Experimental Investigation of two‐
phase flow in microchannels
Co‐current Absorption of Ammonia in Water to
Design an Innovative Bubble Plate Absorber
MSc. thesis
Ali Ammari
Division of Chemical Technology
Department of Chemical Engineering and Technology
School of Chemical Science and Engineering,
KTH Royal Institute of Technology
Stockholm, Sweden
November 2012
Experimental Investigation of two‐
phase flow in microchannels
Co‐current Absorption of Ammonia in Water to
Design an Innovative Bubble Plate Absorber
MSc. thesis
Ali Ammari
Supervisors
Björn Palm
Department of Energy Technology, School of Industrial
Engineering and Management, KTH Royal Institute of Technology
Rolando Zanzi
Department of Chemical Engineering and Technology, Division of
Chemical Technology, KTH Royal Institute of Technology
Gerwin Schmid
Austrian Institute of Technology
Examiner
Krister Sjöström
Division of Chemical Technology, Department of Chemical
Engineering and Technology, School of Chemical Science and
Engineering, KTH Royal Institute of Technology
Stockholm, Sweden
November 2012
i
absorbers because the higher contact surface area provides a higher mass transfer rate.
Furthermore, dispersion of bubbles in the bulk of liquid phase also exhibits better heat transfer
characteristics that facilitate the recovery of dissipated heat of the exothermic absorption.
In this context, plate heat exchangers are believed to be an option to be employed as absorber
in some applications. Commercial plate heat exchangers have only one inlet and outlet for a
working fluid and as a result, gas and liquid should be mixed before supplied to a gap
between the two adjacent plates. The consequence is the high risk of bubble mergence to form
a bigger bubble and to follow the shortest flow paths in vertical direction so that not all the
heat transfer surface can be effectively used. Furthermore this feature makes plate heat
exchangers sensitive to the angle of plate relative to the vertical which would be worst when it
is laid to its side on a horizontal plane.
Austrian Institute of Technology (AIT) develops an efficient Bubble Plate Absorber for
applications in high-pressure absorption systems and this work tries to investigate design
possibility of this Bubble Plate Absorber based on a plate heat exchanger equipped with
microchannels between plates.
Two sets of seven parallel microchannels same in shape and dimension were tested. The first
set had a continuous wall which means fluids could flow independently along the
microchannels; whereas, the other set was benefiting from some linkages between channels
that fluids could cross from one microchannel to another one. Ammonia vapour was injected
via one and two-holed distributors.
It was found that microchannels with continuous wall deliver higher concentration and less
unabsorbed bubbles at the microchannels outlet. In visual analysis by high-speed camera,
changing the vapour distributors from single-hole to double-hole had no significant effect on
the bubble distribution quality in lower flowrates; however, double-hole vapour distributor
showed better performance in higher vapours flowrates.
ii
Khalil Hesam belongs to the first groups of the scientists who graduated from Imperial College of London in the field of Combustion. After several years teaching in leading universities around the world, the sense of commitment to the development of the engineering sciences brought him to one of the smallest towns in Iran to train the young and ambitious students. He is the author of several engineering books which are being taught in different technical universities in Iran. I had the honour of being his student and I would like to share with him this moment of my scientific journey that I started with sitting in his classes.
This work is dedicated to Professor Dr. Khalil Hesam †
iii
LIST OF CONTENTS
Abstract ... i
List of Figures ... v
List of Tables ... vii
Nomenclature ... viii
Preface ... ix
CHAPTER ONE: INTRODUCTION Part I: Two-phase Flow and Microfluidic Systems 1-1 Terminology and Descriptions ... 1
1-2 Flow Characteristics ... 2
1-2-1 Flow Pattern ... 3
1-2-2 Flow Pattern Map ... 4
1-3 Fundamental Parameters ... 5
1-3-1 Dimensionless groups ... 5
1-3-2 Void Fraction and Hold-up ... 6
1-3-3 Pressure Drop ... 7
1-3-4 Bubble Rise Velocity ... 10
Part II: Absorption 1-4 Terminology and Background ... 11
1-5 Absorption Mechanisms ... 11
1-6 Mass Transfer ... 12
1-6-1 Two film Theory ... 12
1-6-2 Penetration Theory ... 14
1-6-3 Regular Surface Renewal ... 15
1-6-4 Other Endeavours ... 16 1-7 Absorption techniques ... 16 1-7-1 Packed Tower ... 17 1-7-2 Spray Tower ... 17 1-7-3 Falling film ... 18 1-7-4 Bubble type ... 18
CHAPTER TWO: EXPERIMENT Part I: Project outline and Motivation 2-1 Facts and Considerations ... 19
2-2 Motivation ... 21
2-2-1 Project Description and Background ... 21
2-2-2 Motivation ... 23
2-2-3 Superiority of design ... 22
2-2-4 Application ... 22
2-2-5 Challenges in Design ... 23
Part II: Experiment 2-3 Equipment and Setup design ... 24
2-3-1 Setup ... 24
2-4 Methodology ... 28
2-4-1 Setup Test ... 28
2-4-2 Pre-experiment ... 30
2-4-3 Experiment ... 32
CHAPTER THREE: RESULTS ANALYSIS AND CONCLUSION Part I: Visual Analysis 3-1 Bubble Behaviour ... 34
3-1-1 Inlet pattern ... 34
3-1-2 Bubble Motion ... 35
3-1-3 Unabsorbed Bubbles at the Outlet ... 38
Part II: Graphical Analysis 3-2 Aims and Scope of Analysis ... 40
3-3 Rich Solution Concentration ... 40
3-3-1 Number of Vapour Distributors’ Hole versus Rich Solution Concentration ... 40
3-3-2 Poor Solution and Cooling Water Temperature Difference versus Rich Solution Concentration ...…. 41
3-3-3 The effect of the Microchannels’ wall on Cooling Rate ... 42
3-3-4 The Effect of the Different Flowrates with Same Ratio on Absorption Quality ... 43
3-4 Mass Transfer ... 43
3-4-1 The Effect of Flowrates ... 43
3-4-2 The Effect of the Different Flowrates with Same Ratio on Diffusivity of Ammonia in Solution ... 44
iv
3-5-1 Momentum Pressure Drop ... 45
3-5-2 Static Pressure Drop ... 46
3-5-3 Frictional Pressure Drop ... 47
3-6 Bubble Rise Velocity ... 50
Part III: Conclusion ... 51
REFERENCES ... 54
APPENDIX Part I: Experimental Data ... 58
A- Experiment Data Sheet ... 58
B- Data Reduction ... 60
B-1 Chauvenet’s Method vs. Arithmetic Average ... 60
B-2 Graphical Method ... 61
Part II: Calculations C- Microchannel Dimensions and Basic Data... 64
D- Concentration of Rich Solution ... 65
E- Mass Transfer: ... 67
E-1 Diffusivity ... 67
E-2 Mass Transfer Coefficient: ... 70
F- Pressure Drop: ... 71
F-1 Pressure drop in the Lower half of the Absorber ... 71
F-2 Pressure drop in the Upper half of the Absorber ... 73
v
LIST OF FIGURES
Figure 1-1 Flow patterns : (a) bubble; (b) slug; (c) churn; (d) annular ... 4
Figure 1-2 Flow regime map for air-water in a vertical channel with s =0.3 mm, and w = 12 mm.adopted from (Cheng and Zhao, 1997). ... 5
