COUPLED FLUID FLOW AND RADIATION MODELING OF A SMALL

PARTICLE SOLAR RECEIVER

_______________

A Thesis

Presented to the

Faculty of

San Diego State University

_______________

In Partial Fulfillment

of the Requirements for the Degree

Master of Science

in

Mechanical Engineering

_______________

by

Adam Winfield Crocker

Fall 2012

iii

Copyright © 2012

by

Adam Winfield Crocker

All Rights Reserved

iv

ABSTRACT OF THE THESIS

Coupled Fluid Flow and Radiation Modeling of a Small Particle Solar Receiver

by Adam Winfield Crocker

Master of Science in Mechanical Engineering San Diego State University, 2012

In recent years, concentrating solar thermal power has emerged as the most promising technology for utility scale solar electricity generation. Central receiver systems, which are one method of concentrated solar power, use a field of sun-tracking mirrors called heliostats to focus light on a receiver. Existing receivers used in these systems have temperature and flux limitations, which prevent the use of advanced power cycles and reduces plant efficiencies compared to fossil fuel power plants. The development of air-cooled receivers and small particle receivers in particular are summarized herein.

A new type of receiver has been proposed, which makes use of small carbon particles and volumetric absorption in a gas-particle mixture to heat air directly. This thesis builds on previous modeling work done in FORTRAN on the San Diego State University small particle receiver project, expanding the Monte Carlo ray-trace model to include the computation fluid dynamics capabilities of ANSYS FLUENT with the use of several user-defined functions. The input flux is modified to more closely match that provided by a real heliostat field, and the geometry is changed to more accurately approximate a real receiver.

The updated model is benchmarked against existing analytical solutions where possible, and compared to the results of the previous model. The new model is run for a variety of gas mass-flow rates, inlet power levels, and power distributions with a baseline target input power of 5 MW. Outlet gas temperatures predicted by the model ranged from 1300 K to 1550 K, and receiver thermal efficiencies ranged from 80% to 91% depending on operating conditions. The highest efficiencies predicted are with the highest mass-flow rate tested of 6 kg/s.

v

TABLE OF CONTENTS

PAGE

ABSTRACT ............................................................................................................................. iv

LIST OF TABLES .................................................................................................................. vii

LIST OF FIGURES ............................................................................................................... viii

NOMENCLATURE ................................................................................................................ xi

ACKNOWLEDGEMENTS .................................................................................................... xii

CHAPTER

1 INTRODUCTION .........................................................................................................1 

1.1 SPHER Receiver ................................................................................................9 

1.2 Monte Carlo Ray-Trace Method ........................................................................9 

1.3 Scope of Thesis ................................................................................................10 

2 BACKGROUND ON SMALL PARTICLE RECEIVERS .........................................12 

3 PREVIOUS WORK AND MODIFICATIONS TO MONTE CARLO RAY-TRACE MODEL .........................................................................................................16 

3.1 Incident Flux ....................................................................................................17 

3.2 Outlet Tube ......................................................................................................20 

3.3 Comparison to Benchmarks .............................................................................21 

4 COUPLING OF THE MONTE CARLO AND FLUENT MODELS .........................29 

4.1 Coupling Scheme & UDF Functions ...............................................................30 

4.2 Solution Controls .............................................................................................34 

4.3 Grid Matching Scheme ....................................................................................36 

5 COLD FLOW & GRID STUDY .................................................................................40 

5.1 Grid and Geometry ..........................................................................................40 

5.2 Cold Flow Verification ....................................................................................43 

5.3 Grid Study ........................................................................................................45 

6 RESULTS AND DISCUSSION ..................................................................................54 

6.1 Gaussian Flux Distribution ..............................................................................55 

6.2 Flow Rate Variation .........................................................................................60 

6.3 Input Power variation .......................................................................................65 

vi

7 CONCLUSIONS AND FUTURE WORK ..................................................................70 

7.1 Conclusions ......................................................................................................70 

7.2 Future Work .....................................................................................................71 

REFERENCES ........................................................................................................................74

APPENDIX

SUMMARY OF TURBULENCE MODEL CONSTANTS USED IN FLUENT .............77 

vii

LIST OF TABLES

PAGE

Table 5.1. Summary of Grid Attributes ...................................................................................46 

Table 6.1. Summary of Model Runs ........................................................................................54 

viii

LIST OF FIGURES

PAGE

Figure 1.1. Schematic of a central receiver solar thermal power plant. .....................................1 

Figure 1.2. Aerial view of the Gemasolar power plant in Spain.. ..............................................2 

Figure 1.3. Rankine cycle diagram and schematic. ...................................................................3 

Figure 1.4. Brayton cycle diagram and schematic. ....................................................................4 

Figure 1.5. Combined Brayton-Rankine cycle diagram and schematic. ....................................5 

Figure 1.6. Diagram of the Solar One receiver. .........................................................................6 

Figure 1.7. Schematic of pressurized high-temperature DLR receiver. ....................................7 

Figure 1.8. Schematic of the Sandia falling particle receiver. ...................................................7 

Figure 1.9. Schematic of cylindrical small particle solar receiver .............................................8 

Figure 2.1. Cross section of the experimental SPHER ............................................................12 

Figure 2.2. Schematic Cross section of the Weizmann Institute receiver. ..............................13 

Figure 3.1. Temperature field output from original MCRT code. Flow is from right to left, as indicated by white arrows. ...............................................................................18 

Figure 3.2. Flux map predicted from MIRVAL. Figure represents 3 m by 3 m region on aperture plane. .........................................................................................................19 

Figure 3.3. Plot of the local incident solar flux applied to the aperture. ..................................20 

Figure 3.4. Schematic of modified receiver geometry in Monte Carlo ray trace model .........21 

Figure 3.5. Schematic of modified receiver geometry in Monte Carlo ray trace model for black end-walls benchmark. ...................................................................................22 

Figure 3.6. Net energy exchange between parallel black plates with no participating medium, both at 500 K. The thick black line at 2.0 m represents the outlet tube. ..............................................................................................................................22 

Figure 3.7. Geometry of Monte Carlo model for concentric cylinders benchmark. ................23 

Figure 3.8. Net energy exchange between concentric cylinders with no participating medium. The inner cylinder is at 500 K and the outer cylinder is at 1000 K. .............24 

Figure 3.9. Plot of non-dimensional emissive power in gray gas plane layer benchmark. ...................................................................................................................25 

Figure 3.10. Plot of non-dimensional emissive power in gray gas between concentric cylinders. Each data set represents the radial temperature profile at a different axial location. ...............................................................................................................27 

ix

Figure 3.11. Plot of non-dimensional emissive power in gray gas between concentric cylinders for three different optical depths. .................................................................28 

Figure 4.1. Simple flow diagram of the coupling scheme used by the model. ........................30 

Figure 4.2. Wall Flux Interpolation Method. Vertical grid lines correspond to wall element divisions in the Monte Carlo solver. A negative net wall flux means that the wall cell is a net emitter of radiation and thus is heated by the fluid, while a positive value means the cell is a net absorber of radiation and is cooled by the gas. .........................................................................................................32 

Figure 4.3. Thermal boundary layers near the outer cylinder wall at four different axial locations. Legend values indicate axial distance from aperture of each data series. ....................................................................................................................35 

Figure 4.4. Net energy gain in the gas flow as a function of model iteration number for two different temperature under-relaxation methods. ............................................37 

Figure 4.5. Net energy gain in the gas flow as a function of model iteration number with no temperature under-relaxation. This scheme appears to be unstable, or at least not converging. ................................................................................................38 

Figure 4.6. Illustration of a FLUENT grid cell (in red) overhanging the boundary of a Monte Carlo grid cell (in black). ..................................................................................38 

Figure 5.1. Schematic of the geometry as it is modeled in FLUENT and the Monte Carlo code. Drawing is not to scale. ............................................................................40 

Figure 5.2. View of the grid as it appears in FLUENT. ...........................................................41 

Figure 5.3. The FLUENT grid (green) with an approximate overlay of the Monte Carlo grid (black). ........................................................................................................42 

Figure 5.4. Plot of the velocity profile on the outlet plane of the receiver for two FLUENT turbulence models as compared to a theoretical turbulent velocity profile. ..........................................................................................................................44 

Figure 5.5. Cold flow pressure contours in FLUENT .............................................................45 

Figure 5.6. Wall temperatures in FLUENT for each of the three grids studied. .....................46 

Figure 5.7. Velocity vectors in the window corner region for Grid 1. Vectors are colored and scaled by magnitude. Color-bar scale represents velocity in m/s, and length scale of the geometry is indicated at the top of the figure. ........................47 

Figure 5.8. Velocity vectors in the window corner region for Grid 2. Vectors are colored and scaled by magnitude. Color-bar scale represents velocity in m/s, and length scale of the geometry is indicated at the bottom of the figure. ..................48 

Figure 5.9. Velocity vectors in the window corner region for Grid 3. Vectors are colored and scaled by magnitude. Color-bar scale represents velocity in m/s, and length scale of the geometry is indicated at the top of the figure. ........................48 

Figure 5.10. Cylinder wall temperatures in the first half-meter of the receiver. The x-axis is scaled to align with the velocity vectors in Figure 5.11. ..................................49 

x

Figure 5.11. Velocity vectors in the window corner region for Grid 2. Vectors are scaled by magnitude of velocity, and colored by temperature in Kelvin. ....................50 

Figure 5.12. Wall fluxes imposed by the Monte Carlo on the FLUENT solver for each of the three grids studied. Positive values of this net wall flux means the wall is absorbing more radiation than it is emitting.............................................................52 

Figure 6.1. Effect of solar flux distribution on energy balance. ..............................................56 

Figure 6.2. Temperature field inside receiver with 5 MW of uniform incident solar flux and 5 kg/s mass-flow rate. Color-map values are in units of Kelvin. ..................57 

Figure 6.3. Temperature field inside receiver with 5 MW of Gaussian distributed incident solar flux and 5 kg/s mass-flow rate. Color-map values are in units of Kelvin. ..........................................................................................................................57 

Figure 6.4. Cylinder wall temperature profiles for the uniform (blue) and Gaussian (red) flux distribution cases. ........................................................................................59 

Figure 6.5. Effect of mass-flow rate on gas outlet temperature and receiver thermal efficiency, with 5 MW collimated Gaussian input. .....................................................60 

Figure 6.6. Receiver, engine, and total efficiencies as a function of operating temperature. .................................................................................................................62 

Figure 6.7. Temperature field in the receiver with a mass-flow rate of 4.0 kg/s and input power of 5 MW. Color-map numbers are in units of Kelvin. .............................62 

Figure 6.8. Temperature field in the receiver with a mass-flow rate of 6.0 kg/s and input power of 5 MW. Color-map numbers are in units of Kelvin. .............................63 

Figure 6.9. Energy balance as a function of iteration number, for five different mass-flow rate cases, 5 MW input power. ............................................................................64 

Figure 6.10. Axial cylinder wall temperature profiles for five different mass-flow rates, 5 MW input power. ............................................................................................65 

Figure 6.11. Gas outlet temperatures and receiver thermal efficiencies for five different inlet power levels. .........................................................................................66 

Figure 6.12. Temperature field inside the receiver with a mass-flow rate of 2 kg/s and input power of 2 MW. Color-bar units are Kelvin .......................................................67 

Figure 6.13. Temperature field inside the receiver with a mass-flow rate of 6 kg/s and input power of 6 MW. Color-bar units are Kelvin. ......................................................68 

Figure 6.14. Axial cylinder wall temperature profiles for five different inlet power levels and mass-flow rates. ..........................................................................................68 

xi

NOMENCLATURE

= spectral absorption coefficient [1/m] = spectral intensity [W/(μm·m2·sr)] = spectral black-body intensity [W/(μm·m2·sr)]

c = FLUENT cell edge length [m] = specific heat capacity [J/(kg K)] = number of photon emissions [ND]

G = irradiation [W/m2] = thermal conductivity [W/(m·K)]

L = receiver length [m] m = mass [g]

= pressure [Pa] = radiative heat flux vector [W/m2]

r = radius [m] = time [s] = velocity vector [m/s]

T = temperature [K] u+ = wall coordinates velocity [ND] u* = friction velocity [m/s] = average velocity [m/s]

y = position [m] y+ = wall coordinates position [ND] z = axial position [m] β = under-relaxation factor [ND] = viscosity [Pa·s] λ = wavelength [μm] σ = standard deviation [ND]

= spectral scattering coefficient [1/m] Φ = spectral phase function [ND]

= non-dimensional emissive power [ND] = wall shear stress [Pa] = solid angle [sr] = density [g/m3]

xii

ACKNOWLEDGEMENTS

I would like to acknowledge first and foremost my advisor for this research, Dr.

Fletcher Miller, whose guidance and experience made this otherwise-massive undertaking

possible, and whose enthusiastic belief in the potential of solar thermal power drew me to

this project in the first place. I would also like to acknowledge my predecessor in this work

Steve Ruther, who developed the original version of the Monte Carlo ray-trace radiation

solver for this receiver. My research would not have been possible without his work to build

upon. Dr. Arlon Hunt, who first proposed the concept of a small particle solar receiver, also

provided valuable insight throughout the progress of my research. The support and aid of my

colleagues in the SDSU Solar Energy and Combustion Laboratory was invaluable,

particularly Pablo del Campo who helped discover the final errors in the Monte Carlo code.

The financial support I received via grants from Google (grant # 32-2008) and the US

Department of Energy (grant # DE-EE0005800) not only made this work possible, but

allowed me to focus on this project full time.

1

CHAPTER 1

INTRODUCTION

Solar power has long been associated with renewable energy hopes, but has seen

significant growth in recent years. While photovoltaics have typically been the most high-

profile collection method, solar thermal power represents the best long term source for

utility-scale solar energy. Central receiver systems, sometimes referred to as power tower

systems due to their prominent collection towers, are one of the major configurations

available today for solar thermal energy production. In this arrangement, sunlight is focused

by a field of mirror clusters called heliostats onto a single receiver atop a tower as illustrated

in Figure 1.1 [1].

Figure 1.1. Schematic of a central receiver solar thermal power plant. Source: Ruther, Steven James. “Radiation Heat Transfer Simulation of a Small Particle Solar Receiver Using the Monte Carlo Method.” Master’s Thesis, San Diego State University, 2010.

