Massoud M. Engineering Thermofluids: Thermodynamics, Fluid Mechanics, and Heat Transfer

XIV Table of Contents

Vb. Boiling ...................................................................................................... 637

1. Definition of Boiling Heat Transfer Terms....................................... 637

2. Convective Boiling, Analytical Solutions......................................... 641

3. Convective Boiling, Experimental Observation................................ 648

4. Pool Boiling Modes .......................................................................... 650

5. Flow Boiling Modes ......................................................................... 658

Questions .................................................................................................. 672

Problems ................................................................................................... 673

Vc. Condensation ............................................................................................ 677

1. Definition of Condensation Heat Transfer Terms............................. 677

2. Analytical Solution ........................................................................... 678

3. Empirical Solution ............................................................................ 682

4. Condensation Degradation................................................................ 684

Questions .................................................................................................. 685

Problems ................................................................................................... 686

VI. Applications............................................................................................. 687

VIa. Heat Exchangers ....................................................................................... 687

1. Definition of Heat Exchanger Terms................................................ 687

2. Analytical Solution ........................................................................... 690

3. Analysis of Shell and Tube Heat Exchanger..................................... 702

4. Analysis of Condensers..................................................................... 710

5. Analysis of Steam Generators........................................................... 716

6. Transient Analysis of Concentric Heat Exchangers.......................... 719

Questions .................................................................................................. 723

Problems ................................................................................................... 723

VIb. Fundamentals of Flow Measurement........................................................ 728

1. Definition of Flow Measurement Terms........................................... 728

2. Repeatability, Accuracy, and Uncertainty ........................................ 729

3. Flowmeter Types .............................................................................. 732

4. Flowmeter Installation ...................................................................... 744

Questions .................................................................................................. 745

Problems ................................................................................................... 745

VIc. Fundamentals of Turbomachines.............................................................. 747

1. Definition of Turbomachine Terms .................................................. 747

2. Centrifugal Pumps ............................................................................ 749

3. Dimensionless Centrifugal Pumps Performance............................... 755

4. System and Pump Characteristic Curves .......................................... 762

5. Analysis of Hydraulic Turbines ........................................................ 769

6. Analysis of Turboject for Propulsion................................................ 777

Questions .................................................................................................. 779

Problems ................................................................................................... 780

Table of Contents XV

VId. Simulation of Thermofluid Systems ......................................................... 784

1. Definition of Terms........................................................................... 784

2. Mathematical Model for a PWR Loop.............................................. 786

3. Simplified PWR Model..................................................................... 791

4. Mathematical Model for PWR Components, Pump.......................... 802

5. Mathematical Model for PWR Components, Pressurizer ................. 811

6. Mathematical Model for PWR Components, Containment .............. 819

7. Mathematical Model for PWR Components, Steam Generator ........ 827

Questions .................................................................................................. 829

Problems ................................................................................................... 829

VIe. Nuclear Heat Generation........................................................................... 841

1. Definition of Some Nuclear Engineering Terms............................... 841

2. Neutron Transport Equation.............................................................. 853

3. Determination of Neutron Flux in an Infinite Cylindrical Core........ 859

4. Reactor Thermal Design ................................................................... 877

5. Shutdown Power Production............................................................. 882

Questions .................................................................................................. 884

Problems ................................................................................................... 884

VII. Engineering Mathematics ...................................................................... 901

VIIa. Fundamentals............................................................................................ 901

1. Definition of Terms........................................................................... 901

VIIb. Differential Equations .............................................................................. 911

1. Famous Differential Equations ......................................................... 911

2. Analytical Solutions to Differential Equations ................................. 919

3. Pertinent Functions and Polynomials................................................ 936

VIIc. Vector Algebra.......................................................................................... 943

1. Definition of Terms........................................................................... 943

VIId. Linear Algebra.......................................................................................... 963

1. Definition of Terms........................................................................... 963

2. The Inverse of a Matrix..................................................................... 968

3. Set of Linear Equations..................................................................... 971

VIIe. Numerical Analysis ................................................................................. 976

1. Definition of Terms .......................................................................... 976

2. Numerical Solution of Ordinary Differential Equations ................... 979

3. Numerical Solutions of Partial Differential Equations...................... 985

4. The Newton–Raphson Method ....................................................... 1004

5. Curve Fitting to Experimental Data ................................................ 1006

XVI Table of Contents

VIII. Appendices ............................................................................................ 1011

I. Unit Systems, Constants and Numbers................................................... 1013

II. Thermodynamic Data ............................................................................. 1023

III. Pipe and Tube Data................................................................................. 1049

IV. Thermophysical Data.............................................................................. 1059

V. Nuclear Properties of Elements .............................................................. 1091

References........................................................................................................ 1097

Index................................................................................................................. 1111