Figure 1-3 Average liquid hold-up (adopted from Coulson and Richardson (1999)) ... 7
Figure 1-4 Relation between and X for two-phase flow (adopted from Coulson and Richardson (1999)) ... 10
Figure 1-5 Schematic illustration of Two-film Theory ... 13
Figure 1-6 Schematic illustration of penetration theory ... 14
Figure 1-7 Schematic illustration the surface renewal theory ... 15
Figure 2-1 Gas-liquid interfaces in different absorber designs; (a) Spray, (b) Falling film, (c) Bubble ... 20
Figure 2-2 microchannel bubble plate absorber concept (a) Two-phase flow in a microchannel ... 21
Figure 2-3 Single-phase plate heat exchanger as an absorber (a)Flow scheme and layout (from GEA)(b) Distribution of gas phase ... 22
Figure 2-4 Process Flow Diagram of Setup ... 25
Figure 2-5 Test rig at the Austrian Institute of Technology (AIT) ... 25
Figure 2-6 Absorber components designed at AIT ... 26
Figure 2-7 schematic view of sight glass and the microchannel grooves on it (a) cross section top view. Front view of microchannels with (b) continuous walls (c) linked microchannels ... 27
Figure 2-8 Inlet vapour distributors designed at AIT ... 27
Figure 2-9 effect of ambient temperature on vapour temperature (a) without insulation (b) with insulation ... 29
Figure 2-10 Different positions for video capturing a) inlet b) middle c) outlet of absorber ... 29
Figure 2-11an example of output data collected by Data Acquisition Device (rich solution density) ... 30
Figure 2-12 Effect of vapour-Solution ratio and Concentration ... 31
Figure 2-13 leakage between channel no. 5 and 4 ... 32
Figure 2-14 failure of front flange of absorber as a result of 40 Nm torque ... 32
Figure 3-1 Single-hole Vapour inlet with vapour/poor solution flow ratio of a) 0.08, b) 0.1 and c) 0.12 ... 35
Figure 3-2 Double-hole Vapour inlet vapour/poor solution flow ratio of a) 0.08, b) 0.1 and c) 0.12 ... 35
Figure 3-3 Process of size shrinkage for different bubble shape (Double-hole ∆TPS-CW =8 ˚C, V/PS=0.08) ... 36
Figure 3-4 slug flow pattern (∆TPS-CW =3˚C) ... 36
Figure 3-5 Slugs cross the linked points and are being divided into small bubbles, (vapour/poor solution =0.08, ∆TPS-CW=3˚C) ... 37
Figure 3-6 bubbles straight upward movement without crossing the link points, vapour/poor solution =0.053, ∆TPS-CW=3˚C ... 37
Figure 3-7 flow pattern at the middle of the linked wall column with single-hole distributor and vapour/poor solution = 0.08 with (a) ∆TPS-CW =8˚C (b) ∆TPS-CW =3˚C ... 38
Figure 3-8 observation of outlet conditions for unabsorbed ammonia bubble introduced by different methods ... 39
Figure 3-9 The effect of no. of vapour inlet on rich solution concentration (Linked channels, ∆TPS-CW ... 41
Figure 3-10 The effect of no. of vapour inlet on rich solution concentration (Linked channels, ∆TPS-CW =8˚C) ... 41
Figure 3-11 The effect of no. of vapour inlet on rich solution concentration (Continuous channels, ∆TPS-CW =3˚C) ... 41
Figure 3-12 The effect of no. of vapour inlet on rich solution concentration (Continuous channels, ∆TPS-CW =8˚C) ... 41
Figure 3-13 the effect of cooling water temperature on rich solution concentration (Single-hole vapour distributor) ... 42
Figure 3-14 the effect of cooling water temperature on rich solution concentration (Double-hole vapour distributor) ... 42
Figure 3-15 Comparison of Heat removal performances in continuous and linked channels at ∆TPS-CW =3˚C ... 42
Figure 3-16 Comparison of Heat removal performances in continuous and linked channels at ∆TPS-CW =8˚C ... 42
Figure 3-17 Different flowrates with same ratio vs. concentration of rich solution in (Linked microchannels) ... 43
Figure 3-18 Different flowrates with same ratio vs. concentration of rich solution (Continuous microchannels) ... 43
Figure 3-19 The effect of vapour and poor Solution flowrates on mass transfer coefficient (Isotherm, Continuous channels) ... 44
Figure 3-20 The effect of vapour and poor solution flowrates on mass transfer coefficient (Isotherm, Linked channels) ... 44
Figure 3-21 Mass transfer coefficients vs. Reynolds numbers of vapour and poor solution (isotherm, one channel, Continuous channels) ... 44
Figure 3-22 Mass transfer coefficients vs. Reynolds numbers of vapour and poor solution (isotherm, one channel, Linked channels) ... 44
Figure 3-23 The effect of different flowrates with constant ratio on diffusivity of ammonia in solution (Double hole) ... 45
Figure 3-24 Momentum pressure drop vs. vapour/poor solution volume flow ratio (Lower half of the absorber, Continuous channels) ... 45
Figure 3-25 Momentum pressure drop vs. vapour/poor solution volume flow ratio (Lower half of the absorber, Linked channels) ... 45
vi
Figure 3-27 Momentum pressure drop vs. vapour Reynolds number (lower half of the absorber, Linked channels) ... 46
Figure 3-28 Figure 3-29 Static pressure drop vs. homogenous Reynolds number of two-phase flow (upper half of the absorber, Linked channels) ... 46
Figure 3-29 Static pressure drop vs. homogenous Reynolds number of two-phase flow (upper half of the absorber, Linked channels) ... 46
Figure 3-30 Static pressured drop vs. void fraction (upper Half of the absorber, Continuous channels) ... 47
Figure 3-31Static pressured drop vs. density of homogenous two-phase (upper Half of the absorber, Continuous channels) ... 47
Figure 3-32 Static pressured drop vs. density of homogenous two-phase (upper Half of the absorber, Linked channels) ... 47
Figure 3-33 Frictional pressure drop vs. homogenous Reynolds number in two-phase flow (upper Half of the absorber, Continuous Channels) ... 48
Figure 3-34 Frictional pressure drop vs. homogenous Reynolds number in two-phase flow (upper Half of the absorber, Linked channels) ... 48
Figure 3-35 Frictional pressure drop vs. two-phase flow homogenous density, (upper Half of the absorber, Continuous channels) ... 48
Figure 3-36 Frictional pressure drop vs. two-phase flow homogenous density, (upper Half of the absorber, Continuous channels) ... 48
Figure 3-37 Frictional pressure drop vs. vapour/poor solution velocity ratio in separated two-phase flow (lower half of the absorber, Continuous channels) ... 49
Figure 3-38 Frictional pressure drop vs. vapour/poor solution velocity ratio rate in separated two-phase flow (lower half of the absorber, Linked channels) ... 49
Figure 3-39 Frictional pressure drop vs. poor solution Reynolds number, separated two-phase flow (lower half of the absorber, Continuous channels) ... 49
Figure 3-40 Frictional pressure drop vs. poor solution Reynolds number, separated two-phase flow (lower half of the absorber, Linked channels) ... 49
Figure 3-41 Frictional pressure drop vs. vapour Reynolds Number, separated Two-phase flow (lower half the absorber, Continuous channels, isotherm) ... 50
