COMPUTATIONAL INVESTIGATION OF SUPERCRITICAL …

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




1,4, and 5. ............................................................................................................................... 53


1,6, and 7. ............................................................................................................................... 55


6,8, and 9. ............................................................................................................................... 56





Figure 3-10. Extrapolated lateral variation of the heat transfer coefficient for

Cases 1 and 10. ...................................................................................................................... 60


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


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 3-6. 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 4-6. 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


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).


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)


𝑊

𝑚2∗𝐾]


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



Case 𝑻𝒑𝒍𝒂𝒕𝒆 = 𝟑𝟏𝟎 𝑲 (𝑪𝒂𝒔𝒆 𝟕) 𝑻𝒑𝒍𝒂𝒕𝒆 = 𝟑𝟑𝟎 𝑲 (𝑪𝒂𝒔𝒆 𝟏) 𝑻𝒑𝒍𝒂𝒕𝒆 = 𝟑𝟕𝟎 𝑲 (𝑪𝒂𝒔𝒆 𝟔)


𝑊

𝑚2∗𝐾]


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



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


𝑊

𝑚2∗𝐾]


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


6 (fine mesh).


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)


𝑚2∗𝐾]


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]


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.


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



Case 𝑯

𝑾= 𝟏 (𝑪𝒂𝒔𝒆 𝟒)

𝑯

𝑾= 𝟐 (𝑪𝒂𝒔𝒆 𝟏)

𝑯

𝑾= 𝟒 (𝑪𝒂𝒔𝒆 𝟓)


𝑊

𝑚2∗𝐾]


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).


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.


Case 𝑻𝒑𝒍𝒂𝒕𝒆 = 𝟑𝟏𝟎 𝑲 (𝑪𝒂𝒔𝒆 𝟕) 𝑻𝒑𝒍𝒂𝒕𝒆 = 𝟑𝟑𝟎 𝑲 (𝑪𝒂𝒔𝒆 𝟏) 𝑻𝒑𝒍𝒂𝒕𝒆 = 𝟑𝟕𝟎 𝑲 (𝑪𝒂𝒔𝒆 𝟔)


𝑊

𝑚2∗𝐾]


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.


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


𝑊

𝑚2∗𝐾]


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)


𝑚2∗𝐾]


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.



80



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





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

Bibliography

[1] B. M. Fronk and A. S. Rattner, “High-Flux Thermal Management with Supercritical Fluids,” J.

Heat Transfer, vol. 138, no. 12, 2016, doi: 10.1115/1.4034053.

[2] R. Viskanta, “Heat transfer to impinging isothermal gas and flame jets,” Exp. Therm. Fluid Sci.,

vol. 6, no. 2, pp. 111–134, 1993, doi: 10.1016/0894-1777(93)90022-B.

[3] E. Pop, S. Sinha, and K. E. Goodson, “Heat generation and transport in nanometer-scale

transistors,” Proc. IEEE, vol. 94, no. 8, pp. 1587–1601, 2006, doi: 10.1109/JPROC.2006.879794.

[4] A. Bar-Cohen, J. J. Maurer, and D. H. Altman, “Embedded Cooling for Wide Bandgap Power

Amplifiers: A Review,” J. Electron. Packag., vol. 141, no. 4, 2019, doi: 10.1115/1.4043404.

[5] H. Baig, K. C. Heasman, and T. K. Mallick, “Non-uniform illumination in concentrating solar

cells,” Renewable and Sustainable Energy Reviews, vol. 16, no. 8. Pergamon, pp. 5890–5909,

2012, doi: 10.1016/j.rser.2012.06.020.

[6] H. M. S. Bahaidarah, A. A. B. Baloch, and P. Gandhidasan, “Uniform cooling of photovoltaic

panels: A review,” Renewable and Sustainable Energy Reviews, vol. 57. Elsevier Ltd, pp. 1520–

1544, 2016, doi: 10.1016/j.rser.2015.12.064.

