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THERMOELECTRIC COOLING DEVICES: THERMODYNAMIC
MODELLING AND THEIR APPLICATION IN ADSORPTION
COOLING CYCLES
ANUTOSH CHAKRABORTY
(B.Sc Eng. (BUET), M.Eng. (NUS))
A THESIS SUBMITTED
FOR THE DEGREE OF DOCTOR OF PHILOSOPHY
DEPARTMENT OF MECHANICAL ENGINEERING
NATIONAL UNIVERSITY OF SINGAPORE
2005
i
Acknowledgements
I am deeply grateful to my supervisor, Professor Ng Kim Choon, for giving me the
guidance, insight, encouragement, and independence to pursue a challenging project.
His contributions to this work were so integral that they cannot be described in words
here.
I would like to thank Associate Professor Bidyut Baran Saha of Kyushu University,
Japan, for the encouragement and helpful technical advice.
I am deeply grateful to Mr. Sai Maung Aye for his assistance in the electro-adsorption
chiller experimentation program and Mr. R Sacadeven for kindly assisting in the
procurement of equipment, and construction of the constant-volume-variable-pressure
(CVVP) experimental test facility.
I would like to extend my deepest gratitude to my parents for their complete moral
support. Finally, I wish to thank my wife, Dr. Antara Chakraborty and my son Amitosh
Chakraborty, for being a constant source of mental support.
Last but not least, I wish to express my gratitude for the honor to be co-author with my
supervisor in six international peer-reviewed journal papers, three international peer-
reviewed conference papers and one patent (US Patent no 6434955). I also thank A*
STAR for providing financial assistance to a patent application on the electro-
adsorption chiller: a miniaturized cooling cycle design, fabrication and testing results. I
extend my appreciation to the National University of Singapore for the research
scholarship during the course of candidature, to the Micro-system technology initiative
(MSTI) laboratory for giving me full support in the setting up of the test facility.
ii
Table of Contents
Acknowledgements i
Table of Contents ii
Summary vi
List of Tables viii
List of Figures x
List of Symbols xv
Chapter 1 Introduction 1
Chapter 2 Thermodynamic Framework for Mass, Momentum,
Energy and Entropy Balances in Micro to Macro Control
Volumes 9
2.1 Introduction 9
2.2 General form of balance equations 13
2.2.1 Derivation of the Thermodynamic Framework 13
2.2.2 Mass Balance Equation 16
2.2.3 Momentum Balance Equation 18
2.2.4 Energy Balance Equation 20
2.2.5 Summary of section 2.2 26
2.3 Conservation of entropy 26
2.4 Summary of Chapter 2 30
Chapter 3 Thermodynamic modelling of macro and micro
thermoelectric coolers 32
3.1 Introduction 32
3.2 Literature review 34
3.3 Thermoelectric cooling 37
iii
3.3.1 Energy Balance Analysis 38
3.3.2 Entropy Balance Analysis 41
3.3.3 Temperature-entropy plots of bulk
thermoelectric cooling device 42
3.4 Transient behaviour of thermoelectric cooler 51
3.4.1 Derivation of the T-s relation 54
3.4.2 Results and discussions 55
3.5 Microscopic Analysis: Super-lattice type devices 60
3.5.1 Thermodynamic modelling for thin-film
thermoelectrics 63
3.5.2 Results and discussions 67
3.6 Summary of Chapter 3 73
Chapter 4 Adsorption Characteristics of Silica gel + water 74
4.1 Characterization of Silica gels 75
4.2 Isotherms of the silica gel + water system 80
4.2.1 Experiment 80
4.2.2 Results and analysis for adsorption isotherms 85
4.3 Summary of Chapter 4 93
Chapter 5 An electro-adsorption chiller: thermodynamic modelling
and its performance 94
5.1 Introduction 94
5.2 Thermodynamic property fields of
adsorbate-adsorbent system 95
5.2.1 Mass balance 96
5.2.2 Enthalpy energy and entropy balances 96
iv
5.2.3 Specific heat capacity 103
5.2.4 Results and discussion 104
5.3 Thermodynamic modelling of an
electro-adsorption chiller 106
5.3.1 Mathematical model 109
5.3.2 COP of the electro-adsorption chiller 116
5.3.3 Results and discussion 120
5.4 Summary of Chapter 5 127
Chapter 6 Experimental investigation of an electro-adsorption chiller 128
6.1 Design development and fabrication 128
6.2 Experiments 138
6.3 Results and discussions 143
6.4 Comparison with theoretical modelling 152
6.5 Summary of Chapter 6 155
Chapter 7 Conclusions and Recommendations 156
References 160
Appendices
Appendix A Gauss Theorem approach 171
Appendix B The Thomson effect in equation (3.7) 180
Appendix C Energy balance of a thermoelectric element 185
Appendix D Programming Flow Chart of the thin film thermoelectric
cooler (Superlattice thermoelectric element) 190
Appendix E Programming Flow Chart of the electro-adsorption chiller
and water properties equations 192
v
Appendix F The energy flow of major components of an electro-
adsorption chiller 201
Appendix G Design of an electro-adsorption chiller (EAC) 207
Appendix H The Transmission band of the fused silica or quartz 215
Appendix I Calibration certificates 216
vi
Summary
This thesis presents a thermodynamic framework, which is developed from the basic
Boltzmann Transport Equation (BTE), for the mass, momentum and energy balances
that are applicable to solid state cooling devices and electro-adsorption chiller.
Combining with the concept of Gibbs law, the thermodynamic approach has been
extended to give the entropy flux and the entropy generation analyses which are crucial
to quantify the impacts of various dissipative mechanisms or “bottlenecks” on the solid
state cooler’s efficiency.
The thesis examines the temperature-entropy (T-s) formulation, which successfully
depicts the energy input and energy dissipation within a thermoelectric cooler and a
pulsed thermoelectric cooler by distinguishing the areas under process paths. The
Thomson heat effect, which has been omitted in literature, is now incorporated in the
present analysis. The simulation shows that the total energy dissipation from the
Thomson effect is about 5-6% at the cold junction. On a micro-scale superlattice
thermoelement level, the BTE approach to thermodynamic framework enables the
temperature-entropy flux formulation to be developed. As the physical scale
diminishes, the collision effects of electrons, holes and phonons become significant
and such effects are accounted as entropy generation sources and the corresponding
energy dissipation due to collision effects is mapped using the T-s diagram.
Extending the BTE to a miniaturized adsorption chiller such as the electro-adsorption
chiller (EAC), the adsorbent (silica gel) properties and the isotherm characteristics of
silica gel-water systems are first investigated experimentally before a full-scale
simulation could be performed. The thermodynamic property fields of adsorbate-
adsorbent systems such as the internal energy, enthalpy and entropy as a function of
pressure (P), temperature (T) and the amount of adsorbate (q) have been developed and
vii
the formulation of the specific heat capacity of adsorbate-adsorbent system is proposed
and verified with available experimental data in the literature. Assuming local
thermodynamic equilibrium, the proposed thermodynamic framework is applied
successfully to model the electro-adsorption chiller. A parametric study of the EAC is
performed to locate its optimal operating conditions.
Based on the electro-adsorption chiller modelling, an experimental investigation is
performed to verify its performances at the optimal and rating conditions. The bench-
scale prototype is the first-ever experimentally built EAC where the dimensions are
based on the earlier simulations. To provide uniform heat flux and the necessary power
level to emulate heat input similar to heat generation of computer processors or CPUs,
infra-red heaters are designed to operate through a four-sided and tapered
kaleidoscope. The optimum COP of the EAC has been measured to be 0.78 at a heat
flux of about 5 W/cm2, and the load surface temperature is maintained below or just
above the ambient temperature, a region that could never be achieved by forced-
convective fan cooling. The experiments and the predictions from mathematical model
agree well.
The performance investigation of EAC’s evaporator provided a successful study of the
pool boiling heat flux. Such pool boiling data, typically at 1.8-2.2 kPa, are not
available in the literature. Using the water properties at the working pressure and the
measured boiling data, a novel and yet accurate boiling correlation for the copper-foam
cladded evaporator has been achieved.
viii
List of Tables Chapter 2 Table 2.1 Examples of the source term, Φ, for two common applications. 20 Table 2.2 Explanation of energy source in different fields. 26
Table 2.3 Example of processes leading to the irreversible production of entropy. 29 Table 2.4 Summary of the conservation equations. 31 Chapter 3 Table 3.1 Physical parameters of a single thermoelectric couple. 43 Table 3.2 Description of close loop a-b-c-d-a (Figure 3.3). 47 Table 3.3 Physical parameters of a pulsed thermoelectric cooler. 60 Table 3.4 the balance equations. 64 Table 3.5 the energy, entropy flux and entropy generation equations of the well and the
barrier. 65 Table 3.6 the thermo-physical properties of bulk SiGe and superlattice Si.8Ge.2/Si
thermoelectric element. 67 Table 3.7 Performance analysis of the superlattice thermoelectric cooler at various
electric field. 71 Chapter 4 Table 4.1 Characterization data for the type ‘RD’ and ‘A’ silica gel. 77 Table 4.2. Thermophysical properties of silica gels. 79 Table 4.3. The measured isotherm data for type ‘A’ silica gel. 86
Table 4.4. The measured isotherm data for type ‘RD’ silica gel. 87
Table 4.5. Correlation coefficients for the two grades of Fuji Davison silica gel + water systems (The error quoted refers to the 95% confidence interval of the least square regression of the experimental data). 92
ix
Chapter 5 Table5.1 Specifications of component and material properties used in the simulation
code. 121 Table 5.2 Energy balance schedule of the electro-adsorption chiller (EAC). 127 Chapter 6 Table 6.1 Energy utilization schedule of an Electro-Adsorption Chiller (Refer to Figure
6.6). 142 Table 6.2 Heat flux calibration table (refer to Figure 6.5). 144 Table 6.3 The heat flux density results from measurements and ray-tracing simulation
(the source have a total power of 3.8 KW, refer to Figure 6.7). 145 Table 6.4 Evaporator temperature and cycle average COP (Experimental and simulated
values) 154
x
List of Figures Chapter 1 Figure 1.1 The dependence of the chip surface temperature with the power intensity of
CPU. 2 Chapter 2 Figure 2.1 Schematic diagram of a thermal transport model showing the applicability
of Boltzmann Transport Equation (BTE) in the thin films. 10 Figure 2.2: The motion of a particle specified by the particle position r and its velocity
v''. 14 Chapter 3 Figure 3.1 The schematic view of a thermoelectric cooler. 37 Figure 3.2 T-α diagram of a thermoelectric pair. 45 Figure 3.3 Temperature-entropy flux diagrams for the thermoelectric cooler at
maximum coefficient of performance identifies the principal energy flows. 46 Figure 3.4 Characteristics performance curve of the thermoelectric cooler. COP is
plotted against entropy flux for the both N and P legs. 49 Figure 3.5 Temperature-entropy flux diagram for the thermoelectric pair at the
optimum current flow by the present study and the other researcher (Chua et al., 2002(b)). 49
Figure 3.6 Temperature-entropy generation diagram of a thermoelectric cooler at
optimum cooling power. Area (abcd) is the total heat dissipation along the thermoelectric element, where area (aefd) is the irreversibility due to heat conduction and area (aghd) indicates the entropy generation due to joule effect. 50
Figure 3.7 Schematic diagram of the first transient thermoelectric cooler. (a) indicates
the normal operation and cold substrate is out of contact, (b) defines the thermoelectric pulse operation and cold substrate is in contact with the cold junction, (c) shows the current profile during normal thermoelectric operation, (d) indicates that the pulse current is applied to the device and (e) the cycle of (tno + tp) is repeated such that the device is operated in an ac mode. 53
Figure 3.8 the cold reservoir temperature of a super-cooling thermoelectric cooler as a
function of time. The small loop indicates the super-cooling during current pulse operation. Temperature drop between the cold and hot junctions
xi
subjected to a step current pulse of magnitude 3 times the DC current and of width 4 s. A indicates the beginning of pulse period, B defines the maximum temperature drop and C represents the end of pulse period, C’ is the beginning of non-contact period, B’ indicates maximum temperature rise due to residual heat. 56
Figure 3.9 T-s diagram of a pulsed thermoelectric cooler during non-pulse operation;
C′ defines the beginning of non-contact, B′ indicates the maximum temperature rise of cold junction during non-pulse period and A′ represents the maximum temperature drop of cold junction with no cooling power. 58
Figure 3.10 Temperature-entropy diagram of a pulsed thermoelectric cooler during
pulsed current operation; A defines the beginning of cooling, B represents the maximum temperature drop and C indicates the maximum cooling power. 59
Figure 3.11 A schematic of the superlattice thermoelectric element (Elsner et al., 1996) 61 Figure 3.12 Schematic diagram of a superlattice thermoelement with electric
conducting wells and electric insulating barriers (Antonyuk et al., 2001). 66 Figure 3.13 the variations of electron temperature in a thin film thermoelectric element
at different electric field; (---) indicates the phonon temperature, (—) defines the electron temperature at 12.5 Kamp/cm2, (---) shows the electron temperature at 25 Kamp/cm2 and () represents the electron temperature at 125 Kamp/cm2. 68
Figure 3.14 The temperature-entropy diagram of a superlattice thermoelectric couple
(both the p and n-type thermoelectric element are shown here). 69 Figure 3.15 the temperature-entropy diagrams of p and n-type thin film thermoelectric
elements. The close loop A-B-C-D shows the T-s diagram when the collision terms are taken into account and the close loop A′-B′-C′-D′ indicates the T-s diagram when collisions are not included. Energy dissipation due to collisions is shown here. 72
Chapter 4 Figure 4.1. Pore size distribution (DFT-Slit model) for two types of silica gel. 78 Figure 4.2. Schematic diagram of the constant volume variable pressure test facility:
T1, T2, T3 = resistance temperature detectors; Ps, Pd, Pe = capacitance manometers. 81
Figure 4.3 The experimental set-up of the constant volume variable pressure (C.V.V.P) test facility. 83
xii
Figure 4.4: Isotherm data for water vapor onto the type ‘RD’ silica gel; for
experimental data points with: () T = 303 K; () T = 308 K; () T = 313 K; () T = 323 K; (+) T = 338 K, for computed data points with solid lines for the Tóth’s equation, for manufacturer’s data points with: (--------) T = 323 K of NACC; 2 (⎯ − − ⎯) T = 338 K of NACC,2 and for one experimental data point with: () T = 298 K for the estimation of monolayer saturation uptake. 89
Figure 4.5: Isotherm data for water vapor onto the type ‘A’ silica gel; for experimental
data points with: () T = 304 K; () T = 310 K; () T = 316 K; () T = 323 K; (+) T = 338 K, for computed data points with solid lines for the Tóth’s equation, and for one experimental data point with: () T =298 K for the estimation of monolayer saturation uptake. 90
Figure 4.6. Isotherm data for water vapor with the silica gel by the present study and
other researchers with: (⎯⎯) type ‘A’ by the present study; (⎯⎯) type ‘A’ by Chihara and Suzuki; (⎯∆⎯) type ‘RD’ by the present study; () type ‘RD’ by Cho and Kim; (-------) type RD by NACC. 92
Chapter 5
Figure 5.1: Different specific heat capacities of a single component type 125 silica gel + water system at 298 K: a data point (---) indicates refers to the liquid adsorbate specific heat capacity of the adsorption system, (----) defines the gas adsorbate specific heat capacity of adsorbent + adsorbate system, () refers to the newly interpreted specific heat capacity with average isosteric heat of adsorption 2560 KJ kg-1, () the specific heat capacity of type 125 silica gel + water system as obtained through DSC and () the specific heat capacity of the adsorption system with the THS technique. 105
Figure 5.2 Schematic showing the principal components and energy flow of the electro-adsorption chiller. 108
Figure 5.3 A block diagram to highlight the sensible and latent heat flow and the
energy balance of a thermoelectrically driven adsorption chiller. 119 Figure 5.4. Temporal history of major components of the Electro-Adsorption Chiller. 123 Figure 5.5 Effects of cycle time on chiller cycle average cooling load temperature,
evaporator temperature and COP. The chosen 500 s cycle time, shown by the solid dotted vertical line, is close to, the minimum cooling load temperature. 124
Figure 5.6 Dűhring diagram for 400 s and 700 s cycles. 125
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Figure 5.7 COP, the cycle average load temperature and the evaporator temperature as functions of the cycle average input current to the thermoelectric modules.
126 Chapter 6 Figure 6.1 The test facility of a bench-scale electro-adsorption chiller. 129 Figure 6.2 Assembly of the reactor; (a) the bottom plate (outside), (b) the bottom plate
(inside view), (c) the heat exchanger, (d) the heat exchanger with bottom plate, (e) the PTFE enclosure and (f) two reactors. 131
Figure 6.3 the condenser unit. 132 Figure 6.4 The evaporator unit; (a) shows the body of the evaporator, (b) indicates the
quartz bottom plate through which infra red heat passes, (c) represents the top part of the evaporator, (d) the IR heater system and (e) is the evaporator. 135
Figure 6.5: Drawing of the kaleidoscope/cone concentrator developed for radiative
heat transfer between the heat source (point 1, four rods of 4 kW total power), and the bottom plate (quartz point 3) of the evaporator (exit at the top). 136
Figure 6.6 Schematic diagram of a prototyped Electro-Adsorption chiller (EAC) to
explain Energy Utilization. 141 Figure 6.7 Tracing 20 rays from the lambertian sources through the kaleidoscope
systems. The heat flux after the source is measured in point 1, then at the exit of the kaleidoscope in point 2, and finally after having passed the fused silica window in point 3. 145
Figure 6.8 The experimentally measured temporal history of the electro-adsorption
chiller at a fixed cooling power of 120 W: () defines the temperature of reactor 1; () indicates the silica gel temperature of reactor 2; () shows the condenser temperature; () represents the load surface (quartz) temperature and () is the evaporator temperature. Hence the switching and cycle time intervals are 100s and 600s, respectively. 146
Figure 6.9 The DC current profile of the thermo-electric cooler for the first half-cycle. 147 Figure 6.10 Effects of cycle time on average load surface (quartz) temperature,
evaporator temperature and cycle average net COP. The constants used in this experiment are the heat flux qevap = 4.7 W/cm2 and the terminal voltage of thermoelectric modules V = 24 volts. 149
Figure 6.11 Experimentally measured load and evaporator temperatures as a function
of heat flux. 150
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Figure 6.12. Heat flux (q flux) versus temperature difference (DT) for water at 1.8 kPa (author’s experiment), 4 kPa (McGillis et al., 1990). and 9 kPa (McGillis et al., 1990). The bottom most data (of 1.8 kPa) is observed to have insignificant boiling, i.e., mainly convective heat transfer by water.
152 Figure 6.13. Experimentally measured temporal history versus the simulated
temperatures of major components of electro-adsorption chiller at a fixed cooling power of 80 W. +++ indicates the simulated load surface temperature using the proposed correlation (Csf = 0.0136, n = 0.33 and m = 0.37) and ×××× shows the simulated load surface temperature using Rohsenow pool boiling correlation (Csf = 0.0132 and n = 0.33).
153
xv
List of Symbols
A area m2
Abed,ads adsorber bed heat transfer area m2
Abed,des desorber bed heat transfer area m2
Acond condenser heat transfer area m2
Aevap Evaporator heat transfer area m2
Ate cross sectional area of a thermoelectric element m2
Atube,ads tube heat transfer area m2
COP the Coefficient of Performance --
COP eac Coefficient of performance of electro-adsorption chiller --
COP net net Coefficient of Performance --
c the mass fraction kg kg-1
cas the proposed specific heat capacity J kg-1 K-1
cgs specific heat capacity at gaseous phase J kg-1 K-1
cls specific heat capacity at liquid phase J kg-1 K-1
cp specific heat capacity J kg-1 K-1
cp,CE specific heat capacity of ceramic J kg-1 K-1
cp,mf specific heat capacity of copper plate J kg-1 K-1
cp,sg specific heat of silica gel J kg-1 K-1
cp,te specific heat capacity of thermoelectric material J kg-1 K-1
cv specific heat capacity at constant volume J kg-1 K-1
DT temperature difference between Quartz surface
and the water vapour temperature at saturation K
Dso a kinetic constant for the silica gel water system
d3 3-D operator
xvi
E electric field volt
Ea activation energy of surface diffusion J kg-1
e the total energy per unit mass J kg-1
F the Faraday constant Coulomb mol-1
F the external force N
f the statistical distribution function
GF the geometric factor m
H the total enthalpy of the system J kg-1
h enthalpy per unit mass J kg-1 K-1
htuin convective heat transfer coefficient (equation 5.33) Wm-2 K-1
I the current amp
J heat flux Wm-2
Js entropy flux W m-2 K-1
JE electric field W m-3
J(=I/A) the current density amp m-2
Jk diffusion flow of component k
Jp the momentum flux N m-3
Js,pn the entropy flux both the p and n leg W m-2 K-1
k the thermal conductivity W m-1 K-1
K thermal conductance W K-1
Ko pre-exponential coefficient kPa-1
kte thermal conductivity of thermoelectric element
(equations 5.28 and 5.31) W m-1 K-1
L length m
Lte characteristics length of thermoelectric element m
xvii
M current pulse magnitude --
Mcond mass of condenser including fins kg
MCE mass of ceramic plate kg
Mevap mass of evaporator including foams kg
Mref,cond mass of condensate in the condenser kg
Mref,evap mass of refrigerant in the evaporator kg
MHX tube weight + fin weight + bottom plate weight kg
Mmf mass of thin copper plate kg
Msg silica gel mass kg
m* the mass of a single molecular particle kg
m mass kg
ma* mass of adsorbate under equilibrium condition kg
mc mass flow rate at the condenser (Figure 5.3) kg s-1
me mass flow rate at the evaporator (Figure 5.3) kg s-1
mw mass flow rate of water (= 0.01 kg s-1) kg s-1
N number
Nte no of thermoelectric couples
n carrier density m-3
nm number of thermoelectric modules
p the momentum vector
P pressure Pa
P1 saturated pressure Pa
ncolliinP , power input with non-collision effect mW
colliinP , power input with collision effect mW
lpr Prandtl number (equation 5.35)
xviii
Q the total heat or energy W or J
Qo the heat transport of electrons/ holes J mol-1
ncolliHQ , heating power at hot side with non-collision effect mW
colliHQ , heating power at hot side with collision effect mW
ncolliLQ , cooling power at cold side with non-collision effect mW
colliLQ , cooling power at hot side with collision effect mW
colliQ power dissipation due to collisions mW
qm the monolayer capacity kg kg-1
q* the adsorbed quantity of adsorbate by the adsorbent
under equilibrium conditions kg kg-1
q fraction of refrigerant adsorbed by the adsorbent kg kg-1
fluxq ′′ Heat flux at the evapotrator Wcm-2
R total resistance ohm
Rp average radius of silica gel particle m
Rg Universal gas constant J kg-1 K-1
r the particle position vector
r′ the new position vector at an infinitesimally time dt
S total entropy J kg-1 K-1
Ste the total Seebeck coefficient V K-1
s entropy per unit mass J kg-1 K-1
T temperature K
Tquartz quartz surface temperature K
Tevap evaporator temperature K
t time s
xix
t1 the töth constant -
tcycle cycle time s
U a peculiar velocity m s-1
U total internal energy of the system J.kg-1
Uads overall heat transfer coefficient of adsorber W m-2 K-1
Ucond condenser heat transfer coefficient W m-2 K-1
Udes Overall heat transfer coefficient of desorber W m-2 K-1
u internal energy per unit mass J kg-1 K-1
V volume of the system m3
V Input voltage to the TE modules volt
v ′′ the particle velocity vector m s-1
v′ the particle velocity vector at an infinitesimally time dt m s-1
v velocity m s-1
υ (= 1/ρ) specific volume m3 kg-1
w energy flow (the combination of thermal and drift energy) J
X any sufficiently smooth vector field
Xtu distance between fins (equation 5.33) m
Y energy dissipation during collision (= ρ e) J m-3
Z thermoelectric Figure of merit K-1
z the total electric charge coulomb
T∆ temperature difference between the hot and
cold junctions K
adsH∆ enthalpy of adsorption kJ kg-1
Φ the source of momentum
Π tensor part of a pressure tensor
xx
π Peltier coefficient V
α the Seebeck coefficient VK-1
Tα thermal expansion factor K-1
β isothermal compressibility factor of the system Pa-1
γ flag which governs condenser transients
δ flag which governs adsorber transients
χ the substantive derive of any variable
ρ the effective density kg m-3
ρte thermoelectric material density kg m-3
τ average collision time fs
Γ the Thomson coefficient VK-1
λte the thermal conductivity Wm-1K-1
λ the phonon wavelength nm
ψ potential energy per unit mass J kg-1
µ the thermodynamic or chemical potential
µl liquid viscosity (equation 5.35) kg/ms
σ the entropy source strength W m-3 K-1
σs Surface tension (equation 5.35) N m-1
σ ′ the electrical conductivity ohm-1 m-1
κ the mass transfer coefficient s-1
θ flag which governs desorber transients
φ the fin efficiency
φ∇ the electric potential gradient V m-1
Subscripts
a adsorbate
xxi
ads adsorption
ads,T adsorption by external cooling and thermoelectrics (Table 5.2)
ads,TL latent heat in the adsorber bed (Table 5.2)
ads,w adsorption by external cooling (Figure 5.3)
ave average temperature
abe adsorbent
b barrier
b,e electrons in the barrier of superlattice
b,h holes in the barrier of superlattice
b,g phonons in the barrier of superlattice
bed,ads adsorption bed
beds, ads, TE bed adsorption by thermoelectric modules
bed,des desorption bed
CE ceramic plate
colli the collision term
colli e-h collision between electrons and holes
colli e-g collision between electrons and phonons
colli h-e collision between holes and electrons
colli h-g collision between holes and phonons
colli g-e collision between phonons and electrons
colli g-h collision between phonons and holes
cond condenser
condL latent heat in the condenser (Table 5.2)
des desorption
des,L latent heat in the desorber bed (Table 5.2)
xxii
dissip dissipation
e electron
evap evaporator
evapL latent heat at the evaporator (Table 5.2)
f liquid phase
fin fin
g phonons or gas
gas gaseous phase
H or hj hot junction
HX heat exchanger
h hole
IR Infrared heater
in inlet
irr irreversible
L or cj cold junction
Load Load surface
l liquid phase
max maximum
mf metal film
min minimum
np non-pulsed period
o outlet
p the pulse period
pr Prediction
q Joulean heat flux
xxiii
quartz quartz (fused silica) surface
r& rate of change of position vector
ref,cond refrigeration (here water) of condenser
ref,evap refrigerant (water vapour) at the evaporator
rev reversible
sys system
TE thermoelectric modules
Tube,ads tube of the adsorption bed
t the total amount
te thermoelectric device
th theoretical
tu tube
v ′′& rate of change of velocity vector
w,e electrons in the well of superlattice
w,h holes in the well of superlattice
w,g phonons in the well of superlattice
w well or water refrigerant
w,i water inlet
w,o water outlet
1
Chapter 1
Introduction
One of the major problems facing the electronics industry is the thermal management
problem where the heat dissipation by conventional fan cooling from a single CPU
(Central Processor Unit) chip has reached a bottleneck situation. With increasing heat
rejection from higher designed clock speeds*, temperatures on the chip surfaces have
reached the thermal design point (TDP) of fan-fins cooling devices, about 73o C. A
single CPU chip containing both power and logic circuits, can no longer sustain the
designed clock speed* because of high thermal dissipation. Consequently, major chip
manufacturers have embarked on two or more processors designed on a single
footprint, distributing the CPU generated heat to a wider area of its casing so as to have
a capability for over-clocking**. A plot of the chip surface temperature with the power
intensity of CPUs is shown in Figure 1.1 (Tomshardware guide, 2005).
A central challenge to the thermal management problem of CPUs is the development
of cooling systems which can handle not only the level of heat dissipation of
computer’s CPU (around 120 W at 3.8 GHz) but they should also have the potential of
being scaled down or miniaturized without being severely bounded by the thermal
bottlenecks of the convective air cooling or boiling.
_____________________________________________________________________
* The clock speed of a CPU is defined as the frequency that a processor executes instructions or that data is processed. This clock speed is measured in millions of cycles per second or megahertz (MHz) and gigahertz (GHz). ** Overclocking is the act of increasing the speed of certain components in a computer other than that specified by the manufacturer. It mainly refers to making the CPU run at a faster rate although it could also refer to making graphics card or other peripherals run faster.
2
A survey of the literature (Phelan et al., 2002) shows that there are two categories of
cooling devices: (i) The passive-type cooling devices in which the load-surface
temperatures are operated well above the ambient temperature and these devices
include forced-convective air cooling, liquid immersion cooling, heat pipe, and
thermo-syphon cooling, and (ii) the active-type cooling device where electricity or
power is consumed. For example, a scale-down vapour-compression refrigeration
system has the capability of lowering the load surface temperature below the ambient.
However, the efficiency of such a vapour-compression type cooler decrease rapidly
when its physical dimensions are down scaled or miniaturized because the dissipative
losses of the refrigeration cycle could not be scaled accordingly. The surface and fluid
friction of the refrigeration cycle increase relatively with reduced volume of devices.
In addition, mechanical devices have high maintenance cost.
Figure 1.1. The dependence of the chip surface temperature with the power intensity of CPU.
50
55
60
65
70
75
20 30 40 50 60 70 80 90 100 110 120
power intensity of CPU (W)
chip
sur
face
tem
pera
ture
(o C)
Heat sink design point
Fan
Heat sink
3
The simplest is forced air convection (Chu, 2004) with the option of an extended heat
sink that effectively increases the heat source surface area for heat exchange and/or the
possibility of introducing ribs or barriers on the surfaces to be cooled to increase air
turbulence so as to realize better heat dissipation. This method is adequate for many
types of current microelectronic cooling applications, but might cease to satisfy the
constraint of compactness for future generations that will require at least an order of
magnitude higher cooling density.
Passive thermo-syphons (Haider et al., 2002) have been proposed. These devices
involve virtually no moving parts except with the possibility of one or more cooling
fans at the condenser. Such a device, however, is highly orientation-dependent as it
relies on gravity to feed condensate from a condenser located at a higher elevation so
as to provide liquid flush back to the evaporator, which is located at a lower elevation.
Thermo-syphons equipped with one or more mini pumps have also been proposed
(Kevin et al., 1999). Instead of relying on gravity, condensate is pumped from the
condenser back to the evaporator. This scheme is orientation-independent and also
allows for the possibilities of forced convective boiling, spraying of condensate or jet-
impingement of condensate at the evaporator, which will effectively enhance boiling
characteristics and therefore cooling performance.
Laid-out heat pipes (Yeh, 1995 and Kim et al., 2003) have found applications
especially in laptop and desktop computers. The evaporating ends of the heat pipes are
judiciously arranged over the CPU while the condensing ends of the same are laid out
so as to effectively increase the surface area of the heat sink.
4
Recently, interest in jet impingement cooling (Lee et al., 1999) of electronics cooling
has been significantly increased. The working fluids can be liquids or gases. In
addition, jet impingement is produced by single or two-phase jets. The advantages are;
direct contact with the hot spot, resulting in a higher heat transfer coefficient, and the
simple structure. In impinging jet cooling, the heat transfer coefficient is found to be
high at the stagnation point only and decreases rapidly away from the stagnation point
with significant temperature gradients over the heating surface.
Mini vapour compression chillers (Roger, 2000) have also found applications. In one
design, the evaporator is arranged over the heat source surface while the mini
condensing unit is positioned away from the heat source. The advantage of such a
system lies in its higher COP. However, many moving parts are involved in the
compressor and they have to be highly reliable. Further scaling down of the
compressor for miniaturized cooling applications may also be a technical challenge,
and this may lead to a sizable loss of compressor efficiency due to high flow leakages
and in turn the low chiller COP.
Thermoelectric chillers (Goldsmid and Douglas, 1954 and Rowe, 1995) are also in use,
but suffer from inherently low COP (typically in the range of 0.1-0.5 for the
temperature ranges characteristic of many microelectronic applications) and high cost.
The low COP means that major increases in cooling density will require unacceptably
high levels of electrical power input and rates of heat rejection to the environment that
will be difficult to satisfy in a compact package, all at increased cost. Thermoelectric
chillers satisfy the requirement of compactness, the absence of moving parts except for
the possibility of one or more cooling fans, and an insensitivity to scale (since energy
5
transfers derive from electron flows). Typically, commercial thermoelectric devices
comprise semiconductors, most commonly Bismuth Telluride. The semiconductor is
doped to produce an excess of electrons in one element (n-type), and a dearth of
electrons in the other element (p-type). Electrical power input drives electrons through
the device. At the cold end, electrons absorb heat as they move from a low energy
level in the p-type semiconductor to a higher energy level in the n-type element. At the
hot side, electrons pass from a high energy level in the n-type element to a lower
energy level in the p-type material, and heat is rejected to a reservoir.
Adsorption chillers (Saha et al., 1995(a), 1995(b), Amar et al., 1996) are also capable
of being miniaturized (Viswanathan et al., 2000) since adsorption of refrigerant into
and desorption of refrigerant from the solid adsorbent are primarily surface, rather than
bulk, processes. A micro-scale reactor consists of micro-channel cavities to house the
adsorbents. The reactor includes a combination of heat exchanger media or adsorbate,
porous adsorbents, and a thin micro-machined contractor media.
The technology of coupling a thermoelectric device (often referred to as a Peltier
device), to an adsorber and a desorber is not new (Edward, 1970 and Bonnissel et al.,
2001). It is typically applied to humidification, dehumidification, gas purification, and
gas detection. In the electro-adsorption chiller (Ng et al., 2002), the two junctions of a
thermoelectric device are separately attached in a thermally conductive but electrically
non-conductive manner to two beds (hot and cold) or reactors. When direct current is
applied to the thermoelectric device, the reactor attached to the cold junction acts as an
adsorber, while the second reactor attached to the hot junction acts as a desorber.
When the direction of flow of direct current through the thermoelectric device is
6
reversed, the original cold junction is switched into a hot junction which in turn also
switches the reactor from an adsorber or absorber to a desorber; concomitantly, the
original hot junction is switched into a cold junction which in turn also switches the
reactor from a desorber to an adsorber or absorber.
This thesis aspires to develop the thermodynamic modelling of solid state cooling
devices namely thermoelectric refrigeration (Rowe, 1995), the pulsed thermoelectric
cooler (Snyder et al., 2002 and Miner et al., 1999), micro thermoelectric refrigeration
using superlattice quantum well structure (Elsner et al., 1996), and electro-adsorption
chiller (Ng et al., 2002) such that their performances can be analyzed and understood.
These modeling equations are derived from the basic Boltzmann Transport Equation
(BTE) to explain both the macroscopic and microscopic thermodynamic concepts.
Based on the Gibbs relationship and the conservation equations, the theoretical
formulations of the temperature-entropy (T-s) for macro and micro scale solid state
semiconductor cooling devices have been developed. These T-s formulations produce
the net energy flow and categorise all types of dissipative mechanisms, which include
(i) the transmission losses, which occur due to input energy sources to the system such
as electrical energy, thermal energy sources, (ii) external losses due to interact with the
environment (iii) the internal losses including friction, mass transfer, internal
regeneration, finite temperature difference heat transfer and (iv) the dissipation due to
collisions of electrons, holes and phonons in solids. This thesis also investigates
experimentally the adsorption isotherms and kinetics to model the electro-adsorption
chiller. The experimental investigation of a bench-scale electro-adsorption chiller is
performed such that the key parameters in evaluating its performance and the energy
7
flow can be understood. The simulations and the experimentally measured results are
compared for verification.
A chapter-wise overview of this report is given below.
Chapter 2 develops the general form of energy balance equations, which are derived
from the Boltzmann Transport Equation (BTE). The balance equations have two terms
“drift” and “collision” and these terms are discussed. This chapter also describes the
conservation of entropy using Gibbs relation and the energy balance equation for
macro and micro-scales and found that the entropy generation in a solid state device
occurs due to irreversible transport processes, collision processes (electrons-holes and
phonons) and free energy changes associated with recombination and generation of
electron-hole pairs are also provided in this chapter.
Based on the conservation laws and the entropy balance equations, Chapter 3 develops
the temperature-entropy formulations of a solid state thermoelectric cooling device, a
pulsed thermoelectric cooler and a micro-scale thermoelectric cooler. The
thermodynamic performances of these solid state coolers are discussed graphically
with T-s diagrams. The mathematical modelling of the thin film superlattice
thermoelement is developed from the basic Boltzmann Transport Equation where the
collision terms are included.
Prior to studying the proposed electro-adsorption chiller, the adsorption characteristics
and isotherms are described briefly in Chapter 4. This chapter starts off with the
experimental measurements of the characterization of two types of silica gel (Fuji
8
Davison type “A” and type “RD”). This chapter also investigates experimentally the
isotherm characteristics of two types of silica gel. The experimental results are
compared with similar experimental data in the literature for verification.
Chapter 5 consists of two subsections. The first sub-section starts with the
development of the full thermodynamic property maps for the adsorbent-adsorbate
system. The property fields express the extensive thermodynamic quantities such as
enthalpy, internal energy and entropy as a function of pressure, temperature and the
amount of adsorbate. In this section, the newly interpreted specific heat capacity of the
adsorbate-adsorbent system has been proposed. Based on the theory of
thermoelectricity, the adsorption characteristics, isotherms, kinetics and the fully
thermodynamic maps of adsorption the second section models the electro-adsorption
chiller. For simplicity, the lumped modelling approach is applied.
Having gone through the analyses of three major chapters (i.e., Chapters 3, 4 and 5),
Chapter 6 investigates a bench-scale electro-adsorption chiller (EAC) experimentally,
where the experimental quantifications are described. The formulations of the EAC
developed in chapter 5 are verified against experimental data. This thesis ends with
conclusions (Chapter 7), where the originality and contributions of the author are
discussed.
9
Chapter 2
Thermodynamic Framework for Mass, Momentum, Energy and
Entropy Balances in Micro to Macro Control Volumes
2.1 Introduction
In the past few decades, significant progress has been made in development of semi-
conductor materials for cooling and heating applications, for example, the
thermoelectric chillers using the Bismuth-Telluride (Bi2Te3), Antimony-Telluride
(Sb2Te3), Silicon-Germanium (SiGe) (Rowe, 1995). With greater needs for
miniaturization of micro or mini-chillers, these semi-conductors are fabricated to
within a few microns thick using the chemical-vapor deposition (CVD) techniques for
making the thin-film materials in layers of the positive-negative or P-N junctions. The
law of equilibrium thermodynamics as well as the conservation laws, which are well
established for analyzing energy and mass transfers of macro control volumes or
systems, may not be sufficient to handle the systems of micro dimensions (Chen, 2001
and 2002), in particular, their inability to handle the dissipation phenomena associated
with high density electrons, holes and phonons collisions or flows and these
dissipations are known to have set real limits to the performance of micro systems.
In this chapter, the Boltzmann Transport Equation (BTE) (Tomizawa, 1993 and
Parrott, 1996) is used to formulate the transport laws for equilibrium and irreversible
thermodynamics. The BTE equations are deemed to be suitable for analyzing micro-
scaled systems because they account for the collisions terms associated with the high
density fluxes of electrons and phonons in the thin films. In Figure 2.1, the relative
merits of applying the classical Gauss theorem (control volume approach), BTE and
10
the molecular dynamics model is elaborated with respect to the length and time scales.
