Post on 13-Nov-2021
The Pennsylvania State University
The Graduate School
COMPUTATIONAL INVESTIGATION OF
SUPERCRITICAL CARBON DIOXIDE SLOT JET
IMPINGEMENT HEAT TRANSFER
A Thesis in
Mechanical Engineering
by
Abdulaziz Alkandari
© 2020 Abdulaziz Alkandari
Submitted in Partial Fulfillment
of the Requirements
for the Degree of
Master of Science
August 2020
ii
The thesis of Abdulaziz Alkandari is to be reviewed and approved by the following:
Alexander S. Rattner
Dorothy Quiggle Career Development Professor, Assistant Professor of Mechanical Engineering,
Thesis Adviser
Robert Kunz
Professor of Mechanical Engineering
Daniel Haworth
Professor of Mechanical Engineering, Head of Graduate Programs and Professor of Mechanical
Engineering
iii
Abstract
Supercritical carbon dioxide (sCO2) has recently been proposed as a promising alternative to
conventional fluids for thermal management applications because of its unique thermophysical properties
near the critical point. Jet impingement is also recognized as one of the most effective configurations for
high intensity heat transfer. Therefore, utilizing the favorable thermophysical properties of sCO2 in
microscale jet impingement may lead to state-of-the-art high-heat-flux thermal management solutions.
However, the effect of the thermophysical property variation in the pseudo-critical temperature range on
the heat transfer behavior of such systems is not well understood. To address this need, computational
simulations of both laminar and turbulent sCO2 microscale jet impingement are conducted. Appropriate
high-fidelity methods, such as Large Eddy Simulations (LES) for turbulent cases, are employed and
validated with relevant data for conventional fluid flows. Following a parametric study, the results obtained
are used to explore the effect of the variation of the thermophysical properties in the pseudo-critical range
on the heat transfer behavior at the stagnation point (x/W=0), stagnation zone (-1<x/W<1), and at various
positions in the vicinity of the stagnation zone (-7<x/W<7). The studied range of conditions span reduced
pressures of 𝑃𝑟 = 1.03 and 1.1, Reynolds numbers 𝑅𝑒𝑖𝑛 = 225 − 11,000, dimensionless jet lengths 𝐻
𝑊=
2 − 4, jet inlet temperatures 𝑇𝑖𝑛 = 294 − 330𝐾, and impingement plate temperatures 𝑇𝑝𝑙𝑎𝑡𝑒 = 270 −
370 K. The turbulent simulations reveal varying spatial distributions of heat transfer coefficient with
different jet and surface temperatures, which can be explained in terms of the temperature-dependent
properties of sCO2. Zones of heat transfer deterioration are observed for both laminar and turbulent flows
due to pseudo-film boiling, in which gas-like fluid concentrates near the heated wall. Finally, enhanced heat
transfer performance is observed when both the jet inlet and impingement plate temperatures are in the
pseudo-critical temperature range.
iv
TABLE OF CONTENTS
LIST OF FIGURES ...................................................................................................................... v LIST OF TABLES ........................................................................................................................ vii LIST OF EQUATIONS ................................................................................................................ viii Acknowledgements ....................................................................................................................... ix
Chapter 1 Introduction and Literature Review ...................................................................................... 1
1.1 Applications for supercritical jet impingement heat transfer................................................... 2 1.2 Supercritical fluid thermophysical properties and transport phenomena................................. 3 1.3 Configurations, transport processes, and trends in jet impingement heat transfer .................. 5 1.4 Summary of investigations on supercritical fluid jets ............................................................. 9 1.5 Objectives and Approach ........................................................................................................ 11
Chapter 2 Computational Framework, Validation, and Numerical Uncertainty .................................... 14
2.1 Governing equations, turbulence modelling, and property algorithms ................................... 15 2.2 Description of domain modelling, discretization, and boundary conditions ........................... 16 2.3 Numerical Approach ............................................................................................................... 25
2.4 Validation and Numerical Errors .................................................................................... 30 2.4.1 Laminar Simulation ..................................................................................................... 30 2.4.2 Turbulent Simulation ................................................................................................... 37
Chapter 3 sCO2 Laminar Slot Jet Impingement ..................................................................................... 44
3.1 Effect of Reynolds Number .................................................................................................... 47 3.2 Effect of Dimensionless Jet Length (H/W).............................................................................. 52 3.3 Effect of Impingement Plate (Tplate) and Jet Inlet (Tin) Temperatures ..................................... 54 3.4 Effect of Reduced Pressure (Pr) .............................................................................................. 59 3.5 Conclusion .............................................................................................................................. 61
Chapter 4 sCO2 Turbulent Slot Jet Impingement .................................................................................. 62
4.1 Effect of Reynolds Number .................................................................................................... 65 4.2 Effect of Dimensionless Jet Length (H/W).............................................................................. 69 4.3 Effect of Impingement Plate (Tplate) and Jet Inlet (Tin) Temperatures ..................................... 73 4.4 Effect of Reduced Pressure (Pr) .............................................................................................. 77 4.5 Conclusion .............................................................................................................................. 79
Chapter 5 Conclusions and Future Research Recommendations ........................................................... 81
5.1 Conclusions ............................................................................................................................. 82 5.2 Future Research Recommendations ........................................................................................ 83
Bibliography .......................................................................................................................................... 84
v
LIST OF FIGURES
Figure 1-1. Specific heat capacity 𝑐𝑝 of sCO2 at 𝑃 = 8.1 MPa compared with
conventional subcritical liquid coolants .................................................................................. 4
Figure 1-2. The flow regions of a jet impingement configuration . ................................................. 6
Figure 2-1. Cross-section of the nozzle . ....................................................................................... 16
Figure 2-2. Schematics of the geometries for the (a) turbulent and (b) laminar
simulations. ............................................................................................................................ 19
Figure 2-3. Detail view of the T-junction zone with (a) Block-Structured, (b)
Rectilinear, and (c) Hybrid meshing approaches. .................................................... 21
Figure 2-4. Schematics of the boundary patches for the (a) turbulent and (b)
laminar simulations. ............................................................................................................... 24
Figure 2-5. buoyantPimpleFoam algorithm flowchart. .................................................................. 26
Figure 2-6. buoyantSimpleFoam algorithm flowchart. .................................................................. 26
Figure 2-7. Lateral variation of the heat transfer coefficient for the (a) steady
symmetric and (b) transient full-domain simulations. ........................................................... 32
Figure 2-8. Comparison of the extrapolated lateral variation of the heat transfer
coefficient for the steady and transient simulations. .............................................................. 33
Figure 2-9. Transient simulation (a) velocity and (b) temperature fields. ..................................... 34
Figure 2-10. Steady simulation (a) velocity and (b) temperature fields. ....................................... 35
Figure 2-11. Comparison of the lateral variation of the heat transfer coefficient
for the Rectilinear and Block-Structured simulations. ........................................................... 38
Figure 2-12. Lateral variation of the heat transfer coefficient of the different
meshes and the extrapolated values with numerical uncertainties. ........................................ 40
Figure 2-13. Lateral variation of the heat transfer coefficient of the different
meshes and the extrapolated values with numerical uncertainties. ........................................ 41
Figure 2-14. Fine simulation instantaneous (a) velocity and (b) temperature
fields. ...................................................................................................................................... 42
Figure 3-1. Extrapolated lateral variation of the heat transfer coefficient for Cases
1-3. ......................................................................................................................................... 48
Figure 3-2. Case 2 (Rein = 225) fine simulation instantaneous (a) velocity, (b)
temperature, and (c) density fields. ........................................................................................ 49
vi
Figure 3-3. Fine mesh constant and variable property simulations lateral variation
of heat transfer coefficient for Case 1. ................................................................................... 51
Figure 3-4. Fine mesh constant and variable property simulations lateral variation
of heat transfer coefficient for Case 3. ................................................................................... 51
Figure 3-5. Extrapolated lateral variation of the heat transfer coefficient for Cases
1,4, and 5. ............................................................................................................................... 53
Figure 3-6. Extrapolated lateral variation of the heat transfer coefficient for Cases
1,6, and 7. ............................................................................................................................... 55
Figure 3-7. Extrapolated lateral variation of the heat transfer coefficient for Cases
6,8, and 9. ............................................................................................................................... 56
Figure 3-8. Fine mesh constant and variable property simulations lateral variation
of heat transfer coefficient for Case 6. ................................................................................... 58
Figure 3-9. Fine mesh constant and variable property simulations lateral variation
of heat transfer coefficient for Case 7. ................................................................................... 58
Figure 3-10. Extrapolated lateral variation of the heat transfer coefficient for
Cases 1 and 10. ...................................................................................................................... 60
Figure 4-1. Extrapolated lateral variation of the heat transfer coefficient for Cases
1-3. ......................................................................................................................................... 66
Figure 4-2. Case 2 (Rein =2750) fine simulation instantaneous (a) velocity, (b)
temperature, and (c) density fields. ........................................................................................ 67
Figure 4-3. Medium mesh constant and variable property simulations lateral
variation of heat transfer coefficient for Case 3. .................................................................... 68
Figure 4-4. Extrapolated lateral variation of the heat transfer coefficient for Cases
1,4, and 5. ............................................................................................................................... 70
Figure 4-5. Fine simulation instantaneous velocity fields for (a) Case 4 (H/W=1),
(b) Case 1 (H/W=2), and (c) Case 5 (H/W=4). ...................................................................... 71
Figure 4-6. Medium mesh constant and variable property simulations lateral
variation of heat transfer coefficient for Case 4. .................................................................... 72
Figure 4-7. Lateral variation of the heat transfer coefficient for Cases 1,6, and 7. ....................... 74
Figure 4-8. Lateral variation of the heat transfer coefficient for Cases 6,8, and 9. ....................... 75
Figure 4-9. Case 6 fine simulation instantaneous (a) temperature and (b) density
fields. ...................................................................................................................................... 76
Figure 4-10. Lateral variation of the heat transfer coefficient for Cases 1 and 10. ....................... 78
vii
LIST OF TABLES
Table 2-1. Dimensions of the computational domains................................................................... 18
Table 2-2. Summary of mesh quality parameters. ......................................................................... 21
Table 2-3. Summary of the boundary conditions. .......................................................................... 23
Table 2-4. Summary of the discretization schemes. ...................................................................... 28
Table 2-5. Summary of the iterative solution and stopping criteria. .............................................. 29
Table 2-6. Summary of steady and transient simulation meshes. .................................................. 31
Table 2-7. Summary of the extrapolated heat transfer coefficients for the steady
and transient simulations. ....................................................................................................... 33
Table 2-8. Summary of the Perfect and Cartesian simulation details. ........................................... 37
Table 2-9. Summary of the heat transfer coefficients for the Perfect and Cartesian
simulations for the baseline mesh. ......................................................................................... 38
Table 2-10. Summary of the validation study meshes. .................................................................. 40
Table 2-11. Summary of the heat transfer coefficients for the different
simulations and the extrapolated values. ................................................................................ 41
Table 3-1. Summary of the laminar sCO2 simulation cases. .......................................................... 46
Table 3-2. Summary of the heat transfer coefficients for Cases 1-3. ............................................ 48
Table 3-3. Summary of the heat transfer coefficients for Cases 1,4, and 5. .................................. 53
Table 3-4. Summary of the heat transfer coefficients for Cases 1,6, and 7. .................................. 55
Table 3-5. Summary of the heat transfer coefficients for Cases 6,8, and 9. .................................. 56
Table 3-6. Summary of the heat transfer coefficients for Cases 1 and 10. .................................... 60
Table 4-1. Summary of the turbulent sCO2 simulation cases. ....................................................... 64
Table 4-2. Summary of the heat transfer coefficients for Cases 1-3. ............................................ 66
Table 4-3. Summary of the heat transfer coefficients for Cases 1,4, and 5. .................................. 70
Table 4-4. Summary of the heat transfer coefficients for Cases 1,6, and 7. .................................. 74
Table 4-5. Summary of the heat transfer coefficients for Cases 6,8, and 9. .................................. 75
Table 4-6. Summary of the heat transfer coefficients for Cases 1 and 10. .................................... 78
viii
LIST OF EQUATIONS
Eqn. 2-1 ………………………………………………………………………(15)
Eqn. 2-2 ………………………………………………………………………(15)
Eqn. 2-3 ………………………………………………………………………(15)
Eqn. 2-4 ………………………………………………………………………(28)
ix
Acknowledgements
I would like to thank Dr. Alexander S. Rattner, who served as my adviser and mentor, for his
insightful guidance, understanding, and warm support during my Masters studies at The Pennsylvania State
University. I would also like to thank all of my current and previous research group members at the
Multiscale Thermal Fluids and Energy (MTFE) Laboratory. Specifically, I would like to thank Mahdi Nabil,
for his assistance in setting up and learning OpenFOAM, Sanjay Adhikari and Christopher Greer for
providing an instructive research environment, and Nosherwan Adil, Ibrahim Elhagali, and Micheal Fair
for being supportive peers. I would also like to thank Professor Robert Kunz for serving as the committee
member and for his insightful suggestions for the project.
I would also like to thank my father, mother, family members, and all friends for their invaluable
support for the duration of my life. This work was supported, in part, through generous support from the
US National Science Foundation (award CBET-160453). This report was prepared as an account of work
sponsored by an agency of the United States Government. Neither the United States Government nor any
agency thereof, nor any of their employees, makes any warranty, express or implied, or assumes any legal
liability or responsibility for the accuracy, completeness, or usefulness of any information, apparatus,
product, or process disclosed, or represents that its use would not infringe privately owned rights.
References herein to any specific commercial product, process, or service-water by trade name, trademark,
manufacturer, or otherwise does not necessarily constitute or imply its endorsement, recommendation, or
favoring by the United States Government or any agency thereof. The views and opinions of authors
expressed herein do not necessarily state or reflect those of the United States Government or any agency
thereof.
