Heat and Mass Transfer in Bubble Column
Dehumidifiers for HDH Desalination
by
Emily Winona Tow
S.B., Massachusetts Institute of Technology (2012)
Submitted to the Department of Mechanical Engineering
in partial fulfillment of the requirements for the degree of
Master of Science in Mechanical Engineering
at the
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
February 2014
c Massachusetts Institute of Technology 2014. All rights reserved.
Author . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Department of Mechanical Engineering
January 17, 2014
Certified by . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
John H. Lienhard V
Samuel C. Collins Professor of Mechanical Engineering
Thesis Supervisor
Accepted by . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
David E. Hardt
Chairman of Graduate Studies
Department of Mechanical Engineering
2
Heat and Mass Transfer in Bubble Column Dehumidifiers for
HDH Desalination
by
Emily Winona Tow
Submitted to the Department of Mechanical Engineering
on January 17, 2014, in partial fulfillment of the
requirements for the degree of
Master of Science in Mechanical Engineering
Abstract
Heat and mass transfer processes governing the performance of bubble dehumidifier trays are studied in order to develop a predictive model and design rules for
efficient and economical design of bubble column dehumidifiers for humidificationdehumidification (HDH) systems. As a result of their high heat transfer coefficients
and large interfacial areas, bubble columns are an inexpensive and compact solution
for dehumidification in HDH, which has promising applications in small-scale desalination and industrial water remediation. Performance parameters for dehumidifier
design for HDH, including a device-specific parallel-flow effectiveness, are explained.
A new model for the performance of single bubble trays is developed based on the
rapid mixing in the column and the approximation of negligible gas-side resistance.
An experiment is performed to measure the heat transfer coefficients outside cooling coils in shallow bubble columns, in which geometric parameters including liquid
height and cylinder diameter, height, and horizontal position relative to the sparger
orifices are varied. The highest heat transfer coefficients are recorded on cylinders
placed in the coalescing region and directly above the sparger orifices. Heat flux and
parallel-flow effectiveness of a bubble column dehumidifier are investigated experimentally to validate the model, which predicts the heat transfer rate well with an
average absolute error of <3%. The independence of heat flux and effectiveness from
liquid depth supports the assumption of negligible gas-side resistance to heat and
mass transfer. Despite the mass exchange, the bubble column dehumidifier performs
like a typical heat exchanger: the heat flux decreases and effectiveness increases with
increasing coil area. The results of this study enable modeling and design of bubble
column dehumidifiers for high heat recovery and low capital cost.
Thesis Supervisor: John H. Lienhard V
Title: Samuel C. Collins Professor of Mechanical Engineering
3
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Acknowledgments
I would like to first acknowledge my advisor, Professor John H. Lienhard V, who has
provided me with direction, encouragement and support whenever I needed it, and let
me do my own thing whenever I didn’t. I greatly appreciate his patience and positive
attitude. I feel that I could not ask for a better advisor.
I would like to thank my parents, Lois and Bruce. When I sent home papers I
had written, I equally appreciated the words of encouragement from my mom and
the discovery of typos by my dad, who tries to follow the math. They have always
made me feel loved and supported.
I must thank Charles for being a phenomenal boyfriend. He has patiently listened
to so much of my heat transfer blather over the years that I think he’d have a good
shot at the qualifying exam. He has supported me during difficult times and made
the rest of the time a lot of fun.
I want to thank Jessie, Betsy, Katharine, Sara and Mollie for being my long-time
friends. <3 ! I should also thank the good people of Beast and the #angrydome for
putting up with me all these years.
I would like to acknowledge the Lienhard Research Group for being fun, but also
being supportive in a way I didn’t expect. I appreciate that we not only celebrate our
successes but support one another through our failures. I would also like to acknowledge Immanuel David Madukauwa-David for his hard work refining the experiment
described in Chapter 3.
I am thankful to all of my teachers and mentors, and I would like to name just a
few who have helped shape my path to the thermal sciences over the last ten years.
At The Urban School, my art teacher Kate Randall encouraged me to make a lot of
work. Her guidance and example have shaped both my art and research practices.
Professor Brisson might not know that his 2.005 lectures inspired my love for the
thermal sciences and his brutal exams boosted my confidence as an engineer. My
undergraduate advisor, Professor Lermusiaux, gave me the incredible opportunity to
TA my favorite class. Finally, I have to thank John Paschkewitz for pushing the
limits of my abilities during my two summers at PARC.
Finally, I would like to thank those who have provided funding for me to pursue
this research with the right combination of direction and freedom. I would like to
acknowledge the King Fahd University of Petroleum and Minerals through the Center
for Clean Water and Clean Energy at MIT and KFUPM (Project #R4-CW-08) for
providing both direction and support in my research. I also like to acknowledge
the Flowers Family Fellowship, the Pappalardo Fellowship, and the National Science
Foundation Graduate Research Fellowship Program under Grant No. 1122374 for
giving me the freedom to pursue projects and directions of my own choosing.
5
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Contents
1 Introduction
1.1 Dehumidification for HDH . . . . . . . . . . . . . . . . . . . . . . . .
1.1.1 Dehumidifier Types . . . . . . . . . . . . . . . . . . . . . . . .
1.1.2 Bubble Column Dehumidifier Performance Parameters . . . .
19
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2 Thermodynamic Model of a Dehumidifying Bubble Tray
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2.1 Heat and Mass Exchanger Model . . . . . . . . . . .
2.2.2 Heat and Mass Transfer Coefficients . . . . . . . . . .
2.2.3 Equivalent Length and Perimeter . . . . . . . . . . .
2.2.4 Mass Fraction Profile . . . . . . . . . . . . . . . . . .
2.2.5 Temperature Profile . . . . . . . . . . . . . . . . . .
2.2.6 Mean Temperature Difference . . . . . . . . . . . . .
2.3 Chapter Conclusions . . . . . . . . . . . . . . . . . . . . . .
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3 Heat Transfer to Horizontal Cylinders in Bubble Trays
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2 Background . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2.1 Existing Heat Transfer Coefficient Correlations . . .
3.2.2 Bulk Flow Regimes . . . . . . . . . . . . . . . . . .
3.2.3 Column Regions . . . . . . . . . . . . . . . . . . . .
3.3 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.3.1 Heat Transfer Coefficient Probe Design . . . . . . .
3.3.2 Fixture Design . . . . . . . . . . . . . . . . . . . .
3.3.3 Experimental Protocol . . . . . . . . . . . . . . . .
3.3.4 Probe Validation . . . . . . . . . . . . . . . . . . .
3.4 Results and Discussion . . . . . . . . . . . . . . . . . . . .
3.4.1 Comparison with Existing Correlations . . . . . . .
3.4.2 Cylinder Diameter . . . . . . . . . . . . . . . . . .
3.4.3 Column Region . . . . . . . . . . . . . . . . . . . .
3.4.4 Flow Regime . . . . . . . . . . . . . . . . . . . . .
3.4.5 Cylinder Height . . . . . . . . . . . . . . . . . . . .
3.4.6 Bubble Impact . . . . . . . . . . . . . . . . . . . .
3.4.7 Empirical Correlation . . . . . . . . . . . . . . . . .
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3.5
3.4.8 Pressure Drop . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.4.9 Design Recommendations . . . . . . . . . . . . . . . . . . . .
Chapter Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . .
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4 Experiments and Modeling of Single-Tray Bubble Column Dehumidifier Performance
73
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
4.2 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
4.2.1 Bubble Size . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
4.2.2 Bubble-Side Resistance . . . . . . . . . . . . . . . . . . . . . . 76
4.2.3 Heat and Mass Transfer Model . . . . . . . . . . . . . . . . . 79
4.2.4 Parallel-Flow Effectiveness . . . . . . . . . . . . . . . . . . . . 83
4.3 Experimental Methods . . . . . . . . . . . . . . . . . . . . . . . . . . 84
4.3.1 Experimental Bubble Column Dehumidifier . . . . . . . . . . 84
4.3.2 Controlling Bubble-on-Coil Impact . . . . . . . . . . . . . . . 86
4.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
4.4.1 Model Agreement . . . . . . . . . . . . . . . . . . . . . . . . . 87
4.4.2 Coil Length . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
4.4.3 Moist Air Temperature . . . . . . . . . . . . . . . . . . . . . . 90
4.4.4 Liquid Height, Sparger Orifice Size, and Bubble-on-Coil Impact 90
4.4.5 Air Gap Heat Transfer . . . . . . . . . . . . . . . . . . . . . . 95
4.4.6 Additional Modeling Results . . . . . . . . . . . . . . . . . . . 97
4.4.7 Modified Model Incorporating Experimental Outside-Coil Heat
Transfer Coefficients . . . . . . . . . . . . . . . . . . . . . . . 99
4.5 Chapter Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
5 Conclusions
5.1 Design for Effective Transport . . . . . .
5.2 Future Work . . . . . . . . . . . . . . . .
5.2.1 Crystallization with HDH . . . .
5.2.2 Solar Heating and Humidification
5.2.3 Dehumidifier Optimization . . . .
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101
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102
102
A Uncertainty analysis of heat transfer coefficient probes
103
B Bubble Column Dehumidifier Model for EES
105
8
List of Figures
1-1 A simple CAOW HDH cycle . . . . . . . . . . . . . . . . . . . . . . .
1-2 Schematic diagram of a single-tray bubble column dehumidifier . . . .
1-3 Multi-tray bubble column dehumidifier designed by G. P. Narayan and
coworkers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1-4 Performance considerations for a bubble column dehumifier for HDH
20
22
2-1 Bubble column dehumidifier . . . . . . . . . . . . . . . . . . . . . . .
2-2 Resistance network model, with temperatures (T), concentrations (C),
and resistances (R) . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2-3 Conservation of energy for air stream with condensation occurring just
outside the control volume . . . . . . . . . . . . . . . . . . . . . . . .
2-4 Conservation of energy on a differential control volume of moist air .
2-5 Dimensionless temperature profile . . . . . . . . . . . . . . . . . . . .
28
3-1 The three heat transfer coefficient probes . . . . . . . . . . . . . . . .
3-2 Schematic diagram showing the heat transfer coefficient probe construction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3-3 Schematic diagram showing the embedding of thermocouples in the
copper tube wall of the heat transfer coefficient probe . . . . . . . . .
3-4 Experimental apparatus: 1. Pressurized dry air inlet; 2. Rotameter
(4-40 cfm); 3. Rotameter (0.4-4 cfm); 4. Tank; 5. Orifice plate sparger;
6. Heat transfer coefficient probe; 7. Thermocouple; 8. Variable autotransformer; 9. Data acquisition unit . . . . . . . . . . . . . . . . . .
3-5 Empty bubble column with a heat transfer coefficient probe secured to
the sparger plate . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3-6 Drawing of the sparger plate with sixteen 3 mm sparger orifices (uncolored). Red fill indicates holes used to hold the probe, and light blue
indicates those used to secure the sparger plate. . . . . . . . . . . . .
3-7 Probe validation in horizontal natural convection in water . . . . . .
3-8 All heat transfer coefficient measurements. The key gives values of the
many variables tested as follows: [probe size (S/M/L)] [impact (Y/N)]
[probe height in cm]/[liquid depth in cm]. “s” denotes that the liquid
was filled to just barely cover the probe . . . . . . . . . . . . . . . . .
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3-9 Experimental data for heat transfer coefficient as a function of superficial velocity over a range of liquid depths are presented along with
several correlations. These results were gathered with the 4.76 mm
probe at a height of 2 cm with bubble-on-coil impact . . . . . . . . .
3-10 Experimental data for heat transfer coefficient as a function of superficial velocity over a range of probe heights and liquid depths are
presented along with several correlations. These results were gathered
using the 9.53 mm probe with impact except where noted. . . . . . .
3-11 Heat transfer coefficient compared to superficial velocity for the three
probe diameters with and without impact . . . . . . . . . . . . . . .
3-12 Results for the three probes presented on the same axes: heat transfer
coefficient in the bulk of the fluid and in the coalescing region, with
and without impact. In each case the height was 2 cm; the region was
changed by varying the liquid depth . . . . . . . . . . . . . . . . . . .
3-13 Regime map for the experimental column showing primary dependence
on liquid depth and secondary dependence on superfical velocity . . .
3-14 Swirl types observed in a short rectangular bubble column, top to
bottom: longitudinal-axis, vertical-axis, and circumferential-axis swirl.
3-15 Swirling regime: clockwise vertical-axis swirl captured with a long exposure to show bubble trajectories . . . . . . . . . . . . . . . . . . . .
3-16 Splashing regime, showing both liquid filaments and drops . . . . . .
3-17 Sloshing regime: images taken 1/4 second apart illustrating sloshing
along the tank’s shortest length . . . . . . . . . . . . . . . . . . . . .
3-18 The heat transfer coefficient varies slightly with changes in flow regime.
These measurements used the 9.5 mm probe at a height of 2 cm . . .
3-19 Heat transfer coefficients on the 9.53 mm probe with impact at a variety
of heights. The fluid is 2 cm over the top of the probe, placing the
probes in the bulk region . . . . . . . . . . . . . . . . . . . . . . . . .
3-20 Heat transfer coefficients in the coalescing region on the 9.53 mm probe
with impact at a variety of probe heights . . . . . . . . . . . . . . . .
3-21 Flow regime map, showing that the liquid depth at the onset of sloshing
is related to the critical height . . . . . . . . . . . . . . . . . . . . . .
3-22 In the coalescing region, the heat transfer coefficient is greater with
impact than without. These measurements were made with all three
probes at a height of 2 cm . . . . . . . . . . . . . . . . . . . . . . . .
3-23 At cylinder heights below 2 cm, impact causes the heat transfer coefficient to increase . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3-24 All experimental heat transfer coefficient measurements compared to
the empirical correlation (Equation 3.12), showing agreement within
about ±20% . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3-25 The gas pressure drop increases with liquid height and superficial velocity
3-26 The pressure drop is always greater than hydrostatic for columns up
to 10 cm in depth, and the difference increases with superficial velocity.
10
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3-27 The ratio of pressure drop to hydrostatic pressure drop, which decreases with liquid height and increases with gas velocity, shows that
the hydrostatic pressure drop assumption fails to estimate blowing
power at low liquid heights. . . . . . . . . . . . . . . . . . . . . . . .
3-28 The ratio of flow work dissipated in the column liquid to the assumed
gravitational potential energy dissipation rate used in dissipation-based
heat transfer theories is found to approach unity at low liquid heights.
3-29 Agreement between pressure drop measurements and Equation 3.14 .
4-1 Resistance network from [1] governing heat and mass transfer in a
bubble column . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4-2 Examples of bubble mushroom formation and departure in the sparger
region . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4-3 Formation of liquid sheets, filaments and drops inside a bubble . . . .
4-4 Simplified thermal resistance network . . . . . . . . . . . . . . . . . .
4-5 Experimental setup: (1) pressurized cooling water inlet, (2, 12) rotameters, (3) fresh water outlet valve, (4-8) thermocouples, (9) plate
sparger, (10) cooling coil, (11) air outlet, (13) cartridge sparger. (14)
resistance heater, (15) pressurized dry air inlet . . . . . . . . . . . . .
4-6 The sparger orifice configurations used to test the effect of bubble-oncoil impact without altering the coil for the small (2.8 mm) orifices .
4-7 Agreement between theoretical and experimental heat transfer rate .
4-8 Agreement between theoretical and experimental parallel-flow effectiveness. It is clear from the cluster around // = 0.85, corresponding
to the 67 cm coil, that the coil size all but determines the effectiveness.
4-9 The effect of coil length on heat flux . . . . . . . . . . . . . . . . . .
4-10 The effect of coil length on effectiveness . . . . . . . . . . . . . . . . .
4-11 The effect of moist air inlet temperature on heat flux . . . . . . . . .
4-12 The effect of moist air inlet temperature on effectiveness . . . . . . .
4-13 For 2.8 mm orifices and liquid height above 4 cm, effectiveness is independent of liquid height . . . . . . . . . . . . . . . . . . . . . . . . .
4-14 For 2.8 mm orifices and small bubbles, the effect of bubble-on-coil
impact on heat flux is small . . . . . . . . . . . . . . . . . . . . . . .
4-15 For 2.8 mm orifices and small bubbles, the effect of bubble-on-coil
impact on effectiveness is small . . . . . . . . . . . . . . . . . . . . .
4-16 The effect on effectiveness of bubble-on-coil impact and liquid height
for 6 mm orifices . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4-17 The percent of the total heat transfer occurring in the air gap increases
with the liquid side temperature pinch, TC − TE,o . . . . . . . . . . .
4-18 The heat flux decreases with increasing coolant temperature, as shown
for three air temperatures . . . . . . . . . . . . . . . . . . . . . . . .
4-19 The effectiveness is nearly constant with changing water temperature,
as shown for three air temperatures. The vertical axis is expanded
to show that there is, however, a slight decrease in effectiveness with
increasing coolant temperature . . . . . . . . . . . . . . . . . . . . . .
11
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4-20 Decreasing the tube diameter at constant coil surface area leads to an
increase in effectiveness which is more pronounced for smaller coils . . 98
4-21 Agreement in heat transfer rate between modified model and experiment 99
4-22 Agreement in parallel-flow effectiveness between modified model and
experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
12
List of Tables
3.1
Selected heat transfer coefficient correlations . . . . . . . . . . . . . .
13
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14
Nomenclature
Roman symbols
A
Relevant area [m2 ]
Ac
Relevant cross-sectional area [m2 ]
C
Heat capacity flow rate [J/K-s]
cp
Specific heat at constant pressure [J/kg-K]
D
Diameter [m]
E
Energy [J]
ė
Specific flow work dissipation [W/kg]
g
Gravitational acceleration [m/s2 ]
H
Column liquid height, measured during bubbling [m]
Hp
Probe height (measured at center) [m]
h
Heat transfer coefficient [W/m2 K]
h(T ) Specific enthalpy [J/kg]
hf g
Latent heat of vaporization [J/kg]
K
Mass transfer coefficient [kg/m2 -s]
K∗
Dimensionless mass transfer coefficient [-]
k
Thermal conductivity [W/m-K]
L
Relevant length [m]
M
Molar mass [kg/kmol]
ṁ
Mass flow rate [kg/s]
m
Water vapor mass fraction [-] and fin parameter [m−1 ]
15
N
Number [-]
P
Perimeter [m]
p
Pressure [Pa]
Q̇
Heat transfer rate [W]
q̇
Heat flux [W/m2 ]
R
Thermal resistance [K/W] and specific gas constant [J/kg-K]
Re
Electrical resistance [Ω]
RF W Flow work ratio [-]
Rm
Mass transfer resistance [s/kg]
r
Radius [m]
T
Temperature [◦ C]
t
Time [s]
U
Overall heat transfer coefficient [W/m2 -K]
U∗
Dimensionless heat transfer coefficient [-]
u
Velocity [m/s]
V
Voltage [V]
V̇
Volumetric flow rate [m3 /s]
ug
Superficial gas velocity [m/s]
v
Velocity [m/s]
x
Water vapor mole fraction [-] and distance [m]
x∗
Dimensionless distance [-]
Z
Thickness ratio [-]
Greek symbols
∆
Mean difference
Θ
Dimensionless temperature difference [-]
α
Thermal diffusivity [m2 /s]
16
Effectiveness [-]
g
Gas holdup [-]
η
Characteristic eddy size
µ
Dynamic viscosity [Pa-s]
ν
Kinematic viscosity [m2 /s]
ρ
Density [kg/m3 ]
σ
Surface tension [N/m]
Subscripts
//
Parallel-flow
1
Generic begin state
2
Generic end state
A
Moist air stream
a
Dry air
atm
Atmospheric
ave
Average
B
Bubble inner surface
b
Bubble
C
Column fluid
c
Cross-sectional
coil
Coil and coolant fluid
cond Condensation
D
Coil metal
d
Distillate
da
Dry air
E
Coolant
E
Probe end cap
17
e
Entry
end
Probe end caps
f
Liquid water
g
Water vapor
h
Sparger orifice
i
In
l
Latent heat
LM
Log mean
ma
Moist air
max Maximum
meas Measured
o
Out
p
Probe
s
Sensible
sat
Saturation
turn Coil turn
w
Water vapor
Named dimensionless ratios
Fo
Fourier number αt/L2 [-]
Fr
Froude number = u2 /gD [-]
K
Dean number = Re (Di /Dturn )1/2 [-]
Lef
Lewis factor U/cp K [-]
Nu
Nusselt number = ht L/k [-]
Pr
Prandtl number = µcp /k [-]
Re
Reynolds number = ρuD/µ [-]
Re?
Modified Reynolds number [-]
St
Stanton number = ht /ρcp u [-]
18
Chapter 1
Introduction
Humidification-dehumidification (HDH) is a thermal desalination method with great
potential in decentralized and high-salinity desalination applications. Although HDH
often requires more energy than many popular processes such as reverse osmosis and
multi-stage flash, it has several advantages. HDH is adaptable to a wide range of water
conditions [2], has low maintenance cost [3] due to its uncomplicated design, and is
compatible with solar thermal energy and other low-temperature energy sources [4].
In its most basic form, a HDH system consists of a heater, a humidifier, a dehumidifier,
and the pumps and piping necessary to move fluid between components. Narayan et
al. [5] describe many HDH system configurations, including the closed-air-open-water
(CAOW) cycle shown in Figure 1-1,which is considered in this work.
Continued research on HDH has the potential to reduce the energy use and capital
cost of HDH desalination. Dehumidification technology, specifically, warrants further
study because the effectiveness of the dehumidifier dominates the energetic performance of the entire HDH system [6] and because the high resistance to diffusion of a
dilute vapor through air requires a large and potentially expensive condenser [7, 8].
This section will identify the unique needs of dehumidifiers in HDH and justify the
choice of bubble column dehumidifiers as the focus of this thesis.
1.1
Dehumidification for HDH
The present approach to modeling is guided by the unique needs of dehumidifiers for
use in HDH desalination. The key restriction in the design of a dehumidifier for HDH
is the need for heat recovery, which prompts the use of salt water as the coolant and
necessitates separation between the coolant and the condensing water. Naturally,
capital cost is another key design consideration.