The first large scale central receiver system to be built and tested was the Solar One

project in the Mojave Desert of California. Operating from 1982-1986, that system used

1,818 heliostats spread over 126 acres to boil water which drove a Rankine steam cycle,

producing up to 10 MW of electrical power [2]. It was redesigned and retrofitted in 1995 to

Sun

Heliostat Field

Receiver

Generator Turbine Tower

2

use molten salt as the receiver heat transfer fluid instead of steam, in the process becoming

Solar Two. The salt could then be stored in tanks before being sent through a heat exchanger

to boil water and drive the steam turbine. The concepts and technology demonstrated by

these two test projects are in use today in half a dozen commercial power plants in Europe

and the United States, as well as several currently under construction. The largest capacity

central receiver system in operation is the 20 MW Gemasolar plant, designed and operated

by Torresol Energy in southern Spain [3] and pictured in Figure 1.2 [4]. This power plant

began operation in 2011, and makes use of molten salt in the receiver and for storage in a

similar manner to Solar Two.

Figure 1.2. Aerial view of the Gemasolar power plant in Spain. Image reproduced from Torresol Energy under the terms of the Free Art License. Source: Torresol Energy. “Gemasolar.” Gemasolar.jpg. Last modified June 21, 2011. http://en.wikipedia.org/wiki/File:Gemasolar.jpg.

Solar thermal power takes advantage of over 100 years of thermodynamics research

and engineering by generating electricity from heat. This means that advances in turbine

design from existing fossil fuel research benefits solar thermal power as well. All existing

solar power plants built to date have made use of the Rankine steam cycle to produce

3

electricity. A temperature-entropy diagram of the Rankine steam cycle is shown in Figure 1.3

[5]. This is a phase-change cycle, which utilizes water and is commonly used at input

temperatures between 400-600 C. the Gemasolar plant for example has a turbine inlet

temperature of 565 C. This presents two major limitations for solar power. Being a phase

change cycle, cooling water is often needed to re-condense the working steam after going

through the turbine. For a technology best suited to dry desert climates, water usage is a

significant concern. Air cooling can and is used in some plants, but this comes with a

reduction in efficiency.

Figure 1.3. Rankine cycle diagram and schematic. Source: Stine, William B. and Michael Geyer. “Power Cycles for Electricity Generation.” Power from the Sun. Last modified October 2004. http://www.powerfromthesun.net/Book/chapter12/chapter12.html#12.4 Brayton Cycle Engines.

Secondly, a peak cycle temperature of 600 C is relatively low compared to modern

fossil fuel power plants, and thus imposes a lower thermodynamic efficiency limit on these

systems.

Both limitations of the Rankine cycle can be overcome by the Brayton cycle. This is

an all gas phase cycle, used in jet engines and some fossil fuel power plants. The Brayton

cycle can operate at much higher temperatures, and thus reach greater thermodynamic

efficiencies as compared to Rankine systems. Additionally, there is no need for cooling water

in an open, all gas phase cycle. A temperature-entropy diagram of an idealized Brayton cycle

is pictured in Figure 1.4 [5].

4

Figure 1.4. Brayton cycle diagram and schematic. Source: Stine, William B. and Michael Geyer. “Power Cycles for Electricity Generation.” Power from the Sun. Last modified October 2004. http://www.powerfromthesun.net/Book/chapter12/chapter12.html#12.4 Brayton Cycle Engines.

An important development in energy production has been combined cycle power

plants, which combine the efforts of Brayton gas turbine with Rankine steam generators to

produce higher efficiencies than either cycle individually. These systems work by using the

exhaust heat from a Brayton cycle to drive a Rankine generator, yielding a larger overall

temperature drop. The state of the art fossil fuel combined cycle generators are capable of

thermodynamic efficiencies as high as 60% today [6]. A diagram of the combined Brayton-

Rankine cycle is shown in Figure 1.5.

One challenge currently preventing the application of Brayton and combined

Brayton-Rankine cycles in solar thermal power is the ability to produce the necessary high

temperature gas. Until recently, the receivers in solar thermal systems have relied on surface

absorption of solar radiation before transferring the thermal input to a working fluid. A

traditional receiver design as used in Solar One is shown in Figure 1.6 [2]. In this

arrangement, incident sunlight from the heliostat field strikes the exterior of one of the many

dark vertical tubes which make up the circumference of the receiver body. The energy is

absorbed by the tube surface there, and is conducted through the tube wall to the interior,

where the cooling fluid carries it away via convection. In current plants, this coolant is

5

Figure 1.5. Combined Brayton-Rankine cycle diagram and schematic. Source: Stine, William B. and Michael Geyer. “Power Cycles for Electricity Generation.” Power from the Sun. Last modified October 2004. http://www.powerfromthesun.net/ Book/chapter12/chapter12.html#12.4 Brayton Cycle Engines.

typically steam or molten salt. A major limitation of this design is the fact that the component

which absorbs incident radiation is not the working fluid itself, but rather an intermediary

tube. Thus, potential temperatures are limited by the material constraints of the absorber.

Additionally, the hottest part of the receiver is the exterior surface of the absorber tubes,

which are exposed directly to the environment. This can result in significant convection and

radiation losses, reducing the receiver’s efficiency. Current designs use absorber materials

with spectrally variant emission properties in an effort to reduce radiation losses, but these

materials can also have temperature limits. Limiting convection losses can be quite difficult

for traditional receiver designs.

The German Aerospace Center (DLR) has conducted research on a volumetric air

cooled receiver for the purposes of driving a gas turbine or combined cycle [7, 8]. The DLR

volumetric receiver operates at pressures up to 15 bar, and uses a porous absorber behind a

curved quartz window to collect concentrated solar energy. Air is forced through the absorber

and is heated to outlet temperatures of 800-1000 C. A diagram of the DLR receiver is shown

in Figure 1.7 [7]. This design was used and tested in conjunction with other lower cost and

lower temperature receivers. The high temperature volumetric receiver was used to heat

6

Figure 1.6. Diagram of the Solar One receiver. Source: Stine, William B., and Michael Geyer. “Central Receiver Systems.” Power from the Sun. Last modified October 2012. http://www.powerfromthesun.net/ Book/chapter10/ chapter10.html.

pre-heated air from the other receivers to the target temperature of 1000 C, and the whole

system was tested in a hybrid scheme with a traditional fossil fuel gas turbine. Receiver

efficiencies averaged 80%, and pressure drop through the receiver was only 120 mbar.

Another potential receiver for high temperatures is the falling particle receiver

investigated by Sandia National Labs [9]. In this arrangement, small carbon particles

approximately 500 μm in diameter are dropped through a beam of concentrated solar

radiation. A schematic of the receiver is shown in Figure 1.8. The particles can then be stored

directly or used to pass heat to a working fluid. At this size, the particles do not exhibit

selective absorption properties, and thus radiation losses can be significant. The falling

particle receiver has shown the potential to produce outlet particle temperatures in excess of

1000 C. Higher temperatures come at the cost of reduced efficiencies however. Another

limitation of this method is the difficulty in extracting the heat from the particles efficiently.

7

Figure 1.7. Schematic of pressurized high-temperature DLR receiver. Source: Buck, R., E. Lüpfert, and F. Téllez. “Receiver for Solar-Hybrid Gas Turbine and CC Systems (REFOS).” Paper presented at the IEA Solar Thermal Conference, Sydney, Australia, 2000.

Figure 1.8. Schematic of the Sandia falling particle receiver.

8

A new receiver design is being investigated here, in which the solar input is absorbed

volumetrically by a gas-particle mixture within the receiver cavity, thus heating the working

fluid directly. The advantages of this method include the ability to accommodate very high

incident flux levels due to volumetric absorption, reduced pressure loss, higher receiver

efficiency, and consequently higher outlet gas temperature. Higher outlet temperatures in

turn allow the possibility of utilizing a Brayton cycle for higher thermodynamic efficiency of

the overall system, ease of daily start-up and shut-down, as well as reduced water usage for

cycle cooling. A schematic of the new receiver design is shown in Figure 1.9.

Figure 1.9. Schematic of cylindrical small particle solar receiver.

Small particle receivers are not limited, however, to electricity generation; the high

flux levels and intimate mixing between gas and particles offer many possibilities for solar

chemistry as well [10].

9

1.1 SPHER RECEIVER

A critical task in designing a small particle solar receiver is modeling the fluid flow

within the cavity, as well as the radiation heat transfer. These two pieces must work together

to give an accurate picture of receiver performance and behavior. The Small Particle Heat

Exchange Receiver (SPHER) was originally proposed independently in 1979 by both Hunt

[11] and Abdelrahman [12]. Its core features derive from the spectral variation in the

absorptivity of the small carbon particles. When controlled for size, the carbon particles

exhibit selective absorptivity, absorbing well in the solar spectrum and emitting poorly at

longer infrared wavelengths. With the proper mass loading of particles (mass of particles per

unit volume of air), the concentrated radiation can be absorbed in the volume of the receiver

before most of the radiation reaches a wall, thus keeping portions of the walls cooler than the

outlet fluid temperature reducing material constraints and heat losses.

1.2 MONTE CARLO RAY-TRACE METHOD

Due to the spectral and directional nature of the radiation source and the particle

properties, calculation of the radiation field in the receiver is difficult with traditional

methods. The Monte Carlo ray-trace (MCRT) method is a statistical approach to solving the

governing equation of radiation exchange in a participating medium, and can be used for

both radiative exchange between surfaces and participating media. It works by simulating

millions of discrete photon bundle emissions from every radiatively participating element in

the domain and tracing each path until the bundle is absorbed or leaves the domain.

Wavelengths, directions and locations of ray emissions are determined according to a

cumulative distribution function (CDF) computed according to the geometry and conditions

of the problem. Likewise, absorption, scattering, and reflection events are also determined

probabilistically based on the spectral and directional properties of surfaces and media in the

domain. Other methods of solving the RTE include the Discrete Ordinates method, which

solves the RTE analytically for a finite number of discrete direction vectors within the

medium. Radiation in directions other than those specifically solved for are lumped in with

the nearest solved direction. Because of this, the method has trouble with strongly directional

radiation. This method works reasonably well in optically thick mediums with uniform

directional radiation, but is less reliable near boundaries and in optically thin cases. The

10

MCRT method was selected for its ability to accurately model spectral property variations in

the system, as well as potentially complex input flux conditions on the receiver window from

the heliostat field. This input will eventually be taken from MIRVAL [13], which is another

Monte Carlo code. MIRVAL uses the MCRT method to simulate the behavior of a specific

the heliostat field, in this case the field at Sandia National Labs where the prototype receiver

will be tested.

1.3 SCOPE OF THESIS

This thesis models the radiation, convection, and conduction heat transfer along with

gas flow in an axisymmetric small particle solar receiver. This research expands on previous

work by coupling the MCRT radiation model developed by Ruther with the more

sophisticated fluid dynamics modeling capabilities of ANSYS FLUENT. This coupling

allows for the inclusion of more realistic inlet and outlet geometries in the receiver, similar to

those represented in Figure 1.9, which will be a critical part of the design of a real prototype.

The use of FLUENT also allows for a turbulence model, convection between the walls and

the gas, and much finer grid sizing for the energy solver. Taken together, these features give

a more complete picture of the heat transfer, mixing, and temperature profiles within the

receiver. The flow and energy solver remains two-dimensional and axisymmetric to save

computation time, while the Monte Carlo ray-trace is conducted in three dimensions. The

window is treated as an open aperture in the MCRT, and as a constant-temperature wall

boundary in the FLUENT solver. Other researchers in the group are working on optical and

cooling models of the window, and their work will be integrated with this model in the

future.

Background on previous experimental and modeling work on small particle receivers

is covered in Chapter two. A more detailed explanation of the Monte Carlo model is shown

in Chapter three, along with modifications to the MCRT model for this work which include a

Gaussian flux distribution on the aperture to more closely match flux maps produced by real

heliostat fields. Chapter three also touches on additional geometry changes to the MCRT

code which did not pass benchmarks and were thus not included in the final model for this

thesis, but have since been fixed and will be used in the future. An unheated flow through the

geometry is modeled in Chapter four in FLUENT, and the velocity profile is validated

11

against theoretical benchmarks. Computational and iteration techniques for coupling the two

solvers are investigated in Chapter five, including enforcement of minimum and maximum

temperature and source term values as well as several under-relaxation schemes. Receiver

performance is modeled at steady state conditions in Chapter six for a variety of mass flow

rates and inlet power levels. Gas outlet and cavity wall temperatures are computed and

compared with those obtained by Ruther’s model, and receiver efficiencies are estimated.

12

CHAPTER 2

BACKGROUND ON SMALL PARTICLE

RECEIVERS

Experimental work on the small particle receiver concept was first done by Hunt and

Brown in 1982 [14]. That work demonstrated the potential of the design, producing outlet gas

temperatures of 1000 K, with a particle mass loading of 1 g/m3 and an incident flux level of

500 kW/m2. A schematic of the receiver built and tested by Hunt and Brown is shown in

Figure 2.1. The aperture window is at the bottom of the receiver as pictured here. The gas-

particle mixture entered the receiver through holes in a manifold on the cylinder wall, visible

in the middle of the diagram. Hot air exited an axial outlet tube upward, leaving the receiver

at the top.

Figure 2.1. Cross section of the experimental SPHER.

More recently in 2003, experiments by Bertocchi et al at the Weizmann Institute of

Science using sub-micron sized carbon particles and a particle mass loading as high as 7 g/m3

produced outlet gas temperatures of 2000 K. [15]. The receiver in those tests was

significantly smaller than that investigated here, with an aperture diameter up to 80 mm and

13

average incident flux levels up to 900 kW/m2, though peak concentration reached 5 MW/m2.

The flow orientation also differed from present work, as the Weizmann receiver had inlets

near the aperture with the bulk of the gas-particle mixture flowing concurrent with the

incident radiation. This flow orientation is illustrated in Figure 2.2 [15] which shows a cross-

sectional schematic of the Weizmann receiver.

Figure 2.2. Schematic Cross section of the Weizmann Institute receiver. Source: Bertocchi, R., J. Karni, and A. Kribus. “Experimental Evaluation of a Non-Isothermal High Temperature Solar Particle Receiver.” Energy 29, no. 5-6 (2004): 687-700.

That work investigated variations in mass flow rate, particle mass loading, and

aperture size, as well as exploring several gases. Receiver efficiencies were estimated

between 80-90%. Further research by Klein, Rubin, and Karni [16] at the Weizmann Institute

in 2008 produced exit gas temperatures up to 1500 K under flux levels of 3 MW/m2. That

work also demonstrated wall temperatures below peak gas temperatures, a finding first seen

in the original tests conducted by Hunt and Brown [14].

