Thermal Stresses in Pipes

Thermal Stresses in Pipes

By Iy ad A l-Z ah arn ah , BSc. & MSc. In Mechanical Engineering

This thesis is subm itted to D ublin C ity U niversity as the fulfillm ent o f the requirem ent for the aw ard o f degree o f

Doctor of Philosophy

Supervisors

P rofessor M .J. H ashm i P rofessor B.S. Y ilbas

School of M echanical & M an u factu rin g E ngineering D ublin C ity U niversity

April 2002

DECLARATION

I hereby certify that this material, which I now submit for assessment on the programe o f study leading to the award o f Doctor o f Philosophy is my own work and has not been taken from the work o f others save and to the extent that such work has been cited and acknowledged within the text o f my work.

I.D No.: 9897021

D ate: ! 3 - 0 S ^ Z 0 o 2

'REF£RENCE

I dedicate this thesis to my beloved parents, brother and sister

ACKNOWLEDGMENT

First and foremost, all praise to Almighty, Allah Who gave me the courage and patience to carry out this work and I Ask to accept my little effort. May He, Subhanahu-ta-Aala, guide the humanity and me to the right path.

My deep appreciation is to my Ph.D., thesis advisors Professor M.S.J. Hashmi (Dublin City University, Ireland) and Professor B.S. Yilbas (King Fahd University of Petroleum and Minerals, Saudi Arabia) for their continuous support and matchless patience throughout this work. Their suggestions were always valuable and working with them was always indeed a wonderful and learning experience. Continuously, they were supportive and kind to me.

I would also like to thank both Dublin City University in Ireland and King Fahd University o f Petroleum and Minerals in Saudi Arabia for providing everything necessary to make this work possible.

Thanks are due to my parents, brother and sister for their prayers and support throughout my academic life. I dedicate this work to them.

Appreciations are due to Mr. Abdallah Al-Gahtani for his kind support before the viva.

Title of thesis:

Thermal Stresses in Pipes

N am e of student:

Iyad Al-Zahamah

S tudent No: 9897021

ABSTRACT T his study presents results about therm al stresses in externally heated pipes th a t are subjected to d ifferen t flow types: lam inar flow , tu rb u len t flow, and pulsating flow . T he effect o f flow R eynolds nu m b er o n therm al stresses in the pipe is studied. To investigate the influence o f fluid and solid properties on the resulting therm al stresses in pipes, tw o solids nam ely; steel and cooper and three fluids nam ely; w ater, coolanol-25, and m ercury are u sed in the study. Pipes w ith d ifferen t diam eters, length to diam eter ratios, and thickness to diam eter ratios are covered in the study to exam ine th e effects on therm al stresses. D ifferen t param eters for p ressu re difference and oscillating frequency are used in the case o f pulsatin g flow to find their in flu en ce on th e resulting therm al stresses in the pipe. T he am ount o f h eat flux at th e outer w all o f the pipe is also included in the study.

In order to account for the turbulence, the k-s model is introduced in the analysis. The numerical scheme employing control volume approach is introduced to discretize the governing equations o f flow and heat transfer. The grid independent tests are carried out to ensure grid independent solutions. To validate the present predictions, data presented in the open literature are accommodated. iv

Table of Contents

Declaration

i

Dedication

ii

Acknowledgment

iii

Abstract

iv

Table o f Contents

V

List o f Tables

viii

List o f Figures

ix

Nomenclature

xiii

CHAPTER 1 Introduction and Scope of the Work 1.1

Introduction

1

1.2

Scope of the present study

5

CHAPTER 2 Literature Review

6

Laminar flow

6

Turbulent flow

13

Pulsating flow

17

Thermal stresses in pipes

25

V

CHAPTER 3 Mathematical Modeling, Numerical Solution and Results Validation 3.1

Introduction

34

3.2

Mathematical Modeling

34

3.2.1

Boundary Conditions

35

3.2.2 Governing Thermal Stress Formulae

37

3.2.3

38

Fully developed Laminar Flow

3.2.4 Turbulent Flow

39

3.2.5 P u lsatin g F lo w

42

3.3

N u m erical S olution and R esu lts v alid atio n

43

3.3.1

Fully Developed Laminar Flow

50

3.3.2 Turbulent Flow

51

3.3.3 Pulsating Flow

53

CHAPTER 4 Results and Discussion 4.1

Introduction

55

4.2

Fully Developed Laminar Flow

55

4.2.1 The Effects o f Diameter, Thickness and Length Size Effects on Thermal Stresses

56

4.2.2 The Material Properties and the Thermal Conductivity Effects on Thermal Stresses

77

4.3

Turbulent Flow

82

4.4

Pulsating Flow

95

4.4.1

The Effects o f Diameter Size and Thickness to Diameter Ratio on Thermal Stresses