1.1.1
Dehumidifier Types
Several dehumidification technologies with various applications are discussed in this
section. This list is by no means exhaustive, and excludes hybrid system types because
they would tend to undermine the simplicity of HDH.
19
Heater
Moist air
Humidifier
COLD
Dehumidifier
HOT
Dry air
Salt water
Fresh water
Figure 1-1: A simple CAOW HDH cycle
Sorbent
Sorbent dehumidifiers use either adsorption of absorption to dehumidify one air
stream while humidifiying another [9]. In adsorbent dehumidification, a rotating
drum is filled with a matrix of a solid dessicant such as silica gel. Absorbent dehumidifiers work similarly, but use a strong salt solution in place of a dessicant. Moist
air is dehumidified as it is blown through a section of the rotating drum, while heated
air is blown through the rest to evaporate the moisture and recharge the drum. This
type of dehumidifier requires external heat input for the recharging stream, and no
liquid water is produced unless the recharging air is recirculated via a condenser.
Refrigerant
Refrigerant dehumidification mimics air refrigeration except that the air is generally
returned it to its original temperature after being cooled and dried [9]. The most basic cycle consists of an evaporator, a condenser, a compressor, and expansion valve,
and a fan. In the evaporator, moist air is blown through a heat exchanger, causing
evaporation of the coolant and condensation of water from the moist air. The refrigerant is then compressed, and it enters the condenser, where it condenses and gives
up its latent heat to the cooled and dried air. The refrigerant is finally expanded
before being returned to the evaporator.
To dehumidify air and collect water for HDH, many people have used dehumidifiers
that are similar to refrigeration dehumidifiers but with salt water as the coolant
20
[10, 11, 12]. Because the coolant does not change phase, the configuration is simpler.
The evaporator is replaced by a shell and tube dehumidifier and cold seawater is run
through the tube. Neither stream is a closed loop. The lack of phase change increases
the thermal resistance inside the tube, but this resistance is still low compared to the
thermal resistance of condensation in the presence of a large fraction of noncondensible
gas outside the tube.
Bubble Column
A bubble column or sieve tray column can be used to reduce the overall resistance to
heat and mass transfer of dehumidification without raising the heat exchanger area.
When used in dehumidification, these reactors allow water to condense from moist air
on the outside of bubbles rather than on a solid surface. Bubbly flows can have very
high specific areas, so water can be condensed at a high rate in a small volume with
low temperature differences. Only a small coil is needed to cool the column while
preheating the salt water. Bubble column dehumidifiers have the potential to reduce
the capital cost of dehumidification by limiting the solid heat exchange surface to a
small coil.
In a bubble column dehumidifier, warm, moist air is bubbled though a column
of fresh water cooled by indirect heat exchange with salt water as shown in Figure
1-2. The concentration gradient from the warm bubble center to the cool bubble
surface drives condensation on the surface of the bubble. The presence of a large
fraction of non-condensible gas leads to a low condensation heat transfer coefficient.
However, the key advantage of the bubble column dehumidifier lies in moving this
resistive condensation process off an expensive solid surface and onto the surfaces
of the bubbles, which leads to a very low gas-side resistance. The heat leaving the
bubbles is then transferred to the saline water at a high heat transfer coefficient
through a coil with a small surface area. Dehumidification in bubble columns has
been shown to reduce device volume and condenser area by an order of magnitude
[13].
Sieve tray columns, in which a fluid flows down a series of trays as gas bubbles
up, are commonly used in distillation and other vapor-liquid reactions. Barrett and
Dunn [14] propose a model for a sieve tray column dehumidifier or humidifier. In
their dehumidifier, cold water enters the top tray and warm, moist air is bubbled in
from the bottom. Given a source of cold water, such a dehumidifier would require no
cooling coils and could be quite inexpensive. However, the cold water source in HDH
is saline water, and dehumidifying moist air from the humidifier by direct contact with
saline water in a tray column would not result in the production of any fresh water.
Sieve tray columns can be used for dehumidification in HDH only if the cooling water
is chilled by the seawater in a separate, indirect-contact heat exchanger. However,
this method does not take advantage of the bubble column’s very high heat transfer
coefficients in transferring heat to the salt water.
Narayan et al. [13] use a multi-stage bubble column dehumidifier for HDH desalination. A multi-stage or multi-tray bubble column, such as the one shown in Figure
1-3, is similar to a sieve tray column but includes coils which snake through the stages
21
Coolant in
Coolant out
Cool, dry air out
Sparger
Fresh
water
out
Warm, moist air in
Figure 1-2: Schematic diagram of a single-tray bubble column dehumidifier
to maintain separation between the direct-contact condensing liquid and the coolant
while making use of the high heat transfer coefficient outside the coils. Multi-stage
bubble columns enhance energy recovery by reducing the temperature drop between
stages (see [15]) and thereby reducing the entropy generated by mixing.
Bubble column dehumidifiers have advantages and disadvantages over shell-andtube dehumidifiers in terms of both entropy generation and cost. As demonstrated
by Kang et al. [16], who compare falling film and bubbling modes of ammonia-water
absorption (a similar heat and mass transfer process), the bubbling mode has significantly higher absorption rates and lower stream-to-stream temperature differences.
The increased heat and mass transfer coefficients and interfacial area in a bubble
column dehumidifier reduce the entropy generation, and the lower (solid) heat exchanger area requirement reduces cost. However, the mixing of streams at different
temperatures within each tray generates entropy. Adding trays has the potential to
reduce the entropy generation due to mixing, but adds additional complexity to the
manufacturing and maintenance of a multi-tray bubble column dehumidifier.
1.1.2
Bubble Column Dehumidifier Performance Parameters
Understanding key performance parameters of HDH systems and dehumidifiers in
particular enables more meaningful interpretation of experimental results and the
identification of directions for improvement. Performance parameters that give insight
into energy use and capital cost are particularly relevant to desalination systems.
Though it offers several advantages, HDH desalination is energy intensive relative
to common processes such as RO and MSF. The importance of energy consumption
22
Figure 1-3: Multi-tray bubble column dehumidifier designed by G. P. Narayan and
coworkers
23
Liquid-side ΔP
Condensation rate
Effectiveness
Gas-side ΔP
≈hydrostatic
Coil area
and cost
Figure 1-4: Performance considerations for a bubble column dehumifier for HDH
necessitates an energy-based performance parameter, the gained output ratio (GOR).
GOR is the ratio of the latent heat of the water produced to the heat input, as shown
in Equation 1.1.
ṁd hf g (Tatm )
(1.1)
GOR =
Q̇
In a thermal system, a GOR much larger than one indicates a high level of energy
recovery. As described in [17], GOR dictates the fuel use (or, in the case of solar
power, collector area) of a desalination system. GOR increases with the specific
entropy generation [17].
GOR is distinct from the efficiency of the system, which compares the actual work
to the least work of separation [18]. Unlike efficiency, GOR does not incorporate the
salinity of the feed. Therfore, GOR is more meaningful for thermal systems, in which
energy use is not a strong function of salinity.
Figure 1-4 illustrates the principal performance-related design considerations for a
bubble column dehumidifier for HDH. It is important to size a HDH system to have the
desired rate of fresh water production, which is equal to the rate of condensation in the
dehumidifier. High effectiveness is critical for high GOR. High heat flux through the
coil must be considered because of the high cost of copper, which causes the coil to be a
significant fraction of the system capital cost. It is important to minimize the pressure
drop in both the liquid and gas streams, as the work required to pressurize those
streams will add to the power consumption of the system. The gas-side pressure drop
includes the hydrostatic pressure drop through the column, so particular attention
will be given in this work to minimizing the height of the column liquid.
Carefully defining effectiveness is important for intuitively understanding column
24
performance. Narayan et al. [19] propose a model for the effectiveness of simultaneous heat and mass exchangers by which bubble columns can be compared to other
dehumidifier types, which are generally counterflow. Compared to counterflow dehumidifiers, the effectiveness of a single-stage bubble column dehumidifier is low because
the interaction of both the coil and bubble streams with the well-mixed column liquid
causes the device to function as if it is in parallel-flow regardless of the physical orientation of the streams. This leads to the need for multi-stage devices. Narayan and
Lienhard [11] demonstrate that combining bubble column stages at different liquid
temperatures into a multi-staged device with an overall counterflow configuration can
achieve effectiveness comparable to conventional dehumidifiers.
However, effective multi-staging requires each stage of the column to have a low enthalpy pinch [20]. As a performance parameter, enthalpy pinch represents an improvement over temperature pinch because of the nonlinearity of the enthalpy-temperature
curve of saturated moist air, but it is still a dimensional quantity. Good heat recovery requires that each stage achieve a large fraction of its maximum single-stage heat
transfer rate. Therefore, to compare the effects of design and operational parameters
of a single column stage, a parallel-flow effectiveness, // , is defined. Equation 1.2
gives // as a function of the actual and maximum possible heat transfer rates of a
single, well-mixed stage, as defined by Tow and Lienhard in [21]:
// =
Q̇
Q̇max//
(1.2)
Effectiveness as defined by [19], which gives insight into the energy use of HDH,
is a function of the number of stages (see [15]), the thermodynamic balancing (see
[20]), and, finally, the parallel-flow effectiveness of each stage. This work will focus
on a single stage of a bubble column dehumidifier and the modeling and experimental
validation of its performance in terms of both heat flux and parallel-flow effectiveness.
25
26
Chapter 2
Thermodynamic Model of a
Dehumidifying Bubble Tray
This chapter is based on a paper by Tow and Lienhard [1].
2.1
Introduction
The development of energy-efficient desalination technologies with low capital and
maintenance costs is critical to combating global fresh water scarcity. Humidificationdehumidification (HDH) is a promising desalination process because of its simple
system design and compatibility with low-grade energy [22, 4]. However, the high
cost of conventional dehumidifiers due to the large amount of copper required inhibits
the use of HDH in poor and remote regions where its low-tech nature could be most
useful.
Bubble column dehumidifiers reduce cost by moving the condensation process
from expensive copper plates to the inner surface of bubbles in fresh water. In a
bubble column dehumidifier, shown in Fig. 2-1, moist gas (usually air) is bubbled
through a sparger into a column of fresh water cooled by a small coil running cold
fluid (usually seawater). The high resistance to water vapor mass diffusion expected in
dehumidifiers due to the high concentration of non-condensible gasses [22] is overcome
by condensing on a very large surface area of bubbles. Heat transfer coefficients
between the liquid in the column and the cooling coil are so high that only a small
copper coil is needed, thereby reducing the cost of the dehumidifier dramatically [13].
Simple and accurate modeling of bubble column dehumidifiers (and bubble column
humidifiers, which may also prove useful in HDH) will enable optimization of column
designs for performance and cost. Developing algebraic equations such as the logmean temperature difference (LMTD) to model heat transfer driving forces in parallelflow and counterflow heat exchangers is useful because it eliminates the need for
integration of differential equations. However, the use of LMTD to approximate the
mean temperature difference relies on the assumption of constant heat capacity in
each stream and does not allow for the thermal energy left in the stream by warm
27
Figure 2-1: Bubble column dehumidifier
water vapor molecules diffusing down a temperature gradient coincident with the
concentration gradient. Mills provides a clear derivation of LMTD [23]. This paper
will follow a similar method to derive mean heat and mass transfer driving forces,
but will account for mass transfer and the resulting change in the heat capacity of
the moist air stream. Many authors have proposed mean temperature differences or
corrections to LMTD for other heat exchanger configurations [24].
Failure to recognize the assumptions made in the derivation of LMTD has led
some researchers to use it in heat and mass exchangers such as dehumidifiers. In
their model of a bubble column dehumidifier, Narayan et al. [13] used a single-stream
LMTD to model the sensible heat transfer driving force from the moist air stream
to the column fluid. Similarly, Chen et al. [25] used LMTD to model the sensible
heat transfer from the moist air in a plate-fin tube dehumidifier. This work will show
that although the standard LMTD is inappropriate for streams with mass exchange,
the error in sensible heat transfer predicted will be on the order of 10%. Since the
majority of the heat removed from the moist air is latent, the error in the total
predicted heat transfer rate due to modeling the moist air stream with LMTD is
small, but if possible, a simple algebraic equation for mean temperature difference in
a dehumidifier that accounts for mass transfer and the corresponding changes in heat
capacity flow rate should be employed.
Both Narayan et al. and Chen et al. used a log-mean humidity (in kg water/kg
dry air) difference to model the mass transfer driving force. Mills [26] uses a mass
fraction driving force for mass transfer in his model of a humidifier which leads to a
mass fraction profile similar to the one developed in this paper for a dehumidifier.
Experiments demonstrate that mixing in the bubble column ensures an essentially
uniform liquid temperature, so the bubble column will be modeled as two single
stream heat exchangers of equal heat transfer rate in contact with the isothermal
column liquid: the seawater side, for which LMTD is appropriate, and the gas side,
28
which has mass exchange. Under conditions typical of these systems, a log-mean
mass fraction difference will be shown to relate the latent heat transfer to the overall
mass transfer coefficient on the air side. An expression for the mean temperature
difference of the moist gas and an algebraic approximation will be presented. Given
knowledge of the heat and mass transfer coefficients of the bubbles and cooling coil,
the model developed in this paper enables calculation of the condensation rate, total
heat transfer rate, and temperature pinch of a single stage bubble column dehumidifier
or humidifier.
2.2
Theory
The dehumidifier will be modeled as two single-stream heat exchangers interacting
with the same isothermal stream, one of which has mass exchange. For the stream
with mass exchange, mean heat and mass transfer driving forces will be found following a method analogous to that used to derive LMTD [23]. The equations and
narrative will assume the device under consideration is a dehumidifier, but the model
applies equally to a humidifier as long as careful attention is paid to signs.
2.2.1
Heat and Mass Exchanger Model
The bubble column as a whole behaves like a parallel-flow device because both the
moist air and coolant streams interact with the column fluid, which is very wellmixed by the bubbles and can be treated as isothermal. Because the coil is small
compared to the volume of bubbles, it will be assumed that the bubbles do not have
significant thermal interactions with the coil that are unmediated by the column
fluid. Similarly, heat transfer between the air stream and the coil in the air gap
above the bubble column will be disregarded by this analysis [21]. Figure 2-2 shows a
simplified resistance network model of the system, where node B is the inner surface
of the bubble on the gas side, node C is the column fluid, D is the average tube
temperature, and A and E represent average stream temperatures of the bubbles and
coolant, respectively. For a steady pool temperature, both the sensible heat transfer
from A to B and the latent heat released by condensation at B are transferred through
the rest of the resistance network to the coolant. The total heat transfer into the
coolant fluid is
Q̇coil = ṁcoil cp,coil [Tcoil,o − Tcoil,i ],
(2.1)
assuming constant specific heat of the coolant liquid, and
Q̇coil = (U A∆TLM )coil
(2.2)
where (U A)coil is based on the forced convection both inside and outside the coil.
The LMTD for a single-stream heat exchanger with no mass exchange, and whose
29
CA
CB
Qcond
Rm
RAB
Bubble
RBC
A
B
RCD
C
TE
TD
TC
TB
TA
D
E
RDE
Coolant
Coil
Figure 2-2: Resistance network model, with temperatures (T), concentrations (C),
and resistances (R)
non-isothermal stream experiences a positive heat transfer is, as usual,
∆TLM =
To − Ti
.
C
ln TToi −T
−TC
(2.3)
It is assumed that the temperature difference across the thin boundary layer outside the bubble is very small compared to the temperature difference inside the bubble
because water has a much greater thermal conductivity and smaller thermal diffusivity (and thus much thinner boundary layer) than air. This will be discussed in
greater detail in the following section, but the result of this assumption is that the resistance between B and C can be neglected, and the moist air stream can be modeled
as interacting directly with the isothermal column fluid. This approximation greatly
simplifies modeling.
Applying conservation of energy to the entire air stream, as in Fig. 2-3,
0 = Q̇s + ṁda [hda (Ti ) − hda (To )] + ṁw,o [hw (Ti ) − hw (To )]
+ (ṁw,i − ṁw,o )[hw (Ti ) − hw (TC )],
(2.4)
and assuming constant specific heat capacities of the air and water vapor such that
hda (T1 ) − hda (T2 ) = cp,da [T1 − T2 ]
(2.5)
hw (T1 ) − hw (T2 ) = cp,w [T1 − T2 ],
(2.6)
and
30
IN
ha (Ti )
hw (Ti )
OUT
m a
ha (To )
m w,o
hw (To )
 w ,i  m
 w, o
m
hw (TC )
Q S
CONDENSING
Figure 2-3: Conservation of energy for air stream with condensation occurring just
outside the control volume
Eqn. (2.7) gives the sensible heat transfer rate into the moist air, which in the case
of a dehumidifier will be negative:
Q̇s = (ṁda cp,da + ṁw,o cp,o )(To − Ti )
+ cp,w (ṁw,i − ṁw,o )(TC − Ti ).
(2.7)
In this equation, the first righthand side term represents the sensible heat lost
by the moist air stream that passes through the column and the second represents
the sensible cooling of water vapor that diffuses to the liquid surface, at TC , and
condenses there. The latent heat of vaporization is not present in Eqn. (2.7) because
the heat released is absorbed on the liquid side of the bubble surface, which is not
part of the air stream. The latent heat transfer rate into the liquid can be computed
from the change in water vapor mass flow rate in the moist air stream, which is equal
to the rate of condensation:
Q̇l = hf g (ṁw,i − ṁw,o ) = hf g (ṁcond ).
(2.8)
Assuming no heat is lost to the environment, the total steady heat transfer rate
into the coolant is the sum of the latent and sensible heat transfers to the column
fluid:
Q̇coil = −Q̇s + Q̇l = ṁcoil cp,coil [Tcoil,o − Tcoil,i ].
(2.9)
2.2.2
Heat and Mass Transfer Coefficients
It is important to verify the assumption of constant heat and mass transfer coefficients
that will be employed in the driving force model. However, detailed modeling of
heat and mass transfer coefficients is beyond the scope of this chapter. Because
bubble columns have primarily been used for gas-liquid reactions where the mass
31
transfer is controlled by the diffusion of the gas into a liquid, many past studies
have addressed the heat and mass transfer coefficients outside a rising bubble and
neglected any resistance inside [27]. To show that the inner transfer coefficient can be
assumed constant for driving force modeling purposes, a scaling argument can be used
to approximate the entry length over which the heat and mass transfer coefficients
inside the bubble reach steady values. Inside the bubble, diffusion and convection
may both contribute to the heat and mass transfer, but a conservative estimate of
entry length will assume that no convection occurs (since convection would shorten
the entry length). Bubble velocity is estimated with Mendelson’s wave analogy to be
around vb = 0.2 m/s [28]. The bubble is within its entry length at short times, around
Fo≤ 0.2, when the thermal boundary layer inside the bubble is still developing [23].
Under typical conditions, the entry length for a bubble of diameter Db = 4 mm can
be approximated by Eqn. (2.10):
Le = vb t ≈
vb FoDb2
≈ 7 mm
4α
(2.10)
Since a typical bubble column is at least 150 mm deep to ensure immersion of the
cooling coil [13], the entry region is a sufficiently small fraction of the column that
the constant heat transfer coefficient assumption is appropriate. Assuming a Lewis
number of order 1 for the moist air, the mass diffusion entry length is comparable, so
a constant mass transfer coefficient can also be assumed.
The heat transfer coefficients inside and outside the coil can also be taken to be
constant along the length of the coil. In laminar flow, secondary flows induced by
the coil curvature significantly reduce the radial length scale for convection (see Mori
and Nakayama [29]) compared to a straight tube, thus shortening the thermal entry
length inside the coil. These secondary flows also significantly raise the inside heat
transfer coefficient above the straight pipe value, scaling as hDE ∼ (Dcoil /Dturn )1/4 ,
where Dturn is the diameter of coil winding. For example, the curved pipe Nusselt
number was nearly ten times the straight pipe value for the cooling coil used in the
bubble column dehumidifier tested in [21]. In turbulent flow, a short entry length
is expected regardless of coil curvature, though the curvature-induced augmentation
of the heat transfer coefficient does extend, to a lesser extent, into turbulent flow
[30]. Outside the coil, the heat transfer coefficient is expected to be approximately
constant so long as the flow conditions are consistent in the vicinity of the entire coil,
e.g. for a single loop placed centrally on a symmetrical sparger.
Estimating the heat and mass transfer coefficients inside and outside the bubble
will help verify the approximation of a negligible temperature gradient outside the
bubble. The bubbles are large enough that the bubble surface can be treated as free,
and the temperature profiles both inside and outside can be approximated as semiinfinite slabspmoving at the bubble terminal velocity. The thermal boundary layer
will grow as παx/vb . Using a characteristic length of the bubble diameter, the heat
transfer coefficient can then be approximated by conduction through the boundary
32
layer thickness as in Eqn. (2.11):
r
hAB ≈ k
vb
παDb
(2.11)
For typical dehumidifier operation temperatures and 4 mm bubbles, hAB ≈ 20
W/m2 -K and hBC ≈ 7000 W/m2 -K, confirming the assumption that hAB hBC .
Even considering that the heat transfer rate outside the bubble is greater due to the
latent heat transferred to the bubble surface, the heat transfer coefficient outside the
bubble should is so much greater that the temperature difference between B and C
can be neglected in the analysis of the mean heat and mass transfer driving forces.
2.2.3
Equivalent Length and Perimeter
For simplicity, the bubble stream will be modeled as a stream having an equivalent
length and perimeter. The equivalent length L is related to the superficial (ug ) and
terminal (vb ) velocities, gas holdup g and column liquid height H by Eqn. (2.12).
v b
g H
(2.12)
L=
ug
A wide array of experimental correlations for holdup can be found in the literature,
depending on the choice of gas and liquid, operating conditions, sparger design, and
column configuration [31].
The equivalent perimeter, P , which satisfies the relationship P L = A , where A
is the total surface area of bubbles entrained in the column, is
P =
6V̇ma,i
,
vb D
(2.13)
assuming spherical bubbles, a negligible change in bubble surface area due to vapor
condensation and temperature change, and a nearly constant rise velocity. The density of the moist air can be calculated by assuming an ideal mixture of air and water
vapor.
