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

Chapter 4: Heat Transfer

Typical Overall Heat-Transfer Coefficients for Air Coolers

System Btu W
hr-ft 2-cF m2 : K
30–300
Finned air cooler/condensing steam 5–50 25–60
Finned air cooler/water 4–10 300–450
Air cooler (fin-fan)/water 50–80 300–700
Air cooler (fin-fan)/light organics 50–125 70–150
Air cooler (fin-fan)/heavy organics 12–25 300–600
Air cooler (fin-fan)/condensing hydrocarbons 50–100
Air cooler (fin-fan)/condensing ammonia 110 650
Air cooler (fin-fan)/condensing Freon 70 400
Air cooler (fin-fan)/gas <5–10 bar/60–130 psig 10–20 60–120
Air cooler (fin-fan)/gas >10–30 bar/130–420 psig 20–50 100–300

Typical Overall Heat-Transfer Coefficients in Exchangers Without Phase
Change ​(Shell-and-Tube Exchangers)

System Btu W
hr-ft 2-cF m2 : K

Gas/gas 2–10 10–50

Water or brine/compressed gas 10–30 60–200

Water/hydrogen with natural gas 80–125 450–700

Water/brine 100–200 600–1200

Water/water 150–300 850–1700

Water/alcohol, organic solvents 50–150 280–850

Water/gasoline 60–90 340–510

Water/gas oil, distillate 35–60 200–340

Water/heavy oil 10–50 60–300

Freon or ammonia/water 40–90 220–510

Light organics/light organics 40–75 220–425

Medium organics/medium organics 20–60 110–340

Heavy organics/heavy organics 10–40 57–220

Heavy organics/light organics 10–60 57–340

Crude oil/gas oil 30–55 170–310

©2020 NCEES 189

Chapter 4: Heat Transfer

Typical Overall Heat-Transfer Coefficients in Water-Cooled Condensers ​
(Shell-and-Tube Exchangers)

Condensing Fluid Btu W
hr-ft 2-cF m2 : K

Alcohol vapors 45–125 250–700
Ammonia vapors 150–250 850–1400
Freon vapors 45–150 250–850
Aqueous vapors 200–1000 1100–5600
Condensing oil 40–100 220–570
Organic vapors 125–175 700–1000
Organic vapors with noncondensables 90–125 500–700
Vacuum condensers 35–90 200–500

Typical Overall Heat-Transfer Coefficients in Heaters With
Condensing Steam ​(Shell-and-Tube Exchangers)

Heated Fluid Btu W
hr-ft2-cF m2 : K

Gas 5–50 30–300

Heavy oil 10–50 60–300

Light oil 35–100 200–600

Kerosene/gasoline 50–200 300–1100

Organic Solvents 90–175 500–1000

Water 250–700 1500–4000

Typical Overall Heat-Transfer Coefficients for Immersed Heating Coils

Immersed Coils Btu W
hr-ft 2-cF m2 : K

Pool Liquid Heating Natural Agitated Natural Agitated
Medium convection convection

Dilute aq. solution Steam 100–200 130–275 500–1000 700–1600

Light oil Steam 35–50 50–100 200–300 300–600

Heavy oil Steam 15–30 50–70 90–170 300–400

Molten sulfur Steam 20–35 35–45 100–200 200–250

Molasses/corn syrup Steam 15–30 60–80 70–170 350–450

Aqueous solution Water 70–100 110–160 400–600 400–650

Light oil Water 20–25 35–50 100–150 200–300

©2020 NCEES 190

Chapter 4: Heat Transfer

Typical Overall Heat-Transfer Coefficients for Plate Exchangers

Plate Exchangers Btu W
hr-ft 2-cF m2 : K
Hot Fluid Cold Fluid

Light organic Light organic 450–900 2500–5000

Light organic Viscous organic 45–90 250–500

Viscous organic Viscous organic 20–35 100–200

Light organic Process water 450–600 2500–3500

Viscous organic Process water 45–90 250–500

Light organic Cooling water 350–800 2000–4500

Viscous organic Cooling water 45–80 250–450

Condensing steam Light organic 450–600 2500–3500

Condensing steam Viscous organic 45–90 250–500

Process water Process water 900–1300 5000–7500

Process water Cooling water 90–1200 500–7000

Dilute aqueous Cooling water 900–1200 5000–7000
solutions

Condensing steam Process water 600–800 3500–4500

Typical Overall Heat-Transfer Coefficients in Evaporators

System Btu W
hr-ft 2-cF m2 : K

Agitated film

Newtonian liquid, m = 1 cP 400 2000

Newtonian liquid, m = 100 cP 300 1500

Newtonian liquid, m = 10,000 cP 120 700

Vertical long tube

Natural circulation 200–600 1000–3500

Forced circulation 400–1000 2000–6000

©2020 NCEES 191

Chapter 4: Heat Transfer

4.4.1.5 Representative Values for Fouling Factors

Representative Values for Fouling Factors
Values for # 125cF/50cC, unless specified otherwise

Material hr - ft2 -cF m2 : K
Btu W

Water 0.0005 0.00009
Seawater, brine, salt water 0.0010 0.00018
Seawater, brine, salt water (> 125°F/50°C) 0.0020 0.00035
River water (brackish) 0.0030 0.00053
River water (muddy, silty) 0.0033 0.00059
Hard water 0.0010 0.00018
City/well water 0.0010 0.00018
Untreated boiler feedwater (> 125°F/50°C) 0.0010 0.00018
Treated boiler feedwater 0.0020 0.00035
Untreated cooling tower water 0.0010 0.00018
Treated cooling tower water 0.0005 0.00009
Distilled water
Hydrocarbons 0.0050 0.00088
Fuel oil 0.0100 0.00176
Asphalt and residue 0.0030 0.00054
Vegetable oil and heavy gas oil 0.0010 0.00018
Light hydrocarbons 0.0040 0.00072
Heavy hydrocarbons
Other 0.0040 0.00070
Quenching liquids 0.0010 0.00018
Refrigerating liquids, brines 0.0010 0.00018
Heat-transfer media 0.0050 0.00090
Polymer forming liquids 0.0020 0.00035
Vaporizing liquids (organic and inorganic) 0.0010 0.00018
Condensing organic liquids 0.0010 0.00018
Organic vapors and liquids (including condensing)
Gases and Vapors 0.0005 0.00009
Steam (clean) 0.0010 0.00018
Steam (oil-bearing) 0.0005 0.00009
Alcohol vapors 0.0020 0.00035
Industrial air or other dirty (oil-bearing) gases 0.0100 0.00176
Diesel exhaust (> 125°F/50°C)

©2020 NCEES 192

Chapter 4: Heat Transfer

4.4.1.6 Nucleate Boiling Heat-Transfer Data

Relative Magnitude of Nucleate Boiling Heat-Transfer
Coefficients at 1 atm, ​Referenced to Value for Water

Fluid h
hwater
Water
Water with 20% sugar 1.0
Water with 10% Na2SO4 0.87
Water with 26% glycerin 0.94
Water with 55% glycerin 0.83
Water with 24% NaCl 0.75
Isopropanol 0.61
Methanol 0.70
Toluene 0.53
Carbon-tetrachloride 0.36
n-Butanol 0.35
0.32

Source: Republished with permission of McGraw-Hill, from Heat Transfer, J.P. Holman, 5th ed.,
New York, 1981; permission conveyed through Copyright Clearance Center, Inc.