[7] C. C. Liu, J. Zhang, A. K. Datta, and S. Tiwari, “Heating effects of clock drivers in bulk, SOI, and

3-D CMOS,” IEEE Electron Device Letters, vol. 23, no. 12. pp. 716–718, 2002, doi:

10.1109/LED.2002.805755.

[8] A. Royne and C. J. Dey, “Design of a jet impingement cooling device for densely packed PV cells

under high concentration,” Sol. Energy, vol. 81, no. 8, pp. 1014–1024, 2007, doi:

10.1016/j.solener.2006.11.015.

[9] D. Chemisana and J. I. Rosell, “Electrical performance increase of concentrator solar cells under

Gaussian temperature profiles,” Prog. Photovoltaics Res. Appl., vol. 21, no. 4, p. n/a-n/a, 2011,

doi: 10.1002/pip.1205.

[10] L. F. Cabeza, A. de Gracia, A. I. Fernández, and M. M. Farid, “Supercritical CO2 as heat transfer

fluid: A review,” Applied Thermal Engineering, vol. 125. Elsevier Ltd, pp. 799–810, 2017, doi:

10.1016/j.applthermaleng.2017.07.049.

[11] J. D. Jackson, “Fluid flow and convective heat transfer to fluids at supercritical pressure,” Nucl.

Eng. Des., vol. 264, pp. 24–40, 2013, doi: 10.1016/j.nucengdes.2012.09.040.

[12] M. Nabil and A. S. Rattner, “Large eddy simulations of high-heat-flux supercritical CO2

convection in microchannels: Mixed convection and non-uniform heating,” Int. J. Heat Mass

Transf., vol. 145, p. 118710, 2019, doi: 10.1016/j.ijheatmasstransfer.2019.118710.

[13] M. Pizzarelli, “The status of the research on the heat transfer deterioration in supercritical fluids: A

review,” Int. Commun. Heat Mass Transf., vol. 95, pp. 132–138, 2018, doi:

10.1016/j.icheatmasstransfer.2018.04.006.

[14] G. Liang and I. Mudawar, “Pool boiling critical heat flux (CHF) – Part 1: Review of mechanisms,

models, and correlations,” International Journal of Heat and Mass Transfer, vol. 117. Elsevier

Ltd, pp. 1352–1367, 2018, doi: 10.1016/j.ijheatmasstransfer.2017.09.134.

[15] I. Pioro and S. Mokry, “Heat Transfer to Supercritical Fluids,” in Heat Transfer - Theoretical

Analysis, Experimental Investigations and Industrial Systems, InTech, 2011.

[16] K. Chen, R. N. Xu, and P. X. Jiang, “Experimental Investigation of Jet Impingement Cooling with

Carbon Dioxide at Supercritical Pressures,” J. Heat Transfer, vol. 140, no. 4, 2018, doi:

10.1115/1.4038421.

[17] K. Chen, P.-X. Jiang, J.-N. Chen, and R.-N. Xu, “Numerical Investigation of Jet Impingement

Cooling of a Flat Plate with Carbon Dioxide at Supercritical Pressures,” Heat Transf. Eng., vol.

39, no. 2, pp. 85–97, 2018, doi: 10.1080/01457632.2017.1288042.

[18] P. Y. Tzeng, C. Y. Soong, and C. D. Hsieh, “Numerical investigation of heat transfer under

confined impinging turbulent slot jets,” Numer. Heat Transf. Part A Appl., vol. 35, no. 8, pp. 903–

924, 1999, doi: 10.1080/104077899274976.

85

[19] J. A. Fitzgerald and S. V. Garimella, “Flow field effects on heat transfer in confined jet

impingement,” J. Heat Transfer, vol. 119, no. 3, pp. 630–632, 1997, doi: 10.1115/1.2824152.