In high orifice velocity gas sparging, bubbles will be neither spherical nor uniform
in size, and correlations from the literature for interfacial area should be used to
compute the effective perimeter [32].
2.2.4
Mass Fraction Profile
The condensation rate is regulated by diffusion of water vapor through the moist air
to the bubble surface, which is assumed to have the temperature of the column fluid.
The partial pressure of water vapor at the bubble surface is equal to the saturation
pressure at that temperature. It will be assumed that no mist forms inside the bubbles
and that all condensation occurs at the bubble surface.
33
Mass transfer is examined through a differential control volume of length dx with
a mass fraction-based mass transfer coefficient K with units of [kg/m2 -s] such that:
dṁcond (x) = KP [m(x) − mC ]dx.
(2.14)
In Eqn. (2.14), a dilute mixture of water vapor in air is assumed such that a mass
fraction difference can represent the mass transfer driving force, as in Mills’ humidifier
model [26]. The saturated bubble surface mass fraction is
mC =
psat (TC )
.
Rw TC ρma (TC )
(2.15)
Steady-state conservation of mass demands that
dṁcond (x) = ṁw (x) − ṁw (x + dx),
(2.16)
so the differential equation for water mass flow rate becomes:
dṁw (x)
= −KP [m(x) − mC ].
dx
(2.17)
Assuming the change in moist air mass flow rate is small,
ṁcond
1,
ṁma,i
(2.18)
ṁw (x) ≈ m(x)ṁma,i
(2.19)
dm(x)
KP
=−
[m(x) − mC ].
dx
ṁma,i
(2.20)
then
and
Solving for m(x) and applying the boundary condition m(x = 0) = mi gives the
water mass fraction profile:
m(x∗ ) = mC + [mi − mC ]e(−K
where
∗ x∗ )
,
(2.21)
x
L
(2.22)
KP L
.
ṁma,i
(2.23)
x∗ =
and the mass transfer NTU, K ∗ , is:
K∗ =
34
 a ha (T ( x  dx))
m
 a ha (T ( x))
m
m w ( x  dx)hw (T ( x  dx))
 w ( x)hw (T ( x))
m
AIR
WATER
x  dx
x
 cond hw (TC ) dQ s ( 0)
dm
Figure 2-4: Conservation of energy on a differential control volume of moist air
Mean Mass Fraction Difference
The mean mass fraction difference, ∆m, is defined by:
∆m =
ṁcond
.
KP L
(2.24)
Evaluation of Eqn. (2.21) at the outlet gives the expected outlet mass fraction:
mo = mC + [mi − mC ]e(−K
∗)
(2.25)
Combining Eqns. (2.19), (2.23), (2.24) and (2.25) gives the mean mass fraction
difference, which in this case is a log mean mass fraction difference:
∆m =
mi − mo
mi −mC .
ln ( m
)
o −mC
(2.26)
The log-mean density difference can be used in Eqn. (2.24) to find the condensation
rate, which can then be used in Eqn. (2.8) to compute the latent heat transfer rate.
2.2.5
Temperature Profile
The sensible heat transfer from the bubbles to the column fluid is regulated by the
temperature difference between the bulk air and the bubble surface. Figure 2-4 illustrates conservation of mass and energy on a differential control volume of moist air
inside the bubble, modeled as a stream with equivalent length and perimeter:
dQ̇s = dṁcond hw (TC ) + ṁa [ha (x + dx) − ha (x)]
+ ṁw (x + dx)hw (T (x + dx)) − ṁw (x)hw (T (x)),
(2.27)
where the sensible heat transfer rate into the differential element is
dQ̇s = −U P [T (x) − TC ]dx
35
(2.28)
and dṁcond is defined by Eqn. (2.16).
The latent heat of vaporization does not appear in the first law for the chosen
control volume because the diffusing water leaves as vapor and condenses just outside
the control volume. The latent heat is then assumed to be carried away across the
thin liquid-side thermal boundary layer into the well-mixed column.
Taking the limit of small dx leads to the differential form of conservation of energy,
assuming, again, constant specific heats.
dṁw
+ U P (T (x) − TC )
dx
dT
dT
+ cp,a ṁa
+ cp,w ṁw (x)
dx
dx
0 =
cp,w
(2.29)
Next we define a dimensionless temperature Θ:
Θ(x∗ ) ≡
T (x∗ ) − TC
.
Ti − TC
(2.30)
Substituting in the water mass flow profile, Eqn. (2.21), and nondimensionalizing
gives a linear, homogeneous, first-order ODE:
i
h
C − C ∗ ∗
i
C
K ∗ e(−K x ) Θ(x∗ )
U∗ −
CC
C(x∗ ) dΘ(x∗ )
,
+
CC
dx∗
0 =
where the heat transfer NTU is
U∗ =
UP L
,
CC
(2.31)
(2.32)
and the heat capacity flow rates are
Ci = ṁda cp,da + mi ṁma,i cp,w ,
(2.33)
CC = ṁda cp,da + mC ṁma,i cp,w ,
(2.34)
and
C(x∗ ) = CC + (Ci − CC )e−K
∗ x∗
(2.35)
Solution of Eqn. (2.31) gives the dimensionless temperature profile of the moist air
along its path through the bubble column:
∗
Θ(x ) = e
(−U ∗ x∗ )
C ( KU ∗∗ +1)
i
.
C(x∗ )
(2.36)
If Eqn. (2.36) excluded the second righthand term, the temperature profile would
be consistent with the profile assumed in the usual derivation of LMTD. However, this
term appears for two reasons: the decreasing heat capacity of the moist air stream
as water condenses, and the thermal energy left in the moist air stream from water
36
vapor cooling as it diffuses to the bubble surface.
2.2.6
Mean Temperature Difference
The relevant mean temperature difference ∆T is defined as the solution to the equation
Q̇s = −U P L∆T.
(2.37)
Combining Eqns. (2.7), (2.23), (2.24), (2.32), (2.33), (2.36), and (2.37) at the air
stream exit, x∗ = 1, leads to an expression for the mean temperature difference, ∆T ,
which drives heat transfer in the moist air stream of the bubble column dehumidifier:
o Θo Ti −TC
∆m
C (1+ Ci −C
) (− Ci −Co Θo Ti −TC )
CC
∆T
mi −mo
i
CC
∆T
Θo =
e
,
Co
(2.38)
Co = ṁda cp,da + ṁw,o cp,w , and
(2.39)
where
Θo =
To − TC
.
Ti − TC
(2.40)
The full solution for ∆T includes the ratio of dimensionless heat and mass transfer
coefficients:
(Ti − TC )(Ci − Co Θo )/CC
.
(2.41)
∆T =
Ci
U∗
(1 + K
∗ ) ln ( C ) − ln(Θo )
o
Solving for ∆T without U ∗ and M ∗ presents a challenge because it appears in
both exponents of Eqn. (2.38). However, Eqn. (2.41) can be modified by relating U ∗
and K ∗ to the Lewis factor, using the specific heat of the saturated mixture near the
bubble surface [33, 26].
U∗
U P L ṁma,i
U
=
≈
= Lef
∗
K
CC KP L
Kcp,ma (TC )
(2.42)
Various Lewis factor correlations based on the Lewis number have been proposed,
but Lewis himself found that for air-water mixtures Lef ≈ 1 [34]. Therefore, for
dehumidifiers condensing water out of air, the first approximation for ∆T is
∆T1 =
(Ti − TC )(Ci − Co Θo )
2 .
CC ln ΘCo Ci o 2
(2.43)
The accuracy can be improved by iterating as follows:
∆Tn =
CC
h
(Ti − TC )(Ci − Co Θo )
i.
Ci −Θo Co Ti −TC ∆m
Ci
+ 1 ln ( Co ) − ln(Θo )
CC
∆Tn−1 mi −mo
(2.44)
The temperature profiles that lead to the computation of ∆TLM , ∆T1 , and ∆T =
∆T∞ are plotted in Fig. 2-5 for a bubble column dehumidifier with typical operating
37
1
ΔT_LM
0.9
ΔT_∞
0.8
ΔT_1
0.7
Θ
0.6
0.5
0.4
0.3
0.2
0.1
0
0
0.02
0.04
0.06
0.08
x*
Figure 2-5: Dimensionless temperature profile
38
0.1
conditions and approximate heat and mass transfer coefficients. The temperature
profile implicitly assumed by using the standard LMTD is consistently lower than
those which take into account changing heat capacity and mass transfer, leading to
approximately a 10% underestimation of the mean temperature difference. The heating due to mass diffusion down a temperature gradient leads to a higher temperature
than predicted by LMTD, and the reduction in heat capacity leads to the steeper
slope at low values of Θ. The temperature profiles of ∆T∞ and ∆T1 are almost
indistinguishable, so as long as Lef ≈ 1, Eqn. (2.43) for ∆T1 can be used to approximate the mean temperature difference, which is presented in dimensional form in
Eqn. (2.45):
Ci (Ti − TC ) − Co (To − TC )
2
.
(2.45)
∆T ≈ ∆T1 =
i −TC )
CC ln CCoi 2 (T
(To −TC )
This equation can be used to find the mean temperature difference driving sensible
heat transfer in the moist air stream of a bubble column dehumidifier or humidifier.
Because of the sign conventions used in this work, ∆T will be negative in the case of a
humidifier, resulting in positive sensible heat transfer into the humidifying air stream.
If the inlet, outlet, and bubble surface heat capacities are set equal, Eqn. (2.45)
reduces to the standard LMTD for a single-stream heat exchanger in which the nonisothermal stream is experiencing a negative heat transfer.
The mean temperature and mass fraction differences are used in the energy balance
for a a column with steady liquid temperature and no heat loss to the environment:
(U A∆TLM )coil = U P L∆T + hf g KP L∆m.
(2.46)
With the heat and mass transfer coefficients and the definitions of effective length
and perimeter, the system consisting of Eqns. (2.1), (2.7), (2.8), (2.15), (2.46), and
Eqn. (2.47),
ṁma,i mi − ṁma,o mo
,
(2.47)
∆m =
KP L
can be solved for the six unknown quantities: Tcoil,o , Ta,o , TC , mo , mC , and finally the
total heat transfer rate, Q̇coil .
2.3
Chapter Conclusions
A model was developed which treats a bubble column dehumidifier as one singlestream heat exchanger and one single-stream heat and mass exchanger in contact
with isothermal column liquid. Algebraic expressions were developed for the mean
heat and mass transfer driving forces. The LMTD commonly used to model the
mean temperature difference in heat exchangers does not apply to the stream with
both heat and mass exchange due to: (a) the changing heat capacity flow rate; and
(b) heating of the moist air stream by diffusion of water vapor down a temperature
gradient. In the stream with both heat and mass exchange, a log-mean mass fraction
difference was shown to be the driving force for mass transfer, and a mean temperature
39
difference was presented which drives sensible heat transfer. With relevant heat and
mass transfer coefficients taken from the literature, these simple algebraic expressions
can be used to model heat and mass exchange in a bubble column dehumidifier or
humidifier.
40
Chapter 3
Heat Transfer to Horizontal
Cylinders in Bubble Trays
This chapter is based on a paper by Tow and Lienhard [35].
3.1
Introduction
Shallow bubble columns are used as dehumidifiers in humidification-dehumidification
(HDH) desalination systems, but their unique geometry limits the applicability of
existing correlations for tall-column heat transfer coefficients. The effect of geometry
on the heat transfer coefficient outside coils in shallow bubble columns, such as those
used in multi-stage bubble column dehumidifiers, is poorly understood. Most of the
literature on heat transfer in bubble columns focuses on the heat transfer coefficient
at the column wall, although some studies address the heat transfer coefficient on
internal heat exchange elements such as cylinders and helical coils. The studies involving internal heat exchange elements (internals), however, disagree on the effects of
the column and heat exchange element diameters. The effects of additional geometric
parameters relevant to shallow columns have not been studied.
Short bubble columns, which are desirable in HDH desalination because their low
gas-side pressure drop reduces the blowing power, have different fluid flow and heat
transfer characteristics than tall columns. Because most bubble column reactors are
orders of magnitude taller than those used for dehumidification [13, 36], the reactor
modeling and design literature generally focuses on the developed flow region in the
middle of the column and neglects to address the entry region near the bottom and the
coalescing region near the free surface. In contrast, a short bubble column may have
no developed (i.e., height-independent) flow. Heat transfer coefficients on internals in
sieve-tray columns, which are similar in height to shallow bubble columns, have not
been studied because sieve trays tend to be used without such elements.The effects
of column region and flow regime on the heat transfer coefficient will be explored in
this work.
Heat transfer in short bubble columns with internals differs from that in tall
bubble columns due to dependence upon additional geometric parameters and the
41
increased importance of the free surface. Among these new parameters, the horizontal
position of the cylinder with respect to the sparger orifices, which can be altered to
control bubble-on-coil impact, was proposed by Narayan et al. [13] as a geometric
parameter of interest in bubble columns with coils. The effects of coil diameter have
been investigated by several authors, although there is disagreement among them
[37, 38, 39]. The depth of the liquid and the distance of the cylinder from the sparger
plate and from the free surface are also shown herein to affect heat transfer.
In this chapter, the heat transfer coefficient between coil and liquid in a shallow
(<10 cm deep) bubble column is measured using horizontal cylindrical probes of three
diameters (5, 10, and 16 mm) over a range of gas superficial velocities. Geometric
parameters relevant to bubble column dehumidifiers including liquid depth, cylinder
height, and horizontal position relative to the sparger orifices are also varied. The
effects of liquid depth and superficial velocity on flow regime and pressure drop are
also explored.
3.2
Background
In this section, a theoretical background on heat transfer between a gas-liquid mixture
in a short bubble column and the surface of an immersed heat exchange element is
presented. Following a discussion of the salient theories of heat transfer in tall bubble
columns, the geometric parameters and flow regimes and regions unique to short
bubble columns are identified with the aim of understanding their effects on the heat
transfer coefficient.
Perhaps the most widely used correlation for heat transfer from the fluid in a
tall bubble column to a large surface such as the column wall is that of Deckwer
[40], which is based on the idea that the bubbles’ flow work is dissipated by small
eddies which interact periodically with the heat transfer surface. The interactions are
modeled as conduction through a semi-infinite slab with a characteristic time equal to
the ratio of a characteristic eddy length and characteristic velocity. The application
of an empirical constant leads to Deckwer’s correlation, Equation 4.21 [40]:
St = 0.1(ReFrPr2 )−1/4
(3.1)
Deckwer’s model assumes all energy dissipated in the column is the bubbles’ gravitational potential energy. Certainly, in a typical tall bubble column, the gravitational potential energy of the entering bubbles dwarfs the kinetic and surface energies
associated with their introduction, but in a sufficiently short bubble column, this
assumption merits investigation. The gravitational potential energy of a spherical
bubble is given by Equation 3.3:
π
ρf Db3 gH
(3.2)
6
The bubbles also have surface and kinetic energy. The surface energy of a bubble
is approximated by Equation 3.3:
Egravitational =
42
Esurf ace = πDb2 σ
(3.3)
The kinetic energy added during the injection of a bubble is estimated by applying
continuity to the displacement of fluid around a growing sphere in Equation 3.4:
∞
dr 2 Z ∞
dr 2
1
b
b
2
2
Ekinetic (rb (t)) ≈
v(r) (4πρr ) dr = 2πρ
rb4
r−2 dr = 2πρ
rb3 .
2
dt
dt
rb
rb
(3.4)
Equation 3.4 depends on the radius and propagation speed of the bubble surface.
The surface propagation speed is estimated by assuming constant volumetric growth
by Equation 3.5:
Z
drb
V̇A
≈
.
dt
4πNh rb2
(3.5)
Combining Equations 3.4 and 3.5, we find that the maximum total kinetic energy
of the liquid surrounding the bubble occurs when the bubble radius is smallest, or
just as the bubble is introduced. The approximate kinetic energy per bubble injected
is then:
Ekinetic ≈
ρV̇A2
.
4πNh2 Dh
(3.6)
Equation 3.6 is approximate because the characteristic bubble radius and interface
velocity were used to evaluate the kinetic energy added during the injection of a
bubble. In actuality, the maximum kinetic energy may be higher or lower because
the bubble interface propagation speed varies with the bubble’s radius and the bubbles
are not really spherical.
This analysis shows that in a short bubble column such as the one tested here,
the gravitational potential energy still dominates despite being significantly reduced.
At a gas flow rate of 2 L/s through a 16-orifice sparger, the initial bubble diameter
is found with a correlation of Akita and Yoshida [41] to be 26 mm. According to
Equations 3.3 through 3.6, in a 5 cm deep tank, the contributions of gravitational,
interfacial, and kinetic energy are then 4.5, 0.15 and 0.4 mJ per bubble, respectively.
Although the contributions of surface and kinetic energy are not sufficient to
distinguish short-column from tall-column heat transfer, geometric effects on the heat
transfer coefficient are expected due to the many parameters needed to define the
geometry of a short column with internals. Along with these new length scales, the
prominence of the free surface directs attention to the fluid dynamics in the coalescing
region. The radial position of the heat exchange element is an important geometric
parameter. For example, several authors find that the heat transfer coefficient on
a cylinder is significantly higher in the center than at the wall [42, 43, 44]. The
horizontal position of the cylinder with respect to the sparger orifices, which can be
altered to encourage bubble-on-coil impact, was proposed by Narayan et al. [13] as a
geometric parameter of interest in bubble columns with coils. The effects of diameter
43
have been investigated by several authors, though there is disagreement among them
[37, 38, 39]. The depth of the liquid and the distance of the cylinder from the sparger
plate and the free surface are shown herein to affect heat transfer. In the present work,
the effects and relative importance of these many parameters are investigated with
the aim of developing heat transfer coefficient correlations for short bubble columns
with internal heat exchange elements.
3.2.1
Existing Heat Transfer Coefficient Correlations
Many correlations have been proposed for the heat transfer coefficient on heat exchange surfaces in bubble columns, but there is significant disagreement between
them [45]. Most are semi-theoretical correlations whose forms depend on the assumed mode of heat transfer. Many correlations echo Deckwer’s (Equation 4.21) [40],
assuming thermal interaction with eddies produced by the dissipation of bubbles’ flow
work. Others consider fluid elements with a different length scale, such as the bubble
diameter or distance between bubbles. Other disparities may be due to differences
in measurement methods and, particularly in the case of correlations for internals,
geometry.
Given that several reviews of bubble column heat transfer coefficient correlations
already exist [36, 39, 44, 46, 47, 48], the goal of this section is not to provide a
thorough review of the subject. Rather, a small selection of correlations with a focus
on those which apply to internals are presented to provide a background against
which to view the experimental results. Table 3.1 gives a variety of correlations from
the last five decades, four of which have one or more geometric parameters. The
included geometric parameters, the relationship between heat transfer coefficient and
superficial velocity, and the magnitude of the predicted heat transfer coefficient (as
shown in [45]) all vary widely.
Authors
Table 3.1: Selected heat transfer coefficient correlations
Year Application Correlation
Konsetov [37]1
Deckwer [40]
Korte [49, 44]
Saxena and Patel [38]
Muroyama et al. [50]
3.2.2
1966
1980
1987
1991
2001
2
h ν
C 1/3 µ 0.14
( ) = 0.18(g Pr D
) ( µp )
Internals
k g
Dp
2 −1/4
Wall
St= 0.1(Re Fr Pr )
C 0.14 µ 0.3
Tube bundle St= 0.139(Re Fr Pr2.26 )−0.28 Af−0.2 ( D
) ( µp )
Dp
DC −Dp
0.21
Internals
h = 14.83( DC )U
4/3
Internals
NuDp = 0.133Pr1/3 (ė1/3 Dp /ν)0.709
Bulk Flow Regimes
Fluid flow in short bubble columns is distinct from the well-studied flow in both
sieve trays and tall bubble columns. Flow in tall bubble columns is generally divided
1
1/3
2/9
The simplification of 0.18g = 0.14ug made by Konsetov based on a correlation by Kutateladze [51] is used in the present evaluation of the heat transfer coefficient.
44
into three regimes: bubbly (or homogeneous), churn-turbulent (or heterogeneous),
and slug flow [46, 52]. Bouaifi et al. [53] further distinguish between perfect and
imperfect bubbly flow. Deckwer et al. [54] map the dependence of flow regime on
superficial gas velocity and column diameter. Flow in sieve trays, in which liquid
flows horizontally, is divided into spray and emulsion (or bubbly) flow regimes by
Zuiderweg [55], depending on whether the gas forms vertical jets or bubbles. Another
regime, froth, is described by Syeda et al. [56] as a combination of jets and bubbles.
The flow regimes observed in the experimental short column are discussed in Section
3.4.4.
3.2.3
Column Regions
Flow in a bubble column is generally divided into vertical regions of similar fluid
dynamics. The regions in a short bubble column differ somewhat from those of both
tall bubble columns and sieve tray columns because of the shallow liquid depth and
lack of cross-flow. However, there are enough similarities to merit discussion of bubble
and sieve tray column regimes. For instance, the sieve tray column regions identified
by Syeda et al. [56] adequately describe the splashing flow in the very short (<4 cm)
bubble columns used in this experiment. As will be discussed in Section 3.4.3, flow
in the taller columns (5-11 cm) tested here is better described by a combination of
the distributor and coalescing regions of tall bubble columns.
Tall Bubble Columns
The fluid dynamics in a bubble column depend on the vertical region. Joshi and Shah
[57] describe three regions of tall bubble columns: “near the bottom, the behavior and
the properties of the bubbles are determined by the sparger design and the gas flow
rate. In the second region, the bubble properties are determined by the liquid flow
pattern. The second region occupies most of the column volume. In the third region
bubble coalescence occurs.” Bubble column reactors used in process engineering
applications tend to be so tall that most experimental and modeling efforts have
focused on the developed (middle) region. However, the distributor region (near the
bottom) and the coalescing region (at the top) are very important to consider in short
columns.
