KHẢO SÁT ĐA DẠNG DI TRUYỀN VÀ TÍNH GÂY BỆNH CỦA NẤM Corynespora cassiicola (Berk. Curt.) Wei PHÂN LẬP TỪ CÂY CAO SU VÀ CÁC CÂY KÝ CHỦ KHÁC

National Chung Hsing University
Department of Mechanical Engineering
Master Thesis

Heat Transfer Effectiveness and Exergy Recovery
Effectiveness of a Spiral Heat Exchanger

Advisor: Jung-Yang San
Student: Nguyen Duc Khuyen

16th June 2011



Acknowledgement

It is my immense pleasure to thank all the people who have helped and
inspired me during my two-year master's study at National Chung-Hsing
University.
I would like to express my deep and sincere gratitude to my thesis advisor,
Professor Jung-Yang San for his guidance and support during my research and
study. With his inspiration, I was able to overcome the problems that I faced
during my two year study in Taiwan.
I sincerely thank all the Professors and members in the Departments of
Mechanical Engineering of NCHU, especially I am obliged to Professor Jau-Huai
Lu, Professor Cheng-Hsiung Kuo, Professor Jerry-Min Chen, Professor JauLiang Chen, Professor Shi-Chang Huang and Professor Quen-Yaw Sheen for
teaching, helping and assisting me in many different ways.
My time at NCHU was largely made enjoyable due to my friends here,
who in a short time became a part of my life. I thank my fellow labmates and
friends: Yuan-Kuei Peng, Hui-Ping Shen, Chih-Hsiang Hsu, Shen-Yang Huang,
Ding-Wei Zhen, Thac Dang, Tung Nguyen, Dony Mathew, You-Chen Huang for

their help and for all the good times, we had in the last two years.
Finally I would like to thank my family for all their love and undying
support. I wish to thank my wife, Tuan Le, whose support was endless and
selfless. I feel very fortunate to have her as my best friend and wife. Thanks to
my son, Tuan Linh, for giving me happiness and joy.

Nguyen Duc Khuyen (

i

)


Abstract

In the present study, a numerical method was developed to investigate the
heat transfer performance of a spiral heat exchanger. In the spiral heat exchanger,
two long metal strips are wound concentrically to create hot-flow channel and coldflow channel. The flow of the two fluids through a spiral heat exchanger was
considered counter-current, the hot-flow circulates counter-clockwise and the coldflow circulates clockwise. The upper surface, lower surface and outer-most side of
the spiral heat exchanger were assumed to be insulated. A heat transfer effectiveness
(  ) and an exergy recovery effectiveness (ex) were defined and evaluated based on
the calculated non-dimensional temperatures of the two counter-flow fluids in the
heat exchanger. At small NTU value, the  value initially increases with the NTU
value; while at higher NTU values, after reaching the maximum heat transfer
effectiveness, the  value starts to slightly decrease. For a set of Nt and NTU values,
the heat transfer effectiveness reaches a minimum value at C*=1.0. As the C*
approaches zero or infinity, the  value would approach the maximum. Conversely,
for a set of NTU, Nt,  c* ,  h* , and FP ,total values, as the C* approaches zero or
infinity, the exergy recovery effectiveness of the spiral heat exchanger is at the
minimum. The exergy recovery effectiveness reaches a maximum as the C* value

nears 1.0. The result also shows that, at small values of Nt (Nt < 40), the  value and
the ex value slightly increase with the Nt value; while these two values remain
almost the same when the number of turns is larger than 40 turns.

Keywords: exergy, spiral heat exchanger, heat transfer effectiveness, exergy
recovery effectiveness.

ii


Table of contents

Acknowledgement ................................................................................................... i
Abstract ...................................................................................................................ii
Table of contents ...................................................................................................iii
List of Tables and Figures ...................................................................................... v
Nomenclature........................................................................................................vii
Chapter 1 Introduction ......................................................................................... 1
1.1 Preface........................................................................................................... 1
1.2 Spiral heat exchanger and main applications................................................ 2
1.3 Survey of literature ....................................................................................... 4
1.4 Objective of the Thesis ............................................................................... 12
Chapter 2