Early numerical modeling of small particle receivers was undertaken by Miller in

1988 [17]. That work modeled an experimental lab scale cylindrical receiver 1 m long and

14

5.9 cm in diameter, powered by a xenon arc lamp with a mostly collimated input. Flow could

be oriented concurrent or opposed to the incident radiation. The model used a four-flux

method to solve the radiative transfer equation in the gas-particle mixture. While specular

variations were considered, the primary limitation of this method was an inability to

accurately capture the effects of scattering and wall boundaries. Additionally, the gas flow

was modeled as either a laminar parabolic profile or a flat slug flow. Temperatures in the gas

flow of the experiment proved difficult to accurately measure, and the model did not produce

good agreement with experimental results.

Numerical modeling of the Weizmann receiver was done by Klein et al. in 2006 [18],

utilizing commercial CFD software coupled with a Monte Carlo ray-tracing method.

However, that work used the Monte Carlo method only for surface radiation exchange, and

used a discrete ordinance method for radiation heat transfer within the gas-particle mixture.

Additionally, the Monte Carlo model was gray and thus unable to capture the spectral

absorption properties of the particle cloud. The flow model in that work was laminar, as the

geometry of the smaller receiver resulted in a low Reynolds number of 400. By contrast, the

Reynolds number in the main cavity of the receiver studied in this thesis is 4000. Good

agreement was found between experimentally observed and numerically computed wall

temperatures.

The Monte Carlo ray-trace method has also been used to model heat transfer in a

solar chemical reactor by Z’Graggen and Steinfeld [10]. The solar reactor shares many

features with small particle solar receivers including a quartz window for incident radiation, a

cylindrical cavity, and direct irradiation of small particles. Unlike small particle receivers

however, in solar chemical reactors the concentrated sunlight is used to drive chemical

reactions instead of simply produce hot gases. Like the Weizmann experiments, the

Z’Graggen reactor includes particle-gas flow concurrent with the incident solar radiation, a

smaller scale design than that investigated here (97mm reactor diameter), and peak flux

levels up to 5 MW/m2. That model also used a finite volume CFD solver coupled with a

Monte Carlo ray-trace code to simulate heat transfer in the reactor. In this case, the Monte

Carlo model included spectral considerations and was used for both surface radiation

exchange as well as in the participating medium. Solar input was used to raise temperatures

to 1300 K where chemistry took over and drove temperatures as high as 2000 K. Again,

15

modeling did a good job of predicting wall temperatures in the reactor, though gas

temperatures in the cavity were less easily matched by experiments.

A concern common to all these various implementations is the structural integrity of

the quartz window, required either to separate gases and reactants from the environment or to

contain pressure. Window investigations have been undertaken in modeling by Röger, Buck,

and Muller-Steinhagen at the German Aerospace Center [19] as well as experimentally by

Karni et al. at the Weizmann Institute [20]. Both experiments used small windows from 12-

32cm in diameter, and found it feasible to maintain a quartz window under pressure up to 15

bar and with internal receiver temperatures in excess of 1300K. Modeling of a larger window

for use in this project was done by Mande [21], though that work was limited to a stress

analysis and did not include temperature considerations. The quartz window is a pivotal but

complex piece to this receiver design, and as such is outside the scope of this thesis. It is a

subject under active research by others in the group at San Diego State University.

16

CHAPTER 3

PREVIOUS WORK AND MODIFICATIONS TO

MONTE CARLO RAY-TRACE MODEL

Previous work on the current receiver has been done by Steve Ruther, using the

Monte Carlo Ray Trace (MCRT) method to model the detailed radiation heat transfer within

a cylindrical receiver with absorbing, emitting, and scattering particles [1]. As introduced in

Chapter 1, the Monte Carlo method is a statistical strategy for solving the radiative transfer

equation (RTE) which is the governing equation of radiation exchange in a participating

medium and is shown in Equation 3.1 [22].

(3.1)

The radiative transfer equation characterizes the change in spectral intensity, , as radiation

travels along a path length through the medium. It includes four terms: energy lost to the

medium via absorption, energy gained via emission from the medium, energy lost due to

scattering, and energy gained due to in-scattering. Each of these terms can have spectral

dependence, and the scattering terms can be highly directional as well. The RTE must be

integrated over a path length to determine the value of spectral intensity at any point in the

domain. That intensity can then be used in an energy balance as shown in Equation 3.2:

(3.2)

This is an energy balance between total energy gained and lost via radiation in a control

volume. The sum is expressed as which is the divergence of the radiative flux.

In order to obtain temperature profiles in the gas-particle mixture, the RTE must be

coupled to the conservation of energy equation. This is accomplished by including the

divergence of the radiative flux in the energy equation of the gas flow as a source term, as

illustrated in Equation 3.3. In this formulation, the divergence of the radiative flux represents

net energy added or removed from a control volume via radiation.

17

· · (3.3)

The full conservation of energy equation includes the total derivative of the

temperature, which in turn includes a velocity term due to advection. Thus it is necessary to

solve for the flow field as well in the model to determine the temperature field. Ruther’s

work included a uni-directional slug flow model of the gas particle mixture in order to obtain

velocity profiles for the energy equation. In that model, the energy equation was simplified to

include only axial advection, radial diffusion, and the volumetric source term for radiation.

Both the Monte Carlo ray-trace and energy solver portions of the model were written in

FORTRAN. The geometry was limited to orthogonal boundaries in order to simplify the

Monte Carlo code and keep computation times relatively short. The window was set at 1.5 m

radius to be at what was considered the upper limit for a quartz window. The total receiver

radius was set at 2.5 m to investigate heating of the gas from indirect insolation (i.e.,

scattered light), and the axial length was set at 5 m to allow for a wide range of optical

thicknesses in the participating medium.

Due to the simplified nature of the energy solver and flow model, Ruther’s work

focused on the radiation heat transfer. Variables studied included particle mass loading,

particle size, mass flow rate, wall optical properties, and flow direction. The greatest receiver

efficiencies of 90% were predicted with opposed flow, entering at the rear of the receiver

cavity and traveling forwards towards the incident radiation, with a particle diameter of 0.2

μm and particle mass loading of 0.30 g/m3. Peak temperatures of 1600 K were predicted,

with a 5MW solar input and a uniform flux level of 707 kW/m2. Figure 3.1 shows an

example of the output from Ruther’s version of the model. It shows the axisymmetric

temperature field in the gas-particle mixture for a 5MW collimated input, with 0.30 g/m3

particle mass loading and 0.5 μm particle diameter.

3.1 INCIDENT FLUX

Previous work included two different arrangements of incoming solar radiation

entering the receiver: uniform collimated radiation as a base case, and a 45-degree cone angle

to more accurately represent what comes from the heliostat field [1]. Both schemes assumed

a uniform incident flux on the aperture, varying only the angle of incidence. This uniform

flux assumption is however not realistic, as real world heliostat fields cannot reasonably

18

Figure 3.1. Temperature field output from original MCRT code. Flow is from right to left, as indicated by white arrows.

provide a uniform flux over the course of a day, and in fact rarely produce a uniform flux

map at all. This is illustrated in Figure 3.2, which shows an example flux map computed by

the MIRVAL code, which is a Monte Carlo simulation of the heliostat field at Sandia

National Labs [13].

In the current absence of a full heliostat field model coupled to the receiver, the

receiver model has therefore been modified to apply a Gaussian flux distribution to the

aperture. The original model computed the bundle energy of each emission from the aperture

according to Equation 3.4 where is the bundle energy, dA is the area of the individual

aperture element, and flux is the incident solar flux on the aperture element.

#

(3.4)

By rearranging this formulation, the number of emissions can be varied based on a bundle

energy consistent with the original model, and dependent on the desired flux on the aperture

element. The aperture plane has five radial divisions, to create five concentric ring-shaped

elements. The uniform flux case imparted 5 MW on a 1.5 m radius aperture, resulting in

1 2 3 4

-0.5

0

0.5

1

1.5

2

2.5

3

Axisymmetric Temperature FieldColor Bar Units K

Length (m)

Rad

uis

(m)

700

800

900

1000

1100

1200

1300

1400

1500

1600

19

Figure 3.2. Flux map predicted from MIRVAL. Figure represents 3 m by 3 m region on aperture plane. Source: Leary, P. L., and J. D. Hankins. A User's Guide for MIRVAL - A Computer Code for Comparing Designs of Heliostat-Receiver Optics for Central Receiver Solar Power Plants. Livermore, CA: Sandia Labs., 1979.

0.707 MW/m2. For the Gaussian distribution, the peak flux is chosen to be just short of thrice

this, at 2.8 MW/m2. This peak was chosen based on simulations of the heliostat test field at

Sandia National Labs using the MIRVAL code. For a Gaussian flux distribution centered at

the middle of the aperture, the probability weighting for a given location is given by equation

3.5 where σ is the standard deviation, and r is the radius from the receiver centerline.

(3.5)

The local flux at each radial division is weighted by this probability, and a standard deviation

of 0.52 yields a total power input equal to the uniform flux case of 5 MW. The flux

distribution on each aperture ring element is plotted in Figure 3.3, along with the MIRVAL

20

Figure 3.3. Plot of the local incident solar flux applied to the aperture.

results for comparison. The wavelength distribution of the incoming radiation is modeled as

blackbody emission at 5780 K to approximate the solar spectrum.

3.2 OUTLET TUBE

The major modification attempted to the original Monte Carlo model is the addition

of an outlet tube to the geometry. The outlet tube exists as a smaller cylinder concentric with

the outer cylindrical wall. It is intended to absorb, emit, and reflect rays in the same manner

as the other receiver walls. To add the outlet tube to the model, an additional radial division

is needed to represent tube wall elements. This radial division has zero thickness, like all wall

elements in the domain. A schematic of the modified geometry with the outlet tube is shown

in Figure 3.4, where r1 is the radius of the outlet tube, r2 is the radius of the outer cylinder

0.0

0.3

0.6

0.9

1.2

1.5

1.8

2.1

2.4

2.7

3.0

0.15 0.45 0.75 1.05 1.35

Loc

al I

nci

den

t F

lux

[MW

/m2]

Radius of Aperture Element [m]

Gaussian Flux Distribution

Approximated Model Flux

Calculated Real Flux

21

Figure 3.4. Schematic of modified receiver geometry in Monte Carlo ray trace model.

wall, and g is the gap distance between the entrance end of the outlet tube and the aperture

plane of the receiver.

3.3 COMPARISON TO BENCHMARKS

To verify the outlet tube addition to the Monte Carlo model, a comparison is made

with existing solutions. The first two benchmark cases investigated examine radiation

exchange between surfaces, with no participating medium between them. The first of these is

meant to model radiation exchange between parallel infinite plates. To simulate this in the

Monte Carlo code, the geometry is modified by extending the outlet tube to span the full

length of the receiver axis. This change will carry over to all the benchmarks described in

this chapter. For the parallel plate case, the end walls are treated as black bodies at 500 K,

while the cylinder walls are perfect specular reflectors as illustrated in Figure 3.5. Results

from this case are plotted in Figure 3.6, with radial position oriented on the vertical axis.

The plot in Figure 3.6 shows the net energy gained or lost by each ring element on

either end wall. The results show a clear pattern, where the net source terms are negative and

of similar magnitude for all end-wall ring elements except those adjacent to a cylindrical

wall, which are significantly greater in magnitude. Although the energy balances between the

22

Figure 3.5. Schematic of modified receiver geometry in Monte Carlo ray trace model for black end-walls benchmark.

Figure 3.6. Net energy exchange between parallel black plates with no participating medium, both at 500 K. The thick black line at 2.0 m represents the outlet tube.

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

4.5

5.0

‐10,000 ‐5,000 0 5,000 10,000 15,000

Rad

ial D

ista

nce

fro

m A

xis

[m]

Net Energy Gain in Wall Element [W]

Energy Exchange Between Black End-Walls

left wallright wall

23

two walls, three elements on each wall show net energy changes significantly removed from

zero. Since both walls are set at a uniform 500 K and the system is completely insulated, the

system is already at equilibrium and thus the net radiation exchange at every wall element

should be exactly zero according to the analytical solution. Because the Monte Carlo method

is a statistical simulation of radiation heat transfer, some deviation from zero is to be

expected. But for the benchmark case to match expectations, these deviations from zero

should be small in magnitude and randomly distributed throughout the domain. The results in

Figure 3.6 show a clear bias, indicating a problem with ray tracing in this case.

The second benchmark case without a participating medium is similar to the first. The

only modification is to the emissive properties of the walls. In this case, the end walls are

modeled as perfect specular reflectors, and the concentric cylinder walls are modeled as

black bodies with an emissivity of one. This is illustrated in Figure 3.7. The inner cylinder is

set at a constant 500 K, while the outer cylinder is set at a constant 1000 K. With the end

walls being mirrored, the geometry should model infinite concentric cylinders.

Figure 3.7. Geometry of Monte Carlo model for concentric cylinders benchmark.

Results from this case are plotted in Figure 3.8, which shows the net energy gained by

each concentric wall element according to axial position. Because of symmetry, there should

not be any axial change in reported energy exchange for this case, but a clear trend is visible

in the net energy gained by the inner cylinder. This again indicates an error in the ray tracing

with the inclusion of the outlet tube.

24

Figure 3.8. Net energy exchange between concentric cylinders with no participating medium. The inner cylinder is at 500 K and the outer cylinder is at 1000 K.

The next benchmark case to consider is radiation in a gray absorbing-emitting medium

between infinite parallel plates [23]. This comparison was also utilized by Ruther [1] to

validate the first version of the MCRT model. In this case, the geometry is similar to the first

case presented in this section, only with the addition of a participating medium. The end

walls are modeled as black, while the cylinder walls are treated as perfect specular reflectors.

The left end wall is set to a constant 1000 K while the right end wall is set to a constant 500

K. Only radiation heat transfer is considered in the domain, and the gas is given a uniform

absorption coefficient. Theory predicts this should result in a linear axial non-dimensional

temperature profile in the gas. The geometry of this benchmark is identical to that illustrated

previously in Figure 3.5. For this case, the geometry is again modified to extend the outlet

tube along the entire axis of the geometry, such that it intersects both end walls. This,

combined with both cylindrical walls being specular with zero emissivity should make the

‐0.8

‐0.6

‐0.4

‐0.2

0.0

0.2

0.4

0.6

0.8

1.0

1.2

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

Net

En

ergy

Gai

n in

Wal

l Ele

men

t [M

W]

Axial Location Along Wall [m]

Energy Exchange Between Concentric Black Cylinders

Inner Cylinder (500 K)

Outter Cylinder (1000 K)

25

heat transfer match the benchmark case of infinite parallel plates. Results of model are

plotted in Figure 3.9, which shows non-dimensional gas temperature as a function of axial

distance from the hot wall for four different radial positions, all of which are between the two

concentric tube walls. This is compared to results from Howell and Perlmutter [23] for an

optical depth of 2. The temperatures in the gas are normalized according to equation 3.6.