Figure 3-42 Frictional pressure drop vs. vapour Reynolds Number, separated Two-phase flow (lower half the absorber, Linked channels, isotherm) ... 50
Figure 3-43 Bubble rise velocity vs. vapour/poor solution velocity ratio ... 50
Figure 3-44 The effect of solution density on the bubble rise velocity (Double hole) ... 50
Figure 3-45 Bubble shape in Continuous channels (double-hole ∆TPS-CW = 3 [˚C] Vapour/Poor Solution Ratio = 0.1) ... 52
Figure 3-46 Bubble shape in Linked channels (double-hole ∆TPS-CW = 3 [˚C] Vapour/Poor Solution Ratio = 0.1) ... 52
Figure B-1 difference between arithmetic average value and Chauvenet’s method ... 61
Figure B-2 Raw data exported by Data Acquisition Device ... 61
Figure B-3 Data sorted ascending ... 61
Figure B-4 first stage of data reduction values <800 was removed ... 62
Figure B-5 second stage of data reduction values <830 was removed ... 62
vii
LIST OF TABLES
Table 1-1 Transition to turbulent reported in different works ... 3
Table 1-2 Equations to calculate mean value of viscosity in gas -liquid two-phase flow ... 9
Table 1-3 “c” values for different flow regimes ... 9
Table 1-4 Mass transfer coefficients for different theories ... 16
Table 2-1 Properties of PMMA used in experiment (adopted from Amsler & Frey Company product datasheet) ... 26
Table 2-2 pre-experiment setting parameters ... 30
Table 2-3 issued values from data acquisition device ... 31
Table 2-4 Algorithm of choosing desirable flow ratio ... 33
Table 3-1 Experiments with lowest unabsorbed ammonia introduced by different methods ... 38
Table 3-2 Experiments with highest unabsorbed ammonia introduced by different methods... 38
Table 3-3 Highest achieved rich solution concentration in constant vapour to poor solution flow ratio ... 52
Table A-1 Uncertainties for different measurement ... 58
Table A-2 Experimental Data Sheet ... 59
Table B-1 Chauvenet’s criterion for rejecting a record ... 60
Table C-1 Microchannel dimensions ... 64
viii
Nomenclature
Roman Suffix
A Absorbate
̇ area m2 A0 bulk concentration
C concentration mole/ m3 b bubble
D diffusivity or coefficient of diffusivity m2/s CW cooling water
d diameter m f frictional
friction factor (Fanning/Darcy) — G gas
friction factor (Moody) — h Hydraulic diameter
G gas mass flux kg/m2.s i interface
g earth gravity=9.8 m/s2 L liquid
mass transfer coefficient m/s m mean value
head loss m mo momentum
L film thickness m PS poor solution
l length m RS rich solution
M molecular mass kg S Solvent
N molar flux mole/m2.s s superficial
P pressure N/m2 st static
R shear stress acting on surface N/m2 T total
S slip rate — TPF two-phase flow
T temperature ˚C V vapour
t time s
time for surface renewal s Setup Components
u velocity m/s CO Coriolis
̅ molecular volume at normal boiling point m3/kg CW cooling water
X Lockhart and Martinelli’s parameter — FM flow meter
x vapour quality — GC gas cylinder
y distance in the direction of transfer m HC heating/cooling unit
HT heater
Greek HX heat exchanger
ρ density kg/m3 PS poor solution;
α void fraction — RS rich solution,
ϵ liquid holdup — RV relief valve
μ viscosity kg/m.s SG sight glass
σ surface tension N/m SV service valve
θ angle degree TC thermocouple
ϕ' friction factor — TK tank
ϕ Chisholm’s parameter — V vapour
ix
PREFACE:
With turning attentions toward sustainability as the common denominator of economy, environment and social welfare, two fundamental concepts have been on the focal of the decision-making circles in the field of energy. “Mitigation and Adoption” are two powerful arms of those policy-makers and think-tankers in confrontation with environmental and development challenges. The structure of the Mitigation is on the foundation of the efficient utilisation of the available energy resources parallel to the reduction of dependency on pollutant fuels. On the flip side, Adoption tries to introduce economically justified renewable energies as the substitutions for depleting fossil fuels and other problematic technologies like hydroelectric dams and nuclear power plants.
Furthermore, even though a high uncertainty in the forecasting the future of energy and its probable consumption patterns(Smil, 2003), the available knowledge and the where todays’ technology has stood, shows that having a so-called key-fuel as a response for all demands in different sectors likely will not happen in the near future. For instance, Hydrogen Economy, which might be one of the dream scenarios, currently has no transparent prospect in terms of the infrastructural feasibility and economic justifiability. On the other hand, after Industrial Revolution in the chase of higher life standards the migration from rural to urban areas boomed. Beside the devastating impacts on environment, urban–industrial societies became highly centralised and radically changed the pattern of population distribution. Some attempts such as empowering local business by distributing energy, particularly electricity was made hopping to reverse this trend which later on it turned out that it was too little too late. In the international scale and after Oil Crisis in 70’s imposed by Arab Petroleum Exporting Countries, those industrialised countries well comprehended that the reliance on the imported energies makes their economy vulnerable and negatively affects their positions in the international bargains. These national and international concerns and consideration as well as environmental aspects of the adventure gave birth to the theory “Decentralised Society”.
Decentralised Society is a self-energy sufficient society from local to nation scale where each energy consumer sector (transportation, building and industrial) is flexible enough to benefit from a rainbow of different forms of energy. In this society energy transmission and distribution costs is as lowest as possible and each so-called Centre has the capability of merging and supporting other Centres which can lead to an integrated network. Based on these considerations, the technological superiority in energy generation and consumption is classified according to the degree of subordination of those technologies from the principles of Mitigation, Adoption and Decentralisation.
As a sustainable energy technology, absorption heat-pumps potentially are able to exhibit a vivid colour among different spectrums of the energy conservation technologies. However, one of the main challenges on the way of adoption of absorption heat-pumps has been the emergence of the high-efficient compression heat-pumps which thanks to the long-lasting compressors driven by cheap electricity pushed absorption heat-pumps to a corner for some decades. For this reason, transition from potentiality into actuality has been driving force of the different investigations and investments in order to improve the performance of the absorbers as the major component in determination of the Coefficient of Performance (COP).
