Numerical Simulations of Supercritical
Water-Hydrocarbon Mixing in a 3-D Cylindrical
Tee Mixer
MASSACHUSETTSIN$OM1E
OF TECHNOLOGY
by
MAYO 8 20
Ashwin Raghavan
LIBRARIES
B.Tech., Mechanical Engineering,
Indian Institute of Technology Bombay (2011)
Submitted to the Department of Mechanical Engineering
in partial fulfillment of the requirements for the degree of
Master of Science in Mechanical Engineering
at the
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
Feb 2014
@
Massachusetts Institute of Technology 2014. All rights reserved.
A uth o r ........................
..........................
ftepartment of Mechanical Engineering
Nov 27th, 2013
I
A
Certified by .................
U
V
Ahmd F. Ghoniem
Professor, Department of Mechanical Engineering
* Theg Supervisor
A ccepted by ...................
t-
.............
David E. Hardt
Professor, Department of Mechanical Engineering
Graduate Officer, Department of Mechanical Engineering
2
Numerical Simulations of Supercritical Water-Hydrocarbon
Mixing in a 3-D Cylindrical Tee Mixer
by
Ashwin Raghavan
Submitted to the Department of Mechanical Engineering
on Nov 27th, 2013, in partial fulfillment of the
requirements for the degree of
Master of Science in Mechanical Engineering
Abstract
Supercritical water upgrading and desulfurization (SCWUDS) is a new concept in
the oil refining industry wherein, crude oil is mixed with supercritical water in a
reactor leading to chemical breakdown of the sulfur containing compounds (desulfurization) and cracking of long chain hydrocarbons to shorter chain compounds closer
to commercial fuel components (upgrading). The focus of the present work is the
development of a numerical tool to investigate the mixing of water and hydrocarbons under supercritical fully-miscible conditions (water and hydrocarbon forming a
single phase) in a realistic 3-D cylindrical tee mixer geometry so as to develop an
understanding of the effects of geometry, flow rates and fluid properties on the mixing dynamics which in turn will influence the rate of thermal cracking reactions of
hydrocarbons and organosulfur compounds as well as the final product distributions.
This work includes a consistent treatment of near-critical thermodynamics and transport property variations of real fluid mixtures. A Peng-Robinson EoS with simple
van der Waals mixing rules is employed to model the near-critical thermodynamic
behavior, with the mixture binary interaction parameter obtained from a Predictive
Peng-Robinson approach using a group contribution method (PPR78). A 2 "d order
accurate finite-volume methodology is used for the numerical solution of the conservation equations.
The developed numerical tool was used to investigate the mixing of supercritical
water and a model hydrocarbon (n-decane) in a small-scale cylindrical tee mixer (pipe
ID = 2.4mm) under fully miscible conditions. For a Reynolds number at the water
inlet of 500 and a AT between the two streams of lOOK (Ten = 800K, Tdn= 700K),
the flow downstream of the mixing joint remained laminar. Most of the mixing and
heat transport occurs due to the circulating action of a counter-rotating vortex pair
(CVP) in the hydrocarbon jet formed due to the reorientation of the vorticity preexisting in the hydrocarbon stream flowing through the vertical pipe. This CVP
gets progressively weaker as it is advected downstream, due to vorticity diffusion
and species and heat transport is dominated by molecular diffusion over small length
scales in the far downstream region. Consequently, the mixing rate plateaus in the far
3
downstream region of the tee mixer. Near-critical property variations were found to
have a negligible impact on the flow field and mixing behavior under these conditions.
However, for a 300K temperature difference between the two streams (TWin =
1000K, Td,in = 700K), the water-HC shear layer becomes unstable and rolls up
downstream of 5 diameter lengths from the mixing joint. The onset of instability
in the shear layer also triggers the stretching and breakdown of the hydrocarbon jet
CVP leading to a significant enhancement in mixing manifested as a jump in the
mixing rate and a thickening of the mean mixing layer. However, water n-decane
mixing under identical inlet conditions but with constant physical properties, showed
a stable shear layer with the flow reaching steady state. For a large AT between the
streams of 300K, the strong density increase (due to cooling of the water component)
and the strong viscosity decrease (due to heating of the n-decane component) leads
to a local increase in the Re within the mixing layer, resulting in the instability of
the shear layer.
SCW n-decane mixing with AT = lOOK was also simulated for increasingly higher
Reynolds numbers up to the transition to turbulence. When the Reynolds number at
the water inlet is increased to 700, the shear layer between the water and n-decane
streams is found to become unstable near x = 6D downstream of the mixing joint
followed by the subsequent roll up of the shear layer. The local increase of Re within
the mixing layer due to mechanisms similar to the AT = 300K case was found to
be the cause of the shear layer instability. At Re.,in = 800 the unsteady small scale
flow structures in the mixing layer and the consequent flow field fluctuations due to
them are much stronger. The stretching and breakdown of the CVP in this case, is
accompanied by stronger streamwise vorticity enhancement resulting in much faster
mixing compared to the case of Re.,i, = 700.
Key words: crude oil upgrading, desulfurization, supercritical water, supercritical fluids, mixing, CFD, mixing tee, intermediate Reynolds number, laminar, shear
layer instability, transition.
Thesis Supervisor: Ahmed F. Ghoniem
Title: Professor, Department of Mechanical Engineering
4
Acknowledgments
I would like to sincerely thank my research advisor, Prof. Ahmed Ghoniem. His timely
advice and guidance in matters of not only research, but also academic life in general
have been priceless. I am really grateful for his efforts in helping me channelize my
thought, interests and enthusiasm towards concentrated research effort. His constant
encouragement and motivation have helped me push myself to broaden and deepen
my knowledge and understanding.
I would like to express my greatest gratitude to our industry collaborator and
sponsor, Saudi Aramco without whose funding and support, this work would not
have been possible. It has indeed been a wonderful experience working with them.
I would also like to thank Prof. William Green, Prof. Michael Timko, Dr. Guang
Wu and all the past and present members of the Supercritical Water Desulfurization
and Upgrading team at MIT, as also our counterparts at Saudi Aramco for their
constant feedback, encouragement, support and fruitful collaboration. I would also
like to express my heartfelt gratitude to Dr. Jose-Sierra Pallares for his invaluable
insights and assistance in the thermodynamic modeling of non-ideal mixtures and the
state-of-the-art equations of state. The completion of this work would truly not have
been possible without him.
Special thanks to Dr.
Santosh Shanbhogue for his efforts in maintaining our
computing cluster and helping us out with all our computing problems, big and small.
I am also grateful to Kushal, Gaurav, Konstantina, Christos and all other members
of the Reacting Gas Dynamics Laboratory. It has been a pleasure to work with them
all and I have learned a lot from each one in the process. I would also like to mention
Pavitra, Kashi, Abhishek, Sudeep, Carl, Sameer, Ujwal, Suvinay, Tapovan, Arun,
Akash, Surekha, Abaya, Snegdha, Shreya, Shobhna, Jhanvi, Kat and the endless list
of my friends who have made my life at MIT worth cherishing forever. My thanks are
also due to Lorraine and Leslie for helping me out with all kinds of administrative
issues.
The acknowledgment cannot be complete without profusely thanking my parents
5
Shri.
T.V. Raghavan and Smt.
Renuka Raghavan, who taught me the value of
hard work, perseverance and learning and showered upon me their constant blessings
and support for all these years. Most of all, I would like to thank my elder brother
Karthikeyan Raghavan for his perennial support and wise guidance. He has truly been
a great mentor and friend throughout my life. I definitely owe my good understanding
of fundamentals in mathematics and science to his clear and lucid teaching during
my early days as a student. As such, he undoubtedly deserves a share of the credit
in all my current and future research endeavors.
Last but not the least, I thank The Brahman (God), the supreme cosmic energy
which is the basis of this vast, incredible universe; in understanding the mysteries of
which, we dedicate our lives.
This work has been sponsored by Saudi Aramco Contract No. 6600023444.
6
Contents
Abstract
4
Acknowledgments
6
1
23
Introduction
1.1
Role of water in desulfurization and upgrading . . . . . . . . . . . . .
26
1.2
Dynam ics of m ixing . . . . . . . . . . . . . . . . . . . . . . . . . . . .
26
1.2.1
Reactor geometry and flow Reynolds number . . . . . . . . . .
27
1.2.2
Near-critical thermodynamic and transport property variations
29
1.2.3
Water-hydrocarbon phase equilibrium . . . . . . . . . . . . . .
31
. . . . . . . . . . . . . . . . . . . . . . .
32
1.3
Thesis objectives and scope
2 Near-critical thermodynamics and transport property variations
2.1
2.2
2.3
3
. . . . . . . . . . . . . . . . . . . . . .
37
Peng-Robinson Equation of State for real fluid mixtures . . . .
39
. . . . . . . . . . . . . . . . . . . .
49
Near-critical thermodynamics
2.1.1
35
Near-critical transport properties
2.2.1
Viscosity and Thermal Conductivity
. . . . . . . . . . . . . .
49
2.2.2
M ass Diffusivity . . . . . . . . . . . . . . . . . . . . . . . . . .
52
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
55
Sum m ary
Problem Formulation and Methodology
57
. . . . . . . . . . . . . . . . . . . . . . . . . . .
57
. . . . . . . . . . . . . . . . . .
60
M eshing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
62
3.1
Governing equations
3.2
Geometry and Boundary Conditions
3.3
7
3.4
3.5
4
Numerical methodology
. . . . . . . . . . . . . . . . . . . . . . . . .
62
3.4.1
Discretization Schemes . . . . . . . . . . . . . . . . . . . . . .
63
3.4.2
Discretized Equations . . . . . . . . . . . . . . . . . . . . . . .
63
3.4.3
Solution Algorithm . . . . . . . . . . . . . . . . . . . . . . . .
65
3.4.4
Numerical Stability . . . . . . . . . . . . . . . . . . . . . . . .
68
3.4.5
Linear System Solvers
. . . . . . . . . . . . . . . . . . . . . .
69
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
69
Sum m ary
Mixing at intermediate Re: Flow dynamics and impact of temperature difference
71
4.1
Validation: Grid convergence tests . . . . . . . . . . . . . . . . . . . .
73
4.2
Case I: Water n-decane mixing with small temperature difference
(Rewin = 500,Twin = 800K,T,in = 700K) . . . . . . . . . . . . . . .
4.3
Case II: Water n-decane mixing with large temperature difference
(Rew,in = 500, Tw,in = 1000K, T
= 700K)
. . . . . . . . . . . . . .
92
4.4
Quantification of mixing . . . . . . . . . . . . . . . . . . . . . . . . .
100
4.5
Impact of near-critical property variations: Cause of shear layer insta-
b ility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
10 5
4.6
Dynamics of flow in the cylindrical tee mixer . . . . . . . . . . . . . .
111
4.7
Vorticity dynamics
116
4.7.1
4.8
5
75
. . . . . . . . . . . . . . . . . . . . . . . . . . . .
Species and thermal transport enhancement due to fluctuations 118
Sum m ary
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
122
Mixing at intermediate Re: Impact of Re
127
5.1
Validation: Grid convergence tests . . . . . . . . . . . . . . . . . . . .
127
5.2
M ixing Results
128
5.3
Quantification of mixing
5.4
Impact of near-critical property variations: Cause of shear layer insta-
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . .
141
b ility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
14 7
5.5
Vorticity dynamics
151
5.6
Species and thermal transport enhancement due to fluctuations
. . . . . . . . . . . . . . . . . . . . . . . . . . . .
8
. . .
154
5.7
6
Summary ...........................................
161
Summary and Future Work
..........
6.1
Summary
6.2
Future Work .........
156
.................................
161
................................
164
9
10
List of Figures
1-1
API Gravity (0) v/s sulfur content (%) of major crude oils in 2005,
. . . . . .
24
1-2
Distribution of proved world crude reserves (1991, 2001 and 2011) [38]
25
1-3
Schematic of the cylindrical tee reactor geometry used at Saudi Aramco,
bubble sizes proportional to 2005 production volumes, [55]
Dhahran, Saudi Arabia . . . . . . . . . . . . . . . . . . . . . . . . . .
28
1-4
Rectangular opposed-flow tee micromixer [9] . . . . . . . . . . . . . .
29
2-1
Water thermodynamic properties: Comparison between ideal-gas equation predictions and NIST data at P = 25MPa (a) p (in kg/M 3 ) and
(b) Cp (in kJ/kg - K ) . . . . . . . . . . . . . . . . . . . . . . . . . .
2-2
Comparison of density (in kg/m
3)
36
predictions by the Peng-Robinson
EoS and the Ideal Gas EoS with NIST data at P = 25MPa (a) water
and (b) n-decane
2-3
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
43
Comparison of density predictions by the Volume-Translated PengRobinson EoS (VT-PR EoS), the simple Peng-Robinson EoS and the
Ideal Gas EoS with NIST data at P = 25 MPa (a) water and (b) n-decane 45
2-4
Comparison of constant pressure specific heat (Cp in kJ/kg - K) predictions by the Peng-Robinson EoS and the Ideal Gas EoS with NIST
data at P = 25 MPa (a) water and (b) n-decane . . . . . . . . . . . .
2-5
48
Comparison of dynamic viscosity (1- in Pa-s)predictions using Chung's
generalized correlations [6] with NIST data at P=25MPa (a) water and
3-1
(b) n-decane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
53
Cylindrical tee mixer geometry and boundary conditions . . . . . . .
61
11
3-2
Cylindrical tee reactor mesh . . . . . . . . . . . . . . . . . . . . . . .
4-1
p (in kg/M 3 ) v/s T (in K) at P = 25 MPa for different mixture compositions of water and n-decane from Yd= 0 (pure water) to
Yd =
1
(pure n-decane) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4-2
62
73
p (in Pa-s) v/s T (in K) at P = 25 MPa for different mixture compositions of water and n-decane from Yd= 0 (pure water) to Yd= 1 (pure
n-decane)
4-3
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
74
u (in m/s) v/s z (in m) for simulated Case I for three different mesh
resolutions (0.10 mm, 0.08 mm and 0.06 mm) at (a) x = 4D (b) x =
6D and (c) x = 8D . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4-4
76
(in m/s) v/s z (in m) for simulated Case II for three different
Umean
mesh resolutions (0.10 mm, 0.08 mm and 0.06 mm) at (a) x = 4D (b)
x = 6D and (c) x = 8D . . . . . . . . . . . . . . . . . . . . . . . . . .
4-5
Tmean
77
(in K) v/s z (in m) for simulated Case II for three different mesh
resolutions (0.10 mm, 0.08 mm and 0.06 mm) at (a) x = 4D (b) x =
6D and (c) x = 8D . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4-6
Yd,mean
v/s z (in m) for simulated Case II for three different mesh
resolutions (0.10 mm, 0.08 mm and 0.06 mm) at (a) x
-
4D (b) x =
6D and (c) x = 8D . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4-7
78
Contour plots on the centerplane
(y=O
79
plane) at steady-state for sim-
ulated Case I of (top) n-decane mass fraction (Yd) field and (bottom)
Temperature (T) field; The white vertical lines indicate the positions
x=2D, x=4D, x=8D, x=10D, x=12D, x=14D downstream of the center
of the mixing joint from left to right
4-8
Profiles along the vertical centerline
. . . . . . . . . . . . . . . . . .
(y=O
81
plane) at steady-state for
simulated Case I of (a) n-decane mass fraction (Yd) field and (b) temperature (T in K) field at different downstream locations (x=2D, x=4D,
x=6D,
x=8D)........
...............................
12
82
4-9
Profiles along the vertical centerline (y=O plane) at steady-state for
simulated Case I of (a) velocity magnitude (Umag in m/s) (b) density (p
in kg/m 3 ) (c) dynamic viscosity (p in Pa-s) and (d) Reynolds number
(Re = pUmagD/p) at different downstream locations (x=2D, x=4D,
83
...............................
x=6D, x=8D)..........
4-10 (top) Mass fraction of n-decane (Yd) contours (middle) Temperature
(T in K) contours and (bottom) Streamwise vorticity (w, in s- 1) con500, T.,in = 800K, T,in= 700K)
tours for simulated Case I (Re, 'in=
at different downstream cross-sections (x=2D, x=4D, x=8D, x=16D,
x=22D)
84
..................................
...........
4-11 Contour plots on the centerplane (y=O plane) at steady-state for the
initially isothermal simulation:
(top) Temperature (T) field (in K)
(middle) Rate of heating/cooling due to heat diffusion
0AQ
1
(in K/s) (bottom) Rate of heating/cooling due to species diffusion
>j pD
along partial enthalpy gradient
hk
(in K/s); The white
vertical lines indicate the positions x=2D, x=4D, x=6D downstream
of the center of the mixing joint from left to right . . . . . . . . . . .
88
4-12 partial enthalpy of water, h, and n-decane, hd (in kJ/kg) v/s T (in
K) at P = 25 MPa for different mixture compositions of water and
n-decane from Yd= 0 (pure water) to Yd= 1 (pure n-decane) . . . . .
89
4-13 Contours of the rate of fluid heating/cooling (in K/s) due to (a) heat
diffusion
(
thalpy gradient(
) and (b) species diffusion along partial en-
A0
i
>j pDk
'
k)
at different downstream locations
(x=2D, x=4D, x=8D, x=16D, x=22D from left to right)
. . . . . . .
4-14 rate of fluid heating/cooling (in K/s) due to (a) heat diffusion ( 1
and (b) species diffusion along partial enthalpy gradient (P
along the vertical centerline at x = 4D for simulated Case I
ZkpDk
90
A
Oyk Ohk)
. . . . .
91
4-15 Lewis number, Le = a/D at different downstream cross-sections (x=2D,
x=4D, x=8D, x=16D,
x=22D).......
4-16 Fourier transform of the
Yd
......................
91
temporal variation at x = 6D, y = 0, z = 0 93
13
4-17 Contour plots on the centerplane (y=O plane) of n-decane mass fraction
(Yd)
field at (a) t = 2.Os (2.5
(b) t = 2.4s (3 tflow-through)
tflow-through)
(c) t = 2.8s (3.5 tflow-through) (d) t = 3.2s (4
tflowthrough)
and (e)
Mean field; The white vertical lines indicate the positions x=4D, x=6D,
x=8D, x=1OD, x=12D and x=16D downstream of the center of the
m ixing joint (left to right) . . . . . . . . . . . . . . . . . . . . . . . .
95
4-18 Profiles along the vertical centerline (y=O plane) at steady-state for
simulated Case II of (a) mean n-decane mass fraction (Yd) field and
(b) mean temperature (T in K) field at different downstream locations
96
......................
(x=2D,x=4D,x=6D,x=8D) ......
4-19 Profiles along the vertical centerline (y=O plane) at steady-state for
simulated Case II of (a) mean velocity magnitude (Umean,mag in m/s)
(b) mean density
(prnean
in kg/m 3 ) (c) mean dynamic viscosity (pmean
in Pa-s) and (d) mean Reynolds number (Remean = PmeanUmean,magD/pmean)
at different downstream locations (x=2D, x=4D, x=6D, x=8D)
. . .
97
4-20 Contours of Yd for simulated Case II at different downstream crosssections (x=2D, x=4D, x=6D, x=8D, x=16D from left to right) at (a)
t = 2.4s (3
tflowthrough)
tflowthrough)
(b) t = 2.8s (3.5
tflowthrough)
(c) t = 3.2s (4
and (d) M ean . . . . . . . . . . . . . . . . . . . . . . . .
98
4-21 Streamwise vorticity (W, in s-') contours for simulated Case II at different downstream cross-sections (x=2D, x=4D, x=6D, x=8D, x=16D
from left to right) at (a) t = 2.4s (3 tflow-through) (b) t = 2.8s (3.5
tflow-through)
(c) t = 3.2s (4
and (d) Mean . . . . . . . .
tflow-through)
99
4-22 Contour plots on the centerplane (y=O plane) of the streamwise velocity (u in m/s) field at (a) t = 2.Os (2.5
tflowthrough)
(c) t = 2.8s (3.5
tflowthrough)
tflowthrough)
(b) t = 2.4s (3
(d) t = 3.2s (4
tflowthrough)
and (e) Mean field; The white vertical lines indicate the positions
X=2D, x=4D, x=6D, x=8D, x=10D, x=12D and x=16D downstream
of the center of the mixing joint (left to right) . . . . . . . . . . . . .
14
101
4-23 Contour plots on the centerplane (y=O plane) of the vertical veloc-
ity (w in m/s) field at (a) t = 2.0s (2.5
tflow-through)
tflow-trough)
(b) t = 2.4s (3
(c) t = 2.8s (3.5 tflow-through) (d) t = 3.2s (4 tjlow-through)
and (e) Mean field; The white vertical lines indicate the positions
x=2D, x=4D, x=6D, x=8D, x=10D, x=12D and x=16D downstream
of the center of the mixing joint (left to right)
4-24
#
. . . . . . . . . . . . .
102
v/s x/D: Variation along the length of the tee of the (a) the species
mixing quality (#species) for simulated Cases I-IV and (b) the thermal
mixing quality (#thermal) for simulated Cases I and II
. . . . . . . . .
104
4-25 Streamwise vorticity (P in s-1) contours for simulated Case I (top)
and Case III (bottom) at different downstream cross-sections (x=2D,
x=4D,
106
......................
x=8D, x=16D, x=22D).......
4-26 n-decane mass fraction (Yd) contours for simulated Case I (top) and
Case III (bottom) at different downstream cross-sections (x=2D, x=4D,
x=8D,
107
..........................
x=16D, x=22D)........
4-27 Profiles along the vertical centerline (y=O plane) at steady-state for
simulated Cases I and III of (a) density (p in kg/rM3 ) and (b) dynamic
viscosity (p in Pa-s) at x=5D
. . . . . . . . . . . . . . . . . . . . . .
108
4-28 Profiles along the vertical centerline (y=O plane) at steady-state for
simulated Cases I and III of the local Reynolds number at x=5D . . .
109
4-29 Y contours on the centerplane for simulated Case IV; The white vertical lines indicate the positions x=4D, x=6D, x=8D, x=10D, x=12D
and x=16D downstream of the center of the mixing joint (left to right) 110
4-30 Yd contours for simulated Case IV at different downstream cross-sections
(x=2D, x=4D, x=6D, x=8D, x=16D from left to right) . . . . . . . .111
4-31 Profiles along the vertical centerline (y=O plane) at steady-state for
simulated Cases II and IV of (a) density (p in kg/m
viscosity (p in Pa-s) at x=5D
3)
and (b) dynamic
. . . . . . . . . . . . . . . . . . . . . .
112
4-32 Profiles along the vertical centerline (y=O plane) at steady-state for
simulated Cases II and IV of the local Reynolds number at x=5D
15
. .
113
4-33 Streamwise vorticity (P, in s-1 ) contours for simulated Case I (Re.,in
=
500, Tw,in= 800K, Td,in= 700K) at different downstream cross-sections
(x=-0.5D, x=-0.25D, x=0, x=0.25D, x=0.5D, x=1D, x=2D) . . . . .
115
4-34 Transverse velocity vectors (velocity component along constant X planes,
Utrans =
v 2 + w 2 ) for simulated Case I (Rew,i, = 500, T.,in= 800K, T,in=
700K) at two downstream cross-sections (x=0.25D, x=2D) Note: The
length of the vectors is not representative of the magnitude in the
figures, the color of the vectors represents the magnitude . . . . . . .
116
4-35 (left) Contour plots on the centerplane (y=O plane) of the velocity
magnitude (Umag = (U2 + v 2 +
w2)1/ 2 )
in the joint region; (right) n-
decane mass fraction profile along the vertical line at x = 2D (the
position indicated by the white line in the left figure)
. . . . . . . . .
117
4-36 Contours of the mean field for simulated Case II at different downstream cross-sections (x=2D, x=4D, x=6D, x=8D, x=16D from left to
right) of (a) cst, in s- 2 (b)
Lb9,,
in s- 2 (c) cs,, in s-2 and (d)
sd,, in s-2119
4-37 Profiles along the vertical line at y=0.0004 (y=D/6) for simulated Case
II of (a)
J st,x,mean
(in s-2) and (b)
b4g,x,mean
(in s-2) at different down-
stream locations (x=4D, x=5D, x=6D, x=8D) . . . . . . . . . . . . . 120
4-38 Profiles along the vertical line at y=0.0004 (y=D/6) for simulated Case
II of
Wx,mean
(in s-1) at different downstream locations (x=4D, x=5D,
x=6D, x=8D).......
4-39 RMS fluctuation of Yd
...............................
(Y2 / 2 )
121
contours for simulated Case II at dif-
ferent downstream cross-sections (x=2D, x=4D, x=6D, x=8D, x=16D
from left to right) . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
122
4-40 Contours for simulated Case II at cross-sections downstream of location
of onset of instability (x=6D, x=8D, x=16D from left to right) of (a)
WmeanY,mean/Uref (b) w'Yd'/uref; Urfe = 0.14 m/s
. . . . . . . . . . .
123
4-41 Contours for simulated Case II at cross-sections downstream of location
of onset of instability (x=6D, x=8D, x=16D from left to right) of (a)
VmeanYd,mean/Uref (b) v'Yd'/uref; uref = 0.14 m/s . . . . . . . . . . . .
