Eindhoven University of Technology MASTER Twophase flow …Eindhoven University of Technology MASTER Twophase flow in cracking furnaces : residence time in annular dispersed flow and

Eindhoven University of Technology

MASTER

Twophase flow in cracking furnaces : residence time in annular dispersed flow and bubbledetachment in turbulent flow

Niestadt, B.J.

Award date:1997

Link to publication

DisclaimerThis document contains a student thesis (bachelor's or master's), as authored by a student at Eindhoven University of Technology. Studenttheses are made available in the TU/e repository upon obtaining the required degree. The grade received is not published on the documentas presented in the repository. The required complexity or quality of research of student theses may vary by program, and the requiredminimum study period may vary in duration.

General rightsCopyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright ownersand it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights.

• Users may download and print one copy of any publication from the public portal for the purpose of private study or research. • You may not further distribute the material or use it for any profit-making activity or commercial gain

https://research.tue.nl/en/studentthesis/twophase-flow-in-cracking-furnaces--residence-time-in-annular-dispersed-flow-and-bubble-detachment-in-turbulent-flow(1cd5872b-2f1c-46ab-b425-9a2e0432cfeb).html

Twophase Flow in Çracking Furnaces

Residence Time in Annular Disper~ed Flow and Bubble Datachment in Turbulent Flow

Masters thesis

B.J.Niestadt Eindhoven Univarsity of Technology

Department of Applied Physics

Pubtic version

~ -

Techni~che Universiteit tU_./ Eindhoven

Twophase Flow in Cracking Furnaces Residence time in Annular Dispersed Flow and Subbie Deiachment in Turbulent Flow

Master's thesis by: Bart J. Niestadt Eindhoven University of Technology

Supervisor: Supervisor TUE:

Committee:

dr. 8. Broers prof. dr. MAJ. Miehels

prof. dr. MAJ.Miehels dr. B. Broers dr. L.P.H. de Goey dr. H.J.H. Clercx dr. ir. L.P.J. Kamp

Eindhoven University of Technology Department of Applied Physics

Den Dolech 2 Eindhoven

Shell International Oil Products Shell Research and Technology Centre Amsterdam ORTET/2

Badhuisweg 3 Amsterdam-Noord

November 1997

NOTIFICA TION This is the public version of a confidential report. Parts of the original report were leftout of this version. Somelimes this had to be done at the costof clarity, through no fault of the author. The author therefore apologises for this to the reader.

Summary

In thermal cracking of heavy oil fractions, one of the steps in the refining of crude oil, twophase flow plays an important role. Twophase flow is the collectiva name of flows of mixtures of gas and liquid. The interaction between the two phases in this type of flows makes it very difficult to model them. An additional problem of the rnadelling of twophase flows in the thermal cracking process is the presence of heat transfer in this process. Th is thesis describes a study of two aspects of the rnadelling of thermal cracking units.

The first aspect involved measuring and rnadelling of the residence time distribution of liquid in annular dispersed flow, one of the flow regimes in twophase flow. Annular dispersed flow is characterised by a liquid film travelling at the wall of the tube and a core of gas with entrained liquid droplets. The liquid film is mainly driven by the shear force between the quickly rnaving gas core and the much slower liquid film. lnstabilities on the surface of the liquid film cause a part of the liquid to entrain into the gas core. On the other hand, entrained draplets deposit continuously from the core back in the film.

The residence time measurements were performed by injection of radioactiva tracer liquid in vertical annular-dispersed flow and dateetion of the radiation coming from the tracer at several locations along the tube. The average velocity and the velocity distribution of the liquid was determined in each section of the tube. The conditions of the flow were chosen such that the average liquid velocity could be evaluated in terms of the gas flow rate and the liquid flow rate. From the residence time distribution, it was found that the film flow behaves in a turbulent manner, the increase of the distribution was small outside the liquid and tracer inlet sections. Further the symmetry of the residence time distribution increased when the entrainment and deposition rates become larger, e.g. at high liquid flow rates. Measurements of the residence time distribution conducted just befare and after U-bends in the experiment, showed that most entrained draplets deposit in the bend because of the centrifugal force.

The measured veloeities were compared to veloeities which were predieled by making use of two models for the entrainment process. The first model, by Miesen [13.1-5], prediets the entrained fraction, the film thickness and the entrainment and deposition rates. Miesen's model is a theoretica! model based on some empirica! relations. The second model, by Asali [2], is an empirica! model predicting the entrained fraction and the film thickness only. With the models it is possible to calculate the cross sectional fraction or holdup of the liquid, which is the volume fraction of liquid in the tube. lt was concluded that the model by Asali gives better results than the model by Miesen.

The average veloeities were also predieled by a numerical simulation in MS Excel ™, based on the mass exchange between the film and the gas core and between elements of a discretised tube. To calculate the mass fluxes, use had to be made of the model by Miesen because Asali does not predict the en trainment and deposition rates. The results of the simulation compared to the measured veloeities turn out to be good. Although the results of the simulation are of similar quality as the predieled veloeities by Asali, the simulation has the advantage of allowing for further refinement, e.g. in the description of tube bends.

The second aspect involved a study of the delachment criteria of bubbles attached to the wall in a flow. The need for this comes from a heat resistance problem in the bubble flow regime of a thermal cracking unit. A thin layer of bubbles is thought to cover the wall rasuiting in an increased heat resistance at the wall which is much higher than predieled by existing models. Th is model was needed to see if this hypothesis could be true.

The model presented here is a model by van Helden [8] which was adapted for the present case. With model the radii of bubbles just befare delachment off the wall were calculated and the influence on the heat resistance at the wall was predicted. In addition the validity of the model was tested by estimating the thickness of the viseaus sublayer in a turbulent boundary layer.

The calculations resulted in an upper and lower limit of the bubble size on detachment. The upper limit is determined by the delachment criteriafora bubble in laminar flow. The lower limit is determined by the thickness of the viseaus sublayer which turned out to be smaller than the calculated bubble size at all times. The predicted heat resistance agrees with the measured heat resistance in real furnaces.

Acknowledgements

This masters thesis is the result of a project I did at the Shell Research and Technology Centre Amsterdam during the last year of my study applied physics. I couldn't wish for a better opportunity to gain experience on setting up and carrying out a research project. I remember sitting in the back of my parents car and watching the huge distillation units, pipelines and starage facilities, every time we passed a factory site, when I was a kid. At that time I could never have guessed that I once would get the beautiful opportunity to work with similar facilities at Shell.

I would like to thank the people who helped me make the project successful. First of all I would like to thank Bart Broers for his dedicated supervision, enthusiasm and pleasant cooperation. I would also like to thank prof. Thijs Miehels for giving me this opportunity. Further I would like to thank Rini Seelen for his technica! advise, interesting conversations and all the time he spend on the project as well. My thanks also goesout to Johan de Jong and all the members of CTANU2 who helped me carrying out the experiments. Furthermore I would like to thank my roommates Niels, Tom, Saeske, Jacko, Geertand Renate and all the other stagiairs at SRTCA, who made my stay in Amsterdam unforgettable.

Bart Niestadt Amsterdam, November 1997.

Contents

Summary

Acknowledgements

Contents

1. Introduetion 1 .1 A brief description of the project and its background 1 .2 Thermal cracking of oil in the refinery process 1 .3 Residence time models in annular dispersed flow 1.4 Heat transfer in the bubble flow regime of TC U's 1.5 Contents of the thesis

2. Theory of annular dispersed flow 2.1 Introduetion 2.2 Characteristics of AD-flow

2.2.1 Different twophase flow regimes 2.2.2 Annular dispersed flow

2.3 Film thickness 2.3.1 The velocity profile in the film 2.3.2 Superficialliquid velocity of the film 2.3.3 Pressure gradient, shear force and film thickness

2.4 Entrained fraction modelling 2.4.1 The entrained fraction model by Miesen 2.4.2 Asali's model for the entrained fraction 2.4.3 Predicted film thickness from Asali's model

2.5 Other flow parameters of AD flow 2.5.1 Liquid, film, gas and entrained holdup 2.5.2 Combined properties of the core 2.5.3 Actual velocity of entrained dropiets 2.5.4 Reynolds number for the film

2.6 Resume of the annular dispersed flow model

3. Modelling and simulation of residence time distribution in AD-flow 3.1 Introduetion

3.1.1 AD-flow simulation 3.1.2 Important flow properties

3.1.2.1 Actual veloeities 3.1.2.2 Entrained fraction 3.1.2.3 Entrainment and deposition rate 3.1.2.4 Liquid, film, gas and entrained holdup

3.2 Basics of the model 3.2.1 Discretisation of the experiment 3.2.2 Mass balance 3.2.3 Modelling the bend 3.2.4 Calculating the signal

3.3 Mathematica! consequences of the discrete model 3.3.1 Determination of location and time

3.3.1.1 The discretisation error in the residence time distribution 3.3.1.2 The discretisation error in the tracer distribution

3.3.2 The accumulated error of the system with liquid exchange

4. Experimental setup 4.1 The vertical flow facility 4.2 The residence time measurements

1

3 3 3 4 5 5

6 6 6 6 8 9 9 10 11 12 12 14 16 17 17 18 18 19 20

21 21 21 21 21 22 22 22 22 22 24 25 26 27 27 28 29 30

33 33 34

4.2.1 The injection system 34 4.2.2 The detectors 35

5. Results of residence time measurements in AD-flow 37 5.1 lntroduction, description of the terminology 37 5.2 Qualitative description of the experimental results 38

5.2.1 Analysis of the detector signa! 38 5.2.2 Velocity measurements compared with liquid and gas flow rates 40

5.3 Velocity spreading; turbulent versus laminar flow 42 5.4 Validation of the predicted velocity 45

5.4.1 Absolute errors of the predieled velocity 45 5.4.2 Relative errors of the predieled velocity 46

5.5 Simulation of AD flow residence times 47 5.5.1 Analysis of raw data 47 5.5.2 Simulated average liquid bulk velocity 49

5.6 Overview of the results 52

6. Bubble datachment in vertical flow 53 6.1 Introduetion 53 6.2 Qualitative description of torces on a bubble 53 6.3 Geometry of the problem 55 6.4 Forces acting on a bubble near the wall 56 6.5 Heat transfer through a bubble-covered wall 57 6.6 Results of the calculation 59 6.7 Discussion of the results 61

6.7.1 Turbulent boundary layers; estimating the viscous sublayer thickness 61 6. 7.2 Consequences for the calculated bubble thickness 62

6.8 Force balance versus boundary layer 64

7. Conclusions and recommendations 65 7.1 Project description 65

7 .1.1 Background 65 7.1.2 Goals 65

7.2 Residence time in AD flow 65 7 .2.1 Approach of the problem 65 7 .2.2 Conclusions 66 7 .2.3 Recommendations 66

7.3 Heat resistance in the bubble flow region of turnaces 67 7 .3.1 Approach of the problem 67 7 .3.2 Conclusions 67 7 .3.3 Recommendations 67

Raferences 69

List of symbols 71

Appendix A; Results of the residence time measurements in AD-flow

Appendix B; Results of the models by Asali and Miesen

Appendix C; Computer code of the numerical residence time simulation

Appendix D; MS Excel TM macro-module for evaluation of the simulated experiment

Appendix E; Results of the simulation

Appendix F; Bubble thickness and temperature anomaly for various conditions (laminar flow model)

2

1. Introduetion

1.1 A brief description of the project and its background

One of the steps in the process of crude oil refining is the cracking of heavy oil fractions coming trom distillation columns. Basically what is done in the cracking process is breaking up large hydracarbon molecules into smaller, lighter ones. These lighter hydrocarbons are the fuels that are valuable, like kerosene and gasoline.

The heavy oil coming trom the vacuum distillation units is also known as short residue. There are various ways of cracking the short residue and converting it to lighter products. One of these is thermal cracking. In a thermal cracking unit {TCU or furnace) the liquid shortresidueis led through a set of vertical tubes facing a heat source. Heating causes the necessary conversion of the oil into lighter products in their gas phase. The amount of conversion is defined as the amount of 165-minus conversion, the fraction of reaction products with an atmospheric boiling point below 165°C.

The turnace is fed with heavy liquid oil fractions, while volumetrie, the output of the turnace is mostly gas. This means that during the process the mass fraction of liquid steadily decreases, while the mass fraction of gas steadily grows. The flow through the turnace is therefore a twophase vertical flow. Dependent on the liquid and gas flow rates, twophase flow can be divided in different flow regimes. Most of these regimes are present in the furnace.

At the Shell Research and Technology Centre Amsterdam {SRTCA), research is done to optimise the thermal cracking process. The processis modelled to see how certain parameterscan be influenced to optimise operation and design. The problem of doing this, however, is that twophase flow with heat transfer is a highly complex system to model. Twophase flow without heat transfer is already governed by many parameters. Taking into account the heat transfer to the flow, the boiling process, and also the conversion due to chemica! reactions, makes it an interesting challenge.

Th is report contains the results of a research project done at SRTCA. The goal of the project was to examine two features of vertical twophase flow in turnaces more extensively. The first of these was to set up a model to predict the residence time of liquid in vertical annular dispersed flow, one of the flow regimes in twophase flow. This problem is introduced insection 1.3 of this chapter.

The second topic which was studied was the role of gas production due to boiling and cracking in the increased heat resistance between the inner side of the tube and the fluid in vertical twophase bubble flow, another typical flow regime of twophase vertical flow. This problem is introduced insection 1.4.

1.2 Thermal cracking of oil in the refinery process

Figure 1. 1 below is an mustration of a typical part of the refinery process including a thermal cracking unit. The crude oil entering the plant is distilled first in the crude distillation unit (CDU). Here a large part of the end products, e.g. gas, petrol, kerosene, diesel, are separated trom the crude oil. What is left after this process is called long residue and consists of 360-plus components, these are oil products with a boiling point above 360°C. This is led to a high-vacuum distillation column (HVU).

The high-vacuum unit is a low-pressure distillation column. The waxy distillate, or 520-minus components in the long residue, consists of oil products with a boiling point between 360°C and 520°C. These can be further processed e.g. in a hydrocracker. What is left, the 520-plus components, is called the short residue and is led to the thermal cracking unit.

The thermal cracking process here consists of two TC U's. TCU I does the primary thermal cracking. The heavy fractions coming trom the HVU are converted to 165-minus components as much as possible. The output of this, a mixture of gas and a small fraction which is stillliquid, is separated in the cyclone. The liquid part leaves the process here as so called visbraken tar. The gas leaving the cyclone is separated in the distillation column.

3

In the distillation process the valuable fuel products are separated as gas. A part of the output of the distillation column leaves the process as waxy distillate. This part is led to TCU 11 where again conversion takes place, after which it is led back to the distillation column. The residue of the distillation column leaves the process as therm al tar.

er oi

u de CDU I

Cl) ::::l

g "C ëii

.Q ~

HVU

Cl) ::::l

/\ t:::"C 0 ëii .s:::. ~ U)

TCU --I

CDU = Crude oil Distillation Unit HVU = High-Vacuum distillation Unit

TCU = Thermal Cracking Unit

j_

c: E :I 0 u c: 0 :p

~ Cyclone :p

~ 111 c

T

Fig 1. 1 The role of thermal cracking in a refinery process.

1.3 Residence time models in annular dispersed flow

~C::J e"S.o e"ot::J 0

~ ç ~ 6,eC::J

~0 ;i-~ ~'(jo

~'(j. 6'C::i

gas

l bulk distillate

waxv distillate

/\ TCU --

11

thermal tar

visbraken tar

The importance of knowledge about the residence time in annular dispersed flow comes from the fact that especially in this flow regime in TCU's the rate of conversion is very high. Annular dispersed flow (henceforward AD-flow) is the twophase flow regime where the ratio between the gas and liquid fraction is the highest This means that in a TCU, where this ratio increases during the process, the AD-flow regime is located at the end of the TCU where the temperatures are the highest in the system. The rate of conversion increases with increasing temperature. The influence of errors in the predicted residence time of liquid is therefore the largestin this section of the TCU. Furthermore AD-flow is the most inhomogeneous flow regime of twophase vertical flow, which adds to the uncertainty in residence times.

At SRTCA, research is done in order to model the whole thermal cracking process in furnaces. The findings of various research projects concerning this subject are put tagether in a computer code HYFIH (Hydrocarbon Flow In process Heaters). Modelling of AD-flow is one of these research projects. The vertical flow facility (VFF) at SRTCA is an experimental plant which was especially built to examine AD-flow. In the current project, the residence time of liquid in AD-flow was measured by injecting radioactiva tracer liquid in the flow. The radiation coming from the tracer

4

liquid was detected at various locations along the tube. The residence time distribution was simulated by modeHing the mass fluxes of liquid between the gas core of the flow and the liquid film at the wall, which are typical features of this flow regime. The experimental data was also compared to existing models tor the entrainment processof draplets in the gas core of the flow.

1.4 Heat transfer in the bubble flow regime of TC U's

A different problem which is encountered at SRTCA is the tact that the HYFIH computer code is unable to predict a sudden increase of the heat resistance at the wall in the bubble flow regime of TCU's. lncreased heat resistance at the wall has been seen in real TCU's and causes premature shut down of the TC U's.

When the heat resistance of the wall in bubble flow suddenly increases, the skin temperature defined as the temperature of the outer side of the tube wall, becomes very high. In addition to the high skin temperatures, the high initial film temperatures (defined as the temperature of the twophase mixture directly adjacent to the inner side of the wall) are responsible tor increased coke formation. Coke formation is the deposition of carbon on the tube walls. This causes the skin temperatures to riseeven more. Because of the danger of darnaging the tube walls (overheating causes material integrity weakening), the TCU has to be shut down prematurely to clean the tubes.

lt is thought that the temperature anomaly is caused by an increased formation of bubbles in this flow regime. The bubbles, which are usually formed at the wall, will stay at the wall too long before detaching into the flow. This forms a thin layer of bubbles at the wall which increases the heat resistance severely.

In the present research project a theoretica! model by van Helden [8] was adapted. With this adapted model the radii of bubbles could be calculated just before they detach into the flow. With the calculated radii the influence on the heat resistance was calculated as well as the unexplained part of the temperature ditterenee between film temperatures and bulk temperatures in the flow. The problem was a lso approached by calculating the thickness of the viscous sublayer in a turbulent boundary layer. The calculated values were compared toeach other.

1.5 Contents of the thesis

The thesis, describing the contents of the project, is organised as follows. Chapter 2 of this report goes deeper into the theory of the annular dispersed flow regime. lt

will start off with a briefdescription of the different flow regimes present in vertical twophase flow. After this a better look is taken at the different features describing annular dispersed flow, and some parameters which are commonly used to model the flow.

In chapter 3 a simulation is presented which was set up to predict the average velocity of liquid in annular dispersed flow. With this simulation, a Visual Basic module in MS Excel n.1, the average residence time of liquid was calculated. The simulation is based on mass flow of liquid between subsequent sections of a discretised tube containing annular dispersed flow.

Chapter 4 is a description of the experiments which were done to measure the residence time of liquid in annular dispersed flow. This chapter describes the experimental setup and contains information about the material, physical and geometrical parameters of the experiments.

In chapter 5 the results of the experiments and the simulation are presented, as well as predieled values from two existing models describing some features in annular dispersed flow.

Chapter 6 describes the theoretica! model of the increased heat resistance problem in bubble flow. This chapter contains a theoretica! description of a model which was set up to predict the bubble size on detachment. lt further contains results and a critica! view on the model, based on calculations of the turbulent boundary layer. Further, with the calculated bubble size, the heat resistance of a layer of bubbles at the wallis calculated.

Chapter 7 reviews the whole project and contains conclusions and recommendations for further study.

5

2. Theory of annular dispersed flow

2.1 Introduetion

The ma in goal of this project was to achieve more insight in the residence time of liquid in vertical annular dispersed flow in tubes. More specific, the dependenee of the residence time of liquid on the flow rates of liquid and gas and, related to that, the average veloeities of the liquid and it's distribution.

The residence time in AD flow is a complex feature to describe because of the complexity of annular dispersed flow itself. Th is chapter will briefly discuss the present models of AD flow used at SRTCA. The following chapters will use these models to present and explain the results from the measurements taken in the vertical flow facility at SRTCA.

In the next section of this chapter a general description of AD flow will be given. From this description different features will be highlighted in the rest of the chapter. Most of the features will turn out to depend strongly on the film thickness and the entrained fraction in AD flow. These two parameters, which are explained later, will therefore be treated more extensively.

2.2 Characteristics of AD flow

2.2.1 Different two phase flow regimes

Multiphase flows play an important role in a lot of industrial applications. But the complexity of these flows make it very hard to understand the different processes which rule the parameters determining the efficiency of these applications. In thermal cracking fumaces, residual oil enters the system as a liquid and leaves the fumace mainly as gas because of conversion and evaporation of the residue. The fluid therefore passes through different flow regimes of the two phase flow.

For a large part the different flow regimes in vertical multiphase flow are determined by the mass flow rates of each phase and the diameter of the tube. But the role of different material properties of the liquid and gas involved, like viscosity and surface tension, cannot be neglected. In general in multiphase flow research, the mass flow rates are translated to superficial veloeities defined as the velocity one phase would have if that phase were to take up all of the tube and so if the other phase were not present. Mathematically the superficial gas and liquid velocities, V59 and V51 , are therefore represented by

and V - 4~ si- 02 trp,

(2. 1)

In (2. 1) W9 and W, are the mass flow rates in kg/s, D is the tube diameter and p1 and p9 the densities of liquid and gas. Note that density is a tunetion of temperature and pressure, which is especially important for gas because it is not incompressible.