Austrian Institute of Technology (AIT) one of the leading research centres in Europe has been involved in an assignment with the aim of designing an innovative Bubble Plate Absorber based on conventional plate heat exchanger and this text will try to cover some aspects of this project based on experiments performed by the author. The behaviour of ammonia bubble inside the microchannels forms the main core of this work; however, some fundamental analysis like pressure drop along the microchannels will be addressed.
The text has been composed in three chapters which each chapter includes two parts. First part of introduction chapter (Chapter I) is dedicated to two-phase flow characteristics in microchannels and gives a general overview on some basic principles of microfluidic systems, second part addresses to the absorption process mainly from mass transfer point of view. Chapter II in first part will embrace the motivations for this project and in the second part explains the setup design and considerations. Chapter III analyses the collected data by experiments visually and graphically. Full series of experimental data can
x be found in the form of a Data Sheet in the Appendix where one sample experiment undergoes different calculations to show the method of the calculations.
Last but not least, I would like to express my sincerest thanks to Austrian Institute of Technology particularly the head of business unit Dr. Michael Monsberger for giving this opportunity for me to be involved in this project and also I extend my gratitude to the all of the engineers and staff in the Department of Energy particularly the project manager Mr Gerwin Schmid who criticized this work and Mr Peter Benovsky for his generous assistances during the experimentation.
1
1-1 Terminology and Descriptions
In physical sciences the term “phase” refers to one of the states of the matter in the standard conditions; namely solid, liquid and gas. The presence of different phases inside the distinct states is also possible and in fact determines some particular properties of the materials, like density, solubility or molecular structure. Examples for these multiphasic materials can be seen in immiscible liquid mixtures or Iron-carbon system which is well-known in metallurgy sciences. However, M. J. Assael et al. (2011) give a more practical definition which says:
“…if, the system has the same composition and density throughout, it is said to be homogeneous and is defined as a phase.” When a gas or vapour is introduced in a channel or tube with certain amount of liquid the structure of flow is called gas/liquid two-phase flow. Unlike flow regimes in single phase which are classified based to the structure of flow into laminar, transition and turbulent, two-phase flow are classified according to their interfacial structure into some different major groups. These groups which also can be called as flow regimes or flow patterns are separated flows, transitional or mixed flow and dispersed flow (Ishii and Hibiki, 2011). The whys and wherefores of different flow patterns lie within the interaction between gas phase and liquid phase. When gas and liquid enters a channel a turbulent region is created which is not necessarily as a result of turbulent flow of individual phases.
After this region the two-phases are separated (if no gas dissolves in liquid) and depends on flowrate and direction of the fluids flow forms one type of flow patterns which will be discussed in upcoming section. The higher velocity of gas due to lower density causes a continuous acceleration of the liquid phase. This difference in velocity is called slip velocity that affects the magnitude of hold-up. Hold-up is defined as
Chapter
One
Introduction
Part I
Two-phase Flow and Microfluidic Systems
"good thoughts, good words, good deeds" Zoroaster (c.628 - c.551)
2 volume of liquid or gas phase in volume of a segment of pipe or duct. Therefore, this can be expressed that the hold-up for liquid ( ) phase must be a value between zero and one; zero for single gas flow and one for single liquid flow. Gas hold-up or void fraction ( ) also can be expressed based on hold-up value as below:
(1.1)
The most well-known application of two-phase flow is in industrial processes like oil and gas industries, food industries etc. However, by emerging new technology based on microfluidic systems the range of application was extended to different fields of sciences like microscale cooling (Tuckerman and Pease 1981), or drug delivery, biotechnical analyses and telecommunication technologies (Henning 1998, Lipman 1999).
Rapid development and interests to understand characteristics of two-phase systems in micro-levels originated from this fact that the precise control over the individual phases and their behaviour maximise the performance of the systems working with different phases (Zhao, Middelberg, 2011). These advantages can be employed in microchannel heat sink technologies where they exhibit competitive features like small coolant and thermal resistance, small flowrate, and stream-wise temperature uniformity.
This must be noted that the term microchannel express a size less than millimetre; however, when it comes to miniaturize a giant industrial heat exchanger into a compact one the prefix micro- can be given to some millimetres in size as a common mistake.
1-2 Flow Characteristics
Investigation on two-phase flow regimes does not seem to be an old scientific trend when Kolev (2005) reports that the first methodical study on two-phase flow goes back to the year 1958 when a Soviet scientist Teletov published the article “On the problem of fluid dynamics of two-phase mixtures”. The first study involving microfluidic system was carried out by Wu and Little in which they were evaluating the performance of Joule Thompson refrigerators (Wu and W. A. Little, 1983).
With a short glimpse at the ratio of published papers to the date as well as new extensive investment on the research projects, two major points can be revealed; first the importance and the demand of the market for this technology and second, existence of high degree of uncertainty in previous achievements. In other words, as Collado (2007) says, there are some serious knowledge shortages in fundamental formulation of two-phase flow even in simple one-dimensional and steady condition. For better comprehension of Collado’s assertion and see how much uncertainty can exist even in works which are in agreement with previous works, let us have a look at one of the fundamentals of the fluid-flows in microchannel namely transition to turbulence flow.
Quest for identifying transition to turbulence in microchannels has led to some controversial results. Many experimental investigations reported an early transition to turbulency less than conventional range (1800-2300). For instance Wu and Little (1983, 1984) studied on a set of microchannels with hydraulic diameter of the 45.46 to 83.08 μm and obtained transition Reynolds number of 350 which was far below the number in macroscale. This became more challenging when Peng et al. (1994) and Peng and Peterson (1996) reported a range between 200 and 700 for a hydraulic diameter of 0.133–0.367 mm. this results were continuously supported with other experiments which summarized in table 1. In almost all experiments, the reason for this abnormal behaviour of fluids was being justified as a result of the higher relative roughness in microchannels.
3
Researcher Transition to Turbulence Dimensions µm
Wu and Little (1983, 1984) Re<350 rectangular microchannels hydraulic diameters ranging from
45.46 to 83.08 Peng et al. (1994),
Peng and Peterson (1996) 200<Re<700 rectangular microchannels 133–367 mm
Mala and Li (1999) Re <1000 microtubes
diameters ranging from 50 to 254 Pfund et al. 2000,
Gui and Scaringe 1995, Wu and Cheng 2003, Qu et al. 2000,
Guo and LI 2003a, b, Sabry 2000)
Re <1000 rectangular microchannels
Hegab et al. (2002) 2000<Re<4000 rectangular microchannels hydraulic diameters ranging from
112 to 210
Qu and Mudawar (2002) No evidence of early transition rectangular microchannels dimensions of 231 and713
Judy et al. (2002) No evidence of early transition rectangular microchannels hydraulic diameters ranging from
15 to150
Sharp and Adrian (2004) 1800<Re<2200 microtubes
diameters ranging from 50 and 247 (Li and Olsen 2006b)
(Li and Olsen 2006a). 1800 <Re< 2000 rectangular microchannels Table 1-1 Transition to turbulent reported in different works
Two recent researches were performed by a quite new experimental technique, microscopic particle image velocimetry (microPIV) (Santiago et al. 1998).