(3.6)

The temperature profile is as expected at three of the radial positions, but not at the innermost

division of r = 0.45 m. This is the radial gas division adjacent to the outlet tube, and a

temperature anomaly here implies unexpected behavior on the outlet tube wall. Further

exploration of this issue is detailed in the Appendix.

Figure 3.9. Plot of non-dimensional emissive power in gray gas plane layer benchmark.

The final benchmark appropriate to the new geometry is radiant transfer between

infinite black concentric cylinders bounding a gray absorbing-emitting medium [24]. This

allows for a test of absorption and emission on the central outlet tube wall, while keeping the

computation time down by excluding scattering. To run this case in the MCRT code, the

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Non

-dim

ensi

onal

Tem

per

atu

re (

)

Non-Dimentional Axial Distance (z/L)

Emissive Power in Gray Plane Layer

r = 0.45

r = 0.75

r = 1.05

r = 1.35

benchmark

26

outlet tube length is again extended to both end walls and the end walls are modeled as

perfect specular reflectors as shown in Figure 3.7. The gas volume is divided into 20 radial

elements by 5 axial elements, in order to provide additional resolution in the radial direction.

This scheme provides improved resolution in the radial direction for comparison to the

benchmark, while reducing axial divisions to save on computation time. The geometry is

modified to match the benchmark cases in Perlmutter [24]. The inner radius (r1) is 0.15m and

the outer radius (r2) is 1.5m, yielding a cylinder radius ratio r1/r2 = 0.1. The cylinder walls are

given an emissivity of 1, and three optical thicknesses (0.1, 2, 10) are modeled. The optical

thickness is defined as τ = a(r2 – r1) where a is the absorption coefficient in the gas. This

results in an absorption coefficient for the three cases of 0.074 m-1, 1.481 m-1, and 7.407 m-1.

The results are plotted as non-dimensional location vs. emissive power. The radial location

(r) in the gas is non-dimensionalized as follows.

(3.7)

The emissive power is again non-dimensionalized by Equation 3.8, where T1 and T2 are the

temperatures in Kelvin of the inner and outer cylinders, 1000K and 500K respectively.

(3.8)

Figure 3.10 shows the radial profile of emissive power in the gas for an optical thickness of

10, at each of the axial divisions in the gas. The results show no variation in the axial

direction, indicating that the specular end wall properties provide a sufficient substitute for

infinite cylinder length.

The results for each of the three optical thicknesses considered are shown in Figure

3.11, denoted by τ in the plot legend. For these data, the emissive power at each radial

division was averaged across all axial divisions to provide a larger statistical basis. The

benchmark data is extracted from an image of the results plotted in Perlmutter [24], with the

aid of the Plot Digitizer program [25]. Agreement with the benchmark is reasonable for an

optical depth of 2, but greater and lesser optical thicknesses yield poor results when

compared to the respective benchmark cases. There appears to be little difference in the

model results between 0.1 and 2 optical thickness cases, while the largest optical depth case

27

Figure 3.10. Plot of non-dimensional emissive power in gray gas between concentric cylinders. Each data set represents the radial temperature profile at a different axial location.

of 10 indicates a trend in the wrong direction. The appendix catalogues more detailed

investigation of these issues, including cases without any participating medium as well as

one-wall emissions and extreme aspect ratios in an attempt to isolate the problem.

Unfortunately, despite considerable effort, the underlying source of these errors was not

found in time for corrected results to be included in thesis1. As a result, the coupled model

described in the remainder of this thesis uses the original version of the MCRT code, without

an outlet tube. In other words, the outlet tube is included in the fluid dynamic calculations,

but is considered transparent to radiation in the MCRT portion of the code. A single

comparison with and without the outlet tube, explained later, showed little difference in

1 As of final publication of this thesis the error was found and corrected and good agreement with the

benchmarks was obtained. The MCRT with outlet tube will be used for future calculations.

0

0.05

0.1

0.15

0.2

0.25

0.3

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Non‐Dim

ensional Emissive

 Power

Non‐Dimentional Distance Across Annulus

Emissive Power in Gray Gas ‐ τ = 10

z = 0.5m

z = 1.5m

z = 2.5m

z = 3.5m

z = 4.5m

28

Figure 3.11. Plot of non-dimensional emissive power in gray gas between concentric cylinders for three different optical depths.

receiver outlet temperature or efficiency. The primary benefit of having the outlet tube in the

radiation model is to determine its temperature (and hence material) for constructing a real

receiver.

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Non‐Dim

ensional Emissive Power

Non‐Dimensional Distance Across Annulus (r/R2)

Emissive Power in Gray Gas ‐ Concentric Cylinders

tau=0.1 (Benchmark)

tau=0.1 (Monte Carlo Results)

tau=2 (Benchmark)

tau=2 (Monte Carlo Results)

tau=10 (Benchmark)

tau=10 (Monte Carlo Results)

29

CHAPTER 4

COUPLING OF THE MONTE CARLO AND

FLUENT MODELS

This chapter describes the methods by which the modified Monte Carlo and FLUENT

Computational Fluid Dynamics models are coupled to solve for the temperature field in the

receiver. A User Defined Function (UDF) was written for this thesis in C and is used in

FLUENT to call custom code during each iteration of the flow solver. This enables the

Monte Carlo code to compute the radiation heat transfer, while FLUENT solves the

conservation equations of mass, momentum, and energy. The User Defined Function (UDF)

utilizes four DEFINE macros in FLUENT: DEFINE_EXECUTE_ON_LOADING,

DEFINE_SOURCE, DEFINE_PROFILE, and DEFINE_EXECUTE_AT_END. The first is

called when the UDF is loaded into FLUENT, and reads initial source term data from a data

file. The second segment is called each time FLUENT solves the energy equation for a

volume cell in the domain. It applies the appropriate source term to the cell based on the

cell’s location. The third subroutine applies source terms to the outer cylinder wall. The final

UDF is run at the end of each Fluent iteration. It queries the temperature field from FLUENT

and puts this data into an array, then passes the array to the FORTRAN code, which executes

the Monte Carlo iteration. An additional subroutine will need to be written to apply source

terms to the outlet tube wall in FLUENT, now that the MCRT code with an outlet tube is

validated. Because the outlet tube has radial thickness in FLUENT, unlike the outer cylinder

wall, it will require its own UDF.

The coupling scheme is illustrated in Figure 4.1. The process begins with the

initialized data in FLUENT. This data consists of an initial flow and temperature field

acquired typically from a previous case. Additionally, the initialized data includes initial

source term estimates loaded from a data file, or computed from a previous case. For new

cases at very different inlet power and mass-flow rates, the initial flow field is acquired by

running the FLUENT solver uncoupled from the MCRT at the target mass flow rate with no

source terms applied and a constant 1000 K gas temperature. The program solves the

30

Figure 4.1. Simple flow diagram of the coupling scheme used by the model.

conservation of energy, momentum, and mass equations to produce temperature, pressure,

and velocity fields in the domain. The temperature field is passed to the Monte Carlo ray

trace program, which uses it to compute the radiation heat transfer. The MCRT computations

produce an array of net volumetric and wall source terms which are passed back to FLUENT.

The flow solver then applies these source terms, re-computes the temperature field (and new

velocity and pressure fields as well), and the two pieces continue to iterate until convergence.

The model is run on a Dell PC running the Windows 7 operating system, with two 64-bit

quad-core Intel Xenon X5560 processors running at 2.80 GHz and 6 GB of RAM. Because

the Monte Carlo code is not set up for parallel computations, each case runs only on a single

processing core, though multiple cases are run on the machine at once.

4.1 COUPLING SCHEME & UDF FUNCTIONS

DEFINE_EXECUTE_ON_LOADING:

This function is run only once for each simulation, when the volumetric and wall source term data is first loaded into FLUENT. It initializes the iteration counter, loads preliminary volume and wall source terms from a data file, and sets the number of Monte Carlo ray emissions to 10,000 per cell for both volume and wall surface elements.

DEFINE_SOURCE:

This function is called once for each volume cell in the FLUENT domain on every iteration. When FLUENT solves the energy equation for a volume cell, this function is called and the cell information is passed to it. The built in FLUENT command C_CENTROID is used to query the centroid location of the cell being solved. Two nested loops and an IF statement then check which MCRT cell the current FLUENT cell is within, based on its centroid location. This set of IF statement checks is hard-coded for the uniform MCRT grid used for this thesis. A non-uniform Monte Carlo grid would require a revision to this function. The

31

corresponding volumetric source term is then applied to the FLUENT volume cell to end the function.

DEFINE_PROFILE:

This function is called once per FLUENT iteration, with the purpose of applying source terms based on the radiation model to the outer cylinder wall in FLUENT. To accomplish this task, the function has two nested loops and a series of IF statements. The outer loop steps through every face element in the FLUENT domain along the outer cylinder wall. Each step, the centroid of that face is queried using the built-in FLUENT function F_CENTROID and its axial location is checked against the MCRT grid boundaries. Since the wall source terms passed from the MCRT solver are a step function with divisions every 0.5 m, linear interpolation is used to apply a continuous flux function to the cylinder wall in FLUENT. Figure 4.2 shows an example of the raw wall flux values from the MCRT (in blue) plotted against the interpolated wall flux values applied in FLUENT (in red).

Equation 4.1 shows how this interpolation is computed within the subroutine, where & are the reference flux values from the MCRT solver on either side of the FLUENT face cell under scrutiny, z is the axial coordinate of the face centroid, and z1 is the axial location of the reference MCRT element.

2 (4.1)

The reference flux values used in this interpolation are computed using an under-relaxation factor detailed in the next section.

DEFINE_EXECUTE_AT_END

This function is the longest of the four UDF subroutines, and accomplishes several tasks. It is executed once at the end of each FLUENT iteration. The first thing this function does is compile the temperature field data and save it in an array that the MCRT solver can use. This is done with three nested loops; the outer two step through each of the 50 MCRT volume cells and the inner most loop steps through every FLUENT volume cell in the gas domain. Within the inner loop, each cell centroid is queried, again using the built-in FLUENT function C_CENTROID. The cell coordinates are checked against the boundaries of the current MCRT cell being looped through. If the FLUENT cell centroid is determined to lie within the present Monte Carlo cell, the built-in FLUENT function C_T is used to query the FLUENT cell’s temperature. The fluent cell’s mass is then computed based on the cell volume and density, both of which are given again by FLUENT functions. Finally, this temperature is added to the running total temperature of the present MCRT cell, weighted by the mass. Equations 4.2 & 4.3 show how the running totals of mass and temperature of each MCRT cell are computed, where and are the temperature and mass of an individual FLUENT cell, respectively. Likewise is the running total mass of an individual MCRT cell which contains the given FLUENT cell. A running

32

Figure 4.2. Wall Flux Interpolation Method. Vertical grid lines correspond to wall element divisions in the Monte Carlo solver. A negative net wall flux means that the wall cell is a net emitter of radiation and thus is heated by the fluid, while a positive value means the cell is a net absorber of radiation and is cooled by the gas.

total sum of the product of each FLUENT cell temperature and mass, represented by ∑ , is computed in Equation 4.3.

, , (4.2)

∑ ∑ 4.3)

At the end of each MCRT cell loop, when all FLUENT cells within it have been queried and added to the running total, the final mass-weighted average temperature of the MCRT cell is computed according to Equation 4.4.

∑ (4.4)

The next task of this UDF subroutine is to aggregate the outer cylinder wall temperatures using a similar method. For the wall elements, the average temperature in each Monte Carlo division is computed based on the area of each FLUENT face since they do not have mass or volume in the model. Again each

33

Monte Carlo division is stepped through, and within each one the subroutine loops through every FLUENT face on the outer cylinder wall, adding the temperatures to the running total of the appropriate MCRT element. At the end of this process however, the temperature array intended for the Monte Carlo must be checked for extremes. This is because the data tables generated by the cumulative distribution functions of the MCRT solver must have finite temperature ranges set in advance of model computations. These limits are set at 200 K and 2200 K, so any average wall temperatures beyond these limits are set to the closest acceptable value. This check was not necessary for the volume cells because FLUENT allows enforcement of temperature limits on volume cells during its calculations, so the UDF does not need to do it. After both wall and volume temperatures are aggregated into an array that can be passed to and read by the MCRT solver, a pair of nested loops steps through each element of this array and applies an under-relaxation scheme to the temperature data, the details of which are discussed in the following section.

The next task of this function is one of managing the iteration process. The iteration counter is incremented, and the number of emissions from each element in the following Monte Carlo ray trace is determined. The first 50 iterations use a static 10,000 emissions per cell, to keep computation time down while the model is still far from the converged solution. After the 50th iteration, the number of emissions is increased by 10% each iteration until reaching a maximum of 1,000,000 per cell. This ramp up of emissions is similar to the method used by Ruther in the first version of the model. Additionally, Ruther showed that 1 million emissions per cell are needed to reach acceptably accurate radiation heat transfer calculations [1]. The present model iterates 5 times once the 1 million emissions threshold has been reached, giving a total of 105 iterations.

The last step before this subroutine calls the FORTRAN MCRT solver is to generate a seed value for the random number generator needed in the Monte Carlo code. This is done by querying the system clock, and using the five-digit time in seconds as the seed. The FORTRAN programing language lacks the built-in functions needed to query the computer system’s clock, but the UDF is written in C which does have this feature. Thus, each iteration the Monte Carlo is given a new seed value for its random number generator. This allows for the model to iterate an indefinite number of times if need be, without the random number sequence produced in the FORTRAN code repeating itself. The eventual repetition of the random number sequence placed a practical limit on the number of iterations Ruther’s version of the model could do while remaining valid [1].

Finally, the UDF is ready to call the FORTRAN subroutine which holds the Monte Carlo solver. The values passed to the MCRT code include the temperature field, the number of emissions per cell, and the random number seed. The Monte Carlo passes back to the UDF volumetric source terms and wall flux values. After these values are computed and passed back to the UDF by the MCRT, they are written to a text file for data keeping purposes. Additionally, the wall flux values are checked for extremes, as described in more detail in the following section.