[20] V. A. Chiriac and A. Ortega, “A numerical study of the unsteady flow and heat tranfer in a

transitional confined slot jet impinging on an isothermal surface,” Int. J. Heat Mass Transf., vol.

45, no. 6, pp. 1237–1248, 2002, doi: 10.1016/S0017-9310(01)00224-1.

[21] H. Martin, “Heat and Mass Transfer between Impinging Gas Jets and Solid Surfaces,” Adv. Heat

Transf., vol. 13, no. C, pp. 1–60, 1977, doi: 10.1016/S0065-2717(08)70221-1.

[22] S. V. Garimella and B. Nenaydykh, “Nozzle-geometry effects in liquid jet impingement heat

transfer,” Int. J. Heat Mass Transf., vol. 39, no. 14, pp. 2915–2923, 1996, doi: 10.1016/0017-

9310(95)00382-7.

[23] A. Dewan, R. Dutta, and B. Srinivasan, “Recent Trends in Computation of Turbulent Jet

Impingement Heat Transfer,” Heat Transf. Eng., vol. 33, no. 4–5, pp. 447–460, 2012, doi:

10.1080/01457632.2012.614154.

[24] James Gaunter, J. Livingood, and P. Hrycak, “Survey of Literature on Flow Characteristics of a

Single Turbulent Jet Impinging on a Flat Plate,” 1970. Accessed: May 16, 2020. [Online].

Available: https://ntrs.nasa.gov/search.jsp?R=19700009658.

[25] G. M. Carlomagno and A. Ianiro, “Thermo-fluid-dynamics of submerged jets impinging at short

nozzle-to-plate distance: A review,” Experimental Thermal and Fluid Science, vol. 58. Elsevier

Inc., pp. 15–35, 2014, doi: 10.1016/j.expthermflusci.2014.06.010.

[26] R. Gardon and J. C. Akfirat, “The role of turbulence in determining the heat-transfer

characteristics of impinging jets,” Int. J. Heat Mass Transf., vol. 8, no. 10, pp. 1261–1272, 1965,

doi: 10.1016/0017-9310(65)90054-2.

[27] M. B. Glauert, “The wall jet,” J. Fluid Mech., vol. 1, no. 06, p. 625, 1956, doi:

10.1017/s002211205600041x.

[28] G. N. Abramovich, The Theory of Turbulent Jets. Cambridge: M.I.T. Press, 1963.

[29] D. Lytle and B. W. Webb, “Air jet impingement heat transfer at low nozzle-plate spacings,” Int. J.

Heat Mass Transf., vol. 37, no. 12, pp. 1687–1697, 1994, doi: 10.1016/0017-9310(94)90059-0.

[30] D. Cooper, D. C. Jackson, B. E. Launder, and G. X. Liao, “Impinging jet studies for turbulence

model assessment-I. Flow-field experiments,” Int. J. Heat Mass Transf., vol. 36, no. 10, pp. 2675–

2684, 1993, doi: 10.1016/S0017-9310(05)80204-2.

[31] P. Aillaud, F. Duchaine, L. Y. M. Gicquel, and S. Didorally, “Secondary peak in the Nusselt

number distribution of impinging jet flows: A phenomenological analysis,” Phys. Fluids, vol. 28,

no. 9, p. 095110, 2016, doi: 10.1063/1.4963687.

[32] H. Hattori and Y. Nagano, “Direct numerical simulation of turbulent heat transfer in plane

impinging jet,” Int. J. Heat Fluid Flow, vol. 25, no. 5, pp. 749–758, 2004, doi:

10.1016/j.ijheatfluidflow.2004.05.004.

[33] Y. M. Chung and K. H. Luo, “Unsteady heat transfer analysis of an impinging jet,” J. Heat

Transfer, vol. 124, no. 6, pp. 1039–1048, 2002, doi: 10.1115/1.1469522.