1
Chapter 1
Introduction and Literature Review
2
Modern high heat-flux thermal management technologies demand reliable, efficient, and compact
systems. Conventional single-phase technologies require high pumping power due to the low fluid thermal
capacity while two-phase technologies are susceptible to boiling instabilities and wall dry-out. Supercritical
carbon dioxide (sCO2) has recently been proposed as a promising alternative because of its high specific
heat capacity, thermal conductivity, and low viscosity at near-critical conditions [1]. Jet impingement is
also recognized as one of the most effective configurations for high intensity heat transfer. Therefore,
leveraging the favorable thermophysical properties of sCO2 in jet impingement may lead to thermal
management solutions that can outperform existing solutions. Conversely, local heat transfer coefficients
can vary significantly in jet impingement systems due the significantly different transport processes in the
stagnation and wall jet regions [2]. At certain conditions, the sharp variation of supercritical fluid properties
with temperature could exacerbate these variations.
Following a brief introduction of supercritical fluid properties and jet impingement heat transfer,
a review of available relevant experimental and computational studies on supercritical fluid jet transports
is presented. Finally, the approach and the research objectives of this dissertation research are discussed.
1.1 Applications for supercritical jet impingement heat transfer
High heat flux thermal management technologies are crucial components of many engineering
systems, including microelectronic devices [3][4] and solar power production [5][6]. Further increasing
power density levels in miniature integrated circuits reduces the performance of these devices unless
accompanied with significant advances in cooling technologies [3]. Furthermore, the performance of
photovoltaic cells is not only governed by the mean device temperature but also the temperature
distribution; hence the need for thermal management technologies that produce high and uniform heat
transfer rates [5].
The current trend of miniaturized integrated circuits and nanoscale electronics has led to hundreds
of millions of transistors on a chip area only a few squared centimeters. In literature, the barrier preventing
3
further scaling of this trend is known as the “power problem”, which is related to increased chip power
densities, heat generation, and temperatures that prevent reliable operation [3]. Therefore, localized
hotspots of higher heat generation and hence temperatures have become the focus of chip designers [7]. In
one investigation into embedded cooling thermal management technologies, microfluidic jet impingement
combined with diamond substrates allowed up to four times higher power output. [4].
Cooling photovoltaic panels is critical for enabling concentrated photovoltaic (CPV) technology.
For CPV systems, there is an uneven distribution of the radiation flux and therefore a non-uniform
temperature distribution across the photovoltaic panel. This elevated and non-uniform panel temperature
significantly decreases the system’s efficiency and can cause structural damage due to thermal stresses [6].
Jet impingement cooling has been used to cool photovoltaic panels, achieving an average heat transfer
coefficient in the range of 105 𝑊𝑚−2𝐾−1 with a temperature reduction between 30 − 70°𝐶 depending on
the concentration ratio [8]. The effect of the temperature profile of the CPV cell has also been studied,
showing that a Gaussian temperature profile improved the electrical performance by 1.52% while an anti-
Gaussian profile decreased the performance by 3% [9].
1.2 Supercritical fluid thermophysical properties and transport phenomena
A supercritical fluid has a thermodynamic state above its critical temperature (𝑇cr,CO2=
304.25 K) and pressure (𝑃cr,CO2= 7.39 MPa) [10]. For a fluid at a constant subcritical pressure, the
thermophysical properties exhibit discontinuities when the saturation temperature is reached; associated
with the phase change process. However, for a fluid at a constant supercritical pressure, the thermophysical
properties have a smooth variation concentrated within a small temperature range called the pseudo-critical
range (Figure 1-1) over which the fluid characteristics vary from liquid-like to gas-like [11]. Outside this
region, the thermophysical properties are less sensitive to temperature changes. The specific heat peaks in
4
this region; the temperature corresponding to the highest specific heat at a given supercritical pressure is
called the pseudo-critical temperature (𝑇𝑝𝑐). The value of 𝑇𝑝𝑐 increases with pressure [12]. At higher
reduced pressures (Pr), defined as the ratio of the operating pressure to the critical pressure, the pseudo-
critical temperature span widens and property variations with temperature are more gradual.
Figure 1-1. Specific heat capacity 𝑐𝑝 of sCO2 at 𝑃 = 8.1 MPa compared with conventional subcritical
liquid coolants [12]
sCO2 has high specific heat capacity and thermal conductivity in the pseudo-critical range with
much lower viscosity than conventional liquid coolants. This may enable high heat flux and near-uniform
temperature thermal management technologies with low pumping power requirements [1]. For example, at
a constant pressure of 8.1MPa, if water is heated from 305–310 K, its specific heat capacity remains almost
constant at 4.17 kJ kg−1K−1, and it would acquire 20.9 kJ kg−1 of thermal energy. However, if sCO2
undergoes the same temperature change, its specific heat capacity varies from 2.9 to 29.2 kJ kg−1K−1,
enabling storage of 80.3 kJ kg−1 of thermal energy. The plot in Figure 1-1 compares the specific heat
capacity of sCO2 at a constant pressure of 8.1MPa with those of subcritical liquids commonly employed in
thermal management applications. CO2 is also nontoxic, inexpensive, and has zero net impact on global
warming [13].
Pseudocritical Region
5
If the temperature of a subcritical fluid exceeds the boiling temperature and the heat flux is greater
than the critical heat flux, a sharp decrease in heat transfer coefficient is encountered. This phenomena,
known as the “boiling crisis”, can lead to major complications including equipment failure [14].
Supercritical convection can present similar, but less severe effects close to the 𝑇𝑝𝑐. Supercritical heat
transfer deterioration can occur due to the concentration of high temperature vapor-like film near walls –
an effect similar to film boiling for subcritical fluids. Under some conditions with significant body forces
(e.g. gravity or centrifugal forces), supercritical heat transfer enhancement can occur due to low-density
gas-like fluid being removed from the heated surface [15]. It is unclear to what extent these processes occur
in jet impingement.
The first sCO2 turbulent round nozzle jet impingement studies have been reported recently and
indicate promising results in terms of enhanced and more uniform heat transfer rates compared with
conventional coolants depending on the pressure, mass flux, heat flux, and inlet temperature [17] [18].
These studies primarily attributed the heat transfer enhancement to the thermophysical property variation
in the pseudocritical region. To the extent of our knowledge, no sCO2 turbulent slot jet impingement heat
transfer studies exist. The simpler quasi-planar geometry of slot jets is more conducive to high order
turbulence-resolving simulations and may assist in isolating unique transport effects due to supercritical
property variation from other hydrodynamic factors, such as streamwise deceleration in round-jet flows.
1.3 Configurations, transport processes, and trends in jet impingement heat transfer
Impinging jet flow systems can be classified as free-surface, submerged and confined, or
unconfined [18]. Free-surface jets are those where the fluid jet is different from the ambient fluid, resulting
in a free surface separating the two fluids. In submerged jet impingement, the impinging fluid is the same
as the ambient fluid. In confined jet impingement, the outflow is bounded between the target plate and a
closed upper boundary [19]. The presence of a confining plate results in a complex flow where the fluid
6
free-jet behavior is coupled to the behavior of the fluid between the two confining plates [20]. The confined
jet configuration is anticipated to be the most relevant for sCO2 applications as the thermal management
devices must be sealed and compact to manage high working pressures.
Impinging jet flows are generally divided into three main regions: the free jet, the stagnation, and
the wall jet regions (Figure 1-2) [21]. The jet exits from the nozzle with a velocity and turbulence
characteristics which mainly depend on the nozzle geometry [22]. After the jet exists the nozzle, the jet can
be far away from the impingement surface so that the jet acts as a free submerged jet. As the free jet travels,
the shear layer widens causing Kelvin-Helmholtz instabilities and roll-up of vortices [23]. However, the jet
interior remains unaffected by the momentum exchange. This region where the original flow velocity is
retained is called the potential core. For both laminar and turbulent flows, the length of the potential core
is around 6-7 nozzle diameters for round jets and 4.7-7.7 nozzle widths for slot jets [24].
Figure 1-2. The flow regions of a jet impingement configuration [25].
7
As the shear layer widens, the potential core vanishes and there is the decaying jet region where
the centerline velocity continuously decreases [25]. The jet finally becomes fully developed and attains a
Gaussian profile. The free jet region is comprised of the potential core, decaying jet, and fully developed
regions. The presence of the different free jet regions depends on the jet exit velocity profile, turbulence
intensity, and the distance from the nozzle to the impingement plate. If the nozzle exit plane is close to the
impingement plate, the fully developed and the decaying jet regions may not be present, and the jet velocity
profile does not significantly evolve [2].
Once the flow reaches the impingement plate, the jet loses its axial velocity and undergoes an abrupt
change of direction in a flow region called the stagnation region. In the stagnation region, the static pressure
sharply increases due to the axial velocity drop and then decreases as the flow accelerates along the
impingement plate. Furthermore, the stagnation region is characterized by very high normal and shear
stresses, making it the most complex flow region in a jet impingement configuration [23]. The stagnation
region typically extends 1 nozzle diameter in each direction along the impingement plate [26]. Also, the
boundary layer in the stagnation region is very thin due to the large stagnation pressure and has nearly
constant thickness; therefore, it is a region of very high heat transfer coefficient [25].
After the jet deflects in the stagnation region, the flow spreads radially parallel to the impingement
plate. The momentum exchange between the stagnation flow, the quiescent fluid, and the impingement
plate leads to the development of the wall jet region [27]. The velocity along the impingement plate
accelerates from zero, at the stagnation point, to a maximum value in the wall jet region. This maximum in
the velocity occurs at a transverse distance of approximately one nozzle diameter [28]. As the wall jet
travels along the impingement plate, it entrains flow causing the jet and the boundary layer to grow in
thickness. This leads to less effective convective heat transfer. The local heat transfer coefficient along the
impingement plate is a complex function of numerous parameters including Reynolds number, nozzle-to-
target spacing, distance from the stagnation point, and the nozzle geometry and exit conditions [23].
Gardon and Akfirat [26] studied the effect of the nozzle-to-plate spacing on the stagnation zone
heat transfer and the lateral variation of the heat transfer coefficient for a confined and submerged jet
8
impingement configuration. They observed that for turbulent flow, the stagnation point heat transfer rate
exhibits a maximum at a dimensionless jet length, defined as the ratio of the nozzle-to-plate spacing to
nozzle width (H/W), of eight and decreases with further increase or decrease of the dimensionless jet length.
This non-monotonic variation is explained by the tradeoff between the centerline turbulence and the
centerline arrival velocity. As the dimensionless jet length increases, the centerline turbulence increases,
which leads to an increase in the heat transfer coefficient, until the jet reaches an almost constant turbulence
level. However, the centerline velocity starts decreasing once the jet is past the potential core at around
(H/W = 5), which leads to a decrease in the heat transfer coefficient. Therefore, the maximum heat transfer
coefficient observed at (H/W = 8) represents a point where effect of the jet’s turbulence on the heat transfer
augmentation is not masked by the decrease of the centerline arrival velocity. At greater dimensionless jet
lengths (H/W > 8), the effect of the diminished velocity dominates the effect of the almost constant jet
turbulence, resulting in a gradually decreasing stagnation point heat transfer coefficient. This also means
that the effect of the nozzle exit condition, such as the velocity profile and the turbulence intensity, on heat
transfer is more prominent at lower dimensionless jet lengths (H/W < 8).
The lateral variation of heat transfer coefficient has also been found to have non-monotonic
variation. In the vicinity of the stagnation point (-1 < x/W < 1), the heat transfer coefficient is approximately
uniform due to the nearly constant thickness of the boundary layer. At higher dimensionless jet lengths
(H/W > 8), the heat transfer coefficient gradually decreases outside the stagnation region. However, at lower
dimensionless jet lengths (H/W<8), a secondary peak in heat transfer coefficient is observed in the vicinity
of (x/W = ±7). Gardon and Akfirat [27] attribute this peak to the transition from laminar to turbulent
boundary layer. The secondary peak in heat transfer coefficient at lower dimensionless jet lengths is also
observed in literature, both experimentally [30][31] and computationally [32][33] .
Vortex dynamics also play a fundamental role in the unsteady heat transfer behavior of jet
impingement configurations. Fluctuations in the stagnation zone heat transfer can be as high as 20 percent
of the time-averaged value [33]. This unsteadiness is due to primary vortices; which originate from the
nozzle exit due to the shear layer instabilities. The primary vortices then deflect from the impingement plate
9
causing an unsteady separation in the wall jet and as a result, secondary vortices are formed. The separation
in the wall jet along with the interaction of the primary and secondary vortices with the impingement plate
dictate the lateral heat transfer coefficient variation near the stagnation zone. Therefore, the secondary peak
in heat transfer coefficient can also be viewed as a product of flow reattachment [34].
1.4 Summary of investigations on supercritical fluid jets
Supercritical jets have only emerged as a subject of interest recently, and have been mainly been
studied in the context of understanding their mixing behavior, typically for supercritical fluid injection, and
in impingement configurations for rock fracture. Only a couple studies have explored the potential for
supercritical jet impingement thermal management. Mixing at supercritical pressures is a key process in
emerging combustion systems [35]. The effect of supercritical pressures on the turbulent mixing, jet
breakup, and the flame structure is an important factor in determining the efficiency of engines [36].
Supercritical jets are also employed in impingement configurations for hydraulic fracturing [37]. The
variation of the thermophysical properties in the pseudo-critical region is proven to have an effect on
particle-carrying ability and perforation performance of pressurized jets [38]. This thermophysical property
variation n is also shown to produce an ultra-high heat flux in jet impingement cooling applications
[16].