16
124
5-1
u (in m/s) v/s z (in m) for simulated Case IV for three different mesh
resolutions (0.10 mm, 0.08 mm and 0.06 mm) at (a) x = 4D (b) x =
6D and (c) x = 8D . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5-2
129
Yd v/s z (in m) for simulated Case IV for three different mesh resolu-
tions (0.10 mm, 0.08 mm and 0.06 mm) at (a) x = 4D (b) x = 6D and
(c) x = 8D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5-3
130
T (in K) v/s z (in in) for simulated Case IV for three different mesh
resolutions (0.10 mm, 0.08 mm and 0.06 mm) at (a) x = 4D (b) x =
6D and (c) x = 8D . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5-4
131
Contour plots on the centerplane (y=O plane) of n-decane mass fraction
(Yd) field at steady-state for (a) Case I and (b) Case II; The white vertical lines indicate the positions x=2D, x=4D, x=6D, x=8D, x=10D,
x=12D and x=16D downstream of the center of the mixing joint (from
left to right) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5-5
133
Contour plots of Yd on cross-sections of the tee at different downstream
locations (x=2D, x=4D, x=8D, x=16D and x=22D from left to right)
at steady-state for (a) Case I and (b) Case II . . . . . . . . . . . . . .
5-6
133
Contour plots of T (in K) on cross-sections of the tee at different downstream locations (x=2D, x=4D, x=8D, x=16D and x=22D from left
to right) at steady-state for (a) Case I and (b) Case II
5-7
. . . . . . . .
134
Contour plots of the streamwise vorticity, w, (in s-1) on cross-sections
of the tee at different downstream locations (x=2D, x=4D, x=8D,
x=16D and x=22D from left to right) at steady-state for (a) Case
I and (b) C ase II
5-8
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
135
Contour plots on the centerplane (y=O plane) of n-decane mass fraction
(Yd) field for simulated Case III at (a) t = 2.Os (4 tflowthrogh) (b) t =
2.2s (4.4
tflowthrough)
(c) t = 2.5s (5 tflow-through) and (d) Mean field;
The white vertical lines indicate the positions x=6D, x=8D, x=10D,
x=12D, x=14D, x=16D, x=18D and x=20D downstream of the center
of the mixing joint (left to right) . . . . . . . . . . . . . . . . . . . . .
17
137
5-9
Fourier transform of the Yd temporal variation at x = 8D, y = 0, z =
0 for C ase III . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
138
5-10 Contours of Yd for simulated Case III at different downstream crosssections (x=2D, x=4D, x=8D, x=12D, x=16D from left to right) at
(a) t = 2.Os (b) t = 2.2s (c) t = 2.5s and (d) Mean
. . . . . . . . . .
139
5-11 Contours of w (in s-1) for simulated Case III at different downstream
cross-sections (x=2D, x=4D, x=8D, x=12D, x=16D from left to right)
at (a) t = 2.0s (b) t = 2.2s (c) t = 2.5s and (d) Mean . . . . . . . . . 140
5-12 Contour plots on the centerplane (y=O plane) for Case III of u'/U,
where U is the reference velcoity used for normalization taken to be
the water inlet average velocity; The white vertical lines indicate the
positions x=2D, x=4D, x=6D, x=8D, x=10D, x=12D, x=14D and
x=16D
downstream of the center of the mixing joint (left to right) . . 141
5-13 Contour plots on the centerplane (y=O plane) of n-decane mass fraction
(Yd) field for simulated Case IV at (a) t = 1.8s (4
= 2.05s
(~
4.5
tflowthrough)
(c) t = 2.25s (5
tflowthrough)
tflowthrough)
(b) t
and (d)
Mean field; The white vertical lines indicate the positions x=4D, x=6D,
x=8D, x=10D, x=12D, x=14D, x=16D and x=18D downstream of the
center of the mixing joint (left to right) . . . . . . . . . . . . . . . . .
142
5-14 Contours of Yd for simulated Case IV at different downstream crosssections (x=4D, x=6D, x=8D, x=12D, x=16D from left to right) at
(a) t = 1.8s (b) t = 2.05s (c) t = 2.25s and (d) Mean . . . . . . . . . 143
5-15 Contours of wx (in s-1) for simulated Case IV at different downstream
cross-sections (x=4D, x=6D, x=8D, x=12D, x=16D from left to right)
at (a) t = 1.8s (b) t = 2.05s (c) t = 2.25s and (d) Mean . . . . . . . 144
5-16
#
v/s x/D: Variation along the length of the tee of the (a) the species
mixing quality (#species) and (b) the thermal mixing quality
(Othermal)
for simulated Cases I-IV . . . . . . . . . . . . . . . . . . . . . . . . .
18
146
5-17 p (in kg/M 3 ) v/s T (in K) at P = 25 MPa for different mixture compositions of water and n-decane from Yd = 0 (pure water) to Yd = 1
(pure n-decane) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
148
5-18 p (in Pa-s) v/s T (in K) at P = 25 MPa for different mixture compositions of water and n-decane from Yd= 0 (pure water) to Yd= 1 (pure
n-decane)
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
148
5-19 Yd contours on the centerplane for simulated Case V; The white vertical
lines indicate the positions x=2D, x=4D, x=6D, x=8D, x=10D, x=12D
and x=14D downstream of the center of the mixing joint (left to right) 149
5-20 Yd contours for simulated Case V at different downstream cross-sections
(x=2D, x=4D, x=8D, x=16D, x=22D from left to right)
. . . . . . .
150
5-21 Profiles along the vertical centerline (y=O plane) at steady-state for
simulated Cases III and V of (a) density (p in kg/m 3 ) and (b) dynamic
viscosity (y in Pa-s) at x=6D
. . . . . . . . . . . . . . . . . . . . . .
152
5-22 Profiles along the vertical centerline (y=O plane) at steady-state for
simulated Cases III and V of the local Reynolds number at x=6D
. .
153
5-23 Contours of the mean field for simulated Case IV at different downstream cross-sections (x=4D, x=6D, x=8D, x=10D, x=16D from left
to right) of (a) ;s,, in --2 (b)
in s- 2
Wbg,y
in s- 2 (c)
,,y in s-2 and (d) Wd,y
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5-24 RMS fluctuation of Yd
155
'21/2) contours for simulated Case IV at dif-
ferent downstream cross-sections (x=2D, x=4D, x=6D, x=8D, x=16D
from left to right) . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
156
5-25 Contours for simulated Case IV at cross-sections downstream of location of onset of instability (x=6D, x=8D, x=10D from left to right) of
(a) WmeanYd,mean/Href (b) w'Y '/uref; uref = 0.13 m/s
. . . . . . . . .
157
5-26 Contours for simulated Case IV at cross-sections downstream of loca-
tion of onset of instability (x=6D, x=8D, x=10D from left to right) of
(a) VmeanYd,mean/Uref (b) v'Y'/uref; u,,f = 0.13 m/s . . . . . . . . . .
19
158
20
List of Tables
2.1
Molecular weight, critical properties and acentric factor for water and
n-decane [32]
2.2
= Bik (in M Pa) [14][39]
. . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . .
51
Function coefficients used for generalized thermal conductivity correlations from Chung et al.[6]
2.7
49
Function coefficients used for generalized viscosity correlations from
Chung et al.[6] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.6
44
Acentric factor, dipole moment and association factor for water and
n-decane [32] [8] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.5
42
Molar critical volume and volume translation parameter for water and
n-decane M athias et al. [27]
2.4
40
Group interaction parameters for the PPR78 model, AkI = Alk and
Bkl
2.3
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . .
54
Lennard-Jones force constants for water and n-decane Liu et al. (1998)
[2 4 ] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
55
4.1
Inlet flow conditions for the cases discussed in this chapter . . . . . .
72
4.2
Properties of water and n-decane at 25MPa and extreme temperatures
of the simulations I and II . . . . . . . . . . . . . . . . . . . . . . . .
74
5.1
Inlet flow conditions for the cases simulated in this study . . . . . . .
128
5.2
Properties of water and n-decane at 25MPa and extreme temperatures
of the simulations in this study
5.3
. . . . . . . . . . . . . . . . . . . . .
147
Inlet flow conditions for the comparison test Case V simulated in this
study........
...
....................................
21
149
22
Chapter 1
Introduction
The crude oil obtained, on an average, from oil wells worldwide continues to become
sourer (higher sulfur mass fraction) and heavier (higher heavy hydrocarbon fraction).
Fig.1-1 shows the API gravity (in ', see Eq.1.1) v/s sulfur content
(%) scatter
for the
most widely produced crudes in the world in 2005 illustrating how the Arab, Ural
and Iran crudes which constitute the bulk of world oil production are of the middle
to heavy and sour grade. Moreover, close to 75% of proved oil reserves in the world
exist in the Middle-East, Ural region and South America (mainly Venezuela) most of
which is heavy, sour crude oil (see Fig. 1-2). On the other hand, regulatory standards
on fuels keep getting tighter with the maximum allowable level of sulfur set by the
US EPA as 15 ppm for diesel [51] and 30 ppm for gasoline [50].
Since a majority
of the crude oil demand comprises of light (for gasoline) and middle (for diesel and
jet fuel) distillates, the increasing heavy fraction of the crude oil presents a serious
demand-supply problem. All this, coupled with the perpetual effort of oil companies
to improve their margins necessitates the development of new efficient cost-effective
technologies for the desulfurization (removal of sulfur compounds) and upgrading
(increasing the light fraction) of crude oil feedstock.
APIgravity =
141.5
141.5
- 131.5
Poin /pwater
(1.1)
One such new concept in the oil industry is supercritical water upgrading and
23
4'_
44 4.
40
38
3'
34
32
~30
as
26
24
22
*#~AftESW
Figure 1-1: API Gravity (0) v/s sulfur content (%) of major crude oils in 2005, bubble
sizes proportional to 2005 production volumes, [55]
desulfurization(SCWUDS) of crude oil wherein, crude oil is mixed with water at supercritical pressure and temperature (Pcriticai = 22. 1MPa, Tcritical = 647K), causing
the sulfur compounds to breakdown, releasing hydrogen sulfide (H 2 S) in the process,
while also resulting in the cracking of long chain heavy hydrocarbons into smaller
chains (which are components of conventional fuels). This technology offers economic
benefits over the conventional desulfurization methodology in which transition metal
catalysts and hydrogen gas are used (hydro-desulfurization).
In order to develop
SCWUDS into a viable alternative to hydro-desulfurization followed by upgrading in
the oil industry, one first needs to understand clearly the physics and chemistry governing the process and the interactions of the two therein. The thermal breakdown
reactions of organic sulfur compounds and heavy hydrocarbons are strongly depen24
r
L!Itribution ofproved
:erves
in 1991 2001 and 2011
0 Middle East
0 S, & Cent. America
- North America
* Europe & Eurasia
Africa
a Asia Pacific
Figure 1-2: Distribution of proved world crude reserves (1991, 2001 and 2011) [38]
dent on the local temperature and concentration of the respective species. Therefore,
understanding and characterizing the mixing dynamics in a realistic SCWUDS reactor
geometry is crucial to be able to estimate conversion rates and product distributions
of the SCWUDS process. The focus of this work and ongoing work is to develop a
computational tool capable of simulating mixing of supercritical water and hydrocarbons at a range of temperature and pressure conditions pertinent to the SCWUDS
process so as to develop insight into the physical mechanisms governing the process.
Consequently, we would be in a position to tweak and optimize the design of the
reactor and/or the process as a whole to make it more effective and efficient.
25
1.1
Role of water in desulfurization and upgrading
When crude oil is heated up to sufficiently high temperatures (typically > 400 C), the
sulfur containing compounds and long hydrocarbon chains breakdown due to thermal
cracking reactions. This typically results in the formation of shorter hydrocarbon
chains, increasing the light fraction. Unfortunately, the cracking reactions also give
rise to coke precursor radicals like olefinic radicals which can further undergo cyclization and condensation reactions to form coke (carbonaceous agglomerates insoluble
even in toluene). Coke formation not only eats into potential light hydrocarbon yields
but also leads to carbonaceous deposits on the reactor walls thus clogging up the reactor. The presence of water is known to suppress coke formation by inhibiting the
cyclization and condensation reactions through the solvation and dispersion of coke
precursors. The solvation and dispersion effect of water prevents coke precursors from
coming together in the medium, thus suppressing coke formation pathways. Water
may also suppress coke formation by capping the coke precursor radicals by acting
as a hydrogen donor. However, this role of water as a hydrogen donor under supercritical conditions is disputed in the literature. The role of water in suppressing coke
formation in the supercritical water upgrading of vacuum residue and bitumen is well
illustrated in the works of Cheng et al. [5] and Vilcaez et al. [53] respectively. Thus
water transports heat to the hydrocarbons in a controlled fashion at the same time
inhibiting coke formation pathways. The downside to the presence of water is the
reduction in the rate of hydrocarbon cracking reactions due to dilution of the species.
Whether water plays an active chemical role in the primary breakdown processes of
large hydrocarbons and sulfur compounds is still uncertain and a topic of continuing
research.
1.2
Dynamics of mixing
Since the rates of the different cracking reactions and coke formation reactions are
strongly dependent on the local temperature and species concentrations, it is im26
portant to understand the transport of heat and water to the hydrocarbons through
mixing. The mixing of supercritical water with hydrocarbons is influenced by a variety of factors which are listed below:
" Reactor geometry and flow Reynolds number
" Near-critical thermodynamic and transport property variations
" Water-hydrocarbon phase equilibrium
1.2.1
Reactor geometry and flow Reynolds number
The present study was focused on a cylindrical tee reactor geometry shown in Fig.13.
Flow in a cylindrical tee mixer exhibits a variety of flow features likely to be
encountered in any general reactor. Also, this reactor geometry is currently being
used by our collaborators at Saudi Aramco for their tests on SCWDS and upgrading
(small scale test reactor, tube ID = 2.4 mm).
As such, it makes sense for us to
begin our modeling efforts with this reactor geometry and study the flow structures
and their effect on the mixing dynamics. The bulk flow Reynolds number (Re) which
depends on the flow rates, densities and viscosities of both the water and hydrocarbon
streams will determine whether the flow downstream of the mixing joint will remain
laminar or undergo transition to turbulence.
Dreher et al. [9] performed CFD simulations of mixing of two streams in a rectangular cross-section opposed-flow tee micromixer with channel width of 300Im (see
Fig. 1-4) for Re up to 1000. They were able to observe the transition of the flow regime
from straight laminar flow for Re < 10 to steady engulfment flow for 10 < Re < 240
(the two fluid streams swap to the opposite side) to periodically fluctuating flow for
240 < Re < 500 and finally to chaotic flow for Re < 500 (turbulence). Hoffmann et
al. [12] experimentally characterized the flow and mixing in a rectangular opposedflow tee micromixer and also observed the flow transition from the straight laminar
to engulfment flow regime. Correia et al. [7] numerically investigated the mixing in
opposed-flow micromixers with a tortuous downstream flow channel and found an unsymmetrical downstream channel configuration results in slightly better mixing than
27
Inlet 2
Cold pressurized
(25MPa) crude oil feed
Inlet 1
Hot SC water
(25MPa)
3D
3D
22D
Outlet
Figure 1-3: Schematic of the cylindrical tee reactor geometry used at Saudi Aramco,
Dhahran, Saudi Arabia
a symmetric configuration. Though these studies focus on the laminar to transition
flow regimes which are of interest to us in the preliminary small-scale tests of SCWDS,
they study mixing of two streams of the same fluid in rectangular opposed-flow tees.
Mixing of hot and cold water in cylindrical mixing tees (similar geometry to ours)
at high Re has been extensively studied in the nuclear industry community using
the Large Eddy Simulation (LES) and Unsteady Reynolds Averaged Navier-Stokes
(URANS) methodologies. LESs were performed by Kuczaj et al. [20, 19], Jayaraju
et al. [15], Westin et al. [54] and Odemark et al. [33] for very high Re (-150,000).
They were all able to obtain satisfactory agreement with the experimental results of
Andersson et al. [2] using appropriate mesh resolutions, inlet boundary conditions
and sub-grid scale models. Frank et al. [11] and Merzari et al. [28] performed URANS
28
of the same flow but were unable to capture the transient behavior as well as the LES
calculations. The focus of all these studies was mainly to validate LES and URANS as
viable predictive techniques for the prediction of thermal fluctuations in tee junctions
in the nuclear industry. As such, they did not investigate the laminar to turbulent
transition in cylindrical tee mixers.
I
b
0
T-shaped
micromixer
C
In I
.5
xJ*"L0
01
direction of view
Figure 1-4: Rectangular opposed-flow tee micromixer [9]
1.2.2
Near-critical thermodynamic and transport property
variations
The peculiar variations of thermodynamic and transport properties near the critical
point can be expected to affect the flow and mixing dynamics. Chapter 2 includes
a detailed discussion of real fluid behavior near the fluid critical point and the near29
critical variation of thermodynamic (density, specific heat) and transport (viscosity,
thermal conductivity, mass diffusivity) properties for both pure real fluids and nonideal mixtures of real fluids. Cubic equations of state which can capture real-fluid
thermodynamics have been formulated by Peng at al.[37] (Peng-Robinson EoS) and
Soave et al.[47] (Redlich-Kwong-Soave EoS). These equations of state are the most
widely used in numerical simulations of real-fluid flows due to their simplicity of implementation and effectiveness in capturing near-critical variations in fluid densities
and specific heats (especially for hydrocarbons). The variations of viscosity and thermal conductivity of fluids and fluid mixtures near the critical point can be modeled
using the correlations developed by Chung et al.[6] for a large number of substances
and a wide range of temperature-pressure conditions. Liu et al.[23][24] and Silva et
al. [46] have developed expressions for the variation of mass diffusivities with temperature and pressure of substances in dense non-ideal mixtures based on the hard-sphere
Lennard-Jones model.
Miller et al. performed Direct Numerical Simulations (DNS) of a supercritical
heptane-nitrogen temporal mixing layer (both two and three dimensional) [29] using a Peng-Robinson EoS to model the near-critical thermodynamic behavior and
studied the effect of density stratification across the shear layer on the transition to
turbulence. Okong'o et al. performed a similar DNS of a supercritical liquid oxygen/hydrogen three dimensional mixing layer [34] and observed a similar effect of
the density stratification causing a suppression of the transition to turbulence. They
investigated this effect by looking at the vorticity budget. The effects of the freestream density ratio on the evolution of incompressible, high Reynolds and Froude
number, confined mixing layer was investigated numerically by Soteriou et al. [48].
They found that a non-unity density ratio alters the flow characteristics significantly,
influencing the entrainment patterns by means of baroclinic generation of vorticity.
Zong et al. [58] have performed the LES of a two-dimensional cryogenic supercritical nitrogen jet at high Re and studied the spatial and temporal evolution of the jet
for different ambient pressures which lead to varying amounts of density stratification.
Their findings seem consistent with previous results of Miller et al. [29] and Okong'o
30
et al. [34] regarding the effect of density stratification leading to the suppression of
shear layer transition to turbulence. A similar study of the dynamics of cryogenic
nitrogen jets at supercritical pressures was performed by Kim et al.
[17] using a
RANS framework with a k-E turbulence model and a presumed probability-density
function for the conserved scalars. Park [36] performed a series of RANS and LES
simulations of a cryogenic liquid nitrogen jet with a variety of turbulence models and
three different equations of state (ideal gas, P-R and SRK) and found that the choice
of an appropriate real-fluid EoS is more important in capturing the flow and mixing
dynamics than the choice of turbulence models. Schmitt et al. [44] analyzed the behavior of a nitrogen coaxial shear jet under supercritical pressures through LES and
experiments and investigated the effect of externally imposed acoustic perturbations
on the jet mixing efficiency.
Narayanan et al.
[31] simulated a supercritical water oxidation (SCWO) hy-
drothermal flame of methanol using RANS and an eddy dissipation combustion model
with single-step kinetics. They were successful in obtaining a fair agreement with their
experiments for the flame position but over-predicted the flame temperature. SierraPallares et al. [45] have also simulated hydrothermal flames in SCWO of methanol
using an eddy dissipation concept along with a micro-mixing model and were able
to predict flame structure and temperature to a fair degree of accuracy. Kim et al.
[18] validated the use of a real-fluid flamelet model in the simulation of gaseous hydrogen/cryogenic oxygen coaxial jet flames under supercritical pressures. All of the
aforementioned numerical studies provide good insight into the flow and mixing behavior of near-critical fluids and were very helpful in formulating the methodology of
the study of 3-D SCW-HC mixing in a tee reactor performed in the present work.
1.2.3
Water-hydrocarbon phase equilibrium
The effect of water-hydrocarbon phase equilibrium on the rate of mixing of the two
fluids is captured very well by Dabiri et al.
[8] in fundamental numerical studies
of the mixing process of a cold hydrocarbon droplet in a vast reservoir of heated
supercritical water. Their studies of water-toluene and water-decane mixing highlight
31
the importance of the upper-critical solution temperature (UCST) of the two fluids
in determining the persistence of the phase interface which greatly retards the mixing
rate. They also observed large abrupt increases in mixing rate for the water-toluene
system as the water temperature is increased from below the UCST to above the
UCST due to the flat nature of the water-toluene phase equilibrium curve. Guang
et al.
[56] extended this numerical framework to ternary water-HC-HC systems of
water-toluene-decane and water-toluene-tetralin.
They demonstrated an interesting
fractionation effect (selective buildup of one of the hydrocarbons in the water phase)
when the water temperature is between the UCSTs of the two different water-HC
binary systems (UCST of water-toluene system is 308'C while UCST of water-decane
system is 359'C). Erriguible et al. [10] simulated jet breakup in a supercritical antisolvent process (SAS) where a is solution injected into CO 2 under conditions below
the mixture critical point (liquid and gaseous phases coexist) using a Volume of Fluid
(VoF) one-fluid method. They neglected the mass transfer between the two phases
across the interface and were able to correlate the jet breakup length to the Weber
number of the liquid phase. However, numerical studies of multiphase flow at nearcritical conditions in general 3-D geometries are not reported in literature. This thesis
focuses on the mixing of water and hydrocarbon under fully miscible conditions (above
the UCST). Development of a robust numerical method to study multiphase mixing
of near-critical fluids in generic 3-D geometries using sophisticated interface-tracking
methods (like VoF and level-set) coupled with mass and heat transfer across the phase
interface is the focus of ongoing and future work in this research project.
1.3
Thesis objectives and scope
The work described in this thesis is the first major step in our efforts to ultimately
simulate the reactive mixing process of supercritical water, multiple hydrocarbons and
organosulfur compounds in the SCWUDS of crude oil. The focus of the present work
was to study the mixing of supercritical water and a model hydrocarbon (chosen to be
n-decane in this study) in a 3-D cylindrical tee reactor geometry under fully miscible
32
conditions (cold hydrocarbon temperature above UCST of water-decane system) at
flow Re up to transition to turbulence. The following was accomplished under the
scope of this work:
" Development of a robust numerical code to perform 3-D near-critical fluid mixing simulations using existing open-source CFD libraries (OpenFOAM
[35])
with a consistent treatment of near-critical fluid thermodynamics and transport property variations (completely new additions to the libraries)
" Characterization of the mixing and flow structures in a cylindrical tee reactor
at intermediate Reynolds numbers under supercritical fully-miscible conditions
" Investigation of the impact of the temperature difference between the streams
on the flow dynamics and mixing behavior under fully miscible conditions
* Investigation of the impact of the Reynolds number on the flow dynamics and
mixing behavior under fully miscible conditions up to the transition to turbulence
The thesis is organized as follows.
Chapter 2 includes a detailed discussion of
real-fluid thermodynamics and transport property variations near the critical point.
This is followed by a detailed discussion of the problem formulation and numerical methodology employed including the governing equations, geometry, meshing,
boundary conditions, flow conditions and numerical methods in Chapter 3. Chapter
4 presents and discusses results of the simulation of water and n-decane mixing at
a water inlet Re of 500 under fully miscible conditions and discusses the flow features and mixing dynamics in a cylindrical tee reactor under laminar flow conditions.
The impact of the temperature difference between the streams on the stability of the
water-HC shear layer and consequently, the mixing dynamics is also discussed in detail in this chapter. Chapter 5 presents mixing results for different inlet Re values in
the range of 500-800 and evaluates the impact of Re on the flow and mixing dynamics.
The Re at which the transition to turbulence occurs is determined and the physical
mechanisms leading to the shear layer instability are explained. Finally, Chapter 6
33
summarizes the work done and discusses the future goals and plans for the research
project.
34
Chapter 2
Near-critical thermodynamics and
transport property variations
While simulating the flow and mixing of fluids, it is crucial to be able to model the
thermodynamic (density, specific heat, enthalpy) and transport (viscosity, thermal
conductivity, mass diffusivity) properties at the relevant temperature and pressure
conditions since they directly affect the different heat and mass transport mechanisms. The distinctive feature of flows of near-critical fluids is the peculiar variation
of these fundamental properties near the critical point as well as the non-linear interactions between the different components in a fluid mixture. Fig.2-1 clearly illustrates
the inability of simple equations of state like the ideal-gas EoS to capture these variations in thermodynamic fluid properties (density, specific heat).
Also, simplified
calculations of transport properties based on the kinetic theory of gases is inadequate. Hence, there is a need to understand the near-critical behavior of pure fluids
and their mixtures and select appropriate equations of state and transport models to
be used in the numerical simulations of the problem at hand. This chapter gives a
brief overview of popular methods of modeling near-critical fluid behavior and discusses the details of the specific equation of state and transport models selected for
our particular application.