For example, at wavelengths ranging from a phonon (1~2 nm) to its mean free path
(300 nm), the molecular dynamic model is usually employed (Lee, 2005). In regions
higher the mean free path of phonons, the Boltzmann Transport Equation (BTE)
models could be utilized to capture the collision contributions of particles and the BTE
model can be extended into the realm of the Gauss theorem or control volume
approach.
Figure 2.1. Schematic diagram of a thermal transport model showing the applicability of Boltzmann Transport Equation (BTE) in the thin films, where τc is the wave interaction time (100 fs), τ defines the average time between collisions, τr is the relaxation time, λ indicates the phonon wavelength (1~2 nm), Λ is the mean free path (~300 nm) and lr is the distance corresponding to relaxation time. (Lee, 2005). Hence when L ~ λ, wave phenomena exhibit, when L ~ Λ and t >> τ, τr, ballistic transport and there is no local thermal equilibrium, when L >>lr and t ~ τ, τr, statistical transport equations are used, when L >> lr and t >> τ, τr, local thermal equilibrium can be applied over space and time, leading to macroscopic transport laws.
τr
Gauss Theorem Approach
(Control volume)
Boltzmann Transport Equation (BTE)
Molecular Dynamics
λ Λ lr >Λ L >> lr Length Scale
t >>τr
τ
τc
Tim
e Sc
ale
11
In this regard, particular attention is paid to the energy, momentum and mass
conservation properties of the collision operator in sub-micron semiconductor devices.
It is noted that the dissipative effects from the collision terms are translated into unique
expressions of entropy generation where one could analysis systems performance
using simply the classical temperature-entropy (T-s) diagrams that capture the rate of
energy input, the dissipation and the useful effects. Another advantage of using the
BTE approach is that when the systems to be analyzed enters into a macro-scale
domain, the collision terms could simply be omitted (as the effects are known to be
additive) and the conservation laws revert back to that of the classical
thermodynamics, a method similar to the direction taken by the works of de Groot and
Mazur. (1962).
Prior to writing down the BTE formulation below, some aspects of classical
thermodynamic development in the recent years are first reviewed. One such topic that
has been greatly discussed in the literature is the finite time thermodynamics (FTT)
(Novikov II, 1958) which has been applied to problems of isothermal transfers to
chiller cycles called the reversible cycle. In the FTT analysis, the processes of heat
transfer (from and to the heat reservoir) are time dependent but all other processes (not
involving heat transfers) are assumed reversible. Take for example, an endo-reversible
chiller that has been commonly modelled (Mey and Vos, 1994). Invariably, they
assumed reversibility for the flow processes of its working fluid but irreversible
processes are arbitrary applied and restricted to the heat interactions with the
environment or heat reservoirs. Owing to the arbitrary assumption of reversibility for
the flow processes of working fluid within the chiller cycle, the finite time
12
thermodynamics (FTT) faces an “uncertain treatment” (Moron, 1998) and hence, it
renders itself totally impractical in the real world.
On the other hand, there has been an excellent development in the general argument of
equilibrium and irreversible that applies to macro systems. It is built upon an approach
following the method of de Groot and Mazur (1962). They proposed the theory of
fluctuations that describes non-equilibrium (irreversible) phenomena, based on the
basic reciprocity relations (Onsager, 1931), and relied on the microscopic as well as
drawing phenomenon about macroscopic behavior. Most non-equilibrium
thermodynamics assumes linear processes occurring close to an equilibrium
thermodynamic state and assumes that the phenomenological coefficients are constant
(Denton, 2002).
For the microscopic and sub-microscopic domains, the transport equations are
developed from the rudiments of Boltzmann Transport Equation (BTE) but
conforming to the First and Second Laws of Thermodynamics. In the sections to
follow, the main objective is to discuss and formulate the basic conservation laws
(mass, momentum and energy balances) for the thin films and based on the specific
dissipative losses encountered in these layers, the entropy generation with respect to
the electrons, holes and phonons fluxes will be formulated.
The organization of this chapter is as follows: Section 2.2 describes the general form of
conservation equations. The equations to be discussed are the mass, momentum and
energy conservation and the properties of the collision operator in sub-micron
semiconductor devices. In section 2.3, the mathematical rigor of the BTE with respect
13
to irreversible production of entropy is presented. Following the Gibbs expression on
entropy, the entropy balance equation is established in relation with the conservation
equations, which comprise a source term or entropy source strength term. The
concluding section of this chapter tabulates a list of the common entropy generation
mechanisms and demonstrates the manner the terms are deployed in the BTE
formulation. An application example of this demonstration is given in Chapter 3.
2.2 General form of balance equations
In this section, the transport processes involving the mass, momentum, and energy are
described from the basic conservation laws, developed together with the transport of
heat or energy by molecular fluxes such as electrons, holes and phonons within the
micro layers in a system.
2.2.1 Derivation of the Thermodynamic Framework
Figure 2.2 shows a schematic of an ensemble or a group of molecular particles, such as
an electrons, holes or phonons, represented initially at a time t where their positions
and velocities are expressed in the range ( ) ( )vr ′′33 , dd near r and v ′′ , respectively.
The operator, 3d indicates the three dimensional spaces of the variables r and v ′′ .
At an infinitesimally small time dt later, the molecular particles move, under the
presence and influence of an external force ) ,( vrF ′′ , to a new position dtrrr &+=′ and
attaining a velocity dtvvv ′′+′′=′ & . Owing to collisions of molecular particles, the
number of molecules in the range vr ′′33 dd could be also changed where particles or
molecules originally outside the range ( ) ( )vr ′′33 dd can be scattered into the domain
under consideration. Conversely, molecules originally inside the domain range could
14
be scattered out. Following the framework of Reif, F. (1965), the generic form of the
conservation laws is additive and is given by,
4444 84444 764444 84444 764444 84444 76"&)_(_
33
&)(),_(_
"33"
&)(),_(_
33''
''''
)"()(),()()(),,()()(),,(
vratMolecules
colli
vrtatMoleculesvrtatMolecules
ddfddtfddtf vrvrvrvrvrvr '"''' +=
(2.1)
where t’= t + dt, the range ( )dtrrr &+=′ and ( )dtvvv ′′+′′=′ & . ( )tf ,, vr is the statistical
distribution function of an ensemble particle, which varies with time t, particle position
vector r, and velocity vector v.
For convenience, we drop the implied 3-D operator, d3, the equation reduces to a more
readable form, i.e.,
Position
0
2
4
6
8
10
12
14
16
18
20
0 2 4 6 8 10 12 14 16 18 20
v'
v''
t
t + dt
r'r
Vel
ocity
Figure 2.2. The motion of a particle specified by the particle position r and its velocity v''.
15
( ) ( ) ( )
( ) ( ) ( )tfdttdtdtfdttfistermcollisiontheor
dttftfdttdtdtf
colli
colli
,,,,,,
,,,,,,
vrvvrrvr
vrvrvvrr
′′−+′′+′′+=′′
′′+′′=+′′+′′+
&&
&&
By invoking the partial derivatives, the basic Boltzmann transport equation is now
expressed for a situation that handles the molecular flux over a thin layer of materials,
i.e.,
tf
tf
tf
grearrangin
tf
tv
vf
tr
rf
tf
colli
drift
colli
∂∂
+⎟⎠⎞
⎜⎝⎛∂∂
−=∂∂
∂∂
+⎟⎟⎠
⎞⎜⎜⎝
⎛∂∂
∂∂
+∂∂
∂∂
=∂
∂ "
"
and the subscript “drift” refers to the flow of flux through space (r) and convective
velocity (v) and the second term is the effect of collisions with time. It is noted that the
term “drift” is a non-equilibrium transport mechanism that associates with the external
force ) ,( vrF ′′ , i.e.,
dtdf
dtdf
tf
Drift
vv
rr
′′′′∂
∂+
∂∂
=⎟⎠⎞
⎜⎝⎛∂∂ , (2.2)
where r is the position vector, v ′′ is the velocity vector, i.e., dtdr
=′′v , and the rate of
change of v ′′ is *mdt
d Fv=′′
, where m* is the mass of a single molecular particle.
The physical meanings of velocity and electric field are described in the drift term of
equation (2.2) is now fully elaborated as,
pF
rv
vF
rv
∂∂
−∂∂′′=
′′∂∂
−∂∂′′=⎟
⎠⎞
⎜⎝⎛∂∂ fff
mf
tf
Drift *,
16
where the momentum *m vp ′′= , is that associated with a molecular particle.
Substituting back into equation (2.2), the general form of the transport equation with
respect to r, p and t becomes
tfff
tf colli
∂∂
+∂∂
+∂∂′′−=
∂∂
pF
rv . (2.3)
Having formulated the basic form of the Boltzmann transport equation or BTE in
short, and two other functions will be used along the BTE and they are: Firstly, the
carrier density of the molecular flux is now incorporated by defining the carrier density
as the integral of the product between the molecular particles and the change in their
momentum (Tomizawa, 1993), i.e.,
∫= pfdn . (2.4)
Secondly, the substantive derive of any variable, χ, that varies both in time and in
space can be expressed in vector notation as χ∇+∂∂
=tdt
d .v = +∂∂t
(v.∇ ) χ, or simply
can be written as ∇⋅+∂∂
= vtdt
d . The following conservation equations can now be
formulated with the BTE format and they are elaborated here below.
2.2.2 Mass Balance Equation:
For a given space and time domain, the basic Boltzman transport equation (2.3) is now
invoked to give
tfff
tf colli
∂∂
+∂∂
+∂∂′′−=
∂∂
pF
rv
where the partials of f to the momentum and space in 3-dimensions can be represented
by applying the “Del” operator, i.e., rp ∂∂
=∇∂∂
=∇ffffp , . Thus, integrating on both
17
sides of equation (2.3) with respect to the momentum space, and noting the definition
of the carrier density n (equation 2.4), yields:
( )t
ndfntn colli
p ∂∂
+∇+′′−∇=∂∂
∫ pFv (2.5)
Should dp be small in the space, the second term of right hand side of equation (2.5)
can be omitted and equation (2.5) reduces to a familiar expression of the form;
tnn
tn colli
∂∂
+′′−∇=∂∂ )(v , (2.6)
where the collision effects from molecular particles are still retained. Defining the
effective density as nm *=ρ , where m* is the mass of single molecular particle. For
simplicity, assuming the particles move with an average velocity, then vv ′′= , and
rearranging gives the mass conservation as
collitt ∂∂
+−∇=∂∂ ρρρ )(v . (2.7)
The advantage of equation (2.7) is that it is perfectly valid for describing electrons or
phonons flow within submicron semiconductor devices and the collision term takes
them into account. In the case of a macro-scale domain, the mass balance omits the
collision term, i.e.,
).( vρρ−∇=
∂∂
t. (2.8)
The form of equation 2.8 can also be derived using the Gauss theorem within a control
volume method (see Appendix A). For example, the rectangular coordinates
nomenclature for the component velocities for kjiv ˆˆˆ wvu ++= would yield the
continuity equation as
( ) ( ) ( ) 0=∂
∂+
∂∂
+∂
∂+
∂∂
zw
yv
xu
tρρρρ . (2.9)
18
Only when steady state is considered, i.e., 0=∂∂
tρ ; and a case of constant-fluid
density, the following simplified continuity equation reduces to
0=∂∂
+∂∂
+∂∂
zw
yv
xu (2.10)
In vectorial notation, the mass conservation for a macro domain and constant density
situation is simply expressed as
( )
v
v
⋅∇−=
⋅−∇=∂∂
ρρ
ρρ
dtd
t . (2.11)
2.2.3 Momentum Balance Equation
Starting from the basic transport equation, t
ffftf colli
∂∂
+∂∂
+∂∂′′−=
∂∂
pF
rv , and multiply
the momentum, p, on both sides before integrating over the momentum space yields;
( ) ( ) pppFppvppp dt
fdfdfdft
collip ∂
∂+∇+∇′′−=
∂∂
∫∫∫∫ .. (2.12)
Invoking the definition of the carrier density, n. and the left hand term of equation
(2.12) can be expanded as, ( ) ( ) ( )tt
nmt
ndft ∂
∂=
∂∂
=∂
∂=
∂∂∫
vvppp ρ* whilst the first term
on the right hand side of (2.12) is reduced to ( ) ( ) pdfdf Jppvpvp ... ∇=′′∇=∇′′ ∫∫ ,
where pJ is a tensor. This flux density of momentum can further be transformed to
( ) ( ) ( ),.... PvvvvppvJ +∇=′′′′∇=′′∇=∇ ∫ ρρdfp
where according to Reif, F. (1965), the velocity vector ( )Uvv +=′′ consists of a mean
velocity v and a peculiar velocity U, and the pressure tensor is given by,
UUUUP ρρ γα == .
19
The second term on the right hand side of equation (2.12) is given by,
( ) ( ) ΦppFppFpFp =∇+∇=∇ ∫∫∫ dfdfdf ppp ... , known as the source of
momentum, and the last term on the right hand side refers to as the collision integral,
( )collicolli t
dtf
∂∂
=⎟⎠⎞
⎜⎝⎛∂∂
∫vpp ρ .Combining these expressions from above, equation (2.7) is
now summarized in terms of the physical variables ( ),, pvρ as
( ) ( ) ( )collitt ∂
∂+++−∇=
∂∂ vΦvvPv ρρρ . . (2.13)
The vector product vv, is also known as the Dyadic product which is given by a set of
matrix functions with respect to the component velocities. Detailed manipulation of the
Dyadic product can be found in many mathematics texts. In general, two kinds of
external forces may act on a fluid: (i) the body forces which are proportional to the
volume such as the gravitational, centrifugal, magnetic, and/or electric fields, and (ii)
the surface forces which are proportional to area, for example, the static pressure drop
due to viscous stresses. From a microscopic viewpoint, the pressure tensor, P, is a
related to short-range interaction between the particles of system, whereas the body
forces, Φ, pertains to the external forces in the system. Equation (2.13) describes a
balance for the momentum density ( vρ ), the momentum flow, ( )vvP ρ+ with a
convective portion ( )vvρ , and Φ, is a source of momentum. Some examples of the
type of source terms employed in two different application scenarios are outlined in
table 2.1.
For a macro scale flow situation, the collision term is omitted and equation (2.13)
reduces to the conventional form as
20
( ) ( )
ΦPv
ΦvvPv
+⋅−∇=
++−∇=∂
∂
dtd
t
ρ
ρρ .. (2.14)
It is noted that the total pressure tensor can be further divided into a scalar hydrostatic
part p, and a tensor part, Π.
Table 2.1 Examples of the source term, Φ, for two common applications
2.2.4 Energy Balance Equation
Associated with the flow of a carrier density, n, there is a corresponding energy flow,
w, such as the thermal energy and the drift energy term. By multiplying both sides of
the basic BTE, i.e., equation (2.3), by w and integrating over the momentum space
(product of mass and velocity) yields
( ) ( ) ∫∫∫∫ ∂∂
+∇+∇′′−=∂∂ pp.p.p d
tfdfdfdf
tcolli
p wFwvww (2.15)
Expanding the left hand side of equation (2.15), eeww ρ===∂∂∫ *p nmndf
t, where
e is the total energy per unit mass and ue +ψ+= 2
21 v . On the right hand side of
equation (2.15), the spatial gradient (i.e., the first term) can be placed outside the
Field Φ
Fluid mechanics (Bird, 1960)
gρ+∇− τ. , the first term indicates the viscous force per unit volume where τ is the stress tensor. The second term shows the gravitational force.
Electric flow (de Groot, and Mazur, 1962)
Ezρ , where z is the total electric charge per unit mass in the control volume and E is the electric field acting on the conductor.
21
integral to give, ( ) wdfdf Jwvvw .p.p. ∇=′′∇=∇′′ ∫∫ , where wJ represents the energy
flux, defined here as
q
n
kkkw df JJvPvwvJ +++=′′= ∑∫
=1
.ep ψρ , and veρ is a convective term, ( vP. ) is
the mechanical work in the system, ∑=
n
kkk J
1ψ equals to a potential energy flux due to
diffusion, qJ is the heat flow. The second term on the right hand side of equation
(2.15) can be expanded as:
( ) vΦvFpwFw .*
.p. ==∇=∇ ∫∫ mdfdfF pp
ρ
The collision term can also be expressed in terms of the energy dissipated Y, i.e.,
( )collicolli
colli
ttnd
tf
∂∂
=∂
∂=
∂∂
∫Yww p .
Combining the above expressions, the energy balance equation is now summarized
that includes the collision terms as;
( )colli
w tt ∂∂
++−∇=∂
∂ YvΦJe ..ρ (2.16)
To further understand the usage of the energy balance equation (i.e., equation 2.16), a
minor digression is needed to demonstrate the presence of kinetic and potential energy
terms. Multiplying the equation of motion [equation (2.13)] by v, a balance equation
for the kinetic energy can be obtained, i.e.,
( ) ( )[ ] vvPvPvvv
vvPvvvv2
.:.21.2
1
.... 2 Φ+∇+⎟⎠⎞
⎜⎝⎛ +∇−=
∂
⎟⎠⎞
⎜⎝⎛∂
⇒Φ++∇−=∂
∂ ρρ
ρρtt
(2.17)
22
which is commonly known as the mechanical energy term. Note that the vectorial
reduction could be performed on equation 2.14, as given by de Groot, (1962),
( )[ ] ⎟⎠⎞
⎜⎝⎛⋅∇=
∂∂
==⎟⎟⎠
⎞⎜⎜⎝
⎛∂∂
=⋅⋅∇ ∑∑==
vvvvv 2
1
23
1 21
21,,1, ρρρρ
n
iji
ij
iji
i
vvx
njivvvx
L
and
( )
( ) v
vv
⋅⋅∇−=⎟⎟⎠
⎞⎜⎜⎝
⎛∂∂
−=∂∂
+⎟⎟⎠
⎞⎜⎜⎝
⎛∂∂
+∂∂
−=
=∂∂
+⎟⎟⎠
⎞⎜⎜⎝
⎛∂∂
−=∇+⋅⋅∇−
∑ ∑∑∑∑∑
∑∑∑ ∑
= == == =
= == =
PPx
vvx
Pvx
PPx
v
njivx
PvPx
PP
n
j
n
iij
ij
n
i
n
jj
iij
n
i
n
jj
iijij
ij
n
i
n
jj
iij
n
i
n
jjij
i
1 11 11 1
1 11 1,,3,2,1,: L
Similar simplifications can also be made on the rate of change of potential energy
density, as shown below;
( )
( ) ( ) JFv
vv
−+−∇=∂
∂
+−⎟⎠
⎞⎜⎝
⎛+−∇=
∂∂ ∑∑∑∑
= ===
Jt
JFJJt
n
k
r
jjkjk
n
kkk
n
kkk
ψρψρψ
ψψρψρψ
.
.1 111 (2.18)
The above results indicate that should ( )rjn
kkjk L,10
1==∑
=
vψ , this implies that the
potential energy is conserved into the chemical reaction situation. JF=∑=
n
kkk FJ
1 which
is referred to the source term where the particles interact with a force field such as the
gravitational field or the electric charge experienced within an electric field. Adding
equations (2.17) and (2.18), the combined effects become
( ) ( )
vJFvPvvPvvv
JFvvvvPvvv
2
2
.:.21.2
1
..:.21.2
1
2
2
Φ+−∇+⎟⎠⎞
⎜⎝⎛ +++−∇=
∂
⎟⎠⎞
⎜⎝⎛ +∂
−+∇−Φ+∇+⎟⎠⎞
⎜⎝⎛ +−∇=
∂∂
+∂
⎟⎠⎞
⎜⎝⎛∂
Jt
JPtt
ψρψρψρ
ψρψρρψρ
23
From the energy balance equation (2.16) and substituting for these derived functions,
one obtains working form of the energy equation which gives in terms of the internal
energy, u, heat flux, Jq, and the associated source and collision terms;
( ) ( )
( ) ( )colli
q
colliw
tu
tu
tt
∂∂
++∇−+−∇=∂
∂∴
∂∂
+Φ+−∇=∂
∂
YJFvPJv
evJe
:.
..
ρρ
ρρ
(2.19)
It is noted that ρuv+Jq is the energy flow due to convection and thermal energy
transfers which contributes to the internal energy change, while the vectors P:∇v + JF
is deemed as the source term associated with the internal energy.
Further simplification of the energy balance Equation (2.19) leads to its concise form
of having only 4 terms and it is applicable to micro or thin layer flow of electrons or
phonons;
( ) ( )
colliq
colliq
tdtdu
tuu
t
∂∂
++⋅−∇=ρ
∂∂
++∇−+ρ⋅−∇=ρ∂∂
YΨJ
YJFvPJv : (2.20)
It is noted that the last term of the right hand side indicates the rate of change of energy
as a result of collisions of the molecular particles such as electrons or phonons.
Collision occurs by four processes in the micro-sublayer, i.e., (i) the increase in carrier
kinetic energy due to interactions between electrons and holes or phonons, (ii) the heat
dissipation arising from electrons or phonons interaction with the lattice vibrations,
(iii) the heat dissipation by the electrons to the holes and vice versa, and (iv) the heat
generation arising from the recombination and generation of the molecular particles.
Equation 2.20 could be extended to macro analysis by simply dropping the collision
term which has its form known conventionally as the energy balance equation;
24
ΨJ +⋅−∇= qdtduρ (2.21)
Another name for equation (2.20) is the equation of internal energy, as it is known in
some texts (Bird, 1960, de Groot and Mazur, 1962).
At this juncture, it is perhaps appropriate to elaborate these terms in terms of some
common application examples. For example, the source term Ψ, in equilibrium and
irreversible thermodynamics has two parts: (i) One aspect is the reversible rate of the
internal energy attributed by effects of compression and (ii) the other is the irreversible
rate of internal energy contributed by any dissipative effect. For most engineering
applications involving real gasses, the internal energy is expressed in terms of the fluid
temperature and the heat capacity. In such cases, the internal energy u may be
considered as a function of υ and T, and assuming constant specific heats, i.e.,
dTcdTpTpdT
Tududu v
T
+⎥⎦
⎤⎢⎣
⎡⎟⎠⎞
⎜⎝⎛∂∂
+−=⎟⎠⎞
⎜⎝⎛∂∂
+⎟⎠⎞
⎜⎝⎛∂∂
= υυυ υυ
and the substantial derivative term becomes
dtdTc
TpTp
dtdTc
dtd
TpTp
dtdu
vv ρρυρρυυ
+∇⎥⎦
⎤⎢⎣
⎡⎟⎠⎞
⎜⎝⎛∂∂
+−=+⎥⎦
⎤⎢⎣
⎡⎟⎠⎞
⎜⎝⎛∂∂
+−= v.
Invoking the energy balance equation (2.21), it becomes
( ) JFvJ +∇⎟⎠⎞
⎜⎝⎛∂∂
−⋅−∇= .V
qv TpT
dtdTcρ (2.22)
Should the gas be assumed ideal, then Tp
Tp
V
=⎟⎠⎞
⎜⎝⎛∂∂
ˆ and the second term of right hand
side equation (2.22) becomes zero, i.e.,
25
JFJ +⋅−∇= qv dtdTcρ . (2.23)
In another example, the internal energy u is considered to be a function of p and T, as
in a real gas. Hence,
dtdp
dtdp
dtdu
dtdhpuh υυυ ++=⇒+= .
Invoking the constant pressure assumption, it can be shown that dtdTc
dtdh
pρρ = .
dtdTc
dtdh
dtdp
dtdu
dtdh
pρρυ=⇒+= (2.24)
From equation (2.24) and substituting for the energy balance equation for dtduρ , it can
be demonstrated that the substantial derivative for the enthalpy, dtdTcpρ , comprise
only two terms, i.e.,
dtdpp
dtdTc
dtdp
dtdu
dtdTc
qp
p
υρρ
υρρρ
++⋅∇−⋅−∇=
+=
JFvJ (2.25)
From the continuity equation (2.11) we show that
v⋅∇= pdtdp υρ
Equation (2.25) is written as,
JFJ +⋅−∇= qp dtdTcρ (2.26)
The right hand side of equation 2.26 comprises (i) the heat flux term, qJ and (ii) the
energy source term JF. Table 2.2 tabulates three scenarios where the source energy
could be elaborated. For example, in the study of adsorption, where many chapters of
the thesis are devoted, the source energy is derived by considering adsorbate mass and
the isosteric heat of adsorption, i.e., adsHq∆& . Other examples in fluid mechanics and
electrically powered semi-conductors are also explained.
26
Table 2.2 Explanation of energy source in different fields.
Field Source of energy JF
Fluid mechanics v∇:τ ; This term indicates the irreversible internal energy increase due to viscous dissipation.
Electrical ρelectJ2; this is the joulean heat.
Chemical / Adsorption adsHq∆& , where q& is the adsorption or desorption rate and adsH∆ is the isosteric enthalpy of adsorption.
It should be emphasized that equations (2.21-2.26) are valid for macro-scale devices
simulation. However, for the analysis of sub-micron devices, the collision terms, which
are provided in equation (2.20), must be included.
2.2.5 Summary of section 2.2
In section 2.2, the conservation laws, i.e., mass, momentum and energy are derived
from the transportation of molecules using the Boltzmann Transport equation. The
thermodynamic framework for describing the conservation laws is applicable to both
the macro-scale and micro-scale systems. This approach differs from the control
volume approach of de Groot and Mazur (1962), Bird, R. B. (1965), etc., where the
Gauss theorem was employed. The latter approach could not embrace the collision
terms, as what has been shown here using the BTE approach. For thin films or micro-
scale systems, the collisions of electrons with electrons, electrons with holes, holes
with holes are deemed to have significant dissipative effects, a source of irreversibility
as quantified in the next section.
2.3 Conservation of entropy
The change of entropy of a system relates not only to the entropy (s) crossing the
boundary between the system and its surroundings, but also to the entropy produced or
generated by processes taking place within the system. Processes inside the system
27
may be either reversible or irreversible (Richard and Fine, 1948). The reversible
process may occur during the transfer of entropy from one portion to another of the
interior without entropy generation. On the other hand the irreversible process
invariably leads to entropy generation inside the system.
The entropy per unit mass is a function of the internal energy u, the specific volume υ
and the mass fractions ck i.e. ),,( kcuss υ= . In equilibrium the total differential of s is
given by the Gibbs relation (de Groot, 1961)
∑=
−+=n
kkk dcpdduTds
1
µυ , where p is the equilibrium pressure and kµ is the
thermodynamic or chemical potential of component k.
∑=
−+=n
k
kk dt
dcdtdp
dtdu
dtdsT
1
µυ (2.27)
From mass balance we get
collitTP
TP
dtd
TP
∂∂
+⋅∇=ρ
ρυρ v
The mass balance equation, equation (2.7), is written as,
colli
kkk
k
tdtd
∂∂
+⋅∇−⋅∇−=ρρρ Jv
where Jk ( )( )vv −= kkρ is the diffusion flow of component k defined with respect to
the ‘barycentric motion’. Defining the component mass fractions ck as
11
== ∑=
n
kk
kk cthatsoc
ρρ ,
and equation (2.7) can be simplified as
28
kk
colli
kkk
k
dtdc
tdtd
J
Jv
⋅−∇=
∂∂
+⋅∇−⋅∇−=
ρ
ρρρ
From equation (2.27) we have,
collicolli
n
k
kkq
n
kkkq
n
k
kk
n
k
kk
tTP
tT
TTT
TT
TT
dtdc
Tdtd
Tp
dtdu
Tdtds
dtdc
dtdp
dtdu
dtdsT
∂∂
+∂∂
+
+⎟⎠⎞
⎜⎝⎛ ∇⋅−∇⋅−
⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜
⎝
⎛−
⋅−∇=
−+=
−+=
∑∑
∑
∑
=
=
=
=
ρρ
µµ
ρµυρρρ
µυ
Y
JFJJJJ
1
1111
21
1
1
(2.28)
where Jk ( )( )vv −ρ= kk is the diffusion flow of component k defined with respect to
the ‘barycentric motion’.
From equation (2.28), one observes that the entropy flow is given by two terms:
⎟⎠
⎞⎜⎝
⎛−= ∑
=
n
kkkqs T 1
1 JJJ µ , (2.29)
and the entropy source strength by
collisionleirreversib
collicolli
n
k
kkq tT
PtTTT
TT
TT
σσσ
ρρ
µσ
+=⇒∂∂
+∂∂
++⎟⎠⎞
⎜⎝⎛ ∇⋅−∇⋅−= ∑
=
YJFJJ 11111
2 (2.30)
The expression of entropy generation (equation 2.30) provides the key to irreversible
thermodynamics. The first two terms indicate entropy generation as a result of
irreversible transport processes whilst the third term represents entropy increase due to
source applied to the system. The last two terms on the right hand side entropy
29
increase due to (1) heat transfers from the other systems via collision processes, (2)
free energy changes associated with recombination and generation of electron-hole
pairs.
From (2.28) the general form of entropy balance equation is written as,
σ+∇= sJdtdS . (2.31)
It is noted here that equation (2.31) has a similar form to the expression for total
entropy, i.e.,
irrrev dtdS
dtdS
dtdS
⎟⎠⎞
⎜⎝⎛+⎟
⎠⎞
⎜⎝⎛= , (2.32)
Comparing the first term of the right hand side of both equations, the reversible
entropy flux is ⎥⎦
⎤⎢⎣
⎡∇=⎟
⎠⎞
⎜⎝⎛
srev
JdtdS . . This is the net sum of quantities carried into the
system by transfers of matter, plus the net sum of quantities of entropy carried in by
the transfer of heat. The second term yields ⎥⎦
⎤⎢⎣
⎡=⎟
⎠⎞
⎜⎝⎛ σ
irrdtdS , which is the entropy
generation rate produced by irreversible processes taking place inside the system.
Table 2.3 shows some common examples of the irreversible processes (hence the
internal entropy generation) of common engineering processes.
Table 2.3 Example of processes leading to the irreversible production of entropy.
Entropy generation irrdt
dS⎟⎠⎞
⎜⎝⎛=σ Equations
Dissipation Mechanical energy (due to friction or viscosity)
⎟⎠⎞
⎜⎝⎛
dtdF
T1 (Richard et. al., 1948), where
dtdF is the rate at which work is done
against force or viscous force.
30
Electrical energy (by the passage of electric current through a resistance)
TR2I , where I is the flow of current
through a resistance R in conductor or semiconductor at temperature T.
Irreversible heat flow in a conducting medium
⎟⎠⎞
⎜⎝⎛∇
T1.q . Here ( )TK∇=q is the vector
which expresses the rate of heat flow. The
final expression is 2TTTk
dtdS
irr
∇∇=⎟
⎠⎞
⎜⎝⎛ .
Free expansion of gas without absorbing heat (Richard et. al., 1948)
The approximate expression for the irreversible increase in entropy accompanying the transfer of any particle element of the gas , consisting of dN moles, from pressure P1 to P2 is
⎟⎟⎠
⎞⎜⎜⎝
⎛=
2
1logPPRdNdSirr
Chemical reaction (Richard et. al., 1948)
dtdx
fBfAfDfcKR
dtdS
ba
dc
pirr
⎟⎟⎠
⎞⎜⎜⎝
⎛−=⎟
⎠⎞
⎜⎝⎛
...
...loglog ,
where Kp is equilibrium constant at temperature T, R is the gas constant, fAfB…fcfd…etc., are the actual instantaneous values of the fugacities, a. b, …, c, d, … are the number of moles and
dtdx is a factor.
Collisions (electron and hole transport in submicron semiconductor or thin film devices)
collicollicolli tTP
tTdtdS
∂ρ∂
ρ+
∂∂
=⎟⎠⎞
⎜⎝⎛ Y1 , where
the first term of right hand side indicates the collision due to change of energy between electrons-electrons, electrons-holes, holes-holes interactions, and the second term defines the collision because of change of carrier density.
2.4 Summary of Chapter 2
The details of the Boltzmann Transport equation for analyzing the mass, momentum,
energy and entropy conservation equations in macro and micro scales have been
discussed in this chapter. This chapter concludes with the summary of all conservation
31
equations as shown in Table 2.4 using Gauss and BTE approaches so that these two
methods could be compared.
Table 2.4 Summary of the conservation equations
Gauss theorem approach (Appendix A) Boltzmann Transport Equation approach
Ω=Ω=∇ ∫∫∫
ΩΩ
ddndVV
... XXX r t
ftf
tf colli
drift ∂∂
+⎟⎠⎞
⎜⎝⎛∂∂
−=∂∂
Mass ).( vρρ−∇=
∂∂
t
collitt ∂∂
+−∇=∂∂ ρρρ )(v
Momentum ( ) Φ++ρ⋅−∇=∂ρ∂ Pvvvt
( ) ( ) ( )
collitt ∂∂
+Φ++−∇=∂
∂ vvvPv ρρρ .
Energy ΨJ +⋅−∇=ρ qdtdu
colliq tdt
du∂∂
++⋅−∇=ρYΨJ
Entropy
JF
JJ
JJ
T
TT
TT
T
Tdtds
n
k
kkq
n
kkkq
1
111
2
1
+
⎟⎠⎞
⎜⎝⎛ µ∇⋅−∇⋅−
⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜
⎝
⎛µ−
⋅−∇=ρ
∑
∑
=
=
collicolli
n
k
kk
q
n
kkkq
tTP
tTT
TT
T
TTTdt
ds
∂ρ∂
ρ+
∂∂
++
⎟⎠⎞
⎜⎝⎛ µ∇⋅−
∇⋅−
⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜
⎝
⎛µ−
⋅−∇=ρ
∑
∑
=
=
YJF
J
JJJ
11
1
1
1
21
32
Chapter 3
Thermodynamic modeling of macro and micro thermoelectric coolers
3.1 Introduction
The reversible thermoelectric effects are the Seebeck, Peltier and Thomson effects
(Burshteyn, 1964). These are related to the transformation of thermal into electrical
energy, and vice versa. The physical nature of the thermoelectric effects is not well
explained in the literature, as well as the dependence of Seebeck coefficient on
temperature and materials. From the electron theory, the absorption of Thomson heat
in the interior of a thermally non-uniform conductor has been reported to be the
additive superposition of two effects: Firstly, a part of Thomson effect is the internal
Peltier effect which is caused by the non-equilibrium electron distribution functions in
a thermally non-uniform conductor. Secondly, the other part is heat absorbed due to
current flow against the drift potential difference. These effects are assumed to be
reversible and hence, the sign of the Thomson heat changes with the reversal of current
direction. The thermoelectric effects in semi-conductors are stronger than those found
in metals. This is attributed to the fact that electron gas in semi-conductors is far from
being degenerated and obeys the classical Boltzmann statistics. The non-equilibrium
changes in the distribution function are more noticeable than in a degenerated gas and
this affects the magnitude of the thermoelectric effects. The magnitude of non-
equilibrium change in the distribution function (and consequently the magnitudes of
the thermoelectric effects) depends on the relative importance of the factors (such as
electric field, thermal non-uniformity) that are responsible for the departure from
equilibrium, as well as the mechanism for re-establishing the equilibrium.
33
When an electric field is applied to a thermoelectric device, the following irreversible
processes are deemed to occur in each elemental volume:
(i) Electric conduction (electric current due to an electric potential gradient).
(ii) Heat conduction (heat flow due to a temperature gradient).
(iii) A cross effect (electric current due to a temperature gradient)
(iv) The appropriate reciprocal effect (heat flow due to an electric potential
gradient).
The last two belongs to a new class of irreversible process (Onsager, 1931).
The organization of this chapter is as follows:
Section 3.2 describes the literature review of the thermoelectric cooling systems. The
thermodynamic modelling of the bulk thermoelectric cooler is discussed in section 3.3.
Although other researchers excluded the Thomson effect, this is included in the thesis.
The explanation of Thomson effect is discussed in Appendix B and in addition, the
entropy analysis of a transient thermoelectric cooler is also discussed. The
thermodynamic behaviour of a thin-film thermoelectric cooler is investigated in
section 3.4 and the analysis is compared with the work of Arenas et al., (2001). The
governing equations employed in this chapter are derived from the Boltzmann
Transport Equation (BTE) that was reported in the previous chapter.
34
3.2 Literature review
Richard and Fine (1948) first generalized the thermodynamic treatment of
thermoelectrics involving the irreversible processes at steady state operation. Applying
the generalized laws of thermodynamics, Lord Kelvin (Thomson) proved the validity
of the relationships dTd
Tππ
−=Γ and Tαπ = where π is the Peltier Coefficient and α
is the Seebeck Coefficient. In 1931, the Onsager symmetry relationships were
developed. These complemented the theory of thermoelectricity and using these
relationships, the thermodynamic foundations for analyzing the irreversible steady
state processes have been laid. A detailed account and justification of the
thermodynamic framework in establishing the connections between all the
thermoelectric effects could be found in the monographs of de Groot, (1962) and
Haase, (1990). The use of semiconductors in thermoelectric refrigeration was proposed
(Goldsmid and Douglas, 1954) as the materials have high mean atomic weights and
high thermoelectric powers. The relationship between cooling capacity and the
Coefficient of Performance (COP) were examined for a single stage unit (Parrott,
1960) with respect to a given temperature difference between the hot and cold
junctions, and the design equations and performance graphs were presented. Recently,
a new approach to design thermoelectric cooling systems optimally has been proposed
by Yamanashi in 1996. He succinctly introduced the dimensionless entropy flow
equations to analyze the thermoelectric cooling performance.
Interface effects in thermoelectric micro-refrigerators were studied by Sungtaek and
Ghoshal (2000). They employed a phenomenological model to examine the behavior
of thermoelectric refrigerators as a function of thermal and electrical contact resistance,
the Seebeck coefficient and heat sink conductance. Ghoshal et al. (2002) described a
35
structured point-contact thermoelectric device, which defines the thermal gradient and
electric fields at the cold junction, and exploited the reduction of thermal conduction,
tunneling properties of point contacts and poor electron-phonon coupling at the
junctions. They have proposed a thermoelectric figure of merit, called the ZT
parameter, which has a range of 1.4-1.7 at room temperature.
In 1961, Landecker and Findlay studied extensively the transient behavior of Peltier
junctions by devising a method for measuring the thermo-junction temperature after
the passage of transient current pulse. They have theoretically demonstrated that the
cold junction temporal profile is a function of both pulse current and duration. Snyder
et al., (2002) reported the possibility of having a super cooler that employs a Peltier
device and exploiting the short time-scale behaviour of the current pulse at a
magnitude several folds higher than that of the non-pulsing period. The Peltier cold
junction breaks contact momentarily with the surface to be cooled prior to the arrival
of the Joulean heat, where the latter is a heat transfer phenomenon with a slower time-
scale. Such a transient operation of the thermoelectrics has enable these cooling
devices to reach an unprecedented low temperature level, typically about 220 - 240K
which widens the application potential of pulsed thermoelectrics, for example, a cryo-
cooler for surgery applications or the cooling of infrared detectors. In 1999, Miner et
al. modified a traditional thermoelectric cooler by using the intermittent contact of a
mechanical element, which was synchronized with an applied pulsed current for a
finite time. The proposed technique effectively increases the thermoelectric figure of
merit by a factor of 1.8 but there is no experimental result to justify the proposed
concept.