34

After this final UDF subroutine is complete, the FLUENT solver begins its next iteration.

4.2 SOLUTION CONTROLS

Because of the coupled nature of the model, combined with the numerical variation

introduced by the Monte Carlo method, several solution controls are necessary to reach a

stable convergence. The first method introduced is clipping of source terms applied to the

outer receiver wall. Early on during model iteration when the total number of Monte Carlo

emissions from each element is relatively low and the temperature field is still developing,

the net wall source terms returned to FLUENT from the MCRT can be of extremely large

magnitude. This presents a problem, as the energy balance at the receiver wall is described

by Equation 4.5, where is the net radiative source term at the wall from the MCRT, in

W/m2.

(4.5)

The thermal conductivity of the gas is proportional to the gas temperature raised to

the 3/4ths power, but this is approximated in the model by a linear function of temperature.

The thermal conductivity is in the range of 0.05-0.15 Wm-1K-1 at temperatures found in the

receiver. This gives an upper limit to the magnitude of the heat flux at the wall, as the gas

temperature beyond the thermal boundary layer is driven by advection in the cavity and

volumetric absorption from incident radiation, and thus is not significantly influenced by

conduction from the wall. This can place significant constraints on the grid near the wall. To

illustrate this, consider Figure 4.3 which plots the gas temperature in the thermal boundary

layer near the outer cylinder wall at four different axial locations. The points on each contour

represent actual grid points in FLUENT.

The green line represents the boundary layer at 4.25m from the aperture, or 0.75m

from the gas inlet. At this location, the wall is heated to 870K by radiation exchange in the

cavity, but the gas is only a bit below 750K, near the inlet temperature of 700K. The Monte

Carlo solver produced the maximum flux value of 5kW/m2 for this wall cell ( in Equation

4.5). The thermal conductivity of the gas (k in Equation 4.5) in this temperature range is

approximately 0.06 W/(m K), and the temperature change through the beginning of the

boundary layer (the first two grid points) is 110 degrees Kelvin. To produce the required

35

Figure 4.3. Thermal boundary layers near the outer cylinder wall at four different axial locations. Legend values indicate axial distance from aperture of each data series.

temperature gradient to balance Equation 4.5, the wall temperature of 870K and gas

temperature of 760K would need to be separated by only 1.3mm. But as Figure 4.3 shows,

the first gas cell adjacent to the wall is located roughly 2.5mm away. Thus, the grid is too

coarse to resolve the boundary layer in this region. Further refinement of the FLUENT grid

in this region may be desirable, but it would come at the cost of increased computational

load. As discussed in Chapter 5.3, while a finer grid is better capable of resolving these

boundary layers, it does not significantly change the outlet temperature and receiver

efficiency. Figure 4.3 also illustrates how the grid is in fact fine enough to resolve the

thermal boundary layers at other axial positions along the outer cylinder wall. Wall heat flux

values an order of magnitude or more than this result in unrealistic solutions from the

FLUENT solver, such as negative temperatures with large enough negative fluxes. Thus, the

incoming wall source terms from the Monte Carlo solver are checked each iteration and

limited to an absolute value of 5 kW/m2. This prevents physically impossible temperature

700

750

800

850

900

950

1000

1050

1100

0 2 4 6 8 10 12 14 16 18 20

Gas

Tem

per

atu

re [

K]

Radial Distance from wall [mm]

z = 0.25

z = 0.75

z = 1.75

z = 4.25

36

results due to the grid sizing, along with limiting large fluctuations that occur early on in the

iteration process.

The second solution control method is the introduction of an under-relaxation factor

for all source terms computed by the Monte Carlo solver. These are applied each iteration,

before the source terms are used by FLUENT to update the temperature field. This damps out

some of the inherently unstable nature of the solution, which is again particularly evident

early on in the iteration process. A relatively aggressive under-relaxation factor of 50% is

used on all source terms. This control method represents one half of the strategy to apply

under-relaxation to the solution of the energy equation, with the other half being applied to

temperature values.

The final solution control used on the model is an under-relaxation factor on the

temperature field. This is applied each iteration to the temperature field produced by

FLUENT, before it is passed to the MCRT. Equation 4.6 shows how the temperature of each

MCRT element is calculated, where Ti is the cell temperature computed by FLUENT during

the current iteration, Ti-1 is the cell temperature applied to the MCRT code on the previous

iteration, and β is the under-relaxation factor.

1 (4.6)

It should be noted that this method is based on T4 and not simply T, because radiation is the

dominant heat transfer mode in the gas and is dependent on T4. The importance of solution

controls on T4 as opposed to T is illustrated in Figure 4.4, which plots the net energy gain by

the receiver (as computed in the MCRT code) for a single case under each scheme. For the

first 20 iterations of this case (in blue), under-relaxation was applied to T4, and from iteration

21 onward (in red) it was applied to simply T. It is clear that the former scheme is more

stable than the later, despite both being applied to a case that was already very near

convergence. Figure 4.5 shows the same case with no under-relaxation on the temperature at

all. Not only are the oscillations in this case much greater, but their magnitude actually grows

over the course of the model run, showing significant instability.

4.3 GRID MATCHING SCHEME

As is discussed in more detail in Chapter 5.1, the FLUENT grid divisions are

generated without concern for the locations of the Monte Carlo grid divisions. The coarsest

37

Figure 4.4. Net energy gain in the gas flow as a function of model iteration number for two different temperature under-relaxation methods.

part of the grid has an average of 200 FLUENT cells in each MCRT cell, while the finest

region has several orders of magnitude more than this. To ensure that the FLUENT and

MCRT grids need not align perfectly, it must be shown that the impact of misaligned grid

cells is small enough to ignore. For this, the worst-case-scenario of grid misalignment is

considered. For an individual FLUENT cell, the worst case is illustrated in Figure 4.6, where

the cell centroid is aligned with the edge of the MCRT cell. The FLUENT cell is assumed to

be square, with a side length of c.

Since the MCRT cell which contains a given FLUENT cell is determined by the

location of the FLUENT cell centroid, the most overhang a FLUENT cell can have is half its

volume. The most extreme miss-match between the two grid schemes then would be if every

FLUENT cell along every border of a single MCRT cell had this much overhang, and the

FLUENT grid is at its most coarse. The total volume of the overhang along the two b-length

sides can be computed from Equation 4.7.

2 (4.7)

4.50

4.55

4.60

4.65

4.70

4.75

4.80

0 10 20 30 40 50 60

Net

En

ergy

Gai

n in

Rec

ieve

r [W

] Millions

Iteration number

Effect of under-relaxation on T4 vs. T

T^4T

38

Figure 4.5. Net energy gain in the gas flow as a function of model iteration number with no temperature under-relaxation. This scheme appears to be unstable, or at least not converging.

Figure 4.6. Illustration of a FLUENT grid cell (in red) overhanging the boundary of a Monte Carlo grid cell (in black).

4.50

4.55

4.60

4.65

4.70

4.75

4.80

0 5 10 15 20 25 30 35

Net

En

ergy

Gai

n in

Rec

eive

r [W

] Millions

Iteration Number

Energy History Without Under-Relaxation

39

Similarly, the total volume of the overhang along the two a-length sides can be computed

from Equation 4.8.

2 (4.8)

Summing these two volumes and dividing by the volume of the MCRT cell yields the total

volume miss-match between the two grid schemes:

% max (4.9)

The largest value of this mismatch occurs in the largest MCRT cell, with the largest

FLUENT cells, where a is 0.5m, b is 0.3m, c is 0.01m, r1 is 1.2m. and r2 is 1.5m, and the

mismatch is 5.33%. In the real grid however, the mismatch is significantly less. On average,

it would be half this, as the FLUENT cells that overhang the edge of an MCRT cell in this

way are equally probable as FLUENT cells that are perfectly aligned with the MCRT cell

edge. Furthermore, many of the FLUENT cells in the outer ring of MCRT cells (i.e. at these

large values of r1 and r2) are made smaller than 0.01m because they are near the wall. It can

also be seen from Equation 4.9 that the worst-case mismatch value also scales linearly with c,

so if further precision is needed it can be had by reducing the maximum grid size of the

FLUENT cells. For regions of high gradients in the receiver, where the grid size is 10-4, this

mismatch value drops to below 0.05%.

40

CHAPTER 5

COLD FLOW & GRID STUDY

5.1 GRID AND GEOMETRY

A schematic of the model receiver geometry is presented in Figure 5.1. Previous work

showed that the gas region between the edge of the aperture and the outer wall of the receiver

experienced minimal heating despite scattered and emitted rays in that region [1]. As a result,

there is little need to model an aperture radius significantly smaller than the receiver radius

itself, thus the receiver geometry has been modified to reduce the radius of the receiver to

that of the window, 1.5 m in this case. Ruther’s work also demonstrated the superior

efficiency of flow opposed to the incident radiation [1]. As a result, this model’s geometry

orients the bulk of the gas flow in the opposed direction. Flow enters the receiver cavity at

the rear wall, travels forward towards the window, and exits down a central outlet tube along

the axis.

Figure 5.1. Schematic of the geometry as it is modeled in FLUENT and the Monte Carlo code. Drawing is not to scale.

41

In the FLUENT Computational Fluid Dynamics (CFD) portion of the model, the

geometry is two-dimensional-axisymmetric, along the central axis of the cylinder. The

FLUENT geometry must match the geometry in the Monte Carlo exactly, because all source

terms calculated in the MCRT are mapped to their corresponding locations in FLUENT as

described previously in Chapter 4.1. Thus, because the current MCRT model is limited to an

orthogonal axisymmetric geometry, so then must the FLUENT model be likewise limited.

This is the reason for poor aerodynamic features such as the outlet tube entrance and the

corner adjacent to the window, both of which are visible in Figure 5.1. Work to expand the

capabilities of the MCRT solver and relax some of these limitations is ongoing. The grid and

model geometry as seen in FLUENT is shown in Figure 5.2. The axis of the receiver runs

along the lower boundary of the domain, while the window is defined by the left edge. The

black line running through the dense grid cells near the axis represents the wall boundaries of

the outlet tube. The grid sizing is varied throughout the geometry to reduce computational

load while still providing needed resolution in areas of interest. In regions that are both far

from any wall and largely uniform in flow, grid cells are up to 2.5 cm on each side. Cells

near the walls or in areas of high velocity or temperature gradients are on the order of 5mm

on each side.

Figure 5.2. View of the grid as it appears in FLUENT.

The geometry in the Monte Carlo ray tracing radiation model is axisymmetric as well,

although rays are traced in three dimensions. Tracing the rays in three dimensions is required,

because circumferential components of a ray’s trajectory can change the path length through

the medium as compared to a strictly two-dimensional computation, and thus have an impact

on absorption and scattering calculations. The grid used is nearly identical to that used

42

previously by Ruther [1]. The volume is partitioned uniformly with 10 divisions in the axial

direction and 5 in the radial direction. Figure 5.3 shows the FLUENT grid in green with an

approximate overlay of the Monte Carlo grid shown in black. The MCRT grid is uniform

because of limitations in the original model, which assumes a uniform grid throughout the

code, though a non-uniform grid would be preferable.

Figure 5.3. The FLUENT grid (green) with an approximate overlay of the Monte Carlo grid (black).

There is an additional axial division at the rear wall of the receiver, to create a set of

five wall elements on that plane with zero thickness. The outer axial wall also has a zero-

thickness radial division, resulting in 10 ring elements at that boundary. These sets of wall

elements are necessary in the MCRT because the ray tracing must have boundaries at these

locations. The back wall elements currently exist only in the MCRT model, because there is

no way to couple them with the FLUENT model at present. The only difference from the

previous model work is the radial dimension of the geometry, which is 1.5 m in this case as

opposed to 2.5m previously. It should be noted that the outlet tube wall does not exist in the

radiation model. Attempts were made to include it, but the modified MCRT code did not

match benchmarks as discussed previously in chapter 2 and thus had to be omitted for this

work. The model therefore behaves as if the outlet tube wall is transparent to all radiation

heat transfer. This is an approximation of a quartz outlet tube, which would be mostly

transparent to incident solar radiation but opaque to longer wavelengths. However, emissions

from the gas within the outlet tube in a real receiver would likely be minimal for two reasons.

First, the gas-particle mixture has a low emissivity in the infrared portion of the spectrum by

design. This is an important feature of the particles’ selective absorption properties. Second,

43

the gas in much of the outlet tube of a real receiver would have minimal particle content, as

they are expected to oxidize fully by the time the mixture leaves the receiver. Thus the

emissivity of outlet tube gasses is likely to be close to zero, so the omission of the tube wall

from the MCRT is not expected to have a significant impact on overall receiver model

behavior. However, without inclusion of the outlet tube wall in the radiation model, the outlet

tube wall temperatures cannot be accurately predicted. These temperatures will be needed for

the prototype receiver design. Future work will add this to the model now that the problem is

remedied.

5.2 COLD FLOW VERIFICATION

Before coupling the FLUENT model to the MCRT, the FLUENT grid is used to run

an unheated case. For this case, the energy solver in FLUENT is still enabled but the inlet gas

is set to 700 K, all walls are set to the zero heat flux boundary condition, and no source terms

are imposed on the gas volume. All other conditions are the same as the baseline scenario: 5

kg/s mass flow rate inlet, 1 MPa operating pressure, no-slip condition at all walls, and

identical geometry as described previously. The turbulence model used in FLUENT is the

shear-stress transport modified κ-ω model with recommended default model parameters.

These parameters are listed in the Appendix. This method was chosen based on FLUENT

user-guide recommendations for internal turbulent flow. The velocity profile at the exit plane

of the outlet tube is compared with a theoretical turbulent velocity profile to validate the cold

flow. Analytical or experimental benchmark cases for the flow in the remainder of the

receiver are not available. The Reynolds number of the flow at the exit plane is roughly

31,000, which is well into the turbulent regime for duct flow. An empirical velocity profile

for turbulent duct flow is given by equation 5.1 in Bejan [26].

2.5 5.5 (5.1)

(5.2)

(5.3)

(5.4)

44

This theoretical profile is plotted in Figure 5.4 against the model results for the cold flow

case at the exit of the outlet tube, for both the κ-ω and κ-ε turbulence models. Both the

FLUENT models show reasonably good agreement with the theoretical velocity profile, but

the κ-ω model does a better job resolving the profile and particularly the boundary layer. The

model under-predicts the turbulent boundary layer thickness slightly compared to theory, but

this is not a focus of the model. Heat transfer across the boundary layer is not a primary

concern of the coupled model, as the dominant modes of heat transfer within the receiver are

radiation, followed by advection within the gas.