Chong et al. [36] used Direct Numerical Simulations (DNS) to study the turbulent mixing and
flame structure of jets at supercritical pressures. Two different inflow configurations were studied: one
including a jet with co-flow and another with a jet and annular with co-flow. They showed that the jet case
had a much lower maximum temperature than the annular case. Furthermore, local hot spots were present
due to inadequate dilation, hence demonstrating the sensitivity of supercritical flames to inflow conditions.
Another experimental study by Segal and Polikhov [35] focused on the effect of the surrounding gas
pressure and temperature on jet breakup of a subcritical, supercritical, and transcritical liquid jet injected in
a quiescent gaseous environment. For subcritical conditions, pronounced ligament formation was observed.
10
Furthermore, the gas inertia and surface tension forces were the controlling factors. However, there was no
ligament formation for the supercritical jet, indicating that surface tension does not contribute to the jet
breakup. Their simulations showed good agreement with experimental results available in literature for
subcritical mixing. However, their approach was unsuccessful for predicting breakup in the supercritical
and transcritical regimes. Further research is needed to understand the breakup mechanisms in supercritical
jets.
sCO2 has been adopted in many oil and gas industry processes due to its unique thermophysical
and chemical properties [39]. Du et al. [40] experimentally investigated the various factors that influence
the rock erosion performance of sCO2 jets. They concluded that there is an optimal nozzle diameter and
spacing between the nozzle exit and the specimen to achieve the optimal rock erosion capacity. An
experimental investigation by Rothenfluh et al. [41] studied the use of supercritical water jet for rock
fragmentation. They sought to investigate the heat transfer performance of the supercritical water jet and
therefore implemented a confined jet impingement configuration in which the jet temperature is higher than
the impingement plate. They found that the specific heat at the 𝑇𝑝𝑐 dominates the heat transfer performance
even if the inlet jet temperature is much greater than 𝑇𝑝𝑐 for an impingement plate temperature below 𝑇𝑝𝑐.
A computational study by the same authors [42] included a RANS simulation of a supercritical water jet
injected into a subcritical water bath. The purpose of the investigation was to assess the ability of RANS
models to capture the sharp thermophysical property variation in the pseudo-critical range. The researchers
discovered that using a constant turbulent Prandtl number led to an overprediction of the thermal
conductivity near the pseudo-critical range. This suggests that higher fidelity simulation approaches (LES
or DNS) may be needed for some turbulent sCO2 flows.
Few studies have been reported on supercritical carbon dioxide jet impingement heat transfer [17].
Joo-Kyun and Toshio [43] computationally studied axisymmetric laminar jet impingement cooling of an
isothermal plate with sCO2. They found that the heat transfer coefficient was higher when the inlet
temperature of the fluid jet was closer to the 𝑇𝑝𝑐. Chen et al. [16] experimentally studied the heat transfer
11
characteristics of round jet impingement cooling with sCO2. They found that the heat transfer coefficient
increases with heat flux up to a limit, but then starts decreasing as the heat flux rises. They also found that
the stagnation point heat transfer coefficient initially increases with increasing inlet temperature and then
sharply decreases due to the thermophysical property variation in the pseudo-critical range. Chen et al.
[17] also performed conjugate heat transfer computations of this configuration using a RANS SST k-𝜔
turbulence model. They showed that radial conduction in the impingement plate cannot be neglected for
higher surface heat fluxes. Furthermore, they concluded that the high specific heat capacity of the fluid near
the 𝑇𝑝𝑐 maximized the average heat transfer coefficient for a given heat flux. To the extent of our
knowledge, the only sCO2 slot jet impingement investigation is a laminar one by Martin et al. [44]. Their
study focused on implementing and validating an equation of state that captures the thermophysical
property variation in the pseudo-critical range.
1.5 Objectives and Approach
In this thesis, the following fundamental research questions about sCO2 slot jet impingement will
be addressed through simulations:
1) What are the effects of variations in fluid thermophysical properties, in the pseudo-critical
range, on slot jet impingement heat transfer performance? A parametric study has been
conducted to quantify the role of Reynolds number (𝑅𝑒 =𝜌∗𝑊∗𝑈
𝜇= 225 − 11,000), impingement
plate temperature (𝑇𝑝𝑙𝑎𝑡𝑒 = 270 − 370 𝐾), fluid inlet temperature (𝑇𝑖𝑛 = 294 − 330 𝐾),
dimensionless jet length (𝐻
𝑊= 2 − 4), and reduced pressure (𝑃𝑟 = 1.03 𝑎𝑛𝑑 1.1) on sCO2 heat
transfer performance in a microscale slot jet impingement configuration. The heat transfer
coefficient is reported for the stagnation point, directly under the impinging jet, the stagnation
zone, the area between (-1 < x/W < 1), and the lateral variation in the wall jet region, at different
12
locations between (-7 < x/W < 7). Following the convention of sCO2 convection heat transfer
performance studies, the dimensional heat transfer data is reported [45]. This is done to avoid
misleading trends in dimensionless numbers due to the sharp variation of thermophysical properties
in the pseudo-critical range. For example, a change in temperature may result in a higher heat
transfer coefficient but a lower Nusselt number if the thermal conductivity has a larger decrease
relative to the heat transfer coefficient.
2) Under what ranges of conditions do we observe, if at all, heat transfer deterioration? As
discussed earlier, heat transfer deterioration has been observed in supercritical convection due to
the concentration of high temperature vapor-like film near walls. This occurs because the
thermophysical properties of supercritical fluids vary sharply in the pseudo-critical range. In this
thesis, I attempt to determine the range of conditions that lead to this phenomenon near the
impingement plate and in the vicinity of the stagnation region (-7 < x/W < 7).
To address the aforementioned questions, simulation approaches are implemented and validated
for laminar and turbulent conditions. For laminar simulations, the quasi-steady and symmetric nature of the
stagnation region and its vicinity is exploited. This enables a significant reduction in the computational cost
with minimal losses in accuracy.
For the turbulent simulations, there is a trade-off between the accuracy and reliability of the
modelling approach and the computational cost. Direct Numerical Simulations (DNS) offer the greatest
accuracy but are limited to lower Reynolds numbers and are not suitable for expansive studies. Reynolds-
Averaged Navier-Stokes (RANS) models are less computationally expensive, but are not suitable for the
complex features present in a jet impingement configuration including vorticial structures, intrinsic
unsteadiness, and strong streamline curvature [46]. Dutta et al. [47] compared different RANS models in
terms of predicting jet impingement heat transfer. They show that some models do not match experimental
data well, some do not predict a secondary spike in Nusselt number, and others show a “false” secondary
13
peak in Nusselt number at higher nondimensional jet lengths. Furthermore, RANS models employ multiple
fitting coefficients that have been developed for flows with relatively uniform fluid properties. Therefore,
the Large Eddy Simulation (LES) turbulence modelling approach is used with a wall-adapting local eddy
viscosity (WALE) sub-grid scale model [48]. The WALE model was reported to have good accuracy for a
jet impingement configuration with an improved prediction of second-order moments [49]. Furthermore,
due to very thin thermal boundary layers in supercritical flows, the first-cell y+ value should be well below
unity [50]. Therefore, a wall-resolved LES turbulence model is implemented for this study.
Due to the lack of computational and experimental studies on sCO2 slot jet impingement heat
transfer, the computational frameworks are validated against air slot jet impingement studies in literature.
For the validation simulations, the fluid is assumed to be incompressible with constant properties.
Following the validation, the sCO2 is assumed to be pseudo-incompressible, meaning that the
thermophysical properties vary with temperature only. This implies that the variation in pressure is small
relative to the mean pressure, as is the case for these studies. To capture the variable properties, curve fits
developed by Nabil [12] for the specific heat capacity, the dynamic viscosity, and Prandtl number were
employed. However, it should be cautioned that there is still significant uncertainty in some sCO2 material
properties at near-critical conditions [51].
14
Chapter 2
Computational Framework, Validation, and
Numerical Uncertainty
15
In this chapter, the details of the laminar and turbulent simulation geometries, domain and
numerical discretization, boundary conditions, governing equations, and solution and property algorithms
are provided. Furthermore, validation and numerical uncertainty studies are presented.
2.1 Governing equations, turbulence modelling, and property algorithms
To study laminar and turbulent sCO2 jet impingement heat transfer, a pseudo-incompressible model
is adopted. The pseudo-incompressible assumption means that the density varies with temperature only.
This implies that the variations in pressure are negligible relative to the mean pressure. The governing
continuity, momentum, and energy equations for velocity (𝑢), pressure (𝑝), and enthalpy (ℎ) are:
𝜕𝜌
𝜕𝑡+
𝜕
𝜕𝑥𝑖(𝜌𝑢𝑖) = 0 (2-1)
𝜕(𝜌𝑢𝑖)
𝜕𝑡+
𝜕
𝜕𝑥𝑗(𝜌𝑢𝑖𝑢𝑗) = −
𝜕𝑝
𝜕𝑥𝑖+
𝜕
𝜕𝑥𝑗[(𝜇 + 𝜇SGS)
𝜕𝑢𝑖
𝜕𝑥𝑗] + 𝐹𝐵
(2-2)
𝜕(𝜌ℎ)
𝜕𝑡+
𝜕(𝜌𝐾)
𝜕𝑡+
𝜕
𝜕𝑥𝑖(𝜌𝑢𝑖ℎ) +
𝜕
𝜕𝑥𝑖(𝜌𝑢𝑖𝐾) −
𝜕𝑃
𝜕𝑡=
𝜕
𝜕𝑥𝑗[(𝛼 + 𝛼SGS)
𝜕ℎ𝑖
𝜕𝑥𝑗] (2-3)
The gravitational body force term is not included in the momentum equation (Equation 2-2), to
isolate the direct effects of thermophysical property variations from buoyancy effects, which are outside
the scope of this investigation. This choice is typical for jet impingement simulations (e.g., [46][52]), as
buoyancy forces are typically small relative to inertial, viscous, and turbulent forces for such flows. The
kinetic energy (K) is included in the energy equation (Equation 2-3) for consistency, although it contributes
only minor effects to the flow. The pressure-work term (𝜕𝑃
𝜕𝑡) is included because an enthalpy-based energy
equation is used rather than an internal-energy-based formulation. For the turbulent simulations, the 𝜇𝑆𝐺𝑆
and 𝛼𝑆𝐺𝑆 terms, which represent sub-grid scale (SGS) eddy diffusivities, are evaluated from the WALE
16
LES model [48]. These terms are zero in the laminar simulations. In Equation 2-2, the 𝐹 𝐵 represents a body
force term. This is a uniform mean pressure gradient applied on the isolated channel domain (see Section
2.2) to achieve a specified mean velocity. This body force drives the flow in the channel domain which is
then mapped to the inlet of the jet impingement arrangement.
The local sCO2 thermophysical properties (𝜌, 𝑐𝑝, 𝜇, 𝑘) are evaluated at each solution time step
explicitly. To capture the variable properties, curve fits developed by Nabil [12] for the specific heat
capacity, dynamic viscosity, and Prandtl number were employed. The thermal conductivity was obtained
from the expression 𝑘 = 𝑐𝑝𝜇 Pr⁄ and the density was obtained from the Peng-Robinson equation of state
[53]. For comparable operating conditions, Nabil found average absolute deviation (AAD) between the
curve fits and reference data for 𝑐𝑝, 𝜇, and Pr of 5%, 1%, and 2%, respectively.
2.2 Description of domain modelling, discretization, and boundary conditions
The computational domains are based on the test section in the single slot impinging two-
dimensional jet on an isothermal flat plate investigation by Gardon and Akfirat [54]. Their test section
consists of a confined air jet, emerging from a 6-inch-long slot nozzle of various widths, impinging on a 6
inch-squared electrically heated plate (Figure 2-1). This nozzle configuration results in a fully-developed
jet velocity profile at the nozzle exit. A temperature difference of 36°𝐹 was maintained between the
impinging jet at the nozzle exit and the impingement plate.
Figure 2-1. Cross-section of the nozzle [54].
17
Converting the physical domain to a simulation domain with acceptable computational costs
requires approximations related to the impingement plate, spanwise, and nozzle lengths. The impingement
plate length dictates the location of the outlets relative to the stagnation point. If the outlets are not
sufficiently far from the stagnation point, backflow might occur, which can have detrimental effects on the
stability of the numerical solution. Therefore, the outlets are kept at a distance (x/W = ±24) from the jet
centerline. However, the required distance to prevent backflow depends on the dimensionless jet length
(H/W), therefore the boundary conditions at the outlets should be specified to handle backflow. The length
of the computational domain in the spanwise direction has been studied by different researchers and was
selected in the range of 2W-2πW for different dimensionless jet lengths (H/W) [54][55]. Since the focus of
this investigation is on lower dimensionless jet lengths (H/W = 2-4), the length of the spanwise direction is
specified to be πW to accommodate the periodic boundary conditions.
The length of the nozzle dictates the jet exit velocity profile, which has a significant effect on the
stagnation zone heat transfer at lower dimensionless jet lengths (H/W < 8). For the purpose of this
investigation, a fully-developed velocity profile is to be ejected from the nozzle exit, similar to the
experimental setup of [54]. Therefore, an isolated domain with periodic boundary conditions is solved
concurrently with the main jet impingement configuration to provide a turbulent inlet condition. This
channel domain flow is driven by a uniform pressure gradient, which is adjusted to achieve a given mean
velocity. The outlet velocity field from the isolated channel domain is mapped to the inlet of the jet
impingement configuration domain. This method is a common strategy for generating a stochastically
varying velocity inlet condition for LES [56].
