EFFECT OF SURFACE SPHERE
ROUGHNESS OR MAIN-
10-18 V d STREAM TURBULENCE

6

4

2 4681 2 4 6 810 2 4 6 810 2 2 4 6 810 3 2 4 6 8104 2 4 6 810 5 2 4 6 810 6 2 4 6 810 7

10-2
2x10 -1

REYNOLDS NUMBER (Re)

6.2.9.2 Terminal Velocity (ut)
For a sphere of diameter Dp, the following equation applies for any Reynolds number (Newton's Law of falling particles):

ut = 4g Dp`tsphere − tf j
3tf CD

For a small sphere of diameter Dp, following Stokes Law:

ut = D 2 g _tsphere − tf i
p

18n

6.2.9.3 Reynolds Numbers for Particles in a Fluid

Reynolds number when particle velocity (ut) is unknown and Dp, ρp, ρf, and μ are known:

Re = ;_14.42 + 1.827 1 2

Ar i2 − 3.798E

where the Archimedes number (Ar) is:

Ar = Dp3 tf g`tp − tf j
n2

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Chapter 6: Fluid Mechanics

Reynolds number when particle diameter (Dp ) is unknown and ut, ρp, ρf , and μ are known:

CD 1
1 Re n2
Re = d0.00433 + 0.203 − 0.0658

where CD = 4n g`tp − tf j
Re
3tf2 u 3
t

Reynolds number when fluid viscosity (μ) is unknown and Dp, ut, ρp, and ρf are known:

Re = e CD 4.8 2
− 0.632
o

Use known quantities to solve for CD.

6.2.9.4 Flow Through Porous Media and Packed Beds
A porous, fixed bed of solid particles can be characterized by:

L = length of particle bed

ds = average particle diameter (diameter of a sphere with the same volume of the particle)
Φ = sphericity of particle (0–­­ 1)

e = porosity or void fraction of the particle bed (dimensionless)

Porosity (e) or void fraction:

e = _Total volume − Volume of solidsi = 1 − Asolid = Avoids
Total volume A A

where

Asolid = area of the solid phase in a cross-section of area A

Avoids = void area in a cross-section of area A

Interstitial velocity (actual velocity of fluid within the pores or voids):

=ui e=VoA u
e

where u = approach velocity (or superficial velocity)

Sphericity of a particle (shape factor):

U = surface area of sphere with same volume as particle
surface area of particle

Friction loss through porous media:

hf = 3 d f L n u2 e _1 − ei o
4 ds g e3

Reynolds number for flow through porous media:

Re = 2 dsnut e _1 1 ei o
3 −

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Chapter 6: Fluid Mechanics

Use the Ergun equation to estimate the pressure loss through a packed bed (ΔP) under laminar and turbulent conditions:

DP = _150u ni _1 − ei2 + `1.75t u2j e _1 − ei o
L `U2 ds2j e3 _U dsi e3

Typical Shape Factors

Particle Φ

Spheres 1.00

Torus 0.89

Ideal cylinder (h = d) 0.87

Octahedron 0.85

Cube 0.81

Sand (average) 0.75

Cylinder (h = 5d) 0.70

Cylinder (h = 10d) 0.58

Tetrahedron 0.67

Berl saddles 0.30–0.37

Raschig rings 0.26–0.53

6.2.9.5 Fluidization
For a fluid passing vertically through a bed of particles, ΔP increases as fluid velocity u increases. The net upward force FB on the
bed is

FB = AΔP
where A = cross-sectional area of the bed

At fluidization, net upward force (fluid drag force) equals the weight of the bed (FB = WB), while the fluid velocity above the bed
is less than the terminal velocity of the particles (ut).
The Reynolds number for a fluidized bed can be approximated by:

Re = C1 + C2 Ar − C1
where

Ar = Archimedes number

C1 = 180_1 − ei
3.5
e3
C2 = 1.75

where ϵ = minimum bed void fraction (porosity) at the point of fluidization

The minimum bed void fraction for bed height H at the first indication of fluidization is

e = 1 − mparticles
H Atp

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Chapter 6: Fluid Mechanics

The minimum fluidization velocity is

= `tp − tf jgd 2particles f3
150 n 1−f
umf

note:

1. usuperficial = umf is the incipient fluidization.

2. For large particles, dparticles ≥ 1 mm, inertial effects are important. Use the Ergun equation.

The maximum fluidization velocity that avoids entrainment is

= `t p − t f jgd 2
particles
usettling (Stokes)
18n

usettling = c 25 m 1−f for common operating condition u = 30 umf
umf 3 f3

6.2.10 Open-Channel Flow
6.2.10.1 Specific Energy (or Specific Heat)

E = u2 + y
2g

where

E = specific energy (or head)

u = fluid velocity

y = depth of liquid

Critical Depth: The depth of flow for a given discharge where the specific energy is at q minimum.

1
q2 3
yc = e g
o

where

yc = critical depth
q = unit discharge cVBo m
Vo = total discharge, volumetric flow rate

B = channel width

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Chapter 6: Fluid Mechanics

Specific Energy Diagram

y

CHANNEL DEPTH

yc

Emin SPECIFIC ENERGY E

6.2.10.2 Froude Number SUBCRITICAL FLOW
(Fr2 < 1)
=Fr g=uy2h Vo B (2)
where g A3

yh = hydraulic depth = A
B

A = cross-sectional area of flow

B = channel width

Supercritical flow: Fr > 1

Subcritical flow: Fr < 1

Critical flow: Fr = 1

6.2.10.3 Hydraulic Jump

SUPERCRITICAL FLOW Hydraulic Jump
(Fr1 > 1)
HYDRAULIC JUMP

FLOW DIRECTION y2
(1)
y1 243

©2020 NCEES

Chapter 6: Fluid Mechanics

y2 = 1 `− 1 + 1 + 8Fr12 j
y1 2

where

y1 = flow depth at upstream supercritical flow location
y2 = flow depth at downstream subcritical flow location
Fr1 = Froude number at upstream supercritical flow location
Fr2 = Froude number at downstream subcritical flow location

6.2.10.4 Manning Equation

vo = l A R 2 1
n 3
H S2

where

vo = discharge volumetric flow rate

k = 1.0 for SI units; 1.49 for U.S. units

A = cross-sectional area of flow

RH = hydraulic radius
S = slope of hydraulic surface

n = Manning's roughness coefficient

Manning's Roughness Coefficients

Material n
Cast iron pipe 0.013
Wrought iron pipe 0.015
Riveted steel pipe 0.016
Corrugated storm pipe 0.024
Glass 0.010
Vitrified sewer pipe 0.014
Concrete pipe 0.013
Excavated canal—earth, uniform 0.023
Natural channel—uniform cross-section 0.050

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Chapter 6: Fluid Mechanics

6.2.11 Two-Phase Flow
6.2.11.1 Flow Patterns

Bubble or Froth Flow: Bubbles of gas are dispersed throughout the liquid. Gas bubbles move at roughly the same velocity as the
liquid.