35
ui)
d0
(q) piie (,,W164 ui) df M~ Md1
-,enba su--pjp
= d me vrp jLIN pue suoipipoad uoi4
:s~wqa~doad 3LutwuXPo'umu4 -104%?M :T~f~
ua@Ak~aq uosixedumoD
dj
000 k
006
OOL
009
(q)
(m) i
oov
009
...................
F--
OL
SH
BMA
BsE) 1espi ...
ISIN ..017
I
1
-1
ILA
d(e)
000
00L
009
006
009
0017
--P
00
00k'
anamonam'u
009
ISIN...
-
-14
41
S
N
-
g
008
000
1I
I0ON
w
2.1
Near-critical thermodynamics
As a fluid is pressurized at a given temperature, the distinction between the liquid
and vapor phases disappears at a particular pressure. This is the critical pressure
of the fluid. Similarly, the temperature above which there is no distinction between
the liquid and vapor phases is the critical temperature of the fluid. So if either the
pressure or the temperature is above the critical value, the fluid is said to be in a
supercritical state and does not undergo a phase transition. However, even at supercritical pressures the properties of the fluid change rapidly in the vicinity of the
critical temperature (rapid drop in density, huge spike in specific heat). Simplified
equations of state like the Ideal Gas EoS cannot capture these variations in thermodynamic properties giving rise to the need for state equations that model more
complex behavior of fluid molecules in order to predict the correct property variation
trends.
The earliest such equation of state that tried to predict the thermodynamic properties of a fluid over a wide range of temperatures and pressures was developed by
van der Waals (1873) [52]. He included the effects of intermolecular forces and the
physical volume occupied by the fluid molecules in developing a cubic equation of
state given by Eq.2.1.
P R T
V -b
a
V2
(2.1)
where P is the pressure, T is the temperature, V is the molar specific volume
and Ru is the universal gas constant (8.314 J/mol-K). a and b are constants for a
given substance with a representing intermolecular interactions and b representing
the physical volume excluded by molecules of the fluid (which are not negligible at
such high pressures). Though this equation correctly represents qualitative trends in
density, it is quantitatively inaccurate with the PVT predictions getting considerably
worse towards higher densities of the fluid.
Moreover, it does not yield accurate
estimates of derived thermodynamics properties like enthalpy and entropy.
Also,
since the EoS involves only two constants for a given substance (both of which are
37
determined to closely satisfy the stability criteria at the critical point), a universal
critical compressibility factor of Zc = 0.375 (see Eq. 2.2) is predicted for all fluids.
Ze =
(2.2)
where, Pc, T, and V, are the critical pressure, critical temperature and critical
specific molar volume respectively. In order to overcome these drawbacks of the van
der Waals EoS, many researchers have proposed complicated equations of state which
are typically formulated in a virial format with exponential terms for the high-density
regime. These include the state equations proposed by Benedict et al. [3], Starling
and Han [49], Martin and Stanford [25] and Lee and Kesler [22]. These equations of
state are cumbersome to use in a CFD calculation and work well only for a specific
set of compounds for which they were developed. A good critical review of these
methods can be found in the dissertation of Kutney [21]. Improved easy-to-use cubic
equations of state were later formulated by Redlich and Kwong [40] (Eq.2.3), Soave
[47] (Eq.2.4) and Peng and Robinson [37] (Eq.2.6).
Redlich-Kwong EoS (1949) [40]:
~R.T
V -b
P = RTa(2.3)
Redlich-Kwong-Soave
a
T'/ V(V + b)
2
EoS (1972) [47]:
~R.T
u
V-b
_
P =
a
V(V+b)
(2.4)
where, a is not a constant but a temperature dependent function (see Eq.2.5) with
the acentric factor (o) as a parameter. The acentric factor for a substance represents
how much its behavior deviates from that of a monoatomic gas.
a = aca(w, Tr)
(2.5)
where ac is the intermolecular force constant at the critical point and T, is the reduced
38
temperature (T/Tc).
Peng-Robinson EoS (1976) [37]:
~R.T
V -b
_a
P = RTa(2.6)
V(V +b)+b(V - b)
with a similar temperature dependence for a as in the RKS EoS. In this study, the
Peng-Robinson EoS was chosen for its simplicity and particular effectiveness in dealing with the thermodynamic behavior of hydrocarbons and water. Specifically, the
extension of the PR EoS to real-fluid mixtures is employed as is explained in detail
in the following section.
2.1.1
Peng-Robinson Equation of State for real fluid mixtures
The cubic Peng-Robinson EoS for real fluid mixtures used in conjunction with standard van der Waals mixing rules is given by:
P =
amn
-,
Z
(2.7)
V 2 +2bmV-b2
V-bm
X
,j(I
bm=
-
(2.8)
kij) /a,a_,
Xi b i
(2.9)
where, X, is the mole fraction of specie i and the parameters ai and bi for each species
in the mixture is calculated using its critical temperature (Tic), critical pressure (Pci)
and acentric factor
ai = 0.457236 R
(wi):
c
1 +
(0.3746 + 1.5423wi - 0.2699w2)
39
I -
(2.10)
substance M.(kg/kmol)
water
18.0
n-decane
142.285
Pe(MPa) Tc(K)
22.06
647
2.123
618.5
V(m/kmol)
0.0571
0.6031
w
0.344
0.484
Table 2.1: Molecular weight, critical properties and acentric factor for water and
n-decane [32]
(2.11)
b = 0 .0 77 7 9 6 1 RuTcj
Pci
The molecular weight (Mw), critical temperature (Tc), critical pressure (Pc), critical
volume (V) and acentric factor (w) for water and n-decane are given in table 2.1.
The molecular weights are required to determine the effective molecular weight of
the mixture, which in turn, is required to calculate the mixture density (p) from the
mixture specific volume (V) using p = Mw/V. In Eq.2.8, kij is a binary interaction
parameter for the species pair i and
j.
In this study, the binary interaction parameter used for the water-n-decane system
is a temperature dependent quadratic polynomial function as given in Eq.2.12 below,
applicable to the range of T from 700K to 1000K at P=25MPa, which encompass the
thermodynamic conditions investigated in this study.
k12 (T) = 1.14558E(-06)T 2 - 2.80487E(-03)T + 1.49148
(2.12)
The Predictive Peng-Robinson or PPR78 approach [14] with a group contribution
method to capture inter-molecular interactions between the components was employed to determine the T dependent function for the binary interaction parameter.
In the PPR78 approach, the same cubic PR EoS is employed with the same linear
mixing rule as in Eq.2.9 for the mixture excluded volume b,. However, the mixing
rule for the parameter am is formulated on a stronger physical basis as given below
in Eq.2.13:
amn = bm *
Xi
ia e
-e
E
(2.13)
where, gre is the residual part of the molar excess Gibbs energy when the pressure
40
goes to infinity and is given by Eq.2.14 below:
C
gsbbEg(T)
2
bm
(2.14)
where, the constant C in Eq.2.13 and Eq.2.14 above is given by:
C
j in (I+
vf2)
0.6232
(2.15)
The interaction parameters Eij (T) in Eq.2.14 are estimated using a group contribution
method (GCM) based on the works of Kehiaian et al. [16] and Abdoul et al. [1], the
equation for which is given below in Eq.2.16:
Ng
EN (T )
=
-
(BkI
Ng
(aik - cYjk) (a 1i -
9
cyjl) AkI (298.15T
(2.16)
R
k=1 1=1
where, Ng is the number of different functional groups defined by the method.
cGk is
the fraction of molecule i occupied by group k, that is:
occurrence of group k in molecule i
total number of groups in molecule i
The constants AkI = Alk and BkI = Bik were determined by Jaubert et al. in their
previous studies by minimizing the deviations between calculated and experimental vapor-liquid equilibrium (VLE) data for numerous binary systems.
The con-
stants Aki and BkI for the functional groups present in the molecules in this study
(-CH3 , -CH 2 andH20 are given (in MPa) in Table2.2. Also, Akk = Bkk= 0. The
capability of the PPR78 model in predicting mixture properties and phase equilibria accurately for systems/conditions outside the date-set used for fitting the above
parameters has been demonstrated extensively, for example in [14], [39]. Due to its
predictive capability, this model was selected for the water-HC system in the present
study for which fitting data under the T,P conditions of interest was not available in
the literature. The advantage of this formulation is that the binary interaction parameter (kij) of Eq.2.8 can be easily recast as a function of temperature using Eq.2.13
41
CH 3 (group 1)
CH 2 (group 2)
H 2 0 (group 16)
CH 2 (group 2)
CH 3 (group 1)
0
A 12 = 74.81
B 1 2 =165.7
A 1 _16 = 3557
BI_ 16 =11195
CH 2 (group 2)
0
H2 0 (group 16)
A 2 - 1 6 = 4324
B 2 - 1 6 =12126
0
-
Table 2.2: Group interaction parameters for the PPR78 model, Aki = Alk and Bk=
BIk (in MPa) [14][39]
to Eq.2.16 as below:
Z
k1(TZ
=1 (
k
--
Gjk )
o ) Aki
(Gil -
(298.15
A k1-1)
(
a (T)
_
_a
T)
kij (T)=
2
a (T) a (T)
bibj
(2.18)
The above equation was used to determine the quadratic variation of kij with T given
in Eq.2.12.
Fig.2-2 shows the variation of density for water and n-decane with temperature
at P = 25MPa (close to the operating pressure of SCWUDS reactor).
It compares
the density predictions of the PR EoS and the Ideal Gas EoS with experimental data
values obtained from NIST [32].
One can see that the PR EoS predicts the density
variation near the critical temperature and above very well while the Ideal Gas EoS
fails to do so. However, the error in density prediction by the PR EoS keeps increasing
towards the high-density (liquid-like) region. This can be corrected using the volume
translation approach of Mathias et al. [27] to correct for the molar specific volumes
predicted by the PR EoS as below:
0.41
Vorr = Vm + tm + fc
6-=
m
RUT
0.41+6,
V
fc = Vcm - (3.946bm + tm)
42
(2.19)
(2.20)
(2.21)
2
1200,
1
1
1
1
1000
800
-- PR EoS
--- NIST
* Ideal Gas EoS
600
CL
400
-
-
400
500
-
S
-
200
9L_
500
600
700
800
900
1000
900
1000
T (K)
(a) water
1600
1400'
1200-PR EoS
- -- NIST
Ideal Gas EoS
1000
0.
800
600
400
400
500
700
600
800
T (K)
(b) n-decane
Figure 2-2: Comparison of density (in kg/M 3 ) predictions by the Peng-Robinson EoS
and the Ideal Gas EoS with NIST data at P = 25MPa (a) water and (b) n-decane
43
substance
V(cm 3 /mol)
t(cm 3 /mol)
water
n-decane
57.1
603.1
-3.40
0.90
Table 2.3: Molar critical volume and volume translation parameter for water and
n-decane Mathias et al. [27]
where, the constant 0.41 was determined by Mathias et al. [27] by regressing data
for many substances and the volume translation parameter (tin) and molar critical
volume (Vcm) for the mixture can be determined using simple mixing rules given
below:
tm =
Xt
Vcm =ZXi
Vci
(2.22)
(2.23)
where, the critical molar volumes (Vcj) and volume translation parameter (ti) for
water and n-decane are given in Table 2.3. The volume translation parameter for
n-decane was not provided in the work of Mathias et al. [27]. The value for n-decane
was roughly tuned to obtain the best fit for predicted densities with NIST data in the
range of 300 K to 650 K at P = 25MPa. Fig.2-3 compares the density variation with
temperature (at P=25MPa) predicted by the PR EoS, the volume translated PR EoS
and the Ideal Gas EoS along with the experimental data values from NIST for both
water and n-decane. The improvement in liquid-like density predictions is evident
from these plots. The VT correction was, however, not employed in the current study
because in the temperature range under investigation (700K to 1000K) the densities
are predicted accurately without it.
Simulating the thermal transport processes
also requires the determination of derived thermodynamic properties like the specific
enthalpy of the mixture (h,), constant pressure specific heat of the mixture (Cp,m)
and partial molar enthalpies of the component species (hi). The functional form
of the mixture specific internal energy (Ur)
is first determined using the departure
function methodology and the rest of the properties can be derived from it as shown
44
1200
-rI
1000
800
-VT-PR EoS
--- PR EoS
--- NIST
Ideal Gas EoS
-
600
F
CL
400
200
-
C0'U
400
500
m
600
700
800
T (K)
900
1000
(a)
1600
1400>
1200 -- VT-PR EoS
-PR
EoS
--. NIST
Ideal Gas EoS
1000
M-
800
600
400
20
I0
400
500
700
600
T (K)
800
900
1000
(b)
Figure 2-3: Comparison of density predictions by the Volume-Translated PengRobinson EoS (VT-PR EoS), the simple Peng-Robinson EoS and the Ideal Gas EoS
with NIST data at P = 25 MPa (a) water and (b) n-decane
45
in Eq.2.24 to Eq.2.30:
where,
Um,IG
(2.24)
Um,IG - AU
Um =
is the specific internal energy of the mixture at the same temperature
but P = 0 or V = oc (where the mixture would behave as an ideal gas mixture) and
Au is the internal energy departure function which can be derived by integrating the
fundamental relation of thermodynamics for the system from V = Vn to V =oc and
using the Maxwell's Relations:
Vm
AU
The term
(O)
)-
[T
=
P] dV
(2.25)
in Eq. 2.25 above can be determined from the PR EoS and inte-
grating gives us the final form of the internal energy functional:
Urn =
U,IG +
- T
a,,
2,b
ln[V +(I-/')b1
da
dT
L
V + (1 + v
/V
) bm
(2.26)
The specific enthalpy functional is easily obtained from Eq. 2.26 keeping in mind
that h = u-+ PV:
hn = hIG + PV - RuT +
1
[(
am2 V-b r, [dT
dam
)
I In
v_
FV±+(1V- ) bn
V +
V + (1 + v)
bm
(2.27)
Differentiating the specific enthalpy with respect to T at a constant P yields the
specific heat capacity at constant pressure (Cp,m)
CPn
OhT
OT )P
IG
(1(+
In
C~r - RU +
-T(VRu
V - bm
as below:
2'-2 bmr
dam 1 )2[
dT V*
m
V + (I - -vF) bm
~
-RuT
(V
V 2 + 2bmV
]
d)2 arn
dT
dT 22
228
(2.28)
2am (V + b)
brn)2 ±*2
-
2
(2.29)
Differentiating the specific enthalpy with respect to the mole fraction of specie i gives
46
the functional form for the partial molar enthalpy of specie i:
Ohm
n
+
2 V2bn
Vm +
)
IG
4
) bm
(P -
Vm + (1 + ,f2) b
dxi )
K-
-P
I
T,P
+ am - T
J±
R
V/ - Vm/bmbi
__
V*
(dT ) ,i
dam)
2
- T d am
OXj OT
-(X ),P
(am
a - T dm
br
bi
( )
(2.30)
2
(2.31)
xj (I - kij)
where, Vi is the partial molar volume of species i given by Eq.2.32 below:
-1
d=]VT,P
RuT
(Vmb
OP
49V
RuT
2
(V
b)
2 am(Vm
- b)
V*2
b
aa3)
TX
(2.32)
(P )
OdVm T,X?
-RuT
(Vm - bm)2
2a, (Vm - bm)
+
V2
The ideal gas mixture properties at temperature T like C
and h
are obtained by
simple mole fraction weighted average of the ideal gas properties of the constituent
species (CfG and h G). These in turn, are calculated using the temperature dependent
polynomials for ideal gas Cp and h variations from Yaws Handbook [57]. Fig. 2-4
shows the variation of Cp with temperature (at P = 25MPa) for water and n-decane
predicted using the PR EoS and using the Yaws polynomials only along with the
experimental data values from NIST. It is seen that the cubic PR EoS satisfactorily
predicts the Cp values while the temperature dependent polynomials alone fail to do
so.
47
60r-
6Q~
I
I
50
U
U
U
S
S
U
U
U
U
U
40
MU
mu
U.
Eu
-PR EoS
--- NIST
Ideal Gas, Yaws HB
30
20
U
a
U
U
U
10
4.
U
UuuuumaUSI~~~*
tw ll1
300
400
500
TTF Tn
M 1,w
...
..
if...
800
700
600
900
F17TrTTrr 177
1
)00
T (K)
(a) water
3.5b
3
-PR
c
EoS
--- NIST
11A11deal Gas, Yaws HB-
2.5
2
1
10
I
I
400
500
I
I
I
i
600
700
800
900
1000
T (K)
(b) n-decane
Figure 2-4: Comparison of constant pressure specific heat (Cp in kJ/kg - K) predictions by the Peng-Robinson EoS and the Ideal Gas EoS with NIST data at P = 25
MPa (a) water and (b) n-decane
48
substance
W
H(Debye)
K
water
n-decane
0.344
0.484
1.8
0.07
0.1
0.0
Table 2.4: Acentric factor, dipole moment and association factor for water and ndecane [32] [8]
2.2
2.2.1
Near-critical transport properties
Viscosity and Thermal Conductivity
The calculation of viscosities and thermal conductivities was done using the correlations developed by Chung et al. [6]. They extended the Chapman-Enskog theory
for dilute gas viscosities [4] by correcting for high pressure dense conditions using
functions involving the acentric factor (w), reduced dipole moment (H.) and association factor (K) of the substance as parameters. The coefficients of these functions
were determined by them by fitting experimental data for a vast number of pure substances. The w, H and K values for water and n-decane are given in Table 2.4. The
fluid viscosity (p) is calculated as follows:
(2.34)
+ Pp
Y = Ik
where the viscosities are in units of Poise
Pk =o
-
36.344 x 10-6 (MT) /
-o =
[G2
7y
(2.35)
+ A6Y
2
A8 + T*+
G 2 exp
4.0785 x 10-5 (MTc)/
2
Fc
T2
(2.36)
(2.37)
V2/39
where, ,uo is the dilute gas viscosity in Poise according to the Chapman-Enskog theory,
M is the molecular weight of the substance in g/mol and V is the molar critical volume
of the substance in cm 3 /mol. Q* is the reduced collision integral given by Eq.2.38
49
and F, is a linear function in w, H' and K given by Eq.2.40.
*=
1 16156
T*B
+
C
exp (DT*)
+
E
+ GT*B sin (ST*
exp (FT*)
- H)
(2.38)
where, A = 1.16145, B = 0.14874, C = 0.52487, D = 0.77320, E = 2.16178, F
2.43787, G = -6.435 x 10-4, H = 7.27371, S = 18.0323 and W = -0.76830.
T* = 1.2593Tc
(2.39)
F, = 1 - 0.2756w + 0.059035rl + s
(2.40)
Y = pVe/6
(2.41)
1 - 0.5Y
3
(1 - Y)
G2 = [A 1
(2.42)
1 - exp (-A 4Y)
1
+ A 2 G1 exp (A 5 Y) + A 3G1I
Y
A1 A 4 + A 2 + A 3
where, A1 to Aio are linear functions in w,
14
and
Ai = ao(i) + a(i)Ow + a 2 (i).
i
(2.43)
given by:
+ a3 (i)
(2.44)
The polynomial coefficients ao(i), a1(i), a 2 (i) and a3 (i) are given in Table 2.5. The reduced dipole moment (H,) can be calculated from the dipole moment of the substance
using Eq. 2.45 below:
Hr = 131.3
(2.45)
(veTC)
where, the dipole moment (1)
1/
is in units of Debye, the critical temperature (Tc) is
in K and the molar critical volume is in cm 3 /mol. A similar procedure is used to
50
i
1
2
3
4
5
6
ao(i)
6.32402
0.12102 x 10-2
5.28346
6.62263
19.74540
-1.89992
a,(i)
50.41190
-0.11536 x 10-2
254.20900
38.09570
7.63034
-12.53670
a 2 (i)
-51.68010
-0.62571 x 10-2
-168.48100
-8.46414
-14.35440
4.98529
a 3 (i)
1189.02000
0.37283 x 10-1
3898.27000
31.41780
31.52670
-18.15070
7
8
24.27450
0.79716
3.44945
1.11764
-11.29130
0.12348 x 10-1
69.34660
-4.11661
9
-0.23816
0.67695 x 10-1
-0.81630
4.02528
10
0.68629 x 10-1
0.34793
0.59256
-0.72663
Table 2.5: Function coefficients used for generalized viscosity correlations from Chung
et al. [6]
determine the thermal conductivity (A) of near-critical fluids as follows:
(2.46)
A = Ak + AP
where the thermal conductivities are in units of cal/(cmsK)
I
IH2
AP = 3.039 x
10-4
(2.47)
+ B6Y
A0
Ak
(TcIM)112 B7 y2H2 (TITc) 1/2
(2.48)
V2/3
C
Ao = 7.452 (po/M) {P
0.215 + 0.2829a 1 - 1.061C2 + 0.2667a 3
0.6366 + OZ2 3 + 1.061aiG 2
Cv
3
2
c1R=
(2.49)
(2.50)
(2.51)
where, C, is the ideal gas heat capacity at constant volume.
OZ2=
2
0.7862 - 0.7109w + 1.3168w
51
(2.52)
3=
H2[= B1
H - exp (-B 4Y)
(2.53)
2.0 + 10.5 (T)
TC
+ B 2 G1 exp (B5Y) + B3G1
1(2.54)
4
2
3
(.B 4B
)B
where, B 1 to B 7 are linear functions in w, H and , given by:
Bi = bo(i) + bi(i)w + b2 (i)r4 + b3 (i)
(2.55)
The polynomial coefficients bo(i), bi(i), b2 (i) and b3 (i) are given in Table 2.6. The
viscosity and thermal conductivity of the water-n-decane mixture is calculated by
a simple mass-fraction weighted average of the properties of the individual species
at the particular temperature and pressure. This methodology to determine transport properties of a real-fluid mixture has also been employed before in the work of
Narayanan et al. [31]. Chung et al. [6] in their work, do propose some complicated
mixing rules to determine the effective properties of the mixture like Vcm,
Mm, nm
and
Km
Tcm, Wn,
which can then be used in Eq.2.34 to Eq.2.54 above to calculate the
viscosity and thermal conductivity of the mixture. However, the application of these
mixing rules yields unphysical negative values for the viscosity and thermal conductivity at certain values of the mixture mass-fractions. This causes artificial numerical
instabilities in the simulation. Hence, the current study employs simple mass-fraction
weighting for the calculation of mixture viscosity and thermal conductivity. Fig.2-5
shows the variation of viscosity with temperature (at P=25MPa) for water and ndecane. The predictions match closely with NIST data with small inaccuracies near
room temperature conditions.
2.2.2
Mass Diffusivity
The mass diffusivity for the water-n-decane binary mixture is calculated using the
Tracer Liu-Silva-Macedo (TLSM) model formulated by Liu et al. [23] [24] [46] as sug52
10-
1X
0.9F
0.8
0.7
0.6
60.51
-Chung
--- NIST
0.4
correlations
0.3
0.2
0.1
I
I
IN
600
500
400
9,00
700
800
-Chung
-- NIST
correlations
700
800
T (K)
900
1000
900
1000
(a) water
1.2 x
13
I
IIII
I
1
0.8f
0.6
0.4
4-
0.2
"I
NOO
400
500
600
T (K)
(b) n-decane
Figure 2-5: Comparison of dynamic viscosity (pu in Pa- s) predictions using Chung's
generalized correlations [6] with NIST data at P=25MPa (a) water and (b) n-decane
53
1
2
3
4
5
6
7
ibo (i)
2.41657
-0.50924
6.61069
14.54250
0.79274
-5.86340
81.17100
bi (i)
0.74824
-1.50936
5.62073
-8.91387
0.82019
12.80050
114.15800
b2()
b3 (i)
-0.91858
-49.99120
64.75990
-5.63794
-0.69369
9.58926
-60.84100
121.72100
69.98340
27.03890
74.34350
6.31734
-65.52920
466.77500
Table 2.6: Function coefficients used for generalized thermal conductivity correlations
from Chung et al.[6]
gested by Kutney in his comprehensive review of near-critical fluid mass diffusivities
[21]. The model is based on the hard-sphere diffusivity theory. The diffusivity of
species 1 in species 2 (D 12 ) is given by Eq. 2.56 below:
DTLSM
12
669.1M12
RUT
pNAT2LSM
M12
TLSM
0.75pNAoTLSM
(
exp
1.2588M
12
- pNAcTLSM
-
0.2786212
kT
(2.56)
2
(2.57)
UTLSM
kTe
( 1 + 1.2
TLSM
6 X1
612
M
12
X2
1_62
k
(7 12
1/3
)SM
X1
1
k
(2.58)
+ X 2 0' 2
(2.59)
(2.60)
= X1 M 1 + X 2 M 2
where, NA =6.023 x 1023 /mol is the Avogadro constant and k = 1.3806 x 10-
23
m 2 kg/s
2
is the Boltzmann constant. In Eq. 2.56, D 12 is in units of cm 2 /s, p is in units of
gm/cm 3, M is in units of gm/mol, o- is in units of Aand 6/k is in units of K. The
Lennard-Jones force constants for water and n-decane o and 6/k were taken from Liu
et al. (1998) [24 and are shown in Table 2.7. They may also be calculated from the
54
substance
o-(A)
e/k(K)
water
n-decane
1.53091
6.71395
3788.51
434.86
Table 2.7: Lennard-Jones force constants for water and n-decane Liu et al. (1998)
[24]
critical properties of the substance using Eq. 2.61 and Eq. 2.62 below.