Several researchers have noted different fluid behavior in the distributor region
[36, 41, 47, 57, 58]. Akita and Yoshida [41] observe that the initial bubble diameter
near the sparger is related to the orifice diameter even though this is not the case
higher in the column. Liebson et al. [58] find that for fast gas flows and large orifices a
“large irregular bubble” forms at an orifice and rises 7.5-10 cm before it is “shattered”
into many small bubbles. Kantarci et al. find that the distributor region, within which
flow depends on height, extends up to 3-4 column diameters in height. The liquid
depths tested in this chapter are all within the various definitions of the distributor
region. In addition to differences due to bubble behavior, the effects of shear due to
the proximity of the sparger plate may affect heat transfer in the distributor region.
In the developed region, which is not present in the short columns tested here,
45
coalescence and splitting regulate the size of bubbles such that the sparger geometry
no longer matters [41] and the flow reaches a gas holdup independent of height. Radial
variations in temperature are negligible [59] but variations in holdup [60] and heat
transfer coefficient [42] may still exist.
The fluid dynamics of the coalescing region at the top of bubble columns are
complicated and the heat transfer coefficient outside internals placed in that region is
unknown. Because the mode of energy dissipation in this region is different, the heat
transfer coefficient correlation may take on a different form than one that applies to
the bulk. In very short columns, splashing and shallow-water waves may also augment
heat transfer in this region.
Sieve Tray Columns
The vertical column regions of sieve-tray columns in the froth regime are fairly similar
to those observed in the present short bubble column. Syeda et al. [56] describe the
regions of trays in sieve-tray columns. In the liquid-continuous region, short jets
form above the sparger orifices and then become bubbles. Bubble coalescence marks
the transition between liquid-continuous and vapor-continuous regions. In the vaporcontinuous region, a layer of splashes is topped off by final layer of drops. This froth
description is independent of the effects of liquid cross-flow and, with the exception
of the jets which were not seen at the gas velocoities tested, matches the present
observations about very short bubble columns.
3.3
Methods
The heat transfer coefficient outside a coil in a short bubble column is measured with
three cylindrical heat transfer coefficient probes of different diameters. Gas superficial
velocity, probe diameter, liquid height, probe height, and horizontal cylinder position
with respect to the sparger holes are varied. Additionally, the pressure drop is measured and the flow regime observed for a range of liquid depths and gas velocities.
3.3.1
Heat Transfer Coefficient Probe Design
The heat transfer coefficient probes, shown in Figure 3-1, dissipate a known power over
a known area and measure the surface and fluid temperatures. Each probe consists of
a cartridge heater encased in a copper tube instrumented with several thermocouples.
The ends are sealed and insulated with acetal caps (kacetal = 0.33 W/m-K [61]). A
separate thermocouple measures the bubble column bulk temperature, T∞ . The heat
transfer coefficient can then be calculated from measurements, taking into account
heat lost through the end caps, using the following equations:
h=
Q̇p − 2Q̇end
,
Ap (Tp,ave − TC )
46
(3.7)
Figure 3-1: The three heat transfer coefficient probes
where the power dissipated is:
V2
Q̇p =
Re
and where the heat lost at each end is:
p
Q̇end = hPp kacetal Ac,p (Tend,ave − TC ).
(3.8)
(3.9)
Tend,ave is the average reading of the two thermocouples closest to the end caps.
The infinitely-long fin approximation of Equation 3.9 can be applied to the end caps
because the fin is much longer than its extinction length, as shown by Equation 3.10:
s
hPp
mL =
≈ 18 to 120 1.
(3.10)
kacetal Ac,p
In bubble column heat transfer coefficient measurements, the heat lost through
the ends is negligible, but it is significant (order 1%) in the natural convection probe
validation tests of Section 3.3.4.
The probes have a 62.2 mm-long heated copper test section with 25.4 mm-long
press-fit acetal end caps. The cylinders were 4.76 mm, 9.53 mm, and 15.88 mm
(3/16”, 3/8”, and 5/8”) in diameter. Cylinders were used to represent coils of large
turn radius compared to the outer diameter of the tube. Then, any effects of curvature
on heat transfer would be dominated by the cylinder diameter. As shown in Figure
3-2, silicone thermal paste was used to fill any air gaps inside the probes, and RTV
silicone was used to seal the space around the heater leads. The thermocouples were
36-gauge K-type with fiberglass insulation.
Thermocouples were distributed in a spiral, covering the probe evenly in both axial
and radial directions. Three, four, and five thermocouples were used on the small,
medium, and large probes, respectively, with the aim of balancing the accuracy of the
average temperature measurement with the risk of altering the heat transfer by adding
resistance and surface roughness. The thermal boundary layer thickness outside the
internals in a bubble column is comparable in thickness to a human hair (≈100 µm
47
RTV
Thermal paste
Thermocouple Copper tube
Cartridge heater
Acetal
Hole
Figure 3-2: Schematic diagram showing the heat transfer coefficient probe construction
Thermocouple
leads
Epoxy
Copper
tube wall
Solder
Thermocouple
bead
Figure 3-3: Schematic diagram showing the embedding of thermocouples in the copper
tube wall of the heat transfer coefficient probe
based on 6000 W/m2 K in water), so any protrusion away from the surface could alter
the heat transfer coefficient at the thermocouple location. The use of even a thin
tape to attach the thermocouple would introduce a conduction resistance of similar
magnitude as the convective resistance to the column fluid, significantly raising the
surface temperature measurement.
To avoid changing the roughness or adding resistance, the thermocouples were
embedded in solder-filled troughs cut into the copper tube. Because solder does not
adhere well to thermocouple metals, the thermocouple was encased in solder in a solid
(but ductile) state. A hand-held butane torch was used to fill the thermocouple bead
pocket with solder, leaving a slight hill on top. A trench was cut into the solidified
solder bead and the thermocouple bead was placed in the bottom of the trench. An
awl was used to press the solder closed around the thermocouple bead. Pressure
above 100 bar (in this case, body weight on a ≈2 mm square) was applied to form
the ductile solder around the thermocouple bead to expel air and reduce the contact
resistance as much as possible. A fine file was used to smooth the cylindrical probe
surface. The thermocouple leads were glued into the trench with epoxy. After curing,
the epoxy was also filed down and the entire probe was sanded and coarsely polished
to discourage outgassing on the probe.
The heat transfer coefficient probes are designed to be accurate within 10-15%
(See Appendix A for uncertainty calculations). Each individual thermocouple has an
uncertainty of 1.1◦ C (except for the thermocouple measuring the bath temperature,
which was calibrated to reduce its uncertainty), and there is additional error related
48
2
3
8
4
6
1
DAQ
7
9
5
Figure 3-4: Experimental apparatus: 1. Pressurized dry air inlet; 2. Rotameter (4-40
cfm); 3. Rotameter (0.4-4 cfm); 4. Tank; 5. Orifice plate sparger; 6. Heat transfer
coefficient probe; 7. Thermocouple; 8. Variable autotransformer; 9. Data acquisition
unit
to calculating the average temperature of the probe surface with only a few measurements. Considering both of these sources of error, the 4.8 mm, 9.5 mm, and 15.9 mm
probes have 95% confidence intervals of 13.7%, 12.4%, and 11.6%, respectively, in the
heat transfer coefficient measurement. A significant fraction of the error is due to the
necessity of keeping a low temperature difference between the probe and the water
to minimize outgassing on the probe.
3.3.2
Fixture Design
The heat transfer coefficient probes fit into an experimental fixture in which the gas
velocity, liquid depth, sparger design, and cylinder diameter, height, and horizontal
position relative to the sparger orifices can be easily varied. Figure 3-4 shows the
experimental setup.
The bubble column is contained by a rectangular polycarbonate (PC) tank, shown
in Figure 3-5. The tank is 157 mm wide by 284 mm long, and can be filled to a
maximum depth of 110 mm above the sparger plate. The tank cross-sectional area
can be considered to be large based on observations about tall bubble columns: at a
hydraulic diameter of 202 mm, the gas holdup is independent of column diameter [36]
and the heat transfer coefficient is within 10% of the large-diameter value [39]. PC was
chosen for its clarity, scratch resistance, and ability to withstand higher temperatures
than the commonly-used PMMA.
The modular gas sparger uses a replaceable PC sparger plate which is held in place
with wing nuts and sealed with a neoprene o-ring. Two sets of holes in the sparger
plate are outfitted with screws and hex nuts to attach the probe at the desired height
and either above or away from the middle two orifices. Two sparger plates were tested,
but all results reported here use the plate shown in Figure 3-6, which has sixteen 3
mm orifices. The other plate, which had 83 orifices, permitted uneven sparging at
lower gas velocities.
49
Figure 3-5: Empty bubble column with a heat transfer coefficient probe secured to
the sparger plate
50
Figure 3-6: Drawing of the sparger plate with sixteen 3 mm sparger orifices (uncolored). Red fill indicates holes used to hold the probe, and light blue indicates those
used to secure the sparger plate.
3.3.3
Experimental Protocol
First, tap water is degassed by boiling and cooling. The probe is polished to remove
oxidation and installed in the desired position. The column is filled with degassed
water to the desired depth during air sparging at 1 cm/s. A wide ruler is positioned a
few millimeters from the front wall of the tank to damp the liquid depth fluctuations
in the vicinity of the depth measurement without causing significant capillary rise.
The heater and DAQ are turned on, and the heater voltage is measured. Ice and/or
hot degassed water are added until the column reaches 20◦ C. The system is allowed
a few minutes to reach a quasi-steady state in which there is a constant temperature
difference between the probe surface and column liquid.
To make each measurement, the air flow rate is set and the system is given about
one minute to return to a quasi-steady state. The air bubbles that accumulate on
the warm probe due to the outgassing of air from the water (which, despite initial
degassing efforts, tends to reabsorb air during bubbling) are brushed off with a curved
pipe cleaner. Because of this bubble-removal procedure, these measurements apply
to heat transfer coefficients in cooling, which is the direction of heat transfer in dehumidification and many chemical processing applications, including Fischer-Tropsch
synthesis. Finally, approximately sixty measurements of each temperature are taken
with the DAQ at half-second intervals. The average temperature of each thermocouple is recorded for use in computing the heat transfer coefficient. This procedure is
repeated for a number of air flow rates for each column-probe configuration.
51
25
Measured Nu
20
15
10
1:1
4.8 mm
5
9.5 mm
15.9 mm
0
0
5
10
15
20
25
Expected Nu
Figure 3-7: Probe validation in horizontal natural convection in water
Throughout the experiment, bulk liquid temperature is maintained as close as
possible to 20◦ C. The standard deviation in bulk temperature was 0.6◦ C, indicating that the relevant liquid properties (notably the viscosity, density, and thermal
conductivity) can be considered constant across all measurements.
3.3.4
Probe Validation
To test the accuracy of the heat transfer coefficient probes, they were used to measure
the well-studied heat transfer coefficient of natural convection on a horizontal cylinder.
Each cylinder was immersed in a tank of degassed water, 8.9 cm deep, at a height
of 3.7 cm. Measurements are compared in Figure 3-7 to a correlation by Churchill
and Chu [62] for natural convection on a horizontal cylinder in a large volume. Heat
losses from the insulated probe ends were accounted for using Equation 3.9. In this
test, the 4.8 mm, 9.5 mm, and 15.9 mm probes have 95% confidence intervals of
5.8%, 5.3%, and 5.0%, respectively (see Appendix A). The probes have a slightly
higher accuracy in the natural convection test than in the heat transfer coefficient
measurement because this test was conducted with a higher temperature difference
(∼ 15◦ C) between the probe and liquid. As shown in Figure 3-7, all three probes
measured heat transfer coefficients with a nearly-constant average deviation of 7.0%
and a maximum deviation of 8.1% from the expected value, both of which are within
the accuracy of the correlation itself, which seems to be accurate within 15% or so in
the Ra∼106 range [62].
52
3.4
Results and Discussion
Superficial velocity, probe diameter, liquid depth, probe height2 , and horizontal probe
position with respect to the sparger holes3 were varied to determine the effects of
geometry and air velocity on the heat transfer coefficient. Figure 3-8 shows all heat
transfer coefficient measurements. Apart from the cylinder diameter, all variables
were observed to have a significant effect on the heat transfer coefficient. In this
section, the effects of different variables will be explored.
3.4.1
Comparison with Existing Correlations
Figure 3-9 compares heat transfer coefficient measurements made using the 4.76 mmdiameter probe at a height of 2 cm with impact to several correlations from the
literature. The shape of the velocity dependence is generally consistent with all three
correlations. The data demonstrate good agreement with the correlation of Saxena
and Patel [38]. In contrast, both Deckwer’s [40] and Konsetov’s [37] correlations
significantly underpredict the present results.
In Figure 3-10, which includes data from the 9.53 mm probe spanning a wide
range of variables, it is clear that even the correlation of Saxena and Patel [38] does
not capture all effects of geometry. In particular, the experimental data at low (<2
cm) probe height is much lower than predicted by their correlation. Clearly, several
additional variables beyond those in the correlation affect the heat transfer coefficient
in a short bubble column. These effects are analyzed in the coming sections.
Most of the heat transfer coefficients measured here are much higher than predicted by Deckwer’s correlation, as well as the many similar tall column correlations
for heat transfer to both column walls and immersed heating elements. To explain
this, we note a subtle difference between heating, which is usually employed during
heat transfer coefficient measurement, and cooling, which is more typically used in industrial bubble columns. Gas bubbles tend to nucleate on heating elements due to the
decrease in the gas solubility of liquids with increasing temperature [63]. Despite efforts to degas the water in this experiment, constant air sparging ensures the presence
of dissolved air. The air bubbles which form on the hot probe add thermal resistance,
decreasing the heat transfer coefficient measurement. In cold coil experiments, however, gas solubility rises near the cold coil and no bubble nucleation is expected. In
the present work, air bubbles were brushed off the probe as described in Section 3.3.3
in order to measure heat transfer coefficients that apply to cooling. However, it may
have been impractical to remove air bubbles before each measurement in the much
taller columns typically used in previous experiments. This difference in experimental
protocol may account for some of the difference between the present measurements,
which apply strictly to cooling, and the most commonly employed correlations.
2
Probe height is measured from the top of the sparger plate to the bottom of the probe.
Bubble-on-coil impact is controlled by changing the horizontal position of the cylinder with
respect to the sparger orifices so that the probe is positioned over the holes in cases of impact.
3
53
Key (see caption)
S Y 2/s
S Y 2/3
S Y 2/4
S Y 2/5
S Y 2/6
S Y 2/7
S Y 2/8
S Y 2/9
S Y 2/10
S N 2/s
S N 2/5
S N 2/10
M N 0/3
M N 0/s
M Y 0/3
M Y 0/s
M N 1/4
M N 1/s
M Y 1/4
M Y 1/s
M N 2/5
M N 2/10
M N 2/s
M Y 2/5
M Y 2/10
M Y 2/s
L N 2/5
L N 2/10
L N 2/s
L Y 2/5
L Y 2/10
L Y 2/s
M Y 4/s
M Y 4/7
M Y 4/10
M Y 6/s
M Y 6/10
M Y 8/s
M Y 8/11
9
8
7
h (kW/m2K)
6
5
4
3
2
1
0
0
1
2
3
ug (cm/s)
4
5
Figure 3-8: All heat transfer coefficient measurements. The key gives values of the
many variables tested as follows: [probe size (S/M/L)] [impact (Y/N)] [probe height
in cm]/[liquid depth in cm]. “s” denotes that the liquid was filled to just barely cover
the probe
54
9000
Depth
8000
2.5 cm
3 cm
4 cm
5 cm
6 cm
7 cm
8 cm
9 cm
10 cm
Theory (Saxena)
Thoery (Konsetov)
Theory (Deckwer)
7000
h (W/m2-K)
6000
5000
4000
3000
2000
1000
0
0
1
2
3
ug (cm/s)
4
5
6
Figure 3-9: Experimental data for heat transfer coefficient as a function of superficial
velocity over a range of liquid depths are presented along with several correlations.
These results were gathered with the 4.76 mm probe at a height of 2 cm with bubbleon-coil impact
3.4.2
Cylinder Diameter
Figure 3-11 shows the insensitivity of heat transfer coefficient to cylinder diameter for
probes between 5 and 16 mm in diameter. This parameter was investigated due to
the disagreement among correlations in the literature on the effect of probe diameter.
It is clear from Figure 3-11 that the effect is not as pronounced as in Konsetov’s
model, in which the heat transfer coefficient is proportional to the -1/3 power of
probe diameter. This result hints at the difference in length scale between the probe
and the relevant fluid structure in the multiphase flow. It is clear that the length
scale of the relevant fluid structure (whose identity is a subject of disagreement) is
much smaller than the diameter of these probes.
It is immediately clear that the effects of probe diameter are insignificant compared
to the effects of other geometric parameters which cause the spread in Figure 3-10.
The 10-15% error in the measurements of the probes, discussed in Section 3.3.1, easily
accounts for the spread in Figure 3-11. For cylinders placed at 2 cm height in 10 cmdeep water, Figure 3-11 also shows that bubble-on-coil impact does not significantly
affect the heat transfer coefficient.
55
9000
Height, Depth
0 cm, 1 cm
8000
0 cm, 3 cm
1 cm, 2 cm
h (W/m2-K)
7000
1 cm, 4 cm
6000
2 cm, 3 cm
5000
2 cm, 5 cm
2 cm, 3 cm *
2 cm, 10 cm
2 cm, 10 cm *
4000
4 cm, 5 cm
4 cm, 7 cm
3000
6 cm, 7 cm
8 cm, 9 cm
2000
8 cm, 11 cm
1000
Theory (Saxena)
Theory (Konsetov)
0
Theory (Deckwer)
0
1
2
3
ug (cm/s)
4
5
6
*No impact
Figure 3-10: Experimental data for heat transfer coefficient as a function of superficial
velocity over a range of probe heights and liquid depths are presented along with
several correlations. These results were gathered using the 9.53 mm probe with
impact except where noted.
3.4.3
Column Region
The splashing flow in the very short (<4 cm) bubble columns used in this experiment
matches the description of sieve trays by Syeda et al. [56] described in Section 3.2.3.
Flow in the taller columns (5-11 cm) tested here is better described by a combination
of the distributor and coalescing regions of tall bubble columns, also described in
Section 3.2.3. For the purpose of predicting heat transfer coefficients, flow in all of
the short columns tested here will be divided into bulk and coalescing regions. The
coalescing region is at the top of the column where the gas holdup spikes and the
bubbles coalesce and burst. The bulk region is everything beneath the coalescing
layer.
Figure 3-12 shows the effects of column region and bubble-on-coil impact on heat
transfer coefficient for all three probes. Although impact does not matter in the bulk,
it clearly affects heat transfer in the coalescing region. The highest heat transfer
56
9000
8000
h (W/m2-K)
7000
6000
5000
4000
5 mm, impact
5 mm, no impact
10 mm, impact
10 mm, no impact
16 mm, impact
16 mm, no impact
3000
2000
1000
0
0
1
2
3
ug (cm/s)
4
5
6
Figure 3-11: Heat transfer coefficient compared to superficial velocity for the three
probe diameters with and without impact
coefficients for all three probes were measured in the coalescing region with impact.
In the case of no impact, the coalescing region proves to perform worse than the
bulk. Clearly, the mode of heat transfer in the coalescing regime is different. Liquid
filaments and droplets, created by the bursting of bubbles in the coalescing region,
may enhance heat transfer in the impact case.
3.4.4
Flow Regime
The character of the multiphase flow in a short bubble column depends on the column
geometry, superficial velocity, and fluid properties. In the column used in this work
with air and water, the dependence of the flow regime on gas superficial velocity
and column depth can be defined by a regime map, Figure 3-13. The map was
contructed by testing 100 velocity/depth combinations (shown in Figure 3-21) in the
experimental column without the probe installed. Figure 3-13 shows that the flow in
the test column can be divided into splashing, sloshing, and swirling regimes. These
flow regimes and their effects on the heat transfer coefficient will be discussed in this
section. In addition, although the regime transitions identified in Figure 3-13 do not
necessarily apply to columns with different areas, aspect ratios, sparger designs, or
fluids, it is clear from this study that the typical tall-column regime map [52] does
not apply generally to short columns.
The swirl regime, which occurs when the liquid is deeper than 7 cm or so, encompasses several swirling flow patterns. Figure 3-14 illustrates swirl around the
longitudinal, vertical, and circumferential axes. Longitudinal- and circumferentialaxis swirl always turned in the direction depicted in Figure 3-14, but vertical-axis
swirl was observed to turn in either direction. Its lack of a preferential direction sug57
9000
5 mm probe
10 mm probe
16 mm probe
h (W/m2-K)
8000
7000
6000
5000
4000
3000
0
1
2
3
4
ug (cm/s)
5
Coalescing, impact
Coalescing, no impact
Bulk, impact
Bulk, no impact
0
1
2
3
4
ug (cm/s)
5
Coalescing, impact
Coalescing, no impact
Bulk, impact
Bulk, no impact
0
1
2
3
4
ug (cm/s)
5
6
Coalescing, impact
Coalescing, no impact
Bulk, impact
Bulk, no impact
Figure 3-12: Results for the three probes presented on the same axes: heat transfer
coefficient in the bulk of the fluid and in the coalescing region, with and without
impact. In each case the height was 2 cm; the region was changed by varying the
liquid depth
gests that vertical-axis swirl is not due to an imbalance in the sparger. The regime
map, Figure 3-13, shows how the swirl type evolves from longitudinal to vertical as
the liquid depth is increased, except at very low superficial velocities. Above a depth
of 10 cm, the flow begins to switch spontaneously between longitudinal, circumferential, and both clockwise and counterclockwise vertical swirl. The time scale of the
switch is on the order of 10 seconds.
According to a review by Joshi and Shah [57], circumferential-axis swirl is common
in tall columns. In this type of swirl, liquid travels up the center of the column and
down the sides. However, circumferential swirl was never observed to be the only
stable mode in the short rectangular column tested, though it was seen intermittently
at a depth of 11 cm. The stability of longitudinal-axis swirl, which is similar to
circumferential-axis swirl, over a range of depths is unsurprising due to the rectangular
shape of the test column.