36
Temperature entropy (T-s) formulation of thermoelectric cycles has been developed
(Chua et al., 2002(b)) within the framework of irreversible thermodynamics and they
have presented the steady state results. Similar work on the T-s diagram of the
thermoelectric module was also performed by Arenas et al. (2000) but they found a
close link between the Seebeck coefficient and the entropy per unit electric current.
Also in 2000 and using a finite time method, Nuwayhid et al. presented a procedure of
minimizing the entropy generation so as to maximize the cooling power of
thermoelectric device.
Solid state superlattice structures could also be used to enhance the thermoelectric
device performance by reducing the thermal conductivity between the hot and cold
junctions and provide selective emission of hot carriers through the thermionic
emission (Mahan, 1994). With much work performed in the study of superlattice
thermoelectric properties, (Antonyuk et. al., 2001), the idea of increasing ZT parameter
could be derived by enhancing the density of electron flow with reduced dimensions.
Recent study shows that superlattice thermal conductivity in the cross-plane direction
is lower than that of in-plane direction and such hetrostructures could be used for
thermionic emission to increase the cooling. The superlattice thermoelectric module
consists of n-type and p-type super-lattices (Shakuri and Bowers, 1997 and Elsner et
al., 1996).
37
3.3 Thermoelectric cooling:
The refrigeration capability of a semiconductor material depends on parameters such
as the Peltier effect, Joulean heat, Thomson heat and material’s physical properties
over the operational temperatures between the hot and cold ends. A thermoelectric
device is composed of P-type and N-type thermoelements such as Bi2Te3 and they are
connected electrically in series and thermally in parallel. These thermoelectric
elements are sandwiched between two ceramic substrates. As shown in Figure 3.1,
thermoelectric cooling is generated by passing a direct current through one or more
pairs of n and p-type thermoelements, the temperature of cold reservoir decreases
because the electrons and holes pass from the low energy level in p-type material
through the interconnecting conductor to the higher energy level in the n-type material.
Similarly, the arrival of electrons and holes in the opposite end results in an increase in
the local temperature forming the hot junction.
P N
J
J
J J
Hot reservoir
Cold reservoir
TH
TL
+ ++ + + +
- -- -- -
J J P-arm
N-arm
Hot reservoir
Cold reservoir
(a) (b)
Figure 3.1. The schematic view of a thermoelectric cooler
38
When a temperature differential is established between the hot and cold junctions, a
Seebeck voltage is generated and the voltage is directly proportional to the temperature
differential. The amount of heat absorbed at the cold-end and dissipated at the hot end
depends on the product of Peltier coefficient (π ), Fourier effect (λte) and the current (I)
flowing through the semiconductor material, whilst the heat generation due to the
Thomson effect occurs along the thermoelectric element. Thermoelectric effects are
caused by coupling between charge transport and heat transport and from which the
basic mass, energy and entropy equations can be formulated to form a thermodynamic
framework.
3.3.1 Energy Balance Analysis:
The Peltier and the Thomson effects enhance the cooling effect in thermoelectric
materials. The Thomson effect is a function of the gradients from the Seebeck
coefficient and the temperature in an operating thermoelectric pallet or element. The
energy balance equation of the thermoelectric cooler, as shown in figure 3.1, follows
the methodology of the Boltzmann transport equation which has been discussed in the
previous chapter, namely:
( ) ( ) ( )colli
q tuEu
tu
∂∂
++∇−+−∇=∂
∂ ρρρ JvPJv :. ,
where u is the internal energy per unit mass, ρ is the density, v defines velocity, Jq
indicates the heat flow, P represents the pressure and JE is the energy field term.
For the bulk thermoelectric material analysis, the contributions from the collision
terms are deemed negligible and can be neglected;
( ) ( ) Eutu
qtete JvPJv +∇−+−∇=∂
∂ :. ρρ .
39
Hence the subscript te indicates the thermoelectric cooler. As there is no velocity and
pressure fields in a thermoelectric device,
Etu
qte JJ +−∇=∂∂ .ρ (3.1)
Invoking the definition of enthalpy;
( )tu
tpV
tu
thpVuh
∂∂
=∂
∂+
∂∂
=∂∂
⇒+=
( )t
Tctp
ph
tT
Th
thpThh te
tepT
te
pte te∂∂
=∂∂
⎟⎟⎠
⎞⎜⎜⎝
⎛∂∂
+∂∂
⎟⎟⎠
⎞⎜⎜⎝
⎛∂∂
=∂∂
⇒= ,,
where the pressure gradient term is zero.
The first term of right hand side of equation (3.1) is the heat current density, qJ (in
W/m2), and the second term is the electric current field, ( )φ∇= JEJ (in W/m3). Using
the thermodynamic-phenomenological treatment (Haase, 1990), the heat current
density
TJFQ
teo
q ∇−−= λJ , (3.2)
where J (in amp/m2) denotes the electric current density, Qo indicates the heat
transport of electrons/ holes, teλ defines the thermal conductivity of the bulk
thermoelectric element and F is the Faraday constant and the electric current density
can be expanded as
tete
otete T
FTQJ ∇′
+∇′=σφσ , (3.3)
where teσ ′ defines the electrical conductance and φ is the electrical potential.
Equation (3.3) is re-arranged as,
tete
o
te
TFTQJ
∇−′
=∇σ
φ (3.4)
40
Hence, ( ) ⎟⎟⎠
⎞⎜⎜⎝
⎛∇+
′+∇∇=+∇−
tete
teteteq T
QTFJJTE
σλ
2
.. JJ
where ⎟⎟⎠
⎞⎜⎜⎝
⎛
teTQ is equivalent to the entropy term.
Now, substitute all these relations into equation (3.1), the expression becomes
( ) tepte
tete
tetete
tetepte T
Ts
FJTJT
tTc ∇⎟⎟
⎠
⎞⎜⎜⎝
⎛∂∂
+′
+∇∇=∂∂
σλρ
2
, . , (3.5)
where ste introduces the transported entropy of the bulk thermoelectric arms. Defining
the Thomson coefficient ( )Γ as (Haase, 1990),
pte
tete
Ts
FT
⎟⎟⎠
⎞⎜⎜⎝
⎛∂∂
≡Γ− ,
the governing thermal transport in the bulk semiconductor arms is given by
( ) tete
tetete
tepte TJJTt
Tc ∇Γ−′
+∇∇=∂∂
σλρ
2
, . (3.6)
Using the Kelvin relations (de Groot, 1961),
=⎟⎟⎠
⎞⎜⎜⎝
⎛∂∂
−−=Γtete TTππ , where ( )Tαπ = is the Peltier coefficient.
Equation (3.6) can now be written as (derivations are shown in Appendix B),
teTte
tetete
tepte TT
JTJTJTt
Tcte
∇⎟⎠⎞
⎜⎝⎛∂∂
−∇+′
+∇=∂∂ αα
σλρ
22
, (3.7)
41
3.3.2 Entropy Balance Analysis
The basic entropy balance equation has been introduced in the previous chapter, as
shown below:
( ) ( ) σρρ
σρ
++−∇=∂
∂
+⋅−∇=
svts
dtds
s
s
J
J
. (3.8)
Introducing the phenomenological relations:
JT
TT
vste
tete
tetetetes
πλρ +∇−=+,J , (3.9)
and invoking the Kelvin laws, one could observe
πα =T . (3.10)
Assuming only the bulk thermoelectric materials, the collision terms are usually
neglected and the entropy source strength is given by (as discussed in the previous
Chapter);
EJTT
TT
TT
n
k
kkq ⋅+⎟
⎠⎞
⎜⎝⎛ ∇⋅−∇⋅−= ∑
=
1111
2
µσ JJ
Hence, the entropy generation of thermoelectric arm can now be modified into a form
that is inclusive of the Peltier coefficient and the local temperature as (de Groot, 1961)
( )tetete
tete
tetete
JTJT
TJT
TTT
σαπλσ ′+∇⋅+∇⋅⎟⎟⎠
⎞⎜⎜⎝
⎛+∇−−= /11
Finally, the entropy balance equation (3.8) becomes;
( ) ( )tetetete
tetetete
TJ
TJ
TT
ts
σπλρ
′+⋅∇−
∇⋅∇=
∂∂ 2
(3.11)
42
The first term on the right-hand side of equation (3.11) represents the change of
entropy due to heat conduction; the second term indicates the entropy flux due to
Seebeck effect and the last term is the entropy generation due to Joulean heat.
A more elegant method of writing the entropy fluxes of thermoelectric arm is to
express them in terms of (i) the heat and carrier fluxes, (ii) the internal dissipation, and
(iii) the entropy generation that comes from conduction and Joulean heat transfer:
( )
⎪⎪
⎭
⎪⎪
⎬
⎫
⎪⎪
⎩
⎪⎪
⎨
⎧
′+⎟⎟
⎠
⎞⎜⎜⎝
⎛∇∇−+⎟⎟
⎠
⎞⎜⎜⎝
⎛−
∇⋅∇=
∂∂
87644 844 76444 8444 76 heatJouleantoduestrengthsourceentropy
tete
ndissipatioernal
tetete
fluxentropy
tete
tetetete
TJ
TT
TJ
TT
ts
σλπλρ 2
int
1.
(3.12)
where the units of entropy flux is in W/m2 K and it is given by,
tete
teteflux T
JT
TS πλ−
∇= ,
(3.13)
and entropy generation per unit volume is in W/m3 K which is expressed as:
tetetetetegen T
JT
TSσ
λ′
+⎟⎟⎠
⎞⎜⎜⎝
⎛∇∇−=
21. .
(3.14)
Equation 3.14 is always a positive term.
3.3.3 Temperature-entropy plots of bulk thermoelectric cooling device
From the work of Carnot and Clasius, Kelvin deduced that the reversible heat flow
discovered by Peltier must have entropy associated with it. It has been shown that the
coefficient discovered by Seebeck was a measure of entropy associated with electric
current (Nolas et. al., 2001). A temperature-entropy (T-s) has been proven to be a
pedagogical tool in analyzing the performance of thermoelectrics undergoing both
43
reversible and irreversible processes. The parameters used in the computation are listed
in Table 3.1.
For an ideal thermoelectric device, the Seebeck (α) coefficient is constant and the heat
flux at the hot and cold junctions are given by αJTi where the subscript i refers either
to the hot or cold junction. Thus, the temperature Seebeck coefficient (T-α) plot is
simply a simple rectangle. However, the temperature effect on the Seebeck (α)
coefficient induces the dissipative losses along the p and n legs of thermoelectrics and
an indication of these losses are shown by the shaded area on the T-α plot, as shown in
Figure 3.2, in which the physical properties of the Bismuth Telluride thermoelectric
pairs are used. The two isotherms are the hot and cold junctions whilst the inclined
lines indicate the effect of temperature on the α values when the current (I) travels
along the p- and n-legs.
Table 3.1 Physical parameters of a single thermoelectric couple
Property value
Hot junction temperature, TH 300 K
Cold junction temperature, TL 270 K
Thermoelectric element length, L 1.15 × 10 -3 m *
Cross-sectional area, A 1.96 × 10-6 m2 *
Electrical conductivity, σ' 97087.38 ohm-1.m-1 *
Seebeck coefficient α (V/K)
α (T) = αo + α1 ln (T/To)
αo = 210 × 10-6 V/K, α1 = 120 × 10-6 V/K
To = 300 K (Seifert et al., 2002)
Thermal conductivity, λ 1.70 W/m K *
* Remarks Melcor thermoelectric catalogue, MELCOR CORPORATION, 1040 Spruce Street, Trenton, NJ 08648, USA. Web site: www.melcor.com
For thermodynamic consistency, the temperature-entropy (or temperature-entropy
flux) diagram is plotted instead using the entropy flux expression derived previously.
44
In an optimum current operation of the thermoelectric cooler, the presence of entropy
loss (due to conduction heat transfer and Joule heat) along the p-n legs is given by
TK∇ /Ti and they are manifested by enclosed areas “k-i-u-r-d-c-k” and “j-i-n-q-a-b-j”
of Figure 3.3. Both mentioned dissipative losses have the consequence of reducing the
cooling effect, represented by area “a-d-r-q”. Within the internal framework (adiabatic
chiller), the cycle “a-b-c-d” operates in an anti-clockwise manner with process with
two isotherms b-c and d-a, as well as two non-adiabats a-b and c-d. Details of these
processes are described in Table 3.2. Thus, the T-s diagram provides an effective
means of analyzing how the real chiller cycle of thermoelectrics is inferior to the ideal
Carnot cycle by capturing the key losses and useful effects in the cycle.
45
265
270
275
280
285
290
295
300
305 -0.0
0025
-0.0
002
-0.0
0015
-0.0
001
-0.0
0005
00.
0000
50.
0001
0.00
015
0.00
020.
0002
5
Seeb
eck
coef
ficie
nt (V
/K)
Temperature (K)
A
BC
D
hot j
unct
ion
cold
junc
tion
P-ar
mN
-arm
J
Figu
re 3
.2. T
-α d
iagr
am o
f a th
erm
oele
ctri
c pa
ir.
46
Figure 3.3. Temperature-entropy flux diagram for the thermoelectric cooler at the maximum coefficient of performance identifies the principal energy flows.
0
50
100
150
200
250
300
350
-500 -400 -300 -200 -100 0 100 200 300 400 500
Entropy flux density (W/m2 K)
Tem
pera
ture
(K)
P-armN-arm
a
bc
d eh i
jk
lJ
mnoqrtuv
TH
TL
Q dissip Q dissip
12
Net cooling power
47
Table 3.2 Description of close loop a-b-c-d-a (Figure 3.3)
Process Description
a--b
This process is developed along P-arm of the thermoelectric pair. Points a and b are the states at the ends of the thermoelement of which the temperatures are TL and TH respectively (as current J flows from P-arm to N-arm). The process a-b can not be adiabatic because it absorbs heat due to the Thomson effect which depends on the Seebeck coefficient of the thermoelectric element. The areas beneath the a-b process represent dissipation due to heat conduction, and heat absorbed by the Thomson effect. Area 2-i-j represents the energy dissipation due to Seebeck coefficient, which varies along the thermoelectric p-arm.
b--c This process represents the heat exchange between system and medium (environment) which occurs at the union between P-arm and N-arm at temperature TH.
c--d
The process c-d is developed along N-arm, where points c and d are at the ends of the thermoelement whose temperature are TH and TL. The areas beneath c-d process indicate heat losses due to dissipation. If this process is adiabatic, the Thomson effect would be null. Area 1-l-k indicates the energy losses due to Seebeck effect along the n-arm.
d--a
This process represents the heat exchange between the system and the heat load at isothermal condition; however it occurs in the physic union between thermoelectric elements at temperature TL.
From temperature-entropy flux diagram, the thermodynamic performance variables
can be easily tallied with the enclosed rectangles. The heat fluxes at the cold (process
d-a) and hot (process b-c) reservoirs are as follows (for details please see Appendix C)
( )
876876
876heatsonT
heatconductionheatJoulean
coolingPeltier
L
cjLL
cjLpnSLL
TITKRIT
TATT
TATJTadrqQ
_hom_
_
2_
,
21
21
∆−∆−−=
∇−=⎟⎟
⎠
⎞
⎜⎜
⎝
⎛ ∇−===
τα
λαλ
α
I
IIArea
, (3.15)
48
( )
876876
876heatsonT
lossConductionheatJoulea
heatPeltier
H
hjHH
hjHhjpnSHH
TITKRIT
TATT
TATJTbctoQ
_hom_
_
2_
,
21
21
∆+∆−+=
∇−=⎟⎟
⎠
⎞
⎜⎜
⎝
⎛ ∇−===
τα
λαλ
α
I
IIArea , (3.16)
where T∆ is the temperature difference across the hot and cold junctions. The first
term of right hand side of both equations (3.15) and (3.16) indicates Peltier cooling and
heating, the second term is the Joulean heat, the third term represents heat conduction
between hot and cold junctions and the heat absorbed by the Thomson effect. The
Thomson heat is transmitted by thermal conduction across the thermoelement and this
effect is not reversible. From the first law of thermodynamics, the required electrical
power input is given by,
( ) ( )
( ) TIRITTP
TITKRITTITKRIT
JTJTadrqbctoP
LH
Q
L
Q
H
coldpnSLhotpnSH
LH
∆++−=
∆+∆++−∆+∆−+=
−=−=
τα
τατα
2
22
,,
21
21
21
21
I
II
AreaArea
44444 844444 7644444 844444 76
(3.17)
The coefficient of performance is defined as PQCOP L= , could be non-
dimensionalized with its maximum COP and plotted against entropy flux of cold
reservoir for both the P and N arms, as shown in Figure 3.4. There are two curves for
the p and n arms of thermoelectrics, and the four points are indicated on each curve
are; (i) Point “A” refers to the low COP at low current limit, (ii) Point “B” is the
maximum COP at optimum current, (iii) Point “C” indicates the maximum entropy
flux or output power and (iv) Point “D” represents the vanishing COP at higher current
limit. It is instructive to compare the present T-s plot with those of the published
reference (Chua et al., 2002 (b)) for the same thermoelectric module.
49
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
-1.0 -0.8 -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 0.8 1.0
Sflux,L/Sflux,L max
CO
P/C
OP m
ax
P-armN-arm
JJ
A
B
C
DA
B
C
D
Figure 3.4. Characteristics performance curve of the thermoelectric cooler. COP is plotted against entropy flux for the both N and P legs.
Figure 3.5. Temperature-entropy flux diagram for the thermoelectric pair at the optimum current flow by the present study and the other researcher (Chua et al., 2002 (b)).
265
270
275
280
285
290
295
300
305
-400 -300 -200 -100 0 100 200 300 400
Entropy flux (W/m2 K)
Tem
pera
ture
(K)
P-armN-arm
a
bc
d a'd'
CO
P/C
OP m
ax
Js/Js,max
Present study Area abcd
Chua et al., 2002 (b) Area a′b′c′d′ ----
50
The recent work of Chua et al. (2002 (b)) assumes the adiabatic process for the T-
entropy flux diagram, as shown in Figure 3.5 where the adiabats are represented by
paths “a-b” and “c-d”. The shaded area (in Figure 3.5) is the heat dissipation due to
Thomson effect. In the Carnot cycle, internal and external dissipations (thermal and
electrical) are zero and the processes a-b and c-d becomes adiabatic. The equivalent
cycle at the same TH and TL is denoted by a rectangle 1-2-j-k (see Figure 3.3), for
which ( ) ( ) ( )[ ]2121 vmAreakvmjAreavmAreaCOPcarnot −= or
( ) ( ) TTjkAreavmAreaCOP Lcarnot ∆== 1221 .
The irreversibility is the heat dissipation, which is shown in Figure 3.3, occurs due to
the Joulean heat and to the heat conduction along the thermoelectric arm. For a better
Figure 3.6. Temperature-entropy generation diagram of a thermoelectric cooler at optimum cooling power. Area (abcd) is the total heat dissipation along the thermoelectric element, where area (aefd) is the irreversibility due to heat conduction and area (aghd) indicates the entropy generation due to joule effect.
265
270
275
280
285
290
295
300
305
0 40000 80000 120000 160000 200000
Entropy generation (W/m3 K)
Tem
pera
ture
(K)
P or N-arm
J J
a b
cd
e
f
g
h
Cold reservoir
Hot reservoirPN
51
understanding of the dissipative effects in the thermoelectric element, the temperature-
entropy generation (T- σ ′ ) plot is used and this is shown in Figure 3.6. Two major
dissipative mechanisms in the thermoelectrics can be observed: Firstly, Joule heat
dissipation occurs (enclosed by area a-g-h-d) evenly in the two sides of thermoelectric
arms and secondly, a smaller loss from the heat conduction (a-e-f-d) affects along the
thermoelectric element. The combined losses are given by the area (a-b-c-d) of Figure
3.6.
3.4 Transient behavior of thermoelectric cooler
The performance of a Peltier or thermoelectric cooler can be enhanced by utilizing the
transient response of a current pulse (Snyder et al., 2002). At fast transient, a high but
short-period (4 seconds) current pulse is injected so that additional Petlier cooling is
developed instantly at the cold junction: The joule heating developed in the
thermoelectric element is a slow phenomenon as heat travels slowly over the materials
and could not reached the cold junction in the short transient. Hence, the cold junction
temperature is at the lowest possible value and heat flows only from the source to the
cold junction. An example of such an application of the short transient cooling is an
infrared detector operating in a ‘winking mode, which needs to be super cold only
during the wink.
Figure 3.7 shows the schematic diagram of a current “pulse train” thermoelectric
cooler with the p- and n- doped semiconductor elements are electrically connected in
series and are thermally in parallel. The hot side of the device is mounted directly onto
a substrate to reject heat to the environment and the thermal contact between the cold
junction of thermoelectric device and the cold substrate (load) is made periodically.
52
There are two periods of operation: Firstly, an “open contact” period with tno and
current flow Jnp to maintain the lowest temperature Tmin at the cold junction but the
cold substrate (load) is not connected with the cold junction. Secondly, a large current
pulse Jp (where Jp=MJnp, M > 1) is applied for a duration tp and concomitantly, the
cold substrate makes contact with the cold junction. Intense Peltier cooling is achieved
during tp, which reduces the cold junction temperature below Tmin. However, Joule and
Thomson heating are developed in the bulk thermoelectric module and diffuse towards
the hot and cold ends of the device. Thermal contact of the cold substrate with the cold
end of the thermoelectric module is maintained over the short current pulse period but
these surface break contact before the cooling could be degraded by the conduction
heat. In the open contact manner, the cold substrate temperature could reach below that
of the cold junction temperature under the normal thermoelectric operation. The total
operation consists of two periods; one is the normal thermoelectric operation and the
other is the pulse or effective operation. The cycle of (normal operation + pulse
operation) is repeated such that the device is continuously produced cooling.
53
Figure 3.7. Schematic diagram of the first transient thermoelectric cooler. (a) indicates the normal operation and cold substrate is out of contact, (b) defines the thermoelectric pulse operation and cold substrate is in contact with the cold junction, (c) shows the current profile during normal thermoelectric operation, (d) indicates that the pulse current is applied to the device and (e) the cycle of (tno + tp) is repeated such that the device is operated in an ac mode.
J np
time t
J p
P N P N P N P N P N P N
Hot substrate H ot junction
Cold junction
Cold substrate (a)
P N P N P N P N P N P N
Hot substrate H ot junction
Cold junction
Cold substrate (b)
(e)
J
t ime t
J np t no
(c)
J
time t
Jp
tp
(d)
J np
J p
54
This section formulates a non-equilibrium thermodynamic model to demonstrate the
energy and entropy balances of a transient thermoelectric cooler. In analyzing the fast
transient cycle in the classical T-s plane, one distinguishes the entropy fluxes of the
working fluid (which is the current flux J) from the external entropy fluxes in the hot
and cold reservoirs and at the interface with the reservoirs. The T-s relation is
formulated that identifies the key heat and work flows, the heat conduction and
electrical resistive dissipative losses.
3.4.1. Derivation of the T-s relation
The methodology of energy and entropy balances of the commercial available
thermoelectric device has been discussed briefly in the previous sections. Starting from
the general entropy balance equation, it can be expressed in terms of the key
parameters such as Peltier effect ( )π , electrical conductivity ( )σ ′ , current density (J)
and Fourier effect (λ), as shown below:
( )4444 84444 76444 8444 76
generationentropy
pp
p
ppp
fluxentropy
tep
p
p
pppp
TJ
TT
TJ
TT
ts
⎪⎭
⎪⎬⎫
⎪⎩
⎪⎨⎧
′+⎟
⎟⎠
⎞⎜⎜⎝
⎛∇∇−+⎟
⎟⎠
⎞⎜⎜⎝
⎛−
∇⋅∇=
∂
∂
σλ
πλρ 2
,
1. , (3.18)
hence the subscript p defines the pulsed thermoelectric cooler, the first term of the
right hand side indicates the entropy flux in a given control volume and the second
terms represent the total entropy generation. The total entropy flux density (in W/m2
K) is expressed by
pp
pp
ptots JT
Tα
λ−
∇=,J , (3.19)
and the entropy generation (in W/m3 K) is written as
pp
p
ppptotp T
JT
Tσ
λσ′
+⎟⎟⎠
⎞⎜⎜⎝
⎛∇∇−=
2
,1. . (3.20)
55
The energy conservation equation is as follows
pp
ppTteppp
ppp
ppp T
TTJTJ
JT
tT
cp
∇⎟⎟⎠
⎞⎜⎜⎝
⎛
∂∂
−∇+′
+∇=∂
∂ αασ
λρ ,
22 . (3.21)
To solve the problem, two boundary conditions are defined: Firstly, during the normal
(open-contact) operation, there is only Peltier cooling at the cold junction
p
cj
cjat
p JTx
Tλ
α=
∂
∂, (3.22)
and at hot junction, a hot side heat sink is represented by, hjhjatp TT = . But during the
pulse (close-contact) operation, at the cold junction the boundary condition becomes
p
subs
p
cj
cjat
p qJTx
Tλλ
α .
+=∂
∂, (3.23)
where .subsq is the cooling load of the cold substrate.
3.4.2 Results and discussions
The transient behavior is demonstrated here using the typical physical and mechanical
properties of a thermoelectric cooler and the simulated results for the hot and cold
junctions are shown in Figure 3.8. A pulse-type cooler is useful not only for testing the
theoretical models but also is in designing the device. Firstly, the thermoelectric cooler
is supplied a steady current until no cooling is produced at the cold end but the
junction temperature of the cold junction reaches a TA’ (at point A’) whilst the hot-
junction is at a constant temperature of TH, as shown in Figure 3.8. A pulse current, at
a magnitude of M (=Ip/Inp) folds from the steady current, is then supplied to the
thermoelectric device which increase the cooling momentarily and depressing the cold
junction temperature further from A’ to B. At which time, the effect of Joulean heat,
56
conducted across the hot to cold junctions, is being felt causing a rise in the cold
junction temperature from B to C. Although the current pulse is terminated at point C,
breaking physical contact to the cold substrate, temperature within the cold junction
continue to rise to point B’ due to the delayed Joulean heat transfer in the legs of the
thermoelectrics.
The parameters used in this modeling are shown in Table 3.3. The energy flows in the
thermoelectric device can be captured by the use of a temperature-entropy diagram, as
shown in Figures 3.9 and 3.10. For the non-pulsing or contact interval, Fig 3.9 shows
220
230
240
250
260
270
280
290
300
310
320
0 20 40 60 80 100 120time (s)
tem
pera
ture
(K)
C
C'
B
B'
A
A'
hot junction
cold junction
Figure 3.8. the cold reservoir temperature of a super-cooling thermoelectric cooler as a function of time. The small loop indicates the super-cooling during current pulse operation. Temperature drop between the cold and hot junctions subjected to a step current pulse of magnitude 3 times the DC current and of width 4 s. A indicates the beginning of pulse period, B defines the maximum temperature drop and C represents the end of pulse period, C’ is the beginning of non-contact period, B’ indicates maximum temperature rise due to residual heat.
Time (s)
Tem
pera
ture
(K)
57
the typical anti-clockwise loop (A’) for the entropy fluxes in the p and n-legs, as
indicated by the process states “a-g-h-a”. The area under the isothermal path “g-h”
indicates the total energy supplied to the thermoelectrics. However, it is noted that the
cold isotherm (T_a) converges to a point at “a” where no area is enclosed under the
isotherm. Thus, no cooling power is generated at the cold junction, i.e., the cooling
generated at the cold junction is totally overwhelmed by the flow of Joulean heat from
the hot junction. Similar T-s plots are also found for points C’ and B’ during the open-
contact operation of the device where the cold isotherms become a single point at “e”
and “f”. It is also noted that the maximum temperature attained within the p and n-legs
might exceed that of the hot junction temperature.
During closed contact operation, as shown in Fig.3.10, point A indicates the start of
contact interval where the imposed current pulse generates substantial cooling,
depressing the cold junction and the cold template temperatures from “f” to “a” (see
the smaller insert diagram of Fig. 3.10). The area below the isotherm b-b' indicates the
amount of cooling on the T-s plot at point A. The isotherms c-c' and d-d' correspond to
the points B and C. It is noted that although the enclosed area for point C is larger than
that of point B, but the temperature of point C has risen due to the conduction of
Joulean heat from the hot junction.
58
220
240
260
280
300
320
340 -3
00-2
00-1
000
100
200
300
entr
opy
flux
(W/m
2 K)
temperature (K)
p-ar
mn-
arm
gh
aef
I np
C'
B'
A'
Figu
re 3
.9.
T-s
dia
gram
of
a pu
lsed
the
rmoe
lect
ric
cool
er d
urin
g no
n-pu
lse
oper
atio
n; C′
defin
es t
he b
egin
ning
of
non-
cont
act,
B′
indi
cate
s th
e m
axim
um t
empe
ratu
re r
ise
of c
old
junc
tion
duri
ng n
on-p
ulse
per
iod
and
A′ r
epre
sent
s th
e m
axim
um te
mpe
ratu
re d
rop
of c
old
junc
tion
with
no
cool
ing
pow
er.
59
220
240
260
280
300
320
340 -3
00-2
00-1
000
100
200
300
entr
opy
flux
(W/m
2 K)
temperature (K)
p-ar
mn-
arm
I p
km
cc'
bb'
dd'
ef
AB
C
225
230
235
240
245
250
255
260
265
-6-4
-20
24
6
cc'
bb'
dd'
e
A
B
C
B
aef
Figu
re 3
.10.
Tem
pera
ture
-ent
ropy
dia
gram
of
a pu
lsed
the
rmoe
lect
ric
cool
er d
urin
g pu
lsed
cu
rren
t op
erat
ion;
A
de
fines
th
e be
ginn
ing
of
cool
ing,
B
re
pres
ents
th
e m
axim
um
tem
pera
ture
dro
p an
d C
indi
cate
s the
max
imum
coo
ling
pow
er.
60
Table 3.3 Physical parameters of a pulsed thermoelectric cooler
Property value Hot reservoir temperature, Thj (K) 308
Thermoelectric element length, Lte (mm) 5.8 (Snyder et al., 2002)
Cross-sectional area, Ate (mm2) 1 (Snyder et al., 2002)
Geometric factor, GFte (mm) 0.1724 (Snyder et al., 2002)
Current, I (amp) 0.675 (Snyder et al., 2002)
Electrical restivity (ohm-cm)
82210 10)( −×++= aveave TT ρρρρ
ρ0 = 5112.0, ρ1 = 163.4, ρ2 = 0.6279
(Melcor)
Seebeck coefficient (V/K)
( ) 92210 10−×++= aveave TT αααα
α0=22224.0, α1= 930.6, α2 = -0.9905
(Melcor)
Thermal conductivity (W/cm K)
( ) 92210 10−×++= aveave TT λλλλ
λ0= 62605.0, λ1= -277.7, λ2 = 0.4131
(Melcor)
Tave = (Thj + Tcj)/2 Melcor thermoelectric catalogue, MELCOR CORPORATION, 1040 Spruce Street, Trenton, NJ 08648, USA. Web site: www.melcor.com
3.5 Microscopic Analysis: Super-lattice type devices
As dimensions of thermoelectric elements (both p and n arms) are reduced to super-
lattice or quantum well scale, they form a category of thin-film thermoelectric cooler
where the conventional macro-scale modeling may not be adequate. Such studies are
of importance because of their potential for achieving high cooling and power
densities. At the micro dimensions, the crystalline structures form the quantum wells
which increase the electron mobility as well as reducing the electrical conductivity. An
example of a super-lattice that is made of silicon as a barrier material and Si0.8Ge0.2 as
conducting material is shown in Figure 3.11.
61
As the silicon (Si) and germanium (Ge) both have the similar crystalline structure, the
Si and solid solution SiGe are fused such that the atoms fit together to form a well-
ordered lattice. The super-lattice quantum well materials are grown by depositing
semiconductors in layers, whose thickness is in the range of a few to up to about 100 Ǻ
(Angstroms), comprising only a few atoms thick. The superlattice structures enhance
the thermoelectric arm performance by reducing the thermal conductivity between the
hot and cold junctions (Elsner et al., 1996) whilst the p-type and n-type superlattice
thermoelectric elements are formed into an “egg-crate” configuration (Elsner et al.,
1996) with metal contacts.
When an electric field is applied to the thin-film thermoelectric cooler, a fraction of
electrons possessing a sufficiently high value of kinetic energy absorbs heat from the
cold reservoir, seeps through the thin film and releases their kinetic energy to the hot
reservoir. Meanwhile, heat is conducted back to the cold reservoir by phonons from the
hot side, which is also heated up the energetic electrons by collisions. Electrons are
…..…..…..
………………
Hot reservoir
Cold reservoir
J
J
Figure 3.11 A schematic of the superlattice thermoelectric element (Elsner et al., 1996)
62
assumed to pass ballistically through the thin film between the hot and cold reservoirs
such that the kinetic energy of electrons is converted to phonons at a minimum level
and a high cooling efficiency is achieved. This would be possible if the thin film
thickness is less than the mean free path of the electrons in the thin film or quantum
well superlattice (Mahan and Woods, 1998). Since the length of the practical device is
typically larger than the electron mean free path, electrons collisions in the thin film
have to be considered in the modeling process.
The efficiency of a thermoelectric module is fundamentally limited by the material
properties of the n- and p-type semiconductors which is determined by a dimensionless
parameter ZT, where T is the temperature and Z characterizes the material’s electrical
and thermal transport properties. Effective thermoelectric materials have a low thermal
conductivity but a high electrical conductivity. The best thermoelectric materials
commercially available today have a ZT ≈ 0.9. For a thermoelectric device to be
competitive to the performance of refrigerators, it would require a ZT ≈ 3 at room
temperature. Recently, several materials with ZT > 1 for both refrigeration and power
generation applications have been reported, in particular the work of
Venkatasubramian et al. (2001), who reported that a high ZT = 2.4 for p-type
superlattices of Bi2Te3/Sb2Te3 at room temperature and ZT = 1.2 for n type
superlattices.
The use of quantum-well structures to enhance the thermoelectric figure of merit was
proposed (Hicks and Dresselhaus, 1993). Since then much work has been done in the
study of superlattice thermoelectric properties, mostly for the in-plane direction where
the enhanced density of electrons (thus, increase in ZT) was achieved by reducing
63
dimensionality. Recent study shows that large ZT improvement is possible for the
cross plane transport as thermal conductivity of cross-plane direction is lower than that
of in-plane direction
This section describes the modeling framework of the superlattice thermoelectric
device to understand the energy transport in semiconductors. The mass, momentum,
energy and entropy balances of the superlattice thermoelectric device are based on the
general arguments from equilibrium and irreversible thermodynamics (de Groot and
Mazur, 1962). These equations contain collision terms that are related to the variations
of the distribution function due to collisions. The Monte Carlo method (Jacoboni and
Reggiani, 1983), which is based on the successive and simultaneous calculation of the
motions of many particles during a small time increment and suitable for the analysis
of transient carrier motion, is used to solve collision terms. Monte Carlo method relies
on the generation of a sequence of random numbers with given distribution
probabilities.
3.5.1 Thermodynamic modeling for thin-film thermoelectrics
The interactions of electrical and thermal phenomena are important to simulate the
superlattice thermoelectric element. The most important thermal effect is the thermal
runaway, in which the electrical dissipation of energy causes an increase in
temperature which in turn causes a greater dissipation of energy. Without repeating the
basic derivation (as derived in Chapter 2), a complete model of electron, hole and
phonon conservation equations for a thin film thermoelectric element is summarized in
Table 3.4
64
Table 3.4 the balance equations
As seen in Table 3.4, the Boltzmann Transport Equation is employed together with (as
discussed in Chapter 2) the phenomenological relationship (Onsager, 1931). The
energy balance equations for electrons, holes and phonon are computed using Gibbs
law. Based on these formulation, the entropy flux and entropy generation for electrons,
holes and phonons in the well and barrier of the superlattice thermoelement are
developed and they are summarized in Table 3.5. The Joule heating in the wells
produces a temperature variation along the x axis; however, the temperature remains
constant along the z axis as shown in Figure 3.12.
Particles Equations
mass ( )colli
eee
e
tnnv
tn
∂∂
+−∇=∂∂ (3.24)
energy colli
eee
e
tUQ
tU
∂∂
++−∇=∂∂ JF. (3.25)Electrons
entropy gecollihecollieirrsee Jt
S−− +++∇=
∂∂
,,,. σσσ (3.26)
mass ( )colli
hhe
h
tnnv
tn
∂∂
+−∇=∂∂ (3.27)
energy colli
hhh
h
tUQ
tU
∂∂
++−∇=∂∂ JF. (3.28)Holes
entropy ghcolliehcollihirrshh Jt
S−− +++∇=
∂∂
,,,. σσσ (3.29)
energy colli
hg
g
tUQ
tU
∂∂
+−∇=∂
∂. (3.30)
Phonons entropy hgcolliegcolligirrgs
g Jt
S−− +++∇=
∂
∂,,,,. σσσ (3.31)
energy collit
UQt
U∂∂
++−∇=∂∂ JF. (3.32)Combined
(Electron + Hole + Phonon) entropy σ+∇= sJ
dtdS . (3.33)
65
Table 3.5 the energy, entropy flux and entropy generation equations of the well and the barrier.
well
Particles Energy balance Entropy flux W/m2 K
Entropy generation W/m3K
Electron ( )
colli
eww
ew
ew
ewewew
w
tT
cJ
Tt
Tc
∂∂
+′
+
∇∇=∂
∂
,
,
2,
,,, .
σ
λ
ew
ewe
ew
ewew
TJ
TT
,
,
,
,, πλ+
∇−
colli
we
we
w
wewe
we
wewewe
tT
Tc
TJ
TT
∂∂
+
′+⎟⎟⎠
⎞⎜⎜⎝
⎛∇∇−
σλ
21.
Hole ( )
colli
hww
hw
hw
hwhwhw
w
tT
cJ
Tt
Tc
∂∂
+′
+
∇∇=∂
∂
,
,
2,
,,, .
σ
λ
hw
hwh
hw
hwhw
TJ
TT
,
,
,
,, πλ+
∇−
colli
wh
wh
w
whwh
wh
whwhwh
tT
Tc
TJ
TT
∂∂
+
′+⎟⎟⎠
⎞⎜⎜⎝
⎛∇∇−
σλ
21.