Befare giving a brief description of the flow regimes present, a classification of the regimes in terms of superficial velocities, as was done by Hewitt and Roberts [1 0] is given in figure 2.1. In tact Hewitt used the momenturn flux to distinguish the different regimes. The parameter G is the mass flux (kg/m2s) which is the mass flow rate (kg/s) divided by the cross section of the tube (m2

). lt should be divided by the density of the fluid (kg/m3) to give the

superficial velocity of that fluid. So dividing the axes by the density and taking the square root would result in the classification in termsof superficial velocity.

6

annular 'Wispy' annular

10 Churn Bubbly

1 Slug

0.1 1 10

Fig. 2. 1 Hewitt and Roberts pattem map for vertical upflow

The regimes of two phase flow are schematically shown in tigure 2.2. After the oil enters the turnace in which it is heated, bubbles are formed within the

liquid which are carried on with the flow. This stage of the flow is called bubble flow. But as the bubbles grow and the vapour mass fraction, also known as the quality x, increases, slugs are formed. Slugs are bullet shaped gas volumes which propagate through the liquid in the flow, causing the liquid to travellocally as a film near the wall. With respect to the slug velocity, the fluid in the film will travel slower and sametimes it willeven travel in the other direction locally. This stage is called slug flow.

The increasing amount of gas present in the flow, relative to the amount of liquid, causes the slug gas pockets to coalesce, forming larger gas pockets. lt seems at first that the size of these bubbles wilt grow to infinity and therefore forming an annular flow. But because of the relatively large liquid flow rate, the gas core is not able to carry all the liquid with the flow. Therefore, liquid accumulates and periodically falls back through the gas core. The flow is highly unstable and the fluid continually pulses up and down. This flow regime is called churn or froth flow.

As the liquid evaporates, the liquid flow rate further down the process becomes small enough to make annular flow possible. This means that a liquid layer near the pipe wall is carried along with the flow by a gas core in the tube. Periodically waves are formed on the surface of the liquid film. The shear stress between the film and the gas, causes small drops to entrain from the tops of the surface waves into the gas core. The resulting flow pattem is referred to as annular dispersed flow and is the final flow regime in most furnaces.

lf the mixture of liquid and gas would be heated too much, the tubes would fall dry and burst due to overheating. lt is therefore important to keep a part of the oil in its liquid phase to cool down the tube walls. This oil, which is not converted into lighter products, will be the residue of the refinery process.

7

Oo 0 0 0

0 0 0

0 0 0 0

0 0 0 0 0 0

0 0

ooO 0 0 0

0 0 0 0

0 0

i Subbieflow

Gas

___ ~iguid ~ -0 G~s __

. Gas

t5 Liqu~d_ _ -· 0

:: :.":.: Liquid . .

i i Slug flow Chumflow

lncreasing gas flow rate

Fig. 2. 2 Flow regimes in two phase upflow

2.2.2 Annular dispersed flow

------------

. . . . . . . . . . . . . . . . . . . . . . . .

:....- . . . . . .. . . . . .

i Annular dispersed

flow

As stated before, annular dispersed flow consists of a liquid film near the tube wall which is carried along through the tube by a relatively fast travelling gas core. The different processes which are going on within this flow regime are illustrated in tigure 2.3. This tigure shows the liquid film near the wall, and the entrained draplets in the gas core of the tube. The arrows illustrate the entrainment and deposition process and the direction of the main flow. Azzopardi [3] showed that the entrainment process comes mostly from the presence of surface waves on the interface of the liquid and gas phase in AD flow.

0

• 0

~: E

0

0

0 • 0

0

to •

0 0

•

VI vg VI

Fig. 2. 3 Entrainment and deposition in AD flow; surface waves

8

Fora description of AD flow, it is necessary to examine the farces acting on the liquid as shown in tigure 2.4 in sectien 2.3.1. Because of the symmetry of the problem and because the film thickness is much smaller than the tube diameter, it is sufficient to describe the flow of a liquid film travelling along a vertical plate. As can be seen, shear stress on the interface of liquid and gas is the driving force which carries the liquidalong with the flow.

x g~

'tint

y .... -

- d .... - -Fig. 2. 4 Farces acting on the film in upflow; v9 and V1 in positive x direction

2.3 Film thickness

The film thickness of the liquid film in AD flow is an important parameter for most of the flow features. Miesen [13.1-5] deduced a set of equations to calculate the film thickness, based on the superficial veloeities of gas and liquid, the entrained fraction, the tube diameter and some material parameters. Below, the calculation of film thickness will be foliowed step by step. The general idea of the model is that the film thickness is determined by the film velocity and the mass flow rate of fluid going through the film. First a velocity profile will be calculated based on the assumption that the film flow is laminar. From this the superficialliquid film velocity is calculated. By substituting the shear stress force at the interface of gas and liquid and the pressure gradient in vertical direction in the tube, an equation for the film thickness is proposed.

2.3.1 The velocity profile in the film

In order to calculate the film thickness, the average velocity of liquid in the film is needed. Therefore it is necessary to calculate the velocity profile in the film first. To do soit is assumed at first that the velocity profile is laminar. From the farces shown in tigure 2.4 the following differential equation is proposed

(dp ) d

2V - -+gp +p-=0 dX I I dy2 (2. 2)

9

Here dp/dx is the pressure gradient in vertical direction, g is the gravitational acceleration which is positive for upflow and negative for downflow in (2.2), p1 and J..l1 are the density and the viscosity of the liquid respectively, and V(y) is the velocity in vertical direction.

The boundary conditions to this differential equation are given by a no-slip condition near the wall,

V(O) = 0 (2. 3)

and continuity of shear stress on the interface of gas and liquid,

(2. 4)

Here 't;nt is the interfacial shear force and d is the film thickness. Equation (2.2) withits conditions (2.3) and (2.4) can be integrated two times to find

the following solution for the velocity profile in the film

{2. 5)

This velocity profile however, is not sufficient to predict the exact velocity V(y) in the film since the interfacial shear stress 't;nt and the pressure gradient still have to be substituted. In order to describe the totalliquid flow through the film the superficialliquid velocity of the film Vslf is calculated.

2.3.2 Superficial liquid velocity of the film

By inlegrating equation (2.5) and dividing by the film thickness d, the average velocity in the film is found to be

- 1 dJ ( \rl (dp ) d2 d'rint V.=- V YJUY=- -+g,q -+-

t d 0 dx 3~ 214 {2. 6)

lf D is the diameter of the tube, the superficialliquid velocity of the film is then defined by

(2. 7)

On the other hand can be described as

(2. 8)

in which Eis the entrained fraction and V51 is the total superficialliquid velocity.

10

From the laminar velocity profile a relation is now found tor the superficial liquid velocity in the film. To calculate the film thickness trom this, it is still necessary to find a relation tor the shear stress at the interface 't;nt , the pressure gradient along the tube dp/dx, and the entrained fraction E.

2.3.3 Pressure gradient, shear force and film thickness

Consider a tube with diameter D, in which annular dispersed flow with film thickness dis present. The three terms that contribute to the pressure loss in the flow are the friction force at the gas-liquid interface, a hydrastatic term and a term which takes into account the effect that entraining dropiets are accelerated by the gas. This is written as

dp = _ 4'Zint _ ( ,qEVs,J _ 4Er (V _V) dx D g Pg + V D sg I

sg

(2. 9)

here V; is the interfacial velocity of the gas-liquid interface, given by

(2. 10)

Th is velocity was found by making use of the laminar velocity profile of equation (2. 5) with y=d, the film thickness.

To complete thesetof equations tor '1nt and dp/dx, the interfacial friction factor f59;

defined on the basis of superficial gas velocity V59 has to be introduced. The dependenee of the interfacial friction on the film thickness originates trom the tact that for thicker films, the surface waves on the interface will have larger amplitudes. Therefore both the roughness of the surface and the shear stress at the surface will increase with increasing film thickness. Hewitt and Hall-Taylor [9] describe the interfacial friction factor with the following relation to the dimensionless film thickness d/D

(2. 11)

in which f9 is the friction factor for a single phase gas flow in a smooth tube and y is a correction factor. Hewitt and Haii-Taylor set y to 90 butbasedon empirica! correlations, Miesen [13.1-5] uses

(2. 12)

By use of the Blasius equation and Reynolds number of the gas Re9 , f9 can be calculated to be

(2. 13)

Using fsgi• the interfacial shear force 't; is defined as

11

'lint = ~ fsgiPg Vs~ + /'). T

!':!..r = Dr(V59 - ~}

where Ll-r is a correction term for impacting droplets coming from the gas core.

(2. 14)

Together (2. 9) and (2. 14) forms a set of equations from which the pressure gradient dp/dx and the interfacial shear force 't;nt are calculated in terms of the superficial gas velocity V 59 and film thickness d.

Except for the entrained fraction E, all the necessary relations needed to set up an equation for the film thickness d are known at this point. As will be shown later in this chapter, the entrained fraction is a parameter which is not easy to predict and therefore forms part of the problem of finding the film thickness. However substituting 't;nt and dp/dx into (2.7) and using (2.8) yields the following equation for the film thickness

(2. 15)

By solving this set of equations, not only the film thickness d but also 't;nt• dp/dx, and the velocity profile in the film are determined.

The model, as presented so far, shows that AD-flow is very difficult to describe as was statedat the beginning of this chapter. Many relations are needed to close thesetof equations. Note that in the above only the first half of the problem is discussed and that still nothing is known about the entrained fraction. This will therefore be the next step in the description of AD flow.

2.4 Entrained fraction modelling

Based on assumptions concerning the velocity profile of the film, an equation was found for the film thickness in terms of the ma in flow parameters V 59, V 51, E and some geometrical and material properties. lt was al ready stated that a prediction of the entrained fraction is still a problem but it was considered known in the former. In the following section some entrainment models will be discussed. Building on the Miesen model for the film thickness, the model for entrainment presenled by Miesen [13.1-5] will be treated first. After that some attention will be paid to a different model by Asali [2] which showed to give some nice results as can be seen in chapter 5.

2.4.1 The entrainment model by Miesen

All present models describing the entrainment process in AD-flow are based on empirica! or semi empirica! calculations. The reason for this is that finding the conditions needed for a droplet to entrain trom a surface wave on the gas-liquid interface, is a problem in itself. As a result of this, most models are only valid within the range of the experiments on which they are based and cannot be extrapolated beyond this range. The model by Miesen [13.1-5] suggests

12

that some extrapolation is possible because it is has relatively strong physical basis. Whether this suggestion is true or not, will be seen in the present experiments.

Miesen's model basically is a set of equations which can be solved for the film thickness and the entrained fraction. Calculation of the film thickness was treated insection 1.3. The equations come trom a proposed empirica! model for the entrainment rate and deposition rate. These are then integrated over the developing flow, giving the entrained fraction. Fora detailed description of the proposed model the reader is referred to [13.2], which describes the Miesen model.

The Miesen model for the entrainment rate is a adapted version of the Hewitt-Govan model for entrainment rate. With f.l1 and f.l9 the dynamic viscosity of the liquid and gas present in the system and the Reynolds number for the gas as defined in (2.12), the entrainment rate is given by the following expression,

E = r c,p,V~[(;:)(;:)'( Re~~e,,)' We r (2. 16)

0 where c1=confidential and C:~= confidential. Further the liquid Reynolds number is here defined as

(2. 17)

and the criticalliquid Reynolds number, is given by

(2. 18)

where f(cr) is an empirica! tunetion which takes into account the dependenee on the surface tension, given by

f(a) = 1+ 0.401n(%w)

in which crw=0.07275 N/m is the surface tension of water at ambient pressure and temperature.

(2. 19)

The idea behind the introduetion of the critica! Reynolds number is that no entrainment occurs if the liquid Reynolds number is below this value, since there are no surface waves present on the film surface below this value. As can be seen later, Asali [2] showed that indeed there is a critica! Reynolds number for the forming of these waves, from which entrainment is thought to originate. Finally, the Weber number is given by

(2. 20)

The model for the deposition rate also follows from the Hewitt-Govan model and can be written as

13

(2. 21)

in which the coefficients are found to be

confidential

Now the entrainment and deposition rates are known, the entrained fraction can be calculated since for developed flow the entrainment rate must equal the deposition rate because of mass balance within the gas core. Doing this yields the following for the entrained fraction in fully developed AD flow,

E=

(2. 22)

Tagether with (2.14), (2.21) farms a complete set of equations trom which the film thickness d and the entrained fraction E can be calculated. Although the entrainment rate and deposition rate were needed in order to find (2.21 ), they are nat important in the calculation of E and d in a fully developed flow. After solving this set of equations, it is possible to calculate the most important parameters of the flow and the Miesen model is complete. Though developing flow is disregarded in the present case, for completeness the integral needed to calculate the entrained fraction in the developing flow will be discussed briefly below. lt is given by

(2. 23)

and it basically calculates the accumulation of draplets in the gas care because of the difference in entrainment rate and deposition rate in the region where no equilibrium has been established in the flow. In (2.22) the entrainment present at Xo. is given by E(x0). Further E(x) is the entrained fraction as a tunetion of the location in the tube.

2.4.2 Asali's model for the entrained fraction

In his artiele [2] Asali compares some existing rnadeis of entrainment and tunes these models to measurements taken in a tube with a diameter of 4.2cm. Although it is stated befare that most of the entrainment models cannot predict entrainment in a range beyond the one in which the experiments for that particular model were done, this model describes the entrainment process in the present case rather well.

The model is based on the assumption that entrainment will nat occur below a certain critica! Reynolds number in the film. lt further states that therefore there must be a limiting value for the entrained fraction, since if the entrained fraction were to exceed this maximum value, the mass flux through the film would correspond to a Reynolds number below the critica! value.

14

Asali further noted that for liquids with a low surface tension the entrained fraction in the developed flow will approximately be equal to the maximum value. For liquids with a high surface tension, like water, Asali gives the following ratio between entrained fraction and the maximum entrained fraction EM

(2. 24)

here k' is a constant related to the entrainment and deposition rates in the experiments. For an exact definition of k', the reader is referred to [2]. Here it should be sufficient to give the empirically found va lues of k' for upflow and downflow; for upflow k'=2.0·1 o-3 s3/kgm914 and for downflow k'=1.2·10"3 s3/kgm914

• V9 is the defined gas velocity, which is related to the superficial gas velocity as

(2. 25)

The maximum allowed entrained fraction EM is defined via the liquid mass flow rate W1 and the critica! liquid mass flow rate through the film W11c corresponding with the critica! Reynolds number Re11c· This yields

(2. 26)

To be able to campare this to the Miesen model of entrainment, the corresponding critica! Reynolds number of the film Re11c is related to the critica! flow rate through the film W11c as

(2. 27)

Asali took measurements of W11c most probably by measuring W1 at the first appearance of roll waves. From this, W11c was calculated using

~fc = (1- E)~. measured (2. 28)

in which wl, measured is the measured liquid flow rate. Unfortunately, Asali is not able to come up with arelation from which W11c could be

calculated without the help of data. lnstead the reader is referred to the stability theory of Andreussi et al. [1]. The KSLA-method, the standard model for flow property predictions in two phase flow, used at SRTCA, makes use of the following corrected relation by Andreussi for the critica! Reynolds number used in the Asali entrainment model

15

Re,fc = confidential (2. 29)

(6. 1

lt is now possible to predict the entrained fraction with the relations given above. To complete the model, the only thing neededis arelation for the film thickness as was deduced in the Miesen model as well.

2.4.3 Predieted film thiekness from Asali's model

So far the entrainment models by Miesen [13.1-5] and Asali [2] were described logether with the considerations on which the models were set up. Both these models have a strong empirica! character because they both resulted from earlier models which had to be tuned on experiments. Further, for the model by Miesen, film thickness calculations were done because the film thickness is an inlegral part of the set of equations. Therefore, to make the Asali model complete, the only thing which remains at this point is a description of the predieled film thickness which follows from Asali. A relation for the film thickness arises from the entrained fraction and a relation which is analogue to (2.27), thus

(2. 30)

in which Wlf is the mass flow rate through the film.

Together with (2.23) this leads to the film thickness

(2. 31)

where V9 is the gas velocity given by (2.24), v9 is the kinematic viscosity of the gas, -rw is the shear stress near the wall as given by Andreussi [1] and fgi is related to fsgt from (2.1 0) by the relation

(2. 32)

With arelation for the film thickness, the Asali model is complete and ready to be compared to the Miesen model.

The calculations of the film thickness, the entrainment rates and the entrained fraction, as were done in this chapter so far, are very important when the veloeities in the film and the gas core need to be estimated. To estimate the residence time of a fluid in AD flow, estimates for these veloeities are unavoidable, which makes the discussion above necessary.

16

2.5 Other flow parameters of AD flow

2.5.1 Liquid, film, gas and entrained holdup

An alternative way of describing liquid fractions in each phase, either in the film or entrained in the gas core of the AD flow, is the description in termsof the cross sectionat fraction or holdup pertinent tothefluid of interest. In AD flow, tour different holdup parameterscan be distinguished; these are the film holdup, the entrained fraction holdup, the liquid holdup which is the sum of the film and the entrained holdup, and the gas holdup. At first these tour parameters seem a bit superfluous, but they will turn out to be quite suitable if the veloeities of gas and liquid are to be estimated.

Suppose in AD flow the gas phase travels with a superficial velocity Vsg· By definition the superficial velocity of the fluid is the velocity of the fluid if it were to take up all of the tube. Th is means that the mean velocity V9 will be given by dividing the superficial velocity Vs9 by the holdup E9 of the particular fluid, because the same amount of fluid does only take up that part of the tube, so

(2. 33)

The average liquid velocity is calculated in the same way, but the resulting velocity will only be an estimate because the average liquid velocity in reality must be a linear combination of the average veloeities of the film and the entrained liquid in the core. However the estimate is good enough to be able to cernpare a certain model with experimental results. Analogue to the considerations above, if E1 is the liquid holdup, the estimated average liquid velocity will therefore be

(2. 34)

The equation above is of importance for the evaluation of the residence time data, because the average velocity of the liquid can be found if use is made of models describing the liquid holdup, like Miesen or Asali. The liquid holdup can be rewritten as the sum of the film and entrained holdup. Because the film thickness is characteristically two orders of magnitude smaller than the tube diameter, a good estimate of the film holdup is given by

4d et=-

D (2. 35)

Assuming further that the droplet velocity of the entrained fraction is given by the superficial gas velocity Vsg• an estimate of the entrained holdup can be found. Note that the superficial velocity of the entrained fraction is given by the product of the entrained fraction E and the superficialliquid velocity V st• yielding

(2. 36)

Then, by realising that the holdup is given by the ratio between the superficial velocity and the real velocity, this results in the following estimate of the entrained holdup Ee,

17

(2. 37)

To calculate the estimated average veloeities of the film, it should be noted that the entrained liquid fraction has tobetaken into account as well. This can be illustrated by making use of the superficial film velocity V511, this yields

(2. 38)

Based on the definition of the holdup as being the partial cross sectionat area of a certain fluid in the tube, the following relation must hold

(2. 39)

The holdup relations are basically the mathematica! representation of the definitions of the superficial velocities. They do not add any new information to the description of the flow. lf no model would be available descrihing the holdup parameters, like the ones presented in the sections before, these relations would be worthless.

2.5.2 Combined properties of the core

In some cases it could be handy to treat the core of the flow, consisting of entrained liquid and gas, as one fluid. With the help of superficial velocities, it is possible to define a superficial core velocity trom which some material parameters of the core can be calculated, yielding

(2. 40)

Now it is possible to define the density of the mixture in the core, this results in

(2. 41)

Note that the holdup parameters are dependent on the entrained fraction and the film thickness. The complete set of independent variables, for AD flow, is given by the film thickness d, the entrained fraction E, the superficialliquid velocity V 51 and the superficial gas velocity V59• Together with the geometrical parameters and fluid properties, these parameters determine all the other flow properties.

2.5.3 Actual velocity of entrained droplets

Consider a flow through a vertical tube in downward direction, the fluid is a mixture of gas and entrained dropiets with diameter o. Suppose the dropiets are homogeneously distributed along the tube in axial and radial direction. The dropiets have density p1 and the density of the gas is p9 and further the viscosity of the gas is J.19 • lf no gravity would be present, the dropiets would have the same velocity as the gas. But under the influence of gravity the velocity of the dropiets will differ slightly trom the gas velocity. The settling velocity of the dropiets with respect to the gas velocity will be

18

(2. 42)

where the terminal settling Reynolds number is given by

Ar Re1 = ..[Ai

18 +0.61 Ar (2. 43)

The Archimedes number gives the ratio between gravitational farces and viseaus farces in the flow.

For AD flow it is necessary to use the average drop size to get the average terminal settling velocity of the droplets. Schellekens [16] did some PDA measurements in the vertical flow facility at SRTCA and came up with a relation for drop size distribution. lt is common in drop size measurements to interpret the size distribution in various ways, dependent on what it will be used for. For example, when the amount of friction on a droplet in a flow is to be calculated, it is suitable to calculate the mean diameter of the droplets. From this the effective cross sectionat area of the drop in the flow can be calculated and from this the friction force on the drop is known.

Here the volume number average diameter has to be used since the gravitational force will be related to the volume of the droplet This diameter is given by

Dv = confidential (2. 44)

lf 8 = f5v is substituted in (2. 42) and (2. 43) and it is realised that for upflow gravity will work in

the opposite direction as for downflow, the velocity of the entrained droptets can be found using

(2. 45)

In (2. 45) the minus sign is used for upflow and the plus sign is used for downflow. The influence of this correction on the residence time of fluid in AD flow is very small,

however it is presented here to complete the presented velocity model of AD flow.