BUBBLE FLOW
Plug Flow: Alternate plugs of liquid and gas move along the upper portion of the pipe, with mostly liquid moving along the lower
portion.

PLUG FLOW
Stratified Flow: Gas flow moves on top and over the liquid forming a distinct, relatively smooth, liquid-gas interface.

STRATIFIED FLOW
Wave Flow: Similar to stratified flow, the fast-moving gas flow creates waves in the liquid phase.

WAVE FLOW
Slug Flow: High-velocity gas picks up waves to form frothy slugs of liquid. These slugs move at higher velocity than the bulk
liquid phase and can create vibrations that can damage equipment.

SLUG FLOW
Annular Flow: As gas velocity increases, liquid forms around the inside of the pipe wall, with the high-velocity gas flowing
through the center.

ANNULAR FLOW
Dispersed Flow (or Spray Flow or Mist Flow): Liquid is entrained as fine droplets in the gas phase.

©2020 NCEES DISPERSED FLOW
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Chapter 6: Fluid Mechanics

6.2.11.2 Flow Regimes Flow Patterns for Horizontal Two-Phase Flow

100,000 DISPERSED

WAVE ANNULAR BUBBLE OR
FROTH
10,000

By STRATIFIED SLUG

1,000

PLUG

100 1 10 100 1,000 10,000
0
Bx
Source: Baker, Ovid, Oil and Gas Journal, Nov. 10, 1958.

Baker parameters for the previous chart:

Bx = 531d mo L n SSSSSTSSRS_tLttL32G 1 WXVWWWWWWW f 1 p
mo G
i2 ncLL3

By = 2.16d mo G nf 1 p
A
1

_tL tGi2

where

A = internal pipe cross-sectional area, ft2

mo G = gas flow rate, lbm
hr
lbm
mo L = liquid flow rate, hr

ρL = liquid density, lbm
ft3
lbm
ρG = gas density, ft3

μL = liquid viscosity, cP

gL = liquid surface tension, dyn
cm

©2020 NCEES 246

Chapter 6: Fluid Mechanics

6.2.12 Compressible Flow

6.2.12.1 Isentropic Flow Relationships
In an ideal gas for an isentropic process, the following relationships exist between static properties at any two points in the flow:

P2 = e T2 k = d tt12 k
P1 T1
k= o^k − 1h n

where ratio of specific heats = cp
cv

The stagnation temperature T0 at a point in the flow is related to the static temperature:

T0 = T + u2
2 cp

Energy relation between two points is

h1 + u12 = h2 + u22
2 2

The relationship between the static and stagnation properties (T0, P0, and r0) at any point in the flow can be
expressed as a function of the Mach number (Ma):

T0 = 1+ k−1 Ma2
T 2

P0 T0 k − k
P T 2
= d n^k − 1h = c1 + k 1 Ma2 m^k − 1h

tt0 T0 k k−1 1
T 2
= d n^k − 1h = c1 + Ma2 m^k − 1h

Compressible flows are often accelerated or decelerated through a nozzle or diffuser. For subsonic flows, the velocity decreases
as the flow cross-sectional area increases and vice versa. For supersonic flows, the velocity increases as the flow cross-sectional
area increases and decreases as the flow cross-sectional area decreases. The point at which the Mach number is sonic is called the
throat; its area is represented by the variable A*.

The following area ratio holds for any Mach number:

A = 1 SSSSTSSSRS1 + 1 _k − 1i Ma2 WWWXWWWWWV2^^kk+−11hh
A* Ma 2
1
2 _k + 1i

where

A = area

A* = area at the sonic point (Ma = 1.0)

In an ideal gas, sonic velocity is:
usound = ^kRT h1 2

when Ma = 1.0

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Chapter 6: Fluid Mechanics

6.2.12.2 Simplified Isothermal Equation

mo = KJ fL tgc A2 P1 NO P12 − P22 o
K D + 2 lne P2 Oe P1
K o O
L P

6.2.12.3 Net Expansion Factors For Gases

Expansion Factors for Compressible Flow Through Orifices and Nozzles

k = 1.3 approximately [CO2, SO2, H2O (steam), k = 1.4 approximately [Air, H2, O2, N2,
H2S, NH3, N2O, Cl2, CH4, C2H2, and C2H4] CO, NO, and HCl]

1.0 1.0

Y — EXPANSION FACTOR0.95 SQUARE 0.95 SQUARE
Y — EXPANSION FACTOREDGE EDGE
0.90 ORIFICE 0.90 ORIFICE
β = 0.2
0.85 β = 0.2 0.85
= 0.5 = 0.5
0.80 = 0.6 0.80 = 0.6
= 0.7 = 0.7
NOZZLE OR = 0.75 NOZZLE OR = 0.75
VENTURI VENTURI
0.75 METER 0.75 METER 0.6

β = 0.2 β = 0.2
0.70 = 0.5 0.70 = 0.5

= 0.6 = 0.6
= 0.7 = 0.7
0.65 = 0.75 0.65 = 0.75

0.60 0.2 0.4 0.6 0.60 0.2 0.4
0 0
PRESSURE RATIO – ∆—P PRESSURE RATIO – ∆—P
P1 P1

where Pl = absolute upstream pressure

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Chapter 6: Fluid Mechanics

Expansion Factors for Compressible Flow Through Pipes

k = 1.3 approximately [CO2, SO2, H2O (steam), H2S, NH3, N2O, Cl2, CH4, C2H2, and C2H4]

1.0 LIMITING FACTORS
FOR SONIC VELOCITY

0.95 k = 1.3

0.90 K KK==41000 K ∆—P Y
P1
0.85 = .612
K 1.2 .525 .631
0.80 1.5 .550 .635
K 20 2.0 .593
Y K= K=1K5=10=8K.60=.K04=.03.0 .658
= 3 .642 .670
0.75 K 4 .678 .685
6 .722
0.70 2.0 .698
=K1=.51.2 8 .750 .705
0.65 10 .773 .718
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 15 .807
0.60 .718
0.55 PRESSURE RATIO – ∆—P 20 .831 .718
P1 40 .877 .718
0 100 .920

k = 1.4 approximately [Air, H2, O2, N2, CO, NO, HCl] LIMITING FACTORS
FOR SONIC VELOCITY
1.0
k = 1.4
0.95

0.90 K ∆—P Y
P1
0.85 .588
1.2 .552 .606
0.80 1.5 .576 .622
2.0 .612
Y .639
3 .662 .649
0.75 4 .697 .671
6 .737
0.70 KKK==K=K24=1000=K1051=0K8.=0K6.0=K4=.03.0 .685
K= 8 .762 .695
0.65 10 .784 .702
15 .818
0.60 K = 1.2 2.0 .710
K= 20 .839 .710
40 .883 .710
0.55 1.5 100 .926
0
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