Phonon ( )
colli
gwwgwgw
gww t
TcT
tT
c∂
∂+∇∇=
∂
∂ ,,,
, . λ
gw
gwgw
TT
,
,, ∇−λ
colli
wg
wg
w
wgwgwg
tT
Tc
TT
∂
∂+
⎟⎟⎠
⎞⎜⎜⎝
⎛∇∇−
1.λ
Combined ( )
colli
ww
w
www
ww
tTc
JTt
Tc
∂∂
+
′+∇∇=
∂∂
σλ
2
.
w
w
w
ww
TJ
TT πλ
+∇
−
colli
w
w
w
ww
w
www
tT
Tc
TJ
TT
∂∂
+
′+⎟⎟⎠
⎞⎜⎜⎝
⎛∇∇−
σλ
21.
barrier Energy balance Entropy flux Entropy generation
Electron ( )colli
ebbebeb
ebb t
TcT
tT
c∂∂
+∇∇=∂∂ ,
,,, . λ
eb
ebeb
TT
,
,, ∇λ−
colli
be
be
b
bebebe
tT
Tc
TT
∂∂
+
⎟⎟⎠
⎞⎜⎜⎝
⎛∇∇−
1.λ
Hole ( )colli
hbbhbhb
hbb t
TcT
tT
c∂
∂+∇λ∇=
∂∂ ,
,,, .
hb
hbhb
TT
,
,, ∇−λ
colli
bh
bh
b
bhbhbh
tT
Tc
TT
∂∂
+
⎟⎟⎠
⎞⎜⎜⎝
⎛∇∇−
1.λ
Phonon ( )colli
gbbgbgb
gbb t
TcT
tT
c∂
∂+∇∇=
∂
∂ ,,,
, . λ gb
gbgb
TT
,
,, ∇−λ
colli
bg
bg
b
bgbgbg
tT
Tc
TT
∂
∂+
⎟⎟⎠
⎞⎜⎜⎝
⎛∇∇−
1.λ
66
Combined ( )colli
bbbb
bb t
TcTt
Tc∂∂
+∇∇=∂∂ λ.
b
bb
TT∇
−λ
colli
b
b
b
bbb
tT
Tc
TT
∂∂
+
⎟⎟⎠
⎞⎜⎜⎝
⎛∇∇−
1.λ
The wells are electric conducting and the barriers are electric non-conducting. At y = 0
the sample is in contact with a cold thermal reservoir of temperature TL, and at y = L
with a hot reservoir of temperature TH. An electric field J is applied parallel to the
thermal gradient. The sample is bulk in size in the xz plane.
. . . . . . … . . . . . . …
Hot reservoir, TH
Cold reservoir, TL
b b
a
barrier barrier well
J
x
y z
Figure 3.12. Schematic diagram of a superlattice thermoelement with electric conducting wells and electric insulating barriers (Antonyuk et al., 2001).
67
3.5.2 Results and discussions
Due to current flow into the superlattice thermoelectric element, the Joulean heat,
which is generated in a well, flows eventually along the thermoelectric element into
the cold and hot reservoirs either via the well or via the barrier. The amount of heat
transported through the well and barrier depends on their cross-sectional areas and
thermal conductivity. The bulk and superlattice thermoelement properties and their
dimensions used in this simulation are taken from the experimental analysis (Elsner et
al., 1996 and Antonyuk et al., 2001), are shown in Table 3.6. The Flow Chart of the
simulation is provided in Appendix D.
Table 3.6 the thermo-physical properties of bulk SiGe and superlattice Si.8Ge.2/Si thermoelectric element.
Sample Electrical resistivity mΩ-cm
Seebeck coefficient
µV/oC
Thermal conductivity
W/m K
Carrier concentration
Figure of merit
Si.8Ge.2/Si 2N 4 -1250 7.66 1019 3101.5 −×
Si.8Ge.2/Si 6P 1.4 850 12.9 20105× 3104 −×
Bulk SiGe P 1 130 5.63 1020 3103.0 −×
Bulk SiGe N 2 -200 6.06 1020 31033.0 −×
Length, m Width, m Thickness, m
Well 6103 −× 6102 −× 101020 −×
Barrier 6103 −× 6102 −× 1010100 −×
The electrons flowing in the well are taken as electron gas. When the hot electron gas
flows through the thin film, it loses energy to holes and phonons, however, it also
gains energy from the external applied voltage. At high electric field the thin film
68
thermoelectric semiconductor produces high energetic electrons and the equilibrium
between electrons and phonons decreases because of high rate of heat generation. At
higher electric field, the electron temperature is generally higher than the phonon
temperature, as shown in Figure 3.13. In addition, the electron and phonon interaction
rate are assumed high enough to saturate the electron drift velocity where the electrons
become so energetic that they can not dissipate energy to the phonons through
collisions. Therefore, they operate far out of equilibrium from the phonons or lattice
vibrations.
260
270
280
290
300
310
320
330
0 0.5 1 1.5 2 2.5 3co-ordinate (micro m)
tem
pera
ture
(K)
260
270
280
290
300
310
320
330
tem
pera
ture
(K)
electron temperature
phonon temperature
Figure 3.13. the variations of electron temperature in a thin film thermoelectric element at different electric field; (---) indicates the phonon temperature, (—) defines the electron temperature at 12.5 Kamp/cm2, (---) shows the electron temperature at 25 Kamp/cm2 and () represents the electron temperature at 125 Kamp/cm2.
69
The micro-scale heat transport phenomena include electron transport, phonon transport
and more importantly, heat generation during electron-phonon interactions. From T-s
diagram, which is shown in Figure 3.14, it is observed that the electron flow in n-type
thermoelectric element is opposite to the current flow whilst the hole flow in p-type
thermoelectric element follows the same direction of current flow. The direction of
phonons is the opposite of electrons and holes. The phonons and the collisions of
electrons and phonons create the inherent dissipation and hence, decrease the device
performance.
The close loop, given by g-h-i-j-k-l-g, establishes the T-s diagram based on electron
temperature and electron current. In this cycle, cooling capacity is equal to (TL x g-l)
Figure 3.14. The temperature-entropy diagram of a superlattice thermoelectric couple (both the p and n-type thermoelectric element are shown here)
250
260
270
280
290
300
310
-800000 -600000 -400000 -200000 0 200000 400000 600000 800000
entropy flux (W/m2 K)
tem
pera
ture
(K)
a
b
cd
e
f a'
b'c'
d' g
h
ij
k
lTL
TH
p-armn-arm
cold reservoir (Isotherm)
hot reservoir (Isotherm)
J
phonon flowphonon flow
electron flow hole flow
70
and the heating power is (TH x i-j). The coefficient of performance due to electron and
hole flow is given by (TL x g-l)/(TH x i-j- TL x g-l), which is large and quite close to the
Carnot COP. This is because the electron thermal conductivity is small as compared
with the phonon thermal conductivity and the total thermal conductivity is the sum of
electron and phonon thermal conductivities.
The energy dissipation due to the transfer of phonons from hot reservoir to cold
reservoir is shown by points a′-b′-c′-d′-a′, as shown in T-s plot of Figure 3.14, and it is
noted that more heat is dissipated at the cold end (a′d′ > b′c′). The performance of the
device is given by state points “a-b-c-d-e-f-a. For example, the net cooling power is
defined by, QL, net = TL(gl - a′d′) = TL (af), where “gl” is the entropy flux at the cold end
due to electrons and holes flowing from cold side to the hot end and “a′d′” is the heat
dissipation due to phonons or lattice vibrations and phonon-electron collisions. The net
heating power is QH, net = TH(ij - b′c′) = TH (cd), where “ij” is the entropy flux at the hot
side due to electron and hole and “b′c′” is the phonon flow at the hot end. So the net
performance is calculated as TL (af)/ TH (cd) - TL (af).
It is informative to show the effect of collisions between electrons and phonons. Figure
3.15 shows the temperature-entropy diagrams for collisions and non collision effects.
Due to collisions, the decrease of cooling power is ( )DDAATSST LcjfluxcjfluxL ′+′=− 1,2, ,
and the increase of heating power is ( )CCBBTSST HhjfluxhjfluxH ′+′=− 1,2, . So the net
amount of input power is increased by ( ) ( )[ ]DDAATCCBBT LH ′+′−′+′ as shown in
Figure 3.15. The net coefficient of performance is dropped by approximately 7.4 %
due to collisions of electrons and phonons, and holes and phonons when the electric
field is 75 KA/cm2. Here it is important to note that the cooling power density of the
71
commercial thermoelectric cooler is limited to several watts per square centimeters,
and the thin film thermoelectric coolers can achieve cooling power densities as high as
several thousands of watts per square centimeters (Vashaee, and Shakouri, 2004).The
performances of the superlattice thermoelectric cooler are furnished in table 3.7.
Table 3.7 Performance analysis of the superlattice thermoelectric cooler at various electric field.
With out collision effects With collision effects J
kA/cm2 Entropy flux at hj W/cm2K
Entropy flux at cj W/cm2K
COP Entropy flux at hj W/cm2K
Entropy flux at cj W/cm2K
COP
Input energy loss
due to collisions
(%) 25 95.8 37.9 0.55 96.6 37.1 0.52 4.8 75 185.9 116.9 1.30 189.7 115.8 1.21 7.4 125 414.9 301.4 1.88 427.2 299.1 1.7 11.8 250 658.2 467.1 1.77 684.3 462.8 1.55 15.3 375 911.8 616.1 1.55 958.1 607.2 1.32 19.4 500 2524.6 1102.6 0.65 2830.7 1063.3 0.51 32.7
For the case of optimum operation 125 kA/cm2
QH, ncolli 0.744 mW QH, colli 0.768 mW
QL, ncolli 0.489 mW QL,colli 0.484 mW Pin, ncolli 0.255 mW
Pin, colli 0.302 mW Loss due to collisions Qcolli = 0.027 mW
72
Figu
re 3
.15.
the
tem
pera
ture
-ent
ropy
dia
gram
s of
p a
nd n
-typ
e th
in fi
lm th
erm
oele
ctri
c el
emen
ts. T
he c
lose
loop
A-B
-C-D
sho
ws
the
T-s
dia
gram
whe
n th
e co
llisi
on t
erm
s ar
e ta
ken
into
acc
ount
and
the
clo
se l
oop
A′-B
′-C′-D
′ in
dica
tes
the
T-s
dia
gram
whe
n co
llisi
ons a
re n
ot in
clud
ed. E
nerg
y di
ssip
atio
n du
e to
col
lisio
ns is
show
n he
re.
240
250
260
270
280
290
300
310
320 -500
000
-400
000
-300
000
-200
000
-100
000
010
0000
2000
0030
0000
4000
0050
0000
entr
opy
flux
(W/m
2 K)
temperature (K)
hot r
eser
voir
(Isot
herm
al)
cold
rese
rvoi
r (I
soth
erm
al)
J
p-ar
mn-
arm
ener
gy d
issi
patio
n du
e to
col
lisio
n
S flu
x, c
j1
S flu
x, c
j2
S flu
x, h
j1
S flu
x, h
j2
AA′
B
B′
C C
′
D
D′
TH
TL
73
3.6 Summary of Chapter 3
From the macro viewpoints, this chapter presents the pedagogical value of a
temperature-entropy flux diagram, where it shows clearly how the energy input in a
thermoelectric cooling device has been consumed by both reversible (Peltier, Seebeck
and Thomson) and irreversible (Joule and Fourier) phenomena. A similarity has been
found between the Seebeck coefficient and entropy flux. T-α diagram indicates the
reversible effect of a thermoelectric pair. T-α is same as the Carnot cycle if the
thermoelectric arms are considered adiabatic. But the temperature dependent seebeck
coefficient indicates the real T-α cycle. The effect of Thomson heat has been
incorporated in the performance estimation of a thermoelectric cooler. This effect,
albeit 5% or so, increases irreversibility at the cold junction. Using the macroscopic
BTE equation approach, the transient behavior of a pulsed-operated thermoelectric
cooling device is accurately presented. The distribution of input power in such a device
for both the generation of useful effects as well as in overcoming the dissipative losses
could be accurately captured by a T-s diagram. The present model predicts accurately
the maximum temperature difference between the hot and cold junctions with respect
to the functions of the pulse current. From the micro viewpoints, this Chapter also
presents the T-s diagram of a superlattice thermoelectric element where the collision
effects of electrons, holes and phonons become significant.
The next Chapter starts with the experimental realization of adsorption isotherms and
characteristics to model an electro-adsorption chiller (EAC).
74
Chapter 4
Adsorption Characteristics of Silica gel + Water
Prior to electro-adsorption chiller (EAC) cycle simulation, it is essential to determine
the adsorption characteristics of water vapor on silica gel, namely the isotherms,
monolayer vapour-uptake capacity and the enthalpy of adsorption. Such data are
needed to determine the energetic performance of adsorption cooling cycles. This
chapter presents (1) the measurement of surface characteristics such as surface areas,
pore size distributions, pore volume, and skeletal densities of silica gels through
nitrogen gas adsorption at liquid nitrogen temperature, (2) the adsorption isotherm
characteristics of water vapor onto two types of silica gel (namely the Fuji Davison
type ‘A’, and ‘RD’) by using a volumetric technique at controlled temperatures
ranging from 303 K to 338 K and at pressures between 500 and 7000 Pa. The
experimental isotherms are then fitted with the Tóth equation which could be used in
the simulation code and it has been found that the predicted isotherms are within the
experimental errors. Extending the Tóth isotherm equation, the monolayer uptake
capacity and the enthalpy of adsorption are also calculated. For verification, these
experimental isotherms and the corresponding computed isosteric enthalpies of
adsorption are compared with published data available from the literature.
75
4.1 Characterization of silica gels
The common types of silica gel found in the commercial chillers are the Fuji Davison
type ‘A’, and ‘RD’. It is well known that the thermophysical properties of silica gels
can directly affect adsorption performance via parameters such as the equilibrium
adsorption capacity and the molecular and surface pore sizes, etc. For an adsorbent to
work well, it should possess a large specific surface area and a relatively good thermal
conductivity. The thermophysical properties of silica gel to be measured are (i) the
surface area, (ii) pore size, (iii) pore volume, (iv) porosity and (v) skeletal density.
Pore size and surface area measurements are performed on the Micrometrics Gas
Adsorption analyzer (model ASAP-2010 with analysis software Version 4) and it
employs the liquid Nitrogen at 77K as the filing fluid. Prior to the experiment, samples
of silica gel are degassed at 413 K for 48 h, to achieve a residual vacuum within the
sample tube of less than 10 mPa. To compute the surface areas, the Brunauer Emmet
Teller (BET) plots of nitrogen adsorption data are selected over the range Pr = 0.03 –
0.3 (following the procedure outlined by IUPAC, 1985) to get the best linear fit - Pr
denotes the relative pressure or the ratio of the prevailing vapor pressure to the
corresponding saturated pressure. A sample of the characterization data are tabulated
in Table 4.1. Using the software provided by the analyzer (Density Functional Theory
DFT- Version 2), the pore volumes and size distributions of the silica gel samples are
computed. The pore size distributions for the two types of silica gel (type “RD” and
type “A”), are depicted in Figure 4.1. It is noted that the Barrett-Joyner-Halenda (BJH)
method has also been investigated for calculating the pore size distributions of the
silica gels but the results are found to be less satisfactory as the method is usually used
to calculate the mesopore structures.
76
The skeletal densities of dry silica gels are determined by the automated Micromeritics
AccuPyc 1330 pycnometer. Helium gas is employed in the test because it behaves
ideally. In the experiments, each silica gel sample is regenerated at 413 K for 24 h and
a dry mass sample is then introduced to the pycnometer. Each sample is tested three
times and the average value is taken. Using the measured skeletal densities and pore
volumes (DFT method), the particle bulk densities of silica gels could then be
calculated. The experimentally determined thermo-physical properties are summarized
in Table 4.2 and one observes that the thermophysical properties of these two types of
silica gel are similar. Although the type 'RD' silica gel has a slightly higher thermal
conductivity than type ‘A’ silica gel, the most significant difference between these two
types of silica gels lies in their water vapor uptake characteristics.
77
Table 4.1 Characterization data for the type ‘RD’ and ‘A’ silica gel
Type RD Type A
Pr 0.048688 0.055774 0.063324 0.071464 0.080186 0.089197 0.098848 0.108946 0.119269 0.130128 0.141184 0.152436 0.164243 0.176378 0.188519 0.201084 0.213885 0.226838
q* (cm3/g STP)
184.2954 188.3066 192.1586 195.9647 199.7403 203.3650 207.0079 210.5666 213.9935 217.4258 220.7799 224.0624 227.3611 230.6294 233.7987 236.9616 240.0596 243.1273
Pr
0.050404 0.059275 0.069060 0.079633 0.091123 0.103225 0.115798 0.129227 0.143242 0.157589 0.172680 0.188005
q* (cm3/g STP)
160.577 164.4663 168.2734 171.9793 175.4915 179.0015 182.3687 185.7076 188.9687 192.0915 195.2137 198.2016
78
Figu
re 4
.1. P
ore
size
dis
trib
utio
n (D
FT-S
lit m
odel
) for
two
type
s of s
ilica
gel
.
Incremental Pore Volume/ cm3 g-1
Pore
wid
th/ n
m
79
Table 4.2. Thermophysical properties of silica gels*
Property Value
Type ‘A’ Type ‘RD’
BET/N2 surface area c
BET constant c
BET Volume STP c
Range of Pr c
(716 ± 3.3) m2⋅g-1
293.8
164.5 cm3⋅g-1
0.05 ~ 0.19
(838 ± 3.8) m2⋅g-1
258.6
192.5 cm3⋅g-1
0.05 ~ 0.23
Pore size c (0.8 ~ 5) nm (0.8 ~ 7.5) nm
Porous volume c
Micropore volume c
Mesopore volume c
0.28 cm3⋅g-1
57%
43%
0.37 cm3⋅g-1
49%
51%
Skeletal density d 2060 kg⋅m-3 2027 kg⋅m-3
Particle bulk density e 1306 kg⋅m-3 1158 kg⋅m-3
Surface area f 650 m2⋅g-1 720 m2⋅g-1
Average pore diameter f 2.2 nm 2.2 nm
Porous volume d 0.36 cm3⋅g-1 0.4 cm3⋅g-1
Apparent density f** 730 kg⋅m-3 700 kg⋅m-3
Mesh Size f 10 - 40 10 – 20
pH f 5.0 4.0
Water content f < 2.0 mass %
Specific heat capacity f 0.921 kJ⋅kg-1⋅K-1 0.921 kJ⋅kg-1⋅K-1
Thermal conductivity f 0.174 W⋅m-1⋅K-1 0.198 W⋅m-1⋅K-1
* Manufacturer's representative chemical composition (dry mass basis): SiO2 (99.7%), Fe2O3 (0.008%), Al2O3 (0.025%), CaO (0.01%), Na2O (0.05%). c Remarks ASAP 2010 d Remarks AccuPyc 1330 e Remarks computed f Remarks Manufacturer ** This value is inclusive of bed porosity.
80
4.2 Isotherms of the silica gel + water
The next step is to determine the isotherm characteristics of silica gel + water, the
isosteric heat of adsorption, and the monolayer uptake capacity which are useful in the
simulation of the adsorption cooling cycle. In the literature, two methods are available
for the above-mentioned measurements, namely (i) the thermal-gravimetric analyzer
(TGA) and (ii) the constant-volume-variable pressure (CVVP) apparatus. Owing to
budget limitation, the CVVP apparatus has been designed and employed for the
experiment. This section investigates experimentally the isotherm adsorption of water
vapor onto Fuji Davison type ‘RD’ and type ‘A’ silica gel.
4.2.1 Experiment
Figure 4.2 shows the schematic diagram of the CVVP which comprises (i) a SS 304
charging tank with a volume of 572.64 ± 27 cm3, inclusive of related piping and
valves, (ii) a SS 304 dosing tank of volume 698.47 ± 32 cm3, inclusive of related
piping and valves, (iii) an evaporator flask with a volume of 2000 cm3 and (iv) the
electro-pneumatic gate valve. The charging tank is designed to have a high aspect
ratio, so that the adsorbent could be spread on the large flat base. Its external wall is
coiled with copper pipes to enable in-situ regeneration of the silica gel using a hot
silicone oil bath supplied at 140 °C.
81
`
The volumes of the charging and dosing tanks is calibrated by using calibrated
standard volume (210.9 ± 0.2 cm3) with a helium gas where the helium is used to fill
the mentioned tanks, sequentially. By measuring the pressures of the combined
volumes at isothermal conditions, the volumes of both the charging and the dosing
tanks are calculated using ideal gas law. Pressures in the two tanks are measured with
two calibrated capacitance manometers (Edwards barocel type 622; 10 kPa span; total
error in Pa: if 90 < P/Pa < 900, total error in Pa = 100⋅[0.0025 + 2.25⋅10-6 + 1.764⋅10-
3⋅P2]1/2, if 900 < P/Pa < 9000, total error in Pa = 100⋅[0.0025 + 2.25⋅10-6 + 7.84⋅10-
4⋅P2]1/2; maximum operating temperature: T = 338 K). Similarly, the temperatures are
measured with three Pt 100 Ω Class A resistance temperature detectors (RTD). The
ARGON GAS CYLINDER
OVERFLOW
DRAIN TEMPERATURE CONTROLLED BATH
TO VACUUM PUMP
PNEUMATIC VALVE TEMPERATURE LINING
+ HEATING TAPE
+ INSULATION
HEATING COILS
SILICA GEL
TANK DOSING TANK
MAGNETIC STIRRER ROD
TEMPERATURE CONTROLLED BATH
DISTILLED WATER IN EVAPORATOR
Pc Pd Pe T1 T2 T3
Figure 4.2. Schematic diagram of the constant volume variable pressure test facility: T1, T2, T3 = resistance temperature detectors; Ps, Pd, Pe = capacitance manometers.
82
RTD for the charging tank, T1 in Figure 4.2, is in direct contact with the silica gel to
give representative adsorbent temperature. Except for the 3.175 mm vacuum adaptors
used to seal the RTD probes, all other inter-connecting piping (SS 316), stainless steel
vacuum fittings, and vacuum-rated stainless steel diaphragm valves (Nupro, Swagelok)
are chosen to be 12.7 mm to ensure good conductance during evacuation. Isothermal
condition in the two tanks is obtained by immersing them in a temperature-controlled
bath (± 0.01 K). Measurement errors caused by a temperature gradient along the
piping, fittings and valves, are mitigated with thermal jacket, whose temperature is
maintained by the same bath. All other exposed piping and fittings are also well
insulated.
A pictorial view of the CVVP apparatus is shown in Figure 4.3. Vacuum environment
is maintained by a two-stage rotary vane vacuum pump (Edwards bubbler pump) with
a water vapor pumping rate of 315 ×10-6 m3 s-1. To prevent back migration of oil mist
into the apparatus, an alumina-packed foreline trap is installed upstream of the vacuum
pump. Argon, with a purity of 99.9995 per cent, is sent through a column of packed
calcium sulphate before being used to purge the vacuum system. Upon evacuation, a
residual argon pressure of 100 Pa could be achieved in the apparatus.
The evaporator flask is immersed in a different temperature-controlled bath (precision
of control: ± 0.01 K). It is first purged with argon with a purity of 99.9995 per cent and
evacuated by the vacuum pump. Distilled water is then charged into the evaporator. Its
temperature could be regulated to supply water vapor at the desired pressure. An
immersion magnetic stirrer is used to ensure uniformity of temperature inside the
evaporator flask.
83
All pressure and temperature readings are continuously monitored by a calibrated 18-
bit ADDA data logger (Fluke, Netdaq 2640A). The calibration of all pressure
transmitters, temperature sensors, and the data logger are traceable to national
standards. In the present test facility, the major measurement bottleneck lies in the
measurement of vapor pressure.
The dry mass of silica gel is determined by the calibrated moisture balance (Satorious
MA40 moisture analyzer, uncertainty: 0.05 percent traceable to DKD standard) at 413
Figure 4.3: The experimental set-up of the constant volume variable pressure (C.V.V.P) test facility
Water bath
Pressure sensor
Evaporator
Charging and dosing tanks
84
K. The mass of silica gel is continuously monitored and recorded when there is no
change in dry silica gel mass and silica gels (range from 0.1 g to 1.25 g) are introduced
into the charging tank.
The dosing tank, the charging tank and all related piping systems are purged by
purified and dried argon, and evacuated. The silica gel is again regenerated in situ at
413 K for 24 h. At the end of regeneration process, the test system is purged with
argon and evacuated. The silica gel is further regenerated at 413 K for another 8 hours,
and the test system is evacuated again. Based on measurements, there is no measurable
interaction between the inert gas and the adsorbent. The effect of the partial pressure of
argon in the tanks is found to be small. Nevertheless, the partial pressure of water
vapor has been adjusted for the presence of argon so as to avoid additional systematic
error.
Water vapor is firstly charged into the dosing tank. Prior to charging, the tank
temperature is initially maintained at about 5 K higher than the evaporator temperature
to eliminate the possibility of condensation. The effect of condensation would cause
significant error in pressure reading. This would lead to significant error in calculating
actual mass of water vapor, which is charged into the dosing tank. The evaporator is
isolated from the dosing tank soon after the equilibrium pressure is achieved inside the
dosing tank. Taking the effect of residual argon, the mass of water vapor is determined
via pressure and temperature. The temperatures of the charging tank and the dosing
tank are subsequently raised to the typical adsorber and desorber temperature of an
adsorption cooling cycle. Once the test system reaches thermodynamic equilibrium at
the desired temperature, the electro-pneumatic gate valve between the dosing and
85
charging tank is opened, and both tanks are allowed to approach equilibrium. The
temperatures of both tanks are adjusted to the desired temperature via water baths
before pressure and temperature measurements are taken for both tanks.
With the known initial mass of dry silica gel adsorbent, the temperature of the test
system is varied to measure the uptake of water vapour at a different temperature and
pressure. The test system is subsequently evacuated. Silica gel is regenerated, and the
apparatus is purged with purified and dried argon before water vapor at different initial
pressure is charged into the dosing tank to enable measurement along the isotherms. A
number of test conditions are repeated at least twice with a fresh batch of silica gel to
ensure that repeatability is achieved to within the reported experimental uncertainty.
4.2.2 Results and analysis for adsorption isotherms
Adsorption equilibrium data of water vapor on these two different types of silica gels
are obtained. The measured isotherm data for type ‘A’ and type ‘RD’ silica gels are
furnished in table 4.3 and table 4.4 respectively. Figures 4.4 and 4.5 depict the
experimental equilibrium data for the type ‘RD’ and ‘A’ silica gel with water vapor,
respectively. The uncertainty of pressure ranges from ( 10± to 200 ) Pa.
86
Table 4.3. The measured isotherm data for type ‘A’ silica gel.
T P1 q* K
kPa
kg kg-1
298.15 5.1054 0.4014 304.15 0.6675 0.0753 0.9523 0.1084 1.3888 0.1593 2.3351 0.2682 3.3496 0.3535 4.1878 0.3651 5.2226 0.3948 310.15 0.9473 0.0717 1.3302 0.1019 1.9090 0.1469 3.0696 0.2376 3.9451 0.3159 4.7446 0.3465 5.6442 0.3860 316.15 1.2730 0.0673 1.7980 0.0946 2.5116 0.1334 3.7915 0.2087 4.7202 0.2705 5.3888 0.2986 323.15 1.7948 0.0624 2.4281 0.0855 3.1658 0.1192 4.7487 0.1725 5.7795 0.2121 338.15 3.2807 0.0489 4.0972 0.0637 5.4151 0.0814
87
Table 4.4. The measured isotherm data for type ‘RD’ silica gel.
T P1 q* K
kPa
kg kg-1
298.15 2.9512 0.4535 304.15 0.5653 0.0916 0.7774 0.1246 1.4679 0.2331 2.2804 0.3507 2.8829 0.4018 3.4849 0.4409 4.4327 0.4465 310.15 0.7591 0.0897 1.0747 0.1206 2.0115 0.2182 2.9984 0.3180 3.5595 0.3640 4.1266 0.4089 4.6402 0.4270 316.15 1.0948 0.0853 1.5080 0.1132 2.6504 0.2012 3.8004 0.2876 4.3298 0.3235 4.7023 0.3454 4.7865 0.3572 5.3335 0.3827 323.15 1.5294 0.0794 2.0491 0.1037 3.4923 0.1744 4.7954 0.2366 5.3003 0.2630 338.15 2.8814 0.0638 3.7069 0.0786 5.3634 0.1191
The performance of type ‘RD’ silica gel water system employed by the adsorption
chiller manufacturer (NACC. PTX data, 1992) available in the form of the Henry's
isotherm (Suzuki, 1990) is also superimposed for comparison. The data of the
88
manufacturer have been obtained for temperatures from 328 to 358 K, and for
pressures from 1000 to 5000 Pa. However, the uncertainties of measurements are not
available from the manufacturer. Apparently the manufacturer is focusing more on the
desorber conditions. Comparing the manufacturer’s data of type ‘RD’ silica gel with
the experimental data for the same type of silica gel, the performance of type ‘RD’
silica gel-water system is found to be consistent in trend with the manufacturer’s data,
as shown in Figure 4.6. From Figures 4.4 and 4.5, the isotherms of T = 304 K and T =
310 K for both type ‘RD’ and ‘A’ silica gel-water systems exhibit signs of onset of
monolayer saturation at higher pressures. The Tóth’s equation (Tóth, 1971 and Suzuki,
1990) is found to fit the experimental data of both type ‘RD’ and type ‘A’ silica gel-
water systems well and is therefore used to find the isotherm parameters and isosteric
heats of adsorption The form of the Tóth’s equation is given as:
( ) ( ) [ ] 11 1
10
1
)(exp1exp
*tt
gadsm
gadso
PTRHqKPTRHK
q⋅⋅∆⋅+
⋅⋅∆⋅= , (4.1)
where q* is the adsorbed quantity of adsorbate by the adsorbent under equilibrium
conditions, qm denotes the monolayer capacity, P1 the equilibrium pressure of the
adsorbate in gas phase, T the equilibrium temperature of gas phase adsorbate, Rg the
universal gas constant, adsH∆ the isosteric enthalpies of adsorption, Ko the pre-
exponential constant, and 1t is the dimensionless Tóth’s constant. Tóth (Tóth, 1971)
observed that the empirical constant 1t , was between the interval of zero to one insofar
as his experiments concerned but 1t could actually be greater than one (Valenzuela and
Myers., 1989).
89
0
0.050.1
0.150.2
0.250.3
0.350.4
0.450.5
01
23
45
67
P 1(k
Pa)
q (kg kg-1
) Figu
re 4
.4.
Isot
herm
dat
a fo
r w
ater
vap
or o
nto
the
type
‘R
D’
silic
a ge
l; fo
r ex
peri
men
tal d
ata
poin
ts w
ith: ()
T =
303
K; ()
T =
308
K; (
) T =
31
3 K
; (
) T
= 32
3 K
; (+
) T
= 33
8 K
, for
com
pute
d da
ta p
oint
s w
ith
solid
lin
es f
or t
he T
óth’
s eq
uatio
n, f
or m
anuf
actu
rer’
s da
ta p
oint
s w
ith: (
----
----
) T =
323
K o
f NA
CC
; 2 (⎯ −
− ⎯
) T =
338
K o
f NA
CC
,2 an
d fo
r on
e ex
peri
men
tal
data
poi
nt w
ith:
()
T =
298
K f
or t
he
estim
atio
n of
mon
olay
er sa
tura
tion
upta
ke.
90
0.00
0.05
0.10
0.15
0.20
0.25
0.30
0.35
0.40
0.45
01
23
45
67
P 1 (k
Pa)
q (kg kg-1
)
Figu
re 4
.5. I
soth
erm
dat
a fo
r w
ater
vap
or o
nto
the
type
‘A
’ si
lica
gel;
for
expe
rim
enta
l dat
a po
ints
with
: ()
T =
304
K; ()
T =
310
K; (
) T =
316
K; ()
T =
323
K; (+)
T =
338
K
, for
com
pute
d da
ta p
oint
s w
ith s
olid
line
s fo
r th
e T
óth’
s eq
uatio
n, a
nd f
or o
ne
expe
rim
enta
l da
ta p
oint
with
: (
) T
=298
K f
or t
he e
stim
atio
n of
mon
olay
er
satu
ratio
n up
take
.
91
The adsorption isotherm parameters and the isosteric enthalpies of adsorption for the
two grades of silica gel (type ‘RD’ and type ‘A’) + water system are summarized in
Table 4.5. Cremer and Davis (1958) first reported a value of 2.98⋅103 kJ⋅kg-1 for
adsH∆ , though neither the experimental uncertainty nor the grade of silica gel was
reported. This value has since been widely used. While Chihara and Suzuki (1983)
reported a adsH∆ that ranged from (2.8⋅103 to 3.5⋅103) kJ⋅kg-1 for the type ‘A’ silica gel,
Sakoda and Suzuki (1984) later adopted the value of 2.8⋅103 kJ⋅kg-1. Referring to Table
4.5, the derived monolayer capacity of the type ‘A’ silica gel is estimated to be 0.4
kg⋅kg-1, which is higher than the value, 0.346 kg⋅kg-1, reported by Chihara and Suzuki
(1983). The derived monolayer capacity of the type ‘RD’ silica gel is similarly
estimated to be 0.45 kg kg-1.
Referring to Figures 4.4 and 4.5, the estimation of the monolayer capacity for both
types of silica gel is based on experimental measurement of water vapor uptake for
both types of silica gel at 298 K near the saturation pressure while maintaining (8 to
10) K of superheat. It is observed that there is a significant difference between the
present measurements and those of references (Chihara et al., 1983 and Sakoda et al.,
1984) for the type ‘A’ silica gel. Moreover, while the Tóth’s equation is employed to
correlate the present experimental data, the Freundlich’s equation is used to correlate
the data in the two said references. The present data for the type ‘RD’ silica gel with
water vapor shows good consistency with the chiller manufacturer’s correlation
(Manufacturer’s Proprietary Data, Nishiyodo Air Conditioning Co Ltd., Tokyo, Japan,
1992).
92
Table 4.5. Correlation coefficients for the two grades of Fuji Davison silica gel -
water systems (The error refers to the 95% confidence interval of the
experimental data)
Ko ∆Hads qm Type (kg⋅kg-1⋅kPa-1)
(kJ⋅kg-1)
(kg⋅kg-1)
t1 Remarks
‘A’ (4.65 ± 0.9)⋅10-10 (2.71 ± 0.1)⋅103 0.4 10 By Tóth
‘RD’ (7.30 ± 2)⋅10-10 (2.693 ± 0.1)⋅103 0.45 12 By Tóth
‘RD’ (Manufacturer) 2.0⋅10-9 2.51⋅103
By Henry
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
1 2 3 4 5 6 7
q* (k
g kg
-1)
P1 (kPa)
T = 303 K
T = 338 K
Figure 4.6. Isotherm data for water vapor with the silica gel by the present study and other researchers with: (⎯⎯) type ‘A’ by the present study; (⎯⎯) type ‘A’ by Chihara and Suzuki; (⎯∆⎯) type ‘RD’ by the present study; () type ‘RD’ by Cho and Kim; (-------) type RD by NACC.
93
It is evident that the water vapor uptake capacity of the type ‘RD’ silica gel, as
presented by Cho and Kim (1992) is much higher than those obtained by other
researchers. However, details of the experimental method and procedures are not
available.
4.3 Summary of Chapter 4
Equilibrium studies of water vapor on the two Fuji Davison type silica gels have been
investigated experimentally. The ranges of pressure and temperature covered in these
experiments are within the operating conditions expected of an adsorption chiller. The
derived monolayer capacities (q*) for water vapor of the type ‘A’ and type ‘RD’ silica
gels have been calculated to be about 0.4 kg⋅kg-1 and 0.45 kg⋅kg-1 respectively. The
Tóth’s equation is found to be able to sufficiently describe the performance of the type
‘A’ and type ‘RD’ silica gels with water vapor. The computed values of the isosteric
heats of adsorption are lower than those found in the literatures (Chihara and Suzuki,
1983, Sakoda and Suzuki, 1984 and Cremer and Davis, 1958).
Based on the measured adsorption isotherms (fitted Tóth correlation) of water vapor on
silica gel at different temperatures and pressures, isosteric enthalpy of adsorption
( adsH∆ ), the mathematical model for the electro-adsorption chiller (EAC) is now fully
equipped with the real behaviour of the silica gel + water pair and details of the
simulation will be described in the chapters to follow.
94
Chapter 5
An electro-adsorption chiller: thermodynamic modelling and its
performance
5.1 Introduction
This chapter presents (1) a general thermodynamic approach for formulating the mass,
energy and entropy balance equations of an adsorbate-adsorbent system and (2) the
theoretical modeling of an electro-adsorption chiller (EAC). Such a modeling approach
enables one to perform detailed parametric study of the key variables that affect the
performance of an EAC and hence, optimizing the EAC with respect to a selected
cooling capacity. The model incorporates the physical characteristics of adsorbent,
adsorption isotherms and adsorption kinetics, expressing them in terms of pressure (P),
the amount of adsorbate(q), and temperature (T) of silica gel + water systems. In
addition, the model also accounts for the behaviour of thermoelectrics and the overall
heat transfer conductance of the reactor beds.
For the theoretical development, this chapter employs the thermodynamic property
fields of adsorbate-adsorbent system that is based on the works of Feuerecker et al.
(1994) where it was originally developed for absorption (liquid-vapour) system. This
method yields the energy and entropy balances, from which a specific heat capacity
expression for silica gel - water as a function of P, T and q, are derived.
95
5.2 Thermodynamic property fields of adsorbate-adsorbent system
The complete thermodynamic property fields for a single-component adsorbent +
adsorbate system is proposed and developed in the sections below. The equations for
computing the actual specific heat capacity, partial enthalpy and entropy are essential
for the analyses of single-component adsorption processes. Using the framework of
adsorption thermodynamics (Lopatkin, 2001), the general equations for the differential
and the integral heat capacities of an adsorbed substance on the surface of solid
adsorbent at constant pressure could be described. It is assumed that the differential
and integral heat capacities are taken to be equal for the case of ideal gases, but
additional terms are needed for them to take into account of temperature increment
dT , i.e., non-equilibrium conditions, within the control volume, and this additional
terms are added only to the heat capacity of gaseous phase.
Recently, the thermodynamic functions such as Gibbs free energy, enthalpy and
entropy on the basis of isothermal immersion of adsorbent in the gas phase for
describing the adsorbate-adsorbent systems have been derived in details by Myers
(Myers, 2002), and followed by defining the heat capacity of the adsorbate-adsorbent
system under condensed (liquid) phase and gaseous phase in terms of the constituents’
properties such as the solid adsorbent and the adsobate, where the latter is assumed to
be perfect gas.
Using the proposed formulation (developed in the thesis) and the experimentally
measured adsorption isotherm data from the literature (Bjurström et al., 1984), the
specific heat capacity of silica gel + water system is analysed and compared to
highlight the major differences between the conventional uncorrected form of the
96
specific heat capacity (Ruthven, 1984, Suzuki, 1990, Saha et. al, 1997) and the
improved simplified expression is also presented.
5.2.1 Mass balance
The mass balance of the adsorbent + adsorbate system can be represented by
gasgastabea Vmmm ρ−=.* , (5.1)
where abem is the mass of adsorbent, tm the total amount of gas in the system, gasV the
gas phase volume, and gasρ the gas phase density. *am is the adsorbed quantity of
adsorbate under equilibrium conditions
5.2.2 Enthalpy Energy and Entropy Balances
The thermodynamic state of the adsorbent-adsorbate control volume can be accurately
expressed in terms of its three measurable properties, namely, the P, T, and ma, where
ma denotes the mass of adsorbate (or q Mabe in kg). Tracking the change of a
thermodynamic variable from an initial to a final state or process, the expressions of an
extensive thermodynamic quantity, such as enthalpy, internal energy or entropy
involving a single component adsorbate + adsorbent system at a given mass of
adsorbent, Mabe, are shown as follows:
( )
( ) ( )∫∫
∫∫
∫ ∫∫
⋅⋅−⋅+⋅⎟⎟⎠
⎞⎜⎜⎝
⎛∂∂
⋅⋅−⋅+
⋅⎟⎟⎠
⎞⎜⎜⎝
⎛∂∂
+⋅⋅=
∂∂
+∂∂
+∂∂
=
P
PT
sys
sysa
m
TaT
sys
sys
m
aTmTPa
T
Tabepabe
aa
a
dPTM
dmmPT
M
dmmHdTcM
dPPHdm
mHdT
THmTPH
a
a
ao
1
1
11
),,(
0
1
0 ,,,
αρ
αρ
,
(5.2)
97
where sysM is the mass, sysρ the density of the adsorbent + adsorbate, and abepc , the
specific heat capacity of the adsorbent.