For the laminar simulations, a common strategy to reduce computational cost is to model the
problem as steady, two-dimensional, and symmetric around a mid-plane [57][58]. The effect of this
simplification on accuracy and its appropriateness is studied in later sections (Section 2.3.1). The
schematics of the simulation geometries are shown in Figure 2-2 and the corresponding dimensions are
listed in Table 2-1.
18
Table 2-1. Dimensions of the computational domains
Section Size
Slot Width (W) 3.175 mm
Domain Height (H=2W) 6.35 mm
Domain Width (𝑍 = 𝜋𝑊) 9.9 mm
Domain Length (𝐿 = 48𝑊) 152.4 mm
Detached Domain Height (HD=4W) 12.7 mm
Inlet Height (HI=2W) 6.35 mm
19
Figure 2-2. Schematics of the geometries for the (a) turbulent and (b) laminar simulations.
20
The simple slot jet domain geometry permits a fully-structured mesh. For the laminar simulations,
there is no strict y+ requirement. However, it is still necessary to have sufficient resolution in the very thin
boundary layer, especially in the stagnation zone, to capture the heat transfer behavior. For the turbulent
simulations, guidelines for relatively high Prandtl number flows [58] suggest a maximum impinging-
surface first cell ∆𝑦1+~0.3, based on the maximum wall shear stress, with a size ratio of 1.05 between
adjacent cells (stretching factor). Based on prior studies, a maximum cell size value of ∆𝑥+~16 and
∆𝑧+~35 should be maintained in the stagnation region. This is a more conservative choice of mesh
resolution which ensures minimal numerical errors due to grid resolution, which is the dominant source of
error in wall-resolved LES [59]. At the other walls, a ∆𝑦1+~1 is maintained with a 1.08 stretching factor.
Although the simple slot jet geometry can be meshed with a simple strategy, a fully regular mesh
leads to an unacceptable computational cost due to the wide range of wall shear stresses on the impingement
plate [60] and very thin cells in the flow direction in the vicinity of the T-junction (Figure 2-3). The broad
range of the wall-normal component of the wall shear stress at the impingement plate means that to maintain
∆𝑦1+~0.3 in the stagnation zone, other areas will have extremely low ∆𝑦1
+. For example, for one case, a
∆𝑦1+~0.3 is sustained at the stagnation region, but the minimum ∆𝑦1
+ for the impingement plate is almost
two orders of magnitudes lower (∆𝑦1+~0.006). A similar variation is also observed in the spanwise (z)
direction.
For a conventional fully-structured mesh (Rectilinear Mesh), cells will become very thin near in
the flow direction near the T-junction, leading to a very low timestep to satisfy the CFL condition relative
to the requirements for the remainder of the domain. For example, for one of the simulations, specifying a
maximum Courant number of one lead to an average courant number an order of magnitude lower (~0.01).
One solution is to slightly modify the domain geometry by introducing a fillet at the edges of the T-junction.
The fillet radius is an order of magnitude below the nozzle width to ensure negligible effects on the jet exit
velocity. This eliminates the corner discontinuity in the domain and allows longer cells in the streamwise
direction and hence a larger timestep. However, this larger timestep is at the cost of reduced mesh quality.
21
The Block-Structured and Hybrid meshes were completed in Salome [61] while the blockMesh utility in
OpenFOAM 1612+ [62] was used for the Perfect mesh. The different meshing approaches are shown in
Figure 2-3 and the corresponding mesh quality parameters are listed in Table 2-2.
Figure 2-3. Detail view of the T-junction zone with (a) Block-Structured, (b) Rectilinear, and (c) Hybrid
meshing approaches.
Table 2-2. Summary of mesh quality parameters.
Mesh Type Maximum Non-Orthogonality Skewness
Rectilinear 0 ~0
Block-Structured ~48 ~0.6
Hybrid ~22 ~0.2
22
Mesh non-orthogonality and skewness reduce the accuracy and stability of numerical solutions
requiring careful selection of discretization schemes (c.f. Section 2.3) [63]. Mesh non-orthogonality is
defined as the angle between the face-normal vector and the vector connecting the two cell centers that
intersect the face. Mesh skewness is a measure of the distance between the face center and the vector
connecting the centroids of the two cells that intersect the face [64]. Comparing the different meshing
approaches, the Rectilinear mesh has optimal mesh quality. However, this comes at great computational
cost due to the larger cell count and significantly smaller timestep requirement. Both the Block-Structured
and Hybrid meshes have poorer mesh quality, but they have a reduced cell number and allow larger timestep
size.
The Rectilinear mesh timestep requirement is set by the ∆𝑦1+ value near the T-junction while the
Block-Structured mesh timestep depends on the ∆𝑥+ value in the stagnation region. The Hybrid mesh
timestep is not directly limited by wall cell size. Therefore, the Block-Structured and Hybrid meshes may
require lower computational costs. Discretization schemes must be carefully selected with the meshes to
ensure stability and accuracy of the numerical solution. Furthermore, the computing time required for each
timestep, which is related to the solution algorithm and discretization, must be balanced with the timestep
size.
In practice, numerical oscillations were observed at the interface between the tetrahedral and
hexahedral cells in the Hybrid mesh. These oscillations were eliminated by using a dissipative second-order
accurate velocity discretization scheme. However, using such schemes is not recommended for LES as they
damp the smaller turbulent scales [65]. Based on these findings, the Block-Structured mesh was selected as
it incurs low-dissipation with acceptable computational cost. More details on the accuracy of the Block-
Structured mesh is provided in Section 2.3.2.
Finally, boundary conditions are defined. The main difference between the turbulent and laminar
simulations is that cyclic boundary conditions are applied in the spanwise (z) direction for the turbulent
simulation while the laminar simulation is two-dimensional. Cyclic boundary conditions are also applied
for the isolated channel-section domain in the streamwise direction. Furthermore, the complete domain is
23
discretized for the turbulent simulation while only half the domain is discretized with a symmetry boundary
condition for the laminar simulation.
The jet inlet plane and impingement plate are specified with constant temperature, while the other
walls and the outlets have zero normal heat flux boundary conditions. For velocity, no slip boundary
conditions are applied at the walls, and the jet inlet is mapped from the nearest outlet plane in the isolated
domain. At the outlets, a pressureInletOutletVelocity velocity boundary condition is applied to cope with
backflow. This boundary condition applies a zero-normal-gradient condition for outflow faces, and
switches to a fixed value based on the continuity solution for inflow [66]. For pressure, a fixedFluxPressure
boundary condition is applied at the inlet and the walls. This boundary condition specifies the pressure
gradient such that the flux on the boundary matches the velocity boundary condition [67]. At the outlets, a
uniform totalPressure boundary condition is applied (𝑝𝑡𝑜𝑡𝑎𝑙 = 𝑝𝑠𝑡𝑎𝑡𝑖𝑐 +1
2𝑢2), which is found to be stable
for cases with local or transient backflow [68]. The boundary patches for the laminar and turbulent cases
are presented in Figure 2-4 and listed in Table 2-3.
Table 2-3. Summary of the boundary conditions.
Patch/Boundary Condition Velocity Pressure Temperature
Outlet Outflow:
𝜕𝑢
𝜕𝑛= 0
Inflow: specify based on
the continuity solution
Total Pressure = 0 𝜕𝑇
𝜕𝑛= 0
Inlet Mapped from the nearest
isolated channel patch Specify the pressure
gradient such that the
flux matches the velocity
boundary condition.
𝑇 = 𝑇𝑖𝑛𝑙𝑒𝑡
Impingement Wall 𝑢 = 0 𝑇 = 𝑇𝑝𝑙𝑎𝑡𝑒
Fixed Walls 𝑢 = 0 𝜕𝑇
𝜕𝑛= 0
24
Figure 2-4. Schematics of the boundary patches for the (a) turbulent and (b) laminar simulations.
25
2.3 Numerical Approach
Simulations are performed using OpenFOAM version 1612 [62]. OpenFOAM is a finite volume,
collocated, parallelizable, open-source software for continuum mechanics. The value of quantities are
approximated as constant over the faces, limiting the accuracy to second-order on structured grids [69]. In
this section, the numerical settings such as discretization schemes and solution algorithms are discussed.
The laminar simulations are computed using buoyantSimpleFoam, a pressure-Poisson equation-
based (PPE) solver for steady flows with heat transfer in OpenFOAM. Turbulent simulations are computed
using a similar unsteady solver: buoyantPimpleFoam. The steady solver implements the Semi-Implicit
Method for Pressure-Linked Equation (SIMPLE) algorithm to solve the coupled momentum and pressure
equations [70]. The unsteady solver implements the PIMPLE algorithm, which is a combination of the
Pressure Implicit Splitting Operator (PISO) [71] and SIMPLE algorithms.
Both algorithms use segregated approaches to solve the pressure and velocity coupling in the
governing equations [63]. Both algorithms start each iteration by updating the momentum equation using
information from the previous step (momentum predictor). The algorithms then update the face fluxes,
solve the Pressure Poisson Equation to enforce continuity, and explicitly update the velocity field (pressure
correction). The energy equation is evaluated and the thermophysical properties are updated after the
momentum predictor but before the pressure correction. After the pressure correction, the density is
obtained from the equation of state and the turbulence properties are updated. The SIMPLE algorithm uses
relaxation factors to stabilize the numerical solution. For the laminar simulations, the relaxation factors are
0.7, 0.3, and 0.7 for the pressure, velocity, and enthalpy fields respectively. The PIMPLE algorithm permits
multiple iterations of the momentum predictor step with relaxation factors. However, for the purpose of this
investigation, only one momentum predictors step is used; this reduces the PIMPLE algorithm to the PISO.
The PISO algorithm involves multiple pressure correction steps per time step; executed 2-3 times for this
investigation. Both algorithms permit the use of explicit non-orthogonal corrector steps to account for
contributions from non-orthogonal neighbor cells. As the laminar simulation involve orthogonal meshes,
26
this feature is not used. For turbulent simulations, 1-2 non-orthogonal correction iterations are performed.
The flowcharts of both solvers are presented in Figures 2-5 and 2-6. The time step size is dynamically
adjusted to maintain a maximum CFL number of one in the turbulent simulations.
Figure 2-5. buoyantPimpleFoam algorithm flowchart.
Figure 2-6. buoyantSimpleFoam algorithm flowchart.
Initialize simulation data
While t < tend Do
1. Update ∆t for stability
2. t+∆t
3. Do PIMPLE loop
3.1 Form and solve the momentum equation (momentum predictor)
3.2 Form and solve the enthalpy equation
3.3 Update thermophysical properties
3.4 PISO Algorithm (pressure correction and nNonOrthogonalCorrectors)
3.4.1 Form and solve Pressure-Poisson Equation
3.4.2 Update face mass fluxes
3.4.3 Correct the velocity field
4. Update turbulence properties
5. Obtain ρ from the equation of state
Initialize simulation data
While (residuals not converged or t < tend) Do
1. t = t + 1
2. Form, relax, and solve momentum equation (momentum predictor)
3. Form, relax, and solve enthalpy equation
4. Update thermophysical properties
5. Form, relax, and solve Pressure-Poisson Equation
6. Update face mass fluxes
7. Correct the velocity field
8. Obtain ρ from the equation of state
9. Update turbulence properties
27
For the temporal discretization, a second-order implicit scheme is used for the turbulent
simulations. For the laminar simulations, second-order accurate central-difference momentum and enthalpy
advection discretization schemes are used. Since the mesh is orthogonal, no corrections are needed for
contributions from non-face-neighbor cells. However, for the turbulent simulations, non-orthogonal
corrections are needed to maintain second-order accuracy due to the reduced mesh quality. Furthermore, a
central-difference scheme is known to produce oscillations unless a very fine mesh requirement is met,
therefore slightly dissipative schemes are used for discretizing the momentum and enthalpy advection
terms. For the momentum advection, a Gauss skewCorrected filteredLinear scheme is used. This is a
second-order accurate scheme which has been shown to correctly map the energy spectrum with a slight
underprediction in the high frequency range, and it produces fewer oscillations than a central-difference
scheme [72]. For the enthalpy advection, a Gauss skewCorrected Gamma 0.1 is used. The Gamma scheme
is a Normal Variation Diminishing (NVD) based bounded second-order accurate scheme [73]. The skew-
correction applied accounts for the mesh skewness, which reduce the accuracy of face integrals to first order
[63].
Gradient terms are discretized using a second-order cell-based linear scheme for the laminar
simulations. For the turbulent simulations, a least squares scheme is utilized; which is second-order accurate
on all mesh types. For the Laplacian terms, second order accurate interpolation schemes are adopted. For
the laminar simulations, no corrections are applied for the Laplacian discretization. However, for the
turbulent simulations, a correction is applied to account for the mesh non-orthogonality, which can violate
the boundedness of the numerical solution [63]. A summary of the discretization schemes is shown in Table
2-4.
28
Table 2-4. Summary of the discretization schemes.
Finally, the solvers and convergence criteria are defined. The choice of pressure iterative solver
depends on the computational resources available. This is because multigrid iterative solvers have excellent
speedup performance but poor parallel scalability [74]. For reference, the larger turbulent simulations were
performed with up to 160-way parallelism, resulting in 60,000-120,000 cells per core. For all simulations,
a Geometric Agglomerated Algebraic Multigrid (GAMG) iterative solver with a DICGaussSeidel smoother
was used for pressure. For momentum and enthalpy, the smoothSolver with a symGaussSeidel smoother
was used. For the turbulent simulations, a time step was assumed to be converged when the absolute
pressure residual norms were of (< 10−5) and the enthalpy and momentum residual norms were (< 10−6).
For laminar simulations, the residuals were 1-2 orders of magnitude larger than those of the turbulent
simulations when the stopping criteria were met.