Energy Equations for a Spiral Heat Exchanger ............................... 13

2.1 Length of a Curve in Polar Coordinates ..................................................... 13
2.2 Length of Archimedes’ spiral in polar Coordinates ................................... 14
2.3 Geometry of a spiral heat exchanger .......................................................... 17
2.4 Mathematical modeling .............................................................................. 20

2.4.1 Energy balance for hot flow ................................................................. 20
2.4.2 Energy balance for cold flow ............................................................. 22
2.4.3 Dimensionless energy equations .......................................................... 24
2.5 Heat transfer effectiveness of heat exchanger ............................................ 28
Chapter 3

Numerical Analysis.......................................................................... 30

3.1 The case for hot-flow capacity rate less than cold flow capacity rate ........ 30

iii


3.1.1 Finite-difference equations for the hot flow: ....................................... 30
3.1.2 Finite-difference equations for the cold flow: ................................... 31
3.2 The case for hot-flow capacity rate lager than cold flow capacity rate ..... 33
3.2.1 Finite-difference equations for the hot flow: ...................................... 33
3.2.2 Finite-difference equations for the cold flow: ................................... 35
3.3 Computer simulation program .................................................................... 36
3.4 Error Analysis of numerical scheme........................................................... 37
3.4.1 Equation for checking the accuracy of numerical scheme................... 37
3.4.2 Error analysis of numerical scheme ..................................................... 38
3.5 Results of heat transfer analysis.................................................................. 39
Chapter 4 Exergy Analysis ................................................................................. 45
4.1 General form of exergy change rate in a flow ............................................ 45
4.1.1 Concept of exergy analysis .................................................................. 45
4.1.2 Exergy change rate for ideal gas flow .................................................. 46
4.1.3 Exergy change rate for incompressible flow........................................ 47
4.1.4 General form of exergy change rate in a flow ..................................... 48
4.2 Exergy analysis for the spiral heat exchanger ............................................ 49

4.3 Exergy recovery effectiveness .................................................................... 50
4.4 Dimensionless exergy recovery effectiveness equations............................ 56
4.4 Results of exergy analysis and discussion .................................................. 57
Chapter 5 Conclusions ........................................................................................ 61
References ............................................................................................................ 63

iv


List of Figures

Figure 2.1 Length of a Curve in Polar Coordinates....................................................... 67
Figure 2.2 Archimedes’ spiral ....................................................................................... 67
Figure 2.3 Archimedes’ spiral is started at   0 ......................................................... 68
Figure 2.4 The constructing principle of a spiral heat exchanger ................................. 68
Figure 2.5 Schematic of a spiral heat exchanger ........................................................... 69
Figure 2.6 One side of spiral heat exchanger ................................................................. 69
Figure 2.7 Infinitesimal segment dof the spiral heat exchanger ................................ 70
Figure 2.8 Control volume of hot flow ........................................................................... 70
Figure 2.9 Control volume of cold flow ......................................................................... 71
Figure 3.1 Flow chart of the program ............................................................................. 72
Figure 3.2 The relationship between NTU and  at Nt=3 ............................................... 73
Figure 3.3 The relationship between NTU and  at Nt=6 ............................................... 74
Figure 3.4 The relationship between NTU and  at Nt=10 ............................................. 75
Figure 3.5 The relationship between NTU and  at Nt=20 ............................................. 76
Figure 3.6 The relationship between NTU and  at Nt=40 ............................................. 77
Figure 3.7 The relationship between NTU and  at Nt=80 ............................................. 78
Figure 3.8 The relationship between NTU and  at Nt=3 ............................................... 79
Figure 3.9 The relationship between NTU and  at Nt=10 ............................................. 80
Figure 3.10 Comparison of heat transfer effectiveness versus NTU at Nt=8 ................. 81