PRESSURE RATIO – ∆—P
P1

where Pl = absolute upstream pressure

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Chapter 6: Fluid Mechanics

6.2.12.4 Critical Pressure Ratio, rc , for Compressible Flow
Critical Pressure Ratio Through Nozzles and Venturi Tubes (Only)

0.64

0.60 β
0.85

0.62
0.80

P2 0.58 0.75
P1 0.56
0.54 0.70
rc =
0.65
1.25 1.30 1.35 1.40 0.60
0.50
0.40
0.20
0

1.45

k= Cp
Cv

where P1 and P2 = absolute pressures upstream and downstream of the nozzle or venturi tube, respectively

6.2.12.5 Choked Flow

Choked flow is a limiting condition where the mass flow will not increase with a further decrease in the downstream pressure
environment while upstream pressure is fixed. Choked flow occurs when the Mach number is 1.0 at the minimum cross-section
area.

Mass flow rate of gas at choked flow:

2 k+1
+ mk − 1
mo = Cd A k t1 P1 gcc
k 1

where

Cd = discharge coefficient of restriction
r1 = density of gas before restriction
P1 = pressure of gas before restriction (absolute)

©2020 NCEES 250

Chapter 6: Fluid Mechanics

6.3 Applications of Fluid Mechanics

6.3.1 Pumps

Types and Subtypes of Pumps

RECIPROCATING STEAM
POWER
CONTROLLED VOLUME
PISTON

POSITIVE DISPLACEMENT BLOW CASE

VANE SCREW

ROTARY FLEXIBLE MEMBER CIRCUMFERENTIAL PISTON

LOBE GEAR

PUMPS

KINETIC CENTRIFUGAL CANNED PUMP
SPECIAL EFFECT OVERHUNG IMPELLER
IMPELLER BETWEEN BEARINGS
TURBINE TYPE
REGENERATIVE TURBINE

REVERSE CENTRIFUGAL
ROTATING CASING

6.3.1.1 Affinity Laws for Pumps, Fans, and Compressors
For small changes in impeller diameter (changes not to exceed 20%):

DD=12 VV=o1o2 H1 and BP1 = D13
H2 BP2 D23

For variations in speed (constant impeller diameter):

SS=12 VV=o1o2 H1 and BP1 = S13
H2 BP2 S23

where
BP = brake power
D = impeller or wheel diameter
H = head (height of fluid)
Vo = volumetric capacity
S = speed (rpm)

©2020 NCEES 251

Chapter 6: Fluid Mechanics

6.3.1.2 Pump Similitude

Predicting Performance of Homologous Pumps
Volume capacity estimate:

=VVo1o2 SS=12 e DD12 o3 e D1 2 e H1 0.5
D2 H2
o o

Pressure or head estimate:

H1 = e S1 2 e D1 2
H2 S2 D2
o o

Brake power estimate:

=BBPP21 tt=12 e SS12 o3 e DD12 o5 tt12 e D1 2 e H1 1.5
D2 H2
o o

Impeller or wheel speed estimate:

=SS12 DD=12 e HH12 o0.5 e Vo2 0.5 e H1 0.75
V1o H2
o o

6.3.1.3 Pump Head

Pump head (Hp) is a variation of the head-basis Bernoulli equation:

Hp = ` Pd − Psjgc + `ud2 − us2j + ` zd − zsj + hf
t g 2g

where

Ps = suction pressure at suction reference point (absolute)
Pd = discharge pressure at discharge reference point (absolute)
us = velocity at the pump suction
ud = velocity at the pump discharge
zs = elevation at the suction reference point
zd = elevation at the discharge reference point
hf = friction loss in the pipe between the reference points

©2020 NCEES 252

PUMP HEAD Chapter 6: Fluid Mechanics DISCHARGE
Ps REFERENCE POINT
Centrifugal Pump
zs SUCTION
REFERENCE POINT

Pd

ud zd
us

Pump Head in Common Units

Pump Head Calculations SI Units
U.S. Units

Component Hp = 2.31_Pd − Psi + `ud2 − us2j + _zd − zsi + hf Hp = `Pd − Psj + `ud2 − us2j + `zd − zsj + hf
SG 2g tg 2g
Hp
P ft m
u
z psi Pa
g m
hf ft s
r sec
ft m

32.2 ft 9.81 m
sec2 s2

ft m

lbm kg
ft3 m3

©2020 NCEES 253

Chapter 6: Fluid Mechanics

6.3.1.4 Pump Curve
A pump curve, head-capacity curve, or H-Q curve is provided by pump manufacturers.
Pump Curve for a Fixed Impeller Diameter and Pump Speed

HEAD

TOTAL HEAD BEP
EFFICIENCY
POWER
BRAKE POWER

NPSHr

VOLUMETRIC CAPACITY
where BEP = best efficiency point (sometimes called best operating point, or BOP)

6.3.1.5 Net-Positive Suction Head (NPSH)
NPSH: Total suction head minus the vapor pressure of the liquid being pumped (units are in height of liquid
(absolute) and the referenced datum is the suction nozzle.)
NPSHa: Net-positive suction head available to the pump
NPSHr: Net-positive suction head required by the pump (provided by the pump manufacturer)
For suction lift:
NPSHa = ha – hvap – hst – hL
For flooded suction:
NPSHa = ha – hvap + hst – hL
where
ha = absolute pressure (in height of liquid) on the surface of the liquid supply level
hvap = vapor pressure (in height of liquid) of the liquid at the temperature being pumped
hst = static height of liquid supply, either above or below the pump centerline or impeller eye
hL = suction line losses in height of liquid

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Chapter 6: Fluid Mechanics

6.3.1.6 Pump Power

Power required to move the fluid, or water power (WP):

Flow rate (gpm) # H (ft) # td lbm n
246, 780 ft3
U.S. units WP (horsepower) =

Metric units WP (watts) = Flow ratec m3 m # H^mh # te kg o # gd m n
s m3 s2

Power required at the pump shaft, or brake power (BP):

BP = WP
h pump

Power required by the pump driver, or supplied power (SP):

SP = WP
h pump h driver h transmission

6.3.1.7 Temperature Rise in a Centrifugal Pump

DT = BP`1 − hpumpj
cpVot

6.3.1.8 Specific Speed (Ns ) at the BEP

Ns = S Vo 0.5
H 0.75

where head (H) and flow rate ^Vo hare taken at the BEP

6.3.1.9 Suction-Specific Speed (Ns ) at the BEP

Ns = S Vo 0.5
_NPSHri 0.75

6.3.1.10 System Curves

System curves are developed from different flow rates through a given system, using the Bernoulli equation.