Figure 5.4. Plot of the velocity profile on the outlet plane of the receiver for two FLUENT turbulence models as compared to a theoretical turbulent velocity profile.

Another metric of performance that can be analyzed from the cold flow case is the

pressure drop across the receiver. Figure 5.5 shows contours of static pressure in the receiver,

displayed in Pascals relative to the inlet. The pressure drop from the inlet to the outlet of the

receiver is less than 100 Pa, and the maximum pressure difference is 200 Pa. This extremely

low pressure drop is encouraging, as the receiver geometry is not optimized for aerodynamics

0

1

2

3

4

5

6

0.00 0.05 0.10 0.15 0.20 0.25 0.30

Axi

al V

eloc

ity

[m/s

]

Distance from Centerline [m]

k-omega (SST)k-epsilonTheory

45

Figure 5.5. Cold flow pressure contours in FLUENT.

at all. Additionally, it is much lower than typical pressure drops found in existing tubular

receiver designs. Furthermore, it is well within limits that would be imposed by a gas turbine

system [27]. Figure 5.4 also illustrates that nearly all of the pressure drop occurs in the

entrance region of the outlet tube. Since the current Monte Carlo model requires orthogonal

boundaries and no curved surfaces, the receiver geometry is not optimized for the gas flow.

This includes the outlet tube entrance, which might use a nozzle entrance in a real receiver

and may have a radius which changes in the axial direction.

5.3 GRID STUDY

A grid study is performed using a modified baseline scenario in order to validate grid

independence of that solution and reduce computational load. The solar input conditions for

this case are 5 MW of total power with a Gaussian flux distribution and collimated incident

rays. The gas flow rate is 5 kg/s and all interior receiver walls are modeled with an

emissivity of 1. To reduce processing time, scattering in the gas-particle mixture is turned

off in the radiation model, leaving only emission and absorption. Three grids are tested

under these conditions. Grid 1 has cells as large as 5 cm away from the walls where

gradients are low, and as small as 1 mm near the walls and in the entrance region of the

outlet tube where gradients are highest. Grid 2 is a refinement of grid 1, with every cell

being roughly half as large as in grid 1. Grid 3 is a refinement of grid 2, with each cell again

46

being half as large as the previous grid 2. Table 5.1 shows a summary of the differences

between each of the three grids.

Table 5.1. Summary of Grid Attributes

Grid 1 Grid 2 Grid 3

Number of Cells 150,000 300,000 600,000

Smallest Cell [m] 0.001 0.0005 0.00025

Largest Cell [m] 0.05 0.025 0.001

Figure 5.6 shows a plot of temperatures on the outer cylinder wall of the receiver for

each of the three grids. Each case was run for 105 iterations with a ramping emissions

scheme as described previously in Section 4.1.

Figure 5.6. Wall temperatures in FLUENT for each of the three grids studied.

600

650

700

750

800

850

900

950

1000

1050

1100

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0

Wall Temperature [K]

Axial Distance from Window [m]

Grid Study ‐ Cylinder Wall Temperatures

grid 1grid 2grid 3

47

Temperatures in the first 0.5 m are variable and unreliable due to the unstable nature

of the flow field in the corner adjacent to the window. In spite of this however, even the most

extreme temperatures are only 1050 K which is manageable for a variety of potential wall

materials. Grids 2 & 3 show a quicker return to expected wall temperatures between 0.5 and

1 m than grid 1 does. The reason for this can be seen in a comparison of the velocity fields

near the corner for each grid. Figure 5.7 shows velocity vectors near the corner of the

window and outer cylinder wall for grid 1. Vectors are colored and scaled to velocity

magnitude. The window is on the left side, the outer cylinder wall is along the top, and the

white vertical line near the right side denotes a distance of 0.5 meters from the window face.

A large recirculation cell is visible in the corner, extending roughly 0.3 meters from the

window. A second smaller recirculation cell can be seen near the wall extending past the 0.5

meter mark. It is this smaller recirculation cell which causes the irregular wall temperatures

seen previously in Figure 5.7 for grid 1, between 0.5 and 1.0 meters.

Figure 5.7. Velocity vectors in the window corner region for Grid 1. Vectors are colored and scaled by magnitude. Color-bar scale represents velocity in m/s, and length scale of the geometry is indicated at the top of the figure.

Figure 5.8 and Figure 5.9 show velocity vectors in the same corner region for grids 2

& 3, respectively. A recirculation cell still exists in the corner, but it is smaller than the one

seen in grid 1, and the flow along the wall remains attached closer to the window than under

48

Figure 5.8. Velocity vectors in the window corner region for Grid 2. Vectors are colored and scaled by magnitude. Color-bar scale represents velocity in m/s, and length scale of the geometry is indicated at the bottom of the figure.

Figure 5.9. Velocity vectors in the window corner region for Grid 3. Vectors are colored and scaled by magnitude. Color-bar scale represents velocity in m/s, and length scale of the geometry is indicated at the top of the figure.

49

the coarser grid scheme. This leads to stable wall temperatures closer to the window for grids

2 and 3, as seen previously in Figure 5.6. There are still some differences between the flow

fields seen in grids 2 and 3, such as the size of the largest recirculation cell and the

magnitude of the velocity in that region. But these differences are not significant enough to

justify the increased computational load needed for the extremely fine grid 3.

Figure 5.10 and Figure 5.11 provide some explanation for the unusual wall

temperatures found in the first 50 centimeters of the receiver, and illustrated previously in

Figure 5.6. Figure 5.10 shows temperatures in this wall region for grid 2, while Figure 5.11 it

shows velocity vectors in the same region. The position axis of Figure 5.10 is scaled to align

with the image in Figure 5.11. The velocity vectors are scaled by magnitude, but colored by

flow temperature. The source term imposed on the wall from the Monte Carlo in this region

is negative, meaning the wall is a net emitter of radiation and thus is heated by the flow.

Figure 5.10. Cylinder wall temperatures in the first half-meter of the receiver. The x-axis is scaled to align with the velocity vectors in Figure 5.11.

From these two plots, it can be seen that the wall temperature drops in regions where

the flow separates from the wall, specifically in three locations: immediately adjacent to the

window, 15 cm away from the window, and 30 cm away from the window. In regions where

the flow is relatively faster and attached to the wall, temperatures are higher because

convection is more effective due to the higher flow velocity. As the flow field in this corner

region changes due to grid resolution or possible transients, so does the wall temperature

600

700

800

900

1000

1100

1200

0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50

Wal

l Tem

per

atu

re [

K]

Axial Distance from Window [m]

50

Figure 5.11. Velocity vectors in the window corner region for Grid 2. Vectors are scaled by magnitude of velocity, and colored by temperature in Kelvin.

profile.

At this point in the analysis of wall temperature and heat flux, it is appropriate to

justify the model assumption that conduction heat losses through the wall to the ambient are

small enough to be neglected. Equation 5.5 shows the energy balance on the outer cylinder

wall, where G is the irradiance on the wall, the second term represents emitted radiation from

the wall, the third term represents convection from the gas adjacent to the wall, and

represents the radial conduction losses through the wall to the ambient. As discussed

previously in Chapter 4.2 and illustrated in Figure 4.2, the magnitude of the heat flux on the

wall due to radiation is limited to 5 kW/m2. This is a net radiation flux, so it is the difference

between the first two terms of Equation 5.5. The FLUENT model then assumes that the

term is zero, leaving the convection term as the only unknown.

0 (5.5)

For conduction losses to be negligible, they should be less than 5% of the net

radiation flux, or less than 200 W/m2. This flux value can be converted to a total heat loss

term by multiplying by the outer receiver wall surface area, which yields 9450 W or roughly

10 kW. This is 0.2% of the total power absorbed by the receiver, showing that the conductive

losses can be made to have a small effect on receiver efficiency. The conduction heat loss

51

through a cylinder wall is expressed by Equation 5.6 [28], where the previously computed

value of 10 kW is represented by , L is the length of the receiver, k is the thermal

conductivity of the insulation around the receiver body, Tw is the inner wall temperature of

the receiver, To is the exterior wall temperature of the insulation, r2 is the outer radius of the

insulation, and r1 is the radius of the receiver wall.

2ln ⁄

(5.6)

A conservative estimate of average receiver wall temperature is 1000 K, and the outer

temperature of the insulation can be taken to be approximately equal to an ambient

temperature of 300 K for a worst-case estimate of the heat loss. The radius of the receiver is

fixed at 1.5 m and the length is fixed at 5 m. A conservative estimate of the average thermal

conductivity of the high-temperature insulating material able to withstand temperatures of

1000K is 0.15 W/(m K) [29]. Solving equation 5.6 above for r2 gives an outer insulation

radius of 2.1m, or an insulation thickness of 60 cm. This is a reasonable worst-case scenario

for needed insulation thickness, and could be further reduced by using a second insulation

material with reduced thermal conductivity at the outer radius ranges where temperatures are

reduced. For example, using the high-temperature insulation for 15 cm radially outward from

the wall would reduce the temperature to 800 K, where a insulation with an average thermal

conductivity of 0.07 W/(m K) could be used [28]. This second layer would need to be 20 cm

thick to reach the ambient temperature, yielding a total insulation thickness of 35 cm.

Returning to the grid study, another metric used to compare grid schemes is the net

radiation heat transfer along the outer cylinder wall. Figure 5.12 shows the wall flux values

applied in FLUENT for each of the three grids. These values are generated in the MCRT

code based on the result of the radiation heat transfer calculations between the walls and the

volume. There is good agreement between grids 2 and 3 for the wall flux profile, while grid 1

fluxes stand apart, particularly at each end. Wall flux values are negative near the front of the

receiver, increasing towards the rear. With a collimated solar at the window, the wall

elements near the front of the receiver do not experience any direct solar irradiation. They are

predominantly irradiated by re-emissions from the gas itself near the wall. The amount of

light illuminating the gas near these early wall elements is itself reduced by the Gaussian

input flux distribution, which imparts only 100 kW/m2 on the outer most ring element of the

52

Figure 5.12. Wall fluxes imposed by the Monte Carlo on the FLUENT solver for each of the three grids studied. Positive values of this net wall flux means the wall is absorbing more radiation than it is emitting.

aperture. Wall elements near the middle of the receiver experience radiation emissions from

gas regions in a wide band of axial locations, while wall elements near the front of the cavity

receive no gas emissions from axial positions to their left. This reduces the magnitude of gas

emissions that reach the wall elements near the window. An additional source of radiative

load that acts on the wall elements near the rear of the cavity more than the window end is

the back “wall” of the receiver. This wall exists in the Monte Carlo radiation model as a

black surface and much of what it emits is absorbed by the cylinder wall. That wall reaches

temperatures of up to 1100K in the Ruther model.

The final comparison between the grids is the average outlet temperature of gas

leaving the receiver. All three grid schemes produced very similar outlet temperatures of

1442 K, 1443 K, and 1445 K for grids 1, 2, and 3, respectively. As the outlet temperature of

‐3000

‐2000

‐1000

0

1000

2000

3000

4000

5000

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0

Net Wall Flux [W

/m2]

Axial Distance from Window [m]

Grid Study - Wall Fluxes

grid 1

grid 2

grid 3

53

the gas is the most important benchmark of the receiver model, it is significant that all three

grids produced such close results in this regard. Grid 2 was chosen for model study moving

forward because the differences between grids 2 and 3 were minimal compared to the

differences between grid 1 and the rest. Grid 2 was chosen over grid 3 because the small

improvements to accuracy did not outweigh the increased computational load of the finest

grid.

54

CHAPTER 6

RESULTS AND DISCUSSION

Once the model was developed and verified, three different parameters were selected

for further study. These are the influence of Gaussian flux distribution as compared to a

uniform solar flux input, the effects of variation in mass-flow rate through the receiver with a

constant solar input, and the effects of varying mass-flow rate proportionally with a varying

flux input. Ten cases in all were conducted to examine these effects, and their differences are

summarized in Table 6.1, along with macro-scale results. For the flux distribution variation,

two cases were run at the baseline conditions of 5 kg/s mass-flow rate and 5 MW of

collimated solar input. For the mass-flow rate study, five cases were run with a 5 MW solar

input and mass-flow rates from 4.0-6.0 kg/s. For the variable input study, five cases were run

with input power varied from 2.0-6.0 MW and mass-flow rate varied proportionally from

2.0-6.0 kg/s.

Table 6.1. Summary of Model Runs

Input Power [MW] 

Gaussian Flux? 

Mass Flow Rate [kg/s] 

Window Losses [kW] 

Net Gain 

(MCRT) [MW] 

Outlet Temperature 

[K] 

Thermal Efficiency 

5  no  5.0 812 4.188 1379 83.8% 

5  yes  4.0 683 4.317 1547 86.3% 5  yes  4.5 587 4.413 1479 88.3% 5  yes  5.0 538 4.462 1420 89.2% 5  yes  5.5 509 4.491 1370 89.8% 5  yes  6.0 477 4.522 1325 90.4% 

6  yes  6.0 597 5.403 1424 90.1% 4  yes  4.0 485 3.515 1409 87.9% 3  yes  3.0 451 2.549 1389 85.0% 2  yes  2.0 393 1.606 1342 80.3% 

For all cases, several parameters of the model were held constant. These include a

particle size of 0.2 μm and particle mass loading of 0.30 kg/m3, along with the flow

55

orientation being kept predominantly opposed to the incident flux. These parameters were

studied in more detail with the original version of the model by Ruther [1], and these values

were chosen based on those results. The boundary conditions are also held constant for all

cases, with the exception of the solar input for certain studies. Constant boundary conditions

include the no-slip condition at all walls, a uniform inlet velocity profile, 700 K constant

temperature inlet gas, and a constant temperature of 1000 K at the window wall. The window

boundary condition was chosen to be a constant temperature because a quartz window in a

real receiver will have a temperature limit near this. By forcing the window temperature to be

1000 K, the computed heat flux at that boundary in FLUENT gives an idea of the convective

heating load on the window. In most cases, this flux was on the order of 50-100 kW. In a real

receiver, the convective load would be far outweighed by the radiative load on the window,

which would largely come in the form of longer wavelength emissions from the gas and

walls. Longer wavelengths of course are likely to be absorbed by a quartz window. Since the

current Monte Carlo model does not include an actual quartz window at the aperture, there is

no way to include radiative loads on the window as of yet, though their magnitude can be

estimated by the window losses shown in Table 6.1. Thermal loads on the window and

potential strategies for reducing them are being investigated by other members of the

research group.