Maximum Heat Flux in Nucleate Boiling (Burnout Heat Flux)

Heat Flux DT Heat Flux DT
°F kW °C
Fluid Surface Btu # 10-3 m2
hr -ft2 23–28
Water 30
Copper 200–270 620–850 —
Benzene —
Propanol Chrome-plated copper 300–400 42–50 940–1260
Butanol 42–50
Ethanol Steel 410 54 1290 33–39

Methanol Copper 43.5 — 130 —
—
Liquid H2 Aluminum 50.5 — 160 —
Liquid N2 —
Liquid O2 Nickel-plated copper 67–110 76–90 210–340 —
2
Nickel-plated copper 79–105 60–70 250–330 11
11
Aluminum 55 — 170

Copper 80.5 — 250

Copper 125 — 390

Chrome-plated copper 111 — 350

Steel 125 — 390

Any metal surface 9.53 4 30

Any metal surface 31.7 20 100

Any metal surface 47.5 20 150

Source: Republished with permission of McGraw-Hill, from Heat Transfer, J.P. Holman, 5th ed., New York,
1981; permission conveyed through Copyright Clearance Center, Inc.

©2020 NCEES 193

Chapter 4: Heat Transfer

4.4.1.7 Solar Radiation Data

Maximum Expected Solar Radiation at Various North Latitudes

Month 30° North Btu 45° North 30° North W 45° North
hr-ft 2 m2
January 40° North 40° North
February
March 24-hr noon 24-hr noon 24-hr noon 24-hr noon 24-hr noon 24-hr noon
April avg. avg. avg. avg. avg. avg.
May
June 65 240 40 170 30 135 205 757 126 536 95 426
July
August 75 270 55 210 45 175 237 852 174 662 142 552
September
October 90 305 75 255 65 230 284 962 237 804 205 726
November
December 110 340 95 300 90 280 347 1073 300 946 284 883

120 360 120 335 115 320 379 1136 379 1057 363 1009

130 365 130 345 130 335 410 1151 410 1088 410 1057

130 365 130 350 130 340 410 1151 410 1104 410 1073

125 360 125 340 120 325 394 1136 394 1073 379 1025

115 350 105 315 100 300 363 1104 331 994 315 946

100 315 80 270 75 245 315 994 252 852 237 773

80 270 60 215 50 185 252 852 189 678 158 584

65 240 45 175 35 140 205 757 142 552 110 442

Source: Langhaar, J.W., "Cooling Pond May Answer Your Water Cooling Problem," Chem.Eng. 60(8), 1953, pages 194–198.

4.4.1.8 Emissivity (f)

Emissivity of Building Materials at Ambient Temperature
(Unless Specified Otherwise)

Material Emissivity

Asbestos 0.96

Brick (building) 0.93

Brick (fireclay) at 2000°F/1100°C 0.75

Enamel (white) 0.90

Glass (smooth) 0.94

Gypsum 0.90

Marble 0.93

Oak 0.90

Oil 0.82

Plaster 0.91

Refractory (good radiator) at 1500°F/800°C 0.85

Refractory (poor radiator) at 1500°F/800°C 0.70

Roofing paper 0.91

Rubber (grey, soft) 0.86

Rubber (hard) 0.95

Water 0.96

©2020 NCEES 194

Chapter 4: Heat Transfer

Emissivity of Metals at Ambient and Elevated Temperatures

Material Emissivity at Ambient Emissivity at
Temperatures ~1000°F/540°C

Aluminum, polished 0.04 0.08
Aluminum, anodized 0.94 0.60
Aluminum, surface roofing 0.22 ­—
Brass, polished 0.10 —
Brass, oxidized 0.61 —
Chromium, polished 0.08 0.26
Copper, polished 0.02 0.18
Copper, oxidized 0.78 0.77
Gold, polished 0.02 0.04
Iron, polished 0.06 0.13
Iron, cast, oxidized 0.63 0.76
Iron, galvanized 0.25 0.6
Iron, oxide 0.90 0.85
Magnesium 0.07 0.18
Stainless steel, polished 0.15 0.22
Stainless steel, weathered 0.85 0.85
Tungsten 0.03 0.10
Zinc, polished 0.05 0.04
Zinc, galvanized 0.25 —

©2020 NCEES 195

Chapter 4: Heat Transfer

4.4.2 Charts with Heat-Transfer Data Btu
hr-ft 2-cF
Overall Heat-Transfer Coefficients for Various Applications (U.S. Units):

Btu °F hr CONDENSATION 600
f2t AQUEOUS VAPOURS 500
PROCESS FLUID COEFFICIENT,
BOILING AQUEOUS

400 ov, fBt2 tu°F hr 400
350
COEFFICIENT, U
300
DILUTE AQUEOUS 300 OVERALL
BOILING ORGANICS 150 250

CONDENSATION ORGANIC VAPORS 300 ESTIMATED

PARAFFINS 200
HEAVY ORGANICS
100
MOLTEN SALTS 200 200

OILS 100
AIR AND GAS 15000
HIGH PRESSURE

RESIDUE

400 500 600 700 800 900

AIR AND GAS THERMAL FLUID
LOW PRESSURE
BRINES CONDENSATE
AIR BOILING STEAM CONDENSING
AND GAS RIVER, WELL, HOT HEAT WATER
SEAWATER TRANSFER OIL
REFRIGERANTS

COOLING TOWER WATER SERVICE FLUID COEFFICIENT, Btu
ft2 °F hr

Source: Reprinted from Chemical Engineering Design, 2nd ed., Gavin Towler and Ray Sinnott, Chapter 19:
Heat Transfer Equipment, © 2013, with permission from Elsevier.

Overall Heat-Transfer Coefficients for Various Applications (SI Units): W
m2 : K

PROCESS FLUID COEFFICIENT, W/m 2°K CONDENSATION
AQUEOUS VAPOURS

BOILING AQUEOUS 2500

2000 OVERALL COEFFICIE1N7T5,0U ov, W/m2 °K 2250
1500 2000 4000
DILUTE AQUEOUS
BOILING ORGANICS

CONDENSATION ORGANIC VAPORS 1500 ESTIMATED
750
PARAFFINS 1250
HEAVY ORGANICS 1500

MOLTEN SALTS 1000 1000

OILS 500 500
AIR AND GAS 250 1000
HIGH PRESSURE

RESIDUE

2000 2500 3000 3500 4500

AIR AND GAS 500 THERMAL FLUID
LOW PRESSURE BRINES
BOILING CONDENSATE STEAM CONDENSING
AIR RIVER, WELL, WATER
AND GAS SEAWATER HOT HEAT

TRANSFER OIL REFRIGERANTS

©2020 NCEES COOLING TOWER WATER SERVICE FLUID COEFFICIENT, W/m2°K

Source: Reprinted from Chemical Engineering Design, 2nd ed., Gavin Towler and Ray Sinnott, Chapter 19:
Heat Transfer Equipment, © 2013, with permission from Elsevier.