2.5.4 Reynolds number for the film

Another parameter which will be helpfut is the Reynolds number for the film. From this Reynolds number some assumptions will be checked for the flow type of the film. By definition the Reynolds number in the film will be

19

(2. 46)

where the factor 4 in the right hand side comes from the hydraulic diameter. This can be rewritten a bit so that u se can be made of the entrained fraction E and the superficial velocity, yielding

Ret = ,qD(1- E)V51

14 (2. 47)

Here use was made of equation (2. 38). Further the film holdup was approximated to get rid of the predieled film thickness, which is a less reliable parameter than the predieled entrained fraction.

2.6 Resume of the annular dispersed flow model

Annular dispersed flow can be modelled by a system consisting of two subsystems, the film and the gas core. These two subsystems interact through the exchange of fluid, known as the entrainment and deposition process. Further, the veloeities in the both subsystems arealso related to each other by the shear stress on the interface between the film and the gas core.

As was said before, the flow properties of AD flow are only determined by the superficial veloeities of the two phases present, the tube diameter and the fluid properties. To calculate the flow parameters, the complete set of independent variables needed is given by the entrained fraction E, the film thickness d, the superficial veloeities and the tube diameter.

Determination of the entrained fraction and the film thickness as a tunetion of the superficial veloeities is key problem in modelling AD flow. Most of the challenges in modelling AD flow come down to this problem, this also applies to predietien of the residence time of liquid in AD flow. Two models for the entrained fraction in the core were presented in this chapter, and the related film thickness was deduced.

20

3. Modelling and simuiatien of residence time distributions in AD-flow

3.1 Introduetion

3.1.1 AD flow simulation

In order to simulate the residence time distribution of liquid in AD flow, use can be made of the mass balance in the flow. Knowing some of the key parameters like film thickness, entrained fraction and the veloeities in film and gas core, it is possible to calculate the mass flux of liquid everywhere in the flow.

With knowledge of the liquid mass flux between the film and the gas core and in the film and the gas core alone, it is possible to simulate the flow. This is done by dividing the whole tube into elements and in each element treating the film and the gas core as separate systems which exchange liquid.

Because the liquid mass flux is known, for each element and both for the film and the gas core, the incoming and outgoing amounts of liquid are known as well. Suppose the concentratien of tracer liquid is known in each element, then it is possible watch the tracer propagate through the flow.

The residence time distribution of liquid in AD flow was simulated using the technique described above. This

In

detector 1

detector 2

I

l.c IÖ. c 1.!! . .:: • .a

detector4

detector 4

Fig. 3. 1 Simulated experiment

was done in order to examina the influence of flow properties and initia! conditions on the residence time of liquid in AD flow. The computer code for the simulation was written in Visual Basic for MS Excel7.0 for Windows 95 and can be found in appendices C and D.

The actual setup which is simulated is illustrated in figure 3.1. lt consists of two tubes, one for down flow and one for up flow. Both tubes are connected by a bend. Note that the location of the detectors and their numbering has nothing to do with the detector locations and their numbering in the Iaberatory experiment described in chapter 4.

3.1.2 Important flow properties

To calculate the fluxes in gas core and film, it is important to know some of the physical properties of the flow. The most important input parameters of the simulation and their influence are briefly discussed below.

3.1.2.1 Actual veloeities

The actual veloeities of the liquid in the film and in the core belong to the key parameters of every residence time model for AD flows. They follow directly from the superficial veloeities if, as an onset, constant veloeities of the liquid in the film and the gas core are assumed and the holdups are known. For a turbulent velocity profile this assumption can be made since the velocity boundary layer will be relatively small compared to the tube diameter in the core or the film thickness. For a laminar flow, this assumption cannot be made.

21

With this assumption, the definition of the film holdup er and the entrained fraction E, the actual velocity in the film V1 is given by equation (2.38). For the velocity of the entrained draplets in the gas core, the superficial gas velocity is a good estimate.

3.1.2.2 Entrained fraction

Another important parameter is the entrained fraction in the gas core. lt not only has direct influence on for instanee the film velocity, but it also affects the actual percentage of the distance that a partiele travels in the gas core. With a smal! entrained fraction, most of the liquid has to travel through the film while for large values of E, the liquid spends a large part in the gas core. Since the veloeities in film and core differ by an order of magnitude, E has a large effect on the final residence time.

Further, the entrained fraction is an important parameter in the calculation of the entrainment and deposition rate, which on their turn are again key parameters in the simulation.

3.1.2.3 Entrainment and deposition rate

The difficulty in determining the residence time in AD flow is that film and gas core interact through the exchange of liquid. The rate at which this exchange takes place delermines the shape of the resulting distribution over time of the detector. lf the entrainment and deposition rates are small, the signa! would clearly show two peaks, one for the liquid in the film and one the liquid in the gas core; the resulting signa! will be asymmetrie. For very large exchange rates, the signa! would converge toa perfect gaussian function. In fact the amount of interaction indicates whether or not the system has to be split up into two subsystems. Note that if the exchange of liquid is very large, the system could be described by one resulting mixture velocity.

3.1.2.4 Liquid, film, gas and entrained holdup

The liquid and film holdup are parameters which have a lot of influence on the resulting mass fluxes. They do not only affect the actual veloeities in the system, but by definition they delermine the cross sectional surfaces of film and core and through that the actual flux in the film and the gas core.

3.2 Basics of the model

3.2.1 Discretisation of the experiment

To be able to simulate the residence time measurements, both the tube and the duration of the experiment in time have to be divided in segments. By doing so, a discrete model is created which can be processed with a computer. As mentioned before, the film and the gas core are treated as independent systems within the flow which communieale only through the exchange of liquid thus simulating the entrainment and deposition process. Figure 3.2 is a schematic drawing of the simulation.

As an initia! condition, the tracer fraction in the first element of the gas core and the film are set to a certain value. The tracer fraction in the rest of the tube is set to zero. By only filling the first element, the initial condition can be treated as the input of a delta peaked tracer injection in the real experiment. This is only valid of course when the length of the tube elements dx is small compared to the totallength of the tube.

With this tracer distribution as an input, the new distribution is calculated after a period of time dt and this will betheinput for the next time cycle. To calculate the propagation of the

22

tracer, use is made of the incoming and outgoing fluxes in each tube element, visualised by the small black arrows in tigure 3.2. Since the fluxes in vertical direction are related to the vertical veloeities V as

<DocV (3. 1)

the length of the tube elements dx must satisfy the condition

dx ~V ·dt (3. 2)

because the amount of incoming liquid must be smaller than the capacity of the element.

Tube 1 ;-~~~~~~~~~~~~~

I

I.É "" I

D detector

Bend

Tube2 ~~~~~~~~~~~~~

Fig. 3. 2 Schematic drawing of mass transport in the discretised AD flow with bend. The black arrows are the mass fluxes between the flow elements, the large grey arrow illustrates the detection process.

Another aspect of discretisation can be illustrated with the following thought experiment. Consider a system like the one described here except for the fact that it wiJl not be discrete, but instead it would be a continuous system. Further no exchange of liquid between the film and the gas care is to be considered and the tracer would be injected partly in the film and partly in the care. The veloeities present in the system would be sharply defined i.e. there wiJl be no spreading in the velocity. Then in the resulting signal of the detector somewhere downstream, two sharp peaks can be recognised, corresponding to the tracer passing the detector in the film and in the gas care.

lf this hypothetical system is discretised, the veloeities cannot be defined sharply because there is an uncertainty of dx and dt in determining the location and the time of a liquid partiele respectively. The resulting detector signal will show a velocity spreading, caused by discretizing the model. This effect wiJl be treated in section 3.3.1.

23

3.2.2 Mass balance

Suppose a liquid of density p1 flows at a velocity V through a pipe with cross section A. The mass flux and the actual amount of mass coming through the pipe over a period of time dt will be

= V. p, (3. 3)

(3. 4)

Also, since there is no liquid accumulating in the system, the incoming and outgoing fluxes have to be in balance. Therefore

 in = out (3. 5)

When the flow is in equilibrium, it is assumed that the rates of entrainment and deposition are equal. lf this would not be the case the entrained fraction in the gas core would be still developing. Mass balance demands all fluxes of the same nature to be equal in this system.

in

, Pvdm ,V film/cell

Fig. 3. 3 Liquid mass fluxes

with

R

R

The total picture of mass balance in each element of the tube, for the film and for the core, is shown in tigure 3.3. Given the film velocity Vr. the droplet velocity in the core Vc, the density of the liquid p1

and the entrainment and deposition rate R, the magnitude of the fluxes will be

 film,in = film,out = ~Pi

 core,in = core,out = Vcp dm

 entr. = dep. = R

(3. 6)

(3. 7)

the density of the droplet mass distributed along the core. Here Et and Et are the liquid and film holdup in the flow.

The mass balance described above doesn't give any information about the propagation of the tracer along the tube. To do so, the tracer fractions in each element in the tube must be taken into account, as well as the cross sectionat surfaces of each the liquid in the film and the liquid entrained in the gas core. The cross sectional surfaces of the film and the core are related to the cross section of the tube as

7r 2 ~lm = &f • Aube = 8 t 4 D

(3. 8)

Acre= {1- &, ) :02 with D the diameter of the tube. Further, to calculate the amount of tracer liquid exchanging between film and gas core, the surface of the gas-film interface is important

24

Ant = 7r~1-&, ·Ddx. (3. 9)

Suppose the total liquid mass m1.elem and mc.elem in each element of the film and the core at all times and the total tracer liquid mass m1•1,.n1 and mc,tr,nl in the film and the core at the nlh element of the tube and after i time cycles are given. Then the tracer mass fractions X1,n1

and Xc.nl in the film and the core at the corresponding element in the tube at that same time are defined as

x - mf,tr,ni f,ni- m

f,elem

with

(3. 10) and

(3. 12) and

m •. X . = e, r.m e, nt

me,elem

me,elem = p dm veell

in which Veen and V film are the volumes of the core and film elements.

{3. 11)

(3. 13)

When the distribution of tracer liquid along the tube in the film is known at a certain moment in time, X1,n1 for all n, the new distribution one time step later can be calculated. For each element in the film the new tracer mass fraction ~.n11•11 wil be a tunetion of the old fraction ~.nl

Xf. n(i+1) = Xf. ni + !0<,, ni (3. 14)

with

(3. 15)

in which Xc.nl is the tracer mass fraction in the core at element n at the ith time cycle. For the tracer mass fraction in the core this will be completely analogue to the above, hence

Xe, n(i+1) =Xe. ni + llXe, ni

llXe, ni = ( <l>e,inXe, (n-1)i- <l>e,outXe. ni }A:oredt +(X,, ni-Xe. ni )RAntdt

(3. 16)

3.2.3 Modelling the bend

The implementation of a bend in the simulation is a highly complex matter. Bends have proven to be complicated components, as was concluded trom former studies of bends in annular dispersed flow [12]. In this model, only the important features tor the exchange of liquid between film and core were taken into account. lt should therefore be kept in mind that the representation of the bend in the simulation is very limited compared to reality. lt was only introduced to study the mixing effect of the bend on the rasuiting signal, to explain some features of the measurements taken after a bend in the vertical flow facility.

lt is known trom former studies that centrifugal force in the bend will cause most of the entrained liquid to deposit in the film, where it is mixed with the rest of the liquid. This process is considered to be of great influence on the residence time. The motivation for this

25

assumption is the tact that liquid exchange between film and core is also a key parameter in straight tube models, since it affects the symmetry of the signal in residence time measurements. In addition it also affects the amount of time spent in the film.

In this model, the bend is represented by a single tube element in which the liquid trom both the film and the gas core just before the bend, is mixed and sent into the next tube. After the bend all the parameters of the flow are left unaffected i.e. the flow is assumed to be in equilibrium state. Though this assumption has influence on the results of the simulation, it is the only description for which no new parameters related to flow redevelopment after the bend have to be introduced.

In order to calculate the mass fractions of tracer in the bend, the total amount of liquid mass present in the bend has to be calculated using

(3. 17)

The tracer fraction in the bend is defined as Xb.l , analogue to the definition of the tracer fractions in the tube in equations (3.1 0) and (3.11 ), thus

x - mb,tr,i b,i- m

bi (3. 18)

To find the fraction at t=(i+1 )dt, it should be kept in mind that Xb.l is also subject to the relation given in (3.14 ). Therefore, if N is the total amount of length elements in one tube, the new fraction xb.l will be

xb, (i+1) = ~ ((mbl- <Df,outAilmdt- <Dc,outA:eudf)Xb, i+ <Df,inAumxf. Nidt + <Dc,inA:euXc. Nidt) bi

(3. 19)

in which X 1.N1 and Xc,NI refer tothelast elementsof the first tube, just before the bend. Note that the outgoing flux to the core of the second tube equals the incoming flux

trom the core in the first tube. This means that the outgoing velocity times the entrained fraction equals the equilibrium velocity times the equilibrium entrained fraction, corresponding to disregarding the developing flow as a first estimate.

3.2.4 Calculating the signal

As is illustrated in tigure 3.2 the detectors in the simulation are defined as devices which monitor the tracer fraction passing one element in the tube. The monitored amounts of tracer fractions are stored in a table and processed later. An advantage above the detectors in a real experiment, is that the modelled detector can distinguish tracer fractions in film and gas core separately. In the Iabaratory experiment, the large ditterenee between the veloeities in the film and in the gas core causes complications. Tracer passing the detector very quickly will have a smaller detection probability then tracer passing the detector slowly. This comes tromthetact that the detector will cover a certain cross sectional volume of the tube. Particles travelling through this volume with a speed of 25 m/s will have their detection probability reduced by a factor 25, compared to particles with a velocity of 1 m/s.

The actual amount of tracer is calculated trom the fractions using (3.1 O) and (3.11 ). The sum of these contributions gives the total amount of tracer mass present in the element at time t=i·dt. lt is desirabie to plot the signal that a real detector would give monitoring the

26

modelled experiment as well. To do so the following sealing is introduced, basedon the dependenee of the detection probability on the velocity

signa/. = mf,tr, ni + mc,tr, ni

I V. V f c

(3. 20)

with V1 and Vc the veloeities in film and gas care.

3.3 Mathematica I consequences of the discrete model

3.3.1 Determination of location and time

As stated in sectien 3.2.1 the exact location of a partiele in the discretised model is undetermined by dx, within an element of length dx. The same can be said about the exact time at which the partiele is located at that specific element of the tube: time is undetermined by a period dt. Th is, of course, limits the determination of the velocity of a partiele in the flow and as aresult discretising the experiment will cause a spreading in the velocity.

Suppose in a real Iabaratory experiment, a hypothetical annular dispersed flow would only be subject to two well defined constant velocities, a film velocity and a core velocity. Further no exchange of liquid between film and core would be present. A delta peaked tracer injection would be done partly in the core, partly in the film. Then the resulting detector signa! at the end of the tube would show two sharp peaks, because all the liquid in each separate subsystem (film or core) would travel with a well-defined velocity.

Figure 3.4 shows the resulting signal of a detector in such a system. The signal for a discretised simuiatien of the same system is shown in tigure 3.5. In the Jatter, two gaussian peaks can be recognised around the predicted average residence times of the both separate systems. Since the magnitude of dx and dt are known, the standard deviation and so the Full Width at Half Maximum (FWHM) can be calculated.

',:Tl, 0.35

0.31.

I 025'

• ''Tl 0.15

Tracer fraclions dMIK:tor 2

--Fig. 3. 4 Resulting signal in a hypothetical AD flow which has no exchange of liquid between the core and the film. The veloeities are sharply defined and there is no spreading around the

average residence times

27

Tf'IICW tt.cta.s det.ctor 2

Fig. 3. 5 Signal of a discretised AD flow where the exchange rate between the core and the film is set to zero. The residence time distribution shows a clear spreading around the

average.

The output of the system from the detectors will give the tracer fraction passing the detector at a certain time, for the whole duration of the experiment. lt is also possible to freeze the system at a certain time and plot the distribution of the tracer along the tubes as a result. In a system without exchange of liquid between the film and the core, both results will show two gaussian peaks for the film and the gas core. Although the mathematica! form of the spreading in both results will be very similar, it has to be calculated in a different way for both.

3.3.1.1 The discretisation error in the residence time distribution

To calculate the standard deviation in the detector signal, one has to realise that the system can be treated as a coneetion of N filters. Each filter will, due to the undetermined location in

each cell (~ ± Y2·dx}, introduce an error

11t = dx rd V (3. 21)

in which ilt,.d is the standard deviation in the residence time and V is either the film velocity V, or the core velocity Vc. The above can beseen from the fact that the accumulated error from both ends of the trajectory will be dx, and V is the defined velocity of the particle. With nd the element of the tube where the detector is located and ~=nd · dx the location of that cell, the standard deviation after nd cells will be

(3. 22)

Proof for this result comes trom the convolution of n Gaussian functions for which this is exactly true. lt is an approximation for general functions. For the case of n filters this is only an estimate of the standard deviation. However for large nd the standard deviation will converge to this result.

Assuming, for the moment, that the resulting output of a block shaped input at the beginning of the tube will be the estimate of a Gaussian function, it is easily calculated that the full width at half maximum (FWHM} will be

28

FWHM = 2 · ~2(-ln~) · (J' = 2.355 · (J' (3. 23)

with cr the standard deviation of the Gaussian. lf !l is the average of the Gaussian distributed population, the above follows trom

f(x) = 1 exp{- (x- ~)

2

} ..[2; a 2a

(3. 24)

in which x is the value of interest, and f(x) gives the odds of this particular x to occur in the population. The FWHM is aften treated as the error when a measured or predicted value is normally distributed i.e. satisfies (3.24 ).

3.3.1.2 The discretisation error in the tracer distribution

As stated befare the FWHM of the gaussians seen in the tracer distribution along the tube, for film and core, is calculated a bit different although the result will be similar. lt is necessary to understand that tor each partiele in one element of the tube, making a transition to the next element is subject to a probable occurrence. Since only a part of the liquid present in the cell will make this transition, this can be treated as the Poisson limit of the binomial distribution.

Suppose again a discretised model tor AD flow without the exchange of liquid. lf the distribution of the tracer along the tube will be plotled after N1 time steps, and p is the probability for makinga transition, the average travelled distance of the particles is

The Poisson distribution tunetion is given by

n

P(n,p) = Lexp(-p) n!

(3. 25)

(3. 26)

with n the number of transitions made and !l the average number of transitions. The varianee in the number of transitions will therefore be

(3. 27)

in which cr is the standard deviation. In the present case this will converge to

(3. 28)

lf V is the velocity of the particle, dx the length of one element and dt the duration of one time step, the odds on making a transition to the next element tor each partiele are

Vdt p=

dx (3. 29)

29

Substituting (3.29) in (3.28) and multiplying this by 2.355-dx, (3.28) gives the FWHM of the distribution of tracer liquid along the tube,

(3. 30)

Of course V=V1 or V=Vc have to be substituted in (3.30) in order to get the errors in the film and in the core.

3.3.2 The accumulated error of the system with liquid exchange

Given the standard deviation for the film and the core separately, introduced by the discretisation of the system, it is now possible to deduce the accumulating error when the both systems are conneeled through the exchange of liquid. The number of transitions between the film and the gas core is assumed to be much smaller than the number of transitions between the elements of the film or of the gas core. Essentially this means that the length of one element dx must be much smaller than the typical exchange length of liquid between the film and the gas core (the average distances that dropiets travel in the core or the film). The influence of these transitions between film and gas core on the accumulated error of both systems will then be smal!, and can be neglected.

Assume a certain partiele in the flow will be detected in element nd at time t. The travelled distance at this element is xd=nd·dx and since the veloeities in both systems are known, the percentages travelled in the core and in the film are known as well. First it is necessary to examine the both limits of t,

with te the time a partiele would need when travelling only in the gas core,

t = nddx c V

g

and ~ the time for a partiele travelling only in the film,

t _ nddx f-

~

(3. 31)

(3. 32)

(3. 33)

With a1 the fraction of the total distance spent in the film and ac the fraction spent in the core the residence time of the partiele in the region of the tube befere the detector will be

(3. 34)

This of course implies

(3. 35)

30

Equations (3.34) and (3.35) are solved tor af and ac, this yields

t-t a=--~

e fe - ff (3. 36)

and

t -t a __ e __ ~-

te- ff (3. 37)

The errors in af and ac are disregarded being of second order. Further (3. 22) is used to calculate the standard deviations in te and 4. lt has to be noted that the liquid travelled af 1 00% of the di stance through the film and ac·1 00% through the gas care. Th is must be taken into account when the partial errors ~4 and ~te are calculated. The errors introduced by the travelled distance through the film and the care are given by

(3. 38)

and

(3. 39)

Now the accumulated standard deviation in the residence time t of the particle, due to discretisation of the experiment, can be calculated by regarding the errors ~4 and ~t.: as standard deviations. The first order estimate of standard deviation in the residence time is

(3. 40)

The error in the resulting gaussian will be the FWHM which is 2.355 times this result. This can be checked using the simulation. Note however that the FWHM ofthe calculated distribution is not only the result of the discretisation. A part of the spreading has a physical origin; it comes from the velocity difference between the care and the film and is dependent on the exchange rate R between the film and the care. To verify the result of (3. 40), the exchange rate between the care and the film, R, has to be set to zero. Now, when the tracer is injected only in the film, it will travel the whole first tube befare the bend in the film because there is no exchange of tracer with the gas care. After the bend, however, a part of the tracer is supplied to the gas care. The tracer fractions passing the last detector at the end of the experiment, will clearly show two separate peaks. The tastest of these two comes from the tracer liquid which travelled 50% of the distance (before the bend) through the film and 50% of the distance through the gas care. Because the veloeities in the film and the gas are known, as well as the location of the last detector, it is possible to calculate the FWHM of this peak and campare it to the measured FWHM. Figure 3. 6 shows the resulting signa! of such an experiment. The gas velocity was 25.44 m/s and the film velocity was 1.65 m/s. The detector is located at 20.1 m and the

31

percentage of the distance travelling through the film was 50.24%. Further the length of the tubeelementsis 0.1 m. Substituting these values in (3. 40) and multiplying by 2.355 to get the FWHM results in a FHWM of 1.42 s. lt can be checked that this is exactly the FWHM of the first peak in tigure 3. 6.