Figure 3.11 Comparison of heat transfer effectiveness at C=1.0 ................................... 82
Figure 3.12 Comparison between spiral heat exchanger at Nt=3
and counter-flow heat exchanger ................................................................ 83

v


Figure 3.13 Comparison between spiral heat exchanger at Nt=10
and counter-flow heat exchanger ................................................................. 84
Figure 3.14 Comparison between spiral heat exchanger at Nt=20
and counter-flow heat exchanger ................................................................ 85
Figure 3.15 Comparison between spiral heat exchanger at Nt=40
and counter-flow heat exchanger ................................................................. 86
Figure 3.16 Temperature distributions at Nt=10, C*=1.0 and NTU=5.0 ....................... 87
Figure 3.17 Temperature distributions at Nt=10, C*=1.0 and NTU=10 ........................ 88
Figure 3.18 Temperature distributions at Nt=10, C*=1.0 and NTU=5.0 ....................... 89
Figure 3.19 Temperature distributions at Nt=10, C*=2.0 and NTU=5.0 ....................... 90
Figure 4.1 The relationship between C* and ex at

FP ,total  0.0 ,  h*  1.2 , Nt=3...................................................................... 91
Figure 4.2 The relationship between C* and ex at

FP ,total  0.0 ,  h*  1.2 , Nt=10.................................................................... 92
Figure 4.3 Effect of  h* on ex at FP ,total  0.0 and Nt=10 .......................................... 93
Figure 4.4 Effect of Nt on ex at FP ,total  0.0 and  h*  1.2 ....................................... 94
Figure 4.5 Effect of  h* on ex,max at Nt=3 ....................................................................... 95
Figure 4.6 Effect of  c* on ex,max at Nt=10 ..................................................................... 96
Figure 4.7 Effect of FP ,total on ex at  h*  1.2 and Nt=10 .......................................... 97
Figure 4.8 Effect of FP ,total on ex at  h*  2.4 and Nt=10 .......................................... 98

vi


Nomenclature

a

constant for the Archimedean spiral, m

At

total heat transfer area of heat exchanger, m2

C

 p ) min /(mc
 p ) max
ratio of capacity rates, (mc

C*

 p )h /(mc
 p )c
modified ratio of capacity rates, (mc

C1

NTU / L*

cp

constant-pressure specific heat, kJ/kg-K

cv

constant-volume specific heat, kJ/kg-K

.

EX

rate of exergy, W

FP

pressure-drop factor

FP ,total

overall pressure drop factor

h

specific enthalpy, kJ/kg

H

height of channel or enthalpy, m or kJ

H0

enthalpy of fluid at ambient state, kJ

k

cp / cv

L

length of a spiral channel, m

L*

non-dimensional length, 2L /(2)2 a

.

m

mass flowrate, kg/s

n

number of segments

NTU

 p ) min
number of transfer units, UA t /(mc

vii


NTUh

 p )h
modified number of transfer units, UA t /(mc

Nt

total number of turns

P

pressure, Pa

P

pressure drop, Pa

Q

heat transfer rate, W

r

radius of spiral, m

r0

minimum radius of spiral, m

rc

maximum radius of spiral, m

rd

maximum radius of dash-line spiral, m

rs

maximum radius of solid-line spiral, m

R

gas constant, kJ/kg-K

s

specific entropy, kJ/kg-K

S

total entropy, kJ/K

S0

entropy of fluid at ambient state, kJ/K

T

temperature, K

T0

ambient temperature or dead-state temperature, K

u

specific internal energy, J/kg

U

overall heat transfer coefficient, W/m2-K

v

specific volume, m3/kg

W

width of hot or cold flow channel, m

Greek symbols

heat transfer effectiveness

viii

 ex

exergy recovery effectiveness



angular position, rad

0

angle at the start point of spiral, rad

d

angle at the end point of dash-line spiral, rad

c

angle at the inlet of cold flow channel, rad

s

angle at the end point of solid-line spiral, rad

h

angle at the outlet of hot flow channel, rad

*

non-dimensional angular position, 

 *

increment in the numerical method, 1/n

c

non-dimensional cold-flow temperature, (Tc  Tc,in ) / (Th,in  Tc,in )

h

non-dimensional hot-flow temperature, (Th  Tc,in ) / (Th,in  Tc,in )

*c

Tc,in / T0

*h

Th,in / T0

Subscript

0

dead state

1

state “1”