196

Chapter 4: Heat Transfer

4.4.3 Heat-Exchanger Design Information

TEMA Heat Exchanger Types

FRONT-END SHELL TYPES REAR-END
STATIONARY HEAD TYPES HEAD TYPES

E L
A ONE-PASS SHELL
FIXED TUBE SHEET
CHANNEL F LIKE "A" STATIONARY HEAD
AND REMOVABLE COVER
PASS SHELL M
B WITH LONGITUDINAL BAFFLE
FIXED TUBE SHEET
BONNET (INTEGRAL COVER) G LIKE "B" STATIONARY HEAD

SPLIT FLOW N

C REMOVABLE H DOUBLE SPLIT FLOW FIXED TUBE SHEET
TUBE J DIVIDED FLOW LIKE ''N" STATIONARY HEAD
BUNDLE
ONLY KETTLE-TYPE REBOILER P

CHANNEL INTEGRAL WITH TUBE OUTSIDE PACKED FLOATING HEAD
SHEET AND REMOVABLE COVER
S
N K
FLOATING HEAD
CHANNEL INTEGRAL WITH TUBE WITH BACKING DEVICE
SHEET AND REMOVABLE COVER
T
DX CROSS FLOW
PULL-THROUGH FLOATING HEAD
SPECIAL HIGH-PRESSURE CLOSURE
U

U-TUBE BUNDLE

W

EXTERNALLY SEALED
FLOATING TUBE SHEET

©2020 NCEES 197

1.2 0.2 0.3 0.4 0.5 1 SHELL PASS t1
1.4 P = TEMPERAT t1
1.5
1.6 t2 –
2.0 T1 –
2.5
3.0 P=

4.0 t2
t1
6.0 T2
8.0
R = 10.0 0.5
15.0 0 0.1
20.0 T1

F = MTD CORRECTION FACTOR

198
1.0 0.6
0.9
0.8
0.7

©2020 NCEES

4.4.4 F-Factor Charts Chapter 4: Heat Transfer

0.1 0.6 0.7 0.8 0.9 1.0 2 OR MORE TUBE PASSES • Q/(Uov A)
TURE EFFICIENCY Δ Tlm
0.2
0.3 MTD CORRECTION FACTOR =
0.4
0.5 F
0.6
0.7 T2
0.8 t1
0.9
1.0 T1 –
t2 –

R=

S

2.0 0.4 2 SHEL t1
P = TEM t1
2.6
3.0 t2 –
T1 –
4.0
P=
6.0
8.0 0.2 0.3
R = 10.0 T1
15.0
20.0 t2
t1
F = MTD CORRECTION FACTOR T2

199 0.1

1.0 0.6 0.5
0.9 0
0.8
0.7

©2020 NCEES

0.1
0.2

Chapter 4: Heat Transfer

0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0

1.2

1.4
1.6
1.8

0.5 0.6 0.7 0.8 0.9 1.0
MPERATURE EFFICIENCY

MTD CORRECTION FACTOR

LL PASS 4 OR MORE TUBE PASSES

•

R= T1 – T2 F = Q/(Uov A)
t2 – t1 Δ Tlm

0.4 3 SHELL P t1
P = TEMP t1

2.5 t2 –
3.0 T1 –

4.0 P=

5.0 0.3 t1
8.0 t2
R = 10.0
15.0 0.2
20.0 T1

F = MTD CORRECTION FACTOR 0.1
T2
200
1.0 0.6 0.5
0.9 0
0.8
0.7

©2020 NCEES

0.2
0.4

0.6 Chapter 4: Heat Transfer
0.8
1.0

1.2
1.4
1.6
1.8
2.0

0.5 0.6 0.7 0.8 0.9 1.0
PERATURE EFFICIENCY

MTD CORRECTION FACTOR

PASSES 6 OR MORE TUBE PASSES

•

R= T1 – T2 F = Q/(Uov A)
t2 – t1 Δ Tlm

0.4 4 SHELL P t1
P = TEMP t1

2.5 t2 –
3.0 T1 –

4.0 P=

6.0 0.3
8.0 t2
R = 10.0 t1
15.0
20.0 0.2
T1
F = MTD CORRECTION FACTOR T2

201 0.5
1.0 0.6 0 0.1
0.9 4
0.8 SHELLS
0.7

©2020 NCEES

0.2
0.4

Chapter 4: Heat Transfer

0.6
0.8
1.0
1.2
1.4
1.6
1.8
2.0

0.5 0.6 0.7 0.8 0.9 1.0
PERATURE EFFICIENCY

MTD CORRECTION FACTOR

PASSES 8 OR MORE TUBE PASSES

•

R= T1 – T2 F = Q/(Uov A)
t2 – t1 Δ Tlm

2.5 0.4 5 SHELL P t1
3.0 P = TEMP t1

4.0 t2 –
T1 –
6.0
8.0 P=
R = 10.0
15.0 0.3
20.0 t2
t1
F = MTD CORRECTION FACTOR
0.2
202 T1

1.0 0.6 0.5
0.9 0 0.1
0.8 5
0.7 SHELLS
T2

©2020 NCEES

0.2
0.4

Chapter 4: Heat Transfer

0.6
0.8
1.0
1.2
1.4
1.6
1.8
2.0

0.5 0.6 0.7 0.8 0.9 1.0
PERATURE EFFICIENCY

MTD CORRECTION FACTOR

PASSES 10 OR MORE TUBE PASSES

•

R= T1 – T2 F = Q/(Uov A)
t2 – t1 Δ Tlm

2.5 0.4 6 SHEL t1
3.0 P = TEM t1

4.0 t2 –
T1 –
6.0
8.0 P=
10.0
15.0 0.3
R = 20.0 t2
t1
F = MTD CORRECTION FACTOR
0.2
203 T1
T2

1.0 0.6 0.5
0.9 0 0.1
0.8 6
0.7 SHELLS

©2020 NCEES

0.2
0.4

Chapter 4: Heat Transfer

0.6
0.8
1.0
1.2

1.4
1.6
1.8
2.0

0.5 0.6 0.7 0.8 0.9 1.0
MPERATURE EFFICIENCY

MTD CORRECTION FACTOR

LL PASSES 12 OR MORE TUBE PASSES

•

R= T1 – T2 F = Q/(Uov A)
t2 – t1 Δ Tlm

1.2 0.4 1 DIVIDED FLOW t1
1.4 P = TEMP t1
1.6
1.8 t2 –
2.0 T1 –
2.5
3.0 P=
5.0
0.3
6.0
8.0 t2
R = 10.0 t1
15.0
20.0 0.1 0.2 T1 T1 T2

F = MTD CORRECTION FACTOR 0

1.0 2040.90.80.7

©2020 NCEES

Chapter 4: Heat Transfer

0.1

0.2

0.3

•

0.4
0.5
0.6
0.7
0.8
0.9
1.0
0.5 0.6 0.7 0.8 0.9 1.0
PERATURE EFFICIENCY

MTD CORRECTION FACTOR

W SHELL PASS 2 OR MORE TUBE PASSES

R= T1 – T2 F = Q/(Uov A)
t2 – t1 Δ Tlm

5 CHEMICAL REACTION ENGINEERING

5.1 Symbols and Definitions

Symbol Symbols Units (U.S.) Units (SI)
CA or [A] Description
lb mole mol
Concentration of component A ft 3 liter
Btu
C p Average heat capacity J
lbm - cF kg : K