0.03

0.025

0.02

I '

)( 0.015

O.Q1

I 1

0.005

0 0 2 4 6

Tracer fractlons detector 4

8

time [sec]

10

LL

12 14 16

Fig. 3. 6 The tracer fractions passing the last element in the tube, after the bend. The first of these two separate peaks has a FWHM of 1.42 s.

32

4. Experimental setup

4.1 The vertical flow facility

The residence timeexperimentsin AD flow are conducted in the so called vertical flow facility, or shortly VFF, which is especially developed to study this regime of two phase flow. The VFF consists of three tubes which are connected by U-bends. Th is is schematically illustrated in tigure 4. 1.

The diameter of the tubes is 0.1 m. The sharpness of the bends is defined as the ratio between the radius of the centre line of the bend and the diameter of the tube, RID. In the vertical flow facility, the ratio of the first two bends is R/0=1.5, the lastbendat the end of the tube has a ratio 1.0.

AD flowloop

tracer

gasliquid

separators

LIQUID

AIR

Radioactiva residue storage

Fig. 4. 1 Schematic drawing of the vertical flow facility

33

From below, air is supplied to the system at a pressure of 1.5 bar. This air is supplied by a compressor and the amount is primarily controlled by a pneumatic control valve, V2 in tigure 4. 1. Just above the beginning of the tube, a water in let is mounted which is developed to supply water as a film on the tube wal I. The water is pumped from the water reservoir, the amount of water supplied can be regulated by pneumatic control valves on the supply line. The water pump operates at a pressure of 4 bar, the compressor supplying the air operates at a pressure of 2 bar.

When both the air and water are being supplied to the system the AD flow is visible in the perspex tube. The conditions of the flow are defined as the desired combination of the pressure p1

and the superficial veloeities of air and water, V59 and V51 , in the tube. These can be fine tuned with 4 manual control valves which areleftout tigure 4. 1 for clarity. Since all three, the superficial veloeities and the pressure, are coupled parameters, it is ditticuit to fine tune the conditions. The superficial veloeities are calculated from the flow rates of gas and liquid. These flow rates are measured by the flow metersFin tigure 4. 1.

At the end of the tube, after the third bend, the water-air mixture is led to two gas-liquid separators. After these separators the water is led to the radioactive residue storage tank because of the radioactive tracer concentrations in the water. The air is released into the environment.

4.2 The residence time measurements

The residence time of liquid in AD flow is measured by quickly injecting a small amount of a radioactive tracer liquid in the flow. At 8 locations along the tube, detectors record the amount of radioactive radialion coming trom the passing flow. The intensity of the radialion is linear dependent on the amount of tracer liquid passing in the tube. Because the tracer liquid in the entrained fraction travels much taster than the film, it has a reduced detection probability.

The tracer liquid, a 49 ln 113m-EDTA complex, has a half life decay time of 99.42 minutes. Th is isotope makes an isomerie transition under theemission of 0.3917 MeV y-radiation. To make this isotope of Indium solvable in water, it is captured in EDTA complex (Ethylene-Diamine-TetraAcetate).

From the signa! coming from the detectors, the average residence time can be found by determining the moments of passage of the peak of the tracer bulk at two locations in the tube. The distance between these locations, divided by period of time between the passage of the tracer at these locations, is the average liquid bulk velocity in the part of the tube between the detection points.

More detailed information about the injection system is given in section 4.2.1. lnformation about the detectors and their location along the tube is given in section 4.2.2.

4.2.1 The injection system

In order to measure the residence time in the flow, the duration of the injection has to be small compared to the total residence time. The injection can then be considered a delta peaked injection. Further, a high reproducibility of the amount of injected tracer liquid is desired. lt is also demanded that the injection does not disturb the main flow too much.

Regarding the distribution of tracer liquid after the injection it is demanded that the tracer is equally distributed in the tangential direction of the tube. ldeally the fractions of tracer injected in the film and in the entrained fraction must represent the exact fractions of the totalliquid entrained and travelling in the film.

To satisfy the conditions mentioned above, an injection system is designed as shown in tigure 4. 2. The injection system is controlled by an 8-way two position valco valve, which is triggered using a comat programmabie micro control unit.

In position 1, the 2 mi tracer loop is tilled through the tracer fill inlet. When the loop is tilled, the left over tracer is colleeled in the tracer residue storage. The high pressure line and the outlet to the experimental tube are closed in this position.

34

Tracerfill

tracer residue slorage

15 bar

8·way two position valco valve

-- pos~ionll

-- pos~ionl

/-

toVFF

1- - - - - - - - - ~

I I I I I

...... 1 I I I I I I _________ ..J

Fig. 4. 2 The tracer injection system. An enlargement of the nozzle is shownon the right.

Fig. 4. 3 The Nozzle

4.2.2 The detectors

In position 2, the high pressure line is conneeled to the 2 mi tracer loop inlet, while the outlet of the loop is conneeled to the VFF line. Th is will blow the tracer directly into the experiment through the nozzle. The tracer fill line is in this position connected to the residue storage.

Because the valve is controlled electronically by the micro control unit the duration of the injection can be adjusted accurately. Assuming the pressure on the high pressure line constant as well, the amount of tracer liquid injected will be constant. The duration of the injection is adjusted to 0.2 sec.

The nozzle, of which a picture is shown in tigure 4. 3, is mounted on a small tube with an inside diameter of 1.0 mm and an outside diameter of 1.59 mm. The hexagonal shaped nozzle itself is 11.05 mm wide. In each of the six sides a small hole is drilled. These six holes are the outlet of the nozzle. Compared with the 50mm wide experimental tube, the cross sectional area of the nozzle is about 25 times smaller.

The locations of the 8 Nal-detectors in the experiment is chosen carefully. The residence time in all three tubes had to be measured, which means that along each tube, at least two detectors were needed. The locations of the detectors are given in table 4.1, tigure 4.1 gives a schematic overview of the location of the detectors.

The goal of the first detector is to check the width of the delta peaked injection and to give the offset for the injection time. The second and the third detector wiJl show the tangential symmetry of the injection. lf the signal in one of these two detectors is signitically larger, or the shape of the signal differs trom the shape in the other detector, the injection cannot be tangential symmetrie. Detector 4 to 8 are the detectors trom which the residence time is determined. The last detector in the tube, detector 8 is placed in the centre of a bend. The reason for this is given by the velocity dependenee of the detection probability, as is discussed below.

35

Detector location description

detector 1 O.OOm 0.21 m above injection point

detector 2 1.40 m height above detector 1

detector 3 1.40 m opposite of detector 2

detector 4 8.23 m 0.13 m before bend 1

detector 5 9.15 m 0.56 m after bend 1

detector 6 14.33 m 0.38 m before bend 2

detector 7 15.82 m 0.88 m after bend 2

detector 8 22.05 m outlet bend I centre

Table 4.1 LocatJon of the detectors

Figure 4. 4 shows schematically the volume of the tube which is seen by a detector. A partiele in the film with velocity V1 will be within the dateetion volume for an average time t1 •

Suppose another partiele travals as an entrained droplet in the gas phase of the flow, its velocity is V2 = 10·V1• This means that since it has to travel the same distance, it will only be within dateetion range during a time t2 = t1 I 10. lts dateetion probability will therefore beten times smaller, and so the signal for particles in the gas flow will be ten times smaller as wel I.

The problem would be solved if the experimental design could be modified such, that the dateetion probabilities would be the same for both the film and the entrained fraction. During the experiment this, unfortunately, can not be done since it would disturb the flow, but the design can be modified at the end of the experiment where the last maasurement is taken .

.. ----tr---~ .. - ..

.. .. .. ..

detector :-5~·-·-·-·-d·-· ... ...

... ... .. ... .. ...

Fig. 4. 4 Dateetion volume

..

...

.. ..

... ...

.. ..

... ...

Therefore, detector 8 is placed at the centre of a sharp bend. lt is known that almost all of the entrained dropJets will depositover the bend in the flow. The rate at which this happens is dependent on the 'sharpness' of the bend, defined as the ratio between the radius of the centre line of the bend and the diameter of the tube i.e. RIO.

36

5. Results of residence time measurements in AD-flow

5.1 lntroduction, description of the terminology

In the following chapter the results of the residence time measurements in the vertical flow facility, will be presented. The residence time is directly determined by the average velocity of liquid in the system and the spreading of this velocity around its average. To maasure the residence time, a radioactiva tracer liquid was injected just after the in let of the experimental tube. Along the experimental tube, which consists of three tubes connected by U-bends, 8 detectors were placed which could piek up the signal coming from the radioactiva tracer when it passed the detectors. The 8 detectors were placed such that the experimental tube as a whole was divided into sections.

Analysing the signal from each detector made it possible to maasure the average velocity of the liquid in each section. The average liquid bulk velocity in a certain region of the experimental tube, is defined here as the length of the region (i.e. the distance between two consecutive detectors) divided by the time shift of the maximum in the signal, associated with the maximum tracer concentratien in the film, between the signals of both detectors.

Distinction has to be made between the veloeities in the different regions of the experimental tube. To do this, the average liquid bulk velocity in the region of the tube between detector number i and detector number j will be written as V ii·

The average liquid bulk veloeities which follow from the experimental data were compared to veloeities predicted by existing models set up by Miesen [13.1-5] and Asali [2]. In fact these are models predicting the entrained fraction and film thickness. With these values the liquid hold-up can be calculated. The predicted average liquid bulk velocity from the modelsis defined as the superficialliquid velocity V51 divided by the predicted liquid hold-up, which follows trom the models. The models are able to predict values for the liquid hold-up in up-flow and down-flow.

Further, the average liquid bulk velocity following trom the experiment is compared to the velocity predicted by a simulation. The simuiatien is basedon mass fluxes between tube elements in two discretised tubes, connected by a simulated U-bend. The simuiatien cannot distinguish up- and down-flow, this has not yet been implemented.

Fig. 5. 1 Classification of the examined veloeities

37

The different veloeities discussed in this chapter can be classified as was done in tigure 5. 1. The measured veloeities will be discussed in sectien 5.2.2. The experimental data are compared to the predieled veloeities from the modelsin sectien 5.4, this will be done in terms of the absolute and the relativa difference between the model and the measurements. In sectien 5.5, the simulated veloeities are compared to the measurements and the values predicted by the models.

Th is chapter will also discuss the spreading in the measured average liquid bulk veloeities in sectien 5.3 and attention will be paid to some of the features which follow from the raw data, i.e. the signal plots from the detectors.

5.2 Qualitative description of the experimental results

As was described in chapter 4, the measurements were done at carefully chosen superficial gas and liquid velocities. The combinations were chosen such that the data could be evaluated at constant superficialliquid velocity and at constant superficial gas velocity.

The number of detectors made it possible to calculate the average veloeities of liquid in up-, down- and developing flow; i.e. the region of the flow just after the inlet before equilibrium flow has been established. lt must be said, however, that interpretation of the measured veloeities is not as straightforward as it seems. For instance, suppose the velocity between the first (right above the injection) and the last detector (at the end of the third tube) is measured. lt cannot be interpreled as the overall average velocity of liquid in annular dispersed flow, since it has tobetaken into account that the flow in the first tube is still a developing flow. lt is unclear over which distance in the first tube the flow is in a developing state, dependent on the gas and liquid flow rates.

In general it is assumed that inlet effects (developing flow) only play a role in the first tube. Apart from the influence of the bends, which is considered to be an inherent part of the experiment, the flow is developed in the second tube (down-flow) and the third tube (up-flow).

5.2.1 Analysis of the detector signal

In tigure 5. 3 the signa I from the detectors is shown for the flow condition determined by Vs9=47 m/s and Vs1=0.0065 m/s. Although these superficial veloeities correspond to extreme conditions for the flow, maximum Vs9 and minimum Vs1, they were especially chosen because this condition clearly shows some interesting features of the flow. Conditions with higher superficial liquid veloeities often do not show these features as clearly because of the higher entrainment and deposition rat es.

The signal plot of the first detector shows the delta peaked tracer injection. From this plot, the offset for the injection time is determined. Note that the width of the peak is much braader than the injection time discussed in chapter 4. An explanation for this is found by realising that a lot of broadening takes place in the developing film.

•

• • • • •••

• • • • • •

water+ tracer The signal plots from the second and the third

detector clearly show identical signals measured from two opposite sides of the tube. The both signals overlap al most completely which is interpreled as a equally distribution of the tracer fluid around the circumference of the tube. The ridge on the ascending slope of the curves in both plots is the effect of tracer liquid travelling as droplets. The peak, which is seen somewhat later in the signal, is caused by the tracer liquid which is travelling with the film. Because of the continuous interaction between the film and the entrained

· ······ water

V

Fig. 5. 2 Tracer distribution in film and co1

38

fraction, tracer fluid is present at all times between the passage of tracer solely travelling as draplets and solely as film. This is shown in tigure 5. 2. The result is the typical asymmetrie farm of the peak.

Note that the entrainment and deposition rates are important for the symmetry of the signal. For the present condition these rates are very small. This results in very asymmetrie detector signals. For very high entrainment and deposition rates, the resulting signal would be one gaussian of which the standard deviation would be determined by the difference between the core and the film velocity.

This effect can be seen in the signal plot from detector 4 as well. lt clearly shows a symmetrie peak from the tracer in the film, which did not participate in the entrainment and deposition process, and a ridge on the ascending slope of the gaussian coming from the entrained tracer fluid.

The fifth detector shows a new feature present in the system. In this plot two peaks can be distinguished, first a small peak passes the detector and after some time the peak from the film can be seen. The small peak in the front comes from the entrained fraction which was deposited over the bend. A known effect of bends is that almast all the entrained draplets deposit in the film (see also Hulsman [12]).

Countrate va time; detector 1 (#47000058)

100000

i 60000

! 40000

120000 I 80000 u 20000

O~o-~

time (a]

Countrate vs time; detector 2 (#47000058)

0 1 2 3 4 5 6 7

time (•)

Countrate vs time; detector 3 (1147000058)

01234567

Countrate V& time; detector 4114700005B)

10 20 JO

time (s)

Countrate vs time; datactor 5 (1147000058)

80000 -

60000-

... 50000 -

!. 40000 -

! 30000 -

70000_ A 20000 - ~ '

100~ _-_1\_--~~-f~/~-~~~=;=;-:_, _____ .-"' ___ ~ 0 10 20 30

Countrate vs. time; detector 8 (114700005B)

60000-

50000 - ,,A 40000 -

i 30000-

~ 20000-

10000-

0 0 10 20 JO "' 50

time [•)

Countrate v& time; detector 71147000058)

50000-

40000 -

·'A i 30000-

~ 20000-

10000-

0 ~

0 10 20 30 40 50 .. tlme(s)

Countrate .,. time; detector a (1147000058)

16000-

16000- ;-\, 14000. 12000-

i 10000 !

i :: \ ::~: ~---~~~/~\

10 20 30

Fig. 5. 3 Detected radioactive radiation intensity versus time measured by each detector

39

The origin of the third peak in the signa! of detector 5 is unclear. Note that the maximum of the peak passes at 27 seconds after the injection. Gomparing this to the location of the main peaks in the signa! of detector 7 and 8 it suggests that the signa! is picked up by detector 5 when the tracer passes in the last tube.

The same might be said about the first smal! peak in the signa! of detector 6. lt passes at 6 seconds after the injection. lt is not clear whether the peak comes tracer entraining from the first peak in the signa! from detector 5, or from tracer which was present between detector 4 and detector 2 and 3. Both hypotheses are possible.

The second smal! peak is associated with the entrained tracer fluid which deposited over the first bend and stayed in the film after that. Proof for this comes from the fact that the peak is shifted about 6 seconds compared with the first peak of detector 5. The main peak in the signa! also shifted 6 seconds compared with the main peak of detector 5, so the both peaks must have travelled with the same velocity. The travelled distance between detector 5 and 6 equals 5.18m so the velocity of both peaks was of the order of 1 m/s which is characteristic for liquid travelling in the film.

Analysing the signa I from detector 8 shows that the first appearance of tracer is detected afterabout 10 seconds. Subtracting the offset of the injection (1.6 sec) shows a residence time minimal value of about 8 seconds, which corresponds to a velocity of 2.5 m/s. This velocity is far below the maximum velocity of the entrained fraction, being approximately the superficial gas velocity (47 m/s). From this one could conclude that the amount of fluid travelling as droplet over the whole distance is negligible. Note however that tracer passing the detector with a velocity of 47 m/s has a deleetion probability which is 47 times smaller than tracer passing with a velocity of 1 m/s, it is possible that this part of the tracer fluid is just not detected or lost in the noise of the signa!.

5.2.2 Velocity measurements compared with liquid and gas flow rates

Figure 5. 4 and 5. 5 show the veloeities which were measured trom the raw data discussed in section 1.1.1. The veloeities shown in tigure 5. 4 are all measured at a superficial liquid velocity of V 51=0.0400 m/s and plotted against increasing superficial gas velocity. The veloeities mapped in tigure 5. 5 represent veloeities measured at a constant superficial gas velocity of V59=25 m/s. The calculated veloeities are allbasedon measurement of the time at which the maximum of the main peak passed the detectors 1 to 8.

Vsl = 0.0400 m/s

3.50 ~----,--------.---------.--------r---.-----.-------.--------:.---------,

3.00 +----+----1----+---t---+-----+----+----+-----1

2.50 +-----+----+----+-----1---+----+----t----+----1

_2.00 +----+----1----+---t---+----+----+----+-----1 i ... "S ~150+----+----+----+-----1---+----+-----t---~-----1

1.00 +-----+----+---+---t---+----+-----+----+--------1

0.50 +----+----+----+----+---+----+----+----+------1

0.00 +----l------+----+-------1---t----+----+----+-------l 0.00 5.00 10.00 15.00 20.00 25.00 30.00 35.00 40.00 45.00

Vsg [m/s)

Fig. 5. 4 Measured average liquid bulk veloeities in different regions of the experimental tube, plottedas a tunetion ofV59• Most datapoints are Jeft out because of confidentiality.

40

The velocity v18 is the average velocity of the bulk liquid throughout the whole experiment, between the first and the last detector. Note that this velocity includes the developing state of the flow. lt will therefore besmaller than the average velocity in a fully developed flow, assuming that most tracer is injected directly in the film. The average velocity in the developing flow is given by v 123, the velocity between the first detector and detector 2 and 3. For all superficial gas and liquid velocities, this velocity is smaller than the other veloeities throughout the experiment. This is probably caused due to inlet effects and the presence of tracer predominantly in the film, with respect to the general liquid distribution in the flow.

For low superficial gas veloeities and high superficialliquid velocities, the veloeities between detector 2, 3 and 4 (being V 234 ) are smaller than the average veloeities throughout the rest of the experiment. This makes interpretation of v18 as being the average velocity of liquid in AD flow questionable because apparently the region between detector 1 and 4 contains the inlet effects of the flow and the injection of tracer.

In order to get good insight in the average veloeities in a fully developed flow it is important to examina the average veloeities in the second and third tube for down-flow and up-flow respectively. The average velocity in down-flow is given by v56, while the average for up-flow is identified with v78 .

Because of the influence of gravity, the velocity in down-flow is expected to begreater than the velocity in up-flow. This turns out to be true at small superficial gas veloeities for all superficial liquid velocities. But the prediction starts to fail for superficial gas veloeities of 30 m/s or higher. For very high liquid flow rates corresponding to superficial liquid veloeities of 0.16 m/s and higher, the prediction still holds at higher gas flow rates.

5.00

4.50

4.00

3.50

3.00 .. ! 2.50 .. :; ~ 2.00

1.50

1.00

0.50

0.00 0.0000 0.0500 0.1000

Vsg = 25 m/s

• .v234

• • V56 x

xV123

xV7B

0.1500 0.2000 0.2500

Vsl[m/s)

Fig. 5. 5 Measured average liquid bulk veloeities as a tunetion ofVst· Most datapoints are leftout because of confidentiality.

An explanation for this tendency is the fact that for larger gas rates, the influence of gravity bacomes relatively small compared to the torces reprasenting the shear between the gas core and the film. Therefore, for larger liquid rates, the prediction will hold longer than for smaller liquid rates. The irrelevance of gravity in AD flow is in fact not a surprise, since this flow region is by definition dominaled by shear forces. lf gravity would be a dominant parameter in the flow, then the flow would be achurn flow.

In the next section it will be shown that the measured veloeities for each condition are very dependent on the film thickness and the entrained fraction. The results of the velocity measurements can be found in appendix A. The shape of the residence time curves,

41

presented in the farmer section, will depend much on the amount of interaction between film and gas core. Th is will be proved with the help of the simuiatien presented in chapter 3.