6.1 GAUSSIAN FLUX DISTRIBUTION

As discussed previously in Chapter 3.1, the Monte Carlo model has been modified to

allow for a Gaussian flux distribution on the aperture. To assess the impact of this change on

the receiver, the baseline coupled case with 5 kg/s mass-flow rate and 5 MW of solar input

was run with a uniform flux distribution and again with the new Gaussian flux distribution.

Figure 6.1 shows the energy balance over the model runs for both of these cases, with the

uniform flux case in blue and the Gaussian flux case in red. The energy balance is computed

by comparing the total energy gained by the flow in FLUENT to the net energy gained in the

Monte Carlo model. The energy gain in the Monte Carlo is simply the input power (5 MW in

this case) less all reported window losses, which are all rays which are traced to exit the

aperture plane. For both these cases (and indeed nearly all coupled cases presented in this

56

Figure 6.1. Effect of solar flux distribution on energy balance.

thesis) the energy balance is several percent less than 100%. Potential sources of this

imbalance are discussed further later in this chapter.

An important feature illustrated in Figure 6.1 is the fact that the uniform flux case

produced an energy balance of 97%, while the Gaussian flux case is less than 96%. One

potential reason for this difference is evident when looking at the temperature fields for each

case, which are shown in Figure 6.2 and Figure 6.3. In the uniform flux case, the peak gas

temperature in the receiver cavity is around 1750 K, seen near the entrance to the outlet tube

in the lower-left corner of Figure 6.2. This value is well below the model-enforced maximum

temperature of 2200 K, and thus FLUENT did not artificially reduce the gas temperature in

any cells for the uniform flux case. By contrast, the maximum gas temperature found in the

Gaussian flux distribution case is 2200 K, visible along the axis near the window. Since the

gas temperature in these cells is limited in FLUENT, some of the energy that the Monte

Carlo solver determined ought to be gained in this region is simply ignored by the model. In

92%

93%

94%

95%

96%

97%

98%

99%

100%

101%

102%

0 10 20 30 40 50 60 70 80 90 100 110

En

ergy

Bal

ance

(F

LU

EN

T/M

CR

T)

Iteration Number

Energy Balance - Gaussian Flux Distribution

Uniform Flux

Gaussian Flux

57

Figure 6.2. Temperature field inside receiver with 5 MW of uniform incident solar flux and 5 kg/s mass-flow rate. Color-map values are in units of Kelvin.

Figure 6.3. Temperature field inside receiver with 5 MW of Gaussian distributed incident solar flux and 5 kg/s mass-flow rate. Color-map values are in units of Kelvin.

other words, if all of the energy sent from the MCRT to FLUENT was accounted for, these

gas cells would have temperatures much higher than 2200 K, and thus the average outlet

temperature of the flow would be higher. This results in an inherent imbalance in the energy

gained by the gas as reported by the two models, and explains the difference in energy

balance found in the uniform and non-uniform input flux cases. In a real receiver, gas at this

temperature would oxidize the small carbon particles extremely quickly, thus making the gas

transparent to the incident radiation and unable to continue gaining energy in this way. But

58

because the current model does not have a way to account for oxidation of the small carbon

particles, this controlling effect is not captured.

This gas temperature is extremely high for several reasons. First, the receiver axis is

the peak of the Gaussian flux distribution, and thus receives nearly 3 MW/m2 on the aperture

at this radial position. Second, there is a stagnation point in on the window boundary at the

axis of symmetry, which limits the ability of the heat here to be carried away by advection in

the flow. This stagnation point is a result of the orthogonal and axisymmetric geometry

restrictions of the current model. The stagnation point is also responsible for the relatively

cooler gas in this same region seen in the uniform flux case. In that case, the cooler gas is in

part a result of the constant temperature boundary condition of 1000 K imposed at the

window plane in FLUENT.

Another significant difference between the two flux distribution cases is the

temperature distribution within the receiver. The uniform flux case has a larger region of

relatively hot gas (1600-1750 K) in the front half-meter of the cavity, while the non-uniform

flux case has gas at this temperature only near the axis. This is again a result of lower flux

levels at the outer radius in the second case. This also produces a slight radial temperature

gradient in the non-uniform flux case that persists nearly all the way to the back of the cavity,

while the uniform flux case has minimal radial temperature change in the bulk of the gas

volume. The hot gas in the front of the uniform flux case produces higher radiation emission

levels in that region than its analog in the Gaussian distribution case. This in turn results in

greater window losses computed by the Monte Carlo solver, as much of the emissions from

hot gas near the front of the cavity simply exit the aperture. Consequently, the uniform flux

case has a lower thermal efficiency (83.9%) than the non-uniform case (89.2%), as well as a

lower outlet temperature (1379 K vs. 1420 K).

An additional source of losses for the uniform flux case is apparent when examining

the outer cylinder wall temperatures for these two model runs. Figure 6.4 shows the wall

temperature profiles for these two variations. Wall temperatures are clearly higher in the

uniform flux case, particularly in the front portion of the receiver. This result is expected, as

incident flux levels are much higher near the wall in the uniform case and in turn more

incident radiation is available to be scattered by the gas-particle mixture and strike the wall.

Since the wall is modeled as a black body, it must re-emit nearly all of the energy absorbed.

59

Figure 6.4. Cylinder wall temperature profiles for the uniform (blue) and Gaussian (red) flux distribution cases.

And since the wall temperature is roughly between 900 and 1200 K, the bulk of its emissions

are at longer wavelengths where the gas-particle mixture does not absorb very well. This

means that most of the emissions from the outer cylinder wall will exit the aperture and be

realized as losses, or be re-absorbed by the wall in another location. Only a limited amount of

the wall’s energy is removed via convection by the flow, as was discussed previously in

Chapter 4.2.

All of these factors together make the Gaussian flux case preferable for receiver

performance from a design standpoint, in addition to this flux distribution being more

realistic than a uniform case. Although the model is still limited in the variety of geometries

it can solve, some of the trends illustrated by these two cases are likely applicable to non-

orthogonal receiver designs. Keeping incident solar flux levels low near the cylinder wall,

700

800

900

1000

1100

1200

1300

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0

Cyl

ind

er W

all T

emp

erat

ure

[K

]

Axial Distance from Window [m]

Wall Temperature Profiles

Uniform Flux

Gaussian Flux

60

even with a collimated radiation input, likely improves thermal efficiency of the receiver and

reduces radiative load on the wall itself. Likewise, keeping the hottest gas region limited to a

smaller volume of fast flowing gas near the entrance to the outlet tube helps to reduce

radiation losses and thus improve receiver thermal efficiency as well.

6.2 FLOW RATE VARIATION

To study the effects of mass-flow rate through the receiver, two cases each were run

above and below the baseline mass-flow rate of 5.0 kg/s, for a total of five different mass-

flow rates with a constant 5 MW of solar input as summarized in Table 6.1. Results of outlet

gas temperature and receiver thermal efficiency for each of the five cases are plotted in

Figure 6.5.

Figure 6.5. Effect of mass-flow rate on gas outlet temperature and receiver thermal efficiency, with 5 MW collimated Gaussian input.

80%

81%

82%

83%

84%

85%

86%

87%

88%

89%

90%

91%

1300

1350

1400

1450

1500

1550

1600

3.0 3.5 4.0 4.5 5.0 5.5 6.0 6.5 7.0

Rec

eive

r E

ffic

ien

cy

Ou

tlet

Tem

per

atu

re [

K]

Gas Mass Flow Rate [kg/s]

Gas Outlet Temperature vs. Mass Flow Rate

Temperature

Efficiency

61

As might be expected, the outlet gas temperature is inversely proportional with the

mass-flow rate through the receiver, at least over this range of mass-flow rates near the

baseline scenario. Outlet temperatures range from 1325 K to 1550 K over a fairly narrow

window of mass-flow rates. This provides a potential design criterion for a real receiver, as

flow rates can be adjusted to reach a target outlet temperature for downstream needs such as

thermal storage or a turbine. Figure 6.6 puts the result from Figure 6.5 into the context of

power plant efficiency and currently existing receivers. Figure 6.6 plots receiver, engine, and

combined efficiency against operating temperatures [2]. For the receivers, operating

temperature refers to outlet temperature, while for the heat engine curve it refers to turbine

inlet temperature. The blue line shows model predicted efficiency of the SDSU small particle

receiver as a function of outlet temperature, at an input power of 5 MW. The red line shows

similar data for existing tubular receivers, which operate at lower temperatures than the small

particle receiver. The green curve shows heat engine efficiency for existing power plants, and

the purple curve shows the combined efficiency of existing receivers & existing heat engines.

Currently we do not have a calculation for the small particle receiver coupled to a suitable

gas turbine to show a combined line for that combination, but that is the subject for future

work. We can conclude that the small particle receiver results show significant improvement

compared to current receiver technology. The ability to maintain receiver efficiency above

85% at higher temperatures allows for higher engine efficiency and thus greater overall

efficiency of the solar thermal power plant.

Figure 6.5 also shows that efficiency improves as mass flow rate goes up over this

range, although the gains are reduced as efficiency approaches 90% or so. This trend is likely

the result of a similar mechanism to that seen in Chapter 6.1. Specifically, lower mass flow

rates result in higher gas temperatures in the receiver, which in turn result on greater

radiation losses out the aperture. This can be seen not only in the outlet temperatures as

shown previously in Figure 6.5, but also in the temperature field. Figure 6.7 and Figure 6.8

show the temperature field inside the receiver at the two extreme cases of 4.0 and 6.0 kg/s

mass-flow rates, respectively. While both cases have regions of model-limited temperatures

of 2200 K near the axis and the window as previously explained in Chapter 6.1, this region is

larger in the lower mass-flow case. Additionally, the gas in the front portion of the receiver

outside of this temperature-limited zone is hotter in the lower mass-flow case than in the

62

Figure 6.6. Receiver, engine, and total efficiencies as a function of operating temperature.

Figure 6.7. Temperature field in the receiver with a mass-flow rate of 4.0 kg/s and input power of 5 MW. Color-map numbers are in units of Kelvin.

0%

10%

20%

30%

40%

50%

60%

70%

80%

90%

100%

400 600 800 1000 1200 1400 1600 1800

Eff

icie

ncy

Operating Temperature [K]

Impact of Receiver Efficiency on Power Plant

SPR @5 MW

Existing Receivers

Engine Efficiency

Total (existing)

63

Figure 6.8. Temperature field in the receiver with a mass-flow rate of 6.0 kg/s and input power of 5 MW. Color-map numbers are in units of Kelvin.

higher mass-flow case. The mass-weighted average temperature in the hottest Monte Carlo

cell (within 0.5 m of the window and 0.3 m of the axis) is 1661 K for the 4.0 kg/s case, as

compared to 1326 K for the 6.0 kg/s case. Thus, this cell emits more radiation in the lower

mass-loading case, much of which exits the receiver aperture and is tallied as a radiative loss.

This trend persists moving radially outward, resulting in overall greater radiative losses for

the lower mass-loading case, thus reducing receiver efficiency.

This issue is also apparent when we examine the energy balance for each of the five

cases. Figure 6.9 shows the energy balance for each case as a function of iteration number.

The colder cases show a better energy balance. The improved energy balance could be a

result of the same mechanism discussed in the previous chapter, that being the relative size of

model-limited temperature regions in the gas. The higher mass-flow rate cases have smaller

volumes of gas at the model limit of 2200 K, thus less of the energy input from the Monte

Carlo model are neglected in the FLUENT model for these cases. Furthermore, it is likely

that the “true” temperatures in this 2200 K region are higher in the low mass-flow cases. This

would result in a greater magnitude of energy neglected by the FLUENT solver. The lower

mass-flow rate cases also show less stability in the energy balance during iteration. This is

possibly due to temperature swings but an explanation has not yet been determined.

Another difference between mass-flow rate cases, visible in Figure 6.7 and Figure 6.8

is the radial temperature gradient in the outlet tube. The lower mass-flow rate case has a

significant temperature gradient which persists all the way to the exit plane. In contrast, the

64

Figure 6.9. Energy balance as a function of iteration number, for five different mass-flow rate cases, 5 MW input power.

higher mass-flow rate case experiences more turbulent mixing and thus has a nearly uniform

gas temperature by the time the flow reaches the outlet plane. This is not an issue of serious

concern, as downstream piping could be used to induce more mixing if needed.

As a result of lower gas temperatures in the higher mass-flow cases, wall

temperatures were also reduced as mass-flow rate increased. This trend can be seen in Figure

6.10, which plots the outer cylinder wall temperature profile for each of the five cases.

Again, as with the wall temperature differences discussed previously in Chapter 6.1, reduced

wall temperatures lead to reduced window losses and greater receiver efficiencies, since

much of the emissions from the black walls exit the aperture. This is especially true for wall

elements near the front of the receiver.

92%

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96%

97%

98%

99%

100%

101%

102%

0 10 20 30 40 50 60 70 80 90 100 110

En

ergy

Bal

ance

(F

luen

t/M

CR

T)

Iteration Number

Energy Balance - Mass Flow Rate Variation

6.0 kg/s

5.5 kg/s

5.0 kg/s

4.5 kg/s

4.0 kg/s

65

Figure 6.10. Axial cylinder wall temperature profiles for five different mass-flow rates, 5 MW input power.

6.3 INPUT POWER VARIATION

To study the effect of varying input power to the receiver, as well as to explore a

potential control mechanism for the receiver, five cases were run with differing mass-flow

rates and input power levels. As summarized at the beginning of this chapter and in Table

6.1, the ratio of mass-flow rate to input power is held constant for these cases at the same

level as the baseline scenario. On the low end, this gives a mass-flow rate of 2.0 kg/s and

input power of 2 MW, while on the high end these values are 6 kg/s and 6 MW respectively.