95

4.4.2

The Effects o f Length Size on Thermal Stresses

104

4.4.3

The Effects o f Reynolds Number on Thermal Stresses

109

4.4.4

The Effects o f Fluid Properties (Prandtl Number) on vi

Thermal Stresses 4.4.5

114

The Effects o f pressure Difference Oscillating Frequency on Thermal Stresses

4.4.6

119

The Effect o f Heating Load Magnitude on Thermal Stresses

124

CHAPTER 5 Conclusions and Future Work 5.1

Conclusions

129

5.1.1

Laminar Flow

129

5.1.2

Turbulent Flow

131

5.1.3

Pulsating Flow

133

5.2

Future Work

135

References

136

Papers in Conference and Journals

147

vii

LIST OF TABLES

Table 3.3.1

The turbulent Prandtl numbers and the resulting Nusselt numbers for the validation and the present study results

Table 4.1.1

Properties of the pipe material and fluids used in the study

Table 4.2.1

Diameters, thicknesses, lengths and grid sizes of

the pipes used in the parametric study for

laminar flow

Table 4.4.1

The pipe diameters and thicknesses at Reynolds

number =500 Table 4.4.2

The pipe lengths at Reynolds number = 500 and

LIST OF FIGURES

Figure 3.1.1

Schematic diagram o f the pipe and coordinates

34

Figure 3.2.1

The turbulent Prandtl numbers for various Reynolds numbers at

41

Different Laminar Prandtl numbers Figure 3.3.1

A finite control volume

44

Figure 3.3.2

Meshes used in the study

49

Figure 3.3.3

Axial distribution o f the dimensionless temperature distribution

50

Figure 3.3.4

The Nusselt numbers for various Reynolds and Prandtl numbers

52

from the present and previous studies Figure 3.3.5

The oscillating pressure difference input and the resulting output

54

Axial velocity Figure 4.2.1

Dimensionless temperature contours for the conditions: D = 0.08

58

m, t = O.lxD, L = 5xD (case 1 in Table 4.2.1) Figure 4.2.2


58

m, t = O.lxD, L = 25xD (case 3 in Table 4.2.1) Figure 4.2.3


59

m, t = 0.5xD, L = 5xD (case 7 in Table 4.2.1) Figure 4.2.4


59

m, t = 0.5xD, L = 25xD (case 9 in Table 4.2.1) Figure 4.2.5

Thermal stresses for cases 1, 2 and 3 in Table 4.2.1

61

Figure 4.2.6


62

Figure 4.2.7


63

Figure 4.2.8


64

Figure 4.2.9


65

Figure 4.2.10 Thermal stresses for cases 16, 17 and 18 in Table 4.2.1

66

Figure 4.2.11


67


68


69

Figure 4.2.14 Dimensionless temperature (T*) contours for the cases o f

71

a) steel and mercury and b) copper and mercury

ix

Figure 4.2.15

Dimensionless temperature (T*) contours for the cases

of

73

of

74

a) steel and water and b) copper and water Figure 4.2.16 Dimensionless temperature (T*) contours for the cases a) steel and coolanol-25 and b) copper and coolanol-25 Figure 4.2.17

The dimensionless outer wall temperature vs. the

76

dimensionless Pipe length for various Prandtl numbers and thermal conductivity ratios Figure 4.2.18

The effective stress distribution vs. the dimensionless pipe

76

radius at the inlet-plane o f the pipe for various Prandtl numbers and thermal conductivity ratios Figure 4.2.19 The effective stress distribution vs. the dimensionless pipe

78

radius at the mid-plane o f the pipe for various Prandtl numbers and thermal conductivity ratios Figure 4.2.20 The effective stress distribution vs. the dimensionless pipe

78

radius at the outlet-plane o f the pipe for various Prandtl numbers and thermal conductivity ratios Figure 4.2.21

The dimensionless temperature ratio vs. the dimensionless

79

distance along the pipe for various Prandtl numbers and thermal conductivity ratios Figure 4.2.22 Location o f the minimum effective stress at the inlet, mid

81

and outlet planes o f the pie for various Prandtl numbers and thermal conductivity ratios Figure 4.3.1

Variation o f the local Nusselt number with the dimensionless

83

length o f the pipe for two fluids at different Reynolds numbers for the cases o f a) steel pipe and b) copper pipe Figure 4.3.2