Vertical-axis swirl, which is shown in Figure 3-15, is a stable flow regime in the test
column but does not seem to occur in tall columns. Gross and Kuhlman [64] measured
the mean circumferential velocity and found it to be zero everywhere in their 21 cm
wide, 18 cm tall hexangonal bubble column. Vertical-axis swirl is identified in the
present experiment by the paths of the large bubbles, which bend around the vertical
axis. The paths of small bubbles indicate that a secondary circumferential-axis swirl
is present during vertical-axis swirl. The vertical swirl occurs because the high gas
holdup near the center of the column [60] reduces the effective density of the mixture
in the center. If the depth of the fluid is relatively uniform, the radial gradient in
effective density leads to a radial gradient in pressure and the formation of a vortex.
58
12
Unstable mixed swirl
10
Liquid depth (cm)
Vertical-axis swirl
8
Longitudinal-axis swirl
6
Sloshing
4
Splashing
2
0
0
1
2
3
4
Superficial velocity (cm/s)
5
6
Figure 3-13: Regime map for the experimental column showing primary dependence
on liquid depth and secondary dependence on superfical velocity
59
Figure 3-14: Swirl types observed in a short rectangular bubble column, top to bottom: longitudinal-axis, vertical-axis, and circumferential-axis swirl.
60
Figure 3-15: Swirling regime: clockwise vertical-axis swirl captured with a long exposure to show bubble trajectories
Consistent with the vortex formation hypothesis, bubbles leaving the sparger at the
center of the column were observed to spin rapidly, forming a stream in the shape of
a tornado.
Splashing is observed when the liquid is very shallow (less than around 4 cm
deep). This regime, depicted in Figure 3-16, is similar to the spray regime noted
by Zuiderweg [55] and the froth regime described by Syeda et al. [56] in sieve trays.
Liquid filaments, some of which break into drops, extend out of the layer of bubbly
liquid matching the description of the imperfect bubbly regime [53] of tall columns.
Sloshing was observed in columns of 4-6 cm in depth. Sloshing caused by gas
injection has been previously noted in argon-oxygen decarburisation [65]. Sloshing
agitated by bubbling is most likely to occur near the column’s natural frequency, which
primarily depends on the tank size [66]. The dominant sloshing mode was across the
shorter length of the rectangular tank, as shown in Figure 3-17, at a frequency of 2
Hz. On several occasions, however, a diagonal sloshing pattern developed in which
the surface of the fluid twisted back and forth.
Figure 3-18 showns that the heat transfer coefficient depends somewhat on the flow
regime. The heat transfer coefficient is usually higher in swirling than in sloshing. As
shown in Figure 3-8, similar results are obtained with the smallest and largest probes.
The highest and lowest heat transfer coefficients are measured in splashing, with and
without impact, respectively. However, because splashing requires a shallow liquid
(see Figure 3-13), it is impossible to place a probe in the bulk of a splashing liquid
without bringing it close to the sparger. Therefore, the measurements of the splashing
regime are taken in the coalescing region, which is discussed in Section 3.4.3, and the
splashing regime cannot be compared directly to the other regimes. However, for a
fixed probe height (as in Figure 3-18), it is shown that changing the depth changes
both the flow regime and the heat transfer coefficient.
61
Figure 3-16: Splashing regime, showing both liquid filaments and drops
Figure 3-17: Sloshing regime: images taken 1/4 second apart illustrating sloshing
along the tank’s shortest length
62
8000
Heat transfer coefficient (W/m2-K)
7000
6000
5000
4000
3000
Splashing, coalescing, impact
Splashing, coalescing, no impact
Sloshing, bulk, impact
Sloshing, bulk, no impact
Swirling, bulk, impact
Swirling, bulk, no impact
2000
1000
0
0
1
2
3
4
5
6
ug (cm/s)
Figure 3-18: The heat transfer coefficient varies slightly with changes in flow regime.
These measurements used the 9.5 mm probe at a height of 2 cm
Other changes to the column geometry might also affect the flow regime and
heat transfer coefficient. Column length and width, for instance, influence frequency
of sloshing and might also influence the heat transfer coefficient in sloshing. The
horizontal position of the probe with respect to the center of the column may also
raise the heat transfer coefficient during swirl compared to the present measurements
taken near the swirl axis.
3.4.5
Cylinder Height
In short bubble columns, the heat transfer coefficient depends on the height of the
cylinder. The effect of cylinder height on heat transfer coefficient is shown in Figs.
3-19 and 3-20 for the bulk and coalescing regions, respectively. In each figure, the
distance from the top of the fluid to the top of the cylinder as held constant. The heat
transfer coefficient increases monotonically with height until reaching a maximum at 4
cm in both regions. Similar but slightly lower heat transfer coefficients are measured
for 6 and 8 cm heights. The drop in heat transfer coefficient as the probe height
is reduced from 4 to 0 cm is unsurprising because the wall acts as a momentum
sink, decreasing the specific kinetic energy in its vicinity. The peak in heat transfer
coefficient around a height of 4 cm is most likely due to the height-dependent bubble
dynamics near the sparger.
it is proposed to express the critical height in terms of a critical modified Reynolds
number based on Deckwer’s [40] characteristic eddy velocity, (ug g/ν)1/4 . Far from the
sparger surface (“the wall”), the flow of heat-carrying eddies is unaffected by the wall
and the heat transfer coefficient is roughly independent of height. However, below
63
9000
8000
7000
h (W/m2-K)
6000
Height (cm)
5000
M0 Y 0/3
M1 Y 1/4
M2 Y 2/5
M4 Y 4/7
M8 Y 8/11
4000
3000
2000
1000
0
0
1
2
3
4
ug (cm/s)
5
6
Figure 3-19: Heat transfer coefficients on the 9.53 mm probe with impact at a variety
of heights. The fluid is 2 cm over the top of the probe, placing the probes in the bulk
region
9000
8000
h (W/m2-K)
7000
6000
Height (cm)
M0Y 0/s
M1Y 1/s
M2Y 2/s
M4Y 4/s
M6Y 6/s
M8Y 8/s
5000
4000
3000
2000
1000
0
0
1
2
3
4
ug (cm/s)
5
6
Figure 3-20: Heat transfer coefficients in the coalescing region on the 9.53 mm probe
with impact at a variety of probe heights
64
12
11
10
Liquid depth (cm)
9
8
Splashing
7
Sloshing
6
Circumferential-axis swirl
5
Vertical-axis swirl
4
Longitudinal-axis swirl
3
Re*=900
2
1
0
0
1
2
3
4
5
Superficial velocity (cm/s)
6
Figure 3-21: Flow regime map, showing that the liquid depth at the onset of sloshing
is related to the critical height
the critical modified Reynolds number, the presence of the wall hinders the heat
transport. Equation 3.11 gives the critical modified Reynolds number, Re? , below
which the heat transfer coefficient is dependent on height:
Re?crit =
1/4 p,crit (ug g)
ν 3/4
H
≈ 900
(3.11)
For example, under conditions typical of this experiment (3 cm/s superficial velocity and 23◦ C water), the critical height is 3.6 cm, which is consistent with the
observation that the heat transfer coefficient is roughly independent of probe height
for heights 4 cm and above.
This critical height is also related to the transition from splashing to sloshing
flow during bubbling in an empty column. Figure 3-21 shows the full results of the
flow regime investigation of Section 3.4.4 with the critical height (Re? =900) plotted
as a function of superficial velocity. The transition to sloshing appears to occur at
the critical height. Intuitively, this makes sense: when the liquid depth is below the
critical height, the effect of the wall is strong enough to damp out any perturbations
and prevent sloshing.
3.4.6
Bubble Impact
Bubble-on-coil impact has a a pronounced effect on heat transfer coefficient in some
geometries. Impact is not important when the cylinder is placed in the bulk region
at a height of at least 2 cm, as shown in Sections 3.4.3 and 3.4.2. However, impact
matters when the cylinder is in the coalescing region or very close to the sparger.
65
9000
8000
h (W/m2-K)
7000
6000
5000
4.8 mm, impact
4.8 mm, no impact
9.5 mm, impact
9.5 mm, no impact
15.9 mm, impact
15.9 mm, no impact
4000
3000
2000
0
1
2
3
4
5
6
ug (cm/s)
Figure 3-22: In the coalescing region, the heat transfer coefficient is greater with
impact than without. These measurements were made with all three probes at a
height of 2 cm
Figure 3-22 shows that the heat transfer coefficient in the coalescing region is raised
by bubble-on-coil impact.
In the bulk region, impact also increases the heat transfer coefficient at very low
probe heights. Figure 3-23 shows how the heat transfer coefficient in the bulk region
with impact (compared to without) is higher at 0 cm, much higher at 1 cm, and only
slightly higher at 2 cm. The improvement in heat transfer coefficient associated with
impact in the coalescing regime extends to low cylinder heights with the exception
of the case when the probe is placed directly on the sparger, which leads to a higher
heat transfer coefficient in the non-impact case (see Figure 3-8).
Any effect of impact on heat transfer coefficient is in disagreement with Deckwer’s
assumption of uniform dissipation of kinetic energy [40]. The fact that impact is
most important near the sparger and in the coalescing region suggests that these are
the regions in which the rate and/or mode of specific energy dissipation are different
than they are in the middle of the short column. The importance of impact near
the sparger and in the coalescing region supports the idea that heat transfer in short
columns should be treated differently than that in tall columns.
66
7000
6000
h (W/m2-K)
5000
2 cm height, impact
2 cm height, no impact
1 cm height, impact
1 cm height, no impact
0 cm height, impact
0 cm height, no impact
4000
3000
2000
1000
0
0
1
2
3
ug (cm/s)
4
5
6
Figure 3-23: At cylinder heights below 2 cm, impact causes the heat transfer coefficient to increase
3.4.7
Empirical Correlation
A correlation of all heat transfer coefficient measurements in this chapter is given in
Equation 3.12.
h
−H (gu )1/4 i
h W s 1/4 i
p
g
h̄ = 16000u1/4
1
−
0.75
exp
×
1
g
450ν 3/4
m2 K m
(3.12)
where H is the height of the center of the cylinder. The form of Equation 3.12 is
based on Deckwer’s [40] superficial velocity dependence and the characteristic height
discussed in Section 3.4.5. Effects of flow regime, flow region, cylinder diameter, and
bubble-on-coil impact are excluded from this correlation. As shown in Figure 3-24,
agreement is within ±20%.
3.4.8
Pressure Drop
It is important to be able to estimate the pressure drop of the short columns used
in bubble column dehumidifiers because the gas-side pressure drop contributes to the
electrical power requirement of HDH desalination. In tall bubble columns, it is safe
to assume that the gas-side pressure drop is hydrostatic. However, in short bubble
columns, the hydrostatic pressure drop is reduced such that other components of
pressure drop such as surface tension, liquid inertia, and minor losses through the
sparger plate cannot be neglected. To quantify the pressure drop in short columns,
a manometer was inserted into a bolt hole to measure the pressure drop between the
sparger cavity and the atmosphere above the column. Figure 3-25 shows the measured
67
10000
+20%
0%
8000
Predicted h (W/m²K)
-20%
6000
4000
2000
0
0
2000
4000
6000
8000
10000
Measured h (W/m²K)
Figure 3-24: All experimental heat transfer coefficient measurements compared to the
empirical correlation (Equation 3.12), showing agreement within about ±20%
pressure drop, which increases with both superficial velocity and liquid depth.
It is clear that the pressure in short columns is always greater than hydrostatic.
Figure 3-26 shows the difference between the pressure drop and its hydrostatic component. The nonzero pressure at zero height is primarily due to minor losses through
the orifices of the dry sparger plate. The negative slope in pressure difference with
increasing depth is due to the gas holdup, which decreases the effective density of
the air-water mixture compared to pure water. The peak around 8 cm may be due
to the transition to a swirling regime (see Section 3.4.4). The exact pressure drop
will depend on the sparger design, and may be reduced by increasing the number of
sparger holes.
To evaluate the validity of the hydrostatic pressure drop assumption employed
in tall bubble columns, the ratio of pressure drop to its hydrostatic component is
plotted in Figure 3-27. This ratio approaches unity near 10 cm in depth, but for the
low column depths and higher gas velocities useful in bubble column dehumidifiers,
68
1400
1200
Pressure (Pa)
1000
800
600
4.3 cm/s
400
3.2 cm/s
2.2 cm/s
200
1.3 cm/s
0
0
2
4
6
8
10
Liquid depth (cm)
Figure 3-25: The gas pressure drop increases with liquid height and superficial velocity
800
4.3 cm/s
3.2 cm/s
Pressure, above hydrostatic (Pa)
700
2.2 cm/s
600
1.3 cm/s
500
400
300
200
100
0
0
(Dry)
2
4
6
Liquid depth (cm)
8
10
Figure 3-26: The pressure drop is always greater than hydrostatic for columns up to
10 cm in depth, and the difference increases with superficial velocity.
69
5
Pressure / hydrostatic pressure
4.3 cm/s
3.2 cm/s
4
2.2 cm/s
1.3 cm/s
3
2
1
0
0
2
4
6
8
10
Liquid depth (cm)
Figure 3-27: The ratio of pressure drop to hydrostatic pressure drop, which decreases
with liquid height and increases with gas velocity, shows that the hydrostatic pressure
drop assumption fails to estimate blowing power at low liquid heights.
the hydrostatic pressure drop assumption is not valid.
Another important aspect of pressure drop is its effect on the specific flow work
dissipation in the column, which, according to most models, affects the heat transfer
coefficient. For example, Deckwer’s model [40] relates heat transfer coefficient to the
one-fouth power of specific hydrostatic flow work dissipation, ė = vg g. To evaluate
the validity of the hydrostatic pressure drop assumption on the flow work dissipation
in the column, a flow work ratio RF W , which estimates the ratio of the pressure
drop leading to energy dissipation in the bulk of the column (excluding minor losses
through the sparger) to the hydrostatic pressure drop, is defined in Equation 3.13.
RF W (vg , H) =
∆p(vg , H) − ∆p(vg , H = 0)
ρf gH
(3.13)
Figure 3-28 shows the flow work ratio, which quickly approaches unity even for short
columns. Therefore, the hydrostatic pressure drop assumption is appropriate for the
purpose of estimating heat transfer coefficients.
It is interesting to note that RF W falls onto two curves depending on the orifice
Reynold’s number. The change in the dependence of RF W on depth may be due to
the transition to turbulent flow through the orifice at Re≈ 2100.
Equation 3.14 gives a correlation for the pressure drop through the sparger used
in this work.
70
2
RFW
1.5
1
4.3 cm/s, Re=3430
3.2 cm/s, Re=2560
0.5
2.2 cm/s, Re=1730
1.3 cm/s, Re=1030
0
0
2
4
6
8
10
Liquid depth (cm)
Figure 3-28: The ratio of flow work dissipated in the column liquid to the assumed
gravitational potential energy dissipation rate used in dissipation-based heat transfer
theories is found to approach unity at low liquid heights.
∆P =
58.2 exp(0.12uh ) + ρgH[0.59H −0.21 ] Reh < 2100
58.2 exp(0.12uh ) + ρgH[0.43H −0.34 ] Reh > 2100
(3.14)
Figure 3-29 shows excellent agreement between Equation 3.14 and the measurements.
3.4.9
Design Recommendations
The results presented here inform the effective and economical design of bubble column dehumidifiers. The cooling coil of a bubble column dehumidifier should be placed
at or just below the critical height (Equation 3.11) with sparger holes placed directly
underneath. In the column used here at 20◦ C and superficial velocity in the 1-5 cm/s
range, the critical height is around 4 cm, which is deep enough for effective gas-liquid
contact (see Chapter 4). Given the importance of limiting the gas-side pressure drop,
it is fortunate that the coalescing region provides the highest heat transfer coefficient.
Therefore, the liquid should be filled to a depth that just barely wets the top of the
coil during bubbling. These recommendations maximize heat transfer coefficients to
promote high effectiveness while maintaining a low gas-side pressure drop.
71
1500
Predicted pressure drop (Pa)
0%
1000
500
0
0
500
1000
1500
Measured pressure drop (Pa)
Figure 3-29: Agreement between pressure drop measurements and Equation 3.14
3.5
Chapter Conclusions
The heat transfer coefficient on a cylinder in a bubble column is measured with horizontal cylindrical probes to elucidate the effects of geometric parameters specific
to shallow bubble columns and develop design rules. This investigation shows that
shallow columns have different heat transfer coefficients and flow regimes than tall
columns. In addition, the hydrostatic pressure drop assumption is shown not to extend to shallow columns. The measured heat transfer coefficients are significantly
higher than predicted by commonly-used correlations for tall columns, although for
sufficiently high cylinder placement, there is good agreement with the correlation of
Saxena and Patel [38]. In short columns, the highest heat transfer coefficients occur when the cylinder is in the coalescing region and aligned over the sparger holes.
Heat transfer coefficient increases with cylinder height until reaching a maximum at
a critical height. Cylinder diameter has little effect on heat transfer. Experimental correlations for heat transfer coefficient and pressure drop were developed based
on these results. Recommendations are made regarding the design of efficient and
economical bubble column dehumidifiers.
72
Chapter 4
Experiments and Modeling of
Single-Tray Bubble Column
Dehumidifier Performance
This chapter is based on two papers by Tow and Lienhard [21, 67].
4.1
Introduction
In HDH desalination, bubble column dehumidifiers recover heat by using the saline
feed water as a coolant in the dehumidifier, thereby preheating it for use in the
humidifier. Multistage bubble columns further enhance this energy recovery by minimizing the stage to stage temperature drop (see [15]). In this chapter, the thermal
performance of a single stage bubble column dehumidifier is investigated experimentally using significantly smaller cooling coils than in previous work, and a predictive
model is developed which shows very good agreement with the experimental data.
The model presented here can be used to predict the performance of a multi-stage
dehumidifier by modeling the performance of each stage.
Sieve tray columns, like multi-stage bubble columns with liquid in cross-flow, are
commonly used in distillation and other vapor-liquid reactions. Barrett and Dunn
[14] proposed a model for a sieve tray column dehumidifier or humidifier. In their
dehumidifier, cold water enters in the top tray and warm, moist air is bubbled in
from the bottom. Given a source of cold water, such a dehumidifier would require no
cooling coils and could be quite inexpensive. However, the cold water source in HDH
is saline water, and dehumidifying moist air from the humidifier by direct contact
with saline water in a tray column would not result in the production of any fresh
water. Because the cooling saline water and condensing fresh water must be kept
separate, the dehumidifier used in the present experimental investigation contains
the saline water within a copper coil, necessitating a new heat transfer model.
The heat and mass transfer processes in bubble column dehumidifiers are not
yet well characterized. Bubble column reactors have been studied extensively as
gas-liquid reactors where the mass diffusion resistance between the bubble surface
73
and the bulk liquid dominates the performance of the column [36]. Additionally,
practical bubble column dehumidifiers for HDH desalination need to be very short
to ensure a minimal gas-side pressure drop, and several researchers have noted that
the gas and liquid phases behave differently near the gas inlet [36, 58, 57, 41]. Most
bubble column reactors are significantly taller than those used for dehumidification
[13, 36], so the entry region is often neglected in the reactor modeling and design
literature. A model by Narayan et al. [13] proposes a thermal resistance network for
the bubble column dehumidifier with transport mechanisms taken from the bubble
column reactor literature. This model predicts the heat flux with moderate accuracy
in simple configurations, but it calls for refinement. Another model [1] proposes a
different resistance network along with mean heat and mass transfer driving forces
in the bubble stream, but does not predict heat and mass transfer coefficients. The
model presented in this paper addresses many of these outstanding issues.
Because the cost of a bubble column dehumidifier is strongly influenced by the
mass of copper used in the coil, the experiment performed in this work uses much
shorter cooling coils than those used by Narayan et al. [13] with the aim of achieving
higher a heat flux on the coil surface. However, changes to the column design (e.g.,
coil length) or operation (e.g., air temperature) that increase the heat flux may reduce
capital cost, but may also reduce the effectiveness of the dehumidifier. By measuring
the parallel-flow effectiveness, the effect of changes to the column design and operation
can be quantified in terms of heat flux and effectiveness to give insight into both
capital and energy costs.
4.2
Theory
Heat and mass transfer in the single-stage bubble column dehumidifier can be described by the resistance network in Figure 4-1. The system is modeled as suggested
in [1] with one critical modification: the gas-side heat and mass transfer resistances
(the left side of Figure 4-1) are considered to be negligible. The dehumidifier is
modeled as a single-stream heat exchanger with the well-mixed column liquid as the
isothermal stream.
Given the assertion by previous authors [13] that the gas-side resistance has a
significant effect on bubble column dehumidifier performance, the presently applied
approximation of negligible gas-side resistance clearly merits justification. In this
section, a brief review of bubble size correlations is performed and the evidence in
support of neglecting the gas-side resistances is presented. A model is then detailed
which predicts heat transfer rate and heat flux in the dehumidifier. Finally, the
parallel-flow effectiveness [21], a performance parameter relevant to individual wellmixed bubble column dehumidifier stages, is defined.
4.2.1
Bubble Size
The length scale of conduction or convection heat transfer in a gas is generally very
important. Therefore, it might be expected that the bubble size influences the per74
CA
CB
Qcond
Rm
RAB
Bubble
RBC
A
B
RCD
C
TE
TD
TC
TB
TA
D
E
RDE
Coolant
Coil
Figure 4-1: Resistance network from [1] governing heat and mass transfer in a bubble
column
formance of a bubble column dehumidifier. In this section, correlations for bubble
size based on sparger design and operation are reviewed in order both to analyze the
present experimental results and to better justify the present assumption of negligible
gas-side resistance.