FA Molar feed of A lb mole mol
sec s

DVg r Gibbs free energy of reaction (molar) Btu J
Dhtr Heat of reaction lb mole mol
K Equilibrium constant
Btu J
lb mole mol

varies varies

lb mole ^1 - nh mol ^1 - nh
ft 3 liter
k Reaction rate constant d n c m

sec s

M Molar ratio of initial reactant concentrations dimensionless
(weighted by the stoichiometric constants)

m Mass of reactor contents lbm kg

mo Mass flow rate of feed lbm kg
sec s
n Moles of reactant or product lb mole
n Reaction order g mol

dimensionless

P Pressure (PA = partial pressure of A) lbf Pa
in2

©2020 NCEES 205

Chapter 5: Chemical Reaction Engineering

Symbol Symbols (con't) Units (U.S.) Units (SI/metric)
q Description Btu J
sec s
rA Heat transfer
SAB
SV Rate of reaction – based on component A lb mole g mol
T ft3-sec L:s
To
Selectivity to A relative to B dimensionless
t
Space velocity = 1 1 1
θA space time sec s
V °F or °R °C or K
XA Temperature
YA
eA Feed temperature °F or °R °C or K
x
Time sec s

Fraction of surface covered by adsorbed spe- dimensionless
cies A
ft3 L
Reactor volume

Fractional conversion of component A dimensionless
dimensionless
Yield of A relative to reactant use

Fractional volume change at full conversion dimensionless
of A

Space time = space 1 sec s
velocity

5.2 Fundamentals

5.2.1 Reaction Rate

5.2.1.1 Rate Constant
A chemical reaction may be expressed by the general equation:

aA + bB * cC + dD

The rate of reaction of any component is defined as the number of moles of that component formed per unit time per unit

volume:

− rA = 1 dnA (negative because A is consumed)
V dt

− rA = − dCA if V is constant
dt

The rate of reaction is frequently expressed as

− rA = k f _CA, CB, ...i

The fractional conversion XA is defined as the moles of A reacted per mole of A fed:

XA = CAo − CA = 1 − CA if V is constant
CAo CAo

©2020 NCEES 206

Chapter 5: Chemical Reaction Engineering

5.2.1.2 Order of Reaction

If − rA = k C x C y , then the reaction is x order with respect to A and y order with respect to B.
A B

The overall order is n = x + y.

5.2.1.3 Temperature Dependence (Arrhenius Equations)

The Arrhenius equation gives the dependence of k on temperature:

k = Ae −Ea
RT

where

A = pre-exponential or frequency factor

Ea = activation energy c J or cal m
mol mol

R = universal gas constant

For values of rate constant ki at two temperatures Ti:

Ea = R T1 T2 ln e k1 o or ln e k1 o = Ea T1 − T2
_T1 − T2i k2 k2 R T1 T2

5.2.1.4 Half-Life

The half-life of a reaction, t1 , is the batch time required to reach 50% conversion.
2
d CA
For − rA =− dt = kCAn t1 occurs when CA = 1 CAo
2 2

For n = 1 (first order) t1 = ln 2
For n Y= 1 2 k

t1 = (n 2n−1 − 1
2 − 1) k CAo(n − 1)

5.2.2 Rate Equations in Differential Form for Irreversible Reactions

5.2.2.1 Zero-Order (A " R)

− rA = − d CA = CAo d XA = k and d XA = k
dt dt dt CAo

5.2.2.2 First-Order ^A " Rh

− rA = − d CA = CAo d XA = k CA and d XA = k CA = k_1 − XAi
dt dt dt CAo

5.2.2.3 Second-Order ^2A " Rh, One Reactant

− rA = − d CA = CAo d XA = k CA2 and d XA = k CA2 = k CAo _1 − XAi2
dt dt dt CAo

©2020 NCEES 207

Chapter 5: Chemical Reaction Engineering

5.2.2.4 Second-Order _A + bB " Ri, Two Reactants

− rA = − d CA = k CACB = k b CAo2 _1 − XAi_M − XAi when M = CBo ! 1
dt b CAo

and

− rA = k b CAo2 _1 − XAi2 when M = 1

Integrated forms of these equations for constant- and variable-volume batch, plug flow, and CSTR reactors are included in
Integrated Reaction Equations for Irreversible Reactions section in this chapter.

5.2.3 Yield and Selectivity

Yield Y is defined as the molar ratio of the desired product formed to the reactant that is consumed.

Selectivity S is defined as the molar ratio of the desired product to undesired product.

5.2.3.1 Two Irreversible Reactions in Parallel

A "kD D (desired) and A "kU U (undesired)

− rA = − d CA = kD CA x + kU CA y
dt

=rD d=dCtD kD CA x

=rU d=dCtU kU CA y d CD
−d CA
YD = instantaneous fractional yield of D =

YD = overall fractional yield of D = ND
NAo − NA

where NA and ND are the final values measured at the reactor outlet
where
S DU o=verall selectivity to D over U ND
NU

ND and NU are the final values measured at the reactor outlet

5.2.3.2 Two First-Order Irreversible Reactions in Series

A "kD D "kU U (D = desired, U = undesired)

− rA = − d CA = kD CA
dt

rD = d CD = kD CA − kU CD
dt

=rU d=dCtU kU CD

The yield and selectivity definitions for series reactions are identical to those for parallel reactions, and the equations for overall
yield and selectivity are the same as those in the previous subsection.

©2020 NCEES 208

Chapter 5: Chemical Reaction Engineering

The maximum concentration of D in a plug flow reactor is

CD, max kU ln e kU o
CAo kD k − k kD
= kU U D x max = 1 =
e o at time klog mean
`kU − kDj

The maximum concentration of D in a CSTR is

CD,max = 1 2 at time x max = 1
CAo kD kU
kU 1 1H
>e kD 2 +

o

5.2.4 Pressure Dependence (Gas Phase Reactions)

All of the equations in the previous sections of this chapter can be written in terms of pressure where

CA = PA
RT

5.3 Reactor Equations

5.3.1 Types of Reactors

For flow reactors, space time t is defined as the reactor volume divided by the inlet volumetric feed rate. Space velocity SV is the
reciprocal of space time, that is, SV = 1/t.

5.3.1.1 Batch Reactor

Constant Volume

For a well-mixed, constant-volume batch reactor:

#− d CA = d XA XA d XA
rA = − dt CAo dt and t = CAo 0 − rA

Variable Volume

For a well-mixed, variable-volume batch reactor:

#−
rA = _1 CAo d XA and t = CAo XA d XA
+ fA XAi dt 0 _− rAi_1 + fA XAi

where eA = fractional volume change at full conversion of A

CA = C Ao e 1 1− XA o or XA = 1 − CA CAo
+f AX 1 + fAXA
A

5.3.1.2 Plug Flow Reactor

For a plug flow reactor, for all values of fA :

#x =
C Ao V = CAo XA d XA
FAo 0 _− rAi

where FAo = moles of A fed per unit time

For a constant volume plug flow reactor (fA = 0):

#x = − CA d CA
C Ao − rA

©2020 NCEES 209

Chapter 5: Chemical Reaction Engineering

5.3.1.3 Continuous Stirred Tank Reactor (CSTR)

For a well-mixed CSTR for all values of eA:

x= C Ao V = CAo XA
FAo _− rAi

where - rA is evaluated at exit stream conditions

For a constant volume CSTR (fA = 0):
x = CAo − CA
_− rA i

5.3.1.4 Equal-Sized Reactors in Series
With a first-order reaction A " R , with no change in volume:

x N−reactors = N x individual

N 1 CAo k x N
k CAo N CA N N
x N - reactors = >e CA N − 1H or = d1 + N n
o

where

N = number of CSTRs (equal volume) in series

CA N = concentration of A leaving the Nth CSTR
N plug flow reactors in series gives the same conversion as a single plug flow reactor with the same total volume.