2

state “2”

avail

available

c

cold flow

gain

exergy gain

h

hot flow

ix


i

ith channel

in

inlet

loss

exergy loss

max

maximum value

mech

mechanical

min

minimum value

out

outlet

opt

optimum value

thermal

thermal exergy

x


Chapter 1

Introduction

1.1 Preface

In this modern world, energy plays an important role in maintaining or
producing different kind of products for the benefit of mankind. Human beings
are living on this planet for millions of years. We have suffered to make our
livings easier. Millions of appliances and machines have been invented to make
our lifetime easier and more comfortable. Some of these machines use electricity
to fulfill our needs, while others use different fossil fuels to fulfill the task. In
industrial processes, the fossil fuels, like oil, coal and natural gas, which are
burned to produce thermal energy. These processes are not reversible and the
exhaust gas is also reusable. The consumption of these fossil fuels involves
discharge of a variety of pollutants like carbon monoxide (CO), carbon dioxide
(CO2), nitrogen oxide (NO) and sulfur dioxide (SO2). These pollutants create fog
and result in the greenhouse effect. The greenhouse effect is responsible for
continually raising the temperature of earth which will eventually make this
planet unfit for habitation. Experts and analysts constantly survey and predict
current energy sources on the earth, according to their reports, these fuels are
going to decrease to a dangerous level if consumption increases in the future or
even remains the same. Prices of these fuels are already getting higher and higher,
and carbon emissions resulting from burning such fuels are damaging clean
environment of the earth. Therefore, we should focus our research and

development on the areas of energy science and technology. In addition to seek
for alternative energy sources (i.e. wind, solar energy and bio-fuel ... etc.), saving
energy and recovering waste energy are very critical solutions for this issue.
Nowadays, Heat exchanger is considered as one of the most common and
important equipment that has been used in industry. Different types of heat

1


exchangers are widely used in various engineering applications, in chemical,
fertilizer, petrochemical, petroleum, power, refrigeration, and others industries.
Thus, design of heat exchangers is a key issue in energy conservation. Spiral heat
exchanger is a special type of heat exchanger which can be used in a wide variety
of applications. For example, these applications include heat recovery processes
in air conditioning and refrigeration systems, chemical reactors, food and dairy
processes, etc. If the characteristics of spiral heat exchanger can be thoroughly
understood, their optimum design and operating conditions will be easily
determined. If so, the raw material and energy resources for manufacturing and
operation of this type of heat exchanger, consequently, the environmental
pollution and costs, will be reduced.

1.2 Spiral heat exchanger and main applications

The geometry of spiral heat exchanger is typically cylindrical in overall
shape. It is usually made of two long metal strips that are wound concentrically
around a split center to create two spiral channels, one for the hot fluid and one
for the cold fluid. The flow of the two fluids through a spiral heat exchanger can
be counter-current flow, co-current flow, or cross flow. For counter-current flow,
one fluid will enter at the center of the spiral and exit at the circumference, while
the other will enter at the circumference and exit at the center. It is usually used

for liquid-liquid, condensing and gas cooling applications. For co-current flow,
both fluids will enter at the center and exit at the circumference, or both will
enter at the circumference and exit at the center. This type of flow is suitable for
handling low density gases to avoid pressure loss. In general, it can be used for
liquid-liquid applications if one liquid has a considerably greater flow rate than
the other. For cross flow, one fluid will flow through the spiral path and the other
will flow through from one end of the cylinder to the other.
When firstly introduced in the 1930s, spiral heat exchanger specifically

2


addressed the needs of the pulp and paper industry. Since then, it has undergone
numerous design refinements and innovations that have made it applicable to
many heat transfer situations. One of the main advantages of spiral plate
exchanger with respect to other heat exchangers is its capacity to handle dirty
fluids, or its lower tendency to fouling. This is due to its particular geometry that
creates a constant change in the flow direction. The change of flow direction
increases local turbulence that eliminates fluid stagnant zones [6]. The high
turbulence created by the continuously curving channels minimizes fouling and
scaling tendencies. This means the spiral design improves heat transfer surface as
compared to common shell-and-tube heat exchanger. Moreover, this also means
the design requires less frequent cleaning than the other designs. Spiral heat
exchanger is also known as an excellent heat exchanger because of its high heat
transfer effectiveness. Due to the ever-changing flow direction in the spiral
channels, the boundary layers are very thin, and hence the heat transfer
coefficient is high, even in a viscous flow situation [1]. In addition, because both
fluids are flowing through a channel from one end to the other, it is possible to
have flows in true counter-current arrangement. This also contributes to a high
heat transfer effectiveness and a high overall heat transfer coefficient. Another