Note: The velocity head terms are usually omitted because the changes in u2 are negligible.
2g

Hs = pressure head + static head (hs ) + pipe losses* (hf )

*Include friction, entrance, and exit losses:

Hs = _ PB − PAigc + hs + hf
t g

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Chapter 6: Fluid Mechanics
Simple Pumping System

PB

STATIC HEAD (HS) PA PUMP

TOTAL HEAD System Curve Plot
SYSTEM CURVE

PRESSURE HEAD
STATIC HEAD (hs )

CAPACITY

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Chapter 6: Fluid Mechanics

6.3.1.11 Pumps in Parallel and Series
Operating point: Centrifugal pumps operate at the intersection of the pump curve and the system curve.
For pumps in parallel, capacities are added horizontally. For pumps in series, heads are added vertically:

Pumps Operating in Parallel

COMBINED (A+B)
PUMP CURVE

PUMP B OPERATING
PUMP A POINT

TOTAL HEAD A

A+B B
SYSTEM CURVE

CAPACITY

Pumps Operating in Series

COMBINED (A+B)
PUMP CURVE

TOTAL HEAD PUMP B OPERATING
PUMP A POINT

SYSTEM
CURVE

A+B
B

A

CAPACITY

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Chapter 6: Fluid Mechanics

6.3.2 Fans, Blowers, Compressors Δp

Fans and Blowers

Typical backward curved fans: POWER

Wo = DP Vo
hf

where

Wo = fan power f

DP = pressure rise CONSTANT N, D, ρ
hf = fan efficiency FLOW RATE

6.3.3 Control Valves

6.3.3.1 Control Valve Flow Characteristics
Flow characteristic of a control valve: The relationship between valve capacity and valve stem travel (or valve lift).

Control Valve Flow Versus Stem Travel

100
QUICK OPENING

80

PERCENT OF MAXIMUM FLOW LINEAR
60

40

MODIFIED
20 PARABOLIC

EQUAL PERCENTAGE
0 20 40 60 80 100

PERCENT OF RATED STEM TRAVEL

Linear: Flow capacity increases linearly with stem travel.

Equal Percentage: Flow capacity increases exponentially with stem travel. Equal increments of stem travel produce equal

percentage changes in the existing CV.

Modified Parabolic: Valve characteristic is approximately midway between linear and equal-percentage characteristics.

It provides fine throttling at low flow capacities and approximately linear characteristics at

higher flow capacities.

Quick Opening: Provides large changes in flow for very small changes in early stem travel.

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Chapter 6: Fluid Mechanics

6.3.3.2 Control Valve Sizing (Traditional Method)

Control Valve Sizing Equations for Liquids (Incompressible Flow)

Equation Use Notes

Vo = CV DP Basic sizing equation; does not CV is the flow coefficient for a control valve. The
SG consider viscosity effects or valve value of CV is dependent on the type of valve and
recovery capabilities also varies with stem travel or percentage of valve
opening. The units and values for the flow coef-
ficient are provided by the manufacturer.

CV = Vo SG Flow coefficient For Newtonian fluids of viscosities similar to
DP water.
Corrected flow coefficient for Use the appropriate FV to predict pressure drop,
CV − Corr = CVFV viscosity select valve size, or predict flow rate.
Maximum allowable differential where:
pressure
Km = valve recovery coefficient (provided by
manufacturer)

DPmax = Km _P1 − rC pvi P1 = valve body inlet pressure (absolute)
pv = liquid vapor pressure (absolute) at the

valve body inlet temperature

rC = 0.96 − 0.28 pv Critical pressure ratio (when manu- rC = critical pressure ratio
pc facturer data is not available)
The critical pressure ratio is provided by the
Vot Control valve Reynolds number manufacturer or, in the absence of correlation data,
n CV the equation below can be used.
Re = 17, 250 pc is the critical pressure of the fluid (absolute).

For engineering units only, where Vo is in gpm, ΔP
is in psi, μ is in cP, and ρ is in lbm .

ft 3

6.3.4 Jet Propulsion

The force produced by jetting action is

F = mo _u2 − u1i JET PROPULSION

m, u1 m, u2

Therefore, according to the conservation of mass:

F = Vo2 t2 u2 − V1o t1 u1
gc

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Chapter 6: Fluid Mechanics

JET FORCES ON PLATES Jet Forces on Plates JeJtETonONanANInINcClLinINeEdDPPlLaAtTeE
JJeEtToOnNaAVVeErRtTicICaAl LPPlaLAteTE JeJtEoTnONa AHHoOrRizIZoOnNtaTAl LPPlaLAteTE

h

Fx = − mo ujet Fy = − mo 3 ujet2 − 2g h F = − mo ujet sin i
gc gc gc

6.3.5 Air Lift

Air Lift Operation

LIQUID
AND AIR

LIQUID LIQUID

NO AIR AIR FLOW
AIR INLET

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Chapter 6: Fluid Mechanics

Common Air Lift Terms

DISCHARGE LEVEL

LIFT

ABOVE

GROUND TOTAL

GROUND LEVEL STARTING

STATIC WATER STATIC LIFT
LEVEL LEVEL TOTAL

PUMPING WATER PUMPING
LEVEL LIFT

DRAW-DOWN

STARTING
SUBMERGENCE

PUMPING
SUBMERGENCE

AIR INLET

Air lifts are used to pump liquids and mixtures of liquids and solids. The volume of air required to pump is

Va = L + 34
C log10c S 34 m

where

Va = quantity of free air required per gallon of liquid pumped e ft3 o
gallon pumped

C = constant found for outside airline (VA) and inside airline (VC) in figure below

S = pumping submergence (%) in figure below

L = total pumping lift (ft)

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Chapter 6: Fluid Mechanics

Constant in Formula for Va

375

350 OUITNSSIIDDEEAAIRIRLLININEE– – VA
325 VC
300
VALUES OF CONSTANT “C” 275

250

225

200

175

150

125
30 35 40 45 50 55 60 65 70 75 80
SUBMERGENCE – PERCENT

Approximate Percent Submergence for Optimum Efficiency
70

SUBMERGENCE – PERCENT 60

50

40

30
30 100 200 300 400 500 600 700 800 900
TOTAL PUMPING LIFT – FEET

Use for either system with straight or tapered pipe. Graphs only available in U.S. units; SI not available.

Source: Gibbs, C.W., New Compressed Air and Gas Data, 2nd ed., Davidson, NC: Ingersoll-Rand Company, 1971, p. 31-3.

©2020 NCEES 262

6.3.6 Solids Handling Chapter 6: Fluid Mechanics
6.3.6.1 Granular Media Storage Vertical Normal Stress Profile in a Silo

BULK SOLIDS
Z

HYDROSTATIC

PRESSURE

Source: Chase, George G., Solids Notes 10, Akron: University of Akron.