Using the definition (U = H-PV), the internal energy for the adsorbent + adsorbate
system is given by
( ) ( ) ( )
( )
( )
( )∫∫
∫
∫∫
∫∫∫∫
⋅⋅−⋅⋅−⋅⎥⎥⎦
⎤
⎢⎢⎣
⎡⎟⎟⎠
⎞⎜⎜⎝
⎛∂
∂⋅−⋅−
⋅⎟⎟⎠
⎞⎜⎜⎝
⎛∂∂
⋅⋅−⋅⋅−
⋅⎟⎟⎠
⎞⎜⎜⎝
⎛∂∂
+⋅⋅=
∂∂
−∂∂
−∂
∂−=
P
PT
sys
sysm
oa
TPa
sys
sys
sys
sys
a
m
TaT
sys
sys
m
aTmTPa
T
Tabepabe
aa
a
dPPTM
dmm
MP
dmmPPT
M
dmmHdTcM
dPP
PVdmmPVdT
TPVdHmTPU
a
a
a
ao
11
1
,21
0
11
0 ,,,
1
),,(
βαρ
ρ
ρρ
βαρ
, (5.3)
and similarly, the entropy is given by
( )
∫
∫∫∫
∫∫∫
⋅⋅−
⋅⎟⎟⎠
⎞⎜⎜⎝
⎛∂∂
⋅⋅−⋅⎟⎟⎠
⎞⎜⎜⎝
⎛∂∂
+⋅⋅=
∂∂
+∂∂
+∂∂
=
P
PT
sys
sys
a
m
TaT
sys
sysm
aTmTPa
T
T
abepabe
aa
a
dPM
dmmPM
dmmSdT
Tc
M
dPPSdm
mSdT
TSmTPS
aa
ao
1
1 0
1
0 ,,
,
),,(
αρ
αρ
(5.4)
where β⎟⎟
⎠
⎞
⎜⎜
⎝
⎛⎟⎟⎠
⎞⎜⎜⎝
⎛∂
∂⋅=
amT
sys
sys P,
1 ρρ
is the isothermal compressibility factor and
Tα ⎟⎟
⎠
⎞
⎜⎜
⎝
⎛⎟⎟⎠
⎞⎜⎜⎝
⎛∂
∂⋅−=
amP
sys
sys T,
1 ρρ
is the thermal expansion factor of the system.
Expressing ( ) TmTPa
a
mH
,,1
⎟⎟⎠
⎞⎜⎜⎝
⎛∂∂ ,
( ) TmTPaa
mS
,,1
⎟⎟⎠
⎞⎜⎜⎝
⎛∂∂ in terms of the P, T, and ma, they
can be further expanded in the following forms:
98
( )
( )
( ) ( )
( )aa
a
m
a
TmTPa
sysag
agTmTPa
TmTP
mV
TmTPT
TmTPhmH
⎟⎠⎞
⎜⎝⎛
∂∂
⋅⎥⎥⎦
⎤
⎢⎢⎣
⎡⎟⎟⎠
⎞⎜⎜⎝
⎛∂
∂−⋅
−=⎟⎟⎠
⎞⎜⎜⎝
⎛∂∂
,,,
,,
1
,,1
1,,
1
1
ν
, (5.5)
( )
( )
( ) ( )
( )aa
a
m
a
TmTPa
sysag
agTmTPa
TmTP
mV
TmTP
TmTPsmS
⎟⎠
⎞⎜⎝
⎛∂
∂⋅⎥⎥⎦
⎤
⎢⎢⎣
⎡⎟⎟⎠
⎞⎜⎜⎝
⎛∂
∂−ν
−=⎟⎟⎠
⎞⎜⎜⎝
⎛∂∂
,,,
,,
1
,,1
1,,
1
1
, (5.6)
It is noted ( )TamP ∂∂ 1 , as found in equations 5.2, 5.3 and 5.4, can be obtained directly
from the isotherm characteristics of Chapter 4.
It is noted in all above equations, integration is performed from an initial reference
temperature, To, to temperature, T, initially at zero adsorbate state, then follows by
isothermal tracking along zero adsorbate state to adsorbate mass, ma, and finally from a
saturated pressure P1(T, ma), to a non-equilibrium pressure, P at constant T and
constant ma. Owing to the cumulative tracking of variables along the mentioned paths,
the equations define accurately the respective instantaneous states of an adsorbent-
adsorbate system insofar as local thermodynamic equilibrium could be assumed.
In equations 5.5 and 5.6, ( )amTP ∂∂ 1 relates to the Clapeyron relation as determined by
the local pressure and temperature, and they could be further expressed respectively as,
( )
( ) adsagTmTPa
HTmTPhmH
a
∆−=⎟⎟⎠
⎞⎜⎜⎝
⎛∂∂ ,,1
,,1
, (5.7)
and
99
( )
( ) THTmTPs
mS ads
agTmTPa
a
∆−=⎟⎟
⎠
⎞⎜⎜⎝
⎛∂∂ ,,1
,,1
. (5.8)
Based on the above equations, the total differential of the extensive enthalpy, internal
energy, and entropy for a single component adsorbate + adsorbent system can therefore
be respectively given by:
( )
( ) ( )
( )( ) ( )
444444444444 8444444444444 76444 8444 76
4444444444444 84444444444444 76
44444444 844444444 76
44444444444444444 844444444444444444 76
44 844 76
pressureduechange
Tsys
sysa
P
P TP
Tsys
sys
a
adsorbateofmasstoduechange
aTmTpa
TPtoduechange
P
Tsys
sysP
P mP
Tsys
sys
amsionthermaltoduechange
m
a
mPTaT
sys
sys
uptakemassalongchange
m
amP
ads
P
g
a
sysg
ads
a
sysg
gadsgp
adsorbent
abepabea
dPTM
dmdPTM
mdm
mH
dTTPT
MdPT
MT
dTdmmPT
MT
dTdmT
HT
mV
H
mVT
HTPc
dTcMmTPdH
a
a
a
a
a
a
⋅⎟⎟⎠
⎞⎜⎜⎝
⎛⋅−⋅+⋅
⎥⎥⎦
⎤
⎢⎢⎣
⎡⋅⎟
⎟⎠
⎞⎜⎜⎝
⎛⋅−⋅
∂∂
+⋅⎟⎟⎠
⎞⎜⎜⎝
⎛∂∂
+
⋅⎥⎥
⎦
⎤
⎢⎢
⎣
⎡
∂∂⋅⎟
⎟⎠
⎞⎜⎜⎝
⎛⋅−⋅−⋅⎟
⎟⎠
⎞⎜⎜⎝
⎛⋅−⋅
∂∂
+
⋅⎥⎥
⎦
⎤
⎢⎢
⎣
⎡⋅⎟
⎟⎠
⎞⎜⎜⎝
⎛⎟⎟⎠
⎞⎜⎜⎝
⎛∂∂
⋅⋅−⋅∂∂
+
⋅
⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢
⎣
⎡
⋅
⎪⎪⎭
⎪⎪⎬
⎫
⎪⎪⎩
⎪⎪⎨
⎧
⎟⎠⎞
⎜⎝⎛
∂∆∂
−⎟⎟⎠
⎞⎜⎜⎝
⎛∂
∂⋅
⎟⎟⎠
⎞⎜⎜⎝
⎛∂
∂−
∆−
⎟⎟⎠
⎞⎜⎜⎝
⎛∂
∂−
⋅∆
++
⋅⋅=
∫
∫
∫
∫
αρ
αρ
αρ
αρ
αρ
ν
νν
ν
11
11
1
),(
),,(
1
11
1
,,,
&
1
,
exp
0 ,
1
0 ,1,
,
, (5.9)
Similarly, the change in the internal energy is given by
100
( )
( ) ( )
( )
( ) ( ) dPPTM
dmdPPTM
m
dmm
MPPdmmH
dTTPPT
MdPPT
MT
dTdmm
MPPT
dTdmmPPT
MT
dTdmTH
TmV
H
mVT
HTPc
dTcMmTPdU
Tsys
sysa
P
P TP
Tsys
sys
a
aTPa
sys
sys
sys
sysa
TmTPa
P
Tsys
sysP
P mP
Tsys
sys
m
a
mPTPa
sys
sys
sys
sys
m
a
mPTaT
sys
sys
m
amP
ads
P
g
a
sysg
ads
a
sysg
gadsgp
abepabea
a
a
a
a
a
a
a
a
⋅⎟⎟⎠
⎞⎜⎜⎝
⎛⋅−⋅⋅−⋅
⎥⎥⎦
⎤
⎢⎢⎣
⎡⋅⎟
⎟⎠
⎞⎜⎜⎝
⎛⋅−⋅⋅
∂∂
−
⋅⎥⎥⎦
⎤
⎢⎢⎣
⎡
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛⎟⎟⎠
⎞⎜⎜⎝
⎛∂
∂⋅
⋅−−⋅⎟⎟
⎠
⎞⎜⎜⎝
⎛∂∂
+
⋅⎥⎥
⎦
⎤
⎢⎢
⎣
⎡
∂∂⋅⎟
⎟⎠
⎞⎜⎜⎝
⎛⋅−⋅⋅−⋅⎟
⎟⎠
⎞⎜⎜⎝
⎛⋅−⋅⋅
∂∂
−
⋅⎥⎥
⎦
⎤
⎢⎢
⎣
⎡⋅
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛⎟⎟⎠
⎞⎜⎜⎝
⎛∂
∂⋅
⋅−
∂∂
−
⋅⎥⎥
⎦
⎤
⎢⎢
⎣
⎡⋅⎟
⎟⎠
⎞⎜⎜⎝
⎛⎟⎟⎠
⎞⎜⎜⎝
⎛∂∂
⋅⋅−⋅⋅∂∂
−
⋅
⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢
⎣
⎡
⋅
⎪⎪⎭
⎪⎪⎬
⎫
⎪⎪⎩
⎪⎪⎨
⎧
⎟⎠⎞
⎜⎝⎛
∂∆∂
−⎟⎟⎠
⎞⎜⎜⎝
⎛∂
∂⋅
⎟⎟⎠
⎞⎜⎜⎝
⎛∂
∂−
∆−
⎟⎟⎠
⎞⎜⎜⎝
⎛∂
∂−
⋅∆
++
⋅⋅=
∫
∫
∫
∫
∫
βαρ
βαρ
ρ
ρρ
βαρ
βαρ
ρ
ρρ
βαρ
ν
νν
ν
1
1
11
1
1
,
,2
11
,,
1
,
0 ,,2
11
0 ,
11
0 ,1,
,
),(
),,(
(5.10)
and the change in the entropy is
101
( )
dPM
dmdPM
mdm
mS
dTTPM
dPM
T
dTdmmPM
T
TdTdm
TH
TmV
H
mVT
HTPc
dTT
cMmTPdS
Tsys
sysa
P
P TP
Tsys
sys
aa
TmTPa
p
Tsys
sysP
P mP
Tsys
sys
a
m
mPTaT
sys
sys
a
m
mP
ads
P
g
a
sysg
ads
a
sysg
gadsgp
abePabea
a
a
a
a
a
a
⋅⎟⎟⎠
⎞⎜⎜⎝
⎛⋅−⋅
⎥⎥⎦
⎤
⎢⎢⎣
⎡⋅⎟
⎟⎠
⎞⎜⎜⎝
⎛⋅
∂∂
−⋅⎟⎟⎠
⎞⎜⎜⎝
⎛∂∂
+
⋅⎥⎥
⎦
⎤
⎢⎢
⎣
⎡
∂∂⋅⎟
⎟⎠
⎞⎜⎜⎝
⎛⋅−⋅⎟
⎟⎠
⎞⎜⎜⎝
⎛⋅
∂∂
−
⋅⎥⎥
⎦
⎤
⎢⎢
⎣
⎡⋅
⎪⎭
⎪⎬⎫
⎪⎩
⎪⎨⎧
⎟⎟⎠
⎞⎜⎜⎝
⎛∂∂
⋅⋅∂∂
−
⋅
⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢
⎣
⎡
⋅
⎪⎪⎭
⎪⎪⎬
⎫
⎪⎪⎩
⎪⎪⎨
⎧
⎟⎠⎞
⎜⎝⎛
∂∆∂
−⎟⎟⎠
⎞⎜⎜⎝
⎛∂
∂⋅
⎟⎟⎠
⎞⎜⎜⎝
⎛∂
∂−
∆−
⎟⎟⎠
⎞⎜⎜⎝
⎛∂
∂−
⋅∆
++
⋅⎥⎦
⎤⎢⎣
⎡ ⋅=
∫
∫
∫
∫
αρ
αρ
αρ
αρ
αρ
ν
νν
ν
1
11
1
,,,
1
,
0 ,
1
0 ,1,
,
),(
),,(
. (5.11)
From equations (5.9-5.11), it is noted that they contain the partial terms of enthalpy,
internal energy and entropy of adsorbate only. The details of these partial terms can be
further expanded in terms of the P and T and these are given as follows:
( ) ( ) dPTm
TTm
MT
mmTPH
mmTPH
P
P a
sysT
TPa
sys
sys
sysT
sys
TPa
a
TPa
a
⋅⎥⎥⎦
⎤
⎢⎢⎣
⎡
⎪⎭
⎪⎬⎫
⎪⎩
⎪⎨⎧
∂⋅∂
∂⋅−⋅⋅−⋅⎟⎟
⎠
⎞⎜⎜⎝
⎛∂
∂⋅−⋅−⋅+
⎭⎬⎫
⎩⎨⎧
∂∂
=⎭⎬⎫
⎩⎨⎧
∂∂
∫1
1
2
,2
,
1
,
2111
),,(),,(
ρα
ρ
ρα
ρ,
(5.12)
where P1 is the pressure at equilibrium (maximum) uptake of the adsorbate. For
simplicity, the contribution from the second term of equation 5.12 (from P1 to P) to the
partial enthalpy with adsorbate uptake is assumed negligible and thus,
102
abeabe MTPa
abea
MTPa
abea
mMmTPH
mMmTPH
,,
1
,, 1
),,,(),,,(
⎭⎬⎫
⎩⎨⎧
∂∂
≈⎭⎬⎫
⎩⎨⎧
∂∂
, (5.13)
and similarly,
( )
dP
PmP
TmT
M
mMPT
mM
P
mmTPH
mmTPU
P
P
a
sys
a
sys
sys
sys
TPa
sys
sys
sys
sys
T
TPa
sys
sys
sys
sys
TPa
a
TPa
a
⋅
⎥⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢⎢
⎣
⎡
⎪⎭
⎪⎬⎫
⎪⎩
⎪⎨⎧
∂⋅∂
∂⋅+
∂⋅∂
∂⋅⋅−
⎪⎭
⎪⎬⎫
⎪⎩
⎪⎨⎧
⎟⎟⎠
⎞⎜⎜⎝
⎛∂
∂⋅−
⋅−⋅
−⎥⎥⎦
⎤
⎢⎢⎣
⎡⎟⎟⎠
⎞⎜⎜⎝
⎛∂
∂⋅−⋅−
⎭⎬⎫
⎩⎨⎧
∂∂
=⎭⎬⎫
⎩⎨⎧
∂∂
∫11
1
22
2
,
,21
,
1
,
21
1
),,(),,(
ρρ
ρ
ρρρ
βα
ρ
ρρ
(5.14)
TPa
a
TPa
a
mmTPH
mmTPU
,
1
, 1
),,(),,(
⎭⎬⎫
⎩⎨⎧
∂∂
≈⎭⎬⎫
⎩⎨⎧
∂∂
, (5.15)
and also
∫ ⋅⎥⎥⎦
⎤
⎢⎢⎣
⎡
∂⋅∂
∂⋅+
⎪⎭
⎪⎬⎫
⎪⎩
⎪⎨⎧
−⎟⎟⎠
⎞⎜⎜⎝
⎛∂
∂⋅⋅+
⎭⎬⎫
⎩⎨⎧
∂∂
=⎭⎬⎫
⎩⎨⎧
∂∂
P
P a
sys
sys
sys
TPa
sys
syssys
T
MTPa
a
MTPa
a
dPTm
Mm
mmTPS
mmTPS
abeabe
1
1
2
2,
,,
1
,,
12
),,(),,(
ρ
ρ
ρρρ
α. (5.16)
abeabe MTPa
abea
MTPa
abea
mMmTPS
mMmTPS
,,
1
,, 1
),,,(),,,(
⎭⎬⎫
⎩⎨⎧
∂∂
≈⎭⎬⎫
⎩⎨⎧
∂∂
(5.17)
103
5.2.3 Specific heat capacity
The specific heat capacity of adsorbent-adsorbate system has two parts; Firstly, the
specific heat capacity derives from the solid adsorbent and secondly, the specific heat
capacity contributes from adsorbate that emanates from different processes associated
with the isothermal conditions. The specific heat capacity of the adsorbate has,
hitherto, assumed to be equal to that of the liquid-phase (Ruthven, 1984 and Chi Tien,
1994 and Saha et al., 1997) that is expressed as;
( )TPcMmcc lp
abe
aabepls ,1,, += (5.18)
where abepc , is the specific heat capacity of solid adsorbent, am is the amount of
adsorbate, abeM is the mass of adsorbent, P1 is the saturated pressure, T is the
temperature and lpc , indicates the specific heat capacity of adsorbate at liquid phase.
Recently some researchers view the specific heat capacity of the adsorbate to be equal
to the gas phase specific heat capacity (Chua et al., 1999, Saha et al., 2003 and Sami et
al., 1996) in the design and simulation of adsorption chiller, which is defined as,
( )TPcMmcc gp
abe
aabepgs ,1,, += , (5.19)
where gpc , is the specific heat capacity at gaseous phase.
In this thesis, the specific heat capacity of a single component adsorbent + adsorbate
system is proposed to be dependent on the isosteric heat of adsorption T
H ads
∂∆∂
as a
function of T, system volume Vsys, specific heat capacity of solid adsorbent and a small
value of the gaseous phase ( ) ( )Tgg ∂ν∂⋅ν1 , where vg is the specific volume at gaseous
phase, and the full expression is given by,
104
( ) ∫ ⋅
⎪⎪⎪
⎭
⎪⎪⎪
⎬
⎫
⎪⎪⎪
⎩
⎪⎪⎪
⎨
⎧
⎟⎠⎞
⎜⎝⎛
∂∆∂
−
⎟⎟⎠
⎞⎜⎜⎝
⎛∂
∂⋅
⎟⎟⎠
⎞⎜⎜⎝
⎛∂
∂−
∆−
⎟⎟⎠
⎞⎜⎜⎝
⎛∂
∂−
⋅∆
++=a
a
m
a
mP
ads
P
g
a
sysg
ads
a
sysg
gads
abeap
abe
aabepas dm
TH
TmV
H
mVT
H
MTPc
Mmcc
0
,
1,,
11,
ν
νν
ν
(5.20)
Assuming that amP
ads
TH
,
⎟⎠
⎞⎜⎝
⎛∂∆∂
is negligible and a constant skeletal system volume sysV ,
the specific heat capacity of adsorbent + adsorbate system is simplified to
( ) ∫ ⋅⎪⎭
⎪⎬⎫
⎪⎩
⎪⎨⎧
⎟⎟⎠
⎞⎜⎜⎝
⎛∂
ν∂⋅
ν∆
−∆
++=am
aP
g
g
adsads
abegp
abe
aabepas dm
TH
TH
MTPc
Mm
cc0
1,,
1
1,
(5.21)
The correction term, namely ∫ ⋅⎪⎭
⎪⎬⎫
⎪⎩
⎪⎨⎧
⎟⎟⎠
⎞⎜⎜⎝
⎛∂
∂⋅
∆−
∆am
aP
g
g
adsads dmT
HTH
0 1
νν
, is found to be
significant, because of the large variation of isosteric heat of adsorption at low
adsorbate uptake and the non-ideality of the gas phase. The effect of the adsorbent in
the adsorbed phase is discussed below.
5.2.4 Results and discussion
Using the proposed specific heat capacity expression, the specific heat capacity of
adsorbate is analyzed with respect to a single-component silica gel + water system,
using the published adsorption isotherms and the isosteric heat of adsorption data of
Bjurström (Bjurström et al., 1984). The specific heat capacity of silica gel + water
system is plotted against the amount of adsorbate uptake (type silica gel - Grace type
125 + water) , as shown in Figure 5.1. The measurement methods employed in their
experiments were the transient hot strip (THS – denoted in square symbols) and the
105
DSC (denoted by filled circles) and these values are superimposed in Figure 5.1 for
comparison. It can be observed that the proposed specific heat capacity agrees better
with the experimentally measured data, whilst the liquid-phase based cls and the
gaseous phase based cgs forms the upper and lower bounds of the experimental data.
0.8
1.0
1.2
1.4
1.6
1.8
2.0
2.2
2.4
0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35
Amount adsorbed (kg kg-1)
Spec
ific
heat
cap
acity
(kJ
kg-1
K-1
)
Figure 5.1: Different specific heat capacities of a single component type 125 silica gel + water system at 298 K: a data point (---) indicates refers to the liquid adsorbate specific heat capacity of the adsorption system, (----) defines the gas adsorbate specific heat capacity of adsorbent + adsorbate system, () refers to the newly interpreted specific heat capacity with average isosteric heat of adsorption 2560 KJ kg-1, () the specific heat capacity of type 125 silica gel + water system as obtained through DSC and () the specific heat capacity of the adsorption system with the THS technique.
106
5.3 Thermodynamic modelling of an electro-adsorption chiller
This section focuses on the thermodynamic modelling of an electro-adsorption chiller
(EAC) (Ng et al., 2002) that symbiotically combines the transient behaviours of
thermoelectric modules and the batch-operated adsorption cooling cycle. The main
processes in an EAC are (i) the electric input to the thermoelectric device that produces
the hot and cold junctions where the heat from the former desorbes the vapor from the
adsorbent (silica gel) into the condenser and the latter adsorbes part of the heat
released by the adsorption of vapour into the silica gel, (ii) useful cooling effect is
achieved at the evaporator, (3) partial heat recovery process is effected by the
thermoelectric devices that occurs mainly during the switching period.
This section consists of three sub-sections; Section one describes the electro-
adsorption chiller, the second section explains the lumped-parameter modeling of the
EAC, usually comprising the energy balance, mass balance and adsorption equilibrium
equations, and the final section discusses the operating conditions on the performance
of electro-adsorption chiller.
Figure 5.2 shows the schematic layout of an electro-adsorption chiller. It comprises the
evaporator, the condenser, the reactors or beds and the thermoelectric modules and
water is used as the refrigerant. For continuous cooling operations, the batch-operated
cycle of an EAC consists of two time intervals, namely the switching and the operation
time intervals. During the operation, the DC power is supplied to thermoelectrics,
producing the hot and cold junctions in relation to the designated desorber and
adsorber. As the thermoelectric device is found to have poorer cooling performance as
107
compared to its heating capacity, external (water) cooling system is incorporated in the
EAC in order to remove the heat during the adsorption process. The air-cooled
condenser condenses the desorbed vapor from the designated desorber and the
condensate (refrigerant) is refluxed back to the evaporator via a pressure reducing
valve. Pool boiling is affected in the evaporator by the vapor uptake at the adsorber,
and thus completing the refrigeration close loop or cycle.
The roles of the reactors (containing the adsorbent) are refreshed by switching which is
performed by reversing the direction of the DC power supply (current flow direction is
reversed). With reversed polarity, what was formerly the cold junction becomes the hot
junction, and vice versa. Concomitantly, the external cooling is also switched to the
designated adsorber and similarly, the evaporator and condenser are also switched to
the respective adsorber and desober. It is noted that during switching interval, no mass
transfers occur between the hot bed and the condenser or the cold bed and the
evaporator.
108
p
n
p
n
p
n
p
n
Heat absorbed from the load surface
Heat rejected to the environment
Control box
Power supply
Electro-magnetic valve
Evaporator
Thermoelectric modules
Copper foam
Reactor or bed Silica-gel + copper fins
Electro-pneumatic gate valve
Condenser
Metering valve
Cooling water loop for external cooling
Figure 5.2. Schematic showing the principal components and energy flow of the electro-adsorption chiller (EAC).
109
5.3.1 Mathematical model
To simplify the mathematical modeling of an electro-adsorption chiller, the following
assumptions are generally made: (i) the temperature is uniform in the adsorbent (silica
gel) layer; (ii) the water vapor is adsorbed uniformly in the reactor; and, (3) both solid
(silica gel) and gas phases exist at a thermodynamic equilibrium. The rate of
adsorption or desorption is calculated by the linear driving force equation and this is
given by (Ruthven, 1984 and Suzuki, 1990),
( )q(t) -*q)( κ=dt
tdq , (5.22)
where κ is the mass-transfer coefficient, expressed in terms of other parameters as
follows (Chihara and Suzuki, 1983)
2
15
p
TRaE
so
ReD g−
=κ , (5.23)
where Rp denotes the average radius of a silica gel particle; R= universal gas constant;
E a = activation energy of surface diffusion; and D so = a kinetic constant for the silica
gel water system. Hence t indicates time in equation (5.22).
It is noted from Chapter 4 that the water vapor uptake is linearly proportional to the
vapor pressure and the Henry’s isotherm equation can be applied (Suzuki, M., 1990).
At high pressure and low temperature, the vapor uptake shows a saturation
phenomenon, and the isotherms are appropriately captured by the Tóth’s equation
(Tóth, 1971), which is given as follows,
( ) ( ) [ ] 11 1
10
1
)(exp1exp
*tt
gadsm
gadso
PTRHqKPTRHK
q⋅⋅∆⋅+
⋅⋅∆⋅= , (5.24)
110
where q* is the adsorbed quantity of adsorbate by the adsorbent under equilibrium
conditions, qm denotes the monolayer capacity, P1 the equilibrium pressure of the
adsorbate in gas phase, T the equilibrium temperature of gas phase adsorbate, Rg the
universal gas constant, adsH∆ the isosteric enthalpies of adsorption, Ko the pre-
exponential constant, and 1t is the dimensionless Tóth’s constant.
At the beginning of the each cycle, the adsorber bed and the desorber bed are isolated
from the evaporator and the condenser by valves connected between them, i.e., the
isosteric phase. The designated valves are opened only when the pressures in the beds
are equalized to those at Pcond and Pevap, and the mass transfer across the bed and
condenser or evaporator begins. The mass balance of refrigerant (here water) is
expressed by neglecting the gas phase as,
( )( )⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
−γ−δ−=dt
dqdt
dqM
dtdM adsbed
returncondensatetodue
desbedabe
evapref ,,, 11
444 8444 76
(5.25)
where δ and γ are the switching functions to designate the role of desorber where the
vapour are condensed at the condenser.
The governing energy balance equation of thermoelectric element, which is discussed
in Chapter 3, is given by
xtxT
cJ
cJ
xtxT
cttxT te
teptetetepte
te
tepte
tete
∂∂Γ
−′
+∂
∂=
∂∂ )),(),(),( 2
2
2
ρσρρλ
. (5.26)
111
The adsorption reactors comprise the silica gel, sandwiched between the heat
exchanger fins. The energy balance for the sorption bed, ignoring heat losses, during
its interaction with the evaporator may be written as follows:
( )
( )
( ) ( ) ( ) ( )[ ]( )( )inwowwwpw
adsadsadsbedevapgevapgadsbed
abeLadsbed
adsbedTEadsbedsadsbed
m
aP
g
g
adsadsgpa
adsbedmfpmfCEpCEHXpHXabepabe
TTTcm
HHTPhThdt
dqMTT
AUdt
dTdm
TH
THTPcm
dtdT
cMcMcMcM
a
,,,
,,
,
,,,,
01,
,,,,,
1,
,1
−−
∆−+∆+−+−
−=⎥⎥⎦
⎤
⎢⎢⎣
⎡⋅
⎪⎭
⎪⎬⎫
⎪⎩
⎪⎨⎧
⎟⎟⎠
⎞⎜⎜⎝
⎛∂
∂⋅
∆−
∆+
++++
∫
&
δδ
νν (5.27)
Here the value of the switching function (δ) depends on the processes in the bed or
reactor. δ would be 1 for adsorption and 0 for desorption. The first term on the left-
hand side represents the amount of sensible heat required to cool the silica gel, and the
copper fins, whereas the second term defines the specific heat capacity of adsorbate at
different isotherms, which has been discussed in the previous section. On the other
hand, the first term on the right hand side denotes the partial absorption of heat from
adsorption reactor to the cold junction of thermoelectric modules, the second term
provides the heat generation by adsorption and the third term denotes the rejection of
heat from the adsorber by external water cooling. As the electro-adsorption chiller is
well insulated, the heat losses to the bed are assumed to be negligible.
The energy balance for the cold end plate (comprising the thermoelectrics and bed) and
is given by,
( ) ( )0
,,,,,, )(=∂
∂+−−=+
x
teteteLteLadsbedadsbedTEadsbeds
LHXpHXLpL x
TAkTtISTTAUdt
dTcMcM .
(5.28)
112
The first term of the left hand side indicates the energy input to the thermoelectric
modules, the second term is the Seebeck effect due to thermoelectric operation and
finally the third term shows the heat conduction at the cold end boundary of the
thermoelectric modules. The outlet temperature of the external cooling source (water)
is calculated by simply log mean temperature difference method and it is given by
( ) ⎟⎟⎠
⎞⎜⎜⎝
⎛ −−+=
wpw
adstubeadstubeadsbedinwadsbedow cm
AUTTTT
,
,,,,,, exp
&. (5.29)
For the regeneration step (when the bed interacts with the condenser) the energy
balance is described as follows:
( )
( )
( ) ( ) ( ) ( ) [ ]adsadsdesbedcondgcondgdesbed
abedesbedH
desbeddesbedsdesbed
m
aP
g
g
adsadsgpa
desbedmfpmfCEpCEHXpHXabepabe
HHTPhThdt
dqMTT
AUdt
dTdm
TH
THTPcm
dtdT
cMcMcMcM
a
∆θ+∆+−θ−+−
=⎥⎥⎦
⎤
⎢⎢⎣
⎡⋅⎪⎭
⎪⎬⎫
⎪⎩
⎪⎨⎧
⎟⎟⎠
⎞⎜⎜⎝
⎛∂
ν∂⋅
ν∆
−∆
+
++++
∫
,,
,
,,,
01,
,,,,,
,1
,1
(5.30)
The value (θ ) depends on adsorption or desorption. During desorption θ equals to 1
and the value of θ is 0 during adsorption. The first term on left-hand side provides the
amount of sensible heat required to heat the silica gel, water, copper fins and the
bottom surface of the chamber, and the second term shows the specific heat capacity of
adsorbate (here water vapor) for different isotherms and isosteric heat of adsorption as
a function of the amount of adsorbate. On the other hand, the first term of the right
hand side represents the heat input due to Seebeck, Joulean and conduction effects of
the thermoelectric modules and finally the second term indicates the transfer of latent
heat from the desorber bed to the condenser.
113
The energy balance for the hot end plate is given by,
( ) ( )Lx
teteteHteDESbedHdesbedTEdesbeds
HHXpHXHpH x
TAkTtISTTAUdt
dTcMcM=∂
∂−+−−=+ )(,,,,,,
(5.31)
The condenser liquefies the refrigerant (water vapor) from the desorber and delivers
the condensate to the evaporator. The influence of condenser is considered at two
levels: (a) how thermal dynamics within the sorption beds depend on condenser
performance; and (b) the energy balance with in the condenser itself. In the current
design, the condenser is of air finned type and has cross airflow. The design has been
performed in such a way that the condenser and the desorber are always maintained at
the refrigerant saturated vapour pressure. The energy balance for the condenser can be
written as
( )[ ] ( )
( ) ( ) ( ) ( ) ( )[ ] ( )acondcondconddesbed
abecondfcondgdesbedcondg
desbedabecondf
condcondcondpcondrefcondfp
TTAUdt
dqMThThTPh
dtdq
MThdt
dTMcMTc
−−−−−+−
=−+
,,
,,,,
11, γθθθ
γ
(5.32)
The first term on the left-hand side represents the sensible heat of condensate from the
condenser and that of condenser material. On the other hand the first term on the right
hand side is the heat generation by condensation, the second term provides the sensible
heat of adsorbate desorbed due to regeneration and finally the third term denotes the
heat transferred to the cooling air. In the current formalism, the micro-fin condenser is
able to contain a certain maximum amount of condensate (M maxrefcond ). Beyond this, the
condensate would flow into the evaporator. Hence, if M refcond < M maxrefcond , γ=1. If
M refcond = M maxrefcond , and (dq desbed , /dt) ≤ 0, γ= 0. But if M refcond = M max
refcond , and (dq desbed , /dt)
> 0, γ=1 (Chua et. al, 1999).
114
The overall heat transfer coefficient of the air-cooled condenser can be described by
the following equation (Sami and Tribes, 1996)
))1(1(
11
ϕ−−++=
cond
finfin
tutuavg
tucond
tuintuin
cond
cond
AA
hKAXA
hAA
U (5.33)
where (A cond ) is the total external area in contact with the air and ϕ represents the fin
efficiency. It is important to note that the fin efficiency is calculated in terms of the
heat exchanger geometry. The air cooled condenser heat transfer coefficient as well as
fin efficiency has been calculated (Briggs and Young, 1962).
The energy balance analysis for the evaporator, and its influence on the sorption bed
dynamics, are quantitatively identical to that of the condenser. The energy balance
equation reads
( )[ ] ( ) ( ) ( )
( ) ( ) ( )[ ] evaploadadsbed
abeadsbedevapgevapgdesbed
abe
condfevapref
evapfevap
evapevappevaprefevapfp
Aqdt
dqMTPhTh
dtdq
M
Thdt
dMTh
dtdT
McMTc
′′+−+−=
−+++
,,
,
,,,,
,1
1
δδ
γθ,(5.34)
where loadq ′′ is the constant heat flux of the cooling components (delivered by IR
device). The left-hand side of this equation represents the total amount of sensible heat
given away by the evaporated liquid and the evaporator material. The second term
shows the enthalpy of evaporated liquid during pool boiling. On the other hand, the
first term on the right hand side calculates the sensible heat required to cool down the
incoming condensate, the second term defines the heat of evaporation, and finally the
third term denotes the required heat generated by the load.
115
The evaporator design is based on the nucleate pool boiling technique where it occurs
at low pressure and this differs greatly from the conventional pool boiling at
atmospheric pressure. In the absence of design data (during the simulation phase of the
research), the well known Rohsenow’s boiling correlation (Rohsenow, 1952) for water
is used but extrapolated where the water properties and the constants Csf= 0.013 and
the index of 0.33 of the correlation are assumed to be valid at the vacuum conditions.
The correlation is given by,
33.0
)(Pr
⎥⎥⎦
⎤
⎢⎢⎣
⎡
−′′
⎟⎟⎠
⎞⎜⎜⎝
⎛+=
gl
s
fgl
load
pf
slfgsf
evapload ghq
chC
TTρρ
σµ
. (5.35)
Rohsenow correlated boiling data shows excellent agreement for pressures ranging
from 101 kPa to 16936 kPa (McGillis et al., 1991). Deviations are expected to be large
at low pressures and heat flux levels. At low heat flux levels, the boiling is not
continuous. The commonly used Rohsenow nucleate boiling correlation agrees well
with pressure variation of the continuous boiling data. In the isolated-bubble regime
(low heat flux and low pressure), the experimental data deviates from the Rohsenow
correlation (McGillis et al., 1991). This is due to the fact that the heat transfer
mechanism comprised of conduction into the fluid during intermittent boiling is not as
effective as continuous boiling heat transfer, resulting in a reduction in heat transfer
performance.
During the switching operation, the roles of reactors or beds are changed (adsorption
bed converts into desorption mode and desorption bed changes in to adsorption mode)
and no latent heat is transferred from the evaporator to the cold bed and from hot bed to
the condenser. When the bed switches to adsorption mode, the governing equation
becomes,
116
( )
( )
( ) ( )( )inwwowwpwLadsbed
adsbedTEadsbedsadsbed
m
aP
g
g
adsadsgpa
adsbedmfpmfCEpCEHXpHXabepabe
TTTcmTT
AUdt
dTdm
TH
THTPcm
dtdT
cMcMcMcM
a
,,,
,,,,
01,
,,,,,
1
,
−−−
−=⎥⎥⎦
⎤
⎢⎢⎣
⎡⋅⎪⎭
⎪⎬⎫
⎪⎩
⎪⎨⎧
⎟⎟⎠
⎞⎜⎜⎝
⎛∂
∂⋅
∆−
∆+
++++
∫
&
νν
(5.36)
When the bed switches to desorption mode, the equation now becomes,
( )
( )
( )desbedH
desbeddesbedsdesbed
m
aP
g
g
adsadsgpa
desbedmfpmfCEpCEHXpHXabepabe
TT
AUdt
dTdm
TH
THTPcm
dtdT
cMcMcMcM
a
,
,,,
01,
,,,,,
1
,
−
=⎥⎥⎦
⎤
⎢⎢⎣
⎡⋅
⎪⎭
⎪⎬⎫
⎪⎩
⎪⎨⎧
⎟⎟⎠
⎞⎜⎜⎝
⎛∂
ν∂⋅
ν∆
−∆
+
++++
∫ (5.37)
During switching the behavior of condenser is described by
( )[ ] ( )
( )acondcondcond
desbedabecondf
condcondcondpcondrefcondfp
TTAUdt
dqMTh
dtdTMcMTc
−−
=+ ,,,, γ (5.38)
And finally the energy balance of evaporator is noted during switching operation.
( )[ ] evaploadevap
evapevappevaprefevapfp Aqdt
dTMcMTc ′′=+ ,,, (5.39)
These energy balance equations are based on the fact that pressure and temperature
change in tandem, the thermal swing of the beds is assumed to be isosteric. t defines
time (equations 5.25-5.39).
5.3.2 COP of the Electro-Adsorption Chiller
The cycle-average Coefficient of Performance (COPnet) of the electro-adsorption
chiller, can be expressed in terms of the cycle average COPs of the individual
adsorption (subscript ads) and thermoelectric (subscript TE) chillers by considering the
overall energy flows, however, COPads is a complicated function of COPTE via the
117
dynamics of the combined cycle. Referring to Figure 5.3, the COPs of the individual
components are defined as the nominal cycle-average cooling rate produced relative to
the particular cycle-average power input.
in
TEadsTE P
QCOP ,= (5.40)
TEdes
evapads Q
QCOP
,= (5.41)
where Qevap denotes the cycle-average cooling rate of the overall device, i.e. the rate of
heat extraction at the evaporator. The cycle-average values of the thermal power
absorbed at the cold junction Qads,TE that drives refrigerant adsorption, as well as the
thermal power rejected at the hot junction Qdes,TE that drives refrigerant desorption for
adsorption cycle. But it is noted that Qads,TE is less than Qdes,TE and the total heat
rejected during adsorption is the sum of Qads,TE plus the amount given by externally
assisted cooling. COPTE will fall below commonly observed thermoelectric COP
values due primarily to losses associated with the large thermal lift (TH- TC).