With gathering all those data, Hetsroni et al. (2005) concluded that there is no difference between smooth and rough microchannels in term of transition to turbulence. He reported that the transition takes place between 1800 and 2200. And early transition which was reported in previous works has been as a result of improper experimental conditions to be compared with theoretical predictions. This proof of similarity in flow regime between micro and macrochannels in single phase flow could be considered as a promising outcome to adopt some formulas for microchannels from macrochannels.
Circumstances are quite different in two-phase flows. Nevertheless, some phenomena can be predicted in two-phase flow like expansion due to upward movement or absorption inside the liquid phase but quantifying some major parameters even in macro-scale is complex. This can be attributed to unpredicted local behaviour of gas and liquid along the tube or channel and the interaction between them. For instance formulation of velocity distribution in two-phase flows is almost unmanageable because of unfeasibility of point to point measurement of velocity for every individual phase.
Despite all those obstacles fluid-mechanics in macroscale is more well-defined than micro-scale and this arises a question whether or not we can employ same formulation and approach toward microchannels. In fact, in large channels and channels with some few millimetres in hydraulic diameter, gravity controls the flow pattern where the effect of surface tension is minor. Instead, in microchannels these are viscosity, surface tension and inertia forces which influence the flow pattern. (Yarin et al, 2009)
1-2-1 Flow Pattern
Prediction of flow pattern in microchannels is a key factor to achieve higher accuracy in microfluidic devices. As a result of working in proportionally smallest scale, interaction between gas and liquid becomes more complicated to be measured. Flow regimes in pipes or ducts highly depend on orientation, which is why they are investigated in horizontal and vertical positions separately. Conversely, flow pattern in microchannels is more independent on direction of flow.
Even though different names given to different flow patterns; four major names are well-defined in term of occurrence and feature which are bubble flow, slug flow, churn flow and annular flow. Other patterns
4 which mostly appear in transition mode between distinct patterns are more arbitrary in naming and obey the definition of the author.
Bubble flow
If gas flowrate is very low compare to that of the liquid phase, gas travels along the channel in bubble form. There is continuity in liquid phase and gas bubbles are dispersed in that. Form shape and size point of view, unlike conventional channels which bubbles tend to be spherical, in microchannels bubbles are more two dimensional circular cylinder (Cheng and Zhao, 1997) or small sphere cap and tail with sizes less or a little larger than diameter of microchannel.(Yarin et al, 2009). [Figure 1(a)]
Slug flow
The “battle” between gas volumetric flow and liquid surface tension leads to a flow pattern called slug flow. This mainly can be observed in experiments the velocity of bubble is not adequate to overcome the strong surface tension force of the liquid bridge between them. It is reported that slug flow is the product of this entrance effect and microchannel characteristics does not have a role. (Serizawa et al, 2002).Furthermore, higher pressure drop in slug flow also goes back to the same fact that the higher surface tension hinders gas slug to freely slip and make a free pass along the wall; therefore, individual slugs are separated by a bulk of liquid called liquid bridge.
The presence of very small and isolated bubbles in the liquid bridge [Figure 1(b)] is due to successive separation of gas from the body of slug. This happens when trace amount of liquid trapped inside the slug tries to escape from the slug (Rouhani and Sohal, 1982).
The presence of these bubbles in microchannels is highly suspicious when Zhao and Bi (2001b) reported that all of their cases almost no isolated gas bubble was tracked particularly when slug size was relatively long.
Churn flow
Higher proportion of gas causes the velocity to overcome to the surface tension and breakdown the individual slugs. A relatively continuity in gas phase can be seen; however, liquid droplets still exist and surrounded by gas phase (Figure 1-1(c)).
Annular flow
In annular flow complete continuous flow of gas pushes the liquid phase completely though the wall in the form of film. In gas phase despite the absence of liquid phase, some small droplets can be observed because of the waves break-up from surface of the liquid film (Figure 1-1-(d))
Figure 1-1 Flow patterns : (a) bubble; (b) slug; (c) churn; (d) annular
1-2-2 Flow Pattern Map
Two-phase flow pattern map is a graphical method to predict the appearance of different flow patterns. They are normally drawn in the form of either mass flux versus vapour quality or superficial liquid
5 velocity versus superficial vapour velocity (Revellin et al. 2006). A typical flow pattern map is presented in figure 2. ( ) (1.2) (1.3)
and superficial velocity and x is quality of gas.
Figure 1-2 Flow regime map for air-water in a vertical channel with s =0.3 mm, and w = 12 mm. adopted from (Cheng and Zhao, 1997).
1-3 Fundamental Parameters 1-3-1 Dimensionless groups
Dimensionless groups are quantities which carry some characteristics of the fluid flows. They facilitate calculations and analysis of data. On the other hand development of these groups helps engineers to characterise a flow component by logical inference (proof of component from a total) or induction (proof of a total by definition of the components). For instance, one famous dimensionless number like Reynolds number is a set of qualities (viscosity) and quantities (length scale) which present flow regime in a channel or surface; whiles, with having this value from mathematical methods (analogy) one can predict those qualities and quantities.
(1.4)
where are density, velocity, characteristic length (which in here is hydraulic diameter)and viscosity of fluid. To take into account the effect of surface tension σ which is one of the most important parameters in calculation in microfluidic system, Weber number is defined as:
(1.5)
Bond or Eötvös number is the ratio of buoyancy force to surface tension force which is defined as below:
( ) (1.6)
This correlation can be expressed in the form of ratio between gravity to surface tension which is identified with Ω:
(1.7)
6
(1.8)
Froud number also can be arranged as:
√
ψ number relates viscose force to surface tension where u is a characteristic velocity which can be bubble rise velocity according to the Suo and Griffith (1964) or fluid velocity.
(1.9)
For prediction and craterisation of bubble shape inside the channel, and Morton numbers can be employed together.
(1.10)
1-3-2 Void Fraction and Hold-up
Magnitude of an area which is occupied by gas phase in two-phase flow systems is defined by void fraction. Void fraction indicates not only the quality of flow in tube but also is a key factor for prediction of pressure drop in microchannels. General expression for void fraction “ ” comes as below:
(1.11)
where is the area of cross section which occupied by gas or vapour and is total area of the
tube occupied by both liquid and gas phase.
Void fraction can be calculated for both homogenous two-phase flows in which velocity of gas and liquid is equal and for separated flow model by the equation:
( ( )
)
(1.12)
where is slip rate which is defined as the ratio of velocity of gas to velocity of liquid phase:
(1.13)
Where and can be found by equations (1.2 and 1.3). For homogenous two-phase flow because , value is 1.0.