The turbulent simulations should be computed for sufficient time interval to obtain statistically
converged quantities. For this investigation, a Flow Through Time (FTT) is defined as the time the fluid
takes to travel along the impingement plate based on the jet exit mean velocity. If the statistics of a turbulent
simulation are allowed to settle for one FTT and then averaged for another FTT, statistical convergence of
Term/Simulation Laminar Turbulent
Temporal - 2nd order fully − implict
Gradient Cell-based linear Least Squares
Momentum Advection 2nd order accurate central difference
filteredLinear with skewness
correction
Enthalpy Advection Gamma 0.1 with skewness correction
Laplacian Linear Linear with corrections for non-
orthogonal neighbor cells
Interpolation Linear
Surface-Normal
Gradient 2nd order accurate central difference
2nd order accurate central difference
with corrections for non-orthogonal
neighbor cells
29
the flow quantities is achieved. This means that the difference between the aforementioned flow quantities
and the quantities obtained from an average twice that interval is less than 0.5% (Equation 2-4). The
relatively short simulation time is permissible as the fluid injected from the isolated channel domain to the
jet impingement configuration has a fully-developed velocity profile with converged turbulence statistics.
This is because the isolated channel has already been separately simulated then mapped to a simulation
with the jet impingement configuration. Furthermore, this investigation focuses on characterizing first-
order moment (mean) quantities, such as average heat transfer coefficients, which are less intermittent than
higher order moment quantities. The turbulent simulations also involve span-wise averaging of the flow
quantities prior to the time averaging. For the steady laminar simulations, the stopping criteria should
consider the flow field, residuals, and the monitored quantities. For this investigation, by the time the flow
travels along the impingement plate and leaves at the outlet, the stagnation point and zone heat transfer
coefficients are converged and the residuals are quasi-steady. Therefore, the stopping criteria is simply
taken as the point where the fluid leaves the domain at the outlet. A summary of the iterative procedure and
the stopping criteria is provided in Table 2-5.
(2-4)
Table 2-5. Summary of the iterative solution and stopping criteria.
Parameter/Simulation Turbulent Laminar
Pressure Solver GAMG
Momentum/Enthalpy Solver smoothSolver
Absolute Pressure Residual
Norm Tolerance 1e-5 1e-4
Absolute
Momentum/Enthalpy
Residual Norm Tolerance
1e-6 1e-5
Stopping Criteria
Settle for 1 FTT then
spanwise and temporal
average for 1 FTT
Fluid leaves the domain
|𝐹𝑇𝑇1−2 − 𝐹𝑇𝑇1−3|
𝐹𝑇𝑇1−3< 0.5%
30
2.4 Validation and Numerical Errors
In this section, validation studies are presented for both the laminar and turbulent simulations and
numerical uncertainties are quantified. In this investigation, the grid-convergence index (GCI) method is
used to asses numerical convergence and uncertainties [75].
2.4.1 Laminar Simulation
Before validating the laminar approach with experimental data [54], the suitability of the steady,
two-dimensional, and symmetric assumptions are assessed. Therefore, two sets of simulations are
performed; one transient, three-dimensional, and with a full domain while the other is steady, two-
dimensional, and with a plane of symmetry. The cases were set up using air with constant thermophysical
properties, evaluated at the jet exit temperature, and a Reynolds number (𝑅𝑒 = 𝜌∗𝑈∗𝑊
𝜇) of 450, a jet exit
temperature of 294K, and an impingement plate temperature of 330K. The characteristic length and velocity
scales are represented by the slot width and the mean jet exit velocity respectively. For the transient
simulations, the stopping criteria and solution algorithms discussed in Section 2.2 for the turbulent
simulations are used.
The numerical solutions on three different meshes were evaluated to quantify the numerical
uncertainties. A grid refinement factor between different meshes of approximately 1.4 was applied in every
direction as suggested by [76] (details in Table 2-6). Plots from both simulations for the lateral variation of
the heat transfer coefficients are shown in Figure 2-7 and then compared in Figure 2-8. The stagnation
point, zone, and mean heat transfer coefficients are compared in Table 2-6. Representative velocity and
temperature fields from the simulations are shown in Figure 2-9 and 2-10 below. Estimating the numerical
uncertainties based on convergence rates of local heat transfer coefficient values, following the approach
of [76], yields high empirical convergence rates (p>2). As the selected numerical methods are theoretically
only second order, (p=2) is assumed in the Richardson extrapolation calculations to be conservative. A
31
safety factor of 2 is used in determining the Grid Convergence Indices (GCIs) and discretization
uncertainties.
Table 2-6. Summary of steady and transient simulation meshes.
Case Steady Transient
Mesh Number of cells
Course 48,872 86,130
Medium 80,507 242,740
Fine 138,460 639,720
32
Figure 2-7. Lateral variation of the heat transfer coefficient for the (a) steady symmetric and (b) transient
full-domain simulations.
33
Figure 2-8. Comparison of the extrapolated lateral variation of the heat transfer coefficient for the steady
and transient simulations.
Table 2-7. Summary of the extrapolated heat transfer coefficients for the steady and transient simulations.
Case Steady Transient Experiment
Region Heat Transfer Coefficient [𝑊
𝑚2∗𝐾]
Stagnation Point
(x/W=0) 133 ± 1 132 ± 1 132
Stagnation Zone
(-1<x/W<1) 112 ± 1 111 ± 1 -
Mean
(-L/2<x/W<L/2) 17 ± 2 22 ± 1 -
34
Figure 2-9. Transient simulation (a) velocity and (b) temperature fields.
(a)
(b)
35
Figure 2-10. Steady simulation (a) velocity and (b) temperature fields.
(a)
(b)
36
For Figure 2-7 and 2-8, and for all plots onward, the numerical uncertainties are not plotted if they
are less than 1% to aid readability. It can be observed that the heat transfer coefficients at the stagnation
point, stagnation zone, and the in the close vicinity of the stagnation point (-4<x/W<4) for both simulations
are essentially equal. The stagnation point heat transfer coefficient from both simulations exactly match the
experimental data (Figure 2-8 and Table 2-6). The deviation between the simulation results is more
significant beyond (x/W=5). This point marks the beginning of the unsteady jet separation which can be
observed in the velocity fields of both simulations (Figure 2-9 and 2-10). For the steady simulation, the
mesh causes the simulation to converge to a different pseudo-steady state for the three different meshes.
This leads to skewed and unphysical extrapolated heat transfer coefficient values for the mean and regions
far away from the stagnation point (5<x/W<-5). Specifically, the extrapolated heat transfer coefficients at
(x/W=6) and at (x/W=7) are negative, therefore they are not shown in the plots to prevent confusion (Figure
2-7 and 2-8). Therefore, it can be concluded that the heat transfer coefficient at the stagnation point, zone,
and regions in the vicinity of the stagnation point (-4<x/W<4) can be regarded as steady, two-dimensional,
and symmetric for comparable laminar flow conditions.
37
2.4.2 Turbulent Simulation
Before validating the turbulent computational framework with experimental data available in
literature [26], the effect of the mesh choice and the discretization schemes on the accuracy of the heat
transfer data is assessed. Therefore, two simulations are performed, one using the Rectilinear mesh and
another using the Block-Structured mesh. The simulation with the Rectilinear mesh has a significantly
larger cell count, but a comparable 𝑦+ value (~0.6). Since the Rectilinear mesh is orthogonal, the same
discretization schemes as for the laminar simulations are used (c.f. Section 2.2). This case requires a smaller
timestep due to thin cells at the T-junction. The cases were set up using air with constant thermophysical
properties, evaluated at the jet exit temperature, and a Reynolds number (𝑅𝑒 = 𝜌∗𝑈∗𝑊
𝜇) of 5,500, a jet exit
temperature of 294K, and an impingement plate temperature of 330K. The characteristic length and velocity
scales are represented by the slot width and the mean jet exit velocity respectively. Details of the Block-
Structured and Rectilinear simulations are presented in Table 2-7 and the results are compared in Table 2-
8 and Figure 2-11.
Table 2-8. Summary of the Rectilinear and Block-Structured simulation details.
Case Block-Structured Rectilinear
Cell Count 12,276,396 61,685,316
Velocity Divergence
Scheme
Gauss skewCorrected
filteredLinear Gauss linear
Temperature
Divergence Scheme
Gauss skewCorrected Gamma
0.1 Gauss linear
Average Timestep 3.22-07 1.27-07
38
Figure 2-11. Comparison of the lateral variation of the heat transfer coefficient for the Rectilinear and
Block-Structured simulations.
Table 2-9. Summary of the heat transfer coefficients for the Rectilinear and Block-Structured simulations
for the baseline mesh.
Case Rectilinear Block-Structured
Region Heat Transfer Coefficient [
𝑊
𝑚2∗𝐾]
Stagnation Point (x/W=0) 406 395
Stagnation Zone
(-1<x/W<1) 385 374
39
Even though the Rectilinear mesh simulation is significantly more computationally expensive than
the Block-Structured simulation, the results are within acceptable limits and therefore justifies the proposed
Block-Structured model. The difference between the heat transfer coefficients at the various locations on
the impingement plate are mainly within 2-3%, however, this difference slightly increases farther away
from the stagnation point. This deviation is mainly due to reduced timestep and the significantly more
refined mesh in the streamwise direction.
For the validation, three simulations with different meshes were run. A refinement factor between
different meshes of approximately 1.4 was applied in every direction as suggested by [76] (details in Table
2-9). The cases were set up using air with constant thermophysical properties, evaluated at the jet exit
temperature, and a Reynolds number (𝑅𝑒 = 𝜌∗𝑈∗𝑊
𝜇) of 11,000, a jet exit temperature of 294K, and
impingement plate temperature of 330K. The characteristic length and velocity scales are represented by
the slot width and the mean jet exit velocity respectively. Plots from the simulations for the lateral variation
of the heat transfer coefficients are shown in Figure 2-12 and then compared with experimental data in
Figure 2-13. The stagnation point, zone, and mean heat transfer coefficients are compared in Table 2-9.
Representative velocity and temperature fields from the simulations are shown in Figure 2-14 below.
Numerical uncertainties for heat transfer coefficients (HTCs) were estimated following the approach of
[76]. The selected numerical methods were theoretically second order accurate, but the empirical
convergence rates for some local HTCs ranged above and below this value. To streamline analysis,
Richardson extrapolation was performed assuming second-order convergence of local quantities with a
conservative safety factor of 2 in the GCI calculations. Here, second order convergence represents an
approximate overall average behavior for different points in the simulation domain. Poor empirical
convergence rates for some local HTCs may be due to their high sensitivity to predicted locations of
separation, reattachment, or laminar-to-turbulent transition, which may shift as the mesh is further refined.
40
Table 2-10. Summary of the validation study meshes.
Figure 2-12. Lateral variation of the heat transfer coefficient of the different meshes and the extrapolated
values with numerical uncertainties.
Case Parameter
Mesh Cell Count 𝑦+ 𝑥+ 𝑧+
Coarse 2,393,160 ~2 ~30 ~72
Medium 6,214,680 ~0.9 ~21 ~48
Fine 15,114,528 ~0.3 ~18 ~37
41
Table 2-11. Summary of the heat transfer coefficients for the different simulations and the extrapolated
values.
Figure 2-13. Lateral variation of the heat transfer coefficient of the different meshes and the extrapolated
values with numerical uncertainties.
Case Coarse Medium Fine Extrapolated
Region Heat Transfer Coefficient [𝑊
𝑚2∗𝐾]
Stagnation Point
(x/W=0) 499 510 541 579 ± 81
Stagnation
Region
(-1<x/W<1)
487 496 512 533 ± 43
Mean
(-L/2<x/W<L/2) 173 183 193 204 ± 25
42
Figure 2-14. Fine simulation instantaneous (a) velocity and (b) temperature fields.
(a)
(b)
43
The deviation of the simulation extrapolated values from the experimental data can be explained in
terms of the effect of nozzle exit turbulence intensities (I) found in the experiments (Figure 2-13). The
impact of jet turbulence levels on stagnation point heat transfer, especially at lower dimensionless jet
lengths (H/W), has been reported in literature [26]. Greater jet turbulence intensity enhances the stagnation
point heat transfer and affects the lateral variation of the heat transfer coefficient. At lower turbulence
intensities, a secondary peak of heat transfer coefficient is observed in the wall jet region. As the turbulence
intensity increases, the secondary peak starts diminishing and eventually disappears (see Figure 2-13,
I=16%). The relatively high turbulence intensity of the simulation (I=8.2%), due to the fully-developed
flow behavior in the isolated channel, may explain the absence of the secondary peak in heat transfer
coefficient in simulations. Local heat transfer coefficient values from the simulation lie between the
experiment results for I=6% and I=18%, suggesting the validity of the computational approach.
44
Chapter 3
sCO2 Laminar Slot Jet Impingement
45
In this chapter, the validated laminar computational framework is employed to conduct a parametric
study of sCO2 slot jet impingement heat transfer. Ten cases are simulated at reduced pressures of 𝑃𝑟 =
1.03 and 1.1, Reynolds numbers 𝑅𝑒𝑖𝑛 = 225 − 900, dimensionless jet lengths 𝐻
𝑊= 2 − 4, jet inlet
temperatures 𝑇𝑖𝑛 = 294 − 330𝐾, and impingement plate temperatures 𝑇𝑝𝑙𝑎𝑡𝑒 = 270 − 370 K (Table 3-1).