From the energy balance of the thermoelectric device (first law analysis applies
irrespective of external cooling to adsorber)
TEadsTEdesin QQP ,, −= (5.42)
The overall or net COP is
( )TEadsin
evapEAC COPCOP
PQ
COP +== 1 (5.43)
The power input from the thermoelectric device is represented by
cycle
t
inputin t
dttIvP
cycle
∫= 0
)( (5.44)
118
where vinput is the fixed terminal voltage per unit thermoelectric elements. The
designed value of vinput is taken from Melcor web site. The time response of control
signal variation I (t) for a fixed terminal voltage can be expressed by
( )[ ]LHtetem
x
tetete
Lx
tetete
Lte
teptete
TTvNn
xtxTA
xtxTAdx
ttxTAcw
tI te
te
−−⎥⎥⎦
⎤
⎢⎢⎣
⎡
∂∂
−∂
∂−
∂∂
===
∫α
λλ00
),(),(),(
)( (5.45)
The maximum COPads of a heat driven chiller is 0.85 (COPads = 85.0≅∆ ads
fg
Hh
), where
hfg is the latent heat of water and adsH∆ is the isosteric heat of adsorption at similar
pressure and temperature conditions. At high thermal-lift, the COPTE of thermo-
electrics is generally low, typically less than 0.3. Based on the experience of operating
the EAC at high thermal lift (TH – TC) of 30 to 35 degrees, the typical COPTE is about
0.2 to 0.4. Thus, at these mentioned operating conditions, the theoretical COP of the
electro-adsorption chiller is estimated to be about
1.1 0.3)(1 0.85 ) 1 ( ≅+=+= TEads COPCOPOPmaxEACC .
119
Figure 5.3 A block diagram to highlight the sensible and latent heat flow and the energy balance of a thermoelectrically driven adsorption chiller
Thermoelectricmodule
Adsorption reactor
Desorption reactor
Condenser
Evaporator
Qevap
Qcond
Pin
Qdes
Qads,TE
Qads,w
)( evapfge Thm&
adsadsbed H
dtdq
∆,
adsdesbed H
dtdq
∆,
( )condfgc Thm&
Latent heat flow
Sensible heat flow
External cooling
[Heat recovery by thermoelectric device]
( )
( ) adsadsbed
condfgc
adsdesbed
evapfge
Hdt
dqThm
Hdt
dqThm
∆+
≅∆+
,
,
&
&
evapdes
condwadsTEads
QQQQQ
+
≅++ ,,
Sensible heat balance
Late
nt e
nerg
y ba
lanc
e
Continuous regeneration
desadsdesbed
evapfge
QHdt
dq
Qhm
<∆
>
,
&
prdes
evapads
thads
fgads
COPQQ
COP
COPHh
COP
==
=∆
=max
prth COPCOP >
120
5.3.3 Results and discussion
The performance predictions of an electro-adsorption chiller are presented here. Using
one dimensional mass and energy conservation equations and the spatially discretized
dependent variables along a control volume, the equation sets are solved by the finite
difference formulations. The updated values of the key system’s variables are obtained
by numerical integration in time. The calculation procedure commenced with the
initialization of variables, the system geometries and the adsorption characteristics of
adsorbent-adsorbate systems. A series of subroutines are used to calculate the
adsorption isotherms and kinetics. To speed up the computation, the current formalism
makes use of polynomial curve fits of the thermodynamic property, data of Wagner
and Kruse (1998). The original thermodynamic property routines of Wagner, W. et al.
(1998) have also been applied to the present formalism. The coupled equations are
solved by the fifth order Gear’s differentiation formulae method in the DIVPAG
subroutine of the IMSL FORTRAN Developer Studio software. Double precision has
been used, and the tolerance is set to 1× 10-6. Here only the initial conditions are
specified, and the total system is allowed to operate from transient to cyclic steady
state conditions. The Flow chart of the EAC modeling is shown in Appendix E.
The key parameters analyzed in the simulation code are (a) the temperature profiles of
the adsorber, desorber, condenser and evaporator, (b) the corresponding heat transfer
conductance and the input voltage of the thermoelectric modules (c) the amount of
adsorbent per reactor of the chiller and (d) the net Coefficient of Performance and the
load surface temperature of the electro-adsorption chiller for a fixed cooling capacity.
The values used in the present simulation are shown in Table 5.1.
121
Table5.1 Specifications of component and material properties used in the simulation code
Parameter or material property Value or descriptive equation, with units
Thermoelectric UT8-12-40-F1 (melcor) units Reference
No of thermoelectric couples/elements in a thermoelectric module 127 / 354 --- Melcor
Specific heat capacity of thermoelectric material 544 J/kg K Rowe, 1995
Thermoelectric material density 3102.7 × kg/m3 Rowe, 1995 Cross sectional area of a thermoelectric element
610619.3 −× m2 Melcor
Characteristics length of thermoelectric element
3103.3 −× m Melcor
Geometric factor 210171.0 −× m Melcor Ceramic plate and very thin copper substrate
Mass of ceramic plate 20 310−× kg Melcor Specific heat capacity of ceramic 775 J/kg K Melcor Mass of thin copper plate 31010 −× kg Melcor Specific heat capacity of copper plate 386 J/kg K Melcor
Sorption thermodynamic properties and sorption beds Kinetic constant 2.54 410−× m2/s Chihara and
Suzuki, 1983 Activation Energy 4.2 410−× J/mole Chihara and
Suzuki, 1983 Average radius of silica gel particle 4107.1 −× m manufacturer Isosteric heat of adsorption 6108.2 × J/kg Chua et al., 2002
Specific heat of silica gel 921 J/kg K Chihara and Suzuki, 1983
Adsorber/desorber bed heat transfer area
0.0169 m2
Adsorber/desorber tube heat transfer area
3101.5 −× m2
Finned surface area 0.10664 m2 Tube weight 0.2 kg Volume of tube 6102.3 −× m3 Bottom plate weight 1.8 kg Fin weight 1.2 Kg Silica gel mass 0.3 kg Overall heat transfer coefficient of adsorber
1090 W/m2K
Author’s experimental
data
Overall heat transfer coefficient of desorber
1850 W/m2K
122
Condenser Condenser heat transfer area 0.5 m2
Mass of condenser including fins 2 kg Author’s
experimental data
Condenser heat transfer coefficient 700 W/m2K Mass of condensate in the condenser 41084.12 −× kg
Evaporator Evaporator heat transfer area 0.0026 m Mass of evaporator including foams 2 kg Mass of refrigerant in the evaporator 0.3 kg
Figure 5.4 shows the prediction of temporal histories for the reactor (adsorber or
desorber), the condenser, the load and the evaporator by using the mathematical model
presented herein. At the very short cycle times, say a few tens of seconds, the COP of
EAC is expected to be low due to incomplete vapour uptake onto the silica gel and the
large amount of sensible heat of the thermoelectric modules as compared to latent heat
transfer. In contrast, a very large cycle time (a few hours) for the cycle would saturate
the adsorbed vapour onto the silica gel and cooling rate would decrease. Thus, there
exist in between these extreme cycle intervals that an optimal COP would be located
for the EAC. Figure 5.5 represents a narrow region of the EAC operation (cycle time
intervals) where the COP shows a monotonic increase (i.e., before the optima is
reached) and yet the load and evaporator temperatures have demonstrated their
minima. These simulations have a fixed input heat flux of 5 watt per cm2, a rate
expected of the IR (Infra-red radiant) heaters in the experiments (described in the
following chapter). From the results, the lowest load temperature is found to be about
24oC and it lies between 400 and 500 seconds, whilst the COP of EAC is computed to
be about 0. 8.
123
1020304050607080
025
050
075
010
0012
5015
0017
5020
0022
5025
0027
5030
00Ti
me
(s)
Temperature (oC)
Des
orpt
ion
bed
Ads
orpt
ion
bed
Con
dens
er
Loa
d
Eva
pora
tor
Figu
re 5
.4.
Tem
pora
l hi
stor
y of
maj
or c
ompo
nent
s of
the
ele
ctro
-ads
orpt
ion
chill
er
obta
ined
by
sim
ulat
ion.
124
Figure 5.6 shows the Dűhring or P-T diagram of the electro-adsorption chiller with two
cycle times (400 s and 600 s), each having the same fixed cooling capacity of 120 W.
At a 700s (denoted by bold line), the difference in the vapor uptake and off-take is
about 15 % and the pressure of the adsorption reactor goes down 1600 Pa, but for a
400s (denoted by dotted line), the difference between the vapour adsorption and
desorption is 10 %. From the P-T profile, one observes the cold-to-hot thermal swing
of the reactor, causing momentary desorption at the beginning of operation and
maintaining the pressure in the hot reactor and the condenser. On the hand, during hot-
to-cold thermal swing of the bed, there is a sudden rise in the bed pressure at the end of
the switching phase, causing momentary adsorption in the reactor.
Figure 5.5. Effects of cycle time on chiller cycle average cooling load temperature, evaporator temperature and COP. The chosen 500 s cycle time, shown by the solid dotted vertical line, is close to, the minimum cooling load temperature.
10
15
20
25
30
35
40
45
50
55
60
200 300 400 500 600 700 800 900Cycle time (s)
Tem
pera
ture
(o C)
0.60
0.65
0.70
0.75
0.80
0.85
Coe
ffici
ent o
f Per
form
ance
(CO
P)
Load
Evaporator
125
Figure 5.7 shows the simulated results of COP and the variations of load
surface/evaporator temperature with different cycle average input currents
⎟⎟⎠
⎞⎜⎜⎝
⎛= ∫
cyclet
cycle
dttIt
I0
)(1~ of the thermoelectric modules. These simulations are conducted
for a fixed input cooling power 5 Watt per cm2 and optimal cycle period 500 seconds.
As expected, the load surface temperature and COP decrease with increasing input
current.
Figure 5.6. Dűhring diagram for 400 s and 700 s cycles
1000
10000
30 35 40 45 50 55 60 65 70 75 80Temperature (oC)
Pres
sure
(Pa)
2%
5%
8%
10%
15%20%25%35%40%45%50%60%70%80%90%100%
j
k
30 %
cycle time 400 s
Cycle time 700 s
qads – qdes = 0.17
0.1
126
For the verification of the EAC modeling, a comparison of the predictions to the
experimental results will be presented in the next chapter. For the moment, the analysis
of energy (both the sensible and latent heat) balances of major components of EAC is
analyzed. The cyclic average energy values for the internal and external balances are
found to be within 0.2% and 2.25%, respectively, as shown in Table 5.2. A full listing
of the energy flows in the major components can be found in Appendix F.
Figure 5.7. COP, the cycle average load temperature and the evaporator temperature as functions of the cycle average input current to the thermoelectric modules.
10
15
20
25
30
35
40
45
50
5 5.5 6 6.5 7 7.5 8 Cycle average current (amp)
Tem
pera
ture
(o C)
0.50
0.55
0.60
0.65
0.70
0.75
0.80
0.85
Coe
ffici
ent o
f Per
form
ance
(CO
P)
Load
Evaporator
of E
AC
127
Table 5.2 Energy balance schedule of the electro-adsorption chiller (EAC)
Sensible heat balance (- indicates heat rejection and + indicates heat addition)
Qevap, (W) Qcond, W Qads,T, W Qdes, W Qevap + Qdes Qcond + Qads,T Error
%
129.05 -183.77 -190.58 237.05 366.10 -374.35 2.25
Latent heat balance
QevapL, W QcondL, W Qads,TL, W QdesL, W QevapL + QdesL QcondL +
Qads,TL Error
%
-136.28 183.95 146.39 -194.74 -331.02 330.35 0.2
5.4 Summary of Chapter 5
The thermodynamic formulations of specific heat capacity, internal energy, enthalpy
and entropy of a single component adsorbent + adsorbate system are essential in the
development of adsorption thermodynamics and the modeling of the electro-adsorption
chiller (EAC).
A theoretical model has been successfully developed incorporating transient modelling
equations with realistic adsorption isotherms and adsorption kinetics, overall heat and
mass transfer correlations, and thermoelectric performances (data as provided by
manufacturer MELCOR). With appropriate initial and boundary conditions and for a
range of cycle time between 400s to 700s, the COP is found to be increasing with cycle
time but there is a corresponding minima for the evaporator and load surface
temperatures. The validity of the simulation will have to shown by comparison with
experimental results, which is described in the following chapter.
128
Chapter 6
Experimental investigation of an electro-adsorption chiller
This chapter describes the experimental investigation of the proposed electro-
adsorption chiller (Ng et al., 2002 and 2005(a)) where its performances at different
operating conditions have been evaluated and analyzed. Experimental procedures and
energy utilization schedules of this chiller are also described. For the verification of
lumped model as presented in chapter 5, the experimentally measured results are
compared with simulation.
Figure 6.1 shows the fully assembled bench scale electro-adsorption chiller comprising
reactors or beds, evaporator and condenser. Based on the simulation results of chapter
5, (i) the expected cooling capacity is about 120-130 W, (ii) each reactor is sized to
hold 0.3 kg of silica gel (Fuji Davison type ‘RD’), (iii) the reactor, evaporator and
condenser surface areas are 0.0169 m2, 0.05 m2 and 0.0026 m2 respectively. The
reactors are connected to an evaporator and a condenser via off on control valves
(electro-pneumatic gate valve). The thermoelectric modules (Melcor, UT8-12-40-F1, 9
pieces, 3 series and 3 parallel ways) are placed between the two reactors.
6.1 Design development and fabrication
The major components of a reactor or bed are (1) two circular copper plates, (2) a heat
exchanger with fins and tubes, (3) the PTFE (Poly-Tetra-Fluoro-Ethylene, thermal
conductivity = 0.25 W/m.K) enclosures (tensile strength 40 MPa). The copper plate as
shown in Figure 6.2 (a and b), has 3 x 3 arrangement of studs and slots (width: 40 mm,
length: 40 mm and depth: 1 mm) on the outer surface and one slot (width: 135 mm,
129
length: 135 mm and depth: 1 mm) at the inner side. The detailed design and drawings
of the electro-adsorption chiller (EAC) are furnished in Appendix G.
The studs and slots are for the positioning of heat exchanger block (packed with silica-
gel) and thermoelectric modules. The heat exchanger block as shown in Figure 6.2 (c
and d), which holds silica gels, composes of 34 slots (slot width is 3 mm and the
thickness of copper wall between two consecutive slots or fin thickness is 1 mm). The
Figure 6.1. The test facility of a bench-scale electro-adsorption chiller
Reactor
Thermoelectric modules
IR heater system
Pressure sensor
Gate valve
Condenser
Cooling loop
U-tube
Evaporator
Vacuum pump
Control system
130
slots are to be filled with silica gel and the silica gels are covered by copper mesh (40
meshes per inch). To measure the silica gel temperature at different points of the bed,
four RTD probes (± 0.15 °C) are placed inside the slot and RTD wires are connected to
the electrical lead through ( 8 pins, TL8K25, Edwards). The heat exchanger is attached
to the inner side of the copper plate with screw tight to maintain the contact resistance
as low as possible. The enclosure is machined from a solid PTFE block. Five short
pipe sockets (DN 10, stainless steel) are placed with viton O’ ring at the outer side of
the chamber (Figure 6.2(e)). Two ports (6 mm diameter) for copper tubes and one port
(25 mm diameter) for electrical lead-through are developed on the top surface of the
enclosure. A circular groove (diameter 250 mm, 3 mm wide and 4 mm deep) for the
location of a centering ring (DN 200) is machined at one surface of the chamber’s
flange. Thermoelectric modules (3 series and 3 parallel connections) are placed at the
outer surface of reactor. The two reactors are attached with thermoelectric devices
where compressive force is applied from four sets of stud and nut at quadrants of the
two reactors [as shown in Figure 6.2 (f)]. The fabrication of two reactors is fully
completed when electro-pneumatic gate valves (normally close, 230 volts, operating
pressure 4.5 to 7 bar), manual valves, pressure transducers (active strain gauge,
accuracy 0.2% full scale, temperature range 30 °C to 130 °C, EDWARDS) and
temperature sensors (Thermister YSI 400 series, ± 0.1 °C) are placed at their positions
on the two reactors.
131
(a) (b)
(e) (f)
Figure 6.2. Assembly of the reactor; (a) the bottom plate (outside), (b) the bottom plate (inside view), (c) the heat exchanger, (d) the heat exchanger with bottom plate, (e) the PTFE enclosure and (f) two reactors.
Thermoelectric modules location
Heat exchanger
Bottom plate
Copper fin
Copper tube
Copper mesh
Short pipe socket
Port for copper pipe
Port for electrical lead-through Gate valve
Pressure sensor
Valve
electrical lead-through PTFE enclosure
40 mm
40
(c) (d)
Silica gel RTD
132
The condenser as shown in Figure 6.3 is an air-cooled type and has a cross air flow
arrangement. It composes of two tube-centered fins bundles, a vapor collector and a
condensate collector tubes. Since all tube-centered fins bundle is machined from a
copper block to achieve 42 parallel fins centered by a 160 mm length tube (inner
diameter 10 mm and outer diameter 14 mm thus the wall thickness is 2 mm), there are
no contact resistances between fins and copper tube. The distance between two fins
(width: 50 mm, length: 50 mm thickness: 1 mm) is 3 mm. The condenser is connected
to the inlet and outlet squared collector tubes (Each collector has outside dimension 20
mm (width), 20 mm (length), 76 mm (height) and inside dimension 16 mm (width), 16
mm (length), 72 mm (height) thus, the overall thickness of the wall is 2 mm) and the
condenser might provide sufficient heat transfer area with the help of a fan (30 Watt,
140 CFM, 230 V).
`
Figure 6.3. The condenser unit.
Inlet port
Water-vapour collector port Pressure
transducer port
Condenser fins
Temperature sensor port
Outlet port
Condensate collector tube
133
Condenser pressure, inlet and outlet temperatures can be monitored through three
copper ports (DN 10), which are machined and brazed to inlet and outlet collector
tubes. For the refrigerant inflow and outflow, two copper ports are machined and
brazed to the top and bottom of two collector tubes. Two YSI (400 series, accuracy
±0.1°C) thermisters are used to measure the temperature of refrigerant at the inlet and
outlet of the condenser. The pressure of the condenser is continuously monitored by a
BOC Edwards pressure transducer (Active strain gauge ± 2% full scale, temperature
range from 30 °C to 130 °C).
The evaporator as shown in Figure 6.4 (a-f) consists of a NW 100 stainless steel tube
body, two NW 100 flanges (one is stainless steel blanking flange and the other is
quartz view port) and standard vacuum fittings. One stainless steel flange is put on top
and the quartz view port is put on the bottom of the tube to form a vacuum enclosure
with the centering O’ rings and claw clamps. Two glass view ports are made on the
tube body of the evaporator to observe the pool boiling and the level of refrigerant.
Four ports (14 mm diameter) and one big port (25 mm diameter) are provided at the
top plate (NW 100, St. Steel, 12 mm thickness) where four short pipe sockets (size DN
10) are welded. The electrical lead through (TL8K25, 8 pins EDWARDS) is placed in
the big port with the viton O’ ring and screw. Two short pipe sockets are connected to
the reactors via electro-pneumatic gate valves (DN 16, pressure range 7101 −× mbar to 2
bar) and flexible hoses. The third one is connected with a temperature sensor
(Thermister, accuracy: ± 0.1°C; YSI 400 Series) and the fourth is connected to a
pressure transducer (Active strain gauge, accuracy ± 0.2 % full scale, BOC Edwards)
with compression fittings. Water refrigerant is charged (or removed) into (or from) the
evaporator chamber by a diaphragm vacuum valve, which is connected to a 4 mm
134
diameter stainless steel tube. The tube is welded at one side of the bottom quartz plate
and connected with another 4 mm diameter, 90 mm long stainless steel tube to allow
warm condensate to flow back to the evaporator. A metering valve with U-bend is used
between the evaporator and the condenser to create a pressure difference during
operation.
The copper foam (5% density, 50 PPI, width 52 mm, length 52 mm and thickness 32
mm) is used as pool boiling enhancement material. The foam structure consists of
ligaments forming a network of inter-connected dodecahedral-like cells and the cells
are randomly oriented and mostly homogeneous in size and shape (provair, 2004).
Metal foam can be produced at various pore size varied from 0.4 mm to 3 mm and net
density from 3% to 15% of a solid of the same material. Copper metal foams have high
surface area to volume ratio and high thermal conductivity. These are potentially
excellent candidates for high heat dissipation applications. Copper foam also has
excellent capillary effect which behaves like a natural pump and has the ability to
generate refrigerant flow far greater than the usual gravity effect. As a result, the foam
is able to draw the surrounding liquid and makes all foam surface areas wet. Foam
material, owing its capillary effect, is used as a liquid transport material. In addition,
open cells of the foam also behave as the fluid re-entrance cavities which play the most
important role in pool boiling applications. Therefore high thermal conductivity copper
foam is the kind of the material that can substitute pool boiling enhancement
structures.
135
(a) (b)
(c) (d)
(e)
Short pipe sockets
View window
Bottom plate (quartz view port)
Quartz plate
Port for electrical leadthrough IR
heater
Power supply
View window
Flexible bellow
Manual valve
Electrical leadthrough
View port
Quartz bottom plate
RTD sensor
Flexible hose (to be connected to reactor)
Pressure transducer
Figure 6.4. The evaporator unit; (a) shows the body of the evaporator, (b) indicates the quartz bottom plate through which infra red heat passes, (c) represents the top part of the evaporator, (d) the IR heater system and (e) is the evaporator.
St Steel body
Blanking flange Claw
clamp
136
An infra-red radiant heater with a tapered homogenizer (kaleidoscope) as shown in
Figure 6.5 is incorporated. The kaleidoscope is used for radiation heat transfer between
a heat source and the evaporator of a bench-scale electro-adsorption chiller. For
constructive reason, the distance between the radiation heat source and the evaporator
is 1 m. The Kaleidoscope is filled with air, its inside surface has a reflectivity of 0.94.
A window made of fused silica (quartz) is the entry aperture of the evaporator. Fused
silica is highly transmissive (τ › 0.94) for radiation up to a wave length of 2500 nm.
The maximum temperature of the source is approximately 1500 K.
Wein’s displacement law yields the wavelength of the peak of the radiative energy
distribution, 2000≈=Tpaekσλ nm, where Boltzmann constant is σ = 2.898 × 10-3 mK.
The peak wavelength is well within the transmission band of the fused silica or quartz
Figure 6.5. Drawing of the kaleidoscope/cone concentrator developed for radiative heat transfer between the heat source (point 1, four rods of 4 kW total power), and the bottom plate (quartz point 3) of the evaporator (exit at the top).
IR-heater
Kaleidoscope 1
1
2 3
137
(Appendix H). The rod sources are Lambertian radiatiors i.e. radiation is equally
distributed in the sphere, the rod looks equally bright from all directions.
The connections between the evaporator and the reactors, and those between the
condenser and the reactors, are performed by the flexible hose (DN 10, stainless steel)
and electro-pneumatic gate valves (normally closed, 240 volt). The flexible hoses are
joined to the reactors (or evaporator/condenser) with standard vacuum fittings such as
centering ring with O’ rings and clamping rings. Two flexible hoses at the outlet of the
reactors are combined and led to condenser through a tee. To vacuum the system, an
Edwards rotary vane pump is used and argon gas cylinder is connected to the test
facility with a manually valve and a special DN 10 stainless steel tee. The outlet port of
the condenser is led to the evaporator via a ¼ mm metering valve and a 4 mm stainless
steel tube. The water vapor quality at the outlet of the condenser can be observed
through a DN 10, 70 mm quartz tube, which is located between the condenser outlet
port and the metering valve, and the metering valve is used to control the flow rate of
liquid refrigerant from the condenser and create a pressure differential between the
condenser and the evaporator.
In this experiment, the data acquisition unit includes an Agilent (34970 A) Data
Acquisition System, BOC Edwards TIC (Turbo Instrument Controller) pressure
acquisition system and a personal computer. The Agilent Data Acquisition unit is used
to accurately capture the required temperatures at the evaporator, reactors and
condenser sections through thermisters and RTD sensors. TIC controller unit is used to
monitor the pressure of evaporator, reactors and condenser. Agilent and TIC controller
138
softwares are installed in a personal computer. The calibration of pressure transmitters,
temperature sensors, and data logger are traceable to national standards.
Power input to the thermoelectric modules is provided by a Agilent 6032A (0-60 Vdc,
0-50A, 1200 W, GBIP auto ranging, programming accuracy: voltage 0.035% or +40
mV and current 0.2% or +85 mA, Readback accuracy: voltage 0.08% and current
0.36%) DC power supplies. As the electro-adsorption chiller is fully automatic, batch
operating machine; a control system is necessary to control the operation sequences of
the machine. A computer program using HP Visual Engineering Environment
(HPVEE) pro version 6.1 is used in conjunction with the electro-adsorption chiller to
(i) control the opening and closing of electro-pneumatic valves and electromagnetic
valves at different time intervals in a batch cycle, (ii) reverse the polarity of the voltage
supplied to the thermoelectric so that the role of the adsorber and desorber reversed
after each batch cycle, (iii) record the power supplied to the thermoelectric modules so
that the performance of the electro-adsorption chiller can be computed. The design and
fabrication of a bench scale electro-adsorption chiller has been successfully completed.
This chiller is relatively compact, fully automatic and free of moving parts (except
fans).
6.2 Experiments
A full scale test plant as shown in Figure 6.1 has been built to test-run an electro-
adsorption chiller (EAC). Prior to experiment, the test facility (evaporator, condenser,
reactors and piping systems) is first evacuated by a two-stage rotary vane vacuum
pump (BOC Edwards pump) with a water vapor pumping rate of 315 ×10-6 m3 s-1. To
prevent back migration of oil mist into the test apparatus, an alumina-packed foreline
139
trap is installed immediately upstream of the vacuum pump. During evacuation, power
is applied to the infra-red radiant heater and thermoelectric modules to heat-up the
copper foam and silica gel to remove moisture and air trapped from the reactors and
the evaporator. Systems pressures are continuously monitored by pressure transducers.
After removing moisture from the reactors, the system is cooled to nearly ambient
temperature and the required vacuum (about 2 mbar) is obtained. The vacuum pump is
switch off and argon, with a purity of 99.9995 per cent, is sent through a column of
packed calcium sulphate into the system to remove any trace of residual air or water
vapor. The argon charging and removing on the system are repeated until the
satisfactory vacuum condition is achieved. Based on measurements involving only
argon and silica gel, it is concluded that there is no measurable interaction between the
inert gas and the adsorbent. The effect of the partial pressure of argon in the reactors is
found to be small. Nevertheless, the partial pressure of water vapor is adjusted for the
presence of argon so as to avoid additional systematic error. The cooling water inlet
temperature is controlled by the room temperature and the electro-pneumatic valves
control the flow of water vapor to the reactors. Air flows to the condenser is controlled
by the power of fan.
The required amount of refrigerant water (about 300 gm) is firstly charged into the
evaporator via a refrigerant charging unit. Prior to charging, the test facility is initially
evacuated and maintained at room temperature. The test facility is fully automatic,
power on is performed to run the test facility. Three time intervals in one cycle of the
electro-adsorption chiller are (i) delay time (time requirement for the control system to
change the polarity of DC power. DC power supply itself also needs to be off before
switching the polarity to prevent electrical shock), (ii) switching time (time allowances
140
for the thermoelectric modules for the purpose of pre-heating and pre-cooling the beds
before the vapor refrigerant entering or leaving the reactors or beds. All electro-
pneumatic gate valves are closed during the switching period) and (iii) operation time
(adsorption/desorption time requirements for the reactors, only the necessary electro-
pneumatic valves and adsorption coolant loop for external sensible cooling, are opened
except the condenser’s fan which is always on to continuously reject heat to the
environment).
Water vapor is generated in the evaporator chamber as the heating power (Qflux, IR)
from the Infra-red radiant heater, is applied to the evaporator by the homogenizer
(kaleidoscope). The control system activates the DC power supply and the electro-
pneumatic valves (off/on control) to start pre-cooling and pre-heating the reactors at
the beginning of switching period. During operation period, one control valve “A” as
shown in Figure 6.6 allows vaporizes refrigerant to enter into pre-cooled reactor
(adsorber bed) from the evaporator, and valve “C” is also opened to release desorbed
refrigerant to the condenser, where the desorbed refrigerant is condensed and heat is
rejected to the environment. Finally, the condensed refrigerant flows back to the
evaporator via the metering valve “E”. At the end of operation period (or the delay
period for 1 second), the control system closes all electro-pneumatic gate valves and
changes the polarity of DC power supply system. The reactor, which is cooled in the
previous cycle, is heated sufficiently, while the other reactor that is acted as a desorber
in the previous cycle is cooled down sufficiently by external cooling and they take the
role of adsorption and desorption again. The control system activates valve “B” and
“C” at the beginning of operation time and the other two valves “A” and “D” are not
activated. After getting cooling effect at the evaporator, the evaporated water vapor is
141
adsorbed onto silica gel and the desorbed water vapor is released to the condenser for
condensation. Heat is rejected from the condenser via a fan and finned heat exchanger.
The warm condensate flows back to the evaporator via the metering valve E. The
energy utilizing schedule of the electro-adsorption chiller is furnished in Table 6.1.
p
n
p
n
p
n
p
n
R-1
R-2
Evaporator
Condenser
A B
C D
V1
V2
V3
V4
PS
Qflux, IR
Qcond
E
Figure 6.6. Schematic diagram of an electro-adsorption chiller (EAC) to explain the energy utilization.
142
All pressures are measured with BOC Edwards TIC (Turbo Instrument Controller)
pressure acquision systems and temperature readings are continuously monitored by a
calibrated data logger (Agilent 34970 Acquisition system). The calibration of pressure
transmitters, temperature sensors, and the data logger are traceable to national
standards. In the present test facility, the major measurement bottleneck lies in the
measurement of vapor pressure and reactor temperatures.
Table 6.1 Energy utilization schedule of an Electro-Adsorption Chiller (Refer to Figure 6.6)
Cycle 1
R-1 SW DES→ADS V1, V2 → ON
R-2 SW ADS→DES V3, V4 → OFF
A, B, C, D CLOSED
PS ON VPS 1
R-1 OP ADS V1, V2 → ON
R-2 OP DES V3, V4 → OFF
A,C → OPEN B,D → CLOSED
PS ON VPS 1
DELAY 1 Second, A, B, C, D Closed PS OFF Cycle 2
R-1 SW ADS→DES V1, V2 → OFF
R-2 SW DES→ADS V3, V4 → ON
A, B, C, D CLOSED
PS ON VPS 2
R-1 OP DES V1, V2 → OFF
R-2 OP ADS V3, V4 → ON
A,C → CLOSE B,D → OPEN
PS ON VPS 2
. . . . .
143
ADS: Adsorption stage, DES: Desorption Stage SW: Switching period, OP: Operation period Cycle = Switching + Operation R-1: Reactor 1 or bed 1, R-2: Reactor 2 or bed 2 PS : Power Supply, VPS: Voltage Polarity Switching A, B, C, D: Electro-pneumatic gate valve (OFF/ON control) V1, V2, V3, V4: Electro-magnetic valve (OFF/ON control)
6.3 Results and discussions
The infra-red radiation heater provides uniform heat flux to (5 cm x 5 cm) window
aperture of evaporator up to 5W/cm2. The heat flux delivery after and before the quartz
window is accurately calibrated using a water cooled heat flux meter (Transducer type:
Circular foil heat flux transducer, model: 1000-0 with amp 11, Sensor Emissivity: 0.94
at 2 microns, Vatell Corporation). The calibrated results are shown in Table 6.2.
In a ray-tracing simulation with the light toolsTM package, the lambertian sources are
set to emit 4000 W at two wavelengths 1800 nm and 2200 nm to represent Wien’s law.
Convection and conduction are neglected, only radiation is considered. The heat flux
after the source, at the exit of the kaleidoscope and after passing through the fused
silica window (points 1, 2 and 3 as shown in Figure 6.7) are predicted and simulated.
The ray tracing results of the tapered kaleidoscope are provided in Table 6.3 and
pictured in Figure 6.7. The correlation between simulated and measured values is good
however the optical efficiency of the system is low, but the heat flux density on the
evaporator meets the target values.
144
Table 6.2 Heat flux calibration table (refer to Figure 6.5)
Sensor Output (mv) *
Power output (W/cm2) Input from
variable transformer
(Volts)
near IR heater point 1
before quartz
window point 2
after quartz
window point 3
near IR heater point 1
before quartz
window point 2
after quartz
window point 3
164 3.6 1.01 0.274 4.49 1.22 0.33
175 4.85 1.27 0.35 6.97 1.95 0.53
185 5.3 1.45 0.49 8.86 2.35 0.8
196 5.72 1.65 0.66 9.68 2.78 1.11
207 6.3 1.85 1.01 11.07 3.25 1.78
218 7.2 2.03 1.22 13.3 3.75 2.26
229 8.1 2.35 1.7 15.55 4.51 3.26
240 8.9 2.52 1.9 17.54 4.97 3.75
Conversion factor x mv = 2.4438 × x -1.2189 W/ cm2
145
The net Infra-Red radient flux is absorbed by the copper foam in the evaporator and
boiling occurs instantly when sufficient superheat is achieved.
Table 6.3 The heat flux density results from measurements and ray-tracing
simulation (the source have a total power of 3.8 KW, refer to Figure 6.7)
Heat flux in W per square cm
Input voltage from variable transformer is 220 Volt
near source (point 1)
exit of kaleidoscope
(point 2)
after fused silica (quartz)
point 3
simulation 8.33 3.78 3.27
measured 13.3 3.75 2.26
Figure 6.7. Tracing 20 rays from the lambertian sources through the kaleidoscope systems. The heat flux after the source is measured in point 1, then at the exit of the kaleidoscope in point 2, and finally after having passed the fused silica window in point 3, the image view of infra-red heater has been pictured from point 3.
Infra-red radiant heater image
Image view from point 3
146
Figure 6.8 shows the temporal histories of the reactors (the adsorber and the desorber),
condenser, evaporator and the load for the cyclic steady state operation at the standard
operating condition (net input voltage 24 volts and average current 6 amp). These are
measured experimentally.
It is observed that the evaporator (vapour) temperature plunges rapidly during the first
half-cycle to 18.7 °C whilst the inner surface temperature of quartz (where IR heat flux
is being directed at) is 23 °C. At the full rating conditions of 4.7 W/cm2, steady state
performances are reached within two full cycles with load surface temperature
stabilizes at 22.5 °C and the evaporator is at 18.2°C. From experimental observation,
Figure 6.8. The experimentally measured temporal history of the electro-adsorption chiller at a fixed cooling power of 120 W: () defines the temperature of reactor 1; () indicates the silica gel temperature of reactor 2; () shows the condenser temperature; () represents the load surface (quartz) temperature and () is the evaporator temperature. Hence the switching and cycle time intervals are 100s and 600s, respectively.
10
20
30
40
50
60
70
80
90
0 450 900 1350 1800 2250 2700 3150 3600
Time (s)
Tem
pera
ture
(o C)
First half cycleswitching
Second half cycle
147
during switching the temperature of the desorber reactor is increased sharply because
of heat recovery from the other bed and continuous regeneration. Figure 6.9 shows the
typical current profile during a half-cycle operation of the electro-adsorption chiller.
A deviation in the control signal (current I (t)) at the beginning is probably due to the
thermal inertia of the cold end heat exchanger and the desorber bed attached to the hot
end. The thermal mass absorbs a relatively large amount of heat at the transient period
and causes an additional heat pumping effect. The current is thus slightly lowered.
After a long operating period the current becomes constant. The current decreases
abruptly at the end of switching. The value of I (t) increases during the switching
Figure 6.9. The DC current profile of the thermo-electric cooler for the first half-cycle.
0
1
2
3
4
5
6
7
8
9
10
1 10 100 1000
Time (s)
Cur
rent
(am
p)
I(t)
Switching period Operation period
Current increase due to regenerative heat transfer
Current due to high thermal lift between hot and cold junctions
148
operation, because of the presence of regenerative heat transfer occurring across the
junctions of thermo-electrics.
For too short a cycle time, the adsorption beds get insufficient time to respond or
saturate. In other words, the bed consumes excessive thermal power due to frequent
cycling. On the other hand, the evaporator as well as the load (quartz) surface
temperature goes higher at very long cycle time, because the adsorbent is saturated in
less than the allotted time. Figure 6.10 represents COPnet (net Coefficient of
Performance) and the load surface temperature as functions of cycle time. For a given
heat flux qevap (4.7 W per cm2) at the evaporator, the pronounced variation of Tload with
time constrains input voltage. COPeac reaches a limiting value at long cycle time,
because in this limit the impact of thermoelectric transient grows negligible. COPeac
can be expressed as dtVtI
tAqcyclet
cycleevapevap
∫0
)(
.. For a long cycle time, I becomes constant (i.e.,
roughly time-independent).
149
A plot of the load and the evaporator temperatures at different input radiant heat flux
(Watt per cm2) is shown in Figure 6.11. These measurements are conducted at
optimum cycle time operation.
Figure 6.10. Effects of cycle time on average load surface (quartz) temperature, evaporator temperature and cycle average net COP. The constants used in this experiment are the heat flux qevap = 4.7 W/cm2 and the terminal voltage of thermoelectric modules V = 24 volts.
17
20
23
26
29
32
300 350 400 450 500 550 600 650 700 750 800
Cycle time (s)
Tem
pera
ture
(o C)
0.60
0.65
0.70
0.75
0.80
0.85
Net
Coe
ffici
ent o
f Per
form
ance
Evaporator
Load
150
The present electro-adsorption chiller test facility is used here to measure the boiling
heat flux as a function of temperature difference between the quartz surface and the
evaporator at low pressure (1.8 kPa) and the measured values are plotted in Figure
6.12. In the experiments, the range of measured heat fluxes varies from 1 to 5 W/cm2
whilst the DT (Tquartz – Tevap = temperature difference between the quartz surface and
the evaporature at saturation) varies up to 5.4 K. It is noted that on the same plot, the
nucleate pool boiling experimental data of water, as performed by McGillis et al.
(1990) at two pressure ranges of 4 and 9 kPa, are also superimposed. However, these
measured heat fluxes of McGillis et al. (1990) were conducted at heat fluxes from 5 to
10 W/cm2. Using the water properties at the working pressures as well as measured
boiling fluxes and DT, a multi-variable regression is conducted to give the following
modified Rohsenow correlation, as shown below, i.e.,
Figure 6.11. Experimentally measured load and evaporator temperatures as a function of heat flux.
10
12
14
16
18
20
22
24
26
0 1 2 3 4 5 6
Heat flux (W/cm2)
Tem
pera
ture
(o C)
Tload
Tevap
151
48476444444 8444444 76 pressureduefactorCorrection
m
atm
nsCorrelatioRohsenow
n
glfgl
load
pf
slfgsf
PP
ghq
chC
DT ⎟⎟⎠
⎞⎜⎜⎝
⎛
⎥⎥⎦
⎤
⎢⎢⎣
⎡
ρ−ρσ
µ′′
⎟⎟⎠
⎞⎜⎜⎝
⎛=
)(Pr
,
where the coefficients Csf and n remain unchanged as in the Rohsenow correlation, i.e.,
0.0136 and 0.33, respectively, but the exponent m for the low pressure correction is
best fitted at 0.37. This heat flux correlation is applicable at low absolute pressures
ranging from 1.8 kPa to 10 kPa and extrapolation beyond these pressures may not be
justifiable as higher system pressures tend to delay the on-set of pool boiling. It should
be highlighted that the lowest data point in Figure 6.12, which falls far from the
modified correction, is observed to have no boiling at low heat flux, i.e., the rise in is
attributed to natural convective heat transfer. Owing to the high Awetted / Abase of copper
foam, the capillary effect is found to be excellent and these effects enhance the boiling
heat transfer rates at higher heat fluxes. Hence, Awetted is the total wetted area of copper
foam and Abase is the base area of foam.