5.3 Velocity spreading; turbulent versus laminar flow

Another parameter which determines the residence time of fluid in the system is the spreading in the average liquid bulk velocity. This spreading determines the spreading in the residence times and therefore determines the longestand shortest residence times in the system. Apart trom the information about the residence time spreading, information about the nature of the flow is revealed.

By looking at figure. 5. 3 , it can be seen that most of the spreading in the system is created in the first region of the experiment, FWHM d8 is only marginally larger than FWHM d4. This initially created residence time spreading is not interesting since it is most probably a result of the developing flow in the first tube. Because of this problem, a different way had to be found to come up with a measure of the spreading created in the last two tubes in the experiment.

The spreading in the residence time can be found by statistica! analysis of the detector signal from one of the detectors. Doing this yields the spreading in residence times in the system at that particular detector. This can be done for two detectors and the spreading in the signa! trom the first detector can be subtracted trom the spreading in the latter detector signal. As a result , a measure for the spreading created in the region between the two detectors is found.

14

12

10

8

....

.!!. ca 6 -'ij " 4

2

0

-2

Spreading rd time vs. Vsg Vsi=0.04m/s

I ~

I

~ ~ j

10 15 20 25 30

Vsg [m/s]

I

~

35 4 0

+ FWHMd8

• FWHMd4

6 dfwhm48

Fig. 5. 6 Spreading in residence time as a tunetion of V59, calculated by subtracting the spreading in the signalof detector 4 tromthespreading in the signal from detector 8.

The best results, using this technique, are found if the signal is a perfect gaussian function. lf the signal is notsymmetrie around its average, it is hard to measure the spreading because it then it is built up trom the spreading in the different components of the signal. For

42

instance, if the spreading is measured in a signal which consists of two peaks referring to two veloeities in the system, the overall spreading in the signal is a combination of the spreading in the two veloeities present and their difference.

Spreading rd time vs. Vsl Vsg=25m/s

20

* • 15 • • ' :

10

~

5 • * -111

A

0 IA A

~ ~

0.05 0.1 0.15 0.2 0.2 5

-5 _L______jl :~--------'---____l___-------'------__ 1

Vsl (m/s)

Fig. 5. 7 Spreading in the residence time as a tunetion of V 51

+ FWHM8

• FWHM4

A dFWHM48

The signal in tigure 5. 3 has the additional problem of changing shape during the whole experiment. Based on the discussion above it is obvious that the measured spreading in two different detector signals would be related to different components in each signal.

Fortunately the measured signals become more and more symmetrie at higher superficial gas and liquid velocities, related to the amount of interaction between the film and the gas core. For small superficial velocities, the error in the measured velocity spreading will be larger but the averages of the measurements are still useful. Using the values for small velocities, it is enough to keep in mind that the error is large.

The spreading in the residence time was measured between detector 4 and 8. The results can beseen in tigure 5. 6 and 5. 7. The data is sorted and plotled as a tunetion ofVsg and vsl·

The measured spreading in the residence time tums out to be very small. This result is quite remarkable because it means that the spreading in the veloeities for the fully developed flow must be small. Therefore it is much more plausible to consider the flow turbulent at all times insteadof laminar. For the gas core this is nota surprise since the Reynolds numbers are of order 1 05 and even larger because of the entrained droplets. However, up until now the film flow was considered a laminar flow.

For free falling film flow it can be found in literature (for instanee Beek [4]) that the transition from laminar to turbulent takes place at Re1=1000, where Re, is the film Reynolds number. In the present case the film is govemed by shear stress at the surface. In addition, because of the large interaction with the gas core, the critica! Reynolds number can be argued to be even smaller than 1000.

Using equation (2. 47), the Reynolds numbers were calculated for the liquid in the film and plottedas a tunetion of V59 and V51 • The plots are shown in tigure 5. 8 and tigure 5. 9 respectively.

43

The Reynolds numbers show a stronger dependency on the superficial liquid velocity than on the superficial gas velocity. This also follows directly from the equation for Rer since it is directly dependent on Vs1 and indirectly dependent on V59 through the entrained fraction. The superficialliquid velocity varies between 0.0065 and 0.2, ditterences of a factor 40, while the entrained fraction varies mostly a factor 10 between 10% and 90%.

.._, Gl

0::

.,_ Gil 0::

8000

7000

6000

5000

4000

3000

2000

1000

0

Reynolds number in the film at constant Vsl

~

~

liC 1-

x 1\ liCK

J J • •

• Vsi=0.0066 m's

• Vsi=0.01 m's

A Vsi=0.02 m's

x Vsi=0.04 m's

x Vs1=0.08 m's

eVsi=0.12m's

+ Vsi=0.16 m's

• Vsi=0.20 m's

0.00 5.00 10.00 15.00 20.00 25.00 30.00 35.00 40.00 45.00 50.00

Vsg [m/s)

Fig. 5. 8 Reynolds number in the film as a tunetion of V59

Reynolds number in the film at constant Vsg

8000

7000 • 6000 • • Vsg=15m's

5000 • Vsg=20 m's

• A A Vsg=25 m's

4000 xVsg=30 m's

:K Vsg=35 m's • A 3000 x • Vsg=40m's • x

A + Vsg=45 m's 2000 ... liC

t liC • 1000 ...

0 0.0000 0.0500 0.1000 0.1500 0.2000 0.2500

Vsl [m/s)

Fig. 5. 9 Reynolds number of the film as a tunetion of V s1

lt should further be stated that the entrained fraction was not measured from the experiments but was calculated using Miesen [13.2]. To calculate the film Reynolds number,

44

the measured actual veloeities could not be used, so the values are not based on experimental data. lt is therefore difficult to delermine the errors.

The lowest Reynolds numbers were found to lie around 300, for the smallest liquid flow rate. Forthese flow rates, the Reynolds numbers were almost independent on V59 because of the small entrained fraction. Compared with the measured spreading in the residence time the film flow behaves turbulent.

The characteristic difference between laminar flow and turbulent flow is that turbulent flows are governed by vortices, while in laminar flow the viscous forces stabilise the circulation. Taking notice of the measurements and the calculated Reynolds numbers for these values, the idea rises to closely watch the features of the film to see which processas might be the cause of the turbulent film. Th is will be discussed at the end of the report.

5.4 Validation of the entrainment models

5.4.1 Absolute errors of the predicted velocity

The measured veloeities for up-flow and down-flow, v78 and v56 respectively, were compared with the veloeities predieled with the model of Miesen [13.1-5] and Asali [2]. The predicted average veloeities were calculated by dividing the superficialliquid velocity by the liquid holdup as predicted by the models. Figure 5. 1 0 is a parity plot of the predieled veloeities for upflow and down-flow against the measured veloeities v78 and v56•

Mieaan model tor upftow vs. exp. data

.. . . .

• o 'o i 2 • .. / •

!t • ._,

> 1 /:,:-·

.. ... •

0.00 1.00 2.00 3.00 4.00 5.00 6 00 7.00

•

" .. ... ' ~ . ~-

~ 2 - • 1 /~t·

0.00 1.00 2.00 3.00 4.00 5 00 6.00 7.00

Y56(m/tl) Cm• .. ured}

Asall model for upftow vs. exp. data

•

• .. -. . ... -. ï 2 •. -•. ~ .... ... >1 ..,.,.,. .... ,'

0.00 1.00 2.00 3.00 4.00 5.00 6.00 7.00

V71 tmlll] (m••ur•d upflow wloo.)

AsaH model fot -nll- V& exp. data

... ,-· 1 "". ...

. ,, ,Jt· '.

•

. .. '/ •

·~--~--~--~--~~~--~~ 0.00 1.00 2.00 3.00 4.00 5.00 6.00 7.00

V56(mls) (m .. •ured)

Fig. 5. 10 Parity plots of the tested models compared with experimental data

45

First the parity plots of Miesen against the measured velocity are examined. A glance at the plots suggests that the model overprediets bath the veloeities for up-flow and down-flow. Further the veloeities for down-flow are slightly more overpredicted than the veloeities for upflow.

The average velocity of the liquid can be written as a linear combination of the film velocity and the velocity of the core, as can beseen from (3. 38). This implies that the maximum value is reached when all the liquid would be entrained and the minimum value is reached when noentrainment would occur. This follows from equations (2. 34) and (2. 35).

Based on the calculation method via the liquid holdup, overprediction of the average velocity means underprediction of the liquid hold-up predicted by the model. Underprediction of the liquid hold-up could be explained by overprediction of either the film velocity or the velocity of the entrained fraction or both. The velocity of the entrained fraction is an order of magnitude larger than the film velocity.

The effect of the film velocity on the average liquid velocity, however, cannot be neglected. When the film velocity is predicted too large it is still difficult to point out exactly where the predietien fails. The film velocity is dependent on the ratio between (1-E), the fraction which is not entrained, and the film thickness d. This means that either the film thickness or the entrained fraction is predicted toa small. In addition the film thickness and the entrained fraction are dependent on each other.

Another explanation would be overestimation of the role of gravity on the film. Gravity plays a much more important role in the film than in the gas core since the veloeities in the gas core are much higher. The terminal settling velocity of the drapletscan al most be neglected (see section 2.5.3).

Compared to the entrainment model by Asali, the results of Asali's model are much better. Asali's entrainment model still overprediets the down-flow liquid velocity, though less dramatically. The predicted up-flow veloeities seem to match much better. Befare makinga statement about the models, it is important to take a closer look at the relativa errors of the model.

5.4.2 Relativa errors of the predicted velocity

Dividing the predicted velocity by the measured velocity yields the relativa error of the models. The rasuiting plot is shown in figure 5. 11, the veloeities are sorted by superficial liquid velocity. This raveals some interesting features about the behaviour of the models at small average bulk velocities.

From the plots it is clear that bath models fail to predict the veloeities for up-flow in the region of average bulk veloeities below 1 m/s. For the set of veloeities found at a superficial liquid velocity of 0.0066 m/s, Asali overprediets the veloeities with relativa errors up to 60%. Miesen still has large discrepancies, up to 40% of the measured value, but prediets better than Asali. On the other hand, the overall spreading is worse with Miesen's entrainment model.

For small superficialliquid velocities, the average bulk velocity seems to depend stronger on the superficial gas velocity than predicted by Miesen and Asali. A possible explanation can be under predicted entrainment in this range. The entrained fraetion velocity stronger depends on the gas velocity, the dependenee of the film velocity on V59 is indirect via shear force. A higher entrained fraction would show a stronger dependenee on V 59 because more liquid is travelling with this velocity. Further, taking into account the developing flow would predict the veloeities smaller as a whole.

Both for Miesen and Asali, the dependenee of the average velocity on V59

impraves for larger V51 • For Miesen, the predieted veloeities are still too largeforthese V5" but the dependenee on V59 is much better. Asali prediets these values better though the dependenee on V59 at very large values of the superficial gas velocity seems to be too strong.

46

1.6

1.6

Re la live error lor different Vsl In Mleoen's predlcted velociUes (upflowl

1.4 • • V$1::0.01 nis

1.2 :· ... • Vsi=0.04 m's

.t. VsF0.12 nis

\ 1 ---·-~~---- . . ---·-··-··--·-··-··-· xVsi=0.20rn's

• Vsi=0.0066m's > 0.8

0.6

0.4

0.2

1.6

1.6

1.2 :g

l 1

! 0.8

0.6

0.4

0.2

V71(m/s)

Relativa error tor different V!llln Miesen's predieled veloeides (downflowl

x

.... . . . .. _'! -~·----- .•.. -~_I!.- .. -.--.-- .. - .. - .. _ .. -- ..

VH(m/s]

• Vsl=0.01 m's

•Vsi=0.04 mts

.t. Vs1=0.12m's

xVs1=0.20rrls

• Vs1=0.0066m's

1.8

1.6

1.4

1.2

0.6

0.4

0.2

1.6

1.6

1.4

Relativa error tor different V!lln Alilll's predlcted veloeities (up flow I

.. . . x ·.:. .. .. . . .

. -.- .. -. ·- .. - .• - .. ;·X-A.--.-··- ·x-.·-.

V71(mls)

• Vsi=0.01 m's

• Vsi=0.04 m's

.tVsi=0.12nVs

xVsi=0.20rn's

• VsJ::0.0066m's

Relativa error for different VfA in Asali's predieled veloeities (downflowl

• VsF0.01 rrfs

1.2 • i ~~:~r..-~.!! .. : .. -~-~-~---)C-"";<··-··-··-··-·· > 0.:

• VsF0.04 rrls

A Vs1=0.12rn'$

xVsi=0.20 m's

• Vs1=0.0066m's

0.6

0.4

0.2

0~--~--~--~----~--~--~ 0

V56(mls)

Fig. 5. 11 Parity plots for relativa error of the tested models eompared with experimental data

The predieled veloeities for down-flow are mueh better for Asali although these values stilleome out a bit too high. Miesen prediets niee at small liquid flow rates, but fails dramatieally at high flow rates. The large ditterenee between the predieted veloeities for up and down-flow suggests that the role of gravity on the entrained fraetion and film thiekness might be a bit overestimated in both models.

Further it must be noted that the measured veloeities for high gas and liquid flow rates have larger errors. This is refleeted in the parity plots and must not be mistaken for spreading in the predieted veloeities. This was already diseussed at the beginning of this chapter.

Based on the parity plots, it is concluded here that the model by Asali gives better results compared to the measured velocities. Both models overpredict the veloeities for small V 51 and V 59, i.e. underprediet the liquid holdup.

5.5 Simulation of AD flow residence times

5.5.1 Analysis of raw data

As described in ehapter 3, the simulation technique is based on mass fluxes of liquid and gas within a diseretised tube i.e. a tube which is divided into small elements of length dx. The veloeities used to caleulate these fluxes are mostry based on general estimates on basis of hold-up, as was discussed in chapter 2. The simulation further makes use of the entrainment rate and film thickness predieled by Miesen.

47

From the former section it could be concluded that Asali's model has a better overall performance. The reason for using Miesen in the simulation is because Asali's model is a pure empirica! model. Asali does not predict the entrainment and deposition rates, it only quantifies the entrained fraction. Miesen's model is basedon an semi-empirica! relation for the entrainment rate by Hewitt-Govan, which is extended with physics to give the film thickness and the entrained fraction. This makes Miesen's model better understood.

0.00012

0.0001

:§ 0.00008

5 0.00006 I! Ë 0.00004

0.00002

0.00007

0.00005

c;; 0.00005

i 0.00004

i! 0.00003

Ë 0.00002

Tracer mass passing detector 1

0 5 10 15 20 25 30

time [sec]


0.00001

0~~~~_,~~--~~ 0 5 10 15 20 25 30

time [sec]

0.00007

0.00006

c;; 0.00005

i 0.00004

~ 0.00003

E 0.00002

Tracer ma ss passing detector 2

0.00001

o~~~r-~--~~~ 0 5 10 15 20 25 30

0.0000451 0.00004 0.000035

:§ 0.00003 :;; 0.000025 ~ 0.00002 ~ 0.000015

time [sec)


0.00001 + 0.000005 t

0~.--r-----~~~~~ 0 5 10 15 20 25 30

time [sec)

Fig. 5. 12 Signal plots from the simulated detectors. Here the grey line represents the signal while the black line represents the tracer mass present. Note that the actual intensity of the

signa! has no significance, only the shape has.

The simulated geometry consistsof two tubes connected by a U-bend. The lengthof the tubes is 10 m. The bend only accounts for the deposition effect usually seen in bends in real experiments. Although the veloeities are not at all affected by the bend, the average bulk velocity will be affected since the residence time of the liquid in the film is increased by the bend. Note that in the real case right after the bend the flow will be in a developing state until equilibrium is established again. This is not modelled in the simulation. The detectors in the simulation are located halfway the first tube (5m), just before the bend (10m), right after the bend (10.1m) and at the end of the second tube (20.1m). As described in chapter 3, a detector is modelled by indexing the tracer concentration in one element of the tube, each time step. One of the goals is to be able to campare the simulated detector signal to the signal from the experiment. Therefore the simulated indexed concentrations in film and gas core, were translated into tracer mass. Then the signal was calculated by sealing the mass in the core and in the film with the inverse of their velocity and taking the sum of these. This is better explained in chapter 3. Platting these values yields a similar graph as the ones shown in tigure 5. 3 for each detector. Figure 5. 12 are the resulting plots tor a simulation of the flow condition with V59=47 m/s and Vs~=0.0066 m/s.

The reason tor sealing the detected tracer mass is to gain insight in the influence of the entrained fraction on the resulting signalof the detector. Since figure 5. 12 does only show the total mass passing the detector, the concentrations in film and gas core arealso needed to be able to campare it to the signa!. These concentration are shown in tigure

The concentration in the gas core is of the same order of magnitude as the concentration in the film. In the signal the actual mass contributes, not the concentration. Since the entrained fraction for this condition is very smal!, the actual mass in the core will be almost two orders of magnitude smaller than in the film. Further its contribution to the signa! is

48

scaled with the inverse of the care velocity which is about 47 m/s while the film velocity for this condition is about 0.7 to 0.8 m/s.

Th is does nat mean that the influence of the entrained fraction on the average velocity will be small as well. In fact the interaction process between the film and the care does increase the average velocity of the liquid significantly, but the fast entrained part of the tracer is largely lost in the signa! from the film. Changing the initia! conditions for the injection of the tracer fluid revealed a lot a bout the sensitivity of the shape of the signa!.

Tracer tractlons at detector 1 Tracer traelions detector 2

0.06 0.035

0.05 0.03

0.04 0.025

0.02 >< 0.03 ><

0.015

0.02 0.01

0.01 0.005

10 15 20 25 30 10 15 20 25 30

u ..... (sec) tlme(•cl

Tracer traelions detector 3 Tracer traelions detector 4

0.035 0.025

0.03 0.02

0.025

0.015

~ >< lm 0.01

0.02 ><

0.015

0.01 0.005

0.005

0

0 0 10 15 20 25 30

time (•cl

Fig. 5. 13 Tracer concentrations in the film and the gas care passing the detectors 1 to 4

lt turned out that the shape of the signa! changed a lot when the tracer fraction in the film was kept very small with respect to the concentratien in the care. The resulting velocity, which is basedon the signa!, is sensitive to this as well. This sensitivity turns out to be very small when the total tracer mass in the film is large compared to the care.

lt is further noticed that the shape of the signa! in the simulation does nat correspond to the shape of the signa! seen in the real experiments (which for this condition is shown in tigure 5. 3 ). The signal trom the simulation is more symmetrie than the signals in the real experiment. A reasans for this could be the described dependency on the initial conditions. Another good reason is the fact that maybe the predicted entrainment and deposition rates are toa high. Less interaction will result in less symmetrie signals. Further it could be the sealing of the signa! related to the local veloeities of the tracer in the gas care and the film. The signal coming from tracer in the gas care might be better visible in reality, than assumed in the model.

The resulting simulated average velocity of the liquid bulk, for this condition, was found to be 0.82 m/s with a spreading due to discretisation of 0.06 mis. The measured average bulk velocity in the Iabaratory experiment for this condition is 0.81 m/s, showing a good agreement of the simulation with the real case for this condition.

5.5.2 Simulated average Jiquid bulk velocity

Figure 5. 14 and 5. 15 are parity plots of the simulated veloeities compared to data from the experiments. Because the simulation does nat account for up and down-flow in particular, the simulated veloeities are compared to the average of the measured veloeities for

49

up-flow and down-flow, V78 and v56. The disadvantage of this is that the spreading in both measurements accumulates.

lgnoring the spreading for this moment, the plot shows an impravement of the predieled veloeities over the original model by Miesen. Especially for higher velocities, the plot show a balanced spreading of the predieled veloeities around the measured velocities. Because the predictions are compared to the average of V78 and V56, more information can be gained by comparison with V78 and V56 individual and by platting the results for each superficialliquid velocity. The results of that comparison are found in appendix E and suggested that the results of the simulation are very good for superficialliquid veloeities higher than 0.08 m/s. For values of V51 below 0.01 m/s, the veloeities for up-flow in the Iabaratory experiment turn out to be signiticantly smaller than for down-flow. The down-flow veloeities match the predieled values for these velocities, resulting in an overprediction for the averages of V78 and V56• This is clearly seen in the parity plots as in tigure 5. 14 and 5. 15 well.

Absolute error In slmuialed average bulk veloeities lor upftow ~o V78)

,.·"' +

0.00 1.00 2.00 3.00 4.00 5 00 6.00

V71(m/s)

• Vsi=0.0066 m's

6 Vsi=0.01 m's

x Vsi=0.02 nYs

:r;Vsi=0.04rrls

• Vsi=0.08 m's

+ Vsi=0.12 m's

• Vsi=0.16 m's

• Vs1=020rn's

Absolute error In slmuialed average bulk veloelilos lor downftow ~ V56)

,-,

.. _, . ,.