This is done in an attempt to model using mass-flow rate variation to regulate receiver

behavior under varying input power levels. Because heliostat fields produce variable power

levels over the course of a day and a year due to changing solar flux levels, it may be

desirable to change receiver operating conditions in response. One potential motivation for

doing so would be to maintain a relatively constant outlet temperature to feed downstream

700

800

900

1000

1100

1200

1300

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0

Wal

l Tem

per

atu

re [

K]

Axial Distance from Window

Wall Temperatures - Mass Flow Rate Variation

6.0 kg/s

5.5 kg/s

5.0 kg/s

4.5 kg/s

4.0 kg/s

66

processes such as a turbine or thermal storage system. Results for outlet gas temperature and

receiver thermal efficiency are presented in Figure 6.11. While temperatures were higher at

higher input levels, the range of outlet temperatures seen here is relatively narrow, ranging

from 1342 K on the low end to 1424 on the high end. Furthermore, power levels between 4

and 6 MW produced outlet temperatures that differ by only 25 degrees Kelvin. This is a

remarkably consistent output considering that input power levels increased by 50% in this

range. Finer tuning of the mass-flow rate variation may provide the ability to maintain nearly

constant outlet temperatures over a broad range of input power levels.

Figure 6.11. Gas outlet temperatures and receiver thermal efficiencies for five different inlet power levels.

This control mechanism may come at some cost to efficiency however, as Figure 6.11

also illustrates that receiver efficiency went down as power levels and mass-flow rates were

reduced. At the low end of input power and mass-flow rate, the receiver efficiency was

barely above 80%, while at the upper end receiver efficiency was over 90%. The reduced

80%

82%

84%

86%

88%

90%

92%

1330

1350

1370

1390

1410

1430

1450

1 2 3 4 5 6 7T

her

mal

Eff

icie

ncy

Ou

tlle

t Tem

per

atu

re [

K]

Mass-Flow Rate [kg/s] & Inlet Power [MW]

Outlet Temperature & Thermal Efficiency

Outlet TemperatureThermal Efficiency

67

receiver efficiency at lower mass-flow rates is attributable to the size of the receiver relative

to the input power level. Though the input power is reduced, the aperture size remains

constant and thus the potential radiative losses remain relatively constant. The temperature

fields for these cases shows minimal differences. Figure 6.12 and Figure 6.13 show the

temperature field in the receiver for the 2 kg/s and 6 kg/s mass-flow rate cases, respectively.

While outlet temperatures are similar between the two, the higher mass-flow rate case has a

small gas region at the model-limited temperature of 2200 K, while the peak temperature in

the lower mass-flow rate case is barely over 1600 K. Despite this, the lower mass-flow case

has a greater mass-weighted average temperature in the hottest Monte Carlo cell by about 50

degrees Kelvin. This trend continues to some extent as we move radially outward, thus

increasing emissions from the gas regions adjacent to the aperture and increasing window

losses accordingly.

Figure 6.12. Temperature field inside the receiver with a mass-flow rate of 2 kg/s and input power of 2 MW. Color-bar units are Kelvin.

Differences in wall temperatures also provide little explanation for the reduced

efficiency at lower power levels. Figure 6.14 shows axial wall temperature profiles for the

five cases of variable inlet power. Though wall temperatures are all within a relatively

narrow band, the lower power level and mass-flow rate cases actually show a trend towards

lower wall temperatures, particularly in the front half of the receiver. As was the case in the

previous chapter, higher wall temperatures, especially near the aperture, produce greater

radiative losses and thus drive down receiver efficiency. Despite the slightly larger radiative

68

Figure 6.13. Temperature field inside the receiver with a mass-flow rate of 6 kg/s and input power of 6 MW. Color-bar units are Kelvin.

Figure 6.14. Axial cylinder wall temperature profiles for five different inlet power levels and mass-flow rates.

700

750

800

850

900

950

1000

1050

1100

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0

Wal

l Tem

per

atu

re [

K]

Axial Distance from Window [m]

Wall Temperature Profiles - Input Power Variation

2 kg/s ‐ 2MW3 kg/s ‐ 3MW4 kg/s ‐ 4MW5 kg/s ‐ 5MW6 kg/s ‐ 6MW

69

losses from the wall in the high power and high mass-flow rate cases, total radiative losses

are still greater in the low power cases. This is a feature that may be important to keep in

mind when designing a real receiver. Although the model has shown wall temperatures that

are relatively low from a material strength standpoint, keeping wall temperatures down and

reducing emissions from the walls can help reduce losses and increase efficiency of the

receiver. Additionally, the receiver size and in particular aperture size must be carefully

designed for a relatively narrow range of power inputs. Decreasing power by a factor of

three, as is the case when going from 6 MW to 2 MW, forces the receiver to operate far from

its design point and thus at reduced efficiency.

70

CHAPTER 7

CONCLUSIONS AND FUTURE WORK

7.1 CONCLUSIONS

The coupling of the Monte Carlo radiation solver to the FLUENT flow solver brings

the model closer to a realizable geometry. On balance, the move to more realistic flow

geometry including the outlet tube did not change receiver efficiency significantly from the

results of the original model by Ruther, but the current model provides more confidence in

the results and will greatly help with actual receiver design such as sizing the outlet tube,

determining wall and window temperatures, and finding the pressure drop through the

receiver. Efficiencies remained between 80-90%. Outlet gas temperatures remained in the

range of 1300 K to 1550 K depending on input conditions. While attempts to include the

outlet tube wall in the Monte Carlo ray-trace solver were not successful until after the

completion of this research, the challenge has recently been overcome and future work will

include it, as discussed in Chapter 7.2.

The introduction of a Gaussian flux distribution on the aperture not only matched

reality better than the previous uniform flux case, but also improved receiver efficiency by

about 5% and increased outlet gas temperature by 40 degrees Kelvin. This improvement

highlights the importance of keeping the bulk of incident radiation focused in the gas volume

and away from the walls. A real receiver design should take this into account when

determining receiver wall geometry, perhaps by re-introducing a slight offset between the

aperture and the outer wall, as well as by angling the outer wall to align parallel with the

most extreme angle of incident radiation. Wall temperatures and radiation losses may also be

kept down by utilizing a more aerodynamic geometry and eliminating sharp corners which

cause recirculation. The recirculation cells which developed in the corner in all cases

produced slightly elevated gas temperatures, which contributed to window losses and made

wall temperatures more difficult to predict.

The model demonstrated that mass-flow rate may be used effectively to adjust

receiver outlet temperature. However, this control mechanism also impacts receiver

71

efficiency significantly. A 50% increase in mass-flow rate from 4 kg/s to 6kg/s reduced

outlet temperatures by more than 200 degrees Kelvin, but increased receiver efficiency by

4%. The highest mass-flow rate modeled at 6 kg/s also produced the greatest receiver

efficiency, which suggests that further investigation of higher mass-flow rates is warranted.

For the first time, a graph of receiver efficiency vs. output temperature was produced that

allows this receiver to be compared to others that might be used to run a gas turbine, as well

as to determine the optimum coupling temperature as shown in Figure 6.6.

Variations in input power along with mass-flow rate showed how sensitive the

receiver design can be to input flux conditions. Reducing the input power level (and therefore

flux concentration) well below the design point of 5 MW significantly reduced receiver

efficiency. This is likely due to radiation losses not being similarly reduced in conjunction

with the reduction in incident flux. It is therefore important to size a real receiver aperture as

small as possible for the intended heliostat field. Additionally, the trend suggests that this

receiver may benefit from an even greater peak flux than that investigated here, which was 3

MW/m2.

7.2 FUTURE WORK

There are many avenues open to improving the current receiver model. The most

immediate improvement, which is currently underway, is the addition of the outlet tube wall

to the Monte Carlo ray-trace model. Preliminary results suggest that the impact of this

addition will not significantly affect receiver outlet temperatures or efficiencies. However, it

is important to include the feature because an estimate of outlet tube wall temperatures is

needed for the design of a real receiver. Emission from this solid surface may put a local

heating load on the window as well that is currently not accounted for.

A second useful change to the model would be a shift to a non-uniform grid in the

Monte Carlo radiation solver. A non-uniform grid in both the axial and radial directions

could be used to provide greater resolution near the front and center of the receiver

respectively, where temperature gradients are higher than in the rear and near the outer edge.

This could be accomplished without increasing the total number of MCRT grid cells, and

thus without a major impact on computation time. Along with adjustments to the MCRT grid,

the FLUENT grid could be refined based on the results presented in this thesis. In particular,

72

finer grid resolution near the walls is needed to fully resolve some of the thermal boundary

layers in the gas. If the refinement is limited to just the first few cells near the wall, it would

not impact computational load significantly. Conversely, it may be possible to further

coarsen the grid in the volume away from the walls to offset the increased load of refinement

near the walls.

The addition to the code likely to impact results of the model most significantly is a

coupled particle oxidation model. As the gas-particle mixture heats up, eventually the small

carbon particles will oxidize into CO2 and CO. As they oxidize, their size will be reduced

until finally the particles disappear altogether. This has a significant effect on the absorption

and emission characteristics of the gas-particle mixture in the receiver. Gas regions with fully

oxidized carbon would have no particles and thus be virtually transparent to incident

radiation. This is expected to eliminate the small overheated gas region found near the center

of the window in many cases, and may also act somewhat as a control mechanism in the

receiver. It may also reduce absorption in the outlet tube gas, which may reduce receiver

efficiency or warrant a modification to the outlet tube geometry. Modeling particle oxidation

in conjunction with the radiation energy exchange and flow solution is a significant

undertaking which may require three-way coupling with a new code. It is also likely to

increase computation time still further.

Another important feature that is still needed in the model is the ability to model non-

orthogonal boundaries in the Monte Carlo. This would allow for the inclusion of a curved

window, as well as an angled or curved outer wall. A real receiver will require a curved

window to contain pressure [21, 30], and these two features will likely have an impact on the

flow field in the receiver. This will require significant changes to the Monte Carlo code,

possibly including a shift to fully three-dimensional modeling, and likely will increase

processing time as a result. It may be necessary to re-structure the Monte Carlo code to allow

for parallel processing to reduce computation time. Should the code transition to a fully

three-dimensional model, natural convection and the effect of swirl velocity should also be

investigated.

Currently, only the radiation flux values applied to the outer cylinder wall are

interpolated, because sensitivity of the solution demanded it and the interpolation scheme

was relatively simple being only one-dimensional. Expanding the source term interpolation

73

scheme to include two-dimensional interpolation of volumetric source terms in the gas would

improve accuracy of the temperature field results. Though the impact may not be all that

significant in the aggregate, local changes might provide important insight. This change

would be fairly straightforward to implement in the current geometry, but would become

significantly more complex with a non-uniform grid, non-orthogonal boundaries, or a fully

three-dimensional Monte Carlo solver.

Aside from expansions to the model, the existing code could be used to examine a

wider range of input parameters including non-collimated incident radiation and a wider

range of input powers and mass-flow rates. It would also be beneficial to scale the model

down to match the size and approximate geometry of a lab-scale receiver currently being

built and tested at San Diego State University [30]. This would serve as an important

validation of the model, and help to quantify the importance of some current approximations.

A wide range of input parameters could then be used in conjunction with an overall cycle

model previously written by Kyle Kitzmiller [27] to characterize system performance with

the receiver coupled to a gas turbine. This would allow for a more robust version of the graph

in Figure 6.6 which showed receiver and combined efficiency as a function of operating

temperature.

74

REFERENCES

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[8] Heller, Peter, Markus Pfänder, Thorsten Denk, Felix Tellez, Antonio Valverde, Jesús Fernandez, and Arik Ring. “Test and Evaluation of a Solar Powered Gas Turbine System.” Solar Energy 80, no. 10 (2006):1225-1230.

[9] Ho, Clifford K., Siri S. Khalsa, and Nathan P. Siegel. “Modeling On-Sun Tests of a Prototype Solid Particle Receiver for Concentrating Solar Power Processes and Storage.” Paper presented at the Proceedings of ASME Energy Sustainability, San Francisco, CA, 2009.

[10] Z'Graggen, A., and A. Steinfeld. “Heat and Mass Transfer Analysis of a Suspension of Reacting Particles Subjected to Concentrated Solar Radiation - Application to the Steam-Gasification of Carbonaceous Materials.” International Journal of Heat and Mass Transfer 52 (2009): 385-395.

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[15] Bertocchi, R., J. Karni, and A. Kribus. “Experimental Evaluation of a Non-Isothermal High Temperature Solar Particle Receiver.” Energy 29, no. 5-6 (2004): 687-700.

[16] Klein, Hanna Helena, Rachamim Rubin, and Jacob Karni. “Experimental Evaluation of Particle Consumption in a Particle Seeded Solar Receiver.” Journal of Solar Energy Engineering 130, no. 1 (2008).

[17] Miller, F. “Radiative Heat Transfer in a Flowing Gas-Particle Mixture.” PhD diss., University of California, Berkely, 1988.

[18] Klein, Hanna Helena, Jacob Karni, Rami Ben-Zvi, and Rudi Bertocchi. “Heat Transfer in a Directly Irradiated Solar Receiver/Reactor for Solid-Gas Reactions.” Solar Energy 81, no. 10 (2007): 1227-1239.

[19] Röger, M., R. Buck, and H. Muller-Steinhagen. “Numerical and Experimental Investigation of a Multiple Air Jet Cooling System for Application in a Solar Thermal Receiver.” Journal of Heat Transfer 127, no. 8 (2005): 863-876.

[20] Karni, J., A. Kribus, B. Ostraich, and E. Kochavi, “A High-Pressure Window for Volumetric Solar Receivers.” Journal of Solar Energy Engineering 120, no. 2 (1998): 101-107.

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[27] Kitzmiller, Kyle, and Fletcher Miller. “Thermodynamic Cycles for Small Particle Heat Exchange Receivers Used in Concentrating Solar Power Plants.” Journal of Solar Energy Engineering 133, no. 3 (2011): 031014. doi:10.1115/1.4004270.

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[29] Foundry Service & Supplies. “Maxfire Blankets (1800ºF - 2600ºF Spun Insulation Blanket).” Blanket and Bulk Products. Accessed October 25, 2012. http://supplies.foundryservice.com/viewitems/blanket-bulk-products/re-blankets-1800-f-2600-f-spun-insulation-blanket-?&bc=100%7C1001%7C1020%7C1058.

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77

APPENDIX

SUMMARY OF TURBULENCE MODEL

CONSTANTS USED IN FLUENT

78

All values are unchanged from the FLUENT defaults for the SST k-ω turbulence

model. For details on the function of these variables, refer to the FLUENT user’s manual

[31].

α*∞ 1

α∞ 0.52

β*∞ 0.09

ζ* 1.5

Mt0 0.25

a1 0.31

βi (inner) 0.075

βi (outer) 0.0828

TKE (inner) 1.176

TKE (outer) 1

SDR (inner) 2

SDR (outer) 1.168

Energy Pr 0.85

Wall Pr 0.85