advantage of spiral heat exchanger is its highly efficient use of space. The
increased heat transfer effectiveness resulting from the fully counter-current flow
means that a particular assignment requires fewer spiral heat exchangers than the
straight-tube alternative. In addition, the geometrical features of the spiral heat
exchanger make it suitable to accommodate a large heat transfer area in a
relatively small volume [7]. Other advantages of spiral heat exchanger are easy
for maintenance, capable of handling slurries and larger particulates, no
differential thermal expansion between the two spirals and no inter-leakage
between the fluids [1]. However, the operating pressures of the fluids in spiral
heat exchanger are limited. For instance, the maximum plate thickness that can
be rolled is 13 mm, which limits the maximum operating pressure to 15 bar [7].
Due to availability of metal sheets, the surface area of spiral heat exchanger is

3


limited to approximately 250 m2 [1]. Spiral heat exchanger is good for
applications in cooling, heating, condensation, evaporation, pasteurization,
digester heating, waste heat recovery and pre-heating processes. Because of the
characteristics of spiral heat exchanger, it is usually used in the following
industries: paper, petrochemical, food, sugar, pharmacy, vegetable oil, water
treatment, steel and mining industries.

1.3 Survey of literature

According to Hewitt et al. [2], the concept of spiral heat exchanger was
first proposed in the late nineteenth century and was reinvented in Sweden in the
1930s. Up to now, much research work relating to the characteristic of spiral heat
exchanger were presented by researchers.
The geometry calculation of a spiral heat exchanger was proposed by Wu

[8]. In the paper, the geometric characteristics of a semi-circled spiral heat
exchanger were thoroughly described. The expressions to calculate the spiral
diameter, number of turns and length of the semi-circled spiral heat exchanger
were also provided.
Picón-Núñez, M. W et al. [7] introduced a methodology for preliminary
sizing of a spiral heat exchanger with single phase processes. In their method, a
set of initial values of plate spacing, internal diameter and plate thickness needs
to be provided. Using an iterative method, the result will be acquired when the
calculated pressure drop and heat duty meet the required specifications of the
design problem. The results were compared with case studies reported in the
literature. A comparison of the temperature profiles between the numerical result
and the analytical solution was shown. The results exhibited the difference of up
to 6 K for the case of the cold stream and of up to 14 K for the case of the hot
stream. However, they showed a similar tendency.

4


A sizing methodology for a compact spiral-plate type heat exchanger was
presented by Picón-Núñez, M. W et al. [9]. They reported a methodology for
designing the spiral-plate exchanger. In their work, the width and spacing of the
channels were considered as continuous variables. Using a set of equations, the
“thermal length” and the “hydraulic length” can be found based on the initial
exchanger geometry. An iterative process was implemented to achieve equality
between the “thermal length” and “hydraulic length” on both streams. The result
was used to select the final dimensions of the heat exchanger.
Sterger et al. [10] examined the use of a double-spiral heat exchanger for
catalytic incineration of contaminated air, such as carbon monoxide,
hydrocarbons, organic compounds, aerosols, and microorganisms. Analytical and
numerical solutions were developed to obtain the maximum value of the thermal

figure of merit (E) and the optimum NTU value for the number of turns up to 18.
Experiments were also carried out to test the validity of the solutions. The result
shows that, the number of transfer units for which the maximum figure of merit
was predicted to occur could not be attained in the experimental apparatus.
Nevertheless, the level of temperature required for catalytic incineration of
ordinary contaminants in air was readily obtained. For example, temperatures of
air as high as 900 K were attained at the core with a total heat input equivalent
to 236 K and a net heat input equivalent to 71 K.
Target et al. [11] used MACSIMA code to obtain closed-form solutions of
the temperature distribution in a double-spiral heat exchanger. In their study, the
spiral heat exchanger was considered to be with one turn and multi-turn, both
with and without heat losses to the surroundings, were considered to find the
optimum number of transfer unit and the maximum value of the thermal figure of
merit (E) at equal rate of flow. The results shown that, at small number of turn,
the temperature profiles of the double-spiral heat exchangers are very irregular.
For case of several turns, an optimal value of the number of transfer units exists
for which the figure of merit is a maximum. For very large NTU, the figure of