5.3.1.5 Equal-Sized Reactors in Parallel

N identical reactors in parallel give the same conversion as a single reactor of the same total volume. (This is equally true for both
plug flow reactors and CSTRs.)

5.3.1.6 Plug Flow Reactors With Recycle

First Order (eA = 0)

kx = ln CAo + R CA
R+1 (R + 1)CA

Second Order (eA = 0)

k CAo x = CAo `CAo − CAj
R+1 CA `CAo + R CAj

where R = recycle ratio, defined as the fraction of the reactor outlet stream that is recycled

Relationship Between Overall Conversion and Single-Pass Conversion

XAs = 1 + XAo XAoj
R`1 −

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Chapter 5: Chemical Reaction Engineering

5.3.2 Integrated Reactor Equations for Irreversible Reactions
5.3.2.1 Zero-Order Reactions _A " R, − rA = ki

Constant Volume
Batch reactor:

k t = CAq XA = CAq − CA
Plug flow reactor or CSTR:

k x = CAq XA = CAq − CA

Variable Volume DV = Vq f A XA
V = Vq _1 + f A XAi ,

Batch reactor:

k t = CfAAo ln _1 + fA XAi = CfAAo ln V
Vo

Plug flow reactor or CSTR:

k x = CAq XA

5.3.2.2 First-Order Reactions _A " R, − rA = k CAi

Constant Volume

Batch reactor:

kt = ln CAo = ln 1 1 = − ln_1 − XAi
CA − XA

Plug flow reactor:

k x = ln CAo = ln 1 1 =− ln _1 − XAi
CA − XA

CSTR:

k x = CAo − CA = 1 XA
CA − XA

Variable Volume
V = Vq _1 + f A XAi, DV = Vq f A XA

Batch reactor:

kt = ln 1 1 = − ln_1 − XAi = − lnd1 − DV n
− XA f AVo

Plug flow reactor:

k x = − _1 + fAiln_1 − XAi − fA XA

CSTR:

kx = XA _1 + f A XAi
1 − XA

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Chapter 5: Chemical Reaction Engineering

5.3.2.3 Second-Order Reactions `2 A " R, − rA = k C 2 j, One Reactant
A

Constant Volume

Batch reactor:

k t = 1 − 1 = XA XAi or CA = 1 + 1
CA CAo CAo _1 − CAo k t CAo

Plug flow reactor:

k x = 1 − 1 = XA XAi or CA = 1 + 1
CA CAo CAo _1 − CAo k x CAo

CSTR:

k x = CAo − CA = XA
CAo _1 − XAi2
C 2
A

Variable Volume

V = Vo (1 + eA XA), DV = Vo eA XA

Batch reactor:

k t = 1 >_1 + fAi XA + fA ln _1 − XAiH
CAo 1 − XA

Plug flow reactor:

k x = 1 =2f A _1 + f A i ln _1 − XAi + fA2 XA + _f A + 1i2 1 XA G
CAo − XA

CSTR:

kx = XA _1 + f A XAi2
CAo _1 − XAi2

5.3.2.4 Second-Order Reactions _A + bB " R, − rA = k CA CBi, Two Reactants

Constant Volume

Batch reactor:

k t b CAo ^M − 1h = ln CB = ln M − XA when M = CBo ! 1
M CA M_1 − XAi b CAo

k t CBo = k t b CAo = CAo − CA = 1 XA when M = 1
CA − XA

Plug flow reactor:

k t b CAo ^M − 1h = ln CB = ln M − XA when M = CBo ! 1
MCA M_1 − XAi b CAo

k x CBo = k x b CAo = CAo − CA = 1 XA when M = 1
CA − XA

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Chapter 5: Chemical Reaction Engineering

CSTR:

kx = CAo − CA CAB = bCAo _1 − XA − XAi when M = CBo ! 1
bCA 8CAo ^M − 1h+ XAi_M b CAo

k x = CAo − CA = XA when M = 1
b CAo _1 −
b C 2 XAi2
A

5.3.3 Complex Reactions

5.3.3.1 First-Order Reversible Reactions (A k1 R)
k2

− rA = − d CA = k1 CA − k2 CR
dt

K=C kk=12 C R eq and M = C Ro
CA eq CAo

d XA = k1 ^ M + 1h a X A eq − XAk
dt M + XA eq

− lnf1 − XA p = − ln CA − CA eq = ^M + 1h k1 t
XA eq CAo − CA eq
a M + XA k

eq

At equilibrium, when XA = XAeq , then -ln(0)"∞ and t"∞.

5.3.3.2 Reactions of Shifting Order
From zero order at high CA to first order at low CA:

− rA = 1 k1 CA
+ k2 CA

ln e CAo o + k2 `CAo − CAj = k1 t
CA

lne CAo o k1 t
CA CAo − CA
= − k2 +
CAo − CA

where

k1 = zero-order rate constant
k2

k1 = first-order rate constant

This form of the rate equation is used for elementary enzyme-catalyzed reactions and for elementary surface-catalyzed
reactions in batch reactor. For plug flow reactor replace time, t, with space time, τ. The equation assumes a constant
density system.

©2020 NCEES 213

Chapter 5: Chemical Reaction Engineering

5.4 Catalytic Reactors

Source: Missen, Ronald W., Charles A. Mims, and Bradley A. Saville, Introduction to Chemical
Reaction Engineering and Kinetics, New York: John Wiley & Sons, Inc., 1999, pp. 191–192.

5.4.1 Key Assumptions

• Catalyst surface contains a fixed number of sites.

• All the catalytic sites are identical.

• Reactivities of the sites depend only on temperature. They do not depend on the nature or amounts of other materials present
on the surface during the reaction.

5.4.2 Surface Reaction Steps A:s " B:s

1. Unimolecular surface reaction:

A • S is a surface-bound species involving A and site S.

Rate is given by: _−rAi = kiA
2. Bimolecular surface reaction: A:s+B:s " C:s+s

Rate: _− rAi = kiA iB
3. Eley-Rideal reaction: A:s+B " C+s

B is a gas-phase species that reacts directly with an adsorbed intermediate.

Rate: _− rAi = kiACB

CB is the gas-phase concentration of B.