Compressive normal stress (Pv) in silos can be calculated by the Janssen equation:

PV = tgD =1 − exp d − 4 nK z nG
4 n K gc D

where

r = granular bulk density

m = solids coefficient of friction

D = silo diameter

K = lateral pressure ratio, where PW = K PV (Janssen's assumption that vertical normal stress is proportional to the lateral
normal stress)

z = bed depth at which pressure is being measured

Sources: Don McGlinchey, editor, Bulk Solids Handling: Equipment Selection and Operation,
and J.M. Rotter, Silo and Hopper Design for Strength, Oxford: Blackwell Publishing Ltd., 2008.

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Chapter 6: Fluid Mechanics

6.3.6.2 Pneumatic Transport
Pneumatic transport (or pneumatic conveying) is using gas to transport particulate solids through a pipeline (such as flour,
pulverized coal, powdered clay).

Flow Regimes:

Dilute Phase—Particles are fully suspended at loadings less than 1%.

Dense Phase—Particles are not suspended (or periodically suspended) with loadings greater than 20%.

Pressure Systems

FEED FILTER FILTER FILTER
HOPPER

BLOWER DISCHARGE HOPPERS
POSITIVE PRESSURE SYSTEM (PUSH)

FEED HOPPERS FILTER
BLOWER

DISCHARGE
HOPPER

NEGATIVE PRESSURE SYSTEM (PULL)

A “PUSH-PULL” SYSTEM USES BLOWERS TO SIMULTANEOUSLY
PUSH (POSITIVE PRESSURE) AND PULL THE SOLIDS (NEGATIVE PRESSURE)

Characteristics of Pneumatic Conveying Flow Regimes

Dilute Phase Dense Phase

High velocity Low velocity

Particles subject to attrition Low particle attrition

Low pressure High pressure

Low cost/simple operation Complex operation

Low loadings High solids loading

©2020 NCEES 264

LOW COST / SIMPLE OPERATION COMPLEX OPERATION
LOW LOADINGS HIGH SOLIDS LOADING

Chapter 6: Fluid Mechanics

Flows in Pneumatic Transport

DENSE PHASE DILUTE PHASE

PRESSURE GRADIENT CONTINUOUS DENSE-
PHASE FLOW

PLUG FLOW DILUTE-PHASE
DISCRETE PLUG FLOW FLOW

DUNE FLOW

DISCONTINUOUS DENSE-
PHASE FLOW

SALTATING FLOW

GAS VELOCITY

Definitions

Saltation—Settling of solid particles in the bottom of the pipe during dilute-phase pneumatic transport

Superficial gas velocity _u gi—The gas volumetric flow `Vogj divided by the pipe cross-sectional area (A):

ug = Vog
A

Superficial solids velocity ^u sh—The solids volumetric flow _Vosi divided by the pipe cross-sectional area:

u s = Vos
A
Vos = mtoss , with mo s and ts as the mass flow rate and density of the solid particles, respectively
where
where Actual gas velocity (ug):

ug = Vog
Ae

e = void fraction

Actual particle velocity (us):

us = Vos
A_1 − ei

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Chapter 6: Fluid Mechanics

Relationships
In vertical pipes, the minimum gas velocity (umin) to suspend particles is when the net upward force on the bed provided by the
gas equals the net weight of the solids bed. (see the Fluidization section in this chapter)

FB = WB
Practical minimum gas velocity:

4 g Dte ts − 1o
tg
u = 2 umin = 2
3CD

where CD = 24
Re

Mass flow rate of the solid particles:

mo s = A us _1 − eits
Mass flow rate of the gas:

mo g = A ug e tg

Solids loading (R):

R = mo s
mo g

Concentration (volume fraction) of solids:

Cs = Vos = us us
Vos + Vog + ug

Dilute-phase pressure drop: The total pressure drop is the sum of the contributions from the carrier-gas pressure drop, acceleration

of the solid particles, the friction of the solid particles against the pipe wall and fittings, the lifting of the solid particles through

the vertical sections, and miscellaneous factors.

DP = DPgf + `DPsa + DPsf + DPsb + DPsvj + DPmisc

Carrier-gas pressure drop ( DPgf ): For the purpose of this equation, compressible flow equations are not used. Treat the gas as an
incompressible fluid:

DPgf = f L ug2 tg
2 gc D

Acceleration of solids pressure drop (DPsa):

DPsa = mo s us
A gc

where A = pipe cross-sectional area

Solids friction in straight pipe pressure drop (DPsf ):
R tg ug2 Lactual
DPsf = ms 2 D gc

where

ls = solids friction factor (if unknown, assume 0.2)
R = solids loading

Lactual = actual length of pipe (not equivalent length)

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Chapter 6: Fluid Mechanics

Solids friction in bends pressure drop (DPsb):

DPsb = Leqe DPs o
Lactual

Vertical lift pressure drop (DPsv):
R g Z tg ug
DPsv = gc us

where Z = total length of vertical pipe where the flow is upwards

Miscellaneous pressure drop (DPmisc):
where DPmisc / additional pressure drop for other components, interferences, and other special conditions

Saltation velocity ( usalt ):

R = 1 f usalt b (Rizk correlation)
10a
gD p

where

D = inside diameter of conveying pipe

a = 1440 Dp + 1.96 (SI units)
= 439 Dp + 1.96 (U.S. units)

b = 1100 Dp + 2.5 (SI units)
= 325 Dp + 2.5 (U.S. units)

Dp = mean particle diameter

6.3.7 Mixing
6.3.7.1 Tank Mixing

Tank Mixing

LIQUID B
LEVEL

JBAFFLE TANK
WN IMPELLER
WHERE:
H BAFFLE T = TANK DIAMETER
D D = IMPELLER DIAMETER
N = ROTATIONAL SPEED
T V = TANK VOLUME
B = BAFFLE WIDTH
W = IMPELLER WIDTH
J = BAFFLE WIDTH

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Chapter 6: Fluid Mechanics

Impeller Reynolds number

Re = D2Nt
n

Flow Number

NQ = q
ND3

where q = volumetric flow rate through the impeller

Power Number

NP = Pgc
N3D5 t

where P = impeller power

Ratio of tangential liquid velocity at blade tips to blade tip velocity (K):

K = r2 NP
NQ

Froude number for tank agitation:

Fr = N2D
g

Power function ( f ) is defined by:

z = NP
Frm

where m = a − log10 Re
b

Examples of Mixing Configurations

Configuration (Unbaffled) a b
Six-blade turbine (vertical blades) 1.0 40.0
Three-blade propeller (pitch 2:1) 1.7 18.0
Three-blade propeller (pitch 1:1) 2.3 18.0

Power delivered to the liquid by an impeller:

P = z Frm N3 D5 t
gc

For tanking mixing where the liquid surface has insignificant wave formation, the Froude number is not a factor:

P = z N3 D5 t
gc

For Re < 10

P = KL N2 D3n
gc

where KL = empirical constant (laminar)

For Re > 10,000

P = KT N3 D5 t
gc

where KT = empirical constant (fully turbulent)

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Chapter 6: Fluid Mechanics

Values of Constants KL and KT for Baffled Tanks

Type of Impeller KL KT
0.32
Propeller, square pitch, 3 blades 41.0 1.00
6.30
Propeller, pitch = 2, 3 blades 43.5 4.80
1.65
Turbine, 6 flat blades 71.0 1.70
1.08
Turbine, 6 curved blades 70.0

Fan turbine, 6 blades 70.0

Flat paddle, 2 blades 36.5

Shrouded turbine, 6 curved blades 97.5

Note: Table is specific to tank configuration and provided as an example only.