Heat flux was shown by Narayan et al. [13] to decrease with increasing bubble
size. In their experiment, the sparger hole pattern was changed to induce variations
in bubble size. However, the relationship between sparger orifice size and bubble
size is complicated. Narayan et al. calculate bubble size from sparger hole size with
Equation 4.1 [13, 32]:
1/3
6σD
h
(4.1)
Db =
g(ρf − ρma )
Equation 4.1 was developed by van Krevelen and Hoftijzer [32] for sufficiently low
gas flow rates that bubbles, affected by neither the presence of preceding bubbles nor
the liquid’s inertial forces [68], depart by their own buoyancy. However, at larger flow
rates, chain bubbling occurs. Miller [68] provides correlations for bubble size in chain
bubbling:
µ
3.22( πg(ρff−ρg ) )1/4 ( V̇Nma
)1/4 Reb < 9
h
Db =
(4.2)
ρ
2.35( π2 g(ρff−ρg ) )1/5 ( V̇Nma
)2/5 Reb > 9
h
where the bubble Reynolds number is defined as follows:
Reb =
ρf ub Db
µf
(4.3)
The terminal bubble velocity in Equation 4.3 is predicted by Miller [68] with
Mendelson’s wave analogy [28], which incorporates buoyant, interfacial, and inertial
75
forces:
s
ub =
2σ
gDb
+
ρf Db
2
(4.4)
When the flow through the orifice becomes turbulent (Reh > 2100), bubbles take
on a range of sizes [58]. The orifice Reynolds number is defined by Equation 4.5 [58]:
Reh =
ρma uh Dh
4ṁma
=
πNh Dh µma
µma
(4.5)
At turbulent Reh , the bubble behavior varies with height. Leibson et al. [58] find
that a “large irregular bubble” forms at an orifice and rises just 7.5-10 cm as a
single entity before it is “shattered” into many small bubbles. These observations are
consistent with those of Joshi and Shah [57], who describe three regions of sparged
bubble columns: “near the bottom, the behavior and the properties of the bubbles
are determined by the sparger design and the gas flow rate. In the second region,
the bubble properties are determined by the liquid flow pattern. The second region
occupies most of the column volume. In the third region bubble coalescence occurs.”
Bubble column reactors used in process engineering applications tend to be sufficiently
tall that most experimental and modeling efforts have focused on the second region.
However, some of the column configurations tested in this work are comparable in
height to the entry region noted by Leibson et al. Akita and Yoshida [41] also observe
that coalescence and splitting cause bubbles in a tall column to gradually approach a
size independent of the orifice size, but find that the initial bubble diameter is related
to the orifice diameter by Equation 4.6:
Db,i
D5 u2 1/6
h h
= 1.88
g
(4.6)
Because the column depths used in this experiment range from below to well above
the entry region proposed by Leibson et al. and the coil is always in the developing
region noted by Joshi and Shah, bubble size will not be predicted in this analysis.
Instead, results will be discussed in terms of sparger hole diameter.
4.2.2
Bubble-Side Resistance
Limited attention has been devoted to the prediction of heat transfer coefficients
inside and outside the bubbles because mass transfer resistance dominates in most
industrial bubble column reactor applications. Because liquid-phase mass transfer in
bubble columns is well-studied, an analogy to mass transfer could easily be used to
approximate the thermal resistance between the bubble surface and the bulk column
liquid (RBC in Figure 4-1). However, the heat and mass transfer resistances inside
the bubble (RAB and Rm in Figure 4-1, respectively) are elusive.
Narayan et al. [13] propose a model for bubble column dehumidifier performance
which treats the heat and mass transfer inside the bubbles. However, the model
relies on several unconfirmed assumptions. The authors give a conservative estimate
76
of the heat transfer coefficient using a Lewis factor [33, 34] mass transfer analogy,
with mass transfer inside the bubble approximated as steady diffusion through a slab
with thickness equal to the bubble radius. This leads to the prediction that heat
flux can be improved by reducing the bubble diameter and that hitting the cooling
coils can improve heat transfer. Finally, they also predict a non-zero mass transfer
resistance to the unphysical [14] diffusion of the condensed liquid water through the
identical column liquid water.
Daous and Al-Zahrani [69] measure the heat and mass transfer resistances experimentally, but the resulting heat transfer coefficient is surprisingly low. They find that
in the homogeneous flow regime (ug < 5 cm/s [36]), the product of the heat transfer
coefficient and specific interfacial area is between 1 and 20 W/m3 -K. Considering that
the specific interfacial area in homogeneous flow is on the order of 100 m−1 [70], the
measured heat transfer coefficients would have to be around 0.01 to 0.2 W/m2 -K, far
lower than the equivalent heat transfer coefficient for conduction through a stagnant
air sphere (order 10 W/m2 -K) of a typical bubble size. In contrast, it would be expected that bubble oscillations, inner circulation, and breakup and coalescence would
raise the heat transfer coefficient well above the conduction-only equivalent.
The dynamics of bubble injection are illustrated by Figures 4-2 and 4-3. To
capture these images, dry air was blown continuously through one cylindrical orifice
of 3 mm diameter and 4.8 mm length into a tank of room temperature tap water.
The clear polycarbonate tank was 10 cm deep and 16×28 cm in cross section. The
velocity of air flow through the orifice was varied within the range of 5-50 m/s. A
Phantom v7.1 monochrome high-speed video camera was used at a frame rates of
8,000-12,000 fps and a resolution of 304×512 pixels. A bright halogen lamp was used
for backlighting. Reflections due to the angled bubble interfaces, particularly the
total internal reflection on parts of the air-water interface, create the dark bubble
images.
As shown in Figure 4-2, high-velocity gas injection causes bubbles to stack up
in mushroom-like forms. The high jet velocity ( 10 m/s) causes bubbles to grow
faster than the previously-released bubble can rise. When the interfaces become close
enough, the mass of fluid above the gas jet becomes small and the jet plunges forward
into the bubble ahead of it. Figure 4-3 shows that it is incorrect to approximate
bubbles as stagnant gas spheres.
Several studies suggest that the bubble-side resistance to heat and mass transfer is
negligible. A review by Kanatarci et al. [36], like much of the bubble column literature,
discusses the liquid-to-wall heat transfer in detail while giving no mention of the
mechanism of heat transfer (or mass transfer, for that matter) through the bubbles.
In their model of a sieve-tray-column dehumidifier, Barrett and Dunn assume that
“both gas and liquid phases are...perfectly mixed so that the conditions of the gas
and liquid streams leaving a tray are representative of conditions on the tray” [14].
Perfect mixing implies that the bubbles take on the temperature of the surrounding
liquid, or that the resistance is essentially zero. Kang et al. [16] use a non-zero
internal heat transfer coefficient proposed by Clift et al. [71] in their comparison of
bubbling and falling-film modes of ammonia-water absorption, but it is clear from
their results that the overall resistance inside the bubbles is very low. When cold
77
1 cm
1 cm
Figure 4-2: Examples of bubble mushroom formation and departure in the sparger
region
78
1 cm
Figure 4-3: Formation of liquid sheets, filaments and drops inside a bubble
gas bubbles are introduced into a vertical channel with hot walls, their simulation
shows that the bubble temperature rises rapidly, becoming almost indistinguishable
from the wall temperature within the first 4-5 cm of rise. Considering that the need
to immerse the coil mandates a minimum height of 4 cm in this work, the results
of Kang et al. suggest that it is reasonable to neglect resistance in the gas phase in
the present model. Finally, as explained in [1], if the gas side resistance RAB can
be neglected, so can the liquid-side resistance just outside the bubbles, RBC , leading
to the assumption of perfect mixing within the short column and enabling simple
modeling of the bubble column dehumidifier.
4.2.3
Heat and Mass Transfer Model
The perfect mixing assumption justified in Section 4.2.2 greatly simplifies the transport modeling. The bubble column dehumidifier acts like a single-stream heat exchanger with the well-mixed column liquid as the isothermal stream. The temperature of the column liquid is dependent on the coil-side thermal resistance—the right
half of Figure 4-1, or the entirety of the simplified resistance network, Figure 4-4—and
the temperatures and flow rates of the moist air and coolant streams. The overall
resistance through the coil can be estimated using correlations found in the literature.
The set of equations that make up the proposed model are not linear, and an equation
solver such as Engineering Equation Solver (EES) [72], used here with the built-in
physical properties of air and water, is recommended. Appendix B contains the EES
setup of this model.
Due to the small hydrostatic head of the short column, all properties are evaluated
at atmospheric pressure. All water is assumed to be pure, although in practice saline
79
TD
TC
RCD
TE
RDE
Figure 4-4: Simplified thermal resistance network
water would generally be used as a coolant (inside the tube) for a dehumidifier used
in HDH desalination.
Assuming the column operates in steady state and is well insulated such that a
negligible quantity of heat is lost to the environment, Equation 4.7 shows that the
total heat transfer rate into the coolant, Q̇CE , is the sum of the sensible and latent
heat transfer rates out of the bubbles, Q̇AB and Q̇cond , respectively:
Q̇CE = Q̇AB + Q̇cond
(4.7)
In the present experiment, the air entering the column is assumed to enter saturated with water. The mass flow rate of dry air and the inlet water vapor mass flow
rate are related to the inlet saturated mass fraction by Equations 4.8 and 4.9:
ṁda = ṁma,i [1 − msat,i ]
(4.8)
ṁw,i = ṁma,i msat,i
(4.9)
The water vapor-air mixture is assumed to behave ideally. As explained in [1],
Equation 4.10 represents the conservation of energy for the entire moist air stream.
The air exit temperature must be equal to the column temperature to satisfy the
approximation of zero gas-side resistance.
Q̇AB = ṁda [hda (TA,i ) − hda (TC )]
+ ṁw,i [hg (TA,i ) − hg (TC )]
(4.10)
Equation 4.11 gives the latent heat released when water vapor condenses at the
bubble surface:
Q̇cond = ṁcond hf g (TC )
(4.11)
Equation 4.12 relates the heat transfer to the coolant to its change in enthalpy.
Q̇CE = ṁE [hE (TE,o ) − hE (TE,i )]
(4.12)
Due to the near-unity Lewis factor of water vapor in air [34], it is expected that
mass transfer rates will keep up with the heat transfer rates such that the air leaves
in a saturated state at the column liquid temperature, TC . Equation 4.13 gives the
80
saturated air outlet mole fraction:
xo =
Psat (TC )
ṁw,o /Mw
=
Patm
ṁw,o /Mw + ṁda /Mda
(4.13)
The condensation mass flow rate, ṁcond , is a parameter of great interest in water
purification applications:
ṁw,i = ṁw,o + ṁcond
(4.14)
Next, the heat transfer driving force is discussed. Due to the excellent mixing in
the bubble column, there is a negligible radial temperature gradient [59] and essentially zero resistance to radial mixing. Although temperature gradients have been
observed in the vertical direction in industrial bubble columns [73], the depth of the
liquid used in this experiment is only about 1-10% of the height of a typical bubble
column. Therefore, the column liquid is expected to be isothermal except very close
to the cooling coil. It is also assumed that due to the good wettability of the copper
coil, the tube only directly contacts the column liquid and not the air bubbles. For
these reasons, the log-mean temperature difference (LMTD) for a single-stream heat
exchanger is used to model the thermal driving force across the coil:
∆TLM,CE =
TE,o − TE,i
(4.15)
T −T
E,i
)
ln ( TCC −TE,o
Equation 4.16 shows the relationship of the total heat transfer Q̇CE to the LMTD
and the total thermal resistance as shown in Figure 4-4:
Q̇CE =
∆TLM,CE
RDE + RCD
(4.16)
It is assumed that all heat transfer occurs within the liquid portion of the column.
Heat transfer from the outgoing air to the exposed portion of the coil in the air gap
(see Figure 2-1) is neglected because in practice, size and cost restrictions will enforce
a short air gap and a small exposed area of coil. Thermal resistance through the
tube wall is neglected because a thin copper tube is used in the experiment discussed
herein, but a third resistance can easily be added in series to account for a more
resistive tube material.
The convective resistances in Equation 4.16 are related to the relevant heat transfer
coefficients and areas by Equation 4.17:
RCD =
1
hCD AD,o
and RDE =
1
hDE AD,i
(4.17)
The heat transfer coefficients inside and outside the coils are relatively wellstudied. Appropriate formulations were developed by Mori and Nakayama [29] for
the heat transfer coefficient inside curved tubes of round cross-section in laminar flow.
Secondary flows induced by the coil curvature significantly reduce the radial length
scale for convection compared to a straight tube. For example, the curved pipe Nus81
selt number in this experiment is predicted to be nearly ten times the straight pipe
value. This curvature-induced augmentation of the heat transfer coefficient extends,
to a lesser extent, into turbulent flow [30]. Equation 4.18 gives the Nusselt number
correlation for laminar flow in a curved tube based on the Dean number K and the
thickness parameter Z [29]:
NuD,DE
K 1/2 hDE DD,i
= 0.8636
=
kE
Z
(4.18)
Equation 4.19 gives the Dean number:
K = ReE
D
D,i
1/2
Dturn
(4.19)
For Pr> 1 (e.g., for water), the thickness parameter Z is given by Equation 4.20:
r
77 −2
2
Z=
1 + 1 + PrE
(4.20)
11
4
Kantarci et al. [36] provide an excellent review of many bubble column design
considerations including correlations for the heat transfer coefficient. Perhaps the
most widely used correlation comes from Deckwer [40] for heat transfer to a large
surface such as the column wall. Several correlations have been proposed for heat
transfer to small cylindrical heat exchange surfaces [38, 37, 50], but there is significant
disagreement among them. Because the smaller length scale of internal heat transfer
equipment is likely, if anything, to augment heat transfer, Deckwer’s model for heat
transfer to an infinitely-large surface is used here as a conservative estimate of the
heat transfer coefficient.
Deckwer’s model is based on the idea that the bubbles’ flow work is dissipated
by small eddies which interact periodically with the heat transfer surface. The interactions are modeled as conduction through a semi-infinite slab with a characteristic
time equal to the ratio of the characteristic eddy length and characteristic velocity.
An empirically-derived constant leads to a Stanton number correlation, Equation 4.21
[40]:
St = 0.1(ReFrPr2 )−1/4
(4.21)
The present authors find the dimensionless form unsatisfactory because both Re
and Fr involve an unspecified length dimension which cancels out when they are
multiplied. The dimensional form is given by Equation 4.22:
1/2 3/4 1/2 −1/4 1/4 1/4
g ug
hCD = 0.1kf ρf cp,f µf
(4.22)
Perhaps a more illustrative representation is given by Equation 4.23, an equivalent
Nusselt number correlation based on Deckwer’s characteristic eddy length η [40]:
Nuη = 0.1Pr1/2
82
(4.23)
where
η=
ν 3 1/4
.
ug g
(4.24)
The superficial velocity ug in the above relationships is the ratio of gas volume
flow rate to column cross-sectional area. In predicting the outcome of the present
experimental results, the superficial velocity is calculated based on the sparger area
because of the coil’s close proximity to the sparger. As mentioned in Section 4.2.1,
the bubble distribution (and, presumably, the energy dissipation distribution) will be
more uniform higher in the column, and the column area should be used to calculate
the superficial velocity for any coils that are located above the developing region.
The heat transfer coefficient outside the coil can also be estimated from the experimental results of Chapter 3. This version of the model is evaluated in Section
4.4.7.
The equations in this section can be solved to find the total heat transfer, Q̇CE ,
and the heat flux through the coil, q̇:
Q̇CE
,
(4.25)
A
where A is the outer surface area of tubing that is immersed in the column liquid.
q̇ =
4.2.4
Parallel-Flow Effectiveness
Tow and Lienhard [21] propose parallel-flow effectiveness as a new performance parameter for individual bubble column dehumidifier stages. In contrast to the effectiveness defined by Narayan et al. [19] for simultaneous heat and mass exchangers, by
which bubble columns can be compared to other (generally counterflow) dehumidifier
types, the parallel-flow effectiveness acknowledges that as long as the column fluid is
well-mixed, each bubble column stage acts like a parallel-flow device.
Compared to counterflow dehumidifiers, the effectiveness of a well-mixed singlestage bubble column dehumidifier is low (around 50%) because of the interaction of
both streams with the well-mixed column liquid. This leads to the need for multistage devices. Narayan and Lienhard [15] demonstrated that combining bubble column stages at different liquid temperatures into a multi-staged device with an overall
counterflow configuration can achieve effectiveness comparable to conventional dehumidifiers.
However, effective multi-staging requires each stage of the column to have a low
enthalpy pinch [20]. Enthalpy pinch represents an improvement over temperature
pinch as a performance parameter for simultaneous heat and mass exchangers because
of the nonlinearity of the enthalpy-temperature curve of saturated moist air, but it
is still a dimensional quantity. Good heat recovery requires that each stage achieve a
large fraction of its maximum single-stage heat transfer rate. Therefore, to compare
the effects of various parameters on the heat recovery of a single column stage, a
parallel-flow effectiveness, // , is defined in [21] by Equation 10:
83
// =
Q̇CE
,
Q̇max,//
(4.26)
where Q̇max,// is the total heat transfer rate Q̇CE from Section 4.2.3, except with all
outlet temperatures equal to the column liquid temperature, or equivalently, with all
resistances evaluated as zero.
Meaningful performance parameters give insight into the cost per unit of fresh
water produced of operating the dehumidifier in a HDH system. The fresh water
production is strongly linked to the heat transfer rate because the majority of the
heat removed from the air stream is latent heat. The coil, generally copper, represents
a capital expense, and the heat flux determines the coil area needed for a system of
a particular capacity. The effectiveness, on the other hand, relates to the energy
use and cost of HDH desalination using a bubble column dehumidifier. Effectiveness
is a function of the number of stages (see [15]), the thermodynamic balancing (see
[20]), and, finally, the parallel-flow effectiveness of each stage. Changes to the column
design and operation will be analyzed in this work in terms of both heat flux and
parallel-flow effectiveness to capture effects on both capital and energy costs.
Because only single-stage bubble columns are considered in this paper, all further
references to effectiveness will denote the parallel-flow effectiveness, // . In tests where
Q̇max,// and coil area, A, are constant (e.g., while varying column liquid height), only
effectiveness will be plotted because the heat flux is related to effectiveness by a
constant as in Equation 4.27:
q̇ = //
4.3
Q̇
max,//
A
(4.27)
Experimental Methods
Heat flux and parallel-flow effectiveness are measured for a variety of conditions. Coil
length, air temperature, column liquid height, and sparger orifice size are varied.
Additionally, in order to make meaningful comparisons to previous studies, the effect
of bubble-on-coil impact is isolated and reported. In total, 24 measurements (each in
terms of heat flux and effectiveness) are made.
4.3.1
Experimental Bubble Column Dehumidifier
Dehumidifier heat flux and effectiveness are measured with an instrumented HDH
system. The 28 cm square by 36 cm high dehumidifier can be filled to any desired
height. The moist air temperature and the air and coolant flow rates are adjustable.
The experimental dehumidifier is shown in Figure 4-5.
Moist air enters the dehumidifier from a humidifier, in which compressed dry air
is forced through a porous stainless steel cartridge sparger into a tank heated by a
submerged resistance heater. The moist air leaves the humidifier close to saturation,
and it cools slightly as it passes through insulated rubber tubing and a rotameter to
84
4
11
5
15
6
2
1
12
7
10
3
9
13
14
8
Figure 4-5: Experimental setup: (1) pressurized cooling water inlet, (2, 12) rotameters, (3) fresh water outlet valve, (4-8) thermocouples, (9) plate sparger, (10) cooling
coil, (11) air outlet, (13) cartridge sparger. (14) resistance heater, (15) pressurized
dry air inlet
the dehumidifier. Condensation in the rotameter establishes that the air entering the
dehumidifier is saturated.
The effect of condensation in the rotameter on the flow rate reading is neglected,
but the difference in density of the warm, moist air from the dry air at STP for which
the rotameter is calibrated is accounted for. Because gas flow in a rotameter is largely
inviscid, the flow rate reading is proportional to the square root of the density, leading
to Equation 4.28, the correction factor from [21]:
V̇ma
=
V̇meas
r
ρda,ST P
ρma (TA,i )
(4.28)
All cooling coils were made from 9.5 mm OD, 8.0 mm ID copper tubes. The two
larger coils had an 8.5 cm turn radius. The medium-sized (67 cm) coil, which was
used in the present work except where noted, was a single loop. An impractically
small (8.7 cm long) “coil” was included to illustrate the trade-off between heat flux
and effectiveness. For reference, the largest coil used here (900 cm2 ) was comparable
in length and equivalent in tube diameter to the coils used by Narayan et al. [3]. The
vertical part of each coil, which connects the immersed portion to the chilled water
source, was insulated by 3.2 mm thick, 9.5 mm ID rubber tubing.
The coolant used in the present experiment was tap water, but it will be referred
to as “coolant” in this paper to provide distinction from the liquid (also tap water)
in the column. The coolant flow rate in all trials was 0.5 L/min, corresponding to
laminar flow at a Reynolds number of 1740.
Equation 4.12 was used to find the heat transfer rate from the coolant stream.
Equations 4.7, 4.8, 4.10, 4.11, 4.13, and 4.14 were used with the temperature and
air flow rate measurements, assuming 100% relative humidity of all air streams, to
85
calculate the heat transfer rate to the air stream. In steady state and with no heat
exchange with the environment, the heat transfer rates from the coolant and to the air
stream should be equal. In practice, however, it was difficult to maintain all streams
and the column at constant temperature and flow rate, and the heat transfer rate was
calculated from the average of the two measurements. There is some measurement
uncertainty, primarily due to the 1.1 K uncertainty of the K-type thermocouples and
the 5-10% uncertainty associated with the rotameter readings. The averaging of airside and coolant-side heat transfer rate measurements reduces the overall uncertainty,
leading to a 95% confidence interval of ±29 W (10-15%) around each heat transfer
rate measurement.
4.3.2
Controlling Bubble-on-Coil Impact
Bubble-on-coil impact is a design parameter proposed by Narayan et al. [13]. Releasing bubbles such that they will hit the cooling coil has several potential benefits.
As suggested by Narayan et al. [13], the gas bubbles might directly contact the cooling coil, introducing a conduction path unmediated by the column liquid. Bubbles
hitting the coil could also change shape, affecting the boundary layer thickness both
inside the bubble and out, or split or slow down, increasing the total interfacial area.
Bubbles rising in the vicinity of the coil may alter the thermal resistance outside the
coil by changing the bulk velocity or by periodically thinning the coil’s boundary
layer. These and other possible phenomena are difficult to isolate. However, by using
different coils for the “impact” and “non-impact” cases, Narayan et al. [13] neglected
to control two additional parameters, one of which significantly affects the heat flux.