5


merit may approach zero or a finite value depending on the configuration of the
inlets and exits.
Naphon and Wongwises [12] experimentally obtained the average in-tube
heat transfer coefficients in a spiral-coil heat exchanger. The experiments were
conducted at various temperatures and flow rates for both working fluids. In their
results, the total heat transfer rate tends to slightly increase with an increase of
water mass flow rate, and it decreases with increasing inlet water temperatures.
Ho and Wijeysundera [13-14] et al. investigated the performance of a
compact spiral-coil heat exchanger. Two theoretical models individually based

on unmixed and mixed air-flow conditions were considered to predict the
performance of the heat exchanger. Experimental studies were also conducted to
verify the predictions of the models. The result shows the conformability
between the computed values and the measured values.
Rennie et al. [15] used the PHOENICS 3.3 commercial software to
numerically evaluate the heat transfer performance of a double-pipe helical heat
exchanger with two different tube diameters. The flow pattern was considered
either parallel flow or counter-flow. The result shows that, the annulus Nusselt
number of the heat exchanger is proportional to the modified Dean number.
Thermal performance of a spiral heat exchanger was studied by Th. Bes
and W. Roetzel [16]. Using mathematical transformations, the energy balance
equations for the main and side spirals were solved to achieve a thermal analysis
of the temperature distribution on both sides. A dimensionless criterion number
(CN) and a log mean temperature difference correction factor (F=ln(1+CN2)/CN2)
were proposed. However, in their study, the influence of outermost turn was
neglected, thus, their theory give a better result for a higher number of turn.
In analysis of heat exchanger, there are two commonly used methods for
calculating the heat transfer, namely, the log mean temperature difference

6


(LMTD) method and effectiveness-NTU ( -NTU) method. If the inlet and outlet
temperatures of both the hot and cold fluids are known, the LMTD method can
be used to solve the problem. In case of the outlet temperatures are not known,
using the LMTD method requires an iterative scheme. In such case, the analysis
can be simplified using the -NTU method [3]. Since W.M. Kays and A.L.
London [4] first developed in 1955, up to now, many researchers have used the NTU method to analyze the performance of heat exchangers.
H.A. Navarro and L.C. Gomez [17-18] provided a numerical method for
analyzing the thermal performance of cross-flow heat exchanger. Firstly, the

governing equations were derived for a single-pass cross-flow heat exchanger
with one fluid mixed and another unmixed. Then, a numerical method was
developed to obtain the temperature distribution and the -NTU relations of the
heat exchanger. The heat exchanger can be with one tube or with several tubes,
including different tube fluid circuiting configurations. A comparison of the heat
transfer effectiveness between model prediction and algebraic relation in the
literature was carried out. The result shows that there is a maximum difference of
about 10%.
Y.H. Cho and H.M. Chang [19] proposed a numerical method for
predicting the effectiveness-NTU relation of a triple-passage counter-flow heat
exchanger. The fourth order Runge-Kutta method was used to solve
dimensionless-form governing equations. The effectiveness of the heat exchanger
was defined as the ratio of the actual heat transfer rate of the cold fluid to the
maximum heat transfer rate obtained at a very large NTU values and a constant
NTU ratio. The effectiveness was also shown in the graphs as a function of
NTU1 , NTU1 / NTU 2 , C2 / C3 , C1 / C3 and  2 (1) .