Table of Contents XVII

Contents

Introduction

Thermodynamics

Fluid Mechanics

Heat Transfer

Two-Phase Flow &

Heat Transfer

Mathematics

Fluid Mechanics (III)

Two-Phase Flow

Fundamentals

Incompressible

Viscous Flow

Compressible

Viscous Flow

Pressure Drop Critical Flow

Thermodynamics

(II)

Fundamentals Power Cycles Mixtures

Condensation

Applications (VI)

Heat

Exchangers

Flow

Measurement

Turbomachines

Simulation

of Systems

Nuclear Heat

Heat Transfer (IV)

Two-phase Heat

Transfer

Conduction Convection Radiation

Boiling

Engineering Mathematics (VII)

Fundamentals

Differential

Equations

Vector

Algenbra

Linear

Algebra

Numerical

Analysis

I

II

III

IV

V

VII

IIa IIb

IIc

IIIa IIIb IIIc

IVa IVb & IVc IVd

VIb VIc

VIa

VId

VIe

VIIa VIIb VIIc VIId VIIe

Applications

VI

Two-Phase Flow &

Heat Transfer (V)

Va Va Vb Vc

Note: Roman numerals refer to the related chapters

Nomenclature

In this book, for the sake of brevity and consistency, as few symbols as possible

are used. Thus, to minimize the number of symbols, yet clearly distinguish vari-

ous parameters, lower case and italic fonts have been used whenever a symbol

represents two or more parameters. For example, while V represents volume, v is

used for specific volume, V for velocity, v for kinematic viscosity, and

V

for

volumetric flow rate. To avoid confusion when solving problems by hand, the

reader may use

 for volume.

Special attention must be paid whenever h representing specific enthalpy and h,

standing for heat transfer coefficient, appear in the same equation. This occurs in

chapters IVe and IVf, dealing with boiling and condensation. Also note that h and

H stand for height. Similarly, In Chapter Va, s represents an element of length as

well as entropy while S stands for slip ratio, respectively.

The units provided below in front of each symbol, are just examples of com-

monly used units. They do not preclude the representation of the same symbol

with different sets of units. The details of the SI units are discussed in Appendix I.

English

symbols Definition SI Unit British Unit

a Acceleration m/s

2

ft/s

2

a Radius m in

A Helmholtz function J Btu

A Area m

2

ft

2

b Width m ft

B Bulk modulus Pa psi

B Buckling cm

–2

in

–2

c Speed of sound m/s ft/s

c

p

Specific heat at constant pressure W/kgC Btu/lbm·F

c

v

Specific heat at constant volume W/kgC Btu/lbm·F

C

d

Discharge Coefficient – –

C

D

Drag coefficient – –

d, D Diameter m (cm) ft (in)

e Specific energy W/kg Btu/lbm

e Uncertainty – –

E Modulus of Elasticity Pa psi

E Total energy J Btu

E Total emissive power W Bu/s

English

XX Nomenclature

symbols Definition SI Unit British Unit

F Force N lbf

F View factor - -

F Peaking factor - -

g Gibbs function J Btu

g Gravitational acceleration m/s

2

ft/s

2

g

c

Conversion constant kgm/Ns

2

slug·ft/lbf·s

2

G Mass flux kg/sm

2

lbm/s·ft

2

G Irradiation W/m

2

Btu/s·ft

2

h Head m ft

ƫ Plank’s constant J·s Btu·s

h Heat transfer coefficient W/m

2

C Btu/h·ft

2

·F

h Specific enthalpy kJ/kg Btu/lbm

H Height m ft

H Enthalpy J Btu

I Geometric inertia m

-1

ft

-1

I Irreversibility J Btu

I Spectral intensity W/m

2

·

P

m·sr Btu/s·ft

2

·

P

m sr

j Conversion factor J/J ft·lbf/Btu

J Radiosity W/m

2

·

P

m -

J Superficial velocity m/s ft/s

J Neutron current density s

–1

cm

–2

s

–1

·ft

–2

J Bessel function of first kind – –

k

s

Spring constant – –

k Boltzmann constant J/K Btu/R

k Thermal conductivity W/mK Btu/h·ft·F

k

f

Infinite medium multiplication factor – –

k

eff

Finite medium multiplication factor – –

K Frictional loss coefficient – –

l Mean free path cm in

L Diffusion length cm in

m Mass kgm lbm

m

Mass flow rate kg/s lbm/s

M Molecular weight kg/kmol lb/lbmol

N

A

Avogadro number – –

P Pressure Pa psi

P Perimeter m ft

q Heat transfer per unit mass J/kg Btu/lbm

'q

Linear heat generation rate W/m kW/ft

q

cc

Heat flux W/m

2

Btu/h·ft

2

q

ccc

Volumetric heat generation rate W/m

3

Btu/h·ft

3

Q Heat transfer J Btu

Q

Rate of heat transfer W Btu/s

R Gas constant kPa·m

3

/kg·K ft·lbf/lbm·R

Nomenclature XXI

English

symbols Definition SI Unit British Unit

r, R Radius m (cm) ft (in)