Two different variations of the Reynolds (𝐺∗𝑊
𝜇), where G is the mass flux, and Nusselt (
ℎ∗𝑊
𝑘) numbers are
reported: one using the thermophysical properties evaluated at the jet inlet temperature (𝑅𝑒𝑖𝑛 , 𝑁𝑢𝑖𝑛) and
another where the thermophysical properties are evaluated at the impingement plate temperature
(𝑅𝑒𝑝𝑙𝑎𝑡𝑒 , 𝑁𝑢𝑝𝑙𝑎𝑡𝑒). This assists in identifying trends in heat transfer as some effects may scale more directly
with variations in the bulk flow or near-wall transport properties. Results are used to assess the effects of
the variation of the thermophysical properties in the pseudo-critical range on heat transfer behavior at the
stagnation point (x/W=0), in the stagnation zone (0<x/W<1), and slightly downstream of the stagnation
zone (1<x/W<4). At instances where unconventional trends of heat transfer are observed, simulations with
constant thermophysical properties, based on the jet exit temperature, are performed to isolate the effects
of the variation of thermophysical properties. The method discussed in Section 2.4.1 is used for Richardson
extrapolation of simulation results and estimating the numerical uncertainties. Therefore, two simulations
are evaluated for each case. The meshes for the laminar validation study discussed in Section 2.3.1 are used.
46
Table 3-1. Summary of the laminar sCO2 simulation cases.
Case Number H/W 𝑅𝑒𝑖𝑛/𝑅𝑒𝑤𝑎𝑙𝑙 Tin [K] Tplate [K] Reduced Pressure
1 (Base) 2 450 / 1,658 294 330 1.1
2 2 225 / 828 294 330 1.1
3 2 900 / 3,313 294 330 1.1
4 4 450 / 1,658 294 330 1.1
5 1 450 / 1,658 294 330 1.1
6 2 450 / 1,642 294 370 1.1
7 2 450 / 1,291 294 310 1.1
8 2 450 / 79 330 270 1.1
9 2 450 / 446 330 370 1.1
10 2 450 / 1,672 294 330 1.03
47
3.1 Effect of Reynolds Number
Three cases were simulated (Cases 1-3) at Reynolds numbers of 450, 225, and 900 respectively.
The Richardson extrapolated lateral variation of the heat transfer coefficients is shown in Figure 3-1 and
the stagnation point and zone extrapolated data are summarized in Table 3-2. The heat transfer coefficients
increase with increasing Reynolds number as seen in Figure 3-1. The average stagnation zone heat transfer
coefficient is approximately 16% lower than that at the stagnation point. Furthermore, the trend for the
lateral variation of the heat transfer coefficients is similar to that of a constant property fluid (see Section
2.4.1). Heat transfer deterioration, which is marked by concentration of low-density vapor-like film near
the wall, is not observed in the vicinity of the stagnation zone (1<x/W<4) even at lower Reynolds numbers
(see Figure 3-2). This phenomenon is observed farther away from the stagnation region due to the transient
jet separation behavior which is prominent at lower Reynolds numbers. As the Reynolds number increases,
the jet separation becomes less severe as does the heat transfer deterioration.
48
Figure 3-1. Extrapolated lateral variation of the heat transfer coefficient for Cases 1-3.
Table 3-2. Summary of the heat transfer coefficients for Cases 1-3.
Case 𝑹𝒆𝒊𝒏 = 𝟐𝟐𝟓 (𝑪𝒂𝒔𝒆 𝟐) 𝑹𝒆𝒊𝒏 = 𝟒𝟓𝟎 (𝑪𝒂𝒔𝒆 𝟏) 𝑹𝒆𝒊𝒏 = 𝟗𝟎𝟎 (𝑪𝒂𝒔𝒆 𝟑)
Region
Heat Transfer Coefficient [𝑊
𝑚2∗𝐾]
𝑁𝑢𝑖𝑛 / 𝑁𝑢𝑝𝑙𝑎𝑡𝑒
Stagnation Point
(x/W=0)
332 ± 83
12 / 35
592 ± 14
21 / 62 844 ± 31
30 / 88
Stagnation Zone
(0<x/W<1)
283 ± 67
10 / 30
497 ± 10
18 / 52
708 ± 20
25 / 74
49
Figure 3-2. Case 2 (Rein = 225) fine simulation instantaneous (a) velocity, (b) temperature, and (c) density
fields.
50
To isolate the effects of thermophysical property variation, simulations with constant properties,
defined based on the nozzle exit temperature, were conducted for Cases 1 and 3 (𝑅𝑒 = 450 , 900). The
lateral variation of the heat transfer coefficients (from the fine meshes) are shown in Figure 3-3 and Figure
3-4. Both cases show degraded heat transfer for variable property simulations. For both cases, the stagnation
point heat transfer coefficient is 16% lower in the varying property simulations. This suggests that for
comparable jet impingement flows, relative heat transfer deterioration effects are not very sensitive to Re.
The relative heat transfer coefficient deterioration decreases farther away from the stagnation point.
This heat transfer performance decrease in these studies can be explained in terms of the jet exit
and impingement plate reduced temperatures; 𝑇𝑟,𝑒𝑥𝑖𝑡 = −14.4𝐾 and 𝑇𝑟,𝑝𝑙𝑎𝑡𝑒 = 21.6 𝐾 respectively. As
𝑇𝑟,𝑝𝑙𝑎𝑡𝑒 has a larger magnitude than 𝑇𝑟,𝑒𝑥𝑖𝑡, this means that for the simulation with variable properties, the
layers of fluid nearest to the impingement plate have gas-like properties and lower specific heat, leading to
a poor heat transfer performance.
51
Figure 3-3. Constant and variable property results for lateral variation of heat transfer coefficient for Case
1 (fine mesh).
Figure 3-4. Constant and variable property results for lateral variation of heat transfer coefficient for Case
3 (fine mesh).
52
3.2 Effect of Dimensionless Jet Length (H/W)
Three cases were simulated (Cases 1,4,5) at dimensionless jet lengths (H/W) of 2, 4, and 1
respectively. The Richardson extrapolated lateral variation of the heat transfer coefficients is shown in
Figure 3-5 and the stagnation point and zone extrapolated data are summarized in Table 3-3. The heat
transfer coefficients slightly decrease with increasing dimensionless jet length as seen in Figure 3-5. The
average stagnation zone heat transfer coefficient is approximately 16% lower than that at the stagnation
point. Furthermore, the trend for the lateral variation of the heat transfer coefficients is similar to that of a
constant property fluid (see Section 2.4.1). In literature, the dimensionless jet length is shown to
significantly affect the heat transfer coefficient in the stagnation zone and its lateral variation [54].
However, this occurs due to the increase in the jet turbulence levels and decreasing jet centerline velocity.
Therefore, for a laminar flow, the heat transfer behavior with varying dimensionless jet length is
significantly different. For a laminar flow, the variation of dimensionless jet length does not significantly
impact the flow field characteristics. Therefore, for a given temperature difference between the jet inlet and
impingement plate temperatures, the heat transfer enhancement associated with supercritical fluids is also
not exploited at different laminar dimensionless jet lengths.
53
Figure 3-5. Extrapolated lateral variation of the heat transfer coefficient for Cases 1,4, and 5.
Table 3-3. Summary of the heat transfer coefficients for Cases 1,4, and 5.
Case 𝑯
𝑾= 𝟏 (𝑪𝒂𝒔𝒆 𝟓)
𝑯
𝑾= 𝟐 (𝑪𝒂𝒔𝒆 𝟏)
𝑯
𝑾= 𝟒 (𝑪𝒂𝒔𝒆 𝟒)
Region Heat Transfer Coefficient [
𝑊
𝑚2∗𝐾]
𝑁𝑢𝑖𝑛 / 𝑁𝑢𝑝𝑙𝑎𝑡𝑒
Stagnation Point
(x/W=0)
599 ± 20
21 / 63
592 ± 14
21 / 62 574 ± 13
21 / 60
Stagnation Zone
(0<x/W<1)
507 ± 9
18 / 53
497 ± 10
18 / 52
452 ± 12
17 / 51
54
3.3 Effect of Impingement Plate (Tplate) and Jet Inlet (Tin) Temperatures
Five cases were simulated (Cases 1,6-9) at two jet inlet temperatures (Tin) of 294 and 330 K and
three impingement plate temperatures (Tplate) of 330, 370, and 310 K. For the cases with Tin below the
pseudo-critical temperature (𝑇𝑝𝑐), the extrapolated lateral variation of the heat transfer coefficients is shown
in Figure 3-6 and the stagnation point and zone extrapolated data are summarized in Table 3-4. The
extrapolated lateral variation of heat transfer coefficients for the other cases with Tin above 𝑇𝑝𝑐 is shown in
Figure 3-7 and the stagnation point and zone extrapolated data are summarized in Table 3-5.
For a Tin below the 𝑇𝑝𝑐, the heat transfer coefficients sharply decrease with increasing Tplate as seen
in Figure 3-4. The stagnation zone heat transfer coefficient is approximately 16% lower than that of the
stagnation point. Additionally, the trend for the lateral variation of the heat transfer coefficients is similar
to that of a constant property fluid (see Section 2.4.1). For a given Tin below 𝑇𝑝𝑐, increasing Tplate above the
𝑇𝑝𝑐 results in a higher temperature gradient near the impingement plate and hence a thinner layer of high
specific heat fluid, leading to a decreased heat transfer coefficient. The extent to which Tplate can be
increased above 𝑇𝑝𝑐 and still experience an increase in the heat transfer coefficient depends on the Reynolds
number; limiting Tplate to be very close to 𝑇𝑝𝑐 for laminar flow. For this investigation, a Reynolds number
of 450 is used to study the effect of Tplate. At this Reynolds number, a Tplate within the pseudo-critical range
(310 K) results in higher heat transfer coefficients than a Tplate slightly above 𝑇𝑝𝑐 (330 K). This indicates
that the Tplate can only be slightly raised above 𝑇𝑝𝑐 before the heat transfer coefficient sharply reduces.
For a given Tplate > 𝑇𝑝𝑐, the heat transfer coefficient reduces as Tin increases above 𝑇𝑝𝑐 as seen in
Figure 3-5. This is because for a Tin above 𝑇𝑝𝑐, the supercritical fluid is above the high specific heat
temperature range. Therefore, the supercritical fluid loses its heat transfer enhancement capabilities for a
Tin and Tplate above or below 𝑇𝑝𝑐. This can be also clearly observed for the higher heat transfer performance
for Case 8, where Tin is above the 𝑇𝑝𝑐 and Tplate is below the 𝑇𝑝𝑐.
55
Figure 3-6. Extrapolated lateral variation of the heat transfer coefficient for Cases 1,6, and 7.
Table 3-4. Summary of the heat transfer coefficients for Cases 1,6, and 7.
Case 𝑻𝒑𝒍𝒂𝒕𝒆 = 𝟑𝟏𝟎 𝑲 (𝑪𝒂𝒔𝒆 𝟕) 𝑻𝒑𝒍𝒂𝒕𝒆 = 𝟑𝟑𝟎 𝑲 (𝑪𝒂𝒔𝒆 𝟏) 𝑻𝒑𝒍𝒂𝒕𝒆 = 𝟑𝟕𝟎 𝑲 (𝑪𝒂𝒔𝒆 𝟔)
Region Heat Transfer Coefficient [
𝑊
𝑚2∗𝐾]
𝑁𝑢𝑖𝑛 / 𝑁𝑢𝑝𝑙𝑎𝑡𝑒
Stagnation Point
(x/W=0)
778 ± 24
28 / 40
592 ± 14
21 / 62 415 ± 8
15 / 47
Stagnation Zone
(0<x/W<1)
655 ± 12
23 / 34
497 ± 10
18 / 52
349 ± 4
12 / 39
56
Figure 3-7. Extrapolated lateral variation of the heat transfer coefficient for Cases 6,8, and 9.
Table 3-5. Summary of the heat transfer coefficients for Cases 6,8, and 9.
Case 𝑻𝒑𝒍𝒂𝒕𝒆 = 𝟐𝟕𝟎 𝑲 (𝑪𝒂𝒔𝒆 𝟖)
𝑻𝒊𝒏 = 𝟑𝟑𝟎 𝑲
𝑻𝒑𝒍𝒂𝒕𝒆 = 𝟑𝟕𝟎 𝑲 (𝑪𝒂𝒔𝒆 𝟗)
𝑻𝒊𝒏 = 𝟑𝟑𝟎 𝑲
𝑻𝒑𝒍𝒂𝒕𝒆 = 𝟑𝟕𝟎 𝑲 (𝑪𝒂𝒔𝒆 𝟔)
𝑻𝒊𝒏 = 𝟐𝟗𝟒 𝑲
Region Heat Transfer Coefficient [
𝑊
𝑚2∗𝐾]
𝑁𝑢𝑖𝑛 / 𝑁𝑢𝑝𝑙𝑎𝑡𝑒
Stagnation Point
(x/W=0)
632 ± 4
66 / 17
191 ± 1
20 / 21
415 ± 8
15 / 47
Stagnation Zone
(0<x/W<1)
526 ± 4
55 / 14
161 ± 1
17 / 18
349 ± 4
12 / 39
57
To isolate the effects of property variation with temperature, simulations with constant properties,
defined based on the nozzle exit temperature, were conducted for Cases 6 and 7 at impingement plate
temperatures of 370K and 310K respectively. The fine mesh lateral variations of the heat transfer
coefficients are shown in Figure 3-8 and Figure 3-9. For Case 6, the simulation with variable properties
shows a heat transfer performance reduction of approximately 47% compared to that of constant properties
at the stagnation point. However, for Case 7, a heat transfer performance enhancement of 6% is observed
at the stagnation point. The heat transfer performance reduction and enhancement decrease farther away
from the stagnation point. This behavior can be explained with respect to the jet exit and impingement plate
reduced temperatures.