L.C. Burmeister [20] derived a formula for the dependence of heat transfer
effectiveness on the NTU value for a spiral-plate heat exchanger with equal flow
capacity rates. They showed that, there is a range of NTU in which the

7


effectiveness reaches a maximum value. However, when the number of turns is
less than 4, the maximum effectiveness will not appear.
San et al. [21] investigated the performance of a cross-flow serpentine
heat exchanger. The -NTU relation was obtained using a numerical analysis.
They found that, in the heat exchanger, the number of tube (N) is a variable
affecting the

value. However, as the N is larger than 11, the effectiveness will

almost independent of the N value. At a fixed NTU value, the heat transfer
effectiveness reached a minimum value at Ct*  1.0. They also found that, at
NTU<6.0 and N>11, the -NTU relation of the heat exchanger is almost the same
as that of straight-through cross-flow heat exchanger.
Th. Bes and W. Roetzel [22] developed an analytical method to calculate
the temperature changes in a counter flow spiral heat exchanger. A set of
equations was derived to mathematically represent the energy balance for each
turn of the spiral heat exchanger. They found that, the effect of outer-most turns
on the temperature distribution is much stronger than that of the inner-most turn.
The larger the value of NTU, the greater the influence of the outer-most turn on
the temperature distribution in the spiral heat exchanger. They also found that,
the influence of minimal radii on the heat transfer effectiveness is small. At large
NTU values, as the NTU increases, depend on the number of turns, the
effectiveness achieves a maximum value and then it decreases with an increase of
the NTU.
Adamski [23] experimentally determined the heat transfer and pressure
drops characteristics of a longitudinal flow spiral recuperator in a ventilation
system. The experiment setup was designed and constructed to measure the flow
rates, the pressure drop and the inlet and outlet temperatures of each fluid. In the
experiment, the cross-sectional area of duct, the length of heat exchanger, the
spiral wall thickness and the width of channel was fixed at 0.0326 m2, 1.35 m,
0.00015 m and 0.003 m respectively. Base on the measured data, the Reynolds

8


number and Nusselt number were determined. The -NTU relation and Fanning

friction factor (f) were also obtained. The result shows that the heat transfer
effectiveness of the recuperator under testing is always greater than 0.85.
Nowadays, the energy demands of the world are accelerating and
increasing at a tremendous rate, while the available natural resources, especially,
the fossil fuel resources are limited. Thus, much effort has been done and focused
on the design of more efficient energy devices and processes. In the heat transfer
industry, engineers and scientists have been traditionally applying the heat
transfer effectiveness to evaluate the performance of heat exchangers. However,
the effectiveness of a heat exchanger only indicates the relative magnitude of the
heat transfer loading in a process. In a waste heat recovery process, the thermal
energy is usually converted directly into work or cooling effect. In this case, the
efficiency of the process in term of exergy can not be fully expressed by the heat
transfer effectiveness. Thus, an evaluation factor based on the second-law of
thermodynamics must be used to evaluate the performance of the heat exchanger.
Up to now, much work on the second-law analysis of heat exchanger has been
conducted by researchers. San et al. [24] analyzed the second-law performance of
a two-dimensional regenerator for various operating conditions. The result shows
that, there is an optimum effectiveness corresponding to the maximum secondlaw efficiency.
Naphon [25] performed a second law analysis for a horizontal concentrictube heat exchanger. The central finite difference method was used to solve the
energy conservation equations. The temperature distribution, entropy generation
and exergy loss in the double-tube heat exchanger were acquired. The effects of
relevant parameters on the entropy generation and exergy loss were also
discussed. An experiment was conducted to verify the prediction of the
theoretical model. The result shows that, the hot water mass flow rate, the cold
water mass flow rate and the inlet hot water temperature have significant effect
on the entropy generation, entropy generation number and exergy loss in the heat

9

exchanger. The result also shows a good agreement between the predicted results
and the measured data.
San et al. [26] performed a second-law analysis for a wet cross-flow heat
exchanger under summer and winter weather conditions. The effectiveness,
exergy recovery factor and second-law efficiency of the heat exchanger were
defined and numerically evaluated under various operating conditions. The result
shows that, the effectiveness depends on NTUc, c , and weather conditions. In
the heat exchanger, there are two optimum operating conditions. One
corresponds to the highest second-law efficiency and the other corresponds to the
maximum exergy recovery factor.
Gupta et al. [27] did a second law analysis for a cross-flow heat exchanger
with axial dispersion in one fluid. An analytical model was used to evaluate the
exergy destruction in the heat exchanger. The result shows that, from the
perspective of the second law of thermodynamics, there is an optimum design for
the heat exchanger. Based on this optimial design, the heat exchanger will
operate with a minimum loss of exergy.
Sarangi et al. [28] analyzed the entropy generation in a counter-flow heat
exchanger. A set of entropy production (Ns) equation for some particular cases
was derived in terms of relevant dimensionless parameters. The result shows that,
the entropy production decreases with an increase of the NTU. For balanced flow
(C=1), the entropy generation reaches a maximum value at =0.5; while at C=0,
the maximum entropy generation is achieved at =1.0.
San et al. [29] studied the second-law efficiency of a cross flow serpentine
heat exchanger. They found that, in the serpentine heat exchanger, the second
law efficiency is a function of number of transfer units (NTU), ratio of heat
capacity rates ( Ct* ), ratio of channel-flow inlet temperature to dead-state
temperature (  * ), number of rectangular tubes (N) and pressure-drop ratio on the