First, coil surface area is shown in Section 4.4.2 to have a very strong influence on
heat flux, especially for long coils with high effectiveness. Therefore, coil surface area
should be kept constant when another parameter is being evaluated. In addition, coil
shape affects the heat transfer coefficient inside the coil due to the secondary flows
that form in curved pipes, which have a more pronounced effect in coils of smaller
turn radius [29], and the high heat transfer coefficients in the thermal entry region
after a sharp bend.
In order to test the effect of bubble-on-coil impact without changing the coil
shape or size, the pattern of sparger holes was modified—without altering the size
or number of holes—to inject streams of bubbles that would or would not hit the
coil. Duct tape, which sticks well to the acrylic sparger plate when wet, was used to
cover unused holes as shown in Figure 4-6. The column was examined visually during
operation to confirm that bubbles were hitting the coil in the impact case. Varying
bubble-on-coil impact by changing only the pattern of sparger holes allows the effect
of bubbles hitting the coils to be distinguished from the effects of changes to the coil
design.
86
Coil
Tape
Figure 4-6: The sparger orifice configurations used to test the effect of bubble-on-coil
impact without altering the coil for the small (2.8 mm) orifices
4.4
Results
In this section, agreement is demonstrated between experimental results and the
proposed model. The effects of variations in coil surface area, air inlet temperature,
and column liquid height on heat flux and parallel-flow effectiveness are shown. The
additional heat transfer that occurs in the air gap above the column liquid is discussed,
and finally, predictions are made regarding the effect of coolant temperature and tube
diameter on performance.
4.4.1
Model Agreement
The model developed herein displays very good agreement with the experimental
data both from this work and from [21] for a well-mixed, coil-cooled, single-stage
bubble column dehumidifier. Figures 4-7 and 4-8 show the agreement in terms of
heat transfer rate and effectiveness, respectively. In no case does the model accurately
predict the performance of the extremely short (8.7 cm long) coil, shown in grey. The
low effectiveness of the short coil forces a significant amount of heat transfer to occur
in the air gap, as will be discussed in Section 4.4.5. In addition, the measured heat
transfer rate leaving the air stream was 64% higher than that entering the coolant
stream, which suggests that leakage of coolant during this measurement was likely.
However, this measurement was included in the experimental results for completeness.
However, all other data is in excellent agreement with an average absolute error of
2.8%.
4.4.2
Coil Length
The relationship between coil length and parallel-flow effectiveness is clear, but even
more marked is the effect of coil length on heat flux. Figures 4-9 and 4-10 show
87
Model heat transfer rate (W)
400
200
Data
1:1
0
0
200
400
600
Measured heat transfer rate (W)
Figure 4-7: Agreement between theoretical and experimental heat transfer rate
1
Data
Model ε//
1:1
0
0
Measured ε//
1
Figure 4-8: Agreement between theoretical and experimental parallel-flow effectiveness. It is clear from the cluster around // = 0.85, corresponding to the 67 cm coil,
that the coil size all but determines the effectiveness.
88
Coil heat flux (kW/m2)
100
Model
Experiment
80
60
40
20
0
0
1
2
3
Coil length (m)
Figure 4-9: The effect of coil length on heat flux
1.2
1
ε//
0.8
0.6
0.4
Model
Experiment
0.2
0
0
1
2
3
Coil length (m)
4
Figure 4-10: The effect of coil length on effectiveness
89
these trends for coils of constant tube diameter using experimental data from [21] at
constant air and coolant temperatures and flow rates. Effectiveness increases with
coil length, asymptotically approaching a value of one when the coil gets very long.
Clearly, the longest (3 m) coil is “very long” for this device. Although a value of
one is within the margin of error for the parallel-flow effectiveness of this coil, the
parallel-flow effectiveness can exceed unity due to thermal interaction between the
air and coolant streams in the air gap above the column liquid. This phenomenon is
neglected in the definition of // but is discussed in Section 4.4.5. The effect of coil
length on effectiveness is not pronounced except at very small lengths because the
heat transfer rate is not strongly dependent on coil length and the maximum singlestage heat transfer rate is independent of coil size. Therefore, heat flux—the heat
transfer rate per unit coil area—rises sharply as the coil size is reduced. Because the
heat flux gained by minimizing the coil area is accompanied by a loss of effectiveness,
the optimal coil size must be determined by analyzing the cost and performance of a
complete HDH system.
The model and experimental data are in good agreement except in the case of
the shortest coil. The low effectiveness of the short coil forces a significant amount of
heat transfer to occur in the air gap, as will be discussed in Section 4.4.5. In addition,
the large difference between the measured heat transfer rates from the air stream and
to the coolant stream suggests that there was significant measurement error. Despite
the likelihood that this measurement is inferior, it is included in the experimental
results for completeness and to illustrate the strong dependence of heat flux on coil
length.
4.4.3
Moist Air Temperature
The effects of varying moist air inlet temperature on heat flux and effectiveness are
shown in Figures 4-11 and 4-12 using experimental data from [21], with which the
model is in excellent agreement. Heat flux increases sharply with moist air temperature because both the higher temperature and the higher concentration of condensible
water vapor contribute to the enthalpy of the warmer moist air. However, effectiveness decreases with increasing air temperature because the heat transfer coefficients
of the coil are near-constant, forcing the much greater heat flux to occur over greater
mean temperature differences, thus widening the temperature pinch and decreasing
the effectiveness.
4.4.4
Liquid Height, Sparger Orifice Size, and Bubble-onCoil Impact
Column liquid height, sparger orifice size and bubble-on-coil impact are discussed
together in this section because of their possible effects on the gas side of the extended
resistance network, Figure 4-1, which was neglected in the present model.
Narayan et al. [13] found that heat flux was independent of liquid height for heights
above 15 cm, the minimum height necessary to cover their large cooling coil. However,
they hypothesize that the critical height, above which heat flux is independent of
90
Coil heat flux (kW/m²)
25
20
15
10
Model
5
Experiment
0
30
40
50
60
Moist air temperature (°C)
70
Figure 4-11: The effect of moist air inlet temperature on heat flux
1.2
1.0
ε//
0.8
0.6
0.4
Model
0.2
Experiment
0.0
30
40
50
60
70
Moist air temperature (°C)
Figure 4-12: The effect of moist air inlet temperature on effectiveness
91
1.0
0.8
ε//
0.6
0.4
Experiment
Model
0.2
0.0
0
5
10
15
Liquid height (cm)
20
Figure 4-13: For 2.8 mm orifices and liquid height above 4 cm, effectiveness is independent of liquid height
liquid height, is on the order of the bubble diameter [13]. The 67 cm coil used here
had only one loop, and therefore the minimum height that could be tested was lower
than the minimum tested in [13], about 4 cm. Liquid height was measured at the
side of the column during bubbling. Although the model presented here includes
no dependence on column height, experiments were conducted at different column
heights to justify the assumption that bubble-side resistance is negligible. Figure
4-13 compares the height-independent model to experimental data from [21]. Figure
4-13 shows that for homogeneous flow through the 2.83 mm orifice plate, any critical
height must be below 4 cm. The air temperature and flow rate were 58.4◦ C and 2.1
L/s and the coolant temperature was 16.6 ± 0.3 ◦ C. The heat flux, not shown, was
just over 23 kW/m2 for all heights.
The absence of any decrease in effectiveness at heights as low as 4 cm suggests
that much of the heat transfer occurs early in each bubble’s residence in the column.
Although it is difficult to verify the claim of Narayan et al. [13] that the critical
height is on the order of the bubble diameter (a few millimeters) these results are
consistent with that hypothesis. Minimizing the liquid height will improve bubble
column performance by reducing the hydrostatic contribution to the air inlet pressure,
thereby reducing the power needed to pump the air. Fortunately a depth of 4 cm
corresponds to a hydrostatic pressure drop of only about 400 Pa per stage, or 2 kPa
for a five-stage dehumidifier. Using narrower tubes so that the liquid height which
just covers the coil can be reduced further lowers the pressure drop, as shown by
Narayan et al. [74] who demonstrate a three-stage bubble column dehumidifier with
an 800 Pa pressure drop.
Using the same sparger with small (2.83 mm) orifices, Figures 4-14 and 4-15
92
20
Coil heat flux (kW/m2)
18
16
14
12
10
Model
8
Impact
6
Non-impact
4
2
0
1
1.5
2
2.5
Air flow rate (L/s)
3
Figure 4-14: For 2.8 mm orifices and small bubbles, the effect of bubble-on-coil impact
on heat flux is small
demonstrate the effect of bubble-on-coil impact for three air flow rates. As discussed
in Section 4.3.2, bubble-on-coil impact was varied by changing the pattern of sparger
orifices rather than changing the coil. The air temperature was 48.8 ± 0.4 ◦ C and
the coolant temperature was 15.8 ± 0.6 ◦ C. The column height was 20 cm.
The effect of bubble-on-coil impact in Figures 4-14 and 4-15 was within the margin
of error of the experiment for all three air flow rates. This is not surprising given the
observation of Narayan et al. [13] that increasing the liquid height beyond what is
required to cover the cooling coil, which would have the effect of decreasing the gasside resistance, does not increase the heat flux. The observation that neither liquid
height nor bubble-on-coil impact can significantly change the effectiveness serves to
justify the assumption of negligible gas-side resistance. Small changes in effectiveness
with bubble-on-coil impact may be due to variation in the outside-coil heat transfer
coefficient.
This finding that bubble-on-coil impact does not significantly raise the heat flux or
effectiveness is in conflict with the observation by Narayan et al. that bubble-on-coil
impact “raises the heat transfer rates to significantly higher values” [13]. Their claim
is supported by a chart which shows that heat flux (in units of kW/m2 ) is about twice
as high with bubble-on-coil impact as without [13]. Neither coil surface area nor inlet
air temperature is specified, though both parameters have been shown in the present
work to have strong effects on heat flux. Thus, it is not impossible that the use of
different coil areas and/or air temperatures led Narayan et al. [13] to obtain very
different heat flux measurements without any significant heat transfer augmentation
from bubble-on-coil impact.
Because all runs in Figure 6 have high effectiveness, it could be argued that bubble93
1.2
1
ε//
0.8
0.6
Model
0.4
Impact
Non-impact
0.2
0
1
1.5
2
2.5
3
Air flow rate (L/s)
Figure 4-15: For 2.8 mm orifices and small bubbles, the effect of bubble-on-coil impact
on effectiveness is small
on-coil impact would have a more significant effect at lower effectiveness. However,
given that bubble impact had no meaningful effect at the lowest effectiveness in Figure
4-15 (87%), a significant difference such as that reported by Narayan et al. [13] would
be unlikely to occur due to bubble-on-coil impact alone in a column designed for high
effectiveness.
Regardless of bubble impact, Figures 4-14 and 4-15 show that heat flux increases
and effectiveness decreases with increasing air flow rate. The increase in heat flux
is due to the increase in maximum single-stage heat transfer rate. The decrease in
effectiveness occurs because the thermal driving force and resistance across the coil
are largely unchanged while the maximum heat transfer rate increases.
Larger, 6.0 mm sparger orifices were tested with the aim of producing larger
bubbles whose gas-side resistance might be greater and also more significantly affected
by changes to the column liquid height and bubble-on-coil impact. The effect of height
and impact for the large orifices is shown in Figure 4-16. The air temperature and
flow rate were 47.8 ◦ C and 1.5 L/s and the coolant temperature was 20.2 ± 0.2 ◦ C.
For the 6.0 mm sparger orifices, bubble-on-coil impact has a small but positive
effect on effectiveness which is more pronounced at low column heights. Due to
the possibility of experimental error, the difference may or may not be meaningful.
However, the correlation from Akita and Yoshida [41] does suggest that the initial
bubble diameter is strongly correlated with the orifice diameter, and it would not be
surprising if the gas-side resistance of a bubble swarm in a short column increases with
increasing bubble diameter. However, no decrease in effectiveness with decreasing
column depth was noticed for the bubbles which impacted the coil. In the impact
case, the coil may cause the large bubbles to slow down, squish or split, all of which
94
1.2
1.0
ε//
0.8
Impact
0.6
Non-impact
0.4
Model
0.2
0.0
0
10
20
Liquid height (cm)
30
Figure 4-16: The effect on effectiveness of bubble-on-coil impact and liquid height for
6 mm orifices
would reduce the resistance to heat and mass transfer to the bubble surface, giving
the effect of a deeper column.
The slight difference in effectiveness between the small and large sparger orifices
(Figures 4-13 and 4-16, respectively) is expected given that the case shown in Figure
4-13 had a higher air inlet temperature and flow rate, both of which are shown in
Figures 4-15 and 4-12 to cause lower effectiveness.
Other effects of bubble-on-coil impact such as changes in the outer coil heat transfer coefficient may also explain why in all cases a slightly higher effectiveness is observed when impact occurs. Because bubble-on-coil impact seems to cause some
improvement, no matter how small, it is worthwhile to consider sparger designs that
facilitate impact so long as they do not raise the system cost. This is especially true
because other design choices which increase the effectiveness tend to increase the capital cost (e.g., raising the coil area) or energy use (e.g., using a narrower tube, which
increases pumping power), whereas the arrangement of holes on a sparger plate is
unlikley to be linked to a significant variation in cost.
4.4.5
Air Gap Heat Transfer
Because the vertical section of the cooling coils used in this experiment was not
perfectly insulated, some heat transfer occurred between the outgoing air and the coils
in the air gap above the column liquid. This fraction, calculated using the method
described by Tow and Lienhard [21], which utilizes the measured pool temperature,
is shown in Figure 4-17 for the same experiments plotted in Figures 4-7 and 4-8. No
model curve is shown because the model neglects heat transfer in the air gap.
If the coil was not exposed in the air gap, the additional heat transfer in the air gap
95
Air gap heat transfer (%)
50
40
30
20
10
0
0
10
20
30
TC-TE,o (°C)
Figure 4-17: The percent of the total heat transfer occurring in the air gap increases
with the liquid side temperature pinch, TC − TE,o
would be zero. However, the reduction in the total heat transfer rate would be much
less than the percent shown in Figure 4-17. Heat transfer occurring in the incoming
exposed coil section reduces the maximum amount of heat transfer that can occur in
the column and reduces the heat transfer driving force (the LMTD). If the potential
for air gap heat transfer is eliminated, the maximum heat transfer rate and LMTD
will rise together, resulting in a comparable effectiveness. However, the heat transfer
between the outgoing air and the outgoing exposed section of coil, which occurs once
both streams have completed their interaction with the column liquid, will be lost.
The remaining maximum possible heat transfer rate between the outgoing air and
outgoing coolant is low due to the low temperature difference between the exiting
air and outgoing coolant and the low absolute humidity of the cool air. Especially
because the low temperature difference also limits the heat transfer driving force,
it can be concluded that the heat transfer rate is not significantly affected by the
possibility of heat transfer in the air gap for reasonably high-effectiveness columns.
In this experiment, there was one exception: the 8.7 cm coil section operating at low
effectiveness whose air exits the liquid at 52.0◦ C may transfer a significant amount
of additional heat to the outgoing section of coil at a cold 24.7◦ C. This is consistent
with the observation that the actual heat flux and effectiveness of the shortest coil
were significantly higher than that predicted by the model, which neglects air gap
heat transfer.
96
18
Coil heat flux (kW/m²)
16
14
12
10
8
6
70°C
60°C
50°C
4
2
0
15
20
25
30
Coolant temperature (°C)
Figure 4-18: The heat flux decreases with increasing coolant temperature, as shown
for three air temperatures
4.4.6
Additional Modeling Results
Given the success (quantified in Section 4.4.1) of the model in predicting the experimental results, it can also be used to predict the effect of additional parameters not
varied in the present experiments. In this section, the effects of coolant temperature
and coolant tube diameter are simulated.
Figure 4-18 shows that heat transfer rate decreases with increasing coolant temperature for an air flow rate is 2 L/s. A coolant temperature of 20◦ C is used with
the same tubing and water flow rate used in the experiment. The coil surface area
is 0.05 m2 . In HDH desalination applications, the coolant temperature will often be
set at the temperature of the water to be treated. However, in cases where brine is
recirculated to increase water recovery and brine concentration, Figure 4-18 shows
that the water should be cooled before recirculation to achieve the highest possible
heat flux.
With changing coolant temperature, effectiveness increases with decreasing water
temperature but only by less than a 1% change over the 15◦ C range shown. This
finding contrasts with the decrease in effectiveness accompanying the increase in heat
flux when air temperature is increased (see Figures 4-11 and 4-12). The difference
is due to the shape of the temperature-enthalpy curve of saturated moist air. The
mass fraction of water vapor in saturated cool air is low, so the heat flux increases
a moderate amount as the coolant temperature is decreased while the temperature
driving force for heat transfer across the coil increases more strongly.
Though tube diameter was not varied experimentally, Figure 4-20 shows that tube
diameter can significantly influence the effectiveness, especially for lower-surface-area
coils. The water flow rate simulated was the same as in the experiment, the air
97
0.975
0.97
ε//
0.965
0.96
50°C
60°C
70°C
0.955
0.95
15
20
25
Coolant temperature (°C)
30
Figure 4-19: The effectiveness is nearly constant with changing water temperature,
as shown for three air temperatures. The vertical axis is expanded to show that there
is, however, a slight decrease in effectiveness with increasing coolant temperature
1
0.98
0.96
0.94
ε//
0.92
0.9
0.88
0.86
900 cm²
600 cm²
300 cm²
0.84
0.82
0.8
6
8
10
12
14
Tube inner diameter (mm)
16
Figure 4-20: Decreasing the tube diameter at constant coil surface area leads to an
increase in effectiveness which is more pronounced for smaller coils
98
Model heat transfer rate (W)
400
200
Data
1:1
0
0
200
400
Measured heat transfer rate (W)
600
Figure 4-21: Agreement in heat transfer rate between modified model and experiment
flow rate was 2 L/s, and the air and coolant temperatures were 60◦ C and 20◦ C,
respectively. Because the tube diameter is linked to the pressure drop, determining
the optimal tube diameter for dehumidification in HDH desalination will require an
analysis of a complete system.
4.4.7
Modified Model Incorporating Experimental OutsideCoil Heat Transfer Coefficients
The model is modified by replacing Equation 4.22 with outside-coil heat transfer coefficients measured in Chapter 3. Because the superficial velocity in the heat transfer
coefficient experiments was calculated using the full cross-sectional area of the column, the same was done in calculating the superficial velocity of the bubble column
dehumidifier in the modified model. Figures 4-21 and 4-22 show the agreement in
terms of heat transfer rate and effectiveness, respectively. The agreement is again
very good, with an average absolute error (excluding, again, the extremely short coil)
of 2.2%.
This decrease in the average absolute error, which is comparable in magnitude
to the experimental error, is slight. Therefore, the simple model presented in Equations 4.7 through 4.27 which uses Deckwer’s correlation [40] for the outside-coil heat
transfer coefficient is found to be sufficient to predict the effectiveness of single-tray
bubble column dehumidifiers. The design rules arising from the experimental results
of Chapter 3 should still be followed in the design of bubble column dehumidifier
geometry.
99
Model effectiveness
1
Data
1:1
0
0
Measured effectiveness
1
Figure 4-22: Agreement in parallel-flow effectiveness between modified model and
experiment
4.5
Chapter Conclusions
A simple and accurate model predicting bubble column dehumidifier performance
was presented and verified with experimental results. Due to the large volumetric
interfacial area in a bubble column, the gas-side resistance is found to be sufficiently
low that it can be neglected in modeling for columns deeper than 4 cm in the homogeneous flow regime. Care should be taken to minimize column height in order
to reduce the gas-side pressure drop across the dehumidifier. As is common for heat
exchangers, a tradeoff exists between heat flux and effectiveness (performance parameters representing capital and energy costs) when sizing the coil. However, it is
shown that high parallel-flow effectiveness can still be achieved with much smaller
coils than those used in previous work. The model developed herein can be used to
predict the performance of each stage of a multistage dehumidifier for the design and
optimization of HDH desalination systems.
100
Chapter 5
Conclusions
A predictive model was developed for the performance of bubble column dehumidifiers
for HDH desalination systems. The experimentally-validated assumption of neligible
gas-side resistance to heat and mass transfer greatly simplifies modeling. Excellent
agreement is demonstrated between the model and experimental results.
In addition, an experimental investigation of the heat transfer coefficient outside
the cooling coil resulted in practical design rules for bubble tray geometry, which are
summarized in the following section.
5.1
Design for Effective Transport
Bubble columns make great dehumidifiers for two reasons: first, the large gas-liquid
interfacial area greatly reduces the heat and mass transfer resistance associated with
condensation of a dilute vapor, and second, the turbulent dissipation of energy necessary for the operation of a bubble column leads to very high heat transfer coefficients
outside the cooling coil. Well-designed tray geometries will take advantage of both
of these aspects by minimizing the pressure drop and maximizing the heat transfer
coefficient.
To maximize the heat transfer coefficient while achieving good gas-liquid contact,
the cooling coil of a bubble column dehumidifier should be placed high enough to be
above the critical height for outside-coil heat transfer. Sparger holes should be placed
directly underneath the coil. To minimize gas-side pressure drop and maximize the
heat transfer coefficient, the liquid should be filled to a depth that just barely wets the
top of the coil during bubbling so long as the depth is sufficient for effective gas-liquid
mixing.
5.2
5.2.1
Future Work
Crystallization with HDH
HDH has great potential for use in crystallization, but this application requires investigation of alternative humidifier designs. Bubble column humidifiers are an interesting
101
alternative to packed beds especially when combined with heating coils to make the
most effective use of solar thermal or waste heat. Alternatively, a foaming humidifier
could potentially use surfactants to create a stable, rising foam as a liquid surface on
which to spray salt water. The lack of solid surfaces will force crystallization to occur
in the bulk, leading to simpler crystal removal than in a packed-bed humidifier.
5.2.2
Solar Heating and Humidification
HDH has the potential to be cleverly integrated with solar energy. Although some
solar-heated HDH systems have been implemented, there is still room for innovation.
Solar ponds, for instance, can be used as a low-cost brine heater. Alternatively, a
solar air heater-humidifier might improve collector efficiency by taking advantage of
phase change.
5.2.3
Dehumidifier Optimization
Improvements to the bubble column dehumidifier may reduce electric power consumption and capital cost while improving effectiveness. Innovations in sparger design may
significantly reduce the pressure drop without lowering the heat transfer coefficient.