For Case 6, with a 𝑇𝑟,𝑝𝑙𝑎𝑡𝑒 = 61.6 K and 𝑇𝑟,𝑒𝑥𝑖𝑡 = −14.4 K, the layer of fluid near the impingement
has a lower specific heat capacity for the variable property simulation compared to that of a constant
property fluid, leading to a severe heat transfer performance reduction. In Section 3.1, a similar simulation
with 𝑇𝑟,𝑝𝑙𝑎𝑡𝑒 = 21.6 𝐾 showed only a 16% heat transfer performance reduction at the stagnation zone,
indicating that for a given jet inlet temperature, the heat transfer performance decrease is more severe as
𝑇𝑟,𝑝𝑙𝑎𝑡𝑒 increases. For Case 7, with a 𝑇𝑟,𝑝𝑙𝑎𝑡𝑒 = 1.6 K and 𝑇𝑟,𝑒𝑥𝑖𝑡 = −14.4 K, the impingement plate
temperature is very close to Tpc, therefore the variation of the thermophysical properties leads to a heat
transfer performance enhancement as the specific heat of the fluid near the impingement plate is much
greater than at the inlet temperature.
58
Figure 3-8. Constant and variable property results for lateral variation of heat transfer coefficient for Case
6 (fine mesh).
Figure 3-9. Constant and variable property results for lateral variation of heat transfer coefficient for Case
6 (fine mesh).
59
3.4 Effect of Reduced Pressure (Pr)
Two cases were simulated (Cases 1,10) at reduced pressures (Pr) of 1.1 and 1.03 respectively. The
Richardson extrapolated lateral variation of the heat transfer coefficients is shown in Figure 3-10 and the
stagnation point and zone extrapolated data are summarized in Table 3-6. The average stagnation zone heat
transfer coefficient is approximately 16% lower than that at the stagnation point. Additionally, the trend for
the lateral variation of the heat transfer coefficients is similar to that of a constant property fluid (see Section
2.4.1). The pressure affects the heat transfer behavior of supercritical fluids by dictating the 𝑇𝑝𝑐 and the
corresponding specific heat value. As the pressure increases, the 𝑇𝑝𝑐 increases, however, the peak specific
heat value decreases. This means that for a given Reynolds number, Tin, and Tplate there is a Pr that provides
the maximum heat transfer performance. The aforementioned parameters lead to a certain temperature field
with fluid layers of different thicknesses and temperatures. The Pr specifies the 𝑇𝑝𝑐 and the maximum
specific heat value. Therefore, the maximum heat transfer performance is provided by a Pr that sets the 𝑇𝑝𝑐
at a temperature which has the thickest layers near the wall. This corresponds to a thick fluid layer with
high specific heat near the wall, hence an enhanced heat transfer behavior. For this investigation at a
Reynolds number of 450, a Tin of 294 K, and a Tplate of 330 K a Pr of 1.1 provides a higher heat transfer
performance than a Pr of 1.03.
60
Figure 3-10. Extrapolated lateral variation of the heat transfer coefficient for Cases 1 and 10.
Table 3-6. Summary of the heat transfer coefficients for Cases 1 and 10.
Case 𝑷𝒓 = 𝟏. 𝟏(𝑪𝒂𝒔𝒆 𝟏) 𝑷𝒓 = 𝟏. 𝟎𝟑(𝑪𝒂𝒔𝒆 𝟏𝟎)
Region Heat Transfer Coefficient [𝑊
𝑚2∗𝐾]
𝑁𝑢𝑖𝑛 / 𝑁𝑢𝑝𝑙𝑎𝑡𝑒
Stagnation Point
(x/W=0)
592 ± 14
21 / 62 550 ± 2
20 / 58
Stagnation Zone
(0<x/W<1)
497 ± 10
18 / 52
453 ± 16
16 / 47
61
3.5 Conclusion
This chapter presented a parametric study of laminar sCO2 slot jet impingement heat transfer. Ten
cases were evaluated at two different reduced pressures 𝑃𝑟 = 1.03 and 1.1, Reynolds numbers 𝑅𝑒𝑖𝑛 =
225 − 900, dimensionless jet lengths 𝐻
𝑊= 2 − 4, jet inlet temperatures 𝑇𝑖𝑛 = 294 − 330, and
impingement plate temperatures 𝑇𝑝𝑙𝑎𝑡𝑒 = 270 − 370 K. Results were used to explore the effect of the
variation of the thermophysical properties in the pseudo-critical range on the heat transfer behavior at the
stagnation point (x/W=0), stagnation zone (0<x/W<1), and at various positions in the vicinity of the
stagnation zone (1<x/W<4). The main findings for this chapter include:
1. The lateral variation of the heat transfer coefficient is similar to that of a constant property fluid
(see Section 2.4.1). The heat transfer coefficient is highest at the stagnation point and decreases
farther away. The shape of the lateral profile is identical to that of a constant property fluid for the
conditions studied.
2. The variation of the thermophysical properties does not produce a more uniform heat transfer
coefficient in the stagnation zone. The stagnation zone heat transfer coefficient is approximately
16% lower than that of the stagnation point, similar to the simulations with constant properties (see
Section 2.4.1).
3. Heat transfer coefficient deterioration can occur in the stagnation region if Tplate >> Tpc due to
variation of fluid properties with temperature. This can be as severe as a 47% reduction in the
stagnation zone heat transfer coefficient for 𝑇𝑟,𝑝𝑙𝑎𝑡𝑒 = 61.6 K and 𝑇𝑟,𝑒𝑥𝑖𝑡 = −14.4 K. Heat transfer
deterioration is minimal in the vicinity of the stagnation region (1<x/W<4). However, it is detected
farther away due the transient jet separation; which decreases as the Reynolds number increases.
4. For enhanced heat transfer performance, it is optimal to have the fluid near the impingement plate
to be as close to the pseudo-critical temperature (𝑇𝑝𝑐) as possible. Therefore, for configurations
where the jet inlet and impingement plate temperatures are both below or above the 𝑇𝑝𝑐, the heat
transfer performance is significantly reduced.
62
Chapter 4
sCO2 Turbulent Slot Jet Impingement
63
In this chapter, the validated turbulent computational framework is employed to conduct a
parametric study of sCO2 slot jet impingement heat transfer. Ten cases are simulated at reduced pressures
of 𝑃𝑟 = 1.03 and 1.1, Reynolds numbers 𝑅𝑒𝑖𝑛 = 2750 − 11000, dimensionless jet lengths 𝐻
𝑊= 2 − 4, jet
inlet temperatures 𝑇𝑖𝑛 = 294 − 330𝐾, and impingement plate temperatures 𝑇𝑝𝑙𝑎𝑡𝑒 = 270 − 370 K (Table
4-1). Two forms of the Reynolds (𝐺∗𝑊
𝜇), where G is the mass flux, and Nusselt (
ℎ∗𝑊
𝑘) numbers are reported:
one using the thermophysical properties evaluated at the jet inlet temperature (𝑅𝑒𝑖𝑛, 𝑁𝑢𝑖𝑛) and another
where the thermophysical properties are evaluated at the impingement plate temperature
(𝑅𝑒𝑝𝑙𝑎𝑡𝑒 , 𝑁𝑢𝑝𝑙𝑎𝑡𝑒). This assists in identifying trends in heat transfer as some effects may scale more directly
with variations in the bulk flow or near-wall transport properties. Results are used to assess the effects of
the variation of the thermophysical properties in the pseudo-critical range on heat transfer behavior at the
stagnation point (x/W=0), in the stagnation zone (-1<x/W<1), downstream of the stagnation zone (-
7<x/W<7), and for the overall impingement plate (-L/2<x/W<L/2). At instances where unconventional
trends of heat transfer are observed, simulations with constant thermophysical properties, based on the jet
exit temperature, are performed to isolate the effects of the variation of thermophysical properties. These
simulations are still under progress; however, they will be in the final document. The method discussed in
Section 2.4.2 is used for quantifying the numerical uncertainties, therefore, two simulations are evaluated
for each case. The meshes for the turbulent validation study discussed in Section 2.4.2 are used. The fine
mesh simulations for Cases 6-10 are still in progress, therefore the data reported is that of the medium mesh
without the associated numerical uncertainties.
64
Table 4-1. Summary of the turbulent sCO2 simulation cases.
Case Number H/W 𝑅𝑒𝑖𝑛/𝑅𝑒𝑤𝑎𝑙𝑙 Tin [K] Tplate [K] Reduced Pressure
1 (Base) 2 5,500 / 20,257 294 330 1.1
2 2 2,750 / 10,125 294 330 1.1
3 2 11,000 / 40,498 294 330 1.1
4 1 5,500 / 20,257 294 330 1.1
5 4 5,500 / 20,257 294 330 1.1
6 2 5,500 / 20,064 294 370 1.1
7 2 5,500 / 15,783 294 310 1.1
8 2 5,500 / 962 330 270 1.1
9 2 5,500 / 5,661 320 370 1.1
10 2 5,500 / 20,436 294 330 1.03
65
4.1 Effect of Reynolds Number
Three cases were simulated (Cases 1-3) at Reynolds numbers of 5500, 2750, and 11000
respectively. The Richardson extrapolated lateral variation of heat transfer coefficients are shown in Figure
4-1 and the stagnation point, zone, and mean extrapolated data are summarized in Table 4-2. The heat
transfer coefficients increase with increasing Reynolds number as seen in Figure 4-1. The average
stagnation zone heat transfer coefficient is approximately 4-7% lower than that at the stagnation point,
similar to that for constant property flows. Furthermore, the trend for the lateral variation of the heat transfer
coefficients is slightly more uniform than that of a constant property fluid (see Section 2.4.2) due to the
thermophysical property variation in the pseudo-critical range. Heat transfer deterioration, which is
marked by concentration of low-density vapor-like film near the wall, is not observed in the vicinity of the
stagnation zone (-7<x/W<7) even at the lowest Reynolds number (see Figure 4-2). This phenomenon is
observed farther away from the stagnation zone as the wall jet transitions dissipates to regular channel flow.
66
Figure 4-1. Extrapolated lateral variation of the heat transfer coefficient for Cases 1-3.
Table 4-2. Summary of the heat transfer coefficients for Cases 1-3.
Case 𝑹𝒆𝒊𝒏 = 𝟐𝟕𝟓𝟎 (𝑪𝒂𝒔𝒆 𝟐) 𝑹𝒆𝒊𝒏 = 𝟓𝟓𝟎𝟎 (𝑪𝒂𝒔𝒆 𝟏) 𝑹𝒆𝒊𝒏 = 𝟏𝟏𝟎𝟎𝟎 (𝑪𝒂𝒔𝒆𝟑)
Region
Heat Transfer Coefficient [𝑊
𝑚2∗𝐾]
𝑁𝑢𝑖𝑛 / 𝑁𝑢𝑝𝑙𝑎𝑡𝑒
Stagnation Point
(x/W=0)
1226 ± 253
44 / 129
1885 ± 3
67 / 198 2814 ± 31
100 / 295
Stagnation Zone
(-1<x/W<1)
1183 ± 195
42 / 124
1775 ± 2
63 / 186
2618 ± 20
93 / 274
Mean
(-L/2<x/W<L/2)
455 ± 20
16 / 48
770 ± 89
28 / 81
1235 ± 226
44 / 129
67
Figure 4-2. Case 2 (Rein =2750) fine simulation instantaneous (a) velocity, (b) temperature, and (c)
density fields.
68
To directly observe the effects of property variation with temperature, a simulation with constant
properties, defined based on the nozzle exit temperature, was conducted for Case 3 (𝑅𝑒 = 11,000). The
medium mesh lateral variation of the heat transfer coefficients is shown in Figure 4-3. The heat transfer
coefficient deterioration due to property variation ~13% at the stagnation point. This performance reduction
is less severe away from the stagnation point (See Section 3.1 for discussion on this performance reduction).
The lateral profile of the heat transfer coefficient is more uniform for the variable property simulations
compared to that with constant properties.
Figure 4-3. Constant and variable property results for lateral variation of heat transfer coefficient for Case
3 (medium mesh).
69
4.2 Effect of Dimensionless Jet Length (H/W)
Three cases were simulated (Cases 1,4,5) at dimensionless jet lengths (H/W) of 2, 1, and 4
respectively. The Richardson extrapolated lateral variation of the heat transfer coefficients is shown in
Figure 4-3 and the stagnation point, zone, and mean extrapolated data are summarized in Table 4-3. The
heat transfer coefficients slightly decrease with increasing dimensionless jet length as seen in Figure 4-3.
The average stagnation zone heat transfer coefficient is approximately 6-8% lower than that at the
stagnation point, similar to that of a constant property fluid. Furthermore, the trend for the lateral variation
of the heat transfer coefficient for (H/W=2,4) is slightly more uniform than that of a constant property fluid
(see Section 2.4.2). The lateral variation of the heat transfer coefficient for (H/W=1) shows a secondary
peak in heat transfer coefficient. Due to the high turbulence intensity at the nozzle exit (I=8.2%), increasing
the dimensionless jet length (H/W) increases the jet turbulence level at the expense of the jet centerline
velocity, leading to the decreased stagnation zone heat transfer. However, the mean heat transfer coefficient
increases with increasing dimensionless jet length due to the length of the wall jet region relative to the
impingement plate length (Figure 4-4). For a fixed length of the impingement plate, increasing the
dimensionless jet length leads to a longer wall jet, hence a shorter channel flow region in which heat transfer
deterioration occurs, therefore, leading to a higher mean heat transfer coefficient.
70
Figure 4-4. Extrapolated lateral variation of the heat transfer coefficient for Cases 1,4, and 5.
Table 4-3. Summary of the heat transfer coefficients for Cases 1,4, and 5.