10

channel-flow side ( Pc / P0 ). The results of the analysis shows that, at Ct*  1.0 ,
the heat transfer effectiveness ( ) reaches a minimum value, while this is the
optimum operating point for acquiring the maximum second-law efficiency.
Wu et al. [30] defined an exergy transfer effectiveness to describe the
performance of heat exchangers operating at above/below ambient temperature
and with (or without) finite pressure drop. Discussion about the effects of NTU,
ratio of heat capacity rates and flow patterns on the exergy transfer effectiveness
was performed, while the working fluids were considered as ideal gas and
incompressible fluid operating with (or without) finite pressure drop. The study
also compared the exergy transfer effectiveness with the heat transfer
effectiveness for heat transfer in parallel flow, counter-flow and cross-flow
arrangements.
Ruan et al. [31] investigated the exergy effectiveness of a three-fluid heat
exchanger. Different forms of exergy effectiveness were defined for each fluid in
the heat exchanger under various operating conditions. The effects of design
parameters on the exergy effectiveness for parallel-flow arrangement were
discussed. They also compared the exergy effectiveness for the parallel-flow
arrangement with that for the counter-flow arrangement. The result shows that,
the exergy effectiveness for the parallel-flow arrangement is always higher than
that for the counter-flow arrangement. However, the trend of the exergy
effectiveness for the parallel-flow arrangement is similar to that for the counterflow arrangement.
San et al. [32] defined an exergy recovery index (  II ) to express the
second-law performance of heat exchangers using in waste heat recovery process.
A general mathematical model was developed to obtain the relation between the
heat transfer effectiveness and exergy recovery index for heat exchangers with
any type of flow arrangement. The exergy recovery index of cross-flow heat
exchanger with both fluids unmixed was established. The result shows that, the

11

exergy recovery index decreases with an increase of the modified overall
pressure-drop factor ( F*P ,total ). The maximum exergy recovery index was found
to be at the ratio of modified heat capacity rates (C*) in the vicinity of 1.0.

1.4 Objective of the Thesis

In this research, the heat transfer performance of a spiral heat exchanger is
numerically investigated. A set of formulas is derived and solved symbolically to
acquire the temperature distributions of the two counter-flow fluids in the heat
exchanger. The relationship between the heat transfer effectiveness and the
number of transfer units ( -NTU) of the spiral heat exchanger with various ratios
of flow-capacity rates are studied. An exergy recovery effectiveness is defined,
and the relationship between the heat transfer effectiveness and exergy recovery
effectiveness is established.

12


Chapter 2

Energy Equations for a Spiral Heat Exchanger

2.1 Length of a Curve in Polar Coordinates

Consider a curve in polar coordinates (Figure 2.1), the distance (r)
between a point on the curve and a reference point “O” is,

r  f   ,

0    c

(2.1)

For the arc length (s) between two points (P and Q) on the curve, by
applying the Pythagorean Theorem, we can get the following relation:

 s 2  (r ) 2   r 

2

If so,
2

s 
r 
2
   r  
  
  

2

Therefore, for an infinitesimal increment of , the arc length “ds” can be
expressed as:

 dr 
ds
 r2  


d
 d 

2

2



 dr 
ds  r  
 d
 d 
2

(2.2)

Integrating the above equation, the arc length formula is then,
c

c

L   ds  
0

0

2

 dr 
r 
 d
 d 
2

13

(2.3)