Also, creative modifications to the stage design using calculated disruptions of mixing
have the potential to raise the parallel-flow effectiveness above the “limit” of one.
102
Appendix A
Uncertainty analysis of heat
transfer coefficient probes
The uncertainty of the heat transfer coefficient measurements using the probes whose
design is described in Chapter 3 is estimated using the method of propagation of
uncertainty. The heat transfer coefficient measurement is:
h̄ =
Q̇
,
Ap ∆T
(A.1)
where the relevant temperature difference is:
∆T = T̄p − TC .
The uncertainty in the surface temperature measurement is:
q
−1
uT̄p,meas = N
N u2T C = uT C N −1/2 ,
(A.2)
(A.3)
where N is the number of thermocouples on the surface.
In addition to the error associated with the individual temperature measurements,
a parameter must be introduced to account for the inaccuracy of calculating the average temperature of a non-isothermal surface with a finite number of measurements.
To estimate this error, the parameter Cparabola is defined. The value of Cparabola is the
error associated with measuring the average value of a symmetric parabola by averaging N evenly-spaced measurements. For the probes with 3, 4, and 5 thermocouples,
Cparabola is 0.0556, 0.0313, and 0.0200, respectively.
The uncertainty in the average probe surface temperature depends on Cparabola
and the maximum temperature difference within the surface:
q
uT̄p = (Cparabola (Tmin − Tmax )p )2 + u2T̄p,meas .
(A.4)
The uncertainty of the temperature difference between the probes and the bath
103
is:
u∆T =
q
u2T̄p + u2TC .
(A.5)
Finally, the fractional uncertainty in heat transfer coefficient can be related to the
fractional uncertainty in temperature difference:
r
2
dh̄
u Q̇
∆T
uh̄ =
u∆T =
.
(A.6)
u∆T = h̄
2
d∆T
Ap ∆T
∆T
The uncertainty associated with the individual K-type thermocouples on the probe
is 1.1 K. The uncertainty of the pool thermocouple (TC ) is taken to be lower (0.55
K, or half the original uncertainty) because of excellent agreement (±0.2 K) between
it and four other thermocouples during calibration in an ice bath. The uncertainty
associated with the area and heat transfer rate measurements are assumed to be
negligible in comparison to that of the thermocouples. The average temperature
difference in the natural convection tests was around 15 K, and in the bubbling tests
it was 6.38 K. The average difference between minimum and maximum probe surface
temperatures was 4.35 K.
Using these average values, the uncertainties of the small-, medium-, and largediameter probes are calculated to be 5.8%, 5.3%, and 5.0 %, respectively, during the
natural convection test. During bubbling tests, the measurement uncertainty of the
small, medium, and large probes are 13.7%, 12.4%, and 11.6%, respectively.
104
Appendix B
Bubble Column Dehumidifier
Model for EES
Note: set guess values based on [67]
$UnitSystem SI C J kg Pa
//inputs, e.g.:
V dot cool=.5[L/min]*Convert(L/min,mˆ3/s)
V dot ai=0.0015[mˆ3/s]
T cooli=22[C]
T ai=61.6[C]
//geometry, e.g.:
A col=11.2[in] ˆ2*Convert(inˆ2,mˆ2)
A po=.05[mˆ2]
D opipe=0.375*Convert(in,m)
A po=pi*D opipe*L pipe
A pi=pi*D ipipe*L pipe
D ipipe=D opipe-0.00078*2[m]
D coil=0.17
//general properties
g=9.81 [m/sˆ2]
P a=101000 [Pa]
sigma C=SurfaceTension(Water,T=T C)
//heat transfers
Q AB+Q cond=Q CE
Q CE=(LMTD CE)/(R DE+R CD)
Q tot=Q CE
105
Q AB=m dot da*(Enthalpy(Air ha,T=T ai,P=P a)
- Enthalpy(Air ha,T=T ao,P=P a))
+ m dot wo*(Enthalpy(Steam IAPWS,T=T ai,x=1)
- Enthalpy(Steam IAPWS,T=T ao,x=1))
+ m dot cond*(Enthalpy(Steam IAPWS,T=T ai,x=1)
- Enthalpy(Steam IAPWS,T=T B,x=1))
Q CE=m dot cool * (Enthalpy(Steam IAPWS,T=T coolo,P=P a)
- Enthalpy(Steam IAPWS,T=T cooli,P=P a))
Q cond=m dot cond * (Enthalpy(Steam IAPWS,T=T B,x=1)
- Enthalpy(Steam IAPWS,T=T B,x=0))
T ao=T C
T B=T C
//LMTD
A 5=T C-T coolo
A 3=(T C-T cooli)/(A 5)
LMTD CE=((T C-T cooli)-(T C-T coolo))/ln(A 3)
//resistances
R CD=1/(htc CD*A po)
R DE=1/(htc DE*A pi)
rho C=Density(Steam IAPWS,T=T C,P=P a)
V dot ai=A col*v g
m dot ai=V dot ai*rho ai
//AB mass transfer
M w=MolarMass(Steam IAPWS)
M a=MolarMass(Air ha)
P sati=Pressure(Steam IAPWS,T=T ai,x=1)
X 0i=P sati/P a
Y 0i=X 0i*M w/(X 0i*M w+(1-X 0i)*M a)
m dot da=(1-Y 0i)*m dot ai
m dot wi=Y 0i*m dot ai
m dot wo=m dot wi-m dot cond
X o=(m dot wo/M w)/(m dot wo/M w+m dot da/M a)
X o=Pressure(Steam IAPWS,T=T C,x=1)/P a
rho ai=Density(Steam IAPWS,T=T ai,x=1)+Density(Air,T=T ai,P=(P a-P sati))
cp C=Cp(Water,T=T C,P=P a)
k C=Conductivity(Water,T=T C,P=P a)
mu C=Viscosity(Water,T=T C,P=P a)
//CD HT to coil
htc CD=0.1*k Cˆ(1/2)*rho Cˆ(3/4)*cP cˆ(1/2)*mu Cˆ(-1/4)*gˆ(1/4)*v gˆ(1/4)
Nu CD=htc CD*D opipe/k C
106
//DE laminar flow through tube
v cool=V dot cool/A cpipe
A cpipe=pi*D ipipeˆ2/4
V dot cool=m dot cool/rho cool
Re cool=rho cool*v cool*D ipipe/mu cool
mu cool=Viscosity(Steam IAPWS,T=T coolo,P=P a)
rho cool=Density(Steam IAPWS,T=T coolo,P=P a)
k cool=Conductivity(Steam IAPWS,T=T coolo,P=P a)
Pr cool=Prandtl(Water,T=T coolo,P=P a)
Nu DE=htc DE*D ipipe/k cool
Z=2/11*(1+(1+77/4*Pr coolˆ(-2))ˆ0.5)
K 1=Re cool*(D ipipe/D coil)ˆ0.5
Nu DE=48/11*0.1979*K 1ˆ0.5/Z “change if turbulent!”
107
108
Bibliography
[1] E. W. Tow and J. H. Lienhard V. Analytical modeling of a bubble column
dehumidifier. In Proceedings of the ASME Summer Heat Transfer Conference,
Minneapolis, MN, July 14-19, 2013, 2013. Paper #HT2013-17763.
[2] David L. Chandler. Purification on the cheap, MIT News Office, February 2013.
[3] Arvind Chari.
A global strategy for the deployment of humidificationdehumidification technology. Master’s thesis, Massachusetts Institute of Technology, 2011.
[4] G. Prakash Narayan, Mostafa H. Sharqawy, Edward K. Summers, John H. Lienhard V, Syed M. Zubair, and M.A. Antar. The potential of solar-driven
humidification-dehumidification desalination for small-scale decentralized water
production. Renewable and Sustainable Energy Reviews, 14(4):1187 – 1201, 2010.
[5] G. P. Narayan, R. K. McGovern, G. P. Thiel, J. A. Miller, M. H. Sharqawy,
S. M. Zubair, M. A. Antar, and J. H. Lienhard V. Status of humidificationdehumidification desalination. In Proceedings of the IDA World Congress on
Desalination and Water Reuse, Perth, Australia, 2011.
[6] G. Prakash Narayan, Mostafa H. Sharqawy, John H. Lienhard V, and Syed M.
Zubair. Thermodynamic analysis of humidification dehumidification desalination
cycles. Desalination and Water Treatment, 16(1-3):339–353, 2010.
[7] M. A. Sievers and J. H. Lienhard V. Design of flat-plate dehumidifiers for humidification dehumidification desalination systems. Heat Transfer Engineering,
34(7):543–561, 2013.
[8] M. A. Sievers and J. H. Lienhard V. Design of plate-fin tube dehumidifiers for
humidification-dehumidification desalination systems. Heat Transfer Engineering, Accepted 2014.
[9] Geoffrey Wilmot Brundrett. Handbook of Dehumidification Technology. Butterworths, 1987.
[10] Steven Lam. Development of a multi-stage bubble column dehumidifier for application in a humidification dehumidification desalination system, Bachelor’s
Thesis, Massachusetts Institute of Technology, 2012.
109
[11] G. Prakash Narayan, Maximus G. St. John, Syed M. Zubair, and John H. Lienhard V. Thermal design of the humidification dehumidification desalination system: An experimental investigation. International Journal of Heat and Mass
Transfer, 58(12):740 – 748, 2013.
[12] Huaigang Cheng and Shichang Wang. Experimental investigation of modified
polypropylene shell-tube column using humidificiation dehumidification desalination process. Transactions of Tianjin University, 13(3):181–185, 2007.
[13] G. Prakash Narayan, Mostafa H. Sharqawy, Steven Lam, Sarit K. Das, and
John H. Lienhard V. Bubble columns for condensation at high concentrations
of noncondensable gas: Heat-transfer model and experiments. AIChE Journal,
59(5):1780–1790, 2013.
[14] Everett C. Barrett and Stephen G. Dunn. Design of direct contact humidifiers
and dehumidifiers using tray columns. Industrial & Engineering Chemistry Process Design and Development, 13(4):353–358, 1974.
[15] G. P. Narayan and J. H. Lienhard V. Thermal design of humidification dehumidification systems for affordable small-scale desalination. IDA Journal, 4(3):24–34,
2012.
[16] Yong Tae Kang, Atsushi Akisawa, and Takao Kashiwagi. Analytical investigation
of two different absorption modes: falling film and bubble types. International
Journal of Refrigeration, 23:430–443, 2000.
[17] Karan H. Mistry, John H. Lienhard V, and Syed M. Zubair. Effect of entropy
generation on the performance of humidification-dehumidification desalination
cycles. International Journal of Thermal Sciences, 49(9):1837 – 1847, 2010.
[18] Karan H. Mistry, Ronan K. McGovern, Gregory P. Thiel, Edward K. Summers,
Syed M. Zubair, and John H. Lienhard V. Entropy generation analysis of desalination technologies. Entropy, 13(10):1829–1864, 2011.
[19] G. P. Narayan, K. H. Mistry, M. H. Sharqawy, S. M. Zubair, and J. H. Lienhard V. Energy effectiveness of simultaneous heat and mass exchange devices.
Frontiers in Heat and Mass Transfer, 1(2):1–13, 2010.
[20] G. Prakash Narayan, Karim M. Chehayeb, Ronan K. McGovern, Gregory P.
Thiel, Syed M. Zubair, and John H. Lienhard V. Thermodynamic balancing of
the humidification dehumidification desalination system by mass extraction and
injection. International Journal of Heat and Mass Transfer, 57(2):756 – 770,
2013.
[21] Emily W. Tow and John H. Lienhard V. Heat flux and effectiveness in bubble
column dehumidifiers for HDH desalination. In Proceedings of the International
Desalination Association World Congress on Desalination and Water Reuse,
Tianjin, China, Oct 20-25, 2013. Paper #TIAN13-121.
110
[22] Hisham Ettouney. Design and analysis of humidification dehumidification desalination process. Desalination, 183(1-3):341 – 352, 2005.
[23] A. F. Mills. Heat Transfer. Prentice Hall, 2nd edition, 1999.
[24] W. M. Nagle. Mean temperature differences in multipass heat exchangers. Industrial & Engineering Chemistry, 25(6):604–609, 1933.
[25] T. D. Chen, J. C. Conklin, and V. D. Baxter. Dehumidification: Prediction of
condensate flow rate for plate-fin tube heat exchangers using the latent j factor.
In Proceedings of the ASME/JSME Joint Thermal Engineering Conference, San
Diego, CA, 1999.
[26] A.F. Mills. Mass Transfer. Prentice Hall, 2001.
[27] N.A. Gumerov. The heat and mass transfer of a vapor bubble with translatory
motion at high Nusselt numbers. International Journal of Multiphase Flow,
22(2):259 – 272, 1996.
[28] Harvey D. Mendelson. The prediction of bubble terminal velocities from wave
theory. AIChE Journal, 13(2):250–253, 1967.
[29] Yasuo Mori and Wataru Nakayama. Study on forced convective heat transfer in
curved pipes: (1st report, laminar region). International Journal of Heat and
Mass Transfer, 8(1):67 – 82, 1965.
[30] Yasuo Mori and Wataru Nakayama. Study of forced convective heat transfer in
curved pipes (2nd report, turbulent region). International Journal of Heat and
Mass Transfer, 10(1):37 – 59, 1967.
[31] Kiyomi Akita and Fumitake Yoshida. Gas holdup and volumetric mass transfer
coefficient in bubble columns: Effects of liquid properties. Industrial & Engineering Chemistry Process Design and Development, 12(1):76–80, 1973.
[32] D. W. van Krevelen and P. J. Hoftijzer. Studies of gas-bubble formation calculation of interfacial area in bubble contactors. Chemical Engineering Progress,
46(1):29–35, 1950.
[33] Johannes C. Kloppers and Detlev G. Kroger. The Lewis factor and its influence
on the performance prediction of wet-cooling towers. International Journal of
Thermal Sciences, 44(9):879 – 884, 2005.
[34] W. K. Lewis. The evaporation of a liquid into a gas. Mechanical Engineering,
44:445–446, 1922.
[35] Emily W. Tow and John H. Lienhard V. Measurements of heat transfer coefficients to cylinders in shallow bubble columns. In Proceedings of the 15th International Heat Transfer Conference, Kyoto, Japan, August 10-15, 2014 (submitted).
111
[36] Nigar Kantarci, Fahir Borak, and Kutlu O. Ulgen. Review: Bubble column
reactors. Process Biochemistry, 40(7):2263–2283, 2005.
[37] V.V. Konsetov. Heat transfer during bubbling of gas through liquid. International Journal of Heat and Mass Transfer, 9(10):1103 – 1108, 1966.
[38] S.C. Saxena and B.B. Patel. Heat transfer investigations in a bubble column
with immersed probes of different diameters. International Communications in
Heat and Mass Transfer, 18(4):467 – 478, 1991.
[39] A. K. Jhawar and A. Prakash. Heat transfer in a slurry bubble column reactor:
A critical overview. Industrial & Engineering Chemistry Research, 51(4):1464–
1473, 2012.
[40] W.-D. Deckwer. On the mechanism of heat transfer in bubble column reactors.
Chemical Engineering Science, 35(6):1341 – 1346, 1980.
[41] K. Akita and F. Yoshida. Bubble size, interfacial area, and liquid-phase mass
transfer coefficient in bubble columns. Industrial & Engineering Chemistry Process Design and Development, 13(1):84–91, 1974.
[42] H. Li and A. Prakash. Heat transfer and hydrodynamics in a three-phase slurry
bubble column. Industrial & Engineering Chemistry Research, 36(11):4688–4694,
1997.
[43] Rahman S. Abdulmohsin and Muthanna H. Al-Dahhan. Impact of internals
on the heat-transfer coefficient in a bubble column. Industrial & Engineering
Chemistry Research, 51(7):2874–2881, 2012.
[44] Moses Odongo O. Kagumba. Heat transfer and bubble dynamics in slurry bubble
columns with internals for Fischer-Tropsch synthesis of clean alternative fuels
and chemicals. PhD thesis, Missouri University of Science and Technology, 2013.
[45] H. Hikita, S. Asai, K. Tanigawa, K. Segawa, and M. Kitao. Gas hold-up in
bubble columns. The Chemical Engineering Journal, 20(1):59 – 67, 1980.
[46] Craig Hulet, Patrice Clement, Patrice Tochon, Daniel Schweich, Nicolas Dromard, and Jerome Anfray. Literature review on heat transfer in two- and threephase bubble columns. International Journal of Chemical Reactor Engineering,
7(1):–, 2009.
[47] S.C. Saxena and N.S. Rao. Heat transfer and gas holdup in a two-phase bubble
column: Air-water system – review and new data. Experimental Thermal and
Fluid Science, 4(2):139 – 151, 1991.
[48] S. Schluter, A. Steiff, and P.-M. Weinspach. Heat transfer in two- and three-phase
bubble column reactors with internals. Chemical Engineering and Processing:
Process Intensification, 34(3):157 – 172, 1995.
112
[49] H. Korte. Heat transfer in bubble columns with and without internals. PhD
thesis, University of Dortmund, 1987.
[50] K. Muroyama, S. Okumichi, Y. Goto, Y. Yamamoto, and S. Saito. Heat transfer
from immersed vertical cylinders in gas-liquid and gas-liquid-solid fluidized beds.
Chemical Engineering & Technology, 24(8):835–842, 2001.
[51] S. S. Kutateladze and M. A. Styrikovich. Hydraulics of gas-liquid systems. Gosenergoizdat, 1959.
[52] Y. T. Shah, B. G. Kelkar, S. P. Godbole, and W.-D. Deckwer. Design parameters
estimations for bubble column reactors. AIChE Journal, 28(3):353–379, 1982.
[53] Mounir Bouaifi, Gilles Hebrard, Dominique Bastoul, and Michel Roustan. A
comparative study of gas hold-up, bubble size, interfacial area and mass transfer
coefficients in stirred gasliquid reactors and bubble columns. Chemical Engineering and Processing: Process Intensification, 40(2):97 – 111, 2001.
[54] Wolf-Dieter Deckwer, Youssef Louisi, Ahmed Zaidi, and Milos Ralek. Hydrodynamic properties of the fischer-tropsch slurry process. Industrial & Engineering
Chemistry Process Design and Development, 19(4):699–708, 1980.
[55] F.J. Zuiderweg. Sieve trays: A view on the state of the art. Chemical Engineering
Science, 37(10):1441 – 1464, 1982.
[56] S.R. Syeda, A. Afacan, and K.T. Chuang. A fundamental model for prediction
of sieve tray efficiency. Chemical Engineering Research and Design, 85(2):269 –
277, 2007.
[57] J.B. Joshi and Y.T. Shah. Invited review: Hydrodynamic and mixing models for
bubble column reactors. Chemical Engineering Communications, 11(1-3):165–
199, 1981.
[58] I. Leibson, E.G. Holcomb, A. G. Cacoso, and J. J. Jacmic. Rate of flow and
mechanics of bubble formation from single submerged orifices ii: Mechanics of
bubble formation. AIChE Journal, 2(3):300–306, 1956.
[59] S.C. Saxena, R. Vadivel, and A.C. Saxena. Gas holdup and heat transfer from
immersed surface in two- and three-phase systems in bubble columns. Chemical
Engineering Communications, 85(1):63–83, 1989.
[60] Jinwen Chen, Fan Li, Sujatha Degaleesan, Puneet Gupta, Muthanna H. AlDahhan, Milorad P. Dudukovic, and Bernard A. Toseland. Fluid dynamic parameters in bubble columns with internals. Chemical Engineering Science, 54(1314):2187 – 2197, 1999.
[61] Pieter Samyn and Patrick De Baets. Friction and wear of acetal: A matter of
scale. Wear, 259(16):697 – 702, 2005.
113
[62] Stuart W. Churchill and Humbert H.S. Chu. Correlating equations for laminar
and turbulent free convection from a horizontal cylinder. International Journal
of Heat and Mass Transfer, 18(9):1049 – 1053, 1975.
[63] Rubin Battino and H. Lawrence Clever. The solubility of gases in liquids. Chemical Reviews, 66(4):395–463, 1966.
[64] R.W. Gross and J.M. Kuhlman. Three-component velocity measurements in a
turbulent recirculating bubble-driven liquid flow. International Journal of Multiphase Flow, 18(3):413 – 421, 1992.
[65] T. M. J. Fabritius, P. T. Kurkinen, P. T. Mure, and J. J. Hrkki. Vibration of
argon-oxygen decarburisation vessel during gas injection. Ironmaking & Steelmaking, 32(2):113–119, 2005.
[66] R. A. Ibrahim. Liquid Sloshing Dynamics: Theory and Applications. Cambridge
University Press, 2005.
[67] Emily W. Tow and John H. Lienhard V. Experiments and modeling of bubble
column dehumidifier performance. International Journal of Thermal Sciences,
Submitted June 2013.
[68] D. N. Miller. Scale-up of agitated vessels gas-liquid mass transfer. AIChE Journal, 20(3):445–453, 1974.
[69] M. Daous and A. Al-Zahrani. A simple approach to measuring the gas phase
heat and mass transfer coefficients in a bubble column. Chemical Engineering &
Technology, 29(12):1438–1443, 2006.
[70] Akhilesh Kumar, T. E. Degaleesan, G. S. Laddha, and H. E. Hoelscher. Bubble
swarm characteristics in bubble columns. The Canadian Journal of Chemical
Engineering, 54(6):503–508, 1976.
[71] R. Clift, J. R. Grace, and M. E. Weber. Bubbles, Drops, and Particles. Academic
Press, 1978.
[72] S.A. Klein. Engineering Equation Solver, F-Chart Software, LLC. 1992.
[73] Serap Kara, Balmohan G. Kelkar, Yatish T. Shah, and Norman L. Carr. Hydrodynamics and axial mixing in a three-phase bubble column. Industrial &
Engineering Chemistry Process Design and Development, 21(4):584–594, 1982.
[74] G. P. Narayan. Thermal design of humidification dehumidification systems for
affordable and small-scale desalination. PhD thesis, Massachusetts Institute of
Technology, 2012.
114