Power required to suspend particles to a maximum height (Z) using a turbine impeller is

P = g tm Vmut _1 − 2 c T 1 e4.35b
D
emi3 m2

where = Z− E − 0.1, with E = clearance between impeller and tank floor
b T

rm , Vm = density and volume, respectively, of solid-liquid suspension, not including the clear liquid
in zone above height Z (also known as cloud height)

ut = terminal velocity of particles

em = volume fraction of liquid in zone occupied by suspension

and

1 = 1 + xsolidsd 1 − 1 n
tm tliquid tsolids tliquids

with xsolids = mass fraction of the solid particles in the solid-liquid suspension

Suspension of Particles in a Tank

Z CLEAR LIQUID
E
SOLID-LIQUID
SUSPENSION

TANK
D

T

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Chapter 6: Fluid Mechanics

6.3.7.2 Blending of Miscible Liquids in a Tank
Correlation of Blending TimeLCsIOQfRUoRIDrESMLAINTisIAOcTNibUORleFBBILNLiEqE-NuADGidIINTsAGTinTEIDMaEBTSAuFFFrObLREinDMeVIS-ECASIgBSLEitELated, Baffled Vessel

1000

100
fT

10

1
1 10 102 103 104 105 106

Re = __N_D__2_p___
μ

Blending time factor (fT) (for miscible Newtonian fluids only):

fT = t _N 2 1 1

D2i3 g6 D2

1 3
H2T2

where t = blend time (sec)

6.4 Flow and Pressure Measurement Techniques

6.4.1 Manometers and Barometers
6.4.1.1 Simple Manometer

Simple Manometer

Patm

PA P2
FLUID 1 z2
(ρfluid 1)

zA FLUID 2
P1 (ρfluid 2)

z1

PA − Patm = PA − P2 = g 9tfluid2 _ z2 − z1 i − tfluid1 _ zA − z1iC
gc

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Chapter 6: Fluid Mechanics

6.4.1.2 Manometer With Multiple Fluids

Manometer With Multiple Fluids

FLUID 1 (ρfluid 1) FLUID 3 (ρfluid 3)
z2P2 z2P2

AzAPA z1P1 B zBPB
z3P3
z1P1 z3P3

FLUID 2 (ρfluid 2) FLUID 4 (ρfluid 4)

PA − PB = _PA − P1i + _P1 − P2i + _P2 − P3j + _P3 − PBj

PA − PB = g 9tfluid1_z1 − zAi + tfluid2_z2 − z1i + tfluid3_z3 − z2j + tfluid4_zB − z3jC
gc

6.4.1.3 Inclined U-Tube Manometer

Inclined U-Tube Manometer P2
P1

x

MANOMETER ∆h
FLUID θ

P1 − P2 = g tm x sini = g tm Dh
gc gc

where

x = difference in tube fill length
rm = density of the manometer fluid (densities of the fluids on each side of the manometer are equal)
q = angle of inclination (horizontal = 0°)

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Chapter 6: Fluid Mechanics

6.4.1.4 Barometers

Another device that works on the same principle as the manometer is the simple barometer.

Patm = PA = Pv + tgh = PB + tgh
gc gc

where Pv = vapor pressure of the barometer fluid

Barometer
PV PB

hρ

PA

6.4.2 Flow Measurement Devices (Summary)

Flow Measurement Devices

Class Meter Type Description Advantages Drawbacks
• High permanent
Mechanical Rotary Rotary piston spins within a chamber of known • Accurate; suitable
piston volume. For each rotation, an amount of fluid for fuel metering pressure drop at
passes through the piston chamber. The rotations are high flows
counted and the flow rate is determined from the rate • Suitable for low
of rotations. volume metering • Clear liquids only
and laboratory or
bench scale testing • High cost

Gear Two rotating gears with synchronized, close-fitting • Accurate; suitable • High permanent

teeth. A fixed quantity of liquid passes through the for fuel metering pressure drop at

meter for each revolution. Permanent magnets in the • Suitable for low high flows
rotating gears transmit a signal to a transducer for volume metering • Clear liquids only
flow measurement. and laboratory or • High cost
bench scale testing

OOPpEeRrAaTtIiOoNnOoFf AaNnOoVvAaLlGgEeAaRrMmETeEteRr

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Chapter 6: Fluid Mechanics

Flow Measurement Devices (cont'd)

Class Meter Type Description Advantages Drawbacks

Mechanical (cont'd) Nutating Also known as a wobbly plate meter. Fluid enters • Accurate and • Accuracy is ad-
Disk a chamber of known volume. When the chamber is repeatable; used versely affected by
filled, the fluid is released, which causes the disk to for water service viscosities below
perform a nutating action (wobble in a circular path metering the meter's desig-
without actually spinning on its axis). The motion is nated threshold
detected by either gearing or magnetic transducers. • Good for hot
The flow rate is determined from the rate of motions. liquids

HOLE

SHAFT NUTATING DISK

INLET OUTLET

Helical Counter-rotation of the gears carries known volumes • Used for heavy • Can only measure
of liquid axially down the length of the gears. The and high-viscous liquids
rotation rate is measured using sensors, which in turn liquids
correlates to flow rate. • Low corrosion al-
• Highest accuracy lowance
of any positive dis-
placement • Cannot handle abra-
flow meter sive fluids

Source: Flowserve Corp., Irving, TX

Rotameter Fluid flows upward through a clear tapered tube and • Simple operation • Must be mounted
(variable suspends a bob. The higher the flow rate, the higher
area) the bob suspends in the tube. The bob is the indicator with few moving vertically
and the reading is obtained from the scale marked on
the tube. parts and no exter- • Changes in fluid
nal power source
FLOW PIPE properties gives

TAPERED TUBE • Inexpensive and erroneous results

BOB widely available • Not suited for large

• Accurate provided pipes (< 6 inches)

the fluid properties • Readout uncertainty
remain unchanged near bottom of the

• Resistant to shock scale

and chemical action • Some fluids may

obscure reading.