152
6.4 Comparison with theoretical modelling
The current formulations of the electro-adsorption chiller (EAC) modeling as
described in chapter 5 makes use of the present experimental data for comparison.
Using the above proposed correlation (with pressure correction), the mathematical
modeling and simulation of EAC are performed. Figure 6.13 compares the predicted
and experimentally measured temperatures of major components of the electro-
adsorption chiller.
Figure 6.12. Heat flux (q flux) versus temperature difference (DT) for water at 1.8 kPa (author’s experiment), 4 kPa (McGillis et al., 1990). and 9 kPa (McGillis et al., 1990). The bottom most data (of 1.8 kPa) is observed to have insignificant boiling, i.e., mainly convective heat transfer by water.
1
10
3 3.5 4 4.5 5 5.5 6 6.5 7 7.5 8DT (K)
q flu
x (W
/cm
2 )
Pool boiling region
Natural convective
heat transfer region
1.8 kPa
4 kPa 9 kPa
∆ Author’s data at 1.8 kPa with copper-foam, Awetted/Abase = 60.
McGillis et al. (1990) data at 4 kPa, with copper fin Awetted / Abase = 10
McGillis et al. (1990) data at 9 kPa, with copper fin Awetted
/ Abase = 10
153
01020304050607080
010
020
030
040
050
060
070
0
Tim
e (s
)
Temperature (oC)
Hot
reac
tor o
r des
orbe
r
Col
d re
acto
r or a
dsor
ber
Con
dens
er
Eva
pora
tor
Load
sur
face
E
xper
imen
tally
mea
sure
d --
- Sim
ulat
ion
Figu
re
6.13
. E
xper
imen
tally
m
easu
red
tem
pora
l hi
stor
y ve
rsus
th
e si
mul
ated
te
mpe
ratu
res
of m
ajor
com
pone
nts
of e
lect
ro-a
dsor
ptio
n ch
iller
at
a fix
ed
cool
ing
pow
er
of
80
W.
+++
indi
cate
s th
e si
mul
ated
lo
ad
surf
ace
tem
pera
ture
usi
ng th
e pr
opos
ed c
orre
latio
n (C
sf =
0.0
136,
n =
0.3
3 an
d m
=
0.37
) an
d ××
×× s
how
s th
e si
mul
ated
loa
d su
rfac
e te
mpe
ratu
re u
sing
R
ohse
now
poo
l boi
ling
corr
elat
ion
(Csf
= 0
.013
2 an
d n
= 0.
33).
Switc
hing
154
The present electro-adsorption chiller modeling is based on some assumptions, which
are defined in the previous Chapter. Upon these limitations on modeling, one may
observe that the current prediction agrees well with the experimental measurements at
the reactors, condenser and evaporator. Table 6.4 shows the average cooling load
temperature and cycle average COP values obtained from for various cycle times (for a
fixed cooling load 4.7 W per cm2 and 100 s switching time) from experiments and
simulations. The experimental values of evaporator temperature are higher than the
simulated values. The experimentally measured COP is lower than its theoretical
values but both show that COP increases with cycle time.
Table 6.4 Evaporator temperature and cycle average COP (Experimental and simulated values)
Simulated values Experimental
Cycle time (s) Tevap
W Tload W
COP
Tevap ± 0.2 °C
Tload ± 0.2 °C
COP ±0.03
300
400
500
600
700
21.95
17.50
16.73
17.13
17.90
30.27
26.50
25.80
26.60
27.30
0.72
0.75
0.78
0.80
0.81
--
20.2
18.3
18.9
19.7
---
24.0
22.2
22.8
23.5
---
0.69
0.74
0.78
0.8
Heat losses from the electro-adsorption chiller are largely due to the sensible heat load
resulting from alternating heating and cooling of silica gel adsorbents, heat exchanger
and the adsorbed or desorbed water vapour. These sensible loads are included in the
system modeling but other sources of heat losses, which are not accounted in the
155
simulation, are interaction with the external environment and the sensible heating and
cooling of reactors.
6.5 Summary of Chapter 6
The experiments show that the bench-scale electro-adsorption chiller (EAC) has been
successfully designed and commissioned. The bench-sized chiller is found to have a
COP of 0.78 with a maximum cooling capacity of 120 W, an improvement of more
than two times when the system is cooled with thermoelectric cooler only. This
cooling capacity is found in many CPUs of today’s digital processing units. For
comparison, the load surface temperature is not maintained below ambient temperature
by a commercially available thermoelectric cooling (TEC) device as TEC is not
suitable for the application at high heat flux. A new boiling correlation is proposed,
which agrees well with the present experimental data. For verification, the new
correlation has been compared with the published data.
156
Chapter 7
Conclusions and Recommendations
In this thesis, a universal thermodynamic framework that is suitable for both the macro
and micro scale modeling of thermoelectric cooler has been derived. It is based on the
Boltzmann Transport Equation (BTE) approach, incorporating the classical Gibbs law
and the energy conservation equation where their amalgamation yields the form of the
temperature-entropy flux (T-s) formulation. The modeling is applied to the energy
flows and performance investigation of the solid state cooling devices such as the
thermoelectric coolers, the pulsed-mode thermoelectric coolers and the thin-film
superlattice thermoelectric device. It demonstrates the regimes of entropy generation
that are associated to irreversible transport processes arising from the spatial thermal
gradients, as well as the contributions from the collisions of electrons, phonons and
holes in a micro- scaled type solid state cooler.
The pedagogical value of the general temperature-entropy flux diagram has been
successfully demonstrated. The area under the process paths of a thermoelectric
element indicates how energy input has been consumed by the presence of both the
reversible (Peltier, Seebeck and Thomson) and the irreversible (Joule and Fourier)
effects. The similarity between the Seebeck coefficient and the entropy flux for these
elements at adiabatic states has been shown using the T-α diagram. For the first time,
the temperature dependency of Seebeck coefficient on the semi-conductor cooler is
revealed thermodynamically. The incorporation of Thomson heat in the performance
estimation of a thermoelectric cooler, as opposed to previous studies, is estimated to be
about 5% (Chakraborty and Ng, 2005(a)).
157
The thesis demonstrates successfully the transient operation of a pulsed thermoelectric
cooler using the same thermodynamic approach and it provides the necessary
expression for the entropy flux, expressed in terms of the basic measurable variables
(Chakraborty and Ng, 2005 (b)). Using the “time-frozen” T-s diagrams, the cyclic
processes of the pulsed and non-pulsed operations of thermoelectric cooler show how
the input energy has been distributed - depicting the useful energy flows and
“bottlenecks” of cycle operation. The thesis presents also the T-s diagrams of a thin
film thermoelectric, where the effects of electrons flow in n-leg and holes flow in p-leg
generate the total cooling effects at the cold junction. However, the phonons flow in
both legs of the semi-conductors contributes to dissipative losses at the cold junction
and the energy dissipation due to collisions is about 5-10%.
Extending the thermodynamic approach to an adsorbent-adsorbate system, the totally
novel properties for describing accurately the specific heat capacity, the partial
enthalpy and the entropy are presented. These equations are essential for the modeling
and analyses of a novel electro-adsorption chiller. Contrary to the assumption of either
a liquid or gaseous phase for the specific heat capacity, the thesis shows the
theoretically defined specific heat capacity of the adsorbate-adsorbate at all
thermodynamic states (Langmuir, 19, 2254-2259, 2003). The adsorption characteristics
(monolayer, vapor uptake capacity, pre-exponential constant and isosteric heat of
adsorption) of water vapor on silica gel are determined experimentally. The
experimental quantifications of these adsorption characteristics are the author’s novel
contributions. Using the improved thermodynamic model as well as the isotherm
characteristics of the silica gel-water pair (performed prior to EAC experiments –
158
JCED, 47, 1177-1181, 2002), the cyclic operation of an EAC could now be accurately
modeled. The predictions have been shown to compare well with the EAC
experiments. For the given dimensions of a bench-scale EAC unit, the model predicted
the exact mass of adsorbent (300 gm), the overall dimensions of reactor beds and the
amount of adsorbate to be used (US Patent no 6434955). One key design of the EAC
experiments is the design of an infra-red radiant heater and a tapered homogenizer or
kaleidoscope (surface reflectivity = 0.94) which enables uniform heat flux supplied to
the quartz-evaporator. It illuminates the foam-metal cladded evaporator up to 5 W/cm2,
corresponding to a total heat load of 120 W, and this heat load is sufficient to simulate
the heat release from a computer CPU.
The design, developments and fabrication of a bench scale electro-adsorption (EAC)
chiller are the author’s original novel contributions. The design of EAC is developed
on the basis of the adsorption isotherms, adsorption property fields and the dynamic
behavior of thermoelectric modules. In the EAC experiments, tests are conducted
across assorted heat input, cycle and switching time intervals for the adsorption cycle.
The measured pressures and temperatures data agree well with the predictions obtained
from the above-mentioned thermodynamic model and they provide a better
understanding of the EAC cooling cycle. One pleasant surprise from the tests of the
evaporator (during the cyclic steady pool boiling) is excellent plot of heat flux versus
the temperature difference (DT= Twall-Tsat) (Ng and Chakraborty, 2005(b)). The boiling
heat flux data at vacuum pressures as low as 1.8 kPa is not available in the literature.
The nearest similar data were those reported by McGillis et al (1990), however, the
heat fluxes and pressures from the latter were conducted at significantly higher.
Improvising the classical Rohsenow correlation, the thesis proposes a new set of
159
coefficients for pool boiling, namely, a power index for pressure ratio correction at low
vacuum pressures. It predicts accurately the boiling q versus DT at pressures ranging
from 1.8 kPa to 10 kPa. The validity of the new correlation is demonstrated by the
good agreement between the predictions and measured load surface temperatures of
evaporator.
Based on the present work, the incremental improvements of the research are
summarized below:
(1) To conduct the experimental investigation of a thin film superlattice thermoelectric
cooler so as to verify the present model inclusive of the collisions terms.
(2) To improve the lumped modelling of EAC, the distributed modelling could be used
and the performances of the miniaturized EAC could be captured.
(3) To conduct the pressurized electro-adsorption chiller on a bench scale where the
problems associated with vacuum operation could be eliminated.
160
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Appendix A Gauss Theorem approach
V is the volume of an element (solids, liquids, vicoelastic materials and rigid bodies)
which is bounded by a closed surface Ω. The differential element of the volume and
surface area is written as dV and dΩ = n dΩ respectively, where n is the unit normal to
Ω and is defined as pointing out of V. with this notation the Gauss’ Theorem is written
as,
Ω=Ω=∇ ∫∫∫ΩΩ
ddndVV
... XXX r (A1)
where X is any sufficiently smooth vector field and is defined on the volume V and the
boundary surfaceΩ.
Balance equations represent the fundamental physical laws and these include the
conservation of mass, momentum, energy and some form of the second law of
thermodynamics, entropy. At any instant, the mass is taken to occupy the volume V
and has a bounding surface given by Ω. The general form of the conservation laws is
given by,
The time rate of change of a quantity = actions of the surroundings on the surface
of V + the actions of the surroundings on the volume itself.
The word “actions” means the change of quantity. Logically, the general form of the
general form of the right hand side of the conservation principle always include all
possible ways to influence the time rate of change of the quantity of interest.
VΩ
172
Conservation of mass
The rate of change of mass of component k within a given volume V is
∫ ∫ ∂∂
=V V
kk dV
tdV
dtd ρ
ρ
dVJddVt
r
j VjkjkkV
k ∑∫∫∫=
Ω+Ω−=
∂∂
1
. vvρρ (A2)
From gauss theorem
∫∫∫∫ΩΩ
Ω=∇⇒Ω=∇ ddVddV kkV
kkkkV
kk .... vvvv ρρρρ
Equation (2) is written as,
( ) ∑
∑∫∫∫
=
=
+−∇=∂∂
⇒
+−∇=∂∂
r
jjkjkk
k
r
j Vjkj
VkkV
k
Jt
dVJdVdVt
1
1
.
.
vv
vv
ρρ
ρρ
(A3)
Total density ∑=
=n
kk
1
ρρ and barycentric velocity ∑=
=n
k
kk
1 ρρ vv
Equation (3) now becomes (neglecting chemical potential)
( )vρρ .−∇=∂∂
t (A4)
Conservation of momentum:
The equation of motion is given by,
( ) ( )
( ) ( ) ZEPt
ZEPt
ZEPdtd
ρρρ
ρρρ
ρρ
++−∇=∂
∂⇒
+−∇=∇+∂
∂⇒
+−∇=
vvv
vvv
v
.
..
.
173
Here vρ = momentum density, P+vvρ = momentum flow and ZEρ = the source of
momentum.
vv = Dyadic product = ji vv (i, j = 1, 2, 3……………..)
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
=232313
322212
312121
vvvvvvvvvvvvvvv
vv
Conservation of energy
In Tensor notation,
( ) ( )
( ) ( )[ ]
vvvvvv
vvvvvv
vvv
2
.:.21.2
1
....
.
2 ZEPPt
ZEPt
ZEPt
ρρρ
ρρρ
ρρρ
+∇+⎟⎠⎞
⎜⎝⎛ +−∇=
∂
⎟⎠⎞
⎜⎝⎛∂
⇒
++∇−=∂
∂⇒
++−∇=∂
∂
(A5)
Note that
( )[ ]
⎟⎠⎞
⎜⎝⎛⋅∇=
∂∂
=
=⎟⎟⎠
⎞⎜⎜⎝
⎛∂∂
=⋅⋅∇
∑
∑
=
=
vv
vvv
2
1
2
3
1
21
21
,,1,
ρ
ρ
ρρ
n
iji
i
ji
jii
vvx
njivvvx
L
and
174
( )
( ) v
vv
⋅⋅∇−=
⎟⎟⎠
⎞⎜⎜⎝
⎛∂∂
−=
∂∂
+⎟⎟⎠
⎞⎜⎜⎝
⎛∂∂
+∂∂
−=
=∂∂
+⎟⎟⎠
⎞⎜⎜⎝
⎛∂∂
−=∇+⋅⋅∇−
∑ ∑
∑∑∑∑
∑∑∑ ∑
= =
= == =
= == =
P
Px
v
vx
Pvx
PPx
v
njivx
PvPx
PP
n
j
n
iij
ij
n
i
n
jj
iij
n
i
n
jj
iijij
ij
n
i
n
jj
iij
n
i
n
jjij
i
1 1
1 11 1
1 11 1,,3,2,1,: L
The rate of change of potential energy density
( )
( )
( ) ( ) JFZEJt
FJFJt
JFJFJt
n
kkk
n
kkk
n
kkk
n
k
r
jjkjk
n
kkk
n
kkk
n
kkk
−−+−∇=∂
∂∴
−−⎟⎠
⎞⎜⎝
⎛+−∇=
∂∂
⇒
+−−⎟⎠
⎞⎜⎝
⎛+−∇=
∂∂
∑∑∑
∑∑∑∑∑
===
= ====
vv
vv
vvv
..
..
..
111
1 1111
ρψρψρψ
ρψρψρψ
ψρψρψρψ
(A6)
Hence, ( )rjn
kkjk L,10
1==∑
=
vψ as potential energy is conserved into chemical
reaction.
∑=
n
kkk FJ
1
= the source term.
Adding equations (A5) and (A6)
( ) ( )
JFPJPt
JFZEJZEPPtt
−∇+⎟⎠⎞
⎜⎝⎛ +++−∇=
∂
⎟⎠⎞
⎜⎝⎛ +∂
−−+∇−+∇+⎟⎠⎞
⎜⎝⎛ +−∇=
∂∂
+∂
⎟⎠⎞
⎜⎝⎛∂
vvvvvv
vvvvvvvv
2
2
:.21.2
1
...:.21.2
1
2
2
ψρψρψρ
ρψρψρρρψρ
175
For total energy
( )eJ
te .−∇=
∂∂ ρ (A7)
where total energy ue ++= ψ2
21 v ,
total energy flux ∑=
+++=n
kqkke JPeJ
1. Jvv ψρ
veρ = a convective term, v.P = Mechanical work in the system, ∑=
n
kkk J
1ψ = a potential
energy flux due to diffusion, qJ = the heat flow.
From equation (A7) we get,
( )
( )t
JPetu
JPetu
t
JPet
u
n
kqkk
n
kqkk
n
kqkk
∂
⎟⎠⎞
⎜⎝⎛ +∂
−⎟⎠
⎞⎜⎝
⎛+++−∇=
∂∂
⇒
⎟⎠
⎞⎜⎝
⎛+++−∇=
∂∂
+∂
⎟⎠⎞
⎜⎝⎛ +∂
⇒
⎟⎠
⎞⎜⎝
⎛+++−∇=
∂
⎟⎠⎞
⎜⎝⎛ ++∂
∑
∑
∑
=
=
=
ψρψρρ
ψρρψρ
ψρψρ
2
1
1
2
1
2
21
..
..21
..21
vJvv
Jvvv
Jvvv
( ) ( )
( )
JFPJP
JPutu
tJPe
tu
n
kqkk
n
kqkk
+∇−⎟⎠⎞
⎜⎝⎛ +++∇+
⎟⎠
⎞⎜⎝
⎛++∇−⎟⎟
⎠
⎞⎜⎜⎝
⎛
⎭⎬⎫
⎩⎨⎧ ++−∇=
∂∂
⇒
∂
⎟⎠⎞
⎜⎝⎛ +∂
−⎟⎠
⎞⎜⎝
⎛++∇−−∇=
∂∂
⇒
∑
∑
=
=
vvvvv
Jvvv
vJvv
:.21.
..21.
21
...
2
1
2
2
1
ψρψρ
ψψρρ
ψρψρρ
( )
JFP
JPJJPutu n
kqkk
+∇−
⎟⎠
⎞⎜⎝
⎛−−−−+++++−∇=
∂∂
⇒ ∑=
v
vvvvvvvvv
:
.21.
21. 2
1
2 ψρψρψρρψρρ
176
( ) ( ) JFPutu
q +∇−+−∇=∂
∂⇒ vJv :. ρρ (A8)
ρuv+Jq is the energy flow due to convection and thermal energy transfer that gives rise
to an internal energy change, while -P:∇v + JF is an internal energy source term.
Equation (A8) is simplified as
( )
( ) ( )
JFpdtdu
JFPuudtdu
JFPuut
q
q
q
+⋅∇−⋅−∇=
+∇−+⋅−∇=⋅∇−
+∇−+⋅−∇=∂∂
vJ
vJvv
vJv
ρ
ρρρ
ρρ
:
:
(A9)
Conservation of entropy
The entropy per unit mass is a function of the internal energy u, the specific volume υ
and the mass fractions ck:
),,( kcuss υ= .
In equilibrium the total differential of s is given by the Gibbs relation:
∑=
−+=n
kkk dcpdduTds
1µυ , where p is the equilibrium pressure and kµ is the
thermodynamic or chemical potential of component k.
∑=
−+=n
k
kk dt
dcdtdp
dtdu
dtdsT
1µυ (A10)
From mass balance we get
177
( )
( )
v
v
v
v
v
v
vvv
vvv
v
⋅∇=
⋅∇=
⋅∇−=−
⋅∇−=⎟⎠⎞
⎜⎝⎛
⋅∇−=⎟⎠⎞
⎜⎝⎛
⋅∇−=
∇⋅−⋅∇−=∇⋅−
⋅−∇=∇⋅−∇⋅+∂∂
⋅−∇=∂∂
TP
dtd
TPdtd
dtd
dtd
d
d
dt
d
dtddtd
t
t
υρ
υρ
ρυυ
ρυυυ
ρυ
ρρ
ρρρρ
ρρρρ
ρρ
2
1
1
1
Equation (A4) is written as
( )
( )
( ) ( )
( ) ( )
( )[ ]
kkk
kkkk
kkkkkk
kkkkk
kkkkkk
kkkkk
kkk
dtddt
ddt
ddt
ddt
dt
t
Jv
vvv
vvvvv
vvvvv
vvv
vvv
v
⋅∇−⋅∇−=
−⋅∇−⋅∇−=
∇⋅−−−⋅∇−⋅∇−=
∇⋅−−+−⋅∇−=
∇⋅−⋅∇−=∇⋅−
⋅−∇=∇⋅−∇⋅+∂∂
⋅−∇=∂∂
ρρ
ρρρ
ρρρρ
ρρρ
ρρρρ
ρρρρ
ρρ
where Jk ( )( )vv −= kkρ is the diffusion flow of component k defined with respect to
the barycentric motion.
178
Defining the component mass fractions ck as
1cthatsoc2
1kk
kk =
ρρ
= ∑=
and making use of equation (A6), equation (A7) can be simplified as
( )
kk
kkkk
kkkk
kk
k
kkk
dtdcdt
dcdtd
dtdc
dt
d
dtd
J
Jvv
Jv
Jv
Jv
⋅−∇=
⋅∇−⋅∇−=⋅∇−+
⋅∇−⋅∇−=+
⋅∇−⋅∇−=⎟⎟⎠
⎞⎜⎜⎝
⎛
⋅∇−⋅∇−=
ρ
ρρρρ
ρ
ρρρρ
ρ
ρρ
ρρ
ρρ
From equation (10) we have,
( )
∑∑
∑∑
∑
∑
∑
∑
=
=
=
=
=
=
=
=
⎭⎬⎫
⎩⎨⎧ −∇⋅−∇⋅−
⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜
⎝
⎛−
⋅−∇=
∇⋅−
⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜
⎝
⎛−
⋅∇−⋅+∇⋅+⋅−∇=
⋅∇−−+⋅+−⋅∇−=
−+=
−+=
n
kk
kkq
kkkq
n
k
kk
n
kkkn
kkkq
q
n
kk
kq
n
k
kk
n
k
kk
FT
TT
TTT
TTF
TTT
Tdtd
TpFJ
Tdtd
Tp
T
dtdc
Tdtd
Tp
dtdu
Tdtds
dtdc
dtdp
dtdu
dtdsT
12
2
1
1
1
1
1
1
1
11
11
11
µµ
µµ
µυρυρ
ρµυρρρ
µυ
JJJJ
JJ
JJJ
JJ
(A11)
where Jk ( )( )vv −ρ= kk is the diffusion flow of component k defined with respect to
the barycentric motion.
179
From equation (A11), one can observe that the entropy flow is given by
⎟⎠
⎞⎜⎝
⎛−= ∑
=
n
kkkqs T 1
1 JJJ µ (A12)
and the entropy source strength by
∑= ⎭
⎬⎫
⎩⎨⎧ −∇⋅−∇⋅−=
n
kk
kkq F
TT
TT
T 12
11 µσ JJ (A13)
180
Appendix B
The Thomson effect in equation (3.7)
The electrical and thermal currents are coupled in a thermoelectric device. The Peltier
coefficient of the junction is a property depends on both the materials and is the ratio
of power evolved at the junction to the current flowing through it. When current is
applied to a thermoelectric element, thermal energy is generated or absorbed at the
junction due to Peltier effect. The Seebeck coefficient depends on temperature and this
is different at different places along the TE material. So the thermoelectric element is
thought of as a series of many small Peltier junctions and each of which is generating
or absorbing heat. This is called Thomson power evolved per unit volume ( )TJ ∇Γ .
In a Thomson effect, heat is absorbed or evolved when current is flow in a
thermoelectric element with a temperature gradient. The heat is proportional to both
the electric current and the temperature gradient. The proportionality constant is
known as Thomson coefficient.
DC power source+ -
J
Heat absorbed at the cold end
p-arm
J
Heat rejected from the hot end
n-arm
J
J
Tcj
Thj
p-arm or
n-arm
Heat absorbed at cold side ( )Jcjπ
Heat involved at the hot side ( )Jhjπ
TJ ∇Γ.Thomson heat
Figure B1 A thermoelectric element showing the transportation of heat and current.
181
If the heat current density of a thermoelectric element is only a result of temperature
gradient and the electric current density occurs due to electric potential gradient. Then,
Tq ∇−= .λJ and φσ ∇′=J . Using equation (3.1) the energy balance equation of a
thermoelectric element is obtained.
( )σ
λρ′
+∇∇=∂∂ 2
. JTtTcp (B1)
The first term of right hand side of equation (B1) is Fourier conduction term and the
second term indicates the Joule heat. But in a thermoelectric cooling device, the
junctions are maintained at different temperatures and an electromotive force will
appear in the circuit. This flow of heat has a tendency to carry the electricity along the
thermoelectric arm.
When two or more irreversible processes take place in a thermodynamic system, they
may interfere with each other. A completely consistent theory of thermoelectricity has
been developed using the Onsager symmetry relationship (Onsager, 1931). When a
current is applied to a thermoelectric device, the heat current of TE arm is developed
as a result of electric current and temperature gradient (cross flow) and the electric
current is generated due to electric potential and the temperature gradient.
Using the thermodynamic-phenomenological treatment, the heat current density is
TJFQo
q ∇−−= λJ , and the electric current density in amp per m2 becomes
TFTQJ
TFT
QJ
o
o
∇−′
=∇
∇′
+∇′=
.σ
φ
σφσ
Equation (3.1) now becomes
182
( )
( ) ⎟⎠⎞
⎜⎝⎛∇+
′+∇∇=
∇−′
+∇+∇∇=
⎟⎠⎞
⎜⎝⎛ ∇−
′+⎟
⎠⎞
⎜⎝⎛ −∇−−∇=
∂∂
TQ
FJTJT
TFTJQJQ
FJT
TFTQJJJ
FQT
tTc
o
oo
oop
σλ
σλ
σλρ
2
2
.
..
..
( ) ⎟⎠⎞
⎜⎝⎛∇+
′+∇∇=
∂∂
TQ
FJTJT
tTc o
p σλρ
2
. (B2)
Hence the third term ⎟⎠⎞
⎜⎝⎛∇
TQ
FJT o occurs due to current flow for temperature gradient
(cross flow) and heat flow for the electric potential gradient (reciprocal effect). This
term indicates the least dissipation of energy (Onsager, 1931)
( )sFJT
TQ
FJT o ∇=⎟
⎠⎞
⎜⎝⎛∇ , where ⎟
⎠⎞
⎜⎝⎛=
TQs o is the entropy (in J/mol.K) along the
thermoelectric arm.
Hence
( )
TTs
FTJ
TTs
FJT
sFJT
TQ
FJT o
∇⎟⎠⎞
⎜⎝⎛
∂∂
=
∇∂∂
=
∇=⎟⎠⎞
⎜⎝⎛∇
.
.
The Thomson heat is defined as
( ) TJQ
TTs
FTJQ
T
T
∇Γ−=
∇⎟⎠⎞
⎜⎝⎛
∂∂
= .,
where Thomson coefficient Ts
FT∂∂
=Γ− .
The temperature gradient is bound to cause a certain degradation of energy by
conduction of heat. The electric potential gradient must cause an additional
183
degradation of energy, making the total rate of increase of entropy along the
thermoelectric arm.
So the electric current due to temperature gradient (cross flow) and the heat current
due to electric potential gradient (reciprocal effect) indicate the Thomson effect, which
will be discussed in the following section. Without Thomson effect, the physics of
thermoelectricity is not completed.
Derivation of equation (3.7)
The energy balance of the thermoelectric arm is
( ) TJJTtTcp ∇Γ−
′+∇∇=
∂∂
σλρ
2
. (B3)
Expression of the third term
Using Kelvin relations ⎥⎦⎤
⎢⎣⎡ =
∂∂
−=Γ TandTT
απππ ,
TTT
JTJ ∇⎟⎠⎞
⎜⎝⎛
∂∂
−=∇Γ .ππ (B4)
TTT∇
∂∂
+∇=∇ .πππ (de Groot and Mazur, 1962)
From equation (B4) we have
184
( )
( )
T
T
T
T
T
JTTT
JT
JTTT
JTTJTJ
JTTJTJ
JJTT
J
JTT
J
TT
JTT
JTJ
αα
αααα
ααα
πππ
πππ
ππ
∇+∇∂∂
−=
∇+∇∂∂
−∇−∇=
∇+∇−∇=
∇+∇−∇=
∇−∇−∇=
∇∂∂
−∇=∇Γ
.
.
.
.
..
Equation (B3) is now written as
( )Tp JTT
TJTJT
tTc αα
σλρ ∇+∇
∂∂
−′
+∇∇=∂∂ ..
2
(B5)
Equation (B5) is same as equation (3.7).
185
Appendix C
Energy balance of a thermoelectric element
Energy balance equation at steady state becomes
022
2
=+− JdxdTJ
dxTd ρτλ , (C1)
where τ indicates Thomson coefficient, λ defines thermal conductivity and ρ is the
electrical resistivity.
The solution becomes
BeJ
AxJT
AeJdxdT
xJ
xJ
++=
+=
⎟⎠⎞
⎜⎝⎛
⎟⎠⎞
⎜⎝⎛
λτ
λτ
τλ
τρ
τρ
and
Putting boundary conditions
( )
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛−
⎟⎠⎞
⎜⎝⎛ −
=⇒=
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛−
⎟⎠⎞
⎜⎝⎛ −
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛−+==−
+=
++=
⎟⎠⎞
⎜⎝⎛
⎟⎠⎞
⎜⎝⎛
⎟⎠⎞
⎜⎝⎛
⎟⎠⎞
⎜⎝⎛
11
1
_________________________________
LJLJ
LJ
LH
L
LJ
H
e
JJLDTAA
eJ
JLDT
eJ
ALJDTTT
BJ
AT
BeJ
ALJT
λτ
λτ
λτ
λτ
λτ
τρ
τλ
τρ
τλ
τρ
τλ
τλ
τρ
Now,
186
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
⎟⎟⎠
⎞⎜⎜⎝
⎛−⎟
⎠⎞
⎜⎝⎛
⎟⎠⎞
⎜⎝⎛ −
⎟⎠⎞
⎜⎝⎛−−=−
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
⎟⎟⎠
⎞⎜⎜⎝
⎛−⎟
⎠⎞
⎜⎝⎛
⎟⎠⎞
⎜⎝⎛ −
⎟⎠⎞
⎜⎝⎛+=
1expexp
1expexp
LJ
JJLDTxJAJA
dxdTA
LJ
JJLDTxJJ
dxdT
λτ
λτ
τρ
λτλ
τρλλ
λτ
λτ
τρ
λτ
τρ
( )
( )
( ) ⎟⎠
⎞⎜⎝
⎛⎟⎠⎞
⎜⎝⎛−−⎟
⎠⎞
⎜⎝⎛ −
⎟⎠⎞
⎜⎝⎛ +−
+−=−
⎟⎠
⎞⎜⎝
⎛⎟⎠⎞
⎜⎝⎛−−⎟
⎠⎞
⎜⎝⎛ −
⎟⎠⎞
⎜⎝⎛ +−
+−=−
⎟⎠
⎞⎜⎝
⎛⎟⎠⎞
⎜⎝⎛−−⎟
⎠⎞
⎜⎝⎛ −
⎟⎠⎞
⎜⎝⎛ +−
+−=−
⎟⎠
⎞⎜⎝
⎛−⎟⎠⎞
⎜⎝⎛
⎟⎠⎞
⎜⎝⎛ +−
⎟⎠⎞
⎜⎝⎛+−=−
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
⎟⎠
⎞⎜⎝
⎛−⎟⎠⎞
⎜⎝⎛
⎟⎠⎞
⎜⎝⎛ −
⎟⎠⎞
⎜⎝⎛−−=−
xAIxL
AI
AIL
LAILA
LADT
AIA
dxdTA
xJxLJ
JLLJLA
LADT
JAdxdTA
xJxLJ
JJLDTAJA
dxdTA
LJ
JJLDTAxJJA
dxdTA
LJ
JJLDTxJAJA
dxdTA
λτ
λτ
λτ
τρλλ
τρλλ
λτ
λτ
λτ
τρλλ
τρλλ
λτ
λτ
λτ
τρλ
τρλλ
λτ
λτ
τρλ
λτ
τρλλ
λτ
λτ
τρ
λτλ
τρλλ
expexp
expexp
expexp
1expexp
1expexp
187
( )
( )
⎟⎟⎠
⎞⎜⎜⎝
⎛⎟⎠⎞
⎜⎝⎛−−⎟⎟
⎠
⎞⎜⎜⎝
⎛⎟⎠⎞
⎜⎝⎛ −
⎟⎠⎞
⎜⎝⎛ +−
+−=−
⎟⎟⎠
⎞⎜⎜⎝
⎛⎟⎠⎞
⎜⎝⎛−−⎟
⎠⎞
⎜⎝⎛ −
⎟⎠⎞
⎜⎝⎛ +−
+−=−
⎟⎟⎠
⎞⎜⎜⎝
⎛⎟⎠⎞
⎜⎝⎛−−⎟
⎠⎞
⎜⎝⎛ −
⎟⎠⎞
⎜⎝⎛ +−
+−=−
Lx
AIL
Lx
AIL
AILI
LADT
IdxdTA
xAIxL
AI
AILI
LADT
IdxdTA
xAIxL
AI
AILI
LADT
IdxdTA
λτ
λτ
λτ
τλρλ
τλρλ
λτ
λτ
λτ
τλρλ
τλρλ
λτ
λτ
λτ
τλρλ
τλρλ
exp1exp
expexp
expexp
(C2)
Defining Fourier heat L
ADTQFλ
= , Joule heat A
LIQJ
2ρ= and Thomson heat
DTIQT τ=
Here, ALI
LADTDTI
F
T
λτ
λτ
==
Equation (2) is written as,
⎟⎟⎠
⎞⎜⎜⎝
⎛⎟⎟⎠
⎞⎜⎜⎝
⎛−−⎟⎟
⎠
⎞⎜⎜⎝
⎛⎟⎠⎞
⎜⎝⎛ −
⎟⎠⎞
⎜⎝⎛ +−
+−=−
Lx
Lx
QQIQ
IdxdTA
F
T
F
T
F
TF
exp1exp
τλρ
τλρλ (C3)
At cold junction (x = 0)
τλρ
τλρτ
λρ
τλρλ IIQ
QQQQ
QQIQ
IdxdTA F
F
T
F
T
F
T
F
TF
x
−⎟⎠⎞
⎜⎝⎛ +−
⎟⎟⎠
⎞⎜⎜⎝
⎛−⎟⎟
⎠
⎞⎜⎜⎝
⎛=
⎟⎟⎠
⎞⎜⎜⎝
⎛−⎟⎟⎠
⎞⎜⎜⎝
⎛
⎟⎠⎞
⎜⎝⎛ +−
+−=−= 1exp1exp0
As we know,
.......................................!642
1!430
1!26
12
11)exp(
642
−+−+−=−
xxxxxx
188
..................21
3601
21
61
211
1exp
42
+⎟⎟⎠
⎞⎜⎜⎝
⎛−⎟⎟
⎠
⎞⎜⎜⎝
⎛+−=
⎟⎟⎠
⎞⎜⎜⎝
⎛−⎟⎟
⎠
⎞⎜⎜⎝
⎛ F
T
F
T
F
T
F
T
F
T
QQQQ
..................21
3601
21
61
211
1exp
42
+⎟⎟⎠
⎞⎜⎜⎝
⎛−⎟⎟
⎠
⎞⎜⎜⎝
⎛+−
×⎟⎠⎞
⎜⎝⎛ +−=
⎟⎟⎠
⎞⎜⎜⎝
⎛−⎟⎟
⎠
⎞⎜⎜⎝
⎛⎟⎠⎞
⎜⎝⎛ +−
F
T
F
T
F
T
F
F
T
F
T
F
IQ
QQQQ
IQτλρ
τλρ
After rearranging,
....................................................................................................................21
21
3601
21
21
61
21
21
3
3
3
42
0
+⎥⎦
⎤⎢⎣
⎡+−⎥
⎦
⎤⎢⎣
⎡++⎥⎦
⎤⎢⎣⎡ −−−=−
= F
TJ
F
T
F
TJ
F
TJTF
x QQQ
QQQ
QQQQQ
dxdTAλ
⎥⎦⎤
⎢⎣⎡ −−−=−
=JTF
x
QQQdxdTA
21
21
0
λ
Similarly at hot junction we can prove that (removing all higher order terms)
⎥⎦⎤
⎢⎣⎡ ++−=−
=JTF
Lx
QQQdxdTA
21
21λ (C4)
Energy analysis
⎟⎟⎠
⎞⎜⎜⎝
⎛⎟⎟⎠
⎞⎜⎜⎝
⎛−−⎟⎟
⎠
⎞⎜⎜⎝
⎛⎟⎠⎞
⎜⎝⎛ −
⎟⎠⎞
⎜⎝⎛ +−
+−=
Lx
Lx
QQIQ
IxITxQ
F
T
F
T
F
TF
exp1exp)()( τ
λρ
τλρα (C5)
Similarly at hot junction
⎥⎦⎤
⎢⎣⎡ ++−+= JTFHH QQQITQ
21
21α (C6)
From (C6) we have,
189
DTIKDTRITQ HH τα21
21 2 +−+= I , where
LAK λ
= , ALR ρ
= .
This is same as (3.16)
At cold junction
⎥⎦⎤
⎢⎣⎡ −−−+= JTFLL QQQITQ
21
21α (C7)
(C7) is written as,
DTIKDTRIITQ LL τα21
21 2 −−−= , which is same as (3.15)
Net power input
( ) TJLH
JTFLJTFHLH
QQTTI
QQQITQQQITQQ
++−=
⎥⎦⎤
⎢⎣⎡ −−−+−⎥⎦
⎤⎢⎣⎡ ++−+=−
α
αα21
21
21
21
(C8)
Entropy flux
)(1
exp1exp)()(
)()(xT
Lx
Lx
QQIQ
xTI
xTxITxJ
F
T
F
T
F
TF
S
⎟⎟⎠
⎞⎜⎜⎝
⎛⎟⎟⎠
⎞⎜⎜⎝
⎛−−⎟⎟
⎠
⎞⎜⎜⎝
⎛⎟⎠⎞
⎜⎝⎛ −
⎟⎠⎞
⎜⎝⎛ +−
+−=τλρ
τλρα (C9)
190
Appendix D
Programming Flow Chart of the thin film thermoelectric cooler (Superlattice thermoelectric element (SLTE))
Figure D1 Ensemble Particle Motion. Here each horizontal solid line shows the trace of each particle. The vertical broken line shows time. The on the solid line shows the scattering or collision time.
τ τ + ∆ τ τ - ∆ τ
Equation Solver
191
Flow chart of SLTE programming and Monte Carlo method to solve collision time.