Void fraction is the hold-up value for gas which means this can be possible to express hold-up based on liquid phase. Lockhart and Martinelli (1949) introduced the parameter X to find hold-up for liquid phase.
√ (1.14)
Where and are the pressures for individual phases with this consideration that pressure drop for each phase must be calculated their own independent characteristic as if no second phase exist.
(1.15) ⁄
7 The correlation between X parameter and Lockhart-Martinelli to measure liquid hold-up is shown in figure 3.
Figure 1-3 Average liquid hold-up (adopted from Coulson and Richardson (1999))
1-3-3 Pressure Drop
Total pressure drop in two-phase flow in tubes or ducts is the sum of the pressure drop due to friction (frictional), pressure drop due to density and elevation (static) and pressure drop due to acceleration (momentum) (Revellin and Thome, 2007) and can be expressed as below:
( ) ( ) ( ) ( ) (1.16)
General form of static pressure drop is expressed as: (
) ( ) (1.17)
where is the angle of channel respect to the horizon . This is seen that static pressure drop is equal to 0.0 for horizontal flow.
The momentum pressure drop brings into account the effect of acceleration of the flow due to flashing or adiabatic effect and can be calculated as follows (Revellin and Thome, 2007):
(
)
( )
(1.18)
Where x indicates vapour quality and G and L are gas and liquid mas flux respectively. For single-phase flow or if vapour quality is constant all along the channel the momentum pressure drop will be equal to 0. Frictional pressure drop is given in the form head loss and is expressed as below:
(1.19)
Due to variety of expressions for friction factor and extreme importance of clear understanding of this term when it comes to calculation therefore a comparative glimpse at their definitions seems to be necessary.
Pressure drop due to friction is produced by shear stress R on the wall of the tube or duct. Two scientists Stanton and Pannell (1914) after measuring pressure drop in pipes with variety of diameters introduced a dimensionless factor and named it friction factor.
(1.20)
Equation 1.8 Lockhart-Martinelli
8 With taking shear stress as a function of velocity , hydraulic diameter , pressure P, viscosity µ, and
magnitude of the surface roughness ε; Moody (1944) expressed friction factor as a function of Reynolds number, and relative roughness of tube.
(1.21)
The Darcy friction factor often is employed in calculations. This friction factor which sometimes is called Fanning factor also is expressed as below:
(1.22)
Therefore based on what have been said the head loss can be rewritten as:
(1.23)
For laminar flow (Re<2000) pressure drop is only a function of Reynolds number and friction factor is defined as:
(1.24)
where is a constant depends on geometry. For circular tubes and pipes this number is 64 (Coulson and Richardson 1999) and for noncircular duct can be found via Shah and London (1978) where is aspect ratio of the duct.
( ) (1.25) In the case of microchannels it is suggested to use Churchill and Usagi (1972), model which is an explicit method for laminar flows. Moody friction factor can be predicted as follows:
[( ) ( ) ] (1.26) where: [( ) ]
There are two different approaches in pressure drop calculation in two-phase flow in channels. First one assumes that the gas and liquid phases are well-mixed and therefore with introducing some average fluid properties, the whole system can be taken as a single homogenous phase flow. This model shows more accuracy for two-phase flow near the critical point where two-phases are well-mixed and some average values can be valid for the flow. For liquid and gas flows with very high mass flowrate where flow pattern becomes more bubbly or misty, this method can be applied as well. Second approach is based on artificially division of two-phase flow into two independent streams. The core assumption is the velocity of each phase is uniform all along the channel. This model shows successful results in power plant industry and promising way to predict pressure drop in refrigeration system (Awad and Muzychka, 2010). Homogenous model
As it mentioned above, some mean values must be introduced in order to substitute in formulas as a single phase. In this section applicable parameters will be shown.
The average density can be calculated by:
( ) (1.27)
To calculate Reynolds number average flow property is used:
9 The value which is average viscosity of two-phase is not well-understood and variety of approaches
in calculation exists. Based on some characteristics of a system, containing gas-liquid two-phase flow one of the equations listed in table 1-2 can be employed.
Researcher Viscosity Average Value Remarks McAdams et al. (1942) ( )
based on the mass averaged value of
reciprocals (1.29)
Cicchitti et al.
(1960) ( ) based on the mass averaged value (1.30)
Dukler et
al.(1964) ( )
for homogeneous two-phase flow, slip velocity is 1.0 therefore, α can be calculated
by equation 1.5 (1.31)
Lin et al.
(1991) ( ) customized version of the McAdams model, based on their experiment with R-12 (1.32)
Bittle and Weis
(2002) ( )
based on weighting volume friction for
viscosity acting in series (1.33) Garcıa et
al.(2003,2007) (
)
( )
based on calculated Reynolds number using kinematic viscosity of liquid flow instead
of the average value because the
frictional resistance of the mixture is due mainly to the liquid
(1.34) Table 1-2 Equations to calculate mean value of viscosity in gas -liquid two-phase flow
Separated flow models
One of the practical ways to calculate pressure drop due to friction was suggested by Lockhart and Martinelli (1949) that is based on calculation of head loss for each individual phases. The term head is an expression that is energy per unit weight of fluid and has dimension of length. After obtaining head loss for every single phase according to the equation 1.19, they are correlated by equation 1.14. Chisholm (1967) developed this method by correlating value and X by:
(1.35)
Finally the total pressure drop of two-phase flow can be obtained by:
(1.36) (1.37) criteria c turbulent/turbulent flow 20 turbulent liquid/streamline gas 10 streamline liquid/turbulent gas 12 and 5 for streamline/streamline flow 5
Table 1-3 “c” values for different flow regimes
As can be seen the value of can be expressed in the form and it is suggested to use this value if . The figure 1-4 illustrates the relation between different parameters of and X.
10 Figure 1-4 Relation between and X for two-phase flow (adopted from Coulson and Richardson (1999))
Frictional pressure drop probably is the most challenging form of pressure drop calculations. Several equations for different conditions and considerations have been presented. However, studies on the range of success of these equations have not led to some sort of certainty yet. Reviewing of previous works clear that researchers have been working on their “Unique” setup and insulations which based on basic principles of methodology of science should not be expected to lead some “universal” outcomes. Therefore, author believes that a set of parameters is required in order to introduce “well-definability” of an experiment to achieve a relatively higher confidence in comparison of outcomes.
At the end of this section this must be mentioned that different other models have been investigated and are being applied in pressure drop calculation i.e. Friedel (1979), Chisholm (1973) and Muller-Steinhagen and Heck (1986) which are being introduced for further study.
1-3-4 Bubble Rise Velocity
In vertical channels the velocity of bubble rise is used for calculations such as equation 1.9. In fact there is no theoretical analysis to predict the velocity of bubble in vertical channels and those empirical equations are mainly developed based on visual studies which has been resulted introduction of different equations; though, it has been tried to bring some most applied equations in different works. Stockes’s equation (1880) expressed the velocity for small bubbles respect to the channel size with relatively perfect spherical shape as bellow:
( )
(1.38) For large bubble size the effect of surface tension and viscosity can be neglected then bubble rise velocity can be expressed as Davies and Taylors’s (1950) equation.