_ ..... ..... + •

• ,'li -* _,· • +

-~-'!i-· -

__ ;·;,~

+.·' ·'

.-'

/

a Vsi=0.0066 rrls

6 Vsi=O.Otrris

x Vaf::0.02m's

:rVsf:o0.04 m's

• VsF0.08 m's

+Vafa0.12rrls _Vstm0.16m's

• Vst=0.20rrls

0.00 1.00 2.00 3.00 4.00 5.00 6.00

V56(m/s)

Fig. 5. 14 Parity plots of the simulated average liquid bulk velocity versus the measured average liquid bulk velocity for all conditions. The data points are sorted by constant V 51 •

Relatlve error in slmulated average bulk veloeities for upftow (to V7S)

2.000

1.600

1.600 •

x +

1.400 • A

1.200 ,. )( x x x •

··'-~~.X-~.~-.- .. -.. -.~: .. : .. -•.- .. -.. -.AA~ e+x•e •

~ 1000

> 0.800

0.600

0.400

0.200

0.000+--~-+---~-~--~ 0.00 1.00 2.00 3.00 4.00 5.00 6.00

V71(mi•J

• Vsi=0.0066 rrls

•Vsi=0.01 rrls

A Vsi=0.02 m's

xVsi=0.04 m's

xVsi=O.OB m's

• Vsi=0.12rrls

+ Vsi=0.16 m's

.Vsi=0.20 rds

Relativa error In slmuialed average bulk veloeities for downflow ~oV56)

2.000

1.800

1.600

1.400

1.200

f 1.000

~ o.eoo

0.600

0.400

0.200

"' "' x "'

-~·""'·.t.-··-x~·-··,~--~·-··-··,··-··-• • .6 ~':,x +. •

+. • + + •

0.000 +----~->--~>----->--~-~ 0.00 1.00 2.00 3.00 4.00 5.00 6.00

Vl(mls)

• Vsi=0.0066 m's

a Vsi::<0.01 m's

A Vst=o.02m's

xV~.04rn's

z Vsi=0.08m's

eVs...0.12rn's

+ VsP0.16rris

.vst=o.20rrls

Fig. 5. 15 Relative difference between the simulation and the experiment in tigure 5. 14, which marks the point for which the simulated values exactly match the measured data.

lt must further be noted that a sudden increase of spreading is seen at veloeities above 1.8 m/s. This spreading is mainly created at superficialliquid veloeities above 0.04 m/s. lt is thought that this spreading is caused for a large part by the spreading of the measured velocities. The reason for this belief is that the individual data series from the simulation, sorted by V 51, do not show any effects which might be the cause of the spreading in this range. The measured veloeities on the other hand did show spreading at larger gas and liquid flow rates.

50

3.5

3

2.5

.!!! 2

.§.

.>o: :i ..0

1.5 >

I I '

0.5 I

0 I 0

6

5

2

0 0

Average bulk velocity, measured and simulated, vs. Vsg (Vs1=0.04 m/s)

5

I

I 10 15

I IV moosured is left out because rf confidentialityl

'

20

Vsg [m/s]

I

I I ~ • ' I : I I

I I

I 25 30

I j

35

Fig. 5. 16 Simuiatien results for V51 = 0.04 m/s

40

Average bulk velocity, measured and simulated, vs. Vsl (Vsg=25 m/s)

a {),. -••

0.05

~~red is left out because f confidentialityl

~ •

0.1

Vsg [m/s]

b.

,.

•

t::.

x

-

0.15

Fig. 5. 17 Sim u lation results for V 59 = 35 m/s

0.2

• Vsirrulated

t::. Vrriesen

x Vasali

• Vsirrulated

b. Vrriesen

x Vasali

The results of the simuiatien were also plotted for different gas and liquid flow rates, tagether with the average of V78 and V56 and the averages of the veloeities for up flow and down flow predieled by Miesen and Asali. Figure 5. 16 and 5. 17 show two of these plots for

51

V51=0.04 m/s and V59=35 m/s. Form these plots it was concluded that for high gas flow rates the simulation prediets the right slope of the dependenee on V59• For large V51 , values above 0.08 m/s, and small V59, the slope of average bulk velocity dependent on V51 is underpredicted. At values ofV59 above 30 m/s, this latter effect has disappeared. Appendix E shows the other results dependent on the superficial velocities.

The veloeities predicted by Miesen, the average for up and down flow, do notshow a decreasing slope for largervalues of V51 • On the other hand, Asali does predict this decreasing slope well. Based on the results of the models by Miesen and Asali, the simulation could be improved by making use of the entrainment model by Asali instead of Miesen. For the moment this was not possible because Asali does not predict the entrainment and deposition rates. Finally it has to be stated that the slope of the average liquid veloeities as a tunetion of V59, for this particular condition, is predicted well withall the used models.

5.6 Overview of the results

In this chapter the results of the residence time measurements were presented and compared to results predieled by two models, of Miesen [13.1-5] and Asali [2], and a simulation model based on mass flow rates between elements of a discretised tube. Also the spreading of the measured veloeities around their average was examined and it was concluded that the film flow and the gas core flow both behave turbulent instead of laminar.

Togainsome insight in the experimentally found veloeities some attempts were made to fit the data to some arbitrary functions of V59 and V51 • The goal of these attempts was to find a fit which could be theoretically explained. Unfortunately no such fit was found.

The comparison of the models to the experiment showed that the models failed to predict the veloeities at small superficial liquid velocities. The down-flow veloeities were better described than the up-flow velocities. Further, it was found that the veloeities predicted with Miesen's model for entrainment, were too large in generaL The model by Asali gave better results in genera!, but had the disadvantage that it was an empirica! model.

lt was shown that the simulation did give better results, compared to the results predieled by the model by Miesen. The simulated veloeities turned out to predict the slope dependent on the superficial veloeities well, compared to the experimental data.

Compared to the model by Asali, it was not clear whether the results of the simulation were better. However, the simulation is based on relatively basic mass flux calculations between the film and the gas core. These mass flux calculations can easily be extended with new features, like calculations of the developing flow after a bend.

52

6. Bubble datachment in vertical flow

6.1 Introduetion

As has already been stated in farmer chapters, heavy oil fractions, the residues of distillation processas in refineries, are led into thermal cracking units (TC U's orthermal cracking furnaces) tor further processing. In TCU's the residual oil is heated up during which conversion into lighter products takes place. A TCU consists of sets of vertical tubes which are connected by U-bends to farm a long chain. The tubes are all grouped around a central heater.

The oil enters the turnace in its liquid phase, and is converted into lighter products in their gas phase until it leaves the turnace again mainly as gas. Only a small part of the oil leaves the turnace in its liquid phase. Same liquid is needed to prevent overheating of the tubes

An undesired side effect of the process is the lay down of carbon on the tube walls, also known as coking. Th is causes the heat resistance of the wall to increase steadily during the process and the walls will heat up slowly. The temperature of the tubes is therefore watched closely to prevent tube bursts due to overheating. When the tubes reach a certain temperature the process is stopped to clean them.

A problem which is seen in TC U's is that in the bubble flow regime of the turnace is that the temperature of the tube wall is larger than predicted by the existing models. Th is means that the temperature of the twophase mixture directly adjacent to the inner side of the tube wall, the film temperature, will be higher as wel I. Because of this the amount of coke formation increases severely in this region of the tube.

Calculation of the amount of vapeur production showed that at the moment the first vapeur is created, the vapeur production is larger than elsewhere in the system. lt is known that the production of vapeur happens at the wall, where bubbles start to farm in small cavities. The high vapeur production probably leads to a high coverage of the wall by bubbles. This creates an isolating thin vapeur film between the wall and the fluid.

As a step towards the salution of this problem, a model for bubble datachment in vertical flow will be treated in this chapter, by examining the farces present on a bubble near the wal I. This model is used to calculate the average size of a bubble just befare it detaches. This can as well be regarded as an estimate of the thickness of the gas layer. Use is made of a datachment model presenled by van Helden [8].

6.2 Qualitative description of forces on a bubble

As a first approach one can look at the farces present on a bubble near the wall of a tube in a flow. Since the size of the bubbles is not expected to be large compared to the diameter of tubes, it is assumed at first that the bubbles are contained in the boundary layer of the flow.

The farces werking on a bubble near the wall can be divided into two groups, farces werking parallel to the wall and torces werking perpendicular to the wal I. Bath of these groups farm a force equilibrium which holds the bubble in place at the wall. The bubble will detach as soon as one of the force balances breaks down.

Parallel to the wall, the balance is formed by the drag force, the buoyancy force (for vertical flow) and a vertical component of the surface tension force. The latter of these torces is created because the altachment angle at the toot of the bubble (the angle between the bubble's surface and the wall) is different at the advancing and raceding side of the bubble with respect to the flow direction. This is explained further in the next section.

The balance of the farces perpendicular to the wall is formed by the surface tension force, the Bernoulli suction force, the thermocapillary force and the expansion force. The last

53

two torces originate from the thermal boundary layer at the walt, which influences the surface tension of the surrounding fluid, and the growth rate of the bubble.

Because the amount of heat supplied to the walt at a constant rate is large, the temperature gradient perpendicular to the walt, at the location of a bubble, is large as welt. This means that the thermocapillary force on the bubble has to be taken into account. Van Helden showed that the expansion force can be neglected in most cases. The expansion force is a dynamic force in the sense that it is dependent on the growth rate. The growth rate is dependent on the time passed since the bubble's moment of creation. Van Helden estimated the expansion force for a vapour bubble in water, growing from it's critica! bubble radius to a radius of 0.25mm in 5 ms. lt was concluded by van Helden that the expansion force was only important in the first millisecond after creation of the bubble. The expansion force rapidly decreased and can be neglected when the bubble radius approaches the datachment radius. The torces which form the perpendicular force balance are thus the Bernoulli suction force, counterbalanced by the surface tension and the thermocapillary force.

Fsuction

V

db

Fig. 6. 1 Geometry and teading torces on a bubble at the walt of a tube

Fora Jaminar vertical up-flow, the force balance perpendicular to the walt is thought to hold much Jonger than the force balance parallel to the walt. Because the streamlines in the flow will curve around the bubble, as if it were a smalt imperfection on the walt, the suction force is thought to be smalt. This means that the datachment process is determined by the force balance parallel to the walt. The same is true when the bubble radius is smalt compared to the viseaus sublayer in a turbulent flow.

In this chapter the force balance parallel to the walt wilt be used to calculate maximum bubble radii for vertical laminar flow or vertical turbulent flow if the viseaus sublayer is assumed to be larger than the bubble radius. This is done under the conditions locally present in the critica! region in bubble flow in a thermal cracking furnace.

Later in this chapter it wilt be discussed whether or not the assumptions for a laminar flow or the viseaus sublayer in a turbulent flow are valid in the present case.

54

6.3 Geometry of the problem

Consider a liquid flow in a tube of arbitrary lengthand with an arbitrary diameter. Forsome reason, this may be due to boiling or a chemica! reaction, steadily growing bubbles are formed at the wall. The bubbles will detach when the volume of the bubble is large enough to destray the force equilibrium acting on the bubble.

A situation sketch of a bubble at the wall is given in tigure 6. 1, the important parameters are shown in this tigure as well. The density of the liquid is p1 and its viscosity is f.l1• The surface tension of the liquid is cr. The velocity near the wall, which is of importance in this case, cannot be considered constant. In tact it will be a tunetion of the distance from the wall. As a first approximation a laminar velocity profile will be used in the model.

The angle between the shell of the bubble and the wall, as seen in the figure, is e. This angle introduces a problem, because at this moment almast no research is done on the magnitude of this angle. The angle is treated as a variabie in the model and is varied between n/4 and n/2.

As an approximation the bubble is considered a truncated sphere with radius Rb. This means, as can be seen in tigure 6. 1 as well, that the thickness of the bubble measured from the wall, db, will be larger than Rb. The thickness of the bubble is given by

db = (1 +cos B)Rb (6. 1)

Another consequence of treating the bubble as a truncated sphere is that the volume of the bubble Vbubble is given by

Vbubble = (%- (1- cos 8)2(% +cos B})lrRb 3 (6. 2)

This result will be used later to calculate the buoyancy force acting on the bubble. Because of drag force acting on the bubble, it will not be a perfect truncated sphere in reality. In tact the bubble will be stretched a bit. This implies that the angle e defined befare will not be constant all around the bubble foot. lt results in a different angle on the receding and advancing side of the bubble. Th is is a very important feature in the geometry of the problem because it results in a vertical component of the surface tension force which acts in the opposite direction of the drag force on the bubble. As will be discussed later, it is this force which counterbalances the other farces acting on the bubble in vertical direction.

The deformation angle d9, the ditterenee between the advancing and receding angle of the bubble is defined as

(6. 3)

Th is angle is later needed to calculate the resulting vertical component of the surface tension force. Finally, the magnitude of the drag force will be a tunetion of the effective surface of the bubble in the flow. What is meant by this is the projection of the bubble (represented by a truncated sphere) on a plane perpendicular to the direction of the flow. This surface is given by

A = (7r- B +cos Bsin B)Rb (6. 4)

55

Calculation of the surface is done by considering the surface as a circle and dividing the circle in three parts, as shown in tigure 6. 2. The biggestof the three surfaces A1, is given by nRb2 minus a (8/n)th part of it. Then the remaining part of circle is divided by the line representing the bubble toot. The part below the foot, A1, is the part of the bubble which is truncated. The other part, A2, can beseen as two straight triangles which could farm a rectangle with it's surface being Rb2(sinecose).

Now, the geometry of the problem has been defined and based on this, a force balance on the bubble can be introduced. This will be done in the next section of this chapter.

' A2

/ /

' Al

) /

Fig. 6. 2 Bubble surface

6.4 Forces acting on a bubble near the wall

Van Helden [8] introduced a model for bubble delachment near the wall of a vertical tube in a flow, basedon the farces acting on the bubble. Because of the orientation of the surface tension, in laminar flow, the farces in vertical direction are much more important than the farces in horizontal direction. lt is known that the ditterenee between the advancing and receding angle of the bubble causes the surface lension to possess a vertical component which is small. This vertical component however is large enough to counterbalance the other farces acting in vertical direction for some time while the bubble is growing.

Figure 6. 1 shows a sketch of the situation, in which the farces acting in vertical direction are illustrated. Magnitude estimation of the farces shows that only three farces are important in the model, these are the drag force F d• the buoyancy force Fb and the component in vertical direction of the surface tension force F 011•

Starting off with the buoyancy force, this force represents the effect of gravity on the bubble. This force can be calculated using

Ft, = (pi - p g )g vbubble (6. 5}

in which g is the gravitational acceleration and p9 is the density of the gas. The drag force represents the force coming trom the flow which pulls the bubble

upward (in case of up-flow). This force is proportional to the square of the velocity of the flow. lt is also directly proportional to the surface of the bubble perpendicular to the flow direction, which is proportional to the square of the bubble radius. These dependencies result in the following equation for the drag force

(6. 6}

in which Cd is the drag force coefficient which is for this case taken to be the drag force coefficient fora sphere in a laminar flow, this is

c- 24 d- Re

b

56

(6. 7}

The bubble Reynolds number is defined as

(6. 8)

Since the bubble radii are expected to be much smaller than the diameter of the tube, the velocity boundary layer has to be taken into account. Gonsidaring the flow laminar or considering the bubble to be completely in the viscous sublayer of the turbulent flow (this is explained later in section 6. 7.1 ), the local velocity at the bubble is given by

(6. 9)

where R1ube is the tubes radius and V"' is the velocity in the middle of the tube. The distance %db is taken in order to get an estimate for the average velocity at the bubble. For precise calculations one should calculate the average velocity at the bubble by integrating the velocity profile over the bubble cross sectionat surface At, in the flow, divided by the magnitude of this surface At,.

The third force acting on the bubble was already mentioned. lt is the vertical component of the surface tension which counterbalances the drag force and the buoyancy force in upward flow. lt originates from the fact that in the flow the bubble is not perfectly spherical but is in fact stretched a bit in the direction of the flow. lt therefore has an advancing and raceding angle which is characterised as a deflection from the equilibrium angle, defined by AS (see equation (6. 3)).

The rasuiting component of the surface tension force is given by

F = _4_1CR-=b:.._a_· _ll_B_si_n_B("_s_in_( B_-_ll_B_)_+_s_in_(_B_+_ll_B....!..)) av (28- 1C)(2B + 1r)

(6. 10}

Now a force balance can be set up. The rasuiting force on the bubble is given by the sum of the three torces mentioned. The bubble will datach when the rasuiting force vanishes. The radius of the bubble for which this happens can be calculated and is the bubble radius on detachment.

6.5 Heat transfer through a bubble-covered wall

Consider a wall which is covered by bubbles to a certain degree. lf the size of the bubbles is known, the temperature difference between the inner side of the wall and the twophase mixture in the main flow can be calculated. The layer of bubbles can be seen as an isolating layer. Suppose the wallis completely covered by small bubbles, then a component of conveelive heat transfer is absent in the bubble layer. The heat transfer through the layer is static and can therefore be described with the heat transfer coefficient Àv and the thickness of the layer.

An additional problem in this case is that it is not likely that the wall is completely covered with bubbles. In fact a dagree of coverage e is introduced to quantify this effect. Reprasenting the ratio bubble covered surface to the total surface e is a dimensionless number between 0 and 1.

For a given constant heat flux through a wall or layer with thickness d and heat transfer coefficient 'A., the temperature difference is given by

57

Qd !'!.T=

/L (6. 11)

Because the wall is not completely covered with bubbles, it is thought that the bubbles create hot spots in the fluid. lt is assumed that the heat transfer in the region of the boundary layer between the bubbles is good. This means that at the surface between the bubbles, the temperature of the liquid is the same as the temperature of the inner side of the wall.

Q !1T1 Q R1 !'!.V1

Q !1T2 Q R2 !'!.V2

R2 !'!.V2

R1 /1V1

R2 !'!.V2

Fig. 6. 3 Electric analogue of the heat transfer problem near the wall. Here Q corresponds with I, R corresponds with d/'A. (where dis the bubble's thickness) and !N corresponds with ~T.

The heat transfer problem can be described by it's electrical analogue, as is sketched in figure 6. 3. The supplied heat remains constant along the walt, this corresponds to a constant current in an electric circuit. Further, a bubble on the walt can be seen as a heat resistance. Further the temperature difference between the waltand the adjacent fluid, ~T. corresponds with a potential difference ~V. Since in electric circuits the potential difference V is calculated with Ohm's law, the heat resistance is given by d/A. corresponding with R.

In the present case, the unexplained temperature difference between the film temperature and the bulk temperature, the temperature of the twophase mixture in the main flow, is the parameter of interest. Here the effect of bubbles on the heat transfer between the walt and the twophase mixture in the bulk is isolated. lt is considered to be an extra con tribution to the heat resistance in addition to e.g. convective heat transfer in the flow. Th is means that the average extra temperature difference between the film temperature and the bulk temperature is given by determined by e and the temperature difference per bubble.

The bubble thickness d has to be corrected since not all the bubbles have the same size. Assuming that the bubbles were created at random and the bubble radius during the growth increases linearly with time, the average bubble thickness is estimated by Y2db. For the extra temperature difference between the film temperature and the bulk temperature this yields

58

(6. 12)

With the estimated bubble thickness, it is now possible to calculate the temperature difference in the furnace. The only unknown parameter is the magnitude of degree of coverage E. This problem is addressed by calculating the temperature difference for variabie E and campare it to the measured temperature differences in the real furnace. Then later the it can be determined whether or nat the found E for which the temperature differences match, is a realistic value.

6.6 Results of the calculation

The model treated in section 6.3 was solved for values of e varying between n/4 and n/2 with ~e varying also between n/4 and n/2. Although nat all combinations of e and ~e are likely, the calculations should give a rough estimate of the size of the bubbles on detachment.

The bubble size on delachment was calculated for a whole range of possible conditions in a furnace. All these conditions are likely to occur in thermal cracking turnaces dependent on the local temperature and pressure and the feed of the furnace. The ranges of possible material parameters are given below.

Parameter Material parameter range min ma x

p, 650 kg/m3 850 kg/m3

Pg 0.4 kg/m3 80 kg/m3

Jll 1·1 o·4 Ns/m2 5·1 o·3 Ns/m2

Jlg 1·1 o·5 Ns/m2 3·1 o·5 Ns/m2

cr 3·10·3 N/m 3·10.2 N/m

Further, the average velocity of the flow was taken 1 m/s, this is very typical for the flow in the critica! region of the furnace. The diameter of the tube was take 0.1 m. For reference, the bubble radii on delachment for vapour bubbles in water were calculated as well. Roughly it can be stated that for the oil in the furnace, the bubble thickness ranges between 0.1 mm and 3mm. The parameters which have a large influence on the bubble size are the parameters which determine the surface tension force and the buoyancy force. These parameters are the difference between the gas and liquid density, p9 and p1, and the surface tension cr. For one of the conditions, the calculated bubble thickness, dependent on e and et~e is plotled in tigure 6. 4.

The deformation angle ~e can never exceed the altachment angle e, tor e is the angle between the bubble's surface near the toot and the wall. The deformation angle is therefore scaled with the altachment angle because the maximum allowable value tor ~e is dependent on e. How e and ~e depend on the bubble radius is probably determined by the equations describing the growth of the bubble. Because these equations are nat known, e and ~e are regarded variabie tor a given bubble radius.

From the calculated bubble thickness, the temperature difference was calculated tor different values of E. With constant heat transfer coefficient Ày, regarding the linearity of (6. 12), it can be seen that the general dependenee on e and ~9/9 of ~ T will be the sa me as in tigure 6. 4. Further, the dependenee on E will be linearas well. Figure 6. 5 shows a plot of the calculated temperature difference between the wall and the fluid in the turnace as a tunetion of e and ~ete.