5.4.3 Enzyme Kinetics: Michaelis-Menten

Source: Missen, Ronald W., Charles A. Mims, and Bradley A. Saville, Introduction to Chemical
Reaction Engineering and Kinetics, New York: John Wiley & Sons, Inc., 1999.

k1 fast
slow
S + E E ES

k−1
kr

ES " P + E

where

S = substrate

E = enzyme

ES = enzyme-substrate complex

P = product

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Chapter 5: Chemical Reaction Engineering

5.4.3.1 Michaelis-Menten Model

Material balance on total enzyme: CE + CES = CE0
Concentration of complex:
CES = k1CsCE = k1CsCE0 = CsCE0
k−1 k1Cs + k−1 k −1
k1 + Cs

Define Michaelis constant: KM = k−1
Rate of production of product P: k1
Initial rate:
rp = krCES = krCE0CS
KM + CS

rP0 = `− rS0j = krCE0CS0
KM + CS0

Limiting Cases CS0 % KM rP0 = `− rS0j = krCE0CS0
Low CS0 KM

High CS0 CS0 & KM rP0, max = krCE0 maximum rate

Intermed=iate CS0 K=M rP0 1 krCE0 = 1 rP0, max
2 2

Michaelis-Menten Equation

Standard form: rP = rP0, maxCS
KM + CS

Initial rate: rP0 = rP0, maxCS0
KM + CS0

5.4.3.2 Estimation of KM and Vmax

Linearized Form 1 = 1 + rPK0, Mmax=C1S0 Intercept r=P01, max Slope KM
Lineweaver-Burk Plot rP0 rP0, max rP0, max

Linearized Form of Integrated Michaelis-Menten Equation

Constant-volume batch reactor:

lne CS o 1 rP0, max t
CS0 KM KM −
= −
CS0 − CS CS0 CS

5.4.3.3 Single-Substrate Inhibition

k1

E + S ? ES

k−1

k2

ES + S ? ESS

k−2

kr

ES " E + P

Rate Law rP = krCE0CS = rP0, maxCS K2 = k −2
k2
KM + CS + CS2 KM + CS + C 2
K2 S
K2

Inhibition occurs due to the term C 2 in the denominator.
S
K2

©2020 NCEES 215

Chapter 5: Chemical Reaction Engineering

Maximum Rate 1

Occurs at CS = _KMK2i2

rP0, max, apparent = krCE0 1 = rP0, max
KM
K2 1
2 KM 2
1 + 2e 1 + 2e K2
o o

The maximum rate from the inhibited reaction is lower than rP0, max for the uninhibited reaction.

5.5 Heat Effects in Reactors

The reactor design equations in the previous sections assume isothermal operation. For non-isothermal operation, both material
and energy balance equations are required.

5.5.1 Batch Reactor

Energy Balance

mcp dT = V_−Dhtri_rAi + q
dt

endothermic reactions q is positive

exothermic reactions q is negative

adiabatic conditions q is zero

5.5.2 Plug-Flow Reactor

mo cp dT = V_−Dhtri_rAi + q
dt

endothermic reactions q is positive

exothermic reactions q is negative

adiabatic conditions q is zero

5.5.3 Continuous Stirred Tank Reactor
mo cp_To − T j = V_−Dhtri_rAi + q

endothermic reactions q is positive
exothermic reactions q is negative
adiabatic conditions q is zero

©2020 NCEES 216

6 FLUID MECHANICS

6.1 Symbols and Definitions

Symbols

Symbol Description Units (U.S.) Units (SI)
A Area ft2 m2
Ar Archimedes diameter dimensionless
C Fitting characteristic dimensionless J
CD Drag coefficient dimensionless kg : K
Cv Valve flow coefficient
dimensionless J
cp Specific heat (constant pressure) kg : K
Btu
cv Specific heat (constant volume) lbm -cF m
m
D Diameter Btu m
DH Hydraulic diameter lbm -cF N
d Diameter (minor) ft or in.
F Force ft or in. m
f Friction factor (Darcy-Weisbach) ft or in. m
Fanning friction factor m
fFanning lbf m
H Total head dimensionless m
h Height dimensionless
hf Head loss J
Head loss in fitting ft
hf, fitting ft
hL Head loss (general) ft
K Loss coefficient ft
KE Kinetic energy
ft
dimensionless

Btu

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Chapter 6: Fluid Mechanics

Symbols (con't)

Symbol Description Units (U.S.) Units (SI)
k
L Ratios of specific heats (cp/cv) dimensionless m
kg
MW Length or thickness ft or in. kmol

Ma Molecular weight lb kg
m lb mole Pa • sn
m
Mach number dimensionless kg
mo Mass lbm s
Apparent Viscosity
n rpm
Ns Mass flow rate lbm m
NPSHa hr m
NPSHr Specific exponent
Specific speed dimensionless Pa
P rpm
m
P Net positive suction head available ft J
PE Pa
Pvap Net positive suction head required ft m
R J
Pressure lbf or psi mol : K
R ft 2
m
Re Wetted perimeter ft rpm
r Potential energy Btu
S Vapor pressure psi °C or K
SG s
T Radius ft or in. m
t s
Universal gas constant Btu or psi-ft 3
u lb mole-cR lb mole-cR

V Reynolds number dimensionless
Radius ft or in.
Vo Rotational speed
Specific gravity rpm
W Temperature dimensionless
Time
Wo °F or °R
X hr or min or sec
x
y Velocity ft
Y sec
z
a Volume ft3 m3

Volumetric flow rate ft3 m3
sec s

Work ft-lbf N•m

Power hp W
m
Distance ft or in. m
Length, distance, or position ft or in. m
Length ft or in.
Expansion factor m
Length or elevation difference dimensionless radian
Angle ft or in.
radian

©2020 NCEES 218

Chapter 6: Fluid Mechanics

Symbols (con't)

Symbol Description Units (U.S.) Units (SI)
β
γ Ratio of small to large diameter dimensionless
d
e Surface tension lbf N = kg
e ft m s2
h Thickness of a film ft
h Absolute roughness ft m
q Porosity, void fraction, or volume
μ fraction ^0 1 e 1 1h dimensionless m
n3 Efficiency
n dimensionless N:s
Fluid viscosity lbm m2
r ft - sec

rf Angle radian radian

rp Dynamic viscosity cP or lbm Pa : s or kg
ft-sec s:m
τ
Infinite, plastic, or high shear cP or lbm Pa : s or kg
xt viscosity ft-sec s:m

t0 Kinematic viscosity ft2 m2
Φ Density hr s
lbm kg
ft3 m3

Density of fluid lbm kg
Density of particles ft3 m3
Stress
lbm kg
ft3 m3

lbf Pa
ft 2

Shear stress lbf Pa
ft 2

Yield stress of fluid lbf Pa
ft 2
Sphericity of particle (0 < Φ ≤ 1,
where Φ = 1 is a perfect sphere) dimensionless

Symbol Value Physical Constants Description
g 32.174 Units
9.8067 Gravitational acceleration
gc 32.174 ft (Earth)
sec2
1 m Gravitational conversion
s2 factor

lbm-ft
lbf -sec 2

kg : m = 1
N : s2

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Chapter 6: Fluid Mechanics

6.2 Fundamentals of Fluid Mechanics

6.2.1 Mechanical Energy Balance

6.2.1.1 Conservation of Mass
Conservation of mass for flow from Point 1 to Point 2 is

mo 1 = mo 2
The continuity equation is

ρ1 A1 u1 = ρ2 A2 u2

For an incompressible fluid, ρ1 = ρ2, therefore:
A1 u1 = A2 u2 and Vo1 = Vo2

6.2.1.2 The Bernoulli Equation s=fetc22 or Nk:gm m2
s2
The Bernoulli equation states, in energy per unit m=ass 32ft.2-llbbfm ,