[34] C. Meola, L. De Luca, and G. M. Carlomagno, “Influence of shear layer dynamics on

impingement heat transfer,” Exp. Therm. Fluid Sci., vol. 13, no. 1, pp. 29–37, 1996, doi:

10.1016/0894-1777(96)00011-8.

[35] C. Segal and S. A. Polikhov, “Experimental study of subcritical to supercritical jet mixing,” Int. J.

Energ. Mater. Chem. Propuls., vol. 8, no. 3, pp. 237–251, 2009, doi:

10.1615/IntJEnergeticMaterialsChemProp.v8.i3.50.

[36] S. T. Chong, Y. Tang, M. Hassanaly, and V. Raman, “Turbulent mixing and combustion of

supercritical jets,” in AIAA SciTech Forum - 55th AIAA Aerospace Sciences Meeting, 2017, doi:

10.2514/6.2017-0141.

[37] Z. He, S. Tian, G. Li, H. Wang, Z. Shen, and Z. Xu, “The pressurization effect of jet fracturing

using supercritical carbon dioxide,” J. Nat. Gas Sci. Eng., vol. 27, pp. 842–851, 2015, doi:

10.1016/j.jngse.2015.09.045.

[38] H. Wang et al., “Experimental investigation on abrasive supercritical CO2 jet perforation,” J. CO2

86

Util., vol. 28, pp. 59–65, 2018, doi: 10.1016/j.jcou.2018.09.018.

[39] Y. Du, R. Wang, H. Ni, and H. Huo, “Application Prospects of Supercritical Carbon Dioxide in

Unconventional Oil and Gas Reservoirs,” 2012, doi: 10.4028/www.scientific.net/AMR.524-

527.1355.

[40] Y. K. Du, R. H. Wang, H. J. Ni, M. K. Li, W. Q. Song, and H. F. Song, “Determination of rock-

breaking performance of high-pressure supercritical carbon dioxide jet,” J. Hydrodyn., vol. 24, no.

4, pp. 554–560, 2012, doi: 10.1016/S1001-6058(11)60277-1.

[41] T. Rothenfluh, M. J. Schuler, and P. Rudolf von Rohr, “Development of a calorimeter for heat flux

measurements in impinging near-critical water jets confined by an annular wall,” J. Supercrit.

Fluids, vol. 73, pp. 141–150, 2013, doi: 10.1016/j.supflu.2012.09.013.

[42] M. J. Schuler, T. Rothenfluh, and P. Rudolf Von Rohr, “Simulation of the thermal field of

submerged supercritical water jets at near-critical pressures,” J. Supercrit. Fluids, vol. 75, pp.

128–137, 2013, doi: 10.1016/j.supflu.2012.12.023.

[43] K. Joo-Kyun and A. Toshio, “A numerical study of heat transfer due to an axisymmetric laminar

impinging jet of supercritical carbon dioxide,” Int. J. Heat Mass Transf., vol. 35, no. 10, pp. 2515–

2526, 1992, doi: 10.1016/0017-9310(92)90093-8.

[44] M. J. Martin, S. Yellapantula, M. S. Day, J. B. Bell, and R. W. Grout, “Impingement of a

supercritical carbon dioxide jet on a planar surface,” in AIAA Scitech 2019 Forum, 2019, doi:

10.2514/6.2019-1559.

[45] S. M. Liao and T. S. Zhao, “An experimental investigation of convection heat transfer to

supercritical carbon dioxide in miniature tubes,” Int. J. Heat Mass Transf., vol. 45, no. 25, pp.

5025–5034, 2002, doi: 10.1016/S0017-9310(02)00206-5.

[46] A. K. Shukla and A. Dewan, “OpenFOAM based LES of slot jet impingement heat transfer at low

nozzle to plate spacing using four SGS models,” Heat Mass Transf. und Stoffuebertragung, vol.

55, no. 3, pp. 911–931, 2019, doi: 10.1007/s00231-018-2470-8.