59

.s "' :g 1

" .c "C

theta

Subbie thlckness trom wall on datachment vs. theta and dtheta;

mul 1.00E·04 Nslm"2 mug 2.00E..()5 Ns1m•2 rhol 700 kg!m•3 rhog 50 kg/m"3 sigma 3.00E..()2 Nlm

dtheta/theta

Fig. 6. 4 Bubble thickness from the wall on delachment of the bubble as a tunetion of the altachment angle 8 and the deformation angle ó8.

theta

Temperature ditterenee between wall and fluid vs. theta and dtheta;

mui1.00E..()4 Nslm"2 mug 2.00E..()5 Ns/m"2 rhol 700 kg/m"3 rhog 50 kg/m"3 sigma 3.00E..()2 Nlm epsilon 0.1 (10%)

Fig. 6. 5 Temperature difference between the wall and the fluid for the condition in tigure 6. 4; the degree of coverage was taken to be 1 0%.

The graphs in figures 6. 4 and 6. 5 reprasent a condition where the surface tension had it's maximum value. This means that the calculated bubble thickness is larger than in the other cases. In real thermal cracking furnaces, the measured additional temperature

60

difference lies between 30K and 50K, corresponding to low values of !::.8/8 and therefore small bubbles. This suggests that the predicted bubble size might be a bit too large. This will be discussed in the section below.

The results of the other conditions which were calculated can be found in appendix F.

6.7 Discussion of the results

6.7.1 Turbulent boundary layers; estimating the viseaus sublayer thickness

A critica! review of the methad of calculation brings up a lot of questions regarding the validity of the model. lt is, for example, not yet proven that the force balance on the bubble perpendicular to the wall is more stabie than the force balance parallel to the wall. The validity of this assumption is dependent on the question whether or not the flow is laminar in the region of interest near the wall. For laminar flow this assumption is valid since the streamlines of the flow will bend over the bubble surface easily. The torces perpendicular to the wall, pointing into the flow, can be neglected in this case.

Turbulent flow through tubes can be divided into two regions dependent on the distance from the wall, the inner and outer region, as can be seen in tigure 6. 6. This subdivision of the turbulent velocity profile comes from the way the velocity parallel to the wall is scaled in each region. In the outer region, the velocity is given by the so called defect law while in the inner region the velocity is governed by the /aw of the wal/. In fact the outer region is the region where the flow is dominated by turbulence while the inner region is the boundary layer of the flow. Since a part of the boundary layer is still dominated by turbulence, the inner and outer region partly overlap. The region where the inner and outer layer overlap is called the inertial sublayer. A more detailed description can be found in most books about turbulence, e.g. Hinze [11].

For now it is important to know that the inner region is further divided into three sublayers: the viseaus sublayer, the buffer layer and the logarithmic layer. The outer region, as can be seen in tigure 6. 6, consists of the core region and again the logarithmic layer. The inertial sublayer therefore matches the logarithmic layer.

In the care region and the logarithmic layer, i.e. the outer region, the flow is completely turbulent. The logarithmic layer, however, differs trom the care region because here the eddies are restricted intheir size by the influence of the wall. In fact, in the logarithmic layer turbulent mixing dominates the flow, but viseaus farces cannot be neglected here. In the viseaus sublayer on the other hand, the flow is dominated by viseaus torces while turbulent mixing does not play an important role anymore.

The buffer layer farms the transitional region between the logarithmic layer and the viseaus sublayer. Viseaus farces and turbulent mixing are of equal importsnee in this region.

In the present case, the thickness of the viseaus sublayer is important because here the flow behaves typically laminar. lf the adapted model by van Helden is valid for laminar flow, it

core region

logarithmic layer

buffer layer

viseaus sublayer

c: 0

-~ .....

~ 0

c: 0

-~ ..... Q.l c: .!::

I ..... -Q.l

~i;' Q.lc:..c ·- ~

Fig. 6. 6 Subdivision of the turbulent boundary layer

will therefore be valid if the bubble thickness does not exceed the thickness of the viseaus sublayer. Note that the flow is not really laminar in this region. Same velocity fluctuations are

61

still present but these fluctuations are caused by turbulence in the layer above the viscous sublayer.

An estimate for the thickness of the viscous sublayer is found with help of the law of the wall and the introduetion of the friction velocity. This friction velocity is not really a velocity but it is a quantity that determines the velocity distribution, having the same dimension as velocity [m/s]. lt is defined as

• fiw u= -p

(6. 13)

where •co is the wall shear stress which is estimated by

(6. 14)

The law of the wall now states that the velocity distribution in the vicinity of the wall is independent on the flow geometry, mathematically represented as

(6. 15)

here u+ is a dimensionless velocity, it is the velocity parallel to the wall, v, divided by the friction velocity. F is a tunetion determined by the sublayer in the inner region of the boundary layer. For the viscous sublayer this is

(6. 16)

This velocity distribution holds until y+ = 5, defining the viscous sublayer. An estimation for the thickness of the layer is therefore found by substitution of y+ = 5 in (6. 16).

Since it is now possible to estimate the thickness of the viscous sublayer, the validity of the presented model for bubble thickness calculations on datachment can be tested. This is done in the next part of the section.

6.7.2 Consequences for the calculated bubble thickness

The Reynolds numbers in the flow have their range between 1.3·1 04 and 8.5·1 05 • The

estimated thickness of the viscous sublayer forthese Reynolds numbers range from 5·10"5 m for the highest Reynolds number, toabout 3-10-3 m for the lowest Reynolds number.

Gomparing these magnitudes to the calculated bubble thickness shows that the model holds only for small bubbles. Most bubbles will exceed the viscous sublayer thickness and will detach as soon as the turbulent forces become large enough to tear the bubble off the wall.

Figure 6. 7 shows the estimated thickness of the viscous sublayer öv as a tunetion of the viscosity of the liquid J.l1• The values were estimated for a flow with an average velocity of 1 m/s. The liquid density was taken 700 kg/m3

, the friction factor f was 0.005 and the tube diameter is 0.1 m.

Using the values shown in figure 6. 7 it is possible to calculate the temperature difference based on this thickness, using (6. 12). The calculated values for this thickness are shown in figure 6. 8. To calculate these values, the degree of coverage & was taken 0.8 so 80% of the wall is covered with bubbles.

62

Estimated thickness of the viscous sublayer

0.000001 +-----------+-----------+--------------j 0.00001 0.0001 0.001 0.01

mu_l

Fig. 6. 7 Estimated thickness of the viscous sublayer near the wall

Temperature ditterenee based on viscous sublayer thickness

delta_T

10+---------~-----~~---+---------~

0.00001 0.0001 0.001 0.01

mu_l

Fig. 6. 8 Temperature ditterenee predicted by using the upper limit of the bubble thickness

Temperature ditterences of 30K to SOK are predicted for Reynolds numbers between 3.5·104 and 7.0·104

• These values for the temperature ditterenee are a bit small compared to the temperature ditterences a seen in real turnaces where the Reynolds numbers are on average larger. The reason for this might bethefact that in reality, the bubbles might become larger than the viscous sublayer. lt is ditticuit to say where exactly in the inner turbulent boundary layer the turbulent torces are large enough to break up the force balance of the bubble. lt may therefore be concluded that further study to the turbulent torces in the boundary

63

layer is needed to solve the problem. The thickness of the viseaus sublayer may be treated as an lower limit of the bubble thickness.

6.8 Force balance versus boundary layer

The discussion above shows that calculation of bubble growth in a flow at the wall is a highly complex matter. Unfortunately it is not possible to set up an easy proper model to predict the bubble thickness on detachment. lt is even more difficult to support a certain model with measurements because thermal cracking units are not developed tor these types of experiments. Therefore the model has to be backed up with only limited knowledge of the important parameters with which the model has to be checked indirectly.

On the other hand, it was shown that the assumptions for the boundary layer and the torces werking on a bubble at the wall give the right order of magnitude tor the temperature difference. The model based on the force balance on a bubble parallel to the wall in a laminar flow slightly overpredicted the temperature differences. Taking the thickness of the viseaus sublayer in a turbulent flow as the maximum thickness of the bubbles, showed that the temperature difference is underpredicted this way. The real bubble thickness will therefore be somewhere between these two predictions; the predieled values can be treated as an upper and lower limit.

More generally to reduce the effect of an isolating layer of bubbles near the wall, it might be considered to reduce the heat flux locally in the furnace, where this problem occurs. By doing this, the amount of vapour production is reduced and the degree of coverage e will be smaller. The amount of vapour production is also reduced by increasing the pressure in the furnace. Also increasing the velocity would be an option since this determines the boundary layer thickness.

64

7. Conclusions and recommendations

7.1 Project description

7.1.1 Background

The whole project was meant to serve a larger goal, namely accurate rnadelling and simulation of a thermal cracking process. This is done by the computer code HYFIH (Hydrocarbon Flow In process Heaters) which is under development at SRTCA.

7.1.2 Goals

The workat SRTCA covered two aspects which had no direct relation toeach other. First of these was to describe the residence time of liquid in annular dispersed flow and give the description experimental backup. In the AD flow regime, at the end of a furnace, the temperatures are the highest in the whole system. In order to describe the conversion process in this regime, an accurate description of the residence time is required, because the conversion rate is high when the temperature is high. Further, AD flow is the most inhomogeneous flow regime of two phase flow. This is reflected in the complexity of a model for the residence time in AD flow.

The first goal for this part of the project was to perfarm measurements of the residence time of liquid in AD flow. Analysis of the residence time measurements, and with this developing a model describing both the average and the distri bution of the residence time, was a second goal.

The second aspect of the project was related to a heat resistance problem in the bubble flow region of thermal cracking furnaces. In the region of the flow where, due to conversion and evaporation, the first vapour bubbles are created, a heat resistance anomaly is seen which is not predieled by the existing models. lt is believed that this anomaly is caused by a layer of bubbles which covers the wall.

The goal of this part of the project was to get insight in the delachment process of bubbles at the wall in a flow. With this insight, calculation of the thickness of the bubble layer could be done and the influence of this layer on the heat transfer between the wall and the fluid could be examined.

7.2 Residence time in AD flow

7.2.1 Approach of the problem

The residence time measurements were performed by injection of radioactive tracer liquid in vertical annular-dispersed flow and detection of the radiation coming from the tracer at severallocations along the tube. In this way data was obtained on the residence time distribution of the liquid in upflow and downflow, resulting in information on both the average velocity of the liquid and the effective spread of velocities. In order to analyse the experimental data, a model was set up describing the liquid mass transfer between the film and the draplets in the gas core, i.e. the deposition and entrainment process. The model was numerically solved in MS Excel resulting in simulated velocity and residence-time distributions which were compared to the data. The average veloeities were also calculated using existing models for annular-dispersed flow by Miesen [13.1-5] and Asali [2], which predict flow parameters like entrained fraction, film thickness and thus the liquid holdup (volume fraction of liquid) in AD flow. From this the velocity was directly calculated as the ratio of superficialliquid velocity and liquid holdup.

65

7.2.2 Conclusions

The ma in condusion to be drawn from our work is the somewhat surprising relatively homogeneaus character of the liquid flow in annular-dispersed flow through straight tubes connected by sharp Ubands. By this I mean that in spite of the two completely different velocity components present in the flow, viz. the film velocity and the much higher droplet velocity, the observed increase of the width of the residence time distribution was small outside the region of developing flow effects after the liquid and the tracer inlet sections.

Various other conclusions are directly linked to the main conclusion. Firstly, from the relatively smalt spreading of the residence times, it must be concluded that the film effectively behaves turbulent instead of laminar. Calculation of the Reynolds numbers of the film showed that this still holds for Reynolds numbers as low as 300. Experimental support for this is found by observation of the film: the surface of the film is rough and continuously disturbed by roll waves which cause entrainment of liquid in the gas care and probably of gas bubbles in the film. These effects clearly promate mixing of the liquid in the film. Secondly, a comparison of the signals from the tracer liquid right befare and after a U-bend shows that entrained draplets effectively deposit on the film by the centrifugal force in the bend. Th is can be interpreled as locally strongly enhanced mixing of the liquid teading to a more homogeneaus character of the flow. Thirdly, high entrainment and deposition rates enhance the homogeneaus character of the flow as is evidenced by the increasing symmetry of the residence-time distribution at e.g. higher liquid flow rates. At relatively low liquid flow rates, where the mass exchange rates between film and draplets are low, the residence-time distribution is asymmetrie and still reflects the two separate contributions from the fast travelling draplets and the slowly moving film. The given signa! in tigure 5.3 of chapter 5 is a good example of this.

From the comparison between the measured data and the simpte velocity predictions based on the holdup as given by the models tor annular-dispersed flow by Miesen and Asali, it is concluded that the Asali model perfarms considerably better for this purpose. However, the predictions of bath models get worse tor upflow at the lowest liquid velocities, i.e., at the combination of relatively low gas flow rates and relatively high liquid flow rates. As in this region of flow the difference between upflow and downflow bacomes noticeable, the validity of the role of gravity in both models may be questioned.

The residence-time predictions based on the model by Miesen improved when the Miesen flow parameters were used in the numerical simulation of the flow. This illustrates the importance of the entrainment and deposition processin annular-dispersed flow. Another conclusion from the simulation was the clear sensitivity of the residence-time signa! on the inlet conditions of the tracer liquid. lt was found that the simulated distribution became more asymmetrie if more of the tracer liquid was assumed to be injected as draplets in the gas care instead of as film. Overall, although the average veloeities of liquid were predicted well, the residence-time distributions were not predicted adequately by the simulations. Apart trom developing-flow effects at the inlet sections, the developing flow right after the bends is thought to be one of the reasons.

Based on the relatively small increase of the spreading in the measured residence-time signa I and the simplicity of the holdup-based predietien of the average velocity, it is concluded that such predietien on basis of Asali's model can be adequate in many cases. In order to calculate the distribution of velocities, the numerical simulation is required, but refinements seem necessary.

7.2.3 Recommendations

For future study of AD flow it is recommended that the model by Miesen should be adapted in order to predict the veloeities better. I think that replacing the laminar velocity profile in the film, on which Miesen's model is based, by a turbulent profile would imprave the model. This is supported by results of the simulation, in which a constant film velocity was assumed and in which the entrainment process was described by Miesen. An alternative to adapting the model by Miesen is to make use of Asali's model tor entrainment, but this later model should be extended with a model tor the liquid mass transfer between film and gas core if it is used in the simulation. Further, the

66

simuiatien in MS Excel could be extended to take into account some effects of the bends, up- and downflow and inlet effects.

7.3 Heat resistance in the bubble flow region of turnaces

7 .3.1 Approach of the problem

The temperature anomaly in bubble flow, which is related to the heat resistance between the tube wall and the fluid, was approached by assuming a thin layer of bubbles covering the wall. The thickness of this layer was calculated by adapting a model for the delachment criterion of bubbles at the wall. To calculate the heat resistance of a layer of bubbles near the wall, a degree of coverage of the wall by bubbles was introduced. The resulting heat resistance was verified with existing temperature measurements in a real furnace. The calculated bubble thickness was compared to calculations of the thickness of the viseaus sublayer (a region of the turbulent boundary layer) in a turbulent flow. Th is is an alternative way of calculating the heat resistance.

7 .3.2 Conclusions

The delachment model was based on a force balance on a bubble at the wall in laminar flow. This yielded a bubble size which was thought to be toa large because of the following reasons. First, calculations of the Reynolds number in the flow showed that it must be a turbulent flow under normal conditions. Second, the predieled temperature anomaly was large compared to the anomaly seen in real furnaces. In order to predict the right temperature anomaly of 30K to SOK, only 20% or less of the surface of the wall was covered with bubbles according to the model. Since there was sufficient confidence in the physical basis of the approach, and since the order of magnitude of the predieled heat resistance was good as well, the outcomes of the model were treated as an upper limit of the bubble size. In a turbulent flow it was believed that the bubbles would become less large because farces related to turbulent mixing would make the bubbles detach sooner than predieled by the force balance in laminar flow. Knowied ge of the importance of turbulent farces over viseaus farces in a turbulent boundary layer would provide the model with a critica! bubble size. Unfortunately this critica! size was nat found. How turbulent farces exactly relate to viseaus farces in the turbulent boundary layer and how these turbulent farces should be taken into account in the force balance described above, turned out to be a complex matter. However, the thickness of the viseaus sublayer in a turbulent boundary layer is the length scale for which turbulent farces are unimportant related to the viseaus farces. This thickness therefore provided a lower limit of the bubble size, since the model based on laminar flow was believed tobevalid at least in the viseaus sublayer. To predict the right temperature anomaly with this lower limit, 80% or more of the surface of the wall had to be covered by bubbles. Since the degree of coverage can hardly become larger, the thickness of the viseaus sublayer is regarded as a lower limit of the bubble size. The order of magnitude of the temperature anomaly seen in turnaces in the bubble flow region is predieled well using realistic values for the conditions in the furnace. In addition the real values of the heat resistance fall nicely between the predieled upper and lower limit. The model should therefore be treated as an onset for further study of this problem.

7.3.3 Recommendations

The model still has to be fine-tuned however. lt could be improved by further study of the torces present. More accurate calculations of the thickness of the viseaus boundary layer should imprave the model as wel I. Also the influence of turbulent farces in the buffer layer and the logarithmic layer (as described in sectien 6.7.1) in the inner region of turbulent boundary layers might give a more narrow range of the predieled bubble thickness at the wall.

67

The model will also improve by study of the degree of coverage of the wall by bubbles. At this point nothing is known about this parameter. Since measurements of this parameter in real turnaces are hard to do, a simple experiment could be done in laboratory, giving better insight in this parameter. Experiments in a glass tube with realistic wall roughness and a homogeneously heated wall could be considered. This would also give information about the average and maximum bubble size.

68

Raferences

[1] Andreussi P., Asali J.C., Hanratty T.J., 1985, lnitiation of Rol/ Waves in Gas-Liquid Flows, Am. lnst. of Chem. Eng. Journal, Vol. 31 #1, pg. 119-126.

[2] Asali J.C., Leman G.W., Hanratty T.J., 1985, Entrainment Measurements and their use in Design Equations, Physico Chemica! Hydrodynamics, Vol. 6 #1/2, pg. 207-221, Pergamon Press Ltd.

[3] Azzopardi B.J., Whalley P.B., Artificial Waves in Annular Two-Phase Flow, lnternal report, Engineering Sciences Division, AERE Harwell, Didcot, Oxon, UK.

[4] Beek W.J., Muttzall K.M.K., 1975, Transport Phenomena, John Wiley & Sons Ltd.

[5] Confidentlal report. Raferenee to this document should only be made in documents having the same, or a higher, security classification. Shell International Oil Products B.V.

[6] Collier J.G., Thome J.R., 1994, Convective Boiling and Condensation, Chapter 4, Introduetion to Pool and Convective Boiling, pg.131-182, Oxford Science Publications, Ciarendon Press, Oxford, UK

[7] Oaemen J.T.F., 1996, Heat Transfer and Residence Times in Annular Dispersed Flow, Masters Thesis, Eindhoven Univarsity of Technology, Dept. of Applied Physics, Eindhoven.

[8] van Helden W.G.J., 1994, On Detaching Bubbles in Upward Flow Boiling, Ph.D. Thesis, Eindhoven Univarsity of Technology, Dept. Machanical Engineering, Eindhoven.

[9] Hewitt G.F., Haii-Taylor N.S., 1970, Annular Two-Phase Flow, Pergamon Press Ltd.

[10] Hewitt G.F., Roberts D.N., 1969, Studies oftwophase flow pattems by simultaneous X-ray and flash photography, AERE-M 2159, HMSO (AERE, see [3])

[11] Hinze J.O., 1959, Turbulence, McGraw-Hill Book Company lnc., US

[12] Hulsman A.J.E.G., 1996, Annular Dispersed Flow in Vertical U-bends, Masters Thesis, Delft Univarsity of Technology, Dept. Mechanica! Engineering, Delft.

[13.1] Confidentlal report. Raferenee to this document should only be made in documents having the same, or a higher, security classification. Shelllntemational Oil Products B.V.



[13.4] Confidentlal report. Raferenee to this document should only be made in documents havlng the same, or a higher, security classification. Shelllntemational Oil Products B.V.


69

[14] Confidential report. Reference to this document should only bemadein documents having the same, or a higher, security classification. Shell International Oil Products B.V.

[15] Nigmatulin R.l., Nigmatulin B.l., Khodzhaev Ya.D., Kroshilin V.E., 1996, Entrainment and Deposition rates in a Dispersed Film Flow, Int. J. Multiphase Flow, Vol 22, No. 1, pg. 19-30

[16] Confidential report. Reference to this document should only be made in documents having the sa me, or a higher, security classification. Shell International Oil Products B.V.

[17] Confidential report. Reference to this document should only be made in documents having the same, or a higher, security classification. Shell International Oil Products B.V.