P gc + u2 + g z = constant
t 2

For one-dimensional flows (with uniform velocity profiles) through conduits with flow from Point 1 to Point 2, expressed in:

Energy Per Unit Mass (Energy Basis)

P1tgc + u12 +g z1 + gc win = P2tgc + u 2 +g z2 + loss
2 2
2

where

win = net shaft work in = power/mass flow rate

Energy Per Unit Volume (Pressure Basis)

P1 + u12 t + t g z1 + t win = P2 + u 2 t + t g z2 + t^lossh
2 gc gc 2 gc
2 gc

Height of Fluid (Head Basis)

P1 gc + u12 + z1 + hs = P2 gc + u 2 + z2 + hL
tg 2g tg 2
2g

where

hs = shaft work head
hL = head loss

6.2.1.3 Energy Line and Hydraulic Grade Line

Energy Line (or Energy Grade Line)
The energy line (EL) represents the total head available to a fluid and can be expressed as:

For inviscid incompressible flow:

EL = P gc + u2 +z = constant along a streamline
tg 2g

©2020 NCEES 220

Chapter 6: Fluid Mechanics

For incompressible flow with losses:

EL = P gc + u2 + z − hL
t g 2g

Hydraulic Grade Line (or Hydraulic Gradient Line)
The hydraulic grade line (HGL) represents the total head available to a fluid, minus the velocity head, and can be expressed as:

For inviscid incompressible flow:

HGL = P gc +z
tg

For incompressible flow with losses:

HGL = P gc +z − hL
t g

Note: The energy or hydraulic grade lines do not represent "sources" or "sinks" of energy such as the effects of pumps or turbines.

Energy Line and Hydraulic Grade Line for Incompressible Fluid Between Two Points (With Losses)

u12 ENERGY LINE hL

2g

HYDRAULIC GRADE LINE u22
2g

P1 gc P2 gc
g g

FLOW 2
DATUM z2

z1 1

6.2.1.4 The Impulse-Momentum Principle

The resultant force in a given direction acting on a fluid equals the rate of change of momentum of the fluid,

where

/ / /F = Vo2 t2 u2 - Vo1 t1 u1

/F = result of all external forces acting on the control volume

/Vo1 t1 u1 = rate of momentum of the fluid flow entering the control volume in the same direction as the force
/Vo2 t2 u2 = rate of momentum of the fluid flow leaving the control volume in the same direction as the force

©2020 NCEES 221

Chapter 6: Fluid Mechanics

6.2.2 Viscosity and Fluid Properties
6.2.2.1 Hydrostatic Head, Stress, Pressure, and Viscosity

Definitions:

Hydrostatic head is

P = tgh
gc

Stress is

x = lim DF
DA
^DA " 0h

where x = surface stress at a point

Pressure is

P = − xn
where xn = stress normal at a point

Newton's Law of Viscosity relates shear stress (τt = stress tangential to the boundary) to the velocity gradient or shear rate
(du/dy), using a constant of proportionality known as the dynamic (absolute) viscosity (μ) of the fluid:

xt = n du
dy

Kinematic viscosity is

v = n
t

6.2.2.2 Fluid Types and Characteristics FLUID TYPES AND CHARACTERISTICS

SHEAR STRESS (τt ) BINGHAM FLUID
τ0 NEWTDOINLAIATNANT

PSEUDOPLASTIC

SHEAR RATE (du/dy)

©2020 NCEES 222

Chapter 6: Fluid Mechanics

Classifications of Fluids

Fluid Fluid Type Behavior Examples
Classification
Viscosity is constant.
Time-Independent
Viscosity Newtonian xt = n du Water, light oil,
dy blood plasma
Time-Dependent Pseudoplastic (shear
Viscosity thinning) The term μ is reserved for Newtonian fluids. Molasses, latex
paint, whole blood
Viscoplastic Apparent viscosity (m) decreases with increased
Viscoelastic
shear stress.

xt = md du n
dy
n

n = power law index, n < 1

m is also known as the consistency coefficient or
consistency index

Apparent viscosity (m) increases with increased

Dilatant (shear thick- shear stress. Corn starch
ening) suspensions
xt = md du n
Thixotropic dy Yogurt, plastisols
n Gypsum paste,
kaolin clay
n = power law index, n > 1 suspensions

Apparent viscosity (m) decreases with duration of Mayonnaise, river
stress. mud, slurries

Rheopectic Apparent viscosity (m) increases with duration of Silicone putty
stress.

Bingham fluid Behaves as a rigid body until a minimum stress
(yield stress) is applied, then reacts as a
Kelvin material Newtonian fluid at shear stresses above the yield
Maxwell material stress.

xt = x0 + h du
dy

h = fluid viscosity
x0 = yield stress

The materials exhibit both viscous and elastic
characteristics during deformation under stress.

6.2.2.3 Surface Tension and Capillary Rise

Surface tension g is the force per unit contact length

c = F
L

where

F = surface force at the interface

L = length of interface

©2020 NCEES 223

Chapter 6: Fluid Mechanics

The capillary rise, h, is approximated by

h = e 4c gc cos a o
tgd

where

h = height of the liquid in the vertical tube

α = angle made by the liquid with the wetted tube wall

d = the diameter of the capillary tube

6.2.3 Velocity

Velocity is defined as the rate of change of position with respect to time

u = dx
dt

where x = position

Velocity of a Newtonian fluid in a thin film is

u^ t h = u y du = u
d dy d

THIN FILM

δu
y

BOUNDARY

The velocity distribution for laminar flow in circular tubes or between planes is

u^r h = umax =1 − c r 2
R
mG

where r = distance from the centerline

R = radius of the tube or half the distance between the parallel planes

u = local velocity at r

umax = velocity at the centerline of the duct
u = average velocity in the duct

Flow Conditions

Fully turbulent flow Circular tubes in Parallel planes in
laminar flow laminar flow

umax = 1.18 2 1.5
u

The shear stress distribution is

x = r
xw R

where t and tw = shear stresses at radii r and R, respectively

©2020 NCEES 224

Chapter 6: Fluid Mechanics

6.2.4 Reynolds Number

Dimensionless number describing flow behavior with the general definition:

Re = inertial forces
viscous forces

6.2.4.1 Hydraulic Diameter

DH = hydraulic diameter (also known as the characteristic length)

=DH 4=# crwoessttseedcptieornimaleatreera 4A
P

Hydraulic Diameters for Various Flow Configurations

Flow Configuration Diagram Hydraulic Diameter
DH =

Through a circular tube D D = inside diameter
Through a square duct u a

a
u

a

Through a rectangular duct a 2ab
ub a+b

Through a circular annulus u D2-D1
D1

D2

©2020 NCEES 225

Chapter 6: Fluid Mechanics

Hydraulic Diameters for Various Flow Configurations (cont'd)