•

FLOW •

©2020 NCEES 273

Chapter 6: Fluid Mechanics

Flow Measurement Devices (cont'd)

Class Meter Type Description Advantages Drawbacks

Mechanical (cont'd) Turbine (or Fluid flows past a turbine wheel positioned in the • Simple and durable • Cannot tolerate

Woltmann center of the pipe with the shaft in line with the pipe. structure; can be cavitation
Type) The rotational speed is proportional to the flow rate.
Shaft rotation is detected electronically. installed vertically • Accuracy adversely
or horizontally
affected by en-

ELECTRONIC • Can be designed to trained gas
PICKUP
detect flow in either • Sensitive to changes
METER direction
HOUSING in fluid viscosity

FLOW • Operates under • Long straight runs
a wide range of of pipe upstream
ROTOR TURBINE temperatures and and downstream
SUPPORT pressures of the meter are

• Low pressure drop needed

across the flow • Bearings are prone
meter to wear (though
some are provided
• Effective in appli- "bearingless")
cations with steady,
high-speed flows • Not suitable for
steam
• Can be used for

gasses but not suit-
able for steam

Paddle Fluid flows past a paddle wheel positioned off-center • Simple and durable • Requires a full pipe

Wheel Type of the pipe with the shaft perpendicular with the pipe. structure; can be of liquid

The rotational speed is proportional to the flow rate. installed vertically • Not suitable for
Shaft rotation is detected electronically. or horizontally steam

FLOW ROTATION • Easy installation • Bearings are prone
into existing sys- to wear
tems for insertion

PADDLE WHEEL models

DETECTOR • Can be designed to
(MOUNTED detect flow in either
EXTERNALLY) direction

Other meters in this class: • Operates under
Single Jet a wide range of
Multi Jet temperatures and
Pelton Wheel pressures

• Low pressure drop
across the flow
meter

• Effective in appli-
cations with steady,
high-speed flows

©2020 NCEES 274

Chapter 6: Fluid Mechanics

Flow Measurement Devices (cont'd)

Class Meter Type Description Advantages Drawbacks

Pressure Venturi The meter constricts the fluid flow and sensors mea- • Highly accurate • Flow must be de-
sure the differential pressure before and within the over a wide range rived from pressure
constriction. The differential pressure is then con- of flows drop
verted to a corresponding flow rate.
• No moving parts • Pipe must be full
(mostly used for
PRESSURE MEASUREMENT • Low pressure drop liquid service)

FLOW • O( DLccuopfiaeps psrpoaxcie-
mately 50)

• Cannot measure flu-
ids in reverse flow

Orifice Flow is restricted using a plate with a hole drilled • Accurate over a • Flow must be de-
Plate (also through it. Sensors measure the differential pressure
square- before and after the meter (two tap configurations are wide range of rived from pressure
edge orifice shown). The differential pressure is then converted to
plate) a corresponding flow rate. flows, but not suit- drop

dP MEASUREMENT able for trade use • Accuracy
(FOR FLANGE TAP OPTION) (2–4% of full scale) reduced at low

dP • No moving parts flows

FLOW • Low cost; price • Plate materials

dP MEASUREMENT dP does not drama-ti- prone to wear and
(FOR VENA CONTRACTA
TAP OPTION) cally increase with corrosion,

Note: Orifices may be drilled in the middle of the pipe size which adversely
plate (concentric) or off-center (eccentric) to accom-
modate certain fluid types and flow regimes. Orifices • Low maintenance effects accuracy
may also be round or segmented.
(orifice plates can • Accuracy effected

be replaced during by high-viscous

maintenance opera- fluids

tions) • Moderate to high

• Easy to convert to permanent pressure

different applica- drop

tions or fluids by • Pipe must be full
replacing the orifice (for liquids)
plate

• In common use

©2020 NCEES 275

Chapter 6: Fluid Mechanics

Flow Measurement Devices (cont'd)

Class Meter Type Description Advantages Drawbacks
• Flow must be de-
Pressure (cont'd) Nozzle Similar to a venturi meter, but the inlet section is in • More accurate than
rived from pressure
the shape of an ellipse and there is no exit section. orifice plates drop
• More expensive
dP MEASUREMENT • High flow capacity than orifice plates
dP and high velocity • Takes up slightly
applications more room than
orifice plates
FLOW • Less susceptible to • Higher permanent
wear and corrosion pressure drop than
than orifice plates venturi meters
• Pipe must be full
• Can operate in (for liquids)
higher turbulence
• More expensive
• Tolerant of fluids than orifice plates
containing sus- or flow nozzle
pended solids meters

• Less expensive • Sensitive to turbu-
than the venturi lence
meter
• More complex to
• Physically smaller manufacture
than the venturi
meter • Accuracy depen-
dent on actual flow
• Can indicate a data
reverse-flow
condition • Cannot indicate a
reverse-flow
Dall Tube Similar to the venturi meter but more compact at • Similar perfor- condition
the expense of some loss in accuracy and additional mance as the
permanent pressure loss. venturi meter

dP • Shorter length than
the venturi meter

FLOW • Low unrecoverable
pressure loss

• Accurate to within
1% of full scale

©2020 NCEES 276

Chapter 6: Fluid Mechanics

Flow Measurement Devices (cont'd)

Class Meter Type Description Advantages Drawbacks

Pressure (cont'd) Wedge Similar in principle to the orifice meter, a wedge • Well suited for • Differential pres-
sure to flow rate
placed in the flow stream creates the differential pres- sludge, slurry, or dependent on em-
pirical data unique
sure element. The fluid is forced downward, similar high-viscous fluid to each model and
application
to a segmented orifice plate, but is guided along a service
• High permanent
sloping wedge shape rather than a sharp edge. The pressure drop

differential pressure is then converted to a corre-

sponding flow rate.

dP

FLOW

WEDGE

Pitot Tube The pitot tube is primarily used for gas or air service. • Essentially no pres- • Low accuracy (dif-
The Pitot tube measures the total pressure (dynamic sure drop ferential pressure
and static pressures combined). The static tube mea- between static and
sures the static pressure only. The difference between • Easy to install and dynamic is small
the two measurements reveals that the dynamic pres- use and therefore prone
sure is converted into the flow rate. to error)
• Instrument can be
dP removed when not • Accuracy depen-
in service dent on placement
STATIC TUBE within the flow
• Can be used to cross-section
FLOW measure gas veloci-
ties and to establish • Low rangeability
a velocity profile
• Requires clean
PITOT TUBE fluids (tube easily
plugs)

Note: The pitot tube (impact tube) and the static tube
are sometimes provided within a single element.