( ) (1.39)
Rise velocity of slug bubble along a microchannel also can be calculated by Nikline’s equation (1962)
( ) ( ) (1.40)
Where, C1=1.2 and C2=0.36. It must be noted that Niklin’s arrangement was for stable slug flow;
therefore, coefficients do not depends on the height of the column. For that reason Infante Ferreira et al.(1984) suggested coefficients C1=1.4 and C2=0.29 for height around 0.5.
11
1-4 Terminology and Background
In chemical engineering science the term absorption is a chemical or physical process in which one or more components of a gas mixture enters, or specifically dissolves, in a bulk of a liquid. From terminology point of view, the gaseous phase is called solute or absorbate whereas liquid phase is called solvent or absorbent. In pollution control technologies the term scrubber is used instead of absorber and absorbate is called scrubbing liquid. In energy and material production industries, it is often necessary to recover the solvent by removing absorbate from absorbent. This recovery process is called generating or desorbing; however, in separation processes is referred as stripping.
Classically, absorption is recognised as a separation method which can be employed in fuel production or emission control. However, if gaseous phase can be absorbed in large quantity in a liquid and be desorbed easily, one can use absorption to convey or circle a gas or vapour along a process. For instance if a process requires circulation of a gas or vapour, due to higher costs of compression compare to that of pumping, this can be justified to pump solution rich of dissolved gas along the process and then recover on the desired side. In almost all applications higher ratio of solute to solvent is favourable in order to consume less power for circulation.
1-5 Absorption Mechanisms
As far as Experimental Sciences show, to the knowledge of the author, all the scientific phenomena merely are physical which means chemical or physical behaviours and reaction of a substances in contact with other substance(s) is originated from their molecular and atomic structure. Therefore, division of absorption into chemical and physical fundamentally is only a different approach toward the mechanism
Part II
12 of the interaction between solute and solvent. However, this report will be loyal to the common sense of engineering sciences and the classification of ammonia absorption in water as a physical process. It means from this section onward, the word absorption will refer to physical absorption and transport phenomena will be main core of the consideration.
Gas absorption is categorised as a transfer of mass across the phase boundary. It means one component from gas phase defuses into the liquid phase until liquid phase approaches to saturation condition. To have a clear shot about this definition let us have a brief view on a basic equation on mass transfer which governed by Fick by 1855. Fick’s law suggests a linear correlation between driving force for mass transfer which is concentration gradient and a coefficient for mass transfer which itself depends on variety of factors like diffusivity and etc. (Coulson and Richardson 1999)
(1.41)
Where is the molar flux, CA is the concentration of element A, y is the distance in the direction of
mass transfer and DAB is the diffusion coefficient of substance A in B that can be predicted empirically by
Gillilands’(1934) suggested formula:
√( ⁄ ) ( )
( ̅ ⁄
̅ ⁄
) (1.42)
Where M is molecular mass and ̅ is molar volume at normal boiling and P is total pressure. According to this equation, the diffusing substance moves from a place of relatively high concentration into that of the low concentration and mass transfer resistance lies within a distance travelled by diffusing substance. For calculation of diffusivity in liquid phase, Wilke and Chang (1955) introduce an empirical formula as follows:
( )√
̅ (1.43)
Where is associated factor (for water =2.6) and in molecular mass of the solvent .In a given condition the diffusivity of an element is an inherent property of that element which means high solubility of a gas, for instance, in a liquid originated from its molecular characteristics. This expression can be a guideline in order to choose proper absorbents for certain absorbate.
In order to lessen operating-costs, in a gas absorption unit, all attempts are to enhance the mass transfer rate. This gives less solvent consumption as well as less energy consumption for circulation and regeneration of the solvent. To increase the magnitude of the NA, one can try to decrease the magnitude
of the y means the diffusion path by broadening contact surface area between phases. William Gossage probably can be named as the inventor of first absorption tower who filled a derelict windmill with brushwood and gorse and introduced the hydrochloric acid gas released as a result of the Leblanc process of alkali production from the bottom, and water at the top he observed that no or little fumes from top. This was protected by British patent of 1836 of course (Danckwerts, 1965). It easily can be seen that how increasing surface area could help William’s absorber to shorten the penetration distance of the gas into the water and relatively full absorption of the hydrochloric acid gas. With this example next part of the introduction is welcomed with focus on the mass transfer theories in gas absorption.
1-6 Mass Transfer
As it was mentioned above, absorption of one gaseous phase in a liquid phase takes place across the phase boundary. In other words, unlike diffusion of components in single phase where continuity exists, in the absorption of gas, the soluble gas diffuses into an interface, dissolves in the liquid and goes into the bulk of liquid (Coulson and Richardson 1999). This continues until system reaches to the saturation condition where equilibrium between gas and liquid establishes. The progression of absorption and the prevailing conditions on either phase has been propounded in three different theories, two-film theory, penetration theory and regular surface renewal theory.
1-6-1 Two film Theory
As Cussler (2007) says, two-film theory probably is the simplest theory for interfacial mass transfer. In fact Whitman’s two-film theory is not only simplest theory but also was the first attempt to understand
13 the occurring conditions during transfer of a component from one phase to another. Never the less, before describing the theory it is necessary to mention that this model does not fully represent most practical equipment but can be applied in experimental data (Coulson and Richardson 1999). In this theory the diffusing component is brought into an interfacial area with driving force of turbulent eddies formed by molecular movements. This turbulency then dies out in interfacial boundary and forms a laminar layer in both gaseous and liquid sides. Due to the turbulency outside of the interfacial area, resistance to the mass transfer is only expected in these layers namely gas phase laminar layer and liquid phase laminar layer. All components are in equilibrium in this region and the absorbate enters the liquid film and afterwards leaves it for the bulk of the absorbent. Figure 1-5 shows two hypothetical layers which are representing resistances to mass transfer area in both sides and concentration distribution. It is obvious that for highly soluble gas-liquid pairs, say ammonia-water, main resistance lies within liquid film and therefore concentration gradient in gas film will be negligible.
G A Phase 1 CA02 C o n ce n tr a ti o n CA01 Phase 2 E Distance y L1 L2 F B D H C CAi1 CAi2 Interface
Figure 1-5 Schematic illustration of Two-film Theory
The main assumption in this model is that the time for establishing equilibrium is sufficiently short in a way that can be called “immediate equilibrium”. For this reason, each film will have characteristics similar to the equimolecular counter diffusion. It means that concentration on both phases will remain same and no net transfer of molecules takes place. Consequently, to obtain overall mass transfer rate for both side first an expression for counter diffusion is required which can be obtained by integrating equation (1.41) at constant temperature and pressure which yields:
( ) (1.44)
This equation can be rewritten in term of a coefficient for mass transfer when driving force is difference in molar concentration.