G, = 0.809VJ/ 3
where, o- is in
A
and V is in cm 3 /mol
Ek
2.3
(2.61)
T -i
1.2593
(2.62)
Summary
The ability to capture the variations of thermodynamic and transport properties
with temperature and pressure is crucial while simulating the heat and mass transfer
processes of near-critical fluids. In this chapter, the details of the consistent treatment
of near-critical thermodynamics of real fluid mixtures using the Peng-Robinson cubic
equation of state were discussed. The use of the PPR78 approach with a group
contribution method for inter-molecular interactions for the purpose of determining
the thermodynamic parameters of the non-ideal mixture was also discussed.
The
calculation of viscosities and thermal conductivities are done using the generalized
correlations of Chung et al. [6]. Mass diffusivity of one component in the other in the
water-n-decane binary mixture is calculated based on the Tracer Liu-Silva-Macedo
(TLSM) model developed by Liu et al. [23] [24] [46].
55
56
Chapter 3
Problem Formulation and
Methodology
3.1
Governing equations
The conservation equations used to simulate the mixing of supercritical water and
hydrocarbon (here, n-decane) are given below in Eq.3.1 to Eq.3.6, in the most general
form in order to accommodate variable density, compressibility and variable transport
property effects. Since the equations are solved numerically using a finite-volume (FV)
formulation, they are expressed in the strong conservative form. The assumptions
employed in each of the equations are also explained below.
Continuity Equation:
Ot
+ O
OXj
= 0
(3.1)
where, p is the bulk mixture density and u = ui is the velocity vector. The Newton's
indicial notation has been used in all the equations.
Momentum Equation (Navier-Stokes):
DPUJUi
&pUi
at+
a
wt
where, P is the pressure field,
OXj
T
=
_
=
P
O
OX
+
&Ti
a~3
oxr
+ pyi
(3.2)
rij is the stress tensor and g = gi is the gravitational
57
acceleration vector. Here, Stokes' formulation of the stress tensor in terms of the
symmetric part of the velocity gradient tensor (the strain tensor Sij) is used to close
the stress terms in the momentum conservation equation (Eq. 3.2) and is given by:
2ij
= 2p
. Sioii
S
3
u ± &uj\
+
Oxj
Oxj
(a,
=)
9u.
2
,
-2
3
Oxi
(3.3)
where, p is the viscosity of the mixture (calculated using the methodology outlined
in Sec.2.2.1) and 6ij is the Kronecker delta tensor. Stokes' assumption of zero bulk
viscosity is inherent in the above formulation of the viscous stress tensor.
Species Transport Equation:
aPYk+
aPU=Yk
at
Ox
where, the subscript k denotes specie k,
a
Oxa
Yk
pD_Dk
ax}
(3.4)
is the mass fraction of specie k in the
mixture and Dk is the mass diffusivity of specie k in the bulk mixture. In general, for
a mixture with n species, one needs to solve (n - 1) species conservation equations like
Eq. 3.4 above and the mass fraction of the n 1h specie is obtained from the constraint
Ek Yk
1. Since this study involves only binary mixtures of water and n-decane,
only 1 species transport equation is solved. In Eq. 3.4, the species diffusive flux vector
has been modeled using the Fick's Law of Diffusion and Soret and Dufour diffusive
effects are neglected. The mass diffusivity of specie k in the bulk mixture (Dk) can
be obtained from its binary diffusivities in each of the other components (DkI) using
Eq. 3.5 below:
Dk =
Xk
(3.5)
where, Xk is the mole fraction of specie k in the mixture and the binary diffusivities
(Dkl) are determined using the TLSM model as explained in Sec.2.2.2. For a binary
mixture, the diffusion coefficient of both species in the bulk mixture simply reduces
to their binary diffusion co-efficient.
58
Enthalpy Transport Equation:
Oph
at
+
Opuh
Oxj
_
=
DP + a
+
Dt axj
A
IA
OT
Oxj
a (Ou
1Yk
±pV
+(
OXj
OXk
B
71
OX
(3.6)
D
where, h is the specific enthalpy of the mixture, T is the temperature, A is the mixture
thermal conductivity and hk is the partial enthalpy of specie k. Term B in the energy
equation is the heat diffusion term modeled using the FourierLaw of Heat Diffusion.
Term A represents the work done by the pressure forces on the fluid. The magnitude
of this pressure work term with respect to the dominant enthalpy advection term
in the energy equation can be estimated using the non-dimensional Eckert number
defined below:
Ec = /2
Ah
(3.7)
where, dh is a characteristic enthalpy difference, which could be taken to be the
enthalpy difference between the water and hydrocarbon streams.
At the low flow
Mach numbers encountered in this study, the Eckert number ~ 10-7 and so, the
pressure work term may be safely neglected. Using a similar scaling argument, we
can also neglect the viscous dissipation term in the energy equation (term D) since
the Brinkmann number (see Eq. 3.8) Br ~ 10-7.
,a2
Br =AAT
(3.8)
where, AT is a characteristic temperature difference in the problem (the temperature
difference between the water and hydrocarbon streams). Term C is the transport of
enthalpy due to species diffusion. Since the water and hydrocarbon have markedly
different specific enthalpies at these conditions, this transport mechanism may be
significant and hence must be considered to make accurate thermal transport predictions. Since the enthalpy of a real-fluid mixture depends on the temperature, pressure
and species mass fractions, the correct formulation of the enthalpy differential must
59
include all of these contributions as shown in the equation below:
dh=CpdT+
v-T (V
dP + EhdY
(3.9)
The enthalpy change due to pressure variation in Eq. 3.9 above can be neglected
since the Eckert number is very small. Recasting the temperature gradient in the
heat diffusion term in Eq.3.6 using Eq.3.9, the enthalpy transport equation can be
reformulated including only the significant terms as:
Modified Enthalpy Transport Equation:
Oph
ata+
Opuh
Oxj
_
&
(A Oh)
E&
[(
A )
OYkl
aOxj KCP Oxj
a)+ kOXj
ax hkPDkc)
.J(3.10)
CP
OXj
The above formulation of the enthalpy equation is contains an implicit treatment of
both the enthalpy advection and heat diffusion terms which has numerical advantages
as explained in Section 3.4. The system of equations is closed using the equation of
state for real-fluid mixtures to determine p and derived thermodynamic properties
like Cp, h as functions of P, T and the mixture composition. Since, the enthalpy
field is calculated by solving the enthalpy transport equation, an iterative procedure
to determine the T field from the enthalpy, pressure and composition fields has to
be employed because of the non-linearity of the enthalpy constitutive relationship for
near-critical fluid mixtures. Typically, 3-4 iterations are sufficient to determine the
temperature with acceptable accuracy (error ~ 10-6).
The viscosity and thermal
conductivity are calculated using the procedure explained in Sec.2.2.1.
3.2
Geometry and Boundary Conditions
The mixer geometry is a cylindrical tee as shown in Fig. 3-1. The geometry corresponds to a small-scale test reactor with cylindrical tube inner diameter (d) of 2.4
mm. The two inlet branches of the tee meet at an angle of 90' with a long downstream section of length 22D. The inlet sections are each of length 3D from the inlet
60
Inlet I
Hot SC waterN-
Parabolic inlet
Inlet 2
Colder SC n-decane
Parabolic inlet
velocity profile
1g
D
3D
e
4ME1
velocity profile
3D
22D
Walls
Adiabatic
No-slip
Outit
Zerb gradient
iriposed for U, T
nd mass fraction
/fields
Fixed value for
back-pressure
Figure 3-1: Cylindrical tee mixer geometry and boundary conditions
to the center of the mixing joint. Hot supercritical water enters from the horizontal
inlet while the colder hydrocarbon (n-decane) stream enters from the vertical inlet.
Since the Reynolds number at both inlets is in the laminar regime for all of the
simulated cases, a parabolic inlet profile velocity boundary conditions is imposed at
both the water and hydrocarbon inlet with specified average bulk velocities for each.
No slip boundary condition is applied at all walls. At the outlet, a zero gradient BC
with flux correction is applied (to ensure that the total mass flux entering the reactor
is same as the total mass flux exiting). A fixed value temperature BC is applied at
both inlets with values corresponding to the specific inlet conditions of the simulation.
All the walls are adiabatic and the outlet is far enough downstream to assume that
the temperature gradient at the outlet is zero. A zero gradient (normal to the face)
boundary condition is imposed for the pressure at both inlets as also on all walls.
This boundary condition is not an intuitive one but can be easily derived from the
momentum equation (in the boundary normal direction) considering that all other
terms are zero. At the outlet, a fixed back-pressure of 25 MPa (the nominal reactor
operating pressure) is specified. Since the flow Mach number is low, one does not
have to worry about pressure waves being reflected back into the domain from the
61
~i ;~ii~;
4;
Figure 3-2: Cylindrical tee reactor mesh
outlet face.
3.3
Meshing
The computational domain is meshed using the meshing software GAMBIT by ANSYS Inc. (see Fig. 3-2). The entire domain has to be smartly decomposed into many
blocks and each block meshed separately so as to obtain the best quality mesh possible
(minimize non-orthogonality of mesh elements). All mesh elements are hexahedra.
The adequacy of the spatial resolution is confirmed a posterioriby grid independence
checks as shown in Sec.4.1 and Sec.5.1.
3.4
Numerical methodology
The conservation equations in Section 3.1 are discretized using the finite-volume
methodology (FVM). The computational code was developed using the set of opensource CFD libraries, OpenFOAM by OpenCFD Ltd.[35]. The underlying use of the
mid-point rule to approximate the integrals of fluxes over the control volume surfaces
62
and the integrals of quantities over cell volumes limits the order of accuracy of the
numerical method to 2nd order at best. The details of the numerical methodology
employed are presented in the following sub-sections 3.4.1 to 3.4.5.
3.4.1
A
2n
Discretization Schemes
order accurate central difference scheme with a correction for mesh non-
orthogonality is used for the gradient of velocity, temperature, pressure and mass
fractions at the face centers. A simple bi-linear interpolation is used to determine
the mass flux per unit area at the face centers. The velocity, temperature and mass
fractions are interpolated to the cell face centers using a
2 nd
order accurate linear inter-
polation scheme with a limiter, which adds upwind scheme based numerical diffusion
in regions of steep gradients to smooth out unphysical oscillations. The transport
properties and pressure are interpolated to the cell face centers using a 2 nd order accurate linear interpolation scheme. Time marching is done using a
2 nd
order accurate
BDF2 (Backward difference formula) semi-implicit scheme.
3.4.2
Discretized Equations
The discretized forms of the continuity, momentum and enthalpy equations are shown
in Eq. 3.11, Eq. 3.12 and Eq. 3.13 respectively.
Discretized form of the Continuity Equation:
3 (3pn+ 1
-
4 pn +
p"-
)+
1
Af pf (Uif)
n
= 0
(3.11)
where, the superscripts n+1, n, n-i denote the new time-step level, old time-step level
and old old time-step level respectively.
Discretized form of the Momentum Equations:
[Ai
(u
[
+1]
_ Apn+1 + [Sin]
]4±1]
Hj [
63
(3.12)
[Ui+l] is the ui solution vector of length N (where N is the total number of grid
cells). Three sets of N equations like the one shown above in Eq. 3.12 need to be
solved; one for each velocity component.
[A ] is
the diagonal part of the co-efficient
matrix of the ui equation and includes contributions of all terms involving un+; the
subscript m referring to the cell number m.
['H'] is the off-diagonal part of the
co-efficient matrix including contributions from all terms involving uz+; where the
subscript nb refers to all neighbor cells of cell m. The co-efficients in both these
NXN matrices are calculated using variable values from the previous time-step level;
hence the superscript n. Aj is the discrete gradient operator. [Si] is the equation
source vector (RHS vector) and includes contributions of all terms involving Um
and
isnb
(the explicitly treated terms) as well as the source terms of the momentum
equation (like gravity force term). The advective transport terms in the momentum
conservation equations are non-linear and are linearized while writing the discrete
form of the equations by treating the mass flux terms explicitly using their values
from the nth time-step level. Therefore, the discretized equations above are not fully
implicit in ui and hence, the time-stepping scheme has been referred to as semiimplicit.
Discretized form of the Enthalpy Equation:
[B] [hn+1] = [gn] [hn+1 _ [Qn]
(3.13)
where, the matrices [B"] and [gn] are analogous to [An] and [WN] in the discretized
momentum equations.
[Qn] is the enthalpy equation source vector and includes the
heat transport due to species diffusion and the part of the temporal derivative of
enthalpy involving previous time-step/iteration values. All the advective and diffusive
terms are treated implicitly in the h equation and their contributions appear in
and [g7].
[B7]
The discrete species conservation equation also has a similar form as Eq.
3.13:
64
Discretized form of the Species Conservation Equation:
[C"] [yn+l]
n [yn+l]
-
[7Zn]
(3.14)
It is clear from Eq. 3.12 that the pressure and velocity fields at the new time-step
level are coupled. In order to decouple the equations, an operator-splitting approach
is adopted based on the Pressure Implicit Splitting of Operators (PISO) algorithm
proposed by Issa [13]. In this approach, an intermediate velocity field u* is predicted
by solving the set of discretized momentum equations by omitting the contribution
of the pressure term. This predicted velocity field in general, will not satisfy mass
conservation.
Therefore, the new time-step pressure field pn+1 is calculated such
that use of this field to correct the velocities will cause both mass and momentum
conservation equations to be satisfied. A discrete equation to determine the correct
pressure field can be formulated by interpolating the velocities in Eq. 3.12 to the face
centers and substituting these in the discrete continuity equation (Eq. 3.11) giving
Eq. 3.15 below:
Discrete Pressure Equation:
]+(
[pn+i (5)
(3pf+14pfp--)+ZAj
f
1
Af pK+ [A f]l
f
=Pnl
If
(3.15)
where,
[An]
1 is the inverse of the diagonal matrix
the face center of face
f
[An] in Eq. 3.12 interpolated to
This technique of interpolating the pressure correction term
to the face centers is called the Rhie-Chow interpolationproposed by Rhie et al. [41]
and prevents odd-even decoupling of the pressure and velocity fields on collocated
grids.
3.4.3
Solution Algorithm
The solution algorithm is constructed using the above uiscrete momentum, species,
temperature and pressure equations and is explained below in a step-wise fashion.
SteD 1 (Velocity Predictor):
65
Predict the velocity-field by solving the discrete momentum equation by omitting
the pressure contribution and using mass flux values (pj (ujf)'f) from the previous
time-step (n):
[l = ([Af] + [7n])- ([Si])
(3.16)
Step 2 (Species Mass Fractions Predictor):
Predict the species mass fraction fields Y* by solving the discrete species conservation
equations using mass flux values (pf
(Uif)nf)
from the previous time-step (n):
[Y*] = ([C] + [ n]) 1 ([R"])
(3.17)
Step 3 (Enthalpy Predictor):
Predict the enthalpy field h* by solving the discrete enthalpy equation using mass
flux values (pf (uif)nf) from the previous time-step (n) and predicted species mass
fractions (Y*):
[h*] = ([B"] + [G"l])-1 ([Q"])
(3.18)
Step 4 (Temperature, Density and Thermodynamic Properties Correction):
Update the temperature field to T* using the enthalpy field calculated in Step 3 using
an iterative procedure. Correct the density field and derived thermodynamic proper-
ties (like Cp) using the EoS and Pn, T* and Y*.
Step 5 (Pressure and Velocity Correction):
Solve the discrete pressure equation to obtain the corrected estimate of the pressure
field (P*):
(3p* -
4
p" + pf-)+E
A
[p* (u*,f) ]+
A
p* [A "]
= 0
(3.19)
Here, the estimate of the new density field p* is used instead of the actual new density
field pn+1. This is the variable density, low Mach number formulation which yields
sufficient accuracy and numerical stability for the low Mach number flows simulated
in this work. For higher Mach number (highly compressible) flows, this term can be
66
easily corrected by accounting for the density change with pressure as below:
pfn+1 =p*
± V (P*
_ pn)
(
(3.20)
OP)T,Yk
This correction to the variable density term though small at low Mach numbers, im-
proves the numerical stability of the pressure equation and hence, has been employed.
In Eq. 3.19, the pressure gradient at the cell face centers has to be corrected for the
non-orthogonality of the cells. This is done using an explicit correction term in the
numerical scheme for the gradient (deferred correction approach). As such, Eq. 3.19
needs to be solved iteratively (typically only 2-3 times) in order to get an accurate
estimate of the pressure field.
The velocity field is corrected using P* to give u**.
Since ui at cell m depends on
the ui values at neighboring cells, the corrected velocity field now does not satisfy
the momentum equation. Hence, a second corrector step for pressure and velocity
must be executed to obtain p** and u***. Issa [13] showed in his work that 2 corrector
steps are sufficient to ensure
formulation itself is only
2
2 ,d
order accuracy for the velocity field. Since, the FVM
nd order accurate at best, further corrector steps will yield
no benefit.
Step 6 (Species Mass Fractions Corrector):
Correct the species mass fraction fields to obtain the new time-step value Yn+1
by solving the discrete species conservation equations corrected mass flux values
(* (U**f*
)
[Yi+1]=
([C***] + [fT***])l
([7***])
(3.21)
Step 7 (Enthalpy Corrector):
Correct the enthalpy field to obtain the new time-step value hn+1 by solving the
discrete enthalpy equation using corrected mass flux values (p* (u *f)rf)
and new
time-step species mass fractions (Ykn+)
[hn+1] = ([B***] + [g***]) 1 ([Q***])
67
(3.22)
Step 8 (Second Temperature, Density and Thermodynamic Properties Correction):
Obtain the new time-step values of the temperature field, density field and derived
thermodynamic properties using the EoS and P**, hn+1 and Yn+1 .
Step 9:
Repeat Step 5 to obtain the new time-step values of the pressure and velocity fields
(pn+1
and u+1).
Step 10 (Transport Properties Correction):
Correct transport properties using
3.4.4
Ykn+l,
Tjn+ and Pjn+.
Numerical Stability
The implicit formulation of the species, enthalpy and pressure equations ensures unconditional numerical stability for these equations. If the diffusion term in the enthalpy equation were treated explicitly, a highly restrictive stability condition on the
time-step size would have had to be followed:
6t < (6X) 2
2a
(3.23)
where a is the thermal diffusivity. This means that if the mesh size is refined by a
factor of 2, the time-step has to be decreased by a factor of 4. This is the reason why
the enthalpy equation was formulated in an implicit fashion as in Eq.3.10 .
As explained previously, the momentum equations are actually semi-implicit with
the mass-flux terms taken from the previous time-step and hence the time-step must
satisfy the Courant-Lewy-Friedrichs(CFL) stability condition:
~t<
6_ <t
1
(3.24)
The time-step size is adjusted dynamically at each time-step to ensure that the above
condition is satisfied at every cell in the domain.
68
3.4.5
Linear System Solvers
Solution of each of the discretized equations involves solving a large linear system of
equations in the respective variable. Appropriate choice of the linear system solver
for each equation set is crucial for computational speed. For the momentum, species
and enthalpy equations, a Preconditioned Bi-Conjugate Gradient (PBiCG) method
with a diagonal-incomplete LU preconditioner for the matrix is used. For the pressure
equation, a Conjugate Gradient (CG) method with a diagonal-incomplete Cholesky
preconditioner is used. Since the size of the equation set is very large (N ~ lmillion),
the equation set is distributed to 48 processor cores (on the Pharos computing cluster)
using the domain decomposition method and solved in parallel with OpenMPI interprocessor communication.
3.5
Summary
In this chapter, the fundamental conservation equations governing the mixing of supercritical water and hydrocarbons are presented and the various simplifying assumptions made are discussed (Sec.3.1).
In Sec.3.2 the cylindrical tee-reactor geometry
is described and the boundary conditions imposed are discussed. Sec.3.3 focuses on
computational mesh generation. This is followed by a detailed discussion of the finitevolume methodology, discretization schemes, discrete forms of conservation equations,
the solution algorithm, linear system solvers and numerical stability in Sec.3.4.
69
70
Chapter 4
Mixing at intermediate Re: Flow
dynamics and impact of
temperature difference
In this chapter, the mixing of supercritical water and n-decane was simulated at a
Reynolds number of 500 at the water inlet for two different inlet temperature conditions as shown in Table 4.1. In both cases, the mass flow rates through the water and
n-decane inlets is the same (a design condition found to be favorable to the process
from the chemical kinetics point of view). The nominal pressure in the tee mixer
was ~ 25MPa (ensured by fixing the back-pressure at the outlet) and the n-decane
inlet temperature was 700 K, which is sufficiently above the Upper Critical Solution
Temperature (UCST) of the water n-decane system
(~ 632 K)
thereby ensuring com-
plete miscibility of the two components. The water inlet temperature in Case I was
800 K and that in Case II was 1000 K. This will help us examine the impact of the
different degrees of variation of properties like density, viscosity with temperature
on the flow and mixing dynamics in a cylindrical tee mixer. In order to understand
the differential impact that the near-critical conditions have on the flow field and the
dynamics of mixing, we also perform a set of simulations (Case II and Case IV) with
the same inlet conditions as in Case I and Case II, but with the physical properties
of the pure fluid components held constant at their inlet values. The thermal trans71
Case
I
II
III
IV
R
500
500
500
500
Redi
rhw,in
204
239
204
239
(kg/hr)
0.11
0.13
0.11
0.13
(kg/hr)
0.11
0.13
0.11
0.13
uw,in
Uw
Udin
T.,in
Td,in
(m/s)
0.0832
0.1416
0.0832
0.1416
(m/s)
0.0144
0.0168
0.0144
0.0168
(K)
800
1000
-
(K)
700
700
-
-
-
Property
variations
yes
yes
no
no
Table 4.1: Inlet flow conditions for the cases discussed in this chapter
port is not solved for in these cases. The effect of gravity forces is not included in
these simulations so as to allow us to study the impact of geometry, Reynolds number
and property variation effects on the flow field and mixing dynamics. The impact of
gravity forces, which could play a significant role considering that the Froude number,
Fr - gD
1, will be the focus of a future study.
The physical properties of the mixture like the density and viscosity can be expected to have a significant impact on the dynamics of the flow. Therefore, it is
important to note at this stage, the variation of the mixture density and viscosity
with both temperature and composition. Fig.5-17 and Fig.5-18 show the variation
of the mixture density (p in kg/m 3 ) and dynamic viscosity (p in Pa - s) with temperature for the range of temperatures pertinent to this study, for different mixture
compositions from pure water to pure n-decane. Table 4.2 shows the p and [t for
the water and n-decane pure components at the extreme temperatures within the tee
mixer for the cases I and II. It can be seen that the mixture density and viscosity
are strong functions of the temperature under these conditions. The density of the
water component increases by around 100% as it gets cooled from 1000K to 700K
within the tee mixer domain in Case II while the density of the n-decane component
decreases by around 30% as it gets heated from 700K to 1000K. The viscosity of
water does not change appreciably in this temperature range. However, the viscosity
of the n-decane component decreases by more than 100% as it gets heated from 700K
to 1000K. These variations in p and p influence the local Reynolds number which in
turn can be expected to have a significant bearing on the local flow dynamics and
stability. These variations in fluid physical properties within the flow domain will be
much smaller in Case I with only a 100K temperature range. The comparison of these
72
500
-- Yd=O -0-Yd=0.2
-+-Yd=0.4 -- Yd=
0.5
Yd=0.6 -Yd=.8
-.- Yd
1
450
400350-
tw
'250
Yd
*
C.
200
100-
00
750
850
800
900
950
1000
T (K)
Figure 4-1: p (in kg/m 3 ) v/s T (in K) at P = 25 MPa for different mixture compositions of water and n-decane from Yd
=
0 (pure water) to Yd
1 (pure n-decane)
two cases (Case I and Case II) will help us evaluate the impact of varying degrees of
property changes on the flow dynamics and mixing in the tee mixer.
4.1
Validation: Grid convergence tests
In order to verify the adequacy of the chosen spatial resolution and validate the
numerics of the code, a grid independence study was performed for both Case I and
Case II. Fig.4-3 shows the streamwise velocity profile along the vertical direction in
Case I at three different axial locations (x=4D, x=6D and x=8D) for three different
mesh resolutions (0.10 mm, 0.08 mm and 0.06 mm). A mesh resolution of 0.10 mm
seems to be sufficient to obtain grid-independence. Fig.4-4 shows the mean streamwise
velocity profile along the vertical direction in Case II at three different axial locations
(x=4D, x=6D and x=8D) for three different mesh resolutions (0.10 mm, 0.08 mm and
0.06 mm). The averages are taken over 4 tflow-through time periods after the initial
transient of 2
tflow-through
time periods.
Grid-independence is attained for a mesh
73
X 10
-a-Yd= 0
-- Yd = 0.2
-Yd
= 0.4
--- Yd = 0.5
8 -Xq
U.
7
~
Yd = 0.6
-- Yd =
'3,..
0.8
Yd = 1
.3,.
0-6
'3.
.3
-. 3..
'3...
"3,.
3
'3.