Case 𝑯
𝑾= 𝟏 (𝑪𝒂𝒔𝒆 𝟒)
𝑯
𝑾= 𝟐 (𝑪𝒂𝒔𝒆 𝟏)
𝑯
𝑾= 𝟒 (𝑪𝒂𝒔𝒆 𝟓)
Region Heat Transfer Coefficient [
𝑊
𝑚2∗𝐾]
𝑁𝑢𝑖𝑛 / 𝑁𝑢𝑝𝑙𝑎𝑡𝑒
Stagnation Point
(x/W=0)
1967 ± 13
70 / 206
1885 ± 3
67 / 198 1921 ± 53
69 / 201
Stagnation Zone
(-1<x/W<1)
1813 ± 9
65 / 190
1775 ± 2
63 / 186
1787 ± 56
64 / 187
Mean
(-L/2<x/W<L/2)
742 ± 106
27 / 78
770 ± 89
28 / 81
810 ± 88
29 / 85
71
Figure 4-5. Fine simulation instantaneous velocity fields for (a) Case 4 (H/W=1), (b) Case 1 (H/W=2),
and (c) Case 5 (H/W=4).
72
The effects of the property variation with temperature are assessed by comparing results with a
simulation with constant properties, defined based on the nozzle exit temperature, for Case 4 (H/W=1). The
medium mesh lateral variation of the heat transfer coefficients is shown in Figure 4-6. The lateral variation
of the heat transfer coefficient shows a secondary peak in heat transfer coefficient for both varying and
constant properties studies, indicating that this secondary peak is due to the trade-off between the jet
centerline velocity and turbulence levels (see Section 1.3 for a detailed discussion).
Figure 4-6. Constant and variable property results for lateral variation of heat transfer coefficient for Case
4 (medium mesh).
73
4.3 Effect of Impingement Plate (Tplate) and Jet Inlet (Tin) Temperatures
Five cases were simulated (Cases 1,6-9) at three jet inlet temperatures (Tin) of 294, 320, and 330 K
and three impingement plate temperatures (Tplate) of 330, 370, and 310 K. For the cases with Tin below the
pseudo-critical temperature (𝑇𝑝𝑐), extrapolated data for Case 1 is used and for Cases 6 and 7 the medium
mesh lateral variation of the heat transfer coefficients is shown in Figure 4-7 and the stagnation point, zone,
and mean data are summarized in Table 4-4. The medium mesh lateral variation of heat transfer coefficients
for the other cases with Tin above 𝑇𝑝𝑐 is shown in Figure 4-8 and the stagnation point, zone, and mean data
are summarized in Table 4-5.
For a Tin below the 𝑇𝑝𝑐, the heat transfer coefficients sharply decrease with increasing Tplate as seen
in Figure 4-5. The average stagnation zone heat transfer coefficient is approximately 6% lower than that of
the stagnation point. The trend for the lateral variation of the heat transfer coefficients when Tplate >> 𝑇𝑝𝑐
(Case 6) shows a sharper decrease due to heat transfer deterioration as a result of the thermophysical
property variation (Figure 4-7). For a given Tin below 𝑇𝑝𝑐, increasing Tplate above the 𝑇𝑝𝑐 results in a higher
temperature gradient near the impingement plate and hence a thinner layer of high specific heat fluid,
leading to a decreased heat transfer coefficient. Similar behavior was also is observed for laminar flow
(Section 3.3), indicating that this trend is independent of Reynolds number.
For a given Tplate > 𝑇𝑝𝑐, the heat transfer coefficient reduces as Tin increases above 𝑇𝑝𝑐 as seen in
Figure 4-6. This is because for Tin above 𝑇𝑝𝑐, the supercritical fluid is above the high specific heat
temperature range. Therefore, the supercritical fluid loses its heat transfer enhancement capabilities for a
Tin and Tplate above or below 𝑇𝑝𝑐. This can be also clearly observed for the higher heat transfer performance
for Case 8, where Tin is above the 𝑇𝑝𝑐 and Tplate is below the 𝑇𝑝𝑐. The mean heat transfer coefficient for
Case 6 is higher than that of Case 8 even though the former has a higher stagnation zone heat transfer
coefficient. This is because the former has a more uniform lateral variation of the heat transfer coefficient
due to the thermophysical property variation.
74
Figure 4-7. Lateral variation of the heat transfer coefficient for Cases 1,6, and 7.
Table 4-4. Summary of the heat transfer coefficients for Cases 1,6, and 7.
Case 𝑻𝒑𝒍𝒂𝒕𝒆 = 𝟑𝟏𝟎 𝑲 (𝑪𝒂𝒔𝒆 𝟕) 𝑻𝒑𝒍𝒂𝒕𝒆 = 𝟑𝟑𝟎 𝑲 (𝑪𝒂𝒔𝒆 𝟏) 𝑻𝒑𝒍𝒂𝒕𝒆 = 𝟑𝟕𝟎 𝑲 (𝑪𝒂𝒔𝒆 𝟔)
Region Heat Transfer Coefficient [
𝑊
𝑚2∗𝐾]
𝑁𝑢𝑖𝑛 / 𝑁𝑢𝑝𝑙𝑎𝑡𝑒
Stagnation Point
(x/W=0)
2497
89 / 130
1885 ± 3
67 / 198 1337
48 / 150
Stagnation Zone
(-1<x/W<1)
2346
84 / 122
1775 ± 2
63 / 186
1263
45 / 142
Mean
(-L/2<x/W<L/2)
830
43 / 30
770 ± 89
28 / 81
622
22 / 70
75
Figure 4-8. Lateral variation of the heat transfer coefficient for Cases 6,8, and 9.
Table 4-5. Summary of the heat transfer coefficients for Cases 6,8, and 9.
Case 𝑻𝒑𝒍𝒂𝒕𝒆 = 𝟐𝟕𝟎 𝑲 (𝑪𝒂𝒔𝒆 𝟖)
𝑻𝒊𝒏 = 𝟑𝟑𝟎 𝑲
𝑻𝒑𝒍𝒂𝒕𝒆 = 𝟑𝟕𝟎 𝑲 (𝑪𝒂𝒔𝒆 𝟗)
𝑻𝒊𝒏 = 𝟑𝟐𝟎 𝑲
𝑻𝒑𝒍𝒂𝒕𝒆 = 𝟑𝟕𝟎 𝑲 (𝑪𝒂𝒔𝒆 𝟔)
𝑻𝒊𝒏 = 𝟐𝟗𝟒 𝑲
Region Heat Transfer Coefficient [
𝑊
𝑚2∗𝐾]
𝑁𝑢𝑖𝑛 / 𝑁𝑢𝑝𝑙𝑎𝑡𝑒
Stagnation Point
(x/W=0)
1868
196 / 49
578
61 / 65
1337
48 / 150
Stagnation Zone
(-1<x/W<1)
1751
184 / 46
549
58 / 62
1263
45 / 142
Mean
(-L/2<x/W<L/2)
547
57 / 14
199
21 / 22
622
22 / 70
76
Figure 4-9. Case 6 fine simulation instantaneous (a) temperature and (b) density fields.
77
4.4 Effect of Reduced Pressure (Pr)
Two cases were simulated (Cases 1,10) at reduced pressures (Pr) of 1.1 and 1.03 respectively. The
Richardson extrapolated data for Case 1 and the medium mesh lateral variation of the heat transfer
coefficients for Case 10 is shown in Figure 4-10 and the stagnation point, zone, and mean data are
summarized in Table 4-6. The average stagnation zone heat transfer coefficient is approximately 6% lower
than that at the stagnation point. Additionally, the trend for the lateral variation of the heat transfer
coefficients is slightly more uniform than that of a constant property fluid (see Section 2.4.2) due to the
thermophysical property variation. For this investigation at a Reynolds number of 5500, a Tin of 294 K, and
a Tplate of 330 K a Pr of 1.1 provides a higher heat transfer performance than a Pr of 1.03 (See Section 3.4
for discussion on the role of reduced pressure in dictating the heat transfer behavior).
78
Figure 4-10. Lateral variation of the heat transfer coefficient for Cases 1 and 10.
Table 4-6. Summary of the heat transfer coefficients for Cases 1 and 10.
Case 𝑷𝒓 = 𝟏. 𝟏(𝑪𝒂𝒔𝒆 𝟏) 𝑷𝒓 = 𝟏. 𝟎𝟑(𝑪𝒂𝒔𝒆 𝟏𝟎)
Region Heat Transfer Coefficient [𝑊
𝑚2∗𝐾]
𝑁𝑢𝑖𝑛 / 𝑁𝑢𝑝𝑙𝑎𝑡𝑒
Stagnation Point
(x/W=0)
1885 ± 3
67 / 198 1769
64 / 197
Stagnation Zone
(-1<x/W<1)
1775 ± 2
63 / 186
1661
60 / 185
Mean
(-L/2<x/W<L/2)
770 ± 89
28 / 81
656
24 / 73
79
4.5 Conclusion
This chapter presented a parametric study of turbulent sCO2 slot jet impingement heat transfer. Ten
cases were evaluated at two different reduced pressures 𝑃𝑟 = 1.03 and 1.1, Reynolds numbers 𝑅𝑒𝑖𝑛 =
225 − 900, dimensionless jet lengths 𝐻
𝑊= 2 − 4, jet inlet temperatures 𝑇𝑖𝑛 = 294 − 330, and
impingement plate temperatures 𝑇𝑝𝑙𝑎𝑡𝑒 = 270 − 370 K. Results were used to explore the effect of the
variation of the thermophysical properties in the pseudo-critical range on the heat transfer behavior at the
stagnation point (x/W=0), stagnation zone (-1<x/W<1), at various positions in the vicinity of the stagnation
zone (-7<x/W<7), and the impingement plate mean (-L/2<x/W<L/2). The main findings for this chapter
include:
1. The relative lateral variation of the heat transfer coefficient is generally less severe if actual
temperature-dependent fluid properties are used rather than fixed properties based on the inlet
conditions. However, if Tplate >> 𝑇𝑝𝑐, with Tin <𝑇𝑝𝑐, the lateral variation becomes sharper due to
pseudo-boiling heat transfer deterioration.
2. The variation of the thermophysical properties with temperature does not produce a more uniform
heat transfer coefficient in the stagnation zone in general. The stagnation zone heat transfer
coefficient is approximately 4-7% lower than that at the stagnation point, similar to simulation
findings with constant properties (see Section 2.4.2).
3. Heat transfer deterioration can occur both in the vicinity of the stagnation region (-7<x/W<7) and
farther downstream, depending on operating conditions. The former occurs if Tplate greatly exceeds
𝑇𝑝𝑐, with Tin <𝑇𝑝𝑐, or while Tplate is much less than 𝑇𝑝𝑐 with Tin >𝑇𝑝𝑐. The latter occurs even for
smaller temperature differences as the wall jet region dissipates to a channel flow.
4. For enhanced heat transfer performance, it is optimal to have the fluid near the impingement plate
to be as close to the pseudo-critical temperature (𝑇𝑝𝑐) as possible. Therefore, for configurations
80
where the jet inlet and impingement plate temperatures are both below or above the 𝑇𝑝𝑐, the heat
transfer performance is significantly reduced.
81
Chapter 5
Conclusions and Future Research
Recommendations
82
In this chapter, the key findings of this Masters thesis on sCO2 microscale slot jet impingement heat
transfer are summarized followed by recommendations for possible extensions of this research.
5.1 Conclusions
In Chapters 3 and 4, results of laminar and turbulent parametric studies of sCO2 microscale slot
jet impingement heat transfer were presented. Results were used to explore the effect of the variation of
the thermophysical properties in the pseudo-critical range on the heat transfer behavior at the stagnation
point (x/W=0), stagnation zone (-1<x/W<1), at various positions in the vicinity of the stagnation zone (-
7<x/W<7), and the impingement plate mean (-L/2<x/W<L/2). Major findings include:
1. Trends which are unique to fluids with variable thermophysical properties of the lateral variation
of the heat transfer coefficient do not exist for laminar flows. However, special trends do exist for
turbulent flows due to the thermophysical property variation. A more uniform lateral variation of
heat transfer coefficient is observed if Tplate is slightly above or below 𝑇𝑝𝑐, however, a rapid
decrease in the lateral variation occurs for larger temperature differences.
2. The variation of the thermophysical properties does not produce a more uniform stagnation zone
heat transfer coefficient, compared with the stagnation point, than that of a constant property fluid
for both laminar and turbulent flows.
3. Heat transfer deterioration is observed in both laminar and turbulent flows. For laminar flows, it
occurs farther away from the stagnation zone due to the unsteady jet separation. For turbulent flows,
this phenomenon occurs in the vicinity of the stagnation region (-7<x/W<7), due to the
thermophysical property variation if Tplate greatly exceeds 𝑇𝑝𝑐, and with smaller temperature
differences farther away from the stagnation region as the wall jet region dissipates to a consistent
channel flow. .
83
4. For enhanced heat transfer performance, it is optimal to have the fluid near the impingement plate
to be as close to the pseudo-critical temperature (𝑇𝑝𝑐) as possible. Therefore, for configurations
where the jet inlet and impingement plate temperatures are both below or above the 𝑇𝑝𝑐, the heat
transfer performance is significantly reduced.
5.2 Future Research Recommendations
More research is needed to fully characterize sCO2 microscale slot jet impingement heat transfer.
My recommendations for future work on this topic include:
1. Jet impingement cooling systems are usually employed in array configurations. The inter-jet
interaction plays an important role in dictating the heat transfer behavior of these systems. The
adopted computational framework in this study can be applied to explore such effects.
2. Investigations for sCO2 microscale slot jet impingement cooling systems of different impingement
plate orientations including: upward, downward, and inclined are needed for applications in which
sudden orientation can vary and mixed convection effects can be significant.
3. Conjugate heat transfer studies can be used to understand the effect of the axial conduction in the
impingement plate on the heat transfer behavior of sCO2 microscale slot jet impingement cooling
systems.
84
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