70

List of symbols

Twophase flow

ao a2 a3 c1 C2 C3 c4 c5 c6

d 0 D, Dv E E, EM fg

fsgi

fgi

g GIGg k' p dp/dx vsl vsg

vslf

vsle

V se

V V1 V9 Ve viv,

v, V bulk

vii W1W9

wlfc

x Xo

y

y 8 Eg &I

Er &e

e Jlg lll Vg

Pg PI Pc cr crw

'tint 't1

'tw

Lh

Ar Re1 Re9

Rele Re11c

polynomial coefficients fit parameters entrainment model film thickness; distance of the detector to the tube tube diameter deposition rate volume number diameter entrained fraction entrainment rate maximum entrained fraction Fanning friction factor for single phase gas flow interfacial Fanning friction factor defined on superficial gas velocity interfacial Fanning friction factor defined on actual gas velocity gravitational acceleration mass fluxes of liquid and gas Asali's constant for entrainment and deposition rates pressure pressure gradient along the tube superficial veloeities of liquid and gas superficial velocity of liquid in the film only superficial velocity of the entrained fraction superficial velocity of the gas care actual velocity actual veloeities of the film, the gas care and the entrained fraction interfacial velocity (actual and approximated) terminal settling velocity velocity of the liquid bulk average bulk velocity between detector i and detector j mass flow rates of liquid and gas critica! liquid mass flow rate through film axial coordinate in the flow initiation point of the AD-flow coordinate in the flow perpendicular to the wall

correction factor droplet diameter gas and liquid holdup film and entrained fraction holdup detection angle dynamic viscosity of gas and liquid kinematic viscosity of the gas densities of gas and liquid density of the gas care (gas and droplets) surface tension and surface tension of water at p=p0 and T=T0

interfacial shear stress shear stress at the wall correction term, influence of deposition on interfacial shear stress

Archimedes number Reynolds number of the liquid and the gas care criticalliquid Reynolds number criticalliquid Reynolds number of the film

71

film Reynolds number basedon liquid mass flow rate in film film Reynolds number terminal settling Reynolds number (droplets) Weber number

AD-flow numerical simulation

Aint

Äti1m Äcore

Ätube

a, ac dx dt FWHM FWHM1

i mq.

mf,elem mc,elem

mf,tr mc,tr

mb,tr

mbl

n nd NI

~t

~tc~Ît

~tro

~Îrd,nd Vmm Vcell

vc xfxc

xb

Xt

<P

<Pentr <Pdep

<Pmm <Pccre

f..l Pdm

cr

surface of the interface between core and film in each tube element cross sectional surface covered by film and by gas core cross sectional surface of the tube fraction of distance travelled through film and gas core length of a tube element duration of a time step full width at half maximum full width at half maximum at t=Nt"dt index of time mass flow rate (based on flux $) actual mass of liquid in each element in the film and the gas core actual mass of tracer liquid in each element in the film and the gas core actual mass of tracer liquid in the bend mass of the liquid in the bend index of length along the tube index number of detection element number of time steps passed at the moment when the tracer distribution in the tube is plotted transition probability equilibrium entrainment and deposition rate simulated signa! coming from the detector time needed to travel through the tube based on core velocity and based on film velocity accumulated error in residence time error in residence time introduced by core and film error in residence time introduced by each element standard deviation in residence time after travelling nd elements volume of film and core element velocity of the core tracer liquid mass fraction in film and core tracer liquid mass fraction in bend travelled distance at t=Nt"dt

liquid mass flux liquid mass flux through interface (entrainment and deposition) liquid mass flux through film and core average of a population drop mass density in the core standard deviation of a population

Bubble detachment; temperature anomaly in bubble flow

72

effective bubble surface in the flow (perpendicular to flow velocity) drag coefficient bubble thickness buoyancy force drag force surface tension force thermocapilary force

Fe

Fsuction

f g Q

Rtube

Rb

ö.T

u v (=v(y)) V vbubble

vb

y y

expansion force suction force Fanning friction factor gravitational acceleration heat flux through wall radius of the tube radius of the bubble temperature anomaly dimensionless velocity friction velocity actual velocity of the flow average flow velocity bubble volume velocity at bubble coordinate perpendicular to the wall dimensionless y

degree of coverage of the wall altachment angle advancing and raceding attachment angle of the bubble heat transfer coefficient heat transfer coefficient of vapour and liquid shear stress at the wall

Reynolds number of the bubble

73

Appendix A Results of the residence time measurements in AD-flow

Confidential data not available

Appendix B Results of the models by Asali and Miesen


Appendix C Computer code of the numerical residence time sim u lation

The simulation based on mass fluxes between elements of a discretised tube and between film and gas core is written in Visual Basic for MS Excel7.0 ™.

Computer code of the simulation:

' Model for annular dispersed flow residence time I Discrete numerical simulation I Written by B.J.Niestadt (TU Eindhoven graduate) I Juli 21 1997

Sub Residence()

Sheets("Residence").Select ActiveWorkbook.DisplayDrawingObjects = xiHide

Range("Z5").SoundNote.Piay

I Reading the input variables

tube_length = Range("C3").Value xsteps% = Range("G3").Value duration = Range("C4").Value tsteps% = Range("G4").Value Show%= Range("D14").Value denom = Fix(tsteps% I Show%) dx = tube_length I xsteps% dt = duration I tsteps% dtube = (Range("C5").Value) I 1000 dfilm = (Range("G5").Value) I 1000 Vg = Range("C6").Value vf = Range("G6").Value densg = Range("C7").Value densl = Range("G7").Value E = Range("CB").Value sigma = Range("GB").Value mug = Range("C9").Value mul = Range("G9").Value Vsg = Range("C10").Value Vsl = Range("G10").Value Pi = Range("Z3").Value epsl = Range("C11").Value epsf = Range("G11").Value epse = epsl - epsf epsg = 1 - epsl xb =0 TDist = Range("U2").Value nt% = Fix(TDist I dt) det1% = Fix(xsteps% * (Range("T13").Value I Range("C3").Value)) det2% = Fix(xsteps% * (Range("T14").Value I Range("C3").Value)) det3% = Fix(xsteps% * (Range("T15").Value I Range("C3").Value)) det4% = Fix(xsteps% * (Range(1'T16").Value I Range("C3").Value)) Range("U2").Value = nt% * dt

Range("M3").Value = dx Range("M4").Value = dt

' Matrix definition and initia! condition

Dim tube() As Single Dim film() As Single Dim tube2() As Single Dim film2() As Single ReDim tube(xsteps%) ReDim film(xsteps%) ReDim tube2(xsteps%) Re Dim film2(xsteps%) film(O) = 0 film(1) = Range("Q3").Value tube(O) = 0 tube(1) = Range("Q4").Value film2(0) = 0 film2(1) = 0 tube2(0) = 0 tube2(1) = 0 For n = 2 To xsteps%

tube(n) = 0 film(n) = 0 tube2(n) = 0 film2(n) = 0

Next n

'Calculate fluxes, velocities, etc .....

We = densg * Vsg " 2 * dtube I sigma densdm = (epse I (epse + epsg)) * densl avds = 140 * (Vsg I (Vsg- 4)) * Exp(-1 * (Vsg I 41.3) * Exp(-1 * (Vsl I 0.09))) * (0.000001) Ar= (9.81 I (mug" 2)) * avds" 3 * densg * (densl- densg) Rets =Ar I (18 + 0.61 * Sqr(Ar)) Vt = Rets * mug I ( densg * avds) Vdg = Vg- Vt

'Force V draplets = Vsg lf Range("V5") = 1 Then

Vdg = Vsg Vt = 0

End lf

fluxm = Vdg * densdm fluxf = vf * densl Aint =Pi* Sqr(epse + epsg) * dtube * dx Afilm = epsf * (Pi I 4) * dtube " 2 Vfilm = Afilm * dx Acell =(Pi I 4) * dtube "2 * (epse + epsg) Veel! = Acell * dx mbl = densdm * Vcell + densl * Vfilm c4 = 0.45 * (0.3)" 0.65 Elimit = 0.3 * (densg * Vsg) I (densl * Vsl)

' Calculating equilibrium entrainment rate

lf E > Elimit Or E = Elimit Then R = c4 * (densl * Vsl * E)" 0.35 * (densg * Vsg)" 0.65 * (1 I (We" 0.5))

El se R = 0.45 * (densl * Vsl * E) I ((We) 11 0.5)

End lf lf Range("V4") = 1 Then R = 0

' Condition check Ved*dt<0.8*dx

check1 = Vdg * dt I dx lf check1 > 0.8 Then

response = MsgBox("Condition V*dt < 0.8*dx not satisfied. lncrease Nt", 48, "ERROR MESSAGE !!!") GoTo finito

End lf

' Output of some calculated parameters

Range("M5").Value = densdm Range("M6").Value = Vt Range("M7").Value = Vdg Range("M8").Value = fluxm Range("M9").Value = avds * 1 000 Range("M1 O").Value = fluxf Range("M11").Value = Aint Range("M12").Value = R Range("M13").Value = Vcell Range("M14").Value = Afilm Range("M15").Value = Vfilm Range("M16").Value = mbl * 1000

Application.Calculation = xiManual

' Actual simuiatien

For i = 1 To tsteps% For n = xsteps% To 1 Step -1

xf = film2(n) xt = tube2(n) film2(n) = ((Vfilm * densl - fluxf * Afilm * dt- R * Aint * dt) * xf + fluxf * Afilm * dt * film2(n - 1) + R * Aint * dt

* xt) I (Vfilm * densl) tube2(n) = ((Vcell * densdm - fluxm * Acell * dt- R * Aint * dt) * xt + fluxm * Acell * dt * tube2(n - 1) + R *

Aint * dt * xf) I (Veen * densdm) Next n

xf = film2(0) xt = tube2(0) xbold = xb film2(0) = ((Vfilm * densl - fluxf * Afilm * dt- R * Aint * dt} * xf + fluxf * Afilm * dt * xbold + R * Aint * dt * xt) I

(Vfilm * densl) tube2(0) =((Veen* densdm- fluxm * Acen * dt- R * Aint * dt) * xt + fluxm * Acen * dt * xbold + R * Aint * dt *

xf) I (Veen * densdm) xb = ((mbl- fluxf * Afilm * dt- fluxm * Acen * dt) * xbold + fluxf * Afilm * dt * film(xsteps%) + fluxm * Acen * dt

* tube(xsteps%)) I mbl

For n = xsteps% To 1 Step -1 xf = film(n) xt = tube(n) film(n) = ((Vfilm * densl - fluxf * Afilm * dt- R * Aint * dt) * xf + fluxf * Afilm * dt * film(n - 1) + R * Aint * dt *

xt) I (Vfilm * densl) tube(n) = ((Vcell * densdm- fluxm * Acen * dt- R * Aint * dt) * xt + fluxm * Acell * dt * tube(n- 1) + R *

Aint * dt * xf) I (Vcell * densdm)

lf i = nt% Then Cells(n + 19, 27) = dx * n Cells(n + 19, 28) = tube(n) Cells(n + 19, 29) = film(n) Cells(xsteps% + 20, 27) = (xsteps% + 1) * dx Cells(xsteps% + 20, 28) = xb Cells(xsteps% + 20, 29) = xb Cells(n + xsteps% + 20, 27) = dx * (n + xsteps% + 1) Cells(n + xsteps% + 20, 28) = tube2(n) Cells(n + xsteps% + 20, 29) = film2(n)

End lf Next n

k = Fix(i I denom)- i I denom lf k = 0 Then

Range("D15").Value =(i I tsteps%) * 100 Cells((i I denom) + 19, 1).Value =i* dt Cells((i I denom) + 19, 2).Value = tube(det2%) Cells((i I denom) + 19, 3).Value = film(det2%) Cells((i I denom) + 19, 6).Value = tube(det1%) Cells((i I denom) + 19, 7).Value = film(det1%) Cells((i I denom) + 19, 10).Value = tube2(det3%) Cells((i I denom) + 19, 11 ).Value = film2(det3%) Cells((i I denom) + 19, 14).Value = tube2(det4%) Cells((i I denom) + 19, 15).Value = film2(det4%)

End lf Next i

Application.Calculation = xiAutomatic Calculate

finito: ActiveWorkbook.DisplayDrawingObjects = xiAII

End Sub

Appendix D MS Excel TM macro-module tor evaluation of the simulated experiment

The computer code in Visual Basic for MS Excel TM for evaluation of the simulated experiments is given below. The module "Eval()" is an extension of the module "Residence(}", shown in appendix C.

Computer code of the evaluation program:

' Evaluation of the experiment 'Written by B.J.Niestadt {TU Eindhoven graduate) 'Juli 29 1997

Sub Eval()

Sheets{"Residence").Select

'initialising parameters

densl = Range{"G7").Value densdm = Range{"M5").Value Vfilm = Range{"M15").Value Vcell = Range{"M13").Value xfinit = Range{"Q3").Value xcinit = Range{"Q4").Value mtrtot = densl * Vfilm * xfinit + densdm * Vcell * xcinit xsteps% = Range{"G3").Value tsteps% = Range{"D14").Value tstmax% = Range{"G4").Value duration = Range("C4").Value length = Range{"C3").Value dt = Range{"M4").Value * {tstmax% I tsteps%) dx = Range{"C3").Value I xsteps%

'Dimensioning of RD-time matrices

Dim sum1() As Single ReDim sum1{tsteps%) sum1{0) = 0 Dim sum2() As Single ReDim sum2{tsteps%) sum2{0) = 0 Dim sum3{) As Single ReDim sum3{tsteps%) sum3{0) = 0 Dim sum4{) As Single ReDim sum4{tsteps%) sum4{0) = 0

max1 = 0 tp1% = 0 max2 = 0 tp2% = 0 max3 =0 tp3% = 0 max4 = 0

tp4% = 0

' Evaluating residence time F-curve, peak time and maximum concentratien

For i= 1 To tsteps% sum1 (i)= sum1 (i- 1) + Cells(i + 19, 8).Value sum2(i) = sum2(i - 1) + Cells(i + 19, 4 ).Value sum3(i) = sum3(i - 1) + Cells(i + 19, 12).Value sum4(i) = sum4(i- 1) + Cells(i + 19, 16).Value lf max1 < Cells(i + 19, 8).Value Then tp1% = i: max1 = Cells(i + 19, 8).Value lf max2 < Cells(i + 19, 4).Value Then tp2% =i: max2 = Cells(i + 19, 4).Value lf max3 < Cells(i + 19, 12).Value Then tp3% =i: max3 = Cells(i + 19, 12).Value lf max4 < Cells(i + 19, 16).Value Then tp4% =i: max4 = Cells(i + 19, 16).Value

Next i

'Piotting the F-curves

Sheets("Evaluation").Select

For i= 1 To tsteps% Cells(i + 9, 1 ).Value =i* dt Cells(i + 9, 2).Value = 100 * sum1 (i) I sum1 (tsteps%) Cells(i + 9, 3).Value = 100 * sum2(i) I sum2(tsteps%) Cells(i + 9, 4).Value = 100 * sum3(i) I sum3(tsteps%) Cells(i + 9, 5).Value = 100 * sum4(i) I sum4(tsteps%) Range("C6").Value = 100 *(i I tsteps%)

Next i

'Print calculated values

Range("K19").Value = dx * Fix(xsteps% * (Sheets("Residence").Range("T13").Value I Sheets("Residence"). Range("C3"). Value))

Range("N19").Value = dx * Fix(xsteps% * (Sheets("Residence").Range("T14").Value I Sheets("Residence").Range("C3").Value))

Range("K24").Value = dx * (xsteps% + 1 + Fix(xsteps% * (Sheets("Residence").Range("T15").Value I Sheets("Residence").Range("C3").Value)))

Range("N24").Value = dx * (xsteps% + 1 + Fix(xsteps% * (Sheets("Residence").Range("T16").Value I Sheets("Residence"). Range("C3"). Value)))

Range("K18").Value = dt * tp1% Range("N18").Value = dt * tp2% Range("K23").Value = dt * tp3% Range("N23").Value = dt * tp4%

End Sub

Appendix E Results of the simuiatien


Appendix F Bubble thickness and temperature anomaly for various conditions

(lam i nar flow model)

The various condition, which can be found on the following pages, are characterised by the following physical properties and dynamic constants:

condition f.ll Ns/m2 f.lg Ns/m2

Pt kg/m3 Po kg/m3 cr N/m E

# 1 1.00. 10-4 2.00. 10"5 650 50 1.00. 10-2 0.1 2 3.00. 10-4 2.00. 10-5 700 10 1.00. 10-2 0.1 3 1.00. 10-4 2.00. 10-5 700 10 3.00. 10-2 0.1 4 1.00. 10-4 2.00. 10-5 700 50 3.00. 10-2 0.1 5 1.00. 10-4 2.00. 10-5 700 50 7.00. 10-3 0.1 6 5.00. 10-4 2.00. 10-5 700 10 7.00. 10-3 0.1 7 1.00. 10-4 2.00. 10-5 700 10 7.00. 10-3 0.1

Here E is the degree of coverage of the tube wall: ratio between the surface of the tube wall covered by bubbles and the total surface of the tube wall.

The conditions are all calculated using the following dynamica! constants:

Vaverage = 1 m/s f = 0.005 Dtube = 0.10 m

average velocity of the fluid in laminar flow Fanning friction factor diameter of the tube

Condition 1

1.80E-03

1.60E-03

1.

8.

6.

theta

theta

Bubble thickness trom wall on datachment vs. theta and dtheta;

mul 1.00E-04 Ns/m"2 mug 2.00E-05 Ns/m"2 rhol 650 kg/mA3 rhog 50 kg/miiJ sigma 1.00E-02 Nlm

dtheta/theta

Temperature difference between walland fluid vs. theta and dtheta;

mul 1.00E-04 Ns/m"2 mug 2.00E-05 Ns/m"2 rhol 650 kgtmA3 rhog 50 kglmA3 sigma 1.00E-02 Nlm epsilon 0.1 (10%)

dtheta/theta

Condition 2

theta

theta

Bubble thickness from wan on datachment vs. theta and dtheta;

mul 3.00E-04 Ns/m"2 mug 2.00E-05 Ns/m"2 rhol 700 kg/m"3 rhog 1 0 kg/m"3 sigma 1.00E-02 N/m

dtheta/theta

Temperature ditterenee between walland fluid vs. theta and dtheta;

mul 3.00E-04 Ns/m"2 mug 2.00E-05 Ns/m"2 rhol 700 kg/m"3 rhog 10 kg/m"3 sigma 1.00E-02 N/m epsilon 0.1 (10%)

dthetaltheta

Condition 3

3.00E-03

2.50E-03

I 2.00E-03

Cll :c .0 :::s .0 "C

1.50E-03

1.00E-03

S.OOE-04

O.OOE+OO

theta

50

g 40

1- 30 J1! Cll

"C 20

10

theta

Bubble thickness from wall on detachment vs. theta and dtheta;

mul 1.00E-04 Ns/m"2 mug 2.00E-05 Ns/m"2 rhol 700 kg/m"3 rhog 10 kg/m"3 sigma 3.00E-02 Nlm

0.719

dtheta/theta


mul 1.00E-04 Ns/m"2 mug 2.00E-05 Ns/m"2 rhol 700 kg/m"3 rhog 10 kg/m"3 sigma 3.00E-02 Nlm epsilon 0.1 (10%)

dtheta/theta

Condition 4

3.00E-03

2.50E-03

§: 2.00E-03

Cll ::ë 1.50E-03 ..0 :::J

..0 1.00E-03 "C

5.00E-04

O.OOE+OO

theta

theta

"'"": "'"":

Subbie thickness from wall on datachment vs. theta and dtheta;

mul 1.00E-04 Ns/m"2 mug 2.00E-05 Ns/m"2 rhol 700 kgfmA3 rhog 50 kg/mA3 sigma 3.00E-02 N!m

dtheta/theta


mui1.00E-04 Ns/m"2 mug 2.00E-05 Ns/m"2 rhol 700 kg/m"3 rhog 50 kg/m"3 sigma 3.00E-02 N!m epsilon 0.1 (10%)

dtheta/theta

Condition 5

theta

30

25

theta

Bubble thickness from wall on datachment vs. theta and dtheta;

mul 1.00E-04 Ns/m"2 mug 2.00E-05 Ns/m"2 rhol 700 kgfmA3 rhog 50 kgfmA3 sigma 7.00E-03 Nlm

dtheta/theta

Temperature ditterenee between walland fluid vs. theta and dtheta;

mul 1.00E-04 Ns/m"2 mug 2.00E-05 Ns/m"2 rhol 700 kgfmA3 rhog 50 kg/mAJ sigma 7.00E-03 Nlm epsilon 0.1 (10"k)

dtheta/theta

Condition 6

:§: Gl :c .c ::l .c

1.40E-03

1.20E-03

1.00E-03

S.OOE-04

6.00E-04

-c 4.00E-04

2.00E-04

O.OOE+OO ~~s~~~~;; r::~:! ......_coo <q-.:~;..,. ~r-.oN~;blD T"""..-....: ..... ·"":~C!e>co~ T"""._T""..-oo 0

theta

theta

Bubble thickness trom wall on delachment vs. theta and dtheta;

mul 5.00E-04 Ns/m"2 mug 2.00E-05 Ns/m"2 rhol 700 kg/m113 rhog 1 0 kg/m"3 sigma 7.00E-03 Nlm

dtheta/theta


mul 5.00E-04 Ns/m"2 mug 2.00E-05 Ns/m"2 rhol 700 kg/m113 rhog 10 kg/m"3 sigma 7.00E-03 Nlm epsilon 0.1 (10%)

dtheta/theta

Condition 7

:§:

1.40E-03

1.20E-03

1.00E-03

S.OOE-04

6.00E-04

theta

theta

Bubble thickness trom wall on delachment vs. theta and dtheta;

mui1.00E-04 Ns/m"2 mug 2.00E-05 Ns/m"2 rhol 700 kg/m"3 rhog 1 0 kg/m"3 sigma 7.00E-03 Nlm

dtheta/theta


mul 1.00E-04 Ns/m"2 mug 2.00E-05 Ns/m"2 rhol 700 kg/m"3 rhog 1 0 kg/m"3 sigma 7.00E-03 Nlm epsilon 0.1 (10%)

dtheta/theta

Documents

Eindhoven University of Technology MASTER Twophase flow …Eindhoven University of Technology MASTER Twophase flow in cracking furnaces : residence time in annular dispersed flow and