( ) (1.45)
Thus, for first phase the rate of transfer can be written as below:
( ) ( ) (1.46)
Similarly for other phase:
( ) ( ) (1.47) According to the assumption of an equimolecular counter diffusion, there is no accumulation of components at the interface. Thus:
(1.48)
Since on both side of the interfacial area equilibrium was assumed, the value of CAi1 and CAi2 can be
14 CA01 and CA02 are fixed and concentration difference is proportional to the mass transfer coefficient which
itself is a function of diffusivity and the diffusion distance in transfer direction. It means if the relative value of the mass transfer coefficient changes, this will affect the concentration at the interface. It is why with increasing turbulency in the fluid decreases the thickness of the laminar layers and eventually increases mass transfer coefficient.
This predictability that bulk flow enhances the mass transfer rate can be unearthed from Stefan’s law which a factor of ⁄ is multiplied in Fick’s equation as the reprehensive of bulk flow effect.
(1.49) where: ( ⁄ ) (1.50)
and CT is total concentration. This ratio is called drift factor and the induced convective flow which
enhances the mass transfer rate is called Stefan’s flow (Cengel, 2006).
1-6-2 Penetration Theory
The attempt to find whether or not resistance to mass transfer exists at the interfacial area, led Higbie to propound the penetration theory in the year 1935. In this model it is suggested that the mass transfer coefficient is inversely proportional to the square root of the exposure time.
√ √ (1.51)
hDm is the mean value for mass transfer coefficient in exposure time . Baehr, and Stephan (2011)
recommend that to obtain useful value for the mass transfer coefficient, time can be calculated via t = d/w
where d is the diameter of the droplet or bubble which is rising or falling and w is the mean velocity of rising or falling. They also reported that contact time determination for a liquid falling through a packing
is more difficult than gas flowing through it. Figure 1-6 is derived from Higbie’s calculations and shows
concentration gradient in a falling film exposed to a soluble gas. As like as two-film theory in the
penetration theory, eddies are the agents of bring the fluid element to the interface. However, the effect of the velocity gradient is neglected and it is assumed that the fluid has a constant velocity at all depths from surface. At the initial exposure time t = 0 concentration is zero in any depth of the liquid film and as time goes by, the concentration gradient is built up and concentration of the liquid reaches saturation after an infinite time. Similarly, other boundary conditions can be defined as below:
t = 0 0<y<∞ CA=CA0 t > 0 y=0 CA=CAi t > 0 y=∞ CA=CA0 CA Falling Liquid Gas CA01 CAi Exposure time (arbitrary units) y=∞ 500 100 1
Figure 1-6 Schematic illustration of penetration theory
y, Distance from surface
t=∞
15
Substitution of the boundary conditions in unsteady state mass transfer equation, namely:
(1.52)
gives an expression for calculating concentration at any depth from surface at given time.
( √ ) ( √ ) (1.53) ( ) ( ) (1.54) ( ) √ ∫ (1.55)
With this correlation the important term effective depth also can be introduced which determines a depth required in liquid phase in order to reach maximum absorption. In other words, the effective depth is an optimum distance from the free surface layer which concentration reaches to the saturation condition. This number is useful for process designers to calculate minimum flowrate of the solvent in absorption process. It must be noted that Higbie in his experiment assumed that the depth of the liquid film is too larger than the effective depth and diffusion is only considered in y direction. Consequently convection is more import than diffusion (Cussler, 2007).
To obtain the mass transfer rate at any distance of y and time t, substituting (1.55) into (1.53) and integrating gives: ( ) ( )√ ⁄ (1.56)
and for y=0 mass transfer per unit of the surface area:
( ) (
) ( )√ (1.57)
√( ) is the point value for the mass transfer coefficient. (the proof of expression 1.51)
1-6-3 Regular Surface Renewal
This theory in fact is an extended model of the penetration theory (Baehr and Stephan, 2011). In penetration theory as Cussler (2007) believes, due to static mode assumed in the interface, the mechanism image (figure 1-6) is unrealistic and “naïve”. Higbie’s model could calculate the mass transfer coefficient by square root dependency on D, but quest for introducing a better correlation led to the Dankwert’s Regular Surface Renewal theory in 1951. The word “better” in here might be interpreted as introduction of a streamlined mechanism which embraces the both steady and unsteady state conditions. In this theory, on the contrary to the two previous theories, the interfacial area is not completely at steady state condition. It means that, a small bulk region close to the interface always is in the exchanging of element with liquid bulk (Figure 1-7).
y, Distance from surface
Gas Interfacial Region Well-mixed bulk Region at CA CAi CA1 CA CA
16 In other words, in the interfacial area mass transfer takes place as the means of penetration theory but a fraction of volume of this region is exchanging with new element from so-called bigger bulk. Consequently, the contact times between phases are not same at all positions as Higbie had assumed and the absorption process falls off as a result of decrease in concentration gradient during the exposure time. If is defined as the time for surface renewal and the number of the transferred moles of component A in area ̇ at time t , the Higbie’s equation (1.57) can be rewritten as:
( )√ ̇ ∫
√ (1.58)
( )√
(1.59)
With calculation per unit area during the exposure time te mass transfer rate can be expressed as:
( )√
(1.60)
Thus, higher mass transfer rate can be expected in shorter surface renewal time. Like the contact time in penetration theory, is not precisely measurable for any equipment (Baehr and Stephan, 2011); however, it can be concluded that the more turbulency the shorter time for surface renewal and higher mass transfer rate.
1-6-4 Other Endeavours
Attempts to find most realistic mechanism for mass transfer in interfacial area has led to several theories e.g. Film Penetration Theory (Dobbins, 1956 and Toor & Marchello, 1958) or Random Surface Renewal.
Levich (1962), Lochiel and Calderbank (1964) could solve mass transfer coefficient via calculation of the concentration distribution close to the interfacial region. They solved convection–diffusion equation based on the characteristics of the fluid in a place near to the interfacial region.
Lamon and Scott (1970) worked on an eddy cell model which was a further development on turbulent theory. Recently, Jajuee et al. (2006) developed a Surface-renewal-stretch (SRS) model. This work is based on penetration theory for fluid–fluid interacting systems, where both turbulence and bulk motion exist. They claim that as a result of the taking into account the turbulence and bulk flow effects as well as disturbed layer at the interfacial region, their developed mathematical model must be more accurate than conventional theories. Table 1-4 gives an overview of mass transfer coefficient in different theories.
Two film Theory
(Whitman’s 1925)
Penetration Theory
(Higbie 1935) ⁄ √
Regular Surface Renewal
(Dankwert’s 1951) ⁄ √
1-7 Absorption techniques
With taking aside the type of absorbers and their applications, an absorption unit consists of at least one gaseous side and one liquid side. Gas phase properties are usually given and proper liquid phase as the absorbent must be decided. Series of predefined criteria for liquid phase which must be met in order to have an efficient operation were suggested by Treybal (1980):