R Thermal resistance C/W h·F/Btu

R

u

Universal gas constant kJ/kmol·K ft·lbf/lbmol·R

s Element of length m ft

s Tube or rod pitch cm in

s Specific entropy J/kg·K Btu/lbm·R

s Volumetric neutron source

strength cm

–3

·s

–1

in

–3

·s

–1

S Entropy J/K Btu/R

S Slip ratio – –

S Surface area m

2

ft

2

S

g

Specific gravity – –

t Time s s

T Temperature C (K) F (R)

T Torque m·N ft·lbf

u Specific internal energy J/kg Btu/lbm

u

Unit vector – –

U Internal energy J Btu

U Overall heat transfer coefficient W/m

2

·K Btu/h·ft

2

·F

v Kinematics viscosity m

2

/s ft

2

/h

v secific volume m

3

/kgm ft

3

/lbm

V Vlocity m/s ft/s

V Volume m

3

ft

3

V Voltage V V

V

Volumetric flow rate m

3

/s ft

3

/s

W Weight kg lbf

w Work per unit mass

of working fluid J/kg Btu/lbm

W Work J Btu

W

Power W Btu/s

x Thermodynamic quality – –

X Flow quality – –

y mole fraction – –

Y Gas expansion factor – –

Z Elevation m ft

Greek

symbols Definition SI Unit British Unit

D

Void fraction, Absorptivity – –

E

Volumetric thermal

expansion coeff. K

–1

R

–1

E

Volumetric flow ratio – –

J

Ratio of c

p

/c

v

– –

J

Shearing strain – –

XXII Nomenclature

Greek

symbols Definition SI Unit British Unit

G Boundary layer thickness mm in

'  Difference in values – –

H

Emissivity – –

H Strain – –

6 Macroscopic cross section cm

–1

–

ȗ Effectiveness – –



K

Efficiency – –

K 

Eta factor – –

T 

Azimuthal angle – –

N

Boltzmann constant J/K Btu/R

N

Isothermal compressibility Pa

–1

psi

–1

N

Thermal conductivity (tensor) W/m· C Btu/h·ft·F

O

System thermal length m ft

O

Mean free path cm in



O 

Wavelength

P

m –

P

Absorption coefficient cm

2

–

P

Dynamic viscosity N·s/m

2

bm/h·ft

Q 

Number of fast neutrons

per fission – –

U

Density kg/m

3

lbm/ft

3

U

Reflectivity – –

V

Surface tension N/m lbf/ft

V

Tensile stress Pa psi

V

Stefan-Boltzmann constant W/m

2

·K

4

Btu/ft

2

·h·R

4

V 

Microscopic cross section cm

–2

–

V

Measure of entropy production J/K Btu/R

W

Shear stress Pa psi

W

Transmissivity – –

I

Relative humidity – –

I

Flux s

–1

·cm

–2

s

–1

·ft

–2

I

Specific availability

(closed system) kJ/kg Btu/lbm

) 

Availability (closed system) kJ Btu

)



Viscous Dissipation function W Btu/s

F 

Fission spectrum of an isotope MeV

–1

Btu

–1

M 

Zenith angel – –

\  Stream function – –

\

 Specific availability

(control volume) kJ/kg Btu/lbm

<

Availability (control volume) kJ Btu

Z

Humidity ratio – –

Ȧ Impeller Speed of a turbomachine rad/s rad/s

: Solid angle sr –

Nomenclature XXIII

Subscripts

B body, buoyancy

CL Centerline

C.V. control volume

e equivalent or hydraulic diameter

f saturated liquid

f friction

f free stream, bulk

g saturated steam

h equivalent or hydraulic diameter

l local

r reduced

max maximum

min minimum

s surface, shaft

sat saturation

v vapor

Abbreviations

# numbers of

1-D one dimensional

Av Avogadro number

b Barns

C Celsius

cm centimeter

eV electron volt

E exponent (Example: 1 u 10

3

= 1E3)

ft foot, feet

F Fahrenheit

g gram

GPM gallon per minute

h hour

hp horsepower

in inch

J Joules

k Kilo

K Kelvin

ln natural logarithm, logarithm to the base e = 2.7182818

log logarithm to the base 10

m meter

min minute

XXIV Nomenclature

mm millimeter

MBtu million Btu

MeV Million electron volt

MWe Mega Watt electric

MWt Mega Watt thermal

R Rankine

s second

S-S steady state

W Watt

Massoud M. Engineering Thermofluids: Thermodynamics, Fluid Mechanics, and Heat Transfer

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