'3.,
3
'3'
~
3
'3.3
~
3.3
4
ik
750
800
850
900
950
1000
T (K)
Figure 4-2: y (in Pa-s) v/s T (in K) at P = 25 MPa for different mixture compositions
of water and n-decane from Yd = 0 (pure water) to Yd = 1 (pure n-decane)
Case
Specie
T (K)
p(kg/m 3 )
I
water
water
n-decane
n-decane
water
water
n-decane
n-decane
700
800
700
800
700
1000
700
1000
125.62
84.22
486.27
424.45
125.62
57.86
486.27
329.94
I
I
I
II
II
II
II
p(Pa - s)
3.1773e
3.3626e
8.2378e
5.8599e
3.1773e
3.9332e
8.2378e
4.1035e
-
05
05
05
05
05
05
05
05
Table 4.2: Properties of water and n-decane at 25MPa and extreme temperatures of
the simulations I and II
74
resolution of 0.06 mm in this case. The velocity field thus seems to be well resolved.
Fig.4-5 and Fig.4-6 show the mean temperature and n-decane mass fraction profile
respectively along the vertical direction in Case II at three different axial locations
(x=4D, x=6D and x=8D) for the three different mesh resolutions (0.10 mm, 0.08 mm
and 0.06 mm). These figures indicate that the mixing layer is well resolved in terms of
the scalar fields. The mixing layer thickness from these plots can be seen to be around
0.8 mm around x = 6D. We will later see that in Case II, we observe that the shear
layer becomes unstable. The wavelengths of the unstable perturbations or waves on
the shear layer are multiples of the mixing layer thickness as stated in literature, for
example [30]. Since the grid size of 0.06 mm gives at least 10 cells within the mixing
layer, we can conclude that the spatial resolution is sufficient to capture the scales of
instability in the shear layer. This gives us reasonable confidence in the consistency
of the numerical methods and code. Unfortunately, there is no experimental data
available in the literature for mixing of near-critical fluids in a cylindrical tee reactor
geometry.
4.2
Case I: Water n-decane mixing with small temperature difference
(Rew,in = 500, T
=,in
800K, Td,in= 700K)
Under these flow conditions, the flow remains laminar and steady state is reached.
Fig.4-7 shows the contour plots of the n-decane mass fraction and temperature fields
on the centerplane (y=O plane) at steady-state. The variations in these scalars can
be seen more clearly (especially in the thin mixing layer), in the plots of Yd and T
along the vertical centerline at different downstream locations (x=2D, x=4D, x=6D,
x=8D) in Fig.4-8. It is clear from these plots, that the mixing layer is slowly advected
downwards as it travels downstream. Based on rough mixing layer thickness estimates
from these plots, we can see that the mixing layer thickness increases from around 0.4
mm at x = 2D to around 0.6 mm at x = 8D. The growth of the mixing layer under
75
0.001
0.06 mm mes h
-
-
0.0005
-
0.08 mm mes h
0.10 mm mes h
-
N O
-0.0005
-0.001
0.05
0.1
U
0.2
0.15
(a) x = 4D
0.001
--
z
0.0005
o
0.06
mm mes h
0.08 mm me h
0.10 mm mes h
0-1
N 0
-0.0005
-
-0.001
0.05
0.1
0.15
0.2
U
(b) x =6D
0.001
-
0.0005
N
-
0.06 mm mes h
0.08 mm mes h
0.10 mm mes h
0
-0.0005
-
00
.
0.05
0.1
-0.001
0.15
U
(c) x = 8D
Figure 4-3: u (in m/s) v/s z (in m) for simulated Case I for three different mesh
resolutions (0.10 mm, 0.08 mm and 0.06 mm) at (a) x = 4D (b) x = 6D and (c) x =
8D
76
0.001
-----
0.0005
-
.---
0.06 mm
0.08 MM
0.10 MM
1
N 0(-0.0005 -
-0.001
0.05
0.1
0.25 0.3
0.2
0.15
Umean
0.001
(a) x
4D
-----
0.06 mm
----
0.0005
----
0.08 MM
0.10 MM
N 0-0.0005 -
-0.001-
0.05
0.1
0.2
0.15
Umean
5
0.3
(b) x = 6D
-0.00.06 mm
rs-
0.08 mm
0.0005
N
-
0-
-0.0005 -
-0.001-
0.05
0.1
0,15
Umean
0.2
6.29
(c) x = 8D
Figure 4-4: um,,,, (in m/s) v/s z (in m) for simulated Case II for three different mesh
resolutions (0.10 mm, 0.08 mm and 0.06 mm) at (a) x =- 4D (b) x =- 6D and (c) x=
8D
77
0.001
---
0.0005
N
-
0.06 mm
0.08 mm
0.10 mm
0
-0.0005
-0.001
750
(a) x
0.001
-
950
4D
0.06 mm
0.08 mm
0.10 MM
0.0005
N
900
8
800
0-
-0.0005-
-0.001
750
800
85.
900
950
(b) x = 6D
0.001
0.0005
----e-a0.10
mm
0.06 mm
0.08 MM
MM
N 0-0.0005
-0.001750
800
85$
900
950
(c) x = 8D
Figure 4-5: Te
(in K) v/s z (in m) for simulated Case II for three different mesh
resolutions (0.10 mm, 0.08 mm and 0.06 mm) at (a) x = 4D (b) x = 6D and (c) x=
8D
78
0.001
-------
0.0005-
0.06
0.08
mm
MM
10 MM
N 0
-0.0005
-0.001
0.2
d 0.6
0.4
0.8
(a) x = 4D
0.001
0.06 mm
0.08 mm
- 0.10 mm
-
.
N 0
-0.0005
-0.001
0.0005-
0.2
0.6
0.4
6D
(b) x
0.0010.0005
N
-------
0.06
0.08
mm
mm
.m
-+-
-
0.8
0-
-0.0005-
-0.001(
0.2
0.4
(c)
Yd
x
0.6
O.8
= 8D
Figure 4-6: Yd,mean v/s z (in m) for simulated Case II for three different mesh resolutions (0.10 mm, 0.08 mm and 0.06 mm) at (a) x = 4D (b) x = 6D and (c) x
8D
79
these laminar conditions is only due to the diffusion of the scalars in the transverse
direction across the mixing layer. While the Yd and T remains fairly unchanged in the
bulk of the water stream except near the thin mixing layer, the T increases and the Yd
decreases by a significant amount in the n-decane stream close to the top of the pipe.
This is due to the advective mixing brought about by the action of a counter-rotating
vortex pair (CVP) in the body of the n-decane jet and a secondary flow of water
around the HC jet as explained in detail in Sec.4.6. Fig.4-9(a) shows the profile of
the velocity magnitude along the vertical centerline at different downstream locations.
It shows a 3 times increase in the velocity of the n-decane stream from x=2D to x=8D.
The peak velocity in the water stream remains fairly constant between these locations
however, the thickness of the water stream as seen on the centerplane is continually
reducing. These geometry dependent flow dynamic effects are explained in detail in
Sec.4.6. The changes in local composition and temperature are reflected in changes in
the local density and viscosity as shown in Fig.4-9(b)&(c) respectively. The reduction
in density of the fluid mixture in the n-decane rich stream at the top of the pipe due to
mixing with water is seen to be stronger than the reduction in viscosity due to the fact
that the local density is a function of the local mole fractions of water and n-decane.
Since water has a much smaller molecular weight than n-decane, a given mass fraction
of water will contribute more moles to the mixture than the same mass fraction of
n-decane. As a result, the mixture density is always strongly biased towards the water
density. The net effect of the flow acceleration of the n-decane rich stream and the
density, viscosity variations on the local flow Reynolds number is shown in Fig.4-9(d).
Here, the pipe diameter is used as the length scale for the Re definition. The Re in
the n-decane rich stream continually increases as we go downstream, while that in
the water rich stream does not change much. This seems to be mainly due to the flow
acceleration rather than the variations in physical properties. The local Re does not
go above 1360 and it seems that the flow is stable to perturbations at these values of
the local Re.
Fig.4-10 shows contour plots of Yd (top), T (middle) and the streamwise component of the vorticity vector, w. (bottom) on cross-sections of the cylindrical tee mixer
80
Yd:
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
T: 700 710 720 730 740 750 760 770 780 790 800
Figure 4-7: Contour plots on the centerplane (y=O plane) at steady-state for simulated Case I of (top) n-decane mass fraction (Yd) field and (bottom) Temperature
(T) field; The white vertical lines indicate the positions x=2D, x=4D, x=8D, x=10D,
x=12D, x=14D downstream of the center of the mixing joint from left to right
(planes of constant X) at different locations downstream of the mixing joint (x=2D,
x=4D, x=8D, x=16D, x=22D). It can be seen that mixing predominantly occurs
due to the large scale transport of species caused by the action of a counter-rotating
vortex pair (CVP) in the hydrocarbon jet coming in from the top (labeled (A) in
Fig.4-10 (bottom)) and the flow of water around the jet towards the top (labeled (B)
in Fig.4-10 (bottom)). Far downstream, as the strength of the CVP diminishes due
to diffusion and the upward flow of water around the HC jet becomes weak, mixing
is dominated by diffusion of species over small length scales and hence, is extremely
slow.
The main role of the supercritical water stream in this process being to heat up
the HC stream in a controlled fashion (along with providing a flux of water into the
HC stream), it is important to understand the physical mechanisms contributing to
the local heating/cooling of the fluid in the tee mixer. This physical insight is difficult
to obtain by looking at the enthalpy transport equation, wince the local enthalpy of
the fluid mixture is a complex function of both the local temperature and the local
81
-
0.001
:
x=2D
x = 4D
-:
x = 6D
x=8D
p
0.0005
N
0
-0.0005
-0.001
i
I
0.2
0.4
__
.
I
I
0.6
0.8
(a) Yd
0.001
x=2D
- x=4D
S-x=6D
0.0005
::
x=8D
N 0
-0.0005
-0.001
I
I
i
I
720
740
760
780
I
T
8 i0
(b) T (in K)
Figure 4-8: Profiles along the vertical centerline (y=-O plane) at steady-state for
simulated Case I of (a) n-decane mass fraction (Yd) field and (b) temperature (T in
K) field at different downstream locations (x=2D, x=4D, x=6D, x=8D)
82
-
0.001
0.001
2
0 x=4D
0 x=4D
=6D
-x
0.0005
9
0.0005
x =8D
O
N 0
-0.0005
-0.0005
-0.001
-0.001
N
0.05
0.2
0.15
b1
_100
0.0005
N
a
-
200
250
300
P
350
400
450
0.001
x=2D
0 x=4D
x=6D
x =8D
D
Sx=4
D
Sx=6
0.0005
N
0
-0.0005
-0.001
-0.001
E0
E0
5E0
e
x=
1000
1200
D
0
-0.0005
4E-05
150
(b) p (in kg/m 3 )
(a) Umag (in m/s)
0.001
x =6D
x = 8D
-
20400
6We
800
(d) Re = PUmagD/A
(c) p (in Pa-s)
Figure 4-9: Profiles along the vertical centerline (y=O plane) at steady-state for
simulated Case I of (a) velocity magnitude (Umag in m/s) (b) density (p in kg/m 3 )
(c) dynamic viscosity (p in Pa-s) and (d) Reynolds number (Re = pUmagD/1p) at
different downstream locations (x=2D, x=4D, x=6D, x=8D)
83
I
2D
8D
4D
16D
22D
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
I
2D
4D
8D
16D
22D
T: 700 710 720 730 740 750 760 770 780 790 800
2D
4D
8D
Co: -50 -40 -30 -20 -10 0
16D
22D
10 20 30 40 50
Figure 4-10: (top) Mass fraction of n-decane (Yd) contours (middle) Temperature (T
in K) contours and (bottom) Streamwise vorticity (P,in s-1) contours for simulated
Case I (Re.,i, = 500, T.,in = 800K, Td,in = 700K) at different downstream crosssections (x=2D, x=4D, x=8D, x=16D, x=22D)
84
composition. As such, an increase in enthalpy of a fluid elements as it moves in the tee
mixer cannot be attributed directly to a corresponding increase in its temperature.
It is therefore necessary to analyze the terms in a transport equation formulated in
terms of the temperature to gain insight into the local heating/cooling of the fluid. A
temperature equation can be obtained by substituting for dh from Eq.3.9 in Eq.3.6:
DT
a
OTi
pCP-- A
Dt = Oxj
_ Oxj
Substituting
pD
k
]
for
8
F
&Yk
Ox
I
O&
[pyk
pDkhk
8Yk +
+
-(
a
Yk
(
(4.1)
from the species conservation equation
+ pU
Eq.3.4 gives:
DT
Dt
1 0
pCp Oxj
OT
1
A
+
Oxj pC
A
YkOhk
pDk
k
& j&xj
(4.2)
(4.2
B
In Eq.4.2 above, term A is the rate of increase/decrease of T (heating/cooling) of a
moving fluid element due to heat diffusion (energy transfer through molecular collisions). Term B represents the rate of heating/cooling of the fluid element due to the
diffusion of species along a gradient of partial enthalpy. As molecules of a particular
species (say, specie k) diffuse along a gradient in partial enthalpy, the difference in
partial enthalpy is taken from/released to the fluid element at that location leading
to the cooling/heating of the fluid element represented by term B in the T equation,
Eq.4.2. This gradient in the partial enthalpy, in general, can arise due to a gradient
in mixture temperature, pressure or composition. The effect of variation of species
partial enthalpies with pressure can be safely neglected due to the very low Mach
numbers
(m
0.0001) and consequently, low P differences
(a
1-2 Pa) in this study.
This heating/cooling mechanism produces an interesting effect for mixing of nonideal mixtures such as the water n-decane system under isothermal conditions. When
ideal-gases mix under initially isothermal conditions, the flow remains isothermal (in
the absence of chemical reactions) since term (A) (heat diffusion) vanishes and term
(B) in Eq.4.2 also vanishes since the partial enthalpies of components in ideal mixtures are only functions of temperature and as such, there are no partial enthalpy
85
gradients in the flow domain. However, for mixing of non-ideal fluids as is the case
with the water n-decane system in this study, the partial enthalpies of the species
(hk) are functions of not only temperature, but also mixture composition. As a result, the heating/cooling term (B) of Eq.4.2 is non-zero even for initially isothermal
flows. Consequently, the flow becomes non-isothermal.
This local cooling/heating of an initially isothermal flow is well illustrated in
Fig.4-11(top) which shows the T contours on the tee centerplane at steady-state for
a simulation with the same flow conditions as Case I, but with both streams entering
at the same Tiiet of 750K. In the mixing joint region, the T drops to as low as 737
K in the mixing layer, where the diffusion of species is strongest.
The thickness
of the fluid layer affected by the cooling continues to grow downstream where it is
also assisted now by heat diffusion which starts to grow. Fig.4-11(bottom) shows
contours of term (B) (
Zkk pDk
O
Ohk)
on the tee centerplane. It is clear that this
term is negative in the mixing layer and is responsible for the T reduction of the fluid
below 750K. Because the near-critical mixture of water and n-decane is a non-ideal
mixture, the partial enthalpy (hk) of both species in the mixture is dependent on not
only the mixture temperature, but also the mixture composition. This fact is clearly
illustrated in Fig.4-12 which shows curves for the variation of the partial enthalpy
of both water and n-decane with T for different mixture compositions. We can see
that the partial enthalpy of water in the mixture increases with higher n-decane mass
fraction of the mixture. Similarly, the partial enthalpy of n-decane in the mixture
increases with higher water mass fraction of the mixture. Water molecules have a
greater affinity for other molecules than for n-decane molecules. Consequently, ndecane molecules taking the place of water molecules in a mixture (increase of Yd)
leads to more repulsive interactions and hence a greater partial enthalpy for the
water component. This is the reason why we observe the trend of increase in
with increasing Yd.
hwater
The same explanation holds for the trend in hn-decane.
The
species (water and n-decane) diffuse from regions in which their concentration is
higher to regions in which their concentration is lower (in accordance with the
2 nd
Law of Thermodynamics). Thus, they are diffusing from towards regions in which
86
their partial enthalpy in the mixture is higher. That is, the gradient in Yk and hk
are in opposite directions. Consequently, the term B of Eq.4.2 is negative (or zero)
everywhere in the domain as observed.
This local cooling of fluid due to species diffusion along a positive partial enthalpy
gradient also affects the heating of the HC stream (when T.is Td,i.) as in Case I.
Fig.4-13 shows term A and Term B of Eq.4.2 on cross-sections of the tee at different
downstream locations for Case I. We can see that the local rate of cooling due to
species diffusion is finite. However, the heat diffusion term is strongly dominant. In
Fig.4-14 we can see that the rate of cooling due to heat diffusion is more than 3 times
that of the rate of cooling due to species diffusion at most points in the mixing layer
along the vertical centerline at x=4D. Thus, this cooling effect only slightly hinders
the process of heating the n-decane stream which is a working objective of the mixing
process in the tee mixer. Also, we can see from Fig.4-13 and Fig.4-14 that the term
B is very small (almost zero) in the water-rich part of the mixing layer even though
significant species diffusion occurs in this region. This is due to the fact that the
partial enthalpy curves for both water and n-decane are very close together in the
low Yd range (The curves corresponding to Yd= 0 and Yd= 0.2 in Fig.4-12 almost
overlap). As a result, the heating/cooling of fluid due to species diffusion is small in
the water-rich region.
The transport of heat between the two streams is seen to be quicker than the transport of species due to molecular heat diffusion being 2-3 times faster than molecular
mass diffusion in regions of high n-decane concentration. This is well illustrated in
Fig.4-15 which shows contours of the Lewis number, the ratio of the thermal diffusivity to the mass diffusivity (Le = o/D) over the cross-section of the tee at different
downstream locations. The Lewis number
2 - 3 in regions rich in n-decane and
1 in water-rich regions. This difference in the rates of thermal and species mixing
is more noticeable at far downstream locations where transport is mainly diffusion
controlled.
87
T (in K)
737 7383 739.6 740.9 74 2.2 743.5 744.8 746.1 747.4 748.7 750
1CP
0
) (in K /s)
(A
-500 -400 -300 -200 -100
pDk
-500 -400 -300 -200 -100
100 200 300 400 500
0
a
a
(in K/s)
100 200 300 400 500
0
Figure 4-11: Contour plots on the centerplane (y=O plane) at steady-state for the
initially isothermal simulation: (top) Temperature (T) field (in K) (middle) Rate of
(A ) (in K/s) (bottom) Rate of heatheating/cooling due to heat diffusion 1
-
ing/cooling due to species diffusion along partial enthalpy gradient
E
PCP
pDkahk
p~kax3
ax3
(in K/s); The white vertical lines indicate the positions x=2D, x=4D, x=6D downstream of the center of the mixing joint from left to right
88
2400
2200
2000
1800
-water, Yd= 0
-+-water, Yd = 0.2
-+-water, Yd = 0.8
1
-water,Yd=
- 0 -n-decane, Yd = 0
n-decane, Yd = 0.2
- n-decane, Yd= 0.8
-@-n-decane, Yd = 1
n-decane
Y
-1600-
01000.0so;water
Yd
1400-
1200
100d-
80'0
750
800
850
900
950
100(
T (K)
Figure 4-12: partial enthalpy of water, h, and n-decane, hd (in kJ/kg) v/s T (in K)
at P
=
25 MPa for different mixture compositions of water and n-decane from Yd = 0
(pure water) to Yd
1 (pure n-decane)
89
I
(a)
(b)
1
a
' E, pDk
-500 -400 -300 -200 -100
(in K/s)
A'
0
a
(in K/s)
100 200 300 400 500
Figure 4-13: Contours of the rate of fluid heating/cooling (in K/s) due to (a) heat
and (b) species diffusion along partial enthalpy gradient
diffusion (g 1 9A a))
a) at different downstream locations (x=2D, x=4D, x=8D, x=16D,
x=22D from left to right)
(P
>kpDkay
90
1
0 [
T|
0.001
-
pD a x
0.0005
N
0
-0.0005
-0.001
-
-500
50
0
DT/Dt
Figure 4-14: rate of fluid heating/cooling (in K/s) due to (a) heat diffu(A O)) and (b) species diffusion along partial enthalpy gradient
sion ( '
(E Zk pDak
h) along the vertical centerline at x = 4D for simulated Case I
I
2D
4D
8D
16D
22D
Le: 1.12 1.3 1.48 1.66 1.84 2.02 2.2 2,38 2.56 2.74 2.92
Figure 4-15: Lewis number, Le = a/D at different downstream cross-sections (x=2D,
x=4D, x=8D, x=16D, x=22D)
91
4.3
Case II: Water n-decane mixing with large temperature difference
(Re,'
= 500, T
1000K, Td in=
700K)
In Sec.4.2, we saw that the shear layer between the water and n-decane streams is
stable and the flow remains laminar for a temperature difference of 1OOK between the
two streams. When, the water enters at a temperature of 1000K (300K greater than
the n-decane inlet), the water n-decane shear layer becomes unstable and the flow
in the cylindrical tee mixer is unsteady. Fig.4-17 shows the n-decane mass fraction
contours on the centerplane of the tee mixer at different time instants (t=2.0s, t=2.4s,
t=2.8s, t=3.2s roughly corresponding to 2.5, 3, 3.5 and 4
and the mean field. The flow was allowed to develop for 2
tflwowthrough
tflowthrough
respectively)
to wipe out
the initial conditions in the tee mixer and thereafter, averaging of field variables was
performed over the next 2
tflowthrough.
Statistically stationary state is achieved after
about 2 to 2.5 tflowthrough. Since the statistics of the field variables do not change with
time after about 2 to 2.5
tflow-through,
we suspect some periodicity in the temporal
variations of the fields. Fig.4-16 shows the Fourier transform of the n-decane mass
fraction field at x = 6D, y = 0, z = 0 just after the shear layer destabilizes. We
can see a cluster of frequency modes around 50 Hz which dominate the flow field
fluctuations. The plots of Fig.4-17 show the location of onset of instability to be close
to 5 diameters downstream of the mixing joint center with the shear layer developing
some waviness at this location. Further downstream, the shear layer starts to roll
up leading to faster mixing through large scale advective transport of species across
the two streams. This effect is manifested in the thickening of the mean mixing layer
beyond x=5D as seen in the mean Yd plot in Fig.4-17(e).
The sudden acceleration in the growth of the mixing layer is more clearly visible
in the Yd and T plots at different downstream locations in Fig.4-18. Fig.4-19(a) shows
the profile of the mean velocity magnitude along the vertical centerline at different
downstream locations. Similar to Case I, the acceleration of the n-decane rich stream
92
2
1.5
I
G)
Iii
1
4-.
w
Lii
I
E
I
0.5
I
0
I
20
I
I
I
I
40
I
I
60
i
Frequency (Hz)
I
80
I
100
Figure 4-16: Fouriertransform of the Yd temporal variation at x = 6D, y = 0, z = 0
93
from x=2D to x=8D is evident.
The peak velocity in the water stream remains
fairly constant between these locations however, the thickness of the water stream
as seen on the centerplane is continually reducing. These geometry dependent flow
dynamic effects are similar to those in Case I and are explained in detail in Sec.4.6.
The changes in local composition and temperature are reflected in changes in the
local density and viscosity as shown in Fig.4-19(b)&(c) respectively. These variations
in local fluid properties are stronger than in Case I due to the larger temperature
difference between the two streams. The net effect of the flow acceleration of the
n-decane rich stream and the density, viscosity variations on the local flow Reynolds
number is shown in Fig.4-19(d). Here, we notice a stark difference to Case I. We see
that the local flow Re in the shear mixing layer increases from x = 2D to x = 8D, so
that the peak Re in the shear layer is above 1500 at x = 8D. This local Re increase
in the shear layer will be shown to be mainly due to the reduction in the viscosity
of the fluid mixture due to the heating up of the n-decane component in Sec.4.5. At
these higher values of local Re, the flow becomes unstable to perturbations leading
to the growth of instabilities and waves followed by the rollup of the shear layer and
finally, the collapse of these structures into small scale turbulence.
Fig.4-20 and Fig.4-21 show the Yd and w, contours respectively, on cross-sections
of the tee at different downstream locations at three different time instants and also
the mean field. Up to x = 4D, fluid mixing over the cross-section is predominantly
caused by the circulating action of the CVP in the body of the HC jet and the
secondary flow of water around the HC jet, similar to Case I. The destabilization of
the shear layer and its subsequent rollup leads to intense stretching and breakdown
of the streamwise vorticity accompanied by a strong enhancement of the same. This
dynamics of the vorticity field is explained in detail in Sec.4.7. The enhanced vorticity
stretches the material line between the two fluids, leading to enhanced mixing as is
evident in Fig.4-20(a)to(d).