Flow Configuration Diagram Hydraulic Diameter
DH =

Through a partially filled pipe cr 28rl - c_r - hiB
(tube) h l

Around a sphere (or sphere l where
through a fluid)
FLUID APPROACH VELOCITY (uo) c = 2 h_2r - hi

Sphere diameter

PROJECTED AREA (Ap) FLUID STREAMLINES

FLUID APPROACH VELOCITY (uO)

Around any object (or an any 4Ap
object through a fluid) P

PROJECTED AREA (Ap) FLUID
STREAMLINES
P = PERIMETER OF SHAPE
PRESENTED NORMAL TO
THE APPROACH VELOCITY

©2020 NCEES 226

Chapter 6: Fluid Mechanics

6.2.4.2 Newtonian Fluid

Re = DH u t
n

where u = approach velocity

Various Forms of Reynolds Numbers and Their Units in Circular Conduits (Pipes)

Reynolds Diameter Fluid Fluid Fluid Volumetric Mass Kinematic
Velocity Density Viscosity Flow rate Flow rate Viscosity
Number Form D
u ρ μ Vo mo ν

Dut ft ft lbm lbm
n sec ft3 ft-sec

Dut m m kg Pa : s orkgNm:2s
n s m3 or m : s

Dut ft ft lbm lbf-sec
32.2n in. sec ft3 ft 2
ft lbm cP
123.9 Du t sec ft3
n

22, 700 Vo t in. lbm cP ft 3
D n ft3 sec

50.6 Vo t in. lbm cP gpm
Dn ft 3

6.31 mo in. cP lbm
Dn in. hr
ft
35.42 Vo t m lbm cP barrels
Dn in. ft 3 hr
in. ft
Du in. sec ft 2
v m/s sec
m2
Du ft s
v sec ft 2
ft sec
Du sec
12v

7740 Du cS
v

1, 419, 000 Vo ft3 cS
Dv sec

3160 Vo in. gpm cS
Dv

©2020 NCEES 227

Chapter 6: Fluid Mechanics

6.2.4.3 Power Law Fluid

Re x = `D n u(2 − n) tj

fK d _3n + 1i n 8 (n − 1) p
4n
n

where

n = power law index

K = consistency index

6.2.4.4 Bingham Fluid

Bingham fluid flow through a pipe:

ReBP = 4Vo t

r D n3f1 + r D3 x0 gc p
24Vo n3

where

n3 = infinite viscosity, or plastic viscosity, or high shear limiting viscosity

x0 = yield stress of the fluid

©2020 NCEES 228

Chapter 6: Fluid MechanicsSAE 10 L

Viscosity as a Function of Temperature for a Variety of Gases and Liquids

100
80
60

40
30

20
G OIL (21° API)

10
8

6

4 35° API DISTILLATE

3 ETHYL ALCOHOL (100%)
2

VISCOSITY, CENTIPOISES (cP) 1

0.8 AMMONIAN(L-IQPBUEEINNDAZT)CEAENNTEECOA(NLREIBQO(ULNIIQDTU)ETIDR)ACHLORWIDATEER GASOLINE
0.6
0.4
0.3
0.2

0.1

0.08

0.06

0.04 AIR AT ATMOSPHERE PRESSURE DIOXIDE OXYGEN (1 ATM) CARBON DIOXIDE
0.03 CHLORINE CARBON WATER VAPOR (1 ATM) METHANE
0.02
AMMONIA VAPOR METHANE HYDROGEN
0.01 WATER VAPOR
0.008
0.006 PROPANE n - PENTANE

0.004 100 200 300 400 500 600 700
0 TEMPERATURE (°F)

Source: Brown, G. G., et. al., Unit Operations, New York: John Wiley & Sons, Inc., 1951.

©2020 NCEES 229

Chapter 6: Fluid Mechanics

6.2.4.5 Critical Reynolds Number
The critical Reynolds number (Rec ) is the minimum Reynolds number at which flow is expected to become turbulent, as shown in
the following table:

Flow Regime Rec
Flow through a pipe 2100
Flow around a sphere 10
1708r1
Circular flow (rotating cylinder,
Taylor-Couette flow) h

where the inner cylinder has a
diameter (r1) and height (h)

6.2.5 Friction

6.2.5.1 Absolute Roughness and Relative Roughness
f
Relative roughness is D .

Absolute Roughness or Specific Roughness (f) of Various Pipes

Material ε mm
ft in. m 0.001
0.0015
PVC and plastic pipes 0.0000033 0.00004 1.0E–06 0.015
0.06
Copper, lead, brass, aluminum (new) 0.000005 0.00006 1.5E–06 0.12
0.15
Stainless steel 0.00005 0.0006 1.5E–05 0.3
0.5
Steel commercial pipe 0.0002 0.0024 6.0E–05 0.5
0.8
Asphalted cast iron 0.0004 0.0048 1.2E–04 1.2
2.0
Galvanized iron 0.0005 0.006 1.5E–04 0.6

Smoothed cement 0.001 0.012 3.0E–04

New cast iron 0.0016 0.019 5.0E–04

Well-planed wood 0.0016 0.019 5.0E–04

Ordinary concrete 0.0026 0.031 8.0E–04

Worn cast iron 0.004 0.048 1.2E–03

Coarse concrete 0.0065 0.078 2.0E–03

Ordinary wood 0.002 0.024 6.1E–04

6.2.5.2 Friction Factors for Laminar Flow

For laminar flow (Re < 2100)

f = 64
Re

6.2.5.3 Friction Factors for Turbulent Flow

The Colebrook equation

1 =− 2 log10 KLKJKKKK f + 2.51 OOPNOOOO
f D Re f
3.7

©2020 NCEES 230

Chapter 6: Fluid Mechanics

The Haaland equation is an empirical approximation of the friction factor that does not require iteration,

1 =− 1.8 log10>6R.e9 +c f 10
f 3.7D
m9 H

for the following conditions

4 # 104 # Re # 108and 0 # f # 0.05
D

For fully turbulent flow

1 = 1.74 − 2 log10 c 2f m
f D

Friction Factor Chart

0.1

0.09 CRITICAL
LAMINAR ZONE TRANSITION
0.08 FLOW
ZONE COMPLETELY TURBULENT REGIME

0.07 f= 64 0.05
Re 0.04
0.03
0.06

0.05 0.02
0.015
0.04 Rcr e
0.01 D
0.03 0.008
f 0.006 RELATIVE ROUGHNESS

0.025 0.004

0.002

0.02 0.001
0.0008
0.015 0.0006

0.0004

0.0002

0.01 SMOOTH PIPES 0.0001
0.009 0.000,05
0.008
0.000,001
79 0.000,005
103
2 3 4 5 67 9 2 3 4 5 67 9 2 3 4 5 67 9 2 3 4 5 67 9 0.000,01
104 105 106 107
2 3 4 5 67 9
108

REYNOLDS NUMBER Re

MOODY DIAGRAM. (FROM L.F. MOODY, TRANS. ASME, VOL. 66, 1944.)

6.2.6 Pressure Drop for Laminar Flow

The Hagen-Poiseuille equation for Vo in terms of the pressure drop DPf is

=Vo r=R84nDLP f r D4 DP f
128n L

This relation is valid only for flow in the laminar region.

©2020 NCEES 231