[47] R. Dutta, A. Dewan, and B. Srinivasan, “Comparison of various integration to wall (ITW) RANS

models for predicting turbulent slot jet impingement heat transfer,” Int. J. Heat Mass Transf., vol.

65, pp. 750–764, 2013, doi: 10.1016/j.ijheatmasstransfer.2013.06.056.

[48] F. Nicoud and F. Ducros, “Subgrid-scale stress modelling based on the square of the velocity

gradient tensor,” Flow, Turbul. Combust., vol. 62, no. 3, pp. 183–200, 1999, doi:

10.1023/A:1009995426001.

[49] G. Lodato, L. Vervisch, and P. Domingo, “A compressible wall-adapting similarity mixed model

for large-eddy simulation of the impinging round jet,” Phys. Fluids, vol. 21, no. 3, p. 035102,

2009, doi: 10.1063/1.3068761.

[50] B. Ničeno and M. Sharabi, “Large eddy simulation of turbulent heat transfer at supercritical

pressures,” Nucl. Eng. Des., vol. 261, pp. 44–55, 2013, doi: 10.1016/j.nucengdes.2013.03.042.

[51] Z. Wang, B. Sun, and L. Yan, “Improved Density Correlation for Supercritical CO2,” Chem. Eng.

Technol., vol. 38, no. 1, pp. 75–84, 2015, doi: 10.1002/ceat.201400357.

[52] A. Sivasamy, V. Selladurai, and P. Rajesh Kanna, “Numerical simulation of two-dimensional

laminar slot-jet impingement flows confined by a parallel wall,” Int. J. Numer. Methods Fluids,

vol. 55, no. 10, pp. 965–983, 2007, doi: 10.1002/fld.1492.

[53] D. Y. Peng and D. B. Robinson, “A New Two-Constant Equation of State,” Ind. Eng. Chem.

Fundam., vol. 15, no. 1, pp. 59–64, 1976, doi: 10.1021/i160057a011.

[54] R. Gardon and J. Cahit Akfirat, “Heat transfer characteristics of impinging two-dimensional air

jets,” J. Heat Transfer, vol. 88, no. 1, pp. 101–107, 1966, doi: 10.1115/1.3691449.

[55] F. Beaubert and S. Viazzo, “Large eddy simulations of plane turbulent impinging jets at moderate

Reynolds numbers,” Int. J. Heat Fluid Flow, vol. 24, no. 4, pp. 512–519, 2003, doi:

10.1016/S0142-727X(03)00045-6.

[56] G. R. Tabor and M. H. Baba-Ahmadi, “Inlet conditions for large eddy simulation: A review,”

Computers and Fluids, vol. 39, no. 4. Pergamon, pp. 553–567, 2010, doi:

10.1016/j.compfluid.2009.10.007.

87

[57] D. Sahoo and M. A. R. Sharif, “Numerical modeling of slot-jet impingement cooling of a constant

heat flux surface confined by a parallel wall,” Int. J. Therm. Sci., vol. 43, no. 9, pp. 877–887,

2004, doi: 10.1016/j.ijthermalsci.2004.01.004.

[58] Y. H. Dong, X. Y. Lu, and L. X. Zhuang, “An investigation of the Prandtl number effect on

turbulent heat transfer in channel flows by large eddy simulation,” Acta Mech., vol. 159, no. 1–4,

pp. 39–51, 2002, doi: 10.1007/BF01171446.

[59] S. Rezaeiravesh and M. Liefvendahl, “Effect of grid resolution on large eddy simulation of wall-

bounded turbulence,” Phys. Fluids, vol. 30, no. 5, p. 055106, 2018, doi: 10.1063/1.5025131.

[60] C. V. Tu and D. H. Wood, “Wall pressure and shear stress measurements beneath an impinging

jet,” Exp. Therm. Fluid Sci., vol. 13, no. 4, pp. 364–373, 1996, doi: 10.1016/S0894-

1777(96)00093-3.