START
CONFIGURATION
Geometry of Device
DATA INPUT
Initial Conditions For mass, energy,
temperature and entropy
EMC (Ensemble Monte-Carlo) For calculating collision time
Using IMSL solver Balance equation of mass (ρ) Balance equation of momentum (v) Balance equation of energy (u and T) Balance equation of entropy (s)
t < tmax
No
END
Start
τ1 = τi τ2 = τs = scattering time
τ2 > τ + ∆ τ
Yes
τcolli = τ + ∆ τ – τ1
Cal. τcolli
τcolli = τ2 – τ1
Drift (τ)
τ1 = τ2 τ2 = new scattering time
192
Appendix E
Programming Flow Chart of the electro-adsorption chiller (EAC) and water properties equations
Initial Conditions for
EAC (t = 0)
Open Files
START
Data Storage
Read Data
Adsorption Bed Tads = Troom qads = 0 Qads = 0
Desorption Bed Tdes = Troom qdes = 0 Qdes = 0
Thermoelectric modules
Tte = Troom Pin = 0
Condenser and evaporator Qcond = 0 Qevap = 0
N
Y
Data Output By
WRITE
Calculation EAC Operation
by Call Function 1 By
DIVPAG in IMSL
Calculation of Adsorption isotherm Adsorption isobar Adsorption kinetics
Calculation of water saturation and superheated properties (Wagner routines) Equations are given in section below
Close
If t ? (Operation time)
Ope
ratio
n pe
riod
193
N
Y
Data Output By
WRITE
START Switching
Initial value of switching
operation (Data from
previous operation period)
ADS - DES DES - ADS TE COND & EVAP
Calculation EAC switching by
Call Function 2
By DIVPAG in IMSL
If ׀Tj – Tj-1 ׀ ? 1× 10-2 N
Y
END
If t ? (switching time)
194
Properties of water and steam
All equations for calculating water and steam thermodynamic properties are covered by the new industrial standard “IAPWS Industrial formulation 1997 for the Thermodynamic Properties of Water and Steam” [Wagner, 1997]. The basic equation for the liquid water is a fundamental equation for the specific Gibbs
free energy g. This equation is expressed in dimensionless form, )/(RTg=γ , and
reads
( ) ( ) ( ) ( )∑=
−−==34
1222.11.7,,
i
JIi
iinRT
Tpg τπτπγ ,
where =π p/p* and =τ T*/T with p*=16.53Mpa and T*=1386K;R=0.461526kJkg-1K-1.
The coefficients ni and exponents Ii and Ji are listed in Table 1.
All thermodynamic properties can be derived from the basic equation by using the
appropriate combinations of the dimensionless Gibbs free energy γ and its derivatives.
Relations between the relevant thermodynamic properties and γ and its derivatives are
summarized in Table 2.
Table E1 Coefficients and exponents of the basic equation i Ii Ji ni
1 0 -2 0.14632971213167
2 0 -2 -0.84548187169114
3 0 0 -.37563603672040×101
4 0 1 0.33855169168385×101
5 0 2 -0.95791963387872
6 0 3 0.15772038513228
7 0 4 -0.16616417199501×10-1
8 0 5 0.81214629983568×10-3
9 1 -9 0.28319080123804×10-3
10 1 -7 -0.60706301565874×10-3
11 1 -1 -0.18990068218419×10-1
12 1 0 -0.32529748770505×10-1
195
13 1 1 -0.21841717175414×10-1
14 1 3 -0.52838357969930×10-4
15 2 -3 -0.47184321073267×10-3
16 2 0 -0.30001780793026×10-3
17 2 1 0.47661393906987×10-4
18 2 3 -0.44141845330846×10-5
19 2 17 -0.72694996297594×10-15
20 3 -4 -0.31679644845054×10-4
21 3 0 -0.28270797985312×10-5
22 3 6 -0.85205128120103×10-9
23 4 -5 -0.22425281908000×10-5
24 4 -2 -0.65171222895601×10-6
25 4 10 -0.14341729937924×10-12
26 5 -8 -0.40516996860117×10-6
27 8 -11 -0.12734301741641×10-8
28 8 -6 -0.17424871230634×10-9
29 21 -29 -0.68762131295531×10-18
30 23 -31 0.14478307828521×10-19
31 29 -38 0.26335781662795×10-22
32 30 -39 -0.11947622640071×10-22
33 31 -40 0.18228094581404×10-23
34 32 -41 -0.93537087292458×10-25
196
Table E2 Relations of thermodynamic properties to the dimensionless Gibbs free
energy γ and its derivatives.
Property Relation
Specific volume
( )Tpgv ∂∂= / ( ) ππγτπ =
RTpv ,
Specific internal energy
( ) ( )Tp pgpTgTgu ∂∂−∂∂−= //
( )πτ πγτγτπ
−=RT
u ,
Specific entropy
( )pTgs ∂∂−= /
( ) γτγτπτ −=
Rs ,
Specific enthalpy
( )pTgTgh ∂∂−= /
( )ττγτπ
=RT
h ,
Specific isobaric heat capacity
( )pp Thc ∂∂= /
( )ττγτ
τπ 2,−=
Rc p
τπ π
γγ ⎟⎟⎠
⎞⎜⎜⎝
⎛∂∂
= ,π
τ τγγ ⎟⎟
⎠
⎞⎜⎜⎝
⎛∂∂
= ,τ
ππ πγγ ⎟⎟
⎠
⎞⎜⎜⎝
⎛∂∂
= 2
2
,τ
πτ τπγγ ⎟⎟⎠
⎞⎜⎜⎝
⎛∂∂
∂=
2
,π
ττ τγγ ⎟⎟
⎠
⎞⎜⎜⎝
⎛∂∂
= 2
2
A sample set of thermodynamic property values are shown below where the basic equations for selected temperatures and pressures are employed to determine these parameters.
Property T=300K, p=0.0035Mpa
v[m3kg-1] 0.100215168×10-2
h[kJkg-1] 0.115331273×103
u[kJkg-1] 0.112324818×103
s[kJkg-1K-1] 0.392294792
cp[kJkg-1K-1] 0.417301218×101
The basic equation for steam is a fundamental equation for the specific Gibbs free
energy g. This equation is expressed in dimensionless form, ( )RTg /=γ , and is
separated into 2 parts, an ideal-gas part 0γ and a residual part rγ , so that
( ) ( ) ( )τπγτπγτπγ ,,),(, 0 r
RTTpg
+== ,
197
The equation for the ideal-gas part 0γ of the dimensionless Gibbs free energy reads
∑=
+=9
1
00 0
lni
Ji
in τπγ ,
The form of the residual part rγ of the dimensionless Gibbs free energy is as follows:
( )∑=
−=43
1
5.0i
JIi
r iin τπγ ,
where */ pp=π and TT /*=τ with *p =1 MPa and *T =540K.The coefficients 0in and
exponents 0iJ are listed in Table 3.The coefficients in and exponents Ii and Ji are listed
in Table 4.
All thermodynamic properties can be derived from the basic equation by using the
appropriate combinations of the ideal-gas part 0γ and the residual part rγ of the
dimensionless Gibbs free energy and their derivatives.
Property Relation
Specific volume
( )Tpgv ∂∂= / ( ) ( )r
RTpv ππ γγπτπ += 0,
Specific internal energy
( ) ( )Tp pgpTgTgu ∂∂−∂∂−= // ( ) ( ) ( )rr
RTu
ππττ γγπγγττπ+−+= 00,
Specific entropy
( )pTgs ∂∂−= / ( ) ( ) ( )rr
Rs γγγγττπ
ττ +−+= 00,
Specific enthalpy
( )pTgTgh ∂∂−= / ( ) ( )r
RTh
ττ γγττπ+= 0,
Specific isobaric heat capacity
198
( )pp Thc ∂∂= / ( ) ( )rp
Rc
ττττ γγττπ
+−= 02,
τπ π
γγ ⎟⎟⎠
⎞⎜⎜⎝
⎛∂∂
=r
r ,π
τ τγγ ⎟⎟
⎠
⎞⎜⎜⎝
⎛∂∂
=r
r ,π
ττ τγγ ⎟⎟
⎠
⎞⎜⎜⎝
⎛∂∂
= 2
2 rr ,
τπ π
γγ ⎟⎟⎠
⎞⎜⎜⎝
⎛∂∂
=0
0 ,π
τ τγγ ⎟⎟
⎠
⎞⎜⎜⎝
⎛∂∂
=0
0 ,π
ττ τγγ ⎟⎟
⎠
⎞⎜⎜⎝
⎛∂∂
= 2
020
Table E3 Coefficients 0in and exponents
0iJ
i 0iJ 0
in
1 0 -0.96927686500217×101
2 1 0.10086655968018×102
3 -5 -0.56087911283020×10-2
4 -4 0.71452738081455×10-1
5 -3 -0.40710498223928
6 -2 0.14240819171444×101
7 -1 -0.43839511319450×101
8 2 -0.28408632460772
9 3 0.21268463753307×10-1
Table E4 Coefficients ni and exponents Ii ,Ji i Ii Ji ni
1 1 0 -0.17731742473213×10-2
2 1 1 -0.17834862292358×10-1
3 1 2 -0.45996013696365×10-1
4 1 3 -0.57581259083432×10-1
5 1 6 -0.50325278727930×10-1
6 2 1 -0.33032641670203×10-4
7 2 2 -0.18948987516315×10-3
8 2 4 -0.39392777243355×10-2
9 2 7 -0.43797295650573×10-1
10 2 36 0.26674547914087×10-4
199
11 3 0 0.20481737692309×10-7
12 3 1 0.43870667284435×10-6
13 3 3 -0.32277677238570×10-4
14 3 6 -0.15033924542148×10-2
15 3 35 -0.40668253562649×10-1
16 4 1 -0.78847309559367×10-9
17 4 2 0.12790717852285×10-7
18 4 3 0.48225372718507×10-6
19 5 7 0.22922076337661×10-5
20 6 3 -0.16714766451061×10-10
21 6 16 -0.21171472321355×10-2
22 6 35 -0.23895741934104×102
23 7 0 -0.59059564324270×10-17
24 7 11 -0.12621808899101×10-5
25 7 25 -0.38946842435739×10-1
26 8 8 0.11256211360459×10-10
27 8 36 -0.82311340897998×101
28 9 13 0.19809712802088×10-7
29 10 4 0.10406965210174×10-18
30 10 10 -0.10234747095929×10-12
31 10 14 -0.10018179379511×10-8
32 16 29 -0.80882908646985×10-10
33 16 50 0.10693031879409
34 18 57 -0.33662250574171
35 20 20 0.89185845355421×10-24
36 20 35 0.30629316876232×10-12
37 20 48 -0.42002467698208×10-5
38 21 21 -0.59056029685639×10-25
39 22 53 0.37826947613457×10-5
40 23 39 -0.12768608934681×10-4
41 24 26 0.73087610595061×10-28
42 24 40 0.55414715350778×10-16
200
43 24 58 -0.94369707241210×10-6
A sample set of thermodynamic property values are shown below where the basic
equations for selected temperatures and pressures are employed to determine these
parameters.
Property T=300K, p=0.0035Mpa
v[m3kg-1] 39.4913866
h[kJkg-1] 2549.91145
u[kJkg-1] 2411.69160
s[kJkg-1K-1] 8.52238967
cp[kJkg-1K-1] 1.91300162
201
Appendix F Energy flow of the major components of an electro-adsorption chiller
Latent heat balance of an electro-adsorption chiller (last half cycle) time Qe,L Qd,L Qa,L Qc,L lhs rhs Error%1.162791 -135.565 -195.282 145.6575 184.4292 -330.847 330.0867 0.229907 5.813953 -135.656 -195.293 145.7517 184.4396 -330.949 330.1914 0.229015 10.46512 -135.734 -195.279 145.8313 184.4265 -331.013 330.2578 0.228068 17.44186 -135.834 -195.255 145.9341 184.4038 -331.089 330.3379 0.226778 23.25581 -135.908 -195.238 146.0095 184.3888 -331.146 330.3983 0.22583 46.51163 -136.149 -195.213 146.2581 184.3663 -331.362 330.6243 0.222726 69.76744 -136.344 -195.227 146.4602 184.3811 -331.571 330.8413 0.220141 93.02326 -136.509 -195.259 146.6328 184.4129 -331.768 331.0457 0.217766 98.83721 -136.546 -195.269 146.672 184.422 -331.815 331.094 0.217191 127.907 -136.709 -195.316 146.8443 184.4688 -332.025 331.3131 0.214415
156.9767 -136.826 -195.36 146.9683 184.5131 -332.185 331.4814 0.211928 180.2326 -136.886 -195.39 147.0333 184.5441 -332.276 331.5774 0.210155 203.4884 -136.921 -195.414 147.072 184.57 -332.335 331.642 0.208538 226.7442 -136.934 -195.432 147.0883 184.5899 -332.366 331.6783 0.207043
250 -136.929 -195.443 147.0854 184.6036 -332.372 331.689 0.205638 279.0698 -136.901 -195.447 147.0589 184.6111 -332.348 331.67 0.203972 308.1395 -136.853 -195.437 147.0112 184.6066 -332.29 331.6178 0.202359 331.3953 -136.804 -195.404 146.9603 184.5796 -332.208 331.54 0.201037 360.4651 -136.73 -195.32 146.8833 184.5042 -332.049 331.3875 0.199305 389.5349 -136.645 -195.187 146.7938 184.3828 -331.832 331.1766 0.197492 418.6047 -136.552 -195.01 146.6939 184.2197 -331.562 330.9137 0.19561 447.6744 -136.452 -194.795 146.5856 184.02 -331.247 330.6056 0.193671 476.7442 -136.346 -194.547 146.4702 183.7887 -330.893 330.2589 0.191689 494.186 -136.28 -194.385 146.3981 183.6369 -330.665 330.035 0.190482
500 -136.258 -194.328 146.3736 183.5844 -330.586 329.958 0.190077 502 -136.238 -194.3 146.3523 183.5576 -330.538 329.9099 0.190077 510 -136.159 -194.187 146.2669 183.4505 -330.345 329.7174 0.190077 520 -136.059 -194.045 146.1603 183.3168 -330.105 329.4771 0.190077 530 -135.96 -193.904 146.0538 183.1833 -329.864 329.2371 0.190077 540 -135.861 -193.763 145.9475 183.0499 -329.624 328.9975 0.190077 550 -135.762 -193.622 145.8414 182.9168 -329.384 328.7582 0.190077 560 -135.664 -193.481 145.7354 182.7839 -329.145 328.5193 0.190077 570 -135.565 -193.341 145.6295 182.6511 -328.906 328.2807 0.190077 580 -135.467 -193.2 145.5239 182.5186 -328.667 328.0425 0.190077 590 -135.369 -193.06 145.4183 182.3862 -328.429 327.8046 0.190077
597.5 -135.295 -192.955 145.3393 182.2871 -328.25 327.6264 0.190077 600 -135.271 -192.92 145.313 182.2541 -328.191 327.567 0.190077
202
Sensible energy balance (Last half chcle) time Qevap Qcond Qadsc Qh Qads,Ex Qads,Te 1.162791 129.05 -184.368 191.0289 236.8934 114.6174 76.41157 10.46512 129.05 -184.265 191.0832 236.9165 114.6499 76.43328 23.25581 129.05 -184.177 191.1482 236.9419 114.6889 76.45929 46.51163 129.05 -184.099 191.2298 236.9799 114.7379 76.49191 63.95349 129.05 -184.085 191.2611 237.0022 114.7567 76.50444
87.2093 129.05 -184.097 191.2676 237.0244 114.7606 76.50705 110.4651 129.05 -184.127 191.2393 237.0392 114.7436 76.4957 168.6047 129.05 -184.217 191.0553 237.0603 114.6332 76.42212 197.6744 129.05 -184.256 190.9215 237.0666 114.5529 76.36858 203.4884 129.05 -184.263 190.8922 237.0675 114.5353 76.35686 209.3023 129.05 -184.27 190.8621 237.0684 114.5173 76.34484 215.1163 129.05 -184.276 190.8313 237.0692 114.4988 76.33253 220.9302 129.05 -184.282 190.7999 237.0699 114.4799 76.31994 226.7442 129.05 -184.287 190.7677 237.0705 114.4606 76.3071 232.5581 129.05 -184.292 190.735 237.0711 114.441 76.294 238.3721 129.05 -184.297 190.7016 237.0716 114.421 76.28066
244.186 129.05 -184.301 190.6677 237.072 114.4006 76.26709 250 129.05 -184.305 190.6333 237.0723 114.38 76.25331
255.814 129.05 -184.309 190.5983 237.0726 114.359 76.23931 261.6279 129.05 -184.312 190.5628 237.0728 114.3377 76.22512 267.4419 129.05 -184.315 190.5268 237.073 114.3161 76.21073 273.2558 129.05 -184.317 190.4904 237.073 114.2942 76.19616 279.0698 129.05 -184.319 190.4536 237.0731 114.2721 76.18142 284.8837 129.05 -184.32 190.4163 237.073 114.2498 76.16651 290.6977 129.05 -184.321 190.3786 237.0729 114.2272 76.15144 296.5116 129.05 -184.322 190.3405 237.0728 114.2043 76.13622 302.3256 129.05 -184.322 190.3021 237.0727 114.1813 76.12085 308.1395 129.05 -184.322 190.2633 237.0729 114.158 76.10534 313.9535 129.05 -184.32 190.2242 237.0731 114.1345 76.08969 319.7674 129.05 -184.318 190.1848 237.0735 114.1109 76.07392 325.5814 129.05 -184.314 190.1451 237.074 114.087 76.05802 331.3953 129.05 -184.309 190.105 237.0745 114.063 76.04201 337.2093 129.05 -184.302 190.0647 237.0752 114.0388 76.02588 343.0233 129.05 -184.293 190.0241 237.0759 114.0145 76.00965 348.8372 129.05 -184.283 189.9833 237.0768 113.99 75.99331 354.6512 129.05 -184.271 189.9422 237.0777 113.9653 75.97688 360.4651 129.05 -184.257 189.9009 237.0786 113.9405 75.96035 366.2791 129.05 -184.241 189.8593 237.0797 113.9156 75.94373
372.093 129.05 -184.224 189.8176 237.0809 113.8905 75.92702 377.907 129.05 -184.204 189.7756 237.0821 113.8653 75.91023
383.7209 129.05 -184.183 189.7334 237.0833 113.84 75.89336 389.5349 129.05 -184.16 189.6911 237.0847 113.8146 75.87642 395.3488 129.05 -184.135 189.6485 237.0861 113.7891 75.8594 401.1628 129.05 -184.109 189.6058 237.0876 113.7635 75.84232 406.9767 129.05 -184.08 189.5629 237.0891 113.7377 75.82516 412.7907 129.05 -184.051 189.5199 237.0908 113.7119 75.80795 418.6047 129.05 -184.019 189.4767 237.0924 113.686 75.79067 424.4186 129.05 -183.986 189.4333 237.0942 113.66 75.77333 430.2326 129.05 -183.952 189.3899 237.096 113.6339 75.75594 436.0465 129.05 -183.916 189.3462 237.0978 113.6077 75.7385 441.8605 129.05 -183.878 189.3025 237.0997 113.5815 75.721
203
447.6744 129.05 -183.839 189.2587 237.1017 113.5552 75.70346 453.4884 129.05 -183.799 189.2147 237.1037 113.5288 75.68587 459.3023 129.05 -183.758 189.1706 237.1058 113.5024 75.66824 465.1163 129.05 -183.715 189.1264 237.1079 113.4759 75.65057 470.9302 129.05 -183.671 189.0822 237.11 113.4493 75.63286 476.7442 129.05 -183.625 189.0378 237.1121 113.4227 75.61512 482.5581 129.05 -183.579 188.9933 237.1142 113.396 75.59734 488.3721 129.05 -183.531 188.9488 237.1166 113.3693 75.57952
494.186 129.05 -183.483 188.9042 237.1189 113.3425 75.56168 500 129.05 -183.433 188.8595 237.1214 113.3157 75.5438
500.5 129.05 -183.43 188.8877 237.1394 113.3326 75.55509 501 129.05 -183.428 188.9158 237.1463 113.3495 75.56633
501.5 129.05 -183.425 188.9438 237.1509 113.3663 75.57751 502 129.05 -183.422 188.9716 237.1542 113.383 75.58863
502.5 129.05 -183.419 188.9993 237.1568 113.3996 75.5997 505 129.05 -183.401 189.1355 237.1617 113.4813 75.65421
507.5 129.05 -183.382 189.2684 237.162 113.561 75.70736 510 129.05 -183.36 189.3979 237.1604 113.6388 75.75917
512.5 129.05 -183.336 189.5242 237.1579 113.7145 75.80967 515 129.05 -183.311 189.6472 237.1547 113.7883 75.85888
517.5 129.05 -183.284 189.7671 237.151 113.8603 75.90684 520 129.05 -183.257 189.8839 237.1469 113.9304 75.95357
522.5 129.05 -183.229 189.9977 237.1427 113.9986 75.99908 525 129.05 -183.2 190.1085 237.1382 114.0651 76.04342
527.5 129.05 -183.17 190.2165 237.1336 114.1299 76.08659 530 129.05 -183.14 190.3216 237.1288 114.1929 76.12863
532.5 129.05 -183.109 190.4239 237.1238 114.2543 76.16955 535 129.05 -183.078 190.5235 237.1185 114.3141 76.20938
537.5 129.05 -183.046 190.6204 237.1131 114.3722 76.24815 540 129.05 -183.015 190.7147 237.1076 114.4288 76.28586
542.5 129.05 -182.983 190.8064 237.102 114.4838 76.32255 545 129.05 -182.95 190.8956 237.0961 114.5374 76.35824
547.5 129.05 -182.918 190.9823 237.09 114.5894 76.39293 550 129.05 -182.886 191.0667 237.0838 114.64 76.42667
552.5 129.05 -182.853 191.1486 237.0775 114.6892 76.45945 555 129.05 -182.82 191.2283 237.071 114.737 76.49131
557.5 129.05 -182.788 191.3056 237.0643 114.7834 76.52225 560 129.05 -182.755 191.3808 237.0575 114.8285 76.55231
562.5 129.05 -182.722 191.4537 237.0505 114.8722 76.5815 565 129.05 -182.689 191.5246 237.0434 114.9147 76.60982
567.5 129.05 -182.656 191.5933 237.0361 114.956 76.63731 570 129.05 -182.623 191.66 237.0287 114.996 76.66398
572.5 129.05 -182.59 191.7246 237.0211 115.0348 76.68985 575 129.05 -182.557 191.7873 237.0134 115.0724 76.71492
577.5 129.05 -182.524 191.8481 237.0055 115.1088 76.73922 580 129.05 -182.491 191.9069 236.9975 115.1441 76.76276
582.5 129.05 -182.458 191.9639 236.9894 115.1783 76.78556 585 129.05 -182.425 192.0191 236.9811 115.2115 76.80764
587.5 129.05 -182.392 192.0725 236.9726 115.2435 76.829 590 129.05 -182.359 192.1241 236.9641 115.2745 76.84966
592.5 129.05 -182.326 192.1741 236.9553 115.3044 76.86963 595 129.05 -182.293 192.2223 236.9465 115.3334 76.88894
597.5 129.05 -182.26 192.269 236.9375 115.3614 76.90758 600 129.05 -182.227 192.314 236.9284 115.3884 76.92558
204
Pin COPte COPT COPnet Qc+Qaes Qe +Qdes %Error COPads 160.4818 0.476139 0.804141 0.804141 365.9434 375.3973 2.583435 0.54476160.4812 0.476157 0.804144 0.804144 365.9455 375.3872 2.580078 0.544755160.4817 0.476173 0.804142 0.804142 365.9487 375.3789 2.576904 0.544748160.4823 0.476189 0.804139 0.804139 365.9521 375.3718 2.574036 0.54474160.4827 0.476204 0.804136 0.804136 365.9553 375.3657 2.571473 0.544732
160.483 0.47622 0.804135 0.804135 365.9583 375.3604 2.569172 0.544725160.4832 0.476237 0.804134 0.804134 365.9612 375.3557 2.567088 0.544719160.4832 0.476253 0.804134 0.804134 365.9639 375.3515 2.565181 0.544713160.4832 0.47627 0.804134 0.804134 365.9665 375.3477 2.563423 0.544707160.4831 0.476286 0.804134 0.804134 365.969 375.3443 2.561791 0.544701
160.483 0.476302 0.804135 0.804135 365.9714 375.3413 2.560267 0.544695160.4829 0.476318 0.804135 0.804135 365.9738 375.3385 2.55884 0.54469160.4828 0.476334 0.804136 0.804136 365.9762 375.336 2.5575 0.544684160.4827 0.476349 0.804136 0.804136 365.9785 375.3338 2.556239 0.544679160.4826 0.476364 0.804137 0.804137 365.9808 375.3318 2.555051 0.544674160.4826 0.476434 0.804137 0.804137 365.9919 375.3248 2.550057 0.544648
160.483 0.476494 0.804135 0.804135 366.0023 375.322 2.546361 0.544624160.4841 0.476545 0.804129 0.804129 366.0121 375.3223 2.543684 0.544602160.4858 0.476588 0.804121 0.804121 366.0213 375.3248 2.541803 0.544581
160.488 0.476621 0.80411 0.80411 366.0299 375.329 2.540529 0.544561160.4907 0.476645 0.804096 0.804096 366.0379 375.3342 2.539703 0.544543
160.494 0.476662 0.80408 0.80408 366.0453 375.3399 2.53919 0.544526160.4977 0.47667 0.804061 0.804061 366.0522 375.3458 2.538871 0.54451
160.502 0.47667 0.80404 0.80404 366.0585 375.3514 2.538649 0.544495160.5067 0.476663 0.804016 0.804016 366.0643 375.3566 2.538438 0.544482160.5118 0.476649 0.803991 0.803991 366.0696 375.3611 2.538168 0.54447160.5174 0.476628 0.803963 0.803963 366.0744 375.3646 2.537777 0.544459160.5233 0.4766 0.803933 0.803933 366.0788 375.367 2.537215 0.544449160.5297 0.476566 0.803901 0.803901 366.0827 375.3681 2.53644 0.54444160.5364 0.476526 0.803868 0.803868 366.0861 375.3679 2.535415 0.544432160.5435 0.47648 0.803832 0.803832 366.0892 375.3663 2.534112 0.544425
160.551 0.476428 0.803795 0.803795 366.0919 375.3631 2.53248 0.544418160.559 0.47637 0.803754 0.803754 366.0945 375.3585 2.530498 0.544413
160.5675 0.476307 0.803712 0.803712 366.097 375.3525 2.528166 0.544407160.5765 0.476239 0.803667 0.803667 366.0993 375.3451 2.525492 0.544402160.5859 0.476166 0.80362 0.80362 366.1014 375.3362 2.52248 0.544397160.5956 0.476089 0.803571 0.803571 366.1034 375.3261 2.519135 0.544392160.6058 0.476007 0.80352 0.80352 366.1053 375.3146 2.515461 0.544388160.6163 0.475922 0.803468 0.803468 366.1071 375.3018 2.511465 0.544384160.6271 0.475833 0.803414 0.803414 366.1088 375.2877 2.50715 0.54438160.6382 0.475741 0.803358 0.803358 366.1103 375.2723 2.502523 0.544376160.6496 0.475645 0.803301 0.803301 366.1118 375.2558 2.497588 0.544373160.6613 0.475546 0.803242 0.803242 366.1131 375.238 2.492352 0.54437160.6733 0.475443 0.803183 0.803183 366.1144 375.219 2.486821 0.544367160.6855 0.475338 0.803121 0.803121 366.1155 375.1988 2.480999 0.544364
160.698 0.47523 0.803059 0.803059 366.1166 375.1776 2.474892 0.544362160.7107 0.47512 0.802996 0.802996 366.1175 375.1552 2.468507 0.54436160.7236 0.475007 0.802931 0.802931 366.1184 375.1317 2.461848 0.544358160.7367 0.474892 0.802866 0.802866 366.1192 375.1071 2.454921 0.544356
160.75 0.474774 0.8028 0.8028 366.1199 375.0815 2.447731 0.544354160.7634 0.474655 0.802732 0.802732 366.1205 375.0549 2.440284 0.544353160.7771 0.474533 0.802664 0.802664 366.1211 375.0273 2.432584 0.544351160.7909 0.474409 0.802595 0.802595 366.1216 374.9987 2.424636 0.54435
205
160.8049 0.474283 0.802525 0.802525 366.122 374.9691 2.416446 0.544349160.819 0.474156 0.802455 0.802455 366.1223 374.9386 2.408018 0.544349
160.8333 0.474027 0.802384 0.802384 366.1226 374.9072 2.399356 0.544348160.8477 0.473896 0.802312 0.802312 366.1228 374.8748 2.390465 0.544348160.8622 0.473764 0.802239 0.802239 366.123 374.8416 2.381347 0.544347160.8769 0.47363 0.802166 0.802166 366.123 374.8075 2.372008 0.544347160.8916 0.473495 0.802093 0.802093 366.1231 374.7725 2.362451 0.544347160.9065 0.473359 0.802019 0.802019 366.123 374.7367 2.352679 0.544347160.9215 0.473221 0.801944 0.801944 366.1229 374.7001 2.342695 0.544347160.9365 0.473082 0.801869 0.801869 366.1228 374.6626 2.332502 0.544348160.9519 0.472942 0.801793 0.801793 366.1227 374.6243 2.322054 0.544348160.9675 0.472799 0.801715 0.801715 366.1229 374.585 2.311292 0.544347160.9834 0.472655 0.801635 0.801635 366.1231 374.5446 2.300167 0.544347160.9996 0.47251 0.801555 0.801555 366.1235 374.5027 2.288619 0.544346
161.016 0.472363 0.801473 0.801473 366.124 374.4591 2.276594 0.544345161.0325 0.472215 0.801391 0.801391 366.1245 374.4138 2.264057 0.544344161.0493 0.472066 0.801307 0.801307 366.1252 374.3666 2.250982 0.544342161.0663 0.471915 0.801223 0.801223 366.1259 374.3174 2.237345 0.54434161.0834 0.471764 0.801138 0.801138 366.1268 374.2663 2.223139 0.544338161.1008 0.471611 0.801051 0.801051 366.1277 374.213 2.20835 0.544336161.1183 0.471457 0.800964 0.800964 366.1286 374.1578 2.19298 0.544334
161.136 0.471302 0.800876 0.800876 366.1297 374.1004 2.177023 0.544332161.1538 0.471146 0.800788 0.800788 366.1309 374.0411 2.160487 0.544329161.1718 0.470989 0.800698 0.800698 366.1321 373.9797 2.143379 0.544326
161.19 0.470832 0.800608 0.800608 366.1333 373.9163 2.125704 0.544323161.2083 0.470673 0.800517 0.800517 366.1347 373.8509 2.107474 0.54432161.2267 0.470514 0.800426 0.800426 366.1361 373.7836 2.088696 0.544317161.2453 0.470354 0.800333 0.800333 366.1376 373.7144 2.069377 0.544314
161.264 0.470193 0.800241 0.800241 366.1391 373.6433 2.049546 0.54431161.2828 0.470031 0.800147 0.800147 366.1408 373.5705 2.029196 0.544306161.3018 0.469869 0.800053 0.800053 366.1424 373.4959 2.008349 0.544302161.3208 0.469706 0.799959 0.799959 366.1442 373.4195 1.987025 0.544299
161.34 0.469542 0.799864 0.799864 366.146 373.3415 1.965222 0.544294161.3593 0.469378 0.799768 0.799768 366.1478 373.2619 1.942963 0.54429161.3787 0.469213 0.799672 0.799672 366.1497 373.1807 1.920259 0.544286161.3982 0.469048 0.799575 0.799575 366.1517 373.098 1.897118 0.544281161.4178 0.468882 0.799478 0.799478 366.1537 373.0139 1.873568 0.544277161.4376 0.468715 0.79938 0.79938 366.1558 372.9282 1.849602 0.544272161.4573 0.468548 0.799282 0.799282 366.1579 372.8412 1.825266 0.544267161.4771 0.468381 0.799185 0.799185 366.16 372.7529 1.800562 0.544262161.4969 0.468214 0.799086 0.799086 366.1621 372.6632 1.77549 0.544257161.5169 0.468046 0.798988 0.798988 366.1642 372.5723 1.750062 0.544252
161.537 0.467877 0.798888 0.798888 366.1666 372.4802 1.724258 0.544247161.5573 0.467708 0.798788 0.798788 366.1689 372.3869 1.698125 0.544242161.5776 0.467539 0.798688 0.798688 366.1714 372.2926 1.671679 0.544236161.5843 0.467589 0.798654 0.798654 366.1894 372.3182 1.673673 0.544195
161.58 0.467671 0.798676 0.798676 366.1963 372.3435 1.678656 0.544179161.5734 0.46776 0.798708 0.798708 366.2009 372.3686 1.684229 0.544168161.5656 0.467851 0.798747 0.798747 366.2042 372.3934 1.690072 0.544161161.5571 0.467944 0.798789 0.798789 366.2068 372.4179 1.69607 0.544155161.5074 0.468426 0.799034 0.799034 366.2117 372.5368 1.727192 0.544144161.4547 0.468908 0.799296 0.799296 366.212 372.6499 1.75797 0.544143161.4013 0.469384 0.79956 0.79956 366.2104 372.7576 1.787801 0.544146161.3482 0.469851 0.799823 0.799823 366.2079 372.8602 1.816525 0.544152
206
161.2958 0.470309 0.800083 0.800083 366.2047 372.9581 1.844157 0.54416161.2441 0.470757 0.800339 0.800339 366.201 373.0516 1.870725 0.544168161.1934 0.471195 0.800591 0.800591 366.1969 373.141 1.896258 0.544177160.8218 0.47435 0.802441 0.802441 366.1576 373.7292 2.067834 0.544268160.7794 0.474704 0.802653 0.802653 366.152 373.7889 2.085741 0.544281160.7378 0.475048 0.80286 0.80286 366.1461 373.846 2.102961 0.544294160.6971 0.475385 0.803064 0.803064 366.14 373.9004 2.11952 0.544308160.6572 0.475713 0.803263 0.803263 366.1338 373.9523 2.135415 0.544322
160.618 0.476033 0.803459 0.803459 366.1275 374.0017 2.15068 0.544337160.5797 0.476345 0.803651 0.803651 366.121 374.0487 2.16533 0.544352160.5421 0.476649 0.803839 0.803839 366.1143 374.0933 2.179382 0.544367160.5052 0.476946 0.804024 0.804024 366.1075 374.1357 2.19285 0.544383
160.469 0.477235 0.804205 0.804205 366.1005 374.1758 2.205753 0.544399160.4336 0.477517 0.804383 0.804383 366.0934 374.2137 2.218113 0.544415160.3988 0.477792 0.804557 0.804557 366.0861 374.2495 2.229923 0.544432160.3647 0.47806 0.804728 0.804728 366.0787 374.2833 2.241204 0.544449160.3313 0.478321 0.804896 0.804896 366.0711 374.3149 2.251973 0.544466160.2985 0.478575 0.805061 0.805061 366.0634 374.3446 2.262238 0.544484160.2663 0.478823 0.805222 0.805222 366.0555 374.3724 2.272018 0.544502160.2347 0.479064 0.805381 0.805381 366.0475 374.3982 2.281322 0.544521160.2038 0.479299 0.805537 0.805537 366.0394 374.4222 2.290154 0.544539160.1734 0.479528 0.805689 0.805689 366.0311 374.4444 2.298528 0.544558160.1436 0.479751 0.805839 0.805839 366.0226 374.4648 2.306459 0.544578160.1144 0.479967 0.805986 0.805986 366.0141 374.4834 2.313938 0.544597160.0857 0.480178 0.806131 0.806131 366.0053 374.5004 2.321006 0.544617160.0576 0.480383 0.806272 0.806272 365.9965 374.5156 2.327653 0.544638
160.03 0.480582 0.806412 0.806412 365.9875 374.5292 2.333878 0.544658160.0029 0.480776 0.806548 0.806548 365.9784 374.5413 2.339704 0.544679
207
Appendix G
Design of an electro-adsorption chiller (EAC)
(1) Amount of silica gel:
When the amount of adsorbent (here silica gel) is very small, heat transfer in the
reactor or bed becomes so rapid that the temperature difference between the hot and
cold beds reaches its maximum value within a very short period and the amount of
refrigerant circulated through the system is small. So, the evaporator temperature
increases, which is shown in the region (1) of Figure G1. Region (3) of Figure G1
corresponds to relatively heavy heat exchangers, where increasing the adsorbent mass
results in reducing the cooling capacity and increasing the evaporator temperature.
Region (2) is effective as the cooling capacity increases or the load temperature
decreases. From the simulation, the optimum amount of silica gel is selected as 300
gm. This amount of silica gel is valid for 120 W at the evaporator. The amount of
Silica gel depends on the applied cooling load. On the basis of the amount of
adsorbent, the size of heat exchanger is designed.
208
(2) Bed Design:
The size of the bed is designed according to the amount of silica gel and cooling load
of the evaporator.
Figure G1: The load surface temperature and the evaporator temperature as a function of silica gel mass for the same input power to the thermoelectric modules (voltage 24 volts and the cycle average current 6.8 amp, cycle time 500 seconds, input power to the evaporator is 5 watt per cm2)
0
10
20
30
40
50
60
70
80
0 100 200 300 400 500 6000
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
Mass of silica gel (gm)
Tem
pera
ture
(C
)
Coe
ffic
ient
of p
erfo
rman
ce
Optimum
(1)
(2)(3)
Copper plate
40 mm × 40 mm × 1 mm depth
Dia 220 mm
Dia 180 mm
135
209
Heat exchanger inside the bed
Heat exchanger (top vies)
Copper tube (1/4” dia)
Fin (thickness 1 mm)
Flexible hose (3/8” dia)
135 mm
Heat exchanger (Isometric view)
32 mm
130 mm
Flexible hose (DN 10)
3 mm
DN 10 centering ring with O ring
¼” cu-tube
Heat exchanger (Front view)
3/8” comp fitting
210
Cover of the bed
75 mm
DN 10 short pipe socket
20 mm
PTFE Chamber (Front view)
φ 220 mm
¼”
φ 25 mm
3 × DN 10 short pipe 2 × DN 16
pipe socket
PTFE Chamber (Top view)
φ 180 mm
211
O’ ring (viton) and PTFE Bed (3-D view)
DN 200 viton O’ ring
Bed
Dia 200 mm Dia 180 mm
Bed for adsorption or desorption
212
(2) Evaporator Design (mat S. steel):
The size of the evaporator depends on the cooling load and the amount of refrigerant
for pool boiling. (the amount of refrigerant hence water must be higher than the total
amount of adsorbate during adsorption)
Size DN 100 Dia 25 mm
4 × DN 10 short pipe socket
Blanking flange (DN 100) for the top cover of the evaporator
DN 100 centering ring
Dia 120 mm
Dia 70 mm Quartz view port
Bottom part of the Evaporator
70 100 mm
DN 25 for window
Body of the Evaporator
213
DN 10 Tee
DN 16 clamping ring
DN 100 centering ring
2 × DN 25 window view port
DN 16 flexible bellow to hold the
copper foam
DN 10 short pipe socket li id
52 mm × 52 mm × 30 mm
Evaporator Design
3-D view of the Evaporator
214
(4) Design of a Kaleidoscope for uniform cooling load:
Kaleidoscope is designed on the basis of cooling load at the evaporator. From the
infra-red radiant heater (4 kW), radiation heat is developed and transferred to the
evaporator through the kaleidoscope and the quartz.
500 mm
500 mm
52 mm × 52 mm
IR heater (4 kW and 1500 K)
Mat: S. steel
Kaleidoscope
215
Appendix H The Transmission band of the fused silica or quartz
216
Appendix I Calibration certificates
217
218