Fig.4-22 and Fig.4-23 show the streamwise (X) and cross-stream (Z) velocity components on the tee centerplane at different time instants (t=2.0s, t=2.4s, t=2.8s,
t=3.2s roughly corresponding to 2.5, 3, 3.5 and 4
94
tflow-through
respectively) and the
(a) t = 2.Os
(b) t = 2.4s
(c) t = 2.8s
(d) t = 3.2s
(e) Mean
Yd:
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Figure 4-17: Contour plots on the centerplane (y=O plane) of n-decane mass fraction
(Yd) field at (a) t = 2.Os (2.5 tflowthrough) (b) t = 2.4s (3 tflow-through) (c) t = 2.8s (3.5
tflow-through) (d) t = 3.2s (4 tflo-trough) and (e) Mean field; The white vertical lines
indicate the positions x=4D, x=6D, x=8D, x=10D, x=12D and x=16D downstream
of the center of the mixing joint (left to right)
95
0.001
0.0005-
x=2D
x = 4D
A
x=6D
e
x = 8D
N 0
-0.0005
-0.001
0.2
0.4
(a)
0.6
d,mean
0.8
1
Ydmean
0.001
x=2D
x=4D
x=6D
x 8D
0.0005
N 0-0.0005-
-0.001
700
800
(b)
Ten900
T
Tmean
1 V0
(in K)
Figure 4-18: Profiles along the vertical centerline (y=O plane) at steady-state for simulated Case II of (a) mean n-decane mass fraction (Yd) field and (b) mean temperature
(T in K) field at different downstream locations (x=2D, x=4D, x=6D, x=8D)
96
0.001
e
0.0005
X=8
D
D
D
D
0.001
Sx
0.0005
N 0
N 0
-0.0005
-0.0005
-0.001
-0.001
i105
0.2
15
mean,mag
0.1
0.25
0.3
-
100
x
x
0
Sx
0.0005
300
400
(b) Pmean (in kg/m 3 )
0
0.001
=2D
=4D
= 6D
x= 2D
4D
x = 6D
x = 8D
0.0005
x = 8D
0
200
Pmean
(a) Umeanmag (in m/s)
0.001
x =2D
= 4D
x =6D
x = 8D
N 0
N 0
-0.0005
-0.0005
-0.001
-0.001
I
I
I
4 :-05
5E-05
6E-05
I
~
7E-05
.....
. ....
1
8E-05
500
-
pmean
(c)
imean
_.
Re mean
(d) Remean = PmeanUmeanmagD/pmean
(in Pa-s)
Figure 4-19: Profiles along the vertical centerline (y=O plane) at steady-state for
simulated Case II of (a) mean velocity magnitude (Umean,mag in m/s) (b) mean density (Pmean in kg/m 3 ) (c) mean dynamic viscosity (pmean in Pa-s) and (d) mean
Reynolds number (Remean = pmeanUmean,magD/Pmean) at different downstream locations (x=2D, x=4D, x=6D, x=8D)
97
I
I
I
(a) t = 2.4s
(b) t
=
2.8s
(c) t
=
3.2s
(d) Mean
Yd:
0
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Figure 4-20: Contours of Yd for simulated Case II at different downstream crosssections (x=2D, x=4D, x=6D, x=8D, x=16D from left to right) at (a) t = 2.4s (3
(d) Mean
tflow-through) (b) t = 2.8s (3.5 tflow-through) (c) t = 3.2s (4 tflowthrough) and
98
(a) t = 2.4s
(b) t
=
2.8s
(c) t
=
3.2s
1
0
7
(d) Mean
(OX
-200-160-120 -80 -40
0
40
80
120 160 200
Figure 4-21: Streamwise vorticity (wx in s') contours for simulated Case II at different downstream cross-sections (x=2D, x=4D, x=6D, x=8D, x=16D from left to
right) at (a) t = 2.4s (3 tflow-through) (b) t = 2.8s (3.5 tflow-through) (c) t = 3.2s (4
tflowthrough)
and (d) Mean
99
mean field. As the stream of water enters the joint region, the jet of HC entering
from the top creates an effective area reduction to the flow of water, thus accelerating
the fluid in the water stream casuing the peak fluid velocity in the stream to reach as
high as 2.5 times the average value at the water inlet. This effect is clearly observable
in the mean streamwise velocity plot in Fig.4-22(e). In Fig.4-23, we see that this
accelerated water stream has a vertically upward velocity. This is due to the motion
of the water from the stream at the bottom, around the HC jet, towards the space
vacated near the top of the pipe cross-section by the downward motion of the HC jet.
These flow effects are expalined in detail in Sec.4.6.
At around x=5D, the water-HC shear layer becomes unstable and develops the
first signs of waviness as can be seen in the instantaneous streamwise velocity plots
of Fig.4-22(a)to(d).
Thereafter, the shear layer undergoes rollup and vortices are
continuously shed from it. The waves on the shear layer induce vertical velocity
fluctuations in the fluid near the shear layer of magnitude comparable to the vertical
velocity of the HC jet in the vertical pipe. This is due to the fact that the induced
vertical velocity fluctuations are components of the streamwise velocity of the water
stream at the shear layer which is much larger than the velocity of the HC stream as
can be seen from the inlet flow conditions in Table 4.1. The eddies on the shear layer
and those shed from it, also cause the velocity profile in the downstream section of
the tee to become more fuller than in the laminar Case I.
4.4
Quantification of mixing
In order to make a quantitative comparative assessment of the mixing performance
at different flow Reynolds numbers, we define a parameter called the 'mixing quality'
(0) for both the species and thermal transport which represents the degree of mixing
achieved at a particular cross-section downstream of the mixing joint (see Eq. 5.1).
This definition is based on the ratio of the mass flux weighted mean square deviation
of the mass fraction/temperature from the well mixed value over the entire crosssection to that over the inlet faces. A mixing quality of 1 indicates completely mixed
100
(a) t = 2.Os
(b) t = 2.4s
(c) t = 2.8s
(d) t = 3.2s
(e) Mean
U -0.007 0.0273 0.0616 0.0959 0.1302 0.1645 0.1988 0.2331 0.2674 0.3017 0.336
Figure 4-22: Contour plots on the centerplane (y=O plane) of the streamwise velocity
(u in m/s) field at (a) t = 2.Os (2.5 tflwo-through) (b) t = 2.4s (3 tflow-through) (c) t
- 2.8s (3.5 tflow-through) (d) t - 3.2s (4 tf low-through) and (e) Mean field; The white
vertical lines indicate the positions X=2D, x=4D, x=6D, x=8D, x=1OD, x=12D and
x=16D downstream of the center of the mixing joint (left to right)
101
(a) t = 2.Os
(b) t = 2.4s
(c) t = 2.8s
(d) t = 3.2s
(e) Mean
W -0.069 -0.0573 -0.0456 -0.0339 -0.0222 -0.0105 0.0012
0.0129 0.0246 0.0363
0.048
Figure 4-23: Contour plots on the centerplane (y=O plane) of the vertical velocity
(w in m/s) field at (a) t = 2.Os (2.5 tflow.through) (b) t = 2.4s (3 tflow-through) (c) t
2.8s (3.5 tflow-through) (d) t = 3.2s (4 tflow-through) and (e) Mean field; The white
vertical lines indicate the positions x=2D, x=4D, x=6D, x=8D, x=10D, x=12D and
x=16D downstream of the center of the mixing joint (left to right)
102
fluid and a mixing quality of 0 indicates completely unmixed fluid.
1
fSx
PU (Yd
ee
tsh
1
jx (T
-
-
SfSlnlets Pux (T
where,
Yd,fullymixed
0.5.
Tfullymixed
2
Ydfullymixed)
PUx (Yd -
Ainlets
Ydfullymixed )
Tfullymixed )
2
- Tfullymixed)
2
A3)
dA.
.dA
Ainlets
.dAs
As(
is 737.68 K in Case I and 826.85 K in Case
II. The well-mixed temperature was calculated by performing an energy balance on
the tee reactor. The well-mixed temperature is lower than the mean of the two inlet
temperatures because of the positive heat of mixing of the water n-decane non-ideal
mixture at these conditions. This means that the enthalpy of the water n-decane
mixture is greater than the mass weighted sum of the enthalpies of the individual
components at the same temperature and pressure. This is possibly due to the repulsive forces between the water and HC molecules. As a result, the water n-decane
mixing process is endothermic. Fig.5-16(a) shows the variation of Ospecies for all four
simulated cases I-IV. Fig.5-16(b) shows the variation of
#thermal
for the simulated
cases I and II. It is evident from the plots that the rate of mixing is very similar
in Cases 1, 111 and IV since the physical mixing mechanism in all three cases is the
same, dominated by the action of the CVP in the HC jet and molecular diffusion
far downstream (where the CVP is very weak). The rate of mixing in Case II also
follows a similar rate until x = 5D, when the flow in the tee is steady. In contrast,
downstream of x = 5D, the onset of waves and rollup of the mixing layer in Case II
leads to intense streamwise vortical structures over the tee cross-section as shown in
Sec.4.7. This leads to a significant enhancement in the rate of mixing after x = 5D
as seen in Fig.5-16(a) and Fig.5-16(b).
Far downstream (beyond x=16D), as these
vortical structures start to weaken due to vorticity diffusion, the rate of mixing starts
to plateau.
103
Species Mixing Quality
0.9
0.8
0.7
0.6
0.5
0.4
----
0.3
0
0.2
0.1
Case IV
+Case Ill
0.0
-~I
0
5
15
10
20
25
(a) #species
Thermal Mixing Quality
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
-0-Case I
-*-ae
H
0.1
0.0
0
5
15
10
20
25
(b) Othermal
Figure 4-24: # v/s x/D: Variation along the length of the tee of the (a) the species
mixing qualit y (#species) for simulated Cases I-IV and ( b) the thermal mixing quality
(Othermal) for simulated Cases I and II
104
4.5
Impact of near-critical property variations: Cause
of shear layer instability
In order to evaluate the impact of the variation of fluid properties of real-fluid mixtures, for a 100K temperature difference between the streams, on the flow and mixing
dynamics in a tee mixer, we performed a simulation of mixing of water and n-decane
with the same inlet conditions as in Case I above, but with the fluid properties of
each pure component held fixed at the inlet values.
The corresponding inlet con-
ditions and fluid properties are given under Case III in Table 4.1. Fig.4-25 shows
the contour plots of the streamwise vorticity on cross-sections of the tee at different
downstream locations for both Case I (top) and Case III (bottom) and Fig.4-26 shows
the contour plots of the n-decane mass fraction on cross-sections of the tee at different
downstream locations for the same. The two sets of plots do not show any noticeable
differences indicating that the density and transport property variations for a 100K
temperature difference between the two streams does not significantly impact the flow
field and mixing. Table 4.2 shows the density and viscosity of the pure components,
water and n-decane at the two extreme temperatures in the domain at 25MPa (the
nominal pressure in the tee mixer).
Fig.4-27(a)&(b) show the variation of the mixture density and dynamic viscosity
along the vertical centerline for the two cases I and III at x = 5D. The temperature
variations in Case I lead to an increase in mixture density in the n-decane rich stream
and within the mixing layer compared to Case III. The density of the water component
of the mixture increases as it cools down (to temperatures lower than the inlet value of
800K) while that of the n-decane component decreases as it heats up (to temperatures
higher than the inlet value of 700K). However, due to the mixture density being
determined by the local mole fractions of the components, the effect of temperature
on the water component density tends to dominate, resulting in the observed increase
in mixture density due to T variations throughout the domain (except in the bulk of
the water stream) compared to Case III. This increase in density due to T variations
is mostly less than 10% of the local mixture density. Fig.4-27(b) shows that the
105
2D
4D
8D
16D
22D
2D
4D
8D
16D
22D
(O: -50 -40 -30 -20 -10 0
Figure 4-25:
Streamwise vorticity (P
10 20 30 40 50
in s-1) contours for simulated Case I (top)
and Case III (bottom) at different downstream cross-sections (x=2D, x=4D, x=8D,
x=16D, x=22D)
mixture viscosity is lower in the n-decane rich stream and the mixing layer in Case
I compared to Case III. This effect can be understood by recalling (from Fig.5-18
and Table 4.2) that the viscosity of the water component decreases with decreasing
temperature and the viscosity of the n-decane component decreases with increasing
temperature at these conditions. Therefore, both the heating up of n-decane and the
cooling of water leads to a decrease in mixture viscosity compared to Case III. The
reduction in mixture viscosity due to T variations is below 10% of the local mixture
viscosity. The net effect of the density and viscosity changes due to T variation on
the local Reynolds number can be observed in Fig.4-28. The Re within the shear layer
in Case I is greater than in Case III (both increase in p and decrease in P lead to
an increase in Re). However, the increase in Re is small and the shear layer remains
stable at these values of the local Re.
In order to evaluate the impact of the variation of the fluid mixture properties,
for the larger AT between the streams (300K), on the flow and mixing dynamics in
106
I
I
2D
4D
8D
16D
22D
2D
4D
8D
16D
22D
Y:
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Figure 4-26: n-decane mass fraction (Yd) contours for simulated Case I (top) and Case
III (bottom) at different downstream cross-sections (x=2D, x=4D, x=8D, x=16D,
x=22D)
the tee mixer, we performed a simulation of mixing of water and n-decane with the
same inlet conditions as in Case II above, but with the fluid properties of each pure
component held fixed at the inlet values. The corresponding inlet conditions and fluid
properties are given under Case IV in Table 4.1. As seen in Fig.5-13, the shear layer
remains stable and the flow remains steady in this case. As a result, mixing occurs
mainly due to the circulating action of the CVP in the HC jet and the flow around
the jet from the water stream at the bottom to the top of the pipe, very similar to
Case I. This can be seen in Fig.5-14 which shows contour plots of Yd on cross-sections
of the tee. Clearly, for a AT of 300K between the two streams, the changes in local
density and viscosity due to temperature variations are large enough to influence the
stability characteristics of the flow.
Fig.4-31(a)&(b) show the variation of the mixture density and dynamic viscosity
along the vertical centerline for the two cases II and IV at x = 5D, the location at
which the shear layer is observed to become unstable in Case II. The temperature
107
0.001
0.0005
Case I
-
- a- -Case
III
N 0 -..
-
-0.0005
-0.001
100
200
300
P
400
500
(a) p (in kg/M 3 )
0.001
0.0005-
Case I
- a- -Case III
-0.0005
-0.001
II
4E-05
I
5E-05
I
6E-05
7E-05
8E-05
(b) y (in Pa-s)
Figure 4-27: Profiles along the vertical centerline (y=O plane) at steady-state for
simulated Cases I and III of (a) density (p in kg/m 3 ) and (b) dynamic viscosity (p
in Pa-s) at x=5D
108
0.001
a
- -h--
Cas el
Cas e II
0.0005
N 0
-0.00051
-0.001
500
Re
1000
I I
1500
Figure 4-28: Profiles along the vertical centerline (y=O plane) at steady-state for
simulated Cases I and III of the local Reynolds number at x=5D
109
Yd:
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Figure 4-29: Yd contours on the centerplane for simulated Case IV; The white vertical lines indicate the positions x=4D, x=6D, x=8D, x=10D, x=12D and x=16D
downstream of the center of the mixing joint (left to right)
variations in Case II lead to an increase in mixture density in the n-decane rich stream
and within the mixing layer compared to Case IV due to reasons similar to Case I.
Since the AT between the streams is 3 times that in Case I, the water within the
mixing layer is cooled to a greater degree from its inlet temperature value. This
results in density increases as much as 20% of the local density within the mixing
layer as compared to Case IV. Fig.5-18 and Table 4.2 indicate that the viscosity of
the n-decane component can reduces more than 100% as it is heated from 700K (ndecane inlet T in case II) to 1000K (water inlet T in Case II). The variation of water
viscosity with temperature is very small in comparison. In accordance with this, the
local viscosity within the mixing layer is lower by as much as 50% in Case II compared
to Case IV due to the heat up of the n-decane component. This large increase in
mixture density and decrease in mixture viscosity within the mixing layer due to the
cooling of water and the heating of n-decane respectively, leads to an increase of the
local flow Reynolds number as shown in Fig.4-32 with the peak value of Re reaching
close to 1500 in case II leading to the shear layer instability. In contrast, the local
Re remains below 1300 within the mixing layer in Case IV just as in cases I and III
and the shear layer remains stable. Thus, though the inlet Re number in Cases I and
II are the same, larger local variations in density and viscosity in Case II due to the
larger AT cause the local Re to increase to values in the unstable flow regime.
110
2D
4D
Yd:
6D
8D
16D
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Figure 4-30: Yd contours for simulated Case IV at different downstream cross-sections
(x=2D, x=4D, x=6D, x=8D, x=16D from left to right)
4.6
Dynamics of flow in the cylindrical tee mixer
In Sec.4.2, we saw that mixing in the cylindrical tee mixer under laminar conditions
is predominantly caused due to the advective action of a CVP in the HC jet and the
flow of the water stream around this jet towards the top of the pipe. The formation
of a CVP in transverse jets has been studied by Marzouk et al. [26]. Schlegel et al.
[43] investigated the CVP formed in transverse jets and the contributions of the wall
boundary layer to its formation. Schlegel et al. [42] also studied the impact of the
CVP formed in high Re reactive transverse jets on the flame front. All these studies
illustrate the strong impact of the CVP on the flow dynamics of a jet in a cross-flow
(the HC stream entering the joint region from the vertical pipe bears resemblances
to this flow configuration). In this section, we investigate the origin of the CVP in
the HC jet and other flow features in the mixing tee and study the dynamics of the
same.
Fig.4-33 shows contour plots of the streamwise vorticity component (w) on crosssections of the tee at different locations withing the mixing joint and slightly downstream of it (x=-0.5D, x=-0.25D, x=0, x=0.25D, x=0.5D, x=D, x=2D). Fig.4-33(a)
shows a snapshot of the water stream just before it comes into contact with HC in
the tee joint. Fig.4-33 (a),(c) and (d) are snapshots of the streamwise vorticity field
within the mixing joint, where the two streams first come into contact. Fig.4-33 (e)
111
0.001
Case I
a
Case IV
- 0.0005-
N
0-
-0.0005
-0.001
, I
,
,
,
,
200
100
I
,
,
300
P
,
400
500
7E-05
8E-05
(a) p (in kg/rn3 )
0.001
Case 1I
--
--
Case IV
0.0005
-0.0005
-0.001
4 -05
5E-05
6E-05
9
(b) I (in Pa-s)
Figure 4-31: Profiles along the vertical centerline (y=O plane) at steady-state for
3
simulated Cases II and IV of (a) density (p in kg/m ) and (b) dynamic viscosity (p
in Pa-s) at x=5D
112
-C
0.001
- --
-
ase II
Case IV
0.0005
N
0
-0.00051
-0.001
I
500
Re
1000
1500
Figure 4-32: Profiles along the vertical centerline (y=O plane) at steady-state for
simulated Cases II and IV of the local Reynolds number at x=5D
113
and (f) are in the downstream pipe. These plots illustrate the process of inception of
the CVP within the bulk of the HC jet entering from the top. In Fig.4-33(b), we can
see the vorticity in a layer of the HC jet in the vertical pipe (anticlockwise (red) on the
right and clockwise (blue) on the left). This vorticity (labeled (C)) was created due
to the action of the vertical pipe wall on the hydrocarbon stream. However, the flow
itself is straight down the vertical pipe without any circulation. As the HC jet flows
into the cross-stream of water, a part of this vorticity gets oriented in the streamwise
direction (X direction). Due to the absence of the viscous stress from side walls after
the HC jet has plunged out of the vertical pipe, this streamwise vorticity leads to
circulation of the fluid (anticlockwise (red) on the right and clockwise (blue) on the
left) and the inception of the CVP (labeled (D) in Fig.4-33(e)). The strength of the
CVP continually decreases as it is advected downstream due to vorticity diffusion and
is less than 10% of its peak value beyond x=10D.
Fig.4-34 shows the transverse velocity vectors (the velocity vector component
along the constant X plane) colored by their magnitude on constant X planes at
two X locations. At x=-0.5D, the water stream just comes in contact with the HC
stream entering the mixing joint from the vertical pipe. In Fig.4-34(a) we can see
the HC jet entering from the top and pushing the stream of water downwards. This
downward motion of the water leads to the creation of the vorticity marked (E) in
Fig.4-33(b) and also causes the water stream to accelerate in the longitudinal direction (X direction) in order to prevent the accumulation of mass near the bottom.
This longitudinal acceleration of the water stream close to the mixing joint region is
clearly visible in the contour plot of the streamwise velocity component in Fig.4-35. A
secondary flow of HC, around the main HC jet, into the water stream is also observed
in the mixing joint region (x=-0.5D to x=0.5D) in Fig.4-34(a). This secondary HC
flow is responsible for the small pockets of well-mixed fluid at the peripheries of the
HC jet, labeled (G) in Fig.4-10(top).
Beyond x=0.5D (the end of the mixing joint), as the HC jet falls further downwards
into the water stream, it vacates space near the top of the pipe cross-section. This
causes fluid from the bulk of the water stream to rush upwards around the HC jet
114
(a) -0.5D
(b) -0.25D
(c) OD
(d) 0.25D
(e) 0.5D
(f) 1D
(g) 2D
Ox: -50 -40 -30 -20 -10 0
10 20 30 40 50
Figure 4-33: Streamwise vorticity (w, in s- 1 ) contours for simulated Case I (Rew,zn
500, Tw,in = 800K, Td,in = 700K) at different downstream cross-sections (x=-O.5D,
x=-0.25D, x-0, x=0.25D, x=0.5D, x=1D, x=2D)
115
tt
t
IlH
111 1
11111
!
fly
(b) 2D
(a) 0.25D
(v
2
1
+ w2) /
0.001 0.0019 0.0028 0.0037 0.0046 0.0055 0.0064 0.0073 0.0082 0.0091
0.01
Figure 4-34: Transverse velocity vectors (velocity component along constant X planes,
2
800K, Td,in= 700K)
+w 2 ) for simulated Case I (Rew,in= 500, T
rans =/v
at two downstream cross-sections (x=0.25D, x=2D) Note: The length of the vectors is
not representative of the magnitude in the figures, the color of the vectors represents
the magnitude
-,in=
into the space at the top and can be observed in Fig.4-34(b). This flow of water to
the top part of the tee is further aided by the circulating action of the CVP. The
addition of mass to the n-decane rich stream at the top causes the acceleration of this
stream as observed in Fig.4-9(a) and Fig.4-19(a) in spite of the slow growth of the
shear layer.
4.7
Vorticity dynamics
The streamwise vorticity, wa, plays a significant role in mixing as explained in Sec.4.6
and hence, we need to look at the evolution of this component of vorticity and the
physical mechanisms affecting it. Eq.5.3 below is the vorticity transport equation
116
NO0
0
.02
OA Yd0.6
0.8
UMg (m/S) 0.001 0.0215 0.042 0.0625 0.083 0.1035 0.124 0.1445 0.165 0.1855 0.206
Figure 4-35: (left) Contour plots on the centerplane (y=O plane) of the velocity mag2
(u 2 + v 2 + w2)l/ ) in the joint region; (right) n-decane mass fraction
nitude (Uma,
profile along the vertical line at x = 2D (the position indicated by the white line in
the left figure)
obtained by applying the curl operator to the momentum transport equation:
at
+ (U.V) W
(
V.)U - W (V.U)+
A
B
(4.5)
(Vp x VP) + V x
)
p
p2&
D
C
where, term (A) is the enhancement of vorticity due to the stretching and tilting of
vortex lines (cZt), term (B) is the vorticity enhancement due to the dilatation of fluid
elements
((ad),
term (C) is the baroclinic vorticity generation
(Cag)
due to the presence
of a density gradient perpendicular to a gradient in pressure and term (D) represents
2
the viscous term (cZ) which includes in it, the vorticity diffusion (pV (v)).
Fig.4-36 shows the mean field of the above-mentioned streamwise vorticity source
terms on different downstream cross-sections for simulated Case II. The formation of
the CVP in the HC jet occurs in a similar fashion to Case I. Thereafter, till x=5D, the
vorticity diffusion dominates as seen in Fig.4-36(c). This leads to a decrease in the
strength of the CVP from the joint region to about x = 5D. This CVP decay process
is clearly visible in Fig.4-21. The destabilization of the shear layer near x=5D triggers
117
stretching and breaking of the CVP into smaller coherent structures. This stretching
and breakdown of the vortices can be clearly observed in Fig.4-21. There is also a
significant enhancement of these streamwise vortices due to the stretching of fluid
elements (6 ,,). Fig.4-38 shows the w, profile along the z-direction at y=0.0004 and
different downstream locations (x=4D, x=5D, x=6D, x=8D). The sudden increase
in the magnitude of wX near z = 0.0001 from x=5D to x=8D can be seen in this
plot. This streamwise vorticity enhancement is due to two effects: (a) the strain
induced in the streamwise direction due to the waviness and rollup of the shear layer
resulting in vorticity enhancement due to fluid stretching as seen in Fig.4-37(a) and (b)
baroclinic generation of vorticity resulting from the average effect over time of strong
instantaneous local density and pressure gradients due to the unsteady motions of
the shear layer, as seen in Fig.4-37(b). The enhanced vorticity stretches the material
line between the two fluids, leading to enhanced mixing beyond x=5D as is evident in
Fig.4-20(a)to(d). As these vortices travel further downstream (beyond x=10D), the
viscous term starts to dominate again and their strength continually reduces.
4.7.1
Species and thermal transport enhancement due to
fluctuations
The advective fluxes of the conserved scalars like Yd and h can be broken down into
contributions due to the mean field and the fluctuations as below, in Eq.5.4:
Ujqo = 'Jjq + Uq'
A
(4.6)
B
where, q denotes a conserved scalar (like Yd or h),
()
denotes mean value and (')
denotes the fluctuation. Term (A) represents the advective flux contribution due
to the mean field; Term (B) represents the advective flux contribution due to the
fluctuations in the velocity and scalar fields.