[61] “SALOME 9.3.0.” 2019.

[62] OpenCFD Ltd, “OpenFOAM V1612+.” 2016.

[63] H. Jasak, “Error Analysis and Estimation for the Finite Volume Method with Applications to Fluid

Flows,” Imperial College of Science, Technology and Medicine, 1996.

[64] B. Fabritius and G. Tabor, “Improving the quality of finite volume meshes through genetic

optimisation,” Eng. Comput., vol. 32, no. 3, pp. 425–440, 2016, doi: 10.1007/s00366-015-0423-0.

[65] R. Mittal and P. Moin, “Suitability of upwind-biased finite difference schemes for large-Eddy

simulation of turbulent flows,” AIAA J., vol. 35, no. 8, pp. 1415–1417, 1997, doi: 10.2514/2.253.

[66] “OpenFOAM: User Guide: Pressure-inlet outlet velocity.”

https://www.openfoam.com/documentation/guides/latest/doc/guide-bcs-outlet-pressure-inlet-

outlet.html (accessed May 28, 2020).

[67] “OpenFOAM: API Guide:

src/finiteVolume/fields/fvPatchFields/derived/fixedFluxPressure/fixedFluxPressureFvPatchScalar

Field.H Source File.”

https://www.openfoam.com/documentation/guides/latest/api/fixedFluxPressureFvPatchScalarField

_8H_source.html (accessed May 28, 2020).

[68] “OpenFOAM: User Guide: Total pressure.”

https://www.openfoam.com/documentation/guides/latest/doc/guide-bcs-inlet-outlet-total-

pressure.html (accessed May 28, 2020).

[69] L. Bricteux, S. Zeoli, and N. Bourgeois, “Validation and scalability of an open source parallel flow

solver,” Concurr. Comput., vol. 29, no. 21, 2017, doi: 10.1002/cpe.4330.

[70] J. H. Ferziger and M. Perić, Computational Methods for Fluid Dynamics. 2002.

[71] R. I. Issa, “Solution of the implicitly discretised fluid flow equations by operator-splitting,” J.

Comput. Phys., vol. 62, no. 1, pp. 40–65, 1986, doi: 10.1016/0021-9991(86)90099-9.

[72] A. S. Epikhin, “Numerical Schemes and Hybrid Approach for the Simulation of Unsteady

Turbulent Flows,” Math. Model. Comput. Simulations, vol. 11, no. 6, pp. 1019–1031, 2019, doi:

10.1134/S2070048219060024.

[73] H. Jasak, H. G. Weller, and A. D. Gosman, “High resolution NVD differencing scheme for

arbitrarily unstructured meshes,” Int. J. Numer. Methods Fluids, vol. 31, no. 2, pp. 431–449, 1999,

doi: 10.1002/(SICI)1097-0363(19990930)31:2<431::AID-FLD884>3.0.CO;2-T.

[74] V. Starikovičius, R. Čiegis, and A. Bugajev, “On Efficiency Analysis of the OpenFOAM-Based

Parallel Solver for Simulation of Heat Transfer in and Around the Electrical Power Cables,”

Inform., vol. 27, no. 1, pp. 161–178, 2016, doi: 10.15388/Informatica.2016.80.

[75] P. J. Roache, “Perspective: A method for uniform reporting of grid refinement studies,” J. Fluids

Eng. Trans. ASME, vol. 116, no. 3, pp. 405–413, 1994, doi: 10.1115/1.2910291.

[76] I. B. Celik, U. Ghia, P. J. Roache, C. J. Freitas, H. Coleman, and P. E. Raad, “Procedure for

estimation and reporting of uncertainty due to discretization in CFD applications,” J. Fluids Eng.

Trans. ASME, vol. 130, no. 7, pp. 0780011–0780014, 2008, doi: 10.1115/1.2960953.

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