Dimensionless temperature (T*) contours for the cases o f copper

85

pipe with water and coolanol-25 at Reynolds number = 10000 Figure 4.3.3

Dimensionless temperature (T*) contours for the cases o f steel

86


Dimensionless temperature (T*) contours for the cases o f copper pipe with water and coolanol-25 at Reynolds number = 30000

x

88

Figure 4.3.5

Dimensionless temperature (T*) contours for the cases o f copper

89



91



92


Mid-plane effective stresses for combinations o f two solids and

94

two fluids at three Reynolds numbers (10000, 30000 and 50000) Figure 4.4.1

Inner and outer wall temperature at the inlet and outlet planes

97

for the cases o f D = 0.04 m and D = 0.08 m and t = 0.125xD Figure 4.4.2

Thermal stresses at inlet and outlet planes for the case o f

98

D = 0.04 m and t = 0.125xD = 0.005 m Figure 4.4.3


99

D = 0.08 m and t - 0.125xD = 0.01 m Figure 4.4.4

Inner and outer wall temperatures at the inlet and outlet planes

101

for the cases o f D = 0.04 m and D = 0.08 m and t = 0.5xD Figure 4.4.5


102



103



106

for the cases o f L =12.5xD and L =17.5xD Figure 4.4.8

Thermal stresses at inlet and outlet planes for

107

the case o f L = 12.5xD Figure 4.4.9

Thermal stresses at inlet and outlet planes for

108

the case o f L = 17.5xD Figure 4.4.10 Inner and outer wall temperatures at the inlet and outlet planes

111

for the cases o f Reynolds number = 1000 and Reynolds number = 1500

Figure 4.4.11 Thermal stresses at inlet and outlet planes for the case o f Reynolds number = 1000

xi

112

Figure 4.4.12 Thermal stresses at inlet and outlet planes for the case o f

113

Reynolds number = 1500 Figure 4.4.13


116

for the cases o f coolanol-25 and mercury Figure 4.4.14 Thermal stresses at inlet and outlet planes for the case o f

117

Coolanol-25 Figure 4.4.15 Thermal stresses at inlet and outlet planes for the case o f

118

Mercury Figure 4.4.16 Inner and outer wall temperatures at the inlet and outlet planes

121

for the cases o f 0.2 Hz and 1 Hz Figure 4.4.17 Thermal stresses at inlet and outlet planes for the case o f 0.2 Hz

122

Figure 4.4.18 Thermal stresses at inlet and outlet planes for the case o f 1.0 Hz

123

Figure 4.4.19 Inner and outer wall temperatures at the inlet and outlet planes

125

for the cases o f 2000 W /m2 and 10000 W/m2 Figure 4.4.20 Thermal stresses at inlet and outlet planes for the case o f

126

2000 W/m2 Figure 4.4.21

Thermal stresses at inlet and outlet planes for the case o f 10000 W/m2

xii

127

Nomenclature A

velocity amplitude

(m/s)

Cp

specific heat o f fluid (J/kgxk)

CpSoiid specific heat o f solid (J/kgxk) D

inside diameter o f the pipe

(m)

E

modulus o f elasticity o f the pipe material

e

constant in law o f wall

h

heat transfer coefficient

J

total flux across a face o f the finite control volume (J/m )

k

turbulent kinetic energy generation variable

(Pa)

(W/m xk) •5

(m /s }

kf

thermal conductivity o f the fluid

(W/mxk)

ks

thermal conductivity o f the solid

(W/mxk)

kfiuid

thermal conductivity o f the fluid

(W/mxk)

ksoiid

thermal conductivity o f the solid

(W/mxk)

L

length of the pipe

n

oscillating frequency o f a pulsating flow

p

pressure

Pr

laminar Prantle number

Prt

turbulent Prantl number

q

heat flux

(m) (Hz)

(Pa)

(W/m2)

Re

laminar Reynolds number

Ret

turbulent Reynolds number

r

radial coordinate

r*

dimensionless radius o f a grid point = (r-r,)/(r0-r[)

r0

pipe outer radius

(m)

ri

pipe inner radius

(m)

S,),

source or sink of variable (j)

t

thickness o f the pipe (m)

T

temperature at a grid point

(m)

(K)

xiii

Tavs

average temperature in the solid

T*

dimensionless temperature at a grid point

T inlet

pipe inlet temperature

Tmean

mean temperature of all grid points (K)

Tsolid

solid-side temperature

Tsw

temperature at the first radial grid point in solid at a certain axial plane

Tfiuid

fluid-side temperature

Tftv

temperature at the last radial grid point in fluid at a certain axial plane

Tinlet

pipe inlet temperature

(K)

ToW

outer-wall temperature

(K)

Tiw

inner-wall temperature

(K)

U

fluid axial velocity

(K)

(m/s)

fluid radial velocity

Vo

resultant radial velocity

X

axial coordinate

(m/s)

(m/s) (m/s)

(m)

Greek Symbols s

turbulent dissipation variable

£u

tangential strain

Sr

radial strain

£x

axial strain

CTV

effective stress

(Pa)

CTU

tangential stress

(Pa)

Or

radial stress

(Pa)

Ox

axial stress

(Pa)

V

gradient/divergence operator

r*

exchange coefficient for