In Sec.4.7, we saw that the rollup of the shear layer triggers intense stretching
and breakdown of the streamwise vorticity into smaller and stronger vortical struc118
j
(a) stretching and tilting
(b) baroclinic
(c) viscous diffusion
(d) dilatation
6)X -5000 -4000 -3000 -2000 -1000
0
1000 2000 3000 4000 5000
Figure 4-36: Contours of the mean field for simulated Case II at different downstream
cross-sections (x=2D, x=4D, x=6D, x=8D, x=16D from left to right) of (a) cst, in
S-2 (b) Cbg,x in S-2 (c) cL,, in s-2 and (d) Wdx in s-2
119
0.001
0.0005
-o--0,......0--
-
NO0
-x
-
-0.0005
-
-- -
- - -.
= 4D
x =5D
x=6D
---
-
x=8D
-0.001
(Do4/Dt
-200
-2000
(a)
Cist,x,mean
2000
stx,mean
(in s- 2 ), Case II
93x
0.001
= 4D
...-. .. x =5D
------ 0---
-T
x=6D
x=8D
0.00051
NO0
-0.0005
-0.001
(13
(LDt
-2000
(b)
Wbg,x,mean
2000
2000
4000
bg,x,mean
(in s-
2
),
Case II
Figure 4-37: Profiles along the vertical line at y= 0 .0 0 0 4 (y=D/6) for simulated Case
II of (a) 6 st,x,mean (in s-2) and (b) Wbg,x,mean (in s-2) at different downstream locations
(x=4D,
x=5D, x=6D, x=8D)
120
0.001
0.0005
S
-0.0005--+---e
=4D
x
x
= 5D
-. x
=8D
-0.00150
100
Cx,mean
Figure 4-38: Profiles along the vertical line at y=0.0004 (y=D/6) for simulated Case
II of wx,rnean (in s- 1 ) at different downstream locations (x=4D, x=5D, x=6D, x=8D)
121
2D
4D
6D
8D
16D
Yd' 0.001 0.0217 0.0424 0.0631 0.0838 0.1045 0.1252 0.1459 0.1666 0.1873 0.208
Figure 4-39: RMS fluctuation of Yd (yd 21/2) contours for simulated Case II at different
downstream cross-sections (x=2D, x=4D, x=6D, x=8D, x=16D from left to right)
tures. This fluctuating streamwise vorticity and the resultant fluid circulation produces strong fluctuations in the scalar variables like Yd over the cross-section of the
tee. This can be seen in Fig.4-39 which shows the RMS fluctuations of Yd on different
cross-sections of the tee for Case II. Fig.4-40 shows the advective flux contributions
due to w on the tee cross-sections. Fig.4-41 shows the advective flux contributions
due to v on the tee cross-sections. The fluctuations in the velocity components are
normalized with uref = 0.14 m/s, the water inlet average velocity. It is clear that
the contribution due to the fluctuations contribute significantly to the transport of
species over the cross-section of the tee. The resultant enhancement in mixing can be
visualized in Fig.4-20 which shows the instantaneous and mean Yd field on different
cross-sections downstream of the mixing joint.
4.8
Summary
In this chapter, simulations of mixing of supercritical water and a model hydrocarbon
(n-decane), under fully-miscible conditions, in a cylindrical tee mixer geometry for a
water inlet Reynolds number of 500, for two different inlet temperatures of the water
stream of 800K (Case I) and 1000K (Case II) were discussed. The mass flow rate of
the n-decane stream was equal to that of the water stream and its inlet temperature
was 700K in both cases.
122
(a) wmeanYd,mean/uref
(b) w'Y'/uref
-0.02 -0.016 -0.012 -0.008 -0.004
0
0.004
0.008
0.012 0.016
0.02
Figure 4-40: Contours for simulated Case II at cross-sections downstream of location of onset of instability (x=6D, x=8D, x=16D from left to right) of (a)
WmeanYd,mean/ref (b) w'Y '/uref; uef = 0.14 m/s
123
(a)
vmeanY,mean/Uref
(b) v'Y '/Uref
-0.02 -0.016 -0.012 -0.008 -0.004
0
0.004 0.008 0.012 0.016
0.02
Figure 4-41: Contours for simulated Case II at cross-sections downstream of location
of onset of instability (x=6D, x=8D, x=16D from left to right) of (a) i)meanYd,mean/Uref
(b) v'Y'/uref; Uref = 0.14 m/s
124
It was found that the flow downstream of the mixing joint remained laminar in
case of the 1OOK temperature difference between the streams (Case I). Most of the
mixing and heat transport occurs due to the circulating action of a counter-rotating
vortex pair (CVP) in the hydrocarbon jet formed due to the reorientation of the
vorticity pre-existing in the hydrocarbon stream flowing through the vertical pipe.
This CVP gets progressively weaker as it flows downstream due to vorticity diffusion
and species and heat transport is dominated by molecular diffusion over small length
scales. Consequently, the mixing rate plateaus in the far downstream region of the tee
mixer. Comparison with mixing of water and n-decane at the same inlet conditions
but without variations of the physical properties with temperature (Case II) suggests
that the near-critical property variations, for a 100K temperature difference between
the streams, have a negligible impact on the flow field and mixing.
For a 300K temperature difference between the two streams (Case II), the waterHC shear layer becomes unstable and starts to form waves near x=5D downstream of
the mixing joint center. Further downstream, the shear layer rolls up and vortices are
shed from it. The onset of instability in the shear layer also triggers the stretching
and breakdown of the CVP in the body of the hydrocarbon jet.
These smaller,
stronger vortical structures cause faster advective transport of species and enthalpy
over the cross-section of the tee leading to a significant enhancement in mixing. This
manifests as a jump in the mixing rate beyond the location of onset of instability and
a thickening of the mean mixing layer.
However, water n-decane mixing under identical inlet conditions but with constant
physical properties (Case IV), shows a stable shear layer with the flow reaching steady
state. In Case II, the temperature of the water component changes strongly (from
1000K to 700K) within the mixing layer and this cooling of the water component leads
to an increase in the density of the fluid mixture within the mixing layer compared to
Case IV. The cooling of water and heating up of n-decane within the mixing layer also
leads to significant reduction in the mixture viscosity (as much as 50%) within the
layer compared to Case IV. Both, the density increase and viscosity decrease lead to
an increase in the local flow Reynolds number within the mixing layer with the peak
125
Re value reaching close to 1500. The shear layer becomes unstable to perturbations
in the flow at these higher local Re. This results in the growth of perturbations in
the shear layer and the consequent waviness and rollup of the layer followed by the
collapse of these structures into small-scale turbulence.
126
Chapter 5
Mixing at intermediate Re: Impact
of Re
In this chapter, the mixing of supercritical water and n-decane is simulated for four
different values of the Reynolds number at the water inlet (Rew,in = 500, 600, 700 and
800) as shown in Table 5.1. In all cases, the mass flow rates through the water and
n-decane inlets is the same (a design condition found to be favorable to the process
from the chemical kinetics point of view). The nominal pressure in the tee mixer was
- 25MPa (ensured by fixing the back-pressure at the outlet) and the n-decane inlet
temperature was 700 K, which is sufficiently above the Upper Critical Solution Temperature (UCST) of the water n-decane system
(a
632 K) thereby ensuring complete
miscibility of the two components. The water inlet temperature was 800 K. The effect
of gravity forces is not included in these simulations so as to allow us to study the
impact of geometry, Reynolds number and property variation effects on the flow field
and mixing dynamics.
5.1
Validation: Grid convergence tests
In order to verify the adequacy of the chosen spatial resolution and verify the numerics
of the code, a grid independence study was performed for the largest Re case, Case IV
(Re,in = 800). Fig.5-1 shows the mean streamwise velocity profile along the vertical
127
Case
I
II
III
IV
Rexi,
500
600
700
800
Red,i,
mh,in
Th,in
Uw,in
Ud,in
T.,in
T,i.
Property
204
245
286
327
(kg/hr)
0.11
0.14
0.16
0.18
(kg/hr)
0.11
0.14
0.16
0.18
(m/s)
0.0832
0.0998
0.1164
0.1331
(m/s)
0.0144
0.0173
0.0202
0.0230
(K)
800
800
800
800
(K)
700
700
700
700
variations
yes
yes
yes
yes
Table 5.1: Inlet flow conditions for the cases simulated in this study
direction for Case IV (the highest Re simulated) at three different axial locations
(x=4D, x=6D and x=8D) for three different mesh resolutions (0.10 mm, 0.08 mm
and 0.06 mm). The averages are taken over 4
initial transient of 2
tflow-through
time periods.
a mesh resolution of 0.06 mm in this case.
tflow-through
time periods after the
Grid-independence is attained for
The velocity field thus seems to be
well resolved. Fig.5-2 and Fig.5-3 show the mean n-decane mass fraction and mean
temperature profiles respectively, at three different axial locations (x=4D, x=6D and
x=8D)
for three different mesh resolutions (0.10 mm, 0.08 mm and 0.06 mm). These
figures indicate that the mixing layer is well resolved in terms of the scalar fields.
The mixing layer thickness from these plots can be seen to be around 0.5 mm around
x = 6D. We will later see that in Case IV, we observe that the shear layer becomes
unstable. The wavelengths of the unstable perturbations or waves on the shear layer
are multiples of the mixing layer thickness as stated in literature, for example [30].
Since the grid size of 0.06 mm gives around 10 cells within the mixing layer, we can
conclude that the spatial resolution is sufficient to capture the scales of instability
in the shear layer. This gives us reasonable confidence in the consistency of the
numerical methods and code. Unfortunately, there is no experimental data available
in the literature for mixing of near-critical fluids in a cylindrical tee reactor geometry.
5.2
Mixing Results
Fig.5-4 shows the n-decane mass fraction (Yd) on the tee centerplane for Case I and
Case II. At these flow Re, the water-HC shear layer remains stable and the flow
in the tee is laminar with a steady-state being reached.
128
Fig.5-5 and Fig.5-6 show
0.001
-
-o-
0.06 mm mesh
0.08 mm mesh
0.10 mm mesh
-
0.0005
N 0
-0.0005
-00
.
01
.
0.1
0.15
0.2
-0.001
0.05
0.25
0.3
Umean
(a) x = 4D
0.001
0.06 mm mesh
-
-
0.08 mm mesh
0.10 mm mesh
-
0.0005
N 0
-0.0005
-0.001
0.05
0.1
0.15
0.2
0.25
0.3
Umean
(b) x
6D
-
0.001
-
-
o-
-
-----
0.06 mm
0.08 mm
mesh
mesh
0.10 mm mesh
0.0005
N 0
-0.0005
-0.001
0.05
0.1
0.15
Umean
0.2
0.25
(c) x = 8D
Figure 5-1: u (in m/s) v/s z (in m) for simulated Case IV for three different mesh
resolutions (0.10 mm, 0.08 mm and 0.06 mm) at (a) x = 4D (b) x = 6D and (c) x
8D
129
0.001
-
-
0.0005
N
-
-
-
0.06 mm mesh
0.08 mm mesh
------
0.10 mm mesh
0.2
0. d,mean 0.6
0
-0.0005
-0.001
0.8
(a) x = 4D
0.001
0.06 mm mesh
-
- --
0.0005
N
-
-----
0.08 mm mesh
0.10 mm mesh
0
-0.0005
-0.001
0.2
A
n
0.6
0.8
Nd,mean
(b) x = 6D
0.0010.06 mm mesh
-a-
- -
0.0005
-------
mmmesh
-
0.10
mm
mesh
N 0
-0.0005-
-0.001
0.2
4
0.6
0.8
d,mean
(c) x = 8D
Figure 5-2: Yd v/s z (in m) for simulated Case IV for three different mesh resolutions
(0.10 mm, 0.08 mm and 0.06 mm) at (a) x = 4D (b) x = 6D and (c) x = 8D
130
0.001
-
0.0005
o-
---
-
-mesh
0.06 mm
0.08 mm mesh
mm mesh
0.10 a--
N 0
-0.0005
-0.001
700
780
mean760
740
720
8 0
(a) x = 4D
0.001
0.06 mm mesh
-
-
0.0005
--
---
-
0.08
-
0.10 mm mesh
mm
mesh
N 0-0.0005-
-0.001
700
720
760
(b) x
6D
-u5 T
fo0.08 mm mesh
0.10 mm mesh
0.001 -
0.06 mm
0.08 m
resoltion(0.1
780
740
d 0
mesh
N 0
-0.0005-.
-0.001-
700
720
740 Tmean 760
780
(c) x = 8D
Figure 5-3: T (in K) v/s z (in m) for simulated Case IV for three different mesh
resolutions (0.10 mm, 0.08 mm and 0.06 mm) at (a) x =- 4D (b) x =- 6D and (c) x=
8D
131
contour plots of the Yd and T fields respectively on different cross-sections of the tee
downstream of the mixing joint, illustrating more clearly, the features of the multidimensional mixing process in the tee mixer. Fig.5-7 shows the streamwise vorticity
field (w,) on these cross-sections for both cases. It is clear that most of the mixing
over large length scales is due to the circulating action of a counter-rotating vortex
pair (CVP) within the body of the HC jet entering through the vertical pipe. This
CVP is formed due to the re-orientation of the pre-existing vorticity in the HC jet
flowing through the vertical pipe as it enters the mixing joint and travels downstream.
The details of the process of inception of the CVP are presented in Sec.4.6. As the HC
jet moves further downwards in the downstream pipe, it vacates space near the top of
the pipe. This causes a motion of water from the stream at the bottom, around the
periphery of the HC jet, towards the space at the top. This secondary flow around the
HC jet is another physical mechanism leading to large scale advective mixing. Fig.5-7
also shows that the strength of the streamwise vorticity and the resultant circulating
motions is higher in Case II compared to Case I. This is due to the fact that, the
pre-existing vorticity in the HC jet in the vertical pipe is larger in Case II due to
the larger flow velocity in this section. The re-orientation of this stronger vorticity
leads to a stronger CVP in the higher Re case. The enhanced advective mixing due
to the stronger streamwise vorticity can be clearly seen in the comparative plots of
the Yd and T fields in Fig.5-5 and Fig.5-6. The strength of these vortices continually
decreases as they travel downstream due to vorticity diffusion by viscous action, falling
to less than 10% of the value just after the mixing joint by around x = 12D. Further
downstream, mixing is mainly controlled by molecular diffusion.
When, Re,, = 700, the shear layer becomes unstable, with waves forming on the
surface followed by the roll up of the shear layer as can be seen in the instantaneous
centerplane Yd plots in Fig.5-8(a)to(c).
The formation of these smaller scale flow
structures on the shear mixing layer leads to faster advective transport of the scalars
resulting in enhanced mixing and heat transport. This is manifested in a thickening
of the mean mixing layer beyond x = 8D as can be seen in the mean Yd field plot in
Fig.5-8(d). In the unsteady cases, the flow is allowed to develop for
132
2
tflow-through
to
(a) Case I, Re.,i, = 500
(b) Case II, Re., a = 600
Yd:
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Figure 5-4: Contour plots on the centerplane (y=O plane) of n-decane mass fraction (Yd) field at steady-state for (a) Case I and (b) Case II; The white vertical
lines indicate the positions x=2D, x=4D, x=6D, x=8D, x=1OD, x=12D and x=16D
downstream of the center of the mixing joint (from left to right)
I
(a) Case I, Rew,in = 500
I
I
I
(b) Case II, Rem,in
Y:
=
I
600
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Figure 5-5: Contour plots of Yd on cross-sections of the tee at different downstream
locations (x=2D, x=4D, x=8D, x=16D and x=22D from left to right) at steady-state
for (a) Case I and (b) Case II
133
(a) Case I, Rew,i,,= 500
(b) Case II, Re ,,j
=
600
T: 700 710 720 730 740 750 760 770 780 790 800
Figure 5-6: Contour plots of T (in K) on cross-sections of the tee at different downstream locations (x=2D, x=4D, x=8D, x=16D and x=22D from left to right) at
steady-state for (a) Case I and (b) Case II
wipe out the initial transients and thereafter, averaging of field variables is performed
over the next 4 tflow-through. Statistically stationary state is typically reached within
2 -
2
.5tflow-through.
Since the statistics of the field variables do not change with
time after about 2 to 2.5 tflow-through, we suspect some periodicity in the temporal
variations of the fields. In Fig.5-8(b) a clear spatial periodicity is observed in the
roll up structures on the water-HC shear layer with a wavelength close to 0.5D.
Fig.5-9 shows the Fouriertransform of the n-decane mass fraction field at x = 8D,
y = 0, z = 0 (a location after the shear layer destabilizes). We can see a cluster of
frequency modes around in the 0-10 Hz and 90-100 Hz bands which dominate the
flow field fluctuations. In addition, a group of frequency modes in the 40-60 Hz range
is also observed but of lesser amplitude. It is not possible to identify from these
plots, an exact location at which the onset of instability occurs in this case, since the
waviness in the shear layer is intermittent in time, with the shear layer being smooth
at certain time instants for the same downstream location (even locations as far as x
134
(a) Case I, Re,,i, = 500
I
(b) Case II, Rem,in = 600
Ox: -50 -40 -30 -20 -10 0
10 20 30 40 50
Figure 5-7: Contour plots of the streamwise vorticity, w. (in s- 1 ) on cross-sections of
the tee at different downstream locations (x=2D, x=4D, x=8D, x=16D and x=22D
from left to right) at steady-state for (a) Case I and (b) Case II
135
= 10D). The effective location of onset of instability is identified to be around x = 6D
from the normalized root-mean square (RMS) velocity fluctuation (u'/U) plot on the
centerplane in Fig.5-12. Fig.5-10 shows the instantaneous and mean Yd contour plots
on cross-sections of the tee at different downstream locations (x=2D, x=4D, x=8D,
x=12D and x=16D). Fig.5-11 shows the instantaneous and mean w, field on the
same cross-sections. Analogous to cases I and II, the flow is steady just downstream
of the mixing joint (up to about x = 6D) and mixing predominantly occurs due to the
circulating action of the CVP in the HC jet and the secondary flow of water around the
jet. This action becomes progressively weaker as the CVP strength decreases due to
vorticity diffusion as it travels downstream. Near x=8D, the unstable motions of the
shear layer trigger the intense stretching and breakdown of the streamwise vorticity
accompanied by an enhancement of the same as seen in Fig.5-11. The development
of these stronger smaller scale vortical structures promotes faster advective transport
of the scalars over the cross-section of the tee. The enhanced vorticity stretches the
material surface between the two fluid streams leading to an enhancement in mixing
as seen in Fig.5-10.
At Rem,, = 800 (Case IV), the water-HC shear layer destabilizes earlier, around x
5D. The unsteadiness in the shear layer is not intermittent as in Case III. The waves
on the shear layer and roll up structures are larger than in Case III (Remi = 700) as
can be seen in the instantaneous Yd plots on the centerplane in Fig.5-13(a)to(c). The
small scale flow structures formed on the shear layer lead to fast advective transport
of the scalars resulting in enhanced mixing. This is manifested in the thickening
of the mean mixing layer beyond x
=
6D and continuing far downstream as seen
in the mean Yd plot on the centerplane in Fig.5-13(d).
This effect is much more
pronounced than in Case III illustrating the efficacy of the small scales in enhancing
the mixing process.
Fig.5-14 shows the instantaneous and mean Yd contour plots
on cross-sections of the tee at different downstream locations (x=4D, x=6D, x=8D,
x=12D and x=16D). Fig.5-15 shows the instantaneous and mean W, field on the
same cross-sections. Near x=6D, the destabilization of the shear layer triggers the
intense stretching and breakdown of the streamwise vorticity accompanied by an
136
(a) t = 2.Os
(b) t = 2.2s
(c) t = 2.5s
(d) Mean
Yd:
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Figure 5-8: Contour plots on the centerplane (y=O plane) of n-decane mass fraction
(Yd) field for simulated Case III at (a) t = 2.Os (4 tflowthrough) (b) t = 2.2s (4.4
tflow-through) (c) t = 2.5s (5 tflowthrough) and (d) Mean field; The white vertical lines
indicate the positions x=6D, x=8D, x=10D, x=12D, x=14D, x=16D, x=18D and
x=20D downstream of the center of the mixing joint (left to right)
137
0.4
0.35
0.3
00.25
0.2
0.15
0.1
0.05
0
20
60
40
80
100
Frequency (Solution Time)'
Figure 5-9: Fourier transform of the Yd temporal variation at x = 8D, y = 0, z = 0
for Case III
138
I
(a) t = 2.Os
(b) t = 2.2s
(c) t
=
2.5s
(d) Mean
Yd:
0
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Figure 5-10: Contours of Yd for simulated Case III at different downstream crosssections (x=2D, x=4D, x=8D, x=12D, x=16D from left to right) at (a) t = 2.Os (b)
t = 2.2s (c) t = 2.5s and (d) Mean
139
I
I
I
(a) t = 2.Os
(b) t
=
2.2s
(c) t
=
2.5s
(d) Mean
Wx -100 -80 -60 -40 -20
0
20
40
60
80 100
Figure 5-11: Contours of w, (in s-') for simulated Case III at different downstream
cross-sections (x=2D, x=4D, x=8D, x=12D, x=16D from left to right) at (a) t = 2.Os
(b) t = 2.2s (c) t = 2.5s and (d) Mean
140
u'/U
u'/U 0.001 0.0449 0.0888 0.1327 0.1766 0.2205 0.2644 0.3083 0.3522 0.3961
0.44
Figure 5-12: Contour plots on the centerplane (y=O plane) for Case III of
u'/U,
where U is the reference velcoity used for normalization taken to be the water inlet
average velocity; The white vertical lines indicate the positions x=2D, x=4D, x=6D,
x=8D, x=10D, x=12D, x=14D and x=16D downstream of the center of the mixing
joint (left to right)
enhancement of the same as seen in Fig.5-15. The development of these smaller scale
vortical structures promotes faster advective transport of the scalars over the crosssection of the tee. The enhanced vorticity stretches the material surface between the
two fluid streams leading to an enhancement in mixing as seen in Fig.5-14. This
stretching and breakdown of the eddies into smaller structures is more vigorous as
compared to Case III and is a classic feature of a turbulent flow field.
5.3
Quantification of mixing
In order to make a quantitative comparative assessment of the mixing performance
at different flow Reynolds numbers, we define a parameter called the 'mixing quality'
(0) for both the species and thermal transport which represents the degree of mixing
achieved at a particular cross-section downstream of the mixing joint (see Eq. 5.1).
This definition is based on the ratio of the mass flux weighted mean square deviation
of the mass fraction/temperature from the well mixed value over the entire crosssection to that over the inlet faces. A mixing quality of 1 indicates completely mixed
fluid and a mixing quality of 0 indicates completely unmixed fluid.
Ospeciesx = 1 -
fs,
fs\
(
du-
djullymixed 32
Pux (Yd - Yd,ullymixed
141
Ainlets
.dAs Asx
.dAs
) 22
(5.1)
(a) t = 1.8s
a
(b) t = 2.05s
(c) t = 2.25s
(d) Mean
Yd:
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Figure 5-13: Contour plots on the centerplane (y=O plane) of n-decane mass fraction
(Yd) field for simulated Case IV at (a) t = 1.8s (4 tflow-through) (b) t = 2.05s (~
4.5 tflowtthrough) (c) t = 2.25s (5 tflow-through) and (d) Mean field; The white vertical
lines indicate the positions x=4D, x=6D, x=8D, x=10D, x=12D, x=14D, x=16D and
x=18D downstream of the center of the mixing joint (left to right)
142
(a) t = 1.8s
1i
(b) t = 2.05s
(c) t
=
2.25s
(d) Mean
Yd:
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Figure 5-14: Contours of Yd for simulated Case IV at different downstream crosssections (x=4D, x=6D, x=8D, x=12D, x=16D from left to right) at (a) t = 1.8s (b)
t = 2.05s (c) t = 2.25s and (d) Mean
143
(a) t = 1.8s
(b) t = 2.05s
(c) t = 2.25s
(d) Mean
(x -200 -160 -120 -80 -40
0
40
80
120 160 200
Figure 5-15: Contours of w_, (in s1) for simulated Case IV at different downstream
cross-sections (x=4D, x=6D, x-=8D, x=12D, x=16D from left to right) at (a) t = 1.8s
(b) t = 2.05s (c) t = 2.25s and (d) Mean
144
Sfs, Pux (T s
thrl-
where,
Yd,fullymixed
= 0.5.
Tfullymixed
Pux (T
Tfullyrnixed)
2
- Tfullymixed)
.dAs
2
Ainiets
(5.2)
.dA 8 Asx
is 737.68 K. The well-mixed temperature was
calculated by performing an energy balance on the tee reactor. The well-mixed temperature is lower than the mean of the two inlet temperatures because of the positive
heat of mixing of the water n-decane non-ideal mixture at these conditions. This
means that the enthalpy of the water n-decane mixture is greater than the mass
weighted sum of the enthalpies of the individual components at the same temperature and pressure. This is possibly due to the repulsive forces between the water
and HC molecules. As a result, the water n-decane mixing process is endothermic.
Fig.5-16(a) shows the variation of
shows the variation of
#thermal
#species for
all four simulated cases I-IV. Fig.5-16(b)
for the simulated cases I-IV. It is evident from the plots
that the rate of mixing is very similar in Cases I and II since the dominant physical
mixing mechanism in both cases is the same, dominated by the action of the CVP in
the HC jet and molecular diffusion far downstream (where the CVP is very weak).