©2020 NCEES 277

Chapter 6: Fluid Mechanics

Flow Measurement Devices (cont'd)

Class Meter Type Description Advantages Drawbacks

Pressure (cont'd) Annubar The annubar or averaging pitot-tube flow meter • Accurate (1% of • Not suitable for
measures the difference between the total pressure full scale) dirty or viscous
(upstream) and the static pressure (downstream) to fluids
derive the flow rate. • Compact design
(sensing lines not • Element must be
dP ANNUBAR required) centered within the
(IMPACT TUBE) pipe

FLOW

UPSTREAM FLOW DOWNSTREAM
SENSING SENSING
PORTS PORTS

SIMPLIFIED CROSS-SECTION

OF SENSING (IMPACT) TUBE

Cone (or Note: Temperature elements can be made integral • Excellent accuracy • Moderate perma-
V-Cone) with the impact tube to provide temperature compen- (0.5% of full scale) nent pressure drop
sation and corrections.
A cone is inserted in the flow stream to create a • Suitable for fluids • Requires exten-
differential pressure similar to a venturi meter or Dall with suspended sive calibration
tube meter, which is then correlated to the flow rate. solids to achieve rated
accuracy
dP • Compact design
(0–2 pipe dia- • Must operate within
FLOW meters) rated β-ratio range

• Suitable for gas
flow measurement

©2020 NCEES 278

Chapter 6: Fluid Mechanics

Flow Measurement Devices (cont'd)

Class Meter Type Description Advantages Drawbacks
• Thermal properties
Thermal Thermal A known amount of heat is applied to the heating • Used primarily for
Mass element. Some of this heat is lost to the flowing fluid. gas service (stack of the gas must be
Meters As flow increases, more heat is lost. The amount of flow measurement known
heat lost is sensed using temperature elements (com- and emissions
paring the upstream and downstream values). The monitoring) • Moderate accuracy
fluid flow is derived from the known heat input and
the temperature measurements. • Low pressure drop • Not for steam
service
• The tempera-
HEATING ELEMENT DOWNSTREAM ture and heating
T1 = UPSTREAM elements come in
T2 = TEMPERATURE a single element
TEMPERATURE assembly for a
ELEMENT compact design
ELEMENT

FLOW • Detects low flows
(laminar flows)

• Can be used as a
velocity meter

Vortex Vortex Vortices (or eddy currents) created by an obstruction • Results are in true • Not suitable for low
Shedding are detected by ultrasonic or optical transducers. The mass flow flow rates
rate of vortex formation and subsequent shedding
caused by the bluff body or obstruction is propor- • Can be used for • Minimum length
tional to the fluid velocity. liquids, gases, and of straight pipe is
steam required upstream
BLUFF BODY RECEIVING and downstream of
(STRUT) TRANSDUCER • Low wear the meter

FLOW EDDYS (VORTICES) • Low cost to install
and maintain

• Low sensitivity to
variations in pro-
cess conditions

• Stable long-term
accuracy and re-
peatability

TRANSMITTING • Applicable to a
TRANSDUCER wide range of pro-
cess temperatures

• Available for a
wide variety of pipe
sizes

©2020 NCEES 279

Chapter 6: Fluid Mechanics

Flow Measurement Devices (cont'd)

Class Meter Type Description Advantages Drawbacks

Magnetic Mag Meter The operation of a magnetic flow meter or mag meter • Ideal for dirty water • Does not work
is based on Faraday's Law, which states that the volt- or other conductive on nonconductive
age induced across any conductor as it moves at right fluids fluids (e.g., hydro-
angles through a magnetic field is proportional to the carbons)
velocity of that conductor. • Suitable for fluids
with two-phase • Expensive
E\u#B#D flow
• Does not correlate
where • No pressure drop to mass flow until
(models are avail- fluid or bulk slurry
E = voltage generated in a conductor able for full pipe density is known
bores)
u = velocity of the conductor
• Accurate

B = magnetic field strength • Measures true volu-
D = length of the conductor metric flow

Ultrasonic The flow meter applies a magnetic field through the • Sufficiently ac- • Expensive
entire cross-section of the flow tube. The curate for custody
velocity is then determined by the meter by transfer • Sensitive to stray
measuring the magnetic strength. vibrations
• Clamp-on systems
For a simple Doppler system, sound waves are used suitable for field • Unwanted
to determine the velocity of a fluid flowing in a pipe. testing and verifi- attenuation can
At zero flow, the frequencies of an ultrasonic wave cation of installed occur
transmitted into a pipe and its reflections from the flow meters
fluid are the same. At flow, the frequency of the • Fluid must be able
reflected wave is different because of the Doppler to transmit ultra-
effect. As fluid velocity increases, the frequency sonic waves
shift increases linearly. A transmitter evaluates the
frequency shift to determine the flow rate.

For a Transit time system, ultrasonic waves are sent
and received between transducers in both direc-
tions in the pipe. At zero flow, it takes the same time
to travel upstream and downstream between the
transducers. At flow, the upstream wave travels more
slowly and takes more time than the downstream
wave. As fluid velocity increases, the difference
between the upstream and downstream times also
increases. A transmitter evaluates the delay times to
determine the flow rate.

Note: Either method can be deployed as a clamp-on
unit (dry) or be installed integral to the fluid (wet).

©2020 NCEES 280

Chapter 6: Fluid Mechanics

Flow Measurement Devices (cont'd)

Class Meter Type Description Advantages Drawbacks

Impulse Coriolis A Coriolis flow meter uses the natural phenomenon • Suitable for highly • Not accurate for
viscous fluids gases at low flow
in which an object begins to "drift" as it travels from rates
• Insensitive to tem-
or toward the center of a rotation occurring in the perature and fluid • High permanent
properties pressure drop
surrounding environment. Coriolis flow meters gen-
• Measures mass
erate this effect by diverting the fluid flow through flow rate directly

a pair of parallel U-tubes with an induced vibration

(by an actuator, not shown) perpendicular to the flow.

The vibration simulates a rotation of the pipe and

the resulting Coriolis "drift" in the fluid causes the

U-tubes to twist and deviate from their parallel align-

ment. The force producing this deviation is propor-

tional to the mass flow rate through the U-tubes.

VIBRATION VIBRATION

FLOW

NO DEFLECTION DEFLECTION

6.4.3 Orifice, Nozzle, and Venturi Meters

6.4.3.1 Square-Edge Orifice Meter (Vena Contracta Taps)

d2 d2 0.66
2

d2 d1 0.64 DISCHARGE COEFFICIENT
0.62 Corifice FOR SQUARE-EDGE
Corifice ORIFICE METERS

FLOW 0.60 β = d—1— = 0.7
SQUARE-EDGE ORIFICE METER 0.58 d2
0.6
104 0.5
0.4
0.2

105 106 107 108
Re

©2020 NCEES 281