The rate of mixing in Case III also follows a similar rate until x = 6D, when the
flow in the tee is steady. In contrast, downstream of x = 6D, the onset of waves and
rollup of the mixing layer in Case III leads to the formation of intense streamwise
vortical structures over the tee cross-section as shown in Sec.5.5.
This leads to a
significant enhancement in the rate of mixing after x = 6D as seen in Fig.5-16(a) and
Fig.5-16(b). Similarly, in Case IV the steep increase in mixing rate when the mixing
layer becomes unstable near x = 5D is evident in Fig.5-16(a) and Fig.5-16(b).
The
streamwise vorticity that stretches the material line between the two streams and
enhances mixing is much stronger in Case IV compared to Case III, due to the higher
flow Re. Hence, the mixing rate after mixing layer transition is also much greater
for Case IV. Far downstream (beyond x=16D), as these vortical structures start to
weaken due to vorticity diffusion, the rate of mixing starts to plateau.
145
Species Mixing Quality
0.9
0.8
0.7
0.6
0.5
0.4
0.3
-UCse
I
,-*-Case 11
0.2
.1Case
0.1
III
-Case
15
20
IV
0.0
5
0
10
25
(a) $species
Thermal Mixing Quality
0.9
0.8
0.7
0.6
0.5
-
0.4
0.3
0.3-*-CAsM
I
,.*-Ca se IH
0.2
0.1
caseICse
12
C_
5a1
0 .01
1 1
5
0
15
10
(b)
I
IV
I I I I
25
20
Othermal
Figure 5-16: 0 v/s x/D: Variation along the length of the tee of the (a) the species
mixing quality
Cases I-IV
(#.Pecies)
and (b) the thermal mixing quality (Othermal) for simulated
146
Case
I to IV
I to IV
I to IV
I to IV
Specie
water
water
n-decane
n-decane
T (K)
p(kg/m 3 )
700
800
700
800
125.62
84.22
486.27
424.45
p(Pa - s)
3.1773e - 05
3.3626e - 05
8.2378e - 05
5.8599e - 05
Table 5.2: Properties of water and n-decane at 25MPa and extreme temperatures of
the simulations in this study
5.4
Impact of near-critical property variations: Cause
of shear layer instability
It is important to note at this stage, the variation of the mixture density and viscosity
with both temperature and composition. Fig.5-17 and Fig.5-18 show the variation
of the mixture density (p in kg/m 3 ) and dynamic viscosity (p in Pa - s) with temperature for the range of temperatures pertinent to this study, for different mixture
compositions from pure water to pure n-decane. Table 4.2 shows the p and P for
the water and n-decane pure components at the extreme temperatures within the tee
mixer for the cases I and II. It can be seen that the mixture density and viscosity are
strong functions of the temperature under these conditions. The density of the water
component can increase by around 50% as it gets cooled from 800K to 700K within
the tee mixer domain while the density of the n-decane component can decrease by
around 10% as it gets heated from 700K to 800K. The viscosity of water does not
change appreciably in this temperature range. However, the viscosity of the n-decane
component can reduce by about 30% as it gets heated from 700K to 800K. These
variations in p and p will influence the local Reynolds number which in turn can be
expected to have a significant bearing on the local flow dynamics and stability. We
intend to evaluate the impact of these property variations (with temperature) within
the mixing tee on the flow field and mixing dynamics.
In order to evaluate the impact of near-critical property variations in the tee
mixer, an additional test simulation was performed (Case V) with inlet conditions
as specified in Table 5.3. In Case V, the inlet conditions are identical to Case III
(Re,
= 700) but with the physical properties (p, p, A, D) held fixed at the inlet
147
-- Yd=0 -+-Yd=0.2
r-^^
-+-Yd=0.4
-- Yd=0.5
-A-Yd=0.6
-+-Yd=0.8
-- Yd=1
450
400350-
2 300
6250
2
1001
0
700
8 00
780
760
740
720
T (K)
Figure 5-17: p (in kg/m 3 ) v/s T (in K) at P = 25 MPa for different mixture compositions of water and n-decane from Yd = 0 (pure water) to Yd = 1 (pure n-decane)
X 10Yd= 0 -+-Yd
=0.2
-Yd
=0.4-
Yd=0.5 -Yd
=0.6
-Yd
=0.8
+Yd
1
4
720
740
Figure 5-18: p (in Pa-s) v/s T (in K) at P
T (K)
=
760
780
800
25 MPa for different mixture compositions
of water and n-decane from Yd = 0 (pure water) to Yd = 1 (pure n-decane)
148
Case
V
Rew,in
700
Red,in
rnw,in
286
(kg/hr)
0.16
Thd,in
Uw,in
(kg/hr) (m/s)
0.16
0.1164
Ud,in
Tw,in
(m/s)
0.0202
(K)
-
Td,in
(K)
-
Property
variations
no
Table 5.3: Inlet flow conditions for the comparison test Case V simulated in this
study
Yd:
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Figure 5-19: Yd contours on the centerplane for simulated Case V; The white vertical
lines indicate the positions x=2D, x=4D, x=6D, x=8D, x=10D, x=12D and x=14D
downstream of the center of the mixing joint (left to right)
values. Though the variation of physical properties with temperature are not allowed
in this simulations, the local variation in the mixture properties due to the mixing
of water and n-decane (since water and n-decane have significantly different physical
properties) are accounted for. Comparison of this case with Case III will tell us how
the near-critical variations of mixture properties affect the flow and mixing.
As seen in Fig.5-13, the shear layer remains stable and the flow remains steady in
Case IV. As a result, mixing occurs mainly due to the circulating action of the CVP
in the HC jet and the flow around the jet from the water stream at the bottom to
the top of the pipe, very similar to the laminar cases I and II. This can be seen in
Fig.5-20 which shows contour plots of Yd on cross-sections of the tee. Clearly, for a
AT of 1OOK between the two streams, the changes in local density and viscosity due
to temperature variations influence the stability characteristics of the flow.
Fig.5-21(a)&(b) show the variation of the mixture density and dynamic viscosity
along the vertical centerline for the two cases III and V at x = 6D, the location at
which the shear layer is observed to become unstable in Case III.. The temperature
variations in Case III lead to an increase in mixture density in the n-decane rich
stream and within the mixing layer compared to Case V. The density of the water
149
2D
I
I
I
4D
Yd
8D
16D
22D
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Figure 5-20: Yd contours for simulated Case V at different downstream cross-sections
(x=2D, x=4D, x=8D, x=16D, x=22D from left to right)
component of the mixture increases as it cools down (to temperatures lower than the
inlet value of 800K) while that of the n-decane component decreases as it heats up
(to temperatures higher than the inlet value of 700K). However, due to the mixture
density being determined by the local mole fractions of the components, the effect
of temperature on the water component density tends to dominate, resulting in the
observed increase in mixture density due to T variations throughout the domain
(except in the bulk of the water stream) compared to Case V. This increase in density
due to T variations is mostly around 10% of the local mixture density. Fig.5-21(b)
shows that the mixture viscosity is lower in the n-decane rich stream and the mixing
layer in Case III compared to Case V. This effect can be understood by recalling
(from Fig.5-18 and Table 4.2) that the viscosity of the water component decreases
with decreasing temperature and the viscosity of the n-decane component decreases
with increasing temperature at these conditions. Therefore, both the heating of ndecane and the cooling of water leads to a decrease in mixture viscosity compared
to Case V. The reduction in mixture viscosity due to T variations is close to 10% of
the local mixture viscosity. The net effect of the density and viscosity changes due
to T variation on the local Reynolds number can be observed in Fig.5-22. The Re
within the shear layer in Case III is greater than in Case V (both increase in p and
decrease in p lead to an increase in Re). This increase in the local Re within the
shear layer makes it unstable to the growth of perturbations resulting in the shear
150
layer instability and roll up observed in Case III.
5.5
Vorticity dynamics
The streamwise vorticity, w, plays a significant role in mixing as explained in Sec.4.6
and hence, we need to look at the evolution of this component of vorticity and the
Eq.5.3 below is the vorticity transport equation
physical mechanisms affecting it.
obtained by applying the curl operator to the momentum transport equation:
1
Ow
?+
at
.V) L= (.V)
A
- W (V.
11
(V.
VP)+ V x
P
B
%V
C
(5.3)
p)
'o
D
where, term (A) is the enhancement of vorticity due to the stretching and tilting of
vortex lines (J't), term (B) is the vorticity enhancement due to the dilatation of fluid
elements
(bGd),
term (C) is the baroclinic vorticity generation (.4)
due to the presence
of a density gradient perpendicular to a gradient in pressure and term (D) represents
the viscous term (c ,) which includes in it, the vorticity diffusion (pV
2
(w)).
In order to understand the impact of the shear layer instability on the streamwise
vorticity field, which plays a crucial role in advective transport, we look at the contributions to the enhancement/production of streamwise vorticity due to cSt,,
, &c, and
d,y
,
bg,y
in Fig.5-23. We look at Case IV in particular, but the same general
phenomena are observed in Case III as well. The formation of the CVP in the HC
jet occurs in a similar fashion to Case I. Thereafter, till x=5D, the vorticity diffusion
dominates leading to a reduction in the strength of the CVP as seen in Fig.5-23(c)
and Fig.5-15. The destabilization of the shear layer near x=5D triggers stretching and
breaking of the CVP into smaller coherent structures with strong vortical structures
spreading over most of the tee cross-section. This stretching and breakdown of the
vortices can be clearly observed in Fig.5-15. There is also a significant enhancement
of these streamwise vortices due to the stretching of fluid elements. This is due to the
strain induced in the streamwise direction due to the waviness and rollup of the shear
layer.
Downstream of x=5D, the unsteady motion of the shear layer also leads to
151
0.001
N
Case III
9
0.0005-
Case V
- ---
0-
-0.0005
-0.001
II
(
100
15 0
I
I
I
200
I
.
I
I
.
I
250
P
I
I
I
I
I
300
I
III
I
I
350
.I
400
(a) p (in kg/mn3)
0.001
N
Case III
CaseV
- -
0.0005-
- -e-
0
-0.0005
-0.001
I
S
4E-05
I
. I I I I
5E-05
I
6E-05
.
i
I
I
I
7E-05
I
I
8E-05
(b) p (in Pa-s)
Figure 5-21: Profiles alon g the vertical centerline (y=O plane) at steady-state for
simulated Cases III and V of (a) density (p in kg/m 3 ) and (b) dynamic viscosity (p
in Pa-s) at x=6D
152
I
0.001
a
- ---
Case III
Case V
0.0005
N
0
-0.00051
-0.001
500
1000
Re
1500
Figure 5-22: Profiles along the vertical centerline (y=O plane) at steady-state for
simulated Cases III and V of the local Reynolds number at x=6D
153
strong density and pressure gradients giving rise to a significant baroclinic generation
of vorticity as seen in Fig.5-23(b). As these vortices travel further downstream, the
viscous diffusion starts to dominate and their strength continually reduces. Even so,
the streamwise vorticity is strong even at far downstream locations (as high as 100s-1
at x=16D) in this case.
5.6
Species and thermal transport enhancement
due to fluctuations
The advective fluxes of the conserved scalars like Yd and h can be broken down into
contributions due to the mean field and the fluctuations as below, in Eq.5.4:
Uio5
-
§iq3 + u'q$'
A
where, 0 denotes a conserved scalar (like
(5.4)
B
Yd
or h), () denotes mean value and (')
denotes the fluctuation. Term (A) represents the advective flux contribution due
to the mean field; Term (B) represents the advective flux contribution due to the
fluctuations in the velocity and scalar fields.
In Sec.5.5, we saw that the rollup of the shear layer triggers intense stretching
and breakdown of the streamwise vorticity into smaller and stronger vortical structures. This fluctuating streamwise vorticity and the resultant fluid circulation produces strong fluctuations in the scalar variables like Yd over the cross-section of the
tee. This can be seen in Fig.5-24 which shows the RMS fluctuations of Yd on different
cross-sections of the tee for Case IV. Fig.5-25 shows the advective flux contributions
due to w on the tee cross-sections. Fig.5-26 shows the advective flux contributions
due to v on the tee cross-sections. The fluctuations in the velocity components are
normalized with uref = 0.13 m/s, the water inlet average velocity. It is clear that
the contribution due to the fluctuations contribute significantly to the transport of
species over the cross-section of the tee. The resultant enhancement in mixing can be
154
I
(a) stretching and tilting term
(b) baroclinic term
(c) viscous term
(d) dilatation term
6)X -5000 -4000 -3000 -2000 -1000
0
1000
2000
3000
4000
5000
Figure 5-23: Contours of the mean field for simulated Case IV at different downstream
cross-sections (x=4D, x=6D, x=8D, x=10D, x=16D from left to right) of (a) Wst,y in
S 2 (b) Cby in s-2 (c) c,,y in s- 2 and (d) ;d,y in S-2
155
2D
4D
d 0.001
6D
8D
16D
0.0259 0.0508 0.0757 0.1006 0.1255 0.1504 0.1753 0.2002 0.2251
0.25
Figure 5-24: RMS fluctuation of Yd (d 2 /) contours for simulated Case IV at different
downstream cross-sections (x=2D, x=4D, x=6D, x=8D, x=16D from left to right)
visualized in Fig.5-14 which shows the instantaneous and mean Yd field on different
cross-sections downstream of the mixing joint.
5.7
Summary
In this chapter, simulations of mixing of supercritical water and a model hydrocarbon (n-decane), under fully-miscible conditions, in a cylindrical tee mixer geometry
for four values of the water inlet Reynolds number in the range of 500 to 800 were
discussed. The mass flow rate of the n-decane stream was equal to that of the water
stream. The inlet temperatures of water and n-decane were 800K and 700K respectively.
It was found that the flow downstream of the mixing joint remained laminar in
Cases I and II (Re,'in = 500 and Re.,in = 600 respectively).
Most of the mixing
and heat transport occurs due to the circulating action of a counter-rotating vortex
pair (CVP) in the hydrocarbon jet formed due to the reorientation of the vorticity
pre-existing in the hydrocarbon stream flowing through the vertical pipe. This CVP
gets progressively weaker as it flows downstream due to vorticity diffusion and species
and heat transport is dominated by molecular diffusion over small length scales. Consequently, the mixing rate plateaus in the far downstream region of the tee mixer.
When the Reynolds number at the water inlet is increased to 700 and the n-decane
156
(a) wmeanYd,mean/uref
(b) w'Y '/uref
-0.05 -0.04 -0.03 -0.02 -0.01 0.00 0.01 0.02 0.03 0.04 0.05
Figure 5-25: Contours for simulated Case IV at cross-sections downstream of location of onset of instability (x=6D, x=8D, x=1OD from left to right) of (a)
WmeanYdmean/Uref (b)
w'Y'/uef; uref = 0.13 m/s
157
(a) VmeanYd,mean/Uref
(b) v'Y'/Uref
-0.05 -0.04 -0.03 -0.02 -0.01 0.00 0.01 0.02 0.03 0.04 0.05
Figure 5-26: Contours for simulated Case IV at cross-sections downstream of location
of onset of instability (x=6D, x=8D, x=10D from left to right) of (a)
(b) v'Y'uref; Uref = 0.13 m/s
158
vmeanYd,mean/uref
inlet Re also increased correspondingly to maintain the same mass flow rate (Case
III), the shear layer between the water and n-decane streams is found to destabilize
near x = 6D downstream of the mixing joint. Further downstream, the shear layer
rolls up and vortices are shed from it. The onset of instability in the shear layer also
triggers the stretching and breakdown of the CVP in the body of the hydrocarbon jet.
These smaller, stronger vortical structures cause faster advective transport of species
and enthalpy over the cross-section of the tee leading to a significant enhancement
in mixing. This manifests as a jump in the mixing rate beyond the location of onset
of instability and a thickening of the mean mixing layer. At Re,,
= 800 (Case IV)
the unsteady small scale flow structures in the mixing layer and the consequent flow
field fluctuations due to them are much stronger. The stretching and breakdown of
the CVP in this case, is accompanied by stronger streamwise vorticity enhancement
resulting in much faster mixing compared to Case III.
However, water n-decane mixing under identical inlet conditions to Case III but
with constant physical properties (Case V), shows a stable shear layer with the flow
reaching steady state. In Case III, the temperature of the water component changes
from 800K to 700K within the mixing layer and this cooling of the water component
leads to an increase in the density of the fluid mixture within the mixing layer compared to Case V. The cooling of water and heating up of n-decane within the mixing
layer also leads to a reduction in the mixture viscosity within the layer compared to
Case IV. Both, the density increase and viscosity decrease lead to an increase in the
local flow Reynolds number within the mixing layer. The shear layer becomes unstable to perturbations in the flow at these higher local Re. This results in the growth
of perturbations in the shear layer and the consequent waviness and rollup of the
layer followed by the collapse of these structures into small-scale turbulence. Thus,
the near-critical variations in mixture properties have a significant impact on the flow
field and mixing dynamics in a tee mixer by influencing the local flow stability.
159
160
Chapter 6
Summary and Future Work
6.1
Summary
A robust numerical tool for simulating mixing of water and hydrocarbons under
near-critical fully miscible conditions in a complicated 3-D mixer geometry was developed using the open-source CFD libraries of OpenFOAM. A consistent treatment
of near-critical fluid thermodynamics and transport property variations is employed
as explained in Chapter 2.
The cubic Peng-Robinson equation of state [37] with
van der Waals mixing rules is used to model the thermodynamic behavior of waterhydrocarbon mixtures at conditions near the critical point.
The Predictive Peng
Robinson (PPR78) approach [14] with a group contribution method (GCM) for modeling inter-molecular interactions is used to determine the binary interaction parameter
for the water-hydrocarbon pair as a function of temperature. Viscosity and thermal
conductivity calculations are performed using the generalized correlations of Chung
et al. [6]. Mass diffusivities are calculated using the Tracer Liu-Silva-Macedo (TLSM)
model by Liu at al. [23] [24] [46]. A
2
nd
order accurate finite-volume methodology is
used for the numerical solution of the conservation equations as detailed in Chapter 3.
An operator-splitting approach based on the PISO algorithm of Issa [13] is employed
to handle the pressure-velocity coupling along with semi-implicit numerical schemes
to ensure the stability of the solution.
The developed numerical tool was used to investigate the mixing of supercritical
161
water and a model hydrocarbon (n-decane) in a small-scale cylindrical tee mixer (pipe
ID = 2.4mm) under fully miscible conditions (TUpper Critical Solution Temperature
of water n-decane system). First, in Chapter 4, mixing simulations were presented
for a water inlet Reynolds number of 500 (the lowest Re simulated in this study),
for two different inlet temperatures of the water stream of 800K and 1000K. The
mass flow rate of the n-decane stream was equal to that of the water stream and its
inlet temperature was 700K in both cases. This being the lowest temperature in the
domain, was carefully chosen to be above the UCST of the water n-decane system
(632K). It was found that the flow downstream of the mixing joint remained laminar
in case of the 100K temperature difference between the streams. Most of the mixing
and heat transport occurs due to the circulating action of a counter-rotating vortex
pair (CVP) in the hydrocarbon jet formed due to the reorientation of the vorticity
pre-existing in the hydrocarbon stream flowing through the vertical pipe. This CVP
gets progressively weaker as it is advected downstream, due to vorticity diffusion
and species and heat transport is dominated by molecular diffusion over small length
scales in the far downstream region. Consequently, the mixing rate plateaus in the far
downstream region of the tee mixer. Comparison with mixing of water and n-decane
at the same inlet conditions but without variations of the physical properties with
temperature suggests that for a 100K temperature difference between the streams,
near-critical property variations have a negligible impact on the flow field and mixing
behavior.
For a 300K temperature difference between the two streams, the water-HC shear
layer becomes unstable and begins to display sustained waviness near x=5D downstream of the mixing joint center. Further downstream, the shear layer rolls up and
vortices are shed from it. The onset of instability in the shear layer also triggers the
stretching and breakdown of the CVP in the body of the hydrocarbon jet. These
smaller, more intense vortical structures cause faster advective transport of species
and enthalpy over the cross-section of the tee leading to a significant enhancement
in mixing. This manifests as a jump in the mixing rate at the location of onset of
instability and a thickening of the mean mixing layer. However, water n-decane mix162
ing under identical inlet conditions but with constant physical properties, showed a
stable shear layer with the flow reaching steady state. In the case of SCW n-decane
mixing with a AT between the streams of 300K, the temperature of the water component changes strongly (from 1000K to 700K) within the mixing layer and this cooling
of water leads to an increase in the density of the fluid mixture within the mixing
layer compared to the simulated case where the properties of the components are held
constant. The cooling of water and heating of n-decane within the mixing layer also
leads to significant reduction in the mixture viscosity (as much as 50%) within the
layer compared to the constant properties case. Both, the density increase and the
viscosity decrease lead to an increase in the local flow Reynolds number within the
mixing layer with the peak Re value reaching close to 1500. The shear layer becomes
unstable to perturbations in the flow at these higher local Re. This results in the
growth of perturbations in the shear layer and the consequent waviness and rollup of
the layer followed by the collapse of these structures into small-scale turbulence.
In Chapter 5, simulations of mixing of supercritical water and n-decane, were
presented for four values of the water inlet Reynolds number in the range of 500 to
800. The mass flow rate of the n-decane stream was equal to that of the water stream.
The inlet temperatures of water and n-decane were 800K and 700K respectively.
It was found that the flow downstream of the mixing joint remained laminar for
Re ,i, = 500 and Re.,,i = 600. For Red,1
= 600, the flow and mixing behavior were
similar to the case of Re.,in = 500 with a slight enhancement in mixing rate due to the
increase in the strength of the CVP with Re. When the Reynolds number at the water
inlet is increased to 700 and the n-decane inlet Re also increased correspondingly to
maintain the same mass flow rate, the shear layer between the water and n-decane
streams is found to destabilize near x = 6D downstream of the mixing joint followed
by subsequent shear layer roll up and vortex shedding. The onset of instability in
the shear layer also triggers the stretching and breakdown of the HC jet CVP into
smaller, stronger streamwise vortices which significantly enhance mixing by stretching
the material surface between the two streams. This manifests as a jump in the mixing
rate at the location of onset of instability and a thickening of the mean mixing layer
163
similar to the case of large AT in Chapter 4. At Re=,2
= 800 the unsteady small
scale flow structures in the mixing layer and the consequent flow field fluctuations
due to them are much stronger. The stretching and breakdown of the CVP in this
case, is accompanied by stronger streamwise vorticity enhancement resulting in much
faster mixing compared to the case of Re.,i, = 700. However, water n-decane mixing
under identical inlet conditions to the Re.,in = 700 case but with constant physical
properties, showed a stable shear layer with the flow reaching steady state. In SCW
n-decane mixing with Re.,in = 700, the temperature of the water component changes
from 800K to 700K within the mixing layer and this cooling of water leads to an
increase in the density of the fluid mixture within the mixing layer compared to
the constant properties case. The cooling of water and heating of n-decane within
the mixing layer also leads to a reduction in the mixture viscosity within the layer
compared to the constant properties case. Both, the density increase and the viscosity
decrease lead to an increase in the local flow Reynolds number within the mixing layer.
The shear layer becomes unstable to perturbations in the flow at these higher local
Re. This results in the growth of perturbations in the shear layer and the consequent
waviness and rollup of the layer followed by the collapse of these structures into
small-scale turbulence. Thus, the near-critical variations in mixture properties have
a significant impact on the flow field and mixing dynamics in a tee mixer by influencing
the local flow stability.
6.2
Future Work
The work presented in this thesis is the first important step in an endeavor to simulate
the reactive mixing of water and crude oil in complicated 3-D reactor geometries.
Though the present work deals with some aspects of this complicated problem, it still
has a number of shortcomings which will be the focus of future work in this project.
The numerical tool developed in this study does not have the ability to track
the phase interface between immiscible phases. In a realistic SCWDS and upgrading
process, the local environment in the reactor may not allow complete miscibility of
164
water and hydrocarbon phases. As such, the ability to track the phase interface and
in general, multiple phase interfaces is essential. This can be done using sophisticated
interface tracking methods like the Level-Set method or the Volume of Fluid method.
In addition, phase equilibrium calculations need to be performed dynamically to
determine the equilibrium compositions of the phases near at the interface. Also, a
consistent treatment of the heat and species transport across the interface is required
to simulate transport and mixing in this multiphase scenario. This development of
the numerical tool is crucial in our efforts to realistically simulate the mixing of water
and hydrocarbons in a range of reactor conditions.
Computational cost constraints currently limit our ability to simulate mixing at
Reynolds numbers higher than 1000. Since it has been found desirable to operate
the reactor under turbulent conditions in this study, we need to develop a framework to simulate high Reynolds number flows with reasonable computational effort.
For this, turbulence modeling using a Large Eddy Simulation methodology must be
implemented. This will involve the development of turbulent mixing models which
incorporate the key physics of near-critical fluids.
Inclusion of chemical reactions in the numerical tool is also an essential future step.
Simplified reaction mechanisms for thermal cracking reactions of hydrocarbons and
sulfur compounds as well as cyclization and condensation reactions leading to coke
formation need to be considered in order to investigate the coupling between transport
and chemical reaction. This will help in obtaining realistic predictions of conversion
rates and product distributions which is the ultimate goal of our collaborative research
effort.
165
166
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